From 0096200283862b273f4dd1e9384541e8d2d505c0 Mon Sep 17 00:00:00 2001
From: Sylvain CALINON <sylvain.calinon@idiap.ch>
Date: Thu, 23 Jan 2025 14:49:02 +0100
Subject: [PATCH] Example with initial state simplified

---
 .../gradientDescent_vs_NewtonMethod01.jpg     | Bin 758911 -> 871824 bytes
 matlab/ergodic_control_SMC_2D.m               |   6 +--
 matlab/iLQR_manipulator_basic.m               |  12 ++---
 matlab/iLQR_manipulator_initStateOptim.m      |  42 +++++++++---------
 4 files changed, 30 insertions(+), 30 deletions(-)

diff --git a/doc/images/gradientDescent_vs_NewtonMethod01.jpg b/doc/images/gradientDescent_vs_NewtonMethod01.jpg
index 8b8bd582068516c0db73405c2852cab9a00a3a20..06f13fffaf327beff511ab47338d7292d7b6a680 100644
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diff --git a/matlab/ergodic_control_SMC_2D.m b/matlab/ergodic_control_SMC_2D.m
index 5653622..3800c73 100644
--- a/matlab/ergodic_control_SMC_2D.m
+++ b/matlab/ergodic_control_SMC_2D.m
@@ -37,7 +37,7 @@ Priors = ones(1,nbGaussian) / nbGaussian; %Mixing coefficients
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 rg = 0:nbFct-1;
 [KX(1,:,:), KX(2,:,:)] = ndgrid(rg, rg);
-Lambda = (KX(1,:).^2 + KX(2,:).^2 + 1)'.^-sp; %Weighting vector (Eq.(15))
+Lambda = (KX(1,:).^2 + KX(2,:).^2 + 1)'.^-sp; %Weighting vector 
 
 %Explicit computation of w_hat by exploiting the Fourier transform properties of Gaussians (optimized version by exploiting symmetries),
 %by enumerating symmetry operations for 2D signal ([-1,-1],[-1,1],[1,-1] and [1,1]), and removing redundant ones -> keeping ([-1,-1],[-1,1])
@@ -49,8 +49,8 @@ w_hat = zeros(nbFct^nbVar, 1);
 for j=1:nbGaussian
 	for n=1:size(op,2)
 		MuTmp = diag(op(:,n)) * Mu(:,j); %Eq.(20)
-		SigmaTmp = diag(op(:,n)) * Sigma(:,:,j) * diag(op(:,n))'; %Eq.(20)
-		w_hat = w_hat + Priors(j) * cos(kk' * MuTmp) .* exp(diag(-.5 * kk' * SigmaTmp * kk)); %Eq.(21)
+		SigmaTmp = diag(op(:,n)) * Sigma(:,:,j) * diag(op(:,n))'; 
+		w_hat = w_hat + Priors(j) * cos(kk' * MuTmp) .* exp(diag(-.5 * kk' * SigmaTmp * kk)); 
 	end
 end
 w_hat = w_hat / L^nbVar / size(op,2);
diff --git a/matlab/iLQR_manipulator_basic.m b/matlab/iLQR_manipulator_basic.m
index 4b91538..c551850 100644
--- a/matlab/iLQR_manipulator_basic.m
+++ b/matlab/iLQR_manipulator_basic.m
@@ -18,11 +18,11 @@ param.nbVarX = 3; %State space dimension (x1,x2,x3)
 param.nbVarU = 3; %Control space dimension (dx1,dx2,dx3)
 param.nbVarF = 2; %Objective function dimension (f1,f2)
 param.l = [2; 2; 1]; %Robot links lengths
-param.r = 1E-6; %Control weight term
+param.q = 1E0; %Tracking weight
+param.r = 1E-6; %Control weight
 param.Mu = [[2; 1], [3; 2]]; %Viapoints 
 
 R = speye((param.nbData-1)*param.nbVarU) * param.r; %Control weight matrix (at trajectory level)
-Q = speye(param.nbVarF * param.nbPoints) * 1E0; %Precision matrix
 
