Study on automatic deduction method of overall transfer equation for branch multibody system

Abstract

The transfer matrix method for multibody system is a new method developed in recent 20 years for studying multibody system dynamics. The new version of transfer matrix method for multibody system and automatic deduction theorem of system overall transfer equation have been formed in 2012. The overall transfer equation plays one of the key roles for transfer matrix method for multibody system. In order to study branch system dynamics with new version of transfer matrix method for multibody system, the automatic deduction theorem of overall transfer equation has been expanded in this article, which made it practicable to deduce automatically the overall transfer equation for branch system with new version of transfer matrix method for multibody system. The transfer equations and transfer matrices of typical elements are developed for the automatic deduction theorem of overall transfer equation. Thereby new version of transfer matrix method for multibody system owns the following features: there is no need to establish the global dynamics equation of system, by which the study on multibody system dynamics will be simplified greatly; the automatic deduction of overall transfer equation for multibody system is realized, consequently the algorithm is simple and highly programmable; moreover, the order of system matrix is always low, which result in high computational speed. The branch multibody system dynamics has been computed with the automatic deduction theorem of overall transfer equation as well as ordinary multibody system dynamics method. The computational results obtained by the two methods have good agreements, which validate the automatic deduction theorem of overall transfer equation for branch multibody system.

Introduction

The dynamics equation of an individual rigid body that established by Newton’s ¹ Law and Euler ² Theorem has promoted the development of rigid body dynamics and mechanical system dynamic design technology. Under strong engineering demand, a new subject branch, multibody system dynamics,^3–9 has been developed step by step since 1960s. The deduction of system dynamics equation and its numerical solution are the two most important parts for studying multibody system dynamics (MSD). Increasing computational speed and accuracy is one of the important research areas for complex multibody system dynamics. The recursive Newton Euler algorithm was presented ¹⁰ to increase the computational speed.

The classic transfer matrix method (TMM) has been developed in 1920s for studying vibration problems of one-dimensional system constructed by elastic components. ¹¹ Holzer ²⁵ (1921) was the very first to solve torsional vibration problem of multi-disk shaft with initial parameter method. Myklestad ²⁶ (1944) solved beam bending vibration model with the method similar to Holzer’s. Proh ²⁷ (1945) successfully generalized Holzer’s method to transverse vibration problem of shaft. Due to the development of computer and application of matrix operation in vibration research, Myklestad’s method has been progressively developed to TMM. Thomson ²⁸ (1950) applied TMM on structure vibration problem of general linear system. Pestel and Leckie ¹² published the first book of TMM “Matrix Method in Elastomechanics.” Rubin ²⁹ (1964) proposed the general treatment process of transfer matrix as well as the relationship between transfer matrix and frequency response matrix with other form. With TMM, Targoff, Lin, Mercer, Lin, Mead, Henderson, McDaniel, and Murthy, respectively, solved statics and dynamics problems of beam, beam-type multiple structure, board with string, ribbed structure, curved multi-span structure, cylindrical shell, rigid ring, and so on. Horner ³⁰ (1978) proposed the numerical stability of Riccati TMM while solving boundary value problem. Combined with TMM and finite element method (FEM), Dokanish ³¹ (1972) proposed the FE-TMM to solve two-dimensional thin board and increased the computational speed of FEM. ¹³ Combined with TMM and numerical integrating method, Kumar and Sankar ¹⁴ developed the discrete-time transfer matrix method (DTTMM) for time-variant system structure dynamics.

In order to meet the requirement for increasing the computational speed of MSD and simplifying the research process of MSD, Rui with his cooperator proposed the transfer matrix method for multibody system (MSTMM):^11,14–24 in 1993, for the first time, the linear MSTMM has been proposed, the huge computation workload of eigenvalue problem of complex linear multibody system as well as the ill condition brought by large stiffness gradient has been solved while the computation efficiency has been increased; in 1997, the new concept of linear multibody system augmented eigenvector and augmented operator have been proposed, multi-rigid-flexible-body system augmented eigenvector with orthogonality has been constructed, the accurate analysis of the dynamics response of complex multi-rigid-flexible-body system has been realized with modal method;^11,15,16 in 1998, the DTTMM for multi-rigid-body system^15,17 has been proposed and improved progressively; through time-variant transfer matrix and state vector with state variable described by physical coordinates, the computational speed of multibody system dynamics has been greatly increased; in 1999, the DTTMM for multi-rigid-flexible-body system has been^15,18 proposed and the high-speed computation of multi-rigid-flexible-body system dynamics has been realized; in 2005, the TMM for two-dimensional system has been proposed, and two-dimensional system dynamics analysis with pure TMM has been realized; ¹⁹ the mixed method with MSTMM and ordinary multibody system dynamics (MSDM), mixed method with MSTMM and FEM, linear controlled MSTMM, controlled discrete-time transfer matrix method for multibody system (MSDTTMM),^20,21 Riccati MSDTTMM, ²² and finite section MSDTTMM have been proposed, respectively; the overall transfer equation is one of the key roles for MSTMM; and in 2012, the automatic deduction theorem ²³ of overall transfer equation has been proposed and the automatic deduction of overall transfer equation has been realized; the new version of MSTMM ²⁴ has been proposed, thereby the research of MSD and the application of new version of MSTMM for numerous engineering technicians have been simplified.

The original automatic deduction theorem of overall transfer equation ²³ is effective for general system for linear MSTMM and MSDTTMM, also effective for the chain system and closed-loop system for the new version of MSTMM. In this article, the automatic deduction theorem of overall transfer equation for branch system for new version of MSTMM has been established. The original automatic deduction theorem of overall transfer equation has been expanded and the corresponding transfer equations and transfer matrices of typical elements have been deduced. The automatic deduction of overall transfer equation for branch multibody system has been realized, also with the result of simple algorithm and high programming. The MSD has been computed using the automatic deduction theorem of overall transfer equation for branch system and ordinary MSDM, respectively, with the good agreements of both results, the effectiveness of automatic deduction theorem of overall transfer equation for branch multibody system has been validated.

Principles and steps of new version of MSTMM

Multibody system dynamics model topology graph and label convention

The topology graph of MSD model is a new graph notation for new version of MSTMM that describes the transfer relationship, position sequence numbers, boundary sequence numbers, and transfer direction among the state vectors of body elements and hinge elements ²³ in MSD. The branch MSD model topology graph shown in Figure 1 is taken as an example for detailed description.

Figure 1.

Topology graph of dynamics model of a branch system.

According to the features of new version of MSTMM, the label conventions in this article are as follows:

1. Mechanics elements are divided into two types—“bodies” and “hinges”—and are numbered uniformly. A body with only two joints connecting with other elements, including boundary, is treated as the element with single-input end and single-output end. A body with more than two joints connecting with other elements is treated as the element with multi-input ends and single-output end. The element j in the transfer path from tip element n to element i and connects directly with element i is named as the inboard element (inboard body or inboard hinge) of element i, the element i is named as the outboard element (outboard hinge or outboard body) of element j.

2. In topology graph, a circle ○ and the number inside it indicate, respectively, a body element and its sequence number; an arrow → and the number beside it indicate, respectively, a hinge element and its sequence number, the arrow direction shows the transfer direction of state vector, and number 0 in the figure means the system boundary.

3. For a non-boundary end, the bold italic small letter ${\mathbf{z}}_{i,j}$ indicates the state vector of joint $P_{i,j}$ in physical coordinates. The first subscript i is the sequence number of body element at end $P_{i,j}$ , while the second subscript j is the sequence number of hinge element at end $P_{i,j}$ . The following state vector for instance

$$z_{i,j}={[\stackrel{··}{x}\stackrel{··}{y}\stackrel{··}{z}{\stackrel{.}{\Omega }}_x{\stackrel{.}{\Omega }}_y{\stackrel{.}{\Omega }}_z{m}_x{m}_y{m}_z{q}_x{q}_y{q}_{z}1]}^T$$

(1)

is the column matrix in global inertial coordinates system oxyz. And $\stackrel{··}{x},\stackrel{··}{y},\stackrel{··}{z}$ , ${\stackrel{·}{\Omega }}_x,{\stackrel{·}{\Omega }}_y,{\stackrel{·}{\Omega }}_z$ , $m_x,m_y,m_z$ , and $q_x,q_y,q_z$ are the acceleration, angular acceleration, internal torque, and internal force of joint point $P_{i,j}$ in the oxyz. The first and the second subscript i and $j(i,j\ne 0)$ in a state vector ${\mathbf{z}}_{i,j}$ denote the sequence numbers of the adjacent body element and hinge element, respectively. For boundary state vector ${\mathbf{z}}_{i,j}$ the second subscript $j=0$ , that is, if the second subscript of the state vector is 0, this end is a boundary end ${\mathbf{z}}_{i,0}$ . The boldface italic capital ${\mathbf{U}}_i$ denotes the transfer matrix of the element whose subscript i is the element sequence number. The boldface italic capital ${\mathbf{U}}_{k,j}$ denotes a partitioned matrix of a transfer matrix, where the subscript k and j are the sequence numbers of row and column of the partitioned matrix in the transfer matrix, respectively. Lowercase $u_{k,j}$ or $u_{\mathrm{kj}}$ denotes an element in a transfer matrix, where k and j represent the numbers of row and column of the element in the transfer matrix, respectively. The transfer matrix ${\mathbf{U}}_{i-k}$ means the successive multiplication of the transfer matrices of all elements along the transfer path from the element i to element k in the system.

• 4. Only one of the boundary ends in a system is called the root, whose state vector is ${\mathbf{z}}_{n,0}$ , and the first subscript n stands for the sequence number of the root element. While the other entire boundary ends are considered as tips. In the state vectors of tips, with state vector ${\mathbf{z}}_{j,0}$ , the first subscript j stands for the sequence number of the tip elements.

• 5. The transfer directions of system state vectors are always from tips to root. Along transfer direction, the nodes entering an element are called input ends of the element, at the meantime are denoted as I, while the node leaving an element is called the output end of the element and is denoted as O. The connection point between elements is the output end of its inboard element and at the same time is the input end of its outboard element.

• 6. Conventions of positive direction for torques and forces: Inboard forces and outboard torques acting on an elements are positive (negative) if their vectors are in positive (negative) directions, outboard forces and inboard torques acting on an elements are positive (negative) if their vectors are in the negative (positive) direction.

• 7. The boldface italic capital ${\mathbf{I}}_n$ denotes an identity matrix with order n. The boldface italic capital $0_{m\times n}$ denotes a null matrix with m rows and n columns, and $0_m$ denotes a null column matrix with m rows.

The motion of every element in the system is described in right-handed inertial Cartesian coordinate system oxyz; x, y, and z are used as the position coordinates of element joint. The origin point of body-fixed coordinate system $I\xi \eta \zeta$ is element input end I, as shown in Figure 2. The first input end $I_1$ will be considered as the origin if there exists more than one input end. The element body-fixed coordinate system $I\xi \eta \zeta$ is unique defined in oxyz. Space-three-angles ⁶ $x-y-z$ are used to describe the orientation of rigid body. Rotating around axes x, y, and z successively, one will obtain ${\theta }_x$ , ${\theta }_y$ , and ${\theta }_z$ , respectively.

Figure 2.

Inertial Cartesian coordinate system oxyz and body-fixed coordinate system $I\xi \eta \zeta$ .

