CN105975733A - Kinematic and dynamic mixed dimensionality reduction solving method for 10-bar underactuated mechanism - Google Patents

Abstract

The invention provides a kinematic and dynamic mixed dimensionality reduction solving method for a 10 bar underactuated mechanism. The method comprises the following steps: step 1, a 10 bar mechanism mathematical model is established according to a Radarsat 2 battery panel deployment mechanism, wherein Radarsat 2 is a high resolution commercial radar satellite carrying a C band sensor; step 2, corresponding geometric equation is created according to the 10 bar mechanism mathematical model; step 3, the geometric equation is solved, and a dynamic model is established through combination with the Euler Lagrange equation; step 4, dynamic parameters of a 10 bar mechanism are solved according to the dynamic model. With adoption of the method, the dynamic parameters of the 10 bar mechanism can be solved, sizes of rod members of the Radarsat 2 battery panel deployment mechanism are optimized, and the battery panel deployment process stability and reliability in actual application satellite deployment process are improved.

Description

The kinesiology of ten bar underactuatuated drive and kinetics mixing solve dimension reduction method

Technical field

The present invention relates to technical field of aerospace control, in particular it relates to the kinesiology of a kind of ten bar underactuatuated drive is with dynamic Mechanics mixing solves dimension reduction method.

Background technology

Method in the present invention, with Radarsat-2 cell panel development mechanism as prototype, studies the some basic of this mechanism Problem, wherein the dynamical foundation configuration of this mechanism's most critical is drive lacking ten bars mechanism therein.Wherein, Radarsat-2 is One high-resolution commercialization radar satellite carrying C-band sensor, is cooperated with MDA company, in 2007 by space administration of Canada On December launches for 14 in Kazakhstan Baikonur base.Underactuated Mechanical Systems is that excitation number becomes less than system The mechanical system of amount number, i.e. accuses the system input number mechanical system less than degree of freedom in system.The drive lacking of under-actuated systems Characteristic is mainly caused by following 4 reasons:

1) dynamics Design of system；Such as aircraft, spacecraft, helicopter, undersea ship and without wheel locomotive etc..

2) in order to reduce expenses or some other practical purpose and design；Such as two angle of rake satellites and flexibility Link robots.

3) drive unsuccessfully；Such as: surface vessel, aircraft.

4) artificial complex lower order nonlinear system is created to further investigate the control of high-order under-actuated systems System；Such as two rank inverted pendulums, ball-and-beam system and rotation pendulum.

In the more than ten years in past, the challenge of under-actuated systems analysis and control design case has attracted substantial amounts of research people Member, the research field of these scholars relates to Non-Linear Control Theory, robot and automatization, the control of automatization's vehicles Control etc. with compliant mechanism.Owing to being limited by rocket load cabin ensemble space, antenna need to be by repeatedly folding when launching Draw in satellite top or both sides.Current most of antenna back side all have employed deployable mechanism design.The expansion of mechanism is not from Drive motor to drive, and multiple degrees of freedom necessarily causes the increased number of motor, so can increase the weight of rocket launching, can drop simultaneously The reliability that low cell panel launches.

Through the most methodical retrieval is found, currently without the correlation technique of ten bar underactuatuated drive.Radarsat-2 Cell panel development mechanism is entirely made up of connecting rod, and the coupling of x and y direction.Due to current Radarsat-2 cell panel Development mechanism does not carry out the exact arithmetic such as equation after designing according to configuration, it is impossible to obtain kinematic parameter accurately, so this structure The development mechanism size of the satellite battery plate under type is not optimum result.Only underactuatuated drive method only has five pole cranks sliding Block mechanism method, and the geometric equation x of crank block and y direction do not couple.So current method can't solve The kinematic problem of Radarsat-2 cell panel development mechanism.The present invention should be to ask in the method utilizing ten bars mechanism drive lacking Solve the kinetic parameter of ten bars mechanism, by described parameter optimization Radarsat-2 each bar of cell panel development mechanism solved The size of part, thus improve cell panel and launch the stability of process, so that satellite launch alleviates quality, improve cell panel exhibition Open procedure reliability.

Summary of the invention

For defect of the prior art, it is an object of the invention to provide the kinesiology of a kind of ten bar underactuatuated drive and move Mechanics mixing solves dimension reduction method.

Kinesiology and the kinetics mixing of the ten bar underactuatuated drive according to present invention offer solve dimension reduction method, including:

Step 1: set up the mathematical model of ten bars mechanism according to satellite battery plate development mechanism, with Radarsat-2 satellite be Example, described Radarsat-2 is the high-resolution commercialization radar satellite carrying C-band sensor；

Step 2: list the geometric equation of correspondence according to the mathematical model of ten bars mechanism；

Step 3: solve geometric equation respectively, and combine Lagrange's equation and set up kinetic model；

Step 4: solve the kinetic parameter of ten bars mechanism according to kinetic model, and according to the actual elastic element of satellite Carry out parameter optimization, obtain the size value of satellite battery plate each rod member of development mechanism.

Preferably, described step 1 includes:

9 motion bars comprised according to satellite battery plate development mechanism, 12 revolute pairs and and celestial body maintain static Sidewall, sets up the mathematical model of ten bars mechanism, wherein:

10 revolute pairs are designated as R_i, wherein i=1,2,3...n, n represent the sum of revolute pair；

Specifically, including revolute pair R₁, revolute pair R₂, revolute pair R₃, revolute pair R₄, revolute pair R₅, revolute pair R₆, rotate Secondary R₇, revolute pair R₈, revolute pair R₉, revolute pair R₁₀；

9 motion bars are designated as l_i,j, wherein j=1,2,3...n, and i ≠ j, n represents the sum of revolute pair, motion bar l_i,j With motion bar l_j,iAll represent revolute pair R_i, revolute pair R_jBetween connecting rod；Represent by revolute pair R_iPoint to revolute pair R_jArrow Amount,Represent by revolute pair R_jPoint to revolute pair R_iVector, be l for compound rod member method for expressing_i,j,k, i.e. represent R_i R_jIt Between connect part and R_j, R_kBetween company's part be composite rod be a connecting rod；

Specifically, including motion bar l₁₂₃, motion bar l_3,4, motion bar l_4,8, motion bar l_8,7, motion bar l_6,7, motion bar l_5,6, motion bar l_2,7, motion bar l_7,9, motion bar l_9,10；Wherein motion bar l₁₂₃It is a compound rod member, revolute pair R₁₀It is welded on l₁₂₃On rod member；Motion bar l_3,4Represent revolute pair R₃, revolute pair R₄Between connecting rod, motion bar l_4,8Represent revolute pair R₄, rotate Secondary R₈Between connecting rod, motion bar l_8,7Represent revolute pair R₈, revolute pair R₇Between connecting rod, motion bar l_6,7Represent revolute pair R₆、 Revolute pair R₇Between connecting rod, motion bar l_5,6Represent revolute pair R₅, revolute pair R₆Between connecting rod, motion bar l_2,7Represent and rotate Secondary R₂, revolute pair R₇Between connecting rod, motion bar l_7,9Represent revolute pair R₇, revolute pair R₉Between connecting rod, motion bar l_9,10Represent Revolute pair R₉, revolute pair R₁₀Between connecting rod；

Wherein, l_1,5Represent revolute pair R₁, revolute pair R₅Between connecting rod, l_1,5Constitute celestial body and maintain static sidewall；Revolute pair R₁With revolute pair R₅For celestial body fixed hinge point；First piece of cell panel is fixed on l_2,3On position, second piece of cell panel is fixed on l_3,4 On position；The computing formula of Radarsat-2 cell panel development mechanism number of degrees of freedom, DOF is as follows:

DOF=3n-2P_L-P_H=3 × 9-2 × 12=3；

In formula: n represents the number of rod member, P_LRepresent the number of kinematic pair lower pair, P_HRepresent the number of kinematic pair higher pair.

