Summary introduction. Lecture #4: Deployment via Geometric Optimization. Coverage optimization. Expected-value multicenter function

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1 Lecture #4: Deployment via Geometric Optimization Summary introduction Francesco Bullo 1 Jorge Cortés Sonia Martínez 1 Department of Mechanical Engineering University of California, Santa Barbara bullo@engineering.ucsb.edu Mechanical and Aerospace Engineering University of California, San Diego {cortes,soniamd}@ucsd.edu Another motion coordination objective: deployment optimal task allocation and space partitioning, optimal placement and tuning of sensors Connection with geometric optimization and basic behaviors Formal definition and analysis of tasks and algorithms Workshop on Distributed Control of Robotic Networks IEEE Conference on Decision and Control Cancun, December 8, 008 / 39 Coverage optimization Expected-value multicenter function DESIGN of performance metrics 1 how to cover a region with n minimum-radius overlapping disks? 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? Objective: Given sensors/nodes/robots/sites (p1,..., pn) 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 [ ] Hexp(p1,..., pn) = Eφ max f( q pi ) i {1,...,n} 3 / 39 4 / 39

2 H exp -optimality of the Voronoi partition Alternative expression in terms of Voronoi partition, Hexp(p1,..., pn) = f( q pi )φ(q)dq Vi(P) for (p1,..., pn) distinct Proposition Let P = {p1,..., pn} F(S). For any performance function f and for any partition {W1,..., Wn} P(S) of S, Hexp(p1,..., pn, V1(P),..., Vn(P)) Hexp(p1,..., pn, W1,..., Wn), and the inequality is strict if any set in {W1,..., Wn} differs from the corresponding set in {V1(P),..., Vn(P)} by a set of positive measure Area problem f(x) = 1 [0,a](x), a R>0 Harea,a(p1,..., pn) = 1 [0,a]( q pi )φ(q)dq Vi(P) = φ(q)dq Vi(P) B(pi,a) = Aφ(Vi(P) B(pi, a)) = Aφ( n B(pi, a)), Area, measured according to φ, covered by the union of the n balls B(p1, a),..., B(pn, a) 5 / 39 Distortion problem f(x) = x Hdistor(p1,..., pn) = q pi φ(q)dq = Jφ(Vi(P), pi) Vi(P ) (Jφ(W, p) is moment of inertia). Note Hdistor(p1,..., pn, W1,..., Wn) = Jφ(Wi, CMφ(Wi)) Aφ(Wi) pi CMφ(Wi) Proposition Let {W1,..., Wn} P(S) be a partition of S. Then, ) Hdistor( CMφ(W1),..., CMφ(Wn), W1,..., Wn Hdistor(p1,..., pn, W1,..., Wn), and the inequality is strict if there exists i {1,..., n} for which Wi has non-vanishing area and pi CMφ(Wi) Mixed distortion-area problem f(x) = x 1 [0,a](x) + b 1 ]a,+ [(x), with a R>0 and b a Hdistor-area,a,b(p1,..., pn) = Jφ(Vi,a(P), pi) + b Aφ(Q \ n B(pi, a)), If b = a, performance f is continuous, and we write Hdistor-area,a. Extension to sets of points and partitions reads Hdistor-area,a(p1,..., pn, W1,..., Wn) ( ) = Jφ(Wi B(pi, a), pi) + a Aφ(Wi (S \ B(pi, a))). Proposition (Hdistor-area,a-optimality of centroid locations) Let {W1,..., Wn} P(S) be a partition of S. Then, ) Hdistor-area,a( CMφ(W1 B(p1, a)),..., CMφ(Wn B(pn, a)), W1,..., Wn Hdistor(p1,..., pn, W1,..., Wn), 6 / 39 7 / 39 and the inequality is strict if there exists i {1,..., n} for which Wi has non-vanishing area and pi CMφ(Wi B(pi, a)). 8 / 39

