Regularizing Numerical Extremals Along Singular Arcs: A Lie-Theoretic Approach

TL;DR

This paper tackles the challenge of finding time-optimal controls for fully actuated mechanical systems in the presence of singular arcs under the Pontryagin Maximum Principle. It develops a Lie-theoretic framework that derives algebraic, analytic expressions for singular controls and demonstrates how to couple these formulas with general-purpose optimal control software to regularize numerical solutions. The authors show that, for a 2-DOF robotic arm, only one actuator can be singular at a time and provide explicit conditions and expressions for the singular control, yielding trajectories that satisfy the PMP and match results from numerical solvers while removing artifacts. The approach enhances understanding of the global optimal structure and provides a practical method to obtain artifact-free, closed-form controls in minimum-time problems, with potential applicability to more complex robotic systems.

Abstract

Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle, performing better when searching for saturated (bang-bang) solutions. In previous work by one of the authors, singular solutions were shown to exist for the time-optimal control problem for fully actuated mechanical systems under hard torque constraints. Explicit formulas, based on a Lie theoretic analysis of the problem, were given for singular segments of trajectories, but the global structure of solutions remains unknown. In this work, we review the aforementioned framework, and show how to effectively combine these formulas with the use of general-purpose optimal control software packages. By using the explicit formula given by the theory in the intervals where the numerical solution enters a singular arc, we not only obtain an algebraic expression for the control in that interval but we are also able to remove artifacts present in the numerical solution. In this way, the best features of numerical algorithms and theory complement each other and provide a better picture of the global optimal structure. We illustrate the technique on a two degree of freedom robotic arm example, using two distinct optimal control numerical software packages running on different programming languages.

Figures (4)

• Figure 1: Diagram of a 2 link planar robotic manipulator. Gravity is assumed to be orthogonal to the plane of movement of the robot. All indicated parameters are given in Table \ref{['tab:arm2dof_param']}.
• Figure 2: Comparison between a $u_1$-singular extremal (dashed black) and the numerical solutions obtained by GPOPS-II (solid blue) and CASADI (solid red). All solutions reach the target state at basically the same time, and follow the same trajectory for the states. Furthermore, they appear to follow the same profile for the input signals, albeit presenting numerical artifacts in the case of the numerical solutions.
• Figure 3: In this figure we can see in the top plot that GPOPS-II (solid blue) recovers the same ratio $\lambda_2/\lambda_4$ than our $u_1$-singular extremal (dashed red). Furthermore, on the bottom plot we see $\phi_1=\left\langle \lambda,g_1 \right\rangle$ (blue) and $\phi_1'=\left\langle \lambda,fg_1 \right\rangle$ (red). We can see that both signals are very close to zero, but $\dot\phi_1$ present very small oscillations which look to correlate to the peaks present in the signal for $u_1$ in Fig. \ref{['fig:arm2dof_u1singsimul']}.
• Figure 4: In this figure we illustrate how the results from this paper can be used to refine and obtain algebraic expressions for optimal control solutions that pass through singularity arcs. In the top plot we have the control signal for the first joint recovered by GPOPS-II (solid blue), CASADI (solid red), and the one predicted by \ref{['eq:uising_u1cont']} using the costates from GPOPS-II (dashed black). On the bottom plot we have the values of $\phi_1=\left\langle \lambda,g_1 \right\rangle$ (blue) and $\phi_1'=\left\langle \lambda,fg_1 \right\rangle$ (red) computed by the costate values returned by GPOPS-II. Notice that when GPOPS-II converges to a singular solution for $\phi_1$, the recovered signal follows the same profile as the one predicted by theory, except for the numerical artifacts. Furthermore, the solution from CASADI struggles to identify the two distinct intervals.

Theorems & Definitions (6)

• lemma 1: sontag1985remarkssontag1986time
• corollary 1
• proposition 1: sontag1985remarkssontag1986time
• theorem 1: sontag1985remarkssontag1986time
• theorem 2: sontag1985remarkssontag1986time
• theorem 3: sontag1985remarkssontag1986time