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

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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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  • Alonso-Mora, J., Beardsley, P., & Siegwart, R. (2018). Cooperative collision avoidance for nonholonomic robots. IEEE Transactions on Robotics, 34(2), 404–420.

    Article  Google Scholar 

  • Anderson, T. W., & Bahadur, R. R. (1962). Classification into two multivariate normal distributions with different covariance matrices. The Annals of Mathematical Statistics, 33(2), 420–431.

    Article  MathSciNet  Google Scholar 

  • Andrews, L. C. (1997). Special functions of mathematics for engineers (Vol. 49). SPIE Press.

  • Arslan, O., & Koditschek, D. E. (2019). Sensor-based reactive navigation in unknown convex sphere worlds. International Journal of Robotics Research, 38(2–3), 196–223.

    Article  Google Scholar 

  • Astolfi, A. (1999). Exponential stabilization of a wheeled mobile robot via discontinuous control. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, 121(1), 121–126.

    Article  Google Scholar 

  • Axelrod, B., Kaelbling, L. P., & Lozano-Pérez, T. (2018). Provably safe robot navigation with obstacle uncertainty. The International Journal of Robotics Research, 37(13–14), 1760–1774.

    Article  Google Scholar 

  • Bareiss, D., & van den Berg, J. (2015). Generalized reciprocal collision avoidance. The International Journal of Robotics Research, 34(12), 1501–1514.

    Article  Google Scholar 

  • Blackmore, L., Ono, M., & Williams, B. C. (2011). Chance-constrained optimal path planning with obstacles. IEEE Transactions on Robotics, 27(6), 1080–1094.

    Article  Google Scholar 

  • Breitenmoser, A., & Martinoli, A. (2016). On combining multi-robot coverage and reciprocal collision avoidance. Springer tracts in advanced robotics (pp. 49–64). Springer Japan.

  • Chen, Y., Cutler, M., & How, J.P. (2015). Decoupled multiagent path planning via incremental sequential convex programming. In 2015 IEEE international conference on robotics and automation (ICRA) (pp. 5954–5961). IEEE.

  • Claes, D., Hennes, D., Tuyls, K., & Meeussen, W. (2012). Collision avoidance under bounded localization uncertainty. In 2012 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 1192–1198). IEEE.

  • Dawson, C., Jasour, A., Hofmann, A., & Williams, B. (2020). Provably safe trajectory optimization in the presence of uncertain convex obstacles. In 2020 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 6237–6244). IEEE.

  • Deits, R., Tedrake, R. (2015). Efficient mixed-integer planning for uavs in cluttered environments. In 2015 IEEE international conference on robotics and automation (ICRA) (pp. 42–49). IEEE

  • Deits, R., & Tedrake, R. (2015). Computing large convex regions of obstacle-free space through semidefinite programming. Springer Tracts in Advanced Robotics, 107, 109–124.

    Article  MathSciNet  Google Scholar 

  • Fiorini, P., & Shiller, Z. (1998). Motion planning in dynamic environments using velocity obstacles. The International Journal of Robotics Research, 17(7), 760–772.

    Article  Google Scholar 

  • Gopalakrishnan, B., Singh, A.K., Kaushik, M., Krishna, K.M., Manocha, D. (2017). Prvo: Probabilistic reciprocal velocity obstacle for multi robot navigation under uncertainty. In 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 1089–1096). IEEE.

  • Hardy, J., & Campbell, M. (2013). Contingency planning over probabilistic obstacle predictions for autonomous road vehicles. IEEE Transactions on Robotics, 29(4), 913–929.

    Article  Google Scholar 

  • Hönig, W., Preiss, J. A., Kumar, T. K., Sukhatme, G. S., & Ayanian, N. (2018). Trajectory planning for quadrotor swarms. IEEE Transactions on Robotics, 34(4), 856–869.

    Article  Google Scholar 

  • Kamel, M., Alonso-Mora, J., Siegwart, R., Nieto, J. (2017). Robust collision avoidance for multiple micro aerial vehicles using nonlinear model predictive control. In 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 236–243). IEEE.

  • Kozlov, M. K., Tarasov, S. P., & Khachiyan, L. G. (1980). The polynomial solvability of convex quadratic programming. USSR Computational Mathematics and Mathematical Physics, 20(5), 223–228.

    Article  MathSciNet  Google Scholar 

  • Liu, S., Watterson, M., Mohta, K., Sun, K., Bhattacharya, S., Taylor, C. J., & Kumar, V. (2017). Planning dynamically feasible trajectories for quadrotors using safe flight corridors in 3-d complex environments. IEEE Robotics and Automation Letters, 2(3), 1688–1695.

    Article  Google Scholar 

  • Luis, C. E., Vukosavljev, M., & Schoellig, A. P. (2020). Online trajectory generation with distributed model predictive control for multi-robot motion planning. IEEE Robotics and Automation Letters, 5(2), 604–611.