Principle of new version of MSTMM

The basic idea of MSTMM is: break up a complicated multibody system into many elements including bodies and hinges whose dynamics properties can be readily expressed in terms of matrices. These transfer matrices of elements are like construct bricks while the library of transfer matrices can be readily developed. When assembled them together according to topology structure of the system, the dynamics properties of entire system can be provided. For chain system, the overall transfer matrix and overall transfer equation of system can be obtained by simply successive multiplying of transfer matrices of elements. According to Newton’s third law, for any two adjacent elements in multibody system, the state vectors of the output end of inboard element and of the input end of outboard element are always equivalent.

It can be proved that there is the direction cosine matrix ⁶ from body-fixed coordinate system to inertial coordinate system

$$\mathbf{A}=\left[\begin{array}{ccc}\hfill c_y{c}_z\hfill & \hfill s_x{s}_y{c}_z-c_x{s}_z\hfill & \hfill c_x{s}_y{c}_z+s_x{s}_z\hfill \\ \multicolumn{1}{c}{c_y{s}_z}& \hfill s_x{s}_y{s}_z+c_x{c}_z\hfill & \hfill c_x{s}_y{s}_z-s_x{c}_z\hfill \\ \multicolumn{1}{c}{-s_y}& \hfill s_x{c}_y\hfill & \hfill c_x{c}_y\hfill \end{array}\right]$$

(2)

The angular velocity $\mathbf{\omega }$ and angular acceleration $\stackrel{·}{\mathbf{\omega }}$ of a rigid body in body-fixed coordinates system can be expressed with space-three-angles ⁶ and its derivative as

$$\left[\begin{array}{c}\hfill {\omega }_{\xi }\hfill \\ \multicolumn{1}{c}{{\omega }_{\eta }}\\ \multicolumn{1}{c}{{\omega }_{\zeta }}\end{array}\right]=\mathbf{H}\left[\begin{array}{c}\hfill {\stackrel{·}{\theta }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_y}\\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_z}\end{array}\right],\left[\begin{array}{c}\hfill {\stackrel{·}{\omega }}_{\xi }\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_{\eta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_{\zeta }}\end{array}\right]=\stackrel{·}{\mathbf{H}}\left[\begin{array}{c}\hfill {\stackrel{·}{\theta }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_y}\\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_z}\end{array}\right]+\mathbf{H}\left[\begin{array}{c}\hfill {\stackrel{··}{\theta }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{\theta }}_y}\\ \multicolumn{1}{c}{{\stackrel{··}{\theta }}_z}\end{array}\right]$$

(3)

where

$$\left[\begin{array}{c}\hfill {\stackrel{··}{\theta }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{\theta }}_y}\\ \multicolumn{1}{c}{{\stackrel{··}{\theta }}_z}\end{array}\right]={\mathbf{H}}^{-1}{\mathbf{A}}^T{\left[\begin{array}{c}\hfill {\stackrel{·}{\Omega }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\Omega }}_y}\\ \multicolumn{1}{c}{{\stackrel{·}{\Omega }}_z}\end{array}\right]}_I-{\mathbf{H}}^{-1}\stackrel{·}{\mathbf{H}}\left[\begin{array}{c}\hfill {\stackrel{·}{\theta }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_y}\\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_z}\end{array}\right]$$

(4)

$$\begin{array}{cc}\multicolumn{1}{c}{}& \mathbf{H}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -s_y\hfill \\ \multicolumn{1}{c}{0}& \hfill c_x\hfill & \hfill s_x{c}_y\hfill \\ \multicolumn{1}{c}{0}& \hfill -s_x\hfill & \hfill c_x{c}_y\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& \stackrel{·}{\mathbf{H}}=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill -c_y{\stackrel{·}{\theta }}_y\hfill \\ \multicolumn{1}{c}{0}& \hfill -s_x{\stackrel{·}{\theta }}_x\hfill & \hfill c_x{c}_y{\stackrel{·}{\theta }}_x-s_x{s}_y{\stackrel{·}{\theta }}_y\hfill \\ \multicolumn{1}{c}{0}& \hfill -c_x{\stackrel{·}{\theta }}_x\hfill & \hfill -s_x{c}_y{\stackrel{·}{\theta }}_x-c_x{s}_y{\stackrel{·}{\theta }}_y\hfill \end{array}\right]\hfill \end{array}$$

(5)

$$s_i=\sin {\theta }_i,c_i=\cos {\theta }_i,i=x,y,z$$

$\stackrel{·}{\mathbf{\Omega }}=[\begin{array}{ccc}\hfill {\stackrel{·}{\Omega }}_x\hfill & \hfill {\stackrel{·}{\Omega }}_y\hfill & \hfill {\stackrel{·}{\Omega }}_z\hfill \end{array}{]}^T$ is the column matrix of the angular acceleration of the element in oxyz.

It should be pointed out that any kind of Euler angle formulation has its inherent singularity, that is why the angular acceleration $\stackrel{·}{\mathbf{\Omega }}$ rather than $\stackrel{··}{\theta }$ is used in the state vector for the new version of MSTMM. Once approaching singular configuration, one can resort to another kind of Euler angle to avoid the singularity problem. Alternatively, many other methods, such as Rodriguez parameters and Euler quaternions, can be also adopted to describe the orientation and avoid singular orientation. This reduces the risk of gimbal lock phenomena.

The state vector, transfer equation, and transfer matrix of element

For convenient description, herein a chain system is taken as an example. The state vector of spatial motion element in joint is defined as equation (1) or

$$\mathit{z}={\left[\begin{array}{ccccc}\hfill {\stackrel{··}{\mathit{r}}}^{\mathrm{T}}\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}^{\mathrm{T}}\hfill & \hfill {\mathit{m}}^{\mathrm{T}}\hfill & \hfill {\mathit{q}}^{\mathrm{T}}\hfill & \hfill 1\hfill \end{array}\right]}^{\mathrm{T}}$$

(6)

where

$$\stackrel{··}{\mathbf{r}}=\left[\begin{array}{c}\hfill \stackrel{··}{x}\hfill \\ \multicolumn{1}{c}{\stackrel{··}{y}}\\ \multicolumn{1}{c}{\stackrel{··}{z}}\end{array}\right],\stackrel{·}{\mathbf{\Omega }}=\left[\begin{array}{c}\hfill {\stackrel{·}{\Omega }}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\Omega }}_y}\\ \multicolumn{1}{c}{{\stackrel{·}{\Omega }}_z}\end{array}\right],\mathbf{m}=\left[\begin{array}{c}\hfill m_x\hfill \\ \multicolumn{1}{c}{m_y}\\ \multicolumn{1}{c}{m_z}\end{array}\right],\mathbf{q}=\left[\begin{array}{c}\hfill q_x\hfill \\ \multicolumn{1}{c}{q_y}\\ \multicolumn{1}{c}{q_z}\end{array}\right]$$

(7)

$\stackrel{·}{\mathbf{r}},\stackrel{·}{\mathbf{\Omega }},\mathbf{m},\mathrm{and}\mathbf{q}$ are, respectively, the column matrices of absolute acceleration, absolute angular acceleration, internal torque, and internal force in joint in oxyz.

Particularly, for the chain system with planar motion

$$\stackrel{··}{\mathbf{r}}=\left[\begin{array}{c}\hfill \stackrel{··}{x}\hfill \\ \multicolumn{1}{c}{\stackrel{··}{y}}\end{array}\right],\stackrel{·}{\mathbf{\Omega }}={{\stackrel{·}{\Omega }}_z}_z,\mathbf{m}=m_z,\mathbf{q}=\left[\begin{array}{c}\hfill q_x\hfill \\ \multicolumn{1}{c}{q_y}\end{array}\right]$$

(8)

Applying above definition of state vector, according to Newton’s second law and Euler theorem, there is linear relationship among the acceleration and angular acceleration of any object and the external force and external torque acting on it in inertial coordinate system, which can be expressed as linear algebraic equation in matrix form, that is, for any element j with two ends, if its definition of the state vector is expressed as equation (1), the following equation has always been strictly satisfied

$${\mathbf{z}}_{j,j+1}(t_i)={\mathbf{U}}_j(t_i){\mathbf{z}}_{j,j-1}(t_i)$$

(9)

Equation (9) described the transfer relation between state vectors in two ends of element j, thus is named as the transfer equation of element j; ${\mathbf{U}}_j(t_i)$ is named as the transfer matrix of element j, which is known at time $t_i$ .

System overall transfer equation and overall transfer matrix

For chain system constructed by n elements, applying repeatedly the element transfer equation (equation (9)), the system overall transfer equation that connects two ends of the system can be obtained

$${\mathbf{z}}_{n,0}={\mathbf{U}}_{1-n}{\mathbf{z}}_{1,0}$$

(10)

where

$${\mathbf{U}}_{1-n}={\mathbf{U}}_n\cdots {\mathbf{U}}_2{\mathbf{U}}_1$$

(11)

is the overall transfer matrix of system.

It can be seen from equations (10) and (11) that the overall transfer matrix ${\mathbf{U}}_{1-n}$ of chain multibody system can be obtained by successive multiplication of transfer matrix of each element along the transfer path from tip to root of the system. Regardless of system scale, the order of overall transfer matrix is constantly equal to the order of element transfer matrix. Therefore, the order is $(13\times 13)$ for spatial motion of multi-rigid-body system or $(7\times 7)$ for planar motion of multi-rigid-body system. Since the matrix order for new version of MSTMM is always small, the requirement for computational time and storage space has been greatly reduced.

Solution of system dynamics

Applying boundary condition and initial condition of the system, such as the initial position coordinate, absolute velocity, rotation angle, and angular velocity, solving overall transfer equation of the system equation (10) and transfer equation of element equation (9) one after another, the state vector of each element at $t_i$ can be obtained. The position coordinate, absolute velocity, rotation angle, and angular velocity at $t_{i+1}$ can be obtained by employing numerical integrating method such as Runge–Kutta method. Then substitute the result of the last step as initial condition, repeat the above process from time $t_{i+1}$ till the end of entire analysis time required.

Transfer equations of typical elements

Transfer equation of spatial motion rigid body with single-input end and single-output end

For spatial motion rigid body j with single-input end and single-output end, as shown in Figure 2, the position vector of rigid body input end I and any point P in the body is ${\mathbf{r}}_I$ and ${\mathbf{r}}_P$ in global inertial coordinate system oxyz. The position vector of P in body-fixed coordinate system $I\xi \eta \zeta$ is ${\mathbf{l}}_{\mathrm{IP}}$ , where the input end I is taken as the origin point of the coordinate $I\xi \eta \zeta$ . ${\mathbf{r}}_{\mathrm{IP}}$ is the column matrix of ${\mathbf{l}}_{\mathrm{IP}}$ in oxyz. The orientation of rigid body j is represented by direction cosine matrix ${\mathbf{A}}_I$ of $I\xi \eta \zeta$ with respect to oxyz. The angular velocity of $I\xi \eta \zeta$ in oxyz and $I\xi \eta \zeta$ is ${\mathbf{\Omega }}_I$ and ${\mathbf{\omega }}_I$ respectively.

