Study on automatic deduction method of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems

Abstract

The transfer matrix method for multibody systems has been developed for 20 years and improved constantly. The new version of transfer matrix method for multibody system and the automatic deduction method of overall transfer equation presented in recent years make it more convenient of the method for engineering application. In this article, by first defining branch subsystem, any complex multibody system may be regarded as the assembling of branch subsystems and simple chain subsystems. If there are closed loops in the system, the loops should be “cut off,” thus a pair of “new boundaries” are generated at each “cutting-off” point. The relationship between the state vectors of the pair of “new boundaries” may be described by a supplement equation. Based on above work, the automatic deduction method of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems is formed. The results of numerical examples obtained by the automatic deduction method and ADAMS software for tree system dynamics as well as mixed system dynamics have good agreements, which validate the features of proposed method such as high computational speed, more effective for complex systems, no need of the system global dynamics equation, highly programmable, as well as convenient popularization and application in engineering.

Keywords

New version of transfer matrix method for multibody systems branch subsystem closed-loop subsystem tree system mixed system automatic deduction method of overall transfer equation

Introduction

Multibody system dynamics methods (MSDMs) that have been developed since 50 years ago^1–7 provide effective methods for dynamics analysis of modern mechanical systems. Consequently, the developments of ships, aircraft, aerospace, traffics, and general machinery industry have been promoted greatly. The main study object of MSDM is the system composed of multiple bodies with relative movements. Establishment of the global dynamics equation of system and its numerical computation are the most important tasks of ordinary MSDM. Meanwhile, improving the speed, precision, and stability of the numerical computation are the key study contents.

With the development and demand of engineering technology, mechanics researchers worldwide are making their effort to present and improve various MSDM creatively. Despite various ordinary MSDMs have different styles, they have two identical features: (1) it is necessary to establish the system global dynamics equation, which is the most brilliant but the most difficult part for each MSDM. (2) For a complex system, the system matrix order of the global dynamics equation is very high. Therefore, it is one of the most important study directions to find effective ways to reduce the order of the system matrix for overcoming the computational difficulty.

For improving the computational speed and simplifying the study procedure of multibody system dynamics (MSD), a lot of researchers have been devoted in developing kinds of algorithms based on classical MSDM. From 1980s to 1990s was the period of rapid development of MSDM. The algorithms proposed by Featherstone,⁸ Brandl et al.,⁹ Rodriguez et al.,¹⁰ Saha¹¹ are all efficient for the number of operations and the storage requirements of these algorithms are proportional to the degrees of freedom of multibody system. The common features of these algorithms are (1) no need to establish the global system dynamics equation and (2) usage of recursive thoughts. According to Featherstone,⁸ Brandl et al.⁹ and Saha,¹¹ the relationship of computational complexities of these algorithms is Saha¹¹ > Featherstone⁸ > Brandl et al.,⁹ which shows that the algorithm by Brandl is the most efficient. Even though the algorithm by Saha is worse than that by Featherstone⁸ in terms of the complexities, it provides a simpler algorithm, which is easier to implement. Transfer matrix method for multibody system (MSTMM) was developed for the exact target with above algorithms. Rui and colleagues^12–19 have put forward MSTMM and tried to perfect it constantly since 20 years ago. The MSTMM has been applied widely in engineering for following aspects: the study procedure of MSD is simplified because of no need to establish the system global dynamics equation; the computational speed is increased greatly because the system matrix order is always low, and it is highly programmable. The new version of MSTMM¹⁶ was presented in 2013, of which the computational complexity is in the order $n$ ¹⁷. Besides, through Riccati transformation, the new version of MSTMM becomes a little more efficient than the algorithm of Brandl et al.⁹, as is evident from Figure 8 in Rui et al.¹⁷ and Table 2 in this paper. Besides, comparing to algorithms in Featherstone,⁸ Brandl et al.⁹, Rodriguez et al.¹⁰ and Saha¹¹, the expression of new version of MSTMM is the most concise and evident since the kinematics and dynamics information are contained in the transfer matrix of each element, different from others that these information are processed separately makes the computational process tedious. Moreover, the consistency in the form of transfer matrices of elements will make it easier to solve rigid-flexible coupling system dynamics¹⁸ with same state vectors. The automatic deduction theorem of overall transfer equation of chain systems and closed-loop systems¹⁹ for the new version of MSTMM was presented in 2014. Subsequently, this theorem was extended for branch systems²⁰ in 2015.

The salient contributions of this article are as follows:

The branch subsystem is first defined.

The general formulae of transfer matrix and transfer equation for branch subsystems are derived, which simplifies the deduction process and the form of overall transfer equation for various complex multibody systems greatly. Thus, it becomes convenient to realize the automatic deduction of overall transfer equation for multibody systems with various complex topology graphs.

By “cutting off” the closed-loop subsystem, any closed-loop-and-branch-mixed systems may be regarded as a tree system with “new boundaries” generated at each “cutting-off” point.

The automatic deduction method of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems is formed, which would provide theoretical support for the software development based on the new version of MSTMM.

Fundamentals of the new version of MSTMM

The basic idea of the new version of MSTMM is “to break up the whole into parts,” that is, to separate a complex multibody system into several body elements and hinge elements whose dynamics characteristics may be expressed by matrices. The transfer equations and transfer matrices of elements are deduced by their dynamics equations strictly. Building up the transfer equation and transfer matrix library of all kinds of elements in advance, the overall transfer matrix and the overall transfer equation of the system may be obtained by assembling them according to topology graph of the system easily. Next, boundary conditions and initial conditions may be utilized to solve the transfer equations of system and all elements, and then the state vectors on the boundaries and elements at present moment may be obtained. The displacement, velocity, rotation angle, angular velocity, and other system motion characteristics at next moment may be determined by numerical integration methods. Thereby, the dynamics characteristics of the system may be gained. Especially for a chain system, the overall transfer matrix can be obtained by successively multiplying transfer matrices of all elements of the system.^12,16

Coordinate system and direction cosine matrix

As shown in Figure 1, the motion of a system is described in the global inertial coordinate system $oxyz$ , which is the Cartesian coordinate system. However, the structure parameters of elements are described in the body-fixed coordinate system $I ξ η ς$ . $I$ is the input of an element with single input while for an element with multiple inputs, the origin is located on the first input $I_{1}$ .

Figure 1.

Global inertial coordinate system $oxyz$ and body-fixed coordinate system $I ξ η ς$ .

It has been proved⁵ that the direction cosine matrix which transforms vectors in the body-fixed coordinates into the inertial coordinates is

A = [\begin{matrix} c_{y} c_{z} & s_{x} s_{y} c_{z} - c_{x} s_{z} & c_{x} s_{y} c_{z} + s_{x} s_{z} \\ c_{y} s_{z} & s_{x} s_{y} s_{z} + c_{x} c_{z} & c_{x} s_{y} s_{z} - s_{x} c_{z} \\ - s_{y} & s_{x} c_{y} & c_{x} c_{y} \end{matrix}]

(1)

where $c_{x} = \cos θ_{x}$ , $c_{y} = \cos θ_{y}$ , $c_{z} = \cos θ_{z}$ , $s_{x} = \sin θ_{x}$ , $s_{y} = \sin θ_{y}$ , $s_{z} = \sin θ_{z}$ . Space-three-axis angles⁵ $θ_{x}$ , $θ_{y}$ , and $θ_{z}$ describe the orientation of the body-fixed coordinate system relative to the global inertial coordinate system.

The relationships among the absolute angular velocity $ω$ of the rigid body in the body-fixed coordinates, the absolute angular velocity $Ω$ in the inertial coordinates, the derivative of space-three-axis angles ${\overset{\cdot}{θ}}_{x}$ , ${\overset{\cdot}{θ}}_{y}$ , ${\overset{\cdot}{θ}}_{z}$ , and their derivatives are given as²⁰

[\begin{matrix} ω_{ξ} \\ ω_{η} \\ ω_{ς} \end{matrix}] = H [\begin{matrix} {\overset{\cdot}{θ}}_{x} \\ {\overset{\cdot}{θ}}_{y} \\ {\overset{\cdot}{θ}}_{z} \end{matrix}]

(2a)

[\begin{matrix} {\overset{\cdot}{ω}}_{ξ} \\ {\overset{\cdot}{ω}}_{η} \\ {\overset{\cdot}{ω}}_{ς} \end{matrix}] = \overset{\cdot}{H} [\begin{matrix} {\overset{\cdot}{θ}}_{x} \\ {\overset{\cdot}{θ}}_{y} \\ {\overset{\cdot}{θ}}_{z} \end{matrix}] + H [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{x} \\ {\overset{\cdot\cdot}{θ}}_{y} \\ {\overset{\cdot\cdot}{θ}}_{z} \end{matrix}]

(2b)

and

ω = A^{T} Ω

(3)

where

H = [\begin{matrix} 1 & 0 & - s_{y} \\ 0 & c_{x} & s_{x} c_{y} \\ 0 & - s_{x} & c_{x} c_{y} \end{matrix}], H = [\begin{matrix} 1 & 0 & - s_{y} \\ 0 & c_{x} & s_{x} c_{y} \\ 0 & - s_{x} & c_{x} c_{y} \end{matrix}]

(4)

Substituting the derivative of equation (3) into equation (2b) yields

[\begin{matrix} {\overset{\cdot\cdot}{θ}}_{x} \\ {\overset{\cdot\cdot}{θ}}_{y} \\ {\overset{\cdot\cdot}{θ}}_{z} \end{matrix}] = H^{- 1} A^{T} [\begin{matrix} {\overset{\cdot}{Ω}}_{x} \\ {\overset{\cdot}{Ω}}_{y} \\ {\overset{\cdot}{Ω}}_{z} \end{matrix}] - H^{- 1} \overset{\cdot}{H} [\begin{matrix} {\overset{\cdot}{θ}}_{x} \\ {\overset{\cdot}{θ}}_{y} \\ {\overset{\cdot}{θ}}_{z} \end{matrix}]

(5)

As long as the initial condition $[\begin{matrix} Θ_{x} & Θ_{y} & Θ_{z} \end{matrix}]^{T}$ is given and $[\begin{matrix} Ω_{x} & Ω_{y} & Ω_{z} \end{matrix}]^{T}$ solved by transfer equation with boundary conditions, equation (5) may be solved by any numerical integration method.

