CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol . 27,aNo. 4,a2014 ·655·

DOI: 10.3901/CJME.2014.0519.098, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Kinematics and Dynamics of Deployable Structures with Scissor-Like-Elements Based on Screw Theory

SUN Yuantao1, *, WANG Sanmin1, MILLS James K2, and ZHI Changjian1

1 School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, 710072, PR China 2 Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Rd., Toronto, Ontario,Canada

Received March 6, 2014; revised May 16, 2014; accepted May 19, 2014

Abstract: Because the deployable structures are complex multi-loop structures and methods of derivation which lead to simpler kinematic and dynamic equations of are the subject of research effort, the and dynamics of deployable structures with scissor-like-elements are presented based on screw theory and the principle of virtual respectively. According to the geometric characteristic of the deployable structure examined, the basic structural unit is the common scissor-like-element(SLE). First, a spatial deployable structure, comprised of three SLEs, is defined, and the constraint topology graph is obtained. The are then derived based on screw theory and the geometric nature of scissor elements. Second, to develop the dynamics of the whole deployable structure, the local coordinates of the SLEs and the Jacobian matrices of the center of of the deployable structure are derived. Then, the equivalent are assembled and added in the equations of motion based on the principle of . Finally, dynamic behavior and unfolded process of the deployable structure are simulated. Its figures of , and input are obtained based on the simulate results. Screw theory not only provides an efficient solution formulation and theory guidance for complex multi-closed loop deployable structures, but also extends the method to solve dynamics of deployable structures. As an efficient mathematical tool, the simper equations of motion are derived based on screw theory.

Keywords: scissor-like-element, screw theory, kinematics, dynamics, the principle of virtual work

tool to both formulate the kinematics of deployable 1 Introduction∗ structures, and then investigate the spatial dynamic properties and behavior. Deployable structures are used widely in many As an efficient analysis tool for mechanism, the applications such as aviation, aerospace and architecture, development of screw theory can be traced back to the 19th and are currently the focus of much research attention in century[1], and more recently in 1978, to HUNT’s work[2], recent years. The kinematic and dynamic characteristics of which initiated the modern development of screw theory. deployable structures play an important role during the MARTÍNEZ, et al[3], studied the kinematics and process of structure deployment. As seen from the existing acceleration of serial mechanisms based on screw theory in literature, deployable structures are complex multi-loop, detail, and the further work[4–8], led to derivation of relevant multi-body systems. Therefore, the development of the screw formulae and various applications. Applied to kinematics and dynamics of deployable structure results in parallel mechanisms, screw theory has also been utilized in very complex and cumbersome kinematics and dynamics the synthesis of the kinematics and dynamics equations of equations. Hence, methods of derivation which lead to these mechanisms[9–13]. Screw theory has been applied to simpler kinematic and dynamic equations of motion have the development of traditional dynamic modeling been the subject of significant research effort. The approaches, such as Newton-Euler method[14–15], application of screw theory to scissor-like deployable Lagrange’s equations[14, 16–17], principle of virtual work[7, 18] structures has been shown to result in simplified and Kane’s equations[19]. Recently, the kinematics and expressions for kinematic modeling of deployable dynamics of deployable structures has been the focus of structures. Utilizing the advantage of this approach, screw great interest by researchers[20–21], however, there are still theory has been shown to be a very effective mathematical fewer papers which present kinematics and dynamics of these deployable structures based on screw theory.

* Corresponding author. E-mail: [email protected] As the basic building unit for a deployable structure, the Supported by National Natural Science Foundation of China(Grant No. SLE is one of the most widely used units found in 51175422) deployable structures[22–23]. As shown in Fig. 1, the SLE © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2014

SUN Yuantao, et al: Kinematics and Dynamics of Deployable Structures with Scissor-Like-Elements Based ·656· on Screw Theory mechanism consists of two bars with a middle hinge, and there are different deployable structures because of the different combination of SLE. The deployable structures, which incorporate the SLE as its basic building unit, have a common characteristic in that they have a single degree of freedom. In contrast, the deployable structure assembled from three SLEs is the research focus of this paper, and is shown in Fig. 2. As shown in Fig. 2, the elements numbered 0–5 and 6–11 represent connecting links and Fig. 3. Constraint topology graph links respectively. Because the forces are seen to act the component center of mass, the relationship between input forces or and all forces or torques of center of mass 2 Kinematic Analysis need to be determined according to the principle of virtual work. Therefore, before deriving the equations of motion, To simplify the following development, the origin of the the Jacobian matrix of center of mass is derived firstly. reference is defined in the point O of connecting link 0 as shown in Fig. 2, and kinematic pairs are identified with

