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Issue
110, September 1999
Internet
Control
by
Jacob Apkarian
Start
• Model Derivation
• Linearization • Control-System
Design • Simulation
•
Coding •
Configurations•
Implementation •
Tuning and Results •
Ready for Takeoff •
Software
and Sources •
Equations PDF
MODEL DERIVATION
The first step
in the process is to develop a mathematical model of the
system. The equations that
I reference in the text are displayed in the sidebar on
pages 39–40 (they are also available for download
in a pdf file).
The tool used
here is the Maple symbolic processing language. The differential
equations of the system are highly nonlinear and coupled,
and are difficult to solve by hand. The process is described
below and is implemented in a Maple script file.
Let’s start
by defining coordinate transformation frames embedded
in the moving bodies of the system. The transformation
matrices between the base frame and travel frame are shown
in equation 1. These coordinate frames have the same origin.
Equation 2 shows
the transformation matrices from the travel frame to the
helicopter body. To go from the travel frame to the counterweight,
the matrices are shown in equation 3. The last two sets
of matrices are for the helicopter body to front motor
(equation 4) and from the helicopter body to the back
motor (equation 5).
The transformations
from the base to the moving bodies are obtained via equation
6. Those transformations are used to obtain the potential
energies of the bodies (equation 7) and the kinetic energy
of each body (equation 8).
The total kinetic
and potential energies are obtained and the Lagrangian
is computed in equation 9. Once the Lagrangians for the
three axes are derived, you can write the Lagrange equations
for each axis.
Each motor exerts
a force normal to the body given by F = KfV,
where Kf is the force constant for the motor/propeller
pair. If the body is horizontal, the two forces result
in a torque about the elevation axis equal to La Kf(Vf
+ Vb), resulting in the generalized Lagrange
equation given in equation 10.
The system rotates
around the travel axis only if the body is pitched and
hovering. The torque around the travel axis resulting
from a given pitch is equal to that component of the total
force of the two motors projected onto the travel axis:
(Vf + Vb) KmLa
sin (pitch(t))
resulting in
equation 11.
The body pitches
because of a difference in the voltage applied to the
motors, which results in a difference in the two forces
generated by the two motors. The difference results in
a torque around the pitch axis given by:
Kf (Vf –
Vb)Lh
Equation 12
results in a set of nonlinear differential equations of
the form given in equation 13. These are solved to obtain
the accelerations around the three axes (equation 14),
where e º elevation, p º pitch, and l º travel.
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