Asymptotics of exponential mapping and limit behavior of ... · Historyoftheproblem 1983J.M.Hammersley: statementoftheplate-ballproblem. 1986A.M.Arthur,G.R.Walsh: integrabilityofHamiltoniansystemofPMP

Asymptotics of exponential mapping and limit behavior ofMaxwell points in the plate-ball problem

A. P. Mashtakov

Program Systems InstituteRussian Academy of Sciences

Pereslavl-Zalessky, Russia

[email protected]

Workshop on Nonlinear Control and Singularities

Toulon, October 24 – 28, 2010

A. P. Mashtakov (PSI RAS) plate-ball problem 25.10.10 1 / 47

The plate-ball problemRolling of sphere on plane without slipping or twisting

Given: A,B ∈ R2, frames (e01, e02, e03), (e11, e12, e13) in R3.

Find: γ(t) ∈ R2, t ∈ [0, t1], — the shortest curve s.t.:γ(0) = A, γ(t1) = B , by rolling along γ(t),orientation of the sphere transfers from (e01, e02, e03) to (e11, e12, e13).

g(t)


History of the problem

1983 J.M. Hammersley: statement of the plate-ball problem.

1986 A.M. Arthur, G.R.Walsh: integrability of Hamiltonian system of PMPin quadratures.

1990 Z.Li, J. Canny: controllability of the system.

1993 V. Jurdjevic:projections of extremal curves (x(t), y(t)) — Euler elasticae,description of different qualitative types of extremal trajectories,differential equations for evolution of Euler angles along extremaltrajectories.


Known results

Yu. Sachkov:Extremal trajectoriesContinuous and discrete symmetries

Fixed points of symmetries (Maxwell points)

Necessary optimality conditionsGlobal structure of the exponential mapping


New results

Asymptotics of extremal trajectories

Limit behavior of Maxwell points for sphere rolling on sinusoids ofsmall amplitude

Upper bound on cut time for the corresponding trajectories


State and control variables

Contact point (x , y) ∈ R2

Orientation of sphere R : ai 7→ ei , i = 1, 2, 3, R ∈ SO(3)

State of the system Q = (x , y ,R) ∈ R2 × SO(3) = MControls u1 = u/2, u2 = v/2, (u1, u2) ∈ R2


Representation of rotations in R3 by quaternions

H = {q = q0 + iq1 + jq2 + kq3|q0, . . . , q3 ∈ R}S3 = {q ∈ H||q|2 = q2

0 + q21 + q2

2 + q23 = 1}

I = {q ∈ H|Re q = q0 = 0}q ∈ S3 ⇒ Rq(a) = qaq−1, a ∈ I ,Rq ∈ SO(3) ∼= SO(I )


Optimal control problem

Control System

x = u1, y = u2,

q0 = 12(q2u1 − q1u2),

q1 = 12(q3u1 + q0u2),

q2 = 12(−q0u1 + q3u2),

q3 = 12(−q1u1 − q2u2),

q ∈ S3, (u1, u2) ∈ R2,

Boundary conditions Q(0) = Q0, Q(t1) = Q1

Cost functional l =

∫ t1

0

√x2 + y2 dt =

∫ t1

0

√u21 + u2

2 dt → min


Existence of solutions

Left-invariant sub-Riemannian problem:

Q = u1X1(Q) + u2X2(Q), (u1, u2) ∈ R2,

Q(0) = Q0, Q(t1) = Q1, Q ∈ M = R2 × SO(3),

l =

∫ t1

0

√u21 + u2

2 dt → min .

Complete controllability by Rashevskii-Chow theorem:

span(X1(q),X2(q),X3(q),X4(q),X5(q)) = TqM ∀ q ∈ M,

X3 = [X1,X2], X4 = [X1, [X1,X2]], X5 = [X2, [X1,X2]].

Filippov’s theorem: ∀Q0,Q1 ∈ M optimal trajectory exists.Q0 = (0, 0, Id) ∈ R2 × SO(3).


