Model-free autonomous control of four-wheel steering using artificial flow guidance

Abstract

Four-wheel steering (4WS) is an effective technique to improve handling performance and lateral stability of road vehicles. Conventionally, controllers utilise the driver's actions and vehicle dynamics to coordinate front and rear-axle steering. This paper proposes a novel approach for 4WS controller design, based on the concept of Artificial Flow Guidance (AFG), which relies on a spatially distributed motion reference through a two-dimensional vector field. This field provides high-level guidance while lower-level steering controllers to control axle centres motions relative to the flow. These flow vectors, computed in real-time via simple geometric construction, can be pre-computed globally to evaluate the guidance algorithm's efficacy. When controlling only the front axle, this same approach can function as an autonomous driving system. Relying solely on a spatial reference field and control targets' velocities enables the controller to work in a simple and robust fashion, without using a reference vehicle dynamics model or lengthy parameter tuning. The proposed approach's effectiveness is validated through co-simulation with MATLAB/Simulink and the CarMaker simulation platform. AFG control performance is found to be at least comparable to that of more complex 4WS controllers using methods such as MPC; in the cases considered, AFG provides superior path-tracking performance.

Keywords:

• Flow guidance
• 4WS
• lateral vehicle dynamics and control
• autonomous driving

1. Introduction

There has been a long history of research into four-wheel steering (4WS) vehicles, for example, see Refs. [1–4]. The additional control arising from rear steering offers the potential for improved handling response, enhanced lateral stability and reduced control effort, for example, when making lane changes at high speed. In most early researches [1–3], the front wheel steering is commanded by the driver, while automatic rear steering assists the manoeuvre or otherwise improves lateral stability. In all cases, the coordination of front and rear steering is based on the vehicle response, not the comparison of position, heading etc. with that of a reference model. Specifically, classical 4WS systems normally utilise front steering input either from a human-driver, or a driver model for autonomous driving, primarily designed for front wheel steering vehicles, such as Pure Pursuit [5,6], Front/Rear wheel based feedback [7–9], linear/nonlinear MPC [10–13], etc.

More specifically, 4WS systems typically use the additional rear steering freedom to simultaneously target desired yaw rate (e.g. inferred from the driver's steering command) together with zero body sideslip angle, to ensure high speed stability. It is also common to include speed-dependence in the controller, so the rear wheels steer in the same direction as the front for stability (high speed) and in the opposite direction for manoeuvrability (low speed).

More recently, with the expansion of environmental perception techniques and on-board computing power, autonomous driving has become extremely popular within academia, industry and also the public at large. For fully autonomous driving, the aforementioned approaches lack a coherent integrated design approach. Hence more recent research has focused on fully autonomous control of the 4WS vehicle [14,15], including for new vehicle architectures where all four wheels are independently controlled [16–18], such as, four-wheel independently steering (4WIS) and four-wheel independently drive (4WID) vehicles.

In [14] Toshihiro et al. transformed the path-tracking problem for 4WS vehicles into tracking of two control points, adjacent to front and rear axle centres, and two steering control inputs were decoupled by the state feedback. This approach has some similarity with the one proposed below, but the control methodologies are fundamentally different.

In [16], Liang et al. proposed a yaw rate tracking-based path-following controller for 4WISD vehicles. The controller is designed using linear matrix inequality theory and includes a coordinating mechanism based on yaw rate prediction to satisfy the conflicting objectives of path-following and dynamics stability. A hierarchical structure is introduced for motion control, and an optimal-based method is adopted for control allocation.

In [18], Wang et al. designed two sliding mode controllers for stability and tracking control, based on a 4WS dynamics model of error, including sideslip angle.

MPC based tracking control has been explored in [17]. A linear time-varying model predictive controller tracks a desired path smoothly while minimising the magnitude of curvature used. The controller uses an extended kinematic model that takes into consideration the vehicle crabbing capability. Ackermann steering geometry is used to transform the control requests, curvature and crabbing angle to wheel angles.

Essentially, these optimisation based driver models have migrated from conventional FWS to handle 4WS, the additional rear steering input being used to enhance the path tracking functionality. It is shown that MPC can be made to work and achieve excellent performance (and even be feasible in real time) but it is computationally expensive and will require more sophisticated micro-controllers than are currently in common industrial use. Besides, MPC requires a significant number of parameter choices (for example sample time, prediction horizon, control horizon and constraints and weights) which makes the test and development a time-consuming process. Also, questions about the needs for high model accuracy, and hence robustness when model parameters change (surface friction, added load, tyre wear/pressure, etc.), are not trivial to answer. This motivates a simpler approach based directly on the 4WS architecture.

The current paper presents a new approach to fully automate 4WS vehicle control, whereby front/rear steering coordination comes from a unified motion reference. This is achieved by defining a spatially distributed motion reference in the form of a vector field (artificial flow field), which is motivated by the stationary flow of a fluid. An example is shown in Figure 1, where each vector is a velocity reference for a particle located an any point on the map. This Artificial Flow Guidance (AFG) control approach was first proposed by Gordon et al. [19], and later realised by physically modelled (CFD) flow fields in [4]. It is worth noting that the reference field is distributed continuously in the feasible region on the road; hence references of multiple targets can be determined uniformly, which makes simultaneous control of both front and rear steering actuators relatively easy. The AFG control architecture can serve purposes beyond autonomous 4WS control; for example, recent research on heavy vehicles has demonstrated AFG as a viable chassis control architecture for steerable axles on long combination trucks [20].