Topology graph of MSD model and label convention

The topology graph of MSD model, as shown in Figure 2, is a new diagramming method for MSTMM to describe the transfer information. To be noted that in the topology graph, circles denote body elements (could be rigid bodies, beams, rods or other flexible bodies), while arrows denote hinge elements. 0 presents the boundary while the other numbers are the sequence number of elements. Usually, one boundary is chosen as the output of the whole system that is defined as root boundary while others are inputs as tip boundaries. Form tips to the root, the sequence numbers gradually increase.

Figure 2.

Topology graph of a closed-loop-and-branch-mixed system.

Details about label convention are in Rui et al.²⁰

State vector, transfer matrix, and transfer equation

In the new version of MSTMM, the definitions of state vector, transfer matrix and transfer equation of elements, system overall transfer matrix and transfer equation as well as the principle for solution may be found in Rui et al.^16,17 Here, only the expressions are given.

The state vector of elements with spatial motion is

z = {[\begin{matrix} {\overset{\cdot\cdot}{r}}^{T} & {\overset{\cdot}{Ω}}^{T} & m^{T} & q^{T} & 1 \end{matrix}]}^{T}

(6)

where

\overset{\cdot\cdot}{r} = [\begin{matrix} \overset{\cdot\cdot}{x} \\ \overset{\cdot\cdot}{y} \\ \overset{\cdot\cdot}{z} \end{matrix}], \overset{\cdot}{Ω} = [\begin{matrix} {\overset{\cdot}{Ω}}_{x} \\ {\overset{\cdot}{Ω}}_{y} \\ {\overset{\cdot}{Ω}}_{z} \end{matrix}], m = [\begin{matrix} m_{x} \\ m_{y} \\ m_{z} \end{matrix}], q = [\begin{matrix} q_{x} \\ q_{y} \\ q_{z} \end{matrix}]

(7)

$\overset{\cdot\cdot}{r}$ , $\overset{\cdot}{Ω}$ , $m$ , and $q$ are the column matrices of absolute acceleration, absolute angular acceleration, internal moment, and internal force of elements in the global inertial coordinates.

For element $j$ , the transfer equation can be expressed as¹⁶

z_{j, O} = U_{j} z_{j, I}

(8)

where $U_{j}$ is the transfer matrix.

The overall state vector of system boundaries is

z_{all} = {[\begin{matrix} z_{root}^{T} & z_{tip}^{T} \end{matrix}]}^{T}

(9)

where $z_{root}$ is the state vector of the root element, while $z_{tip}$ are the state vectors of all tip elements.

The overall transfer equation of system is

U_{all} z_{all} = 0

(10)

where $U_{a l l}$ is the overall transfer matrix of the system.

Transfer equations and transfer matrices of branch subsystems and typical elements

Definition of branch subsystems

The subsystem composed of a multi-input body and all inboard hinges of the body, as dashed part in Figure 3, is defined as a branch subsystem, where $j$ is the sequence number of the body, $O$ is the output, $I_{1}, I_{2}, \dots, I_{L}$ are the sequence numbers of all hinges, and $L$ is the sum of hinges.

Figure 3.

Topology graph of a branch subsystem.

Transfer equation and geometric equation of branch subsystem

Transfer equation of a rigid body with multiple inputs and single output

According to the description in Rui et al.²⁰, for a spatial motion body with multiple inputs and single output, like body $j$ in Figure 3, the definition of state vector of the output is identical to in equation (6). However, the comprehensive state vector of inputs is defined as²⁰

\begin{matrix} z_{j, I_{1} - I_{L}} = [\begin{matrix} {\overset{\cdot\cdot}{r}}_{j, I_{1}}^{T} & {\overset{\cdot}{Ω}}_{j, I_{1}}^{T} & m_{j, I_{1}}^{T} & q_{j, I_{1}}^{T} & m_{j, I_{2}}^{T} & q_{j, I_{2}}^{T} \end{matrix} \\ {\begin{matrix} \dots & m_{j, I_{L}}^{T} & q_{j, I_{L}}^{T} & 1 \end{matrix}]}^{T} \end{matrix}

(11)

Similar to equation (8), the transfer equation of the spatial motion body with multiple inputs is

z_{j, O} = U_{j, I_{1} - I_{L}} z_{j, I_{1} - I_{L}}

(12)

and its transfer matrix is²⁰

\begin{matrix} U_{j, I_{1} - I_{L}} = \\ [\begin{matrix} I_{3} & E_{1} & O_{3 \times 3} & O_{3 \times 3} & \dots & O_{3 \times 3} & O_{3 \times 3} & \dots & E_{2} \\ O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 3} & \dots & O_{3 \times 3} & O_{3 \times 3} & \dots & O_{3 \times 1} \\ u_{3, 1} & u_{3, 2} & I_{3} & O_{3 \times 3} & \dots & I_{3} & u_{3, (2 i + 2)} & \dots & u_{3, (2 L + 3)} \\ E_{3} & E_{4} & O_{3 \times 3} & E_{6} & \dots & O_{3 \times 3} & O_{3 \times 3} & \dots & E_{5} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & \dots & O_{1 \times 3} & O_{1 \times 3} & \dots & 1 \end{matrix}] \end{matrix}

(13)

where

\begin{matrix} u_{3, 1} = E_{6} E_{3} + E_{4}, u_{3, 2} = E_{6} E_{4} + E_{7} \\ u_{3, (2 i + 2)} = E_{I_{i}} + E_{6}, u_{3, (2 L + 3)} = E_{6} E_{5} + E_{8} \end{matrix}

(14)

{\begin{matrix} E_{1} = - {\tilde{r}}_{I_{1} O}, E_{2} = {\tilde{Ω}}_{I_{1}} {\tilde{Ω}}_{I_{1}} r_{I_{1} O}, E_{3} = - m I_{3} \\ E_{4} = m {\tilde{r}}_{I_{1} C}, E_{5} = f_{C} - m {\tilde{Ω}}_{I_{1}} {\tilde{Ω}}_{I_{1}} r_{I_{1} C} \\ E_{6} = {\tilde{r}}_{I_{1} O}, E_{7} = A_{I_{1}} J_{I_{1}} A_{I_{1}}^{T}, E_{I_{i}} = - {\tilde{r}}_{I_{1} I_{i}} \\ E_{8} = - m_{C} - {\tilde{r}}_{I_{1} C} f_{C} + A_{I_{1}} {\tilde{Ω}}_{I_{1}} J_{I_{1}} ω_{I_{1}} \end{matrix}

(15)

$r_{I_{1} C}$ and $r_{I_{1} O}$ are the position vectors from the first input $I_{1}$ to mass center $C$ and to output $O$ , respectively, in inertial coordinates. $A_{I_{1}}$ is the direction cosine matrix from $I_{1} ξ η ς$ to $oxyz$ . $Ω_{I_{1}}$ and $ω_{I_{1}}$ are the column matrices of absolute angular velocity in $oxyz$ and $I_{1} ξ η ς$ , respectively. $m$ is the mass of rigid body. $f_{C}$ and $m_{C}$ are respectively external forces and external moments acting on mass center of the rigid body. $J_{I_{1}}$ is the moment of inertial with respect to $I_{1}$ .

For rigid body 21 with two inputs and single output in Figure 2, the transfer equation and transfer matrix are

z_{21, 22} = U_{21, 19 - 20} z_{21, 19 - 20}

(16)

\begin{array}{l} U_{21, 19 - 20} = \\ [\begin{matrix} I_{3} & E_{1} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & E_{2} \\ O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ u_{3, 1} & u_{3, 2} & I_{3} & E_{6} & I_{3} & u_{3, 6} & u_{3, 7} \\ E_{3} & E_{4} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & I_{3} & E_{5} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}] \end{array}

(17)

where the corresponding parameter matrices are defined in equations (14) and (15).

Transfer equation of multi-hinge subsystem

The concept of multi-hinge subsystem is first put forward in Rui et al.²⁰, which is composed of multiple hinges those have the same outboard body, like $I_{1} - I_{L}$ in Figure 3. Obviously, the outputs of multi-hinge subsystem are just the inputs of the outboard body, that is, the comprehensive state vector at outputs of the multi-hinge subsystem is just that at inputs of the multi-input body, see equation (11).

