CN107957682B - Enhanced fast power-order approach law sliding mode control method of quad-rotor unmanned aerial vehicle system - Google Patents

Enhanced fast power-order approach law sliding mode control method of quad-rotor unmanned aerial vehicle system Download PDF

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CN107957682B
CN107957682B CN201710532397.2A CN201710532397A CN107957682B CN 107957682 B CN107957682 B CN 107957682B CN 201710532397 A CN201710532397 A CN 201710532397A CN 107957682 B CN107957682 B CN 107957682B
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陈强
陈凯杰
叶艳
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Zhejiang University of Technology ZJUT
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Abstract

An enhanced fast power approximation law sliding mode control method of a four-rotor unmanned aerial vehicle system is designed by combining a fast power approximation law sliding mode control method aiming at the four-rotor unmanned aerial vehicle system. The design of the enhanced fast power approach law is to ensure that the sliding mode of the system can reach the sliding mode surface more quickly, and meanwhile, the buffeting phenomenon of the system is not increased, so that the fast and stable control of the system is realized.

Description

Enhanced fast power-order approach law sliding mode control method of quad-rotor unmanned aerial vehicle system
Technical Field
The invention relates to an enhanced fast power-law approach sliding mode control method of a four-rotor unmanned aerial vehicle system.
Background
The four-rotor aircraft is one of the rotor aircraft, and attracts wide attention of universities, research institutions and companies at home and abroad due to the advantages of small volume, good maneuverability, simple design, low manufacturing cost and the like. The rotor unmanned aerial vehicle is very suitable for civil and military fields such as monitoring and reconnaissance. In the civil field, the rotor unmanned aerial vehicle is mainly applied to disaster relief, ground monitoring, high-altitude aerial photography and the like; because its concealment is high, the good reliability also is used for military fields such as battlefield control, military reconnaissance. In the aspect of scientific research, the quad-rotor unmanned aerial vehicle has the dynamic characteristics of nonlinearity, under-actuation and strong coupling, and is often used as an experimental carrier for theoretical research and method verification by researchers. Aiming at the control problem of a four-rotor unmanned aerial vehicle system, a plurality of control methods exist, such as PID control, self-adaptive control, sliding mode control and the like.
The method for approaching law sliding mode control can improve the rapidity and the robustness of the quad-rotor unmanned aerial vehicle, and greatly weakens the buffeting problem caused by the traditional sliding mode control. The sliding mode can be designed according to the requirement, and the sliding mode movement of the system is irrelevant to the parameter change of a control object and the external interference, so that the robustness of the sliding mode variable structure control system is stronger than that of a common conventional continuous system. However, the conventional sliding mode variable structure causes a singularity problem and a buffeting phenomenon. Compared with the traditional fast power approximation law sliding mode control, the enhanced fast power approximation law sliding mode control has the advantages that the approximation speed can be self-adjusted, the approximation speed is higher, and the arrival time is shorter.
Disclosure of Invention
In order to overcome the defects of too low approach speed and too long arrival time of the existing four-rotor unmanned aerial vehicle system, the invention provides an enhanced fast power approach law sliding mode control method of the four-rotor unmanned aerial vehicle system, and the system is ensured to arrive at a sliding mode surface more quickly.
