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Research Papers

# Special Coordinates Associated With Recursive Forward Dynamics Algorithm for Open Loop Rigid Multibody Systems

[+] Author and Article Information
Sangamesh R. Deepak

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiasangu.09@gmail.com

Ashitava Ghosal1

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiaasitava@mecheng.iisc.ernet.in

We have used the term coordinates to mean the quantities that independently and completely describe a subset or all of the rigid bodies making up a multibody system.

The symbol $y$ actually represents coordinates stacked in the form of a vector.

The coordinates could be pseudocoordinates. So $ẏi$ could be familiar translational and angular velocities, while $yi$ itself is symbolic.

There may be problems where $fj$ may depend linearly or nonlinearly on $λ$. Those situations arise when dry friction is modeled into the equations. We will not consider such cases here.

Appendix shows how to find such an $Ek$ and $Dk$.

Descendants of node $i$ can always be arranged in a sequence. For example, the descendants of Node 2 in Fig. 1 can be arranged as 4, 5, 7, 10, 11, and 12. For this order, $h(2,3)=7$.

If the driving constraint is holonomic, then the constraint can be written as $Φ¯(yj,yk,qk(t),t)=0$. Its differentiation gives $(∂Φ¯∕∂yj)ẏj+(∂Φ¯∕∂yk)ẏk=−∂Φ¯∕∂t−(∂Φ¯∕∂qk)k̇k$. The above equation is the motivation for Eq. 1.

1

Corresponding author.

J. Appl. Mech 75(5), 051003 (Jul 02, 2008) (11 pages) doi:10.1115/1.2936923 History: Received March 12, 2007; Revised April 04, 2008; Published July 02, 2008

## Abstract

The recursive forward dynamics algorithm (RFDA) for a tree structured rigid multibody system has two stages. In the first stage, while going down the tree, certain equations are associated with each node. These equations are decoupled from the equations related to the node’s descendants. We refer them as the equations of RFDA of the node and the current paper derives them in a new way. In the new derivation, associated with each node, we recursively obtain the coordinates, which describe the system consisting of the node and all its descendants. The special property of these coordinates is that a portion of the equations of motion with respect to these coordinates is actually the equations of RFDA associated with the node. We first show the derivation for a two noded system and then extend to a general tree structure. Two examples are used to illustrate the derivation. While the derivation conclusively shows that equations of RFDA are part of equations of motion, it most importantly gives the associated coordinates and the left out portion of the equations of motion. These are significant insights into the RFDA.

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## Figures

Figure 1

A typical tree structure

Figure 2

A simple example given in Ref. 2

Figure 3

Two planar rigid bodies with a revolute joint at point Pj=Pk

Figure 4

Visualization of various coordinates for body k: (a), (b), and (c) indicate small changes in xPk, yPk, and ϕk, respectively, as a part of coordinate y¯k (see Eq. 24); (d)–(f) indicate a small change in components of y̆kc (see Eq. 39)

Figure 5

Changes in system due to small changes in the coordinate [yjTỹkfT]T defined in Eq. 48

Figure 6

Changes in system due to small changes in the coordinate y¯, defined in equation 20

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