CN111958606A - Distributed robust tracking control method applied to multi-degree-of-freedom mechanical arm - Google Patents

Distributed robust tracking control method applied to multi-degree-of-freedom mechanical arm Download PDF

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CN111958606A
CN111958606A CN202010945143.5A CN202010945143A CN111958606A CN 111958606 A CN111958606 A CN 111958606A CN 202010945143 A CN202010945143 A CN 202010945143A CN 111958606 A CN111958606 A CN 111958606A
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mechanical arm
disturbance
joint
theta
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孙振兴
杨俊�
王珒
杨明星
董浩
缪琪卿
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Zhejiang Xieli Robot Technology Co ltd
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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
    • 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

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Abstract

The invention discloses a distributed robust tracking control method applied to a multi-degree-of-freedom mechanical arm, and belongs to the field of robot control. Aiming at the disturbance and uncertainty of the multi-degree-of-freedom mechanical arm, the invention provides a distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm, the method decomposes the whole mechanical arm into single joints and lumped disturbance, an extended high-gain observer is adopted to observe the lumped disturbance of each joint, a coupling term obtained by estimation is compensated through a localized design method, the stability of the internal state of each joint is kept, and the relative independence between the single joints is realized through a virtual decoupling method; when the joints are connected with each other, the stability of the whole mechanical arm can be realized only by ensuring the local stability of a single joint. The control method has better tracking performance and anti-interference capability, can ensure the global stability of a closed-loop system, has moderate calculation amount and is easy to realize.

Description

Distributed robust tracking control method applied to multi-degree-of-freedom mechanical arm
Technical Field
The invention relates to the field of robot control, in particular to a distributed robust tracking control method applied to a multi-degree-of-freedom mechanical arm.
Background
The multi-connecting-rod mechanical arm has the characteristics of high precision, strong reliability, high response speed and the like, and is widely applied to the fields of industrial robots, artificial hands, humanoid robots, smart robot hands and the like. In recent decades, with the development of industrial applications, tracking control of robots has been receiving more and more attention. The trajectory tracking control is to move each joint of the robot arm along a desired trajectory. The PID control (proportional-integral-derivative control) becomes a main control strategy for robot control due to its characteristics of simple control structure, high reliability, strong practicability, etc. However, in practical application, because the robot has the characteristics of strong nonlinearity, inevitable uncertainty, load change and the like, the PID control is often difficult to realize high-precision position tracking control. In order to improve the tracking performance of the robot, it is an effective measure to seek an advanced control strategy. The predecessors have many research results, such as optimal control, adaptive control, robust control, variable structure control, neural network, etc.
The control methods are all based on a centralized control structure, and require complex hardware configuration and huge calculation amount which is difficult to realize. In contrast, a decentralized control approach using only local information for a single joint is highly desirable. The invention has the following patent: a reconfigurable mechanical arm distributed control system and a control method adopting position measurement are disclosed in publication No. CN105196294A, publication No. 2015, 12, 30, and by providing a joint speed observer, a torque observer, expected dynamic information and a system dynamic model, a distributed controller is designed by adopting dynamic information of local joints, compensation is carried out on a model determination item, a friction modeling error and an inter-joint coupling item, the controller chattering is inhibited, and the mechanical arm joints are enabled to accurately track an expected track. However, for the case of unknown disturbance, the scheme cannot perform good control, and each joint is analyzed and compensated, so that a large number of parameters are involved, which is not beneficial to the implementation of the algorithm. The invention has the following patent: a recursion distributed rapid convergence robust control method of a mechanical arm is disclosed in publication No. CN110480641A, published in 2019, 11 and 22, and the method comprises the steps of respectively establishing kinematic and dynamic recursion relations of two adjacent arms in the mechanical arm to obtain generalized velocity derivatives of the arms and expressions of interaction force between the two adjacent arms; and then, deducing a state error equation by using an expression of the generalized velocity and the interaction force, and designing a recursive distributed rapid convergence robust controller of the mechanical arm by combining a finite time control method and a self-adaptive robust control method. But this approach also leads to the occurrence of buffeting while introducing limited time control.
For robotic arm tracking control, control inputs and state variables cannot be completely decoupled due to coupling between joints, such as moment of inertia and coriolis forces. Furthermore, the robot arm is often subjected to different types of disturbances, such as load variations, friction forces, gravity forces, external disturbances, etc., which adversely affect the positioning accuracy and repeatability. The prior art also provides a sliding mode control method to solve the problem, but the sliding mode control needs a large control gain to offset the mutual relation, so that a buffeting phenomenon can be generated.
