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) Optimal control and locomotion Porquerolles, October 2010 1 / 30
Outline Inverse optimal control 1 Modelling goal oriented human locomotion 2 Analysis of P k ( L ) 3 General results Asymptotic analysis Stability results F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 2 / 30
Inverse optimal control Outline Inverse optimal control 1 Modelling goal oriented human locomotion 2 Analysis of P k ( L ) 3 General results Asymptotic analysis Stability results F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 3 / 30
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 ( 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) Optimal control and locomotion Porquerolles, October 2010 4 / 30
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 Modelling step: propose a class of optimal control problems 1 Analysis step: enhance qualitative properties of the optimal synthesis 2 (e.g. inactivations) → reduce the class of problems Comparison step: numerical methods 3 → choice of the best fitting L F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 5 / 30
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 Modelling step: propose a class of optimal control problems 1 Analysis step: enhance qualitative properties of the optimal synthesis 2 (e.g. inactivations) → reduce the class of problems Comparison step: numerical methods 3 → choice of the best fitting L F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 5 / 30
Modelling goal oriented human locomotion Outline Inverse optimal control 1 Modelling goal oriented human locomotion 2 Analysis of P k ( L ) 3 General results Asymptotic analysis Stability results F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 6 / 30
Modelling goal oriented human locomotion Trajectories of the Human Locomotion (x , y , ) 1 θ 1 1 (x , y , ) 0 θ 0 0 Goal-oriented human locomotion (Berthoz, Laumond et al.) Initial point ( x 0 , y 0 , θ 0 ) → Final point ( x 1 , y 1 , θ 1 ) ( 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
Modelling goal oriented human locomotion Trajectories of the Human Locomotion HYPOTHESIS: the chosen trajectory is solution of a minimization problem � y, ˙ min L ( x, y, θ, ˙ x, ˙ 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
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! � ˙ x = v cos θ ( Dubins ) − → v = tangential velocity . y = v sin θ ˙ F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 9 / 30
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
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 � T L ( x, y, θ, ˙ θ, . . . , θ ( k ) ) dt min C L ( u ) = 0 x = cos θ ˙ u ∈ L p , among all trajectories of y = sin θ ˙ θ ( k ) = u s.t. ( x, y, θ )(0) = ( x 0 , y 0 , θ 0 ) and ( x, y, θ )( T ) = ( x 1 , y 1 , θ 1 ) . F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 11 / 30
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 C 2 ) ∂ 2 L L is strictly convex w.r.t. u and ∂u 2 > 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
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 � T L ( ˙ θ, . . . , θ ( k ) ) dt min C L ( u ) = ( k = 1 or 2) 0 x = cos θ ˙ P k ( L ) y = sin θ ˙ subject to θ ( k ) = u with ( x, y, θ )(0) = ( x 0 , y 0 , θ 0 ) and ( x, y, θ )( T ) = ( x 1 , y 1 , θ 1 ) . REMARKS : The time T is not fixed The initial point can be chosen as X 0 = (0 , 0 , π/ 2) The target X 1 = ( x 1 , y 1 , θ 1 ) is far from X 0 : | ( x 1 , y 1 ) | “large” F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 13 / 30
Analysis of P k ( L ) Outline Inverse optimal control 1 Modelling goal oriented human locomotion 2 Analysis of P k ( L ) 3 General results Asymptotic analysis Stability results F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 14 / 30
Analysis of P k ( L ) General results Outline Inverse optimal control 1 Modelling goal oriented human locomotion 2 Analysis of P k ( L ) 3 General results Asymptotic analysis Stability results F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 15 / 30
Analysis of P k ( L ) General results Analysis of P k ( L ) – General Results Proposition For every target X 1 , there exists an optimal trajectory of P k ( 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 ( x 1 , y 1 ) 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
Analysis of P k ( L ) General results CASE k = 2 : H = p 1 cos θ + p 2 sin θ + p 3 ˙ θ + p 4 ¨ θ − νL ( ˙ θ, ¨ θ ) ≡ 0 optimal traj. are C ∞ ( ν = 1 ) No strictly abnormal extremals → Adjoint Equation : ( p 1 , p 2 ) are constant and using ∂H ∂u = 0 , the adjoint equation writes as an ODE: θ (4) = F L θ, ˙ θ, ¨ θ, θ (3) ; ( p 1 , p 2 ) � � , with initial data: ( θ, θ (3) )(0) = ( π 2 , 0) [transversality condition] CASE k = 1 : H = p 1 cos θ + p 2 sin θ + p 3 ˙ θ − νL ( ˙ θ ) ≡ 0 optimal traj. are C ∞ ( ν = 1 ) No strictly abnormal extremals → Adjoint Equation : ( p 1 , p 2 ) are constant and θ (0) = π ¨ θ, ˙ � � θ = G L θ ; ( p 1 , p 2 ) , 2 F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 17 / 30
Analysis of P k ( L ) Asymptotic analysis Outline Inverse optimal control 1 Modelling goal oriented human locomotion 2 Analysis of P k ( L ) 3 General results Asymptotic analysis Stability results F. Jean (ENSTA ParisTech) Optimal control and locomotion Porquerolles, October 2010 18 / 30
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