CN114397812A - Quaternion output method in motion attitude control - Google Patents

Quaternion output method in motion attitude control Download PDF

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CN114397812A
CN114397812A CN202111359056.2A CN202111359056A CN114397812A CN 114397812 A CN114397812 A CN 114397812A CN 202111359056 A CN202111359056 A CN 202111359056A CN 114397812 A CN114397812 A CN 114397812A
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quaternion
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张鸿燕
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Hainan Normal University
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Abstract

The invention relates to the technical field of attitude control, in particular to a method for outputting quaternion in motion attitude control; the method comprises the following steps: sampling angular velocity according to the moving body to obtain discrete time angular velocity vector omegak(ii) a Determining precision regulation parameter l, angular speed discrete sampling step length tau and initial time t of moving body0And initial quaternion
Figure RE-DDA0003525390420000013
Constructing an initialized quaternion differential equation; calculating a key auxiliary parameter c and a key auxiliary parameter beta; using discrete time angular velocity vector omegakDetermining a system matrix omega; constructing a sine matrix G by utilizing the linear combination of a key auxiliary parameter c, a key auxiliary parameter beta, an angular speed discrete sampling step length tau, an identity matrix I and a system matrix omega, and obtaining a discrete time tkSine matrix of
Figure RE-DDA0003525390420000011
Using initial values
Figure RE-DDA0003525390420000014
And discrete time tkSine matrix of
Figure RE-DDA0003525390420000012
And calculating the quaternion by adopting an iterative algorithm so as to obtain a quaternion sequence. The invention aims to solve the problem that the existing attitude control parameters of a moving body cannot meet the requirements of high-precision real-time calculation and precision regulation.

Description

Quaternion output method in motion attitude control
Technical Field
The invention relates to the technical field of attitude control, in particular to a quaternion output method in motion attitude control.
Background
In the technical fields of spacecraft and aircraft control, torpedo control, robotics and automation, human attitude capture, computer graphics and game design, molecular dynamics, flight simulation, and the like, in order to control the attitude of a moving body, it is necessary to solve a linear time-varying Quaternion Differential Equation (QKDE) proposed by a.c. robinson in 1958.
In actual Measurement and control of the attitude of a moving body, an Inertial Measurement Unit (IMU) or a functionally equivalent sensor can be used to acquire data in real time to obtain three components of angular velocity. However, resolving QKDE with high accuracy is difficult, mainly for the following reasons:
1) the angular velocity component varies with time and the severity of the variation is unpredictable (e.g., supersonic fighter aircraft flying by maneuver), which tends to cause stiffness problems and unstable numerical performance of the QKDE.
2) Numerical cumulative errors result in computational inaccuracies.
3) The cumulative effect of numerical errors results in a normalization condition for quaternions:
Figure RE-GDA0003525390410000011
it is difficult to satisfy continuously.
4) The requirements for real-time computation must be met and the temporal and spatial complexity of the computation must be sufficiently low.
5) In order to improve the resolution precision, the existing resolution method needs higher-performance IMU hardware equipment, which increases the cost of system implementation and even causes a phenomenon of being "clipped".
6) It is difficult to find an explicit difference format and corresponding parameters that easily control the numerical accuracy.
7) Numerical methods with different precisions (different orders) have different theoretical forms and structural complexity, and corresponding differential formats have no uniform formal description, which brings much trouble to the design and implementation of the algorithm and increases the complexity of calculation.
For example, the conventional numerical algorithm and the conventional sine-geometric algorithm are used for solving the attitude of the moving body, and the conventional numerical algorithm cannot solve the performance problems listed in items 1 to 5 listed above; the simmering geometry algorithm is subdivided into two subclasses, wherein the simmering difference format given by the implicit simmering geometry algorithm cannot meet the requirement of real-time calculation and is difficult to regulate and control in precision, and the existing simmering difference format given by the explicit simmering geometry algorithm has the problems pointed out in item 5 and item 7 above. Therefore, in view of the difficulty in solving the QKDE, the existing attitude control of the moving body cannot meet the requirements of high-precision real-time calculation and precision regulation.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides an output method of quaternion in motion attitude control, which aims to solve the problem that the existing motion body attitude control parameters cannot meet the requirements of high-precision real-time calculation and precision regulation.
