CN109015657B - Final state network optimization method for redundant mechanical arm repetitive motion planning - Google Patents

Final state network optimization method for redundant mechanical arm repetitive motion planning Download PDF

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CN109015657B
CN109015657B CN201811042159.4A CN201811042159A CN109015657B CN 109015657 B CN109015657 B CN 109015657B CN 201811042159 A CN201811042159 A CN 201811042159A CN 109015657 B CN109015657 B CN 109015657B
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mechanical arm
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孔颖
张瑞阳
唐青青
陆凯杭
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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
    • 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
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

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Abstract

Redundant mechanical arm repetitive motion planning orientedThe final state network optimization method comprises the following steps: 1) determining a desired target trajectory r of an end effector of a redundant robotic arm*(t) and the desired joint angle θ*(0) (ii) a 2) Designing a final attraction optimization index to form a mechanical arm repetitive motion planning scheme, 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 forming a repeated motion planning scheme with a final state attraction optimization characteristic by taking the theta (0) as a motion starting point; 3) constructing a final state network model of a finite value activation function, and solving a time-varying matrix equation by using the final state network; 4) and the result obtained by the solution is used for controlling each joint motor and driving the mechanical arm to execute the task. The invention provides a redundant mechanical arm repeatable movement method with high precision and limited time convergence.

