Vector Field-Based Guidance for Planar Smooth Path Following of a Small Unmanned Helicopter

. Tis study investigates a guidance method that will allow the small unmanned helicopter to follow the predefned horizontal smooth path. A stable nonlinear guidance law, which needs the information of the inertial position and groundspeed of the helicopter and the implicit function of the desired path, is designed to generate the reference course rate command based on the concept of the vector feld. Te asymptotic approximation to the desired path with bounded following errors has also been demonstrated by using the Lyapunov stability arguments. Some conditions for guaranteeing the stability have been extended. Simulations on following four types of planar paths, i.e., the square, circular, elliptic, and cubic curve paths, have verifed the efectiveness of the proposed method. Te predefned square path following performance of the proposed vector feld-based method is compared with another two guidance laws, which are based on the PD-like and fuzzy logic control. Te comparison shows that the proposed method can guide the helicopter to follow the predefned path most smoothly. Te maximum overshoot by using the proposed method is less than 0.016 meters, while those by using the other two methods are all larger than 0.80meters. Moreover, the vector feld-based method will cost the least time for the vehicle to converge to the predefned path.


Introduction
Small unmanned aerial vehicles including the fxed-wing [1], quadrotors [2], and helicopter [3] have been studied and applied in military and civilian felds in past decades.Te unmanned helicopters have attracted many researchers because they can fy vertically and manoeuvre in narrow spaces, especially hovering over interesting areas [3], such as the scene of the residential structure fre, the high-voltage transmission tower, and the damage position of the bridge.In the aspects of research, the vision-aided navigation and control of unmanned helicopters [4,5], confict-free navigation [6], monocular vision based indoor fight [7], and cooperative control of multiple unmanned helicopters [8] have appeared in recent years.It needs the vehicle to have the ability to follow (or track) the predefned path (or trajectory) precisely to realize the autonomous or automatic control of its own.According to the model-based controller structure, the approaches to realize the path following or trajectory tracking control of a small unmanned helicopter can be divided into two main categories.One approach uses an integrated method to solve the helicopter's guidance and control problem simultaneously.Te alternative method adopts the hierarchical control architecture to separate the above problem into an inner loop for stabilizing its dynamics and an outer loop for guidance.
Te integrated method to realize the path following or trajectory tracking mostly appears on the nonlinear control techniques.Zhou et al. applied the concept of backstepping control for the helicopter in continuous time [9] to track the predefned position and yaw reference trajectories.Razzaghian and Kardehi Moghaddam applied the nonlinear fuzzy sliding mode control method to the trajectory control of the helicopter with the help of a dynamic inverter [10].Tsai et al. combined the fuzzy basis function networks and the backstepping technique to reach the design of an intelligent adaptive tracking controller, where its performance was also compared with a numerical neural network controller [11,12].Benitez-Morales et al. [13] applied the feedback linearization control methodology for the trajectory tracking of a small helicopter's nonlinear longitudinal dynamics in simulation.Kim and Shim used the nonlinear model predictive control (MPC) method to track the predefned trajectory where the nonlinear model contains the kinematics and the system specifc dynamics [14].Te application of the MPC method for realizing the path tracking in unknown and cluttered environments was also discussed in [15].While applying the nonlinear control methods, it needs good knowledge of the vehicle's nonlinear model.Furthermore, the backstepping and fuzzy technique will mostly introduce new parameters and transformed variables.Te computation of the neural network and model predictive control is large.All of the above make them difcult to be used in the low-cost avionics systems.
