CN107972030A - A kind of initial position modification method in redundant mechanical arm repeating motion - Google Patents

A kind of initial position modification method in redundant mechanical arm repeating motion Download PDF

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CN107972030A
CN107972030A CN201711101684.4A CN201711101684A CN107972030A CN 107972030 A CN107972030 A CN 107972030A CN 201711101684 A CN201711101684 A CN 201711101684A CN 107972030 A CN107972030 A CN 107972030A
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theta
mechanical arm
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CN107972030B (en
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孔颖
黄奕筱
朱佳超
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Zhejiang Lover Health Science and Technology Development Co Ltd
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    • 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
    • B25J9/1643—Programme controls characterised by the control loop redundant 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
    • B25J9/1612—Programme controls characterised by the hand, wrist, grip 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
    • B25J9/1653—Programme controls characterised by the control loop parameters identification, estimation, stiffness, accuracy, error analysis

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  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Health & Medical Sciences (AREA)
  • General Health & Medical Sciences (AREA)
  • Orthopedic Medicine & Surgery (AREA)
  • Feedback Control In General (AREA)
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Abstract

A kind of initial position modification method in redundant mechanical arm repeating motion, comprises the following steps：1) redundant mechanical arm end effector expectation target track r is determined*(t) the joint angles θ to back with expectation*(0), design final state and attract optimizing index, initial joint angle when forming mechanical arm repeating motion programme, wherein redundant mechanical arm actual motion can be arbitrarily designated, and not require end effector to be on desired trajectory；2) the initial joint angles θ (0) during redundant mechanical arm actual motion is given, is movement starting point with θ (0), new repeating motion programme is converted into corresponding quadratic programming problem；3) the final state neural network model of finite value activation primitive, finite value final state Neural Networks Solution time-varying matrix equation are built；4) obtained result will be solved for controlling each joint motor, driving mechanical arm performs task.The present invention provides a kind of precision height, the redundant mechanical arm repeating motion initial position fix method of finite time convergence control.

Description

Initial position correction method in redundant mechanical arm repeated movement
Technical Field
The invention relates to a repetitive motion planning and control technology of a redundant mechanical arm, in particular to an inverse kinematics solving method of the redundant mechanical arm under the condition of initial offset and limited time convergence performance indexes.
Background
Mechanical arms have been a research hotspot in the field of robots. A robot arm refers to a mechanical device with active end, and the end tasks include transportation, welding, painting and assembling, and the like, and is widely used in the fields of industrial manufacturing, medical treatment, entertainment service, fire fighting, military affairs, space exploration and the like. Depending on how many Degrees of Freedom (DOF) the robot arms can be divided into redundant robot arms and non-redundant robot arms. Redundant robotic arms refer to robotic arms that possess more DOF than is required to complete a given end task. Correspondingly, a non-redundant robotic arm refers to a robotic arm that does not have an unnecessary DOF when performing a given end task. Obviously, the redundant robot arm is more flexible and advantageous because of its redundant DOF compared to the non-redundant robot arm, which is one of the reasons why it is increasingly important in practical engineering applications.
One fundamental problem with redundant robotic arms in real-time motion control is the redundancy resolution problem, also known as inverse kinematics. And solving the value of each joint angle of the corresponding redundant mechanical arm by knowing the position and the posture of the end effector. The classical approach is a redundancy resolution scheme based on pseudo-inverse. Consider a mechanical arm operating in m-dimensional space with n degrees of freedom, the relationship between the trajectory of the tip and the displacement of the joint (i.e. positive kinematic problem)
r(t)＝f(θ(t))
Where r (t) represents the displacement of the end effector of the robotic arm in a cartesian coordinate system in the workspace, and θ (t) represents the joint displacement. The differential motion relation between the terminal Cartesian space and the joint space is
Wherein,is the time derivative of r and is,is a vector of the velocity of the joint,is the jacobian matrix of the mechanical arm.
For redundant mechanical arms, the traditional method is to solve Moore-Penrose generalized inverse (pseudo-inverse), and the least square solution of the joint variable speed can be obtained
Here, J+＝JT(JJT)-1Is the pseudo-inverse of the jacobian matrix J.
