CN111459188A - Multi-rotor nonlinear flight control method based on quaternion - Google Patents

Multi-rotor nonlinear flight control method based on quaternion Download PDF

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CN111459188A
CN111459188A CN202010358434.4A CN202010358434A CN111459188A CN 111459188 A CN111459188 A CN 111459188A CN 202010358434 A CN202010358434 A CN 202010358434A CN 111459188 A CN111459188 A CN 111459188A
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quaternion
rotor
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CN111459188B (en
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张永
谢志鸿
徐贤鹏
李军
马国梁
黄成�
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Nanjing University of Science and Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/10—Simultaneous control of position or course in three dimensions
    • G05D1/101—Simultaneous control of position or course in three dimensions specially adapted for aircraft
    • G05D1/102—Simultaneous control of position or course in three dimensions specially adapted for aircraft specially adapted for vertical take-off of aircraft
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
    • G05D1/0816—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability
    • G05D1/0825—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability using mathematical models
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
    • G05D1/0858—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft specially adapted for vertical take-off of aircraft

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Abstract

The invention discloses a quaternion-based multi-rotor nonlinear flight control method which comprises the following steps of firstly, assuming environmental variables and motion conditions of multiple rotors to simplify multi-rotor modeling, secondly, establishing a multi-rotor position change equation according to Newton's second law, establishing an attitude change equation according to a momentum moment theorem and a kinematics equation of quaternion, and finally, decomposing the control of the multiple rotors into position control and attitude control of an inner ring structure and an outer ring structure, and designing an outer ring position controller and an inner ring attitude controller of a multi-rotor aircraft by combining a backstepping method with an L yapunov stability theorem.

Description

Multi-rotor nonlinear flight control method based on quaternion
Technical Field
The invention belongs to the technical field of multi-rotor flight control, and particularly relates to a quaternion-based multi-rotor nonlinear flight control method.
Background
In many miniature unmanned aerial vehicles, many rotor crafts structure with control simply, the rotational speed that only needs the control motor just can realize the aircraft complicated motion such as vertical take-off landing, and is little to the influence of environment when taking off and land moreover, the small in noise that sends has many advantages in comparison with other aircraft: the aircraft has independent take-off and landing capability and environmental adaptation capability, and can fly in various modes, such as hovering, forward flying, side flying, backward flying, yawing rotation and the like. In the field of flight control, the multi-rotor aircraft is a nonlinear model and has the characteristics of strong coupling, under-actuation and the like. Therefore, the multi-rotor aircraft has wide application prospect and research significance.
At present, the attitude motion description of the multi-rotor is mostly based on Euler angles. The euler angles are relatively well understood, but the attitude change cannot be intuitively represented, and the attitude change of a relatively complex gesture is generally obtained by decomposing the gesture change into multiple rotations of the body around each coordinate axis. The rotation represented by the unit quaternion is one of the shaft angle methods, and for the attitude change of the body, the quaternion can be used for visually representing the dead-axle rotation of multiple rotors, so that the problems of singularity and universal joint deadlock are avoided.
In engineering, the control of the multiple rotors is mostly traditional controllers designed based on a PID control method, the controllers do not depend on modeling precision, influence of external environment on the multiple rotors is ignored, and the flight of the multiple rotors is assumed to be approximately linear. However, this can only be achieved in a low-speed flight or hovering state without external disturbance, and when high-speed flight is performed or the external disturbance is strong, the controller cannot effectively control the stability of the multiple rotors. Because the design steps of these controllers are simple, it is difficult to control under-actuated, strong coupling, the many rotor systems that are easily influenced by the external world, when many rotors are carrying out track tracking flight, it is difficult to consider the interference killing feature when requiring tracking accuracy and response speed.
Disclosure of Invention
The invention aims to provide a quaternion-based multi-rotor nonlinear flight control method capable of stably controlling the position and the attitude of a plurality of rotors.
The technical solution for realizing the purpose of the invention is as follows: a quaternion-based multi-rotor nonlinear flight control method is characterized by comprising the following steps:
step 1, making assumptions on external environment and aircraft motion conditions to obtain a four-rotor aircraft kinetic equation;
step 2, establishing a control model: establishing a multi-rotor position control model according to Newton's second law, and establishing an attitude control model according to a momentum moment theorem and a quaternion kinematics equation;
step 3, decomposing the flight control of the multiple rotors into outer ring position control and inner ring attitude control, and establishing a conversion relation between an Euler angle and a unit quaternion;
and 4, respectively constructing an outer ring position controller and an inner ring attitude controller by utilizing a backstepping method and combining the L yapunov principle, and controlling the positions and the attitudes of the multiple rotors.
