CN114851196B - Mechanical arm track tracking control method based on fuzzy self-adaptive global sliding mode - Google Patents
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Abstract
The invention discloses a mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode, which comprises the following steps of: step 1, designing a real variable parameter sliding mode surface structure based on fuzzy control; step 2, designing a sliding mode variable s; step 3, designing the input of the fuzzy controller as an error q e The output is a time-varying coefficient alpha 1 (t)、α 2 (t), and designing a corresponding membership function and fuzzy logic; step 4, designing a global sliding mode variable sigma of an integral form; and 5, designing a mechanical arm track tracking control law and a self-adaptive law based on a fuzzy self-adaptive sliding mode to realize track tracking control. The method adopts a Lagrange dynamics model of the mechanical arm, considers model parameter uncertainty and external environment interference of the mechanical arm, designs a time-varying parameter dynamic sliding mode surface and an integral global sliding mode surface based on fuzzy control, further designs a control law and a self-adaptive law, and realizes the track tracking control of the mechanical arm.
Description
Technical Field
The invention belongs to the field of nonlinear system control, relates to a mechanical arm trajectory tracking control algorithm, and particularly relates to a mechanical arm trajectory tracking control method based on a fuzzy self-adaptive global sliding mode.
Background
The mechanical arm is a highly nonlinear system which is difficult to accurately model, and in addition, other external unknown disturbances exist when the mechanical arm works, which bring great difficulty to the problem of trajectory tracking control of the mechanical arm.
Sliding mode control is widely used in the control of nonlinear systems. Once a traditional fixed parameter sliding mode surface structure is determined, the convergence process of the system on the sliding mode surface is movement on a fixed plane, and the system is difficult to ensure that errors have higher convergence speed in different positions and areas of the plane.
Due to the fact that uncertainty of a model when the mechanical arm works and uncertainty of external interference are considered, the requirement for the robustness of the system is increased.
Disclosure of Invention
The invention aims to provide a mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode.
The purpose of the invention is realized by the following technical scheme:
a mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode comprises the following steps:
s(t)＝f(x(t),k(t))；
k(t)＝α(t)k；
wherein alpha (t) is a time-varying coefficient, and k is a fixed parameter;
K 1 (t)＝α 1 (t)K 1 ；
K 2 (t)＝α 2 (t)K 2 ；
wherein, K 1 ,K 2 ＞0，
λ 2 ＞1，0＜α 1 (t),α 2 (t)＜1；q e Is the angular position error of the mechanical arm;
the fuzzy logic is: when inputting q e At a higher value, output alpha 2 (t) is a large value, α 1 (t) is a smaller value; when inputting q e When the size is proper, alpha is output 2 (t) is the appropriate median value, α 1 (t) is a moderate value; when inputting q e When smaller, output alpha 2 (t) is a smaller value, α 1 (t) is greater;
wherein, K 3 ＞0，γ＜1；
And 5, aiming at the sliding mode variables, the fuzzy controller and the global sliding mode variables designed in the steps 2, 3 and 4, designing a mechanical arm track tracking control law and a self-adaptive law based on a fuzzy self-adaptive sliding mode to realize track tracking control, wherein:
the mechanical arm track tracking control law based on the fuzzy self-adaptive sliding mode is as follows:
wherein k is more than 0,q d In order to be the desired trajectory of the robotic arm,
is an adaptive law.
The adaptive law is:
wherein λ is 0 ,λ 1 ,λ 2 > 0,q is the angular position of the robotic arm.
Compared with the prior art, the invention has the following advantages:
the invention designs a controller based on a fuzzy self-adaptive global sliding mode aiming at the problem of track tracking of a mechanical arm, wherein a model of a control object is a rigid mechanical arm considering model uncertainty and external interference, and a control law and a self-adaptive law are designed by adopting a self-adaptive sliding mode control method to realize track tracking control. Meanwhile, fuzzy control is introduced, sliding mode parameters are adjusted in real time according to the distance between the state variable of the system and a balance point, the whole convergence process of the system on a sliding mode surface has high convergence speed, torque waste is reduced, the robustness of the system is improved through the global sliding mode, the system can effectively overcome model uncertainty and external interference, meanwhile, the upper bound specific value of an uncertainty item does not need to be known, and the normal work of the mechanical arm system under different aging degrees and different environments is facilitated.
