ORBIT DETERMINATION

1
ORBIT DETERMINATION

• ASEN 5070
• LECTURES 910
• 9/18/06

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Alternate Derivation for using Taylor
Series Expansion
Assume that we can write the solution for
based on initial conditions
.
Expand the true solution about
and retain 1st order terms
Define , then
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Alternate Derivation for using Taylor
Series Expansion
For example, assume that
then
and
Note that the last row will be
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State Transition Matrix Differential Equation for
a General State Vector
Let represent a vector of force model
parameters and a vector of measurement
model parameters. and are constants.
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State Transition Matrix Differential Equation for
a General State Vector
Hence, may be written as
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State Transition Matrix Differential Equation for
a General State Vector
Then yields
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State Transition Matrix Differential Equation for
a General State Vector
Hence, we need only to integrate the
matrix of differential equations,
within I.C.
The remaining elements of simply are the
elements of an identity matrix.
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Sympletic Property of
Under certain conditions on A(t) the state
transition matrix may be inverted analytically
(Battin, 1987). Under these conditions is
referred to as being sympletic.
If the matrix A(t) can be partitioned in the form
(4.2.12)
where the submatrices have the properties that
,
, and
(4.2.13)
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Sympletic Property of
Then can be similarly partitioned
as
and may be written as
(4.2.14)
If then Eq. (4.2.13) is true
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Sympletic Property of
In this case (consider a 2-D case for simplicity)
and
,
,
Because
,
is symplectic
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Example, Problem 14, Chapter 4 of Text
14a) Generate the and matrix for
the pendulum problem. Assume that we wish to
estimate , , and at some epoch time.
Derive the equations of motion from the free body
diagram.
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Example, Problem 14, Chapter 4 of Text
In component form
(1)
(2)
Eq. (2) gives us , and Eq. (1)
gives the tension in the cord,
Hence, we need to solve
,
writing Eqs. in 1st order form
Then,
,
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Example, Problem 14, Chapter 4 of Text
Can we use LaPlace Transforms to solve for
?
Where indicates that is evaluated
on a reference solution for .
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Example, Problem 14, Chapter 4 of Text
Choose initial conditions , and
generate the reference trajectory
i.e.
while simultaneously integrating
Note that we do not need to integrate equations
for
since it is a constant
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Example, Problem 14, Chapter 4 of Text
,
IC
,
i.e.,
,
To do this in Matlab we would use the Reshape
command. Which would write a matrix as a vector
and vice versa. (see hints under handouts on web
Matlab help for solving problem 4.10). The
vector derivatives is
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Example, Problem 14, Chapter 4 of Text
Compute
From the law of cosines
Hence,
is evaluated on the reference solution
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Example, Problem 14, Chapter 4 of Text
The observations are related to a reference state
deviation vector by,
. . .
Defining
Then
and
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Example, Problem 14, Chapter 4 of Text
Here
at reference time,
and observed -
computed
14 b) Assume small oscillations, i.e.,
, . Then the equations of motion
become
Define
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Example, Problem 14, Chapter 4 of Text
This is the equation for a harmonic oscillator
which has the solution
The constants are evaluated by noting that
Hence,
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Example, Problem 14, Chapter 4 of Text
We may now write the state transition matrix
directly by differentiating the solution for
and , i.e.,
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Example, Problem 14, Chapter 4 of Text
Alternatively write as a 1st
order system
Taking the inverse Laplace Transform gives us
14c) The assumption that is small restricts
this solution to small values of . However,
if we linearize about a reference
solution, we do not require that be
small, only that the deviation of from the
reference, , be small.
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Least Squares with apriori Information
If an apriori value is available for (call
it ) and an associated symmetric weighting
matrix , the weighted least squares
estimate of can be obtained.
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Least Squares with apriori Information
Given
Where is the error in and its
influence on is reflected in the
weighting matrix
and
Choose to minimize the performance index
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Least Squares with apriori Information
Writing explicitly in terms of

(4.3.24)
Results in (See Eq B.7.4)
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Least Squares with apriori Information
Solving for yields
(4.3.25)
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Least Squares with apriori Information
Note that is symmetric
also
which will be positive definite if H and/or
is full rank. Hence, minimizes .
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Concept Test
True or False

• If the relationship between the observations and
  the state is linear we
• do not have to iterate the Newton-Raphson
  equation. ____________

2) Given observations of range rate ,
j1----5, all elements of the following
state vectors can be solved for (indicate T or F
for case A and B)
Case A T or F __.
Case B T or F __.
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Concept Test
3) The state vector in case 2.B could be solved
for uniquely with one observation each of
at one instant in time. _______
4) For problem 2.B we may use any initial guess
for the state but it may take many
iterations to converge. _______
5) Given
Since the observation-state and state propagation
equations are linear we do not have to use a
state deviation vector. ________
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Concept Test
6) The differential equation in each column of
is independent of the equations
in other columns. _________
7) The least squares solution minimizes the sum
of the residuals. ______

• For the equation we always have
  more unknowns than
• equations. ________

9) If the determinant of a symmetric matrix is
negative a) It is not positive
definite _______ b) Its
inverse does not exist. _______
10) The derivative of a scalar WRT a vector is a
scalar. _____
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Concept Test
11) The rank of is
2. _______
12) What is the one topic or concept you are
having the most trouble with in this
class?