CN103593524A - Dynamics modeling and analyzing method for aerospace vehicle - Google Patents

Abstract

The invention discloses a dynamics modeling and analyzing method for an aerospace vehicle and belongs to the technical field of flight dynamics modeling and simulation analysis of aircrafts. According to the dynamics modeling and analyzing method, the variable-span and variable-sweepback aerospace vehicle is simplified and is composed of a fuselage, a left wing inner side, a left wing outer side, a right wing inner side and a right wing outer side, wherein the fuselage, the left wing inner side, the left wing outer side, the right wing inner side and the right wing outer side are independent rigid bodies. The deformation movement of wings relative to the fuselage is represented through a set of constraint equations, so that the freedom degree of a dynamics model is reduced; the aerodynamic force exerted on the aerospace vehicle is simplified to act on the fuselage only, and the dynamics model is established through the Kane method; related items derived from the deformation movement are extracted from the dynamics model to form additional force and additional force moment so that the influence on the dynamics characteristics of the aerospace vehicle by the inertia force and inertia moment caused by the deformation moment can be represented. The dynamics model of the aerospace vehicle is simplified through the dynamics modeling and analyzing method, and integral items and derivation terms of the inertia moment do not appear in the equations. In this way, dynamics modeling and simulating can be easily performed on the aerospace vehicle, and meanwhile accuracy is high.

Description

A kind of Dynamic Modeling of morphing aircraft and analytical approach

Technical field

The invention belongs to aircraft flight mechanical modeling and simulation analysis technical field, is a kind of Dynamic Modeling for morphing aircraft and analytical approach specifically.

Background technology

Dual-use aviation has in recent years proposed more and more higher requirement to aircraft performance, aircraft should adapt to variation, the execution different task of flight environment of vehicle, guarantee again flying quality, and will meet economy requirement, and current vehicle technology cannot meet these requirements simultaneously.Morphing aircraft technology is a kind of potential, technological approaches that can effectively address this problem.Morphing aircraft be a kind of can large scale change aerodynamic configuration so that realize the aviation aircraft of multitask flight.The research of morphing aircraft has had quite long history, proposes the patented claim of " Variable Geometry Wing " as far back as the existing people of the 1916 Nian， U.S..In recent years, the fast development in the fields such as new material, new drive unit and new control technology has further excited people to study the enthusiasm of intelligent morphing aircraft, and in the past few decades, a large amount of research has been carried out morphing aircraft is technical in countries in the world.

Under different flying conditions, in order to obtain optimal performance, morphing aircraft need to change aerodynamic configuration in sizable scope, therefore, can not as orthodox flight device, morphing aircraft be carried out to Dynamic Modeling as single rigid body, and will set up a kind of kinetic model that comprises distressed structure.

At present, when morphing aircraft is carried out to Dynamic Modeling, mostly adopt classical Newton mechanics method, aircraft is regarded as to an integral body, ask for its momentum and its momentum moment to barycenter, then to time differentiate, and then set up aircraft make a concerted effort outside translation motion under F effect and the rotational motion equation under resultant moment M effect outside.In this process, consider the distortion of aircraft, need to ask for whole aircraft about the statical moment of reference point by integration, need moment of inertia differentiate to be out of shape to solve aircraft the problem of the moment of inertia variation bringing simultaneously, can find that this method calculated amount is larger, and need to carry out accurate modeling to the profile of aircraft and mass distribution.

In addition, when morphing aircraft is carried out to dynamic analysis, be difficult at present analyze outside the variation of deacration power the impact by the kinetic inertia of variant on vehicle dynamics characteristic.

Summary of the invention

The object of the invention is in order to solve existing Modeling of Vehicle method, to be difficult to set up the problem of the accurate model of morphing aircraft, and cannot the problem of Accurate Analysis amoeboid movement to the kinetic effect of morphing aircraft.The Dynamic Modeling and the analytical approach that the present invention proposes a kind of morphing aircraft based on Kane method, the present invention has simplified the kinetic model of morphing aircraft, and modeling method is simple, and the degree of accuracy of the kinetic model of setting up is higher; The kinetic model that application is described, the invention allows for the analytical approach of a kind of amoeboid movement on vehicle dynamics impact, can analyze outside the variation of deacration power the impact by the kinetic inertial force of variant for the dynamics of aircraft.

The modeling method of kinetic model provided by the invention is as follows:

The morphing aircraft at variable length, variable angle of sweep is reduced to a multi-rigid-body system, by five parts such as fuselage, left wing inner side, left wing outside, right flank inner side and right flank outsides, formed, each part is reduced to an independent rigid body, fuselage is reduced to homogeneous cylinder, and the inner side of each wing and outside are reduced to homogeneous thin bar; Wing is reduced to the motion that can control effectively by variant control gear with respect to the amoeboid movement of fuselage, utilize one group of equation of constraint to represent this motion, reduced degree of freedom to amoeboid movement relevant in kinetic model, morphing aircraft can be represented with the kinetic model of a six degree of freedom, then utilize Kane method to set up the kinetic model of aircraft; In the process of modeling, broad sense active force is simplified: the interaction force between wing and fuselage and between wing medial and lateral is used as to constraining force, by equation of constraint, is represented; The suffered aerodynamic force of morphing aircraft is reduced to and is only acted on fuselage, and other parts are only subject to the effect of self gravitation.

After establishing the six-degree-of-freedom dynamic model of morphing aircraft, can obtain six kinetics equations, correspond respectively to selected six-freedom degree.

The analytical approach of the amoeboid movement proposing based on modeling method of the present invention on vehicle dynamics impact, when analyzing, from kinetic model, extract the continuous item come from amoeboid movement and form additional force, additional moment, with it, represent the impact on vehicle dynamics characteristic of mass force and touqhe that wing amoeboid movement causes; Utilize the computing formula of additional force and additional moment to calculate its size in variant process, the size that in the data result obtaining and variant process, the suffered aerodynamic force of morphing aircraft and aerodynamic moment change compares, if both differ one more than the order of magnitude, can ignore wherein less one, and affect hardly the dynamics of aircraft, but but can simplify the kinetic model of aircraft.

The invention has the advantages that:

(1) in the kinetic model that the present invention sets up, there will not be integration item and the differentiate item to moment of inertia, the equation obtaining is easy to carry out numerical evaluation and emulation;

(2) in the kinetic model that the present invention sets up, utilize equation of constraint to substitute the degree of freedom of amoeboid movement, the degree of freedom of system has been reduced, reduced the complexity of equation;

(3) the present invention has provided the analytical approach of a kind of easy analysis variant motion on the impact of morphing aircraft dynamics, and the method can be analyzed additional force that variant motion the produces kinetic effect to morphing aircraft exactly.

Accompanying drawing explanation

Fig. 1: in the present invention, variable length becomes the designs simplification schematic diagram of the morphing aircraft at angle of sweep;

Fig. 2: the coordinate system schematic diagram of morphing aircraft in the present invention;

Fig. 3 A～3D: the Si Zhong aerodynamic arrangement of morphing aircraft in the present invention;

Fig. 4: the lift coefficient in the present invention under morphing aircraft Si Zhong aerodynamic arrangement;

Fig. 5: the resistance coefficient in the present invention under morphing aircraft Si Zhong aerodynamic arrangement;

Fig. 6: the aerodynamic center position in the present invention under morphing aircraft Si Zhong aerodynamic arrangement;

Fig. 7 A and Fig. 7 B: the simulation result figure of additional force size in morphing aircraft deformation process in the present invention.

Embodiment

Below in conjunction with drawings and Examples, the present invention is described in further detail.

The concrete implementation step of the dynamic modeling method of a kind of morphing aircraft that the present invention proposes is as follows:

(1) physical model of morphing aircraft is simplified.

