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CN113927596A - Time-varying output constraint robot teleoperation finite time control method based on width neural learning - Google Patents

Time-varying output constraint robot teleoperation finite time control method based on width neural learning Download PDF

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CN113927596A
CN113927596A CN202111167440.2A CN202111167440A CN113927596A CN 113927596 A CN113927596 A CN 113927596A CN 202111167440 A CN202111167440 A CN 202111167440A CN 113927596 A CN113927596 A CN 113927596A
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黄攀峰
李陇南
马志强
鹿振宇
常海涛
陈海飞
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Abstract

本发明公开了一种宽度神经学习的时变输出约束机器人有限时间控制方法。基于积分障碍李雅普诺夫函数、宽度神经学习算法和有限时间理论，创新性地提出了一种基于宽度神经学习的时变输出约束有限时间控制器。直接积分障碍李雅普诺夫函数保证了系统的输出在时变边界内；宽度神经学习算法融合了传统神经网络和宽度学习的优点，基于逆动力学观测器，利用宽度学习算法解决了外力感知难题，同时消解了控制率uxj设计过程中的模型不确定性；有限时间理论保证了机器人对参考信号的高精度和快速性追踪。综上所述，该算法确保了机器人与环境间稳定、安全与高效交互，提升了遥操作系统的可靠性和高效性。

Figure 202111167440

The invention discloses a limited-time control method for a time-varying output-constrained robot with width neural learning. Based on the integral barrier Lyapunov function, breadth neural learning algorithm and finite time theory, a time-varying output constraint finite time controller based on breadth neural learning is innovatively proposed. The direct integration obstacle Lyapunov function ensures that the output of the system is within the time-varying boundary; the breadth neural learning algorithm combines the advantages of traditional neural networks and breadth learning, based on the inverse dynamics observer, and uses the breadth learning algorithm to solve the external force perception problem, At the same time, the model uncertainty in the design process of the control rate u xj is eliminated; the finite time theory ensures the high precision and fast tracking of the reference signal by the robot. In summary, the algorithm ensures the stable, safe and efficient interaction between the robot and the environment, and improves the reliability and efficiency of the teleoperating system.

Figure 202111167440

Description

Time-varying output constraint robot teleoperation finite time control method based on width neural learning
Technical Field
The invention belongs to the technical field of robot control, and particularly relates to a time-varying output constraint robot teleoperation limited time control method based on width neural learning.
Background
Teleoperation technology makes full use of human intelligence and the operational capabilities of robots. The method greatly extends the perception and behavior ability of human beings in remote, unstructured and dangerous environments, and is an indispensable key technology in deep space, deep sea and deep ground exploration. Compared with an intelligent robot, the teleoperation technology fully considers the defects of the current intelligent technology, such as decision problems involved in emergency situations and safety and constraint problems in operation. The technology combines human perception and decision-making capability with the operation capability of the robot, integrally enhances the processing capability of a teleoperation system in the case of emergency under an unstructured environment, and is the most practical human-computer hybrid intelligent strategy at present.
The composition of the remote operation system is complex, and the generation mechanism of the operation instruction, the remote transmission of the instruction and the like cannot avoid causing uncertain time delay to the system. The uncertain time delay seriously affects the stability of the system and degrades the performance of the system; model uncertainty also poses a threat to the stability of the system. At the same time, due to the constraints of operation time and working space, the robotic end effector needs to complete an operation task within a desired time while satisfying physical constraints. If in deep space exploration, the robot is limited by the on-orbit operation time, and when the robot executes an inspection task (such as through a narrow space), the on-orbit operation task needs to be completed within a limited time under the condition of ensuring the safety.
Disclosure of Invention
The technical problem solved by the invention is as follows: based on the difficult problems, the patent provides a time-varying output-limited robot teleoperation limited time control method, and a feasible scheme is provided for a teleoperation system to carry out actual work.
