Page 59 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

34 A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS

link i
center of mass
r * i

origin of
p * i
origin of link i + 1,
link i, joint i
joint i − 1

r i p i
p i − 1

arm base

Figure 2.5 Balance of forces and torques acting on a single link.

the accelerations ¨ r 1 and ¨ r 2 of the links’ centers of mass by Newton’s sec-
ond law,

f 1 = m 1 ¨ r 1
(2.9)
f 2 = m 2 ¨ r 2

From these equations, accelerations ¨ r i of the centers of mass can be derived.

Let ω i be the angular velocity vector of the center of mass of link i.
Let ˙ω i be the corresponding angular acceleration.
Let I i be the inertia matrix of link i.

Then torques are related to angular velocities and accelerations by Euler’s
equations,

n 1 = I 1 ˙ω 1 + ω 1 × I 1 ω 1
(2.10)
n 2 = I 2 ˙ω 2 + ω 2 × I 2 ω 2
For our planar two-link manipulator shown in Figure 2.1, the torque is normal
to the arm’s plane. Rotary inertia through the centers of mass of links 1 and 2
are [7]

2
2
I 1 = m 1 l /12 + m 1 R /4
1 (2.11)
2
2
I 2 = m 2 l /12 + m 2 R /4
2