[Paper Review] Prioritized Inverse Kinematics: Nonsmoothness, Trajectory Existence, Task Convergence, Stability

Sang-ik An, Dongheui Lee|arXiv (Cornell University)|May 9, 2019

Robotic Mechanisms and Dynamics35 references4 citations

TL;DR

This paper establishes theoretical foundations for prioritized inverse kinematics (PIK) by analyzing nonsmoothness, trajectory existence, task convergence, and stability. It introduces an alternative existence and uniqueness theorem for joint trajectories under PIK solutions, proves task convergence without assuming constant desired trajectories, and demonstrates asymptotic stability of equilibrium points, enabling reliable control design for robotic systems with prioritized tasks.

ABSTRACT

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 (output regulation or 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 a PIK solution cannot be guaranteed by the classical theorems. So, we construct an alternative existence and uniqueness theorem that uses structural information of PIK solutions. 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 task convergence. The study goes further to analyze stability of equilibrium points of the differential equation whose right hand side is a 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 a joint trajectory generated from a PIK solution.

Motivation & Objective

• To address theoretical gaps in prioritized inverse kinematics (PIK), particularly regarding nonsmoothness and trajectory existence.
• To establish conditions under which joint trajectories exist and are unique despite nonsmooth PIK solutions.
• To analyze task convergence when tasks are designed to follow time-varying desired trajectories, not just fixed positions.
• To investigate the stability of equilibrium points in the differential equation defined by a PIK solution.
• To provide a theoretical framework that supports reliable design of PIK-based control laws for robotic systems.

Proposed method

• Developed tools to analyze nonsmoothness of PIK solutions by relaxing the representation property of objective functions.
• Proposed an alternative existence and uniqueness theorem using structural information of PIK solutions, bypassing classical theorems like Peano’s.
• Formulated PIK solutions as control laws for dynamical systems with prioritized outputs, enabling analysis of trajectory behavior.
• Analyzed task convergence under time-varying desired trajectories using integral inequalities and limit arguments.
• Established asymptotic stability of equilibrium points via Lyapunov-like analysis, proving convergence to desired task positions.
• Validated findings using a two-link manipulator example, demonstrating trajectory existence, convergence, and stability under nonsmooth PIK.

Experimental results

Research questions

• RQ1 Under what conditions is a PIK solution nonsmooth, and how can this be formally characterized?
• RQ2 Can joint trajectories satisfying a PIK solution be guaranteed to exist and be unique when classical theorems fail?
• RQ3 Do PIK solutions ensure task convergence when tasks follow time-varying desired trajectories, and for which feedback gains?
• RQ4 Are equilibrium points of the PIK-induced differential equation asymptotically stable when tasks aim for fixed desired positions?
• RQ5 How can theoretical findings be applied to ensure reliable behavior in practical robotic systems?

Key findings

• A sufficient condition for nonsmoothness of PIK solutions is derived, showing that classical existence theorems cannot guarantee trajectory existence.
• An alternative existence and uniqueness theorem is constructed using structural properties of PIK solutions, ensuring trajectory existence even when PIK is nonsmooth.
• Task convergence to time-varying desired trajectories is proven for all positive feedback gains, without requiring constant desired trajectories.
• When tasks aim for fixed desired positions, the equilibrium points of the PIK-induced differential equation are asymptotically stable.
• The limit of the joint trajectory converges to a point in the feasible set only if the final configuration satisfies the kinematic constraint, ensuring task error convergence.
• The two-link manipulator example confirms that trajectory existence, task convergence, and stability can be analytically guaranteed using the proposed framework.