Effect of the coordinate frame on high-order expansion of serial-chain displacement

Abstract

An algorithmic differentiation technique gives a simpler, faster power series expansion of the finite displacement of a closed-loop linkage. It accomplishes this by using a higher order than what has been implemented by complicated prior formulas for kinematic derivatives. In this expansion, the joint rates and axis lines generate the instantaneous screw of each link. Constraining the terminal link to have a zero instantaneous screw satisfies closure. In order to maintain closure over a finite displacement, it is necessary to track the spatial trajectory of each joint axis line, which in turn is directed by the instantaneous screw of a link to which it is attached. Prior algorithms express these screws in a common ground-referenced coordinate frame. Motivated by the kinematics solver portion of the recursive Newton–Euler algorithm, an alternative formulation uses sparse matrices to update the instantaneous screw between successive link-local frames. The recursive Newton–Euler algorithm, however, conducts the expansion to only second order, where this paper shows local coordinate frames that are only instantaneously aligned with their respective links give identical expressions to those in frames that move with the links. Moving frames, however, require about 40% of the operations of the global-frame formulation in the asymptotic limit. Both incrementally translated (Java) and statically compiled (C++) software implementations offer more modest performance gains; execution profiling shows reasons in order of importance (1) balance of calculation tasks when below the asymptotic limit, (2) Java array bounds checking, and (3) hardware acceleration of loops.

Optimal path crossing the orientation exclusion zone of a robot with offset wrist

“…Of the four methods enforcing tool position, Milenkovic [1] is unique in finding for a spherical wrist (1) a path optimized to give the least amount of the peak magnitude of tool orientational deviation occurring along this path (i.e., the least maximum) and (2) the least interval of the traversal over which the tool deviates from its desired orientation when satisfying limits on both joint rate and acceleration. The procedures in the path-planning and execution stages are derived from a high-order path-following algorithm, which was found to have a 10-fold reduction in calculation time [27] over a prior secondorder predictor-corrector [28]. Meeting these objectives for an offset wrist motivates the current study.…”

“…High-order path following employs algorithmic differentiation [33] for directly determining the series coefficients of the inverse kinematic transform. These coefficients have a concise derivation and simple expression leading to rapid calculation at high orders [27,34]. Conditions on a virtual displacement are applied to this process.…”

“…Setting one or more of these independent variables to zero locks those joints. When the calculated loop closure error [49] accumulates at the end of a path-following step, a nonzero T BCn+1 [0] is applied to the next step in a combined predictor-corrector procedure [27,52,53]. Equation (A2) tracks the joint screws over a finite displacement giving…”

“…The robot joint numbers start at 1 at the robot base. To keep the same notation used in prior work [27,34,48], the grounded base is numbered link 1, which is connected to initial moving link 2 in the robot by joint 1. A nonredundant serial robot is extended with a virtual joint representing tool motion that closes a kinematic chain.…”

“…Reducing the step length to meet this limit under-applies any correction combined into the expansion of f (x). The non-reversion method is therefore conducted with a separate, second-order corrector, the calculation time penalty being under 20% [27]. The algorithm below estimates the first zero of θ {d} z , where d is the order of the derivative of the angle of a passive joint.…”