A non-overshooting controller for vehicle path following

Abstract

Unmanned pavement construction is of great significance in China, and one of the most important issues is how to follow the designed path near the boundary of the pavement construction area to avoid curbs or railings. In this paper, we raise a simple yet effective controller, named the proportional-integral-radius and improved particle swarm optimization (PIR-IPSO) controller, for fast non-overshooting path-following control of an unmanned articulated vehicle (UAV). Firstly, UAV kinematics model is introduced and segmented UAV steering dynamics model is built through field experiments; then, the raw data collected by differential global positioning system (DGPS) is used to build the measurement error distribution model that simulates positioning errors. Next, line of sight (LOS) guidance law is introduced and the LOS initial parameter is assigned based on human driving behavior. Besides, the initial control parameters tuned by the Ziegler-Nichols (ZN) method are used as the initial iterative parameters of the PSO controller. An improved PSO fitness function is also designed to achieve fast non-overshoot control performance. Experiments show that compared with the PSO, ZN and ZN-PSO controller, the PIR-PSO-based controller has significantly less settling time and almost no overshoot in various UAV initial states. Furthermore, compared with other controllers, the proposed PIR-IPSO-based controller achieves precise non-overshoot control, relatively less settling time and centimeter-level positioning error in various initial deviations.

Keywords

Non-overshooting control unmanned articulated vehicle line of sight guidance improved particle swarm optimization Ziegler-Nichols PI control settling time differential global positioning system

Introduction

The motivation of this paper comes from path following in asphalt pavement construction, as shown in Figure 1, where an unmanned articulated vehicle (UAV), such as the road roller, follows the planned path from the starting point in numerical order. In addition to general vehicle trajectory tracking requirements that needs to be met, the following two actual engineering requirements also need to be considered. Firstly, to achieve good asphalt pavement construction, UAV must work at a constant speed during pavement construction. Secondly, if there is an overshoot when UAV turns to the first or the last path, it is very likely to hit the curbstone on one side of the road and this may cause construction accidents, which highlights the significance of non-overshoot control. Thirdly, since the starting point is usually near the construction area, the slow convergence of the lateral deviation to zero is not permitted. In short, a fast non-overshooting and uniform speed UAV path following control system needs to be designed.

Figure 1.

Asphalt pavement construction process. The construction area is surrounded by two straight segments and two curved areas representing the curb and the articulated vehicle follows the planned paths in sequence from ① to ⑧, where ①, ④ and ⑦ means the vehicle is moving forward, ②, ⑤ and ⑧ means the vehicle is driving in reverse, and ③ and ⑥ means the vehicle is turning to the next path.

UAV path following control is the most important issue and it usually involves three subsystems, that is, the navigation subsystem, the guidance subsystem and the control subsystem. The navigation subsystem obtains real-time UAV status parameters including position, heading angle and velocity through various sensors (Wang et al., 2020; Zhang et al., 2020), the guidance subsystem calculates UAV desired status parameters based on the current status parameters obtained by the navigation subsystem and the designed guidance algorithm, and then outputs these parameters to the control subsystem (Yu et al., 2015), and the control subsystem controls the steering wheel angle and the deflection of the gear lever through various controllers (Ding et al., 2013; Khalaji, 2019). Since the performance of the navigation subsystem mainly depends on the accuracy of the sensors, the most critical issue for path following is to choose an appropriate guidance method and control strategy according to specific experimental scenarios and experimental requirements.

In recent years, theoretical research on non-overshoot control strategy has flourished. Du et al. (2019) proposes a linear non-overshooting control system based on posicast control. Lu et al. (2017) presents zero-overshoot PID control for second-order plants in companion form. Lu and Liu (2018) addresses an integral variable structure control scheme to avoid overshooting responses. In Xavier et al. (2018), robust non-overshooting tracking using continuous control for linear multi-variable systems is proposed. Aiming at overshoot performance limitation when the plant transfer function has unstable poles, Zhao et al. (2019) proposes a generalized first order reset controller with a high-pass filter. Babu et al. (2020) designs robust non-overshooting controllers for multi-input multi-output state feedback-based descriptor system, which helps to track a time-varying reference by rejecting the matched disturbances. In Wu and Jayasuriya (2013), the circle criterion, the description function and non-overshooting conditions are used for neutral stable, uncertain, single-input single-output plants under input amplitude saturation. Babu et al. (2018) introduces an integral sliding mode technique-based tracking controller to provide robustness against matched disturbances. In Zhu and Zhao (2013), a controller using feedback linearization and global coordinate transformation techniques is presented to be applicable to the cases with a variety of reference inputs and possibly non-zero initial conditions. Aiming at the key defect of sliding mode control, namely chattering effect caused by discontinuous switching control, González et al. (2017) use a combination of several specialized sliding mode controller for fast non-overshooting response.

