Optimal control models of the goal-oriented human locomotion (with - - PowerPoint PPT Presentation

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Optimal control models of the goal-oriented human locomotion (with - - PowerPoint PPT Presentation

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Optimal control models of the goal-oriented human locomotion (with Y. Chitour, F. Chittaro, and P. Mason) Fr ed eric Jean (ENSTA ParisTech, Paris) Nonlinear Control and Singularities October 24-28, 2010 F. Jean (ENSTA ParisTech)

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Optimal control models

  • f the goal-oriented human locomotion

(with Y. Chitour, F. Chittaro, and P. Mason) Fr´ ed´ eric Jean

(ENSTA ParisTech, Paris)

Nonlinear Control and Singularities – October 24-28, 2010

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 1 / 30

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SLIDE 2

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 2 / 30

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SLIDE 3

Inverse optimal control

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 3 / 30

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Inverse optimal control

Inverse optimal control

Analysis/modelling of human motor control → looking for optimality principles Subjects under study:

Arm pointing motions (with J-P. Gauthier, B. Berret) Saccadic motion of the eyes Goal oriented human locomotion

Mathematical formulation: inverse optimal control Given ˙ X = φ(X, u) and a set Γ of trajectories, find a cost C(Xu) such that every γ ∈ Γ is solution of inf{C(Xu) : Xu traj. s.t. Xu(0) = γ(0), Xu(T) = γ(T)}.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 4 / 30

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Inverse optimal control

Difficulties:

Dynamical model not always known (← hierarchical optimal control) Limited precision of both dynamical models and costs ⇒ necessity of stability (genericity) of the criterion Non-unicity of the cost that is solution of the problem. No general method

Validation method: a program in three steps

1

Modelling step: propose a class of optimal control problems

2

Analysis step: enhance qualitative properties of the optimal synthesis (e.g. inactivations) → reduce the class of problems

3

Comparison step: numerical methods → choice of the best fitting L

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 5 / 30

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SLIDE 6

Inverse optimal control

Difficulties:

Dynamical model not always known (← hierarchical optimal control) Limited precision of both dynamical models and costs ⇒ necessity of stability (genericity) of the criterion Non-unicity of the cost that is solution of the problem. No general method

Validation method: a program in three steps

1

Modelling step: propose a class of optimal control problems

2

Analysis step: enhance qualitative properties of the optimal synthesis (e.g. inactivations) → reduce the class of problems

3

Comparison step: numerical methods → choice of the best fitting L

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 5 / 30

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SLIDE 7

Modelling goal oriented human locomotion

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 6 / 30

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Modelling goal oriented human locomotion

Trajectories of the Human Locomotion

0 θ0

(x , y , )

1 1 θ1

(x , y , )

Goal-oriented human locomotion (Berthoz, Laumond et al.) Initial point (x0, y0, θ0) → Final point (x1, y1, θ1)

(x, y position, θ orientation of the body)

QUESTIONS : Which trajectory is experimentally the most likely? What criterion is used to choose this trajectory?

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 7 / 30

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Modelling goal oriented human locomotion

Trajectories of the Human Locomotion

HYPOTHESIS: the chosen trajectory is solution of a minimization problem          min

  • L(x, y, θ, ˙

x, ˙ y, ˙ θ, . . . )dt among all “possible” trajectories joining the initial point to the final one. → TWO QUESTIONS: What are the possible trajectories ? dynamical constraints? How to choose the criterion ? (inverse optimal control problem)

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 8 / 30

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Modelling goal oriented human locomotion

Dynamical Constraints

[Arechavaleta-Laumond-Hicheur-Berthoz, 2006] (=[ALHB]) ↓ 1st experimental observation: if target far enough, the velocity is perpendicular to the body

v

→ ˙ x sin θ − ˙ y cos θ = 0 nonholonomic constraint!

(Dubins)

− → ˙ x = v cos θ ˙ y = v sin θ v = tangential velocity.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 9 / 30

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Modelling goal oriented human locomotion

Dynamical Model

2nd experimental observation in [ALHB] ↓ the velocity has a positive lower bound, v ≥ a > 0, and is a function (almost constant) of the curvature ⇒ The trajectories may be parameterized by arc-length (we are only interested by the geometric curves) → v ≡ 1.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 10 / 30

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Modelling goal oriented human locomotion

Dynamical Model

Previous observations: L = L(x, y, θ, ˙ θ, . . . ) trajectories obtained through a minimization procedure ⇒ trajectories in a complete functional space, θ ∈ W k,p, and L = L(x, y, θ, ˙ θ, . . . , θ(k)). → the trajectories are solutions of the optimal control problem min CL(u) = T L(x, y, θ, ˙ θ, . . . , θ(k))dt among all trajectories of    ˙ x = cos θ ˙ y = sin θ θ(k) = u u ∈ Lp, s.t. (x, y, θ)(0) = (x0, y0, θ0) and (x, y, θ)(T) = (x1, y1, θ1).