Figure 1. Example of a distributed flow field for guidance towards the central lane position for the Hockenheim race track.

The main contributions of this paper, compared to other 4WS controllers and earlier Flow Guidance based controllers, are (i) the proposed autonomous 4WS controller does not require explicit vehicle models, it only needs the location and velocity information of the control targets, (ii) this controller adopts a two-in-two-out (2I2O) configuration, with two axle centres as control targets and two axle steering as outputs, (iii) the implementation of the proposed controller does not require additional yaw moment control, as the two-point motion tracking is sufficient to constrain both yaw and sideslip dynamics, simultaneously providing global motion guidance for the vehicle.

The remainder of this paper is structured as follows: In Section 2, the flow guidance approach is introduced. Section 3 is devoted to its implementation in a 4WS passenger vehicle. Simulations and results in Matlab/Simulink and Simulink/CarMaker are presented in Section 4. Conclusions are drawn in Section 5, including discussion and conjectures about future applicability.

2. Flow guidance and 4WS

Typically, a decision-making and execution system for autonomous driving can be decomposed into three stages:

• Route planning, working at the highest level, searches and sets an optimised scheduled route through feasible road networks to the goal position.

• Motion planning utilises enhanced local maps, along with environment information from a perception system, to generate detailed motion planning, including speed, direction and curvature, etc., in the estimated pose and collision free space. Motion planning at this stage typically exists in form of reference trajectory or path. Also, motion planning within a local map involves behavioural decision, in order to coordinate with traffic rules and other traffic participants.

• Vehicle control at the final stage is to track the generated reference trajectory with on-board state estimation, environment awareness and actuation capability.

This paper has primary focus on the vehicle control layer, assuming a reference path has been planned, and essential sensor information is available, such as vehicle CoG location, speed, yaw angle etc. A curvilinear coordinate system [21] is adopted for describing the reference path. As shown in Figure 2(a), s is the longitudinal travel distance, κ is the curvature, $\mathit{t}$ is the path tangent vector (equivalent to track local heading angle Θ) and n is the nominal lateral offset from the target path.

A reference flow field based on the concept of Flow Guidance [4,19] is computed on demand to guide the vehicle converging to the path. Paper [19] presented early research using Flow Guidance for high-speed automated vehicles. The idea was to pre-compute a flow field (reference vector field) such as the one shown in Figure 1. The field could be analysed as a whole and, provided the lower level steering controller was capable of tracking the flow within certain tolerance, stable and accurate path-following could be guaranteed [19]. Analysis assumed minimal yaw dynamics, so the vehicle was modelled as a particle, and implementation was limited to FWS.

Figure 2. Definition of (a) vehicle path tracking in track coordinate and (b) key vectors for flow vector generation. ${\mathit{p}}_t$ is the control point, ${\mathit{p}}^{\prime }$ is the reference point on the path $\mathit{P}$, ${\mathit{p}}^{″}$ is the preview point and $\mathit{L}$ is the preview distance as measured along the desired path.

In [4] the application was for low-speed guidance in general environments (e.g. including roads, parking structures etc.) where a fluid (or thermal) flow field was used to explore the environment and provide a nature-inspired guidance reference. A particular contribution of [4] is the use of a multi-point tracking, where the lower level controller sought to match the flow at many points on the vehicle chassis. Multi-point tracking offers additional benefits for high-speed motion control. On the other hand, a CFD based field calculation limits the real-time feasibility, and only low-speed scenarios were considered in that paper.

As introduced in [19], one algorithm for reference vector generation uses a linear combination of simple geometric vectors, as illustrated in Figure 2(b). There ${\mathit{p}}_t$ is the current position of the point of interest. $\mathit{P}$ is the target path, ${\mathit{p}}^{\mathrm{\prime }}$ is the projection (nearest) point on the target path, and ${\mathit{p}}^{″}$ is the preview point with a certain preview distance L. A detailed derivation of the preview distance L can be found in Appendix. ${\hat{\mathit{t}}}_1$ and ${\hat{\mathit{t}}}_2$ are the unit tangent vectors at ${\mathit{p}}^{\prime }$ and ${\mathit{p}}^{″}$, while ${\hat{\mathit{t}}}_3$ is the unit vector pointing from ${\mathit{p}}_t$ to ${\mathit{p}}^{″}$. The signed angle between ${\hat{\mathit{t}}}_1$ and $\widehat{{\mathit{t}}_2}$ is found from the cross-product, $2\theta ={\sin }^{-1}({\hat{t}}_1\times {\hat{t}}_2)$.

From this the flow vector direction is determined via (1) $\hat{\mathit{w}}={\hat{\mathit{t}}}_3+\frac{{\hat{\mathit{t}}}_1-{\hat{\mathit{t}}}_2}{2\cos \theta }$(1) This equation for $\hat{\mathit{w}}$ determines the directional information of the velocity only, and it is normalised to unit length before being applied. The magnitude is set equal to the desired speed, set manually or pre-calculated based on curvature, ground adhesion coefficient, vehicle limit, etc.

