Safe coordination of robots in cyclic paths

Highlights

• Efficient offline coordination strategy for multiple robots in cyclic paths by using a MILP formulation.

• Assurance of minimum distance between the robots and incorporation of acceleration and velocity limits, including positive minimum speed.

• Online controller that keeps the system synchronized over the presence of disturbances on velocities and positions.

• Simulations with up to 48 robots and a real experiment with 3 e-puck robots.

Abstract

This paper presents a MILP (mixed integer linear programming) based formulation for the coordination of multiple robots. We consider robots that must follow closed intersecting paths persistently. We propose an off-line planning of velocity profiles preventing the need of online collision avoidance maneuvers or path replanning. Our robot model considers minimum and maximum speed constraints, which allows our strategy to be applied to fixed-wing aerial robots. We also deal with three important other issues which are usually disregarded in the related literature: minimum spatial separation; acceleration limits; and uncertainties on the speeds and positions. Simulations with up to 48 robots show the efficiency of our strategy. A real experiment with 3 actual e-puck robots is presented in order to demonstrate the robustness of our formulation in a real world scenario.

Introduction

This work addresses the problem of safely coordinating a group of robots to cover closed paths that have intersection points with each other. This work is highly motivated by persistent coverage tasks, in which the team of robots have to keep covering (visiting) some target regions or points periodically. As pointed out by Palacios-Gasós et al. [1], the problem of covering an environment with a team of robots, commonly called multi-robot coverage problem, can fall in three different categories: static, dynamic or persistent. In static coverage, fixed positions for a set of robots are defined in order to statically monitor an environment. Dynamic coverage is related to exploration missions, where an area must be explored until some level of knowledge is reached. This work fits in the persistent coverage scenario. Different from the previous scenarios, in persistent coverage a given area or specific points inside the area must be continuously kept under monitoring, which means that robots must keep periodically revisiting determined places in the environment. The persistent problem is usually associated to surveillance missions. Such persistent coverage tasks, in which the robots must keep visiting periodically specific points of the environment, may be modeled as the problem of controlling the robots to traverse periodically predefined closed paths [2]. In general, such paths might have intersections with each other. This work aims at allowing the robots to navigate along these paths without colliding with each other while crossing these intersections in the presence of velocity and acceleration constraints and position and velocity uncertainty.

The problem we tackle in the present work was earlier addressed by Gonçalves et al. [3], who achieved the objective of planning a collision-free velocity profile for point robots by solving a MILP (mixed integer linear programming) problem considering minimum and maximum speed limits as physical constraints. In this journal paper, we further extend our results by considering some practical issues which are usually disregarded in multi-robot coverage literature. We take into account real size robots, acceleration limits and uncertainties in the individual robot controllers. Minimum and maximum speed limits are also taken into account, which allows for the application of our strategy to fixed-wing aerial vehicles. Our objective is to maximize safety. We maximize the minimum spatial distance between the robots by extending the sizes of what we call dangerous regions. The formulation can be easily changed in order to optimize other objectives such as the traversing time. In order to illustrate the efficiency of our approach, we present several simulations, among which scenarios with up to $48$ robots. In this example, the path of a single robot intersects up to $4$ other paths. We also present an experiment with $3$ actual e-puck robots that shows the robustness of the solution and the stability of the online computation of reference speed that keeps the robots synchronized. This last experiment was only possible due to the improvements we develop in the present work.

The strategy proposed in this work applies when the robots must move along previously specified paths in persistent coverage scenarios [4]. These scenarios are commonly associated with the idea of keeping a desired coverage level, which in turn might be associated with the idea of accumulation functions. In general, these functions are defined over the environment in such a way that their values keep increasing over time at points not covered by a robot sensor footprint within a time interval. Although in the present work we do not consider the problem of keeping an accumulation function bounded, we formally treat the problem of visiting points of the environment periodically and ensure the absence of collisions among the robots. Lin and Cassandras [5] use the idea of accumulation function and propose a method to compute elliptic trajectories in order to cover a rectangular area. Here we do not tackle the problem of defining the curves that the robots must follow. Thus, the method presented by Lin and Cassandras [5], which neglects the inter-robot collision issue, could be used to generate appropriate closed paths while our strategy could generate a collision free robot coordination.

In the literature, the collision problem is commonly addressed with stopping policies [6], [7], or sensing–repelling strategies [8], [9], many times associated with repulsive potential fields [10]. The former is not applicable when the robots are fixed-wing UAVs (Unmanned Aerial Vehicles), since a minimum velocity is needed for aerodynamic lift. The latter one is not usually applicable when the robots must be kept on their planned paths, to avoid other fixed obstacles for example. Besides, in general there is no guarantee of absence of collisions and deadlocks. The collision avoidance problem may also be tackled inside the control law of the robots by considering a null space based behavioral approach [11].

