Distributed robotic networks: rendezvous, connectivity, and deployment

Transcription

1 Distributed robotic networks: rendezvous, connectivity, and deployment Sonia Martínez and Jorge Cortés Mechanical and Aerospace Engineering University of California, San Diego 28th Benelux Meeting on Systems and Control Spa, Belgium, March 17, 2009 Acknowledgments: Francesco Bullo Anurag Ganguli

2 What we have seen in the previous lecture Cooperative robotic network model proximity graphs control and communication law, task, execution time, space, and communication complexity analysis agree and pursue algorithm Complexity analysis is challenging even in 1 dimension! Blend of math geometric structures distributed algorithms stability analysis linear iterations Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

3 What we will see in this lecture Basic motion coordination tasks: get together at a point, stay connected, deploy over a region Design coordination algorithms that achieve these tasks and analyze their correctness and time complexity Expand set of math tools: invariance principles for non-deterministic systems, geometric optimization, nonsmooth stability analysis Robustness against link failures, agents arrivals and departures, delays, asynchronism Image credits: jupiterimages and Animal Behavior Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

4 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

5 Rendezvous objective Objective: achieve multi-robot rendezvous; i.e. arrive at the same location of space, while maintaining connectivity r-disk connectivity visibility connectivity Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

6 We have to be careful... Blindly getting closer to neighboring agents might break overall connectivity Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

7 Network definition and rendezvous tasks The objective is applicable for general robotic networks S disk, S LD and S -disk, and the relative-sensing networks Sdisk rs and Srs vis-disk We adopt the discrete-time motion model p [i] (l + 1) = p [i] (l) + u [i] (l), i {1,..., n} Also for the relative-sensing networks p [i] fixed (l + 1) = p[i] fixed (l) + R[i] fixed u[i] i (l), i {1,..., n} Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

8 The rendezvous task via aggregate objective functions Coordination task formulated as function minimization Diameter convex hull Perimeter relative convex hull Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

9 The rendezvous task formally Let S = ({1,..., n}, R, E cmm ) be a uniform robotic network The (exact) rendezvous task T rendezvous : X n {true, false} for S is T rendezvous (x [1],..., x [n] ) { true, if x [i] = x [j], for all (i, j) E cmm (x [1],..., x [n] ), = false, otherwise For ɛ R >0, the ɛ-rendezvous task T ɛ-rendezvous : (R d ) n {true, false} is T ɛ-rendezvous (P ) = true p [i] avrg ( { p [j] (i, j) E cmm (P )} ) 2 < ɛ, i {1,..., n} Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

10 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

11 Constraint sets for connectivity Design constraint sets with key properties Constraints are flexible enough so that network does not get stuck Constraints change continuously with agents position r-disk connectivity visibility connectivity Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

12 Enforcing range-limited links pairwise Pairwise connectivity maintenance problem: Given two neighbors in G disk (r), find a rich set of control inputs for both agents with the property that, after moving, both agents are again within distance r If p [i] (l) p [j] (l) r, and remain in connectivity set, then p [i] (l + 1) p [j] (l + 1) r Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

13 Enforcing range-limited links w/ all neighbors Definition (Connectivity constraint set) Consider a group of agents at positions P = {p [1],..., p [n] } R d. The connectivity constraint set of agent i with respect to P is X disk (p [i], P ) = { Xdisk (p [i], q) q P \ {p [i] } s.t. q p [i] 2 r } Same procedure over sparser graphs means fewer constraints: G LD (r) has same connected components as G disk (r) and is spatially distributed over G disk (r) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

14 Enforcing range-limited line-of-sight links pairwise For Q δ = {q Q dist(q, Q) δ} δ-contraction of compact nonconvex Q R 2 Pairwise connectivity maintenance problem: Given two neighbors in G vis-disk,qδ, find a rich set of control inputs for both agents with the property that, after moving, both agents are again within distance r and visible to each other in Q δ visibility region of agent i visibility pairwise constraint set Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