Roman letters. The ωA of A is assumed to be given. The constraint topology graph is shown in Fig. 3, with circles and lines used to represent links and revolute joints respectively. From Fig. 3, it is seen that there are three kinematic closed-loops, denoted as closed-loop I, II and III. A common revolute joint is shared amongst every pair of closed-loops. Every closed-loop is

composed of n=6 links and g=6 revolute joints. If fi and v is the number of the degree of freedom(DOF) of ith Fig. 1. Three deployable structures revolute joint and over-constraint separately, based on the modified formula[24–25] of degree of freedom m=6(n–g–1)+

Σ fi+v=1, each closed-loop has one DOF. Since the number of DOF of the overall deployable structure is equal to one, a single known input is required to determine its motion. Therefore, there is one independent input amongst the previous three closed-loop inputs, and the motion of the remaining links can be determined through the solution of the closed-loops within the deployable structure. Here, the connecting link 0 of closed-loop I is chosen as the fixed frame, hence the kinematic equations of closed-loop II and III may be derived from the moving platform. In addition, the kinematic equations, which describe paths shown by dotted and dot-dashed line in Fig. 3, not only express the relationship between the closed-loop, but also lead to the Fig. 2. Cartesian coordinate system, motion pairs, solution of the kinematics of the moving platforms. These connecting links and links are represented by connecting link 3 and connecting link 5, shown in Fig. 2. This paper is organized as follows. Utilizing a screw In the following, the velocity and acceleration kinematic theory formulation, the kinematic equations of a deployable equations are first derived for the deployable structure, and structure are first derived. In this derivation, the deployable then discussed subsequently. structure kinematic equation is assembled from three individual SLEs, in which the structure is sub-divided into 2.1 Velocity analysis three independent closed-loops. The constraint topology First, the kinematic equation of closed-loop I, a parallel graph, shown in Fig. 3, combined with the closed-loop mechanism with two branches is developed. Since motion equations and the constraints amongst three connecting link 0 is the fixed frame, it follows that closed-loops, permits the kinematic equations to be derived. connecting link 1 is located on the moving platform. The dynamics of the deployable structure is then derived. Assuming that the velocity of connecting link 1 is v1 , The component twists are computed and the dynamics based on screw theory, it follows that equations are then derived based on the principle of virtual ωωω work. AAξ BB ξ CC ξ v1, (1)

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Since the angular velocity ωi is obtained through solution ωωξ ξ ω ξ v . (2) EE WW HH 1 of the above equations, the velocity is ωiξi and its coordinate component may be expressed as That is,

ωi ωω ii ξ , (11) ωωω ωω ω AAξ BB ξ CC ξ EE ξ WW ξ  HH ξ , (3) where ω() is a matrix representing the first three screw where ωi and ξi represent the angular velocity and screw of . kinematic pair i respectively. In the analysis of closed-loop Since these screw velocities in the above represent II, connecting link 3 is chosen as reference body in this angular velocity, we may now write the linear velocity of paper, but it is a moving platform in relation to the fixed the motion joint i as frame. Therefore, the velocity of connecting link 3 is initially calculated, as shown in Fig. 3. Since the dotted line v vωiiξ  ωω  ii ξ  ρ, (12) shown in Fig. 3 represents a serial mechanism, the velocity of connecting link 3 is given as where v() is a vector representing the last three elements of the velocity, ρ is a radius vector from the point to the basis ωωω AAξ BB ξ KK ξ v3 , (4) point. Based on the previous analysis, we note that the key where v3 is the velocity of connecting link 3. Since aspect of this approach is to solve the relevant component connecting link 3 has zero angular velocity, Eq. (4) may velocity for the linear velocity component. From Fig. 2, it then be solved. can be seen that closed-loop I has a fixed frames, while Utilizing the same approach to the solution of the closed-loops II and III do not have a fixed frame. Therefore, velocity for closed-loop I, the following equations may be the approach utilizing the velocity is discussed separately. obtained: As seen in closed-loop I, there is a fixed reference frame and the linear velocity of connecting link 1 is computed

ωKKξ ωω BB ξ LL ξ 32v , (5) below as an example. Beginning with frame 0, connecting link 1 will move due to the revolute joint A, B and C or