3 Smoothness properties of H exp Some proof ideas Dscn(f) (finite) discontinuities of f f and f+, limiting values from the left and from the right Theorem Expected-value multicenter function Hexp : S n R is 1 globally Lipschitz on S n ; and continuously differentiable on S n \ Scoinc, where Hexp (P ) = f( q pi )φ(q)dq Vi(P) + ( f (a) f+(a) ) n (q)φ(q)dq out,b(pi,a) Vi(P) B(pi,a) a Dscn(f) Consider the case of smooth performance f, Hexp (P ) = f ( q pi ) φ(q)dq Vi(P ) + f ( q pi ) ni(q), q φ(q)dq Vi(P ) + f ( q pj ) nji(q), q φ(q)dq j neigh i Vj(P ) Vi(P ) = integral over Vi + integral along arcs in Vi Therefore, the gradient of Hexp is spatially distributed over GD 9 / / 39 Some proof ideas Particular gradients Distortion problem: continuous performance, Hexp (P ) = f ( q pi ) φ(q)dq = Aφ(Vi(P))(CMφ(Vi(P)) pi) Vi(P ) }{{} for f(x)=x + f ( q pi ) ni(q), q φ(q)dq Vi(P ) f ( q pi ) ni(q), q φ(q)dq Vi(P ) Hdistor (P ) = Aφ(Vi(P))(CMφ(Vi(P)) pi) Area problem: performance has single discontinuity, Harea,a (P ) = n (q)φ(q)dq out,b(pi,a) Vi(P) B(pi,a) Mixed distortion-area: continuous performance (b = a ), Hdistor-area,a (P ) = Aφ(Vi,a(P))(CMφ(Vi,a(P)) pi) 11 / 39 1 / 39

4 Tuning the optimization problem Geometric-center laws Gradients of Harea,a, Hdistor-area,a,b are distributed over GLD (a) Uniform networks SD and SLD of locally-connected first-order agents in a polytope Q Rd with the Delaunay and r-limited Delaunay graphs as communication graphs Robotic agents with range-limited interactions can compute gradients of Harea,a and Hdistor-area,a,b as long as r a Proposition (Constant-factor approximation of Hdistor ) Let S Rd be bounded and measurable. Consider the mixed distortion-area problem with a ]0, diam S] and b = diam(s). Then, for all P S n, Hdistor-area,a,b (P ) Hdistor (P ) β Hdistor-area,a,b (P ) < 0, where β = a diam(s) [0, 1] 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 Similarly, constant-factor approximations of Hexp 13 / 39 Vrn-cntrd algorithm 14 / 39 Simulation Optimizes distortion Hdistor Robotic Network: SD in Q, with absolute sensing of own position Distributed Algorithm: Vrn-cntrd Alphabet: A = Rd {null} function msg(p, i) function ctl(p, y) T 1: V := Q {Hp,prcvd for all non-null prcvd y} : return CMφ (V ) p gradient descent For ε R>0, the ε-distortion deployment task ( true, if p[i] CMφ (V [i] (P )) Tε-distor-dply (P ) = false, otherwise, 15 / 39 ε, i {1,..., n}, 16 / 39

5 Voronoi-centroid law on planar vehicles Algorithm illustration Robotic Network: Svehicles in Q with absolute sensing of own position Distributed Algorithm: Vrn-cntrd-dynmcs Alphabet: A = R {null} function msg((p, θ), i) function ctl((p, θ), (psmpld, θsmpld ), y) T 1: V := Q {Hpsmpld,prcvd for all non-null prcvd y} : v := kprop (cos θ, sin θ) (p CMφ (V )) ( sin θ, cos θ) (p CMφ (V )) 3: ω := kprop arctan (cos θ, sin θ) (p CMφ (V )) 4: return (v, ω) 18 / / 39 Lmtd-Vrn-nrml algorithm Simulation Optimizes area Harea, r Robotic Network: SLD in Q with absolute sensing of own position and with communication range r Distributed Algorithm: Lmtd-Vrn-nrml Alphabet: A = Rd {null} function msg(p, i) gradient descent 19 / 39 function ctl(p, y) T 1: V :=RQ {Hp,prcvd for all non-null prcvd y} : v := V B(p, r ) nout,b(p, r ) (q)φ(q)dq 3: λ R := max{λ δ 7 φ(q)dq is strictly increasing on [0, λ]} V B(p+δv, r ) 4: return λ v 0 / 39

