1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 383

0.25

0.2

0 .1 5

0 .1

0.05

E .._.. 0

""-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

-0.5 ·0.4 ·0.3 -0(2 -0.1

x m)

0 0.1

Figure 3.8. Position of the center of mass for the "rotate gait".

-0.1

-0.2

E

;: ·0.3

-0.4

-0.5

·0.6 '----'----'---~-~~-~-~-~-~

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 ·0.05 0

x (m)

Figure 3.9. Position of the center of mass for the "parallel parking gait".

presence of momentum, however, there is a tight coupling between the geometry of
the connection and the effect of the inertia and momentum. This suggests a very
strong relationship between the geometric and dynamic phases as encoded by the
connection, and the generation of gaits.

Stability, controllability, and optimal control.

Control theory adds to the study of dynamical systems the idea that in many in-
stances, one can directly intervene in the dynamics rather than passively watching.
For example, while Newton's equations govern the dynamics of a satellite, we can
intervene in these dynamics by controlling onboard gyroscopes, thrusters, or rotors.
Quite often, control engineers are tempted to overwhelm the intrinsic dynamics of
a system with the controls. However, in many circumstances (fluid control is an