CN114488804A - Robot variable-gain supercoiled sliding mode control method and system based on event triggering - Google Patents

Robot variable-gain supercoiled sliding mode control method and system based on event triggering Download PDF

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CN114488804A
CN114488804A CN202210064143.3A CN202210064143A CN114488804A CN 114488804 A CN114488804 A CN 114488804A CN 202210064143 A CN202210064143 A CN 202210064143A CN 114488804 A CN114488804 A CN 114488804A
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谢远龙
吴昊
王书亭
蒋立泉
刘宇
胡倚铭
熊体凡
段佳旺
吴航
李逸飞
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Huazhong University of Science and Technology
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Abstract

The invention belongs to the field of robot control, and particularly discloses a robot variable gain supercoiled sliding mode control method and system based on event triggering, which comprises the following steps: s1, constructing a robot multi-mode unified kinematics error model; s2, designing a PI sliding mode surface related to the error; s3, designing a flutter-free supercoiled sliding mode approach law, constructing a gain adaptive function, and switching to a variable gain based on a barrier function when a sliding variable is far away from a sliding mode face by adopting high gain linearly related to time; s4, designing a controller based on a robot multi-mode unified kinematic error model and a flutter-free supercoiled sliding mode approach law to control the robot to track; a sliding variable far away from the sliding mode surface adopts a time triggering mode of periodic sampling; for a sliding variable that reaches a predetermined neighborhood, an event trigger form is employed. The invention can realize rapid convergence and realize jitter-free tracking control in the absence of boundary information of disturbance derivatives.

Description

Robot variable-gain supercoiled sliding mode control method and system based on event triggering
Technical Field
The invention belongs to the field of robot control, and particularly relates to a robot variable-gain supercoiled sliding mode control method and system based on event triggering.
Background
High-precision trajectory tracking is one of the most critical problems in the field of mobile robot control and is also a subject of attention in the robot industry, and in general, a better method is to determine a kinematics model to realize kinematics control. Due to the simplicity of sliding mode control, the method is insensitive to disturbance and is widely applied to the field of mobile robots. The sliding mode control is variable structure control essentially, large-amplitude trembling can occur in the system due to high switch gain, gain reduction is usually selected to avoid the trembling, tracking precision and response speed are sacrificed in the method, and in addition, methods such as an integral sliding mode and an ultra-spiral sliding mode are also provided, so that the trembling when the sliding mode reaches the surface can be effectively reduced. At the same time, in a complex operating environment, the construction of an accurate model is bound to unknown disturbances, the Sliding Mode Control (SMC) generally adopted can provide closed-loop insensitivity to the unknown disturbances, most SMC designs need to implement an upper bound on the bounded disturbances or derivatives, enhance the robustness of the system by adaptively adjusting the corresponding control gain, and can guarantee limited time convergence to the required sliding surface.
Although SMC has shown its effectiveness in the industrial field, there are the following areas to be improved when applying the SMC method to the motion control of robots:
1) although the supercoiled sliding mode has obvious advantages in the aspect of improving flutter, the approaching law of the supercoiled sliding mode is essentially variable speed approaching, when an output variable is near the sliding mode surface, if fixed gain is adopted, if the gain is too large, a system reaching the sliding mode surface has higher speed, larger buffeting can be caused, and if the gain is too small, the adjusting time is long;
2) when the upper disturbance bound is unknown, adaptive control cannot be realized, and when the boundary information of a disturbance derivative is lacked, how to realize a jitter-free SMC scheme to perform tracking control on the mobile robot becomes an important problem.
3) The traditional periodic sampling control increases the load operation of a controller, and the system convergence cannot be effectively supervised when the disturbance and the jump are frequent.
Therefore, a tracking control method capable of performing adaptive gain adjustment when the disturbance boundary is unknown and having fast response is needed.
Disclosure of Invention
In view of the above defects or improvement needs in the prior art, the present invention provides a method and a system for controlling a variable gain supercoiled sliding mode of a robot based on event triggering, and aims to realize fast and jitter-free adaptive tracking control of a mobile robot in the absence of boundary information of disturbance derivatives.
