The UDUT Decomposition of Manipulator Inertia Matrix Subir Kumar Saha RD Center Toshiba Corporation  Ukishimacho Kawasakiku Kawasaki  Jalpan Email sahaCmel
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The UDUT Decomposition of Manipulator Inertia Matrix Subir Kumar Saha RD Center Toshiba Corporation Ukishimacho Kawasakiku Kawasaki Jalpan Email sahaCmel

ukirdctoshibacojp Abstract In this paper the UDUT decomposition of the gen eralized inertia matrix of an nlink serial manipulator is presented in symbolic form where U and D respec tively are the upper triangular and diagonal matrices To render the d

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The UDUT Decomposition of Manipulator Inertia Matrix Subir Kumar Saha RD Center Toshiba Corporation Ukishimacho Kawasakiku Kawasaki Jalpan Email sahaCmel

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The UDUT Decomposition of Manipulator Inertia Matrix Subir Kumar Saha R&D Center, Toshiba Corporation 4-1 Ukishima-cho, Kawasaki-ku, Kawasaki 210, Jalpan E-mail: Abstract In this paper, the UDUT decomposition of the gen- eralized inertia matrix of an n-link serial manipulator is presented in symbolic form, where U and D, respec- tively, are the upper triangular and diagonal matrices. To render the decomposition, the elementary upper tri- angular matrices, associated to a modified Gaussian elimination, are introduced, whereas each element of the

inertia matrix is written as an expression, instead offinding it as a number with the aid of an algorithm. The resulting UDUT decomposition shows recursive relations among the elements of the associated matri- ces. Thus, algorithms of order ‘n can be developed not only for the inverse but also for the forward dynam- ics. As an illustration, a forward dynamics algorithm is presented here. 1 Introduction In robotics, the inverse and forward dynamics of robotic manipulators, associated to their control and simulation, respectively, are well-known prob- lems. While, based on Euler-Lagrange

equations [l], Newton-Euler equations [a], and other techniques, e.g., [3], many efficient recursive inverse dynamics al- gorithms of order n, O(n), i.e., whose complexities are linear in the number of links, n, exist, an O(n) al- gorithm for forward dynamics was not available until very recently [4, 51. The reason lies in the difficulties of deriving the n x n generalized inertia matrix (GIM) of the manipulator, and the solution of the set of n lin- ear algebraic equations in joint accelerations, namely, the dynamic equations of motion. If a straightforward approach is undertaken, i.e., an

algorithm is used to calculate the GIM and decompose it numerically, us- ing, for example, the Cholesky decomposition [B], be- fore the joint accelerations are solved by forward and backward substitutions, the complexity of 0(n3) is in- evitable. Hence, a different look: into the problem was sought, which resulted in an approach called articulated-body inertia [4]. The new concept allowed to calculate the joint accelerations with O(n) computations. Later, in [5] and consequent publications, U(n) forward dy- namics algorithms for different robotic systems are reported, which are based on Kalman

filtering and smoothing techniques, arising in the state estimation theory. This approach provides a deeper insight to the manipulator dynamil-s. It was possible due to the establishment of the equivalency of the discrete-time state space systems to the spatially recursive state space model in which the distance between two succes- sive joints plays the role cif time interval of the discrete- time models. In this paper, the elements of the GIM are derived as expressions, as opposed to numbers in an algorith- mic approach. Such representation enables us to per- form the Gaussian elimination

(GE) [6] of the GIM symbolically. Note, however, the modifications in the introduced GE, as carried out in SA, which is called here the reverse Gaussian elimination (RGE). The symbolic RGE results in the desired UDUT decompo- sition of the GIM, where U and D are the upper tri- angular and the diagonal matrices, respectively, whose elements have symbolic recursive representations. The recursiveness is not recognizable in the algorithmic ap- proach, where the operations are performed on num- bers. Thus, using the proposed approach, U(n) algo- rithms for both the inverse and forward dynamics of

the system at hand can be developed. As an illustration of the proposed decomposition, an U(n) forward dynamics algorithm is presented in SB. The efficiency of the scheme, compared to that of an efficient (n3) algoriithm, e.g., [7], is appreciable while n 2 12, as in [4]. Thus, the proposed forward dynamics scheme is suitable only for highly redun- dant manipulators (n 2 12), which may be required to - 2829 - I E E E Inter na t I on a I Confer en c 8 on Robotics and Automatton 0-7803-1965-6/95 $4.00 01995 IEEE
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Fig. 1 A system of coupled links. achieve non-stop operation in the

