This paper examines the control continuity in hierarchical task-space controllers. While the continuity is ensured for any a priori fixed number of tasks-even in ill-conditioned configurations-, the control resulting from a hierarchical stack-of-task computation may not be continuous under some discrete events. In particular, we study how the continuity of the stack-of-task control computation is affected under discreet scheduling operations such as on-the-fly priority switching between tasks, or tasks insertion and removal, which changes the number of tasks in the stack controller. Different ways to formulate a hierarchy of tasks are presented together with their continuity properties, which is thoroughly analyzed under such discreet scheduling operations.

This article presents a comparison between dense and sparse model predictive control (MPC) formulations, in the context of walking motion generation for humanoid robots. The former formulation leads to smaller, the latter one to larger but more structured optimization problem. We put an accent on the sparse formulation and point out a number of advantages that it presents. In particular, motion generation with variable center of mass (CoM) height, as well as variable discretization of the preview window, come at a negligible additional computational cost. We present a sparse formulation that comprises a diagonal Hessian matrix and has only simple bounds (while still retaining the possibility to generate motions for an omnidirectional walk). Finally, we present the results from a customized code used to solve the underlying quadratic program (QP).