I'm modeling a single 3D rigid body in preparation for some more complicated modeling in order to gain a better understanding of Euler angles, the angular velocity vector and the rotating coordinate system.
The body is rotated in inertial frame by an intrinsic ZXZ rotation, with respective state variables ##\alpha##, ##\beta##, and ##\gamma##. Thus, I have the rotation matrix ##R## that will transform points in the body frame coordinates into fixed frame coordinates...and vise versa.
I have composed the angular velocity of the body frame relative to the fixed frame described in fixed frame coordinates, then transformed this into body frame coordinates by ##\vec{\omega}=R^T{\vec{\omega}}_{fixed}##. My moment of inertia tensor for the body is diagonal, and described in body frame coordinates (lets just call it a sphere).
The kinetic energy of the body is then ##T = \frac{1}{2}\vec{\omega}^TI\vec{\omega}##
I formulated the dynamics using Lagrange's equations, and included torque inputs on each of the rotation variables: ##\tau_{\alpha}##, ##\tau_{\beta}##, and ##\tau_{\gamma}## (which can be zeroed during simulation). I simulated and plotted the behavior in Matlab with various initial conditions and inputs. I also plotted angular velocity vector in both frames.
If I mess with one state variable at a time, the model behaves exactly how I would expect. For example, I can set an initial velocity to ##\beta## and it will spin about the X axis. Or apply a torque ##\tau_{\gamma}## and it will spin about the Z axis.
But, if I include two initial velocities (no inputs), say ##\alpha## and ##\beta##, I would expect it to spin about the Z axis while also spinning about the intermediate X axis. And if the speeds were the same, I'd expect them each to complete one rotation at the same time. Well, the body does have an off axis movement, but not like I would expect. Maybe it is just hard to visualize the body moving about two axes.
I would also expect (I think) the angular velocity vector to maintain the same magnitude and direction; it does maintain the magnitude, but not the direction (in the body frame). Is this not right?
Either my modeling is wrong or I do not fully understand Euler angles. As it is right now with my model, there is no intuitive sense to what the model will do given the inputs. For example, if I did a yaw-pitch-roll rotation, it seems like I could control each maneuver independently from each other...like, say make it roll while yawing. Is this not correct?
If you could shed any light on my throught process for the modeling, that would be great! One thing I was unsure of was what coordinate system (fixed or body) to calculate the rotational KE. I did it in the body frame, but didn't know if I was suppose to transform everything to the fixed frame.
The body is rotated in inertial frame by an intrinsic ZXZ rotation, with respective state variables ##\alpha##, ##\beta##, and ##\gamma##. Thus, I have the rotation matrix ##R## that will transform points in the body frame coordinates into fixed frame coordinates...and vise versa.
I have composed the angular velocity of the body frame relative to the fixed frame described in fixed frame coordinates, then transformed this into body frame coordinates by ##\vec{\omega}=R^T{\vec{\omega}}_{fixed}##. My moment of inertia tensor for the body is diagonal, and described in body frame coordinates (lets just call it a sphere).
The kinetic energy of the body is then ##T = \frac{1}{2}\vec{\omega}^TI\vec{\omega}##
I formulated the dynamics using Lagrange's equations, and included torque inputs on each of the rotation variables: ##\tau_{\alpha}##, ##\tau_{\beta}##, and ##\tau_{\gamma}## (which can be zeroed during simulation). I simulated and plotted the behavior in Matlab with various initial conditions and inputs. I also plotted angular velocity vector in both frames.
If I mess with one state variable at a time, the model behaves exactly how I would expect. For example, I can set an initial velocity to ##\beta## and it will spin about the X axis. Or apply a torque ##\tau_{\gamma}## and it will spin about the Z axis.
But, if I include two initial velocities (no inputs), say ##\alpha## and ##\beta##, I would expect it to spin about the Z axis while also spinning about the intermediate X axis. And if the speeds were the same, I'd expect them each to complete one rotation at the same time. Well, the body does have an off axis movement, but not like I would expect. Maybe it is just hard to visualize the body moving about two axes.
I would also expect (I think) the angular velocity vector to maintain the same magnitude and direction; it does maintain the magnitude, but not the direction (in the body frame). Is this not right?
Either my modeling is wrong or I do not fully understand Euler angles. As it is right now with my model, there is no intuitive sense to what the model will do given the inputs. For example, if I did a yaw-pitch-roll rotation, it seems like I could control each maneuver independently from each other...like, say make it roll while yawing. Is this not correct?
If you could shed any light on my throught process for the modeling, that would be great! One thing I was unsure of was what coordinate system (fixed or body) to calculate the rotational KE. I did it in the body frame, but didn't know if I was suppose to transform everything to the fixed frame.