demo_TPGMR_LQR02.m 7.24 KB
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function demo_TPGMR_LQR02
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% Demonstration a task-parameterized probabilistic model encoding movements in the form of virtual spring-damper 
% systems acting in multiple frames of reference. Each candidate coordinate system observes a set of 
% demonstrations from its own perspective, by extracting an attractor path whose variations depend on the 
% relevance of the frame through the task. This information is exploited to generate a new attractor path 
% corresponding to new situations (new positions and orientation of the frames), while the predicted covariances 
% are exploited by a linear quadratic regulator (LQR) to estimate the stiffness and damping feedback terms of 
% the spring-damper systems, resulting in a minimal intervention control strategy.
%
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% This demo presents the results for an infinite horizon LQR.
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%  
% Author:	Sylvain Calinon, 2014
%         http://programming-by-demonstration.org/SylvainCalinon
%
% This source code is given for free! In exchange, I would be grateful if you cite  
% the following reference in any academic publication that uses this code or part of it: 
%
% @inproceedings{Calinon14ICRA,
%   author="Calinon, S. and Bruno, D. and Caldwell, D. G.",
%   title="A task-parameterized probabilistic model with minimal intervention control",
%   booktitle="Proc. {IEEE} Intl Conf. on Robotics and Automation ({ICRA})",
%   year="2014",
%   month="May-June",
%   address="Hong Kong, China",
%   pages="3339--3344"
% }

%% Parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
model.nbStates = 3; %Number of Gaussians in the GMM
model.nbFrames = 2; %Number of candidate frames of reference
model.nbVar = 3; %Dimension of the datapoints in the dataset (here: t,x1,x2)
model.dt = 0.01; %Time step 
nbRepros = 8; %Number of reproductions with new situations randomly generated
rFactor = 1E-1; %Weighting term for the minimization of control commands in LQR


%% Load 3rd order tensor data
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('Load 3rd order tensor data...');
% The MAT file contains a structure 's' with the multiple demonstrations. 's(n).Data' is a matrix data for 
% sample n (with 's(n).nbData' datapoints). 's(n).p(m).b' and 's(n).p(m).A' contain the position and 
% orientation of the m-th candidate coordinate system for this demonstration. 'Data' contains the observations 
% in the different frames. It is a 3rd order tensor of dimension D x P x N, with D=3 the dimension of a 
% datapoint, P=2 the number of candidate frames, and N=200x4 the number of datapoints in a trajectory (200)
% multiplied by the number of demonstrations (5).
load('data/DataLQR01.mat');


%% Tensor GMM learning
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fprintf('Parameters estimation of tensor GMM with EM:');
model = init_tensorGMM_timeBased(Data, model); %Initialization 
model = EM_tensorGMM(Data, model);


%% Reproduction with LQR for the task parameters used to train the model
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('Reproductions with LQR...');
DataIn = [1:s(1).nbData] * model.dt;
for n=1:nbSamples
  %Retrieval of attractor path through task-parameterized GMR
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  a(n) = estimateAttractorPath(DataIn, model, s(n));  
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  r(n) = reproduction_LQR_infiniteHorizon(DataIn, model, a(n), a(n).currTar(:,1), rFactor);
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end


%% Reproduction with LQR for new task parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('New reproductions with LQR...');
for n=1:nbRepros 
  for m=1:model.nbFrames
    %Random generation of new task parameters 
    id=ceil(rand(2,1)*nbSamples);
    w=rand(2); w=w/sum(w);
    rTmp.p(m).b = s(id(1)).p(m).b * w(1) + s(id(2)).p(m).b * w(2);
    rTmp.p(m).A = s(id(1)).p(m).A * w(1) + s(id(2)).p(m).A * w(2);
  end
  %Retrieval of attractor path through task-parameterized GMR
  anew(n) = estimateAttractorPath(DataIn, model, rTmp);
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  rnew(n) = reproduction_LQR_infiniteHorizon(DataIn, model, anew(n), anew(n).currTar(:,1), rFactor);
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end


%% Plots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure('position',[20,50,1300,500]);
xx = round(linspace(1,64,nbSamples));
clrmap = colormap('jet');
clrmap = min(clrmap(xx,:),.95);
limAxes = [-1.2 0.8 -1.1 0.9];
colPegs = [[.9,.5,.9];[.5,.9,.5]];