15 Enforcing range-limited line-of-sight links w/ all neighbors Definition (Line-of-sight connectivity constraint set) Consider a group of agents at positions P = {p [1],..., p [n] } in a nonconvex allowable environment Q δ. The line-of-sight connectivity constraint sets of agent i with respect to P is X vis-disk (p [i], P ; Q δ ) = { Xvis-disk (p [i], q; Q δ ) q P \ {p [i] } } Fewer constraints can be generated via sparser graphs with the same connected components and spatially distributed over Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

16 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

17 Circumcenter control and communication law For X = R d, X = S d or X = R d1 S d2, d = d 1 + d 2, circumcenter CC(W ) of a bounded set W X is center of closed ball of minimum radius that contains W Circumradius CR(W ) is radius of this ball [Informal description:] At each communication round each agent performs the following tasks: (i) it transmits its position and receives its neighbors positions; (ii) it computes the circumcenter of the point set comprised of its neighbors and of itself. Between communication rounds, each robot moves toward this circumcenter point while maintaining connectivity with its neighbors using appropriate connectivity constraint sets. Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

18 Circumcenter control and communication law Illustration of the algorithm execution Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

19 Circumcenter control and communication law Formal algorithm description Robotic Network: S disk with a discrete-time motion model, with absolute sensing of own position, and with communication range r, in R d Distributed Algorithm: circumcenter Alphabet: L = R d {null} function msg(p, i) 1: return p function ctrl(p, y) 1: p goal := CC({p} {p rcvd for all non-null p rcvd y}) 2: X := X disk (p, {p rcvd for all non-null p rcvd y}) 3: return fti(p, p goal, X ) p Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

20 Simulations z y x Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

21 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

22 Some bad news... Circumcenter algorithms are nonlinear discrete-time dynamical systems x l+1 = f(x l ) To analyze convergence, we need at least f continuous to use classic Lyapunov/LaSalle results But circumcenter algorithms are discontinuous because of changes in interaction topology Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

23 Alternative idea Fixed undirected graph G, define fixed-topology circumcenter algorithm f G : (R d ) n (R d ) n, f G,i (p 1,..., p n ) = fti(p, p goal, X ) p Now, there are no topological changes in f G, hence f G is continuous Define set-valued map T CC : (R d ) n P((R d ) n ) T CC (p 1,..., p n ) = {f G (p 1,..., p n ) G connected} Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

24 Non-deterministic dynamical systems Given T : X P(X), a trajectory of T is sequence {x m } m N0 X such that x m+1 T (x m ), m N 0 T is closed at x if x m x, y m y with y m T (x m ) imply y T (x) Every continuous map T : R d R d is closed on R d A set C is weakly positively invariant if, for any p 0 C, there exists p T (p 0 ) such that p C strongly positively invariant if, for any p 0 C, all p T (p 0 ) verifies p C A point p 0 is a fixed point of T if p 0 T (p 0 ) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

25 LaSalle Invariance Principle set-valued maps V : X R is non-increasing along T on S X if V (x ) V (x) for all x T (x) and all x S Theorem (LaSalle Invariance Principle) For S compact and strongly invariant with V continuous and nonincreasing along closed T on S Any trajectory starting in S converges to largest weakly invariant set contained in {x S x T (x) with V (x ) = V (x)} Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

26 Correctness T CC is closed and diameter is non-increasing Recall set-valued map T CC : (R d ) n P((R d ) n ) T CC (p 1,..., p n ) = {f G (p 1,..., p n ) G connected} T CC is closed: finite combination of individual continuous maps Define V diam (P ) = diam(co(p )) = max { p i p j i, j {1,..., n}} diag((r d ) n ) = { (p,..., p) (R d ) n p R d} Lemma The function V diam = diam co: (R d ) n R + verifies: 1 V diam is continuous and invariant under permutations; 2 V diam (P ) = 0 if and only if P diag((r d ) n ); 3 V diam is non-increasing along T CC Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

27 Correctness via LaSalle Invariance Principle To recap 1 T CC is closed 2 V = diam is non-increasing along T CC 3 Evolution starting from P 0 is contained in co(p 0 ) (compact and strongly invariant) Application of LaSalle Invariance Principle: trajectories starting at P 0 converge to M, largest weakly positively invariant set contained in {P co(p 0 ) P T CC (P ) such that diam(p ) = diam(p )} Have to identify M! In fact, M = diag((r d ) n ) co(p 0 ) Convergence to a point can be concluded with a little bit of extra work Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