Morphing aircraft as shown in Figure 1, suppose that whole morphing aircraft is comprised of five parts: fuselage 1, left wing inner side 2, left wing outside 3, right flank inner side 4 and right flank outside 5, each part is reduced to an independent rigid body, and quality is respectively m _b, m ₁, m ₂, m ₁, m ₂, the gross mass of morphing aircraft is m _t.Fuselage 1 is reduced to homogeneous cylinder, and section radius is R, and length is l ₀, barycenter is a C _b.Described left wing inner side 2, left wing outside 3, right flank inner side 4 and right flank outside 5 are all reduced to homogeneous thin bar, and the annexation between described left wing inner side 2 and left wing outside 3, outside right flank inner side 4 and right flank between 5 is being slidably connected along pole length direction; Between described left wing inner side 2 and fuselage 1, right flank inner side 4 is and is rotationally connected with annexation between fuselage 1, rotation axis lay respectively at into an A with put B.

Take an A and relative to fuselage 1, rotate as axle in left wing inner side 2, left wing outside 3 is done linear telescopic along bar length direction with respect to left wing inner side 2 and is moved, take a B and relative to fuselage 1, rotate as axle in right flank inner side 4, right flank outside 5 is done linear telescopic along bar length direction with respect to right flank inner side 4 and moved.If left wing inner side 2 is l with the pole length of right flank inner side 4 ₁, left wing outside 3 is l with the pole length in right flank outside 5 ₂.Left wing angle of sweep is θ ₁, right flank angle of sweep is θ ₂, the extension elongation of the relative left wing in left wing outside 3 inner side 2 is Δ ₁, the extension elongation of relative right flank inner side, right flank outside 54 is Δ ₂.If the bar after two wings are simplified and the axis of symmetry of fuselage 1 are positioned at same plane, this plane is also longitudinal principal axis of inertia plane of fuselage 1.

(2) Coordinate system definition.

In the Dynamic Modeling process of morphing aircraft, it is very crucial selecting suitable coordinate system, both can set up the mathematical model of simplification, can make again the explicit physical meaning of equation.Each coordinate system of setting up in the present invention as shown in Figure 2.

O _gx _gy _gz _gfor earth axes, be assumed to be inertial system, z _gaxle points to the earth's core vertically downward.

Ox _by _bz _bfor the body axis system being connected with fuselage 1, the true origin that the mid point O of an A, some B line of take is body axis system, because the position of wing rotating shaft on fuselage 1 fixed, the distance that set up an office O and A, B are ordered is that normal value a(is shown in Fig. 1), center of gravity (barycenter) C of set up an office O and fuselage 1 _bdistance be b.X _baxle is in morphing aircraft symmetrical plane and be parallel to the orientation of its axis head of morphing aircraft, x _baxle is the center inertia principal axis of fuselage 1, y _bit is right-hand that axle points to fuselage perpendicular to morphing aircraft symmetrical plane, z _baxle is in the symmetrical plane of morphing aircraft and x _baxle is vertical and point to fuselage below.

Consider that morphing aircraft is in flight course, for the aerodynamic characteristic of balance morphing aircraft, may introduce Centroid Adjustment System, so set up coordinate system C _bx ₁y ₁z ₁with fuselage 1 barycenter C _bfor initial point, coordinate axis x ₁y ₁z ₁direction and coordinate system Ox _by _bz _bcorresponding coordinate axle x _by _bz _bdirection is consistent.

O ₂x ₂y ₂z ₂barycenter O with left wing inner side 2 ₂for initial point, by Ox _by _bz _baround z _bturn-θ of axle ₁angle obtains, x ₂y ₂z ₂x after axle difference corresponding rotation _by _bz _baxle.O ₃x ₃y ₃z ₃barycenter O with left wing outside 3 ₃for initial point, x ₃y ₃z ₃the direction of axle and O ₂x ₂y ₂z ₂the direction of each corresponding axis is identical.In like manner can be able to the barycenter O of right flank inner side 4 ₄coordinate system O for initial point ₄x ₄y ₄z ₄(Ox _by _bz _baround z _baxle turns θ ₁angle obtains, x ₄y ₄z ₄x after axle difference corresponding rotation _by _bz _baxle), with the barycenter O in right flank outside 5 ₅coordinate system O for initial point ₅x ₅y ₅z ₅(x ₅y ₅z ₅the direction of axle and O ₂x ₂y ₂z ₂the direction of each corresponding axis is identical).

Position relationship between each coordinate system can be used coordinate conversion matrix A _ija is described _ijthe coordinate conversion matrix of expression from coordinate system j to coordinate system i.

(3) equation of constraint of amoeboid movement represents.

Amoeboid movement wing with respect to fuselage 1 is assumed to be and can control effectively by variant control gear, and position, speed and acceleration that the length of wing, angle of sweep change can reach setting value by control.So given deformation process can utilize one group of equation of constraint to represent:

X＝f(X _r,t) (1)

In formula: X is state vector, represent each state parameter of wing amoeboid movement, described state parameter comprises the deformable parameter (Δ of wing ₁, Δ ₂, θ ₁, θ ₂) size with and the speed and the acceleration that change; X _rfor the set-point vector of each state parameter, t is the time.

Above equation of constraint can represent the relevant degree of freedom of morphing aircraft wing amoeboid movement effectively.Given equation of constraint (1), the independence and freedom number of degrees of morphing aircraft are just decided by the degree of freedom of fuselage 1 completely.

(4) generalized coordinate and general velocity chooses.

The motion of morphing aircraft fuselage is the spatial movement of a six degree of freedom, is utilizing Kane method to set up in the process of morphing aircraft system dynamics model, gets the six-freedom degree of fuselage 1 as the degree of freedom of morphing aircraft.

Choose coordinate x, y, z and the body axis system Ox of O point in earth axes on fuselage 1 _by _bz _bwith respect to earth axes O _gx _gy _gz _gpitching angle theta, roll angle six variablees of crab angle ψ are as the generalized coordinate of morphing aircraft system.Choose O point at earth axes O _gx _gy _gz _gthe speed V of middle translation motion is at body axis system coordinate axis x _b, y _b, z _bon component u, v, w, and body axis system with respect to the angular velocity omega of earth axes rotational motion at body axis system coordinate axis x _b, y _b, z _bon component p, q, these six variablees of r as the general velocity u of morphing aircraft system _k.

(5) construct triumphant grace equation.

Triumphant grace equation in the inner expression formula of inertial reference system (inertial system) is:

${\tilde{F}}_k+{\tilde{F}}_k^{*}=0(k=1,...,f)---\left(2\right)$

In formula:

with

be respectively general velocity u _kcorresponding broad sense active force and broad sense inertial force, the number of degrees of freedom, that f is corresponding system.And have:

${\tilde{F}}_k=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}[F_i\·v_{\mathrm{ci}}^{\left(k\right)}+M_i\·{\mathrm{\ω}}_i^{\left(k\right)}]---\left(3\right)$

${\tilde{F}}_k^{*}=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}[F_i^{*}\·v_{\mathrm{ci}}^{\left(k\right)}+M_i^{*}\·{\mathrm{\ω}}_i^{\left(k\right)}]---\left(4\right)$

In formula: the rigid body quantity that N is corresponding system;

the inclined to one side speed of k barycenter that is called rigid body i;

the k drift angle speed that is called rigid body i; F _iand M _ibe respectively the main square of the relative barycenter of main resultant of the active force acting on rigid body i;

with

the inertia that the is respectively rigid body i main square of the relative barycenter of resultant of advocating.

with

can obtain with following formula:

$F_i^{*}=-m_i{a}_{\mathrm{ci}}---\left(5\right)$

$M_i^{*}=-J_i\·{\stackrel{\·}{\mathrm{\ω}}}_i-{\mathrm{\ω}}_i\×(J_i\·{\mathrm{\ω}}_i)---\left(6\right)$

In formula: m _iquality for rigid body i; a _cibarycenter acceleration for rigid body i; J _ifor the inertial tensor of rigid body i, ω _iwith

be respectively rotational angular velocity and the angular acceleration of rigid body i.