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 11 / 30

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Modelling goal oriented human locomotion

Admissible costs

ELEMENTARY REMARKS : k ≥ 3 not reasonable ⇒ k = 1 or k = 2. The whole problem is invariant by rototranslations → L = L( ˙ θ, . . . , θ(k)) independent of (x, y, θ) 0 is the unique minimum of L Normalization: L(0) = 1 ⇒ cost of a straight line = its length L is convex w.r.t. u TECHNICAL ASSUMPTIONS : L is smooth (at least C2) L is strictly convex w.r.t. u and ∂2L ∂u2 > 0 L( ˙ θ, . . . , θ(k−1), u) ≥ C|u|p for |u| > R.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 12 / 30

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Modelling goal oriented human locomotion

To summarize

Inverse Optimal Control Problem Given recorded experimental data (e.g. [ALHB])), infer a cost function L such that the recorded trajectories are optimal solutions of Pk(L)          min CL(u) = T L( ˙ θ, . . . , θ(k))dt (k = 1 or 2) subject to    ˙ x = cos θ ˙ y = sin θ θ(k) = u with (x, y, θ)(0) = (x0, y0, θ0) and (x, y, θ)(T) = (x1, y1, θ1). REMARKS : The time T is not fixed The initial point can be chosen as X0 = (0, 0, π/2) The target X1 = (x1, y1, θ1) is far from X0: |(x1, y1)| “large”

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 13 / 30

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SLIDE 15

Analysis of Pk(L)

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 14 / 30

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Analysis of Pk(L) General results

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 15 / 30

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Analysis of Pk(L) General results

Analysis of Pk(L) – General Results

Proposition For every target X1, there exists an optimal trajectory of Pk(L). Every optimal trajectory satisfies the Pontryagin Maximum Principle. REMARKS : The control system is controllable The proof of existence uses standard arguments (cf. Lee & Markus) The optimal control does not belong a priori to L∞([0, T]). However it is possible to prove that, for (x1, y1) far away from 0, the optimal control is uniformly bounded. → not necessary to put an a priori bound on the control

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 16 / 30

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Analysis of Pk(L) General results

CASE k = 2: H = p1 cos θ + p2 sin θ + p3 ˙ θ + p4¨ θ − νL( ˙ θ, ¨ θ) ≡ 0 No strictly abnormal extremals →

  • ptimal traj. are C∞ (ν = 1)

Adjoint Equation: (p1, p2) are constant and using ∂H

∂u = 0, the adjoint equation writes as an ODE:

θ(4) = FL

  • θ, ˙

θ, ¨ θ, θ(3); (p1, p2)

  • ,

with initial data: (θ, θ(3))(0) = ( π

2 , 0)

[transversality condition] CASE k = 1: H = p1 cos θ + p2 sin θ + p3 ˙ θ − νL( ˙ θ) ≡ 0 No strictly abnormal extremals →

  • ptimal traj. are C∞ (ν = 1)

Adjoint Equation: (p1, p2) are constant and ¨ θ = GL

  • θ, ˙

θ; (p1, p2)

  • ,

θ(0) = π 2

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 17 / 30

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SLIDE 19

Analysis of Pk(L) Asymptotic analysis

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 18 / 30

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Analysis of Pk(L) Asymptotic analysis

Analysis of P2(L)

We present only the analysis for k = 2; same thing for k = 1. Simplifying assumption: L( ˙ θ, ¨ θ) = 1 + ψ( ˙ θ) + φ(¨ θ) REMIND: Adjoint equation ⇒ fourth-order ODE on θ parameterized by (p1, p2): θ(4) = FL

  • θ, ˙

θ, ¨ θ, θ(3); (p1, p2)

  • ,

with initial data: (θ, θ(3))(0) = ( π

2 , 0)

H = p1 cos θ + p2 sin θ + p3 ˙ θ + p4¨ θ − L( ˙ θ, ¨ θ) ≡ 0

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 19 / 30

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Analysis of Pk(L) Asymptotic analysis

Notations: (x1, y1) = ρ(cos α, sin α), Θ = (θ, ˙ θ, ¨ θ, θ(3)).

( )

1 y1

, y x x α

Proposition Given ν > 0, |Θ(t) − (α, 0, 0, 0)| < ν for t ∈ [τν, T − τν]. Proposition As |(x1, y1)| → ∞, (p1, p2) ∼ (x1, y1) |(x1, y1)| = (cos α, sin α). (α, 0, 0, 0) equilibrium of the “limit” equation θ(4) = FL

  • Θ;

(x1,y1) |(x1,y1)|

  • Consequence: an optimal trajectory has a stable behaviour near the

equilibrium Yeq = (α, 0, 0, 0).