Description

Final state network optimization method for redundant mechanical arm repetitive motion planning
Technical Field
The invention relates to a repetitive motion planning and control technology of a redundant mechanical arm, in particular to a solution method for inverse kinematics of the redundant mechanical arm under the condition that the initial position of each motion joint angle of the mechanical arm deviates an expected track by final attraction optimization performance index.
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. By adopting a specific redundancy analysis scheme, the redundant mechanical arm can freely move in a larger joint subspace while completing a given end task, and avoids obstacles in the environment and joint physical limitation of the redundant mechanical arm. 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
Figure BDA0001792385860000021
Wherein the content of the first and second substances,
Figure BDA0001792385860000022
is the time derivative of r and is,
Figure BDA0001792385860000023
is a vector of the velocity of the joint,
Figure BDA0001792385860000024
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
Figure BDA0001792385860000025
Here, J+＝JT(JJT)-1Is a Jacobian matrixThe pseudo-inverse of J.
When the end effector does repeated motion in the cartesian operating space, the closed end effector motion trajectory may not generate a closed joint angle trajectory, resulting in a joint angle deviation phenomenon. The deviations from this non-repetitive motion problem may cause the robot arm to perform an unpredictable number of tasks during repetitive operations. In most cases, the self-motion method is used to replace the commonly used pseudo-inverse control method for improvement, because the pseudo-inverse control method cannot obtain the repeatability of completing the original repeated motion. But the tuning efficiency using the self-method is often not high.
Disclosure of Invention
In order to overcome the defects of the existing terminal state network optimization method, such as no finite time convergence, low convergence speed and low convergence precision, the invention discloses a recurrent neural network with finite time convergence performance, which is used for solving a time-varying problem. 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. In order to realize the repeated movement of the redundant mechanical arm when each joint angle of the redundant mechanical arm deviates from an expected position at an initial position, the invention provides a redundant mechanical arm track planning method which has higher precision, limited time convergence and easy realization and is based on a final state attraction optimization index.
In order to solve the technical problems, the invention provides the following technical scheme:
a redundant mechanical arm repetitive motion planning-oriented final state network optimization method comprises the following steps:
1) determining a desired target trajectory r of an end effector of a redundant robotic arm*(t) and the desired joint angle θ*(0)；
2) Designing a final attraction optimization index to form a mechanical arm repetitive motion planning scheme, 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:
Figure BDA0001792385860000031
wherein the content of the first and second substances,
Figure BDA0001792385860000032
θ、
Figure BDA0001792385860000033
respectively representing joint angle and joint angular velocity, theta, of the redundant robot arm*(0) Is a desired initial value of each joint angle, p > 0, betaθMore than 0, and more than 0 and more than delta and less than 1 are design parameters used for forming the dynamic performance of joint displacement; 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,
Figure BDA0001792385860000034
a velocity vector representing a desired velocity of the end effector; by reducing the error (r) between the desired path and the actual path of motion of the end effector, the initial position of the robotic arm may not be on the desired path*-f (θ)), changing the direction of movement of the end effector, βrA parametric gain for > 0 representing position, for adjusting the rate at which the end effector moves to a desired path; j (theta) is a Jacobian matrix of the redundant manipulator, and f (theta) is an actual motion track of the redundant manipulator;
3) constructing a final state network model of a finite value activation function, wherein the dynamic characteristics of the final state network model are described by the following equations
Figure BDA0001792385860000035
The joint angular displacement deviation E (t) -theta (0) is equal to theta (t), the finite time of the system expressed by the dynamic equation converges to zero, 0 & ltdelta & lt 1, rho & gt 0, betaE> 0, the required convergence time T is:
Figure BDA0001792385860000041
when the index function reaches the minimum value, all joint angles of the redundant mechanical arm can be drawn back to the expected target track;
establishing a Lagrangian function for solving the quadratic programming in the step 2):
Figure BDA0001792385860000042
wherein λ (t) is a Lagrange multiplier vector, λTIs the transpose of the λ (t) vector; by deriving each variable with a Lagrange function and making it zero, the following time-varying matrix equation can be obtained
WY＝v (3)
Wherein the content of the first and second substances,
Figure BDA0001792385860000043
i is an identity matrix
Figure BDA0001792385860000044
Figure BDA0001792385860000045
The finite final state network described by the formula (2) is used for solving the time-varying matrix equation (3), and the solving equation of the obtained system is as follows
Figure BDA0001792385860000046
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 has the following beneficial effects: the invention provides a final attraction optimization index, so that the positions of the initial joint angles are not considered by a redundant mechanical arm, and the joint angles can still return to the expected initial positions after passing through a given track, thereby realizing a repeated motion planning task. Compared with the existing motion planning scheme, the scheme has the characteristic of limited time convergence, and the provided final attraction optimization index can be beneficial to improving the calculation precision.
Compared with the traditional repeated motion planning scheme and the asymptotic convergence dynamic network convergence effect, the repeated motion planning scheme with the final state attraction convergence characteristic has the finite time convergence characteristic, and the corresponding final state network solving method not only accelerates the convergence speed of the dynamic network, but also improves the convergence precision of the network. The solution of the network is realized by the aid of a finite-value activation function, a calculation tool is provided for solving the related time-varying problem, and the method is easy to realize in practical application.
Drawings
Fig. 1 is a flow chart of a re-planning scheme provided by the present invention.
FIG. 2 is a view of a redundant robotic arm ER6B-C60 employing an iterative planning scheme of the present invention, wherein (a) is a front view of the fixed robotic arm ER6B-C60, and (b) is a side view of the fixed robotic arm ER6B-C60
FIG. 3 is a diagram of the motion trajectory of the end effector of the redundant robotic arm ER 6B-C60.
FIG. 4 is an error trajectory when solving with a net-state network and a recurrent neural network.
FIG. 5 is a graph of error trajectories on the x, y, z axes when solving for a net of final states.
FIG. 6 is a movement track of each joint angle of the redundant manipulator ER6B-C60 when the final state network is used for solving.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 6, a redundant manipulator repetitive motion planning-oriented final state network optimization method comprises the following 4 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 attraction optimization index, solving a quadratic programming problem by using a finite-value final state network, and obtaining each joint angle track. 4) And driving a motor to operate according to the result obtained by the solution, so that the mechanical arm completes the track task. The specific process is as follows:
1) determining the desired trajectory
The desired joint angle of the redundant robotic arm ER6B-C60 is set. Setting the target path as an eight-leaf rose track with the leaf length of 0.04m, and determining an expected track equation r of an eight-leaf rose shape*。
Figure BDA0001792385860000061
Wherein alpha is 1, beta is 4,
Figure BDA0001792385860000062
the track time T of the tail end of the mechanical arm to finish the eight-leaf rose is 2.25 pi S, and the area of the track time T is 2 x alpha2*π≈0.01m2. The eight-leaf rose surface is parallel to the Z-axis plane, and the joint angle of the redundant mechanical arm ER6B-C60 expected to be closed is set
θ*(0)＝[0,-π/4,0,π/2,-π/4,0]T
Since the initial position of each joint angle of the ER6B-C60 of the redundant mechanical arm may not be on the expected motion track, the six joint angles of the mechanical arm are set to be theta (0) — [0, -pi/4, 0, pi/2, -pi/4 +2,0]TWith the 5 th joint angle offset by 2rad from the desired position.
2) Secondary planning scheme for establishing repeated motion of 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
Figure BDA0001792385860000063
Figure BDA0001792385860000064
Wherein the content of the first and second substances,
Figure BDA0001792385860000065
θ、
Figure BDA0001792385860000066
respectively representing joint angle and joint angular velocity, theta, of the redundant robot arm*(0) Is a desired initial value, β, of each joint angleθ> 0, ρ > 0, 1 > δ > 0 are design parameters used to develop the dynamic behavior of joint displacement. 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,
Figure BDA0001792385860000067
representing a desired velocity vector for the end effector. By reducing the error (r) between the desired path and the actual path of motion of the end effector, the initial position of the robotic arm may not be on the desired path*-f (θ)), changing the direction of movement of the end effector, βrA parametric gain for > 0 representing position, for adjusting the rate at which the end effector moves to a desired path; j (theta) is a jacobian matrix of the redundant manipulator, and f (theta) is an actual motion track of the redundant manipulator.
3) Solving the quadratic programming problem with finite-valued final state network
By deriving each variable of the lagrange function and making it zero, the following time-varying matrix equation can be obtained
WY＝v
Figure BDA0001792385860000071
I is an identity matrix
Figure BDA0001792385860000072
Figure BDA0001792385860000073
And recording the error E of the time-varying matrix equation as WY-v. Constructing a neural network model according to a finite value final state network dynamic equation (2)
Figure BDA0001792385860000074
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 ER6B-C60 for implementing the present iterative planning scheme is shown in FIG. 2. The mechanical arm is composed of 1 base and 7 connecting rods and is composed of a joint a1, a joint a2, a joint a3, a joint a4, a joint a5 and a joint a 6. The redundant mechanical arm has the length L of the connecting rod as shown in the figure1＝0.563m，L2＝0.9m，L3＝0.16m，L4＝1.0135m，L5＝0.195m，L6＝0.22m，L7＝0m。
The motion trajectory of the end effector of the robot arm in space is shown in fig. 3. The target eight-leaf rose track and the motion track of the end effector of the mechanical arm are shown in the figure. It can be seen that the initial position of the end effector is not on the desired trajectory. As time increases, the actual trajectory coincides with the desired trajectory. When solving a repetitive motion planning scheme with a recursive network, the closing speed of the end effector is slower than the solving speed of the net-state network. In order to further compare the angle convergence effect when different networks solve the repetitive motion planning scheme, fig. 4 shows an error convergence effect diagram obtained by applying different networks.
Defining a calculation error JE(t)＝||W(t)y(t)-v(t)||2. FIG. 4 shows the final limit valueAnd solving the error convergence track of the quadratic programming problem by the state network. It can be seen from the figure that when the finite-value final state network is used for solving, when the time t is close to 1 second, the error convergence precision reaches 4 x 10-3。
Errors in the XYZ three directions of the end effector of the redundant robot arm when performing the trajectory task are shown in fig. 5. As can be seen from the figure, when t is 5s, the error accuracy in three directions reaches 10-6。
When the redundant mechanical arm completes the track task, the motion track of each joint angle is shown in fig. 6. The movement process of the two joint angles can be clearly seen in the figure, and finally the initial expected position of the movement is returned.