Disclosure of Invention
1. Technical problem to be solved
Aiming at the problems that a distributed control method in the prior art is complex in algorithm and has disturbance uncertainty, and unknown disturbance and buffeting phenomena cannot be avoided, the invention provides a novel distributed robust tracking control method which can realize better tracking performance and anti-interference capability, simultaneously ensures the global stability of a closed-loop system, is moderate in calculated amount and is easy to realize.
2. Technical scheme
A distributed robust tracking control method applied to a multi-degree-of-freedom mechanical arm comprises the following steps:
the method comprises the following steps: establishing a mechanical arm dynamic equation by using an Euler-Lagrange equation;
step two: writing a disturbance equation corresponding to the mechanical arm dynamic equation in the step one, and performing equation transformation;
step three: writing the transformed equation into a state equation form;
step four: designing a high-gain observer according to a state equation in the step three;
step five: and constructing a distributed robust output feedback controller according to the observed values of the state and the disturbance.
Furthermore, the method for deriving the rigid robot dynamic model mainly comprises a Newton-Euler equation and a Lagrange-Euler equation. In the control field, the lagrange-euler equation is the first choice. Considering an n-link rigid mechanical arm, the dynamic equation can be expressed by using an Euler-Lagrange function:
Figure BDA0002675054870000021
where θ is the joint position vector, θ ═ θ1(t),…,θn(t)]T∈Rn；
Figure BDA00026750548700000211
In the form of a velocity vector, the velocity vector,
Figure BDA0002675054870000022
Figure BDA0002675054870000023
Figure BDA0002675054870000024
in order to be the vector of the acceleration,
Figure BDA0002675054870000025
g (theta) is a gravity vector, and G (theta) belongs to Rn；
Figure BDA0002675054870000026
Being the centrifugal force and the coriolis force vector,
Figure BDA0002675054870000027
m (θ) is an inertia matrix, M(θ)＝MT(θ)∈Rn×n；Rn×nIs an n x n dimensional matrix; u is the input torque vector, u ═ u1,…,un∈Rn](ii) a D (t) is system disturbance (including internal disturbance (such as friction and parameter disturbance) and external disturbance (such as load torque)), D (t) ([ d ═ d)1(t),…,dn(t)]T∈Rn。RnIs an n-dimensional column vector. M (theta) is a positive definite matrix, and any two constants are mu2>μ1>0, presence of μ1I≤M(θ)≤μ2I; when constant μ3>Time 0, matrix
Figure BDA0002675054870000028
Satisfy the requirement of
Figure BDA0002675054870000029
When constant μ4>When 0, the gravity moment G (theta) satisfies | | G (theta) | luminance2≤μ4(ii) a For any t is more than or equal to 0, the time-varying matrix
Figure BDA00026750548700000210
Always obliquely symmetrical, micro-signals theta: [0, ∞]→Rn。
Further, for the two-joint mechanical arm, a dynamic equation obtained by adopting a lagrange-euler formula is as follows:
Figure BDA0002675054870000031
wherein:
Figure BDA0002675054870000032
in the above formula θx1Is the position of the joint 1, thetay1Is the position of the joint 2; thetax2Being the first derivative of the position of the joint 1, thetay2As the first derivative of the position of the joint 2, m1And m2Respectively, mass of two segments of the arm, /)1And l2Respectively, the length of two segments of the arm, dx(t)、dy(t) represents the lumped disturbance of the mechanical arm and g represents gravity.
The kinetic equation of the above equation can be written as:
Figure BDA0002675054870000033
wherein:
Figure BDA0002675054870000034
Figure BDA0002675054870000035
the following coordinate changes were used:
Figure BDA0002675054870000036
wherein
Figure BDA0002675054870000037
And
Figure BDA0002675054870000038
is a desired position trajectory, and
Figure BDA0002675054870000039
transforming two joint kinetic equations into:
Figure BDA00026750548700000310
wherein
Figure BDA00026750548700000311
And
Figure BDA0002675054870000041
zx1＝ex1,
Figure BDA0002675054870000042
y1＝ey1,
Figure BDA0002675054870000043
>1 is a direct scaling factor to be determined, and the 2 nd subsystem in the interconnected system (2) may be equivalent to the following:
Figure BDA0002675054870000044
the above equation can be written in the form of a state space:
Figure BDA0002675054870000045
wherein
Figure BDA0002675054870000046
C＝[1 0],
Figure BDA0002675054870000047
And
Figure BDA0002675054870000048
further, in the second step, the influence of coupling, uncertainty and unknown disturbance of each joint is considered, and a corresponding disturbance equation is written:
by mixing dx(t) and dy(t) is extended to a new state variable, i.e.
