Safe Coverage of Compact Domains For Second Order Dynamical Systems
aa r X i v : . [ ee ss . S Y ] N ov Safe Coverage of Compact Domains ForSecond Order Dynamical Systems
Juan Chacon ∗ Mo Chen ∗∗ Razvan C. Fetecau ∗∗∗∗
Department of Mathematics, Simon Fraser University, Burnaby BC,Canada, V5A 1S6, juan chacon [email protected] ∗∗ School of Computing Science, Simon Fraser University, BurnabyBC, Canada, V5A 1S6, [email protected] ∗∗∗
Department of Mathematics, Simon Fraser University, BurnabyBC, Canada, V5A 1S6, [email protected]
Abstract:
Autonomous systems operating in close proximity with each other to cover aspecified area has many potential applications, but to achieve effective coordination, two keychallenges need to be addressed: coordination and safety. For coordination, we propose alocally asymptotically stable distributed coverage controller for compact domains in the planeand homogeneous vehicles modeled with second order dynamics with bounded input forces.This control policy is based on artificial potentials designed to enforce desired vehicle-domainand inter-vehicle separations, and can be applied to arbitrary compact domains includingnon-convex ones. We prove, using Lyapunov theory, that certain coverage configurations arelocally asymptotically stable. For safety, we utilize Hamilton-Jacobi (HJ) reachability theoryto guarantee pairwise collision avoidance. Rather than computing numerical solutions ofthe associated HJ partial differential equation as is typically done, we derive an analyticalsolution for our second-order vehicle model. This provides an exact, global solution ratherthan an approximate, local one within some computational domain. In addition to considerablyreducing collision count, the collision avoidance controller also reduces oscillatory behaviour ofvehicles, helping the system reach steady state faster. We demonstrate our approach in threerepresentative simulations involving a square domain, triangle domain, and a non-convex movingdomain.
Keywords:
Autonomous robots; swarm intelligence; decentralized control; Hamilton-Jacobireachability; artificial potentials; coverage control1. INTRODUCTIONAutonomous systems have great potential to positivelyimpact society. However, these systems still largely operatein controlled environments in the absence of other agents.Autonomous systems cooperating in close proximity witheach other has the potential to improve efficiency. Specifi-cally, in this paper we consider the problem of controllingmultiple autonomous systems to cover a desired area ina decentralized and safe manner. To achieve effective co-ordination in this context, two key challenges need to beaddressed: coordination and safety.In coverage control problems, the objective is to deployagents within a target domain such that they can achievean optimal sensing of the domain of interest. A commonapproach to the coverage problem is by means of Voronoidiagrams (Cort´es et al. (2004); Gao et al. (2012)), wherethe goal is to minimize a coverage functional that involvesa tessellation of the domain and the locations of vehicleswithin the tesselation. This results in a high-dimensionaloptimization problem that has to be solved in real time.In our approach we achieve coverage (alternative termi-nologies are balanced or anti-consensus configurations)through swarming by artificial potentials (see Leonard andFiorelli (2001); Sepulchre et al. (2007) and also, the recentreview by Chung et al. (2018)). In a related problem,artificial potentials have been used for containment offollower agents within the convex hull of leaders (Ren andCao (2011); Cao et al. (2012)). (a) (b)
