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avoidance, by starting from $\bm{x}_1$ and reaching $\bm{\mu}_T$ while avoiding the two obstacles in red. } \label{fig:iLQR_obstacle} -\vspace{-80pt} +\vspace{-20pt} \end{wrapfigure} By taking as example a point-mass system, iLQR can be used to solve an ellipsoidal obstacle avoidance problem with the cost (typically used with other costs, see Fig.~\ref{fig:iLQR_obstacle} for an example). We first define a ReLU-like function and its gradient as (see also Fig.~\ref{fig:fcut}-\emph{right}) -\begin{align*} - f^\tp{avoid}(d) &= \begin{cases} 0, & \text{if}\ d>1 \\ 1-d, & \text{otherwise} \end{cases}, \\ - \bm{g}^\tp{avoid}(d) &= \begin{cases} 0, \hspace{6mm} & \text{if}\ d>1 \\ -1, & \text{otherwise} \end{cases}, -\end{align*} +\begin{equation*} + f^\tp{avoid}(d) = \begin{cases} 0, & \text{if}\ d>1 \\ 1-d, & \text{otherwise} \end{cases}, \quad\quad + g^\tp{avoid}(d) = \begin{cases} 0, \hspace{6mm} & \text{if}\ d>1 \\ -1, & \text{otherwise} \end{cases}, +\end{equation*} that we exploit to define $\bm{f}(\bm{x}_t)$ and $\bm{J}(\bm{x}_t)$ in \eqref{eq:du} as \begin{align*} - \bm{f}(\bm{x}_t) &= f^\tp{avoid}\Big( e(\bm{x}_t) \Big), \\ - \bm{J}(\bm{x}_t) &= 2 \; \bm{g}^\tp{avoid}\Big( e(\bm{x}_t) \Big) \; {(\bm{x}_t-\bm{\mu})}^\trsp \; \bm{\Sigma}^{-1} ,\\ + \bm{f}(\bm{x}_t) &= f^\tp{avoid}\Big( e(\bm{x}_t) \Big), \quad\quad + \bm{J}(\bm{x}_t) = 2 \; g^\tp{avoid}\Big( e(\bm{x}_t) \Big) \; {(\bm{x}_t-\bm{\mu})}^\trsp \; \bm{\Sigma}^{-1} ,\\ \text{with}\quad e(\bm{x}_t) &= {(\bm{x}_t-\bm{\mu})}^\trsp \bm{\Sigma}^{-1} (\bm{x}_t-\bm{\mu}). \end{align*} -In the above, $\bm{f}(\bm{x})$ defines a continuous and smooth function that is $0$ outside the obstacle boundaries and $1$ at the center of the obstacle. The ellipsoid is defined by a center $\bm{\mu}$ and a shape $\bm{\Sigma}=\bm{V}\bm{V}^\trsp$, where $\bm{V}$ is a horizontal concatenation of vectors $\bm{V}_i$ describing the principal axes of the ellipsoid, see Fig.~\ref{fig:iLQR_obstacle}. +In the above, $\bm{f}(\bm{x})$ defines a continuous function that is $0$ outside the obstacle boundaries and $1$ at the center of the obstacle. The ellipsoid is defined by a center $\bm{\mu}$ and a shape $\bm{\Sigma}=\bm{V}\bm{V}^\trsp$, where $\bm{V}$ is a horizontal concatenation of vectors $\bm{V}_i$ describing the principal axes of the ellipsoid, see Fig.~\ref{fig:iLQR_obstacle}. %A similar decomposition on the corresponding precision matrix is expressed as $\bm{\Sigma}^{-1}=\bm{U}\bm{U}^\trsp$. A similar principle can be applied to robot manipulators by composing forward kinematics and obstacle avoidance functions. @@ -1954,6 +1954,27 @@ A similar principle can be applied to robot manipulators by composing forward ki % Alternatively, an avoidance cost c = (1 - f'*f) * wQ (with wQ a scalar weigh) can be used in practice because 0<=(1 - f'*f)<=1 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsubsection{Maintaining a desired 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{equation*} + f^\tp{dist}(d) = 1-d, \quad\quad + g^\tp{dist}(d) = -1, +\end{equation*} +that we exploit to define $\bm{f}(\bm{x}_t)$ and $\bm{J}(\bm{x}_t)$ in \eqref{eq:du} as +\begin{align*} + \bm{f}(\bm{x}_t) &= f^\tp{dist}\Big( e(\bm{x}_t) \Big), \quad\quad + \bm{J}(\bm{x}_t) = -\frac{2}{r} {(\bm{x}_t-\bm{\mu})}^\trsp ,\\ %\; g^\tp{dist}\Big( e(\bm{x}_t) \Big) + \text{with}\quad e(\bm{x}_t) &= \frac{1}{r^2} {(\bm{x}_t-\bm{\mu})}^\trsp (\bm{x}_t-\bm{\mu}). +\end{align*} + +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. + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Manipulability tracking} \begin{flushright} diff --git a/matlab/iLQR_distMaintenance.m b/matlab/iLQR_distMaintenance.m index 13439f4..46130fa 100644 --- a/matlab/iLQR_distMaintenance.m +++ b/matlab/iLQR_distMaintenance.m @@ -89,9 +89,9 @@ 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)) / param.dist; - f = 1 - sum(e.^2, 1)'; %Residuals - Jtmp = -e' * repmat(eye(param.nbVarU)/param.dist, 1, param.nbData); + 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); J = Jtmp .* kron(eye(param.nbData), ones(1,param.nbVarU)); %Jacobians end @@ -100,10 +100,10 @@ end %function [f, J] = f_dist(x, param) % f=[]; J=[]; % for t=1:size(x,2) -% e = (x(:,t) - param.Mu(1:2)) / param.dist; -% ftmp = 1 - e' * e; +% e = x(:,t) - param.Mu(1:2); +% ftmp = 1 - e' * e / param.dist^2; % f = [f; ftmp]; %Residuals -% Jtmp = -e' / param.dist; +% Jtmp = -e' / param.dist^2; % J = blkdiag(J, Jtmp); %Jacobians % end %end -- GitLab