Correspondingly, non-overshooting control has also been widely used in various application scenarios. In De Pascali et al. (2016), a LMI-based controller is investigated, which ensures fast rise time and non-overshooting behavior for oil pressure control. In Tabatabaei and Barati-Boldaji (2017), non-overshooting PD and PID controllers are designed for DC servomechanism system. Due to the load disturbance, a disturbance observer in Qian et al. (2016) is put forward to estimate the mismatched uncertainties for permanent magnet synchronous motor servo system. In Roveda et al. (2018), robot industrial assembly control scheme is used to avoid mechanical force overshoot.

However, due to the lack of the guidance subsystem, the control lag caused by the large inertia of UAV cannot be solved. Line of sight (LOS) guidance is the most widely used among guidance methods due to its simplicity and ease of implementation and many control strategies combined with LOS has been proposed to meet various practical needs, including unmanned surface vehicle (Jiang et al., 2020; Liu et al., 2015; Rout et al., 2020; Woo et al., 2019) underwater vehicles (Sahu and Subudhi, 2017; Wang et al., 2020) and unmanned aerial vehicle (Chen et al., 2016; Chen et al., 2019; He et al., 2017; Hu et al., 2020; Wang et al., 2019; Zheng and Zou, 2016; Zuo et al., 2019), these LOS-based control strategies can compensate for control lag, but the optimized indicators are generally designed as the shortest settling time and the minimum path deviation, which inevitably leads to overshoot of the following path. This paper therefore introduces a novel guidance and control system based on the IPSO algorithm, the LOS guidance law and the PI controller, which can meet the needs of fast non-overshoot control. The parameter search process is as follows. Firstly, the appropriate initial guidance and the control parameters, including the LOS guidance radius, the proportional parameter and the integral parameter, are selected through the Ziegler-Nichols (ZN) method (Yucelen et al., 2006) and human driving experience. Secondly, these three parameters are optimized using the improved PSO (IPSO) algorithm at the same time. In addition, a novel PSO fitness function is designed to meet the requirements of our project (Chiou and Liu, 2009; Dai et al., 2018; Ye et al., 2017).

In short, this paper proposes a non-overshooting controller for articulated vehicle path following, with contributions in the following three aspects: (1) UAV steering wheel data in field experiments is used to obtain UAV steering dynamics model parameters and differential global positioning system (DGPS) precious single-point positioning in field tests is used to acquire DPGS measurement error distribution model, and the agitation error is added to the simulation experiments. (2) Only the control parameters are optimized in the existing literature. However, both the guidance parameters and the control parameters are optimized. (3) The Rectified Linear Unit (ReLU) function is introduced to the IPSO fitness function to achieve precise non-overshoot control under various initial parameters. The rest of this paper is organized as follows. Section 2 introduces UAV control system and every part of it in detail. The experimental results are given in Section 3 and the conclusion of this study is summarized in Section 4. Additionally, Table 1 describes all symbols used in this paper.

Table 1.

Description of all symbols.

Name	Description
Θ_D,Θ_C,ΔΘ	Desired UAV heading angle, current UAV heading angle and angle deviation between desired and current UAV heading angle
x, y,	Position in WE and NS directions relative to any fixed point
γ, ω_γ	UAV articulated steering angle and UAV articulated steering angular velocity
ξ_f,ξ_r,ξ_c	Front wheel viewed as state reference point (x_f, y_f, Θ_f, γ_f), rear wheel viewed as state reference point (x_r, y_r, Θ_r, γ_r) and current UAV status parameters (x_c, y_c, Θ_c, γ_c)
$\tilde{ξ}$ _, ${\tilde{ξ}}_{c}$	Estimated ξ and estimated current ξ_c
θ_cs,θ_csc	UAV current steering wheel angle and UAV current steering wheel control angle
K_ZN, L_ZN, T_ZN	Static amplification factor, capacity lag time and time constant of ZN tuned method
ψ _LOS, R_LOS	LOS angle and radius
Δl	Lateral deviation
ω ₁ to ω₅	Weight coefficient of improved PSO fitness function
l _AR, v	Distance from UAV front or rear axle to the hing point and UAV speed

PIR-IPSO-based UAV control system

As shown in Figure 2, the UAV control system based on proportional-integral-radius and improved particle swarm optimization (PIR-IPSO algorithm includes the following six parts: UAV kinematic model, UAV steering dynamic model, DGPS, LOS guidance, PI controller and IPSO algorithm. The details of these six parts are shown in sections 2.1 to 2.6 respectively.)

