1 Introduction

Deployable structures are used for the ease of storage and transportation, and they are deployed into their operational configuration when required. Deployable structure presents contractility and developability characters, which are widely applied in various areas for ease of storage and transportation, such as aviation, aerospace, civil engineering, and so on^[1-3]. Particularly on aerospace, since the space of vehicle is limited, and the deployable structures that need to carry become increasingly large, so deployable structures must be fold up to compression state for transportation to specified space location, after that develop to expectant work configuration for task. Since some special dynamic characteristics are presented during development process, it is necessary to research kinematics and dynamics behaviors of deployable structures.

Along with the development of aerospace technology, the dynamics of deployable institutions have became to be one of the hot research attracts that attracts many scholars^[4-13]. Li et al.^[14]established the stiffness model of deployable mechanism and carried out its simulation, Zhao et al.^[15] investigated mechanism theory and application based on deployable mechanism of scissors unit. Chen et al.^[16] executed the research on kinematics respect of deployable institutions consisted of Bricard fundamental unit. Based on the Moore-Penrose generalized matrix theory, Zhao^[17] took the node's natural coordinates under Cartesian coordinates as unknown quantity, established dynamics general equation for deployable structure system, combined with generalized inverse matrix and multi-body dynamics theory to analyze the develop movement process of deployable institutions. Zhang et al.^[18] adopted multi-body dynamics theory, combined with finite element method and Lagrange equation, established multi flexible-body dynamic equation for deployable structure. Yang et al.^[19] utilized matrix condensation method to carry out statics analysis for a cylindrical folding mechanism, and using SIMP method to execute topology optimization design, so as to reduce the structure weight. Sun et al.^[20] established dynamic equations of deployable structure based on Cartesian coordinate system, and using Runge-Kutta method to analyze the movement regularity of deployable institutions during develop process. Research stress of above references majorly focused on design theory and kinetic analysis of deployable mechanism, and at the same time, research related to the application of influence coefficient method on kinetic analysis process is relatively rare. In this paper, the influence coefficient method is adopted to analyze movement variation of planar deployable mechanism during motion process, the dynamic analysis procedure is simplified, of the process, the application range of this method is expanded, so as to display higher practical application value.

The displacement of unit-mechanism under generalized coordinates is solved by employing coordinate transformation, then the first order and second order influence coefficient are derived. The principle of virtual work is utilized to complete the analysis of dynamic characteristics, the explicit time formula related to displacement, velocity and acceleration under generalized coordinate is deduced, the explicit time formula related to displacement, velocity and acceleration under generalized coordinate is deduced, and based on D'alembert principle, using dynamic static method to calculate the constraint force of each hinge point. The displacement, velocity, acceleration and constraint force of all the hinged points are calculated, moreover, motion curve and constraint force curve are depicted. A numerical example is proposed and validated.

2 Dynamic Model of Deployable Structure

This paper takes plane unit that consists of four scissors units as research object. Fig. 1 shows the plane deployable structure including four scissors units. The plane unit of experimental deployable structure is presented in figures, and dynamic model of scissors unit deployable structure is built on the basis of Cartesian coordinate.

Scissor unit consists of two central mutually hinged and rotatable link members, a number of basic scissors link units are linked each other on terminal hinged point to compose basic piece unit plane deployable structure, and in this paper, taking a hinged planar deployable structure that has four scissors unit as an example to illustrate the deduction.

The schematic diagram of Scissor unit planar deployable mechanism is shown in Fig. 1, planar unit and serial number that assigned to hinged points of deployable structure is depicted.

Taking the planar deployable mechanism that consists of four scissor units as an example, serial number of each hinged point is shown in Fig. 2, the first term of i or j subscript represents the scissors unit number, the second term of i or j subscript represents the hinged point number, α and β are geometric parameters, the first term of link length l subscript represents scissors unit number, the second term of link length l subscript represents link number.