Preferably, described step 2 includes:

List the geometric equation of ten bars mechanism:

That is:

$l_{1,2}{e}^{{\mathrm{j\α}}_1}+l_{2,7}{e}^{{\mathrm{j\θ}}_7}=l_{1,5}{e}^{{\mathrm{j\θ}}_{10}}+l_{5,6}{e}^{{\mathrm{j\θ}}_6}+l_{6,7}{e}^{{\mathrm{j\θ}}_5}$

$l_{2,10}{e}^{j({\mathrm{\α}}_1-{\mathrm{\β}}_2-{\mathrm{\θ}}_1)}+l_{10,9}{e}^{{\mathrm{j\θ}}_9}+l_{9,7}{e}^{{\mathrm{j\θ}}_8}=l_{2,7}{e}^{{\mathrm{j\θ}}_7}$

$l_{10,9}{e}^{{\mathrm{j\θ}}_{91}}+l_{9,7}{e}^{{\mathrm{j\θ}}_8}=l_{10,3}{e}^{j({\mathrm{\α}}_1+{\mathrm{\θ}}_1-{\mathrm{\θ}}_2)}+l_{3,4}{e}^{{\mathrm{j\θ}}_2}+l_{4,8}{e}^{{\mathrm{j\θ}}_3}+l_{8,7}{e}^{{\mathrm{j\θ}}_4}$

Analyze above three equation, obtain following expression:

l_1,2cosα₁+l_2,7cosθ₇=l_1,5cosθ₁₀+l_5,6cosθ₆+l_6,7cosθ₅

l_1,2sinα₁+l_2,7sinθ₇=l_1,5sinθ₁₀+l_5,6sinθ₆+l_6,7sinθ₅

l_2,10cos[α₁-(θ₁+β₂)]+l_10,9cosθ₉+l_9,7cosθ₈=l_2,7cosθ₇

l_2,10sin[α₁-(θ₁+β₂)]+l_10,9sinθ₉+l_9,7sinθ₈=l_2,7sinθ₇

l_10,9cosθ₉+l_9,7cosθ₈=l_10,3cos(α₁+θ₁-θ₂)+l_3,4cosθ₂+l_4,8cosθ₃+l_8,7cosθ₄

l_10,9sinθ₉+l_9,7sinθ₈=l_10,3sin(α₁+θ₁-θ₂)+l_3,4sinθ₂+l_4,8sinθ₃+l_8,7sinθ₄

In formula: θ₁Represent l_2,3And formed angle, θ along clockwise direction between horizontal plane₂Represent l_3,4And between horizontal plane The most formed angle, θ₃Represent l_4,8And formed angle, θ along clockwise direction between horizontal plane₄Represent l_7,8With Formed angle, θ in the counterclockwise direction between horizontal plane₅Represent l_6,7And formed angle along clockwise direction between horizontal plane, θ₆Represent l_5,6And formed angle, θ in the counterclockwise direction between horizontal plane₇Represent to be become in the counterclockwise direction with between horizontal plane Angle, θ₈Represent l_7,9And formed angle, θ in the counterclockwise direction between horizontal plane₉Represent l_9,10And along edge between horizontal plane The most formed angle, θ₁₀Represent l_1,5And formed angle along clockwise direction between horizontal plane；l_1,2Corner be α₁, l_3,4With l_2,3Relative rotation is β₂。

Preferably, described step 3 includes:

Step 3.1: solve byFive bar underactuatuated drive forms of motion of the loop of composition, its Middle number of degrees of freedom: DOF=3n-2P_L-P_H=3 × 4-2 × 5=2, only revolute pair R₁Place is provided with motor, so being drive lacking five Linkage, makes to be reduced to 2 degree of freedom by 3 degree of freedom by splitting elementary cell, lists geometric equation；

Specifically, as illustrated in fig. 2, it is assumed that O is the initial point of fixed coordinate system, then OD represents x direction, is the most upwards Y direction；The angle of OA with OD isThe angle of AB with OD isThe angle of BC with OD isThe angle of CD with OD is

X direction: l₁cosθ₁+l₂cosθ₂=l₅+l₃cosθ₃+l₄cosθ₄

Y direction: l₁sinθ₁+l₂sinθ₂=l₃sinθ₃+l₄sinθ₄；

In formula: l₁、l₂、l₃、l₄、l₅Represent l respectively_1,2、l_2,7、l_7,6、l_6,5、l_1,5；

Solving simultaneous equation, wherein θ₁=ω t, ω represent hinge point R₁The rotational angular velocity of position motors, t express time, And to calculated θ₂、θ₄Carry out second order derivation；

Step 3.2: mechanism is modeled according to Lagrange First Law；

$\left\{\begin{array}{c}M\stackrel{\·\·}{q}+C_q^T\mathrm{\λ}=Q\\ C=0\end{array}\right.;$

In formula: the mass matrix of M outgoing mechanism,Represent the second dervative i.e. generalized acceleration of generalized coordinates,Represent Mechanism's position shape constraint equation transposed matrix to the Jacobian matrix of mechanism's generalized coordinates, λ represents Lagrange multiplier vector λ, Q The generalized force vector of outgoing mechanism, the position shape constraint equation of C outgoing mechanism；

I.e. list concrete formula, here draw Suzanne Lenglen day multiplier vector

λ=[λ₁ λ₂...λ₆]^T；

In formula: λ_iRepresent that in Lagrange multiplier vector, i-th is vectorial, i=1,2,3...6；

Wherein, M is:

$M=\left[\begin{array}{cccccccc}{J}_1& 0& 0& 0& 0& 0& 0& 0\\ 0& J_2& 0& 0& 0& 0& 0& 0\\ 0& 0& m_2& 0& 0& 0& 0& 0\\ 0& 0& 0& m_2& 0& 0& 0& 0\\ 0& 0& 0& 0& J_3& 0& 0& 0\\ 0& 0& 0& 0& 0& m_3& 0& 0\\ 0& 0& 0& 0& 0& 0& m_3& 0\\ 0& 0& 0& 0& 0& 0& 0& J_4\end{array}\right];$

In formula: J₁Represent l₁Rotary inertia, J₂Represent l₂Rotary inertia, J₃Represent l₃Rotary inertia, J₄Represent l₄'s Rotary inertia, m₂Represent l₂Quality, m₃Represent l₃Quality；

Listing a shape Constrained equations, the concrete form of C is:

$\left\{\begin{array}{c}{x}_2=l_1{\mathrm{cos\θ}}_1+\frac{1}{2}{l}_2{\mathrm{cos\θ}}_2\\ y_2=l_1{\mathrm{sin\θ}}_1+\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2\\ x_3=l_4{\mathrm{cos\θ}}_4+\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3\\ y_3=l_4{\mathrm{sin\θ}}_4+\frac{1}{2}{l}_3{\mathrm{sin\θ}}_3\\ l_1{\mathrm{cos\θ}}_1+l_2{\mathrm{cos\θ}}_2=1_5+l_3{\mathrm{cos\θ}}_3+l_4{\mathrm{cos\θ}}_4\\ l_1{\mathrm{sin\θ}}_1+l_2{\mathrm{sin\θ}}_2=l_3{\mathrm{sin\θ}}_3+l_4{\mathrm{sin\θ}}_4\end{array}\right\}.$

Concrete form be:

$\left[\begin{array}{cccccc}-l_1{\mathrm{sin\θ}}_1& l_1{\mathrm{cos\θ}}_1& 0& 0& -l_1{\mathrm{sin\θ}}_1& l_1{\mathrm{cos\θ}}_1\\ -\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2& \frac{1}{2}{l}_2{\mathrm{cos\θ}}_2& 0& 0& -l_2{\mathrm{sin\θ}}_2& l_2{\mathrm{cos\θ}}_2\\ -1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& \frac{1}{2}{l}_3{\mathrm{sin\θ}}_3& -\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3& l_3{\mathrm{sin\θ}}_3& -l_3{\mathrm{cos\θ}}_3\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& l_4{\mathrm{sin\θ}}_4& -l_4{\mathrm{cos\θ}}_4& l_4{\mathrm{sin\θ}}_4& -l_4{\mathrm{cos\θ}}_4\end{array}\right]$

The computing formula of Q is as follows:

Q=[M_r 0 0 0 0 0 0 0]^T

In formula: M_rIt is applied to the driving moment on driving link OA for motor；

Obtaining after arrangement:

$\left[\begin{array}{c}-l_1{\mathrm{sin\θ}}_1{\mathrm{\λ}}_1+l_1{\mathrm{cos\θ}}_1{\mathrm{\λ}}_2-l_1{\mathrm{sin\θ}}_1{\mathrm{\λ}}_5+l_1{\mathrm{cos\θ}}_1{\mathrm{\λ}}_6+J_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\\ -\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2{\mathrm{\λ}}_1+\frac{1}{2}{l}_2{\mathrm{cos\θ}}_2{\mathrm{\λ}}_2-l_2{\mathrm{sin\θ}}_2{\mathrm{\λ}}_5+l_2{\mathrm{cos\θ}}_2{\mathrm{\λ}}_6+J_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}\\ -{\mathrm{\λ}}_1+m_2\left(-l_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\right){\mathrm{sin\θ}}_1-l_1{\left(\frac{{\mathrm{d\θ}}_1}{dt}\right)}^2{\mathrm{cos\θ}}_1-\frac{1}{2}{l}_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}{\mathrm{sin\θ}}_2-\frac{1}{2}{l}_2{\left(\frac{{\mathrm{d\θ}}_2}{dt}\right)}^2{\mathrm{cos\θ}}_2\\ -{\mathrm{\λ}}_2+m_2\left(l_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\right){\mathrm{cos\θ}}_1-l_1{\left(\frac{{\mathrm{d\θ}}_1}{dt}\right)}^2{\mathrm{sin\θ}}_1-\frac{1}{2}{l}_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}{\mathrm{cos\θ}}_2-\frac{1}{2}{l}_2{\left(\frac{{\mathrm{d\θ}}_2}{dt}\right)}^2{\mathrm{sin\θ}}_2\\ \frac{1}{2}{l}_3{\mathrm{sin\θ}}_3{\mathrm{\λ}}_3-\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3{\mathrm{\λ}}_4+l_3{\mathrm{sin\θ}}_3{\mathrm{\λ}}_5-l_3{\mathrm{cos\θ}}_3{\mathrm{\λ}}_6+J_3\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\\ -{\mathrm{\λ}}_3+m_3\left(l_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\right){\mathrm{sin\θ}}_4-l_4{\left(\frac{{\mathrm{d\θ}}_4}{dt}\right)}^2{\mathrm{cos\θ}}_4+\frac{1}{2}{l}_3\frac{d^2{\mathrm{\θ}}_3}{{\mathrm{dt}}^2}{\mathrm{sin\θ}}_3+\frac{1}{2}{l}_3{\left(\frac{{\mathrm{d\θ}}_3}{dt}\right)}^2{\mathrm{sin\θ}}_3\\ -{\mathrm{\λ}}_4+m_3\left(-l_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\right){\mathrm{cos\θ}}_4-l_4{\left(\frac{{\mathrm{d\θ}}_4}{dt}\right)}^2{\mathrm{sin\θ}}_4-\frac{1}{2}{l}_3\frac{d^2{\mathrm{\θ}}_3}{{\mathrm{dt}}^2}{\mathrm{cos\θ}}_3+\frac{1}{2}{l}_3{\left(\frac{{\mathrm{d\θ}}_3}{dt}\right)}^2{\mathrm{sin\θ}}_3\\ l_4{\mathrm{sin\θ}}_4{\mathrm{\λ}}_3-l_4{\mathrm{cos\θ}}_4{\mathrm{\λ}}_4+l_4{\mathrm{sin\θ}}_4{\mathrm{\λ}}_5-l_4{\mathrm{cos\θ}}_4{\mathrm{\λ}}_6+J_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\end{array}\right];$

Willλ₁,λ₂,...λ₈,M_rAs unknown quantity, the coupling terms containing speed is put on the right of equation, by equation arrangement is

A f=P

Form, wherein unknown quantifier is:

$f={({\stackrel{\·\·}{\mathrm{\θ}}}_2,{\mathrm{\λ}}_1,{\mathrm{\λ}}_2,\mathrm{...}{\mathrm{\λ}}_8,M_r)}^T$

In formula:Represent θ₃Second dervative and the second dervative of asking of angular displacement be angular acceleration,Represent θ₂Second order lead The second dervative of asking of number and angular displacement is angular acceleration, λ_iRepresenting i-th vector in Lagrange multiplier vector, f represents equation The vector being made up of unknown quantity after arrangement, A represents the matrix of known coefficient item composition in addition to unknown number, and P represents that equation arranges During equation coupling terms moved on to the matrix of composition behind the right；

Equation group A f=P is multiplied by A in both sides simultaneously^-1, obtain f=A^-1P, i.e. tries to achieve unknown quantity；

According to first five-rodClosed-loop, it is thus achieved that revolute pair R₂, revolute pair R₇'s Displacement, speed, acceleration, and kinesiology and the kinetic parameter of closed-loop is obtained according to four-bar linkage negative vector method, Closed-loop is as follows:

According to second four-bar mechanismObtain revolute pair R₃, revolute pair R₁₀, revolute pair R₉, rotate Secondary R₇Kinetics and kinematics parameters, i.e.For known quantity, list the equation of the 3rd closed-loop:

For known quantity, obtain the kinematics parameters that residual rotation is secondary.

Preferably, step 5 is also included: the size value obtaining satellite battery plate each rod member of development mechanism according to step 4 is adjusted Whole satellite battery plate development mechanism, or manufacture satellite battery plate development mechanism.

