CN111975779A - Random number-based calculation method for minimum kinetic parameter set of multi-joint mechanical arm - Google Patents

Random number-based calculation method for minimum kinetic parameter set of multi-joint mechanical arm Download PDF

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CN111975779A
CN111975779A CN202010854346.3A CN202010854346A CN111975779A CN 111975779 A CN111975779 A CN 111975779A CN 202010854346 A CN202010854346 A CN 202010854346A CN 111975779 A CN111975779 A CN 111975779A
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mechanical arm
matrix
random
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卢剑伟
陈新法
朱汉子
任远凯
杨凡
韩建辉
陈佳枫
钱钧
曹剑
董方方
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Hefei University of Technology
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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
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
    • 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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  • Robotics (AREA)
  • Mechanical Engineering (AREA)
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Abstract

The invention discloses a method for calculating a minimum kinetic parameter set of a multi-joint mechanical arm based on random numbers, which comprises the following steps: 1. simulating joint motion parameters of the mechanical arm by using random numbers, 2, designing a transformation matrix to perform column transformation on a regression matrix of the kinetic parameters, 3, analyzing the independence and the correlation of each column of the regression matrix, and 4, calculating and solving a minimum kinetic parameter set of the multi-joint mechanical arm. The method can determine the minimum dynamic parameter set of the multi-joint mechanical arm in a short time, and the acquired minimum dynamic parameter set is accurate and effective and has better identification, so that a foundation is laid for further improving the dynamic performance of the multi-joint mechanical arm.

Description

Random number-based calculation method for minimum kinetic parameter set of multi-joint mechanical arm
Technical Field
The invention relates to a method for calculating a minimum kinetic parameter set of a multi-joint mechanical arm based on random numbers.
Background
With the increase of the application field of the mechanical arm, high speed and high precision are important technical performances pursued by the mechanical arm, and the key for researching and developing the high-performance mechanical arm is to comprehensively master the dynamic performance and accurately control the dynamic performance. The identification of the mechanical arm dynamics parameters is an important prerequisite for mastering and controlling the dynamics performance of the mechanical arm. However, the multi-joint mechanical arm dynamic model is complex, and not all parameters can be identified. Therefore, the minimum kinetic parameter set is screened and determined from the standard kinetic parameters, so that the complexity of kinetic parameter identification can be reduced, and the rapidity and the robustness of mechanical arm control based on a kinetic model can be improved.
External researchers began earlier to focus on the problem of clustering the minimum kinetic parameters of multi-joint robotic arms. Among the many methods of determining the minimum set of kinetic parameters, the common idea is: the dynamic parameters are fused into a mechanical arm dynamic model to construct a new formula, and a required dynamic parameter combination is solved by using a recursive closed-loop relation method; and summarizing corresponding dynamic parameter combination rules by analyzing the mechanical arm dynamic parameter regression matrix, and determining the minimum dynamic parameter set according to the combination rules according to the motion relation among the mechanical arm joints. However, the above method for obtaining the minimum kinetic parameter set has disadvantages, which are shown in the following: on one hand, the regression matrix of the kinetic parameters needs to be analyzed for a long time to determine the combination relationship of the kinetic parameters; on the other hand, the process of determining the minimum parameter set according to the combination rule in the actual application process is complicated.
Disclosure of Invention
The invention aims to solve the defects in the prior art, and provides a random number-based calculation method for the minimum kinetic parameter set of the multi-joint mechanical arm, so that the minimum kinetic parameter set of the multi-joint mechanical arm can be determined in a short time, the obtained minimum kinetic parameter set is accurate and effective, and the method has good identification performance, thereby laying a foundation for further improving the kinetic performance of the multi-joint mechanical arm.
