WO2020124938A1 - 柔性关节机械臂分数阶滑模优化控制方法 - Google Patents

柔性关节机械臂分数阶滑模优化控制方法 Download PDF

Info

Publication number
WO2020124938A1
WO2020124938A1 PCT/CN2019/086551 CN2019086551W WO2020124938A1 WO 2020124938 A1 WO2020124938 A1 WO 2020124938A1 CN 2019086551 W CN2019086551 W CN 2019086551W WO 2020124938 A1 WO2020124938 A1 WO 2020124938A1
Authority
WO
WIPO (PCT)
Prior art keywords
fractional
sliding mode
order
joint
control
Prior art date
Application number
PCT/CN2019/086551
Other languages
English (en)
French (fr)
Inventor
汪允鹤
王继虎
夏正仙
王杰高
Original Assignee
南京埃斯顿机器人工程有限公司
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by 南京埃斯顿机器人工程有限公司 filed Critical 南京埃斯顿机器人工程有限公司
Publication of WO2020124938A1 publication Critical patent/WO2020124938A1/zh

Links

Images

Classifications

    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G05B13/045—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance using a perturbation signal
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1635—Programme controls characterised by the control loop flexible-arm control

Definitions

  • the invention relates to a flexible joint mechanical arm control method, in particular to a flexible joint mechanical arm fractional order sliding mode optimization control method, which belongs to the field of robot control.
  • fuzzy sliding mode control has strong robustness and does not depend on the system model, it can make full use of the advantages of expert information, but the fuzzy control system has a large static error.
  • fractional sliding mode control method of a flexible joint manipulator discloses a flexible joint manipulator system
  • the fractional-order sliding mode control method aims at the problems of buffeting and trajectory tracking control in the integer-order sliding mode dynamic control of the flexibility of the mechanical arm joint, combined with the advantages of sliding mode variable structure control, the introduction of fractional order calculus theory, and Taking advantage of the fast convergence, information memory and heredity of the fractional differential operator, a fractional sliding mode variable structure controller is proposed to realize the design of the fractional sliding mode variable structure control method.
  • the traditional indirect discretization method is adopted in the method of the invention, which has better robustness and anti-interference characteristics than traditional integer-order sliding mode variable structure control, and can weaken the chattering of sliding mode motion to a certain extent Phenomenon, but the indirect discretization of the conventional fractional differential operator has certain flaws in the low-frequency and high-frequency processing of the control signal. Therefore, the control effect of this method is still not ideal when the application requires high control accuracy.
  • the present invention proposes a flexible joint robot arm fractional order
  • the sliding mode optimization control method adopts the optimized indirect discretization method to improve the approximation accuracy of the fractional order differential operator, and is applied to the fractional order sliding mode control method of the flexible joint manipulator system, which can effectively weaken the chattering and strong vibration of the sliding mode motion. Robust and anti-jamming characteristics.
  • the invention provides a fractional-order sliding mode optimization control method for a flexible joint manipulator, and the steps include:
  • Step 1 Establish a dynamic mathematical model of the servo system of the flexible joint manipulator.
  • Step 2 Calculate the tracking error and sliding mode surface of the servo control system.
  • Re( ⁇ ) represents the real part of ⁇ .
  • the Gamma function is defined as
  • the RL type fractional calculus form has the following properties:
  • n is an integer and N is a set of integers.
  • Step 3 Establish the mathematical model of the discrete filter of the fractional differential operator
  • the mathematical model of the optimized indirect discretization algorithm is established by optimizing the improved indirect discretization algorithm.
  • the discretization filter of the conventional fractional differential operator adopts the indirect discretization method to realize the approximation of the ⁇ -order fractional differential operator s ⁇ in the frequency range of ( ⁇ b , ⁇ h ) by using rational function cascade,
  • the rational function of the cascade is
  • N is an integer
  • the greater the n the higher the approximation accuracy.
  • the improvement of the conventional indirect discretization algorithm is to approximate the fractional differential operator with a fractional transfer function, ie
  • This step optimizes the conventional and improved indirect discretization algorithm and uses the optimization algorithm to determine the filter parameters. Compared with the above formula, adding a filter by Taylor's first-order approximation brings a problem of reducing accuracy. After optimization, The indirect discretization algorithm can effectively improve the approximation accuracy.
  • the structure of the improved indirect discretization algorithm is used to approximate the fractional calculus s ⁇ to
  • G is the filter and Ge is the indirect discretization filter.
  • ⁇ 1 , ⁇ 2 , L, ⁇ n+1 are constant coefficients of the numerator
  • ⁇ 1 , ⁇ 2 , L, ⁇ n+1 are constant coefficients of the denominator.
  • the parameters of the filter G are determined after tuning, and then the approximate effect that the filter G reaches the optimal amplitude and phase frequency in the frequency band is obtained, indirectly
  • the fractional differential operator s ⁇ is implemented to avoid the precision loss caused by the truncation error.
  • Step 4 Calculate the fractional sliding mode approach law and control amount of the robot arm servo control system
  • t is a time variable
  • u(t) is the output control quantity
  • K is the rigidity coefficient of the flexible joint
  • D is the damping coefficient of the flexible joint
  • J is the motor inertia
  • d(t) is the bounded external disturbance (defined as
  • d max is the maximum boundary value ).
  • the joint position error e(t) qq d
  • q d is the expected value of the joint position
  • q is the actual value of the joint position
  • Step 5 Update the mechanical arm joint state parameters
  • the method of the present invention is optimized on the basis of the conventional indirect discretization algorithm. Compared with the traditional indirect discretization method and the improved indirect discretization method, it has a better approximation effect at low and high frequencies, and has a wider approximate frequency band. .
  • a fractional-order sliding mode variable structure controller with strong robustness, anti-interference and weakening chattering effect is proposed, which can make the machine
  • the flexible joint dynamics control system of the arm has better continuity, fastness, robustness and good anti-interference.
  • the design of the optimal fractional sliding mode variable structure control method is realized.
  • FIG. 1 is a schematic flow chart of a method for optimally controlling a fractional-order sliding mode of a flexible joint manipulator of the present invention.
