demo_TPbatchLQR02.m 11.4 KB
Newer Older
Sylvain Calinon's avatar
Sylvain Calinon committed
1 2
function demo_TPbatchLQR02
% Batch solution of a linear quadratic optimal control (unconstrained linear MPC) acting in multiple frames,
3
% which is equivalent to TP-GMM combined with LQR.
Sylvain Calinon's avatar
Sylvain Calinon committed
4
%
5 6 7 8
% Writing code takes time. Polishing it and making it available to others takes longer! 
% If some parts of the code were useful for your research of for a better understanding 
% of the algorithms, please reward the authors by citing the related publications, 
% and consider making your own research available in this way.
Sylvain Calinon's avatar
Sylvain Calinon committed
9
%
10
% @article{Calinon16JIST,
Sylvain Calinon's avatar
Sylvain Calinon committed
11
%   author="Calinon, S.",
12 13
%   title="A Tutorial on Task-Parameterized Movement Learning and Retrieval",
%   journal="Intelligent Service Robotics",
14 15 16 17 18 19
%		publisher="Springer Berlin Heidelberg",
%		doi="10.1007/s11370-015-0187-9",
%		year="2016",
%		volume="9",
%		number="1",
%		pages="1--29"
Sylvain Calinon's avatar
Sylvain Calinon committed
20
% }
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
% 
% Copyright (c) 2015 Idiap Research Institute, http://idiap.ch/
% Written by Sylvain Calinon, http://calinon.ch/
% 
% This file is part of PbDlib, http://www.idiap.ch/software/pbdlib/
% 
% PbDlib is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License version 3 as
% published by the Free Software Foundation.
% 
% PbDlib is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
% 
% You should have received a copy of the GNU General Public License
% along with PbDlib. If not, see <http://www.gnu.org/licenses/>.
Sylvain Calinon's avatar
Sylvain Calinon committed
38 39 40 41 42 43

addpath('./m_fcts/');


%% Parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Sylvain Calinon's avatar
Sylvain Calinon committed
44
model.nbStates = 6; %Number of Gaussians in the GMM
Sylvain Calinon's avatar
Sylvain Calinon committed
45 46 47 48 49 50 51 52 53 54 55 56 57 58
model.nbFrames = 2; %Number of candidate frames of reference
model.nbVarPos = 2; %Dimension of position data (here: x1,x2)
model.nbDeriv = 2; %Number of static & dynamic features (D=2 for [x,dx])
model.nbVar = model.nbVarPos * model.nbDeriv; %Dimension of state vector
model.rfactor = 1E-1;	%Control cost in LQR
model.dt = 0.01; %Time step duration
nbData = 200; %Number of datapoints in a trajectory
nbRepros = 5; %Number of reproductions

%Control cost matrix
R = eye(model.nbVarPos) * model.rfactor;
R = kron(eye(nbData-1),R);


Sylvain Calinon's avatar
Sylvain Calinon committed
59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
%% Dynamical System settings (discrete version)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %Integration with Euler method 
% Ac1d = diag(ones(model.nbDeriv-1,1),1); 
% Bc1d = [zeros(model.nbDeriv-1,1); 1];
% A = kron(eye(model.nbDeriv)+Ac1d*model.dt, eye(model.nbVarPos)); 
% B = kron(Bc1d*model.dt, eye(model.nbVarPos));

%Integration with higher order Taylor series expansion
A1d = zeros(model.nbDeriv);
for i=0:model.nbDeriv-1
	A1d = A1d + diag(ones(model.nbDeriv-i,1),i) * model.dt^i * 1/factorial(i); %Discrete 1D
end
B1d = zeros(model.nbDeriv,1); 
for i=1:model.nbDeriv
	B1d(model.nbDeriv-i+1) = model.dt^i * 1/factorial(i); %Discrete 1D
end
A = kron(A1d, eye(model.nbVarPos)); %Discrete nD
B = kron(B1d, eye(model.nbVarPos)); %Discrete nD

% %Conversion with control toolbox
% Ac1d = diag(ones(model.nbDeriv-1,1),1); %Continuous 1D
% Bc1d = [zeros(model.nbDeriv-1,1); 1]; %Continuous 1D
% Cc1d = [1, zeros(1,model.nbDeriv-1)]; %Continuous 1D
% sysd = c2d(ss(Ac1d,Bc1d,Cc1d,0), model.dt);
% A = kron(sysd.a, eye(model.nbVarPos)); %Discrete nD
% B = kron(sysd.b, eye(model.nbVarPos)); %Discrete nD


