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Taking the derivative of q t with respect to t gives us
d q t
= d exp( t log q )
=log q exp( t log q )
= log q ) q t .
At t =0,thisisjust
d q
=log q
= 0 , θ
2 r .
Pulling out the 2
term, we get
2 (0 r )= 1
2 w .
Multiplying this quantity by the quaternion q gives the change relative to q ,just
as it did for matrices.
1.6.6 Numerical Integration for Angular Velocity
As angular velocity and torque/angular acceleration are both vectors, we might
think we could perform the followoing:
ω i +1 = ω i + h I 1 τ.
However, as
τ = I ω + ω
I ω,
we cannot simply multiply τ by the inverse of I anddotheEulerstep.
One solution is to ignore the ω
I ω term and perform the Euler step as written
anyway. This term represents the precession of the system—for example, a tipped,
spinning top will spin about its local axis but will also slowly precess around its
vertical axis as well. Removing this term will not be strictly accurate but can add
some stability.
The alternative is to do the integration in a different way. Consider the angular
momentum L instead, which is I ω . The derivative L = I ω = I α = τ . Hence we
can do the following:
L i +1
= L i + hτ,
I i L i +1 .
ω i +1
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