Borkar et al. [12] planned the motion of multiple robots over a single previously defined Lissajous curve. An algorithm is used to define a curve and the number of robots to cover a rectangular area. The motion is then planned by maximizing the size of the robots such that collision free trajectories are still feasible. In contrast to methodology presented by Borkar et al. [12], our approach considers robots with known physical dimensions and then maximizes the size of the enlargement, further defined as $\Delta s$, of the so called dangerous regions, where the presence of multiple robots is forbidden at a given instant of time. In our formulation it was possible to deal with the collision problem by avoiding multiple robots simultaneously in the same dangerous region. The problem is solved by considering a MILP formulation in which the necessary constraints for collision avoidance are the only ones that required the use of integer variables. Another type of curve that is also considered in the trajectory planning of multiple robots is the Bernstein–Bézier curve. For instance, Škrjancc and Klančar [13] plan the motion of multiple robots, each one between a pair of predefined points, while avoiding collisions. Škrjancc and Klančar [13] used the continuity property of the Bézier curves to establish maximum acceleration constraints. In our work, we also incorporate acceleration limits, which is an important feature to ensure that the planned velocity profile will be feasible by actual physical robots.

Different works have used MILP formulations in the coordination of multi-robot systems. Avellar et al. [14] presented a MILP solution that provides the number of UAVs and the paths they need to follow in order to cover an area in minimum time. Richards and How [15] proposed a theory that plans the paths through waypoints in a sequence that minimizes the elapsed time. Collision with other robots and with obstacles are avoided. In order to take into account dynamic limits the authors use a conservative strategy writing the constraints in a linear form. Here we use a similar idea that will allow us to include the acceleration limits in our set of constraints. Richards and How [16] use Decentralized Model Predictive Control (DMPC) together with a MILP formulation to generate collision free trajectories for multiple UAVs with linearized dynamics. The problem is solved online at each time step, a methodology that is different from the one in the present work, since our problem is solved offline and only once. Altché et al. [17] used an offline MILP formulation to plan the motion of multiple actual size robots in a cross section, which has its main application in the autonomous driving through intersections.

Peng and Akella [18] used MILP formulations to coordinate a set of robots from initial to final points along predefined paths without collisions. They consider “collision zones” defined by entrance and exit points in the robots’ predefined paths similarly to what we will develop in this paper. In fact, their MILP formulation is very similar to ours in the way they consider collision zones, and in the way they use a conservative strategy to consider dynamic constraints. The main difference of our work is that we consider cyclic paths while they consider fixed finite paths. Thus, our method required a strategy to solve this infinite horizon problem with a finite number of variables and constraints. In addition, we consider limited uncertainties on the velocities executed by the physical robots with respect to the ones computed from the solution of the optimization problem. Another related work is presented by Shahriari and Biglarbegian [19], who consider the idea of collision zones to deal with the collision issue around intersections of multiple paths. Although the problem of minimum positive velocities is not considered by Shahriari and Biglarbegian [19], the “motion liveness” of the robots is ensured, i.e. their conflict resolution method does not require the robots to stop. Regarding these collision zones, another contribution of our work is the formal definition of these regions by constructing an equivalence relation.

In contrast to the MILP formulation, the motion planning of multiple underwater vehicles between sets of initial and final points is done by Zhuang et al. [20], who use a Legendre pseudospectral method for optimization. The planning is made offline and dynamical obstacles are considered in a second replanning stage, which is online. In the present work we also have an online computed second stage, which is used to maintain the robots properly synchronized on their respective cyclic paths. Abichandani et al. [21] also deal with the motion planning of multiple underwater vehicles, but using a mixed integer nonlinear programming formulation instead. Assurance of communication between the robots is also incorporated. In these works, the robots build local communication networks and execute, each one, a multi robot planner, usually probabilistic, in order to find feasible trajectories to a goal position. The best solution is then spread through the network. In this context, the implementation of the solution provided by our strategy, which is centralized and computed offline, has an important advantage: it does not require any communication between the robots. In addition, an offline strategy has also the advantage of providing a predictable behavior for the entire system without the need of complex real time computations during the navigation.

Another issue that is also of concern in persistent multi-robot coverage tasks is the energy supply limitation [22]. Mathew et al. [23] planned the motion of a set of charging robots to refuel a set of task robots. The problem is tackled with a DAG (direct acyclic graph) formulation and solved with MILP. Scherer and Rinner [24] decomposed the environment in cells and proposed an algorithm to compute the path for each robot of a set. The robots move between base stations and points of interest, while communication with the basis and enough energy to return are ensured. The consideration of this energy issue is out of the scope of this paper. However, the techniques just discussed may be directly or indirectly incorporated in our framework. The predictability of our solution may be an important feature to easy the application of a refueling strategy.

The majority of works discussed until now considers the point-to-point problem, when multiple robots must follow, each one, a route (predefined or not) from an initial to a final point avoiding collisions. The coordination of multiple robots moving persistently in cyclic paths is not as frequently addressed as the point-to-point problem. Sabattini et al. [11] use an independent subgroup of robots to control a subgroup of dependent robots. The independent robots are controlled directly in such a way that the dependent robots converge to and circulate predefined closed curves. The task is performed by establishing a consensus-based interconnection between the robots and is computed online.

