Hybrid gradient vector fields for path-following guidance

Yi-yng Zho , Zhen Yng ,b,*, Wei-ren Kong , Hi-yin Pio , Ji-hun Hung ,Xio-feng Lv , De-yun Zhou

a School of Electronics and Information, Northwestern Polytechnical University, Xi’An, 710072, China

b School of Automation Science and Electrical Engineering, Beihang University, Beijing,100191, China

c The 93147th Unit of Chinese PLA Air Force, Chengdu, 610091, China

Keywords:Unmanned aerial vehicle (UAV)Path-following guidance (PFG)Hybrid gradient vector field (HGVF)Switching strategy

ABSTRACT Guidance path-planning and following are two core technologies used for controlling un-manned aerial vehicles(UAVs)in both military and civilian applications.However,only a few approaches treat both the technologies simultaneously.In this study,an innovative hybrid gradient vector fields for path-following guidance(HGVFs-PFG)algorithm is proposed to control fixed-wing UAVs to follow a generated guidance path and oriented target curves in three-dimensional space, which can be any combination of straight lines, arcs, and helixes as motion primitives.The algorithm aids the creation of vector fields (VFs) for these motion primitives as well as the design of an effective switching strategy to ensure that only one VF is activated at any time to ensure that the complex paths are followed completely.The strategies designed in earlier studies have flaws that prevent the UAV from following arcs that make its turning angle too large.The proposed switching strategy solves this problem by introducing the concept of the virtual way-points.Finally, the performance of the HGVFs-PFG algorithm is verified using a reducedorder autopilot and four representative simulation scenarios.The simulation considers the constraints of the aircraft, and its results indicate that the algorithm performs well in following both lateral and longitudinal control,particularly for curved paths.In general,the proposed technical method is practical and competitive.

1.Introduction

Fixed-wing unmanned aerial vehicles (UAVs) are becoming increasingly popular in research and other applications such as reconnaissance,surveillance,search,and rescue.A vast majority of military and civilian missions usually require UAVs to fly to the mission area and follow given paths autonomously,which requires the UAVs to employ the critical technology of path-following guidance (PFG) [1].

Despite extensive research efforts, path planning and control design are still recurrent problems when handling UAVs.In an obstacle-free environment, the flight path, which usually consists of some discrete waypoints that are wirelessly relayed to the UAV’s autopilot for low-level control,can be generated algorithmically or artificially by the ground station[2,3].The PFG algorithm tuned to a UAV can guide it to the mission area and make it follow these waypoints.In general,a UAV following a path consisting of straight lines and arcs is sufficient for most missions.However, certain applications require that the curved paths, including landing, formation flight, and low-altitude penetration, be followed precisely so that the PFG algorithm is able to exploit the full range of the UAV’s turning and maneuvering capabilities.

The application of the vector field (VF) method in aircraft motion control has become a popular research topic in recent years.It is used to solve the problems of standoff tracking [4,5], obstacle avoidance[2,6,7],formation flight[8,9],landing[10,11],and so on.The VF method is gradually becoming a popular translation framework for autonomous systems owing to its ease of implementation and stable performance, with the ability to translate high-level specifications into low-level descriptions of movement.The potential of the VF method is that the system can be guided to achieve the desired behavior through appropriate design of the guidance equations while guaranteeing provable stability and convergence.Using the Lyapunov stability criterion, Nelson et al.[12]reported that controlling the heading rate of the vehicle using VF allows it to converge asymptotically to a predetermined path and allows stable path-following under high-wind conditions.

PFG can be achieved through geometric or control methods[13].Typical geometric methods include line-of-sight(LOS)[14-17]and nonlinear path-following guidance (NPFG) [18] algorithms.These methods use a concept similar to that of virtual distance to obtain a point on the target curves that is continuously updated, and the vehicle is made to follow it.The stability of the methods depends heavily on the choice of the virtual distance parameter.Chen et al.[17] proposed a guidance method with impact time and angle constraints using a range-based LOS shaping strategy.The guidance laws are designed based on LOS shaping to meet the impact time and impact angle constraints.Compared with several other trajectory shaping based guidance laws,the proposed one is easier to design and can be applied in broader engagement scenarios.Control methods are usually integrated with the characteristics of the UAV to generate control commands to ensure that the crosstracking error converges to zero.Some well-known control techniques are VF [12], linear quadratic regulator (LQR) [19], sliding mode control [20], model predictive control [21], backstepping control[22],and adaptive control[23].Sujit et al.[13]compared the path-following performances of VF,LQR,pure pursuit with LOS,and NPFG methods using a 6-DOF aircraft simulation model and constant wind and gusts in random directions.The Monte Carlo method illustrates that the VF technique can implement pathfollowing more accurately than other methods; further, it requires the least control effort.

