Locally-optimal multi-robot navigation under delaying disturbances using homotopy constraints

Abstract

We study the problem of reliable motion coordination strategies for teams of mobile robots when any of the robots can be temporarily stopped by an exogenous disturbance at any time. We assume that an arbitrary multi-robot planner initially provides coordinated trajectories computed without considering such disturbances. We are interested in designing a control strategy that handles delaying disturbance such that collisions and deadlocks are provably avoided, and the travel time is minimized. The problem is analyzed in a coordination space framework, in which each dimension represents the position of a single robot along its planned trajectory. We demonstrate that to avoid deadlocks, the trajectory of the system in the coordination space must be homotopic to the trajectory corresponding to the planned solution. We propose a controller that abides this homotopy constraint while minimizing the travel time. Besides being provably deadlock-free, our experiments show that travel time is significantly smaller with our method than than with a reactive method.

Appendices

Appendix A: Completion of the obstacle region

In this section, we construct the maximal set $\chi ^\delta \subset {\chi ^\mathrm {free}}$. We start by defining $\Delta ^{NW}$ and $\Delta ^{SE}$ as follows:

$$\begin{aligned} \Delta ^{NW}:= & {} \left\{ (a,b)\in [0,T]^2:b>a\right\} \end{aligned}$$

(10)

$$\begin{aligned} \Delta ^{SE}:= & {} \left\{ (a,b)\in [0,T]^2:b<a\right\} . \end{aligned}$$

(11)

Then, we build the completed obstacle region ${\chi ^\mathrm {obs}}^\delta$ as follows:

$$\begin{aligned} C^{NW}_{ij}:= & {} \left( (C_{ij}\cap \Delta ^{NW})+(\mathbb {R}_-\times \mathbb {R}_+) \right) \cap [0,T]^2 \end{aligned}$$

(12)

$$\begin{aligned} C^{SE}_{ij}:= & {} \left( (C_{ij}\cap \Delta ^{SE})+(\mathbb {R}_+\times \mathbb {R}_-)\right) \cap [0,T]^2 \end{aligned}$$

(13)

where $A+B$ denotes the set of $a+b$ with $(a,b)\in A\times B$, $\mathbb {R}_+:=\{x\in \mathbb {R}:x\ge 0\}$ and $\mathbb {R}_-:=\{x\in \mathbb {R}:x\le 0\}$. The completion process is depicted in Fig. 2.

We use $\min (x,y)$ and $\max (x,y)$ operators on two points x, y in the coordination space, defined component-wise as:

$$\begin{aligned} (\min (x,y))_i:= & {} \min (x_i,y_i)\end{aligned}$$

(14)

$$\begin{aligned} (\max (x,y))_i:= & {} \max (x_i,y_i). \end{aligned}$$

(15)

Property 2

(Invariance through $\min$ and $\max$ operators) For all $x,y\in \chi ^\delta$, we have $\min (x,y)\in \chi ^\delta$ and $\max (x,y)\in \chi ^\delta$.

Fig. 10

Illustration of the transformation of $\varphi ^1$ into $\max (\varphi ^1,\varphi ^2)$. Two cases may appear, and in both cases, the transformation remains within $\chi ^\delta$

