CN114516054B - Mechanical arm time delay estimation control method - Google Patents

Mechanical arm time delay estimation control method Download PDF

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CN114516054B
CN114516054B CN202210294070.7A CN202210294070A CN114516054B CN 114516054 B CN114516054 B CN 114516054B CN 202210294070 A CN202210294070 A CN 202210294070A CN 114516054 B CN114516054 B CN 114516054B
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gain
time delay
control
matrix
mechanical arm
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CN114516054A (en
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王尧尧
单忠德
孙正
谭荣凯
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Nanjing University of Aeronautics and Astronautics
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Nanjing University of Aeronautics and Astronautics
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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
    • B25J9/1679—Programme controls characterised by the tasks executed

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Abstract

The invention discloses a mechanical arm time delay estimation control method. The method comprises the following steps: delay estimation, fractional order nonsingular terminal sliding mode dynamics and nonlinear self-adaptive gain. The dynamics of the lumped system is estimated by a compact form in time delay estimation, a fractional order nonsingular terminal sliding mode is used for guaranteeing high control performance, a nonlinear self-adaptive gain has a novel nonlinear structure and a simple and effective updating algorithm, noise influence is effectively restrained when the control performance is good, and high self-adaptive performance is guaranteed when the control performance is poor. The time delay control method provided by the method has no model, high precision and good robustness, thereby further improving the control effect and the operation performance of the rope-driven mechanical arm.

Description

Mechanical arm time delay estimation control method
Technical Field
The invention belongs to the technical field of robot system control, and particularly relates to a track tracking control method for a mechanical arm.
Background
The rope-driven mechanical arm receives more and more attention by the characteristics of small motion inertia, low energy consumption, high flexibility, good interaction safety, large load-weight ratio and the like. But the application of rope drive technology also presents challenges for high performance control. Therefore, the problem of how to accurately control the robotic arm in complex uncertainties and time-varying disturbances is of interest to many students. Over the past several years, several methods of controlling flexible robotic arms have been reported [ W.Shang, B.Zhang, B.Zhang, F.Zhang, and S.Cong, "Synchronization control in the cable space for cable-driven parallel robots," IEEE Trans.Ind.Electron., vol.66, no.6, pp.4544-4554, jun.2019 ]. However, it generally requires complex estimation algorithms and is not suitable for practical applications, especially under time-varying uncertainties and disturbances. In practical applications, the key requirements of the control scheme are simplicity, efficiency and robustness.
Time delay control is a simple and effective method and is an advantageous tool for solving the above problems. Delay control is typically divided into two parts: delay estimation and desired dynamics. The former is used for estimating the dynamics of the lumped system and ensuring a model-free scheme; the latter ensures a certain control performance. Time delay control is widely applied to a plurality of practical systems, such as bipedal robots, exoskeleton robots and the like due to the advantage of no model. But the use of delay estimation also creates errors, i.e., delay estimation errors. Thus, the dynamics section typically employs robust Control strategies to suppress potential delay estimation errors, such as sliding mode Control and its variations [ S.Lee, and P.H. Chang, "Control of a heavy-duty robotic excavator using time delay Control with integral sliding surface," Control Eng. Practice., vol.10, no.7, pp.697-711,2002 ], [ M.Ghafarian, B.Shirinzadeh, A.Al-Jodah and T.K. das, "Adaptive fuzzy sliding mode Control for high-precision motion tracking of a multi-DOF micro/nano manger," IEEE Robot. Autom. Lett., vol.5, no.3, pp.4313-4320, jul.2017 ], etc. While the above-described delay control scheme achieves significant results, most are still limited by a constant control gain, i.e., may not be applicable to time-varying uncertainties and disturbances.
Y Wang et al propose a delay control scheme using nonlinear adaptive gain adjustment [ Y.Wang, L.Liu, D.Wang, F.Ju, and B.Chen, "Time-delay control using a novel nonlinear adaptive law for accurate trajectory tracking of cable-drive-robot," IEEE Trans. Ind. Information, vol.16, no.8, pp.5234-5243,2020 ], which can improve the adaptive performance while effectively suppressing the noise effect by creating a new buffer. The method can also be improved in two aspects: 1) The nonlinear adaptive law is improved, and the adaptive performance and the noise effect are further improved. 2) The adaptation law should be effectively simplified in view of its complex form. The adaptive law of its design uses seven additional parameters, which are obviously disadvantageous for practical applications.