The following minimum velocity norm performance indicators with equality constraints were proposed in 1969 as target functions for motion planning, D.E. Whitney (Resolved motion control of simulations and humanprostheses, IEEE trans. Man-Machine Syst.,1969,10(2): 47-53):
in the formula, A is a positive definite weighting matrix. Solving the above-mentioned planning problem, the demand solves the following system of equations
It is solved into
Formula (1) is a special case of formula (3) when a ═ I. It can also be seen that the planning problem is solved by solving equation set (2).
When the end effector does repeated motion in the cartesian operating space, the closed end effector motion trajectory may generate a non-closed joint angle trajectory, resulting in a joint angle deviation phenomenon. This non-repetitive motion problem may cause unpredictable situations in the repetitive operation of the robotic arm. Generally, the most widely used pseudo-inverse control method cannot obtain repeatability in order to complete the original repetitive motion. Usually, the compensation is made by self-movement, which is often inefficient to adjust (see Klein C A and Huang C, Review of Pseudo Inverse Control for use with kinematic recovery managers, IEEE Trans. Syst. Man. Cybern.1983,13(2): 245-.
Quadratic Optimization (QP) -based redundancy resolution schemes are of interest, f.t. cheng in 1994 proposes joint unbiased performance indicators (f. -t.cheng, t. -h.chen, and y. -y.sun, resolution manager redundancy requirements, ieee trans. robotics automation, 1994,10(1): 65-71):
in order to efficiently execute the repetitive motion task, y.zhang introduces a repetitive motion index as an optimization criterion. A Repetitive Motion Planning (RMP) scheme is formed, and redundancy resolution (Zhang Y, Wang J, Xia Y. adaptive Neural network for redundancy resolution, IEEE Trans Neural network, 2003,14(3): 658-. The repetitive motion index is an asymptotic convergence performance index AOC (asymptotic-convergent Optimality Criterion) as follows:
the recurrent neural network solver is an effective method for solving the redundancy analysis problem based on quadratic optimization description. The general neural network solver has asymptotic convergence performance, can obtain an effective solution after the calculation time is long enough, and can be applied to real-time online solution of various time-varying problems.
Recently, recurrent neural networks with limited time convergence performance have been used to solve time-varying problems. Compared with a recurrent neural network with asymptotic convergence dynamic characteristics, the final state convergence dynamic characteristics have finite time convergence, and not only can the convergence speed be improved, but also higher convergence precision is achieved. The finite time neural networks of the documents do not employ an activation function, i.e. a linear activation function, or have an infinite activation function, i.e. when the input tends to infinity, the activation function also tends to infinity. In practical implementation, due to limited energy, a neural network solver with a finite activation function is required, and the implementation of the infinite activation function neural network has essential difficulties.
Disclosure of Invention
In order to overcome the defects that the existing redundant mechanical arm repetitive motion control method is low in precision and cannot achieve limited time convergence, the invention provides the initial position correction method in the redundant mechanical arm repetitive motion, which is high in precision and limited time convergence.
In order to solve the technical problems, the invention adopts the following technical scheme:
a method for correcting an initial position in repeated movement of a redundant mechanical arm comprises the following steps:
1) determining an equation r of an expected trajectory of an end effector of a redundant manipulator*(t) setting a joint angle θ at which the joint is expected to be closed*(0)；
2) Designing a final state attraction optimization index to form a secondary planning scheme for repeated motion of the mechanical arm, wherein an initial joint angle of the redundant mechanical arm in actual motion can be randomly specified, and an end effector is not required to be on an expected track; giving an initial joint angle theta (0) when the redundant mechanical arm actually moves, and taking the theta (0) as a motion starting point, wherein a formed repeated motion planning scheme is described as a quadratic plan with a final state attraction optimization index:
wherein, theta,Respectively representing joint angles and angular velocities, theta, of redundant robot arms*(0) Is the desired initial value of each joint angle, β is a coefficient matrix used to form the dynamic performance of the joint displacement, γEIs a correction vector whose value is βERelated, theta (t) -theta*(0) Representing deviation of each joint angle from the initial expected joint angular displacement, r*Representing a desired motion profile of the end effector of the robotic arm,a velocity vector representing a desired velocity of the end effector;
because the initial position of the mechanical arm is not on the expected track, the error (r) between the expected path and the actual motion track position of the end effector is reduced*-f (θ)), changing the direction of motion of the end effector, βrThe parameter gain of the position is represented by more than 0 and is used for adjusting the speed of the end effector moving to the expected path, J (theta) is a Jacobian matrix of the redundant mechanical arm, f (theta) is the actual movement track of the redundant mechanical arm, βEIs a coefficient matrix, which is used to form the dynamic behavior of joint displacement, gammaEIs a correction vector whose value is βEIn connection with this, the present invention is,t0is the time of the initial moment of time,
and satisfy
3) Constructing a final state neural network model with a limit activation function, the dynamic characteristics of which are described by the following equations
Wherein E is an error variable, βEMore than 0 is a design parameter, E is the convergence error of the dynamic equation, gammaEIs a correction vector whose value is βE(ii) related;
establishing a Lagrange function for solving the quadratic programming problem in the step 2)
Wherein λ (t) is a Lagrange multiplier vector, λTIs the transpose of the λ (t) vector; for each by lagrange functionThe individual variables are derived and made zero to obtain the following time-varying matrix equation
WY＝v (3)
Wherein,
i is an identity matrix
The finite final state neural network described by the formula (2) is used for solving the time-varying matrix equation (3) to obtain a system solution equation as follows
4) And 3) using the result obtained in the step 3) to control each joint motor and drive the mechanical arm to execute the task.
