A Geometric Study of Single Gimbal Control Moment Gyros — Singularity Problems and Steering Law —
Haruhisa Kurokawa
Mechanical Engineering Laboratory
Report of Mechanical Engineering Laboratory, No. 175, p.108, 1998. A Geometric Study of Single Gimbal Control Moment Gyros
— Singularity Problems and Steering Law —
by
Haruhisa Kurokawa
Abstract
In this research, a geometric study of singularity of single gimbal CMGs clarified a more serious problem characteristics and steering motion of single gimbal of continuous steering, that is, no steering law can follow Control Moment Gyros (CMGs) was carried out in order all command sequences inside a certain region of the to clarify singularity problems, to construct an effective angular momentum space if the command is given in steering law, and to evaluate this law’s performance. real time. Based on this result, a candidate steering law Passability, as defined by differential geometry effective for rather small space was proposed and verified clarified whether continuous steering motion is possible not only analytically, but also using ground experiments in the neighborhood of a singular system state. which simulated attitude control in space. Topological study of general single gimbal CMGs Similar evaluation of other steering laws and clarified conditions for continuous steering motion over comparison of various system configurations in terms a wider range of angular momentum space. It was shown of the allowed angular momentum region and the that there are angular momentum vector trajectories such system’s weight indicated that the pyramid type single that corresponding gimbal angles cannot be continuous. gimbal CMG system with the proposed steering law is If the command torque, as a function of time, results in one of the most effective candidate torquer for attitude such a trajectory in the angular momentum space, any control, having such advantages as a simple mechanism, steering law neither can follow the command exactly a simpler steering law, and a larger angular momentum nor can be effective. space. A more detailed study of the symmetric pyramid type
Keywords Attitude control, Singularity, Momentum exchange device, Inverse kinematics, Steering law
––– i ––– Acknowledgments
This research work is a result of projects conducted from cover-to-cover, providing constructive criticism. at the Mechanical Engineering Laboratory, Agency of I would also like to thank my colleague Akio Suzuki Industrial Technology and Science, Ministry of who constructed most of the experimental apparatus, and International Trade and Industry, Japan. Related projects designed and installed controllers for the attitude control. are, “Development of Attitude Control Equipment Prof. Tsuneo Yoshikawa of Kyoto University helped (FY1982–1987)“, “Attitude Control System for Large me when we started the project of attitude control by Space Structures (FY1988–1993)”, and “High Precision CMGs. Discussions held with Dr. Nazareth Bedrossian Position and Attitude Control in Space (FY1993–1997)”. and Dr. Joseph Paradiso of the Charles Stark Draper Laboratory (CSDL) were invaluable. They gave me I wish to acknowledge my debt to many people. Prof. valuable suggestions with various research papers in this Nobuyuki Yajima of the Institute of Space and field. Astronautical Science (ISAS) are earnestly thanked for Dr. Mark Lee Ford as a visiting researcher of our inspiring me with this theme, as well as for collaborations laboratory spent his precious hours for me to correct during his tenure as a division head of our laboratory. I expressions in English. would extend thanks to the late Prof. Toru Tanabe, I would like to thank all the above people, other formerly of the University of Tokyo for his guidance in colleagues sharing other research projects, and the the culmination of this work into a dissertation. In Mechanical Engineering Laboratory (MEL) and the finishing this work, the following professors guided me, directors especially the Director General Dr. Kenichi Assoc. Prof. Shinichi Nakasuka of the University Tokyo, Matuno and the former Department Head Dr. Kiyofumi Prof. Hiroki Matsuo of ISAS, Prof. Shinji Suzuki, Prof. Matsuda for allowing me to continue this research. Yoshihiko Nakamura,Assoc. Prof. Ken Sasaki of the Finally, I thank my wife and daughters for their patience University of Tokyo. particularly during some hectic months. Many discussions with Dr. Shigeru Kokaji of our laboratory proved invaluable. He patiently listened to my abstract explanation of geometry and provided Haruhisa Kurokawa valuable suggestions. Furthermore, he assisted me by soldering and checking circuits, and reviewed this paper June 7, 1997
ÐÐÐ ii ÐÐÐ Contents
Abstract ...... i Acknowledgments ...... ii Terms ...... viii Nomenclature ...... ix List of Figures ...... x List of Tables ...... xiii
Chapter 1 Introduction ...... 1 1.1 Research Background ...... 1 1.2 Scope of Discussion ...... 3 1.3 Outline of this Thesis ...... 4
Chapter 2 Characteristics of Control Moment Gyro Systems ...... 5 2.1 CMG Unit Type ...... 5 2.2 System Configuration ...... 5 2.2.1 Single Gimbal CMGs ...... 6 2.2.2 Two Dimensional System and Twin Type System ...... 7 2.2.3 Configuration of Double Gimbal CMGs...... 7 2.3 Three Axis Attitude Control ...... 7 2.3.1 Block Diagram ...... 8 2.3.2 CMG Steering Law ...... 8 2.3.3 Momentum Management ...... 8 2.3.4 Maneuver Command ...... 8 2.3.5 Disturbance ...... 8 2.3.6 Angular Momentum Trajectory ...... 8 2.4 Comparison and Selection ...... 9 2.4.1 Performance Index ...... 9 2.4.2 Component Level Comparison ...... 9 2.4.3 System Level Comparison ...... 9 2.4.4 Work Space Size and Weight ...... 9
Chapter 3 General Formulation ...... 11 3.1 Angular Momentum and Torque ...... 11 3.2 Steering Law ...... 12 3.3 Singular Value Decomposition and I/O Ratio...... 12 3.4 Singularity ...... 13
––– iii ––– 3.5 Singularity Avoidance ...... 13 3.5.1 Gradient Method ...... 14 3.5.2 Steering in Proximity to a Singular State ...... 14
Chapter 4 Singular Surface and Passability...... 15 4.1 Singular Surface ...... 15 4.1.1 Continuous Mapping ...... 15 4.1.2 Envelope ...... 16 4.1.3 Visualization Method of the Surface ...... 16 4.2 Differential Geometry ...... 17 4.2.1 Tangent Space and Subspace...... 17 4.2.2 Gaussian Curvature ...... 17 4.3 Passability ...... 18 4.3.1 Quadratic Form ...... 18 4.3.2 Signature of Quadratic Form ...... 19 4.3.3 Passability and Singularity Avoidance ...... 19 4.3.4 Discrimination ...... 20 4.4 Internal Impassable Surface ...... 21 4.4.1 Impassable Surface of an Independent Type System ...... 21 4.4.2 Impassable Surface of a Multiple Type System ...... 21 4.4.3 Minimum System ...... 22
Chapter 5 Inverse Kinematics...... 23 5.1 Manifold...... 23 5.2 Manifold Path ...... 24 5.3 Domain and Equivalence Class ...... 24 5.4 Terminal Class and Domain Type ...... 25 5.5 Class Connection ...... 25 5.5.1 Type 2 Domain ...... 25 5.5.2 Type 1 Domain ...... 26 5.5.3 Class Connection Rules...... 27 5.5.4 Continuous Steering over Domains ...... 28 5.5.5 Manifold Selection ...... 28 5.5.6 Discussion of the Critical Point...... 29 5.6 Topological Problem ...... 29
Chapter 6 Pyramid Type CMG System ...... 31 6.1 System Definition ...... 31 6.2 Symmetry ...... 31 6.3 Singular Manifold for the H Origin ...... 33
––– iv ––– 6.4 Singular Surface Geometry ...... 35
Chapter 7 Global Problem, Steering Law Exactness and Proposal ...... 41 7.1 Global Problem ...... 41 7.1.1 Control Along the z Axis ...... 41 7.1.2 Global problem...... 45 7.1.3 Details of the Problem ...... 45 7.1.4 Possible Solutions ...... 47 7.2 Steering Law with Error ...... 47 7.2.1 Geometrical Meaning ...... 47 7.2.2 Exactness of Control ...... 48 7.3 Path Planning ...... 49 7.4 Preferred Gimbal Angle ...... 49 7.5 Exact Steering Law ...... 51 7.5.1 Workspace Restriction...... 51 7.5.2 Repeatability and Unique Inversion ...... 51 7.5.3 Constrained Control ...... 52 7.5.4 Reduced Workspace ...... 52 7.5.5 Characteristics of Constrained Control ...... 54
Chapter 8 Ground Experiments ...... 57 8.1 Attitude Control ...... 57 8.1.1 Dynamics...... 57 8.1.2 Exact Linearization ...... 57 8.1.3 Control Method ...... 58 8.2 Experimental Facility and Procedure ...... 58 8.2.1 Facility...... 58 8.2.2 Design of Control Command Sequence ...... 59 8.2.3 Experimental Procedure ...... 59 8.3 Experimental Results ...... 60 8.3.1 Attitude Keeping under Constant Disturbance ...... 60 8.3.2 Rotation About the z Axis ...... 64 8.3.3 Maneuver after Momentum Accumulation ...... 67 8.3.4 Mode Selection and Switching...... 69 8.4 Summary of Experiments ...... 69
Chapter 9 Evaluation ...... 71 9.1 Conditions for Comparison ...... 71 9.2 Spherical Workspace ...... 71 9.3 Evaluation by Weight ...... 72
––– v ––– 9.4 Ellipsoidal Workspace ...... 73 9.5 Summary of Evaluation ...... 75
Chapter 10 Conclusions ...... 77
Appendix A Double Gimbal CMG System...... 79 A.1 General Formulation ...... 79 A.2 Singularity ...... 79 A.3 Steering Law and Null Motion ...... 80 A.4 Passability ...... 80 A.4.1 Two Unit System ...... 80 A.4.2 Three Unit System ...... 81 A.5 Workspace ...... 81
Appendix B Proofs of Theories ...... 83 B.1 Basis of Tangent Spaces ...... 83 B.2 Gaussian Curvature ...... 83 B.3 Inverse Mapping Theory ...... 84 B.4 Impassable condition for two negative signs ...... 85
Appendix C Internal Impassability of Multiple Type Systems ...... 87 C.1 Roof Type System M(2, 2) ...... 87 C.1.1 Evaluation of Singular Surface (2) ...... 87 C.1.2 Evaluation of Singular Surface (3) ...... 87 C.1.3 Evaluation of Singular Surface (5) ...... 88 C.1.4 Evaluation of Singular Surface (7) ...... 88 C.1.5 Conclusion ...... 88 C.2 M(3, 2): M(2, 2)+1 ...... 88 C.2.1 Condition (3) of M(2,2) ...... 88 C.2.2 Condition (5) of M(2,2) ...... 89 C.3 M(3, 3): M(2, 2)+2 ...... 89 C.4 M(2, 2, 1): M(2, 2)+1 ...... 89 C.5 M(2, 2, 2): M(2, 2)+2 ...... 89 C.6 Minimum System ...... 89
Appendix D Six and Five Unit Systems ...... 91 D.1 Symmetric Six Unit System S(6) ...... 91 D.1.1 System Definition ...... 91
––– vi ––– D.1.2 Fault Management ...... 91 D.1.3 Four out of Six Control ...... 92 D.2 Five Unit Skew System ...... 92
Appendix E Specification of Experimental Apparatus and Experimental Procedure ...... 95 E.1 Experimental Apparatus ...... 95 E.2 Specifications ...... 97 E.3 Attitude Control System ...... 97 E.4 Steering Law Implementation ...... 99 E.5 Code Size and Calculation Time ...... 99 E.6 Parameter Estimation ...... 99
Appendix F General kinematics ...... 101 F.1 Analogy with a Spatial Link Mechanism ...... 101 F.2 Spatial Link Mechanism Kinematics ...... 101 F.3 Singularity ...... 102 F.4 Passability ...... 102
References ...... 105
––– vii ––– Terms
Class : A set of manifolds which correspond to a certain Null motion : Gimbal angle motion which keeps the domain and are equivalent to each other. angular momentum vector constant.
Domain : A region in the angular momentum space which Single gimbal CMG : Fig. 2–1 is surrounded by singular surfaces and does not contain any singular surface. Singular surface : A surface formed by the total angular momentum vector point, H, which corresponds Double gimbal CMG : Fig. 2–1 to singular point.
Gimbal vector : A unit vector of gimbal direction. Singular vector : A unit vector to the plane spanned by all torque vectors when the system is singular. Independent type : A single gimbal CMG system without parallel gimbal direction pair. Skew type : A single gimbal CMG system with gimbal directions axially symmetric about one direction. Manifold : A connected subspace of gimbal angle space whose element corresponds to the same total Symmetric type : A single gimbal CMG system with angular momentum. gimbal directions arranged normal to surfaces of a regular polyhedron. Manifold equivalence : Two manifolds corresponding to a certain domain are equivalent if there is an Torque vector : A unit vector of a component CMG to angular momentum path which corresponds to a which direction the CMG can generate an output continuous manifold path between these two torque. manifolds. Workspace: Allowed region of the angular momentum Multiple type : A single gimbal CMG system composed vector of a CMG system. of groups each of which elements possess identical gimbal direction.
––– viii ––– Nomenclature
Symbol Definition Section number Symbol Definition Section number ––––––––––––––––––––––––––––––––––––––––––– ––––––––––––––––––––––––––––––––––––––––––– α : Skew angle of the symmetric pyramid type M(2, 2): Roof type system 2.2.1 system 6.1 Mi : Manifold 5.1 β : Euler parameter of satellite orientation 8.1.1 MSj : Singular manifold 5.1 β* : Vector part of β 8.1.1 n : Number of CMG units in the system 3.1 B : Strip like surface of impassable surface called ⋅ pi : = 1 / (u hi) 4.1.3 branch 6.4 P: Diagonal matrix of pi . 4.1.3 c* : = cosα 6.1 θ : Gimbal angle of ith CMG unit 3.1 × i ci := gi hi. Torque vector 3.1 θ θ :=(i.). A state variable of the system. Point of n C : Jacobian matrix of the kinematic function, dimensional torus T (n) 3.1 θ H = f ( ) 3.1 Θ θ S: Singularly constrained tangent space of the θ ∈Θ d S S. 4.2.1 space (two dimensional). 4.2.1 θ ∈Θ Θ − d N N 4.2.1 N: Null space of C (n 2 dimensional). 4.2.1 θ ∈Θ Θ Θ d T T 4.2.1 T: Complementary subspace of N (two dimensional). 4.2.1 D : Domain in the H space surrounded by singular θ surfaces 5.3 rg : Symmetric transformation in the space. 6.2 ε ε := {i }. Sign parameter of the singular surface. Rg : Symmetric transformation in the H space. 6.2 4.1.1 s* : = sinα 6.1 gi : Gimbal vector 3.1 S(n) : Symmetric type single gimbal CMG system. G : Equivalence class in a domain. 5.3 2.2.1 : ε hi : Normalized angular momentum vector 3.1 Sε A region of the singular surface of sign . 4.1.1 Σ θ H := hi.= f ( ). Total angular momentum vector. T : Total output torque of the system 3.1 3.1 u : Singular vector. Unit vector normal to all torque vectors. 3.4 κ : Gaussian curvature of the singular surface.4.2.2
t ω : Gimbal rate vector. Time derivative of θ. 3.1 LA : Segment included by a manifold of H=(0,0,0) ω : Null motion, 3.2 6.3 N
––– ix ––– List of Figures
Chapter 2 5–1 Manifolds in the neighborhood of a singular point. 2–1 Two types of CMG units 5–2 Continuous change of manifolds. 2–2 Configurations of single gimbal CMGs 5–3 An example of a continuous manifold 2–3 Twin type system path. 2–4 Block diagram of three axis attitude 5–4 Relations between H space, manifold control space and θ space. Chapter 3 5–5 Domains and manifolds of the pyramid type system 3–1 Orthonormal vectors of a CMG unit 5–6 Class connection graph around domains 3–2 Gimbal angle and vectors 5–7 An illustration of class connection rule 3–3 Input ⁄Output ratio (1). 3–4 Singularity condition and singular vector 5–8 An illustration of class connection rule 3–5 Typical vector arrangement for a 2D (2). system 5–9 An illustration of motion by the gradient 3–6 Steering at a singular condition method. 5–10 Manifold relations around critical point Chapter 4
4–1 Vectors at a singularity condition Chapter 6 4–2 Examples of the singular surfaces for the 6–1 Schematic of a pyramid type system pyramid type system. 6–2 Transformation in H space and in θ space 4–3 Envelope of a roof type system M(2, 2). 6–3 Line segments for singular manifold 4–4 Cross sections of a singular surface of the 6–4 Definition of the cross sectional plane and pyramid type system. the distance d 4–5 Infinitesimal motion from a singular point 6–5 Saddle like part of the envelope of 2D system. 6–6 Cross sections of singular surface 4–6 Second order infinitesimal motion from 6–7 Internal impassable singular surface singular surface. 6–8 Analytical line on an impassable surface 4–7 Possible motions in both direction of u at 6–9 Equilateral parallel hexahedron of a singular point. impassable branches 4–8 Local shape of an impassable singular 6–10 Overall structure of impassable branches surface. 6–11 Internal impassable surface with envelope 4–9 Impassable surface of S(6) cutaway 4–10 Impassable surface of Skew(5) with skew 6–12 Cross section through the xz plane angle α = 0.6 rad. 6–13 Cross section through the xy plane 4–11 Impassable surface of another Skew(5), α with skew angle = 1.2 rad. Chapter 7 4–12 Impassable surface of S(4). 7–1 Candidate of workspace Chapter 5 7–2 Cross section nearly crossing P
––– x ––– 7–3 Manifold bifurcation and termination 8–12 Command sequence of Experiment J
from DA 8–13 Results of Experiment J 7–4 Simplified class connection diagram
around domain DA Chapter 9 7–5 Manifolds of eight domains around the z 9–1 System configurations for comparison axis 9–2 Spherical workspace size for various 7–6 Singular manifold of a point U on the z− system configurations axis 9–3 Trade-off between workspace size and 7–7 Manifold of H near the origin system weight 7–8 Continuos change of manifold for H 9–4 Definition of ellipsoidal workspace nearly along the z axis 9–5 Average radius vs. skew angle 7–9 Manifold connection over several 9–6 Workspace radius as a function of aspect domains ratio 7–10 Cross sections of domains 9–7 Combined plot of radii as a function of 7–11 Possible motion following an example of aspect ratio singular surface 9–8 Converted weight as a function of aspect 7–12 Illustration of H trajectory of the CMG ratio system for the example maneuver 9–9 Radius as a function of aspect ratio for a 7–13 Avoidance of an impassable surface degraded system with one faulty unit 7–14 Problems of movement on an impassable surface Appendix A 7–15 Change in manifolds for H moving along A–1 Vectors and variables relevant to a double the x axis gimbal CMG 7–16 Estimation of reduced workspace for A–2 Vectors at singularity conditions exact steering A–3 Infinitesimal motion at a singular point 7–17 Discontinuity in the maximum of of condition (b) det(CCt) 7–18 Cross section of possible workspace by Appendix D constrained steering law 7–19 Reduced workspace of the constrained D–1 Envelopes of S(6) and degraded systems system D–2 Four unit subsystem of MIR type system 7–20 Reduced workspace of three modes D–3 Restricted workspace of a constrained MIR-type system Chapter 8 D–4 Concept of singularity avoidance by an 8–1 Experimental test rig showing the center− additional torquer mount suspending mechanism Appendix E 8–2 Target trajectory 8–3 Block diagram of the control system E–1 Experimental apparatus 8–4 Results of Experiment A E–2 Block diagram of experimental apparatus 8–5 Results of Experiment B E–3 Three axis gimbal mechanism 8–6 Results of Experiment C E–4 Single gimbal CMG 8–7 Results of Experiment D E–5 Balance adjuster 8–8 Results of Experiment E E–6 Onboard computer 8–9 Results by Experiment F E–7 Block diagram of the model matching 8–10 Results of Experiment G controller. 8–11 Results of Experiment H E–8 Block diagram of the tracking controller.
––– xi ––– E–9 Block diagram of the gradient method. E–10 Block diagram of the constrained method.
Appendix F
F−1 Analogy to a parallel link mechanism
––– xii ––– List of Tables
Chapter 2 8–2 Condition and Results of Experiments (2)
2–1 Component Level Comparison Appendix E 2–2 System Level Comparison E–1 Specification of experimental apparatus Chapter 6 E–2 Code size and calculation time of process
6–1 Symmetric Transformations Appendix F 6–2 Segment Transformation Rule F–1 Similarity between CMGs and link Chapter 8 mechanism
8–1 Condition and Results of Experiments (1)
––– xiii ––– ––– xiv ––– –– 2. Characteristics of Control Moment Gyro Systems ––
Chapter 1
Introduction
A Control Moment Gyro (CMG) is a torque generator simple calculation5). If another system was chosen, a for attitude control of an artificial satellite in space. It simple computation scheme was required using an analog rivals a reaction wheel in its high output torque and rapid circuit. For example, a method using an approximation response. It is therefore used for large manned satellites, with some feedback was proposed6, 7, 8). For the three such as a space station, and is also a candidate torquer double gimbal CMG system9) applied to the Skylab, an for a space robot. approximated inverse using the transposed Jacobian was There are two types of CMGs, single gimbal and used10). This CMG system successfully completed its double gimbal. Though single gimbal CMGs are better mission, though one of the CMGs became nonfunctional in terms of mechanical simplicity and higher output during the flight11). After that, studies of double gimbal torque than double gimbal CMGs, the control of single CMGs have continued for eventual application to the gimbal CMGs has inherent and serious singularity space shuttle and the space station “Freedom” which is problem. At a singularity condition, a CMG system now called ISS12, 13). cannot produce a three axis torque. Despite various Another CMG type, i.e., a single gimbal one, was efforts to overcome this problem, the problem still studied for use in satellites such as the “High Energy remains, especially in the case of the pyramid type CMG Astronomical Observatory (HEAO)” and the “Large system. Space Telescope (LST)”. One of the configurations This research aims to elucidate this singularity intensively studied was a pyramid type, which consists problem. Detailed study of the pyramid type system of four single gimbal CMGs in a skew configuration. leads to a global problem of singularity. The final Comparing six different independently developed objective of this work involves evaluation of various steering laws indicated that an exact inverse calculation steering laws and the proposal of an effective steering was necessary14). It was also observed from various law. As all the geometric studies are either theoretical simulations that the singularity problem could not be or analytical and based on computer calculations, ground ignored. It was concluded that some sort of singularity experiments were carried out to support those results. avoidance control using system redundancy was required for this type system. 1.1 Research Background A roof type system, which is another four unit system of single gimbal CMGs, was also a candidate for the HEAO. As its mathematical formulation is simpler than Research of CMG systems started in the mid 1960s. that of the pyramid type, singularity avoidance was This was intended for later application to the large originally included in a steering law15). An improvement satellite of the USA, “Skylab”, and its high precision of this law involved a new approach in which the nature component, ”Apollo Telescope Mount (ATM)” 1, 2, 3). of numerical calculation and discrete time control were The studies included hardware studies of a gyro bearing utilized16). and gyro motor, and software studies for attitude control Singularity avoidance has been studied for all CMG and CMG steering control. Evaluation of various types types. This was a simple matter for double gimbal CMG and configurations was made in terms of weight and systems17)–20). Typically used was a gradient method, power consumption4). At that time, an onboard computer which maximized a certain objective function by using lacked the ability to perform real time matrix inversion redundancy21). While this method was effective in the calculation. One of the candidates was a twin type evaluation of double gimbal CMG systems, it was not system made of two single gimbal CMGs driven in successful for single gimbal CMG systems. For opposite directions. Control of this system requires only
––– 1 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– example, optimization of a redundant variable resulted system. The reason this type was selected was because in discontinuity16) or an optimized value became a six unit system was considered too large and too singular15) in the case of a roof type system. In the case complicated. Many proposals suggested a type of of a pyramid type system, various problems were found gradient method34, 35, 36). The method utilized for the in computer simulations even when a gradient method four unit subsystem of the “MIR” was also of this was used. kind27). Another method used global optimization28), Margulies was the first to formulate a theory of and nearly all methods showed some problems in singularity and control22). His paper included geometric computer simulations. theory of a singular surface, a generalized solution of Passability is defined locally and its problem reported the output equation and null motion, and the possibility first was a kind of local problem28). Later, Bauer showed of singularity avoidance for a general single gimbal CMG difficulty in steering as a global problem37). He found system. Also, some problems of the gradient method two different command sequences, both of which could were pointed out using an example of a two dimensional not be realized by the same steering method. After this, system. Vadali proposed a method to overcome this problem Works by the Russian researcher, Tokar, were using a preferred state38). Finally, the problem by Bauer published in the same year, and included a description was formulated exactly, stating that no steering law can of the singular surface shape23), the size of the follow an arbitrary command sequence inside certain workspace24) and some considerations of the gimbal wide region of the workspace39). Under this limitation, limits25). In his next paper26), passability of a singular an effective method was proposed. surface was introduced. It was made clear that a system The research described above dealt with exact such as a pyramid type has an ‘impassable’ surface inside control, but other research has also been carried out. One its workspace. Moreover, the problems of steering near research effort permitted an error in the output if required. such an impassable surface were described. In spite of Generalized inverse Jacobian22) minimizes the error. those important results, his work was not widely Extension of this method, called the SR inverse method, received, because the original papers were published in was first proposed for control of a manipulator and later Russian. Even though an English translation appeared, applied to CMG control40,41). Another research type several terms were used for a CMG, such as “gyroforce”, dealt with path planning. If the command sequence in “gyro stabilizer” and “gyrodyne”. His conclusion was the near future is given, steering can be planned that a system with no less that six units would provide beforehand which realizes not only singularity avoidance an adequately sized workspace including no impassable but also some degree of optimization42, 43, 44). In one surfaces. After this work, a six unit symmetric system of the research papers43), some paths were chosen by was designed for the Russian space station “MIR”27). Kurokawa in consideration of impassable surfaces. Since Some years later after Tokar’s studies, Kurokawa all these tended to take a heuristic approach, evaluation formulated passability again28) in terms of the geometric was made by computer simulation considering attitude theory given by Margulies. Most of these results control of a given satellite. coincided with Tokar’s work. In addition, the existence More realistic studies have also been made which of impassability in the roof type system was clarified29) dealt with attitude control using a CMG system, and a discrimination method using the surface curvature considering disturbance and other torquers. The largest was presented30,31,32,33). In the last paper, the theory problem may be a precision control using a CMG system. was expanded to a general system including a double Since a CMG system can generate a large output torque gimbal CMG system. It was made clear that multiple and its output resolution is critical for precision control, systems of no less than six units do not have any internal various analyses and simulations have shown that impassable surface, while any system of less that six pointing control by a CMG system can result in a limit units must have such a surface. Various configurations, cycle because of friction in gimbal motion45, 46, 47). In even containing faulty units, were compared with regard spite of efforts such as improvement of motor control48) to their workspace size as an extension of Tokar’s work. and torque cancellation by additional reaction wheels49), Along with these theoretical and general research the problem of precision control has not been overcome. works, intensive efforts continued to find an effective For application to the space station, another studies were steering law regarding the passability problem as a local carried out such as an effective combination of a CMG problem. Most of these dealt with the pyramid type and RCS50) and integration of CMGs and power
––– 2 ––– –– 2. Characteristics–– of1. Control Introduction Moment –– Gyro Systems –– storage51). In order to evaluate its attitude control alone, but it is made in consideration with the attitude performance, not only numerical simulations, but also control of artificial satellites. Exactness and strict real some experiments using real mechanisms have been time feature of steering laws are essential for the real- made, such as a platform supported by a spherical air time attitude control. bearing44, 52). The author also developed ground test For this aim, a geometric approach was taken. As equipment using normal ball bearings53) and attempted described above, there have been various research works robust attitude control using a CMG system54,55). dealing with singularity and steering laws. Most used The motion of a CMG system with regard to the computer simulations to evaluate their steering laws, for motion of the angular momentum vector is similar to lack of other methods. As simulations alone cannot the motion of a link mechanism22). Analysis of the guarantee the performance of a system as nonlinear as a motion and control of such a mechanism has been widely CMG system, it is necessary to clarify the problem of studied. Those results were, therefore, used for CMG singularity by other means. A geometrical approach is a control40, 41). On the other hand, some researchers first more effective way of simplification and qualitative studied CMG control and then applied their results to a comprehension. The theoretical portion of this work robot control56, 57, 58). In spite of various researches in aims for general formulation of singularity problems. robot kinematics59, 60, 61), generalized theory for Under consideration of these general results, singularity and inverse kinematics has not been extensive study was made for a specific type of system, formulated yet. that is, the pyramid type. The reasons why this system was chosen are: 1.2 Scope of Discussion 1) A three-unit system does not need further study because it has no redundancy. Systems with no less than six units also do not need detailed study for This research effort deals with the following subjects: singularity avoidance, a fact described in more detail (1) General formulation of an arbitrarily configured in this work. Thus, four and five unit systems remain CMG system, especially of single gimbal CMGs. for further study. (2) Geometric study of the singularity problem of a 2) Most previous research works dealt with this general single gimbal CMG system. pyramid type system. Four units are the minimum (3) Problem of exact and real-time steering of the having one degree of redundancy. The number of pyramid type CMG system. units is important in the real situation. By a (4) Proposal and evaluation of steering laws for the simplified evaluation, a system with fewer units is pyramid type CMG system. lighter for a given total storage of angular (5) Evaluation of various CMG systems. momentum. Also, steering law calculation is less The main purposes of this work are to clarify the complicated for a system with fewer units. singularity problems, to construct an exact and strictly 3) The pyramid type system has symmetry, which real-time steering law, and to specify and evaluate its enables easier analysis. Numerical data and performance. Among all, singularity problems are the analytical expression of some geometric most important relating to the others. A singularity can characteristics can be reduced by using this degrade a CMG system, even causing the system to loose symmetry. This fact is useful for actual control, and this situation might be fatal for an artificial implementation. satellite. Therefore, a CMG system must have redundancy and it must be controlled to avoid As geometric study is more qualitative rather than singularities by using an appropriate steering law. quantitative, ground experiments were performed to Problems include whether such singularity avoidance is demonstrate the performance of the steering laws. Also globally possible and which steering law can realize such for evaluation, various system types are compared in control. Even if a steering law cannot avoid all the terms of the size of the possible angular momentum singularities, the system’s working range of the angular vector operational space and the systems’ weight. momentum must be specified in which singularity As mentioned above, specific studies of an attitude avoidance is strictly guaranteed because such control are beyond the scope of this work. Such studies specification is necessary for designing the total attitude involve optimal maneuvering and angular momentum control system. Thus, this work deals with CMG systems management, which are possible only after the
––– 3 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– specification of CMG systems are given by using the surface will be defined. Passability and surface geometry results of this work. In addition, this work does not treat will be related by the curvature of the surface. in detail steering laws of double gimbal CMG systems, Chapter 5 will introduce a way of understanding the combinations of single and double gimbal CMGs, steering motion as to whether continuous control is combination of CMGs and other torquers, passive type possible, or how the impassable situation can be avoided, CMGs62), and systems with different size or controllable if possible. size CMGs63). Also, the effect of gimbal limit is not From Chapter 6 to 8, the pyramid type system will considered in general except in the proposed method for be detailed. Chapter 6 will offer analytical and geometric the pyramid type system. system results without considering a steering law. The impassable singular surface of this system will be fully 1.3 Outline of this Thesis defined. Chapter 7 will prove the ‘global’ problem. After various proposals are evaluated based on this result, a Chapter 2 will represent a general description of a new proposal will be offered. CMG unit and CMG systems. The difference between Chapter 8 will demonstrate the performance of the three types of torquers, reaction wheels, single gimbal proposed method by using a ground test apparatus. CMGs and double gimbal CMGs, will be described. Chapter 9 will offer evaluation not only of the Also an important parameter termed ‘workspace’ in this proposed method for the pyramid type system but also paper will be defined in terms of an attitude control of various system configurations. system. Chapter 10 will conclude this work. Chapter 3 will represent a general formulation of an Because double gimbal CMG systems and various arbitrary system of single gimbal CMGs, which includes systems other than the pyramid type system will not be the kinematic equation and the torque equation. The detailed in the main text, Appendices A and D will general steering law, singularity and singularity provide these details. Appendices B and C present avoidance will be outlined. This chapter is analytical detailed proofs of some theories given in Chapter 4. while the following chapters, from Chapter 4 to 7, are Appendix E will give the specifications and mainly geometrical. implementation of the ground test apparatus. Appendix Chapter 4 will detail singularity. A singular surface F will detail the kinematics of a general spatial link which includes the angular momentum envelope will mechanism which is analogous to the CMG kinematics. be examined. For this surface, ‘passability’ which is one of the most important characteristics of a singular
––– 4 ––– –– 2. Characteristics of Control Moment Gyro Systems ––
Chapter 2
Characteristics of Control Moment Gyro Systems
A control moment gyro (CMG) system is a torquer axes in the case of single gimbal CMGs and the outer for three axis attitude control of an artificial satellite. gimbal axes in the case of double gimbal CMGs. In the There are two types of CMG units and various following figures, these principal axes are indicated by configurations of three axis torquer systems. Designing arrows denoted by gi . a CMG system therefore includes a process of selecting The system of each configuration is named as a system a unit type and a system type defined by configuration. type such as twin type system or the pyramid type system. Among two unit types and various system types, a single gimbal CMG system of pyramid configuration is mainly described in this work. For the simple Gyro Effect Torque comparison, this chapter gives an outline of CMG system T Gimbal Motor characteristics with consideration paid to its use in an Gyro Motor attitude control system. The angular momentum ω workspace, torque output, steering law and singularity Flywheel problems are the important factors for evaluation of a AA CMG system. GimbalAA Mechanism 2.1 CMG Unit Type Angular Momentum Vector A CMG consists of a flywheel rotating at a constant speed, one or two supporting gimbals, and motors which (a) Single gimbal CMG drive the gimbals. A rotating flywheel possesses angular momentum with a constant vector length. Gimbal Outer Gimbal motion changes the direction of this vector and thus Motor generates a gyro−effect torque. A
There are two types of CMG units, as shown in Fig. AA Outer 2–1, single gimbal and double gimbal. A single gimbal Gimbal Flywheel CMG generates a one axis torque and a double gimbal CMG generates a two axis torque. In both cases, the direction of the output torque changes in accordance with gimbal motion. For this reason, a system composed of Inner several units is usually required to obtain the desired Gimbal torque. Gyro AAA Motor Motor 2.2 System Configuration Inner Gimbal (b) Double gimbal CMG Typical system configurations will now be discussed. The configuration is defined by a set of principal axes Fig. 2–1 Two types of CMG units of all the component CMG units, which are the gimbal
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2.2.1 Single Gimbal CMGs parallel to each other are considered and because a tetrahedron and hexahedron are complementary or Typical single gimbal CMG systems have certain kinds “dual” to each other. The three, four, six, and ten unit of symmetries, which can be classified into two types, systems are denoted as S(3), S(4), S(6) and S(10). The ‘independent’ and ‘multiple’. They are somewhat four unit or S(4) system, shown in Fig. 2–2(a), is called different in their mathematical description. the symmetric ‘pyramid type’. Most of this work deals with this type of system. An example of the six unit or (1) Independent Type S(6) system, shown in Fig. 2–2(b), is now in use on the Independent type CMGs have no parallel axis pairs. Russian space station “MIR”. Two categories of independent type CMGs, ‘symmetric Skew Type All individual units are arranged types’ and ‘skew types’, have been mainly studied. in axial symmetry about a certain axis as depicted in Symmetric Type Gimbal axes are arranged Fig. 2–2(c). Skew three and four unit systems of certain symmetrically according to a regular polyhedron. There skew angles are the same as the S(3) and the S(4). are five regular polyhedrons with 4, 6, 8, 12 and 20 surfaces. Possible configurations of this type are three, (2) Multiple Type four, six and ten unit systems, because only surfaces not In this type some number of individual units possess
θ g g4 3 3 h3 g4 θ 4 h4 g5 α g3 Z h4 h h3 α 2 θ h5 2 π Y g6 2 ⁄n h X g2 g2 2 h1 θ 1 h6 g1 g1
h1 (a) Pyramid type S(4) (c) Skew type
θ13 g1
g2 g g1 h2 6 θ12 g2 h6 θ23
g h1 1 g θ11 2 h g 3 g 5 1 θ22 h 5 g g3 2
θ21 h4 g4
(b) Symmetric type S(6) (d) Multiple type M(3, 3)
Fig. 2–2 Configurations of single gimbal CMGs
––– 6 ––– –– 2. Characteristics of Control Moment Gyro Systems ––
identical gimbal directions. These are denoted as M(m1, m2, ...) hereafter, where mi is the number of the units with the same gimbal direction. As an example, the system in Fig. 2–2(d) is denoted by M(3,3). A similar Gimbal Motor system called ‘roof type’15, 16) would be denoted as M(2,2) with this notation.
