﻿ 轮边电机驱动差动转向车辆动力学控制
 文章快速检索
 同济大学学报(自然科学版)  2018, Vol. 46 Issue (5): 631-638.  DOI: 10.11908/j.issn.0253-374x.2018.05.010 0

### 引用本文

YU Zhuoping, GAO Letian, ZHANG Renxie, XIONG Lu. Dynamic Control of Electric Motor Driven Skid-steered Vehicles[J]. Journal of Tongji University (Natural Science), 2018, 46(5): 631-638. DOI: 10.11908/j.issn.0253-374x.2018.05.010

### 文章历史

1. 同济大学 汽车学院，上海 201804;
2. 同济大学 新能源汽车工程中心，上海 201804

Dynamic Control of Electric Motor Driven Skid-steered Vehicles
YU Zhuoping 1,2, GAO Letian 1,2, ZHANG Renxie 1,2, XIONG Lu 1,2
1. School of Automotive Studies, Tongji University, Shanghai 201804, China;
2. Clean Energy Automotive Engineering Center, Tongji University, Shanghai 201804, China
Abstract: A dynamic controller based on anti-windup proportional-integral (PI) algorithm for skid-steered vehicles was designed. The dynamic characteristics of skid-steered vehicles were analyzed, and a double closed-loop lateral dynamic control method was proposed to improve the control performance which was influenced by the complex and varied tire-road adhesion conditions. The simulation and experiment results show that the proposed dynamic controller has good transient and steady tracking performance, which can track the driver's intention accurately.
Key words: skid-steered vehicles    2-degree of freedom (DOF) reference model    dynamic characteristics    anti-windup proportional-integral (PI) algorithm

1 车辆动力学模型 1.1 车身动力学模型

 图 1 车身动力学模型 Fig.1 Vehicle body dynamic model

 $\left\{ \begin{array}{l} m{{\dot v}_x} = \sum\limits_{i = 1}^3 {\left( {{F_{x{\rm{l}}i}} + {F_{x{\rm{r}}i}}} \right) + m\gamma {v_y}} \\ m{{\dot v}_y} = \sum\limits_{i = 1}^3 {\left( {{F_{y{\rm{l}}i}} + {F_{y{\rm{r}}i}}} \right) - m\gamma {v_x}} \\ {I_z}\dot \gamma = \left( {\sum\limits_{i = 1}^3 {{F_{x{\rm{r}}i}}} - \sum\limits_{i = 1}^3 {{F_{x{\rm{l}}i}}} } \right)\frac{B}{2} + \left( {{F_{y{\rm{r1}}}} - {F_{y{\rm{l1}}}}} \right)a + \\ \;\;\;\;\;\;\left( {{F_{y{\rm{r2}}}} - {F_{y{\rm{l2}}}}} \right)b + \left( {{F_{y{\rm{r3}}}} - {F_{y{\rm{l3}}}}} \right)c \end{array} \right.$ (1)

1.2 车轮动力学模型

 图 2 车轮动力学模型 Fig.2 Wheel dynamics model

 $3{J_\omega }{{\dot \omega }_j} = - R\sum\limits_{i = 1}^3 {{F_{xji}}\left( \lambda \right)} - {i_{{\rm{tr}}}}{T_{{\rm{b}}j}} + {i_{{\rm{tr}}}}{T_j}$ (2)

 $3{J_\omega }{{\dot \omega }_j} = - R\sum\limits_{i = 1}^3 {{F_{xi}}\left( {{\omega _j}} \right)} - {i_{{\rm{tr}}}}{T_{{\rm{b}}j}} + {i_{{\rm{tr}}}}{T_j}$ (3)
1.3 轮胎模型

 ${f_\lambda }\left( {{\lambda _{\rm{c}}}} \right) = {\theta _1} - {\theta _1}\exp \left[ { - \frac{{{\theta _2}}}{{{\theta _1}}}\left( {{\lambda _{\rm{c}}} + {\theta _5}\lambda _{\rm{c}}^2} \right)} \right] - {\theta _3}{\lambda _{\rm{c}}} + {\theta _4}\lambda _{\rm{c}}^2$ (4)

 图 3 轮胎模型特性曲线 Fig.3 Characteristics curve of tire model
2 控制器设计

 图 4 动力学控制器结构 Fig.4 Architecture of dynamic controller
2.1 驾驶员意图解析 2.1.1 加速意图解析

 ${L_{\rm{D}}} = 100\% \alpha$ (5)

 ${T_{\rm{D}}} = \left\{ \begin{array}{l} {L_{\rm{D}}}{T_{\max }},\;\;\;\;\;\;\;n \le {n_{\rm{b}}}\\ \frac{{{L_{\rm{D}}}{T_{\max }}{n_{\rm{b}}}}}{n},\;\;\;n > {n_{\rm{b}}} \end{array} \right.$ (6)