The invention provides a quadratic optimization method for forming redundant mechanical arm track planning by using a repetitive motion performance optimization index, namely a final attraction optimization index. And taking a final state neural network with a finite value activation function as a solver, and realizing a repetitive motion planning task of finite time convergence of the redundant mechanical arm under the condition of initial position deviation.
The invention provides a quadratic optimization method for forming redundant mechanical arm track planning by using a repetitive motion performance optimization index, namely a final attraction optimization index. And taking a final state neural network with a finite value activation function as a solver, and realizing a repetitive motion planning task of finite time convergence of the redundant mechanical arm under the condition of initial position deviation.
In order to achieve the purpose, the optimization index of the redundant mechanical arm track planning is designed to be a final state attraction optimization index TOC (terminal optimization criterion), namely
Wherein c ═ θ (t) - θ*(0)，θ、Respectively representing joint angles and angular velocities, theta, of redundant robot arms*(0) Is the desired initial value for each joint angle, β is a coefficient matrix, i.e., the root of the feature is negative, γ is a correction vector, whose value is related to β,t0is the time of the initial moment.
And satisfy
The design idea of the above-described repeatable motion optimization scheme is to expect the following dynamic equations
The joint angular displacement deviation ∈ (t) — θ (t) - θ (0). The finite time of the system expressed by this dynamic equation converges to zero.
When the final attraction index based on the secondary optimization reaches the minimum value, the motion end effector of the redundant mechanical arm realizes repeated motion on the speed layer.
In order to solve the problem of repetitive motion planning under the optimization index TOC, a final state neural network model with a finite value activation function is adopted, and the dynamic equation of the network is as follows:
wherein, βE0 is coefficient matrix, E is convergence error of dynamic equation, gammaEIs a correction vector whose value is βEIn connection with this, the present invention is,
the invention has the beneficial effects that: the precision is high, the time is limited to converge, when the initial joint angles of the mechanical arm deviate from the expected positions, the redundant mechanical arm can effectively complete the work task, and whether the initial joint angles are on the expected task track or not is not considered.
Drawings
FIG. 1 is a flow chart of a re-planning scheme provided by the present invention.
FIG. 2 illustrates a redundant robotic arm KATANA6M180 employing the iterative planning scheme of the present invention.
Figure 3 is a motion profile of the redundant robotic arm KATANA6M180 end effector.
FIG. 4 redundant robot arm KATANA6M180 individual joint error trajectories.
Figure 5 redundant robot arm KATANA6M180 end effector various position error trajectories.
FIG. 6 shows the variation locus of the angular velocity of each joint of the redundant robot arm KATANA6M 180.
FIG. 7 shows error trajectories when solving with finite-value final state neural networks and recurrent neural networks.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 7, an initial position correction method in a repetitive motion of a redundant manipulator includes the following four steps: 1. determining an expected target track of the end effector of the redundant manipulator and an expected convergence of each joint angle 2, establishing a redundant manipulator repetitive motion quadratic programming scheme 3 with a final state attraction optimization index, and solving a quadratic programming problem by using a finite value final state neural network to obtain each joint angle track. 4. And using the result obtained by the solution for driving the motor to operate so as to enable the redundant mechanical arm to execute the task.
