Rotations: SO(3) is the special orthogonal group for 3 dimensions, which is the set of rotation matrices. Rotation matrices have the determinant 1, which means

Rotations: SO(3) is the special orthogonal group for 3 dimensions, which is the set of rotation matrices. Rotation matrices have the determinant 1, which means that they keep the length of a vector the same (forcing it to change direction). Orthogonal group means that the rotation R times its transpose R^T yields the identity matrix aka R^T R=RR^T=I. This means that rotations are trivially reversible. You can obtain rotation matrices either by explicit use of cosine/sine along your desired axis or you can use the Rodrigues rotation formula to describe a rotation along an axis with an angle. so(3) is the Lie algebra to the Lie group SO(3). The capitalization here is critical. In particular, so(3) is the set of skew symmetric matrices. 3 by 3 skew-symmetric matrices essentially describe the cross product a x b for a constant a. Given a skew symmetric matrix and a rotation angle theta you can use the exponential function to map so(3) to SO(3) through e^(theta S). Note that the exponential is a general mapping from a Lie algebra to its Lie group. Transformations: SE(3) is the special euclidean group for 3 dimensions. Its structure consists of T = [[R, t], [0, 1]]. if R = I, then T is a pure translation by t. If R != I and t = 0, then it is a pure rotation. When you have both a rotation R and a translation t transformation then translating first and rotating second will give you T, but rotating first and translating second will give you T' = [[R, Rt], [0, 1]]. se(3) is the Lie algebra to the Lie group SE(3). One way to represent so(3) is through twists. Revolute joints are represented by a vector Xi = [ omega, q cross-product omega] where omega is a vector that represents the parameters of a skew symmetric matrix and q is a 3d point on the axis of rotation. Prismatic joints are represented by Xi = [0, v] where v is pointing in the direction of the movement. The transformation of a twist can be obtained as usual through the exponential function: exp(theta Xi)=T. https://en.wikipedia.org/wiki/3D_rotation_group https://en.wikipedia.org/wiki/3D_rotation_group https://en.wikipedia.org/wiki/Orthogonal_group https://en.wikipedia.org/wiki/Orthogonal_group https://en.wikipedia.org/wiki/Euclidean_group https://en.wikipedia.org/wiki/Euclidean_group https://en.wikipedia.org/wiki/Screw_theory https://en.wikipedia.org/wiki/Screw_theory