Further, the making of the assumptions about the external environment and the motion conditions of the aircraft in step 1 includes:
1) in the flying process, the gravity acceleration g is not changed, and the multiple rotors are six-degree-of-freedom rigid bodies with unchanged mass;
2) the multi-rotor wing is symmetrical about the center, the materials of all parts are the same, and the mass center is positioned in the center of the machine body;
3) neglecting the curvature of the earth under an inertial coordinate system;
4) without considering the influence of revolution and rotation of the earth.
Further, step 2, controlling and establishing a control model, including a position control model of the multiple rotors in a navigation system with north-east-ground as a coordinate axis and an attitude control model under the navigation system:
the position control model is as follows:
Figure BDA0002474259090000021
wherein
Figure BDA0002474259090000022
Representing the velocity in the x, y, z directions respectively,
Figure BDA0002474259090000023
acceleration in x, y and z directions is respectively expressed, phi represents a rolling angle, theta represents a pitching angle, psi represents a yaw angle, T represents a rotor wing pulling force, m represents a body mass, g represents gravity acceleration, and k represents air resistance;
the theorem of moment of rigid body momentum is as follows:
Figure BDA0002474259090000024
wherein J ═ diag { J ═ Jbx,Jby,JbzDenotes a moment of inertia matrix, τ ═ τp,τq,τr]TRepresenting the drive torque provided by the motor, ω ═ p, q, r]TIndicates the rotational angular velocity of the multi-rotor]×Representing an antisymmetric matrix, τrotorRepresenting the gyroscopic moment of the rotor;
Figure BDA0002474259090000025
Jrotorexpressing the rotational inertia of the rotor, wherein omega is the sum of the rotating speeds of the rotors of the motors;
the kinematic equation of the quaternion is as follows:
Figure BDA0002474259090000031
wherein P (ω) ═ 0 ωT]TIs a quaternion form of the angular velocity vector ω, q ═ q0q1q2q3]T＝[ηT]TUnit quaternion attitude, q, for multiple rotors0、q1、q2、q3Four parameters each being a unit quaternionNumber η is a scalar representing the real part of the quaternion, vector, representing the imaginary part of the quaternion,
Figure BDA0002474259090000032
representing a quaternion multiplication; multiplication by quaternion to obtain quaternion kinematic equation
Figure BDA0002474259090000033
Wherein T (q) ∈ R4×3The expression is:
Figure BDA0002474259090000034
therefore, the attitude control model of the multi-rotor is obtained by combining the momentum moment theorem (2) and the quaternion kinematics equation (5):
Figure BDA0002474259090000035
further, in step 3, the flight control of the multiple rotors is decomposed into outer ring position control and inner ring attitude control, and a conversion relationship between the euler angle and the unit quaternion is established, specifically as follows:
the basis for decomposing the flight control of multiple rotors into outer ring position control and inner ring attitude control is as follows: according to the equation of the position change of the multiple rotors in the formula (1), the position change of the multiple rotors is not only related to the tension T provided by the rotors, but also related to the Euler angles, namely postures, of the multiple rotors; as known by the attitude change equation of the multiple rotors in the formula (7), the attitude motion can be realized by directly controlling the rotating moment of the multiple rotors around the center of mass; the inner ring controls the postures of the multiple rotors, and the position of the outer ring controls the positions of the multiple rotors under a navigation coordinate system;
the conversion relation between the Euler angle and the unit quaternion is as follows: in the navigation coordinate system, phi represents a roll angle, theta represents a pitch angle, and psi represents a yaw angle, and the corresponding unit quaternion q (phi, theta, psi) is expressed by euler angle:
Figure BDA0002474259090000036
further, an outer ring position controller is constructed in step 4, specifically as follows:
firstly, the position control only considers the thrust, gravity and air resistance of the rotor, and a state vector x and a position virtual control quantity u are established according to a position dynamic equation (1) of the multiple rotors:
Figure BDA0002474259090000041
then, the inverse step method is combined with L yapunov principle to obtain the virtual feedback control quantity u in x, y and z directionsx、ux、uxRespectively comprises the following steps:
Figure BDA0002474259090000042
Figure BDA0002474259090000043
Figure BDA0002474259090000044
wherein x1、x3、x5And x2、x4、x6Are each xn、yn、znDeviation of direction position tracking and deviation of velocity tracking, x1d、x3d、x3d，
Figure BDA0002474259090000047
Are respectively xn、yn、znDesired position and desired velocity in direction, kx1、kx2、kx3、kx4、kx5、kx6Is a control parameter greater than zero;
the magnitude of the multi-rotor input tension T is obtained by the formulas (9) and (13):
Figure BDA0002474259090000045
further, an inner ring attitude controller is constructed in the step 4, and the specific steps are as follows:
first, the desired attitude angle φ of the multi-rotor is obtained from equation (9)d、θd，φd、θdIs less than 45 °:
Figure BDA0002474259090000046
desired yaw angle psidGiven by the control command, the desired attitude angle is then converted into the desired unit quaternion q by equation (8)d；
Then, the attitude is q ═ η by the actual quaternionT]TAnd target quaternion attitude of qd＝[ηd d T]TObtaining deviation quaternion of qe：
Figure BDA0002474259090000051
Finally, the method is obtained by combining a reverse step method with L yapunov principle
Figure BDA0002474259090000053
Then, the feedback control quantity tau is obtained by the formula (7)d：
Figure BDA0002474259090000052
Wherein k ise1＞0，Ke2＝diag{ke21,ke22,ke2}，ke21、ke22、ke23＞0，e1Is a quaternion position error, e2Is the angular velocity error, omega is the actual angular velocity, omegadIs the desired angular velocity.
