Reaching movements with robust or stochastic optimal control models (Crevecoeur et al 2019)

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"We explored the hypothesis that compensation for unmodelled disturbances was supported by a robust neural control strategy. We studied the predictions of stochastic optimal control (LQG) (Linear Quadratic Gaussian) (Todorov, 2005) and a robust control design that can equivalently be described as a “min-max” or worst-case strategy (Basar and Bernhard, 1991) applied to linear models of planar reaching movements. The robust controller displayed an increase in control gains, resulting in faster movements towards the target and more vigorous responses to perturbations. Our experimental results supported these predictions: the occurrence of unexpected force field disturbances evoked both faster movements and more vigorous responses to perturbations. Thus, the neural controller was more robust in the sense that the feedback responses reduced the impact of the perturbations (step and force field). Thus the compensation for disturbances involved a “model-free” component. ..."
1 . Crevecoeur F, Scott SH, Cluff T (2019) Robust control in human reaching movements: a model-free strategy to compensate for unpredictable disturbances. J Neurosci [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type:
Brain Region(s)/Organism:
Cell Type(s):
Gap Junctions:
Simulation Environment: MATLAB;
Model Concept(s): Motor control;
Implementer(s): Crevecoeur, Frédéric ;
function [L,K,newCost,Sx,Se,s,JC] = extLQG(A,B,C,D,H,Q,R,cxi,comega,ceta,ce,cxhat,xInit)

% [L,K,newCost,Sx,Se,s,sall,JC] = extLQG(A,B,C,D,H,Q,R,cxi,comega,ceta,ce,cxhat,cxe,xInit)
% Calculates a series of feedback gains and Kalman gains for extended LQG
% control. The routine follows (Todorov, 2005, Neural Comp, 17, 1084-1108).
%   [A, B, C, D, H]     matrices for the state pace representation
%   [Q, R]              cost matrices
%   CXI, COMEGA, CETA   motor, sensory and internal noise
%                       covariance matrices
%   CE, CXHAT           initial covariance matrices for the error
%                       and the estimation
%   XINIT               initial state vector 
% The output matrices are:
%   L                   time series of optimal feedback gains
%   K                   time series of non adaptive Kalmen gains
%   NEWCOST             expected cost after stopping the iterations
%   SX, SE              series of cost matrices determined by the backwards
%                       recurrences for the state (SX) and error (SE) terms
%   S                   scalar component of the total expected cost
%   JC                  covariance matrix
%   Uses: > computeOFC
%         > computeExtKalman
%   Writtent by F. Crevecoeur - Spet. 6, 2019
%   Used in: Robust control in human reaching movements: a model free
%   strategy to compensate for unpredictable disturbances. 
%   Crevecoeur F., Scott S. H., Cluff T. 
%   DOI:

n = size(A,1);
m = size(B,2);
p = size(H,1);
c = size(C,3);
step = size(R,3);

K = zeros(n,p,step);

tol = 10^-14;
current = 10^6;
itmax = 100;
count = 0;
found = false;

% The optimal control and Kalman gains are calculated iteratively (no more
% than itmax times if it does not converge)
while ~found && count < itmax
    [L,Sx,Se,s] = computeOFC(A,B,C,D,H,Q,R,K,cxi,comega,ceta);
    [K,JC] = computeExtKalman(A,B,C,D,H,cxi,comega,ceta,L,cxhat,ce);

    % Expected cost
    newCost = xInit'*Sx*xInit + trace((Sx+Se)*ce)+ s;
    % Relative improvement
    dCost = abs(current - newCost)/newCost;
    current = newCost;

    % Check: of the relative improvement is small, the solution is found
    if dCost > tol
        found = false;
        found = true;
    count = count + 1;

fprintf('Number of iterations (ext LQG): %d\n', count); % Number of iterations 

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