VFs have various forms, including the Lyapunov VFs (LVFs) [4]and gradient VFs (GVFs) [24].The LVF can be used to generate heading guidance for vehicle convergence and adherence to straight and circular paths using the fundamental sets of Lyapunov functions.Various LVF guidance laws are activated throughout the flight to achieve complete adherence to the desired path [12].The GVF consists of individually weighted convergence and circulation components that generate guidance vectors toward and along the paths, respectively.Both the LVF and GVF methods can generate similar VFs for basic shapes such as lines or circles.However,GVFs allow for time-varying scenarios and the ability to support more advanced surfaces than the LVFs used so far [2].For example, the GVF method can generate an n-dimensional VF, which facilitates adherence to dynamic paths and avoidance of obstacles.

Nelson et al.[12] created VFs considering that wind disturbances can make the vehicle converge to a straight line or a circular path.Frew et al.[4] proposed a VF-based control structure to achieve the standoff tracking of moving target vehicles.Lawrence et al.[25] created provably stable VFs for a class of star-shaped, threedimensional curves.Beard et al.[26] created GVF guidance equations from two intersecting two-dimensional surfaces to follow the straight-line and helical paths of the Dubins airplane path (DAP).Liang et al.[27] generated VFs to study the problem of following arbitrary,twice differentiable curved paths in both two-and threedimensional spaces.Kapitanyuk et al.[28]proposed a VF encoding method to allow the desired path to be an arbitrary smooth curve in its implicit form.Wilhelm et al.[2]focused on optimizing the GVF method for the path-following, obstacle-avoidance problem.They found that the heading angle guidance generated by the GVF method is a candidate solution for the implementation of obstacle avoidance under high-wind conditions.Guo et al.[10]reported that following the heading-angle-based VF path is more suitable for a large-scale UAV with a differential throttle.The authors verified by simulation that this method can improve the probability of successful landing of the full-wing solar-powered UAV.Chen et al.[29]proposed an optimization-based VF coding method.The online optimization procedure is used to estimate the error so that the path generated by the VF is close to the shortest path from the current position to the target path.Goncalves et al.[24] proposed the GVF generation technique to obtain robustness to wide ranges of paths in n-dimensional space.However, it is only applicable to bounded closed curves and cannot be directly applied to unbounded desired paths.Pereira et al.[7]developed a composite VF to guide a UAV along closed curves and avoid unknown dynamic obstacles.Wang et al.[8] proposed a nonuniform VF method to implement the formations of UAVs.Comparisons with standard formation algorithms indicate that the proposed method is effective in achieving formations in various path scenarios.

Depending on the requirements of the problem at hand, the guidance law design for VFs includes direct and indirect methods.The advantage of the direct method[2,6,8,28,29]is that it can solve the path-planning, -following, and -control problems simultaneously.The indirect method [10,26,30,31] decouples the pathplanning, -following, and -control problems.It usually employs existing curve models to generate guidance and target paths among waypoints and implements path-following and -control by designing the VF control laws for these curve models.In this method, the quality of the desired path-following and UAV performance is guaranteed using an appropriate curve model,and the VF guidance laws enable these paths to be followed accurately.The advantage of the indirect method is that it can directly utilize the effective characteristics of some well-established curve models such as the Dubins car path (DCP) [32], B-spline [31], B′ezier [33],and Pythagorean-hodograph(PH)curve[30]models.These models can reduce the design complexity of the guidance law.In contrast,there is no need to redesign the guidance law when the guidance path switches between the two configurations of the UAV.Zhao et al.[31] proposed a VF-based method for UAV curved pathfollowing using the concept of input-to-state stability and the Bspline curves as the reference path.The method can be extended to include multiple UAVs, allowing the scheme to be used for formation path-following control.Lee et al.[30]designed the VF method for PH curves as reference paths and introduced angle constraints in the vector space.

At present,the DCP model has already matured in the autopilot industry for the path planning of cars,robots,and so on.It consists of arcs and lines,and it is the minimum-distance path between two two-dimensional configurations and has curvature constraints[32]that can be adapted to the UAV by assuming a fixed altitude [34].The DAP model is more complex than the DCP model because of the altitude component.Chitsaz et al.[35] obtained the minimum distance path for Dubins aircraft using the Pontryagin minimum principle.Based on it, McLain et al.[26] added an altitude component to the DCP, considered the curvature and flight-path angle constraints, and proved that the minimum path taken by a UAV between two configurations consists of straight lines and helical paths.In particular, the literature mentions a DAP that complies with the flight-path angle limits of the UAV based on the altitude difference between the start and end configurations, for which extra arcs and helical paths are introduced.The DAP model retains the excellent characteristics of the DCP model and makes the model more consistent with fixed-wing UAVs.