First step, determining a desired trajectory
Setting the desired joint angle of the redundant mechanical arm KATANA6M180
θ*(0)＝[0,0,0,0,0]T
The side length of the parallelogram is determined to be 0.04 meter, the tracking time required by each side is 1.5s, the included angle between the surface of the parallelogram and the X axis is pi/6 rad, and the time T of the end effector for completing the quadrilateral track is 6 s. Considering that the initial position of the redundant robot arm KATANA6M180 may not be on the desired motion trajectory, the initial values of the 5 joint angles of the robot arm are set to [0,0,0, -pi/4, 0 ], (0) }]T。
Secondly, establishing a quadratic programming scheme of the repeated motion of the redundant mechanical arm
In order to realize the repetitive motion planning of the finite time convergence of the redundant mechanical arm, the repetitive motion trajectory planning of the redundant mechanical arm is described as the following quadratic planning problem, and the optimization index of the final attraction is
Wherein c ═ θ (t) - θ*(0) Representing deviation of each joint angle from the initial expected joint angular displacement, r*Indicating the end of a robot armThe desired motion profile of the actuator is then,representing a desired velocity vector for the end effector. Because the initial position of the mechanical arm is not on the expected track, the error (r) between the expected path and the actual motion track position of the end effector is reduced*-f (θ)), changing the direction of motion of the end effector, βrParameter gain for > 0 for position to adjust the rate at which the end effector moves to the desired path βEIs a coefficient matrix, which is used to form the dynamic behavior of joint displacement, gammaEIs a correction vector whose value is βEIt is related.
J (theta) is a jacobian matrix of the redundant manipulator, and f (theta) is an actual motion track of the redundant manipulator.
Constructing a final state neural network model of the finite value activation function, wherein the dynamic characteristics of the model are described by the following equations
Wherein E is an error variable, βEMore than 0 is a design parameter, E is the convergence error of the dynamic equation, gammaEIs a correction vector whose value is βE(ii) related;
establishing a Lagrange function for solving the quadratic programming problem in the step 2)
Wherein λ (t) is a Lagrange multiplier vector, λTIs the transpose of the λ (t) vector; each variable is differentiated through a Lagrange function and is made to be zero, and the following time-varying matrix equation is obtained
WY＝v (3)
I is an identity matrix
Recording the error E of the time-varying matrix equation to be WY-v, and constructing a neural network model according to the finite value final state neural network dynamic equation (3)
4) And 3) using the result obtained in the step 3) to control each joint motor and drive the mechanical arm to execute the task.
A redundant robotic arm KATANA6M180 for implementing the iterative planning scheme of the present invention is shown in fig. 2. The mechanical arm is composed of 1 base and 4 connecting rods and is composed of a joint 1, a joint 2, a joint 3, a joint 4 and a joint 5. The redundant mechanical arm has the length L of the connecting rod as shown in the figure10.2035 m, L20.19 m, L30.139 m, L40.315 m.
The motion trajectory of the end effector of the robot arm in space is shown in fig. 3. The target circle trajectory (black dashed line) and the robot arm end effector motion trajectory (black solid line) are given in the figure. It can be seen that the initial position of the end effector is not on the desired trajectory. With the increase of time, the actual track is matched with the expected track, and the final value position error precision of the end effector reaches 10 on the XYZ axes in three directions-4As shown in fig. 5.
When each joint angle of the KATANA6M180 redundant mechanical arm passes through 10s, the final value error of each joint angle reaches 10-5All joint trajectories are substantially closed, the trajectories being shown in fig. 4.
In order to verify the effectiveness of the final attraction optimization index TOC in the repetitive motion planning, a joint angular velocity transient trajectory obtained in the process of completing a quadrilateral trajectory by the mechanical arm KATANA6M180 end effector is shown in fig. 6. When T is 10s, the maximum deviation between each joint angle before and after the robot arm movement and its desired joint angle position is 2.005 × 10-5The maximum deviation of the final value error of each joint angle obtained by solving through a recurrent neural network is 6.113 x 10-4As shown in table 1.
TABLE 1
To compare convergence properties of asymptotically converging networks with those of the final neural network, a calculation error J is definedE(t)＝||W(t)y(t)-v(t)||2. Fig. 7 shows the error convergence trajectories for solving the quadratic programming problem with the finite-valued final state neural network and the recurrent neural network, respectively. It can be seen from the figure that when the finite-value final state neural network is used for solving, when the time t is close to 0.006 second, the error convergence precision reaches 0.3 x 10-3When the recurrent neural network is used for solving, the convergence speed of the error is slow.

