Unilateral Constraints for Torque-based Whole-Body Control

Hierarchical Incremental MPC for Redundant Robots: A Robust and Singularity-Free Approach

“…in [16], Cartesian forces calculated using the equivalent Cartesian mass matrix (JM −1 J ) −1 are involved in stage costs, and kinematic singularity occurs when the Jacobian matrix loses its rank. Also, different from the approaches in [14], [15], Hessian matrices of the constrained OCPs are always positive definite (see Lemma 1). This allows to relax the common assumption that singularity-free tasks are defined and gives more flexibility in defining tasks.…”

“…is an intermediate variable for time instance (the same hereinafter), N is the length of the prediction horizon, and • k+j|k stands for the predicted variables, in particular, • k|k := •(k), and x i,k+j+1|k is calculated using the discrete-time incremental system (15). That is, for all j ∈ I [0,N −1] ,…”

“…Without using the null-space projection method, the authors in [14] implement the task hierarchy by solving the cascaded QPs in lexicographic order, while the authors in [15], [16] propose that equality constraints are imposed to guarantee strict task hierarchy. Thus, in [14]- [16], algorithmic singularity is avoided.…”

“…Thus, in [14]- [16], algorithmic singularity is avoided. However, once the Jacobian matrix becomes singular, Hessian matrices of QPs in [14], [15] are no longer positive definite, and kinematic singularity occurs. This is because, for QP solvers, numerical weakness increases with the rise of the condition number of the Hessian matrix.…”

“…Then, similar to (15), let q(k + 1) := col (q(k + 1), q(k)), q := col(q, q) and again a canonical linear equation is obtained:…”