Wheeled Mobile Robot State Estimation Model Using Odometry - Robotics Stack Exchange most recent 30 from robotics.stackexchange.com 2019-12-14T16:18:41Z https://robotics.stackexchange.com/feeds/question/16519 https://creativecommons.org/licenses/by-sa/4.0/rdf https://robotics.stackexchange.com/q/16519 2 Wheeled Mobile Robot State Estimation Model Using Odometry joe.dinius https://robotics.stackexchange.com/users/21359 2018-10-09T20:24:26Z 2018-10-09T20:24:26Z <p>I have a process model update that follows the form:</p> <p><span class="math-container">$$\mathbf{x}_{k|k-1} = \mathbf{x}_{k-1|k-1} + \delta \mathbf{x}_k,$$</span></p> <p>where <span class="math-container">$\mathbf{x}_{k-1|k-1}$</span> is the posterior estimate from the last filter update and <span class="math-container">$\delta \mathbf{x}_k$</span> is the increment due to odometry (which is treated as a control step).</p> <p>My question is this: When trying to implement this as a predict step in a state estimator (e.g. Kalman filter), is it enough to consider <span class="math-container">$\mathbf{x}_{k-1|k-1}$</span> and <span class="math-container">$\delta \mathbf{x}_k$</span> as being <em>independent</em> Gaussian random variables drawn from distributions (<span class="math-container">$\mathcal{N}(\mathbf{x}_{k-1|k-1}, P_{k-1|k-1})$</span> and <span class="math-container">$\mathcal{N}(\delta \mathbf{x}_k, Q)$</span>, <span class="math-container">$Q$</span> is the odometry model covariance and <span class="math-container">$\mathcal{N}(m, \sigma^2)$</span> is a Gaussian distribution with mean <span class="math-container">$m$</span> and variance <span class="math-container">$\sigma^2$</span>)? In this case, I would have the following model covariance update during the prediction step:</p> <p><span class="math-container">$$\mathbf{x}_{k|k-1} = \mathbf{x}_{k-1|k-1} + \delta \mathbf{x}_k \\ P_{k|k-1} = P_{k-1|k-1} + Q$$</span></p> <p>My hangup is that this does not follow the normal Kalman filter prediction paradigm:</p> <p><span class="math-container">$$P_{k|k-1} = J P_{k-1|k-1} J^T + Q.$$</span></p> <p>Fundamentally, I believe my update equation is correct because of the fact that the mean and covariance of the sum of two Gaussian distributions is the sum of the mean and the sum of the covariance, respectively, but I am having difficulty reconciling this fact with the Kalman filter equations. Does my reasoning make sense? Am I correct?</p>