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parts of $\bm{x}_1$ need to be estimated, the corresponding diagonal entries in $\bm{\tilde{R}}$ can be set with arbitrary high values, by reformulating the minimization problem as -\begin{align} - \min_{\bm{\tilde{u}}} \bm{f}(\bm{x})^{\!\trsp} \bm{Q} \bm{f}(\bm{x}) - + {(\bm{\tilde{u}}-\bm{\mu})}^\trsp \! \bm{\tilde{R}} (\bm{\tilde{u}}-\bm{\mu}), -\end{align} -where $\bm{\mu}$ is a vector containing the initial elements in $\bm{x}_1$ that do not need to be estimated, where the other elements are zero. +The approach can for example be used to estimate the optimal placement of a robot manipulator. For a planar robot, this can for example be implemented by defining the kinematic chain starting with two prismatic joints along the two axes directions, providing a way to represent the location of a free floating platform as two additional prismatic joints in the kinematic chain. If we would like the base of the robot to remain static, we can request that these first two prismatic joints will stay still during the motion by removing them from the list of controlled variables updated by iLQR, see Figure \ref{fig:iLQR_initStateOptim} and the corresponding source code example. + +%If only some parts of $\bm{x}_1$ need to be estimated, the corresponding diagonal entries in $\bm{\tilde{R}}$ can be set with arbitrary high values, by reformulating the minimization problem as +%\begin{align} +% \min_{\bm{\tilde{u}}} \bm{f}(\bm{x})^{\!\trsp} \bm{Q} \bm{f}(\bm{x}) +% + {(\bm{\tilde{u}}-\bm{\mu})}^\trsp \! \bm{\tilde{R}} (\bm{\tilde{u}}-\bm{\mu}), +%\end{align} +%where $\bm{\mu}$ is a vector containing the initial elements in $\bm{x}_1$ that do not need to be estimated, where the other elements are zero. -The above approach can for example be used to estimate the optimal placement of a robot manipulator. For a planar robot, this can for example be implemented by defining the kinematic chain starting with two prismatic joints along the two axes directions, providing a way to represent the location of a free floating platform as two additional prismatic joints in the kinematic chain. If we would like the base of the robot to remain static, we can request that these first two prismatic joints will stay still during the motion. By using the above formulation, this can be done by setting arbitrary large control weights for these corresponding joints, see Figure \ref{fig:iLQR_initStateOptim} and the corresponding source code example. +%By using the above formulation, this can be done by setting arbitrary large control weights for these corresponding joints \newpage diff --git a/matlab/LQT_recursive.m b/matlab/LQT_recursive.m index 0eaf800..e10ca11 100644 --- a/matlab/LQT_recursive.m +++ b/matlab/LQT_recursive.m @@ -75,12 +75,12 @@ end for n=1:2 x = [zeros(param.nbVar,1); 1]; %Augmented state space for t=1:param.nbData - Ka = (B' * P(:,:,t) * B + Ra) \ B' * P(:,:,t) * A; %Feedback gain - u = -Ka * x; %Feedback control on augmented state (resulting in feedback and feedforward terms on state) - x = A * x + B * u; %Update of state vector if t==25 && n==2 x = x + xn; %Simulated noise on the state end + Ka = (B' * P(:,:,t) * B + Ra) \ B' * P(:,:,t) * A; %Feedback gain + u = -Ka * x; %Feedback control on augmented state (resulting in feedback and feedforward terms on state) + x = A * x + B * u; %Update of state vector r(n).x(:,t) = x; %Log data end end @@ -92,7 +92,9 @@ h = figure('position',[10 10 800 800],'color',[1 1 1]); axis off; hold on; plot(r(1).x(1,1), r(1).x(2,1), 'k.','markersize',30); plot(r(1).x(1,:), r(1).x(2,:), 'k:','linewidth',2); plot(r(2).x(1,:), r(2).x(2,:), 'k-','linewidth',2); -plot(r(2).x(1,24:25), r(2).x(2,24:25), 'g.','markersize',20); +plot(r(2).x(1,:), r(2).x(2,:), 'k.','markersize',12); +plot(r(2).x(1,24:25), r(2).x(2,24:25), 'g-','linewidth',2); %Perturbation +plot(r(2).x(1,24:25), r(2).x(2,24:25), 'g.','markersize',20); %Perturbation plot(Mu(1,:), Mu(2,:), 'r.','markersize',40); axis equal; diff --git a/matlab/LQT_recursive_LS.m b/matlab/LQT_recursive_LS.m index 10aa423..9cd3db8 100644 --- a/matlab/LQT_recursive_LS.m +++ b/matlab/LQT_recursive_LS.m @@ -94,11 +94,14 @@ end for n=1:2 x = [zeros(param.nbVar,1); 1]; %Augmented state space for t=1:param.nbData-1 - u = -Ka(:,:,t) * x; %Feedback control on augmented state (resulting in feedback and feedforward teRas on state) - x = A * x + B * u; %Update of state vector if t==25 && n==2 x = x + xn; %Simulated noise on the state end +% u = -Ka(:,:,t) * x; %Feedback control on augmented state (resulting in feedforward part and feedback on state) + u_ff = -Ka(:,end,t); %Feedforward part on state + u = u_ff - Ka(:,1:end-1,t) * x(1:end-1); %Feedforward and feedback part on state + + x = A * x + B * u; %Update of state vector r(n).x(:,t) = x; %Log data end end @@ -110,7 +113,9 @@ h = figure('position',[10 10 800 800],'color',[1 1 1]); axis off; hold on; plot(r(1).x(1,1), r(1).x(2,1), 'k.','markersize',30); plot(r(1).x(1,:), r(1).x(2,:), 'k:','linewidth',2); plot(r(2).x(1,:), r(2).x(2,:), 'k-','linewidth',2); -plot(r(2).x(1,24:25), r(2).x(2,24:25), 'g.','markersize',20); +plot(r(2).x(1,:), r(2).x(2,:), 'k.','markersize',12); +plot(r(2).x(1,24:25), r(2).x(2,24:25), 'g-','linewidth',2); %Perturbation +plot(r(2).x(1,24:25), r(2).x(2,24:25), 'g.','markersize',20); %Perturbation plot(Mu(1,:), Mu(2,:), 'r.','markersize',40); axis equal; diff --git a/matlab/ergodic_control_SMC_2D.m b/matlab/ergodic_control_SMC_2D.m index 3800c73..8c742d8 100644 --- a/matlab/ergodic_control_SMC_2D.m +++ b/matlab/ergodic_control_SMC_2D.m @@ -8,6 +8,8 @@ function ergodic_control_SMC_2D %% This file is part of RCFS <https://rcfs.ch> %% License: GPL-3.0-only +%pkg('load','statistics'); %if mvnpdf() is used + %% Parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -55,17 +57,18 @@ for j=1:nbGaussian end w_hat = w_hat / L^nbVar / size(op,2); -% %Alternative computation of w_hat by discretization -% nbRes = 100; -% xm1d = linspace(xlim(1), xlim(2), nbRes); %Spatial range for 1D -% [xm(1,:,:), xm(2,:,:)] = ndgrid(xm1d, xm1d); %Spatial range -% g = zeros(1,nbRes^nbVar); -% for k=1:nbGaussian -% %Spatial distribution (you can call pkg('load','statistics') in octave if the package is missing) -% g = g + Priors(k) * mvnpdf(xm(:,:)', Mu(:,k)', Sigma(:,:,k))'; -% end -% phi_inv = cos(KX(1,:)' * xm(1,:) * om) .