The position vector of any point P in rigid body

$${\mathbf{r}}_P={\mathbf{r}}_I+{\mathbf{r}}_{\mathrm{IP}}$$

(12)

taking the derivative and the second derivative with respect to time, one obtains

$${\stackrel{·}{\mathbf{r}}}_P={\stackrel{·}{\mathbf{r}}}_I+{\tilde{\mathbf{\Omega }}}_I{\mathbf{r}}_{\mathrm{IP}}$$

(13)

$${\stackrel{··}{\mathit{r}}}_P={\stackrel{··}{\mathit{r}}}_I-{\tilde{\mathit{r}}}_{\mathrm{IP}}{\stackrel{·}{\mathbf{\Omega }}}_I+{\tilde{\mathbf{\Omega }}}_I{\tilde{\mathbf{\Omega }}}_I{\mathit{r}}_{\mathrm{IP}}$$

(14)

The angular velocity and angular acceleration of any point P and input end I are respectively equal for rigid body, that is

$${\mathbf{\Omega }}_P={\mathbf{\Omega }}_I$$

(15)

$${\stackrel{·}{\mathbf{\Omega }}}_P={\stackrel{·}{\mathbf{\Omega }}}_I$$

(16)

Considering label convention, the rigid body dynamics equation in oxyz can be expressed as

$$m{\stackrel{··}{\mathbf{r}}}_I-m{\tilde{\mathbf{r}}}_{\mathrm{IC}}{\stackrel{·}{\mathbf{\Omega }}}_I+m{\tilde{\mathbf{\Omega }}}_I{\tilde{\mathbf{\Omega }}}_I{\mathbf{r}}_{\mathrm{IC}}={\mathbf{q}}_I-{\mathbf{q}}_O+{\mathbf{f}}_C$$

(17)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{A}}_I{\tilde{\mathbf{\omega }}}_I{\mathbf{J}}_I{\mathbf{\omega }}_I+{\mathbf{A}}_I{\mathbf{J}}_I{\mathbf{A}}_I^T{\stackrel{·}{\mathbf{\Omega }}}_I+m{\tilde{\mathbf{r}}}_{\mathrm{IC}}{\stackrel{··}{\mathbf{r}}}_I\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{m}}_O-{\mathbf{m}}_I-{\tilde{\mathbf{r}}}_{\mathrm{IO}}{\mathbf{q}}_O+{\mathbf{m}}_C+{\tilde{\mathbf{r}}}_{\mathrm{IC}}{\mathbf{f}}_C\hfill \end{array}$$

(18)

Rewrite equation (14) and dynamics equations of rigid body equations (17) and (18)

$${\stackrel{··}{\mathbf{r}}}_O={\stackrel{··}{\mathbf{r}}}_{I_1}+{\mathbf{E}}_1{\stackrel{·}{\mathbf{\Omega }}}_{I_1}+{\mathbf{E}}_2$$

(19)

$${\mathbf{q}}_O={\mathbf{q}}_I+{\mathbf{E}}_3{\stackrel{··}{\mathbf{r}}}_I+{\mathbf{E}}_4{\stackrel{·}{\mathbf{\Omega }}}_I+{\mathbf{E}}_5$$

(20)

$${\mathbf{m}}_O={\mathbf{m}}_I+{\mathbf{E}}_6{\mathbf{q}}_O+{\mathbf{E}}_7{\stackrel{··}{\mathbf{r}}}_I+{\mathbf{E}}_8{\stackrel{·}{\mathbf{\Omega }}}_I+{\mathbf{E}}_9$$

(21)

Note the state vector definition equation (6), the transfer equation, and transfer matrix of spatial motion rigid body j with single-input end and single-output end can be obtained by rewriting system dynamics equation

$${\mathbf{z}}_O={\mathbf{U}}_j{\mathbf{z}}_I$$

(22)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{U}}_j=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{ccccc}\hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_1\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{E}}_2\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_6{\mathbf{E}}_3+{\mathbf{E}}_7}& \hfill {\mathbf{E}}_6{\mathbf{E}}_4+{\mathbf{E}}_8\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_6\hfill & \hfill {\mathbf{E}}_6{\mathbf{E}}_5+{\mathbf{E}}_9\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_3}& \hfill {\mathbf{E}}_4\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_5\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 3}}& \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array}$$

(23)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{E}}_1=-{\tilde{\mathbf{r}}}_{\mathrm{IO}},{\mathbf{E}}_2={\tilde{\mathbf{\Omega }}}_I{\tilde{\mathbf{\Omega }}}_I{\mathbf{r}}_{\mathrm{IO}},{\mathbf{E}}_3=-m{\mathbf{I}}_3,\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_4=m{\tilde{\mathbf{r}}}_{\mathrm{IC}},{\mathbf{E}}_5={\mathbf{f}}_C-m{\tilde{\mathbf{\Omega }}}_I{\tilde{\mathbf{\Omega }}}_I{\mathbf{r}}_{\mathrm{IC}}\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_6={\tilde{\mathbf{r}}}_{\mathrm{IO}},{\mathbf{E}}_7=m{\tilde{\mathbf{r}}}_{\mathrm{IC}},{\mathbf{E}}_8={\mathbf{A}}_I{\mathbf{J}}_I{\mathbf{A}}_I^T\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_9=-{\mathbf{m}}_C-{\tilde{\mathbf{r}}}_{\mathrm{IC}}{\mathbf{f}}_C+{\mathbf{A}}_I{\tilde{\mathbf{\omega }}}_I{\mathbf{J}}_I{\mathbf{\omega }}_I\hfill \end{array}$$

(24)

${\mathbf{z}}_O$ and ${\mathbf{z}}_I$ are the state vectors of output end O and input end I, respectively, m is the mass of rigid body, $J_I$ is the matrix of moment of inertia with respect to input end I in $I\xi \eta \zeta$ , ${\mathbf{f}}_c$ and ${\mathbf{m}}_c$ are the column matrices of external force and external torque acting on center of mass of rigid body, respectively.

Transfer matrix of spatial large motion rigid body with multiple inputs and single output

For rigid body j with multi-input ends and single-output end as shown in Figure 3, one end is regarded as output end, all other ends are regarded as input ends; $I_k(k=1,2,\dots ,L)$ , O, and C are the input ends, output end, and center of mass, respectively; L is the sum of input ends. The position vector of P in inertial coordinate system oxyz and body-fixed coordinate system $I_1\xi \eta \zeta$ are ${\mathbf{r}}_P$ and ${\mathbf{l}}_{I_{1}P}$ , respectively. The column matrix of ${\mathbf{l}}_{I_{1}P}$ in oxyz is ${\mathbf{r}}_{I_{1}P}$ . ${\mathbf{A}}_{I_1}$ is the direction cosine matrix of $I_{I_1}\xi \eta \zeta$ with respect to oxyz, ${\mathbf{\Omega }}_{I_1}$ and ${\mathbf{\omega }}_{I_1}$ are its angular velocities in $I_{I_1}\xi \eta \zeta$ and oxyz.

Figure 3.

Spatial motion rigid body with multi-input ends and single-output end.

Note the derivative and the second derivative of ${\stackrel{·}{\mathbf{A}}}_{I_1}$ with respect to time are

$${\stackrel{·}{\mathbf{A}}}_{I_1}={\mathbf{A}}_{I_1}{\tilde{\mathbf{\omega }}}_{I_1}={\tilde{\mathbf{\Omega }}}_{I_1}{\mathbf{A}}_{I_1}$$

(25)

$${\stackrel{··}{\mathbf{A}}}_{I_1}={\mathbf{A}}_{I_1}({\widetilde{\stackrel{·}{\mathbf{\omega }}}}_{I_1}+{\tilde{\mathbf{\omega }}}_{I_1}{\tilde{\mathbf{\omega }}}_{I_1})=({\widetilde{\stackrel{·}{\mathbf{\Omega }}}}_{I_1}+{\tilde{\mathbf{\Omega }}}_{I_1}{\tilde{\mathbf{\Omega }}}_{I_1}){\mathbf{A}}_{I_1}$$

(26)

and

$${\mathbf{r}}_{I_{1}P}={\mathbf{A}}_{I_1}{\mathbf{l}}_{I_{1}P},{\mathbf{\Omega }}_{I_1}={\mathbf{A}}_{I_1}{\mathbf{\Omega }}_{I_1},{\stackrel{·}{\mathbf{\Omega }}}_{I_1}={\mathbf{A}}_{I_1}{\stackrel{·}{\mathbf{\omega }}}_{I_1}$$

(27)

According to rigid body kinematics, the relationship between acceleration of the first input end and output end of rigid body

$${\mathbf{\Omega }}_O={\mathbf{\Omega }}_{I_1},{\stackrel{·}{\mathbf{\Omega }}}_O={\stackrel{·}{\mathbf{\Omega }}}_{I_1}$$

(28)

$${\stackrel{··}{\mathbf{r}}}_O={\stackrel{··}{\mathbf{r}}}_{I_1}-{\tilde{\mathbf{r}}}_{I_{1}O}{\stackrel{·}{\mathbf{\Omega }}}_{I_1}+{\tilde{\mathbf{\Omega }}}_{I_1}{\tilde{\mathbf{\Omega }}}_{I_1}{\mathbf{r}}_{I_{1}O}$$

(29)

The geometric equation of the first inboard end $I_1$ and the other inboard ends $I_k(k=2,3,\dots ,L)$

$${\stackrel{·}{\mathbf{\Omega }}}_{I_k}={\stackrel{·}{\mathbf{\Omega }}}_{I_1}(k=2,3,\dots ,L)$$

(30)

$${\stackrel{··}{\mathbf{r}}}_{I_k}={\stackrel{··}{\mathbf{r}}}_{I_1}-{\tilde{\mathbf{r}}}_{I_1{I}_k}{\stackrel{·}{\mathbf{\Omega }}}_{I_1}+{\tilde{\mathbf{\Omega }}}_{I_1}{\tilde{\mathbf{\Omega }}}_{I_1}{\mathbf{r}}_{I_1{I}_k}(k=2,3,\dots ,L)$$

(31)

Considering label convention, the rigid body dynamics equation in oxyz can be expressed as

$$m{\stackrel{\mathrm{··}}{r}}_{I_1}-m{\tilde{r}}_{I_{1}C}{\stackrel{.}{\Omega }}_{I_1}+m{\tilde{\Omega }}_{I_1}{\tilde{\Omega }}_{I_1}{r}_{I_{1}C}=q_{I_1}-q_O+f_C+{\displaystyle \sum _{k=2}^L{q}_{I_k}}$$

(32)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{A}}_{I_1}{\tilde{\mathbf{\omega }}}_{I_1}{\mathbf{J}}_{I_1}{\mathbf{\omega }}_{I_1}+{\mathbf{A}}_{I_1}{\mathbf{J}}_{I_1}{\mathbf{A}}_{I_1}^T{\stackrel{·}{\mathbf{\Omega }}}_{I_1}+m{\tilde{\mathbf{r}}}_{I_{1}C}{\stackrel{··}{\mathbf{r}}}_{I_1}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{m}}_O-{\mathbf{m}}_{I_1}-{\tilde{\mathbf{r}}}_{I_{1}O}{\mathbf{q}}_O+{\mathbf{m}}_C+{\tilde{\mathbf{r}}}_{I_{1}C}{\mathbf{f}}_C\hfill \\ \multicolumn{1}{c}{}& +\sum _{k=2}^k{\tilde{\mathbf{r}}}_{I_1{I}_k}{\mathbf{q}}_{I_k}-\sum _{k=2}^L{\mathbf{m}}_{I_k}\hfill \end{array}$$

(33)

Rewrite equation (29) and dynamics equations of rigid body equations (32) and (33)

$${\stackrel{··}{\mathbf{r}}}_O={\stackrel{··}{\mathbf{r}}}_{I_1}+{\mathbf{E}}_1{\stackrel{·}{\mathbf{\Omega }}}_{I_1}+{\mathbf{E}}_2$$

(34)

$${\mathbf{q}}_O={\mathbf{q}}_{I_1}+{\mathbf{E}}_3{\stackrel{··}{\mathbf{r}}}_{I_1}+{\mathbf{E}}_4{\stackrel{·}{\mathbf{\Omega }}}_{I_1}+{\mathbf{E}}_5+\sum _{j=2}^L{\mathbf{q}}_{I_j}$$