The multi-hinge subsystem composed of multiple smooth ball and socket hinges is taken as an example to demonstrate the transfer equation and transfer matrix of the multi-hinge subsystem. For each smooth ball-and-socket hinge $I_{1}, I_{2}, \dots, I_{L}$ , the corresponding accelerations, internal moments, and internal forces are identical at input and output

{\begin{matrix} {\overset{\cdot\cdot}{r}}_{I_{k}, O} = {\overset{\cdot\cdot}{r}}_{I_{k}, I} \\ m_{I_{k}, O} = m_{I_{k}, I} \\ q_{I_{k}, O} = q_{I_{k}, I} \end{matrix} (k = 1, 2, \dots, L)

(18)

If the output of the outboard body also connects to a smooth ball-and-socket hinge, the main transfer equation²⁰ of multi-hinge subsystem is

z_{j, I_{1} - I_{L}} = {\hat{U}}_{I_{1}} z_{i_{1}, I_{1}} + \sum_{k = 2}^{L} {\hat{U}}_{i_{k}, I_{k}} z_{i_{k}, I_{k}}

(19)

where ${\hat{U}}_{I_{1}}$ is the expanded transfer matrix of the first hinge, while ${\hat{U}}_{i_{k}, I_{k}}$ is the force expanding extraction matrix at the non-first input. $i_{k}$ is the sequence number of the inboard body of the $k th$ hinge

\begin{array}{l} {\hat{U}}_{I_{1}} = \\ [\begin{matrix} I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ - u_{3, 2}^{- 1} u_{3, 1} & O_{3 \times 3} & - u_{3, 2}^{- 1} u_{3, 3} & - u_{3, 2}^{- 1} u_{3, 4} & - u_{3, 2}^{- 1} u_{3, 3 + 2 L} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}] \end{array}

(20)

\begin{array}{l} {\hat{U}}_{i_{k}, I_{k}} = \\ \frac{\begin{matrix} 1 & 2 & 3 & 4 & 5 \end{matrix}}{[\begin{matrix} O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & - u_{3, 2}^{- 1} u_{3, 1 + 2 k} & - u_{3, 2}^{- 1} u_{3, 2 + 2 k} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 0 \end{matrix}] \begin{matrix} 1 \\ 2 \\ 3 \\ 4 \\ 2 k + 1 \\ 2 k + 2 \\ 2 L + 3 \end{matrix} (k = 2, 3, \dots, L)} \end{array}

(21)

where $u_{m, n} (m = 3, n = 1, 2, \dots, 3 + 2 L)$ is the partition matrix of transfer matrix $U_{j, I_{1} - I_{L}}$ given in equation (13)

For the multi-hinge subsystem composed by smooth ball-and-socket hinges 19 and 20 in Figure 2, the main transfer equation and associated transfer matrices are

z_{21, 19 - 20} = {\hat{U}}_{19} z_{17, 19} + {\hat{U}}_{18, 20} z_{18, 20}

(22)

{\hat{U}}_{19} = [\begin{matrix} I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ - u_{3, 2}^{- 1} u_{3, 1} & O_{3 \times 3} & - u_{3, 2}^{- 1} u_{3, 3} & - u_{3, 2}^{- 1} u_{3, 4} & - u_{3, 2}^{- 1} u_{3, 7} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}]

(23)

{\hat{U}}_{18, 20} = [\begin{matrix} O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & - u_{3, 2}^{- 1} u_{3, 5} & - u_{3, 2}^{- 1} u_{3, 6} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 0 \end{matrix}]

(24)

where according to equation (17), the associated partition matrices are

{\begin{matrix} u_{3, 1} = E_{6} E_{3} + E_{4}, u_{3, 2} = E_{6} E_{4} + E_{7} \\ u_{3, 3} = I_{3}, u_{3, 4} = E_{6}, u_{3, 5} = I_{3} \\ u_{3, 6} = E_{I_{2}} + E_{6}, u_{3, 7} = E_{6} E_{5} + E_{8} \end{matrix}

(25)

Main transfer equation of branch subsystem

Based on equations (12) and (19), the main transfer equation of the branch subsystem in Figure 3 is

\begin{matrix} z_{j, O} = U_{j, I_{1} - I_{L}} z_{j, I_{1} - I_{L}} \\ = U_{j, I_{1} - I_{L}} ({\hat{U}}_{I_{1}} z_{i_{1}, I_{1}} + \sum_{k = 2}^{L} {\hat{U}}_{i_{k}, I_{k}} z_{i_{k}, I_{k}}) \\ = \underset{{\hat{U}}_{m, i_{1}}}{\underset{︸}{U_{j, I_{1} - I_{L}} {\hat{U}}_{I_{1}}}} z_{i_{1}, I_{1}} + \sum_{k = 2}^{L} \underset{{\hat{U}}_{m, i_{k}}}{\underset{︸}{U_{j, I_{1} - I_{L}} {\hat{U}}_{i_{k}, I_{k}}}} z_{i_{k}, I_{k}} \end{matrix}

(26)

where $m$ is the sequence number of the whole branch subsystem, $U_{m, i_{1}}$ is the coefficient matrix of state vector of the first input, while ${\hat{U}}_{m, i_{k}}$ $(k = 2, 3, \dots, L)$ is of the $k th$ input in the branch subsystem.

According to equation (26), for branch subsystem made up of rigid body 21, smooth ball-and-socket hinges 19 and 20 in Figure 2, the main transfer equation is

z_{21, 22} = U_{21, 19 - 20} {\hat{U}}_{19} z_{17, 19} + U_{21, 19 - 20} {\hat{U}}_{18, 20} z_{18, 20}

(27)

Geometric equations of branch subsystem

For a branch subsystem, only the main transfer equation is not enough to get the unknowns in the state vectors. Thus, it is necessary to introduce geometric equations. Geometric equation is used to describe the relationship of accelerations and internal moments between the output and non-first inputs in branch subsystem.

For the multi-input body $j$ , the relationship of acceleration between the first input ${\overset{\cdot\cdot}{r}}_{j, I_{1}}$ and $k th$ input ${\overset{\cdot\cdot}{r}}_{j, I_{k}}$ is

{\overset{\cdot\cdot}{r}}_{j, I_{k}} = {\overset{\cdot\cdot}{r}}_{j, I_{1}} + E_{1, I_{k}} {\overset{\cdot}{Ω}}_{j, I_{1}} + E_{2, I_{k}}, (k = 2, 3, \dots, L)

(28)

where

E_{1, I_{k}} = - {\tilde{r}}_{j, I_{1} I_{k}}, E_{2, I_{k}} = {\tilde{Ω}}_{j, I_{1}} {\tilde{Ω}}_{j, I_{1}} r_{j, I_{1} I_{k}}

(29)

The relationship of acceleration between the first input ${\overset{\cdot\cdot}{r}}_{j, I_{1}}$ end and output end ${\overset{\cdot\cdot}{r}}_{j, O}$ of multi-input body is

{\overset{\cdot\cdot}{r}}_{j, O} = {\overset{\cdot\cdot}{r}}_{j, I_{1}} + E_{1, O} {\overset{\cdot}{Ω}}_{j, I_{1}} + E_{2, O}

(30)

where

E_{1, O} = - {\tilde{r}}_{j, I_{1} O}, E_{2, O} = {\tilde{Ω}}_{j, I_{1}} {\tilde{Ω}}_{j, I_{1}} r_{j, I_{1} O}

(31)

In equations (28)–(31), $r_{j, I_{1} I_{k}}$ and $r_{j, I_{1} O}$ are the position vectors from the first input to the $k th$ input and to the output $O$ of body $j$ in the inertial coordinates.

For one smooth ball-and-socket hinge, the accelerations at input end and output end are equal, that is

{\overset{\cdot\cdot}{r}}_{i_{k}, I_{k}} = {\overset{\cdot\cdot}{r}}_{j, I_{k}}, (k = 2, 3, \dots, L)

(32)

where ${\overset{\cdot\cdot}{r}}_{i_{k}, I_{k}}$ is the acceleration of the $k th$ input of the branch subsystem.

Substituting equations (30) and (32) into equation (28) yields

{\overset{\cdot\cdot}{r}}_{i_{k}, I_{k}} = {\overset{\cdot\cdot}{r}}_{j, O} + E_{1, O I_{k}} {\overset{\cdot}{Ω}}_{j, I_{1}} + E_{2, O I_{k}}, (k = 2, 3, \dots, L)

(33)

where

E_{1, O I_{k}} = {\tilde{r}}_{j, I_{1} O} - {\tilde{r}}_{j, I_{1} I_{k}}, E_{2, O I_{k}} = {\tilde{Ω}}_{j, I_{1}} {\tilde{Ω}}_{j, I_{1}} (r_{j, I_{1} I_{k}} - r_{j, I_{1} O})

(34)

Moreover, for an arbitrary smooth ball-and-socket hinge, the internal moments at inputs are zero, which may be expressed as

m_{i_{k}, I_{k}} = 0, (k = 2, 3, \dots, L)

(35)

Therefore, rewriting equations (32) and (34) in the compact form yields the geometric equation of branch subsystem

H_{\overset{\cdot\cdot}{r}, m} z_{i_{k}, I_{k}} = H_{O, I_{k}} z_{j, O}, (k = 2, 3, \dots, L)

(36)

where

H_{\overset{\cdot\cdot}{r}, m} = [\begin{matrix} I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 1} \end{matrix}]

(37)

H_{O, I_{k}} = [\begin{matrix} I_{3} & E_{1, O I_{k}} & O_{3 \times 3} & O_{3 \times 3} & E_{2, O I_{k}} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \end{matrix}]

(38)

$H_{\overset{\cdot\cdot}{r}, m}$ is the acceleration and moment extraction matrix, while $H_{O, I_{k}}$ is the associated matrix. For the branch subsystem made up of rigid body 21, smooth ball-and-socket hinges 19 and 20 in Figure 2, its geometric equation is

H_{\overset{\cdot\cdot}{r}, m} z_{18, 20} = H_{22, 20} z_{21, 22}

(39)

where

H_{22, 20} = [\begin{matrix} I_{3} & E_{1, O I_{2}} & O_{3 \times 3} & O_{3 \times 3} & E_{2, O I_{2}} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \end{matrix}]

(40)

E_{1, O I_{2}} = {\tilde{r}}_{21, I_{1} O} - {\tilde{r}}_{21, I_{1} I_{2}}, E_{2, O I_{k}} = {\tilde{Ω}}_{21} {\tilde{Ω}}_{21} (r_{21, I_{1} I_{2}} - r_{21, I_{1} O})

(41)

Transfer equation and transfer matrix of typical elements

Spatial motion body with single input and single output

The transfer equation of spatial motion rigid body $j$ with single input and single output is similar as equation (8).

