CN109159121A - Using redundant robot's repetitive motion planning method of parabolic type final state neural network - Google Patents

Abstract

A redundant robot repetitive motion planning method using a parabolic final state neural network. The expected trajectory r _d (t) of the robot end-effector is given in Cartesian space, and the expected retraction angle θ _d of each joint is given. (0); For the repetitive motion of the robot, the asymptotic convergence performance index is adopted, and the quadratic optimization problem of redundant robot trajectory planning is transformed into a time-varying matrix equation solving problem, and the parabolic final state neural network is used as the solver. A repetitive motion planning task that achieves finite-time convergence of redundant robots in the presence of an initial position offset. The invention provides a redundant robot motion planning method using a parabolic final state neural network with finite time convergence, high calculation accuracy and easy implementation.

Description

A redundant robot repetitive motion planning method using parabolic final state neural network

technical field

The invention relates to a repetitive motion planning technology for an industrial robot, and in particular, proposes a redundant robot repetitive motion planning method using a parabolic final state neural network under the condition that the initial position deviates from the desired trajectory with limited time convergence.

Background technique

The redundant robot has good flexibility and fault tolerance. It can use the redundant degrees of freedom to enhance obstacle avoidance without affecting the operation of the end effector, and can complete variable tasks in a complex working environment. At present, redundant robots have played an important role in many application fields, such as manufacturing, medical equipment, logistics and transportation, military defense, etc.

The number of joints n of the redundant robot is greater than the degree of freedom m required by the end effector to perform the expected task, which makes the redundant robot have greater flexibility and fault tolerance, and the redundant degree of freedom makes the redundant solution problem infinite. a solution. Therefore, how to obtain the inverse kinematics solution in real time and accurately, that is, by knowing the position and attitude of the end effector, to solve the value of each joint angle of the corresponding redundant robot is the difficulty of redundant robot motion planning. The conventional method is Pseudo-inverse-based parsing scheme. Consider the relationship between the joint angle of the n-degree-of-freedom robot and the displacement of the end effector

r(t)=f(θ(t))

Among them, r(t) is the pose variable of the end effector in the Cartesian coordinate system, and θ(t) represents the joint angle. The relationship between the differential of the terminal Cartesian space and the joint space is

in, and are their respective time derivatives, is the robot Jacobian matrix. The minimum norm solution of the joint velocity variable is obtained by computing the pseudo-inverse of J(θ)

where J ⁺ =J ^T (JJ ^T ) ^-1 is the pseudo-inverse of the Jacobian matrix.

Minimum Velocity Norm Performance Metrics with Equality Constraints as Objective Functions for Motion Planning (D.E.Whitney, Resolved motion rate control of manipulators and human prostheses, IEEE Trans.Man-Machine Syst. , 1969, 10(2):47-53) for

where A is a positive definite weighting matrix. To solve the above planning problem, the following equations need to be solved

It is solved as

It can be seen that formula (1) is a special case of formula (3) when A=I.

The redundant resolution scheme based on quadratic optimization (QP) has attracted attention, and F.T. Y.Sun, Resolving manipulator redundancy under inequality constraints (redundant robot trajectory planning method under inequality constraints), IEEE Trans.Robotics Automat., 1994, 10(1): 65-71):

The performance of redundant robot trajectory planning is directly related to whether the robot can achieve a given end task. When the motion trajectory of the end effector is closed, after the robot completes the end work task, the trajectory of each joint angle variable in the motion space is not necessarily closed. This non-repetitive problem may result in undesired joint configurations, which may lead to unexpected and dangerous situations in the repetitive operation of closed trajectories at the ends of redundant robots. The most widely used pseudo-inverse control method cannot guarantee the repeatability of motion. In order to complete the repetitive motion, the method of self-motion is usually used to make up, and the adjustment of self-motion is often inefficient (see Klein C A and Huang C, Review of Pseudo Inverse Control for use with Kinematically Redundant Manipulators (redundant based on pseudo-inverse control method). Robotic motion planning. IEEE Trans.Syst.Man.Cybern.1983,13(2):245-250; Tchon K, Janiak M. Repeatable approximation of the Jacobian pseudo-inverse ). Systems and Control Letters, 2009, 58(12):849-856).

In order to perform repetitive motion tasks, repetitive motion indicators are introduced as optimization criteria to form a repetitive motion planning (RMP) scheme (Zhang Y, Wang J, Xia Y. A dual neural network for redundancy resolution of kinematically redundant manipulators subject to joint limits and joint velocity limits (A method for redundant robot trajectory planning based on joint angle and angular velocity limits. IEEE Trans Neural Netw., 2003, 14(3):658-667). The commonly used repetitive motion indicators are the following asymptotic convergence performance indicators AOC (Asympototically-Convengent OptimalityCriterion):

The quadratic programming (QP) here is solved by the recurrent neural network (RNN) calculation method. The usual neural network solvers have asymptotic convergence performance, and can obtain efficient solutions as long as the computation time is long enough.