%Construct Su and Sx matrices, see Eq. (35)
Su = zeros(model.nbVar*nbData, model.nbVarPos*(nbData-1));
Sx = kron(ones(nbData,1),eye(model.nbVar)); 
M = B;
for n=2:nbData
	%Build Sx matrix
	id1 = (n-1)*model.nbVar+1:nbData*model.nbVar;
	Sx(id1,:) = Sx(id1,:) * A;
	%Build Su matrix
	id1 = (n-1)*model.nbVar+1:n*model.nbVar; 
	id2 = 1:(n-1)*model.nbVarPos;
	Su(id1,id2) = M;
	M = [A*M(:,1:model.nbVarPos), M];
end


Sylvain Calinon's avatar
Sylvain Calinon committed
105 106 107 108 109 110 111 112 113 114 115 116
%% 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 DC x P x N, with D=2 the dimension of a
% datapoint, C=2 the number of derivatives (incl. position), P=2 the number of candidate frames, and N=TM 
% the number of datapoints in a trajectory (T=200) multiplied by the number of demonstrations (M=5).
load('data/DataWithDeriv01.mat');


Sylvain Calinon's avatar
Sylvain Calinon committed
117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
% %Modify the data to make the system move back and forth
% for n=1:nbSamples
% 	s(n).Data0 = [s(n).Data0, fliplr(s(n).Data0)];
% end	
% nbData = nbData * 2;

%Recompute 3rd order tensor data 
D = (diag(ones(1,nbData-1),-1)-eye(nbData)) / model.dt;
D(end,end) = 0;
Data = zeros(model.nbVar, model.nbFrames, nbSamples*nbData);
for n=1:nbSamples
	s(n).Data = zeros(model.nbVar,model.nbFrames,nbData);
	DataTmp = s(n).Data0;
	for k=1:model.nbDeriv-1
		DataTmp = [DataTmp; s(n).Data0*D^k]; %Compute derivatives
	end
	for m=1:model.nbFrames
		s(n).Data(:,m,:) = s(n).p(m).A \ (DataTmp - repmat(s(n).p(m).b, 1, nbData));
		Data(:,m,(n-1)*nbData+1:n*nbData) = s(n).Data(:,m,:);
	end
end
%Recompute R
R = eye(model.nbVarPos) * model.rfactor;
R = kron(eye(nbData-1),R);


Sylvain Calinon's avatar
Sylvain Calinon committed
143 144 145 146
%% TP-GMM learning
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fprintf('Parameters estimation of TP-GMM with EM...');
%model = init_tensorGMM_kmeans(Data, model); %Initialization
Sylvain Calinon's avatar
Sylvain Calinon committed
147
model = init_tensorGMM_kbins(s, model);
Sylvain Calinon's avatar
Sylvain Calinon committed
148

Sylvain Calinon's avatar
Sylvain Calinon committed
149 150 151 152 153 154 155 156 157 158 159 160
% %Initialization based on position data
% model0 = init_tensorGMM_kmeans(Data(1:model.nbVarPos,:,:), model);
% [~,~,GAMMA2] = EM_tensorGMM(Data(1:model.nbVarPos,:,:), model0);
% model.Priors = model0.Priors;
% for i=1:model.nbStates
% 	for m=1:model.nbFrames
% 		DataTmp = squeeze(Data(:,m,:));
% 		model.Mu(:,m,i) = DataTmp * GAMMA2(i,:)';
% 		DataTmp = DataTmp - repmat(model.Mu(:,m,i),1,nbData*nbSamples);
% 		model.Sigma(:,:,m,i) = DataTmp * diag(GAMMA2(i,:)) * DataTmp';
% 	end
% end
Sylvain Calinon's avatar
Sylvain Calinon committed
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190

model = EM_tensorGMM(Data, model);


%% Reproduction with LQR for the task parameters used to train the model
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('Reproductions with batch LQR...');
for n=1:nbSamples
	
	%GMM projection, see Eq. (5)
	for i=1:model.nbStates
		for m=1:model.nbFrames
			s(n).p(m).Mu(:,i) = s(n).p(m).A * model.Mu(:,m,i) + s(n).p(m).b;
			s(n).p(m).Sigma(:,:,i) = s(n).p(m).A * model.Sigma(:,:,m,i) * s(n).p(m).A';
		end
	end
	
	%Compute best path for the n-th demonstration
	[~,s(n).q] = max(model.Pix(:,(n-1)*nbData+1:n*nbData),[],1); %works also for nbStates=1	
	
	%Build a reference trajectory for each frame, see Eq. (27)
	invSigmaQ = zeros(model.nbVar*nbData);
	for m=1:model.nbFrames
		s(n).p(m).MuQ = reshape(s(n).p(m).Mu(:,s(n).q), model.nbVar*nbData, 1);  
		s(n).p(m).SigmaQ = (kron(ones(nbData,1), eye(model.nbVar)) * reshape(s(n).p(m).Sigma(:,:,s(n).q), model.nbVar, model.nbVar*nbData)) ...
			.* kron(eye(nbData), ones(model.nbVar));
		invSigmaQ = invSigmaQ + inv(s(n).p(m).SigmaQ);
	end
	
