C H A P T E R
2
The
Simple
Pendulum
2.1
INTRODUCTION
Our goals for this chapter are modest: we’d like to understand the dynamics of a pendulum.
Why a pendulum? In part, because the dynamics of a majority of our multilink robotics
manipulators are simply the dynamics of a large number of coupled pendula. Also, the
dynamics of a single pendulum are rich enough to introduce most of the concepts from
nonlinear dynamics that we will use in this text, but tractable enough for us to (mostly)
understand in the next few pages.
g
θ
m
l
FIGURE
2.1
The Simple Pendulum
The Lagrangian derivation (e.g, [35]) of the equations of motion of the simple pen
dulum yields:
Iθ
¨
(
t
) +
mgl
sin
θ
(
t
) =
Q,
where
I
is the moment of inertia, and
I
=
ml
2
for the simple pendulum. We’ll consider
the case where the generalized force,
Q
, models a damping torque (from friction) plus a
control torque input,
u
(
t
)
:
Q
=
−
bθ
˙
(
t
) +
u
(
t
)
.
2.2
NONLINEAR DYNAMICS W/ A CONSTANT TORQUE
Let us first consider the dynamics of the pendulum if it is driven in a particular simple way:
a torque which does not vary with time:
Iθ
¨
+
bθ
˙
+
mgl
sin
θ
=
u
0
.
(2.1)
These are relatively simple equations, so we should be able to integrate them to obtain
θ
(
t
)
given
θ
(0), θ˙(0)... right? Although it is possible, integrating even the simplest case
(0)
.
Let’s start by plotting
x
˙
vs
x
for the case when
u
0
= 0
:
x
˙
x
mgl
b
π

π
The first thing to notice is that the system has a number of
fixed points
or
steady
∗
states
, which occur whenever
x
˙ = 0
. In this simple example, the zerocrossings are
x
=
{
...,
−
π,
0
,
π,
2
π,
...
}
. When the system is in one of these states, it will never leave that
state. If the initial conditions are at a fixed point, we know that
x
(
∞
)
will be at the same
fixed point.
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 Fall '13
 BertrandI.Halperin
 Physics, Chaos Theory, Energy, Bifurcation theory, Bifurcation diagram, Russ Tedrake