Mathematical Problems in Engineering

Volume 2018, Article ID 3104397, 17 pages

https://doi.org/10.1155/2018/3104397

## Optimal Path-Following Guidance with Generalized Weighting Functions Based on Indirect Gauss Pseudospectral Method

^{1}School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China^{2}Xi’an Institute of Modern Control Technology, Xi’an 710065, China

Correspondence should be addressed to Qi Chen; nc.ude.tsujn@nehciq

Received 15 May 2018; Revised 16 July 2018; Accepted 25 July 2018; Published 9 August 2018

Academic Editor: Renato Vidoni

Copyright © 2018 Qi Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An indirect Gauss pseudospectral method based path-following guidance law is presented in this paper. A virtual target moving along the desired path with explicitly specified speed is introduced to formulate the guidance problem. By establishing a virtual target-fixed coordinate system, the path-following guidance is transformed into a terminal guidance with impact angle constraints, which is then solved by using indirect Gauss pseudospectral method. Meanwhile, the acceleration dynamics are modeled as the first-order lag to the command. Using the receding horizon technique a closed-loop guidance law, which considers generalized weighting functions (even discontinuous) of both the states and the control cost, is derived. The accuracy and effectiveness of the proposed guidance law are validated by numerical comparisons. A STM32 Nucleo board based on the ARM Cortex-M7 processor is used to evaluate the real-time computational performance of the proposed indirect Gauss pseudospectral method. Simulations for various types of desired paths are presented to show that the proposed guidance law has better performance when compared with the existing results for pure pursuit, a nonlinear guidance law, and trajectory shaping path-following guidance and provides more degrees of freedom in path-following guidance design applications.

#### 1. Introduction

During the last decade, there is a growing interest in unmanned aerial vehicles (UAVs) in both civilian and military applications like geological surveys, power line patrol, reconnaissance, etc. In most of these applications, the UAVs are usually required to follow a desired path accurately. The desired paths are commonly planned as straight lines or circular lines with specified constraints. To obtain a satisfactory path-following performance, a robust and efficient path-following guidance law is needed.

In recent years, a variety of path-following guidance laws have been developed for UAVs. Nelson et al. [1] and Lawrence et al. [2] proposed approaches based on a notion of vector field. The vector field-based path-following approach uses vector fields to represent the desired headings to drive the UAV onto the defined path. This approach has high robustness but is complicated in construction of the vector fields and also difficult in implementation. A waypoint-based path-following law was developed by Tsourdos et al. [3], in which a number of waypoints are selected on the desired path for the vehicle to pass through. However, the results based on this approach usually have low accuracy, and, specifically, a large path-following error may occur when the path is highly curved. Another waypoint-based guidance law was proposed by Liang et al. [4] for entry vehicle. In this study, the prescribed waypoints and the expected heading angle are imposed on the vehicle as additional constraints. The proposed guidance can successfully generate a lateral trajectory that satisfies the waypoint constraint and thus guarantee that the vehicle is able to reach all the waypoints. Yang et al. [5] studied the path tracking problem for a fixed-wing unmanned aircraft using the error-regulation philosophy, in which an adaptive nonlinear model predictive controller is designed to minimize both the mean and the maximum error between the reference trajectory and the UAV, and thus provides accurate tracking performance.

Virtual target-based approaches lead to another class of path-following guidance laws. The main objective of these approaches is to chase a virtual target point moving along the desired path, which is ahead of the UAV. The virtual target is initially placed at the beginning of the desired path. Once the vehicle starts to track the desired path, a virtual speed related to the vehicle’s speed and the separation between the vehicle and the virtual target is generated and imposed on the virtual target. By using the virtual speed and the curvature of the desired path, the states of the virtual target can be explicitly propagated along the desired path till the end of the engagement [6–8]. As a consequence, the position of the virtual target is always available during the entire guidance process. The line-of-sight guidance [9] and proportional navigation guidance [10] were used to drive the vehicle to chase the virtual target, which eventually drives the vehicle onto the path, and the same problem has been considered by Medagoda and Gibbens [11] using pure pursuit guidance. However, a heading error will be caused for curved paths because pure pursuit guidance consistently compels the vehicle to head toward the target. Therefore, a path-following error will occur. Further improvement to pure pursuit guidance was proposed by Cho et al. [12], in which differential geometry of space curves is used to extend the pure pursuit method for 3D path following. A nonlinear path-following guidance law adapted from pure pursuit based methods was proposed by Park et al. [13, 14]. This method is prominent due to its robustness of convergence for all initial geometries, the simple guidance command, and the so-called “look-ahead effect” which guarantees accurate following of curved paths. However, the lateral acceleration is undefined when the initial position of the vehicle is outside of the specified look-ahead distance from the desired path. Moreover, an overshoot response in the initial phase is another issue. Exploiting the concept of terminal missile guidance law with impact angle constraint, Ratnoo et al. [15] proposed a new path-following law based on trajectory shaping guidance. The advantages of this method are the fast rate of convergence, the negligible path-following error, and the strong robustness with respect to the minimum distance.

