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Decentralized probabilistic multi-robot collision avoidance using buffered uncertainty-aware Voronoi cells

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Abstract

In this paper, we present a decentralized and communication-free collision avoidance approach for multi-robot systems that accounts for both robot localization and sensing uncertainties. The approach relies on the computation of an uncertainty-aware safe region for each robot to navigate among other robots and static obstacles in the environment, under the assumption of Gaussian-distributed uncertainty. In particular, at each time step, we construct a chance-constrained buffered uncertainty-aware Voronoi cell (B-UAVC) for each robot given a specified collision probability threshold. Probabilistic collision avoidance is achieved by constraining the motion of each robot to be within its corresponding B-UAVC, i.e. the collision probability between the robots and obstacles remains below the specified threshold. The proposed approach is decentralized, communication-free, scalable with the number of robots and robust to robots’ localization and sensing uncertainties. We applied the approach to single-integrator, double-integrator, differential-drive robots, and robots with general nonlinear dynamics. Extensive simulations and experiments with a team of ground vehicles, quadrotors, and heterogeneous robot teams are performed to analyze and validate the proposed approach.

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Correspondence to Hai Zhu.

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This work was supported in part by the Netherlands Organization for Scientific Research (NWO) domain Applied Sciences (Veni 15916) and the U.S. Office of Naval Research Global (ONRG) NICOP-grant N62909-19-1-2027. We are grateful for their support. A video of the experimental results is available at https://youtu.be/5F3fjjgwCSs

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Appendices

Appendix

A Proofs of Lemmas and Theorems

1.1 A.1 Proof of Lemma 1

Proof

First we can write the random variable \(\mathbf {d}_o\) in an equivalent form \(\mathbf {d}_o = \Sigma _o^\prime \mathbf {d}_o^\prime \), where \(\mathbf {d}_o^\prime \sim \mathcal {N}(0,I) \in \mathbb {R}^d\) and \(\Sigma _o^{\prime }\Sigma _o^{\prime T} = \Sigma _o\). Note that \(\mathbf {d}_o^{\prime T}\mathbf {d}_o^{\prime }\) is a chi-squared random variable with d degrees of freedom. Hence, there is

$$\begin{aligned} \text {Pr}(\mathbf {d}_o^{\prime T}\mathbf {d}_o^{\prime } \le F^{-1}(1-\epsilon )) = 1 - \epsilon . \end{aligned}$$

Also note that \(\Sigma _o^{-1} = (\Sigma _o^{\prime }\Sigma _o^{\prime T})^{-1} = \Sigma _o^{\prime T^{-1}}\Sigma _o^{\prime -1}\), thus \(\mathbf {d}_o^T\Sigma _o^{-1}\mathbf {d}_o = \mathbf {d}_o^{\prime T}\Sigma _o^{\prime T}\Sigma _o^{\prime T^{-1}}\Sigma _o^{\prime -1}\Sigma _o^\prime \mathbf {d}_o^\prime = \mathbf {d}_o^{\prime T}\mathbf {d}_o^{\prime }\). Hence, it follows that \(\text {Pr}(\mathbf {d}_o^T\Sigma _o^{-1}\mathbf {d}_o \le F^{-1}(1-\epsilon )) = 1-\epsilon \). Thus, let \(\mathcal {D}_o = \{ \mathbf {d}: \mathbf {d}^T\Sigma _o^{-1}\mathbf {d}\le F^{-1}(1-\epsilon ) \}\), there is \(\text {Pr}(\mathbf {d}_o \in \mathcal {D}_o) = 1 - \epsilon \). \(\square \)

1.2 A.2 Proof of Theorem 1

Proof

We need to prove that the set \(\mathcal {S}_o\) contains the set \(\mathcal {O}_o\) with probability \(1-\epsilon \). It is equivalent to that for any point in \(\mathcal {O}_o\), the set \(\mathcal {S}_o\) contains this point with probability \(1-\epsilon \). Recall the definition of \(\mathcal {O}_o\), every \(\mathbf {y}\in \mathcal {O}_o\) can be written as \(\mathbf {x}+ \mathbf {d}_o\) with some \(\mathbf {x}\in \hat{\mathcal {O}}_o\). Also note the definition \(\mathcal {S}_o = \{\mathbf {x}+ \mathbf {d}~|~\mathbf {x}\in \hat{\mathcal {O}}_o,\mathbf {d}\in \mathcal {D}_o\}\). Hence the probability that \(\mathcal {S}_o\) contains \(\mathbf {y}\) is equal to the probability that \(\mathcal {D}_o\) contains \(\mathbf {d}_o\). That is, \(\text {Pr}(\mathbf {y}\in \mathcal {S}_o) = \text {Pr}(\mathbf {d}_o \in \mathcal {D}_o)= 1 - \epsilon , \forall \mathbf {y}\in \mathcal {O}_o\). Thus, \(\text {Pr}(\mathcal {O}_o \subseteq \mathcal {S}_o) = 1-\epsilon \). \(\mathcal {S}_o\) is a maximal \(\epsilon \)-shadow of \(\mathcal {O}_o\).

