I'd like to start messing around programming and building something with an Arduino board, but I can't think of any great ideas on what to build. Do you have any suggestions?
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I show kids, who have never programmed, or done any electronics before, to make a simple 'Phototrope', a light sensitive robot, in about a day. It costs under £30 (GBP) including Arduino, electronics and off-the-shelf mechanics. If folks really get into mobile robots, the initial project can grow and grow (which I feel is part of the fun).
There are international robot competitions which require relatively simple mechanics to get started, e.g. in the UK http://www.tic.ac.uk/micromouse/toh.asp
Ultimate performance require specially built machines (for lightness) , but folks would get creditable results with an Arduino Nano, the right electronics, and a couple of good motors.
A line following robot is the classic mobile robot project. The track can be as simple as electrical tape. Pololu have some fun videos about their near-Arduino 3PI robot. The sensors are about £1, and there are a bunch of simple motor+gearbox kits from lots of places for under £10. Add a few £ for motor control, and you have autonomous robot mechanics, in need of programming! Add an Infrared Remote receiver (about £1), and you can drive it around using your TV remote. Add a small solar cell, use an Arduino analogue input to measure voltage, and it can find the sun. With a bit more electronics, it can 'feed' itself. And so it gets more sophisticated. Each step might be no more than a few hours to a few days effort, and you'll find new problems to solve and learn from.
IMHO, the most interesting (low-cost) competitions are maze solving robots. The international competition rule require the robot to explore a walled maze, usually using Infrared sensors, and calculate their optimal route. The challenges include keeping track of current position to near-millimeter accuracy, dealing with real world's unpredictably noisy environment and optimising straight-line speed with shortest distance cornering.
All that in 16K of program, and 1K RAM, with real-time interrupt handling (as much as 100K interrupts/second for some motor systems), sensor sampling, motor speed control, and maze solving is an interesting programming challenge. (You might make it 'easy' with 32K of program, and 2K RAM :-)
I'm working on a 'constrained' robot challenge (based on Arduino) so that robot performance is mainly about programming rather than having a big budget.
Path finding/obstacle navigation is typically a good project to start with. If you want something practical, take a look at how iRobot vacuums the floor and come up with a better scheme.
Start small and build up to something more complex. Control servos. Blink LEDs. Debounce inputs. Read analog sensors. Display text on an LCD. Then put it together.
Despite the name, I like the "Evil Genius" book for PIC microcontrollers because of the small, easily digestible projects that tend to build on one another. It is, of course, aimed at PIC programmers rather than the Arduino, but the material covered will be useful no matter what you're developing on.
I know Arduino is trendy right now, but I also like the Teensy++ development board because of its low price-point ($24), breadboard-compatible PCB, relatively high pin count, Linux development environment, USB connectivity, and not needing a programmer. Worth considering for smaller projects.
If you come up with something cool, let me know. I need an excuse to do something fun :)
A POV device could be fun to build (just google for POV Arduino). POV means persistence of vision.
Depends on your background and if you want practical or cool. On the practical side, a remote control could be a simple starting point. It's got buttons and lights but isn't too demanding.
For a cool project maybe a Simon-style memory game or anything with lights & noises (thinking theremin-style).
I don't have suggestions or perhaps something like a line follower robot. I could help you with some links for inspiration
Arduino tutorials
Top 40 Arduino Projects of the Web
20 Unbelievable Arduino Projects