While constructing triumphant grace equation, first, with body axis system Ox _by _bz _bfor reference frame solves each rigid body with respect to systemic velocity and the rotational angular velocity of inertial system, then it is carried out to differentiate and obtain the barycenter acceleration of each rigid body, the angular acceleration of rotation around center of mass.Next, calculate the inclined to one side speed of barycenter, the drift angle speed of each rigid body.

The simplification of the present invention to broad sense active force: the control system of supposition deformation mechanism can be controlled the amoeboid movement of wing as required, do not consider the motion control problem of deformation mechanism, interaction force between wing and fuselage and between wing medial and lateral can be used as to the constraining force of morphing aircraft kinetic model, by equation of constraint (1), represented, according to Kane method principle, the not aobvious constraining force containing system in triumphant grace equation.

For the present invention for morphing aircraft for, its suffered outside active force comprises gravity, aerodynamic force and motor power three parts.Because the degree of freedom of the morphing aircraft system of setting up in the present invention is the six-freedom degree of fuselage, for these degree of freedom, the suffered aerodynamic effect of morphing aircraft can be simplified on fuselage completely, and do not affect the correctness of these degree of freedom motion, therefore the suffered aerodynamic force of morphing aircraft can be reduced to and only act on fuselage, other parts are only subject to the effect of self gravitation.

The broad sense inertial force that each general velocity is corresponding can calculate according to formula (4), and the mass force and touqhe of each rigid body can calculate according to formula (5), (6) respectively.

According to formula (2～6), finally can draw six kinetics equations of morphing aircraft shown in Fig. 1, i.e. formula (7～12).

$0=T-\mathrm{DC\αC\β}-\mathrm{YC\αS\β}+\mathrm{LS\α}-m_t[\mathrm{gS\θ}+(\stackrel{\·}{u}-\mathrm{rv}+\mathrm{qw})]+m_{b}b(r^2+q^2)-$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}\{(\stackrel{\&CenterDot;}{r}-\mathrm{pq}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1){\mathrm{C\&theta;}}_1+[{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2+q^2]{\mathrm{S\&theta;}}_1-(\stackrel{\&CenterDot;}{r}-\mathrm{pq}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2){\mathrm{C\&theta;}}_2+[{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2+q^2\left]{\mathrm{S\&theta;}}_2\right\}----\left(7\right)$

$m_2\left\{\right[q^2{L}_1-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2]{\mathrm{S\&theta;}}_1+[(\stackrel{\&CenterDot;}{r}-\mathrm{pq}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1){\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)]{\mathrm{C\&theta;}}_1+$

$[q^2{L}_2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2]{\mathrm{S\&theta;}}_2+[(\mathrm{pq}-\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)L_2-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)]{\mathrm{C\&theta;}}_2\}$

$0=m_t[\mathrm{gC\&theta;S\&phi;}-(\stackrel{\&CenterDot;}{v}+\mathrm{ri}-\mathrm{pw})]-\mathrm{DS\&beta;}+\mathrm{YC\&beta;}-m_b(b\stackrel{\&CenterDot;}{r}+\mathrm{bpq})-$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}\{({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1-\stackrel{\&CenterDot;}{r}-\mathrm{pq}){\mathrm{S\&theta;}}_1+[{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2+p^2]{\mathrm{C\&theta;}}_1-({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2+\stackrel{\&CenterDot;}{r}+\mathrm{pq}){\mathrm{S\&theta;}}_2-[{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2+p^2\left]{\mathrm{C\&theta;}}_2\right\}----\left(8\right)$

$m_2\{-[{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1\mathrm{pq}+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)+L_1(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)]{\mathrm{S\&theta;}}_1+[L_1{p}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2]{\mathrm{C\&theta;}}_1-$

$[{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_2\mathrm{pq}+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)+L_2(\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)]{\mathrm{S\&theta;}}_2-[L_2{p}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2]{\mathrm{C\&theta;}}_2\}$

$0=m_t[\mathrm{gC\&theta;C\&phi;}-(\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv})]-\mathrm{DS\&alpha;C\&beta;}-\mathrm{YS\&alpha;S\&beta;}-\mathrm{LC\&alpha;}-m_b(-b\stackrel{\&CenterDot;}{q}+\mathrm{bpr})-$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}[(\stackrel{\&CenterDot;}{q}+2p{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1-\mathrm{pr})S{\mathrm{\&theta;}}_1+(2q{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1-\stackrel{\&CenterDot;}{p}-\mathrm{qr})C{\mathrm{\&theta;}}_1+(\stackrel{\&CenterDot;}{q}-2p{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2-\mathrm{pr}){\mathrm{S\&theta;}}_2+(2q{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2+\stackrel{\&CenterDot;}{p}+\mathrm{qr}){\mathrm{C\&theta;}}_2]----\left(9\right)$

$m_2\left\{\right[2q{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1+(\stackrel{\&CenterDot;}{q}-\mathrm{pr}){\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+2\mathrm{<figure-callout\; id="1"\; label="p\; L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">p</figure-callout>}{L}_{}{}_1{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1]{\mathrm{S\&theta;}}_1+(2\mathrm{<figure-callout\; id="1"\; label="q\; L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">q</figure-callout>}{L}_{}{}_1{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1-\stackrel{\&CenterDot;}{p}{L}_1-\mathrm{qr}{L}_1-2p{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1){\mathrm{C\&theta;}}_1+$

$[2q{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2+(\stackrel{\&CenterDot;}{q}-\mathrm{pr})L_2-2\mathrm{<figure-callout\; id="2"\; label="p\; L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">p</figure-callout>}{L}_{}{}_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2]{\mathrm{S\&theta;}}_2+(2\mathrm{<figure-callout\; id="2"\; label="q\; L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">q</figure-callout>}{L}_{}{}_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2+\stackrel{\&CenterDot;}{p}{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_2+\mathrm{<figure-callout\; id="2"\; label="qr\; L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">qr</figure-callout>}{L}_{}{}_2+2p{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2){\mathrm{C\&theta;}}_2\}$

$0=[\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{m}_1({\mathrm{C\&theta;}}_2-{\mathrm{C\&theta;}}_1)+m_2({\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_2{\mathrm{C\&theta;}}_2-{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1{\mathrm{C\&theta;}}_1)]\mathrm{gC\&theta;C\&phi;}+({\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_2+J_3)[{\mathrm{S\&theta;}}_1{\mathrm{C\&theta;}}_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p+\stackrel{\&CenterDot;}{q}-\mathrm{pr})+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">C</figure-callout>}}^2{\mathrm{\&theta;}}_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})]+$

$M_x-J_x\stackrel{\&CenterDot;}{p}-(J_z-J_y)\mathrm{qr}+({\mathrm{<figure-callout\; id="4"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_4+J_5)[{\mathrm{S\&theta;}}_2{\mathrm{C\&theta;}}_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\stackrel{\&CenterDot;}{q}+\mathrm{pr})-{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">C</figure-callout>}}^2{\mathrm{\&theta;}}_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})]+$

$m_1(a+\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_1)[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p+\stackrel{\&CenterDot;}{q}-\mathrm{pr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_1+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_1]+---\left(10\right)$

$m_2(a+{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1{\mathrm{C\&theta;}}_1)\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}q+L_1(\stackrel{\&CenterDot;}{q}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p-\mathrm{pr})]{\mathrm{S\&theta;}}_1+[L_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}p\left]{\mathrm{C\&theta;}}_1\right\}-$

$m_1(a+\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_2)[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+(-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p+\stackrel{\&CenterDot;}{q}-\mathrm{pr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_2+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_2]-$

$m_2(a+{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_2{\mathrm{C\&theta;}}_2)\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}+\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}q+L_2(\stackrel{\&CenterDot;}{q}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\mathrm{pr})]{\mathrm{S\&theta;}}_2+[L_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}p\left]{\mathrm{C\&theta;}}_2\right\}$

