Trans(an,0,0) a translation along rotated Xn.1 = xn a length an, and
Rot(xn,a) = rotation about x,, the twist angle cn.
By multiplying these rotation and translation matrices, the general A matrix for any two
successive links is calculated as
cos sin 8 cos a sin 0 sin a a cos 8
n-1n sin cos cos a -cos sin a a sin
0 sina cos a d
0 0 0 1 (3-6)
(3-6)
In this work, the term frames refers to the coordinate frames assigned to the links of the
manipulator. Coordinate frames represent a three-dimensional space by three orthonormal
vectors; X, Y, and Z. The origin of each frame is located at the intersection of these vectors.
When referring to these vectors, a trailing subscript will be used to identify the frame in
concern. Links are the physical components which make up the manipulator. The motion of the
links is accomplished by the joints. A joint can be rotational revolutee) or translational
(prismatic). The coordinate frames are fixed to the links, and homogeneous transformations are
used to relate the position and orientation of the frames. A joint axis is the vector about which
joint motion takes place. Link coordinate frames are assigned such that the frame's Z axis
defines the vector about which joint motion occurs. Joint motion will be either around or along
the respective Z axis.
Controls Background
The use of a Bode plot, in which the frequency response of a system can be analyzed,
enables the control system designer to evaluate closed-loop system characteristics based on
knowledge of the open-loop system. A controller can then be designed to achieve the desired
system characteristics. Palm (1983) presented some of the basic considerations used in the
frequency-response methods of control system design. First, the cosed-loop system's steady-
state error can be minimized by maintaining a high, open-loop gain in the low-frequency range.
Second, a slope of -1 in the gain curve near the crossover frequency will help to provide an