Compared with the prior art, the invention has the following remarkable advantages: (1) the postures of the multiple rotors are described by quaternions, and the controller is designed by combining a backstepping method, so that the possible singular problem of describing the posture angle by the Euler angle is effectively avoided; (2) the device has better performance when dealing with the nonlinear problem, and can stably control the position and the posture of the multiple rotors.
Drawings
FIG. 1 is a flow chart of a quaternion-based multi-rotor nonlinear flight control method in an example of the invention.
Fig. 2 is a block diagram of an example multi-rotor in an example of the invention.
Fig. 3 is a schematic diagram of an inner and outer loop control strategy in an embodiment of the present invention.
FIG. 4 is a graph of attitude response at a step input used in simulations in an example of the present invention.
FIG. 5 is a graph of position response at a step input used in simulations in an example of the present invention.
FIG. 6 is a graph of position response at the time of input with a unit slope in a simulation in an example of the present invention.
FIG. 7 is a graph illustrating the effect of multiple rotor tracking position "loop" shaped trajectories in a position tracking flight test in an example of the present invention.
FIG. 8 is a graph illustrating the effect of multiple rotors tracking the "helix" trajectory in a attitude tracking flight test in accordance with an embodiment of the present invention.
DETAILED DESCRIPTION OF EMBODIMENT (S) OF INVENTION
A quaternion-based multi-rotor nonlinear flight control method is characterized by comprising the following steps:
step 1, making assumptions on external environment and aircraft motion conditions to obtain a four-rotor aircraft kinetic equation;
step 2, establishing a control model: establishing a multi-rotor position control model according to Newton's second law, and establishing an attitude control model according to a momentum moment theorem and a quaternion kinematics equation;
step 3, decomposing the flight control of the multiple rotors into outer ring position control and inner ring attitude control, and establishing a conversion relation between an Euler angle and a unit quaternion;
and 4, respectively constructing an outer ring position controller and an inner ring attitude controller by utilizing a backstepping method and combining the L yapunov principle, and controlling the positions and the attitudes of the multiple rotors.
Further, the making of the assumptions about the external environment and the motion conditions of the aircraft in step 1 includes:
1) in the flying process, the gravity acceleration g is not changed, and the multiple rotors are six-degree-of-freedom rigid bodies with unchanged mass;
2) the multi-rotor wing is symmetrical about the center, the materials of all parts are the same, and the mass center is positioned in the center of the machine body;
3) neglecting the curvature of the earth under an inertial coordinate system;
4) without considering the influence of revolution and rotation of the earth.
