REPRESENTATION OF 3D MOTION BY PROJECTIVE ANGLES
Giovanni Legnani
Department of Mechanical and
Industrial Engineering
University of Brescia
Brescia, Italy
Email: giovanni.legnani@unibs.it
Irene Fassi
Institute of Industrial
Technologies and Automation
National Research Council
Milan, Italy
Email: irene.fassi@itia.cnr.it
ABSTRACT
Angular motion in 3D space has been represented
using several approaches, such as cardanic angles,
quaternions, Euler parameters. Notwithstanding that in some
applications some of them are more convenient, none of these
solutions is completely satisfactory as a general tool. After a
critical review of the main advantages and drawbacks of each
notation, the paper defines and discusses the major
characteristics of a new angular convention that has recently
been proposed. The known results are summarized in a clear
and systematic way and the analysis is extended to the analysis
of the angular velocity and acceleration.
INTRODUCTION
The methodology to represent angular position (attitude) of
a body in 3D space is a subject that has been widely discussed
in literature [1, 2, 3]. Different notations have been proposed to
describe body attitude, rotations and the corresponding angular
velocity and acceleration during general 3D motion, such as:
Cardanic angles, Euler angles, Euler parameters, Rodriguez-
Hamilton parameter, quaternions, and others. The high number
of available notations highlights the fact that none of them is
fully satisfactory and applicable to any situation.
One of the major characteristics of 3D angular motion is
the so called "non integrability of the angular velocity" [4]. In
practice, it is well known that it does not exist any set of angular
coordinates whose time derivative coincide with the angular
velocity. In other words, the time integral of the angular
velocity is not a valid set of angular coordinates. This concept is
illustrated by the example of Figure 1. A rigid body whose
initial attitude is illustrated in the top left part of the figure is
first rotated by 90° degrees around Z axis, and then by 90°
degrees around Y in order to reach the attitude (1). If the two
rotations are applied in the reverse order (first Y, and then Z),
the body reaches the attitude (2). The angular velocities for both
cases are also plotted in Figure 1. It is evident that in both
cases, while the final angular position is different, the time
integral is identical:
90 dt a
y y
=
90 dt a
z z
.
FIGURE 1. EXAMPLE TO ILLUSTRATE THE CONCEPT OF NON
INTEGRABILITY OF THE ANGULAR VELOCITY
In many cases (e.g. Euler angles or Cardan angles) three
parameters are utilized, in other cases (e.g. quaternions) the 3D
x y
z
z
z
y
y
z
z
y
y
(2)
(1)
(1) (2)
t
t
t
t
Proceedings of the ASME 2015 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2015
August 2-5, 2015, Boston, Massachusetts, USA
DETC2015-46242
1 Copyright © 2015 by ASME