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Unformatted text preview: = 4, S = 0.8, V to = 40, L = 300. Plot pt , vt , ft , and bt versus t. Comment on what you see. Report the takeoﬀ time and
takeoﬀ position for the proﬁle you ﬁnd.
14.8 Optimal spacecraft landing. We consider the problem of optimizing the thrust proﬁle for a spacecraft
to carry out a landing at a target position. The spacecraft dynamics are
mp = f − mge3 ,
¨
where m > 0 is the spacecraft mass, p(t) ∈ R3 is the spacecraft position, with 0 the target landing
position and p3 (t) representing height, f (t) ∈ R3 is the thrust force, and g > 0 is the gravitational
acceleration. (For simplicity we assume that the spacecraft mass is constant. This is not always
a good assumption, since the mass decreases with fuel use. We will also ignore any atmospheric
friction.) We must have p(T td ) = 0 and p(T td ) = 0, where T td is the touchdown time. The
˙
spacecraft must remain in a region given by
p3 (t) ≥ α (p1 (t), p2 (t)) 2 ,
where α > 0 is a given minimum glide slope. The initial position p(0) and velocity p(0) are given.
˙
The thrust force...
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 Fall '13
 F.Borrelli
 The Aeneid

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