28 Correctness Theorem (Correctness of the circumcenter laws) For d N, r R >0 and ɛ R >0, the following statements hold: 1 on S disk, the law CC circumcenter (with control magnitude bounds and relaxed G-connectivity constraints) achieves T rendezvous ; 2 on S LD, the law CC circumcenter achieves T ɛ-rendezvous Furthermore, 1 if any two agents belong to the same connected component at l N 0, then they continue to belong to the same connected component subsequently; and 2 for each evolution, there exists P = (p 1,..., p n) (R d ) n such that: 1 the evolution asymptotically approaches P, and 2 for each i, j {1,..., n}, either p i = p j, or p i p j 2 > r (for the networks S disk and S LD) or p i p j > r (for the network S -disk ). Similar result for visibility networks in non-convex environments Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

29 Correctness Time complexity Theorem (Time complexity of circumcenter laws) For r R >0 and ɛ ]0, 1[, the following statements hold: 1 on the network S disk, evolving on the real line R (i.e., with d = 1), TC(T rendezvous, CC circumcenter ) Θ(n); 2 on the network S LD, evolving on the real line R (i.e., with d = 1), TC(T (rɛ)-rendezvous, CC circumcenter ) Θ(n 2 log(nɛ 1 )); and Similar results for visibility networks Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

30 Robustness of circumcenter algorithms Push whole idea further!, e.g., for robustness against link failures topology G 1 topology G 2 topology G 3 Look at evolution under link failures as outcome of nondeterministic evolution under multiple interaction topologies P {evolution under G 1, evolution under G 2, evolution under G 3 } Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

31 Rendezvous Corollary (Circumcenter algorithm over G disk (r) on R d ) For {P m } m N0 synchronous execution with link failures such that union of any l N consecutive graphs in execution has globally reachable node Then, there exists (p,..., p ) diag((r d ) n ) such that P m (p,..., p ) as m + Proof uses T CC,l (P ) = {f Gl f G1 (P ) l s=1 G i has globally reachable node} Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

32 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

33 Deployment Objective: optimal task allocation and space partitioning optimal placement and tuning of sensors What notion of optimality? What algorithm design? top-down approach: define aggregate function measuring goodness of deployment, then synthesize algorithm that optimizes function bottom-up approach: synthesize reasonable interaction law among agents, then analyze network behavior Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

34 Coverage optimization DESIGN of performance metrics 1 how to cover a region with n minimum-radius overlapping disks? 2 how to design a minimum-distortion (fixed-rate) vector quantizer? (Lloyd 57) 3 where to place mailboxes in a city / cache servers on the internet? ANALYSIS of cooperative distributed behaviors 4 how do animals share territory? what if every fish in a swarm goes toward center of own dominance region? Barlow, Hexagonal territories, Animal Behavior, what if each vehicle goes to center of mass of own Voronoi cell? 6 what if each vehicle moves away from closest vehicle? Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

35 Expected-value multicenter function Objective: Given sensors/nodes/robots/sites (p 1,..., p n ) moving in environment Q achieve optimal coverage φ: R d R 0 density f : R 0 R non-increasing and piecewise continuously differentiable, possibly with finite jump discontinuities [ ] maximize H exp (p 1,..., p n ) = E φ max f( q p i ) i {1,...,n} Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

36 H exp -optimality of the Voronoi partition Alternative expression in terms of Voronoi partition, H exp (p 1,..., p n ) = for (p 1,..., p n ) distinct n f( q p i 2 )φ(q)dq i=1 V i(p ) Proposition Let P = {p 1,..., p n } F(S). For any performance function f and for any partition {W 1,..., W n } P(S) of S, H exp (p 1,..., p n, V 1 (P ),..., V n (P )) H exp (p 1,..., p n, W 1,..., W n ), and the inequality is strict if any set in {W 1,..., W n } differs from the corresponding set in {V 1 (P ),..., V n (P )} by a set of positive measure Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