Based on quadratic programming to describe the redundancy analysis problem, the published literature often uses recurrent neural network to solve it. Compared with the recurrent neural network with asymptotically convergent dynamic characteristics, the dynamic characteristics of the parabolic final state neural network have finite time convergence, fast convergence speed and high calculation accuracy. It is worth pointing out that the redundant robot trajectory planning problem boils down to a time-varying computational problem, and an effective way to solve the time-varying problem is to use a recursive neural network computational method with finite time convergence performance.

SUMMARY OF THE INVENTION

In order to overcome the shortcomings of the existing redundant robot repetitive motion planning methods that cannot converge in limited time, have low calculation accuracy, and are difficult to implement, the present invention provides a parabolic final state with limited time convergence, high calculation accuracy, and easy implementation. A redundant robot motion planning method with neural networks. The invention adopts the asymptotic convergence performance index, transforms the quadratic optimization problem of redundant robot trajectory planning into the time-varying matrix equation solving problem, and uses the parabolic final state neural network as the solver. Repetitive motion planning tasks to achieve finite-time convergence of redundant robots.

In order to achieve this purpose, the present invention provides following technical scheme:

A redundant robot repetitive motion planning method using a parabolic final state neural network, comprising the following steps:

1) The expected trajectory r _d (t) of the robot end effector is given in Cartesian space, and the expected retraction angle θ _d (0) of each joint is given;

2) For the repetitive motion robot, its initial joint angle is θ(0)=θ ₀ , and the initial joint angle θ ₀ is different from the expected joint angle, that is, θ ₀ ≠θ _d (0);

3) The redundant robot repetitive motion planning is described as a quadratic programming problem, and the asymptotic convergence performance index is given to form the repetitive motion planning scheme:

in,

g(θ)=-ρ ₀ (θ(t)-θ _d (0))

ρ ₀ > 0, θ(t)-θ _d (0) represents the deviation of the joint angle from the desired joint angle. Since the initial position of the robot is not on the desired trajectory, the left side of the relationship between the end effector velocity and the joint velocity needs to be added. The last feedback deviation is r _d -f(θ), which represents the error between the expected trajectory and the actual trajectory; due to the artificially constructed convergence relationship, this error will continue to shrink until it is zero; the parameter β It is used to adjust the rate at which the end effector moves to the desired trajectory, β>0; J(θ) is the Jacobian matrix calculated according to the DH parameters of the robot, and f(θ) is the actual trajectory of the robot end effector;

4) Construct the following dynamic characteristic equation based on parabolic final state neural network

Among them, E is the error matrix variable, E _ij is the element of matrix E, ρ>0, σ>0, ε>0,

Equation (5) is stable in finite time, and its finite time convergence needs to be explained in two cases:

When |E _ij (0)|<σ, the convergence time is

When |E _ij (0)|≥σ, the convergence time is

When ρ=0, equation (5) becomes the case without acceleration term

Equation (8) is also stable in finite time, and its finite time convergence is also described in two cases:

When |E _ij (0)|<σ, the convergence time is

When |E _ij (0)|≥σ, the convergence time is

(5) Define the Lagrangian function

where λ is the Lagrange multiplier vector, about and λ to find the partial derivatives respectively, and make them zero, after finishing

W(t)Y(t)=v (12)

in,

I is the 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 (13)

According to the dynamic characteristic equation (5) of the parabolic final state neural network, the parabolic final state neural network equation is given

Through the neural network calculation process, the angles of each joint of the robot are obtained.

The technical idea of the present invention is: adopt the performance index of asymptotic convergence

in,

g(θ)=-ρ ₀ (θ(t)-θ _d (0))

ρ ₀ &gt; 0, θ(t)-θ _d (0) represents the deviation of the joint angle from the expected joint angle. Due to the convergence relationship of the artificial structure, this error will continue to shrink until it is zero. The parameter β is used to adjust the rate at which the end effector moves to the desired trajectory, β &gt; 0.

Constructing a dynamic characteristic equation based on parabolic final state neural network for repetitive motion planning of redundant robots

Among them, E is the error matrix variable, E _ij is the element of matrix E, ρ>0, σ>0, ε>0,

Equation (5) is stable in finite time, and its finite time convergence needs to be explained in two cases:

When |E _ij (0)|<σ, the convergence time is

When |E _ij (0)|≥σ, the convergence time is

When ρ=0, equation (5) becomes the case without acceleration term

Equation (8) is also stable in finite time, and its finite time convergence is also described in two cases:

When |E _ij (0)|<σ, the convergence time is

When |E _ij (0)|≥σ, the convergence time is

When E _ij (t) converges to zero in finite time, it means that the end effector is moving along the desired trajectory.