ωωωQQξ PP ξ N ξ N 32v , (6) revolute joint E, W and H. Therefore, its velocity and linear velocity can be obtained as where throughout this paper, v is defined as the velocity ji ωωωξ ξ ξ v , (13) of component i relative to component j. Combining Eq. (5) AA BB CC 1 and Eq. (6), then we obtain ω vv1v 1   v 11  ρ . (14)

ωωBBξ KK ξ  ωωωω LL ξ QQ ξ  PP ξ NN ξ . (7) Linear velocities of the left hand side components of

closed-loop I will be solved similarly. However, the SLE Since the number of DOF of closed-loop II is equal to 1, consisted of components 8 and 9 are treated somewhat Eq. (7) can be solved, given ω determined from Eq. (4). B differently. To solve its linear velocity, the reference body, Similarly, the equations of closed-loop III may then be such as connecting link 3 or 5, shown in Fig.2 must first be derived to give chosen. Without loss of generality, the linear velocity of

revolute joint P in closed-loop II is derived below, with ωωξ ξ  ω ξ v , (8) TT WW UU 54 connecting link 3 regarded as a reference body.

The velocity of connecting link 3 is ωωω   SSξ PP ξ RR ξ 54v , (9)

ωωωAAξ BB ξ KK ξ v3. (15) ωωTTξ WW ξ  ω UU ξ  ωωω SS ξ PP ξ RR ξ . (10) And based on the vector of velocity of relative motion, the Note that the direction of the computation of the angular velocity of revolute joint P is given as velocity is from connecting link 0 to connecting link 1 in closed-loop I. Hence, the angular velocity of revolute joint vv3 QP v, (16) B and W is identical to the angular velocity of link 6 relative to link 7, and the angular velocity of link 11 where vQ is velocity of revolute joint. relative to link 10 respectively, as shown in Fig. 3. When Based on Eq. (12), the linear velocity of revolute joint P

ωB and ωW, the solution of Eq. (3), are chosen as the known is obtained as quantity for solving the remaining equations, the remaining angular velocities will be solved. vvPv P  ω v PP  ρ . (17)

SUN Yuantao, et al: Kinematics and Dynamics of Deployable Structures with Scissor-Like-Elements Based ·658· on Screw Theory

Since the derivation of the remaining linear velocity where terms is identical, these derivations are omitted. 5 ω  ωω   ω  ω SLie  TTξ WW ξ UU ξ  WW ξ UU ξ , 2.2 Acceleration analysis ε 6 In this paper, vectors , a and scalar α represent the SLie  ωωSSξ PP ξ  ω RR ξ  ω PP ξ  ω RR ξ . relevant acceleration, linear acceleration and respectively. In the following, ε is defined as j i Since the angular acceleration of revolute joint A has the screw acceleration of component i relative to been obtained, the angular acceleration of the remaining component j. Analyzing closed-loop I, shown in Fig. 2, the revolute joints can be obtained by combining the above screw acceleration is obtained as equations. Note that the direction of the angular acceleration is in the same sense as angular velocity. 1 αααAAξ BB ξ  CC ξ SLie 0ε 1, (18) Computing the linear acceleration, the relevant screw acceleration may be obtained simply as a linear velocity. 2 Then, based on the formula for screw acceleration, the ααEEξ WW ξ  α HH ξ SLie 0ε 1, (19) linear acceleration may then be determined.

In closed-loop I, connecting link 1 is used to illustrate αααξ ξ  ξ S12 ααξ  ξ  α ξ S , AA BB CC Lie EE WW HH Lie the above process. The screw acceleration is easily solved, (20) and its corresponding linear acceleration is given by

1 where SLie ωωωAAξ BB ξ  CC ξ  ωω BB ξ CC ξ , aa1 1 α 11 ρ ω 1  ω 11   ρ , (27)

2 SLie ωωEEξ WW ξ  ω HH ξ  ω WW ξ ω HH ξ . where a1aε 1   ω 11v , α1  αα 11 ξ , ω1  ωω 11 ξ . a() and α() is two vectors representing the last and the The notation “[ ]” represents the Lie bracket operation in first three elements of the acceleration respectively. The this paper. meaning of ω() is the same before. Since neither closed-loop II nor III is attached to a fixed Based on the analytical method used to determine the frame, closed-loop II or III may be represented as a parallel linear velocity of closed-loop II and III, the screw mechanism without a fix reference body relative to acceleration of revolute joint P is closed-loop I. For closed-loop II the screw equations are given as εP ε 3 3P ε vv 3 3P . (28)