37 Distortion problem f(x) = x 2 and the inequality is strict if there exists i {1,..., n} for which W i has non-vanishing area and p i CM φ (W i ) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72 n n H dist (p 1,..., p n ) = q p i 2 2φ(q)dq = J φ (V i (P ), p i ) i=1 V i(p ) i=1 (J φ (W, p) is moment of inertia). Note H dist (p 1,..., p n, W 1,..., W n ) n n = J φ (W i, CM φ (W i )) area φ (W i ) p i CM φ (W i ) 2 2 i=1 i=1 Proposition Let {W 1,..., W n } P(S) be a partition of S. Then, ( ) H dist CMφ (W 1 ),..., CM φ (W n ), W 1,..., W n H dist (p 1,..., p n, W 1,..., W n ),

38 Area problem f(x) = 1 [0,a] (x), a R >0 H area,a (p 1,..., p n ) = = = n 1 [0,a] ( q p i 2 )φ(q)dq i=1 n i=1 V i(p ) V i(p ) B(p i,a) φ(q)dq n area φ (V i (P ) B(p i, a)) = area φ ( n i=1b(p i, a)), i=1 Area, measured according to φ, covered by the union of the n balls B(p 1, a),..., B(p n, a) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

39 Mixed distortion-area problem f(x) = x 2 1 [0,a] (x) + b 1 ]a,+ [ (x), with a R >0 and b a 2 and the inequality is strict if there exists i {1,..., n} for which W i has non-vanishing area and p i CM φ (W i B(p i, a)). Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72 H dist-area,a,b (p 1,..., p n ) = n J φ (V i,a (P ), p i ) + b area φ (Q \ n i=1b(p i, a)), i=1 If b = a 2, f is continuous, we write H dist-area,a. Extension reads H dist-area,a (p 1,..., p n, W 1,..., W n ) n ( ) = J φ (W i B(p i, a), p i ) + a 2 area φ (W i (S \ B(p i, a))). i=1 Proposition (H dist-area,a -optimality of centroid locations) Let {W 1,..., W n } P(S) be a partition of S. Then, H dist-area,a ( CMφ (W 1 B(p 1, a)),..., CM φ (W n B(p n, a)), W 1,..., W n ) H dist (p 1,..., p n, W 1,..., W n ),

40 Smoothness properties of H exp Dscn(f) (finite) discontinuities of f f and f +, limiting values from the left and from the right Theorem Expected-value multicenter function H exp : S n R is 1 globally Lipschitz on S n ; and 2 continuously differentiable on S n \ S coinc, where H exp (P ) = p i V i(p ) + a Dscn(f) p i f( q p i 2 )φ(q)dq ( f (a) f + (a) ) n (q)φ(q)dq out,b(pi,a) Vi(P ) B(pi,a) = integral over V i + integral along arcs in V i Therefore, the gradient of H exp is spatially distributed over G D Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

41 Particular gradients Distortion problem: continuous performance, H dist p i (P ) = 2 area φ (V i (P ))(CM φ (V i (P )) p i ) Area problem: performance has single discontinuity, H area,a (P ) = n p (q)φ(q)dq out,b(pi,a) i V i(p ) B(p i,a) Mixed distortion-area: continuous performance (b = a 2 ), H dist-area,a p i (P ) = 2 area φ (V i,a (P ))(CM φ (V i,a (P )) p i ) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

42 Tuning the optimization problem Gradients of H area,a, H dist-area,a,b are distributed over G LD (r)2a Robotic agents with range-limited interactions can compute gradients of H area,a and H dist-area,a,b as long as r 2a Proposition (Constant-factor approximation of H dist ) Let S R d be bounded and measurable. Consider the mixed distortion-area problem with a ]0, diam S] and b = diam(s) 2. Then, for all P S n, where β = H dist-area,a,b (P ) H dist (P ) β 2 H dist-area,a,b (P ) < 0, a diam(s) [0, 1] Similarly, constant-factor approximations of H exp Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

