James R. Forbes

Zi Cong Guo, Vassili Korotkine, James R. Forbes et al.

We present the Koopman State Estimator (KoopSE), a framework for model-free batch state estimation of control-affine systems that makes no linearization assumptions, requires no problem-specific feature selections, and has an inference computational cost that is independent of the number of training points. We lift the original nonlinear system into a higher-dimensional Reproducing Kernel Hilbert Space (RKHS), where the system becomes bilinear. The time-invariant model matrices can be learned by solving a least-squares problem on training trajectories. At test time, the system is algebraically manipulated into a linear time-varying system, where standard batch linear state estimation techniques can be used to efficiently compute state means and covariances. Random Fourier Features (RFF) are used to combine the computational efficiency of Koopman-based methods and the generality of kernel-embedding methods. KoopSE is validated experimentally on a localization task involving a mobile robot equipped with ultra-wideband receivers and wheel odometry. KoopSE estimates are more accurate and consistent than the standard model-based extended Rauch-Tung-Striebel (RTS) smoother, despite KoopSE having no prior knowledge of the system's motion or measurement models.

Charles C. Cossette, Mohammed Shalaby, David Saussié et al.

For a team of robots to work collaboratively, it is crucial that each robot have the ability to determine the position of their neighbors, relative to themselves, in order to execute tasks autonomously. This letter presents an algorithm for determining the three-dimensional relative position between two mobile robots, each using nothing more than a single ultra-wideband transceiver, an accelerometer, a rate gyro, and a magnetometer. A sliding window filter estimates the relative position at selected keypoints by combining the distance measurements with acceleration estimates, which each agent computes using an on-board attitude estimator. The algorithm is appropriate for real-time implementation, and has been tested in simulation and experiment, where it comfortably outperforms standard estimators. A positioning accuracy of less than 1 meter is achieved with inexpensive sensors.

Timothy D. Barfoot, James R. Forbes, Gabriele M. T. D'Eleuterio

Robotics and computer vision problems commonly require handling rigid-body motions comprising translation and rotation - together referred to as pose. In some situations, a vectorial parameterization of pose can be useful, where elements of a vector space are surjectively mapped to a matrix Lie group. For example, these vectorial representations can be employed for optimization as well as uncertainty representation on groups. The most common mapping is the matrix exponential, which maps elements of a Lie algebra onto the associated Lie group. However, this choice is not unique. It has been previously shown how to characterize all such vectorial parameterizations for SO(3), the group of rotations. Some results are also known for the group of poses, where it is possible to build a family of vectorial mappings that includes the matrix exponential as well as the Cayley transformation. We extend what is known for these pose mappings to the 4 x 4 representation common in robotics, and also demonstrate three different examples of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem. In the pointcloud alignment problem our results lead to a new algorithm based on the Cayley transformation, which we call CayPer.

Timothy D. Barfoot, James R. Forbes, David Yoon

We present a Gaussian Variational Inference (GVI) technique that can be applied to large-scale nonlinear batch state estimation problems. The main contribution is to show how to fit both the mean and (inverse) covariance of a Gaussian to the posterior efficiently, by exploiting factorization of the joint likelihood of the state and data, as is common in practical problems. This is different than Maximum A Posteriori (MAP) estimation, which seeks the point estimate for the state that maximizes the posterior (i.e., the mode). The proposed Exactly Sparse Gaussian Variational Inference (ESGVI) technique stores the inverse covariance matrix, which is typically very sparse (e.g., block-tridiagonal for classic state estimation). We show that the only blocks of the (dense) covariance matrix that are required during the calculations correspond to the non-zero blocks of the inverse covariance matrix, and further show how to calculate these blocks efficiently in the general GVI problem. ESGVI operates iteratively, and while we can use analytical derivatives at each iteration, Gaussian cubature can be substituted, thereby producing an efficient derivative-free batch formulation. ESGVI simplifies to precisely the Rauch-Tung-Striebel (RTS) smoother in the batch linear estimation case, but goes beyond the 'extended' RTS smoother in the nonlinear case since it finds the best-fit Gaussian (mean and covariance), not the MAP point estimate. We demonstrate the technique on controlled simulation problems and a batch nonlinear Simultaneous Localization and Mapping (SLAM) problem with an experimental dataset.