[Paper Review] Feedback Regularization and Geometric PID Control for Trajectory Tracking of Coupled Mechanical Systems: Hoop Robots on an Inclined Plane

T. W. U. Madhushani, D. H. S. Maithripala+1 more|arXiv (Cornell University)|Sep 30, 2016

Control and Dynamics of Mobile Robots13 references3 citations

TL;DR

This paper proposes a coordinate-independent geometric PID control framework for trajectory tracking in underactuated coupled mechanical systems, specifically hoop robots on an inclined plane. By introducing feedback regularization to transform the system into a mechanical structure, and combining it with geometric PID control, the method achieves almost-semiglobal, locally exponential tracking with robustness to constant disturbances and parameter uncertainties, including unknown surface inclination.

ABSTRACT

This paper applies geometric PID control for asymptotic tracking of a desired trajectory by a hoop robot in the presence of disturbances and uncertainties. The hoop robot, consisting of a circular body rolling without slip along a one-dimensional surface, is a planar analog of a spherical robot. A variety of coupled mechanical system may be used to actuate the hoop robot. This paper specifically considers two different actuators, one a simple pendulum and the other an internal cart. The geometric PID controller requires the plant to be a mechanical system, and the hoop robot does not satisfy this condition. Therefore a geometric inner loop is presented that gives the hoop robot the required structure. This procedure is here referred to as feedback regularization. Feedback regularization--in contrast to feedback linearization--is coordinate independent, and hence reflects the fundamental system structure. Note also that the resulting mechanical system is nonlinear and underactuated. Subsequently, the geometric PID outer loop guarantees almost-semiglobal tracking with locally exponential convergence, and the integral action of the PID guarantees robustness to constant disturbances and parameter uncertainties, including constant inclination of the rolling surface. The complete tracking controller is the composition of the two coordinate-independent loops, and therefore is also coordinate independent.

Motivation & Objective

• To address the challenge of trajectory tracking in underactuated mechanical systems like hoop robots, which are inherently nonlinear and lack full actuation.
• To overcome the limitation that geometric PID control typically requires a mechanical system structure, which the hoop robot does not naturally satisfy.
• To ensure robustness against constant disturbances and parameter uncertainties, including unknown surface inclination, through integral action in the control design.
• To develop a coordinate-independent control framework that preserves system structure and avoids coordinate patch singularities.
• To demonstrate the effectiveness of the approach on two specific actuation mechanisms: a simple pendulum and an internal cart.

Proposed method

• Feedback regularization is applied to transform the hoop robot into a mechanical system by redefining control inputs, ensuring the system satisfies the structural requirements for geometric control.
• The geometric PID controller is designed using the Levi-Civita connection for velocity differentiation, enabling proper formulation of integral action on Riemannian manifolds.
• The controller is decomposed into two coordinate-independent loops: a geometric inner loop (feedback regularization) and an outer loop (geometric PID) for trajectory tracking.
• Lyapunov-based stability analysis is used, with a composite energy-like function $ W_s $ to prove local exponential convergence of the tracking error to zero.
• The analysis accounts for cubic and quadratic velocity terms arising from coupling and Riemannian structure, bounding their influence via gain selection.
• Sufficiently large PID gains are selected to ensure $ ho_{ ext{min}}(Q_s) $ is arbitrarily large, guaranteeing exponential convergence and boundedness of actuator velocities.

Experimental results

Research questions

• RQ1 Can geometric PID control be extended to output tracking in underactuated mechanical systems that are not fully actuated?
• RQ2 How can a non-mechanical system like a hoop robot be transformed into a mechanical system suitable for geometric control?
• RQ3 Can feedback regularization provide a coordinate-independent alternative to feedback linearization for control design?
• RQ4 What conditions ensure almost-semiglobal, locally exponential convergence in the presence of constant disturbances and parameter uncertainties?
• RQ5 How can the controller maintain bounded actuator velocities while ensuring convergence, despite nonlinear coupling and underactuation?

Key findings

• The proposed feedback regularization successfully transforms the hoop robot into a mechanical system, enabling the application of geometric control techniques.
• The geometric PID controller ensures almost-semiglobal, locally exponential convergence of the tracking error to zero, even with bounded parametric uncertainty and constant disturbances.
• The integral action of the PID controller provides robustness to constant disturbances, including unknown surface inclination, without requiring prior knowledge of the inclination angle.
• Sufficiently large PID gains can be selected to make $ ho_{ ext{min}}(Q_s) $ arbitrarily large, ensuring exponential convergence and enabling boundedness of actuator velocities.
• The controller maintains $ ||v_a(t)|| < k_a $ for all $ t > 0 $, provided initial conditions and gains are chosen appropriately, ensuring stability and convergence.
• The closed-loop system converges to the largest invariant set where $ \dot{W}_s \equiv 0 $, which corresponds to the desired equilibrium point $ (\bar{y}_0, 0) $, confirming asymptotic tracking.