One key variable for vector generation is the preview distance, that is, the distance between ${\mathit{p}}^{\prime }$ and ${\mathit{p}}^{″}$, along the reference path. Similar to human driving, the preview distance must be long enough for stability, but not too long as this might affect the accuracy of the path following. Hence, the following function is adopted to adjust the preview distance: (2) $L=max(\frac{|n|U}{\sqrt{2a|n|+b}},L_0)$(2) $|n|$ being the magnitude of lateral offset. Thus L is modulated according to the current speed, U, lateral offtrack, n (in track coordinates), plus an acceleration limit a. Parameter $b\ge 0$ is a bias offset and $L_0$ is a minimum preview distance; b and $L_0$ avoid singularity as $n\to 0$. Effects of these parameters are discussed further in Appendix.

Then Figure 3(a) shows the integral curves (streamlines or particle trajectories) of the field, assuming particles can follow the field ideally. Nominal lane boundaries $n=\pm 10\mathrm{m}$ are assumed in these figures. The reference speed for this example is 60 km/h. It can be observed that all particle trajectories starting at a road boundaries converge towards the central reference path smoothly. (Note: no attempt has been made to create a ‘racing line’ here).

The convergence and tracking performance analysis of Flow Guidance approach was discussed in [19]. It can be examined more intuitively through the streamline pattern in curvilinear coordinates, shown here in Figure 3(b) for the same track segment. The streamlines show motion tracking in the case where perfect flow following is achieved, starting here from road boundaries and all with the same initial velocity. Clearly, the vehicle will converge to the desired path provided reference points are able to track the flow. In the event of initial deviations from the flow, convergent path following is guaranteed provided directional errors are systematically reduced via low-level feedback control. The effect of changing curvature is see in Figure 3(b) as a distortion in the flow pattern, for example, around $s\in [1250,1300]$; this affects the required acceleration for flow following (flow acceleration) but not the overall convergence property of the streamlines.

Figure 3. Vehicle convergence path for Hockenheim track in (a) Cartesian coordinates and (b) track coordinate.

Figure 4. Control architecture of the AFG based autonomous 4WS controller.

3. Methodology

AFG creates a velocity flow field that can guide chosen control point(s) to the reference path by following the local flow field. For FWS vehicles, a single control point is required, and this can be conveniently chosen at the centre of front axle. In this case, steering action strongly couples to the direction of the vehicle velocity vector at the control point, making the task of following the flow field as simple as possible. For 4WS vehicles, two control points may be chosen, to simultaneously influence the lateral and yaw motions of the vehicle, which is the fundamental design concept underlying the proposed controller. Here again, it is natural to select these at the centre of each of the steerable axles. Other locations for the control points could have been chosen, for example at the centres of percussion (see [22]), so that for example rear-wheel steering does not directly influence lateral acceleration at the front control point. Indeed, in heavy truck dynamics, it has been shown to be advantageous to adjust certain control point locations to improve yaw stability and tracking [23] depending on vehicle speed. However, in this work we restrict attention to using the axle centres.

The control architecture is shown in Figure 4. The axle-centre reference points are converted from Cartesian to curvilinear coordinates, so that preview points and unit vectors ${\hat{\mathit{t}}}_i$ in Equation (1(1) $\hat{\mathit{w}}={\hat{\mathit{t}}}_3+\frac{{\hat{\mathit{t}}}_1-{\hat{\mathit{t}}}_2}{2\cos \theta }$(1) ) can be determined. The flow references $\hat{\mathit{w}}$ are then found in inertial Cartesian coordinates, which are then conveniently used within the control algorithm. Hence, yaw angle is used to convert steer and axle-centre velocities (especially their directions) to Cartesian coordinates within the AFG controller – see below. The speed controller outputs drive/brake torques corresponding to required speed profile, and in this paper the speed controller works in parallel to the directional AFG steering controller.

To be more specific, once the velocity reference for each axle is calculated, a local feedforward and proportional-integral feedback controller is adopted to direct the motion of the axle centres to follow the target flow. Although the controller is quite simple and generic, it will be applied across a range of vehicle speeds and scenarios without changing any of the control parameters.

At axle i, where $i\in$(front, rear), the steering angle ${\delta }_i$ is the sum of the feedforward ${\delta }_{\mathrm{ff}}$ and feedback ${\delta }_{\mathrm{fb}}$ terms: (3) $\delta ={\delta }_{\mathrm{ff}}+{\delta }_{\mathrm{fb}}$(3) The feedforward term steers the wheels at axle i to be parallel to the flow field at the axle centre location: (4) ${\delta }_{\mathrm{ff},i}={\psi }_{\mathrm{ref},i}-{\psi }_u$(4) Here ${\psi }_u$ is the yaw angle of the vehicle, and ${\psi }_{\mathrm{ref}}$ is the angle of the reference flow vector at the centre of the axle. At very low speeds, feedforward is expected to be sufficient for vehicle guidance. However, feedback is required to compensate for non-zero slip angles when cornering under dynamic conditions.