In order to achieve the purpose, the invention adopts the following technical scheme: a method for outputting quaternion in motion attitude control comprises the following steps:
sampling angular velocity according to angular velocity measuring sensor in moving body to obtain continuous time angular velocity signal, and converting the continuous time angular velocity signal into discrete time angular velocity signal to obtain discrete time angular velocity vector omegak；
Determining precision regulation parameter l, angular speed discrete sampling step length tau and initial time t of moving body0And initial quaternion
Figure RE-GDA0003525390410000021
Constructing an initialized quaternion differential equation;
according to the angular velocity vector omega at the discrete momentkCalculating a key auxiliary parameter c and a key auxiliary parameter beta according to the precision regulation parameter l and the angular speed discrete sampling step length tau;
by discrete time angular velocity vector omegakDetermining a system matrix omega;
constructing a sine matrix G by utilizing the linear combination of a key auxiliary parameter c, a key auxiliary parameter beta, an angular speed discrete sampling step length tau, an identity matrix I and a system matrix omega, and calculating the sine matrix G according to 2l order precision to obtain a discrete time tkSine matrix of
Figure RE-GDA0003525390410000022
Using the initial quaternion
Figure RE-GDA0003525390410000023
And discrete time tkSine matrix of
Figure RE-GDA0003525390410000024
And an iterative algorithm is adopted for calculation, so that a quaternion sequence is obtained.
Preferably, the precision regulation parameter l is a positive integer, the angular velocity discrete sampling step τ satisfies the sampling theorem, and the initialized quaternion differential equation satisfies the normalization condition.
Preferably, the calculating steps of the key auxiliary parameter c and the key auxiliary parameter β are as follows:
1) calculating the order s of two polynomials by using the precision regulation parameter l1And s2The method specifically comprises the following steps:
Figure RE-GDA0003525390410000031
2) iterative computation of polynomial coefficients using parallel, serial or interleaved modes
Figure RE-GDA0003525390410000032
And
Figure RE-GDA0003525390410000033
wherein a is0＝1/2，b 01, specifically:
Figure RE-GDA0003525390410000034
3) discrete sampling step length tau and discrete moment angular velocity vector omega are utilizedkCalculating an auxiliary parameter c;
c＝||ωk||2τ2/4；
4) calculating two polynomials n(s)1C) and d(s)2-c); the method specifically comprises the following steps:
Figure RE-GDA0003525390410000035
and
Figure RE-GDA0003525390410000036
5) calculating a key parameter beta by using a polynomial; the method specifically comprises the following steps:
Figure RE-GDA0003525390410000037
preferably, the system matrix Ω has an antisymmetric relationship, since Ω ═ ΩkNamely, the concrete construction formula is as follows:
Figure RE-GDA0003525390410000038
preferably, the matrix is a sine matrix
Figure RE-GDA0003525390410000039
The specific construction steps are as follows:
1) the linear combination construction formula of the sine matrix G with 2l order precision is as follows:
Figure RE-GDA00035253904100000310
wherein I is an identity matrix of order 4;
2) at tk＝t0At time + k τ, the corresponding sine matrix
Figure RE-GDA00035253904100000311
Comprises the following steps:
Figure RE-GDA0003525390410000041
preferably, the resolution steps of the quaternion sequence are as follows:
the quaternion iterative computation adopts a sine difference format:
Figure RE-GDA0003525390410000042
using initial quaternion
Figure RE-GDA0003525390410000043
And matrix with sine
Figure RE-GDA0003525390410000045
The quaternion sequence with each specific numerical value vector is sequentially calculated as the time beat increases.
