Dimensional Synthesis of Robots using a Double Quaternion Formulation of the Workspace

Abstract

This paper presents a synthesis procedure for a robot that guides an end-effector as close as possible to a user-specified trajectory. The technique maps the goal workspace from the group of spatial displacements, SE(3), to the group of 4 × 4 rotations, SO(4), in order to obtain a bi-invariant metric for the design procedure. Double quaternions are used to provide a convenient parameterization for SO(4). An example is presented that compares designs obtained using dual quaternions to those for double quaternions, for varying locations of the fixed frame, in order to demonstrate the technique.

Geometric Design of Linkages

“…For CC chains, we use [77,78] and Murray [85]. See Ahlers [79] for the design of a CC robot that approximate a specified trajectory. Tsai and Roth [135] solved the design equations for spatial RR chains.…”

Approaching Dual Quaternions From Matrix Algebra

“…Then, q 1 + eq 2 = ξ(q 1 + q 2 ) + η(q 1 − q 2 ). Since ξ 2 = ξ, η 2 = η and ξη = 0, the terms (q 1 + q 2 ) and (q 1 − q 2 ) operate independently in the double quaternion product which has been found quite convenient when manipulating kinematic equations expressed in terms of double quaternions [23]. A third possible representation for double quaternions consists in having two quaternions expressed in different bases of imaginary units whose product is commutative.…”

“…M is closer to a 4D rotation matrix as δ tends to infinity (it can be verified that det( M ) = 1 − 5/δ 2 ). This is the approach pioneered in [44] and [45] to approximate 3D homogeneous transformations by 4D rotation matrices and used, for example, in [23] in dimensional synthesis, or in [46] to solve the inverse kinematics of a 6R robot. This kind of approximation introduces a tradeoff between numerical stability and accuracy of the approximation (see [45] for details).…”

Dimensional Synthesis of Spatial RR Robots

“…Therefore, we map the desired trajectory and the workspace of the RR chain to the group of rotations in four dimensional space SO( 4) that approximates the group of spatial displacements SE(3) over a specified region of space. McCarthy and Ahlers (1999) demonstrate this design approach for CC chains and show that it is coordinate-frame invariant to a specified accuracy within the task space. This paper combines this approach with a new solution for RR chain synthesis.…”