\(\square \)

B Procedure to compute the best linear separator between two Gaussian distributions

The objective is to solve the following minimax problem:

$$\begin{aligned} (\mathbf {a}_{ij}, b_{ij}) = \arg \underset{\mathbf {a}_{ij}\in \mathbb {R}^d,b_{ij}\in \mathbb {R}}{\min \max }(\text {Pr}_i, \text {Pr}_j), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \text {Pr}_i(\mathbf {a}_{ij}^T\mathbf {p}> b_{ij})&= 1 - \Phi ((b_{ij} - \mathbf {a}_{ij}^T\hat{\mathbf {p}}_i)/\sqrt{\mathbf {a}_{ij}^T\Sigma _i\mathbf {a}_{ij}}), \\ \text {Pr}_j(\mathbf {a}_{ij}^T\mathbf {p}\le b_{ij})&= 1 - \Phi ((\mathbf {a}_{ij}^T\hat{\mathbf {p}}_j - b_{ij})/\sqrt{\mathbf {a}_{ij}^T\Sigma _j\mathbf {a}_{ij}}). \end{aligned} \end{aligned}$$

Let \(u_1 = \frac{b_{ij} - \mathbf {a}_{ij}^T\hat{\mathbf {p}}_i}{\sqrt{\mathbf {a}_{ij}^T\Sigma _i\mathbf {a}_{ij}}}\), \(u_2 = \frac{\mathbf {a}_{ij}^T\hat{\mathbf {p}}_j - b_{ij}}{\sqrt{\mathbf {a}_{ij}^T\Sigma _j\mathbf {a}_{ij}}}\). As the function \(\Phi (\cdot )\) is monotonic, the original minimax problem is equivalent to

$$\begin{aligned} (\mathbf {a}_{ij}, b_{ij}) = \arg \underset{\mathbf {a}_{ij}\in \mathbb {R}^d,b_{ij}\in \mathbb {R}}{\max \min }(u_1, u_2). \end{aligned}$$

We can write \(u_1\) in the following form for a given \(u_2\),

$$\begin{aligned} u_1 = \frac{\mathbf {a}_{ij}^T\hat{\mathbf {p}}_{ij} - u_2\sqrt{\mathbf {a}_{ij}^T\Sigma _j\mathbf {a}_{ij}}}{\sqrt{\mathbf {a}_{ij}^T\Sigma _i\mathbf {a}_{ij}}}, \end{aligned}$$

where \(\hat{\mathbf {p}}_{ij} = \hat{\mathbf {p}}_j - \hat{\mathbf {p}}_i\). For each given \(u_2\), \(u_1\) needs to be maximized. Hence, we can differentiate the above equation with respect to \(\mathbf {a}_{ij}\) and set the derivative to equal to zero, which leads to

$$\begin{aligned} \mathbf {a}_{ij} = [t\Sigma _i + (1-t)\Sigma _j]^{-1}\hat{\mathbf {p}}_{ij}, \end{aligned}$$
(48)

where \(t \in (0,1)\) is a scaler. Thus according to definition of \(u_1\) and \(u_2\), we have

$$\begin{aligned} b_{ij} = \mathbf {a}_{ij}^T\hat{\mathbf {p}}_i + t\mathbf {a}_{ij}^T\Sigma _i\mathbf {a}_{ij} = \mathbf {a}_{ij}^T\hat{\mathbf {p}}_j - (1-t)\mathbf {a}_{ij}^T\Sigma _j\mathbf {a}_{ij}. \end{aligned}$$
(49)

It is proved that \(u_1 = u_2\) must be hold for the solution of the minimax problem (Anderson and Bahadur 1962), which leads to

$$\begin{aligned} \mathbf {a}_{ij}^T[t^2\Sigma _i - (1-t)^2\Sigma _j]\mathbf {a}_{ij} = 0. \end{aligned}$$
(50)

Thus, one can first solve for t by combining Eqs. (48) and (50) via numerical iteration efficiently. Then \(\mathbf {a}_{ij}\) and \(b_{ij}\) can be computed using Eqs. (48) and (49).