2.1.2 期望横摆角速度计算

 $\left\{ \begin{array}{l} m\left( {{{\dot v}_y} + {{\dot v}_x}\gamma } \right) = - \frac{2}{{{v_x}}}\left[ {\left( {{k_{y1}} + {k_{y2}} + {k_{y3}}} \right){v_y} + } \right.\\ \;\;\;\;\;\;\;\left. {\left( {a{k_{y1}} + b{k_{y2}} + c{k_{y3}}} \right)\gamma } \right]\\ {I_z}\dot \gamma = \frac{{B\left( {{k_{x1}} + {k_{x2}} + {k_{x3}}} \right)}}{2}\frac{{\Delta {v_x}}}{{{v_x}}} - \\ \;\;\;\;\;\;\;\frac{2}{{{v_x}}}\left( {a{k_{y1}} + b{k_{y2}} + c{k_{y3}}} \right){v_y} - \\ \;\;\;\;\;\;\;\frac{1}{{{v_x}}}\left[ {\frac{1}{2}{B^2}\left( {{k_{x1}} + {k_{x2}} + {k_{x3}}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {2\left( {{a^2}{k_{y1}} + {b^2}{k_{y2}} + {c^2}{k_{y3}}} \right)} \right]\gamma \end{array} \right.$ (7)

 $\begin{array}{l} \gamma = B\left( {{k_{x1}} + {k_{x2}} + {k_{x3}}} \right)\left( {{k_{y1}} + {k_{y2}} + {k_{y3}}} \right)\Delta {v_x}/\left\{ {{B^2}\left( {{k_{x1}} + } \right.} \right.\\ \;\;\;\;\;\;\left. {{k_{x2}} + {k_{x3}}} \right)\left( {{k_{y1}} + {k_{y2}} + {k_{y3}}} \right) + 4\left( {{k_{y1}} + {k_{y2}} + } \right.\\ \;\;\;\;\;\;\left. {{k_{y3}}} \right)\left( {{a^2}{k_{y1}} + {b^2}{k_{y2}} + {c^2}{k_{y3}}} \right) - 4\left( {a{k_{y1}} + } \right.\\ \;\;\;\;\;\;\left. {{{\left. {b{k_{y2}} + c{k_{y3}}} \right)}^2} - 2\left( {a{k_{y1}} + b{k_{y2}} + c{k_{y3}}} \right)mv_x^2} \right\} \end{array}$ (8)

 $\frac{{\Delta {v_x}}}{{{v_x}}} = {k_{\rm{s}}}{\delta _{{\rm{dr}}}}$ (9)

 $\frac{\gamma }{{\Delta {v_x}/{v_x}}} = \frac{{A{v_x}}}{{C + Dv_x^2}} = \frac{{{v_x}A/C}}{{1 + D/Cv_x^2}}$ (10)

 $A = B\left( {{k_{x1}} + {k_{x2}} + {k_{x3}}} \right)\left( {{k_{y1}} + {k_{y2}} + {k_{y3}}} \right)$
 $\begin{array}{l} C = {B^2}\left( {{k_{x1}} + {k_{x2}} + {k_{x3}}} \right)\left( {{k_{y1}} + {k_{y2}} + {k_{y3}}} \right) + \\ \;\;\;\;\;\;4\left( {{k_{y1}} + {k_{y2}} + {k_{y3}}} \right)\left( {{a^2}{k_{y1}} + {b^2}{k_{y2}} + {c^2}{k_{y3}}} \right) - \\ \;\;\;\;\;\;4{\left( {a{k_{y1}} + b{k_{y2}} + c{k_{y3}}} \right)^2} \end{array}$
 $D = - 2\left( {a{k_{y1}} + b{k_{y2}} + c{k_{y3}}} \right)m$

 ${\left( {\frac{\gamma }{\delta }} \right)_{\rm{s}}} = \frac{{{v_x}/L}}{{1 + Kv_x^2}}$ (11)

 $\frac{\gamma }{{\Delta {v_x}/{v_x}}} = \frac{{A{v_x}}}{C}$ (12)

 ${\gamma _{{\rm{des}}}} \le \frac{{0.8\mu g}}{{{v_x}}}$ (13)

 ${\gamma _{{\rm{des}}}} = \min \left( {\frac{{A{v_x}}}{C}{k_{\rm{s}}}{\delta _{{\rm{dr}}}},\frac{{0.8\mu g}}{{{v_x}}}} \right)$ (14)
2.2 轮速差跟踪控制

 ${\gamma _{{\rm{ref}}}} = {\gamma _{{\rm{des}}}}$ (15)

