Scalable Formation Control of Multi-Robot Chain Networks using a PDE Abstraction

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1 Salable Formation Control of Multi-Robot Chain Networks using a PDE Abstration Karthik Elamvazhuthi and Spring Berman Abstrat This work investigates the appliation of boundary ontrol of the wave equation to ahieve leader-indued formation ontrol of a multi-robot network with a hain topology. In ontrast to previous related work on ontrolling formations of single integrator agents, we onsider a model for double integrator agents. For trajetory planning, we use the flatness based method for assigning trajetories to leader agents so that the agents trajetories and ontrol inputs are omputed in a deentralized way. We show how the approximation greatly simplifies the planning problem and the resulting synthesized ontrols are bounded and independent of the number of agents in the network. We validate our formation ontrol approah with simulations of and agents that onverge to onfigurations on three different type of target urves. Introdution A onsiderable amount of effort has been applied in reent years to problems of ahieving onsensus, overage, task alloation, and oordinated motion in multirobot systems. In partiular, ertain multi-robot appliations will require formation ontrol of the robots to positions along a speified losed or open urve within a ertain amount of time. For instane, the urve ould lie along an objet to be transported or a struture to be monitored, or it ould ontain target formations or floking trajetories for aerial vehiles and spaeraft. Moreover, formation ontrol provides useful benhmarks for investigating the range of oordinated behaviors that an be ahieved under the onstraints that are typial to multi-robot systems. These onstraints inlude unreliable or absent ommuniation and global information, limited resoures for sensing and omputation, and the presene of unpreditable envi- Karthik Elamvazhuthi Spring Berman Shool for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, USA; karthikevaz@asu.edu,spring.berman@asu.edu

2 Karthik Elamvazhuthi and Spring Berman ronmental disturbanes. In addition, ontrol shemes for oordinating large multirobot systems should be salable to arbitrary robot population sizes. The virtual struture method is a well-known approah to formation ontrol [8]. It is based on a ombination of onsensus and graph rigidity onepts, in whih agents know the target distane to be maintained from their neighbors on a graph that defines their interation topology. While maintaining these inter-agent distanes, they must also simultaneously reah onsensus on the enter of the formation. Other approahes to this problem inlude potential field methods [] and ontrol Lyapunov funtions for multi-agent oordination [5]. Alternative approahes to multi-robot formation ontrol have been derived from partial differential equation (PDE) models of the system. The appliability of many of these PDE models is based on the fat that finite differene approximations of differential operators on ontinuous domains have the same struture as analogous operators defined on graphs (e.g., the Laplaian) and also provide intuition on the use of analogous operators on graphs suh as advetion-based oordination [5]. In [3], finite differene approximations of PDEs used in image proessing are applied to a ooperative boundary overage problem. The work in [7] used another numerial solution method for PDEs, the smoothed partile hydrodynamis method, to model finite-sized robots as an inompressible fluid, inorporating nonholonomi onstraints and obstale avoidane. PDE models were used in [] to analyze the string stability of large vehile platoons, and PDE approximations of vehile platoons were applied in [] to study the saling behavior of system stability margins with the number of agents. Linear and nonlinear advetion-diffusion models are used in [8] to enable deployment of multiple agents into formations using a boundary ontrol methodology. Similar work was done by [3] on planar deployments of multiple agents using flatness-based trajetory planning of the Burgers equation. Fluid flow models are also extensively used in traffi flow problems [7]. Closely related to partial differential equations is the onept of partial differene equations on graphs, whih has also been a subjet of a number of studies. Spatially invariant systems are another type of infinite-dimensional approximation of large-sale networks. These have proven to be quite insightful in understanding saling laws and struture-dependent performane limitations of large vehile networks [], []. In this work, we address a formation ontrol problem for a multi-agent network with an undireted hain graph topology. None of the agents have ommuniation apabilities, and two of the agents (the leader agents) have global position information while the remaining robots (the follower agents) have only loal sensing and annot measure their global positions. Eah agent s motion is governed by double integrator dynamis, whih integrate fore atuation as the loal ontrol parameter. For agents with double integrator dynamis, the wave equation serves as a useful abstration. This is in ontrast with the single integrator agent models studied in [8, 3], whih employed advetion-diffusion equation abstrations for formation ontrol. We demonstrate that trajetory planning based on the wave equation an impliitly aount for delays in system ontrollability that are inherent to this type of model. In doing so, our approah enables the synthesis of bounded ontrol inputs that drive hain networks with arbitrarily large numbers of agents to on-