The geometrical conditions(the length between point i₁₁ to point j₁₁) that this plane scissor units deployable structure needs to satisfy (in V direction) are:

$ {i_{11}}{j_{11}} = {i_{12}}{j_{12}} = {i_{21}}{j_{21}} = {i_{22}}{j_{22}} $

(1)

Supposing that the links are rigid body on confi-guration design, thus

$ {i_{12}}{j_{12}} = {i_{21}}{j_{21}} $

(2)

Then it is concluded as:

$ l_{12}^2 + l_{14}^2 - 2{l_{12}}{l_{14}}\cos \alpha = l_{21}^2 + l_{23}^2 - 2{l_{21}}{l_{23}}\cos \alpha $

(3)

Due to the arbitrariness of α during developing process, it is derived:

$ \left\{ \begin{array}{l} l_{12}^2 + l_{14}^2 = l_{21}^2 + l_{23}^2\\ {l_{12}}{l_{14}} = {l_{21}}{l_{23}} \end{array} \right. $

(4)

The similar operation can be done and obtained the following expressions, giving by:

$ \left\{ \begin{array}{l} l_{32}^2 + l_{34}^2 = l_{41}^2 + l_{43}^2\\ {l_{32}}{l_{34}} = {l_{41}}{l_{43}} \end{array} \right. $

(5)

$ \left\{ \begin{array}{l} l_{13}^2 + l_{14}^2 = l_{31}^2 + l_{32}^2\\ {l_{13}}{l_{14}} = {l_{31}}{l_{32}} \end{array} \right. $

(6)

$ \left\{ \begin{array}{l} l_{23}^2 + l_{24}^2 = l_{41}^2 + l_{42}^2\\ {l_{23}}{l_{24}} = {l_{41}}{l_{42}} \end{array} \right. $

(7)

Simultaneous solving the above four equations, it is obvious that there are multiple solutions for 16 unknown variables and 8 equations. This article selects a case as an example to carry out analysis, namely the lengths l of each link are equivalent conditions.

3 Dynamics Investigation 3.1 Influence Coefficient Method

Mechanism kinematic influence coefficient is proposed by Prof. Tesar and Dufy etc.^[21-22], and the influence coefficient method involves establishing 1-step exercise coefficient matrix and 2-step exercise coefficient matrix, the former refers to Jacobian matrix, and the latter one refers to Hessian matrix. The Kinematic influence coefficient highly abstractly reflects the essence of mechanism kinematics and dynamics, and expresses velocity analysis, acceleration analysis, error analysis and stress analysis in the form of a simple style. Moreover, it is convenient to in-depth research mechanism's special configuration, the working space of the robot arm and operability research, and do not need a great deal of calculation.

As a kind of mechanism with complex geometrical shape, the planar deployable mechanism not only is multi-closed-loop structure that contains a great deal of redundant constraints, but also is a series-parallel structure. The fact that a linkage simultaneously joins with multiple linkages causes numerous over-constraints exist in a planar deployable mechanism. If adopting traditional dynamic analysis method, then the building process of constraint equation is more complicated. When using the influence coefficient method, it is convenient to analyze direct kinetic relationship between any two linkages, to find out the speed-acceleration expression, and various forms of solution can be mutually transformed. The more complex of mechanism, the more superior of influence coefficient method can be displayed. Therefore, the influence coefficient method could be a relatively effective mathematic tool for the research of planar deployable mechanism, which could clarify and simplify the analysis process of planar deployable mechanism^[23].

For the deployable structure showed in Fig. 1, if exercise is input by single-degree-of-freedom kinematic pair, which can be revolute pairs or movement pairs, then the position of mechanism's any pole is determined. Pole's position can be described by hinged point coordinates and pole's angular position, using U_i[Φ_i, X_i, Y_i]^T and Φ_i, X_i, Y_i to respectively represent the angular position Φ under reference frame selected for defining the ith linkage's position and the x, y coordinate of reference points, they are the function of input motion parameters φ₁, φ₂, …, φ_N, then derive:

$ U = f\left( {{\varphi _1}{\varphi _2} \cdots {\varphi _N}} \right) $

(8)

Derivative of U with respect to time, obtain any point's speed of component:

$ \mathit{\boldsymbol{\dot U = }}\left[ \mathit{\boldsymbol{J}} \right]\mathit{\boldsymbol{\dot \varphi }} $

(9)

where [J] is called the first order influence coefficient matrix or Jacobian matrix.

