Keywords

1 Introduction

For a autonomous vehicle, the control system utilized the real-time position and the kinematic model to track the target path which is given by a planning system precisely. The control system is composed of the lateral control and longitudinal control system, and the lateral control attracts more attention due to its difficulty in the previous researches and most of the researches for lateral control are proposed based on classical control theory [1,2,3,4]. In [1], Fenton proposed a lateral control system based on the root-locus. In [5], Kosecka proposed a vehicle sensing system model and estimate the dynamic property of the lateral error and heading-angle error based on the look ahead distance, which is widely used in later researches [5,6,7,8]. In [9], a time-variant look ahead distance is proposed to adapt to different speed. In [10], a fuzzy control strategy is proposed to handle the nonlinear characteristics and uncertainty of the system.

In most of the previous researches, a certain moving direction (forward or backward) was considered or constrained and only one look ahead distances was proposed to estimate the control errors. In this paper, multiple look ahead distances are calculated for lateral error and heading-angle error and multiple fuzzy inference engines are proposed with vehicle speed and the above-mentioned errors. Moreover, a kinematic model of mixer truck was proposed in this paper to address the length and width of the concrete mixer truck.

2 Kinematic Modeling

In previous researches, a linear model was proposed as a simplified model of the real kinematic model of the vehicle. But in most cases, the simplified linear model can not capture the real dynamic properties of the vehicle. Especially when the length of the vehicle is very large (e.g. a concrete truck), the errors of the simplified linear model is unacceptable.

In this paper, a Unicycle model is proposed to describe the vehicle which is shown in Fig. 1. X axis refers to the east and the Y axis refers to the North; P0(X0, Y0) is the center point of the rear axis of the vehicle; P0(X0, Y0) is the center point of the front axis of the vehicle; L is the length between different axis; φ is the heading angle of the vehicle and θ is the steering angle of the vehicle.

Fig. 1
A graph of Y versus X plots a vehicle of length L, heading at an angle phi from the vertical axis. The front wheels of the vehicle are inclined at an angle theta from the direction of the vehicle towards the horizontal axis.

Unicycle model of the vehicle

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As such, the dynamic property of the vehicle can be described as follows:

$$ \left[ {\begin{array}{*{20}l} {\dot{X}_{0} } \\ {\dot{Y}_{0} } \\ {\dot{X}_{1} } \\ {\dot{Y}_{1} } \\ {\dot{\varphi }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\nu \sin \varphi } \\ {\nu \cos \varphi } \\ {\nu \sin (\varphi + \theta )} \\ {\nu \cos (\varphi + \theta )} \\ {\frac{\nu \tan \theta }{{\text{L}}}} \\ \end{array} } \right] $$
(1)

3 Control

3.1 Match Point

The match point Pnow(Xnow, Ynow) is the point on the reference path which meets the following requirements: (1) the heading of Pnow on the path should match the heading angle of vehicle; (2) the distance between the match point and the real position should be the smallest.

3.2 Look Ahead Points

Multiple look ahead points are calculated in this section to estimate the corresponding look-ahead distances.

The lateral error look ahead point Ppos(Xpos, Ypos) and the heading-angle look ahead point Pang(Xang, Yang) are defined as follows:

$${N}_{pos}=\mathrm{argmin }\left|\sum_{i={N}_{now}}^{{i}^{*}}\sqrt{{{(x}_{i+1}-{x}_{i})}^{2}+{{(y}_{i+1}-{y}_{i})}^{2}}-{d}_{p}\right|$$
$${N}_{ang}={\text{argmin}} \left|\sum_{i={N}_{now}}^{{i}^{*}}(arc{\text{tan}}\frac{{x}_{i+1}-{x}_{now}}{{y}_{i+1}-{y}_{now}})-{d}_{a}\right|$$
(2)

dp and da are lateral error look ahead and heading-angle look ahead distance respectively, and Npos, Nang are the sequence number of Ppos and Pang in the reference path respectively.

