CN109159121B - Redundant robot repetitive motion planning method adopting parabolic final state neural network - Google Patents

Redundant robot repetitive motion planning method adopting parabolic final state neural network Download PDF

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CN109159121B
CN109159121B CN201811061891.6A CN201811061891A CN109159121B CN 109159121 B CN109159121 B CN 109159121B CN 201811061891 A CN201811061891 A CN 201811061891A CN 109159121 B CN109159121 B CN 109159121B
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孙明轩
张钰
吴雨芯
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Zhejiang University of Technology ZJUT
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
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    • B25J9/16—Programme controls
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Abstract

A redundant robot repetitive motion planning method adopting a parabolic final state neural network gives an expected track r of a robot end effector in a Cartesian spaced(t) and giving the desired pull-in angle θ for each jointd(0) (ii) a For the repeated motion of the robot, asymptotic convergence performance indexes are adopted, a quadratic optimization problem of redundant robot trajectory planning is converted into a time-varying matrix equation solving problem, and a parabolic final state neural network is used as a solver. And under the condition of initial position deviation, realizing a repeated motion planning task of limited time convergence of the redundant robot. The invention provides a redundant robot motion planning method which has limited time convergence and high calculation precision and is easy to realize and adopts a parabolic final state neural network.

Description

Redundant robot repetitive motion planning method adopting parabolic final state neural network
Technical Field
The invention relates to a repetitive motion planning technology for an industrial robot, and particularly provides a repetitive motion planning method for a redundant robot by adopting a parabolic final state neural network under the conditions of limited time convergence and deviation of an initial position from an expected track.
Background
The redundant robot has good flexibility and fault tolerance, can utilize redundant degrees of freedom to enhance obstacle avoidance without influencing the operation of an end effector, and can complete variable operation tasks in a complex working environment. At present, redundant robots have played an important role in a number of application areas, such as manufacturing, medical devices, logistics transportation, military and national defense, etc.
The redundant robot has a number of joints n greater than the number of degrees of freedom m required by the end effector to perform a desired task, which allows the redundant robot to have greater flexibility and fault tolerance, while the redundant degrees of freedom allow an infinite number of solutions to the redundancy solution problem. Therefore, how to accurately obtain an inverse kinematics solution in real time, that is, by knowing the position and the posture of the end effector, solving the values of the joint angles of the corresponding redundant robot is a difficult point of redundant robot motion planning, and the conventional method is an analysis scheme based on a pseudo-inverse. Considering the relation between the angle of each joint of the n-degree-of-freedom robot and the displacement of the end effector
r(t)＝f(θ(t))
Where r (t) is a pose variable of the end effector in a cartesian coordinate system, and θ (t) represents a joint angle. The relationship between the differential of the terminal Cartesian space and the joint space is
Figure GDA0002719837610000011
Wherein,
Figure GDA0002719837610000012
and
Figure GDA0002719837610000013
respectively, are the respective time derivatives of the same,
Figure GDA0002719837610000014
is the robot jacobian matrix. Obtaining a minimum norm solution of the joint velocity variable by calculating the pseudo-inverse of J (theta (t))
Figure GDA0002719837610000015
Wherein, J+＝JT(JJT)-1Is the pseudo-inverse of the jacobian matrix.
Minimum speed norm performance indicators with equality constraints are the objective functions of motion planning (D.E. Whitney, Resolved motion control of manipulators and human prosthetics), IEEE trans. Man-Machine Syst.,1969,10(2): 47-53)
Figure GDA0002719837610000021
Figure GDA0002719837610000022
In the formula, A is a positive definite weighting matrix. Solving the above-mentioned planning problem, the demand solves the following system of equations
Figure GDA0002719837610000023
It is solved into
Figure GDA0002719837610000024
It can be seen that formula (1) is a special case of formula (3) when a ═ I.