  • FIG. 2 is a block diagram of a control structure of a flexible joint manipulator with a fractional sliding mode in the present invention.
  • the upper picture is the amplitude frequency characteristic
  • the lower picture is the phase frequency characteristic.
  • FIG. 4 is a schematic diagram of a trajectory of a fractional sliding mode surface of a robot arm joint controlled by fractional sliding mode according to the present invention.
  • FIG. 5 is a schematic diagram of the trajectory of an integer order sliding mode surface of a robot arm joint controlled by fractional order sliding mode of the present invention.
  • FIG. 6 is a schematic diagram of the output control amount trajectory of the joint step response of the robot arm controlled by the fractional sliding mode of the present invention.
  • FIG. 7 is a schematic diagram of a sliding mode surface trajectory of a step response of a robot arm joint controlled by fractional sliding mode according to the present invention.
  • the method for optimizing the control of the fractional sliding mode of the flexible joint manipulator proposed by the present invention has the following steps:
  • Step 1 Establish a dynamic mathematical model of the servo system of the flexible joint manipulator
  • This step is to calculate the dynamic mathematical model of the robot arm servo system, and its expression is
  • M(q) is the inertia matrix of the manipulator
  • G(q) is the gravity matrix of the manipulator
  • ⁇ ext is the external load disturbance of the manipulator
  • u is the output control quantity.
  • Step 2 Calculate the tracking error and sliding mode surface of the servo control system
  • This step will use the Riemann-Liouville (RL) fractional integral form of the integrable function f(t), which is defined as
  • Re( ⁇ ) represents the real part of ⁇ .
  • the Gamma function is defined as
  • the RL type fractional calculus form has the following properties:
  • n is an integer and N is a set of integers.
  • This step is to calculate the tracking error and fractional sliding mode surface of the robot arm servo control system.
  • q d is the desired joint position value
  • q is the actual value of the joint position
  • the sliding surface S(t) is designed as
  • ⁇ is the adjustment coefficient
  • D ⁇ represents the fractional calculus operator
  • D 1- ⁇ e(t) and D 2- ⁇ e(t) are expressed as 1- ⁇ , 2- ⁇ order differential, their expressions are
  • the ⁇ -order differential can be obtained from equation (5), and the fractional sliding mode surface can be organized as
  • Step 3 Establish the mathematical model of the discrete filter of the fractional differential operator
  • the discretization filter of the conventional fractional order differential operator adopts the indirect discretization method to realize the approximation of the fractional order differential operator s ⁇ in the frequency range of ( ⁇ b , ⁇ h ) by using the cascading of rational functions.
  • the rational function is
  • N is an integer
  • the greater the n the higher the approximation accuracy.
  • the improvement of the conventional indirect discretization algorithm is to approximate the fractional differential operator with a fractional transfer function, ie
  • This step optimizes the conventional and improved indirect discretization algorithm and uses the optimization algorithm to determine the filter parameters. Compared with the above formula, adding a filter by Taylor's first-order approximation brings a problem of reducing accuracy. After optimization, The indirect discretization algorithm can effectively improve the approximation accuracy.
  • the structure of the improved indirect discretization algorithm is used to approximate the fractional calculus s ⁇ to
  • G is the filter and Ge is the indirect discretization filter.
  • ⁇ 1 , ⁇ 2 , L, ⁇ n+1 are constant coefficients of the numerator
  • ⁇ 1 , ⁇ 2 , L, ⁇ n+1 are constant coefficients of the denominator.
  • the parameters of the filter G are determined after tuning, and then the approximate effect that the filter G reaches the optimal amplitude and phase frequency in the frequency band is obtained, indirectly
  • the fractional differential operator s ⁇ is implemented to avoid the precision loss caused by the truncation error.
  • the filter parameters used in the improved indirect discretization algorithm are the parameters
  • the optimized indirect discretization algorithm has a better approximation effect in low frequency and high frequency bands, and has a wider approximate frequency band.
  • Step 4 Calculate the sliding mode approach law and control amount of the robot arm servo control system
  • This step is to calculate the fractional sliding mode approach law and control law of the robot arm servo control system.
  • t is a time variable
  • u(t) is the output control quantity
  • K is the rigidity coefficient of the flexible joint
  • D is the damping coefficient of the flexible joint
  • J is the motor inertia
  • d(t) is the bounded external disturbance (defined as
  • d max is the maximum boundary value ).
  • the fractional approach law can change the speed of the system state when it reaches the sliding mode surface by adjusting the exponential approach coefficient k, constant velocity approach coefficient ⁇ , and differential order ⁇ .
  • the expression is
  • Step 5 Update the mechanical arm joint state parameters
  • the angle sensor installed at the joint of the robot arm is used to collect joint parameters and feed back.
  • fractional-order sliding mode optimization control method of the flexible joint manipulator proposed by the present invention is shown in FIG. 2.
  • a fractional-order sliding mode approach is derived based on the traditional integer order sliding mode approach law and sliding mode surface Law and fractional order reaching law, optimized on the basis of the conventional indirect discretization algorithm to improve the approximate accuracy of the fractional differential operator, the effect is shown in Figure 3, in order to study the fractional order calculus and sliding mode variable structure
  • Lyapunov theory and fractional stability theory are used to prove the stability of the entire system.
  • the single flexible joint of the manipulator adopts fractional sliding mode variable structure control
  • the obtained fractional sliding mode surface curve is shown in Fig. 4
  • the control curve of fractional sliding mode and integer order sliding mode control variable under step signal excitation is as follows
  • the trajectory comparison curve of fractional sliding mode and integer order sliding mode control sliding mode surface under step signal excitation is shown in Fig. 7, by comparing Fig.
  • the optimized indirect discretization algorithm approximates the fractional differential calculation Compared with the traditional integer-order sliding mode control, the fractional order reaching law and the fractional sliding mode surface of the sub can effectively weaken the chattering phenomenon of the sliding mode motion, and it shows stronger robust characteristics and better effect under the excitation of the step signal. Anti-jamming characteristics, and good performance improvement in trajectory tracking motion control.

