Systems & Control Letters 60 (2011) 101–110 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle On the observability of linear motion quantities in navigation systems Pedro Batista ∗ , Carlos Silvestre, Paulo Oliveira Instituto Superior Técnico, Institute for Systems and Robotics Av. Rovisco Pais, 1049-001 Lisboa, Portugal article info Article history: Received 24 November 2008 Received in revised form 14 September 2010 Accepted 6 November 2010 Available online 10 December 2010 Keywords: Linear time-varying systems Observability 3-D linear motion kinematics Navigation systems abstract Navigation systems are a key element in a large variety of mobile platforms, where the correct knowledge of their position and attitude is essential in most applications. This paper focuses on the observability of linear motion quantities (position, linear velocity, linear acceleration, and accelerometer bias). It presents necessary and sufficient conditions, with a clear physical insight, for the observability of these variables in 3-D. The analysis provided is based on kinematic models, which are exact and intrinsic to the motion of a rigid-body, and different cases are presented depending on the assumptions made on the sensor suite that is available on-board. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The design of Integrated Navigation Systems arises naturally in the development of a large variety of mobile platforms, whether manned or unmanned, autonomous or human-operated, as the knowledge of the position and attitude of the vehicle is a basic requirement for its successful operation. Moreover, for control purposes, other quantities such as the linear and angular velocities are also often required. Dead-reckoning navigation systems such as Inertial Navigation Systems (INS) provide all these quantities. However, the estimation of the position and attitude of the vehicle is necessarily obtained in this type of system by integrating higher-order derivatives such as the linear acceleration and the angular velocity, which are measured using, e.g., an Inertial Measurement Unit (IMU). As such, and regardless of the accuracy and precision of the IMU, the errors in the position and attitude estimates grow unbounded due to non- idealities such as noise and bias that affect the IMU’s readings [1]. These intrinsic limitations of dead-reckoning navigation systems are usually tackled by using aiding devices such as position and attitude sensors, e.g., the popular Global Positioning System (GPS), inclinometers, and magnetometers. However, even with the inclusion of aiding devices, not all states are always observable, in particular, if biases are considered and the acceleration of gravity is not known with enough accuracy. This paper investigates the observability of linear motion quantities of mobile platforms. ∗ Corresponding author. E-mail addresses: pbatista@isr.ist.utl.pt (P. Batista), cjs@isr.ist.utl.pt (C. Silvestre), pjcro@isr.ist.utl.pt (P. Oliveira). Previous work on the study of observability of navigation systems can be found in the literature. In [2] the observability of INS during initial alignment and calibration at rest is analyzed. The nominal nonlinear navigation system dynamics are perturbed yielding linearized error dynamics which are then shown not to be completely observable. In [3] the observability of a linearized INS error model is also examined for a stationary vehicle and it is reported, among other results concerning the leveling errors, that the unobservable states, which are distributed in two decoupled subspaces, can be systematically determined. In-flight alignment of INS is studied in [4] where it is shown that its observability can be improved by adequate maneuvering. In [5] sufficient conditions for the observability of stationary Strapdown Inertial Navigation Systems (SDINS) are analytically derived. In [6] an observability analysis of a GPS/INS system during two types of maneuvers, linear acceleration and steady turn, is presented. The analysis is based on a perturbation model of the INS and it is shown that the observability is improved when the vehicle maneuvers. Observability properties of the errors in an integrated navigation system are studied in [7], where the authors show that acceleration changes improve the estimates of attitude and rate-gyro bias and changes of the angular velocity enhance the lever arm estimate. However, no theoretical results for non-trivial trajectories are given and only simulation results are provided, which confirms that the degree of observability of the system increases with the richness of the trajectories described by the vehicle. To the best of the authors knowledge, in the literature only local observability results are known, most of them obtained in the context of navigation systems designed around the Extended Kalman Filter (EKF). These results, that reflect the continued adoption of EKF techniques to solve the Navigation problem, are very intuitive and were fundamental to motivate the need for the analysis presented 0167-6911/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2010.11.002