Due Wednesday, January 28
1) Draw a rough floorplan of the room you are sitting in, complete with furniture and other obstacles. Draw a triangular robot. Assume the robot can translate and rotate, and attach a frame to the robot; let the configuration space be parameterized by the location and orientation of that frame 
with respect to some world frame. Draw the slice of configuration space with 
, and a slice of configuration space with 
.
2) Imagine you are writing software to cause a robot to gift-wrap presents. You put the end effector of the robot arm at the two front bottom corners of the box and measure the locations: 
, 
. You also measure the location of a point along the bottom face of the box: 
Using a homogeneous transform matrix, write an expression for an orthogonal frame with the following properties: a is the origin, the first axis points from a towards b, the second axis is in the plane of the bottom face of the box, and makes a positive dot product with the vector from a to c, and the third axis is chosen to satisfy the right-hand rule. Each of the axes should be unit vectors orthogonal to each other, and don't forget to include the translation component in your matrix.
3) The point 
is a point on the front of the box, with coordinates measured with respect to the box frame. In the world frame, what are the coordinates of this point?
4) Find an object that is rigid and mostly planar. Draw it. Place four point frictionless fingers around the boundary so as to immobilize it, and use Reuleaux's method to prove that it is immobilized.
5) Consider the homogeneous transform matrix
(1)![^0_1T = \left[ \begin{array}{ccc} R & & p \ 0 & 0 & 1 \end{array}. \right]](/local--math/eqs/4a5ee2eea4fcc60232f179d269e1f291.png)
that describes a displacement from frame 0 to frame 1. In terms of 
and 
, give the inverse transformation matrix 
that "undoes" the effect of the displacement. Remember that 
.
