Continuous-Time Trajectory Estimation for Differentially Flat Systems

Continuous-time estimation using splines on Lie groups has been gaining traction in the literature due to the ability to incorporate high-frequency sensor data without introducing new optimization parameters. However, evaluating time derivatives and Jacobians of Lie group splines is computationally expensive, limiting their use mainly to offline applications. Motivated by the trajectory planning literature, we develop a new estimation technique that leverages the differential flatness property of many dynamic systems to define the spline in the system's flat output space, which is often Euclidean. Doing so has the added benefit of providing a simple and effective way to include system inputs in the estimation process. We show an example of flatness-based estimation for the unicycle dynamic model. We then show that this new method can achieve similar performance as Lie group spline estimation with significantly less computation time, and validate its use in hardware using a differential-drive robot.