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Path Tracking Control Algorithm of Tractor-Implement

  • LIU Zhiyong , 1, 2 ,
  • WEN Changkai 2, 3 ,
  • XIAO Yuejin 2, 3 ,
  • FU Weiqiang 2, 3 ,
  • WANG Hao 2, 3 ,
  • MENG Zhijun , 1, 2, 3
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  • 1. School of Agricultural Engineering, Jiangsu University, Zhenjiang 212013, China
  • 2. National Research Center of Intelligent Equipment for Agriculture, Beijing 100097, China
  • 3. State Key Laboratory of Intelligent Agricultural Power Equipment, Beijing 100097, China
MENG Zhijun, E-mail:

Received date: 2023-08-10

  Online published: 2023-12-20

Supported by

National Key Research and Development Program of China(2022YFD200150302)

National Major Agricultural Science and Technology Project(NK202216010303)

Beijing postdoctoral work funding project(2023-ZZ-112)

Copyright

copyright©2023 by the authors

Abstract

[Objective] The usual agricultural machinery navigation focuses on the tracking accuracy of the tractor, while the tracking effect of the trailed implement in the trailed agricultural vehicle is the core of the work quality. The connection mode of the tractor and the implement is non-rigid, and the implement can rotate around the hinge joint. In path tracking, this non-rigid structure, leads to the phenomenon of non-overlapping trajectories of the tractor and the implement, reduce the path tracking accuracy. In addition, problems such as large hysteresis and poor anti-interference ability are also very obvious. In order to solve the above problems, a tractor-implement path tracking control method based on variable structure sliding mode control was proposed, taking the tractor front wheel angle as the control variable and the trailed implement as the control target. [Methods] Firstly, the linear deviation model was established. Based on the structural relationship between the tractor and the trailed agricultural implements, the overall kinematics model of the vehicle was established by considering the four degrees of freedom of the vehicle: transverse, longitudinal, heading and articulation angle, ignoring the lateral force of the vehicle and the slip in the forward process. The geometric relationship between the vehicle and the reference path was integrated to establish the linear deviation model of vehicle-road based on the vehicle kinematic model and an approximate linearization method. Then, the control algorithm was designed. The switching function was designed considering three evaluation indexes: lateral deviation, course deviation and hinged angle deviation. The exponential reaching law was used as the reaching mode, the saturation function was used instead of the sign function to reduce the control variable jitter, and the convergence of the control law was verified by combining the Lyapunov function. The system was three-dimensional, in order to improve the dynamic response and steady-state characteristics of the system, the two conjugate dominant poles of the system were assigned within the required range, and the third point was kept away from the two dominant poles to reduce the interference on the system performance. The coefficient matrix of the switching function was solved based on the Ackermann formula, then the calculation formula of the tractor front wheel angle was obtained, and the whole control algorithm was designed. Finally, the path tracking control simulation experiment was carried out. The sliding mode controller was built in the MATLAB/Simulink environment, the controller was composed of the deviation calculation module and the control output calculation module. The tractor-implement model in Carsim software was selected with the front car as a tractor and the rear car as the single-axle implement, and tracking control simulation tests of different reference paths were conducted in the MATLAB/Carsim co-simulation environment. [Results and Discussions] Based on the co-simulation environment, the tracking simulation experiments of three reference paths were carried out. When tracking the double lane change path, the lateral deviation and heading deviation of the agricultural implement converged to 0 m and 0° after 8 s. When the reference heading changed, the lateral deviation and heading deviation were less than 0.1 m and less than 7°. When tracking the circular reference path, the lateral deviation of agricultural machinery tended to be stable after 7 s and was always less than 0.03 m, and the heading deviation of agricultural machinery tended to be stable after 7 s and remained at 0°. The simulation results of the double lane change path and the circular path showed that the controller could maintain good performance when tracking the constant curvature reference path. When tracking the reference path of the S-shaped curve, the tracking performance of the agricultural machinery on the section with constant curvature was the same as the previous two road conditions, and the maximum lateral deviation of the agricultural machinery at the curvature change was less than 0.05 m, the controller still maintained good tracking performance when tracking the variable curvature path. [Conclusions] The sliding mode variable structure controller designed in this study can effectively track the linear and circular reference paths, and still maintain a good tracking effect when tracking the variable curvature paths. Agricultural machinery can be on-line in a short time, which meets the requirements of speediness. In the tracking simulation test, the angle of the tractor front wheel and the articulated angle between the tractor and agricultural implement are kept in a small range, which meets the needs of actual production and reduces the possibility of safety accidents. In summary, the agricultural implement can effectively track the reference path and meet the requirements of precision, rapidity and safety. The model and method proposed in this study provide a reference for the automatic navigation of tractive agricultural implement. In future research, special attention will be paid to the tracking control effect of the control algorithm in the actual field operation and under the condition of large speed changes.