$ \begin{array}{*{20}{c}} {\left[ \mathit{\boldsymbol{J}} \right] = {{\left[ {\frac{{\partial U}}{{\partial {\varphi _1}}}\frac{{\partial U}}{{\partial {\varphi _2}}} \cdots \frac{{\partial U}}{{\partial {\varphi _N}}}} \right]}_{1 \times N}} = }\\ {\left[ {\begin{array}{*{20}{c}} {\frac{{\partial {f_1}}}{{\partial {\varphi _1}}}}& \cdots &{\frac{{\partial {f_1}}}{{\partial {\varphi _N}}}}\\ \vdots &{}& \vdots \\ {\frac{{\partial {f_3}}}{{\partial {\varphi _1}}}}& \cdots &{\frac{{\partial {f_3}}}{{\partial {\varphi _N}}}} \end{array}} \right] \in {{\bf{R}}^{3 \times N}}}\\ {\mathit{\boldsymbol{\dot \varphi }} = \left[ {{{\dot \varphi }_1}{{\dot \varphi }_2} \cdots {{\dot \varphi }_N}} \right]_{N \times 1}^{\rm{T}}} \end{array} $

Derivative of $\dot {\mathit{\pmb{U}}}$ with time, obtain any point's acceleration of component:

$ \mathit{\boldsymbol{\ddot U = }}{{\mathit{\boldsymbol{\dot \varphi }}}^{\rm{T}}}\left[ \mathit{\boldsymbol{H}} \right]\mathit{\boldsymbol{\dot \varphi }} + \left[ \mathit{\boldsymbol{J}} \right]\mathit{\boldsymbol{\ddot \varphi }} $

(10)

where [H] is called the second order influence coefficient matrix or Hessian matrix, and

$ \begin{array}{*{20}{c}} {\left[ \mathit{\boldsymbol{H}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}U}}{{\partial {\varphi _1}\partial {\varphi _1}}}}& \cdots &{\frac{{{\partial ^2}U}}{{\partial {\varphi _1}\partial {\varphi _N}}}}\\ \vdots &{}& \vdots \\ {\frac{{{\partial ^2}U}}{{\partial {\varphi _N}\partial {\varphi _1}}}}& \cdots &{\frac{{{\partial ^2}U}}{{\partial {\varphi _N}\partial {\varphi _N}}}} \end{array}} \right]}\\ {\mathit{\boldsymbol{\ddot \varphi = }}\left[ {{{\ddot \varphi }_1}{{\ddot \varphi }_2} \cdots {{\ddot \varphi }_N}} \right]_{N \times 1}^{\rm{T}}} \end{array} $

3.2 Kinematics Analysis

Before the research of kinematics and dynamics of mechanism, proper fixed coordinate system and movement coordinate system should be established. If the selected coordinate system is appropriate, not only the mechanism motion can be described clearly and concisely for the further analysis, and the derivation process of motion equation can also be simplified.

The simplified form of kinematics and dynamics equations and the convenience of solving process can be used to reduce the complicity of equation solving, and shorten the calculation time.

Taking the scissors unit deployable structure shown in Fig. 3 as an example, assuming pole's length are equivalent, and when deploying the mechanism, which connects the support structure through revolute pairs at i₁₂, i₂₂, j₂₂ and j₄₂, and articulates with a slide block K₀ by revolute pairs, then the slide block moves along the OY axis, and mechanism deploys along the OX direction. The origin of fixed coordinate system ∑XOY is located in the center of articulated point i₂₂, OX axis is consistent with the direction of i₂₂j₂₂, OY axis is established according to the principle of Cartesian coordinates. The fixed coordinate system of movement unit ∑x_toy_t is established for each unit mechanism t(t=2, 4, 6, …, N), OX axis is consistent with the direction of i_t2j_t2, OY axis is established according to principle of Cartesian coordinates. According to above principle of Cartesian coordinates, when t=2, coordinate system of movement unit ∑x_toy_t and fixed coordinate system ∑XOY are coincided.

