汽车技术

e	c_e
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

参数	取值	参数	取值
α₁	1	n	7
β₁	1	g	5
p	13	h	3
q	11	k_P0	1.2
k	1	k_I0	0.05
r	1	k_D0	0.002
m	9

参数	取值	参数	取值
α₁	1	n	7
β₁	1	g	5
p	13	h	3
q	11	k_P0	1.2
k	1	k_I0	0.05
r	1	k_D0	0.002
m	9

参数	取值	参数	取值
M/kg	33 860	h_g/m	1.25
M_b/kg	31 860	R/m	0.52
m_wf/kg	80	J_f/kg∙m²	28
m_wm/kg	130	J_m/kg∙m²	34
m_wr/kg	130	J_r/kg∙m²	34
I_z/kg∙m²	268 000	B_f/m	2.11
a/m	3.64	B_m/m	1.85
b/m	1.26	B_r/m	1.85
c/m	2.63	i_w	25
μ	0.015	α_G/(°)	0
ρ/kg∙m^-3	1.225	C_d	0.76
k_t	1.564 5	A_f/m²	8.2

参数	取值	参数	取值
M/kg	33 860	h_g/m	1.25
M_b/kg	31 860	R/m	0.52
m_wf/kg	80	J_f/kg∙m²	28
m_wm/kg	130	J_m/kg∙m²	34
m_wr/kg	130	J_r/kg∙m²	34
I_z/kg∙m²	268 000	B_f/m	2.11
a/m	3.64	B_m/m	1.85
b/m	1.26	B_r/m	1.85
c/m	2.63	i_w	25
μ	0.015	α_G/(°)	0
ρ/kg∙m^-3	1.225	C_d	0.76
k_t	1.564 5	A_f/m²	8.2

基于全轮转向的三轴车辆横向稳定性控制研究^*

PDF下载

董高尚 , 孙晓强 , 全振强 , 陈龙

汽车技术 | 2025,(8): 49-55

收起

汽车技术 | 2025, (8): 49-55

基于全轮转向的三轴车辆横向稳定性控制研究^*

全屏

董高尚, 孙晓强, 全振强, 陈龙

作者信息

江苏大学汽车工程研究院，镇江 212013

通讯作者:

孙晓强（1989—），教授，主要研究方向为车辆系统动力学控制等，sxq@ujs.edu.cn。

Research on Lateral Stability Control of Three-Axle Vehicle Based on All-Wheel Steering

Gaoshang Dong, Xiaoqiang Sun, Zhenqiang Quan, Long Chen

Affiliations

Institute of Automotive Engineering, Jiangsu University, Zhenjiang 212013

出版时间: 2025-08-24 doi: 10.19620/j.cnki.1000-3703.20240093

文章导航

摘要

收起

为了解决三轴车辆在复杂行车环境急转弯时的横向失稳问题，提出一种前馈控制与反馈控制相结合的全轮转向控制策略。首先，考虑轮胎非线性力学特性和车辆转向过程中垂直载荷转移及各轴垂向刚度差异，构建9自由度三轴车辆动力学模型，并基于牛顿插值法动态校正参考模型的轮胎侧偏刚度，提出基于阿克曼原理的零质心侧偏角比例前馈控制策略；然后，设计基于非奇异快速终端滑模控制的前轮转向反馈控制器和基于模糊PID控制的中、后轮转向反馈控制器，建立前馈和状态反馈跟踪理想模型的全轮转向控制策略；最后，分别在鱼钩和双移线测试工况下进行车辆转向性能仿真验证。结果表明：与前馈控制策略相比，所设计的全轮转向控制策略使车辆轨迹跟踪能力提高34%，车辆侧向稳定性提升26%。

关键词

三轴车辆 / 全轮转向 / 非奇异快速终端滑模 / 模糊PID / 操纵稳定性

Abstract

收起

In order to improve the active safety of three-axle vehicle under special driving conditions, an all-wheel steering control strategy combining feedforward control and feedback control is proposed. Firstly, considering the nonlinear mechanical properties of tires and the difference of vertical stiffness of each axle, a nine-degree-of-freedom three-axle vehicle dynamics model is established, and the tire cornering stiffness in the reference model is dynamically corrected in real time based on Newton interpolation method. On the basis of this model, a zero-centroid sideslip angle proportional feedforward control based on Ackerman principle is proposed to cooperate with the front wheel steering feedback controller based on Nonsingular Fast Terminal Sliding Mode (NFTSM) and the middle and rear wheel steering feedback controller based on Fuzzy PID to form an all-wheel steering control strategy. Finally, the vehicle is simulated and verified under the condition of fish hook and double lane change. The results show that the designed all-wheel steering control system improves the vehicle's trajectory tracking performance by 34% and lateral stability by 26% over the feedforward control strategy.

