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 yield a solution suitable for real-time implementations. The algorithm is applied to realistic angle-only tracking data, and shows promising results when compared to the Monte Carlo solution.

We consider probabilistic methods for detecting conflicts as a function of predicted trajectory. A conflict is an event representing collision or imminent collision between vehicles or objects. The computations use state estimate and covariance from a target tracking filter based on sensor readings. Existing work is primarily 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 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 often require. Instead, we propose a sound numerical approximation that leads to evaluating a one-dimensional integral which is suitable for real-time implementations.

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. Methods: Bias and precision of workload scores (NCR11 vs. TISS or NEMS) were assessed for 6126 consecutive admissions (23910 ICU-days) at three intensive care units. Inter-rater reliability was analyzed by having nurses at nine ICUs score workload using NCR11 for three dummy intensive care patient cases presented over a 3-year period. Variability in scoring was analyzed using the coefficient of variation. Results: Agreement between NCR11 and TISS or NEMS was poor and limits of agreement were wide. Linear relationships between NCR11 and TISS or NEMS scores differed between units. Variability in NCR11 scoring decreased significantly from 10.4% to 5.9% between dummy cases 1 and 2 and remained low for patient case 3. Conclusion: The NCR11 does not measure the same elements of workload in the ICU as do TISS and NEMS. Inter-rater reliability with NCR11 is good, showing little variation in scoring between nurses.

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.

Positioning of road vehicles in wireless radio networks is a highly non-linear multi-sensor problem. The radio measurements deliver snapshot information of position as circle, radial and hyperbolic lines with various reliability, and possibly also complex power attenuation maps. Further, fading and absence of line of sight give complicated disturbances on these measurements. Temporal and spatial prior knowledge may include maximal velocity and acceleration of the mobile unit, and for automotive applications also the constraint that most of the time is spent on roads, which are stored in a digital map. We outline a framework where all this information can be incorporated, and the true a posteriori distribution of position can be approximated with arbitrary accuracy to be traded o with real-time requirements. The algorithm is based on the particle filter, and we demonstrate it on a few simulation examples.

Particle filters (sequential Monte Carlo methods) handle recursive state estimation of arbitrary systems. However, a direct application shows that often a vast number of particle is needed for the filter to work well. This paper developes a combined particle and Kalman filter for recursive state estimation of nonlinear systems, where the state vector can be partitioned into one linear and Gaussian part and one nonlinear and/or non-Gaussian. The linear and Gaussian part is estimated using the Kalman filter and there maining part is estimated using the particle filter. Based on Bayes rule the solutions from the two filters are blended together. This two-filter approach is compared to a standard particle filter by applying both filters to a simple navigation problem. The result shows that using the combined filter the number of particles needed to achieve the same performance as the particle filter is much less.

In future versions of Saab Gripen, the mechanical artifical 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 sensors used are a three-axis magnetic detector, true airspeed, angle of attack, barometric altitude, flight control rate gyros and load factor. The sensor data are 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 navigational 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 of an INS or data bus failure. The system is called synthetic attitude and heading reference system (SAHRS), and is a Saab patent. A similar system is developed for and is operational in the Saab Viggen.

The overall purpose with this thesis is to investigate and provide computationally efficient methods for estimation and detection. The focus is on airborne applications, and we seek estimation and detection methods which are accurate and reliable yet effective with respect to computational load. In particular, the methods shall be optimized for terrain-aided navigation andcollision avoidance respectively. The estimation part focuses on particle filtering and the in general much more efficient marginalized particle filter. The detection part focuses on finding efficient methods for evaluating the probability of extreme values. This is achieved by considering the, in general, much easier task to compute the probability of level-crossings.

The concept of aircraft navigation using terrain height information is attractive because of the independence of external information sources. Typicallyterrain-aided navigation consists of an inertial navigation unit supported by position estimates from a terrain-aided positioning (TAP) system. TAP integrated with an inertial navigation system is challenging due to its highly nonlinear nature. Today, the particle filter is an accepted method for estimation of more or less nonlinear systems. At least when the requirements on computational load are not rigorous. In many on-line processing applications the requirements are such that they prevent the use of theparticle filter. We need more efficient estimation methods to overcome this issue, and the marginalized particle filter constitutes a possible solution. The basic principle for the marginalized particle filter is to utilize linear and discrete substructures within the overall nonlinear system. These substructures are used for efficient estimation by applying optimal filters such as the Kalman filter. The computationally demanding particle filter can then be concentrated on a smaller part of the estimation problem.

The concept of an aircraft collision avoidance system is to assist or ultimately replace the pilot in order to to minimize the resulting collision risk. Detection is needed in aircraft collision avoidance because of the stochastic nature of thesensor readings, here we use information from video cameras. Conflict is declared if the minimum distance between two aircraft is less than a level. The level is given by the radius of a safety sphere surrounding the aircraft.We use the fact that the probability of conflict, for the process studied here, is identical to the probability for a down-crossing of the surface of the sphere. In general, it is easier to compute the probability of down-crossings compared to extremes. The Monte Carlo method provides a way forward to compute the probability of conflict. However, to provide a computationally tractable solution we approximate the crossing of the safety sphere with the crossing of a circular disc. The approximate method yields a result which is as accurate as the Monte Carlo method but the computational load is decreased significantly.

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 yield a solution suitable for real-time implementations. The algorithm is applied to realistic angle-only tracking data, and shows promising results when compared to the Monte Carlo solution.

We consider probabilistic methods for detecting conflicts as a function of predicted trajectory. A conflict is an event representing collision or imminent collision between vehicles or objects. The computations use state estimate and covariance from a target tracking filter based on sensor readings. Existing work is primarily 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 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 often require. Instead, we propose a sound numerical approximation that leads to evaluating a one-dimensional integral which is suitable for real-time implementations.

The recently developed particle filter offers a general numerical tool to approximate the state a posteriori density in nonlinear and non-Gaussian filtering problems with arbitrary accuracy. Because the particle filter is fairly easy to implement and tune, it has quickly become a popular tool in signal processing applications. Its main drawback is that it is quite computer intensive. For a given filtering accuracy, the computational complexity increases quickly with the state dimension. One remedy to this problem is what in statistics is called Rao-Blackwellization, where states appearing linearly in the dynamics are marginalized out. This leads to that a Kalman filter is attached to each particle. Our main contribution here is to sort out when marginalization is possible for state space models, and to point out the implications in some typical signal processing applications. The methodology and impact in practice is illustrated on terrain navigation for aircrafts. The marginalized particle filter for a state-space model with nine states is evaluated on real aircraft data, and the result is that very good accuracy is achieved with quite reasonable complexity.

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 non-linear 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 non-linear 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 aircraft's elevation profile to a digital elevation map, and a car's 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 to 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.