CN113219840B - Self-adaptive sliding mode cross-coupling contour control method for three-axis motion platform - Google Patents

Self-adaptive sliding mode cross-coupling contour control method for three-axis motion platform Download PDF

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CN113219840B
CN113219840B CN202110644963.5A CN202110644963A CN113219840B CN 113219840 B CN113219840 B CN 113219840B CN 202110644963 A CN202110644963 A CN 202110644963A CN 113219840 B CN113219840 B CN 113219840B
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李国柱
段颖妮
白潇
孟雪
雷俊红
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Shaanxi Tonglida Intelligent Technology Co ltd
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Xian University
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Abstract

The invention discloses a self-adaptive sliding mode cross-coupling contour control method for a three-axis motion platform, which comprises the steps of constructing and initializing a self-adaptive sliding mode controller; inputting an initial interpolation command signal to the initialized self-adaptive sliding mode controller to obtain an initial output result of the servo controller; updating a self-adaptive sliding mode control law and a contour error of a self-adaptive sliding mode controller according to the initial interpolation command signal and an initial output result of the servo controller to respectively obtain a current self-adaptive sliding mode control law and a current contour error; sending the current contour error into a cross coupling controller, and obtaining a compensation control quantity component of the current contour error through a PID control algorithm; and controlling the actual motion of the three-axis motion platform by using the superposition of the current adaptive sliding mode control law and the compensation control quantity component of the current contour error as the input of a servo driver. The invention improves the self-adaptability and the control strategy effectiveness of the sliding mode controller.

Description

Self-adaptive sliding mode cross-coupling contour control method for three-axis motion platform
Technical Field
The invention relates to the field of mechanical control, in particular to a self-adaptive sliding mode cross-coupling contour control method for a three-axis motion platform.
Background
In the modern manufacturing and processing technology, high-precision multi-axis motion control always belongs to a research hotspot problem, and two errors, namely a tracking error and a contour error, exist in the multi-axis track tracking motion process. The tracking error is the error between the actual position and the desired position and the contour error is the distance of the actual position to the desired trajectory. For the machining motion, the contour error can reflect the contour machining precision, so the control of the contour error is more emphasized by people.
For many years, scholars at home and abroad make a great deal of research on contour error control. For reducing the contour error, the conventional strategy is to reduce the tracking error of a single axis, such as PID control, neural network, iterative learning control, sliding mode control, and the like. However, in a multi-axis motion system, due to the mismatch of dynamic synchronization performance between the axes, only the improvement of single-axis tracking accuracy is considered, and the profile error cannot be effectively reduced, so a multi-axis coordination mechanism must be introduced to effectively suppress the profile error, and cross-coupled control (CCC) is the most widely applied method at present. The basic idea of cross coupling is to establish a contour error estimation model by using given and feedback information of each coordinate axis, seek a contour error compensation control rule and send compensation quantity to each coordinate axis, and achieve the purpose of improving the precision of contour errors. The cross coupling controller mainly comprises two parts of contour error estimation and control compensation strategy.