The technical scheme proposed for solving the technical problems is as follows:
an enhanced fast power-law approach sliding mode control method of a four-rotor unmanned aerial vehicle system comprises the following steps:
step 1, determining a transfer matrix from a body coordinate system based on a quad-rotor unmanned aerial vehicle to an inertial coordinate system based on the earth;
Figure RE-GDA0001501536530000021
psi, theta and phi are respectively the yaw angle, pitch angle and roll angle of the unmanned aerial vehicle, and represent the rotation angle of the unmanned aerial vehicle around each axis of the sequential inertial coordinate system, and TψTransition matrix, T, representing psiθA transition matrix, T, representing thetaφA transition matrix representing phi;
step 2, analyzing the unmanned aerial vehicle dynamic model according to a Newton Euler formula, wherein the process is as follows:
2.1, the translation process comprises the following steps:
Figure RE-GDA0001501536530000022
wherein x, y, z represent unmanned aerial vehicle position under the inertial coordinate system respectively, and m represents unmanned aerial vehicle's quality, and g represents acceleration of gravity, and mg represents the gravity that unmanned aerial vehicle receives, and resultant force U that four rotors producedr；
2.2, the rotation process comprises the following steps:
Figure RE-GDA0001501536530000023
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzRespectively representing the rotational inertia component of each axis on the coordinate system of the machine body, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,
Figure RE-GDA0001501536530000024
respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the unmanned aerial vehicle generally flies at a low speed or hovers at a low speed and has small change of the attitude angle, the unmanned aerial vehicle is considered to be
Figure RE-GDA0001501536530000025
Then the formula (3) is represented as the formula (4) in the rotation process
Figure RE-GDA0001501536530000026
The unmanned aerial vehicle dynamics model obtained through the joint type (1), (2) and (4) is shown as a formula (5)
Figure RE-GDA0001501536530000031
Wherein
Figure RE-GDA0001501536530000033
Ux、Uy、UzThe input quantities of the three position controllers are respectively;
2.3, according to the formula (5), performing decoupling calculation on the position and posture relationship, wherein the result is as follows:
Figure RE-GDA0001501536530000035
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidFor desired signal values of ψ, the arcsin function is an arcsine function and the arctan function is an arctangent function;
step 3, calculating the tracking error of the position, the position sliding mode surface and the first derivative thereof at each sampling moment, and decoupling out the combined external force U according to the formula (6)rAnd expected value of attitude angle phid、θdCalculating a tracking error of the attitude angle, a sliding mode surface of the attitude angle and a first derivative thereof, and designing a position controller and an attitude angle controller, wherein the process is as follows:
3.1, defining the position tracking error and its first and second differentials:
Figure RE-GDA0001501536530000036
wherein i is 1, 2, 3, X1＝x，X2＝y，X3＝z，X1dRepresenting the desired signal of X, X2dThe desired signal, X, representing y3dRepresenting the desired signal of z, e1Indicating the position tracking error of x, e2Indicating the position tracking error of y, e3A position tracking error representing z;
3.2, slip form surfaces defining position:
Figure RE-GDA0001501536530000037
wherein c isiIs a normal number, s1Sliding form face of x, s2Sliding form face of y, s3A slip form face of z;
3.3, respectively carrying out derivation on two sides of the formula (8) to obtain a first derivative of the sliding mode surface as
Figure RE-GDA0001501536530000038
Substituting formula (7) for formula (9) to obtain
Substituting formula (5) for formula (10) to obtain
Wherein U is1＝Ux，U2＝Uy，U3＝Uz；
3.4, selecting the approximation law sliding mode
Figure RE-GDA0001501536530000043
Wherein
Figure RE-GDA0001501536530000044
0<δi<1，γi>0，piIs a positive integer, k1i>0，k2i>0，0<βi<1，αi>1, sign function is a sign function;
the joint type (10) and the formula (11) obtain the input of the position controller:
Figure RE-GDA0001501536530000045
3.5 decoupling out of external force U according to equation (6)rAnd the expected value of attitude angle phid、θdThe tracking error defining the attitude angle and its first and second differentials:
Figure RE-GDA0001501536530000046
wherein j is 4, 5, 6, X4＝φ，X5＝θ，X6＝ψ，X4dRepresenting the desired signal of phi, X5dThe desired signal, X, representing theta6dThe desired signal, e, representing psi4Indicating a tracking error of phi, e5Denotes the tracking error of theta, e6A tracking error representing ψ;
3.6, slip form surfaces defining attitude angles:
Figure RE-GDA0001501536530000047
wherein c isjIs a normal number, s4Phi slip form face, s5Sliding form surface of theta, s6A slip-form face of psi;
3.7, respectively carrying out derivation on two sides of the formula (14) to obtain a first derivative of the sliding mode surface of the attitude angle as
Figure RE-GDA0001501536530000048
Substituting formula (13) for formula (15) to obtain
Substituting formula (5) for formula (16) to obtain
Figure RE-GDA00015015365300000410
