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RESEARCH PAPERS

# Constraint Wrench Formulation for Closed-Loop Systems Using Two-Level Recursions

[+] Author and Article Information
Himanshu Chaudhary

M.L.V. Textile and Engineering College Bhilwara, Pratap Nagar, Bhilwara (Rajasthan) 311001, Indiahimanshubhl@rediffmail.com

Subir Kumar Saha

Department of Mechanical Engineering, I.I.T. Delhi, Hauz Khas, New Delhi 110016, Indiasaha@mech.iitd.ernet.in

Note that n-vector represents n dimensional vector.

J. Mech. Des 129(12), 1234-1242 (Jan 12, 2007) (9 pages) doi:10.1115/1.2779890 History: Received August 22, 2006; Revised January 12, 2007

## Abstract

In order to compute the constraint moments and forces, together referred here as wrenches, in closed-loop mechanical systems, it is necessary to formulate a dynamics problem in a suitable manner so that the wrenches can be computed efficiently. A new constraint wrench formulation for closed-loop systems is presented in this paper using two-level recursions, namely, subsystem level and body level. A subsystem is referred here as the serial- or tree-type branches of a spanning tree obtained by cutting the appropriate joints of the closed loops of the system at hand. For each subsystem, unconstrained Newton–Euler equations of motion are systematically reduced to a minimal set in terms of the Lagrange multipliers representing the constraint wrenches at the cut joints and the driving torques/forces provided by the actuators. The set of unknown Lagrange multipliers and the driving torques/forces associated to all subsystems are solved in a recursive fashion using the concepts of a determinate subsystem. Next, the constraint forces and moments at the uncut joints of each subsystem are calculated recursively from one body to another. Effectiveness of the proposed algorithm is illustrated using a multiloop planar carpet scraping machine and the spatial RSSR (where R and S stand for revolute and spherical, respectively) mechanism.

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## Figures

Figure 7

Subsystems of spanning tree for the carpet scraping mechanism

Figure 8

Comparison of driving torque

Figure 1

Closed loops of a multiloop system: (a) the closed loops and (b) its open loops

Figure 2

A spanning tree

Figure 3

Open systems for four-bar mechanism

Figure 4

Coupled bodies

Figure 5

Constraint wrenches calculation using two-level recursive subsystem approach

Figure 6

Carpet scraping mechanism

Figure 9

Figure 10

Spanning tree of the 7R mechanism

## Errata

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