Fig. 1. Initial (a) and steady (b) states for covering asquare domain, with a square number of vehicles( N = 16).Reachability analysis as a safety verification tool has beenstudied extensively in the past several decades (Althoff andDolan (2011); Frehse et al. (2011); Chen et al. (2013)). Inparticular, Hamilton-Jacobi (HJ) reachability (Yang et al.(2013); Chen and Tomlin (2018)) has seen success in appli-cations such as collision avoidance (Gattami et al. (2011);Chen et al. (2016)), air traffic management (Margellos andLygeros (2013); Chen et al. (2018)), and forced landing(Akametalu et al. (2018)). HJ reachability analysis is basedon dynamic programming, and involves solving an HJpartial differential equation (PDE) to compute a backwardreachable set (BRS) representing the set of states fromhich danger is inevitable. Safety can therefore be guar-anteed, despite the worst-case actions of another agent, byusing the derived optimal controller when the system stateis on the boundary of the BRS. Contributions : In this paper we develop a new approach toself-collective coordination of autonomous agents/vehiclesthat aim to reach and cover a target domain. The mainfactors that we consider in our approach are: i) reachand spread over the target domain without having set apriori the coverage configuration and the final state of eachvehicle, ii) have a distributed control of vehicles in theabsence of any leaders, that allows for self-organizationand intelligence to emerge at the group level, and (iii)guarantee collision-avoidance throughout the coordinationprocess.In this aim, we consider a control system that includesboth a coverage and a safety controller. The coveragecontroller is designed to bring the vehicles inside andspread them over the target domain, while the secondguarantees collision avoidance of vehicles. In particular,the coverage controller uses artificial potentials for theinter-individual forces which are designed to achieve acertain desired inter-vehicle spacing (Leonard and Fiorelli(2001)). Such controllers enable emergent self-collectivebehaviour of the vehicles, similar to the highly coordinatedmotions observed in biological groups (e.g., flocks of birds,schools of fish); see Camazine et al. (2003).We emphasize that the coverage controller proposed here(which also includes the approach to the target) is donethrough agent swarming; there is no leader and no orderamong the agents. This feature offers robustness to thecontroller, as it does not have to rely on the well func-tioning of each individual agent. Self-collective and coop-erative behaviour in systems of interacting agents havebeen of central interest in physics and biology literature(see Vicsek et al. (1995); Couzin et al. (2002); D’Orsognaet al. (2006); Cucker and Smale (2007); Fetecau and Guo(2012)). A collaborative robot search and target location(without coverage) based on a swarming model was donein Liu et al. (2010).Our model is second-order, where agents are controlledthrough their acceleration; this is to be contrasted withfirst-order models, where the control inputs are the agents’velocities. We set a priori bounds on the control forces,making our controller more realistic than previous ap-proaches, where infinite forces were needed to guaranteecollision avoidance (Leonard and Fiorelli (2001); Husseinand Stipanovi´c (2007)). For an illustration, we show inFig. 1 the initial and final states for a simulation using ourcontroller for N = 16 vehicles that cover a square domain.The safety controller is derived from HJ reachability anal-ysis. Unlike the typical approach of numerically solving anassociated HJ PDE, we derive the analytical solution tothe PDE to eliminate numerical errors and computationbounds. While multi-vehicle collision avoidance is in gen-eral intractable, we observe drastically reduced collisionrate by just considering pairwise interactions.2. BACKGROUND An oriented measure of how far a point x is from a givendomain Ω is the signed distance, defined by b ( x ) = min x p ∈ ∂ Ω k x − x p k , if x / ∈ Ω − min x p ∈ ∂ Ω k x − x p k , if x ∈ Ω . If ∇ b ( x ) exists, then there exists a unique P ∂ Ω ( x ) ∈ ∂ Ω,called the projection of x on ∂ Ω, such