Energy conservation based fuzzy tracking for unmanned aerial vehicle missions under a priori known wind information

Abstract

The aim of this work is to include the navigation step for the waypoint-based guidance of a UAV system and to illustrate aspects such as tracking of the reference trajectory under wind presence, while conserving total energy requirements. The mission is represented utilising graph theory tools. The mathematical modelling of an aircraft controlled by an actuator surface is presented in terms of simple analytic relationships in order to simulate the actual horizontal motion of the vehicle. Its equivalence with a Tagaki–Sugeno (T-S) fuzzy system is illustrated that can aid the control methodology involved. Additionally, the advantages of utilising such an analysis is also stressed. The model formulated is an error posture model, that depends on current and reference posture. The control law is designed through parallel distributed compensation (PDC) and the gains are computed with the help of linear matrix inequalities (LMIs). Hence stability for the system is also guaranteed provided that the state variables are bounded in a priori known compact space. Moreover the energy requirements are described.

This article is contributing towards energy enhancing a UAV mission and generating safely-flyable trajectories to meet mission objectives. The control law used is calculated in the pre-flight planning and can be used in real time for any trajectory to be tracked under any environmental conditions. Provided that angular and linear velocities are bounded, the latter is feasible under the assumption that the magnitude of air speed is small compared to the ground velocity of the aerial vehicle. The methodology offers an improved visualisation to aid an analyst with the representation of a UAV mission through graph theory tools utilising energy requirements for the mission and fast computational schema using matrix analysis.

A simulation example of a UAV deployed from a source to reach a destination node under windy conditions is included to illustrate the analysis. The reference trajectory used is a piecewise Bezier–Bernstein curve referred to as the Dubins path.

Introduction

On board guidance and control systems are used to monitor flight performance. Apart from the time of arrival and endurance of flight they are also utilised to maintain updated fuel reserves. Presently, overall flight planning is performed on ground dispatch computers. However, it is crucial to be able to estimate, real-time, the fuel profile on board in order to increase automation and performance of the preflight planning when disturbances occur. In other words, wasting fuel due to unforecast disturbances can degrade the performance of the preflight planning mechanism. At the same end, if the time of arrival is available by a particular strategy then a fuel optimal trajectory can be possibly generated on-line with manipulation of flight parameters like speed, propulsion magnitude, angle, etc.

Existing research till recently, focused heavily on methods which improve the path planning algorithms. In particular, the optimum paths based on minimum distance or time travelled has been addressed from researches in Richards and How (2002) and Rathinam and Sengupta (2004a), among other interesting works. In practice, the optimum distance path solutions do not take into account the energy constraints. Without the energy utilisation the selected path may not be feasible or realisable. In other words, when disturbances occur in a path, such as an unforecast unfavorable wind, then the shortest distance path may not guarantee an energy optimum one. The latter was shown in work (Economou et al., 2007b) where the authors decoupled the problem of a shortest path from the Euclidean perspective and the energy one. In addition they proposed a methodology utilising energy graphs and pseudo-boolean expressions to describe the overall energy requirements. Consequently motivated by such an analysis there is a need to investigate the problem of navigation from the energy perspective rather than the shortest distance one.

Apart from the Navigation phase the pre-flight planning involves also the Guidance phase. In fact if energy routes are of interest the problem becomes threefold. Firstly, a predefined trajectory needs to be followed, the ground speed of the vehicle should be regulated and decision making should be performed in order to find a minimum sense path to follow that will fulfil mission objectives. Thus energy requirements should be described and then the path of optimum sense to reach a goal should be calculated. In the proposed analysis a graph theory formulation of the problem, introduced in Economou et al., 2007a, Economou et al., 2007b and Kladis et al. (2007), is combined with the physical and functional limitations of the vehicle with a focus on energy conservation of the involved flight path.

The tracking problem remains even nowadays a challenging area. Depending on the application domain, an adequate control law needs to be designed in order to fulfil mission objectives. Among many interesting approaches solving the tracking problem are those of Fliess et al. (1995) and Laumond (1998). Those are focused on the motion planning problem. However, their limitation lies in asymptotic tracking when the ground velocity of the vehicle becomes zero. Moreover, other works by Kanayama et al. (1991) and Klančar and Skrjanc (2007) suggested the combination of state feed-forward and feed-back control actions. The stability of the system was proved through a Lyapunov function and the gains were computed by linearisation of the tracking-error model to zero. Although there is potential in such approaches a serious drawback is with respect to their performance in real-time. The latter occurs due to the fact that the gains calculated that stabilise the system need to be updated and recalculated when the trajectory to be tracked is altered due to an event while in flight.

The approach proposed in this article, which is coupled with the routing problem, is based on the equivalent Takagi–Sugeno fuzzy model to the nonlinear error posture model of the vehicle's kinematics, introduced in earlier work (Tanaka et al., 1996). The control law used (parallel distributed compensation, PDC, Sugeno and Kang, 1986 and later improved by Wang et al., 1995) is computed utilising liner matrix inequality (LMI) techniques (Tanaka and Wang, 2001). The methodology proposed can be implemented for any type of reference trajectory provided that linear and angular velocities are bounded and $\left|V_w\right|⪡\left|V_g\right|$. $\left|V_w\right|$, $\left|V_g\right|$ are the magnitude of the wind and ground velocity, respectively. The latter can be interpreted with respect to the physical and functional limitations of the vehicle. The trajectory used in this work is the Dubins (1957) path. Additionally, disturbance rejection conditions are included in order to accommodate real world conditions. Stability is proven for any initial condition of the system for a priori known compact space of the state variables.

This article is contributing towards energy-enhancing a UAV mission and generating safely-flyable trajectories to meet mission objectives. The control law used is calculated in the pre-flight planning and can be used in real time for any trajectory. Additionally, it offers an improved visualisation to aid an analyst with the representation of a UAV mission through graph theory tools utilising energy requirements for the mission and fast computational schema using matrix analysis.

In this work preliminaries are addressed in Section 2. Those involve the graph theory tools. The vehicle's kinematics, the trajectory to be used, the model of environmental conditions and the description of the energy requirements are also included. Thereafter, in Section 3 the error posture model used and its equivalence to the Takagi–Sugeno (T–S) model are described. Additionally, the architecture of the controller and LMI conditions that stabilise the system are also included. Finally through a simulation example the analysis is illustrated in Section 4.2. In Section 5 concluding remarks are stated. Lastly an Appendix is added that includes the gains calculated for the control law for the T–S fuzzy model.