Figure BDA0002675054870000049
And
Figure BDA00026750548700000410
the above state space equation can be extended to the following form:
Figure BDA00026750548700000411
further, in the third step, the method for establishing the state equation is as follows:
rewrite equation (8) to a state space form:
Figure BDA00026750548700000412
wherein
Figure BDA0002675054870000051
Figure BDA0002675054870000052
And
Figure BDA0002675054870000053
further, in the fourth step, the high-gain observer extending the state space of the above equation (9) may be designed as follows:
Figure BDA0002675054870000054
wherein the content of the first and second substances,
Figure BDA0002675054870000055
is zx1、zx2、zx3、zy1、zy2、zy3An estimate of (d). O isx＝[ax1ax2 ax3]TAnd Oy＝[ay1 ay2 ay3]TIs the observer gain, whose components correspond to the coefficients of the helvetz polynomial.
pi(s)＝s3+ai3s2+ai2s+ai1 (i＝x，y) (11)
By definition
Figure BDA0002675054870000056
Figure BDA0002675054870000057
And
Figure BDA0002675054870000058
subsystem ExAnd EyThe error dynamics of (c) can be written as:
Figure BDA0002675054870000059
further, in the fifth step, a robust dispersion output feedback controller is designed according to the estimated state and disturbance information observed by the extended high-gain state observer:
Figure BDA00026750548700000510
wherein k isx1，kx2，ky1And ky2For feedback control gain of subsystems x and y, let kx＝[kx1 kx2]，ky＝[ky1ky2]Satisfies A-BKi(i ═ x, y) is a Helveltz matrix.
3. Advantageous effects
Compared with the prior art, the invention has the advantages that:
the invention provides a distributed robust tracking control method applied to a multi-degree-of-freedom mechanical arm to process disturbance, uncertainty and the like among the mechanical arms. For each joint, the designed high-gain observer can estimate the coupling, uncertainty and disturbance of the robot, the method can offset the coupling, each node actually works in a relatively independent mode, and the stability of the whole mechanical arm can be decomposed into the stability of a single joint. Finally, the stability of the global system can be obtained as long as the internal state of each node is properly stable.
Drawings
FIG. 1 is a functional block diagram of the present invention;
FIG. 2 is a schematic structural diagram of a dual link robotic arm of the present invention;
3-5 are first joint physical trajectory tracking curves of the present invention;
FIGS. 6-8 are second joint physical trajectory tracking curves of the present invention;
fig. 9 to 11 are first joint physical trajectory tracking curves under the condition of step disturbance;
fig. 12 to 14 are second joint physical trajectory tracking curves in case of step perturbation;
FIGS. 15-18 are outputs of an observer at step perturbations;
fig. 19 to 21 are track following curves of the first joint under sine wave disturbance;
fig. 22 to 24 are track-following curves of the second joint under sine wave disturbance;
fig. 25 to 28 are outputs of the observer when a sine wave disturbance is generated.
Detailed Description
The invention is described in detail below with reference to the drawings and specific examples.