Disclosure of Invention
Aiming at the defects in the prior art, the self-adaptive sliding mode cross-coupling contour control method for the three-axis motion platform provided by the invention solves the problems of large tracking error, poor adaptability and low effectiveness in the control process of the three-axis motion platform.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that:
the method for controlling the self-adaptive sliding mode cross coupling profile of the three-axis motion platform comprises the following steps:
s1, constructing and initializing a self-adaptive sliding mode controller, and connecting the self-adaptive sliding mode controller with a servo controller;
s2, inputting an initial interpolation command signal to the initialized self-adaptive sliding mode controller to obtain an initial output result of the servo controller;
s3, updating the self-adaptive sliding mode control law and the contour error of the self-adaptive sliding mode controller according to the initial interpolation command signal and the initial output result of the servo controller, and respectively obtaining the current self-adaptive sliding mode control law and the current contour error;
s4, sending the current contour error into a cross coupling controller, obtaining the compensation control quantity of the current contour error through a PID control algorithm, and simultaneously calculating the cross coupling gain of the compensation control quantity of the current contour error;
s5, inputting a new interpolation command signal to the current self-adaptive sliding mode controller, overlapping the current self-adaptive sliding mode control law and the current contour error through the compensation control quantity cross coupling gain generated by the cross coupling controller, and inputting the overlapping result into the servo controller to obtain the output result of the servo controller, and completing the first self-adaptive sliding mode cross coupling contour control;
s6, judging whether to continue the self-adaptive sliding mode cross coupling contour control, if so, entering the step S7; otherwise, ending the control;
s7, updating the parameters and contour error of the adaptive sliding mode controller according to the interpolation command signal of the previous round and the output result of the servo controller, and returning to the step S5.
Further, the specific method for updating the adaptive sliding mode control law according to the initial interpolation command signal and the initial output result of the servo controller in step S3 includes the following sub-steps:
s3-1-1, according to the formula:
Figure BDA0003108868890000031
e＝ri-pi
Figure BDA0003108868890000032
Figure BDA0003108868890000033
Figure BDA0003108868890000034
Figure BDA0003108868890000035
obtaining an initial adaptive sliding mode control law ui(ii) a Wherein r isiFor the initial interpolation of the command signal, piG is an initial parameter estimation value, f is an actual true value in an initial neural network state, e is an initial tracking error, sat(s) is a saturation function, s is a sliding mode surface, lambda, eta and k are normal numbers, W is a weight matrix of the initial RBF neural network and comprises l elements, W is an initial parameter estimation valueTIs the transposition of the initial matrix, and H is the l hidden matrices of the initial RBF neural networkA transposed matrix formed by output results of the Gaussian basis functions including the layer nodes, hjThe output result of the Gaussian function of the j-th hidden layer node of the initial RBF neural network is obtained, X is the input value of the initial RBF neural network, cjAs the center vector of the jth hidden layer node, bjFor a base width parameter of a hidden layer node, | | · | | is an operation of solving a vector mode, Δ is a linear region width of a saturation function, two points above a symbol represent second-order derivation, one point above the symbol represents first-order derivation, and i ═ represents three axes of a three-axis platform;
s3-1-2, according to the formula:
Figure BDA0003108868890000041
Figure BDA0003108868890000042
obtaining the self-adaptation law of the current RBF neural network
Figure BDA0003108868890000043
And adaptation law of current parameter estimation
Figure BDA0003108868890000044
Wherein, gmaxAnd gminUpper and lower boundary values, gamma, respectively, for the current parameter estimate1And gamma2For different adaptive learning rates;
s3-1-3, adaptive law of current RBF neural network
Figure BDA0003108868890000045
Adding the weight matrix W of the initial RBF neural network to obtain the weight matrix of the current RBF neural network, and estimating the self-adaptation law of the current parameters
Figure BDA0003108868890000046
Adding the initial parameter estimation value g to obtain a current parameter estimation value;
and S3-1-4, substituting the weight matrix of the current RBF neural network and the current parameter estimation value into the formula same as that in the step S3-1 for calculation to obtain the current adaptive sliding mode control law.
