Kinematics of redundantly actuated closed chains

The instantaneous kinematics of a hybrid manipulation system, which combines the traditional serial chain geometry with parallelism in actuation, and the problem of coordination is discussed. The indeterminacy and singularities in the inverse kinematics and statics equations and measures of kinematic performance are analyzed. Finally, coordination algorithms that maintain an optimal force distribution between the actuators while avoiding or exploiting singularities are presented. Disciplines Engineering | Mechanical Engineering Comments Suggested Citation: Kumar, V. and J.F. Gardner. (1990). "Kinematics of Redundantly Actuated Closed Chains." IEEE Transactions on Robotics and Automation, Vol. 6(2)2. pp. 269 274. ©1990 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/248 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION. V O L 6 . NO 7 . APRII. I Y Y O 269 Kinematics of Redundantly Actuated Closed Chains VlJAY K U M A R . \ t F \ l H I R . IFrF. A\n JOHN F GARDNER Abstract-The instantaneous kinematics of a hybrid manipulation system, which combines the traditional serial chain geometry with parallelism in actuation, and the problem of coordination is discussed. The indeterminacy and singularities in the inverse kinematics and statics equations and measures of kinematic performance are analyzed. Finally, coordination algorithms that maintain an optimal force distribution between the actuators while avoiding or exploiting singularities are presented.The instantaneous kinematics of a hybrid manipulation system, which combines the traditional serial chain geometry with parallelism in actuation, and the problem of coordination is discussed. The indeterminacy and singularities in the inverse kinematics and statics equations and measures of kinematic performance are analyzed. Finally, coordination algorithms that maintain an optimal force distribution between the actuators while avoiding or exploiting singularities are presented. I . IUTROI>UCTION Broadly speaking, there are two types of geometries for robot manipulators: serial chain and parallel chain linkages. However, robotic systems such as two cooperating arms, walking vehicles. and multifingered grippers consist of several actively controlled articulations (serial linkages), which act in parallel on an objectiendeffectoriground. Unlike serial manipulators. they include one or more closed kinematic chains in their structure. and in addition, unlike completely parallel manipulators, there is more than one actuator in a particular chain. and the number of actuators typically exceeds the mobility typically. In this paper. such devices are called hybrid manipulators. Examples of robots with completely in-parallel actuation and the kinematic and dynamic analysis of such parallel systems have been presented in [ I ] , 121, [7], and [12]. The dualities that exist between serial and parallel geometries and between kinematics and statics have been pointed out in [17]. The control problem for such dynamically constrained systems has also been analyzed (see, e .g . , 141, [5], [16]. Hybrid systems are characterized by redundantly-actuated closed chains. The actuator rates are uniquely determined by the specified trajectory, but the actuator forces are underdetermined. The redundancy in such systems is dual to the kinematic redundancy in serial chain manipulators in which the number of actuators exceeds the dimension of the task space [ 111. The presence of redundancy engenders a need for techniques that will resolve (or even exploit) the redundancy in the system. Redundancy in manipulation systems with parallelism has been studied with reference to multifingered grippers [SI, [ 15) and walking vehicles [3], [ IO], [ 141. Since all these systems involve interaction between several actively controlled arms with a passive object [9]. most reports have described attempts to optimize contact conditions 131. [S]-[lO], [ 141. These optimization efforts have largely ignored the performance of actuators at the joint level and have treated each articulation (legifingeriarm) independently rather than considering the entire system. In research on multiple arm systems (see. e . g . , [13], [16], [18]. [2O]). the force distribution problem has not been addressed directly. Instead. in most proposed control schemes, the force distribution is automatically determined by trajectory errors, which results in unacceptably large interaction forces, thus affecting the system performance adversely. Even in studies in which this problem has been addressed, the load distribution has been a priori specified [13], [ 161. This paper addresses the issue of coordination in robotic systems with redundantly actuated closed chains. First. an instantaneous kinematic analysis of hybrid devices is presented. The singularities in the t201). Manuscript received October 18. 1988: revised October 31. 1989. V. Kumar IS with the Department of Mechanical Engineering and Applied J . F. Gardner is with the Department of Mechanical Engineering. PennIEEE Log Number 9035792. Mechanics, University of Pennsylvania. Philadelphia. PA 19104. sylvania State University, University Park. PA 16802. kinematics and statics equations. which are characteristic of systems with closed chains, are analyzed. We next discuss the kinematic characterization of the actuators to improve the understanding of the nature of redundancy in hybrid systems. Coordination schemes, which avoid singularities and compute optimal force distributions, are derived from this characterization. Simulations are used to demonstrate the efficacy of the proposed schemes. 11. I N S T 4 h T A N t O t i S KINI:MATIC ANALYSIS The analysis in this paper is restricted to symmetric configurations (same number of links. joints. and actuators on each chain). It is assumed that the joints are frictionless and that inertial forces are negligible since the main objective of the paper is to gain an insight into the problem rather than work with a perfect model. Let the manipulation system be comprised of n parallel chains. each consisting of m serially connected links (and possessing m degrees of freedom), and let the task space of d-dimensional (where d 5 m ) . Let a ( <m) be the number of actuators in each of the n chains. There are r = (n x a ) actively controlled joints. The special case a = 1 ( r = n) corresponds to a scheme of actuation that is completely parallel; an extensive survey of such mechanisms can be found in [ 7 ] , whereas n = 1 ( a must equal m in this case) denotes the standard serial arm. For the ith serial chain, the end-effector rate '8 is given by