For feedback, the angle error $e_i$ is used; this equals the angular deviation between the velocity vector at the axle centre and the flow vector: (5) $e_i={\varphi }_i-{\psi }_{\mathrm{ref},i}$(5) ${\varphi }_i$ being the polar angle of the velocity vector at the ith axle centre. Here a simple PI controller is adopted: (6) ${\delta }_{\mathrm{fb},i}(t)=-K_{P,i}e(t)-K_{I,i}{\int }_{t_0}^{t}e\left(t^{\mathrm{\prime }}\right)\mathrm{d}{t}^{\mathrm{\prime }}$(6) where $K_{P,i}$ and $K_{I,i}$ are the PI controller gains, which can be set to be tuned separately for front and rear axles, depending on the physical capabilities of their respective steering mechanisms, for example, actuator response and maximum steering angle. Note that, PID controllers have been used previously in 4WS system design, for example, [15,24].

The PI controller gains were tuned using a simple vector field, where all vector are parallel to the centre line on a wide open straight track, so only directional guidance is imposed (lateral position is ignored). An example is shown in Figure 5, where an initial constant steering angle of 0.002 rad is applied to the front axle to accumulate angle errors. Then, at t=5s, the AFG controller is turned on; the performance of the lower level controller can be examined and tuned on this basis. For this case, the parameters were $K_{P,f}=0.7,K_{I,f}=0.2$ for front axle, and $K_{P,r}=0.5,K_{I,r}=0.1$ for the rear axle. Parameters for the high-level flow guidance are fixed throughout, with a, b and $L_0$ in Equation (2(2) $L=max(\frac{|n|U}{\sqrt{2a|n|+b}},L_0)$(2) ) set to 0.3, 1 and 3, respectively.

For the longitudinal speed controller, a simple PID controller is used, with a speed reference $v_{\mathrm{ref}}$ and a speed feedback $v_{\mathrm{fb}}$, where $v_{\mathrm{fb}}$ is the speed of the vehicle centre of mass. The speed controller output is the drive/brake torque $T_{\mathrm{db}}$, which is then used to calculate the required torque $T_{\mathrm{req}}$ for each axle.

Figure 5. Tuning for the lower level PI controllers, with initial accumulated angle errors.

4. Simulation and results

To validate and examine the performance of the AFG approach, we model a 4WS vehicle (a light vehicle, 2 axles) whose parameters are shown in Table 1.

Table 1. Vehicle parameters.

Symbol	Value	Units	Description
m	1644.8	kg	Vehicle mass
$l_a$	1.24	m	Distance of front axle to CoG
$l_b$	1.51	m	Distance of rear axle to CoG
h	0.48	m	CoG height
l	$1.63/1.51$	m	Track width front/rear
${\mathit{I}}_{\mathrm{xx}}$	670.94	kgm${}^2$	Roll moment of inertia
${\mathit{I}}_{\mathrm{yy}}$	2111.19	kgm${}^2$	Pitch moment of inertia
${\mathit{I}}_{\mathrm{zz}}$	2488.89	kgm${}^2$	Yaw moment of inertia
$r_w$	0.33	m	Rolling radius
$T_{d,max}$	480	Nm	Max. drive torque on either axle
$T_{b,max}$	2000	Nm	Max. brake torque on either axle
${\delta }_{f,max}$	36	deg	Max. front axle steering angle
${\delta }_{r,max}$	10	deg	Max. rear axle steering angle

Simulations are carried out with co-simulation of MATLAB/Simulink & CarMaker. As an integrated virtual testing environment, CarMaker offers an adaptive driver model with artificial intelligence, which can be used for comparison. Several test scenarios are considered for testing the AFG 4WS controller, in the form of a spiral transition, a Figure-of-8 track, and lane changes.

4.1. Test scenario 1: Euler spiral track

The first scenario uses an Euler spiral track with linearly changing curvature, which provides quasi-steady-state conditions allowing lateral acceleration to increase smoothly. This type of road/track geometry is often used in highway/rail to provide a smooth transition between sections of different curvature. The curve for this transitions from straight case has a maximum curvature of $1/30{\mathrm{m}}^{-1}$, while the target speed is $60\mathrm{km}/\mathrm{h}$ ($16.7\mathrm{m}/\mathrm{s}$). This corresponds to a very challenging maximum acceleration of $9.26{\mathrm{ms}}^{-2}$, at the adhesion limit of the vehicle.

Figure 6 shows the vehicle trajectory (every $0.5\mathrm{s}$) along the track, and the solid blue arrows indicate the CoG acceleration vector for each instance. The corresponding steering angles and vehicle responses are shown in Figure 7, with the first 5s removed as it accelerates along a straight track to reach the reference speed. With continuously increasing curvature, the steering angles change smoothly, as shown in Figure 7(a). During the short interval of constant curvature there is a slight change in steer angles as the body sideslip angle is gradually reduced Figure 7(e). A small oscillation in steering also occurs, due to coupling with the speed controller with the vehicle on the limits of tyre adhesion. Further, the lateral acceleration in Figure 7(c) and yaw rate in Figure 7(d) are steady and smooth throughout the spiral entry and exit. In spite of the very high lateral accelerations, lateral off-tracking remains within a few cm throughout the manoeuvre Figure 7(b) and body sideslip angle remains less than 1${}^{\circ }$ throughout. Note that zero body sideslip is not a target for the AFG controller; instead, body sideslip angle emerges from – and is constrained by – the flow following algorithm applied to the axle centres.