The technical scheme of the invention is as follows: a time-varying output constraint robot teleoperation finite time control method based on width neural learning is characterized by comprising the following steps:
step 1: performing dynamic modeling on the mechanical arm;
step 2: estimating the force of an operator and the environmental force by a width neural learning algorithm and an inverse dynamics observer;
and step 3: and designing a time-varying output constraint finite time control law, eliminating uncertain influence of a model, and realizing high-precision tracking and rapid convergence of the teleoperation robot.
The further technical scheme of the invention is as follows: the system is a pair of n-degree-of-freedom mechanical arms, and the model expression is as follows:
Figure BDA0003291951410000021
wherein j belongs to { m, s }, and is respectively a master-slave robot identifier.
Figure BDA0003291951410000022
qj∈Rn×1Acceleration, velocity and position of the joint space, respectively;
Figure BDA0003291951410000023
xj∈Rn×1acceleration, velocity and position of the operating space, respectively; mxjIs a matrix of inertia quantities, CxjIs a matrix of centrifugal and Coriolis forces, gxjIs a gravity term matrix, uxjRepresenting a control input, Fmν＝FhIndicating operator applied force, Fsν＝FeRepresenting the contact force between the slave robot and the environment.
The further technical scheme of the invention is as follows: the step 2 comprises the following substeps:
step 2.1: the model was linearized as:
Figure BDA0003291951410000024
in the formula
Figure BDA0003291951410000025
Is a linear regression matrix, eta is a parameter vector related to the mechanical arm, theta belongs to Rn×1Is the product of the two;
step 2.2: evaluating external force by using a deep neural learning algorithm:
Figure BDA0003291951410000026
wherein q isj,
Figure BDA0003291951410000027
Is an input to the neural network;
step 2.3: based on the inverse dynamics observer, the dynamics model (15) of the mechanical arm is further linearized
Figure BDA0003291951410000028
In the formula
Figure BDA0003291951410000029
Is a linear regression matrix, eta is a parameter vector related to the mechanical arm, theta belongs to Rn×1Is the product of the two;
step 2.4: the external force is estimated by designing a deep neural learning algorithm:
Figure BDA0003291951410000031
the input of the neural network is qj,
Figure BDA0003291951410000032
The estimation of the interaction force can be realized through a width neural learning algorithm.
The further technical scheme of the invention is as follows: the step 3 comprises the following substeps:
step 3.1: by designing the controller uxjRealize the reference track
Figure BDA0003291951410000033
Tracking for a limited time while guaranteeing an output xη1Within a restricted area, i.e.
Figure BDA0003291951410000034
The control law of the master-slave robot is
Figure BDA0003291951410000035
In formula (22):
Figure BDA0003291951410000036
is composed of
Figure BDA0003291951410000037
Satisfies the following conditions
Figure BDA0003291951410000038
The update law in equation (22) is selected as follows:
Figure BDA0003291951410000039
error variable e in equation (22)j1,ej2J is as { m, s }
Figure BDA00032919514100000310
In the formula:
Figure BDA00032919514100000311
is a reference track of the master and slave terminals, alphaj1Is a virtual control quantity to be designed.
