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# Calculus in Robotics

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## Master Jellee

on 17 June 2014

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#### Transcript of Calculus in Robotics

Calculus in Robotics!
PID Controller

Integral Control
Structure:
use the integral to find sum of these errors
multiply integral value by constant, iGain (determined experimentally)

Motor Speed = (iGain)(Integral value)
Topics We Will Be Covering:
Fourier Series/Transform
Vectors
PID Controller
Fourier Transform
Introduction
Vectors!
Remember...
Some Review
What is the integral for?
Calculus plays a significant role in many aspects of robotics and machinery, whether it be through motion control, signal processing, or programming
Our goal for today is to introduce several topics of calculus that hold important applications in the field of robotics
By Karen Malacon & Janice Lee
Proportional Control
Basic, Easy to Use
Difference between setpoint and process variable is error.
Motor Speed=(pGain)(error)
Proportional Gain determined experimentally
Proportional Control

pGain=0.3
error=setpoint-process variable
While(error not negligible)(error=setpoint-processVariable
motorspeed=error*pGain)
The Fourier Transform is a tool that breaks a waveform (function or signal) into an alternate representation, characterized by sine and cosine graphs.
Shows that any waveform can be rewritten as the sum of sinusoidal functions
One of the Fundamental Secrets of the Universe
: All waveforms, no matter what you scribble or observe in the universe, are actually just the sum of simple sinusoids of different frequencies.
Applications
Used in MRI (Magnetic Resonance Imaging) to determine what the frequencies and amplitudes are of the signals measured from the object being imaged.
Audio Compression: Take a sound, expand its fourier series, but converges quickly; MP3 Format
Fourier Series
Way to represent a wave-like function using a combination of simple sine waves.
It decomposes any periodic function or period signal into the sum of a set of simple oscillating functions.
magnitude and direction
cartesian coordinates (x,y)
polar coordinates (magnitude and angle)
Example
Find the fourier series of the function
Formulas We Need to Know:
Fourier Series Notation
Euler-Fourier Formulas
http://www.sosmath.com/fourier/fourier1/fourier1.html
Solution
Since f(x) is odd, then an = 0, for n greater than or equal to 0. We turn our attention to the coefficients bn. For any n greater than or equal to 1, we have
We deduce
Therefore
Proportional
Integral
Derivative
PID Elements
Process box changes the speed of the motor
Pseudo Code
Offset
Derivative Control
What is Derivative?
Hopefully all of you know the answer to this one.
The Derivative will measure how fast the robot arm changes position or how fast the error increases and decreases.
Time
Error
Motor Speed=(dGain)( e)
Derivative Gain
PID Control
Pseudo Code
pGain=0.3
iGain=0.2
dGain=0.1

error=setPoint-processVariable

While(error NOT negligible){
error=setPoint-processVariable
error=errorSum+error
calculate e

motorSpeed=(error*pGain)+(errorSum*iGain)+( e*dGain)}
Vector calculus useful for control of omnidirecional/holonomic robot
Can move sideways and rotate, not just forward backward
Area under the curve in a graph that measures TIME (x) by ERROR (y)
Error = difference between current and final position
Error decreases
as
time increases
(robot arm moves closer to final position as time passes)
pGain = 0.3;
iGain = 0.2;

errorSum = 0;

while(error not negligible)
{
error = setPoint - processVariable;
errorSum = errorSum + error;

motorSpeed = error*pGain + errorSum*iGain;
}
Full transcript