Wherein U isjAs input to the attitude angle controller, U4＝τx，U5＝τy，U6＝τz,
Figure RE-GDA0001501536530000051
Figure RE-GDA0001501536530000052
B4(x)＝b1，B5(x)＝b2，B5(x)＝b3；
3.8, selecting the approximation law sliding mode
Figure RE-GDA0001501536530000053
Wherein
Figure RE-GDA0001501536530000054
0<δj<1，γj>0，pjIs a positive integer, k1j>0，k2j>0，0<βj<1，αj>1；
A joint type (17) and an equation (18) for obtaining an input of the attitude angle controller:
further, the enhanced fast power-law approach sliding mode control method further comprises the following steps:
step 4, proving that the sliding mode can reach the vicinity of the balance zero point in limited time, and simultaneously verifying that the arrival time of the enhanced fast power approximation law is less than that of the traditional fast power approximation law, wherein the process is as follows:
4.1, design Lyapunov function
Figure RE-GDA0001501536530000056
The derivation is performed on both sides of the function to obtain:
Figure RE-GDA0001501536530000057
wherein0<δ<1，γ>0, p is a positive integer, s is a slip form face, 1>β>0，α>1，k1>0，k2>0，
Figure RE-GDA0001501536530000059
Due to D(s)>0, thenTherefore, according to the accessibility of the sliding mode, the sliding mode can reach the vicinity of the equilibrium point in a limited time;
4.2 comparing the arrival time with the traditional fast power approach law sliding mode control method, the process is as follows:
for the enhanced fast power approach law, when the initial position s (0) >1, the first term plays a dominant role in the process of s (0) → s ═ 1, and thus, there is formula (19)
Figure RE-GDA00015015365300000512
Figure RE-GDA00015015365300000513
When the initial position s (0) < -1, the first term dominates for the process of s (0) → s ═ 1
Simultaneous (20) and (21) in the process of s (0) → sign [ s (0) ], obtaining (22)
Figure RE-GDA0001501536530000063
Wherein | s (0) | > 1;
for the conventional fast power-of-approximation law, the arrival time in the process of s (0) → sign [ s (0) ] is
Figure RE-GDA0001501536530000065
Thus, in the process of s (0) → sign [ s (0) ], the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law;
similarly, the second term dominates the sign [ s (0) ] → 0 process
Figure RE-GDA0001501536530000066
The arrival time of the enhanced fast power is
Figure RE-GDA0001501536530000067
The conventional fast power approximation law has an arrival time of
Figure RE-GDA0001501536530000068
Figure RE-GDA0001501536530000069
Thus, in sign [ s (0) ] → 0, the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law;
in summary, the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law.
The sliding mode control method of the four-rotor unmanned aerial vehicle system is designed based on enhanced fast power approximation law control, stable control of the system is achieved, and time for a sliding mode to reach a sliding mode surface is shortened, so that time for fixed-point flight of the unmanned aerial vehicle is shortened.
The technical conception of the invention is as follows: aiming at a four-rotor unmanned aerial vehicle system, an enhanced fast power approximation law sliding mode control method of the four-rotor unmanned aerial vehicle system is designed by combining a fast power approximation law sliding mode control method. The design of the enhanced fast power approach law is to ensure that the sliding mode of the system can reach the sliding mode surface more quickly, and meanwhile, the buffeting phenomenon of the system is not increased, so that the fast and stable control of the system is realized.
The invention has the advantages that: the robustness of the system is enhanced, and compared with the traditional fast power approach law sliding mode control, the arrival time of a sliding mode is shortened under the condition of not increasing buffeting, so that the system can realize stable convergence more quickly.
Drawings
Fig. 1 is a schematic diagram of the position tracking effect of a quad-rotor drone, where the dotted line represents the traditional double power approximation law control and the dotted line represents the enhanced double power approximation law control.
Fig. 2 is a schematic diagram of position tracking error for a quad-rotor drone, where the dashed line represents traditional double power approximation law control and the dotted line represents enhanced double power approximation law control.
Fig. 3 is a schematic diagram of input of a position controller under control of a conventional double power approach law of a quad-rotor drone.
Fig. 4 is a schematic diagram of the input of a position controller under the control of the enhanced double power law of approach for a quad-rotor drone.
Fig. 5 is an input schematic diagram of an attitude angle controller under the control of a conventional double power approach law of a quad-rotor unmanned aerial vehicle.
Fig. 6 is an input schematic diagram of an attitude angle controller under enhanced double power approximation law control of a quad-rotor drone.
Fig. 7 is a schematic diagram of a position sliding mode surface, wherein a dotted line represents a conventional double power approximation law control and a dotted line represents an enhanced double power approximation law control.
Fig. 8 is a schematic diagram of a position sliding mode surface, wherein a dotted line represents a conventional double power approximation law control and a dotted line represents an enhanced double power approximation law control.