Key words

Three-axle vehicle / All-wheel steering / Nonsingular Fast Terminal Sliding Mode (NFTSM) / Fuzzy PID / Handling stability

引用本文

董高尚, 孙晓强, 全振强, 陈龙. 基于全轮转向的三轴车辆横向稳定性控制研究^*. 汽车技术, 2025 , (8) : 49 -55 . DOI: 10.19620/j.cnki.1000-3703.20240093

Gaoshang Dong, Xiaoqiang Sun, Zhenqiang Quan, Long Chen. Research on Lateral Stability Control of Three-Axle Vehicle Based on All-Wheel Steering[J]. Automobile Technology, 2025 , (8) : 49 -55 . DOI: 10.19620/j.cnki.1000-3703.20240093

正文

收起

1 前言

收起

三轴车辆在公路物流、矿产运输中广泛应用^[1]。在高速急转弯、急制动或外界干扰等特殊工况下，车辆常因轮胎侧向力饱和而发生失稳、侧翻等事故^[2]。因此，在复杂恶劣的环境下，三轴车辆的主动安全技术对于驾驶安全具有重要意义。

目前，全轮主动转向技术通过施加一个独立于转向盘转角输入的附加转角，对各车轮的转向角进行调整，同时改善车辆低速行驶的机动灵活性和高速行驶的操纵稳定性^[3-4]。王保华等^[5]基于线性模型提出比例前馈与横摆角速度反馈相结合的控制策略；Xu等^[6]通过稳态增益重构与超螺旋滑模控制，实现低速路径跟踪与高速稳定性的动态平衡；陆海英等^[7]在三轴车辆建模中引入阿克曼（Ackerman）原理，根据运动学约束关系设计全轮转向控制策略，有效抑制了轮胎打滑。

现有研究多采用简化模型或固定参数，未能全面考虑轮胎非线性力学特性、载荷转移对侧偏刚度的影响，且控制器多偏重中、后轮设计，缺乏多轴结构差异性建模与控制器协同设计。此外，前馈与反馈控制融合不充分，难以兼顾动态响应与稳态稳定性。为此，本文建立非线性9自由度三轴车辆动力学模型，提出基于阿克曼原理的零质心侧偏角比例前馈控制，协同基于非奇异快速终端滑模（Nonsingular Fast Terminal Sliding Mode，NFTSM）的前轮主动转向反馈控制器和基于模糊PID的中、后轮主动转向反馈控制器的全轮主动转向控制策略，并使用牛顿插值法动态校正轮胎侧偏刚度^[8]。最后，通过仿真验证该策略的有效性。

2 三轴车辆模型分析

收起

2.1 动力学模型

为了符合三轴车辆操纵稳定性动态特性和转向控制系统的设计需求，本文建立9自由度非线性车辆动力学模型，包括车身的横向运动、纵向运动、横摆运动和6个车轮的旋转运动，如图1所示。

根据牛顿定律，相关的动力学方程为^[9]：

(1)

$\begin{array}{l}M\left({\dot{v}}_{x}-{v}_{y}\gamma \right)=\left({F}_{xfl}+{F}_{xfr}\right)cos{\delta }_{f}-\left({F}_{yfl}+{F}_{yfr}\right)sin{\delta }_{f}+\\ \left({F}_{xml}+{F}_{xmr}\right)cos{\delta }_{m}-\left({F}_{yml}+{F}_{ymr}\right)sin{\delta }_{m}+\\ \left({F}_{xrl}+{F}_{xrr}\right)cos{\delta }_{r}-\left({F}_{yrl}+{F}_{yrr}\right)sin{\delta }_{r}-\\ Mg\mu -Mgsin{\alpha }_{G}-\frac{1}{2}\rho {C}_{d}{A}_{f}{v}_{x}^{2}\end{array}$

(2)

$\begin{array}{l}M\left({\dot{v}}_{y}+{v}_{x}\gamma \right)=\left({F}_{xfl}+{F}_{xfr}\right)sin{\delta }_{f}+\left({F}_{yfl}+{F}_{yfr}\right)cos{\delta }_{f}+\\ \left({F}_{xml}+{F}_{xmr}\right)sin{\delta }_{m}+\left({F}_{yml}+{F}_{ymr}\right)cos{\delta }_{m}+\\ \left({F}_{xrl}+{F}_{xrr}\right)sin{\delta }_{r}+\left({F}_{yrl}+{F}_{yrr}\right)cos{\delta }_{r}\end{array}$

(3)

