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 take-oﬀ time and
take-oﬀ 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
- The Aeneid