FIG. 9 is a control flow diagram of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 9, an enhanced fast power-law approach sliding mode control method for a quad-rotor unmanned aerial vehicle system includes the following steps:
step 1, determining a transfer matrix from a body coordinate system based on a quad-rotor unmanned aerial vehicle to an inertial coordinate system based on the earth;
psi, theta and phi are respectively the yaw angle, pitch angle and roll angle of the unmanned aerial vehicle, and represent the rotation angle of the unmanned aerial vehicle around each axis of the sequential inertial coordinate system, and TψTransition matrix, T, representing psiθA transition matrix, T, representing thetaφA transition matrix representing phi;
step 2, analyzing the unmanned aerial vehicle dynamic model according to a Newton Euler formula, wherein the process is as follows:
2.1, the translation process comprises the following steps:
Figure RE-GDA0001501536530000082
wherein x, y, z represent unmanned aerial vehicle position under the inertial coordinate system respectively, and m represents unmanned aerial vehicle's quality, and g represents acceleration of gravity, and mg represents the gravity that unmanned aerial vehicle receives, and resultant force U that four rotors producedr；
2.2, the rotation process comprises the following steps:
Figure RE-GDA0001501536530000083
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzRespectively representing the rotational inertia component of each axis on the coordinate system of the machine body, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,
Figure RE-GDA0001501536530000084
respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the unmanned aerial vehicle generally flies at a low speed or hovers at a low speed and has small change of the attitude angle, the unmanned aerial vehicle is considered to be
Figure RE-GDA0001501536530000085
Then the formula (3) is represented as the formula (4) in the rotation process
The unmanned aerial vehicle dynamics model obtained through the joint type (1), (2) and (4) is shown as a formula (5)
Figure RE-GDA0001501536530000091
Wherein Ux、Uy、UzThe input quantities of the three position controllers are respectively;
2.3, according to the formula (5), performing decoupling calculation on the position and posture relationship, wherein the result is as follows:
Figure RE-GDA0001501536530000095
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidFor the desired signal value of ψ, the arcsin function is an arcsine function and the arctan function isAn arctangent function;
step 3, calculating the tracking error of the position, the position sliding mode surface and the first derivative thereof at each sampling moment, and decoupling out the combined external force U according to the formula (6)rAnd expected value of attitude angle phid、θdCalculating a tracking error of the attitude angle, a sliding mode surface of the attitude angle and a first derivative thereof, and designing a position controller and an attitude angle controller, wherein the process is as follows;
3.1, defining the position tracking error and its first and second differentials:
Figure RE-GDA0001501536530000096
wherein i is 1, 2, 3, X1＝x，X2＝y，X3＝z，X1dRepresenting the desired signal of X, X2dThe desired signal, X, representing y3dRepresenting the desired signal of z, e1Indicating the position tracking error of x, e2Indicating the position tracking error of y, e3A position tracking error representing z;
3.2, slip form surfaces defining position:
Figure RE-GDA0001501536530000097
wherein c isiIs a normal number, s1Sliding form face of x, s2Sliding form face of y, s3A slip form face of z;
3.3, respectively carrying out derivation on two sides of the formula (8) to obtain a first derivative of the sliding mode surface as
Figure RE-GDA0001501536530000101
Substituting formula (7) for formula (9) to obtain
Substituting formula (5) for formula (10) to obtain
Wherein U is1＝Ux，U2＝Uy，U3＝Uz；
3.4, selecting the approximation law sliding mode
Wherein
Figure RE-GDA0001501536530000105
0<δi<1，γi>0，piIs a positive integer, k1i>0，k2i>0，0<βi<1，αi>1, sign function is a sign function;
the joint type (10) and the formula (11) obtain the input of the position controller:
Figure RE-GDA0001501536530000106
3.5 decoupling out of external force U according to equation (6)rAnd the expected value of attitude angle phid、θdThe tracking error defining the attitude angle and its first and second differentials:
wherein j is 4, 5, 6, X4＝φ，X5＝θ，X6＝ψ，X4dRepresenting the desired signal of phi, X5dThe desired signal, X, representing theta6dThe desired signal, e, representing psi4Indicating a tracking error of phi, e5Denotes the tracking error of theta, e6A tracking error representing ψ;
3.6, slip form surfaces defining attitude angles:
Figure RE-GDA0001501536530000108
wherein c isjIs a normal number, s4Phi slip form face, s5Sliding form surface of theta, s6A slip-form face of psi;
3.7, respectively carrying out derivation on two sides of the formula (14) to obtain a first derivative of the sliding mode surface of the attitude angle as
Figure RE-GDA0001501536530000109
Substituting formula (13) for formula (15) to obtain
Figure RE-GDA00015015365300001010
Substituting formula (5) for formula (16) to obtain
Wherein U isjAs input to the attitude angle controller, U4＝τx，U5＝τy，U6＝τz,
Figure RE-GDA0001501536530000113
B4(x)＝b1，B5(x)＝b2，B5(x)＝b3；
3.8, selecting the approximation law sliding mode
Figure RE-GDA0001501536530000114
Wherein
Figure RE-GDA0001501536530000115
0<δj<1，γj>0，pjIs a positive integer, k1j>0，k2j>0，0<βj<1，αj>1；
A joint type (17) and an equation (18) for obtaining an input of the attitude angle controller:
the enhanced fast power-law approach law sliding mode control method further comprises the following steps:
step 4, proving that the sliding mode can reach the vicinity of a balance zero point in limited time, and simultaneously verifying that the arrival time of the enhanced rapid power approximation law is less than that of the traditional rapid power approximation law;
4.1, design Lyapunov function
Figure RE-GDA0001501536530000117
The derivation is performed on both sides of the function to obtain:
Figure RE-GDA0001501536530000118
wherein d(s) δ + (1- δ) e-γ|s|p，0<δ<1，γ>0, p is a positive integer, s is a slip form face, 1>β>0，α>1，k1>0，k2>0，
Figure RE-GDA0001501536530000119
Due to D(s)>0, then
Figure RE-GDA00015015365300001110
Therefore, according to the accessibility of the sliding mode, the sliding mode can reach the vicinity of the equilibrium point in a limited time;
4.2 comparing the arrival time with the traditional fast power approach law sliding mode control method, the process is as follows:
for the enhanced fast power approach law, when the initial position s (0) >1, the first term plays a dominant role in the process of s (0) → s ═ 1, and thus, there is formula (19)
Figure RE-GDA00015015365300001111
Figure RE-GDA0001501536530000121
When the initial position s (0) < -1, the first term dominates for the process of s (0) → s ═ 1
Figure RE-GDA0001501536530000122
Figure RE-GDA0001501536530000123
Simultaneous (20) and (21) in the process of s (0) → sign [ s (0) ], obtaining (22)
Figure RE-GDA0001501536530000124
Wherein | s (0) | > 1;
for the conventional fast power-of-approximation law, the arrival time in the process of s (0) → sign [ s (0) ] is
Figure RE-GDA0001501536530000125
Figure RE-GDA0001501536530000126
Thus, in the process of s (0) → sign [ s (0) ], the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law;
similarly, the second term dominates the sign [ s (0) ] → 0 process
Figure RE-GDA0001501536530000127
The arrival time of the enhanced fast power is
The conventional fast power approximation law has an arrival time of
Figure RE-GDA00015015365300001210
Thus, in sign [ s (0) ] → 0, the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law;
in summary, the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law.
In order to verify the effectiveness of the method, the invention provides a comparison between an enhanced fast power approximation law sliding mode control method and a traditional fast power approximation law sliding mode control method:
for more efficient comparison, all parameters of the system are consistent, i.e. X1d＝X2d＝X3d＝ 2，X6d0.5, 10 g; parameters of the slip form surface: c. C1＝c2＝c3＝1，c4＝c5＝c6＝2， k1＝k2＝k3＝1，k4＝k5＝k6＝0.2，β1＝β2＝β3＝0.7，β4＝β5＝β60.3; parameter of D(s) in enhanced fast power approximation law: delta4＝δ5＝δ6＝0.5，γ1＝γ2＝γ3＝γ4＝γ5＝γ6＝2，p1＝p2＝p3＝p4＝p5＝p 61 is ═ 1; parameters of quad-rotor unmanned aerial vehicle: m ═0.625，L＝0.1275,Ixx＝2.3×10-3，Iyy＝2.4×10-3，Izz＝2.6×10-3， L＝0.1275，KF＝2.103×10-6,KM＝2.091×10-8(ii) a Sampling parameters: t is ts＝0.007， N＝1000；
From fig. 5, we can see that the enhanced fast power approximation law can reach the sliding mode surface faster than the traditional fast power approximation law; with reference to fig. 1 and 2, it can be seen that the quad-rotor drone under the control of the enhanced fast power-law approach reaches a designated position faster than the quad-rotor drone under the control of the conventional fast power-law approach.