* cos(KX(2,:)' * xm(2,:) * om) / L^nbVar / nbRes^nbVar; -% w_hat = phi_inv * g'; %Fourier coefficients of spatial distribution +%%Alternative computation of w_hat by discretization +%nbRes = 40; +%xm1d = linspace(xlim(1), xlim(2), nbRes); %Spatial range for 1D +%[xm(1,:,:), xm(2,:,:)] = ndgrid(xm1d, xm1d); %Spatial range +%g = zeros(1,nbRes^nbVar); +%for k=1:nbGaussian +%%Spatial distribution (you can call pkg('load','statistics') in octave if the package is missing) +%g = g + Priors(k) * mvnpdf(xm(:,:)', Mu(:,k)', Sigma(:,:,k))'; +%end +%phi_inv = cos(KX(1,:)' * xm(1,:) * om) .* cos(KX(2,:)' * xm(2,:) * om) / L^nbVar / nbRes^nbVar; +%%w_hat2 = phi_inv / g; %Fourier coefficients of spatial distribution +%w_hat = phi_inv * g'; %Fourier coefficients of spatial distribution %Fourier basis functions (for a discretized map) xm1d = linspace(xlim(1), xlim(2), nbRes); %Spatial range for 1D @@ -74,7 +77,7 @@ phim = cos(KX(1,:)' * xm(1,:) * om) .* cos(KX(2,:)' * xm(2,:) * om) * 2^nbVar; % [xx, yy] = ndgrid(1:nbFct, 1:nbFct); hk = [1; 2*ones(nbFct-1,1)]; -HK = hk(xx(:)) .* hk(yy(:)); +HK = hk(xx(:)) .* hk(yy(:)); phim = phim .* repmat(HK,[1,nbRes^nbVar]); %Desired spatial distribution @@ -110,7 +113,7 @@ end % Plot %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -h = figure('position',[10,10,1800,600]); +h = figure('position',[10,10,1600,600]); %x ax(1) = subplot(1,3,1); hold on; axis off; title('Path generated by ergodic control','fontsize',20); diff --git a/matlab/iLQR_manipulator_recursive.m b/matlab/iLQR_manipulator_recursive.m index 544ad1b..629c52e 100644 --- a/matlab/iLQR_manipulator_recursive.m +++ b/matlab/iLQR_manipulator_recursive.m @@ -130,7 +130,7 @@ for t=1:param.nbPoints msh(:,:,t) = param.A(:,:,t) * msh0 + repmat(param.Mu(1:2,t), 1, size(msh0,2)); end -h = figure('position',[10,10,800,800],'color',[1,1,1]); hold on; axis off; +h = figure('position',[10,10,800,800],'color',[1,1,1]); hold on; %axis off; colMat = lines(param.nbPoints); ftmp = fkin0(x(:,1), param); @@ -148,9 +148,11 @@ end ftmp = fkin(x, param); plot(ftmp(1,:), ftmp(2,:), 'k-','linewidth',2); +plot(ftmp(1,:), ftmp(2,:), 'k.','markersize',12); plot(ftmp(1,[1,tl]), ftmp(2,[1,tl]), 'k.','markersize',20); plot(ftmp(1,tn-1:tn), ftmp(2,tn-1:tn), 'g.','markersize',20); %Perturbation plot(ftmp(1,tn-1:tn), ftmp(2,tn-1:tn), 'g-','linewidth',3); %Perturbation +%plot(xref(1,:), xref(2,:), 'b.','markersize',12); axis equal; waitfor(h); diff --git a/matlab/spline1D.m b/matlab/spline1D.m index e20e2ee..481a9cf 100644 --- a/matlab/spline1D.m +++ b/matlab/spline1D.m @@ -94,7 +94,7 @@ w_vis = kron(param.C, eye(param.nbOut)) * w; %% Plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -h = figure('position',[10 10 2500 1800]); +h = figure('position',[10 10 1500 800]); set(gca,'linewidth',2); clrmap = lines(param.nbFct * param.nbSeg); %Colors for total number of control points @@ -124,7 +124,7 @@ for n=1:param.nbSeg tSeg_n = linspace(0, 1/param.nbSeg, param.nbFct) + (n-1)/param.nbSeg; tSeg = [tSeg, tSeg_n]; end -plot(tSeg, w_vis, '.', 'markersize',38,'color',[0 0 0]); +plot(tSeg, w_vis, '.', 'markersize',18,'color',[0 0 0]); %Plot segments between control points for n=1:param.nbSeg id = (n-1)*param.nbFct + [1,2]; -- GitLab