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精品论文Kinematics and Dynamics Hessian Matrixes ofManipulators Based on Lie Bracket5ZHAO Tieshi1,2, GENG Mingchao1,2, LIU Xiao1,2, YUAN Feihu1,2(1. Parallel Robot and Mechatronic System Laboratory of Hebei Province,Yanshan University,066004;2. Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry ofNational Education,Yanshan University, 066004)10Abstract: The inertia forces and inertia coupling among the components of a manipulator must be considered in the design and control. However, as the traditional representations of accelerations haveno coordinate invariance, the corresponding inertia and Hessian matrices are not coordinate invariant, which causes the complication for analysis and control of kinemics and dynamics of a manipulator.Based on the investigation into the physical meaning of Lie bracket between two twists, the Lie bracket15representation of kinemics Hessian matrix of the manipulator is deduced, the acceleration of the end-effector is expressed as linear-bilinear forms in this paper. Further, Newton-Eulers' equations arerewritten as linear-bilinear forms, from which the dynamics Hessian matrix is deducted. The formulaeand Hessian matrixes are proved to be coordinate invariant and convenient to program. An index of dynamics coupling based on dynamics Hessian matrixes is presented, and it is applied to a foldable20parallel manipulator, which demonstrates the convenience of the kinematics and dynamics Hessian matrices.Key words: Lie Bracket, dynamics Hessian matrix, coupling index, parallel manipulator0Introduction25In robotic representation theories, Denavit and Hartenberg 1 notation for definition of spatial mechanisms and the homogeneous transformation of points introduced by Maxwell 2 are the most popular ones. Alternative methods with screw theory 3, Lie algebra 4 and dual quaternion 5 have been used. In kinematics, screw theory was employed with a great deal of contributions 6-7. The work was extended to the acceleration of an unconstrained rigid body and the end-effector of a30serial manipulator 8-10, where the acceleration of the end effector is in terms of the direction andmoment parts of the same screw coordinates. In addition, a linear-bilinear form with the influence coefficient matrices has been proposed to represent the acceleration of a manipulator 11.Besides, robotics researchers developed many of the most efficient algorithms in dynamics, andLagrange and Newton-Euler's formulations are two main streams 12-14. However, both of them35become complex to analyze and calculate when applied to robot dynamics. Numerous formulations and recursive algorithms 15-16 have therefore been proposed that attempt to reduce their symbolic and numerical complexity. A systemic theory of robotic mathematics based on homogeneous transformation and screw theory was presented 17, which enhanced the application of Lie groups and Lie algebra in robotics. Frank C. Park 18 investigated the modeling and40computational aspects of the (POE) product-of-exponentials formula for robot kinematics with Lie groups and Lie algebra; J. M. Rico and J. Gollardo 19 studied the application of Lie algebra to the mobility analysis of kinematic chains. The geometrical foundations 20 of robotics have been presented, in which a succinct dynamics analysis using the derivatives of twist representing rigid body accelerations was given. The spatial operator algebraic formulation 21 of dynamics has been45developed by identifying structural similarities in an open chain dynamics. The formulations ofrobot dynamics with Lie groups and Lie algebras 22-25 have drawn much attention, in which LieFoundations: National Science Foundation of China Grant(No.50975244)Brief author introduction:ZHAO Tieshi(1963-) is currently a professor with the Department of MechatronicsEngineering, Yanshan University. His research interests include parallel mechanisms, stabilized platforms, sensor technology, and robotics technology. E-mail: tszhaoysu.edu.cn- 19 -groups and Remannian geometry were used, link velocities and accelerations were expressed in terms of standard operations on the Lie algebra of