6.4: Angular Velocity Sensors (Rate Gyro)

Learning Objectives

After completing this section, you will be able to:

1. Describe the basic working principle of a rate gyro sensor.
2. Explain the process of implementing a mechanical rate gyro sensor into a MEMS device.
3. Provide the basic working principle of a piezoelectric rate gyro sensor.

This section discusses a family of angular velocity sensors (also called rate gyro) used to measure the angular velocity of a given axis. In general, the gyroscope is an inertial sensor that is used to measurement or control the orientation and rotational velocity of a body. A rate gyro is a type of gyroscope. Instead of indicating direction, it measures the angular rate of change with time. If a gyroscope has only one gimbal ring, with consequently only one degree of freedom, it can be adapted for use as a rate gyro to measure a rate of angular movement. That is, a gyroscope measures multidimensional angular velocity, whereas a rate gyro is for one direction. There are mainly two types of rate gyros: mechanical sensors (e.g., piezoelectric, MEMS) [26] and optical laser gyro sensors [6], [25]. Mechanical rate gyros are based on the principle of Coriolis force [27], defined below. In addition, both piezoelectric and MEMS-based rate gyro sensors are vibration-based, to be explained below.

Definition 6.2

In physics, the Coriolis force is defined as an inertia (or fictitious) force that applies to objects in motion within a reference coordinate (frame) that rotates with respect to the inertia coordinate. For example, Figure 6.19 shows a reference coordinate with anticlockwise rotation, where the force acts to the left of the motion of the object.

Consider the reference coordinate, also called the body coordinate, denoted by \(x_b\) and \(y_b\) and the inertia coordinate denoted by \(x_i\) and \(y_i\) on a two-dimensional plane.

In Figure 6.19, the reference frame (coordinate) is represented by the gray box with a proof mass \(m\) suspended by four springs. Note that a proof mass is a known, suspended, or inertial mass used in sensors to measure external acceleration or vibration by deflecting relative to a housing. The reference coordinate rotates around origin \(O\) in the inertia frame (coordinate \(x_i\) and \(y_i\)) with an angular speed of \({\dot{θ}}_z\). This results in a downward linear speed of \(v=R{\dot{θ}}_z\). Based upon the Coriolis force principle, the resulting Coriolis force is shown below.

\[ f_c=2mv{\dot{θ}}_z\ \label{6.35}\]

From equation \ref{6.35}, we can see that the reference frame rotational speed \({\dot{θ}}_z\) can be found if both the Coriolis force \(f_c\) and linear velocity \(v=R{\dot{θ}}_z\) can be measured. The Coriolis force can be measured by the displacement of mass \(m\) due to two horizontal springs and applied Coriolis force \(f_c\). However, the linear speed is difficult to measure under constant speed, especially when it is combined with the linear motion velocity of the reference frame (that is, \(v\neq R{\dot{θ}}_z\)), leading to the following concept of vibration-based rate gyros.

Most single-axis vibrating rate gyros have a proof mass \(m\) (same as that in Figure 6.19) with two orthogonal degrees of freedoms (vertical and horizontal); see Figure 6.20. Note that the vertical axis (\(y_b\)) is the driving axis to create a vibration motion in that direction. The horizontal axis (\(x_b\)) is the sensing axis to detect the Coriolis force. When the rate gyro continuously vibrates at a resonant frequency in one direction (\(y_b\)), the oscillation remains absent in the orthogonal (horizontal) direction (\(x_b\)) until the external rotation has occurred. When the external rotation occurs, the rotation rate \({\dot{θ}}_z\ \)triggers energy shifting from the vertical vibration towards the horizontal direction due to the Coriolis effect [27].

The two degrees of freedom equations of motion for vibrating rate gyros based upon the Newton and Coriolis’ principles are shown in equations \ref{6.36} to \ref{6.38}.

\[\ m\dfrac{d^2y_b}{dt^2}+c\dfrac{dy_b}{dt}+ky_b=f_d+2m{\dot{θ}}_z\dfrac{dx_b}{dt} \label{6.36}\]

For equation \ref{6.36}, \(m\) is the proof mass, \(k\) is the spring stiffness in \(y_b\) direction shown in Figure 6.20, \(c\) is the friction coefficient of the proof mass, and \(f_d\) is the driving force used to create the oscillation motion of proof mass. The response due to rotation \({\dot{θ}}_z\) \(\left(2m{\dot{θ}}_z\left(dx_b/dt\right)\right)\) is very small in magnitude compared to the driving response due to a relatively large driving force \(f_d\). As a result, the Coriolis force in the vertical direction (\(y_b\)) can be ignored. Equation \ref{6.36} can then be rewritten as follows.

\[m\dfrac{d^2y_b}{dt^2}+c\dfrac{dy_b}{dt}+ky_b=f_d \label{6.37}\]

In turn, the equation of motion in the horizontal direction can be expressed as follows.

\[m\dfrac{d^2x_b}{dt^2}+c\dfrac{dx_b}{dt}+kx_b=-2m{\dot{θ}}_z\dfrac{dy_b}{dt} \label{6.38}\]

where it is assumed that the spring stiffness \(k\) and friction coefficient in both \(x_b\) and \(y_b\) directions are the same. Furthermore, \(2m{\dot{θ}}_z\left(dy_b/dt\right)\) is the Coriolis force applied in the \(x_b\) direction due to rotation \({\dot{θ}}_z\) in the \(z\) direction (see Figure 6.19). The free body diagram of the proof mass can be found in Figure 6.21, where \(f_c=2m{\dot{θ}}_z\left(dy_b/dt\right)\).

