Technical Note: CRT with Hypothesis Testing

GPS-INS Outlier Detection & Elimination using a Sliding Window Filter Paul F. Roysdon † Abstract— Many applications require reliable, high precision navigation (sub-meter accuracy) while using low-cost inertial and global navigation satellite systems (GNSS). Success requires optimal state estimate while mitigating measurement outliers. Common implementations use an Extended Kalman Filter (EKF) combined with the Receiver Autonomous Integrity Mon- itoring (RAIM) on a single epoch. However, if the linearization point of the EKF is incorrect or if the number of residuals is too low, outlier detection decisions may be incorrect. False alarms result in good information not being incorporated. Missed detections result in incorrect information being incorporated. Either case can cause subsequent incorrect decisions in the future, possibly causing divergence, due to the state and covariance now being incorrect. This article formulates a sliding window estimator that solves the full-nonlinear Maximum A Posteriori estimate in real-time. By leveraging the resulting window of residuals, an improved fault detection and removal strategy is implemented. Sensor data is used to demonstrate the interval RAIM (iRAIM) performance improvement. Jay A. Farrell ‡ Data redundancy is critical to successful outlier detection and removal and can be enhanced by considering all GNSS and inertial measurement unit (IMU) data within a sliding temporal window. The resulting full nonlinear Maximum A Posteriori (MAP) estimator, without outlier detection and re- moval, is presented in [6]. This article extends that approach with methods to detect and remove outlier measurements within the temporal interval and is therefore referred to as interval RAIM (iRAIM). Because this approach allows real-time analysis of numerous fault scenarios, with real- time error correction, outlier detection and removal can be improved. This estimator is demonstrated in using real-world data involving urban canyons and overhead foliage. II. B ACKGROUND AND N OTATION This section introduces Global Positioning System (GPS) aided inertial navigation system (INS) background [7]. I. I NTRODUCTION The past decade has seen the rapid rise and adoption of navigation systems on automobiles, unmanned vehicles, and personal mobile devices such as smartphones. These systems can exhibit very good accuracy (e.g. sub-meter error). However, further improvements in the reliability and continuity of this accuracy are required to fully support autonomous vehicle operations, especially in urban environ- ments, where variations in the operating conditions and direct signal path can have critical effects. To design a reliable, high-performance system, it is critical to detect and remove outlier measurements before they degrade performance. In GNSS applications such outlier measurements can be caused by multi-path, non-line of sight signals, or overhead foliage. RAIM is a set of techniques to cope with GNSS receiver outlier measurements, based on measurement residual gener- ation techniques equivalent to least-squares [1], [2]. Integrity is a measure of the trust that can be placed in the correctness of the information supplied by the total system. Often, RAIM is designed assuming only one outlier occurs and that there is enough measurement redundancy to detect and identify the source [1]. The principle of multiple outlier detection has also been well developed over several decades [3], [4]. The authors of [5] included an inertial measurement unit and a Kalman filter to “extend” the RAIM capabilities, a method called eRAIM. However, both RAIM and eRAIM are based on measurements from a single epoch, limiting data redundancy. Furthermore, the residual generation algorithm in RAIM and eRAIM assumes a linear system. † Ph.D. student, ‡ Professor at the Dept. of Electrical & Computer Engi- neering, UC Riverside. {proysdon, farrell}@ece.ucr.edu. A. Aided Inertial Navigation Let x ∈ R n s denote the rover state vector, where x(t) = [p | (t), v | (t), q | (t), b | a (t), b | g (t)] | ∈ R n s , where p, v, b a , b g each in R 3 represent the position, ve- locity, accelerometer bias and gyro bias vectors, respectively, q ∈ R 4 represents the attitude quaternion (n s = 16). The kinematic equations for the rover state are x(t) = f (x(t), u(t)), where f : R n s × R 6 7→ R n s represents the kinematics, and u ∈ R 6 is the vector of specific forces and angular rates. The function f is accurately known (see eqn. 11.31-11.33 in [7], and [8]). Nature integrates eqn. (1) to produce x(t). Let τ i denote the time instants at which IMU measure- ments are valid. Assume there is a prior for the initial state: x(t 0 ) ∼ N (x 0 , P 0 ). Given the initial condition x 0 and the ˜ (τ i ) = u(τ i )+b(τ i )+ω u (τ i ) of u(τ i ), IMU measurements u with additive stochastic errors ω u (τ i ) ∼ N (0, Qd) and b = [b | a , b | g ] | , a navigation system propagates an estimate of the vehicle state as the solution of ˆ ˙ (t) = f ( x(t), ˜ (t)), x u where x(t) denotes the real-time estimate of x(t). The solution of (2) over the interval t ∈ [τ i−1 , τ i ] from the initial condition x i−1 is represented as the operator: Z τ i φ(x i−1 , u i−1 ) = x i−1 + f (x(τ ), u(τ ))d τ (3) τ i−1