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 Inverse optimal control 1 Arm pointing motions 2 Modelling Necessary and sufficient conditions for inactivation Validation/Simulations Goal oriented human locomotion 3 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 ( X u ) such that every γ ∈ Γ is solution of inf { C ( X u ) : X u traj. s.t. X u (0) = γ (0) , X u ( 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 Modelling step: propose a class of optimal control problems 1 Analysis step: enhance qualitative properties of the optimal synthesis 2 → reduce the class of problems (using geometric control theory) Comparison step: numerical methods 3 → 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 Modelling step: propose a class of optimal control problems 1 Analysis step: enhance qualitative properties of the optimal synthesis 2 → reduce the class of problems (using geometric control theory) Comparison step: numerical methods 3 → choice of the best fitting L (identification) F. Jean (ENSTA ParisTech) Inverse optimal control PICOF ’12 4 / 34

Inverse optimal control Outline Inverse optimal control 1 Arm pointing motions 2 Modelling Necessary and sufficient conditions for inactivation Validation/Simulations Goal oriented human locomotion 3 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 Inverse optimal control 1 Arm pointing motions 2 Modelling Necessary and sufficient conditions for inactivation Validation/Simulations Goal oriented human locomotion 3 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 stick diagram stick diagram velocity 0 0.2 0.2 0.4 0 0.4 DA DP EMGs BI TR Time (s) Time (s) F. Jean (ENSTA ParisTech) Inverse optimal control PICOF ’12 8 / 34

Arm pointing motions Typical experimental data for 2 dof stick diagram stick diagram velocity 0 0.2 0.4 0.2 0.4 0 DA DP EMGs BI TR Time (s) Time (s) 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 Inverse optimal control 1 Arm pointing motions 2 Modelling Necessary and sufficient conditions for inactivation Validation/Simulations Goal oriented human locomotion 3 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: x ∈ R n , → M ( x )¨ x = ψ ( x, ˙ x ) + u, u = action of the muscles (torques); ψ ( x, ˙ x ) = gravity + frictions + Coriolis; M ( x ) = inertia matrix (positive definite); ˙ x ) ∈ R 2 n , u ∈ R n . ⇔ X = φ ( X, u ) , X = ( x, ˙ Bounds on the torque u : 1 , u + n , u + i < 0 < u + u ∈ [ u − 1 ] × ... × [ u − u − n ] , i F. Jean (ENSTA ParisTech) Inverse optimal control PICOF ’12 12 / 34

Arm pointing motions Modelling Optimal control problem � T Criterion: J ( u ) = f ( X, u ) dt . 0 u �→ f ( X, u ) strictly convex. Hyp. Initial data: X s = ( x s , 0) , target: X t = ( x t , 0) . The time T > 0 is fixed. Optimal control problem ( P ) minimise the integral cost J ( u ) among the trajectories of ˙ X = φ ( X, u ) joining X s to X t 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 Inverse optimal control 1 Arm pointing motions 2 Modelling Necessary and sufficient conditions for inactivation Validation/Simulations Goal oriented human locomotion 3 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 u i 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: n n � � � � � w = udx = u i dx i = u i ˙ x i dt. i =1 i =1 Measure of the energetic consumption = absolute work: n � ˙ ˙ � Aw = Aw ( X, u ) , o` u Aw ( X, u ) = | u i ˙ x i | , X = ( x, ˙ x ) i =1 ˙ → Aw non differentiable w.r.t. u when one component u i = 0 . F. Jean (ENSTA ParisTech) Inverse optimal control PICOF ’12 16 / 34

Arm pointing motions Necessary and sufficient conditions for inactivation � T Form of the costs: J ( u ) = f ( X, u ) dt with 0 ∂ϕ f ( X, u ) = ϕ ( ˙ Aw, X, u ) , � = 0 ∂ ˙ Aw 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 ∈ ∂ u H = − ˙ x∂ u | u ∗ | + g ( t ) , where g continuous. � g = ˙ x if u ∗ > 0 , x if u ∗ < 0 , g = − ˙ → x ] if u ∗ = 0 . g ∈ [ − ˙ x, ˙ 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 Inverse optimal control 1 Arm pointing motions 2 Modelling Necessary and sufficient conditions for inactivation Validation/Simulations Goal oriented human locomotion 3 Modelling Analysis of the direct problem Locomotion depends only on ˙ θ F. Jean (ENSTA ParisTech) Inverse optimal control PICOF ’12 19 / 34