    Article  Google Scholar 

  • Luo, W., Sun, W., & Kapoor, A. (2020). Multi-robot collision avoidance under uncertainty with probabilistic safety barrier certificates. In 2020 advances in neural information processing systems (NeurIPS) (Vol. 33).

  • Lyons, D., Calliess, J., & Hanebeck, U.D. (2012). Chance constrained model predictive control for multi-agent systems with coupling constraints. In 2012 American control conference (ACC) (pp. 1223–1230). IEEE.

  • Morgan, D., Subramanian, G. P., Chung, S. J., & Hadaegh, F. Y. (2016). Swarm assignment and trajectory optimization using variable-swarm, distributed auction assignment and sequential convex programming. International Journal of Robotics Research, 35(10), 1261–1285.

    Article  Google Scholar 

  • Nägeli, T., Meier, L., Domahidi, A., Alonso-Mora, J., & Hilliges, O. (2017). Real-time planning for automated multi-view drone cinematography. ACM Transactions on Graphics, 36(4), 1–10.

    Article  Google Scholar 

  • Okabe, A., Boots, B., Sugihara, K., & Chiu, S. N. (2009). Spatial tessellations: Concepts and applications of Voronoi diagrams. Wiley.

  • Pierson, A., Schwarting, W., Karaman, S., & Rus, D. (2020). Weighted buffered voronoi cells for distributed semi-cooperative behavior. In 2020 IEEE international conference on robotics and automation (ICRA) (pp. 5611–5617). IEEE.

  • Schmerling, E., Pavone, M. (2017). Evaluating trajectory collision probability through adaptive importance sampling for safe motion planning. In Robotics: Science and systems (Vol. 13).

  • Serra-Gómez, A., Brito, B., Zhu, H., Chung, J.J., Alonso-Mora, J. (2020). With whom to communicate: Learning efficient communication for multi-robot collision avoidance. In 2020 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 11770–11776). IEEE.

  • Shim, D., Kim, H., & Sastry, S. (2003). Decentralized nonlinear model predictive control of multiple flying robots. In 2003 IEEE conference on decision and control (CDC) (pp. 3621–3626). IEEE.

  • Tordesillas, J., Lopez, B.T., & How, J.P. (2019). Faster: Fast and safe trajectory planner for flights in unknown environments. In 2019 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 1934–1940). IEEE

  • Van Den Berg, J., Guy, S. J., Lin, M., & Manocha, D. (2011). Reciprocal n-body collision avoidance. Springer Tracts in Advanced Robotics, 70, 3–19.

    Article  Google Scholar 

  • Van den Berg, J., Lin, M., & Manocha, D. (2008). Reciprocal velocity obstacles for real-time multi-agent navigation. In 2008 IEEE international conference on robotics and automation (ICRA) (pp. 1928–1935). IEEE.

  • Wang, M., & Schwager, M. (2019) Distributed collision avoidance of multiple robots with probabilistic buffered voronoi cells. In 2019 international symposium on multi-robot and multi-agent systems (MRS) (pp. 169–175). IEEE.

  • Zanelli, A., Domahidi, A., Jerez, J., & Morari, M. (2020). FORCES NLP: An efficient implementation of interior-point methods for multistage nonlinear nonconvex programs. International Journal of Control, 1, 13–29.

    Article  MathSciNet  Google Scholar 

  • Zhou, L., Tzoumas, V., Pappas, G. J., & Tokekar, P. (2018). Resilient active target tracking with multiple robots. IEEE Robotics and Automation Letters, 4(1), 129–136.

    Article  Google Scholar 

  • Zhou, D., Wang, Z., Bandyopadhyay, S., & Schwager, M. (2017). Fast, on-line collision avoidance for dynamic vehicles using buffered Voronoi cells. IEEE Robotics and Automation Letters, 2(2), 1047–1054.

    Article  Google Scholar 

  • Zhu, H., & Alonso-Mora, J. (2019). B-uavc: Buffered uncertainty-aware Voronoi cells for probabilistic multi-robot collision avoidance. In 2019 international symposium on multi-robot and multi-agent systems (MRS) (pp. 162–168). IEEE.

  • Zhu, H., Juhl, J., Ferranti, L., Alonso-Mora, J. (2019). Distributed multi-robot formation splitting and merging in dynamic environments. In 2019 international conference on robotics and automation (ICRA) (pp. 9080–9086). IEEE.

  • Zhu, H., & Alonso-Mora, J. (2019). Chance-constrained collision avoidance for MAVs in dynamic environments. IEEE Robotics and Automation Letters, 4(2), 776–783.

    Article  Google Scholar 

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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

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Supplementary material 1 (mp4 16260 KB)



A Proofs of Lemmas and Theorems

1.1 A.1 Proof of Lemma 1


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


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:

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


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

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

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

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

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

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

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

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.

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

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).

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