Pontryagin maximum principle

Abnormal extremal trajectories: rolling of sphere along straight lines.Normal extremals:

θ = c , c = −r sin θ, α = r = 0, (1)x = cos(θ + α), y = sin(θ + α), (2)

q0 =12

(q2 cos(θ + α)− q1 sin(θ + α)),

q1 =12

(q3 cos(θ + α) + q0 sin(θ + α)),

q2 =12

(−q0 cos(θ + α) + q3 sin(θ + α)),

q3 =12

(−q1 cos(θ + α)− q2 sin(θ + α)),

(1) mathematical pendulum,(1), (2) Euler elasticae.


Mathematical pendulum θ = c , c = −r sin θ

(θ, c , r) ∈ C = {θ ∈ S1, c ∈ R, r ≥ 0}Energy E = c2/2− r cos θ = const ∈ [−r ,+∞).

Decomposition C = ∪7i=1Ci :

C1 = {λ ∈ C | E ∈ (−r , r), r > 0},C2 = {λ ∈ C | E ∈ (r ,+∞), r > 0},C3 = {λ ∈ C | E = r > 0, c 6= 0},C4 = {λ ∈ C | E = −r , r > 0},C5 = {λ ∈ C | E = r > 0, c = 0},C6 = {λ ∈ C | r = 0, c 6= 0},C7 = {λ ∈ C | r = 0, c = 0}.


Optimality of extremal trajectories

Short arcs of extremal trajectories Q(s) are optimalCut time along Q(s):

tcut = sup{t > 0 | Q(s), s ∈ [0, t], is optimal }.

Maxwell time:

∃ Q(s) 6≡ Q(s), Q(0) = Q(0) = Q0,

Q(t) = Q(t) Maxwell point,t = tMax Maxwell time.

Upper bound on cut time:

tcut ≤ tMax.


Reflections and symmetries of exponential mapping

Yu. Sachkov:reflections ε1, ε2

?

��

��

��

��

��+

ε1

ε2

ε3θ

cγ

γ1

γ2

γ3

j

�Y

*

pc

(xs, ys)

(x1s, y

1s)

�

� l⊥

(xs, ys)

(x2s, y

2s)

-

*

λ = (θ, c , α, r) ∈ C , s > 0Exp(λ, s) = Qs = (xs , yy ,Rs) ∈ M = R2 × SO(3)

symmetries of exponential mapping:εi ◦ Exp(λ, s)) = Exp ◦εi (λ, s)

fixed point: Exp(λi , t) = Exp(λ, t)⇔ εi (Qt) = Qt


Necessary optimality conditions in terms of MAX1

MAXi = {(λ, t) | λi 6= λ, Qt = Q it}

(λ, t) ∈ MAXi ⇒ Qs = Exp(λ, s) not optimal for s > t.

Theorem (Yu. Sachkov)

Let Qs = (xs , ys ,Rs) = Exp(λ, s), t > 0 satisfy the conditions:(1) q3(t) = 0,(2) elastica {(xs , ys) | s ∈ [0, t]} is nondegenerate and not centered at

inflection point.Then (λ, t) ∈ MAX1, thus for any t1 > t the trajectory Qs , s ∈ [0, t1], isnot optimal.


Visualization of MAX1

Rolling of the sphere (Video)


Necessary optimality conditions in terms of MAX2

Theorem (Yu. Sachkov)

Let Qs = (xs , ys ,Rs) = Exp(λ, s), t > 0 satisfy the conditions:(1) (xq1 + yq2)(t) = 0,(2) elastica {(xs , ys) | s ∈ [0, t]} is nondegenerate and not centered at

vertex.Then (λ, t) ∈ MAX2, thus for any t1 > t the trajectory Qs , s ∈ [0, t1], isnot optimal.


Visualization of MAX2



Asymptotics of extremal trajectories near (θ, c) = (0, 0)

s = mt, d = c/m, m =√r ,(

xy

)= A(α)

(xy

),(

q1q2

)= A(α)

(q1q2

)A(α) =

(cosα − sinαsinα cosα

), q0 = q0, q3 = q3.

d

qp-p

ρ0 =√θ20 + d2

0 → 0asymptotics of pendulum (harmonic oscilla-tions)

θ(s) = θ0 cos s + d0 sin s + O(ρ20),

d(s) = −θ0 sin s + d0 cos s + O(ρ20).