Compared with prior art, the present invention has a following beneficial effect:

The kinesiology of the ten bar underactuatuated drive that the present invention provides and kinetics mixing solve dimension reduction method and can solve the problem that satellite The kinematic problem of cell panel development mechanism, solves the kinetic parameter of ten bars mechanism, is defended by the described parameter optimization solved The size of star cell panel each rod member of development mechanism, thus improve cell panel and launch the stability of process, so that satellite is sent out Penetrate and alleviate quality, improve cell panel and launch process reliability.Additionally it is possible to by simplified model, improve satellite and manufacture field Desin speed.

Accompanying drawing explanation

By the detailed description non-limiting example made with reference to the following drawings of reading, the further feature of the present invention, Purpose and advantage will become more apparent upon:

Fig. 1 is the Radarsat-2 cell panel development mechanism schematic diagram that the present invention applies；

Fig. 2 is first five bar drive lacking closed-loop schematic diagram after dimensionality reduction.

Detailed description of the invention

Below in conjunction with specific embodiment, the present invention is described in detail.Following example will assist in the technology of this area Personnel are further appreciated by the present invention, but limit the present invention the most in any form.It should be pointed out that, the ordinary skill to this area For personnel, without departing from the inventive concept of the premise, it is also possible to make some changes and improvements.These broadly fall into the present invention Protection domain.

Kinesiology and the kinetics mixing of the ten bar underactuatuated drive according to present invention offer solve dimension reduction method, solve such as figure The kinetic parameter of the Radarsat-2 cell panel development mechanism shown in 1.

To being similar to the drive lacking model of Radarsat-2 cell panel development mechanism, the present invention is to connecting rods more than five bars Mechanism for implementing dimension reduction method solves kinesiology and the dynamics problem of the machinery based on complicated linkages underactuatuated drive.Pass through Set up the geometric model of complicated drive lacking linkage, set up Lagrange first kind kinetics equation, then close according to geometry Be entirety is divided into several with four-bar mechanism etc. or the elementary cell of five bar underactuatuated drive, then to five bar underactuatuated drive Carry out geometric model foundation and set up the Lagrange first kind equation of motion, reducing dimension, alleviate the meter of synchronization Calculation amount, then utilize high-performance computer solve elementary cell kinematics parameters, then using the parameter solved as it is known that Solve the kinematic parameter of adjacent cells around, until solving the parameter of all unit of whole system.

Specifically, whole cell panel development mechanism totally 9 motion bars, 10 revolute pairs, wherein this mechanism is segmented into three Individual closed-loop, first ring can be byForm five bar drive lacking, second ring byGroup Become, the 3rd ring byComposition, solves successively and can obtain last solution.

In the present embodiment, this method starts firstly the need of mechanism, and motor is arranged on revolute pair R₁, revolute pair R₃Position, electricity Machine can start simultaneously, it is also possible to orderliness starts, as two pieces of cell panel supporting member l_2,3、l_3,4In alignment, and and The when that celestial body being vertical, motor quits work, and mechanism is the most fully deployed.

The step of the computational methods that the present invention proposes is as follows:

Specifically, including motion bar l₁₂₃, motion bar l_3,4, motion bar l_4,8, motion bar l_8,7, motion bar l_6,7, motion bar l_5,6, motion bar l_2,7, motion bar l_7,9, motion bar l_9,10；Wherein motion bar l₁₂₃It is a coupled pole, revolute pair R₁₀It is welded on l₁₂₃On rod member；Motion bar l_3,4Represent revolute pair R₃, revolute pair R₄Between connecting rod, motion bar l_4,8Represent revolute pair R₄, rotate Secondary R₈Between connecting rod, motion bar l_8,7Represent revolute pair R₈, revolute pair R₇Between connecting rod, motion bar l_6,7Represent revolute pair R₆、 Revolute pair R₇Between connecting rod, motion bar l_5,6Represent revolute pair R₅, revolute pair R₆Between connecting rod, motion bar l_2,7Represent and rotate Secondary R₂, revolute pair R₇Between connecting rod, motion bar l_7,9Represent revolute pair R₇, revolute pair R₉Between connecting rod, motion bar l_9,10Represent Revolute pair R₉, revolute pair R₁₀Between connecting rod；

Wherein, l_1,5Represent revolute pair R₁, revolute pair R₅Between connecting rod, l_1,5Constitute celestial body and maintain static sidewall；Revolute pair R₁With revolute pair R₅For celestial body fixed hinge point；l_1,2And l_2,3Position is first piece of cell panel, l_3,4Position is second piece of cell panel； The computing formula of Radarsat-2 cell panel development mechanism number of degrees of freedom, DOF is as follows:

DOF=3n-2P_L-P_H=3 × 9-2 × 12=3；

In formula: n represents the number of rod member, P_LRepresent the number of kinematic pair lower pair, P_HRepresent the number of kinematic pair higher pair.

Described step 2 includes:

List the geometric equation of ten bars mechanism:

That is:

$l_{1,2}{e}^{j{\mathrm{\α}}_1}{+l}_{2,7}{e}^{{\mathrm{j\θ}}_7}=l_{1,5}{e}^{{\mathrm{j\θ}}_{10}}+l_{\mathrm{5,6}}{e}^{j{\mathrm{\θ}}_6}+l_{6,7}{e}^{{\mathrm{j\θ}}_5}$

$l_{2,10}{e}^{j({\mathrm{\α}}_1-{\mathrm{\β}}_2-{\mathrm{\θ}}_1)}+l_{10,9}{e}^{{\mathrm{j\θ}}_9}+l_{9,7}{e}^{{\mathrm{j\θ}}_8}=l_{2,7}{e}^{{\mathrm{j\θ}}_7}$

$l_{10,9}{e}^{{\mathrm{j\θ}}_{91}}+l_{9,7}{e}^{{\mathrm{j\θ}}_8}=l_{10,3}{e}^{j({\mathrm{\α}}_1+{\mathrm{\θ}}_1-{\mathrm{\θ}}_2)}+l_{3,4}{e}^{{\mathrm{j\θ}}_2}+l_{4,8}{e}^{{\mathrm{j\θ}}_3}+l_{8,7}{e}^{{\mathrm{j\θ}}_4}$

Analyze above three equation, obtain following expression:

l_1,2cosα₁+l_2,7cosθ₇=l_1,5cosθ₁₀+l_5,6cosθ₆+l_6,7cosθ₅

l_1,2sinα₁+l_2,7sinθ₇=l_1,5sinθ₁₀+l_5,6sinθ₆+l_6,7sinθ₅

l_2,10cos[α₁-(θ1₁+β₂)]+l_10,9cosθ₉+l_9,7cosθ₈=l_2,7cosθ₇

l_2,10sin[α₁-(θ₁+β₂)]+l_10,9sinθ₉+l_9,7sinθ₈=l_2,7sinθ₇

l_10,9cosθ₉+l_9,7cosθ₈=l_10,3cos(α₁+θ₁-θ₂)+l_3,4cosθ₂+l_4,8cosθ₃+l_8,7cosθ₄

l_10,9sinθ₉+l_9,7sinθ₈=l_10,3sin(α₁+θ₁-θ₂)+l_3,4sinθ₂+l_4,8sinθ₃+l_8,7sinθ₄

In formula: θ₁Represent l_2,3And formed angle, θ along clockwise direction between horizontal plane₂Represent l_3,4And between horizontal plane The most formed angle, θ₃Represent l_4,8And formed angle, θ along clockwise direction between horizontal plane₄Represent l_7,8With Formed angle, θ in the counterclockwise direction between horizontal plane₅Represent l_6,7And formed angle along clockwise direction between horizontal plane, θ₆Represent l_5,6And formed angle, θ in the counterclockwise direction between horizontal plane₇Represent to be become in the counterclockwise direction with between horizontal plane Angle, θ₈Represent l_7,9And formed angle, θ in the counterclockwise direction between horizontal plane₉Represent l_9,10And along edge between horizontal plane The most formed angle, θ₁₀Represent l_1,5And formed angle along clockwise direction between horizontal plane；l_1,2Corner be α₁, l_3,4With l_2,3Relative rotation is β₂。