Example 1
The present invention is applied to a double link robot arm shown in fig. 2, which is mainly composed of a joint 1 and a joint 2, in the figure, θx1Indicates the position of the joint 1, thetay1Indicates the position of the joint 2,/1And l2The lengths of the two segments of the robot arm are shown separately. As shown in fig. 1, a schematic block diagram of the present invention is shown, and the distributed robust tracking control method is completed by 5 steps of establishing a dynamic model of a mechanical arm system, coordinate transformation, state equation construction, designing a high-gain observer, and constructing a distributed robust tracking controller. Specifically, the distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm comprises the following steps:
establishing a mechanical arm dynamic equation by using an Euler-Lagrange equation;
the method for deriving the rigid robot dynamic model mainly comprises a Newton-Euler equation and a Lagrange-Euler equation. In the control field, the lagrange-euler equation is the first choice. Considering an n-link rigid mechanical arm, the dynamic equation can be expressed by using an Euler-Lagrange function:
Figure BDA0002675054870000061
where θ is the joint position vector, θ ═ θ1(t),…,θn(t)]T∈Rn；
Figure BDA0002675054870000071
In the form of a velocity vector, the velocity vector,
Figure BDA0002675054870000072
Figure BDA0002675054870000073
Figure BDA0002675054870000074
in order to be the vector of the acceleration,
Figure BDA0002675054870000075
g (theta) is a gravity vector, and G (theta) belongs to Rn；
Figure BDA0002675054870000076
Being the centrifugal force and the coriolis force vector,
Figure BDA0002675054870000077
m (theta) is an inertia matrix, and M (theta) is MT(θ)∈Rn×n(ii) a u is the input torque vector, u ═ u1,…,un∈Rn](ii) a D (t) is system disturbance (including internal disturbance (such as friction and parameter disturbance) and external disturbance (such as load torque)), D (t) ([ d ═ d)1(t),…,dn(t)]T∈Rn。
M (theta) is a positive definite matrix, and any two constants are mu2>μ1>0, presence of μ1I≤M(θ)≤μ2I；
When constant μ3>Time 0, matrix
Figure BDA0002675054870000078
Satisfy the requirement of
Figure BDA0002675054870000079
When constant μ4>When 0, the gravity moment G (theta) satisfies | | G (theta) | luminance2≤μ4；
For any t is more than or equal to 0, the time-varying matrix
Figure BDA00026750548700000710
Always obliquely symmetrical, micro-signals theta: [0, ∞]→Rn。
For the two-joint mechanical arm shown in fig. 2, the dynamic equation obtained by using the lagrange-euler formula is as follows:
Figure BDA00026750548700000711
wherein:
Figure BDA00026750548700000712
in the formula [ theta ]x1Is the position of the joint 1, thetay1Is the position of the joint 2; thetax2Being the first derivative of the position of the joint 1, thetay2As the first derivative of the position of the joint 2, m1And m2Respectively, mass of two segments of the arm, /)1And l2Respectively, the length of two segments of the arm, dx(t)、dy(t) represents the lumped disturbance of the mechanical arm and g represents gravity.
The kinetic equation of the above equation can be written as:
Figure BDA00026750548700000713
wherein:
Figure BDA00026750548700000714
Figure BDA0002675054870000081
the following coordinate changes were used:
Figure BDA0002675054870000082
wherein
Figure BDA0002675054870000083
And
Figure BDA0002675054870000084
is a desired position trajectory, and
Figure BDA0002675054870000085
transforming two joint kinetic equations into:
Figure BDA0002675054870000086
wherein
Figure BDA0002675054870000087
And
Figure BDA0002675054870000088
zx1＝ex1,
Figure BDA0002675054870000089
zy1＝ey1,
Figure BDA00026750548700000810
>1 is a direct scaling factor to be determined, and equation (5) above may be equivalent to the following system:
Figure BDA00026750548700000811
the above equation can be written in the form of a state space:
Figure BDA00026750548700000812
wherein
Figure BDA00026750548700000813
C＝[1 0],
Figure BDA00026750548700000814
And
Figure BDA00026750548700000815
step two, considering the coupling, uncertainty and unknown disturbance influence of each joint, writing out a disturbance equation corresponding to the formula (6), and connecting
General will ofx(t) and dy(t) is extended to a new state variable, i.e.
Figure BDA00026750548700000816
And
Figure BDA00026750548700000817
the above state space equation can be extended to the following form:
Figure BDA0002675054870000091
writing the transformed equation into a state equation form: rewrite equation (8) to a state space form:
Figure BDA0002675054870000092
wherein
Figure BDA0002675054870000093
Figure BDA0002675054870000094
And
Figure BDA0002675054870000095
step four, designing a high-gain observer according to the state equation, wherein the high-gain observer which expands the state space of the above formula (9) can be designed into the following form:
Figure BDA0002675054870000096
wherein the content of the first and second substances,
Figure BDA0002675054870000097
is zx1、zx2、zx3、zy1、zy2、zy3An estimate of (d). O isx＝[ax1ax2 ax3]TAnd Oy＝[ay1 ay2 ay3]TIs the observer gain, whose components correspond to the coefficients of the helvetz polynomial.
pi(s)＝s3+ai3s2+ai2s+ai1 (i＝x,y) (11)
By definition
Figure BDA0002675054870000098
Figure BDA0002675054870000099
And
Figure BDA00026750548700000910
subsystem ExAnd EyThe error dynamics of (c) can be written as:
Figure BDA0002675054870000101
step five, designing a robust dispersion output feedback controller according to the estimation state and disturbance information observed by the extended high-gain state observer:
Figure BDA0002675054870000102
wherein k isx1,kx2,ky1And ky2For feedback control gain of subsystems x and y, let kx＝[kx1 kx2]，ky＝[ky1ky2]Satisfies A-BKi(i ═ x, y) is a Helveltz matrix.