Sinusoids of small amplitude:

x(s) =sm

+ O(ρ20),

y(s) =1m

(θ0 sin s + d0(1− cos s)) + O(ρ20).

x

y

?0/m

d0/m

(?0 /m+2p)

r0/m

θ0 = ρ0 cosχ0, d0 = ρ0 sinχ0, ρ0 → 0



q0(s) = coss2m

+ O(ρ20),

q1(s) =1

2(m2 − 1)(m cos

s2m

sin s − (1 + cos s) sins2m

)θ0+

+1

2(m2 − 1)(m(1− cos s) cos

s2m− sin s sin

s2m

)d0 + O(ρ20),

q2(s) =− sins2m

+ O(ρ20),

q3(s) =1

2(m2 − 1)((−1 + cos s) cos

s2m

+ m sin s sins2m

)θ0+

+1

2(m2 − 1)(sin s cos

s2m−m(1 + cos s) sin

s2m

)d0 + O(ρ20).


Limit behavior of Maxwell set MAX1

p = s2

q3 =d0 cos p − θ0 sin p

m2 − 1· g1(p,m) + O(ρ2

0)

g1(p,m) = cos pm sin p −m cos p sin p

m

p1(m) = min{p > 0|g1(p,m) = 0}

Qt = Exp(λ, t), λ = (θ0, d0,m, α) ∈ C1,ρ0 → 0 ⇒ ∃t ∈ [0, t1 + ε] such that (λ, t) ∈ MAX1

⇒ trajectory Qt not optimal at t ∈ [0, t1 + ε], t1 = 2p1(m)/m.


Bounds of the first root of equation g1(p,m) = 0

p(m) =

mρ, for m ∈ (0, 1

2 ],

mπ([ m1−m ] + 1), for m ∈ (1

2 , 1),

π([ 1m−1 ] + 1), for m ∈ (1, 2],

ρ, for m > 2, ρ = tan ρ, ρ ∈(π, 3π

2

)p(m) =

{mπ([ m

1−m ] + 2), for m ∈ (0, 1),

π([ 1m−1 ] + 2), for m ∈ (1,+∞).

TheoremFor any m ∈ (0, 1) ∪ (1,+∞),

p(m) ≤ p1(m) < p(m).



0.0 0.5 1.5 2.0 2.5 3.0m

5

10

15

20

25

p

p(m) p1(m) p(m)


Properties of the first root of equation g1(p,m) = 0

Theorem

a) limm→1

p1(m) = +∞;

b) p1(m) grows for m ∈ (0, 1) and decreases for m ∈ (1,+∞);c) p1(1 + 1

n ) = π(n + 1), p1( nn+1) = πn, p1(1 + 2

2n+1) = π(n + 32),

p1(2n+12n+3) = π(n + 1

2);d) p1(m) is continuous for m ∈ (0, 1) ∪ (1,+∞);e) p1(m) is smooth for

m ∈ (0, 1) ∪ (1,+∞)\({ nn + 1

|n ∈ N} ∪ {1 +1n|n ∈ N}).

∀ n ∈ N : p′1( nn+1) = +∞, p′1(n+1

n ) = −∞.



Hyperbola p0(m) = π m1−m , m ∈ (1/2, 1)

Plot of p1(m) crosses plot of p0(m) at points(1− 1

r , π(r − 1)), 2r ∈ N, r > 1New coordinates δ = 1−m

m (p − p0(m)) + π,s = 1+m

1−m , s > 3.

g1(δ, s) = 1s+1(sin(δs)− s sin δ)

δ1(s) = min{δ > 0|g1(δ, s) = 0}

TheoremFor any s > 1,

π − 1s − 1

< δ1(s) < π +1

s − 1



For any m ∈ (12 , 1)

p0(m)− 12< p1(m) < p0(m) +

12

symmetry: p1(m) = mp1( 1m )

p(m) =

{π m

1−m − 1/2, for m ∈ (12 , 1),

m( πm−1 −

12), for m ∈ (1, 2),

p(m) =

{π m

1−m + 1/2, for m ∈ (12 , 1),

m( πm−1 + 1

2), for m ∈ (1, 2).

