CN114488804B - Method and system for controlling variable-gain supercoiled sliding mode of robot based on event triggering - Google Patents

Method and system for controlling variable-gain supercoiled sliding mode of robot based on event triggering Download PDF

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CN114488804B
CN114488804B CN202210064143.3A CN202210064143A CN114488804B CN 114488804 B CN114488804 B CN 114488804B CN 202210064143 A CN202210064143 A CN 202210064143A CN 114488804 B CN114488804 B CN 114488804B
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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 variable-gain supercoiled sliding mode control method and a variable-gain supercoiled sliding mode control system for a robot based on event triggering, wherein the method comprises the following steps: s1, constructing a multi-mode unified kinematic error model of the robot; s2, designing a PI sliding mode surface related to errors; s3, designing a flutter-free supercoiled sliding mode approach law, constructing a gain self-adaptive function, enabling a sliding variable to adopt high gain linearly related to time when the sliding variable is far away from a sliding mode surface, and switching to variable gain based on a barrier function when the sliding variable reaches a preset neighborhood; s4, designing a controller based on a robot multi-mode unified kinematic error model and a flutter-free supercoiled sliding mode approach law so as to control the robot to track; for a sliding variable far away from a sliding mode surface, a time triggering mode of periodic sampling is adopted; for sliding variables reaching a predetermined neighborhood, an event-triggered form is employed. The invention can realize rapid convergence and jitter-free tracking control when the boundary information of disturbance derivative is lacking.

Description

Method and system for controlling variable-gain supercoiled sliding mode of robot based on event triggering
Technical Field
The invention belongs to the field of robot control, and particularly relates to a variable-gain supercoiled sliding mode control method and system for a robot 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 of the robot industry, and in general, a better method is to realize kinematic control by determining a kinematic model. Because of the simplicity of sliding mode control, it is insensitive to disturbances and is widely used in the field of mobile robots. The sliding mode control is basically variable structure control, the high switching gain can cause large tremble in the system, the gain is usually selected to be reduced in order to avoid the tremble, the tracking precision and the response speed are sacrificed by the method, and in addition, the method such as an integral sliding mode, a supercoiled sliding mode and the like is also presented, so that the tremble when reaching the sliding mode surface can be effectively reduced. At the same time, in complex operating environments, building an accurate model necessarily has unknown disturbances, and the commonly employed Sliding Mode Control (SMC) can provide closed loop insensitivity to unknown disturbances, and most SMC designs need to implement upper limits of 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 desired sliding surface.
Although SMC has shown its effectiveness in the industrial field, there are the following aspects to be improved when the SMC method is applied to motion control of a robot:
1) Although the supercoiled sliding mode has obvious advantages in the aspect of improving flutter, the supercoiled sliding mode approach law is basically variable speed approach, when an output variable is near a sliding mode surface, if a fixed gain is adopted, the system reaching the sliding mode surface has a larger speed, larger buffeting can be caused, and if the gain is too small, the adjusting time is long;
2) When the disturbance upper bound is unknown, self-adaptive control cannot be realized, and when the boundary information of disturbance derivative is lacking, how to realize the jitter-free SMC scheme to carry out tracking control on the mobile robot becomes an important problem.
3) The traditional periodic sampling control increases the load operation of the controller, and can not effectively monitor the system convergence when disturbance jump is frequent.
Therefore, there is a need for a tracking control method that can perform gain adaptive adjustment when the disturbance boundary is unknown, and that responds quickly.
Disclosure of Invention
Aiming at the defects or improvement demands of the prior art, the invention provides a variable-gain supercoiled sliding mode control method and system for a robot based on event triggering, which aim to realize rapid and jitter-free self-adaptive tracking control of a mobile robot when boundary information of disturbance derivative is lacking.