presence of actuator failures, to maneuver in a highly constrained environ- ment, etc. The decomposition, due to the availability of the recursive symbolic relations, is, nevertheless, very useful for analytic investigation of the robot dy- namics or to verify the intermediate steps of the for- ward dynamics calculations. 2 Decoupled NOC Since the natural orthogonal complement (N0C)- a matrix that relates the velocities of the links of a me- chanical system to its joint rates, and used to reduce the Newton-Euler equations of motion of uncoupled links of the system to an independent set of

Euler- Lagrange equations-in its present form, i.e., as ob- tained in [$, 91, is not suitable to write the elements of the GIM as expressions, it is derived here as two decozlpled matrices. This allows to perform the sym- bolic RGE, as done in $4. Now, let us define the 6- dimensional vector of the twist, or the spatial velocity [4, 51, of the ith link, Fig. 1, as (1) ti E [UT, Ci TT ] where w; and C; are the 3-dimensional vectors of an- gular velocity and the velocity of the mass center of the ith link, Ci, respectively. Moreover, for the system of coupled links, Fig. 1, wi and Ci are written

as wi = wj + Biei ci = cj + wj x rj + wi x di (2.) (2b) where wj and Cj are the angular velocity and the ve- locity of Cj of the jth link, respectively, whereas the Cartesian vectors, rj and di, are shown in Fig. 1. Combining eqs.(2a) and (2b), the twist, ti, is ex- pressed as a function of tj and ii, i.e., ti = Bijtj + piBi (3) where the 6 x 6 matrix, Bij, and the 6-dimensional vector, pi, are given by B..= [ "1 and pi z [ ei ] (4) *I - cij 1 ei x di 1 and 0 being the 3 x 3 identity and zero matrices, respectively, which, henceforth, should be understood as of dimensions compatible to the

size of a matrix where they appear. Moreover, Cij is the 3 x 3 cross- product tensor, associated to the vector, cij = cj - ci. A 3 x 3 cross-product tensor, associated to the %dimensional vector, z, denoted by Z, is defined by Z~zX1~- (5) a(Z x x) ax for any arbitrary %dimensional vector, x. Further- more, ei is the unit vector parallel to the axis of the ith revolute pair. It is pointed out here that the matrix, Bij, and the vector, pi , have the following interpretations: while the former multiplied to tj gives ti, if there is no ith joint, the latter takes into account the effect of the ith

joint motion. Also, from the definition of BQ, eq.(4), and Fig. 1, the following properties are derived: BijBj~, = Bik and Bii = 1 (6) Now, for the manipulator consisting of n links, as denoted in Fig. 2 by #1, . . 8 , #n, coupled by n revolute pairs, namely, 1, . . . , n, the 6n-dimensional generalized twist, t, is defined as t =. [tT, ..') ty (7) where ti, for i = 1, "*, n, is given in eq.(l). Using eq.(3) and the properties given by eq.(6), vector t is represented as t = TO, where Tr TlTd (8) In eq.(8), the 6n x n matrix, T, is the natural orthog- onal complement (NOG) of the system at

hand, where TI and Td are the 6n x 6n lower block triangular and the 6n x n block diagonal matrices, respectively, i.e., r 1 0 ... 01 - 2830 -
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....... .... .... articulated body i Fig. 2 An n-link n-DOF manipulator. whereas the n-dimensional vector of joint rates, 6, is as follows: Note, the foregoing derivations are also possible for manipulators with joints other than revolute, e.g., prismatic, in which the matrix, Td, will change. e= [&,...,B,]T (9c) 3 Generalized Inertia Matrix If 6 of eq.($) denotes the vector of independent gen- eralized speeds of the manipulator under

study, Fig. 2, the n x n generalized inertia matrix (GIM), I, as de- fined in [$, 91, is given as I z T~MT T~T~MT~T~ (10) M diag[Ml, . . , M,] (11) where the 6n x 6n generalized mass matrix, M, is de- fined by The 6 x 6 matrix, Mi, for i = 1,. . . , n, being the ex- tended mass of the ith link with respect to its mass center, Ci, which is also referred to as the spatial in- ertaa matriz: [4, 51. Matrix Mi is where mi and Ii are the mass and the 3 x 3 inertia ten- sor about, Ci, of the ith link, respectively. Upon sub- stitution of eq.(9a) into eq.(lO), the GIM, I, is rewrit- ten as I =: TzMTd,