In several studies, to verify the performance of VF algorithms,the switching strategy is often neglected [2,8,24,27,28].Two purposes are served through this.On the one hand, the strategy establishes termination criteria for VFs that converge to an unbounded curve to allow the UAV to stop following when it reaches the end of the target curve.On the other hand, the strategy provides switching or transition schemes among multiple VF guidance equations.Yao et al.[6] used a composite VF to simultaneously achieve collision avoidance and path following.They generated two VFs, one for path following and the other for collision avoidance.Bump functions were used to combine these two VFs smoothly.Pereira et al.[7]proposed a VF strategy that switches between two forms:a VF to converge and circulate the target curve and a VF to avoid obstacles by circulating the closest one.Some complicated target curves can usually be pieced together from predesigned basic paths.Such pieces of paths that can be combined sequentially to produce more complicated paths are referred to as motion primitives [35].Thus, complicated target curves can be followed by creating VF guidance equations for each motion primitive.In this situation,a suitable switching strategy needs to be designed so that the UAV transitions to the subsequent motion primitive at the appropriate position during the following process.

Guidance path-planning and -following are two significant functionalities of UAVs, but only a few approaches are able to handle both simultaneously.The main contribution of this study is the development of an innovative hybrid GVFs path-following guidance (HGVFs-PFG) algorithm to provide a unified solution to both these problems.The algorithm is implemented to control fixed-wing UAVs to follow a generated guidance path and oriented target curves in three-dimensional space, which can be any combination of straight lines,arcs,and helixes as motion primitives.In this study,we modify the DAP[26]to a high-altitude-type version to avoid the drawbacks of the earlier method, of using bisection search to determine the shared turning radii of arcs and helixes.Furthermore, we use the improved DAP to describe the guidance path and target curves.The proposed Dubins airplane guidance path (DAGP) generation algorithm is also used to generate the guidance path.To our knowledge, no VF method that provides a complete solution for following a DAP has been proposed so far.Guo et al.[10] and Beard and McLain [1] employed the same strategy to switch among multiple guidance laws used to describe motion primitives to achieve adherence to the DAP.This strategy performs appropriately in most cases, but it is flawed in that it cannot completely follow an arc path that increases the turning angle of the UAV beyond 180°.The switching strategy proposed in this study solves this problem by introducing the concept of the virtual waypoint.This strategy ensures that only one VF is activated at any time to allow the UAV to follow the DAPs constructed among multiple waypoints in a designed sequence, thus achieving complete adherence to the guidance path and target curves.Finally,the performance of the HGVFs-PFG algorithm is verified in our simulation using a reduced-order autopilot and representative scenarios.

This paper is organized as follows.Section 2 describes the research problem and architecture as well as the HGVFs-PFG algorithm proposed in this paper to solve the PFG problem.Section 3 details the DAGP generation algorithm for planning guidance paths for UAVs.Section 4 describes the HGVFs algorithm and switching strategy for following the guidance paths and oriented target curves.Section 5 designs four representative scenarios to verify the guidance-following performance of the HGVFs-PFG algorithm.Finally, Section 6 concludes the paper.

2.Problem formulation

2.1.Problem description

We consider a UAV represented by a configuration vector c=[cn,ce,cd]Tin the configuration space R3and assume that it is modeled as a single integrator, c˙= u, in which u is the control input.

In this study, an oriented target curve τ⇀refers to the curve to which the UAV needs to be guided and which it follows.Depending on the task requirements, τ⇀can be closed or nonclosed.For a closed curve, the UAV needs to keep following it, and for a nonclosed curve, the UAV needs to follow it to its end.The set of all points on the oriented target curve τ⇀can be individually represented by τ.

2.2.System architecture

To solve the PFG problem, we designed an architecture that included the HGVFs-PFG algorithm, as depicted in Fig.1.

The HGVFs-PFG algorithm generates the guidance path and output commands for the attitude control system to follow the guidance path and the target curves.The path planner module is responsible for the generation of the guidance path,which outputs the DAGP based on the initial position of the UAV and the oriented target curves.The path manager module is responsible for storing the DAGP and waypoints for the target curves.It outputs the path to be followed to the path-following module based on the current position.Finally, the path-following module is responsible for generating control commands to the low-level autopilot.In practice, the low-level autopilot converts the control commands into servo commands to control the motion of the UAV.In this study,we use a reduced-order controller instead of the low-level autopilot and do not model the state estimation; this is sufficient for verifying the high-level path-following algorithm.