that b ( x ) = (cid:26) k P ∂ Ω ( x ) − x k , if x / ∈ Ω − k P ∂ Ω ( x ) − x k , if x ∈ Ω (1)and ∇ b ( x ) = x − P ∂ Ω ( x ) b ( x ) . (2) Consider the two-player differential game described by thejoint system ˙ z ( t ) = f ( z ( t ) , u ( t ) , d ( t )) (3) z (0) = x, where z ∈ R n is the joint state of the players, u ∈ U is the control input of Player 1 (hereafter referred to as“control”) and d ∈ D is the control input of Player 2(hereafter referred to as “disturbance”) .We assume f : R n × U × D → R n is uniformly continuous,bounded, and Lipschitz continuous in z for fixed u and d ,and u ( · ) ∈ U , d ( · ) ∈ D are measurable functions. Underthese assumptions we can guarantee the dynamical system(3) has a unique solution.In this differential game, the goal of player 2 (the dis-turbance) is to drive the system into some target setusing only non-anticipative strategies, while player 1 (thecontrol) aims to drive the system away from it.We introduce the time-to-reach problem as follows. (Time-to-reach) Find the time to reach a target Γ from any initial state x , in a scenario where player 1maximizes the time, while player 2 minimizes the time.Player 2 is restricted to using non-anticipative strategies,with knowledge of player 1’s current and past decisions.Such a time is denoted by φ ( x ) . Following Yang et al. (2013), the time to reach a closedtarget Γ ⊂ R with compact boundary, given u ( · ) and d ( · ) is defined as T x [ u, d ] = min { t | z ( t ) ∈ Γ } , and the set of non-anticipative strategies asΘ = { θ : U → D| u ( τ ) = ˆ u ( τ ) ∀ τ ≤ t ⇒ θ [ u ] ( τ ) = θ [ˆ u ] ( τ ) ∀ τ ≤ t } . Given the above definitions, the
Time-to-reach problem isequivalent to the differential game problem as follows: φ ( x ) = min θ ∈ Θ max u ∈U T x [ u, θ [ u ]] . The collection of all the states that are reachable in a finitetime is the capturability set R ∗ = { x ∈ R n | φ ( x ) < + ∞} .Applying the dynamic programming principle, we can ob-tain φ as the viscosity solution for the following stationaryHJ PDE:min u ∈U max d ∈D {−∇ φ ( z ) · f ( z, u, d ) − } = 0 , in R ∗ \ Γ (4) φ ( x ) = 0 , on Γ . reviously, this PDE has typically been solved using finitedifference methods such as the Lax-Friedrichs method(Yang et al. (2013)).From the solution φ ( x ) we can obtain the optimal avoid-ance control as u ∗ ( x ) = arg min u ∈U max d ∈D {−∇ φ ( z ) · f ( z, u, d ) − } . (5)3. PROBLEM FORMULATIONWe consider a group of N vehicles, each of them denotedas Q i , i = 1 , . . . , N , with dynamics described by˙ p i = v i , ˙ v i = u i , k v i k ≤ v max , k u i k ≤ u max . (6)Here, p i = ( p i,x , p i,y ) and v i = ( v i,x , v i,y ) are the positionand velocity of Q i respectively, and u i = ( u i,x , u i,y ) is thecontrol force applied to this mobile agent.We consider a vehicle safe if there is no other vehicle closerthan a predefined collision radius c r , in other words, if k p i − p j k > c r , for any j = i. (7) Definition 1. (r-Subcover).
A group of agents is an r -subcover for a compact domain Ω ⊆ R if:(1) The distance between any two vehicles is at least r .(2) The signed distance from any vehicle to Ω is less thanequal to − r . Definition 2. (r-Cover). An r -subcover for Ω is an r -cover for Ω if its size is maximal (i.e., no larger numberof agents can be an r -subcover for Ω).The r -subcover definition is closely related to finding a wayto pack circular objects of radius r inside of a containerwith shape Ω. Having an r -cover implies the container isfull and there is no room for more of such objects.We are interested in the following safe domain coverageproblem. (Safe-domain-coverage) Consider a compact domain Ω in the plane and N vehicles each with dynamics describedby (6) , starting from safe initial conditions. Find themaximal r > and a control policy that leads to a stablesteady state which is an r -cover for Ω , while satisfying thesafety condition (7) at any time.