3 Salable Formation Control of Multi-Robot Chain Networks using a PDE Abstration 3 figurations on target open and losed urves. For our trajetory planning approah, we use the flatness based method, a well-known method in the ontext of finitedimensional systems []. Namely, using the so-alled flat output, we ahieve an expliit parametrization of the system states and ontrol input. Sine its introdution, the method has been extended to infinite-dimensional systems as well [9, ]. This approah serves as a useful alternative to optimal ontrol methods, whih require numerous yles of integration and, in the ase of PDE models, lead to illonditioning issues arising from numerial disretization. Problem Formulation We onsider a group of N non-ommuniating agents that move in the spae R n, n {,,3}. The position and ontrol input of agent j at time t are denoted by z j (t)=[z j (t)... zn j (t)]t R n and u j (t)=[u j (t)... un j (t)]t R n, respetively. We assume that agent j an measure its distane from two other agents j and j + at all times, or in other words, that agent j is onneted to agents j and j +. The agent interonnetion topology forms a one-dimensional undireted hain graph. No agent an measure its global position z j (t) exept for agents and N, whih we all the leader agents. The positions of the leader agents evolve aording to speified trajetories: z i (t)=ui (t), zi N(t)=u i N(t), i =,...,n. () The dynamis of the follower agents j =,...,N with ontrol inputs on their aeleration: are given by double-integrators u i j(t)= d z i j (t) dt = (z i j+ (t) zi j(t)) (z i j(t) z i j (t)) + f i j, i =,...,n, () where and f j i are onstants. The ontrol objetive is to drive the agents positions to points along a target open or losed urve, g : []! R n, at equilibrium. We will show that we an ahieve this objetive by designing the leader agents position ontrol inputs, u (t) and u N (t), and the follower agents onstant aeleration inputs f j i. Toward this end, we define h = /N and rewrite Eq. () as u i j(t)= d z i j (t) dt =(h) (zi j+ (t) zi j (t)+zi j (t)) h + f i j, i =,...,n. (3) As N!, Eq. (3) onverges to n one-dimensional partial differential equations (PDE s) that evolve in time over a ontinuous spatial domain, whih we normalize to the interval [ ]. The agent population is represented as a ontinuum with a spatial distribution in dimension i given by z i (x,t), x [ ]. Note that the positions of the leader agents are then z i (,t) and z i (,t), whih are defined at the endpoints of the domain. We assume zero initial onditions, so the entire agent population begins at

4 4 Karthik Elamvazhuthi and Spring Berman rest at the origin. The spatiotemporal evolution of z i (x,t) is then governed by the following set of n fored wave equations with time-dependent Dirihlet boundary onditions: z i (x,t) t =(h) z i (x,t) x + f i (x) (4) z i (,t) =u i (t), zi (,t)=u i N(t) (5) z i (x,) =, dz i (x,) dt =. (6) The time-independent soure funtion f i (x) an be designed to speify the target urve g to whih the agent positions onverge at equilibrium. The produt h defines the speed of wave propagation over the domain. We will use the nonhomogeneous boundary value problem Eq. (4) Eq. (6) to plan trajetories for the multi-agent system. We will disuss how this ontinuous approximation simplifies the planning problem and takes into aount some inherent limitations that are not obvious in the original disrete ontrol system Eq. (), Eq. (). 3 Limitations on Controllability The ontinuous PDE approximation, whih is an infinite-dimensional system, shares ertain ontrollability limitations with the original system, whih has a finitedimensional state spae. If a finite-dimensional system is ontrollable at a partiular time, the Kalman rank ondition an be used to show that the system is ontrollable at any time. This result implies that leader-follower protools on grid networks are ontrollable for any number of agents in the network. Analysis of the ontrollability gramian shows that as the agent population inreases, there is a rise in the minimum energy required to drive the network to a hosen target state [3, 6]. However, there is a more fundamental problem in ontrolling large networks of double-integrator agents than the need for a orrespondingly large amount of ontrol energy. This problem is the onstraint on the minimum time needed to drive the network to a target state. In ontrast to finite-dimensional systems, infinite-dimensional systems have muh more varied notions of ontrollability. In partiular, for a system desribed by a wave equation whose Laplaian operator has oeffiient k, a minimum time of T = /k is required to attain exat ontrollability of the system [9]. This is due to the finite speed of propagation of a wave over a one-dimensional spatial domain. Even if a leader agent introdues an infinite amount of ontrol effort, information annot travel faster than this speed through the network. Moreover, high-frequeny disturbanes over the network will take even longer to stabilize due to the relatively lower speed of the wave pakets. Conversely, systems desribed by the heat equation, whih an be onsidered to be an infinite-dimensional version of the single-

