Differential Rotational
Author
I learned that rotational velocity and rotational acceleration can be mathematically modelled by a single vector.
Its direction for the axis and its magnitude for the anticlockwise angle it represents.
Q1
Is it safe to integrate rotational acceleration to get rotational velocity?
Q2
Is it safe to integrate rotational velocity to get rotation vector V?
Such that V/|V| is the axis and |V| is the angle?
NOTE:
After testing I found out that Q2 is probably wrong (need an explanation though)...
I'm somewhat of a beginner in physics myself, but I'll tell you what I've learned, and maybe some of the experts here can offer corrections or clarifications.
Angular acceleration can be integrated to get angular velocity. (Alternately, you can integrate torque to get angular momentum.)
Orientation, however, is usually represented by a quaternion or 3x3 matrix. Integrating from angular velocity is based on the observation that given a point p relative to the center of mass, the instantaneous velocity due to rotation is wXp, where w is angular velocity.
The instantaneous rotational velocity of the three orthonormal basis vectors can be found in the same way. This can be compactly expressed as star(w)*R, where R is the 3x3 orientation matrix, and star() is a skew-symmetric matrix. Therefore:
R-dot = star(w)*R
R-dot can then be submitted as the derivative of R to any 'black box' numerical integrator. (There is an equivalent derivation for q-dot, if your orientation is represented as a quaternion.)
That's about as much as I'm sure of. R-dot and q-dot are only approximations, and I believe they become increasingly inaccurate at higher angular velocities (at least it seems this would be the case). Also, R or q will probably need to be normalized after each update, as even a small change will take them out of orthonormality.
I'd be interested to know if there are other commonly accepted ways of integrating angular velocity. Perhaps others will provide some insight on that...
Angular acceleration can be integrated to get angular velocity. (Alternately, you can integrate torque to get angular momentum.)
Orientation, however, is usually represented by a quaternion or 3x3 matrix. Integrating from angular velocity is based on the observation that given a point p relative to the center of mass, the instantaneous velocity due to rotation is wXp, where w is angular velocity.
The instantaneous rotational velocity of the three orthonormal basis vectors can be found in the same way. This can be compactly expressed as star(w)*R, where R is the 3x3 orientation matrix, and star() is a skew-symmetric matrix. Therefore:
R-dot = star(w)*R
R-dot can then be submitted as the derivative of R to any 'black box' numerical integrator. (There is an equivalent derivation for q-dot, if your orientation is represented as a quaternion.)
That's about as much as I'm sure of. R-dot and q-dot are only approximations, and I believe they become increasingly inaccurate at higher angular velocities (at least it seems this would be the case). Also, R or q will probably need to be normalized after each update, as even a small change will take them out of orthonormality.
I'd be interested to know if there are other commonly accepted ways of integrating angular velocity. Perhaps others will provide some insight on that...
Author
Well what I have up and running till now is as such:
The integral of the rotational velocity (taken alone, as if the initial rotational position if I might say) must lie on the same axis so it's just a linear integration.
The result of such rotation is let to be DeltaR..
The real initial rotation (orientation) is R_n, R_n+1 = R_n * DeltaR
It is visually satisfying, however is this mathematically correct (not necesarily exact)...
In other words how can this be compared to the correctness of an euler integration of positional vectors
-----
Another question: Am making a spaceship orientation control..
There is a roll, and pitch acceleration... Is it correct to add these vectorially to attain the resulting acceleration and integrate that to acquire rotational veclocity..
Maybe next time i'll post the project..
The integral of the rotational velocity (taken alone, as if the initial rotational position if I might say) must lie on the same axis so it's just a linear integration.
The result of such rotation is let to be DeltaR..
The real initial rotation (orientation) is R_n, R_n+1 = R_n * DeltaR
It is visually satisfying, however is this mathematically correct (not necesarily exact)...
In other words how can this be compared to the correctness of an euler integration of positional vectors
-----
Another question: Am making a spaceship orientation control..
There is a roll, and pitch acceleration... Is it correct to add these vectorially to attain the resulting acceleration and integrate that to acquire rotational veclocity..
Maybe next time i'll post the project..
This topic is closed to new replies.
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