Landscapes

  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Software Systems (AREA)
  • Evolutionary Computation (AREA)
  • Medical Informatics (AREA)
  • Health & Medical Sciences (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Artificial Intelligence (AREA)
  • Feedback Control In General (AREA)

Abstract

一种柔性关节机械臂分数阶滑模优化控制方法，针对机械臂关节柔性的整数阶滑模动力学控制中存在的抖振和轨迹跟踪控制的问题，结合滑模变结构控制的优点，引入分数阶微积分理论，以分数阶微分算子的快速收敛性、信息记忆性和遗传性特点，在常规分数阶间接离散化Oustaloup算法的基础上进行优化提升分数阶微分算子的近似精度，从而提出一种新型的具有强鲁棒性、抗干扰性和削弱抖振效果更好的分数阶滑模变结构控制器，在机械臂的关节柔性动力学控制系统具有更好的连续性、快速性、鲁棒性和良好的抗干扰性。最终，实现分数阶滑模变结构控制方法的设计。

Description

柔性关节机械臂分数阶滑模优化控制方法 技术领域
本发明涉及一种柔性关节机械臂控制方法，具体说是一种柔性关节机械臂分数阶滑模优化控制方法，属于机器人控制领域。
背景技术
在柔性关节机械臂的动力学控制和轨迹跟踪控制方面，现已有利用整数阶滑模变结构控制方法对参数时变和外部扰动的强鲁棒特性等优点，但传统整数阶滑模控制系统存在抖振较大的问题 [1]。当前比较常用的削减滑模抖振的方法有：边界层内的正侧化方法 [2]、基于观测器的调节方法 [3，4]、高阶滑模控制算法等方法 [5]可在一定程度上削弱抖振，其中前两种方法不具有传统滑模控制器的鲁棒特性，使得系统存在稳态误差，后一种算法比较复杂，在一阶或二阶的低阶系统中控制律存在控制输出信号与其导数的耦合，不利于滑模控制律的设计。模糊滑模控制虽然具有强鲁棒特性且不依赖系统模型，能充分利用专家信息等优点，但模糊控制系统存在较大的静差。
现有分数阶滑模控制方法中仍存在一定的缺点，如中国发明专利申请文献《一种柔性关节机械臂的分数阶滑模控制方法》(公开号CN108181813A)公开了一种柔性关节机械臂系统的分数阶滑模控制方法，针对机械臂关节柔性的整数阶滑模动力学控制中存在的抖振和轨迹跟踪控制的问题，结合滑模变结构控制的优点，引入分数阶微积分理论，并利用分数阶微分算子的快速收敛性、信息记忆性和遗传性，提出了一种分数阶滑模变结构控制器，实现分数阶滑模变结构控制方法的设计。该发明方法中采用传统间接离散化的方法，相比于传统整数阶滑模变结构控制而言具有较好的鲁棒特性和抗干扰特性、并可以在一定程度上削弱滑模运动的抖振现象，但是常规分数阶微分算子的间接离散化对控制信号的低频和高频段处理具有一定的缺陷，因此在应用要求控制精度较高的场合该方法控制效果仍然不够理想。
发明内容
为克服现有的机械臂运动控制系统中未考虑柔性关节机械臂本体结构和外部负载扰动的不确定性，以及存在滑模控制抖振等问题，本发明提出了一种柔性关节机械臂分数阶滑模优化控制方法，采用优化间接离散化方法提升分数阶微分算子的逼近精度，并应用于柔性关节机械臂系统的分数阶滑模控制方法中，能有效削弱滑模运动的抖振、强鲁棒和抗干扰特性。
本发明为了解决上述技术问题提出的一种柔性关节机械臂分数阶滑模优化控制方法，其步骤包括：
步骤1.建立柔性关节机械臂伺服系统的动力学数学模型。
步骤2.计算伺服控制系统的跟踪误差和滑模面。
采用可积函数f(t)的Riemann-Liouville(RL)分数阶积分形式，其定义为
Figure PCTCN2019086551-appb-000001
式中，
Figure PCTCN2019086551-appb-000002
表示求函数f(t)的α阶次积分，a和t为运算的上下限，τ为积分变量，Γ(.)为Gamma函数。
分数阶微积分
Figure PCTCN2019086551-appb-000003
定义为
Figure PCTCN2019086551-appb-000004
式中，Re(α)表示α的实部。
Gamma函数定义为
Figure PCTCN2019086551-appb-000005
RL型分数阶微积分形式，具有如下性质：
Figure PCTCN2019086551-appb-000006
式中，m为整数，N为整数集。
步骤3.建立分数阶微分算子离散化滤波器数学模型；
本步骤通过对改进间接离散化算法进行优化建立优化间接离散化算法的数学模型。常规分数阶微分算子的离散化滤波器是采用间接离散化方法利用有理函数级联的方式在(ω b，ω h)频率范围内实现α阶次的分数阶微分算子s α的逼近，该级联的有理函数为
Figure PCTCN2019086551-appb-000007
式中，
0＜α＜1
Figure PCTCN2019086551-appb-000008
Figure PCTCN2019086551-appb-000009
Figure PCTCN2019086551-appb-000010
N为整数，滤波器阶次n＝2N+1，且n越大逼近精度越高。
对常规间接离散化算法进行改进，是将分数阶微分算子用分数阶传递函数近似，即
Figure PCTCN2019086551-appb-000011
式中，取s＝jω(ω为频率变量，j为频域符号)，λ＞0，
Figure PCTCN2019086551-appb-000012
b＞0，d＞0，b和d为调整因子。
在(ω b，ω h)频率段内，将上式用泰勒公式展开，并取一阶近似，可得
Figure PCTCN2019086551-appb-000013
将上式右侧的K(s)用间接离散化递推展开，继而得到近似公式
Figure PCTCN2019086551-appb-000014
