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\usepackage{graphicx,amsmath,amssymb,bm,xcolor,soul,nicefrac,listings,algorithm2e,dsfont,caption,subcaption,wrapfig,sidecap} \usepackage[hidelinks]{hyperref} \usepackage[makeroom]{cancel} +\usepackage[export]{adjustbox} %for valign in figures %pseudocode \newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{lightgray}{#1}} @@ -2691,12 +2692,23 @@ A similar principle can be applied to robot manipulators by composing forward ki %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{Maintaining a desired distance to an object} %(1-e'*e)'*(1-e'*e) version, where 1-e'*e is a scalar +\subsubsection{Distance to an object} %(1-e'*e)'*(1-e'*e) version, where 1-e'*e is a scalar \begin{flushright} \filename{iLQR\_distMaintenance.*} \end{flushright} -The obstacle example above can easily be extended to the problem of maintaining a desired distance to an object, which can also typically used with other objectives. We first define a function and a gradient +\begin{wrapfigure}{r}{.38\textwidth} +%\vspace{-20pt} +\centering +\includegraphics[width=.14\textwidth,valign=c]{images/iLQR_distMaintain01.png}\hspace{6mm} +\includegraphics[width=.14\textwidth,valign=c]{images/iLQR_distMaintain02.png} +\caption{\footnotesize +\emph{Left:} Optimal control problem to generate a motion by considering a point mass starting from an initial point (in black), with a cost asking to reach or maintain a desired distance to a point (in red). \emph{Right:} Similar problem for a contour to reach/maintain defined as a covariance matrix. +} +\label{fig:distMaintain} +\end{wrapfigure} + +The obstacle example above can easily be extended to the problem of reaching/maintaining a desired distance to an object, which can also typically used with other objectives. We first define a function and a gradient \begin{equation*} f^\tp{dist}(d) = 1-d, \quad\quad g^\tp{dist}(d) = -1, @@ -2710,6 +2722,10 @@ that we exploit to define $\bm{f}(\bm{x}_t)$ and $\bm{J}(\bm{x}_t)$ in \eqref{eq In the above, $\bm{f}(\bm{x})$ defines a continuous function that is $0$ on a sphere of radius $r$ centered on the object (defined by a center $\bm{\mu}$), and increasing/decreasing when moving away from this surface in one direction or the other. +Similarly, the error can be weighted by a covariance matrix $\bm{\Sigma}$, by using $e(\bm{x}_t) = {(\bm{x}_t-\bm{\mu})}^\trsp \bm{\Sigma}^{-1} (\bm{x}_t-\bm{\mu})$ and $\bm{J}(\bm{x}_t) = -2 {(\bm{x}_t-\bm{\mu})}^\trsp \bm{\Sigma}^{-1}$, which corresponds to the problem of reaching/maintaining the contour of an ellipse. + +Figure \ref{fig:distMaintain} presents examples with a point mass. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Manipulability tracking} @@ -2800,7 +2816,7 @@ Figure \ref{fig:iLQR_decoupling} presents a simple example within a 2D reaching \centering \includegraphics[width=.32\textwidth]{images/iLQR_curvature01.png} \caption{\footnotesize -Point mass control problem starting form an initial point (in black), with a cost asking to pass through a set of viapoints (in red), together with a cost on curvature (black path). +Optimal control problem to generate a motion by considering a point mass starting from an initial point (in black), with a cost asking to pass through a set of viapoints (in red), together with a cost on curvature (black path). } \label{fig:curvature} \end{wrapfigure} @@ -2840,6 +2856,8 @@ With this formulation, by using the derivative property $(fg)'=f'g+fg'$, the der \label{eq:diff_curvature} \end{equation} +Figure \ref{fig:curvature} presents an example with a point mass. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{iLQR with control primitives} diff --git a/matlab/iLQR_distMaintenance.m b/matlab/iLQR_distMaintenance.m index dee8be9..82a25b9 100644 --- a/matlab/iLQR_distMaintenance.m +++ b/matlab/iLQR_distMaintenance.m @@ -17,7 +17,12 @@ param.nbIter = 100; %Maximum number of iterations for iLQR param.nbVarX = 2; %State space dimension (x1,x2) param.nbVarU = 2; %Control space dimension (dx1,dx2) param.Mu = [1.0; 0.3]; %Object location -param.dist = .4; %Distance to maintain + +%param.dist = .4; %Distance to maintain +%param.Sigma = eye(param.nbVarX) * param.dist^2; %Covariance matrix +vtmp = [1; 1]; %Main axis of covariance matrix +param.Sigma = vtmp * vtmp' + eye(2) * 1E-1; %Covariance matrix + param.q = 1E0; %Distance maintenance weight term param.r = 1E-3; %Control weight term @@ -63,14 +68,19 @@ disp(['iLQR converged in ' num2str(n) ' iterations.']); %% Plot state space %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -al = linspace(-pi, pi, 50); h = figure('position',[10,10,800,800],'color',[1,1,1]); hold on; axis off; -msh = param.dist * [cos(al); sin(al)] + repmat(param.Mu(1:2), 1, 50); +al = linspace(-pi, pi, 50); +%msh = param.dist * [cos(al); sin(al)] + repmat(param.Mu(1:2), 1, 50); +[V,D] = eig(param.Sigma); +msh = V * D.^.5 * [cos(al); sin(al)] + repmat(param.Mu(1:2), 1, 50); + patch(msh(1,:), msh(2,:), [1 .8 .8],'linewidth',2,'edgecolor',[.8 .4 .4]); plot(param.Mu(1), param.Mu(2), '.','markersize',25,'color',[.8 0 0]); plot(x(1,:), x(2,:), '-','linewidth',2,'color',[0 0 0]); -plot(x(1,[1,end]), x(2,[1,end]), '.','markersize',25,'color',[0 0 0]); +plot(x(1,1), x(2,1), '.','markersize',25,'color',[0 0 0]); +plot(x(1,end), x(2,end), '.','markersize',25,'color',[0 .6 0]); axis equal; +%print('-dpng','iLQR_distMaintain02.png'); waitfor(h); end @@ -79,8 +89,11 @@ end %Residuals f and Jacobians J for maintaining a desired distance to an object (fast version with computation in matrix form) function [f, J] = f_dist(x, param) e = x - repmat(param.Mu(1:2), 1, param.nbData); - f = 1 - sum(e.^2, 1)' / param.dist^2; %Residuals - Jtmp = repmat(-e'/param.dist^2, 1, param.nbData); +% f = 1 - sum(e.^2, 1)' / param.dist^2; %Residuals +% Jtmp = repmat(-e'/param.dist^2, 1, param.nbData); + f = 1 - sum(e .* (param.Sigma\e), 1)'; + Jtmp = repmat(-param.Sigma\e, param.nbData,1 )'; + J = Jtmp .* kron(eye(param.nbData), ones(1,param.nbVarU)); %Jacobians end -- GitLab