Trajectory tracking of caster-type omnidirectional mobile platform based on MPC and PID

doi:10.11772/j.issn.1001-9081.2023071003

Journal of Computer Applications ›› 2024, Vol. 44 ›› Issue (7): 2285-2293.DOI: 10.11772/j.issn.1001-9081.2023071003

• Frontier and comprehensive applications • Previous Articles Next Articles

Trajectory tracking of caster-type omnidirectional mobile platform based on MPC and PID

Huaxia LI¹, Xiaorong HUANG¹^,²^,³, Anlin SHEN¹^,²^,³, Peng JIANG¹, Yiqiang PENG¹^,²^,³(), Liqi SUI⁴

^1.School of Automobile and Transportation，Xihua University，Chengdu Sichuan 610039，China
^2.Vehicle Measurement，Control and Safety Key Laboratory of Sichuan Province （Xihua University），Chengdu Sichuan 610039，China
^3.Provincial Engineering Research Center for New Energy Vehicle Intelligent Control and Simulation Test Technology of Sichuan （Xihua University），Chengdu Sichuan 610039，China
^4.Yibin Fengchuan Powertrain Technology Company Limited，Yibin Sichuan 644600，China

Received:2023-07-25 Revised:2023-09-26 Accepted:2023-09-28 Online:2023-10-26 Published:2024-07-10
Contact: Yiqiang PENG
About author:LI Huaxia， born in 1998， M. S. candidate. His research interests include motion control of wheeled mobile robot.
HUANG Xiaorong， born in 1990， Ph. D.， associate professor. Her research interests include electric vehicle drive and electronic control system.
SHEN Anlin， born in 1985， M. S.， engineer. His research interests include vehicle dynamics.
JIANG Peng， born in 1997， M. S. candidate. His research interests include structural design and motion control of wheeled omnidirectional mobile robot.
SUI Liqi， born in 1986， Ph. D.， engineer. His research interests include optimal design and shift control of electric-driven mechanical transmission.
First author contact:PENG Yiqiang， born in 1963， Ph. D.， professor. His research interests include automotive electronic control， electric vehicle control， mechatronics.
Supported by:
Key Research and Development Project of Science and Technology Department of Sichuan Province(2020YFG0211);Science and Technology Innovation Base Construction Project of Sichuan Province(2022ZYD0125);International Cooperation Project of Science and Technology Department of Sichuan Province(2023YFH0065)

基于MPC和PID的脚轮式全向移动平台轨迹跟踪

李华夏¹, 黄晓蓉¹^,²^,³, 沈安林¹^,²^,³, 蒋鹏¹, 彭忆强¹^,²^,³(), 隋立起⁴

^1.西华大学汽车与交通学院, 成都 610039
^2.汽车测控与安全四川省重点实验室(西华大学), 成都 610039
^3.四川省新能源汽车智能控制与仿真测试技术工程研究中心(西华大学), 成都 610039
^4.宜宾丰川动力科技有限公司, 四川宜宾 644600

通讯作者: 彭忆强
作者简介:李华夏（1998—），男，四川成都人，硕士研究生，主要研究方向：轮式移动机器人运动控制；
黄晓蓉（1990—），女，四川成都人，副教授，博士，主要研究方向：电动汽车驱动与电控系统；
沈安林（1985—），男，四川南充人，工程师，硕士，主要研究方向：车辆动力学；
蒋鹏（1997—），男，四川资阳人，硕士研究生，主要研究方向：轮式全向移动机器人的结构设计和运动控制；
隋立起（1986—），男，山东潍坊人，工程师，博士，主要研究方向：电驱动机械变速器的优化设计和换档控制。
第一联系人：彭忆强（1963—），男，四川成都人，教授，博士，主要研究方向：汽车电子控制、电动汽车控制、机电一体化；
基金资助:
四川省科技厅重点研发项目(2020YFG0211);四川省科技创新基地建设项目(2022ZYD0125);四川省科技厅国际合作项目(2023YFH0065)

Abstract

Abstract:

Aiming at the problem that existing motion control strategies cannot guarantee high-precision control for independently driven caster-type omnidirectional mobile platform， a double closed-loop trajectory tracking control strategy was proposed by combining Model Predictive Control （MPC） and Proportion Integral Differential （PID） control. Firstly， kinematic geometric relationship was used to establish three-degree-of-freedom kinematic model of independently driven caster-type omnidirectional mobile platform in world coordinate system， and based on orthogonal decomposition method， inverse kinematic model of platform in robot coordinate system was established to reflect the relationship between center point speed of platform and rotation speed of each caster. Secondly， MPC was used to design position controller based on three-degree-of-freedom kinematic model， so that platform could track positions of desired trajectory， and the optimal control quantity was solved through the position controller while taking multi-objective constraints into account. Finally， PID was used to design speed controller to track desired speed output by position controller. Desired caster speed was calculated through the inverse kinematic model of platform， thereby driving platform to achieve omnidirectional motion. The effectiveness of the proposed control strategy was verified through simulation， and the platform could effectively track linear trajectories and circular trajectories. Simulation results show that compared with position single-loop trajectory tracking control strategy that decouples drive caster speed through angle inverse kinematic model of platform， the system overshoot is reduced by 97.23% and the response time is shortened by 36.84% after adding speed inner loop.

Key words: independently driven caster-type omnidirectional mobile platform, Model Predictive Control (MPC), trajectory tracking, Proportional Integral Derivative (PID) control, kinematic model

摘要：

针对现有运动控制策略无法保证独立驱动脚轮式全向移动平台位姿的高精度控制问题，提出一种基于模型预测控制（MPC）和PID控制相结合的双闭环轨迹跟踪控制策略。首先，利用运动学几何关系建立独立驱动脚轮式全向移动平台在世界坐标系下的三自由度运动学模型，基于正交分解法建立平台在机器人坐标系下的逆运动学模型，以反映平台中心点速度与各个脚轮转速间的关系；其次，采用MPC并基于三自由度运动学模型设计位姿控制器，使平台对期望轨迹进行位姿跟踪，并在考虑多目标约束条件的情况下通过位姿控制器求解出最优控制量；最后，采用PID设计速度控制器，用于跟踪位姿控制器输出的期望速度，通过平台逆运动学模型计算得到期望轮速，从而驱动平台实现全向运动。通过仿真验证了所提控制策略的有效性，平台能有效跟踪直线轨迹和圆形轨迹。仿真结果表明，与通过平台转角逆运动学模型解耦驱动轮速的位置单环轨迹跟踪控制策略相比，加入速度内环后系统超调量下降97.23%，响应时间缩短36.84%。

关键词: 独立驱动脚轮式全向移动平台, 模型预测控制, 轨迹跟踪, PID控制, 运动学模型

CLC Number:

TP273.4

Huaxia LI, Xiaorong HUANG, Anlin SHEN, Peng JIANG, Yiqiang PENG, Liqi SUI. Trajectory tracking of caster-type omnidirectional mobile platform based on MPC and PID[J]. Journal of Computer Applications, 2024, 44(7): 2285-2293.

李华夏, 黄晓蓉, 沈安林, 蒋鹏, 彭忆强, 隋立起. 基于MPC和PID的脚轮式全向移动平台轨迹跟踪[J]. 《计算机应用》唯一官方网站, 2024, 44(7): 2285-2293.

Figures/Tables 23

Fig. 1 Three-degree-of-freedom kinematic model of platform under world coordinate system

Tab. 1 Explanation of symbolic parameters for three-degree-of-freedom kinematic model of the platform

参数	含义
$X W ‑ O ‑ Y W$	世界坐标系
$X R ‑ O ‑ Y R$	机器人坐标系
$x$	脚轮式全向移动平台的横坐标位置
$y$	脚轮式全向移动平台的纵坐标位置
$φ$	脚轮式全向移动平台的朝向角
$v x$	沿 $X R$ 轴方向的速度
$v y$	沿 $Y R$ 轴方向的速度
$ω$	脚轮式全向移动平台自转角速度

Tab. 1 Explanation of symbolic parameters for three-degree-of-freedom kinematic model of the platform

参数	含义
$X W ‑ O ‑ Y W$	世界坐标系
$X R ‑ O ‑ Y R$	机器人坐标系
$x$	脚轮式全向移动平台的横坐标位置
$y$	脚轮式全向移动平台的纵坐标位置
$φ$	脚轮式全向移动平台的朝向角
$v x$	沿 $X R$ 轴方向的速度
$v y$	沿 $Y R$ 轴方向的速度
$ω$	脚轮式全向移动平台自转角速度

Fig. 2 Schematic diagram of platform structure

Fig. 3 Schematic diagram of caster structure

Tab. 2 Kinematic model parameters of platform under robot coordinate system

符号	参数含义	取值
$A$	车轮中心
$B$	转向轴中心
$C$	轮地接触点
$θ ˙ x i i ∈ 1,2, 3$	脚轮 $i$ 绕车轮中心的角速度
$θ ˙ s i i ∈ 1,2, 3$	脚轮 $i$ 绕转向轴中心的角速度
$θ s i i ∈ 1,2, 3$	脚轮 $i$ 绕转向轴中心的旋转角度
$θ ˙ c i i ∈ 1,2, 3$	脚轮 $i$ 绕轮地接触点的角速度
$r$	脚轮半径	0.1 m
$d$	转向轴中心与轮心的偏置距离	0.15 m
$h$	转向轴中心与平台中心连线长度	0.5 m
$α = α 1 α 2 α 3$	转向轴中心与平台中心连线与X轴的夹角	$30 ° 150 ° 270 °$
$V$	平台中心点的速度矢量

