Page 67 - Dynamic Vision for Perception and Control of Motion

2.2 Objects 51

control time history for a moving car from just the trajectory observed, but, of
course, with a motion model “in mind” (see Section 14.6.1).
The translations of the center of gravity (cg) and the rotations around this cg de-
scribe the motion of objects. For articulated objects also, the relative motion of the
components has to be represented. Usually, the modeling step for object motion re-
sults in a (nonlinear) system of n differential equations of first order with n state
components X, q (constant) parameters p and r control components U (for subjects
see Chapter 3).

2.2.5.1 Definition of State and Control Variables
A set of
x State variables is a collection of variables for describing temporal processes,
which allows decoupling future developments from the past. State variables
cannot be changed at one time. (This is quite different from “states” in computer
science or automaton theory. Therefore, to accentuate this difference, sometimes
use will be made of the terms s-state for systems dynamics states and a-state for
automaton-state to clarify the exact meaning.) The same process may be de-
scribed by different state variables, like Cartesian or polar coordinates for posi-
tions and their time derivatives for speeds. Mixed descriptions are possible and
sometimes advantageous. The minimum number of variables required to com-
pletely decouple future developments from the past is called the order n of the
system. Note that because of the second-order relationship between forces or
moments and the corresponding temporal changes according to Newton’s law,
velocity components are state variables.
x Control variables are those variables in a dynamic system, that may be changed
at each time “at will”. There may be any kind of discontinuity; however, very
frequently control time histories are smooth with a few points of discontinuity
when certain events occur.
Differential equations describe constraints on temporal changes in the system.
Standard forms are n equations of first order (“state equations”) or an n-th order
system, usually given as a transfer function of nth order for linear systems. There
are an infinite variety of (usually nonlinear) differential equations for describing
the same temporal process. System parameters p allow us to adapt the representa-
tion to a class of problems
dX / dt f ( , ,t) . (2.26)
p
X
Since real-time performance, usually, requires short cycle times for control, lin-
earization of the equations of motion around a nominal set point (index N) is suffi-
ciently representative of the process if the set point is adjusted along the trajectory.
With the substitution
X X N x , (2.27)
one obtains

dX / dt dX / dt dx dt . (2.28)
/
N
The resulting sets of differential equations then are for the nominal trajectory:
dX N / dt ( f X N , ,t) ; (2.29)
p