5 Salable Formation Control of Multi-Robot Chain Networks using a PDE Abstration 5 integrator based Laplaian models [4], are not subjet to a delay in ontrollability. These models have an infinite speed of information propagation over the domain, and hene ontrollability at any time, albeit only approximately in the ontinuous ase. Consequently, while a single-integrator agent network with a grid topology an be ontrolled to any state at any time, the minimum time T needed to ontrol a double-integrator network with ontinuous approximation Eq. (4) inreases with the number of agents N as T = /(h)=n/. For large N, it therefore takes a long time for the ontrol effort of a leader agent to propagate through the entire network, resulting in a large set of states that are unreahable for times t apple T. This issue an be identified with the problem of numerially approximating optimal ontrollers of the wave equation by onstruting optimal ontrollers of the orresponding semidisrete system [4]. When the ontrols are onstruted in this manner, they diverge as the number of mesh points (equivalent to agents in our ase) inreases, in spite of the onvergene of the disrete model to the ontinuous model. This divergene has been attributed to (a) the finite time needed for ontrollability, and (b) the mismath between the wave speeds of high-frequeny perturbations in disrete and ontinuous media [4]. Owing to the riher dynamis of the semi-disrete system, the minimum time required to reah a target state might be even higher than that indiated by the ontinuum approximation. 4 Trajetory Planning In this setion, we show that trajetory planning based on the wave equation an impliitly aount for the delay in system ontrollability that is desribed in Set. 3. In doing so, this approah enables the synthesis of bounded ontrol inputs that drive hain networks with arbitrarily large numbers of agents to target onfigurations. We modify the follower agent ontrol inputs defined in Eq. () by saling the onstant by the agent population N, whih hanges the oeffiient to (N) = (/h). Then, as N!, Eq. () onverges to the wave equation in Eq. (4) with the oeffiient (h) replaed by. The orresponding speed of wave propagation is, whih is independent of the number of agents. The ontrol effort per agent defined in Eq. () remains bounded as N! due to the onvergene of the semi-disrete system to its ontinuous approximation. Stritly speaking, this argument requires strong onvergene, whih an be easily ahieved by introduing a small amount of damping in the system using veloity-based ompensation [4]. To simplify the onstrution of the ontrol inputs, we deompose the boundary value problem Eq. (4) Eq. (6), with (h) replaed by, into two omponents. This deomposition is possible beause the wave equation Eq. (4) is a linear PDE. The first omponent is a boundary value problem for an unfored wave equation (note that the supersript i has been suppressed to simplify the notation):

6 6 Karthik Elamvazhuthi and Spring Berman z(x,t) t = z(x,t) x (7) z(,t) =, z(,t)=u a (t) (8) z(x, ) =, z(x,) = t (9) The seond omponent is a boundary value problem for a fored wave equation: z(x,t) t = z(x,t) x + f (x) () z(,t) =u (t), z(,t)=u b (t) () z(x, ) =, z(x,) = t () Here, u a (t) drives the system to equilibrium, while u (t) and u b (t) shift the datum of the solution depending on the desired target state. 4. Unfored Wave Equation Component Following the approah of [] for flatness based trajetory generation, we take the Laplae transform of Eq. (7) in the time variable and obtain The general solution of this equation is Z(x, s) =A(s) osh s Z(x,s)= d Z(x,s) dx (3) xs xs + B(s)sinh where A(s) and B(s) are arbitrary funtions of s. Applying the boundary ondition Z(,s)= to Eq. (4), we find that A(s)=. Now define the funtion r(t)= z(,t)/ x. The Laplae transform of this funtion is R(s) =dz(,s)/dx, whih is the derivative of Eq. (4) with x =. This derivative is R(s) =B(s) s osh() = B(s) s, whih yields B(s) =R(s) s. Let y(t) denote the flat output, and define its Laplae transform as Y (s)=r(s) s. Then, Eq. (4) beomes Z(x,s) =Y (s)sinh xs (4) = Y (s)exs/ Y (s)e xs/ (5) Applying the boundary ondition Z(,s)=U a (s) to Eq. (5), we obtain s U a (s) =Y (s)sinh = Y (s)es/ Y (s)e s/. (6)

