# A Deep Reinforcement Learning Approach for Active SLAM
## 1 Introduction  
Simultaneous Localization and Mapping (SLAM) refers to the problem of incrementally building the map of a previously unseen environment while at the same time locating the robot on it. Since its statement, more than three decades ago, it has significantly drawn the attention of the robotics community and numerous approaches have been developed, either based on novel filter techniques or optimization methods where inertial and visual measurements may be jointly optimized. See [1; 2; 3; 4] and references there in. Active localization was first introduced by [PERSON] et al. [5], where it was proven that picking actions to minimize the localization's uncertainty would result in a better localization than using a passive approach (e.g., Markov localization or adaptive Monte Carlo localization). Active SLAM augments this approach to the SLAM problem, and it can be defined as the paradigm of controlling a robot which is performing SLAM so as to reduce the uncertainty of its localization and the map's representation [6; 7]. Typically, it consists of three stages [8]: (i) the identification of all possible locations to explore (ideally infinite), (ii) the computation of the utility or reward generated by the actions that would take the robot from its current position to each of those locations and (iii) the selection and execution of the optimal action. In the second step, the utility of each action is traditionally computed by quantifying the uncertainty in the estimation of the two target random variables: the robot's pose and the map's representation. This matrix quantification can be done on the basis of either Theory of Optimal Experimental Design (TOED) or Information Theory (IT). According to the first one, four optimality criteria can be inferred from the generic covariance matrix \(\Sigma\in\mathbb{R}^{\ell\times\ell}\) of the state vector \(x=(x_{1}...x_{\ell})^{T}\in\mathbb{R}^{\ell}\), with eigenvalues \(\lambda_{1}...\lambda_{\ell}\):* T-optimality criterion: captures the average variance, \[\text{T-}opt\triangleq\frac{1}{\ell}\sum_{k=1}^{\ell}\lambda_{k}\] (1)
* A-optimality criterion: captures the harmonic mean variance, which is sensitive to a lower-than average value, \[\text{A-}opt\triangleq\left(\frac{1}{\ell}\sum_{k=1}^{\ell}\lambda_{k}^{-1} \right)^{-1}\] (2)
* D-optimality criterion: captures the whole covariance (hyper) ellipsoid, \[\text{D-}opt\triangleq\exp\left(\frac{1}{\ell}\sum_{k=1}^{\ell}\log(\lambda_{ k})\right)\] (3)
* E-optimality criterion: captures the minimum eigenvalue, \[\text{E-}opt\triangleq\min(\lambda_{k}:k=1...\ell)\] (4)  
Within IT, Shannon's entropy (or its variation) is the most commonly used metric, which can be expressed for an \(\ell\)-dimensional multivariate Gaussian distribution, \(X\sim\mathcal{N}^{\ell}(\mu,\Sigma)\), as follows:  
\[\mathcal{H}(X)\triangleq\frac{\ell}{2}(1+\log(2\pi))+\frac{1}{2}\log(\det( \Sigma)) \tag{5}\]  
The proper choice of the criterion is still an open research issue. Since each of them captures a different metric of \(\Sigma\), they do not lead to the same behavior neither require the same computational resources. It is worth noticing that every criterion is right-unbounded and that the following relationship holds for every well-defined covariance matrix: \(\text{E-}opt\leq\text{A-}opt\leq\text{D-}opt\leq\text{T-}opt\). Moreover, the robot's state space representation (using quaternions, rotation matrices or Lie algebras) and the uncertainty representation have recently been shown to be of utmost importance in the criteria properties and therefore in the algorithm performance [9]. While differential state representations allow the use of any criterion, absolute representations can only rely on D-opt (or Shannon's entropy, equivalently); being the other criteria unable to capture a true metric of the covariance matrix nor to ensure certain key properties such as their monotonicity during exploratory trajectories.