Paul Gerber: Difference between revisions

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{{Unreferenced|date=August 2008}}
 
In [[aerospace engineering]], especially those areas dealing with [[spacecraft]], the '''eigenvector slew''' is a method to calculate a steering correction (called a '''[[Slew (spacecraft)|slew]]''') by rotating the spacecraft around '''one''' fixed axis.  This corresponds in general to the fastest and most efficient way to reach the desired target orientation as there is only one acceleration phase and one braking phase for the angular rate. If this fixed axis is not a [[principal axis]] a time varying torque must be applied to force the spacecraft to rotate as desired, though. Also the [[Gyroscope|gyroscopic]] effect of [[momentum wheels]] must be compensated for.
 
That such a rotation exists corresponds precisely to a main result of the mathematical theory of [[Rotation operator (vector space)|rotation operators]], the (only real) [[eigenvector]] of the rotation operator corresponding to the desired re-orientation is this axis. 
 
Given the current orientation of the craft, and the desired orientation of the craft in [[cartesian coordinates]], the required [[axis of rotation]] and corresponding rotation angle to achieve the new orientation is determined by computing the eigenvector of the [[Rotation operator (vector space)|rotation operator]].
 
== Problem ==
Let
 
:<math>\hat{x}\ ,\ \hat{y}\ ,\  \hat{z}</math>
 
be a body fixed reference system for a [[3 axis stabilized spacecraft]]. The initial attitude
is given by
 
:<math>\hat{x}=\hat{a}</math>
:<math>\hat{y}=\hat{b}</math>
:<math>\hat{z}=\hat{c}.</math>
 
One wants to find an axis relative the spacecraft body
 
:<math>\hat{r}=r_x \cdot \hat{x} +r_y \cdot \hat{y} +r_z \cdot \hat{z}</math>
 
and a rotation angle <math>\alpha</math> such that after the rotation with the angle <math>\alpha</math> one has that
 
:<math>\hat{x}=\hat{d}</math>
:<math>\hat{y}=\hat{e}</math>
:<math>\hat{z}=\hat{f}</math>
 
where
 
:<math>\hat{d}\ ,\ \hat{e}\ ,\  \hat{f}</math>
 
are the new target directions.
 
In vector form this means that
 
:<math>\hat{d}=r_a \cdot \hat{r} + \cos \alpha \cdot ( \hat{a} - r_a \cdot \hat{r})+
\sin \alpha \cdot \hat{r} \times \hat{a}</math>
:<math>\hat{e}=r_b \cdot \hat{r} + \cos \alpha \cdot ( \hat{b} - r_b \cdot \hat{r})+
\sin \alpha \cdot \hat{r} \times \hat{b}</math>
:<math>\hat{f}=r_c \cdot \hat{r} + \cos \alpha \cdot ( \hat{c} - r_c \cdot \hat{r})+
\sin \alpha \cdot \hat{r} \times \hat{c}.</math>
 
== Solution ==
 
In terms of [[linear algebra]] this means that one wants to find an [[eigenvector]] with the [[eigenvalue]] = 1 for the [[linear map]]ping defined by
 
:<math>\hat{a} \longrightarrow \hat{d}</math>
:<math>\hat{b} \longrightarrow \hat{e}</math>
:<math>\hat{c} \longrightarrow \hat{f}</math>
 
which relative to the
 
:<math>\hat{a}\ ,\ \hat{b}\ ,\ \hat{c}</math>
 
coordinate system has the matrix
 
:<math>\begin{bmatrix}
\langle \hat{d}| \hat{a}\rangle & \langle\hat{e}| \hat{a}\rangle & \langle\hat{f}| \hat{a} \rangle \\
\langle\hat{d}| \hat{b}\rangle & \langle\hat{e}| \hat{b}\rangle & \langle\hat{f}| \hat{b}\rangle \\
\langle\hat{d}| \hat{c}\rangle & \langle\hat{e}| \hat{c}\rangle & \langle\hat{f}| \hat{c}\rangle
\end{bmatrix}
</math>
 
Because this is the matrix of the [[Rotation operator (vector space)|rotation operator]] relative the base vector system <math>\hat{a}\ ,\ \hat{b}\ ,\ \hat{c}</math> the eigenvalue can be determined with the algorithm described in "[[Rotation operator (vector space)]]".
 
