{"title": "Global Regularization of Inverse Kinematics for Redundant Manipulators", "book": "Advances in Neural Information Processing Systems", "page_first": 255, "page_last": 262, "abstract": null, "full_text": "Global Regularization of Inverse Kinematics for Redundant \n\nManipulators \n\nDavid DeMers \n\nDept. of Computer Science & Engr. \nInstitute for Neural Computation \nUniversity of California, San Diego \n\nLa Jolla. CA 92093-0114 \n\nKenneth Kreutz-Delgado \n\nDept. of Electrical & Computer Engr. \n\nInstitute for Neural Computation \nUniversity of California, San Diego \n\nLa Jolla, CA 92093-0407 \n\nAbstract \n\nThe inverse kinematics problem for redundant manipulators is ill-posed and \nnonlinear. There are two fundamentally different issues which result in the need \nfor some form of regularization; the existence of multiple solution branches \n(global ill-posedness) and the existence of excess degrees of freedom (local ill(cid:173)\nposedness). For certain classes of manipulators, learning methods applied to \ninput-output data generated from the forward function can be used to globally \nregularize the problem by partitioning the domain of the forward mapping into \na finite set of regions over which the inverse problem is well-posed. Local \nregularization can be accomplished by an appropriate parameterization of the \nredundancy consistently over each region. As a result, the ill-posed problem can \nbe transformed into a finite set of well-posed problems. Each can then be solved \nseparately to construct approximate direct inverse functions. \n\n1 \n\nINTRODUCTION \n\nThe robot forward kinematics function maps a vector of joint variables to the end-effector \nconfiguration space, or workspace, here assumed to be Euclidean. We denote this mapping \nby f(\u00b7) : en -+ wm ~ X m , f(O) I-t x, for 0 E en (the input space or joint space) \nand x E wm (the workspace). When m < n, we say that the manipulator has redundant \ndegrees--of -freedom (dot). \nThe inverse kinematics problem is the following: given a desired workspace location x, \nfind joint variables 0 such that f(O) = x. Even when the forward kinematics is known, \n\n255 \n\n\f256 \n\nDeMers and Kreutz-Delgado \n\nthe inverse kinematics for a manipulator is not generically solvable in closed form (Craig. \n1986). This problem is ill-posedl due to two separate phenomena. First. multiple solution \nbranches can exist (for both non-redundant as well as redundant manipulators). The second \nsource of ill-posedness arises because of the redundant dofs. Each of the inverse solution \nbranches consists of a submanifold of dimensionality equal to the number of redundant \ndofs. Thus the inverse solution requires two regularizations; global regularization to select \na solution branch. and local regularization. to resolve the redundancy. In this paper the \nexistence of at least one solution is assumed; that is. inverses will be sought only for points \nin the reachable workspace. i.e. desired x in the image of 1(\u00b7). \nGiven input-output data generated from the kinematics mapping (pairs consisting of joint \nvariable values & corresponding end-effector location). can the inverse mapping be learned \nwithout making any a priori regularizing assumptions or restrictions? We show that the \nanswer can be \"yes\". The approach taken towards the solution is based on the use of \nlearning methods to partition the data into groups such that the inverse kinematics problem. \nwhen restricted to each grouP. is well-posed. after which a direct inverse function can be \napproximated on each group. \nA direct inverse function is desireable. For instance. a direct inverse is computable quickly; \nif implemented by a feedforward network were used. one function evaluation is equivalent to \na single forward propagation. More importantly. theoretical results show that an algorithm \nfor tracking a cyclic path in the workspace will produce a cyclic trajectory of joint angles \nif and only if it is equivalent to a direct inverse function (Baker. 1990). That is. inverse \nfunctions are necessary to ensure that when following a closed loop the ann configurations \nwhich result in the same end-effector location will be the same. \nUnfortunately. topological results show that a single global inverse function does not exist \nfor generic robot manipulators. However. a global topological analysis of the kinematics \nfunction and the nature of the manifolds induced in the input space and workspace show \nthat for certain robot geometries the mapping may be expressed as the union of a finite set \nof well-behaved local regions (Burdick. 1991). In this case. the redundancy takes the form \nof a submanifold which can be parameterized (locally) consistently by. for example. the \nuse of topology preserving neural networks. \n\n2 TOPOLOGY AND ROBOT KINEMATICS \n\nIt is known that for certain robot geometries the input space can be partitioned into disjoint \nregions which have the property that no more than one inverse solution branch lies within \nanyone of the regions (Burdick. 1988). We assume in the following that the manipulator in \nquestion has such a geometry. and has all revolute joints. Thus en = 1'\" \u2022 the n-torus. The \nredundancy manifolds in this case have the topology of T n - m \u2022 n - m-dimensional torii. \nFor en a compact manifold of dimensionality n. wm a compact manifold of dimensionality \nm. and I a smooth map from en to wm . let the differential del be the map from the tangent \nspace of en at (J E en to the tangent space of wm at I( (J). The set of points in en which \n1 lli-posedness can arise from having either too many or too few constraints to result in a unique \nand valid solution. That is, an overconstrained system may be ill-posed and have no solutions; such \nsystems are typically solved by finding a least-squares or some such minimum cost solution. An \nunderconstrained system may have multiple (possibly infinite) solutions. The inverse kinematics \nproblem for redundant manipulators is underconstrained. \n\n\fGlobal Regularization of Inverse Kinematics for Redundant Manipulators \n\n257 \n\nmap to x E wm is the pre-image of x, denoted by 1-1 (x). The differential del has a \nnatural representation given by an m by n Jacobian matrix whose elements consist of the \nfirst partial derivatives of I w.r.t. a basis of en. Define S as the set of critical points of I, \nwhich are the set of all 0 E en such that de/(O) has rank less than the dimensionality of \nwm. Elements of the image of S, I(S) are called the critical values. The set'R!.Wm\\S \nare the regular values of f. For 0 E en, if 30\u00b7 E 1-1(/(0)), O\u00b7 E S, we call 0 a \nco-regular point of I. \nThe kinematic ~napping of certain classes of manipulators (with the geometry herein as(cid:173)\nsumed) can be decomposed based on the c