43 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

44 Geometric-center laws Uniform networks S D and S LD of locally-connected first-order agents in a polytope Q R d with the Delaunay and r-limited Delaunay graphs as communication graphs All laws share similar structure At each communication round each agent performs the following tasks: it transmits its position and receives its neighbors positions; it computes a notion of geometric center of its own cell determined according to some notion of partition of the environment Between communication rounds, each robot moves toward this center Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

45 Vrn-cntrd algorithm Optimizes distortion H dist Robotic Network: S D in Q, with absolute sensing of own position Distributed Algorithm: Vrn-cntrd Alphabet: L = R d {null} function msg(p, i) 1: return p function ctrl(p, y) 1: V := Q ( {H p,prcvd for all non-null p rcvd y} ) 2: return CM φ (V ) p Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

46 Simulation initial configuration gradient descent final configuration For ɛ R >0, the ɛ-distortion deployment task { true, if p [i] CM φ (V [i] (P )) T ɛ-distor-dply (P ) = 2 ɛ, i {1,..., n}, false, otherwise, Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

47 Voronoi-centroid law on planar vehicles Robotic Network: S vehicles in Q with absolute sensing of own position Distributed Algorithm: Vrn-cntrd-dynmcs Alphabet: L = R 2 {null} function msg((p, θ), i) 1: return p function ctrl((p, θ), (p smpld, θ smpld ), y) 1: V := Q ( { H psmpld,p rcvd for all non-null p rcvd y } ) 2: v := k prop (cos θ, sin θ) (p CM φ (V )) 3: ω := 2k prop arctan ( sin θ, cos θ) (p CM φ(v )) (cos θ, sin θ) (p CM φ (V )) 4: return (v, ω) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

48 Algorithm illustration Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

49 Simulation initial configuration Mart ınez & Cort es (UCSD) gradient descent final configuration Distributed robotic networks March 17, / 72

50 Lmtd-Vrn-nrml algorithm Optimizes area H area, r 2 Robotic Network: S LD in Q with absolute sensing of own position and with communication range r Distributed Algorithm: Lmtd-Vrn-nrml Alphabet: L = R d {null} function msg(p, i) 1: return p function ctrl(p, y) 1: V := Q ( {H p,prcvd for all non-null p rcvd y} ) 2: v := V B(p, r 2 ) n out,b(p, r )(q)φ(q)dq 2 3: λ := max {λ δ } V B(p+δv, r2 ) φ(q)dq is strictly increasing on [0, λ] 4: return λ v Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

51 Simulation initial configuration gradient descent final configuration For r, ɛ R >0, T ɛ-r-area-dply (P ) { true, if V = (P ) B(p [i], r 2 ) n out,b(p [i], r )(q)φ(q)dq 2 2 ɛ, i {1,..., n}, false, otherwise. Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

52 Lmtd-Vrn-cntrd algorithm Optimizes H dist-area, r 2 Robotic Network: S LD in Q with absolute sensing of own position, and with communication range r Distributed Algorithm: Lmtd-Vrn-cntrd Alphabet: L = R d {null} function msg(p, i) 1: return p function ctrl(p, y) 1: V := Q B(p, r 2 ) ( {H p,prcvd for all non-null p rcvd y} ) 2: return CM φ (V ) p Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

53 Simulation initial configuration gradient descent final configuration For r, R>0, T -r-distor-area-dply (P ) ( [i] true, if p[i] CMφ (V r (P ))) 2, i {1,..., n}, 2 = false, otherwise. Mart ınez & Cort es (UCSD) Distributed robotic networks March 17, / 72

54 Optimizing Hdist via constant-factor approximation Limited range run #1: 16 agents, density φ is sum of 4 Gaussians, time invariant, 1st order dynamics initial configuration gradient descent of H r 2 final configuration initial configuration gradient descent of Hexp final configuration Unlimited range run #2: 16 agents, density φ is sum of 4 Gaussians, time invariant, 1st order dynamics Mart ınez & Cort es (UCSD) Distributed robotic networks March 17, / 72

