5. The Rotation matrices

A rotation matrix for a newbie is a matrix which helps us determine a vector in a particular co-ordinates system when the vector or the co-ordinate frame is rotating. To add more clarity, let’s assume we have a particular point “P” in a 3-D space, represented by a vector \overrightarrow{P_{1}}  (say the three components of \overrightarrow{P_{1}} are P1X, P1Y, and P1Z) in a given co-ordinate frame C1. Further, if we rotate this co-ordinate frame by an angle \theta  about the Z axis and refer to this new orientation of C1 as C2, then the vector representation of P in C2 won’t be \overrightarrow{P_{1}}  but it will be given by the following equation

\left[ \begin{matrix} { { P }_{ 2X } } \\ { P }_{ 2Y } \\ { P }_{ 2Z } \end{matrix} \right] =\left[ \begin{matrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{matrix} \right] *\left[ \begin{matrix} { P }_{ 1X } \\ { P }_{ 1X } \\ { P }_{ 1X } \end{matrix} \right]

Where P2X, P2Y, and P2Z are components of the vector \overrightarrow{P_{2}} , which is a vector representation of point P in the co-ordinate frame C2.

After this, say the co-ordinate frame C2 undergoes another rotation about the Y axis by an angle \phi and attains an orientation which is denoted by C3. So the vector representation of P in C3 will be given by

\left[ \begin{matrix} { { P }_{ 3X } } \\ { P }_{ 3Y } \\ { P }_{ 3Z } \end{matrix} \right] =\left[ \begin{matrix} cos\phi & 0 & sin\phi \\ 0 & 1 & 0 \\ -sin\phi & 0 & cos\phi \end{matrix} \right] *\left[ \begin{matrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{matrix} \right] *\left[ \begin{matrix} { P }_{ 1X } \\ { P }_{ 2X } \\ { P }_{ 3X } \end{matrix} \right]

where P3X, P3Y, and P3Z are component of the vector \overrightarrow{P_{3}} , which is a vector representation of point P in the co-ordinate frame C3.

In case, we rotate the co-ordinate frame C 3 along X axis by an angle \psi , it will attain another orientation which we can refer by C4. The vector representation of P in C4 is given by

\left[ \begin{matrix} { P }_{ 4X } \\ { P }_{ 4Y } \\ { P }_{ 4Z } \end{matrix} \right] =\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\psi & -sin\psi \\ 0 & sin\psi & cos\psi \end{matrix} \right] *\left[ \begin{matrix} { P }_{ 3X } \\ { P }_{ 3Y } \\ { P }_{ 3Z } \end{matrix} \right]

Where P4X, P4Y, and P4Z are component of the vector \overrightarrow{P_{4}} , which is a vector representation of point P in the co-ordinate frame C4.

These rotation matrix can be used further in case the co-ordinate frame undergoes any more rotations. If the co-ordinate frame is rotated about the Z axis by an angle \alpha , then the vector in the newly oriented frame is obtained by multiplying the matrix \left[ \begin{matrix} cos\alpha & -sin\alpha & 0 \\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1 \end{matrix} \right] with the vector observed in the co-ordinate frame before it got rotated about the Z axis by \alpha . In this case, the point P can be defined as a vector in the co-ordinate frame C5 (C4 after being rotated by angle \alpha about the Z axis) by the following equation

\left[ \begin{matrix} { P }_{ 5X } \\ { P }_{ 5Y } \\ { P }_{ 5Z } \end{matrix} \right] =\left[ \begin{matrix} cos\alpha & -sin\alpha & 0 \\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1 \end{matrix} \right] *\left[ \begin{matrix} { P }_{ 4X } \\ { P }_{ 4Y } \\ { P }_{ 4Z } \end{matrix} \right]

Where P5X, P5Y, and P5Z are component of the vector \overrightarrow{P_{5}} , which is a vector representation of point P in the co-ordinate frame C5

Here, if C4 would have been rotated by an angle \alpha about Y axis instead of Z, then we would have multiplied with \left[ \begin{matrix} cos\alpha & 0 & sin\alpha \\ 0 & 1 & 0 \\ -sin\alpha & 0 & cos\alpha \end{matrix} \right] to \left[ \begin{matrix} { P }_{ 4X } \\ { P }_{ 4Y } \\ { P }_{ 4Z } \end{matrix} \right] get the vector in the newly oriented co-ordinate frame, on the other hand had it been rotated by X axis then we would have multiplied  \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha \end{matrix} \right]  with \left[ \begin{matrix} { P }_{ 4X } \\ { P }_{ 4Y } \\ { P }_{ 4Z } \end{matrix} \right]  for the same purpose.

The rotation matrix we used in the above case are examples of passive rotation matrix, i.e., rotation matrix used to determine a vector when the frame is rotating.

When we need to make vector observation of a point which is changing its location, and if the co-ordinate frame of the observer is fixed in space, then we use active rotation matrix.

