Trajectory Planning for Constrained Parallel Manipulators paper presents an algorithm for generating trajectories for multi-degree of freedom

Hai-Jun Su e-mail: suh@eng.uci.edu Robotics and Automation Laboratory, University of California, Irvine, Irvine, CA 92697 Peter Dietmaier e-mail: dietmaier@mech.tu-graz.ac.at Institute of General Mechanics, TU-Graz J. Michael McCarthy e-mail: jmmccart@uci.edu Robotics and Automation Laboratory, University of California, Irvine, Irvine, CA 92697 Trajectory Planning for Constrained Parallel Manipulators This paper presents an algorithm for generating trajectories for multi-degree of freedom spatial linkages, termed constrained parallel manipulators. These articulated systems are formed by supporting a workpiece, or end-effector, with a set of serial chains, each of which imposes a constraint on the end-effector. Our goal is to plan trajectories for systems that have workspaces ranging from two through five degrees-of-freedom. This is done by specifying a goal trajectory and finding the system trajectory that comes closest to it using a dual quaternion metric. We enumerate these parallel mechanisms and for- mulate a general numerical approach for their analysis and trajectory planning. Ex- amples are provided to illustrate the results. 关DOI: 10.1115/1.1623187兴 Introduction In this paper we formulate a trajectory planning algorithm for parallel manipulators that have less than six degrees-of-freedom. For our purposes, we assume that each supporting serial chain of the system imposes a constraint on the movement of the work- piece, or end-effector. Thus, no supporting chain has six-degrees- of-freedom. The constraints imposed by each chain can be de- signed to provide structural resistance to forces in one or more directions, while allowing the system to move in other directions. There are six basic joints used in the construction of these sup- porting chains, and we enumerate their various combinations. This allows us to count the large number of assemblies available for constrained parallel manipulators. We then formulate a general algorithm for the analysis of these systems which uses the Jacobians the supporting chains. Of im- portance is the ability to compute a trajectory for the end-effector of the system that approximates a specified trajectory while main- taining its kinematic constraints. The algorithm has been inte- grated into SYNTHETICA 关1兴, a Java based kinematic synthesis soft- ware. And examples are provided to illustrate the results. the 3-RPS system by Huang et al. 关12兴, the 3-PSP by Gregorio and Parenti-Castelli 关13兴, and the double tripod by Hertz and Hughes 关14兴. Our work has a goal similar to that of Merlet’s 关15兴 ‘‘Trajectory Verifier’’ in that we define a trajectory and determine whether the system can reach it. However, we also determine the closest ap- proaching system movement. This closest approaching trajectory is also described in Fluckiger’s 关16兴 CINEGEN, however, we fo- cus on constrained systems with parallel structure. Our approach is to solve for the constrained parallel manipulator configuration that comes closest at each frame in a specified trajectory. The key-frame interpolation scheme we use is presented in 关17兴, and based on double quaternion formulation of Etzel and McCarthy 关18兴 and Ge et al. 关19兴. See also 关20兴. Kinematics of Constrained Robots The kinematic analysis of a constrained robot begins with the kinematics equations of its supporting serial chains. Each chain can be modelled using 4⫻4 homogeneous transformations and the Denavit-Hartenberg convention 关21兴 to obtain the kinematic equa- tion 关 K 共 ␪ ជ 兲兴 ⫽ 关 Z 共 ␪ 1 ,d 1 兲兴关 X 共 ␣ 12 ,a 12 兲兴 Literature Review This research arises in the context of efforts to develop a soft- ware system for the kinematic synthesis of spatial linkages 关1兴. Kinematic synthesis theory yields designs for serial chains that guide a workpiece through a finite set of positions and orienta- tions, see McCarthy 关2兴. These chains necessarily have less than six degrees-of-freedom, and are often termed ‘‘constrained robotic systems.’’ Also see 关2–5兴. The design process yields multiple serial chains that can reach the prescribed goal positions and provides the opportunity to as- semble systems with parallel architecture. Analysis and simulation allows interactive evaluation these candidate designs. This frame- work for linkage design was introduced by Rubel and Kaufman 关6兴, Erdman and Gustafson 关7兴 and Waldron and Song 关8兴, and later followed by Ruth and McCarthy 关9兴 and Larochelle 关10兴. Our focus is the challenge of animating the broad range of linkage systems that are not constrained to one degree-of- freedom, but do not have full six degrees-of-freedom of the usual parallel manipulator. Joshi and Tsai 关11兴 call these systems ‘‘lim- ited DOF parallel manipulators.’’ Examples are the recent study of ⫻ 关 Z 共 ␪ 2 ,d 2 兲兴 . . . 关 X 共 ␣ n⫺1,n ,a n⫺1,n 兲兴关 Z 共 ␪ n ,d n 兲兴 , where 2⭐n⭐5. 关 Z(•,•) 兴 and 关 X(•,•) 兴 denote screw displace- ments about the z and x-axes, respectively. The parameters ( ␪ ,d) define the movement at each joint and ( ␣ ,a) are the twist angle and length of each link, collectively known as the Denavit- Hartenberg parameters. Notice that a serial chain robot is usually defined in terms of revolute 共R兲 and prismatic 共P兲 joints which have the kinematics equations, revolute: 关 R 共 ␪ 兲兴 ⫽ 关 Z 共 ␪ ,⫺ 兲兴 , prismatic: 关 P 共 d 兲兴 ⫽ 关 Z 共 ⫺,d 兲兴 . The hyphen denotes parameters that are constant. For our pur- poses, we include four additional joints that are special assemblies of R and P joints, Table 1. They are: i. the cylindric joint, denoted by C, which is a PR chain with parallel axes, such that 关 C 共 ␪ ,d 兲兴 ⫽ 关 Z 共 ␪ ,d 兲兴 ; Contributed by the Mechanisms and Robotics Committee for publication in the J OURNAL OF M ECHANICAL D ESIGN . Manuscript received Aug. 2002; rev. April 2003. Associate Editor: M. Raghavan. Journal of Mechanical Design ii. the universal joint, denoted by T, which consists of two revolute joints with axes that intersect in a right angle, that is Copyright © 2003 by ASME DECEMBER 2003, Vol. 125 O 709