Cite this article

LIU Zhiyong , WEN Changkai , XIAO Yuejin , FU Weiqiang , WANG Hao , MENG Zhijun . Path Tracking Control Algorithm of Tractor-Implement[J]. Smart Agriculture, 2023 , 5(4) : 58 -67 . DOI: 10.12133/j.smartag.SA202308012

0 引 言

牵引式农机具通过牵引装置挂接在拖拉机后面,由拖拉机动力牵引完成各种作业。农机导航技术多以拖拉机为控制目标进行轨迹跟踪,牵引式农机具因其铰接式连接的结构特点,会出现拖拉机和农机具轨迹不重合的现象,降低路径跟踪精度1
针对上述问题,诸多学者基于现有的拖拉机-悬挂式农机具自动导航,以拖拉机为控制对象,以农机具为控制目标,开展了非悬挂式农机具导航的研究2。目前应用较多的控制策略有比例积分微分(Proportional Integral Derivative,PID)控制、滑模控制、模糊控制、最优控制和模型预测控制。Astolfi等3利用李雅普诺夫方法研究了拖拉机-挂车系统直线运动和圆形运动的渐进镇定问题,设计了有界非线性控制律。Naderolasli等4针对拖拉机-牵引式机器人,结合李雅普诺夫方法和动态曲面控制设计了自适应神经网络约束控制器。Thanpattranon等5提出了一种利用激光测距的控制方法,用于拖拉机-牵引式挂车在果园的曲线行驶和停车控制。冯雷6建立了拖拉机-牵引式农机具动力学模型,并给出侧偏刚度的测量方法。Backman等7、Kayacan等8建立了改进的拖拉机-牵引式农机具运动学模型,并设计了基于非线性模型预测控制的路径跟踪控制器,在中低速下农机具的跟踪偏差在0.1 m以内。Karkee和Steward9建立了拖拉机-牵引式农机具运动学模型、动力学模型和轮胎松弛张度模型,设计了线性二次型最优控制器,仿真结果验证了控制器的快速性和精度。
农机田间作业环境复杂,参数扰动、负载变化、地面平整度、车轮滑转以及测量误差等因素都会影响车辆的跟踪精度,牵引式农机具灵活性差、自身重量大且结构复杂,这些特点增加了跟踪控制的难度。滑模控制与扰动和被控对象的参数无关,具有响应快、抗干扰、设计简单等优点10,该控制方法的结构是可变的,会根据系统的状态而变化,使其达到期望的目标,适用于农机的路径跟踪控制。赵翾等11针对两轴农用铰接车设计了滑模变结构控制器,前后车体由液压驱动进行折腰转向,在跟踪圆形路径时,偏差约为0.1 m。张培培等12设计了滑模控制器用于牵引式农机具的直线和圆形路径跟踪,为简化运算,所建立的模型假设铰接点位于拖拉机后轴中心。Ji等13设计了自适应二阶滑模控制器,提出了拖拉机预瞄横向偏移模型,通过对比仿真验证了控制方法的优越性。张硕等14设计了横向偏差和航向偏差双目标联合滑模控制器,拖拉机在直线路径跟踪时平均绝对偏差保持在0.049 m以内。Alipour等15首次建立了考虑车轮横向滑移和纵向滑移的非完整动力学模型,基于此设计了滑模控制器实现轮式移动机器人的跟踪控制。上述研究多以悬挂式农机具为研究对象,其挂载方式与本研究的牵引式农机具有本质区别。牵引式农机具与拖拉机的非刚性连接方式,导致农机具可绕铰接点转动,在自动导航中存在农机具与拖拉机轨迹不重合的问题,直接影响路径跟踪的精度。
本研究建立拖拉机-牵引式农机具运动学模型,采用近似线性化的方法求解偏差状态方程,设计滑模变结构控制器对牵引式农机具进行路径跟踪控制,最后通过仿真手段验证算法的准确性和快速性。

