### Homework 2

```COMS W4733, Computational Aspects of Robotics, Fall 2014
HOMEWORK 2, DATE DUE: Thursday, Oct. 9. Individual part due in class, programming team part due electronically 11:59PM.
Note: This is a two part assignment. Part I is an INDIVIDUAL assignment, to be done only by yourself. Part II is a a
team problem to work on with your lab partner on the Create robot.
PART I:
1. (8 pts) Given 3 coordinate frames, o1 x1 y1 z1 , o2 x2 y2 z2 , o3 x3 y3 z3 . If

R21
1

=  0
0
0

0√
1
√2
3
2
−
3
2
1
2

0

1
 ; R3 =  0
1
0
1
0

−1
0 
0
(1)
Find the matrix R32 .
2. (8 pts) Consider the following rotation matrix. What is the axis of rotation (X,Y, or Z) and what is the angle of
rotation?


0 −0.866

1
0
0
0.5
0.5
 0
0.866
(2)
3. (4 pts) Draw a standard world coordinate frame w. Then draw the resulting coordinate frame described by the
transform below:

0 1 2
1 0 4 

0 0 5 
0 0 1

0
 0

 −1
0
(3)
4. (4 pts) Fill out the missing items in the 4x4 homogeneous transform below:

?
 ?

 ?
0
0
0
−1
0

−1 5
0 3 

0 2 
0 1
(4)
5. Below is the Cartesian coordinate frame of a robot’s final position:

0.354 −0.674
 0.505
0.722

 −0.788 0.160
0
0
0.649
0.475
0.595
0

4.33
2.5 

8 
1
(a) (8 pts) Show the ZYZ Euler angles and translational components that will achieve this position.
(b) (8 pts) Show the Equivalent axis rotation and axis direction that will achieve this position.
1
(5)
6. (10 pts) Consider the diagram of fig. 2.14. A robot base frame o0 x0 y0 z0 is set up 1 meter from a table. The table
top is 1 meter high and 1 meter square. A table frame o1 x1 y1 z1 is fixed to the edge of the table as shown. A cube
measuring 20 cm on a side is placed in the center of the table with cube frame o2 x2 y2 z2 established at the center
of the cube as shown. A camera is situated directly above the center of the block 2 meters above the table top with
camera frame o3 x3 y3 z3 attached as shown.
(a) Find the homogeneous transformations relating each of these frames to the base frame o0 x0 y0 z0 :
base
base
base
Ttable
, Tcube
, Tcamera
(b) Find the homogeneous transformation relating the cube frame o2 x2 y2 z2 to the camera frame o3 x3 y3 z3 :
cube
Tcamera
PART II:
Implement a BUG2 algorithm to move the create from a designated starting point to a goal point in an environment.
Your robot should start out along the straight line path from start to goal (the M Line). If it hits any obstacles along
the way, it will invoke a wall following behavior until the M Line is reacquired and the robot again starts out for the
goal point along the M Line. When goal point is reached, robot will stop.
(30 points) Test your code in the simulator. Assume a trajectory from a start point to a goal point exactly 4 meters
in front of the robot. Add test obstacles in the simulator that impede the path, requiring the create to invoke BUG2
behavior.
(20 points) Now, test your code on physical obstacles in a real environment. Assume the goal point is exactly 4
meters in front to the robot’s starting point.
(5 points) Extra Credit: Map out your robot’s progress graphically, showing posiiton and robot orientation as it
moves. Note: this may slow down your communication and affect performance of your robot. If you are experiencing
this, then cache the pose information and map it out after the robot stops.
2
3
```