$0=M_y+b(\mathrm{DS\&alpha;C\&beta;}+\mathrm{YS\&alpha;S\&beta;}+\mathrm{LC\&alpha;})+[-{\mathrm{bm}}_b+\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}({\mathrm{S\&theta;}}_1+{\mathrm{S\&theta;}}_2){\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">m</figure-callout>}}_1+(L_{1}S{\mathrm{\&theta;}}_1+L_{2}S{\mathrm{\&theta;}}_2)m_2]\mathrm{gC\&theta;C\&phi;}+$

$m_{b}b(\stackrel{\&CenterDot;}{w}-b\stackrel{\&CenterDot;}{q}-\mathrm{qu}+\mathrm{pv}+\mathrm{bpr})-J_y\stackrel{\&CenterDot;}{q}-(J_x-J_z)\mathrm{pr}+({\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_2+J_3){\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">S</figure-callout>}}^2{\mathrm{\&theta;}}_1(-\stackrel{\&CenterDot;}{q}+\mathrm{pr}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p)+$

$({\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_2+J_3){\mathrm{S\&theta;}}_1{\mathrm{C\&theta;}}_1(\stackrel{\&CenterDot;}{p}+\mathrm{qr}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q)+({\mathrm{<figure-callout\; id="4"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_4+J_5){\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">S</figure-callout>}}^2{\mathrm{\&theta;}}_2(-\stackrel{\&CenterDot;}{q}+\mathrm{pr}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p)-({\mathrm{<figure-callout\; id="4"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_4+J_5)S{\mathrm{\&theta;}}_{2}C{\mathrm{\&theta;}}_2(\stackrel{\&CenterDot;}{p}+\mathrm{qr}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q)-$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_1[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+(\stackrel{\&CenterDot;}{q}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p-\mathrm{pr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_1+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_1]----\left(11\right)$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_2[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}+\mathrm{aqr}+(\stackrel{\&CenterDot;}{q}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\mathrm{pr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_2+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_2]-$

$m_2{L}_{1}S{\mathrm{\&theta;}}_1\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}q+L_1(\stackrel{\&CenterDot;}{q}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p-\mathrm{pr})]{\mathrm{S\&theta;}}_1+[L_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}p\left]C{\mathrm{\&theta;}}_1\right\}-$

$m_2{L}_{2}S{\mathrm{\&theta;}}_2\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}+\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}q+L_2(\stackrel{\&CenterDot;}{q}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\mathrm{pr})]{\mathrm{S\&theta;}}_2+[L_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}p\left]C{\mathrm{\&theta;}}_2\right\}$

$0=M_z+b(-\mathrm{DS\&beta;}+\mathrm{YC\&beta;})+[\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}({\mathrm{C\&theta;}}_2-C{\mathrm{\&theta;}}_1){\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">m</figure-callout>}}_1+({\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_2{\mathrm{C\&theta;}}_2-{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1{\mathrm{C\&theta;}}_1)m_2]\mathrm{gS\&theta;}-m_{b}b(\stackrel{\&CenterDot;}{v}+b\stackrel{\&CenterDot;}{r}+\mathrm{ru}-\mathrm{pw}+\mathrm{bpq})+$

$[{\mathrm{bm}}_b-\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}({\mathrm{S\&theta;}}_1+S{\mathrm{\&theta;}}_2)m_1-(L_{1}S{\mathrm{\&theta;}}_1+L_{2}S{\mathrm{\&theta;}}_2)m_2]\mathrm{gC\&theta;S\&phi;}-J_z\stackrel{\&CenterDot;}{r}-(J_y-J_x)\mathrm{pq}-({\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_2+J_3)(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)+$

$({\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_2+J_3)[({\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">C</figure-callout>}}^2{\mathrm{\&theta;}}_1-{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">S</figure-callout>}}^2{\mathrm{\&theta;}}_1)\mathrm{pq}+S{\mathrm{\&theta;}}_{1}C{\mathrm{\&theta;}}_1(p^2-q^2)]+({\mathrm{<figure-callout\; id="4"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_4+J_5)[-(\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)+({\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">C</figure-callout>}}^2{\mathrm{\&theta;}}_2-{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">S</figure-callout>}}^2{\mathrm{\&theta;}}_2)\mathrm{pq}+S{\mathrm{\&theta;}}_{2}C{\mathrm{\&theta;}}_2(q^2-p^2)]-$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}S{\mathrm{\&theta;}}_1\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}+a{r}^2-\mathrm{pw}+a{p}^2-(\mathrm{pq}+\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}S{\mathrm{\&theta;}}_1+[{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">p</figure-callout>}}^2+{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2\left]\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}C{\mathrm{\&theta;}}_1\right\}-$

$m_1\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}S{\mathrm{\&theta;}}_2\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}-a{r}^2-\mathrm{pw}-a{p}^2-(\mathrm{pq}+\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}S{\mathrm{\&theta;}}_2-[{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">p</figure-callout>}}^2+{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2\left]\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}C{\mathrm{\&theta;}}_2\right\}+---\left(12\right)$

$m_2{L}_{1}S{\mathrm{\&theta;}}_1\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}+{\mathrm{ar}}^2-\mathrm{pw}+a{p}^2-[L_1(\mathrm{pq}+\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)]{\mathrm{S\&theta;}}_1+[L_1{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">p</figure-callout>}}^2+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1\left]{\mathrm{C\&theta;}}_1\right\}+$

$m_2{L}_{2}S{\mathrm{\&theta;}}_2\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}-{\mathrm{ar}}^2-\mathrm{pw}-a{p}^2-[L_2(\mathrm{pq}+\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)]{\mathrm{S\&theta;}}_2-[L_2{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">p</figure-callout>}}^2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2\left]{\mathrm{C\&theta;}}_2\right\}+$

$m_1(a+\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_1)\{\stackrel{\&CenterDot;}{u}+a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}-\mathrm{apq}+[{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">q</figure-callout>}}^2+{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2]\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_1+(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1-\mathrm{pq})\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_1\}+$

$m_1(a+\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_2)\{\stackrel{\&CenterDot;}{u}-a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}+\mathrm{apq}+[{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">q</figure-callout>}}^2+{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2]\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{S\&theta;}}_2+(\mathrm{pq}-\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1}{2}{\mathrm{C\&theta;}}_2\}-$

$m_2(a+L_{1}C{\mathrm{\&theta;}}_1)\{\stackrel{\&CenterDot;}{u}+a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}-\mathrm{apq}+[L_1{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">q</figure-callout>}}^2+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1]S{\mathrm{\&theta;}}_1+[L_1(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1-\mathrm{pq})+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)\left]C{\mathrm{\&theta;}}_1\right\}+$

$m_2(a+L_{2}C{\mathrm{\&theta;}}_2)\{\stackrel{\&CenterDot;}{u}-a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}+\mathrm{apq}+[L_2{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">q</figure-callout>}}^2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2]S{\mathrm{\&theta;}}_2+[L_2\left({\mathrm{pq}-\stackrel{\&CenterDot;}{r}-\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2\right)-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)\left]C{\mathrm{\&theta;}}_2\right\}$

In above equation, D is resistance, and Y is side force, and L is lift, M _xfor rolling moment, M _yfor pitching moment, M _zfor yawing, α is the angle of attack, and β is yaw angle; J _x, J _y, J _zfor fuselage is with respect to coordinate system C _bx ₁y ₁z ₁the moment of inertia of each coordinate axis, ${\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_2={\mathrm{<figure-callout\; id="4"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">J</figure-callout>}}_4=\frac{m_1{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1^2}{12},$ ${\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000032.png"\; state="\{\{state\}\}">J</figure-callout>}}_3={\mathrm{<figure-callout\; id="5"\; label="J"\; filenames="HDA0000413072380000012.png,HDA0000413072380000041.png"\; state="\{\{state\}\}">J</figure-callout>}}_5=\frac{m_2{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_2^2}{12};$ ${\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1+{\mathrm{\&Delta;}}_1-\frac{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_2}{2},$ ${\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">L</figure-callout>}}_2={\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_1+{\mathrm{\&Delta;}}_2-\frac{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA0000413072380000012.png,HDA0000413072380000031.png"\; state="\{\{state\}\}">l</figure-callout>}}_2}{2};$ Point above variable represents the number of times of variable to time differentiate; For formula of reduction, express, with C, replace cos, with S, replace sin.