本步骤通过对常规及改进间接离散化算法进行优化，采用最优化算法寻优确定滤波参数，相比于上式由泰勒一阶近似展开增加一个滤波器带来降低精度等问题而言，优化后的间接离散化算法可有效提高近似精度。
采用改进间接离散化算法的结构，将分数阶微积分s α近似为
s α≈G×G e
式中，G为滤波器，G e为间接离散化滤波器。
滤波器G的传递函数形式为
Figure PCTCN2019086551-appb-000015
并可以通过扩展G的阶次来达到更高的精度需求，即
Figure PCTCN2019086551-appb-000016
式中，α 1，α 2，L，α n+1为分子的常系数，β 1，β 2，L，β n+1为分母的常系数。
利用ITAE参数寻优方法经对比超调量、调节时间等性能指标，经整定后确定滤波器G的参数，继而获得滤波器G在频率段内达到幅频及相频最优的近似效果，间接实现分数阶微分算子s α，避免了截断误差造成的精度损失。
步骤4.计算机械臂伺服控制系统的分数阶滑模趋近律和控制量；
柔性关节机械臂关节动力学模型：
Figure PCTCN2019086551-appb-000017
式中，t为时间变量，
Figure PCTCN2019086551-appb-000018
(
Figure PCTCN2019086551-appb-000019
为电机转速)，u(t)为输出控制量，
Figure PCTCN2019086551-appb-000020
K为柔性关节的刚性系数，D为柔性关节阻尼系数，J为电机转动惯量，d(t)为有界外部扰动量(定义为|d(t)|≤d max，d max为最大边界值)。
分数阶 滑模趋近律通过调节指数趋近系数k、等速趋近系数ε以及微分阶次α可以改变系统状态到达滑模面S(t)时的速度，其表达式为
D αS(t)＝-k·S(t)-ε·sign(S(t))
式中，k＞0，ε＞0，sign()为符号函数即
Figure PCTCN2019086551-appb-000021
根据以上步骤，设计相应的分数阶滑模控制器的控制律方程：
Figure PCTCN2019086551-appb-000022
式中，关节位置误差e(t)＝q-q d，q d为关节位置的期望值，q为关节位置的实际值。
步骤5.更新机械臂关节状态参数；
通过安装在机械臂关节处的角度传感器，采集机械臂关节参数，并反馈。
本发明方法，在常规间接离散化算法的基础上进行优化，相比于传统间接离散化方法及改进间接离散化方法在低频和高频具有更好的近似效果，并且具有更宽的近似频率段。利用分数阶微分算子的快速收敛性、信息记忆性和遗传性，从而提出了具有强鲁棒性、抗干扰性和削弱抖振效果更好的分数阶滑模变结构控制器，可使机械臂的柔性关节动力学控制系统具有更好的连续性、快速性、鲁棒性和良好的抗干扰性。最终，实现最优分数阶滑模变结构控制方法的设计。
附图说明
图1为本发明柔性关节机械臂分数阶滑模优化控制方法流程示意图。
图2为本发明中分数阶滑模的柔性关节机械臂控制结构框图。
图3为本发明分数阶微分算子频率响应示意图。其中：上图是幅频特性，下图为相频特性。
图4为本发明分数阶滑模控制的机械臂关节分数阶滑模面轨迹示意图。
图5为本发明分数阶滑模控制的机械臂关节整数阶滑模面轨迹示意图。
图6为本发明分数阶滑模控制的机械臂关节阶跃响应的输出控制量轨迹示意图。
图7为本发明分数阶滑模控制的机械臂关节阶跃响应的滑模面轨迹示意图。
具体实施方式
为了能够更好的了解本发明的技术特征、技术内容及其达到的技术效果，现结合实施例和附图对本发明方法作更加详细的说明。
实施例：
如图1所示，本发明提出的柔性关节机械臂分数阶滑模优化控制方法，其步骤如下：
步骤1.建立柔性关节机械臂伺服系统的动力学数学模型；
本步骤所述为计算机械臂伺服系统的动力学数学模型，其表达式为
Figure PCTCN2019086551-appb-000023
式中，M(q)为机械臂惯量矩阵，
Figure PCTCN2019086551-appb-000024
为离心力和哥式力项，G(q)为机械臂重力矩阵，τ ext为机械臂外部负载扰动，u为输出控制量。
采用以下参数为例建立柔性关节机械臂的动力学模型：
机械臂负载转动惯量：J a＝1.6×10 -5kg.m 2；
电机转动惯量：J＝2.56×10 -4kg.m 2；
关节刚度系数：K＝1.29Nm/rad；
阻尼系数：D＝3.6×10 2Nm/(rad/s)；
机械臂质量：m＝5.3kg；
机械臂质心：l c＝0.15m；
机械臂长度：l＝0.3m。
步骤2.计算伺服控制系统的跟踪误差和滑模面；
该步骤将采用可积函数f(t)的Riemann-Liouville(RL)分数阶积分形式，其定义为
Figure PCTCN2019086551-appb-000025
式中，
Figure PCTCN2019086551-appb-000026
表示求函数f(t)的α阶次积分，a和t为运算的上下限，τ为积分变量，Γ(.)为Gamma函数。
分数阶微积分
Figure PCTCN2019086551-appb-000027
定义为
Figure PCTCN2019086551-appb-000028
式中，Re(α)表示α的实部。
Gamma函数定义为
Figure PCTCN2019086551-appb-000029
RL型分数阶微积分形式，具有如下性质：
Figure PCTCN2019086551-appb-000030
式中，m为整数，N为整数集。
本步骤所述为计算机械臂伺服控制系统的跟踪误差和分数阶滑模面。
跟踪误差表达式为
e(t)＝q-q d   (4)
式中，q d为期望的关节位置值，q为关节位置的实际值。
滑模面S(t)设计为
S(t)＝λe(t)+D 1-αe(t)+D 2-αe(t)，0＜α＜1   (5)
式中，λ为调整系数，D α表示分数阶微积分算子，则D 1-αe(t)与D 2-αe(t)表示为求跟踪误差e(t)的1-α、2-α阶微分，其表达式分别为
