## Abstract

Potential field-based collision avoidance algorithms for mobile robots frequently assume vehicles and obstacles to have circular or spherical shapes. This assumption not only simplifies the analysis but also limits the mobility of agents in confined spaces, particularly for vehicles with elongated or irregular shapes. To increase mobility, this letter presents a decentralized collision avoidance framework for nonholonomic systems of unicycle type that considers the non-circular shape and relative orientation of vehicles and obstacles. The framework builds on the concepts of potential field and avoidance functions. However, it proposes using a non-constant minimum safe distance radius that changes based on the shape, relative position, and relative orientation of agents. The control framework is proven to guarantee collision avoidance at all times and is shown, via simulation, to increase the ability of agents to navigate through narrow spaces safely.

## 1 Introduction

A critical task of mobile robots is the ability to follow a desired trajectory or path while avoiding collisions with other vehicles and obstacles. Methods for collision avoidance include the use of velocity obstacle [1] and its variants [2,3], the concept of collision cones [4,5], the use of potential field functions [6–9], and the implementation of barrier functions [10,11], to name a few (see Refs. [12–14] for comprehensive reviews). A common assumption among most of these methods is the generalization of agents (namely, vehicles and obstacles) as having circular or spherical shapes. Although this assumption simplifies the analysis and implementation, it is also overly conservative, particularly for agents with elongated shapes. For instance, consider the two vehicles in Fig. 1, which have nearly rectangular shapes. The circular body assumption will require both vehicles to keep a distance from each other equal to or larger than the summation of their circumradii, $rijmax=hi+hj$, regardless of their relative orientation. That is, the circular shape assumes the worst-case scenario and makes the vehicles occupy a larger space than necessary, which can impede the motion of agents in cluttered spaces and narrow corridors [15]. Therefore, for confined environments, it is more suitable to design avoidance control strategies that take the vehicle’s and the obstacle’s shape into consideration.

Examples of control algorithms that explicitly consider the non-circular shape of agents include methods based on reachability concepts. For instance, Ref. [4] introduces a collision cone approach for dynamic obstacles of arbitrary shape, which is later extended to obstacles of deformable shape [15] and to nonholonomic vehicles [5]. In Ref. [16], a distance-projection method is formulated using reachable sets that take into account the shape and kinematics of vehicles and obstacles. A common drawback of the aforementioned methods is that, with a large number of dynamic obstacles, they either cannot guarantee collision avoidance or the problem can quickly become intractable.

Collision avoidance with objects of non-circular shape has also been approached using potential field functions, which have the typical advantage of generating methods that can be rigorously proven to be safe for an arbitrarily large number of obstacles. However, most of these approaches rely on computing the closest distance to the obstacle [6] or a set of points in its boundary [17], which can generate non-smooth control inputs, be computationally expensive, or not suitable for non-convex obstacles. Others have approximated the shape of elongated agents using ellipsoids [7,18,19], which can still introduce conservatism for objects of irregular shape. Another common approach is representing the obstacles as an assemblage of $n$-dimensional small spheres [20] and ellipsoids [18]. However, these methods would require a collision avoidance action for each individual surface.

Recently, gradient-based optimization methods have been proposed using signed distance functions [21,22] and control barrier functions [10,11,23]. For instance, the work in Ref. [21] introduces an optimization scheme to determine collision-free trajectories by approximating the gradient of signed distance functions to obstacles of primitive shapes. Similarly, Ref. [11] introduces the use of control barrier functions to guarantee the safety of vehicles with limited actuation. Yet, many of these approaches do not yield controllers in closed form or may yield discontinuous control inputs due to the approximations.

In this letter, we present a decentralized, reactive potential field-based collision avoidance method for an arbitrarily large group of nonholonomic vehicles of unicycle type and arbitrary shape. We use the concept of avoidance functions presented in Ref. [7] and extend it to vehicles and obstacles of arbitrary shape by introducing a non-constant minimum safe distance. The minimum safe distance (or avoidance radius) is a function of the relative orientation between the obstacle and the vehicle, as well as their shapes. Unlike previous potential field-based frameworks, ours only requires a single function between pairs of agents and generates analytical, smooth control inputs. To prevent deadlocks, a common drawback of potential field methods [17], we include an almost everywhere continuous and smooth control perturbation aimed to break symmetries in the potential field. The overall control strategy is rigorously proven to avoid collision among cooperative vehicles and static obstacles at all times. We validate the control approach via a numerical example with vehicles and obstacles of rectangular shape. To the best of our knowledge, this is the first time that a time-varying avoidance radius or minimum safe distance is assumed within the avoidance functions framework [7].