Further, the specific method for updating the contour error according to the initial interpolation command signal and the initial output result of the servo controller in step S3 includes the following sub-steps:
s3-2-1, acquiring current reference point RnThe current actual position point P is set to the reference point RnDistance of | PRnL is taken as an initial minimum distance;
s3-2-2, traversing the current actual position point P and the reference point RnAnd a reference point RnFront R*The distance between reference points, and whether there is a coincidence of | PRm-1|>|PRm|、|PRm|<|PRnI and I PRm|<|PRm+1If yes, acquiring the first reference point meeting the conditions as the current nearest reference point RmAnd proceeds to step S3-3; otherwise, judging the current reference point RnAs the current nearest reference point RmAnd stops the judgment and proceeds to step S3-4; wherein PRm-1L is the current actual position point P to the current nearest reference point RmDistance of last reference point, | PRmL is the current actual position point P to the current nearest reference point RmDistance, | PRm+1L is the current actual position point P to the current nearest reference point RmThe distance of the next reference point;
s3-2-3, updating the value of the initial minimum distance to | PRmAnd go to step S3-4;
s3-2-4, according to the formula:
Figure BDA0003108868890000051
Figure BDA0003108868890000052
obtain the current contour error epsilonc(ii) a Wherein epsilonx、εyAnd εzThe components of the contour error vector on the x, y and z axes respectively, | PQ | is the actual position point P to the current nearest reference point RmThe vertical distance of the corresponding tangent line PQ is from the actual position point P to the current nearest reference point RmPerpendicular vector of corresponding tangent line, (x)m,ym,zm) As the current nearest reference point Rm(Δ R) of (C)x,ΔRy,ΔRz) As the current nearest reference point RmThe coordinate difference between two adjacent reference points, (a, b, c) is the coordinate of the actual position point P.
Further, the specific method in step S4 includes the following sub-steps:
s4-1, according to the formula:
Figure BDA0003108868890000053
obtaining the compensation control quantity u of the current contour errorc(k) (ii) a Where k is the kth sampling period, εc(k) Is the profile error of the kth sampling period, epsilonc(K-1) is the profile error for the (K-1) th sampling period, Kp、KiAnd KdThree parameters of PID respectively;
s4-2, according to the formula:
Figure BDA0003108868890000061
obtaining the cross coupling gain C of the current contour error on the X axis, the Y axis and the Z axisx、CyAnd Cz；
S4-3, respectively multiplying the compensation control quantity of the current contour error by the cross coupling gain C of the current contour error on the X axis, the Y axis and the Z axisx、CyAnd CzAnd obtaining the component of the compensation control quantity of the current contour error.
The invention has the beneficial effects that: updating the self-adaptive sliding mode control law in real time through a feedback signal so as to reduce the tracking error and improve the accuracy of the actual motion track; the controlled object model and parameters are estimated through an RBF neural network and a parameter identification technology, buffeting of the sliding mode controller is reduced, uncertainty of the model and the parameters is overcome, a cross-coupling controller is designed on the basis of a three-dimensional contour error estimation algorithm based on a nearest reference point, and control strategy effectiveness is improved.
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FIG. 1 is a control flow diagram of the present invention;
FIG. 2 is a block diagram of a cross-coupling control system of a self-adaptive sliding mode control three-axis motion platform;
FIG. 3 is a schematic diagram of a three-axis motion platform profile error;
FIG. 4 is a graph of X-axis tracking error;
FIG. 5 is a Y-axis tracking error plot;
FIG. 6 is a Z-axis tracking error plot;
FIG. 7 is a graph of the controller adjusted X-axis tracking error;
FIG. 8 is a graph of the controller adjusted Y-axis tracking error;
FIG. 9 is a comparison of different profile error estimation methods;
FIG. 10 is a graph comparing the control effect of different profile errors.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
As shown in fig. 1 and fig. 2, the adaptive sliding mode cross-coupling contour control method for a three-axis motion platform includes the following steps:
s1, constructing and initializing a self-adaptive sliding mode controller, and connecting the self-adaptive sliding mode controller with a servo controller;
s2, inputting an initial interpolation command signal to the initialized self-adaptive sliding mode controller to obtain an initial output result of the servo controller;
s3, updating the self-adaptive sliding mode control law and the contour error of the self-adaptive sliding mode controller according to the initial interpolation command signal and the initial output result of the servo controller, and respectively obtaining the current self-adaptive sliding mode control law and the current contour error;
s4, sending the current contour error into a cross coupling controller, obtaining the compensation control quantity of the current contour error through a PID control algorithm, and simultaneously calculating the cross coupling gain of the compensation control quantity of the current contour error;
s5, inputting a new interpolation command signal to the current self-adaptive sliding mode controller, superposing the cross coupling gain of the compensation control quantity generated by the cross coupling controller on the current self-adaptive sliding mode control law and the current contour error, and inputting the superposed result into the servo controller to obtain the output result of the servo controller and complete the first self-adaptive sliding mode cross coupling contour control;
s6, judging whether to continue the self-adaptive sliding mode cross coupling contour control, if so, entering the step S7; otherwise, ending the control;
s7, updating the parameters and contour error of the adaptive sliding mode controller according to the interpolation command signal of the previous round and the output result of the servo controller, and returning to the step S5.