 $\Delta {\omega _{{\rm{ref}}}} = B{\gamma _{{\rm{ref}}}}$ (16)

 $3{J_\omega }{{\dot \omega }_{\rm{i}}} = - 3R{F_x}\left( {{\omega _{\rm{i}}}} \right) - {i_{{\rm{tr}}}}{T_{\rm{b}}} + {i_{{\rm{tr}}}}{T_{\rm{i}}}$ (17)
 $3{J_\omega }{{\dot \omega }_{\rm{o}}} = - 3R{F_x}\left( {{\omega _{\rm{o}}}} \right) - {i_{{\rm{tr}}}}{T_{\rm{b}}} + {i_{{\rm{tr}}}}{T_{\rm{o}}}$ (18)

 $\begin{array}{l} 3{J_\omega }\Delta \dot {\tilde \omega} = - 3R\left[ {{F_x}\left( {\omega + \Delta \omega } \right) - {F_x}\left( {\omega - \Delta \omega } \right)} \right] + \\ \;\;\;\;\;\;\frac{{2{i_{{\rm{tr}}}}R}}{B}{M_z} - 3{J_\omega }\Delta {{\dot \omega }_{{\rm{ref}}}} \end{array}$ (19)

 $\left| {\frac{{{F_x}\left( {\omega + \Delta \omega } \right) - {F_x}\left( {\omega - \Delta \omega } \right)}}{{2\Delta \omega }}} \right| \le \left\| {\frac{{\partial {f_\lambda }\left( {{\lambda _{\rm{c}}}} \right)}}{{\partial {\lambda _{\rm{c}}}}}} \right\| \le {L_1}$ (20)

 $\left\{ \begin{array}{l} {M_z} = - {\eta _1}{\rm{sat}}\left( {\frac{s}{{{\eta _3}}}} \right)\\ \dot \varepsilon = - {\eta _2}\varepsilon + {\eta _3}{\rm{sat}}\left( {\frac{s}{{{\eta _3}}}} \right)\\ s = \Delta \tilde \omega + {\eta _2}\varepsilon \end{array} \right.$ (21)

(1) 当s>η3时，积分运算饱和，此时$\varepsilon \to \frac{{{\eta _3}}}{{{\eta _2}}}$，控制律(21)可改写为如下形式：

 ${M_z} = - {\eta _1}$ (22)

 $\begin{array}{l} \dot s = \Delta \dot {\tilde \omega} + {\eta _2}\dot \varepsilon = \\ \;\;\;\;\; - \frac{R}{{{J_\omega }}}\left[ {{F_x}\left( {\omega + \Delta \omega } \right) - {F_x}\left( {\omega - \Delta \omega } \right)} \right] - \frac{{2{i_{{\rm{tr}}}}R}}{{3B{J_\omega }}}{\eta _1} - \\ \;\;\;\;\;\Delta {{\dot \omega }_{{\rm{ref}}}} \le - \frac{{R{L_1}}}{{{J_\omega }}} - \frac{{2{i_{{\rm{tr}}}}R}}{{3B{J_\omega }}}{\eta _1} - \Delta {{\dot \omega }_{{\rm{ref}}}} \end{array}$ (23)

 ${\eta _1} > \left( { - \frac{{R{L_1}}}{{{J_\omega }}} - \Delta {{\dot \omega }_{{\rm{ref}}}}} \right)\frac{{3B{J_\omega }}}{{2{i_{{\rm{tr}}}}R}}$ (24)

(2) 当积分未饱和时，控制律(21)可写为如下形式：

 ${M_z} = - \frac{{{\eta _1}}}{{{\eta _3}}}\Delta \omega \left( t \right) - \frac{{{\eta _1}{\eta _2}}}{{{\eta _3}}}\int {\Delta \omega \left( t \right){\rm{d}}t}$ (25)

2.3 参考轮速差修正

 $\left\{ \begin{array}{l} {\gamma _{{\rm{ref}}}} = - {\eta _4}{\rm{sat}}\left( {\frac{{\tilde \gamma + {\eta _5}{\varepsilon _\gamma }}}{{{\eta _6}}}} \right) + {\gamma _{{\rm{des}}}}\\ {{\dot \varepsilon }_\gamma } = - {\eta _5}{\varepsilon _\gamma } + {\eta _6}{\rm{sat}}\left( {\frac{{\tilde \gamma + {\eta _5}{\varepsilon _\gamma }}}{{{\eta _6}}}} \right) \end{array} \right.$ (26)

2.4 转矩分配

 ${T_{\rm{l}}} = {T_{\rm{r}}} = \frac{{{T_{\rm{D}}}}}{2}$ (27)