Figure PCTCN2019086551-appb-000031
Figure PCTCN2019086551-appb-000032
根据分数阶微积分性质，可对式(5)求α阶微分，分数阶滑模面可整理为
Figure PCTCN2019086551-appb-000033
步骤3.建立分数阶微分算子离散化滤波器数学模型；
常规分数阶微分算子的离散化滤波器是采用间接离散化方法利用有理函数级联的方式在(ω b，ω h)频率范围内实现分数阶微分算子s α的逼近，该级联的有理函数为
Figure PCTCN2019086551-appb-000034
式中，
0＜α＜1
Figure PCTCN2019086551-appb-000035
Figure PCTCN2019086551-appb-000036
Figure PCTCN2019086551-appb-000037
N为整数，滤波器阶次n＝2N+1，且n越大逼近精度越高。
对常规间接离散化算法进行改进，是将分数阶微分算子用分数阶传递函数近似，即
Figure PCTCN2019086551-appb-000038
式中，取s＝jω(ω为频率变量，j为频域符号)，λ＞0，
Figure PCTCN2019086551-appb-000039
b＞0，d＞0，b和d为调整因子。。
在(ω b，ω h)频率段内，用泰勒公式展开，并取一阶近似，可得
Figure PCTCN2019086551-appb-000040
将上式右侧的K(s)用间接离散化递推展开，继而得到近似公式
Figure PCTCN2019086551-appb-000041
本步骤通过对常规及改进间接离散化算法进行优化，采用最优化算法寻优确定滤波参数，相比于上式由泰勒一阶近似展开增加一个滤波器带来降低精度等问题而言，优化后的间接离散化算法可有效提高近似精度。
采用改进间接离散化算法的结构，将分数阶微积分s α近似为
s α≈G×G e  (12)
式中，G为滤波器，G e为间接离散化滤波器。
滤波器G的传递函数形式为
Figure PCTCN2019086551-appb-000042
并可以通过扩展G的阶次来达到更高的精度需求，即
Figure PCTCN2019086551-appb-000043
式中，α 1，α 2，L，α n+1为分子的常系数，β 1，β 2，L，β n+1为分母的常系数。
利用ITAE参数寻优方法经对比超调量、调节时间等性能指标，经整定后确定滤波器G的参数，继而获得滤波器G在频率段内达到幅频及相频最优的近似效果，间接实现分数阶微分算子s α，避免了截断误差造成的精度损失。
根据上述算法，在频率段(10 -3，10 3)内，取N＝4，计算分数阶微分算子s α(α＝0.5)，在改进间接离散化算法中使用的滤波器参数为参数寻优的初始化值，经参数寻优后，继而获得一组最优滤波器G的参数(α 1，α 2，α 3)＝(0.902，1438，-0.0012)，(β 1，β 2，β 3)＝(0.24，1438，+0.6312)，使得滤波器G在频率段内达到幅频和相频的最优近似，并分别与常规间接离散化算法、改进间接离散化算法进行频率响应的对比，如图3所示，相比于常规间接离散化算法，优化间接离散化算法在低频及高频段具有更好的近似效果，并且具有更宽的近似频率段。
步骤4.计算机械臂伺服控制系统的滑模趋近律和控制量；
本步骤所述为计算机械臂伺服控制系统分数阶滑模趋近律和控制律。
根据柔性关节机械臂关节动力学模型：
Figure PCTCN2019086551-appb-000044
式中，t为时间变量，
Figure PCTCN2019086551-appb-000045
(
Figure PCTCN2019086551-appb-000046
为电机转速)，u(t)为输出控制量，
Figure PCTCN2019086551-appb-000047
K为柔性关节的刚性系数，D为柔性关节阻尼系数，J为电机转动惯量，d(t)为有界外部扰动量(定义为|d(t)|≤d max，d max为最大边界值)。
分数阶趋近律通过调节指数趋近系数k、等速趋近系数ε以及微分阶次α可以改变系统状态到达滑模面时的速度，其表达式为
D αS(t)＝-k·S(t)-ε·sign(S(t))，k＞0，ε＞0  (15)
由式(14)、(15)设计相应的控制律为
Figure PCTCN2019086551-appb-000048
经Lyapunov稳定性验证，可得控制律的状态将在有限时间内收敛于滑模面S(t)＝0，而渐近稳定。
步骤5.更新机械臂关节状态参数；
本步骤所述为通过安装在机械臂关节处的角度传感器，采集关节参数，并反馈。
综上，本发明提出的柔性关节机械臂分数阶滑模优化控制方法，如图2所示，一种以传统整数阶滑模趋近律、滑模面为基础推导出分数阶滑模趋近律和分数阶趋近律，开在常规间接离散化算法的基础上进行优化提升分数阶微分算子的近似精度，效果如图3所示，以此研究了分数阶微积分与滑模变结构控制结合在柔性关节机械臂控制中的作用，并利用Lyapunov理论与分数阶稳定性理论对整个系统的稳定性进行证明。根据上述方法，对机械臂的单个柔性关节采用分数阶滑模变结构控制，优化间接离散化方法采用以下参数：ω b＝10 -3、ω h＝10 3、N＝4、α＝0.9，得到分数阶滑模面曲线如图4所示，整数阶(α＝1.0)滑模面曲线如图5所示，在阶跃信号激励下分数阶滑模与整数阶滑模控制量对比曲线如图6所示，在阶跃信号激励下分数阶滑模与整数阶滑模控制滑模面轨迹对比曲线如图7所示，通过对比图4-7，优化间接离散化算法近似分数阶微分算子的分数阶趋近律和分数阶滑模面相比于传统整数阶滑模控制可有效削弱滑模运动的抖振现象，在阶跃信号的激励下显示具有效果更好的强鲁棒特性和抗干扰特性，以及在轨迹跟踪运动控制中具有较好的性能提升作用。
以上阐述的是本发明给出的最优分数阶滑模的柔性关节机械臂控制方法，显然本发明不仅仅是限于上述实施例，在不偏于本发明基本精神及不超出本发明实质内容所涉及范围的前提下对其可作种种变形加以实施。