2.2.2 Two Dimensional System and Twin Type System
A single gimbal CMG system of an arbitrary number of units all having a common gimbal direction will be − called a two dimensional system in this work. In such a Fig. 2 3 Twin type system system, the angular momentum vector and output torque vector are always on a certain plane normal to the gimbal 2.2.3 Configuration of Double Gimbal CMGs direction. Though this type of system is not ordinarily used by itself for attitude control, it can easily be Two typical configurations of double gimbal CMGs visualized and understood. It is, therefore, used for some are an orthogonal type and a parallel type. The examples in this work. orthogonal type consists of three orthogonally positioned If a pair of single gimbal CMG units with a common units. This type of system was used for the ‘Skylab’ gimbal direction are driven in opposite directions by the space vehicle. The parallel type consists of an arbitrary same angle, the direction of the output torque is always number of units all having a common axis20). kept constant, as shown in Fig. 2–3. This type of system is called a ‘twin type’ or a ‘V−pair’ system5). 2.3 Three Axis Attitude Control Though a three axis system is easily designed by combining several twin type CMGs, such a system is The design requirement of a CMG system is − not so much advantageous. A three or more V pair determined by the specification of a spacecraft attitude system is identical to a multiple system, M(2, 2, ..., 2), control. There are various kinds of attitude control whose state variables are constrained, but its workspace techniques such as spin stabilization, bias momentum is smaller than that of the original multiple system. stabilization and zero momentum active control. The − Though a V pair system is the easiest to control, a last is also called three axis attitude control. Reaction multiple system can be also simply controlled as will be wheels and CMGs are commonly used torquers for this described later.
Disturbance A B C Tcom TCMG + Maneuver Vehicle CMG CMG Command Control Steering System Spacecraft Generator - Law Law
D Momentum Unloading Management Torquers Control Logic
Attitude & Rate Sensors
Fig. 2–4 Block diagram of three axis attitude control
––– 7 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– attitude control. management control block, D, because such torquers have their own limitations, i.e., a gas jet does not have 2.3.1 Block Diagram enough resolution and it have a limit of storage, and a magnetic torquer’s output depends on orbit position. A functional block diagram of a three axis attitude For effective management of angular momentum, the control is shown in Fig. 2–4. Most of the blocks are the space of allowed angular momentum of a CMG system same when either reaction wheels or CMGs are used. must be defined beforehand. This space is termed The attitude and rotational velocity commands are ‘workspace’ in this paper. The workspace must be generated by a maneuver command generator denoted included by the possible angular momentum space of by A in Fig. 2–4. The command and sensor information the CMG itself. Moreover, a simple shaped space such are the inputs to the vehicle control law block, B. This as a sphere tends to result in more simplified block calculates the torque necessary for control. The management. next block, C, shows the CMG steering law which calculates the CMG motion for the torque calculated by 2.3.4 Maneuver Command block B. In this manner the actual CMG system is driven and an output torque to the satellite is generated. The The command issued by a maneuver command blocks relating to CMG control are the CMG steering generator depends on the mode of operation. Typical law, C, and the momentum management block, D. Those operational modes are pointing, maneuvering, scanning two blocks are described first in the following sections. and tracking. In the pointing mode, precision is of Then, relating subjects, i.e., maneuver commands, primary importance and is affected by disturbances, disturbances and the motion of angular momentum torque response and resolution. The speed of vector will be explained. maneuvering as well as momentum accumulation while pointing is a matter of workspace size of the torquer. 2.3.2 CMG Steering Law 2.3.5 Disturbance The steering law block computes a set of gimbal angle rates which produce the required torque. The steering The time dependence of disturbances vary according law is usually realized in two parts, one being simply a to orbit parameters and a mission type, such as earth solution to a linear equation and the other for singularity pointing or inertial pointing. In any case, a disturbance avoidance by using system redundancy. may have cyclic terms and offset terms. The following This block is usually designed independent of the function is an example of disturbance used for the particulars of the total attitude control system. This simulation of HEAO with a pyramid type CMG implies that the vehicle control law (B in Fig. 2–4) is system14); designed under the assumption that the output of the ω ω − ω t Tg = (Txsin t, Ty(cos t 1), Tzsin t) , CMG system corresponds exactly to the command. The CMG steering law must satisfy this requirement. The where ω denotes orbital angular rate. Because there is meaning of this exactness is described in a later chapter. an offset in the y direction, angular momentum will be accumulated in this direction while pointing. 2.3.3 Momentum Management 2.3.6 Angular Momentum Trajectory A CMG and a reaction wheel are called momentum exchange devices because they don’t actually “produce” The size and shape of the workspace determines the angular momentum but rather exchange it with the maximum accumulation of disturbances or the maximum satellite. Such torquers have limits to their accumulation speed of maneuvering. A disturbance or a maneuvering of angular momentum, because the rotational speed of a command can be expressed as a function of time by a flywheel is limited. Therefore, another type torquer is trajectory of the angular momentum vector of the needed when it becomes necessary to offload excess satellite. Since the total angular momentum of the system accumulated momentum. This unloading is usually done is equal to the time integral of the disturbance, the angular by gas jets or magnetic torquers. The unloading process momentum trajectory of a CMG system can be expressed must be carefully managed by the momentum using the spacecraft’s momentum and disturbance. The
––– 8 ––– –– 2. Characteristics of Control Moment Gyro Systems –– workspace of a CMG system must include any possible Table 2–1 Component Level Comparison angular momentum trajectory when the unloading –––––––––––––––––––––––––––––––––––––––––––– torquers are not operating. Angular Momentum Torque Reaction Wheel 1 to 1000 1 2.4 Comparison and Selection Double Gimbal CMG 1000 to 3000 100 Single Gimbal CMG 10 to 2000 1000 –––––––––––––––––––––––––––––––––––––––––––– CMG systems and a reaction wheel system are all examples of the same type of torquers. In order to design a two axis torque. an attitude control system, some sort of selection criteria Maximum output torque is much different. A single is needed. By using the following performance indices, gimbal CMG can produce more output torque than a a brief comparison will be made, first at the component double gimbal CMG. The reason is as follows. The level then at the system level. output torque of a single gimbal CMG appears on the flywheel and is then transferred directly to the satellite 2.4.1 Performance Index across the gimbal bearings. The output torque can be much larger than the gimbal motor torque required to The performance of a CMG systems depends not only drive the gimbal. This is called ‘torque amplification’. on elements of hardware design, such as the CMG unit By contrast, some part of the output torque of a double type and the system configuration, but also on the design gimbal CMG must be balanced by the gimbal motors. of the steering law. These factors all affect the maximum Thus, in this case, the output toque is limited by the motor workspace and the magnitude of the output torque, two torque limit. nonscalar performance indices. Another performance index is the steering law complexity, which affects the 2.4.3 System Level Comparison attitude control cycle time and the capacity of an onboard computer. Table 2−2 shows a system level comparison for the three types of torquers being compared. Difference in 2.4.2 Component Level Comparison the first two indices, torque and weight, are derived from component level differences. The other two indices Table 2−1 clarifies the main differences among these relate to each other. The steering law of any reaction three torquers64). A reaction wheel has only one motor wheel system is linear and no singularity problems arise. which is used not only for accumulation of angular Steering law complexity and singularity problems of momentum but also for generation of torque. On the CMG systems, especially single gimbal CMGs, can be other hand, the CMGs use either two or three motors, serious and thus form the main subject of the present one for accumulation of angular momentum and the work. others for torque generation. Since the torque of a motor depends on its speed and the same maximum torque 2.4.4 Work Space Size and Weight cannot be generated over the motor’s working speed range, both angular momentum and output torque of a The size and shape of the maximum workspace are reaction wheel are much smaller than for CMGs. not compared in the above table because they depend Size and weight of a CMG depends on the size of the on the number of units and system configuration. flywheel and complexity of the mechanism. A double Workspace size as a scalar value, and the weigh of the gimbal CMG is the most complicated at the unit level, CMG system can be roughly evaluated in terms of the but less so at the system level because this unit generates number of units. Let’s consider similarly shaped
Table 2–2 System Level Comparison –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Torque Weight Steering Law Singularity Reaction Wheel 1 1 simple none Double Gimbal CMG 100 2 not simple slight Single Gimbal CMG 1000 2 most complex serious ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
––– 9 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– flywheels of diameter d and thickness t. Similarity but can still realize the same workspace size. Despite implies t ∝ d. Then, the weight and the size of maximum the fact that other factors are ignored in estimating the workspace of an n unit system, denoted as W and H, weight, it can generally be concluded that the systems follow the following relation if the rotational rate of the of less units have advantages in weight. gyro is the same: In this evaluation, it is assumed that the size of the work space is proportional to the number of units by the W ∝ n d2 t ∝ n d3, (2–1) same multiplier for any system. From the comparison H ∝ n ∫ (t d d2) dr ∝ n d5 . (2–2) in Chapter 9, this is almost true for systems of no less than 6 units in the case of single gimbal CMGs. This, If H is set constant, W is given by; however, is not true in the case of less that 6 units. W ∝ n d3 ∝ n 2/5 . (2–3) Therefore it is better to evaluate some configuration composed of 4 to 6 units. This implies that the system with fewer units is lighter
––– 10 ––– –– 3. General Formulation ––
Chapter 3
General Formulation
θ This chapter first defines vectors, variables and dependent upon the gimbal angle i. Once the initial parameters of a single gimbal CMG system in an vectors are defined as in Fig. 3–2, the other vectors are arbitrary configuration, after which a basic obtained as follows; mathematical description of several system h = h cosθ + c sinθ , characteristics are made. These characteristics are the i i0 i i0 i − θ θ kinematic equation, the steering law, the torque output ci = hi0sin i + ci0cos i . (3–2) performance index, and singularity avoidance. The The total angular momentum is the sum of all h shape of the maximum workspace and singularity i multiplied by the unit’s angular momentum value which problem are described in the next chapter. Similar is denoted by h. In this work, H denotes the total angular descriptions for double gimbal systems are given in momentum without the multiplier h: Appendix A. Σ H = hi . (3–3) 3.1 Angular Momentum and Torque This relation is simply written as a nonlinear mapping θ from the set of i to H; A generalized system is considered consisting of n θ identically sized single gimbal CMG units. The number H = f ( ) . (3–4) n is not less than 3 to enable three axis control. The θ θ , θ θ The variable, =( 1 2, ..., n), is a point on an n system configuration is defined by the relative dimensional torus denoted by T(n) which is the domain arrangement of the gimbal directions. The system state of this mapping. The mapping range is a subspace of is defined by the set of all gimbal angles, each of which the physical Euclidean space and is denoted by H. This θ are denoted by i. Three mutually orthogonal unit space is the maximum workspace. vectors are shown in Fig. 3–1 and defined as follows: By the analogy of this relation with a spatial link mechanism, this relation will be called “kinematics” or gi : gimbal vector, “kinematic equation” in this work (see Appendix F). hi : normalized angular momentum vector, The output torque without the multiplier h is obtained ci : torque vector, by taking the time derivative as follows. where T = − dH / dt = − Σ ∂h /∂θ dθ /dt . (3–5) ∂ ∂θ × i i i ci = hi / i = gi hi . (3–1) Any additional gyro effect torques generated by the The gimbal vectors are constant while the others are satellite motion are omitted because they are usually
θ gi hi0
θ i g hi h ci0 c c i
Fig. 3–1 Orthonormal vectors of a CMG unit Fig. 3–2 Gimbal angle and vectors
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− treated in the overall satellite system dynamics, which [c3 c1 c2], [c1 c2 c3]) , (3–11) includes the CMG system (see Chapter 8). where [a b c] denotes the vector triple product, a⋅(b×c). Because the total output torque is a sum of output of each unit, it is also given as, − Σ ω 3.3 Singular Value Decomposition and T = ci i = − C ω , (3–6) I/O Ratio where The magnitude of the total output torque is not a ω θ ω ω ω ..... ω t simple sum of the output of each unit. An elements of i = d i/dt, and = ( 1, 2, , n) . each output, ω c , normal to T cancels each other. The (3–7) i i ratio of input and output norms, |ω|/|T|, can be evaluated ω by a singular value of the matrix C. The variable i is the rotational rate of each gimbal. The vector ω is a component vector of a tangent space The matrix C can be decomposed into a diagonal × of T(n). The matrix C is a Jacobian of Eq. 3–4 and is matrix by two orthonormal matrices, Q (3 3) and R given by, (n×n) as follows;
C = (c c .... c ) . (3–8) 1 2 n σ 1 000.. 0 As the unit’s angular momentum value is omitted in = σ QCR 00002 .. , (3–12) Eqs. 3–5 and 3–6, the real output is obtained by σ 003 0.. 0 multiplying h. σ where i is called a singular value of C. As shown in 3.2 Steering Law Fig. 3–3, the maximum ratio of the input and output norms is given by the radius of the ellipsoid whose The ‘steering law’ functions to compute the gimbal principal diameters are the singular values. Thus, the ω rates, , necessary to produce the desired torque, Tcom, and is generally given as a solution of the linear equation ω ω 3 ... n given in Eq. 3–6: ω − t t −1 − t t −1 = C (CC ) Tcom + (I C (CC ) C) k . (3–9) |ω|=1 ω 2 where I is the n × n identity matrix and k is an arbitrary ω 1 vector of n elements. The first term has the minimum norm among all n - sphere −1 solutions to the equation. The matrix Ct(CCt) is called (a) Gimbal rate a pseudo-inverse matrix. The second term, denoted by ω N, is a solution of the homogeneous equation; ω C N = 0 . (3–10) σ 3 ω This implies that the motion by this N does not generate a torque (T) and keeps the angular momentum (H) H constant. In this sense, this term is called a ‘null motion’. σ The null motion has n−3 degrees of freedom because it 1 σ is an element of the kernel of the linear transformation 2 represented by C. An effective method of calculating a null motion is given in Ref. 22. For example, a null motion of a four (b) Angular momentum ellipsoid unit system is generally given as, ω − N = ([c2 c3 c4], [c3 c4 c1], Fig. 3–3 Input ⁄ Output ratio
––– 12 ––– –– 3. General Formulation –– size of this ellipsoid represents the performance index of the output torque. The following relations are derived h3 from the fact that all row vectors of C are unit vectors. h2 σ 2 + σ 2 + σ 2 = Trace(CCt) = n , (3–13) 1 2 3 H t σ ⋅ σ ⋅ σ 2 h S det(CC ) = ( 1 2 3) . (3–14) 1 S0 1 Singular Line O A B 3.4 Singularity
Angular Momentum The steering law function in Eq. 3–9 is invalid at lower Envelope ranks of C where the following condition is satisfied:
det(CCt) =0 . (3–15) (a) Angular momentum Referring to Fig. 3–4, degeneration of rank implies that all the possible output, T of Eq. 3–6, does not span three dimensional space. Since all the row vectors, ci, of O matrix C become coplanar, the output T does not have a component normal to this plane. Let u denote the unit normal vector of this plane and be called a ‘singular A vector’. It is defined by ⋅ S u ci = 0, where i = 1, 2, ...., n , (3–16) 0 and may also be written in the matrix form as; B ut C = 0 . (3–17)
The rank of C does not generally reduce to 1. The S1 rank is unity only when all ci are aligned in the same direction. This can only happen if all the gi are on the (b) Vector arrangement same plane, as the case for a roof type system. When the system is singular, one of the singular Fig. 3–5 Typical vector arrangement values reduces to 0. In this sense, the minimum singular for a 2D system value can be used as a singularity measure. But the determinant, det(CCt), is also useful as such a measure and is more easily calculated. envelope’ is clearly singular. The singular H other than Figure 3–5 shows two types of singularity of a two this envelope are called ‘internal’. dimensional, three unit system. The border of the maximum workspace, termed the ‘angular momentum 3.5 Singularity Avoidance
Singular Vector u Any steering law is based on the solution of Eq. 3– 9. Among all solutions, the pseudo-inverse solution with no null motion was regarded effective. However, the c1 fact that the pseudo-inverse solution has a minimum norm implies that once the torque vector is nearly normal to the required output then this unit hardly moves. If c2 the required torque maintains its direction, such a unit
cn keeps its state so the system sometimes approaches a singular state. In order to avoid such a situation, singularity avoidance is usually included in the steering Fig. 3–4 Singularity condition and singular vector law.
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3.5.1 Gradient Method pyramid type single gimbal CMG systems, various simulations showed that a gradient method is not A system containing more than three units possesses effective. Details of this problem is described in null motion redundancy. Freedom in determining null Chapters 4, 5 and 7. motion can realize singularity avoidance while keeping the output torque exactly equal to the command. The 3.5.2 Steering in Proximity to a Singular State gradient method is a general method in which some objective function is maximized. The following There is no solution to Eq. 3–6 in a singular state formulation of a gradient method is taken from Ref. 21. except when Tcom is orthogonal to the singular vector The objective function, W(θ), is chosen as a u. Even when Tcom is normal to u, the solution is not continuous function of θ. It is zero in the singular state given by Eq. 3–9 because the linear equation is and otherwise positive. The dependence of W on CMG mathematically singular. A generalized solution can be motion is: obtained which is the exact solution when Tcom is normal to u otherwise minimizes the output error. The ∆ Σ ξ ω W = i i , (3–18) minimum error is realized when the output is equal to where the projection of the torque command onto the plane normal to the singular direction (Fig. 3–6). Such motion ξ ∂ ∂θ 22) i = W/ i . (3–19) is given as : In order to obtain the objective function extremum, ω − t t t −1 = C (CC + k uu ) Tcom . (3–22) the motion ω should be determined so that ∆W is positive. This ∆W has two parts, one given by the Derivation of this is explained by supposing that there pseudo-inverse solution and the other by a null motion. is a virtual CMG unit whose torque vector c equals u. Another method called the SR (Singularity Robust) The first depends on the command torque Tcom, while the latter depends on the selection of a null motion. inverse steering law is proposed as a smooth extension 41) Though the first part cannot be changed, the latter can of this . This method minimizes the weighted sum of ω be freely determined. The latter part is evaluated as the input norm, | |, and the norm of the error. The SR follows; solution is given as: ω − t t −1 ∆ ξt − t t −1 = C (CC + W) Tcom, WN = (I C (CC ) C) k . (3–20) where W is a n×n matrix . (3–23) It is easily observed that the matrix (I − Ct(CCt)− 1C) is semi-positive symmetric. If the vector k is In both methods, the solution is zero if the command, selected as: Tcom, is either zero or parallel to the u direction. This method, therefore, cannot always guarantee avoidance k = k ξ, where k >0 , (3–21) of a singular state nor can it escape from one. Moreover, ∆ this kind of control is effective only if the attitude control then WN becomes a semi-positive quadratic form. Thus, the null motion by this k results in non-negative is not totally degraded by the error in torque. Details ∆ are described in Section 7.2. WN, so it is expected that singularity is avoided. Various objective functions have been proposed, such as: Possible Output Tcom
(1) (det(CCt))−1/2, 21) u σ 36) (2) min( i), (3) min(1/|d |), i c1 where di is a row vector of the matrix Ct(CCt)−1, 35) c2 c Σ × 2 27) n (4) i,j |ci cj | , .
This gradient method has been successful for double gimbal CMG systems21). However, in the case of Fig. 3–6 Steering at a singular condition
––– 14 ––– –– 4. Singular Surface and Passability ––
Chapter 4
Singular Surface and Passability
ε ⋅ Angular momentum vectors in a singular condition i = sign( u hi) . (4–1) form a smooth surface which includes the angular Thus there are 2n combinations of singular points momentum envelope. This chapter first summarizes the for the given direction u. This combination is denoted geometric theory of the singular surface of a general by ε or by a set of signs, such as {+ + − + ... +}. single gimbal CMG system by following the research For the given singular direction u and the given set work in Ref. 22. It includes a definition of a singular of signs, each torque vector in the singular condition is surface, a mapping from a sphere to the surface, and determined by: techniques for drawing the surface by computer calculation. By using these techniques, the workspace ε × × cSi = i gi u / |gi u | . (4–2) is visualized for various system configurations. Also, geometric characteristics such as Gaussian curvature of From this point, variables subscripted by S denote a singular surface is defined. singular point values. The total angular momentum HS The passability of a singular surface is then defined. is obtained as follows: The existence of an impassable surface explains why Σ ε × × × HS = i (gi u) gi / |gi u | . (4–3) most steering laws fail to generate output starting from certain initial states. A gradient method works well for This defines a continuous mapping from u to HS ε avoiding passable singular points but not for avoiding while the i are fixed as parameters. The domain of u is ± impassable ones. a unit sphere except gi direction, because the ± The passability can be determined by the curvature denominator of Eq. 4–3 is zero when u = gi . Thus HS ε of the singular surface. It is demonstrated that any with fixed i form a two dimensional surface with u independent type system has an internal impassable covering this sphere. This surface is denoted as Sε. If ε − surface while multiple type systems of no less than six all the i are reversed and the vector u is changed to u, units have no internal impassable surfaces. HS remains the same. This implies that the surface of ε ε { i} and the surface of all the i reversed are identical. For example S{− + +} is the same as S{+ − −}. One may 4.1 Singular Surface ε thus suppose that no less than half of the i are positive. Thus, the number of different surfaces is 2n−1. 4.1.1 Continuous Mapping ± In case that u = gi, any state of this ith unit satisfies
Let’s examine all the singular points and their H vectors. First, an independent type system is assumed u h in the following discussion. S g The torque vectors, ci, satisfy the condition given by Eq.(3–16) when the system is singular. On each singular ε = 1 point, a singular vector u is defined. As a reverse relation c of this, singular points are obtained from a given u vector. S –c Given any singular vector u, there are two S ε = – 1 possibilities of singularity condition for each unit as hS and –h in Fig. 4–1. The two cases are distinguished by S – hS the following sign variable;
Fig. 4−1 Vectors at a singularity condition
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z the singular condition. As the vector hi rotates about gi, g 1 these singular H form a unit circle which appears as a g2 hole or a window of the surface Sε as shown in Fig. 4–2 − (b). As there is a hole for each gi, or gi, direction, the ε surface has 2n holes in total. Surfaces of different { i} are connected by these unit circles (for example, C1 in Fig. 4–2 (b) and (c)). Thus all the surfaces form a closed surface. This closed surface is called a ‘singular surface’. It may be noted that the same kind of continuous mapping xyθ θ is defined from u to S with all the S forming a two dimensional surface in the n dimensional torus of θ. Such a surface, however, is not termed a singular surface in this paper. –g3 –g4 An independent type system is assumed in the above (a) Lattice points of the unit sphere of vector u discussion. In the case of a multiple type system, the number of different singular surface is 2m−1 where m is the number of groups. Each surface has 2m holes of Unit Circle C1 diameter of several values which is determined by the g Envelope 1 number of units in a group and sign ε. In case that u = ± z gi, any state of units of this group satisfies the singular condition. Thus, all singular H of this u form a circular plate which fills the hole. Another singular surface of different sign connects to this plate by a circle of different xy diameter.
4.1.2 Envelope
The angular momentum envelope, which is the border of the maximum workspace, is most definitely (b) Singular surface of all sign positive denoted by ε S{++++} singular. The surface corresponding to all i positive is clearly a part of the envelope. Surfaces with one negative Unit Circle C1 Envelope Portion sign which is connected to this surface by the holes share Internal Portion the envelope surface in the case of an independent type system. Unit Circle C 2 The envelope of a multiple type system consists of a singular surface of all positive signs and circular plate which fills 2m holes22). The one negative sign surfaces do not share the envelope surface and is fully internal. The singular surface of a M(2, 2) roof type system shown in Fig. 4–3 is part of an envelope of all positive signs. There are four circular holes of diameter 2. The circular (c) Singular surface of one minus sign denoted by S{−+++} plates filling these four circles share the envelope. The singular surface of one negative sign is connected at the center of these plates. Fig. 4–2 Examples of the singular surfaces for the pyramid type system. Each dot of Figs. (b) & (c) corresponds to the lattice 4.1.3 Visualization Method of the Surface point of Fig. (a). The unit circle indicated by C1 connects two singular surfaces S{+ + + +} & S{– + + +}. Other The singular surface and envelope are visualized by circles of the surface S{– + + +}, C2 for example, are con- taking θ at each lattice points of the unit sphere and nections to other singular surfaces such as S{– – + +} calculating the angular momentum using Eq. 4–3.
