Merge pull request #16 from AlWiVo/master

only small typos found on chapter SLAM

Author: correll

book.tex

@@ -224,14 +224,14 @@ \subsection{SLAM as a Maximum-Likelihood Estimation Problem}
 \begin{equation}
 x^* = \arg \min_{x}\sum_{<i,j>\in \mathcal{C}}e_{ij}^T\Omega_{ij}e_{ij}
 \end{equation}
-with $ e_ij(x_i,x_j)=z_{ij}-\hat{z}_{ij}(x_i,xj)$ the error between measurement and expected value. Note that the sum actually needs to be minimized as the individual terms are technically the negative log-likelihood.
+with $ e_{ij}(x_i,x_j)=z_{ij}-\hat{z}_{ij}(x_i,xj)$ the error between measurement and expected value. Note that the sum actually needs to be minimized as the individual terms are technically the negative log-likelihood.
 
 \subsection{Numerical Techniques for Graph-based SLAM}
-Solving the MLE problem is non-trivial, especially if the number of constraints, i.e., observations that relate one feature to another, provided is large. A classical approach is to linearize the problem at the current configuration and reducing it to a problem of the form $ Ax=b$. The intuition here is to calculate the impact of small changes in the positions of all nodes on all $ e_{ij}$. After performing this motion, linearization and optimization can be  repeated until convergence. 
+Solving the MLE problem is non-trivial, especially if the number of constraints provided, i.e., observations that relate one feature to another, is large. A classical approach is to linearize the problem at the current configuration and reducing it to a problem of the form $ Ax=b$. The intuition here is to calculate the impact of small changes in the positions of all nodes on all $ e_{ij}$. After performing this motion, linearization and optimization can be  repeated until convergence. 
 
-More recently, more powerful numerical methods have been developed. Instead of solving the MLE, on can employ a stochastic gradient descent algorithm. A gradient descent algorithm is an iterative approach to find the optimum of a function by moving along its gradient. Whereas a gradient descent algorithm would calculate the gradient on a fitness landscape from all available constraints, a stochastic gradient descent picks only a (non-necessarily random) subset. Intuitive examples are fitting a line to a set of $n$ points, but taking only a subset of these points when calculating the next best guess. As gradient descent works iteratively, the hope is that the algorithm takes a large part of the constraints into account. For solving Graph-based SLAM, a stochastic gradient descent algorithm would not take into account all constraints available to the robot, but  iteratively work on one constraint after the other. Here, constraints are observations on the mutual pose of nodes $i$ and $j$. Optimizing these constraints now requires moving both nodes $i$ and $j$ so that the error between where the robot thinks the nodes should be and what it actually sees gets reduced.  As this is a trade-off between multiple, maybe conflicting observations, the result will approximate a Maximum Likelihood estimate.
+Recently, more powerful numerical methods have been developed. Instead of solving the MLE, one can employ a stochastic gradient descent algorithm. A gradient descent algorithm is an iterative approach to find the optimum of a function by moving along its gradient. Whereas a gradient descent algorithm would calculate the gradient on a fitness landscape from all available constraints, a stochastic gradient descent picks only a (non-necessarily random) subset. Intuitive examples are fitting a line to a set of $n$ points, but taking only a subset of these points when calculating the next best guess. As gradient descent works iteratively, the hope is that the algorithm takes a large part of the constraints into account. For solving Graph-based SLAM, a stochastic gradient descent algorithm would not take into account all constraints available to the robot, but  iteratively work on one constraint after the other. Here, constraints are observations on the mutual pose of nodes $i$ and $j$. Optimizing these constraints now requires moving both nodes $i$ and $j$ so that the error between where the robot thinks the nodes should be and what it actually sees gets reduced.  As this is a trade-off between multiple, maybe conflicting observations, the result will approximate a Maximum Likelihood estimate.
 
-More specifically, with $ e_ij$ the error between an observation and what the robot expects to see, based on its previous observation and sensor model, one can distribute the error along the entire trajectory between both features that are involved in the constraint. That is, if the constraint involves features $i$ and $j$, not only $i$ and $j$'s pose will be updated but all points inbetween will be moved a tiny bit.
+More specifically, with $ e_{ij}$ the error between an observation and what the robot expects to see, based on its previous observation and sensor model, one can distribute the error along the entire trajectory between both features that are involved in the constraint. That is, if the constraint involves features $i$ and $j$, not only $i$ and $j$'s pose will be updated but all points inbetween will be moved a tiny bit.
 
 %This approach is cumbersome and quickly gets out of control if a robot is mapping an environment over multiple hours --- leading to millions of nodes in the graph and constraints. To overcome this problem, [Gris07] propose to (1) merge nodes of a graph as it is build up by relying on accurate localization of the robot within the existing map and (2) to chose a different graph representation.