Digest for "Flatness-based Finite-Horizon Multi-UAV Formation Trajectory Planning and Directionally Aware Collision Avoidance Tracking"

Abstract

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Optimal collision-free formation control of the unmanned aerial vehicle (UAV) is a challenge. The state-of-the-art optimal control approaches often rely on numerical methods sensitive to initial guesses. This paper presents an innovative collision-free finite-time formation control scheme for multiple UAVs leveraging the differential flatness of the UAV dynamics, eliminating the need for numerical methods. We formulate a finite-time optimal control problem to plan a formation trajectory for feasible initial states. This optimal control problem in formation trajectory planning involves a collective performance index to meet the formation requirements to achieve relative positions and velocity consensus. It is solved by applying Pontryagin’s principle. Subsequently, a collision-constrained regulating problem is addressed to ensure collision-free tracking of the planned formation trajectory. The tracking problem incorporates a directionally aware collision avoidance strategy that prioritizes avoiding UAVs in the forward path and relative approach. It assigns lower priority to those on the sides with an oblique relative approach, disregarding UAVs behind and not in the relative approach. The high-fidelity simulation results validate the effectiveness of the proposed control scheme.

Assumption 1

(Connectivity) The formation graph is directed, with its underlying undirected graph being connected and containing at least one globally reachable node, i.e., a node \(i\) such that there is a directed path from \(i\) to node \(j\) for all \(j\in\mathcal{V}\) with \(j\neq i\) [30].

Assumption 2

The communication graph is a complete directed graph to account for all inter-UAV collisions (thus, \( \mathcal{G}_f\subseteq\mathcal{G}_c\) ), where every pair of UAVs is connected by directed edges in both directions.

Assumption 3

\( \Lambda(\mathbf{p}(0))=\Phi(\mathbf{p}(0))=\Lambda(\mathbf{p}(t_f))=\Phi(\mathbf{p})(t_f)= \emptyset\) .

Problem 1

(Multi-UAV Formation Control in State Space with Collision Avoidance) (a) Consider a multi-UAV system on the directed connected graph \( \mathcal{G}_f(\mathcal{V},\mathcal{E}_f)\) with UAV dynamics (2)-(3). The multi-UAV formation control problem is to determine control strategies that drive all UAVs from their initial state to a desired formation specified by \( \mathbf{d}_{ij},\forall(i,j)\in\mathcal{E}_f\) in the state space in a finite time horizon \( t_f>0\) . (b) All UAVs must avoid mutual collisions, i.e., \( \Lambda(\mathbf{p})\) and \( \Phi(\mathbf{p})\) must be \( \emptyset\) for all \( t\in(0,t_f)\) .

Problem 2

(Multi-UAV Formation Control in Flat Space with Collision Avoidance) (a) Consider a multi-UAV system on the directed connected graph \( \mathcal{G}_f(\mathcal{V},\mathcal{E}_f)\) in flat space with dynamics (6). The multi-UAV formation control problem is to determine control strategies for the UAV team that drive all UAVs from their initial state to a desired formation specified by \( \mathbf{d}_{ij},\forall(i,j)\in\mathcal{E}_f\) in the flat state space in a finite time horizon \( t_f\) . (b) Additionally, \( \Lambda(\mathbf{p})\) and \( \Phi(\mathbf{p})\) must be \( \emptyset\) for all \( t\in(0,t_f)\) .

Theorem 1

Consider the multi-UAV formation control problem in flat coordinates defined by (6) and (7). The unique globally optimal control input and corresponding optimal formation trajectory for any initial state \( \mathbf{r}_0\) are given by

\[ \begin{align} &\mathbf{u}_{\mathbf{r}}(t)=-\mathbf{R}^{-1}\mathbf{B}^\top\mathbf{G}(t_f-t)\mathbf{H}^{-1}(t_f)\mathbf{r}_0, \end{align} \]

(8)

\[ \begin{align} &\mathbf{r}(t)=\mathbf{H}(t_f-t)\mathbf{H}^{-1}(t_f)\mathbf{r}_0, \end{align} \]

(9)

where

\[ \begin{align} &\mathbf{H}(t)=\begin{bmatrix} \mathbf{I} & \mathbf{0} \end{bmatrix}\exp{(-t\mathbf{M})}\begin{bmatrix} \mathbf{I} \\ \mathbf{Q}_f \end{bmatrix}, \end{align} \]

(10)

\[ \begin{align} &\mathbf{G}(t)=\begin{bmatrix} \mathbf{0} & \mathbf{I} \end{bmatrix}\exp{(-t\mathbf{M})}\begin{bmatrix} \mathbf{I} \\ \mathbf{Q}_f \end{bmatrix}, \end{align} \]

(11)

\[ \begin{align} &\mathbf{M}=\begin{bmatrix} \mathbf{A} & -\mathbf{B}\mathbf{R}^{-1} \mathbf{B}^\top \\ -\mathbf{Q} & -\mathbf{A}^\top \end{bmatrix}. \end{align} \]

(12)

Lemma 1

If the initial conditions of the UAVs satisfy

\[ \begin{equation} ||\mathbf{p}_i(0) - \mathbf{p}_j(0)|| \geq r_i + r_j + \epsilon, \end{equation} \]

(27)

for all \(i,j \in \mathcal{V}\), \(i \neq j\), and some \(\epsilon > 0\), then the UAVs can achieve \(\frac{d\hat{V}}{dt} \geq 0\) using bounded control effort, thereby avoiding collisions.

Theorem 2

Given the conditions of Lemma 1, any trajectory that results in a collision is sub-optimal.

Remark 1

The weighting factors \( \alpha_{ij},\beta_{ij},\xi_{ij} \) are introduced to dynamically adjust the sensitivity of the penalty function \( v_{ij} \) based on the relative motion of the agents. In the context of directionally aware collision avoidance, \( v_{ij} \) quantifies the collision risk between agents based on their proximity, and \( \alpha_{ij},\beta_{ij},\xi_{ij} \) modulate the impact of \( v_{ij} \) by accounting for the agents’ forward paths and relative approach. This distinction ensures that the weighting factors do not alter the fundamental role of \( v_{ij} \) in the collision avoidance mechanism. Rather, they dynamically scale its effect based on directional considerations.

Assumption 4

(Constancy of weighting factors in derivatives of the penalty function) Given the distinct roles of the weighting factors \( \alpha_{ij}, \beta_{ij}, \xi_{ij} \) in dynamically adjusting the sensitivity of the penalty function \( v_{ij} \), as stated in Remark 1, we assume that these factors are treated as constants when calculating the time and partial derivatives of \( v_{ij} \).

Remark 2

Assumption 4 simplifies the computation of the penalty function derivatives and improves computational efficiency, robustness, and smoothness of control responses. However, as demonstrated in Appendix 10, the control strategy in (25), utilizing the forward-path-aware penalty function \( v_{ij}^F \), ensures bounded control inputs even without Assumption 4, thereby guaranteeing stability. Analogous stability conditions can be derived for the approach-aware penalty function \( v_{ij}^A \) and the unified penalty function \( v_{ij}^U \), reinforcing the robustness of the directionally aware collision avoidance framework.

References

[1] C. Li and Z. Qu, ``Distributed finite-time consensus of nonlinear systems under switching topologies,'' Automatica, vol. 50, no. 6, pp. 1626--1631, 2014.

Table of contents

Digest for "Flatness-based Finite-Horizon Multi-UAV Formation Trajectory Planning and Directionally Aware Collision Avoidance Tracking"

Abstract

References