However, the research works mentioned above have not considered the autopilot delay, which cannot be ignored in practice, especially for the UAVs with low control authority. Most of the existing works assume that the actual lateral acceleration is the same as the command, which means that the acceleration response is instantaneous. However, this is not practical as it always takes some time to achieve the desired guidance command in practical system. This delay may degrade the overall path-following performance and even cause instability. To obtain satisfactory path-following performance, it is of necessity to take the autopilot delay into account. In addition, previous works mentioned above generally utilize simple guidance laws to chase the virtual target and do not take the weighting functions into account to improve the path-following guidance performance. As can be shown that through appropriate selections of the weighting functions, the vehicle’s trajectory and acceleration profile can be shaped as desired for achieving different path-following objectives. Furthermore, if the weighting functions can be chosen arbitrarily, then the flexibility of the path-following guidance design could be largely enhanced. Various weighting functions, such as constant function [16], Gaussian function [17], time-to-go function [18], exponential function [19], hyperbolic tangent function [20], and sinusoidal function [21], have been used to devise terminal guidance laws for different guidance objectives. These weighting functions have their corresponding advantages, for example, reducing sensitivity with respect to initial heading error, extending the operational margin to cope with the external disturbances in the terminal phase, or alleviating the acceleration command at the initial phase. Discontinuous functions that consist of the combination of the above-mentioned weighting functions give rise to attractive and prominent types of the weighting functions, in which different weighting factors are applied to weigh the states and/or controls in different guidance phases. Consequently, discontinuous weighting functions can take advantages of several weighting functions during the entire guidance process and thus significantly enhance the guidance performance. But, meanwhile, these discontinuous weighting functions are usually intractable for most of the guidance laws in existing literatures. However, the method proposed in this paper can easily cope with these discontinuous weighting functions. The results presented in this paper are the first attempts in the literature to derive the optimal path-following guidance with generalized weighting functions using the indirect Gauss pseudospectral method. This novel approach can handle complex weighting functions (even though they are discontinuous) which are intractable for most of the guidance laws in previous studies and thus provides more degrees of freedom in path-following guidance design applications. Similar to the use of terminal missile guidance law for path following in [15], a novel guidance logic with impact angle constraint considering generalized weighting functions as well as the autopilot delay is proposed in this paper to follow the virtual target on a planar path. Detailed numerical comparisons are presented to demonstrate the high path-following performance of the proposed guidance.

This paper is organized as follows. In Section 2, the path-following problem based on virtual target pursuit is formulated. In Section 3, an indirect Gauss pseudospectral method based approach is derived, and then a closed-loop path-following guidance law is proposed. The validation of the Gauss pseudospectral method based approach and the performance of the proposed guidance law are presented by numerical simulations in Section 4. Conclusions are given in Section 5.

#### 2. Problem Formulation

Consider the path-following guidance geometry shown in Figure 1. Here,* M* denotes a vehicle; the curve is the desired path;* T* is a virtual target moving along the path and governed by the curvature of the desired path. The vehicle pursues the virtual target and reduces the distance* R* to converge to the desired path.* V*_{m}, *γ*_{m}, and *γ*_{t} denote the vehicle velocity, vehicle heading angle, and the virtual target heading angle, respectively.* a*_{m} is the vehicle acceleration perpendicular to the velocity vector to change the heading angle *γ*_{m}. Motivated by [11, 15], the speed of the virtual target is chosen as a function of the vehicle speed* v*_{m} and the closing distance* R* as follows:where is a design parameter within the guidance algorithm and represents the minimum allowed separation between the vehicle and the virtual target, because the virtual target’s speed is inversely proportional to the closing distance. Therefore, the speed of the virtual target increases as the vehicle approaches the virtual target, which makes the vehicle always in pursuit of the virtual vehicle. This constraint links the dynamics of the vehicle and the virtual target and implies that the vehicle can never be closer to the virtual target than the minimum separation (e.g., ). The choice of and the guidance law of the vehicle affect the vehicle following performance. In order to achieve better following performance, the vehicle is expected to approach the virtual target in tail chase. To this end, the guidance law of the vehicle had better maintain a capacity of impact angle control (the expected impact angle is the virtual target heading angle *γ*_{t}). Trajectory shaping guidance law [16] is analyzed in [15] and is proven to be an effective logic for virtual target following on a planar path. However, when taking the autopilot dynamics into account, this method will have some limitation (for instance, oscillation and instability). Moreover, weighting functions are not considered in the previous studies. To eliminate this limitation and enhance the flexibility of the design of the path-following guidance, a new guidance law with impact angle constraint is proposed in this paper.