Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generation of Conservative Dynamical Systems Based on Stiffness Encoding

Tengyu Hou, Hanming Bai, Ye Ding, Han Ding

TL;DR

A stiffness encoding framework to modulate DS properties by embedding specific stiffnesses is proposed, which effectively slows down the decay rate of energy in the energy tank and improves the stability margin of the control system.

Abstract

Dynamical systems (DSs) provide a framework for high flexibility, robustness, and control reliability and are widely used in motion planning and physical human-robot interaction. The properties of the DS directly determine the robot's specific motion patterns and the performance of the closed-loop control system. In this paper, we establish a quantitative relationship between stiffness properties and DS. We propose a stiffness encoding framework to modulate DS properties by embedding specific stiffnesses. In particular, from the perspective of the closed-loop control system's passivity, a conservative DS is learned by encoding a conservative stiffness. The generated DS has a symmetric attraction behavior and a variable stiffness profile. The proposed method is applicable to demonstration trajectories belonging to different manifolds and types (e.g., closed and self-intersecting trajectories), and the closed-loop control system is always guaranteed to be passive in different cases. For controllers tracking the general DS, the passivity of the system needs to be guaranteed by the energy tank. We further propose a generic vector field decomposition strategy based on conservative stiffness, which effectively slows down the decay rate of energy in the energy tank and improves the stability margin of the control system. Finally, a series of simulations in various scenarios and experiments on planar and curved motion tasks demonstrate the validity of our theory and methodology.

Generation of Conservative Dynamical Systems Based on Stiffness Encoding

TL;DR

A stiffness encoding framework to modulate DS properties by embedding specific stiffnesses is proposed, which effectively slows down the decay rate of energy in the energy tank and improves the stability margin of the control system.

Abstract

Dynamical systems (DSs) provide a framework for high flexibility, robustness, and control reliability and are widely used in motion planning and physical human-robot interaction. The properties of the DS directly determine the robot's specific motion patterns and the performance of the closed-loop control system. In this paper, we establish a quantitative relationship between stiffness properties and DS. We propose a stiffness encoding framework to modulate DS properties by embedding specific stiffnesses. In particular, from the perspective of the closed-loop control system's passivity, a conservative DS is learned by encoding a conservative stiffness. The generated DS has a symmetric attraction behavior and a variable stiffness profile. The proposed method is applicable to demonstration trajectories belonging to different manifolds and types (e.g., closed and self-intersecting trajectories), and the closed-loop control system is always guaranteed to be passive in different cases. For controllers tracking the general DS, the passivity of the system needs to be guaranteed by the energy tank. We further propose a generic vector field decomposition strategy based on conservative stiffness, which effectively slows down the decay rate of energy in the energy tank and improves the stability margin of the control system. Finally, a series of simulations in various scenarios and experiments on planar and curved motion tasks demonstrate the validity of our theory and methodology.

Paper Structure

This paper contains 35 sections, 6 theorems, 65 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Learning DS from demonstrations
  3. Research on impedance control
  4. Contribution
  5. Related work
  6. Preliminaries
  7. Impedance Control
  8. Stiffness Encoding
  9. Passivity Analysis
  10. Gaussian Processes
  11. Generation of Conservative DS
  12. Generation of the Integral Path
  13. Generation of the Integral Path
  14. Construction of the Potential Function
  15. Solution of the Optimization Problem
  16. ...and 20 more sections

Key Result

Lemma 1

The conservativeness of the stiffness matrix ${\mathbf{K}}({\boldsymbol{\xi }})$ in a linear space is equivalent to the simultaneous satisfaction of symmetry and exactness: where ${k_{ij}}$ means the $i$th row and $j$th column of ${\mathbf{K}}$, and ${\partial \Sigma }$ is an arbitrary closed path.

Figures (23)

  • Figure 1: Schematic diagram of the stiffness encoding. The color of each point in the figure indicates the value of the angular velocity of the vector field at the corresponding point. When encoding a specific stiffness, the corresponding DS exhibits particular properties. (1) When the stiffness matrix satisfies the exactness, a well-defined DS is guaranteed, i.e., a continuous vector field in space. (2) The DS is conservative when the stiffness is exact and symmetric. Conservativeness is shown in the graph as a constant angular velocity of zero. In the figure, we show a conservative DS with symmetric attraction behavior, and it is not contractive. (3) For the DS to be contractive, the encoded stiffness must satisfy both exactness and negative definiteness. It can be seen that all trajectories of the DS converge to each other. (4) The DS is conservative and contractive when the stiffness matrix is exact, symmetric, and negative definite.
  • Figure 2: Schematic of integral path generation in Section \ref{['subsubsec:Step1']}. (a)-(c) show the construction process of ${V_1}\left( {\boldsymbol{\xi }} \right)$. The red dots represent demonstration trajectories. (d) shows ${{\mathbf{f}}_1}({\boldsymbol{\xi }}) = - \nabla {V_1}({\boldsymbol{\xi }})$, where the black curve is the integration path from the average start point of the demonstration trajectories. (a) The initial function ${V_0}\left( {\boldsymbol{\xi }} \right)$. (b) The notch function ${V_{\text{tank}}}\left( {\boldsymbol{\xi }} \right)$. (c) The composite function ${V_1}\left( {\boldsymbol{\xi }} \right)$. (d) The vector field ${{\mathbf{f}}_1}({\boldsymbol{\xi }})$ corresponding to ${V_1}\left( {\boldsymbol{\xi }} \right)$.
  • Figure 3: Schematic of GP-Ramp function ${V_2}\left( {\boldsymbol{\xi }} \right)$ generation in Section \ref{['subsubsubsec:Step21']}. The green dots in (a) denote the expanded point set ${\boldsymbol{X}_{{\text{SimAll}}}}$ and the black line denotes the integration path. (b) and (c) are the essential functions that make up (d). (a) The expanded point set ${\boldsymbol{X}_{{\text{SimAll}}}}$. (b) The function ${V_{21}}\left( {\boldsymbol{\xi }} \right)$. (c) The modulation function ${V_{22}}\left( {\boldsymbol{\xi }} \right)$. (d) The GP-Ramp function ${V_{2}}\left( {\boldsymbol{\xi }} \right)$.
  • Figure 4: Simulation of DS generation on the L-shaped trajectory. The process and details of the DS generation based on GP are displayed in (a), (b) and (c). The red dots represent demonstration trajectories. (a) The total potential function ${V_p}$ based on GP. (b) DS ${\boldsymbol{\dot \xi }} = {{\mathbf{f}}_p}\left( {{\boldsymbol{\xi }},{{\boldsymbol{\theta }}_2},{\mathbf{y}}} \right)$ based on GP. (c) The velocity comparison. The actual velocity represents the result of learning through the GP.
  • Figure 5: Schematic diagram of the iterative optimization process. Update ${\mathbf{y}}$ by solving the QCQP optimization problem ${{\mathbf{P}}_1}$ (\ref{['opt:optim1']}) and subsequently update ${\boldsymbol{\theta} _2}$ by solving the optimization problem (\ref{['opt:optim2']}).
  • ...and 18 more figures

Theorems & Definitions (6)

  • Lemma 1: Conservative stiffness matrix hou2024conserstiff
  • Lemma 2: Contraction theory lohmillerContractionAnalysisNonlinear1998
  • Lemma 3: Kronan2016passinter
  • Proposition 1
  • Lemma 4: Consistency of stiffness conservativeness hou2024conserstiff
  • Proposition 2