Symmetries and Maxwell points in the plate-ball problem and other - PowerPoint PPT Presentation

Symmetries and Maxwell points in the plate-ball problem and other invariant optimal control problems on Lie groups governed by the pendulum Yuri L. Sachkov Program Systems Institute Russian Academy of Sciences Pereslavl-Zalessky, Russia sachkov@sys.botik.ru Workshop on Nonlinear Control and Singularities Toulon, October 24 – 28, 2010

The plate-ball problem Rolling of sphere on plane without slipping or twisting Given: A , B ∈ R 2 , frames ( a 1 , a 2 , a 3 ) , ( b 1 , b 2 , b 3 ) in R 3 . Find: γ ( t ) ∈ R 2 , t ∈ [ 0 , t 1 ] , — the shortest curve s.t.: γ ( 0 ) = A , γ ( t 1 ) = B , by rolling along γ ( t ) , orientation of the sphere transfers from ( a 1 , a 2 , a 3 ) to ( b 1 , b 2 , b 3 ) .

State and control variables • Contact point ( x , y ) ∈ R 2 • Orientation of sphere R : a i �→ e i , i = 1 , 2 , 3, R ∈ SO ( 3 ) • State of the system Q = ( x , y , R ) ∈ R 2 × SO ( 3 ) = M • Boundary conditions Q ( 0 ) = Q 0 , Q ( t 1 ) = Q 1 • Controls u 1 = u / 2, u 2 = v / 2 • Cost functional � t 1 � t 1 � � x 2 + ˙ y 2 dt = u 2 1 + u 2 l ( γ ) = ˙ 2 dt → min 0 0

Control system ( x , y ) ∈ R 2 , ( u 1 , u 2 ) ∈ R 2 , x = u 1 , ˙ y = u 2 , ˙   0 − ω 3 ω 2 ˙  ,  R ∈ SO ( 3 ) , − ω 1 R = R Ω , Ω = ω 3 0 − ω 2 ω 1 0   ω 1   angular velocity vector. ω = ω 2 ω 3 No twisting ⇒ ω 3 = 0. No slipping ⇒ ω 1 = u 2 , ω 2 = − u 1 .   0 0 − u 1   − u 2 Ω = 0 0 u 1 u 2 0

History of the problem 1894 H. Hertz: rolling sphere as a nonholonomic mechanical system. 1983 J.M. Hammersley: statement of the plate-ball problem. 1986 A.M. Arthur, G.R.Walsh: integrability of Hamiltonian system of PMP in quadratures. 1990 Z. Li, E. Canny: complete controllability of the control system. 1993 V. Jurdjevic: - projections of extremal curves ( x ( t ) , y ( t )) — Euler elasticae, - description of qualitative types of extremal trajectories, - quadratures for evolution of Euler angles along extremal trajectories.

New results • Parameterization of extremal trajectories • Continuous and discrete symmetries • Fixed points of symmetries (Maxwell points) • Necessary optimality conditions • Global structure of the exponential mapping • Asymptotics of extremal trajectories and limit behavior of Maxwell points for sphere rolling along sinusoids of small amplitude (Next talk by Alexey Mashtakov)

Existence of solutions • Left-invariant sub-Riemannian problem: ˙ ( u 1 , u 2 ) ∈ R 2 , Q = u 1 X 1 ( Q ) + u 2 X 2 ( Q ) , Q ∈ M = R 2 × SO ( 3 ) , Q ( 0 ) = Q 0 , Q ( t 1 ) = Q 1 , � t 1 � u 2 1 + u 2 l = 2 dt → min . 0 • Complete controllability by Rashevskii-Chow theorem: span Q ( X 1 , X 2 , X 3 , X 4 , X 5 ) = T Q M ∀ Q ∈ M , X 3 = [ X 1 , X 2 ] , X 4 = [ X 1 , X 3 ] , X 5 = [ X 2 , X 3 ] . • Filippov’s theorem: ∀ Q 0 , Q 1 ∈ M optimal trajectory exists. • Q 0 = ( 0 , 0 , Id ) ∈ R 2 × SO ( 3 ) .

Pontryagin maximum principle • Abnormal extremal trajectories: rolling of sphere along straight lines. • Normal extremals: ˙ c = − r sin θ, θ = c , ˙ α = ˙ ˙ r = 0 , (1) x = cos ( θ + α ) , ˙ y = sin ( θ + α ) , ˙ (2) ˙ R = R ( sin ( θ + α ) A 1 − cos ( θ + α ) A 2 ) ,     0 0 0 0 0 1   ,   , A 1 = 0 0 − 1 A 2 = 0 0 0 0 1 0 − 1 0 0   0 − 1 0   . A 3 = [ A 1 , A 2 ] = 1 0 0 0 0 0 ( 1 ) mathematical pendulum, ( 2 ) Euler elasticae.

