CN116872219A - Robot control method, electronic device and storage medium based on U-K equation - Google Patents
Robot control method, electronic device and storage medium based on U-K equation Download PDFInfo
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Abstract
Description
技术领域Technical field
本发明涉及机器人控制技术领域,特别涉及基于U-K方程的机器人控制方法、电子设备及存储介质。The present invention relates to the field of robot control technology, and in particular to a robot control method, electronic equipment and storage medium based on the U-K equation.
背景技术Background technique
目前,工业领域常用的电机控制方法包括基于PID调节器的控制、改进PID控制方法、鲁棒控制、自适应控制、滑膜控制等。其中,PID控制是最常见,应用也较多的控制方法,通常由比例环节(P)、积分环节(I)、微分环节(D)三个部分组成,调整这三个部分的参数可以让电机的速度稳定在设定的速度附近。目前的研究中,复杂的多提机构/机械系统自身的机械特性会不断变化,更多的,在医疗康复领域,康复机械常常会和患者的肢体接触,这会导致传统的控制方法适应不了多变的工作情况,同时也无法保证高速度高精度的工作要求。At present, commonly used motor control methods in the industrial field include control based on PID regulators, improved PID control methods, robust control, adaptive control, synovial membrane control, etc. Among them, PID control is the most common control method with many applications. It usually consists of three parts: proportional link (P), integral link (I), and differential link (D). Adjusting the parameters of these three parts can make the motor The speed is stable near the set speed. In current research, the mechanical characteristics of complex multi-lift mechanisms/mechanical systems will continue to change. Furthermore, in the field of medical rehabilitation, rehabilitation machinery often comes into contact with the patient's limbs, which makes traditional control methods unable to adapt to multi-lift mechanisms. The working conditions change, and the high-speed and high-precision working requirements cannot be guaranteed.
同时,由于传统的机械系统的动力学建模都是采用以达朗贝尔原理为基础的拉格朗日动力学建模法,采用这种方法会对系统的相关条件和限制进行简化和理想化,从而造成误差,影响模型的精度。在工业生产、机器人控制、医疗康复等领域,实际复杂多变的环境状况和机械结构也会给传统的轨迹跟踪带来更多不确定因素。当系统存在非理想约束时,传统的基于达朗贝尔原理的动力学建模方法将无法解决该问题。At the same time, since traditional dynamic modeling of mechanical systems uses the Lagrangian dynamic modeling method based on D'Alembert's principle, using this method will simplify and idealize the relevant conditions and limitations of the system. , thus causing errors and affecting the accuracy of the model. In the fields of industrial production, robot control, medical rehabilitation and other fields, the actual complex and changeable environmental conditions and mechanical structures will also bring more uncertain factors to traditional trajectory tracking. When the system has non-ideal constraints, the traditional dynamic modeling method based on D'Alembert's principle will not be able to solve the problem.
发明内容Contents of the invention
为了实现本发明的上述目的和其他优点,本发明的第一目的是提供基于U-K方程的机器人控制方法,包括以下步骤:In order to achieve the above objects and other advantages of the present invention, the first object of the present invention is to provide a robot control method based on the U-K equation, which includes the following steps:
对机械臂进行正向运动学建模和动力学建模,得到动力学方程;Perform forward kinematics modeling and dynamics modeling on the robotic arm to obtain the dynamics equation;
对所述动力学方程进行处理,得到无约束状态下系统的加速度与系统的外力之间的关系;Process the dynamic equations to obtain the relationship between the acceleration of the system and the external force of the system in the unconstrained state;
对约束状态下系统的加速度与系统的外力之间的关系添加系统约束;Add system constraints to the relationship between the acceleration of the system and the external force of the system in the constrained state;
通过Udwadia-Kalaba方程求解满足系统约束的约束力,得到约束力的精确表达式;Solve the binding force that satisfies the system constraints through the Udwadia-Kalaba equation and obtain the exact expression of the binding force;
通过约束力的精确表达式计算得到系统运动的精确运动方程,实现对机械系统的精确控制。The precise motion equation of the system motion is calculated through the precise expression of the binding force, thereby achieving precise control of the mechanical system.
进一步地,所述对机械臂进行正向运动学建模和动力学建模包括以下步骤:Further, the forward kinematics modeling and dynamics modeling of the robotic arm includes the following steps:
采用基于旋量代数、李群和李代数的方法对机械臂进行正向运动学建模;Methods based on spinor algebra, Lie groups and Lie algebras are used to model the forward kinematics of the robotic arm;
采用拉格朗日动力学方法进行动力学建模,得到动力学方程。The Lagrangian dynamics method is used for dynamic modeling and the dynamic equations are obtained.
