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# Quaternion Integration

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It appears that NASA has a quaternion integration method that does not require normalization between integration steps. Does anyone have any information on this technique? Their webpage (http://ettc.usc.edu/dryden/tos/closedform.html) states: "NASA''s technology represents a substantial improvement over traditional methods of integrating 4-D kinematics equations. Those methods solve the quaternion kinematic equations in a way that does not automatically preserve the unit normalization of the quaternion. These methods are therefore inherently prone to error and difficult to correct. Such traditional methods are computation-intensive because of the substantial amount of error correction needed to compensate for analytical drift. NASA''s integration method preserves the mathematics of the quaternion space that it creates and therefore introduces no error associated specifically with non-preservation of the quaternion norm. As a result, error correction associated with normalization is not needed in the NASA method"

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Since quaternions are 4-d complex number , just transform a quaternion with Laplace integration from a+jb to a-jb using dt as differential term , integrate the quaternion and revert back to 3d at lest this is what i would do , correct me if i''m wrong :-)

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I can''t believe they patented a method for integrating quaternion equations.

You''re probably not allowed to use it without paying a licence.

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quote:
Original post by v71
Since quaternions are 4-d complex number , just transform a quaternion with Laplace integration from a+jb to a-jb using dt as differential term , integrate the quaternion and revert back to 3d at lest this is what i would do , correct me if i''m wrong :-)

Do you know what the date of that patent is? If it is very recent, there may very well be prior art that invalidates the patent.

What you suggest is true, but that doesn''t seem to be what NASA has done. But what NASA seems to have done is to transform the kinematic equations, originally developed based on Euler angle rotations, into quaternion space. The entire equation set not just the rotation itself. They may have used a higher-dimensional form of the Laplace transform to get there. My head hurts thinking about the algebra involved. But the work basically should be just a bunch of ugly algebraic manipulations with perhaps something like Laplace transforms thrown in.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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You can learn what they mean by licensing opportunities here:

http://tech-transfer.larc.nasa.gov/patent.htm

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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The US patents office has a lengthy description here.

It describes prior art as solving quaternion kinematices equation using numerical integration schemes that "violate the mathematics of quaternion space" requiring the renormalisation of quaternions after each (integration) time step. This renormalisation "introduces error into the integration in the form of analytical drift."

The new and improved method "constructs 4-D quaternion kinematics equations and integrates them while preserving the norm of the quaternion." Basically it sounds like he has found some integrating factor that enables a closed form solution to the quaternion kinematics equation.

Unfortunately the details are buried in bad formatting and extraneous junk related to including this in some navigation system. I might try to decipher it tonight, or you might immediately be able to tell by looking at the patent.

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Patenting math ?
Pardon me, but that sounds really cheesy and bad form and all that.
Sorry, but in science you shouldn`t patent things like that.

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no need to get all freaked out over patents of math. from what i read the patent is specific to the integration of this method into "INS of flight vehicles. The NASA method and device performs three functions.."

device + math = patent.

that''s pretty much as fitting to the ideology of the patent system as you can get. idea + specific implementation.

-me

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Excuse me, but why would one want to integrate quaternions anyway? I figured it must have something to do with kinematics but where exactly comes quaternion integration into play?

Sorry if the question might sound a little stupid but I''m just curious.

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I have to integrate quaternions for my physics engine. The problem is that I have to normalize after each integration step which induces errors. I found the patent doing a google search for quaternion integration. Unfortunately I do not, at present, know how to solve the error problem when integrating quaternions. If anyone has an algorithm for this it would be appreciated.

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