Per-Johan Nordlund

Particle filters for positioning, navigation, and tracking

A framework for positioning, navigation, and tracking problems using particle filters (sequential Monte Carlo methods) is developed. It consists of a class of motion models and a general nonlinear measurement equation in position. A general algorithm is presented, which is parsimonious with the particle dimension. It is based on marginalization, enabling a Kalman filter to estimate all position derivatives, and the particle filter becomes low dimensional. This is of utmost importance for high-performance real-time applications. Automotive and airborne applications illustrate numerically the advantage over classical Kalman filter-based algorithms. Here, the use of nonlinear models and non-Gaussian noise is the main explanation for the improvement in accuracy. More specifically, we describe how the technique of map matching is used to match an aircrafts elevation profile to a digital elevation map and a cars horizontal driven path to a street map. In both cases, real-time implementations are available, and tests have shown that the accuracy in both cases is comparable with satellite navigation (as GPS) but with higher integrity. Based on simulations, we also argue how the particle filter can be used for positioning based on cellular phone measurements, for integrated navigation in aircraft, and for target tracking in aircraft and cars. Finally, the particle filter enables a promising solution to the combined task of navigation and tracking, with possible application to airborne hunting and collision avoidance systems in cars.

Marginalized particle filters for mixed linear/nonlinear state-space models

The particle filter offers a general numerical tool to approximate the posterior density function for the state in nonlinear and non-Gaussian filtering problems. While the particle filter is fairly easy to implement and tune, its main drawback is that it is quite computer intensive, with the computational complexity increasing quickly with the state dimension. One remedy to this problem is to marginalize out the states appearing linearly in the dynamics. The result is that one Kalman filter is associated with each particle. The main contribution in this paper is the derivation of the details for the marginalized particle filter for a general nonlinear state-space model. Several important special cases occurring in typical signal processing applications will also be discussed. The marginalized particle filter is applied to an integrated navigation system for aircraft. It is demonstrated that the complete high-dimensional system can be based on a particle filter using marginalization for all but three states. Excellent performance on real flight data is reported.

Marginalized Particle Filter for Accurate and Reliable Terrain-Aided Navigation

This paper details an approach to the integration of INS (inertial navigation system) and TAP (terrain-aided positioning). The solution is characterized by a joint design of INS and TAP, meaning that the highly nonlinear TAP is not designed separately but jointly with the INS using one and the same filter. The applied filter extends the theory of the MPF (marginalized particle filter) given by. The key idea with MPF is to estimate the nonlinear part using the particle filter (PF), and the part which is linear, conditional upon the nonlinear part, is estimated using the Kalman filter. The extension lies in the possibility to deal with a third multimodal part, where the discrete mode variable is also estimated jointly with the linear and nonlinear parts. Conditionally upon the mode and the nonlinear part, the resulting subsystem is linear and estimated using the Kalman filter. Given the nonlinear motion equations which the INS uses to compute navigation data, the INS equations must be linearized for the MPF to work. A set of linearized equations is derived and the linearization errors are shown to be insignificant with respect to the final result. Simulations are performed and the result indicates near-optimal accuracy when compared with the Cramer-Rao lower bound.

Sequential Monte Carlo filtering techniques applied to integrated navigation systems

This paper addresses the problem of integrated aircraft navigation, more specifically how to integrate inertial navigation with terrain aided positioning. This is a highly nonlinear and non-Gaussian recursive state estimation problem which requires state of the art methods. We propose an algorithm based on the particle filter with particular attention to the complexity of the problem. The proposed algorithm takes advantage of linear and Gaussian structure within the system and solves these parts using the Kalman filter. The remaining parts suffering from severe nonlinear and/or non-Gaussian structure are solved using the particle filter. The proposed filter is applied to a simplified integrated navigation system. The result shows that very good performance is achieved for a tractable computational load.

Multicentre study of validity and interrater reliability of the modified Nursing Care Recording System (NCR11) for assessment of workload in the ICU.

Background:  Reliable assessment of nursing workload is necessary for the quantitative approach to staffing of intensive care units. The Nursing Care Recording System (NCR11) scores both the nursing contribution to patient care and those related to medical procedures. The purpose of the present work was to compare NCR11 scoring with the Therapeutic Intervention Scoring System (TISS) and Nine Equivalents of Nurse Manpower use Score (NEMS) and to examine the interrater reliability of NCR11 scoring.

