Rhombic Triacontahedron

Another magnetic tile 3d printed polyhedron for the collection. This time, the rhombic triacontahedron.

http://www.thingiverse.com/thing:1593892
https://github.com/eshira/polyhedra/tree/master/Rhombic-Triacontahedron

Happy printing!

Three wheeled robot simulation

I made a simple simulation to figure out how to orient the wheels of a 3 wheeled robot with independently steering and driving wheels based on a desired instantaneous center of curvature (ICC).  It makes a lot of assumptions (perfect point contacts of the wheels with the ground, no slippage allowed, etc) but is a useful conceptual tool.

Get the latest version here:
https://github.com/eshira/wheelsim (uses Python, numpy, and matplotlib)

The triangle represents the frame. The view is a bird's eye view. Each wheel is mounted on short extensions that pivot about the frame at the vertex of the triangle.  The colored lines are the perpendicular lines to the wheel. Where they meet, a red dot denotes the instantaneous center of curvature, or ICC. The black circle denotes the motion of a point in the middle of the robot.

Below the ICC is at the bottom of the robot frame, marked by the red dot. Points on the robot that fall on the perimeter of the black circle, will move along the black circle if the wheels are driven at the appropriate velocities.

To describe the motion of any particular point on the robot frame, just imagine a circle concentric with the red dot that passes through that point on the frame. The relative linear velocities of the wheels can be described by the relative perimeter lengths of the circles that pass through their center.

Below, we restrict the wheels to only 170 degrees steering rotation (remove 5 degrees from end of range on each side). This creates the red areas, which are places the ICC cannot be without forcing the wheels into the disallowed steering rotations.

The red lines pass through the pivot point of the steering of each wheel. To avoid intersections with the robot frame, and because typical servos only support 180 degrees of rotation anyway, the wheels are mirrored if they pass that boundary. To avoid discontinuous driving caused by having to stop to flip the wheel over to the other side, avoid having the ICC cross this boundary.

Here we have parallel line driving (near vertical direction, along the black line). In this situation, the ICC is very far away. The perimeter of the black circle therefore appears to be completely straight. The wheels are all parallel. The colored lines represent the normals to the wheels, which intersect at the ICC, as they are not quite parallel.

Note that the purple wheel is probably going to collide with the frame if we try to drive straight in the direction of the top tip of the frame. It is probably better to drive as shown below instead.

Above the robot drives along the black line shown, horizontal to the image.

Corrections can be made by turning 'left' and 'right' as shown below.