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
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
The matrices for the remaining six bonds are
Steven M LaValle 2020-08-14