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z
x y
Unit Circle C Fig. 4–3 Envelope of a roof type system M(2, 2). 1 Dots are obtained from Eq. 4–3 for lattice points Fig. 4–4 Cross sections of a singular surface of of a u sphere with all positive signs. Circles are the pyramid type system. filled by plates. The outermost unit circle is the same as C1 in Fig. 4–2. The other lines are cross sections of planes orthogonal to the gimbal axis g1. Figures 4–2 and 4–3 are such examples. A singular surface and an envelope may also be visualized using various cross sections. The following inverse mapping 4–4 is such an example. theory22) is available to obtain a cross section of the The proof of this theory is given in Appendix B.3. singular surface. 4.2 Differential Geometry
Inverse Mapping Theory Geometric theory presented in Ref. 22 formulated θ Suppose that is constrained singular and V is an fundamental forms of the singular surface and clarified arbitrary vector normal to u. If the differential dH geometric characteristics. Other than Gaussian along the singular surface satisfies, curvature, details are given in the original paper. dH = V × u , (4–4) 4.2.1 Tangent Space and Subspace then the differential of u is given by Suppose that θ is on a singular point. The differential du = κ ( CPCtV) × u , (4–5) dθ is a tangent vector of the θ space. The following where κ is the Gaussian curvature of the singular three subspaces are defined in the tangent space of the θ surface, which is described in Section 4.2.2. The space32): Θ θ matrix P is a diagonal matrix whose nonzero element S: Singularly constrained tangent space of the Pii is given by: space (two dimensional). Θ : Space of null motion, i.e., the null space of C (n− P = p = 1 / (u ⋅ h ) . (4–6) N ii i i 2 dimensional). Θ Θ T: Complementary subspace of N (two Using this theory, a cross section of the singular dimensional). The solution given by Eq. 3–22 surface is calculated by the following procedure. First, for all Tcom belongs to this space. obtain a singular point on the cross sectional plane and The elements of these three subspaces are denoted by θ θ θ its u vector by some means. Second, obtain dH on the d S, d N and d T. These are illustrated in Fig. 4–5 for intersection of the surface tangential plane and the cross a two dimensional three-unit system, for example. The sectional plane. Third, obtain V by Eq. 4–4 and du by general bases of subspaces are given in Appendix B.1. Eq. 4–5 after which dθ is obtained by the relation dθ = ⋅ 4.2.2 Gaussian Curvature pi ci du (Appendix B). Finally, H on the cross sectional θ plane is obtained by numerical integration of d . Figure The Gaussian curvature, κ, of a singular surface is
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By using the relation as, Singular Direction ∂ ∂θ u B H / i = ci , dθ =(∆, 0, ∆)t ∂2 ∂θ ∂θ N1 H / i j = – hi, if i = j otherwise 0 ,(4–9) the difference ∆H as h B 1 h2 ∆ θ θ − θ H = H( S+d ) H( S) , (4–10) B θ ∆ ∆ t h3 d N2=(0, , ) is expressed as,
(a) Singular state Θ ∆ Σ θ − Σ θ 2 (b) Null motion N H = ci d i 1 / 2 i hi(d i) , (4–11) where the third order terms are omitted. The first order difference is a linear combination of ci and has no component in the u direction. On the other hand, the second order term may have a component to this direction. More specifically, B B ∆ ⋅ ⋅ Σ θ 2 θ ∆ ∆ ∆ t θ ∆ ∆ ∆ t H u = 1 / 2 u (– ihi (d i) ) , d S=( , , ) d T=( , , – ) Θ Θ − Σ θ 2 (c) S (d) T = 1 / 2 i(d i) / pi . (4–12)
Fig. 4–5 Infinitesimal motion from a singular This may also be expressed in matrix form as: point of 2D system. ∆H⋅u = −1 / 2 dθt P−1 dθ . (4–13) θ θ θ Four independent motions, d N1, d N2, d S and θ Θ Θ This is a quadratic form of dθ . d T, are members of three subspaces, N, S i Θ If any dθ are decomposed as follows, and T. θ θ θ d = d S + d N , (4–14) given by: the quadratic form (4–12) is also similarly decomposed: 1 / κ = 1/2 Σ Σ p p [c c u ]2 . (4–7) i j i j i j ∆ ⋅ − θ t −1 θ H u = 1 / 2 d S P d S The proof of this is detailed in Appendix B.2. − 1 / 2 dθ t P−1 dθ . (4–15) The sign of Gaussian curvature has an important role N N in determining the following passability of the surface. −1 θ This is derived by the fact that P d S is an element Θ θ t −1 θ of T hence d N P d S = 0 (See Appendix B.1). 4.3 Passability Let QS and QN denote the two quadratic forms on the right of Eq. 4–15: − θ t −1 θ 4.3.1 Quadratic Form QS = 1 / 2 d S P d S , Q = −1 / 2 dθ t P−1 dθ . (4–16) θ N N N Suppose that the system state is singular, that is, S θ θ is a singular point and HS is on the singular surface. It The vectors, d S and d N, are elements of the tangent θ Θ Θ is instructive to examine an infinitesimal change in subspace S and N, and they can be represented by from this singular point and the resulting infinitesimal using bases of each subspaces: change in H. A second order Taylor’s series expansion dθ = φ e + φ e , θ θ S 1 S1 2 S2 of H( ) in the neighborhood of the S is given by: dθ = ψ e + ... + ψ e , (4–17) θ θ N 1 N1 n-2 Nn-2 H( S+d ) θ Σ ∂ ∂θ θ where e and e are bases of Θ and Θ . These = H( S) + i H / i d i Si Ni S N expressions are expressed simply as, Σ Σ ∂2 ∂θ ∂θ θ θ + 1 / 2 i j( H / i j)d id j θ φ d S = ES , + O(dθ 3) . (4–8) i × φ × where ES : n 2, : 1 2 ,
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θ ψ d N= EN , limited to a certain side of the surface. Thus, no motion × ψ × from this side of the surface to the other side is possible. where ES : n n–2, : 1 n–2 . (4–18) With indefinite forms, some motions result on one side Substituting these into two quadratic forms in Eq. 4–16 of the surface while others appear on the other side. results in the following: The signature is the characteristics of the form itself, − φt t −1 φ independent of the variables which is dθ in this case. QS = 1 / 2 ES P ES , N Thus, any singular point is categorized by this − ψt t −1 ψ QN = 1 / 2 EN P EN . (4–19) characteristics such as definite or indefinite. The first quadratic form is of order 2 and expresses θ 4.3.3 Passability and Singularity Avoidance the curvature of the singular surface, because d S represents a motion on the surface. The second quadratic − Since the quadratic form and its derivatives are form is of order n 2. By the definition, this QN is ∆ θ ⋅ continuous with respect to θ, its eigenvalues which H(d N) u. Therefore, if this quadratic form is not zero, this null motion moves the vector H away from the determine the signature are also continuous along the surface as shown in Fig. 4–6. surface. This implies that if a point has a definite form Note that the decomposition in Eq. 4–15 is not always then its neighborhood likewise does. The points of θ definite form make up a certain area of the surface, near possible, for example if Cd S = 0. This case, however, can be treated by similarity with another neighborhood. which it is not possible to pass from one side to the other if θ is in the neighborhood of this singular point. In this u sense, such an area is called ‘impassable’, while that of an indefinite form is termed ‘passable’. This notation Q N follows that of Tokar26) who pioneered this work. Other ∆ θ ∆ H( N) notation used in other references are elliptic/ 28, 65) 32) H hyperbolic and definite /indefinite . Another aspect of this form category is as follows. QS If the singular point is passable, i.e., having an indefinite ∆H( θ form, a certain value of dθ results in a zero value of S) N Singular surface θ the quadratic form. The motion by this d N keeps H on the singular surface but θ does not stay singular. This Fig. 4−6 Second order infinitesimal motion implies that escape from the passable singular point is from singular surface possible while keeping H the same. On the contrary, no motion can keep H at an impassable singular point. The internal singular point of a two dimensional 4.3.2 Signature of Quadratic Form system is passable. Figure 4–7 shows two motions in opposite directions at the singular point. The null motion Σ Any quadratic form, aij xixj, can be transformed to Σb y 2 by using a regular transformation from {x } to i i i Singular direction u {yi}. The set of two numbers of positive and negative bi are called “signature” of the quadratic form. Any quadratic form has a unique signature, that is, the signature does not depend on the transformation, as is Sylvester’s law of inertia. B B By the signature, a quadratic form is categorized as dθ=dθ – dθ dθ =dθ + dθ definite, semi-definite or indefinite. Definite form have N1 N2 N1 N2 =( ∆, – ∆, 0)t =(∆, ∆, 2∆)t only the same signs, while an indefinite form has both positive and negative signs. A semi-definite form has (a) ∆H⋅u < 0 (b) ∆H⋅u > 0 only the same sign but their number is less than the order of the form. Fig. 4–7 Possible motions in both direction of u at a singular point. This is the case of an internal singular If the quadratic form, QN is definite or semi-definite, θ state of a 2D system. Infinitesimal null motions d N1 it implies that any motion away from the surface is θ and d N2 are defined in Fig. 4–5.
––– 19 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– in Fig. 4–5 (b) allows the system to escape from the From these results, passability is discriminated for ε singular point while keeping H the same. any singular point as follows. First obtain the sign { i}. From the above discussion, it is clear that no steering Reverse all the sign if required so that the number of law can avoid an impassable singularity if the command negative signs is less than that of positive signs. If they is to approach the surface and the initial θ is in the are all positive, this surface is impassable. If more than neighborhood of the impassable singular point. On the two signs are negative, this surface is passable. The other hand, a steering law such as the gradient method remaining cases may correspond to the above two cases is effective in avoiding a passable singularity because (2) or (3). The next procedure is to calculate the Gaussian escape from such a singular point is always possible even curvature κ by Eq. 4–7. If there is only one negative when H is on the singular surface. sign, it is impassable when κ is negative otherwise passable. If there is two negative signs and κ is positive, 4.3.4 Discrimination we need additional calculation to determine passability.
Passability of a surface is defined by the signature. u The following discussion gives a discrimination method of this by the sign {ε } and the curvature of the surface. i Singular Surface Equations 4–18 represents a basis change for each subspace. As mentioned above, the signature is conserved by any basis change. Thus the signature of the total quadratic form is conserved and is simply ε obtained by the signs of pi, which is i, because of Eq. 4–12. Thus, passability which is defined by the signature (a) ε = {+ + + ... +} of QN, is determined by the total signature and the signature of Q . The signature of Q indicates S S u characteristics such as concavity/convexity of the surface because this quadratic form expresses curvature of the singular surface. Thus the following three conditions for an impassable surface are obtained in terms of the sign and the curvature of the surface32).
Condition (1) ε={+ + ... +}. Both QN and QS have only positive signs. This (b) ε = {– + + ... +} singular point is on the surface S{+ + ... +} which is a part of the envelope and is trivially impassable. Gaussian curvature is positive and the surface is convex to u. (Fig. 4–8(a)) u ε Condition (2) All the i but one are positive and the − signature of QS is { +}. This surface is partially on the envelope and impassable. Some part of this impassable surface is possibly inside the envelope. The Gaussian (c) ε = {– – + ... +} curvature is negative and the surface is a hyperbolic saddle point. (Fig. 4–8(b)) ε Condition (3) All the i but two are positive and the Fig. 4−8 Local shape of an impassable singular − − signature of QS is { }. surface. The surface is fully inside the envelope. The A side of the surface in –u direction is an allowed surface is concave to u and the Gaussian curvature H region while another side to u direction is unreachable through the surface. (a) is a concave is positive. Note that positive κ is not a necessary part of the envelope, (b) is a saddle point of the condition for this because there is a possibility that envelope and internal surface, and (c) is convex the signature of QS is {+ +}. (Fig. 4–8(c)) and fully internal.
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By removing one unit whose sign is negative and by In Fig. 4–9 it is clear that although the symmetric six checking the above condition (2) for this subsystem, unit system S(6) has internal impassable surfaces, they passability of the original system is determined. This is are very near the envelope. On the contrary, Figs. 4– proven in Appendix B. 4. 10, 4–11 and 4–12 show that the skew five unit system and the S(4) system have internal impassable surface 4.4 Internal Impassable Surface considerably further inside their envelope. If the workspace of these systems is defined such that it does not include these impassable surfaces in order to assure 4.4.1 Impassable Surface of an Independent Type singularity avoidance, it becomes much smaller than the System workspace given by the envelope.
In the case of an independent type system, the 4.4.2 Impassable Surface of a Multiple Type singular surface with all ε positive except one joins i System smoothly into the envelope, as depicted in Fig. 4–4. Because the surface and the curvature are continuous, The analysis of a multiple type system is quite any impassable portion also goes into the envelope. different from the above discussion. For multiple Thus, any independent type system has an impassable systems, every surface corresponding to all ε positive surface distinct from the envelope. i except one are totally inside the envelope. It is instructive Figures 4–9, 4–10, 4–11 and 4–12 show examples to examine passability conditions (2) and (3) of section of internal impassable surfaces, along with the envelopes. 4.3.4. Here, each variable is subscripted by the group number, because all the variables of the same group can be represented by only one member. The number of units in the ith group is denoted by mi.
z
z
x y
x
(a) Envelope. Singular surface S{+ + + + +} (a) Envelope. Singular surface S{+ + + + + +}
Envelope Envelope Internal Part Internal Part z z
x x
Unit of H Unit of H
(b) Impassable surface S{− + + + +}
(b) Impassable surface S{− + + + + +} Fig. 4–10 Impassable surface of Skew(5) Fig. 4–9 Impassable surface of S(6) with skew angle α = 0.6 rad.
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the right is the Gaussian curvature of the subsystem envelope excluding the ith group, and thus is positive. The first term is zero if mi=2 and otherwise positive. Thus the overall curvature is positive and condition (2) z is not satisfied.
Condition (3) Suppose that two signs are negative. x If the two units corresponding to these two negative signs belong to different groups, condition (3) simply results in condition (2) of the subsystem by removing one of the two units, so the condition is not satisfied. Suppose then that the two units corresponding to the two negative signs belong to the same group, this being the ith one. If the unit number, mi, is larger than two, the above (a) Envelope. reasoning is applied and the condition is not satisfied. Singular surface S{+ + + + +} If mi=2, Envelope κ − − Σ 2 Internal Part 1 / = 1/2 ( pi pi ) j pj [ci cj u ] 2 + Σ ≠ Σ ≠ p p [c c u ] . (4–21) z j i k i j k j k If the overall system is definite, the subsystem without one unit of the ith group is also definite and κ x condition (2) is satisfied for this subsystem, so 1 / is negative. In this case, Eq. 4–21 in its entirety is also negative, so the condition is not satisfied.
The discussion presented here does not hold for (b) Impassable surface S{− + + + +} systems of fewer units, such as M(2,2). Further details are examined in Appendix C and the results lead to the Fig. 4–11 Impassable surface of another Skew(5), with skew angle α = 1.2 rad. following conclusion
4.4.3 Minimum System Condition (2) Suppose that only one of the signs is negative and it is in the ith group. The Gaussian The conclusion is as follows. curvature of (4–7) is written as: Any multiple type system with no less than six units κ − Σ 2 1/ = 1/2( pi +pi +...+pi) j pj [ci cj u ] has no impassable singular surface other than the envelope, while any independent type system has internal +Σ Σ p p [c c u ]2 , (4–20) j≠i k≠i j k j k impassable surfaces. where pi and all the pj are positive. The second term on
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Chapter 5
Inverse Kinematics
The impassable surfaces defined in the previous The inverse image is a sum of manifolds and singular chapter cause steering law problems. It is possible to manifolds, which are denoted by Mi and MSj leave this problem unsolved and define a workspace respectively. Note that this manifold should be termed which excludes the impassable surfaces, but the resulting rather ‘null motion manifold’ or ‘self-motion space would be much smaller in the case of a four or manifold’60). In this work however, no other manifold five unit system. Impassability, however, is defined is used and it is simply called ‘manifold’. locally only in the neighborhood of an impassable The shape of manifolds in the neighborhood of a singular point. There is a possibility to avoid an singular point is characterized by the quadratic impassable situation by using some kind of global relationship given by Eq. 4–12. Suppose that H is in the control. For this, a geometric approach was taken in neighborhood of a singular surface where H = HS + eu. order to understand the CMG control qualitatively by By the same discussion as Eqs. from 4–10 to 4–15, θ θ ignoring the factor of time. The sequence of torque possible in the neighborhood of the singular point, S, commands is represented as a trajectory of the angular satisfies the following quadratic relation; momentum vector, while the possible gimbal angles are − Σ θ 2 ≈ 1⁄2 (d Ni) ⁄ pi e, represented by a manifold. By using equivalence θ θ θ relations of manifolds and their connections, conditions where = S + d N . (5–1) necessary for continuous control are formulated. θ In this equation, the motions, d N, is a tangent vector θ at the singular point S. In the case of an impassable 5.1 Manifold singular state, this quadratic form is definite, so this manifold resembles a super-ellipsoid. The quadratic A steering law is a method to obtain gimbal rates form of an impassable singular point is indefinite, so which corresponds to a given torque command. If we the shape of the manifold resembles super-hyperbolic ignore the factor of time, the steering law is regarded as surfaces in the neighborhood of this singular point. This a method to obtain gimbal angles by a given change of is illustrated in Fig. 5–1 for a four unit system, for which the angular momentum. This is the reverse relation of the manifolds are loops in the four dimensional torus. the kinematic equation 3–4. The (forward) kinematics is a one-to-one mapping but the reverse relation, which MS(H ) = {θ } M (H –e u) S S S 01 S M (HS) is called an ‘inverse kinematics’, is generally a one-to- (H +e u) θ M0 S S multi mapping. Therefore, possible θ having the same H is given by an inverse image of this mapping. The inverse image from H to θ is a set of sub-spaces disjoint to each other. Supposing that a sub-space has (H +2 e u) (H +e u) no singular state, an n−3 dimensional tangent space is M0 S M0 S M1(HS+e u) defined at each point of this space as a linear space of (a) Ellipsoidal (b) Manifolds crossing null motion. Thus, this sub-space is a n−3 dimensional manifolds around near a passable manifold. Supposing that a sub-space has singular an impassable singular point. singular point. points, no tangent is defined there, but even in this case, tangent spaces are defined at all other points of this space. Fig. 5−1 Manifolds in the neighborhood of a Thus, this sub-space is nearly the same as a manifold singular point. and in this work will be termed a ‘singular manifold’. These manifolds are one-dimensional loops if the number of units is four.
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5.2 Manifold Path H HS–e u
As H changes continuously, each manifold changes HS its shape continuously as shown in Fig. 5–2. A manifold may deform its shape into a point when H crosses an HS+e u impassable surface and may bifurcate when H crosses a θ passable surface. These continuous change of a manifold M0 M1 can be simplified as a continuous path in the manifold (c) Continuous change of manifolds in the space, where each manifold is regarded as a point (Fig. neighborhood of impassable H. 5–3). If we need an exact definition of manifold continuity, it can be made based on the distance d H M0 M1 M2 between manifolds which is defined as follows; HS–e u
d(MA, MB) H | θ − φ ∀θ ∈ ∀φ ∈ S = max( min( ); MA ) ; MB ) , H +e u (5–2) S θ | θ − where MA and MB are two manifolds and the norm (d) Continuous change of manifolds in the φ | is defined appropriately in the gimbal angle space. neighborhood of passable H. By this definition, a manifold becomes discontinuous at a bifurcation point. Fig. 5Ð2 Continuous change of manifolds The meaning of a continuous manifold path can be corresponding to H path across a singular point. thought of in the following terms. If the manifold (M1 in Fig. 5–3 for example), including an initial θ, is on a
continuous manifold path for a given H path (the path H 3
2 M 2 = 6 θ D S2 H0H1 in Fig. 5–3), then any θ of the manifold on the m other side of the path (M2 in Fig. 5–3) can be reached by
some continuous steering method using an appropriate H1 2
M4 M5 3 = θ M2 D null motion, while any of another manifold (M4 in m Fig. 5–3 for example) cannot be reached. If the manifold path θ H S1 path bifurcates, path selection (from M3 either to M4 or Domain
to M5) depends on the null motion hence on the steering
1 = 2 = θ H D method. If the manifold path including the initial 0 m M M3 θ 1 terminates somewhere for a given H path ( S2 for the path H1H2 in Fig. 5–3 for example), no steering method Manifold Paths can realize this motion. Fig. 5−3 An example of a continuous manifold path. θ A passable singular point S1 is a bifurcating 5.3 Domain and Equivalence Class θ point and an impassable point S2 is a terminal of the path. The angular momentum space is divided into several domains by one or more continuous singular surfaces. These will hereafter be simply termed ‘domains’. Let A manifold equivalence relation is defined as follows: each domain have no singular surface inside and its border be a set of singular surfaces. Each domain will definition: Two manifolds of a domain are considered be denoted by Di. Any continuous path of H inside a ‘equivalent’ when there is a path from one manifold domain corresponds to a finite number of continuous to the other which corresponds to an H path inside manifold paths with neither bifurcation nor termination. the domain. Thus, the number of manifolds for each point in the domain is constant. All the equivalent manifolds form a domain in the
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Singular Surface Equivalence Class Manifold Path H path G1 θ MA MA A HA θ MB MB B
H θ B ’B D2
M ’B Domain D1 G2 θ H space Manifold space space
Fig. 5−4 Relations between H space, manifold space and θ space. manifold space which is isomorphic to the original a neighbor domain which results in termination from domain of H. As the representation of this set of this class. By this definition, the class of the domain equivalent manifolds, an ‘equivalence class’ is defined. just inside the envelope is not a terminal class even The number of the classes is called the ‘order’ of the though it terminates on the envelope, because an H path domain. Let Gi and m denote the class and order, exiting the envelope has no meaning. respectively. The order of each domain is obtained in a Each domain is classified into one of the following step by step fashion. The outermost domain next to the three types by considering the order and number of envelope is of order 1. Two domains facing each other terminal classes, k: have an order difference of 1, because only one manifold Type 1: m = k > 0, path either bifurcates or terminates. Type 2: m > k > 0, The definition of equivalence class may be extended Type 3: k = 0. to different domains connected by a certain H path. The outermost domain nearest the envelope is Type 3 Classes of different domains are termed equivalent when by the above definition. Type 3 domains have no the H path connecting the domains corresponds to a terminal class, and as such, no difficulty arises as far as continuous manifold path which includes these classes. steering inside of itself and its neighboring domains. In Fig. 5–3 for example, the path from M1 to M6 via M2 is continuous through domains D1, D2 and D3, so these 5.5 Class Connection manifolds and classes on this path are equivalent. On the other hand, in domain is not equivalent to M3 D1 Class connection around Type 1 and 2 domains is any manifold of the domain . The relationship D2 described by examples in the following sections. A Type between manifold and class is illustrated in Fig. 5–4. 2 domain is examined first. By introducing a graph of The equivalence among two domains implies that class connection, a Type 1 domain is next examined. the manifold path is continuous. If there is bifurcation on the manifold path, classes are not equivalent. In this 5.5.1 Type 2 Domain case, classes can be termed ‘connected’, because a continuous θ path can be chosen. The following examples are obtained by computer calculations for the S(4) system. Figure 5–5(a) shows 5.4 Terminal Class and Domain Type a part of a cross section of a singular surface near the envelope. The curved triangle is where the surface was For continuous steering, it is important to know cut, and the bold line indicates an impassable edge. This whether or not each class has equivalent or connected triangle divides the H space into two parts; the domain classes for any H path exiting of the domain. The class outside is denoted by D0 and the domain inside by D1. is called a ‘terminal class’ if there is an H path exiting to Domain D0 is a Type 3 domain just inside the envelope
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Impassable Surface S0 and D1 is a Type 2 domain of m = 2. Figures 5–5(b) and (c) show manifolds of some points in these two domains. S The manifolds are drawn using a two-dimensional θ C projection of the skew coordinates. While the actual space is a four-dimensional torus, the illustration is made Domain D1 B Q in a quadrilateral whose edges parallel to each other are (Type 2) P R A regarded as the same points. Consider now the angular momentum path PAQBR, Domain D0 which traverses the domain D1 in Fig. 5–5(a). There is Passable Surface S1 no equivalent class for this path which traverses three domains, from D0 back to D0 via D1. The manifold path (a) Cross section of an internal singular surface and H path bifurcates at points A and B. Though the classes are not equivalent, they are connected and continuous steering Bifurcation at A is very much possible by some gradient method because R such bifurcation points (passable singular points) are easily avoidable. Consider next the path PAQCS penetrating the ∈ M M2 G2 1 impassable surface. Figure 5–5(c) shows that the Q manifold M2 is of the terminal class in this domain. Once H this class is selected when going into D1 from P through A, there is no continuous way to reach another manifold, θ Bifurcation at B 2 such as M1, so steering fails. On the contrary, if manifold P M1 is selected, continuous motion egressing this domain along QCS is strictly guaranteed without any special θ 1 steering methods. In this case, the main question is how (b) Manifolds for path PQR to select an appropriate class. The above discussion is more easily understood by Impassable utilizing a class connection graph, as shown in Fig. 5– Singular Point S 6(a). The jagged lines represent the cross section of a singular surface obtained from numerical computation. Various circles drawn inside each domain represent C equivalence classes of domains and the color of the circles (white and gray) indicates whether they are a terminal class or not. Circles drawn on the edge of the M domain represent a class of singular manifolds. Curved 2 Q H M1 lines connecting the circles represent class connections. This connection graph makes it easily understood that all classes would be connected even after the omission θ 2 P of the terminal class G2. Therefore, necessary class selection is unique for any H path crossing this domain. θ 1 (c) Manifolds for path QCS 5.5.2 Type 1 Domain
Fig. 5-5 Domains and manifolds of the pyramid Figure 5–6(b) shows another cross section of a type system singular surface and a class connection graph. The A cross section of a Type 2 domain and manifolds triangular domain D3 is Type 1. In this case, no class for several points are obtained by computer remains if we omit the terminal classes of this domain. calculation. Manifolds are drawn as a two θ θ Thus, there is no contiguous connection of classes for dimensional projection on ( 1, 2) coordinates in a quadrilateral whose right and left edges (space an H path such as FG. This implies that no steering law 2π apart) are regarded as the same points. can realize this angular momentum path. On the other
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Domain D3 (Type 1)
G2
G1
Domain D1 (Type 2)
Domain D2
Domain D4 (a) Type 2 domain (c) Another type 1 domain (Type 1) Domain D3 (Type 1)
D' D Impassable surface (Bold)
Passable surface E E' Class of manifold
Class connection F G Bifurcation
Termination
H path Envelope part (b) Type 1 domain
Fig. 5−6 Class connection graph around domains
hand, continuous motion along the path DE is possible Since the singular point θ along the surface is continuous, by selecting the appropriate class prior to bifurcation. passable points θP and θQ smoothly change to impassable θ θ This selection, however, is not always effective. If points S and T. This implies that after bifurcation by another path, D’E’ for example, is taken, the class to be the passable surface, one of the two manifolds must selected is different. This implies that there is no unique terminate at the impassable surface. Thus, class selection rule for entering a Type 1 domain. Figure 5– connections such as those drawn in Fig. 5–6(a) are 6(c) shows another Type 2 domain example including general for this type of domain even if its order is not 2. various class connections. The above discussion also Suppose that two impassable surfaces cross as shown holds in this case. in Fig. 5–8. Singular points are continuous along each surface. Manifolds MP and MQ which are the terminal 5.5.3 Class Connection Rules manifolds of each surface are equivalent and manifolds MR and MS also. Therefore, there can be no connection The class connections in Fig. 5–6 can be derived between two manifolds of the different group, MP and without calculation of manifolds but by considering MR for example. Thus, class connections such as those continuity. In the cross section shown in Fig. 5–7, the drawn in Fig. 5–6(b) are derived. With increased sharp point R represents the borderline between an complexity in surface crossings however, finding class impassable and a passable sides of the singular surface. connections is more difficult.
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Impassable Surface 5.5.4 Continuous Steering over Domains
S R T From the above two examples, the following general facts are observed. Q (1) Continuous steering around a Type 2 domain A depends upon manifold selection prior to Passable Surface bifurcation. If any class other than a terminal P class is selected, continuous control is guaranteed. (a) H space (2) Selecting a manifold other than the terminal classes is impossible while entering a Type 1 domain. (3) Some paths of H which cross a Type 1 domain do MR θ not have a connected manifold path. R θ Q The item (3) implies that there is no continuous θ path for a certain H path. The item (2) implies that even if a θ MS S continuous θ path exists for any given H path, real-time MQ θ steering is not guaranteed when H path is not given T MA2 beforehand. Those two items, therefore, implies that no MT steering law can maintain continuous steering over the entire work space if the system contains Type 1 domains. θ P MA1 On the other hand, an impassable surface of a Type 2 domain does not cause any problem if an appropriate (b) θ space MP manifold is selected before bifurcation as the item (1).