Further, step 2, controlling and establishing a control model, including a position control model of the multiple rotors in a navigation system with north-east-ground as a coordinate axis and an attitude control model under the navigation system:
the position control model is as follows:
Figure BDA0002474259090000061
wherein
Figure BDA0002474259090000062
Representing the velocity in the x, y, z directions respectively,
Figure BDA0002474259090000063
acceleration in x, y and z directions is respectively expressed, phi represents a rolling angle, theta represents a pitching angle, psi represents a yaw angle, T represents a rotor wing pulling force, m represents a body mass, g represents gravity acceleration, and k represents air resistance;
the theorem of moment of rigid body momentum is as follows:
Figure BDA0002474259090000067
wherein J ═ diag { J ═ Jbx,Jby,JbzDenotes a moment of inertia matrix, τ ═ τp,τq,τr]TRepresenting the drive torque provided by the motor, ω ═ p, q, r]TIndicates the rotational angular velocity of the multi-rotor]×Representing an antisymmetric matrix, τrotorRepresenting the gyroscopic moment of the rotor;
Figure BDA0002474259090000064
Jrotorexpressing the rotational inertia of the rotor, wherein omega is the sum of the rotating speeds of the rotors of the motors;
the kinematic equation of the quaternion is as follows:
Figure BDA0002474259090000065
wherein P (ω) ═ 0 ωT]TIs a quaternion form of the angular velocity vector ω, q ═ q0q1q2q3]T＝[ηT]TUnit quaternion attitude, q, for multiple rotors0、q1、q2、q3Four parameters, respectively, of unit quaternions, η scalar, representing the real part of the quaternion, vector, representing the imaginary part of the quaternion,
Figure BDA0002474259090000066
representing a quaternion multiplication; multiplication by quaternion to obtain quaternion kinematic equation
Figure BDA0002474259090000071
Wherein T (q) ∈ R4×3The expression is:
Figure BDA0002474259090000072
therefore, the attitude control model of the multi-rotor is obtained by combining the momentum moment theorem (2) and the quaternion kinematics equation (5):
Figure BDA0002474259090000073
further, in step 3, the flight control of the multiple rotors is decomposed into outer ring position control and inner ring attitude control, and a conversion relationship between the euler angle and the unit quaternion is established, specifically as follows:
the basis for decomposing the flight control of multiple rotors into outer ring position control and inner ring attitude control is as follows: according to the equation of the position change of the multiple rotors in the formula (1), the position change of the multiple rotors is not only related to the tension T provided by the rotors, but also related to the Euler angles, namely postures, of the multiple rotors; as known by the attitude change equation of the multiple rotors in the formula (7), the attitude motion can be realized by directly controlling the rotating moment of the multiple rotors around the center of mass; the inner ring controls the postures of the multiple rotors, and the position of the outer ring controls the positions of the multiple rotors under a navigation coordinate system;
the conversion relation between the Euler angle and the unit quaternion is as follows: in the navigation coordinate system, phi represents a roll angle, theta represents a pitch angle, and psi represents a yaw angle, and the corresponding unit quaternion q (phi, theta, psi) is expressed by euler angle:
Figure BDA0002474259090000074
further, an outer ring position controller is constructed in step 4, specifically as follows:
firstly, the position control only considers the thrust, gravity and air resistance of the rotor, and a state vector x and a position virtual control quantity u are established according to a position dynamic equation (1) of the multiple rotors:
Figure BDA0002474259090000075
then, the inverse step method is combined with L yapunov principle to obtain the virtual feedback control quantity u in x, y and z directionsx、ux、uxRespectively comprises the following steps:
Figure BDA0002474259090000081
Figure BDA0002474259090000082
Figure BDA0002474259090000083
wherein x1、x3、x5And x2、x4、x6Are each xn、yn、znDeviation of direction position tracking and deviation of velocity tracking, x1d、x3d、x3d，
Figure BDA0002474259090000088
Are respectively xn、yn、znDesired position and desired velocity in direction, kx1、kx2、kx3、kx4、kx5、kx6Is a control parameter greater than zero;
the magnitude of the multi-rotor input tension T is obtained by the formulas (9) and (13):
Figure BDA0002474259090000084
further, an inner ring attitude controller is constructed in the step 4, and the specific steps are as follows:
first, the desired attitude angle φ of the multi-rotor is obtained from equation (9)d、θd，φd、θdIs less than 45 °:
Figure BDA0002474259090000085
desired yaw angle psidGiven by the control command, the desired attitude angle is then converted into the desired unit quaternion q by equation (8)d；
Then, the attitude is q ═ η by the actual quaternionT]TAnd target quaternion attitude of qd＝[ηd d T]TTo obtain a deviation quaternion ofqe：
Figure BDA0002474259090000086
Finally, the method is obtained by combining a reverse step method with L yapunov principle
Figure BDA0002474259090000087
Then, the feedback control quantity tau is obtained by the formula (7)d：
Figure BDA0002474259090000091
Wherein k ise1＞0，Ke2＝diag{ke21,ke22,ke2}，ke2、ke22、ke2＞0，e1Is a quaternion position error, e2Is the angular velocity error, omega is the actual angular velocity, omegadIs the desired angular velocity.
The technical solutions in the embodiments of the present invention will be described in detail below with reference to the accompanying drawings in the embodiments of the present invention.
Examples
With reference to fig. 1, an embodiment of the present invention provides a quaternion-based multi-rotor nonlinear flight control method, including:
s1, making assumptions about the motion process of the aircraft and the external environment;
s2, establishing a multi-rotor position control model and an attitude control model;
s3, decomposing the flight control of the multiple rotors into outer ring position control and inner ring attitude control;
s4, designing an outer ring position controller by utilizing a backstepping method and combining the L yapunov principle;
and S5, designing the inner ring attitude controller by utilizing a backstepping method and combining the L yapunov principle.
The invention relates to a quaternion-based multi-rotor flight control method, which comprises the steps of firstly, making assumptions on the motion process of an aircraft and the external environment in order to obtain the aircraft dynamic equation of a multi-rotor, then establishing a multi-rotor position control model according to Newton's second law, and establishing an attitude control model according to a data momentum moment theorem and a quaternion kinematic equation.
