Contrôle optimal de trajectoires locomotrices humaines Quang-Cuong Pham 21 janvier 2010 Laboratoire de Physiologie de la Perception et de l’Action Collège de France, Paris, France
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Outline Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Influence of vision on the average trajectories Influence of vision on the variability profiles “Desired-trajectory” versus optimal feedback control Stochastic optimal control models Conclusions 1 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Outline Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Influence of vision on the average trajectories Influence of vision on the variability profiles “Desired-trajectory” versus optimal feedback control Stochastic optimal control models Conclusions 1 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Redundancy in the control of locomotion Neural commands Target Sequence of foot positions Locomotor path Whole−body trajectory (path + velocity profile) Foot positions Whole−body path Task Starting position 2 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Redundancy in the control of arm movements Neural commands Muscle activations Target Hand path Joint angles kinematics Hand trajectory (path + velocity profile) Starting position Hand path Joint angle Task Jordan and Wolpert, in The Cognitive Neuroscience , 1999 3 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Spatial control of arm movements Straight hand paths Bell-shaped velocity profiles Morasso, Exp Brain Res , 1981 Atkeson and Hollerbach, J Neurosci , 1985 ◮ Stereotypy observed only for hand trajectories in Cartesian coordinates ◮ Control in terms of Cartesian coordinates of the hand, not in terms of e.g. joint angles or muscle activity 4 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Optimal control of arm movements ◮ Humans may select the hand trajectories that minimize a certain cost ◮ One popular model is the minimum jerk model developped by Flash and Hogan Z 1 „ d 3 x « 2 « 2 ! „ d 3 y min + d t d t 3 d t 3 x , y 0 Typical features: ◮ Straight, smooth, hand paths ◮ Bell-shaped velocity profiles ◮ Inverse relationship between velocity and curvature (via-points) 5 / 41 Flash and Hogan, J Neurosci , 1985
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Questions ◮ Is human locomotion controlled at the level of whole-body trajectories? ◮ Are locomotor trajectories optimal? According to what criteria? ◮ What mechanisms underly the formation of locomotor trajectories? 6 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Outline Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Influence of vision on the average trajectories Influence of vision on the variability profiles “Desired-trajectory” versus optimal feedback control Stochastic optimal control models Conclusions 6 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions General experimental methods ◮ Motion capture system: infrared cameras + light reflective markers ◮ Body position defined by shoulders’ midpoint Light-reflective marker Shoulders' midpoint Locomotor trajectory 7 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Average trajectories and variabilities ◮ Time rescaling so that t 0 = 0 and t 1 = 1 ◮ Definition of average trajectories and variabilities Instantaneous Sample trajectory deviation velocity profile Sample Average trajectory Instantaneous trajectory velocity deviation Average velocity profile 8 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Experiment 1: stereotypy of locomotor trajectories ◮ Reminder: control of arm movements in terms of Cartesian coordinates of the hand ◮ What is planned and controlled in goal-oriented locomotion? ◮ Step-level: plan and execute sequences of precise foot positions (FP), resulting in a whole-body trajectory ◮ Trajectory-level: plan a whole-body trajectory (in Cartesian space) and implement it by appropriate sequences of foot positions ◮ Variability of the sequences of FP versus variability of whole-body trajectories 9 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Experiment 1: methods ◮ Protocol: walking towards and through a distant doorway (Arechavaleta et al, 2006) ◮ Constraints on Initial and final positions and walking directions ◮ 40 targets (a target = position × orientation) ◮ 6 subjects × 40 targets × 3 repetitions = 720 trajectories 7 6 5 Y axis (in meters) 4 3 2 1 0 Starting position and orientation −1 −3 −2 −1 0 1 2 3 10 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Experiment 1: results (trajectory stereotypy) Max Traj Deviation (in m) 0.2 Straight Low curvature Medium curvature High curvature 0.15 0.1 0.05 Geometric paths 0 ST LC MC HC Categories 1m Max Norm Velo Deviation 1m 1m 0.1 1m 0.08 0.06 Normalized 1 1 1 1 0.04 velocity profiles 0.02 0 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 ST LC MC HC Scaled time Categories Hicheur, Pham et al, Eur J Neurosci , 2007 Even for HC, maximum variability was ≤ 17 cm 11 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Experiment 1: results (foot positions variability) 7 Rig ht Ste p 6 45 Lef t St ep Trajectory Deviation Traj and Step Deviations (in %) Step Deviation 40 5 35 4 30 3 25 2 20 15 1 10 0 5 7 0 ST LC MC HC 6 Category 5 4 ◮ variability of the sequences of FP 3 ( ≥ 20% of step length) 2 ◮ variability of whole-body 1 trajectories ( ≤ 5% of trajectory 0 length) 1 - 3 - 2 - 1 0 1 2 3 12 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Experiment 1: conclusions ◮ Goal-oriented locomotion is not planned and controlled as a sequence of precise “foot pointings” ◮ Rather, it is likely planned and controlled at the level of whole-body trajectories ◮ This is reminiscent of the concept of spatial control of hand movements (Morasso, Exp Brain Res , 1981) 13 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Outline Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Influence of vision on the average trajectories Influence of vision on the variability profiles “Desired-trajectory” versus optimal feedback control Stochastic optimal control models Conclusions 13 / 41
Context Stereotypy of locomotor trajectories Deterministic optimal control models Control mechanisms underlying the formation of trajectories Stochastic optimal control models Conclusions Context ◮ Reminder: minimum jerk model for hand trajectories ◮ Common features of hand and locomotor trajectories: ◮ smoothness ◮ straightness for locomotor “reaching” ◮ inverse relationship between velocity and curvature ◮ Can the minimum jerk model also simulate locomotor trajectories? 14 / 41
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