双驱动球形机器人及其运动控制的研究
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摘要
球形机器人是一种以球形或近似球形为其外壳形状的独立运动体,在运动方式上以滚动为主,隶属于移动机器人门类。与其它移动机器人相比,球形机器人在结构形状和运动机理上较为新颖,在实际应用中也表现出很多独特的优势,所以从该类机器人出现以来,迅速得到各国机器人研究人员的广泛关注。
本文的内容主要是在所设计的双驱动球形机器人基础上,通过理论分析,仿真模拟和实验验证等主要手段,对机器人在力学方程建模和控制器设计方法两个方面展开研究,具体如下:
在总结现有各种球形机器人样机构型的基础之上,提出一种新型的双驱动球形机器人构型方案。该构型方案主要是将两个直流电机在球体内呈水平共线的方式放置。由于在球体内部缺乏一个相对稳定平台,所以驱动电机的输出力矩将会导致机器人的重心偏离球体的形心,此时机器人相对接触点的重力力矩就成为驱动球体前向滚动的直接力矩来源。该球形机器人的转向电机采用齿轮齿条啮合的方式来带动球体内部构件实现转向。
为了更好的了解和认识所设计的双驱动球形机器人,需要对其在任意时刻位置和姿态的描述方法进行研究。当描述球形机器人在某一平面的运动状态时,可以将独立变量分为对外部特征和内部特征的描述两部分:外部特征确定平面-球系统的具体位姿,内部特征确定内部构件的位置关系。描述这些特征的独立变量之间可以通过转换矩阵进行相互转换。
由于球形机器人是一个受非完整约束、强耦合、欠驱动、非线性的系统,全面而准确的描述该系统全部特征的动力学方程冗长而复杂,所以在该动力学方程基础上进行机器人的分析和控制器的设计难度很大,这极大的限制球形机器人在工程实际中的应用。为此,本文提出将机器人内部构件的转向和球体的前向滚动分步执行的运动策略,此时,一个受非完整约束的系统被转变为两个受完整约束的子系统,对机器人控制的难度得以降低。在这种分步执行策略下,通过点对点直线插补式的路径规划方法,球形机器人可以实现某一任意长直线段的运动轨迹,进而也可以实现某一封闭曲线的近似逼近。为验证所提出方法的有效性,相应的构建实验平台来对某一椭圆轨迹进行逼近,实验结果表明,当对行走轨迹跟踪精度要求不是很高的情况下,完全可以利用分步执行的运动策略和直线插补式的路径规划方法,来使双驱动球形机器人完成一般性的探测任务。
前向滚动运动是双驱动球形机器人的主要运动方式,同时利用重力力矩进行驱动的运动机理与以往的移动方式也有很多不同,因此需要着重对其动力学方程进行研究。在分析中,首先,使用基于能量耗散形式下的拉格朗日方程建立该动力学模型,利用仿真手段对状态变量的变化曲线进行分析,对理论方程的正确性进行验证;其次,根据临界摆角的概念将滚动摩阻作用下和爬坡状态下的机器人动力学方程在形式上进行统一,使方程更具通用性;再次,将临界摆角和系统的平衡点联系起来,在新的平衡点处使系统的动力学方程更有效的转化为线性定长的状态空间形式。
基于状态空间形式的方程,本文展开一系列控制器设计方法和仿真分析的研究。其中主要包括:对双驱动球形机器人的镇定性进行分析;利用扩展状态变量的方法,来实现系统对任意期望输入的跟踪问题;分析利用状态观测器来对不可量测状态进行估计的控制器设计方法;开展针对具体的性能指标构建目标函数,设计最优控制器的研究。在构建的实验中,具体的测试数据也有力的证明临界摆角的存在和状态增益法设计控制器的有效性。
另外,本文还对非完整系统的光滑时变控制器设计方法和滚动摩阻的作用机理等相关问题进行相应的理论分析和研究。
Spherical robot, belonging to mobile robot category, is an independent movement unit with sphere or approximate sphere shape, and its main motion is characterized by rolling style. Spherical robot attracts robot researchers’universal attention at once since it appeared in the world, not only because of its novel configuration and movement mechanism, but also because its special advantages in engineering application with respect to the other mobile robots.
This study develops mechanic modeling and controller design of a new dual-drive spherical robot. The methods applied in research are theoretical analysis, simulation and experimental verification. The contents can be expressed as follow,
Based on summarizations of the present spherical robots in view of structure, a new dual-drive spherical robot has been proposed. The main character is that the axial lines of two DC motors have been placed on the same straight line. Without a relative stable platform in the robot, the output torque of driving motor would lead robot’s gravity center to deviate from sphere’s center, and then, the gravity torque with regard to point of contact would cause the robot rolling forward. The turning motor leads inner components to change direction steering by rack and pinion to transfer torque.