Claims (1)

  1. 一种柔性关节机械臂分数阶滑模优化控制方法，其步骤包括：
    步骤1.建立柔性关节机械臂伺服系统的动力学数学模型；
    步骤2.计算伺服控制系统的跟踪误差和滑模面
    采用可积函数f(t)的Riemann-Liouville(RL)分数阶积分形式，其定义为
    Figure PCTCN2019086551-appb-100001
    式中，
    Figure PCTCN2019086551-appb-100002
    表示求函数f(t)的α阶次积分，a和t为运算的上下限，τ为积分变量，Γ(.)为Gamma函数。
    步骤3.建立分数阶微分算子离散化滤波器数学模型；
    采用改进间接离散化算法的结构，将分数阶微积分s α近似为
    s α≈G×G e
    式中，G为滤波器，G e为间接离散化滤波器。
    滤波器G的传递函数为
    Figure PCTCN2019086551-appb-100003
    式中，α 1，α 2，L，α n+1为分子的常系数，β 1，β 2，L，β n+1为分母的常系数；
    步骤4.计算机械臂伺服控制系统的分数阶滑模趋近律和控制量
    柔性关节机械臂关节动力学模型：
    Figure PCTCN2019086551-appb-100004
    式中，t为时间变量，
    Figure PCTCN2019086551-appb-100005
    (
    Figure PCTCN2019086551-appb-100006
    为电机转速)，u(t)为输出控制量，
    Figure PCTCN2019086551-appb-100007
    K为柔性关节的刚性系数，D为柔性关节阻尼系数，J为电机转动惯量，d(t)为有界外部扰动量(定义为|d(t)|≤d max，d max为最大边界值)；
    分数阶 滑模趋近律通过调节指数趋近系数k、等速趋近系数ε以及微分阶次α可以改变系统状态到达滑模面S(t)时的速度，其表达式为
    D αS(t)＝-k·S(t)-ε·sign(S(t))
    式中，k＞0，ε＞0，sign()为符号函数即
    根据以上步骤，设计相应的分数阶滑模控制器的控制律方程：
    Figure PCTCN2019086551-appb-100008
    步骤5.更新机械臂关节状态参数
    通过安装在机械臂关节处的角度传感器，采集机械臂关节参数，并反馈。
PCT/CN2019/086551 2018-12-21 2019-05-13 柔性关节机械臂分数阶滑模优化控制方法 WO2020124938A1 (zh)

