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Adaptive Ankle Torque Control for Bipedal Humanoid Walking on Surfaces with Unknown Horizontal and Vertical Motion

Jacob Stewart, I-Chia Chang, Yan Gu, Petros A. Ioannou

Abstract

Achieving stable bipedal walking on surfaces with unknown motion remains a challenging control problem due to the hybrid, time-varying, partially unknown dynamics of the robot and the difficulty of accurate state and surface motion estimation. Surface motion imposes uncertainty on both system parameters and non-homogeneous disturbance in the walking robot dynamics. In this paper, we design an adaptive ankle torque controller to simultaneously address these two uncertainties and propose a step-length planner to minimize the required control torque. Typically, an adaptive controller is used for a continuous system. To apply adaptive control on a hybrid system such as a walking robot, an intermediate command profile is introduced to ensure a continuous error system. Simulations on a planar bipedal robot, along with comparisons against a baseline controller, demonstrate that the proposed approach effectively ensures stable walking and accurate tracking under unknown, time-varying disturbances.

Adaptive Ankle Torque Control for Bipedal Humanoid Walking on Surfaces with Unknown Horizontal and Vertical Motion

Abstract

Achieving stable bipedal walking on surfaces with unknown motion remains a challenging control problem due to the hybrid, time-varying, partially unknown dynamics of the robot and the difficulty of accurate state and surface motion estimation. Surface motion imposes uncertainty on both system parameters and non-homogeneous disturbance in the walking robot dynamics. In this paper, we design an adaptive ankle torque controller to simultaneously address these two uncertainties and propose a step-length planner to minimize the required control torque. Typically, an adaptive controller is used for a continuous system. To apply adaptive control on a hybrid system such as a walking robot, an intermediate command profile is introduced to ensure a continuous error system. Simulations on a planar bipedal robot, along with comparisons against a baseline controller, demonstrate that the proposed approach effectively ensures stable walking and accurate tracking under unknown, time-varying disturbances.

Paper Structure

This paper contains 16 sections, 3 theorems, 23 equations, 4 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. METHOD
  4. Discrete Footstep Planner Based on Commanded State
  5. Continuous Stabilizing Controller Based on Actual State
  6. Continuous Adaptive Controller
  7. Observer design
  8. Compensator design
  9. Parameter estimation
  10. CASE STUDY
  11. Full-Order Model of Robot Dynamics
  12. High-Level Controller
  13. Mid-Level Motion Generation
  14. Low-Level Controller
  15. Validation Results
  16. ...and 1 more sections

Key Result

Lemma 1

The step lengths chosen according to (LQRsteplengthcontrollaw) drives $\mathbf{e}^c(t) \to 0$ and $\mathbf{x}^c(t)\to \mathbf{x}^d(t)$ as $t\to\infty$.

Figures (4)

  • Figure 1: (a) A linear inverted pendulum model on a platform moving along the $x$ and $z$ axes of the world frame {$w$}. (b) A seven-link robot whose state is $\mathbf{q} = [\mathbf{x}_{base}^T, \mathbf{q}_{j}^T]^T$, with $\mathbf{x}_{base} = [x_b, z_b, \theta_b]^T$ and $\mathbf{q}_j = [q_{1L}, q_{2L}, q_{3L}, q_{1R}, q_{2R}, q_{3R}]^T$. The variables $x_b$, $z_b$, $\theta_b$, and the elements of $\mathbf{q}_j$ are defined as illustrated in subplot (b).
  • Figure 2: Performance of the HT-LIP (blue), PD+FF (red), and proposed (black) controllers in the absence of a disturbance under Case 1.
  • Figure 3: Performance of the HT-LIP (blue), PD+FF (red), and proposed (black) controllers for small amplitude disturbances under Case 2.
  • Figure 4: Performance of the HT-LIP (blue), PD+FF (red), and proposed (black) controllers for time-varying disturbances under Case 3.

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof : Proof of Lemma \ref{['lem:stepCtrl']}
  • proof : Proof of Lemma \ref{['lem:stable']}
  • proof : Proof of Theorem \ref{['thm:adaptiveCtrlPerformance']}