Disclosure of Invention
The invention aims to provide a more effective self-adaptive control algorithm, and aims to improve the control effect of a mechanical arm and meet the actual application requirements of engineering.
In order to solve the problems, the invention provides a novel nonlinear structure and a simple and effective self-adaptive gain updating algorithm, provides a time delay control scheme based on nonlinear self-adaptive gain, ensures control precision and robustness, and utilizes a Lyapunov method to consider self-adaptive gain and fractional order nonsingular terminal sliding mode dynamics, thereby proving the stability of the algorithm.
The invention discloses a control method for estimating time delay of a mechanical arm, which is used for controlling a serial mechanical arm driven by a rope with n degrees of freedom, wherein n is a positive integer, and comprises the following steps of:
(1) Establishing a dynamics equation of the n-degree-of-freedom rope driving mechanical arm:
Figure BDA0003562568090000021
Figure BDA0003562568090000022
Figure BDA0003562568090000023
in θ t Is the position vector of the robot motor, q t Is the position vector of the robot joint, t is the moment,
Figure BDA0003562568090000024
for theta t First derivative of>
Figure BDA0003562568090000025
For theta t Second derivative of>
Figure BDA0003562568090000026
Is q t First derivative of>
Figure BDA0003562568090000027
Is q t D represents the damping matrix of the motor, J represents the inertia matrix of the motor, τ link,t Is the moment of the connecting rod tau t Is the motor torque, M (q t ) Representing an inertia matrix of the device,
Figure BDA0003562568090000028
representing the Coriolis/centrifugal matrix, G (q t ) Is a gravity vector, +.>
Figure BDA0003562568090000029
Is the friction vector τ d,t For describing lumped uncertainty, K p Matrix representing joint stiffness, K d A matrix representing joint damping;
(2) The dynamics equation of the rope driving mechanical arm given in the step (1) is rewritten into the following form:
Figure BDA00035625680900000210
Figure BDA00035625680900000211
wherein the method comprises the steps of
Figure BDA00035625680900000212
To adaptively control parameters H t Representing lumped system dynamics at time t;
(3) Obtaining H in the step (2) by adopting a time delay estimation method t Approximation of (2)
Figure BDA0003562568090000031
And use the approximation +.>
Figure BDA0003562568090000032
Instead of the true value, the control signal is calculated:
Figure BDA0003562568090000033
wherein Deltat represents the delay time, H t-Δt Represents H t The value of the delay delta t moment tau t-Δt Denoted τ t Delay delta t time value, q t-Δt Represents q t The value of the time delay deltat is,
Figure BDA0003562568090000034
represents q t-Δt Is a second derivative of (2);
(4) Based on the time delay estimation, the classical time delay control method is obtained as follows:
Figure BDA0003562568090000035
q in d,t In order to achieve the desired joint angle,
Figure BDA0003562568090000036
is q d,t E t ＝q d,t -q t For controlling error +.>
Figure BDA0003562568090000037
Is e t Is the first derivative of k p And k d Is a diagonal matrix for adjusting the desired control effect;
(5) Substituting the step (4) into the step (2), and obtaining the following error dynamics:
Figure BDA0003562568090000038
in the method, in the process of the invention,
Figure BDA0003562568090000039
representing a delay estimation error;
(6) At delta t In the bounded case, by selecting the appropriate k p And k d Maintain stable error dynamics while ensuring delta t The following condition is satisfied, where I is the identity matrix:
Figure BDA00035625680900000310
(7) The following slip form surfaces are adopted:
Figure BDA00035625680900000311
Figure BDA00035625680900000312
wherein s is t Is a sliding-mode surface,
Figure BDA00035625680900000313
to approach the law, D β [●],-1<β<1 is a fractional order operator, lambda 1 ，λ 2 ，α 1 ，α 2 ，β 1 ，β 2 ，k 1 ，k 2 Mu are all constant gains, where alpha 1 ，α 2 ，β 1 ，β 2 Mu is positive gain and satisfies 0<α 1 ,0<α 2 ，0<β 1 <1，0<β 2 <1,0<μ<1，/>
Figure BDA00035625680900000314