A redundancy resolution scheme based on Quadratic Optimization (QP) is concerned, and f.t.cheng, t. -h.cheng proposes a joint unbiased performance index (f.t.cheng, t. -h.chen, and y. -y.sun, resolution robot reduction integrity constraints (redundant robot trajectory planning method under inequality constraint), IEEE trans.robotics automation, 1994, 10 (1): 65-71) in 1994:
Figure GDA0002719837610000025
the performance of redundant robot trajectory planning is directly related to whether the robot can achieve a given end task. When the motion trail of the end effector is closed, the trail of each joint angle variable in the motion space is not necessarily closed after the robot finishes the end work task. This non-repetitive problem may create undesirable joint configurations that may cause unexpected and even dangerous situations for repeated operation of the redundant robot tip closure trajectory. The most widely used pseudo-inverse control method cannot ensure the repeatability of the movement. To accomplish the repetitive motion, it is usually compensated by a self-motion method, and the adjustment of the self-motion is often inefficient (see Klein C A and Huang C, Review of Pseudo Inverse Control for use with kinematic Renderstandable Manipulators (Redundant robot motion planning based on Pseudo-Inverse Control method), IEEE Trans. Syst. Man. Cybern.1983,13(2): 245-.
In order to execute the Repetitive motion task, a Repetitive motion index is introduced as an optimization criterion, and a Repetitive Motion Planning (RMP) scheme (Zhang Y, Wang J, Xia Y.A dual Neural network for a redundant resolution of kinematic vectors sub-object to joint limits and joint variables limits (redundant robot trajectory planning method based on joint angle and angular velocity limits). IEEE Trans Neural network, 2003,14(3): 658-. A common repetitive motion index is the asymptotic convergence performance index AOC (asymptotic-convergent Optimality Criterion) as follows:
Figure GDA0002719837610000031
the Quadratic Programming (QP) here is solved using a Recurrent Neural Network (RNN) computation method. The general neural network solver has asymptotic convergence performance, and an effective solution can be obtained as long as the calculation time is long enough.
And describing a redundancy analysis problem based on quadratic programming, and solving published documents by adopting a recurrent neural network. Compared with a recurrent neural network with asymptotic convergence dynamic characteristics, the dynamic characteristics based on the parabolic final state neural network have limited time convergence, and are high in convergence speed and calculation accuracy. It is worth pointing out that the redundant robot trajectory planning problem is resolved into a time-varying computation problem, and an effective method for solving the time-varying problem is to use a recurrent neural network computation method with limited time convergence.
Disclosure of Invention
In order to overcome the defects that the existing redundant robot repetitive motion planning method cannot be converged in limited time, has low calculation precision and is difficult to realize, the invention provides the redundant robot motion planning method which has the advantages of limited time convergence, high calculation precision and easy realization and adopts the parabolic final state neural network. The invention adopts asymptotic convergence performance indexes, converts the quadratic optimization problem of redundant robot trajectory planning into a time-varying matrix equation solving problem, takes a parabolic final state neural network as a solver, and realizes the repeated motion planning task of the redundant robot with limited time convergence under the condition of initial position deviation.