Described step 3 includes:

Step 3.1: solve byFive bar underactuatuated drive forms of motion of the loop of composition, its Middle number of degrees of freedom: DOF=3n-2P_L-P_H=3 × 4-2 × 5=2, only revolute pair R₁Place is provided with motor, so being drive lacking five Linkage, makes to be reduced to 2 degree of freedom by 3 degree of freedom by splitting elementary cell, lists geometric equation；

Assume the initial point that O is fixed coordinate system, then OD represents x direction, is the most upwards y direction；The folder of OA Yu OD Angle isThe angle of AB with OD isThe angle of BC with OD isThe angle of CD with OD is

X direction: l₁cosθ₁+l₂cosθ₂=l₅+l₃cosθ₃+l₄cosθ₄

Y direction: l₁sinθ₁+l₂sinθ₂=l₃sinθ₃+l₄sinθ₄；

In formula: l₁、l₂、l₃、l₄、l₅Represent l respectively_1,2、l_2,7、l_7,6、l_6,5、l_1,5；

Solving simultaneous equation, wherein θ₁=ω t, ω represent hinge point R₁The rotational angular velocity of position motors, t express time, And to calculated θ₂、θ₄Carry out second order derivation；

Step 3.2: mechanism is modeled according to Lagrange First Law；

$\left\{\begin{array}{c}M\stackrel{\·\·}{q}+C_q^T\mathrm{\λ}=Q\\ C=0\end{array}\right.;$

In formula: the mass matrix of M outgoing mechanism,Represent the second dervative i.e. generalized acceleration of generalized coordinates,Represent Mechanism's position shape constraint equation transposed matrix to the Jacobian matrix of mechanism's generalized coordinates, λ represents Lagrange multiplier vector λ, Q The generalized force vector of outgoing mechanism, the position shape constraint equation of C outgoing mechanism；

I.e. list concrete formula, here draw Suzanne Lenglen day multiplier vector

λ=[λ₁ λ₂...λ₆]^T；

In formula: λ_iRepresent that in Lagrange multiplier vector, i-th is vectorial, i=1,2,3...6；

Wherein, M is:

$M=\left[\begin{array}{cccccccc}{J}_1& 0& 0& 0& 0& 0& 0& 0\\ 0& J_2& 0& 0& 0& 0& 0& 0\\ 0& 0& m_2& 0& 0& 0& 0& 0\\ 0& 0& 0& m_2& 0& 0& 0& 0\\ 0& 0& 0& 0& J_3& 0& 0& 0\\ 0& 0& 0& 0& 0& m_3& 0& 0\\ 0& 0& 0& 0& 0& 0& m_3& 0\\ 0& 0& 0& 0& 0& 0& 0& J_4\end{array}\right];$

In formula: J₁Represent l₁Rotary inertia, J₂Represent l₂Rotary inertia, J₃Represent l₃Rotary inertia, J₄Represent l₄'s Rotary inertia, m₂Represent l₂Quality, m₃Represent l₃Quality；

Listing a shape Constrained equations, the concrete form of C is:

$\left\{\begin{array}{c}{x}_2=l_1{\mathrm{cos\θ}}_1+\frac{1}{2}{l}_2{\mathrm{cos\θ}}_2\\ y_2=l_1{\mathrm{sin\θ}}_1+\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2\\ x_3=l_4{\mathrm{cos\θ}}_4+\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3\\ y_3=l_4{\mathrm{sin\θ}}_4+\frac{1}{2}{l}_3{\mathrm{sin\θ}}_3\\ l_1{\mathrm{cos\θ}}_1+l_2{\mathrm{cos\θ}}_2=l_5+l_3{\mathrm{cos\θ}}_3+l_4{\mathrm{cos\θ}}_4\\ l_1{\mathrm{sin\θ}}_1+l_2{\mathrm{sin\θ}}_2=l_3{\mathrm{sin\θ}}_3+l_4{\mathrm{sin\θ}}_4\end{array}\right\}$

Concrete form be:

$\left[\begin{array}{cccccc}-l_1{\mathrm{sin\θ}}_1& l_1{\mathrm{cos\θ}}_1& 0& 0& -l_1{\mathrm{sin\θ}}_1& l_1{\mathrm{cos\θ}}_1\\ -\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2& \frac{1}{2}{l}_2{\mathrm{cos\θ}}_2& 0& 0& -l_2{\mathrm{sin\θ}}_2& l_2{\mathrm{cos\θ}}_2\\ -1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& \frac{1}{2}{l}_3{\mathrm{sin\θ}}_3& -\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3& l_3{\mathrm{sin\θ}}_3& -l_3{\mathrm{cos\θ}}_3\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& l_4{\mathrm{sin\θ}}_4& -l_4{\mathrm{cos\θ}}_4& l_4{\mathrm{sin\θ}}_4& -l_4{\mathrm{cos\θ}}_4\end{array}\right]$

The computing formula of Q is as follows:

Q=[M_r 0 0 0 0 0 0 0]^T

In formula: M_rIt is applied to the driving moment on driving link OA for motor；

Obtaining after arrangement:

