A robust integration of GPS and MEMS-INS through trajectory-constrained adaptive Kalman filtering

Micro-Electro-Mechanical Sensor (MEMS) inertial navigation sensors (INS) and GPS are attractive technologies for low-cost navigation applications. In the loosely-coupled integration schemes, GPS provides the position and velocity for real-time correction of the inertial sensor errors as well as the navigation parameters of the MEMS-INS. As an efficient estimation algorithm, the Kalman filter (KF) is widely used in integrated navigation systems. The traditional KF can produce optimal estimates under the assumption that both observation errors and prediction errors are zero-mean and independent of each other. Unfortunately, this is not true in some scenarios, such as severe multipath environments or urban canyon areas. In such cases, adaptive factors can be applied to suppress the influence of the abnormal states. These adaptive factors are usually constructed under the assumption that the present information should be reliable and accurate. However this assumption is not always true because an operational system often has unexpected errors due to either state or observation models. Based on a Conventional Kalman filter (CKF) for the loosely-coupled integration of GPS and MEMS-INS, this paper introduces an efficient Adaptive Kalman filter (AKF) to control the influence of the state model errors. Since the efficiency of the AKF strongly depends on the precision of the GPS solution of the present epoch, a trajectory-constrained algorithm is used to improve the GPS solution’s reliability and accuracy. The basic idea is to use the information of several previous epochs to predict the solution at the present epoch, which is further fused with the present GPS output to generate a reliable GPS solution at the present epoch. The research focuses on the establishment of the trajectory model using a discrete time movingwindow implementation. This paper introduces the Newton Forward Extrapolation (NFE) model, which is comparatively simple and suitable for constraining the trajectories of moving platforms, thus improving the reliability and accuracy of GPS position and velocity solutions. A GPS/MEMS-INS vehicle experiment was conducted in order to study the performance and efficiency of the proposed algorithms. Three algorithms were tested: CKF, AKF without trajectory constraint, and AKF with trajectory constraint (TCAKF). The results show that: i) the CKF has the worst performance since it is influenced by either state model errors of the MEMSINS or GPS observation errors; ii) the AKF can control the influence of abnormal state model errors, however its precision is dramatically decreased when GPS observations are contaminated by outliers, and iii) TCAKF can control both the state model and the observation model errors. The TCAKF can therefore be used for loosely-coupled GPS/MEMS-INS integration. INTRODUCTION GPS can provide real-time precise position and velocity information suitable for most land, marine, and aircraft navigation applications [1]. However, it still faces challenges in ensuring continuity of service in urban areas due to, e.g., blockages of GPS signals resulting in insufficient measurements, which would limit GPS’s ability to deliver the required level of availability, accuracy and reliability of positioning due to poor GDOP, low carrier-to-noise density and multipath (including reflection and diffraction) [2]. Therefore, the integration of GPS with other complementary navigation technologies, such as INS, is undertaken mainly to overcome the limitations of each navigation sensor and improve the overall (integrated) system performance. The most attractive advantage of integrated GPS/INS is that it is complementary with respect to the error characteristics of each sensor. INS may have a small error over the short term but its error grows significantly with time, while GPS has a stable accuracy which is almost uncorrelated with time. Since MEMS inertial sensors are compact and inexpensive, they are attractive technologies for integrating with GPS in either looselyor tightly-coupled integration schemes for many vehicle navigation applications. However, there are still many problems in using MEMS-INS compared with conventional INS sensors, e.g., scale factor nonlinearities, misalignment, high noise and temperature-dependent biases. Regular calibration using external aiding sources is essential to control the growth of INS errors [3]. These problems may bias the estimated INS errors, thus compromising system accuracy and integrity [4]. There are three ways to enhance the accuracy and reliability: i) use an auxiliary sensor system, such as an odometer or magnetometer [5]. Obviously, those external sensors increase the cost and complexity of the system. ii) Apply the non-holonomic constraint to the GPS/INS integration by the mechanical limits on the steering angle. Two kinds of constraint measurements – velocity constraints and height constraints can be introduced into INS navigation computations. This can provide reliable solutions for longer periods in the absence of GPS. iii) Another approach for preventing the errors from growing under adverse GPS conditions is the use of backward smoothing techniques, e.g. the Rauch-Tung-Striebel (RTS) smoother [6]. However, it is an offline processing algorithm and very difficult to implement in real-time GPS/INS navigation. The Kalman filter (KF) is widely used for performing the fusion solution in GPS/INS integrated systems. The KF operation relies on the correct definition of a dynamic model and a stochastic model [7]. The dynamic model describes the propagation of system states over time. The stochastic model describes the stochastic properties of the system process noise and observation errors. Optimal solutions are obtained when both observation errors and prediction errors are assumed to be zero-mean and independent of each other. Unfortunately this is not true in some scenarios, such as severe multipath environments or urban canyon areas. Unknown modelled errors in either the state or observation model may result in a sub-optimal solution even divergence of the filter. The Fading KF was proposed to overcome the shortcomings of conventional Kalman filtering [8-10]. Accordingly, without the need to identify the process errors and state errors, adaptive Kalman filters have been developed to estimate the optimal adaptive factors by innovation estimation in order to make the filter residuals covariance consistent with their theoretical covariance estimated with present information and residuals [11-14]. Popular adaptive methods can be roughly classified into state covariance scaling, multi-model adaptive estimation and adaptive stochastic modelling [15-16]. In such cases adaptive factors can be applied to suppress the influence of the abnormal states. In general, adaptive factors are constructed under the assumption that the present information is reliable and accurate, which is not always true for real-time navigation applications. This paper introduces an adaptive Kalman filter (AKF) to control the influence of the state model errors. Since the efficiency of the AKF strongly depends on the precision of the GPS solution for the present epoch, a trajectory-constrained KF (TCAKF) algorithm is proposed in order to improve the GPS solution’s reliability and accuracy. This paper is organised as follows: firstly the state and observation models for loosely-coupled GPS/INS integration are presented; the KF, AKF and TCAKF estimation algorithms are described; and a trajectoryconstraint method for the TCAKF to enhance GPS solutions is proposed; finally, the three algorithms are compared using experimental data. STATE AND OBSERVATION MODELLING The most common, and simplest, GPS/INS integration scheme is referred to as loosely-coupled integration, in which the integration KF uses the GPS position and velocity to estimate the inertial sensor errors as well as the navigation parameters of the INS. Both the state and the observation models for the integration KF are established below. STATE EQUATION The integration KF has a 15 dimension error state vector consisting of P (position), V (velocity) and A (attitude), gyro drift and accelerator bias in three components. For convenience in expression and calculation, an ECEF (Earth Centered Earth Fixed) coordinate frame is employed. The following need to be defined: i) Differential error equations for the position vector [ ] e X Y Z δ δ δ δ = r : e e δ δ = r V & (1) where the superscript e denotes the ECEF frame, simplified as e-frame hereafter. ii) Differential error equations for the velocity vector X Y Z T e e e e V V V δ δ δ δ ⎡ ⎤ = ⎣ ⎦ V : 2 e e e e e e e e ie b δ δ δ δ = − + − + V F θ N r Ω V R b & (2) where 2