Known according to construction process, six kinetics equations (formula 7～12) of morphing aircraft correspond respectively to general velocity u, v, w, p, q, the r of system, this kinetics equation (formula 7～12) builds under body axis system, and each in kinetics equation all has actual physical significance.The equilibrium equation that is morphing aircraft corresponding to the equation of u, v, w, represents that active force that morphing aircraft is suffered and inertial force sum are at body axis system x _b, y _b, z _bbeing projected as on axle is zero; The torque equilibrium equation that is morphing aircraft corresponding to the equation of p, q, r, represent active moment that morphing aircraft is suffered and moment of inertia and at body axis system x _b, y _b, z _bbeing projected as on axle is zero.According to traditional flight mechanics sorting technique, corresponding to the equation of u, v, q, be longitudinal dynamics equation, corresponding to the equation of w, p, r, be horizontal lateral dynamics equation.Check that in equation, the dimension of each is all corresponding strength guiding principle or moment dimension.

The variant motion that the present invention proposes is as follows on the concrete implementation step of the analytical approach of morphing aircraft dynamics impact:

(1) additional force, additional moment are defined.

When the morphing aircraft at variable length, variable angle of sweep carries out amoeboid movement, can produce the dynamics that is different from Fixed Wing AirVehicle.Except the suffered aerodynamic force of morphing aircraft can change with the variation of aerodynamic configuration, the impact that also can produce the additional force and the additional moment that come from amoeboid movement.Additional force refer in the equilibrium equation of morphing aircraft all due to amoeboid movement introduce item (not comprising aerodynamic force) and, contain

or

item and, it is at body axis system x _b, y _b, z _bcomponent on axle is denoted as respectively Δ F _x, Δ F _y, Δ F _z.Additional moment refer in the torque equilibrium equation of morphing aircraft all due to amoeboid movement introduce item (not comprising aerodynamic couple) and, contain or item and, it is at body axis system x _b, y _b, z _bcomponent on axle is denoted as respectively Δ M _x, Δ M _y, Δ M _z.Analyze knownly, additional force and additional moment are exactly the mass force and touqhe of the wing that causes of amoeboid movement, and this is also that morphing aircraft kinetics equation is compared with conventional Fixed Wing AirVehicle kinetics equation, remove the key distinction place of aerodynamic force outside changing.

(2) additional force, additional moment are carried out to simulation analysis.

Set the state of flight of morphing aircraft, variant is moved through to formula (1) to be expressed, utilize the computing formula of additional force and additional moment to calculate additional force and the additional moment in variant process, calculate the suffered aerodynamic force of morphing aircraft in variant process and the changing value of aerodynamic moment simultaneously, then these two kinds of result of calculations are compared respectively, big or small on the dynamic (dynamical) impact of morphing aircraft in variant process by relatively analyzing additional force and additional moment, if two kinds of results (comprise additional force and aerodynamic comparative result, the comparative result of additional moment and aerodynamic moment) differ one more than the order of magnitude, can ignore wherein less one, and affect hardly the dynamics of morphing aircraft, but but can greatly simplify the kinetic model of morphing aircraft.

Embodiment mono-:

For the variable length shown in Fig. 1, become the morphing aircraft at angle of sweep, the concrete steps of the Dynamic Modeling providing according to the invention described above, set up the kinetic model of a morphing aircraft, and the design parameter of this morphing aircraft is as following table (Ma is Mach number):

The setting of table 1 simulation parameter

Deformation process is: wing be deformed into symmetrical distortion, the first step, aerodynamic arrangement (1), as Fig. 3 A, launches completely from wing outside, angle of sweep is zero; Aerodynamic arrangement (2), as Fig. 3 B, is at the uniform velocity deformed to wing and shrinks completely, and angle of sweep is 45 degree, and deformation time is respectively 1s and 15s; Second step, aerodynamic arrangement (3) and startup layout (4), as Fig. 3 C and Fig. 3 D, carry out the contrary reversal deformation of same first step deformation process.

So amoeboid movement can represent with following equation of constraint:

$\left.\begin{array}{c}{\mathrm{\&theta;}}_1={\mathrm{\&theta;}}_2=t\&CenterDot;\frac{{\mathrm{\&theta;}}_{\mathrm{end}}-{\mathrm{\&theta;}}_{\mathrm{start}}}{T}\\ {\mathrm{\&Delta;}}_1={\mathrm{\&Delta;}}_2=l_2-t\&CenterDot;\frac{{\mathrm{\&Delta;}}_{\mathrm{end}}-{\mathrm{\&Delta;}}_{\mathrm{start}}}{T}\end{array}\right\}---\left(13\right)$

In formula: t is the time, T is distortion T.T. (being respectively 1s and 15s), θ _startangle of sweep while starting for distortion, θ _endangle of sweep while finishing for distortion, Δ _startthe length that while starting for distortion, stretch out in wing outside, Δ _endthe length that while finishing for distortion, stretch out in wing outside.

Fig. 3 has provided four kinds of aerodynamic configurations in deformation process, according to aerodynamic data simulation calculation software, calculate the result of lift coefficient, resistance coefficient and aerodynamic center of morphing aircraft under Zhe Sizhong aerodynamic arrangement as shown in Fig. 4, Fig. 5, Fig. 6, can find that in deformation process, the suffered aerodynamic variation of aircraft is very large.

As seen from Figure 4, the lift coefficient of aircraft before and after distortion changes greatly, and as under 2 ° of angles of attack, aircraft is deformed to aerodynamic arrangement (4) from aerodynamic arrangement (1), and lift coefficient reduces 27.3%.

As seen from Figure 5, under Low Angle Of Attack condition, while being 2 ° as the angle of attack, before and after distortion, resistance coefficient is substantially constant; Under large angle of attack condition, the variation of resistance coefficient is larger, if the angle of attack is that under 10° Shi，Si Zhong aerodynamic arrangement, resistance coefficient maximum differs 8.5%.

As seen from Figure 6, along with the variation of aerodynamic arrangement, the variation of aerodynamic center is very obvious.From aerodynamic arrangement (1), be deformed to aerodynamic arrangement (4), Center of Pressure along fuselage Axial changes about 0.76m, and morphing aircraft barycenter has only correspondingly moved 0.14m backward, therefore the longitudinal aerodynamic moment with respect to barycenter that morphing aircraft is subject to alters a great deal, along with the nose-down pitching moment that morphing aircraft is subject to that carries out of distortion strengthens gradually, this makes morphing aircraft carry out trim and control longitudinally more difficult, and under the large angle of attack, longitudinal aerodynamic moment that the moment that elevating rudder produce cannot trim morphing aircraft.

According to the definition of additional force, additional moment, from formula (7)～(12), extract the computing formula of additional force, additional moment, then can be calculated the additional force that in morphing aircraft deformation process, morphing aircraft is subject to, the size of additional moment, symmetry is at the uniform velocity out of shape additional force result of calculation and is seen Fig. 7 A; Reverse symmetry is at the uniform velocity out of shape additional force result of calculation and is seen Fig. 7 B.

Simulation result shows: only having u corresponding equation (7) is x _bthere is distortion additional force Δ F in direction of principal axis _x, amoeboid movement does not have the impact of additional force or moment on all the other equations; The process of distortion is shorter is that speed is larger, and additional force is just larger, in deformation process, is that in 15s situation, additional force is negligible, in deformation process, is that in 1s situation, additional force increases to some extent, but is only 1.6% of the suffered gravity of morphing aircraft; The additional force of initial deformation is less than the additional force in the situation of large angle of sweep.Comparison diagram 7A and Fig. 7 B are known, and the size of the additional force that in deformation process, same position produces is identical, and forward and reverse at the uniform velocity additional force size that distortion produces is identical.

