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Dedicated interface-subroutines that will optimize communications between the offline
SIMULINK-environment and the on-line flightsimulator environment could help
streamline these conversions. This concept of ‘portable’ flight control laws has been
illustrated in figure 1.4, assuming that the language C is used to implement the flight
control laws in the flightsimulator and FCCs.
The Mathworks offers several solutions which could be helpful for this purpose,
such as the REALTIME WORKSHOP and the REALTIME WORKSHOP EMBEDDED CODER,
the latter of which targets real-time embedded processors, DSP boards, realtime
operating systems, and PCs. Other useful third-party tools that could be applied
to interface with the FDC models are STATEFLOW, which can e.g. be used to create
very complex mode-controller systems, the DIALS & GAUGES BLOCKSET, which
allows a.o. graphical display of cockpit instruments during SIMULINK simulations,
10 Chapter 1. Flight control system development
and the VIRTUAL REALITY TOOLBOX, which provides three-dimensional animation
facilities for SIMULINK models. There are also several third-party products available
that allow SIMULINK models to be coupled to specific experimental hardware setups.
By integrating MATLAB, SIMULINK, and other tools that allow easy interfacing
with external simulation devices or FCC’s, quick prototyping of flight control laws
becomes feasible, and the transitions between off-line simulations, real-time simulations,
and actual flight could be streamlined enormously.
It is not likely that these functions will soon be integrated in future versions of the
FDC toolbox, no matter how exciting those prospects may be. Any future work on
this software will probably mainly be concentrated on optimizing and improving the
existing tools and models, and widening the application areas of the toolbox within
its current scope. However, it is hoped that this far-reaching vision of FDC’s future
will inspire others to develop their own add-ons or variants of the software, so that
maybe one day some parts of this vision will be realized.
Chapter 2
Rigid body equations of motion
Before we can start building the mathematical model of the aircraft, some fundamental
knowledge about the equations of motion is needed. In this chapter, the equations
of motion of a rigid aircraft will be derived and expressed in the state-space form; a
summary of these state equations will be given in section 2.5. The next chapter will
build upon these equations to construct the nonlinear six-degree-of-freedom aircraft
model. 1
2.1 General rigid-body equations of motion
The aircraft equations of motion will be derived from Newton’s laws, which state
the connection between force and motion. We start by deriving the general force and
moments equations for a rigid body and defining the relations for the angular momentum.
This results in six ordinary differential equations, representing the linear
and angular accelerations in the body-fixed reference frame.
2.1.1 General force equation for a rigid body
Consider a mass point dm that moves with time-varying velocity V under the influence
of a force F. Both V and F are measured relatively to a right-handed orthogonal
reference frame OXYZ. This reference frame may be moving with a constant linear
velocity, relative to the fixed position of the stars (a.k.a. ‘inertial space’), but it may
not accelerate or rotate. Applying Newton’s second law yields:
dF = dm · ˙V (2.1)
Applying this equation to all mass points of a rigid body and summing all contributions
across this body yields:
ådF = ådm
dV
dt
= d
dt åVdm (2.2)
Let the center of gravity of the rigid body have a velocity Vc.g. with components
u, v, and w along the X, Y, and Z-axes of the right-handed reference frame. The
velocity of each mass point within the rigid body then equals the sum of Vc.g. and
1The derivation of the rigid-body equations has been extensively discussed in literature. This chapter
has been inspired by refs.[11, 14, 15, 16, 19, 20, 25, 25, 26], and [33] (to name just a few).
12 Chapter 2. Rigid body equations of motion
the velocity of the mass point with respect to this center of gravity. If the position of
the mass point with respect to the c.g. is denoted by the vector r, the following vector
equation is found:
V = Vc.g. + ˙r (2.3)
therefore:
åVdm = å(Vc.g. + ˙r)dm = mVc.g. + d
dt årdm (2.4)
In this equation, m denotes the total mass of the rigid body. In the center of gravity
we can write:
år dm = 0 (2.5)
so the equation for the total force F acting upon the rigid body, becomes:
　
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