αξ  αα ξ  ξ S 3 ε , (21) KK BB LL Lie 3 2 According to Eq. (27), the linear acceleration of revolute joint P is obtained as 4 αααQQξ  PP ξ  N ξ N SLie 3ε 2 , (22)

aa2 2 α 22 ρ ω 2  ω 22   ρ , (29) 34 ααBBξ KK ξ  α LL ξ SLie αααQQξ  PP ξ  NN ξ SLie ,

(23) where a2aε 2   ω 22v , the solution of α2 and ω2 is the same as Eq. (27). where S 3  ωξ  ωω ξ  ξ   ω ξ  ω ξ , 3 Jacobian Matrix of the Component Center Lie KK BB LL BB LL of Mass

S 4 ωξ  ωω ξ  ξ  ω ξ  ω ξ . Lie QQ PP NN PP NN The Jacobian matrix of the component center of mass plays an important role in the development of the dynamic Similarly, the equations of closed-loop III are derived as equations of motion of the deployable structure. To derive the dynamic equations of motion of this deployable 5 ααTTξ  WW ξ  α UU ξ SLie 5ε 4 , (24) structure, the Jacobian matrix at the center of mass of each link is first developed in this section. Since there are two types of components in this deployable structure, namely a αααξ ξ ξ S 6 ε , (25) SS PP RR Lie 5 4 connecting link and link, here we develop the relevant Jacobian matrices based on their characteristics in the 56 ααTTξ WW ξ  α UU ξ SLie ααSSξ PP ξ α RR ξ SLie , motion process. Since not all connecting links rotate during (26) the folding and unfolding process, the attitude angle of the

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T connecting links are given by ϕi=(0 0 0) , i=1, 2, 3, 4, 5. The radius vector of each link is ri (i=6, 7, , 11), then Assuming that the angle between link 7 and the z-axis is β the location of each link in the inertial coordinate frame is T as shown in Fig. 2, the length of connecting link and link is given as: pi=(ϕi ri) , i=6, 7,  , 11. Therefore, the li and l respectively, the radius vector between center of location of each component may be combined to give mass of connecting link and origin of coordinates is T respectively pii φ ri  , i 1, 2,  , 11. (30)

  0   0   0  With the derivative of Eq. (30) given as       r   0, r 2ll 2 sinβ , r 2ll 2 sinβ , 1   21  31         2l cos β 2l cos β 0 pJii (),β β (31)

 (2ll1  2 sinβ )sin(π / 3)   where Ji(β) is the ith Jacobian matrix of the center of mass.  β π  r41(2ll 2 sin )cos( / 3) ,    2l cos β  4 Dynamics Analysis

(2ll 2 sinβ )sin(π / 3)  1  In this section, the inverse dynamics of this deployable  β π  r51(2ll 2 sin )cos( / 3) . structure is derived. To solve the inverse dynamics, first    0  each twist is obtained through analysis, and then the dynamic equations of motion are derived based on the Therefore, the location of each connecting link in the principle of virtual work. Since the connecting links do not T inertial coordinate frame is pi=(ϕi ri) , i=1, 2, 3, 4, 5. rotate during the motion process of the deployable structure, Based on the motion characteristic of this deployable their moments of are not considered in the structure, three SLEs (6B7, 8P9, and 10W11) move in their derivation of the equations of motion. Therefore, there are respective planes, with local coordinate frames xoyand different expressions between the connecting links and the xoy   as shown in Fig. 4. From the above, this deployable links. structure motion in will be seen as three SLEs motion The dynamics of the ith connecting link is given as in their respective planes. As shown in Fig. 4, two local coordinate frames can be obtained by rotation by an angle Ggii m , (32) ϕ, with transformation matrix given as faim ii, i 1, 2,  , 5, (33) cosφφ sin 0     where Gi—Gravity of component i, R  sinφφ cos 0 .   f —The ith body inertia force,  0 01 i mi—Mass of component i. Combining Eq. (32) and Eq. (33), the twist of the

connecting link Fi is given as

fG F   ii, i 1, 2,  , 5. (34) i    031

The expression of the gravity and inertia force applied to each link is similar to the connecting link. Note that each link has a of inertia, hence the dynamics of each link is given by

τ jRIR jj jjε  ω  j (RIR jj jω j ), j  6, 7,  , 11, (35)

where Rj —Transformation matrix of component j,

Ij—Rotational inertia of component j. Fig. 4. Local coordinate frames of deployable structure Therefore, the link twist is

Assume that the rotation angle of each link is ϕ ( i=6, 7, fGjj i    Fj   , j  6, 7, , 11, (36)  , 11) with respect to the local coordinate frame, the  τ j  rotation angle in the global coordinate frame is given as

φii Rφ. where fi and Gj is the inertia force and gravity force.