Equation (10) then generates a reference trajectory for the master based on the operator behavior,
Figure BDA00032919514100000312
formula (27) wherein
Figure BDA00032919514100000313
Representing estimates of operator force and environmental force, respectively, wherein:
Figure BDA00032919514100000314
acceleration, speed and position of the main end reference track respectively;
Figure BDA00032919514100000315
scaling factors for the operator force estimate and the environmental force estimate, respectively; mr,Cr,grTarget impedance parameters for operator behavior, respectively;
step 3.2: the reference trajectories of the slave end are:
Figure BDA00032919514100000316
Tf(t) network transmission delay from the master to the slave; designing a virtual control quantity alphaj1：
Figure BDA00032919514100000317
In the formula, 0 < xij＜1，
Figure BDA0003291951410000046
And gammaj1Respectively as follows:
Figure BDA0003291951410000041
Figure BDA0003291951410000042
in formula (22):
Figure BDA0003291951410000043
respectively operator force and contact force between robot and environment, kj1,kj2,χj,bjIs a normal number, and is,
Figure BDA0003291951410000044
is opposite to the angular array. The weight vector of the neural network is W obtained by the step 2 width neural learning algorithmj，
Figure BDA0003291951410000045
Effects of the invention
The invention has the technical effects that: the invention focuses on solving the problems of system instability caused by time delay and uncertainty in a teleoperation system, difficulty in external force perception of interaction between a robot and an unknown environment, limited operation space and operation time of the robot and the like, and provides a time-varying output constraint robot teleoperation limited time control method based on width neural learning. The wide neural learning algorithm effectively combines incremental learning and an RBF neural network, realizes the estimation of operator force and environmental force, and simultaneously resolves the negative influence caused by the uncertainty of a system model; the time-varying output constraint algorithm ensures that the tail end position of the robot does not exceed a time-varying boundary, and the operation safety is improved; the limited time control method ensures the rapid tracking capability of the robot track. The method does not need a force sensor, the model of the system does not need to be known accurately, and the high-precision tracking and the rapidity control of the teleoperation robot are realized on the premise of ensuring the safe operation of the system.
Compared with the prior art, the invention has the following advantages:
(1) the invention designs a width neural learning algorithm to realize the estimation of operator force and environmental force; while eliminating the effect of model uncertainty in the controller design. Compared with the traditional neural network, the network can ensure higher estimation accuracy, simultaneously saves a large amount of calculation pressure and does not need sufficient learning conditions;
(2) a limited time controller with strong robustness is designed, and limited time control of the robot under the condition that the operation time is limited is realized. The invention has faster convergence speed and achieves the aim of efficient interaction between the robot and the environment.
(3) The invention considers the safety problem of the operation process, constrains the output of the robot, and the constraint boundary is a time-varying boundary, while the constant boundary is a special form of the time-varying boundary, so the method has stronger practicability and practical significance.
Drawings
FIG. 1 is a diagram of a teleoperational system control framework;
FIG. 2 a breadth neural learning algorithm framework;
FIG. 3 is a broad neural learning algorithm based on inverse dynamics;
FIG. 4 is a simulation effect diagram (x, y, z axes are examples); (a) illustrating the tracking effect on the x-axis and y-axis
(b) Illustrating the tracking effect on the z-axis and the operation space
Detailed Description
Referring to fig. 1-4, the method comprises the following steps:
the method comprises the following steps: dynamic modeling of a system
Step two: designing a width neural learning algorithm, and then estimating the force of an operator and the environmental force based on an inverse dynamics observer;
step three: designing a time-varying output constraint finite time control law, eliminating uncertain influence of a model, realizing high-precision tracking and rapid convergence of the teleoperation robot (the master end is similar to the slave end control law, so that a variable j is in accordance with { m, s } in a unified way)
The steps are integrated, so that stable, safe and efficient interaction between the teleoperation system and the environment can be realized.
Teleoperation systems are complex and an overview of the overall framework is necessary to more clearly expand the subsequent discussion.
The method comprises the following steps: the system consists of a pair of n-degree-of-freedom mechanical arms, the dynamics form in an operation space is as follows, and for the sake of simplicity, the dynamics of a master end and a slave end are uniformly written as follows:
Figure BDA0003291951410000051
in the formula: j belongs to { m, s }, and is respectively a master-slave robot identifier.
Figure BDA0003291951410000061
qj∈Rn×1Acceleration, velocity and position of the joint space, respectively.
Figure BDA0003291951410000062
xj∈Rn×1Acceleration, velocity and position of the operating space, respectively. MxjIs a matrix of inertia quantities, CxjIs a matrix of centrifugal and Coriolis forces, gxjIs a gravity term matrix, uxjRepresenting a control input, Fmν＝FhIndicating operator applied force, Fsν＝FeRepresenting the contact force between the slave robot and the environment.
