Page 69 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

44 A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS

Suppose that the line of compliant motion forms an angle δ with the horizontal.
Define a unit vector along that line as u(δ). Then:

• The position loop projection becomes u(δ)(p e · u(δ)).
π
π
• The force loop projection becomes u(δ + )(f e · u(δ + )).
2 2
These operations will be implemented properly if matrix Q is defined to align
the constraint frame with the known compliance line, and vector s differentiates
the directions of control loop actions:

cos δ − sin δ
Q =
sin δ − cos δ
(2.19)

1
s =
0
2.6 TRAJECTORY MODIFICATION

Robot trajectories (equivalently, robot paths) are generated in many ways. For
example, as explained in Section 1.2, not rarely a path is obtained manually: A
technician brings the arm manipulator to one point at the time, he or she presses
a button, and the point goes into the trajectory database. A sufficient number of
those points makes for a path. Or, the path can be obtained automatically via some
application-specific software. Either way, if the robot goes through the obtained
path, it is very possible that the motion would be less than perfect; for example,
it may be jerky or make corners that are too sharp. For some applications, path
smoothness may be very critical. Then, the set of collected path points has to
be further processed into a path that satisfies additional requirements, such as
smoothness.
Depending on the application, more requirements to the path quality may
appear: a continuity of its second and even third derivatives (which relates to
the path smoothness), precision of its straight line segments, and so on. That
is, techniques used for modifying the robot path often emphasize appropriate
mathematical properties of the path curves. The path preprocessing will likely
include both position and orientation information of the path. If, for example,
such work is to be done for a six-DOF arm manipulator, the desired properties
of the path are expected from all DOF curve components.
Common trajectory modification techniques are polynomial trajectories,which
amount to the satisfaction of appropriate constraints, and straight-line interpola-
tion.
Polynomial Trajectories (Satisfaction of Constraints). Consider an example in
Figure 2.12. We want to obtain a mathematical expression for a simple path that
would bring this two-link planar arm from its initial point (position) p a to the
destination point p b . Positional constraints are defined by the joint angle vectors