The objective of this experiment is to design a controller that maintains the direction of the gyroscope module while the top base plate is rotated relative to the bottom base plate. While the disk spins, the SRV02 is used to apply the correct amount of counter torque and maintain the gyroscope heading in the event of disturbances (i.e., rotation of the bottom support plate).
Gyroscopes are used in many different platforms, e.g., airplanes, large marine ships, submarines, and satellites.
In the next sections, we will develop a model of the gyroscope device from first principles in order to obtain a transfer function of the plant, which we will later use in a control scheme. To do so, however, we need some preliminaries on angular momentum and the gyroscopic effect.
Let's consider an rigid body of mass , moving with any general motion in space. Assume that the axes x-y-z are attached to the body with origin at its center of mass . Now, let be a mass element of the body situated by position vector with respect to the center of mass .
Given the body's linear velocity (i.e., the velocity of its center of mass) and its angular velocity , the total angular momentum of the body with respect to the center of mass is given as
The first term is since the integral by definition of the center of mass. By using the triple product rule for the cross product in the second term, it can be shown that
where is the inertia tensor of the body about the center of mass .
Now, since the body's reference frame is rotating, the derivative of the total angular momentum about the center of mass can be written as
where is the time derivative of the angular momentum as seen in the rotating x-y-z frame. We can view the term as the part due to the change in magnitude of and the second term as the part due to the change in direction of . By virtue of the Newton-Euler equation, we have
and we notice that either a change in magnitude of the angular moment or a change in direction of the angular momentum generates a torque. Inversely, a torque can generate a change in either magnitude or direction of the angular momentum, depending about which axis it is applied.
Assume now that the total angular momentum of the gimbaling gyroscope is given by
where is the angular velocity of the gyro module and is the angular velocity of the flywheel, is the inertia tensor of the gyro module - without the flywheel - and is the inertia tensor of the flywheel.
Assuming that the deflection of the flywheel gimbal axis is small and slow, we now have that, when expressed in the rotating frame of the flywheel gimbal coordinates,
and
Further assuming that the rotary base motor has achieved steady-state (, ), and that the flywheel's velocity is constant (i.e. ), we have that
Assuming that the gimbal frame is aligned with the principal axes of he gyro module assembly and ignoring the contribution of the wheel inertia to the principal moment of inertia about the vertical axis of the gyro module, then the second term is 0, and we are left with the equations
corresponds to the torque due to the gyroscopic effect and will be hereon denoted as . We will further neglect the second equation pertaining to the vertical axis torque by assuming that is typically small.
The Servo Base Unit (SRV02) open-loop transfer function is given by
where is the load gear position and is the applied motor voltage. The system steady-state gain and time constant are given by:
and
Consider the simplified model shown in Figure 1.1.
The inertial disc, flywheel, spins at a relatively constant velocity, . When the base rotates at a speed of , the resulting gyroscopic torque about the sensitive axis is
where is the angular momentum of the flywheel and is its moment of inertia. The springs mounted on the gyroscope counteract the gyroscopic torque, , by the following amount
where is the rotational stiffness of the springs.
Given that the spring torque equals the gyroscopic torque, , we can equate equations (1.2) and (1.3) to obtain the expression
Assume that the base speed is proportional to the deflection angle through the gain , then
By examining equations (1.4) and (1.5), we find the gyroscopic sensitivity gain is given by
Thus, the deflection angle can be used to measure , the rotation rate of the platform relative to the base, without a direct measurement. NOTE: the dynamics in the sensitive axis (i.e., deflection axis) are ignored. A more complete model would include these dynamics as the transfer function .
The two springs affecting the sensitive axis are shown in Figure 1.2. The stiffness at the axis of rotation is derived in the following fashion. Assume the springs have a spring constant and an un-stretched length . The length of the springs at the normal position, i.e. , is given by . If the axis is rotated by an angle , then the two forces about the sensitive axis are given by (for a small )
and
The spring torque about the pivot due to the two forces is
The rotational stiffness is given by
The block diagram shown in Figure 2.1 is a general unity feedback compensator with compensator (controller) and a transfer function representing the plant, . The measured output , is supposed to track the reference and the tracking has to match certain desired specifications.
