Reduced multibody system transfer matrix method using decoupled hinge equations

In the multibody system transfer matrix method (MSTMM), the transfer matrix of body elements may be directly obtained from kinematic and kinetic equations. However, regarding the transfer matrices of hinge elements, typically information of their outboard body is involved complicating modeling and even resulting in combinatorial problems w.r.t. various types of outboard body's output links. This problem may be resolved by formulating decoupled hinge equations and introducing the Riccati transformation in the new version of MSTMM called the reduced multibody system transfer matrix method in this paper. Systematic procedures for chain, tree, closed‐loop, and arbitrary general systems are defined, respectively, to generate the overall system equations satisfying the boundary conditions of the system during the entire computational process. As a result, accumulation errors are avoided and computational stability is guaranteed even for huge systems with long chains as demonstrated by examples and comparison with commercial software automatic dynamic analysis of the mechanical system.


| INTRODUCTION
Multibody system simulation has been developed and gradually become one of the most important foundations of dynamic system design. [1][2][3] Especially in China, the multibody system transfer matrix method 4 (MSTMM) as a rather new modeling method gained great attention and has been applied to many engineering problems. The principle of MSTMM is modular modeling of every element by utilizing linear relationships among kinematic and kinetic quantities acting on it. Any general multibody system can be broken up into two kinds of elements, bodies and hinges, whose dynamics properties can be easily expressed in a transfer matrix form. Both share a parallel status, which avoids the difficulty caused by a special hinge treatment as a constraint as required in ordinary multibody system dynamics. The transfer matrix of each element is regarded as a building block, and a generic library of element transfer matrices 5 can be developed in advance, which then can be assembled according to a specific system topology. 6 The transfer matrix method was originally developed for linear However, there still remain two problems to be resolved in this paper. In principle, the overall transfer matrix of a system is obtained by multiplying its element transfer matrices. For a large system, the accumulative error resulting from successive multiplication of a large number of transfer matrices in a long chain may lead to computational stability problems, 8 which may be resolved by drawing lessons from the Riccati transfer matrix method 9 characterized by the Riccati transformation. 10 A second problem is that hinge transfer equations depend on the information of its outboard body and even the next hinge which hinders its applicability. This can be avoided by a specific description of hinges in the context of Riccati transformation. 11 In this paper, these two ideas are generalized by defining various reduced transformations to enhance the element library for modeling any type of system, such as chains, trees, closed-loops, and any general system with the features of being time-variant, nonlinear, and involving large motion, Figure 1. After deducing basic element equations in Section 2, systematic procedures for setting up the overall system equations will be discussed in  Numerical examples in Section 7 will validate the proposed method called the reduced multibody system transfer matrix method (RMSTMM). It should be pointed out that the paper focuses on deriving acceleration equations in a recursive manner similar to classical approaches like Brandl, 12 but requiring only a single sweep.
For further numerical integration, well-known schemes are used and not discussed in the paper. respectively. The interacting internal forces q and torques m in Figure 2A are defined such that they fulfill Newton's third law. Connection of two adjacent elements i and j according to O I ≡ i j then yields the identities

| BASIC ELEMENT EQUATIONS
between output end O i and input end I j , which are later essential for recursively sweeping through the system and connecting the elements.
In the inertial frame Sxyz { }, the position, velocity, and acceleration of an arbitrary point P on a body in Figure 2B can be denoted as follows: I  I P  I  I  P  I  I I IP  I  I IP  I  IP I   P  I  I  I I I Further, rotation is identical for every point on a rigid body, especially Newton's second law a F m = C relates the mass m of a rigid body, absolute acceleration a r = C C of its mass center and resultant force F. With P C = in Equation (2) we get from Figure 2B F I G U R E 1 Dynamics models of chain (A), tree (B), closed-loop (C), and general system (D) The moment of momentum is given as follows: .   I  I  I  I  I  I  I  I I  I I  I  I  I  I  I I  I  I  I   I I  I  I  I  I  I  I  I  I  I   I I  I  I  I  I  I   I  I  I   I  I  I   I  I   T  T  T   T  T T  T   T where z a k , refers to forces and torques acting on input end I k and Further, kinematics Equations (3) and (4) may be summarized as follows: where For multi-input bodies, rigid body kinematics (3), (4) is also valid between the first input end I I ≡ 1 and remaining input ends I k , which may be written as consistency equations where According to the definitions (11), these equations may be summarized as follows: Such sets of three hinge equations may be also found for pin hinges and translational hinges, 13 where the first describes the force/torque equilibrium, the second vanishing forces/torques in free moving directions due to smoothness, and the third motion consistency in constrained directions of the hinge. In the following, these basic body and hinge transfer relations will be utilized as building blocks for obtaining the modeling procedure for the multibody system with various topologies.