Our work deals with persistent tasks. The proposed solution considers an offline planner to coordinate the robots and an online controller to maintain the synchronism among them. A difficulty that appears when offline planners are used to deal with persistent tasks is the mathematical formulation of a model that satisfies the constraints for all instants of time. The work presented by Gonçalves et al. [3] introduces a strategy to solve this infinite horizon cyclic problem with a finite number of variables and constraints. As we have shown, this is possible due to the assumed periodicity and commensurablility properties of the cycles. In fact, this is one of the main contributions of our formulation. The problem is formulated as a MILP, whose constraints directly ensures our problem’s requirements satisfaction. These are related to safety, periodicity, commensurability and speed limitation. In this paper we extend the previous results by considering constraints related to finite size robots, speed and acceleration limits and uncertainties, which are most of the time disregarded in previous works. The consideration of finite size robots makes our guarantee of collision free trajectories more realistic, since the robots are no longer represented as single points. The maintenance of minimum and maximum speed limits allows the method to be applied to fixed wing UAVs. Acceleration limits ensures the achievement of feasible trajectories. Taking into account uncertainties is also essential to ensure robustness in realistic scenarios. Despite the fact that it is unusual to consider bounded disturbances in multi-robot systems literature, some works [25] take these into account to provide robustness. In the present work, our method to deal with uncertainties is based on the definition of an online step responsible for setting the appropriate reference speeds that keeps the cycles of the robots synchronized, which is necessary to the veracity of our constraints. In fact, our strategy can be divided in two stages. The first one is a discrete high level strategy (offline) solved by the MILP. The second one is a continuous lower level stage (online) which, according to the dynamics of our robot, ensures the performance planned by the MILP. The latter is decentralized and consists on an online feedback based computation of average speeds for each robot.

It should be clear that an important advantage of our collision avoidance treatment is that the robots do not need to execute unpredictable online computed maneuvers. Their trajectories are defined in a way that multiple robots will never traverse a given intersection point very close to each other on time. We maximize the minimum possible distance between the robots to improve safety and coverage efficiency. Safety is improved because the robots will be more distant apart. Coverage efficiency will be implicitly improved because if the robots are not very close to each other they are not covering the same area at the same time. In addition, in the case of aerial robots this separation might help to minimize aerodynamic interference. Tang et al. [26] use a multi-robot planner to solve a point-to-point problem even when the robots are at different heights. They experimentally show the better performance when the planner is used and justify it with the decrease in aerodynamic interference.

Besides coverage tasks, we also envision other important applications for the framework proposed in this work. In fact, our solution might be used in any context where we have multiple robots executing cyclic motion in a shared workspace with the risk of collisions. Direct examples are: planning the motion of multiple robots transporting products in a warehouse; and the planning of the schedule of multiple trains connecting multiple cities.

In Table 1 we summarize qualitative properties of some cited related works. On the first column we place the related works, while in the remaining columns we show if the work has or does not have a given property. Citations marked with an asterisk ($\ast$) indicate the use of a MILP formulation in those works. A check mark (

) is used to indicate that the associated property is incorporated in the cited work while an x mark (

) is used to indicate the opposite. More specifically, we identify if the cited work considers: (i) cyclic paths; (ii) some type of uncertainty; (iii) vehicle’s physical size, i.e. the robots are not represented as single points; (iv) off-line planning; (v) centralized approach; (vi) acceleration dynamics; and (vii) non-zero minimum velocity constraint (liveness), i.e. robots never need to stop.

From Table 1 it is clear the position of our work in the current state of the art. Note that among the papers that consider cyclic paths, our work is the only one that considers both acceleration dynamics and motion liveness, thus it is the most suitable for fixed-wing UAVs. Note also that uncertainties are explicitly considered in only very few works. In addition, among the other works that use a MILP formulation (marked with $\ast$) cyclic paths are only considered by Gonçalves et al. [3], in the earlier conference version of this paper.

The predominance of centralized approaches in the related literature is also evident in Table 1. This centralization, together with off-line computation, is appropriate in some scenarios. Basically, this is important in scenarios where it is desirable to have: (i) predictability; (ii) guarantee of absence of collisions without any online strategy to do so; (iii) low on-board computation capacity; and (iv) absence of communication. One must also remember that our method can be applied in the presence of uncertainty and minimum and maximum velocity constraints, which makes it hard to be solved online. Several online methods have already been proposed in the absence of uncertainty and in the case where the robots are allowed to stop. It is not straight forward to adapt these methods to consider such features online. In our strategy, predictability comes from the cyclic nature of our solution, which allow us to know the state of the system for all instants in the future. Online computation of collision avoidance maneuvers, which could originate a high probability of collision, are not necessary since the motion is preplanned to avoid such situations. Since online maneuvers are not computed no powerful onboard computing power is required. Finally, no communication is needed because each robot needs to keep itself synchronized with time, only.

In the next section we formally define our problem and its specifications. We also present the structure of the adopted solution. In Section 3 we formulate the MILP problem. In that section we also develop the low-level controller that deals with the uncertainties. Discussions regarding practical issues and variations of the formulation are presented in Section 4. In Section 5 we present some details on how we implemented our coordination strategy. Simulations, a timing analysis and a real world experiment are shown in Section 6. Finally, we conclude the paper and present future works in Section 7.