 $\left\{ \begin{array}{l} {T_{\rm{i}}} = - \frac{R}{{{i_{{\rm{tr}}}}}}\frac{{{M_z}}}{B} + \frac{1}{2}{T_{\rm{D}}}\\ {T_{\rm{o}}} = \frac{R}{{{i_{{\rm{tr}}}}}}\frac{{{M_z}}}{B} + \frac{1}{2}{T_{\rm{D}}} \end{array} \right.$ (28)

 $\left\{ \begin{array}{l} {T_{\rm{i}}} = - \frac{{2R}}{{{i_{{\rm{tr}}}}}}\frac{{{M_z}}}{B} + {T_{{\rm{max}}}}\\ {T_{\rm{o}}} = {T_{{\rm{max}}}} \end{array} \right.$ (29)
3 仿真及试验验证 3.1 仿真验证

 $\left\{ \begin{array}{l} {T_{\rm{t}}} = {K_{\rm{p}}}\tilde u\left( t \right) + {K_{\rm{i}}}\int {\tilde u\left( t \right){\rm{d}}t} \\ \tilde u = u - {u_{{\rm{ex}}}} \end{array} \right.$ (30)

3.1.1 转向盘转角阶跃输入

 图 5 转向盘转角阶跃输入仿真结果 Fig.5 Simulation results of step input of steering wheel angle

3.1.2 转向盘转角正弦输入

 图 6 转向盘转角正弦输入仿真结果 Fig.6 Simulation results of sine input of steering wheel angle

3.2 试验验证

 图 7 车辆结构简图 Fig.7 System structure diagram of vehicle

 图 8 轮边电机外特性 Fig.8 External characteristics of motor
 图 9 试验场地 Fig.9 Test site
3.2.1 稳态圆周工况

 图 10 稳态圆周工况试验结果 Fig.10 Experiment results of steady circling test

3.2.2 蛇行工况

 图 11 蛇行工况试验结果 Fig.11 Experiment results of slalom test
4 结语

 [1] WEISS K R. Skid-steering[J]. Automobile Engineer, 1971, 61(4): 22 [2] 孙勇, 李文哲. 双流传动履带车辆转向机构的研究现状及发展趋势[J]. 农机化研究, 2008(3): 235 SUN Yong, LI Wenzhe. Actualities and developing tendency of turning mechanism of tracked vehicles with twin driving[J]. Journal of Agricultural Mechanization Research, 2008(3): 235 [3] 张韬懿, 王田苗, 吴耀, 等. 全地形无人车的设计与实现[J]. 机器人, 2013, 35(6): 657 ZHANG Taoyi, WANG Tianmiao, WU Yao, et al. Design and realization of an all-terrain unmanned ground Vehicle[J]. Robot, 2013, 35(6): 657 [4] D'ANDREA-NOVEL B, CAMPION G, BASTIN G. Control of nonholonomic wheeled mobile robots by state feedback linearization[J]. International Journal of Robotics Research, 1995, 14(6): 543 DOI:10.1177/027836499501400602 [5] 董文杰, 霍伟. 链式系统的轨迹跟踪控制[J]. 自动化学报, 2000, 26(3): 310 DONG Wenjie, HUO Wei. Trajectory tracking control of chained systems[J]. Acta Automatica Sinica, 2000, 26(3): 310 [6] 张扬名. 非完整约束下的移动机器人跟踪控制问题的研究[D]. 湘潭: 湘潭大学, 2013. ZHANG Yangming. Tracking control for nonholonomic mobile robots[D]. Xiangtan: Xiangtan University, 2013. [7] BENAOUMEUR I, LAREDJ B, REDA H E A, et al. Backstepping approach for autonomous mobile robot trajectory tracking[J]. Indonesian Journal of Electrical Engineering and Computer Science, 2016, 2(3): 478 DOI:10.11591/ijeecs.v2.i3 [8] KANG J, KIM W, LEE J, et al. Skid steering-based control of a robotic vehicle with six in-wheel drives[J]. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2010, 1(D11): 1 [9] JIN C, XIONG L, YU Z, et al. Path following control for skid steering vehicles with vehicle speed adaption[C]// SAE World Congress. Detroit: [s. n. ], 2014: 1-11. [10] 熊璐, 黄少帅, 陈远龙, 等. 轮式差动转向无人车运动跟踪控制的研究[J]. 汽车工程, 2015, 37(10): 1109 XIONG Lu, HUANG Shaoshuai, CHEN Yuanlong, et al. A research on motion tracking control for unmanned ground vehicle with wheeled skid-steering[J]. Automotive Engineering, 2015, 37(10): 1109 DOI:10.3969/j.issn.1000-680X.2015.10.002 [11] ZHANG Y, HU J, LI X. Steady-state characteristics of skid steering for wheeled vehicles[J]. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2014, 228(9): 1095 DOI:10.1177/0954407013516105