Input-Output Feedback Linearization Preserving Task Priority for Multivariate Nonlinear Systems Having Singular Input Gain Matrix

We propose an extension of the input-output feedback linearization for a class of multivariate systems that are not input-output linearizable in a classical manner. The key observation is that the usual input-output linearization problem can be interpreted as the problem of solving simultaneous linear equations associated with the input gain matrix: thus, even at points where the input gain matrix becomes singular, it is still possible to solve a part of linear equations, by which a subset of input-output relations is made linear or close to be linear. Based on this observation, we adopt the task priority-based approach in the input-output linearization problem. First, we generalize the classical Byrnes-Isidori normal form to a prioritized normal form having a triangular structure, so that the singularity of a subblock of the input gain matrix related to lower-priority tasks does not directly propagate to higher-priority tasks. Next, we present a prioritized input-output linearization via the multi-objective optimization with the lexicographical ordering, resulting in a prioritized semilinear form that establishes input output relations whose subset with higher priority is linear or close to be linear. Finally, Lyapunov analysis on ultimate boundedness and task achievement is provided, particularly when the proposed prioritized input-output linearization is applied to the output tracking problem. This work introduces a new control framework for complex systems having critical and noncritical control issues, by assigning higher priority to the critical ones.

Prioritized Inverse Kinematics: Desired Task Trajectories in Nonsingular Task Spaces

A prioritized inverse kinematics (PIK) solution can be considered as a (regulation or output tracking) control law of a dynamical system with prioritized multiple outputs. We propose a method that guarantees that a joint trajectory generated from a class of PIK solutions exists uniquely in a nonsingular configuration space. We start by assuming that desired task trajectories stay in nonsingular task spaces and find conditions for task trajectories to stay in a neighborhood of desired task trajectories in which we can guarantee existence and uniqueness of a joint trajectory in a nonsingular configuration space. Based on this result, we find a sufficient condition for task convergence and analyze various stability notions such as stability, uniform stability, uniform asymptotic stability, and exponential stability in both continuous and discrete times. We discuss why the number of tasks is limited in discrete time and show how preconditioning can be used in order to overcome this limitation.

Prioritized Inverse Kinematics: Nonsmoothness, Trajectory Existence, Task Convergence, Stability

In this paper, we study various theoretical properties of a class of prioritized inverse kinematics (PIK) solutions that can be considered as a class of (regulation or output tracking) control laws of a dynamical system with prioritized multiple outputs. We first develop tools to investigate nonsmoothness of PIK solutions and find a sufficient condition for nonsmoothness. It implies that existence and uniqueness of a joint trajectory satisfying the PIK solution cannot be guaranteed by the classical theorems. So, we construct an alternative existence and uniqueness theorem that uses structural information of the PIK solution. Then, we narrow the class of PIK solutions down to the case that all tasks are designed to follow some desired task trajectories and discover a few properties related to convergence. The study goes further to analyze stability of equilibrium points of the differential equation whose right hand side is the PIK solution when all tasks are designed to reach some desired task positions. Finally, we furnish an example with a two-link manipulator that shows how our findings can be used to analyze the behavior of the joint trajectory generated from the PIK solution.