The beneficial effects of the invention are mainly manifested in that a parabolic final state neural network method is applied to the redundant robot repetitive motion planning, and the repetitive motion planning task of the redundant robot's limited time convergence is realized under the condition of initial position offset. Compared with the asymptotic neural network repetitive motion planning method, the parabolic final state neural network method has faster convergence characteristics, and the final return angle of each joint of the robot is more accurate. It is suitable for solving time-varying problems (equation (12) is a time-varying matrix equation), and is easy to implement in engineering applications, with low cost and meeting practical engineering needs.

Description of drawings

FIG. 1 is a flow chart of the repetition planning scheme provided by the present invention.

Figure 2 shows the function F(E _ij (t),σ) when taking different values of σ.

FIG. 3 is the redundant robot PA10 of the repeated planning scheme of the present invention. (From Y Zhang, Z Zhang. Repetitive motion planning and control of redundant robot manipulators. Berlin: Springer, 2013: 23-24)

Figure 4 shows the motion trajectory of the redundant robot PA10 end effector.

Figure 5 shows the motion trajectories of each joint of the redundant robot PA10.

Figure 6 shows the joint angles of the redundant robot PA10.

Figure 7 shows the speed of each joint of the redundant robot PA10.

Figure 8 shows the position errors of the end effector of the redundant robot PA10.

Figure 9 shows the error when solving with asymptotic neural network and parabolic final state neural network.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings.

Referring to Figures 1 to 9, a redundant robot repetitive motion planning method using a parabolic final state neural network, Figure 1 is a flowchart of a redundant robot repetitive motion planning scheme, which consists of the following three steps: 1. Determine the redundant The expected trajectory of the end-effector of the redundant robot and the expected angle of each joint to be closed; 2. The asymptotic convergence performance index is adopted, and the redundant robot repetitive motion quadratic programming scheme is formed; 3. The parabolic final state neural network is used to solve the quadratic programming problem , to obtain the trajectory of each joint angle, including the following steps:

1) Determine the desired trajectory

Set the expected retraction of redundant robot PA10 Determine the coordinates of the center of the circle trajectory (x=0.2m, y=0, z=0), set the radius of the circle to 0.2m, and the angle between the circular surface and the xy plane is The time for the end effector to complete the circular trajectory is T=10s. Considering that the initial position of the redundant robot is not on the desired trajectory, the initial values of the robot's seven joint angles are set as

2) Establish a quadratic programming scheme for repetitive motion of redundant robots

The redundant robot repetitive motion trajectory planning is described as the following quadratic programming problem, and the asymptotic convergence performance index is given

in,

g(θ)=-ρ ₀ (θ(t)-θ _d (0))ρ ₀ &gt; 0, θ(t)-θ _d (0) represents the deviation of the joint angle from the desired joint angle, due to the initial position of the robot is not on the desired trajectory, so a feedback deviation needs to be added to the left of the relationship between the speed of the end effector and the joint speed, that is, r _d -f(θ), which represents the error between the expected trajectory and the actual trajectory; Due to the artificially constructed convergence relationship, this error will continue to shrink until zero; the parameter β is used to adjust the rate at which the end effector moves to the desired trajectory, β>0; J(θ) is the Jacques obtained from the DH parameters of the robot ratio matrix, f(θ) is the actual trajectory of the robot end effector;

3) Constructing a parabolic final state neural network to solve the above quadratic programming problem

A parabolic final state neural network model is constructed, and its dynamic characteristic equation is

Among them, E is the error matrix variable, E _ij is the element of matrix E, ρ>0, σ>0, ε>0,

Equation (5) is stable in finite time, and its finite time convergence needs to be explained in two cases:

1) When |E _ij (0)|<σ, the convergence time is

2) When |E _ij (0)|≥σ, the convergence time is

When ρ=0, equation (5) becomes the case without acceleration term

Equation (8) is also stable in finite time, and its finite time convergence is also described in two cases:

1) When |E _ij (0)|<σ, the convergence time is

2) When |E _ij (0)|≥σ, the convergence time is

Define the following Lagrangian function

where λ is the Lagrange multiplier vector, about and λ to find the partial derivatives respectively, and make them zero, after finishing

W(t)Y(t)=v (12)

in,

I is the identity matrix;

The error is defined by equation (12)

E(t)=W(t)Y(t)-v (13)

According to the dynamic characteristic equation (5) of the parabolic final state neural network, the parabolic final state neural network equation is given

Through the neural network calculation process, the angles of each joint of the robot are obtained.

Figure 2 shows the function F(E _ij (t),σ) when different σ values are taken. It can be seen from the figure that the parabolic final state neural network error dynamic equation can converge quickly when different σ values are taken.