%DEMOS
subplot(1,3,1); hold on; box on; title('Demonstrations');
for n=1:nbSamples
  %Plot frames
  for m=1:model.nbFrames
    plot([s(n).p(m).b(2) s(n).p(m).b(2)+s(n).p(m).A(2,3)], [s(n).p(m).b(3) s(n).p(m).b(3)+s(n).p(m).A(3,3)], '-','linewidth',6,'color',colPegs(m,:));
    plot(s(n).p(m).b(2), s(n).p(m).b(3),'.','markersize',30,'color',colPegs(m,:)-[.05,.05,.05]);
  end
  %Plot trajectories
  plot(s(n).Data0(2,1), s(n).Data0(3,1),'.','markersize',12,'color',clrmap(n,:));
  plot(s(n).Data0(2,:), s(n).Data0(3,:),'-','linewidth',1.5,'color',clrmap(n,:));
end
axis(limAxes); axis square; set(gca,'xtick',[],'ytick',[]);

%REPROS
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subplot(1,3,2); hold on; box on; title('Reproductions with infinite horizon LQR');
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for n=1:nbSamples
  %Plot frames
  for m=1:model.nbFrames
    plot([s(n).p(m).b(2) s(n).p(m).b(2)+s(n).p(m).A(2,3)], [s(n).p(m).b(3) s(n).p(m).b(3)+s(n).p(m).A(3,3)], '-','linewidth',6,'color',colPegs(m,:));
    plot(s(n).p(m).b(2), s(n).p(m).b(3),'.','markersize',30,'color',colPegs(m,:)-[.05,.05,.05]);
  end
  %Plot Gaussians
  plotGMM(r(n).Mu(2:3,:,1), r(n).Sigma(2:3,2:3,:,1), [.7 .7 .7]);
end
for n=1:nbSamples
  %Plot trajectories
  plot(r(n).Data(2,1), r(n).Data(3,1),'.','markersize',12,'color',clrmap(n,:));
  plot(r(n).Data(2,:), r(n).Data(3,:),'-','linewidth',1.5,'color',clrmap(n,:));
end
axis(limAxes); axis square; set(gca,'xtick',[],'ytick',[]);

%NEW REPROS
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subplot(1,3,3); hold on; box on; title('New reproductions with infinite horizon LQR');
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for n=1:nbRepros
  %Plot frames
  for m=1:model.nbFrames
    plot([rnew(n).p(m).b(2) rnew(n).p(m).b(2)+rnew(n).p(m).A(2,3)], [rnew(n).p(m).b(3) rnew(n).p(m).b(3)+rnew(n).p(m).A(3,3)], '-','linewidth',6,'color',colPegs(m,:));
    plot(rnew(n).p(m).b(2), rnew(n).p(m).b(3), '.','markersize',30,'color',colPegs(m,:)-[.05,.05,.05]);
  end
  %Plot Gaussians
  plotGMM(rnew(n).Mu(2:3,:,1), rnew(n).Sigma(2:3,2:3,:,1), [.7 .7 .7]);
end
for n=1:nbRepros
  %Plot trajectories
  plot(rnew(n).Data(2,1), rnew(n).Data(3,1),'.','markersize',12,'color',[.2 .2 .2]);
  plot(rnew(n).Data(2,:), rnew(n).Data(3,:),'-','linewidth',1.5,'color',[.2 .2 .2]);
end
axis(limAxes); axis square; set(gca,'xtick',[],'ytick',[]);

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%print('-dpng','outTest1.png');
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%Plot additional information 
figure; 
%Plot norm of control commands
subplot(1,2,1); hold on;
for n=1:nbRepros
  plot(DataIn, rnew(n).ddxNorm, 'k-', 'linewidth', 2);
end
xlabel('t'); ylabel('|ddx|');
%Plot strength of the stiffness term
subplot(1,2,2); hold on;
for n=1:nbRepros
  plot(DataIn, rnew(n).kpDet, 'k-', 'linewidth', 2);
end
xlabel('t'); ylabel('|Kp|');

%Plot accelerations due to feedback and feedforward terms
figure; hold on;
n=1; k=1;
plot(r(n).FB(k,:),'r-','linewidth',2);
plot(r(n).FF(k,:),'b-','linewidth',2);
legend('ddx feedback','ddx feedforward');
xlabel('t'); ylabel(['ddx_' num2str(k)]);

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%print('-dpng','outTest2.png');
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%pause;
%close all;