In the above-mentioned embodiments of the quaternion-based multi-rotor flight control method, optionally, four rotors are selected as specific implementation examples according to the control complexity and the practical application. Following the requirement of the control model and the mathematical precision of many rotors, the mathematical model of many rotors of establishing includes:
the parameters of the multi-rotor aircraft are selected from OS4 of EPF L (a cross-shaped four-rotor), the structure of the body and related parameters are shown in figure 2 and table 1, the control quantities provided by the motor are rotor tension T and driving torque tau respectivelyp，τq，τrThe expression is as follows:
Figure BDA0002474259090000101
wherein, ω is1，ω2，ω3，ω4Representing the rotational speed of four rotors, K, respectivelyTExpressing the thrust coefficient, KQDenotes the coefficient of drag,/x、lyRespectively showing the x-axis and y-axis arm lengths of the machine body, and omega shows the sum of the rotating speeds of the rotors.
In the embodiment of the invention, based on Newton's second law and aerodynamics, the multi-rotor is subjected to stress analysis, and a position control model of the multi-rotor can be obtained:
Figure BDA0002474259090000102
wherein
Figure BDA0002474259090000103
Respectively, speed and acceleration in x, y and z directions, phi a roll angle, theta a pitch angle, psi a yaw angle, T a rotor drag, m a body mass, g a gravitational acceleration, and k an air resistance.
Table 1 OS4 aircraft parameters for EPF L provided by embodiments of the present invention
Figure BDA0002474259090000104
In the embodiment of the invention, the attitude control model of the quaternion is obtained by utilizing the theorem of moment of momentum and the kinematics equation of the quaternion:
Figure BDA0002474259090000111
wherein q is [ q ]0q1q2q3]T＝[ηT]TUnit quaternion attitude, q, for multiple rotors0、q1、q2、q3Four parameters, respectively quaternions, η, are scalar, representing the real part of the quaternion, vector, representing the imaginary part of the quaternion,
Figure BDA0002474259090000112
ω＝[p,q,r]Tindicating the angular speed of rotation of the multiple rotors,
Figure BDA0002474259090000113
for angular acceleration, the moment of inertia matrix J ═ diag { J ═ Jbx,Jby,JbzDriving torque τ ═ τ [ τ ]pτqτr]T，τrotorGyroscopic moment for multiple rotors, JrotorIs the moment of inertia of the rotor, then:
Figure BDA0002474259090000114
and obtaining a control strategy schematic diagram 3 by combining an example according to the coupling relation of the multi-rotor position and attitude control model analysis system.
According to the mathematical model and the control strategy diagram of the multi-rotor, a feedback controller based on a backstepping method and the L yapunov stability principle is designed.
The input quantity of the multiple rotors is as follows: tau isp，τq，τrT, the state of the multi-rotor is classified into an attitude state and a position state, and the attitude of the multi-rotor is defined by the input vector τ ═ τ [ τ ] as shown in equations (2) and (3)pτqτr]TTo control, the position of the multiple rotors can be controlled by both the input T and the attitude (euler angle) of the multiple rotors.
In the embodiment of the invention, according to the coupling relation among the positions, postures and inputs of the multiple rotors and the characteristics of under-actuation, nonlinearity, easy external interference and the like of the multiple rotors, in order to design a control system with high precision and strong anti-interference performance, the control of the multiple rotors is decomposed into outer ring position control and inner ring posture control based on the idea of double-ring control.
In the present example, the detailed procedure for obtaining the position feedback compensation amount based on the back-stepping method and the L yapunov stability principle is as follows:
the position control only considers the thrust, gravity and air resistance of the rotor wings, and a state vector x and a virtual control quantity u are established according to a position dynamic equation (2) of the multiple rotor wings:
Figure BDA0002474259090000121
first, a position controller in the x-direction is designed, and its subsystems can be expressed as:
Figure BDA0002474259090000122
first, suppose μ1Is a virtual control quantity of x speed, x1Position tracking error in the x-direction, x2Velocity tracking error in the x-direction and x2(t0)＝0，x1dTo a desired position, V1Is about x1L yapunov's energy function,
Figure BDA0002474259090000123
to V1The derivation can be:
Figure BDA0002474259090000124
setting a control parameter kx1> 0, let the virtual control quantity mu1Satisfies the following conditions:
μ1＝x2d+kx1x1(9)
then there are:
Figure BDA0002474259090000125
wherein the first term is a positive number and the sign of the second term is unknown. To make it possible to
Figure BDA0002474259090000126
To be negative and introduce the actual input, L yapunov energy function V needs to be designed2。
The following equations (7) and (9) can be obtained: x is the number of2＝x2-μ1＝x2-x2d-kx1x1And, combining (7) and (9) to obtain:
Figure BDA0002474259090000127
let V2(x1,x2)＝V1+x2 2Then to the energy function V2Taking the derivative, we can get:
Figure BDA0002474259090000128
make acceleration
Figure BDA0002474259090000129
kx2If > 0 is a control parameter, then:
Figure BDA00024742590900001210
Figure BDA00024742590900001211
from the L yapunov stability theorem, the resulting equilibrium point is the global stable point.