$\begin{array}{l}{I}_{z}\dot{\gamma }={L}_{f}\left({F}_{xfl}+{F}_{xfr}\right)sin{\delta }_{f}+\frac{{B}_{f}}{2}\left({F}_{xfr}-{F}_{xfl}\right)cos{\delta }_{f}+\\ {L}_{f}\left({F}_{yfl}+{F}_{yfr}\right)cos{\delta }_{f}+\frac{{B}_{f}}{2}\left({F}_{yfl}-{F}_{yfr}\right)sin{\delta }_{f}-\\ {L}_{m}\left({F}_{xml}+{F}_{xmr}\right)sin{\delta }_{m}-\frac{{B}_{m}}{2}\left({F}_{xml}-{F}_{xmr}\right)cos{\delta }_{m}-\\ {L}_{m}\left({F}_{yml}+{F}_{ymr}\right)cos{\delta }_{m}+\frac{{B}_{m}}{2}\left({F}_{yml}-{F}_{ymr}\right)sin{\delta }_{m}-\\ {L}_{r}\left({F}_{xrl}+{F}_{xrr}\right)sin{\delta }_{r}-\frac{{B}_{r}}{2}\left({F}_{xrl}-{F}_{xrr}\right)cos{\delta }_{r}-\\ {L}_{r}\left({F}_{yrl}+{F}_{yrr}\right)cos{\delta }_{r}+\frac{{B}_{r}}{2}\left({F}_{yrl}-{F}_{yrr}\right)sin{\delta }_{r}\end{array}$

式中：F_xij、F_yij分别为轮胎纵向力和侧向力，B_i、L_i分别为车辆的轮距和轴距，δ_i为轮胎转向角，i=f,m,r分别为车辆前、中、后轴，j=l,r分别为车辆的左、右轮，M为整车质量，I_z为车辆横摆转动惯量，v_x、v_y分别车辆纵向速度和横向速度，γ为车辆横摆角速度，μ为滚动阻力系数，α_G为坡度角，ρ为空气密度，C_d为空气阻力系数，A_f为迎风面积。

各车轮旋转运动的动力学方程为：

（4）

${J}_{ij}{\dot{\omega }}_{ij}={T}_{ij}-{F}_{xij}R-\mu {F}_{zij}R$

式中：T_ij、F_zij(i=f,m,r， j=l,r)分别为作用于车轮的扭矩和垂向力，ω_ij为轮胎的滚动角速度，J_ij为车轮转动惯量，R为车轮滚动半径。

由于魔术公式轮胎模型在其试验范围内精度和置信度较高，能够反映联合工况下轮胎的非线性力学性能，故在车辆控制系统设计中，可通过该模型计算轮胎的纵向力和横向力：

（5）

$\left\{\begin{array}{l}{F}_{x}={F}_{x0}{G}_{x\alpha }(\alpha,s,{F}_{z})\\ {F}_{y}={F}_{y0}{G}_{y\lambda }(\alpha,s,\varphi,{F}_{z})+{S}_{vy\lambda }\end{array}\right.$

式中：G_xα( )、G_yλ( )为权重函数，F_x₀为纯纵滑工况下的轮胎纵向力，F_y₀为纯侧偏工况下的轮胎侧向力，α、s、F_z分别为轮胎的侧偏角、滑移率和垂直载荷，φ为车轮外倾角，S_vyλ为垂直偏移量。

在车辆转向行驶过程中，各车轮的垂直载荷受到横、纵向加速度的影响，车辆中、后轴的载荷分布相近且均大于前轴的载荷分布。假定车辆中、后轴相对于前轴的等效垂向刚度系数为k_t，车辆重力以及横、纵向惯性力在车辆各轴上的分配比例系数t_i和p_i分别为：

（6）

$\left\{\begin{array}{l}{t}_{i}=\frac{{\eta }_{i}}{2{L}_{f}^{2}+\left(1+{k}_{t}\right){L}_{m}^{2}+\left(1+{k}_{t}\right){L}_{r}^{2}+2{L}_{f}{L}_{m}+2{L}_{f}{L}_{r}-2{k}_{t}{L}_{m}{L}_{r}}\\ {p}_{i}=\frac{{\xi }_{i}}{2{L}_{f}^{2}+{k}_{t}^{2}{L}_{m}^{2}+2{k}_{t}^{2}{L}_{r}^{2}+4{k}_{t}{L}_{f}{L}_{m}-3{k}_{t}{L}_{m}{L}_{r}}\end{array}\right.$

其中，

$\left\{\begin{array}{l}{\eta }_{f}={L}_{m}^{2}+{L}_{r}^{2}+{L}_{f}{L}_{m}+{L}_{f}{L}_{r}，{\xi }_{f}=-2{L}_{f}-{k}_{t}{L}_{m}-{k}_{t}{L}_{r}\\ {\eta }_{m}={L}_{f}^{2}+{k}_{t}{L}_{r}^{2}+{L}_{f}{L}_{m}-{k}_{t}{L}_{m}{L}_{r}，{\xi }_{m}={L}_{f}+2{k}_{t}{L}_{m}-{k}_{t}{L}_{r}\\ {\eta }_{r}={L}_{f}^{2}+{k}_{t}{L}_{r}^{2}+{L}_{f}{L}_{m}-{k}_{t}{L}_{m}{L}_{r}，{\xi }_{r}={L}_{f}-{k}_{t}{L}_{m}+2{k}_{t}{L}_{r}\end{array}\right.$

