Minimization principles in human motions: the inverse optimal - - PowerPoint PPT Presentation

Minimization principles in human motions: the inverse optimal control approach

Fr´ ed´ eric Jean

ENSTA ParisTech, Paris (and Team GECO, INRIA Saclay)

PICOF ’12 Ecole Polytechnique, April 2–4, 2012

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 1 / 34

Inverse optimal control

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 2 / 34

Inverse optimal control

Inverse optimal control

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

Arm pointing motions Goal oriented human locomotion Saccadic motion of the eyes

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)

Inverse optimal control PICOF ’12 3 / 34

Inverse optimal control

Difficulties:

Γ = experimental data (noise, feedbacks, etc) Dynamical model not always known (← hierarchical optimal control) Limited precision of both dynamical models and costs ⇒ necessity of stability (genericity) of the criterion Non well-posed inverse 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 → reduce the class of problems (using geometric control theory)

3

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

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 4 / 34

Inverse optimal control

Difficulties:

Γ = experimental data (noise, feedbacks, etc) Dynamical model not always known (← hierarchical optimal control) Limited precision of both dynamical models and costs ⇒ necessity of stability (genericity) of the criterion Non well-posed inverse 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 → reduce the class of problems (using geometric control theory)

3

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

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 4 / 34

Inverse optimal control

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 5 / 34

Arm pointing motions

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 6 / 34

Arm pointing motions

Arm pointing motions

with B. Berret, C. Papaxanthis, T. Pozzo (INSERM Dijon), J.-P. Gauthier (Univ. Toulon) and C. Darlot (CNRS - Telecom ParisTech)

Pointing motions in a vertical plane (1, 2, or 3 degrees of freedom) Fast motions in fixed time

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 7 / 34

Arm pointing motions

Typical experimental data for 1 dof

DA DP BI TR

0.2 0.4

Time (s) Time (s)

EMGs

velocity

stick diagram stick diagram

0.4 0.2

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 8 / 34

Arm pointing motions

Typical experimental data for 2 dof

velocity

Time (s) Time (s) DA DP BI TR

EMGs

0.2 0.4 0.2 0.4 stick diagram stick diagram

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 9 / 34

Arm pointing motions

Main characteristics

Some strong qualitative characteristics: simultaneous inactivations of opposing muscles; asymmetric velocity profile (acceleration phases shorter than the deceleration ones); ... and more quantitative ones: (for 2 et 3 dof) curvature of the finger trajectory; (for 3 dof) final configuration of the arm.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 10 / 34

Arm pointing motions

Main characteristics

Some strong qualitative characteristics: simultaneous inactivations of opposing muscles; asymmetric velocity profile (acceleration phases shorter than the deceleration ones); ... and more quantitative ones: (for 2 et 3 dof) curvature of the finger trajectory; (for 3 dof) final configuration of the arm.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 10 / 34

Arm pointing motions Modelling

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 11 / 34

Arm pointing motions Modelling

Modelling

Arm = controlled mechanical system described by: → M(x)¨ x = ψ(x, ˙ x) + u, x ∈ Rn, u = action of the muscles (torques); ψ(x, ˙ x) = gravity + frictions + Coriolis; M(x) = inertia matrix (positive definite); ⇔ ˙ X = φ(X, u), X = (x, ˙ x) ∈ R2n, u ∈ Rn. Bounds on the torque u: u ∈ [u−

1 , u+ 1 ] × ... × [u− n , u+ n ],

u−

i < 0 < u+ i

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 12 / 34

Arm pointing motions Modelling

Optimal control problem

Criterion: J(u) = T f(X, u)dt. Hyp. u → f(X, u) strictly convex. Initial data: Xs = (xs, 0), target: Xt = (xt, 0). The time T > 0 is fixed. Optimal control problem (P) minimise the integral cost J(u) among the trajectories

• f ˙

X = φ(X, u) joining Xs to Xt in time T.

