On the Robot Compliant Motion Control The work presented here is a nonlinear approach for the control and stability analysis of manipulative systemsin compliant maneuvers. Stability of the environ- H. Kazerooni ment and the manipulator taken as a whole hasbeen investigatedusing unstructured MechanicalEngineering Department, modelsfor the dynamic behavior of the robot manipulator and the environment, Universityof Minnesota, and a bound for stable manipulation has been derived. We show that for stability of Minneapolis,Minn. 55455 the robot, there must be some initial compliancy either in the robot or in the en- vironment. The general stability condition has been extended to the particular case where the environment is very rigid in comparison with the robot stiffness. A fast, light-weight, active end-effector (a miniature robot) which can be attached to the end-point of large commercial robots has been designed and built to verify the con- trol method. The deviceis a planar, five-bar linkage which is driven by two direct drive, brush-lessDC motors. The control method makes the end-effector .to behave dynamically as a two-dimensional, Remote Center Compliance (RCC).
1 Introduction Most assemblyoperations and manufacturing tasks require method is a subclassof the condition given by the Small Gain mechanicalinteractions with the environment or with the ob- Theorem. For a particular application, one can replace the ject being manipulated, along with "fast" motion in un- unstructured dynamic models with known models and then a constrained space. In constrained maneuvers,the interaction tighter condition can be achieved. The stability criterion force' must be accommodated rather than resisted. Two revealsthat there must be some initial compliancy either in the methods have been suggested for development of compliant robot or in the environment. The initial compliancy in the motion. The first approach is aimed at controlling force and robot can be obtained by a passive compliant element such as position in a nonconflicting way [10, II, 12, l8J. In this an RCC (Remote Center Compliance) or compliancy within method, force is commanded along those directions constrain- the positioning feedback. Practitioners always observed that ed by the environment, while position is commanded along the system of a robot and a stiff environment can always be those directions in which the manipulator is unconstrained stabilized when a compliant element (e.g., piece of rubber or and free to move. The second approach is focused on develop- an RCC) is installed betweenthe robot and environment. The ing a relationship between the interaction force and the stability criterion also shows that no compensator can be manipulator position [1, 4, 5, 13J. By controlling the found to stabilize the interaction of the ideal positioning manipulator position and specifying its relationship with the system (very rigid tracking robot) with an infinitely rigid en- interaction force, a designer can ensure that the manipulator vironment. In this casethe robot and environment both resem- will be able to maneuver in a constrained space while main- ble ideal sources of flow (defined in bond graph theory) and taining appropriate contact force. This paper describes an they do not physically complement each other. A fast, light- analysis on the control and stability of the robot in constrain- weight, active end-effector (a miniature robot) whcih can be ed maneuverswhen the secondmethod is employed to control attached to the end-point of a commercial robot manipulator the robot compliancy. has beendesigned and built to experimentally verify the con- We start with modeling the robot and the environment with trol method. unstructured dynamic models. To arrive at a general stability criterion, we avoid using structured dynamic models such as 2 Dynamic Model of the Robot With Positioning first or second order transfer functions as general representa- Controllers tions of the dynamic behavior of the components of the robot (such as actuators). Using unstructured models for the robot In this section, a general approach will be developed to and environment, we analyzethe stability of the robot and en- describe the dynamic behavior of a large class of industrial vironment via the Small Gain Theorem and Nyquist Criterion. and researchrobot manipulators having positioning (tracking) We show that the stability criterion achieved via the Nyquist controllers. The fact that most industrial manipulators already have some kind of positioning controller is the motivation Iln this article, "force" implies force and torque and "position" implies behind our approach. Also, a number of methodologies exist position and orientation. for the development of robust positioning controllers for Contributed by the Dynamic Systems and Control Division for publication in direct and nondirect robot manipulators (14, 17J. the JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL. Manuscript In general, the end-point position of a robot manipulator received by the Dynamic Systems and Control Division February 1987; revised manuscript received May 2, 1988. Associate Editor: R. Shoureshi. that has a positioning controller is a dynamic function of its
416/Vol.111,SEPTEMBER 1989 Transactions of the ASME with small gain. (The gain of an operator is defined in Appen- dix A.) No assumption on the internal structures of G(e) and S(d) is made. Figure I shows the nature of the mapping in equation (I). We define G(e) and S(d) as stable, nonlinear operations in e Lp-space to represent the dynamic behavior of the closed-loop robots. G(e) and S(d) are such that G:Lpn -Lpn, S:Lpn-Lpn and also there exist constants (x" (3., (X2' and (32 such that IIG(e) Ilpsa, Ilellp+(31 and IIS(d) lip Fig. 1 The dynamics of the manipulator with the positioning controller (all the operators of the block diagrams of this paper are unspecified sa21Idllp+(32. (The definition of stability in Lp-sense is and may be transfer function matrices or time domain Input-output given in Appendix A.) relationships) A similar modeling method can be given for the analysis of the linearly treated robots.s The transfer function matrices, G input trajectory vector, e, and the external force, d. Let G and and S in equation (2) are defined to describe the dynamic S be two functions that showthe robot end-point position in a behavior of a linearly treated robot manipulator with position- global coordinate frame, y, is a function of the input trajec- ing controller. tory, e, and the external force, d.2 (d is measuredin the global YU"') = GU",)eU",) +SU",)dU",) (2) coordinate frame also.) In equation (2), S is called the sensitivity transfer function y=G(e) +S(d) (1) matrix and it maps the external forces to the end-point posi- The motion of the robot end-point in response to imposed tion. GU",) is the closed-loop transfer function matrix that forces, d, is caused by either structural compliance in the maps the input trajectory vector, e, to the robot end-point robot) or by the compliance of the positioning controller. position. y. For a robot with a "good" positioning controller, Robot manipulators with tracking controllers are not infinite- within the closed loop bandwidth; SU"') is "small" in the ly stiff in responseto external forces (also called disturbances). singular value sense,6 while GU",) is approximately a unity Even though the positioning controllers of robots are usually matrix. designedto foOowthe trajectory commandsand reject distur- bances,the robot end-point will move somewhat in response 3 Dynamic Behavior of the Environment to imposed forces on it. S is called the sensitivity function and The environment can be very "soft" or very "stiff." We do it maps the external forces to the robot end-point position. For a robot with a "good" positioning controller, S is a mapping -r-:r-hroughout this paper, for the benefit of clarilY. we develop the frequency domain Iheory for linearly trealed robots in parallel with Ihe nonlinear analysis. The linear analysis is useful not only for analysis of robols wilh inherently linear 2T"he assumplion Ihatlinearsuperposition (in eQuation(I» holds for the ef- dynamics, bur also for robots with locally linearized dynamic behavior. In the fects of d and e is useful in understanding the nature of the interaction belween Jalter case, the analysis is correct only in the neighborhood of the operating the robot and the environment. This interaction is in a feedback form and will point. be clarified with the help of Fig. 2. We will note in Section 4 that the results of 6The maximum singular value of a matrix A, amax (A) is defined as: the nonlinear analysis do not dependon this assumplion. and one can extend the IAzl obtained resulls to cover the casewhen G(e) and SId) do not superimpose. amax(A)=max- 31n a simple example, if a Remote Center Compliance (RCC) with a linear Izi dynamic behavior is installed at the endpoint of Ihe robol. then S is eQual10 the where z is a nonzero vector and 1.1 denotes Ihe Euclidean norm of a vector or an reciprocal of stiffness (impedance in the dynamic sense)of the RCC. scalar.
