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Bearing-Constrained Leader-Follower Formation of Single-Integrators with Disturbance Rejection: Adaptive Variable-Structure Approaches

Thanh Truong Nguyen, Dung Van Vu, Tuynh Van Pham, Minh Hoang Trinh

Abstract

This paper studies the problem of stabilizing a leader-follower formation specified by a set of bearing constraints and being disturbed by some unknown uniformly bounded disturbance{s}. A set of leaders are positioned at their desired positions, while each follower is modeled by a single integrator with an additive time-varying disturbance. Adaptive variable-structure control laws using displacements or only bearing vectors are provided to stabilize the desired formation. Thanks to the adaptive mechanisms, the proposed control laws require neither information of the bearing Laplacian nor the disturbances' directions and upper bounds. It is further proved that when the leaders are moving with a same bounded uniformly continuous velocity, the moving target formation can still be achieved under the proposed control laws. Simulation results are also given to support the stability analysis.

Bearing-Constrained Leader-Follower Formation of Single-Integrators with Disturbance Rejection: Adaptive Variable-Structure Approaches

Abstract

This paper studies the problem of stabilizing a leader-follower formation specified by a set of bearing constraints and being disturbed by some unknown uniformly bounded disturbance{s}. A set of leaders are positioned at their desired positions, while each follower is modeled by a single integrator with an additive time-varying disturbance. Adaptive variable-structure control laws using displacements or only bearing vectors are provided to stabilize the desired formation. Thanks to the adaptive mechanisms, the proposed control laws require neither information of the bearing Laplacian nor the disturbances' directions and upper bounds. It is further proved that when the leaders are moving with a same bounded uniformly continuous velocity, the moving target formation can still be achieved under the proposed control laws. Simulation results are also given to support the stability analysis.
Paper Structure (17 sections, 9 theorems, 56 equations, 6 figures)

This paper contains 17 sections, 9 theorems, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Bearing rigidity theory
  4. Problem formulation
  5. Displacement-based formation control
  6. Proposed control law
  7. Stability analysis
  8. Bearing-only based formation control
  9. Proposed control law
  10. Stability analysis
  11. Application in formation tracking
  12. Simulation results
  13. Bearing-based formation control with disturbance rejection
  14. Bearing-only formation control with disturbance rejection
  15. Bearing-based formation tracking
  16. ...and 2 more sections

Key Result

Lemma 1

Consider the Problem problem:1. If $\gamma_{ij}(0)>\gamma_0:=\beta \sqrt{\frac{dn}{\lambda_{\min}(\mathbf{L}_{ff}^*)}}$, $\forall (i,j) \in \mathcal{E}$. Under the control law eq:bearing_based_control_law, $\mathbf{p}(t) \to \mathbf{p}^*$ in finite time.

Figures (6)

  • Figure 1: An infinitesimally bearing rigid framework $(\mathcal{G},\mathbf{p}^*)$ in $\mathbb{R}^3$. (a) the graph $\mathcal{G}$; (b) a desired configuration $\mathbf{p}^*$ where $\mathbf{p}_i^*, i=1,\ldots, 20,$ are located at the vertices of a dodecahedron.
  • Figure 2: Simulation 1a: the 20-agent system under the control law \ref{['eq:bearing_based_control_law']}. (a) Trajectories of agents from 0 to 40 seconds (leaders are marked with $\Delta$, followers' initial and final positions ($t=40$ sec) are marked with '${\rm x}$' and '${\rm o}$', respectively); (b) Trajectories of agents from 40 to 80 seconds; (c) Formation's error versus time; (d) A subset of the adaptive gains $\gamma_{ij}$ versus time.
  • Figure 3: Simulation 1b: the 20-agent system under the control law \ref{['eq:bearing_based_control_law2']}. (a) Trajectories of agents from 0 to 30 seconds (leaders are marked with $\Delta$, followers' initial and final positions are marked with '${\rm x}$' and '${\rm o}$', respectively); (b) Trajectories of agents from 30 to 90 seconds; (c) Formation's error versus time; (d) A subset of the adaptive gains $\gamma_{ij}$ versus time.
  • Figure 4: Simulation 2: the 20-agent system under the bearing-only control law \ref{['eq:Bearing_OnlyC']}. (a) Trajectories of agents from 0 to 5 seconds (leaders are marked with $\Delta$, followers' initial and final positions are marked with '${\rm x}$' and '${\rm o}$', respectively); (b) Trajectories of agents from 5 to 15 seconds; (c) Trajectories of agents from 15 to 25 seconds; (d) Formation's error versus time; (e) A subset of the adaptive gains $\gamma_{i}$ versus time. (f) Magnitude of control input versus time.
  • Figure 5: Simulation 3: the 20-agent system with moving leaders under the control law \ref{['eq:bearing_based_control_law']}. (a) Trajectories of agents (leaders' trajectories are colored blue, followers' positions at $t=0$ and $t=20$ sec. are marked with '${\rm x}$' and '${\rm o}$', respectively); (b) Formation's error versus time; (c) A subset of the adaptive gains $\gamma_{ij}$ versus time; (d) The magnitude of the control input versus time.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Definition 1: Desired formation
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 6 more