(35)

$$m_O={\displaystyle \sum _{j=1}^L{m}_{I_j}}+{\displaystyle \sum _{j=2}^L{E}_{I_j}}{q}_{I_j}+E_6{q}_O+E_7{\stackrel{\mathrm{··}}{r}}_{I_1}+E_8{\stackrel{.}{\Omega }}_{I_1}+E_9$$

(36)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{E}}_1=-{\tilde{\mathbf{r}}}_{I_{1}O},{\mathbf{E}}_2={\tilde{\mathbf{\Omega }}}_{I_1}{\tilde{\mathbf{\Omega }}}_{I_1}{\mathbf{r}}_{I_{1}O},{\mathbf{E}}_3=-m{\mathbf{I}}_3\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_4=m{\tilde{\mathbf{r}}}_{I_{1}C},{\mathbf{E}}_5={\mathbf{f}}_C-m{\tilde{\mathbf{\Omega }}}_{I_1}{\tilde{\mathbf{\Omega }}}_{I_1}{\mathbf{r}}_{I_{1}C}\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_{I_j}=-{\tilde{\mathbf{r}}}_{I_1{I}_j},{\mathbf{E}}_6={\tilde{\mathbf{r}}}_{I_{1}O},{\mathbf{E}}_7=m{\tilde{\mathbf{r}}}_{I_{1}C}\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_8={\mathbf{A}}_{I_1}{\mathbf{J}}_{I_1}{\mathbf{A}}_{I_1}^T,{\mathbf{E}}_9=-{\mathbf{m}}_C-{\tilde{\mathbf{r}}}_{I_{1}C}{\mathbf{f}}_C+{\mathbf{A}}_{I_1}{\tilde{\mathbf{\omega }}}_{I_1}{\mathbf{J}}_{I_1}{\mathbf{\omega }}_{I_1}\hfill \end{array}$$

(37)

The state vector of rigid body outboard end is defined as equation (6), and the comprehensive state vector of input ends is defined as

$$z_{j,I_1-I_L}={[{\stackrel{··}{r}}_{I_1}^T{\stackrel{.}{\Omega }}_{I_1}^T{m}_{I_1}^T{q}_{I_1}^T{m}_{I_2}^T{q}_{I_2}^T\dots m_{I_L}^T{q}_{I_L}^{T}1]}^T$$

(38)

where the first subscript j of comprehensive state vector of the input end denotes the sequence number of outboard body of multiple hinges subsystem, the second subscript $I_1-I_L$ denotes the sequence numbers of all hinges associated with input ends of the rigid body, that is, $I_1,I_2,\dots ,I_L$ respectively, L is the sum of input ends of rigid body; the state vector contains absolute acceleration ${\stackrel{··}{\mathbf{r}}}_{I_1}$ , absolute angular acceleration ${\stackrel{·}{\mathbf{\Omega }}}_{I_1}$ , internal torque ${\mathbf{m}}_{I_1}$ , internal force ${\mathbf{q}}_{I_1}$ of the first input end $I_1$ of the rigid body and internal torques ${\mathbf{m}}_{I_2},\dots ,{\mathbf{m}}_{I_L}$ , internal forces ${\mathbf{q}}_{I_1},\dots ,{\mathbf{q}}_{I_L}$ of all the other input ends $I_1,\dots ,I_L$ .

According to the acceleration and internal force equations (35) and (36) and the definition equations (6) and (38) of state vector of output end and comprehensive state vector of input ends of rigid body, the transfer equation and the transfer matrix of spatial motion rigid body with multi-input ends and single-output end can be obtained

$${\mathbf{z}}_{j,O}={\mathbf{U}}_{j,I_1-I_L}{\mathbf{z}}_{j,I_1-I_L}$$

(39)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{U}}_{j,I_1-I_L}=\left[\begin{array}{cccccccccc}\hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_1\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill \cdots \hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{E}}_2\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_3}& \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill \cdots \hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_6{\mathbf{E}}_3+{\mathbf{E}}_7}& \hfill {\mathbf{E}}_6{\mathbf{E}}_4+{\mathbf{E}}_8\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_6\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_{I_2}+{\mathbf{E}}_6\hfill & \hfill \cdots \hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_{I_N}+{\mathbf{E}}_6\hfill & \hfill {\mathbf{E}}_6{\mathbf{E}}_5+{\mathbf{E}}_9\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_3}& \hfill {\mathbf{E}}_4\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill \cdots \hfill & \hfill {\mathbf{O}}_3\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{E}}_5\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 3}}& \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array}$$

(40)

Transfer equation of rigid body with planar large motion

Notice the definition equations (8)–(10) of the state vector of planar motion multibody system, similarly, the transfer equation and transfer matrix of planar motion rigid body j with single-input end and single-output end can be obtained

$${\mathbf{z}}_O={\mathbf{U}}_j{\mathbf{z}}_I$$

(41)

$${\mathbf{U}}_j=\left[\begin{array}{ccccc}\hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{E}}_1\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 2}\hfill & \hfill {\mathbf{E}}_2\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 2}}& \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_6{\mathbf{E}}_3+{\mathbf{E}}_7}& \hfill {\mathbf{E}}_6{\mathbf{E}}_4+{\mathbf{E}}_8\hfill & \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{E}}_6\hfill & \hfill {\mathbf{E}}_6{\mathbf{E}}_5+{\mathbf{E}}_9\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_3}& \hfill {\mathbf{E}}_4\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{E}}_5\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 2}}& \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill 1\hfill \end{array}\right]$$

(42)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{E}}_1=\left[\begin{array}{c}\hfill -y_{\mathrm{IO}}\hfill \\ \multicolumn{1}{c}{x_{\mathrm{IO}}}\end{array}\right],{\mathbf{E}}_2=\left[\begin{array}{c}\hfill -x_{\mathrm{IO}}{\Omega }_I^2\hfill \\ \multicolumn{1}{c}{-y_{\mathrm{IO}}{\Omega }_I^2}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_3=\left[\begin{array}{cc}\hfill -m\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill -m\hfill \end{array}\right],{\mathbf{E}}_4=\left[\begin{array}{c}\hfill m{y}_{\mathrm{IC}}\hfill \\ \multicolumn{1}{c}{-m{x}_{\mathrm{IC}}}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_5=\left[\begin{array}{c}\hfill m{x}_{\mathrm{IC}}{\Omega }_I^2+f_{x,C}\hfill \\ \multicolumn{1}{c}{m{y}_{\mathrm{IC}}{\Omega }_I^2+f_{y,C}}\end{array}\right],{\mathbf{E}}_6=\left[\begin{array}{cc}\hfill -y_{\mathrm{IO}}\hfill & \hfill x_{\mathrm{IO}}\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_7=\left[\begin{array}{cc}\hfill -m{y}_{\mathrm{IC}}\hfill & \hfill m{x}_{\mathrm{IC}}\hfill \end{array}\right],{\mathbf{E}}_8=\left[J_I\right]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_9=[-m_C+y_{\mathrm{IC}}{f}_{x,C}-x_{\mathrm{IC}}{f}_{y,C}]\hfill \end{array}$$

(43)

And, the transfer matrix of planar motion rigid body with multiple input ends and single-output end can be obtained

$${\mathbf{z}}_O={\mathbf{U}}_{j,I_1-I_L}{\mathbf{z}}_{j,I_1-I_L}$$

(44)

where ${\mathbf{z}}_O$ is defined by equation (6)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{z}}_{j,I_1-I_L}=\hfill \\ \multicolumn{1}{c}{}& {\left[\begin{array}{ccccc}\hfill {\stackrel{··}{\mathbf{r}}}_{I_1}^T\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}_{I_1}^T\hfill & \hfill {\mathbf{m}}_{I_1}^T\hfill & \hfill \begin{array}{cccccc}\hfill {\mathbf{q}}_{I_1}^T\hfill & \hfill {\mathbf{m}}_{I_2}^T\hfill & \hfill {\mathbf{q}}_{I_2}^T\hfill & \hfill \dots \hfill & \hfill {\mathbf{m}}_{I_L}^T\hfill & \hfill {\mathbf{q}}_{I_L}^T\hfill \end{array}\hfill & \hfill 1\hfill \end{array}\right]}^T\hfill \end{array}$$

(45)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{U}}_{j,I_1-I_L}=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{cccccccccc}\hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{E}}_1\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 2}\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 2}\hfill & \hfill \dots \hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 2}\hfill & \hfill {\mathbf{E}}_2\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 2}}& \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill \dots \hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_6{\mathbf{E}}_3+{\mathbf{E}}_7}& \hfill {\mathbf{E}}_6{\mathbf{E}}_4+{\mathbf{E}}_8\hfill & \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{E}}_6\hfill & \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{E}}_{I_2}+{\mathbf{E}}_6\hfill & \hfill \dots \hfill & \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{E}}_{I_N}+{\mathbf{E}}_6\hfill & \hfill {\mathbf{E}}_6{\mathbf{E}}_5+{\mathbf{E}}_9\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_3}& \hfill {\mathbf{E}}_4\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{I}}_2\hfill & \hfill \dots \hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{E}}_5\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 2}}& \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill \dots \hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array}$$

(46)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{E}}_1=\left[\begin{array}{c}\hfill -y_{I_{1}O}\hfill \\ \multicolumn{1}{c}{x_{I_{1}O}}\end{array}\right],{\mathbf{E}}_2=\left[\begin{array}{c}\hfill -x_{I_{1}O}{\Omega }_{I_1}^2\hfill \\ \multicolumn{1}{c}{-y_{I_{1}O}{\Omega }_{I_1}^2}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_3=\left[\begin{array}{cc}\hfill -m\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill -m\hfill \end{array}\right],{\mathbf{E}}_4=\left[\begin{array}{c}\hfill m{y}_{I_{1}C}\hfill \\ \multicolumn{1}{c}{-m{x}_{I_{1}C}}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_5=\left[\begin{array}{c}\hfill m{x}_{I_{1}C}{\Omega }_{I_1}^2+f_{x,C}\hfill \\ \multicolumn{1}{c}{m{y}_{I_{1}C}{\Omega }_{I_1}^2+f_{y,C}}\end{array}\right],{\mathbf{m}}_O=[m_O]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{m}}_{I_j}=[m_{I_j}],{\mathbf{E}}_{I_j}=[\begin{array}{cc}\hfill y_{I_1{I}_j}\hfill & \hfill -x_{I_1{I}_j}\hfill \end{array}]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_6=[\begin{array}{cc}\hfill -y_{I_{1}O}\hfill & \hfill x_{I_{1}O}\hfill \end{array}],{\mathbf{E}}_7=[\begin{array}{cc}\hfill -m{y}_{I_{1}C}\hfill & \hfill m{x}_{I_{1}C}\hfill \end{array}]\hfill \\ \multicolumn{1}{c}{}& {\mathbf{E}}_8=[J_{I_1}],{\mathbf{E}}_9=[-m_C+y_{I_{1}C}{f}_{x,C}-x_{I_{1}C}{f}_{y,C}]\hfill \end{array}$$

(47)

Transfer equation of smooth ball-and-socket hinge with spatial large motion

For smooth ball-and-socket hinge j with spatial motion inboard body $j-1$ and outboard body $j+1$ . Notice that, on two ends of the hinge, position coordinate, acceleration, internal force are equivalent and internal torque is equal to zero. It can be proved that the transfer equation and transfer matrix of smooth ball-and-socket hinge j, with outboard body whose outboard hinge is also smooth ball-and-socket hinge, are

$${\mathbf{z}}_{j+1,j}={\mathbf{U}}_j{\mathbf{z}}_{j-1,j}$$

(48)

$${\mathbf{U}}_j=\left[\begin{array}{ccccc}\hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,1}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,4}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,5}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 3}}& \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill 1\hfill \end{array}\right]$$

(49)

where ${\mathbf{u}}_{m,n}$ is the partitioned matrix of transfer matrix ${\mathbf{U}}_j$ of outboard body given in equation (23).