Figure 2.

PIR-IPSO-based UAV control system.

UAV kinematic model

UAV kinematic model can be simplified with two wheels and an articulated frame, as shown in Figure 3, $ξ_{c} = (x_{c}, y_{c}, Θ_{c}, γ_{c})$ indicate UAV current horizontal, vertical coordinates, heading angle and articulated steering angle. When UAV is moving forward, the midpoint of the front wheel is regarded as the current state reference point, and the expression is $ξ_{f} = ξ_{c}$ . When UAV is reversing, the midpoint of the rear wheel is regarded as the current state reference point, and the expression is $ξ_{r} = ξ_{c}$ . We take the forward movement of UAV as an example and the relationship between these four parameters is shown in formula (1)

\begin{matrix} Θ_{r} = Θ_{f} + γ \\ x_{r} = x_{f} - l_{AR} [\cos Θ_{f} + \cos (Θ_{f} + γ)] \\ y_{r} = y_{f} - l_{AR} [\sin Θ_{f} + \sin (Θ_{f} + γ)] \end{matrix}

(1)

where $γ$ is articulated steering angle and $l_{AR}$ is the distance from the front or rear axle to the hinge point. UAV is longitudinally symmetrical in structure. Assuming that the wheels are in point contact with the ground, then the UAV kinematics model is simplified to plane motion and there is no lateral and longitudinal slip. In this case, the speed perpendicular to $v_{f}$ and $v_{r}$ direction is 0, and the expression is

\begin{matrix} \frac{d x_{f}}{dt} \sin Θ_{f} - \frac{d y_{f}}{dt} \cos Θ_{f} = 0 \\ \frac{d x_{r}}{dt} \sin Θ_{r} - \frac{d y_{r}}{dt} \cos Θ_{r} = 0 \end{matrix}

(2)

Where $\frac{d x_{f}}{dt} 、$ $\frac{d y_{f}}{dt} 、$ $\frac{d x_{r}}{dt} 、$ $\frac{d y_{r}}{dt}$ are the position changes of the midpoint of the front and rear wheels with time. When UAV is driving at a constant speed $v_{f}$ in the construction area, the kinematic equation can be inferred from equations (1) and (2)

[\begin{matrix} \frac{d x_{f}}{dt} \\ \frac{d y_{f}}{dt} \\ \frac{d Θ_{f}}{dt} \\ \frac{d γ}{dt} \end{matrix}] = [\begin{matrix} v_{f} \cos Θ_{f} \\ v_{f} \sin Θ_{f} \\ \frac{v_{f} \sin γ}{l_{AR} (\cos γ + 1)} \\ 0 \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ \frac{l_{AR}}{l_{AR} (\cos γ + 1)} \\ 1 \end{matrix}] ω_{γ}

(3)

where $ω_{γ} = \frac{d γ}{d t}$ is heading angular velocity.

Figure 3.

UAV kinematic model.

UAV steering dynamics model

Field experiments are used to construct UAV steering dynamics model. We turn the steering wheel in place and record the change of the steering wheel angle over time. Open-loop transfer function can be expressed as Xinjing et al. (2017) and Ma et al. (2013). As shown in Figure 4, the least squares fitting is used to find the optimal parameters. T is 3.3s and $κ$ is 198

θ_{s} = min {\frac{K}{s (1 + T s)}, \frac{κ}{s^{2}}}

(4)

Figure 4.

UAV steering wheel angle and its fitting curves.

Differential global positioning system positioning error model

Since global positioning system-real time kinematic (GPS-RTK) is widely used for precise point position (Xu et al., 2020; Yan et al., 2020), this part uses the raw data in field experiments to determine DGPS measurement error distribution model.