SE(3). T.S. Zhao 26-27 study the dynamic and coupling actuation of elastic underactuated manipulators based on Lie groups. However, as most50of dynamics formulae have no coordinate invariance, the analysis, design and control of manipulators are still complicated, especially for the analysis on the dynamic coupling performances. When kinematics and dynamics Hessian matrixes are involved, the design and analysisare more complicated for a serial or parallel manipulator.Based on the above work, this paper investigates the physical meaning of Lie bracket between55two twists. And then, the accelerations of the end-effector of manipulators are expressed as linear-bilinear forms, from which the kinemics Hessian matrixes are expressed as the Lie brackets of joint screws. Further, Newton-Eulers' equations are rewritten as linear-bilinear forms, from which the dynamics Hessian matrix is deducted. The formulae and Hessian matrixes are proved to be coordinate invariant and convenient to program. In the end, an index of dynamics coupling60based on dynamics Hessian matrixes is presented, and it is applied to a foldable parallel manipulator, which demonstrates the convenience of the kinematics and dynamics Hessian matrixes.1Kinematics Hessian Matrixes of Manipulators Based on LieBrackets65A Lie bracket is a linear operator, also a derivative operator 28-31. This section gives an investigation into the physical meaning of the Lie brackets between two twists, and then formulates acceleration mappings of manipulators with Jacobin and kinematics Hessian matrixes represented in Lie brackets.1.1Lie Bracket Representation of Acceleration70LetS = (s ; s 0 )andS = (s; s 0 )be two screws, the exponential mapping ofS with ai i ijjjimagnitude jiirepresents a rigid body motion of rigid body j with respect to i from time t=0 to t)é s03 3 ùS j (t ) = exp(ji Si ) S j (0),(1)where,S i = êis o´s úis the adjoint representation of screwS i ,si ands oare theë ii ûskew-symmetrical representations of three-dimensional vectors si75the time derivative of the above equation at t=0 is)j i i j i i jS& (0) = j& S S (0) = j& S , S (0) .andis o . Settingj&i (0) = 0 ,When choose the measure time from any instant, the following is given)j i i j i i jS& = j& S S = j& S , S .It can be seen that the Lie bracket is also the partial derivative of S j¶ S jwith respect toji , i.e.80¶ji= Si , S j .(2)LetVi = j&i SiandFig. 1 The Ship-based stabilized platform and helicopterV j = j&i Si + j& j S j be the twist of rigid bodies i and j with respect to frameo-xyz shown in Fig.1,the time derivative of V jis given85Aj = Ai + j&&j S j + j&ij& j Si , S j .(3)where,Ai = j&&i Si + j&i S& i . SinceV j = ( j , v j )is the twist of the rigid body j, jis the angularvelocity vector, and v jis the velocity of the point in the rigid body j that corresponds with theorigin of the fixed frame.A j = ( j , a j )represents the corresponding angular acceleration andinstantaneous linear acceleration of point o, and is called the acceleration of a rigid body in this90paper. Equation (3) gives a Lie algebra representation of the acceleration of a rigid body. Letjj jAr = j&& Sji jijandAc = j& j& S , S , it can be proved thatrAAjjis reciprocal tocby theirreciprocal product, i.e.rcA j × A j = j&& j S j j& i S i , j& j S j = j&& j& j&s o × s s+ s (s s o + s o s )= 0j i jji jj i jj iwhere,s o × s s= -s× s s o . Since A rkeeps the direction and pitch of twistV , it is the relativeji jj i j jjAj95acceleration of rigid body j; andccan be called the Coriolis acceleration of rigid body j causedby the bulk motion described with velocityVi .Ai is the corresponding bulk acceleration.1.2Kinematics Hessian Matrix of Serial ManipulatorLet j , j& and j&&be the joint displacement, velocity and acceleration vectors of a serialmanipulator with n links, respectively,kVk = Gj j&be the twist of link k. The time derivative of100twistVk isAk = Gj j&& + G& j j& ,(4)kkj1k6´nwhere,G k = S ,L, S ,L,0,L0 , k = 1,2,K, nis the Jacobian matrix of the kth link of themanipulator, andS i ,i =1,2,K, nare the screws, represented in frame o-xyz,describing the105joint axes of the manipulator. From (2), the following is