Gyroscope

Objective

The purpose of this experiment is to design a controller that maintains the direction of a gyroscope under base excitation. The controller can also be used to rotate the gyro platform to a desired orientation using the gyroscopic principle.

Equipment Required

• Gyroscope (Electromechanical plant)

• Input/Output Electronic ECP Model 750 box

Part A: Modeling

The Gyroscope Model

This experiment will be performed using the Model 750 Control Gyroscope. The system, shown in Fig. 3.1, consists of an electromechanical plant and a full complement of control hardware and software. The user interface to the system is via an easy-to-use PC-based environment that supports a broad range of controller specification, trajectory generation, data acquisition and plotting features. A picture of the setup including the DSP card and input/output electronics is shown in Fig. 3.2.

Description of the Gyroscope

The gyroscope consists of a high inertia brass rotor suspended in an assembly with four angular degrees of freedom, as seen in Fig. 3.3. The rotor spin torque is provided by a rare Earth magnet type DC motor (Motor #1), whose angular position is measured by an optical encoder (Encoder #1) with a resolution of 2,000 counts per revolution. The motor drives the rotor through a 10:3 gear reduction ratio, which amplifies both the torque and encoder resolution by this factor. The first transverse gimbal assembly (body C) is driven by another rare Earth magnet motor (Motor #2) to effect motion about Axis #2. The motor drives a 6.1:1 capstan to amplify the torque between the adjoining bodies C and B. A 1000 line encoder (Encoder #2) with 4x interpolation is mounted on the motor to provide feedback of the relative position between bodies C and B with resolution of 24,400 counts per revolution.

The subsequent gimbal assembly, body B, rotates with respect to body A about Axis #3. There is no active torque applied about this axis. A brake, which is actuated via a toggle switch on the Controller box, may be used to lock the relative position between bodies A and B and hence reduce the system's degrees of freedom. The relative angle between A and B is measured by Encoder #3 with a resolution of 16,000 counts per revolution. Finally, body A rotates without actively applied torque relative to the base frame (inertial ground) along Axis #4. The Axis #4 brake is controlled similar to the Axis #3 brake and the relative angle between body A and the base frame is measured by an optical encoder (Encoder #4) with a resolution of 16,000 counts per revolution.

Inertial switches or “g-switches” are installed on bodies A, B, and C to sense any over-speed condition in the gimbal assemblies. The switches are set to actuate at 2.1 g’s. For Axis #2, limit switches and mechanical stops are provided at the safe limit of travel. When any of these normally closed switches sense a high angular rate condition, they open and thereby cause a relay to turn off power to the controller box. When this power is lost, the fail-safe brakes (power-on-to-release type) at Axes #3 and #4 engage. Also, upon loss of power, the windings of Motor #1 and #2 have shorted, thereby causing electromechanical damping. Thus, all axes are actively slowed and stopped whenever an over-speed or over-travel condition is detected.

Metal slip rings are included at each gimbal axis to allow continuous angular motion. These low noise, low friction, slip rings pass all electrical signals including those of the motors, encoder, g-switches, limit switches, and brakes to the control box.

Mathematical Model of the Gyroscope

In the configuration used in the experiment, gimbal Axis #3 is locked ( = 0), so that bodies A and B become one and the same (see Fig. 3.4). The resulting plant is useful for demonstrations of gyroscopic torque action where the position and rate, and , may be controlled by rotating gimbal #2 while the rotor is spinning. Figure 3.5 shows the coordinate system used for this model.

Basic Equations

Figure 3.5 shows the definitions of the coordinate axes used for the model development. Let (x = A, B, C and D) denote the scalar moments of inertia respectively in the bodies A, B, C, and D. Referring to Fig. 3.5, the inertia matrices are given as follows:

The basic gyroscope equation can be written as:

where is the vector of applied torque. Referring to Fig. 3.5, the components of applied torque can be written as

where and are the applied torques about Axes #1 and #2, respectively (see Fig. 3.5).

The components of the angular momentum vector and the angular velocity vector can be written as

Substituting equations 3.2, 3.3 and 3.4 into Eq. 3.1 results in the following set of nonlinear equations:

Linearized Model

In this section, we proceed to linearize the nonlinear gyroscopic equations (Eq. 3.5) by imposing certain assumptions.

Assumptions

1. The angle of rotation of the rotor disk D about the gimbal axis 2 is small.

2. are small, where is the spin speed of the rotor disk D (rotational speed of the rotor) which is given as 400 RPM or 41.89 rad/sec.