To enable the intuitive understanding of positioning accuracy, 121 single-point positioning data after coordinate system transformation are plotted in meters in Gaussian plane format in Figure 5. In the case the average value is taken as the true value and it is set to 0, the error is no more than 0.01 meter at the 95% confidence interval. (-3.34×10⁻⁴∼3.34×10⁻⁴ meter for x-axis and -3.72×10⁻⁴∼3.72×10⁻⁴ meter for y-axis)

Figure 5.

121 single-point positioning data plotted in meters in Gaussian plane format.

The Lilliefors test can check whether the collected data come from a normally distributed population and experiments show that these raw data conform to a normal distribution with a variance of 3.85 × 10⁻⁶, as shown in Figure 6.

Figure 6.

Histogram of the raw data and normal distribution curve of DGPS.

LOS proportional guidance law

LOS guidance claims that vehicle trajectory converges to the preset path if vehicle heading is aligned with so-called LOS angle ψ_LOS. The expression is

ψ_{LOS} = \arctan \frac{y_{LOS} - y}{x_{LOS} - x}

(5)

Where the position of the LOS coordinate $(x_{LOS}, y_{LOS})$ at each moment is given by formula (6)

(y_{LOS} - y)^{2} + (x_{LOS} - x)^{2} = (n L_{pp})^{2}

(6)

Where nL_pp means a distance of n times vehicle length L_pp.,nL_pp is replaced with R_LOS and its initial value is set to 4 meters according to human driving experience. As shown in Figure 7, $ψ_{p}$ and $ψ$ are defined as rotate clockwise in the north direction to the preset path and along the direction of the master-slave antenna, respectively. $ψ_{d} = ψ - ψ_{p}$ . Fossen and Pettersen (2014) proves the lateral tracking error $e$ is exponentially asymptotically stable when LOS proportional guidance law is $ψ_{d} = ψ_{LOS} = \arcsin (e / R_{LOS})$ .

Figure 7.

The principle of line-of-sight guidance.

Proportional-integral controller

Due to the control lag caused by the steering mechanism, UAV cannot obey the LOS proportional guidance law in real time. Therefore, P controller cannot meet the needs of UAV trajectory tracking. PI、PD and PID can all be alternative control strategies. However, the drum of the vibratory roller may produce high-frequency vibration during pavement construction, and the error will be sharply amplified under the differential parameters, which is not allowed during trajectory tracking. Finally, the PI controller is adopted.

Three parameters, including the static amplification factor K_ZN, capacity lag time L_ZN and time constant T_ZN, can be read through ZN-tuned method, in which an open loop step response can be obtained from a bump test. The proportional and integral parameters can be obtained by formulae (7)– (9)

K_{p} = \frac{0.9 T_{ZN}}{K_{ZN} L_{ZN}}

(7)

T_{i} = \frac{L_{ZN}}{0.3}

(8)

T_{d} = \infty

(9)

Since UAV steering wheel angle is almost [-820º, +820º] and the articulated steering angle is [-14º, +14º], the static amplification factor is set as $K_{s γ} = θ_{s max} / γ_{max} \approx 70$ . Accordingly, $K_{p}$ and $T_{i}$ are approximately 1.22 and 0.15 according to Figure 8.

Figure 8.

Proportional and integral parameters selected by the Ziegler-Nichols method.

Improved particle swarm optimization

Integral square error (ISE), integral absolute error (IAE), integral time absolute error (ITAE) and integral time square error (ITSE) are often used to evaluate error performance index. IAE is selected as the fitness function in Khoud et al. (2018)

J = \int_{0}^{\infty} (ω_{1} | Δ l (t) | + ω_{2} | Δ Θ (t) |) dt

(10)

where $Δ l$ is the lateral deviation. Weight coefficient $ω_{1}$ , $ω_{2}$ are two constant parameters and they are satisfied with the constraint $ω_{1} > > ω_{2} \approx ω_{3}$ . Although the fitness function mentioned in formula (10) has good performance in settling time and accuracy, it causes overshoot. In this paper, the ReLU function is introduced to the original fitness function

\begin{matrix} J = \int_{0}^{\infty} (ω_{1} | Δ l (t) | + ω_{2} | Δ Θ (t) | + ω_{3} | γ | + ω_{4} ReLU (Δ l (t)) + ω_{5} ReLU (Δ = 5 %)) dt \end{matrix}

(11)

Where a large weight coefficient $ω_{4} > > ω_{1}$ is given to avoid overshoot during path following. A small weight coefficient $ω_{5}$ is introduced to reduce settling time and Δ=5% means 5% error band of the target value. When the optimization of LOS radius causes no solution to the cost function, the fitness function value is set close to approach infinity. In this paper, PSO with the proposed ReLU-integrated fitness function is defined as improved PSO (IPSO).