In order to achieve the above object, according to an aspect of the present invention, a method for controlling a variable gain supercoiled sliding mode of a robot based on event triggering is provided, comprising the steps of:
s1, constructing a multi-mode unified kinematic error model of the robot;
s2, designing a PI sliding mode surface related to errors;
s3, designing a flutter-free supercoiled sliding mode approach law, constructing a gain self-adaptive function in the approach law, enabling sliding variables to adopt high gain linearly related to time when the sliding variables are far away from a sliding mode surface, and switching to variable gain based on a barrier function when the sliding variables reach a preset neighborhood;
s4, designing a controller based on a robot multi-mode unified kinematic error model and a flutter-free supercoiled sliding mode approach law so as to control the robot to track; the sliding variable far from the sliding mode surface adopts a time triggering mode of periodic sampling; for sliding variables reaching a predetermined neighborhood, an event-triggered form is employed.
As a further preferable mode, in step S1, the constructed robot multimode unified kinematic error modelThe following are provided:
wherein f (q e )、g(q e ) For the error matrix, u is the control law, i.e., the control input, expressed as:
wherein q e ＝(x e ,y e ,θ e ) T Is an error state space vector, q e ＝q r -q，q r ＝(x r ,y r ,θ r ) T 、q e ＝(x e ,y e ,θ e ) T The actual state space vector and the reference state space vector of the robot under the map coordinate system are respectively; x, y and θ are the abscissa and ordinate of the map coordinate system in the actual state of the robot and the rotation angle of the robot coordinate system relative to the map coordinate system; x is x r ,y r ,θ r Respectively the rotation angles of the robot coordinate system relative to the map coordinate system in the horizontal coordinate, the vertical coordinate and the map coordinate system under the robot reference state; v r Indicating the desired axial velocity omega r Indicating the expected rotation speed of the robot body, θ f Is the front wheel rotation angle theta r Is the rear wheel rotation angle, the anticlockwise angle is positive, and the clockwise angle is negative; v denotes an axial velocity, ω denotes a rotational speed of the robot body, and L denotes an axial length of the robot; coefficient r=tan (θ r )/tan(θ f )。
As a further preferred, r is specifically: when the double Ackerman model is adopted, r= -1; when the ackerman model is adopted, r=0; when a translation model is adopted, r=1; when the variant ackerman model is adopted, r= (-1, 0) U (0, 1).
As a further preferred aspect, in step S2, the PI slip plane S associated with the error is as follows:
wherein s is E R 3 ，q e ∈R 3 ，R 3 Representing a three-dimensional space vector; sign is a sign function, t is time, k, λ, γ are constants, and k > 0, λ > 0, γ ε (0, 1).
As a further preferable mode, in the step S3, the non-flutter super-spiral sliding mode approaches lawThe method comprises the following steps:
wherein L (s (t)) is a gain adaptive function, delta is a positive number satisfying 0 < delta < 1, tanh(s) is a hyperbolic tangent function,is->S (t) is a sliding variable, < ->Is a time-varying disturbance of an unknown boundary.
As a further preferred, the gain adaptive function is specifically:
wherein F is a positive constant, which is thatEnsuring as small as possible under the condition of convergence; v is any positive number, v.fwdarw.0 + The method comprises the steps of carrying out a first treatment on the surface of the sin (v) represents a neighborhood near zero point, alpha is any positive number, t switch Is the gain switching time.
As a further preferred feature, in step S4, the controller is designed as follows:
when t is more than or equal to 0 and less than t switch And when the controller for periodic sampling is adopted:
wherein u is the control law,u r ＝(v r ,ω r ) T to reference the input signal v r 、ω r Respectively representing a reference axial speed and a reference rotating speed of the robot;
when t is greater than or equal to t switch And adopting an event-triggered controller:
wherein t is i For the time of the ith sample, u (t i )、s(t i )、L(s(t i ))、q e (t i ) Respectively t i Control inputs for time, sliding variables, gain, error state space vector, g (q e (t i ))、f(q e (t i ) Are t respectively i G (q) of time e )、f(q e )。
As a further preference, in step S4, when 0.ltoreq.t < t switch And when the control law is updated in real time, sampling at fixed time intervals is adopted for approaching.