While applying the separate inner and outer loop approach, the inner loop control usually uses the simple and well-established design methods.For guidance design, the traditional proportion integration diferentiation (PID) controllers are used due to its simple structure and efectiveness [16,17].And it is mostly used in the lateral/longitudinal manoeuvres by reducing the lateral deviation from a desired fght path.Two Mamdani-type fuzzy controllers were used to handle the lateral/longitudinal control, respectively, in [18].Te robust control techniques have also been applied in the guidance loop design.A Kalmanflter based linear quadratic integral (LQI) controller was used for the position controller design by considering the characteristics of the inner attitude closed-loop dynamics [19].Bergerman et al. adopted the FLC method together with the simple PD controller for the position and heading control of a helicopter [20].Te authors of reference [21] developed a composite nonlinear feedback control technique to achieve a high-performance position control.Part of the technique is based on the linear H 2 /H∞ optimization method.Te robust control techniques are useful for the outer loop design.However, it needs much more calculation due to its higher order compared with the simple PID technique.Marantos et al. proposed a robust control scheme that is decomposed into a position and an attitude control module, operating in a cascaded form with low complexity [22].Ma and Huo applied the hierarchical inner-outer loop structure to realize the singularity-free path following control of an unmanned helicopter.Te outer-loop position controller is constructed with the hyperbolic tangent function [23].
Considering the controller structure and the purpose of the horizontal path following of a small unmanned helicopter with low-cost avionics, we adopted the hierarchical control architecture for the fight control system design.Te unstable dynamics of the helicopter have been stabilized by the setpoint tracking LQG control technique [24].Te LQG technique can also track the reference heading rate and velocities in the body-fxed coordinate frame.
Te core part of the outer loop is based on the concept of the vector feld which has been successfully applied in miniature aerial vehicles to generate the commanded course rate [25].Te method calculates a vector feld around the path to be followed.Te vectors in the feld are directed toward the path to be followed and represent the desired direction of fight.And it has realized the waypoint path following and collision avoidance for the vehicles in the simulation and real fight tests [26,27].In [25], the straightline and circular orbit paths following approaches have been provided.However, in some cases, the route of an aircraft is a more complex smooth curve.In order to follow the general horizontal smooth paths that can be expressed by implicit functions, including the ellipse and cubic curve, a general smooth path following guidance law based on the concept of the vector feld has been produced for a small unmanned helicopter in this study.
Te main contributions of this study are summarized as follows: First, a guidance law based on the vector feld for following a class of horizontal paths is presented in the Cartesian coordinate system.Te stability of the path following with bounded following errors is demonstrated by the Lyapunov stability arguments.Second, some conditions for guaranteeing the stability have been extended by comparing with those appeared in [24].Te simulations about tracking the square, quadratic, and cubic curves were presented to evaluate the efectiveness of proposed method.Te following performance of the square path was compared with those by using another tow guidance laws.Tird, the control results were compared with another two guidance laws based on the PD-like and fuzzy logic control, which illustrates the efectiveness and better performance of the proposed control strategies.
Te remainder of the paper is organized as follows: Te formal problem description for a small unmanned helicopter to follow a predefned smooth path was given in Section 2. In Section 3, the course rate command is generated based on the concept of the vector feld.Te detailed stability analysis is also given.In Section 4, the simulation details on four types of planar paths are presented.Te conclusion and future work are given in Section 5.

Problem Description
Te helicopter adopts the hierarchical control architecture for the planar smooth path following.Figure 1 shows the structure of the control system.Te inner-loop controller is used not only to stabilize the helicopter dynamics but also track the reference signals from outputs of the outer-loop controller.Te outer-loop controller design is based on the inertial positions and velocity of the helicopter and the desired path from the path generator.And it is used to generate the desired heading rate and velocities.
In Figure 1, [p I N , p I E , p I D ] denotes the position of the center of gravity of the helicopter in the local north-eastdown, Cartesian coordinates [20].u ref , v ref , w ref , and r ref denote the reference velocities and the heading rate in the body-fxed coordinates, respectively.