Fig. 3 to 10 are simulation results based on the control method, as shown in fig. 3, a distributed robust tracking control curve fits a reference trajectory, while a normal PID has a large error with the reference trajectory, as shown in fig. 4, the normal PID error curve fluctuates around zero, and the distributed control can maintain the error curve to be zero well, and has a smaller error curve, as shown in fig. 5, the normal PID suddenly has a large torque output at the start stage of the simulation, while the distributed control has a torque rise process, and has a more accurate force control.
Fig. 6 to 8 are second joint physical trajectory tracking curves, as shown in fig. 6, a distributed robust tracking control curve fits a reference trajectory, while a normal PID has a large error with the reference trajectory, as shown in fig. 7, the normal PID error curve fluctuates around zero, and the distributed control can maintain the error curve to be zero well, and has a smaller error curve, as shown in fig. 8, the normal PID suddenly has a large torque output at the start stage of simulation, while the distributed control has a torque rise process, and has more accurate force control.
Fig. 9 to 11 are first joint physical trajectory tracking curves under the condition of step disturbance, as shown in fig. 9, when step disturbance occurs when t is 4s, the distributed control curve can still fit with the reference trajectory and can better track the reference trajectory compared with a normal PID deviating from the reference trajectory, as shown in fig. 10, the normal PID control error is large, and the distributed control error curve returns to zero immediately after fluctuating once, and compared with PID control, the algorithm is less affected by the step disturbance. As shown in fig. 11, the ordinary PID suddenly has a large torque output at the beginning of the simulation, while the distributed control has a torque rise process, and when the disturbance occurs, the distributed control has a smaller torque jump and has more accurate force control.
Fig. 12 to 14 are second joint physical trajectory tracking curves under the condition of step disturbance, as shown in fig. 12, when step disturbance occurs when t is 4s, the distributed control curve can still fit with the reference trajectory and can better track the reference trajectory compared with the normal PID deviating from the reference trajectory, as shown in fig. 13, the normal PID control error is large, and the distributed control error curve returns to zero immediately after fluctuating once, and compared with PID control, the algorithm is less affected by step disturbance. As shown in fig. 14, the ordinary PID suddenly has a large torque output at the beginning of the simulation, while the distributed control has a torque rise process, and when the disturbance occurs, the distributed control has a smaller torque jump and has more accurate force control.
Fig. 15 to 18 show the output of the observer during step disturbance, and as shown in fig. 15, when step disturbance occurs, the first joint occurs when t is 4s
Figure BDA0002675054870000111
The zero returning is carried out after the fluctuation, and the convergence speed is higher; as shown in fig. 16, when t is 4s, the first joint
Figure BDA0002675054870000112
After the fluctuation, fast convergence returns to zero. As shown in fig. 17, when t is 4s, the second joint
Figure BDA0002675054870000113
The zero returning is carried out after the fluctuation, and the convergence speed is higher; as shown in fig. 18, when t is 4s, the second joint
Figure BDA0002675054870000114
After the fluctuation, fast convergence returns to zero. It is shown that the estimated values are very close to the actual values and the estimator estimates the state of the system and disturbance very well.
Fig. 19 to 21 are track tracking curves of the first joint under sine wave disturbance, as shown in fig. 19, when t is 3s and sine disturbance with frequency and amplitude of 20Hz and 0.1rad/s occurs, the distributed robust tracking control curve fits the reference track, while the normal PID has a larger error with the reference track, as shown in fig. 20, the distributed robust control fluctuates up and down at zero value, and there is a more ideal error curve than the normal PID control which fluctuates up and down around the disturbance signal. As shown in fig. 21, the ordinary PID suddenly has a large torque output at the beginning of the simulation, and the distributed control has a torque rising process, when a sinusoidal disturbance signal is added, the ordinary PID has a situation that the torque is suddenly changed, and the distributed control has a transition process, and the distributed control has a more accurate force control.