From the geometrical relationship of scissors unit deployable structure and coordinate, which is established according to principle of Cartesian coordinates, it can be seen that since the articulated points connects i₁₁i₁₂, i₂₁i₂₂, i₃₁i₃₂, i₄₁i₄₂… are parallel to each other, then coordinate axis o₀x₀, o₁x₁, o₂x₂… are parallel to each other too, and since the articulated points i₂₂, j₂₂, i₄₂, j₄₂… coincide with OX axis, then coordinate axis o₀x₀, o₁x₁, o₂x₂… coincide with OX axis. For any deploy angle θ, the coordinate transformation matrix from coordinate system of movement unit coordinate system ∑x_toy_t to fixed coordinate system can be described by formula (11):

$ {\mathit{\boldsymbol{R}}_{ot}} = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right] $

(11)

Because the configuration of planar deployable structure's each unit is the same, thus:

$ {i_{\left( {2t} \right)2}}{j_{\left( {2t} \right)2}} = l\cos \left( \theta \right)\left( {t = 1,2, \cdots ,N} \right) $

(12)

The origin of movement unit coordinate system ∑x_toy_t corresponding to the x point of fixed coordinate system ∑XOY, and x's coordinate is:

$ {x_{ot}} = t{i_{\left( {2t} \right)2}}{j_{\left( {2t} \right)2}} = lt\cos \theta $

(13)

The coordinate transformation matrix from coordinate system ∑x_toy_t to fixed coordinate system ∑XOY is:

$ {\mathit{\boldsymbol{T}}_{ot}} = \left[ {\begin{array}{*{20}{c}} {{x_{o\left( {t - 1} \right)}}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {2tl\cos \theta }\\ 0 \end{array}} \right] = t{\mathit{\boldsymbol{T}}_o} $

(14)

where, T_o=[2lcos θ  0]^T, T_o's the first and second order derivative derivation of θ respectively is:

$ {\left[ {{{\mathit{\boldsymbol{\dot T}}}_o}} \right]_\theta } = - 2l{\left[ {\begin{array}{*{20}{c}} {\sin \theta }&0 \end{array}} \right]^{\rm{T}}} $

(15)

$ {\left[ {{{\mathit{\boldsymbol{\ddot T}}}_o}} \right]_{\theta \theta }} = - 2l{\left[ {\begin{array}{*{20}{c}} {\cos \theta }&0 \end{array}} \right]^{\rm{T}}} $

(16)

Using formula (14) can acquire T_ot's first and second order derivative derivation of θ respectively, given as:

$ {\left[ {{{\mathit{\boldsymbol{\dot T}}}_{ot}}} \right]_\theta } = t{\left[ {{{\mathit{\boldsymbol{\dot T}}}_o}} \right]_\theta } $

(17)

$ {\left[ {{{\mathit{\boldsymbol{\ddot T}}}_{ot}}} \right]_{\theta \theta }} = t{\left[ {{{\mathit{\boldsymbol{\ddot T}}}_o}} \right]_{\theta \theta }} $

(18)

Assuming ^or_M^t represents M point's radius vector of unit mechanism in fixed coordinate system ∑x_toy_t, and ^tr_M^t represents M point's radius vector of unit mechanism in movement unit coordinate system, then

$ {}^o\mathit{\boldsymbol{r}}_M^t = {\mathit{\boldsymbol{T}}_{ot}} + {\mathit{\boldsymbol{R}}_{ot}}{}^t\mathit{\boldsymbol{r}}_M^t = {\mathit{\boldsymbol{T}}_{ot}} + {}^t\mathit{\boldsymbol{r}}_M^t $

(19)

Using formula (19) to get the first and second order derivation of time, can obtain the velocity and acceleration of point M:

$ {}^ov_M^t = {}^o\dot r_M^t = {{\dot T}_{ot}} + {}^o\dot r_M^t $

(20)

$ {}^oa_M^t = {}^o\ddot r_M^t = {{\ddot T}_{ot}} + {}^t\ddot r_M^t $

(21)

According to coordinate translation variables and M point's coordinate, it is available that

$ v_M^t = {}^oJ_{Mt}^t\omega $

(22)

