Contents

Physical setup

In this section we will alternatively show how to use physical system modeling to build the nonlinear ball and beam model implemented by mathematical model in Ball & Beam: Simulink Modeling. We will also implement the control algorithm developed in the Ball & Beam: Root Locus Controller Design page to control the ball's motion.

The configuration of the physical system is shown in the schematic below. As the servo gear turns through an angle , the lever changes the angle of the beam by . When the angle is changed from the horizontal position, gravity causes the ball to roll along the beam.

For this problem, we will ensure that the ball rolls without slipping and the friction between the beam and ball is negligible.

The constants and variables for this example are defined as follows:

(m)      mass of the ball              0.11 kg

(R)      radius of the ball            0.015 m

(d)      lever arm offset              0.03 m

(g)      gravitational acceleration    9.8 m/s^2

(L)      length of the beam            1.0 m

(J)      ball's moment of inertia      9.99e-6 kg.m^2

(r)      ball position coordinate

(alpha)  beam angle coordinate

(theta)  servo gear angle

The design criteria for this problem are:

Create world frame and basic configuration

Open a new Simscape Multibody model by typing smnew in the MATLAB command window. A new model, as shown below, opens with a few commonly used blocks already in the model. The PS-Simulink and Simulink-PS blocks define the boundary between Simulink input/output models where the blocks are evaluated sequentially and Simscape models where the equations are evaluated simultaneously. Furthermore, you will need to define the associated model parameters in the MATLAB workspace. This can be accomplished by running the file ball_beam_properties.m which can be downloaded by right-clicking and selecting Save link as ....

To configure basic settings in the model, carry out the following:

Assemble the gear

The gear needs a pin at its center and another pin at a radial distance of d from the center where the lever is attached.

Tips for adding blocks:

Add the following blocks:

* Revolute Joint, rename this "Revolute Gear"
* Transfer Fcn
* Gain
* Step Input

Define the degree of freedom for the gear.

Create a subsystem to define the gear and pins:

Now we will add the pins:

Now we will define the gear:

Here is the Gear subsystem:

We will drive the gear with a motion input. The motion input must be filtered to avoid an algebraic loop. We must also ensure that the motion control signal and its first two derivatives are continuous, so we will apply a critically damped filter in the Simulink-PS converter block.

We will measure the angle of the gear and use that angle to estimate the angle of the beam upon which the ball is resting.

The overall model will look like the following:

Running the simulation (CTRL-T or press the green arrow run button) will produce the following plot.

Add the lever

For the given dimensions of the Gear and the Beam the lever that connects Gear and the beam needs to be 0.35 m in length. In this section we will add the lever to the model. Add the following items to the model:

We will define a lever with one solid and two connection points. First we define the connection points.

Now we define the lever

The Lever subsystem should look like the following:

The whole model should appear as follows:

Simulating the model shows the results as the lever swings around:

Add the beam

We will now define the connection between the beam and lever.

We will define the beam inside this subsystem. First we define the connection points for the end of the beam.

Next we define the beam.

We will define pins to visually represent the connection between the beam and the lever and a fixed point in space.

Now we define the reference frame for the ball.

The Beam subsystem should like the following:

Now ee need to define the connection between the beam and a fixed point in space.

The whole model should then appear as follows:

Simulating the model then produces the following plot as the lever swings around.

Add the ball

Now we will add the ball to our model. The ball must be constrained to translate along the beam, and a separate constraint is used to ensure that the ball rolls without slipping on the beam. A few blocks are used to define the axes and constraints.

Inside this subsystem, we will define the constraints for the ball. First we will define the constraint for rolling without slipping.

Next, we will add the constraint for the ball to translate along the beam.

Connect the Prismatic Joint to the reference frame and sense the ball position.

We need to add a degree of freedom so that the ball can rotate as it rolls. We first need to define the axis of rotation.

Now we can add the degree of freedom.

The Ball Beam Constraints subsystem should look like the following:

Finally, we can add the ball.

The resulting complete model should look something like the following:

If you simulate this system, you should then see the following response from the scopes. As well as the following animation.

Implementing the controller

We have now successfully modeled the physical system. We will convert this into a subsystem and add the controller.

Now we will close the loop with a controller.

As this is now a stiff system, we need to change the solver settings.

The resulting closed-loop system should appears as follows:

Running the simulation then produces the following outputs as observed on the Scope.

You can download the final Simscape model created here by right-clicking here and then selecting Save link as ....

Published with MATLAB® 9.7