。

式中：t_i和p_i(i=f,m,r)分别为车辆重力以及横、纵向惯性力在各轴的分配比例系数。

车辆的横向加速度a_x、纵向加速度a_y会引起垂向载荷转移，则各车轮的垂向载荷为：

（7）

$\left\{\begin{array}{l}{F}_{zi(l,r)}={m}_{wf}g+(\frac{1}{2}{M}_{b}g-{M}_{b}{a}_{y}\frac{{h}_{g}}{{B}_{i}}){t}_{i}+{M}_{b}{a}_{x}\frac{{h}_{g}}{2}{p}_{i}\\ {a}_{x}={\dot{v}}_{x}-{v}_{y}\gamma \\ {a}_{y}={\dot{v}}_{y}+{v}_{x}\gamma \end{array}\right.$

式中：m_w_i（i=f,m,r）为各轮胎的质量，M_b、h_g分别为车辆的簧载质量和质心高度。

基于轮胎动力学分析，各轮胎的侧偏角为：

（8）

${\alpha }_{i(l,r)}={\delta }_{i}-arctan\left(\frac{{v}_{y}+{L}_{i}\gamma }{{v}_{x}\mp {B}_{i}\gamma /2}\right)$

根据车辆状态量，各车轮的中心速度为：

（9）

${v}_{wi(l,r)}=\left({v}_{x}\mp \frac{{B}_{i}\gamma }{2}\right)cos{\delta }_{i}+\left({v}_{y}+{L}_{i}\gamma \right)sin{\delta }_{i}$

因此，结合车轮角速度和车轮中心速度，各轮胎的纵向滑移率、横向滑移率和总滑移率为：

（10）

$\left\{\begin{array}{l}{s}_{ijx}=\left\{\begin{array}{l}\frac{{\omega }_{ij}R-v{w}_{ij}cos{\alpha }_{ij}}{{\omega }_{ij}R}, {\omega }_{ij}Rv{w}_{ij}cos{\alpha }_{ij}\\ \frac{v{w}_{ij}cos{\alpha }_{ij}-{\omega }_{ij}R}{{v}_{ij}cos{\alpha }_{ij}}, {\omega }_{ij}R\le v{w}_{ij}cos{\alpha }_{ij}\end{array}\right.\\ {s}_{ijy}=\left\{\begin{array}{l}\left(1-{s}_{ijx}\right)tan{\alpha }_{ij}, {\omega }_{ij}Rv{w}_{ij}cos{\alpha }_{ij}\\ tan{\alpha }_{ij}, {\omega }_{ij}R\le v{w}_{ij}cos{\alpha }_{ij}\end{array}\right.\\ {s}_{ij}=\sqrt{{s}_{ijx}^{2}+{s}_{ijy}^{2}}\end{array}\right.$

2.2 参考模型

由于线性二自由度车辆动力学模型在车辆转向过程中能够保证车身的运动轨迹和姿态^[10]，故将其用于车辆稳定性控制器的参考模型。根据力矩平衡关系，车辆的线性二自由度运动微分方程可表示为：

（11）

$\left\{\begin{array}{l}\dot{\beta }=\left({k}_{f}+{k}_{m}+{k}_{r}\right)M{v}_{x}^{-1}\beta +\left[\left(a{k}_{f}-b{k}_{m}-c{k}_{r}\right)M{v}_{x}^{-2}-1\right]\gamma -\\ {k}_{f}M{v}_{x}^{-1}{\delta }_{f}-{k}_{m}M{v}_{x}^{-1}{\delta }_{m}-{k}_{r}M{v}_{x}^{-1}{\delta }_{r}\\ \dot{\gamma }=\left(a{k}_{f}-b{k}_{m}-c{k}_{r}\right){I}_{z}^{-1}\beta +\left({a}^{2}{k}_{f}+{b}^{2}{k}_{m}+{c}^{2}{k}_{r}\right){I}_{z}^{-1}{v}_{x}^{-1}\gamma -\\ a{k}_{f}{I}_{z}^{-1}{\delta }_{f}+b{k}_{m}{I}_{z}^{-1}{\delta }_{m}+c{k}_{r}{I}_{z}^{-1}{\delta }_{r}\end{array}\right.$

式中：β为车辆质心侧偏角，k_f、k_m、k_r分别为前、中、后轴轮胎的侧偏刚度。

使用一阶惯性环节降低转向盘输入突变的影响，鉴于车辆行驶中最大横向加速度a_y受到道路摩擦因数μ限制，可通过消除横向加速度中的相位滞后，则理想横摆角速度为：

（12）

${\gamma }_{d}=min \{\left|{k}_{\gamma }{\delta }_{f}\right|,\left|\mu g{v}_{x}^{-1}\right|\}sgn\left({\delta }_{f}\right)(1+{\tau }_{\gamma }{s)}^{-1}$

