Numerical integration and differentation in the PID algorithm
Introduction
PID is elaborated algorithm to use in numerous ways. Though it might difficult to set up the parameters correctly, it makes the robot move more smoothly and thus more rapid. The P is for Proportional, I is the Integral and D is Differential. We use PID in line following.
Aim
Proportion, Integration and Derivation.
Plotting, Error function.
Robot
Almost any robot will do, but we will use Asimov 2/ Verne robot.
Sensors
A light sensor is used.
Example Video
Theory
If applying only proportional coding, the steering is affected by the amount of difference in reflectance from the average. The integral term is a sum (or integral) of the previous (historical) errors, and differential is used to forecast the next error based on the previous errors.
| Term | Meaning | Math |
|---|---|---|
| Error | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} | Sensor value - target value |
| Integral | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} | Integral + error |
| Derivative | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} | Error - previous error |
We use three different parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_p} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_d} . Furthermore, the integral needs to forget the very old data, and thus we need an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} term (which can be multiplied together to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_i} already). Of course, the parameters depend on the line and robot.
Exercises
References
See [PID Controller For Lego Mindstorms Robots]
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