Fig. 5−7 An illustration of class connection rule (1). 5.5.5 Manifold Selection
There has been no simple method to select an Impassable Surface appropriate manifold prior to bifurcation. Although a gradient method avoids a passable singular point (a bifurcation point), judicious manifold selection depends R P on the control values of θ before bifurcation. The T gradient method is unsuitable because of the following reason. The objective function of a gradient method is Q S defined zero at a singular point and otherwise positive. (a) H space Thus the singular point is the minimum of the objective function along the manifold. There must therefore be local maxima on both side of the singular point, such as Lines of Singular Point at A and B in Fig. 5–9. Knowing that the objective function is continuous, the manifolds before and after MR MP bifurcation have local maxima A’, B’, A”, and B” in the neighborhood of A and B. The gradient method only θ θ maintains the local maximum and its motion may be T1 T2 either A’AA” or B’BB”. Even if one of the manifolds MQ after bifurcation belongs to a terminal class, this method MS can not move θ from one maximum point to the other. If sufficient time computing power were available, a θ (b) space method like a path planning42) could be utilized (See Fig. 5−8 An illustration of class connection rule (2). Chap. 8). If a number of possible H paths of a certain If two impassable surfaces cross each other, both length are assumed, calculations along those paths may terminal classes are different. then be carried out in order to determine whether there
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Passable Singular Point Terminal Class A Impassable
B" Passable V B A"
A C B' B Local Minimum A' (a) Cross section orthogonal to gi direction. Fig. 5−9 An illustration of motion by the gradient
method. Manifolds 4
A, A', A", B, B' and B" are the local maxima of an θ
, , 3
objective function. Motion may be either A'AA" or θ
, , 2 B'BB" in the neighborhood of the passable θ A singular surface. θ 0 1 2 are any bifurcations or intersections with impassable Ｖ Singular surfaces. It would then be possible to determine an Manifold appropriate motion. For this strategy, a question still remains whether such manifold selection can be consistent for all possible H path. This will be discussed S in Chapter 7. B C M V Terminal 5.5.6 Discussion of the Critical Point Manifold (b) Manifold of points A, B and C The above discussion pertains to manifolds and classes within domains. An arbitrary angular momentum Fig. 5–10 Manifold relations around critical point not on a singular surface is discussed. It was point. assumed that anything on the singular surface and on The distance between the origin and the cross the singular manifold is qualitatively the same as that in sectional plane of (a) is 0.875 of the maximum the H neighborhood and in the neighbor manifold. There distance. is however some exceptions. If the H path starts at an intersection of singular surfaces, there is a problem of manifold selection. Three of Type 1 domains that there is no such mapping. This triangular domains in Fig. 5–10 are Type 2 of order 2. fact is explained directly by the topology of kinematic Having the same kind of class relations as the Type 2 mapping. domain in Fig. 5–6(a) means any trajectory across one Consider the circle on the envelope which of them can be continuously realized by an appropriate corresponds to the case u=gi. Suppose there is a control. However, if one were to commence at the candidate mapping from H to θ. The image of the circle crossing point V, the possibility of selecting a terminal by this mapping is a loop on the torus where θi changes class cannot be omitted once the initial θ is selected on from 0 to 2π. Consider a deformation of the circle in the H space and the image in the θ space. The circle can the manifold. This situation is depicted in Fig. 5–10(b). be deformed to a point in the H space but their image in the θ space cannot, because continuity requires θi to vary 5.6 Topological Problem from 0 to 2π. (Note: A similar statement for robot kinematics was generally proven by topological theories A steering law can be represented by a mapping from in Ref. 59.) the H space to the θ space. It would be nice if there is a The above is true as long as the mapping covers the continuous mapping which uniquely determines θ from H space in its entirety. If a small enough region of H H. However, it is clearly observed from the examples space, a domain for example, is considered, any
––– 29 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– continuous steering law can obviously realize such a not be serious. On the other hand, the discussion in this continuous mapping. Even if such a region includes a chapter only suggested that there is a possibility of Type 2 domain, we can realize such a mapping by using continuous steering in the presence of a Type 2 domain. an appropriate manifold selection. Thus, the question is The discussion in the above pertains to the local area how large a region of H space can be covered by such a around one domain. In actuality, the candidate continuous mapping, and what steering law actually can workspace may involve various domains, so a study of realize such a mapping. global class connections is necessary. In the following It is to be noted that the examples of Type 1 domains chapters, a relatively specific problem will be studied and critical points in the previous section are only found for the symmetric pyramid type system using geometric near the envelope by various computer calculations of tools given in this chapter. the S(4) system. Therefore, the above problems may
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Chapter 6
Pyramid Type CMG System
This chapter and the next two deal with a symmetric s* = sinα = 23⁄ , pyramid type system of single gimbal CMGs. This chapter describes system characteristics obtained by c* = cosα = 13⁄ . (6–1) analysis and computer calculations. These are kinematic equations, system symmetry, expression of the gimbal Angular momentum vectors and torque vectors of all angles when the angular momentum is at its origin, the units are given as: internal singular surface details, and impassable surface − θ − θ geometry. All of them will be utilized for the analysis c * sin 1 cos 2 = θ = − θ of steering motion in the next chapter. hh1 cos 1 , 2 c * sin 2 , θ θ s * sin 1 s * sin 2 6.1 System Definition c * sinθ cosθ 3 4 =− θ = θ hh3 cos 3 , 4 c * sin 4 , The pyramid type CMG system consists of four θ θ s * sin 3 s * sin 4 single gimbal CMGs in a skew configuration, as depicted (6–2) in Fig. 6–1. An example of this type is the S(4) symmetric system, where each gimbal axis lies in the − θ − θ c * cos 1 sin 2 direction normal to each surface of a regular octahedron. = − θ = − θ cc1 sin 1 , 2 c * cos 2 , The pyramid shown in Fig. 6–1 is the upper half of an θ θ s * cos 1 s * cos 2 octahedron. The skew angle of this type denoted by α c * cosθ − sinθ α , 3 4 is given as cos = 13⁄ and is about 53.7 degrees. It = θ = θ cc3 sin 3 , 4 c * cos 4 , is expedient to define additional parameters: θ θ s * cos 3 s * sin 4 (6–3)
θ g g4 3 3 where the origin and the direction of each gimbal angle c 4 h3 are defined by Fig. 6–1. c3 θ h4 4 6.2 Symmetry Z α h2 θ c1 2 The pyramid type CMG system has symmetry in its Y kinematics. This symmetry is useful for understanding X g 2 the geometry of the singular surface and will be used h1 c2 θ for deriving a global problem in the next chapter. This 1 g1 symmetry is derived by the rotational transformations of regular octahedron, which is a well-known example of finite group theory. There are 48 symmetric − Fig. 6 1 Schematic of a pyramid type system. transformations of an octahedron including the mirror The origin of each θ is defined when h is on the i i transformations. square in the xy plane. The symmetric type S(4) is the case where the skew angle α is set as The symmetry of the pyramid type system is cosα = 1/ 3 . represented by two groups of transformations and an
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Z Z θ θ 3 X 4 θ θ 4 1 θ θ 3 2
Y θ θ Y 2 X 1
(a) Rotation about z axis
Z Y' θ θ 3 2 θ 4 θ 3 X' θ 2 Z θ 1 θ 1
New θ 4 Coordinate Y' X' System (b) Rotation about g1 axis
Fig. 6−2 Transformation in H space and in θ space. A new coordinate system (X', Y', Z) is defined in (b) for a simple expression.
equivalence relationship of those. Transformations in Another example is a 1 ⁄3 rotation about a gimbal the H space represent rotation of a vector, while the axis shown in Fig. 6–2 (b). Two transformations are as others in the θ space represent permutation with follows: translation, i.e., a kind of an affine transformation. The θ = θ π −θ − π rg ( ) ( 1+2 ⁄ 3, 3 2 ⁄ 3, meaning of the equivalence is as follows. As all four −θ π θ − π CMG units are arranged on the surfaces of the 4 +2 ⁄ 3, 2 2 ⁄ 3) , (6–7) hexahedron, a certain rotation in the H space preserves t = t Rg((x’, y’, z) ) (z, x’, y’) . (6–8) the hexahedron and thus results in the exchange of four units. Therefore such a rotation in the H space is The latter transformation is expressed in a different equivalent to the transformation in the θ space. coordinate system from the original; one which is rotated ° An example of the H transformation is a 1 ⁄4 reverse 45 about the z axis as in Fig. 6–2 (b). This is because rotation about the z axis shown in Fig. 6–2 (a), after expressions based on these new coordinates are simpler which CMG unit i is replaced by unit i +1. This than those based on the original coordinates. The transformation of θ is expressed by: equivalence Eq. 6–6 is also maintained by the transformations rg and Rg. Applicable notation of all θ θ θ θ θ θ θ θ θ rz( =( 1, 2, 3, 4)) = ( 2, 3, 4, 1) ,(6–4) will now be defined. Rotations about g are first defined. The identical and the transformation of the angular momentum vector 1 transformation and the g -z plane reflection are denoted by: 1 by Re1 and RE1, respectively. A 1 ⁄ 3 rotation about the t = − t Rz((x, y, z) ) (y, x, z) . (6–5) g1 axis after Re1 (or RE1) is denoted by Rr1 (or RR1). A reverse 1 ⁄ 3 rotation after R (or R ) is denoted by By those two transformations, the following equivalence e1 E1 R (or R ). Thus six transformations are defined. relationship is satisfied. q1 Q1 Successive 1 ⁄ 4 rotations about the z axis are simply θ θ denoted by increasing the indices. For example, R is Rz(H( )) = H(rz( )) . (6–6) e2 a 1 ⁄ 4 rotation about the z axis and Rr3 is a 1 ⁄ 2 rotation
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Table 6−1 Symmetric Transformations Notation* H θ H Transformation (A) Transformation Transformation ** of Mirror (MA) ______θ θ θ θ − − − e1 ( x, y, z) (1, 2, 3, 4) ( x, y, z) π−θ π−θ π−θ π−θ − − − E1 ( y, x, z) ( 1, 4, 3, 2) ( y, x, z) σ θ −2σ−θ σ−θ −2σ θ − − − r1 ( z, x, y) (2+ 1, 3, 2 4, + 2) ( z, x, y) −σ−θ σ+θ −σ+θ σ−θ − − − R1 ( z, y, x) ( 1, 3, 2, 4) ( z, y, x) −2σ+θ σ+θ −2σ−θ σ−θ − − − q1 ( y, z,x) ( 1, 2 4, 2, 2 3) ( y, z, x) σ−θ −σ−θ σ+θ −σ+θ − − − Q1 ( x, z, y) ( 1, 2, 4, 3) ( x, z, y)
− θ θ θ θ − − e2 ( y, x, z) ( 4, 1, 2, 3) ( y, x, z) − π−θ π−θ π−θ π−θ − − E2 ( x, y, z) ( 2, 1, 4, 3) ( x, y, z) − −2σ+θ 2σ+θ − 2σ−θ 2σ−θ − − r2 ( x, z, y) ( 2, 1, 3, 4) ( x, z, y) − σ−θ −σ−θ σ+θ −σ+θ − − R2 ( y, z, x) ( 4, 1, 3, 2) ( y, z, x) − σ−θ − σ+θ σ+θ − σ−θ − − q2 ( z, y, x) (2 3, 2 1, 2 4, 2 2) ( z, y, x) − −σ+θ σ−θ −σ−θ σ+θ − − Q2 ( z, x, y) ( 3, 1, 2, 4) ( z, x, y)
− − θ θ θ θ − e3 ( x, y, z) ( 3, 4, 1, 2) ( x, y, z) − − π−θ π−θ π−θ π−θ − E3 ( y, x, z) ( 3, 2, 1, 4) ( y, x, z) − − σ−θ − σ+θ σ+θ − σ−θ − r3 ( z, x, y) (2 4, 2 2, 2 1, 2 3) ( z, x, y) − − −σ+θ σ−θ −σ−θ σ+θ − R3 ( z, y, x) ( 2, 4, 1, 3) ( z, y, x) − − − σ−θ σ−θ − σ+θ σ+θ − q3 ( y, z, x) ( 2 2, 2 3, 2 1, 2 4) ( y, z, x) − − σ+θ −σ+θ σ−θ −σ−θ − Q3 ( x, z, y) ( 4, 3, 1, 2) ( x, z, y)
− θ θ θ θ − − e4 ( y, x, z) ( 2, 3, 4, 1) ( y, x, z) − π−θ π−θ π−θ π−θ − − E4 ( x, y, z) ( 4, 3, 2, 1) ( x, y, z) − − σ−θ σ−θ − σ+θ σ+θ − − r4 ( x, z, y) ( 2 3, 2 4, 2 2, 2 1) ( x, z, y) − σ+θ −σ+θ σ−θ −σ−θ − − R4 ( y, z, x) ( 3, 2, 4, 1) ( y, z, x) − σ+θ − σ−θ σ−θ − σ+θ − − q4 ( z, y, x) (2 4, 2 2, 2 3, 2 1) ( z, y, x) − −σ−θ σ+θ −σ+θ σ−θ − − Q4 ( z, x, y) ( 2, 4, 3, 1) ( z, x, y) ______Note; *: Each transformation is represented only by its suffix. **: σ = π ⁄ 3 about z after a 1 ⁄ 3 rotation about g1. So far, these total 6.3 Singular Manifold for the H Origin 24 transformations. Subsequent point symmetric transformations by the origin are denoted by adding M The origin of H, (0, 0, 0)t, is used as a nominal state to the left of the original notation, MRe1 for example. of control. This H corresponds to one singular manifold After including these, all 48 transformations are defined. with 6 singular points. The 6 singular points divides the Before continuing, it should be noted that Rr2 is not a singular manifold into 12 line segments. The 12 simple rotation about the g2 axis. segments are classified into two groups. These segments − Table 6 1 presents a list of all 48 symmetric and groups are used to explain the global problem and a transformations. The first row shows the notation, the steering law in the next chapter. second row gives the H transformation, the third shows The singular manifold for this H origin has analytical θ the transformation and the last row gives the H expressions. It consists of four lines, which are straight transformation of the point symmetric image. Both the lines but closed in the torus space. Two of them are θ H transformation and the transformation are expressed given by, as the right hand side only. So, for example, the (1) θ = ( φ, −φ, φ, −φ), expressions 6–4 and 6–5 are given by Re2 and 6–7 and − 6–8 by Rr1 in Table 6 1. Note again that all H where −π < φ ≤ π, (6–9) transformations are expressed in the new coordinate θ φ π φ−π φ+π φ−π t system rotated 45° for simplicity. Τransformation of all (2) = ( + ⁄2, ⁄2, ⁄2, ⁄2) , θ the point symmetric images in space is omitted in Table where −π < φ ≤ π . (6–10) − π θ 6 1 but is simply accomplished by adding to each i.
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π π l e Singular k (h) f (k) a (g) 4
θ 0 0 j θ ,
2 g θ b (l) c h d −π −π −π 0 π −π 0 π θ θ θ 1, 3 i (b) (θ , θ )=(φ+ψ, φ−ψ) (a) (θ , θ ) =(φ, −φ) i j i j 0≤φ≤π, ψ= π⁄2, –5π⁄6
Fig. 6−3 Line segments for singular manifold. Arrows a to l are parametric line segments of a pair of coordinates.
The remaining two lines can be obtained as symmetric expressed by the combination of two coordinate sets. images of the latter. Let’s define notation of total 12 Referring to Fig. 6–3 (a), each segment will have a line segments and derive their symmetric relations. parameter along it and the direction in which the The first line by Eq. 6–9 includes 6 singular points parameter increases will be expressed by an arrow. Six θ hence 6 line segments. Let line segment of be line segments from LA to LF in a four-dimensional torus
Table 6−2 Segment Transformation Rule ______Transformation A Transformation MA Α G H K L M N G H K L M N ------e1 G H K L M N −H −G −L −K −N −M E1 H G M N K L −G −H −N −M −L −K r1 L K N M G H −K −L −M −N −H −G R1 K L G H N M −L −K −H −G −M −N q1 M N H G L K −N −M −G −H −K −L Q1 N M L K H G −M −N −K −L −G −H
e2 −H −G −M −N −L −K G H N M K L E2 −G −H −L −K −M −N H G K L N M r2 −N −M −Κ −L −H −G M N L K G H R2 −M −N −H −G −K −L N M G H L H q2 −L −K −G −H −N −M K L H G M N Q2 −K −L −N −M −G −H L K M N H G
e3 G H L K N M −H −G −K −L −M −N E3 H G N M L K −G −H −M −N −K −L r3 K L M N G H −L −K −N −M −H −G R3 L K G H M N −K −L −H −G −N −M q3 N M H G K L −M −N −G −H −L −K Q3 M N K L H G −N −M −L −K −G −H
e4 −H −G −N −M −K −L G H M N L K E4 −G −H −K −L −N −M H G L K M N r4 −M −N −L −K −H −G N M K L G H R4 −N −M −H −G −L −K M N G H K L q4 −K −L −G −H −M −N L K H G N M Q4 −L −K −M −N −G −H K L N M H G ______Note:Each transformation and each segment are represented by their suffices.
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are then given by a pair of those segments (from a to f) transformation of LA, while any segment from LG to as follows: LN by some transformation of LG. This implies that any characteristics of the segments must accordingly be {LA, LB, LC, LD, LE, LF} derived from characteristics of either LA or LG. = { {a,a}, {b,b}, {c,c}, {d,d}, {e,e}, {f,f}; θ θ θ θ {( 1, 2),( 3, 4)} } . (6–11) 6.4 Singular Surface Geometry
Segment LA in this expression, for example, is given as follows: Singular surface has been described by its curvature or by an example of a cross section in the previous θ ϕ, −ϕ, ϕ, −ϕ −π ≤ ϕ π LA={ :=( ), ⁄ 6 < ⁄ 6} . chapters. Now the total geometry of the singular surface especially of the impassable surface will be examined (6–12) by using a series of cross sections66). The line by Eq. 6–10 orthogonally crosses the first Here, cross sectional planes orthogonal to the g1 axis line at two singular points. The two singular points divide are mainly used as shown in Fig. 6–4. Each plane has a the line into two segments. Referring to Fig. 6–3 (b), parameter d which is a distance from the H origin to the the two segments are defined as follows: plane. All distances will henceforth be normalized by θ θ θ θ its maximum value, which is the distance to the unit circle LG = {( 1, 2)=g, ( 3, 4)=g} , θ θ θ θ LH = {( 1, 2)=h, ( 3, 4)=h} . (6–13) Sectional Plane d : Distance from Other two lines, i.e., four line segments are defined Orthogonal to g1 the H Origin similarly: A θ θ θ θ LK = {( 1, 4)=k, ( 3, 2)=l} , θ θ θ θ LL = {( 1, 4)=l, ( 3, 2)=k} , θ θ θ θ LM = {( 2, 1)=k, ( 4, 3)=l} , θ θ θ θ LN = {( 2, 1)=l, ( 4, 3)=k} . (6–14)
The set of segments from LA to LF are transformed to the same set of the segments by any symmetric transformation. The followings are examples of transformed results, where a minus sign before the Envelope segment implies that the direction is reversed. Portion of Internal Impassable Surface RE1(LA, LB, LC, LD, LE, LF) − − − − − − (a) Envelope and sectional plane =( LD, LC, LB, LA, LF, LE) ,
Rr1(LA, LB, LC, LD, LE, LF)
= (LC, LD, LE, LF, LA, LB) , g g2 Re2(LA, LB, LC, LD, LE, LF) 4 − − − − − − =( LA, LF, LE, LD, LC, LB) , g1 MRe1(LA, LB, LC, LD, LE, LF) g3 = (LD, LE, LF, LA, LB, LC) . (6–15) (b) Cross section on plane A Similarly, the other segments from LG to LN are also transformed to the same segments. The results of all Fig. 6–4 Definition of the cross sectional plane such transformations are listed in Table 6−2 in which and the distance d. each segment is represented by its suffix. The distance d is divided by 2 2 for From the above analysis, it is observed that any normalization such that d=1 for the unit circle segment from LA to LF can be obtained by some on the envelope.
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Internal A impassable part Domain D1 D1
A B A B Internal passable Envelope part part (b)d=0.8919 (a)d=0.9 d = 1.0 Unit Circle D1 d = 0.99 A B d = 0.92 A B
Fig. 6–5 Saddle like part of the envelope. Curves are various cross sections of S{− + + +}. For d<0.94, some portions of surfaces are inside the D2 envelope. (c)d=0.885 (d)d=0.88 on the envelope connecting two singular surface of D A D3 B different signs. A 3 B Figure 6–5 illustrates a singular surface of one negative sign, a portion of which is shared with the envelope. The outermost unit circle is the case of u=g i D2 and d=1. Other curves are cross sections with d<1. It is of interest to observe what happens to this surface when d is made smaller and smaller. (e)d=0.87 (f)d=0.857 As d decreases in size, the curve deforms like a A B triangle. After sharp edges appear, folding and then D3 small triangles appear on the curve as shown by A in A B Fig. 6–5. As a result of this folding, each curve is divided into six smoothly curved segments. As the D4 curvature changes sign at the folding point, the Gaussian curvature also changes its sign. This folding is therefore a connecting point between a passable and an impassable surface. From continuity to the envelope, passability of each curve is determined as (g)d=0.85 (h)d=0.64 shown in Fig. 6–6. Observing the cross sections of A B various d in this figure, it can be seen that both passable and impassable surfaces penetrate the envelope like “strips” or “belts”. Folding also appears in the cross sectional line at d ≈ 0.65 and as d becomes smaller, the strip bifurcates. By repetitive calculation for various d values, impassable portions of the singular surface are obtained as in Fig. 6–7. This figure does not show all of the (i)d=0.6 internal impassable surfaces. All surfaces are obtained Fig. 6–6 Cross sections of singular surface. by successive 1⁄4 rotations about the z axis. Bold curves are impassable and thin curves are Let’s make a simplification of these impassable passable. All are drawn at the same scale except the last two, (h)d=0.64 and (i)d=0.6. The impassable surfaces. The following analytical expressions are curve segments AB have envelope parts as shown in found29) which corresponds to a smooth line shown (a), are totally internal as shown in (b) and are divided into two as shown in (h).
––– 36 ––– –– 6. Pyramid Type CMG System ––
Z A Z
g1 P (0,0,2s*) C Q B
α D E F D’ Y P’ O X X C’ Y 4c* A B’ F’ Q’
Fig. 6–8 Analytical line on an impassable surface. Fig. 6–7 Internal impassable singular surface. The surface near this line is called a branch. Cross sections orthogonal to g1 are drawn at a d step size of 0.05. The detail of the region indicated by A is in Fig. 6−6.
(4) Straight line DE in Fig. 6–8. This line is on the impassable surface shown − t φ − t H = ( c*, c*, s*) + c*sin (1, 1, 2 ) , in Fig. 6–7 hence can represent the surface. This line has the following four parts: (6–19) where: θ φ π π π φ −π (1) Elliptic arc AB = ( + ⁄ 3, ⁄ 6, 5 ⁄ 6, ⁄ 3 ) , π ≤ φ ≤ π H = (2(c*− cosφ ), 0, 2s*sinφ )t , (6–16) (D) 5 ⁄ 6 (E) , − φ φ where: u = ( ( c*cos + sin ), φ − φ φ t θ = (− π ⁄ 2, φ , π ⁄ 2, π − φ ) , ( c*cos s*cos ), c*cos ) . π ≥ φ ≥ π ⁄ 2 (B in Fig. 6–8) , Let point F on the arc AB be the location at which u = ( − s*cosφ , 0, sinφ )t . the line AB is divided to an envelope side (AF) and an internal side (FB). The line FBCDE is connected and continuous both in the H space and in the θ space. By (2) Straight line BC the transformation MRR1 in the notation of Sec. 6.2, a t H = (2c*sinφ , 0, 2s*) , (6–17) line F’B’C’D’E can be obtained which is continuously where: connected to the original line. Thus, referring to Fig. 6– 8, the line FBCDED’C’B’F’ connects two points on θ = (φ , π ⁄ 2, − φ , π ⁄ 2 ) , opposite sides of the envelope. − π ≥ φ ≥ − π (B) ⁄ 2 5 ⁄ 6 (C) , The surface of Fig. 6–7 is composed of several strips u = ( 0, − s*cosφ , − sinφ )t . of impassable surface and a portion of it is represented by this line. This particular strip will be called an impassable branch and be denoted by B . In the (3) Circular arc CD e1 following figures, the analytical line FBCDED’C’B’F’ t H =(c*−cosφ, c*(1−sinφ ), c*(1+sinφ )) , is simplified by using a broken line QPP’Q’, as shown (6–18) in Fig. 6–8. This branch, along with its symmetric images by where: transformations R , R , R , R and R , form a θ − π φ π π r1 q1 E1 R1 Q1 = ( 5 ⁄ 6, , 5 ⁄ 6, ⁄ 2 ) , frame of parallel hexahedron shown in Fig. 6–9, which (C) π ⁄ 2 ≥ φ ≥ π ⁄ 6 (D) , fits all the surface shown in Fig. 6–7. Each branch is t denoted by B and the subscript denotes the u = g2 = ( 0, s*, c*) .
––– 37 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 ––
Z transformation, Br1 for example. In this figure, some
, BE1 P lines are shared by two branches. This is partially the Be1 result of simplification of the curved line. The other Q B B reason is that two branches share the same surface near q1 E1
R1 , B B B e1 the envelope and this surface bifurcates as depicted in
, , Q1 B r1
r1 B Fig. 6–6. B E1 , B By the successive rotation about the z axis, the stellar r1 B
e1 hexahedron in Fig. 6–10 is obtained. In this figure, only
, X B B Q0 B the suffix is shown for each branch. The same stellar R1 Y Q1 hexahedron is drawn in Fig. 6–11 with a cut envelope to Bq1 , Bq1 BR1 reveal the size and the shape of the internal impassable surface. In summary, all impassable surfaces of the
Fig. 6–9 Equilateral parallel hexahedron of pyramid type CMG system are described by the envelope impassable branches.
q4, Q4 q4, e4, E4 Z r4, R4 e3, E3 Z e2, E2 g3 q3, Q3 g1 e1, E1 E1, e2 E4, e1 q2, Q2 q1, Q1 E2, e3
r1, R1 r3, R3
e4, E3 Q3, r1 Q2, r4 r2, Q4 E1, R1 O X
r2, R2 e1, Q1
X E2, r2 r3, Q1
R2, q1 Q2, q4,e2 R1 Y q2, R3
R4, q3 r1, q1
e3, Q3 r2, q2 E4 r4, e4, Q4 E3, r3 R4, q4
q3, R3 Envelope
Fig. 6–10 Overall structure of impassable branches Fig. 6–12 Cross section through the xz plane. Cutaway of Envelope
Z Envelope
P Passable
P’
Y X
Y
Impassable
Simplified Branches X Fig. 6–11 Internal impassable surface with envelope cutaway. Fig. 6–13 Cross section through the xy plane.
––– 38 ––– –– 6. Pyramid Type CMG System –– and the frame like structure of the branches. surfaces cannot be ignored because they surround the Figures 6–12 and 6–13 are examples of other cross origin which is the nominal point for the control. sections which are not orthogonal to the gimbal axes. Moreover, some of the impassable surfaces crosses z In these figures, all singular surface, passable and axis and others lie on the x-y plane. As the CMG impassable, are drawn. Two figures show that there are system’s axes coincide with those of an attitude control relatively large region with no singular surface and system, angular momentum of the CMG system tend to impassable surfaces are narrow strips compared with the travel near such axes. maximum workspace. Nevertheless, impassable
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––– 40 ––– –– 7. Global Problem, Steering Law Exactness and Proposal ––
Chapter 7
Global Problem, Steering Law Exactness and Proposal
This chapter deals with the question whether it is First, an H path and its deviations along the z axis possible to steer the pyramid type CMG system to avoid from O to P are considered. For continuous control on any terminal class of Type 2 domains. In Chapter 5, it is these paths, the necessary condition of θ at the origin O clarified that any steering law will fail if it aims to cover is obtained. Then, by consideration of an H path from the workspace in its entirety. Moreover in this chapter, O to Q, the impossibility of continuous control is derived. it will be shown by examples that any steering law fails continuous and real-time control for the wide variation 7.1.1 Control Along the z Axis of command inputs even if an appropriate manifold selection is tried. Let’s find necessary conditions for continuous real- Based on this global problem, various steering laws time steering along the z axis as shown by OP in Fig. 7– are evaluated. In so doing, the CMG motion by each 1. Consider now only H in the neighborhood of P. Fig. steering law is analyzed geometrically. Three groups of 7–2 (a) shows a cross section of the singular surface by steering laws are examined and their performance and a plane orthogonal to z axis and which crosses near the problems are clarified. The first of those permits errors point P. In the close-up view of Fig. 7–2(b), it can be in the output. The second is realized as a path planning. seen that there are eight domains around the center and The third one is effective for a certain fixed direction. four pairs of impassable branches cross each other. By those evaluation, importance of steering law One of the eight domains, the domain DA in Fig. 7– exactness is clarified. 2(b), has 2 equivalence classes whose elements (which Finally, a new type steering law will be proposed are manifolds of those classes) are MA0 and MA1 in Fig. which assures exact and real time control inside a reduced 7–3. The class GA0 including MA0 bifurcates into two workspace. This steering law uses a simple constraint classes G00 and G01 when entering the neighbor domain and determines uniquely the system state from the D1 in Fig. 7–2(b). Two classes are represented by two angular momentum. The reduced workspace is larger manifolds M00 and M01 in Fig. 7–3. The classes G00 than the spherical one which excludes all impassable surfaces, but has the same length in one direction as the original maximal workspace. Candidate workspace Z 7.1 Global Problem P
Following the discussion of Section 5.7, the workspace size must be slightly reduced from the T maximum to enable continuous control. Here, it is shown Q O S that continuous control is impossible if the workspace X R includes certain domains. An example of the workspace is a sphere around the H origin O in Fig. 7–1 including Y the hexahedron made of impassable branches which is a part of the stellar hexahedron in Fig. 6–11. Domains around two vertices, P and Q, are relevant for the following discussion. Fig. 7–1 Candidate of workspace
––– 41 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 ––
Y around these domains, as in Fig. 7–4. The situation is similar to that around a Type 2 domain described in
Envelope Passable Chapter 5 and hence, only one class GA0 must be selected in this domain DA. Impassable As illustrated in Fig. 7–5, domains and manifolds symmetric about the z axis are obtained by successive X 1/4 rotations. Though there are two equivalence classes to each of the four domains, i.e., DA to DD, one of them must be selected with the consideration above. |H| = 1 Manifolds to be selected are MA1, MB1, MC1, and MD1 in Fig. 7–5. All the four domains are connected with each other by a line segment lying on the z axis which is shown as a cross point U in Fig. 7–5 (b). Figure 7–6 shows a singular manifold for this point U. By (a) Cross section normal to the z axis. comparing the manifolds MA1, MB1, MC1, and MD1 in Y Fig. 7–5 and the singular manifold MU in Fig. 7–6, it is observed that all the manifolds are continuously φ φ φ φ connected by two curved line segments, 1 2 and 3 4, shown bold in Fig. 7–6. Thus, the necessary condition
Domain D for continuous steering from the point U to either DA, 1 θ DB, DC or DD is controlling on one of the two φ φ φ φ segments, either 1 2 or 3 4, when H is at U. Branch BE4 X If a cross sectional plane orthogonal to the z axis is moved towards the H origin, the topology of domain H path 1 connections, the intersection of the singular surface and class connections over domains become different from those in Figs. 7–2 and Fig. 7–4. However, two segments Branch Be3 Domain DA of the singular manifold in Fig. 7–6 are analytically HH path path 2 2 t defined for any H= (0,0,Hz) on the z axis as follows: (b) Magnified view inside the dotted square θ φ ψ φ−ψ φ ψ φ−ψ shown in (a). = ( + , , + , ), ψ φ where Hz = 4 s* cos sin . (7–1) Fig. 7–2 Cross section nearly crossing P . The distance from O to the plane is 1.4 (not One of the segments includes the point (φ, φ, φ, φ) normalized). This plane crosses the z axis φ -1 where = sin (Hz ⁄(4 s*)) and both of its edges are nearer the origin than P, as OP = 2 s* ≈ 1.63. singular. The other segment can be obtained as a mirror image of this. Because the segment (and its edges) given by Eq. 7– and G01 are equivalent to the terminal classes of the θ 1 are continuous with respect to Hz, must be located impassable branches BE4 and Be3 respectively. This on this segment, or on its mirror image, for any Hz. At implies that continuously changes to a singular point M00 the H origin, this segment and its mirror image take the (θ in Fig. 7–3(c)) when H follows the path 1 in Fig. E4 following simplified form: θ 7–2, and M00 changes to another singular point ( e3) by θ (ψ, −ψ, ψ, −ψ) the H path 2. (H=(0,0,0)) = , Thus the class containing MA1 in Fig. 7–3 should be where −π/6 ≤ ψ ≤ π/6 and 5π/6 ≤ ψ ≤ 7π/6 . selected in the domain DA for continuous steering in consideration of these H paths. This is more easily (7–2) understood by making a simplified class connection map
––– 42 ––– –– 7. Global Problem, Steering Law Exactness and Proposal ––
–3π⁄2 –3π⁄2
∈ MA0 GA0 2 2 θ θ MA1
0 0
–π⁄2 –π⁄2 π –3π⁄2 π π – ⁄2 0 θ – ⁄20 θ –3 ⁄2 1 1
− t (a) Two manifold of H = ( 0.02, 0.02, 1.4) in domain DA.
–3π⁄2 –3π⁄2
M01 M00 ∈ θ G01 e3 2 2 ∈ θ θ G00 θ E4
0 0
–π⁄2 –π⁄2 π π –π⁄2 –3π⁄2 – ⁄20 θ –3 ⁄2 0 θ 1 1 (b) Two manifolds of H = (−0.05, 0.05, 1.4)t in (c) Impassable singular points of branches BE4 and B . domain D1. Both are connected with MA0 in (a). e3 θ is connected with M in (b) through Path 1 Another manifold connected with MA1 is not E4 00 in Fig. 7−2 and θ is connected with M in (b) drawn. e3 01 through Path 2.
Fig. 7–3 Manifold bifurcation and termination from DA. Manifolds are drawn using (θ1, θ2) coordinates. The θ origin is not on the center to avoid a manifold drawn separately.
G01 GA0
Domain DA BE4 D1 GA1
G00 G12
Be3
Fig. 7–4 Simplified class connection diagram around domain DA. For clarity, this figure of domains has been simplified by omitting some singular surfaces.