$\left[\begin{array}{c}-l_1{\mathrm{sin\θ}}_1{\mathrm{\λ}}_1+l_1{\mathrm{cos\θ}}_1{\mathrm{\λ}}_2-l_1{\mathrm{sin\θ}}_1{\mathrm{\λ}}_5+l_1{\mathrm{cos\θ}}_1{\mathrm{\λ}}_6+J_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\\ -\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2{\mathrm{\λ}}_1+\frac{1}{2}{l}_2{\mathrm{cos\θ}}_2{\mathrm{\λ}}_2-l_2{\mathrm{sin\θ}}_2{\mathrm{\λ}}_5+l_2{\mathrm{cos\θ}}_2{\mathrm{\λ}}_6+J_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}\\ -{\mathrm{\λ}}_1+m_2\left(-l_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\right){\mathrm{sin\θ}}_1-l_1{\left(\frac{{\mathrm{d\θ}}_1}{dt}\right)}^2{\mathrm{cos\θ}}_1-\frac{1}{2}{l}_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}{\mathrm{sin\θ}}_2-\frac{1}{2}{l}_2{\left(\frac{{\mathrm{d\θ}}_2}{dt}\right)}^2{\mathrm{cos\θ}}_2\\ -{\mathrm{\λ}}_2+m_2\left(l_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\right){\mathrm{cos\θ}}_1-l_1{\left(\frac{{\mathrm{d\θ}}_1}{dt}\right)}^2{\mathrm{sin\θ}}_1-\frac{1}{2}{l}_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}{\mathrm{cos\θ}}_2-\frac{1}{2}{l}_2{\left(\frac{{\mathrm{d\θ}}_2}{dt}\right)}^2{\mathrm{sin\θ}}_2\\ \frac{1}{2}{l}_3{\mathrm{sin\θ}}_3{\mathrm{\λ}}_3-\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3{\mathrm{\λ}}_4+l_3{\mathrm{sin\θ}}_3{\mathrm{\λ}}_5-l_3{\mathrm{cos\θ}}_3{\mathrm{\λ}}_6+J_3\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\\ -{\mathrm{\λ}}_3+m_3\left(l_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\right){\mathrm{sin\θ}}_4-l_4{\left(\frac{{\mathrm{d\θ}}_4}{dt}\right)}^2{\mathrm{cos\θ}}_4+\frac{1}{2}{l}_3\frac{d^2{\mathrm{\θ}}_3}{{\mathrm{dt}}^2}{\mathrm{sin\θ}}_3+\frac{1}{2}{l}_3{\left(\frac{{\mathrm{d\θ}}_3}{dt}\right)}^2{\mathrm{sin\θ}}_3\\ -{\mathrm{\λ}}_4+m_3\left(-l_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\right){\mathrm{cos\θ}}_4-l_4{\left(\frac{{\mathrm{d\θ}}_4}{dt}\right)}^2{\mathrm{sin\θ}}_4-\frac{1}{2}{l}_3\frac{d^2{\mathrm{\θ}}_3}{{\mathrm{dt}}^2}{\mathrm{cos\θ}}_3+\frac{1}{2}{l}_3{\left(\frac{{\mathrm{d\θ}}_3}{dt}\right)}^2{\mathrm{sin\θ}}_3\\ l_4{\mathrm{sin\θ}}_4{\mathrm{\λ}}_3-l_4{\mathrm{cos\θ}}_4{\mathrm{\λ}}_4+l_4{\mathrm{sin\θ}}_4{\mathrm{\λ}}_5-l_4{\mathrm{cos\θ}}_4{\mathrm{\λ}}_6+J_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\end{array}\right];$

Willλ₁,λ₂,...λ₈,M_rAs unknown quantity, the coupling terms containing speed is put on the right of equation, by equation arrangement is

A f=P

Form, wherein unknown quantifier is:

$f={({\stackrel{\·\·}{\mathrm{\θ}}}_2,{\mathrm{\λ}}_1,{\mathrm{\λ}}_2,\mathrm{...}{\mathrm{\λ}}_8,M_r)}^T$

In formula:Represent θ₃Second dervative and the second dervative of asking of angular displacement be angular acceleration,Represent θ₂Second order lead The second dervative of asking of number and angular displacement is angular acceleration, λ_iRepresenting i-th vector in Lagrange multiplier vector, f represents equation The vector being made up of unknown quantity after arrangement, A represents the matrix of known coefficient item composition in addition to unknown number, and P represents that equation arranges During equation coupling terms moved on to the matrix of composition behind the right；

Equation group A f=P is multiplied by A in both sides simultaneously^-1, obtain f=A^-1P, i.e. tries to achieve unknown quantity；

According to first five-rodClosed-loop, it is thus achieved that revolute pair R₂, revolute pair R₇'s Displacement, speed, acceleration, and kinesiology and the kinetic parameter of closed-loop is obtained according to four-bar linkage negative vector method, Closed-loop is as follows:

According to second four-bar mechanismObtain revolute pair R₃, revolute pair R₁₀, revolute pair R₉, rotate Secondary R₇Kinetics and kinematics parameters, i.e.For known quantity, list the equation of the 3rd closed-loop:

For known quantity, obtain the kinematics parameters that residual rotation is secondary.

Above the specific embodiment of the present invention is described.It is to be appreciated that the invention is not limited in above-mentioned Particular implementation, those skilled in the art can make a variety of changes within the scope of the claims or revise, this not shadow Ring the flesh and blood of the present invention.In the case of not conflicting, the feature in embodiments herein and embodiment can any phase Combination mutually.

Claims (5)

1. kinesiology and the kinetics mixing of a bar underactuatuated drive solves dimension reduction method, it is characterised in that including:

Step 1: set up the mathematical model of ten bars mechanism according to satellite battery plate development mechanism；

Step 2: list the geometric equation of correspondence according to the mathematical model of ten bars mechanism；

Step 3: solve geometric equation respectively, and combine Lagrange's equation and set up kinetic model；

Step 4: solve the kinetic parameter of ten bars mechanism according to kinetic model, and carry out according to the actual elastic element of satellite Parameter optimization, obtains the size value of satellite battery plate each rod member of development mechanism.

The kinesiology of ten bar underactuatuated drive the most according to claim 1 and kinetics mixing solve dimension reduction method, its feature Being, described step 1 includes:

9 motion bars comprised according to satellite battery plate development mechanism, 12 revolute pairs and and celestial body maintain static sidewall, Set up the mathematical model of ten bars mechanism, wherein:

10 revolute pairs are designated as R_i, wherein i=1,2,3...n, n represent the sum of revolute pair；

Specifically, including revolute pair R₁, revolute pair R₂, revolute pair R₃, revolute pair R₄, revolute pair R₅, revolute pair R₆, revolute pair R₇、 Revolute pair R₈, revolute pair R₉, revolute pair R₁₀；

9 motion bars are designated as l_i,j, wherein j=1,2,3...n, and i ≠ j, n represents the sum of revolute pair, motion bar l_i,jWith work Lever l_j,iAll represent revolute pair R_i, revolute pair R_jBetween connecting rod；Represent by revolute pair R_iPoint to revolute pair R_jVector, Represent by revolute pair R_jPoint to revolute pair R_iVector, be l for compound rod member method for expressing_i,j,k, i.e. represent R_i R_jBetween connect Part and R_j, R_kBetween company's part be composite rod be a connecting rod；

Specifically, including motion bar l₁₂₃, motion bar l_3,4, motion bar l_4,8, motion bar l_8,7, motion bar l_6,7, motion bar l_5,6、 Motion bar l_2,7, motion bar l_7,9, motion bar l_9,10；Wherein motion bar l₁₂₃It is a compound rod member, revolute pair R₁₀It is welded on l₁₂₃ On rod member；Motion bar l_3,4Represent revolute pair R₃, revolute pair R₄Between connecting rod, motion bar l_4,8Represent revolute pair R₄, revolute pair R₈ Between connecting rod, motion bar l_8,7Represent revolute pair R₈, revolute pair R₇Between connecting rod, motion bar l_6,7Represent revolute pair R₆, rotate Secondary R₇Between connecting rod, motion bar l_5,6Represent revolute pair R₅, revolute pair R₆Between connecting rod, motion bar l_2,7Represent revolute pair R₂、 Revolute pair R₇Between connecting rod, motion bar l_7,9Represent revolute pair R₇, revolute pair R₉Between connecting rod, motion bar l_9,10Represent and rotate Secondary R₉, revolute pair R₁₀Between connecting rod；

Wherein, l_1,5Represent revolute pair R₁, revolute pair R₅Between connecting rod, l_1,5Constitute celestial body and maintain static sidewall；Revolute pair R₁With Revolute pair R₅For celestial body fixed hinge point；First piece of cell panel is fixed on l_2,3On position, second piece of cell panel is fixed on l_3,4Position Put；The computing formula of satellite battery plate development mechanism number of degrees of freedom, DOF is as follows:

DOF=3n-2P_L-P_H=3 × 9-2 × 12=3；

In formula: n represents the number of rod member, P_LRepresent the number of kinematic pair lower pair, P_HRepresent the number of kinematic pair higher pair.