进一步地,系统的约束方程为:Furthermore, the constraint equation of the system is:
; ;
其中,矩阵为系统的约束矩阵,/>为加速度矢量,/>为约束矢量。Among them, matrix is the constraint matrix of the system,/> is the acceleration vector,/> is the constraint vector.
进一步地,所述通过Udwadia-Kalaba方程求解满足系统约束的约束力包括以下步骤:Further, solving the binding force that satisfies the system constraints through the Udwadia-Kalaba equation includes the following steps:
对于六自由度机械臂,其动力学方程的表达式为:For a six-degree-of-freedom manipulator, the expression of its dynamic equation is:
; ;
其中,为系统的惯性矩阵,/>为科氏力,/>为重力,/>为系统控制输入;in, is the inertia matrix of the system,/> is the Coriolis force,/> is gravity,/> Input for system control;
系统所受的约束为:The constraints on the system are:
; ;
基于Udwadia-Kalaba方程给出的控制器和控制力矩为:The controller and control torque given based on the Udwadia-Kalaba equation are:
; ;
其中,表示矩阵的Mooore-Penrose逆。in, Represents the Mooore-Penrose inverse of a matrix.
进一步地,还包括以下步骤:Further, the following steps are included:
在基于Udwadia-Kalaba方程给出的控制器中添加不确定性参数;Add uncertainty parameters in the controller given based on the Udwadia-Kalaba equation;
将含有不确定性参数的矩阵和向量及逆进行分解;Decompose matrices, vectors and inverses containing uncertainty parameters;
设计基于Udwadia-Kalaba方程的鲁棒控制器。Design a robust controller based on the Udwadia-Kalaba equation.
进一步地,所述在基于Udwadia-Kalaba方程给出的控制器中添加不确定性参数,得到控制器为:Further, by adding uncertainty parameters to the controller given based on the Udwadia-Kalaba equation, the obtained controller is:
; ;
其中,是未知量。in, is an unknown quantity.
进一步地,所述将含有不确定性参数的矩阵和向量及逆进行分解,得到:Further, by decomposing the matrix, vector and inverse containing uncertainty parameters, we obtain:
; ;
其中,、 />、 />为名义部分,且/>,/>、/>、/>为含不确定部分;in, , /> , /> is the nominal part, and/> ,/> ,/> ,/> It contains an uncertain part;
还包括规定以下公式:It also includes the following formula:
; ;
。 .
进一步地,所述设计基于U-K方程的鲁棒控制器包括:Further, the design of a robust controller based on the U-K equation includes:
; ;
其中,in,
; ;
; ;
其中,是标量,为待设计参数;选择函数/>,使/>;in, is a scalar, a parameter to be designed; select function/> ,make/> ;
假设三个条件,对于任意满秩:Assuming three conditions, for any Full rank:
存在一个函数,使所有的/>满足;There is a function , making all/> satisfy ;
对于给定的且/>,使/>;for a given and/> , make/> ;
存在一个常数,使/>。There is a constant , make/> .
进一步地,还包括系统稳定性分析步骤:Furthermore, it also includes system stability analysis steps:
利用Lyapunov渐近稳定性函数的选取要求选取合法的Lyapunov函数;Use the selection requirements of Lyapunov asymptotic stability function to select a legal Lyapunov function;
结合Lyapunov稳定性判定定理进行分析,证明鲁棒控制器满足一致有界性和一致最终有界性,控制系统渐近稳定。Analysis based on Lyapunov's stability determination theorem proves that the robust controller satisfies consistent boundedness and consistent ultimate boundedness, and the control system is asymptotically stable.
本发明的第二目的是提供一种电子设备,包括:存储器,其上存储有程序代码;处理器,其与所述存储器联接,并且当所述程序代码被所述处理器执行时,实现基于U-K方程的机器人控制方法。A second object of the present invention is to provide an electronic device, including: a memory having program code stored thereon; a processor coupled to the memory, and when the program code is executed by the processor, the Robot control method based on U-K equation.
本发明的第三目的是提供一种计算机可读存储介质,其上存储有程序指令,所述程序指令被执行时实现基于U-K方程的机器人控制方法。The third object of the present invention is to provide a computer-readable storage medium on which program instructions are stored. When the program instructions are executed, a robot control method based on the U-K equation is implemented.
