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Motion planning for manipulators under task space constraints is difficult asit constrains the joint configurations to always lie on an implicitly definedmanifold. It is possible to view task constrained motion planning as anoptimization problem with non-linear equality constraints which can be solvedby general non-linear optimization techniques. In this paper, we present anovel custom optimizer which exploits the underlying structure present in manytask constraints.  At the core of our approach are some simple reformulations, which whencoupled with the \emph{method of alternating projection}, leads to an efficientconvex optimization based routine for computing a feasible solution to the taskconstraints. We subsequently build on this result and use the concept ofAugmented Lagrangian to guide the feasible solutions towards those which alsominimize the user defined cost function. We show that the proposed optimizer isfully distributive and thus, can be easily parallelized. We validate ourformulation on some common robotic benchmark problems. In particular, we showthat the proposed optimizer achieves cyclic motion in the joint spacecorresponding to a similar nature trajectory in the task space. Furthermore, asa baseline, we compare the proposed optimizer with an off-the-shelf non-linearsolver provide in open source package SciPy. We show that for similar taskconstraint residuals and smoothness cost, it can be upto more than three timesfaster than the SciPy alternative.

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