$ {}^oa_M^t = {}^oJ_{Mt}^t\delta + {}^oH_{Mt}^t{\omega ^2} $

(23)

where

$ \begin{array}{l} {}^oJ_{Mt}^t = v_M^t/\omega = {\rm{d}}\left( {{{\dot T}_{ot}} + {}^t\dot r_M^t} \right)/{\rm{d}}\theta = \\ \;\;\;\;\;\;\;\;\;\;{\left[ {{{\dot T}_{ot}}} \right]_\theta } + {\left[ {{}^t\dot r_M^t} \right]_\theta } = t{\left[ {{{\dot T}_o}} \right]_\theta } + {\left[ {{}^t\dot r_M^t} \right]_\theta } \end{array} $

(24)

$ \begin{array}{l} {}^oH_{Mt}^t = {\rm{d}}\left( {{}^oJ_{Mt}^t} \right)/{\rm{d}}\theta = {{\rm{d}}^2}\left( {{{\dot T}_{ot}} + {}^t\dot r_M^t} \right)/{\rm{d}}{\theta ^2} = \\ \;\;\;\;\;\;\;\;\;\;{\left[ {{{\dot T}_{ot}}} \right]_{\theta \theta }} + \left[ {{}^t\dot r} \right]_{M\theta \theta }^t = t{\left[ {{{\dot T}_o}} \right]_{\theta \theta }} + {\left[ {{}^t\dot r_M^t} \right]_{\theta \theta }} \end{array} $

(25)

^oJ_Mt^t is the first order influence coefficient of line movement, ^oH_Mt^t is the second order influence coefficient of line movement. Where,

$ {\left[ {{}^t\dot r_M^t} \right]_\theta } = {\rm{d}}\left( {{}^t\dot r_M^t} \right)/{\rm{d}}\theta ,{\left[ {{}^t\dot r_M^t} \right]_{\theta \theta }} = {{\rm{d}}^2}\left( {{}^t\dot r_M^t} \right)/{\rm{d}}{\theta ^2} $

^oJ_Mt^t represents the sum of coordinate translation transformation matrix T_ot and the coordinate ^tr_M^t of point M, M is located at movement unit coordinate system, ^oJ_Mt^t and ^oH_Mt^t are function of θ, and are irrelative with velocity and acceleration of mechanism.

According to the above reasons, the displacement, velocity and acceleration of planar deployable structure's any point can be obtained. In other words, it only needs to calculate the displacement, velocity and acceleration of five points: i₁₁, i₁₂(i₂₁), i₂₂, j₃₂(j₂₁) and j₂₂.

3.3 Dynamic Analysis

The planar deployable structure is articulated with a slide block K₀ by revolute pairs at i₁₁, the slide block can move along OY axis, and the planar deployable structure can deploy along OX direction.

Assume the center of mass of component i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ is located at K₂₁, K₂₂ point respectively, then inertia force Q_F^t and moment of inertia Q_T^t of unit mechanism t can be represented by:

$ \left[ {Q_F^t} \right] = - {m^t}\left[ {{}^oa_K^t} \right] $

(26)

$ \left[ {Q_{\rm{T}}^t} \right] = - {\mathit{\boldsymbol{J}}^t}\left[ {{\delta ^t}} \right] $

(27)

where, [Q_F^t]=[Q_x^t Q_y^t]^T and Q_x^tQ_y^t are the inertia force's components of unit mechanism t in x and y direction, m^t=m₁^t+m₂^t is the mass of unit mechanism t, m₁^tm₂^t are the mass of component i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ respectively. [Q_T^t]=[Q_T1^t, Q_T2^t]^T and Q_T1^t, Q_T2^t are the moment of inertia of component i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ respectively, ${\mathit{\pmb{J}}^t} = \left[ {\begin{array}{*{20}{c}} {J_1^t} & 0\\ 0 & {J_2^t} \end{array}} \right]$ is the rotational inertia matrix of unit mechanism t, J₁^t, J₂^t are the rotational inertia of component i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ about points K₂₁, K₂₂. When the virtual displacement Δθ occurs on unit mechanism, the virtual displacement of unit t's center of mass K₂ correspond to the variation of K's displacement to θ, which can be expressed by:

$ \Delta r_{K2}^t = {}^oG_{K2r}^t\Delta \theta $

(28)

The virtual displacement Δφ^t of unit t's component i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ can be expressed by:

$ \Delta {\varphi ^t} = {}^oG_\theta ^t\Delta \theta $

(29)

The virtual work calculated by inertia force and mo-ment of inertia of unit t can be represented by:

$ \mathit{\boldsymbol{W}}_Q^t = {\left[ {Q_{\rm{T}}^t} \right]^{\rm{T}}}\Delta {\varphi ^t} + {\left[ {Q_F^t} \right]^{\rm{T}}}\Delta r_{K2}^t $

(30)

Through the relation between the displacementin y direction of slide block at hinging point i₁₁, which is solved earlier in this paper, and θ, then the variation is:

$ {\Delta ^o}y_i^0 = 2l\cos \left( {\theta /2} \right)\Delta \theta $

(31)

If signing the force exerted by slide block as F, then the virtual work is:

$ {W_F} = F{\Delta ^o}y_i^0 $

(32)

According to the principle of virtual work,

$ \sum\limits_{t = 0}^{N - 1} {W_Q^t + {W_F} = 0} $

(33)

Substitute equations in formula (33), it can be obtained and given by:

$ \delta = \frac{{2Fl\cos \left( {\theta /2} \right) - {\omega ^2}\sum\limits_{t = 0}^{N - 1} {{m^t}\left( {t\left[ {{T_o}} \right]_{\theta \theta }^{\rm{T}} + \left[ {{}^tr_{K2}^t} \right]_{\theta \theta }^{\rm{T}}} \right)\left( {t{{\left[ {{T_o}} \right]}_\theta } + {{\left[ {{}^tr_{K2}^t} \right]}_\theta }} \right)} }}{{\sum\limits_{t = 0}^{N - 1} {{{\left[ {{}^oG_\theta ^t} \right]}^{\rm{T}}}{J^{t0}}G_\theta ^t} + \sum\limits_{t = 0}^{N - 1} {{m^t}\left( {t\left[ {{T_o}} \right]_\theta ^{\rm{T}} + \left[ {{}^tr_{K2}^t} \right]_\theta ^{\rm{T}}} \right)\left( {t{{\left[ {{T_o}} \right]}_\theta } + {{\left[ {{}^tr_{K2}^t} \right]}_\theta }} \right)} }} $

(34)

Since ⁰r_K2⁰=¹r_K2¹=…=^N-1r_K2^N-1, so unifying them by r_K2, assume the mass and rotational inertia of all unit mechanism are equivalent, and represent by m and J respectively. Substitute equations in formula (34), then

$ \delta = \frac{{24Fl\cos \left( {\theta /2} \right) - N\left( {N + 1} \right)\left( {N - 1} \right)m{\omega ^2}{l^2}\sin \theta }}{{3NJ + 3mN{l^2} + 4N\left( {N + 1} \right)\left( {N - 1} \right)m{l^2}{{\sin }^2}\left( {\theta /2} \right)}} = \frac{{24Fl\cos \left( {\theta /2} \right) - A{\omega ^2}\sin \theta }}{{B + 4A{{\sin }^2}\left( {\theta /2} \right)}} $

(35)

where, A=N(N+1)(N－1), B=3 NJ. Formula (35) is a second-order differential equation, after the reduction of order, using Runge-Kutta method to execute numerical calculation and obtain the time curve of generalized coordinates and generalized velocity, substitute them in above equations, and calculate the displacement, velocity and acceleration of each component and hinge point.

Dynamics analysis of planar deployable structure is completed in foregoing part, and the next step is to calculate constraint force of each hinge point during deploy process. For the sake of research convenience, selecting the scissors unit mechanism of force planar deployable structure i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ as research object, in which, the constraint force of each hinge point is shown in Fig. 4, in order to simple marking, using F_x^jt represents the constraint force of unit mechanism in x direction that sustained on hinge j_t, and other symbols are taken same manner. The symbols F_x^kt, F_y^kt represent the force that component j₂₂K₂₂i₂₁ exerts on component i₂₂K₂₁j₂₁ in unit mechanism.