Nomenclature A = the closed-loop map- ping from r to e in j = complex number nota- ment position before Fig. 4 tion ..r:::-i contact d = vector4 of the external J" = Jacobian 0 = vector of the joint force on the robot I;, mi = length and mass of angles of the robot end-point each link T = vector of the robot e = input trajectory vector Mo = inertia matrix torques E = environment dynamics r = input-command vector Ee.Ed. /l.. 'Y = positive scalars f = vector of the contact n = degreesof the freedom "'0 = frequency range of force, of the robot n ~ 6 operation (bandwidth) ifl.f2 fn)T S = robot manipulator ai' /;';. II = positive scalars f ~ = the limiting value of sensitivity (l/stiffness) a = small perturbation of the contact force for T = positive scalar On in the infinitely rigid V = the forward loop map- neighborhood of environment ping from e to f in On= 90 deg G = robot dynamics with Fig. 4 oe = end-point deflection in positioning controller x = vector of the environ- yn-direction H = compensator ment deflection oYn. Oy, = end-point deflection in (operating on the con- Xi = location of the center the direction normal tact force. j) of mass and tangential to the In = identity matrix y = vector of the robot part jj = moment of inertia of end-point position ofn' oft = contact force in the each link relative to y ~ = the limiting value of direction normal and the end-point of the the robot position for tangential to the part link rigid environment "'d = dynamic ~II vectors in this paper are n x 1 vectors. Xo = vector of the environ- manipulability
Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 1989, Vol. 111/417 To simplify the block diagram of Fig. 2, we introduce a map- ping from e to f. f= V(e) (9)
If all the operators in Fig, 2 are transfer function matrices, then V=E(ln +S E)-IO. Vis assumed to be a stable operator in Lp-sense; therefore V:Lnp-Lnp and also IIV(e)llp scx41Iellp+t34' With this assumption, we basically claim that a robot with stable tracking controller remains stable when it is in contact with an environment. Note that one can still define V without assuming the superposition of effects of not restrain ourselvesto any geometry or to any structure. If e and d in equation (5) (or equation (I», one point on the environment is displaced as vector of x, with force vector, J, then the dynamic behavior of the environment 5 The Architecture of the Closed-Loop System is given by equation (3). We proposethe architecture of Fig. 3 to develop compliancy /=E(x) (3) for the robot. The compensator, H, is considered to operate If Xois the initial location of the point of contact on the en- on the contact force, f. The compensator output signal is be- vironment before deformation occurs then, x=y-xo. E is ing subtracted from the input command vector, r, resulting in assumed to be stable in Lp-sense; E:Lnp-Lnp and the input trajectory vector, e, for the robot manipulator. The I IE (x) II saJllxllp +fJJ. Confining equation (3) to cover readersshould be reminded that the robot in Fig. 3 can be con- the linearly treated environment, equation (4) representsthe sidered a weak tracking robot (open loop robot without any dynamic behavior of the environment. feedbackon the position and velocity) when the gain of S is a /U"') = EU",)xU",) (4) large number. There are two feedback loops in the system; the inner loop EU"') is a transfer function matrix that maps the displacement (which is the natural feedback loop), is the same as the one vector, x, to the contact force, /. Matrix E is a n x n transfer shown in Fig. 2. This loop shows how the contact force affects function matrix. E is a singular matrix when the robot in- the robot in a natrual way when the robot is in contact with the teracts with the environment in only some directions. For ex- environment. The outer feedback loop is the controlled feed- ample, in grinding a surface, the robot is constrained by the back loop. If the robot and the environment are not in con- environment in the direction normal to the surface only. tact, then the dynamic behavior of the systemreduces to the Readers can be convinced of the truth of equation (4) by one representedby equation (I) (with d = 0), which is a plain analyzing the relationship of the force and displacementof a positioning system. When the robot and the environment are mass, spring and damper as a simple model of the environ- in contact, then the values of the contact force and the end- ment. E resemblesthe impedance of a system. References[4 point position of robot are given by f and y where the follow- and 5] represent (Js2 +Ds+K) for E where J, D, and K are ing equationsare true: symmetric matrices and s=j", (4). y=G(e) +S( -/) (10) 4 Nonlinear Dynamic Behavior of the Robot and f=E(x) where x=y-Xo (II) Environment e=r-H(/) (12) Supposea manipulator with dynamic equation (1) is in con- If the operators in equations (10), (II), and (12) are con- tact with an environment given by equation (3); then!= -d. sidered as transfer function matrices, equations (13) and (14) Figure 2 shows the dynamics of the robot manipulator and en- can be obtained to represent the interaction force and the vironment when they are in contact with eachother. Note that robot end-point position for linearly treated systems when in some applications, the robot will have only unidirectional xo=O, force on the environment. For example, in the grinding a sur- f=E(ln+SE+GHE)-IGr (13) face by a robot, the robot can only push the surface. If one considers positive!; for "pushing" and negative!; for "pull- y= (In+SE+G HE)-IGr (14) ing," then in this class of manipulation, the robot The objective is to choose a class of compensators, H, to con- manipulator and the environment are in contact with each trol the contact force with the input command r. By knowing other only along those directions where !i > 0 for S, G, E, and choosing H, one can shapethe contact force. The i= I, ..., n. In some applications such as screwing a bolt, value of H is the choice of designer and, depending on the the interaction force can be positive and negative. This means task, it can have various values in different directions. A large the robot can have clockwise and counterclockwise interaction value for H develops a compliant robot while a small H torque. The nonlinear discriminator block diagram in Fig. 2 is generates a stiff robot. Reference [7] describes a micro drawn with dashed-lineto illustrate the above concept. manipulator in which the compliancy in the systemis shaped Using equations (I) and (3), equations (5) and (6) represent for metal removal application. Note that Sand GH add in the entire dynamic behavior of the robot and environment equation (13) to develop the total compliancy in the system. taken as a whole. GH representsthe electronic compliancy in the robot while S models the natural hardware compliancy (such as RCC or the y=G(e) +S( -/) (5) robot structural compliancy) in the system.7 Equation (13) J=E(x) where x=y-'X"o (6) also shows that a robot with good tracking capability (small If all the operators in Fig. 2 are considered linear transfer ~quation (13) can be rewritten as f = (£-1 + S + GH) -I Gr. Note that the function matrices, equations (7) and (8) can be obtained to environment admittance (I/impedance in the linear domain), £-1, the robot represent the end point position and the contact force when sensitivity (l/stiFFness in the linear domain), S, and the electronic compliancy, xo=O. GH, add together to Form the total sensitivity of the system. IF H=O,then only the admittance of the environment and the robot add together to Form the com. y=(ln+8E)-IOeJ=E(ln (7) pliancy for the system. By closing the loop via H, one can not only add to the +8 E) -10 e (8) total sensitivity but also shape the sensitivity of the system.