The transfer matrix is²⁰

U_{j} = [\begin{matrix} I_{3} & E_{1} & O_{3 \times 3} & O_{3 \times 3} & E_{2} \\ O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ E_{6} E_{3} + E_{7} & E_{6} E_{4} + E_{8} & I_{3} & E_{6} & E_{6} E_{5} + E_{9} \\ E_{3} & E_{4} & O_{3 \times 3} & I_{3} & E_{5} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}]

(42)

where

{\begin{matrix} E_{1} = - {\tilde{r}}_{IO}, E_{2} = {\tilde{Ω}}_{I} {\tilde{Ω}}_{I} r_{IO}, E_{3} = - m I_{3} \\ E_{4} = m {\tilde{r}}_{IC}, E_{5} = f_{C} - m {\tilde{Ω}}_{I} {\tilde{Ω}}_{I} r_{IC} \\ E_{6} = {\tilde{r}}_{IO}, E_{7} = A_{I} J_{I} A_{I}^{T} \\ E_{8} = - m_{C} - {\tilde{r}}_{IC} f_{C} + A_{I} {\tilde{Ω}}_{I} J_{I} ω_{I} \end{matrix}

(43)

Spatial motion smooth ball-and-socket hinge

The transfer equation of smooth ball-and–socket hinge $j$ is identical to equation (8). A necessary and sufficient condition o derive the transfer matrix of ball-and-socket hinge is that the moments in three orientations in body-fixed coordinate system are zero at the output of the outboard body hinge $j$ .

The transfer matrix is²⁰

U_{j} = [\begin{matrix} I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ - u_{3, 2}^{- 1} u_{3, 1} & O_{3 \times 3} & - u_{3, 2}^{- 1} u_{3, 3} & - u_{3, 2}^{- 1} u_{3, 4} & - u_{3, 2}^{- 1} u_{3, 5} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}]

(44)

where $u_{m, n} (m = 3, n = 1, 2, \dots, 5)$ is the partition matrix of transfer matrix of outboard body.

Spatial motion smooth sliding hinge

The transfer equation of smooth sliding hinge $j$ is the same as equation (8). The body-fixed coordinate system at the output of the inboard body coincides with that at the input of the sliding hinge. Similarly, the coordinate system at the input of the outboard body coincides with that at the output of the corresponding sliding hinge. Obviously, $ξ$ axes of the body-fixed coordinate systems at input and output of the sliding hinge are coincident, and sliding direction is along $ξ$ axis, so that the internal force is zero along this direction, and the relative slip distance is denoted as $l_{s, j}$ .

For sliding hinge $j$ , the angular accelerations and internal forces at input and output are satisfied with

{\overset{\cdot}{Ω}}_{j, I} = {\overset{\cdot}{Ω}}_{j, O}

(45)

q_{j, O} = q_{j, I}

(46a)

H_{3}^{T} A_{j, I}^{T} q_{j, I} = 0

(46b)

where

H_{3} = {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T}

(47)

$H_{3}$ is used to extract the force along $ξ$ axis.

The acceleration at output is

r_{j, O} = r_{j, I} + A_{j, I} l_{j, IO}

(48)

where

l_{j, IO} = {[\begin{matrix} 0 & 0 & l_{s, j} \end{matrix}]}^{T}

(49)

m_{j, O} = m_{j, I} + {\tilde{r}}_{j, IO} q_{j, I}

(50)

Taking the first and second time derivatives of equation (48) results in

{\overset{\cdot}{r}}_{j, O} = {\overset{\cdot}{r}}_{j, I} + {\tilde{Ω}}_{j, I} A_{j, I} l_{j, IO} + A_{j, I} {\overset{\cdot}{l}}_{j, IO}

(51)

\begin{matrix} {\overset{\cdot\cdot}{r}}_{j, O} = {\overset{\cdot\cdot}{r}}_{j, I} + {\tilde{r}}_{j, IO}^{T} {\overset{\cdot}{Ω}}_{j, I} + A_{j, I} {\tilde{Ω}}_{j, I} {\tilde{Ω}}_{j, I} l_{j, IO} \\ + 2 A_{j, I} {\tilde{Ω}}_{j, I} {\overset{\cdot}{l}}_{j, IO} + A_{j, I} {\overset{\cdot\cdot}{l}}_{j, IO} \end{matrix}

(52)

The vectors in equation (52) are projected onto the body-fixed coordinate system as

\begin{matrix} A_{j, I}^{T} {\overset{\cdot\cdot}{r}}_{j, O} = A_{j, I}^{T} {\overset{\cdot\cdot}{r}}_{j, I} + A_{j, I}^{T} {\tilde{r}}_{j, IO}^{T} {\overset{\cdot}{Ω}}_{j, I} \\ + ({\tilde{Ω}}_{j, I} {\tilde{Ω}}_{j, I} l_{j, IO} + 2 {\tilde{Ω}}_{j, I} {\overset{\cdot}{l}}_{j, IO}) + {\overset{\cdot\cdot}{l}}_{j, IO} \end{matrix}

(53)

Equation (53) includes three equations. Here, in order to eliminate the term of relative sliding acceleration, the first two equations are extracted

\begin{matrix} H_{1 - 2}^{T} A_{j, I}^{T} {\overset{\cdot\cdot}{r}}_{j, O} = H_{1 - 2}^{T} A_{j, I}^{T} {\overset{\cdot\cdot}{r}}_{j, I} + H_{1 - 2}^{T} A_{j, I}^{T} {\tilde{r}}_{j, IO}^{T} {\overset{\cdot}{Ω}}_{j, I} \\ + H_{1 - 2}^{T} ({\tilde{Ω}}_{j, I} {\tilde{Ω}}_{j, I} l_{j, IO} + 2 {\tilde{Ω}}_{j, I} {\overset{\cdot}{l}}_{j, IO}) \end{matrix}

(54)

where

H_{1 - 2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}]

(55)

Assuming the output of the outboard body of the sliding hinge $j$ is also a sliding hinge with parallel sliding direction as sliding hinge $j$ or a free boundary, then

H_{3}^{T} A_{j^{+}, O}^{T} q_{j^{+}, O} = 0

(56)

According to the transfer equation of outboard body of the sliding hinge $z_{j^{+}, O} = U_{j^{+}} z_{j^{+}, I}$ , where transfer matrix $U_{j^{+}}$ is identical to equation (42), $q_{j^{+}, O}$ is extracted as

q_{j^{+}, O} = u_{4, 1} {\overset{\cdot\cdot}{r}}_{j^{+}, I} + u_{4, 2} {\tilde{Ω}}_{j^{+}, I} + u_{4, 3} m_{j^{+}, I} + u_{4, 4} q_{j^{+}, I} + u_{4, 5}

(57)

where

u_{4, 1} = E_{3}, u_{4, 2} = E_{4}, u_{4, 3} = O_{3 \times 3}, u_{4, 4} = I_{3}, u_{4, 5} = E_{6} E_{5} + E_{9}

(58)

Substituting equation (57) into equation (56), it becomes

\begin{matrix} 0 = H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 1} {\overset{\cdot\cdot}{r}}_{j^{+}, I} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 2} {\overset{\cdot}{Ω}}_{j^{+}, I} \\ + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 3} m_{j^{+}, I} \\ + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 4} q_{j^{+}, I} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 5} \\ = H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 1} {\overset{\cdot\cdot}{r}}_{j, O} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 2} {\overset{\cdot}{Ω}}_{j, O} \\ + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 3} m_{j, O} \\ + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 4} q_{j, O} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 5} \end{matrix}

(59)

Considering equations (45), (46a), (50), and (59) could be rewritten as

\begin{array}{l} 0 = H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 1} {\overset{\cdot\cdot}{r}}_{j, O} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 2} {\tilde{Ω}}_{j, I} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 3} m_{j, I} \\ + H_{3}^{T} A_{j^{+}, O}^{T} (u_{4, 4} + u_{4, 3} {\tilde{r}}_{j, I O}) q_{j, I} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 5} \end{array}

(60)

In view of equations (46b), (60) may be changed into

\begin{matrix} - H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 1} {\overset{\cdot\cdot}{r}}_{j, O} = \\ H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 2} {\overset{\cdot}{Ω}}_{j, I} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 3} m_{j, I} \\ + H_{3}^{T} A_{j^{+}, O}^{T} (u_{4, 4} + u_{4, 3} {\tilde{r}}_{j, IO}) A_{j, I} {\bar{H}}_{1 - 2} \\ A_{j, I}^{T} q_{j, I} + H_{3}^{T} A_{j^{+}, O}^{T} u_{4, 5} \end{matrix}

