Page 96 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

EXERCISES 71

2.10 EXERCISES

1. Develop direct and inverse kinematics equations, for both position and veloc-
ity, for a two-link planar arm manipulator, the so-called RP arm, where
R means “revolute joint” and P means “prismatic” (or sliding) joint (see
Figure 2.E.1). The sliding link l 2 is perpendicular to the revolute link l 1 ,and
has the front and rear ends; the front end holds the arm’s end effector (the
hand). Draw a sketch. Analyze degeneracies, if any. Notation: θ 1 = [0, 2π],
l 2 = [l 2min , l 2max ]; ranges of both joints, respectively: l 2 = (l 2max − l 2min );
l 1 = const > 0 − lengths of links.

l 2

J 1

l 1

q 1

J o
Figure 2.E.1

2. Design a straight-line path of bounded accuracy for a planar RR (revo-
lute–revolute) arm manipulator, given the starting S and target T positions,
(θ 1S ,θ 2S ) and (θ 1T ,θ 2T ):
θ 1S = π/4, θ 2S = π/2, θ 1T = 0, θ 2T = π/6

3. The lengths of arm links are l 1 = 50 and l 2 = 70. Angles θ 1 and θ 2 are mea-
sured counterclockwise, as shown in Figure 2.E.2.
Find the minimum number of knot points for the path that will guarantee that
the deviation of the actual path from the straight line (S, T ) will be within
the error δ = 2. The knot points are not constrained to lie on the line (S, T )
or to be spread uniformly between points S and T . Discuss significance of
these conditions. Draw a sketch. Explain why your knot number is minimum.
4. Consider the best- and worst-case performance of Tarry’s algorithm in a planar
graph. The algorithm’s objective is to traverse the whole graph and return
to the starting vertex. Design a planar graph that would provide to Tarry
algorithm different options for motion, and such that the algorithm would