Fig. 22 to 24 are track tracking curves of the second joint under sine wave disturbance, as shown in fig. 22, when the sine wave disturbance occurs, the distributed control can better track the reference track, while the normal PID control curve has a larger deviation from the reference track, as shown in fig. 23, the distributed robust control fluctuates around a zero value, and compared with the normal PID control error curve which fluctuates largely and irregularly, there is a more ideal error curve. As shown in fig. 24, the ordinary PID suddenly has a large torque output at the beginning of the simulation, and the distributed control has a torque rise process, and the distributed control has more precise force control.
Fig. 25 to 28 are outputs of the observer when a sine wave disturbance is generated. As shown in fig. 25, after the sinusoidal perturbation signal is added, t is 3s, the first joint
Figure BDA0002675054870000115
Fluctuation is carried out on the zero value, the variation range is small, and the observer can well estimate the first joint disturbance value; as shown in fig. 26, when t is 3s, the first joint
Figure BDA0002675054870000116
Fluctuation is carried out above and below a zero value, and the same range is smaller; as shown in fig. 27, when t is 3s, the second joint
Figure BDA0002675054870000117
Fluctuation is carried out around a zero value, and the variation range is small; as shown in fig. 28, when t is 3s, the second joint
Figure BDA0002675054870000118
Fluctuating up and down around zero and varying over a small range. It is shown that the estimated values are very close to the actual values and the estimator estimates the state of the system and disturbance very well.
In conclusion, the invention provides a novel distributed accurate tracking control method for a multi-degree-of-freedom planar mechanical arm with disturbance and uncertainty. The main idea of this method is to break up the whole manipulator into individual joints and collective perturbations. An Extended High Gain Observer (EHGO) was used to observe the collective disturbances of each joint, including coupling, load disturbances, uncertainty, etc., which we introduced into the design of the controller. The method compensates the estimated coupling terms through a localized design method, and keeps the internal state of the single joint stable. Thus, each joint can be separated into individual joints in a relatively independent manner through a virtual decoupling method. When the joints are connected with each other, the stability of the whole mechanical arm can be realized only by ensuring the local stability of a single joint. Simulation results show that the control method has effectiveness, compared with the traditional PID control, the control method has better tracking performance and anti-interference capability, the method can also ensure the global stability of a closed-loop system, the calculation amount is moderate, the control performance is superior, and the control method is easy to realize.
The invention and its embodiments have been described above schematically, without limitation, and the invention can be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The representation in the drawings is only one of the embodiments of the invention, the actual construction is not limited thereto, and any reference signs in the claims shall not limit the claims concerned. Therefore, if a person skilled in the art receives the teachings of the present invention, without inventive design, a similar structure and an embodiment to the above technical solution should be covered by the protection scope of the present patent. Furthermore, the word "comprising" does not exclude other elements or steps, and the word "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. Several of the elements recited in the product claims may also be implemented by one element in software or hardware. The terms first, second, etc. are used to denote names, but not any particular order.

Claims (7)

1. A distributed robust tracking control method applied to a multi-degree-of-freedom mechanical arm is characterized by comprising the following steps:
the method comprises the following steps: establishing a mechanical arm dynamic equation by using an Euler-Lagrange equation;
step two: constructing a disturbance equation corresponding to a mechanical arm dynamic equation, and carrying out coordinate transformation on the disturbance equation;
step three: constructing the transformed disturbance equation in a state equation form;
step four: designing a high-gain observer according to a state equation in the step three;
step five: and constructing a distributed robust output feedback controller according to the observed values of the state and the disturbance.
