CN102323990A - Pneumatic model for rigid body space motion - Google Patents

Abstract

The invention discloses a pneumatic model for rigid body space motion, which obtains an expression for motion attitude of a rigid body by defining ternary numbers, avoids a problem of a bizarre attitude equation, further obtains a pneumatic model expression of speed according to the ternary numbers, and obtains pneumatic force expressions for an angle of attack of air flow and an angle of sideslip, and then obtains the model for the main motion state of the rigid body; the air flow axial rigid body motion equation is simplified by introducing the the ternary numbers, therefore, no bizarre points appear in the rigid body motion equation so as to be convenience for use in engineering.

Description

A kind of aerodynamic model of rigid space motion

Technical field

The present invention relates to the spatial movement rigid model, particularly the big motor-driven aerodynamic model problem of aircraft.

Background technology

Axis is that the rigid motion differential equation is a fundamental equation of describing spatial movements such as aircraft, torpedo, spacecraft.Usually, in data processing etc. was used, the state variable of axon system mainly comprised the X of 3 speed components, three Eulerian angle and earth axes _E, Y _E, Z _EDeng, because Z _EBe defined as vertical ground and point to ground ball center, so Z _EActual flying height for bearing; X _E, Y _EUsually main dependence GPS, GNSS, the Big Dipper etc. directly provide; Eulerian angle are represented the rigid space athletic posture, and the differential equation of portrayal rigid body attitude is a core wherein, is that pitching, lift-over and crab angle are described with three Eulerian angle usually.When the angle of pitch of rigid body was ± 90 °, roll angle and crab angle can't definite values, and it is excessive that error is found the solution in the zone of closing on this singular point simultaneously, caused intolerable error on the engineering and can not use; For fear of this problem, people at first adopt the method for restriction angle of pitch span, and this makes equation degenerate, attitude work entirely, thereby be difficult to be widely used in engineering practice.Along with the research to the aircraft extreme flight, people have adopted direction cosine method, equivalent gyration vector method, hypercomplex number method etc. to calculate the rigid motion attitude again in succession.

Direction cosine method has been avoided " unusual " phenomenon of Eulerian angle describing methods, and calculating attitude matrix with direction cosine method does not have the equation degenerate problem, attitude work entirely; But need find the solution 9 differential equations; Calculated amount is bigger, and real-time is relatively poor, can't satisfy the engineering practice requirement.Equivalence gyration vector method such as list appearance recursion, Shuangzi appearance gyration vector, three increment gyration vectors and four increment rotating vector methods and various correction algorithms on this basis and recursive algorithm etc.When studying rotating vector in the document, all be based on the algorithm that rate gyro is output as angle increment.Yet in actual engineering, the output of some gyros is angle rate signals, like optical fibre gyro, dynamic tuned gyroscope etc.When rate gyro was output as angle rate signal, the Algorithm Error of rotating vector method obviously increased.The hypercomplex number method is that the function of 4 Eulerian angle of definition calculates the boat appearance; Can effectively remedy the singularity of Eulerian angle describing method; As long as separate 4 differential equation of first order formula groups; Analogy has tangible minimizing to cosine attitude matrix differential equation calculated amount, can satisfy in the engineering practice requirement to real-time.Its The common calculation methods has the card of finishing approximatioss, second order, fourth-order Runge-Kutta method and three rank Taylor expansion methods etc.Finishing card approximatioss essence is list appearance algorithm, can not compensate by exchange error what limited rotation caused, and the algorithm drift under high current intelligence in the attitude algorithm can be very serious.When adopting fourth-order Runge-Kutta method to find the solution the hypercomplex number differential equation,, the trigonometric function value can occur to exceed ± 1 phenomenon, disperse thereby cause calculating along with the continuous accumulation of integral error; The Taylor expansion method also is restricted because of the deficiency of computational accuracy.When rigid body is big when motor-driven, angular speed causes the error of said method bigger more greatly; Moreover, the error that attitude is estimated usually can cause the error of 4 components of speed, highly output sharply to increase, and causes aerodynamic model and parameter estimation inaccurate.

Summary of the invention

In order to overcome the existing big problem of rigid motion model output error; The present invention provides a kind of aerodynamic model of rigid space motion, and this model obtains the expression of rigid motion attitude through definition ternary number; Avoided attitude equation singular problem; Further obtain the aerodynamic model expression formula of speed according to the ternary number, and obtained the aerodynamic expression of the air-flow angle of attack and yaw angle, thereby obtain the model of the main motion state of rigid body.