## 2 Problem Formulation

### 2.1 Multi-Vehicle Dynamics.

*i*th robot (see Fig. 1). In what follows, we will omit the time argument of signals unless deemed necessary.

*a*)

*b*)

*b*) can only be shown to be Lagrange stable [12], the linear dynamics of the reference point (4

*a*) are controllable. That is, for any desired trajectory $zid\u2208R2$, one can design a state feedback control law $ui$ such that $zi\u2192zid$ as $t\u2192\u221e$.

### 2.2 Minimum Safe Distance.

*i*th vehicle and

*j*th agent as

### 2.3 Control Objective.

The *i*th vehicle’s ultimate control objective is to follow a desired trajectory, given as $zid\u2208R2$, while maintaining a safe distance $rij$ from other vehicles and obstacles at all times. To this end, we assume that each vehicle can detect, either via communication or onboard sensors, the relative position and orientation of other agents within a bounded detection radius $R$, where $R>supi,j\u2260i{rij}=rijmax$. Then, we can formulate the control objective as follows. Design a control strategy $ui$ such that $zi\u2192zid$ as $t\u2192\u221e$ and $\Vert zi\u2212zj\Vert >rij$$\u2200i,j\u2260i,t\u22650$.

## 3 Trajectory Tracking With Collision Avoidance Control

### 3.1 Avoidance Functions.

*i*th vehicle and

*j*th agent as

### 3.2 Control Law.

*a*)

*b*)

*c*)

*i*th agent, $\u03d1i$ is the angle between $uia$ and $z\u02d9id\u2212z\u02d9i$, and $R(\u22c5)$ is the $2\xd72$ rotational matrix. The first three terms in (11a) comprise the trajectory tracking control law, with $Kv$ and $Kp$ regulating the convergence rate (to be shown next). The term $uia$ is the cooperative collision avoidance strategy and $uip$ is an optional control perturbation aimed to reduce the occurrence of unwanted local minima (i.e., deadlocks). Note that $uia$ is only active when another vehicle or obstacle is within the reaction distance $Rij$. In addition, note that $uip$ is upper bounded by $\lambda 1$, almost everywhere continuous (except at $uia=0$ or $z\u02d9id\u2212z\u02d9i=0$), null if there is no collision threat, and always perpendicular to $z\u02d9id\u2212z\u02d9i$. To prove the latter, it is sufficient to show that $(z\u02d9id\u2212z\u02d9i)TR(\pi /2)\u2212\u03d1i)uia=0$. Therefore, let $z\u02d9id\u2212z\u02d9i=[a,b]T$ and $uia=[c,d]T$. Then,

Similar to Ref. [8], we now make the following assumption about the desired trajectory.

*The desired trajectory satisfies the following constraint*: $z\u02d9idTuia\u2264Kv\Vert z\u02d9id\u2212z\u02d9i\Vert 2$*for all*$t\u22650$.

Note that Assumption 1 is trivially satisfied when the desired trajectory is constant $z\u02d9id=z\xa8id=0$. In the case of non-constant trajectory, we can assume that the *i*th robot can momentarily freeze the desired trajectory while trying to resolve a collision conflict, similar to Ref. [8], or when $z\u02d9idTuia>Kv\Vert z\u02d9id\u2212z\u02d9i\Vert 2$.

#### Collision avoidance

#### Trajectory tracking

*Assume that*$\u2203t0\u22650$*such that*$\Vert zi(t)\u2212zj(t)\Vert \u2265Rij$$\u2200j\u2260i$*and*$\u2200t\u2265t0$. *Then*, $(zi(t),z\u02d9i(t))$*converges to*$(zid(t),z\u02d9id(t))$*exponentially as*$t\u2192\u221e$.

