Kinematics of Wheeled Mobile Robots Pt. 3

Our team finally got the robot moving and steering! Which means it's finally time for me to start the implementation process of the kinematic and control equations. To further verify and help prepare for the implementation, I made a little simulator in MATLAB to help visualize how the robot will move given certain (steady state) commands. Transient commands are a whole other issue, which I may discuss later. 

The structure of the top level code is simple. It defines the parameters of the robot and of the simulation, and calls the swerve_command function. This function performs the inverse kinematics and spits out the wheel angles and speeds, as well as the position (x, y, theta) information for that time step. Over several time steps, the robot will move. If steady state inputs are held, the wheel angles and speeds will remain constant, even if the robot itself is executing some curvilinear motion. The robot is plotted at each time step. To verify the wheel angles are correct, you need to visually confirm that the wheels are tangent to the path taken by the wheels in each snapshot. 

Confirming the wheel speeds is more difficult. For any type of pure translation all the wheel speeds just need to be the same. This has been confirmed. If there is any type of rotation involved, the wheels will orbiting some central point. If the location of the point is known, then the radius to each wheel can be found and using the omega = v/r equation you can confirm that all the angular speeds about that central point are equal. This is hard to do currently because we are normalizing all our speeds because they are simply throttling values which will be sent to the motors. We do not know exactly what speeds they will spin at. However, the ratios of the speeds are telling. What we know is that the wheels on the largest radius of the orbit must have the highest linear velocity, and the wheels on the smallest radius of the orbit must have the smallest linear velocity. This allows all the wheels to have the same angular velocity around that central point. As you can see from the photos, this phenomenon is evident by the colored velocity vectors. 

There is something funky going on with the way that the motion of the robot is plotted. The wheel angles and speeds look okay, but when I try to plot the motion some of the components of the motion (X and rotation) have the wrong sign and it looks like the robot is skidding across the floor. I haven't been able to figure out why that is yet. I have a feeling it's some kind of sign error but I haven't found it yet.

Open Issues:

1. Figuring out how to update the wheel speeds and angles while moving (transient command inputs)

2. Normalizing the wheel speed outputs to prevent speed commands to the motor from saturating

3. Finding the little sign errors that are messing things up

Bonus:

We also made a little case for the webcam BreakerBot will be using for the First Person View mode for remote operation. BreakerBot is watching you...

MATLAB Project: String Vibration

We recently covered vibrations of strings in my mechanical vibrations class. I decided to try and make a GUI to animate the vibration of the string. There are lots of options for changing the string properties and simulation parameters, as well as subjecting the string to various end conditions and combinations of initial conditions.

I hope to expand this simulation to animate 2-D membranes as well. It would be very cool to simulate how a drum head or the body of a violin vibrates in MATLAB.

Learning to Use Simulink

For our senior design project, we are trying to implement a swerve drive system to achieve holonomic robot motion. None of us really know much about controls, but from the information we have gathered online, it seems advantageous to model our control system in Simulink before trying to implement it with code on our robot. Simulink seems pretty simple to use, at least for basic PID loops. If you have a block diagram for the system you want to model, you just drag and drop the blocks of the system and connect them together. You can change the parameters of the blocks simply by double clicking. 

I made a really simple one here of a basic PI loop responding to a step input. You can change the proportional and integral gains to tune the controller, if you wish. 

Here are my concerns regarding the Simulink model:

1. I don't know how to model my inverse kinematics with the standard blocks. Will I have to make my own custom block to do all the kinematics, or make each step of the equation a separate block?

2. How to get the angle of the robot with respect to the ground? In our robot, this will come from a gyro, but how can I model this in Simulink?

3. Step inputs and sinusoid inputs are useful, but it would be really helpful to model the input from our actual joystick. I need to figure out how to do that. There is a joystick block, but I have no idea how to use it.

4. We currently have no idea how to model the robot dynamics (the 'plant', in control theory lingo). This is the most important part. If your plant is inaccurate, your control will not work.

5. Where does the plant begin/end? Should we attempt to model the motor behavior in Simulink? Or make the motor part of the plant? I've found sources online that go either way.

So after a couple hours of playing with Simulink, I think I have solved issue #1. I ended up writing a custom Simulink block from a MATLAB function. It then compiles the MATLAB code into the block. I am making a smaller system to test the kinematics independently. I will check them with the graphs I generated earlier in the semester.

I sort of found a way around #2 for now by integrating the angular rate signal. But the angle from this is before the plant, so it is not really correct. But it may work for now, just to get the system to run.

The main problem I have right now is compiling the code from the MATLAB Function Block. The Simulink-compatible C-compiler is was not installed on my computer, so I had to download the Xcode app and use the compiler. But it's still returning an error, so there is something else I have to fix evidently.