Claims (1)

1. A method for correcting an initial position in the repetitive motion of a redundant mechanical arm is characterized in that: the method comprises the following steps:
1) determining an equation r of an expected trajectory of an end effector of a redundant manipulator*(t) setting a joint angle θ at which the joint is expected to be closed*(0)；
2) Designing a final state attraction optimization index to form a secondary planning scheme for repeated motion of the mechanical arm, wherein an initial joint angle of the redundant mechanical arm in actual motion can be randomly specified, and an end effector is not required to be on an expected track; giving an initial joint angle theta (0) when the redundant mechanical arm actually moves, and taking the theta (0) as a motion starting point, wherein a formed repeated motion planning scheme is described as a quadratic plan with a final state attraction optimization index:
<mrow> <mtable> <mtr> <mtd> <munder> <mrow> <mi>min</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> </mrow> <mrow> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </munder> </mtd> <mtd> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>&amp;beta;</mi> <mi>E</mi> </msub> <mi>c</mi> <mo>+</mo> <mi>&amp;gamma;</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>&amp;beta;</mi> <mi>E</mi> </msub> <mi>c</mi> <mo>+</mo> <msub> <mi>&amp;gamma;</mi> <mi>E</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mi>u</mi> <mi>b</mi> <mi>j</mi> <mi>e</mi> <mi>c</mi> <mi>t</mi> <mi> </mi> <mi>t</mi> <mi>o</mi> </mrow> </mtd> <mtd> <mrow> <mi>J</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msup> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>&amp;beta;</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>*</mo> </msup> <mo>-</mo> <mi>f</mi> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
wherein, theta,Respectively representing joint angles and angular velocities, theta, of redundant robot arms*(0) Is the desired initial value of each joint angle, β is a coefficient matrix used to form the dynamic performance of the joint displacement, γEIs a correction vector whose value is βERelated, theta (t) -theta*(0) Representing deviation of each joint angle from the initial expected joint angular displacement, r*Indicating an end effector of a robotic armThe desired trajectory of the movement is such that,a velocity vector representing a desired velocity of the end effector;
because the initial position of the mechanical arm is not on the expected track, the error (r) between the expected path and the actual motion track position of the end effector is reduced*-f (θ)), changing the direction of motion of the end effector, βrThe parameter gain of the position is represented by more than 0 and is used for adjusting the speed of the end effector moving to the expected path, J (theta) is a Jacobian matrix of the redundant mechanical arm, f (theta) is the actual movement track of the redundant mechanical arm, βEIs a coefficient matrix, which is used to form the dynamic behavior of joint displacement, gammaEIs a correction vector whose value is βEIn connection with this, the present invention is,t0is the time of the initial moment of time,
and satisfy
3) Constructing a final state neural network model with a limit activation function, the dynamic characteristics of which are described by the following equations
<mrow> <mover> <mi>E</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>&amp;beta;</mi> <mi>E</mi> </msub> <mi>E</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;gamma;</mi> <mi>E</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
Wherein E is an error variable, βE＞0Is a design parameter, E is the convergence error of the dynamic equation, γEIs a correction vector whose value is βE(ii) related;
establishing a Lagrange function for solving the quadratic programming problem in the step 2)
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Wherein λ (t) is a Lagrange multiplier vector, λTIs the transpose of the λ (t) vector; each variable is differentiated through a Lagrange function and is made to be zero, and the following time-varying matrix equation is obtained
WY＝v (3)
Wherein,
i is an identity matrix
<mrow> <mi>Y</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mi>&amp;lambda;</mi> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
<mrow> <mi>v</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;beta;</mi> <mi>E</mi> </msub> <mi>c</mi> <mo>-</mo> <msub> <mi>&amp;gamma;</mi> <mi>E</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>&amp;beta;</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>*</mo> </msup> <mo>-</mo> <mi>f</mi> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>
The finite final state neural network described by the formula (2) is used for solving the time-varying matrix equation (3) to obtain a system solution equation as follows
<mrow> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>-</mo> <mover> <mi>W</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>Y</mi> <mo>-</mo> <mi>W</mi> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>&amp;beta;</mi> <mi>E</mi> </msub> <mrow> <mo>(</mo> <mi>W</mi> <mi>Y</mi> <mo>-</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;gamma;</mi> <mi>E</mi> </msub> <mo>+</mo> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
4) And 3) using the result obtained in the step 3) to control each joint motor and drive the mechanical arm to execute the task.
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