Neuromechanics & Motor Control - Lecture 15

Kalman filter

Estimation technique that combines internal model predictions with sensory observations to provide an optimal estimate of a system’s state

Forward model

Predictive model that estimates the outcome of motor commands before sensory feedback is available

Equation for fast prediction of state using internal model and control input

\hat{x}_{n}=A_{i}\hat{x}_{n-1}+B_{i}u_{n}+w_{n}

Predicted sensory feedback based on current state estimate

\hat{y}_{n}=C_{i}\hat{x}_{n}+v_{n}

Actual sensory observation of the system’s state

y_{n}

Predicted next state in Kalman filtering

\hat{x}_{\bar{k}}=A\hat{x}_{k-1}+Bu_{k}

Actual sensory feedback in Kalman filtering

z_{k}

Updated state estimate using Kalman gain and sensory error

\hat{x}_{k}=\hat{x}_{\bar{k}}+K_{k}\left(z_{k}-H\hat{x}_{\bar{k}}\right)

K_{k} (Kalman gain)

Factor that determines the weighting between prediction and observation in state estimation

Cerebellum

Brain region involved in motor control, coordination, and motor learning, functioning as a comparator of intended and actual movements

Motor control functions of cerebellum

Equilibrium, posture, and reaching

Coordination functions of cerebellum

Timing, precision, accuracy, and reflex adjustment

Motor learning in cerebellum

Learning motor skills and calibrating sensorimotor relationships

Cerebellar lesion effects

Impaired coordination and learning of new motor tasks without affecting muscle strength or perception

Formula for combining two noisy estimates into an optimal estimate

x_{\text{new}} = \frac{\sigma_2^2 x_1 + \sigma_1^2 x_2}{\sigma_1^2 + \sigma_2^2}

Formula for variance of the combined estimate

\sigma_{\text{new}}^2 = \frac{\sigma_1^2 \sigma_2^2}{\sigma_1^2 + \sigma_2^2}

Prediction equation with process noise in scalar Kalman filter

x_{n}=ax_{n-1}+w_{n}

Observation equation with measurement noise in scalar Kalman filter

y_{n}=cx_{n}+v_{n}

Predicted state estimate in scalar Kalman filter

\tilde{x}_{n}^{\text{pred}} = a\hat{x}_{n-1}

Observation-based estimate of the state

\tilde{x}_{n}^{\text{obs}} = \frac{y_n}{c}

Predicted variance based on previous variance and process noise

\tilde{P}_{n}^{\text{pred}} = aP_{n} + Q

Variance of the observation-based estimate

\tilde{P}_{n}^{\text{obs}} = \frac{R^2}{c}

Optimal estimate in scalar Kalman filter

\hat{x}_n = a \hat{x}_{n-1} + \frac{K}{c} \left( y_n - c a \hat{x}_{n-1} \right)

Formula for Kalman gain based on prediction variance and measurement noise

K=\frac{c \tilde{P}_{n}^{\text{pred}}}{R + c^2 \tilde{P}_{n}^{\text{pred}}}

Updated variance of the state estimate after applying Kalman gain

P_n = (1 - Kc) \tilde{P}_{n}^{\text{pred}}

Bayesian formulation of optimal state estimation using prediction and observation

p(\hat{x}_n, y_n) = p(y_n | \hat{x}_n) p(\hat{x}_n)

Model identification with Kalman filter

Estimating model coefficients θn using system observations and Kalman filtering

Coefficients representing system dynamics in model identification

θn = [a1, a2, b0, b1, c1, c2]

Input vector used for model identification in Kalman filtering

ϕn = [yn−1, yn−2, rn, rn−1, vn−1, vn−2]