(61)

where

{\bar{H}}_{1 - 2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]

(62)

Rewriting equations (54) and (61) in a compact form yields

\begin{matrix} E_{j, O} {\overset{\cdot\cdot}{r}}_{j, O} = E_{j, I, \overset{\cdot\cdot}{r}} {\overset{\cdot\cdot}{r}}_{j, I} + E_{j, I, \overset{\cdot}{Ω}} {\overset{\cdot}{Ω}}_{j, I} \\ + E_{j, I, m} m_{j, I} + E_{j, I, q} q_{j, I} + E_{j, 0} \end{matrix}

(63)

Based on the transfer equation of a sliding hinge $j$ , taking equations (45), (46a), (50), and (63) into consideration, the transfer matrix of sliding hinge is

\begin{matrix} U_{j} = \\ [\begin{matrix} E_{j, O}^{- 1} E_{j, I, \overset{\cdot\cdot}{r}} & E_{j, O}^{- 1} E_{j, I, \overset{\cdot}{Ω}} & E_{j, O}^{- 1} E_{j, I, m} & E_{j, O}^{- 1} E_{j, I, q} & E_{j, O}^{- 1} E_{j, 0} \\ O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & I_{3} & {\tilde{r}}_{j, IO} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & I_{3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}] \end{matrix}

(64)

where

\begin{matrix} E_{j, O} = [\begin{matrix} H_{1 - 2}^{T} A_{j, I}^{T} \\ - H_{3}^{T} A_{j^{+}, O}^{T} u_{41} \end{matrix}], E_{j, I, \overset{\cdot\cdot}{r}} = [\begin{matrix} H_{1 - 2}^{T} A_{j, I_{1}}^{T} \\ O_{1 \times 3} \end{matrix}] \\ E_{j, I, \overset{\cdot}{Ω}} = [\begin{matrix} H_{1 - 2}^{T} A_{j, I}^{T} {\tilde{r}}_{j, IO}^{T} \\ H_{3}^{T} A_{j^{+}, O}^{T} u_{42} \end{matrix}], E_{j, I, m} = [\begin{matrix} O_{2 \times 3} \\ H_{3}^{T} A_{j^{+}, O}^{T} u_{43} \end{matrix}] \\ E_{j, I, q} = [\begin{matrix} O_{2 \times 3} \\ H_{3}^{T} A_{j^{+}, O}^{T} (u_{44} + u_{43} {\tilde{r}}_{j, IO}) A_{j, I} {\bar{H}}_{1 - 2} A_{j, I}^{T} \end{matrix}] \\ E_{j, 0} = [\begin{matrix} H_{1 - 2}^{T} ({\tilde{Ω}}_{j, I} {\tilde{Ω}}_{j, I} l_{j, IO} + 2 {\tilde{Ω}}_{j, I} {\overset{\cdot}{l}}_{j, IO}) \\ H_{3}^{T} A_{j^{+}, O}^{T} u_{45} \end{matrix}] \end{matrix}

(65)

Overall transfer equation for tree systems

Any tree multibody system may be treated as the assembling of branch subsystems and simple chain subsystems. For instance, in the tree multibody system shown in Figure 4, there are three branch subsystems totally with dotted lines, which are numbered as 24, 25, and 26, respectively.

Figure 4.

Topology graph of the tree multibody system.

For each branch subsystem, equation (26) may be adapted directly as

\begin{matrix} z_{21, 22} = {\hat{U}}_{24, 17} z_{17, 19} + {\hat{U}}_{24, 18} z_{18, 20} \\ z_{17, 19} = {\hat{U}}_{26, 9} z_{9, 13} + {\hat{U}}_{26, 10} z_{10, 14} \\ z_{18, 20} = {\hat{U}}_{25, 11} z_{11, 15} + {\hat{U}}_{25, 12} z_{12, 16} \end{matrix}

(66)

where

\begin{matrix} {\hat{U}}_{24, 17} = U_{21, 19 - 20} {\hat{U}}_{19}, {\hat{U}}_{24, 18} = U_{21, 19 - 20} {\hat{U}}_{18, 20} \\ {\hat{U}}_{26, 9} = U_{17, 13 - 14} {\hat{U}}_{13}, {\hat{U}}_{26, 10} = U_{17, 13 - 14} {\hat{U}}_{10, 14} \\ {\hat{U}}_{25, 11} = U_{18, 15 - 16} {\hat{U}}_{15}, {\hat{U}}_{25, 12} = U_{18, 15 - 16} {\hat{U}}_{12, 16} \end{matrix}

(67)

For the rest chain subsystems, the following transfer relationships exist

\begin{matrix} z_{23, 0} = U_{23} U_{22} z_{21, 22} \\ z_{9, 13} = U_{9} U_{5} U_{1} z_{1, 0}, z_{10, 14} = U_{10} U_{6} U_{2} z_{2, 0} \\ z_{11, 15} = U_{11} U_{7} U_{3} z_{3, 0}, z_{12, 16} = U_{12} U_{8} U_{4} z_{4, 0} \end{matrix}

(68)

Substitution of equation (66) into equation (68) results in the main transfer equation of system

\begin{matrix} z_{23, 0} = U_{23} U_{22} {\hat{U}}_{24, 17} {\hat{U}}_{26, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + U_{23} U_{22} {\hat{U}}_{24, 17} {\hat{U}}_{26, 10} U_{10} U_{6} U_{2} z_{2, 0} \\ + U_{23} U_{22} {\hat{U}}_{24, 18} {\hat{U}}_{25, 11} U_{11} U_{7} U_{3} z_{3, 0} \\ + U_{23} U_{22} {\hat{U}}_{24, 18} {\hat{U}}_{25, 12} U_{12} U_{8} U_{4} z_{4, 0} \end{matrix}

(69)

For each of them with two inputs, there are three geometric equations, which are

\begin{matrix} H_{\overset{\cdot\cdot}{r}, m} z_{18, 20} = H_{22, 20} z_{21, 22} \\ H_{\overset{\cdot\cdot}{r}, m} z_{10, 14} = H_{19, 14} z_{17, 19} \\ H_{\overset{\cdot\cdot}{r}, m} z_{12, 16} = H_{20, 16} z_{18, 20} \end{matrix}

(70)

Substitution of equation (66) in equation (70) results in

\begin{matrix} H_{\overset{\cdot\cdot}{r}, m} z_{18, 20} = H_{22, 20} ({\hat{U}}_{24, 17} z_{17, 19} + {\hat{U}}_{24, 18} z_{18, 20}) \\ H_{\overset{\cdot\cdot}{r}, m} z_{10, 14} = H_{19, 14} ({\hat{U}}_{26, 9} z_{9, 13} + {\hat{U}}_{26, 10} z_{10, 14}) \\ H_{\overset{\cdot\cdot}{r}, m} z_{12, 16} = H_{20, 16} ({\hat{U}}_{25, 11} z_{11, 15} + {\hat{U}}_{25, 12} z_{12, 16}) \end{matrix}

(71)

Substituting equation (68) into equation (71), the geometric equations may be expressed by the boundary sate vectors

\begin{matrix} {\begin{matrix} H_{22, 20} {\hat{U}}_{24, 17} {\hat{U}}_{26, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + H_{22, 20} {\hat{U}}_{24, 17} {\hat{U}}_{26, 10} U_{10} U_{6} U_{2} z_{2, 0} \\ + (H_{22, 20} {\hat{U}}_{24, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{25, 11} U_{11} U_{7} U_{3} z_{3, 0} \\ + (H_{22, 20} {\hat{U}}_{24, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{25, 12} U_{12} U_{8} U_{4} z_{4, 0} = 0 \end{matrix} \\ \begin{matrix} {\begin{matrix} H_{19, 14} {\hat{U}}_{26, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + (H_{19, 14} {\hat{U}}_{26, 10} - H_{\overset{\cdot\cdot}{r}, m}) U_{10} U_{6} U_{2} z_{2, 0} = 0 \end{matrix} \\ {\begin{matrix} H_{20, 16} {\hat{U}}_{25, 11} U_{11} U_{7} U_{3} z_{3, 0} \\ + (H_{20, 16} {\hat{U}}_{25, 12} - H_{\overset{\cdot\cdot}{r}, m}) U_{12} U_{8} U_{4} z_{4, 0} = 0 \end{matrix} \end{matrix} \end{matrix}

(72)

where $H_{\overset{\cdot\cdot}{r}, m}$ is the same as equation (37), while $H_{22, 20}$ , $H_{19, 14}$ , and $H_{20, 16}$ have the identical form as equation (38).

According to equation (9), the overall state vector of system boundaries in Figure 4 is

z_{all} = {[\begin{matrix} z_{23, 0}^{T} & z_{1, 0}^{T} & z_{2, 0}^{T} & z_{3, 0}^{T} & z_{4, 0}^{T} \end{matrix}]}^{T}

(73)

The overall transfer equation of system is consistent with equation (10).