Therefore, the virtual control amount u in the x directionxComprises the following steps:
Figure BDA0002474259090000131
likewise, the y direction:
Figure BDA0002474259090000132
wherein x is3d、x3Is the desired position and position deviation in the y direction, x4Is the speed x in the y direction4Tracking deviation of (k)x3＞0,kx4> 0 are control parameters.
The z direction:
Figure BDA0002474259090000133
wherein x is5d、x5Is the desired position and position deviation in the z direction, x6Is the z-direction velocity x6Tracking deviation of (k)x5＞0,kx6> 0 are control parameters.
Through equation (5), the rotor tension T of the multiple rotors is obtained:
Figure BDA0002474259090000134
in the present example, the feedback control amount τ is obtained based on the back-stepping method and the L yapunov stability principledThe detailed process is as follows:
firstly, listing a multi-rotor control model under the condition of no external interference:
Figure BDA0002474259090000135
secondly, obtaining the expected attitude angle of the multiple rotors through the position control rate and the position control equation in the step 4, and needing to limit phi to ensure significance in the process of inverting the attitude angle and stability in the process of flying the multiple rotorsd、θdIs less than 45 ° in absolute value.
Figure BDA0002474259090000136
Desired yaw angle psidGiven by a position control command, and then converting the expected attitude angle into an expected unit quaternion qd。
The conversion formula is as follows:
Figure BDA0002474259090000141
and the actual quaternion posture is q ═ ηT]TAnd target quaternion attitude of qd＝[ηd d T]TObtaining deviation quaternion of qe。
Figure BDA0002474259090000142
The attitude controller was next designed using a back-stepping approach in conjunction with the L yapunov principle:
let λ be1Tracking a virtual quantity, e, for angular velocity omega2Is the tracking error of angular velocity and e2(t0)＝0，e1Tracking real parts of quaternionsError, ωdTo desired angular velocity, V3Is an energy function of L yapunov,
Figure BDA0002474259090000143
to V3The derivation can be:
Figure BDA0002474259090000144
setting a control parameter ke1> 0, let track the virtual quantity lambda1Satisfies the following conditions:
λ1＝ωd+ke1 e(24)
then there is a change in the number of,
Figure BDA0002474259090000145
the first term is negative and the second term is not sign-determined. To make it possible to
Figure BDA0002474259090000146
For negative numbers, and introducing actual inputs, it is necessary to design L yapunov energy function V4。
From (22) (24) can be obtained: e.g. of the type2＝ω-λ1＝ω-ωd-ke1 eThe following equations (22) and (24) are combined:
Figure BDA0002474259090000147
is provided with
Figure BDA0002474259090000148
To V4(e1,e2) Taking the derivative, we can get:
Figure BDA0002474259090000149
order to
Figure BDA00024742590900001410
Wherein Ke2＝diag{ke2,ke22,ke23And has ke2、ke22、ke23> 0, for control parameters, will
Figure BDA0002474259090000151
Bringing in
Figure BDA0002474259090000152
Figure BDA0002474259090000153
Therefore, the method comprises the following steps:
Figure BDA0002474259090000154
the resulting equilibrium point is therefore globally stable. Handle
Figure BDA0002474259090000155
The belt-in formula (14) can be:
Figure BDA0002474259090000156
Figure BDA0002474259090000157
and (4) integrating (14), (15), (16), (17) and (29), and designing a controller of multiple rotors.
In the embodiment of the invention, the nonlinear controller is designed by using the method, the simulation model of the multiple rotors is built on the Simulink simulation platform, and in theoretical research and actual engineering, whether the designed nonlinear controller can adjust the system performance is analyzed by adjusting the parameters of the controller and analyzing the attitude and position response of the multiple rotors, so as to meet the expected requirements. The system performance is evaluated mainly by indexes such as response speed, overshoot, steady-state error and the like of the controller under an ideal Simulink simulation environment without external interference.
In the present example, the inner loop nonlinear control is first verifiedWhether the controller can realize the attitude control is set as the initial state of the attitude angle and the angular speed of the multi-rotor aircraft to be 0, and the position and the speed thereof to be 0 (q)0＝[1 0 0 0]T). The roll, pitch, and yaw angles are all expected to be 30 ° (q)0＝[0.92 0.18 0.31 0.18]T). When k ise1＝17，Ke2＝[10；10；10]The attitude angle response curve is shown in fig. 4, the attitude angle adjustment time is about 0.5 second, the adjustment time is less than 1 second, and no overshoot occurs.