Claims (1)

1. A redundant manipulator repetitive motion planning-oriented final state network optimization method is characterized by comprising the following steps:
1) determining a desired motion trajectory r of an end effector of a redundant robotic arm*(t) and desired initial values of respective joint angles θ*(0)；
2) Designing a final state attraction optimization index to form a redundant mechanical arm repeated motion planning scheme, wherein an initial joint angle of the redundant mechanical arm in actual motion is randomly specified, and an end effector of the redundant mechanical arm is not required to be on an expected motion 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:
Figure FDA0003189597170000011
wherein the content of the first and second substances,
Figure FDA0003189597170000012
θ、
Figure FDA0003189597170000013
each representing a redundant robot armJoint angle and joint angular velocity, θ*(0) Is a desired initial value of each joint angle, p > 0, betaθMore than 0, and more than 0 and more than delta and less than 1 are design parameters used for expressing the dynamic performance of joint displacement; theta (t) -theta*(0) Representing the displacement deviation, r, of each joint angle from the desired initial value for each joint angle*Representing a desired motion profile of the redundant robotic arm end effector,
Figure FDA0003189597170000015
representing a desired velocity vector for the redundant robotic arm end effector; because the initial position of the redundant mechanical arm is not on the expected motion track, the error r between the expected motion track and the actual motion track position of the end effector of the redundant mechanical arm is reduced*-f (θ), changing the direction of motion of the redundant robotic arm end effector, βrThe parameter gain of the position is represented by more than 0 and is used for adjusting the speed of the redundant mechanical arm end effector to move to a desired motion track; j (theta) is a Jacobian matrix of the redundant manipulator, and f (theta) is an actual motion track of the redundant manipulator;
3) constructing a final state network model of a finite value activation function, wherein the dynamic characteristics of the final state network model are described by the following equations
Figure FDA0003189597170000014
Wherein the joint angular displacement deviation e (t) ═ θ (t) - θ*(0) The finite time of the system expressed by the dynamic characteristic equation (2) converges to zero, 0 & ltdelta & lt 1, rho & gt 0, betaE> 0, the required convergence time T is:
Figure FDA0003189597170000021
when the index function (1) reaches the minimum value, all joint angles of the redundant mechanical arm are drawn back to the expected motion track;
establishing a Lagrangian function for solving the quadratic programming in the step 2):
Figure FDA0003189597170000022
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 the content of the first and second substances,
Figure FDA0003189597170000023
i is an identity matrix
Figure FDA0003189597170000024
Figure FDA0003189597170000025
The time-varying matrix equation (3) is solved by the final state network of the finite-value activation function described by the formula (2) with the notation of E being WY-v and E being the time-varying matrix equation error,
the solution equation is obtained as follows
Figure FDA0003189597170000026
4) And 3) using the result obtained in the step 3) to control each joint motor and drive the redundant mechanical arm to execute the task.
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CN109940615B (en) * 2019-03-13 2021-02-23 浙江科技学院 Terminal state network optimization method for synchronous repeated motion planning of double-arm manipulator
CN110695994B (en) * 2019-10-08 2021-03-30 浙江科技学院 Finite time planning method for cooperative repetitive motion of double-arm manipulator
CN111687844B (en) * 2020-06-19 2021-08-31 浙江大学 Method for completing unrepeatable covering task by using mechanical arm to lift up for minimum times

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