Lecture 1, July 26: History of
navigation
Lecture 2, July 27: History of navigation, dead reckoning, piloting
Lecture 3, July 31: Position fixing
using bearing and range
measurements, VOR
Lecture 4, August 2: ADF, DME,
position fixing using pseudo range
measurements, hyperbolic positioning, LORAN, Omega
Lecture 5, August 3: Decca, position
fixing using difference in
bearings, indirect ranging, nonsimultaneous measurements (running
fixes),
courses -- rhumb lines, Mercator charts
Lecture 6, August 7: Great circle
routes, rhumb line approximations,
gnomonic charts, celestial navigation -- terminology
Lecture 7, August 9: Celestial
navigation -- lines of position,
complete solution for position fix
Lecture 8, August 10: Tutorial 1 (Solutions)
Lecture 9, August 14: Not held, compensated by quiz on Aug. 29.
Lecture 10, August 16: Tutorial 2
(Solutions)
Lecture 11, August 17: Random
variables, PDF, pdf, mean, variance
Lecture 12, August 21: Gaussian
distribution, pdf of a function of a
random variable
Lecture 13, August 23: pdf of a
function of a random variable, multiple
random variables, joint PDFs and pdfs
Lecture 14, August 24: Covariance matrix, Gaussian distribution for two
random variables, analysis of navigational error
Lecture 15, August 28: Error bounds
in terms of ellipses, mean square
error as trace of the covariance, BLUE for combining
estimates/measurements
Quiz 1, August 29: Solutions
Lecture 16, August 30: Solution for
BLUE, dead reckoning example
Lecture 17, August 31: BLUE for
vector measurements, INS --
stabilized platforms
Lecture 18, September 4: INS --
strapdown
Lecture 19, September 6: Solutions of Quiz 1, Tutorial
3 (Solutions)
Lecture 20, September 7: Tutorial 4 (Solutions)
Midsem, September 13: Solutions
September 17, 20, 21: Lectures postponed due to works visit
Lecture 21, September 25:
Classification of missiles and missile guidance, homing guidance, 2D
tactical engagement geometry
In this lecture, we saw different
classifications of missiles based on missions and maneuvering
mechanisms, and guidance systems.We also listed various kinds of homing
guidance stategies, and looked at a general 2D tactical engagement
geometry.
Lecture 22, September 26: Types
of
command guidance, pursuit guidance (Compensatory lecture.)
In this lecture, we outlined two kinds
of command guidance schemes, and began a detailed study of pursuit
guidance in the case where the target moves uniformly along a straight
line. We wrote down range and range rate equations for pursuit
guidance and concluded that the LOS rotates except in a tail chase or a
head on intercept. We solved the equations to obtain the range as a
function of the LOS angle, and concluded that intercept is achieved
only if the missile has a speed advantage over the target, in
which case the engagement always ends in a tail chase. We derived an
expression for the LATAX as a function of the LOS angle, and concluded
that the LATAX requirement remains bounded only if the missile is not
more than twice as fast as the target. Finally, we derived an integral
expression for the time to intercept as a function of the initial LOS
and range.
Lectures 23 (September 27)
and 24 (September 28): Linearized
terminal phase analysis of pursuit guidance, linear miss distance
analysis under turn rate constraint, collision triangle
In this lecture, we began by
considering the terminal phase of an ideal pursuit trajectory, in which
the LOS angle remains small. Using small angle approximations, we
linearized the range-rate and LOS-rate equations for pursuit guidance.
The linearized equations were then solved in closed form. The closed
form solutions confirmed our previous conclusions about the terminal
LATAX requirement. Next, we used the closed-form solutions of the
linearized terminal phase pursuit guidance equations to estimate the
miss distance that results if the missile is unable to follow the ideal
pursuit trajectory because of limited LATAX capability. Finally, we
introduced the concept of a collision triangle.
Lecture 25 (October 3)
Linearized miss distance equations for
proportional guidance (Compensatory lecture.)
We began by noting that the
proportional guidance command is zero if the missile and the target are
on a collision triangle. To study the effect of a small launch error
from the nominal collision triangle situation, we derived linearized
equations describing perturbations from the nominal collision triangle.
The linearized equations were used to obtain an equation for the miss
distance in terms of perturbations in the target and missile
accelerations.
Lecture 26 (October 4) Homing
loop, analytical solution of
linearized
miss distance equations in the case of initial launch error
The equations derived in the previous
lecture were used to draw the homing loop. We noted that the loop
contains a time varying element, and also contains the nominal flight
time along the original collision triangle as a parameter. We obtained
closed-form solutions for the miss distance (as a function of time)
resulting from an initial launch error assuming the target does not
manuever. We found that the value of the miss distance at the nominal
intercept time is zero. We also obtained a closed-form expression for
the missle LATAX, which showed that the guidance constant has to be
sufficiently large in order to keep the terminal LATAX requirement
bounded.
Lecture 27 (October 5) Method of
adjoints
We began by noting that the use of the
homing loop to study the dependance of the miss distance on the nominal
flight time as well as other factor influencing miss (such as launch
error, target maneuvers etc) requires considerable simulation effort
since the homing loop involves the nominal flight time as a parameter.