2
Journal of Robotics Te design of the inner-loop controller is based on the helicopter's dynamics.And a setpoint LQG technique [24] with the combination of an LQR and a linear quadratic estimator is used not only to stabilize the internal stability of the unmanned helicopter but also to track the reference signals u ref , v ref , w ref , and r ref .
Te outer-loop controller design is separated into two parts which are used to produce the reference velocities and the heading rate command, respectively.Te vertical reference velocity design is based on the PI method.In this paper, we only consider the reference heading rate command generator.Te following two assumptions are adopted: Assumption 1. Te set altitude of the helicopter was well controlled.It means that the fight path of the vehicle is on the horizontal plane with a certain height in the local coordinate system.Assumption 2. Te heading of the fying vehicle ψ is assumed to be equal to its course χ (see Figure 2).If the helicopter is working in the low-speed fight mode, it has _ ψ ≈ r [20].Ten, the reference heading rate r ref can be generated as the course rate command u cmd shown in (2), i.e., r ref � u cmd .
In this study, the desired horizontal smooth path is presented by a twice continuously diferentiable implicit function f: R 2 ⟶ R as follows: where x � p I N and y � p I E .To generate the heading (course) rate command, the kinematics of the helicopter in the inertial north-east frame is used, and it is given as follows: where the groundspeed V g, max are positive constants.
Considering that the desired path ( 1) is the generic curve, it is difcult to express the explicit Euclidean distance between the position of the helicopter and the generic curvebased path mostly.Te value f(x, y) when the UAV is in (x, y) is used as the distance value.In this study, the nominal distance function d(x, y) is defned as d(x, y) � f(x, y).Te value of d(x, y) can represent the position of the helicopter relative to the desired path.If d(x, y) � 0, it means that the helicopter is on the path.Moreover, it can distinguish whether the helicopter is within the curve when the curve is closed or the helicopter is on the right half-plane of the curve when the curve is not closed.
Te following is to design the heading (course) rate command u cmd which will make the helicopter fy along the desired path in the right direction.

Vector Field-Based Nonlinear Guidance Law
Te magnitude gradient of d(x, y) in point (x, y) is denoted as ‖∇f‖ � , where f x and f y are the partial derivatives of d(x, y) or f(x, y)) with respect to x and y, respectively.For some given f(x, y), the value of ‖∇f‖ may be zero in some points (x, y).If the helicopter's initial position is at these points, the guidance to be designed below will fail, and the helicopter will lose control.In order to avoid the helicopter being trapped at the center point of the closed curve (such as the center of a circle or ellipse), the fight domain for the helicopter is defned as D int � x, y|x,  y ∈ R, ‖∇f‖ ≥ λ}, where λ > 0 defnes the size of the nofy zone.
Another property of g(x) is that its derivative, i.e., g ′ (x), is monotonic decreasing over [0, +∞].Ten, it has the fact that g ′ (x) is monotonic increasing over [− ∞, 0], g(0) � 0, and Journal of Robotics 3 Te desired course shown in (3) has two parts: g(d(x, y)) and ξ(x, y).If the helicopter is on the desired path f(x, y) � 0, then g(d(x, y)) � g(0) � 0, and ξ(x, y) is the tangent direction of the desired path.If the helicopter is far away from the desired path, then g(d(x, y)) ⟶ ± π/2.Together with the property of g ′ (x), the desired course is to guide the helicopter approach the desired path with the course angle from ± π/2 + ξ(x, y) to ξ(x, y).For example, if the helicopter is far away from a desired straight-line path, then ξ(x, y) � 0. In this case, the desired course is ± π/2, which is to make the helicopter fy vertically to the desired straight-line path.
Te state variables are defned as (d(x, y),  χ) T , where  χ � χ − χ d .Te dynamics of the variables can be derived as follows: ( Te main purpose of this study is to fnd the course rate command u cmd , which will make the errors d(x, y) and  χ converge to zeros as the time goes to infnity, i.e., make the helicopter fy along the desired path with bounded errors in fnite time.
Theorem 1.If the fying helicopter's initial position is not on the desired path but within the fight domain D int , the course rate command shown as (6), which is a nonlinear combination of the course error  χ and the time derivative of the desired course shown as (3), will make the helicopter approach the desired path with the desired course and fy along the desired path with bounded errors in fnite time.
Let W 1 �  χ 2 /2, and the boundary region around the sliding surface with respect to ε is defned as S ε � | χ| ≤ ε  .Ten, the derivative of W 1 is as follows: and it gets that  χ will reach S ε in fnite time.It remains to show that, inside S ε , system (8) will converge to the equilibrium point (0, 0) T .Te Lyapunov candidate function is defned as follows: Diferentiating W with respect to time, it has Inside the boundary region S ε , it has | χ| ≤ ε.So, +d(x, y)V g ‖∇f‖(sin (g(d(x, y))) Let (x) � x sin(g(x)), and Journal of Robotics where μ 1 > 0, μ 2 > 0, and d is arbitrary.It has the fact that both ϕ(x) and ξ(x) are symmetric functions in x, and ϕ(0) � ξ(0) � 0. It is noted that the function ξ(x) defned here is diferent from that appeared in [24], which will extend the condition for the stability.When 0 ≤ x ≤ d, then ξ ′ (x) � μ 1 x, and Figure 3 shows the plot comparison of the function ϕ(x) and ξ(x) with g(x) � atan(0.4x),d � 2, and μ 1 � 0.18.Te blue solid line is the plot of function ϕ(x) with respect to x. Te red dashed line is the plot of function ξ(x) as shown in [24].Te black dotted line is the plot of one of feasible functions ξ(x) with μ 2 � 0.155 in this paper.In fact, ϕ(x) ≥ ξ(x), and the function ξ(x) in [24] is the special case of (13).
where the matrix is positive defnite.It has the fact that the function W 3 (d(x, y),  χ) is continuous positive defnite.Ten, it derives that the equilibrium point of the nonlinear system ( 8) is uniformly asymptotically stable [28].
), and εV g, max /2kρ < μ 1 < cos (g(d))g ′ (d), μ 2 ≤ sin (g(d)) with the arbitrary chosen value d.It can state that the control law (6) will make the helicopter fy along the desired path with bounded errors in fnite time.And the path following errors (the distance and course error) will converge to zeros as the time goes to infnity.