式中：τ_γ为一阶惯性环节时间常数，k_γ为理想横摆角速度的稳态增益，sgn( )为符号函数。

参照横摆角速度的设计思想，理想质心侧偏角为：

（13）

${\beta }_{d}={k}_{\beta }(1+{\tau }_{\beta }{s)}^{-1}{\delta }_{f}$

式中：τ_β=0.1、k_β=0.1分别为理想质心侧偏角的响应时间常数和稳态增益。

当参考模型中轮胎侧偏刚度为固定值时，无法反映车辆转向过程中垂向载荷转移对轮胎侧偏特性的影响，故本文对参考模型的前、中、后轴轮胎侧偏刚度进行实时动态校正，计算6种不同载荷下轮胎侧偏角与侧向力的关系，如图2所示。

基于牛顿插值法可估计任何载荷的轮胎侧偏刚度：

（14）

$k=\frac{{F}_{z}-{F}_{zmin}}{{F}_{zmax}-{F}_{zmin}}{k}_{min}+\frac{{F}_{zmax}-{F}_{z}}{{F}_{zmax}-{F}_{zmin}}{k}_{max}$

式中：k_min、k_max为相邻载荷下的轮胎侧偏刚度。

由于三轴车辆中、后轴的轮胎为双排配置，考虑到垂向载荷转移、内外轮胎与质心间的横向距离差，本文对外侧和内侧轮胎的权重系数分别设置为0.6和0.4。

3 三轴车辆转向控制器设计

收起

为了提升车辆在高速急转弯、急制动等特殊工况下的操纵稳定性，将参考模型的理想横摆角速度和零质心侧偏角作为控制目标，采用牛顿插值法动态校正轮胎侧偏刚度，提出比例前馈控制协同主动转向反馈控制的全轮转向控制策略，其系统框架如图3所示。

3.1 基于零质心侧偏角的比例前馈控制器

在车辆行驶中，质心侧偏角直接影响车辆的稳定性和安全性。为了降低轮胎磨损，最大限度地减小车辆质心侧偏角，通过调整车辆转向状态和纵向速度中后轴转向比例系数，从而实现车辆质心侧偏角为零的控制目标。

三轴车辆的中轮转向角为

${\delta }_{mff}={G}_{21}{\delta }_{f}$

，后轮转向角

${\delta }_{rff}={G}_{31}{\delta }_{f}$

，其中，G₂₁、G₃₁分别为中、后轴转向比例系数。基于Ackerman定理，结合车辆几何学模型和动力学模型可知，车辆转向到达稳态时，必须满足β=0、

$\dot{\beta }=0$

、γ=0。联立车辆线性二自由度模型的运动微分方程，比例转向系数G₂₁、G₃₁与车辆纵向速度间关系如图4所示。

3.2 基于NFTSM的主动前轮转向反馈控制器

NFTSM能够在有限的时间内实现对系统预期状态的精确跟踪，并解决普通滑模抖振、收敛速度慢等问题^[11]。本文对前轮施加一个独立于转向盘转角输入的附加转角，通过调整前轮转向角，改善车辆稳定性控制。

将横摆角的偏差作为控制器的跟踪误差，

$e=\varphi -{\varphi }_{d}$

，

$\dot{e}=\gamma -{\gamma }_{d}$

，

$\ddot{e}=\dot{\gamma }-{\dot{\gamma }}_{d}$

。因此，非奇异快速终端滑模控制器的非线性滑模面为：

（15）

$\left\{\begin{array}{l}s=e+{a}_{1}^{-1}{e}^{g/h}+{\beta }_{1}^{-1}{\dot{e}}^{p/q}\\ {\alpha }_{1}\in {R}^{+}, {\beta }_{1}\in {R}^{+}, 1g/h2, 1p/q2\end{array}\right.$

式中：g、h、p和q为正奇数。

为了缓解抖振现象，保证系统控制输入的连续性，采用带负指数项的终端吸引子设计趋近律:

（16）

$\left\{\begin{array}{l}\dot{s}=\left(-ks-r{s}^{m/n}\right){\dot{e}}^{p/q-1}\\ k\in {R}^{+}, r\in {R}^{+}, 0m/n1\end{array}\right.$

式中：m、n为正奇数。

非奇异快速终端滑模控制可表示为：

（17）

$\begin{array}{l}\dot{e}+\frac{g}{{\alpha }_{1}h}{e}^{g/h-1}\dot{e}+\frac{p}{{\beta }_{1}q}{\dot{e}}^{p/q-1}\left[{a}_{21}\beta +{a}_{22}\gamma +\right.\\ \left.\left({b}_{21}+{b}_{22}{G}_{21}+{b}_{23}{G}_{31}\right){\delta }_{f}-{\dot{\gamma }}_{d}\right]=\left(-ks-r{s}^{m/n}\right){\dot{e}}^{p/q-1}\end{array}$