Theorem 2 guarantees the exponential convergence of agents to the desired trajectory after the resolution of all conflicts. Yet, similar to other potential field-based methods, deadlocks can still occur if a conflict persists. The following section addresses this issue.

### 3.3 Deadlocks.

A common drawback of potential field methods is the occurrence of deadlocks, that is, when a vehicle cannot reach its desired destination due to persistent interaction with other agents. These situations are due to symmetries between the tracking and avoidance control, i.e., when $uia\u2192\u2212z\xa8id\u2212Kv(z\u02d9id\u2212z\u02d9i)\u2212Kp(zid\u2212zi)$. For damped systems with constant desired trajectories, this condition reduces to $uia\u2192\u2212Kp(zid\u2212zi)$ [29]. To break the symmetry and aid in conflict resolution, we propose adding the perpendicular perturbation (11c). Note that $uip\u22480$ when the vehicle is still in motion (particularly for large $\lambda 2$) and is only active when the avoidance control is active (i.e., under the presence of a collision threat). In Ref. [9], it is shown that these perpendicular perturbations can aid in conflict resolution. Finally, it is worth mentioning that different from Refs. [9,29], the perturbation control proposed in (11c) is almost everywhere continuous and smooth.

Even with this control perturbation, we cannot guarantee that the vehicles will always converge to $zid$, particularly under the presence of static obstacles. For instance, vehicles may be trapped between large obstacles, walls, or dead-end corridors, which is a common drawback of all previous approaches [9]. In decentralized scenarios where agents have limited information about their surroundings, such as the problem presented herein, the vehicles may need to apply other heuristic measures (e.g., exploration).

## 4 Numerical Example

In this section, we present an example with vehicles and obstacles of rectangular shape.

### 4.1 Minimum Safe Distance.

*i*th and

*j*th vehicles can be approximated by rectangles with length $\u2113i$, $\u2113j$ and width $wi$, $wj$. Without loss of generality, let their lengths be aligned with the $x$-axis (as shown in Fig. 1). Define the following orientation-dependent functions:

*a*)

*b*)

*a*)

*b*)

*c*)

*j*th agent and using the continuous differentiable approximation of the minimum function [30], one can obtain a smooth function for $rij$

*j*th agent can come from the

*i*th vehicle, which is generally shorter than the constant minimum distance scenario.

### 4.2 Simulations.

*a*)

*b*)

We first simulate the system assuming the conventional potential field-based approach of a constant minimum safe distance. Since the circumradius for the vehicles and the obstacles is $1.25m$ and $2m$, respectively, we choose $rij=rijmax=2.24m$ as the minimum safe distance with other vehicles and $rij=rijmax=2.53m$ with obstacles. We also choose $\Delta R=2m$, for a constant reaction distance of $Rij\u2208{4.24m,4.53m}$. The system’s response can be seen in Fig. 3, where all agents navigate safely but cannot reach the other side of the domain despite the walls having a separation of $4m$. Figure 4 illustrates the position errors, which keep increasing over time.

The response of the multi-vehicle system with the proposed control strategy is illustrated in Fig. 5, where we approximated the rectangular shapes using (22) and (23), for $\epsilon =0.01$ and $\delta =6$. Observe that all vehicles are able to track the desired trajectory and transit through a narrow passage safely. In addition, Fig. 6 shows that the agents eventually converge to the desired trajectory once collision threats are over. Finally, the control input force and torque are given in Figs. 7 and 8, respectively, where it can be observed that the control inputs are bounded and continuous.

## 5 Conclusions

This letter presents a decentralized, reactive collision avoidance framework for nonholonomic systems of unicycle type and arbitrary shape. The framework is built upon the concept of avoidance functions but exploits the use of a novel non-constant minimum safe distance radius that explicitly considers the shape of vehicles and obstacles, as well as their relative position and orientation. In contrast to previous potential field-based methods, the proposed framework only requires a single function between pairs of agents and generates analytical, smooth control inputs. The overall control strategy is rigorously proven to avoid collisions at all times and to converge exponentially to the desired trajectory once all conflicts have been resolved. To the authors’ knowledge, this is the first provably safe approach to use a time-varying avoidance radius or minimum safe distance within the avoidance functions framework.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## References

*Differential Geometric Control Theory*