Laplacian Cooperative Attitude Control of Multiple Rigid Bodies
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1 Laplacian Cooperative Attitue Control of Multiple Rigi Boies Dimos V. Dimarogonas, Panagiotis Tsiotras an Kostas J. Kyriakopoulos Abstract Motivate by the fact that linear controllers can stabilize the rotational motion of a rigi boy, we propose in this paper a control strategy that exploits graph theoretic tools for cooperative control of multiple rigi boies. The control objective is to stabilize the system to a configuration where the rigi boies will have a common orientation an common angular velocity. The control law respects the limite information each rigi boy has with respect to the rest of the team. Specifically, each rigi boy is equippe with a control law that is base on the Laplacian matrix of the communication graph, which encoes the limite communication capabilities between the team members. Similarly to the linear case, the convergence of the multi-agent system relies on the connectivity of the communication graph. I. INTRODUCTION Cooperative istribute control of multiple vehicles has gaine increase attention in recent years in the control community, ue to the fact that it provies feasible solutions to large-scale multi-agent problems, in terms both of complexity an computational loa. Among the various specifications the control esign aims to impose on the multi-agent team is the state-agreement or consensus problem, i.e. convergence of the multi-agent system to a common configuration. This esign objective has been extensively pursue in the last few years. In most cases, vehicle motion is moelle by a single integrator [5],[], while ouble integrator moels have also been consiere [9]. A recent review of the various approaches for solving the consensus problem when the unerlying ynamics are linear is foun in [7]. A common analysis tool that is frequently use to moel these istribute systems is algebraic graph theory []. Motivate by the fact that linear controllers can stabilize a rigi boy [], in this paper we propose a control strategy that exploits graph theoretic tools for cooperative control of multiple rigi boies. The control objective is to stabilize the system to a configuration where all the rigi boies have a common orientation an common angular velocity. When the esire angular velocity is set to zero for all agents, all the satellites en up at relative orientations which can be efine a priori (an which can also be zero for the case of the same esire orientation for all the satellites). The propose control law for each agent respects the limite information each rigi boy has with respect to the rest of the team. Dimos Dimarogonas an Kostas Kyriakopoulos are with the Control Systems Lab, Department of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou 578, Greece {imar,kkyria@mail.ntua.gr}. Panagiotis Tsiotras is with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA -5, p.tsiotras@ae.gatech.eu Cooperative control of multiple rigi boies has been aresse recently by many authors [], [], []. While these papers use istribute consensus algorithms to achieve the esire objective, they are not irectly relate to the algebraic graph theoretic framework encountere in this work. Specifically, here we equip each rigi boy with a control law that is base on the Laplacian matrix of the communication graph, which encoes the limite communication capabilities between the team members. Similarly to the linear case, the convergence of the multi-agent system relies on the connectivity of the communication graph. The results are also extene to the case when each rigi boy converges to a esire not necessarily zero orientation with respect to each of the agents with which it can communicate. We shoul note that similar results to the ones presente in this paper, were erive at almost the same time in the recent paper [6]. The rest of the paper is organize as follows: Section II escribes the system an the three problems treate in this paper. Assumptions regaring the communication topology between the agents are also presente, an moelle in terms of an unirecte graph. Section III begins with some backgroun on algebraic graph theory that is use in the sequel, an procees with the evelopment of the propose istribute feeback control strategy. This strategy rives the multi-agent team to a common configuration. In the same section the stability analysis for each of the problems introuce in Section II is given. Computer simulations are inclue in Section IV to illustrate the success of the propose approach. Section V summarizes the results of this paper an inicates some current research efforts. II. SYSTEM AND PROBLEM DEFINITION Consier a team of N rigi boies (henceforth calle agents) inexe by N = {,..., N}. The ynamics of agent i are given by []: J i ω i = S (ω i ) J i ω i + u i, i N, () where ω i R is the angular velocity vector in each satellite s boy fixe frame, u i R is the acting torque vector, an J i is the symmetric inertia matrix of agent i. The matrix S( ) enotes a skew-symmetric matrix representing the cross prouct between two vectors, i.e. S(v )v = v v. In this paper, the orientation of the rigi boies with respect to the inertial frame are escribe in terms of the Moifie Roriguez Parameters (MRPs)[8], []. The kinematics of agent i in terms of the MRPs, are given by: σ i = G i (σ i ) ω i, i N, ()