Drawings
FIG. 1 is a block diagram of a mechanical arm trajectory tracking control system based on a fuzzy adaptive sliding mode;
FIG. 2 is a membership function of a fuzzy control input;
FIG. 3 is a membership function of the fuzzy control output;
fig. 4 is an angular position error convergence curve when d = 0.1;
fig. 5 is an angular velocity error convergence curve at d = 0.1;
fig. 6 is a moment curve at d = 0.1;
fig. 7 is an angular position error convergence curve when d =0.1 and d = 0.2;
fig. 8 is an angular position error convergence curve when d =0.3 and d = 0.4;
fig. 9 is an angular velocity error convergence curve when d =0.1 and d = 0.2;
fig. 10 is an angular velocity error convergence curve when d =0.3 and d = 0.4;
FIG. 11 is a graph of angular position, angular velocity error convergence for fixed plus sinusoidal perturbations;
FIG. 12 is a graph of angular position, angular velocity error convergence for a fixed disturbance whitening noise;
FIG. 13 is an angular position, angular velocity error convergence curve for a fixed parameter sliding mode structure;
FIG. 14 is an angular position, angular velocity error convergence curve of a time varying parameter sliding mode structure;
fig. 15 is a moment curve for a fixed parameter (left) and time varying parameter (right) slip-form configuration.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings, but not limited thereto, and any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.
The invention provides a mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode, which comprises the following steps of:
In slip form control, the configuration of the slip form face will determine the movement of the system on the slip form face. For the design of the fixed parameter sliding mode surface structure, the following form can be written:
s(t)＝f(x(t),k)；
wherein s (t) is a sliding mode variable, x (t) is the state of the system, and k is a parameter of a sliding mode surface.
When the system state converges on the sliding mode surface s (t) =0=0, the kinetic equation of the system is:
f(x(t),k)＝0。
will converge to an equilibrium point on the fixed slip-form face. The traditional fixed parameter sliding mode surface structure can only ensure that the system state has higher convergence speed in certain range, thereby limiting the integral convergence process on the sliding mode surface.
Changing the fixed parameters of the slip form surface, and changing the structure of the slip form surface into:
s(t)＝f(x(t),k(t))；
k(t)＝α(t)k；
where α (t) is a time-varying coefficient and k is a fixed parameter. The structure of the sliding mode surface is continuously changed along with the convergence of the system, so that the effect of accelerating the convergence speed is achieved. The time-varying coefficient α (t) is output by the fuzzy controller.
And 2, designing a specific sliding mode surface by combining a mechanical arm mathematical model aiming at the time-varying parameter sliding mode surface structure in the step 1.
A sliding mode variable s is designed, which has the form shown below:
K 1 (t)＝α 1 (t)K 1 ；
K 2 (t)＝α 2 (t)K 2 ；
wherein, K 1 ,K 2 ＞0，
λ 2 ＞1，0＜α 1 (t),α 2 (t) < 1 is the output of the fuzzy controller; q. q.s e Is the angular position error of the robotic arm.
When the system is on the sliding mode surface s =0, the system state dynamics equation is shown as follows:
when the system error q e When the size of the particles is larger than the required size,
plays a leading role in the convergence of errors; when the system error q e When the value is close to 0, the value is,
the convergence of the error is dominant.
And 3, designing a fuzzy controller aiming at the time-varying parameters in the step 2.
In step 2, the convergence dominant term of the system on the sliding mode surface is subjected to an error q e The influence of (c). The input to the design fuzzy controller is the error q e The output is a time-varying coefficient alpha 1 (t)、α 2 (t), designing a corresponding membership function, wherein the specific expression is as follows:
membership function of input:
membership function of output:
where i =1,2.
The fuzzy logic of the fuzzy controller is as follows:
when inputting q e When larger, output alpha 2 (t) is a large value, α 1 (t) is a smaller value; when inputting q e When the size is moderate, alpha is output 2 (t) is the appropriate median value, α 1 (t) is a moderate value; when inputting q e When smaller, output alpha 2 (t) is a smaller value, α 1 (t) is a large value.
The fuzzy controller plays a role in further strengthening the dominant item and weakening the non-dominant item, so that the convergence speed of the system is accelerated. Meanwhile, because each parameter influences the output of the control torque, the waste of the torque can be reduced.
