1. Basic Definition
The Gyroscope Gimbal Motion usually refers to its rotational motion around the axis of the gimbal. As for the high-speed rotation of the rotor around its own axis, it is done to enable the MEMS gyroscope to acquire a larger angular momentum. Moreover, when the gyroscope is in a normal operating state, its speed basically remains a constant value, and we no longer need to study the rotation motion of the rotor around its own axis.
So how do we describe the rotation of the gyroscope around the inner and outer ring axes?
2. Coordinate System Definitions and Initial Setup
As shown in the figure below, there are two coordinate systems. One is the reference coordinate system \(\boldsymbol{Ox_oy_oz_o}\), which serves as the reference base for the gyroscope’s movement. The other is the inner ring coordinate system \(\boldsymbol{Oxyz}\), which can represent the position of the rotation axis but does not participate in the rotation of the rotor. The origins of both coordinate systems are all coincident with the ring frame support point \(\boldsymbol{O}\).
3. Describing Rotation via the Inner Ring Coordinate System
When the two coordinate systems are at their initial positions, all their axes are coincident. When the gyroscope rotates around the inner and outer ring axes, the inner ring coordinate system rotates relative to the reference coordinate system. When the gyroscope rotates around the outer ring axis with angular velocity \( \boldsymbol{\dot{\alpha}}\) and through an angle \( \boldsymbol{\alpha} \), the inner ring coordinate system rotates from its original position of coinciding with the reference coordinate system \( \boldsymbol{O x_0 y_0 z_0} \) to the position \( \boldsymbol{O x_1 y_1 z_1} \). When the gyroscope rotates around the inner ring axis with angular velocity \( \boldsymbol{\dot{\beta}} \) and through an angle \( \boldsymbol{\beta} \), the inner ring coordinate system rotates from the position \( \boldsymbol{O x_1 y_1 z_1} \) after the first rotation to the position \( \boldsymbol{O x y z} \). Where, \( \boldsymbol{\dot{\alpha}}\) is usually assumed to be in the positive direction along the outer ring axis, and \( \boldsymbol{\dot{\beta}} \) can be assumed to be in the positive direction along the inner ring axis, or in the negative direction along the inner ring axis, but we here prefer the former.
From this, it can be seen that the rotation of the gyroscope around the inner and outer ring axes can be represented by the rotation of the inner ring coordinate system relative to the reference coordinate system. In fact, as long as the rotation of the \( \boldsymbol{z} \) axis of the inner ring coordinate system (representing the gyroscope’s spin axis) relative to the reference coordinate system is determined, the rotation of the gyroscope around the inner and outer ring axes can be completely determined. In other words, the rotation of the gyroscope around the inner and outer ring axes can be represented by the rotation of the self-rotation axis relative to the reference coordinate system. However, the rotation angle and angular velocity of the self-rotation axis relative to the reference coordinate system still must be described in the inertial space. The trajectory of the self-rotation axis relative to the reference coordinate system is a conical surface with the pivot point \( \boldsymbol{O} \) of the ring frame as the vertex. This figure is not only difficult to depict but also inconvenient to use. Therefore, it is necessary to find a simple and practical motion representation method.
4. Simplifying Motion Description: The Tip Trajectory Method
Assuming that a spherical surface is fixed to the reference coordinate system, with its center coinciding with the gimbal pivot point \( \boldsymbol{O} \) and a radius of unit length. The intersection point of this spherical surface with the \(\boldsymbol{z_o}\) axis of the reference coordinate system is \( \boldsymbol{O’} \), which is called the pole or tip of the spin axis. Thus, the rotation of the spin axis relative to the reference coordinate system can be represented by the motion of this pole or tip on the spherical surface. This method is called the tip trajectory method. However, representing motion on a spherical surface is still not very convenient.
5. Further Simplification: The Phase Plane Representation
To further simplify the representation of gyroscope motion, a plane tangent to the spherical surface is constructed at the origin \( \boldsymbol{O’} \). Two coordinate axes are defined on this tangent plane: \(\boldsymbol{O’ \alpha}\) parallel to \(\boldsymbol{O y_0}\), and \(\boldsymbol{O’ \beta}\) parallel to \(\boldsymbol{O x_0}\). As shown in the above Figure, this tangent plane is called the phase plane. When the rotation angles \( \boldsymbol{\alpha} \) and \( \boldsymbol{\beta} \) of the spin axis relative to the reference coordinate system around the inner and outer gimbal axes are small, the displacement of the tip on the phase plane relative to the origin \( \boldsymbol{O’} \) —namely its coordinate values \( \boldsymbol{\alpha} \) and \( \boldsymbol{\beta} \) —approximately represents the rotation angles \( \boldsymbol{\alpha} \) and \( \boldsymbol{\beta} \) of the spin axis relative to the reference coordinate system. The linear velocities \( \boldsymbol{\dot{\alpha}}\) and \( \boldsymbol{\dot{\beta}} \) of the tip moving on the phase plane approximately represent the rotational angular velocities \( \boldsymbol{\dot{\alpha}}\) and \( \boldsymbol{\dot{\beta}} \) of the spin axis relative to the reference coordinate system. Furthermore, the trajectory of the tip on the phase plane approximately represents the spatial trajectory of the spin axis. In this way, the motion of the tip on the spherical surface can be simplified to the motion of the tip on the phase plane.
6. Summary of Correspondences in the Phase Plane Method
This method of representing gyroscope motion is frequently used in discussions about gyroscope dynamics. The correspondence between the tip’s movement on the phase plane and the gyroscope’s rotation must be clearly understood. On the phase plane, the coordinate value \( \boldsymbol{\alpha} \) and linear velocity \( \boldsymbol{\dot{\alpha}}\) along the positive direction of the \( \boldsymbol{\alpha} \) -axis correspond to the rotation angle \( \boldsymbol{\alpha} \) and angular velocity \( \boldsymbol{\dot{\alpha}}\) of the gyroscope around the positive direction of the outer gimbal axis. The coordinate value \( \boldsymbol{\beta} \) and linear velocity \( \boldsymbol{\dot{\beta}} \) along the positive direction of the \( \boldsymbol{\beta} \) -axis correspond to the rotation angle \( \boldsymbol{\beta} \) and angular velocity \( \boldsymbol{\dot{\beta}} \) of the gyroscope around the positive direction of the inner gimbal axis. The trajectory of the tip on the phase plane reflects the motion trajectory of the spin axis in inertial space.