1 拖拉机-牵引式农机具系统偏差模型建立

1.1 拖拉机-牵引式农机具运动学模型建立

图1所示,在二维平面简化拖拉机-牵引式农机具系统,只考虑农机具的纵向、横向、横摆角和铰接角四个自由度。同轴的两个车轮合并为一个车轮,忽略车辆的侧向力和前进过程中的滑移,建立拖拉机-牵引式农机具运动学模型。图中XOY为平面直角坐标系,P为拖拉机与农机具相连的铰接点。
图1 拖拉机-牵引式农机具运动学模型

Fig. 1 Kinematic model for tractor-implement

根据图1建立拖拉机-牵引式农机具系统运动学微分方程16,具体算法见公式(1)
φ ˙ 1 = v t a n   δ L 1
式中:φ 1表示拖拉机的航向,(°);v表示拖拉机后轴中心速度,m/s;δ表示拖拉机前轮转角,(°);L 1表示拖拉机轴距,m;L 2表示拖拉机后轴到铰接点的距离,m。
农机具的纵向速度和横摆角速度分别由铰接点P的速度沿农机具前进方向和垂直于前进方向分解获得17,如公式(2)公式(3)所示。
v i = v c o s γ + v L 2 t a n δ s i n γ L 1
φ ˙ 2 = v s i n γ L 3 - v L 2 t a n δ c o s γ L 1 L 3
式中:v i为农机具后轴中心速度,m/s;φ 2为农机具的航向,(°);γ为拖拉机和农机具间的铰接角,(°);L 3为农机具车轴到铰接点的距离,m。
铰接角的变化率为拖拉机航向变化率和农机具航向变化率之差,铰接角变化率如公式(4)所示。
γ ˙ = φ ˙ 1 - φ ˙ 2    = v t a n δ L 1 - v s i n γ L 3 + v L 2 c o s γ t a n δ L 1 L 3

1.2 车辆-道路偏差模型建立

建立车辆-道路运动学模型描述行进过程中农机具和参考路径间的偏差变化,如图2所示,xoy为与车辆固连的动态坐标系,x轴始终为农机具前进方向,y轴垂直于x轴,o为农机具车轴中心,点Ao点前方前视距离L q处参考路径上的对应点。
图2 车辆-道路偏差模型

