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Data-Driven Optimal Control of Tethered Space Robot Deployment with Learning Based Koopman Operator

Ao Jin, Fan Zhang, Panfeng Huang

TL;DR

Simulation results verify the effectiveness of proposed data-driven optimal control framework with an improved deep learning based Koopman operator and show that it could deploy tethered space robot more quickly with less swing of in-plane angle.

Abstract

To avoid complex constraints of the traditional nonlinear method for tethered space robot (TSR) deployment, this paper proposes a data-driven optimal control framework with an improved deep learning based Koopman operator that could be applied to complex environments. In consideration of TSR's nonlinearity, its finite dimensional lifted representation is derived with the state-dependent only embedding functions in the Koopman framework. A deep learning approach is adopted to approximate the global linear representation of TSR. Deep neural networks (DNN) are developed to parameterize Koopman operator and its embedding functions. An auxiliary neural network is developed to encode the nonlinear control term of finite dimensional lifted system. In addition, the state matrix A and control matrix B of lifted linear system in the embedding space are also estimated during training DNN. Then three loss functions that related to reconstruction and prediction ability of network and controllability of lifted linear system are designed for training the entire network. With the global linear system produced from DNN, Linear Quadratic Regulator (LQR) is applied to derive the optimal control policy for the TSR deployment. Finally, simulation results verify the effectiveness of proposed framework and show that it could deploy tethered space robot more quickly with less swing of in-plane angle.

Data-Driven Optimal Control of Tethered Space Robot Deployment with Learning Based Koopman Operator

TL;DR

Simulation results verify the effectiveness of proposed data-driven optimal control framework with an improved deep learning based Koopman operator and show that it could deploy tethered space robot more quickly with less swing of in-plane angle.

Abstract

To avoid complex constraints of the traditional nonlinear method for tethered space robot (TSR) deployment, this paper proposes a data-driven optimal control framework with an improved deep learning based Koopman operator that could be applied to complex environments. In consideration of TSR's nonlinearity, its finite dimensional lifted representation is derived with the state-dependent only embedding functions in the Koopman framework. A deep learning approach is adopted to approximate the global linear representation of TSR. Deep neural networks (DNN) are developed to parameterize Koopman operator and its embedding functions. An auxiliary neural network is developed to encode the nonlinear control term of finite dimensional lifted system. In addition, the state matrix A and control matrix B of lifted linear system in the embedding space are also estimated during training DNN. Then three loss functions that related to reconstruction and prediction ability of network and controllability of lifted linear system are designed for training the entire network. With the global linear system produced from DNN, Linear Quadratic Regulator (LQR) is applied to derive the optimal control policy for the TSR deployment. Finally, simulation results verify the effectiveness of proposed framework and show that it could deploy tethered space robot more quickly with less swing of in-plane angle.
Paper Structure (14 sections, 1 theorem, 25 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 1 theorem, 25 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. System Dynamics and Koopman Operator
  3. Dynamics of Tether Space Robot
  4. Preliminaries on Koopman Operator
  5. Koopman Operator Learning and Controller Design
  6. A Data Driven Framework for learning Koopman Operator
  7. Network Performance Index
  8. Linear Quadratic Regulator
  9. Simulation and Results
  10. Data Sampling
  11. Network Parameters Setting
  12. Accuracy of reconstruction and prediction
  13. Control Simulation for TSR deployment
  14. CONCLUSION

Key Result

Theorem 1

2022arXiv220712132C Given a nonlinear system (koopman) and a state-dependent only observable ${\Phi}$ where ${\Phi} (f(x,0)) \in span \{{\Phi}\}$ with ${\Phi}: X \rightarrow \mathbb{R}^K$ and $X$ is convex, then an exact finite dimensional lifted form of the nonlinear system is defined as: with $A\in \mathbb{R}^{K \times K}$ and

Figures (11)

  • Figure 1: Schematic diagram of tether space robot
  • Figure 2: Illustration of Koopman operator for nonlinear system with control input
  • Figure 3: Framework for learning Koopman operator.
  • Figure 4: Schematic of optimal control for TSR deployment using deep learning based Koopman operator
  • Figure 5: The framework of data sampling using multi-thread
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1