2.3.Coordinate frames

In this study, we assume a flat, nonrotating Earth.The defined inertial, vehicle, and body frames are depicted in Fig.2.

Fig.1.Architecture of the HGVFs-PFG algorithm.

Fig.2.Inertial, vehicle, and body frames.

The inertial frame used in this study is referred to as the North-East-Down (NED) coordinate system.The configuration vector of the UAV,that is,its position,is represented by c=(cn,ce,cd)T,where cnand cedenote the positions along the north-axis iiand east-axis ji, respectively, in the inertial frame.cddenotes the vertical axis kithat complies with the right-hand rule, which results in negative values for the altitude gain.

The origin of the vehicle frame is at the center of mass of the UAV, and its axes iv, jv, and kvare aligned with ii, ji, and kiof the inertial frame,respectively.

The origin of the body frame is at the center of mass of the UAV,ibis along the longitudinal direction of the UAV and points to the head as positive,kbis in the symmetry plane of the UAV and points from the cockpit cover to the seat direction,and jbis determined by the right-hand rule and points to the right wing.

The orientation of the UAV is specified in terms of Euler angles.To translate the vehicle frame onto the body frame,we first rotate a heading angle around kv, then a pitch angle around the intermediate jv, and finally a roll angle around the intermediate iv.

Several studies have reported that path-following based on VFs is very effective at rejecting wind disturbances [2,10,12].Because this study is focused on validating the high-level HGVFs- PFG algorithm, the effect of wind is not considered.In this case, the airspeed Vais equal to the ground speed Vg,the heading angle ψ is equal to the course angle,and the flight-path angle γ is equal to the pitch angle.

2.4.Notations

To avoid confusion and misunderstanding,we need to highlight the implication of some of the notations used in this paper.

Bold letters indicate the vector or position in the inertial coordinate system.For example,c=[cn,ce,cd]Tdenotes the position of the UAV, that is, the configuration vector.

3.Optimal guidance path generation

In this section, we improve the DAP model and propose the DAGP generation algorithm based on it to solve the guidance path problem described in Subsection 2.1.The algorithm is employed to generate the path from the current position c0of the UAV to the oriented target curves τ⇀.

3.1.Improved DAP model

The generated guidance path that maximizes the average altitude from the energy perspective is a reasonable solution.In Ref.[26],the authors consider this factor as well as the constraints of curvature and flight-path angle to construct the DAP model.As a three-dimensional extension of the DCP model, the DAP model retains its excellent characteristics.The DAP model can be appropriately used as a guidance path model because it is consistent with fixed-wing UAVs.In this study,we improve the DAP model for path generation to obtain its high-altitude-type.

The physical capabilities of the UAV limit the commanded roll angle φcand flight-path angle Yc, the constraints of which are represented as follows:

The length calculation and path generation for all DAP types rely on the well-established DCP model.See Refs.[32,36] for details.It should be noted that we obtain the minimum turning radius of the DAP based on the coordinated-turn condition as follows:

where Vadenotes the airspeed and g denotes gravity.

From the DCP model, 42 types of DAP models can be derived,consisting of five elementary paths: straight line (S), left turn (L),right turn (R), left helix (HL), and right helix (HR).Three types of altitude components are introduced: low altitude, medium altitude,and high altitude,which are calculated based on the length of the DCP(Lcar),the altitude difference δ=wid-cdbetween the start altitude cdand the end altitude wid,and the flight-path angle limit.Therefore, if δ < 0, an extra maneuver path is generated at the beginning for the medium- and high-altitude types.Similarly, if δ>0,an extra maneuver path is generated at the end,as depicted in Fig.3.The DAP model ensures that the UAV remains at as high an altitude as possible for most of the path to save energy.

(1) The DAP is of the low-altitude type when

where floor [x] is a function that rounds x down to the nearest integer.

Fig.3.Dubins airplane path.

The length of this path is calculated as follows:

For the high-altitude type, the turning radius of the generated arc path is set to the minimum turning radius,whereas that of the helix is obtained through calculation.This helps avoid the drawbacks of the earlier model [26], of using bisection search to determine the shared turning radii of the arcs and helixes.