Transactions of the ASME 418/Vol 11, SEPTEMBER 1989 Fig. 4 Simplified version of Fig. 3. In the linear domain, V= E(SE+In)-IG.
given in Appendix A. Substituting for I !fll p from inequality 16 into inequality 18 results in inequality 20. (Note that f= V(e)) gain for S) may generatea large contact force in a particular IIH(V(e»llpsa4asllellp+asfJ4+fJs (20) contact. One cannot choose arbitrarily large values for H; the stability of the closed-loop system of Fig. 3 must be a4aS in inequality 20 represents the gain of the loop mapping, guaranteed. The trade-off between the closed-loop stability H( V(e». The third stability condition requires that H be and the size of H is investigated in Section6. chosen such that the loop mapping, H(V(e», is linearly When the robot is not in contact with the environment (i.e., bounded with less than a unity slope. The following corollary the outer feedback loops in Fig. 3 do not exist), the actual develops a stability bound if H is selected as a linear transfer position of the robot end-point is governed by equation (1) function matrix. (with d = 0). When the robot is in contact with the environ- Corollary. The key parameter in the proposition is the size ment, then the contact force follows r according to equations of a4aS. According to the proposition, to guarantee the (10), (II), and (12). The input command vector, r, is used dif- stability of the system, H must be chosen such that norm of ferently for the two categories of maneuverings; as an input HV(e) is linearly bounded with a slope that is smaller than trajectory command in unconstrained space(equation (I) with unity. If H is chosen as a linear operator (the impulse respone) d = 0) and as a command to control force in constrained space. while all the other operators are still nonlinear, then: There is no hardware or software switch in the control system when the robot travels betweenunconstrained spaceand con- IIHV(e)llps-yIIV(e)llp (21) strained space.The feedback loop on the contact force closes where: -y = Uma.(N) (22) naturally when the robot encounters the environment. Um.. indicates the maximum singular value, and N is a matrix whose ijth entry is IIHij III. In other words, each member of 6 Stability Analysis N is the LI norm of each corresponding member of H. Con- The objective of this sectionis to arrive at a sufficient condi- sidering inequality 16, inequality 21 can be rewritten as: tion for stability of the systemshown in Fig. 3. This sufficient IIHV(e)llps-yIIV(e)llps-ya41Iellp+-yfJ4 (23) condition leadsto the introduction of a classof compensators, Comparing inequality 23 with inequality 20, to guarantee the H, that can be used to develop compliancy for the family of closed loop stability, -YCX4must be smaller than unity, or, robot manipulators with dynamic behavior represented by equation (I). Using operator V defined by equation (9), the equivalently: block diagram of Fig. 4 is constructed as a simplified version -y< -=- (24) of the block diagram in Fig. 3. First we use the Small Gain CX4 Theorem to derive the general stability condition. Then, with To quarantee the stability of the closed-loop system, H must the help of a corollary, we showthe stability condition when H be chosensuch its "size" is smaller than the reciprocal of the is chosenas a linear operator (transfer function matrix) while "gain" of the forward loop mapping in Fig. 4. Note that "y V is a nonlinear operator. Finally, if all the operators in Fig. 3 representsa "size" of H in the singular value sense. are transfer function matrices, then the stability bound is When all the operators of Fig. 4 are linear transfer function shown by inequality 25. Section 7 is devoted to stability matrices one can use Multivariable Nyquist Criterion (9) to ar- analysis of the linearly treated systems,8when the environ- rive at the sufficient condition for stability of the closed loop ment is infinitely rigid in comparison with the robot stiffness. system.This sufficient condition leads to the introduction of a The following proposition (using the Small Gain Theorem class of transfer function matrices, H, that stabilize the family in references [IS, 16]) states the stability condition of the of linearly treated robot manipulators and environment using closed-loop system shown in Fig. 4. dynamic equations (2) and (4). The detailed derivation for the If conditions I, II, and III hold: stability condition is given in Appendix C. Appendix D shows I. V is a Lp-stable operator, that is that the stability condition given by Nyquist Criterion is a subsetof the condition given by the Small Gain Theorem. Ac- (0) V(e): Lnp-Lnp (IS) cording to the results of Appendix C, the sufficient condition (b) IIV(e)llp,:SQ41Iellp+t34 (16) for stability is given by inequality 25. II. His chosensuch that mapping H(/) is Lp-stable, that is umax(GHE) of Dynamic Systems, Measurement, and Control SEPTEMBER 1989, Vol 1/419 Journal stability criterion. The stability criterion when n = I is given by inequality 27. IHGI 420 I Vol. 111, SEPTEMBER 1989 Transactions of the ASME tain any spring or dampers. The compliancy in the active end- sisting of all functions/= (/1,12, .f.)T: (O,CXI)-R' such effector is developed electronically and therefore can be that: modulated by an on-line computer. Satisfying a kinematic constraint for this end-effector allows for uncoupled dynamic behavior