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On the relationship between dynamics and complexity in multi-agent collision avoidance


This work examines how dynamics and complexity are related in multi-agent collision avoidance. Motivated particularly by work in the field of automated driving, this work considers a variant of the reciprocal n-body collision avoidance problem. In this problem, agents must avoid collision while moving according to individual reward functions in a crowded environment. The main contribution of this work is the result that there is a quantifiable relationship between system dynamics and the requirement for agent coordination, and that this requirement can change the complexity class of the problem dramatically: from P to NEXP or even \(\hbox {NEXP}^{\text {NP}}\). A constructive proof is provided that demonstrates the relationship, and potential practical applications are discussed.

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  1. Indicator lights are a common channel of communication, but they are notoriously unreliable. Horns also provide a form of communication, but are limited by context. Relative positions and speeds can convey intent, but, as channels of communication, these are very low bandwidth.

  2. Section 5 gives an overview of efficient methods for the various types of dynamics computations Problem 1 entails.

  3. The problem of choosing when and with whom to communicate, while also difficult, is not a focus of this paper.

  4. Breaking this assumption weakens the logical connection in Lemma 3 from a biconditional (if and only if) to a material condition (if).

  5. There is even a special tool, the horn, for alerting those around us that someone’s behavior is aberrant.

  6. To avoid problems in dealing with dynamic constraints (Wilkie et al. 2009) defined generalized velocity obstacles that are derived in control space rather than velocity space.


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The author would like to acknowledge the reviewers for providing many insightful comments and corrections, as well as Philipp Robbel and Elmar Mair for comments and feedback, and Valerie Aquila for assistance with figures and editing.

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This is one of several papers published in Autonomous Robots comprising the “Special Issue on Robotics Science and Systems”.



The first portion of the appendix will prove that the VO representation cannot conservatively approximate inertially constrained systems. It will also show that the VO representation belongs to the family of ICS representations using the inevitable collision obstacle (ICO) concept.

The last portion of the appendix will describe a conjecture about the problem complexity of finding a unique set of, collision-free, non-disjoint SPs.

A.1 The inevitable collision obstacle

Definition 15

An inevitable collision obstacle (ICO) is the set of states of an agent \(A\) that result in collision with \(\mathcal {B}_i\) for any control sequence \(\phi \) is applied to \(A\):

$$\begin{aligned} \text {ICO}(\mathcal {B}_i)=\{\mathbf{x}\;|\;\forall \phi ,\exists t::A(\phi (\mathbf{x},t))\cap \mathcal {B}_i\ne \emptyset \} \end{aligned}$$

The ICO is closely related to the ICS concept, both of which were introduced by Fraichard and Asama (2004).

A.2 The velocity obstacle

This section recalls the velocity obstacle and relevant properties. We use the definitions from Fiorini and Shiller (1998), and the reader is referred to that work for more detail.Footnote 6

In this section, assume \(t\in T\), where \(T=[0,\infty )\) is a finite time horizon. Let \(\varPhi _v\) be the set of feasible velocity commands for \(A\), and let \(\phi _v(\mathbf{x}, t)\) denote the state of \(A\) after constant velocity v is applied to initial state \(\mathbf{x}\) for a time t.

Definition 16

The velocity obstacle for \(A\) due to \(O_i\), written \(\text {VO}_{A|O_i}\), is the set of velocities such that \(A\) at some point enters into a collision state with \(O\). In other words, given initial state \(\mathbf{x}\), and for all feasible velocity commands \(v\in \varPhi _v\) there is a collision at some time \(t\in T\) between \(A(\mathbf{x})\) and the state space obstacle \(\mathcal {B}_i\) due to \(O_i\):

$$\begin{aligned} \text {VO}_{A|O_i}=\{v\;|\;\exists t::A(\phi _v(\mathbf{x},t))\cap \mathcal {B}_i\ne \emptyset \} \end{aligned}$$

A.3 Velocity obstacles and inertially constrained systems

Lemma 9

The VO representation cannot guarantee collision avoidance in inertially constrained systems.


By Definition 16, the complement of the velocity obstacle is exactly the set of all velocities that, when instantaneously applied, would avoid collision. However, controlling to a velocity instantaneously is impossible in an inertially constrained system. Therefore, the complement of the velocity obstacle is unreachable, and by Lemma 1, it cannot be used to guarantee non-collision. \(\square \)

Velocity obstacle and inevitable collision obstacle equivalence

The reader will note the similarity between Definition 15 and 16, and work by Shiller et al. (2010) suggests that a deeper relationship exists. The proof proceeds by exploiting the similarity and showing that ICO computations are both necessary and sufficient in order to compute the VO.

Definition 17

A velocity \(\text {ICO}\) (ICO\(_v\)) for a given state space obstacle \(\mathcal {B}_i\) is an \(\text {ICO}\) computed over the velocity control trajectory set \(\varPhi _v\):

$$\begin{aligned} \text {ICO}(\mathcal {B}_i)_v=\{\mathbf{x}\;|\forall \phi _v,\exists t::A(\phi _v(\mathbf{x},t))\cap \mathcal {B}_i\ne \emptyset \} \end{aligned}$$

Lemma 10

Computing a velocity obstacle is exactly equivalent to computing an inevitable collision obstacle over a restricted control space.


For a given obstacle \(O_i\) and corresponding state space obstacle \(\mathcal {B}_i\), use Definition 16 to perform a variable rewrite on the definition of a velocity ICO (Definition 17):

$$\begin{aligned} \text {ICO}(\mathcal {B}_i)_v&=\{\mathbf{x}\;|\;\forall \phi _v,\exists t::A(\phi _v(\mathbf{x},t))\cap \mathcal {B}_i\ne \emptyset \}\\&=\{\mathbf{x}\;|\;\forall \phi _v, v\in \text {VO}_{A|O_i}\} \end{aligned}$$

Thus, the ICO(\(\mathcal {B})_v\) and \(\hbox {VO}_{A|O}\) are equivalent, which means that the velocity obstacle representation is equivalent to the ICO representation over a restricted control space. \(\square \)

The result of Lemma 10 provides a simple but formal unification of two common techniques for collision avoidance under the same theoretical framework: that velocity obstacles are exactly inevitable collision obstacles over a restricted set of the inputs.

A.4 A special case of coordination

The proof of Lemma 3 asserts through Definition 3 that finding a unique set of collision-free, non-disjoint SPs induces a coordination requirement. Invoking this type of coordination has interesting complexity implications because the general problem of identifying a unique assignment of such SPs is likely reducible to a Unique-SAT problem, which is coNP-Hard (Blass and Gurevich 1982). The following conjecture captures this:

Conjecture 2

There is no efficiently computable (i.e. P-time) solution to identifying a unique set of collision-free stopping paths in a system that does not exhibit SP disjointness.

Investigation of Conjecture 2 would be an interesting point for future work.

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Johnson, J.K. On the relationship between dynamics and complexity in multi-agent collision avoidance. Auton Robot 42, 1389–1404 (2018).

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  • Complexity
  • Dynamics
  • Collision avoidance