4. METHODOLOGYThe controller we design and present has two components.First, each vehicle in the group evolves according to acoverage controller that consists in interaction forces withthe rest of the vehicles and with the boundary of thetarget domain, as well as in braking forces in the currentdirection of motion. Second, a safety controller, basedon Hamilton-Jacobi reachability theory, activates whentwo vehicles come within an unsafe region with respectto each other. The desired distance r is built into thecoverage controller by means of artificial potentials. Forcertain setups we prove that our proposed coverage controlstrategy is asymptotically stable, which leads to an r -coverfor the domain when this is admissible. For notational simplicity, let p ij := p i − p j , denote by h i the shortest vector connecting p i with ∂ Ω, i.e. h i = p i − Fig. 2. Illustration of control forces acting on two vehicleslocated at p i and p j . P ∂ Ω ( p i ), and let [[ h i ]] := b ( p i ). The proposed control forceis given as u i = − N X j = i f I ( k p ij k ) p ij k p ij k − f h ([[ h i ]]) h i [[ h i ]] (8)+ f v i ( k v i k ) v i k v i k , where the three terms in (8) represent inter-vehicle,vehicle-domain, and braking forces, respectively. Figure 2illustrates the control forces for two generic vehicles lo-cated at p i and p j . Shown there are the unit vectors in thedirections of the inter-vehicle and vehicle-domain forces(yellow and blue arrows, respectively), along with theresultant that gives the overall control force (red arrows).Note that due to the nonsmoothness of the boundary,different points may have different types of projections: p i projects on the foot of the perpendicular to ∂ Ω, while p j projects on a corner point of ∂ Ω.We assume the following forms for the functions f I , f h and f v i that appear in the various control forces in Equation(8). Figure 3 shows the inter-vehicle force f I and thevehicle-domain force f h . Note that f I ( r ) is negative for r < r d , and zero otherwise. This means that for twovehicles within distance 0 < r < r d from each other, theirinter-vehicle interactions are repulsive, while two vehiclesat distance larger than r d apart do not interact at all. Thevehicle-domain force f h ( r ) is zero for r < − r d , and positivefor r > − r d . For a vehicle i outside the target domain,i.e., with [[ h i ]] >
0, this results in an attractive interactionforce toward ∂ Ω. On the other hand, for a vehicle insidethe domain, where [[ h i ]] <
0, one distinguishes two cases:i) the vehicle is within distance r d to the boundary, inwhich case it experiences a repulsive force from it, or ii)the vehicle is more than distance r d from the boundary,in which case it does not interact with the boundary atall. Finally, we take f v i ( k v i k ) to be negative to result in abraking force; a specific form of f v i will be chosen belowfor analytical considerations (see Equation (10)).An important ingredient of our controller is that one canassociate a Lyapunov function to it and hence, investigateanalytically the stability of its solutions. We address theseconsiderations now. Lemma 3.
The vehicle-domain force − f h ([[ h i ]]) h i [[ h i ]] andthe inter-vehicle force − f I ( k p ij k ) p ij k p ij k are conservative.ig. 3. Inter-vehicle and vehicle-domain control forces. Proof.
Let us consider the potential V h ( p i ) = Z [[ h i ]] − rd f h ( s ) ds, which satisfies ∇ i V h ( p i ) = f h ([[ h i ]]) ∇ ([[ h i ]])= f h ([[ h i ]]) h i [[ h i ]] . where we have used Equations (1) and (2) for [[ h i ]] and ∇ ([[ h i ]]) respectively (see Theorem 5.1(iii) in Delfour andZol´esio (2001)).Similarly, it can be shown that the inter-vehicle force isthe negative gradient of the potential V I ( p ij ) = Z k p ij k r d f I ( s ) ds. ✷ Using Lemma 3, the control given in Equation (8) becomes u i = N X j = i − ∇ i V I ( p ij ) − ∇ i V h ( p i ) + f v i ( k v i k ) v i k v i k . (9) Asymptotic behaviour of the controlled system.
Considerthe following candidate for a Lyapunov function, consist-ing in kinetic plus (artificial) potential energy:Φ = 12 N X i =1 (cid:16) ˙ p i · ˙ p i + N X j = i V I ( p ij ) + 2 V h ( p i ) (cid:17) . Note that each term in Φ is non-negative, and Φ reachesits absolute minimum value when the vehicles are totallystopped. Also, at the global minimum Φ = 0, the equilib-rium configuration is an r d -subcover of Ω; in particular,all vehicles are inside the target domain.The derivative of Φ with respect to time can be calculatedas: ˙Φ = N X i =1 ˙ p i · (cid:16) u i + N X j = i ∇ i V I ( p ij ) + ∇ i V h ( p i ) (cid:17) = N X i =1 ˙ p i · f v i ( k v i k ) v i k v i k = N X i =1 f v i ( k v i k ) k v i k , where we used the dynamics (6) and equation (9). Thus,if we choose f v i ( k v i k ) = − a i k v i k , with a i > , i = 1 , . . . , N, (10)then ˙Φ is negative semidefinite and equal to zero if andonly if ˙ p i = 0 for all i (i.e., all vehicles are at equilib-rium). By LaSalle Invariance Principle we can conclude that the controlled system approaches asymptotically anequilibrium configuration.For certain simple setups (e.g., a square number of vehiclesin a square domain or a triangular number of vehicles ina triangular domain – see Figs. 1b and 5d), the r d -coversare isolated equilibria. Hence, together with the fact thatsuch equilibria are global minimizers for Φ, their localasymptotic stability can be inferred. The formal result isgiven by the following proposition. Proposition 4.