Next, assume that a closed-loop control system is employed so that the resulting oscillation in the vertical direction is a cosine signal, as below with magnitude \(m_y\) and frequency \(ω_b\).

\[y_b=m_y{\mathrm{cos} ω_bt\ }\ \to \ {\dot{y}}_b=-m_yω_b{\mathrm{sin} ω_bt\ } \label{6.39}\]

If \(ω_b\) is way greater than the maximum frequency component of \({\dot{θ}}_z\), we can consider \({\dot{θ}}_z\) as a constant because \({\dot{θ}}_z\) is much slower than \(ω_b\). Recall that Theorem 3.2 (Section 3.2) states that for a sinusoidal input \(A{\mathrm{sin} ωt\ }\), the steady-state output of a stable system described by transfer function \(G(s)\) is in the form of \(A\left|G(jω)\right|{\mathrm{sin} \left(ωt+\angle G(jω)\right). }\) We can derive the transfer function of the sensing system in Figure 6.20 described by equation \ref{6.38}, resulting in the following equation, assuming that \(\dot{θ}_z\) is a constant (much slower than \(ω_b\)).

\[G_{rg}(s)=\dfrac{X_b(s)}{U_b(s)}=\dfrac{2mm_yω_b{\dot{θ}}_z}{ms^2+cs+k} \label{6.40}\]

where \(X_b(s)\) is the Laplace transform of \(x_b\left(t\right),\ \)and \(U_b(s)\) is the Laplace transform of the oscillation signal \({\mathrm{sin} ω_bt\ }\). Again, using Theorem 3.2 yields the following steady-state response of \(x_b\).

\[x_{b-ss}\left(t\right)=\dfrac{2mm_yω_b{\dot{θ}}_z}{k}{\mathrm{sin} \left(ω_bt+\angle G_{rg}(jω)\right)\ } \label{6.41}\]

If the magnitude of the steady-state response, \(2mm_yω_b{\dot{θ}}_z/k\), can be detected, the angular velocity \(\dot{θ}_z\) can be obtained from the detected magnitude.

Example 6.7

Let a rate gyro proof mass \(m=0.05\ kg\), sinusoid vibration displacement signal \(y_b\left(t\right)=-{10}^{-5}{\cos {2π10}^5t\ }\), and rotational speed \(\dot{θ}_z=10\ rad/s\). Find the Coriolis force. In this case, we can find the Coriolis force using the following equation with \(m_y={10}^{-5}\) and \(ω_b=2π{10}^5\).

\[ f_c=2m{\dot{θ}}_z\dfrac{dy_b}{dt}=2m{\dot{θ}}_zm_yω_b \sin ω_bt\  \label{6.42}\]

Substituting known parameters, we have the following.

\[ f_c=2\times 0.005\times 10\times {10}^{-5} \times 2π\times {10}^5 \sin {2π10}^5t =0.2π \sin {2π10}^5t (Nm) \label{6.43}\]

There are two widely used implementations of the rate gyro sensors based on vibration principle. The first one is MEMS based. Figure 6.22 shows the simplest structure of a MEMS vibration rate gyro sensor. Here, a proof mass is placed on top of the substrate representing the reference frame. Four support pillars are fixed on to the substrate through four suspension beams to generate the spring forces in two directions. Both top-bottom driving electrodes (orange) and side sensing electrodes (green) are fixed to the substrate. The driving electrodes apply force to the proof mass, causing the proof mass to vibrate vertically without angular movement. In addition, the angular movement of the substrate leads to horizontal vibrations due to the generated Coriolis force. Therefore, the capacitance-based sensing electrodes can measure the vibration magnitude and convert it to the angular speed information. Note that the working principle of the vibration rate gyro was addressed earlier.

Piezoelectric rate gyros are sensors using piezoelectric oscillators to capture the rotational movements of objects; see Figure 6.23. They are widely used for compensating hand shaking in video and digital still cameras. The principle of piezoelectric rate gyros (gyroscopes) is the same as that of the Foucault pendulum. The right plot of Figure 6.23 shows a piezoelectric element vibrating up and down (green arrow in \(y_b\) direction) in the middle section of the piezoelectric rod (see left plot). This is due to applied AC drive voltage \(v_d\) to the upper and lower electrodes (see the right drawing).

When there is no angular velocity \({\dot{θ}}_z\) in the \(z\) direction, the applied voltage only causes the up and down vibration. However, when the angular velocity \({\dot{θ}}_z\) is present, the Coriolis force is generated and applied to the middle section of the piezoelectric rod, causing the oscillation horizontally (blue arrow in the \(x_b\) direction in the right plot). When the rod oscillates in the \(x_b\) direction, AC voltages (\(v_{s+}\) and \(v_{s-}\)) are generated, where the magnitude of AC voltages (\(v_{s+}\) and \(v_{s-}\)) are proportional to the angular velocity. In summary, the principle of a piezoelectric rate gyro is the same as that of a MEMS rate gyro. The only difference is how the electric signal is collected. See reference [28] for a detailed review of piezoelectric rate gyros.

A ring laser gyroscope (RLG) [29] consists of a ring laser having two independent counter-propagating resonant modes over the same path [30]. The difference in phase is used to detect rotational speed. It operates based on the principle of the Sagnac effect [31] which shifts the nulls of the internal standing wave pattern in response to angular rotation. Interference between the counter-propagating beams (frequency difference between the two beams) can be observed externally, resulting in motion of the standing wave pattern used to calculate rotational speed.

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