2 where the matrix G i is given by G i (σ i ) = ( ) σ T i σ i I S i (σ i ) + σ i σi T, an has the following properties [] ( ) + σ T σig T i (σ i ) ω i = i σ i σ T iω i, () ( ) + σ T G i (σ i ) G T i (σ i ) = i σ i I. () Each agent is assigne a subset N i N from the rest of the team, calle agent i s communication set, that inclues the agents with which it can communicate in orer to achieve the esire objective. The limite inter-agent communication is encoe in terms of a communication graph: Definition : The communication graph G = {V, E} is an unirecte graph that consists of a set of vertices V = {,..., N} inexe by the team members, an a set of eges, E = {(i, j) V V : j N i } containing pairs of noes that represent inter-agent communication specifications. We assume that the formation graph is unirecte, in the sense that i N j j N i, i, j N, i j.. It is obvious that (i, j) E if an only if i N j j N i. The control law is of the form u i = u i (ω i, σ i, ω j, σ j ), j N i (5) representing the limite communication capabilities of each agent. The three problems treate in this paper can now be state as follows (enote by P, P, P): P Derive istribute control laws of the form (5) that rive the team of N rigi boies to a common configuration with respect to both orientation an angular velocities. P Derive istribute control laws of the form (5) that rive the team of N rigi boies to a common zero angular velocity an common orientation. P Derive istribute control laws of the form (5) that rive the team of N rigi boies to a configuration where all rigi boies have the same angular velocity, while their final relative orientations are prescribe a priori. III. CONTROL DESIGN AND STABILITY ANALYSIS A. Tools from Algebraic Graph Theory In this subsection we review some tools from algebraic graph theory [] that we use in the sequel. For an unirecte graph G with n vertices, the ajacency matrix A = A(G) = (a ij ) is the n n symmetric matrix given by a ij =, if (i, j) E an a ij =, otherwise. If there is an ege connecting two vertices i, j, i.e. (i, j) E, then i, j are calle ajacent. A path of length r from a vertex i to a vertex j is a sequence of r + istinct vertices starting with i an ening with j such that consecutive vertices are ajacent. If there is a path between any two vertices of G, then G is calle connecte (otherwise it is calle isconnecte). The egree i of vertex i is the number of its neighboring vertices, i.e. i = {#j : (i, j) E} = N i. Let be the n n iagonal matrix of i s. The (combinatorial) Laplacian of G is the symmetric positive semiefinite matrix L = A. The Laplacian matrix L captures many topological properties of the graph. Of particular interest is the fact that for a connecte graph, the Laplacian has a single zero eigenvalue an the corresponing eigenvector is the vector of ones, enote by. B. Propose Control Strategy-Problem For Problem we propose the feeback control strategy for agent i as follows: u i = G i (σ i ) (σ i σ j ) (ω i ω j ). (6) This control strategy respects the limite communication ruling between the members of the multi-agent team. Uner this control strategy the following theorem hols: Theorem : Assume that the communication graph is connecte. Then the control strategy (6) is a solution to Problem. Proof: Let u, ω, σ R N be the stack vectors of all the control inputs, the angular velocities an the orientations of the multi-agent team, respectively. Then it is easily erive from (6) that where u = G T (σ) (L I ) σ (L I ) ω, G (σ) = blockiag (G (σ ),..., G N (σ N )), where L enotes the Laplacian of the associate communication graph, an enotes the stanar Kronecker prouct between two matrices. Let us now choose V (σ, ω) = ( ωt ij i ω i ) + σt (L I ) σ as a caniate Lyapunov function. Function V is positive semiefinite. The level sets of V efine compact sets in the prouct space of agents angular velocities an relative orientations. Specifically, the set Ω c = {(ω, σ) : V (σ, ω) c} for c > is close by continuity of V. For all (ω, σ) Ω c we have ω T ij i ω i c ω i Furthermore, we also have σ T (L I ) σ c σ i σ j c, (i, j) E. c λ min (J i ). σ i σ j c Connectivity of G ensures that the maximum length of a path connecting two vertices of the graph is at most N. Hence σ i σ j c (N ), for all i, j N. Differentiating now V with respect to time, we get V (σ, ω) = (ωij T i ω i ) + σ T (L I ) σ = u T ω + σ T (L I ) G (σ) ω.