To achieve the above object, according to an aspect of the present invention, a robot variable gain supercoiled sliding mode control method based on event triggering is provided, which includes the following steps:
s1, constructing a robot multi-mode unified kinematics error model;
s2, designing a PI sliding mode surface related to the error;
s3, designing a flutter-free supercoiled sliding mode approach law, constructing a gain adaptive function in the flutter-free supercoiled sliding mode approach law, and switching to a variable gain based on a barrier function when a sliding variable is far away from a sliding mode face by adopting high gain linearly related to time;
s4, designing a controller based on a robot multi-mode unified kinematic error model and a flutter-free supercoiled sliding mode approach law to control the robot to track; the sliding variable far away from the sliding mode surface adopts a time triggering mode of periodic sampling; for a sliding variable that reaches a predetermined neighborhood, an event trigger form is employed.
As a further advantageOptionally, in step S1, the robot multi-mode unified kinematic error model is constructed
Figure BDA0003479578550000021
The following were used:
Figure BDA0003479578550000022
wherein, f (q)e)、g(qe) To be an error matrix, u is the control law, i.e., the control input, expressed as:
Figure BDA0003479578550000031
Figure BDA0003479578550000032
wherein q ise＝(xe,ye,θe)TIs an error state space vector, qe＝qr-q，qr＝(xr,yr,θr)T、qe＝(xe,ye,θe)TRespectively an actual state space vector and a reference state space vector of the robot under a map coordinate system; x, y and theta are respectively an abscissa and an ordinate of the map coordinate system in the actual state of the robot and a rotation angle of the robot coordinate system relative to the map coordinate system; x is the number ofr,yr,θrRespectively representing the abscissa and ordinate of the map coordinate system and the rotation angle of the robot coordinate system relative to the map coordinate system in the robot reference state; v. ofrIndicating the desired axial velocity, ωrIndicating the desired rotational speed of the robot body, thetafIs the angle of rotation of the front wheel, thetarIs the corner of the rear wheel, and the counterclockwise angle is specified to be positive and the clockwise angle is specified to be negative; v represents the axial velocity, ω represents the rotational speed of the robot body, and L represents the robot axial length; coefficient r is tan (θ)r)/tan(θf)。
As a further preferred example, r specifically is, according to different motion models adopted by the robot: when a double Ackerman model is adopted, r is-1; when an ackerman model is adopted, r is 0; when a translation model is adopted, r is 1; when the Ackermann model is used, r ═ 1,0 @ U (0, 1).
Further preferably, in step S2, the PI slip form surface S related to the error is as follows:
Figure BDA0003479578550000033
wherein s ∈ R3，qe∈R3，R3Representing a three-dimensional space vector; sign is a sign function, t is time, k, lambda and gamma are all constants, k is greater than 0, lambda is greater than 0, and gamma belongs to (0, 1).
More preferably, in step S3, the flutter-free supercoiled sliding mode approximation law
Figure BDA0003479578550000035
The method comprises the following specific steps:
Figure BDA0003479578550000034
wherein L (s (t)) is a gain adaptive function, δ is a positive number satisfying 0 < δ < 1, tanh(s) is a hyperbolic tangent function,
Figure BDA0003479578550000041
is composed of
Figure BDA0003479578550000042
S (t) is a slip variable,
Figure BDA0003479578550000043
a time-varying perturbation of an unknown boundary.
As a further preferred, the gain adaptive function is specifically:
Figure BDA0003479578550000044
wherein, F is a normal number which is as small as possible under the condition of ensuring convergence; v is any positive number, v → 0+(ii) a sin (v) denotes the neighborhood around the zero point, α is an arbitrary positive number, tswitchIs the gain switching time.