For the smooth ball-and-socket hinge whose outboard hinge is fictitious hinge (end free), the internal torque at two ends of its outboard body are both zero, thus the transfer equation and transfer matrix of such smooth ball-and-socket hinge are the same as equations (48) and (49).

Transfer equation of smooth pin with planar large motion

For smooth pin j with planar motion inboard body $j-1$ and outboard body $j+1$ , the transfer equation is

$${\mathbf{z}}_{j+1,j}={\mathbf{U}}_j{\mathbf{z}}_{j-1,j}$$

(50)

and the transfer matrix is

$${\mathbf{U}}_j=\left[\begin{array}{ccccc}\hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 2}\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill \\ \multicolumn{1}{c}{-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,1}}& \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,4}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,5}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 2}}& \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{I}}_1\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{2\times 2}}& \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill & \hfill {\mathbf{I}}_2\hfill & \hfill {\mathbf{O}}_{2\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 2}}& \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 1}\hfill & \hfill {\mathbf{O}}_{1\times 2}\hfill & \hfill 1\hfill \end{array}\right]$$

(51)

where ${\mathbf{u}}_{m,n}$ is the partitioned matrix of transfer matrix ${\mathbf{U}}_j$ of outboard body given in equation (42).

For the smooth pin whose outboard hinge is fictitious hinge (end free), the transfer equation and transfer matrix of such smooth pin are the same as equations (50) and (51).

Automatic deduction method of overall transfer equation for branch multibody system

In Rui et al., ²³ the automatic deduction theorem of overall transfer equation has been proposed, which is effective to any system for linear MSTMM and MSDTTMM, and is also effective to chain system and closed-loop system for new version of MSTMM. It is because that the smooth hinge can be treated as special elastic hinge with very small torsional stiffness for any topology structure, the transfer matrices of all elements along transfer path and direction exit. For new version of MSTMM, chain system and closed-loop system, the transfer matrices of all elements exit along the transfer path and transfer direction. The automatic deduction theorem of overall transfer equation for new version of MSTMM has been developed in this article so that effective to branch system.

Main transfer equation of multi-hinge subsystem with spatial motion

For the subsystem of multiple smooth ball-and-socket hinges with spatial motion or pin hinges with planar motion in the branch system as shown in Figure 1, due to the outboard body is the same rigid body with spatial motion as shown in Figure 4 in terms of thick arrow, it is treated as subsystem with multiple input ends and multiple output ends. Its transfer equation includes two parts which are the main transfer equation and geometric equations. The main transfer equation describes the relation among the acceleration, angular acceleration, internal torque, and internal force acting on element. While geometric equations describe the relation among the acceleration and angular acceleration between the first input end and the kth $(k=2,3)$ output end.

Figure 4.

Subsystem of multiple smooth ball-and-socket hinges with the same outboard body.

The definition of state vector of each input end in multi-hinge subsystem is

$${\mathbf{z}}_{i,I_k}=[\begin{array}{ccccc}\hfill {\stackrel{··}{\mathit{r}}}_{I_k}^T\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}_{I_k}^T\hfill & \hfill {\mathit{m}}_{I_k}^T\hfill & \hfill {\mathit{q}}_{I_k}^T\hfill & \hfill 1\hfill \end{array}{]}_j^T(k=1,2,\dots ,L)$$

(52)

The definition of the comprehensive state vector of all output ends in multi-hinge subsystem is the same as the comprehensive state vector of input ends of the rigid body with multi-input ends as follows

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{z}}_{j,I_1-I_3}=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{ccccccccc}\hfill {\stackrel{··}{\mathbf{r}}}_{I_1}^T\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}_{I_1}^T\hfill & \hfill {\mathbf{m}}_{I_1}^T\hfill & \hfill {\mathbf{q}}_{I_1}^T\hfill & \hfill {\mathbf{m}}_{I_2}^T\hfill & \hfill {\mathbf{q}}_{I_2}^T\hfill & \hfill {\mathbf{m}}_{I_3}^T\hfill & \hfill {\mathbf{q}}_{I_3}^T\hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array}$$

(53)

where the first subscript i and j of state vector denote, respectively, the sequence numbers of inboard body and outboard body of the hinge, while the second subscript $I_k$ is the sequence number of hinge, $I_1-I_L$ represents sequence numbers of all hinges $I_1,I_2,\dots ,I_L$ in subsystem with multi-input ends and multi-output ends, and L is the sum of hinges in the subsystem, whose state variables include the entire state vector of the first output end $I_1$ and internal torques and internal forces ${\mathbf{m}}_{I_2},\dots ,{\mathbf{m}}_{I_L}$ , ${\mathbf{q}}_{I_2},\dots ,{\mathbf{q}}_{I_L}$ of all other output ends $I_2,\dots ,I_L$ , respectively.

It can be proved that the main transfer equation describes the relationship among the comprehensive state vector of output ends and state vectors of each input end in the multi-hinge subsystem is

$${\mathbf{z}}_{j,I_1-I_L}={\hat{\mathbf{U}}}_{I_1}{\mathbf{z}}_{i_1,I_1}+{\hat{\mathbf{U}}}_{i_2,I_2}{\mathbf{z}}_{i_2,I_2}+{\hat{\mathbf{U}}}_{i_3,I_3}{\mathbf{z}}_{i_3,I_3}$$

(54)

where the coefficient matrix ${\hat{\mathbf{U}}}_{I_1}$ of the state vector ${\mathbf{z}}_{i_1,I_1}$ of the input end of the inboard smooth hinge of the first input end of multi-input body is an expanded transfer matrix of hinge, composed on the basis of the transfer matrix of the hinge whose outboard body is the body with single-input end and single-output end, and complemented by null matrices with n rows (n equals to the sum of internal torque and internal force of all other input ends except the first input end). The coefficient matrix ${\hat{\mathbf{U}}}_{i_k,I_k}$ of the state vector ${\mathbf{z}}_{i_k,I_k}(k=2,3)$ of the input end of the inboard smooth hinge of other input ends of multi-input body, is a force expanded extraction transfer matrix that extract internal torque and internal force from the state vector of the hinge, in which partitioned submatrices ${\mathbf{U}}_{2,3}$ and ${\mathbf{U}}_{2,4}$ are $-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,1+2k}$ and $-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,2+2k}$ , $(k=2,3)$ respectively.

Then, we deduce the coefficient matrix of state vector in the main transfer equation (54). In the multi-hinge subsystem with multi-input ends and multi-output ends with the same outboard body, accelerations, internal torques, and internal forces are equivalent at input and output ends for each hinge

$${\stackrel{··}{\mathbf{r}}}_{O_1}={\stackrel{··}{\mathbf{r}}}_{I_1}$$

(55)

$${\mathbf{m}}_{O_1}={\mathbf{m}}_{I_1}$$

(56)

$${\mathbf{q}}_{O_1}={\mathbf{q}}_{I_1}$$

(57)

$${\mathbf{m}}_{O_k}={\mathbf{m}}_{I_k}(k=2,3)$$

(58)

$${\mathbf{q}}_{O_k}={\mathbf{q}}_{I_k}(k=2,3)$$

(59)

If the output end of outboard body of the hinge is also connected with a smooth hinge, according to the transfer matrix of the outboard body, we have

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathbf{m}}_{j^{+},O}=}& {\mathbf{u}}_{3,1}{\stackrel{··}{\mathbf{r}}}_{j^{+},I_1}+{\mathbf{u}}_{3,2}{\stackrel{·}{\mathbf{\Omega }}}_{j^{+},I_1}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{u}}_{3,3}{\mathbf{m}}_{j^{+},I_1}+{\mathbf{u}}_{3,4}{\mathbf{q}}_{j^{+},I_1}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{u}}_{3,5}{\mathbf{m}}_{j^{+},I_2}+{\mathbf{u}}_{3,6}{\mathbf{q}}_{j^{+},I_2}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{u}}_{3,7}{\mathbf{m}}_{j^{+},I_3}+{\mathbf{u}}_{3,8}{\mathbf{q}}_{j^{+},I_3}+{\mathbf{u}}_{3,9}\hfill \end{array}$$

(60)

therefore

$$\begin{array}{cc}\multicolumn{1}{c}{0=}& {\mathbf{u}}_{3,1}{\stackrel{··}{\mathbf{r}}}_{j,O_1}+{\mathbf{u}}_{3,2}{\stackrel{·}{\mathbf{\Omega }}}_{j,O_1}+{\mathbf{u}}_{3,3}{\mathbf{m}}_{j,O_1}+{\mathbf{u}}_{3,4}{\mathbf{q}}_{j,O_1}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{u}}_{3,5}{\mathbf{m}}_{j,O_2}+{\mathbf{u}}_{3,6}{\mathbf{q}}_{j,O_2}+{\mathbf{u}}_{3,7}{\mathbf{m}}_{j,O_3}+{\mathbf{u}}_{3,8}{\mathbf{q}}_{j,O_3}+{\mathbf{u}}_{3,9}\hfill \\ \multicolumn{1}{c}{=}& {\mathbf{u}}_{3,1}{\stackrel{··}{\mathbf{r}}}_{j,I_1}+{\mathbf{u}}_{3,2}{\stackrel{·}{\mathbf{\Omega }}}_{j,I_1}+{\mathbf{u}}_{3,3}{\mathbf{m}}_{j,I_1}+{\mathbf{u}}_{3,4}{\mathbf{q}}_{j,I_1}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{u}}_{3,5}{\mathbf{m}}_{j,I_2}+{\mathbf{u}}_{3,6}{\mathbf{q}}_{j,I_2}+{\mathbf{u}}_{3,7}{\mathbf{m}}_{j,I_3}+{\mathbf{u}}_{3,8}{\mathbf{q}}_{j,I_3}+{\mathbf{u}}_{3,9}\hfill \end{array}$$

(61)

after transposing, one obtains

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\mathbf{\Omega }}}_{j,O_1}=}& -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,1}{\stackrel{··}{\mathbf{r}}}_{j,I_1}-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,3}{\mathbf{m}}_{j,I_1}\hfill \\ \multicolumn{1}{c}{}& -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,4}{\mathbf{q}}_{j,I_1}-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,9}\hfill \\ \multicolumn{1}{c}{}& -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,5}{\mathbf{m}}_{j,I_2}-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,6}{\mathbf{q}}_{j,I_2}\hfill \\ \multicolumn{1}{c}{}& -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,7}{\mathbf{m}}_{j,I_3}-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,8}{\mathbf{q}}_{j,I_3}\hfill \end{array}$$

(62)

expressing as the form of equation (54), one obtains

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\hat{\mathbf{U}}}_{I_1}=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{ccccc}\hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{-{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,1}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,4}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,9}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 3}}& \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array}$$

(63)

For multi-hinge subsystem with planar motion, the column quantities of null matrix and identity matrix in coefficient matrices ${\hat{\mathbf{U}}}_{I_1}$ , ${\hat{\mathbf{U}}}_{i_k,I_k}$ should be adjusted corresponding to the row number of the state vector which it multiplies

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\hat{\mathbf{U}}}_{i_2,I_2}=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{ccccc}\hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,5}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,6}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 3}}& \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill 0\hfill \end{array}\right]\hfill \end{array},$$

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\hat{\mathbf{U}}}_{i_3,I_3}=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{ccccc}\hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,7}\hfill & \hfill -{\mathbf{u}}_{3,2}^{-1}{\mathbf{u}}_{3,8}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{3\times 3}}& \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{O}}_{3\times 3}\hfill & \hfill {\mathbf{I}}_3\hfill & \hfill {\mathbf{O}}_{3\times 1}\hfill \\ \multicolumn{1}{c}{{\mathbf{O}}_{1\times 3}}& \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill {\mathbf{O}}_{1\times 3}\hfill & \hfill 0\hfill \end{array}\right]\hfill \end{array}$$

(64)

For the branch system shown as Figure 4, the coefficient matrix of state vector ${\mathbf{z}}_{i_1,I_1}({\mathbf{z}}_{4,7})$ is ${\hat{\mathbf{U}}}_{i_1}({\hat{\mathbf{U}}}_7)$ . The coefficient matrices of state vectors ${\mathbf{z}}_{i_2,I_2}$ and ${\mathbf{z}}_{i_3,I_3}$ ( ${\mathbf{z}}_{5,8}$ and ${\mathbf{z}}_{6,9}$ ) are ${\hat{\mathbf{U}}}_{i_2,I_2}$ and ${\hat{\mathbf{U}}}_{i_3,I_3}$ ( ${\hat{\mathbf{U}}}_{5,8}$ and ${\hat{\mathbf{U}}}_{6,9}$ ).