Substituting the main transfer equation (69) and geometric equation (72) into equation (10), the overall transfer matrix may be obtained

U_{a l l} = [\begin{matrix} - I_{13} & T_{1 - 23} & T_{2 - 23} & T_{3 - 23} & T_{4 - 23} \\ O_{6 \times 13} & G_{1 - 19, 14} & G_{2 - 19, 14} & O_{6 \times 13} & O_{6 \times 13} \\ O_{6 \times 13} & O_{6 \times 13} & O_{6 \times 13} & G_{3 - 20, 16} & G_{4 - 20, 16} \\ O_{6 \times 13} & G_{1 - 22, 20} & G_{2 - 22, 20} & G_{3 - 22, 20} & G_{4 - 22, 20} \end{matrix}]

(74)

where

\begin{matrix} T_{1 - 23} = U_{23} U_{22} {\hat{U}}_{24, 17} {\hat{U}}_{26, 9} U_{9} U_{5} U_{1} \\ T_{2 - 23} = U_{23} U_{22} {\hat{U}}_{24, 17} {\hat{U}}_{26, 10} U_{10} U_{6} U_{2} \\ T_{3 - 23} = U_{23} U_{22} {\hat{U}}_{24, 18} {\hat{U}}_{25, 11} U_{11} U_{7} U_{3} \\ T_{4 - 23} = U_{23} U_{22} {\hat{U}}_{24, 18} {\hat{U}}_{25, 12} U_{12} U_{8} U_{4} \\ G_{1 - 19, 14} = H_{19, 14} {\hat{U}}_{26, 9} U_{9} U_{5} U_{1} \\ G_{2 - 19, 14} = (H_{19, 14} {\hat{U}}_{26, 10} - H_{\overset{\cdot\cdot}{r}, m}) U_{10} U_{6} U_{2} \\ G_{3 - 20, 16} = H_{20, 16} {\hat{U}}_{25, 11} U_{11} U_{7} U_{3} \\ G_{4 - 20, 16} = (H_{20, 16} {\hat{U}}_{25, 12} - H_{\overset{\cdot\cdot}{r}, m}) U_{12} U_{8} U_{4} \\ G_{1 - 22, 20} = H_{22, 20} {\hat{U}}_{24, 17} {\hat{U}}_{26, 9} U_{9} U_{5} U_{1} \\ G_{2 - 22, 20} = H_{22, 20} {\hat{U}}_{24, 17} {\hat{U}}_{26, 10} U_{10} U_{6} U_{2} \\ G_{3 - 22, 20} = (H_{22, 20} {\hat{U}}_{24, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{25, 11} U_{11} U_{7} U_{3} \\ G_{4 - 22, 20} = (H_{22, 20} {\hat{U}}_{24, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{25, 12} U_{12} U_{8} U_{4} \end{matrix}

(75)

Overall transfer equation for closed-loop-and-branch-mixed systems

Closed-loop subsystem transform

For the closed-loop subsystem composed of bodies 1, 2, 5, 6, 9 and hinges 3, 4, 7, 8, 11 as shown in Figure 5, the loop is “cut off” at point $P_{2, 11}$ , then becomes into a branch system. Herein, a pair of “new boundaries” is generated. It is worthy to be noted that the state vectors $z_{11, 0}$ and $z_{2, 0}$ of the pair of “new boundaries” are both the state vector of point $P_{2, 11}$ , but transfer along the opposite directions, which results in that accelerations and angular accelerations in these two state vectors are equal, whereas internal forces and internal moments are opposite equal, that is

z_{11, 0} = C z_{2, 0}

(76)

where

C = [\begin{matrix} I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & I_{3} & O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & - I_{3} & O_{3 \times 3} & O_{3 \times 1} \\ O_{3 \times 3} & O_{3 \times 3} & O_{3 \times 3} & - I_{3} & O_{3 \times 1} \\ O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & O_{1 \times 3} & 1 \end{matrix}]

(77)

Figure 5.

(a) Closed-loop subsystem and (b) branch system after cutting off the closed loop.

The overall transfer equation for closed-loop-and-branch-mixed systems

For the closed-loop-and-branch-mixed system in Figure 2, after “cutting off” the closed loop at point $P_{3, 24}$ that is the connection point of body 3 and hinge 24, the mixed system becomes a tree system as shown in Figure 6 with a pair of “new boundaries,” of which state vectors $z_{24, 0}$ and $z_{3, 0}$ satisfy equation (76).

Figure 6.

Topology graph of a closed-loop-and-branch-mixed system after cutting off closed loop.

Along the transfer path, the main transfer equation is

\begin{matrix} z_{23, 0} = U_{23} U_{22} {\hat{U}}_{25, 17} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + U_{23} U_{22} {\hat{U}}_{25, 17} {\hat{U}}_{27, 10} U_{10} U_{6} U_{2} U_{24} z_{24, 0} \\ + U_{23} U_{22} {\hat{U}}_{25, 18} {\hat{U}}_{26, 11} U_{11} U_{7} U_{3} z_{3, 0} \\ + U_{23} U_{22} {\hat{U}}_{25, 18} {\hat{U}}_{26, 12} U_{12} U_{8} U_{4} z_{4, 0} \end{matrix}

(78)

and geometric equations for three branch subsystems are

\begin{array}{l} {\begin{cases} H_{22, 20} {\hat{U}}_{25, 17} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + H_{22, 20} {\hat{U}}_{25, 17} {\hat{U}}_{27, 10} U_{10} U_{6} U_{2} U_{24} z_{24, 0} \\ + (H_{22, 20} {\hat{U}}_{25, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{26, 11} U_{11} U_{7} U_{3} z_{3, 0} \\ + (H_{22, 20} {\hat{U}}_{25, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{26, 12} U_{12} U_{8} U_{4} z_{4, 0} = 0 \end{cases} \\ {\begin{cases} H_{19, 14} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + H_{22, 20} {\hat{U}}_{25, 17} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} z_{1, 0} \\ + (H_{19, 14} {\hat{U}}_{27, 10} - H_{\overset{\cdot\cdot}{r}, m}) U_{10} U_{6} U_{2} U_{24} z_{24, 0} = 0 \end{cases} \\ {\begin{cases} H_{20, 16} {\hat{U}}_{26, 11} U_{11} U_{7} U_{3} z_{3, 0} \\ + (H_{20, 16} {\hat{U}}_{26, 12} - H_{\overset{\cdot\cdot}{r}, m}) U_{12} U_{8} U_{4} z_{4, 0} = 0 \end{cases} \end{array}

(79)

where

\begin{matrix} {\hat{U}}_{25, 17} = U_{21, 19 - 20} {\hat{U}}_{19}, {\hat{U}}_{25, 18} = U_{21, 19 - 20} {\hat{U}}_{18, 20} \\ {\hat{U}}_{27, 9} = U_{17, 13 - 14} {\hat{U}}_{13}, {\hat{U}}_{27, 10} = U_{17, 13 - 14} {\hat{U}}_{10, 14} \\ {\hat{U}}_{26, 11} = U_{18, 15 - 16} {\hat{U}}_{15}, {\hat{U}}_{26, 12} = U_{18, 15 - 16} {\hat{U}}_{12, 16} \end{matrix}

(80)

as well as the supplement equation for closed loop is

z_{24, 0} = C z_{3, 0}

(81)

The overall state vector of system boundaries is

z_{all} = {[\begin{matrix} z_{23, 0}^{T} & z_{1, 0}^{T} & z_{24, 0}^{T} & z_{3, 0}^{T} & z_{4, 0}^{T} \end{matrix}]}^{T}

(82)

Substituting the main equation (78), geometric equation (79), and the supplement equation (81) into equation (10), the overall transfer matrix is

U_{a l l} = [\begin{matrix} {- I}_{13} & T_{1 - 23} & T_{24 - 23} & T_{3 - 23} & T_{4 - 23} \\ O_{6 \times 13} & G_{1 - 19, 14} & G_{24 - 19, 14} & O_{6 \times 13} & O_{6 \times 13} \\ O_{6 \times 13} & O_{6 \times 13} & O_{6 \times 13} & G_{3 - 20, 16} & G_{4 - 20, 16} \\ O_{6 \times 13} & G_{1 - 22, 20} & G_{24 - 22, 20} & G_{3 - 22, 20} & G_{4 - 22, 20} \\ O_{13 \times 13} & O_{13 \times 13} & {- I}_{13} & C & O_{13 \times 13} \end{matrix}]

(83)

where

\begin{array}{l} T_{1 - 23} = U_{23} U_{22} {\hat{U}}_{25, 17} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} \\ T_{24 - 23} = U_{23} U_{22} {\hat{U}}_{25, 17} {\hat{U}}_{27, 10} U_{10} U_{6} U_{2} U_{24} \\ T_{3 - 23} = U_{23} U_{22} {\hat{U}}_{25, 18} {\hat{U}}_{26, 11} U_{11} U_{7} U_{3} \\ T_{4 - 23} = U_{23} U_{22} {\hat{U}}_{25, 18} {\hat{U}}_{26, 12} U_{12} U_{8} U_{4} \\ G_{1 - 19, 14} = H_{19, 14} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} \\ G_{24 - 19, 14} = (H_{19, 14} {\hat{U}}_{27, 10} - H_{\overset{\cdot\cdot}{r}, m}) U_{10} U_{6} U_{2} U_{24} \\ G_{3 - 20, 16} = H_{20, 16} {\hat{U}}_{26, 11} U_{11} U_{7} U_{3} \\ G_{4 - 20, 16} = (H_{20, 16} U_{18, 15 - 16} {\hat{U}}_{12, 16} - H_{\overset{\cdot\cdot}{r}, m}) U_{12} U_{8} U_{4} \\ G_{1 - 22, 20} = H_{22, 20} {\hat{U}}_{25, 17} {\hat{U}}_{27, 9} U_{9} U_{5} U_{1} \\ G_{24 - 22, 20} = H_{22, 20} {\hat{U}}_{25, 17} {\hat{U}}_{27, 10} U_{10} U_{6} U_{2} U_{24} \\ G_{3 - 22, 20} = (H_{22, 20} {\hat{U}}_{25, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{26, 11} U_{11} U_{7} U_{3} \\ G_{4 - 22, 20} = (H_{22, 20} {\hat{U}}_{25, 18} - H_{\overset{\cdot\cdot}{r}, m}) {\hat{U}}_{26, 12} U_{12} U_{8} U_{4} \end{array}

(84)

Matrix $C$ is determined by equation (77).