Simulation

Parameter settings

Parameters for UAV control system and PSO are shown in Table 2.

Table 2.

Parameter settings.

Parameters	Values	Parameters	Values
l _AR	3.69 m	ω ₁	0.99
UAV speed v	2 m/s	ω ₂	0.01
γ	[-14º, +14º]	ω ₃	0.01
Sample time Δt	0.1s	ω ₄	100000000
Population size	40	ω ₅	0.1
Number of iterations	40	The range of K_p	[0.001,100]
The range of Radius	[0.001,10]	The range of T_i	[0.001,100]

Comparison of PSO, ZN, ZN-PSO and PIR-PSO under a set of experiment

2 meters and 0 degree deviation is set and random initial three parameters using the PSO algorithm, appropriate initial PI control parameters with fixed radius using the ZN-PSO algorithm (Khanduja and Bhushan, 2017), appropriate initial LOS guidance and PI control parameters using the PIR-PSO algorithm are tested and the changes of parameters and fitness value in the process of multi-parameter optimization are shown in Figure 9.

Figure 9.

Parameter and fitness iterations of PSO (a), ZN-PSO (b) and PIR-PSO (c).

The four final parameters shown in Table 3 are set as the UAV’s control system parameters to compare the actual control effects of these four algorithms. As shown in Figure 10, the PSO algorithm cannot follow the desire path in 40 seconds, the pure ZN tuned algorithm, the ZN tuned PSO algorithm and the PIR-PSO algorithm take about 12.0 seconds, 8.0 seconds and 5.2 seconds to reach settling time, respectively. (Settling time refers to the response reaches and stays within 5% of the final value.)

Table 3.

Fitness value and final parameters of PSO, ZN, ZN-PSO and PIR-PSO.

Algorithm	Initial parameters(Kp, Ti and Radius)	Average fitness value	Final parameters(Kp, Ti and Radius)
PSO	Random	649.1	[71,100,10]
ZN	[1.22, 0.15, 4]	151.3	[1.22, 0.15, 4]
ZN-PSO	[1.22, 0.15, 4]	69.3	[1.81,0.027, 4]
PIR-PSO	[1.22, 0.15, 4]	67.6	[2.87,0.034,2.13]

Figure 10.

Path following performance of PSO, ZN, ZN-PSO, PIR-PSO.

Comparison of five algorithms in multiple experiments

Comparison of PIR-PSO and PIR-IPSO in multiple experiments

To further explore the effect of the proposed algorithm, various initial deviations are set, the performance of steady-state errors, overshoot and settling time is evaluated and the quantitative results are recorded and analyzed. Five groups of different initial angles and lateral deviations, that is, 1 meter and 0 degree initial deviation, 2 meters and -5 degrees initial deviation, 2 meters and 0 degree initial deviation, 2 meters and 5 degrees initial deviation and 3 meters and 0 degree initial deviation, are set to verify the performance of the proposed PIR-IPSO algorithm and the PIR-PSO algorithm, each group is performed 20 times. The box plot of these two algorithms is shown in Figure 11, marked in black and blue, respectively. In Figure 11(a), it is observed that all the maximum deviations of the PIR-IPSO algorithm are less than 0 meters, that means there is no overshoot. However, the PIR-PSO algorithm inevitably produces a few centimeters of overshoot. Figure 11(b) shows the steady-state error, which is caused by the added agitation error. It can be inferred from Figure 11(c) that the settling time of the PIR-IPSO algorithm is slightly longer than the PIR-PSO algorithm.

Figure 11.

(a) Maximum deviation, (b) final deviation and (c) settling time under different initial deviations between PIR-PSO and PIR-IPSO (20 experiments per group).

Comparison of five algorithms in multiple experiments

As shown in Table 4, 95% confidence interval is used to evaluate the overshoot, the steady-state error and the settling time of the five algorithms. The PSO algorithm is unstable when UAV path following. Due to the agitation error, the settling time of the pure ZN tuned algorithm varies greatly. Besides, ZN tuned PSO algorithm has relatively large overshoot. Compared with the above three algorithms, the PIR-PSO algorithm and the PIR-IPSO algorithm perform better in overshoot, steady-state error and settling time. In addition, although the settling time of the PIR-IPSO algorithm is longer than that of the PIR-PSO algorithm, the PIR-IPSO algorithm achieves precise non-overshoot control under various initial deviations.

