Force distribution in closed kinematic chains

The problem of force distribution in systems involving multiple frictional contacts between actively coordinated mechanisms and passive objects is examined. The special case in which the contact interaction can he modeled by three components of forces (zero moments) is particularly interesting. The Moore-Penrose Generalized Inverse solution for such a model (point contact) is shown to yield a solution vector such that the difference between the forces at any two contact points projected along the line joining the two points vanishes. Such a system of contact forces is described by a helicoidal vector field which is geometrically similar to the velocity field in a rigid body twisting about an instantaneous screw axis. A method to determine this force system is presented. The possibility of superposing another force field which constitutes the null system is also investigated. Disciplines Engineering | Mechanical Engineering Comments Suggested Citation: Kumar, V. and K.J. Waldron. (1988). "Force Distribution in Closed Kinematic Chains." IEEE Journal of Robotics and Automation, Vol. 4(6), p. 657 664. ©1988 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/meam_papers/250 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4 , NO. 6. DECEMBER 1988 657 Force Distribution in Closed Kinematic Chains VIJAY R. KUMAR, MEMBER, IEEE, AND KENNETH J. WALDRON Abstract-The problem of force distribution in systems involving multiple frictional contacts between actively coordinated mechanisms and passive objects is examined. The special case in which the contact interaction can he modeled by three components of forces (zero moments) is particularly interesting. The Moore-Penrose Generalized Inverse solution for such a model (point contact) is shown to yield a solution vector such that the difference between the forces at any two contact points projected along the line joining the two points vanishes. Such a system of contact forces is described by a helicoidal vector field which is geometrically similar to the velocity field in a rigid body twisting about an instantaneous screw axis. A method to determine this force system is presented. The possibility of superposing another force field which constitutes the null system is also investigated.The problem of force distribution in systems involving multiple frictional contacts between actively coordinated mechanisms and passive objects is examined. The special case in which the contact interaction can he modeled by three components of forces (zero moments) is particularly interesting. The Moore-Penrose Generalized Inverse solution for such a model (point contact) is shown to yield a solution vector such that the difference between the forces at any two contact points projected along the line joining the two points vanishes. Such a system of contact forces is described by a helicoidal vector field which is geometrically similar to the velocity field in a rigid body twisting about an instantaneous screw axis. A method to determine this force system is presented. The possibility of superposing another force field which constitutes the null system is also investigated. INTRODUCTION HIS PAPER addresses the problem of force distribution in T systems with closed kinematic chains involving multiple frictional contacts between an actively controlled structure and an object. Such systems are statically indeterminate and active coordination demands optimal solutions for force control [ 131. One example of such a redundant system can be found in walking vehicles [6], [ 1 11, [ 151 in which the legs of the vehicle and the terrain form closed loops (see Fig. 1). A similar situation exists in multifingered grippers [ 11-[3], [5], [7], [14], [17]. It has been shown that the redundancy in such systems can be resolved by linear programming techniques [5], [ 1 11 or by the application of the Moore-Penrose Generalized Inverse [6], [8]. The nature of the generalized inverse or the pseudo-inverse solution, which in turns leads to a decomposition of the force field, is explored in this paper for the special case in which the contact interaction is limited to a pure force (or zero pitch wrench [4]) through a contact center or contact point. This point-contact model is valid even for distributed contacts provided contact moments can be neglected and the contact center is known. It is convenient to decompose the system of contact forces or the force field consisting of (only) the contact forces, into an equilibrating force field and an interaction force field. The interaction force between any two contact points is defined as the component of the difference of the contact forces along the line joining the two contact points. This condition may be mathematically expressed as (F; F,) . (ri rj ) = 0 (1) Manuscript received September 24, 1987; revised March 7, 1988. This work was supported in part by DARPA under Contract DAAE 07-81-K-ROOl. Part of the material in this paper was presented at the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, PA, Apr. 25-29. V. R. Kumar is with the Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104. K. J . Waldron is with the Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210. IEEE Log Number 8822989. \ (b) Fig. 1. Examples of closed-loop kinematic chains in an actively coordinated mechanisms. (a) A walking vehicle. (b) A multifingered robotic gripper. (r, is the position vector and F, is the contact wrench at the ith contact point; w is the load wrench (fand care the associated force and couple), which is the resultant of the weight of the object, and inertial forces and moments. Any convenient object-fixed or vehicle-fixed reference frame can be used.) where Fi and Fj are the contact forces, and ri and r, are the position vectors at the ith and j th contacts (in any convenient body-fixed or object-fixed reference frame), respectively. This is illustrated through examples for a twoand a threecontact case in Fig. 2. The equilibrating force field consists of equilibrating forces, which are the forces required to maintain equilibrium against an external load. Further, these forces have no interaction force components. Thus the interaction force field consists of forces which must have a zero net resultant. It includes force components which squeeze the body (in the case of multifingered grippers) or the terrain (in the case of walking vehicles). It has been shown [8] that the pseudo-inverse solution for the force system belongs to the equilibrating force field. Further, the interaction force field was shown to be the set of forces belonging to the null space. This result is presented here 0882-4967/88/1200-0657$01.00 O 1988 IEEE 6 5 8 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 6, DECEMBER 1988 (b) Fig. 2. The zero interaction force condition for (a) two and (b) three contacts (F, is the contact force at the ith contact). in the form of a theorem. The relationship between the generalized inverse solution and the decomposition of the force field is analyzed in greater detail, and the nature of the two force fields is explored. In particular, it is shown that the equilibrating force field is mathematically isomorphic to the velocity field in a rigid body. A computationally efficient, analytical method to obtain this solution is also presented. THE PSEUDO-INVERSE SOLUTION It has been assumed here that the contact interaction is such that moments cannot be transmitted, which means that there is a total of three force components at each contact. The equilibrium equations for the grasped object, or for the walking vehicle, may be written in the form G q = w (2) where w is the 6 x 1 external load vector consisting of the inertial forces and torques, and the weight of the object (vehicle body), q is the unknown 3n x 1 force vector, and n is the number of contact interactions. G is the 6 x 3n coefficient matrix which is analogous to the Jacobian matrix encountered in the kinematics of serial chain manipulators. Each 6 x 1 column vector is a zero-pitch screw through a point of contact in the screw (in this case, line) coordinates (see Hunt [4] for a definition of screw coordinates). If S,, Si,,, and SjZ are the zero-pitch screw axes parallel to the x, y, and z axes (of any convenient coordinate system) passing through the ith contact point Alternatively, the same expression may be written as where 1 3 is the 3 x 3 identity matrix and R; is a skewsymmetric 3 x 3 matrix. 0 -z; y; R;= [ ?Yi Zi -4 where (xi , y;, z;) are the coordinates of the ith point of contact. In general, the Moore-Penrose Generalized Inverse or the pseudo-inverse of G , G + , seeks to find the minimum norm, least squares solution [12] for the force vector q. In this problem, if the screw system defined by the 3n zero-pitch wrenches is a sixth-order screw system or a six-system, w always belongs to the column space of G. It is assumed that this is the case here. The pseudo-inverse, then, is a right inverse which yields a minimum norm solution which must belong to the row space of G .