CN116383574A - Humanoid upper limb robot inverse kinematics solving method based on high-order differentiator - Google Patents

Humanoid upper limb robot inverse kinematics solving method based on high-order differentiator Download PDF

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CN116383574A
CN116383574A CN202310290747.4A CN202310290747A CN116383574A CN 116383574 A CN116383574 A CN 116383574A CN 202310290747 A CN202310290747 A CN 202310290747A CN 116383574 A CN116383574 A CN 116383574A
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upper limb
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金弘哲
印鸿
赵杰
鞠枫嘉
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Harbin Institute of Technology
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Abstract

A method for solving inverse kinematics of a humanoid upper limb robot based on a higher-order differentiator, the method comprising: establishing a mathematical model in the form of a time-varying linear equation set; obtaining an error signal and an integral signal of the tail end position of the mechanical arm according to the mathematical model; solving a higher derivative of the error signal according to a higher differentiator, and establishing a dynamic error model; designing a time-varying linear equation set solver based on dynamic errors, and outputting the optimal action of the mechanical arm aiming at the equation set; and establishing a state space model according to the optimal action to obtain an optimal solution of the joint variable. The invention has the advantages of faster convergence speed, higher solving precision, low energy consumption and higher operability when being applied to the inverse kinematics calculation of the mechanical arm.

Description

Humanoid upper limb robot inverse kinematics solving method based on high-order differentiator
Technical Field
The invention belongs to the technical field of computers, and relates to a human-simulated upper limb robot inverse kinematics solving method based on a high-order differentiator.
Background
The most commonly encountered problems in the fields of control theory, engineering and the like of the linear equation set are how to solve the problems, and are key technologies for solving the problems related to practical application such as inverse kinematics, image reconstruction, signal processing and the like of a robot.
Generally, the solving methods of the problem include a direct method and an iterative method. The direct method mainly comprises a Gaussian elimination method, and although the direct method theoretically has absolute accurate solution, once the number of variables is increased, the solution cost of the method is high; the iteration method is effective for a system with more variables, and is usually a Newton iteration method and a gradient neural network, and the method has a good effect when solving the static constant problem, and can ensure that the tracking error is as small as possible and the operation speed is high.
However, the practical application system is a time-varying system essentially, the method can not guarantee the decline of the error function due to the lag error of the time-varying parameter, the traditional method is only suitable for solving the linear steady system, and when the method is applied to solving the time-varying linear equation set, the lag error of the time-varying parameter can cause larger error, so the method is not suitable for solving the time-varying linear equation set problem. In recent years, scholars have proposed a zero-change neural network (ZNN) to solve the time-varying problem, but the model still has the problems of slow convergence speed and insufficient precision in the deep exploration stage at present.
Disclosure of Invention
The invention provides a method for solving inverse kinematics of a humanoid upper limb robot based on a high-order differentiator for overcoming the prior art. The method
The method for solving the inverse kinematics of the humanoid upper limb robot based on the high-order differentiator comprises the following steps:
s1, establishing a mathematical model in the form of a time-varying linear equation set;
s2, obtaining an error signal and an integral signal of the tail end position of the mechanical arm according to a mathematical model;
s3, solving a higher derivative of the error signal according to the higher differentiator, and establishing a dynamic error model;
s4, designing a time-varying linear equation set solver based on dynamic errors, and outputting the optimal action of the mechanical arm aiming at the equation set;
and S5, establishing a state space model according to the optimal action to obtain an optimal solution of the joint variable.
Compared with the prior art, the invention has the beneficial effects that:
the invention can solve the time-varying linear equation system, and can not only be used for solving the linear steady system as the traditional method.
Compared with the emerging return-to-zero neural network method, the method has the advantages of higher convergence speed and higher solving precision.
When the method is applied to inverse kinematics calculation of the humanoid upper limb robot, the consumed energy is lower, the operability is higher, and the obtained solution has the globally optimal characteristic.
The technical scheme of the invention is further described below with reference to the accompanying drawings and examples:
drawings
FIG. 1 is a flow chart of an implementation of a method for solving a time-varying linear system of equations based on a high order differentiator;
FIG. 2 is a design diagram of a time-varying linear system of equations solution based on a high order differentiator established by the method of the present invention;
FIG. 3 is a graph of the tracking position error of the end of the robot arm obtained when solving the inverse kinematics problem of the robot using the method of the present invention in an embodiment;
FIG. 4 is a graph showing the joint angles of a first manipulator obtained when solving the inverse kinematics problem of a robot using the method of the present invention in an embodiment;
FIG. 5 is a graph showing the second arm joint angle obtained when solving the inverse kinematics problem of the robot using the method of the present invention in the example.
Detailed Description
Embodiments of the technical scheme of the present invention will be described in detail below with reference to the accompanying drawings. Unless otherwise defined, technical or scientific terms used herein should be given the ordinary meaning as understood by one of ordinary skill in the art to which this invention belongs.
Referring to fig. 1-2, the method for solving inverse kinematics of a humanoid upper limb robot based on a higher-order differentiator according to the present embodiment includes:
s1, establishing a mathematical model in the form of a time-varying linear equation set;
s2, obtaining an error signal and an integral signal of the tail end position of the mechanical arm according to a mathematical model;
s3, solving a higher derivative of the error signal according to the higher differentiator, and establishing a dynamic error model;
s4, designing a time-varying linear equation set solver based on dynamic errors, and outputting the optimal action of the mechanical arm aiming at the equation set;
and S5, establishing a state space model according to the optimal action to obtain an optimal solution of the joint variable.
The method of the embodiment overcomes the defect that the traditional method is only suitable for solving a linear steady system, and when a time-varying linear equation system is used for solving, a larger error is caused by a hysteresis error of a time-varying parameter; the scheme of the implementation mode solves the problems of low convergence speed and low precision existing in the ZNN method at present.
Further, in the further aspect, the method comprises the steps of,
in step S1, a mathematical model of an actual physical system or a numerical solution system in the form of a time-varying system of linear equations is built, for example: p (t) Q (t) =u (t), where,
Figure BDA0004141351510000021
and->
Figure BDA0004141351510000022
Is based on a known time-varying parameter of the mathematical model,/->
Figure BDA0004141351510000031
Is a time-varying vector to be solved, and obtains a time-varying parameter matrix P (t) and a vector U (t) in a mathematical model through the attribute of the system and a system sensor,
in step S2, an error function equation E (t) =p (t) Q (t) -U (t) is designed, and an error integral signal is obtained
Figure BDA0004141351510000032
In step S3, the time higher derivative of the residual error E (t) is solved according to the higher-order differentiator, wherein E (t) =p (t) Q (t) -U (t), and a new multi-objective optimization model-dynamic error is built according to the error, the integral signal of the error and the higher derivative information of the error;
Figure BDA0004141351510000033
wherein, c 0 ,c 1 ,…,c L Is a parameter to be designed.
In the step, the multi-objective optimization model considers the integral information of errors, which is beneficial to improving the precision of solving the time-varying linear equation set; and the differential information of each order of errors is considered, so that the stability of a solving model and the global optimal searching energy are improved, and the establishment of the multi-objective optimization model is not realized in other methods at present.
Step S4 is to design a time-varying linear equation set solver based on a high-order differentiator based on the dynamic error value obtained by the differentiator, and output the optimal action as:
Figure BDA0004141351510000034
wherein, alpha and beta are to be designed into solver parameters, beta epsilon [ delta ] max ,∞),α∈(β,∞),
Figure BDA0004141351510000035
Sigma (&) is a class of satisfaction + & gt>
Figure BDA0004141351510000036
Sigmoid-type function of property, +.>
Figure BDA0004141351510000037
Is a linear differential equation and epsilon (t) is an decay function with 0 as an asymptote.
Step S5 is designing an optimal action Q IN Then using a state space model defined as follows;
Figure BDA0004141351510000038
and obtaining the joint variable optimal solution with high convergence speed and high precision.
The invention is further elucidated in the form of examples on the basis of the above inventive concept:
examples
According to a time-varying linear equation system solving method based on a high-order differentiator, the method is applied to the inverse kinematics solving of the humanoid upper limb robot, and each joint variable of the two arms is solved.
In the embodiment, a humanoid upper limb robot with two 8-degree-of-freedom mechanical arms and a 3-degree-of-freedom waist is selected, and a time-varying linear equation set solver based on a high-order differentiator is applied to inverse kinematics solution of the humanoid upper limb robot;
firstly, acquiring a positive kinematics equation of the humanoid upper limb robot:
Y k =F kk ,Θ B )
in the method, in the process of the invention,
Figure BDA0004141351510000041
is the angle vector of the joint with dimension 3 of the waist, < >>
Figure BDA0004141351510000042
Is the joint angle vector with the dimension of 8 of the two arms, the subscript k is used for distinguishing the left arm from the right arm,/>
Figure BDA0004141351510000043
Is the end position vector of the two arms with the dimension of 3. Because the kinematic formula of the position layer is strong in nonlinearity, the two sides of the equation are simultaneously derived from time to time, and the kinematic formula of the speed layer can be obtained as follows:
Figure BDA0004141351510000044
in the method, in the process of the invention,
Figure BDA0004141351510000045
and->
Figure BDA0004141351510000046
The jacobian matrix representing the humanoid upper limb robot is a time variable which changes with the motion state of the random robot, and is +.>
Figure BDA0004141351510000047
Is the joint angular velocity vector of two 8-degree-of-freedom mechanical arms,>
Figure BDA0004141351510000048
is a joint angular velocity vector of the waist with 3 degrees of freedom.
The inverse kinematics of the robot is solved to obtain the expected track of the tail end of the given mechanical arm, and then corresponding angle values and angular velocity values of all joints are obtained according to the mapping relation of the kinematics.
Definition symbol
Figure BDA0004141351510000049
Is the desired end position trajectory, which sums Θ B All are smooth known time-varying functions, so the inverse kinematics solution of the humanoid upper limb robot can be equivalent to the solution of the following formula and the following formula is converged to 0;
Figure BDA00041413515100000410
let P k =J k
Figure BDA00041413515100000411
It can be seen that the velocity layer inverse kinematics solution equation is consistent with the error function equation obtained in step S2 of the above embodiment. On the basis, an integral signal of the error, namely the position tracking error of the tail end of the double mechanical arm, can be calculated, as shown in fig. 3:
Figure BDA00041413515100000412
because the positive kinematic model of the upper limb of the humanoid robot and the expected tail end position track are known, the tail end position tracking error of the mechanical arm can be obtained, and then the high-order differential information such as the speed tracking error, the acceleration tracking error, the jerk tracking error and the like of the tail end of the mechanical arm can be obtained according to the high-order differentiator. In order to improve the tracking precision of the mechanical arm and the stability of an inverse kinematics solving model, reduce the energy consumption of the mechanical arm, a new multi-objective optimizing model (dynamic error) is established to restrict the position error of the tail end of the mechanical arm and the differential information of each order, and the position error and the differential information are called dynamic error. The expression is:
Figure BDA00041413515100000413
wherein, c 0 ,c 1 ,…,c L Gain coefficient of double-arm position tracking error and differential information of each order, c for ensuring tracking accuracy 0 ,c 1 Setting larger parameter values, and simultaneously reducing energy consumed by the mechanical arm, namely restraining acceleration tracking error and jerk error, in order to ensure the stability of the model, c 2 ,…,c L Setting smaller parameter values.
According to the dynamic error value, the optimal action of the mechanical arm is designed as:
Figure BDA0004141351510000051
wherein alpha and beta are solver parameters to be designed, and beta is epsilon [ delta ] max ,∞),α∈(β,∞),
Figure BDA0004141351510000052
Sigma (&) is a class of satisfaction + & gt>
Figure BDA0004141351510000053
Sigmoid-type function of property, ε (t) is an decay function with 0 as the asymptote, ++>
Figure BDA0004141351510000054
The method is characterized in that the method is to solve the joint angle, angular velocity, acceleration and other linear combination values of the mechanical arm by using a solving mode of a linear differential equation set, namely, a solution of a time-varying linear equation set with high convergence speed and high precision is obtained by using a state space equation defined as follows:
Figure BDA0004141351510000055
wherein X is 1 =Q (0) ,X 2 =Q (1) ,…,X L =Q (L-1) The obtained information such as the joint angle vector, the joint velocity vector, the joint acceleration vector and the like is respectively represented, and the energy consumed by the mechanical arm is smaller through constraint acceleration, jerk and the like.
It can be seen that it is practical to solve the inverse kinematics problem of the humanoid upper limb robot based on the time-varying linear equation system solving method of the high-order differentiator, and the simulation results are shown in fig. 3 and 4.
The present invention has been described in terms of preferred embodiments, but is not limited to the invention, and any equivalent embodiments can be made by those skilled in the art without departing from the scope of the invention, as long as the equivalent embodiments are possible using the above-described structures and technical matters.

Claims (10)

1. The inverse kinematics solving method of the humanoid upper limb robot based on the high-order differentiator is characterized by comprising the following steps of: comprising:
s1, establishing a mathematical model in the form of a time-varying linear equation set;
s2, obtaining an error signal and an integral signal of the tail end position of the mechanical arm according to a mathematical model;
s3, solving a higher derivative of the error signal according to the higher differentiator, and establishing a dynamic error model;
s4, designing a time-varying linear equation set solver based on dynamic errors, and outputting the optimal action of the mechanical arm aiming at the equation set;
and S5, establishing a state space model according to the optimal action to obtain an optimal solution of the joint variable.
2. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 1, which is characterized by comprising the following steps of: in step S1 a mathematical model of the actual physical system in the form of a time-varying system of linear equations is built.
3. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 1, which is characterized by comprising the following steps of: in step S1, a mathematical model in the form of a time-varying linear equation set is established, and a time-varying parameter matrix P (t) and a vector U (t) in the mathematical model are obtained as follows:
P(t)Q(t)=U(t)
in the method, in the process of the invention,
Figure FDA0004141351490000011
and->
Figure FDA0004141351490000012
Is based on a known time-varying parameter of the mathematical model,/->
Figure FDA0004141351490000013
Is the time-varying vector to be solved.
4. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 1, which is characterized by comprising the following steps of: the error function equation designed in the step S2 is E (t) =P (t) Q (t) -U (t), and an integral signal of the error is obtained
Figure FDA0004141351490000014
5. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 1, which is characterized by comprising the following steps of: in step S3, the time higher derivative of the residual error E (t) is solved according to the higher-order differentiator, where E (t) =p (t) Q (t) -U (t), and a new dynamic error model is built according to the error, the integral signal of the error, and the higher derivative information of the error;
Figure FDA0004141351490000015
wherein, c 0 ,c 1 ,…,c L Is a parameter to be designed.
6. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 5, which is characterized by comprising the following steps of: step S4, designing a time-varying linear equation set solver based on a higher-order differentiator according to the dynamic error value obtained based on the differentiator in step S3, and outputting the optimal motion of the mechanical arm as:
Figure FDA0004141351490000016
wherein, alpha and beta are to be designed into solver parameters, beta epsilon [ delta ] max ,∞),α∈(β,∞),
Figure FDA0004141351490000021
Sigma (&) is a class of satisfaction + & gt>
Figure FDA0004141351490000022
Sigmoid-type function of property, +.>
Figure FDA0004141351490000023
Is a linear differential equation and epsilon (t) is an decay function with 0 as an asymptote.
7. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 6, which is characterized by comprising the following steps of: step S5 according to the optimal action Q IN Using a state space model defined as follows;
Figure FDA0004141351490000024
and obtaining the robot joint variable.
8. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 1, which is characterized by comprising the following steps of: the humanoid upper limb robot is provided with two 8-degree-of-freedom mechanical arms and a 3-degree-of-freedom waist.
9. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 8, which is characterized by comprising the following steps of: step S1, obtaining a kinematic formula of a speed layer as follows
Figure FDA0004141351490000025
In (1) the->
Figure FDA0004141351490000026
And->
Figure FDA0004141351490000027
Jacobian matrix representing humanoid upper limb robot,>
Figure FDA0004141351490000028
is the joint angular velocity vector of the 8-degree-of-freedom mechanical arm,>
Figure FDA0004141351490000029
the joint angular velocity vector of the waist with 3 degrees of freedom is used for distinguishing the left arm from the right arm, and the values of k=1 and k=2 are respectively given in the following table k.
10. The method for solving the inverse kinematics of the humanoid upper limb robot based on the higher-order differentiator according to claim 1, which is characterized by comprising the following steps of: a mathematical model in the form of a time-varying system of linear equations is built based on the desired target state input by the external environment and the external sensor data.
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