More preferably, in step S4, the controller is specifically designed as follows:
when t is more than or equal to 0 and less than tswitchIn time, a controller that employs periodic sampling:
Figure BDA0003479578550000045
wherein u is a control law,
Figure BDA0003479578550000046
ur＝(vr,ωr)Tfor reference to the input signal, vr、ωrRespectively representing a reference axial speed and a reference rotating speed of the robot;
when t is more than or equal to tswitchAnd then, adopting an event-triggered controller:
Figure BDA0003479578550000047
wherein, tiAt the time of the ith sample, u (t)i)、s(ti)、L(s(ti))、qe(ti) Are each tiControl input at time, sliding variable, gain, error state space vector, g (q)e(ti))、f(qe(ti) Respectively is tiG (q) of timee)、f(qe)。
More preferably, in step S4, when t is 0. ltoreq.t < tswitchAnd updating the control law in real time, and approaching by adopting sampling at fixed time intervals.
As a further preference it is possible to use,in step S4, when t is larger than or equal to tswitchThe event triggering conditions are specifically as follows:
Figure BDA0003479578550000051
Figure BDA0003479578550000052
wherein e isΔFor the difference of the error state space vector between two time instants, ψ ═ g (q)e(t)){[g(qe(ti))]-1；K1,K2,K3If the value is more than 0, the values are all Lipschitz constants; Γ is a positive number approaching 0; and C and D are respectively a stream set and a hop set, and when the system enters the hop set, the control law is updated.
According to another aspect of the invention, an event trigger-based robot variable gain supercoiled sliding mode control system is provided, which comprises a processor, wherein the processor is used for executing an event trigger-based robot variable gain supercoiled sliding mode control method.
Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. the method establishes a kinematics tracking error model containing external disturbance, designs a flutter-free super-spiral sliding mode approach law, linear self-adaptive gain far away from the sliding mode surface and barrier function variable gain close to the sliding mode surface by constructing a PI sliding mode surface related to the error, and ensures that the control gain is changed along with the absolute value of the disturbance, thereby ensuring that a track tracking control error approaches to zero in limited time and improving the robustness and reliability of the control; meanwhile, convergence supervision is carried out by adopting an event triggering idea, so that the burden of a controller in the later process can be reduced; the method can realize rapid convergence, realize jitter-free tracking control in the absence of boundary information of disturbance derivatives, and can be used for tracking control of the mobile robot in different modes.
2. The invention designs a specific superspiral sliding mode approach law which greatly weakens the flutter reaching the sliding mode surface, thereby reducing the flutter and improving the tracking precision; meanwhile, a gain self-adaptive function is constructed according to the gain of the supercoil, and high gain linearly related to time is adopted when the gain is far away from the sliding mode surface, so that an output variable can rapidly enter a preset neighborhood within limited time and enters the preset neighborhood to be switched to variable gain based on a barrier function, and the use of the barrier function strategy ensures that the supercoil gain is increased once the derivative of disturbance is increased, thereby ensuring that an output value belongs to the required neighborhood.
3. The invention adopts the idea of event triggering to carry out convergence supervision, and the event triggering method requires less transmission required by a stable system, thereby reducing energy consumption and network load; meanwhile, the constructed trigger condition is used as a supervision criterion, and the system convergence is effectively supervised when the disturbance jumps.
Drawings
FIG. 1 is a schematic diagram of a kinematic model of a four-wheel omnidirectional mobile robot according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a variable gain supercoiled sliding mode control system of a robot based on event triggering according to an embodiment of the present invention;
fig. 3 is a variable gain function in the sliding mode approach process according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The embodiment of the invention provides a robot variable-gain supercoiled sliding mode control method based on event triggering, which comprises the following steps as shown in fig. 2:
s1, constructing a robot multi-mode unified kinematics error model;
s2, designing a PI sliding mode surface related to the error to obtain an expression form of the output quantity;
s3, constructing a gain adaptive function, designing a flutter-free super-spiral sliding mode approach law, adopting high gain linearly related to time when the super-spiral sliding mode approach law is far away from a sliding mode surface, and switching to variable gain based on a barrier function when the super-spiral sliding mode approach law is close to the sliding mode surface so that the mobile robot tracks along an expected track;
s4, designing a controller based on event triggering, adopting a time triggering mode of periodic sampling for the sliding variable far away from the sliding mode surface, and adopting event triggering for the sliding variable reaching a preset neighborhood to reduce the load of the controller.