givenS& i = Vi-1 , Si = j&1 S1 , Si + j&2 S2 , Si + L + j&i-1 Si-1 , Si where,i =1,2,L, k . It can be rewritten asS& = j& T S , S , S, S ,L,S, S , 0,L0T .Hence,i 1 i 2& kij¶G ki -1 iT k110where,Gj = j&=j&T¶j THj ,(5)é0êê0kê MS1 , S2 L0LMOS1 , Sk 0MMSk -1 , Sk 0L 0ùMúMúL 0úHj = êê0 0L0ê MMMMêêë0 0L0ú00 LúM O M úú0 L 0úû n´nis a n´n matrixwhose elements are six-dimensional column vectors. It is called as kinematicsHessian matrix of the manipulator described by Lie brackets. The rth row cthcolumn element of115jH k can be given byH k ì Sr , Sc forr < c.Substituting (5) into (4) proves thatj ( rc) = íî 0forr ³ ckjjA = G k j&& + j&T H kj& .(6)This linear-bilinear formulation can be further proved to be coordinate invariant. With similar derivation, (6) can be given in frame j-xyz,jj kT j k120Ak = Gj j&& +j&Hj j& ,(7)j k j j j jkwhere,Gj = S1 ,L, S i ,L, S k ,0,L,06´n ,and the rth row cthcolumn element ofH j canalso be expressed asj H kì j S ,jSc forr < c .rj ( rc) = íî 0forr ³ c125130The above two equations demonstrate the coordinate invariance of the linear-bilinear formulation of accelerations of a manipulator, and also prove the coordinate invariance of the kinematics Hessian matrix of a manipulator described by Lie brackets. It is very convenient in acceleration analysis, and its elements have clear physical meanings, can be obtained directly from the joint axes.1.3 Kinematics Hessian Matrix of Parallel ManipulatorSuppose a parallel manipulator with six degrees of freedom consists of a base, a moving platform and m limbs. Each limb has six one-degree-of-freedom joints and connects the base andthe moving platform. Suppose the rth joint of the ith limb is the actuated joint of the parallelmanipulator, andq& = (q& q&L q&) T = (j& (1) j& ( 2) Lj& ( m) ) T and= ( ) T = (1)1 2( 2)m r r r( m) ) Tq&&q&&1 q&&2 L q&&mj&&rj&&rLj&&rare the generalized velocity and acceleration respectively.135Based on the principle given in the above section, the acceleration of the moving platform isAG P (i )j (i )j T (i ) H P (i )j (i )P = j&& + &j & ,(8)where,G P (i ) = S (i )， S (i )L S (i ) is the Jacobian matrix of the ith limb,H P (i ) is thej126jkinematics Hessian matrix with the rth row cthcolumn element,HP (i ) =rì S (i ) , S (i ) , forr < c140i.e.j ( rc)íî 0cfor,r ³ c12é0 S (i ) , S (i ) 1 S (i ) , S (i ) S (i ) , S (i ) S (i ) , S (i ) S (i ) , S (i ) ùê (i )3(i )1(i )4(i )1(i )5(i )1(i )(i ) úê00ê00 S 2 , S 3 0 S 2 , S 4 S (i ) , S (i ) S 2 , S 5 S (i ) , S (i ) S 2 , S 6 ú6 S (i ) , S (i ) újH P (i ) = ê3 4 353 6 úê000 0 S (i ) , S (i ) S (i ) , S (i ) ú4 5 4 6ê0ê0000 S (i ) , S (i )úúê00000úû5 6Then the joint acceleration of the ith limb is given byj&&(i ) = Gj (i ) ( A- j& T (i ) H P (i )j& (i ) ) ,(9)P P jwhere,PjGj (i ) = G P (i ) -1 . The rth joint acceleration of the ith limb can be expressed as145&&(i ) = Gj (i ) ( A- & T (i ) H P (i ) & (i ) ),(10)j r P ( r ) P j j jGwhere,j (i )P ( r )is the rth column of matrixPG j (i ) . With the same steps, the other actuated jointaccelerations can be obtained. The accelerations of all actuated joints can be combined into a matrix formrìq&T L(1) q&üï ïïq&T L( 2) q& ïq&& = G q A - ír ý ,(11)P Pïïîq&TML( m)rïþq&ï150GPwhere,qmatrixis the Jacobian matrix of the moving platform on the generalized coordinates.L(i ) = Gj T (i ) Gj T (i ) * H P (i ) Gj (i ) ,Lr is arqP ( r )jqwhere,Gj (i ) = Gj (i )G Pis the Jacobian matrix of the ith limb on the generalized coordinates, “*”q P q155denotes the generalized scalar product of matrixes considering the former matrix as a constant and to multiple each element in the later one. Equation (11) can further be rewritten asPPpq&& = G q A- q& T H q q& ,(12)pwhere, H qeach limb,is acubic matrix with the element consists of the corresponding elementin Lr ofq(1)( 2)( m ) TH P (ij ) = ( Lr (ij )Lr (ij ) L Lr (ij ) ).160The acceleration of the moving platform is obtained from (12),A = G p (q&& + q& T H q q&) ,(13)and furtherpqpA = G p q&& + q& T H p q& .(14)pqq165Hence, the kinematics Hessian matrix of the moving platform on the generalized coordinates is given byPPqSubstituting (13) into (9)H q = Gq * H P .j&&(i ) = Gj (i ) (G P (q&& + q&T H q q&) - j& T