Figure 6. Trajectory and acceleration of vehicle along the spiral track, with maximum $\kappa =1/30{\mathrm{m}}^{-1}$ and target speed $16.7\mathrm{m}/\mathrm{s}$. Each small car indicates the vehicle position and attitude, and the time gap between every two cars is $0.5\mathrm{s}$.

Figure 7. Vehicle response along the spiral track, with maximum $\kappa =1/30{\mathrm{m}}^{-1}$ and target speed $16.7\mathrm{m}/\mathrm{s}$.

4.2. Test scenario 2: Figure-of-8 track

The Figure-of-8 track is designed to test the dynamic stability of the proposed 4WS steering controller, especially on the crossing segments where the road curvature jumps from κ to $-\kappa$ or vice versa. Discontinuities in curvature are challenging since sharp changes will be observed in steering angle, lateral acceleration and yaw rate, etc. It is worth noting that this kind of curvature change is not regularly seen in modern road/rail design. In this case, the curvature is set to be $1/15{\mathrm{m}}^{-1}$ with target speed $8\mathrm{m}/\mathrm{s}$, equivalent to a nominal steady state lateral acceleration of $4.27{\mathrm{ms}}^{-2}$ – not as large as for test scenario 1 but still taking the vehicle well beyond the linear region of the tyres during sharp transitions.

The Figure-of-8 track itself and the vehicle trajectory (CoG) along the track is shown in Figure 8. The solid blue line indicates the vehicle trajectory and the dashed red line indicates the track centre, that is, target path. It can be observed that the vehicle follows the target path well with limited off-tracking.

Figure 8. The Figure-of-8 track (dashed line for track centre) and vehicle trajectory (solid line).

Steering angles and vehicle responses are shown in Figure 9. Figure 9(a) shows steering angles at the front and rear axles, both staying within the physical limits. As expected, both experience a step change during the crossing segments, and there is also some overshoot upon transition, which is caused by the simultaneous actions of feedforward and PI feedback control at the sharp changes in curvature. These spikes in steering angle lead to corresponding peaks in lateral acceleration, yaw rate, etc., and these are not experienced when curvature change is smooth (scenario 1). The lateral deviations of the axle centres from the target path are shown in Figure 9(b), with off tracking mostly within the range of $[-0.1,0.1]\mathrm{m}$ at the front axle and $[-0.05,0.05]\mathrm{m}$ at the rear. The short duration spikes of $\sim \pm 0.15\mathrm{m}$ occur when there is a sharp change in curvature, but overall the lateral deviations are controlled predictably under these dynamic conditions.

Figure 9. Vehicle response on Figure-of-8 track with $\kappa =1/15{\mathrm{m}}^{-1}$ and target speed $8\mathrm{m}/\mathrm{s}$.

Comparatively, in [25] Fnadi et al. proposed an MPC based dynamic path tracking approach for 4WS vehicles, for a circular path of radius of $25\mathrm{m}$, the maximum lateral deviation reached $0.5\mathrm{m}$ at speed of $10\mathrm{m}/\mathrm{s}$. Though the result was based on experiments, and hence subject to additional limitations and disturbances, we see the predicted tracking performance here is comparable to that of MPC, at the very least.

4.3. Test scenario 3: lane change

In this case a lane change test is carried out. This is conveniently performed using an artificially spliced track with a lateral offset of $3.5\mathrm{m}$ and longitudinal transition length of $20\mathrm{m}$. For simplicity, the central path is composed of two arcs (radius $30\mathrm{m}$) rather than a smoother polynomial or spiral definition. The reference speed is $50\mathrm{km}/\mathrm{h}$ ($13.89\mathrm{m}/\mathrm{s}$) corresponding to a nominal lateral acceleration of $6.43{\mathrm{ms}}^{-2}$. Hence we expect a sharp steering deviation as the curvature changes.

The steering angles and vehicle responses are shown in Figure 10. Figure 10(a) shows the steering angles of the front and rear axles, in opposite directions. Lateral deviations remain small Figure 10(b) and the vehicle remains stable, as seen from the yaw rate Figure 10(d). After the lane change is complete, the vehicle settles into equilibrium within one second, as seen from the yaw rate and lateral acceleration Figure 10(d,c). The maximum lateral acceleration, approaching $9{\mathrm{ms}}^{-2}$ can be observed around the point of curvature switch, but in spite of this large transient acceleration, the lateral motion is well controlled. Note that this is not proposed as an optimal lane change – the path can be designed to maintain quasi-steady-state conditions; rather, it shows that the 4WS controller is simple, robust and effective.