3) The DAP belongs to the medium-altitude type when

In this case,the path is extended by generating an extra turning arc.The length of this path is calculated as follows:

where Lcardenotes the length of the DCP corresponding to this DAP and φ denotes the central angle of the extra turning arc.Lcaris calculated as follows:

3.2.DAGP generation algorithm

4.HGVFs

In this section,we propose the HGVFs algorithm and switching strategy to solve the pathfollowing problem described in Section 2.1.The algorithm follows the guidance path obtained using the DAGP generation algorithm and target curves to produce reference commands for the low-level attitude controller.Using the HGVFs algorithm,we redesign the encoding scheme for straight lines and curved paths and establish the VF control equations.Furthermore,we present the HGVFs switching strategy to implement complete adherence to the DAGP and oriented target curves.

Given a configuration vector c,a VF F (c)is defined such that F(c): R3→Tc(R3), where Tc(R3) is the tangent space of R3.Let αi(c): R3→R be a surface is a level set of a function with bounded second-partial derivatives and with linearly independent gradients at the target curve τ.

We construct the intersection of the two two-dimensional manifolds represented by α1(c) and α2(c) as follows:

such that the target curve τ ∊τα.

The VF F（c）=（Fn，Fe，Fd）Tcan be decomposed into two components:one that converges to the target curve τ and one that circulates it.

Fig.4.Guidance path obtained using the DAGP generation algorithm for the holding path.

Fig.5.Structure of the DAGP generation algorithm.

The controller converts the VF F（c） into the flight-path angle command Ycand roll angle command φcbased on Eq.(15)and Eq.(17), respectively

where sata[x] denotes the saturation function and is defined as follows:

be a differentiable, positive-definite scalar function with bounded second-order partial derivatives.Therefore,

where -∇V is a vector that points toward the path; following the vector will transition the UAV onto the path.Thereafter,∇α1×∇α2ensures that the UAV would stay along the path, and the direction of its motion is perpendicular to ∇α1and ∇α2.That is, Fconv= 0 when it is on the target curve, allowing Fcircto dominate.

More precisely, we can create a VF F（c） such that if u = F（c）,then as t →∞, c →τ(t); and once in τ, this set is continuously traversed by c in a designated direction.When c is continuous and ǁc˙ǁ> 0 (∀t ≥0), the abovementioned statements are satisfied.

In this study,we construct VFs for the two motion primitives of the DAP.All motion primitives are constructed through the intersection of two two-dimensional manifolds α1(c) = 0 and α2(c) = 0 in the configuration space R3,where α1(c)and α2(c)have bounded second-order partial derivatives and ∇α1(c)and ∇α2(c)are linearly independent.The abovementioned conditions guarantee that the following target curve τ, denoted by the intersection,is connected and one-dimensional [24].

The potential function is employed as follows:

4.1.Switching strategy

Because the curve obtained by the intersection of two manifolds α1(c) = 0 and α2(c) = 0 has an infinite length in R3, a switching strategy needs to be designed to make the UAV switch to the next motion primitive at the appropriate position or stop following when it crosses the endpoint of the target curves.In this section,the switching strategy is established by the maneuver space, and the concept of the virtual waypoints is introduced to implement the complete adherence to DAGP and oriented target curves.

By referring to the half-plane concept[1],the maneuver space is expressed as follows:

where c denotes the configuration vector and widenotes the next waypoint to be followed.nHIindicates the normal vector of the half-plane and is expressed as

Fig.6 depicts the top view of a UAV following the DAPs by employing the HGVFs switching strategy.During the entire process of following the path, only one maneuver space （wi，nHI） is activated at each moment(as represented by the yellow area in Fig.6,switching to the next maneuver space when the UAV crosses the yellow area).

4.2.Encoding method

The five fundamental paths constituting the DAP described in Section 3.1 can be composed using two motion primitives, a straight line and helix, where an arc is a special case of a helix.

HGVFs establish VF guidance laws for the motion primitives to cover all types of DAPs.

4.2.1.Straight line

The two two-dimensional surfaces α1(c)and α2(c)are defined as

where cs=（cos ψscos Ys，sin ψscos Ys，-cos Ys）Tdenotes the direction vector of the desired straight line.ψsand Ysdenote the desired heading and flight-path angle of the straight line, respectively.n1and n2represent the unit vector perpendicular to the lateral plane α1(c) and longitudinal plane α2(c), respectively, and are expressed as

Fig.6.Complete process of following the target curves using the switching strategy.

Fig.7.Without the maneuver space constructed by the virtual waypoint, the target curve cannot be followed completely.

As illustrated in Fig.9, the intersection of these two surfaces α1(c) and α2(c) and the maneuver space M（cs1，nH） defines the desired straight line path Psas follows:

where cs0denotes an arbitrary point on the straight line M（cs1，nH） indicates the maneuver space, where cs1denotes the end point of the desired straight line,nHdenotes the normal vector of the half-plane, and, in this scenario, nH= cs.

