Continuous-Time Estimation of Attitude Using B-Splines on Lie Groups

Filtering algorithms are the workhorse of spacecraft attitude estimation, but recent research has shown that the use of batch estimation techniques can result in higher accuracy per unit of computational cost. This paper presents an approach for singularity-free batch estimation of attitude in continuous time using B-Spline curves on unit-length quaternions. It extends existing theory of unit-length quaternion B-splines to general Lie groups and arbitrary B-spline order. It is shown how to use these curves for continuous-time batch estimation using Gauss-Newton or Levenberg-Marquardt, including efficient curve initialization, a parameter update step that preserves the Lie group constraint within an unconstrained optimization framework, and the derivation of Jacobians of the B-spline's value and its time derivatives with respect to an update of its parameters. For unit-length quaternion splines, the equations for angular velocity and angular acceleration are derived. An implementation of this algorithm is evaluated on two problems: spacecraft attitude estimation using a three-axis magnetometer, a sun sensor, and 1) a three-axis gyroscope, and 2) a continuous-time vehicle dynamics model based on Euler's equation. Its performance is compared against a standard multiplicative extended Kalman filter and a recently published batch attitude estimation algorithm. The results show that B-splines have equal or superior performance over all test cases and provide two key tuning parameters, the number of knots and the spline order, that an engineer can use to trade off accuracy and computational efficiency when choosing a spline representation for a given estimation problem.