The present invention solves the technical scheme that its technical matters adopts, a kind of aerodynamic model of rigid space motion, and its characteristic may further comprise the steps:

1, definition ternary number:

Wherein: refers to lift-over, the angle of pitch respectively, and

Further obtain:

$\left\{\begin{array}{c}{\stackrel{\·}{s}}_1={\mathrm{qs}}_3-{\mathrm{rs}}_2\\ {\stackrel{\·}{s}}_2={\mathrm{ps}}_3+{\mathrm{rs}}_1\\ s_3=-{\mathrm{ps}}_2-{\mathrm{qs}}_1\end{array}\right.$

Wherein: p, q, r are respectively lift-over, pitching, yaw rate; Parameter-definition is identical in full;

2, the aerodynamic model of speed is respectively:

${\stackrel{\·}{V}}_0=V_0[(\frac{\mathrm{QS}}{m{V}_0}{C}_x-\frac{{\mathrm{gs}}_1}{V_0})\cos \mathrm{\α}\cos \mathrm{\β}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_y+\frac{{\mathrm{gs}}_2}{V_0})\sin \mathrm{\β}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z+\frac{{\mathrm{gs}}_3}{V_0})\sin \mathrm{\α}\cos \mathrm{\β}]$

Wherein: V ₀Be rigid body center of mass motion speed, g is an acceleration of gravity, and α is the air-flow angle of attack, and β is a yaw angle, and Q is dynamic pressure, and S is a wing area, and m is an Aircraft Quality, C _x, C _y, C _zBe respectively vertically, side direction, normal direction aerodynamic force;

3, the air-flow angle of attack and yaw angle aerodynamic model are:

$\stackrel{\·}{\mathrm{\α}}=q-p\cos \mathrm{\α}\tan \mathrm{\β}-r\sin \mathrm{\α}\tan \mathrm{\β}+\frac{\cos \mathrm{\α}}{\cos \mathrm{\β}}(\frac{{\mathrm{gs}}_3}{V_0}+\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z)+\frac{\sin \mathrm{\α}}{\cos \mathrm{\β}}(\frac{{\mathrm{gs}}_1}{V_0}-\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_x)$

$\stackrel{\·}{\mathrm{\β}}=-r\cos \mathrm{\α}+p\sin \mathrm{\α}+\cos \mathrm{\β}(\frac{\mathrm{QS}}{m{V}_0}{C}_y+\frac{{\mathrm{gs}}_2}{V_0})-\sin \mathrm{\β}[(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_x-\frac{{\mathrm{gs}}_1}{V_0})\cos \mathrm{\α}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z+\frac{{\mathrm{gs}}_3}{V_0})\sin \mathrm{\α}]$

Usually yaw angle is usually less than 90 °, and cos β can not be zero.

The invention has the beneficial effects as follows: to have simplified the air-flow axle be the rigid motion equation through introducing the ternary number, makes singular point no longer to occur in the rigid motion attitude equation, is convenient to engineering and uses.

Below in conjunction with embodiment the present invention is elaborated.

Embodiment

1, definition ternary number:

Wherein:

refers to lift-over, the angle of pitch respectively, and

Further obtain:

$\left\{\begin{array}{c}{\stackrel{\·}{s}}_1={\mathrm{qs}}_3-{\mathrm{rs}}_2\\ {\stackrel{\·}{s}}_2={\mathrm{ps}}_3+{\mathrm{rs}}_1\\ s_3=-{\mathrm{ps}}_2-{\mathrm{qs}}_1\end{array}\right.$

Wherein: p, q, r are respectively lift-over, pitching, yaw rate; Parameter-definition is identical in full;

2, the aerodynamic model of speed is respectively:

${\stackrel{\·}{V}}_0=V_0[(\frac{\mathrm{QS}}{m{V}_0}{C}_x-\frac{{\mathrm{gs}}_1}{V_0})\cos \mathrm{\α}\cos \mathrm{\β}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_y+\frac{{\mathrm{gs}}_2}{V_0})\sin \mathrm{\β}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z+\frac{{\mathrm{gs}}_3}{V_0})\sin \mathrm{\α}\cos \mathrm{\β}]$

