Combining Method of Alternating Projections and Augmented Lagrangian for Task Constrained Trajectory Optimization

Motion planning for manipulators under task space constraints is difficult as it constrains the joint configurations to always lie on an implicitly defined manifold. It is possible to view task constrained motion planning as an optimization problem with non-linear equality constraints which can be solved by general non-linear optimization techniques. In this paper, we present a novel custom optimizer which exploits the underlying structure present in many task constraints. At the core of our approach are some simple reformulations, which when coupled with the \emph{method of alternating projection}, leads to an efficient convex optimization based routine for computing a feasible solution to the task constraints. We subsequently build on this result and use the concept of Augmented Lagrangian to guide the feasible solutions towards those which also minimize the user defined cost function. We show that the proposed optimizer is fully distributive and thus, can be easily parallelized. We validate our formulation on some common robotic benchmark problems. In particular, we show that the proposed optimizer achieves cyclic motion in the joint space corresponding to a similar nature trajectory in the task space. Furthermore, as a baseline, we compare the proposed optimizer with an off-the-shelf non-linear solver provide in open source package SciPy. We show that for similar task constraint residuals and smoothness cost, it can be upto more than three times faster than the SciPy alternative.