7 Salable Formation Control of Multi-Robot Chain Networks using a PDE Abstration 7 Taking the inverse Laplae transform of Eq. (5) and Eq. (6) yields a parametrization of the state and input trajetories in terms of the output: z(x,t) = y t + x u a (t) = y t + y t y t x The input is therefore defined by the output values at times t and t +. Sine time must be nonnegative, (t ), whih implies that (t + ). Hene, a minimum time of t = / is needed to drive the system from its initial state to its final state. This is onsistent with the ontrollability results for the wave equation that were disussed in Set. 3. Let T be the time at whih the system is to be driven to the target state. We define the funtions g(x)=y(t + x x ) and h(x)=y(t ) and denote the target state and its desired time derivative by z (x) and zt (x), respetively. We obtain the following expressions for z (x) and zt (x): (7) (8) z (x) =z(x,t )= g(x) h(x) (9) zt (x) = z(x,t ) = dg(x) + dh(x) t dx dx () Solving Eq. (9) and Eq. () for g(x) and h(x) yields: Z x g(x) = zt (s)ds + z (x) () h(x) = Z x z t (s)ds z (x) () We set y(t) = for t apple T. Then, a boundary ontrol trajetory u a (t) that drives system to the desired target state an be onstruted as: 8 ><, t [,T ) u a (t)= >: h(t t), t [T,T ) (3) g(t T + ), t [T,T ] 4. Fored Wave Equation Component The boundary ontrol inputs Eq. () in the fored wave equation Eq. () are defined as u (t)=z () and u b (t)=z () for t T. These ontrol inputs drive the leader agents to their target final loations and anhor them there. Their design aounts for the fat that the left boundary in the unfored wave equation Eq. (7) is always fixed at zero. Using the superposition priniple, the ontrol input Eq. (3)

8 8 Karthik Elamvazhuthi and Spring Berman to the unfored equation an be designed to simultaneously drive the system to the target state and negate the undesirable transient effet of the ontrol inputs to the fored equation. This an be done by modifying Eq. () and Eq. () to be: Z x g(x)= (zt (s) z pt (s,t ))ds +(z (x) z p (x,t )), (4) h(x)= Z x (z t (s) z pt (s,t ))ds (z (x) z p (x,t )), (5) where z p (x,t) is the solution of the boundary value problem Eq. ()-Eq. () and z pt (x,t) is its time derivative. Another role of the fored omponent is to enode the target equilibrium state through the funtion f (x). The system equilibrium state z eq (x), obtained by setting the seond time derivative to zero in Eq. (), is the solution to the resulting boundary value problem, d z(x) dx = f (x), z()=u e, z()=u e (6) where u e and u e are the equilibrium values of u (t) and u b (t), respetively. When f (x)=n p sin(npx), n Z +, and u e = u e, we see that z eq (x)= sin(npx)+u e. Sine the sine series forms a omplete basis of L [,], the set of square integrable funtions over the domain [], any funtion in L [,] an therefore be designed as a target state. For example, for the desired target state z d (x)=sin(px)+sin(6px), we ould assign f (x)= (p) sin(px)+ (6px) sin(6px) and u e = u e =. In order to implement nonzero boundary onditions, u e and u e may be nonzero to shift the datum of the sinusoids from the origin. 5 Simulation Results We validated our formation ontrol approah for populations of N = and N = agents. The agents start at the origin and are required to reah their final equilibrium onfiguration in time T = s. The agent equilibrium onfigurations were speified along three target urves: a line, a irle, and a 3D losed urve in the form of a Lissajous knot. Fig. and Fig. show the time evolution of the agent positions for both population sizes and eah type of target urve. The plots in Fig. demonstrate that for populations of both N = and N =, the agent positions remain near the target urve when t T = s. For both target urves, the population of agents exhibits smaller osillations around the desired equilibrium positions than the population of agents. This is beause the network with the larger number of agents more losely approximates the ontinuum PDE model from whih the ontrol inputs are derived. The snapshots in Fig. show that the networks of and agents have similar transient dynamis (i.e., similar agent position distributions at t =.