(8) Through step (7), the design controller is:
Figure BDA00035625680900000315
Figure BDA00035625680900000316
in the middle of
Figure BDA00035625680900000317
And ψ is defined as follows:
Figure BDA0003562568090000041
Figure BDA0003562568090000042
where i represents a row or column of vectors, alpha 1 ,α 2 ,β 1 ,β 2 ,k 4 ,k 5 Sigma is a constant gain, where k 4 Sigma is k 4 Normal number gain of > 0, sigma > 1,
Figure BDA0003562568090000043
-1＜β 1 ,β 2 < 1 is fractional order operator, < ->
Figure BDA0003562568090000044
Is an adaptive gain, wherein->
Figure BDA0003562568090000045
Is defined as follows:
Figure BDA0003562568090000046
where i represents a row or column of vectors, k 4 ,
Figure BDA0003562568090000047
Sigma is the normal number gain, ">
Figure BDA0003562568090000048
For the gain updated over time, use is made of +.>
Figure BDA0003562568090000049
Integral acquisition->
Figure BDA00035625680900000410
The definition is as follows:
Figure BDA00035625680900000411
in omega i ，Δ i Is a positive gain, wherein delta i Is a small gain to avoid at s i,t An infinite update rate around =0,
Figure BDA00035625680900000412
for positively limiting->
Figure BDA00035625680900000413
And->
Figure BDA00035625680900000414
A range;
(9) Finally, the controller is designed to:
Figure BDA00035625680900000415
the invention has the advantages that: the control algorithm of the invention adopts a new nonlinear structure and a simple and effective self-adaptive gain updating algorithm, and the new self-adaptive updating algorithm only needs 3 parameters. The proposed delay control scheme is modeless, highly accurate and robust.
Drawings
FIG. 1 is a view of a rope-driven mechanical arm polar-I employed in an embodiment of the present invention;
FIG. 2 is a graph comparing control performance of a controller embodying the present invention with other controllers; the control performance is compared, the controller provided by the invention is a black solid line; the controller 1 of Wang, the controller 2 of Wang, the controller of Baek and the controller of Jin are other gray solid lines or broken lines; (a) and (b) are tracking performance, (c) and (d) are control errors, (e) and (f) are amplified peak phase control errors, and (g) and (h) are control moments;
FIG. 3 is a graph showing forward gain contrast of a controller embodying the present invention with other controllers; adaptive gain
Figure BDA0003562568090000051
The controller provided by the invention is a black solid line; the controller 1 of Wang, the controller 2 of Wang, the controller of Baek and the controller of Jin are other gray solid lines or broken lines; (a) And (b) adaptive gain for the proposed controller of the present invention>
Figure BDA0003562568090000052
And Wang proposed adaptive gain of controllers 1 and 2 +.>
Figure BDA0003562568090000053
(c) And (d) is an amplified +.>
Figure BDA0003562568090000054
(e) And (f) adaptive gain of controller Baek and Jin->
Figure BDA0003562568090000055
Detailed Description
The invention is further described in conjunction with the drawings, the following examples are provided for the purpose of illustrating the invention and not for the purpose of limiting the scope of the invention, and various equivalent modifications to the invention are intended to be encompassed by the claims of the present invention. The specific implementation steps are as follows:
the invention discloses a control method for estimating time delay of a mechanical arm, which is used for controlling a n-degree-of-freedom rope-driven serial mechanical arm, wherein n is a positive integer, and the control method comprises the following steps of:
(1) Establishing a dynamics equation of the n-degree-of-freedom rope driving mechanical arm:
Figure BDA0003562568090000056
Figure BDA0003562568090000057
Figure BDA0003562568090000058
in θ t Is the position vector of the robot motor, such as theta in figure 1 1 ,θ 2 。q t Is the position vector of the robot joint, as q in fig. 1 1 ,q 2 T is the time of day, and,
Figure BDA0003562568090000059
for theta t First derivative of>
Figure BDA00035625680900000510
For theta t Second derivative of>
Figure BDA00035625680900000511
Is q t First derivative of>