Geometric equation of spatial motion multi-hinge subsystem

The output ends of the multi-hinge subsystem are corresponding to the input ends of the outboard body. The relationship among the acceleration of the first input end and other input ends of the outboard rigid body satisfies acceleration composition theorem. The accelerations of input end and output end of individual smooth hinge are equivalent, so the relationship among acceleration of hinge corresponding to the first input end of outboard rigid body and the accelerations of input ends of other hinges satisfy acceleration composition theorem. The relationship among acceleration of hinge corresponding to the first input end of outboard rigid body and the accelerations of input ends of other hinges, is expressed by comprehensive state vector and state vectors of input ends of other hinges corresponding to non-first input end, complemented with the parts that torques are zero at input and output ends of smooth hinge, and is named as the geometric equation of multi-hinge subsystem

$${\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}{\mathbf{z}}_{i,I_k}={\mathbf{H}}_{I_1,I_k}{\mathbf{z}}_{j,I_1-I_L}(k=2,3)$$

(65)

where ${\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}$ is the acceleration and internal torque extraction matrix of the state vector, and ${\mathbf{H}}_{I_1,I_k}$ is the acceleration associated matrix of $I_1$ input end and $I_k(k=2,3,\dots )$ input end. The meaning of the state vector and its subscript has been elaborated below equations (52) and (53), especially notice that the first subscript i and j of the state vector denote the sequence number of the inboard body and outboard body of hinge, respectively; the second subscript $I_k$ is the hinge sequence number and $I_1-I_N$ express the sequence numbers of all hinges in multi-hinge subsystem.

Then solve extraction matrix ${\mathit{H}}_{\stackrel{··}{\mathit{r}},\mathit{m}}$ of acceleration and internal torque and associated matrix ${\mathbf{H}}_{I_1,I_k}$ of the state vector.

Each input end and the first output end in multi-hinge subsystem satisfy the following relationship

$${\stackrel{··}{\mathbf{r}}}_{I_k}={\stackrel{··}{\mathbf{r}}}_{O_1}+{\mathbf{E}}_{1,I_k}{\stackrel{·}{\mathbf{\Omega }}}_{O_1}+{\mathbf{E}}_{2,I_k}$$

(66)

$${\mathit{m}}_{I_k}=0$$

(67)

where

$${\mathit{E}}_{1,I_k}=-{\tilde{\mathit{r}}}_{I_1{I}_k},{\mathit{E}}_{2,I_k}={\tilde{\mathbf{\Omega }}}_{I_1}{\tilde{\mathbf{\Omega }}}_{I_1}{\mathit{r}}_{I_1{I}_k}(k=2,3,\dots )$$

(68)

Rewriting as the form of equation (65), one obtains

$${\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}=\left[\begin{array}{ccccc}\hfill \mathbf{I}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill \\ \multicolumn{1}{c}{\mathbf{O}}& \hfill \mathbf{O}\hfill & \hfill \mathbf{I}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill \end{array}\right]$$

(69)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{H}}_{I_1,I_k}=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{cccccccccc}\hfill \mathbf{I}\hfill & \hfill {\mathbf{E}}_{1,I_k}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \dots \hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill {\mathbf{E}}_{2,I_k}\hfill \\ \multicolumn{1}{c}{\mathbf{O}}& \hfill \mathbf{O}\hfill & \hfill \mathbf{I}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \dots \hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& (k=2,3,\dots )\hfill \end{array}$$

(70)

Transfer equation of planar motion multiple smooth pins subsystem with the same outboard body

Note that for planar motion multiple smooth pins subsystem with the same outboard body

$${\stackrel{·}{\mathbf{\Omega }}}_O=[{\stackrel{·}{\Omega }}_O],{\stackrel{·}{\mathbf{\Omega }}}_{I_1}=[{\stackrel{·}{\Omega }}_{I_1}]$$

(71)

Equation (68) becomes

$${\mathbf{E}}_{1,I_k}=\left[\begin{array}{c}\hfill -y_{I_1{I}_k}\hfill \\ \multicolumn{1}{c}{x_{I_1{I}_k}}\end{array}\right],{\mathbf{E}}_{2,I_k}=\left[\begin{array}{c}\hfill -x_{I_1{I}_k}{\Omega }_{I_1}^2\hfill \\ \multicolumn{1}{c}{-y_{I_1{I}_k}{\Omega }_{I_1}^2}\end{array}\right]$$

(72)

Then main transfer equation and its coefficient matrix of state vector, geometric equation, acceleration and internal torque extraction matrix, associated matrix of the comprehensive state vector corresponding to $I_k$ input end in subsystem composed by multiple planar motion smooth pins with multi-input ends and multi-output ends and the same outboard body, are equations (54), (63), (64), (65), (69), and (70), respectively.

Overall transfer matrix of branch multibody system

As shown in Figure 4, for the branch system where smooth hinges 7, 8, and 9 connect with the same outboard rigid body 10, the outboard body 10 is regarded as body with multi-input and single-output end, the transfer equation between comprehensive state vector of input ends and state vector of output end have been developed. Smooth hinges 7, 8, and 9 are regarded as a multi-input and multi-output subsystem, develop the main transfer equation and geometric equation among state vector of each input end and comprehensive state vector of output end; then the overall transfer matrix of branch system can be obtained.

For multi-hinge subsystem, the state vectors of each input end ${\mathbf{z}}_{4,7}$ , ${\mathbf{z}}_{5,8}$ , ${\mathbf{z}}_{6,9}$ and the state vector of output end ${\mathbf{z}}_{10,11}$ of body 10 are defined as equation (6). There are main transfer equations

$${\mathbf{z}}_{10,7-9}={\hat{\mathbf{U}}}_7{\mathbf{z}}_{4,7}+{\hat{\mathbf{U}}}_{5,8}{\mathbf{z}}_{5,8}+{\hat{\mathbf{U}}}_{6,9}{\mathbf{z}}_{6,9}$$

(73)

and geometric equations

$${\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}{\mathbf{z}}_{5,8}={\mathbf{H}}_{7,8}{\mathbf{z}}_{10,7-9},{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}{\mathbf{z}}_{6,9}={\mathbf{H}}_{7,9}{\mathbf{z}}_{10,7-9}$$

(74)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{z}}_{10,7-9}=\hfill \\ \multicolumn{1}{c}{}& {\left[\begin{array}{ccccccccc}\hfill {\stackrel{··}{\mathbf{r}}}_{10,7}\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}_{10,7}\hfill & \hfill {\mathbf{m}}_{10,7}\hfill & \hfill {\mathbf{q}}_{10,7}\hfill & \hfill {\mathbf{m}}_{10,8}\hfill & \hfill {\mathbf{q}}_{10,8}\hfill & \hfill {\mathbf{m}}_{10,9}\hfill & \hfill {\mathbf{q}}_{10,9}\hfill & \hfill 1\hfill \end{array}\right]}^T\hfill \end{array}$$

(75)

is the comprehensive state vector of output end of multi-hinge subsystem. The state vectors of all input ends of rigid body with multi-input ends and single-output end are uniformly expressed by a comprehensive state vector ${\mathbf{z}}_{10,7-9}$ , the first subscript 10 is the sequence number of body, the second one 7–9 indicates the sequence numbers 7, 8, 9 of all inboard hinges of body 10.

One can deduce the main transfer equation of rigid body 3 with multi-input ends and single-output end

$${\mathbf{z}}_{10,11}={\mathbf{U}}_{10,7-9}{\mathbf{z}}_{10,7-9}$$

(76)

where ${\mathbf{U}}_{10,7-9}$ is the main transfer matrix of rigid body 10 with multi-input and one output.

It is easy to obtain the transfer equations of other elements in Figure 4

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{z}}_{12,0}={\mathbf{U}}_{12,}{\mathbf{z}}_{12,11}={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{z}}_{10,11}\hfill \\ \multicolumn{1}{c}{}& {\mathbf{z}}_{4,7}={\mathbf{U}}_{4,}{\mathbf{z}}_{4,1}={\mathbf{U}}_4{\mathbf{U}}_1{\mathbf{z}}_{1,0}\hfill \\ \multicolumn{1}{c}{}& {\mathbf{z}}_{5,8}={\mathbf{U}}_5{\mathbf{z}}_{5,2}={\mathbf{U}}_5{\mathbf{U}}_2{\mathbf{z}}_{2,0}\hfill \\ \multicolumn{1}{c}{}& {\mathbf{z}}_{6,9}={\mathbf{U}}_6{\mathbf{z}}_{6,3}={\mathbf{U}}_6{\mathbf{U}}_3{\mathbf{z}}_{3,0}\hfill \end{array}$$

(77)

According to system dynamics model topology and the transfer equations of elements and subsystem equations (73), (76), and (77), the system main transfer equation is obtained

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathbf{z}}_{12,0}}& ={\mathbf{U}}_{12}{\mathbf{z}}_{12,11}={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{z}}_{10,11}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\mathbf{z}}_{10,7-9}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}({\hat{\mathbf{U}}}_7{\mathbf{z}}_{4,7}+{\hat{\mathbf{U}}}_{5,8}{\mathbf{z}}_{5,8}+{\hat{\mathbf{U}}}_{6,9}{\mathbf{z}}_{6,9})\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_7{\mathbf{z}}_{4,7}+{\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_{5,8}{\mathbf{z}}_{5,8}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_{6,9}{\mathbf{z}}_{6,9}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_7{\mathbf{U}}_4{\mathbf{U}}_1{\mathbf{z}}_{1,0}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_{5,8}{\mathbf{U}}_5{\mathbf{U}}_2{\mathbf{z}}_{2,0}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_{6,9}{\mathbf{U}}_6{\mathbf{U}}_3{\mathbf{z}}_{3,0}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{T}}_{1-12}{\mathbf{z}}_{1,0}+{\mathbf{T}}_{2-12}{\mathbf{z}}_{2,0}+{\mathbf{T}}_{3-12}{\mathbf{z}}_{3,0}\hfill \end{array}$$