Simulations
Tis section presents the simulations of the proposed vector feld-based guidance law for the path following of an ALIGN T-Rex 600 RC model helicopter (see Figure 4).Te helicopter was instrumented with avionics weights 4.9 kg.For simulation, the detailed inner-loop controller design together with the linear model of the helicopter can be seen in [24].And we choose g(x) � atan(0.4x),k � π/2, and ε � 0.15, respectively.Te local earth-fxed coordinate system is defned to follow the north-east-down convention.Te forward and sideway reference velocities along the body axis of the helicopter were set as u ref � 3m/s and v ref � 0m/s, respectively.Te simulation sample time was set as 20 ms.
While conducting the simulations, we applied the proposed method to follow four types of planar paths, i.e., the square, circular, elliptic, and cubic curve paths in the Matlab/ Simulink GUI environment, respectively.Te helicopter fied in a counter clockwise direction.
4.1.Square Path Following.In this section, another two guidance laws have also been applied to follow the square path.Te results will be compared to show the performance of the proposed vector feld-based method.Te frst one for comparison is a PD-like nonlinear control law (PD_PFC) [29] shown as follows: 6 Journal of Robotics where K 1 and K 2 are the constant parameters, and the signed value D(x, y) denotes the vertical distance from the current position of the vehicle to the desired straight line.
Te other guidance law has the same expression as PD_PFC, but the parameter K 2 is adjusted by the fuzzy logic (FC_PFC).Te structure of the FC_PFC is shown in Figure 5, and the control parameter K 2 is given as follows: where K 20 is a constant, and ∆k 2 is the tuning part for k 2 .In the design of fuzzy control logic, the domains of discourse for the two inputs (D(x, y) and _ D(x, y)) and the output ∆k 2 are set as [− 6, 6], [− 3, 3], and [− 0.09, 0.09], respectively.Te details of the membership functions and the rules for the design of the fuzzy logic unit can be seen in [29].In this study, we choose K 1 � 0.05 and K 20 � 0.075, respectively.Ten, K 2 > 0.018, according to the fuzzy control mechanism shown in [29].In this case, the control law shown FC_PFC can make the helicopter fy along the predefned straightline path.
Figure 7 shows the comparison of the deviation D(x, y) when the helicopter follows the square path with the methods PD_PFC, FC_PFC, and the vector feld.While doing the waypoint switch, the helicopter will switch route if the distance between the position of the vehicle and the next waypoint is less than 8 m.Tis mechanism causes the abrupt changes of D(x, y) with about 8 meters.Figure 7 also shows that the proposed vector feld method will make the helicopter reach much smoother transition of course with smaller overshoot and oscillation amplitude of D(x, y) than those by using the other two methods, PD_PFC and FC_PFC.
Figure 8 shows the comparison of the course rate command u cmd with the methods PD_PFC, FC_PFC, and the vector feld.When the route is switched, compared with the other two methods, PD_PFC and FC_PFC, the course rate command u cmd generated by using the method vector feld can converge to zero most quickly, which means that the method can make the helicopter fy along the target route most quickly.Figures 9 and 10 show the g(d(x, y)) and ξ(x, y) in (3) while following the predefned square path.Te corresponding value ranges are (0, 1.27) and (0, 4.71), respectively.
