3D translation

The robot, $ {\cal A}$, is translated by some $ x_t,y_t,z_t\in {\mathbb{R}}$ using

$\displaystyle (x,y,z) \mapsto (x+x_t, y+y_t, z+z_t).$ (3.36)

A primitive of the form

$\displaystyle H_i = \{ (x,y,z) \in {\cal W}\;\vert\; f_i(x,y,z) \leq 0 \}$ (3.37)

is transformed to

$\displaystyle \{ (x,y,z) \in {\cal W}\;\vert\; f_i(x-x_t,y-y_t,z-z_t) \leq 0 \}.$ (3.38)

The translated robot is denoted as $ {\cal A}(x_t,y_t,z_t)$.

Figure 3.8: Any three-dimensional rotation can be described as a sequence of yaw, pitch, and roll rotations.

Steven M LaValle 2012-04-20