3 With the choice of the control law in (6) we get V (σ, ω) = ω T (L I ) ω. By LaSalle s invariance principle, the system converges to the largest invariant set insie the set M = {(σ, ω) : ω T (L I ) ω = }. Since L I is positive semiefinite, if follows that (L I )ω = which implies that Lω = Lω = Lω =, (7) where ω, ω, ω R N are the stack vectors of the three coefficients of the agents angular velocities, respectively. Connectivity of the communication graph implies that L has a simple zero eigenvalue with corresponing eigenvector. Equation (7) now implies that ω, ω, ω are eigenvectors of L corresponing to the zero eigenvalue, thus they belong to span{ }. Hence ω i = ω j for all i, j N, implying that all ω i s converge to a common value at steay state. We next procee to show that this common value ω is constant. Insie the set M, we have (L I ) ω = hence also (L I ) ω =, which yiels L ω = L ω = L ω = an following the same argument as before, that ω, ω, ω span{ }. As a result, we have shown that ω i = ω for all i N, that is, the angular accelerations converge to a common value as well. Insie the set M, we also have J i ω i = S (ω i ) J i ω i + u i ωij T i ω i = ωi T u i ωi T J i ω i + G i (σ i ) =, or in stack vector form, (σ i σ j ) ω T (J ω + G T (σ) (L I ) σ) =, where J = blockiag (J,..., J N ). Now since ω, ω, ω span{ } the last equation implies that J i ω i + G i (σ i ) (σ i σ j ) =, or ( N ) ω i = ω = J i N G i (σ i ) (σ i σ j ), ( for all i N. Using J N ) = J i we now have ω i = ω = N J G i (σ i ) (σ i σ j ) ( N ) ω i = N J N G i (σ i ) (σ i σ j ). The fact that the communication graph is unirecte implies (σ i σ j ) =, an hence ω i = ω = ω =. It follows that the common angular acceleration of the rigi boies is zero, an therefore the common angular velocity for all agents, ω, is in fact, constant. This, in turn, implies that ω i = for all i N an from equation () the control inputs of each rigi boy must also be zero. Hence we have u = for all trajectories insie the set M, which implies or G T (σ) (L I ) σ = G (σ) G T (σ) (L I ) σ = (Σ I ) (L I ) σ = (ΣL I ) σ =, (8) where ( ( ) + σ T ( ) ) Σ = iag σ + σ T,..., N σ N. It follows from (8) that ΣLσ = ΣLσ = ΣLσ =, where σ, σ, σ R N are the stack vectors of the three coefficients of the agents orientations, respectively. The spectral properties of L are retaine uner the multiplication with the positive efinite iagonal matrix Σ. Hence the σ i s converge to a common value as well as t. Remark : It shoul be note at this point that while the control law (6) guarantees that the agents will converge to a configuration where ω (t) =... = ω N (t) = ω, with ω constant, an σ (t) =... = σ N (t) = σ (t), as t, it is nonetheless not guarantee that ω will be equal to zero. Consecutively, it is not guarantee that σ (t) will reach a constant value. The latter is guarantee if we a the aitional constraint that ω =. This is achieve with the treatment of Problem, which is iscusse in the sequel. C. Propose Control Strategy-Problem Theorem guarantees that the team of rigi boies will converge to a common constant angular velocity, while their orientations will eventually have a common value, which may not remain constant. In fact, it will not be constant unless the common, final angular velocity all satellites converge to is zero (see Remark ). In orer to ensure that all agents converge to the same constant orientation, in this section we show that it is sufficient that one agent has a amping element on the angular velocity. Without loss of generality, we assume that this is agent. The following theorem is the main result of this section: Theorem : Assume that the communication graph is connecte. Then the control strategy u i = G T i (σ i ) (σ i σ j ) (ω i ω j ) a i ω i, (9)
4 where i =,..., N an a i =, if i =, an a i =, otherwise, is a suitable solution to Problem. Proof: We choose again V (σ, ω) = ( ) ωt ij i ω i + σt (L I ) σ as a caniate Lyapunov function. Differentiating with respect to time an after some manipulation we get V (σ, ω) = ω T (L I ) ω ω. It follows that ω remains boune. By LaSalle s invariance principle, the system converges to the largest invariant set insie the set M = {(σ, ω) : (ω T (L I ) ω = ) (ω = )}. Similarly to the proof of Theorem, the conition ω T (L I ) ω = guarantees that all ω i s converge to a common value. Since ω =, this common value is zero. Furthermore, the orientations of the agents converge to a common value, which is constant, ue to the fact that ω i = for all i N. D. Propose Control Strategy-Problem In the iscussion thus far, it has been assume that it is esirable that all rigi boies converge to the same orientation. For some applications (i.e., Earth monitoring or stellar observation using a satellite