Automatic deduction theorem of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems

Considering equations (10), (73)–(75) and equations (82)–(84) comprehensively, following features of the new version of MSTMM may be summarized, which forms automatic deduction theorem of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems.

Tree systems

The overall transfer equation for tree multibody system is composed of two parts: the main transfer equation and geometric equations:

The first row of the overall transfer matrix corresponds to the main transfer equation, where the coefficient matrix of root state vector is minus identity matrix. The coefficient matrices of tip state vectors (such as $T_{1 - 23}$ , $T_{2 - 23}$ , $T_{3 - 23}$ , $T_{4 - 23}$ ) are equal to the successive premultiplication of the transfer matrices of all elements along the transfer path from one tip to the root.

The quantity of geometric equations is the sum of non-first inputs of all branch subsystems in the tree multibody system. The structure of coefficient matrices corresponding to tip state vectors obeys: the coefficient matrix from one tip till the output of a branch subsystem, via the first input hinge of the branch subsystem (such as $G_{1 - 19, 14}$ , $G_{3 - 20, 16}$ , $G_{1 - 22, 20}$ , $G_{24 - 22, 20}$ ) is equal to successive premultiplication of the transfer matrices of all elements along the transfer path, and then premultiplied by acceleration associated matrix (such as $H_{19, 14}$ , $H_{20, 16}$ , $H_{22, 20}$ ), whereas via the non-first input hinge of the branch subsystem (such as $G_{24 - 19, 14}$ , $G_{4 - 20, 16}$ , $G_{3 - 22, 20}$ , $G_{4 - 22, 20}$ ) is equal to successive premultiplication of the transfer matrices of all the elements along the transfer path, then premultiplied by acceleration associated matrix (such as $H_{19, 14}$ , $H_{20, 16}$ , $H_{22, 20}$ ), where it should be subtracted that the product of the acceleration and moment extraction matrix $H_{\overset{\cdot\cdot}{r}, m}$ and the transfer matrices of the elements from the inboard body of the branch subsystem till the tip.

Closed-loop-and-branch-mixed system

The overall transfer equation for closed-loop-and-branch-mixed systems is composed of three parts: the main transfer equation, geometric equations, and supplement equations. The statements of coefficient matrices in the main transfer equation and geometric equations are similar as those for tree systems.

As for the supplement equations, the quantity of supplement equations is equal to the number of closed-loop subsystems. After a closed-loop subsystem is “cut off,” a pair of “new boundaries” will be generated at the “cutting-off” point, of which the state vectors are equivalent in theory, however, transmitted in opposite directions. Therefore, according to label convention, the accelerations and angular accelerations are equal, while the internal forces and internal moments are opposite equal. Consequently, the relationship of the pair of “new state vectors” may be described by a supplement equation (such as equation (54)), that is, one state vector is equal to Matrix $C$ premultiplies the other state vector. Above-mentioned automatic deduction theorem is effective to various chain multibody systems, closed-loop multibody systems, tree systems, and closed-loop-and-branch-mixed systems.

Numerical examples

Tree MSD

In Figure 4, body 23 is hang on ceiling by a smooth ball-and-socket hinge, that is, the boundary at body 23 is simply supported. Boundaries at bodies 1, 2, 3, and 4 are free. System boundary conditions are given as

\begin{matrix} z_{23, 0} = [\begin{matrix} 0^{T} & {\overset{\cdot}{Ω}}_{O}^{T} & 0^{T} & q_{O}^{T} & 1 \end{matrix}]_{23, 0}^{T} \\ z_{k, 0} = [\begin{matrix} {\overset{\cdot\cdot}{r}}_{I}^{T} & {\overset{\cdot}{Ω}}_{I}^{T} & 0^{T} & 0^{T} & 1 \end{matrix}]_{k, 0}^{T} (k = 1, 2, 3, 4) \end{matrix}

and the initial conditions are

\begin{matrix} θ_{23} = {[\begin{matrix} 0 & 0 & π / 6 \end{matrix}]}^{T} rad, θ_{4} = {[\begin{matrix} 0 & π / 6 & 0 \end{matrix}]}^{T} rad \\ θ_{2} = θ_{1} = {[\begin{matrix} π / 6 & 0 & 0 \end{matrix}]}^{T} rad, θ_{k} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} rad \\ (k = 3, 9, 10, 11, 12, 17, 18, 21) \\ {\overset{\cdot}{θ}}_{k} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} rad s^{- 1}, (k = 1, \dots, 23) \end{matrix}

All hinges are smooth ball-and-socket hinges, and structure parameters of all rigid bodies are listed in Table 1, where m is the mass of rigid body, $J_{I}$ the moment of inertia with respect to the input. $L_{IC}$ , $L_{IO}$ , and $L_{I_{1} I_{2}}$ are position vectors of mass center, output, and the second input of the rigid body in the body-fixed coordinates. It is required to solve the motion of the system acted under gravity. $y$ axis is set opposite to the gravity direction. Gravity acceleration is $g = 9.8 m s^{- 2}$

Table 1.

Structure parameters of rigid bodies.

Sequence number	1	2	3
$m (kg)$	1.0	1.0	1.0
$J_{I} (kg m^{2})$	$J_{1}$	$J_{2}$	$J_{2}$
$L_{IC} (m)$	$[\begin{matrix} 0 & 0.5 & 0 \end{matrix}]^{T}$	$[\begin{matrix} - 0.5 & 0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0.5 & 0 & 0 \end{matrix}]^{T}$
$L_{IO} (m)$	$[\begin{matrix} 0 & 1.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} - 0.5 & 0.5 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0.5 & 0.5 & 0 \end{matrix}]^{T}$
Sequence number	4	9	10
$m (kg)$	1.0	1.0	1.0
$J_{I} (kg m^{2})$	$J_{1}$	$J_{1}$	$J_{3}$
$L_{IC} (m)$	$[\begin{matrix} 0 & 0.5 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0 & 0.5 & 0 \end{matrix}]^{T}$	$[\begin{matrix} - 0.5 & 0.5 & 0 \end{matrix}]^{T}$
$L_{IO} (m)$	$[\begin{matrix} 0 & 1.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0 & 1.0 & 0 \end{matrix}]^{T}$	${[\matrix{ { - 0.5} & {1.0} & 0 \cr } ]^{\rm{T}}}$
Sequence number	11	12	17
$m (kg)$	1.0	1.0	1.0
$J_{I} (kg m^{2})$	$J_{3}$	$J_{3}$	$J_{4}$
$L_{IC} (m)$	$[\begin{matrix} 0.5 & 0.5 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0 & 0.5 & 0 \end{matrix}]^{T}$	${[- 0.5 1.0 0]}^{T}$
$L_{IO} (m)$	$[\begin{matrix} 0.5 & 1.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0 & 1.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 1.0 & 2.0 & 0 \end{matrix}]^{T}$
$L_{I_{1} I_{2}} (m)$			$[\begin{matrix} 2.0 & 0 & 0 \end{matrix}]^{T}$
Sequence number	18	21	23
$m (kg)$	1.0	1.0	1.0
$J_{I} (kg m^{2})$	$J_{4}$	$J_{5}$	$J_{1}$
$L_{IC} (m)$	$[\begin{matrix} 1.0 & 1.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 2.0 & 2.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0.5 & 0.5 & 0 \end{matrix}]^{T}$
$L_{IO} (m)$	$[\begin{matrix} 1.0 & 2.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 2.0 & 4.0 & 0 \end{matrix}]^{T}$	$[\begin{matrix} 0 & 1.0 & 0 \end{matrix}]^{T}$
$L_{I_{1} I_{2}} (m)$	$[\begin{matrix} 2.0 & 0 & 0 \end{matrix}]^{T}$	${[4.0 0 0]}^{T}$

where

J_{1} = [\begin{matrix} 5 & 0 & 0 \\ 0 & 4.75 & 0 \\ 0 & 0 & 5 \end{matrix}], J_{2} = [\begin{matrix} 4.75 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}], J_{3} = [\begin{matrix} 5 & - 0.25 & 0 \\ - 0.25 & 5 & 0 \\ 0 & 0 & 5.25 \end{matrix}], J_{4} = [\begin{matrix} 9 & - 1 & 0 \\ - 1 & 9 & 0 \\ 0 & 0 & 10 \end{matrix}], J_{5} = [\begin{matrix} 16 & - 4 & 0 \\ - 4 & 16 & 0 \\ 0 & 0 & 20 \end{matrix}]

The tree MSD are computed by the new version of MSTMM and ADAMS software, respectively. Time history of rotation angles in three orientations of four rigid bodies are shown in Figure 7. It can be seen that the results obtained by these two methods have good agreements, which validates the automatic deduction method of overall transfer equation for tree multibody systems.

