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Dynamical models for Terrain-Aided Navigation (TAN) with closed-form ground maps
We consider the scenario of a drone evolving in a 3D space without direct measurements of its 3D position. This scenario can occur after a GPS failure or simply because GPS devices are not used as they are easy to jam. It is assumed instead that the drone is equipped with a radar-altimeter. A radar-altimeter gives a measurement of the vertical distance between the drone and the ground. It is also assumed that the terrain flied over by the drone is known. The issue of the following is to determine if these two pieces of information are sufficient to reconstruct the horizontal and the vertical position of the drone. This problem is called Terrain-Aided Navigation and is represented in Figure 2.1. Figure 2.2 represents the test map that has been used throughout the thesis. For the sake of simplicity, the drone is represented by a point in 3 dimensions and we suppose that either its 3D speed or its 3D acceleration is controlled. In any case, the inertial speed i.e. the speed in the inertial frame is supposed to be known. It is a substantial simplification as in many applications the inertial speed is not measured but the speed in the body frame is. Pose estimation is then generally
needed to reconstruct the inertial speed. However, one can imagine that an external estimation loop is run such that the inertial speed is reconstructed precisely. Occasionally, we will assume that an additional measurement of altitude is available mainly to simplify the design of some observers. It is not a restrictive assumption as baro-altimeters are very common in aircrafts. We make the previous simplifying assumptions because we want to focus on the main difficulty of the problem of TAN which is the nature of the available information on the position. Actually, the available information depends on the nature of the area that is flied over by the drone and consequently on the input. For example, let us assume that the drone flies over an ambiguous area, like a flat or a periodic one, with constant altitude then one measurement of height matches a whole horizontal area. The resulting estimation error on the horizontal position is of the order of magnitude of the size of the area which can be very large. On the contrary, if the drone flies over a rough terrain, then one measurement of height corresponds to a smaller area on the ground and the estimation error is reduced. Intuitively, the more perturbed the ground flied over is, the more information it contains. The idea of the following is to verify this intuition by studying the dynamical model of the drone with different types of maps.
Local weak observability of Terrain-Aided Navigation models
In this section, we fix a position X, a speed V and an acceleration U. Depending on the dynamics considered in the following, X is a point in the statespace and V a constant input or (X; V ) is a point in the statespace and U a constant input. To check if a model is locally weakly observable, we evaluate the observability rank condition. To do so, The idea is to compute all the functions from the observation spaceOU(h) (orOV (h)) by successively applying the Lie derivative on the component of the observation function.
Nonlinear observer design by Immersion and Invariance (I&I) for Terrain-Aided Navigation
Generally, an observer is defined as a dynamical system designed to asymptotically approach the unknown state of the original system. The main difficulty is that it must be constructed using only the available information.There is no explicit form of observers that handles any system so it must be adapted to the particular class of systems treated. The most popular nonlinear observer are the nonlinear Luenberger and Kalman observers. They are especially well suited for a large class of nonlinear systems that still exhibit some linear structure (see [Besançon, 2007]). Another family of nonlinear observers are high-gain observers. They are based on the idea of compensating the nonlinearity buy using a high gain on the innovation term. This also requires in general that the original system contains a linear detectable part (see [Khalil and Praly, 2014]). In the sequel, we prefer to use a more general method, called Immersion and Invariance (I&I), that is known to be able to deal with very nonlinear systems. Besides, it gives one many degrees of freedom to choose the estimation error equation. In this section, we first recall the general concepts of I&I applied to observer design. Then, we present the design of observers for the systems with a quadratic map, a cubic map, a Gaussian map and a trigonometric map.
Table of contents :
1 Introduction
I Terrain-Aided Navigation with analytical ground maps
2 Terrain models and observability conditions
2.1 General nonlinear observability conditions
2.1.1 General nonlinear controlled system
2.1.2 General definitions of observability
2.2 Observability properties in Terrain-Aided Navigation
2.2.1 Dynamical models for Terrain-AidedNavigation (TAN) with closed-formground maps
2.2.2 Local weak observability of Terrain-Aided Navigation models
3 Observer and controller design for TAN
3.1 Nonlinear observer design by I&I for TAN
3.1.1 Description of I&I for the design of observers of a general nonlinear system
3.1.2 Application of I&I design to Terrain-Aided Navigation
3.2 Persistent output-feedback control and stochastic excitation
3.2.1 Output-feedback control
3.2.2 An example of a piecewise constant almost surely persistent processes
II Terrain-Aided Navigation with real ground maps
4 Elements of stochastic control and stochastic filtering
4.1 Discrete-time Stochastic Optimal Control
4.1.1 Deterministic case
4.1.2 Stochastic case with perfect information
4.1.3 Stochastic case with imperfect information
4.2 State Estimation, Nonlinear Filtering and Particle Filters
4.2.1 Optimal estimation
4.2.2 Linear Kalman filtering
4.2.3 Nonlinear filtering
4.3 Design of suboptimal policies
4.3.1 Dual effect and classification of suboptimal policies
4.3.2 Model Predictive Control
5 Modelling of joint problem of control and estimation
5.1 Definition of the coupling optimisation problem
5.1.1 Setup of the problem of optimal control and estimation
5.1.2 Dynamic programming principle
5.2 Near-optimal nonlinear filtering
5.2.1 Statement of the problem
5.2.2 Error bounds between the optimal MSE and the empirical MSE
5.3 Stochastic optimal control with an estimation-based cost
6 Dual Particle Fisher SMPC and its application to TAN
6.1 Design of the Dual Particle Fisher SMPC schemes
6.1.1 Implementation of the guiding objective in Explicit Dual SMPC
6.1.2 Penalising Fisher Dual Particle SMPC scheme
6.1.3 Lyapunov/Fisher Dual Particle SMPC scheme
6.2 Modelling of TAN with a real ground map
6.3 Numerical simulations
7 Conclusion