In order to achieve the purpose, the invention provides the following technical scheme:
a redundant robot repetitive motion planning method adopting a parabolic final state neural network comprises the following steps:
1) given a desired trajectory r of a redundant robotic end effector in Cartesian spaced(t) and giving the desired joint angle θ of each jointd(0)；
2) For the redundant robot with repeated motion, the initial joint angle is theta (0) ═ theta0Initial joint angle θ0Different from the desired joint angle, i.e. theta0≠θd(0)；
3) Describing the redundant robot repetitive motion planning as a quadratic planning problem, providing an asymptotic convergence performance index, and forming a repetitive motion planning scheme:
Figure GDA0002719837610000041
wherein,
g(θ(t))＝-ρ0(θ(t)-θd(0))
ρ0>0，θ(t)-θd(0) the deviation of the joint angle from the expected joint angle is shown, and since the initial position of the redundant robot is not on the expected track, a feedback deviation amount, namely r, needs to be added to the right side of the relation between the speed of the redundant robot end effector and the speed of the jointd(t) -f (θ (t)), the deviation amount representing an error existing between the desired trajectory and the actual trajectory; due to the artificially constructed convergence relationship, this error will be continuously reduced to zero; the parameter beta is used to adjust the rate at which the redundant robot end effector moves to a desired trajectory, beta>0; j (theta (t)) is a Jacobian matrix obtained according to a DH parameter of the redundant robot, and f (theta (t)) is an actual track of the end effector of the redundant robot;
4) constructing a dynamic characteristic equation based on a parabolic final state neural network as follows
Figure GDA0002719837610000042
Wherein E is an error matrix variable, EijIs the element of the matrix E, rho > 0, sigma>0,>0，
Figure GDA0002719837610000043
Figure GDA0002719837610000044
Equation (5) is time-limited stable, and its time-limited convergence needs to be illustrated in two cases:
when | Eij(0) < σ, convergence time of
Figure GDA0002719837610000045
When | Eij(0) | ≧ σ, convergence time of
Figure GDA0002719837610000051
When ρ is 0, equation (5) becomes the case without the acceleration term
Figure GDA0002719837610000052
Equation (8) is also time-limited stable, and its time-limited convergence is also illustrated in two cases:
when | Eij(0) When | < σ, the convergence time is
Figure GDA0002719837610000053
When | Eij(0) | ≧ σ, convergence time of
Figure GDA0002719837610000054
(5) Defining lagrange functions
Figure GDA0002719837610000055
Wherein λ is a Lagrange multiplier vector,
Figure GDA0002719837610000056
about
Figure GDA0002719837610000057
And lambda are respectively calculated to obtain the partial derivatives, and the partial derivatives are made to be zero and are obtained by sorting
W(t)Y(t)＝v(t) (12)
Wherein,
Figure GDA0002719837610000058
Figure GDA0002719837610000059
Figure GDA00027198376100000510
i is an identity matrix;
6) to solve the quadratic programming problem in step 3), the error is defined by equation (12)
E(t)＝W(t)Y(t)-v(t) (13)
Giving out the parabolic final state neural network equation according to the parabolic final state neural network dynamic characteristic equation (5)
Figure GDA0002719837610000061
And obtaining the angle of each joint of the redundant robot through a neural network calculation process.
The technical conception of the invention is as follows: performance indicators with asymptotic convergence
Figure GDA0002719837610000062
Wherein,
g(θ(t))＝- ρ0(θ(t)-θd(0))
ρ0>0，θ(t)-θd(0) which represents the deviation of the joint angle from the desired joint angle, this error will decrease continuously until it is zero due to the artificially constructed convergence. The parameter beta is used to adjust the rate at which the redundant robot end effector moves to a desired trajectory, beta>0。
Dynamic characteristic equation based on parabolic final state neural network for redundant robot repetitive motion planning
Figure GDA0002719837610000063
Wherein E is an error matrix variable, EijIs the element of the matrix E, rho > 0, sigma>0,>0，
Figure GDA0002719837610000064
Figure GDA0002719837610000065
Equation (5) is time-limited stable, and its time-limited convergence needs to be illustrated in two cases:
when | Eij(0) < σ, convergence time of
Figure GDA0002719837610000071
When | Eij(0) | ≧ σ, convergence time of
Figure GDA0002719837610000075
When ρ is 0, equation (5) becomes the case without the acceleration term
Figure GDA0002719837610000072
Equation (8) is also time-limited stable, and its time-limited convergence is also illustrated in two cases:
when | Eij(0) When | < σ, the convergence time is
Figure GDA0002719837610000073
When | Eij(0) | ≧ σ, convergence time of
Figure GDA0002719837610000074
When E isij(t) the finite time converges to zero, indicating that the redundant robotic end effector is moving along the desired trajectory.
