Pierre M. Larochelle
e-mail: pierrel@fit.edu
Agnes M. Agius
Mechanical & Aerospace Engineering
Department,
Florida Institute of Technology, Melbourne, FL
Interactive Visualization of the
Coupler Surfaces of the Spatial
4C Mechanism
This paper presents a methodology for generating and displaying the coupler surfaces of
the spatial 4C mechanism. The 4C mechanism is a two-degree-of-freedom spatial four-
bar linkage. The path traced by a point attached to the coupler or floating link defines a
surface in space. It is shown that the coupler surface of the spatial 4C mechanism is a
ruled surface with 16th degree directrices. An interactive graphical user interface has
been created and integrated with kinematic analysis routines to generate and interac-
tively display these coupler surfaces. The result is a user-friendly and efficient means of
generating and visualizing the coupler surfaces of the spatial 4C four-bar
mechanism. DOI: 10.1115/1.2049067
1 Introduction
In this paper we present a parametrized representation of the
coupler surface of the general spatial 4C four-bar mechanism see
see Fig. 1 devised to facilitate interactive visualization. The
methodology utilizes an analytic representation of the linkage’s
coupler point position that is parametrized by the mechanism’s
joint variables i.e., rotation angles and translational distances.
The kinematic analysis of the kinematic closed chain is utilized to
eliminate all but two joint variables. The result is a representation
of the coupler surface associated with a coupler point on a spatial
four-bar mechanism that is parametrized by its two driving joint
variables.
Recently, there have been some significant efforts made to ad-
dress the challenge of designing useful spatial mechanisms. A
novel idea of serially coupling spatial joints to facilitate the gen-
eration of spatial motion was presented in 1. In 2, a Burmester-
Theory-based computer-aided design program for spatial 4C
mechanisms was reported. Efforts were made to address circuit
and branch defects in 3. Approximate motion synthesis was ad-
dressed in 4 and 5, whereas function generation has been ad-
dress in 6. Point paths of the RCCC mechanism were studied in
7, and the design of these mechanisms to avoid jamming move-
ments was addressed in 8. The invariant properties of the mo-
tions that spatial mechanisms generate has been studied in 9.A
study of the effects of joint clearance in spatial mechanisms is
reported in 10. The exploration of utilizing virtual reality tech-
niques to address the inherent visualization and interaction chal-
lenges was reported in 11. The design of a passively balanced
spatial linkage for use in a haptic interface was presented by 12.
A study of fabrication errors in the manufacture of spatial mecha-
nisms was reported in 13. Moreover, the work reported here was
motivated to a large extent by the study of the coupler curves of
the RCCC mechanism reported by Marble and Pennock in 14.
They used a dual-number approach to generate parametric equa-
tions of the coupler curves of RCCC spatial four-bar mechanisms.
Furthermore, Marble and Pennock proved that the polynomial de-
scribing the coupler curve is 16th degree. Here, we build upon
their work and examine the coupler surfaces of the 4C mecha-
nism.
The immediate goal here was to perform a kinematic study of
the coupler surfaces of the spatial 4C mechanism and to create an
interactive visualization tool to present these surfaces. This is one
step toward the long-term goal of creating an interactive
computer-aided-design software tool to synthesize spatial 4C
mechanisms for prescribed coupler surfaces. Such mechanisms
would be applicable wherever complex surface motions are re-
quired and/or manufactured; such as in free-form modeling and
manufacturing, mold and pattern making, surface finishing, etc.
Consider this conceptual example of free-form manufacturing, a
NURBS-based software e.g., Alias’ SurfaceStudio™ or McNeel
& Associates’ Rhinoceros™ could be used to design and pre-
scribe a desired complex free-form surface. A spatial 4C mecha-
nism could then be synthesized such that its coupler surface in-
cludes the prescribed surface. Finally, the free-form surface could
be manufactured via material removal from raw stock by attach-
ing a cutting tool to the coupler of the mechanism such that the tip
of a cutting tool is coincident with the coupler point. See 15 for
a study of the rapid-prototyping manufacturability of ruled sur-
faces. Let us consider another conceptual example, the design and
manufacture of a complex automobile quarter panel. These sur-
faces are often generated with free-form surface-design software
programs. A spatial 4C mechanism could be designed such that its
coupler point traces the prescribed surface. Such a mechanism
could be applicable to 1 making the patterns/dies used to manu-
facture the quarter panels, 2 surface finishing of the quarter pan-
els grinding, sanding, painting, etc., and, 3 inspecting the fin-
ished quarter panels e.g., cameras, touch probes, etc.. In 2001,
16 proposed a framework for the interactive design of rigid-
body motions. Here, one could envision future work to address
interactive design of spatial mechanism for desired coupler sur-
faces.
The paper proceeds as follows. First, the necessary kinematic
analyses of the spatial 4C mechanism are performed. Next, we
utilize the results of this analysis to reveal the properties and
structure of the coupler surface. We continue by utilizing the re-
sults of the analyses to generate the parametric representation of
the coupler surface of the spatial 4C mechanism. Finally, the in-
teractive visualization program SPASUR spatial surfaces17 is
used to present images of example coupler surfaces.
1
2 Spatial 4C Mechanism
A spatial 4C mechanism has four cylindrical or C joints, each
joint permitting relative rotation and translation along a line, see
Fig. 1. These four C joints connect four rigid bodies to form a
Contributed by Mechanisms and Robotics Committee for publication in the JOUR-
NAL OF MECHANICAL DESIGN. Manuscript received December 16, 2004; revised re-
ceived March 14, 2005. Associate Editor: Gordon R. Pennock.
1
Source code for SPASUR is available on request from the author.
1122 / Vol. 127, NOVEMBER 2005 Copyright © 2005 by ASME Transactions of the ASME