C Deadlock resolution heuristic

We detect and resolve deadlocks in a heuristic way in this paper. Let \(\left\| \Delta \mathbf {p}_i\right\| \) be the position progress between two consecutive time steps of robot i, and \(\Delta \mathbf {p}_{\min }\) a predefined minimum allowable progress distance for the robot in \(n_{\text {dead}}\) time steps. If the robot has not reached its goal and \(\Sigma _{n_{\text {dead}}}\left\| \Delta \mathbf {p}_i\right\| \le \Delta \mathbf {p}_{\min }\), we consider the robot as in a deadlock situation. For the one-step controller, each robot must be at the “projected goal” \(\mathbf {g}_i^*\) when the system is in a deadlock configuration (Zhou et al. 2017). In this case, each robot chooses one of the nearby edges within its B-UAVC to move along. For receding horizon planning of high-order dynamical systems, the robot may get stuck due to a local minima of the trajectory optimization problem. In this case, we temporarily change the goal location \(\mathbf {g}_i\) of each robot by clockwise rotating it along the z axis with \(90^\circ \), i.e.

$$\begin{aligned} \mathbf {g}_{i,\text {temp}} = R_Z(-90^\circ )(\mathbf {g}_i - \hat{\mathbf {p}}_i) + \hat{\mathbf {p}}_i, \end{aligned}$$
(51)

where \(R_Z\) denotes the rotation matrix for rotations around z-axis. This temporary rotation will change the objective of the trajectory optimization problem, thus helping the robot to recover from a local minima. Once the robot recovers from stuck, its goal is changed back to \(\mathbf {g}_i\).

Similar to most heuristic deadlock resolutions, the solutions presented here can not guarantee that all robots will eventually reach their goals since livelocks (robots continuously repeat a sequence of behaviors that bring them from one deadlock situation to another one) may still occur.

D Quadrotor dynamics model

We use the Parrot Bebop 2 quadrotor in our experiments. The state of the quadrotor is

$$\begin{aligned} \mathbf {x}= [\mathbf {p}^T, \mathbf {v}^T, \phi , \theta , \psi ]^T \in \mathbb {R}^9, \end{aligned}$$

where \(\mathbf {p}= [p_x, p_y, p_z]^T \in \mathbb {R}^3\) is the position, \(\mathbf {v}= [v_x, v_y, v_z]^T \in \mathbb {R}^3\) the velocity, and \(\phi , \theta , \psi \) the roll, pitch and yaw angles of the quadrotor. The control inputs to the quadrotor are

$$\begin{aligned} \mathbf {u}= [\phi _c, \theta _c, v_{z_c}, \dot{\psi }_c]^T \in \mathbb {R}^4, \end{aligned}$$

where \(\phi _c\) and \(\theta _c\) are commanded roll and pitch angles, \(v_{z_c}\) the commanded velocity in vertical z direction, and \(\dot{\psi }_c\) the commanded yaw rate.

The dynamics of the quadrotor position and velocity are

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\mathbf {p}}} = \mathbf {v}, \begin{bmatrix} \dot{v}_x \\ \dot{v}_y \end{bmatrix} = R_Z(\psi ) \begin{bmatrix} \tan \theta \\ -\tan \phi \end{bmatrix} g -\begin{bmatrix} k_{D_x}v_x \\ k_{D_y}v_y \end{bmatrix}, \\ \dot{v}_z = \frac{1}{\tau _{v_z}}(k_{v_z}v_{z_c} - v_z), \end{array}\right. } \end{aligned}$$

where \(g = 9.81~\text {m}/\text {s}^2\) is the Earth’s gravity, \(R_Z(\psi ) = \begin{array}{ll} \cos \psi &{}-\sin \psi \\ \sin \psi &{}\cos \psi \end{array}\) is the rotation matrix along the z-body axis, \(k_{D_x}\) and \(k_{D_y}\) the drag coefficient, \(k_{v_z}\) and \(\tau _{v_z}\) the gain and time constant of vertical velocity control.

The attitude dynamics of the quadrotor are

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\phi }= \frac{1}{\tau _{\phi }}(k_{\phi }\phi _c - \phi ), \\ \dot{\theta }= \frac{1}{\tau _{\theta }}(k_{\theta }\theta _c - \theta ), \\ \dot{\psi }= \dot{\psi }_c, \end{array}\right. } \end{aligned}$$

where \(k_{\phi }, k_{\theta }\) and \(\tau _{\phi }, \tau _{\theta }\) are the gains and time constants of roll and pitch angles control respectively.

We obtained the dynamics model parameters \(k_{D_x} = 0.25\), \(k_{D_y} = 0.33,\) \(k_{v_z}=1.2270\), \(\tau _{v_z}=0.3367\), \(k_{\phi }=1.1260\), \(\tau _{\phi }=0.2368\), \(k_{\theta }=1.1075\) and \(\tau _{\theta }=0.2318\) by collecting real flying data and performing system identification.

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Zhu, H., Brito, B. & Alonso-Mora, J. Decentralized probabilistic multi-robot collision avoidance using buffered uncertainty-aware Voronoi cells. Auton Robot 46, 401–420 (2022). https://doi.org/10.1007/s10514-021-10029-2

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