Global Journal of Engineering Sciences (GJES)
Discussion
of Solutions for Combining Rotations
Authored by Shuh Jing Ying
Abstract
Combining
two rotations into one is described in detail in the book Advanced Dynamics by
Shuh Jing Ying. The solutions are available and are verified as presented in
the 2019 ASEE Annual Conference. The repeated use of the combination of two
rotations into one allows the combination of many rotations into one. This is
especially useful when time is of the essence, such as for a guided missile
shooting enemy’s missile in the mid-air. So, this is a very important subject.
However, when carefully looking into details, it is discovered that the
solutions are not unique. Since this is a very important subject, it is
worthwhile to be discussed.
Introduction
If
this is the first time that you have seen how two rotations are combined into
one rotation, it is not surprising since Ying’s Advanced Dynamics has a limited distribution.
So, the ideas expounded below may seem innovative and new, yet they are not.
The purpose of this paper is to demonstrate the use of rotation operators to
solve this problem and to inspire readers to create other innovative solutions.
This is the educational purpose of this paper. Since this solution is based on
rotation operators, and rotation operators are often overlooked in dynamics,
let us begin with a brief historical overview of rotation operator. Then, I
will start from the definition of rotation operator, provide examples of
operations, and then verify the solutions.
Rotation
operator was first introduced by JW Gibbs in 1901 as mentioned in Ying’s Advanced Dynamics [1]. A search of the
online scientific literature revealed no papers directly related to this study.
However, there are some papers parallel to the rotation operator in this paper,
for example, “Beyond Euler angles: Exploiting the angle- axis parametrization
in a multipole expansion of the rotation operator” by Mark Siemens et al. [2]
uses angle and rotating axis as the arguments for the operator, similar to the
rotation operator in this paper but the application is in a totally different
field, quantum mechanics. Other useful references related to the vector and
tensors analyses include the following: Advanced Dynamics by D Greenwood [3];
Vector analysis and Cartesian Tensors by D Bourne [4] and From Vectors to
Tensors by JR Ruiz-Tolosa [5].
Review of Rotation Operators
Definition of rotation operator
The
notations used in the equations are as follows: bold letters represent vectors,
a double arrow on the top of a letter indicates a dyad or dyadic. A pair of
vectors written in a definite order, such as ij, is called dyad and a linear
combination of dyads is known as a dyadic. Now, consider that a position vector
r is rotated with respect to vector n by angle β to r’. The angle β is measured
in the plane perpendicular to n, containing the ends of vectors r and r’ in
that plane as shown in Figure 1. Let a be a vector with the direction of n and
the magnitude of the component of r along n, so that
a =
n(r.n)
Let b
and c be vectors in the circular plane, which is the top view of Fig. 1a
looking down directly along – n. Hence
r' = a
+ b + c
The
radius of the circle is r sinθ where θ is the angle between r and n, or
Combination of Two Successive Rotations about Different Axes by
One Rotation
Suppose a rigid body to be
rotated by two steps. First it is rotated about the k axis by an angle of φ and
then it is rotated about k’ axis by an angle of ψ. The directions of k and k’
are known, and the plane containing them is determined. Choose the x axis
perpendicular to the plane. Suppose the true angle between k and k’ is θ, as
shown in Figure 2, then
n Eq (29) because of
the missing of sin(β⁄2) or cos(β⁄2), the vector n is not a unit vector, but the
vector is in the same direction as in the other two possible solutions. But β =
sin(-1)(A). The rotated angle is not the same as in the first solution. It
cannot be the right solution. To make it clear a numerical example is given
below.
Numerical Example
Let us consider the
coordinates as follows: taking the X axis parallel to latitude with the
positive X pointing east, Y axis parallel to longitude with positive Y pointing
north and Z axis perpendicular to the surface of Earth, the origin of the
coordinates at the center of the ball joint. The center line of the missile is
along the position vector
r=10 i+20 j+50 k
• Suppose that it is required
to rotate π/36 with respect to i axis and then rotate π/8 with respect to k
axis. Find the final position vector of missile.
• To illustrate the use of the
formulas for the combined rotation, calculate the rotation of the missile from
the original position by angle of β about the axis n. And compare the results.
(a) Calculate β and n according to first set of the solutions as given in Eqs
(24) and (25); (b) according to second set of solutions as given in Eqs (26)
and (27); (c) according to third set of the solutions as given in Eqs (28) and
(29).
n ∙ r = (0.2141998 i +
0.04260687 j + 0.9758599 k) ∙ (10 i + 20 j + 50 k)
=2.141998 + 0.8521374
+ 48.792995 = 51.7871304
n × r = (0.2141998 i +
0.04260687 j+0.9758599 k) × (10 i +20 j + 50 k)
=4.283996 k - 10.70999
j - 0.4260687 k + 2.1303435 i + 9.758599 j - 19.517198 i
= -17.3865545 i -
0.951391 j + 3.8579273 k
r2= (1-0.920219059) (51.7871304) (0.2141998 i + 0.04260687 j+ 0.9758599
k) + 0.920219059 (10 i +20 j + 50 k) + 0.391403733 (-17.3865545 i-0.951391
j+3.8579273 k)
=0.884993461
i+0.176035651 j+4.03188813 k + 9.20219059 i+18.40438118 j+46.01095295 k
- 6.805162335
i-0.372377988 j+1.510007147 k
=3.282021716
i+18.20803884 j+51.55284823 k
=3.282 i+ 18.208
j+51.553 k (31)
Compare Eq. (31) with
Eq. (30), we find that they are identical. So, they reached the same results.
b. According to the
second set of solutions,
n ∙ r = 0.8556254 +
0.340388 + 19.49046 = 20.6864734
n x r = k (0.08556254
x 20) – j (0.08556254 x 50) - k (0.0170194 x 10)
+ i (0.0170194 x 50) +
j (0.3898092 x 10) – i (0.3898092 x 20)
= - 6.945214 i -
0.380035 j + 1.5410568 k
Conclusion
It is found that there are
three possible solutions for the combining rotations mathematically. One is
already confirmed to be the true solution however the other two solutions
proved not to be the solution. In this report the first set of the solutions is
the right one with the right β and n for the combining rotations. In the second
set, the unit vector n for the axis of the combining rotations is the same as n
in the first set but the value of β is much larger so the result of the
combining rotations must be different from the true value. The numerical
example confirms this situation. In the third set of solutions, although the
vector n is in the same direction as n in the first set but the magnitude is
different so it will not be a unit vector and the value of β is different from
β in the first set, so it is expected that the result of the combining
rotations will not match the first set. Numerical example just confirms this
argument. There is no surprise in this study. This is just to report that the
solution for the combining rotations is further studied
For
more about Iris Publishers
please click on: http://irispublishersgroup.com/submit-manuscript.php.
To read more about this article - https://irispublishers.com/gjes/fulltext/discussion-of-solutions-for-combining-rotations.ID.000628.php
Indexing List of Iris Publishers: https://medium.com/@irispublishers/what-is-the-indexing-list-of-iris-publishers-4ace353e4eee
Iris
publishers google scholar citations: https://scholar.google.co.in/scholar?hl=en&as_sdt=0%2C5&q=irispublishers&btnG=
Comments
Post a Comment