Substituting Eqs.(27) and (28) into Eq.(20), we get the VF guidance equation for the desired straight line as follows

4.2.2.Helical path

The helical path is specified by the following parametric equation:

where ψhand Yhdenote the desired heading and flight-path angle of the helix,respectively.h0=(hn,he,hd)Tdefines the center of the helix, which can be obtained from the initial position of the helix c(0) as follows:

Fig.8.Two different strategies for constructing maneuver space.

where α1(c) = 0 ∩α2(c) = 0 represents the set of helix points obtained by the intersection of the two surfaces α1(c) and α2(c).M（ch1，nH）denotes the maneuver space,where ch1represents the end point of the desired helical path and ，nHdenotes the normal vector of the half-plane.

The desired VF of the helical path is obtained from Eq.(20) as follows:

4.2.3.Arc path

The arc path is a special case in which the desired flight-path angle Yhin Eq.(31) is equal to 0.Thus, each component in Eq.(37) is expressed as follows:

Unlike in the case of following a helical path,when the turning angle of a UAV is greater than 180◦,two maneuver spaces have to be constructed for the arc path by introducing the virtual waypoint,as described in Section 4.1, to ensure that the path is followed.

These two half-planes are illustrated in Fig.10.

5.Simulation

5.1.Comparison algorithms

To demonstrate the performance of the HGVFs-PFG algorithm,numerical simulations are performed, which are compared with the results obtained using the following three methods.

1) The NPFG algorithm [18] is a classical following method for smooth trajectories based on waypoint navigation, and it has been verified through flight experiments.

2) The vector-field guidance (VFG) algorithm [30] generates a reference path and VF in a designed virtual vector space so that the vehicle follows the generated path in the vector space.

3) The GVFSG algorithm is a representative GVF waypoint following guidance method using which the VF equations are constructed based on Eq.(30).

The NPFG algorithm sets a virtual target on the desired path to guide the UAV.The virtual target is recalculated at each time step.

In the VFG method, the designed VF control equation is as follows:

where s（τ*） denotes the nearest point on the target curve τ.S denotes the current position,which is defined by the vector space S and is expressed as follows:

where R and ς denote the normalized radial distance and polar angle defined in the spherical coordinate system,respectively,and Vsrepresents the normalized velocity.

The VF-weighting coefficients of the HGVFs, VFG, and GVFSG algorithms are set to kconv= 0.1 and kcirc= 1.The GVFSG and VFG algorithms switch the guidance law among waypoints using the same switching strategy as that used in the HGVFs algorithm.

5.2.Reduced-order controller

Table 1Scenario parameters (all distances are in meters).

In this study, a reduced-order controller for the autopilot was developed as follows:equations to model the closed-loop response of the commanded state, and the coordinated-turn condition is used to eliminate the lift force from the equations of motion.This controller updates the position and attitude of the UAV based on the commanded roll angle φcand the commanded flight-path angle Yc.It should be noted that although the reduced-order controller captures only the approximate dynamic behavior of the system under specific conditions, it is acceptable as a validation of the high-level PFG algorithm.

Various gain parameters are considered in experiments to verify the effectiveness of the proposed HGVFs-PFG algorithm.Finally,the best performing set is selected as KPψ= 8, KDψ= 5, KY= 2, and Kψ= 1.

where KPφ, KDφ, Kγ, and Kψ are gain coefficients.

The controller employs first- and second-order differential

Fig.11.Simulation results obtained using the HGVFs-PFG algorithm in typical scenarios.

Fig.12.Results of simulation using the HGVFs-PFG and other algorithms in typical scenarios.

5.3.Scenarios

The smooth attitude change of the UAV for path following can improve flyability and flight safety by avoiding damage to the airframe and external stores from extreme maneuvers.In this study, four representative scenarios are designed and compared with other algorithms to verify the ability of HGVFs-PFG to mitigate command saturation and to evaluate the path following performance through lateral steady-state errors.These simulation scenarios cover most of the PFG requirements, and their parameters are listed in Table 1.

Scenario 1 represents a simple class of PFG problems that require guiding the UAV from an arbitrary position to a predetermined straight-line path.The path starts at winitand ends at wend,as indicated in Table 1.Command saturation may occur when the initial heading and pitch deviations are large.During guidance,we usually want the UAV to approach and converge to a predetermined heading with attitude changes as smooth as possible.Fig.11(a) depicts the guidance and following paths generated by the HGVFs-PFG algorithm in this scenario.