According to the D'alembert's principle, ad-opting dynamic static method to create the balance equation respectively for the component i₂₂K₂₁j₂₁, j₂₂K₂₂i₂₁ and solving, get formula

$ {F^t} = J{L^t} + q{Q^t} $

(36)

where,

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{F}}^t} = {{\left[ {\begin{array}{*{20}{c}} {F_x^{jt}}&{F_y^{jt}}&{F_x^{it}}&{F_y^{it}} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{L}}^t} = {{\left[ {\begin{array}{*{20}{c}} {F_x^{j,t + 1}}&{F_y^{j,t + 1}}&{F_x^{i,t + 1}}&{F_y^{i,t + 1}} \end{array}} \right]}^{\rm{T}}}} \end{array} $

and F^t represent the driven force of unit mechanism t, which is the load of unit t－1, and L^t represents the load of unit mechanism t, which is the driven force of unit t+1, they need to satisfy the relationship below:

$ {\mathit{\boldsymbol{L}}^{t - 1}} = - {\mathit{\boldsymbol{F}}^t} $

(37)

Q^t=[Q_x1^t  Q_y1^t  Q_x2^t  Q_y2^t  Q_T1^t  Q_T2^t]^T is the inertia force vector of unit mechanism, J and q are coefficient matrix,

$ \mathit{\boldsymbol{J = }}\left[ {\begin{array}{*{20}{c}} { - 1}&{ - \cot \left( {\frac{\theta }{2}} \right)}&0&{ - \cot \left( {\frac{\theta }{2}} \right)}\\ { - \tan \left( {\frac{\theta }{2}} \right)}&{ - 1}&{ - \tan \left( {\frac{\theta }{2}} \right)}&0\\ 0&{\cot \left( {\frac{\theta }{2}} \right)}&{ - 1}&{\cot \left( {\frac{\theta }{2}} \right)}\\ {\tan \left( {\frac{\theta }{2}} \right)}&0&{\tan \left( {\frac{\theta }{2}} \right)}&{ - 1} \end{array}} \right] $

$ \mathit{\boldsymbol{q = }} - \frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1&{\cot \left( {\frac{\theta }{2}} \right)}&1&{\cot \left( {\frac{\theta }{2}} \right)}&{\csc \left( {\frac{\theta }{2}} \right)/l}&{\csc \left( {\frac{\theta }{2}} \right)/l}\\ {\tan \left( {\frac{\theta }{2}} \right)}&1&{\tan \left( {\frac{\theta }{2}} \right)}&1&{ - \sec \left( {\frac{\theta }{2}} \right)/l}&{\sec \left( {\frac{\theta }{2}} \right)/l}\\ 1&{ - \cot \left( \theta \right)}&1&{ - \cot \left( {\frac{\theta }{2}} \right)}&{ - \csc \left( {\frac{\theta }{2}} \right)/l}&{ - \csc \left( {\frac{\theta }{2}} \right)/l}\\ { - \tan \left( {\frac{\theta }{2}} \right)}&1&{ - \tan \left( {\frac{\theta }{2}} \right)}&1&{\sec \left( {\frac{\theta }{2}} \right)/l}&{ - \sec \left( {\frac{\theta }{2}} \right)/l} \end{array}} \right] $

In hinge k, the force on component i₂₂K₂₁j₂₁ exerted by component i₂₂K₂₁j₂₁ is:

$ \mathit{\boldsymbol{F}}_k^t = {\mathit{\boldsymbol{J}}_k}{\mathit{\boldsymbol{L}}^t} + {\mathit{\boldsymbol{q}}_k}{\mathit{\boldsymbol{Q}}^t} $

(38)

where, J_kq_k are coefficient matrix, meet formula:

$ {\mathit{\boldsymbol{J}}_k} = \left[ {\begin{array}{*{20}{c}} 1&{\cot \left( {\theta /2} \right)}&{ - 1}&{\cot \left( {\theta /2} \right)}\\ {\tan \left( {\theta /2} \right)}&1&{\tan \left( {\theta /2} \right)}&{ - 1} \end{array}} \right] $