Table 4.

Quantitative path following performance of the five algorithms.

Algorithm	Initial lateral deviation	Initial angle deviation	Stability	95% confidence interval of 20 experiments
Algorithm	Initial lateral deviation	Initial angle deviation	Stability	Overshoot	Steady-state error	Settling time
PSO	1m/2m/3m	−5°/0°/5°	Unstable	\	\	\
ZN	1 m	0°	Stable	(0.00%, 0.71%)	(0.000m, 0.011m)	(5.96s, 9.10s)
	2 m	−5°	Stable	(0.16%, 0.79%)	(-0.001m, 0.009m)	(5.80s, 5.97s)
	2 m	0°	Stable	(0.00%, 0.46%)	(0.007m, 0.015m)	(6.82s, 9.94s)
	2 m	5°	Stable	(0.00%, 0.00%)	(0.013m, 0.023m)	(10.30s, 10.48s)
	3 m	0°	Stable	(1.74%, 2.47%)	(-0.019m, -0.010m)	(6.72s, 6.90s)
ZN-PSO	1 m	0°	Stable	(0.61%,1.27%)	(0.000m, 0.858m)	(6.75s, 7.08s)
	2 m	−5°	Stable	(4.05%, 5.33%)	(-0.051m, -0.035m)	(7.06s, 7.26s)
	2 m	0°	Stable	(3.55%, 4.81%)	(-0.045m, -0.032m)	(7.09s, 7.29s)
	2 m	5°	Stable	(2.04%, 3.06%)	(-0.030m, -0.016m)	(7.17s, 7.39s)
	3 m	0°	Stable	(8.37%, 9.74%)	(-0.095m, -0.080m)	(7.00s, 7.20s)
PIR-PSO	1 m	0°	Stable	(0.00%, 0.93%)	(0.000m, 0.010m)	(4.02s, 4.79s)
	2 m	−5°	Stable	(1.26%, 2.43%)	(-0.007m, 0.003m)	(5.10s, 5.21s)
	2 m	0°	Stable	(0.00%, 1.19%)	(0.002m, 0.014m)	(5.53s, 6.20s)
	2 m	5°	Stable	(1.20%, 3.00%)	(-0.003m, 0.003m)	(5.06s, 5.18s)
	3 m	0°	Stable	(0.00%, 0.72%)	(-0.002m, 0.009m)	(6.65s, 7.29s)
PIR-IPSO	1 m	0°	Stable	(0.00%, 0.00%)	(0.007m, 0.010m)	(5.34s, 6.60s)
	2 m	−5°	Stable	(0.00%, 0.00%)	(0.012m, 0.019m)	(5.38s, 6.66s)
	2 m	0°	Stable	(0.00%, 0.00%)	(0.011m, 0.017m)	(5.68s, 6.66s)
	2 m	5°	Stable	(0.00%, 0.00%)	(0.006m, 0.014m)	(5.80s, 7.34s)
	3 m	0°	Stable	(0.00%, 0.00%)	(0.012m, 0.020m)	(7.43s, 8.80s)

Conclusions

A simple and effective path following control method, named the PIR-IPSO algorithm, has been raised for UAV fast non-overshooting control. Raw data from field experiments is used to obtain UAV steering dynamics model parameters and DGPS measurement error distribution model. The guidance and control laws are developed using the LOS guidance law and PI controller, respectively. The LOS guidance law adopted has been proved to exponentially asymptotically stable using Lyapunov stability criterion and the reasons for using PI controller instead of P, PD or PID controller are given. The guidance and control parameters are optimized by the PIR-PSO algorithm, and to achieve fast non-overshooting control, a new fitness function is designed, namely the PIR-IPSO algorithm. Implementing the developed guidance and control laws, simulations are carried out, and the results are illustrated and explained. Comparing the desired and actual paths, it is observed that the lateral deviation can quickly converge to zero. Compared with the PSO algorithm, the ZN algorithm and the ZN tuned PSO algorithm, the PIR-PSO-based algorithm exhibits a superior performance in settling time. The results obtained from among overshoot, settling time and steady-state error are compared and it is observed that, of these controllers, the PIR-IPSO-based controller achieves precise non-overshoot control, relatively less settling time and centimeter-level positioning error in various initial deviations.

