Page 60 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

DYNAMICS 35

Angular velocities and accelerations are

˙
ω 1 = θ 1
˙ ˙
ω 2 = θ 1 + θ 2
(2.12)
¨
˙ ω 1 = θ 1
¨ ¨
˙ ω 2 = θ 1 + θ 2
Substituting those into Euler’s equations (and taking into the account that ω i ×
I i ω i = 0), we obtain

¨
n 1 = I 1 θ 1
(2.13)
n 2 = I 2 (θ 1 + θ 2 )
¨
¨
Finally, Newton–Euler equations are combined with static equations [Eq. (2.8)]
to produce the torques at arm joints—that is, to do inverse dynamics. After
simplifications, these become (details can be found in Refs. 7 and 8)

2 2
m 2 l 1 l 2 m 2 l 2 m 2 l 2
¨
¨
n 1,2 = θ 1 I 2 + cos θ 2 + + θ 2 I 2 +
2 4 4
m 2 l 1 l 2 2 m 2 l 2 g 2
+ θ ˙ 1 sin θ 2 + cos (θ 1 + θ 2 )
2 2
− l 2 sin (θ 1 + θ 2 )f 2,3x − l 2 cos (θ 1 + θ 2 )f 2,3y + n 2,3 (2.14)
m 1 l + m 2 l
2 2
¨
n 0,1 = θ 1 I 1 + I 2 + m 2 l 1 l 2 cos θ 2 + 1 2 + m 2 l 1 2
4
2
m 2 l 2 m 2 l 1 l 2
¨
+ θ 2 I 2 + + cos θ 2
4 2
m 2 l 1 l 2 2
˙ ˙
− θ ˙ 2 sin θ 2 − m 2 l 1 l 2 θ 1 θ 2 sin θ 2
2

m 2 l 2 m 1
+ cos (θ 1 + θ 2 ) + l 1 + m 2 cos θ 1 g 2
2 2
− (l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ))f 2,3y + n 2,3
There are three types of terms that appear in such equations. Taking as an example
the above equation for n 1,2 , these are:

Dynamic Torques (Terms 1, 2, and 3). These arise from the arm movement,
and depend on velocities and accelerations.
Gravity Torques (Term 4). These are due to the (vertical) gravity force.