$ {\mathit{\boldsymbol{q}}_k} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} { - 1}&{\cot \left( {\frac{\theta }{2}} \right)}&1&{\cot \left( {\frac{\theta }{2}} \right)}&{\csc \left( {\frac{\theta }{2}} \right)/l}&{\csc \left( {\frac{\theta }{2}} \right)/l}\\ {\tan \left( {\frac{\theta }{2}} \right)}&{ - 1}&{\tan \left( {\frac{\theta }{2}} \right)}&1&{ - \sec \left( {\frac{\theta }{2}} \right)/l}&{\sec \left( {\frac{\theta }{2}} \right)/l} \end{array}} \right] $

The magnitude |F_k^t| of F_k^t represents the con-straint force of hinge k_t.

When θ=0, π, 2π, 3π, …, tan (θ/2), cot(θ/2), sec(θ/2) and csc(θ/2) are infinity, matrix J, q, J_k, q_k are pointless, thus formula (36) and (38) are no longer valid, mechanism respectively corresponds to two different kinds of singular configuration, and leads to the degrees of freedom to be greater than 1, movement is uncertain. When θ=0, 2π, …, hinge point i_t and j_t coincide, two components overlap in the same unit, all units connect successively and coincide with x axis; when θ=π, 3π, …, hinge point i₀, i₁, i₂, …, i_N－1 coincide, j₀, j₁, j₂, …, j_N－1 coincide, k₁, k₂, …, k_N－1 coincide, two components of each unit both coincide with y axis.

If the external applied load of unit N-1 on hinge i_N, j_N is represented by L^N－1, substitute it in formula (36) and (38), solve and obtain constraint force F^N－1, F_k^N－1 of hinge i_N－1, j_N－1, substitute F^N－1 in formula (36), (37) and (38), get the solution of F^N－2, F_k^N－2. Execute recursive successively and solve constraint force of each hinge.

4 Numerical Example

Taking the scissors unit planar deployable structure as an example, adopt pole length l=500 mm, r=2 mm, pole mass m=1 kg, density ρ=7 800 kg/m³, driven force acting in y axis direction through hinge point i₁₁ is F=-40 N, the initial value of fixed coordinate system is θ₀=(π-0.1) rad, displace-ment, velocity and acceleration of each hinge point and constraint force of articulated link chain are shown in Figs. 5-8.

From the analysis of Fig. 8, after exerting a driving force F on hinge point i₁₁ along y axis, surface deployable institutions complete development with varying acceleration. When configuration angle θ ≫ 0, the angular acceleration of each component approximate to constant, the development approximate to uniform angular acceleration, the linear acceleration of each hinge point is nearly unchanged, approximate to uniformly accelerated motion, the constraint force of each hinge point approximates to be constant.

When the configuration angle θ of unit mechanism is approaching to π, angular acceleration of each component increases sharply, the angular velocity changes quickly, linear acceleration of each hinge point also increases sharply, linear velocity changes quickly, and causes comparatively large constraint force on each hinge.

It can be concluded that, for the scissors unit institutions, the variation of displacement, velocity and acceleration of hinge point trend to be consistent, this phenomenon explains that the planar deployable unit of scissors mechanism displays consistent dynamic characteristics.

5 Conclusions

This paper adopts influence coefficient method to analyze the dynamic characteristics for planar deployable mechanism.

Based on the cartesian coordinate system, the dynamics analysis model of deployable mechanism is established, through coordinate transformation, the displacement and movement influence coefficient under generalized coordinates of each unit mechanism are obtained; the principle of virtual work is applied to complete dynamic characteristics analysis; based on the D'Alembert's principle, combined with dynamic static method, calculated the constraint force for each hinge; finally, dynamic analysis and calculating examples are presented to verify the effectiveness of proposed method in this paper. Utilizing the method presented in this paper can simplify dynamic analysis procedure, and expand the application range of this method, thus can be provided with higher practical application value.

From above research and analysis, the dynamic characteristics of surface deployable institutions during develop process can be effectively predicted, and this research would facilitate engineering application. In the following study, the varying driving force can be considered to help avoid motion mutation during the moving process of orbiting spacecraft, which can affect motion attitude, so as to prolong the service life of spacecraft.