For verification of the outer loop controller, its response to the step signal is verified a priori, and the response to the ramp signal is verified a second time. And obtaining control parameters of the outer ring controller through multiple times of adjustment: k is a radical ofx1＝kx2＝kx3＝kx4＝1.3，kx5＝2，kx6＝1.6。
The response of the outer-loop controller to the step signal is shown in fig. 5, and it can be seen through a simulation curve that the step response errors of all positions approach 0 without overshoot, and the adjustment time is about 2s, which indicates that the steady-state performance and the dynamic performance of the system both meet the design index.
The response of the outer ring controller to the unit slope signal is as shown in fig. 6, through a simulation curve, the outer ring position controller can accurately track the unit slope signal, after the system is stable, the steady-state error is 0.0015, the steady-state error is basically overlapped with the expected track, and the hysteresis phenomenon does not exist. Simulation results show that: the designed controller can effectively track the unit slope signal.
In an embodiment of the invention, the tracking characteristics of the controller for a multi-rotor aircraft flying in a given flight path are verified. Take "return" and "spiral" shaped tracks as examples.
The flying point of the 'go back' shaped track is (0,0,0), the middle passes through (0,0,5), (5,5,5), (0,5,5) and (0,0,5) in sequence, and finally returns to the flying point (0,0,0), and the flight point flight response curve is shown in fig. 7. By analyzing the trajectory tracking curve in the three-dimensional space, the aircraft can track the target trajectory rapidly and accurately, and the real-time error convergence time in each direction is less than 2.5 s. When the horizontal direction is switched to the command, the absolute value of the error of the height z direction is maximum is less than 0.08 m. Simulation results show that: the designed nonlinear controller can effectively track the 'loop' shaped track.
As can be seen from the simulation graph 8, the multiple rotors are able to accurately track the "helical" trajectory and reach steady state. The starting position of the multiple rotors is the origin of coordinates. Wherein the height instruction is a unit slope instruction (z ═ t), the height steady-state error can track to the track instruction in a very short time (5s), and the steady-state error is reduced to 0.0015 m. The track command in the horizontal direction is a sinusoidal signal (x is 5sin (ω t), y is 5cos (ω t), and ω is pi/40), and it can be seen from the figure that the absolute value of the error due to the phase difference is less than 0.003m after the system is stabilized in the x direction.
To sum up, the above-mentioned simulation experiment preliminary verification many rotors nonlinear control method based on quaternion can carry out effectual control to the flight of many rotors, and the control performance index includes: response time, steady state error, and immunity to interference.

Claims (6)

1. A quaternion-based multi-rotor nonlinear flight control method is characterized by comprising the following steps:
step 1, making assumptions on external environment and aircraft motion conditions to obtain a four-rotor aircraft kinetic equation;
step 2, establishing a control model: establishing a multi-rotor position control model according to Newton's second law, and establishing an attitude control model according to a momentum moment theorem and a quaternion kinematics equation;
step 3, decomposing the flight control of the multiple rotors into outer ring position control and inner ring attitude control, and establishing a conversion relation between an Euler angle and a unit quaternion;
and 4, respectively constructing an outer ring position controller and an inner ring attitude controller by utilizing a backstepping method and combining the L yapunov principle, and controlling the positions and the attitudes of the multiple rotors.
2. The quaternion-based multi-rotor nonlinear flight control method of claim 1, wherein the making of the assumptions about the external environment and the aircraft motion conditions of step 1 comprises:
1) in the flying process, the gravity acceleration g is not changed, and the multiple rotors are six-degree-of-freedom rigid bodies with unchanged mass;
2) the multi-rotor wing is symmetrical about the center, the materials of all parts are the same, and the mass center is positioned in the center of the machine body;
3) neglecting the curvature of the earth under an inertial coordinate system;
4) without considering the influence of revolution and rotation of the earth.