Appendix B: Proof of Theorem 1

Proof

(Necessary condition) We first prove that taking values in $\chi ^\delta$ is a necessary condition for being homotopic to $\delta$ by contradiction. Consider a solution $\varphi$. Assume that $\varphi$ is homotopic to $\delta$, but it does not take only values in $\chi ^\delta$. As a consequence, it takes some value $x^c\in {\chi ^\mathrm {obs}}^\delta$ at some point $\tau ^c$ such that $(\varphi _i,\varphi _j)(\tau ^c)=(x^c_i,x^c_j)\in C_{ij}^{NW}$ for some $i,j\in \{1\ldots n\}$ (or, equivalently, $(x^c_j,x^c_i)\in C_{ji}^{SE}$). By construction of $C_{ij}^{NW}$, there exists $(x^0_i,x^0_j)\in C_{ij}$ distinct from $(x^c_i,x^c_j)$ (because $\varphi$ is collision-free) such that $x^0_i\ge x^c_i$ and $x^0_j\le x^c_i$. Consider the maximal segment $\Sigma \subset [0,T]^2$ going through points $(x^0_i,x^0_j)$ and $(x^c_i,x^c_j)$. As $\varphi$ is assumed to be homotopic to $\delta$, there exists a continuous transformation $H:[0,1]\rightarrow \varPhi$ such that $H(0)=\varphi$ and $H(1)=\delta$. For all $\alpha \in [0,1]$, $H(\alpha )$ intersects $\Sigma$. $H(\alpha )=\varphi$ intersects at configuration $(x^c_i,x^c_j)$ and H(1) intersects $\Sigma$ at a configuration on the image of $\delta$. As a consequence, by continuity, $H(\alpha )$ goes through $(x^0_i,x^0_j)$ for some $\alpha$, which is absurd as $H(\alpha )$ should be a solution for all $\alpha \in [0,1]$ (solution $H(\alpha )$ should be collision-free in particular). $\square$

Proof

(Sufficient condition) Now, we prove that taking values in $\chi ^\delta$ is a sufficient condition. Consider two arbitrary solutions $\varphi ^1, \varphi ^2\in \varPhi$ taking values in $\chi ^\delta$ and the following continuous transformation H defined as follows for all $\alpha \in [0,1],~\tau \in [0,1]$:

$$\begin{aligned} H(\alpha )(\tau ):=\min \left( \varphi ^1(\tau +\alpha ), \max \left( \varphi ^1(\tau ), \varphi ^2(\tau )\right) \right) \end{aligned}$$

(16)

where $\varphi ^1(\tau +\alpha ) \equiv (T\ldots T)$ if $\tau +\alpha > 1$ by convention.

We have $H(0)(\tau )=\min (\varphi ^1(\tau ), \max (\varphi ^1(\tau ), \varphi ^2(\tau )))= \varphi ^1(\tau )$, so that $H(0)=\varphi ^1$. Moreover,

$$\begin{aligned} H(1)(\tau )= & {} \min (\varphi ^1(1+\tau ),\max (\varphi ^1(\tau ), \varphi ^2(\tau )))\end{aligned}$$

(17)

$$\begin{aligned}= & {} \min (T,\max (\varphi ^1(\tau ), \varphi ^2(\tau )))\end{aligned}$$

(18)

$$\begin{aligned}= & {} \min (T,\max (\varphi ^1(\tau ), \varphi ^2(\tau )))\end{aligned}$$

(19)

$$\begin{aligned}= & {} \max (\varphi ^1(\tau ), \varphi ^2(\tau ))\text {.} \end{aligned}$$

(20)

As a result, $H(1)=\max (\varphi ^1,\varphi ^2)$. Finally, by Property 2 (see also Fig. 10), $\varphi ^1$ and $\varphi ^2$ taking values in $\chi ^\delta$ implies that for all $\alpha \in [0,1]$, $H(\alpha )$ takes values in $\chi ^\delta$. Moreover, $H(\alpha )$ is non-decreasing as $\min$ and $\max$ operators do not affect that property. As a result, H continuously transforms $\varphi ^1$ into $\max (\varphi ^1, \varphi ^2)$ while remaining in $\varPhi$. By symmetry of the roles played by $\varphi ^1$ and $\varphi ^2$, there also exists a continuous transformation transforming $\varphi ^2$ into $\max (\varphi ^1, \varphi ^2)$ while remaining in $\varPhi$. As a result, $\varphi ^1$ and $\varphi ^2$ are both homotopic to $\max (\varphi ^1, \varphi ^2)$, so that $\varphi ^1$ and $\varphi ^2$ are homotopic solutions. In particular, choosing for $\varphi ^1$ an arbitrary solution taking values in $\chi ^\delta$ and $\varphi ^2 \equiv \delta$, we obtain that any solution taking values in $\chi ^\delta$ is homotopic to $\delta$. $\square$