cluster with a large baseline) it may be necessary for the rigi boies to acquire an maintain a certain (perhaps nonzero) relative orientation among themselves. The relative orientation for each pair of rigi boies may be ifferent, an can be ictate by the mission requirements. In this section, we thus impose the specification that for each pair (i, j) E, there exists a esire relative orientation σij R to which we wish the pair of rigi boies to converge. Next, we show how to moify the control law (6) in orer to achieve this objective. Throughout this section, we assume that there are no conflicting inter-agent objectives. Hence, we assume that σij = σ ji, i, j N, i j. The next theorem proposes a control law to achieve the objective state previously. Theorem : Assume that the communication graph is connecte. Then the control strategy u i = G T i (σ i ) ( σi σ j σij) (ω i ω j ), () where i =,..., N is a suitable solution for Problem. Proof: For each agent i, we efine the cost function an we introuce γ i (σ) = V (σ, ω) = σ i σ j σij, ( ) ωt ij i ω i + γ i (σ) as a caniate Lyapunov function. We then have V (σ, ω) = ωij T i ω i + γ i σ. With a slight abuse of notation we rewrite the last term in the previous equation as [ ] γ i = T γ i σ... T γ i, σ N where, = (σ i σ j ) + σii, i = j, σ i + σ j + σij, j N i, j i,, j / N i. where we have efine σ ii = σ ij. Hence, = γ j + i N j = (σ j σ i ) + σjj + ( σ i + σ j + σij) i N j i N j = σ j σ i + σjj i N j i N j = j σ j σ i + σjj. i N j It follows that [ γ i = γi σ ] σ N = [ ] [ σ N σ N σ j j N + [ ] σ σnn an finally, j N N σ j ] γ i = ((L I ) σ + c l ) T, () where c l = [ σ σ NN ] T. Using (), V can be written as V (σ, ω) = u T ω + ((L I ) σ + c l ) T G (σ) ω. From () it can be easily erive that u = G T (σ) ((L I ) σ + c l ) (L I ) ω. With the previous choice of the control law we get V (σ, ω) = ω T (L I ) ω. By LaSalle s invariance principle, the system converges to the largest invariant set insie the set M = {(σ, ω) : ω T (L I ) ω = }. The conition ω T (L I ) ω = guarantees again that all ω i s converge to a common value. Following the proof of Theorem, we euce that the invariance of M, along with u = implies that G T (σ) ((L I ) σ + c l ) = or G (σ) G T (σ) ((L I ) σ + c l ) =, which yiels
5 (Σ I ) ((L I ) σ + c l ) =. Finally, it follows that (L I ) σ + c l = ue to the positive efiniteness of Σ. For all i N, let σi enote the esire orientation of agent i with respect to the global coorinate frame. It is then obvious that σij = σ i σ j for all (i, j) E for all possible esire final orientations. Define σ i σ j σij = σ i σ j (σi σ j ) = σ i σ j. Then we have (L I )σ + c l = (L I ) σ = L σ = L σ = L σ =, where σ, σ, σ are the stack vectors of each of the three coefficients of σ of the agents orientations, respectively. The fact that the communication graph is connecte implies that L has a simple zero eigenvalue with corresponing eigenvector the vector of ones,. This guarantees that each one of the vectors σ, σ, σ are eigenvectors of L belonging to span{ }. Therefore all σ i are equal to a common vector value, say c. Hence σ i = c for all i N which implies that σ i σ j = σ ij j N i, i, j. We conclue that the agents converge to the esire, specifie configuration of relative orientations. Remark : Similarly to Remark, it shoul be note that while the control law () guarantees that the agents will converge to a configuration where ω (t) = = ω N (t) = ω, where ω is a constant value, an σ i (t) σ j (t) = σ ij for all (i, j) E, it is not guarantee that ω will be equal to zero. Consecutively, it is not guarantee that each σ i (t) will reach a constant value. The latter is guarantee if we a the aitional constraint that ω tens to zero. This is accomplishe with the extension of Theorem in this case. The treatment is straightforwar an is omitte here. IV. SIMULATIONS To verify the results of the previous section we provie next computer simulations of the propose control esigns. The first simulation involves four rigi boies evolving uner the control strategy (6). The Laplacian of the communication graph encoing the static communication ruling has been selecte to be of the form L = The initial conitions on the angular velocities an orientations were chosen as: σ () = [.,., ] T, σ () = [.,., ] T, σ () = [.,., ] T, σ () = [,, ] T, ω () = [,, ] T, ω () = [,., ] T, ω () = [,,.] T, ω () = [,,.] T. The inertia matrix of the four rigi boies was chosen as J = iag (, 5, ). Figure shows the plots of the angular velocities an orientations of the four rigi boies with respect to time uner the feeback law (6). We observe that the system behaves as expecte. The angular velocities converge to a common non-zero value. Consequently, the orientations converge