The beneficial effect that this scheme produced is:
1. by reasonably configuring the precision control parameter l in the form of a positive integer, the discrete state transition matrix for resolving the quaternion parameter is a sine matrix G, the resolving precision is 2l orders, the function of freely configuring and controlling the calculation precision by the algorithm parameter is realized, the algorithms with different resolving precision orders have a uniform form, and the posture of the moving body is conveniently controlled and configured.
2. The sine matrix G is constructed by utilizing the linear combination of the key auxiliary parameter c, the key auxiliary parameter beta, the angular velocity discrete sampling step length tau, the unit matrix I and the system matrix omega, the sine matrix G is an orthogonal matrix and does not change the unit mode characteristic of the quaternion, when the mode of the initial quaternion is 1, the subsequent calculation process automatically keeps the mode unchanged, and the sine matrix G can ensure that the phenomenon of accumulated calculation errors does not exist.
3. Compared with the prior art which needs to perform calculation on a trigonometric function, the method does not involve any complex calculation and only uses the simplest real numbers of addition, subtraction, multiplication and division.
4. Compared with the prior art, the method for resolving different accuracy orders in different forms has a unified simple form, the accuracy grade is set only by the value of the accuracy control parameter l, and the numerical error epsilon of the resolved quaternion meets the condition
Figure RE-GDA0003525390410000044
Different accuracy regulation and control parameters l are set, the increase of calculation cost can be completely ignored, and meanwhile, the calculation accuracy can be obviously improved.
Drawings
In order to more clearly illustrate the embodiments of the present invention, the drawings, which are required to be used in the embodiments, will be briefly described below. In all the drawings, the elements or parts are not necessarily drawn to actual scale.
FIG. 1 is a flow chart of a quaternion differential equation solving method in the quaternion output method in motion attitude control according to the present invention;
FIG. 2 is a flowchart illustrating the calculation of a key auxiliary parameter c and a key auxiliary parameter β in the method for outputting quaternion in motion attitude control according to the present invention;
FIG. 3 is a graph showing the relationship between the variation of numerical value error with the discrete sampling step τ of angular velocity in the method for outputting quaternion in motion attitude control according to the present invention;
fig. 4 is a graph showing the relationship between the numerical error and the accuracy control parameter l in the method for outputting quaternion in motion attitude control according to the present invention.
Detailed Description
Embodiments of the present invention will be described in detail below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and therefore are only examples, and the protection scope of the present invention is not limited thereby.
The invention relates to a quaternion input method in motion attitude controlThe quaternion differential equation involved in the method is derived from a quaternion differential equation (QKDE) in the form of a first order time varying coefficient with initial conditions. Such as: suppose the instantaneous angular velocity of the motion of the moving body is
Figure RE-GDA0003525390410000056
Wherein ω is1(t)，ω2(t)，ω3(t) are the three components of the instantaneous angular velocity ω (t),
Figure RE-GDA0003525390410000055
is a vector representation of a quaternion, then QKDE can be expressed as:
Figure RE-GDA0003525390410000051
the solution condition being given an initial quaternion
Figure RE-GDA0003525390410000052
And satisfies the constraint of a die length of 1, i.e.
Figure RE-GDA0003525390410000053
The coefficient matrix of QKDE is uniquely determined by angular velocity, which is of the form shown by the equation:
Figure RE-GDA0003525390410000054
the coefficient matrix of QKDE is essentially a 4-dimensional matrix representation of a scalar part being 0 and the vector part being a special quaternion of angular velocity.
Meanwhile, the moving body attitude control system to which the quaternion output method of the present invention is applicable includes, but is not limited to: attitude control of various airplanes, various four-rotor aircrafts and various spacecrafts, such as space stations, satellites and airships; flight control and guidance systems for various torpedoes and missiles; various robot motion control systems using an inertial navigation unit, and an automatic driving system using an inertial navigation unit, and the like.