The contrast changing by above simulation result and aerodynamic force can draw to draw a conclusion, under the little symmetric deformation speed of wing (if deformation process is 15s, even longer), the impact of amoeboid movement on the dynamics of morphing aircraft, mainly come from aerodynamic variation, now can neglect the impact of additional force, additional moment, also these can be removed from equation (7)～(12), this can great reduced mechanism.

The present invention not detailed description is known to the skilled person technology.

Claims (5)

1. the Dynamic Modeling of morphing aircraft and an analytical approach, is characterized in that: the dynamic modeling method and the analytical approach two parts content of variant motion on the impact of morphing aircraft dynamics that comprise morphing aircraft.

2. the Dynamic Modeling of a kind of morphing aircraft according to claim 1 and analytical approach, is characterized in that: the dynamic modeling method of described morphing aircraft comprises the steps,

The first step, simplifies the physical model of morphing aircraft;

Suppose that whole morphing aircraft is comprised of five parts: fuselage, left wing inner side, left wing outside, right flank inner side and right flank outside, each part is reduced to an independent rigid body, and quality is respectively m _b, m ₁, m ₂, m ₁, m ₂, the gross mass of morphing aircraft is m _t; Fuselage is reduced to homogeneous cylinder, and section radius is R, and length is l ₀, barycenter is a C _b; Inside described left wing, outside left wing, inside right flank and right flank outside is all reduced to homogeneous thin bar, and the annexation inside described left wing and between outside left wing, inside right flank and between outside right flank is being slidably connected along pole length direction; Annexation inside described left wing and between fuselage, inside right flank and between fuselage is and is rotationally connected, and rotation axis lays respectively at as an A and some B;

Second step, Coordinate system definition;

O _gx _gy _gz _gfor earth axes, be assumed to be inertial system, z _gaxle points to the earth's core vertically downward;

Ox _by _bz _bfor the body axis system being connected with fuselage, the true origin that the mid point O of an A, some B line of take is body axis system, x _baxle is in morphing aircraft symmetrical plane and be parallel to the orientation of its axis head of morphing aircraft, x _baxle is the center inertia principal axis of fuselage, y _bit is right-hand that axle points to fuselage perpendicular to morphing aircraft symmetrical plane, z _baxle is in the symmetrical plane of morphing aircraft and x _baxle is vertical and point to fuselage below;

Coordinate system C _bx ₁y ₁z ₁with fuselage barycenter C _bfor initial point, coordinate axis x ₁y ₁z ₁direction and coordinate system Ox _by _bz _bcorresponding coordinate axle x _by _bz _bdirection is consistent;

O ₂x ₂y ₂z ₂barycenter O with left wing inner side ₂for initial point, by Ox _by _bz _baround z _bturn-θ of axle ₁angle obtains, x ₂y ₂z ₂x after axle difference corresponding rotation _by _bz _baxle; O ₃x ₃y ₃z ₃barycenter O with left wing outside ₃for initial point, x ₃y ₃z ₃the direction of axle and O ₂x ₂y ₂z ₂the direction of each corresponding axis is identical; In like manner be able to the barycenter O of right flank inner side ₄coordinate system O for initial point ₄x ₄y ₄z ₄, with the barycenter O in right flank outside ₅coordinate system O for initial point ₅x ₅y ₅z ₅;

The 3rd step, the equation of constraint of amoeboid movement represents;

For a given deformation process, utilize one group of equation of constraint to represent:

X＝f(X _r,t) (1)

In formula: X is state vector, represent each state parameter of wing amoeboid movement, described state parameter comprises the deformable parameter Δ of wing ₁, Δ ₂, θ ₁, θ ₂size with and the speed and the acceleration that change; X _rfor the set-point vector of each state parameter, t is the time;

The 4th step, the choosing of generalized coordinate and general velocity;

Get the six-freedom degree of fuselage as the degree of freedom of morphing aircraft;

Choose coordinate x, y, z and the body axis system Ox of O point in earth axes on fuselage _by _bz _bwith respect to earth axes O _gx _gy _gz _gpitching angle theta, roll angle

six variablees of crab angle ψ are as the generalized coordinate of morphing aircraft system; Choose O point at earth axes O _gx _gy _gz _gthe speed V of middle translation motion is at body axis system coordinate axis x _b, y _b, z _bon component u, v, w, and body axis system with respect to the angular velocity omega of earth axes rotational motion at body axis system coordinate axis x _b, y _b, z _bon component p, q, these six variablees of r as the general velocity u of morphing aircraft system _k;

The 5th step, constructs triumphant grace equation;

The expression formula of triumphant grace equation in inertial reference system is:

${\tilde{F}}_k+{\tilde{F}}_k^{*}=0(k=1,...,f)---\left(2\right)$

In formula: with

be respectively general velocity u _kcorresponding broad sense active force and broad sense inertial force, the number of degrees of freedom, that f is corresponding system, and have:

${\tilde{F}}_k=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}[F_i\&CenterDot;v_{\mathrm{ci}}^{\left(k\right)}+M_i\&CenterDot;{\mathrm{\&omega;}}_i^{\left(k\right)}]---\left(3\right)$

${\tilde{F}}_k^{*}=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}[F_i^{*}\&CenterDot;v_{\mathrm{ci}}^{\left(k\right)}+M_i^{*}\&CenterDot;{\mathrm{\&omega;}}_i^{\left(k\right)}]---\left(4\right)$

In formula: the rigid body quantity that N is corresponding system;

the inclined to one side speed of k barycenter that is called rigid body i;

the k drift angle speed that is called rigid body i; F _iand M _ibe respectively the main square of the relative barycenter of main resultant of the active force acting on rigid body i;

with

the inertia that the is respectively rigid body i main square of the relative barycenter of resultant of advocating.

3. the Dynamic Modeling of a kind of morphing aircraft according to claim 1 and 2 and analytical approach, it is characterized in that: described in the dynamic modeling method of morphing aircraft, broad sense active force is made to following simplification: the control system of supposition deformation mechanism can be controlled the amoeboid movement of wing as required, do not consider the motion control problem of deformation mechanism, the constraining force of interaction force between wing and fuselage and between wing medial and lateral being used as to morphing aircraft kinetic model, is represented by equation of constraint (1).

4. the Dynamic Modeling of a kind of morphing aircraft according to claim 1 and 2 and analytical approach, is characterized in that: in the dynamic modeling method of morphing aircraft, and six kinetics equations of described morphing aircraft, as shown in the formula (7～12):

$0=T-\mathrm{DC\&alpha;C\&beta;}-\mathrm{YC\&alpha;S\&beta;}+\mathrm{LS\&alpha;}-m_t[\mathrm{gS\&theta;}+(\stackrel{\&CenterDot;}{u}-\mathrm{rv}+\mathrm{qw})]+m_{b}b(r^2+q^2)-$

$m_1\frac{l_1}{2}\{(\stackrel{\&CenterDot;}{r}-\mathrm{pq}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1){\mathrm{C\&theta;}}_1+[{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2+q^2]{\mathrm{S\&theta;}}_1-(\stackrel{\&CenterDot;}{r}-\mathrm{pq}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2){\mathrm{C\&theta;}}_2+[{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2+q^2\left]{\mathrm{S\&theta;}}_2\right\}----\left(7\right)$

$m_2\left\{\right[q^2{L}_1-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2]{\mathrm{S\&theta;}}_1+[(\stackrel{\&CenterDot;}{r}-\mathrm{pq}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)L_1+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)]{\mathrm{C\&theta;}}_1+$

$[q^2{L}_2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2]{\mathrm{S\&theta;}}_2+[(\mathrm{pq}-\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)L_2-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)]{\mathrm{C\&theta;}}_2\}$