for a bounded range. Two state-of-the-art miniature Jo~if;IPdt<~ fori=1,2,...,n actuators power the end-effector directly. A miniature force Definition 2: For all Te(O,~), the functionfr defined by: cell measuresthe forces in two dimensions. The tool holder can maneuvera very light pneumatic grinder in a linear work- [I Osts T spaceof about 0.3 x 0.3 in. A bound for the global stability of fr= the manipulator and environment has been derived. For 0 T Acknowledgment This research work is supported by the National Science Foundation grant under Contract NSF/DMC-8604123. where II/; lip is defined as: [r = ] lip References IVillp= Jo (lJjVjlPdt I Hogan, N., "Impedance Control: An Approach to Manipulation," where (lJj is the weighting factor. (lJj is particularly useful for ASME JOURNAL Of DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL,Vol. 107, 1985. scaling forces and torques of different units. 2 Hollis, R. L., "A Planar XY Robotic Fine Positioning Device," IEEE In- Definition 5; Let v(.) :Lnpe-Lnpe. We say that the operator lernational Conference on Robotics and Automation," St. Louis, MO, Mar. 1985. V(. ) is Lp-stable, if: 3 Kazerooni, H., and Houpt, P. K., "On The Loop Transrer Recovery," Int. J. of c.'ontro/, Vol. 43, No.3, Mar. 1986. (0) v(.): Lnp-Lnp 4 Kazerooni, H., "Fundamentals or Robust Compliant Motion for (b) there exist finite real constants CX4and fJ4 such that: Manipulators," IEEE J. of Robotics and Automation, Vol. 2, No.2, June 1986. IIV(e)llpsCX41Iellp+(34 veELnp 5 Kazerooni, H., "A Design Method for Robust Compliant Motion of According to this definition we first assume that the Manipulators," IEEE J. of Robotics and Automation, Vol. 2, No.2, June operator maps Lnpe to Lnpe. It is clear that if one does not 1986. 6 Kazerooni, "An Approach to Automated Deburring by Robot show that v(.):Lnpe-Lnpe, the satisfaction of condition (0) is Manipulators," ASME JOURNAL Of DYNAMIC SYSTEMS, MEASUREMENT, AND impossible since Lnpe contains Lnp. Once mapping, v('), from CONTROL, Dec. 1986. Lnpe to Lnpe is established, then we say that the operator v(.) 7 Kazerooni, H., "Robotic Deburring of Parts with Unknown Geometry," is Lp-stable if, whenever the input belongs to Lnp, the Proceedings of American Control Conference, Atlanta, GA, June 1988. 8 Kazerooni, H., "Direct Drive Active Compliant End-effector," IEEE J. resulting output belongs to Lnp. Moreover, the norm of the of Robotics ond Automation, Vol. 4, No.3, June 1988. output is not larger than CX4times the norm of the input plus 9 Lehtomaki, N. A., Sandell, N. R., and Athans, M., "Robustness Results the offset constant (34' in Linear-Quadratic Gaussian Based Multivariable Control Designs," IEEE Trans. on Automatic Control, Vol. 26, No. I, Feb. 1981, pp. 75-92. Definition 6; The smallest cx4 such that there exist a (34 so 10 Mason, M. T., "Compliance and Force Control for Computer Controlled that inequality b of Definition 5 is satisfied is called the gain of Manipulators," IEEE Transaction on Systems, Man, and Cybernetics, the operator v('). SMC-II(6), June 1981, pp. 418-432. II Paul, R. P. C., and Shimano, B., "Compliance and Control," Pro- Definition 7; Let v(.): Lnpe-Lnpe. The operator V(.) is ceedings of the Joint Automatic Control Conference, San Francisco, 1976, pp. said to be causal if: 694-699. 12 Raibert, M. H., and Craig, J. J., "Hybrid Position/Force Control of V(e)r= V(er) V T«X) and V eELnpe Manipulators," ASME JOURNAL Of DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 102, June 1981, pp. 126-133. Proof of the Nonlinear Stability Proposition. Define the 13 Salisbury, K. J., "Active Stiffness Control or Manipulator in Cartesian Coordinates," Proceedings of the 19th IEEE Conference on Decision and Con- closed-loop mapping A :r-e (Fig. 4). trol, pp. 95-100, IEEE, Albuquerque, N. Mex., Dec. 1980. 14 Siotine, J. J., "The Robust Control or Robot Manipulators," The Inter- e=r-H(V(e» (AI) national J. of Robotics Research, Vol. 4, No.2, 1985. For each finite T, inequality (A2) is true. 15 Vidyasagar, M., Nonlinear SystenlS Analysis, Prentice-Hall, 1978. 16 Vidyasagar, M., and Desoer, C. A., Feedback Systems: Input-Output Ilerllps Ilrrllp+ IIH(V(e»rllp forall tE(O,T) (A2) Properties, Academic Press, 1975. Since H( V(e» is Lp-stable. Therefore, inequality (A3) is 17 Vidyasagar, M., Spong, M. W., "Robust Nonlinear Control or Robot Manipulators," IEEE Conference on Decision and Control, Dec. 1985. true. 18 Whitney, D. E., "Force-Feedback Control of Manipulator Fine Mo- IleTJlps I IrTI Ip+QSQ4 I leT I Ip+QS(34+(3S tions," ASME JOURNAL Of DYNAMtC SYSTEMS,MEASUREMENT, AND CONTROL, June 1977, pp. 91-97. for all tE(O, T) (A3) 19 Yoshikawa, T., "Dynamic Manipulability of Robot Manipulators," Jour- nal of Robotic Systems, Vol. 2, No. I, 1985. Since QSQ4is less than unity: IleTllps I IrTI Ip + QS(34+i3s for all tE(O,7) (A4) APPENDIX A I-QsQ4 I-QsQ4 Inequality (A4), shows that e(.) is bounded over (O,T). Definitions 1 to 7 will be used in the stability proof of the Because this reasoning is valid for every finite T, it follows closed-loop system (Vidyasagar, 1978, Vidyasagar and that e(. )ELnpe, i.e., that A:Lnpe-Lnpe' Next we show that the Desoer, 1975). mapping A is Lp-stable in the sense of definition 5. Since Definition 1: For all pE(I,QI), we label as Lnp the set con- rELnp' therefore Ilrllp<~ for all tE(O,~), therefore in- Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 1989, Vol.111/423 ~ ~ equality (AS) is true. 