6 Lmtd-Vrn-cntrd algorithm Simulation Optimizes Hdistor-area, r Robotic Network: SLD in Q with absolute sensing of own position, and with communication range r Distributed Algorithm: Lmtd-Vrn-cntrd Alphabet: A = Rd {null} gradient descent For r, ε R>0, Tε-r-area-dply (P ) ( R true, if V [i] (P ) B(p[i], r ) nout,b(p[i], r ) (q)φ(q)dq = false, otherwise. function msg(p, i) function ctl(p, y) T 1: V := Q B(p, r ) {Hp,prcvd for all non-null prcvd y} : return CMφ (V ) p ε, i {1,..., n}, 1 / 39 / 39 Optimizing Hdistor via constant-factor approximation Simulation Limited range run #1: 16 agents, density φ is sum of 4 Gaussians, time invariant, 1st order dynamics gradient descent For r, ε R>0, Tε-r-distor-area-dply (P ) ( [i] true, if p[i] CMφ (V r (P ))) = false, otherwise. Unlimited range run #: 16 agents, density φ is sum of 4 Gaussians, time invariant, 1st order dynamics ε, i {1,..., n}, 3 / 39 gradient descent of H r gradient descent of Hexp 4 / 39

7 Correctness of the geometric-center algorithms Time complexity of CC Lmtd-Vrn-cntrd Theorem For d N, r R>0 and ε R>0, the following statements hold. 1 on the network SD, the law CCVrn-cntrd and on the network Svehicles, the law CCVrn-cntrd-dynmcs both achieve the ε-distortion deployment task Tε-distor-dply. Moreover, any execution of CCVrn-cntrd and CCVrn-cntrd-dynmcs monotonically optimizes the multicenter function Hdistor; on the network SLD, the law CCLmtd-Vrn-nrml achieves the ε-r-area deployment task Tε-r-area-dply. Moreover, any execution of CCLmtd-Vrn-nrml monotonically optimizes the multicenter function Harea, r ; and 3 on the network SLD, the law CCLmtd-Vrn-cntrd achieves the ε-r-distortion-area deployment task Tε-r-distor-area-dply. Moreover, any execution of CCLmtd-Vrn-cntrd monotonically optimizes the multicenter function Hdistor-area, r. Assume diam(q) is independent of n, r and ε 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 ε R>0, on the network SLD TC(Tε-r-distor-area-dply, CCLmtd-Vrn-cntrd) O(n 3 log(nε 1 )) 5 / 39 6 / 39 Deployment: basic behaviors Deployment: 1-center optimization problems move away from closest Equilibria? Asymptotic behavior? Optimizing network-wide function? move towards furthest smq(p) = min{ p q q Q} Lipschitz 0 smq(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 (xi) xi x, xi ΩV S 7 / 39 8 / 39

