Representin Position and Orientation

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Example:

%The toolbox also supports symbolic mathematics
>> syms theta
>> R=rot2(theta)
R = 
[ cos(theta), -sin(theta)]
[ sin(theta),  cos(theta)]

%  simplify :Symbolic simplification
>> simplify(R*R)
ans =
[ cos(2*theta), -sin(2*theta)]
[ sin(2*theta),  cos(2*theta)]
>> simplify(det(R))
ans =
1

Lie Group and Lie Algebra

引用一下SLAM 十四讲中的这个图，总结了李群与李代数之间的转换关系。

Example:

>> R = rot2(0.3)
R =
    0.9553   -0.2955
    0.2955    0.9553
    
>> s = logm(R)
s =
         0   -0.3000
    0.3000         0
>> vex(s)
ans =
    0.3000

% the result is, as expected, our original rotation matrix, 
 >> expm(s)
ans =

    0.9553   -0.2955
    0.2955    0.9553
    
% skew-symmetric matrix
>> syms w
>> skew(w)
ans =
[ 0, -w]
[ w,  0]

In fact the command:

>> R = rot2(0.3)

is equivalent to :

>> R = expm ( skew(0.3) )

R =

    0.9553   -0.2955
    0.2955    0.9553

Note:

1. logm Matrix logarithm.
   L = logm(A) is the principal matrix logarithm A, the inverse of expm(A).
2. vex Convert skew-symmetric matrix to vector.
   V = vex(S) is the vector which has the corresponding skew-symmetric
   matrix S. In the case that S (2x2) then V is 1x1

           S = | 0  -v |
               | v   0 |

Formally we can write
$R = e^{[\theta]_x} \in SO(2)$R=e[θ]x​∈SO(2)
where $\theta$θ is the rotation angle, and the notation $[.]_x: R \longmapsto R^{2X2}$[.]x​:R⟼R2X2 indicates a mapping from a scalar to a skew-symmentric matrix.