Consider a group of N vehicles with dy-namics defined by (6), and the control law given by (9)and (10). Let the equilibrium of interest be of the form˙ p i = 0, k p ij k ≥ r d and [[ h i ]] ≤ − r d for i, j = 1 , · · · , N (see Definitions 1 and 2), and assume that this equilib-rium configuration is isolated. Also assume that there is aneighborhood about the equilibrium in which the controllaw remains smooth. Then, the equilibrium is a globalminimum of the sum of all the artificial potentials andis locally asymptotically stable.Choosing an adequate r d when solving the safe-domain-coverage problem leads us to a nonlinear optimizationproblem (see L´opez and Beasley (2011)) which can bedifficult in itself. Our heuristic approach for picking thisparameter relies on the premise that any vehicle is coveringroughly the same square area, i.e., r d = r Area (Ω)
N . (11)Note that (11) gives the exact formula for the maximalradius when both the number of vehicles and the domainare square, making the r d -covers isolated equilibria, a keyassumption in Proposition 4. Also, this particular choicefor r d leads to the desired cover for several domains withregular geometries (see Section 5). The dynamics between two vehicles Q i , Q j can be definedin terms of their relative states p r,x = p i,x − p j,x p r,y = p i,y − p j,y v r,x = v i,x − v j,x v r,y = v i,y − v j,y , where the vehicle Q i is the evader and Q j is the pursuer,the latter being considered as the model disturbance. Inthis case the relative dynamical system is given by˙ p r,x = v r,x ˙ p r,y = v r,y ˙ v r,x = u i,x − u j,x ˙ v r,y = u i,y − u j,y k u i k , k u j k ≤ u max , (12)where u i = ( u i,x , u i,y ) and u j = ( u j,x , u j,y ) are the controlinputs of the agents Q i and Q j , respectively. From theperspective of agent Q i , the control inputs of Q j , u j =( u j,x , u j,y ), are treated as worst-case disturbance.According to (7), the unsafe states are described by thetarget set Γ = (cid:8) z : p r,x + p r,y ≤ c r (cid:9) . Consider the time it takes for the solution of the dynamicalsystem (12), with starting point z in R ∗ \ Γ, to reachΓ when the disturbance and control inputs are optimal.sing a geometric argument one can show that this timeis the minimum of the two solutions of the followingquadratic equation: (cid:0) v r,x + v r,y (cid:1) ψ ( z ) + 2 ( p r,x v r,x + p r,y v r,y ) ψ ( z ) (13)+ (cid:0) p r,x + p r,y − c r (cid:1) = 0 . Let us verify first that the minimum of the two solutionssatisfies indeed the HJ PDE (4).
Proposition 5.
Consider the function ψ ( z ) defined as ψ ( z ) := − ( p r,x v r,x + p r,y v r,y ) − √ ∆ v r,x + v r,y in R ∗ \ Γ , where∆ = ( p r,x v r,x + p r,y v r,y ) − (cid:0) v r,x + v r,y (cid:1) (cid:0) p r,x + p r,y − c r (cid:1) . Also define ψ ( z ) to be 0 on Γ. Then ψ ( z ) satisfies equation(4). Proof.
