Six degree of freedom with oblate Earth: angular acceleration issue!

[Xwang]

Hi to all!

I'm developing a six degree of freedom which uses the oblate rotating Earth and which wil be used to simulate the motion and the trajectory of airplane and spacecrafts around the world (so no simplification is actuable).

My work is based on Steven's "Aircraft Control and Simulation" and is pretty similar to this one:

http://www.mathworks.com/access/helpdesk/help/toolbox/aeroblks/6dofecefquaternion.html

One of the requirements is that it is possible to use the same model also for calculate the trim condition.

To do that I need the variation of velocity vector expressed in body axis and the relative angular acceleration ([math]\dot{\vec{V_{b}}}[/math] and [math]\dot{\vec{\omega_{rel}}}[/math]).

The first one is already calculated, but the angular relative acceleration is unknown since only the absolute angular acceleration is given (it contains the effects given by the motion of NED system above an oblate Earth).

How can I calculate it?

Thank you,

Xwang

[D H]

You have two rotations here: The NED frame with respect to the Earth-fixed frame, and the Earth-fixed frame with respect to inertial. Your rate gyros (if you have them) measure rotation with respect to inertial, of course. The angular velocity with respect to NED is somewhat easy because angular velocity vectors are additive. Angular acceleration is a horse of a different color.

 

The angular velocity of the plane with respect to inertial and the plane with respect to NED are related vectorially. Denoting, for short, I as the inertial frame, E as the earth-fixed frame, L as the local NED frame, and B as the plane's body frame,

 

[math]\boldsymbol{\omega}_{I \to B:B} =

\boldsymbol{\omega}_{I \to E:B} +

\boldsymbol{\omega}_{E \to L:B} +

\boldsymbol{\omega}_{L \to B:B}[/math]

 

The plane body frame is not the natural frame for expressing either the earth rotation rate ([math]\boldsymbol{\omega}_{I \to E}[/math]) or the NED frame rotation rate ([math]\boldsymbol{\omega}_{E \to L}[/math]). With appropriate representations, the above becomes

 

[math]\boldsymbol{\omega}_{I \to B:B} =

\boldsymbol{\omega}_{L \to B:B} +

\mathbf T_{L \to B}(

\boldsymbol{\omega}_{E \to L:L} +

\mathbf T_{E \to L}\,

\boldsymbol{\omega}_{I \to E:E})

[/math]

 

Differentiating with respect to time,

 

[math]\dot{\boldsymbol{\omega}}_{I \to B:B} =

\dot{\boldsymbol{\omega}}_{L \to B:B} +

\dot{\mathbf T}_{L \to B}(

\boldsymbol{\omega}_{E \to L:L} +

\mathbf T_{E \to L}\,

\boldsymbol{\omega}_{I \to E:E}) +

\mathbf T_{L \to B}(

\dot{\boldsymbol{\omega}}_{E \to L:L} +

\dot{\mathbf T}_{E \to L}\,

\boldsymbol{\omega}_{I \to E:E})

[/math]

 

Note: I intentionally omitted the earth's angular acceleration from the above, as this term is very, very small.

 

The time derivative of a transformation matrix is given by

 

[math]\frac{d\mathbf T_{A \to B}}{dt} =

-\,\mathbf X(\boldsymbol{\omega}_{A \to B:B})\mathbf T_{A \to B}[/math]

 

where

 

[math]\mathbf T_{A \to B}[/math] is the transformation matrix from frame A to frame B,

[math]\boldsymbol{\omega}_{A \to B:B}[/math] is the angular velocity of frame B with respect to frame A expressed in frame B coordinates, and

[math]\mathbf X(\mathbf u)[/math] is the skew symmetric cross product matrix generated from the vector u.

 

With this,

 

[math]\dot{\boldsymbol{\omega}}_{I \to B:B} =

\dot{\boldsymbol{\omega}}_{L \to B:B} -

\boldsymbol{\omega}_{L \to B:B} \times

(\mathbf T_{L \to B}(

\boldsymbol{\omega}_{E \to L:L} +

\mathbf T_{E \to L}\,

\boldsymbol{\omega}_{I \to E:E})) +[/math][math]

\mathbf T_{L \to B}(

\dot{\boldsymbol{\omega}}_{E \to L:L} -

\boldsymbol{\omega}_{E \to L:L}\times

(\mathbf T_{E \to L}\,\boldsymbol{\omega}_{I \to E:E}))[/math]

 

And with this result, perhaps you might want to rethink whether you should be doing your calculations in the inertial frame and transforming to NED when finished.

Edited November 26, 2008 by D H
Something went bump in the night.

[Xwang]

...

And with this result, perhaps you might want to rethink whether you should be doing your calculations in the inertial frame and transforming to NED when finished.

 

Thanks you for your help.

What i don't understand is how doing calculation in the inertial frame would resolve the problem. in such a way I still have all the acceleration given in absolute terms and not only the "relative to the air" ones (considering the air still, of course). Moreover computing all the equation in ECi, I will have to recompute the inertial tensor every step.

Do you agree?

Xwang

[D H]

Thanks you for your help.

What i don't understand is how doing calculation in the inertial frame would resolve the problem.

 

Translational states and rotational states decouple in an inertial frame. This is not the case in a rotating frame. Moreover, the equations of motion are vastly simpler in an inertial frame. Finally, your inertial navigation system measures state derivatives (acceleration and angular rate) with respect to inertial. Just because you want to control state in the NED frame doesn't mean you have to do physics in the NED frame.

 

Moreover computing all the equation in ECi, I will have to recompute the inertial tensor every step.

Do you agree?

I disagree. Attitude rate measurements are with respect to inertial but are expressed in INS case frame. Since the transformation from the vehicle body frame to the INS case frame is a fixed (non-varying) transformation, it is just as valid to say that attitude rate measurements are respect to inertial but are expressed in body frame coordinates. This is the natural frame for propagating rotational state. It's a rotating frame, so you will need to incorporate the inertial torque [math]-\boldsymbol{\omega} \times ({\mathbf I} \boldsymbol{\omega})[/math] into your rotational equations of motion. Note well: That is the only pseudo-torque you need to incorporate into your equations of motion if you represent rotational state with respect to inertial. Choose some other rotating frame as the basis and you get an absolute mess, and you get the added disadvantage of coupled translational and rotational states.