因此，基于误差反馈计算控制器的滑模控制律为：

（18）

$\begin{array}{l}{\delta }_{f}^{*}=\frac{1}{{b}_{21}+{b}_{22}{G}_{21}+{b}_{23}{G}_{31}}\times \left\{{\dot{\gamma }}_{d}-{a}_{21}\beta -{a}_{22}\gamma +\right.\\ \left.\frac{{\beta }_{1}q}{p}\left[\left(\frac{g}{{\alpha }_{1}h}{e}^{{g}_{1}/{h}_{1}-1}-1\right){\dot{e}}^{-2p/q}-ks-r{s}^{m/n}\right]\right\}\end{array}$

为了验证该算法下控制系统的稳定性，基于李雅普诺夫（Lyapunov）理论，Lyapunov函数可定义为V=0.5 s²，并对该函数求一阶导数

$\dot{V}=s\dot{s}=s\left(-ks-r{s}^{m/n}\right){\dot{e}}^{p/q-1}$

，分别讨论以下2种情况：

a. 当

$\dot{e}\ne 0$

时，系统满足

${\dot{e}}^{p/q-1}$

>0，此时系统Lyapunov函数满足

$\dot{V}=s\dot{s}=\left(-{k}_{1}{s}^{2}-{r}_{1}{s}^{{m}_{1}/{n}_{1}+1}\right){\dot{e}}^{{p}_{1}/{q}_{1}-1}0$

。

b. 当

$\dot{e}=0$

时，则满足

$\ddot{e}=-\frac{{\beta }_{1}q}{p}\left(ke+r{e}^{m/n}\right)0$

。

因此，控制系统的Lyapunov函数的时间导数也满足

$\dot{V}=s\dot{s}=\left(p/q-1\right)\left(-k{s}^{2}-r{s}^{m/n+1}\right){\dot{e}}^{p/q-2}\ddot{e}0$

。

根据Lyapunov稳定性准则，上述分析过程表明系统状态可在有限时间内收敛到平衡点，证明该算法的控制系统具有较好的稳定性。

3.3 基于模糊PID的主动中、后轮转向反馈控制器

模糊PID控制结合传统PID和模糊控制，通过模糊规则表对控制参数实时迭代，自适应调整PID控制参数，有效提升参考模型的跟踪精度，以实现跟踪期望轨迹^[12]。模糊PID控制器的输入量为：理想参考模型的横摆角速度、质心侧偏角、实际动力学模型的偏差e和偏差变化率c_e，其论域设置为[-6,6]；输出量为PID控制参数的修正量Δk_P、Δk_I、Δk_D，论域设置为[-1,1]；根据车辆模型输入与输出量数值范围，量化因子分别设定为K_e=300、K_c=200、K_ΔkP=0.8、K_ΔkI=0.03、K_ΔkD=0.001。

将模糊化等级均选取为7级，即模糊子集为{负大（Negtive Big，NB），负中（Negative Medium，NM），负小（Negative Small，NS），零（Zero，ZO），正小（Positive Small，PS），正中（Positive Medium，PM），正大（Positive Big，PB）}，分别以高斯型（gaussmf）和三角形（trimf）作为输入量和输出量的隶属度函数，模糊推理方法采用Mamdani法。根据所设定的参数，制定模糊控制规则，相应的模糊规则如表1~表3所示。

根据模糊规则表可以实现对k_P、k_I、k_D进行动态整定，模糊PID反馈控制器在比例前馈控制器基础上引入补偿转角，因此，中、后轴车轮主动转向角的控制输出量可表示为

${\delta }_{i\left(m,r\right)}={\delta }_{i\left(m,r\right)ff}+\Delta {\delta }_{i\left(m,r\right)fb}$

。

4 控制系统仿真分析

收起

通过设计鱼钩和对开双移线工况的仿真条件，验证本文控制策略对三轴车辆操纵稳定性的有效性。为了尽可能减小转向机械机构间隙的影响，线性化转向盘转角和车轮转向角关系，设置转向传动比为i_w，转角输入如图5所示，车辆参数和控制参数如表4、表5所示。

4.1 鱼钩工况仿真分析

使用鱼钩工况验证三轴车辆在高速急转条件下的操纵稳定性。为了模拟实际行驶环境，车辆纵向速度设置为60 km/h，路面附着系数为0.85。对比基于零质心侧偏角的比例前馈控制器和考虑误差反馈的协同控制器，仿真结果如图6所示。