TheoremFor any m ∈ (1/2, 1) ∪ (1, 2),

p(m) < p1(m) < p(m).



1

m

p

p(m) p(m)p(m)1 0p(m)



d = ρ cosχ, θ = ρ sinχχ ∈ S1, m > 0, m 6= 1

TheoremLet λ = (ρ, χ,m, α) ∈ C1 and m > 0, m 6= 1. Then

limρ→0,m→m

tcut(λ) ≤ t1(m).

TheoremLet λ = (ρ, χ,m, α) ∈ C1. Let K ⊂ {m ∈ R | m > 0, m 6= 1} be acompact. Then

limρ→0,m∈K

tcut(λ) ≤ maxm∈K

t1(m).



xq1 + yq2 =d0 cos p − θ0 sin p

m(m2 − 1)· 2g2(p,m) + O(ρ2

0)

g2(p,m) = mp cos pm sin p − (p cos p + (m2 − 1) sin p) sin p

m

p2(m) = min{p > 0|g2(p,m) = 0}

Qt = Exp(λ, t), λ = (θ0, d0,m, α) ∈ C1,ρ0 → 0 ⇒ ∃t ∈ [0, t2 + ε] such that (λ, t) ∈ MAX2 ⇒trajectory Qt not optimal at t ∈ [0, t2 + ε], t2 = 2p2(m)/m.numerical computationp1(m) ≤ p2(m) < p1(m)


Conjugate time as (θ, c)→ 0

Local chartq0 =

√1− q2

1 − q22 − q2

3

Conjugate time

tconj = min{t > 0| det(∂(x , y , qi )

∂(λ, t)) = 0}

Upper bound on cut timetcut ≤ tconj

numerical computation2p1(m)

m ≤ tconj <2p1(m)

m


Maxwell times and conjugate time as (θ, c)→ 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

m

5

10

15

20

25

30

35

t

t1(m) t2(m)tconj(m)


Motion planning for nonholonomicfive-dimensional systems


The plate-ball problem

Optimal control problemGiven: Q0,Q1 ∈ R2 × SO(3),Find: Q(t) ∈ R2 × SO(3), t ∈ [0, t1], such thatQ(0) = Q0,Q(t1) = Q1, where projection (Qx(t),Qy (t)) is theshortest curve in R2. (open problem)Motion planning problemGiven: Q0,Q1 ∈ R2 × SO(3), ε > 0Find: any trajectory Q(t) ∈ R2 × SO(3), t ∈ [0, t1], such thatQ(0) = Q0, dist(Q(t1),Q1) < ε.


Motion planning for nonholonomic five-dimensional systems

State and control variables:Q ∈ M ∼ R5, (u1, u2) ∈ R2

Control system:Q = u1(t)X1(Q) + u2(t)X2(Q)

Boundary conditionsQ(0) = Q0, dist(Q(t1),Q1) < ε

Growth vector (2,3,5):

dim span{X1(Q),X2(Q)} = 2,dim span{X1(Q),X2(Q),X3(Q)} = 3,dim span{X1(Q),X2(Q),X3(Q),X4(Q),X5(Q)} = 5,

where X3 = [X1,X2], X4 = [X1,X3], X5 = [X2,X3].


Motion planning for nonholonomic five-dimensional systems

Classes of controls:piecewise constantui = αis , for t ∈ [ts−1, ts ],where s ∈ 1, k , t0 = 0, tk = t1.

optimal∫ t10

√u21 + u2

2 dt → min


Motion Planing Iterative Algorithm

Resulting control:

ui = usi ,

for t ∈ [1−sk t1, s

k t1],where s ∈ 1, k ,k — number of iteration.

Approximating system: Q = u1(t)X1(Q) + u2(t)X2(Q)

trajectory of approximated system

trajectory of original system


Nilpotent Approximation

Nilpotent approximation in privileged coordinates (algorithm by A.Bellaiche).Canonical nilpotent five-dimensional system:

x = u1,

y = u2,

z = 12(xu2 − yu1),

v = 12(x2 + y2)u2,

w = −12(x2 + y2)u1.