Further, step S1 is to construct a robot multi-mode unified kinematic error model, which specifically includes:
as shown in fig. 1, the mobile robot kinematics model is represented as:
Figure BDA0003479578550000071
wherein the mobile robot state is defined as q, and (x, y, theta) represents a robot pose space,
Figure BDA0003479578550000072
the pose spatial derivative of the mobile robot is shown, v represents the axial speed, and omega represents the rotating speed of the robot body.
The mobile robot is a four-wheel full-steering independent drive mobile robot, can operate in different modes, and continuously inputs u as follows:
Figure BDA0003479578550000073
wherein, thetafIs the corner of the front wheel, and the counterclockwise angle is specified to be positive and the clockwise angle is specified to be negative; l ═ Lf+LrRepresenting the axial length of the robot, r being the angle theta of rotation of the front wheelfAnd rear wheel steering angle thetarThe coefficient determined is represented by r ═ tan (θ)r)/tan(θf). r is expressed as:
Figure BDA0003479578550000074
in all modes, a reference state vector q is definedr＝(xr,yr,θr)TReference input signal ur＝(vr,ωr)TThe unified motion state equation of the mobile robot in different modes is written as follows:
Figure BDA0003479578550000075
further obtain an error state space vector qeComprises the following steps:
Figure BDA0003479578550000076
error state space vector qeThe derivative of (d) is expressed as:
Figure BDA0003479578550000081
further robot multi-mode unified kinematic error model
Figure BDA0003479578550000082
Expressed as:
Figure BDA0003479578550000083
wherein the content of the first and second substances,
Figure BDA0003479578550000084
further, the PI sliding mode surface S related to the error in step S2 specifically includes:
Figure BDA0003479578550000085
wherein s ∈ R3，qe∈R3(ii) a λ and γ are constants, λ > 0, γ ∈ (0, 1). Taking the derivative of the formula:
Figure BDA0003479578550000086
the calculation equation for the control law is thus expressed in the form:
Figure BDA0003479578550000087
further, the design of the sliding mode approach law in step S3 specifically includes:
Figure BDA0003479578550000088
wherein L (s (t)) is a gain adaptive function, δ is a positive number satisfying 0 < δ < 1, tanh(s) is a hyperbolic tangent function,
Figure BDA0003479578550000089
is composed of
Figure BDA00034795785500000810
The derivative of (a) of (b),
Figure BDA00034795785500000811
a time-varying perturbation of an unknown boundary. In addition, in the present embodiment, δ is preferably 1/2, that is, 0.5. The adaptive gain coefficients are:
Figure BDA00034795785500000812
wherein, F > 0 is a normal number, which is as small as possible under the condition that convergence should be ensured in order to avoid overestimating the gain F; v is a positive number v → 0 which is arbitrarily small+Sin (v) denotes the neighborhood around the zero point, α is an arbitrary positive number, tswitchIs the gain switching time. Sliding variable sAnd (t) when the sliding mode surface s is far away from 0, high gain linearly related to time is adopted, and when the sliding variable approaches the sliding mode surface, the variable gain based on the barrier function is switched. Let ε equal sin (v), s (t) ε (- ε, ε) a variable gain supercoil controller using a barrier function can solve unknown interference for any ε>0 and for all t ≧ 0, the inequality | s (t) luminance<ε holds true.
This strategy allows the adaptive gain to be increased and decreased according to the current value of the output variable, decreasing as the output variable approaches zero until a value that allows the disturbance to be compensated for; on the other hand, when the disturbance increases and the control gain is less than the absolute value of the disturbance, the output variable increases, and the control gain may be increased as needed until it is ensured that the system solution never leaves the neighborhood around ε of 0.
Further, the design of the event trigger controller in step S4 specifically includes:
when t is more than or equal to 0 and less than tswitchIn time, a controller that employs periodic sampling:
Figure BDA0003479578550000091
wherein k and lambda are more than 0, gamma is more than 0 and less than 1, and k, lambda and gamma are constants.
