From b93bb401e4c1aa9c362e0a07a973a00b5db36c8d Mon Sep 17 00:00:00 2001
From: Sylvain CALINON <sylvain.calinon@idiap.ch>
Date: Thu, 11 Aug 2022 13:43:20 +0200
Subject: [PATCH] Section 8.5.8 dded

---
 doc/rcfs.pdf                  | Bin 3083298 -> 3084851 bytes
 doc/rcfs.tex                  |  37 ++++++++++++++++++++++++++--------
 matlab/iLQR_distMaintenance.m |  12 +++++------
 3 files changed, 35 insertions(+), 14 deletions(-)

diff --git a/doc/rcfs.pdf b/doc/rcfs.pdf
index 0103c220c8070f0a22935d03adf128895f82f1b7..b3def33d782aad8d0a3ec87e6c1f2916ab4ebe6b 100644
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diff --git a/doc/rcfs.tex b/doc/rcfs.tex
index a87f986..aa904d1 100644
--- a/doc/rcfs.tex
+++ b/doc/rcfs.tex
@@ -1928,22 +1928,22 @@ with the corresponding Jacobian matrix $\bm{J}^\tp{CoM}\in\mathbb{R}^{2\times 5}
 Reaching task with obstacle 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
-- 
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