(78)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{T}}_{1-12}={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_7{\mathbf{U}}_4{\mathbf{U}}_1\hfill \\ \multicolumn{1}{c}{}& {\mathbf{T}}_{2-12}={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_{5,8}{\mathbf{U}}_5{\mathbf{U}}_2\hfill \\ \multicolumn{1}{c}{}& {\mathbf{T}}_{3-12}={\mathbf{U}}_{12}{\mathbf{U}}_{11}{\mathbf{U}}_{10,7-9}{\hat{\mathbf{U}}}_{6,9}{\mathbf{U}}_6{\mathbf{U}}_3\hfill \end{array}$$

(79)

Express the state vector in geometric equation (equation (74)) with the state vector of system boundary ends

$$\begin{array}{cc}\multicolumn{1}{c}{0}& ={\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_7{\mathbf{z}}_{4,7}+({\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_{5,8}-{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}){\mathbf{z}}_{5,8}+{\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_{6,9}{\mathbf{z}}_{6,9}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_7{\mathbf{U}}_4{\mathbf{U}}_1{\mathbf{z}}_{1,0}+({\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_{5,8}-{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}){\mathbf{U}}_5{\mathbf{U}}_2{\mathbf{z}}_{2,0}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_{6,9}{\mathbf{U}}_6{\mathbf{U}}_3{\mathbf{z}}_{3,0}\hfill \end{array}$$

(80)

$$\begin{array}{cc}\multicolumn{1}{c}{0}& ={\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_7{\mathbf{z}}_{4,7}+{\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_{5,8}{\mathbf{z}}_{5,8}+({\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_{6,9}-{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}){\mathbf{z}}_{6,9}\hfill \\ \multicolumn{1}{c}{}& ={\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_7{\mathbf{U}}_4{\mathbf{U}}_1{\mathbf{z}}_{1,0}+{\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_{5,8}{\mathbf{U}}_5{\mathbf{U}}_2{\mathbf{z}}_{2,0}\hfill \\ \multicolumn{1}{c}{}& +({\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_{6,9}-{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}){\mathbf{U}}_6{\mathbf{U}}_3{\mathbf{z}}_{3,0}\hfill \end{array}$$

(81)

or write as

$${\mathit{G}}_{1-7,8}{\mathbf{z}}_{1,0}+{\mathbf{G}}_{2-7,8}{\mathbf{z}}_{11,0}+{\mathbf{G}}_{3-7,8}{\mathbf{z}}_{3,0}=0$$

(82)

$${\mathbf{G}}_{1-7,9}{\mathbf{z}}_{1,0}+{\mathbf{G}}_{2-7,9}{\mathbf{z}}_{11,0}+{\mathbf{G}}_{3-7,9}{\mathbf{z}}_{3,0}=0$$

(83)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{G}}_{1-7,8}={\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_7{\mathbf{U}}_4{\mathbf{U}}_1\hfill \\ \multicolumn{1}{c}{}& {\mathbf{G}}_{2-7,8}=({\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_{5,8}-{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}){\mathbf{U}}_5{\mathbf{U}}_2\hfill \\ \multicolumn{1}{c}{}& {\mathbf{G}}_{3-7,8}={\mathbf{H}}_{7,8}{\hat{\mathbf{U}}}_{6,9}{\mathbf{U}}_6{\mathbf{U}}_3\hfill \\ \multicolumn{1}{c}{}& {\mathbf{G}}_{1-7,9}={\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_7{\mathbf{U}}_4{\mathbf{U}}_1\hfill \\ \multicolumn{1}{c}{}& {\mathbf{G}}_{2-7,9}={\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_{5,8}{\mathbf{U}}_5{\mathbf{U}}_2\hfill \\ \multicolumn{1}{c}{}& {\mathbf{G}}_{3-7,9}=({\mathbf{H}}_{7,9}{\hat{\mathbf{U}}}_{6,9}-{\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}){\mathbf{U}}_6{\mathbf{U}}_3\hfill \end{array}$$

(84)

The system overall transfer equation is composed by system main transfer equation (78) and the geometric equations (equations (82) and (83)) of all multi-hinge subsystems with the same outboard body

$$\left[\begin{array}{cccc}\hfill -\mathbf{I}\hfill & \hfill {\mathbf{T}}_{1-12}\hfill & \hfill {\mathbf{T}}_{2-12}\hfill & \hfill {\mathbf{T}}_{3-12}\hfill \\ \multicolumn{1}{c}{\mathbf{O}}& \hfill {\mathbf{G}}_{1-7,8}\hfill & \hfill {\mathbf{G}}_{2-7,8}\hfill & \hfill {\mathbf{G}}_{3-7,8}\hfill \\ \multicolumn{1}{c}{\mathbf{O}}& \hfill {\mathbf{G}}_{1-7,9}\hfill & \hfill {\mathbf{G}}_{2-7,9}\hfill & \hfill {\mathbf{G}}_{3-7,9}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\mathbf{z}}_{12,0}\hfill \\ \multicolumn{1}{c}{{\mathbf{z}}_{1,0}}\\ \multicolumn{1}{c}{{\mathbf{z}}_{2,0}}\\ \multicolumn{1}{c}{{\mathbf{z}}_{3,0}}\end{array}\right]=0$$

(85)

or write as

$${\mathit{U}}_{\mathrm{all}}{\mathit{z}}_{\mathrm{all}}=\mathit{0}$$

(86)

where

$${\mathbf{z}}_{\mathrm{all}}={\left[\begin{array}{cccc}\hfill {\mathbf{z}}_{12,0}^T\hfill & \hfill {\mathbf{z}}_{1,0}^T\hfill & \hfill {\mathbf{z}}_{2,0}^T\hfill & \hfill {\mathbf{z}}_{3,0}^T\hfill \end{array}\right]}^T$$

(87)

$${\mathbf{U}}_{\mathrm{all}}=\left[\begin{array}{cccc}\hfill -\mathbf{I}\hfill & \hfill {\mathbf{T}}_{1-12}\hfill & \hfill {\mathbf{T}}_{2-12}\hfill & \hfill {\mathbf{T}}_{3-12}\hfill \\ \multicolumn{1}{c}{\mathbf{O}}& \hfill {\mathbf{G}}_{1-7,8}\hfill & \hfill {\mathbf{G}}_{2-7,8}\hfill & \hfill {\mathbf{G}}_{3-7,8}\hfill \\ \multicolumn{1}{c}{\mathbf{O}}& \hfill {\mathbf{G}}_{1-7,9}\hfill & \hfill {\mathbf{G}}_{2-7,9}\hfill & \hfill {\mathbf{G}}_{3-7,9}\hfill \end{array}\right]$$

(88)

Rewrite equations (87) and (88) as the following form

$${\mathit{z}}_{\mathrm{all}}={\left[\begin{array}{cc}\hfill {\mathit{z}}_{\mathrm{root}}^T\hfill & \hfill {\mathit{z}}_{\mathrm{tip}}^T\hfill \end{array}\right]}^T$$

(89)

$${\mathit{U}}_{\mathrm{all}}=\left[\begin{array}{cc}\hfill -\mathit{I}\hfill & \hfill \mathit{T}\hfill \\ \multicolumn{1}{c}{\mathit{O}}& \hfill \mathit{G}\hfill \end{array}\right]$$

(90)

where

$${\mathbf{z}}_{\mathrm{root}}={\mathbf{z}}_{12,0}$$

(91)

$${\mathbf{z}}_{\mathrm{tip}}={\left[\begin{array}{ccc}\hfill {\mathbf{z}}_{1,0}^T\hfill & \hfill {\mathbf{z}}_{2,0}^T\hfill & \hfill {\mathbf{z}}_{3,0}^T\hfill \end{array}\right]}^T$$

(92)

$$\mathbf{T}={\left[\begin{array}{ccc}\hfill {\mathbf{T}}_{1-12}\hfill & \hfill {\mathbf{T}}_{2-12}\hfill & \hfill {\mathbf{T}}_{3-12}\hfill \end{array}\right]}^T$$

(93)

$$\mathbf{G}=\left[\begin{array}{ccc}\hfill {\mathbf{G}}_{1-7,8}\hfill & \hfill {\mathbf{G}}_{2-7,8}\hfill & \hfill {\mathbf{G}}_{3-7,8}\hfill \\ \multicolumn{1}{c}{{\mathbf{G}}_{1-7,9}}& \hfill {\mathbf{G}}_{2-7,9}\hfill & \hfill {\mathbf{G}}_{3-7,9}\hfill \end{array}\right]$$

(94)

Define the state vector ${\mathit{z}}_{\mathrm{all}}$ of all boundary ends as the system boundary state vector, such as equation (89); the coefficient matrix ${\mathit{U}}_{\mathrm{all}}$ of the system boundary state vector as the system overall transfer matrix, such as equations (88) or (90); the coefficient matrix $\mathit{T}$ of tip state vector as the coefficient matrix of main transfer equation, such as equation (93); the coefficient matrix $\mathit{G}$ of tip state vector as the geometric equation coefficient matrix, such as equation (94). There is one body with four ends in the branch system as shown in Figure 4, the sum of ends that greater than two counts as 2, so there are two geometric equations.

Automatic deduction theorem of overall transfer equation of branch multibody system

From equation (86), the following characteristics of overall transfer equation for new version of MSTMM can be obtained, which form the automatic deduction theorem of overall transfer equation for branch multibody system:

1. The state vector involved in the overall transfer equation is system boundary state vector ${\mathit{z}}_{\mathrm{all}}$ . The coefficient matrix of system boundary state vector ${\mathit{z}}_{\mathrm{all}}$ is system overall transfer matrix ${\mathit{U}}_{\mathrm{all}}$ .

2. The overall transfer equation is composed of main transfer equation and geometric equations. In the main transfer equation, the coefficient matrix of the system boundary state vector corresponds to the first row of the overall transfer matrix; the coefficient matrix of root state vector is minus identity matrix. The structure of the coefficient matrix T of the tip state vector such as equation (93) obeys following laws: the coefficient matrix from tip, through the first input end of body with multi-input ends, to root (such as ${\mathbf{T}}_{1-12}$ ) is the successive premultiplication of the transfer matrices of all elements along the transfer path, where the involved transfer matrix of inboard smooth hinge connecting the first input end of the body with multi-input ends and one output end is the expanded transfer matrix of this hinge (such as ${\hat{\mathbf{U}}}_7$ ); the transfer matrix of the body with multi-input ends is the transfer matrix of the body in the situation with multi-input ends and single-output end (such as ${\mathbf{U}}_{10,7-9}$ , the following is the same); the coefficient matrix from tip, through the non-first input end, to root is the successive premultiplication of transfer matrices of all elements along the transfer path (such as ${\mathbf{T}}_{2-12}$ , ${\mathbf{T}}_{3-12}$ ), where involved transfer matrix of input end of multi-hinge subsystem is the force expanding extraction matrix of non-first input ends (such as ${\hat{\mathbf{U}}}_{5,8}$ and ${\hat{\mathbf{U}}}_{6,9}$ ). The quantity of geometric equations is equal to the sum of ends greater than two in each element in the system.

