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Limited Information Shared Control: A Potential Game Approach

Balint Varga, Jairo Inga, Soeren Hohmann

TL;DR

The proposed systematic design uses a novel class of games to model human–machine interaction: the near potential differential games (NPDG), and provides a necessary and sufficient condition for the existence of an NPDG and derive an algorithm for finding a N PDG, which completely describes a given differential game.

Abstract

This paper presents a systematic method for the design of a limited information shared control (LISC). LISC is used in applications where not all system states or reference trajectories are measurable by the automation. Typical examples are partially human-controlled systems, in which some subsystems are fully controlled by automation while others are controlled by a human. The proposed systematic design method uses a novel class of games to model human-machine interaction: the near potential differential games (NPDG). We provide a necessary and sufficient condition for the existence of an NPDG and derive an algorithm for finding a NPDG that completely describes a given differential game. The proposed design method is applied to the control of a large vehicle-manipulator system, in which the manipulator is controlled by a human operator and the vehicle is fully automated. The suitability of the NPDG to model differential games is verified in simulations, leading to a faster and more accurate controller design compared to manual tuning. Furthermore, the overall design process is validated in a study with sixteen test subjects, indicating the applicability of the proposed concept in real applications.

Limited Information Shared Control: A Potential Game Approach

TL;DR

The proposed systematic design uses a novel class of games to model human–machine interaction: the near potential differential games (NPDG), and provides a necessary and sufficient condition for the existence of an NPDG and derive an algorithm for finding a N PDG, which completely describes a given differential game.

Abstract

This paper presents a systematic method for the design of a limited information shared control (LISC). LISC is used in applications where not all system states or reference trajectories are measurable by the automation. Typical examples are partially human-controlled systems, in which some subsystems are fully controlled by automation while others are controlled by a human. The proposed systematic design method uses a novel class of games to model human-machine interaction: the near potential differential games (NPDG). We provide a necessary and sufficient condition for the existence of an NPDG and derive an algorithm for finding a NPDG that completely describes a given differential game. The proposed design method is applied to the control of a large vehicle-manipulator system, in which the manipulator is controlled by a human operator and the vehicle is fully automated. The suitability of the NPDG to model differential games is verified in simulations, leading to a faster and more accurate controller design compared to manual tuning. Furthermore, the overall design process is validated in a study with sixteen test subjects, indicating the applicability of the proposed concept in real applications.
Paper Structure (31 sections, 2 theorems, 47 equations, 8 figures, 4 tables)

This paper contains 31 sections, 2 theorems, 47 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modelling Shared Controls by Differential Games
  4. Concept of the Limited Information Shared Control
  5. Exact LQ Potential Differential Games
  6. Design Problem and the Solution Steps
  7. Near Potential Differential Games
  8. Distance to LQ NPDG
  9. Dynamics of the LQ NPDGs
  10. Method for Finding a LQ NPDG
  11. Systematic Design of LISC by means of NPDG
  12. Design of the CS
  13. Design of a LISC
  14. Verification of the Design
  15. Model of the vehicle-manipulator
  16. ...and 16 more sections

Key Result

Lemma 1

Let an LQ differential game with NE state trajectories (eq:sol_NE) be given. Furthermore, let a PDG with NE state trajectories (eq:sol_pot) be given. The game $\Gamma_d$ is a LQ-NPDG, if holds, where ${\boldsymbol{x}_\mathrm{max} = \mathrm{max }\left(\left\lVert\boldsymbol{x}^*(t)\right\rVert_2, \left\lVert \boldsymbol{x}^{(p)}(t)\right\rVert_2\right)}$ is the maximum magnitude of the state vecto

Figures (8)

  • Figure 1: Illustration of the design steps.
  • Figure 2: An example of a vehicle-manipulator 2019_ModelPredictiveControl_varga
  • Figure 3: The resulting noisy trajectories of the original game (OG), with solid lines and the NPDG (PG) with dashed lines with SNR$=5\,$dB.
  • Figure 4: The dynamics of Hamiltonian functions, the blue solid lines are the right side of (\ref{['eq:def_e_pot']}) with SNR$=5\,$dB and the purple dashed lines are the left side of (\ref{['eq:def_e_pot']}).
  • Figure 5: Picture of the test bench with the GUI and the joystick
  • ...and 3 more figures

Theorems & Definitions (11)

  • Definition 1: Cooperation state 2020_ValidationCooperativeSharedControl_vargab
  • Definition 2: Exact LQ PDG
  • Definition 3: Differential Distance
  • Definition 4: Near Potential Differential Game
  • Lemma 1: LQ-NPDG
  • proof
  • Definition 5: Closed-Loop System Matrix Error
  • Lemma 2: Boundedness of NPDGs
  • proof
  • Remark
  • ...and 1 more