Suppose that
Figure BDA0003479578550000092
For the moment of demarcation of uncertainty and approach of the variable s, at
Figure BDA0003479578550000093
The change of the sliding variable s over a period of time is indeterminate
Figure BDA0003479578550000094
In the approach process, the variable | s | is in
Figure BDA0003479578550000095
In the middle and downward decrease, wherein
Figure BDA0003479578550000096
Is composed of
Figure BDA0003479578550000097
The value of the variable corresponding to the time. Therefore, the control law needs to be updated in real time in the whole process, and the process is approached by sampling with a time interval T ═ τ.
When t is more than or equal to tswitchTime, sampling event triggered controller:
Figure BDA0003479578550000098
the self-adaptive gain is switched to a barrier function, the output sliding variable s enters a relatively stable state | s (t) | < epsilon, and event triggering is adopted as a supervision condition for updating the control law, so that the burden of the controller can be reduced in the long following tracking process.
Specifically, the event triggering conditions are as follows:
Figure BDA0003479578550000099
Figure BDA00034795785500000910
wherein e isΔFor error state space vector q between two time instantseThe difference value C and D are respectively a stream set and a jump set, and when the system enters the jump set, the control law is updated; gamma is a positive number approaching 0, K1,K2,K3More than 0, is a Lipschitz constant and is self-adaptive gain
Figure BDA0003479578550000101
Because of the presence of the minimum value Γ, the process does not have the phenomenon of sesno and all states of the system are bounded.
Finally, the stability and event triggering conditions of this embodiment are verified, and in the following verification process, let ∈ sin (ν), F > 0 is an arbitrary positive number, and F should be as small as possible under the condition of ensuring convergence in order to avoid overestimating gain. In addition, considering that the tanh function replaces the conventional sign function and the derivation is complicated, the derivation of the proving process using the sign function is continued for the proving of the present embodiment.
When t is more than or equal to 0 and less than tswitchIt turns out that the arrival switching time is limited. According to the expression form of the superspiral flutter-free superspiral sliding mode approach law of the step S3:
Figure BDA0003479578550000102
wherein the content of the first and second substances,
Figure BDA0003479578550000103
the boundary of the perturbation derivative is unknown. It can be seen that, at the beginning
Figure BDA0003479578550000104
While the gain l (t) ═ at is maintained,
Figure BDA0003479578550000105
will always meet in a limited time
Figure BDA0003479578550000106
Followed by
Figure BDA0003479578550000107
Begin to fall, say at
Figure BDA0003479578550000108
At a moment have
Figure BDA0003479578550000109
At this time
Figure BDA00034795785500001010
So that s (t) is in
Figure BDA00034795785500001011
Is decreasing;
in the same way, the method starts from the beginning
Figure BDA00034795785500001012
While the gain l (t) ═ at is maintained,
Figure BDA00034795785500001013
always meet
Figure BDA00034795785500001014
Followed by
Figure BDA00034795785500001015
Begin to increase, suppose that
Figure BDA00034795785500001016
At a moment have
Figure BDA00034795785500001017
At this time
Figure BDA00034795785500001018
So that s (t) is in
Figure BDA00034795785500001019
Is incremental;
thus, s (t) can converge in a finite time, of course arriving in a finite time, with a finite gain switching time,
Figure BDA00034795785500001020
according to Lipschitz's theory, if a constant K exists, such that
Figure BDA00034795785500001021
Then Φ is the Lipschitz condition, where the constant K is referred to as the Lipschitz constant for f over the interval Θ.
Thus, define:
||f(qe(t))-ψf(qe(ti))||≤K1||qe(t)-qe(ti)||
||qe(t))-ψqe(ti)||≤K2||qe(t)-qe(ti)||
|||qe(t)|γsign(qe(t))-ψ|qe(ti)|γsign(qe(ti))||≤K3||qe(t)-qe(ti)|
wherein ψ ═ g (q)e(t)){[g(qe(ti))]-1，K1，K2，K3Greater than 0, is a Lipschitz constant, and t is an element (t)i，ti+1) T is the current time, tiThe time of the last trigger.