In future research, field experiments are carried out to verify the effectiveness of our proposed algorithm. Specifically, CC6200 articulated road roller is used as the controlled plant, the master-slave antenna is installed on the top of the vehicle to measure the real-time status of the roller, and the proposed algorithm is applied to the main controller based on Beckhoff CPU Module.

Footnotes

Acknowledgements

The authors would like to thank the reviewers for their valuable suggestions for improving the quality of the paper. In addition, Tong Xu is especially grateful to TingTing Chen of University College London for providing linguistic suggestions during the preparation of this manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Tong Xu

Dong Wang

References

Babu

Xavier

Bandyopadhyay

(2018) Robust nonovershooting tracking controller for descriptor systems. European Journal of Control 43(9): 57–63.

Babu

Xavier

Bandyopadhyay

(2020) Robust output regulation for state feedback descriptor systems with nonovershooting behavior. European Journal of Control 52(3): 19–25.

Chen

Lee

Liao

Kung

(2016) The design of particle swarm optimization guidance using a line-of-sight evaluation method. Computers and Electrical Engineering 54(C): 159–169.

Chen

Wang

Yang

Wang

(2019) Trajectory-following guidance based on a virtual target and an angle constraint. Aerospace Science and Technology 87(3): 448–458.

Chiou

Liu

(2009) Numerical simulation for Fuzzy-PID controllers and helping EP reproduction with PSO hybrid algorithm. Simulation Modelling Practice and Theory 17(10): 1555–1565.

Dai

Taghia

Lam

Katupitiya

(2018) Integration of sliding mode based steering control and PSO based drive force control for a 4WS4WD vehicle. Autonomous Robots 42(3): 553–568.

Ding

Mikaric

(2013) Design of an active trailer-steering system for multi-trailer articulated heavy vehicles using real-time simulations. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 227(5): 643–655.

(2019) Non-overshooting control of linear system by a simple asymptotic gain scheduling method. Transactions of the Institute of Measurement and Control 41(15): 4187–4196.

Fossen

Pettersen

(2014) On uniform semiglobal exponential stability (USGES) of proportional line-of-sight guidance laws. Automatica 50(11): 2912–2917.

10.

González

Barreiro

Dormido

Baños

(2017) Nonlinear adaptive sliding mode control with fast non-overshooting responses and chattering avoidance. Journal of the Franklin Institute, 354(7): 2788–2815.

11.

Wang

(2017) A novel sliding mode guidance law without line-of-sight angular rate information accounting for autopilot lag. International Journal of Systems Science 48(16): 3363–3373.

12.

Han

Xin

(2020) Three-dimensional guidance for various target motions with terminal angle constraints using twisting control. IEEE Transactions on Industrial Electronics 67(2): 1242–1253.

13.

Jiang

Peng

Wang

Chen

CLP

(2020) Line-of-sight target enclosing of an underactuated autonomous surface vehicle with experiment results. IEEE Transactions on Industrial Informatics 16(2): 832–841.

14.

Khalaji

(2019) PID-based target tracking control of a tractor-trailer mobile robot. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(13): 4776–4787.

15.

Khanduja

Bhushan

(2017) Intelligent control of CSTR using IMC-PID and PSO-PID controller. In: 1st IEEE International Conference on Power Electronics, Intelligent Control and Energy Systems, ICPEICES 2016.

16.

Khoud

K Ben

Bouallègue

Ayadi

(2018) Design and co-simulation of a fuzzy gain-scheduled PID controller based on particle swarm optimization algorithms for a quad tilt wing unmanned aerial vehicle. Transactions of the Institute of Measurement and Control 40(14): 3933–3952.

17.

Liu

Sun

Zou

(2015) Integrated line of sight and model predictive control for path following and roll motion control using rudder. Journal of Ship Research 59(2): 99–112.

18.

Liu

(2017) Initial-value compensation of a proportional-integral-derivative controller for non-overshooting motion. Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering 231(8): 626–637.

19.

Liu

(2018) An integral variable structure control scheme for non-overshooting responses. Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering 232(4): 442–452.

20.

Chirarattananon

Fuller

Wood

(2013) Controlled flight of a biologically inspired, insect-scale robot. Science 340(6132): 603–607.

21.

Pascali

Cocetti

Biral

, et al. (2016) LMI-based non-overshooting pressure control design for a wet clutch. 2016 IEEE Conference on Control Applications (CCA). IEEE. pp. 329–334.

