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A Spatial Operator Algebra International Journal of Robotics Research vol. 10, pp. 371-381, Aug. 1991 A Spatial Op erator Algebra for Manipulator Mo deling and Control 1 2 1 G. Ro driguez , K. Kreutz{Delgado , and A. Jain 2 1 AMES Department, R{011 Jet Propulsion Lab oratory Univ. of California, San Diego California Institute of Technology La Jolla, CA 92093 4800 Oak Grove Drive Pasadena, CA 91109 Abstract A recently develop ed spatial op erator algebra for manipulator mo deling, control and tra- jectory design is discussed. The elements of this algebra are linear op erators whose domain and range spaces consist of forces, moments, velo cities, and accelerations. The e ect of these op erators is equivalent to a spatial recursion along the span of a manipulator. Inversion of op erators can be eciently obtained via techniques of recursive ltering and smo othing. The op erator algebra provides a high-level framework for describing the dynamic and kinematic b ehavior of a manipu- lator and for control and tra jectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract op erator expressions by insp ection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of sp eci c algorithms is greatly simpli ed. 1 Intro duction: A Spatial Op erator Algebra A new approach to the mo deling and analysis of systems of rigid b o dies interacting among them- selves and their environment has recently b een develop ed in Ro driguez 1987a and Ro driguez and Kreutz-Delgado 1992b . This work develops a framework for clearly understand- ing issues relating to the kinematics, dynamics and control of manipulators in dynamic interaction with each other, while keeping the complexity involved in analyzing such systems to manageable prop ortions. The analysis given in Ro driguez 1987a and Ro driguez and Kreutz-Delgado 1992b has shown that certain linear op erators are always present in the dynamical and kinematical equations of rob ot arms. These op erators are called \spatial op erators" since they showhow forces, velo cities, and accelerations propagate through space from one rigid body to the next. Not only do the op erators haveobvious physical interpretations, but they are implicitly equivalent to tip-to-base or base-to-tip recursions which, if needed, can b e immediately turned into implementable algorithms 1 by pro jecting them onto appropriate co ordinate frames. Comp ositions of spatial op erators, when allowed to op erate on functions of the jointvelo c- ities and accelerations, result in the dynamical equations of motion which arise from a Lagrangian analysis. The fact that the op erators have equivalent recursive algorithms is a generalization of the well-known equivalence describ ed in Silver 1982 between the Lagrangian and recursive Newton-Euler approaches to manipulator dynamics. The op erator-based formulation of rob ot dy- namics leads to an integration of these two approaches, so that analytical expressions can b e shown to almost always have implicit, and obvious, recursive equivalents which are straightforward to mechanize. The essential ingredients of the op erator algebra are the op erations of addition and multi- plication see Roman 1975 , Rudin 1973 . There is also an \adjoint," or \*", op erator which can op erate on elements of the spatial algebra. If a spatial op erator A is \causal," in the sense that it implies an inward recursion, then its adjoint A is \anticausal." An anticausal op eration implies an outward recursion. Op erator inversion is also de ned in the spatial op erator algebra. For an arbitrary nite dimensional linear op erator, inversion is achieved by the traditional tech- niques of linear algebra. However, many imp ortant spatial op erators encountered in multib o dy dynamics b elong to a class that can be factored as the pro duct of a causal op erator, a diagonal op erator, and an anticausal op erator. For these op erators, inversion can often b e achieved using the inward/outward sweep solutions of spatially recursive Kalman ltering and smo othing describ ed in Ro driguez 1987a , Ro driguez and Kreutz-Delgado 1992b and Anderson and Mo ore 1979 . That the equations of multib o dy dynamics can be completely describ ed by an algebra of spatial op erators is certainly of mathematical interest. However, the signi cance of this result go es b eyond the mathematics and is useful in a very practical sense. The spatial op erator algebra