A Bi-Level Optimization Approach to Joint Trajectory Optimization for Redundant Manipulators

TL;DR

This work addresses the time-optimal tracking of a Cartesian path by a redundant manipulator. It introduces a bi-level optimization where a convex inner subproblem computes the maximum feasible path speed given a trajectory, and a high-level outer problem optimizes the trajectory using directional derivatives and a primal–dual scheme. The method yields bijective equivalence with the original problem, offers closed-form solutions in key special cases, and demonstrates substantial improvements in traversal time while maintaining Cartesian accuracy, validated through both simulations and real UR10e experiments. The approach leverages the problem structure to achieve efficient local optima, with practical implications for industrial robotics where fast, precise path traversal is essential.

Abstract

In this work, we present an approach to minimizing the time necessary for the end-effector of a redundant robot manipulator to traverse a Cartesian path by optimizing the trajectory of its joints. Each joint has limits in the ranges of position, velocity and acceleration, the latter making jerks in joint space undesirable. The proposed approach takes this nonlinear optimization problem whose variables are path speed and joint trajectory and reformulates it into a bi-level problem. The lower-level formulation is a convex subproblem that considers a fixed joint trajectory and maximizes path speed while considering all joint velocity and acceleration constraints. Under particular conditions, this subproblem has a closed-form solution. Then, we solve a higher-level subproblem by leveraging the directional derivative of the lower-level value with respect to the joint trajectory parameters. In particular, we use this direction to implement a Primal-Dual method that considers the path accuracy and joint position constraints. We show the efficacy of our proposed approach with simulations and experimental results.

Figures (14)

• Figure 1: Reference task: A 3R Planar Robot manipulator to trace the 2D leading edge of a fanblade for a material deposit application. The free variable for this particular simulated scenario is the rotation along the curve edge.
• Figure 2: Final Cartesian path $\chi$ and error with the reference path for the trajectory obtained with Algorithm \ref{['alg:comp']} for ten different initial conditions. For all experiments, the error has stayed under the designed threshold.
• Figure 3: Evolution of the objective function and constraint tolerance usage of Algorithm \ref{['alg:comp']} with the number of iterations for ten different initial conditions. Though improvement exists for all experiments, one can see that performance is doubtlessly dependant on initial conditions.
• Figure 4: Initial and Final configuration of the free variable $\phi$ over the path length for all initial conditions. Even relatively small changes in the free variable can drastically improve performance.
• Figure 5: Comparison between the optimal joint trajectories obtained for $\phi_5$ (solid lines) and $\phi_9$ (dashed lines). Though they trace the same Cartesian curve, experiment 9 ended with a final time less than half of experiment 5. Note that the joint paths in experiment 9 have gentler slopes and less curvature.

Theorems & Definitions (6)

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