Recursive estimation of three-dimensional aircraft position using terrain-aided positioning

As a part of aircraft navigation, three-dimensional position must be computed continuously. For accuracy and reliability reasons, several sensors are integrated together, and here we are dealing with dead-reckoning integrated with terrain-aided positioning. Terrain-aided positioning suffers from severe nonlinear structure, meaning that we have to solve a nonlinear recursive Bayesian estimation problem. This is not possible to do exactly, but recursive Monte Carlo methods, also known as particle filters, provide a promising approximate solution. To reduce the computational load of the normally rather computer intensive particle filter we present an algorithm which takes advantage of linear structure. The algorithm is based on a Rao-Blackwellisation technique, meaning that we marginalise the full conditional posterior density with respect to the linear part. The linear part of the state vector is estimated using multiple Kalman filters, and the particle filter is then used for the remaining part. Simulations show that the computational load is reduced significantly.

Conflict Detection Metrics for Aircraft Sense and Avoid Systems

The task of an airborne collision avoidance system is to continuously evaluate the risk of collision and in the case of too high risk initiate an evasive action. The traditional way to assess risk is to focus on a critical point of time. A recently proposed alternative is to evaluate the cumulated risk over time. It is the purpose of this contribution to evaluate the difference between these two concepts and also to validate an approximate method for computing the cumulated risk, suitable for real-time implementations. For this purpose, random scenarios are generated from stochastic models created from observed conflicts. A realistic tracking filter, based on angle-only measurements, is used to produce uncertain state estimates which are used for risk assessment. It is shown that the cumulated risk is much more robust to estimation accuracy than the maximum of the instantaneous risk. The intended application is for unmanned aerial vehicles to be used in civilian airspace, but a real mid-air collision scenario between two traffic aircraft is studied as well.

Probabilistic Noncooperative Near Mid-Air Collision Avoidance

We propose a probabilistic method to compute the near mid-air collision risk as a function of predicted flight trajectory. The computations use state estimate and covariance from a target tracking filter based on angle-only sensors such as digital video cameras. The majority of existing work is focused on risk estimation at a certain time instant. Here we derive an expression for the integrated risk over the critical time horizon. This is possible using probability for level-crossing, and the expression applies to a three-dimensional piecewise straight flight trajectory. The Monte Carlo technique provides a method to compute the probability, but a huge number of simulations is needed to get sufficient reliability for the small risks that the applications require. Instead we propose a method which through sound geometric and numerical approximations yields a solution suitable for real-time implementations. The algorithm is applied to realistic angle-only tracking data, and shows promising results when compared with the Monte Carlo solution.

Synthetic attitude and heading reference for Saab Gripen

In future versions of Saab Gripen, the mechanical artificial horizon will be replaced by a computer calculated attitude and heading, independent of the inertial navigation system (INS). The system uses data from sensors already existing in the aircraft, which are easily available in a highly integrated, 4th generation combat aircraft such as the Gripen. The sensor information used is a three-axis magnetic detector, true airspeed, angle of attack, barometric altitude, flight control rate gyros and load factor. The sensor data is fused together in an extended Kalman filter (EKF). Each sensor by itself is of relatively poor quality. For instance, the accuracy of the rate gyros is in the order of degrees per second, rather than degrees per hour as is the case in gyros dedicated for navigation use. However, when all data are combined, they provide an attitude and heading estimate with sufficient quality for its purpose; to cross-monitor the INS, and to serve as a backup in case the INS fails or data can not be displayed. The system is called synthetic attitude and heading reference system (SAHRS), and is a Saab patent. A similar system is developed and operational in the Saab Viggen.

The probability of near midair collisions using level-crossings

We consider probabilistic methods to compute the near midair collision risk using state estimate and covariance from a target tracking filter based on angle-only sensors such as digital video cameras. Existing work is only concerned with risk estimation at a certain time instant, while the focus here is to compute the integrated risk over the critical time horizon. This novel formulation leads to evaluating the probability for level-crossing. The analytic expression for this involves a multi-dimensional integral which is hardly tractable in practice. Further, a huge number of Monte Carlo simulations would be needed to get sufficient reliability for the small risks that the applications require. Instead, we propose a sound numerical approximation that leads to a one-dimensional integral which is suitable for real-time implementations.