与现有技术相比,本发明的有益效果是:Compared with the prior art, the beneficial effects of the present invention are:
本发明提供基于U-K方程的机器人控制方法、电子设备及存储介质,在系统动力学求解部分加入了基于U-K方程的U-K控制器,提高了系统的控制精度、效果以及快速响应。同时加入了可对内在不确定因素收敛的U-K鲁棒控制器;鲁棒控制器的设计,不仅提高了系统的控制效果,还可以面对复杂变化的机械系统保持控制效果不变,大大减小了不同患者使用时电机的控制成本,无需重新设计参数。The invention provides a robot control method, electronic equipment and storage medium based on the U-K equation. A U-K controller based on the U-K equation is added to the system dynamics solution part, thereby improving the control accuracy, effect and rapid response of the system. At the same time, a U-K robust controller is added that can converge to the inherent uncertain factors; the design of the robust controller not only improves the control effect of the system, but also maintains the control effect unchanged in the face of complex changes in the mechanical system, greatly reducing the It reduces the control cost of the motor when used by different patients, and there is no need to redesign parameters.
上述说明仅是本发明技术方案的概述,为了能够更清楚了解本发明的技术手段,并可依照说明书的内容予以实施,以下以本发明的较佳实施例并配合附图详细说明如后。本发明的具体实施方式由以下实施例及其附图详细给出。The above description is only an overview of the technical solutions of the present invention. In order to have a clearer understanding of the technical means of the present invention and implement them according to the contents of the description, the preferred embodiments of the present invention are described in detail below with reference to the accompanying drawings. Specific embodiments of the present invention are given in detail by the following examples and accompanying drawings.
附图说明Description of the drawings
此处所说明的附图用来提供对本发明的进一步理解,构成本申请的一部分,本发明的示意性实施例及其说明用于解释本发明,并不构成对本发明的不当限定。在附图中:The drawings described here are used to provide a further understanding of the present invention and constitute a part of this application. The illustrative embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the attached picture:
图1为实施例1的基于U-K方程的机器人控制方法流程图;Figure 1 is a flow chart of the robot control method based on the U-K equation in Embodiment 1;
图2为实施例2的电子设备示意图;Figure 2 is a schematic diagram of the electronic device of Embodiment 2;
图3为实施例3的存储介质示意图。Figure 3 is a schematic diagram of the storage medium in Embodiment 3.
具体实施方式Detailed ways
下面,结合附图以及具体实施方式,对本发明做进一步描述,需要说明的是,在不相冲突的前提下,以下描述的各实施例之间或各技术特征之间可以任意组合形成新的实施例。Below, the present invention will be further described with reference to the accompanying drawings and specific embodiments. It should be noted that, on the premise that there is no conflict, the various embodiments or technical features described below can be arbitrarily combined to form new embodiments. .
实施例1Example 1
基于U-K方程的机器人控制方法,如图1所示,包括以下步骤:The robot control method based on the U-K equation, as shown in Figure 1, includes the following steps:
S1、对机械臂进行正向运动学建模和动力学建模,得到动力学方程;具体地,包括以下步骤:S1. Perform forward kinematics modeling and dynamic modeling on the robotic arm to obtain the dynamic equation; specifically, the following steps are included:
采用基于旋量代数、李群和李代数的方法对机械臂进行正向运动学建模,为后续的轨迹规划提供数学基础;Methods based on spinor algebra, Lie groups and Lie algebras are used to model the forward kinematics of the robotic arm, providing a mathematical basis for subsequent trajectory planning;
采用拉格朗日动力学方法进行动力学建模,得到基本的动力学方程。The Lagrangian dynamics method is used for dynamic modeling and the basic dynamic equations are obtained.
S2、对得到的动力学方程进行处理,得到无约束状态下系统的加速度与系统的外力之间的关系;S2. Process the obtained dynamic equations to obtain the relationship between the acceleration of the system and the external force of the system in the unconstrained state;
S3、对约束状态下系统的加速度与系统的外力之间的关系添加系统约束;系统的约束一般来源于系统自身的机械结构和满足运动学方程的轨迹产生的轨迹约束。系统的理想约束和非理想约束都可以写成如下的约束方程的形式。S3. Add system constraints to the relationship between the acceleration of the system and the external force of the system in the constrained state; the constraints of the system generally come from the system's own mechanical structure and the trajectory constraints generated by the trajectory that satisfies the kinematic equations. Both the ideal constraints and non-ideal constraints of the system can be written in the form of the following constraint equations.