The structure of coefficient matrix of tip state vector in geometric equations of branch system such as equations (85) and (86) obeys the following rules: the coefficient matrix from tip to the first input end of body with multi-input ends (such as ${\mathbf{G}}_{1-7,8}$ and ${\mathbf{G}}_{1-7,9}$ ) is the successive premultiplication of transfer matrices of all elements along the transfer path, and then premultiplied by associated matrix (such as ${\mathbf{H}}_{7,8}$ and ${\mathbf{H}}_{7,9}$ ), where involved transfer matrix of inboard smooth hinge connecting the first input end of body with multi-input ends is the expanded transfer matrix of this hinge (such as ${\hat{\mathbf{U}}}_7$ ); the coefficient matrix from tip to non-first input ends of body with multi-input ends (such as ${\mathbf{G}}_{3-7,8}$ and ${\mathbf{G}}_{2-7,9}$ ) is the successive premultiplication of transfer matrices of all elements along the transfer path, and then premultiplied by the acceleration associated matrix among the I₁th input end and the I_kth $(k=2,3,\dots )$ input end (such as ${\mathbf{H}}_{7,8}$ and ${\mathbf{H}}_{7,9}$ ), where involved transfer matrix of hinge element in multi-hinge subsystem is the force expanded extraction matrix of input ends of the hinge (such as ${\hat{\mathbf{U}}}_{5,8}$ , ${\hat{\mathbf{U}}}_{6,9}$ ); the coefficient matrix from tip to non-first input ends of body with multi-input ends (such as ${\mathbf{G}}_{2-7,8}$ and ${\mathbf{G}}_{3-7,9}$ ), where the force extraction matrix and acceleration associated matrix correspond to the same subscript of hinge (such as ${\mathbf{H}}_{7,8}$ and ${\hat{\mathbf{U}}}_{5,8}$ , ${\mathbf{H}}_{7,9}$ and ${\hat{\mathbf{U}}}_{6,9}$ ), is the successive premultiplication of transfer matrices of all elements in the transfer path, and then premultiplied by associated matrix, however, the acceleration and internal torque extraction matrix ${\mathbf{H}}_{\stackrel{··}{\mathbf{r}},\mathbf{m}}$ should be subtracted.

• 3. For chain system and closed-loop system, automatic deduction theorem of overall transfer equation and the form of overall transfer matrix are the same for new version of MSTMM and for MSTMM.

Above-mentioned automatic deduction theorem is effective for various multi-rigid-body system, chain multi-rigid-flexible-body system, closed-loop multi-rigid-flexible-body system, and tree multi-rigid-flexible-body system that the multi-end bodies in the system are rigid bodies.

Algorithms for dynamics analysis

The algorithm using the proposed method for dynamics analysis is shown as follows:

1. Decide the initial condition, system parameters and boundary conditions of the multibody system.

2. Set $i=1$ .

3. Knowing the initial conditions such as position coordinates, velocities, rotation angles, and angular velocities $\mathbf{r}(t_{i-1})$ , $\stackrel{·}{\mathbf{r}}(t_{i-1})$ , $\mathbf{\theta }(t_{i-1})$ , and $\mathbf{\Omega }(t_{i-1})$ and system parameters at $t_{i-1}$ , formulate the transfer matrices of each element and subsystem, and the system overall transfer matrix obtained by the automatic deduction theorem presented in this article.

4. Apply system boundary conditions and overall transfer equation, compute unknown state variables in every boundary state vector ${\mathbf{z}}_{n,0}$ , ${\mathbf{z}}_{j,0}(j=1,2,\dots )$ , which is the function of element of the overall transfer matrix.

5. After the entire state vectors in boundary have been solved, state vector of each element can be obtained by solving the element transfer equation successively.

6. Apply initial conditions $\mathbf{r}(t_{i-1})$ , $\stackrel{·}{\mathbf{r}}(t_{i-1})$ , $\mathbf{\theta }(t_{i-1})$ , and $\mathbf{\Omega }(t_{i-1})$ , accompany with the computed valued of acceleration $\stackrel{··}{\mathbf{r}}(t_{i-1})$ and angular acceleration $\stackrel{·}{\mathbf{\Omega }}(t_{i-1})$ at $t_{i-1}$ , compute position coordinates, velocities, rotation angles, angular velocities $\mathbf{r}(t_i)$ , $\stackrel{·}{\mathbf{r}}(t_i)$ , $\mathbf{\theta }(t_i)$ , and $\mathbf{\Omega }(t_i)$ at $t_i$ , with Runge–Kutta method or other numerical integrating methods.

7. Let $i=i+1$ , use the computed values of the last step $\mathbf{r}(t_i)$ , $\stackrel{·}{\mathbf{r}}(t_i)$ , $\mathbf{\theta }(t_i)$ , and $\mathbf{\Omega }(t_i)$ at time $t_i$ as initial condition and return step (4).

8. Repeat above steps, until the time required for complete analysis.

Numerical examples

As shown in Figure 4, body 12 of the branch system is hang on ceiling by smooth ball-and-socket hinge, in other words, the boundary of body 12 is simply support. Boundaries of body 4, 5, and 6 are free. That is

$${\mathbf{z}}_{12,0}=[\begin{array}{ccccc}\hfill 0^T\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}^T\hfill & \hfill 0^T\hfill & \hfill {\mathbf{q}}^T\hfill & \hfill 1\hfill \end{array}{]}_{12,0}^T$$

$${\mathbf{z}}_{k,0}=[\begin{array}{ccccc}\hfill {\stackrel{··}{\mathbf{r}}}^T\hfill & \hfill {\stackrel{·}{\mathbf{\Omega }}}^T\hfill & \hfill 0^T\hfill & \hfill 0^T\hfill & \hfill 1\hfill \end{array}{]}_{k,0}^T,k=4,5,6$$

There are system initial conditions: the initial rotation angles and angular velocities of rigid body 12, 10, 4, 5, 6

$${\theta }_{12}=\pi /3\mathrm{rad},{\theta }_{10}={\theta }_4={\theta }_5={\theta }_6=0\mathrm{rad}$$

$${\stackrel{·}{\theta }}_{12}={\stackrel{·}{\theta }}_{10}={\stackrel{·}{\theta }}_4={\stackrel{·}{\theta }}_5={\stackrel{·}{\theta }}_6=0\mathrm{rad}{\mathrm{s}}^{-1}$$

solve the system motion under gravity.

Set axis y is opposite with the gravity direction. In the figure, hinge 11, 7, 8, and 9 are all smooth ball-and-socket hinges; body 12, 4, 5, and 6 are rigid bodies with single-input end and single-output end and the same structure parameters, rigid body mass $m_{12}=m_4=m_5=m_6=1.0\mathrm{kg}$ , the moment of inertia with respect to each input end of rigid body $J_{12}=J_4=J_5=J_6=5.0\mathrm{kg}{\mathrm{m}}^2$ , the position coordinates of mass center and input end in body-fixed coordinate system are ${\mathbf{L}}_{\mathrm{IC}}=[\begin{array}{cc}\hfill 0\hfill & \hfill 0.5\hfill \end{array}{]}^T\mathrm{m}$ and ${\mathbf{L}}_{\mathrm{IO}}=[\begin{array}{cc}\hfill 0\hfill & \hfill 1.0\hfill \end{array}{]}^T\mathrm{m}$ respectively. Body 10 is a rigid body with three input ends and one output end, mass $m_{10}=1.0\mathrm{kg}$ , the moment of inertia with respect to the first input end $P_{10,7}$ is $J_{I_1}=10.0\mathrm{kg}{\mathrm{m}}^2$ , the coordinates of mass center, output ends, the second input end and the third input end in body-fixed coordinate system of the first input end are ${\mathbf{L}}_{I_{1}C}=[\begin{array}{cc}\hfill 1.5\hfill & \hfill 1.5\hfill \end{array}{]}^T\mathrm{m}$ , ${\mathbf{L}}_{I_{1}O}=[\begin{array}{cc}\hfill 1.5\hfill & \hfill 3.0\hfill \end{array}{]}^T\mathrm{m}$ , ${\mathbf{L}}_{I_1{I}_2}=[\begin{array}{cc}\hfill 1.5\hfill & \hfill 0.0\hfill \end{array}{]}^T\mathrm{m}$ , and ${\mathbf{L}}_{I_1{I}_3}=[\begin{array}{cc}\hfill 3.0\hfill & \hfill 0.0\hfill \end{array}{]}^T\mathrm{m}$ , respectively. Gravity acceleration $g=9.8\mathrm{m}{\mathrm{s}}^{-2}$ .

The overall transfer equation, state vector, and overall transfer matrices of the system are shown as equations (86)–(88), respectively. The transfer equations and transfer matrices of rigid body with single-input end and single-output end, of smooth hinge with single-input end and single-output end, of rigid body with three input ends and single-output end, and of three-smooth ball-and-socket hinges subsystem are shown as equations (22), (23), (45), (46), (50), (51), (54), (63), and (64), respectively. The computational results of time history of rotation angles of the five rigid bodies that obtained by new version of MSTMM with automatic deduction theorem of overall transfer equation and automatic dynamic analysis of mechanical systems (ADAMS) are shown in Figure 5. The results obtained by the two methods have good agreements, which validate the automatic deduction theorem of overall transfer equation for branch multibody system.

Figure 5.

Rotation angle of each rigid body obtained by new version of MSTMM and ADAMS.

The computational speed is very important for multibody system with large degrees of freedom (DOF). Herein, a new system with a chain subsystem composed by the same 245 rigid bodies and hinge as body 4, 5, 6, and 12 in the above numeric example is connected with body 12. The initial rotation angles and angular velocities of all the new rigid bodies are the same

$${\theta }_k=0\mathrm{rad},{\stackrel{·}{\theta }}_k=0\mathrm{rad}{\mathrm{s}}^{-1},(k=14,16,\dots ,502)$$

The DOF of the system is 750. The dynamics of the system is computed respectively with both the proposed method and ADAMS.

The computational results of time history of rotation angles of the four boundary rigid bodies as well as the body with multi-input ends obtained by new version of MSTMM with automatic deduction theorem of overall transfer equation and by ADAMS are shown in Figure 6, while the computational time is shown in Table 1.

Figure 6.

Rotation angles of tip and root rigid body obtained by new version of MSTMM and ADAMS.

Table 1.

Computational time comparison.

Method	New version of MSTMM	ADAMS
Time cost/s	70.19	193

MSTMM: transfer matrix method for multibody system; ADAMS: automatic dynamic analysis of mechanical systems.

The time measurement of the proposed method and ADAMS are made with the same CPU environment on Windows. Both the results are picked as average of 10 times of computations, in order to reduce the effect of background tasks. The settings of the solver are shown in Figure 7.

Figure 7.

Solver settings.

The results obtained by the two methods have good agreements, which validate the automatic deduction theorem of overall transfer equation for branch multibody system. The computational speed of new version of MSTMM is much higher than ADAMS.

Conclusion

On the research work basis of new version of MSTMM, ²⁴ the principle equation of new version of MSTMM has been elaborated and expanded in this article. According to the foundational principle, transfer equation and transfer matrix of element, overall transfer equation, and overall transfer matrix of system, in new version of MSTMM, are accurate and boundary conditions satisfied; it has been elaborated that new version of MSTMM, in principle, belongs to accurate analysis method of MSD. Proceed from the deduction of the basic transfer equations and transfer matrices of rigid body with multiple input ends, multi-hinge subsystem with multiple input ends and multiple output ends, the concept of topology graph of system dynamics model has been expanded, the automatic deduction theorem of overall transfer equation for branch system has been formed for new version of MSTMM, by which the sphere of application of the automatic deduction theorem ²³ of overall transfer equation for branch multibody system has been expanded from linear MSTMM, MSDTTMM to new version of MSTMM.

The computational results of the multibody system dynamics obtained by both automatic deduction theorem of overall transfer equation for branch system and ordinary multibody system dynamics method have good agreements, which validate the automatic deduction theorem of overall transfer equation for new version of MSTMM.