Figure BDA00035625680900000512
Is q t D represents the damping matrix of the motor, J represents the inertia matrix of the motor, τ link,t Is the moment of the connecting rod tau t Is the motor torque, M (q t ) Representing an inertial matrix>
Figure BDA00035625680900000513
Representing the Coriolis/centrifugal matrix, G (q t ) Is a gravity vector, +.>
Figure BDA00035625680900000514
Is the friction vector τ d,t For describing lumped uncertainty, K p Matrix representing joint stiffness, K d A matrix representing joint damping;
(2) The dynamics equation of the rope driving mechanical arm given in the step (1) is rewritten into the following form:
Figure BDA00035625680900000515
Figure BDA00035625680900000516
wherein the method comprises the steps of
Figure BDA00035625680900000517
To adaptively control parameters H t Indicated at time tLumped system dynamics;
(3) Obtaining H in the step (2) by adopting a time delay estimation method t Approximation of (2)
Figure BDA00035625680900000518
And use the approximation +.>
Figure BDA00035625680900000519
Instead of the true value, the control signal is calculated: />
Figure BDA00035625680900000520
Wherein Deltat represents the delay time, H t-Δt Represents H t The value of the delay delta t moment tau t-Δt Denoted τ t Delay delta t time value, q t-Δt Represents q t The value of the time delay deltat is,
Figure BDA0003562568090000061
represents q t-Δt Is a second derivative of (2);
(4) Based on the time delay estimation, the classical time delay control method is obtained as follows:
Figure BDA0003562568090000062
q in d,t In order to achieve the desired joint angle,
Figure BDA0003562568090000063
is q d,t E t ＝q d,t -q t For controlling error +.>
Figure BDA0003562568090000064
Is e t Is the first derivative of k p And k d Is a diagonal matrix for adjusting the desired control effect;
(5) Substituting the step (4) into the step (2), and obtaining the following error dynamics:
Figure BDA0003562568090000065
in the method, in the process of the invention,
Figure BDA0003562568090000066
representing a delay estimation error;
(6) At delta t In the bounded case, by selecting the appropriate k p And k d Maintain stable error dynamics while ensuring delta t The following condition is satisfied, where I is the identity matrix:
Figure BDA0003562568090000067
(7) The following slip form surfaces are adopted:
Figure BDA0003562568090000068
Figure BDA0003562568090000069
wherein s is t Is a sliding-mode surface,
Figure BDA00035625680900000610
is an approach law. D (D) β [●],-1<β<1 is a fractional order operator, lambda 1,2 ,λ 1 ，λ 2 ，α 1 ,α 2 ，β 1 ，β 2 ，k 1 ，k 2 Mu are all constant gains, where alpha 1 ，α 2 ，β 1 ，β 2 Mu is positive gain and satisfies 0<α 1 ,0<α 2 ，0<β 1 <1，0<β 2 <1,0<μ<1，/>
Figure BDA00035625680900000611
(8) Through step (7), the design controller is:
Figure BDA00035625680900000612
Figure BDA00035625680900000613
in the middle of
Figure BDA00035625680900000614
And ψ is defined as follows:
Figure BDA00035625680900000615
Figure BDA00035625680900000616
where i represents a certain row (column) of the vector, α 1 ,α 2 ,β 1 ,β 2 ,k 4 ,k 5 Sigma is a constant gain, where k 4 Sigma is k 4 Normal number gain of > 0, sigma > 1,
Figure BDA0003562568090000071
-1＜β 1 ,β 2 < 1 is fractional order operator, < ->
Figure BDA0003562568090000072
Is an adaptive gain, wherein
Figure BDA0003562568090000073
Is defined as follows:
Figure BDA0003562568090000074
wherein i represents eachA certain row (column) of vectors, k 4 ,
Figure BDA0003562568090000075
Sigma is the normal number gain, ">
Figure BDA0003562568090000076
For the gain updated over time, use is made of +.>
Figure BDA0003562568090000077
Integral acquisition->
Figure BDA0003562568090000078
The definition is as follows:
Figure BDA0003562568090000079
in omega i ，Δ i Is a positive gain, wherein delta i Is a small gain to avoid at s i,t An infinite update rate around =0,
Figure BDA00035625680900000710
for positively limiting->
Figure BDA00035625680900000711
And->
Figure BDA00035625680900000712
A range;
(9) Finally, the controller is designed to:
Figure BDA00035625680900000713
and (3) performing stability analysis on the invented controller and the self-adaptive algorithm.
