Page 91 - Dynamic Vision for Perception and Control of Motion

3.4 Behavioral Capabilities for Locomotion 75

3.4.1.2 Transition Matrices for Single Step Predictions

Equation 3.6 with matrices F and G may be transformed into a difference equation
with the cycle time T for grid point spacing by one of the standard methods. (Pre-
cise numerical integration from 0 to T for v = 0 may be the most convenient one for
complex right-hand sides.) The resulting general form then is
˜
[(k x 1) ] T A x [kT ] B u [kT ] [kT ]

˜
v
(3.7)
˜
˜

or in short-hand notation, x A x B u ,
v
k 1 k k k
where the matrices A, B have the same dimensions as F, G. In the general case of
local linearization, all entries of these matrices may depend on the nominal state
and control variables (X N, U N). The procedures for computing the elements of A
and B have to be part of the “4-D knowledge base” for the application at hand.
Software packages for these transformations are standard in control engineering.
For deeper understanding of motion processes of subjects observed, a knowl-
edge base has to be available linking the actual state and its time history to goal-
oriented behaviors and to stereotypical control outputs on the time line. This will
be discussed in Section 3.4.3.
Once the initial conditions of the state are fixed or given, the evolving trajectory
will depend both on this state (through matrix A, the so-called homogeneous part)
and on the controls applied (the non-homogeneous part). Of course, this part also
has to take the initial conditions into account to achieve the goals set in a close-to-
optimal way. The collection of conditions influencing the decision for control out-
put is called the “situation” (to be discussed in Chapters 4 and 13).
3.4.2 Control Variables for Ground Vehicles
A wheeled ground vehicle has three control variables, usually, two for longitudinal
control and one for lateral control, the steering system. Longitudinal control is
achieved by actuating either fuel injection (for acceleration or mild decelerations)
í2
or brakes (for decelerations up to §í1 g (Earth gravity acceleration § 9.81 m s )).
Ground vehicles are controlled through proper time histories of these three control
variables. In synchronization with the video signal this is done 25 (PAL-imagery)
or 30 times a second (NTSC). Characteristic maneuvers require corresponding
stereotypical temporal sequences of control output. The result will be correspond-
ing time histories of changing state variables. Some of these can be measured di-
rectly by conventional sensors, while others can be observed from analyzing image
sequences.
After starting a maneuver, these expected time histories of state variables form
essential knowledge for efficient guidance of the vehicle. The differences between
expectations and actual measurements give hints on the situation with respect to
perturbations and can be used to apply corrective feedback control with little time
delay; the lower implementation level does not have to wait for the higher system
levels to respond with a change in the behavioral mode running. To a first degree
of approximation, longitudinal and lateral control can be considered decoupled (not
affecting each other). There are very sophisticated dynamic models available in
automotive engineering in the car industry and in research for simulating and ana-