For z ∈ R ∗ \ Γ, ψ ( z ) is the minimum solution ofthe quadratic equation (13). By using implicit differentia-tion on (13) one can show ∂ψ∂p r,x = − v r,x ψ ( z ) + p r,x (cid:0) v r,x + v r,y (cid:1) ψ ( z ) + ( p r,x v r,x + p r,y v r,y ) ∂ψ∂p r,y = − v r,y ψ ( z ) + p r,y (cid:0) v r,x + v r,y (cid:1) ψ ( z ) + ( p r,x v r,x + p r,y v r,y ) . To put system (12) in the general form (3) from Sec. 2.2,let z = ( p r,x , p r,y , v r,x , v r,y ), u = ( u x , u y ) := ( u i,x , u i,y ), d = ( d x , d y ) := ( u j,x , u j,y ), and let f ( z, u, d ) represent theright-hand-side of (12). Then,min u ∈U max d ∈D {−∇ ψ ( z ) · f ( z, u, d ) − } = min u ∈U max d ∈D (cid:26) − ∂ψ ( z ) ∂p r,x v r,x − ∂ψ ( z ) ∂v r,x ( u x − d x ) − ∂ψ ( z ) ∂p r,y v r,y − ∂ψ ( z ) ∂v r,y ( u y − d y ) − (cid:27) = − (cid:18) ∂ψ ( z ) ∂p r,x v r,x + ∂ψ ( z ) ∂p r,y v r,y (cid:19) −
1= ( v r,x ψ ( z ) + p r,x ) v r,x + ( v r,y ψ ( z ) + p r,y ) v r,y (cid:0) v r,x + v r,y (cid:1) ψ ( z ) + ( p r,x v r,x + p r,y v r,y ) −
1= 0 , where we have used the fact that ψ ( z ) is differentiable in R ∗ \ Γ and that the minimum and maximum in the secondequal sign are attained at u ∗ = u max (cid:16) ∂ψ ( z ) ∂v r,x , ∂ψ ( z ) ∂v r,y (cid:17)(cid:13)(cid:13)(cid:13) ∂ψ ( z ) ∂v r,x , ∂ψ ( z ) ∂v r,y (cid:13)(cid:13)(cid:13) (14a) d ∗ = u max (cid:16) ∂ψ ( z ) ∂v r,x , ∂ψ ( z ) ∂v r,y (cid:17)(cid:13)(cid:13)(cid:13) ∂ψ ( z ) ∂v r,x , ∂ψ ( z ) ∂v r,y (cid:13)(cid:13)(cid:13) (14b)and therefore cancel out.For z ∈ Γ the equation (4) is satisfied by the definition of ψ . ✷ Similarly as before, by implicit differentiation of (13) onecan also show ∂ψ∂v r,x = − v r,x ψ ( z ) + p r,x ψ ( z ) (cid:0) v r,x + v r,y (cid:1) ψ ( z ) + ( p r,x v r,x + p r,y v r,y )(15a) ∂ψ∂v r,y = − v r,y ψ ( z ) + p r,y ψ ( z ) (cid:0) v r,x + v r,y (cid:1) ψ ( z ) + ( p r,x v r,x + p r,y v r,y ) . (15b)Now, by using (14a) we can derive a closed expression forthe optimal avoidance controller.For applications using the static HJ PDE (4) or similarequations, the solution is commonly approximated viafinite difference methods such as the one presented inYang et al. (2013); however, using an analytic solutionleads to two main advantages. First, refinements in theresolution when using uniform grid point spacing mayrequire a large amount of memory and long computationaltimes that scale very poorly. Second, these methods areonly able to compute the solution in a bounded domain.Having an analytical solution allows us to have the bestpossible resolution in an unbounded domain. Practically,this allows us to predict and react to possible collisionsarbitrarily far into the future. We have presented two controllers. First, a controller basedon virtual potentials which leads to coverage, but withouttaking into consideration safety; and second, a safetycontroller that guarantees pairwise collision avoidance.The main objective of this subsection is to describe howto switch between these two controllers.We will consider that vehicle Q i is in potential conflict withvehicle Q j if the time to collision ψ ( z i ) (time to reach Γ),given the relative current state z i , is less than or equal toa specified time horizon t safety . In other words, a dangeris defined when a vehicle is susceptible to collision in thenext t safety seconds. In such a case Q i must use the safetycontroller, otherwise, the coverage controller is used.In the case that a vehicle detects more than one conflict,it will apply the control policy of the first conflict detectedat that particular time. Algorithm 1 describes the overallcontrol logic for every vehicle Q i .In Algorithm 1, lines 6 and 7 can be obtained from equa-tions (15a), (15b) and (14a) (also note the normalizationstep in line 14), while line 12 comes from the explicitcoverage control (8).5. NUMERICAL SIMULATIONSIn this section, we show numerical simulations with var-ious domains using the coverage and safety controllersdiscussed above. We first consider the problem in which several vehiclescover a square domain. Here we present two strategies:while both of them use the coverage controller describedin Sec. 4.1, only the second strategy switches to the safetycontroller when necessary, according to Sec. 4.3. In bothcases 16 vehicles start from a horizontal line setup outsideof the target square domain, as shown in Fig. 1a.The left column of Fig. 4 illustrates different time stepsfor the scenario when the safety controller is not used, nput:
