Convergence of trajectory splitting in general non-convex settings

Establish theoretical convergence guarantees for the ADMM-based trajectory splitting algorithm that splits a robot trajectory into multiple segments and enforces continuity via consensus variables and augmented Lagrangian updates, in the general non-convex multi-jointed robotic motion planning setting characterized by nonlinear forward kinematics and semi-convex signed-distance collision-avoidance constraints.

Background

The paper proposes a distributed trajectory optimization method that splits a trajectory into segments solved in parallel and then enforces continuity via ADMM-style consensus updates. Convergence is guaranteed for convex cases and under restricted prox-regular conditions, but multi-joint robotic motion planning with nonlinear forward kinematics falls outside these regimes.

In the general non-convex case relevant to multi-jointed robots, the authors note that convergence results for ADMM remain an active area of research and explicitly state that there is no theoretical proof of convergence for their trajectory splitting algorithm, despite empirical evidence of reasonable convergence in simulations and experiments. Establishing formal guarantees would close this gap.