Step two: based on an inverse dynamics observer, a wide neural learning algorithm is designed to estimate the interaction force (the contact force between a main-end operator and a main-end robot, and the contact force between a slave-end robot and the environment). The dynamic model (15) of the mechanical arm can be linearized
Figure BDA0003291951410000063
In the formula
Figure BDA0003291951410000064
Is a linear regression matrix, eta is a parameter vector related to the mechanical arm, theta belongs to Rn×1Is the product of the two. Because the mechanical arm cannot be accurately modeled and the model has certain deviation, the traditional inverse dynamics observer cannot realize accurate estimation of the external force. Therefore, designing a deep neural learning algorithm enables estimation of external forces:
Figure BDA0003291951410000065
the input of the neural network is qj,
Figure BDA0003291951410000066
The estimation of the interaction force can be realized through a width neural learning algorithm. Due to the change of the external environment and the uncertainty of the system, the traditional algorithm needs to adjust and retrain the neural network parameters, but the wide neural learning algorithm designed by the invention combines RBF and incremental learning, and obtains a better training network by setting threshold deviation and increasing nodes in a self-adaptive manner, as shown in figure 2. Based on an inverse dynamics observer, the estimation of the interaction force can be realized by utilizing the wide neural network algorithm; while the broad neural learning algorithm can compensate for uncertainty in the system model. The pseudo code is shown in table one based on the width neural learning algorithm.
TABLE-learning algorithm based on breadth nerve
Figure BDA0003291951410000067
Figure BDA0003291951410000071
In the table: x is heel qj,
Figure BDA0003291951410000072
Related input vector, W, beta are weight vector and radial basis of RBF neural network respectivelyVector, AmAnd Am+1Are all defined mode matrices.
Figure BDA0003291951410000073
In the formula: d ═ am)+Hm+1，
Figure BDA0003291951410000074
Wherein C is Hm+1-AmD, so the weight can be:
Wei＝(λI+(Am)TAm)-1(Am)TY (20)
Figure BDA0003291951410000081
step three: in order to ensure the safety of the operation, the position output of the robot needs to be limited, namely the end position of the robot is within the time-varying boundary. The method adopts direct barrier Lyapunov function (IBLF), a breadth neural learning algorithm and a finite time theory, and firstly provides a time-varying output constraint finite time control method based on breadth neural learning.
A control target: by designing the controller uxjRealize the reference track
Figure BDA0003291951410000082
Tracking for a limited time while guaranteeing an output xη1Within a restricted area, i.e.
Figure BDA0003291951410000083
The control law of master-slave robot is designed as
Figure BDA0003291951410000084
In formula (22):
Figure BDA0003291951410000085
is composed of
Figure BDA0003291951410000086
Satisfies the following conditions
Figure BDA0003291951410000087
The update law in equation (22) is selected as follows:
Figure BDA0003291951410000088
error variable e in equation (22)j1,ej2J is as { m, s }
Figure BDA0003291951410000089
In the formula:
Figure BDA00032919514100000810
is a reference track of the master and slave terminals, alphaj1Is a virtual control quantity to be designed.
Equation (26) then generates a reference trajectory for the master based on the operator behavior,
Figure BDA00032919514100000811
formula (27) wherein
Figure BDA00032919514100000812
Representing the estimated values of the operator force and the environmental force, respectively (without force measuring means, estimated in step one using a neuro-learning algorithm, see step two), in which:
Figure BDA00032919514100000813
acceleration, speed and position of the main end reference track respectively;
Figure BDA0003291951410000091
scaling factors for the operator force estimate and the environmental force estimate, respectively; mr,Cr,grRespectively, target impedance parameters for operator behavior.