Suppose 1: desired trajectory q d,t Is smooth and its derivative
Figure BDA00035625680900000714
And->
Figure BDA00035625680900000715
Exist and are bounded.
Let the li-apunov equation be as follows:
Figure BDA00035625680900000716
in the method, in the process of the invention,
Figure BDA00035625680900000717
differentiating the formula (1) by combining the controller in the step (11) can obtain:
Figure BDA00035625680900000718
wherein for brevity use
Figure BDA00035625680900000719
We then bring the formulas in steps (9) - (11) to formula (2), resulting in:
Figure BDA0003562568090000081
in the method, in the process of the invention,
Figure BDA0003562568090000082
is a bounded delay estimation error, u i,t Given in step (8), the transformation of equation (3) is available:
Figure BDA0003562568090000083
in the method, in the process of the invention,
Figure BDA0003562568090000084
/>
Figure BDA0003562568090000085
for the following
Figure BDA0003562568090000086
When (1)/(1)>
Figure BDA0003562568090000087
(2)/>
Figure BDA0003562568090000088
And |s i,t |≥Ω i Can meet the requirement, can obtain:
Figure BDA0003562568090000089
for the following
Figure BDA00035625680900000810
When->
Figure BDA00035625680900000811
Can meet the requirement of +.>
Figure BDA00035625680900000812
When->
Figure BDA00035625680900000813
And |s i,t |＜Ω i When kept unchanged, can ensure +.>
Figure BDA00035625680900000814
Thus equation (3) can be given as:
Figure BDA00035625680900000815
by comparing equation (5) with equation (6), it can be seen thatThe latter is more restrictive. In addition, for equation (5) and equation (6), V t At |s i,t In the case of =0, it seems to be divergent, but in both cases the slip form surface s i,t Has been limited to |s i,t |<Ω i This means that the system will remain stable. When the system diverges to |s i,t |≥Ω i When, should be considered
Figure BDA00035625680900000816
Is the case in (a). The following analysis was thus performed on equation (5):
when (when)
Figure BDA00035625680900000817
In the case of formula (5) becomes +.>
Figure BDA00035625680900000818
Thereafter V t Continuously decreasing until s is reached i,t ＝0。
When (when)
Figure BDA00035625680900000819
When equation (5) can be expressed as the following two inequalities:
Figure BDA00035625680900000820
Figure BDA0003562568090000091
in the method, in the process of the invention,
Figure BDA0003562568090000092
for equation (7) and equation (8), at s i,t On the premise of not equal to 0, when +.>
Figure BDA0003562568090000093
V while remaining unchanged t Will continue to decrease. Thus, the slide surface s i,t Converging to:
Figure BDA0003562568090000094
thereby proving the slip form surface s i,t Based on the theoretical result of fractional order nonsingular terminal sliding mode surface, obtaining control error e t And
Figure BDA0003562568090000095
may be bounded for a limited time. So far, the certification ends.
In order to verify the effectiveness of the control method, the method provided by the invention and other 4 delay control methods existing in the prior art are tested in the embodiment. Specifically, the controller in the step (11) provided by the invention is included, and the other four controllers are respectively: the controller 1[Y.Wang,L.Liu,D.Wang,F.Ju,and B.Chen, "Time-delay control using a novel nonlinear adaptive law for accurate trajectory tracking of cable-drive-robots," IEEE trans.ind.information, "vol.16, no.8, pp.5234-5243,2020.] and the controller 2[Y.Wang,F.Yan,J.Chen,F.Ju,and B.Chen," A new adaptive Time-delay control scheme for cable-driven manipulators, "IEEE trans.ind.information, vol.15, no.6, pp.3469-3481, jun.2019.], baek-designed controller [ j.baek, s.cho, and s.han," Practical Time-delay control with adaptive gains for trajectory tracking of robot manipulators, "IEEE trans.ind.electron., vol.65, no.7, pp.5682-5692, jul.2018.], and Jin-designed controller [ m.jin, j.le, n.g.tsagaris," Model-free robust adaptive control of humanoid robots with flexible joints, "IEEE trans.ind.electron, no.6, pp.6, fe.1716, vol.1716.