系统的约束方程为:The constraint equation of the system is:
; ;
其中,矩阵为系统的约束矩阵,/>为加速度矢量,/>为约束矢量。Among them, matrix is the constraint matrix of the system,/> is the acceleration vector,/> is the constraint vector.
S4、通过Udwadia-Kalaba方程求解满足系统约束的约束力,得到约束力的精确表达式;S4. Use the Udwadia-Kalaba equation to solve the binding force that satisfies the system constraints and obtain the precise expression of the binding force;
S5、通过约束力的精确表达式计算得到系统运动的精确运动方程,实现对机械系统的精确控制。S5. Calculate the precise motion equation of the system motion through the precise expression of the binding force to achieve precise control of the mechanical system.
其中,通过Udwadia-Kalaba方程求解满足系统约束的约束力包括以下步骤:Among them, solving the binding force that satisfies the system constraints through the Udwadia-Kalaba equation includes the following steps:
Udwadia-Kalaba方程的主要内容是,在满足约束的任意时刻的系统中,/>个质点的/>维加速度矢量由下式方程给出:The main content of the Udwadia-Kalaba equation is that at any time when the constraints are satisfied In the system,/> a particle/> The dimensional acceleration vector is given by the following equation:
。 .
对于六自由度机械臂,其动力学方程的表达式为:For a six-degree-of-freedom manipulator, the expression of its dynamic equation is:
; ;
其中,为系统的惯性矩阵,/>为科氏力,/>为重力,/>为系统控制输入;in, is the inertia matrix of the system,/> is the Coriolis force,/> is gravity,/> Input for system control;
系统所受的约束为:The constraints on the system are:
; ;
基于Udwadia-Kalaba方程给出的控制器和控制力矩为:The controller and control torque given based on the Udwadia-Kalaba equation are:
; ;
其中,表示矩阵的Mooore-Penrose逆,是一种广义逆矩阵,对于可逆方阵就是逆矩阵,对于一个/>维秩为/>矩阵/>,其/>维矩阵且满足:in, Represents the Mooore-Penrose inverse of a matrix, which is a generalized inverse matrix. For a reversible square matrix, it is an inverse matrix. For a /> The dimension rank is/> matrix/> , its/> dimensional matrix and satisfies:
; ;
对于任意矩阵A,其MP逆存在且唯一,零矩阵的MP逆就是零矩阵。For any matrix A, its MP inverse exists and is unique, and the MP inverse of a zero matrix is the zero matrix.
为了解决系统不确定因素,添加U-K鲁棒控制器。具体还包括以下步骤:In order to solve the system uncertainties, a U-K robust controller is added. Specifically, it also includes the following steps:
S6、对于六自由度的上肢康复训练机械臂来说,实际应用于不同的患者需要考虑各种不确定因素,包括机械臂的质量和长度都会发生变化,需要在基于Udwadia-Kalaba方程给出的控制器中添加不确定性参数,得到控制器为:S6. For the six-degree-of-freedom upper limb rehabilitation training robotic arm, various uncertain factors need to be considered when it is actually used on different patients, including changes in the mass and length of the robotic arm. It needs to be based on the Udwadia-Kalaba equation. Add uncertainty parameters to the controller, and the controller is:
; ;
; ;
其中,是未知量。in, is an unknown quantity.
S7、此时考虑到系统要保持稳定,将含有不确定性参数的矩阵和向量及逆进行分解,得到:S7. At this time, considering that the system must remain stable, decompose the matrix, vector and inverse containing the uncertainty parameters to obtain:
; ;
其中,、 />、 />为名义部分,且/>,/>、/>、/>为含不确定部分;in, , /> , /> is the nominal part, and/> ,/> ,/> ,/> It contains an uncertain part;
为了方便控制器的设计,规定以下公式:In order to facilitate the design of the controller, the following formula is specified:
; ;
。 .
S8、设计基于Udwadia-Kalaba方程的鲁棒控制器:S8. Design a robust controller based on the Udwadia-Kalaba equation:
; ;
其中,in,
; ;
其中,是标量,为待设计参数;选择函数/>,使/>;in, is a scalar, a parameter to be designed; select function/> ,make/> ;
假设三个条件,对于任意满秩:Assuming three conditions, for any Full rank:
存在一个函数,使所有的/>满足;There is a function , making all/> satisfy ;
对于给定的且/>,使/>;for a given And/> , make/> ;
存在一个常数,使/>。There is a constant , make/> .
