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16.346 Astrodynamics
Fall 2008
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Lecture 32 Powered Flight Guidance to Maximize Final Energy
Lagrange Multipliers Example
Consider a simple example of the use of Lagrange Multipliers:
Find the point on the curve x2 y = 2 which is nearest the origin.
Here we must make x2 + y 2 a minimum subject to the constraint x2 y − 2 = 0.
Solution: Find the minimum of the function f (x, y) = x2 + y 2 − λ(x2 y − 2) when x and
y are unconstrained:
∂f
∂f
= 2x − λ2xy = 0
= 2y − λx2 = 0
∂x
∂y
√
from which we find: λ = 1 √ x = ± 2√ y = 1 so that the two points at minimum
distance from the origin are 2, 1 and − 2, 1.
Thrust Vector Attitude Control to Maximize Total Energy
State Equations
⎡
⎤ ⎡
⎤
x
vx
d
⎢ y
⎥ ⎢
vy
⎥
⎣
⎦
=
⎣
⎦
⇐⇒
v
cos
β(t)
a
dt
x
T
aT sin β(t) − g
vy
dx
= f [x(t), β(t)]
dt
Performance Index
J = gy(t1 ) +
1
2
[vx2 (t1 ) + vy2 (t1 )]
⇐⇒
J = gx2 (t1 ) +
1
2
[x23 (t1 ) + x24 (t1 )]
Admissible Variations
x(t) = xm (t) + α(t)
β(t) = βm (t) + αγ(t)
with
(t0 ) = 0
Lagrange Multipliers
Introduce the vector Lagrange Multiplier λ(t) (also called the Co-State) and write
� t1
�
�
dx
λT(t)
− f [x(t), β(t)] dt = 0 I=
dt
t0
The Problem
To maximize J − I as a function of α
dJ ��
= g2 (t1 ) + x3m (t1 )3 (t1 ) + x4m (t1 )4 (t1 )
�
dα α=0
�
t1
�
d
∂f
∂f
�
dI
��
T
=
λ (t)
−
−
γ dt
�
∂x
∂β
dα
α=0
dt
t0
16.346 Astrodynamics
Lecture 32
Integration by Parts
� t1
�
d
∂f
∂f
�
dI
��
=
λT(t)
γ dt
−
−
�
dt
∂x
∂β
dα
α=0
t0
�t1 �
t1 �
T
�
t1
�
�
∂f
∂f
dλ
T
T
� −
= λ (t)(t)�
+ λ (t)
(t) dt −
λT(t) γ(t) dt
∂β
dt
∂x
t0
t0
t0
�
t1
∂f
dJ ��
= λT(t1 )(t1 ) −
λT(t) γ(t) dt
must equal
�
∂β
dα α=0
t0
Here we require the Co-State λ (t) to satisfy the differential equation
In our case
⎡
0
∂f
⎢ 0
=
⎣
0
∂x
0
∂f
dλ T
= −λ T
dt
∂x
⎤
0 1 0 0 0 1⎥
⎦
0 0 0 0 0 0 so that
⎡
λ1 (t) = c1
λ2 (t) = c2
Also
λ3 (t) = c1 t + c3
λ4 (t) = c2 t + c4
⎤
0
∂f
⎢
0
⎥
=
⎣
⎦
−aT sin β
∂β
aT cos β
Choose the constants c1 , c2 , c3 , c4 so that
λ1 (t) = 0
λ2 (t) = g
Then, if we are to have
dI
��
dJ
��
=0
�
−
�
dα
α=0 dα
α=0
λ3 (t) = vxm (t1 )
λ4 (t) = g(t1 − t) + vym (t1 )
�
t1
λ T(t)
we must require that
t0
∂f
γ(t) dt = 0 ∂β
From the Fundamental Lemma of the Calculus of Variations it follows that
λ T(t)
∂f
=0
∂β
which is called the Optimality Condition.
In our case, we have
−λ3 (t) sin βm (t) + λ4 (t) cos βm (t) = 0
Thus, the optimum program for β(t) is
g(t1 − t) + vym (t1 )
λ4 (t)
= tan βm (t) =
λ3 (t)
vxm (t1 )
called the Linear-Tangent Law.
This result formed the basis of the so-called Iterated Guidance Mode used by the
Saturn launch vehicle’s guidance system to place the Apollo spacecraft in an earth parking
orbit prior to its voyage to the moon.
16.346 Astrodynamics
Lecture 32