Fig. 2 Vehicle-road deviation model

根据车辆和参考路径间的运动学关系,得到前视距离处横向偏差的变化率和航向偏差变化率18,如公式(5)公式(6)所示。
d ˙ e = v φ e + L q φ ˙ 2     = v φ e + L q ( v s i n γ e L 3 - v L 2 t a n δ L 1 L 3 )
φ ˙ e = φ ˙ 2 - ρ v     = ( v s i n γ e L 3 - v L 2 t a n δ L 1 L 3 ) - ρ v
式中:d e表示农机具横向偏差,m;φ e表示农机具航向偏差,(°);γ e表示铰接角偏差,(°); L q表示前视距离,m;ρ表示参考路径上点A处的曲率,m-1
由于参考路径的曲率往往不太大(通常小于0.05 m-1),并且考虑稳定性和安全性,因此,设置铰接角参考值为0,车辆的实际铰接角即为铰接角偏差。所以,铰接角偏差的变化率计算如公式(7)所示。
γ ˙ e = γ ˙     = v t a n δ L 1 - v s i n γ L 3 + v L 2 c o s γ t a n δ L 1 L 3
基于车辆本身的结构和安全性19,可知,30°>γ>-30°,45°>δ>-45°,简化上述运动关系,重新定义状态方程如公式(8)所示。
x ˙ L = A x L + B u + E ρ v
式中:u为系统输入,u=δ xL 为系统误差状态量, xL =[d e φ e γ e T公式(8)中各系数矩阵的物理意义如公式(9)~公式(11)所示。
A = 0 v v L q L 3 0 0 v L 3 0 0 - v L 3
B = - v L 2 L q L 1 L 3 - v L 2 L 1 L 3 v ( L 2 + L 3 ) L 1 L 3 T
E = 0 - 1 0 T
结合公式(8)~公式(11),进行之后的滑模变结构控制器设计。

2 滑模变结构控制算法设计

拖拉机-牵引式农机具系统具有很强的非线性,第1节采用近似线性化的方法建立了线性状态方程,本节基于该状态方程进行滑模变结构控制算法设计。主要包括两部分:一是切换函数的设计,使系统具有良好的动态响应品质和稳态性能20;二是趋近律的选择,使系统快速平稳地到达滑模面上。

2.1 切换函数设计和趋近律的选择

对拖拉机-牵引式农机具自动导航系统,取滑模状态的切换函数,如公式(12)所示。
s = C x = c 1 x 1 + c 2 x 2 + + c n x n
式中:s表示切换状态函数;n表示状态变量的维数; C 表示滑模状态控制律n阶系数矩阵。
为了加快系统的响应速度,选择指数趋近律21作为趋近模态,如公式(13)所示。
s ˙ = - ε s g n ( s ) - k s
式中:ε表示等速趋近速率;k表示指数趋近律系数;sgn(s)表示符号函数。
但是采用符号函数进行仿真试验时,控制器的输出会产生剧烈的抖振,如图3(a)所示,丧失了实际应用的意义。用饱和函数sat(s)代替符号函数sgn(s)可以消除这一现象22,结果如图3(b)所示。
图3 符号函数和饱和函数下的车辆前轮转角输出

Fig. 3 Vehicle’s front wheel angle output under symbol function and saturation function

因此,趋近律选择的计算方法为公式(14)
s ˙ = - ε s a t ( s ) - k s
利用李雅普诺夫函数验证控制律收敛性,定义状态函数为公式(15)
V = 1 2 s 2
结合公式(14)公式(15),得切换函数的一阶导数,见公式(16)
V ˙ = s s ˙    = - ε s - k s 2 , s > 1           - ε s 2 - k s 2 , 1 s 0 0 , s = 0                            - ε s 2 - k s 2 , 0 s - 1 ε s - k s 2 , - 1 > s           
公式(16)根据李雅普诺稳定性理论证明闭环控制系统是渐近稳定的23