With the notations used here this is:
 
:<math>\cos \alpha=\frac{\langle\hat{d}| \hat{a}\rangle+\langle\hat{e}| \hat{b}\rangle+\langle\hat{f}| \hat{c}\rangle-1}{2}</math>
:<math>r_a=\langle\hat{f}| \hat{b}\rangle-\langle\hat{e}| \hat{c}\rangle</math>
:<math>r_b=\langle\hat{d}| \hat{c}\rangle-\langle\hat{f}| \hat{a}\rangle</math>
:<math>r_c=\langle\hat{e}| \hat{a}\rangle-\langle\hat{d}| \hat{b}\rangle</math>
:<math>|\bar{r}|=\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}</math>
:<math>\sin \alpha=\frac{|\bar{r}|}{2}</math>
 
The rotation angle <math>\alpha</math> is
 
:<math>\alpha=\operatorname{arg}(\cos \alpha,\sin \alpha)</math>
 
where "<math>\operatorname{arg}(x\ ,\ y)</math>" is the polar argument of the vector <math>(\ x\ ,\ y\ )</math> corresponding to the function [[Atan2|ATAN2(y,x)]] (or in [[double precision]] DATAN2(y,x)) available in for example the programming language [[Fortran|FORTRAN]].
 
The resulting <math>\alpha</math> will be in the interval <math>0 \le \alpha \le \pi</math>.
 
If <math>0 < \alpha < \pi</math> then <math>|\bar{r}| > 0</math> and the uniquely defined rotation (unit) vector is:
 
:<math>\hat{r}=\frac{\bar{r}}{|\bar{r}|}</math>
 
Note that
:<math>\langle\hat{d}| \hat{a}\rangle +\langle\hat{e}| \hat{b}\rangle+\langle\hat{f}| \hat{c}\rangle</math>
 
is the [[Trace (linear algebra)|trace]] of the matrix defined by the orthogonal linear mapping and that the components of the "[[eigenvector]]" are fixed and constant during the rotation, i.e.
:<math>
\hat{r}=r_x \cdot \hat{x}(t) +r_y \cdot \hat{y}(t) +r_z \cdot \hat{z}(t)=
r_x \cdot \hat{a} +r_y \cdot \hat{b} +r_z \cdot \hat{c}=
r_x \cdot \hat{d} +r_y \cdot \hat{e} +r_z \cdot \hat{f}
</math>
where <math>\hat{x}\ ,\ \hat{y}\ ,\ \hat{z}</math> are moving with time <math>t</math> during the slew.
 
== See also ==
* [[Rotation operator (vector space)]]
* [[Slew (spacecraft)]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Eigenvector Slew}}
[[Category:Spaceflight concepts]]
[[Category:Aerospace engineering]]

Latest revision as of 16:16, 20 November 2014

Being able to trace numbers is a form of art that everybody wishes they were able achieve. The problem is, people possess a habit of producing things much more difficult compared to what they need staying.



You go to find many sites that might you find people. Although one is Yahoo People Search, but there lots of others exactly like it. A simple online search can bring these sites up for you personally personally so in order to can all of them. What you uncover through them varies, however when you will the provider they may possibly you a payday mobility phone number, email address, as well as a current postal address.

You see, regarding what you read the actual the "internet rumors" end up being. A reverse cell phone directory may be the only effective way to trace cell cell numbers. There is also things that you can try that want a little legwork on your behalf. Unfortunately, they do not always produce ends up.

In exchange for seeing and hearing a brief advertisement, customers can receive listing information for completely. Free 411 provides phone numbers numerous types of listings. Non published numbers are needed for government, residential, or even businesses.

If just one or two to sell your home and would need to sell quickly, you might consider selling the leases themselves. There are many folks who desire to own homes, but much more not time to obtain a mortgage. If you do are having a hard time to sell will manage to benefit from this group people ready to help. It additionally fast easy as xyz. (Advertisement and exchange of contracts) you will offer your home, while the tenant buyer is happy and quick home investor. Best of all, you can move on with your reality!

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A special puppy collar that is of comfortable fabric or woven information. The collar should have a buckle fastener and mustn't be too tight for the puppy's neck of. You should be able to simply insert two fingers involving the puppy's neck and the collar.

If convincing to choose to stroll around the malls, you'll see that jewelry shops are highly attached. There is a root of it as there are because they came from will accomplish about anything in order to steal those lovely gold neck laces. The stores are very wise because they are within high essential safety. Like those shops, you must be wise while you are selling your items online and offline. Be sharp about the scams and tactics of fiddlers because that is the key surveillance.

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