Referring to fig. 1-4, a method for outputting quaternion in motion attitude control includes the following steps:
first, a moving body is acquired, angular velocity sampling is performed based on an angular velocity measurement sensor in the moving body, a continuous-time angular velocity signal is acquired, and a discrete-time angular velocity vector ω is acquired by converting the continuous-time angular velocity signal into a discrete-time angular velocity signalk. The angular velocity measuring sensor can select a conventional Inertial Measurement Unit (IMU) or a sensor with equivalent function, so as to realize the angular velocity sampling of the moving body.
Secondly, determining a precision regulation parameter l, an angular speed discrete sampling step length tau and an initial time t of the moving body0And initial quaternion
Figure RE-GDA0003525390410000061
And constructing an initialized quaternion differential equation. Wherein all selected conditions meet the following requirements:
1) the precision control parameter l must be a positive integer. The method can be freely configured according to the performance requirement of an actual system, for example, any one of 1 to 10 is selected as the precision regulation parameter l.
1) The discrete sampling step length tau adopting the angular speed must be positive real number, and the basic requirement is that 0 < tau < 1. The high-precision resolving requirement can be met by a cheap angular velocity sensor (such as an IMU) without adopting an expensive angular velocity measuring sensor.
2) Starting time t adopted0It is simply the start of time and can be any practical value.
3) Initial quaternion
Figure RE-GDA0003525390410000062
Can be set arbitrarily according to requirements, and only needs to meet normalization conditions
Figure RE-GDA0003525390410000063
Thirdly, please refer to fig. 2 together, according to the angular velocity direction at discrete timeQuantity omegakAnd calculating a key auxiliary parameter c and a key auxiliary parameter beta according to the precision regulation parameter l and the angular speed discrete sampling step length tau. The specific calculation steps of the key auxiliary parameter c and the key auxiliary parameter beta are as follows:
C1) calculating the order s of two polynomials by using the precision regulation parameter l1And s2The method specifically comprises the following steps:
Figure RE-GDA0003525390410000071
C2) iterative computation of polynomial coefficients using parallel, serial or interleaved modes
Figure RE-GDA0003525390410000072
And
Figure RE-GDA0003525390410000073
wherein a is0＝1/2，b 01, specifically:
Figure RE-GDA0003525390410000074
C3) discrete sampling step length tau and discrete moment angular velocity vector omega are utilizedkCalculating an auxiliary parameter c;
c＝||ωk||2τ2/4。
C4) calculating two polynomials n(s)1C) and d(s)2-c); the method specifically comprises the following steps:
Figure RE-GDA0003525390410000075
and
Figure RE-GDA0003525390410000076
for the calculation of the polynomial, different calculation methods may be selected, for example, direct calculation according to the polynomial definition may be adopted, or calculation using a fast algorithm may be adopted.
C5) Calculating a key parameter beta by using a polynomial; β is essentially a function of l and c, and can be expressed as β (l, c), specifically:
Figure RE-GDA0003525390410000077
beta is essentially a function of the precision control parameter l and the auxiliary parameter c, since c is represented by wkDepending on τ, it can be seen that the ultimate dependency is the initially set parameter l, τ, t0And angular velocity omega at discrete timekTogether, determine the beta value required in the actual calculation process. For typical specific values of l, a specific expression of β (l, c) can be obtained as shown in table 1:
Figure RE-GDA0003525390410000078
Figure RE-GDA0003525390410000081
TABLE 1
Fourth, using the discrete-time angular velocity vector ωkAnd determining a system matrix omega. At discrete time tk＝t0+ k τ, since c ═ ck，β＝βk，Ω＝Ωk。
I.e. omegak＝Ω(ω(tk))＝Ω(ωk)。
The system matrix omega has an antisymmetric relation, and the specific construction steps are as follows:
Figure RE-GDA0003525390410000082
fifthly, constructing a sine matrix G by utilizing the linear combination of a key auxiliary parameter c, a key auxiliary parameter beta, an angular speed discrete sampling step length tau, an identity matrix I and a system matrix omega, calculating the sine matrix G according to 2l order precision, and obtaining a discrete time tkSine matrix of
Figure RE-GDA0003525390410000083
Sine matrix
Figure RE-GDA0003525390410000084
The specific construction steps are as follows:
1) the linear combination of the sine matrix with 2l order precision is constructed as follows:
Figure RE-GDA0003525390410000085
where I is an identity matrix of order 4.