• Theorem. The minimum of (P) is reached by some optimal trajectory
• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 13 / 34

Arm pointing motions Necessary and sufficient conditions for inactivation

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 14 / 34

Arm pointing motions Necessary and sufficient conditions for inactivation

Necessary condition

Definition u contains an inactivation if one of its components ui is ≡ 0 on a non-empty interval. Not◦: SC = set of functions f(X, u) such that u → f(X, u) is strictly convex and differentiable. Theorem For a generic cost f ∈ SC, no minimizing control of (P) contain inactivation. ⇒ the cost f is necessarily non differentiable w.r.t. u (Proof: Pontryagin Maximum Principle + Thom transversality)

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 15 / 34

Arm pointing motions Necessary and sufficient conditions for inactivation

Sufficient condition

Constraints on the cost: necessarily non differentiable w.r.t. u related to energetic consumption Candidates: functions of the absolute work of the controlled forces. Work of the controlled forces: w =

• udx =
• n
• i=1

uidxi =

• n
• i=1

ui ˙ xidt. Measure of the energetic consumption = absolute work: Aw =

• ˙

Aw(X, u),

• `

u ˙ Aw(X, u) =

n

• i=1

|ui ˙ xi|, X = (x, ˙ x) → ˙ Aw non differentiable w.r.t. u when one component ui = 0.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 16 / 34

Arm pointing motions Necessary and sufficient conditions for inactivation

Form of the costs: J(u) = T f(X, u)dt with f(X, u) = ϕ( ˙ Aw, X, u), ∂ϕ ∂ ˙ Aw = 0 Theorem (Inactivation Principle) Minimizing such a cost J(u) implies the occurrence of inactivations in every optimal trajectory of (P) when T is small enough.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 17 / 34

Arm pointing motions Necessary and sufficient conditions for inactivation

Explanation (n = 1 and f(X, u) = | ˙ xu|+ differentiable fn)

• Pontr. Max. Principle: every minimizing control u∗(·) maximizes

the Hamiltonian H = −f(X, u) + P T φ(X, u). ⇒ 0 ∈ ∂uH = − ˙ x∂u|u∗| + g(t), where g continuous. → g = ˙ x if u∗ > 0, g = − ˙ x if u∗ < 0, g ∈ [ − ˙ x, ˙ x] if u∗ = 0. Thus, when the sign of u∗ changes, g passes from ˙ x to − ˙ x continuously = ⇒ inactivation!!

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 18 / 34

Arm pointing motions Validation/Simulations

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 19 / 34

Arm pointing motions Validation/Simulations

Simulations (2 dof)

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 20 / 34

Arm pointing motions Validation/Simulations

Typical experimental data for 2 dof

velocity

Time (s) Time (s) DA DP BI TR

EMGs

0.2 0.4 0.2 0.4 stick diagram stick diagram

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 21 / 34

Goal oriented human locomotion

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 22 / 34

Goal oriented human locomotion

Goal-oriented human locomotion

with Y. Chitour, F. Chittaro, and P. Mason (LSS)

0 θ0

(x , y , )

1 1 θ1

(x , y , )

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)

Inverse optimal control PICOF ’12 23 / 34

Goal oriented human locomotion Modelling

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 24 / 34

Goal oriented human locomotion Modelling

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)

Inverse optimal control PICOF ’12 25 / 34

Goal oriented human locomotion Modelling

Modelling

Using: experimental observations, mostly from [Arechavaleta-Laumond-Hicheur-Berthoz, 2006] (=[ALHB]) symetries of the problem; the existence of solutions; we obtain a class of optimal control models.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 26 / 34

Goal oriented human locomotion Modelling

Inverse Optimal Control Problem Given experimental data (e.g. [ALHB]), infer a cost function L ∈ Lk such that the recorded trajectories are optimal solutions of Pk(L)           min CL(u) = T L( ˙ θ, . . . , θ(k))dt subject to    ˙ x = cos θ ˙ y = sin θ θ(k) = u u ∈ Lp with (x, y, θ)(0) = 0 and (x, y, θ)(T) = X1, T not fixed. Class of admissible costs Lk = convex w.r.t. θ(k) + technical conditions MAIN QUESTIONS Stability of the direct problem What is the value of k? (k = 1 or k = 2)

