Contents

Physical Setup

In this section we show how to build the inverted pendulum model using the physical modeling blocks of Simscape Multibody. The blocks in the Simscape library represent actual physical components; therefore, complex multibody dynamic models can be built without the need to build mathematical equations from physical principles as was done by applying Newton's laws to generate the model implemented in Inverted Pendulum: Simulink Modeling.

The system parameters are defined as follows:

(M)       mass of the cart                         0.5 kg

(m)       mass of the pendulum                     0.2 kg

(b)       coefficient of friction for cart         0.1 N/m/sec

(l)       length to pendulum center of mass        0.3 m

(I)       mass moment of inertia of the pendulum   0.006 kg.m^2

Create world frame and basic configuration

Open a new Simscape Multibody model by typing smnew in the MATLAB command window. A new model opens, as shown below, 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.

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

Assemble base plane and cart

We will model the cart as a point mass moving along an axis. We use the World Frame to define the axis along which the cart will travel.

Tips for adding blocks:

Add the following blocks:

* Prismatic Reference
* Scope
* Pulse Generator

For the Pulse Generator, double-click on the block and set Period to "10", Amplitude to "1000", Pulse Width to "0.01", and Phase delay to "1".

The Prismatic Joint allows only one translational degree of freedom. The Prismatic Joint will be actuated by a force input.

The Prismatic Joint next needs to be connected to the rest of the model.

Since we are modeling the sliding cart as a point mass, only the mass will affect the simulation results.

The model at this point should now appear as follows.

Running a simulation (type CTRL-T or press the green arrow run button), the resulting plot shows both the distance traveled by the cart as well as its velocity.

Pendulum subsystem and connecting the cart to the pendulum

Add the following blocks to the model:

To define the axis of rotation for the pendulum:

To define the degree of freedom of rotation for the pendulum:

To model the pendulum:

To model the connection point to the cart:

The resulting model should appear as follows:

Running a simulation (type CTRL-T or press the green arrow run button), the following plot is generated, where one can see that the addition of the pendulum alters the cart behavior both in its distance traveled as well as its velocity.

Selecting outputs for controller and angle conversion

We now need to measure the angle and angular velocity of the pendulum:

We need to limit the measured angle to stay between -pi and pi radians.

In this subsystem we will add pi radians to the measurement, find the remainder when the signal is divided by 2*pi, and then subtract pi radians.

Here you can see the resulting subsystem for wrapping the angle

We will plot the new signals on a Scope

The resulting model should appear as follows:

Running a simulation (type CTRL-T or press the green arrow run button), the followings plot are generated. The motion of the cart is the same as before, but now we can see the motion of the pendulum. The associated animation shown below is also generated.

Create subsystem for pendulum and cart

We have now successfully created all the elements of the inverted pendulum system. We will convert this into a subsystem.

Your model should now appear as shown:

Closed-loop setup

We will now add blocks for open and closed-loop testing.

Add the following blocks:

The Disturbance also will be added to the control signal.

We wish to manually select open or closed-loop behavior.

We will close the loop to control the pendulum angle.

Your model should now appear as follows. The output of the simulation is unchanged from prior results when in open-loop mode.

Controller implementation

We will now implement the PID control gains developed in the Inverted Pendulum: PID Controller Design page.

Double-click on Manual Switch until the input from the PID Controller is selected.

Running a simulation produces the following two plots show the controlled response of the system. After the initial impact the controller was able to quickly bring down the pendulum angle to zero and the pendulum velocity is also zero. The cart moves slowly and with a constant velocity in the negative X direction to keep the pendulum balanced.

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

Published with MATLAB® 9.7