2) Thus, at tk＝t0+ k τ times, corresponding sine matrix
Figure RE-GDA0003525390410000086
Comprises the following steps:
Figure RE-GDA0003525390410000087
sixth, using initial quaternion
Figure RE-GDA0003525390410000088
And discrete time tkSine matrix of
Figure RE-GDA0003525390410000089
And calculating the quaternion by adopting an iterative algorithm so as to obtain a quaternion sequence.
The resolution steps of the quaternion sequence are as follows:
the quaternion iterative computation adopts a sine difference format:
Figure RE-GDA0003525390410000091
using initial values
Figure RE-GDA0003525390410000092
And matrix with sine
Figure RE-GDA0003525390410000093
Sequentially calculating vectors with various specific numerical values as the time beat increases
Figure RE-GDA0003525390410000094
Such that the solved quaternion has a constant modulus property, i.e.
Figure RE-GDA0003525390410000095
For any integer k. And converting the quaternion sequence into Euler angles so as to realize the attitude control of the moving body.
The steps from one to six of the invention can be changed or integrated again according to the realization requirement of the actual physical system. Such as: step three and step four are fused into step five or step one and step two can be exchanged.
The method samples the angular velocity of the moving body, selects a precision regulation parameter l in a positive integer form to control the quaternion parameter resolving process in the attitude control, and transfers the resolved quaternion parameter discrete state to a matrix G which is a sine matrix, and the calculating precision is 2l orders.
The invention realizes the functions of free configuration of algorithm parameters and convenient regulation and control of calculation precision, enables quaternion calculation algorithms in attitude control with different calculation precision orders to have a unified form, realizes convenient regulation and control and configuration of attitude control precision of a moving body, and simultaneously meets the aims of high precision, real-time calculation and reduction of software and hardware cost through good algorithms.
The method is characterized in that a sine matrix G is constructed by utilizing the linear combination of a key auxiliary parameter c, a key auxiliary parameter beta, an angular velocity discrete sampling step length tau, an identity matrix I and a system matrix omega, the sine matrix G is an orthogonal matrix and does not change the identity of the unit module of the quaternion, when the identity of the initial quaternion is 1, the identity of the subsequent calculation process is automatically kept unchanged, and meanwhile, the phenomenon of error accumulation does not exist in the process of controlling the attitude of the moving body due to the sine property.
The method does not involve any complex calculation, does not contain any calculation related to trigonometric functions in the existing method, and only uses the simplest real number for addition, subtraction, multiplication and division. The quaternion calculation complexity is low, and the requirement of real-time calculation of the attitude control of the moving body is met.
From the perspective of real-time computation, please refer to FIG. 3, the requirement of the method for the storage space is a constant complexity
Figure RE-GDA0003525390410000096
The complexity for the computation time is a linear complexity
Figure RE-GDA0003525390410000097
Where n is the length of time t of the solutionf-t0Determination of the ratio to the discrete sampling step τ of the angular velocity, i.e.
Figure RE-GDA0003525390410000101
The performance requirement of the attitude control of the moving body can be met by utilizing various existing cheap angular velocity measuring sensors.
For the case of the precision control parameter l, please refer to fig. 4, the numerical error e of the quaternion satisfies the condition
Figure RE-GDA0003525390410000102
The characteristics of no error accumulation, extremely low upper bound of calculation error and real-time calculability ensured by the pungent characteristic indicate that the method can realize high-precision regulation and control on the real-time control of the posture of the moving body.
The above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; such modifications and substitutions do not depart from the spirit and scope of the present invention, and they should be construed as being included in the following claims and description.