The invention has the following beneficial effects: the parabolic final state neural network method is applied to redundant robot repetitive motion planning, and a repetitive motion planning task of redundant robot limited time convergence is realized under the condition of initial position deviation. Compared with an asymptotic neural network repetitive motion planning method, the parabolic final state neural network method has the characteristic of faster convergence, and finally, the error precision of each joint of the redundant robot deviating from the expected joint angle is higher. The method is suitable for solving the time-varying problem (the formula (12) is a time-varying matrix equation), is easy to realize in engineering application, has low cost and meets the actual engineering requirements.
Drawings
Fig. 1 is a flow chart of a repetitive motion planning scheme provided by the present invention.
FIG. 2 is a diagram of the function F (E) when taking different values of σij(t),σ)。
Fig. 3 is a diagram of a redundant robot PA10 of the repetitive motion planning scheme of the present invention. (taken from Y Zhang, Z Zhang, Repetitive motion planning and control of redundant robot repeated motion planning and control.) Berlin: Springer,2013:23-24)
Fig. 4 is a motion trajectory of the redundant robot PA10 end effector.
Fig. 5 shows the motion trajectory of each joint of the redundant robot PA10, where 1 is a PA10 robot arm base, 2 is an elbow trajectory, 3 is a wrist trajectory, and 4 is an end effector motion trajectory.
Fig. 6 shows the respective joint angles of the redundant robot PA 10.
Fig. 7 shows the joint speeds of the redundant robot PA 10.
Fig. 8 shows the position errors of the redundant robot PA10 end effector.
FIG. 9 shows the error of solving with the asymptotic neural network and the parabolic final state neural network.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 9, a redundant robot repetitive motion planning method using a parabolic final state neural network, fig. 1 is a flowchart of a redundant robot repetitive motion planning scheme, which is composed of the following three steps: 1. Determining a desired trajectory and a desired joint angle of a redundant robot end effector; 2. adopting asymptotic convergence performance indexes and forming a redundant robot repetitive motion quadratic programming scheme; 3. solving a quadratic programming problem by using a parabolic final state neural network to obtain each joint angle track, and comprising the following steps of:
1) determining a desired trajectory
Setting a desired joint angle for a redundant robot PA10
Figure GDA0002719837610000081
Determining the center coordinates (x is 0.2m, y is 0, z is 0) of the circle locus, setting the radius of the circle to be 0.2m, and setting the included angle between the circular surface and the xy plane as
Figure GDA0002719837610000082
The time T of the redundant robot end effector completing the circular track is 10 s. The seven joint angle initial values of the redundant robot PA10 are set to take into account that the initial position of the redundant robot is not on the desired motion trajectory
Figure GDA0002719837610000083
2) Secondary planning scheme for establishing repeated motion of redundant robot
Describing the redundant robot repetitive motion trajectory planning as the following quadratic planning problem, and providing an asymptotic convergence performance index
Figure GDA0002719837610000084
Wherein,
g(θ(t))＝-ρ0(θ(t)-θd(0))
ρ0>0，θ(t)-θd(0) the deviation of the joint angle from the expected joint angle is shown, and since the initial position of the redundant robot is not on the expected track, a feedback deviation amount, namely r, needs to be added to the right side of the relation between the speed of the redundant robot end effector and the speed of the jointd(t) -f (θ (t)), the deviation amount representing an error existing between the desired trajectory and the actual trajectory; due to the artificially constructed convergence relationship, this error will be continuously reduced to zero; the parameter beta is used to adjust the rate at which the redundant robot end effector moves to a desired trajectory, beta>0; j (theta (t)) is a Jacobian matrix obtained according to a DH parameter of the redundant robot, and f (theta (t)) is an actual track of the end effector of the redundant robot;