In conclusion, compared with the traditional fast power approximation law sliding mode control, the enhanced fast power approximation law sliding mode control has shorter arrival time, so that the system enters stable convergence more quickly.
While the foregoing has described a preferred embodiment of the invention, it will be appreciated that the invention is not limited to the embodiment described, but is capable of numerous modifications without departing from the basic spirit and scope of the invention as set out in the appended claims.

Claims (2)

1. An enhanced fast power-law approach sliding mode control method of a four-rotor unmanned aerial vehicle system comprises the following steps:
step 1, determining a transfer matrix from a body coordinate system based on a quad-rotor unmanned aerial vehicle to an inertial coordinate system based on the earth;
Figure FDA0002282470930000011
psi, theta and phi are respectively the yaw angle, pitch angle and roll angle of the unmanned aerial vehicle, and represent the rotation angle of the unmanned aerial vehicle around each axis of the sequential inertial coordinate system, and TψTransition matrix, T, representing psiθA transition matrix, T, representing thetaφA transition matrix representing phi;
step 2, analyzing the unmanned aerial vehicle dynamic model according to a Newton Euler formula, wherein the process is as follows:
2.1, the translation process comprises the following steps:
Figure FDA0002282470930000012
wherein x, y, z represent unmanned aerial vehicle position under the inertial coordinate system respectively, and m represents unmanned aerial vehicle's quality, and g represents acceleration of gravity, and mg represents the gravity that unmanned aerial vehicle receives, and resultant force U that four rotors producedr；
2.2, the rotation process comprises the following steps:
Figure FDA0002282470930000013
wherein tau isx、τy、τzRespectively representing the axial moment components, I, in the coordinate system of the machine bodyxx、Iyy、IzzRespectively representing the rotational inertia component of each axis on the coordinate system of the machine body, x represents cross product, wp、wq、wrRespectively representing the attitude angular velocity components of each axis on the coordinate system of the body,respectively representing the attitude angular acceleration components of all axes on the coordinate system of the machine body;
considering that the unmanned aerial vehicle generally flies at a low speed or hovers at a low speed and has small change of the attitude angle, the unmanned aerial vehicle is considered to be
Figure FDA0002282470930000016
Then the formula (3) is represented as the formula (4) in the rotation process
Figure FDA0002282470930000014
The unmanned aerial vehicle dynamics model obtained through the joint type (1), (2) and (4) is shown as a formula (5)
Wherein
Figure FDA0002282470930000025
Figure FDA0002282470930000026
Ux、Uy、UzThe input quantities of the three position controllers are respectively;
2.3, according to the formula (5), performing decoupling calculation on the position and posture relationship, wherein the result is as follows:
Figure FDA0002282470930000022
wherein phidIs the desired signal value of phi, thetadDesired signal value of theta, psidFor desired signal values of ψ, the arcsin function is an arcsine function and the arctan function is an arctangent function;
step 3, calculating the tracking error of the position, the position sliding mode surface and the first derivative thereof at each sampling moment, and decoupling out the combined external force U according to the formula (6)rAnd expected value of attitude angle phid、θdCalculating a tracking error of the attitude angle, a sliding mode surface of the attitude angle and a first derivative thereof, and designing a position controller and an attitude angle controller, wherein the process is as follows:
3.1, defining the position tracking error and its first and second differentials:
Figure FDA0002282470930000023
wherein i is 1, 2, 3, X1＝x，X2＝y，X3＝z，X1dRepresenting the desired signal of X, X2dThe desired signal, X, representing y3dRepresenting the desired signal of z, e1Bit representing xSet tracking error, e2Indicating the position tracking error of y, e3A position tracking error representing z;
3.2, slip form surfaces defining position:
Figure FDA0002282470930000024
wherein c isiIs a normal number, s1Sliding form face of x, s2Sliding form face of y, s3A slip form face of z;
3.3, respectively carrying out derivation on two sides of the formula (8) to obtain a first derivative of the sliding mode surface as