And 4, designing a global sliding mode variable aiming at the time-varying parameter sliding mode surface structure designed in the step 2.
In a sliding mode control system, a system convergence process is divided into two stages, wherein one stage is an arrival stage, namely a stage that the system arrives at a sliding mode surface from an initial position; the second is the sliding phase, i.e. the system is on the sliding surface, and the convergence is limited only by the sliding surface itself and converges to the equilibrium point along the sliding surface. The global sliding mode enables the system to be on the sliding mode surface from the beginning, and enables the original arrival section to have the strong robustness of sliding mode control.
Designing a global sliding mode variable:
wherein, K 3 ＞0，γ＜1。
Meanwhile, due to the introduction of the integrator, the change of the sliding mode variable sigma is more gradual than that of the variable s, and the buffeting of the control moment can be reduced when the variable s is used as a control quantity.
And 5, designing a mechanical arm track tracking control law and an adaptive law aiming at the sliding mode variables and the fuzzy controllers designed in the steps 2, 3 and 4.
The track tracking control law of the mechanical arm based on the fuzzy self-adaptive sliding mode is designed as follows:
wherein k is more than 0,q d In order to be the desired trajectory of the robotic arm,
is an adaptive law.
The self-adaptation law is designed as follows:
wherein λ is 0 ,λ 1 ,λ 2 > 0,q is the angular position of the robotic arm.
Example (b):
for a typical rigid mechanical arm, the lagrangian dynamics model can be represented by the following second-order nonlinear differential equation:
wherein,
respectively representing angular position, angular velocity and angular acceleration of the joint; m (q) is belonged to R n×n Is a symmetric positive definite matrix, representing inertia;
representing centrifugal force and coriolis force torque; g (q) epsilon R n Representing gravitational torque; tau epsilon to R n The control torque is indicated.
Writing the above equation in the form of an error:
wherein q is d Angular position of desired joint, q e ＝q-q d Is the joint angular position error.
The model uncertainty of the mechanical arm is taken into account and expressed in the equation as the uncertainty of the parameters. Considering the external disturbance and friction torque of the mechanical arm, set to tau d And
the kinetic equation becomes:
wherein M is 0 (q)、
G 0 (q) is a known nominal fraction, Δ M (q)、
Δ G (q) model uncertainty, τ d And
respectively a fixed disturbance and a time-varying disturbance.
Further simplifying as follows:
wherein ρ 0 、ρ 1 、ρ 2 For constants greater than 0, | | · | | represents a two-norm of the vector, satisfying the assumption:
the friction torque is considered so that the following equation holds:
wherein, γ 0 、γ 1 、γ 2 A constant greater than 0, satisfying the assumption that:
the final written error is of the form:
a sliding mode variable s is designed, which has the form shown below:
wherein, K 1 ,K 2 ＞0，
λ 2 ＞1，0＜α 1 (t),α 2 (t) < 1 is the output of the fuzzy controller; q. q.s e Is the angular position error of the robotic arm.
When the system is on the sliding mode surface s =0, the system state dynamics equation is shown as follows:
when the system error q e When the size of the composite material is larger,
plays a leading role in the convergence of errors; when the system error q e When the value is close to 0, the value is,
the convergence of the error is dominant.
According to the logic idea of the dominant control, the two norms of the system state error are | | | q e And | l is used as the input of fuzzy control, so as to measure the distance between the system and the balance point. The fuzzy controller is designed as follows:
fuzzy control input q e The membership function of | is shown in FIG. 2, and the fuzzy control outputs alpha 0 (t),α 1 (t),α 2 (t) membership function as shown in FIG. 3, fuzzy rule as shown in Table 1, and center of gravity method for defuzzification.
TABLE 1 fuzzy rules
| ||q e || | α 1 (t) | α 2 (t) |
| S | B | S |
| M | M | M |
| B | S | B |
Note: s, B, M is a fuzzy set, representing small, large, medium.