The specific method for updating the adaptive sliding mode control law according to the initial interpolation command signal and the initial output result of the servo controller in step S3 includes the following sub-steps:
s3-1-1, according to the formula:
Figure BDA0003108868890000081
e＝ri-pi
Figure BDA0003108868890000082
Figure BDA0003108868890000083
Figure BDA0003108868890000084
Figure BDA0003108868890000085
obtaining an initial adaptive sliding mode control law ui(ii) a Wherein r isiFor the initial interpolation of the command signal, piG is an initial parameter estimation value, f is an actual true value in an initial neural network state, e is an initial tracking error, sat(s) is a saturation function, s is a sliding mode surface, lambda, eta and k are normal numbers, W is a weight matrix of the initial RBF neural network and comprises l elements, W is an initial parameter estimation valueTIs the transposition of initial matrix, H is the transposition matrix formed by the output results of the Gaussian basis functions of l hidden layer nodes of the initial RBF neural network, HjThe output result of the Gaussian function of the j-th hidden layer node of the initial RBF neural network is obtained, X is the input value of the initial RBF neural network, cjAs the center vector of the jth hidden layer node, bjFor a base width parameter of a hidden layer node, | | · | | is an operation of solving a vector mode, Δ is a linear region width of a saturation function, two points above a symbol represent second-order derivation, one point above the symbol represents first-order derivation, and i ═ represents three axes of a three-axis platform;
s3-1-2, according to the formula:
Figure BDA0003108868890000091
Figure BDA0003108868890000092
obtaining the self-adaptation law of the current RBF neural network
Figure BDA0003108868890000093
And adaptation law of current parameter estimation
Figure BDA0003108868890000094
Wherein, gmaxAnd gminUpper and lower boundary values, gamma, respectively, for the current parameter estimate1And gamma2For different adaptive learning rates;
s3-1-3, adaptive law of current RBF neural network
Figure BDA0003108868890000095
Adding the weight matrix W of the initial RBF neural network to obtain the weight matrix of the current RBF neural network, and estimating the self-adaptation law of the current parameters
Figure BDA0003108868890000096
Adding the initial parameter estimation value g to obtain a current parameter estimation value;
and S3-1-4, substituting the weight matrix of the current RBF neural network and the current parameter estimation value into the formula same as that in the step S3-1 for calculation to obtain the current adaptive sliding mode control law.