Therefore, the sliding mode approach law is rewritten as a time-dependent expression:
Figure BDA0003479578550000111
the Lyapunov function was designed as:
Figure BDA0003479578550000112
derivation of the Lyapunov function:
Figure BDA0003479578550000113
while
Figure BDA0003479578550000114
Thus:
Figure BDA0003479578550000121
order:
Figure BDA0003479578550000122
wherein the content of the first and second substances,
Figure BDA0003479578550000123
Figure BDA0003479578550000124
thus, by solving a quadratic equation
Figure BDA0003479578550000125
Its positive solution is to satisfy | s2(t)|1/2< ε. To ensure
Figure BDA0003479578550000126
Namely, the requirements are as follows:
Figure BDA0003479578550000127
if it is
Figure BDA0003479578550000128
When s is2≤|s(ti) When | < epsilon, V2(s(ti) Is less than 0, and the error variable forces | s (t)i) I is reduced to s (t)i)|≤|s2If | < ε, convergence of the variable at t > 0 is demonstrated. Thus, it is only necessary to ensure
Figure BDA0003479578550000129
Namely:
Figure BDA0003479578550000131
therefore, the temperature of the molten metal is controlled,
Figure BDA0003479578550000132
Figure BDA0003479578550000133
error e between momentsΔIf | | l exceeds the range, the control law needs to be updated, namely the event trigger condition is satisfied. At this time, the process of the present invention,
Figure BDA0003479578550000134
ψ＝g(qe(t)){[g(qe(ti))]-1。
the time-line variable gain ensures that the time-line variable gain approaches quickly when the time-line variable gain is far away from the sliding mode surface, the barrier function ensures that the super-helical gain is increased and then decreased when the derivative of disturbance is increased, and then the convergence supervision is achieved by utilizing the idea of event triggering, so that the output value is limited in an expected neighborhood.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (10)

1. A robot variable gain supercoiled sliding mode control method based on event triggering is characterized by comprising the following steps:
s1, constructing a robot multi-mode unified kinematics error model;
s2, designing a PI sliding mode surface related to the error;
s3, designing a flutter-free supercoiled sliding mode approach law, constructing a gain adaptive function in the flutter-free supercoiled sliding mode approach law, and switching to a variable gain based on a barrier function when a sliding variable is far away from a sliding mode face by adopting high gain linearly related to time;
s4, designing a controller based on a robot multi-mode unified kinematic error model and a flutter-free supercoiled sliding mode approach law to control the robot to track; the sliding variable far away from the sliding mode surface adopts a time triggering mode of periodic sampling; for a sliding variable that reaches a predetermined neighborhood, an event trigger form is employed.
2. The event-triggered-based robot variable-gain supercoiled sliding mode control method of claim 1, characterized in that in step S1, the constructed robot multimode unified kinematic error model
Figure FDA0003479578540000011
The following were used:
Figure FDA0003479578540000012
wherein, f (q)e)、g(qe) To be an error matrix, u is the control law, i.e., the control input, expressed as:
Figure FDA0003479578540000013
Figure FDA0003479578540000014
wherein q ise＝(xe,ye,θe)TIs an error state space vector, qe＝qr-q，qr＝(xr,yr,θr)T、qe＝(xe,ye,θe)TRespectively an actual state space vector and a reference state space vector of the robot under a map coordinate system; x, y and theta are respectively an abscissa and an ordinate of the map coordinate system in the actual state of the robot and a rotation angle of the robot coordinate system relative to the map coordinate system; x is the number ofr,yr,θrRespectively representing the abscissa and ordinate of the map coordinate system and the rotation angle of the robot coordinate system relative to the map coordinate system in the robot reference state; v. ofrIndicating the desired axial velocity, ωrIndicating the desired rotational speed of the robot body, thetafIs the angle of rotation of the front wheel, thetarIs the corner of the rear wheel, and the counterclockwise angle is specified to be positive and the clockwise angle is specified to be negative; v represents the axial velocity, ω represents the rotational speed of the robot body, and L represents the robot axial length; coefficient r is tan (θ)r)/tan(θf)。
3. The event trigger-based robot variable gain supercoiled sliding mode control method according to claim 2, characterized in that according to different motion models adopted by the robot, r is specifically: when a double Ackerman model is adopted, r is-1; when an ackerman model is adopted, r is 0; when a translation model is adopted, r is 1; when the Ackermann model is used, r ═ 1,0 @ U (0, 1).
