Optimal Time-Invariant Distributed Formation Tracking for Second-Order Multi-Agent Systems

TL;DR

This paper addresses the optimal time-invariant formation tracking problem with the aim of providing a distributed solution for multi-agent systems with second-order integrator dynamics by formalizing and proposing a solution to an optimization problem that encapsulates trajectory tracking, distance-based formation control and input energy minimization.

Abstract

This paper addresses the optimal time-invariant formation tracking problem with the aim of providing a distributed solution for multi-agent systems with second-order integrator dynamics. In the literature, most of the results related to multi-agent formation tracking do not consider energy issues while investigating distributed feedback control laws. In order to account for this crucial design aspect, we contribute by formalizing and proposing a solution to an optimization problem that encapsulates trajectory tracking, distance-based formation control and input energy minimization, through a specific and key choice of potential functions in the optimization cost. To this end, we show how to compute the inverse dynamics in a centralized fashion by means of the Projector-Operator-based Newton's method for Trajectory Optimization (PRONTO) and, more importantly, we exploit such an offline solution as a general reference to devise a stabilizing online distributed control law. Finally, numerical examples involving a cubic formation following a chicane-like path in the 3D space are provided to validate the proposed control strategies.

Figures (3)

• Figure 1: Potential function $\sigma_{d_{ij}}(s_{ij})$ with $\beta_{ij}=3$, $\alpha_{ij}=0.5$, $k_{a_{ij}}>k_{r_{ij}}$ and its derivatives w.r.t. $s_{ij}$ up to the second order. Attractive and repulsive behaviors can be associated to the values in the dark green and yellow areas respectively.
• Figure 2: Comparison between the offline solution provided by PRONTO and the system dynamics governed by the online distributed feedback controller. (a)-(b): trajectories of the positions, in dark green; trajectory of the centroid position, in light green; desired straight path, in black; cubic shape formation, depicted progressively with ordered shades blue-indigo-purple-red, beginning with blue. (c)-(d): total input energy, in magenta and average input energy, in green, spent by the system; the dashed lines show their time-averaged values over the interval $[0,T]$. (e)-(f): settling times at $10\%$, $1\%$, $0.1\%$, (i.e. $\delta = 0.1,0.01,0.001$ respectively) depicted by the vertical lines, and established evaluating a quantitative weighted trade-off between formation and tracking instantaneous costs.
• Figure 3: Distributed control law seen as a consensus protocol. (a)-(c)-(e): distance-based potential error functions \ref{['eq:potential']} and their derivatives w.r.t. the corresponding current squared distance converging to $0$ before final time $T$; agents are guaranteed to reach the required geometrical shape. (b): relative velocities converging to $0$, ensuring the agents to follow the desired path with straight trajectories. (d)-(f): response of the centroid positions and velocities (first, second and third components in blue, orange and yellow respectively) converging to the given specifications.

Theorems & Definitions (10)

• Definition 1: Hamiltonian associated to \ref{['eq:dyn_sys']}-\ref{['eq:minimizationproblemOIFT']}
• Theorem 1: PMP applied to \ref{['eq:dyn_sys']}-\ref{['eq:minimizationproblemOIFT']}
• proof
• Proposition 1: Local first-order centroid estimator
• proof
• Remark 1
• Corollary 1: Local second-order centroid estimator
• proof
• Remark 2
• Remark 3