provides a convenient means to manipulate the equations describing multib o dy b ehavior at a very high level of abstraction. This lib erates the user from the excruciating detail involved in more traditional approaches to multib o dy dynamics where often one \can't see the forest for the trees." Furthermore, at any stage of an abstract manipulation of equations, spatially recursive algorithms to implement the op erator expressions can b e readily obtained by insp ection. Therefore the transition from abstract op erator mathematics to practical implementation is straightforward to p erform and often requires only a simple mental exercise. When applied to the dynamical analysis of an n link manipulator, the algebra typically leads to O n recursive algorithms. However, numerical eciency is not the main motivation for its development. What the algebra primarily o ers is a powerful mathematical framework that b ecause of its simplicity is b elieved to have great p otential for addressing advanced control and motion planning problems Ro driguez 1989c . To illustrate the use of the spatial op erators, several applications of the algebra to rob otics will b e presented: 1 an op erator representation of the manipulator Jacobian matrix; 2 the rob ot dynamical equations formulated in terms of the spatial algebra, showing the equivalence b etween the recursive Newton-Euler and Lagrangian formulations of rob ot dynamics in a far more transparent way than b efore; 3 the op erator factorization and inversion of the manipulator mass matrix which immediately results in O n recursive forward dynamics algorithms for an n link serial manipulator; 2 4 the joint accelerations of a manipulator due to a tip contact force; 5 the recursive computation of the equivalent mass matrix as seen at the tip of a manipulator, referred to by Khatib 1985 as the op erational space inertia matrix; 6 recursive forward dynamics of a closed chain system. Finally,we discuss additional applications and researchinvolving the spatial op erator algebra. 2 The Jacobian Op erator Consider an n link serial chain manipulator. After de ning a link spatial velo city to be V k = 6 _ col[! k ;vk] 2 R , the recursion which describ es the relationship b etween joint angle rates, = _ _ col[ 1; ; n], and link velo cities, V = col [V 1; ;V n] is see Ro driguez and Kreutz-Delgado 1992b , Craig 1986 : 8 > > V n +1= 0 > > > > > > > < for k = n 1 > > _ > V k = k +1;kV k +1+H k k > > > > > > : end lo op V 0 = 1; 0V 1 3 th H k = [h k 0 0 0] where hk 2 R is the unit vector in the direction of the k joint axis. k +1;k is de ned as 0 1 ~ I lk+1;k B C k +1;k= @ A 0 I th th where l k +1;k is the vector from the k +1 joint to the k joint. Thus, k +1;k is the Jacobian which transforms velo cities across a rigid link. This recursion represents a base-to-tip recursion which shows how link velo cities propagate outward to the tip, p oint \0" on link 1, from the base \link n + 1." This assumes for simplicity that the base has zero velo city. Note that the link numb ering convention used here, and in Ro driguez 1987a and Ro driguez and Kreutz-Delgado 1992b , increases from the tip to the base unlike the numb ering convention describ ed in most rob otics textb o oks suchas Craig 1986 . This convention makes it easier to describ e the recursive algorithms presented in this pap er. Summation of the ab ove recursion leads to n X _ V k = i; k H i i i=k 3 where the facts that i; i = I and i; j j; k = i; k have b een used. Also note that 1 i; j =j; i. This naturally suggests that we de ne the \op erators" H = diag[H 1; ;H n], B =[ 1; 0; 0; ;0] and 0 1 I 0 0 0 B C B C B C B C 2; 1 I 0 0 4 B C = B C . B C . B C . 0 B C @ A n; 1 n; 2 I _ This results in V 0 = B H or _ V 0 = J ; where J = B H 2.1 The Jacobian op erator J in 2.1 is seen to be the pro duct of three op erators B , and H . The op erator H , b eing blo ck diagonal, is called \memoryless" or nonrecursive. The op erator B pro jects out the link 1 velo city V 1 of the comp osite velo city V and propagates it to the tip lo cation at p oint 0. The op erator is lower blo ck triangular, which we denote as \causal," and is upp er blo ck triangular and hence \anticausal." represents a propagation of link velo cities from the base to the tip, which is viewed as the anticausal direction, as opp osed to the tip-to-base recursion represented by which is denoted as causal.
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