As a further preference, in step S4, when t.gtoreq.t switch When the event triggering conditions are specifically:
wherein e Δ Is the difference of the error state space vector between the two moments, ψ=g (q e (t)){[g(q e (t i ))] -1 ；K 1 ,K 2 ,K 3 > 0, both Lipschitz constants; Γ is a positive number approaching 0; c, D is the flow set and jump set respectively, when the system enters the jump set, the control law is updated.
According to another aspect of the invention, a robot variable gain supercoiled sliding mode control system based on event triggering is provided, which comprises a processor, wherein the processor is used for executing a robot variable gain supercoiled sliding mode control method based on event triggering.
In general, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. the invention establishes a kinematic tracking error model containing external disturbance, designs a flutter-free supercoiled sliding mode approach law by constructing a PI sliding mode surface related to the error, and a linear self-adaptive gain far away from the sliding mode surface and a barrier function variable gain near the sliding mode surface, ensures that the control gain follows the absolute value change of the disturbance, thereby ensuring that the track tracking control error approaches zero in a limited time and improving the robustness and reliability of control; meanwhile, the thought of event triggering is adopted for convergence supervision, so that the burden of a controller in the following process can be reduced; the method can realize rapid convergence, realize jitter-free tracking control when the boundary information of disturbance derivative is lacking, and can be used for tracking control of the mobile robot in different modes.
2. The invention designs a specific super-spiral sliding mode approach law, and the super-spiral approach law 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 built aiming at the gain of the supercoiled, and high gain linearly related to time is adopted when the supercoiled is far away from a sliding mode surface, so that an output variable can quickly enter a preset adjacent area in limited time, enter the preset adjacent area and be switched to a variable gain based on a barrier function, and the use of a barrier function strategy ensures that once the derivative of disturbance is increased, the supercoiled gain is also increased, and therefore the output value is ensured to belong to a required adjacent area.
3. The invention adopts the concept of event triggering to perform convergence supervision, and the event triggering method requires less transmission required for stabilizing the system, so that the energy consumption and the network load can be reduced; meanwhile, by taking the constructed triggering condition as a supervision criterion, the convergence of the system 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 robot variable gain supercoiled sliding mode control system based on event triggering according to an embodiment of the present invention;
FIG. 3 shows a variable gain function during sliding mode approach according to an embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
The embodiment of the invention provides a robot variable gain supercoiled sliding mode control method based on event triggering, which is shown in figure 2 and comprises the following steps:
s1, constructing a multi-mode unified kinematic error model of the robot;
s2, designing a PI sliding mode surface related to errors to obtain an expression form of output quantity;
s3, constructing a gain self-adaptive function, designing a flutter-free supercoiled sliding mode approach law, adopting high gain linearly related to time when the flutter-free supercoiled sliding mode approach law is far away from a sliding mode surface, and switching to variable gain based on a barrier function when the flutter-free supercoiled 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, wherein a sliding variable far from a sliding mode surface is in a time triggering mode of periodic sampling, and the sliding variable reaching a preset neighborhood is subjected to event triggering to reduce the burden of the controller.
Further, step S1 builds a multi-mode unified kinematic error model of the robot, which specifically includes:
as shown in fig. 1, the mobile robot kinematic model is expressed as:
wherein the mobile robot state is defined as q, (x, y, θ) represents the robot pose space,the pose space derivative of the mobile robot is represented, v represents the axial velocity, ω represents the rotational speed of the robot body.