3) constructing a parabolic final state neural network to solve the quadratic programming problem
Constructing a parabolic final state neural network model with a dynamic characteristic equation of
Figure GDA0002719837610000091
Wherein E is an error matrix variable, EijIs the element of the matrix E, rho > 0, sigma>0,>0，
Figure GDA0002719837610000092
Figure GDA0002719837610000093
Equation (5) is time-limited stable, and its time-limited convergence needs to be illustrated in two cases:
1) when | Eij(0) < σ, convergence time of
Figure GDA0002719837610000094
2) When | Eij(0) | ≧ σ, convergence time of
Figure GDA0002719837610000095
When ρ is 0, equation (5) becomes the case without the acceleration term
Figure GDA0002719837610000101
Equation (8) is also time-limited stable, and its time-limited convergence is also illustrated in two cases:
1) when | Eij(0) When | < σ, the convergence time is
Figure GDA0002719837610000102
2) When | Eij(0) | ≧ σ, convergence time of
Figure GDA0002719837610000103
Defining the following Lagrangian function
Figure GDA0002719837610000104
Wherein λ is a Lagrange multiplier vector,
Figure GDA0002719837610000105
about
Figure GDA0002719837610000106
And lambda are respectively calculated to obtain the partial derivatives, and the partial derivatives are made to be zero and are obtained by sorting
W(t)Y(t)＝v(t) (12)
Wherein,
Figure GDA0002719837610000107
Figure GDA0002719837610000108
Figure GDA0002719837610000109
i is an identity matrix;
error is defined by equation (12)
E(t)＝W(t)Y(t)-v(t) (13)
Giving out the parabolic final state neural network equation according to the parabolic final state neural network dynamic characteristic equation (5)
Figure GDA00027198376100001010
And obtaining the angle of each joint of the redundant robot through a neural network calculation process.
FIG. 2 is a diagram of the function F (E) when taking different values of σij(t), sigma), it can be seen from the figure that the parabolic final state neural network error dynamic equation can be converged quickly when different sigma values are taken.
The redundant robot PA10 shown in fig. 3 is used to implement the repetitive motion planning scheme of the present invention. The redundant robot has seven degrees of freedom (four axes of rotation and three pivots), the main specifications: the total length of the robot is 1.345m, the weight of the robot is 343N, the maximum combined speed of all axes is 1.55m/s, and the payload weight is 98N. The length of the arm is d 3-0.450 m, d 5-0.5 m, and d 7-0.08 m.
Fig. 4 is a motion trajectory of a redundant robotic end effector in space. The target circular track and the motion track of the redundant robot end effector are given in the figure. It can be seen that the initial position of the redundant robotic end effector is not on the desired trajectory. As shown in fig. 8, as time increases (T ═ 10s), the final position error accuracy of the redundant robot end effector reaches 2 × 10 in the three directions of XYZ axes-4m, the actual trajectory coincides with the desired trajectory.
Fig. 5 shows the motion trail of each joint of the redundant robot. As can be seen from the figure, the trajectories of the respective joints are closed after one cycle of the operation, and repetitive motion control is realized.
In order to verify the effectiveness of the parabolic final state neural network method in the repetitive motion planning, a joint angle transient trajectory and a joint angular velocity transient trajectory obtained in the process of completing a circular trajectory by the redundant robot PA10 end effector are shown in FIGS. 6 and 7. It can be seen that the joint angles of the redundant robot eventually converge to the desired joint angle positions.
