A New Method of Task Specification for Spherical Mechanism Design

In this paper we present a novel method for motion generation task specification for spherical mechanisms. This is accomplished with a new methodology for determining the optimal design sphere and the orientations on this design sphere for a finite set of desired spatial positions. In addition, we include a modification to the method which enables the designer to require that one of the n desired spatial positions be exactly preserved. The result is that designers can now specify spherical mechanism motion generation tasks without having to introduce into the design space an artificial design sphere. They are now free to work in unconstrained three-dimensional space. The application of this new task specification technique is discussed in a design case study. INTRODUCTION Spherical mechanisms are linkages which generate motion on concentric spheres and are the simplest mechanisms which provide spatial movement. In contrast, planar mechanisms generate two-dimensional motion. For this reason their design is compatible with using conventional drafting tools while the synthesis of spherical mechanisms is three-dimensional and is not well suited for drafting techniques. It is essential that the spherical mechanism designer be able visualize the entire problem in three-dimensions. Computer graphics can be an effective tool for Address all correspondence to this author. 1 providing this necessary visualization of the problem to the designer. Efforts have been made to create computer graphics based software packages for spherical four-bar mechanism design. SPHINX was the first spherical mechanism computer-aided design(CAD) program written by Larochelle et al 1993 for use on Silicon Graphics workstations. SPHINX begins by displaying a design sphere. The design sphere defines the surface in space upon which the workpiece is to be moved. The relative displacements between the positions on the design sphere are purely rotational and are called orientations. Orientations are defined by their longitude, latitude, and roll angles(Larochelle and McCarthy 1995). In SPHINX orientations are displayed to the designer as coordinate frames on the surface of the design sphere, see Fig. 1. The current version of SPHINX has modules for performing synthesis for three or four position rigid body guidance. It is important to note that in SPHINX the design sphere is of arbitrary radius and its location in space is undefined. SPHINXPC (Ruth and McCarthy 1997) is a CAD program for personal computers which like SPHINX utilizes a design sphere with orientations displayed on the sphere’s surface. With this software spherical mechanisms can be designed for four orientations. SPHINXPC also has the capability to design planar mechanisms for four position rigid body guidance. In 1995 Osborn and Vance developed the first virtual reality(VR) based approach to spherical mechanism design, entitled SPHEREVR. This initial exploration of the use of VR for spherical mechanism design has led to the development of a 3rd generation of VR based spherical mechanism design software called Copyright  1998 by ASME Figure 1. SPHINX DESIGN SPHERE ISIS, see Larochelle, Vance, and McCarthy 1998. The program utilizes the compute engine of SPHINX1.2 and provides virtual objects in the design environment so that the design process takes place in a virtual representation of the physical workspace. This new approach to mechanism design has demonstrated a need for new and efficient means for specifying the design task in the actual physical workspace of the mechanism. To synthesize a spherical mechanism, the designer must first define the task to be accomplished. Here we are concerned with task specification for moving a workpiece through a sequence of prescribed orientations in space. This task is referred to as rigidbody guidance by Suh and Radcliffe 1978 and as motion generation by Erdman and Sandor 1997. An example of a rigid body guidance task is shown in Fig. 2. The desired positions of the workpiece are defined in space. A coordinate frame is attached to the workpiece and its location, in each of the desired positions, is recorded. To date, when designing spherical mechanisms the designer must determine an appropriate design sphere, i.e. its center and radius, from the desired spatial positions. Moreover, the sets of angles which define the orientations of the body with respect to that design sphere must also be determined. Currently, no methodologies exist to facilitate this process. It is only after determining the design sphere and the orientations that the designer can utilize CAD tools such as SPHINX and SPHINXPC. In this paper, one method of determining the optimal design sphere and orientations from a desired set of spatial positions is presented. First, the spatial positions are approximated with orientations in four-dimensional Euclidean space(E4). Biquaternions are then used to represent these orientations. Next, the distance between the spatial positions and the orientations on a candidate design sphere are calculated using a bi-invariant metric on biquaternions. Finally, an optimization method is used to minimize the distances between the spherical orientations on the candidate design sphere and the spatial positions. The result 2 Figure 2. A DESIRED TASK is a procedure which numerically determines the optimal design sphere and orientations for a finite set of desired spatial positions. ORIENTATIONS IN E4 AND BIQUATERNIONS In 1995 Larochelle and McCarthy presented an algorithm for approximating a set of n positions in planar Euclidean space (E2) with n spherical orientations in three-dimensional Euclidean space (E3). By utilizing a bi-invariant metric on the image space of spherical displacements they arrived at an approximate biinvariant metric for planar positions in which the error induced by the spherical approximation is of the order 1 R 2 , where R is the radius of the approximating sphere. In this paper we extend their methodology to the general spatial case and utilize the results to provide a novel method of specifying motion generation tasks for spherical mechanisms. It was shown in Larochelle and McCarthy 1995 that orientations in E3 may be used to approximate positions in a bounded region of a two-dimensional plane. We utilize the contributions of Etzel and McCarthy 1996 and extend that idea by using orientations in E4 to approximate positions in a bounded region of three-dimensional space. This can be done by using a small portion of a four-dimensional hypersphere, a wedge, to approximate a bounded region of space. Orientations on the surface of this wedge, which we represent with biquaternions, can be used to approximate the spatial positions. See Ge 1994 in which he examines the theory of biquaternions as representations of orientations on a hypersphere. We proceed by briefly reviewing quaternions and biquaternions. Recall that an orientation in E3 can be represented by a quaternion q = [q1 q2 q3 q4] . The four components of the Copyright  1998 by ASME quaternion q, sometimes referred to as Euler parameters are, q1 = sx sin θ 2 = sxs θ 2 q2 = sy sin θ 2 = sys θ 2 q3 = sz sin θ 2 = szs θ 2 q4 = cos θ 2 = c θ 2 (1) where s and θ are the rotation axis and the angle of rotation associated with the orientation, respectively. Note that the components of q satisfy the following constraint equation, q1 +q 2 2 +q 2 3 +q 2 4 1 = 0 (2) and lie on a unit hypersphere which we denote as the image space of spherical displacements. Recall that the position of a body in E3 has six degrees of freedom (three to define orientation and three to define location) and can be represented by a 4x4 homogeneous transform(Paul 1981): T = 2 64 [R(θ;φ;ψ)] .. d . . . . . . . . . . . . . . . . 0 0 0 .. 1 3 75 (3) [R(θ;φ;ψ)] = Roty(θ)Rotx( φ)Rotz(ψ) where d is a 3x1 translation vector. The angles θ, φ, and ψ are the longitude, latitude, and roll angles respectively (see Larochelle and McCarthy 1995). In 1996 Etzel and McCarthy showed that a 4x4 homogeneous transform in E3 can be approximated by a pure rotation in E4: [D] = [J(α;β;γ)][K(θ;φ;ψ)] (4)