to a common value which varies with time. Angular Velocity Norms Orientation Norms Fig.. Plots of the angular velocities an orientations the four rigi boies with respect to time uner the feeback law (6). In the secon simulation we a a amping element to the controller of agent, as in the feeback strategy (9). Figure shows the plots of the angular velocities an orientations of the four rigi boies with respect to time uner the feeback law (9). We observe that all angular velocities converge to zero, while the corresponing boy orientations converge to a common, an constant this time, zero value. Angular Velocity Norms Orientation Norms Fig.. Plots of the angular velocities an orientations the four rigi boies with respect to time uner the feeback law (9). The thir simulation involves four rigi boies evolving uner the control strategy (). The same Laplacian matrix an the same initial conitions for the angular ve-
6 locities as in the previous simulations were use, while the initial conitions for the orientations were chosen as σ () = [.6,.,.8] T, σ () = [,., ] T, σ () = [,,.] T, σ () = [, 6,.] T. The esire relative orientations between the members of the team were chosen as σ = [,, ] T, σ = [,, ] T, σ = [,., ] T. Note that these specifications correspon to the following esire configuration: σ = σ = σ = σ, σ = σ = σ = σ., σ = σ, σ = σ an σ = σ +. Figure shows the plots of the angular velocities along with the an the cost functions γ i for each rigi boy evolving uner the control law (). We observe that the angular velocities converge to a common value, while the γ i of each rigi boy ten to zero. Therefore, the esire relative orientations between each agent with the other agents belonging to its communication set are achieve. This is also verifie in Figure where the relative orientation coefficients converge to the esire values. st Orientation coefficient n Orientation coefficient Angular velocity norms.5..5 r Orientation coefficient Fig.. Plots of the orientations of the four rigi boies with respect to time for each of the three coefficients uner the feeback law ().. Cost Functions Fig.. Plots of the angular velocities an the cost function γ i of the four rigi boies with respect to time uner the feeback law (). V. CONCLUSIONS We propose a istribute control strategy that exploits graph theoretic tools for cooperative control of multiple rigi boies. The control objective was the stabilization of the overall system to a configuration where all the rigi boies have a common orientation an common angular velocity. Similarly to the linear case, the convergence of the multiagent system was shown to rely on the connectivity of the communication graph. We also extene our results to the case where each rigi boy aims to converge to a esire (not necessarily zero) orientation with respect to each of the agents with which it can communicate. Further research efforts involve the cases of switching interconnection topology, as well as the case of uniirectional communication ruling. REFERENCES [] J. Cortes, S. Martinez, an F. Bullo. Robust renezvous for mobile autonomous agents via proximity graphs in imansions. IEEE Transactions on Automatic Control, submitte for publication,. [] C. Gosil an G. Royle. Algebraic Graph Theory. Springer Grauate Texts in Mathematics # 7,. [] J. R. Lawton an R. Bear. Synchronize multiple spacecraft rotations. Automatica, 8(8):59 6,. [] M. Mesbahi an F.Y. Haaegh. Formation flying control of multiple spacecraft via graphs, matrix inequalities, an switching. Journal of Guiance,Control an Dynamics, ():69 77,. [5] R. Olfati-Saber an R.M. Murray. Consensus problems in networks of agents with switching topology an time-elays. IEEE Transactions on Automatic Control, 9(9):5 5,. [6] W. Ren. Distribute attitue alignment among multiple networke spacecraft. 6 American Control Conference, to appear. [7] W. Ren, R. W. Bear, an E. M. Atkins. A survey of consensus problems in multi-agent coorination. 5 American Control Conference, pages [8] M. D. Shuster. A survey of attitue representations. Journal of Astronautical Sciences, ():9 57, 99. [9] H.G. Tanner, A. Jababaie, an G.J. Pappas. Stable flocking of mobile agents. st IEEE Conf. Decision an Control,. [] P. Tsiotras. New control laws for the attitue stabilization of rigi boies. In th IFAC Symposium on Automatic Control in Aerospace, pages 6, Sept. -7, 99. Palo Alto, CA. [] P. Tsiotras. Further passivity results for the attitue control problem. IEEE Transactions on Automatic Control, ():597 6, 998. [] P.K.C. Wang, F.Y. Haaegh, an K. Lau. Synchronize formation rotation an attitue control of multiple free-flying spacecraft. Journal of Guiance,Control an Dynamics, ():8 5, 999.
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