Re: Low pass filter on quaternions

Note that with any kind of filtering you will introduce some 
sluggishness since abrupt changes will be muffled. An intuitive way to 
get this done is through a dynamics simulation.

Define inertia for the controlled object and control its orientation 
through a damped spring. Let Q0 be the current orientation of the object 
and Q1 the desired orientation (the orientation of the input device). 
First make sure that Q0 and Q1 are in the same hemisphere (of a 4D unit 
sphere). Thus if the 4-component dot product of Q0 and Q1 is negative 
simply set Q1 to -Q1. Next take the relative rotation, which is Qrel = 
Q0 Q1*, where Q1* is the conjugate of Q1. We will use log(Qrel) the log 
of Qrel for computing the torque. The log of a quaternion [xyz,w]  is 
the 3-vector xyz * acos(w) / sqrt(1 - w * w) , (log([0, 0, 0: 1]) = (0, 
0, 0)). Instead of the log you could also use the vector part (xyz) as a 
cheap and dirty log, since acos(w) / sqrt(1 - w * w) approaches 1 for w 
around 1.

The torque can now be defined as log(Qrel) * Cspring, where Cspring is 
the spring constant. You will need to damp out oscillations, by 
multiplying the relative angular velocity (also a 3-vector) by a damping 
constant and subtracting it from the torque. If the input device does 
not provide an angular velocity, you can take a linear approximation of 
the angular velocity or simply assume a zero angular velocity. By 
setting the angular damping to 2 sqrt(inertia Cspring), you will get a 
critically damped spring (just enough damping to remove all 
oscillations). In an isolated setting you can use inertia = 1 (identity) 
and tweak the response of the spring damper by changing the spring 
constant (keeping the damping critical).

Just a thought...

Gino











On 3/28/2010 5:32 PM, John McCutchan wrote:
> Hi,
>
> Each quaternion represents the orientation of an input device. I want
> to remove jitter that comes from holding it in a human hand.
>
> John
>
> On Sun, Mar 28, 2010 at 12:56 AM, Marc B. Reynolds
> <[email protected]>  wrote:
>    
>> Also, what do they represent?
>>
>> -----Original Message-----
>> From: Robin Green [mailto:[email protected]]
>> Sent: Sunday, March 28, 2010 8:36 AM
>> To: Game Development Algorithms
>> Subject: Re: [Algorithms] Low pass filter on quaternions
>>
>>
>> Do you want to lowpass the signal, or do you want to fit a smooth
>> curve to the samples?
>> Are the samples regularly spaced?
>>
>> - Robin Green
>>
>>
>>
>> On Sat, Mar 27, 2010 at 7:14 PM, John McCutchan<[email protected]>
>> wrote:
>>      
>>> Hi,
>>>
>>> I have a sequence of quaternions and want to filter out any high
>>> frequency changes. I'm wondering if anyone has a paper or some code
>>> for performing a low pass filter on quaternions?
>>>
>>> Thanks,
>>> --
>>> John McCutchan<[email protected]>
>>>
>>>
>>>        
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