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Y
Ε3 e1 DD DC Ε2 e2 U DA DB X U(0, 0, 1.4) Ε4 4 e (b) Four domains around the point U
e3 Ε1 DA
(a) Branches and domains
MD1 ∈ MC0 GC0 MD0 ∈ GD0 MC1
DD DC
D A DB
M ∈ M A1 MA0 GA0 B1
MB0 ∈ GB0
(c) Manifolds of eight domains.
Fig. 7−5 Manifolds of eight domains around the z axis. In Fig. (a), each branch is indicated by its suffix such as E1 for BE1. In Fig. (c), each θ θ domain, from DA to DD, has two manifolds. They are drawn using ( 1, 2) coordinates. Though four manifolds, from MA1 to MD1, are congruent, they look different in two dimensional projections.
––– 44 ––– –– 7. Global Problem, Steering Law Exactness and Proposal ––
These crossings are not singular points This is a more general conclusion than the difficulty but only due to the 2D projection . reported in the former research work37), which dealt only with the specific examples of motion along the z axis. By the geometric analysis made above, it is understood φ 4 that not only the H path on the z axis but also a variety of other paths cannot be realized by any steering law. φ 3 Singular Point φ 7.1.3 Details of the Problem 1 2
θ φ 2 Suppose that θ is on the latter segment of Eq. 7–3, θ MU when H = 0. This segment is denoted by LF after the 1 notation of Section 6.3. Referring to Fig. 7–7, an infinitesimal motion of H towards (1, 1, 1)t moves H Segment away from a singular surface and θ moves onto a manifold which is originally a rectangle outlined by Fig. 7–6 Singular manifold of a point U on the z-axis. segments LF, LM, LC and LH when H = 0. As H moves φ φ closely along the z axis in the same domain, the manifold Two curved line segments drawn bold, 1 2 and φ φ 3 4, are the segments continuously connected changes equivalently as shown in Fig. 7–8, with neither to manifolds, MA1 to MD1 in Fig. 7–5. bifurcation nor termination. Finally, near the point U in Fig. 7–5 (b) the manifold connects with either MA0, MB0 or MD0 (Fig. 7–9). Because the impending H path is 7.1.2 Global problem not given, there is a possibility of these manifolds being θ selected once is determined on manifold MV in Fig. The same discussion can be made for the H path from 7–9. The three manifolds inevitably bifurcate into O to Q. All θ and H are simply transformed by a terminal classes if the H path crosses certain branches, rotational transformation such as the 1 ⁄3 rotation about for example branches BE4 and Be3 if manifold MA0 is selected, branches B and B for M and branches the g1 axis. This transformation is denoted by Rr1 in the E1 e4 B0 notation of the previous chapter. By the corresponding BE3 and Be2 for MD0. θ transformation, the above conditions, Eq. 7–2, is now The manifolds in Fig. 7–8 are all inside one domain, transformed to the following segment: denoted by DV. Though Fig. 7–10 in the next page indicates that this domain is not large by itself, some θ(H=(0,0,0)) = (ψ, −ψ, ψ, −ψ) ,
where π/2 ≤ ψ ≤ 5π/6 and −π/2 ≤ ψ ≤ −π/6 . M (H=(0.02, 0.02, 0.02)t) (7–3) V θ 2 U Since the two sets of segments given by Eqs. 7–2 LH and 7–3 have no common θ, continuous control from O LC T to P and from O to Q cannot be satisfied simultaneously. It is clear that once the condition imposed by Eq. 7–3 is satisfied, the system will meet an impassable singularity V L on the H path nearly along the z axis while crossing M θ some of the impassable branches and in Fig. 7– 1 BEi Bei LF S 5. Thus it is concluded that continuous steering over this entire workspace, including O, P and Q is not possible. These two sets of segments defined by the above two equations are ( , ) and ( , ) in the LA LD LC LF Fig. 7–7 Manifold of H near the origin. notation of Section 6.3. The remaining segments LB MV is one of the manifolds for H = (0.02, 0.02, and LE are the condition for the continuous control in 0.02)t which continuously deformed from the rect- the OR direction. angle STUV. Four edges of the rectangle are LF, LM, LC and LH by the notation of Section 6.4.
––– 45 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 ––
a = 0.02 Y
a = 0.25 DV
a = 0.5
a = 0.75 X a = 1.0
a = 1.3
|H|=1
Fig. 7–8 Continuous change of manifold for H Envelope nearly along the z axis. (a) HZ= 0.1 These six manifolds are for H=(0.02, 0.02, a)t where a = 0.02, 0.25, 0.5, 0.75, 1.0 and 1.3. The last manifold for a = 1.3 is MV of the next figure. Filled circles and blank circles are maximum and D minimum of det(CCt) along the manifolds. V
MV
θ 2
Envelope θ 1 (b) HZ= 0.5
(a) Manifold MV
DV
MD0
MB0
θ 2 MA0 Envelope θ 1 (c) HZ= 1.0
(b) Manifolds in the neighborhood Fig. 7−10 Cross sections of domains.
Domain DV corresponding to Fig. 7−9 Manifold connection over several manifolds in Fig. 7−8. domains. The manifold MV connects partially with MA0, These domains have manifolds equivalent to MB0 and MD0. MV.
––– 46 ––– –– 7. Global Problem, Steering Law Exactness and Proposal –– neighboring domains have equivalent classes to this surface. Because a passable surface is generally avoided manifold. Once the segments at H = 0 given by Eq. 7– by steering laws using a gradient method, such a steering 3 are selected, H then moves inside these domains, there motion will take place on an impassable surface. Its is no way to escape from the manifold equivalent to this solution is obtained so that the output torque lies on a MV. The branches mentioned above pass along edges plane tangential to the singular surface. Therefore, this of the “top” half of an octahedron, namely PQ, PR, PS steering motion can be imagined as a ‘sliding’ motion and PT in Fig. 7–1. Since it is safe to assume continuity of H on the singular surface. of the surfaces and manifolds, it can be expressed that As depicted in Fig. 7–11, there are four possibilities some parts of this manifold connect to manifolds of motion along the singular surface when the command, belonging to terminal classes of these branches. Tcom, is fixed. The case (d) can be ignored straight off Therefore there is a possibility of termination for any H because it is not stable. The case (c) is possible when path crossing such branches. Moreover, these domains the surface is convex to u and this is the case of an are so large that this problem cannot be neglected. envelope (see Section 4.3.4 and Fig. 4–8). For an internal Manifold MV, however, is not connected to MC so it surface, only the cases (a) and (b) are possible. Thus, a does not present a problem with respect to branches BE2 motion is always possible in response to the command and Be1 when this manifold is selected. as long as the command is fixed. This discussion ignores how large the torque error 7.1.4 Possible Solutions is. If the area of the impassable surface is excessively large, this steering law is not effective in practical use. The above discussion is made without consideration As shown previously in Figs. 4–10, 11 and 12, the for any specific steering law. The problem applies to impassable surface of a 4 or 5 unit CMG system is shaped any steering law which aims an exact and strictly real like a narrow strip and the curvature of the surface is time control. Exactness implies that an output is always negative to its narrow direction. Because of this, such equal to the command input. Strict real time feature implies that information of future command is not used. Possible methods to overcome the problem could (3) involve either of the following. Singular Surface (3) 1) relaxing an exactness condition. 2) relaxing a real time condition. 3) restricting the workspace. (2) In the following sections, from Section 7.2 to 7.4, various (2) proposals of the former two kinds will be evaluated. By (1) (1) these evaluations, importance of steering law exactness and real time feature will be clarified. Then, a new (a) Smooth Break Away (b) Folding steering law using workspace restriction will be proposed in Section 7.5.
7.2 Steering Law with Error
(2) Stop The steering laws described in Section 3.5.2 enable Stop calculation of the inverse Jacobian even on a singular (1) point. This is made possible by permitting a minimum error in output torque. This kind of method has so far been evaluated only by a limited number of simulations. (c) Stop at Convex (d) Unstable Stop
7.2.1 Geometrical Meaning Fig. 7−11 Possible motion following an example of singular surface. The CMG motion by a steering method accompanied Motions may be (1) reach the singular surface, by error is understood from the shape of an impassable (2) go along the surface, then (3) break away from the surface, if possible.
––– 47 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– escape motion as in the cases (a) and (b) will take place H motion due to approximately to the narrow direction. Possible error, Vehicle's Rotation therefore, can be roughly estimated by the width of the singular suface stip. H Trajectory If a faster ‘sliding’ motion is required, judicious Acceleration knowledge of the direction of narrow width is very Deceleration useful. This direction can be approximately obtained as an eigenvector of the negative curvature of the surface. H0 Supposing that this direction is obtained as v, the θ motion, dθ , that will realize this ‘sliding’ motion is S H obtained by movement along the singular surface as 1 follows: Fig. 7−12 Illustration of H trajectory of the − dθS = PCt (CCt + k u ut ) 1 v . (7–4) CMG system for the example maneuver. This is derived by Eq. 3–22 and Eq. B–9 in Appendix B. Impassable Surface 7.2.2 Exactness of Control Desired Path As mentioned previously, the steering of a CMG system is similar to the kinematic control of a multi- joint manipulator (also see Appendix F). If a CMG is Detour used by itself and the objective is to realize a certain H trajectory, the steering law problem is analogous to the kinematic control of a manipulator. In this situation, the H above method gives a possible solution whose H deviates 0 slightly from the desired trajectory. The difference H1 between CMG control and control of a manipulator is that a CMG is used for the attitude control. The objective Fig. 7−13 Avoidance of an impassable surface is not to control the actual CMG but rather to control the vehicle’s attitude. If there is an output torque error, not only is there deviation in the path of H but also the coordinates fixed on the CMG system. The H trajectory attitude of the satellite changes from that intended. This of the CMG system will be some path from H0 to H1, as attitude error changes the command issued by the depicted in Fig. 7–12. Though the exact path varies for feedback control and then the desired H path also different control methods, H0 and H1 will not vary if changes. The above method should therefore be the control is successful and if there are no disturbances. evaluated in consideration of the attitude control. Suppose that there is an impassable surface somewhere Suppose that the angular momentum of the satellite along this path. If this surface is located sufficiently far is zero and the angular momentum of the CMG system from the goal and the surface are small enough, it might on board is not zero. Suppose further that the control be possible to make a ‘detour’ as shown in Fig. 7–13, command is to maneuver the satellite and finally stop and reach the goal by the above steering law. the rotation. This implies that the final angular If the surface is near enough the goal, H will stay on momentum of the satellite is zero and the angular the singular surface (at the point A in Fig. 7–14(a)) momentum of the CMG system is not zero. By the despite the negative surface curvature, because the point conservation law, the initial and final angular momentum A is the nearest to H1. Since the residual angular − must be the same in the inertial coordinates. Since the momentum H H1 of the spacecraft is not zero, there coordinate frame of the CMG system rotates, the initial remains some rotation of the spacecraft when H is on and final angular momentum will be different in the this surface. Though this rotation depends on the inertia rotating coordinate frame. matrix of the body and the direction of the residual angular momentum, H of the CMG may stay inside some Let H0 and H1 denote the initial and final H in area of the impassable surface, as shown in Fig. 7–14(b).
––– 48 ––– –– 7. Global Problem, Steering Law Exactness and Proposal ––
No sliding motion possible 43, 44). This problem is similar to the path planning and its realization of a robot manipulator. H1 In Section 7.1, it was concluded that some of the various possible command sequences cannot be realized
A simultaneously by the same steering law. This is true as long as the future H path is not specified. On the contrary, manifold selection and continuous θ path is possible for a given H path, even for one of the two Impassable paths in Fig. 7–2 for example. Of course, continuous Surface control is not possible if the H path starts from the singular surface as described in Section 5.5.6 or if the H H Trajectory path crosses a Type 2 domain as described in Section (a) Impassability 5.5. Thus, conditions of successful path planning can be clarified by this geometric study. If we permit a This residual angular momentum minimum error in the solution42,43), path planning is causes rotation of spacecraft always possible because of the nature of such motions. Geometric study also reveals some problem of this H1 manner of path planning. Since different manifolds may be selected for different H paths, the θ path may be completely different even though the H path is very similar, which may degrade the robustness of the control system. Moreover, optimization is limited only to the given H path but no future situations are considered. Moreover, this method is too complicated for actual Possible H Trajectory implementation.
(b) Motion on the impassable surface 7.4 Preferred Gimbal Angle
Fig. 7−14 Problems of movement on an 38) impassable surface. Another method is similar to path planning but supposes that the direction of a near future maneuver can be known, and this direction is one of certain predefined possibilities. This method introduces In this case, it is impossible to reach the goal and the ‘preferred gimbal angles’ from the maneuver direction spacecraft will continue its rotation. and adjusts the system to the preferred angles before the If an attitude keeping problem under some maneuver motion. An examples of preferred gimbal disturbance is considered, such a situations as in Fig. 7– angles and their corresponding maneuver directions are 14 is not avoided by this steering law. Because of these, given as follows38): it is better not to use the above steering law and better to keep steering law exactness. –––––––––––––––––––––––––––––––––––––––––– Direction of Maneuver Preferred Gimbal Angles t 7.3 Path Planning z-axis, (1, 1, 1) direction (0, 0, 0, 0) x-axis, (4, 2, 0)t direction (−π⁄3, π⁄3, 2π⁄3, −2π⁄3 ) y-axis, (2, 4, 0)t direction (−2π⁄3, −π⁄3, π⁄3, 2π⁄3 ) Another steering law approach takes advantage of –––––––––––––––––––––––––––––––––––––––––––– rapid maneuvering to enable off-line planning. If the maneuver occurs fast enough, the period of maneuver It is guaranteed from the discussion of Section 7.1.1 will be short enough that any disturbance can be that an initial gimbal angle of (0, 0, 0, 0) is suitable for neglected and hence the maneuver trajectory and H path z-axis maneuvers. However, evaluation of the others can be designed beforehand. For this given path, the are not simple. CMG motion can be planned by off-line calculations42,
––– 49 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 ––
A(θ ), B θ 0 4 θ 3
θ θ 1 2 HX=1.0 HX=1.0 B HX=0 Hx=0 θ A( 0)
≤ ≤ (a) 0.0 HX 1.0
HX=1.0 B, C HX=1.2 HX=1.4 HX=1.0
HX=1.2 C HX=1.4 B Connection
≤ ≤ (b) 1.0 HX 1.4
HX=1.4
D
HX=2.6
HX=2.6 C HX=1.4
(c) 1.4 ≤ HX ≤ 2.6
Fig. 7−15 Change in manifolds for H moving along the x axis. t θ θ θ θ (H = (HX, 0, 0) , HX step size = 0.2). Manifolds are drawn in ( 1, 3) and ( 2, 4) coordinates as shown in (a). Manifold bifurcations are observed in (b). Motion of θ from the preferred θ −π π π − π angles 0=( ⁄ 3, ⁄ 3, 2 ⁄ 3, 2 ⁄ 3) follows the line ABCD, which is a trace of the maxima of det(CCt) and is indicated by dots.
In the (1, 1, 1)t direction, there are two impassable H on the x axis. A gradient method is used and θ is t branches, i.e., BE2 and Be1. The discussion in Section maintained at the local maximum of det(CC ). Starting θ θ 7.1.3 suggests that the θ on the segments LF, LM, LC or with set to the values in line 2 of the above list, LH may be better than preferred angle (0, 0, 0, 0). The subsequently follows the path ABCD. There are only second and third θ are on segments LL and LM. The two bifurcations in this manifold path, as shown in Fig. second preferred θ in the above list will next be evaluated 7–15(b), and these correspond to passable surfaces in by observing the change in manifolds as H moves along the neighborhood of two impassable branches. The bold the x axis. curve in the left plot of Fig. 7–15(b) indicates a Figure 7–15 shows manifolds corresponding several connection, and this shows that the manifolds are
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connected by this part before and after the bifurcations. 4 s* The robustness of this motion is evaluated by the length Possibly Passable of this part, which is more than π ⁄2 in this case. The preceding discussion verifies the performance Impassable of this steering method but also clarifies its limitations. Z This method is valid only when H is initially on the P origin. No method was specified to obtain θ when H is not zero. If a gradient method is applied, this method is T valid as long as the maneuver is carried out exactly along the defined direction. But if the maneuvering path Q S deviates, various gradient method problems may occur, which will be described in the next section. X R Y Conceptually, this method may be effective for exact and real-time steering as long as the system does not meet impassable singularity before H returns to the 2 c* origin. The main question is whether this can be assured. By extension of the first set of preferred angles and by making the motion exact, a more effective method will Estimated be proposed in the following section. Workspace
Fig. 7–16 Estimation of reduced workspace for 7.5 Exact Steering Law exact steering. Branches drawn by bold lines are impassable but those drawn by thin lines may be The evaluation above clarified that steering law made passable. exactness and real time feature are important for the real usage of the CMG system in the attitude control of a satellite. In this section, a new steering law is proposed 7–16. The shape of this workspace is not defined by the which assures its exactness and real time feature39). discussion in Section 7.1, however it does include the z Though, the idea of manifold selection prior to axis and its neighborhood, and an area on the xy plane bifurcation is important for analyzing continuous control, slightly smaller than the square QRST as shown in Fig. suitable algorithms for actually doing this have not been 7–16. developed. Segments defined by Eqs. 7–1 and 7–2 only defines θ where H is on the z axis. While geometric 7.5.2 Repeatability and Unique Inversion concepts such as class connection around domains are useful for evaluating the steering law, the actual steering It is required that θ remains on the segment given by law algorithm must determine the θ value at any H point Eq. 7–1 whenever H is on the z axis. This is a matter of so that the desired manifold selection is made. repeatability of inverse kinematics. Generally, repeatability is realized only when the system has an 7.5.1 Workspace Restriction inverse mapping67). The following example illustrates that an ordinary gradient method does not possess Because of the problem in Section 7.1, the workspace repeatability over the workspace in Fig. 7–16. must be restricted in order to keep exact steering. One Figure 7–17 shows manifolds of several H points way of workspace restriction is to exclude all impassable on the line from O to Q. Each jagged-edge rounded surfaces from the workspace. This however is a too strict rectangle is a computer output of the manifold drawn in way of restriction. The condition imposed by Eq. 7–1 is θ θ ( 1, 2) coordinates. Dots on the manifolds indicate local effective with regard to motion nearly along the z axis, maxima of det(CCt) along each manifold. This implies while it is not applicable for control in the neighborhood that θ may be controlled on these points by a gradient of Q. Thus, a new workspace of a two-lobe shape may method. If the initial θ meets the condition of Eq. 7–2, be obtained by excluding some of the impassable θ will follow line AB for commands on the H path branches crossing near Q, R, S and T as shown in Fig. approaching Q from O. The line linking the local
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θ − θ θ − θ Local Maxima of det(CCt) 1 2 + 3 4 = 0 , (7–5) θ 2 because of the following reasons. When H is on the z axis, this condition gives the center of the segment de- scribed by Eq. 7–1 which has the maximum det(CCt) on Manifold of the segment. This condition preserves some of the H=(0,0,0)t system’s z axis symmetry. Moreover, it is simple and θ A 1 the constrained kinematics are also simple. Finally, it D will be shown in the next section that the workspace by this constraint is an appropriate realization of the ex- pected workspace in Section 7.5.1. B C The constrained kinematics has a following analytical E form;
Manifold of H=(0.5,0.5,0)t −+c * cosφψ sin sin φγ sin H = 2−−sinφψ sinc * cos φγ sin , Fig. 7–17 Discontinuity in the maximum of s * sinφψ (cos+ cos γ ) det(CCt). Rounded rectangles are parts of the manifold θ φ+ψ, φ+γ, φ−ψ, φ−γ for H on the z axis. Dots indicate the local where = ( ) .(7–6) maxima of det(CCt).
7.5.4 Reduced Workspace maxima is discontinuous at point B where H ≈ (0.3, 0.3, 0)t. After passing this H, θ approaches another The allowed workspace of this system is defined by maximum, either C or E. Suppose the case of C here. If keeping unique inversion feature within the domain of φ ψ γ −π π after this motion the command path of H is reversed three variables, , , , to [ ⁄2, ⁄2]. Figure 7–18 back to O, θ never goes back to B but follows CD, the shows possible regions of H in several cross sectional other line of maxima. Finally, θ does not satisfy Eq. 7– plane orthogonal to the z axis. As an envelope of each 2 when H returns to O. Thus, such a method is not a region is not simple enough to be handled by the possible candidate. momentum management procedure of a controller, an This problem is derived from the fact that an approximation is required. An example approximation ≤ equilibrium point by a gradient method, i.e., nominal θ is made where |Hz| 2s*, which is shown by rounded for a given H is neither unique nor continuous. The squares in the same figure. These rounded squares are condition of Eq. 7–1 requires that θ must be uniquely defined by the following equation; determined by an inversion from H to θ. Thus, this unique inversion is required for the exact steering law. −−2(*ccpsq ) H = −−2(*)sp c cq , (7–7) 7.5.3 Constrained Control Hz
In kinematics, characteristics of unique inversion are where 68) realized by utilizing direct constraints of variables . |Hz| ≤ 2s*, s = Hz ⁄ (2s*), c = (1 − s2)−1⁄2, By using some algebraic relation of variables as −1⁄2 −1⁄2 constraints, inversion of the constrained kinematics p = sin s and q = cos s . becomes a one-to-one and continuous inside of some By using this approximation, a workspace of the range, which specifies the workspace. For a four constrained steering law is defined as illustrated in 3- dimensional system, it is adequate to constrain one dimension in Fig. 7–19. degree of freedom. The reduced workspace has the same maximum Though various constraining conditions were length as the maximum workspace in the z axis, but the possible, the following was applied39):
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y Angular Momentum Envelope y
Internal Singular Surface
2c*
x x
2(1+c*) Cross Section of Allowed (a) H =0.0 Workspace z (b) Hz=0.4 y y Approximated Workspace by Eq. 7–7
x x
(d) H =1.0 (c) Hz=0.75 z y y
y
x x x
(e) Hz=1.4 (f) Hz=2.0 (g) Hz=2.6
Fig. 7−18 Cross section of possible workspace by constrained steering law. Possible region of angular momentum given by Eq. 7–5 is drawn by two parameter net of (ψ, γ) under condition that Hz is constant and the determinant of the Jacobian is positive. Approximated workspace is defined by Eq. 7–7.
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z type of mechanism but flexible wires can be used for 4s* power supply.
(3) Simplicity of Calculation Even though the proposed steering law involves calculation of an inverse Jacobian, which in turn involves inversion of a 3 × 3 matrix, this is simpler than calculation of the pseudo-inversion of a 3 × 4 matrix. Also, no complex calculations for the gradient method are needed. 2c* 2c* The actual implementation is also simple, though it does x y use some feedback. Implementation details are described in Appendix E. Because of the simplicity, this method actually installed on the experimental computer needed
about 2⁄ 3 memory storage and about 1 ⁄ 2 calculation time compared with the gradient method with an objective function det(CCt) (see Appendix E.5).
Fig. 7−19 Reduce workspace of the constrained (4) Modes and Mode Changing system. The constraint of Eq. 7–5, the kinematics of Eq. 7–6 and the workspace of Eq. 7–7 defines one constrained system. As the original unconstrained system has minor diameter on the xy plane is only about 1 ⁄3 that of symmetry, this constrained system can be symmetrically the maximum workspace. transformed. There are six possible transformations whose representations in the H space are the identical 7.5.5 Characteristics of Constrained Control transformation, a mirror transformation about x-z plane and ±2/3π rotation about g1 with or without the mirror The followings are characteristics of this method. transformation. By those transformations, six Most of all are useful for the real usage in the attitude constrained systems are defined which have their own control system. constraint condition and own workspace, and have the similar properties such as exactness. (1) Exactness and Repeatability The six constrained systems makes three pairs. These The inversion of Eq. 7–6 is not exactly one-to-one. pairs are called “modes” and termed M1, M2 and M3. In order to maintain a one-to-one feature, H must be The workspace of each pair has a shape similar that kept inside the previously defined workspace. By shown in Fig. 7–19, and the dominant direction lies along adhering to this limitation, continuous control over this the z-axis for the M1 mode, along (1, 1, 0)t for the M2 space is strictly guaranteed. Moreover, unique inversion mode, and along (1, –1, 0)t for the M3 mode as shown characteristic of the steering law assures repeatability. in Fig. 7–20. The nominal gimbal angles, which correspond to H=(0, 0, 0)t are of the form of (ψ, −ψ, ψ, (2) Gimbal Limits −ψ), where ψ=0 or π for the M1 mode, 1/3π or −2/3π Because of the uniqueness, each gimbal angle is for the M2 mode, and −1/3π or 2/3π for the M3 mode. φ ψ exactly within a certain domain. The domains of , , Since the dominant directions of all the workspaces are γ [−π π and are included in the domain ⁄2, ⁄2]. Each orthogonal to each other, attitude control performance θ −π π gimbal angle, i, is therefore inside the domain [ , ]. will be improved by introducing mode switching. This is very advantageous compared with other steering Different modes share a region in H space inside of laws. With a gradient method for example, the domain which we can select and change modes. When it is θ of is not defined. Gimbal angles greater than one required to change the steering law mode, gimbal angles revolution are observed in results of some computer must be changed to satisfy another constraint while simulations. As a result, mechanisms such as a slip ring keeping the same H. There is, however, no continuous is needed to permit free rotation of the gimbal. In path from θ of one mode to θ of another mode without a contrast, the method described here does not require this
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z z z (1, -1, 0)t
y y y x x x
(1, 1, 0)t
(a) M1 (b) M2 (c) M3
Fig. 7−20 Reduce workspaces of three modes.
change in H, except for H=0. (When H=0, there exists the constraint of Eq. 7–5 for any value of the skew angle a mode connection path given by θ ∝ (1, −1, 1, −1).) α as long as the four units are set symmetrically about Therefore, operations like feedback attitude control the z axis. For any α, the workspace size to the z direction should be deferred until the switching process is is 4 sinα, while that of x or y direction is a little less than completed. 2 cosα as shown in Fig. 7–19. If a smaller skew angle α In the experiments, the following simple method was is used, the workspace becomes shorter in the z direction applied. Here, one specifies a condition such that H is and wider in the x and y directions. In this manner an on the dominant direction of the newer mode (e.g. the z- arbitrary design of the workspace shape can be obtained. axis of the M1 mode). The gimbal angle for H along the Of course, the original symmetry of the regular z-axis is acquired by a direct inverse calculation of Eq. octahedron is lost in an arbitrary skew angle α and only 7–6, as follows: one mode in the item (4) is available. If the skew angle α = tan−1(1 ⁄2), the size of the φ = sin−1(H ⁄ 2s*) , ψ = γ = 0 . (7–8) z workspace along the x, y and z axes is almost identical. The simplest way of changing θ from the current to the This configuration therefore gives the maximum above is a motion along a line. unidirectional workspace size. If a spherical workspace This gimbal motion causes undesired torque but its is desired for convenience of the attitude control, this is effect can be made small. Since H is the same for the the best configuration of four unit systems. initial and the final θ in the CMG coordinate frame, the Application of the constrained control is not limited initial and the final angular momentum of the spacecraft to skew type systems. Any four unit system can be alone may be similar when this motion is made fast controlled using one constraint. If the system does not enough. Thus, this motion will result in small deviation have symmetry, however, a simple constraint as Eq. 7– of the spacecraft’s orientation and this deviation can be 5 may not be effective. In Appendix D, the same easily corrected by the feedback attitude control once constraint as Eq. 7–5 will be applied to the four unit the mode is changed. subsystem of the MIR-type, i.e., S(6) system69). Though mode changing while H is not on any principal axes cannot be specified by an analytical (6) Performance solution, iterative numerical solution can be applied to Performance of the proposed steering law was find a goal gimbal angles by using the solution 7–8. demonstrated by using ground-based test equipment. The results are detailed in the following chapter. Also, (5) General Skew Case and the Maximum Spherical the pyramid type system controlled by this steering law Workspace was evaluated by comparing with other type CMG The proposed method does not depend on a specific systems in terms of the workspace size. The results are configuration symmetry. Equation 7–6 is satisfied with detailed in Chapter 9.