Claims (6)

1. A method for outputting quaternion in motion attitude control is characterized by comprising the following steps:
sampling angular velocity according to angular velocity measuring sensor in moving body to obtain continuous time angular velocity signal, and converting the continuous time angular velocity signal into discrete time angular velocity signal to obtain discrete time angular velocity vector omegak；
Determining precision regulation parameter l, angular speed discrete sampling step length tau and initial time t of moving body0And initial quaternion
Figure RE-FDA0003525390400000011
Constructing an initialized quaternion differential equation;
according to the angular velocity vector omega at the discrete momentkCalculating a key auxiliary parameter c and a key auxiliary parameter beta according to the precision regulation parameter l and the angular speed discrete sampling step length tau;
by discrete time angular velocity vector omegakDetermining a system matrix omega;
constructing a sine matrix G by utilizing the linear combination of a key auxiliary parameter c, a key auxiliary parameter beta, an angular speed discrete sampling step length tau, an identity matrix I and a system matrix omega, and calculating the sine matrix G according to 2l order precision to obtain a discrete time tkSine matrix of
Figure RE-FDA0003525390400000012
Using the initial quaternion
Figure RE-FDA0003525390400000013
And discrete time tkSine matrix of
Figure RE-FDA0003525390400000014
And using an iterative algorithmAnd calculating to obtain a quaternion sequence.
2. The method for outputting quaternion in motion attitude control according to claim 1, wherein the precision regulation parameter l is a positive integer, the angular velocity discrete sampling step τ satisfies a sampling theorem, and the initialized quaternion differential equation satisfies a normalization condition.
3. The method for outputting quaternion in motion attitude control according to claim 1 or 2, wherein the calculating steps of the key auxiliary parameter c and the key auxiliary parameter β are as follows:
1) calculating the order s of two polynomials by using the precision regulation parameter l1And s2The method specifically comprises the following steps:
Figure RE-FDA0003525390400000015
2) iterative computation of polynomial coefficients using parallel, serial or interleaved modes
Figure RE-FDA0003525390400000016
And
Figure RE-FDA0003525390400000017
wherein a is0＝1/2，b01, specifically:
Figure RE-FDA0003525390400000018
3) discrete sampling step length tau and discrete moment angular velocity vector omega are utilizedkCalculating an auxiliary parameter c;
c＝||ωk||2τ2/4；
4) calculating two polynomials n(s)1C) and d(s)2-c); the method specifically comprises the following steps:
Figure RE-FDA0003525390400000021
and
Figure RE-FDA0003525390400000022
5) calculating a key parameter beta by using a polynomial; the method specifically comprises the following steps:
Figure RE-FDA0003525390400000023
4. the method of claim 3, wherein the system matrix Ω has an anti-symmetric relationship, and Ω is ΩkNamely, the concrete construction formula is as follows:
Figure RE-FDA0003525390400000024
5. the method as claimed in claim 4, wherein the matrix is a sine matrix
Figure RE-FDA0003525390400000025
The specific construction steps are as follows:
1) the linear combination construction formula of the sine matrix G with 2l order precision is as follows:
Figure RE-FDA0003525390400000026
wherein I is an identity matrix of order 4;
2) at tk＝t0At time + k τ, the corresponding sine matrix
Figure RE-FDA0003525390400000027
Comprises the following steps:
Figure RE-FDA0003525390400000028
6. the method for outputting quaternion in motion attitude control according to claim 5, wherein the solving of the quaternion sequence comprises the following steps:
the quaternion iterative computation adopts a sine difference format:
Figure RE-FDA0003525390400000029
using initial quaternion
Figure RE-FDA00035253904000000210
And matrix with sine
Figure RE-FDA00035253904000000211
The quaternion sequence with each specific numerical value vector is sequentially calculated as the time beat increases.
CN202111359056.2A 2021-11-17 2021-11-17 Quaternion output method in motion attitude control Withdrawn CN114397812A (en)

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