Through fuzzy self-adaptive control, the parameters of the sliding mode surface are self-adaptively adjusted according to the position of the distance balance point in the convergence process of the system. When the system is far from the equilibrium point, | q e The greater the value of the | l,
the term plays a leading role in error convergence, and the output coefficient alpha of the fuzzy adaptive controller is output at the moment 2 (t) is greater, α 1 (t) is smaller, thereby enhancing the dominance of the term. Similarly, when the system is at a moderate distance from the balance point, the output alpha is output 2 (t) is the appropriate median value, α 1 (t) is a moderate value of (a),
and
collectively as a leading item; working systemOutput alpha when the distance from the balance point is small 2 (t) is a smaller value, α 1 (t) is a larger value, enhanced
The dominant role of the term. The dominant term action of different stages is enhanced through a fuzzy controller, so that the convergence speed is improved. Since these parameters directly affect the magnitude of the control torque, the introduction of such adaptive control can also reduce the waste of control torque.
In order to further improve the robustness of the system and reduce buffeting of control moment, a global sliding mode variable is designed:
wherein, K 3 ＞0，γ＜1。
Due to the introduction of the integrator, the system is located on the sliding mode surface σ =0 from the beginning, and the system is always located on (near) σ =0 through designing a control law, so that the sliding mode variable s moves according to the following rule:
where | ε | is finite and very small.
According to the method, the obtained control law can be designed as follows:
wherein, k is more than 0,
is an adaptive law.
The self-adaptive law is designed as follows:
wherein λ is 0 ,λ 1 ,λ 2 > 0, |, is a norm of the vector.
The stability analysis of the designed control law and adaptive law is as follows:
from formula (7):
the kinetic equation for the variable s obtained by substituting equations (11) and (12) into equation (6) is:
from formula (9):
the kinetic equation for the variable σ available in equation (15) is taken from equation (16):
consider the Lyapunov function V, which is of the form:
derivation of this can yield:
bringing (17) into (19) yields:
by the nature of the robotic arm
Being an antisymmetric matrix, it is further available:
in conclusion, the designed fuzzy adaptive global sliding mode-based control law can enable the system to be always on the sliding mode surface σ =0, so that the system state error moves on the sliding mode surface and converges to the balance point.
Selecting a two-degree-of-freedom rigid mechanical arm for simulation, wherein a mathematical model is shown as a formula (21):
wherein the form of each matrix is:
the parameter expressions in the coefficient matrix are as follows:
consider the model uncertainty parameter d times, i.e.: m is 1 ＝(1+d)m 10 ，m 2 ＝(1+d)m 20 ，l 1 ＝(1+d)l 10 ，l 2 ＝(1+d)l 20 ，J 1 ＝(1+d)J 10 ，J 2 ＝(1+d)J 20 。
The physical quantity parameters of the mechanical arm are as follows:
quality: m is 10 ＝0.5kg，m 20 ＝1.5kg。
Length: l 10 ＝1m，l 20 ＝0.8m。
Moment of inertia: j. the design is a square 10 ＝J 20 ＝5kg·m 2 。
Acceleration of gravity: g =9.8N/kg.
Setting a desired track:
the uncertainty interference in setting (4) is: tau is d ＝5，
Parameters of sliding mode variables in the set types (7) and (9) are as follows: k 1 ＝5，K 2 ＝2，K 3 ＝10，λ 1 ＝0.6，λ 2 ＝2，γ＝0.7。
The control law and adaptive law parameters shown in the set (11) and (12) are as follows: k =5, λ 0 ＝λ 1 ＝λ 2 =1 and limits the maximum value of the control torque to 150N · m.
The simulation results with d taken to be 0.1 are shown in fig. 4-6. As can be seen from fig. 4 to 6, the controller is designed to make the system converge to the equilibrium point in a short time while controlling the magnitude of the torque within a suitable range. To further illustrate the superiority of the present invention, the following comparative simulation experiments were conducted.
The convergence curves under different degrees of uncertainty are shown in fig. 7-10, and it can be seen from fig. 7-10 that, as the degree of uncertainty increases, the convergence curves of the angular position and the angular velocity error do not change significantly, the convergence speed is basically unchanged, and the strong robustness of the system to the model uncertainty is verified.
Setting the uncertainty coefficient of the model to be 0.1 again, and carrying out simulation experiments on different types of environmental interference, wherein the results are as follows:
the external disturbance is set to the following form: tau is d ＝5，
For the superposition of the stationary and sinusoidal perturbations, the convergence curve is shown in FIG. 11.