$0=m_t[\mathrm{gC\&theta;S\&phi;}-(\stackrel{\&CenterDot;}{v}+\mathrm{ru}-\mathrm{pw})]-\mathrm{DS\&beta;}+\mathrm{YC\&beta;}-m_b(b\stackrel{\&CenterDot;}{r}+\mathrm{bpq})-$

$m_1\frac{l_1}{2}\{({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1-\stackrel{\&CenterDot;}{r}-\mathrm{pq}){\mathrm{S\&theta;}}_1+[{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2+p^2]{\mathrm{C\&theta;}}_1-({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2+\stackrel{\&CenterDot;}{r}+\mathrm{pq}){\mathrm{S\&theta;}}_2-[{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2+p^2\left]{\mathrm{C\&theta;}}_2\right\}----\left(8\right)$

$m_2\{-[L_1\mathrm{pq}+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)+L_1(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)]{\mathrm{S\&theta;}}_1+[L_1{p}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2]{\mathrm{C\&theta;}}_1-$

$[L_2\mathrm{pq}+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)+L_2(\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)]{\mathrm{S\&theta;}}_2-[L_2{p}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2]{\mathrm{C\&theta;}}_2\}$

$0=m_t[\mathrm{gC\&theta;C\&phi;}-(\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv})]-\mathrm{DS\&alpha;C\&beta;}-\mathrm{YS\&alpha;S\&beta;}-\mathrm{LC\&alpha;}-m_b(-b\stackrel{\&CenterDot;}{q}+\mathrm{bpr})-$

$m_1\frac{l_1}{2}[(\stackrel{\&CenterDot;}{q}+2p{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1-\mathrm{pr})S{\mathrm{\&theta;}}_1+(2q{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1-\stackrel{\&CenterDot;}{p}-\mathrm{qr})C{\mathrm{\&theta;}}_1+(\stackrel{\&CenterDot;}{q}-2p{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2-\mathrm{pr}){\mathrm{S\&theta;}}_2+(2q{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2+\stackrel{\&CenterDot;}{p}+\mathrm{qr}){\mathrm{C\&theta;}}_2]----\left(9\right)$

$m_2\left\{\right[2q{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1+(\stackrel{\&CenterDot;}{q}-\mathrm{pr})L_1+2p{L}_1{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1]{\mathrm{S\&theta;}}_1+(2q{L}_1{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1-\stackrel{\&CenterDot;}{p}{L}_1-\mathrm{qr}{L}_1-2p{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1){\mathrm{C\&theta;}}_1+$

$[2q{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2+(\stackrel{\&CenterDot;}{q}-\mathrm{pr})L_2-2p{L}_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2]{\mathrm{S\&theta;}}_2+(2q{L}_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2+\stackrel{\&CenterDot;}{p}{L}_2+\mathrm{qr}{L}_2+2p{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2){\mathrm{C\&theta;}}_2\}$

$0=[\frac{l_1}{2}{m}_1({\mathrm{C\&theta;}}_2-{\mathrm{C\&theta;}}_1)+m_2(L_2{\mathrm{C\&theta;}}_2-L_1{\mathrm{C\&theta;}}_1)]\mathrm{gC\&theta;C\&phi;}+(J_2+J_3)[{\mathrm{S\&theta;}}_1{\mathrm{C\&theta;}}_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p+\stackrel{\&CenterDot;}{q}-\mathrm{pr})+C^2{\mathrm{\&theta;}}_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})]+$

$M_x-J_x\stackrel{\&CenterDot;}{p}-(J_z-J_y)\mathrm{qr}+(J_4+J_5)[{\mathrm{S\&theta;}}_2{\mathrm{C\&theta;}}_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\stackrel{\&CenterDot;}{q}+\mathrm{pr})-C^2{\mathrm{\&theta;}}_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})]+$

$m_1(a+\frac{l_1}{2}{\mathrm{C\&theta;}}_1)[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p+\stackrel{\&CenterDot;}{q}-\mathrm{pr})\frac{l_1}{2}{\mathrm{S\&theta;}}_1+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})\frac{l_1}{2}{\mathrm{C\&theta;}}_1]+---\left(10\right)$

$m_2(a+L_1{\mathrm{C\&theta;}}_1)\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}q+L_1(\stackrel{\&CenterDot;}{q}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p-\mathrm{pr})]{\mathrm{S\&theta;}}_1+[L_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}p\left]{\mathrm{C\&theta;}}_1\right\}-$

$m_1(a+\frac{l_1}{2}{\mathrm{C\&theta;}}_2)[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+(-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p+\stackrel{\&CenterDot;}{q}-\mathrm{pr})\frac{l_1}{2}{\mathrm{S\&theta;}}_2+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})\frac{l_1}{2}{\mathrm{C\&theta;}}_2]-$

$m_2(a+L_2{\mathrm{C\&theta;}}_2)\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}+\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}q+L_2(\stackrel{\&CenterDot;}{q}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\mathrm{pr})]{\mathrm{S\&theta;}}_2+[L_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}p\left]{\mathrm{C\&theta;}}_2\right\}$

$0=M_y+b(\mathrm{DS\&alpha;C\&beta;}+\mathrm{YS\&alpha;S\&beta;}+\mathrm{LC\&alpha;})+[-{\mathrm{bm}}_b+\frac{l_1}{2}({\mathrm{S\&theta;}}_1+{\mathrm{S\&theta;}}_2)m_1+(L_{1}S{\mathrm{\&theta;}}_1+L_{2}S{\mathrm{\&theta;}}_2)m_2]\mathrm{gC\&theta;C\&phi;}+$

$m_{b}b(\stackrel{\&CenterDot;}{w}-b\stackrel{\&CenterDot;}{q}-\mathrm{qu}+\mathrm{pv}+\mathrm{bpr})-J_y\stackrel{\&CenterDot;}{q}-(J_x-J_z)\mathrm{pr}+(J_2+J_3)S^2{\mathrm{\&theta;}}_1(-\stackrel{\&CenterDot;}{q}+\mathrm{pr}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p)+$

$(J_2+J_3){\mathrm{S\&theta;}}_1{\mathrm{C\&theta;}}_1(\stackrel{\&CenterDot;}{p}+\mathrm{qr}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q)+(J_4+J_5)S^2{\mathrm{\&theta;}}_2(-\stackrel{\&CenterDot;}{q}+\mathrm{pr}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p)-(J_4+J_5)S{\mathrm{\&theta;}}_{2}C{\mathrm{\&theta;}}_2(\stackrel{\&CenterDot;}{p}+\mathrm{qr}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q)-$

$m_1\frac{l_1}{2}{\mathrm{S\&theta;}}_1[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+(\stackrel{\&CenterDot;}{q}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p-\mathrm{pr})\frac{l_1}{2}{\mathrm{S\&theta;}}_1+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})\frac{l_1}{2}{\mathrm{C\&theta;}}_1]----\left(11\right)$

$m_1\frac{l_1}{2}{\mathrm{S\&theta;}}_2[\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}+\mathrm{aqr}+(\stackrel{\&CenterDot;}{q}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\mathrm{pr})\frac{l_1}{2}{\mathrm{S\&theta;}}_2+(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})\frac{l_1}{2}{\mathrm{C\&theta;}}_2]-$

$m_2{L}_{1}S{\mathrm{\&theta;}}_1\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}-a\stackrel{\&CenterDot;}{p}-\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}q+L_1(\stackrel{\&CenterDot;}{q}+2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}p-\mathrm{pr})]{\mathrm{S\&theta;}}_1+[L_1(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{1}q-\stackrel{\&CenterDot;}{p}-\mathrm{qr})-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{1}p\left]C{\mathrm{\&theta;}}_1\right\}-$