For comparison, the same vehicle, but with only conventional FWS executes the same lane change track using the IPG-Driver (internal CarMaker driver model). Results are shown in Figure 11. As can be observed in Figure 11(a), the rear axle is not steered, while the front steering is smoother and has smaller amplitude than with the 4WS controller. This may be preferable in practice, but not if the task is to follow the desired path, as being proposed for the autonomous 4WS controller. As seen in Figure 11(b), the axle centres deviate by up to $0.4\mathrm{m}$ with the driver model and FWS only; this compares with $0.07\mathrm{m}$ for the AFG 4WS controller, so around 6 times larger. Since the single lane change is a typical test case for path tracking problems, a quantitative benchmark for state-of-the-art path tracking controllers, that is, MPC, LQR, Sliding Mode Control and yaw-rate (r) tracking-based controllers versus the proposed AFG approach is presented in Table 2. From the table, it can be seen that under similar – though not identical – working conditions, AFG offers reduced maximum lateral deviation from the target path than the other methods. It is quite possible that this is partially due to the conditions no being identical, and it might be a future controller, such as MPC or LQR, can be carefully tuned to further improve lateral deviation performance, so we merely claim that AFG has comparable performance, in spite of the fact that it is a considerably simpler control method and requires minimal parameter tuning.

Figure 10. Vehicle response during a lane change test, with target speed $50\mathrm{km}/\mathrm{h}$.

To extend the comparisons in Table 2, a further lane change test was performed, with lateral distance of 7m and longitudinal velocity $25\mathrm{m}/\mathrm{s}$ ($90\mathrm{km}/\mathrm{h}$). A pair of arcs was again used for the reference path, the radius being $100\mathrm{m}$, corresponding to a nominal lateral acceleration of $6.25{\mathrm{ms}}^{-2}$. The vehicle response was found to be similar to the previous $3\mathrm{m}$ ($50\mathrm{km}/\mathrm{h}$) case. At the higher speed the steering action was found to be more dynamic, this also being attributed to the sudden switch in curvature at the middle of the lane change, plus there was no retuning of control parameters. In spite of this, the tracking performance was significantly better than the comparison cases in the table – $0.06\mathrm{m}$ compared to $0.23\mathrm{m}$ and $0.75\mathrm{m}$. For brevity, we do not present further plots for this case.

Figure 11. Vehicle response during the lane change with a traditional driver model and FWS vehicle.

Table 2. Quantitative benchmark for MPC, SMC and LQR versus AFG.

Approach	Velocity	Lane distance	Max. lateral deviation
MPC [17]	$2.5\mathrm{m}/\mathrm{s}$	3.5 m	0.85 m
LQR [18]	$25\mathrm{m}/\mathrm{s}$	7 m	0.75 m
SMC [18]	$25\mathrm{m}/\mathrm{s}$	7 m	0.23 m
r tracking [16]	$33.3\mathrm{m}/\mathrm{s}$	3.5 m	0.17 m
AFG	$25\mathrm{m}/\mathrm{s}$	7 m	0.06 m
AFG	$13.8\mathrm{m}/\mathrm{s}$	3.5 m	0.07 m

4.4. Test scenario 4: switched lane change

The final test scenario is called a ‘switched’ lane change; when a lane change is required, the reference path is shifted to the adjacent lane and the reference flow from the second lane is used to guide the vehicle. Hence, in contrast to the lane change of Section 4.3, there is no pre-planned transition path; all guidance is obtained from the flow structure (here for a straight track). The reference speed of 100 km/h is higher than in previous scenarios, but again there has been no change to any flow or control parameters. The scenario is mostly to test the capability of AFG to perform such a manoeuvre, not to perform a smooth or optimal lane change. We expect the sharp switch of flow field to generate large initial velocity errors, so large steering angles are expected to arise from PI feedback.

The resulting trajectory is shown in Figure 12; with lane width set to 3m and a vehicle width of $1.9\mathrm{m}$, it is found that the lane change is successfully achieved over a distance of $36\mathrm{m}$ (and $1.1\mathrm{s}$ duration) to entirely move out of the current lane. So the simple switched lane change could be used for autonomous collision avoidance by 4WS, even though the underlying design was developed for precise path following – again, there was no parameter change for flow guidance or PI flow tracking. And since there was no path planning, the method is considerably simpler than traditional approaches to collision avoidance by steering.

More detailed responses are shown in Figure 13. It can be seen from Figure 13(c) that the lateral accelerations are relatively high and in fact the tyre adhesion limits are occasionally reached (see the front slip angles in Figure 13(e)). In Figure 13(a) it is seen that front and rear steering angles are mostly in-phase, which is a ‘rational’ approach to suppress yaw motion during the lane change – even though this was not specifically designed into the controller. Rear steering amplitudes are seen to be smaller than the front, due to the lower PI gains for rear steering. Note that, even for this aggressive manoeuvre, the front and rear steer actuators do not reach their maximum limits. The lateral acceleration and yaw rate do experience some unwanted oscillations during settling, but this is partially due to the speed controller which applies a driving torque during $t=5\sim 7\mathrm{s}$.

It is worth emphasising that the tuning of PI gains was carried out only once – see Section 3 – and applied to all test cases. For the switched lane change, conducted at a higher speed, some performance improvement would be expected after some re-tuning; but, as mentioned, the test was carried out more as a robustness check than a realistic performance test.

Figure 12. Vehicle trajectory for a switched lane change.