Mathematical pendulum ˙ θ = c , ˙ c = − r sin θ s ❙ ❙ L θ ❙ ❙ m s ❄ mg • λ = ( θ, c , r ) ∈ C = { θ ∈ S 1 , c ∈ R , r ≥ 0 } , • Energy E = c 2 / 2 − r cos θ = const ∈ [ − r , + ∞ ) , • r = g / L ≥ 0.

Stratificaion of the phase cylinder C of pendulum C = ∪ 7 i = 1 C i , C i ∩ C j = ∅ , i � = j c 3 C 1 = { λ ∈ C | E ∈ ( − r , r ) , r > 0 } , C 2 C 3 C 2 = { λ ∈ C | E ∈ ( r , + ∞ ) , r > 0 } , ❥ 2 ❘ ❅ C 3 = { λ ∈ C | E = r > 0 , c � = 0 } , C 1 1 C 5 C 4 = { λ ∈ C | E = − r , r > 0 } , ☛ ✁ θ -3 -2 -1 1 2 3 − π ■ ❅ π C 5 = { λ ∈ C | E = r > 0 , c = 0 } , C 4 -1 C 6 = { λ ∈ C | r = 0 , c � = 0 } , ❨ � ✒ C 7 = { λ ∈ C | r = 0 , c = 0 } . -2 C 3 C 2 -3

Euler elasticae ˙ x = cos ( θ + α ) , ˙ y = sin ( θ + α ) C 1 (oscillations of pendulum): inflectional elasticae 3.5 3 3.5 3 2.5 3 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 1 2 3 4 -3 -2 -1 0 C 2 (rotations of pendulum): non-inflectional elasticae 1.2 1 0.8 0.6 0.4 0.2 0 -2 -1.5 -1 -0.5 0 0.5

Euler elasticae ˙ x = cos ( θ + α ) , ˙ y = sin ( θ + α ) C 3 (separatrix motions of penulum): critical elasticae 2 1.5 1 0.5 0 -2 -1 0 1 2 C 4 , C 5 , C 7 (equilibria of pendulum): straight lines C 6 (uniform rotation of pendulum under zero gravity): circles

Integration of normal Hamiltonian system of PMP ˙ θ = c , c = − r sin θ, ˙ x = cos ( θ + α ) , ˙ y = sin ( θ + α ) , ˙ ˙ R = R ( sin ( θ + α ) A 1 − cos ( θ + α ) A 2 ) . • θ t , c t , x t , y t : Jacobi’s functions cn, sn, dn, E, cn ( u , k ) = cos ( am ( u , k )) , sn ( u , k ) = sin ( am ( u , k )) , � ϕ dt ϕ = am ( u , k ) ⇐ ⇒ u = � = F ( ϕ, k ) . 1 − k 2 sin 2 t 0 • R ( t ) = e ( α − ϕ 0 3 ) A 3 e − ϕ 0 2 A 2 e ϕ 1 ( t ) A 3 e ϕ 2 ( t ) A 2 e ( ϕ 3 ( t ) − α ) A 3 , ϕ i ( t ) : Jacobi’s functions + elliptic integral of the 3-rd kind � u dt Π( n , u , k ) = � . ( 1 − n sin 2 t ) 1 − k 2 sin 2 t 0

Parameterization of trajectories of oscillating pendulum and inflectional Euler elasticae ( ϕ, k ) — coordinates rectifying the flow of pendulum , ϕ t = ϕ + t , sin ( θ t / 2 ) = k sn ( √ r ϕ t , k ) , cos ( θ t / 2 ) = dn ( √ r ϕ t , k ) , c t = 2 k √ r cn ( √ r ϕ t , k ) , x t = ¯ x t cos α − ¯ y t sin α, y t = ¯ x t sin α + ¯ y t cos α, x t = ( 2 ( E ( √ r ϕ t , k ) − E ( √ r ϕ, k )) − √ rt ) / √ r , ¯ y t = 2 k ( cn ( √ r ϕ, k ) − cn ( √ r ϕ t , k )) / √ r , ¯

Parameterization of the matrix of rotation for the case of oscillating pendulum � √ √ M − c 2 cos ϕ 2 ( t ) = c t / M , sin ϕ 2 ( t ) = ± t / M , � M − c 2 cos ϕ 3 ( t ) = ∓ sin θ t / t , � M − c 2 sin ϕ 3 ( t ) = ± ( r − cos θ t ) / t , √ √ 2 √ r ( 1 − r )(Π( l , am ( √ r ϕ t , k ) , k ) M M ( 1 + r ) ϕ 1 ( t ) = 2 t + − Π( l , am ( √ r ϕ, k ) , k )) , 4 k 2 r M = 1 + r 2 + 2 E , l = − ( 1 − r ) 2 .

