CYCLIC PURSUIT WITHOUT COORDINATES: CONVERGENCE to REGULAR POLYGON FORMATIONS. Daniel Liberzon

Transcription

1 CYCLIC PURSUIT WITHOUT COORDINATES: CONVERGENCE to REGULAR POLYGON FORMATIONS Daniel Liberzon Joint work with Maxim Arnold (now at UT Dallas) and Yuliy Baryshnikov University of Illinois, Urbana-Champaign, U.S.A. CDC, LA, Dec of 11

2 MINIMALISTIC CONTROL & SENSING MODEL y φ agent i α x Assume forward speed is constant & fixed (say ) Angular speed w i { w, 0, w} ẋ i = u i cos φ i ẏ i = u i sin φ i φ i = w i u i 1 Can detect presence of another agent in windshield sector (unlimited sensing range) ( α, α) i i+1 w Agents cyclically arranged, with each agent seeing its target agent in its windshield (connectivity) can be maintained if is large enough System behavior depends critically on windshield angle α 2of 11

3 SMALL WINDSHIELD ANGLE: RENDEZVOUS α < π/n where n = # of agents In this case all agents converge to a single point, with perimeter P (t) of the formation polygon serving as a Lyapunov function [Yu LaValle L, IEEE TAC, 2012] 3of 11

4 LARGE WINDSHIELD ANGLE: REGULAR POLYGONS? α > π/n where n = # of agents Agents diverge but tend to form regular polygons Our goal: theoretically justify this empirically observed phenomenon 4of 11

5 RELATION to PRIOR WORK Many works on convergence of multi-agent formations to regular shapes, using different tools and different modeling assumptions: Behroozi Gagnon (1980s), Richardson (2001), Marshall Broucke Francis (2004, 2006), Pavone Frazzoli (2007), Sinha Ghose (2007), Arnold Baryshnikov LaValle (2012), Galloway Justh Krishnaprasad (2013), Sharma Ramakrishnan Kumar (2013) Unique feature of our model (same as in Yu LaValle L, 2012): neither inter-agent distance nor heading error available for feedback As in Marshall et al., we will use eigenvalue properties of block-circulant matrices to show convergence But unlike in Marshall et al., here regular shape is attractive while formation size grows, which makes analysis different 5of 11

6 a i 1 SYSTEM EQUATIONS α i 1 `i a i ψ i α i `i+1 a i+1 Angles are subject to constraint P ni=1 ψ i = π(n 2k) ψ i =(sinα i 1 +sin(ψ i + α i ))/`i (sin α i +sin(ψ i+1 + α i+1 ))/`i+1 `i = cos α i 1 cos(ψ i + α i ) Constant-bearing case: assume α i α i, i.e., each a i maintains its target exactly on its windshield boundary a i+1 This is sliding regime for angular velocity w i { w, 0, w} 6of 11

7 STATIONARY SHAPES ARE REGULAR POLYGONS a i 1 α `i a i a i+1 ψ i α `i+1 A shape is an equivalence class of polygons w.r.t. scaling & rigid motions A shape is stationary if it is invariant under system dynamics For a shape to be stationary we must have ψ i 0 i `i(t) =`i(0) t (cos α +cos(ψ+α)) Case of interest here is when α is large enough s.t. cos α +cos(ψ+α) < 0 ψ i =(sinα +sin(ψ i + α))/`i (sin α +sin(ψ i+1 + α))/`i+1 `i = cos α cos(ψ i + α) Can show that then all angles must be equal: ψ i ψ i so all edges have the same derivative: Since `i(t) and `i(t)/`j(t) 1, stationary shape must have equal edges: Example: convex regular ψ ψ+α `i(t) =`j(t) i, j n-gon ψ = π 2π/n α Need α>π/n 7of 11

8 SPACE TIME COORDINATE CHANGE Similarly to [Galloway et al., 2013], instead of `i(t) ρ i (t):= `i(t) P P (t):= n edge lengths consider where `i(t) is perimeter P (t) i=1 In rescaled coords, stationary shapes (regular polygons) stationary points, i.e., equilibria: ψ i ψ, ρ i 1/n i τ(t) :=t/p (t) Also introduce time rescaling which allows us to write the system in (ψ i, ρ i ) coords If equilibria are locally exponentially stable for this new system, then regular polygons are locally attractive for original system 8of 11

9 EIGENVALUE ANALYSIS of LINEARIZED SYSTEM Rescaled coords: (ψ 1, ρ 1,...,ψ n, ρ n ) where 2n 2n Jacobian matrix at equilibria takes block-circulant form ψ i = ψ, ρ i =1/n J = ρ i (t):= `i(t)/p (t) A 0 A 1 A 2 A n 1 A n 1 A 0 A A 1 A 2 A n 1 A 0 It always has one 0 eigenvalue with eigenvector (d, 0,...,d,0) not an admissible direction as all angles ψ i cannot increase k J 2 2 -th pair of eigenvalues of is eigenvalues of matrix A 0 +A 1 χ k +A 2 χ 2 k +...+A n 1χ n 1 k where χ k := e 2πik/n By Routh-Hurwitz criterion for polynomials with complex coefficients can show that these eigenvalues have Re(λ) < 0 if and only if (3C 2 +AC +1+B)cos(2π/n) < (6C 2 +A 2 +5AC 1 B) where A:= cos α, B := cos ψ, C:= cos(ψ+α) Sufficient condition for local attractivity of regular polygons 9of 11

10 DISCUSSION We have convergence to regular polygons if A + C<0and (3C 2 +AC +1+B)cos(2π/n) < (6C 2 +A 2 +5AC 1 B) where A:= cos α, B := cos ψ, C:= cos(ψ+α) ( ) n For convex regular -gon we have ψ = π(1 2/n) and can show that ( ) holds for α>π/n as long as is not too large (recall that α<π/n gives rendezvous) α Example: n =4 Pink region is where A + C>0 Blue region is where ( ) fails White region is the good region Square ( ψ = π/2) is locally attractive for α (π/4, π) Ψ Α 10 of 11

11 DISCUSSION A + C<0 We have convergence to regular polygons if and (3C 2 +AC +1+B)cos(2π/n) < (6C 2 +A 2 +5AC 1 B) where A:= cos α, B := cos ψ, C:= cos(ψ+α) ( ) n For convex regular -gon we have ψ = π(1 2/n) and can show that ( ) holds for α>π/n as long as is not too large (recall that α<π/n gives rendezvous) α Example: n =5 Pink region is where A + C>0 Blue region is where ( ) fails White region is the good region Ψ 1.5 Both pentagon ( ψ =3π/5) and pentagram ( ψ = π/5) are locally attractive for some range of (separatrix?) Α 10 of 11

12 DISCUSSION We have convergence to regular polygons if A + C<0and (3C 2 +AC +1+B)cos(2π/n) < (6C 2 +A 2 +5AC 1 B) where A:= cos α, B := cos ψ, C:= cos(ψ+α) ( ) Using 1 instead of cos(2π/n) in, we obtain 3C 2 +A 2 +4AC 2 2B >0 ( ) which can be simplified to µ cos 2α + ψ µ +3cos 2α + 3ψ 2 2 > 0 This sufficient condition is more conservative but works for any n 10 of 11

13 CONCLUSIONS Agents in cyclic pursuit with large fixed relative heading angle diverge but tend to form regular patterns Stationary shapes are regular polygons Block-circulant structure of linearization matrix gives a sufficient condition for convergence Need to better understand system behavior, especially far away from stationary shapes Extensions to non-constant heading angles and non-cyclic formations are also of interest 11 of 11