In order to understand and analyze the dual-drive spherical robot better, it is necessary to study its situation and orientation description on arbitrary time. The individual variables can be divided into two parts, inner and outer character, when describing movement states of the robot rolling on a plane. The outer character variables define postures of plate-ball system, and the inner character variables decide positional relationship of the component. Those variables can be transformed each other by transition matrix.
Because the spherical robot involves some problems of nonholonomic constraints, strongly coupled manner, underactuated and nonlinear property, dynamic equations described the whole character variables entirely appear extraordinary complex and tedious. Based on those dynamic equations, the problems of system analysis and controller design become more difficult, which would greatly limit application of the robot. Therefore, a movement strategy, executing turning direction and rolling forward alternatively, has been proposed. Here, the system suffered by nonholonomic constraints has been transformed into two subsystems suffered only by holonomic constraints, and then, difficulty of controlling design has been decreased. Furthermore, the spherical robot can realize a straight arbitrary-length path, and can approach an enclosed trajectory approximately through the proposed trajectory planning method, named‘point-to-point straight-line interpolation’. At last, in order to verify the proposed method, experiment platform has been built up to tracking an ellipse trajectory. The results illustrate that when tracking precision doesn’t be restricted strictly, the dual-drive spherical robot can fulfill some generally explore assignments applied the above strategies and methods.
Rolling forward is the main movement of the dual-drive spherical robot. Hence, the dynamic of this motion should be investigated carefully because its driving mechanism using gravity torque is more differently than other mobile manners. Firstly, the dynamic equation has been obtained from Lagrangian function based on energy dissipation. Then, changeable curves of state variables have been analyzed by simulation method. Secondly, the dynamic equation forms, suffered by rolling frictional resistance and climbing state, have been unified. Such makes the derived equation more universality. Lastly, dynamic equation has been transformed into state space form with linear invariant property at new equilibrium point of system, which is relative to critical pendulum angle.
A series of research work about controller design and simulation analysis has been developed base on the derived state-space form equation. The contents are as followings: analysis about stabilization of the robot system; resolve tracking a reference input problem by means of expending state variables; study controller design method to estimate unmeasurable state using state observer; develop optimal controller for a special objective function constructed from some performance indexes. Experiment data prove existence of the critical pendulum angle and validity of state gain controller.
In addition, this paper has also studied some other problems related to the spherical robot, such as, smooth time-variable controller to nonholonomic system, mechanism of rolling friction, and so on.
本文的内容主要是在所设计的双驱动球形机器人基础上,通过理论分析,仿真模拟和实验验证等主要手段,对机器人在力学方程建模和控制器设计方法两个方面展开研究,具体如下:
在总结现有各种球形机器人样机构型的基础之上,提出一种新型的双驱动球形机器人构型方案。该构型方案主要是将两个直流电机在球体内呈水平共线的方式放置。由于在球体内部缺乏一个相对稳定平台,所以驱动电机的输出力矩将会导致机器人的重心偏离球体的形心,此时机器人相对接触点的重力力矩就成为驱动球体前向滚动的直接力矩来源。该球形机器人的转向电机采用齿轮齿条啮合的方式来带动球体内部构件实现转向。
为了更好的了解和认识所设计的双驱动球形机器人,需要对其在任意时刻位置和姿态的描述方法进行研究。当描述球形机器人在某一平面的运动状态时,可以将独立变量分为对外部特征和内部特征的描述两部分:外部特征确定平面-球系统的具体位姿,内部特征确定内部构件的位置关系。描述这些特征的独立变量之间可以通过转换矩阵进行相互转换。
由于球形机器人是一个受非完整约束、强耦合、欠驱动、非线性的系统,全面而准确的描述该系统全部特征的动力学方程冗长而复杂,所以在该动力学方程基础上进行机器人的分析和控制器的设计难度很大,这极大的限制球形机器人在工程实际中的应用。为此,本文提出将机器人内部构件的转向和球体的前向滚动分步执行的运动策略,此时,一个受非完整约束的系统被转变为两个受完整约束的子系统,对机器人控制的难度得以降低。在这种分步执行策略下,通过点对点直线插补式的路径规划方法,球形机器人可以实现某一任意长直线段的运动轨迹,进而也可以实现某一封闭曲线的近似逼近。为验证所提出方法的有效性,相应的构建实验平台来对某一椭圆轨迹进行逼近,实验结果表明,当对行走轨迹跟踪精度要求不是很高的情况下,完全可以利用分步执行的运动策略和直线插补式的路径规划方法,来使双驱动球形机器人完成一般性的探测任务。
前向滚动运动是双驱动球形机器人的主要运动方式,同时利用重力力矩进行驱动的运动机理与以往的移动方式也有很多不同,因此需要着重对其动力学方程进行研究。在分析中,首先,使用基于能量耗散形式下的拉格朗日方程建立该动力学模型,利用仿真手段对状态变量的变化曲线进行分析,对理论方程的正确性进行验证;其次,根据临界摆角的概念将滚动摩阻作用下和爬坡状态下的机器人动力学方程在形式上进行统一,使方程更具通用性;再次,将临界摆角和系统的平衡点联系起来,在新的平衡点处使系统的动力学方程更有效的转化为线性定长的状态空间形式。
基于状态空间形式的方程,本文展开一系列控制器设计方法和仿真分析的研究。其中主要包括:对双驱动球形机器人的镇定性进行分析;利用扩展状态变量的方法,来实现系统对任意期望输入的跟踪问题;分析利用状态观测器来对不可量测状态进行估计的控制器设计方法;开展针对具体的性能指标构建目标函数,设计最优控制器的研究。在构建的实验中,具体的测试数据也有力的证明临界摆角的存在和状态增益法设计控制器的有效性。
另外,本文还对非完整系统的光滑时变控制器设计方法和滚动摩阻的作用机理等相关问题进行相应的理论分析和研究。
