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Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability

Federico Girlanda, Lasse Shala, Shivesh Kumar, Frank Kirchner

TL;DR

The paper tackles robust co-design for underactuated systems by jointly optimizing design parameters, nominal trajectories, and stabilizing controllers. It introduces RTC-D, a bi-level, CMA-ES-based framework where an inner RTC layer optimizes a DIRTRAN trajectory and a TVLQR controller for fixed design, while an outer CMA-ES layer adjusts design variables to maximize the time-varying region of attraction (ROA) volume, using SOS or simulation-based ROA estimation. The approach yields substantial increases in ROA volume on a cart-pole and a torque-limited pendulum, with both simulation and real-world experiments validating improved robustness to disturbances. This framework enables explicit stability guarantees via SOS-based ROA for simpler systems and provides a scalable pathway toward robust, design-aware control of underactuated platforms in practical settings.

Abstract

Optimal behaviours of a system to perform a specific task can be achieved by leveraging the coupling between trajectory optimization, stabilization, and design optimization. This approach is particularly advantageous for underactuated systems, which are systems that have fewer actuators than degrees of freedom and thus require for more elaborate control systems. This paper proposes a novel co-design algorithm, namely Robust Trajectory Control with Design optimization (RTC-D). An inner optimization layer (RTC) simultaneously performs direct transcription (DIRTRAN) to find a nominal trajectory while computing optimal hyperparameters for a stabilizing time-varying linear quadratic regulator (TVLQR). RTC-D augments RTC with a design optimization layer, maximizing the system's robustness through a time-varying Lyapunov-based region of attraction (ROA) analysis. This analysis provides a formal guarantee of stability for a set of off-nominal states. The proposed algorithm has been tested on two different underactuated systems: the torque-limited simple pendulum and the cart-pole. Extensive simulations of off-nominal initial conditions demonstrate improved robustness, while real-system experiments show increased insensitivity to torque disturbances.

Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability

TL;DR

The paper tackles robust co-design for underactuated systems by jointly optimizing design parameters, nominal trajectories, and stabilizing controllers. It introduces RTC-D, a bi-level, CMA-ES-based framework where an inner RTC layer optimizes a DIRTRAN trajectory and a TVLQR controller for fixed design, while an outer CMA-ES layer adjusts design variables to maximize the time-varying region of attraction (ROA) volume, using SOS or simulation-based ROA estimation. The approach yields substantial increases in ROA volume on a cart-pole and a torque-limited pendulum, with both simulation and real-world experiments validating improved robustness to disturbances. This framework enables explicit stability guarantees via SOS-based ROA for simpler systems and provides a scalable pathway toward robust, design-aware control of underactuated platforms in practical settings.

Abstract

Optimal behaviours of a system to perform a specific task can be achieved by leveraging the coupling between trajectory optimization, stabilization, and design optimization. This approach is particularly advantageous for underactuated systems, which are systems that have fewer actuators than degrees of freedom and thus require for more elaborate control systems. This paper proposes a novel co-design algorithm, namely Robust Trajectory Control with Design optimization (RTC-D). An inner optimization layer (RTC) simultaneously performs direct transcription (DIRTRAN) to find a nominal trajectory while computing optimal hyperparameters for a stabilizing time-varying linear quadratic regulator (TVLQR). RTC-D augments RTC with a design optimization layer, maximizing the system's robustness through a time-varying Lyapunov-based region of attraction (ROA) analysis. This analysis provides a formal guarantee of stability for a set of off-nominal states. The proposed algorithm has been tested on two different underactuated systems: the torque-limited simple pendulum and the cart-pole. Extensive simulations of off-nominal initial conditions demonstrate improved robustness, while real-system experiments show increased insensitivity to torque disturbances.
Paper Structure (15 sections, 9 equations, 5 figures, 1 table)

This paper contains 15 sections, 9 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Co-Design Problem Description
  4. Trajectory Optimization
  5. Trajectory Stabilization
  6. Region of Attraction Estimation
  7. Sum of Squares (SOS)
  8. Simulation based
  9. Proposed Co-design Framework
  10. RTC
  11. RTC-D
  12. Results and Discussion
  13. Linear Inverted Pendulum
  14. Simple pendulum
  15. Conclusion and Outlook

Figures (5)

  • Figure 1: Robust co-optimization for the optimal fitness of the desired motion.
  • Figure 2: RTC-D algorithm scheme.
  • Figure 3: Experimental Systems: Cart Pole (left) and Simple Pendulum (right)
  • Figure 4: Funnel volume increasing due to RTC for Cart-pole (left) and comparison between RTC and RTC-D optimization for Simple pendulum (right).
  • Figure 5: Experimental verification of stability guarantee (green funnel) given by the RTC for Cart-pole (left) and RTC-D for Simple pendulum (right). The optimal configuration (RTC and RTC-D) manages to achieve the desired final stabilization where the initial one does not.