We introduced the method of adjoints as a more computationally
economical alternative. We saw the block diagram operations required to
form the adjoint of a given system whose description is available as a
block diagram. We applied the operations to form the adjoint of the
homing loop, and saw how the adjoint of the homing loop can be used to
generate miss distances as functions of the flight time resulting from
various different factors.
October 9: Lecture missed
Lecture 28 (October 11) Justification
for the method of adjoints,
analytical miss distance in the presence of lags using the method of
adjoints
We considered a simple first-order time
varying linear system to understand the relationship between the
impulse response of a given system and that of its adjoint. This
relationship helped us see why the adjoint method is useful, especially
in studying the performance of a guidance system. Next, we considered
applying the method of adjoints to obtain analytical expressions for
the miss distance as a function of the flight time in the case where
the missile flight control system has a single lag. For this purpose,
we rearranged the homing loop before constructing its adjoint.
Lecture 29 (October 12) Analytical
miss distance in the
presence of lags using the method of adjoints
In this lecture, we completed the
analysis of miss distance due to flight control system lag that was
started in the previous lecture. We found that the miss distance due to
initial launch errors as well step target maneuvers reduces as the
flight time increases in comparison to the lag.
Lecture 30 (October 16) Augmented
proportional guidance, proportional
command guidance
We began by rewriting the LATAX command
generated by proportional guidance in terms of the zero effort miss
(ZEM). We found that the proportional guidance LATAX command is
proportional to the ZEM, where the ZEM at any time t is calculated as the terminal
miss distance that results when the missile and target velocity vectors
are frozen at their values at time t.
One way to refine the guidance startegy is to re-define the ZEM at time
t as the terminal miss
distance that results when the missle velocity and target acceleration are frozen at their
respective values at time t.
This leads to the augmented proportional guidance strategy in which the
guidance command involves the LOS rate as well as the target
acceleration. Next, we briefly described a command guidance system in
which a command station tracks the missile and the target using radar.
The radar measurements are used to estimate the LOS rate, which is then
used to generate and uplink the proportional guidance command to the
missile. We noted that though the guidance command in homing as well as
proportional command guidance is based on proportional guidance, a
proportional command guidance system is more susceptible to measurement
noise in the terminal phase.
Lecture
31 (October 17) Strategic missiles, ballistic missile trajectory
(Compensatory lecture)
We listed the phases of flight of a
strategic missile, and began an analysis of the ballistic phase of the
trajectory of such a missile assuming earth to be perfectly spherical
and homogeneous, and atmospheric drag to be absent. We applied Newton's
law in polar coordinates and obtained the polar equation of the missile
trajectory.
Lecture 32 (October 18) Tutorial 5,
problem 1 and problem 3
Lecture 33 (October 19) Tutorial 5,
problems 2 and 4
Lecture 34 (October 23) Tutorial 6,
problems 1, 2 and 3
Quiz 2, October 24, Solutions
October 25, Lecture missed, compensated by quiz on October 24
October 26, Lecture missed
Lecture 35 (October 30) Polar
equation of a ballistic trajectory
We compared the polar equation obtained
in lecture 31 with the polar equation of an ellipse (with the origin at
its focus) and deduced parameters describing the shape, size and
orientation of the missile trajectory in terms of the missile position
and velocity at the end of the boost phase.
Lecture 36 (October 31)
Hit equation (Compensatory lecture)
We used the polar equation of the
missile trajectory to obtain the hit equation giving the cutoff speed
required to hit a given target in terms of the cutoff position and
flight path angle. Next, we found an expression for the time of flight
between any two angular locations on an elliptic ballistic trajectory.
We noted that the hit equation gives several trajectories that achieve
an intercept starting from a given location. These trajectories have
different cutoff velocity vectors and ballistic flight times.
Lecture 37 (November 1)
Lambert's problem, Lambert guidance
We stated Lambert's problem and
observed how its solution reduces to solving a scalar algebraic
equation relating the flight time to the cutoff flight path angle. Next
we discussed Lambert guidance or velocity-to-be gained guidance, in
which the velocity-to-be-gained is calculated by solving the Lambert
problem in every guidance cycle during the boost phase. The
acceleration command is then along the velocity-to-be-gained. We
performed a linearized analysis to estimate the miss distance resulting
from perturbations in the cutoff conditions. This miss distance
analysis allows us to compute the dispersion in intercept location (in
terms of variance of the miss distance) in terms of dispersions in the
cutoff conditions (expressed in terms of the covariance matrix of the
perturbation vector).
Lecture 38 (November 2) Solution of two point boundary
value problems such as Lambert's problem
We outlined the main ideas involved in
numerically solving two point boundary value problems such as Lambert's
problem in cases where analytical expressions for the solutions of the
dynamical equations are not possible. We saw that such problems can be
solved using Newton's method, for which the required sensitivities
(partial derivatives of terminal conditions to cutoff conditions) can
be obtained by numerically integrating the dynamical equations along
with the corresponding variational equation.
Lecture 39 (November 6) and Lecture 40 (November
13) Tutorial 7, problems 1 and 2.
Lecture 41 (November 14), Lecture 42 (November 15) and Lecture 43
(November 16) Tutorial 8