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I am currently studying to become an aerospace engineer. As such, most of the guides I'm posting here in the North Pacific University are engineering/physics/aerospace related. I'd like to note that while I will focus on spacecraft dynamics in this guide, that rotation matrices have much broader applications, from video games and virtual reality to mathematics.
Introduction
One of the things that can make aerospace engineering so difficult is that spacecraft dynamics problems generally involve the use of multiple reference frames, which will be described in more detail in a subsequent guide. Due to this reality, it is necessary to be able to convert vectors in one reference frame to an equivalent vector in another via a transformation. In order to do this, we need to be able to describe the orientation of one reference frame with respect to another. Doing so will involve some nasty notation, and the result will be a rotation matrix.
Let's define two arbitrary reference frames that we want to find a rotation matrix for: F1 and F2. F1 is defined by the unit vectors x1, y1, and z1. F2 is defined by the unit vectors x2, y2, and z2. Additionally, let's define an arbitrary vector r. r can be expressed in each reference frame as r = F1Tr1 = F2Tr2, where r1 and r2 are column matrices containing the coordinates of the vector r. The superscript T in this case denotes the transpose of the given matrix.
Transformation
The next step is to find the coordinates of the unit vectors defining the reference frame F1 in the reference frame F2. To achieve this, we take the dot product of each unit vector of F1 with each unit vector of F2. This results in the rotation matrix C21, which again is F1 dotted with F2T.
C21 =
[x2·x1 , x2·y1 , x2·z1;
y2·x1 , y2·y1 , y2·z1;
z2·x1 , z2·y1 , z2·z1]
Since the dot products result in scalar cosines, this matrix is also sometimes referred to as the Direction Cosine Matrix. To transform r2 into r1, we use the relationship r1 = C21Tr2 = C12r2.
Identity Matrix
The matrix:
[1 , 0 , 0;
0 , 1 , 0;
0 , 0 , 1]
is defined as the identity matrix, which can also be denoted by 1. Note that C21TC21 = 1, and that C21-1 = C21T. This means that C21 is an orthonormal matrix, because its inverse is equal to its transpose.
Principal Rotations
Principal rotations are rotations about one of the coordinate axes of a given reference frame.
I am currently studying to become an aerospace engineer. As such, most of the guides I'm posting here in the North Pacific University are engineering/physics/aerospace related. I'd like to note that while I will focus on spacecraft dynamics in this guide, that rotation matrices have much broader applications, from video games and virtual reality to mathematics.
Introduction
One of the things that can make aerospace engineering so difficult is that spacecraft dynamics problems generally involve the use of multiple reference frames, which will be described in more detail in a subsequent guide. Due to this reality, it is necessary to be able to convert vectors in one reference frame to an equivalent vector in another via a transformation. In order to do this, we need to be able to describe the orientation of one reference frame with respect to another. Doing so will involve some nasty notation, and the result will be a rotation matrix.
Let's define two arbitrary reference frames that we want to find a rotation matrix for: F1 and F2. F1 is defined by the unit vectors x1, y1, and z1. F2 is defined by the unit vectors x2, y2, and z2. Additionally, let's define an arbitrary vector r. r can be expressed in each reference frame as r = F1Tr1 = F2Tr2, where r1 and r2 are column matrices containing the coordinates of the vector r. The superscript T in this case denotes the transpose of the given matrix.
Transformation
The next step is to find the coordinates of the unit vectors defining the reference frame F1 in the reference frame F2. To achieve this, we take the dot product of each unit vector of F1 with each unit vector of F2. This results in the rotation matrix C21, which again is F1 dotted with F2T.
C21 =
[x2·x1 , x2·y1 , x2·z1;
y2·x1 , y2·y1 , y2·z1;
z2·x1 , z2·y1 , z2·z1]
Since the dot products result in scalar cosines, this matrix is also sometimes referred to as the Direction Cosine Matrix. To transform r2 into r1, we use the relationship r1 = C21Tr2 = C12r2.
Identity Matrix
The matrix:
[1 , 0 , 0;
0 , 1 , 0;
0 , 0 , 1]
is defined as the identity matrix, which can also be denoted by 1. Note that C21TC21 = 1, and that C21-1 = C21T. This means that C21 is an orthonormal matrix, because its inverse is equal to its transpose.
Principal Rotations
Principal rotations are rotations about one of the coordinate axes of a given reference frame.