The mobile robot is a four-wheel full-steering independent driving mobile robot, can operate in different modes, and the continuous input u is expressed as:
wherein θ f Is the front wheel rotation angle, the anticlockwise angle is positive, and the clockwise angle is negative; l=l f +L r Represents the axial length of the robot, and r is the rotation angle theta of the front wheel f And rear wheel steering angle theta r The determined coefficient is expressed as r=tan (θ r )/tan(θ f ). r is represented as:
in all modes, a reference state vector q is defined r ＝(x r ,y r ,θ r ) T Reference to input signal u r ＝(v r ,ω r ) T The unified motion state equation of the mobile robot in different modes is written as follows:
thereby obtaining the error state space vector q e The method comprises the following steps:
error state space vector q e The derivative expression of (c) is:
further, the robot multimode unified kinematic error modelExpressed as: />Wherein,
further, in step S2, the PI slip form surface S related to the error specifically is:
wherein s is E R 3 ，q e ∈R 3 The method comprises the steps of carrying out a first treatment on the surface of the Lambda and gamma are constants, lambda > 0, gamma epsilon (0, 1). Deriving the following:
the expression of the calculation equation of the control law is thus:
further, in step S3, the design of the sliding mode approach law is specifically:
wherein L (s (t)) is a gain adaptive function, delta is a positive number satisfying 0 < delta < 1, tanh(s) is a hyperbolic tangent function,is->Derivative of>Is a time-varying disturbance of an unknown boundary. In addition, in the present embodiment, it is preferable to take δ as 1/2, that is, as 0.5. The adaptive gain coefficients are:
wherein F > 0 is a normal number, and F should be as small as possible under the condition of ensuring convergence in order to avoid overestimation of the gain; v is an arbitrarily small positive number v.fwdarw.0 + Sin (v) represents a neighborhood near zero point, alpha is any positive number, t switch Is the gain switching time. The sliding variable s (t) is switched to a variable gain based on a barrier function when approaching the sliding surface, with a high gain linearly dependent on time when moving away from the sliding surface s=0. Let epsilonThe variable gain supercoiled controller using a barrier function when =sin (v), s (t) e (- ε, ε) can solve the unknown interference for any ε>0 and for all t.gtoreq.0, the inequality |s (t) | for<Epsilon holds.
This strategy allows the adaptive gain to increase and decrease according to the current value of the output variable, decreasing as the output variable goes to zero until a value that allows compensation for disturbances is reached; 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 can be increased as needed until it is ensured that the system solution never leaves the epsilon neighborhood of 0.
Further, in step S4, the design of the event trigger controller is specifically:
when t is more than or equal to 0 and less than t switch And when the controller for periodic sampling is adopted:
wherein k, lambda is more than 0,0 < gamma is less than 1, and k, lambda and gamma are constants.
Assume thatBoundary moments for uncertainty and approach of variable s, at +.>The variation of the sliding variable s in the time period is uncertain, in +.>In the approach, the variable |s| is +.>Decrease in middle, wherein>Is->The variable value corresponding to the moment. The control law needs to be updated in real time throughout the process, which is approximated by sampling at time interval t=τ.
When t is greater than or equal to t switch At this time, the controller triggered by the sampling event:
the adaptive gain is switched to the 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 following long tracking process.
Specifically, the event triggering conditions are:
wherein e Δ For two time error state space vectors q e C, D are respectively a stream set and a jump set, and when the system enters the jump set, the control law is updated; Γ is a positive number approaching 0, K 1 ,K 2 ,K 3 > 0, lipschitz constant, adaptive gainBecause of the presence of the minimum Γ, the process does not suffer from the gano phenomenon and all states of the system are bounded.
Finally, the stability and the event triggering condition of this embodiment are demonstrated, and in the following proving process, epsilon=sin (v), 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 overestimation of the gain. Furthermore, considering that the tan function replaces the conventional sign function and the derivation is more complex, the derivation of the attestation process will be performed along with the sign function for attestation of the present embodiment.