'd Ilell p< 0) for all tE(O,O) (A5) inequality (AS) implies e belongs to Lp-space whenever r belongsto Lp-space. With the samereasoning from equations (AI) to (A5), it can be shown that inequality (A6) is true. Ilelips Ilrll p + CXS.B4+.BS for all tE(O,O) (A6) Fig. C1 Simplified block diagram of the system In Fig. 3 l-cxscx4 l-cxscx4 Inequality (A6) shows the linear boundednessof e (condition b of definition 5). Inequalities (AS) and (A6) taken together, Therefore the augmented loop transfer function (GHE + SE) guarantee that the closed-loop mapping A is Lp-stable. has the same number of unstable poles that SE has. Note that in many casesSE is a stable system. 3) Number of poles on jtll axis for both loop SE and APPENDIX B (GHE + SE) are equal. A very rigid environment generates a very large force for a Considering that the systemin Fig. C I is stable when H = 0, small displacement. We choose the minimum singular value of we plan to find how robust the systemis when GHE is added E to represent the size of E. The following proposition states to the feedbackloop. If the loop transfer function SE (without the limiting value of the force when the robot manipulator is compensator, H) develops a stable closed-loop system, then in contact with a very rigid environment. we are looking for a condition on H such that the augmented loop transfer function (GHE + SE) guaranteesthe stability of If O,nin(E) > Mo. where Mo is an arbitrarily large number, then the closed-loop system. According to the Nyquist Criterion, the value of the force given by equation (J 3) will approach to the systemin Fig. CI remains stable if the anti-clockwise en- the expression given by equation (BI) circlement of the det(SE+ GHE + In) around the center of the fr»=(S+GH)-IGr (81) s-plane is equal to the number of unstable poles of the loop Proof: We will prove that !f r» -fl approaches a small number transfer function (GHE + SE). According to conditions 2 and 3, the loop transfer functions SE and (GHE + SE) both have as Mo approaches a large number. the same number of unstable poles. The closed-loop system f= -f= (S+GH) -I [In -(S+GH) E (In +SE+GHE)-I]Gr when H = 0 is stable according to condition I; the en- (82) circlements of det(SE+In) is equal to unstable poles of SE. Factoring (In + SE + GHE) -I to the right-hand side: When GHE is added to the system, for stability of the closed- loop system, the number of the encirclements of I~ -1= (S+GH)-I (In +SE+GHE) -jar (B3) det(SE + GHE + In) must be equal to the number of unstable I/~ -/1< I1max(S + GH) -I X I1ma.(In + SE poles of the (GHE + SE). Since the number of unstable poles of (SE + GHE) and SE are the same,therefore the stability of +GHE)-I X I1ma.(G) Irl (84) the systemdet(SE + GHE + In) must have the same number of I I I1ma.(G) Irl encirclementsthat det(SE + In) has. A sufficient condition to lf~ -I < (B5) Umin(8 + GH) X IUmin(8E + GHE) -II guarantee the equality of the number of encirclements of det(SE+GHE+In) and det(SE+In) is that the det(SE+ umax(G)lrl GHE + In) does not pass through the origin of the s-plane for 1/=-/1< I1mi"(8 + GH) X Il1mi"(8 + GH) x I1mi..(E) -1' all possiblenonzero but finite values of H, or (86) det(SE+GHE+In)~O for all tIIE(O,~) (CI) Urn.. (0) and umin (8+ OH) are bounded values. If If inequality CI does not hold then there must be a nonzero Umin(E) > Mo. then it is clear that the left-hand side of ine- vector Z suchthat: quality (86) can be an arbitrarily small number by choosing (SE+GHE+/n)z=O (C2) Mo to be a large number. The proof for y ~ = 0 is similar to the above. or: GHEz=-(SE+/n)z (C3) A sufficient condition to guarantee that equality (C3) will not happenis given by inequality (C4). APPENDIX C oma.(GHE) 424/Vol.111, SEPTEMBER 1989 Transactions of the ASME K where: p. = Umax(Q), and Q is the matrix whose ijth entry is given by (Q)ij=supwl(V)ijl, ,,=umax(R), and R is the matrix whose ijth entry is given by (R)ij =suPwl(H)ijl Substituting inequality (D2) in (DI): I IHV(e) I Ipsp.,,1 lei Ip (D3) According to the stability condition, to guarantee the closed loop stability p."< I or: 11<- (D4) p. Note that the followings are true: Omax(V)sp. forall"'E(O,~) (05) Fig. E1 The five bar linkage In the general form omax(H)sJI forall "'E(O,~) (06) Substituting (05) and (06) into inequality (04) which JI2 guaranteesthe stability of the system,the following inequality (El) is obtained: Umax(H) < . where: for all ",E(O,a» (07) Umax( V) JI1 = -II sin(81)+a 15 sin(82), J21 =/lcos(81)-a 15 cos(82) JI2 = b 15 sin(82), In = b 15 cos(82) n ( 1-/\ <: . -max'--' -umax(E(ln+SE)-IG) for all wE(O.~) (08) The mass matrix is given by equation (E2). Inequality (08) is identical to inequality (26), This shows that [Mil M12 the linear condition for stability given by the multivariable Ny- M= (E2) quist Criterion is a subsetof the generalcondition given by the M21 Mn ] Small Gain Theorem. where: MI. =J. +m2 112+J2 a2 +J] c2 +2 X2 I. cos(8. -8Ja m2 M12=J2 ab+bcos(81-82>X2/. m2+J]cd +c cos(84 -8])x] 14 m] M21 =M12 APPENDIX E M22 = 2 m] 14 X3 d cos(84 -8]) + J3 d" + J4 + m3 142+ J2 b2 This Appendix is dedicated to deriving of the Jacobian and the mass matrix of a general five-bar linkage. In Fig. El. Ji. Ii. a, b, c, dare given below. Xi' mi' and 8i representthe moment of the inertia relative to a = I. sin(81 -83)/(/2 sin(82 -83» the end-point, length, location of the center of mass,mass and b = 14 sin(84 -83)/(/2 sin(82 -83» the orientation of each link for i = 1, 2, 3 and 4. Using the standard method, the Jacobian of the linkage can c= I. sin(81 -8J/(/3 sin(82 -83» be represented by equation (El). d = 14 sin(8. -8J/(/3 sin(82 -83» Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 1989, Vol.111/425 l 10 L I .Doto -Simulotion ~ .16Y,ljc.»