State x i of a vehicle Q i ; states { x j } j = i ofother vehicles { Q j } j = i ; a domain Ω tocover. Parameter:
A time horizon for safety check t safety ; Output:
A control u i for Q i . saf e ← True; for j = i do z ← x i − x j ; if ψ ( z ) ≤ t safety then saf e ← False; U ix = − v r,x ψ ( z )+ p r,x ψ ( z ) ( v r,x + v r,y ) ψ ( z )+( p r,x v r,x + p r,y v r,y ) ; U iy = − v r,y ψ ( z )+ p r,y ψ ( z ) ( v r,x + v r,y ) ψ ( z )+( p r,x v r,x + p r,y v r,y ) ; break for end end if safe then ( U ix , U iy ) =- P Nj = i f I ( k p ij k ) p ij k p ij k - f h ([[ h i ]]) h i [[ h i ]] + f v i ( k v i k ) v i k v i k ; end u i = u max ( U ix ,U iy ) k ( U ix ,U iy ) k ; return u i ; Algorithm 1:
Overall control logic for a generic vehicle Q i .while the right column shows results in the presence ofthe safety controller. The large coloured dots representthe position of the vehicles, the dashed tails are pasttrajectories (shown for the previous 5 seconds), and thearrows indicate the movement direction. Note howeverthat we have not shown the arrows when the velocitiesare very small, as the tails are more meaningful in thiscase.At t = 0 (s) there are no contributions from the inter-vehicle forces or from the safety controller as the vehiclesare far from each other and the initial speed is zero. Theonly contribution comes from the vehicle-domain forces,which pull the mobile agents toward the interior of thesquare; see initial trajectory tails in Figs. 4a and 4b. At t = 4 . t = 60 (s) both control strategies reach the steadystate shown in Fig. 1b. We note however that the systemwith collision avoidance reaches the equilibrium faster, asshown by Figs. 4e and 4f.A collision event starts when the distance between twovehicles is less than or equal to the collision radius c r ,and ends when the distance becomes greater than c r . Thecollision event count for the square domain coverage withand without the safety controller, for various number of (a) (b)(c) (d)(e) (f) Fig. 4. Square domain coverage at different time instants,without (left) and with (right) safety controller, when N = 16, c r = 2 (m), v max = 10 (m/s), u max =3 (m/s ), t safety = 5 (s), side length l = 20 (m),domain area A = l = 400 (m ) and r d = q AN =5 (m). Vehicles start in a horizontal line configurationand reach a square grid steady state which is an r d -cover of the domain (see Definition 2). The use of thesafety controller reduces both the collision count andthe overshoot, and helps reach the steady state faster.vehicles, is shown in Table 1. It is noteworthy that in theabsence of the safety controller the collision count increasessignificantly with the number of vehicles, while it remainszero when the safety controller is used.Table 1. Square coverage collision count. number of vehicles without avoidance with avoidance9 12 016 39 025 124 0 Now we consider the scenario in which an equilateraltriangular domain is covered by a triangular number of a) (b)(c) (d)
Fig. 5. Equilateral triangular domain coverage at differenttimes instants, with safety controller, when N = 15, c r = 2 (m), v max = 10 (m/s), u max = 3 (m/s ), t safety = 5 (s), side length l = √ (m), domain area A = √ l = √ (m ) and r d = q AN = √ √ (m).The vehicles start in a horizontal line configurationand reach a steady state formation that is an r d -coverof the domain.vehicles, i.e. N = n ( n +1)2 , n ∈ N . At the start of thesimulation the vehicles lie on a horizontal line outside thedomain. The evolution for a group of N = 15 agents, eachof them using the coverage and pairwise safety strategiesdiscussed earlier, is illustrated in Fig. 5. The tails representthe 5-second history of the vehicle positions.The triangle coverage simulation was run for 6, 10 and15 vehicles. Table 