The external disturbance is set to the following form: tau is d ＝5，
A gaussian white noise signal with a mean value of 0 and a variance of 1, which is a superposition of fixed disturbance plus white noise, and the convergence curve is shown in fig. 12. As can be seen from FIG. 12, the simulation curve of the system has no obvious change under different environmental interferences, the convergence rate is basically unchanged, and the robustness of the system to the environmental interferences is verified.
And resetting the external disturbance into a superposition form of fixed disturbance and sinusoidal disturbance, and setting the output of the fuzzy controller as: alpha (alpha) ("alpha") 1 ＝α 2 =0.55, the system will be degraded to a fixed parameter sliding mode structure, and the result is shown in fig. 13 to 14 by comparing it with the convergence process under the time-varying parameter sliding mode structure.
The convergence rate is directly influenced by the magnitude of the control torque, and the comparison torque curve is shown in fig. 15 while considering the convergence rate. As can be directly seen from the convergence curve in fig. 15, the convergence speed under the time-varying parameter sliding mode structure is faster than that of the fixed parameter sliding mode structure, and the magnitude of the moment cannot be directly obtained from the curve. Defining the average torque index in the convergence process t time as:
wherein I (τ) ∈ R n ，τ∈[τ 1 τ 2 … τ n ] T ∈R n 。
The average torque indexes of all joints under the fixed parameter sliding mode structure are as follows:
the average torque indexes of all joints under the time-varying parameter sliding mode structure are as follows:
under the condition that the time-varying parameter sliding mode structure can be obtained, the average torque of each joint in the convergence process is smaller than that of the fixed parameter sliding mode structure, and the superiority of the time-varying parameter sliding mode structure based on fuzzy control is verified, namely the convergence speed is accelerated, and the torque waste is reduced.
Claims (3)
1. A mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode is characterized by comprising the following steps:
step 1, aiming at the track tracking control problem of the rigid mechanical arm considering model uncertainty and external interference, designing a real variable parameter sliding mode surface structure based on fuzzy control:
s(t)＝f(x(t),k(t))；
k(t)＝α(t)k；
wherein x (t) is the state of the system, alpha (t) is a time-varying coefficient, and k is a fixed parameter;
step 2, aiming at the time-varying parameter sliding mode surface structure in the step 1, a mechanical arm mathematical model is combined to design a sliding mode variable s, and parameters of the sliding mode variable s are adaptively adjusted through the output of a fuzzy controller, wherein the specific form is as follows:
K 1 (t)＝α 1 (t)K 1 ；
K 2 (t)＝α 2 (t)K 2 ；
wherein, K 1 ,K 2 ＞0，
λ 2 ＞1，0＜α 1 (t),α 2 (t)＜1；q e Is the angular position error of the mechanical arm;
step 3, designing the input of the fuzzy controller as an error q e The output is a time-varying coefficient alpha 1 (t)、α 2 (t) and designing corresponding membership function and fuzzy logic, wherein:
the specific expression of the membership function is as follows:
membership function of input:
output membership function:
wherein i =1,2;
the fuzzy logic is: when inputting q e When larger, output alpha 2 (t) is a large value, α 1 (t) is a smaller value; when inputting q e When the size is moderate, alpha is output 2 (t) is the appropriate median value, α 1 (t) is a moderate value; when inputting q e When smaller, output alpha 2 (t) is a smaller value, α 1 (t) is greater;
step 4, aiming at the sliding mode variable designed in the step 2, designing a global sliding mode variable sigma of an integral form:
wherein, K 3 ＞0，γ＜1；
And 5, aiming at the sliding mode variables, the fuzzy controller and the global sliding mode variables designed in the steps 2, 3 and 4, designing a mechanical arm track tracking control law and a self-adaptive law based on a fuzzy self-adaptive sliding mode to realize track tracking control.
2. The method for tracking and controlling the trajectory of the mechanical arm based on the fuzzy adaptive global sliding mode according to claim 1, wherein the trajectory tracking and controlling law of the mechanical arm based on the fuzzy adaptive sliding mode is as follows:
wherein k is more than 0,q d In order to be the desired trajectory of the robotic arm,
is an adaptive law.
3. The mechanical arm trajectory tracking control method based on the fuzzy adaptive global sliding mode according to claim 1, characterized in that the adaptive law is as follows:
wherein λ is 0 ,λ 1 ,λ 2 > 0,q is the angular position of the robotic arm.
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