	%Batch LQR (unconstrained linear MPC), see Eq. (37)
Sylvain Calinon's avatar
Sylvain Calinon committed
191
	Rq = Su' * invSigmaQ * Su + R;
Sylvain Calinon's avatar
Sylvain Calinon committed
192 193 194 195 196 197 198 199
	X = [s(1).Data0(:,1) + randn(model.nbVarPos,1)*0E0; zeros(model.nbVarPos,1)];
 	rq = zeros(model.nbVar*nbData,1);
	for m=1:model.nbFrames
		rq = rq + s(n).p(m).SigmaQ \ (s(n).p(m).MuQ - Sx*X);
	end
	rq = Su' * rq; 
 	u = Rq \ rq; %can also be computed with u = lscov(Rq, rq);
	r(n).Data = reshape(Sx*X+Su*u, model.nbVar, nbData);
Sylvain Calinon's avatar
Sylvain Calinon committed
200
	
Sylvain Calinon's avatar
Sylvain Calinon committed
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237
end


%% Reproduction with LQR for new task parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('New reproductions with batch LQR...');
for n=1:nbRepros
	
	%Random generation of new task parameters
	for m=1:model.nbFrames
		id=ceil(rand(2,1)*nbSamples);
		w=rand(2); w=w/sum(w);
		rnew(n).p(m).b = s(id(1)).p(m).b * w(1) + s(id(2)).p(m).b * w(2);
		rnew(n).p(m).A = s(id(1)).p(m).A * w(1) + s(id(2)).p(m).A * w(2);
	end
	
	%GMM projection, see Eq. (5)
	for i=1:model.nbStates
		for m=1:model.nbFrames
			rnew(n).p(m).Mu(:,i) = rnew(n).p(m).A * model.Mu(:,m,i) + rnew(n).p(m).b;
			rnew(n).p(m).Sigma(:,:,i) = rnew(n).p(m).A * model.Sigma(:,:,m,i) * rnew(n).p(m).A';
		end
	end
	
	%Compute best path for the 1st demonstration (HSMM can alternatively be used here)
	[~,rnew(n).q] = max(model.Pix(:,1:nbData),[],1); %works also for nbStates=1
	
	%Build a reference trajectory for each frame, see Eq. (27)
	invSigmaQ = zeros(model.nbVar*nbData);
	for m=1:model.nbFrames
		rnew(n).p(m).MuQ = reshape(rnew(n).p(m).Mu(:,rnew(n).q), model.nbVar*nbData, 1);  
		rnew(n).p(m).SigmaQ = (kron(ones(nbData,1), eye(model.nbVar)) * ...
			reshape(rnew(n).p(m).Sigma(:,:,rnew(n).q), model.nbVar, model.nbVar*nbData)) .* kron(eye(nbData), ones(model.nbVar));
		invSigmaQ = invSigmaQ + inv(rnew(n).p(m).SigmaQ);
	end
	
	%Batch LQR (unconstrained linear MPC), see Eq. (37)
Sylvain Calinon's avatar
Sylvain Calinon committed
238
	Rq = Su' * invSigmaQ * Su + R;
Sylvain Calinon's avatar
Sylvain Calinon committed
239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307
	X = [s(1).Data0(:,1) + randn(model.nbVarPos,1)*0E0; zeros(model.nbVarPos,1)];
 	rq = zeros(model.nbVar*nbData,1);
	for m=1:model.nbFrames
		rq = rq + rnew(n).p(m).SigmaQ \ (rnew(n).p(m).MuQ - Sx*X);
	end
	rq = Su' * rq; 
 	u = Rq \ rq; %can also be computed with u = lscov(Rq, rq);
	rnew(n).Data = reshape(Sx*X+Su*u, model.nbVar, nbData);
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(1) s(n).p(m).b(1)+s(n).p(m).A(1,2)], [s(n).p(m).b(2) s(n).p(m).b(2)+s(n).p(m).A(2,2)], '-','linewidth',6,'color',colPegs(m,:));
		plot(s(n).p(m).b(1), s(n).p(m).b(2),'.','markersize',30,'color',colPegs(m,:)-[.05,.05,.05]);
	end
	%Plot trajectories
	plot(s(n).Data0(1,1), s(n).Data0(2,1),'.','markersize',12,'color',clrmap(n,:));
	plot(s(n).Data0(1,:), s(n).Data0(2,:),'-','linewidth',1.5,'color',clrmap(n,:));
end
axis(limAxes); axis square; set(gca,'xtick',[],'ytick',[]);

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

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

%print('-dpng','graphs/demo_TPbatchLQR02.png');
%pause;
%close all;