As shown in fig. 3, the specific method for updating the contour error according to the initial interpolation command signal and the initial output result of the servo controller in step S3 includes the following sub-steps:
s3-2-1, acquiring current reference point RnThe current actual position point P is set to the reference point RnDistance of | PRnL is taken as an initial minimum distance;
s3-2-2, traversing the current actual position point P and the reference point RnAnd a reference point RnFront R*The distance between reference points, and whether there is a coincidence of | PRm-1|>|PRm|、|PRm|<|PRnI and I PRm|<|PRm+1If yes, acquiring the first reference point meeting the conditions as the current nearest reference point RmAnd proceeds to step S3-3; otherwise, judging the current reference point RnIs the most currentNear reference point RmAnd stops the judgment and proceeds to step S3-4; wherein PRm-1L is the current actual position point P to the current nearest reference point RmDistance of last reference point, | PRmL is the current actual position point P to the current nearest reference point RmDistance, | PRm+1L is the current actual position point P to the current nearest reference point RmThe distance of the next reference point;
s3-2-3, updating the value of the initial minimum distance to | PRmAnd go to step S3-4;
s3-2-4, according to the formula:
Figure BDA0003108868890000101
Figure BDA0003108868890000102
obtain the current contour error epsilonc(ii) a Wherein epsilonx、εyAnd εzThe components of the contour error vector on the x, y and z axes respectively, | PQ | is the actual position point P to the current nearest reference point RmThe vertical distance of the corresponding tangent line PQ is from the actual position point P to the current nearest reference point RmPerpendicular vector of corresponding tangent line, (x)m,ym,zm) As the current nearest reference point Rm(Δ R) of (C)x,ΔRy,ΔRz) As the current nearest reference point RmThe coordinate difference between two adjacent reference points, (a, b, c) is the coordinate of the actual position point P.
The specific method in step S4 includes the following substeps:
s4-1, according to the formula:
Figure BDA0003108868890000103
obtaining the compensation control quantity u of the current contour errorc(k) (ii) a Where k is the kth sampling period, εc(k) Is the profile error of the kth sampling period, epsilonc(K-1) is the profile error for the (K-1) th sampling period, Kp、KiAnd KdThree parameters of PID respectively;
s4-2, according to the formula:
Figure BDA0003108868890000111
obtaining the cross coupling gain C of the current contour error on the X axis, the Y axis and the Z axisx、CyAnd Cz；
S4-3, respectively multiplying the compensation control quantity of the current contour error by the cross coupling gain C of the current contour error on the X axis, the Y axis and the Z axisx、CyAnd CzAnd obtaining the component of the compensation control quantity of the current contour error.
In one embodiment of the present invention, the design process of the single-axis adaptive sliding mode control law is as follows:
in the time domain that the servo driver works in the speed mode, the dynamic model of the three-axis motion platform is as follows:
Figure BDA0003108868890000112
wherein p isiIs the output of the ith axis of the servo controller; u. ofiIs the input of the ith axis of the servo controller; a gain coefficient which is a control amount of the i-th axis; mass of the ith axis; is the viscous friction coefficient of the i-th axis; disturbance force of the ith axis comprises load disturbance and unmodeled disturbance; i ═ x, y, z.
Omitting the axis subscripts in the notation, the dynamic model is transformed to:
Figure BDA0003108868890000113
wherein d is the disturbance amount.
Inputting an interpolation command signal r, defining a sliding mode function:
Figure BDA0003108868890000114
wherein e is a tracking error, and e is r-p, s is a sliding mode surface, and λ is a normal number.
Designing an approximation rule, and approximating the sliding mode surface s by adopting an exponential law, wherein the approximation rule is as follows:
Figure BDA0003108868890000121
where η and k are both normal numbers and sgn(s) is a sign function with respect to s.
At f0And g0In the known case, the sliding mode control law is:
Figure BDA0003108868890000122
due to f of the actual system0And g0With uncertainty in modeling, the present invention utilizes RBF neural networks to replace f0While substituting g with the parameter estimate0(ii) a The RBF network can approximate any nonlinear function with the network input of
Figure BDA0003108868890000123
The number of hidden layer nodes is l, and the output h of the Gaussian function of the jth hidden layer nodejComprises the following steps:
Figure BDA0003108868890000124
wherein, cjAs the center vector of the jth hidden layer node, bjIs the base width parameter of the hidden layer node.
Outputting 1 node of the output layer, and outputting as RBF network pair f0The approximation of f, then:
Figure BDA0003108868890000125
wherein, W is the weight matrix of the RBF neural network.