Upon linearization (), we further obtain the following linearized equations:

In these equations, the rotor spin dynamics (Eq. 3.6) are decoupled from those of the second and fourth gimbals (Eqs. 3.7 and 3.8). Since the rotor speed may be independently controlled, the salient dynamics become those involving motion at the gimbal locations. The gimbal angular positions and are related to and as

Taking the Laplace transform of Eqs. 3.7, 3.8 and 3.9 and eliminating and from the resulting equations results in the following transfer functions for and :

Part B: Controller Design

Control System Overview

The objective is to design a controller for regulation and control of the angular position using the gimbal torque . This mimics the following problem: Given a spacecraft with a momentum gyro, we would like to regulate and control the spacecraft attitude () using the gimbal torque () of the gyroscope. In this lab, the position of Axis #4, , is controlled by torquing the Axis #2 motor. This is accomplished here with the use of a technique known as successive loop closure. First, a rate feedback loop around is closed to damp the nutation mode of the gyroscope. Subsequently, an outer loop is closed to control . The block diagram for this process can be seen in Figure 3.6.

The equations defining the numerators and denominators in the block diagram are as follows.

Note that the and shown in Fig. 3.6 represent measurements using encoders at Axes #2 and #4, respectively. The values and represent the encoder gains for Encoders #2 and #4, respectively. Finally, is a control effort gain and is the rotational speed of the rotor in rad/sec (given above). The numerical values of various parameters are given in Tables 1 through 3 in the next page.

System Parameters

Table 1: Gyroscope Inertia Values

Table 2: Encoder Gains

Table 3: Control Effort Gains

Specifications

1. All closed-loop poles must be in the left half of the complex plane

2. The peak time in response to a unit step command input must be less than 0.2 sec.

Procedure

1. Select the inner loop derivative gain to be equal to 0.08.

2. Open a new m-file and compute the transfer function using equations (3.12) through (3.14) and Fig. 3.6. Save your m-file.

Part C: Controller Implementation & Evaluation

Experimental Setup

These steps should be performed primarily by the TA, but the students should follow along to gain a deeper understanding of how the experiment works.

Functionality Test

This test ensures if everything is sound mechanically, electrically, and in the software.

1. Hardware checks

   1. Ensure the rotor cover panel is secure.

   2. Switch on the ECP control box and disengage Axis 3 and Axis 4 brakes.

Orientation setting

1. With the ECP control box powered on, all brakes off, and flywheel not turning, roughly orient the gyroscope according to Figure 3.7 (as viewed from the front such that labels are the correct way up).

2. Apply all brakes and turn off the ECP control box.

3. Ensure Axis 3 is parallel with the platform base by using a spirit level or phone app, making small adjustments by hand (with the brakes on).

Controller Setup

Configure Data Logging

1. Go to the Data menu and click on Setup Data Acquisition.

2. Be sure that the selected variables to log match those below. When finished, hit OK to exit this menu.

Initialize Rotor

1. Switch on the ECP control box.

2. Brake all axes so that the accelerating rotor does not cause the axes to move from the initialized positions:

   1. Set Axis #3 and Axis #4 on the ECP control box to ON.

Configure Control Algorithm

1. Go to the Setup menu and clicking on Control Algorithm.

2. Click the Load From Disk button and select the default controller file, GyroControl_Gimbal4_PID.alg. The original file can be found at C:\AE4610-ControlsLab\Gyro_Controller but, since this version is read-only, it should be loaded from a locally saved copy on the TA's login profile.

Controller Evaluation

With the rotor initialized and the closed-loop controller active, you will now evaluate your calculated gains in three different trajectories.

Trajectory 1: Small Step Input

1. Select the Command menu and then Trajectory 1.

2. Select the Step Input and then click the Setup button. Enter the following parameters for this case:

Trajectory 2: Large Step Input

1. Repeat steps 1-8 from Trajectory 1, but this time using the following trajectory settings:

   1. Step size = 1000 counts.

   2. Dwell time = 1000 ms.

Trajectory 3: Ramp Input

The step response is useful for system characterization but is seldom used for an actual in-service trajectory because it is excessively harsh (high acceleration and jerk ("rate of change of acceleration")). A more common trajectory used for tracking applications is a ramp.

1. Repeat steps 1-8 from Trajectory 1, this time using the following trajectory settings:

   1. Ramp input (Unidirectional Moves not checked).

   2. Distance = 6000 counts.

Trajectory 1 Re-Run

Only if time permits at the end of your session, go back to Trajectory 1 and see if you can improve the controller's response by manually adjusting the gains using your intuition and knowledge of individual gain effects. This task is intended only to allow you an opportunity to get a real-life feel for the effect of gain changes, it will not be graded.

Analysis & Lab Report

• Include an overview of the controller design from Part B.

• Using the SIMULINK block diagram you have developed of Fig. 3.6 in Part B, simulate the system response for the same step and ramp commands you have used in Part C. Compare your simulated responses with the corresponding experimental results. What is the rise time and % overshoot in the experiment and simulation responses to the step command used?