The adopted experimental platform is polar-I shown in figure 1, the polar-I is driven by an ECMA-CA0604SS motor, a driver is ASD-A2-042-L, a sensor E6B2-CWZ1X 2000P/R with resolution of 0.045 degrees is adopted for joint position measurement, matlab/xPC is built by a PCI-6229 board card, and the execution period is 1ms.
(1) The parameters of the controller provided by the invention are set as follows:
λ 1，2 ＝diag(0.8，0.8)，α 1，2 ＝diag(0.9，0.9)，β 1 ＝10 -2 ×(1，1)，β 2 ＝(0.99，0.99)，k 1 ＝diag(1，1)，k 2 ＝diag(2，2)，μ＝0.8，Ω＝10 -3 ×(8，8)，Δ＝(3，6)，
Figure BDA0003562568090000101
k 3 ＝10 2 ×diag(4，4)，k 4 ＝10 4 ×diag(1.2，1.2)，
Figure BDA0003562568090000102
σ＝1.8，Δt＝2ms.
(2) The parameters of the controller 1 proposed by Wang are set as follows:
η 1 ＝(2，2)，η 2 ＝10 -2 ×(2，2)，Ω＝10 -2 ×(1，1)，k 5 ＝diag(3，3)，k 6 ＝diag(5，5)，k 7 ＝diag(2，2)，γ＝(1.8，1.8)
the other parameters are consistent with the parameter settings of the controller provided by the invention.
(3) The parameters of the controller 2 proposed by Wang are set as follows:
ε 1 ＝ε 2 ＝diag(3，3)，
Figure BDA0003562568090000103
the remaining parameters were consistent with the parameters of the controller 1 as proposed by Wang.
(4) The controller parameters proposed by Baek are set as:
K s ＝diag(0.3，0.8)，
Figure BDA0003562568090000104
Ω＝10 -2 ×(6.5，6.5)，β＝(3，3)，
k 8 diag (0.13). The other parameters are consistent with the parameters of the controller provided by the invention.
(5) The set of controller parameters proposed by Jin are:
α 3 ＝1 -2 ×(3，5)，σ 1 ，＝(1.14，1.15)，σ 2 ，＝(0.8，0.8)，ω，＝(0.26，0.4)，
Figure BDA0003562568090000105
the comparative results are shown in fig. 2 and 3, and in order to make the comparison clearer and more accurate, the present invention calculates a Root Mean Square Error (RMSE), a Maximum Absolute Error (MAE) and an Average Absolute Error (AAE), and the results are shown in table 1 below. Thanks to the delay estimation and the FONTSM dynamics and the newly designed adaptive gain, the proposed controller works best in five delay control methods.
Table 1: control performance of five controllers
Figure BDA0003562568090000106
The delay control scheme provided by the invention establishes a model-free structure by applying delay estimation, and ensures the simplicity of practical application. Fractional order nonsingular terminal sliding mode dynamics are utilized to ensure high control performance under complex time-varying uncertainty and disturbance. Based on a simple and effective updating algorithm, a new nonlinear self-adaptive gain is designed to improve the performance. By introducing a new nonlinear structure, the adaptive gain ensures better overall performance than proposed by y.wang et al. The comprehensive updating algorithm provided by the invention only needs three parameters.