The reference trajectories of the slave end are:
Figure BDA0003291951410000092
Tfand (t) is the network transmission delay from the master end to the slave end. Equation (22), designed virtual control quantity αj1
Figure BDA0003291951410000093
In the formula, 0 < xij＜1，
Figure BDA0003291951410000094
And gammaj1Respectively as follows:
Figure BDA0003291951410000095
Figure BDA0003291951410000096
in formula (22):
Figure BDA0003291951410000097
respectively operator force and contact force between robot and environment, kj1,kj2,χj,bjIs a normal number, and is,
Figure BDA0003291951410000098
is opposite to the angular array. Here the weight vector of the neural network is step 2 width neurologyW obtained by learning algorithmj，
Figure BDA0003291951410000099
Aiming at the teleoperation system (15), virtual control quantity (28), control quantity (22) and updating law (24) are selected, so that the closed-loop stability of the teleoperation system is ensured, and meanwhile, the safe and efficient interaction between the robot and the environment is realized.
Overall process framework of the system: a teleoperational system control framework based on time-varying output constraints is shown in fig. 1. Position signal x of the masterm(t) transmitting to the slave end through the communication link to obtain the reference signal of the slave end
Figure BDA00032919514100000910
Then designing a time-varying output constraint finite time controller u based on the width neural learningxs(see step 3), the reference signal can be realized by the slave robot
Figure BDA00032919514100000911
High precision and fast tracking. Meanwhile, the environment force of the slave end is estimated by using a width neural learning algorithm (see step 2), and virtual environment parameters are transmitted to the master end. And at the master end, reconstructing the environment force of the slave end at the master end by utilizing the motion information of the master end and the virtual environment parameters of the slave end. In order to provide better force perception for the operator, the reference trajectory of the main terminal is generated based on the behavior of the operator. Then designing a time-varying output constraint finite time controller u based on the width neural learning at the main endxm(see step 3), realize the master robot to the master reference signal
Figure BDA0003291951410000101
High precision and fast tracking. (due to master control law uxmAnd slave control law uxsThe same form, so unify the expression)
In conclusion, the invention discloses a finite time control method of a time-varying output constraint robot for width neural learning. Lyapunov function based on integral obstacle and breadth neural learningAn algorithm and a finite time theory creatively provide a time-varying output constraint finite time controller based on the width neural learning. The direct integral barrier Lyapunov function ensures that the output of the system is in a time-varying boundary; the width neural learning algorithm integrates the advantages of the traditional neural network and the width learning, based on the inverse dynamics observer, solves the external force perception problem by using the width learning algorithm, and simultaneously resolves the control rate uxjModel uncertainty in the design process; the finite time theory ensures high precision and rapidity tracking of the reference signal by the robot. In conclusion, the algorithm ensures stable, safe and efficient interaction between the robot and the environment, and improves the reliability and the efficiency of the teleoperation system.

Claims (4)

1.一种基于宽度神经学习的时变输出约束机器人遥操作有限时间控制方法，其特征在于，包括以下步骤：1. a time-varying output constraint robot teleoperation limited-time control method based on breadth neural learning, is characterized in that, comprises the following steps: 步骤1：对机械臂进行动力学建模；Step 1: Dynamic modeling of the robotic arm; 步骤2：通过宽度神经学习算法和逆动力学观测器实现对操作者力和环境力的估计；Step 2: Realize the estimation of operator force and environmental force through the breadth neural learning algorithm and inverse dynamics observer; 步骤3：设计时变输出约束有限时间控制律，消解模型不确定影响，实现遥操作机器人的高精度追踪和快速性收敛。Step 3: Design a time-varying output constraint finite-time control law to eliminate the uncertainty of the model and achieve high-precision tracking and rapid convergence of the teleoperated robot. 2.如权利要求1所述的一种基于宽度神经学习的时变输出约束机器人遥操作有限时间控制方法，其特征在于，所述系统为一对n自由度机械臂，模型表达式为：2. a kind of time-varying output constraint robot teleoperation limited-time control method based on width neural learning as claimed in claim 1, is characterized in that, described system is a pair of n-degree-of-freedom mechanical arms, and the model expression is:
Figure FDA0003291951400000011
Figure FDA0003291951400000011
 其中，j∈{m,s},分别为主从机器人标识。
Figure FDA0003291951400000012
分别为关节空间的加速度，速度和位置；
Figure FDA0003291951400000013
分别为操作空间的加速度，速度和位置；Mxj为惯性量矩阵，Cxj为离心力和哥氏力矩阵，gxj为重力项矩阵，uxj表示控制输入，Fmν＝Fh表示操作者施加力，Fsν＝Fe表示从端机器人和环境间的接触力。Among them, j∈{m,s}, respectively, the master and slave robot identification.