2.2 滑模控制律求解

由2.1节可知,闭环系统是渐近稳定的,然而为了更好地控制性能,还要求系统具有优良的动态响应特性。通过极点配置的方式可以使系统快速平稳地追踪到期望轨迹。
系统为三阶闭环系统,故有三个极点,其中两个主导极点距离虚轴较近,对系统影响较大,第三个极点远离两个主导极点,对系统的影响较小,可以将闭环系统近似为一对共轭极点主导的二阶系统24
设闭环主导极点为λ 1λ 2,如公式(17)所示。
λ 1,2 = - ξ w n ± j w n 1 - ξ 2
式中:λ 1,2表示两个闭环主导极点;wn 表示系统固有频率;ζ表示系统阻尼比。
为获得良好的动态响应品质,系统的超调量和峰值时间应满足以下条件,如公式(18)公式(19)所示。
σ = e x p ( - ξ π / 1 - ξ 2 ) 5 %
t p = π w n 1 - ξ 2 10
式中:σ表示系统超调量;tp 表示系统峰值时间。
求解公式(18)公式(19)wn ≥0.5,ξ≥0.7。取wn =0.5,ξ=0.8,则主导极点25公式(20)所示。
λ 1,2 = - 0.4 ± j 0.48
第三个极点应该远离两个主导极点,满足 3 |>5 1 |,取λ 3=-5。
已知指定极点,利用Ackermann公式可以求出系数向量 C,见公式(21)~公式(23)
C = e T P ( A )
e T = 0 , , 0,1 B , A B , , A n - 1 B - 1
P ( A ) = ( A - λ 1 E ) ( A - λ 2 E ) ( A - λ n - 1 E ) ( A - λ n E )
式中:n表示系统维数;λn 表示系统的闭环极点; E 表示n阶单位矩阵。
L 1=2 m,L 2=0.5 m,L 3=1.2 m,L q=2 m,配置的极点位于车速v=2 m/s附近,解得c 1=0.425,c 2=0.969,c 3=1.863。则切换函数如公式(24)所示。
s = 0.425 d e + 0.969 φ e + 1.863 γ e
ε=0.5,k=2,结合公式(8)公式(14)公式(21),得到控制律u计算公式(25)
u = 1 47.2 - 10.2 φ e + 0.9 γ e - 24 s - 6 s a t ( s ) - 0.23 ρ

3 路径跟踪控制算法联合仿真分析

3.1 仿真环境搭建

3.1.1 Carsim车辆模型设置

Carsim是针对车辆动力学的仿真软件,可以连接MATLAB/Simulink、Labview等外部环境。首先建立车体模型,从模块已有的数据库中选择带有挂车的车型,设置牵引车的车型为拖拉机,挂车的车型为单轴牵引式农机具,如图4所示。拖拉机和农机具的车身参数设置如表1所示。
图4 仿真试验的车型选择

Fig. 4 Vehicle model selection of simulation test

表1 车身参数

Table 1 Body parameter setting

车身参数 长度/m 宽度/m 质量/kg 转动惯量/(kg·m2
拖拉机 2.5 1.8 3 500 126 000
农机具 1.2 1.6 1 200 30 000
之后对仿真工况进行设定。本研究不使用Carsim软件中的转向和制动模式,是以控制器求解的前轮转角作为Carsim模型的输入,速度基本维持在2 m/s,所以将这两个模式置空。农机具初始位置和初始航向根据后续试验需要设定。路面系数和空气动力系数等相关系数按照模块的默认参数设置。

3.1.2 联合仿真环境

通过搭建MATLAB/Carsim联合仿真环境,分析系统的输出结果以验证控制算法的各项性能。在MATLAB环境中进行控制器的模型搭建。控制器的输入为Carsim软件输出的农机具的位置、航向和铰接角等状态信息,偏差计算函数用以计算实际路径与参考路径间的偏差,滑模变结构算法根据偏差信息求解前轮转角。路径跟踪控制器模型如图5所示。
图5 车辆路径跟踪控制器模型

Fig. 5 Vehicle path tracking controller model

在搭建的联合仿真环境中进行路径跟踪控制仿真试验,联合仿真环境如图6所示。车速v保持在2 m/s左右,前视距离L q为2 m,仿真步长为0.01 s。进行双移线、圆形和S形曲线三种工况下的仿真试验。
图6 MATLAB/Carsim联合仿真环境

Fig. 6 MATLAB/Carsim co-simulation environment

3.2 双移线路径跟踪仿真

双移线试验是模仿实际车辆行驶中超车或避障后要回到原有的位置上的行动轨迹,路径跟踪控制初始条件如下:农机具的起点坐标(-2,1),农机具初始航向φ 0=0°,初始铰接角γ 0=0°,初始速度v 0=2 m/s,仿真时间为100 s,结果如图7所示。
图7 车辆的双移线路径仿真结果

