Integration?
That is, if the instrument provides a continuous signal proportional to $\frac{dx}{dt}$, you can find the total change in $x$ of $[t_1,t_2]$ from
$$ \Delta x = \int_{t_1}^{t_2} dt \frac{dx}{dt} \approx \sum_{i=0}^n \Delta t \left. \frac{dx}{dt} \right |_i . $$
Of course, as anyone working on inertial guidance knows, that approximation is makes for real difficulty as does the matter of noise and calibration drifts on the instrument. But if you can sanity check the result from time to time (say by comparing with GPS) this will work quite well.
Since in my last post I made a lot of mistakes, I'll try to improve that answer starting from scratch.
First of all: lagrangian coordinates.
It first seemed clear that two coordinates suffice in describing the body orientation. Of course this is false, since in order to describe every possible orientation one needs three coordinates. We'll call them $(\theta, \phi, \psi)$.
Any more coordinates? I don't think so. The only external forces acting of the body are the rope tension and gravity. They are both directed along the vertical, so no oscillations should develop.
Next: rotation description. Refer to this link. You could pick Euler angles or Tait-Bryan, your choice. Given the angles, you then compute the rotation matrix. With the matrix you can compute any vector rotation regarding the body. Will call this $R(\theta (t), \phi (t), \psi (t)) = R(t)$.
Body description: your model depends on many quantities. They essentially are
- Initial body orientation and initial angular velocity
- Initial relative positions of the three gyroscopes
- Initial orientations of the gyroscopes and angular velocities
- Initial center of gravity position
- $\theta(t), \phi(t), \psi(t)$
Given any of these vectors $\vec{x}_0$, you can compute the corresponding vector at time $t$ as $\vec{x}(t) = R(t) \cdot \vec{x}_0 = R(\theta, \phi, \psi) \cdot \vec{x}_0 = \vec{x}(\theta, \phi, \psi)$.
What's next? Compute the kinetic energy of the body. It is given by
$$ T(\theta, \phi, \psi, \dot{\theta}, \dot{\phi}, \dot{\psi}) = \frac{1}{2} I_1 \omega_1^2(\theta, \phi, \psi) + \frac{1}{2} I_2 \omega_2^2(\theta, \phi, \psi) +\frac{1}{2} I_3 \omega_3^2(\theta, \phi, \psi) + \frac{1}{2} I_{TOT} \omega_{body}^2(\theta, \phi, \psi, \dot{\theta}, \dot{\phi}, \dot{\psi}) $$.
Again note that the gyroscopes' angular velocities are constant in magnitude (if you admin no friction and only forces normal to the axis, but they rotate with the body: their relative angle to the body has remain fixed.
Finally compute the potential energy as
$$ U(\theta, \phi, \psi) = mgh(\theta, \phi, \psi)$$
The total lagrangian is
$L = T-U$.
Now you should be able to apply the Euler Lagrange equations and solve numerically. Good luck! You might need to help yourself with mathematica or any other sort of algebra package in order not to damage your brain with the derivatives and scalar products.
Best Answer
The term gyroscope is more often associated with sensors that can sense angular velocities, and these can be used in a control feedback system with actuators such as reaction wheels or control moment gyros that provide the application of torque to the body of the spacecraft and thus control attitude.
Thrusters indeed can provide translational force (delta-v) but if used in pairs can also provide torque for attitude control. Typically a spacecraft will use both thrusters and reaction wheels to steer the spacecraft to a rotational rate of zero, and if other sensors are available, to point in a specific direction. Thrusters using on-off or rather bang-bang control larger torque and acceleration, and when the spacecraft is within a smaller range, the reaction wheel control system will take over.
Reaction wheels are usually clustered as 4 along different axes to provide yaw pitch and roll torques. Only three are required, but the forth provides redundancy in the event any one of the other three fail. The geometry is such that any three can be used to provide three axes of orthogonal torque.
Reaction wheel or CMG attitude control systems are not without their limits. To obtain torque, the wheels must spin-up to high velocities creating a high level of angular momentum. At some point they saturate (reach their limit of speed) and require some means to dump their momentum. For low earth orbit satellites this can sometimes be done by using atmospheric drag or the Earth's magnetic field, but sometimes fuel must be consumed by the reaction control system (thrusters) to dump the angular momentum. Once momentum is lowered the wheels can be used again.