/6ftljc.»1./ in/lbf db / 16Ynlj""J/6fnlj""JI .01 10 100I 1000 rad/sec Fig. 11 The end-point admittance (1/lmpendance) 104 ll'f/Oiv :lDr - Olbf. - ~D T -1 ~C. Fig. 12 Force In the Yn.dlrectlon Increases from zero to 2 Ibf the input command representsthe magnitude of G at each fre- quency. For measurementof the sensitivty transfer function 0.11tbf/Dlv matrix, the input excitation was supplied by the rotation of an eccentric massmounted on the tool bit. The rotating mass ex- erts a centrifugal, sinusoidal force on the tool bit. The fre- 1-0; quency of the imposed force is equal to the frequency of rota- tion of the mass. By varying the frequency of the rotation of - the mass,one can vary the frequency of the imposed force on T the end-effector. Figure 10 depicts the sensitivity transfer O.II,lbt- function. The values of the sensitivity transfer functions along the normal and tangential directions, within their bandwidths, are 0.7 in/lbf and 0.197 in/lbf, respectively. The nature of compliancy for the end effector is given by equation (31). H was chosen such that (S+H)-I in each direction is equal to the desired stiffness. H must also guaranteethe stability of the closed-loopsystem. The stability Fig. 13 Force In the y,.dlrectlon increases from zero to 0.16 Ib criteria for a one-degree-of-freedomsystem is given by ine- qualities 32and 33. Inequality 33 shows that the more rigid the the force in the y,-direction remains at zero. Next the end- environment is, the smaller H must be chosento guaranteethe effector was moved 0.5 in. beyond the edge of the part in the stability of the closed-loop system. In the case of a rigid en- y,-direction. Figure 13shows the contact forces. The force in vironment ("large" E) and a "good" positioning system y,-direction increases from zero to 0.161bf while the force in ("small" 8), H must be chosen as a very small gain. The Yn-direction remains at zero. In both casesthe end-effector values for H along the normal and tangential directions within was moved as 0.5 in. beyond the edge of the stiff wall. Since their bandwidths are0.01 in/lbf and 0.194 in/lbf, respectively. the stiffness of the end-effector in Yn-direction is larger than These values result in 0.39 in/lbf and 0.7 in/lbf for (S + H) thelarger stiffness than the in y,-direction,contact force the in Yt-direction.contact force in. Yn-direction is within the bandwidth of the system. Figure II shows the ex- perimental and theoretical values of the end-point compliancy (Fig. II actually shows the end-point admittance where it is 11 Summary and Conclusion reciprocal of the impedance in the linear case.) In another set of experiments, the whole end-effector was A new controller architecture for compliance control has moved in two different directions to encounter a edge of a been investigated using unstructured models for dynamic behavior of robot manipulators and environment. This part. The objective was to observethe uncoupled time-domain dynamic behavior of the end-effector when the end-effector is unified approach of modeling robot and environment in contact with the hard environment. The controller was dynamics is expressed in terms of sensitivity functions. The designedsuch that the values of (S + H) -I in tangential and control approach allows not only for tracking the input- command vector, but also for compliancy in the constrained normal direction are 0.32 Ibf/in and 4.0 lb/in, respectively. First the end-effector was moved 0.5 in. beyond the edge of maneuverings. An active end-effector has been designed, built, and tested for verification of the control method. The the part in Yn-direction. Figure 12 shows the contact forces. active end-effector (unlike the passive system)does not con- The force in Yn-direction increasesfrom zero to 2.0 lbf while Transactions of the ASME 422/Vol.111, SEPTEMBER 1989 ~FUIN. , "' tain any spring or dampers. The compliancy in the active end- sisting of all functionsf= (/I,f2' ...,fn)T: (O, modulated by an on-line computer. Satisfying a kinematic 0. constraint for this end-effector allows for uncoupled dynamic J If; I p dt < actuators power the end-effector directly. A miniature force Definition 2: For all TE(O, stability of the envi~°.n~ent and. the ro.bot t~ken as a whol~, is called the truncation of f to bhe interval (0,1). there must be some InItIal compliancy eIther m the robot or m the environment. The initial compliancy in the robot can be Definition 3: The set of all functions f = h, f2' ...,f n) T: obtained by a nonzero sensitivity function for the positioning «O, controller or a passive compliant element. Lnpe. fby itself mayor may not belong to Lnp. Definition 4: The norm onLnp is defined by: Acknowledgment I lfll = [t I If; II 2] 1/2 This research work is supported by the National Science P ;= I p Foundation grant under Contract NSF/DMC-8604123. II . wereh 1111"Vi pIS d e fime d as: r 0. [ ] lip References I If;llp= Jo "'i If;IPdt 1 Hogan, N., "Impedance Control: An Approach to Manipulation," where "'i is the weighting factor. "'i is particularly useful for ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 107, scalin 1985. . g forces and tor q ues of different units 2 Hollis, R. L., "A Planar XY Robotic Fine Positioning Device," IEEE In- Definition 5: Let v ( .):L n -Ln. We say that the operator ~~;5~tional Conference on Robotics and Automation," St. Louis, MO, Mar. V(.) is Lp-stable, if: pe pe 3 Kazerooni, H., and Houpt, P. K., "On The Loop Transfer Recovery," (0) V(.): Lnp-Lnp Int. J. of Cont~o/, Vol. 43, No.3, Mar. 1986. " (b) there exist finite real constants a and {3 such that: 4 Kazeroom, H., "Fundamentals of Robust Compliant MotIon for 4 4 Manipulators,"IEEEJ.ofRoboticsondAutomation,Vol.2,No.2,June IIV(e)11 ~a lie II +(:J4 VeELn 1986. p 4 p p 5 Kazerooni, H., "A Design Method for Robust Compliant Motion of According to this definition we first assume that the Manipulators," IEEE J. of Robotics and Automation, Vol. 2, No.2, June operator maps Lnpe to Lnpe. It is clear that if one does not 198 6.6 K ." A A h t At ted Deb . azeroom, n pproac 0 u oma urnng y 0 o. ..y;; Y' .n . b Rbt showthatv(.):Ln~-Ln~,thesatisfactionofcondition(o)is Manipulators," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND ImpossIble sInce L pe contains L p' Once mappIng, v('), from CoNTROL, Dec. 1986. Lnpe to Lnpe is established, then we say that the operator v(.) 7 Kazerooni, H., "Robotic Deburring of Parts with Unknown Geometry," is L -stable if whenever the in '. g s to Ln the Proceedmgs of.Amerlcan. Contr~1 Conf~rence, At~anta, GA, June. 1988 resultingp output , belongs to Ln .Moreoverp ut belon the norm p'of the 8 Kazeroom, H., "DIrect Dnve ActIve Compliant End-effector," IEEE J.. .p '. of Robotics and Automation, Vol. 4, No.3, June 1988. output IS not larger than a4 tImes the norm of the Input plus 9 Lehtomaki, N. A., Sandell, N. R., and Athans, M., "Robustness Results the offset constant P'4. inTrans. Linear-Quadratic on Automatic GaussianControl, Vol.Based 26, MultivariableNo. I, Feb. 1981,Control pp. Designs,"75-92. IEEE Defmltlon 6: The smallest a4 such that there exIst a P'4 so 10 Mason, M. T., "Compliance and Force Control for Computer Controlled that inequality b of Definition 5 is satisfied is called the gain of Manipulators," IEEE Transaction on Systems, Man, ond Cybernetics, the operator V('). SMC-II(6), June 1981, pp. 418-432. II Paul, R. P. C., and Shimano, B., "Compliance and c;ontrol," Pro- Definition 7: Let V(.): Lnpe-Lnpe. The operator V(.) is ceedings of the Joint Automotic Control Conference, San FrancIsco, 1976, pp. said to be causal if: 694-699. 12 Raibert, M. H., and Craig, J. J., "Hybrid Position/Force Control of V(e)T= V(eT) V T«x. and V eELnpe Manipulators,'" ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 102, June 1981, pp. 126-133. Proof of the Nonlinear Stability Proposition. Define the 13 ~alisbury, K. J., :'Active Stiffness Control of Manipulat<>.r.in Cartesian closed-loop mapping A:r-e (Fig. 4). Coordmates," Proceedmgs of the 19th IEEE Conference on DecISIon and Con- trol, pp. 95-100, IEEE, Albuquerque, N. Mex., Dec. 19~0. e=r-H( V(e» (AI) national14 Slotine, J. of J.Robotics J., "The Research,Robust ControlVol. 4, No.2,of Robot 1985. Mampulators," The Inter- For each fInIte T, InequalIty (A2) IS true. 15 Vidyasagar, M., Nonlinear Systems Analysis, Prentice-Hall, 1978. II II < II II IIH 16 Vidyasagar, M., and Desoer, C. A., Feedback Systems: Input-Output eT p- rT p+ ( V ( e » T II p f,or all tE(O , T) (A2) Properties, Academic Press, 1975. .Since H( V(e» is L -stable. Therefore, inequality (A3) is 17 Vidyasagar, M., Spong, M. W., "Robust Nonlinear Control of Robot t p Manipulators," IEEE Conference on Decision and Control, Dec. 1985. rue. 18 Whitney, D. E., "Force-Feedback Control of Manipulator Fine Mo- lie II ~ I Ir II + a a lie II + a,P'4 + P's tions," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, T P T P S 4 T P .June 1977, pp. 91-97. for all tE(O,T) (A3) I 19 Yoshikawa, T., "Dynamic Manipulability of Robot Manipulators," Jour-. ... I " nal of Robotic Systems, Vol. 2, No. I, 1985. SInce a,a4 IS less than unIty. IleTllps IlrTllp + aSP'4+P'S for all tE(O, 1) (A4) A P PEN D I X A I -a,a. I -a,a4 Inequality (A4), shows that e(.) is bounded over (O,T). Defmltlons I to 7 wlll.be used m the stablll~y proof of the Because this reasoning is valid for every finite T, it follows closed-loop system (Vldyasagar, 1978, Vldyasagar and that e(. )ELnpe, i.e., that A:Lnpe-Lnpe. Next we show that the Desoer, 1975). mapping A is L -stable in the sense of definition 5. Since p . Definition 1: F<>r all pE(I, J9urnal of Dynamic Systems, Measurement, and Control SEPTEMBER 1989, Vol. 111/423 '",,' I equality (A5) is true. Ilellp 424IVol.111, SEPTEMBER 1989 Transactions of the ASME where: !1 = am_x(Q), and Q is the matrix whose ijth entry is given by (Q)ij=sup",I(V)ijl, II F amax(R), and R is the matrix whose ijth entry is given by (~)ij=sup", I(H)ijl S~bstituting inequality (D2) in (Dl): I IIHV(e)llp~!1l1llellp (D3) e ~cording to the stability condition, to guarantee the closed ldop stability jJ.II< lor: 1 11< -(D4) jJ. Note that the followings are true: amax(V)~!1 for all CJJE(O,(X) (D5) Fig. E1 The five bar linkage in the general form amax(H)~1I forall CJJE(O,(X) (D6) Substituting (D5) and (D6) into