 %Time occurrence of viapoints
 tl = linspace(1, param.nbData, param.nbPoints+1);
@@ -43,16 +43,16 @@ Su = Su0(idx,:);
 for n=1:param.nbMaxIter	
 	x = reshape(Su0 * u + Sx0 * x0, param.nbVarX, param.nbData); %System evolution
 	[f, J] = f_reach(x(:,tl), param); %Residuals and Jacobians
-	du = (Su' * J' * Q * J * Su + R) \ (-Su' * J' * Q * f(:) - u * param.r); %Gauss-Newton update
+	du = (Su' * J' * J * Su * param.q + R) \ (-Su' * J' * f(:) * param.q - u * param.r); %Gauss-Newton update
 	
 	%Estimate step size with backtracking line search method
 	alpha = 1;
-	cost0 = f(:)' * Q * f(:) + norm(u)^2 * param.r; %Cost
+	cost0 = norm(f(:))^2 * param.q + norm(u)^2 * param.r; %Cost
 	while 1
 		utmp = u + du * alpha;
 		xtmp = reshape(Su0 * utmp + Sx0 * x0, param.nbVarX, param.nbData); %System evolution
 		ftmp = f_reach(xtmp(:,tl), param); %Residuals
-		cost = ftmp(:)' * Q * ftmp(:) + norm(utmp)^2 * param.r; %Cost
+		cost = norm(ftmp(:))^2 * param.q + norm(utmp)^2 * param.r; %Cost
 		if cost < cost0 || alpha < 1E-3
 			break;
 		end
@@ -60,7 +60,7 @@ for n=1:param.nbMaxIter
 	end
 	u = u + du * alpha; %Update
 	
-	if norm(du * alpha) < 1E-2
+	if norm(du * alpha) < 1E-3
 		break; %Stop iLQR when solution is reached
 	end
 end
diff --git a/matlab/iLQR_manipulator_initStateOptim.m b/matlab/iLQR_manipulator_initStateOptim.m
index 313b2f7..1d14d0f 100644
--- a/matlab/iLQR_manipulator_initStateOptim.m
+++ b/matlab/iLQR_manipulator_initStateOptim.m
@@ -15,54 +15,54 @@ param.nbData = 30; %Number of datapoints in a trajectory
 param.nbPoints = 2; %Number of viapoints
 param.dt = 1E0; %Time step size
 param.nbIter = 50; %Number of iterations for iLQR
-param.nbDOFs = 5; %Number of articulated links
-param.nbVarX = param.nbDOFs * 2; %State space dimension (q01,q02,q1,q2,q3, x1,x2,l1,l2,l3), with 5 rotary and 5 prismatic joints
-param.nbVarU = param.nbDOFs * 2; %Control space dimension (dq01,dq02,dq1,dq2,dq3  dx1,dx2,dl1,dl2,dl3)
+param.nbDOFs = 5; %Number of revolute joints
+param.nbVarX = param.nbDOFs * 2; %State space dimension (q01,q02,q1,q2,q3, x1,x2,l1,l2,l3), with 5 revolute and 5 prismatic joints
+param.nbVarU = 3; %Control space dimension (we only control dq1,dq2,dq3)
 param.nbVarF = 3; %Task space dimension (f1,f2,f3, with f3 as orientation)
 
-param.Mu = [[2; 1; -pi/6], [3; 2; -pi/3]]; %Viapoints
-param.Mu(1:2,:) = param.Mu(1:2,:) + 5;
-
-Rx0 = diag([ones(1,2)*1E0, ones(1,param.nbDOFs-2)*1E0, ...
-            zeros(1,2), ones(1,param.nbDOFs-2)*1E0]); %Weights to modify pose (q01,q02,q1,q2,q3, x1,x2,l1,l2,l3) -> here, only x1,x2 can change
-R = kron(speye(param.nbData-1), diag([ones(1,2)*1E0, ones(1,param.nbDOFs-2)*1E-6, ...
-            ones(1,2)*1E0, ones(1,param.nbDOFs-2)*1E0])); %Weights for (dq01,dq02,dq1,dq2,dq3  dx1,dx2,dl1,dl2,dl3) -> here, only dq1,dq2,dq3 can change
-Ra = blkdiag(Rx0, R); %Weights for augmented control commands
+param.Mu = [[7; 6; -pi/6], [8; 7; -pi/3]]; %Viapoints
 