55 Correctness of the geometric-center algorithms Theorem For d N, r R >0 and ɛ R >0, the following statements hold. 1 on the network S D, the law CC Vrn-cntrd and on the network S vehicles, the law CC Vrn-cntrd-dynmcs both achieve the ɛ-distortion deployment task T ɛ-distor-dply. Moreover, any execution of CC Vrn-cntrd and CC Vrn-cntrd-dynmcs monotonically optimizes the multicenter function H dist ; 2 on the network S LD, the law CC Lmtd-Vrn-nrml achieves the ɛ-r-area deployment task T ɛ-r-area-dply. Moreover, any execution of CC Lmtd-Vrn-nrml monotonically optimizes the multicenter function H area, r 2 ; and 3 on the network S LD, the law CC Lmtd-Vrn-cntrd achieves the ɛ-r-distortion-area deployment task T ɛ-r-distor-area-dply. Moreover, any execution of CC Lmtd-Vrn-cntrd monotonically optimizes the multicenter function H dist-area, r 2. Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

56 Time complexity of CC Lmtd-Vrn-cntrd Assume diam(q) is independent of n, r and ɛ Theorem (Time complexity of Lmtd-Vrn-cntrd law) Assume the robots evolve in a closed interval Q R, that is, d = 1, and assume that the density is uniform, that is, φ 1. For r R >0 and ɛ R >0, on the network S LD TC(T ɛ-r-distor-area-dply, CC Lmtd-Vrn-cntrd ) O(n 3 log(nɛ 1 )) Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

57 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

58 Deployment: basic behaviors move away from closest move towards furthest Equilibria? Asymptotic behavior? Optimizing network-wide function? Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

59 Deployment: 1-center optimization problems sm Q (p) = min{ p q q Q} Lipschitz 0 sm Q (p) p IC(Q) lg Q (p) = max{ p q q Q} Lipschitz 0 lg Q (p) p = CC(Q) Locally Lipschitz function V are differentiable a.e. Generalized gradient of V is V (x) = convex closure lim i V (x i) x i x, x i Ω V S Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

60 Deployment: 1-center optimization problems + gradient flow of sm Q ṗ i = + Ln[ sm Q ](p) move away from closest gradient flow of lg Q ṗ i = Ln[ lg Q ](p) move toward furthest For X essentially locally bounded, Filippov solution of ẋ = X(x) is absolutely continuous function t [t 0, t 1] x(t) verifying ẋ K[X](x) = co{ lim i X(x i) x i x, x i S} For V locally Lipschitz, gradient flow is ẋ = Ln[ V ](x) Ln = least norm operator Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

61 Nonsmooth LaSalle Invariance Principle Evolution of V along Filippov solution t V (x(t)) is differentiable a.e. d dt V (x(t)) L X V (x(t)) = {a R v K[X](x) s.t. ζ v = a, ζ V (x)} }{{} set-valued Lie derivative LaSalle Invariance Principle For S compact and strongly invariant with max L X V (x) 0 Any Filippov solution starting in S converges to largest weakly { invariant set contained in x S 0 L } X V (x) E.g., nonsmooth gradient flow ẋ = Ln[ V ](x) converges to critical set Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

62 Deployment: multi-center optimization sphere packing and disk covering move away from closest : ṗ i = + Ln( sm Vi(P ))(p i ) at fixed V i (P ) move towards furthest : ṗ i = Ln( lg Vi(P ))(p i ) at fixed V i (P ) Aggregate objective functions! H sp (P ) = min i H dc (P ) = max i sm Vi(P )(p i ) = min i j lg Vi(P )(p i ) = max q Q [ 1 2 p i p j, dist(p i, Q) ] [ min i q p i ] Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

63 Deployment: multi-center optimization Critical points of H sp and H dc (locally Lipschitz) If 0 int H sp (P ), then P is strict local maximum, all agents have same cost, and P is incenter Voronoi configuration If 0 int H dc (P ), then P is strict local minimum, all agents have same cost, and P is circumcenter Voronoi configuration Aggregate functions monotonically optimized along evolution min L Ln( smv(p ) )H sp (P ) 0 max L Ln( lgv(p ) )H dc (P ) 0 Asymptotic convergence to center Voronoi configurations via nonsmooth LaSalle Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