The kinesiology of ten bar underactuatuated drive the most according to claim 2 and kinetics mixing solve dimension reduction method, its feature Being, described step 2 includes:

List the geometric equation of ten bars mechanism:

That is:

$l_{1,2}{e}^{{\mathrm{j\α}}_1}+l_{2,7}{e}^{{\mathrm{j\θ}}_7}=l_{1,5}{e}^{{\mathrm{j\θ}}_{10}}+l_{5,6}{e}^{{\mathrm{j\θ}}_6}+l_{6,7}{e}^{{\mathrm{j\θ}}_5}$

$l_{2,10}{e}^{j({\mathrm{\α}}_1-{\mathrm{\β}}_2-{\mathrm{\θ}}_1)}+l_{10,9}{e}^{{\mathrm{j\θ}}_9}+l_{9,7}{e}^{{\mathrm{j\θ}}_8}=l_{2,7}{e}^{{\mathrm{j\θ}}_7}$

$l_{10,9}{e}^{{\mathrm{j\θ}}_{91}}+l_{9,7}{e}^{{\mathrm{j\θ}}_8}=l_{10,3}{e}^{j({\mathrm{\α}}_1+{\mathrm{\θ}}_1-{\mathrm{\θ}}_2)}+l_{3,4}{e}^{{\mathrm{j\θ}}_2}+l_{4,8}{e}^{{\mathrm{j\θ}}_3}+l_{8,7}{e}^{{\mathrm{j\θ}}_4}$

Analyze above three equation, obtain following expression:

l_1,2cosα₁+l_2,7cosθ₇=l_1,5cosθ₁₀+l_5,6cosθ₆+l_6,7cosθ₅

l_1,2sinα₁+l_2,7sinθ₇=l_1,5sinθ₁₀+l_5,6sinθ₆+l_6,7sinθ₅

l_2,10cos[α₁-(θ₁+β₂)]+l_10,9cosθ₉+l_9,7cosθ₈=l_2,7cosθ₇

l_2,10sin[α₁-(θ₁+β₂)]+l_10,9sinθ₉+l_9,7sinθ₈=l_2,7sinθ₇

l_10,9cosθ₉+l_9,7cosθ₈=l_10,3cos(α₁+θ₁-θ₂)+l_3,4cosθ₂+l_4,8cosθ₃+l_8,7cosθ₄

l_10,9sinθ₉+l_9,7sinθ₈=l_10,3sin(α₁+θ₁-θ₂)+l_3,4sinθ₂+l_4,8sinθ₃+l_8,7sinθ₄

In formula: θ₁Represent l_2,3And formed angle, θ along clockwise direction between horizontal plane₂Represent l_3,4And along inverse between horizontal plane Angle formed by clockwise, θ₃Represent l_4,8And formed angle, θ along clockwise direction between horizontal plane₄Represent l_7,8With level Formed angle, θ in the counterclockwise direction between face₅Represent l_6,7And formed angle, θ along clockwise direction between horizontal plane₆Table Show l_5,6And formed angle, θ in the counterclockwise direction between horizontal plane₇Represent formed in the counterclockwise direction between horizontal plane Angle, θ₈Represent l_7,9And formed angle, θ in the counterclockwise direction between horizontal plane₉Represent l_9,10And along along inverse between horizontal plane Angle formed by clockwise, θ₁₀Represent l_1,5And formed angle along clockwise direction between horizontal plane；l_1,2Corner be α₁, l_3,4With l_2,3Relative rotation is β₂。

The kinesiology of ten bar underactuatuated drive the most according to claim 3 and kinetics mixing solve dimension reduction method, its feature Being, described step 3 includes:

Step 3.1: solve byFive bar underactuatuated drive forms of motion of the loop of composition, wherein certainly By the number of degrees: DOF=3n-2P_L-P_H=3 × 4-2 × 5=2, only revolute pair R₁Place is provided with motor, so being drive lacking five bar machine Structure, makes to be reduced to 2 degree of freedom by 3 degree of freedom by splitting elementary cell, lists geometric equation；

l₁cosθ₁+l₂cosθ₂=l₅+l₃cosθ₃+l₄cosθ₄

l₁sinθ₁+l₂sinθ₂=l₃sinθ₃+l₄sinθ₄；

In formula: l₁、l₂、l₃、l₄、l₅Represent l respectively_1,2、l_2,7、l_7,6、l_6,5、l_1,5；

Solving simultaneous equation, wherein θ₁=ω t, ω represent hinge point R₁The rotational angular velocity of position motors, t express time, and right Calculated θ₂、θ₄Carry out second order derivation；

Step 3.2: mechanism is modeled according to Lagrange First Law；

$\left\{\begin{array}{c}M\stackrel{\·\·}{q}+C_q^T\mathrm{\λ}=Q\\ C=0\end{array}\right.;$

In formula: the mass matrix of M outgoing mechanism,Represent the second dervative i.e. generalized acceleration of generalized coordinates,Outgoing mechanism position The shape constraint equation transposed matrix to the Jacobian matrix of mechanism's generalized coordinates, λ represents Lagrange multiplier vector λ, and Q represents machine The generalized force vector of structure, the position shape constraint equation of C outgoing mechanism；

I.e. list concrete formula, here draw Suzanne Lenglen day multiplier vector

λ=[λ₁ λ₂ ... λ₆]^T；

In formula: λ_iRepresent that in Lagrange multiplier vector, i-th is vectorial, i=1,2,3...6；

Wherein, M is:

$M=\left[\begin{array}{cccccccc}{J}_1& 0& 0& 0& 0& 0& 0& 0\\ 0& J_2& 0& 0& 0& 0& 0& 0\\ 0& 0& m_2& 0& 0& 0& 0& 0\\ 0& 0& 0& m_2& 0& 0& 0& 0\\ 0& 0& 0& 0& J_3& 0& 0& 0\\ 0& 0& 0& 0& 0& m_3& 0& 0\\ 0& 0& 0& 0& 0& 0& m_3& 0\\ 0& 0& 0& 0& 0& 0& 0& J_4\end{array}\right];$

In formula: J₁Represent l₁Rotary inertia, J₂Represent l₂Rotary inertia, J₃Represent l₃Rotary inertia, J₄Represent l₄Rotation Inertia, m₂Represent l₂Quality, m₃Represent l₃Quality；

Listing a shape Constrained equations, the concrete form of C is:

$\left\{\begin{array}{c}{x}_2=l_1{\mathrm{cos\θ}}_1+\frac{1}{2}{l}_2{\mathrm{cos\θ}}_2\\ y_2=l_1{\mathrm{sin\θ}}_1+\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2\\ x_3=l_4{\mathrm{cos\θ}}_4+\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3\\ y_3=l_4{\mathrm{sin\θ}}_4+\frac{1}{2}{l}_3{\mathrm{sin\θ}}_3\\ l_1{\mathrm{cos\θ}}_1+l_2{\mathrm{cos\θ}}_2=l_5+l_3{\mathrm{cos\θ}}_3+l_4{\mathrm{cos\θ}}_4\\ l_1{\mathrm{sin\θ}}_1+l_2{\mathrm{sin\θ}}_2=l_3{\mathrm{sin\θ}}_3+l_4{\mathrm{sin\θ}}_4\end{array}\right\}$