$m_2{L}_{2}S{\mathrm{\&theta;}}_2\{\stackrel{\&CenterDot;}{w}-\mathrm{qu}+\mathrm{pv}+a\stackrel{\&CenterDot;}{p}+\mathrm{aqr}+[2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}q+L_2(\stackrel{\&CenterDot;}{q}-2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}p-\mathrm{pr})]{\mathrm{S\&theta;}}_2+[L_2(2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{2}q+\stackrel{\&CenterDot;}{p}+\mathrm{qr})+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_{2}p\left]C{\mathrm{\&theta;}}_2\right\}$

$0=M_z+b(-\mathrm{DS\&beta;}+\mathrm{YC\&beta;})+[\frac{l_1}{2}({\mathrm{C\&theta;}}_2-C{\mathrm{\&theta;}}_1)m_1+(L_2{\mathrm{C\&theta;}}_2-L_1{\mathrm{C\&theta;}}_1)m_2]\mathrm{gS\&theta;}-m_{b}b(\stackrel{\&CenterDot;}{v}+b\stackrel{\&CenterDot;}{r}+\mathrm{ru}-\mathrm{pw}+\mathrm{bpq})+$

$[{\mathrm{bm}}_b-\frac{l_1}{2}({\mathrm{S\&theta;}}_1+S{\mathrm{\&theta;}}_2)m_1-(L_{1}S{\mathrm{\&theta;}}_1+L_{2}S{\mathrm{\&theta;}}_2)m_2]\mathrm{gC\&theta;S\&phi;}-J_z\stackrel{\&CenterDot;}{r}-(J_y-J_x)\mathrm{pq}-(J_2+J_3)(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)+$

$(J_2+J_3)[(C^2{\mathrm{\&theta;}}_1-S^2{\mathrm{\&theta;}}_1)\mathrm{pq}+S{\mathrm{\&theta;}}_{1}C{\mathrm{\&theta;}}_1(p^2-q^2)]+(J_4+J_5)[-(\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)+(C^2{\mathrm{\&theta;}}_2-S^2{\mathrm{\&theta;}}_2)\mathrm{pq}+S{\mathrm{\&theta;}}_{2}C{\mathrm{\&theta;}}_2(q^2-p^2)]-$

$m_1\frac{l_1}{2}S{\mathrm{\&theta;}}_1\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}+a{r}^2-\mathrm{pw}+a{p}^2-(\mathrm{pq}+\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)\frac{l_1}{2}S{\mathrm{\&theta;}}_1+[p^2+{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2\left]\frac{l_1}{2}C{\mathrm{\&theta;}}_1\right\}-$

$\begin{array}{c}{m}_1\frac{l_1}{2}{\mathrm{S\&theta;}}_2\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}-{\mathrm{ar}}^2-\mathrm{pw}-{\mathrm{ap}}^2-(\mathrm{pq}+\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)\frac{l_1}{2}{\mathrm{S\&theta;}}_2-[p^2+{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2\left]\left(\frac{l_1}{2}{\mathrm{C\&theta;}}_2\right)\right\}+\\ m_2{L}_1{\mathrm{S\&theta;}}_1\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}+{\mathrm{ar}}^2-\mathrm{pw}+{\mathrm{ap}}^2-[L_1(\mathrm{pq}+\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1)+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)]{\mathrm{S\&theta;}}_1+[L_1{p}^2+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1\left]{\mathrm{C\&theta;}}_1\right\}+\end{array}---\left(12\right)$

$m_2{L}_{2}S{\mathrm{\&theta;}}_2\{\stackrel{\&CenterDot;}{v}+\mathrm{ru}-{\mathrm{ar}}^2-\mathrm{pw}-a{p}^2-[L_2(\mathrm{pq}+\stackrel{\&CenterDot;}{r}+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)]{\mathrm{S\&theta;}}_2-[L_2{p}^2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2\left]{\mathrm{C\&theta;}}_2\right\}+$

$m_1(a+\frac{l_1}{2}{\mathrm{C\&theta;}}_1)\{\stackrel{\&CenterDot;}{u}+a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}-\mathrm{apq}+[q^2+{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2]\frac{l_1}{2}{\mathrm{S\&theta;}}_1+(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1-\mathrm{pq})\frac{l_1}{2}{\mathrm{C\&theta;}}_1\}+$

$m_1(a+\frac{l_1}{2}{\mathrm{C\&theta;}}_2)\{\stackrel{\&CenterDot;}{u}-a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}+\mathrm{apq}+[q^2+{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2]\frac{l_1}{2}{\mathrm{S\&theta;}}_2+(\mathrm{pq}-\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2)\frac{l_1}{2}{\mathrm{C\&theta;}}_2\}-$

$m_2(a+L_{1}C{\mathrm{\&theta;}}_1)\{\stackrel{\&CenterDot;}{u}+a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}-\mathrm{apq}+[L_1{q}^2+L_1{(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_1]S{\mathrm{\&theta;}}_1+[L_1(\stackrel{\&CenterDot;}{r}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1-\mathrm{pq})+2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_1(r-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_1)\left]C{\mathrm{\&theta;}}_1\right\}+$

$m_2(a+L_{2}C{\mathrm{\&theta;}}_2)\{\stackrel{\&CenterDot;}{u}-a\stackrel{\&CenterDot;}{r}-\mathrm{rv}+\mathrm{qw}+\mathrm{apq}+[L_2{q}^2+L_2{(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)}^2-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Delta;}}}_2]S{\mathrm{\&theta;}}_2+[L_2\left({\mathrm{pq}-\stackrel{\&CenterDot;}{r}-\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2\right)-2{\stackrel{\&CenterDot;}{\mathrm{\&Delta;}}}_2(r+{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_2)\left]C{\mathrm{\&theta;}}_2\right\}$

In above equation, D is resistance, and Y is side force, and L is lift, M _xfor rolling moment, M _yfor pitching moment, M _zfor yawing, α is the angle of attack, and β is yaw angle; J _x, J _y, J _zfor fuselage is with respect to coordinate system C _bx ₁y ₁z ₁the moment of inertia of each coordinate axis, $J_2=J_4=\frac{m_1{l}_1^2}{12},$ $J_3=J_5=\frac{m_2{l}_2^2}{12};$ $L_1=l_1+{\mathrm{\&Delta;}}_1-\frac{l_2}{2},$ $L_2=l_1+{\mathrm{\&Delta;}}_2-\frac{l_2}{2};$ Point above variable represents the number of times of variable to time differentiate; C represents cos, and S represents sin.

5. the Dynamic Modeling of a kind of morphing aircraft according to claim 1 and analytical approach, is characterized in that: described variant motion is defined as follows additional force, additional moment the described kinetics equation of analytical approach application of morphing aircraft dynamics impact:

When the morphing aircraft at variable length, variable angle of sweep carries out amoeboid movement, additional force refer in the equilibrium equation of morphing aircraft all due to amoeboid movement introduce item and, contain

or

item and, it is at body axis system x _b, y _b, z _bcomponent on axle is denoted as respectively Δ F _x, Δ F _y, Δ F _z; Additional moment refer in the torque equilibrium equation of morphing aircraft all due to amoeboid movement introduce item and, contain

or item and, it is at body axis system x _b, y _b, z _bcomponent on axle is denoted as respectively Δ M _x, Δ M _y, Δ M _z; Additional force and additional moment are exactly the mass force and touqhe of the wing that causes of amoeboid movement;

Described variant motion on the analytic process of the analytical approach of morphing aircraft dynamics impact is: the state of flight that sets morphing aircraft, variant is moved through to formula (1) to be expressed, utilize the computing formula of additional force and additional moment to calculate additional force and the additional moment in variant process, calculate the suffered aerodynamic force of morphing aircraft in variant process and the changing value of aerodynamic moment simultaneously, then these two kinds of result of calculations are compared respectively, by comparative analysis go out additional force and additional moment in variant process on the dynamic (dynamical) impact size of morphing aircraft, if two kinds of results differ one more than the order of magnitude, ignore wherein less one, simplify the kinetic model of morphing aircraft.

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