5. Conclusions

This paper has presented a novel, model-free, Artificial Flow Guidance (AFG) controller for an autonomous four wheel steer (4WS) vehicle. The design intent has been for robust and precise path following, plus inherent lateral stability arising from the two-point flow-following algorithm.

Figure 13. Vehicle response of a switched lane change.

AFG generates a reference velocity vector field using a simple geometric algorithm, which is easy to compute in real-time. AFG uses the positions and velocities of front and rear axle centres and a simple lower level ‘feedforward+feedback’ controller. The resulting flow field properties can be visualised globally, for example to confirm streamline convergence; this implies that local flow-following automatically leads global path-following.

Simulations were conducted for four scenarios using a high-fidelity vehicle model. These show precise path following under quasi-steady-state conditions, even when the vehicle reaches very high lateral accelerations. Sudden changes in path curvature do cause sharp steering responses, seemingly due to the flow variations near the curvature transitions – see for example in Figure 3(b). This will be considered in future work.

High precision was shown on tight curves in the first two scenarios, and also in the pre-planned lane change of scenario 3, where off-tracking was improved by a factor of 5 compared to a standard FWS vehicle. Robustness was demonstrated for a ‘switched lane change’ where the vehicle was to respond to a $3\mathrm{m}$ lateral deviation from the intended lane, with no path planning for the transition between lanes. Vehicle speed was set higher in this case and with no change in control parameters. The lane change was successfully achieved in just over 1 second, suggesting this can be a simple and effective approach to collision avoidance by steering.

The present paper has, in the interests of simplicity, focused attention on the potential for a simple fixed-gain autonomous 4WS controller to perform a range of complex driving tasks in a precise and robust manner. Future work will focus on exploring possible performance improvements from (1) adapting the flow algorithm to different driving conditions and (2) improving the low-level velocity tracking algorithm. For example, in (1), it may be desirable to reduce flow acceleration demands at higher speeds – equivalent to increasing preview distance (beyond the speed dependency already introduced). For (2) it will be important to address the sometimes high bandwidth and amplitude demands arising from the PI steering controller. It may be that introducing speed-dependent steering rate and amplitude saturation will make the actuator demands more appropriate for real-world implementation; or a more sophisticated controller could entirely replace the simple PI controller used in the above.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix. Flow vector and variable preview distance verification

To start, as shown in Figure A1(a), with a control target point ${\mathit{p}}_t$, deviated from target path with distance n, without loss of generality, assume ${\mathit{p}}_t$ is on the left side of the path ($n>0$), and set the s position and the heading angle Θ of ${\mathit{p}}_t$ to be zero. And within the range of preview distance L, which will not be too far ahead, the curvature between ${\mathit{p}}_t$ and ${\mathit{p}}^{{}^{″}}$ can be considered as a constant κ.

Given the transform between track coordinate and Cartesian coordinate, (A1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}s}\Theta & =\kappa (s)\end{array}$(A1) (A2) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}s}x& =\cos \mathrm{\Theta }\end{array}$(A2) (A3) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}s}y& =\sin \mathrm{\Theta }\end{array}$(A3) with κ being constant, (A4) $\begin{array}{rl}\Theta & =\mathrm{\kappa s}\end{array}$(A4) (A5) $\begin{array}{rl}x& =\frac{1}{\kappa }\cos (\mathrm{\kappa s})\end{array}$(A5) (A6) $\begin{array}{rl}y& =\frac{-1}{\kappa }(\sin (\mathrm{\kappa s})-1)\end{array}$(A6) then, (A7) $\begin{array}{rl}{\hat{\mathit{t}}}_1& =(1,0)\end{array}$(A7) (A8) $\begin{array}{rl}{\hat{\mathit{t}}}_2& =(\cos (\mathrm{L\kappa }),\sin (\mathrm{L\kappa }))\end{array}$(A8) (A9) $\begin{array}{rl}{\mathit{t}}_3& =(\frac{1}{\kappa }\sin (\mathrm{L\kappa }),\frac{1-\cos (\mathrm{L\kappa })}{\kappa }-n)\end{array}$(A9) The trigonometric non-linearity makes the normalisation and partial derivative quite complex and trivial. Hence let us consider a special case when $\kappa \to 0$, ${\hat{\mathit{t}}}_2\mathrm{and}{\hat{\mathit{t}}}_3$ will become ${\hat{\mathit{t}}}_2\approx (1,0)$ and, (A10) $\begin{array}{rl}{\mathit{t}}_3& \approx \underset{\kappa \to 0}{lim}(\frac{1}{\kappa }\sin (\mathrm{L\kappa }),\frac{1-\cos (\mathrm{L\kappa })}{\kappa }-n)\\ & =(L,-n)\end{array}$(A10) which is equivalent to a segment of straight track, as shown in Figure A1(b).

Figure A1. Geometry and definition of variables of AFG (a) on a regular curve path and (b) simplification for a straight path.