Table 1 presents the comparison of the maximum overshoots, rise time, and convergence time displayed in the elliptic areas, (a), (b), (c), (d), (e), and (f ) of Figure 7. Te rise time refers to the length of time that the helicopter fies toward the straight line before |D(x, y)| the frst time becomes less than 0.4 meters.Te convergence time is defned as the time from the helicopter following the desired path until |D(x, y)| is less than 0.4 meters.It can be seen that the maximum overshoot by using the method vector feld is less than 0.016 meters, while that by using the method FC_PFC is within the range of 0.81∼0.87meters.Te maximum overshoot produced by using the method PD_PFC is about 1.9 meters, which is the largest.Te method PD_PFC has the shortest rise time, which is about 2.64 s.While using the methods, the vector feld and FC_PFC, the corresponding rise times are about 3.66 s and 3.5 s, respectively.However, the convergence times by using the method vector feld are all less than 3.72 s.Te corresponding convergence times by using the methods FC_PFC and PD_PFC are all lager than 7.30 s.
Table 2 shows the average and the standard deviation values of the distance D(x, y) of the three methods, respectively.Te comparison of the standard deviation values and the maximum overshoots show that the vector feld method makes the helicopter to follow the predefned path most smoothly.Te mean value of D(x, y) by using the Journal of Robotics 7 method vector feld is the largest.Tis is because almost all of the helicopter's fight is inside the square by using the method vector feld.While using the methods PD_PFC and FC_PFC, the helicopter will fy across the path during its turning.
Table 3 shows the comparison of the fight time of each route with the three methods.It can be shown that the method PD_PFC will cost the most time to complete each route.Table 4 shows the comparison of the moment when the vehicle starts to turn with the three methods.It shows that the method vector feld will make the helicopter switch the waypoints frstly.
It means that it takes the least time for the helicopter with the method of the vector feld to reach the waypoints.
Table 3 shows the comparison of the fight time of each route with the three methods.It can be shown that the method PD_PFC will cost the most time to complete each route.Table 4 shows the comparison of the moment when the vehicle starts to turn with the three methods.It shows that the method vector feld will make the helicopter switch the waypoints frstly.It means that it takes the least time for the helicopter with the method of the vector feld to reach the waypoints.
Te above indicates that the method vector feld can achieve the best path following of the predefned square path than those by using the methods PD_PFC and FC_PFC.Although the method PD_PFC has the performance of the shortest rise time, it has the largest overshoot, convergence time, and oscillation amplitude.With the parameter K 2 adjusted by the fuzzy logic, the method FC_PFC can follow the square path better than that by using the method PD_PFC.But the design of the fuzzy rules of the method FC_PFC is limited.Te fuzzy rules cannot be fully considered.It makes the FC_PFC still have a larger overshoot than those by using the method vector feld.And it takes more fight time for the vehicle to reach the goal with the  8 Journal of Robotics method FC_PFC than that by using the method vector feld.In comparison, the control law designed by using the method vector feld is based on the smooth transition to the predefned path frstly.And it makes the method vector feld have the best advantage.Journal of Robotics approach 0.1 rad/s while the helicopter is fying along the circle path with a radius of 30 meters at a speed of 3 m/s.Since the curvature of each point on the ellipse is not the same, Figure 12(c) shows that the course rate command generated for the ellipse path following varies periodically while the helicopter is fying along the ellipse round and round.
For cubic curve path following, the function of the predefned cubic curve is f(x, y): 3 − y � 0. Te initial position of the small unmanned helicopter was set as (− 30,     − 15, − 20) in the local north-east-down coordinate.Figure 13(a) shows that the helicopter will follow the predefned cubic path after it has reach the path.Te nominal distance d(x, y) shown in Figure 13(b) indicates that there is no overshoot when the helicopter fies towards the cubic path.