+Rx0 = zeros(2); %we only search for initial x1,x2 in the state vector (q01,q02,q1,q2,q3, x1,x2,l1,l2,l3)
+R = speye(3*(param.nbData-1)) * 1E-6; %We only control dq1,dq2,dq3 in the velocity vector (dq01,dq02,dq1,dq2,dq3  dx1,dx2,dl1,dl2,dl3)
+Ra = blkdiag(Rx0, R); %Weights for list of decision variables (control commands augmented with initial state)
 Q = kron(speye(param.nbPoints), diag([1E0, 1E0, 0])); %Precision matrix (by removing orientation constraint for the viapoint tracking task)
 
 %Time occurrence of viapoints
 tl = linspace(1, param.nbData, param.nbPoints+1);
 tl = round(tl(2:end));
-idx = (tl - 1) * param.nbVarX + [1:param.nbVarX]';
+idx = (tl - 1) * param.nbVarX + [1:param.nbVarX]'; %indices listing viapoints to reach
 
 
 %% Iterative LQR (iLQR)
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 x0_hat = [0; pi/2; pi/4; -pi/2; -pi/4;  .1; .1; 3; 2; 1]; %Initial robot pose estimate (q01,q02,q1,q2,q3, x1,x2,l1,l2,l3)
-ua = [x0_hat; zeros(param.nbVarU*(param.nbData-1), 1)]; %Initial augmented control commands
+ua = [x0_hat(6:7); zeros(param.nbVarU*(param.nbData-1), 1)]; %Initialization of augmented control commands with x1,x2 as initial state and dq1,dq2,dq3 as list of control commands 
+ua_full = [x0_hat; zeros(param.nbVarX*(param.nbData-1), 1)];
+
+idu = [12:10:size(ua_full,1)] + [1:3]'; 
+idu = [6,7,idu(:)']; %indices listing decision variables to update
 
 %Transfer matrices (for linear system as single integrator)
 Sx0 = kron(ones(param.nbData,1), eye(param.nbVarX));
 Su0 = [zeros(param.nbVarX, param.nbVarX*(param.nbData-1)); kron(tril(ones(param.nbData-1)), eye(param.nbVarX)*param.dt)];
 Sa0 = [Sx0, Su0]; 
-Sa = [Sx0(idx,:), Su0(idx,:)];
+Sa = Sa0(idx,idu);
 
 for n=1:param.nbIter
-	x = reshape(Sa0 * ua, param.nbVarX, param.nbData); %System evolution
+	ua_full(idu) = ua;
+	x = reshape(Sa0 * ua_full, param.nbVarX, param.nbData); %System evolution
 	[f, J] = f_reach(x(:,tl), param); 
-	fua = ua - [x0_hat; zeros(param.nbVarU*(param.nbData-1),1)];
-	dua = (Sa' * J' * Q * J * Sa + Ra) \ (-Sa' * J' * Q * f(:) - Ra * fua); %Gradient for initial pose and control commands
+	dua = (Sa' * J' * Q * J * Sa + Ra) \ (-Sa' * J' * Q * f(:) - Ra * ua); %Gradient for initial pose and control commands
 	
 	%Estimate step size with line search method
 	alpha = 1;
-	cost0 = f(:)' * Q * f(:) + fua' * Ra * fua;
+	cost0 = f(:)' * Q * f(:) + ua' * Ra * ua;
 	while 1
 		uatmp = ua + dua * alpha;
-		xtmp = reshape(Sa0 * uatmp, param.nbVarX, param.nbData);
+		ua_full(idu) = uatmp;
+		xtmp = reshape(Sa0 * ua_full, param.nbVarX, param.nbData);
 		ftmp = f_reach(xtmp(:,tl), param);
-		fuatmp = uatmp - [x0_hat; zeros(param.nbVarU*(param.nbData-1),1)];
-		cost = ftmp(:)' * Q * ftmp(:) + fuatmp' * Ra * fuatmp;
+		cost = ftmp(:)' * Q * ftmp(:) + uatmp' * Ra * uatmp;
 		if cost < cost0 || alpha < 1E-4
 			break;
 		end
-- 
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