The homogeneous transformation matrix

This can be considered as the 3D counterpart to the 2D transformation matrix, (3.52). The following four operations are performed in succession:

- Translate by along the -axis.
- Rotate counterclockwise by about the -axis.
- Translate by along the -axis.
- Rotate counterclockwise by about the -axis.

As in the 2D case, the first matrix, , is special. To represent any position and orientation of , it could be defined as a general rigid-body homogeneous transformation matrix, (3.50). If the first body is only capable of rotation via a revolute joint, then a simple convention is usually followed. Let the parameters of be assigned as (there is no -axis). This implies that from (3.55) is the identity matrix, which makes .

The transformation for gives the relationship between the body frame of and the body frame of . The position of a point on is given by

For each revolute joint, is treated as the only variable in . Prismatic joints can be modeled by allowing to vary. More complicated joints can be modeled as a sequence of degenerate joints. For example, a spherical joint can be considered as a sequence of three zero-length revolute joints; the joints perform a roll, a pitch, and a yaw. Another option for more complicated joints is to abandon the DH representation and directly develop the homogeneous transformation matrix. This might be needed to preserve topological properties that become important in Chapter 4.

The parameters from Figure 3.17 may be substituted into the homogeneous transformation matrices to obtain

and

A point in the body frame of the last link appears in as

(3.64) |

Note that the bonds correspond exactly to the axes of rotation. This
suggests that the axes should be chosen to coincide with the
bonds. Since consecutive bonds meet at atoms, there is no distance
between them. From Figure 3.15c, observe that this makes for all . From Figure 3.15a, it can be seen that each
corresponds to a *bond length*, the distance between
consecutive carbon atoms. See Figure 3.20. This
leaves two angular parameters, and . Since the
only possible motion of the links is via rotation of the -axes,
the angle between two consecutive axes, as shown in Figure
3.15d, must remain constant. In chemistry, this is referred
to as the *bond angle* and is represented in the DH
parameterization as . The remaining parameters
are the variables that represent the degrees of freedom. However,
looking at Figure 3.15b, observe that the example is
degenerate because each -axis has no frame of reference because
each . This does not, however, cause any problems. For
visualization purposes, it may be helpful to replace and
by and , respectively. This way it is easy
to see that as the bond for the -axis is twisted, the observed
angle changes accordingly. Each bond is interpreted as a link,
. The origin of each
must be chosen to coincide with the
intersection point of the - and -axes. Thus, most of
the points in
will lie in the direction; see Figure
3.20.

The next task is to write down the matrices. Attach a world frame to the first bond, with the second atom at the origin and the bond aligned with the -axis, in the negative direction; see Figure 3.20. To define , recall that from (3.54) because is dropped. The parameter represents the distance between the intersection points of the - and -axes along the axis. Since there is no -axis, there is freedom to choose ; hence, let to obtain

(3.65) |

The application of to points in causes them to rotate around the -axis, which appears correct.

The matrices for the remaining six bonds are

for . The position of any point, , is given by

(3.67) |

Steven M LaValle 2020-08-14