Applications Claiming Priority (2)

Application Number Priority Date Filing Date Title
CN201811568816.9A CN109683478A (zh) 2018-12-21 2018-12-21 柔性关节机械臂分数阶滑模优化控制方法
CN201811568816.9 2018-12-21

Publications (1)

Publication Number Publication Date
WO2020124938A1 true WO2020124938A1 (zh) 2020-06-25

Family

ID=66188550

Family Applications (1)

Application Number Title Priority Date Filing Date
PCT/CN2019/086551 WO2020124938A1 (zh) 2018-12-21 2019-05-13 柔性关节机械臂分数阶滑模优化控制方法

Country Status (2)

Country Link
CN (1) CN109683478A (zh)
WO (1) WO2020124938A1 (zh)

Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112987566A (zh) * 2021-02-08 2021-06-18 南京工业大学 一种气动热的超螺旋非线性分数阶滑模无模型控制方法
CN113325711A (zh) * 2021-05-27 2021-08-31 广东工业大学 一种用于柔性机械臂高精度定位的智能控制方法
CN113467245A (zh) * 2021-07-15 2021-10-01 北京信息科技大学 飞行器的分数阶滑模控制方法、装置及系统
CN113927592A (zh) * 2021-08-24 2022-01-14 盐城工学院 一种基于自适应降阶滑模算法的机械臂力位混合控制方法
CN114237051A (zh) * 2021-12-16 2022-03-25 江苏理工学院 基于分数阶滑模反步法的动力翼伞高度控制方法
CN116702575A (zh) * 2023-08-08 2023-09-05 石家庄铁道大学 一种柔性互联装置分数阶建模与优化控制方法

Families Citing this family (11)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109683478A (zh) * 2018-12-21 2019-04-26 南京埃斯顿机器人工程有限公司 柔性关节机械臂分数阶滑模优化控制方法
CN110058525B (zh) * 2019-05-17 2021-10-01 金陵科技学院 分数阶离心飞轮调速系统有限时间镇定控制方法
CN110187723B (zh) * 2019-06-04 2021-05-25 南京航空航天大学 一种刚-柔耦合机电伺服系统的振动控制方法
CN110450154A (zh) * 2019-07-18 2019-11-15 广州弘度信息科技有限公司 一种基于动力学的机械臂柔性控制方法
CN111037573B (zh) * 2020-01-08 2022-09-30 山东理工大学 一种仿人柔性关节手臂的振动抑制方法
CN111240191B (zh) * 2020-01-16 2022-09-02 西北工业大学 变时延条件下机器人系统的自适应模糊阶次稳定操控方法
CN112207823B (zh) * 2020-09-22 2023-07-25 北京石油化工学院 一种康复机器人等速运动控制方法及系统
CN112327620B (zh) * 2020-10-29 2022-05-17 广东省智能机器人研究院 兼顾绕障的移动机器人鲁棒控制方法及系统
CN113867136A (zh) * 2021-11-22 2021-12-31 浙江工业大学 基于遗传算法与模糊分数阶pid的钢结构损伤检测机器人姿态控制方法
CN116317775B (zh) * 2023-03-17 2023-12-08 铜陵学院 一种电动助力车驱动系统双分数阶积分滑模控制方法
CN116619372B (zh) * 2023-05-31 2024-02-23 南京埃斯顿机器人工程有限公司 一种机械臂轨迹精度提升方法、设备、产品和介质

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2009181392A (ja) * 2008-01-31 2009-08-13 Omron Corp モデル予測制御方法およびモデル予測制御装置
CN102591207A (zh) * 2012-03-01 2012-07-18 北京航空航天大学 一种基于干扰观测器的柔性关节机械臂的滑模控制方法
CN106354962A (zh) * 2016-08-02 2017-01-25 电子科技大学 基于分频表征的磷酸铁锂电池分数阶等效电路模型建立方法
CN108181813A (zh) * 2017-12-28 2018-06-19 南京埃斯顿机器人工程有限公司 一种柔性关节机械臂的分数阶滑模控制方法
CN109683478A (zh) * 2018-12-21 2019-04-26 南京埃斯顿机器人工程有限公司 柔性关节机械臂分数阶滑模优化控制方法

