26
Intended Learning Outcomes for This Topic
2
➌ Define the action and Lagrangian for a system
By the end of this topic, you should be able to:
➋ Define generalized velocities for a system
➍ Derive equations of motion for a system in terms of its generalized coordinates
➊ Define generalized coordinates for a system
➎ Apply the concept of generalized coordinates to measurands in embedded systems
26
Coordinate Systems
4
z
x
y
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Generalized Coordinates
5
If we consider red box’s location and temperature, its generalized coordinates
would be latitude, longitude, elevation, and temperature:
Generalized coordinate space for the TI SensorTag:
{✓,a
x
,a
y
,a
z
,g
x
,g
y
,g
z
,m
x
,m
y
,m
z
,h,p,l,M,spl}
z
x
y
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Definitions
6
Configuration
Set of sensors for the measurands of interest
Configuration space
Set of possible configurations
Dimension / degrees of freedom
Smallest number of sensors needed to recreate a given configuration
Coordinate space
Set of possible measurand values
Generalized velocity
Rate of change of measurand coordinates
26
Example: Ball with Embedded Sensors
7
time
time
y-velocity height, h(t)
Measurements for a real instance:
The measurements are a path in the generalized
coordinate space with configuration
{x-accel., y-accel. z-accel.}
…
26
An Example
8
26
Example: Ball with Embedded Sensors
9
- - -
-
-
( )
()
- - -
-
-
-
( )
()
time
time
y-velocity height, h(t)
Recall: measurements are a path in the
generalized coordinate space
26
Feasible Paths in Generalized Coordinate Space
11
For any physical system, there will be certain paths that do not occur
time
height, h(t)
true h(t)
h
1
(t)
h
2
(t)
t
0
t
1
For the ball, looking at a single coordinate (height) as a function of time:
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Feasible Paths Minimize the “Action”, S
12
S =
Z
r
[K(t) U (t)] dt
=
Z
t
2
t
1
[K(t) U (t)] dt.
Let K(t) be the generalized kinetic energy
and let U(t) be the generalized potential energy
For physically feasible paths, S is stationary
K(t) − U(t) is called the Lagrangian
26
Extremal Values of the Functional S
13
time
height, h(t)
path, h(t), that
minimizes the
functional S
h
1
(t)
t
0
t
1
h
2
(t)
x
f(x)
x
min
x
min
−∆x
x
min
+∆x
value of x that
minimizes the
function f(x)
S is a functional: A mapping from functions to real values
Extremals of functionals are the analogs of maxima and minima of functions