2. The distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm according to claim 1, wherein in the first step, the method for establishing the mechanical arm dynamics equation is as follows:
assuming an n-link rigid mechanical arm, expressing a dynamic equation of the rigid mechanical arm by using an Euler-Lagrange function:
Figure FDA0002675054860000011
wherein θ is a joint position vector, [ θ ]1(t),…,θn(t)]T∈Rn；
Figure FDA0002675054860000012
In the form of a velocity vector, the velocity vector,
Figure FDA0002675054860000013
Figure FDA0002675054860000014
Figure FDA0002675054860000015
in order to be the vector of the acceleration,
Figure FDA0002675054860000016
g (theta) is a gravity vector, and G (theta) belongs to Rn；
Figure FDA0002675054860000017
Being the centrifugal force and the coriolis force vector,
Figure FDA0002675054860000018
m (theta) is an inertia matrix, and M (theta) is MT(θ)∈Rn×n(ii) a u is the input torque vector, u ═ u1,…,un∈Rn](ii) a D (t) is the system disturbance, D (t) ═ d1(t),…,dn(t)]T∈Rn(ii) a M (theta) is a positive definite matrix, and any two constants are mu2>μ1>0, presence of μ1I≤M(θ)≤μ2I; when constant μ3>Time 0, matrix
Figure FDA0002675054860000019
Satisfy the requirement of
Figure FDA00026750548600000110
When constant μ4>When 0, the gravity moment G (theta) satisfies | | G (theta) | luminance2≤μ4(ii) a For any t is more than or equal to 0, the time-varying matrix
Figure FDA00026750548600000111
Is obliquely symmetrical, theta: [0, ∞ ]]→Rn。
3. The distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm according to claim 2, wherein for the two-joint mechanical arm, a dynamic equation obtained by adopting a Lagrange-Euler formula is as follows:
Figure FDA00026750548600000112
wherein:
Figure FDA00026750548600000113
in the formula, thetax1Is the position of the first joint, thetay1Is the position of the second joint; thetax2Is the first derivative of the first joint position, θy2Is the first derivative of the second joint position, m1And m2Respectively, mass of two segments of the arm, /)1And l2Respectively, the length of two segments of the arm, dx(t)、dy(t) represents the lumped disturbance of the mechanical arm, g represents gravity;
the equation for dynamics of equation (2) can be written as:
Figure FDA0002675054860000021
wherein:
Figure FDA0002675054860000022
Figure FDA0002675054860000023
the following coordinate changes were used:
Figure FDA00026750548600000213
wherein
Figure FDA0002675054860000024
And
Figure FDA0002675054860000025
is a desired position trajectory, and
Figure FDA0002675054860000026
through coordinate transformation, the two-joint dynamics equation is converted into:
Figure FDA0002675054860000027
wherein
Figure FDA0002675054860000028
And
Figure FDA0002675054860000029
zx1＝ex1,
Figure FDA00026750548600000210
zy1＝ey1,
Figure FDA00026750548600000211
>1 is a direct scaling factor to be determined, and equation (5) is equivalent to the following system:
Figure FDA00026750548600000212
equation (6) above is written in state space form:
Figure FDA0002675054860000031
wherein
Figure FDA0002675054860000032
C＝[1 0],
Figure FDA0002675054860000033
And
Figure FDA0002675054860000034
4. the distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm according to claim 1, wherein in the second step, the disturbance equation corresponding to the mechanical arm dynamics equation is as follows:
by mixing dx(t) and dy(t) is extended to a new state variable, i.e.
Figure FDA0002675054860000035
And
Figure FDA0002675054860000036
equation (7) the state space equation is extended to the following form:
Figure FDA0002675054860000037
5. the distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm according to any one of claims 1 to 4, characterized in that in step three, the establishment method of the state equation is as follows:
rewrite equation (8) to a state space form:
Figure FDA0002675054860000038
wherein
Figure FDA0002675054860000039
Figure FDA00026750548600000310
And
Figure FDA00026750548600000311
6. the distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm according to claim 5, wherein in step four, the high-gain observer expanding the state space of the above formula (9) is designed in the following form:
Figure FDA0002675054860000041
wherein the content of the first and second substances,
Figure FDA0002675054860000042
is zx1、zx2、zx3、zy1、zy2、zy3Estimated value of, Ox＝[ax1 ax2ax3]TAnd Oy＝[ay1 ay2 ay3]TIs the observer gain, whose components correspond to the coefficients of the Helverz polynomial;
pi(s)＝s3+ai3s2+ai2s+ai1(i＝x,y) (11)
by definition
Figure FDA0002675054860000043
Figure FDA0002675054860000044
And
Figure FDA0002675054860000045
subsystem ExAnd EyThe error of (2) is dynamically written as:
Figure FDA0002675054860000046
7. the distributed robust tracking control method applied to the multi-degree-of-freedom mechanical arm according to claim 6, wherein in the fifth step, a robust dispersion output feedback controller is designed according to the estimated state and disturbance information observed by the extended high-gain state observer:
Figure FDA0002675054860000047
wherein k isx1,kx2,ky1And ky2For feedback control gain of subsystems x and y, let kx＝[kx1 kx2]，ky＝[ky1 ky2]Satisfies A-BKi(i ═ x, y) is a Helveltz matrix.
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