Page 65 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

40 A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS

of e 2 but not vice versa, which is understandable given the links’ sequential
connection. It can be shown that if the gains L 01 and L 12 are zeroes, then joint
values θ 1 and θ 2 differ from their desired values by amounts proportional to the
disturbance values and inversely proportional to the feedback gains K 01 and K 12 .
If, in turn, both gains K are zero, then the system’s steady state is independent
of the commands and reflects only the balance of gravitational and disturbance
forces. This of course indicates the importance of the feedback.
Dynamic Stability Analysis. This is done to verify the stability hypothesis.
Suppose that n 2,3 = 0, f 2,3x = 0, and f 2,3y = 0. Assume a small initial error
δθ 1 (t),

δθ 1 (0) = θ 1 − θ 1d
δθ 1 (0) = θ 1 − θ 1d

and constant θ 1d and θ 2d for t> 0. Then, assuming that stability can be achieved
via feedback, linearized equations in terms of δθ 1 (t) and δθ 2 (t) are written.
Stability of those equations can be assessed by applying to them the Laplace
transform and studying the characteristic polynomial [7]. Tests for stability are
in general difficult to apply, so simpler necessary conditions are used, followed
by a detail experimental verification.

2.5 COMPLIANT MOTION
When the robot is expected to physically interact with other objects, additional
care has to be taken to ensure a smooth operation. Imagine, for example, that
the robot has to move its hand along a straight line, on a flat surface, say a
table. It is easy to program such motion, but what if the table has a tiny bump
right along the robot’s path? The robot will attempt to produce a straight line,
effectively trying to cut through the bump. Serious forces will develop, with a
likely unfortunate outcome. What is needed is some mechanism for the robot to
“comply” with deviations of the table’s surface from the expected surface. Two
types of motion are considered in such cases:

• Guarded motion, when the arm is still moving in free space, before it con-
tacts an object. Position control similar to the one above is used.
• Compliant motion, when the arm is in continuous contact with the object’s
surface. Position control and force control are then used simultaneously.

Consider an example: Let us say that our task requires the robot to grasp an
object A (see Figure 2.9) that is initially positioned on top of object B 1 , move
it first into contact with the surface of table T , then slide it along T until it
contacts an object B 2 , and stop there. Let us assume that the grasping operation
itself presents no difficulty and that the grasp is a rigid grasp; that is, for all