3D translation

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

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

A primitive of the form

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

is transformed to

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