J. Michael McCarthy

Dual quaternion synthesis of constrained robotic systems

Constrained robotics systems are serial or parallel robots with less than six degrees of freedom. Dimensional synthesis is defined as the process of dimensioning a robot, that is, designing the link dimensions for a given task or set of tasks. In finite-position synthesis, we define the task as a series of positions that the robot must reach. Dimensional synthesis of planar mechanisms was first solved using graphic methods, and later those methods were transformed into algebraic equations that described the constraints on the movement of the mechanism. This approach was successfully applied to spherical mechanisms and simple cases of spatial mechanisms. The methodology was not extended to general constrained robots due to the difficulty in stating the geometric constraints for robots with more than three links. A systematic approach for the synthesis of spatial robots was developed based on using the kinematics equations of the robot. The kinematics equations are spatial transformations from a fixed frame to the end-effector of the robot, parameterized by both the dimensions of the links and the joint variables. In this dissertation, a method for the kinematic synthesis of constrained robots is presented. It is based on the use of dual quaternions to construct the kinematics equations of the robot from a reference position and to equate them to a set of task positions. A calculation was devised to compute the maximum number of task positions for each robot topology, and a classification of constrained robots was obtained according to this. The design equations produced using this methodology have been solved numerically for both the link dimensions and the joint variables, and also a scheme has been introduced to eliminate the joint variables in order to obtain algebraic equations. These have been further simplified to closed algebraic expressions in several cases. The dual quaternion synthesis methodology provides with a tool for the systematic design of constrained robots. Some of these results have been implemented in computer-aided design systems.

A Polynomial Homotopy Formulation of the Inverse Static Analysis of Planar Compliant Mechanisms

This paper formulates the inverse static analysis of planar compliant mechanisms in polynomial form. The goal is to find the equilibrium configurations of the system in response to a known force/moment applied to the mechanism. The geometric constraint of the linkage defines a set of kinematics equations which are combined with equilibrium equations obtained from partial derivatives of the potential-energy function. In order to apply polynomial homotopy solver to these equations, we approximate the linear torsion spring torque at each joint by using sine and cosine functions. The results obtained from the homotopy solver are then refined using Newton-Raphson iteration. To demonstrate the analysis steps, we study two example planar compliant mechanisms, a four-bar linkage with two torsional springs, and a parallel platform supported by three linear springs. Numerical examples are provided together with plots of the potential energy during a movement between selected equilibrium positions.

Algorithm 857: POLSYS_GLP—a parallel general linear product homotopy code for solving polynomial systems of equations

Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked and handling singular solutions have made probability-one homotopy methods even more practical. POLSYS_GLP consists of Fortran 95 modules for finding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95-derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS_PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although finding the optimal GLP structure is a difficult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS_GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.

Synthesis of Bistable Compliant Four-Bar Mechanisms Using Polynomial Homotopy

In this paper we formulate and solve the synthesis equations for a compliant four-bar linkage with three specified equilibrium configurations in the plane. The kinematic synthesis equations as for rigid-body mechanisms are combined with equilibrium constraints at the flexure pivots to form design equations. These equations are simplified by modeling the joint angle variables in the equilibrium equations using sine and cosine functions. Polynomial homotopy continuation is applied to compute all of the design candidates that satisfy these design equations, which are refined using a Newton-Raphson technique. A numerical example demonstrates design methodology in which the homotopy solver obtained eight real solutions. Two of them provide two stable and one unstable equilibrium, and hence, can be used as the prototype of bistable compliant mechanisms.

Trajectory Planning for Constrained Parallel Manipulators

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

Geometric Design of Cylindric PRS Serial Chains

This paper considers the design of cylindric PRS serial chains. This five degree-of-freedom robot can be designed to reach an arbitrary set of eight spatial positions. However, it is often convenient to choose some of the design parameters and specify a task with fewer positions. For this reason, we study the three through eight position synthesis problems and consider various choices of design parameters for each. A linear product decomposition is used to obtain bounds on the number of solutions to these design problems. For all cases of six or fewer positions, the bound is exact and we give a reduction of the problem to the solution of an eigenvalue problem. For seven and eight position tasks, the linear product decomposition is useful for generating a start system for solving the problems by continuation. The large number of solutions so obtained contraindicates an elimination approach for seven or eight position tasks, hence continuation is the preferred approach.

A planar quaternion approach to the kinematic synthesis of a parallel manipulator

In this paper we present a technique for designing planar parallel manipulators with platforms capable of reaching any number of desired poses. The manipulator consists of a platform connected to ground by RPR chains. The set of positions and orientations available to the end-effector of a general RPR chain is mapped into the space of planar quaternions to obtain a quadratic manifold. The coefficients of this constraint manifold are functions of the locations of the base and platform R joints and the distance between them. Evaluating the constraint manifold at each desired pose and defining the limits on the extension of the P joint yields a set of equations. Solutions of these equations determine chains that contain the desired poses as part of their workspaces. Parallel manipulators that can reach the prescribed workspace are assembled from these chains. An example shows the determination of three RPR chains that form a manipulator able to reach a prescribed workspace.

Clifford Algebra Exponentials and Planar Linkage Synthesis Equations

This paper uses the exponential defined on a Clifford algebra of planar projective space to show that the standard-form design equations used for planar linkage synthesis are obtained directly from the relative kinematics equations of the chain. The relative kinematics equations of a serial chain appear in the matrix exponential formulation of the kinematics equations for a robot. We show that formulating these same equations using a Clifford algebra yields design equations that include the joint variables in a way that is convenient for algebraic manipulation. The result is a single formulation that yields the design equations for planar 2R dyads, 3R triads, and nR single degree-of-freedom coupled serial chains and facilitates the algebraic solution of these equations including the inverse kinematics of the chain. These results link the basic equations of planar linkage design to standard techniques in robotics.

Kinematic Synthesis With Contact Direction and Curvature Constraints on the Workpiece

In this paper, we consider the synthesis of a planar RR chain that guides a rigid body, or workpiece, such that it does not violate normal direction and curvature constraints imposed by contact with objects in the environment. These constraints are transformed into conditions on the velocity and acceleration of points in the moving body. We use this to formulate the synthesis equations for an RR chain, which are solved by algebraic elimination. An example of the design of a planar RR linkage and a four-bar chain in which the coupler maintains in contact with two objects in two locations is used to illustrate the results.Copyright

Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

© 2014 by ASME. This paper presents a design procedure for six-bar linkages that use eight accuracy points to approximate a specified input-output function. In the kinematic synthesis of linkages, this is known as the synthesis of a function generator to perform mechanical computation. Our formulation uses isotropic coordinates to define the loop equations of the Watt II, Stephenson II, and Stephenson III six-bar linkages. The result is 22 polynomial equations in 22 unknowns that are solved using the polynomial homotopy software BERTINI. The bilinear structure of the system yields a polynomial degree of 705,432. Our first run of BERTINI generated 92,736 nonsingular solutions, which were used as the basis of a parameter homotopy solution. The algorithm was tested on the design of the Watt II logarithmic function generator patented by Svoboda in 1944. Our algorithm yielded his linkage and 64 others in 129 min of parallel computation on a Mac Pro with 12±2.93 GHz processors. Three additional examples are provided as well.