A Parameter-Linear Formulation of the Optimal Path Following Problem for Robotic Manipulator

TL;DR

This work reformulates time-optimal path following for robotic manipulators by shifting from time minimization to maximizing the path speed along a prescribed path, thereby avoiding singularities associated with $z(\sigma)=0$. The authors derive a parameterized OPF with dynamics $\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{h}(\mathbf{q},\dot{\mathbf{q}})=\boldsymbol{\tau}$ and introduce $z(\sigma)=\dot{\sigma}^2$, enabling a discretization that yields a parameter-linear optimization problem. They present two discretization strategies—piecewise-polynomial and B-spline representations of $z(\sigma)$—and show that, neglecting jerk constraints and viscous friction, the problems are linear in the optimization variables; with jerk and friction constraints included via SCP, these formulations remain computationally efficient. Numerical results on a 6-DOF manipulator demonstrate comparable terminal times to conventional approaches but with substantial reductions in computation time, especially for the piecewise-linear $z$ formulation, and offer smoother trajectories when using B-splines. The approach shows promise for online applications and motivates future integration with MPC-style online computation frameworks.

Abstract

In this paper the computational challenges of time-optimal path following are addressed. The standard approach is to minimize the travel time, which inevitably leads to singularities at zero path speed, when reformulating the optimization problem in terms of a path parameter. Thus, smooth trajectory generation while maintaining a low computational effort is quite challenging, since the singularities have to be taken into account. To this end, a different approach is presented in this paper. This approach is based on maximizing the path speed along a prescribed path. Furthermore, the approach is capable of planning smooth trajectories numerically efficient. Moreover, the discrete reformulation of the underlying problem is linear in optimization variables.

Figures (2)

• Figure 1: Arbitrary path in the joint space ${\mathbf{q}(\sigma)}$.
• Figure 2: Optimal path speed $z^*$ as well as first and second geometric derivative for following the arbitrary joint path. Blue circles: optimal solution with \ref{['eq:MaxZ_OPF1']} using a piecewise-linear approach in $z$. Orange crosses: optimal solution with \ref{['eq:MaxZ_OPF2']} using B-splines.