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 20 / 30

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Analysis of Pk(L) Asymptotic analysis

Asymptotic Trajectories

The equilibrium Yeq = (α, 0, 0, 0) is not stable: its stable manifold Ws is 2-dimensional.

Z(0)

eq

Y

eq

u s u s

Y

→ the limit optimal trajectory must be “contained” in Ws. → Close to Yeq, the trajectory is tangent to the stable space

  • f the linearized ODE
  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 21 / 30

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Analysis of Pk(L) Asymptotic analysis

Asymptotic Trajectories

Ws contains a 1-parameter family of trajectories: we compute them numerically and select the one starting with θ(0) = π/2 and θ(3)(0) = 0.

  • 8
  • 6
  • 4
  • 2

2 4 6 8

  • 8
  • 6
  • 4
  • 2

2 4 6 8

We obtain also the limit value of p(0) → initialization of a shooting algorithm. We can recover dL from the phase portrait

  • f Θ

(Figure: L = 1 + ˙ θ2 + ¨ θ2)

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 22 / 30

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Analysis of Pk(L) Asymptotic analysis

5 10 15 20 5 10 15 20 25 30

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 23 / 30

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Analysis of Pk(L) Stability results

Outline

1

Inverse optimal control

2

Modelling goal oriented human locomotion

3

Analysis of Pk(L) General results Asymptotic analysis Stability results

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 24 / 30

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SLIDE 26

Analysis of Pk(L) Stability results

Stability results

Let L0 be a cost admissible for k = 1 or k = 2. Let Lε be a family of costs admissible for k = 2 s.t. Lε → L0 in the following sense: |Lε( ˙ θ, ¨ θ) − L0( ˙ θ, ¨ θ)| or |Lε( ˙ θ, ¨ θ) − L0( ˙ θ)| ≤ C(ε)|¨ θ|p Proposition The optimal trajectories (xε, yε, θε, ˙ θε) of P2(Lε) converge uniformly to the optimal trajectories of Pk(L0), i.e., dunif

  • (xε, yε, θε, ˙

θε), T0

  • → 0,

T0 = {(x, y, θ, ˙ θ) s.t. (x, y, θ, ˙ θ) or (x, y, θ) is optimal for Pk(L0)}. With additional technical hypothesis, the adjoint vectors (and so ¨ θ and θ(3)) also converge uniformly.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 25 / 30

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Analysis of Pk(L) Stability results

Stability results

CONSEQUENCES The optimal synthesis of a problem Pk(L0) is stable under perturbations of the cost. Our modelling is ”reasonable” (physiological costs) A solution of the inverse problem = a cost and his perturbations → we will look for the simplest cost in this class Question: k = 1 for the simplest one? Adjoint equation of a perturbation = perturbation of the adjoint equation

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 26 / 30

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Analysis of Pk(L) Stability results

Analysis of the case k = 1

H = p1 cos θ + p2 sin θ + ˙ θL′( ˙ θ) − L( ˙ θ) ≡ 0 Remark: If ˙ θ(t0) = 0, then p1 cos θ(t0) + p2 sin θ(t0) = 1, → depends on one parameter Proposition Let T be the set of trajectories s.t. ˙ θ = 0 at some time. To any trajectory in T , with ˙ θ(t0) = 0, we apply the transformation:    ¯ t = t − t0 ¯ θ(¯ t) = θ(t) − θ(t0) (¯ x(¯ t), ¯ y(¯ t)) = Rot(−θ(t0))

  • (x(t), y(t)) − (x(t0), y(t0))
  • Then, for every fixed ¯

t, the set of (¯ x(¯ t), ¯ y(¯ t), ¯ θ(¯ t)), for all trajectories in T , is a curve in R3.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 27 / 30

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Analysis of Pk(L) Stability results

Numerical test

Numerical test: apply the transformation to the recorded curves. Does it give a curve? Transformation applied at three different times ¯ t1, ¯ t2, ¯ t3 to ∼700 recorded trajectories

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 28 / 30

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SLIDE 30

Analysis of Pk(L) Stability results

Validity of the test (work in progress)

Numerically: same test applied to the solutions of some P2(L) Transversality argument: for k = 2, a generic cost L should not give such a curve through the transformation Analysis of the optimal synthesis using the asymptotic trajectories

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 29 / 30

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Analysis of Pk(L) Stability results

Conclusion

Models with k = 1 should be sufficient.

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 30 / 30

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Analysis of Pk(L) Stability results

Conclusion

Models with k = 1 should be sufficient. A deep explanation: recorded trajectories theoretical solutions

  • F. Jean (ENSTA ParisTech)

Optimal control and locomotion Porquerolles, October 2010 30 / 30