22.

Pascali

Cocetti

Biral

, et al. (2016) LMI-based non-overshooting pressure control design for a wet clutch. In: 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, Argentina, 19–22 September 20016, pp. 329–334. IEEE.

23.

Qian

Luo

Sun

(2016) Improved nonlinear sliding mode control based on load disturbance observer for permanent magnet synchronous motor servo system. Advances in Mechanical Engineering 8(4): 1–12.

24.

Rout

Cui

Han

(2020) Modified line-of-sight guidance law with adaptive neural network control of underactuated marine vehicles with state and input constraints. IEEE Transactions on Control Systems Technology 28(5): 1902–1914.

25.

Roveda

Pedrocchi

Beschi

Molinati Tosatti

(2018) High-accuracy robotized industrial assembly task control schema with force overshoots avoidance. Control Engineering Practice, 71(October): 142–153.

26.

Sahu

Subudhi

(2017) Potential function-based path-following control of an autonomous underwater vehicle in an obstacle-rich environment. Transactions of the Institute of Measurement and Control 39(8): 1236–1252.

27.

Tabatabaei

Barati-Boldaji

(2017) Non-overshooting PD and PID controllers design. Automatika 58(4): 400–409.

28.

Wang

Yao

, et al. (2020) A novel SINS/DVL tightly integrated navigation method for complex environment. IEEE Transactions on Instrumentation and Measurement 69(7): 5183–5196.

29.

Wang

Meng

(2019) Path following of the autonomous airship with compensation of unknown wind and modeling uncertainties. Aerospace Science and Technology 93(10): 1–12.

30.

Wang

Yao

Zhang

(2020) Path planning under constraints and path following control of autonomous underwater vehicle with dynamical uncertainties and wave disturbances. Journal of Intelligent and Robotic Systems: Theory and Applications 99(3): 891–908.

31.

Woo

Kim

(2019) Deep reinforcement learning-based controller for path following of an unmanned surface vehicle. Ocean Engineering, 183(November): 155–166.

32.

Jayasuriya

(2013) A robust synthesis methodology for neutrally stable uncertain SISO plants under input amplitude saturation. International Journal of Systems Science 44(3): 461–470.

33.

Xavier

Bandyopadhyay

Schmid

(2018) Robust non-overshooting tracking using continuous control for linear multivariable systems. IET Control Theory and Applications 12(7): 1006–1011.

34.

Xinjing

Yibo

Fei

Shijiu

(2017) Horizontal path following for underactuated AUV based on dynamic circle guidance. Robotica 35(4): 876–891.

35.

Chen

Wang

Zhang

(2020) Bidirectional sliding window for boundary recognition of pavement construction area using GPS-RTK. Applied Sciences 10(4): 1–14.

36.

Yan

Wang

Zhang

, et al. (2020) Analysis of GNSS clock prediction performance with different interrupt intervals and application to real-time kinematic precise point positioning. Advances in Space Research 65(3): 978–996.

37.

Yin

Gong

Zhou

(2017) Position control of nonlinear hydraulic system using an improved PSO based PID controller. Mechanical Systems and Signal Processing 83: 241–259.

38.

Chen

Allgöwer

(2015) Nonlinear model predictive control for path following problems. International Journal of Robust and Nonlinear Control 25(8): 1168–1182.

39.

Yucelen

Kaymakci

Kurtulan

(2006) Self-tuning PID controller using Ziegler-Nichols method for programmable logic controllers. Ifac Proceedings 39(14): 11–16.

40.

Zhang

Wang

, et al. (2020). A high-order coning error compensation algorithm under high rate maneuvering. IEEE Sensors Journal 20(1): 208–218.

41.

Zhao

Nešić

Tan

Hua

(2019) Overcoming overshoot performance limitations of linear systems with reset control. Automatica 101(3): 27–35.

42.

Zheng

Zou

(2016) Adaptive integral LOS path following for an unmanned airship with uncertainties based on robust RBFNN backstepping. ISA Transactions 65(11): 210–219.

43.

Zhu

Zhao

(2013) Non-overshooting output tracking of feedback linearisable nonlinear systems. International Journal of Control 86(5): 821–832.

44.

Zuo

Cheng

Wang

Sun

(2019) Three-dimensional path-following backstepping control for an underactuated stratospheric airship. IEEE Transactions on Aerospace and Electronic Systems 55(3): 1483–1497.