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Chapter 8
Ground Experiments
The previous chapters dealt only with CMG systems The time derivative of β, i.e., dβ ⁄dt is expressed by ω while aiming primarily at a geometric understanding of angular velocity denoted by V: the CMG motion and leaving quantitative matters dβ* ⁄ dt = 1 ⁄ 2 (β + β* × ω ) , (8–3) neglected. In this chapter, a CMG-based total attitude 0 V control system is briefly formulated and in order to where quantitatively verify the steering law performance, β* = (β , β , β )t . (8–4) ground experiments were carried out. Test results 1 2 3 showed clearly the problems found with typical steering This β* is called a vector part of β and regarded as a laws and the performance of the steering law proposed usual vector in three-dimensional physical space. in the previous chapter. The attitude dynamics of a rigid body is represented in the body’s coordinates by the following Euler 8.1 Attitude Control equation: ω ⁄ τ − ω × IV d dt = V p , (8–5) 8.1.1 Dynamics τ where IV denotes satellite’s moment of inertia and The attitude, namely the orientation of a body can denotes the torque applied to the satellite. This torque be represented by various ways70) such as the direction comes from both the outside as a disturbance torque and cosines, the Euler angles, Roll-Pitch-Yaw angles, from the inside by the CMG system. The vector p is the Rodrigues parameters and Euler parameters. In this total angular momentum of both the satellite and the work, representation is made using Euler parameters. CMG system and is given by: Any attitude is defined as is caused by a single p = I ω + H . (8–6) β = (β β β β V V CMG rotation. The Euler parameters 0, 1, 2, 3) represent an attitude caused by a single rotation of angle This total angular momentum is conserved in the φ t inertial coordinates if there is no disturbance torque. By about the axis e = (e1, e2, e3) : substituting Eq. 8–6 into Eq. 8–5, the ω ×H term β = cos(φ / 2), (8–1) V CMG 0 appears. This term is omitted in Eq. 3–5, i.e., the output β φ i = ei sin( / 2) , equation of the CMG but is evaluated here. Both the kinematic equation and the dynamic equation, Eqs. 8–3 where i = 1, 2, 3 and |e|=1. As the rotation has three and 8–5, are the describing functions of the system. degree of freedom, there is a constraining condition that Σ β 2 = 1. i 8.1.2 Exact Linearization Any attitude, which is the result of a rotation a = (a , a , a , a ) after a rotation b = (b , b , b , b ), is 0 1 2 3 0 1 2 3 The system has six independent variables, three ⋅ expressed as a multiplication by a b in the sense of a β ω components of * and three components of V. The Hamiltonian quarternion as follows: ω × ω dynamics is nonlinear as seen by the term V (IV V) ⋅ − a b = (a0b0 – a1b1 a2b2 – a3b3, when Eq. 8–6 is substituted into Eq. 8–5. Nevertheless, − it is well known that this nonlinear system can be exactly a0b1 + a1b0 + a2b3 a3b2, linearized with a suitable feedback71, 72). a b − a b + a b + a b , 0 2 1 3 2 0 3 1 Let (β*, dβ* ⁄ dt) be state variables and let v be a a0b3 + a1b2 – a2b1 + a3b0) . (8–2) new input variable. If a real input τ is given as:
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Chapter 8
Ground Experiments
The previous chapters dealt only with CMG systems The time derivative of β, i.e., dβ⁄d t is expressed by ω while aiming primarily at a geometric understanding of angular velocity denoted by V: the CMG motion and leaving quantitative matters dβ* ⁄ dt = 1⁄ 2β ω + β*×ω , (8–3) neglected. In this chapter, a CMG-based total attitude 0 V V control system is briefly formulated and in order to where corrected July, 2011 quantitatively verify the steering law performance, β* = (β ,β ,β )t . (8–4) ground experiments were carried out. Test results 1 2 3 showed clearly the problems found with typical steering Thisβ* is called a vector part ofβ and regarded as a laws and the performance of the steering law proposed usual vector in three-dimensional physical space. in the previous chapter. The attitude dynamics of a rigid body is represented in the body’s coordinates by the following Euler 8.1 Attitude Control equation: ω⁄ τ − ω × IV d dt = V p , (8–5) 8.1.1 Dynamics τ where IV denotes satellite’s moment of inertia and The attitude, namely the orientation of a body can denotes the torque applied to the satellite. This torque be represented by various ways70) such as the direction comes from both the outside as a disturbance torque and cosines, the Euler angles, Roll-Pitch-Yaw angles, from the inside by the CMG system. The vector p is the Rodrigues parameters and Euler parameters. In this total angular momentum of both the satellite and the work, representation is made using Euler parameters. CMG system and is given by: Any attitude is defined as is caused by a single p = I ω + H . (8–6) β = ( β β β β V V CMG rotation. The Euler parameters 0, 1, 2, 3) represent an attitude caused by a single rotation of angle This total angular momentum is conserved in the φ t inertial coordinates if there is no disturbance torque. By about the axis e = (e1, e2, e3) : substituting Eq. 8–6 into Eq. 8–5, the ω ×H term β = cos(φ / 2), (8–1) V CMG 0 appears. This term is omitted in Eq. 3–5, i.e., the output β φ i = ei sin( / 2) , equation of the CMG but is evaluated here. Both the kinematic equation and the dynamic equation, Eqs. 8–3 where i = 1, 2, 3 and |e|=1. As the rotation has three and 8–5, are the describing functions of the system. degree of freedom, there is a constraining condition that Σβ2 = 1. i 8.1.2 Exact Linearization Any attitude, which is the result of a rotation a = (a , a , a , a ) after a rotation b = (b , b , b , b ), is 0 1 2 3 0 1 2 3 The system has six independent variables, three ⋅ expressed as a multiplication by a b in the sense of a β ω components of * and three components of V. The Hamiltonian quarternion as follows: ω × ω dynamics is nonlinear as seen by the term V (IV V) ⋅ − a b = (a0b0 – a1b1 a2b2 – a3b3, when Eq. 8–6 is substituted into Eq. 8–5. Nevertheless, − it is well known that this nonlinear system can be exactly a0b1 + a1b0 + a2b3 a3b2, linearized with a suitable feedback71, 72). a b − a b + a b + a b , 0 2 1 3 2 0 3 1 Let (β*, dβ*⁄ dt) be state variables and let v be a a0b3 + a1b2 – a2b1 + a3b0) . (8–2) new input variable. If a real inputτ is given as:
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τ ω × ω = V IV V Support Rod + (2 ⁄ β ) I (β 2µ + β*β*tµ − β β*×µ) , g2 0 V 0 0 Three Axis Rate Gyroscope (8–7) Gimbal AAAA where g3 CMG3 t CMG2 µ = v + 1 ⁄ 4 ωV ωV β* , (8–8) AA the system becomes a set of second order linear systems described by: 750mm
d2β* ⁄ dt 2 = v . (8–9)
g 8.1.3 Control Method 1 g4 Onboard z Computer CMG1 Several controllers given by linear control theory can Rotary be applied to the above linearized system. In the ground Balance Encoder Adjusters y tests done in the experimental portion of this work, a h4 CMG4 x model matching controller and a tracking controller were used. Design of the model matching controller was made by applying a model transfer function in order to satisfy Fig. 8–1 Experimental test rig showing the center-mount suspending mechanism. specified steady and transient properties54). This figure shows that the pyramid The tracking controller allowed an appropriate configuration can be realized so that all four motion of the angular momentum vector to be designed. units fit the surfaces of a rectangular Tracking PD control of a given trajectory is realized by parallelepiped. the following input:
v = f ·(β* – r) + f ·(dβ* ⁄ dt – r ) + r ,(8–10) 1 2 1 2 the ceiling by a three axis gimbal mechanism in its center. If the center of the gimbal coincides with the body’s where r(t) is the trajectory to be followed and r1 and r2 are its time derivatives, defined by: center of gravity, no torque appears at any orientation due to the gravitational force. In this way motion in ⁄ 2 ⁄ 2 r1(t)=dr dt , r2(t)=d r dt . (8–11) space can be simulated. This situation was realized by Details of both are described in Appendix E. using three balance adjusters, which could control their weight along three orthogonal axes, which allowed the center of gravity to be controlled. These mechanisms 8.2 Experimental Facility and were also used for initial set up without CMG control, Procedure and generation of disturbance torque and unloading. The orientation and angular rate of the body were measured by rotary encoders at the three axis gimbal In order to quantitatively demonstrate the problems and rate gyroscopes. In actual satellites these quantities and performance of steering laws, a ground test facility are measured by various sensors such as star/sun/earth was constructed54, 55) and a set of ground tests was sensors and rate gyroscopes. carried out. The main torquer was a pyramid type single gimbal CMG system. All attitude control and the steering law 8.2.1 Facility processes were installed in an onboard computer. A wireless link was used for command transfer from the The ground test facility shown in Fig. 8–1 simulates stationary computer. All power was supplied from the the attitude dynamics of a spacecraft. The main structure laboratory by a pair of thin wires which caused little is a cubic frame made of steel pipes and joints. Triangular disturbance force. Additional details are presented in plates on several surfaces holds such devices as the CMG Appendix E. units, the CMG driver circuits, balance adjusters and a computer. The system in its entirety was suspended from
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8.2.2 Design of Control Command Sequence (3) (5) φ (4)
Figure 8–2 shows a typical tracking control trajectory. This function is continuous with regard to (2) (6) the first time derivative. It consists of a constant (1) (7) (8) acceleration, a constant speed rotation, a constant t1 t2 t1 t3 t1 t2 t1 t3 deceleration, a constant attitude and then the same Rotational angle sequence in reverse. time The maximum rotation of this trajectory was set in Fig. 8–2 Target trajectory. consideration with the limit of rotation of the supporting This trajectory has eight parts as, gimbal. The magnitude of the acceleration and the (1) constant acceleration by d2φ⁄dt2 = a, φ deceleration which are almost proportional to the CMG (2) constant rate rotation, d ⁄dt=at1, 2φ 2 − output torque was set as large as possible so that the (3) constant deceleration by d ⁄dt = a, (4) pointing control at φ =at 2+at t , friction torque of the supporting gimbal can be neglected. 1 1 2 (5) to (8) are the reverse of (1) to (4).
8.2.3 Experimental Procedure SR is represented by Eq. 3–23. The CM method was Tests were conducted using the software whose block introduced in Section 7.5 and allows one free parameter, diagram is shown in Fig. 8–3. Additional details of each a feedback gain denoted by k. Further details of the GM block are described in Appendix E. The control and the CM implementations are given in Appendix E. command sequence for each experiment was a sequence The test procedure can be outlined as follows: First, of a number of maneuver motions given either as the body was controlled at a nominal attitude of β*=(0, reference attitude or as a trajectory in Fig. 8–2. After 0, 0)t by using only the balance adjusters along with the each maneuver, the attitude returned to the original PID controller. This control mode was done in order to position and rotation of the body ceased. wait until the pendulum motion of the body and the To allow comparison, three types of steering laws supporting rod stopped. Then the CMG control started were tested: a gradient method (abbreviated to GM with a sequence of attitude commands consisting of hereafter), a SR inverse method (abbreviated to SR) and either an attitude reference in the model matching the constrained method (abbreviated to CM) proposed controller case, or a trajectory in the tracking controller in Chapter 7. The GM uses procedure described in case. The balance adjusters were controlled so that they Section 3.4.1. Its objective function is det(CCt) and the generated an expected disturbance torque during the free parameter is the gain k defined in Eq. 3–21. The experiment, but which was zero otherwise.
Momentum and Disturbance Balance Adjusters Reference Attitude Torque Management or Command Desired Torque Target Trajectory TCOM Motion Output ω T Attitude CMG Pro- Pyramid Attitude portional Command Controller Steering Type Body Generator Law Limiter CMG System
Attitude and Rotational Rate
Fig. 8–3 Block diagram of the control system. ω Proportional limiter is used to limit the gimbal rate, com, to the maximum gimbal rate so that the real rate vector is proportional to the desired vector. By using this, the real output becomes proportional to the torque command, i.e., Tcom ⁄⁄ T.
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The third experiment was made from another initial 8.3 Experimental Results gimbal angles, which are one of the preferred angles for this direction. The results in Fig. 8–6 shows that this The experiments were carried out to demonstrate the initial angles are appropriate for this situation. following problems and performance characteristics of the CMG steering laws69). (1) Performance and problems of a singularity Table 8-1 Condition and Results of Experiments (1) avoidance steering law such as SR-inverse Experiment Initial θ Steering Results law. law —————————————————————— (2) Problem of z axis rotation and advantage of Experiment A ( 0, 0, 0, 0) GM Fig. 8–4 preferred gimbal angles Experiment B ( 0, 0, 0, 0) SR Fig. 8–5 (3) The gradient method’s inability to keep a Experiment C (−π/3, π/3, −π/3, π/3)GM Fig. 8–6 nominal condition —————————————————————— (4) Performance in various modes of constrained control (5) Advantages of mode switching. The results of the following experiments are drawn in five graphical forms with time on the horizontal axis. The first graph shows the ideal and measured attitude variation on the vertical axis. Other variables on the vertical include the measured gimbal angles, the output torque level and the angular momentum of the CMG system. Also shown is the determinant det(CCt), which provides information regarding the proximity of the system to a singular point. Note that the output torque T here is an actual value with the multiplier h but the angular momentum H is still without the multiplier (see Section 3.1).
8.3.1 Attitude Keeping under Constant Disturbance
By the following three experiments, the item (1) was demonstrated. The H path along (−1, 1, 0)t direction was planned and pointing control under the constant disturbance about this direction was carried out by the model matching controller. The conditions of the three experiments are listed in Table 8–1. The reference attitude and the disturbance were kept constant and the angular momentum of the CMG system, i.e., H continuously increased along (−1, 1, 0)t direction when the pointing control was successful. The first two experiments clarifies the impassable singularity problem of the gradient method and performance of singularity avoidance by the SR-inverse steering law. By the GM, the system became singular (AB in Fig. 8–4 (f)), after that no torque was generated and the body rotated by the disturbance torque (AA in Fig. 8–4 (a)). On the contrary, the SR worked well with slight degradation of pointing accuracy (BA in Fig. 8– 5).
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0.3AAAAAAAAAAAAAA A A A β 0.2 2 0.1AAAAAAAAAAAAAA A A β A 3
β 0 -0.1AAAAAAAAAAAAAA A A A β -0.2 AA 1 -0.3AAAAAAAAAAAAAA A A A 0 102030 (a) attitude time(s)
2 θ AAAAAAAAAAAAAAA 4 A A θ 1 AAAAAAAAAAAAAA A A1 A θ 0 θ 3 AAAAAAAAAAAAAAA θ A A -1 2 AAAAAAAAAAAAAA0A 102030A A (b) gimbal angle time(s)
1 T AAAAAAAAAAAAAA com ATout A A 0.5 0 AAAAAAAAAAAAAA A A A T (N m) (N T -0.5AAAAAAAAAAAAAA A A A -1 0 102030time(s) (c) torqueAAAAAAAAAAAAAA command & outputA A A AAAAAAAAAAAAAA A A A 2 Hy 1 AAAAAAAAAAAAAA A A A Hz H 0 AAAAAAAAAAAAAA A A A -1 Hx -2 AAAAAAAAAAAAAA A A A 0 102030time(s) (d) CMG momentum (normalized)
1.5AAAAAAAAAAAAAA A A A 1 AAAAAAAAAAAAAA A A A
det AB 0.5AAAAAAAAAAAAAAA A A
0 AAAAAAAAAAAAAA0A 102030A A (e) determinant time(s)
Fig. 8−4 Results of Experiment A. The attitude keeping by the gradient method under constant disturbance torque about (−1,1,0)t direction from initial θ of (0, 0, 0, 0).
––– 61 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– AAAAAAAAAAAAAAA A A 0.01 β BA β 3 0.005AAAAAAAAAAAAAA 2 A A A
β 0 β AAAAAAAAAAAAAA 1 A A A -0.005 -0.01AAAAAAAAAAAAAA A A A 0102030 (a) attitude time(s)
2 AAAAAAAAAAAAAAA θ A A 4 1 θ AAAAAAAAAAAAAAA A1 A θ 0 θ 2 -1 AAAAAAAAAAAAAA A A θ A 3 -2 0102030AAAAAAAAAAAAAA A A A (b) gimbal angle time(s)
2 1 AAAAAAAAAAAAAA A ToutA A 0 AAAAAAAAAAAAAA A A A
T (N m) (N T -1 -2 AAAAAAAAAAAAAA A A Tcom A -3 0102030 AAAAAAAAAAAAAA A time(s) A A (c) torque command & output
2 AAAAAAAAAAAAAA A A A Hy 1 AAAAAAAAAAAAAA A A A Hz H 0 AAAAAAAAAAAAAA A A A -1 Hx -2 AAAAAAAAAAAAAA A A A 0102030time(s) (d) CMG momentum (normalized)
1.5 AAAAAAAAAAAAAA A A A
1 AAAAAAAAAAAAAA A A A det 0.5 AAAAAAAAAAAAAA A A A 0 0102030AAAAAAAAAAAAAA A A A time(s) (e) determinant
Fig. 8−5 Results by Experiment B. The attitude keeping by the SR method under constant disturbance torque about (−1,1,0)t direction from initial θ of (0, 0, 0, 0).
––– 62 ––– –– 8. Ground Experiments ––
0.0025 β β 0.00125AAAAAAAAAAAAAA2 3 A A A 0 β AAAAAAAAAAAAAA A A A -0.00125 β 1 -0.0025AAAAAAAAAAAAAA A A A 0102030 (a) attitudeAAAAAAAAAAAAAA A time(s) A A
2 AAAAAAAAAAAAAAA A A θ θ 1 2 4 0 AAAAAAAAAAAAAAA A A
θ θ -1 1 θ AAAAAAAAAAAAAAA A 3 A -2 -3 AAAAAAAAAAAAAA A A A 0102030 (b) gimbal angle time(s) 1.5AAAAAAAAAAAAAAA A A 1 Tcom Tout 0.5AAAAAAAAAAAAAA A A A
T (N m) (N T 0 -0.5AAAAAAAAAAAAAA A A A -1 AAAAAAAAAAAAAA0102030A A A (c) torque command & output time(s)
3 H 2 AAAAAAAAAAAAAA A Ay A 1 H
H z 0 AAAAAAAAAAAAAA A A A H -1 AAAAAAAAAAAAAA A A x A -2 0102030time(s) (d) CMG AAAAAAAAAAAAAAmomentum (normalized)A A A
2 AAAAAAAAAAAAAA A A A 1.5AAAAAAAAAAAAAA A A A 1 det AAAAAAAAAAAAAA A A A 0.5 0 AAAAAAAAAAAAAA A A A 0102030 (e) determinant time(s)
Fig. 8−6 Results of Experiment C. The attitude keeping by the gradient method under constant disturbance torque about (−1,1,0)t direction from initial θ of (−π ⁄ 3, π ⁄ 3, −π ⁄ 3, π ⁄ 3).
––– 63 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 ––
8.3.2 Rotation About the z Axis the nominal θ = (0, 0, 0, 0), while there is a problem of impassability from another initial θ. The conditions of By the next three experiments, the above item (2) the three experiments are listed in Table 8–2. was demonstrated. The H path along the z axis was In the experiments, a command sequence consisting planned which is symmetric to the H path of the previous of two maneuver motions was used when the model three experiments. This time, maneuvering motion was matching controller was operating. The reference performed. For attitude control about the z axis, the H attitude to the controller was changed twice, at first to trajectory is also on the z axis. This H path intersects a an orientation rotated 30 degrees about the z axis for t ≤ singular surface. There is no singularity problem from 10 seconds then to the initial orientation for 10 seconds
0.1 β β AAAAAAAAAAAAAA1, 2 A A 0 AAAAAAAAAAAAAA DA A A
β -0.1 β 3 Reference -0.2AAAAAAAAAAAAAA A A -0.3 AAAAAAAAAAAAAA01020A A (a) attitude time(s)
1AAAAAAAAAAAAAA A A
θ 0AAAAAAAAAAAAAA A A -1AAAAAAAAAAAAAA A A 01020 (b) gimbalAAAAAAAAAAAAAA angle time(s)A A AAAAAAAAAAAAAA A A 20 T AAAAAAAAAAAAAA com A Tout A 0 T (N m) (N T AAAAAAAAAAAAAA A A -20
AAAAAAAAAAAAAA01020A A (c) torqueAAAAAAAAAAAAAA command & output time(s)A A 2AAAAAAAAAAAAAA Hz A A DB
H 0AAAAAAAAAAAAAA A A Hx Hy -2AAAAAAAAAAAAAA A A 01020 time(s) (d) CMG momentum (normalized) 2AAAAAAAAAAAAAAA A 1.5AAAAAAAAAAAAAAA A
1 det 0.5AAAAAAAAAAAAAAA A 0AAAAAAAAAAAAAA A A 01020 (e) determinant time(s)
Fig. 8−7 Results of Experiment D. The z-axis maneuver from initial θ of (0, 0, 0, 0) by the gradient method.
––– 64 ––– –– 8. Ground Experiments ––
≤ t. Table 8-2 Condition and Results of Experiments (2) As the initial gimbal angles of Experiment D is Experiment Initial θ Steering Results preferred gimbal angles for this direction, the system law did not approach any singular point as shown in Fig. 8– —————————————————————— Experiment D ( 0, 0, 0, 0) GM Fig. 8–7 7 and smooth maneuvering was performed as is the Experiment E (−π/3, π/3, −π/3, π/3) GM* Fig. 8–8 analytical result of Section 7.1. On the contrary, in Experiment F (−π/3, π/3, −π/3, π/3)SR Fig. 8–9 Experiment E, the CMG system became singular at t ≈ —————————————————————— 2.3 second (EC in Fig. 8–8 (e)). For this reason, GM (GM* is a modified GM)
0.1AAAAAAAAAAAAAAA A 0 β , β -0.1AAAAAAAAAAAAAAβ 1 2A A β 3 EA -0.2 -0.3AAAAAAAAAAAAAAReference A A -0.4 AAAAAAAAAAAAAA01020A A (a) attitude time(s) AAAAAAAAAAAAAA4 A A θ 4 θ AAAAAAAAAAAAAA2 2 A A
θ θ 3 AAAAAAAAAAAAAA0 A A θ 1 -2AAAAAAAAAAAAAA A A 01020 (b) gimbal angle time(s)
20AAAAAAAAAAAAAA Tcom A A Tout
AAAAAAAAAAAAAA0 A A T (N m) (N T -20AAAAAAAAAAAAAA A A
AAAAAAAAAAAAAA01020A A (c) torqueAAAAAAAAAAAAAA command & output Atime(s) A H AAAAAAAAAAAAAA2 z A A
EB H AAAAAAAAAAAAAA0 A A Hx Hy -2AAAAAAAAAAAAAA A A 01020 (d) CMG momentum (normalized) time(s) AAAAAAAAAAAAAA2 A A 1.5 EC
AAAAAAAAAAAAAA1 A A det 0.5AAAAAAAAAAAAAA A A 0 01020 (e) determinantAAAAAAAAAAAAAA Atime(s) A
Fig. 8−8 Results of Experiment E. The z-axis maneuver from initial θ of (−π ⁄ 3, π ⁄ 3, −π ⁄ 3, π ⁄ 3) by the modified gradient method.
––– 65 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– was modified so that the transpose Jacobian was used D (EA in Fig. 8–8 (a) and DA in Fig. 8–7 (a)). on behalf of the pseudo-inverse Jacobian when the This time, the SR was not effective as shown in Fig. system was nearly singular. By this, the above singular 8–9. The CMG system approached a singular point state was avoided but the element of H to the z axis, i.e., similar to the above experiment (FC in Fig. 8–9 (e)). Hz once saturated at smaller value than that of the Moreover, no singularity avoidance motion was realized maximum in Experiment D (EB in Fig. 8–8 (d) and DB because the command changed faster than that in in Fig. 8–7 (d)). As the result, the transient motion of Experiment B. the control was degraded compared with the Experiment
0.1AAAAAAAAAAAAAAA A 0 β β AAAAAAAAAAAAAAβ FA 1, 2 A A -0.1 3 β -0.2AAAAAAAAAAAAAAA A -0.3 Reference -0.4AAAAAAAAAAAAAA A A 01020 (a) attitude time(s)
4 θ AAAAAAAAAAAAAA 2 A A 2 θ 4 θ AAAAAAAAAAAAAA θ A A 0 3 θ AAAAAAAAAAAAAA 1 A A -2 01020 (b) gimbalAAAAAAAAAAAAAA angle time(s)A A AAAAAAAAAAAAAA A A 20 Tcom AAAAAAAAAAAAAA ATout A 0 T (N m) (N T AAAAAAAAAAAAAA A A -20 AAAAAAAAAAAAAA01020A A (c) torque command & output time(s) 2 1AAAAAAAAAAAAAA Hz A A FB
H 0 AAAAAAAAAAAAAA H A A Hx y -1AAAAAAAAAAAAAA A A -2 01020 (d) CMGAAAAAAAAAAAAAA momentum (normalized) time(s)A A 2AAAAAAAAAAAAAAA A 1.5AAAAAAAAAAAAAA A A det 1 FC 0.5AAAAAAAAAAAAAA A A 0 AAAAAAAAAAAAAA01020A A (e) determinant time(s)
Fig. 8−9 Results of Experiment F. The z-axis maneuver from initial θ of (−π ⁄ 3, π ⁄ 3, −π ⁄ 3, π ⁄ 3) by the SR method.
––– 66 ––– –– 8. Ground Experiments ––
8.3.3 Maneuver after Momentum Accumulation Two experiments were carried out, i.e., Experiment G by CM and Experiment H by GM for the same control In Section 7.1, it was concluded that a gradient sequence. The results of these are shown in Fig. 8–10 method cannot always keep the nominal θ given by Eq. and Fig. 8–11. The control sequence consists of five 7–2 after H travels around and back to the origin. In parts. The reference orientation for the regulator is order to demonstrate this problems, i.e., the above item shown in Fig. 8–10(a). The first part of the sequence is (3), a control sequence including maneuvering motions a maneuver about the z axis, shown as the part A in Fig. as well as momentum accumulation was used while the 8–10(a). This command is similar to that of Section system was controlled by the model matching controller. 8.3.2. In the second part B, a disturbance torque was AAAAAAAAAAAAAAA A A A A 0.1 AB C D E
0 AAAAAAAAAAAAAAA A A A A β β 3 -0.1 AAAAAAAAAAAAAAA β A A A A reference(reference(3) GA -0.2 AAAAAAAAAAAAAA A A A A A 0 20406080100time(s) (a) attitude
1 AAAAAAAAAAAAAA θA A A A A 0.5 3 θ 0 AAAAAAAAAAAAAA A A4 A A A θ -0.5 GB -1 AAAAAAAAAAAAAA Aθ Aθ A A A 2 1 -1.5 0 20 40 60 80 100 AAAAAAAAAAAAAA A A time(s)A A A (b) gimbalAAAAAAAAAAAAAA angle A A A A A 20 Tcom Tout 10 AAAAAAAAAAAAAA A A A A A 0
T (N m) (N T A A A A A A -10 -20 AAAAAAAAAAAAAA A A A A A 0 20 40time(s) 60 80 100 (c) torque command & output 2 H AAAAAAAAAAAAAA A x A AHz A A 1 AAAAAAAAAAAAAAA A A A A
H 0 -1 AAAAAAAAAAAAAAA A A A A Hy GC -2 0AAAAAAAAAAAAAA A 20A 40A 60A 80 100A time(s) (d) CMG momentum (normalized) 1.5 AAAAAAAAAAAAAAA A A A A 1
det AAAAAAAAAAAAAAA A A A A 0.5 AAAAAAAAAAAAAAA A A A A 0 0 20406080100time(s) (e) determinantAAAAAAAAAAAAAA A A A A A
Fig. 8−10 Results of Experiment G. Control by the proposed constrained method.
––– 67 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– applied to move H to (0.6, 0.4, 0.)t while the body’s the motions of the first three parts of the two experiments orientation was fixed (see GC in Fig. 8–10(d)). This are almost the same. This implies that both the steering motion is similar to that of Section 8.3.1. The third part, laws have similar control performances. In the fourth indicated by C, is the same as the first part. Then in the part, however, motions of θ are different. The gimbal part D a reversed disturbance was applied to move H angle θ returned to the original in the case of CM (GB back to its origin similarly to the part B. Both processes in Fig. 8–10) but to the different point in the case of GM B and D resulted in a similar H path as described in (HB in Fig. 8–11). This is because GM controlled the Section 7.5.2. Finally, the same maneuver as the part A CMGs to another local maximum of det(CCt) as was tried (E in Fig. 8–10(a)). described in 7.5.2. As the result of this, the last part was Those results in Fig. 8–10 and Fig. 8–11 show that successful in the case of CM (GA in Fig. 8–10), while it
0.1AAAAAAAAAAAAAAA A A A A A B CDE 0AAAAAAAAAAAAAA A A A A A
β β3 -0.1AAAAAAAAAAAAAA A A A A A reference HA -0.2AAAAAAAAAAAAAA A A A A A 0 20 40time(s) 60 80 100 (a) attitude 4AAAAAAAAAAAAAA A A A A A θ HB θ 4
2AAAAAAAAAAAAAA A3 A A A A θ 0 AAAAAAAAAAAAAA θA A A A A 2 θ1 -2 AAAAAAAAAAAAAA0A 20406080100A A A A (b) gimbal angle time(s)
20AAAAAAAAAAAAAA A A A A A T Tcom out 10AAAAAAAAAAAAAA A A A A A 0
T (N m) (N T AAAAAAAAAAAAAA A A A A A -10
-20AAAAAAAAAAAAAA A A A A A 0 20406080100 (c) torque command & output time(s)
2 AAAAAAAAAAAAAA A HxA AH A AHC 1 z
0AAAAAAAAAAAAAA A A A A A H -1AAAAAAAAAAAAAAA A A A A Hy -2 AAAAAAAAAAAAAA0A 20406080100A time(s)A A A (d) CMG momentum (normalized)
1.6
1.2AAAAAAAAAAAAAAA A A A A
det 0.8AAAAAAAAAAAAAA A A A A A 0.4 HD AAAAAAAAAAAAAA A A A A AHE 0 0 20 40time(s) 60 80 100 (e) determinantAAAAAAAAAAAAAA A A A A A
Fig. 8−11 Results of Experiment H. Control by the gradient method.
––– 68 ––– –– 8. Ground Experiments –– was not in the case of GM (HA in Fig. 8–11). In the 0.8, 0.)t and finally in the part G the maneuver about the case of GM, the determinant went to zero and H saturated (1,1,0)t direction was repeated. (HC in Fig. 8–11) by hitting or approaching some The results are shown in Fig. 8–13. The first three singular surface (HD and HE in Fig. 8–11). parts A, B and C, show almost the same motion as in the These two results clarifies that the unique inversion previous experiment. In the part C there was a slight characteristic of CM is important even for such a simple degradation of control compared with the response in maneuver. the part A. In the part D, the mode of the CM was changed 8.3.4 Mode Selection and Switching directly, without considering the attitude control and by using the fastest direct path to the other mode as The next test was carried out using only the CM in described in Section 7.5.5 (4). This motion inevitably order to demonstrate the above items (4) and (5). The generated an undesired torque and there was some different modes of the constrained method have different attitude deviation of the body, as indicated by JA in Fig. workspaces of H, as described in Section 7.5.4(4). In 8–13. However, because the CMG was moved as fast Experiment J, a maneuver motion resulting in two kinds as possible, this deviation was somewhat minimized. of the H paths were planned for which different control The attitude control immediately following this motion modes were required and mode switching was easily corrected such a deviation. performed. As the theory in item (4) of Section 7.5.5 predicts, In this experiment, trajectory tracking control was the two maneuver motions in the M2 mode, shown in used for the attitude control. The target trajectory for the parts E and G of Fig. 8–13, were successful and did this tracking control is shown in Fig. 8–12. This consisted not meet a singularity. of the following motions: A maneuver about the z axis This experiment demonstrated the advantages of the shown by the part A in Fig. 8–12 was planned by a constrained method proposed in the previous chapter. trajectory defined in Section 8.2.2. After this maneuver, In addition, trajectory tracking control was used this time. a disturbance was applied in the part B so that the angular The results in Fig. 8–13(c) show that almost constant momentum was accumulated to H=(0.5, 0.5, 0.)t. Then torque was realized during the period of constant in the part C, the z axis maneuver was repeated. As the acceleration or deceleration. The proposed method can H path for these three parts are in the workspace of the also cope with a change of maneuver direction by M1 mode, this mode is selected. The maneuver switching between modes. Even by using a direct change command of the latter parts was designed so that the of the mode, deviation in attitude can be made small resulting H path went out of the M1 mode workspace enough to be corrected by the attitude control. but was inside the M2 mode one. For this reason, mode switching in the part D was conducted before the next 8.4 Summary of Experiments maneuver. Then, the new principal axis was in the (1,1,0)t direction. In the part E, a maneuver was From experiments A to F, it is observed that an performed about this new direction, then in the part F appropriate combination of the initial gimbal angles and another disturbance was applied to accumulate H to (1.2, the maneuver direction (or momentum accumulation
AB CDEFG AAAAAAAAAAAAAAAA0.4 A A A A A A 0.3 β β 3 3 AAAAAAAAAAAAAAAA0.2 A A A β Aβ A β , βA β β 1, 2 1 2 β 1, 2 β 0.1 3 AAAAAAAAAAAAAAAA0 A A A A A A -0.1 AAAAAAAAAAAAAAAA0A 20A 40A 60A 80A 100A 120 time(s)
Fig. 8−12 Command sequence of Experiment J. Maneuver motions in A and C are the same rotations about the z axis. Maneuver motions in E and G are the same rotation about the (1,1,0)t axis. In periods B and F, a disturbance torque was applied so that the final H becomes (0.5,0.5,0.)t in B and (1.2, 0.8,0.)t in F. The CM mode is changed at D.