Optimality of extremal trajectories • Short arcs of extremal trajectories Q ( s ) are optimal • Cut time along Q ( s ) : t cut = sup { t > 0 | Q ( s ) , s ∈ [ 0 , t ] , is optimal } . • Maxwell time: ∃ ˜ Q ( 0 ) = ˜ Q ( s ) �≡ Q ( s ) , Q ( 0 ) = Q 0 , Q ( t ) = ˜ Q ( t ) Maxwell point , t = t Max Maxwell time . • Upper bound on cut time: t cut ≤ t Max .

Rotations Φ β , β ∈ S 1 ( θ, c , r , α ) �→ ( θ, c , r , α + β ) , � x s � cos β � � x s � � − sin β �→ , y s sin β cos β y s R s �→ e β A 3 R s e − β A 3 .

Reflections ε i c ( x s , y s ) ✒ γ 2 γ ✛ ✯ ❥ ✑ ε 2 ✑ ✑ ✑ ✑ p c θ ✑ ✑ ε 3 ε 1 ✑ ✑ ❄ ✰ ✑ ✙ ❨ ( x 1 s , y 1 s ) ✒ γ 3 γ 1 ε 1 : ( θ s , c s ) �→ ( θ t − s , − c t − s ) , s ∈ [ 0 , t ] ( x s , y s ) �→ ( x 1 s , y 1 s ) = ( x t − s − x t , y t − s − y t ) R s �→ ( R t ) − 1 R t − s

Reflections ε i ( x s , y s ) ( x s , y s ) ✲ ✯ ( x 2 s , y 2 s ) ✯ l ✯ l ⊥ ( x 3 s , y 3 s ) ε 2 : ( θ s , c s ) �→ ( − θ t − s , c t − s ) , s ∈ [ 0 , t ] ( x s , y s ) �→ ( x 2 s , y 2 s ) = ( x t − s − x t , y t − y t − s ) R s �→ I 2 ( R t ) − 1 R t − s I 2 , I 2 = e π A 2 . ε 3 : ( θ s , c s ) �→ ( − θ s , − c s ) , s ∈ [ 0 , t ] ( x s , y s ) �→ ( x 3 s , y 3 s ) = ( x s , − y s ) R s �→ I 2 R s I 2 .

� � � � � � Exponential mapping and its symmetries • Group of symmetries G = � Φ β , ε 1 , ε 2 , ε 3 � = { Φ β , Φ β ◦ ε i | β ∈ S 1 , i = 1 , 2 , 3 } • Exponential mapping Exp ( λ, s ) = Q s = ( x s , y s , R s ) ∈ M = R 2 × SO ( 3 ) , λ = ( θ, c , α, r ) ∈ C , s > 0 . • Symmetries of exponential mapping Exp Exp C × R + ( λ, t ) � Q t M � � ε i ◦ Φ β ε i ◦ Φ β ε i ◦ Φ β ε i ◦ Φ β Exp � M ( λ i ,β , t ) � Exp � Q i ,β C × R + t

Maxwell sets corresponding to reflections • MAX i = { ( λ, t ) | ∃ β ∈ S 1 : λ i ,β � = λ, Q t = Q i ,β t } , i = 1 , 2 , 3. • Necessary optimality conditions: ( λ, t ) ∈ MAX i ⇒ Q s = Exp ( λ, s ) not optimal for s > t , t cut ( λ ) ≤ t .

Representation of rotations in R 3 by quaternions • H = { q = q 0 + iq 1 + jq 2 + kq 3 | q 0 , . . . , q 3 ∈ R } • S 3 = { q ∈ H || q | 2 = q 2 0 + q 2 1 + q 2 2 + q 2 3 = 1 } • I = { q ∈ H | Re q = q 0 = 0 } • q ∈ S 3 ⇒ R q ( a ) = qaq − 1 , R q ∈ SO ( 3 ) ∼ a ∈ I , = SO ( I ) • lift of the system ˙ R = R Ω from SO ( 3 ) to S 3 :   q 0 = 1 2 ( q 2 u 1 − q 1 u 2 ) , ˙     q 1 = 1 ˙ 2 ( q 3 u 1 + q 0 u 2 ) , q ∈ S 3 , ( u 1 , u 2 ) ∈ R 2 , q 2 = 1  ˙ 2 ( − q 0 u 1 + q 3 u 2 ) ,     q 3 = 1 ˙ 2 ( − q 1 u 1 − q 2 u 2 ) , q ( 0 ) = 1 .