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 27 / 34

Goal oriented human locomotion Analysis of the direct problem

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 28 / 34

Goal oriented human locomotion Analysis of the direct problem

Proposition For every target X1, there exists an optimal trajectory of Pk(L). Notation: T (L, X1) = the set of trajectories (x, y, θ, ˙ θ) s.t. (x, y, θ) is optimal for P1(L) with final point X1 if L ∈ L1; (x, y, θ, ˙ θ) is optimal for P2(L) with final point X1 if L ∈ L2. Proposition If Xε

1 → X1, Lε converges to L0, and (xε, yε, θε, ˙

θε) ∈ T (Lε, Xε

1), then

dunif

• (xε, yε, θε, ˙

θε), T (L0, X1)

• → 0

→ Optimal trajectories are stable under perturbations of the cost and of the target.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 29 / 34

Goal oriented human locomotion Analysis of the direct problem

Stability results

Theorem The optimal synthesis of a problem Pk(L) is stable under perturbations of the cost in L1 ∪ L2 (robustness). Consequences Stability of the direct problem Our modelling is compatible with the physiology A solution “up to perturbations” is sufficient Question Can we choose, up to perturbations, a cost in L1? (i.e. a cost that depends only on the curvature)

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 30 / 34

Goal oriented human locomotion Locomotion depends only on ˙ θ

Outline

1

Inverse optimal control

2

Arm pointing motions Modelling Necessary and sufficient conditions for inactivation Validation/Simulations

3

Goal oriented human locomotion Modelling Analysis of the direct problem Locomotion depends only on ˙ θ

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 31 / 34

Goal oriented human locomotion Locomotion depends only on ˙ θ

Remark on the problem P1(L): Up to a canonical rotation + translation Φ, the minimizers of P1(L) form a one-parameter class of curves. → Set Φ : a curve (x, y, θ)(·) s.t. ˙ θ = 0 once → a curve (¯ x, ¯ y, ¯ θ)(·), and Φt(x, y, θ) = (¯ x(t), ¯ y(t), ¯ θ(t)) → For L ∈ Lk, ML = {minimizers (x, y, θ) of Pk(L) s.t. ˙ θ = 0 once} Proposition For every fixed t, the set Φt(ML): is a curve in R3 if L ∈ L1; contains at least a surface in R3 if L ∈ L2.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 32 / 34

Goal oriented human locomotion Locomotion depends only on ˙ θ

Numerical test: apply the transformation Φt to the recorded curves. Does it give a curve? YES!

• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 angle T=0.25 T=0.50 T=0.75 T=1.00 T=1.25 T=1.50 T=1.75 T=2.00 x y angle

Φt applied at different times to ∼ 200 recorded trajectories Φt applied to computed trajectories with k = 2 Conclusion Models with k = 1 should be sufficient to describe human locomotion

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 33 / 34

Goal oriented human locomotion Locomotion depends only on ˙ θ

References

• F. Chittaro, F. Jean, and P. Mason. On the inverse optimal control

problems of the human locomotion: stability and robustness of the

• minimizers. Journal of Mathematical Sciences, to appear.
• Y. Chitour, F. Jean, and P. Mason. Optimal control models of the

goal-oriented human locomotion. SIAM J. Control Optim., 2012.

• B. Berret, F. Jean, and J.-P. Gauthier. A biomechanical inactivation
• principle. Proceedings of the Steklov Institute of Mathematics, 2010.
• B. Berret, C. Darlot, F. Jean, T. Pozzo, C. Papaxanthis, and

J.-P. Gauthier. The Inactivation Principle: Mathematical Solutions Minimizing the Absolute Work and Biological Implications for the Planning of Arm Movements, PLoS Computational Biology, 2008.

• F. Jean (ENSTA ParisTech)

Inverse optimal control PICOF ’12 34 / 34