Hierarchical Tracking Control With Arbitrary Task Dimensions: Application to Trajectory Tracking on Submanifolds

Hierarchical impedance control has been recently shown to effectively allow trajectory tracking, while guaranteeing the order of priorities during the execution. Nevertheless, the choice of the tasks is required to be such that, after being properly decoupled, they are all feasible and lead to an invertible Jacobian matrix. In this work, a modification is proposed that removes both these restrictions. The user is free to specify as many tasks as desired and especially without necessarily guaranteeing in advance that none of the tasks will become singular during the execution. Whenever tasks with higher priority use-up all the degrees of freedom, all the other tasks are naturally ignored. Still, as soon as some of the tasks with higher priority become singular, then the freed-up controllability is used to execute the next task in the stack. This is realized automatically, without any rearrangement of the tasks in the priority stack. As an application, the case of trajectory tracking on a submanifold of the workspace is considered, in which multiple charts of the atlas are used for the tasks. Simulations are used to validate the stability analysis.