TU Graz - Signal Processing and Speech Communication Laboratory Bayesian Decision Theory in Sensorimotor Control Matthias Freiberger, Martin ¨ Ottl Signal Processing and Speech Communication Laboratory Advanced Signal Processing Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 1/88
TU Graz - Signal Processing and Speech Communication Laboratory Outline Introduction Definition Challenges of sensorimotor control Bayesian Integration in motor control Cost Functions Optimal Estimation Optimal Feedback Control Introduction Linear Quadratic Gaussian Framework (LQG) LQG + Kalman Filter LQG Multiplicative Noise Minimal Intervention Principle Hierarchical Optimal Controller Conclusion References Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 2/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - What is sensorimotor control? ”sen · so · ri · mo · tor: (adj.) Of, relating to, or involving both sensory and motor activity: sensorimotor nerve centers; sensorimotor pathways.” The American Heritage Dictionary of the English Language, Fourth Edition ”Movement is the only way for humans to interact with the world. All communication including speech, sign language, gestures and writing, is mediated by the motor system.” http://www.pom.cam.ac.uk/research/sensorimotor.html Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 3/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - What is sensorimotor control? We want to understand/describe by application methods from computer science and control theory how ◮ .. human beings are able to play back a tennis ball ◮ .. or grab a bottle of water and drink ◮ .. birds of prey are capable of catching a mouse in flight ◮ .. basically how any kind of physical interaction with the environment is performed by biological systems, pursuing a certain objective while permanently performing corrections using sensor input Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 4/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Challenges ◮ Action selection is a fundamental decision process ◮ CNS sends constantly sends motor commands to the muscles ◮ At each point in time: the appropriate motor command needs to be selected ◮ Knowledge about the environment needs to be combined with actual observation data and knowledge about cost/reward of currently possible actions to make optimal decisions. Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 5/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Schematic Control Flow Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 6/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Uncertainty of human sensorium ◮ Human sensorium is plagued by noise ◮ Muscle output is noisy as well ◮ Therefore state of environment/body needs to be estimated ◮ Additionally the “cost” of each movement shall be minimized ◮ Bayesian statistics come in as a powerful way to deal with the uncertainty of the human sensorium Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 7/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Bayesian integration ◮ CNS needs to integrate prior knowledge about environment with knowledge obtained from sensory data to estimate the state of the environment optimally ◮ When estimating bounce location of a tennis ball: ball might be more likely to bounce off at edges of court Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 8/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Bayesian Cue Combination Combination of sensor signals for better estimates ◮ Combination of different sensor modalities (e.g. Vision and Proprioception) ◮ Combination of signal of same modality (several visual cues to a stereo image... ) ◮ Cues need to be weighted against each other Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 9/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Bayesian Cue Combination Given a set of observations from different cues d 1 , d 2 , d 3 , ..., d n under the assumption that cues are independent from each other we can rewrite the likelihood P ( d 1 , d 2 , d 3 , ..., d n ) as n � P ( d 1 , d 2 , d 3 , ..., d n | s ) = P ( d k | s ) (1) k =1 Therefore we can rewrite the corresponding posterior probability: P ( s | d 1 , d 2 , d 3 , ..., d n ) = P ( s ) · � n k =1 P ( d k | s ) (2) P ( d 1 , d 2 , d 3 , ..., d n ) Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 10/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Cost Functions ◮ Model how good or bad the outcome of a particular move is ◮ Seems reasonable to minimize consumed energy and strain on muscles ◮ Several cost functions have been proposed (smoothness,precision) ◮ CNS also adapts very well to external cost functions Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 11/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Cost Functions ◮ Actual cost function of human movement can be inferred using indifference lines ◮ Utility function can be found from these lines : compare points from lines,and assigning utilities to lines Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 12/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Cost Functions Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 13/88
TU Graz - Signal Processing and Speech Communication Laboratory Intro - Cost Functions Given a set of possible actions X and a set of possible outcomes O , as well as a utility function U ( o ) : O → R , for any x ∈ X we can compute the expected utility � E{ U } = P ( o | x ) · U ( o ) (3) O Therefore the optimal decision in respect to the cost function U ( o ) is considered to be the one which maximizes the expected utility E{ U } . Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 14/88
TU Graz - Signal Processing and Speech Communication Laboratory Outline Introduction Definition Challenges of sensorimotor control Bayesian Integration in motor control Cost Functions Optimal Estimation Optimal Feedback Control Introduction Linear Quadratic Gaussian Framework (LQG) LQG + Kalman Filter LQG Multiplicative Noise Minimal Intervention Principle Hierarchical Optimal Controller Conclusion References Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 15/88
TU Graz - Signal Processing and Speech Communication Laboratory Optimal Estimation – Intro Until now Find the optimal action for a finite amount of actions But the world is continious... Actual continuos state of our body parts has to be estimated permanently, optimal actions according to state estimation need to be found. In control terms... We need to model ourselves an observer, which estimates the inner state (e.g the position and velocity) of our limbs Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 16/88
TU Graz - Signal Processing and Speech Communication Laboratory Optimal Estimation – Experiment Experiment setup ◮ Test subjects had to estimate the location of their thumb after moving their arm ◮ Resistive or assistive force has been added by torque motors ◮ Hand is constrained to move on a straight line ◮ Arm is illuminated for 2s, to give an initial state ◮ After that, participants have to rely solely on proprioception Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 17/88
TU Graz - Signal Processing and Speech Communication Laboratory Optimal Estimation – Experiment Experiment setup Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 18/88
TU Graz - Signal Processing and Speech Communication Laboratory Optimal Estimation – Models A system that mimics the behavior of a natural process, is called an internal model Internal models are an important concept in motor control Basically, two classes of internal models can be distinguished: forward models and backward models Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 19/88
TU Graz - Signal Processing and Speech Communication Laboratory Optimal Estimation – Internal models: forward vs. backward Forward models ◮ Mimic the causal flow of a process by predicting its next state ◮ Comes up natural since delays in most sensorimotor loops are large,feedback control may be too slow for rapid movements ◮ Key indegredient in systems that use motor outflow (efference copy) Backward models ◮ Estimate the appropriate motor command which caused a particular state transition Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 20/88
TU Graz - Signal Processing and Speech Communication Laboratory Optimal Estimation – Internal models: forward vs. backward How do we optimally model our limbs now? ◮ Wolpert et. al. used a forward model incorparating a correction term for the given problem. ◮ State estimation for a system containing noise is a complex task ◮ We will follow an intuitive approach by modeling an observer for a deterministic system first ◮ From our deterministic observer, we will perform the transition to a Probabilistic Observer ( Kalman Filter) Matthias Freiberger, Martin ¨ Ottl Advanced Signal Processing page 21/88
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