The technical scheme adopted by the invention for solving the technical problems is as follows:
the invention relates to a method for calculating a minimum kinetic parameter set of a multi-joint mechanical arm based on random numbers, which is characterized by comprising the following steps of:
step 1, establishing a dynamic model of a multi-joint mechanical arm by using a Newton-Euler method, and carrying out linearization processing on a nonlinear item in the dynamic model to obtain a linearized dynamic model of the multi-joint mechanical arm;
step 2, solving a column transformation rotation matrix;
step 2.1, setting generalized observation matrix functions of n acquisition points according to the linearized dynamic model of the multi-joint mechanical arm;
step 2.2, generating a group of random numbers for simulating the motion parameters of the mechanical arm joint, thereby establishing a random generalized observation matrix psirandomAnd the random generalized observation matrix is a non-column full rank matrix;
step 2.3, carrying out generalized observation on the random observation matrix psirandomPerforming QR decomposition to obtain a first Q matrix and a first R matrix, and constructing a column transformation original vector theta by using a formula (1):
Figure BDA0002645853110000021
in the formula (1), the reaction mixture is,
Figure BDA0002645853110000022
for column positions where the diagonal value of the first R matrix is zero,
Figure BDA0002645853110000023
column positions for which the first matrix diagonal values are non-zero;
step 2.4, defining a column transformation unit vector set [ E1,E2,…,Ei,…,Em]；EiA column transformation unit vector in which the ith element is '1' and the other elements are '0';
step 2.5, solving a column transformation rotation matrix p of the random generalized matrix by using the formula (2):
p＝[EΘ1 EΘ2 ... EΘi ... EΘm] (2)
in equation (2), Θ i is the ith element of the column-transformed original vector Θ, EΘiA column transformation unit vector in which the theta i element is '1' and the other elements are '0';
and 3, solving the minimum kinetic parameter set of the mechanical arm according to the column transformation rotation matrix.
The method for calculating the minimum kinetic parameter set of the multi-joint mechanical arm is also characterized in that the step 3 is carried out according to the following process:
step 3.1, utilizing the formula (3) to carry out random generalized observation on the matrix psirandomAnd (3) performing column transformation:
ψrandomp＝[ψ1,ψ2] (3)
in formula (3), phi1For k independent column vectors, psi2M-k column vectors to be deleted and combined;
step 3.2, solving the basic parameter phi of the minimum dynamic parameter set of the multi-joint mechanical arm by using the formula (4)1Parameter phi linearly combinable with a base parameter2：
Figure BDA0002645853110000024
In the formula (4), phi is the original dynamic parameter set of the multi-joint mechanical arm;
step 3.3, for psirandomCarrying out QR decomposition on p to obtain a second Q matrix and a second R matrix [ R ]k Rm-k]；
Step 3.4, solving the minimum parameter set phi of the multi-joint mechanical arm dynamics by using the formula (5)min：
φmin＝φ1+βφ2 (5)
In the formula (5), beta is a basic parameter phi of the multi-joint mechanical arm dynamics1With a linearly combinable parameter phi2And has the following combination coefficients:
β＝Rk -1Rm-k (6)。
compared with the prior art, the invention has the beneficial effects that:
1. the method utilizes the generated random numbers to simulate the motion parameters of the multi-joint mechanical arm to form a random generalized observation matrix, designs a column transformation rotation matrix of the random generalized observation matrix to carry out linear transformation on the original kinetic parameter set of the multi-joint mechanical arm, thereby determining the minimum kinetic parameter set of the multi-joint mechanical arm and solving the problems of long time consumption and complex process for determining the kinetic parameter combination relation of the multi-joint mechanical arm.
2. The minimum kinetic parameter set of the multi-joint mechanical arm obtained by calculation is accurate and effective, and can be well applied to the identification of the kinetic parameters of the multi-joint mechanical arm.
Drawings
FIG. 1 is a diagram of the relationship between a six-joint robot arm and a connecting rod in the invention and an MDH coordinate system;
FIG. 2 is a comparison of the moment before and after model simplification in accordance with the present invention;
FIG. 3a is a comparison graph of simulated values and calculated identification values of the joint moment of the joint 1;
FIG. 3b is a comparison graph of simulated values and calculated identification values of joint torque of the joint 2;
FIG. 3c is a comparison graph of simulated values and calculated identification values of the joint torque of the joint 3;
FIG. 4 is a difference diagram of simulated values and identification calculations of joint moments.
Detailed Description
In this embodiment, a method for calculating a minimum kinetic parameter set of a multi-joint mechanical arm based on a random number is performed as follows:
step 1, establishing a dynamic model of the multi-joint mechanical arm by using a Newton-Euler method, wherein the dynamic model is shown as a formula (1):
Figure BDA0002645853110000031
in the formula (1), τ represents the joint moments of the robot arm, q,
Figure BDA0002645853110000032
Respectively represent the joint angle position, joint angular velocity and joint angular acceleration of the mechanical arm, M represents an inertia matrix, C represents coriolis force and centrifugal force, and G represents gravity.
Nonlinear terms in the dynamic model are linearized, and the linearized dynamic model of the multi-joint mechanical arm is obtained as shown in the formula (2):
Figure BDA0002645853110000033
in the formula (2), W is an observation matrix related to the joint motion parameters of the mechanical arm, and phi is a standard kinetic parameter set of the mechanical arm.
Step 2, solving a column transformation rotation matrix;
step 2.1, setting generalized observation matrix functions of n acquisition points according to a linearized dynamic model of the multi-joint mechanical arm as shown in a formula (3):
Figure BDA0002645853110000041
in the formula (3), psi is a generalized observation matrix of n acquisition points, z is the number of the kinematic joints, and m is the number of the standard kinetic parameters.
The smallest linearly independent group of psi corresponds to the base parameter space of the standard kinetic parameter set phi. The minimum number of linearly independent columns is also the spatial dimension k of ψ, while the m-k columns are required to be deleted and linearly integrated. The spatial dimension k of ψ corresponds to the number of basis parameters of the standard kinetic parameter set φ, and this k column contributes most. The screened m-k columns correspond to parameters with contribution degree of almost zero and parameters with contribution degree of correlation in the standard parameter set phi, wherein the parameters with contribution degree of zero need to be deleted, and the parameters with contribution degree of correlation need to be linearly combined based on the base parameters.
Step 2.2, generating a group of random numbers for simulating the motion parameters of the mechanical arm joint, thereby establishing a random generalized observation matrix psirandomAnd the random generalized observation matrix is a non-column full rank matrix;
step 2.3, carrying out random generalized observation on the matrix psirandomPerforming QR decomposition to obtain a first Q matrix and a first R matrix as shown in a formula (4):
Figure BDA0002645853110000042
by aligning psirandomAnd decomposing and solving the contribution degree of the kinetic parameters. However, the regression matrix psi due to the dynamic parameters of the multi-joint manipulatorrandomIs a non-column full rank matrix with no uniqueness to its standard QR decomposition. But QR decomposition by Householder can solve ψrandomIs not the only problem.
The original vector Θ is transformed using the construction column of equation (5):
Figure BDA0002645853110000043
in the formula (5), the reaction mixture is,
Figure BDA0002645853110000044
for column positions where the diagonal value of the first R matrix is zero,
Figure BDA0002645853110000045
column positions for which the diagonal value of the first R matrix is non-zero;
step 2.4, defining a column transformation unit vector set [ E1,E2,…,Ei,…,Em]；EiA column transformation unit vector in which the ith element is '1' and the other elements are '0';
step 2.5, solving a column transformation rotation matrix p of the random generalized matrix by using the formula (6):
p＝[EΘ1 EΘ2 ... EΘi ... EΘm] (6)
in equation (6), Θ i is the ith element of the column-transformed original vector Θ, EΘiA column transformation unit vector in which the theta i element is '1' and the other elements are '0';
step 3, solving the minimum parameter set phi of the multi-joint mechanical arm dynamicsmin；
Step 3.1, utilizing the formula (7) to carry out random generalized observation on the matrix psirandomAnd (3) performing column transformation:
ψrandomp＝[ψ1,ψ2] (7)
in formula (7), phi1For k independent column vectors, psi2M-k column vectors to be deleted and combined;
step 3.2, solving the basic parameter phi of the minimum dynamic parameter set of the multi-joint mechanical arm by using the formula (8)1Parameter phi linearly combinable with a base parameter2：
Figure BDA0002645853110000051
Step 3.3, to ψ using the formula (9)randomCarrying out QR decomposition on p to obtain a second Q matrix [ Q ]1,Q2]And a second R matrix [ R ]kRm-k]；
Figure BDA0002645853110000052
In the formula (9), RkFor k independent column vectors, R, in a second R matrixm-kAre m-k column vectors in the second R matrix.
Step 3.4, solving the minimum parameter set phi of the multi-joint mechanical arm dynamics by using the formula (10)min：
φmin＝φ1+βφ2 (10)
In the formula (10), beta is a basic parameter phi of multi-joint mechanical arm dynamics1With a linearly combinable parameter phi2And has the following combination coefficients:
β＝Rk -1Rm-k (11)
example 1:
the method for calculating the minimum kinetic parameter set of the multi-joint mechanical arm based on the random number comprises the following steps:
step 1, establishing a dynamic model of the six-joint mechanical arm, wherein FIG. 1 is a diagram of an MDH (minimization drive h) of the six-joint mechanical arm and a relation diagram of a connecting rod and an MDH coordinate system, and the dynamic model is subjected to linearization processing. The robot MDH parameter values are shown in table 1:
table 1: robot MDH parameter value
Figure BDA0002645853110000061
And 2, generating a group of random number simulation mechanical arm joint motion parameters through a computer, establishing a random generalized observation matrix, carrying out QR decomposition on the random generalized observation matrix, constructing a column transformation original vector, and solving a column transformation rotation matrix.
And 3, solving the minimum parameter set of the mechanical arm dynamics. The symbolic combination solution of the minimum kinetic parameter set of the mechanical arm obtained by the method is as follows:
Lzz1+Lyy2+Lyy3+0.487lz2+0.3002lz3
+0.5929m2+0.209(m3+m4+m5+m6),
Lxx2-Lyy2-0.186(m3-m4-m5-m6),
Lxy2,
Lxz2-0.4318lz3+0.0403(m3+m4+m5+m6),
Lyz2,
Lzz2+0.186(m3+m4+m5+m6),
lx2+0.4318(m3+m4+m5+m6),
ly2,
Lxx3-Lyy3+Lyy4+0.8662lz4+0.1871(m4+m5+m6),
Lxy3-0.0203l4z-0.0088(m4-m5-m6),
Lxz3,Lyz3,
Lzz3+Lyy4+0.8663lz4+0.188(m4+m5+m6),
lx3-0.0203(m4-m5-m6),
ly3-lz4-0.4331(m4-m5-m6),
Lxx4-Lyy4+Lyy5,
Lxy4,Lxz4,Lyz4,
Lzz4+Lyy5,lx4
ly4+lz5,
Lxx5-Lyy5+Lyy6,Lxy5,
Lyz5,Lzz5+Lyy6,lx5,
ly5-lz6,
Lxx6-Lyy6,Lxy6,Lxz6,
Lyz6,Lzz6,lx6,ly6
in order to verify the accuracy and the effectiveness of the minimum parameter set calculated by the method on dynamics, a mechanical arm simulation model is established through Adams, an identification excitation track is obtained by adopting finite term Fourier series, and the torque values of all joints of the mechanical arm before and after simplification are compared, as shown in figure 2. The minimum dynamic parameter set of the front 3 joints of the six-joint mechanical arm is identified by using a least square method, joint moments of the front three joints are calculated, the simulation moment and identification moment error values of the 3 joints are shown in fig. 4, and the simulation moment and identification moment error values of the front 3 joints under the excitation track are shown in fig. 3a, fig. 3b and fig. 3 c. The result shows that the minimum kinetic parameter set calculation method of the invention completely reflects the kinetic characteristics of the original mechanical arm; meanwhile, through the analysis of the identification result, the minimum kinetic parameter set calculated by the method has better identification.

Claims (2)

1. A method for calculating the minimum kinetic parameter set of a multi-joint mechanical arm based on random numbers is characterized by comprising the following steps:
step 1, establishing a dynamic model of a multi-joint mechanical arm by using a Newton-Euler method, and carrying out linearization processing on a nonlinear item in the dynamic model to obtain a linearized dynamic model of the multi-joint mechanical arm;
step 2, solving a column transformation rotation matrix;
step 2.1, setting generalized observation matrix functions of n acquisition points according to the linearized dynamic model of the multi-joint mechanical arm;
step 2.2, generating a group of random numbers for simulating the motion parameters of the mechanical arm joint, thereby establishing a random generalized observation matrix psirandomAnd the random generalized observation matrix is a non-column full rank matrix;
step 2.3, carrying out generalized observation on the random observation matrix psirandomPerforming QR decomposition to obtain a first Q matrix and a first R matrix, and constructing a column transformation original vector theta by using a formula (1):
Figure FDA0002645853100000011
in the formula (1), the reaction mixture is,
Figure FDA0002645853100000012
for column positions where the diagonal value of the first R matrix is zero,
Figure FDA0002645853100000013
column positions for which the first matrix diagonal values are non-zero;
step 2.4, defining a column transformation unit vector set [ E1,E2,…,Ei,…,Em]；EiA column transformation unit vector in which the ith element is '1' and the other elements are '0';
step 2.5, solving a column transformation rotation matrix p of the random generalized matrix by using the formula (2):
p＝[EΘ1 EΘ2 ... EΘi ... EΘm] (2)
in equation (2), Θ i is the ith element of the column-transformed original vector Θ, EΘiA column transformation unit vector in which the theta i element is '1' and the other elements are '0';
and 3, solving the minimum kinetic parameter set of the mechanical arm according to the column transformation rotation matrix.
2. The method for calculating the minimum kinetic parameter set of the multi-joint mechanical arm according to claim 1, wherein the step 3 is performed as follows:
step 3.1, utilizing the formula (3) to carry out random generalized observation on the matrix psirandomAnd (3) performing column transformation:
ψrandomp＝[ψ1,ψ2] (3)
in formula (3), phi1For k independent column vectors, psi2M-k column vectors to be deleted and combined;
step 3.2, solving the basic parameter phi of the minimum dynamic parameter set of the multi-joint mechanical arm by using the formula (4)1Parameter phi linearly combinable with a base parameter2：
Figure FDA0002645853100000021
In the formula (4), phi is the original dynamic parameter set of the multi-joint mechanical arm;
step 3.3, for psirandomCarrying out QR decomposition on p to obtain a second Q matrix and a second R matrix [ R ]k Rm-k]；
Step 3.4, solving the minimum parameter set phi of the multi-joint mechanical arm dynamics by using the formula (5)min：
φmin＝φ1+βφ2 (5)
In the formula (5), beta is a basic parameter phi of the multi-joint mechanical arm dynamics1With a linearly combinable parameter phi2And has the following combination coefficients:
β＝Rk -1Rm-k (6)。
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CN110531707A (en) * 2019-09-16 2019-12-03 无锡信捷电气股份有限公司 The friction model of SCARA robot improves and dynamic parameters identification method
CN111267105A (en) * 2020-03-18 2020-06-12 无锡砺成智能装备有限公司 Kinetic parameter identification and collision detection method for six-joint robot
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