$\stackrel{\·}{h}={\mathrm{us}}_1-{\mathrm{vs}}_2-{\mathrm{ws}}_3$

Wherein: V ₀Be rigid body center of mass motion speed, h is a height, and g is an acceleration of gravity, and α is the air-flow angle of attack, and β is a yaw angle, and Q is dynamic pressure, and S is a wing area, and m is an Aircraft Quality, C _x, C _y, C _zBe respectively vertically, side direction, normal direction aerodynamic force;

3, the air-flow angle of attack and yaw angle aerodynamic model are:

$\stackrel{\·}{\mathrm{\α}}=q-p\cos \mathrm{\α}\tan \mathrm{\β}-r\sin \mathrm{\α}\tan \mathrm{\β}+\frac{\cos \mathrm{\α}}{\cos \mathrm{\β}}(\frac{{\mathrm{gs}}_3}{V_0}+\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z)+\frac{\sin \mathrm{\α}}{\cos \mathrm{\β}}(\frac{{\mathrm{gs}}_1}{V_0}-\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_x)$

$\stackrel{\·}{\mathrm{\β}}=-r\cos \mathrm{\α}+p\sin \mathrm{\α}+\cos \mathrm{\β}(\frac{\mathrm{QS}}{m{V}_0}{C}_y+\frac{{\mathrm{gs}}_2}{V_0})-\sin \mathrm{\β}[(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_x-\frac{{\mathrm{gs}}_1}{V_0})\cos \mathrm{\α}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z+\frac{{\mathrm{gs}}_3}{V_0})\sin \mathrm{\α}]$

Usually yaw angle is usually less than 90 °, and cos β can not be zero.

Claims (1)

1. the aerodynamic model of rigid space motion, its characteristic may further comprise the steps:

A) definition ternary number:

Wherein:

refers to lift-over, the angle of pitch respectively, and

Further obtain:

$\left\{\begin{array}{c}{\stackrel{\·}{s}}_1={\mathrm{qs}}_3-{\mathrm{rs}}_2\\ {\stackrel{\·}{s}}_2={\mathrm{ps}}_3+{\mathrm{rs}}_1\\ s_3=-{\mathrm{ps}}_2-{\mathrm{qs}}_1\end{array}\right.$

Wherein: p, q, r are respectively lift-over, pitching, yaw rate; Parameter-definition is identical in full;

B) aerodynamic model of speed is respectively:

${\stackrel{\·}{V}}_0=V_0[(\frac{\mathrm{QS}}{m{V}_0}{C}_x-\frac{{\mathrm{gs}}_1}{V_0})\cos \mathrm{\α}\cos \mathrm{\β}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_y+\frac{{\mathrm{gs}}_2}{V_0})\sin \mathrm{\β}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z+\frac{{\mathrm{gs}}_3}{V_0})\sin \mathrm{\α}\cos \mathrm{\β}]$

Wherein: V ₀Be rigid body center of mass motion speed, g is an acceleration of gravity, and α is the air-flow angle of attack, and β is a yaw angle, and Q is dynamic pressure, and S is a wing area, and m is an Aircraft Quality, C _x, C _y, C _zBe respectively vertically, side direction, normal direction aerodynamic force;

C) the air-flow angle of attack and yaw angle aerodynamic model are:

$\stackrel{\·}{\mathrm{\α}}=q-p\cos \mathrm{\α}\tan \mathrm{\β}-r\sin \mathrm{\α}\tan \mathrm{\β}+\frac{\cos \mathrm{\α}}{\cos \mathrm{\β}}(\frac{{\mathrm{gs}}_3}{V_0}+\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z)+\frac{\sin \mathrm{\α}}{\cos \mathrm{\β}}(\frac{{\mathrm{gs}}_1}{V_0}-\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_x)$

$\stackrel{\·}{\mathrm{\β}}=-r\cos \mathrm{\α}+p\sin \mathrm{\α}+\cos \mathrm{\β}(\frac{\mathrm{QS}}{m{V}_0}{C}_y+\frac{{\mathrm{gs}}_2}{V_0})-\sin \mathrm{\β}[(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_x-\frac{{\mathrm{gs}}_1}{V_0})\cos \mathrm{\α}-(\frac{\mathrm{QS}}{{\mathrm{mV}}_0}{C}_z+\frac{{\mathrm{gs}}_3}{V_0})\sin \mathrm{\α}].$