Family Cites Families (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN100576200C (zh) * 2008-05-15 2009-12-30 北京理工大学 一种基于多抽样的分数阶傅立叶变换实现方法
US10488827B2 (en) * 2012-02-15 2019-11-26 Nri R&D Patent Licensing, Llc Adaptive multi-level control for variable-hierarchy-structure hierarchical systems
CN102591306B (zh) * 2012-03-08 2013-07-10 南京埃斯顿机器人工程有限公司 双系统组件式的工业机器人控制器
CN104260107B (zh) * 2014-09-04 2016-03-30 北京邮电大学 一种实现空间机械臂柔性关节齿隙补偿的方法

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2009181392A (ja) * 2008-01-31 2009-08-13 Omron Corp モデル予測制御方法およびモデル予測制御装置
CN102591207A (zh) * 2012-03-01 2012-07-18 北京航空航天大学 一种基于干扰观测器的柔性关节机械臂的滑模控制方法
CN106354962A (zh) * 2016-08-02 2017-01-25 电子科技大学 基于分频表征的磷酸铁锂电池分数阶等效电路模型建立方法
CN108181813A (zh) * 2017-12-28 2018-06-19 南京埃斯顿机器人工程有限公司 一种柔性关节机械臂的分数阶滑模控制方法
CN109683478A (zh) * 2018-12-21 2019-04-26 南京埃斯顿机器人工程有限公司 柔性关节机械臂分数阶滑模优化控制方法

Cited By (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112987566A (zh) * 2021-02-08 2021-06-18 南京工业大学 一种气动热的超螺旋非线性分数阶滑模无模型控制方法
CN113325711A (zh) * 2021-05-27 2021-08-31 广东工业大学 一种用于柔性机械臂高精度定位的智能控制方法
CN113467245A (zh) * 2021-07-15 2021-10-01 北京信息科技大学 飞行器的分数阶滑模控制方法、装置及系统
CN113467245B (zh) * 2021-07-15 2023-06-02 北京信息科技大学 飞行器的分数阶滑模控制方法、装置及系统
CN113927592A (zh) * 2021-08-24 2022-01-14 盐城工学院 一种基于自适应降阶滑模算法的机械臂力位混合控制方法
CN114237051A (zh) * 2021-12-16 2022-03-25 江苏理工学院 基于分数阶滑模反步法的动力翼伞高度控制方法
CN116702575A (zh) * 2023-08-08 2023-09-05 石家庄铁道大学 一种柔性互联装置分数阶建模与优化控制方法
CN116702575B (zh) * 2023-08-08 2023-10-20 石家庄铁道大学 一种柔性互联装置分数阶建模与优化控制方法

Also Published As

Publication number Publication date
CN109683478A (zh) 2019-04-26

Similar Documents

Publication Publication Date Title
WO2020124938A1 (zh) 柔性关节机械臂分数阶滑模优化控制方法
CN108181813B (zh) 一种柔性关节机械臂的分数阶滑模控制方法
CN107577147B (zh) 一种基于自适应Smith预估器的遥操作双边PID控制方法
CN108875253B (zh) 基于干扰观测器的欠驱动吊车系统的终端滑模消摆控制方法及系统
CN110071676B (zh) 一种柔性机器人关节伺服系统的振动抑制方法和装置
CN108333939B (zh) 一种基于神经网络的时标分离飞行器弹性体智能控制方法
CN108052008B (zh) 一种倾转旋翼飞行器过渡态切换过程的几何最优控制器设计方法
Abdul-Adheem et al. From PID to nonlinear state error feedback controller
Chang et al. Adaptive fuzzy backstepping output constraint control of flexible manipulator with actuator saturation
CN107544256A (zh) 基于自适应反步法的水下机器人滑模控制方法
CN110471438B (zh) 一种刚性飞行器的固定时间自适应姿态跟踪控制方法
CN106113040B (zh) 基于串并联估计模型的柔性机械臂系统模糊控制方法
CN104300863A (zh) 一种变负载永磁同步电机调速的自适应滑模控制方法
CN109848990B (zh) 基于pso的膝关节外骨骼增益可变无模型角度控制方法
CN108527372A (zh) 一种变刚度串联弹性驱动器的机器人关节自适应控制方法
CN109828468B (zh) 一种针对磁滞非线性机器人系统的控制方法
CN112631318B (zh) 一种cmg框架伺服系统高次谐波干扰补偿控制方法
CN113183154A (zh) 一种柔性关节机械臂的自适应反演控制方法
Zhang et al. On low-complexity control design to spacecraft attitude stabilization: An online-learning approach
CN110488854B (zh) 一种基于神经网络估计的刚性飞行器固定时间姿态跟踪控制方法
CN108415253B (zh) 基于马尔可夫跳变理论的飞轮电池磁轴承系统的控制方法
Ghorbel et al. Stability analysis of adaptively controlled flexible joint manipulators
CN108656111B (zh) 双机械臂系统有限时间参数辨识与位置同步控制方法
CN110687796A (zh) 基于神经网络的自适应有限时间命令滤波反步控制方法
CN114421835A (zh) 一种基于模糊偏差耦合反步滑模策略的多电机控制方法

Legal Events

Date Code Title Description
121 Ep: the epo has been informed by wipo that ep was designated in this application

Ref document number: 19900756

Country of ref document: EP

Kind code of ref document: A1

NENP Non-entry into the national phase

Ref country code: DE

122 Ep: pct application non-entry in european phase

Ref document number: 19900756

Country of ref document: EP

Kind code of ref document: A1