由图6可知，在前馈控制协同误差反馈控制策略下，车辆的质心侧偏角、横摆角速度、侧向加速度的峰值均有降低。在图6b中，协同控制策略下的车辆质心侧偏角能够较快收敛到稳态值，过冲更小，可有效限制车辆的横向偏移，相较于前馈控制器，车辆的轨迹跟随能力提高了34%。在图6c中，在车辆横摆角速度方面，协同控制策略下的稳态值更小，能够纠正车身横摆姿态。同时，车辆侧向加速度也相对较小，较前馈控制器，车辆的侧向稳定性提高了23%。因此，本文的协同控制策略可以保证三轴车辆在高速转向过程的轨迹跟随性和横向稳定性。

4.2 双移线工况仿真分析

使用对开双移线工况验证三轴车辆在连续转向条件下的操纵稳定性。车辆的纵向速度设置为60 km/h，路面附着系数为0.85，两种控制策略下仿真结果如图7所示。

由图7可知，与前馈控制器相比，本文协同控制策略的车辆横摆角速度和质心侧偏角能够较快地收敛到稳态值，超调量更小，车辆的轨迹跟随能力提高了31%，车辆的侧向稳定性提高了26%。根据仿真结果，基于协同控制策略的全轮主动转向器极大地提高了三轴车辆在复杂行驶工况下的操纵稳定性和主动安全性。

5 结束语

收起

本文以三轴车辆全轮主动转向横向稳定性控制技术为研究对象，建立9自由度三轴车辆动力学模型，基于此模型，设计了基于阿克曼转向原理的零质心侧偏角比例前馈控制器、基于非奇异快速终端滑模的前轮主动转向反馈控制器和基于模糊PID的中、后轮主动转向反馈控制器，并在鱼钩工况和双移线工况下仿真验证了该策略能够提高车辆侧向稳定性和行驶安全性。为了进一步提升车辆的性能，未来可结合车路协同控制、智能轮胎等技术车开展底盘集成控制研究。

基金

收起

*国家自然科学基金项目(U20A20331)

参考文献

收起

文献

收起

参考文献引证文献

排序方式：

[1]

颜龙, 杨晓峰, 沈钰杰, 等. 变车载质量下重型车辆ISD悬架平顺性与道路友好性分析[J]. 江苏大学学报(自然科学版), 2022, 43(4): 407-415.

YAN

, YANG

X F

, SHEN

Y J

, et al. Analysis of Ride Smoothness and Road Friendliness of Heavy Vehicles with ISD Suspension under Variable Vehicle Mass[J]. Journal of Jiangsu University (Natural Science Edition), 2022, 43(4): 407-415.

[2]

袁磊, 史辉, 刘洋. 基于非线性模型的全轮转向重型车辆相平面特性分析[J]. 汽车技术, 2020(4): 50-56.

YUAN

, SHI

, LIU

. Analysis of Phase Plane Characteristics of All-Wheel Steering Heavy Duty Vehicle Based on Nonlinear Models[J]. Automobile Technology, 2020(4): 50-56.

[3]

付翔, 杨凤举, 黄斌, 等. 主动后轮转向四轮独立驱动车辆的协调控制[J]. 江苏大学学报(自然科学版), 2021, 42(5): 497-505.

, YANG

F J

, HUANG

, et al. Coordinated Control of Four-Wheel Independent Drive Vehicle with Active Rear Wheel Steering[J]. Journal of Jiangsu University (Natural Science Edition), 2021, 42(5): 497-505.

[4]

JIANG

, MENG

, CHEN

G P

, et al. Differential-Steering Based Path Tracking Control and Energy-Saving Torque Distribution Strategy of 6WID Unmanned Ground Vehicle[J]. Energy, 2022, 254.

[5]

王保华, 喻超. 某四轴车辆全轮转向最优补偿控制策略研究[J]. 系统仿真学报, 2019, 31(3): 528-537.

WANG

B H

, YU

. Research on the Optimal Compensation Control Strategy for All Wheel Steering of a Certain Four Axis Vehicle[J]. Journal of System Simulation, 2019, 31(3): 528-537.

[6]

, LIU

X X

, ZHENG

, et al. A Sliding Mode Control Scheme for Steering Flexibility and Stability in All-Wheel-Steering Multi-Axle Vehicles[J]. International Journal of Control, Automation and Systems, 2023(21): 1926-1938.

[7]

陆海英, 张成铭, 郭艳秋, 等. 基于阿克曼原理的三轴车辆全轮转向最优控制[J]. 广西大学学报(自然科学版), 2020, 45(3): 558-568.

H Y

, ZHANG

C M

, GUO

Y Q

, et al. All-Wheel Steering Optimal Control of Three-Axis Vehicle Based on Ackerman Principle[J]. Journal of Guangxi University (Natural Science Edition), 2020, 45(3): 558-568.

[8]

ZOU

, SONG

L T

, WANG

X F

, et al. A New Approach to Newton-Type Polynomial Interpolation with Parameters[J]. Mathematical Problems in Engineering, 2020(7): 1-15.