Spherical robot, belonging to mobile robot category, is an independent movement unit with sphere or approximate sphere shape, and its main motion is characterized by rolling style. Spherical robot attracts robot researchers’universal attention at once since it appeared in the world, not only because of its novel configuration and movement mechanism, but also because its special advantages in engineering application with respect to the other mobile robots.
This study develops mechanic modeling and controller design of a new dual-drive spherical robot. The methods applied in research are theoretical analysis, simulation and experimental verification. The contents can be expressed as follow,
Based on summarizations of the present spherical robots in view of structure, a new dual-drive spherical robot has been proposed. The main character is that the axial lines of two DC motors have been placed on the same straight line. Without a relative stable platform in the robot, the output torque of driving motor would lead robot’s gravity center to deviate from sphere’s center, and then, the gravity torque with regard to point of contact would cause the robot rolling forward. The turning motor leads inner components to change direction steering by rack and pinion to transfer torque.
In order to understand and analyze the dual-drive spherical robot better, it is necessary to study its situation and orientation description on arbitrary time. The individual variables can be divided into two parts, inner and outer character, when describing movement states of the robot rolling on a plane. The outer character variables define postures of plate-ball system, and the inner character variables decide positional relationship of the component. Those variables can be transformed each other by transition matrix.
Because the spherical robot involves some problems of nonholonomic constraints, strongly coupled manner, underactuated and nonlinear property, dynamic equations described the whole character variables entirely appear extraordinary complex and tedious. Based on those dynamic equations, the problems of system analysis and controller design become more difficult, which would greatly limit application of the robot. Therefore, a movement strategy, executing turning direction and rolling forward alternatively, has been proposed. Here, the system suffered by nonholonomic constraints has been transformed into two subsystems suffered only by holonomic constraints, and then, difficulty of controlling design has been decreased. Furthermore, the spherical robot can realize a straight arbitrary-length path, and can approach an enclosed trajectory approximately through the proposed trajectory planning method, named‘point-to-point straight-line interpolation’. At last, in order to verify the proposed method, experiment platform has been built up to tracking an ellipse trajectory. The results illustrate that when tracking precision doesn’t be restricted strictly, the dual-drive spherical robot can fulfill some generally explore assignments applied the above strategies and methods.
Rolling forward is the main movement of the dual-drive spherical robot. Hence, the dynamic of this motion should be investigated carefully because its driving mechanism using gravity torque is more differently than other mobile manners. Firstly, the dynamic equation has been obtained from Lagrangian function based on energy dissipation. Then, changeable curves of state variables have been analyzed by simulation method. Secondly, the dynamic equation forms, suffered by rolling frictional resistance and climbing state, have been unified. Such makes the derived equation more universality. Lastly, dynamic equation has been transformed into state space form with linear invariant property at new equilibrium point of system, which is relative to critical pendulum angle.
A series of research work about controller design and simulation analysis has been developed base on the derived state-space form equation. The contents are as followings: analysis about stabilization of the robot system; resolve tracking a reference input problem by means of expending state variables; study controller design method to estimate unmeasurable state using state observer; develop optimal controller for a special objective function constructed from some performance indexes. Experiment data prove existence of the critical pendulum angle and validity of state gain controller.
In addition, this paper has also studied some other problems related to the spherical robot, such as, smooth time-variable controller to nonholonomic system, mechanism of rolling friction, and so on.
引文
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