The UKF and CDKF for Low-cost SDINS / GPS In-motion Alignment

In-motion alignment of a SDINS/GPS integrated system, determining the angular relationship between the navigation frame and the body frame is a necessary process. This paper describes the in-motion alignment procedure for a low-cost GPS/INS integrated system using the Unscented Kalman Filter (UKF) and Central Difference Kalman Filter (CDKF). The UKF and CDKF are used to implement the in-motion alignment based on the model of sensor biases and errors in position, velocity and attitude. The UKF and CDKF can tolerate large initial angle errors and do not require coarse alignment. Compared to the Extended Kalman Filter (EKF) methods, the UKF and CDKF give better approximations. The most obvious shortcomings of the EKF are, however, that it requires the computation of Jacobian matrices and the linear approximations of the EKF can be very inaccurate in some scenarios, leading to filter instability. In contrast, the proposed UKF and CDKF state distributions do not require Jacobian matrices, and instead use a set of weighted sample points to capture the true mean and covariance of the Gaussian Random Variable to 3 rd order accuracy. Simulation results with an in-motion MEMS-IMU/GPS integrated system based on loose-coupling show that the alignment can converge in a short time. The UKF and CDKF work well even with large initial attitude error values of 10°, 20°, and 30° for east, north and heading, respectively. Moreover, the UKF and CDKF can efficiently eliminate the sensor biases. In summary, the UKF and CDKF appear to be better approachs to handling large and small attitude errors of a SDINS/GPS integrated system compared to the conventional EKF. INTRODUCTION The alignment of a Strapdown Inertial Navigation System (SDINS) is an important process for determining the angular relationship between the navigation frame and the body frame. The navigation error of a SDINS increases with time. In the case of a stationary alignment process, the initial errors of a SDINS are usually estimated and compensated for. However, when the body frame is fixed with respect to the navigation frame, it leads to poor observability of the heading angle error. Therefore, correction of navigation errors during in-motion alignment is a necessary process and can be accomplished by integrating SDINS with non-inertial sensors, such as GPS. This problem has been extensively explored [1-4]. To resolve this problem, a Kalman Filter (KF) or Extended Kalman Filter (EKF) has traditionally been used, based on either linear or nonlinear error models. However the EKF is limited by the linearization performed on the model. To overcome this shortcoming, the so-called Unscented Kalman Filter (UKF) and Central Difference Kalman Filter (CDKF) have been proposed. They do not require the computation of Jacobian matrices, and instead use a set of weighted sample points to capture the true mean and covariance of the Gaussian Random Variable to 3 rd order accuracy. This paper investigates the problem of in-motion alignment of a SDINS/GPS integrated system using the UKF and CDKF. The main feature of the UKF or CDKF is that the unscented transformation (UT) is developed to overcome the deficiencies of linearization, by providing a more direct and explicit mechanism for transforming the mean and covariance information [5][6]. This method has been extensively studied and used in applications [7-9]. Nonlinear error equations have been derived for the inmotion alignment problem, which is simulated as a SDINS error model described by nonlinear equations and linear measurement. This paper describes the nonlinear error model using the UKF and CDKF so as to provide an improved solution to this problem. In [1,10,11], the integration was designed in the full state space. Such approach usually combines the strapdown computation with the KF’s time updating. As a result, the KF’s time-updating must have the same updating rate as the INS to keep the integrated system’s high output rate. Taking into account the time-consuming unscented transformations at the UKF’s time-updating procedure, the computation load of the integrated system would be high through this approach. The integration Kalman filter is designed in the error state space in this paper. Through this approach, the strapdown computation can be separated from the KF’s timeupdating procedure. The time-updating of the KF can be implemented at a relatively low updating rate i.e. the same rate as the GPS updating rate, meanwhile the standalone strapdown computation can be performed at a high updating rate. This approach therefore can reduce the overall computation load of the system. SDINS/GPS ERROR MODEL An INS can be classified into two classes, GINS (Gimballed Inertial Navigation System) and SDINS, based on its structure and mechanisation. In GINS, gyros and accelerometers are mounted on a mechanical platform. In a SDINS however, the mechanical platform is replaced by a mathematical process, executed in a computer, to implement the navigation. Hence, in a SDINS/GPS system, the error model contains the coordinate transformation matrix from a body frame (bframe) to a navigation frame (NED) (n-frame) and the GPS navigation result. In this paper, the SDINS/GPS state space model (mechanics equation) in the local level frame (L-frame) is given by: