CN112454349A - Mechanical arm control transformation method considering variable stiffness joint delay characteristics - Google Patents

Mechanical arm control transformation method considering variable stiffness joint delay characteristics Download PDF

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CN112454349A
CN112454349A CN202010574622.0A CN202010574622A CN112454349A CN 112454349 A CN112454349 A CN 112454349A CN 202010574622 A CN202010574622 A CN 202010574622A CN 112454349 A CN112454349 A CN 112454349A
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mechanical arm
rigidity
joint
control law
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CN112454349B (en
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胡勇
魏春岭
刘磊
张海博
徐拴锋
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Beijing Institute of Control Engineering
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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
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1646—Programme controls characterised by the control loop variable structure system, sliding mode control
    • 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
    • B25J9/1607—Calculation of inertia, jacobian matrixes and inverses
    • 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
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control

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  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Physics & Mathematics (AREA)
  • Mathematical Physics (AREA)
  • Automation & Control Theory (AREA)
  • Feedback Control In General (AREA)

Abstract

The invention relates to a mechanical arm control conversion method considering variable stiffness joint delay characteristics, which is realized by the following steps: (1) considering the time delay characteristic of a variable-stiffness joint installed on the mechanical arm, and establishing an n-degree-of-freedom mechanical arm dynamic model; (2) reducing the time delay in the kinetic model to the original
Figure DDA0002550879790000011
Establishing a new dynamic model; m is more than 1; (3) designing a small-delay control law for stabilizing the closed loop of the new dynamic model; (4) and (3) on the basis of the small-delay control law, considering the scaling relation of the two dynamic models in the steps (1) and (2), carrying out scaling transformation, designing a control law which enables the closed loop in the step (1) to be stable, and realizing the control of the mechanical arm by using the control law.

Description

Mechanical arm control transformation method considering variable stiffness joint delay characteristics
Technical Field
The invention belongs to the field of mechanical arm control, and relates to a mechanical arm control transformation method considering the large delay characteristic of a variable stiffness joint.
Background
In an unstructured environment, the application of the variable-stiffness joint to an industrial robot or a household robot is more and more extensive, so that the problem of environment interaction completeness is solved, and in the prior art, some robots or people adopting the variable-stiffness joint directly ignore the variable-stiffness delay characteristic, so that the closed-loop stability of an actual system cannot be ensured; or the complex variable stiffness dynamics and the robot dynamics are coupled and designed to be controlled, so that the design difficulty of the controller is greatly increased, and the direct popularization and application of the existing control method are limited.
In terms of control input of the mechanical arm, besides joint control of the mechanical arm, rigidity is additionally added as input, and online adjustment of the rigidity enables the application of the robot to be more flexible. Because the dynamics of the robot is very complex, when the rigidity is taken as a control quantity, if the dynamic characteristic of variable rigidity is taken into consideration, the design difficulty of the controller is greatly increased, and the dynamic characteristic is generally neglected for direct design. However, in the variable-stiffness joint, the compression of the spring is generally changed or the stiffness is adjusted by changing the size of the moment arm by a physical method, so that the response speed of stiffness change is delayed greatly, which brings a large influence on the stability of control.
Disclosure of Invention
The technical problem solved by the invention is as follows: the defects of the prior art are overcome, and the mechanical arm control conversion method considering the variable stiffness joint delay characteristic is provided.
The technical scheme of the invention is as follows: a mechanical arm control conversion method considering variable stiffness joint delay characteristics is realized by the following steps:
(1) considering the time delay characteristic of a variable-stiffness joint installed on the mechanical arm, and establishing an n-degree-of-freedom mechanical arm dynamic model;
(2) reducing the time delay in the kinetic model to the original
Figure BDA0002550879770000011
Establishing a new dynamic model; m is more than 1;
(3) designing a small-delay control law for stabilizing the closed loop of the new dynamic model;
(4) and (3) on the basis of the small-delay control law, considering the scaling relation of the two dynamic models in the steps (1) and (2), carrying out scaling transformation, designing a control law which enables the closed loop in the step (1) to be stable, and realizing the control of the mechanical arm by using the control law.
Preferably, before the step (2), analyzing the time delay characteristic of the mechanical arm, and if the time delay is greater than the control period of the mechanical arm, starting to execute the step (2); otherwise, directly designing a small-delay control law which enables the dynamic model in the step (1) to be closed-loop and stable, and controlling the mechanical arm by using the control law.
Preferably, the small delay control law designed in the step (3) is assumed to be:
Figure BDA0002550879770000021
Figure BDA0002550879770000022
the control law designed in the step (4) is as follows:
Figure BDA0002550879770000023
Figure BDA0002550879770000024
in the formula, u is an n-dimensional torque vector and corresponds to the joint torque of the mechanical arm with n degrees of freedom, k is a l-dimensional rigidity vector and corresponds to the rigidity input of l rigidity-adjustable joints of the mechanical arm with n degrees of freedom, and u is0() And k0() Control functions, q and q, respectively, referring to the existing moment and stiffness
Figure BDA0002550879770000025
A joint angle vector and a joint angular velocity vector corresponding to the mechanical arm with n degrees of freedom, G (q) is a gravity gradient vector in a Lagrange equation, and δ q is a deformation angle of l rigidity-adjustable joints; t is time, and B is an n multiplied by n matrix, which describes the effect of the joint moment on the generalized coordinates.
Preferably, the kinetic model in step (1) is as follows:
Figure BDA0002550879770000026
in the formula, q is a generalized coordinate vector of a structural space, and the dimension is n; d (q) is an inertia matrix;
Figure BDA0002550879770000027
the method is characterized in that the method is a Coriolis matrix, G (q) is a gradient vector of a gravity potential energy field, B is an n multiplied by n matrix and describes the effect of joint torque on a generalized coordinate, E (delta q) is an n multiplied by l matrix, the joint deformation effect of a joint with variable rigidity is mapped on the generalized coordinate, l is the number of joints with adjustable rigidity and meets the condition that l is less than or equal to n, the ith row of the E (delta q) is all zero if the rigidity of the ith joint of an n-degree-of-freedom mechanical arm is not adjustable, the ith row of the E (delta q) is assumed to be adjustable, the ith joint rigidity of the n-degree-of-freedom mechanical arm is adjustable, and the jth row and the j column of the E (delta q) are delta qjThe remaining elements in row i are 0; u (T) represents n-dimensional moment vector, k (T) is the rigidity input of the joints with adjustable rigidity, and T is [ T ═ T1,T2,…Tl]TThe rigidity adjusting time delay respectively corresponding to the one joint with adjustable rigidity is k (T-T) ═ k1(t-T1) k2(t-T2) … kl(t-Tl)]T。
Preferably, the new kinetic model in step (2) is as follows:
Figure BDA0002550879770000031
preferably, m is determined according to the actual time delay T ═ T1,T2,…Tl]TSelecting, wherein generally, it needs to be ensured that the time delay of the established new dynamic model is less than the time length delta T of one control period of the planning controller of the upper computer of the mechanical arm, wherein l is the number of the adjustable rigidity joints in the mechanical arm.
The value range of m is preferred
Figure BDA0002550879770000032
Wherein, TmaxIs time delay T ═ T1,T2,…Tl]TOf (d) is the maximum fraction of (d).
Compared with the prior art, the invention has the beneficial effects that:
the variable stiffness joint adopted by the mechanical arm generally changes the tension force of a spring or changes the magnitude of a force arm of a supporting point by a physical method to adjust the stiffness, the response speed of the stiffness change is delayed greatly, if the dynamic characteristic of the variable stiffness is taken into consideration, the design difficulty of a controller is greatly increased, and the dynamic characteristic of the existing control law design is generally neglected and is directly designed. However, in practical applications, the larger delay may make the system unstable. In order to guarantee stability and operation precision when the mechanical arm performs relevant operation, the time scaling method is provided, dynamic characteristics of the variable-rigidity joint are taken into consideration, a control law directly designed by neglecting the dynamic characteristics can be converted into practical control through corresponding control transformation, and stability and steady-state precision of the mechanical arm with the variable-rigidity joint in the operation process are guaranteed by sacrificing certain dynamic performance.
The invention provides a control transformation method considering variable stiffness regulation delay, which can directly popularize and apply some existing control laws to the condition of considering variable stiffness large delay characteristics and can ensure the stability and steady-state precision of the mechanical arm in the operation process.
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FIG. 1 is a flow chart of the present invention.
Detailed Description
The invention is further illustrated by the following examples.
The basic principle of the invention is as follows: by adopting a time scaling method, the dynamic characteristics of the variable-stiffness joint are considered in the control design of the mechanical arm, and the stability and the steady-state precision of the system can be ensured under the condition of sacrificing a certain response speed.
As shown in fig. 1, a mechanical arm control transformation method considering variable stiffness joint delay characteristics is realized by the following steps:
(1) considering the time delay characteristic of a variable-stiffness joint installed on the mechanical arm, and establishing an n-degree-of-freedom mechanical arm dynamic model;
(2) reducing the time delay in the kinetic model to the original
Figure BDA0002550879770000041
Establishing a new dynamic model; m is more than 1;
(3) designing a small-delay control law for stabilizing the closed loop of the new dynamic model:
Figure BDA0002550879770000042
(4) based on the small delay control law, considering the scaling relation of the two dynamic models in the steps (1) and (2), carrying out scaling transformation, and designing a control law which enables the closed loop in the step (1) to be stable:
Figure BDA0002550879770000043
the control law is utilized to realize the control of the mechanical arm.
Scheme verification
Considering a general model of an n-degree-of-freedom mechanical arm, q is a generalized coordinate vector of a structural space, the dimension is n, u is an n-dimensional moment vector, and a mathematical model of the robot can be described as a controlled Lagrange system
Figure BDA0002550879770000044
Where D (q) is the inertial matrix of the system.
Figure RE-GDA0002915262240000046
Is a coriolis matrix that is formed by a matrix of coriolis,
Figure RE-GDA0002915262240000045
is the gradient vector of the gravitational potential energy field, V (q) is the gravitational potential energy of the system, B is an n multiplied by n matrix which describes the effect of the joint moment on the wide-sense coordinate, and u is the input n-dimensional moment vector.
When a certain joint or certain joints of the mechanical arm are provided with variable-rigidity joints, dynamics become
Figure BDA0002550879770000051
k is the rigidity input of l rigidity-adjustable joints, E (delta q) is an n multiplied by l matrix, the joint deformation effect of the rigidity-variable joints is mapped to a generalized coordinate, l is the rigidity-adjustable joint number, l is equal to or less than n, the ith joint rigidity of the n-freedom-degree mechanical arm is not adjustable, the ith row of the E (delta q) is all zero, the ith joint rigidity of the n-freedom-degree mechanical arm is adjustable, the jth joint in the l rigidity-adjustable joints is correspondingly arranged, and the ith row and the jth column of the E (delta q) are delta qjThe remaining elements of row i are 0.
Then both u and k can be used as system inputs to adjust the motion of the robotic arm. In practice, the joint stiffness k of the robot arm is ═ k1 k2 … kl]TThe change of the variable stiffness motor is obtained by adjusting the angular position of the variable stiffness motor, the response speed of the stiffness has larger delay compared with the response speed of the input u, the delay is related to the joint position, the deformation quantity of the deformation element and the dynamic response speed of the variable stiffness motor, modeling is quite complicated, the duration delta T of one control period of the planning controller of the upper computer of the mechanical arm is set, the value of the time delay of the variable stiffness joint is larger than the delta T, and the vector T is set to be [ T ═ T [ T [ ]1,T2,…Tl]TRespectively corresponding to rigidity adjusting time delay of the joints with adjustable rigidity, a system dynamics model can be written (called a system (3) below)
Figure BDA0002550879770000052
Wherein k (T-T) ═[k1(t-T1) k2(t-T2) … kl(t-Tl)]T。
Now a new variable is introduced
Figure BDA0002550879770000053
Wherein m is>1, defining variables
s(τ)＝q(t) (4)
Is provided with
Figure BDA0002550879770000054
Then can obtain
Figure BDA0002550879770000061
When neglecting the system (3) delay, consider the following control law,
Figure BDA0002550879770000062
note u0And k0The expression is a function of the system state and time, has generality, and can represent any state feedback or output feedback control law designed by adopting the existing theoretical method under the condition of neglecting the joint delay characteristic.
New variable
Figure BDA0002550879770000063
(4) Substitution of equations (3) into (5) and (6)
Figure BDA0002550879770000064
If (7) is replaced by the following control law
Figure BDA0002550879770000065
Substituted into formula (3) and changed into new variable
Figure BDA0002550879770000066
Defining a correlation system (hereinafter system (11))
Figure BDA0002550879770000067
It can be seen that the variable stiffness dynamic time delay is smaller and is the time delay of the original system (3)
Figure BDA0002550879770000068
If the initially defined control law (7) is substituted, a closed system loop of
Figure BDA0002550879770000071
It can be observed that equations (10) and (12) have identical dynamic evolution behavior except that variables s and τ are replaced by q and t, respectively, that is, if the system (11) with small dynamic time delay of system stiffness is closed-loop stable under the action of the control law (7), the dynamic equation (10) is closed-loop stable. Note that the closed-loop behavior of the system (3) with large system stiffness and dynamic time delay under the control law (9) is equivalent to equation (10), which is obtained by applying a time scaling transform. It can be concluded that if the control law (7) stabilizes the system (11), the control law (9) stabilizes the system (3). Therefore, for the control problem of the system (3) with large variable stiffness dynamic time delay, the method provided by the application can be converted into the control problem of the system (11) with small variable stiffness dynamic time delay.
For the mechanical arm joint tracking problem, if the small time delay system (11) can asymptotically and stably track the reference output q under the action of the control law (7)d(mt), the large time delay system (3) can track the reference output q under the action of the control law (9)d(T), in such scaling, the value of m may be [ T ] according to the actual time delay T1,T2,…Tl]TSelected as appropriate, let TmaxIs time delay T ═ T1,T2,…Tl]TThe maximum component of (1) is selected empirically
Figure BDA0002550879770000072
In time, the system can be generally stabilized by setting other control parameters, but the system robustness is limited; selecting
Figure BDA0002550879770000073
When the value of m is selected to be too large, the dynamic response of the system is too slow. In general, selecting
Figure BDA0002550879770000074
Therefore, the design of the controller (7) of the system (11) is very easy, most of the existing control laws are designed according to the assumed conditions, the control law (9) can be obtained by applying the transformation relation provided by the invention and transforming according to the control law (7), and the control law (9) can be suitable for the actual system (3) with larger variable-stiffness time delay characteristics.
The invention provides a method based on time scaling transformation, which can convert a control law directly designed by neglecting dynamic characteristics into practical control considering the larger time delay characteristics of variable rigidity through corresponding control transformation, and ensure the stability and steady-state precision of a mechanical arm in the operation process by sacrificing certain dynamic performance.
The control transformation method provided by the invention can be used for controlling various mechanical arms adopting variable-stiffness joints, has clear results and generality, and can solve the problem of variable-stiffness delay characteristics in the existing control law design. The invention has wide application range and strong practicability, can be widely applied to the motion control of industrial robots or household robots with variable-rigidity joints in unstructured environments, and has better application prospect.
The invention has not been described in detail in part in the common general knowledge of a person skilled in the art.

Claims (7)

1. A mechanical arm control transformation method considering variable stiffness joint delay characteristics is characterized by being realized by the following modes:
(1) considering the time delay characteristic of a variable-stiffness joint installed on the mechanical arm, and establishing an n-degree-of-freedom mechanical arm dynamic model;
(2) reducing the time delay in the kinetic model to the original
Figure FDA0002550879760000011
Establishing a new dynamic model; m is more than 1;
(3) designing a small-delay control law for stabilizing the closed loop of the new dynamic model;
(4) and (3) on the basis of the small-delay control law, considering the scaling relation of the two dynamic models in the steps (1) and (2), carrying out scaling transformation, designing a control law which enables the closed loop in the step (1) to be stable, and realizing the control of the mechanical arm by using the control law.
2. The method of claim 1, wherein: analyzing the time delay characteristic of the mechanical arm before the step (2), and starting to execute the step (2) if the time delay is greater than the control period of the mechanical arm; otherwise, directly designing a small-delay control law which enables the dynamic model in the step (1) to be closed-loop and stable, and controlling the mechanical arm by using the control law.
3. The method according to claim 1 or 2, characterized in that: assuming that the small delay control law designed in the step (3) is as follows:
Figure FDA0002550879760000012
Figure FDA0002550879760000013
the control law designed in the step (4) is as follows:
Figure FDA0002550879760000014
Figure FDA0002550879760000015
in the formula, u is an n-dimensional torque vector and corresponds to the joint torque of the mechanical arm with n degrees of freedom, k is a l-dimensional rigidity vector and corresponds to the rigidity input of l rigidity-adjustable joints of the mechanical arm with n degrees of freedom, and u is0() And k0() Control functions, q and q, respectively, referring to the existing moment and stiffness
Figure FDA0002550879760000016
A joint angle vector and a joint angular velocity vector corresponding to the mechanical arm with n degrees of freedom, G (q) is a gravity gradient vector in a Lagrange equation, and δ q is a deformation angle of l rigidity-adjustable joints; t is time, and B is an n multiplied by n matrix, which describes the effect of the joint moment on the generalized coordinates.
4. The method according to claim 1 or 2, characterized in that: the kinetic model in step (1) is as follows:
Figure FDA0002550879760000021
in the formula, q is a generalized coordinate vector of a structural space, and the dimension is n; d (q) is an inertia matrix;
Figure FDA0002550879760000022
is a Coriolis matrix, G (q) is the gradient vector of the gravitational potential energy field, B is n × nThe matrix describes the effect of joint torque on a generalized coordinate, E (delta q) is an n multiplied by l matrix, the joint deformation effect of a rigidity-variable joint is mapped to the generalized coordinate, l is the number of rigidity-adjustable joints and satisfies that l is less than or equal to n, the ith row of the E (delta q) is all zero if the ith joint rigidity of an n-freedom-degree mechanical arm is not adjustable, the ith row of the E (delta q) is assumed to be adjustable, the jth joint rigidity of the n-freedom-degree mechanical arm is adjustable, and the jth joint in the l-rigidity-adjustable joints is correspondingly arranged, and the ith row and the jth column of the E (delta q) are delta qjThe remaining elements in row i are 0; u (T) represents n-dimensional moment vector, k (T) is the rigidity input of the joints with adjustable rigidity, and T is [ T ═ T1,T2,…Tl]TThe rigidity adjusting time delay respectively corresponding to the one joint with adjustable rigidity is k (T-T) ═ k1(t-T1) k2(t-T2) … kl(t-Tl)]T。
5. The method of claim 4, wherein: the new kinetic model in step (2) is as follows:
Figure FDA0002550879760000023
6. the method according to claim 1 or 2, characterized in that: m is determined according to the actual time delay T ═ T1,T2,…Tl]TSelecting, wherein generally, it needs to be ensured that the time delay of the established new dynamic model is less than the time length delta T of one control period of the planning controller of the upper computer of the mechanical arm, wherein l is the number of the adjustable rigidity joints in the mechanical arm.
7. The method of claim 6, wherein: the value range of m is preferred
Figure FDA0002550879760000024
Wherein, TmaxIs time delay T ═ T1,T2,…Tl]TThe largest component of (a).
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