本实施例还包括系统稳定性分析步骤:This embodiment also includes system stability analysis steps:
S9、利用Lyapunov渐近稳定性函数的选取要求选取合法的Lyapunov函数;S9. Use the selection requirements of Lyapunov asymptotic stability function to select a legal Lyapunov function;
S10、结合Lyapunov稳定性判定定理进行分析,可以证明该鲁棒控制器满足一致有界性和一致最终有界性,该控制系统渐近稳定。S10. Combined with Lyapunov stability determination theorem analysis, it can be proved that the robust controller satisfies consistent boundedness and consistent final boundedness, and the control system is asymptotically stable.
本发明在系统动力学求解部分加入了基于U-K方程的U-K控制器,提高了系统的控制精度和效果以及快速响应。同时加入了可对内在不确定因素收敛的U-K鲁棒控制器。鲁棒控制器的设计,不仅提高了系统的控制效果,还可以面对复杂变化的机械系统保持控制效果不变,大大减小了不同患者使用时电机的控制成本,无需重新设计参数。The present invention adds a U-K controller based on the U-K equation in the system dynamics solution part, thereby improving the control accuracy, effect and rapid response of the system. At the same time, a U-K robust controller that can converge to intrinsic uncertainties is added. The design of the robust controller not only improves the control effect of the system, but also maintains the control effect unchanged in the face of complex changes in the mechanical system, greatly reducing the cost of controlling the motor when used by different patients without the need to redesign parameters.
实施例2Example 2
一种电子设备200,如图2所示,包括但不限于:存储器201,其上存储有程序代码;处理器202,其与存储器联接,并且当程序代码被处理器执行时,实现基于U-K方程的机器人控制方法。关于方法的详细描述,可以参照上述方法实施例中的对应描述,在此不再赘述。An electronic device 200, as shown in Figure 2, includes but is not limited to: a memory 201, on which program code is stored; a processor 202, which is connected to the memory, and when the program code is executed by the processor, the U-K equation is implemented robot control method. For a detailed description of the method, reference may be made to the corresponding description in the above method embodiment, which will not be described again here.
实施例3Example 3
一种计算机可读存储介质,如图3所示,其上存储有程序指令,程序指令被执行时实现的基于U-K方程的机器人控制方法。关于方法的详细描述,可以参照上述方法实施例中的对应描述,在此不再赘述。A computer-readable storage medium, as shown in Figure 3, has program instructions stored thereon, and when the program instructions are executed, a robot control method based on the U-K equation is implemented. For a detailed description of the method, reference may be made to the corresponding description in the above method embodiment, which will not be described again here.
还需要说明的是,术语“包括”、“包含”或者其任何其他变体意在涵盖非排他性的包含,从而使得包括一系列要素的过程、方法、商品或者设备不仅包括那些要素,而且还包括没有明确列出的其他要素,或者是还包括为这种过程、方法、商品或者设备所固有的要素。在没有更多限制的情况下,由语句“包括一个……”限定的要素,并不排除在包括要素的过程、方法、商品或者设备中还存在另外的相同要素。It should also be noted that the terms "comprises," "comprises," or any other variation thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that includes a list of elements not only includes those elements, but also includes Other elements are not expressly listed or are inherent to the process, method, article or equipment. Without further limitation, an element qualified by the statement "comprises a..." does not exclude the presence of additional identical elements in the process, method, good, or device that includes the element.
本说明书中的各个实施例均采用递进的方式描述,各个实施例之间相同相似的部分互相参见即可,每个实施例重点说明的都是与其他实施例的不同之处。Each embodiment in this specification is described in a progressive manner. The same and similar parts between the various embodiments can be referred to each other. Each embodiment focuses on its differences from other embodiments.
以上仅为本说明书实施例而已,并不用于限制本说明书一个或多个实施例。对于本领域技术人员来说,本说明书一个或多个实施例可以有各种更改和变换。凡在本说明书一个或多个实施例的精神和原理之内所作的任何修改、等同替换、改进等,均应包含在本说明书一个或多个实施例的权利要求范围之内。The above are only embodiments of this specification and are not intended to limit one or more embodiments of this specification. For those skilled in the art, various modifications and transformations may be made to one or more embodiments of this specification. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of one or more embodiments of this specification shall be included in the scope of the claims of one or more embodiments of this specification.
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