Figure FDA0003291951400000012
are the acceleration, velocity and position of the joint space, respectively;
Figure FDA0003291951400000013
are the acceleration, velocity and position of the operating space, respectively; M xj is the inertia matrix, C xj is the centrifugal force and Coriolis force matrix, g xj is the gravity term matrix, u xj represents the control input, F mν = F h represents the operator exerted Force, F sν = Fe represents the contact force between the slave robot and the environment. 3.如权利要求1或2所述的一种基于宽度神经学习的时变输出约束机器人遥操作有限时间控制方法，其特征在于，所述步骤2包括以下子步骤：3. a kind of time-varying output constraint robot teleoperation limited time control method based on breadth neural learning as claimed in claim 1 or 2, is characterized in that, described step 2 comprises the following sub-steps: 步骤2.1：将模型线性化为：Step 2.1: Linearize the model as:
Figure FDA0003291951400000014
Figure FDA0003291951400000014
 式中
Figure FDA0003291951400000015
为线性回归矩阵，η为有关机械臂的参数向量，Θ∈Rn×1为二者乘积；in the formula
Figure FDA0003291951400000015
is the linear regression matrix, η is the parameter vector about the manipulator, Θ∈R n×1 is the product of the two; 步骤2.2：利用深度神经学习算法对外力评估：Step 2.2: Use deep neural learning algorithms to evaluate external forces:
Figure FDA0003291951400000016
Figure FDA0003291951400000016
 其中
Figure FDA0003291951400000017
为神经网络的输入；in
Figure FDA0003291951400000017
is the input of the neural network; 步骤2.3：基于逆动力学观测器，对机械臂的动力学模型(1)进一步线性化为Step 2.3: Based on the inverse dynamics observer, the dynamic model (1) of the manipulator is further linearized as
Figure FDA0003291951400000021
Figure FDA0003291951400000021
 式中
Figure FDA0003291951400000022
为线性回归矩阵，η为有关机械臂的参数向量，Θ∈Rn×1为二者乘积；in the formula
Figure FDA0003291951400000022
is the linear regression matrix, η is the parameter vector about the manipulator, Θ∈R n×1 is the product of the two; 步骤2.4：设计深度神经学习算法实现对外力进行估计：Step 2.4: Design a deep neural learning algorithm to estimate the external force:
Figure FDA0003291951400000023
Figure FDA0003291951400000023
 该神经网络的输入为
Figure FDA0003291951400000024
通过宽度神经学习算法可实现对交互力的估计。The input to this neural network is
Figure FDA0003291951400000024
The estimation of the interaction force can be achieved by a breadth neural learning algorithm. 4.如权利要求1所述的一种基于宽度神经学习的时变输出约束机器人遥操作有限时间控制方法，其特征在于，所述步骤3中，包括以下子步骤：4. a kind of time-varying output constraint robot teleoperation limited time control method based on breadth neural learning as claimed in claim 1, is characterized in that, in described step 3, comprises the following sub-steps: 步骤3.1：通过设计控制器uxj,实现对参考轨迹
Figure FDA0003291951400000025
有限时间追踪，同时要保证输出xη1在约束区域内，即
Figure FDA0003291951400000026
Step 3.1: By designing the controller u xj , realize the reference trajectory
Figure FDA0003291951400000025
Finite time tracking, while ensuring that the output x η1 is within the constraint region, that is
Figure FDA0003291951400000026
 主从机器人的控制律为The control law of the master-slave robot is