Let W*The ideal weight is then, under the ideal condition:
f0＝W*TH+εf；
wherein epsilonfThe error is approximated for the network.
For g0Definition of g is g0Then:
g0＝g+εg；
wherein epsilongAn error is estimated for the parameter.
Using f and g instead of f0And g0And then, the sliding mode control law is as follows:
Figure BDA0003108868890000131
accordingly:
Figure BDA0003108868890000132
according to the lyapunov function:
Figure BDA0003108868890000133
wherein, γ1And gamma2For different adaptive learning rates, then:
Figure BDA0003108868890000134
the adaptive law of W is:
Figure BDA0003108868890000135
taking the self-adaptation law of g as follows:
Figure BDA0003108868890000136
wherein, gmaxAnd gminRespectively an upper boundary value and a lower boundary value for the parameter estimation.
Because the approximation error of the RBF network is very small positive real number, the eta is not less than epsilonN+ D, then
Figure BDA0003108868890000137
Wherein D is the maximum value of the disturbance quantity, and the stability of the single-shaft servo system is ensured.
To suppress buffeting, the sign function sgn(s) is replaced with a saturation function sat(s) using fixed boundary layer sliding mode control:
Figure BDA0003108868890000141
where Δ is the linear region width of the saturation function.
The single-axis adaptive sliding mode control law is as follows:
Figure BDA0003108868890000142
in another embodiment of the present invention, the simulation analysis of the method is:
establishing a mathematical model of each axis of the three-axis motion platform:
Figure BDA0003108868890000143
and selecting a saddle-shaped curve to simulate the saddle-shaped curve, wherein the curves of the interpolation command signal on an X axis, a Y axis and a Z axis are respectively as follows: r isx＝0.1cos(2πT)、ry0.1sin (2 π T) and rx＝0.05cos(4πT)。
Firstly, single-axis control performance is verified, an X axis, a Y axis and a Z axis are independently controlled, and traditional sliding mode control and self-adaptive sliding mode control of the invention are respectively adopted. In the algorithm, the RBF neural network structure is double-input single-output, the number of neurons in an implicit layer is 5, namely l is 5; adaptive learning rate of gamma10.1 and γ220; x-axis and Y-axis parameter estimation gxAnd gyThe value range is [0.4, 2 ]]Z-axis parameter estimation gzThe value range is [2, 6 ]]。
As shown in fig. 4, 5, and 6, the adaptive sliding mode control algorithm needs to perform approximation of an unknown model and parameters in an early stage, and at this time, the tracking error is large, and after the algorithm converges rapidly, the tracking error of the adaptive sliding mode control is significantly smaller than that of the conventional sliding mode control.
As shown in fig. 7 and 8, in the simulation process, the adaptive sliding mode control algorithm is the same for the controllers designed for the X axis and the Y axis, and in order to compare the adaptability of the adaptive sliding mode control algorithm provided by the present invention and the conventional sliding mode control algorithm, the different sliding mode controllers designed for the X axis and the Y axis are exchanged and the experiment is repeated.
The traditional sliding mode controller has certain dependence on a controlled object mathematical model, when the mathematical model is not accurately established, the control effect can be greatly reduced, and the adaptive capacity of the self-adaptive sliding mode controller to unknown models and parameters is enhanced.
As shown in fig. 9, after the actual running track is obtained by applying the PID control method to each axis of the three-axis motion platform, the three-axis contour error estimation algorithm and the straight line contour error estimation algorithm based on the nearest reference point proposed by the present invention are respectively used to calculate the contour error estimation value.
The maximum value of the estimation error of the linear contour error estimation algorithm is 1.172 multiplied by 10-6Meter, mean value of estimation error 2.8983 × 10-7Metric, the maximum value of the estimation error of the contour error track algorithm based on the nearest reference point is 6.5131 multiplied by 10-7Meter, mean value of estimation error 9.9048 × 10-8And (4) rice. The simulation results show that the contour error estimation algorithm based on the nearest reference point can realize more accurate contour error estimation.