The redundant robot PA10 shown in FIG. 3 is used to implement the repetitive planning scheme of the present invention. The robot has seven degrees of freedom (four rotation axes and three pivot axes), main specifications: the overall length of the manipulator is 1.345m, the weight of the manipulator is 343N, the maximum combined speed of all axes is 1.55m/s, the payload The weight is 98N. The lengths of the arms are d3=0.450m, d5=0.5m, and d7=0.08m.

Figure 4 shows the motion trajectory of the robot end effector in space. The figure shows the target circle trajectory and the motion trajectory of the robot end effector. It can be seen that the initial position of the end effector is not on the desired trajectory. As shown in Fig. 8, with the increase of time (T=10s), the final value position error accuracy of the end effector reaches 2× ^10-4 m in the three directions of XYZ axis, and the actual trajectory is consistent with the expected trajectory.

Figure 5 shows the motion trajectories of each joint of the redundant robot. As can be seen from the figure, the trajectory of each joint is closed after running for one cycle, realizing repetitive motion control.

In order to verify the effectiveness of the parabolic final state neural network method in repetitive motion planning, the joint angle transient trajectory and joint angular velocity transient trajectory obtained when the robot PA10 end effector completes the circular trajectory are shown in Figures 6 and 7. It can be seen that each joint angle of the redundant robot finally converges to the desired joint angle position.

When T=10s, the maximum deviation of the final value error of each joint angle obtained by the asymptotic neural network solution is 6.8740×10 ^-4 rad, and the maximum deviation of the final value error of each joint angle obtained by the parabolic final state neural network solution is 9.1348×10 ^-5 rad, see Table 1 for details:

Table 1 Among them, the dynamic characteristics of asymptotic neural network

Dynamic characteristics of parabolic final state neural network

in,

ρ=1, ε=1, σ=1

To compare the convergence performance of the asymptotic neural network and the parabolic final state neural network, the error norm ||E(t)|| _F = ||W(t)Y(t)-v(t)|| _F is calculated. Figure 9 shows the error norm convergence trajectories for solving quadratic programming problems with asymptotic neural network and parabolic final state neural network, respectively. It can be seen from the figure that when the norm of E(t) converges to 0.01, it takes time T(E _ij (0))=0.861s to solve with the parabolic final state neural network, and it takes time to solve with the asymptotic neural network T(E _ij (0))=6.932s.

Claims (1)

1. a robot repetitive motion planning method employing a parabolic final state neural network, is characterized in that, described method comprises the following steps: 1) The expected trajectory r _d (t) of the robot end effector is given in Cartesian space, and the expected retraction angle θ _d (0) of each joint is given; 2) For the repetitive motion robot, its initial joint angle is θ(0)=θ ₀ , and the initial joint angle θ ₀ is different from the expected joint angle, that is, θ ₀ ≠θ _d (0); 3) The redundant robot repetitive motion planning is described as a quadratic programming problem, and the asymptotic convergence performance index is given to form the repetitive motion planning scheme: in, g(θ)=-ρ ₀ (θ(t)-θ _d (0)) ρ ₀ &gt; 0, θ(t)-θ _d (0) represents the deviation of the joint angle from the desired joint angle. Since the initial position of the robot is not on the desired trajectory, the left side of the relationship between the end effector velocity and the joint velocity needs to be added. The last feedback deviation is r _d -f(θ), which represents the error between the expected trajectory and the actual trajectory; due to the artificially constructed convergence relationship, this error will continue to shrink until it is zero; the parameter β It is used to adjust the rate at which the end effector moves to the desired trajectory, β>0; J(θ) is the Jacobian matrix calculated according to the DH parameters of the robot, and f(θ) is the actual trajectory of the robot end effector; 4) Construct the following dynamic characteristic equation based on parabolic final state neural network Among them, E is the error matrix variable, E _ij is the element of matrix E, ρ>0, σ>0, ε>0, Equation (5) is stable in finite time, and its finite time convergence needs to be explained in two cases: When |E _ij (0)|<σ, the convergence time is When |E _ij (0)|≥σ, the convergence time is When ρ=0, equation (5) becomes the case without acceleration term Equation (8) is also stable in finite time, and its finite time convergence is also described in two cases: When |E _ij (0)|<σ, the convergence time is When |E _ij (0)|≥σ, the convergence time is 5) Define the Lagrangian function where λ is the Lagrange multiplier vector, about and λ to find the partial derivatives respectively, and make them zero, after finishing W(t)Y(t)=v (12) in, I is the 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 (13) According to the dynamic characteristic equation (5) of the parabolic final state neural network, the parabolic final state neural network equation is given Through the neural network calculation process, the angles of each joint of the robot are obtained.