| RECURSIVE EQUATIONS FOR CHAIN SYSTEMS
A chain system as shown in Figure 1A with its corresponding topology graph in Figure 3A always has at least one free boundary end, which is regarded as the tip corresponding to the first element of the system. In such a case, the Riccati transformation 9 assumes that one part of state vectors (11) can be expressed as a function of the other, for example, as follows: Due to vanishing forces and torques at a free boundary, this is obvious for the tip body 1 reading as follows: Applied to connection points between the elements of the chain, the reduced transformation (21) which allows to transfer from tip to root with identities ≡ .
By sweeping through the chain along the transfer path in Figure

| Body treatment
The second equation yields Substituting the first equation of (23) and Equation (27) into the first equation of (26) yields

| Hinge treatment
The reduced transfer equation of a hinge can be obtained by substituting the reduced transformation (23) into the set of basic Equation (19): By assembling the last two equations in matrix form and with desired hinge transfer equations S e S e ( , ) → ( , )

| Recursive procedure for chain systems
With the above relations we may now summarize the complete recursive procedure for a chain system as follows:  (25). (v) Extract unknown accelerations for feeding a numerical integration scheme.
(vi) For the chain system, it is obvious that the number of multiplications of matrices scales linearly with the number of elements where the transfer matrices have a rather low order independent of the DOF of the multibody system.

| TREE STRUCTURES
Next, we require a procedure to obtain output quantities S e , from input information S e , By substituting into transfer Equation (12) yields where Substituting consistency Equation (16), that is, into Equation (37) yields From Equation (14), being identical to Equations (27) where After obtaining matrices for output end of element 11 from Equation (43), we may proceed along the output chain in Figure 3B from element 12 to the output end of the element (2n−1) with the procedure in Section 3:

| TREATMENT OF ISOLATED CLOSED-LOOPS
The treatment of a closed-loop system as shown in Figure 1C Then, solution of the second equation of body transfer Equation (1) , This indicates that in a closed-loop the reduced transformation (21) should be generally expanded by the unknown cut-state z a C , as follows: Especially for the input end of the element i = 1 of the spanning chain in Figure 1C, the first equation of identity (46) may be then written as follows: (1) , As will be shown later, a recursive procedure similar to that for chains can be established for the spanning chain finally starting from Equation Noting that the second part of the closed-loop condition (46) has not been taken into account yet, substituting Equation (47) into the second equation of (46) yields , (1) , , We may now assume that state vectors of all connecting points in a closed-loop satisfy such a relation between actual state z b and cut state z a C , , z b C , of the spanning chain, that is, which may be interpreted as a complementary equation for a closedloop system. Especially for the input end of element i = 1 of the spanning chain in Figure 3C, this reads as follows: , , due to closing condition (46) and for the output end of element n as Combining it with Equation (51) yields where now the number of equations equals that of unknowns and allows to compute the unknown cut state z a C , and z b C , . Next, the reduced transfer matrices of Equations (49) and (53) for bodies and hinges need to be determined to complete the concept.

| Body treatment
Equations (49) and (53) for the input of any single-input body i in a closed-loop read as The second body transfer equation still results in Equation (27) and (28) which, however, may be extended artificially as follows: Substitution into Equation (58) with recursive formulas

| Hinge treatment
The relations (57) are also applicable here. Substitution into hinge Combining the last two equations and solving them for Finally, substituting Equation (59) into the second equation of (57) where The kinematics relationships (2) and (4) of rigid body 1 between cutoff point P I (1) and attachment point P (1) where  The remaining part of the system in Figure 3D is a tree sub- 6×1 for the free tip P I (11) and S e , For hinges, Equations (66), (67), and (64) have to be supplemented by The whole procedure for a general system may be summarized as follows:

| VALIDATION BY NUMERICAL EXAMPLES
To validate the proposed strategies and demonstrate their efficiency, three types of systems will be simulated: a tree system, an isolated closed loop, and a combination of loop and tree as shown in Figure 1.

| Huge spatial tree systems
To see the computational efficiency of the proposed approach called RMSTMM, numerical simulations of the spatial tree system in Figure 1B

| Some closed-loop simulations
Next, the planar closed-loop system in Figure 1C after each time-step the positions are corrected with a Newton-Raphson iteration solving Equation (86) according to Zhang. 15 Results of all three methods agree well up to n/2 = 24 bodies, see Figure 5. However, for n/2 = 25 bodies, computational instability

| Simulation of a general system with a closed-loop subsystem
Finally, the planar general system composed of a closed-loop and a tree as shown in Figure 1D is computed with RMSTMM, NV-MSTMM, 13  . The initial position is shown in Figure 1D, all initial speeds are zero. The simulation results for some angular speeds shown in Figure 8 have good agreement validating the concept.
It may be pointed out that the same scheme as in Section 7.2 was used for RMSTMM. Obviously, the position correction of (86) works well as the total energy stays constant in Figure 9A and position violations are small in Figure 9B (green and black curves). Without this correction, position errors are higher and may grow (blue and red curves in Figure 9B).

| CONCLUSIONS
The paper systematically develops reduced transformations and recursive schemes, which make RMSTMM generally and easily applicable to multibody systems with chain, closed-loop, tree, and general topology. Compared with NV-MSTMM, the method not only benefits from high algorithmic stability but is much easier to use due to fully independent hinge equations and ensures strict

CONFLICT OF INTEREST
The authors declare that there are no conflict of interest.