Finally, as shown in fig. 10, the effect of contour error control is compared, and the control strategy of the cross-coupled controller combined with the single-axis sliding mode controller is adopted in the comparison algorithm.
Under the action of the algorithm, after the system converges the estimation of the unknown model and the parameters, the profile error is obviously reduced and is obviously smaller than that under the SMC + CCC method after 0.2 s. The maximum value of the profile error is 1.0156 multiplied by 10 under the SMC + CCC method-5Rice, mean value 6.0514X 10-6Rice; the maximum value of the profile error is 3.6159 multiplied by 10 under the SMC + CCC method-6Rice, mean value 1.7206X 10-6And (4) rice. Therefore, the algorithm can obviously improve the contour machining accuracy of the three-axis motion platform.
The self-adaptive sliding mode control law is updated in real time through the feedback signal so as to reduce the tracking error and improve the accuracy of the actual motion track; the controlled object model and parameters are estimated through an RBF neural network and a parameter identification technology, buffeting of the sliding mode controller is reduced, uncertainty of the model and the parameters is overcome, a cross-coupling controller is designed on the basis of a three-dimensional contour error estimation algorithm based on a nearest reference point, and control strategy effectiveness is improved.

Claims (1)

1. A three-axis motion platform self-adaptive sliding mode cross coupling contour control method is characterized by comprising the following steps:
s1, constructing and initializing a self-adaptive sliding mode controller, and connecting the self-adaptive sliding mode controller with a servo controller;
s2, inputting an initial interpolation command signal to the initialized self-adaptive sliding mode controller to obtain an initial output result of the servo controller;
s3, updating the self-adaptive sliding mode control law and the contour error of the self-adaptive sliding mode controller according to the initial interpolation command signal and the initial output result of the servo controller, and respectively obtaining the current self-adaptive sliding mode control law and the current contour error;
s4, sending the current contour error into a cross coupling controller, obtaining the compensation control quantity of the current contour error through a PID control algorithm, and simultaneously calculating the cross coupling gain of the compensation control quantity of the current contour error;
s5, inputting a new interpolation command signal to the current self-adaptive sliding mode controller, superposing the cross coupling gain of the compensation control quantity generated by the cross coupling controller on the current self-adaptive sliding mode control law and the current contour error, and inputting the superposed result into the servo controller to obtain the output result of the servo controller and complete the first self-adaptive sliding mode cross coupling contour control;
s6, judging whether to continue the self-adaptive sliding mode cross coupling contour control, if so, entering the step S7; otherwise, ending the control;
s7, updating the parameters and contour error of the adaptive sliding mode controller according to the interpolation command signal of the previous round and the output result of the servo controller, and returning to the step S5;
the specific method for updating the adaptive sliding mode control law according to the initial interpolation command signal and the initial output result of the servo controller in step S3 includes the following sub-steps:
s3-1-1, according to the formula:
Figure FDA0003411986460000021
e＝ri-pi
Figure FDA0003411986460000022
Figure FDA0003411986460000023
Figure FDA0003411986460000024
Figure FDA0003411986460000025
obtaining an initial adaptive sliding mode control law ui(ii) a Wherein r isiFor the initial interpolation of the command signal, piG is an initial parameter estimation value, f is an actual true value in an initial neural network state, e is an initial tracking error, sat(s) is a saturation function, s is a sliding mode surface, lambda, eta and k are normal numbers, W is a weight matrix of the initial RBF neural network and comprises l elements, W is an initial parameter estimation valueTIs the transposition of initial matrix, H is the transposition matrix formed by the output results of the Gaussian basis functions of l hidden layer nodes of the initial RBF neural network, HjThe output result of the Gaussian function of the j-th hidden layer node of the initial RBF neural network is obtained, X is the input value of the initial RBF neural network, cjAs the center vector of the jth hidden layer node, bjFor a base width parameter of a hidden layer node, | | · | | is an operation of solving a vector mode, Δ is a linear region width of a saturation function, two points above a symbol represent second-order derivation, one point above the symbol represents first-order derivation, and i ═ represents three axes of a three-axis platform;