Conclusions
Te guidance of the helicopter to realize the horizontal desired path following has been designed based on the concept of the vector feld.Te guidance law only needs the inertial information of the helicopter including the position and velocity in the north-east frame and the implicit function of the desired path.Te asymptotic stability analysis of the horizontal smooth path following with bounded errors has also been given.Compared with the work in [25], the fight paths have been extended to a class of the smooth curve which can be expressed by implicit functions.Compared with the work in [24], the conditions for the stability of the proposed vector feld guidance method have been extended.
Simulations on following four types of planar paths, i.e., the square, circular, elliptic, and cubic curve paths, have verifed the efectiveness of the proposed method.Furthermore, another two PD-like nonlinear control laws, PD_PFC and FC_PFC, were also applied to the square path following.Te comparison shows that the method vector feld has the best path following performance.Te helicopter will arrive each waypoint and converge to the next route with the shortest arrival time by using the method vector feld.Additionally, the simulation fight path produced by using  the method vector feld is the smoothest with the smallest overshoot.Te guidance law designed in this paper can also be applied in the guidance of UGV and other types of UAV working in the horizontal plane.
It can be found that the function g(x) in (3) will infuence the process that the helicopter reaches the sliding surface.In the future, a suitable function with satisfactory parameters designed will be studied to make the helicopter follow a serial of smooth paths better.Additionally, since the ground velocity in the horizontal plane V g can vary within [V g, min , V g, max ], another control law to adjust the velocity V g considering the convergence rate of diferent kinds of path will also be invested.
When there is external interference, especially the wind is larger enough, the assumption that the heading of the fying vehicle ψ is equal to its course χ cannot be guaranteed.In this case, the kinematic model shown as ( 2) is not suitable for generating the heading rate command.Te kinematic model which is the function of the heading, airspeed, and wind speed [25] will be used to generate the heading rate command.

Figure 1 :
Figure 1: Te hierarchical fight control structure for the helicopter.

Figure 6 :
Figure 6: Comparison of the square path following with the methods PD_PFC, FC_PFC, and the vector feld (a) in the north-east-down 3D frame and (b) in the horizontal plane.

Figure 5 :
Figure 5: Te structure of the FC_PFC course rate command generator for the helicopter.

Table 1 :
Comparison of the maximum overshoot and rise time about D(x, y) with the methods the vector feld, FC_PFC, and PD_PFC.

Table 2 :
Comparison of the statistical result about D(x, y) with the three methods.

Table 3 :
Comparison of the fight time of each route with the three methods.

Table 4 :
Comparison of the moment when the vehicle starts to turn with the three methods.