Q = (x , y , z , v ,w) ∈ R5, (u1, u2) ∈ R2,Boundary conditions Q(0) = Q1, Q(1) = 0.


Piecewise constant control

Theorem

For any Q1 ∈ R5 exist (αi , βi , γi , δi ) ∈ R4, i ∈ {1, 2} and control

ui =

αi , for t ∈ [0, 1

4 ],

βi , for t ∈ (14 ,

12 ],

γi , for t ∈ (12 ,

34 ],

δi , for t ∈ (34 , 1],

such that Q(0) = Q1, Q(1) = 0

algebraic equations for parameters (αi , βi , γi , δi )

nonunique solutionfinal fixing of parameters by criterion max |ui | → min


Optimal control

Canonical nilpotent five-dimensional system∫ t10

√u21 + u2

2 dt → min

Nilpotent sub-Riemannian problem with growth vector (2,3,5) (Yu.Sachkov):

extremal trajectories

bounds on cut time

global structure of exponential mapping

symmetries

reduction to system of 3 algebraic equations in Jacobian functions of 3variables

numerical solution in Wolfram Mathematica


Convergence of algorithm

optimal controllocal controllability∀x1 ∈ M ∀O1 ⊂ M ∃O2 ⊂ O1 ∀x0 ∈ O2 ∃u(·) ∈ U :x(0) = x0, x(t1) = x1, x(·) ⊂ O1

piecewise constant controlRball < 10−2, Rmach < 10−4 (see next)

x1

O2

O1

R

x0

Trajectories of approximating system

Trajectories of original system

optimal control

piecewise constant control

optimal controlpiecewise constant control


Motion Planning for Sphere Rolling on a Plane

local chartq0 =

√1− q2

1 − q22 − q2

3 > 0.

control system

x = u1,

y = u2,

q1 = 12(q3 u1 +

√1− q2

1 − q22 − q2

3 u2),

q2 = 12(−

√1− q2

1 − q22 − q2

3 u1 + q3 u2),

q3 = 12(−q1 u1 − q2 u2).


Motion Planning for Sphere Rolling on a Plane

Boundary conditions:Q0 = (0, 0, 0, 0, 0), Q1 = (−1.525, 1.475, 0.346, 0.626, 0.242)Error: ε = 10−5

Iterations: 8

0.2 0.4 0.6 0.8 1.0

t

- 3

- 2

- 1

1

x,y,

q1,q2,q3

1x - x1y - y

q11 - q1

q21 - q2

q31 - q3


Visualization of motion planing algorithm



Motion Planning for Car with Two Trailers

state variables ξ = (x , y , θ, φ1, φ2)ξ ∈ R2 × S1 × (S1 − {π})2

control system

x = cos θ u1,

y = sin θ u1,

θ = u2,

φ1 = − sinφ1 u1+

+(−1− cosφ1) u2,

φ2 = (sin(φ1 − φ2) + sinφ1) u1+

+(cos(φ1 − φ2) + cosφ1)u2.


Motion Planning for Car with Two Trailers

Boundary conditions: ξ0 = (0, 0, 0.785, 0.785,−0.785),ξ1 = (−0.252,−0.339, 1.085, 0.514,−1.281)Error: ε = 10−6

Iterations: 7

0.2 0.4 0.6 0.8 1.0t

- 0.5

0.5

1.0

1x - x1y - y

1q - q1j - j111j - j2 2

x y qjj1 2, , , ,


Visualization of motion planing algorithm

Car with Two Trailers (Video)


Results and plans

asymptotics of extremal trajectories for the plate-ball problem near(θ, c) = (0, 0),

upper bound on cut time near (θ, c) = (0, 0),bound on Maxwell time t1 (proved)bound on Maxwell time t2 (numerics)bound on conjugate time tconj (numerics)

motion planing algorithm for nonholonomic five-dimensional systemsbased on nilpotent approximation

peacewise constant controloptimal control (work in progress)


Documents

Asymptotics of exponential mapping and limit behavior of ... · Historyoftheproblem 1983J.M.Hammersley: statementoftheplate-ballproblem. 1986A.M.Arthur,G.R.Walsh: integrabilityofHamiltoniansystemofPMP