Figure FDA0002282470930000031
Substituting formula (7) for formula (9) to obtain
Figure FDA0002282470930000032
Substituting formula (5) for formula (10) to obtain
Wherein U is1＝Ux，U2＝Uy，U3＝Uz；
3.4, selecting the approximation law sliding mode
Figure FDA0002282470930000034
Wherein
Figure FDA00022824709300000310
0＜δi＜1，γi＞0，piIs a positive integer, k1i＞0，k2i＞0，0＜βi＜1，αiThe sign function is a sign function when the sign function is more than 1;
the joint type (10) and the formula (11) obtain the input of the position controller:
Figure FDA0002282470930000035
3.5 decoupling out of external force U according to equation (6)rAnd the expected value of attitude angle phid、θdThe tracking error defining the attitude angle and its first and second differentials:
Figure FDA0002282470930000036
wherein j is 4, 5, 6, X4＝φ，X5＝θ，X6＝ψ，X4dRepresenting the desired signal of phi, X5dThe desired signal, X, representing theta6dThe desired signal, e, representing psi4Indicating a tracking error of phi, e5Denotes the tracking error of theta, e6A tracking error representing ψ;
3.6, slip form surfaces defining attitude angles:
Figure FDA0002282470930000037
wherein c isjIs a normal number, s4Phi slip form face, s5Sliding form surface of theta, s6A slip-form face of psi;
3.7, respectively carrying out derivation on two sides of the formula (14) to obtain a first derivative of the sliding mode surface of the attitude angle as
Figure FDA0002282470930000038
Substituting formula (13) for formula (15) to obtain
Figure FDA0002282470930000039
Substituting formula (5) for formula (16) to obtain
Figure FDA0002282470930000041
Wherein U isjAs input to the attitude angle controller, U4＝τx，U5＝τy，U6＝τz,
Figure FDA0002282470930000045
Figure FDA0002282470930000046
B4(x)＝b1，B5(x)＝b2，B5(x)＝b3；
3.8, selecting the approximation law sliding mode
Figure FDA0002282470930000042
Wherein0＜δj＜1，γj＞0，pjIs a positive integer, k1j＞0，k2j＞0，0＜βj＜1，αj＞1；
A joint type (17) and an equation (18) for obtaining an input of the attitude angle controller:
2. the enhanced fast power-law approach sliding-mode control method of the quad-rotor unmanned aerial vehicle system according to claim 1, characterized in that: the enhanced fast power-law approach law sliding mode control method further comprises the following steps:
step 4, proving that the sliding mode can reach the vicinity of the balance zero point in limited time, and simultaneously verifying that the arrival time of the enhanced fast power approximation law is less than that of the traditional fast power approximation law, wherein the process is as follows:
4.1, design Lyapunov functionThe derivation is performed on both sides of the function to obtain:
Figure FDA0002282470930000044
wherein
Figure FDA0002282470930000049
Delta is more than 0 and less than 1, gamma is more than 0, p is a positive integer, s is a sliding mode surface, 1 is more than β and more than 0, α is more than 1, k is1＞0，k2＞0，
Due to D(s)>0, thenTherefore, according to the accessibility of the sliding mode, the sliding mode can reach the vicinity of the equilibrium point in a limited time;
4.2 comparing the arrival time with the traditional fast power approach law sliding mode control method, the process is as follows:
for the enhanced fast power approach law, when the initial position s (0) >1, the first term plays a dominant role in the process of s (0) → s ═ 1, and thus, there is formula (19)
Figure FDA0002282470930000051
Figure FDA0002282470930000052
Figure FDA0002282470930000053
When the initial position s (0) < -1, the first term dominates for the process of s (0) → s ═ 1
Figure FDA0002282470930000054
Figure FDA0002282470930000055
Simultaneous (20) and (21) in the process of s (0) → sign [ s (0) ], obtaining (22)
Figure FDA0002282470930000056
Wherein | s (0) | > 1;
for the conventional fast power-of-approximation law, the arrival time in the process of s (0) → sign [ s (0) ] is
Figure FDA0002282470930000058
Thus, in the process of s (0) → sign [ s (0) ], the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law;
similarly, the second term dominates the sign [ s (0) ] → 0 process
Figure FDA0002282470930000059
The arrival time of the enhanced fast power is
Figure FDA00022824709300000510
Traditional Chinese medicineThe arrival time of the fast power approximation law is
Figure FDA00022824709300000512
Thus, in sign [ s (0) ] → 0, the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law;
in summary, the arrival time of the enhanced fast power approximation law is shorter than that of the conventional fast power approximation law.
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