Claims (1)

1. The mechanical arm time delay estimation control method is used for controlling n degrees of freedom rope driven serial mechanical arms, wherein n is a positive integer, and is characterized by comprising the following steps of:
(1) Establishing a dynamics equation of the n-degree-of-freedom rope driving mechanical arm:
Figure FDA0003562568080000011
Figure FDA0003562568080000012
Figure FDA0003562568080000013
in θ t Is the position vector of the robot motor, q t Is the position vector of the robot joint, t is the moment,
Figure FDA0003562568080000014
for theta t First derivative of>
Figure FDA0003562568080000015
For theta t Second derivative of>
Figure FDA0003562568080000016
Is q t First derivative of>
Figure FDA0003562568080000017
Is q t D represents the damping matrix of the motor, J represents the inertia matrix of the motor, τ link，t Is the moment of the connecting rod tau t Is the motor torque, M (q t ) Representing an inertial matrix>
Figure FDA0003562568080000018
Representing the Coriolis/centrifugal matrix, G (q t ) Is a gravity vector, +.>
Figure FDA0003562568080000019
Is the friction vector τ d，t For describing lumped uncertainty, K p Matrix representing joint stiffness, K d A matrix representing joint damping;
(2) The dynamics equation of the rope driving mechanical arm given in the step (1) is rewritten into the following form:
Figure FDA00035625680800000110
Figure FDA00035625680800000111
wherein the method comprises the steps of
Figure FDA00035625680800000112
To adaptively control parameters H t Representing lumped system dynamics at time t;
(3) Obtaining H in the step (2) by adopting a time delay estimation method t Approximation of (2)
Figure FDA00035625680800000113
And use the approximation +.>
Figure FDA00035625680800000114
Instead of the true value, the control signal is calculated:
Figure FDA00035625680800000115
wherein Deltat represents the delay time, H t-Δt Represents H t The value of the delay delta t moment tau t-Δt Denoted τ t Delay delta t time value, q t-Δt Represents q t The value of the time delay deltat is,
Figure FDA00035625680800000116
represents q t-Δt Is a second derivative of (2);
(4) Based on the time delay estimation, the classical time delay control method is obtained as follows:
Figure FDA00035625680800000117
q in d，t In order to achieve the desired joint angle,
Figure FDA00035625680800000118
is q d，t E t ＝q d，t -q t For controlling error +.>
Figure FDA00035625680800000119
Is e t Is the first derivative of k p And k d Is a diagonal matrix for adjusting the desired control effect;
(5) Substituting the step (4) into the step (2), and obtaining the following error dynamics:
Figure FDA00035625680800000120
in the method, in the process of the invention,
Figure FDA0003562568080000021
representing a delay estimation error;
(6) At delta t In the bounded case, by selecting the appropriate k p And k d Maintain stable error dynamics while ensuring delta t The following condition is satisfied, where I is the identity matrix:
Figure FDA0003562568080000022
(7) The following slip form surfaces are adopted:
Figure FDA0003562568080000023
Figure FDA0003562568080000024
/>
wherein s is t Is a sliding-mode surface,
Figure FDA0003562568080000025
to approach the law, D β [·]Beta is more than 1 and less than 1 is fractional order operator, lambda 1 ，λ 2 ，α 1 ，α 2 ，β 1 ，β 2 ，k 1 ，k 2 Mu are all constant gains, where alpha 1 ，α 2 ，β 1 ，β 2 Mu is positive gain and satisfies 0 < alpha 1 ，0＜α 2 ，0＜β 1 ＜1，0＜β 2 ＜1，0＜μ＜1，/>
Figure FDA0003562568080000026
(8) Through step (7), the design controller is:
Figure FDA0003562568080000027
Figure FDA0003562568080000028
in the middle of
Figure FDA0003562568080000029
And ψ is defined as follows:
Figure FDA00035625680800000210
Figure FDA00035625680800000211
where i represents a certain row or line of vectorsOne column, alpha 1 ，α 2 ，β 1 ，β 2 ，k 4 ，k 5 Sigma is a constant gain, where k 4 Sigma is k 4 Normal number gain of > 0, sigma > 1,
Figure FDA00035625680800000212
-1＜β 1 ，β 2 < 1 is fractional order operator, < ->
Figure FDA00035625680800000213
Is an adaptive gain, wherein->
Figure FDA00035625680800000214
Is defined as follows:
Figure FDA00035625680800000215
where i represents a row or column of vectors, k 4 ，
Figure FDA00035625680800000216
Sigma is the normal number gain, ">
Figure FDA00035625680800000217
For the gain updated over time, use is made of +.>
Figure FDA00035625680800000218
Integral acquisition->
Figure FDA00035625680800000219
The definition is as follows:
Figure FDA00035625680800000220
in omega i ，Δ i Is positive gainWherein delta is i Is a small gain to avoid at s i，t An infinite update rate around =0,
Figure FDA0003562568080000031
for positively limiting->
Figure FDA0003562568080000032
And->
Figure FDA0003562568080000033
A range;
(9) Finally, the controller is designed to:
Figure FDA0003562568080000034
/>
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