Tab. 2 Kinematic model parameters of platform under robot coordinate system

符号	参数含义	取值
$A$	车轮中心
$B$	转向轴中心
$C$	轮地接触点
$θ ˙ x i i ∈ 1,2, 3$	脚轮 $i$ 绕车轮中心的角速度
$θ ˙ s i i ∈ 1,2, 3$	脚轮 $i$ 绕转向轴中心的角速度
$θ s i i ∈ 1,2, 3$	脚轮 $i$ 绕转向轴中心的旋转角度
$θ ˙ c i i ∈ 1,2, 3$	脚轮 $i$ 绕轮地接触点的角速度
$r$	脚轮半径	0.1 m
$d$	转向轴中心与轮心的偏置距离	0.15 m
$h$	转向轴中心与平台中心连线长度	0.5 m
$α = α 1 α 2 α 3$	转向轴中心与平台中心连线与X轴的夹角	$30 ° 150 ° 270 °$
$V$	平台中心点的速度矢量

Tab. 3 Speed orthogonal decomposition table for caster-type omnidirectional mobile platform

方向	$v x$	$v y$	$ω$
平行方向	$- v x s i n θ s i$	$v y c o s θ s i$	$ω h c o s α i - θ s i$
垂直方向	$v x c o s θ s i$	$v y s i n θ s i$	$ω - h s i n α i - θ s i + d$

Tab. 3 Speed orthogonal decomposition table for caster-type omnidirectional mobile platform

方向	$v x$	$v y$	$ω$
平行方向	$- v x s i n θ s i$	$v y c o s θ s i$	$ω h c o s α i - θ s i$
垂直方向	$v x c o s θ s i$	$v y s i n θ s i$	$ω - h s i n α i - θ s i + d$

Fig. 4 MPC and PID double closed-loop control block diagram

Tab. 4 Role description of key parameters of MPC and PID double closed-loop control strategy

控制器	参数	作用
位姿控制器	$N p$	影响控制系统对轨迹跟踪的精确度和行驶稳定性
	$N c$	影响系统的响应速度
	$T$	影响控制系统执行算法的速率
	$Q$	保证控制系统对参考轨迹的跟踪精度
	$R$	保证控制系统较为平稳的控制平台跟踪上参考轨迹
速度控制器	$K p$	减少系统的偏差
	$T i$	消除静差，提高系统的无差度
	$T d$	反映系统偏差的变化趋势，加快系统的响应

Tab. 4 Role description of key parameters of MPC and PID double closed-loop control strategy

控制器	参数	作用
位姿控制器	$N p$	影响控制系统对轨迹跟踪的精确度和行驶稳定性
	$N c$	影响系统的响应速度
	$T$	影响控制系统执行算法的速率
	$Q$	保证控制系统对参考轨迹的跟踪精度
	$R$	保证控制系统较为平稳的控制平台跟踪上参考轨迹
速度控制器	$K p$	减少系统的偏差
	$T i$	消除静差，提高系统的无差度
	$T d$	反映系统偏差的变化趋势，加快系统的响应

Fig. 5 Linear tracking trajectory of body

Fig. 6 Tracking curve of linear trajectory

Fig. 7 Tracking position error of linear trajactory

Fig. 8 Velocity tracking curve of of linear trajactory

Fig. 9 Castor steering angle curve of linear trajactory

Fig. 10 Caster speed curve of linear trajectory

Fig. 11 Circular tracking trajectory of body

Fig. 12 Tracking curve of circular trajectory

Fig. 13 Tracking position error of circular trajectory

Fig. 14 Speed tracking curve of circular trajectory

Fig. 15 Caster steering angle curve of circular trajectory

Fig. 16 Caster speed curve of circular trajectory

Fig. 17 Comparison of trajectory tracking under linear conditions

Fig. 18 Comparison of trajectory tracking error under linear conditions

Fig. 19 Comparison of speed tracking curves under linear conditions

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Trajectory tracking of caster-type omnidirectional mobile platform based on MPC and PID

基于MPC和PID的脚轮式全向移动平台轨迹跟踪

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