Imagine a vector \overrightarrow{V_{1}}  with vector components V1x , V1Y , and V1Z observed in a fixed coordinate frame C1. If this vector is rotated about the Z axis by an angle \theta , then the vector will change from \overrightarrow{V_{1}}   to \overrightarrow{V_{2}} , and the vector component of vector \overrightarrow{V_{2}}   in C1 co-ordinate frame is given by

\left[ \begin{matrix} { V }_{ 2X } \\ { V }_{ 2Y } \\ { V }_{ 2Z } \end{matrix} \right] =\left[ \begin{matrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{matrix} \right] *\left[ \begin{matrix} { V }_{ 1X } \\ { V }_{ 1Y } \\ { V }_{ 1Z } \end{matrix} \right]

where V2X, V2Y and V2Z are components of vector \overrightarrow{V_{2}} after being rotated about the Z axis.  Further if vector \overrightarrow{V_{1}} after being rotated about the Z axis, is rotated about the Y axis by an angle \phi , then its new vector component V3X, V3Y and V3Z  are given by

\left[ \begin{matrix} { V }_{ 3X } \\ { V }_{ 3Y } \\ { V }_{ 3Z } \end{matrix} \right] =\left[ \begin{matrix} cos\phi & 0 & -sin\phi \\ 1 & 0 & 1 \\ sin\phi & 0 & cos\phi \end{matrix} \right] *\left[ \begin{matrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{matrix} \right] *\left[ \begin{matrix} { V }_{ 1X } \\ { V }_{ 1Y } \\ { V }_{ 1Z } \end{matrix} \right]

Again, if this vector is given another rotation about X axis by an angle \psi , then the new vector components  V4x, V4Y, and V4Z can be obtained by

\left[ \begin{matrix} { V }_{ 4X } \\ { V }_{ 4Y } \\ { V }_{ 4Z } \end{matrix} \right] =\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\psi & -sin\psi \\ 0 & sin\psi & cos\psi \end{matrix} \right] *\left[ \begin{matrix} { V }_{ 3X } \\ { V }_{ 3Y } \\ { V }_{ 3Z } \end{matrix} \right]

These matrices can be further used to map a vector which is changing its orientation about any of the axis when observed from a fixed co-ordinate system. For example, if the above vector \overrightarrow{V_{4}} rotates about the Z axis by an angle \alpha , then the new components V5X, V5Y and V5Z can be obtained by

\left[ \begin{matrix} { V }_{ 5X } \\ { V }_{ 5Y } \\ { V }_{ 5Z } \end{matrix} \right] =\left[ \begin{matrix} cos\alpha & -sin\alpha & 0 \\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1 \end{matrix} \right] *\left[ \begin{matrix} { V }_{ 4X } \\ { V }_{ 4Y } \\ { V }_{ 4Z } \end{matrix} \right]

On the other hand had \overrightarrow{V_{4}} been rotated by an angle \alpha , about the Y axis, then V5X, V5Y and V5Z would have been obtained by

\left[ \begin{matrix} { V }_{ 5X } \\ { V }_{ 5Y } \\ { V }_{ 5Z } \end{matrix} \right] =\left[ \begin{matrix} cos\alpha & 0 & -sin\alpha \\ 0 & 1 & 0 \\ sin\alpha & 0 & cos\alpha \end{matrix} \right] *\left[ \begin{matrix} { V }_{ 4X } \\ { V }_{ 4Y } \\ { V }_{ 4Z } \end{matrix} \right]

In the other case, had the rotation taken place along Z axis, by an angle \alpha , then the vector components V5X, V5Y and V5Z can be calculated by

\left[ \begin{matrix} { V }_{ 5X } \\ { V }_{ 5Y } \\ { V }_{ 5Z } \end{matrix} \right] =\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\alpha & sin\alpha \\ 0 & -sin\alpha & cos\alpha \end{matrix} \right] *\left[ \begin{matrix} { V }_{ 4X } \\ { V }_{ 4Y } \\ { V }_{ 4Z } \end{matrix} \right]

With this we complete our discussion on active and passive rotation matrices.  We can also observe that, the passive rotation matrices can be a way to define orientation of one co-ordinate frame with respect to the other.

With this, let’s get back to our Satellite Reference Frame. The passive rotation matrix to determine the orientation of orbit reference frame with respect to the ECI frame is given by

\left[ \begin{matrix} -sin(u)cos(\Omega )-cos(u)cos(i)sin(\Omega ) & -sin(u)sin(\Omega )+cos(u)cos(i)cos(\Omega ) & cos(u)sin(i) \\ -sin(i)sin(\Omega ) & sin(i)cos(\Omega ) & -cos(i) \\ -cos(u)cos(\Omega )+sin(u)cos(i)sin(\Omega ) & -cos(u)sin(\Omega )-sin(u)cos(i)cos(\Omega ) & -sin(u)sin(i) \end{matrix} \right]

Where u stands for the sum of argument of perigee and the true anomaly , i stands for the orbital inclination and \Omega  stands for the right ascension of the ascending node.

Since we know the orientation of the ECI frame, this relative orientation of orbit reference frame with respect to ECI frame can give us a great deal of information on how is the orbit reference frame oriented in space. So, in a similar way, if we can able to calculate the rotation matrix which can give us the orientation of the Satellite Body Frame with respect to the Satellite Reference Frame, with this we will not only know the orientation of Satellite Body Frame in space but we will also get to know how much is the satellite misaligned from its idea orientation. If we analyse these relative rotation parameters a bit more, we can calculate the rate at which satellite is deviating from its ideal orientation and this can help us to calculate the body rates of the satellite. With this we have answered the why’s, what‘s and how’s of the questions we raised in the beginning of the post.

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