2 shows the collision count, with andwithout the safety controller. Again, the use of the pairwisesafety strategy reduces the number of collisions consider-ably. We note here that safety issues may arise when avehicle needs to avoid more than two vehicles at the sametime. In this case the pairwise safety approach used in thispaper does not guarantee collision avoidance. Guaranteedcollision avoidance for more than two vehicles is exploredin Chen et al. (2016).Table 2. Triangle coverage collision count number of vehicles without avoidance with avoidance6 9 010 23 015 72 2 Finally, we consider the scenario in which vehicles coverand follow a non-convex moving domain. While the do-main preserves its shape, it moves with a constant velocity v Ω = (0 . , . (a) (b)(c) (d) Fig. 6. Vehicles covering and following a moving, non-convex domain, when N = 9, c r = 2 (m), v max =10 (m/s), u max = 3 (m/s ), t safety = 5 (s), domainarea A = 225 (m ) and r d = q AN = 5 (m). Thevehicles start in linear formation, approach and coverthe domain, while following it. The vehicles laggingbehind exhibit oscillations due to a bouncing effect inthe narrow corners.positions during the last 30 seconds of the simulation.Initially, all the 9 vehicles lie on a line perpendicular tothe movement direction of the target domain, as shown inFig. 6a.We distinguish two main behaviours: during a first phaseof the simulation (Figs. 6b and 6c) the vehicles cover thedomain approximately evenly, adopting the arrow shape,while in a second phase (Fig. 6d), a clearer domain-following behaviour is observed. The oscillations of thetwo vehicles that are lagging behind are the effect oftheir proximity to the corners. Indeed, as one of the linesegments of the boundary wedge gets closer to the vehiclenear the corner, it pushes it towards the other segmentof the wedge, a back-and-forth motion that causes thezigzagging.Unlike the convex case, in non-convex domains the pro-jection on the boundary for points outside of the domainmay not be unique; this is the case for instance of thegreen vehicle in the middle of the initial setup – see Fig.6a. Although the chance for a vehicle to lie in one of thesestates is extremely unlikely (the set of points where thishappens has zero measure), this fact may yield ambiguityin the definition of the domain-vehicle force. We mitigatethis issue by considering the contribution from only one ofthe multiple projection points; consequently, the numeri-cal time evolution may depend on the chosen projectionmethod.. CONCLUSIONIn this paper, we proposed a method for safe multi-vehiclecoordination which allows a swarm of vehicles to cover anycompact planar shape. Vehicles are modeled using second-order dynamics with bounded acceleration, which is morerealistic than the first-order models commonly used forcoverage problems. The coverage controller is based on ar-tificial potentials among vehicles, and between each vehicleand the domain of interest. Using Lyapunov analysis, weprove that our algorithm guarantees coverage. The safetycontroller is based on Hamilton-Jacobi reachability, whichguarantees pairwise collision avoidance. Besides drasticallyreducing collision count, the safety controller also helpsbreak symmetries and lead to faster convergence to steadystates. We demonstrate our approach on three represen-tative simulations involving a square domain, a triangledomain, and a non-convex moving domain.Immediate future work includes parameter tuning toreduce oscillations in the vehicles’ movement, study-ing three-dimensional coverage, investigating geometricalproperties of steady states, investigating scenarios involv-ing partial information, and implementing our approachon robotic platforms.REFERENCESAkametalu, A.K., Tomlin, C.J., and Chen, M. (2018).Reachability-Based Forced Landing System. Journal ofGuidance, Control, and Dynamics , 41(12), 2529–2542.doi:10.2514/1.G003490.Althoff, M. and Dolan, J.M. (2011). Set-based com-putation of vehicle behaviors for the online verifica-tion of autonomous vehicles. In
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