Saturday, February 11, 2017

The math behind the vertical plotter

Many people are asking after the math behind the rope plotter. I used only second grade high school math to do it. Here's how it works. 

Target motor positions

For all calculations I am using a coordinate system with the top left of the door as (0,0) and with cm as units. So the top right of the door is (90,0). The positive y-axis is pointing down so I have only positive numbers. The first goal is now to calculate the correct lengths of the ropes L and R in terms of the coordinates (x,y) of the target location for the robot. For this you can use Pythagoras. The left rope is the easiest: there is a triangle with sides x,y and L where `L = (x**2 + y**2)**0.5` (In python the double asterisk means a power). The right rope is also part of a triangle. This triangle has y, 90-x and R as sides. Therefore `R = ( (90-x)**2 + y**2 )**0.5`. Now that we have the target lengths we have to calculate the degrees the motor has to rotate to reach our desired coordinate. The diameter of the bush is about 0,9 cm. A complete 360 degree rotation results therefore in a 0,6 cm * 3,14 change in length. That is about 170 degrees of rotation to move one centimeter. I mounted the motors so that forward rotation is upwards. Therefore the motor target for the left motor is L * -170, the right target is R * -170. 

Navigating the drawing canvas

Now we have a formula to navigate the complete door. But what we really want is to navigate a piece of paper in the middle of the door. I wanted the robot to be usable independently of the space between the attachment points or the size of the paper. Therefore I defined a second coordinate system that has (0,0) in the top left of the paper and is normalised. This means all coordinates are between 0 and 1. This coordinate system is easy to scale on different plotting surfaces and paper sizes. It works like this: let m be the margin left of the paper, n be the margin on top and C the with of the canvas. The coordinates (x,y) in terms of the normalised coordinates (u,v) become simply: `x = C * u + m` and `y = C * v + n`.

At this point it's simply a matter of reading two files, x.rtf and y.rtf with normalised coordinates and moving the robot towards each of these coordinates. 

Ev3 program

You can also download the Lego Mindstorms Ev3 program to see the details. The ev3 main program expects 2 rtf files (robot text files), on named x.rtf and the other y.rtf. Both contain target coordinates. x.rtf has the number of coordinates as it's first line. Note that you can't open these files with a Rich Text File editor!