Figure 7.

Time history of rotation angles of body 1, 2, 21, and 23 in tree system.

Dynamics of closed-loop-and-branch-mixed systems

In the closed-loop-and-branch-mixed system shown in Figure 2, body 23 is hang on ceiling by a smooth ball-and-socket hinge. A pair of “new boundaries” are generated after “cutting off” the closed loop at point $P_{3, 24}$ , of which the state vectors are $z_{24, 0}$ and $z_{3, 0}$ . Then the topology graph becomes into Figure 6. The boundary conditions are

\begin{matrix} z_{23, 0} = [\begin{matrix} 0^{T} & {\overset{\cdot}{Ω}}_{O}^{T} & 0^{T} & q_{O}^{T} & 1 \end{matrix}]_{23, 0}^{T} \\ z_{k, 0} = [\begin{matrix} {\overset{\cdot\cdot}{r}}_{I}^{T} & {\overset{\cdot}{Ω}}_{I}^{T} & 0^{T} & 0^{T} & 1 \end{matrix}]_{k, 0}^{T}, (k = 1, 4) \\ z_{k, 0} = [\begin{matrix} {\overset{\cdot\cdot}{r}}_{I}^{T} & {\overset{\cdot}{Ω}}_{I}^{T} & {m_{I}}^{T} & {q_{I}}^{T} & 1 \end{matrix}]_{k, 0}^{T}, (k = 3, 24) \end{matrix}

and the initial conditions of the system are

\begin{matrix} θ_{23} = {[\begin{matrix} 0 & 0 & π / 3 \end{matrix}]}^{T} rad, θ_{4} = {[\begin{matrix} π / 6 & 0 & 0 \end{matrix}]}^{T} rad \\ θ_{1} = {[\begin{matrix} 0 & π / 6 & 0 \end{matrix}]}^{T} rad, θ_{k} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} rad \\ (k = 2, 3, 9, 10, 11, 12, 17, 18, 21) \\ {\overset{\cdot}{θ}}_{k} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} rad s^{- 1}, (k = 1, \dots, 23) \end{matrix}

Hinge 24 is a smooth ball-and-socket hinge, while the structure parameters of all the other elements are identical to the corresponding elements in the example of tree multibody system. The dynamics of the mixed system are computed by the new version of MSTMM and ADAMS software. The computational results are given in Figure 8, from which results by these two methods are in good agreements. This validates the automatic deduction method of overall transfer equation for closed-loop-and-branch-mixed systems.

Figure 8.

Time history of rotation angles of body 1, 2, 21, and 23 in mixed system.

Dynamics of large closed-loop-and-branch-mixed systems

The large closed-loop-and-branch-mixed system containing 250 rigid bodies is shown in Figure 9. Body 499 is hang on ceiling by a smooth ball-and-socket hinge. Boundaries of body 1 and body 4 are free. The closed loop is “cut off” at point $P_{3, 500}$ . In summary, the boundary conditions are given as

\begin{matrix} z_{499, 0} = [\begin{matrix} 0^{T} & {\overset{\cdot}{Ω}}_{O}^{T} & 0^{T} & q_{O}^{T} & 1 \end{matrix}]_{499, 0}^{T} \\ z_{k, 0} = [\begin{matrix} {\overset{\cdot\cdot}{r}}_{I}^{T} & {\overset{\cdot}{Ω}}_{I}^{T} & 0^{T} & 0^{T} & 1 \end{matrix}]_{k, 0}^{T}, (k = 1, 4) \\ z_{k, 0} = [\begin{matrix} {\overset{\cdot\cdot}{r}}_{I}^{T} & {\overset{\cdot}{Ω}}_{I}^{T} & m_{I}^{T} & q_{I}^{T} & 1 \end{matrix}]_{k, 0}^{T}, (k = 3, 500) \end{matrix}

The system initial conditions are

\begin{array}{l} θ_{23} = {[\begin{matrix} 0 & 0 & π / 3 \end{matrix}]}^{T} rad, θ_{1} = θ_{9} = {[\begin{matrix} 0 & π / 6 & 0 \end{matrix}]}^{T} rad \\ θ_{4} = {[\begin{matrix} π / 6 & 0 & 0 \end{matrix}]}^{T} rad, θ_{k} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} rad \\ (k = 1, 2, 3, 9, 10, 11, 13, 15, 17, 19, 21, 25, \dots, 499) \\ {\tilde{θ}}_{k} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} rad s^{- 1}, (k = 2, 3, 4, 9, 10, 11, 13, 15, 17, \dots, 499) \end{array}

All hinges are smooth ball-and-socket hinges, and the structure parameters of body 1, 2, 3, 4, 9, 10, 11, 12, 17, 18, 21, and 23 are the same with that of the same sequence number in Table 1. The structure parameters of body 25, 27, 29,…, 499 are the same with those of body 23.

Figure 9.

Topology graph of the large mixed system with 250 bodies.

The dynamics of large closed-loop-and-branch-mixed system are computed by new version of MSTMM with Riccati transformation^10,11 and ADAMS software. The rotation angles varied with time of four rigid bodies are depicted in Figure 10, from which the results obtained by these two methods have good agreements.

Figure 10.

Time history of rotation angles of body 1, 2, 18, and 21 in large mixed system.

The overall transfer matrix of chain system is equal to the successive multiplication of transfer matrices of all elements. Due to the constant order of transfer matrix (the order of transfer matrix is 7 with planar motion while 13 with spatial motion), the system works with $O (n_{B})$ operations, where $n_{B}$ is the number of bodies in the system. Table 2 shows the comparison of computational complexity among the new version of MSTMM, the new version of MSTMM through Riccati transformation, and the recursive algorithm proposed by Brandl et al.⁹ on computing a chain system composed of $n / 3$ spatial rigid bodies and connected with smooth ball-and-socket hinges. The result implies that the new version of MSTMM through Riccati transformation is the best on computational complexity. Although the new version of MSTMM is not as good as the other two in terms of computational complexity, however, provides a much more concise method on programming. The system works with $O (n_{B}) + O (n_{L})$ operations for closed-loop system in the best case that the closed loops are not interconnected heavily, and with $O (n_{B}) + O (n_{B} n_{L} + n_{L}^{2})$ in the worst case. As for the mixed system, the part of computing the transfer matrices of elements always works with $O (n_{B})$ . However, for the part of overall transfer matrix, the computational complexity depends on the number of branches, the input number of each multi-input body, and the connecting way of the branches. Computational precision of the new version of MSTMM is discussed in Rui et al.¹⁷, see Table 1, which demonstrates that the new version of MSTMM has higher precision than discrete time transfer matrix method for MSD.

Table 2.

Computational complexity of dynamical methods.

Dynamical method	M	A
New version of MSTMM	$1757 n - 863$	$1570 n - 794$
Recursive algorithm of Brandl	$777 n - 223$	$650 n - 197$
New version of MSTMM through Riccati transformation	$731 n - 297$	$602 n - 258$

Conclusion

By defining branch subsystems, the general transfer equation and transfer matrix for branch subsystems are developed, which simplifies the deduction process and form of overall transfer equation of system greatly for complicated multibody systems.

Accordingly, the tree multibody system may be regarded as the assembling of branch subsystems and simple chain subsystems. Thus, it is convenient to establish the overall transfer equation and to realize programming using the automatic deduction of overall transfer equation for tree multibody systems.

Meanwhile, by “cutting off” the closed-loop subsystem, any closed-loop-and-branch-mixed system may be treated as a tree multibody system with “new boundaries,” further, as the assembling of branch subsystems and simple chain subsystems. It becomes convenient to establish overall transfer equation for closed-loop-and-branch-mixed system, and automatic deduction of overall transfer equation for closed-loop-and-branch-mixed system is realized.

In comparison of the proposed method and ADAMS software, the results show that the proposed method is valid and has following features: no need of global dynamics equation; high computational speed, such as, for the third numerical example, it takes 6.52 s by the proposed method to compute the system dynamics, while 238 s by ADAMS; highly programmable and convenient to be applied in engineering.

In comparison of the new version of MSTMM to recursive method by Brandl et al.⁹, the new version of MSTMM through Riccati transformation is a little more efficient than the nth order recursive method.

In general, any multibody system, according to its topology graph, may always be categorized into one of the following systems: chain systems, closed-loop systems, branch systems, tree systems, or closed-loop-and-branch-mixed systems. Automatic deduction theorem of overall transfer equation has been given for chain systems and closed-loop systems of new version of MSTMM in Rui et al.¹⁹ and for branch systems of new version of MSTMM has been given in Rui et al.²⁰ Hence, automatic deduction theorems of overall transfer equation for general systems are composed by and Rui et al.^19,20, and this paper together.

Footnotes

Appendix 1 Acknowledgements

The authors owe special thanks to Prof. Dieter Bestle from Brandenburgische Technische Universität, for his instructive suggestions and offering lots of important discussion on this article during his visit to Institute of Launch Dynamics, Nanjing University of Science and Technology.

Handling Editor: Crinela Pislaru

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research in this article is supported by Research Fund for the Doctoral Program of Higher Education of China (20113219110025, no. 20133219110037), Nature Science Foundation of China Government (11472135), and Program for New Century Excellent Talents (NCET-10-0075).

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