Scenario 2 represents a more general class of cases in which the UAV needs to be guided to follow multiple waypoints sequentially.The waypoints listed in Table 1 create an 8-shape path.When the UAV performs reconnaissance, surveillance, and patrol missions, it often needs to perform an 8-shape hovering flight over a certain area to achieve long-term and comprehensive coverage.For nonholonomic vehicles, following the C0curves is a challenge.In particular, the 8-shape path makes it easy for the UAV to saturate the lateral control commands while following the next waypoint.Under coordinated-turn conditions,the aircraft needs to adjust the roll angle continuously to switch among the three attitudes of the left turn, right turn, and level flight to achieve stable following to the desired path.Fig.11(b) depicts the guidance and following paths generated by the HGVFs-PFG algorithm in this scenario.

In Scenario 3,the UAV needs to guide and follow a holding path composed of straight lines and curves,which is widely used in civil aviation, mainly for regulating aircraft flow during approach and landing.The fixed-point position wfixand coordinate offsets Xoffsetand Yoffsetof the holding path are presented in Table 1.The implications of the coordinate offsets are depicted in Fig.4.Fig.11(c)depicts the guidance and following paths generated by the HGVFs-PFG algorithm in this scenario.

In Scenario 4, for the performances of the algorithms to be evaluated, the UAV needs to follow and be guided to an S-shaped path consisting of three curves with constant curvatures.The starting points winitand ending points wendof the curved paths and their corresponding commanded heading angles ψcinitand ψcendare listed in Table 1.Fig.11(d)depicts the guidance and following paths generated by the HGVFs-PFG algorithm in this scenario.

Because the GVFSG algorithm can only be applied to PFG problems involving waypoints, the algorithm cannot be used in Scenarios 3 and 4.

5.4.Analysis of simulation results

Fig.13.Flight attitude plots generated by the four algorithms in Scenario 1.

The simulation is implemented in Matlab,where the airspeed is set to Va=50 m/s,the flight-path angle range is limited to Yc=［-28.65°，28.65°],and the roll angle range is limited to φc=［-45°，45°].The simulation frequency of the reduced-order controller,path manager module, and path following module is set to 10 Hz,where the path planner is performed by the ground station to generate the guidance path in these scenarios.

The feasibility of using the HGVFs-PFG algorithm to follow guidance paths and oriented target curves can be determined from the preliminary results of the simulation presented in Fig.11.The yellow dashed line represents all the candidate guidance paths,where the shortest path is denoted as the optimal DAGP represented by a blue dashed line.For implementation, the lengths of these paths have to be calculated based on the DAGP generation algorithm presented in Fig.5.

In Fig.12, the paths generated by the four algorithms in the various scenarios are distinguished using different colors.

The attitudes and their rates of change for the HGVFs-PFG and other algorithms in Scenario 1 are depicted in Fig.13.

Fig.14.Flight attitude plots for the four algorithms in Scenario 2.

As depicted in Fig.13(c),(d),(f),a large initial heading deviation can cause roll angle saturation.The generated DAGP can effectively mitigate this situation, thus allowing the HGVFs-PFG algorithm to guide the UAV to approach the target curves with a smoother attitude by following the pre-planned DAGP.As depicted in Fig.13(b)and(e),the longitudinal control capability of the HGVFs-PFG algorithm is also good, with smaller overshoots and faster convergence when transitioning from the guidance path to the target curves.Because the vector space designed by VFG considers only the lateral control constraint, it leads to the most aggressive maneuver in the longitudinal direction.However, Fig.13(f) shows that its lateral control is more stable when compared with the NPFG and GVFSG algorithms in terms of change in attitude.Further, the straight-line path to be followed generated by the NPFG algorithm is approximated as proportional-derivative control.The GVFSG algorithm provides the most aggressive control of the initial attitude when compared with other algorithms because it uses a pure straight-line VF for guidance.It cannot satisfy the orientation constraint of the optimal guidance position.In Scenario 2, the PFG performance of the algorithms is evaluated when the target curves are constructed from a set of waypoints.The attitudes and rates of change for the four algorithms in Scenario 2 are presented in Fig.14.

As depicted in Fig.12(b),there is a large deviation between the UAV path and the desired curve when switching waypoints.This is because the UAV needs to roll out quickly after completing a level flight to follow the next path.This bias is caused by the inherent delay in the dynamic system and the limited roll angular velocity of the UAV.As seen in Fig.14(f),the sudden change in heading at the next waypoint causes the roll angle in all the algorithms to saturate to reduce the heading error as soon as possible, among which the HGVFs-PFG and GVFSG algorithms have the smallest overshoot for the rate of change of roll angle.

Scenario 3 evaluates the PFG performance of the algorithms when a target curve includes both straight lines and curves.The attitudes and their rates of change for the HGVFs-PFG and other algorithms in Scenario 3 are presented in Fig.15.

