Sigma-Point Kalman Filtering for Integrated GPS and Inertial Navigation

Abstract

A sigma-point Kalman filter is derived for integrating GPS measurements with inertial measurements from gyros and accelerometers to determine both the position and the attitude of a moving vehicle. Sigma-point filters use 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 position/attitude estimation problems. The filter formulation is based on standard inertial navigation equations. 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 used to guarantee that quaternion normalization is maintained in the filter. Simulation and experimental results are shown to compare the performance of the sigma-point filter with a standard extended Kalman filter approach.

Marginalized Unscented Quaternion Estimator for Integrated INS/GPS

“…According to Crassidis (2006) and Crassidis and Markley (2003), when a = ℓ = 1, Equation (35) gives the standard vector of Modified Rodrigues Parameters (MRP), and when a = 0 and ℓ = 1, Equation (35) gives the Gibbs vector. In Crassidis (2006) and Crassidis and Markley (2003) ℓ = 2(a + 1) is chosen and in this case the norm of δℜ is equal to the angle of rotation.…”

“…Therefore, using the 4 × 4 covariance matrix for the quaternion will cause a numerical instability problem. To overcome this problem, Crassidis (2006) and Crassidis and Markely (2003) developed the so-called USQUE with the consideration of dimensional mismatch for spacecraft attitude estimation and integrated navigation. In the USQUE, the GRP is used for local attitude representation and filtering and the quaternion is used for global nonsingular attitude propagation.…”

“…However, these error equation-based models suffer more or less from model error due to external disturbances and artificial approximation. In contrast, the Standard Inertial Navigation Equations (SINE)-based process model with no model simplification has drawn incremental research interest in recent years (Crassidis, 2006;Zhou et al, 2010;Zhou et al, 2011). Any vehicle's manoeuvre and external disturbances can be captured by the SINE-based process model using an Inertial Measurement Unit (IMU).…”

“…Generally, this research can be partitioned into two main aspects: one is research on the architectures of integration, such as the loosely-coupled, tightly-coupled and ultra-tightly-coupled INS/GPS integration and the other is research on integrated filtering algorithm design Miller and Campbell, 2012;Sun et al, 2013;Crassidis, 2006;Zhang et al, 2015;Crassidis et al, 2007). This paper focuses on the second aspect and takes the loosely-coupled INS/GPS integration system as the representative application object.…”

“…However, when applied to the quaternionbased model, the UKF may destroy the unit-norm constraint of the quaternion due to the averaging operation on sigma points. With this consideration, a quaternion form of the traditional UKF called UnScented QUaternion Estimator (USQUE) is developed to cope with the quaternion normalization constraint for the quaternionbased attitude kinematics (Crassidis, 2006;Crassidis and Markley, 2003). In the USQUE, a hierarchy considering the dimensional mismatch is implemented, where the four-dimensional quaternion is used for global non-singular attitude representation and a set of three-dimensional unconstrained parameters, such as the Generalised Rodrigues Parameter (GRP), for local attitude representation and filtering.…”