Homework #4 Solution

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Description

Notice:

 

Please upload 1 le for this homework (there are no programming assignment this week). In your assignment include the names of the other students you have collaborated with to do the homework problems. The Collaboration Policy for this course is detailed in the syllabus.

 

 

Homework exercises:

 

  1. 1 from the Lecture Notes on Robotics Planning and Kinematics.

 

  1. Suppose we build a new robot by placing and interconnecting a xed-base Scara robot (see Lecture Notes, gure 315) on top of a at disk robot (such as a roomba). What is the con guration space of this new robot?

 

  1. Consider the con guration space of the SCARA manipulator, represented as the square [   ;  ]  [   ;], with appropriate point identi cations to de ne a torus. De ne a graph using the following nodes given in
coordinates of the square: n1 = ( ;   ), n2 = ( ;       ), n3 = (   ; 0), n4 = ( ;   ), n5 = ( ;  ),
2  
                                                                2              
n6 = (   ;   ), n7  = (       ; ), n8 = ( ; 0), n9 = ( ; ), n10 = ( ; ), n11  = ( ;  ),
2 2 2 2 2 2 2 2
n12 = (0;   ), n13 = (0;       ), n14 = (0; 0), n15 = (0;   ), n16 = (0;  ), n17 = (   ; ), n18 = (   ;     ),
      2 2 2 2 2
n19 = (   ; 0), n20 = (   ;   ), n21 = (   ;  ), n22 = ( ;     ), n23 = ( ;     ), n = ( ; 0), n24 = ( ;   ),
    2       2  
  2     2                 2                                             2  
n25 =(;).                                                                                  
  • Which nodes of the above represent the same point on the torus?

 

  • What is the shortest path between the points n7and n19? (Draw it in a square representing the torus)

 

  • Now de ne a graph over the previous set of points, fn1; : : : ; n24g where edges are de ned as

 

follows: Given ni = (a; b) and nj = (c; d), there is a connection between them if and only if ja cj + jb dj 2 . List the set of edges of the graph and draw the graph edges and nodes over

a square.

 

  • What are the neighbors of node n1? and of n22?

 

  • Implement a BFS algorithm by hand over the previous graph starting from n7. Indicate in your drawing the layers of nodes (layer 0, layer 1, layer 2, : : : ) and the edges chosen by the BFS.

 

 

 

 

 

 

 

 

 

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