3.3 Lateral Error and Heading-Angle Error

The lateral error and heading-angle error are calculated as follows:

$${E}_{pos}=\frac{1}{{{N}_{pos}-N}_{now}}\sum_{i={N}_{now}}^{{N}_{pos}}{x}_{i}$$
(3)
$${E}_{ang}=\sum_{i={N}_{now}}^{{N}_{ang}}arc \tan(\frac{{x}_{i}-{x}_{now}}{{y}_{i}-{y}_{now}})$$
(4)

3.4 Fuzzy Inference Engines

The fuzzy inference engines are utilized to calculate the target steering with three input (speed v, heading angle error Eang, lateral error Epos) and two outputs (θang and θpos), and every parameter is divided into 5 fuzzy sets according to their own value set as shown in Fig. 2.

Fig. 2
4 line graphs titled Angle error, Speed, Lateral offset, and Steering wheel angle are labeled a to d. They plot the degree of membership versus E subscript a n g, degree of membership versus V, degree of membership versus E subscript p o s, and degree of membership versus theta. The plotted curves include N B, N S, Z E, P S, and P B in different trends.

Membership function of 5 parameters

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And the target wheel angle of the vehicle is given below:

$$\theta =\gamma *{\theta }_{ang}+(1-\gamma ){\theta }_{pos}$$
(5)

4 Experiments

4.1 Simulations

In this section, two simulations are conducted to show the effectiveness of our work. The original one look ahead distance method is compared with our two look ahead distances method in the first simulation; And the PID control method is compared with our fuzzy method to show the effectiveness of the proposed work.

In the first simulation, the reference path is shown as below, which includes a S-shape curve and a high-frequency sine noise to estimate the precision of the proposed control method. And the simulation is shown in Fig. 3.

Fig. 3
2 sets of 5 graphs each highlight simulation results for one look ahead and proposed two look ahead distance methods. They include graphs of y versus x and 2 insets plotting reference and simulation curves, E subscript p o s versus time, E subscript a n g versus time, speed versus time, and a subscript b t versus time.

Simulation result of one look ahead distance method (left) and the proposed two look ahead distance method (right)

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In the simulation, two special condition (S-shape curve and a high-frequency sine noise) are zoomed in to show the precision of the proposed work. The simulation results shows that the original one look ahead distance method can’t handle the noise quite precisely and a cut-corner is observed when the look ahead distance is quite big. However, our work can handle the noise quite well and the precision is also acceptable.

In the second simulation, the comparison results of PID control method and our method is given. The above two look ahead distance method is used for both control method. The parameter of both control method are fine-tuned with our best effort. The reference path of the second simulation include an S-shape curve, a sharp turn and a straight-way. The simulation result is shown in Fig. 4.

Fig. 4
2 sets of 5 graphs each highlight simulation results for the P I D and proposed fuzzy control methods. They include graphs of y versus x and insets plotting reference and simulation curves, E subscript p o s versus time, E subscript a n g versus time, speed versus time, and a subscript b t versus time.

Simulation result of PID control method (left) and the proposed fuzzy control method (right)

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4.2 On-Board Experiments

The proposed lateral control method has been experimented on our unmanned concrete mixer truck (Fig. 5) for about 500 km and the maximum lateral error for a straight-way is less than 30 cm and for a sharp curve is less than 80 cm, which is acceptable for most conditions of the concrete mixer truck, and the results are shown in Figs. 6 and 7.

Fig. 5
A photo of a concrete mixer truck in a large, enclosed area with traffic cones placed near it.

Our unmanned concrete mixer truck

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Fig. 6
A graph titled Figure 1 plots frequency on both axes and an almost overlapping set of 2 curves indicating the real trajectory of errors, and a reference line for the same. The curves indicate an asymmetric shape in an increasing trend, overlapping in the initial and final stages.

Tracking errors of our model for a U-shape curve (blue line is the reference path, red line is the real trajectory)

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Fig. 7
A graph titled Figure 1 plots frequency on both axes and an almost overlapping set of 2 curves in increasing trends indicating the real trajectory of errors, and a reference line for the same. The curves in the initial stages are slightly apart but overlap during the steady increase.

Tracking errors of our model for a backward reference path (blue line is the reference path, red line is the real trajectory)

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5 Conclusions and Future Work

In this paper, multiple look ahead distances are calculated for lateral error and heading-angle error and multiple fuzzy inference engines are proposed with vehicle speed and the above-mentioned errors. Moreover, a kinematic model of mixer truck was proposed in this paper to address the length and width of the concrete mixer truck. Satisfactory simulation and on-board experiments results are obtained for our model.

Future work may include longitudinal control model and an analysis of the effect of speed for our model with more on-board experiments.