When t is more than or equal to 0 and less than t switch When the arrival switching time is to be proven to be limited. According to the expression form of the supercoiled non-flutter supercoiled sliding mode approach law in the step S3:
wherein,is unknown for perturbing the boundaries of the derivative. It can be seen that at the beginning +.>At the time, under the gain L (t) =αt hold, ++>Always meet +.>Subsequently->Start to descend, assuming +.>At the moment there is +.>At this timeSo s (t) is +.>Is decreasing;
similarly, at the beginningAt the time, under the gain L (t) =αt hold, ++>Meet at will->Subsequently->Start to increase, assume->At the moment there is +.>At this time->So s (t) is +.>Is incremental;
s (t) can therefore converge in a finite time, of course arrive in a finite time, the gain switching time is finite,
according to Lipschitz theory, if a constant K is present, such thatΦ is the Lipschitz condition, where the constant K is called the Lipschitz constant for f over the interval Θ.
Thus, define:
||f(q e (t))-ψf(q e (t i ))||≤K 1 ||q e (t)-q e (t i )||
||q e (t))-ψq e (t i )||≤K 2 ||q e (t)-q e (t i )||
|||q e (t)| γ sign(q e (t))-ψ|q e (t i )| γ sign(q e (t i ))||≤K 3 ||q e (t)-q e (t i )|
wherein ψ=g (q e (t)){[g(q e (t i ))] -1 ，K 1 ，K 2 ，K 3 > 0, all Lipschitz constants, t.epsilon.t i ，t i+1 ) T is the current time, t i Is the time of the last trigger.
Therefore, the sliding mode approach law is rewritten as a time-dependent expression:
the Lyapunov function is designed as:
derivative the Lyapunov function:
whileThus:
and (3) making:
wherein,
thus, by solving the quadratic equationIts positive solution is to satisfy |s 2 (t)| 1/2 < ε. To ensure->Namely, the requirements are as follows: />
If it isWhen s is 2 ≤|s(t i ) When < ε, V 2 (s(t i ) Less than 0) and the error variable forces |s (t) i ) The I is reduced to be I s (t i )|≤|s 2 I < ε, thus proving convergence of the variables at t > 0. Therefore, only need to ensureNamely:
thus, the first and second substrates are bonded together,
error of time Δ And when the I exceeds the range, updating the control law, namely an event triggering condition. At this time, the liquid crystal display device,ψ＝g(q e (t)){[g(q e (t i ))] -1 。
the time linear variable gain ensures that the speed approaches when the distance from the sliding mode surface is far, the barrier function ensures that the supercoiled gain is increased and then decreased when the derivative of disturbance is increased, and the aim of convergence supervision is achieved by utilizing the thought of event triggering, so that the output value is ensured to be limited in the expected neighborhood.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.

Claims (6)

1. The method for controlling the variable-gain supercoiled sliding mode of the robot based on event triggering is characterized by comprising the following steps:
s1, constructing a multi-mode unified kinematic error model of a robot
Wherein f (q e )、g(q e ) For the error matrix, u is the control law, i.e., the control input, expressed as:
wherein q e ＝(x e ,y e ,θ e ) T Is an error state space vector, q e ＝q r -q，q r ＝(x r ,y r ,θ r ) T 、q e ＝(x e ,y e ,θ e ) T The actual state space vector and the reference state space vector of the robot under the map coordinate system are respectively; x, y and θ are the abscissa and ordinate of the map coordinate system in the actual state of the robot and the rotation angle of the robot coordinate system relative to the map coordinate system; x is x r ,y r ,θ r Respectively the rotation angles of the robot coordinate system relative to the map coordinate system in the horizontal coordinate, the vertical coordinate and the map coordinate system under the robot reference state; v r Indicating the desired axial velocity omega r Indicating the expected rotation speed of the robot body, θ f Is the front wheel rotation angle theta r Is the rear wheel rotation angle, the anticlockwise angle is positive, and the clockwise angle is negative; v denotes an axial velocity, ω denotes a rotational speed of the robot body, and L denotes an axial length of the robot; coefficient r=tan (θ r )/tan(θ f )；
S2, designing a PI sliding mode surface S related to errors:
wherein s is E R 3 ，q e ∈R 3 ，R 3 Representing a three-dimensional space vector; sign is a sign function, t is time, k, lambda and gamma are constants, k is more than 0, lambda is more than 0, gamma epsilon (0, 1);
s3, designing a flutter-free supercoiled sliding mode approach law, constructing a gain self-adaptive function in the approach law, enabling sliding variables to adopt high gain linearly related to time when the sliding variables are far away from a sliding mode surface, and switching to variable gain based on a barrier function when the sliding variables reach a preset neighborhood;
non-flutter super-spiral sliding mode approach lawThe method comprises the following steps:
wherein L (s (t)) is a gain adaptive function, delta is a positive number satisfying 0 < delta < 1, tanh(s) is a hyperbolic tangent function,is thatS (t) is a sliding variable, < ->Time-varying perturbations that are unknown boundaries;
the gain adaptive function is specifically:
wherein F is a positive constant which is as small as possible while ensuring convergence; v is any positive number, v.fwdarw.0 + The method comprises the steps of carrying out a first treatment on the surface of the sin (v) represents a neighborhood near zero point, alpha is any positive number, t switch Is the gain switching time;
s4, designing a controller based on a robot multi-mode unified kinematic error model and a flutter-free supercoiled sliding mode approach law so as to control the robot to track; the sliding variable far from the sliding mode surface adopts a time triggering mode of periodic sampling; for sliding variables reaching a predetermined neighborhood, an event-triggered form is employed.
2. The method for controlling the variable-gain supercoiled sliding mode of the robot based on the event triggering according to claim 1, wherein r is specifically as follows according to different motion models adopted by the robot: when the double Ackerman model is adopted, r= -1; when the ackerman model is adopted, r=0; when a translation model is adopted, r=1; when the variant ackerman model is adopted, r= (-1, 0) U (0, 1).
3. The event-triggered based control method for the variable-gain supercoiled sliding mode of a robot according to claim 1, wherein in step S4, the controller is designed as follows:
when t is more than or equal to 0 and less than t switch And when the controller for periodic sampling is adopted:
wherein u is the control law,u r ＝(v r ,ω r ) T to reference the input signal v r 、ω r Respectively representing a reference axial speed and a reference rotating speed of the robot;
when t is greater than or equal to t switch And adopting an event-triggered controller:
u(t i )＝[g(q e (t i ))] -1 [-L(s(t i ))|s(t i )| 12 sign(s(t i ))+θ-kλq e (t i )-λ|q e (t i )| γ sign(q e (t i ))-f(q e (t i ))]
wherein t is i For the time of the ith sample, u (t i )、s(t i )、L(s(t i ))、q e (t i ) Respectively t i Control inputs for time, sliding variables, gain, error state space vector, g (q e (t i ))、f(q e (t i ) Are t respectively i G (q) of time e )、f(q e )。
4. The event-triggered based control method of variable-gain supercoiled sliding mode of a robot according to claim 3, wherein in step S4, when t is 0.ltoreq.t < t switch And when the control law is updated in real time, sampling at fixed time intervals is adopted for approaching.
5. The event-triggered based method for controlling a variable-gain supercoiled slipform of a robot according to claim 3, wherein in step S4, when t is not less than t switch When the event triggering conditions are specifically:
wherein e Δ Is the difference of the error state space vector between the two moments, ψ=g (q e (t)){[g(q e (t i ))] -1 ；K 1 ,K 2 ,K 3 > 0, both Lipschitz constants; Γ is a positive number approaching 0; c, D is the flow set and jump set respectively, when the system enters the jump set, the control law is updated.
6. An event trigger based robot variable gain supercoiled sliding mode control system comprising a processor for executing the event trigger based robot variable gain supercoiled sliding mode control method according to any of claims 1 to 5.
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