3D Transformations

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http://planning.cs.uiuc.edu/node101.html

http://planning.cs.uiuc.edu/node102.html

3D translation

The robot, ${\cal A}$ , istranslated 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. $\begin{figure}\centerline{\psfig{file=figs/yawpitchroll.eps,width=1.7in}}\end{figure}$

A 3D body can be rotated about three orthogonal axes, as shown in Figure 3.8. Borrowing aviation terminology, these rotations will be referred to as yaw, pitch, and roll:

A yaw is a counterclockwise rotation of $\alpha$ about the-axis. The rotation matrix is given by
$\displaystyle R_z(\alpha) = \begin{pmatrix}\cos\alpha & -\sin\alpha & 0 \sin\alpha & \cos\alpha & 0 0 & 0 & 1 \end{pmatrix} .$ (3.39)

Note that the upper left entries of $R_z(\alpha)$ form a 2D rotation applied to the and coordinates, whereas the coordinate remains constant.
A pitch is a counterclockwise rotation of $\beta$ about the-axis. The rotation matrix is given by
$\displaystyle R_y(\beta) = \begin{pmatrix}\cos\beta & 0 & \sin\beta 0 & 1 & 0 -\sin\beta & 0 & \cos\beta \end{pmatrix} .$ (3.40)
A roll is a counterclockwise rotation of $\gamma$ about the-axis. The rotation matrix is given by
$\displaystyle R_x(\gamma) = \begin{pmatrix}1 & 0 & 0 0 & \cos\gamma & -\sin\gamma 0 & \sin\gamma & \cos\gamma \end{pmatrix} .$ (3.41)

Each rotation matrix is a simple extension of the 2D rotation matrix, (3.31). For example, the yaw matrix, $R_z(\alpha)$ , essentially performs a 2D rotation with respect to the

and

coordinates while leaving the

coordinate unchanged. Thus, the third row and third column of $R_z(\alpha)$ look like part of the identity matrix, while the upper right portion of $R_z(\alpha)$ looks like the 2D rotation matrix.

The yaw, pitch, and roll rotations can be used to place a 3D body in any orientation. A single rotation matrix can be formed by multiplying the yaw, pitch, and roll rotation matrices to obtain

$\begin{displaymath}\begin{split}R(\alpha,& \beta,\gamma) = R_z(\alpha) R_y(\b......\sin\gamma & \cos\beta \cos\gamma \end{pmatrix}. \end{split}\end{displaymath}$ (3.42)

It is important to note that $R(\alpha,\beta,\gamma)$ performs the roll first, then the pitch, and finally the yaw. If the order of these operations is changed, a different rotation matrix would result. Be careful when interpreting the rotations. Consider the final rotation, a yaw by $\alpha$ . Imagine sitting inside of a robot ${\cal A}$ that looks like an aircraft. If $\beta = \gamma = 0$ , then the yaw turns the plane in a way that feels like turning a car to the left. However, for arbitrary values of $\beta$ and $\gamma$ , the final rotation axis will not be vertically aligned with the aircraft because the aircraft is left in an unusual orientation before $\alpha$ is applied. The yaw rotation occurs about the

-axis of the world frame, not the body frame of ${\cal A}$ . Each time a new rotation matrix is introduced from the left, it has no concern for original body frame of ${\cal A}$ . It simply rotates every point in ${\mathbb{R}}^3$ in terms of the world frame. Note that 3D rotations depend on three parameters, $\alpha$ , $\beta$ , and $\gamma$ , whereas 2D rotations depend only on a single parameter, $\theta$ . The primitives of the model can be transformed using $R(\alpha,\beta,\gamma)$ , resulting in ${\cal A}(\alpha,\beta,\gamma)$ .

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