––– 69 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– direction) is the most important. This implies that the proposed constraint method. This method not only method of ‘preferred gimbal angles’ described in Section preserves the merit of the above preferred gimbal angle 7.4 is effective. Though the experiments showed the method but also realizes continuous steering and capability of singularity avoidance of the SR inverse repeatability. Though the workspace of each mode is method, its performance was worse than that when the restricted, it can maintain nearly continuous steering for initial gimbal angles were set appropriately. Moreover, various directional maneuver and/or momentum its performance of singularity avoidance depends on the accumulation events by using the proposed mode speed of momentum accumulation and it sometimes fails. switching operation. This was successfully demonstrated Experiments G and H clarified the advantage of the by Experiment J.
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θ AAAAAAAAAAAAAA A AA A θA A -1 4 θ -2AAAAAAAAAAAAAA A AA A A2 A -3 AAAAAAAAAAAAAA0 20A 40AA 60time(s)A 80A 100A 120 (b) gimbalAAAAAAAAAAAAAA angleA AA A A A 20 AAAAAAAAAAAAAA ATcom AA A A A 10
T (N m) (N T 0AAAAAAAAAAAAAA A AA A A A Tout -10AAAAAAAAAAAAAA A AA A A A 0 20 40 60time(s) 80 100 120 (c) torque command & output
3 AAAAAAAAAAAAAA A AHx A A A A 2 Hz 1
H 0AAAAAAAAAAAAAA A AA A A A -1 H -2AAAAAAAAAAAAAA A AA y A A A -3 AAAAAAAAAAAAAA0 20A 40AA 60time(s)A 80A 100A 120 (d) CMG momentum (normalized) 1.6AAAAAAAAAAAAAAA AA A A A 1.2AAAAAAAAAAAAAAA AA A A A
det 0.8 0.4AAAAAAAAAAAAAAA AA A A A 0AAAAAAAAAAAAAA A AA A A A 0 20 40 60time(s) 80 100 120 (e) determinant
Fig. 8−13 Results of Experiment J. Tracking control for the command given in Fig. 8−7, illustrating use of the proposed constrained method.
––– 70 ––– –– 9. Evaluation ––
Chapter 9
Evaluation
The previous chapters dealt mainly with a specific S(4), S(6) and S(10). (b) Skew Type of 5 and 6 units, pyramid type CMG system. In order to evaluate its denoted by Skew(n). (c) Multiple Type M(m,m) and performance, comparison with regard to the workspace M(m,m,m) with orthogonal gimbal axes. and weight was made for various system configurations. In addition to those, the following systems were selected for comparison. A system denoted by × 9.1 Conditions for Comparison 2 Skew(n) is a doubled skew configuration. The system denoted by 1+Skew(n) and shown in Fig. 9–1 (a) indicates that one unit is added to the symmetric axis of The previous chapters revealed that it is generally a skew type system. The system denoted by S(3,4) is a difficult to avoid impassable singularities. Most steering combination of two symmetric configurations, S(3) and laws have various problems. The only exception is the S(4) as shown in Fig. 9–1 (b). The units are arranged in constrained steering law proposed in Section 7.5, whose the surface directions and the vertex directions of a performance is verified within a certain workspace. But regular octahedron. this method is only effective for the pyramid type CMG system, and if another configuration is used, a gradient method is the only candidate. Thus, evaluation of various systems was made under the assumption that a gradient method is used and the work space is determined so that it does not include any g4 impassable surfaces. g 5 α g3 gn+1 9.2 Spherical Workspace g 2π⁄n 6 g2 Several CMG systems were examined including g1 double gimbal CMGs32, 73) with the following criteria r and χ : (a) Example of 1+Skew(n). r : Maximum radius of a sphere in the angular g7 momentum space, centered on the H origin, and g g3 including no impassable surfaces. 4 χ := r ⁄n.
Obviously, r = n for all double gimbal CMG systems g6 because their work space is a unit sphere of radius n (Appendix A). The radius of any multiple type system g2 is obtained simply, because its envelope has circular g1 plates which the maximum sphere touches. Thus, the radius r of M(m, m) with orthogonal axes is m and that g5 of M(m, m, m) with orthogonal axes is 2m. (b) S(3, 4). Various CMG configurations of up to ten units were examined. These included: (a) Symmetric Type , S(3), Fig. 9−1 System configurations for comparison.
––– 71 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 ––
10. 2×SKEW(5) 2×S(4) S(6) χ=0.75 1+SKEW(6) SKEW(6) S(3,4) χ=0.67 M(2,2,2) S(10) Double M(3,3,3) 5. Gimbal CMG M(5,5)
Envelope 1+SKEW(7) of S(4) M(4,4) : Radius of Spherical Workspace r M(3,3) 1+SKEW(4) 0. S(4) SKEW(5) 0510 n : Number of Units
Fig. 9−2 Spherical workspace size for various system configurations. Filled circles indicates the workspace size of the original system, while attached open circles indicate the workspace size of degraded systems. As a reference, a square indicates the envelope size of the S(4) system.
The radius r was commonly obtained by computer ≈ 0.765 , (9–1) calculation. It was searched by calculating |H| for impassable surfaces of all lattice points of u on the unit which is for an infinite number of units arranged equally sphere at a given increment. In the case of skew type in all directions. system, calculations were made using various values of b) Most systems of no less than six units have the skew angle and the maximum radius r is sought. The respectable χ values ranging from 0.67 to 0.75. This is various possibilities of unit’s breakdown are also because although such systems have internal impassable considered and the worst cases are taken, except in the surfaces, they are only near the envelope. On the case of multiple systems for which all possible cases are contrary, four and five unit systems have smaller χ values considered. because they have internal impassable surfaces much The results are shown in Fig. 9–2 as a graph of the further inside. number of units n versus workspace radius r. The ratio c) Although any multiple type system composed of χ is the slope of the line from the origin. Filled circles no less than six units has no internal impassable surfaces, indicate values of original systems, while blank circles its radius r is considerably smaller than that of other connected to them by straight lines indicate the independent systems. performance of degraded systems. Conclusion from these results are as follows: d) Degradation of the system due to unit’s break a) As the number of units increases, the shape of the down becomes smaller as the number of units increases. angular momentum envelope approaches a sphere and χ also approaches a limiting value given by: 9.3 Evaluation by Weight
⋅ ∫ ()huS dS u∈S2 The workspace size and system weight will be χ∞ = ∫ dS evaluated in light of the preceding results. Suppose that 2 u∈S the work space size H and system weight W satisfy the ∝ π⁄ similar relation as Eqs. 2–1 and 2–2, which are W n = ∫ 22sinφφ d 0 d3 and H ∝ r d5 where d is the diameter of the flywheel. = π / 4 Then, the following relationship is obtained by setting
––– 72 ––– –– 9. Evaluation ––
W as a parameter while H is set constant: Z
r 3 W 5 ⁄ n 5 = constant . (9–2)
µ Figure 9–3 shows results of this comparison. Dotted r2 = r1 curves indicate the relationship of n and r which satisfy the condition 9–2. While the W = a curve passes the r r S(6) point, the other points are under this curve. This 1 1 implies that the S(6) is the lightest for the same spherical X workspace size among all systems evaluated above. As Y the W = 1.5a curve passes the S(4) point, the S(4) system is 50% heavier than the S(6) system. α The data point labeled Skew(4, opt) shows the Fig. 9−4 Definition of ellipsoidal workspace. µ workspace of a skew type four unit system with the skew Aspect ratio is defined as r2 ⁄r1. angle α = tan-1(1 ⁄2) described in Section 7.5.4(5). As the W = 1.15a curve passes this data point, this particular system with the proposed steering law can realize r1 and r2 (=µr1) are the minimum and maximum radii Skew(4) system only 15% heavier than S(6). of the ellipsoid. Skew type systems of 4, 5 and 6 units were examined at various skew angle values. As we 9.4 Ellipsoidal Workspace have two parameters, i.e., the skew angle α and the aspect ratio µ, comparison will be made by keeping one parameter constant. An ellipsoidally shaped workspace may be required By keeping the aspect ratio constant, the radii r and when the attitude control has a certain principal axis. In 1 r were calculated with respect to the skew angle α as this section, skew type CMG systems of 4, 5 and 6 units 2 shown in Fig. 9–5. In these figures, the radii are are evaluated in terms of their workspace size. Systems represented by an average radius defined as follows; of more than 6 units are omitted because they have disadvantages in weight as the results above. 3 2 An evaluation similar to that done in Section 9.2 was rA = rr1 2 . (9–3) made under the same condition that the workspace does not include any impassable surface. The shape of the This value represents the radius of a sphere which has workspace is defined axially symmetric with a fixed the same volume as the ellipsoid. aspect ratio µ as shown in Fig. 9–4. Evaluation criteria For four unit systems, each resulting curve has a maximum at a different skew angle, as shown in Fig. 9– 5 (a). Noteworthy is the system corresponding to the 10 maximum of the µ=1.0 case, the S(4) symmetric type r 3W 5 ⁄ n 5 = const system. For the other aspect ratios, the optimum skew W = a angle increases as the ratio increases. The results of W = 1.15a W = 1.5a five unit skew systems are different. Though there are S( 6) local maxima, the global maxima for any aspect ratio 5 are given when the skew angle is π ⁄2. At this skew angle, all gimbal axes are on the same plane. In the case of six unit systems, we can find two groups of candidates result in the largest workspace, one has a SKEW(5) skew angle of π ⁄2 and the other 0.26≤α≤0.33. SKEW(4, α ) S(4) op t : Radius of Spherical Workspace Next, the radii r1 and r2 were calculated with respect r 0 0246810 to the aspect ratio by keeping the skew angle constant n : Number of Units (Fig. 9–6). In each figure, the maximum and minimum radii are plotted with respect to the aspect ratio, for − Fig. 9 3 Trade-off between workspace size and various skew angles. Fig. 9–6 (a), for the Skew(4) system weight. Dotted curves indicate equal workspaces with arrangements, shows that there is an optimal skew angle equal weight (W).
––– 73 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– AAAAAAAAAAA A A A
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r α 0.375 r2 0.4 3 =0.3 0.35 µ=1.5 AAAAAAAAAAA A A A 1.5AAAAAAAAAA0.325A A A A 2 µ=1.0 µ=2.0 r µ =2.5 1 0.45 0.475
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Average Radius Radius Average 0 2 0.1 0.2 0.3 0.4 0.5 r1 AAAAAAAAAA ASkew AngleA α (radian)A A AAAAAAAAAA A A A A (b) Skew(5, α) 0 1 1.5 2 2.5 3 AAAAAAAAAA A AspectA Ratio µA A α A (b) Skew(5, ) r AAAAAAAAAA A A µ=2.5A A A A A A A 4 0.5 0.45 0.4 r µ 2 AAAAAAAAAA A A A =1.0A AAAAAAAAAA6 α=0.32A A A A 2 r µ=2.0 µ A A A A =1.5A AAAAAAAAAA4 A A 0.26A A
0 radius
Average Radius Radius Average 0.1 0.2 0.3 0.4 0.5 2 AAAAAAAAAA ASkew AngleA α (radian)A A AAAAAAAAAA A A A A r1 α (c) Skew(6, ) AAAAAAAAAA0 1A 1.5A 2A 2.5A 3 Aspect Ratio µ Fig. 9−5 Average radius vs. skew angle. (c) Skew(6,α)
Fig. 9−6 Workspace radius as a function of giving the largest workspace for each value of the aspect aspect ratio. ratio. For example, the optimal skew angle is 0.35 π at an aspect ratio of 1.75. On the contrary, a π ⁄2 skew α=0.5 Skew(6, α)
angle is optimum for any aspect ratio in the case of 2 6 r α=0.32 Skew(5, π ⁄ 2) Skew(5). For Skew(6), a 0.32 π skew angle is optimum r2 4 π and for aspect ratio less than 1.4, while ⁄2 is optimum for 1 M(2, 2)=Roof(4) r r1 Skew(4, α) larger aspect ratios. These optimum values are selected 2 and plotted in Fig. 9–7. 0.35 0.375 0.4 radius α=0.3 Radius values in this figure can be converted to 0 0.325 1 1.5 2 2.5 3 indicate the system weight as discussed in Section 9.3. Aspect Ratio µ This ‘converted weight’ W is equivalent to the weight of a system whose workspace size is a certain fixed value. Fig. 9−7 Combined plot of radii as a function of aspect ratio. By a relationship similar to Eq. 9–2, this converted Envelope size of multiple systems, M(2, 2), is weight is defined as follows; drawn in addition to the results in Fig. 9−6.
2 −1 / 5 W = n ⁄(r1 r 2) . (9–4) The results are plotted in Fig. 9–8. The converted (See Fig. 7–19). The results in Fig. 9–8 show that the weight of the Skew(4, α) system, controlled by the weight is much larger in the case of the original Skew(4) proposed constrained method, is also included. In this system than the other systems. All the other systems, evaluation, the workspace of this system is approximated including the 4-unit skew system with the constrained by an ellipsoid whose radii are given by 2cosα and 4sinα steering law, have similar weight values.
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4 A A A A A Skew(6, π ⁄ 2) 4 3 AAAAAAAAAAAA A A A r Skew(4, 0.4 π ) Skew(5, π ⁄ 2) 2 Skew(6) α AAAAAAAAAAAWeight 2 A A A A Skew(4, )
Skew(5) radius Skew(4) 1 A A A ASkew(4)’ A 0 1 1.5 2 2.5 3 01 1.5 2 2.5 3 AAAAAAAAAAA A AAspect Ratio µ A A Aspect Ratio µ − Fig. 9−8 Converted weight as a function of Fig. 9 9 Radius as a function of aspect ratio aspect ratio. for a degraded system with one faulty unit. The original Skew(4) system is shown by the solid line and the Skew(4) system controlled by the proposed constrained method is shown by the heavy dashed line of Skew(4)'. of 4, 5 and 6 units with optimal skew angle, and with constrained control in the case of the 4 unit system, have similar workspace size with the same weight. The next figure, Fig. 9–9, shows a relation similar to (3) If fault tolerance is required, the Skew(6) system Fig. 9–7, when one unit becomes nonfunctional. This is much better than Skew(4) and Skew(5) in terms of figure shows that degradation of the Skew(6) system is degradation of the workspace due to loss of one unit. much less than the others. On the contrary, degradation In the evaluation of this chapter, the three modes of α of the Skew(4, ) with the constrained method is serious. a symmetric pyramid type system were not considered. Since the workspace of each mode is a similarly shaped 9.5 Summary of Evaluation ellipsoid, this pyramid type system becomes more promising by considering the three modes. In addition (1) For the spherical workspace, the S(6) system is to this, other factors are also important, such as superior in terms of the system weight. The Skew(4) mechanical complexity and steering law complexity. By system driven by the constrained method is only 15 % considering these, the Skew(4) system with the proposed heavier in the simplified comparison. constrained method is advantageous for actual use, (2) For the ellipsoidal workspace, skew type systems especially the S(4) system.
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––– 76 ––– –– 9. Evaluation ––
Chapter 10
Conclusions
In this paper, the control of a single gimbal CMG evaluated. This evaluation showed that steering law system has been investigated, with emphasis on the exactness is the most important. An alternative steering symmetric pyramid type. Specifically, the singularity law was proposed which maintains exactness but which problems have been examined using geometric theories is valid in a restricted workspace. Then in Chapter 8, and computer calculations. The “global” problem of the this proposed method was evaluated using ground pyramid type system has been clarified, and a new experiments. First, the problems described in Chapter 7 steering law approach has been proposed and verified were demonstrated. Then, the proposed method and using ground experiments. other steering laws were tested using some attitude In Chapter 2, single gimbal CMGs were described in control sequences. The performance of the proposed comparison with double gimbal CMGs and reaction method was verified, especially for a realistic sequence wheels. Then in Chapter 3, an analytical formulation of including maneuvering and pointing under a specified general single gimbal CMGs was presented. directional disturbance. In Chapter 4, the singularity problem was described. In Chapter 9, a pyramid type system with the proposed Methods for obtaining singular surfaces, especially the steering law was compared with other types of CMG envelope, were presented. The passability of a singular systems. Evaluation according to workspace size surface was defined. Then, examples of some impassable showed that the symmetric six unit system was superior surfaces of various CMG systems were given. The in terms of weight. However, the proposed method, with results showed that impassable singularity is a serious spherically shaped workspace, showed significant problem for the steering law of 4 and 5 unit CMG improvement. Moreover, it was shown that the systems. In Chapter 5, continuous steering under the workspace size was almost equal to that of the five or existence of an impassable singular surface has been six unit skew system when an ellipsoidal workspace is generally examined by using a topological study. A considered. Because of this result and the fact that a method to overcome some types of impassable symmetric pyramid type system has three modes, as well singularities were described in a geometric manner. as mechanical and steering law simplicity, it was Some example conditions were presented in which no concluded that the pyramid type CMG system with the steering law can realize continuous motion. proposed steering law would be an ideal candidate for In Chapter 6, the symmetric pyramid type CMG three axis attitude control. system was defined. Analytical results including This paper does not include studies of more realistic symmetry and singular surface structure were presented. attitude control problems, which should be investigated Chapter 7 clarified the global steering problem that in consideration with the results of this paper. Evaluation continuous real-time steering cannot be realized over may require more detailed characteristics with regard to most of the workspace. By the consequences of this mission requirement, disturbance profiles and unloading and by geometric theories, typical steering laws were torquer specifications.
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––– 78 ––– Appendix A
Double Gimbal CMG System
∂ ∂θ × Formulation of an arbitrarily configured double cOi = hi/ Oi = gOi hi, gimbal CMG system is made in accordance with the ∂ ∂θ × cIi = hi/ Ii = gIi hi, formulation presented in Chapters 3 and 4.
Note that the vector cIi is a unit vector while the vector c is not. A.1 General Formulation Oi The system configuration is then defined by the set of {g }. The dependent system variables, namely the Consider a system of n equally sized double gimbal Oi total angular momentum H and the output torque T are CMGs in an arbitrary configuration. For each unit, one given by: fixed vector, two variable gimbal angles, and four other Σ θ vectors are defined. These are diagrammed in Fig. A–1 H = i hi = H( ) and defined by: Σ ω ω T = i (cOi Oi + cIi Ii )
gOi: Fixed unit vector along the outer gimbal axis. = C ω, θ Oi: Outer gimbal angle with origin located as θ shown in Fig. A–1 where is a point on a 2n dimensional torus whose coordinates are given by: θ Ii: Inner gimbal angle with origin located as shown in Fig. A–1 θ θ θ θ θ θ θ = ( O1 I1 O2 I2 . . . On In). g : Unit vector along inner gimbal axis (a function Ii θ The 2n dimensional vector ω is defined by: of Oi) h : Angular momentum vector, normalized to i ω = (dθi/dt). |hi|=1 The difference between these expressions and those cOi: Outer gimbal torque vector for the single gimbal system are that the vector c is not c : Inner gimbal torque vector Oi Ii a unit vector and that some vector variables are not always independent to another θ variable as follows: where both torque vectors are defined as follows: ∂ ∂θ − cIi / Ii = hi , ∂ ∂θ × × × × cIi / Oi = (gOi gIi) hi + gIi (gOi hi), ∂ ∂θ × × cOi / Ii = gOi (gIi hi), ∂ ∂θ × × cOi / Oi = gOi (gOi hi). θ O ∂ ∂θ gI The last expression implies that the vector cOi / Oi is cI parallel to gIi .
g O A.2 Singularity cO
θ h When the system is singular, the following relation I is satisfied.
det(CCt) = 0 Fig. A−1 Vectors and variables relevant to a double gimbal CMG
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previous section, suppose further that some unit is in θ condition (b), then any d Oi results in the same H. This implies that outer gimbal motion of this unit alone is a h = u null motion. When there are two units both in condition (a) but different signs of ε, the angular momentum vectors of these two units are in opposite directions and g gO I they move on the unit sphere. Motion of the two units can be chosen exactly canceling each other. Thus, this motion is a null motion.
(a) Condition (a). A.4 Passability
Passability of a singular surface can be defined by a quadratic form similar to Eq. 4–15. However, another g I u approach is also possible.
A.4.1 Two Unit System h = g O First, a two unit system will be considered. For an
arbitrary configuration of gO1 and gO2, there are five spherical singular surfaces. One surface is of diameter 2 and is an angular momentum envelope. The remaining (b) Condition (b). − four are unit spheres with their centers at gO1, gO1, gO2 and −g . It is adequate to check the following two cases: − O2 Fig. A 2 Vectors at singularity conditions. The first case, CASE I, is that one unit satisfies condition (b). Singular surface for this case is a unit sphere. The Geometric comprehension of this fact is easier than the second case, CASE II, is that both units satisfy condition case of a single gimbal system. Since both torque vectors (a). In this case H is at its origin. are orthogonal to each other and to h, a singular vector u is determined as u parallels h. An exception arises CASE I Infinitesimal motion of the unit satisfying (a) when the vector hi is parallel to the vector gOi, where cIi is a zero vector. Thus, there are four possible conditions is exactly on the singular surface. On the other hand, of singularity for each unit when u is specified: infinitesimal motion of another unit, say unit 1 for example, includes a motion out of the surface with regard (a) h = ε u, where ε =1 or −1 i i i to the second order differential. Second order (b) h = ε g , and (g × g ).u = 0, θ θ i i Oi Ii Oi differentials such as d I1 d O1 are orthogonal to gO1 but where εi=1 or −1. These are drawn in Fig. A–2. Condition (b) is called ‘gimbal lock’ because such a unit looses one degree of freedom autonomously. g Because the domain of u is a unit sphere, the total I u angular momentum H forms a number of spheres in accordance with the set of ε . i h = g O θ θ d Id O A.3 Steering Law and Null Motion θ d I
Any steering law has the same expression as in Eq. 3–9. When the system is singular, some null motions Fig. A−3 Infinitesimal motion at are easily obtained as follows. Referring back to the a singular point of condition (b).
––– 80 ––– –– A. Double Gimbal CMG System –– not always to u in Fig. A–3. This motion therefore can whose center is on the origin and additional spheres of realize a second order motion in both directions away diameter 2. The unit sphere corresponds to the case that from the singular surface. Thus, internal part of this all the units satisfy condition (a). In this case, the surface is passable. The exception is the case that u = following motion of h vectors is realized by null motions. ± gO1 but this is either the case that H is on the envelope − (1) dh1 = dh2 = dh3/2, or it is regarded as the following CASE II. (2) dh1 = − dh2, and dh3 = 0, ε CASE II This condition is simply expressed as: where it is supposed that the third unit’s is negative. Clearly, the second order motion by (1) is in the direction − − h1 = h2 = u or u. of −u and that by (2) is in the direction of u. Therefore, this singular point is passable. Any null motion satisfies the following: The sphere of diameter 2 represents the case that one − dh1 = dh2, of the units satisfies condition (b). In this case, the two unit subsystem is equivalent to CASE I in the above and so the differential is exactly zero for any null motion. this singular point is passable. Thus, a three (or more) This implies that the quadratic form is exactly zero here. unit system has no internal impassable surface. This is similar to the H origin of the roof type system M(2,2) (see Section C.1). A.5 Workspace A.4.2 Three Unit System There is no internal impassable surface for a double The internal singular surface of an arbitrarily gimbal CMG system of no less than three units. The configured three unit system consists of a unit sphere available work space is a sphere of diameter n.
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––– 82 ––– Appendix B
Proofs of Theories
A full description and proof of the formulation given The two candidates of eSi given by Basis B–1 satisfy in Chapter 4 is made here. Eq. B–5 with the above dui for i value in the same order. Thus, the basis candidates B–1 constitutes the bases of Θ B.1 Basis of Tangent Spaces S. The following is a general relationship for four arbitrary vectors in three dimensional space: For an independent type system in a singular state, there are two independent torque vectors. Suppose that a[ b c d ] – b[ c d a ] + c[ d a b ] – c[ a b c ]=0 , they are described by c and c . The three sets of bases, 1 2 (B–7) Θ Θ Θ ei, of the subspaces S, N and T, are obtained as follows. where [ ] is a box product of three vectors. By substituting c , c , c and u into these four vectors, the Basis of Θ 1 2 i S following relationship is obtained: t eSi = ( p1 qi, 1, p2 qi, 2,. . . ,pn qi, n) , (B–1) q1, 2 ci + q2, i c1 + qi, 1c2 = 0 . (B–8) where i = 1 or 2. Hence, the candidates for e in Eq. B–2 satisfy the Θ Ni Basis of N definition of null motion since CeNi = 0. Furthermore, t they are independent, because q is not zero by eN1 = ( q2, 3, q3, 1, q1, 2, 0, . . . ,0) , 1, 2 definition. e = ( q , q , 0, q , 0, . . . ,0)t , N2 2, 4 4, 1 1, 2 The scalar product of a null motion and n dimensional . . . vector in Eq. B–5 is described by: t e − = ( q , q , 0, . . . ,0, q ) , Σ θ θ Nn 2 2, n n, 1 1, 2 i d Ni (d Si ⁄ pi) (B–2) Σ θ ( ⋅ = i d Ni cSi du) Basis of Θ θ t t T = (d N) C du = ( θ t t C d N) du eT2 = ( qi, 1, qi, 2, . . . ,qi, n) , (B–3) = 0 . (B–9) where qi, j is defined as the following vector triple product; θ Thus, the n dimensional vector d Si ⁄ pi is orthogonal to the null motion and belongs to Θ . Basis B–3 is obtained q = [ c c u ] . (B–4) T i, j i j by simple substitution. These are derived as follows. θ The definition of d S is given by differentiation of B.2 Gaussian Curvature the singularity relation (Eq. 3–16) as,
⋅ ⋅ θ Gaussian curvature of a surface in three dimensional cSi du = dcSi u = d Si ⁄ pi . (B–5) space is defined by the second fundamental form of a Since the vector du lies on a plane orthogonal to u, there surface, for which there are various definitions. Among are two independent vectors du1 and du2. These can be them is the area ratio of the surface and a Gaussian defined as: sphere. By this ratio, the Gaussian curvature κ is defined × as: dui = ci u, where i = 1 and 2 . (B–6)
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κ 22) 1 ⁄ = [u dH(du1) dH(du1)] ⁄ [u du1 du2] , paper requires knowledge of the theory of dyadics. Here this is proven in vector form. (B–10) The differential of H by du given by Eq. 4–5 is obtained by substituting this du into Eq. B–12, giving: where du1 and du2 are selected such that they are independent, as Eq. B–6, for example. The two dH(du) = CPCt du differentials dθ and dθ corresponding to these du S1 S2 κ t t × are defined by Eq. B–5 as: = CPC ((CPC V) u) . (B–17) θ ⋅ A part of the term on the right-hand side can be rewritten d Sj = {pi (cSi duj)} , (B–11) t as follows: = P C duj , Ct((CPCtV)×u) where j = 1 or 2 . t t × = (ci ) ((CPC V) u) Therefore: t t = (|ci (CPC V) u|) dH(du ) = C dθ (B–12) × t t i Si = ( ci u ) (CPC V) = CPCt du i = ( c × u )tC (PCtV) , (B–18) Σ ⋅ i = j pj (cSj dui)cj , where expressions (xi) and (vi) denote a row vector and where i = 1 or 2 . a matrix,
Substituting this into the first triple product on the (xi) = (x1 x2 .... xn) , (B–19) right-hand side of Eq. B–10 results in the following: (vi) = (v1 v2 ..... vn) . [u dH(du1) dH(du2)] This notation as well as a matrix notation such as (xij) Σ Σ ⋅ ⋅ = i jpipj(ci du1)(cj du2)[ci cj u] , (B–13) are used from this point in this section. The first term of Eq. B–18, ( c × u )tC, is the From the four vector relationship, the following is i following matrix: obtained; × t × t ⋅ ⋅ − ⋅ ⋅ ( ci u ) C =(( ci u ) (cj)) (ci du1)(cj du2) (ci du2)(cj du1) × ⋅ × = ([ci cj u]) , (B–20) = (ci cj) (du1 du2) . (B–14) Multiplying both sides by P, a new matrix R is defined. As vectors such as ci and dui are orthogonal to the unit vector u, the expression on the right with both two terms R = P([ci cj u])P = (pi pj [ci cj u]) . (B–21) multiplied by u is unchanged. Thus, the right hand side of Eq. B–17 can be rewritten × ⋅ × (ci cj) (du1 du2) as: = (c ×c )⋅u (du ×du )⋅u i j 1 2 κ CPCt((CPCtV)×u) = [ci cj u] [u du1 du2] . (B–15) = κ CRCtV, κ t By substituting this into Eq. B–13 and using [ci cj u] = C(rij )C V. = −[c c u], the following result is obtained: κ ⋅ j i = ( ci )(rij )(cj V)) [u dH(du1) dH(du2)] κ Σ ⋅ = ij rij (cj V) ci , (B–22) Σ Σ 2 = 1⁄2 i jpipj[u du1 du2][ci cj u] .(B–16) where rij is the element of R given by the preceding Thus, the expression of Gaussian curvature in Eq. 4–7 equation. Taking advantage of the fact that matrix B– − is obtained. 21 is skew symmetric (rji = rij): κ Σ ⋅ ij rij (cj V) ci B.3 Inverse Mapping Theory Σ ⋅ − ⋅ = 1⁄2 ij rij ( (cj V) ci (ci V) cj ) .(B–23)
Proof of the inverse mapping theory in the original From the vector product rule and the fact that u is normal
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× to (ci cj), and ⋅ − ⋅ (cj V) ci (ci V) cj aa11 12 ... a1k = (c × c ) × V j i aa... a = 21 22 2k Ak = [c c u] (u × V ) . (B–24) : ::: . j i aa... a From Eqs. B–21 and 4–7, kk12 kk κ CPCt((CPCtV)×u) A is positive definite if and only if the ‘minor’, i.e., detAk κ Σ × = 1⁄2 ij rij [ci cj u] ( V u ) is positive for all k = 1, 2, ..., m. κ Σ 2 × = 1⁄2 ij pi pj [ci cj u] ( V u ) Though all the minors must be examined in general, = V × u . (B–25) in case of checking passability, only two of them should be examined because we can suppose that ε and ε From Eq. B–17, n n−1 are negative without loosing generality, . dH = V×u . (B–26) Suppose that the condition (3) in Section 4.3.4 is satisfied, that is, the Gaussian curvature κ is positive. In Thus, Eq. 4–4 is derived and the theory is proven. this case, there are two possibilities where the quadratic form QN is positive definite or its signature has two B.4 Impassable condition for two negative terms. In both cases, determinant of the matrix A is positive because this determinant is a product of negative signs N all eigenvalues and its sign is determined by the signature. Passability is defined by the signature of the quadratic Consider the subsystem without the nth unit which form, QN, in Eq. 4–19. Let AN denote the matrix of this has the negative sign. For this subsystem, ANn−3 is the quadratic form as; matrix of the quadratic form, because the original matrix t −1 A is n−2×n−2. As there is one negative sign, the AN = EN P EN . (B–27) N condition (2) in Section 4.3.4 that the Gaussian curvature Let’s find a condition of impassable state in case that is negative is equal to that the matrix ANn−3 is positive ε two of the signs, i, are negative and remaining signs definite. Thus, only this condition is enough to assure are positive. that minors for all k = 1, 2, ..., n−3 are positive. We have the following linear algebra theory for From the above theory, we have the following definite matrix. conclusion.