3. The quaternion-based multi-rotor nonlinear flight control method according to claim 1 or 2, wherein the control establishing step 2 is a control model comprising a position control model of the multi-rotor under a navigation system with north-east-ground as a coordinate axis and an attitude control model under the navigation system:
the position control model is as follows:
Figure FDA0002474259080000011
wherein
Figure FDA0002474259080000012
Representing the velocity in the x, y, z directions respectively,
Figure FDA0002474259080000013
acceleration in x, y and z directions is respectively expressed, phi represents a rolling angle, theta represents a pitching angle, psi represents a yaw angle, T represents a rotor wing pulling force, m represents a body mass, g represents gravity acceleration, and k represents air resistance;
the theorem of moment of rigid body momentum is as follows:
Figure FDA0002474259080000014
wherein J ═ diag { J ═ Jbx，Jby，JbzDenotes a moment of inertia matrix, τ ═ τp，τq，τr]TIndicating motor lifterSupplied drive torque, ω ═ p, q, r]TIndicates the rotational angular velocity of the multi-rotor]×Representing an antisymmetric matrix, τrotorRepresenting the gyroscopic moment of the rotor;
Figure FDA0002474259080000021
Jrotorexpressing the rotational inertia of the rotor, wherein omega is the sum of the rotating speeds of the rotors of the motors;
the kinematic equation of the quaternion is as follows:
Figure FDA0002474259080000022
wherein P (ω) ═ 0 ωT]TIs a quaternion form of the angular velocity vector ω, q ═ qoq1q2q3]T＝[ηT]TUnit quaternion attitude, q, for multiple rotors0、q1、q2、q3Four parameters, respectively, of unit quaternions, η scalar, representing the real part of the quaternion, vector, representing the imaginary part of the quaternion,
Figure FDA0002474259080000023
representing a quaternion multiplication; multiplication by quaternion to obtain quaternion kinematic equation
Figure FDA0002474259080000024
Wherein T (q) ∈ R4×3The expression is:
Figure FDA0002474259080000025
therefore, the attitude control model of the multi-rotor is obtained by combining the momentum moment theorem (2) and the quaternion kinematics equation (5):
Figure FDA0002474259080000026
4. the quaternion-based multi-rotor nonlinear flight control method of claim 3, wherein the decomposing of the multi-rotor flight control into outer loop position control and inner loop attitude control in step 3 establishes the conversion relationship between euler's angle and unit quaternion as follows:
the basis for decomposing the flight control of multiple rotors into outer ring position control and inner ring attitude control is as follows: according to the equation of the position change of the multiple rotors in the formula (1), the position change of the multiple rotors is not only related to the tension T provided by the rotors, but also related to the Euler angles, namely postures, of the multiple rotors; as known by the attitude change equation of the multiple rotors in the formula (7), the attitude motion can be realized by directly controlling the rotating moment of the multiple rotors around the center of mass; the inner ring controls the postures of the multiple rotors, and the position of the outer ring controls the positions of the multiple rotors under a navigation coordinate system;
the conversion relation between the Euler angle and the unit quaternion is as follows: in the navigation coordinate system, phi represents a roll angle, theta represents a pitch angle, and psi represents a yaw angle, and the corresponding unit quaternion q (phi, theta, psi) is expressed by euler angle:
Figure FDA0002474259080000031
5. the quaternion-based multi-rotor nonlinear flight control method of claim 4, wherein the outer ring position controller is constructed in step 4 as follows:
firstly, the position control only considers the thrust, gravity and air resistance of the rotor, and a state vector x and a position virtual control quantity u are established according to a position dynamic equation (1) of the multiple rotors:
Figure FDA0002474259080000032
then, by a reverse step methodL yapunov principle, obtaining virtual feedback control amount u in x, y and z directionsx、ux、uxRespectively comprises the following steps:
Figure FDA0002474259080000033
Figure FDA0002474259080000034
Figure FDA0002474259080000035
wherein x1、x3、x5And x2、x4、x6Are each xn、yn、znDeviation of direction position tracking and deviation of velocity tracking, x1d、x3d、x3d，
Figure FDA0002474259080000036
Are respectively xn、yn、znDesired position and desired velocity in direction, kx1、kx2、kx3、kx4、kx5、kx6Is a control parameter greater than zero;
the magnitude of the multi-rotor input tension T is obtained by the formulas (9) and (13):
Figure FDA0002474259080000037
6. the quaternion-based multi-rotor nonlinear flight control method of claim 5, wherein an inner ring attitude controller is constructed in step 4, and the method comprises the following specific steps:
first, the desired attitude angle φ of the multi-rotor is obtained from equation (9)d、θd，φd、θdIs less than 45 °:
Figure FDA0002474259080000041
desired yaw angle psidGiven by the control command, the desired attitude angle is then converted into the desired unit quaternion q by equation (8)d；
Then, the attitude is q ═ η by the actual quaternionT]TAnd target quaternion attitude of qd＝[ηd d T]TObtaining deviation quaternion of qe：
Figure FDA0002474259080000042
Finally, the method is obtained by combining a reverse step method with L yapunov principle
Figure FDA0002474259080000043
Then, the feedback control quantity tau is obtained by the formula (7)d：
Figure FDA0002474259080000044
Wherein k ise1＞0，Ke2＝diag{ke21，ke2，ke23}，ke21、ke22、ke23＞0，e1Is a quaternion position error, e2Is the angular velocity error, omega is the actual angular velocity, omegadIs the desired angular velocity.
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