Figure FDA0003291951400000027
Figure FDA0003291951400000027
 式(8)中：
Figure FDA0003291951400000028
为
Figure FDA0003291951400000029
的广义逆，满足如下条件In formula (8):
Figure FDA0003291951400000028
for
Figure FDA0003291951400000029
The generalized inverse of , which satisfies the following conditions
Figure FDA00032919514000000210
Figure FDA00032919514000000210
 式(8)中更新律选取如下：The update law in formula (8) is selected as follows:
Figure FDA00032919514000000211
Figure FDA00032919514000000211
 式(8)中误差变量ej1,ej2,j∈{m,s}为In formula (8), the error variables e j1 , e j2 , j∈{m,s} are
Figure FDA00032919514000000212
Figure FDA00032919514000000212
 式中：
Figure FDA00032919514000000213
为主从端的参考轨迹，αj1为待设计的虚拟控制量。where:
Figure FDA00032919514000000213
It is the reference trajectory of the master and slave, and α j1 is the virtual control quantity to be designed. 式(12)则基于操作者行为可生成主端的参考轨迹，Equation (12) can generate the reference trajectory of the master based on the operator's behavior,
Figure FDA00032919514000000214
Figure FDA00032919514000000214
 式(13)其中
Figure FDA00032919514000000215
分别代表操作者力和环境力的估计值，式中：
Figure FDA00032919514000000216
分别为主端参考轨迹的加速度、速度和位置；
Figure FDA0003291951400000031
分别为操作者力估计和环境力估计的比例系数；Mr,Cr,gr分别为操作者行为的目标阻抗参数；Formula (13) where
Figure FDA00032919514000000215
represent the estimated values of operator force and environmental force, respectively, where:
Figure FDA00032919514000000216
are the acceleration, velocity and position of the reference trajectory of the main terminal, respectively;
Figure FDA0003291951400000031
are the proportional coefficients of operator force estimation and environmental force estimation, respectively ; Mr , Cr , and gr are the target impedance parameters of operator behavior; 步骤3.2：从端的参考轨迹为：
Figure FDA0003291951400000032
Tf(t)为主端到从端的网络传输时延；设计虚拟控制量αj1：Step 3.2: The reference trajectory of the slave is:
Figure FDA0003291951400000032
T f (t) network transmission delay from master to slave; design virtual control quantity α j1 :
Figure FDA0003291951400000033
Figure FDA0003291951400000033
 式中0＜ξj＜1，
Figure FDA0003291951400000039
和γj1分别为：where 0 < ξ j < 1,
Figure FDA0003291951400000039
and γ j1 are:
Figure FDA0003291951400000034
Figure FDA0003291951400000034
Figure FDA0003291951400000035
Figure FDA0003291951400000035
 式(8)中：
Figure FDA0003291951400000036
j∈{m,s}分别为操作者力和机器人与环境间的接触力，kj1,kj2,χj,bj为正常数，
Figure FDA0003291951400000037
为正对角阵。此处神经网络的权重向量为步骤2宽度神经学习算法得到的Wj，
Figure FDA0003291951400000038
In formula (8):
Figure FDA0003291951400000036
j∈{m,s} are the operator force and the contact force between the robot and the environment, respectively, k j1 , k j2 , χ j , b j are positive numbers,
Figure FDA0003291951400000037
is a positive diagonal matrix. Here, the weight vector of the neural network is W j obtained by the width neural learning algorithm in step 2,
Figure FDA0003291951400000038
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