From 5b4b1353e375c47e6574d5a65b98d19e6315a724 Mon Sep 17 00:00:00 2001
From: Sylvain CALINON <sylvain.calinon@idiap.ch>
Date: Fri, 23 Aug 2024 12:00:00 +0200
Subject: [PATCH] push before change of SMC
---
doc/rcfs.tex | 8 ++++++++
1 file changed, 8 insertions(+)
diff --git a/doc/rcfs.tex b/doc/rcfs.tex
index ee8bf99..afc1e2b 100644
--- a/doc/rcfs.tex
+++ b/doc/rcfs.tex
@@ -155,6 +155,13 @@ can be seen as a product of Gaussians $\prod_{k=1}^K\mathcal{N}(\bm{\mu}_k,\bm{W
$\bm{\hat{\mu}}$ and $\bm{\hat{W}}$ are the same as the solution of~\eqref{eq:quad_costs_multiple} and its Hessian, respectively. Viewing the quadratic cost probabilistically can capture more information about the cost function in the form of the covariance matrix $\bm{\hat{W}}^{-1}$.
+In case of Gaussians close to singularity, to be numerically more robust when computing the product of two Gaussians $\mathcal{N}(\bm{\mu}_1, \bm{\Sigma}_1)$ and $\mathcal{N}(\bm{\mu}_2, \bm{\Sigma}_2)$, the Gaussian $\mathcal{N}(\bm{\hat{\mu}}, \bm{\hat{\Sigma}})$ resulting from the product can be alternatively computed with
+\begin{equation*}
+ \bm{\hat{\mu}} = \bm{\Sigma}_2 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\mu}_1 +
+ \bm{\Sigma}_1 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\mu}_2 , \quad
+ \bm{\hat{\Sigma}} = \bm{\Sigma}_1 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\Sigma}_2.
+\end{equation*}
+
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{images/PoG01.png}
@@ -1273,6 +1280,7 @@ Figure \ref{fig:Bezier_2D_eikonal} presents an example in 2D.
\section{Linear quadratic tracking (LQT)}\label{sec:LQT}
\begin{flushright}
\filename{LQT.*}
+\filename{LQT\_nullspace.*}
\end{flushright}
Linear quadratic tracking (LQT) is a simple form of optimal control that trades off tracking and control costs expressed as quadratic terms over a time horizon, with the evolution of the state described in a linear form. The LQT problem is formulated as the minimization of the cost
--
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