Robotics

Helper page for the Peter Corke: Introduction to robotics MOOC.

L3.10 2-axis representation

Check understanding: The rotation matrix

 0.9363 −0.2896  0.1987
 0.3130  0.9447 −0.0978
−0.1593  0.1538  0.9752

is equivalent to the 2-vector representation:

• A=(−1.8833,6.1427,1.0000),O=(1.0000,−0.4925,4.9085)
• A=(1.0000,−0.4925,4.9085),O=(2.9914,1.0000,−0.5091)
• A=(1.0000,−0.4925,4.9085),O=(−1.8833,6.1427,1.0000) (CORRECT)
• A=(−1.8833,6.1427,1.0000),O=(2.9914,1.0000,−0.5091)

Solution:

o=[-1.8833, 6.1427, 1.0000];     %Orientation vector
a=[1.0000, -0.4925, 4.9085];     %Approach vector
R=oa2r(o,a)    %To create rotation matrix R from o and a vectors

R =

 0.9363         -0.2896         0.1987
 0.3130          0.9447        -0.0978
-0.1593         0.1538         0.9752
 
n=cross(o,a)     %To calculate normal vector n
n =
30.6439       10.2442       -5.2152

r=[n' o' a']     %To create rotation matrix r with columns of n, o, a vectors

# Wrong, n, o and a are unit vectors: 

r=[unit(n') unit(o') unit(a')]     %To create rotation matrix r with columns of n, o, a vectors

tr2rpy(R)     %To calculate roll, pitch and yaw angles from rotation matrix R
tr2rpy(r)     %To calculate roll, pitch and yaw angles from rotation matrix r

Can you please explain:

1. How can matrices R=oa2r(o,a) and r=[n' o' a'] proved to be equivalent? 2. Why is a difference appearing in the yaw angle calculated from matrix R and from matrix r?

Explanation

We can really only do this by trial error, try each of the potential answers and see if it gives the desired rotation matrix. Note that the arguments to the oa2r() function are given in the order O then A, the reverse of what's above.

>> oa2r( [1.0000,   -0.4925,  4.9085], [-1.8833,  6.1427, 1.0000])
>> oa2r( [2.9914, 1.0000,  -0.5091], [1.0000,   -0.4925,  4.9085])
>> oa2r( [-1.8833,  6.1427, 1.0000], [1.0000,   -0.4925,  4.9085])
>> oa2r( [2.9914, 1.0000,  -0.5091], [-1.8833,  6.1427, 1.0000])