9 Salable Formation Control of Multi-Robot Chain Networks using a PDE Abstration Leaders Followers 6 4 Leaders Followers t (s) t (s) z (x, t).5.5 z (x, t) z (x, t).5.5 z (x, t) (a) Line, N = agents (b) Line, N = agents Leaders Followers 4 Leaders Followers t (s) t (s) z (x, t) 4 z (x, t) z (x, t) 4 z (x, t) () Cirle, N = agents (d) Cirle, N = agents Fig. Evolution of agent trajetories from the origin to a target line or a target irle at equilibrium. For larity, the trajetories of only agents are shown in eah plot. s and t =.5 s), but that by T = s, the larger network has onverged muh loser to the target 3D urve than the smaller network. Fig. 3 shows the time evolution of the absolute errors between the atual agent positions and their designed equilibrium positions along a irle for both N = and N =. The plots show that the agent position errors derease markedly after T = s, indiating that the agent positions approah the desired equilibria in the required amount of time. The population of agents learly displays smaller osillations about these equilibria than the population of agents.

10 Karthik Elamvazhuthi and Spring Berman z 3 (x, t).5 z 3 (x, t).5 z 3 (x, t).5 z (x, t).5.5 z (x, t) z (x, t).5.5 z (x, t) z (x, t).5.5 z (x, t) (a) N =, t =. s (b) N =, t =.5 s () N =, t = s z 3 (x, t).5 z 3 (x, t).5 z 3 (x, t).5 z (x, t).5.5 z (x, t) z (x, t).5.5 z (x, t) z (x, t).5.5 z (x, t) (d) N =, t =. s (e) N =, t =.5 s (f) N =, t = s Fig. Snapshots of agent positions (blue markers) at t =. s,.5 s, and s as they onverge from the origin to a target 3D urve (blak line). The positions of agents are shown in eah plot..5.5 Error.5 Error t (s) (a) Cirle, N = agents t (s) (b) Cirle, N = agents Fig. 3 Time evolution of the agent position errors when the target urve is the irle shown in Fig. (), Fig. (d). For larity, the position errors of only agents are shown in both plots.

11 Salable Formation Control of Multi-Robot Chain Networks using a PDE Abstration 6 Conlusions and Future Work We have presented a trajetory planning methodology for a formation ontrol problem on a hain network of agents with double integrator dynamis. Our approah is based on a wave equation abstration of the system dynamis, and it is salable with the number of agents and produes bounded ontrol inputs. Despite the marginal stability of the system, the resulting open-loop ontrol laws suessfully drive the system to a target equilibrium state. Due to the open-loop nature of the ontrol strategy and the approximation error between the ontinuous and disrete models, our method shows higher auray in ahieving the desired state as the number of agents in the network inreases. Future work is needed on inluding feedbak stabilization or other ompensation shemes so that a wider lass of systems is ontrollable using this approah. It was also observed that double integrator networks have ertain fundamental limitations for one-dimensional agent interonnetion topologies. The wave equation has similar limitations in higher dimensions. Hene, graph topologies that have equivalent ontinuum approximations an be expeted to have similar limitations. It would be interesting to see how the minimum time for ontrollability is refleted in double integrator networks with arbitrary topologies. In addition, we plan to extend our methodology to multi-robot systems with realisti onstraints and limitations. For instane, the robots motion may be subjet to holonomi onstraints, whih ould be addressed by using infinite-dimensional equivalents of suh networks with non-holonomi agents [9]. Robust ontrol tools for distributed parameter systems [6] ould be applied to systems with stohastiity and unertainty in the robot dynamis. We will also need to aount for possible loss of network onnetivity and dynamially hanging network topologies. Furthermore, we would like to develop methods for formation ontrol of systems that an be modeled by a wider lass of linear and nonlinear PDEs. Additional variability an be inorporated either by using more leader agents or hanging the ontrol gains. For this reason, ontrolling a multi-agent system using nonlinear interonnetion shemes an inrease the set of reahable states at equilibrium. Another diretion for future work is modeling grid networks of higher dimensions, and thus enabling deployment to formations on two-dimensional and three-dimensional manifolds. Referenes. Bamieh, B., Jovanovi, M.R., Mitra, P., Patterson, S.: Coherene in large-sale networks: Dimension-dependent limitations of loal feedbak. IEEE Transations on Automati Control 57(9), (). Barooah, P., Mehta, P.G., Hespanha, J.P.: Mistuning-based ontrol design to improve losedloop stability margin of vehiular platoons. IEEE Transations on Automati Control 54(9), 3 (9)

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