Large maneuvers may cause the vehicle to reach an unrecoverable state in the longitudinal plane.As depicted in Figs.15(b) and 15(e), the guidance path provided by the DAGP generation algorithm can produce a smaller initial pitch angle velocity, which effectively reduces the saturation of the longitudinal control command, while the NPFG and VFG algorithms generate a saturated pitch attitude.When transitioning from the guidance path to the holding path, the results of the HGVFs-PFG algorithm are more stable when compared with the other algorithms in terms of longitudinal control.Figs.15(c),15(d),15(f)indicates that the NPFG and VFG algorithms generate oscillations in the lateral plane when following the curve.Because the VF can converge exactly to the path created by the intersection of the two surfaces, the HGVFs algorithm is more efficient than the NPFG and VFG algorithms in the implementation of following the path of a constant curvature curve.

Fig.15.Flight attitude plots generated by the three algorithms in Scenario 3.

The attitudes and their rates of change for the three algorithms in Scenario 4 are presented in Fig.16.

As depicted in Fig.16(d)and(f),the attitude jitter generated by the HGVFs-PFG algorithm during waypoint switching is much less when compared with the other algorithms.Moreover, similar to Scenario 3, the HGVFs-PFG algorithm is more stable in terms of lateral control when following a path composed of multiple curves.The NPFG algorithm generates roll and pitch commands by employing look-ahead distances associated with virtual target points.Shorter look-ahead distances allow the UAV to follow a more aggressive path.However, very short look-ahead distances can cause instability in following owing to the inherent latency of the system.

Table 2 summarizes the steady-state errors in the lateral plane for each algorithm in the various scenarios.In following the straight-line path, the desired heading angle is fixed, and the steady-state error for each algorithm is very close to zero.When compared with the NPFG and VFG algorithms, the steady-state error for the HGVFs-PFG algorithm is also significantly competitive in curved paths,where the desired heading angle is constantly changing.

The nonlinear scheme of the NPFG algorithm demonstrates good performance for the PFG of waypoints and curves,although its control commands are more likely to saturate as compared with the HGVFs-PFG algorithm.When following the curves, stability may not be guaranteed.The VFG algorithm demonstrates good lateral control, but its longitudinal behavior is more aggressive when compared with other algorithms.The HGVFs-PFG algorithm allows arbitrary initial and terminal directions to be set,which is superior to the capabilities of the NPFG and VFG algorithms.When compared with the other algorithms,the guidance path generated by the HGVFs-PFG algorithm allows better performance in lateral control, which avoids sudden changes in the heading angle commands with less overshoot during the guidance process and makes the command less saturated.The HGVFs-PFG algorithm demonstrated similar performance in longitudinal control, and the DAP allows the pitch angle to change smoothly during the process of following.In general, the proposed technical approach is practical and competitive.Through extensive simulations, we demonstrate that for both the waypoints and curves of PFG, the HGVFs-PFG algorithm provides effective and robust following performance.

Fig.16.Flight attitude plots for the three algorithms in Scenario 4.

Table 2Steady-state errors for HGVFs-PFG and other algorithms in typical scenarios(meters).

6.Conclusions

In this study, we investigate the PFG problem for fixed-wing UAVs.An innovative HGVFs- PFG algorithm is proposed to control the UAV to follow a guidance path and oriented target curves in three-dimensional space.Based on the results and analysis of the simulations per-formed, the following main conclusions can be drawn:

1) The DAGP generation algorithm proposed in this study con

siders the curvature and flight-path angle constraints.In the case of large initial lateral and longitudinal deviations,the DAGP can make the algorithm used for following have lower heading and pitch change rates and effectively reduce the saturation of control commands,which is more suitable for a UAV with weak lateral control.

2) The target curve is oriented, and the initial direction of the starting point can be set.The generated path guides the UAV to approach the target curve with an appropriate heading attitude.

3) The lateral and longitudinal controls generated by the HGVFs have less overshoot and faster convergence when transitioning from the DAGP to the target curves.

4) Because the VF can converge exactly to the path created by the intersection of the two surfaces,the HGVFs are more effective in following curved paths.

Our future studies will include extending the algorithm to follow and guide multiple UAVs and considering the effects of wind.Furthermore,the algorithm must be loaded into the controller of a fixed-wing UAV for practical testing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China under Grant No.62076204 and Grant No.62006193,in part by the Postdoctoral Science Foundation of China under Grants No.2021M700337, and in part by the Fundamental Research Funds for the Central Universities under Grant No.3102019ZX016.