Conclusion Theory Consider a symmetric m×m matrix denoted by A singular surface of all the ε but two are positive is A and its sub-matrix A . i k impassable if and only if κ > 0 and κ’ < 0, where κ’ is the Gaussian curvature of the subsystem without one aa... a 11 12 1m unit of negative sign. aa... a A = 21 22 2m :::: , aamm12... a mm
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––– 86 ––– Appendix C
Internal Impassability of Multiple Type Systems
Θ In this appendix, the minimum systems with no Though the subspace N of the null motion is internal impassable surfaces are found which are M(3, generally two-dimensional, it is one-dimensional when 3) and M(2, 2, 2). First, discrimination of singularity u lies on the plane spanned by the two gimbal vectors for the four unit roof type system is made. Then, the g1 and g2. In this case, there are three independent null minimum systems are searched by adding units to this vectors. From this point, each variable pi is defined for ≥ system. each group so that the condition pi 0 is satisfied.
C.1 Roof Type System M(2, 2) C.1.1 Evaluation of Singular Surface (2) The four unit roof type system is shown in Fig. 2–2. Independent null motions are; All singular surfaces are given as follows. φ − 1 = (1, 1, 0, 0), (C–1) ≠ ≠ u g1, u g2 φ − 2 = (0, 0, 1, 1) . (1) θ = θ , θ = θ , ε={+ + + +}: 11 12 21 22 θ t −1 θ θ φ φ The quadratic form d N P d N by d N = a1 1+a2 2 Envelope is; θ θ θ θ (2) 11 = 12, 21 = 22, t −1 dθN P dθN ε − − − − ={+ + } or { + +}: 2 2 2 2 = −1⁄2(a1 ⁄p1+a1 ⁄p1−a2 ⁄p2 − a2 ⁄p2) Internal Surface − 2 2 = a1 ⁄p1 + a2 ⁄p2 , (C–2) (3) θ11 = θ12+π, θ21 = θ22, ε={+ − + +}: Thus, this is indefinite and passable. Circle orthogonal to g1 centered on H origin. θ θ θ θ π ε − (4) 11 = 12, 21 = 22+ , ={+ + + }:
Circle orthogonal to g2 centered on H origin. C.1.2 Evaluation of Singular Surface (3) θ θ π θ θ π ε − − (5) 11 = 12+ , 21 = 22+ , ={+ + }: Independent null motions are; H origin φ1 = (1, 1, 0, 0) , (C–3) u = g1 φ − 2 = (0, 0, 1, 1) . θ θ (6) 21 = 22: Envelope The quadratic form dθ tP−1dθ by dθ = a φ +a φ θ θ π N N N 1 1 2 2 (7) 21 = 22+ : Inside Circle (3) is; dθ tP−1dθ u = g2 N N θ θ = −1⁄2(a 2⁄p −a 2⁄p +a 2⁄p + a 2⁄p ) (8) 11 = 12: Envelope 1 1 1 1 2 2 2 2 θ θ π − 2 ≤ (9) 11 = 12+ : Inside Circle (4) = a2 ⁄p2 0 , (C–4) Thus, this is semi-definite and impassable. It is only necessary to evaluate internal surfaces (2), If u happens to be on the plane spanned by g and g , (3), (5) and (7), because pairs such as (3) & (4) or (7) & 1 2 another null vector φ (dependent on φ and φ ) is given (9) are identical when the group numbers are changed. 3 1 2
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φ = (1, 0, −1, 0) (C–5) 3 C.1.4 Evaluation of Singular Surface (7) θ φ φ φ The quadratic form by d N = a1 1+a2 2+a3 3 is; Independent null motions are; θ t −1 θ d N P d N φ − 2 − 2 − 2 1 = (0, 0, 1, 1) , (C–11) = 1⁄2{(a1+a3) ⁄p1 a1 ⁄p1 +(a2 a3) ⁄p2 2 φ = (a, b, 1, −1) . + a2 ⁄p2} 2 − 2 2 θ φ φ = 1⁄2{ a3 ⁄p1 + 2a1a3⁄p1 + 2a2 ⁄p2 The quadratic form by d N = a1 1+a2 2 is; − 2 t −1 2 2 2a2a3⁄p2 + a3 ⁄p2 } dθN P dθN = −1⁄2{(a1+a2) ⁄p2− (a1−a2) ⁄p2} , − 2 − 2 = 1⁄2(ra3 + 2rsa1a3 + 2(a2 a3⁄2) ⁄p2 ) (C–12) = −1⁄2{r(a 2 + sa )2 − rs2a 2 3 1 1 Thus, this is indefinite and passable. − 2 + 2(a2 a3⁄2) ⁄p2 },
where r = 1⁄p1+3⁄(4p2) >0, 2rs = 2⁄p1 . C.1.5 Conclusion (C–6)
Thus, this is indefinite and passable. The roof type system M(2, 2) has internal impassable surfaces given by (3) and (5) with u not on the plane spanned by g1 and g2. Both are not fully definite but C.1.3 Evaluation of Singular Surface (5) the quadratic form of (3) is semi-definite and that of (5) is zero. Independent null motions are; φ C.2 M(3, 2): M(2, 2)+1 1 = (1, 1, 0, 0) , (C–7) φ = (0, 0, 1, 1) . 2 The system M(3, 2), which results from adding an θ φ φ The quadratic form by d N = a1 1+a2 2 is; additional unit to M(2, 2), will now be analyzed. The passability of sub-system M(2, 2) was evaluated above, dθ tP−1dθ N N and so only those impassable conditions should be tested. 2 2 2 2 = −1⁄2(a1 ⁄p1−a1 ⁄p1+a2 ⁄p2 −a2 ⁄p2) = 0 . (C–8) C.2.1 Condition (3) of M(2,2)
Thus, this is zero for any null motion and impassable. System M(3, 2) is asymmetric and so both conditions If u is on the plane spanned by g1 and g2, another (3) and (4) should be tested. The singular points are φ φ φ null vector 3 (dependent on 1 and 2) is given by; given by; φ − θ θ θ π θ θ 3 = (1, 0, 1, 0) . (C–9) (3') 13 = 11 = 12+ , 21 = 22, θ φ φ φ ε={+ − + + +} , The quadratic form by d N = a1 1+a2 2+a3 3 is; θ θ θ θ θ π θ t −1 θ (4') 13 = 11 = 12, 21 = 22+ , d N P d N − 2 − 2 − 2 ε={+ + + + −} . = 1⁄2{(a1+a3) ⁄p1 a1 ⁄p1+(a2 a3) ⁄p2 − 2 a2 ⁄p2} Independent null motions are then; − 2 − φ ± = 1⁄2 {(1⁄p1+1⁄p2)a3 + 2(a1⁄p1 a2⁄p2)a3} 1 = (1, 1, 0, 0, 0) , − φ ± = 1⁄2 [(1⁄p1+1⁄p2) 2 = (1, 1, 1, 0, 0) , (C–13) ⋅ − 2 {a3+(a1⁄p1 a2⁄p2)⁄(1⁄p1+1⁄p2) } φ −(±1) 2 = (0, 0, 0, 1, ) . − − 2 (a1⁄p1 a2⁄p2) ⁄(1⁄p1+1⁄p2) ] . where upper and lower sign of the multiple sign ± (C–10) correspond to (3') and (4') respectively. The quadratic
––– 88 ––– –– C, Internal Impassability of Multiple Type Systems ––
− θ φ φ φ dθ tP 1dθ form by d N = a1 1+a2 2+a3 3 is; N N − 2 − 2 2 2 (3') = 1⁄2 {a1 ⁄p1 (a1+a2) ⁄p1 + a2 ⁄p1 + a3 ⁄p2 θ t −1 θ − 2 d N P d N a3 ⁄p2} = −1⁄2{a 2⁄p − (a +a )2⁄p + a 2⁄p − − 2 − 2 1 1 1 2 1 2 1 = 1⁄4( (a1+a2) + (a1 a2) )⁄p1 , (C–17) + a 2⁄p + a 2⁄p } 3 2 3 2 Thus, this is passable. − − 2 = 1⁄2( 2a1a2⁄p1 + 2a2 ⁄p2) It is concluded that M(3, 2) has an impassable surface − − 2 − 2 = 1⁄2{ 1⁄2(a1+a2) ⁄p1 + 1⁄2(a1 a2) ⁄p1 corresponding to condition (4'). 2 + 2a2 ⁄p2} , C.3 M(3, 3): M(2, 2)+2 (C–14)
(4') Even if a certain sub-system M(3,2) of M(3, 3) t −1 satisfies condition (4'), the same condition will be (3') in dθN P dθN the other sub-system. This implies that there is no − 2 2 2 2 = 1⁄2{a1 ⁄p1 + (a1+a2) ⁄p1 + a2 ⁄p1 + a3 ⁄p2 internal impassable surface. 2 − a3 ⁄p2} 2 2 2 = −1⁄2(a1 ⁄p1 + (a1+a2) ⁄p1 + a2 ⁄p1) . C.4 M(2, 2, 1): M(2, 2)+1 (C–15) This configuration is not a multiple system, so there Thus, (3') is passable and (4') is impassable. The singular is an internal impassable surface. H of (4') forms a circle centered on the origin and having a diameter of 3. C.5 M(2, 2, 2): M(2, 2)+2
C.2.2 Condition (5) of M(2,2) If u is not parallel to gi, all internal surfaces are passable from the discussion of Section 3.2.5. If u is The singular points are given by; parallel to gi, condition (7) is satisfied by sub-system θ θ θ π θ θ π (5') 13 = 11 = 12+ , 21 = 22+ , M(2, 2) which includes the ith group. Thus all internal ε={+ − + + −} . surfaces are passable.
Independent null motions are then; C.6 Minimum System φ 1 = (1, 1, 0, 0, 0) , φ 2 = (0, 1, 1, 0, 0) , (C–16) The multiple systems M(3, 3) and M(2, 2, 2) are the minimum systems with no internal impassable singular φ = (0, 0, 1, 1) . 3 surfaces. Both systems consist of six units. θ φ φ φ The quadratic form by d N = a1 1+a2 2+a3 3 is;
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––– 90 ––– Appendix D
Six and Five Unit Systems
D.1 Symmetric Six Unit System S(6)
D.1.1 System Definition z
The S(6) system is a symmetric type with six units arranged in the surface directions of a regular dodecahedron. Its work space possesses symmetry and y can be approximated by a sphere. Being an independent x type, the system has internal impassable surfaces, which are very near the envelope. Control over most of the entire workspace shown in Fig. D–1 (a) can be accomplished using a gradient method. The diameter of the controllable spherical workspace is about 4.27 (a) Original S(6) system times larger than the angular momentum of the unit (see Fig. 9–2 in Chapter 9).
D.1.2 Fault Management z
(1) Loss of One Unit The S(6) system without any one unit is a congruent five unit skew type system with a different major axis y direction. Thus, we need only one steering law for this x type of failure. The original and degraded system envelopes are shown in Fig. D–1. The original envelope is similar to a sphere but that of the degraded system is more similar to an ellipsoid. Figure D–1 (b) corresponds (b) After loss of one unit to the failure of the unit arranged in the z direction. If another unit fails, the envelope has the same shape but z its major axis is different. As the skew angle of this system is not optimized, there is a more serious internal impassable surface problem than the optimized system described in Section . y 9 4 has. Moreover, even though we can use this x workspace, its major axis is unknown before the accident, and there are six possibilities. Thus, it is safe to consider all possible situations and to evaluate a spherical workspace which is included by all six possible (c) After loss of two units envelopes (see Fig. 9–2). Fig. D–1 Envelopes of S(6) and degraded systems. All are drawn in the same scale.
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(2) Loss of Two Units A gradient method applied to the MIR system uses 27) Any failure of two units also results in a congruent the following objective function : configuration of four units. However, the system is not W = Σij |ci × cj |2 . (D–1) at all symmetric and the envelope is like a skew ellipsoid, as shown in Fig. D–1 (c). Because of reasons similar to As described in Section 7.5.5, the concept of the those given above, the workspace size must be evaluated constrained control can be applied to this subsystem69). by a spherical workspace. The four unit subsystem can be regarded as a deformed pyramid configuration in which two units have a skew D.1.3 Four out of Six Control − angle α1 (= sin1 121 / (+ cos(π / 5 )) ) and the other α π/2 − α The CMG system installed on the space station two, a skew angle 2 (= 1) (Fig. D–2). As the “MIR” is a S(6) system but only four units out of six are kinematic equation of this system is similar to that of operating simultaneously. The subsystem of four units the pyramid type system, the same constraining condition is the same as that in the above section. As mentioned as Eq. 7–5 can be applied and nonredundant kinematics there, any four unit subsystem is geometrically similar to Eq. 7–6 can be obtained. This constrained congruent. Therefore, any fault up to two units can be system has a restricted workspace (Fig. D–3), inside simply covered by exchange of faulty unit with a backup which exact steering is assured. Of course, this unit without change of the steering law. configuration is not rotationally symmetric about any gimbal axis, so there is no additional mode. g4 Z g3 D.2 Five Unit Skew System g 1 α α2 1 As described in Chapter 9, various ellipsoidal X Y g2 workspaces can be designed by selecting the skew angle. The workspace size is given in the figures of Chapter 9. The fault management is similar to that of the S(6) system described in Appendix D.1.1, except that the skew angle
g6 is different. g5 As this type of systems have not studied well, no effective steering laws have been proposed except the Fig. D–2 Four unit subsystem of MIR type system. gradient method. Application of the constraint method is possible with two independent constraining equations, Constrained workspace Original Workspace Impassable Surface
Actual Motion of H
H'
Additional Angular Momentum Imposed by Another Torquer Desired Motion H of H H path
Fig. D−3 Restricted workspace of a constrained Fig. D−4 Concept of singularity avoidance by MIR-type system. an additional torquer
––– 92 ––– –– D. Six and Five Unit Systems –– but the symmetry of this system cannot be preserved by impassable surface strip, the system can avoid the using any two linear equations. Therefore, finding surface, as shown in Fig. D–4. This mechanism is very appropriate constraints may need exhaustive calculations simple and has the following characteristics. and evaluations with some criteria of work space size and shape. (1) The avoidance movement can be planned so that it is Here, a potential steering law with an additional towards the narrower direction of the impassable torquer will be briefly outlined, which may be effective strip. Thus, the required angular momentum for for four or five unit system. avoidance can be minimized. Moreover, this motion Impassable surfaces of these systems are shaped like only aims to avoid the surface, therefore an ON/OFF surface strips as shown in Figs. 4–10 to 4–12 and 6–7 to type torquer such as a gas jet would be adequate. 6–10. In Section 6.4, these strips were called ‘impassable (2) This motion must take place only once to avoid a branches’ for the S(4) system. They can be approximated singular surface. Thus it may be accomplished by a by the analytical expression given by Eqs. 6–16 to 6– gas jet system. 19. Though there is no such expression for 5 unit (3) Sometimes, no singularity avoidance is necessary systems, they can be expressed by some numerical look- when H is approaching an impassable surface, as is up table. This look-up table can be reduced in its size described in Sections 7.1 and 7.4 on manifold with the aid of system symmetry. By this knowledge, connections. The method described here, however, we can distinguish whether H is approaching an can not use this knowledge of manifold connections, impassable surface. and hence is very simple but requires additional If appropriate angular momentum is added using torquing more often than necessary. another torquer when H is nearly crossing some
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––– 94 ––– Appendix E
Specification of Experimental Apparatus and Experimental Procedure
E.1 Experimental Apparatus
The experimental apparatus, as shown in Figs. 8–1 and E–1, is composed of a body structure, a three axis gimbal, attitude sensors and related circuitry, a CMG system, balance adjusters and an onboard computer. The block diagram is shown in Fig. E–2. The body is a truss structure made of steel pipes. It is designed sufficiently stiff so that deformation caused by its weight can be neglected when the system changes orientation. The three axis gimbal, which uses normal ball bearings, permits free rotation of the body (Fig. E– 3). A precision rotary encoder is installed on each gimbal axis. The encoder’s output pulses are converted to an angle value by a decoder circuit, then supplied to the onboard computer. The rotational speed of the body is measured by rate gyroscopes, whose outputs are analog signals that are converted to digital values by an Analog − to Digital (A/D) conversion circuit. Fig. E 1 Experimental apparatus
Three Axis RE P/D Gimbal Wireless Rate Modem Gyroscope A/D
TG DCM Balance RE P/D Adjusters Wireless Rate Servo D/A Modem Circuit Onboard Computer TG DCM CMGs RE P/D
Rate Servo D/A Circuit Stationary Computer RE: Rotary Encoder D/A: Digital to Analog Converter P/D: Pulse Decoder DCM: DC Servo Motor A/D: Analog to Digital Converter TG: Tachogenerator
Fig. E–2 Block diagram of experimental apparatus
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Fig. E−3 Three axis gimbal mechanism Fig. E−4 Single gimbal CMG
Fig. E−5 Balance adjuster Fig. E−6 Onboard computer
The CMG system is a S(4) type system composed of by a rotary encoder. The rotational speed of each gimbal four single gimbal CMGs and a unit CMG is shown in motor is controlled by a servo driver circuit. Fig. E–4. A wheel motor and a driver circuit installed The balance adjusters are composed of a moving inside the casing drives the flywheel at constant speed. weight driven by a linear ball screw mechanism as shown Slip rings installed through the gimbal axis enable free in Fig. E–5. The speed of the moving weight is rotation of the gimbal. The gimbal angle is measured controlled by a DC motor and a rate servo circuit, and
––– 96 ––– –– E. Specification of Experimental Apparatus and Experimental Procedure –– its position is measured by a rotary encoder and a decoder E.3 Attitude Control System circuit. The onboard computer is composed of a 32 bit Two types of controllers were installed in the onboard microprocessor, an interface circuit for the decoder computer. One was a model matching controller and circuits, an A/D converter board, a D/A converter board, the other was a PD tracking controller. The block and a wireless modem driver (Fig. E–6). The wireless diagram of the model matching controller is shown in modem enables serial communication with the stationary Fig. E–7. Each parameter was set by a kind of a pole computer. assignment method called the ‘model matching method54, 55). These were obtained so that the overall E.2 Specifications transfer function matched the given function. In the experiments, the pole of the function was set to 1.0. The The specifications of the experimental apparatus are block diagram of the tracking controller is shown in Fig. listed in Table E–1. E–8. Each parameter was set by a similar method above.
Table E–1 Specification of experimental apparatus ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Test Rig Size 750mm cube Moment of inertia { 38, 38, 42 } Kg m2 about x, y and z axes Weight Approx. 250 Kg
CMG made at the Mechanical Engineering Laboratory Flywheel Diameter 130 mm Rotational Rate 5,000 rpm Angular Momentum 3.8 Nms Gimbal Motor ESCAP – 34HL11-219E/204-2 Gimbal Motor Reduction Gear P42 (17.7:1) + 62:13 Gimbal Rotation Sensor Heiden Hein–ROD456.015B3600 + EXE601/5F Resolution 3600×5 pulse⁄rev (0.02 deg ⁄1 pulse)
Attitude Sensor Optical Rotary Encoder Canon – R10 Resolution 81,000 pulse ⁄rev. × 4 (1pulse ⁄10 arc sec) Rate gyroscope JAE – DARS Resolution ±0.5 deg⁄sec
Onboard computer CPU i80386SX16MHz with i80387 Memory 640KB Operating System MS-DOS Ver.3.3 in ROM Peripherals A/D, D/A, Pulse Decoder, Wireless Modem Cycle time 10 ms – 12 ms Software Development MS-DOS Ver.3.0, Optimizing C Compiler, Turbo C Environment Compiler –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
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k Body dynamics 1 . + r + v β β 1 + 1 1 k s − s − − s
model transfer function f2 1 (s+p) f1
Fig. E–7 Block diagram of the model matching controller.
Body dynamics ⋅ . r r + β β 1 1 + + v 1 1 f1 ⋅⋅ s s − + s s r +− f2
Fig. E–8 Block diagram of the tracking controller.
ω t t -1 Tcom –C (CC ) +
C ω θ N 1 (3-11) × k det(CCt) ξ (3-19)
Fig. E–9 Block diagram of the gradient method.
E.4 Steering Law Implementation Fig. E–9. The SR inverse method is defined in Section 3.5.2. Three types of steering laws were installed on the The constraint method used the kinematic equation onboard computer, the gradient method, the SR inverse 7–6. In actual implementation, numerical inversion of method, and the constraint method proposed in Chapter this equation is inappropriate because of nonlinearity. 7. The gradient method was exactly the same as Therefore, the steering law was realized as a solution of described in Chapter 2. Its block diagram is shown in linear equations which are obtained by differentiation:
cc* sinφψ sin+− cos φγ sin * cos φ cos ψ sin φ cos γ 1 T =− −+cosφψ sincc * sin φγ sin − sin φ cos ψ − * cos φ cos γω * h (E–1) φψ+− γ φψ − φγ sss* cos (cos cos ) * sin sin * sin sin
––– 98 ––– –– E. Specification of Experimental Apparatus and Experimental Procedure ––
where h is angular momentum of each CMG unit and Fig. E–10. ω∗ is speed of three variable vector that is (dφ/dt, dψ/dt, γ t d /dt) . E.5 Code Size and Calculation Time In real situations, the constraint condition is not guaranteed because the values of the gimbal rates derived Control laws and steering laws were implemented from the above equation are used for a finite sampling in the onboard computer. All the programs are coded by time and are not renewed continuously. If the constraint C language and compiled by the Turbo-C compiler condition is not satisfied, neither the following variable version 2.0. Their code size and calculation time are transformation nor the constrained kinematics is valid. listed in Table E–2. The constrained method needed about 2 ⁄ 3 memory storage and about 1 ⁄ 2 calculation θ = (φ+ψ, φ+γ, φ−ψ, φ−γ) , (E–2) time of the gradient method. To cope with this, an approximated solution with feedback was adopted in which null motion of the Table E–2 Code size and calculation time original system was added to make residual (i.e., the of process left of Eq. 7–5) vanish. An approximation of (φ, ψ, γ) is defined by the following equation. process code size calculation time (bytes)* (ms) φ = (θ + θ + θ + θ )/4 , 1 2 3 4 –––––––––––––––––––––––––––––––––––––––––– ψ θ − θ = ( 1 3)/2 , (E–3) MM-Controller 6,700 0.85
γ = (θ2 − θ4)/2 . Tracking Controller 8,800 1.1 Gradient Method 3,800 3.8 By using this, an approximated motion is obtained as a Constrained Method 2,800 1.8 solution of Eq. E–1. The Jacobian matrix in Eq. E–1 is –––––––––––––––––––––––––––––––––––––––––– a 3×3 matrix and its inverse can easily be obtained. With * Code size is an approximate value this inverse matrix and the command torque Tcom, the transformed gimbal rate ω∗ is obtained. By the coordinates transformation given by Eq. E–2, the real gimbal rates are obtained. E.6 Parameter Estimation After that, feedback terms are added. Null motion has one degree of freedom and is generally obtained as The system has various parameters. In order to ω ω k N where | N| = 1 (normalized after Eq. 3–11). For design an attitude controller, the inertia matrix of the the stable feedback, the multiplier k is determined with body and the size of the angular momentum of each an appropriate feedback gain a as follows: CMG unit must be given. Since precise evaluation of k = −a (θ1 − θ2 + θ3 − θ4) such parameters by calculation was not enough, they were estimated by experiments. ⋅ ( ω − ω +ω −ω ) . (E–4) N1 N2 N3 N4 First, the weight of moving mass of each balance The block diagram of this steering law is shown in adjuster was estimated by measuring the torque with a
d(φ, ψ, γ) ω ω -1 dt 1 Tcom J Transform + (E-1) J Jacobian of (E-3) ω N × Transform (3-11) (E-2) k θ (E-4)
Fig. E-10 Block diagram of the constrained method.
––– 99 ––– –– Technical Report of Mechanical Engineering Laboratory No.175 –– scale when the position of the mass was moved in a step- Then, the body was rotated by the CMG system, by-step manner. Then a certain fixed torque was applied generating a constant torque on the principal axis of the by this mechanism while the body was stabilized by the body. It was presumed that the principal axes (eigenaxes CMG system. The total angular momentum of the CMG of the inertia matrix) were the same as the structure’s system linearly increased by the constant disturbance frame directions. Trials about three axes were then made. torque. From the kinematic relation and measured By comparing the measured angular velocity of the body gimbal angles, the size of angular momentum of each with the CMG angular momentum, the moment of inertia unit was thereby estimated. about each axis was estimated. The estimated values are included in the specification of Table E–1.
––– 100 ––– Appendix F
General kinematics
F.1 Analogy with a Spatial Link Mechanism
The total angular momentum of a CMG system, H, is a three dimensional vector and is given as the sum of all hi by Eq. 3–3. Each hi has unit length and rotates g about g . This is then very similar to a spatial link θ 3 i 3 22, 33, 40) mechanism such as a multi-joint manipulator . h3 The total angular momentum, H, corresponds to the point θ h2 of the “hand”, i.e., the tip of the manipulator and the 2 θ gimbal angles, i, correspond to the joint angles. g2 H − A parallel link mechanism shown in Fig. F 1 Z h1 θ corresponds exactly to a single gimbal CMG system. In 1 the case of a link mechanism, the study of the relationship g X Y 1 between the input joint angle and the output hand point (a) Parallel link is called kinematics, since it is an instantaneous (b) CMG mechanism vectors relationship and thus does not explicitly include time. In this sense, the system equation giving H from θ (in Fig. F–1 Analogy to a parallel link mechanism Eq. 3–4) is called a kinematic equation of a CMG system. Table F−1 shows the similarity of a CMG system Orientation has various representation, such as and a manipulator. orthogonal matrix, quarternion and Euler angles. Any not redundant representation is enough for describing F.2 Spatial Link Mechanism Kinematics local geometry here, because discussion is limited in the neighborhood of a singular point. Thus, output variable State variable of a link mechanism is a set of n joint can be represented as follows; displacements denoted by q = {qi}. Output variable is a ∈ 3 γ ∈ p set of a position vector p( R ) and orientation ( a γ γ γ t x = ( p1 p2 p3 1 2 3) = γ , (F–1) subset of SO(3)).
Table F-1 Similarity between CMGs and link mechanism ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– CMG Link mechanism State variable θ : gimbal angles q : joint angles Output variable H : angular momentum x : end point location (and orientation) Kinematics H = H(θ) x = x(q) Kinematics nonlinear without cross nonlinear with cross coupling of qi θ complexity coupling of each i –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
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Kinematic equation is a nonlinear equation from joint x(q+dq) − x(q) displacements q to the output x. Exact expression of Σ ∂ ∂ 1/2 Σ ∂ 2 ∂ ∂ = i f/ qi dqi + ij f / qi qj dqi dqj this equation is found in most literature. 3 + O(dq ) . (F–9) x = f(q) . (F–2) Then, component of the singular direction is extracted Differentiation of the equation leads to, and the 3rd and higher order terms are omitted.
dx = dq = J dq . (F–3) As taking scalar product with ζ, first term vanishes,
Usual definition of Jacobian with axial velocity of ∆x = ζt {x(q+dq) − x(q) } the end-effector ω is given as follows, 1/2 Σ ζ ⋅ ∂ 2 ∂ ∂ = ij ( f / qi qj ) dqi dqj t dp / dt = 1/2 dq Q dq . (F–10) ω = J*dq / dt . (F–4) where matrix Q is, ζt ∂ 2 ∂ ∂ From assumption above, there is a non-singular Q = ( f / qi qj ) . (F–11) transformation between ω and dγ/dt as, This matrix is not so simple as a diagonal matrix P ω = Γ dγ/dt . (F–5) in the case of CMGs. Moreover, signature is not explicitly obtained from this matrix Q. Therefore, The next step is to decompose this quadratic form I3 0 into two sub-quadratic forms. One rises from the curved ∂f / ∂q = J = Γ∗ -1 J∗ , where Γ∗ = . 0 G hyper surface of singular state and the other from the displacement from this hyper surface. This is obtained S (F–6) by decomposing dq into two parts, dq which keeps singularity and homogeneous motion dqN: Keeping generality, we can take the origin where the N S transformation Γ∗ is identity hence J = J∗. dq = dq + dq . (F–12) The definition of dqN and dqS is, F.3 Singularity det ( J( q+dqS) )=0 , N Singular state is a case where the matrix J does not J dq = 0 . (F–13) have full rank. This means that, If a singular state is characterized by one direction det ( J ) = 0 , (F–7) ζ, the dimension of the singular hyper surface is 5 and so is the vector space of dqS. In this case the rank of J is ζ and there exists at least one direction, denoted as , in 5. The kernel of J, that is a vector space of dqN, is n-5 the tangent space of the x space, which satisfies, dimensional. ζt J = 0 , where | ζ | = 1 . (F–8) Substituting (F–12) into (F–10) leads to, ∆x = 1/2d(qS)tQdqS + 1/2 (dqN)tQdqN This direction can be called a singular direction. The difference between CMGs and a manipulator is that there + (dqS)tQdqN . (F–14) is no explicit expression which gives singular state The third term in the right will be shown zero. In order variable q from this singular direction ζ as in Eqs. 4–2 to prevent complication, let partial derivatives of the and 4–3. function f = (fi) be denoted by index with comma, such as, F.4 Passability ∂ ∂ fi,j = fi / qj . (F–15) In order to classify singularity, small displacement Singular direction is defined again as, from the singular point is expressed as a Taylor series,
––– 102 ––– –– F. General Kinematics ––
Σ ζ i i fi,j = 0 , (F–16) hyper surface. The second quadratic form gives passability. There is a restriction for the above and its differential is discussion, i.e., the expression (F–12) is not always Σ ζ Σ ζ Σ S possible. It is neither possible nor the last equation be i d i fi,j + i i j fi,jk dq k = 0 . (F–17) assured, when the product space of two linear spaces Then, spanned by {dqS} and {dqN} do not cover whole tangent (dqN)t Q dqS space. 1/2 Σ ζ Σ Σ N S A similar expression of passability is thus obtained = i i j k fi,jk dq j dq k generally as CMGs but the following differences and 1/2 Σ Σ ζ Σ S N = j{ i i k fi,jk dq k }dq j problems are remaining for further study; − 1/2 Σ Σ ζ N = j i d i fi,j dq j (1) No expression of Gaussian curvature is obtained = − 1/2 Σ dζ {Σ f , dqN } i i j i j j (2) Neither simple description of whole quadratic form t N = − 1/2 dζ (J dq ) nor whole signature is obtained. This is because there is ζ = 0 . (F–18) no general solution of singular state from the direction ε and sign { i}. Thus the quadratic form ∆x is divided into two as, (3) Dimension of the singular hyper surface (= 5) is − ∆x = 1/2(dqS)tQdqS + 1/2 (dqN)tQdqN greater than that of remaining space ( =n 5), in usual case, because the number of the joints is not greater than = 1/2 Σ f ζ dqS dqS ijk i,jk i j k 10. Therefore it seems easier to deal with the quadratic Σ ζ N N + 1/2 ijk fi,jk i dq j dq k . (F–19) form directly. The first term is a kind of curvature of the singular
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