8 Deployment: 1-center optimization problems Nonsmooth LaSalle Invariance Principle + gradient flow of smq ṗi = + Ln[ smq](p) move away from closest gradient flow of lg Q ṗi = Ln[ lg Q ](p) move toward furthest For X essentially locally bounded, Filippov solution of ẋ = X(x) is absolutely continuous function t [t0, t1] x(t) verifying ẋ K[X](x) = co{ lim i X(xi) xi x, xi S} For V locally Lipschitz, gradient flow is ẋ = Ln[ V ](x) Ln = least norm operator Evolution of V along Filippov solution t V (x(t)) is differentiable a.e. d dt V (x(t)) LXV (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 LXV (x) 0 Any Filippov solution starting in S converges to largest weakly invariant set contained in {x S 0 LXV (x)} E.g., nonsmooth gradient flow ẋ = Ln[ V ](x) converges to critical set 9 / 39 Deployment: multi-center optimization sphere packing and disk covering Deployment: multi-center optimization 30 / 39 move away from closest : ṗi = + Ln( sm Vi(P ))(pi) at fixed Vi(P ) move towards furthest : ṗi = Ln( lg Vi(P ))(pi) at fixed Vi(P ) Critical points of Hsp and Hdc (locally Lipschitz) If 0 int( Hsp(P )), then P is strict local maximum, all agents have same cost, and P is incenter Voronoi configuration If 0 int( Hdc(P )), then P is strict local minimum, all agents have same cost, and P is circumcenter Voronoi configuration Aggregate functions monotonically optimized along evolution Aggregate objective functions! Hsp(P ) = min i Hdc(P ) = max i sm Vi(P )(pi) = min i j lg Vi(P )(pi) = max q Q [ 1 pi pj, dist(pi, Q)] [ min q pi ] i min L Ln( smv())hsp(p ) 0 max L Ln( lgv() )Hdc(P ) 0 Asymptotic convergence to center Voronoi configurations via nonsmooth LaSalle 31 / 39 3 / 39

9 Voronoi-circumcenter algorithm Voronoi-incenter algorithm Robotic Network: SD in Q with absolute sensing of own position Distributed Algorithm: Vrn-crcmcntr Alphabet: A = R d {null} function msg(p, i) function ctl(p, y) 1: V := Q ( for all non-null prcvd y}) {Hp,prcvd : return CC(V ) p Robotic Network: SD in Q, with absolute sensing of own position Distributed Algorithm: Vrn-ncntr Alphabet: A = R d {null} function msg(p, i) function ctl(p, y) 1: V := Q ( for all non-null prcvd y}) {Hp,prcvd : return x IC(V ) p 33 / / 39 Correctness of the geometric-center algorithms For ε R>0, the ε-disk-covering deployment task { true, if p [i] CC(V [i] (P )) ε, i {1,..., n}, Tε-dc-dply(P ) = false, otherwise, For ε R>0, the ε-sphere-packing deployment task { true, if dist(p [i], IC(V [i] (P ))) ε, i {1,..., n}, Tε-sp-dply(P ) = false, otherwise, Theorem For d N, r R>0 and ε R>0, the following statements hold. 1 on the network SD, any execution of the law CCVrn-crcmcntr monotonically optimizes the multicenter function Hdc; on the network SD, any execution of the law CCVrn-ncntr monotonically optimizes the multicenter function Hsp. Summary and conclusions Aggregate objective functions 1 variety of scenarios: expected-value, disk-covering, sphere-packing smoothness properties and gradient information 3 geometric-center control and communication laws Technical tools 1 Geometric optimization Geometric models, proximity graphs, spatially-distributed maps 3 Systems theory, nonsmooth stability analysis 35 / / 39

10 References Voronoi partitions Deployment scenarios and algorithms: J. Cortés, S. Martínez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 0():43--55, 004 J. Cortés, S. Martínez, and F. Bullo. Spatially-distributed coverage optimization and control with limited-range interactions. ESAIM. Control, Optimisation & Calculus of Variations, 11: , 005 Let (p1,..., pn) Q n denote the positions of n points The Voronoi partition V(P ) = {V1,..., Vn} generated by (p1,..., pn) Vi = {q Q q pi q pj, j i} = Q j HP(pi, pj) where HP(pi, pj) is half plane (pi, pj) Nonsmooth stability analysis: J. Cortés. Discontinuous dynamical systems -- a tutorial on solutions, nonsmooth analysis, and stability. IEEE Control Systems Magazine, 8(3):36--73, 008 Geometric and combinatorial optimization: P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Computing Surveys, 30(4): , generators 5 generators 50 generators Return 37 / / 39 Distributed Voronoi computation Assume: agent with sensing/communication radius Ri Objective: smallest Ri which provides sufficient information for Vi For all i, agent i performs: 1: initialize Ri and compute Vi = j: pi pj RiHP(pi, pj) : while Ri < max q Vi b pi q do 3: Ri := Ri 4: detect vehicles pj within radius Ri, recompute Vi Return 39 / 39