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Reduction of Velocity-Dependent Terms in Total Energy Shaping Approach

M. Reza J. Harandi, Mehrzad Namvar

TL;DR

The paper tackles actuator constraints in stabilization of underactuated mechanical systems using IDA-PBC, highlighting kinetic-energy shaping terms as a key practical bottleneck. It introduces a simultaneous IDA-PBC (SIDA-PBC) framework that designs the free generalized forces through an $\ell_\infty$-norm optimization to minimize kinetic-energy shaping, without altering the matching PDEs, enabling applicability to single-actuator systems. The authors derive an analytical, closed-form solution for the optimization, facilitating construction of the free-forces matrix and demonstrating substantial reductions in peak actuation in both simulations on a Pendubot and experiments on a 3-DOF haptic device. This work broadens the practical reach of energy-shaping control under actuator constraints, offering implementable gains with no significant computational burden and preserving the stability structure of IDA-PBC.

Abstract

Total energy shaping through interconnection and damping assignment passivity-based control (IDA-PBC) provides a powerful and systematic framework for stabilizing underactuated mechanical systems. Despite its theoretical appeal, incorporating actuator limitations into total energy shaping remains a largely open problem, with only limited results reported in the existing literature. In practice, the closed-loop behavior of energy-shaping controllers is strongly affected by the kinetic energy shaping terms. In this paper, a simultaneous IDA-PBC (SIDA-PBC) framework is employed to systematically attenuate the kinetic energy shaping terms by exploiting generalized forces, without altering the matching partial differential equations (PDEs). The free component of the generalized forces is derived analytically via an $\ell_\infty$-norm optimization formulation. Although a reduction in kinetic energy shaping terms does not necessarily guarantee a decrease in the overall control effort, the proposed approach effectively suppresses kinetic energy shaping components and achieves a reduced control magnitude whenever such a reduction is structurally feasible. Unlike existing approaches based on gyroscopic terms, which require multiple actuators, the proposed method is applicable to mechanical systems with a single actuator. Simulation and experimental results are provided to validate the effectiveness of the proposed approach.

Reduction of Velocity-Dependent Terms in Total Energy Shaping Approach

TL;DR

The paper tackles actuator constraints in stabilization of underactuated mechanical systems using IDA-PBC, highlighting kinetic-energy shaping terms as a key practical bottleneck. It introduces a simultaneous IDA-PBC (SIDA-PBC) framework that designs the free generalized forces through an -norm optimization to minimize kinetic-energy shaping, without altering the matching PDEs, enabling applicability to single-actuator systems. The authors derive an analytical, closed-form solution for the optimization, facilitating construction of the free-forces matrix and demonstrating substantial reductions in peak actuation in both simulations on a Pendubot and experiments on a 3-DOF haptic device. This work broadens the practical reach of energy-shaping control under actuator constraints, offering implementable gains with no significant computational burden and preserving the stability structure of IDA-PBC.

Abstract

Total energy shaping through interconnection and damping assignment passivity-based control (IDA-PBC) provides a powerful and systematic framework for stabilizing underactuated mechanical systems. Despite its theoretical appeal, incorporating actuator limitations into total energy shaping remains a largely open problem, with only limited results reported in the existing literature. In practice, the closed-loop behavior of energy-shaping controllers is strongly affected by the kinetic energy shaping terms. In this paper, a simultaneous IDA-PBC (SIDA-PBC) framework is employed to systematically attenuate the kinetic energy shaping terms by exploiting generalized forces, without altering the matching partial differential equations (PDEs). The free component of the generalized forces is derived analytically via an -norm optimization formulation. Although a reduction in kinetic energy shaping terms does not necessarily guarantee a decrease in the overall control effort, the proposed approach effectively suppresses kinetic energy shaping components and achieves a reduced control magnitude whenever such a reduction is structurally feasible. Unlike existing approaches based on gyroscopic terms, which require multiple actuators, the proposed method is applicable to mechanical systems with a single actuator. Simulation and experimental results are provided to validate the effectiveness of the proposed approach.
Paper Structure (6 sections, 1 theorem, 42 equations, 8 figures, 1 table)

This paper contains 6 sections, 1 theorem, 42 equations, 8 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Overview of SIDA-PBC
  3. Main Results
  4. Simulation Results
  5. Experimental Results
  6. Conclusion

Key Result

Theorem 1

Consider the following optimization problem in which $A\in\mathbb{R}^{n\times n}, x\in\mathbb{R}^n, b\in\mathbb{R}^n$ and $x\neq 0$. The solution of (opt) is $A^*=A_{s}^*+A_w^*$ in which the symmetric matrix $A_{s}^*$ and the skew-symmetric matrix $A_w^*$ are as follows where the $i$-th element of $\xi^*\in\mathbb{R}^n$ is:

Figures (8)

  • Figure 1: Schematic of the Pendubot.
  • Figure 2: Configuration variables of the Pendubot under the three controllers. Both $q_1$ and $q_2$ converge toward the reference values corresponding to the upright position.
  • Figure 3: Comparison of the kinetic energy shaping terms for the three controllers. According to Theorem \ref{['thm:linf_opt']}, $u_{ov-ki}$ remains almost zero throughout the response, whereas under (\ref{['red']}) it is zero only during the initial moments.
  • Figure 4: Control signals generated by the three controllers. The control input resulting from the control law in (\ref{['red']}) exhibits a smaller magnitude compared to the others.
  • Figure 5: Schematic of the Geomagic Touch robot.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3