Concrete form be:

$\left[\begin{array}{cccccc}-l_1{\mathrm{sin\θ}}_1& l_1{\mathrm{cos\θ}}_1& 0& 0& -l_1{\mathrm{sin\θ}}_1& l_1{\mathrm{cos\θ}}_1\\ -\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2& \frac{1}{2}{l}_2{\mathrm{cos\θ}}_2& 0& 0& -l_2{\mathrm{sin\θ}}_2& l_2{\mathrm{cos\θ}}_2\\ -1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& \frac{1}{2}{l}_3{\mathrm{sin\θ}}_3& -\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3& l_3{\mathrm{sin\θ}}_3& -l_3{\mathrm{cos\θ}}_3\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& l_4{\mathrm{sin\θ}}_4& -l_4{\mathrm{cos\θ}}_4& l_4{\mathrm{sin\θ}}_4& -l_4{\mathrm{cos\θ}}_4\end{array}\right]$

The computing formula of Q is as follows:

Q=[M_r 0 0 0 0 0 0 0]^T

In formula: M_rIt is applied to the driving moment on driving link OA for motor；

Obtaining after arrangement:

$\left[\begin{array}{c}-l_1{\mathrm{sin\θ}}_1{\mathrm{\λ}}_1+l_1{\mathrm{cos\θ}}_1{\mathrm{\λ}}_2-l_1{\mathrm{sin\θ}}_1{\mathrm{\λ}}_5+l_1{\mathrm{cos\θ}}_1{\mathrm{\λ}}_6+J_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\\ -\frac{1}{2}{l}_2{\mathrm{sin\θ}}_2{\mathrm{\λ}}_1+\frac{1}{2}{l}_2{\mathrm{cos\θ}}_2{\mathrm{\λ}}_2-l_2{\mathrm{sin\θ}}_2{\mathrm{\λ}}_5+l_2{\mathrm{cos\θ}}_2{\mathrm{\λ}}_6+J_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}\\ -{\mathrm{\λ}}_1+m_2\left(-l_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\right){\mathrm{sin\θ}}_1-l_1{\left(\frac{{\mathrm{d\θ}}_1}{dt}\right)}^2{\mathrm{cos\θ}}_1-\frac{1}{2}{l}_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}{\mathrm{sin\θ}}_2-\frac{1}{2}{l}_2{\left(\frac{{\mathrm{d\θ}}_2}{dt}\right)}^2{\mathrm{cos\θ}}_2\\ -{\mathrm{\λ}}_2+m_2\left(l_1\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\right){\mathrm{cos\θ}}_1-l_1{\left(\frac{{\mathrm{d\θ}}_1}{dt}\right)}^2{\mathrm{sin\θ}}_1-\frac{1}{2}{l}_2\frac{d^2{\mathrm{\θ}}_2}{{\mathrm{dt}}^2}{\mathrm{cos\θ}}_2-\frac{1}{2}{l}_2{\left(\frac{{\mathrm{d\θ}}_2}{dt}\right)}^2{\mathrm{sin\θ}}_2\\ \frac{1}{2}{l}_3{\mathrm{sin\θ}}_3{\mathrm{\λ}}_3-\frac{1}{2}{l}_3{\mathrm{cos\θ}}_3{\mathrm{\λ}}_4+l_3{\mathrm{sin\θ}}_3{\mathrm{\λ}}_5-l_3{\mathrm{cos\θ}}_3{\mathrm{\λ}}_6+J_3\frac{d^2{\mathrm{\θ}}_1}{{\mathrm{dt}}^2}\\ -{\mathrm{\λ}}_3+m_3\left(l_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\right){\mathrm{sin\θ}}_4-l_4{\left(\frac{{\mathrm{d\θ}}_4}{dt}\right)}^2{\mathrm{cos\θ}}_4+\frac{1}{2}{l}_3\frac{d^2{\mathrm{\θ}}_3}{{\mathrm{dt}}^2}{\mathrm{sin\θ}}_3+\frac{1}{2}{l}_3{\left(\frac{{\mathrm{d\θ}}_3}{dt}\right)}^2{\mathrm{sin\θ}}_3\\ -{\mathrm{\λ}}_4+m_3\left(-l_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\right){\mathrm{cos\θ}}_4-l_4{\left(\frac{{\mathrm{d\θ}}_4}{dt}\right)}^2{\mathrm{sin\θ}}_4-\frac{1}{2}{l}_3\frac{d^2{\mathrm{\θ}}_3}{{\mathrm{dt}}^2}{\mathrm{cos\θ}}_3+\frac{1}{2}{l}_3{\left(\frac{{\mathrm{d\θ}}_3}{dt}\right)}^2{\mathrm{sin\θ}}_3\\ l_4{\mathrm{sin\θ}}_4{\mathrm{\λ}}_3-l_4{\mathrm{cos\θ}}_4{\mathrm{\λ}}_4+l_4{\mathrm{sin\θ}}_4{\mathrm{\λ}}_5-l_4{\mathrm{cos\θ}}_4{\mathrm{\λ}}_6+J_4\frac{d^2{\mathrm{\θ}}_4}{{\mathrm{dt}}^2}\end{array}\right];$

WillAs unknown quantity, the coupling terms containing speed is put on the right of equation, equation is arranged as A f= P

Form, wherein unknown quantifier is:

$f={({\stackrel{\·\·}{\mathrm{\θ}}}_2,{\mathrm{\λ}}_1,{\mathrm{\λ}}_2,\mathrm{...}{\mathrm{\λ}}_8,M_r)}^T$

In formula:Represent θ₃Second dervative and the second dervative of asking of angular displacement be angular acceleration,Represent θ₂Second dervative and The second dervative of asking of angular displacement is angular acceleration, λ_iRepresenting i-th vector in Lagrange multiplier vector, f represents that equation arranges Later the vector being made up of unknown quantity, A represents the matrix of known coefficient item composition in addition to unknown number, and P represents that equation arranges process Middle equation coupling terms is moved on to the right after composition matrix；

Equation group A f=P is multiplied by A in both sides simultaneously^-1, obtain f=A^-1P, i.e. tries to achieve unknown quantity；

According to first five-rodClosed-loop, it is thus achieved that revolute pair R₂, revolute pair R₇Displacement, Speed, acceleration, and kinesiology and the kinetic parameter of closed-loop, closed-loop is obtained according to four-bar linkage negative vector method As follows:

According to second four-bar mechanismObtain revolute pair R₃, revolute pair R₁₀, revolute pair R₉, revolute pair R₇ Kinetics and kinematics parameters, i.e.For known quantity, list the equation of the 3rd closed-loop:

For known quantity, obtain the kinematics parameters that residual rotation is secondary.

The kinesiology of ten bar underactuatuated drive the most according to claim 1 and kinetics mixing solve dimension reduction method, its feature It is, also includes step 5: the size value obtaining Radarsat-2 cell panel each rod member of development mechanism according to step 4 adjusts Radarsat-2 cell panel development mechanism, or manufacture satellite battery plate development mechanism.