Unscented Filtering for Spacecraft Attitude Estimation

A new spacecraft attitude estimation approach based on the Unscented Filter is derived. For nonlinear systems the Unscented Filter uses a carefully selected set of sample points to more accurately map the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The filter formulation is based on standard attitude-vector measurements using a gyro-based model for attitude propagation. The global attitude parameterization is given by a quaternion, while a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is derived from the local attitude error, which guarantees that quaternion normalization is maintained in the filter. Simulation results indicate that the Unscented Filter is more robust than the Extended Kalman Filter under realistic initial attitude-error conditions. INTRODUCTION The Extended Kalman Filter (EKF) is widely used in attitude estimation. Several parameterizations can be used to represent the attitude, such as Euler angles, quaternions, modified Rodrigues parameters, and even the rotation vector. Quaternions are especially appealing since no singularities are present and the kinematics equation is bilinear. However, the quaternion must obey a normalization constraint, which can be violated by the linear measurementupdates associated with the standard EKF approach. The most common approach to overcome this shortfall involves using a multiplicative error quaternion, where after neglecting higher-order terms the fourAssociate Professor, Associate Fellow AIAA. Email: [email protected] Aerospace Engineer, Fellow AIAA. Email: [email protected] component quaternion can effectively be replaced by a three-component error vector. Under ideal circumstances, such as small attitude errors, this approach works extremely well. One interesting fact of the formulation presented in Ref. [2] is that the 4 × 4 quaternion covariance is assumed to have rank three, i.e., the 4 × 4 matrix can be projected onto a 3 × 3 matrix without any loss in information. But, this is only strictly valid when the constraint is linear, which is not the case for the quaternion. However, the covariance is nearly singular, and a linear computation such as the EKF can make it exactly singular. This approach is justifiable for small estimation errors, but may cause difficulties outside the valid linear region, e.g., during the initialization stage of the EKF. Several approaches have addressed the issue of initialization for attitude estimation. Reference [6] explicitly includes the quaternion constraint in the measurement update. This works well for large initial condition errors; however, linearizations about the previous error estimate are required for the covariance propagation and for the measurement updates, which may produce biased estimates. Reference [7] breaks the measurement update into two steps. The linear first step uses a transformation of the desired states, while the second step uses a non-recursive minimization to recover the desired states. This also works well; however, local iterations at each time-step on a constrained minimization problem are required in the second step. In this paper a new attitude estimation approach, based on a filter developed by Julier, Uhlmann and Durrant-Whyte, is shown as an alternative to the EKF. This filter approach, which they call the Unscented Filter (UF), has several advantages over the EKF, including: 1) the expected error is lower than the EKF, 2) the new filter can be applied to nondifferentiable functions, 3) the new filter avoids the derivation of Jacobian matrices, and 4) the new filter