s3-1-2, according to the formula:
Figure FDA0003411986460000031
Figure FDA0003411986460000032
obtaining the self-adaptation law of the current RBF neural network
Figure FDA0003411986460000033
And adaptation law of current parameter estimation
Figure FDA0003411986460000034
Wherein, gmaxAnd gminUpper and lower boundary values estimated for the current parameter, respectivelyValue, gamma1And gamma2For different adaptive learning rates;
s3-1-3, adaptive law of current RBF neural network
Figure FDA0003411986460000035
Adding the weight matrix W of the initial RBF neural network to obtain the weight matrix of the current RBF neural network, and estimating the self-adaptation law of the current parameters
Figure FDA0003411986460000036
Adding the initial parameter estimation value g to obtain a current parameter estimation value;
s3-1-4, substituting the weight matrix of the current RBF neural network and the current parameter estimation value into the formula same as that in the step S3-1 for calculation to obtain the current adaptive sliding mode control law;
the specific method for updating the profile error according to the initial interpolation command signal and the initial output result of the servo controller in step S3 includes the following sub-steps:
s3-2-1, acquiring current reference point RnThe current actual position point P is set to the reference point RnDistance of | PRnL is taken as an initial minimum distance;
s3-2-2, traversing the current actual position point P and the reference point RnAnd a reference point RnFront R*The distance between reference points, and whether there is a coincidence of | PRm-1|>|PRm|、|PRm|<|PRnI and I PRm|<|PRm+1If yes, acquiring the first reference point meeting the conditions as the current nearest reference point RmAnd proceeds to step S3-3; otherwise, judging the current reference point RnAs the current nearest reference point RmAnd stops the judgment and proceeds to step S3-4; wherein | PRm-1L is the current actual position point P to the current nearest reference point RmDistance of last reference point, | PRmL is the current actual position point P to the current nearest reference point RmDistance, | PRm+1L is the current actual position point P to the current nearest reference point RmThe distance of the next reference point;
s3-2-3, updating the value of the initial minimum distance to | PRmAnd go to step S3-4;
s3-2-4, according to the formula:
Figure FDA0003411986460000041
Figure FDA0003411986460000042
obtain the current contour error epsilonc(ii) a Wherein epsilonx、εyAnd εzThe components of the contour error vector on the x, y and z axes respectively, | PQ | is the actual position point P to the current nearest reference point RmThe vertical distance of the corresponding tangent line PQ is from the actual position point P to the current nearest reference point RmPerpendicular vector of corresponding tangent line, (x)m,ym,zm) As the current nearest reference point Rm(Δ R) of (C)x,ΔRy,ΔRz) As the current nearest reference point RmThe coordinate difference value of two adjacent reference points, (a, b and c) is the coordinate of the actual position point P;
the specific method in step S4 includes the following substeps:
s4-1, according to the formula:
Figure FDA0003411986460000043
obtaining the compensation control quantity u of the current contour errorc(k) (ii) a Where k is the kth sampling period, εc(k) Is the profile error of the kth sampling period, epsilonc(K-1) is the profile error for the (K-1) th sampling period, Kp、KiAnd KdThree parameters of PID respectively;
s4-2, according to the formula:
Figure FDA0003411986460000051
obtaining the cross coupling gain C of the current contour error on the X axis, the Y axis and the Z axisx、CyAnd Cz；
S4-3, respectively multiplying the compensation control quantity of the current contour error by the cross coupling gain C of the current contour error on the X axis, the Y axis and the Z axisx、CyAnd CzAnd obtaining the component of the compensation control quantity of the current contour error.
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