4. The event-triggered based variable-gain supercoiled sliding mode control method of robot of claim 2, characterized in that in step S2, the error-related PI sliding mode surface S is as follows:
Figure FDA0003479578540000021
wherein s ∈ R3，qe∈R3，R3Representing a three-dimensional space vector; sign is a sign function, t is time, k, lambda and gamma are all constants, k is greater than 0, lambda is greater than 0, and gamma belongs to (0, 1).
5. The event-triggered based variable-gain supercoiled sliding mode control method of robot of claim 4, characterized in that in step S3, the flutter-free supercoiled sliding mode approximation law
Figure FDA0003479578540000025
The method comprises the following specific steps:
Figure FDA0003479578540000022
wherein L (s (t)) is a gain adaptive function and δ is a positive numberSatisfies 0 < delta < 1, tan h(s) is hyperbolic tangent function,
Figure FDA0003479578540000023
is composed of
Figure FDA0003479578540000026
S (t) is a slip variable,
Figure FDA0003479578540000024
a time-varying perturbation of an unknown boundary.
6. The event trigger-based robot variable-gain supercoiled sliding mode control method according to claim 5, wherein the gain adaptive function is specifically:
Figure FDA0003479578540000031
wherein, F is a normal number which is as small as possible under the condition of ensuring convergence; v is an arbitrary positive number, v → 0+(ii) a sin (v) denotes the neighborhood around the zero point, α is an arbitrary positive number, tswitchIs the gain switching time.
7. The event-triggered-based variable-gain supercoiled sliding mode control method for the robot as claimed in claim 6, wherein in step S4, the controller is designed as follows:
when t is more than or equal to 0 and less than tswitchIn time, a controller that employs periodic sampling:
Figure FDA0003479578540000032
wherein u is a control law,
Figure FDA0003479578540000033
ur＝(vr,ωr)Tfor reference to the input signal, vr、ωrRespectively representing a reference axial speed and a reference rotating speed of the robot;
when t is more than or equal to tswitchAnd then, adopting an event-triggered controller:
Figure FDA0003479578540000034
wherein, tiAt the time of the ith sample, u (t)i)、s(ti)、L(s(ti))、qe(ti) Are each tiControl input at time, sliding variable, gain, error state space vector, g (q)e(ti))、f(qe(ti) Respectively is tiG (q) of timee)、f(qe)。
8. The event-based triggering-based variable-gain supercoiled sliding mode control method of robot as claimed in claim 7, wherein in step S4, when t is more than or equal to 0 and less than tswitchAnd updating the control law in real time, and approaching by adopting sampling at fixed time intervals.
9. The event trigger-based robot variable gain supercoiled sliding mode control method of claim 7, wherein in step S4, when t ≧ tswitchThe event triggering conditions are specifically as follows:
Figure FDA0003479578540000041
Figure FDA0003479578540000042
wherein e isΔFor the difference of the error state space vector between two time instants, ψ ═ g (q)e(t)){[g(qe(ti))]-1；K1,K2,K3If the value is more than 0, the values are all Lipschitz constants; Γ is a positive number approaching 0; and C and D are respectively a stream set and a hop set, and when the system enters the hop set, the control law is updated.
10. An event trigger based robot variable gain supercoiled sliding mode control system, characterized in that it comprises a processor for executing the event trigger based robot variable gain supercoiled sliding mode control method according to any of claims 1-9.
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