When T is 10s, the maximum deviation of the final value error of each joint angle obtained by solving by using an asymptotic neural network is 6.8740 multiplied by 10-4rad, the final value error of each joint angle obtained by solving by adopting a parabolic final state neural networkLarge deviation 9.1348X 10-5rad, detailed in table 1:
Figure GDA0002719837610000121
TABLE 1 wherein the asymptotic neural network dynamics
Figure GDA0002719837610000122
Dynamic characteristic of parabolic final state neural network
Figure GDA0002719837610000123
Wherein,
Figure GDA0002719837610000124
Figure GDA0002719837610000125
ρ＝1,＝1,σ＝1
to compare convergence performance of the asymptotic neural network and the parabolic final state neural network, the error norm | | | E (t) | torry is calculatedF＝||W(t)Y(t)-v(t)||F. Fig. 9 shows the error norm convergence trajectories for solving the quadratic programming problem with the asymptotic neural network and the parabolic final state neural network, respectively. As can be seen, when the norm of E (T) converges to 0.01, the time T (E) is required for solving the parabolic final state neural networkij(0) 0.861s, time T (E) required to solve with the asymptotic neural networkij(0))＝6.932s。

Claims (1)

1. A redundant robot repetitive motion planning method adopting a parabolic final state neural network is characterized by comprising the following steps:
1) given a desired trajectory r of a redundant robotic end effector in Cartesian spaced(t) and giving the desired joint angle θ of each jointd(0)；
2) For the redundant robot with repeated motion, the initial joint angle is theta (0) ═ theta0Initial joint angle θ0Different from the desired joint angle, i.e. theta0≠θd(0)；
3) Describing the redundant robot repetitive motion planning as a quadratic planning problem, providing an asymptotic convergence performance index, and forming a repetitive motion planning scheme:
Figure FDA0002719837600000011
wherein,
g(θ(t))＝-ρ0(θ(t)-θd(0))
ρ0>0，θ(t)-θd(0) the deviation between the joint angle theta (t) and the expected joint angle is shown, and since the initial position of the redundant robot is not on the expected track, a feedback deviation amount, namely r, needs to be added to the right side of the relation between the speed of the redundant robot end effector and the speed of the jointd(t) -f (θ (t)), the deviation amount representing an error existing between the desired trajectory and the actual trajectory; due to the artificially constructed convergence relationship, this error will be continuously reduced to zero; the parameter beta is used to adjust the rate at which the redundant robot end effector moves to a desired trajectory, beta>0; j (theta (t)) is a Jacobian matrix obtained according to a DH parameter of the redundant robot, and f (theta (t)) is an actual track of the end effector of the redundant robot;
4) constructing a dynamic characteristic equation based on a parabolic final state neural network as follows
Figure FDA0002719837600000012
Wherein E is an error matrix variable, EijIs the element of the matrix E, rho > 0, sigma>0,>0，
Figure FDA0002719837600000013
Figure FDA0002719837600000021
Equation (5) is time-limited stable, and its time-limited convergence needs to be illustrated in two cases:
when | Eij(0) < σ, convergence time of
Figure FDA0002719837600000022
When | Eij(0) | ≧ σ, convergence time of
Figure FDA0002719837600000023
When ρ is 0, equation (5) becomes the case without the acceleration term
Figure FDA0002719837600000024
Equation (8) is also time-limited stable, and its time-limited convergence is also illustrated in two cases:
when | Eij(0) When | < σ, the convergence time is
Figure FDA0002719837600000025
When | Eij(0) | ≧ σ, convergence time of
Figure FDA0002719837600000026
5) Defining lagrange functions
Figure FDA0002719837600000027
Wherein λ is a Lagrange multiplier vector,
Figure FDA0002719837600000028
about
Figure FDA0002719837600000029
And lambda are respectively calculated to obtain the partial derivatives, and the partial derivatives are made to be zero and are obtained by sorting
W(t)Y(t)＝v(t) (12)
Wherein,
Figure FDA0002719837600000031
Figure FDA0002719837600000032
Figure FDA0002719837600000033
i is an identity matrix;
6) to solve the quadratic programming problem in step 3), the error is defined by equation (12)
E(t)＝W(t)Y(t)-v(t) (13)
Giving out the parabolic final state neural network equation according to the parabolic final state neural network dynamic characteristic equation (5)
Figure FDA0002719837600000034
And obtaining the angle of each joint of the redundant robot through a neural network calculation process.
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