Then the reference velocity vector $\mathit{w}$ for ${\mathit{p}}_t$ can be calculated as, (A11) $\mathit{w}=\frac{V}{\sqrt{L^2+n^2}}(L,-n)$(A11) where V is the reference longitudinal speed. To simplify the problem even further, assume (A12) ${\mathit{w}}_s=\frac{\mathrm{VL}}{\sqrt{L^2+n^2}}=U=\mathrm{const}$(A12) which is equivalent to $n\ll L$. Then we have $\mathit{w}=(U,\frac{-\mathrm{nU}}{L})$. And the tangent and normal acceleration can be calculated as, (A13) $\begin{array}{rl}\mathit{a}& =({\mathit{w}}_s\frac{\mathrm{\partial }}{\mathrm{\partial }s}+{\mathit{w}}_n\frac{\mathrm{\partial }}{\mathrm{\partial }n})\cdot \mathit{w}\\ & =(\frac{\mathrm{\partial }{\mathit{w}}_s}{\mathrm{\partial }s},\frac{\mathrm{\partial }{\mathit{w}}_n}{\mathrm{\partial }s}){\mathit{w}}_s+(\frac{\mathrm{\partial }{\mathit{w}}_s}{\mathrm{\partial }n},\frac{\mathrm{\partial }{\mathit{w}}_n}{\mathrm{\partial }n}){\mathit{w}}_n\\ & =(0,0)U+(0,\frac{-U}{L})\frac{-\mathrm{Un}}{L}=(0,\frac{U^{2}n}{L^2})\end{array}$(A13) Since the tangent acceleration is zero, while (A14) ${\mathit{a}}_n=\frac{U^{2}n}{L^2}$(A14) suppose,

• $L=\mathrm{const}.$, then the normal acceleration converge to zero linearly.

• ${\mathit{a}}_n=\mathrm{const}.$ To explore a variable $L(n)$, integrate both sides of Equation (A14(A14) ${\mathit{a}}_n=\frac{U^{2}n}{L^2}$(A14) ) with respect to normal deviation n,

(A15) ${\int }_0^n{\mathit{a}}_n\mathit{n}={\mathit{a}}_{n}n+\frac{b}{2}=\frac{U^2{n}^2}{L^2}$(A15) where $\frac{b}{2}$ is a constant of integration. Hence, L can be expressed as (A16) $L(n)=\frac{\mathrm{Un}}{\sqrt{2{\mathit{a}}_{n}n+b}}$(A16) where n is the normal (or lateral) deviation from the target path, ${\mathit{a}}_n$ is a normal acceleration reference (or limit).

Slope of the convergence path of the control point to target path is (A17) $\tan \Phi =\frac{-n}{L(n)}=\frac{-\sqrt{2{\mathit{a}}_{n}n+b}}{U}$(A17) With $n\to 0$,

• if $b>0$, $\tan \Phi \to \frac{-\sqrt{b}}{U}$, that is, control point will converge to target path with negative residual slope (for $n>0$), indicating a more aggressive integration into the target path.

• if b = 0, $\tan \Phi$ will converge to 0 sub-linearly with respect to n, same as L, but L = 0 should be avoided, which will be discussed later.

• if $b<0$, $\tan \Phi$ will converge to 0 with a positive $n=\frac{-b}{{\mathit{a}}_n}$, and $L\to \mathrm{\infty }$. Without introducing a separate $L\in \mathbb{R}$ for $n<\frac{-b}{{\mathit{a}}_n}$ and letting $L\to \mathrm{\infty }$ (preventing L jumping into $\mathbb{C}$) means path of control point will be parallel to target path with deviation of $n=\frac{-b}{{\mathit{a}}_n}$, indicating a boundary tolerance for lateral deviation, this is not good for path tracking. Hence $b<0$ should be avoided in principle.

In case $b<0$, a separate minimum L should be introduced to avoid singularity and jumping into complex domain; for b = 0 and $b>0$, the purpose of a minimum monotonic L is to avoid extremely small preview distance, which can cause unnecessary over-reaction, same as for daily driving, where human drivers' preview time normally varies from $0.5\mathrm{s}$ to $2\mathrm{s}$ depending on the speed. A minimum preview distance $L_0$ can be set according to a boundary layer of the lateral deviation, similar to what is observed for $b<0$, but with a minimum preview distance, control point will continue to converge, with slope smaller (quicker) than $b<0$ and bigger than (slower) than $b\geqq 0$, when L is monotonic. (A18) $L_0=\frac{U{n}_0}{\sqrt{2{\mathit{a}}_n{n}_0+b}}$(A18) For example, with ${\mathit{a}}_n=1{\mathrm{ms}}^{-2}$, $U=20\mathrm{m}/\mathrm{s}$, a boundary of $0.1\mathrm{m}$ will give a minimum $L_0$ of $1.83\mathrm{m}$ when b = 1, and $4.47\mathrm{m}$ when b = 0. And it should be noted that shorter preview distance leads to more aggressive correction to target path, which agree with what the slope suggests. And, with the minimum $L_0$, the residual negative slope for $b>0$ is also eliminated, hence discontinuity is avoided.

The above discussion is based on $n>0$, that is, control target on the left side of target path, the conclusion holds for $n<0$ because only the magnitude of n influences, which leads to, (A19) $L(n)=max(\frac{U\left|n\right|}{\sqrt{2{\mathit{a}}_n\left|n\right|+b}},L_0)$(A19)

Figure A2. Different convergence patterns corresponding to the choices of variable preview distance factor b.