[9]

孙艺珊, 吴志成. 三轴车悬架动力学参数的匹配[J]. 江苏大学学报(自然科学版), 2018, 39(3): 249-253.

SUN

Y S

, WU

Z C

. Matching of Dynamic Parameters of Three-Axle Vehicle Suspension[J]. Journal of Jiangsu University (Natural Science Edition), 2018, 39(3) : 249-253.

[10]

张佩, 孙文新, 胡杰, 等. 四轮转向车辆后轮转角控制方法研究[J]. 汽车工程, 2023, 45(12): 2242-2250.

ZHANG

, SUN

W X

, HU

, et al. Research on Rear Wheel Angle Control Method for Four-Wheel Steering Vehicle[J]. Automotive Engineering, 2023, 45(12): 2242-2250.

[11]

辛林杰, 王庆林, 李原. 基于非奇异快速终端滑模的一类二阶系统控制[C]// 第35届中国控制会议. 成都, 中国: 第35届中国控制会议论文集, 2016: 376-380.

XIN

L J

, WANG

Q L

, LI

. Control of a Class of Second-Order Systems Based on Nonsingular Fast Terminal Sliding Mode[C]// The 35th Chinese Control Conference (CCC2016). Chengdu, China: Chinese Control Conference, 2016: 376-380.

[12]

薛文平, 张春玲. 基于遗传算法的汽车主动悬架变论域模糊PID控制[J]. 江苏大学学报(自然科学版), 2024, 45(1): 8-15.

XUE

W P

, ZHANG

C L

. Variable Universe Fuzzy PID Control of Automobile Active Suspension Based on Genetic Algorithm[J]. Journal of Jiangsu University (Natural Science Edition), 2024, 45(1): 8-15.

2025年第卷第8期

PDF下载

481

221

引用本文

BibTeX

文章信息

doi: 10.19620/j.cnki.1000-3703.20240093

首发时间：2025-10-28
出版时间：2025-08-24

补充材料

相关文章

文章信息

作者

出版历史

修回日期：2024-03-11

基金

*国家自然科学基金项目(U20A20331)

作者信息

江苏大学汽车工程研究院，镇江 212013

通讯作者:

孙晓强（1989—），教授，主要研究方向为车辆系统动力学控制等，sxq@ujs.edu.cn。

参考文献

分享链接

https://castjournals.cast.org.cn/joweb/qcjs/CN/10.19620/j.cnki.1000-3703.20240093

分享至

全文二维码

扫描看全文

引用本文

BibTeX

本文的引用情况

2种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

关闭全屏

BibTeX
EndNote
RefWorks
TxT

e	c_e
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

c_e

e	c_e
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

参数	取值	参数	取值
α₁	1	n	7
β₁	1	g	5
p	13	h	3
q	11	k_P0	1.2
k	1	k_I0	0.05
r	1	k_D0	0.002
m	9

参数

取值

参数

取值

α₁

β₁

k_P0

1.2

k_I0

0.05

k_D0

0.002

参数	取值	参数	取值
M/kg	33 860	h_g/m	1.25
M_b/kg	31 860	R/m	0.52
m_wf/kg	80	J_f/kg∙m²	28
m_wm/kg	130	J_m/kg∙m²	34
m_wr/kg	130	J_r/kg∙m²	34
I_z/kg∙m²	268 000	B_f/m	2.11
a/m	3.64	B_m/m	1.85
b/m	1.26	B_r/m	1.85
c/m	2.63	i_w	25
μ	0.015	α_G/(°)	0
ρ/kg∙m^-3	1.225	C_d	0.76
k_t	1.564 5	A_f/m²	8.2

参数

取值

参数

取值

M/kg

33 860

h_g/m

1.25

M_b/kg

31 860

R/m

0.52

m_wf/kg

J_f/kg∙m²

m_wm/kg

130

J_m/kg∙m²

m_wr/kg

130

J_r/kg∙m²

I_z/kg∙m²

268 000

B_f/m

2.11

a/m

3.64

B_m/m

1.85

b/m

1.26

B_r/m

1.85

c/m

2.63

i_w

0.015

α_G/(°)

ρ/kg∙m^-3

1.225

C_d

0.76

k_t

1.564 5

A_f/m²

8.2

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	ZO	ZO	NS
NS	PM	PM	PS	PS	ZO	NS	NS
ZO	PM	PM	ZO	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NM	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	ZO	ZO	ZO	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PM
PM	ZO	ZO	PS	PS	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

e	c_e
e	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NM	ZO
NS	ZO	NS	NM	NS	NS	NS	ZO
ZO	ZO	NS	NS	ZO	NS	NS	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PM	NS	PS	PS	PS	PS	PM
PB	PB	PM	PM	PM	PS	PS	PB