The output of the system can be written as
By solving for we get the closed-loop transfer function
When a second-order system is placed in series with a proportional compensator in the feedback loop as in Figure 1.2, the resulting closed-loop transfer function can be expressed as
where is the natural frequency and is the damping ratio. This is called the standard second order transfer function. Its response properties depend on the values of and .
The desired time-domain specifications for stabilizing the gyroscope are:
or 3Hz, and
To stabilize the heading of the gyroscope, we will develop a Proportional-Derivative (PD) controller depicted in Figure 2.2.
Assuming that the support plate (and servo) rotates relative to the base by the angle (not measured) and that the gyro module rotates relative to the servo module by the angle (measured), the total rotation angle of the gyro module relative to the base plate can be expressed by
We want to design a controller that maintains the gyro heading, i.e. keeps , independent of and we can only use the measurement from the gyro sensor, . In other terms, we want to stabilize the system such that . Differentiating equation (2.4) gives
Given that and the gyro gain definition in equation (1.5), this becomes
Taking the Laplace and solving for , we get
Introducing the new variable such that
which is the integral of the deflection angle, the gyro transfer function can be changed into the form
Add the SRV02 dynamics given in Section 1.1 into to introduce our control variable to get
Adding the PD control
and solving for we obtain the closed-loop transfer function
Find the steady-state speed of the flywheel, , given the motor equation
where is the nominal current, is the nominal voltage, is the motor resistance, and is the back-emf constant.
2. Find the value of the gyroscope sensitivity gain, . The flywheel moment of inertia is . The radius and spring stiffness parameters, are respectively and .
3. The closed-loop transfer function was found in equation (2.6). Find the PD control gains, and , in terms of and . (HINT: Remember the standard second-order system equation).
4. Based on the nominal SRV02 model parameters, and given in Section 1.1, calculate the control gains needed to satisfy the time-domain response requirements given in Section 2.2.
In this section, the gyroscopic control developed in Section 2.3 is implemented on the actual system. The goal is to see if the gyro module can maintain its heading when a disturbance is added by the user, i.e., the base plate is rotated.
The q_gyro Simulink diagram shown in Figure 3.1 is used to run the PD control on the Quanser Rotary Gyroscope system. The SRV02 Gyroscope subsystem contains QUARC blocks that interface with the DC motor and sensors of the system.
0. Download the experiment files:
Verify that the amplifier is turned ON and the disc is rotating. Ask TA if there's an issue.
Run the setup_gyro.m script
Open the q_gyro simulink diagram.
Briefly describe the main goal of the experiment
Briefly describe the experimental procedure in steps 7 and 9 of section 3.
Briefly describe the experimental procedure in step 11 and 12 of section 3.
To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.
Press Connect button under Monitor & Tune and Press Start .
While the system is running, manually rotate the bottom base plate about 45 degrees. The GYRO module should be maintaining its heading. Verify your response by viewing the scopes in your experiment and comparing against the provided scope examples.
Stop the controller once you have obtained a representative response.
Plot the responses from the theta (deg), alpha (deg), and Vm (V) scopes in a MATLAB figure. The response data is saved in variables data_theta , data_alpha, data_vm.
Start the controller again, but this time with the Manual Switch set in the upward position, which turns off the PD controller.
Rotate the bottom base plate by the same amount as previously done, in an attempt to reproduce the motion as previously executed.
Plot the responses.
Examine how the GYRO module responds when you rotate the base plate. Does this make sense? Explain the result when the PD control is ON and OFF. Based on your observations, explain what the PD control is actually doing and how it relates to gyroscopes.
Explain the effect of having the PD control on and off.
Briefly explain how does this relate to an actual gyroscope system?