Fig. 7 Simulation results of vehicle’s double lane change

分析图7可知,农机具在仿真时间内与参考双移线路径基本吻合,跟踪效果较好。横向偏差和航向偏差在8 s收敛到0 m和0°。在35、50、65和80 s时,参考航向发生改变,农机具横向偏差和航向偏差分别保持在0.1 m和7°以内。前轮转角最大为23.4°,铰接角最大为10.8°,符合安全性和稳定性的要求。

3.3 圆形路径跟踪仿真

参考路径为圆心坐标(0,0),半径为25 m的圆形。路径跟踪控制初始条件如下:农机具的起点坐标(-2,-25),农机具初始航向φ 0=0°,初始铰接角γ 0=0°,初始速度v 0=2 m/s,仿真时间为80 s,车辆圆形路径仿真结果如图8所示。
图8 车辆的圆形路径仿真结果

Fig. 8 Simulation results of vehicle's circular path

分析图8可知,农机具在仿真时间内与参考圆形路径基本吻合,轨迹较平滑。横向偏差经7 s趋于稳定,保持在0.03 m以内,航向偏差经7 s收敛到0°。前轮转角稳定在4.5°,铰接角稳定在2.8°,符合安全性和稳定性的要求。综上可知,控制器可以跟踪圆形路径。

3.4 S形曲线路径跟踪仿真

S形曲线由三条直线段和两个半径相同的半圆组成,起点为(0,0),半圆的半径为25 m,圆心分别为(50,25)和(25,75)。路径跟踪控制初始条件如下:农机具的起点坐标(-2,0.2),农机具初始航向φ 0=0°,初始铰接角γ 0=0°,初始速度v 0=2 m/s,仿真时间为130 s,S形曲线路径仿真结果如图9示。
图9 车辆的S形曲线路径仿真结果

Fig. 9 Simulation results of vehicle's s-curve path

分析图9可知,S形路径跟踪仿真试验在直线段和圆形处的仿真效果分别与4.2节、4.3节相近。在直线跟踪时,农机具横向偏差从初始的0.2 m经8 s收敛到0 m;在两个半圆处,农机具的横向偏差保持在0.03 m以内。在25、70、80和115 s处,参考曲率发生改变,农机具横向偏差产生保持在0.05 m以内,并在经过曲率变化处后迅速跟踪到参考路径。综上可知,在跟踪变曲率路径时,控制器依然保持良好的精度和快速性。

4 结 论

为提高拖拉机-牵引式农机具的路径跟踪精度,本研究提出了一种基于滑模变结构控制的路径跟踪方法,主要结论如下。
1)本研究建立了拖拉机-牵引式农机具运动学模型,并通过近似线性化建立了车辆-道路偏差模型。采用Ackermann公式进行极点配置,构建了基于指数趋近律的滑模变结构控制器,用以牵引式农机具的路径跟踪控制。
2)本研究在联合仿真环境中对控制算法进行验证。结果表明,在跟踪双移线路径时,设计的控制器使得农机具横向偏差和航向偏差经8 s收敛到0 m和0°,当参考航向发生改变时,农机具横向偏差小于0.1 m,航向偏差小于7°。在跟踪圆形路径时,农机具横向偏差和航向偏差经7 s趋于稳定,横向偏差保持在0.03 m以内,航向偏差保持在0°。在跟踪变曲率路径时,农机具横向偏差保持在理想范围内。三种仿真试验中,拖拉机前轮转角和拖拉机与农机具间的铰接角都保持在较小范围内,降低了事故发生的可能性。试验结果验证了控制算法具有良好的跟踪精度和响应速度,满足安全性的要求。
本研究提出的模型与方法为牵引式农机具自动导航提供了思路参考。

利益冲突声明

本研究不存在研究者以及与公开研究成果有关的利益冲突。

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