SUN Yuantao, et al: Kinematics and Dynamics of Deployable Structures with Scissor-Like-Elements Based ·660· on Screw Theory

Based on the principle of virtual work and Eq. (31), the and the simulation results are obtained as the following inverse dynamics equation of this deployable structure, figures as shown in Fig. 6–Fig. 10. In Fig. 6–Fig. 10, the comprised of three SLEs, is obtained as various plots are labeled with x, y and z which implies the relevant coordinate components respectively, with the 5 11 T JFTT  JF, (37) subscript, i.e. Roman letters A, B, etc. refer to the relevant ii jj ij16 components or motion pairs. where T is an input moment. Table 1. Basic parameter Parameter Value 5 Simulation Results Length of link l/m 2 Width of link’s section b/cm 2 Based on the above analysis, a simple model of this Height of link’s section h/cm 5 deployable structure is derived utilizing its geometrical Input angular velocity ω/(rad • s–1) sin t characteristics. The unfolding process of the deployable Mass of connecting link mj/g 25 structure, as shown in Fig. 5 is simulated, with the dynamic Mass of link mi /kg 0.5 –2 equations of motion solved, using MATLAB software. In Gravity g/(m • s ) [0 0 –9.8] Fig. 5, the black point and line stands for the connecting links and the link respectively.

Fig. 6. Angular velocity

Fig. 7. Linear velocity

Note that due to the characteristic of this deployable structure with scissor-like-elements, the relevant z-axis rotation components are not exhibited in Fig. 6. During the unfolding process, the linear velocities of all components which are not in the xoy plane are negative, as shown in Fig. Fig. 5. Process of configuration 7. Fig. 8 and Fig. 9 illustrate that all which include angular accelerations and linear accelerations will The relevant system parameters of deployable structure meet at a null point. To obtain the required unfolding are given in Table 1. The input of revolute joint A is given velocity, an input torque is obtained as shown in Fig. 10.

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(2) Since the spatial deployable structure assembled by three SLEs is a serial-parallel mechanism, it is decomposed into three closed-loop mechanisms and two serial mechanisms between two of three closed-loops, before solving for the linear velocity. From the kinematics simulation results, it is seen that the angular velocity between two links is twice as large as the angular velocity between link and connecting link, and the linear velocity of same coordinate component is equal in the coordinate component. (3) In the derivation of the system dynamics, the Jacobian matrix of the center of mass is determined by utilizing transformation matrices. The dynamic equations of deployable structure are derived based on the principle Fig. 8. Angular acceleration of virtual work. In the dynamics analysis of each

component twist, the acceleration results are obtained. The approach proposed here utilizing a screw theory approach can also be used in the serial-parallel mechanisms.

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331–336. WANG Sanmin, born in 1961, is a professor at School of [18] TSAI L W. Solving the inverse dynamics of a Stewart-Gough Mechanical Engineering, Northwestern Polytechnical University, manipulator by the principle of virtual work[J]. Journal of China. His research interests include mechanism, mechanical Mechanical Design. 2000, 122(1): 3–9. and control and mechanical dynamics. [19] HUANG Y, LIAO Q, WEI S, LEI G. Research on dynamics of a E-mail: [email protected] bicycle robot with front-wheel drive by using Kane equations based on screw theory[C]//2010 International Conference on Artificial Intelligence and Computational Intelligence. Sanya, China, Oct MILLS James K, is a professor at Mechanical & Industrial 23–24, 2010: 546–551. Engineering, University of Toronto. His research interests include [20] CHEN W J, FU G Y, GONG J H, et al. Dynamic deployment robot control, control of multi-, design of actuators, simulation for pantographic deployable masts[J]. Mechanics of localization, development of fixtureless assembly technology, Structures and , 2002, 30(2): 249–277. design and control of high machines. [21] HU M, CHEN C, PAN J, et al. Dynamics analysis on deployment E-mail: [email protected] for the multi-deployment lock solar array[C]//2010 International Conference on E-Product E-Service and E-Entertainment(ICEEE). ZHI Changjian, born in 1984, is a PhD candidate at School of Henan, China: IEEE Computer Society, Nov 7–9. 2010: 1–5. Mechanical Engineering, Northwestern Polytechnical University, [22] ZHAO J S, DAI J S. WANG J Y, et al. Mechanism synthesis of a China. foldable stair[J]. Journal of Mechanisms and Robotics-Transactions E-mail: [email protected] of the ASME, 2012, 4(0145021).