64 Outline 1 Rendezvous and connectivity maintenance The rendezvous objective Maintaining connectivity Circumcenter algorithms Correctness analysis via nondeterministic systems 2 Deployment Expected-value deployment Geometric-center laws Disk-covering and sphere-packing deployment Geometric-center laws 3 Conclusions Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

65 Voronoi-circumcenter algorithm Robotic Network: S D in Q with absolute sensing of own position Distributed Algorithm: Vrn-crcmcntr Alphabet: L = R d {null} function msg(p, i) 1: return p function ctrl(p, y) 1: V := Q ( {H p,prcvd for all non-null p rcvd y} ) 2: return CC(V ) p Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

66 Voronoi-incenter algorithm Robotic Network: S D in Q with absolute sensing of own position Distributed Algorithm: Vrn-ncntr Alphabet: L = R d {null} function msg(p, i) 1: return p function ctrl(p, y) 1: V := Q ( {H p,prcvd for all non-null p rcvd y} ) 2: return x IC(V ) p Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

67 Correctness of the geometric-center algorithms For ɛ R >0, the ɛ-disk-covering deployment task { true, if p [i] CC(V [i] (P )) 2 ɛ, i {1,..., n}, T ɛ-dc-dply (P ) = false, otherwise, For ɛ R >0, the ɛ-sphere-packing deployment task { true, if dist 2 (p [i], IC(V [i] (P ))) ɛ, i {1,..., n}, T ɛ-sp-dply (P ) = false, otherwise, Theorem For d N, r R >0 and ɛ R >0, the following statements hold. 1 on the network S D, any execution of the law CC Vrn-crcmcntr monotonically optimizes the multicenter function H dc ; 2 on the network S D, any execution of the law CC Vrn-ncntr monotonically optimizes the multicenter function H sp. Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

68 Summary and conclusions Examined three basic motion coordination tasks 1 rendezvous: circumcenter algorithms 2 connectivity maintenance: flexible constraint sets in convex/nonconvex scenarios 3 deployment: gradient algorithms based on geometric centers Correctness and (1-d) complexity analysis of geometric-center control and communication laws via 1 Discrete- and continuous-time nondeterministic dynamical systems 2 Invariance principles, stability analysis 3 Geometric structures and geometric optimization Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

69 Motion coordination is emerging discipline Literature is full of exciting problems, solutions, and tools we have not covered Formation control, consensus, cohesiveness, flocking, collective synchronization, boundary estimation, cooperative control over constant graphs, quantization, asynchronism, delays, distributed estimation, spatial estimation, data fusion, target tracking, networks with minimal capabilities, target assignment, vehicle dynamics and energy-constrained motion, vehicle routing, dynamic servicing problems, load balancing, robotic implementations,... Too long a list to fit it here! Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

70 Book coming out in June 2009 Freely available online (forever) at Self-contained exposition of graph-theoretic concepts, distributed algorithms, and complexity measures Detailed treatment of averaging and consensus algorithms interpreted as linear iterations Introduction of geometric notions such as partitions, proximity graphs, and multicenter functions Detailed treatment of motion coordination algorithms for deployment, rendezvous, connectivity maintenance, and boundary estimation Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

71 Voronoi partitions Let (p 1,..., p n ) Q n denote the positions of n points The Voronoi partition V(P ) = {V 1,..., V n } generated by (p 1,..., p n ) V i = {q Q q p i q p j, j i} = Q j HP(p i, p j ) where HP(p i, p j ) is half plane (p i, p j ) 3 generators 5 generators 50 generators Return Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72

72 Distributed Voronoi computation Assume: agent with sensing/communication radius R i Objective: smallest R i which provides sufficient information for V i For all i, agent i performs: 1: initialize R i and compute V i = j: pi p j R i HP(p i, p j ) 2: while R i < 2 max q b Vi p i q do 3: R i := 2R i 4: detect vehicles p j within radius R i, recompute V i Return Martínez & Cortés (UCSD) Distributed robotic networks March 17, / 72