where M E TTMTl (13) Matrix M is the 6n x 6n symmetric matrix, i.e., I where the 6 x 6 matrix, Mi, for i = 1,. . s, n, is as follows: I I Mi E Mi + Mi,i+l (154 in which the matrix, Mi,i+l, is defined by that can be calculated recursively as Mi,i+l = B:+i, In eq.(l5c), if i = n, M,+l = 0, because there is no (n + 1)st link in the kinematic chain. Hence, M,,,+l vanish, and from eq.(l5a), M, E M,. The interpretations, associated to matrix Mi, eq.(l5a), are provided below: (A) For i = n, M, G M,, and the expression, M,t,, denotes the 6-dimensional vector, whose first three components are the

angular momen- tum of the nth link about C,, and the rest are its linear momentum, which is obvious from eqs.(l) and (12). (B) For i = n - 1, using eqs.(l5a) and (15c), the expression, Mn-1t,+1, can be verified as the 6- dimensional momenta vector of the system con- sisting of two rigidly connected links, #(n - 1) and #n, i.e., composite body (n - l), whose angu- lar momentum is calculated about Cn-l_ Thus, similar to the definiition of M,, matrix M,-l is the extended mass of composite body (n - 1) with respect to Cn-l. (C) Extending the explanations, (B), to Mi, matrix Mi is defined as the

extended mass of composide body i that consisting of rigidly connected links, #i,. . , #n, with respect to the mass center of the ith link, Ci. Now, eq.(9b) is substituted into the expression for I, eq.(13), which leads ito the desired expression for the GIM, i.e., I= [ i,, . .. ’71 , where iij E pTMiBijpj 2,1 ’.. 2nn (16) for i = 1, e * . , j; j = 1, . s , n. The word, “sym, de- notes the symmetric elements of matrix I. - 2831
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4 The UDUT Decomposition where 1 is the n x n identity matrix and matrices TI and Td are defined in eqs.(9a) and (9b), respectively, whereas the 6n x

6n matrix, M, is as follows: The steps to find the desired decomposition are given below: Based on eqs.(30) and (31), the reverse Gaussian elimination (RGE) is performed on the GIM, I, eq.(16), while k = n, . . , 2, i.e., EI=L2, where EEE~...E, (17) The matrices, E and L2, are, respectively, being the n x n upper and lower triagular matrices. An essential property of the EUTM, as intro- duced in eqs.(26)-(29), is stated here as E;' e (1 - akX:)-l = 1 + CX~XZ (18) where at and Xk are defined in eqs.(27) and (as), respectively. Using eq.( IS), the GIM, I, is written from eq.(17) as I = UL2,

where U E E-' (19) In eq.(19), matrices U and L2 are the n x n up- per and lower triangular matrices, respectively. From the inverse of the EUTM, eq.(18), it is clear that the diagonal elements of U are unity and the above-diagonal elements are the components of vectors, ak, for k = 2,. . . ,n, which are evaluated from eq.(31). Since the decomposition of I given by eq.(19) is not unique [6], a unique decomposition is ob- tained by normalizing the elements of L2 as L2 = DL, where D E diag[rizl, ..I, hn] (20) D being the n x n matrix whose non zero diagc- nal elements are those of matrix Lz, as

calculated using eq.(32). Hence, the diagonal elements of matrix L are unity. Finally, for the symmetric matrix, I, L e UT [6]. Therefore, the desired decomposition of the manipulator GIM, I, eq.(l6), is I = UDUT (21) where the elements of the matrices, U and D, are evaluated using eqs.(31)-(35). N e diag[-, MI2 a-., -1 an n+l (22b) ml mn in which M2,i+l, for i = n,. 1 , 1, is obtained from eq.(35). The decomposition of matrix U, eq.(22a), is simi- lar to that of [5], which is based on the Kalman fil- tering. Moreover, contrary to the definition of the ex- tended mass of composite body i, M;,