inequality (D4) which JII J12 gmarantees the stability of the system, the following inequality J" = [ J J. ] (El) is obtained: 21 22 : 1 where: alax(H) < for all CJJE(0, (X) (D7) .. T amax(V) JII=-/lsm(91)+alssm(92), J21=/lcos(91)-alscos(92) i 1 JI2 = b Is sin(92), J22 = b Is cos(92) al'l'ax (H)< amax(E(In+SE)-IG) for all CJJE(O(X), (D8) Th emass matrlX IS gIven by equatIon (E2). Inequality (D8) is identical to inequality (26). This shows that [Mil M12 the linear condition for stability given by the multivariable Ny- M = ] (E2) q~ist Criterion is a subset of the general condition given by the M21 M22 SlJ1all Gain Theorem. where: MIl =JI+m2 112+J2 a2+J3 c2+2 X2 II cos(91-92)a m2 MI2 =J2 a b+b cos(91 -9JX2 II m2 +J3cd + c cos(94 -93)x3 14 m3 A P PEN D I X E M 21-12-M hIS AppendIx IS dedIcated to denvmg of the JacobIan and M22=2m3/4x3dcos(94-93)+J3d2+J4+m3/42+J2b2. th mass matrix of a general five-bar linkage. In Fig. El, Ji, Ii, a, b, c, d are gIven below. Xi mi, and 9i represent the moment of the inertia relative to a = II sin(91 -93)/(/2 sin(92 -93» th en?-poi~t, length, lo~ation o~ the center of mass, mass and b = I sin(9 -9 )/(1 sin(9 -9 » th~ OrIentatIon of each link for 1= 1,2,3 and 4. 4 4 3 2 2 3 Using the standard method, the Jacobian of the linkage can c = I. sin(91 -9J/(/3 sin(92 -93» b~ represented by equation (El). d= 14 sin(91 -9J/(/3 sin(92 -93» .- i J urnal of Dynamic Systems, Measurement, and Control SEPTEMBER 1989, Vol.1111425 f3 = 1.0,~=6.0.r = 1.0 {J = 1.0,~ =6.0, r = 1.0 ~ '" on roo N ~ on N N 0 4 8 12 16 20 24 28 t (sec) Fig. 6 8 12 16 20 24 28 t (sec) Flg.S References Craig, J. J., 1986,Introduction to Robotics: Mechanicsand Control, Addison-Wesley,Reading, Mass. All parameters were initially assumed to be set to1heir true Craig, J. J., Hsu, P., and Sastry,S. S., 1987,"Adaptive Control of value, but the value of mass at the end of the second link, m2 MechanicalManipulators," Int. J.RoboticsRese~~ch, Vol. 6,.N~. 2, pp. 16-28. . d f 2 k 3 k h . 5F . 3 Dubowsky, S... and DesForges, D. T., 1979, The ApplicatIon of Model- was Increase rom g to gat! e tIme t= sec.. Igures Referenced Adaptive Control to Robotic Manipulators," ASME JOURNAL OF to 6 show the results of the trackIng error response and the DYNAMICSYSTEMS, MEASUREMENT ANDCONTROL, Vol. 101,pp. 193-200. parameter identification process. r 33= I was used for all.. the Horowitz,R., and Tomizuka,M., 1980,"An AdaptiveControl Schemefor cases shown. Note that the state variables (Figs. 3, 4) have Mechani~~1Manipulators-Compensation of Nonlinearity and Decoupling .. llh . I . h. h d Control, ASME Paper80-WA/DSC-6. essentla y come to t. elf correct va ues WIt m t r~e secons. Hsu,P., Bodson,M., Sastry,S., andPaden, B., 1987,"Adaptive Identifica- As expected, the estImates of the second mass (FIg. 6) show tionand Control for Manipulatorswithout UsingJoint Accelerations," IEEE the most severe estimation errors; however this estimate has InternationalConference on Roboticscan Automation,1987, pp. 1210-1215. also converged to the actual mass value after three seconds. Khosla, P., and Kanade,T., 1985,"Parameter Identification of Robot The estlm~te o! the first mass (FIg. 5) shows a maxImum error Dynamics,"Lee, C. S. IEEE G., andConf. Chung, Decision M. J.,and 1984, Control, "An Adaptive Fort Lauderdale, Control Fla.Strategy for of approXImately 10 percent from the actual value. Mechanical Manipulator," IEEE Trans. Automatic Control, Vol. AC-29, No. 9. Luh, J. Y. S., Walker, M. W., and Paul, R. P. C., 1980,"On-Line Computa- tional Scheme for Mechanical Manipulators," ASME JOURNALOF DYNAMIC 5 C onc I'uslons SYSTEMS,Middleton,MEASUREMENT R. H., andANDGoodwin, CONTROL, G. C.,Vol. "Adaptive I, pp. 60-70.Computed Torque Con- From the above developments, we can draw the following trol for Rigid Link Manipulators,"Proc. 25th Conf. Doc. Contr., Athens, I . F. h d t t d t Greece,pp. 68-73. conc uslons lfSt, tea ap .Ive pure compu e orque Orin, D. E., McGhee,R. B., Vukobratovic,M., and Hartock, G., 1979, algorithm is easy to implement, computationally very efficient "Kinematicand Kinetic Analysisof Open-ChainLinkage Utilizing NeWton- and is stable under moderate feedback gains. The algorithm EulerMethods," Mathematical Biosciences, Vol. 43,pp. 107-130. allows accurate estimations of system parameters and ex- Paul, R. P., 1981,Robot.Manipulators: Mathematics, Programming, and f h . bl Th d ..Control, MIT Press, CambrIdge, MA. cellent control 0 t e system state vana es. e eslgn IS Sadegh,N.,andHorowitz,R.,1987,"StabilityAnalysisofanAdaptiveCon- made very simple by the explicit expression of the parameter troller for RoboticManipulators," IEEE InternationalConference on Robotics range allowed for stability. Second, if no additional filters are and A~tomation,pp. 1223:1229." . used and a Lyapunov function method is assumed then this SlotIne, J. J. E., and LI, W., 1987, On the AdaptIve Control of Robot , , Manipulators," Int. J. Robotics Research, Vol. 6, No.3, pp. 49-59. paper seems to have accounted for many of the re~sonable Wang,T., andKohli, D., 1985,"Closed and Expanded Form of Manipulator choices of the control laws of the form (5) and adaptatIon laws DynamicsUsing LagrangianApproach," ASME Journal of Mechanisms, in!the form of (9). Transmissions,and Automation in Design,Vol. 107,pp. 223-225. 4 6.J Vol. 112, SEPTEMBER 1990 Transactions of the ASME