matrix Mj,j+l implies the extended mass of articulated body i-a sys- tem comprising of links #i, 3 s, #n that are coupled by joints i + 1, '. . , n, as shown in Fig. 2-with re- spect to the mass center of the ith link, Ci. Thus, matrix Mi,;+l is the articulated-body inertia of link i, as termed in [4], and, referred to [5], as the state estimation error covariance, which satisfies the dis- crete Riccati equations. The scalar, hi, is, however, interpreted in this paper as the moment of inertia of articulated body i about the axis of rotation of the ith revolute joint. I 5 Forward Dynamics An

application of the proposed decomposition of the manipulator inertia matrix is the development of an O(n) recursive algorithm for forward dynamics, where the joint accelerations are solved from the dy- namic equations of motion. Let the equations of mo- tion of the n-link n-degrees of freedom (DOF) manip- ulator, as shown in Fig. 2, is given by 18 = T (23) where I is the GIM, eq.(16), and 8 is the n-dimensional vector of joint accelerations, the time derivative of 8, as defined in eq.(9c), whereas T denotes the n- dimensional vector of joint torques or forces due to known external applied

moments and forces, and those, resulting from the gravity, centrifugal and cori- olis accelerations. Vector T is assumed to be efficiently calculated from an O(n) inverse dynamics algorithm, e.g., [7], while 8 = 0, and the known moments and forces. Thus, in order to render an O(n) forward dynamics algorithm, it is necessary to solve 8 from eq.(23) with O(n) computations. This is done by solv- ing the following three sets of equations, which are obtained by substituting eq.(21) into eq.(23), i.e., Note also that the matrix, U, can be decomposed as UT=r, D+=T, and UT6=+ (24) U = 1 + T;(T~ -

I)~NT~ (22a) The recursive schemes to obtain 6 are shown in §B. 2832 -
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6 Conclusions Based on the knowledge of elementary Linear Alge- bra, the symbolic UDUT decomposition of the gener- alized inertia matrix (GIM) of a serial manipulator is proposed, where U and D are the unit upper triangu- lar and diagonal matrices, respectively. The symbolic operations were possible due to the representation of the elements of the GIM as expressions that owes to the decoupling of the natural orthogonal complement (NOC), as done eq.(8). Since the decomposition shows recursive relations among

the elements of the associ- ated matrices, an U(.) forward dynamics algorithm could be developed. Moreover , several physical inter- pretations are provided, as after eqs.(l5c) and (22b). Without the decoupled NOC, as in [8, 91, the GIM can only be calculated using an algorithm, resulting in a numerical representation. This calls for numer- ical decomposition, which prohibits to recognize the recursive relations. The proposed O(n) forward dynamics scheme re- quires (20171 - 335) multiplications or divisions and (193n - 361) additions or subtractions, which, com- pared to the U(n3) algorithm

[7], is not efficient while n < 12. Thus, like the forward dynamics scheme in [4], the algorithm is only appreciable for those manip- ulators having redundancy of six or more. The UDUT decomposition is, nevertheless, useful for analytical investigation of the manipulator dynam- ics. For example, the effects of individual link param- eters on the composite bodies of the system can be evaluated from eq.(l5a), which can help a robot de- signer. Also, the recursive equations can be used to verify the intermediate steps of forward dynamics cal- culations. The similar findings are also reported in

[4] and [5], which are based on the definition of the articulated-body inertia and the Kalman filtering, re- spectively. The UDUT decomposition, on the other hand, uses simple rules of the Gaussian elimination, which are easy to understand and follow. References [l] J.Y.S. Luh, M.W. Walker, and R.P.C. Paul, “On- line computational scheme for mechanical manip- ulators, ASME J. of Dyn. Sys., Meas., and Con- trol, Vol. 102, pp. 69-76, 1980. [2] J .M. Hollerbach, “A recursive Lagrangian formu- lation of manipulator dynamics and a comparative study of dynamics formulation complexity, IEEE Trans. on

Sys., Man, and Cybernatics, Vol. SMC- 10, pp. 730-736, NOV , 1980. T.R. Kane, and D.A. Levinson, “The use of Kane’s dynamical equations in robotics, Int. J. of Rob. Res., Vol. 2, No. 3, pp. 3-21, 1983. R. Featherstone, “The calculation of robot dynam- ics using articulated-body inertias, Int. J. of Rob. Res., Vol. 2, No. 1, pp. 13-30, 1983. G. Rodriguez, “Kalmtan filtering, smoothing, and recursive robot arm forward and inverse dynamics, IEEE Trans. on R & A, Vol. RA-3, No. 6, pp. 624-639, 1987. G.E. Stewart, Introduction to Matrix Computa- tions, Academic Press, Inc., NY, 1973. M.W. Walker,

and D.E. Orin, “Efficient dy- namic computer simulation of robotic mecha- nisms, ASME J. of Dyn. Sys., Meas., and Con- trol, Vol. 104, Sept., pp. 205-211, 1982. J. Angeles, and S. Lee, “The formulation of dy- namical equations of holonomic mechanical sys- tems using a natural orthogonal complement, ASME J. of Appl. Mech., Vol. 55, March, pp. 243- 244, 1988. S.K. §aha, and J. Angeles, “Dynamics of nonhole nomic mechanical systems using a natural orthog- onal complement, ASME J. of Appl. Mech., Vol. 58, March, pp. 238-243, 1991. A. Reverse Gaussian Elimination Conventionally, the Gaussian

elimination (GE) [6] begins from the first column of the matrix under in- terest. In the proposed elimination, however, it is as- sumed that the GE of matrix I, eq.(16), starts from the nth column. Thus, the name reverse Gaussian elimination (RGE) is usled. In the RGE, after the annihilation of the first (n - 1) elements of the nth column, the modified inertia matrix, denoted by L,, is given as ”1 where inn is the pivot [a] and i$) are the modified elements of I, whereas “sym denotes the symmetric - 2833 -
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elements of the (n - 1) x (n - 1) matrix, resulting from the deletion

of the nth row and column of matrix L, . Equation (25) is realized by premultiplying matrix I with the elementary upper triangular matrix (EUTM) of order n and index n, as done in GE with elementary lower trianguzar matrix (ELTM) [6]. An EUTM of order TI and index k, denoted by Ek , is defined similar to an ELTM as (26) T Ek E 1 - olkxk where 1 is the n x n identity matrix and the n- dimensional vectors, ak and Ak, are defined by Ek E in which Mi is given in eq.(l5a). Note that the 6 x 6 matrix, $ik, has a recursive relation, whose sub- stitution, along with that of Mi,i+l, eq.(l5c), into

eq.(34), leads to a recursive relation for Mik, i.e., for k=n,..s,2; i=k-1,.**,1 Mik = Mi + B:+l,iMi+i,kBi+i,i (35) where, if i = k - 1, Mi+l,k E Mkk = Mk,k+l- '+k'+k, and M,,,+1 G M,. AT -1 0 . . . -alk . . . 0- 1. Solution for i: The solution, i = U-l-r, is evalu- ated as, i = Ea, . S. , En7, which has the following recursion: For i = n - 1,. . . , 1 .. .. .. .. (29) 1 -ak-l,k "' 0 .., 0 .. .i; = 7% - ppj. a,a+1 . (36) - 1- where .in =: r,, and the 6-dimensional vector, 1 0's Lk = EkLk+i (30) where, if k = n and L,+1 SE I, the matrix, LE of eq.(25), immediately follows. Furthermore, the ele-

ments of Ek and Lk, aik and i$', respectively, are computed from the following scheme, i.e., e For k = n,...,2; Do i = k- 1,...,1; Do j = i, ... , 1 c~ik = pr$ik and 2:;) = pTMikBijpj (31) end do j; end do i; end for IC. in eq.(37), %+I = '+i+l%+l + %+1,i+z; 6n,n+l = 0. 2. Solution for 7: The solution of the equation, D+ = i, involves the inversion of the diagonal matrix, D of eq.(20), which is simple, namely, D-' has only nonzero diagonal elements that are the reciprocal of the corresponding elements of D. Vector T is obtained below: For i = 1,. , n 7;. = In eq.(31), matrix Bij, and vectors

pi and pj are de- fined in eq.(4), whereas the 6-dimensional vector Gik 3. Soluiion for 6: In this step, 6 = U-T+, is calcu- lated as ET+, i.e., for i = 2, + - . , n and the terms associated to it are written as $k 3 Mk,k+lpk; $ik f BFi+k; kk E pr$k (32) where O1 71, and the 6-dimensional vector, fii,i- 1, is obtained from (33) where the 6 x 6 matrix, Mi%, eq.(31), or Mk,k+1, eq.(32), is evaluated from the following relation: n Mi, = Mi - @ik, where @ik (34) l=k - 2834 -