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Unformatted text preview: iven by p(k ) ∈ R2 , and the velocity by v (k ) ∈ R2 , for k = 1, . . . , K . Here h > 0 is
the sampling period. These are related by the equations
p(k + 1) = p(k ) + hv (k ), v (k + 1) = (1 − α)v (k ) + (h/m)f (k ), k = 1, . . . , K − 1, where f (k ) ∈ R2 is the force applied to the vehicle at time kh, m > 0 is the vehicle mass, and
α ∈ (0, 1) models drag on the vehicle; in the absense of any other force, the vehicle velocity decreases
by the factor 1 − α in each discretized time interval. (These formulas are approximations of more
accurate formulas that involve matrix exponentials.)
The force comes from two thrusters, and from gravity:
f (k ) = cos θ1
sin θ1 u1 (k ) + cos θ2
sin θ2 u2 (k ) + 0
−mg , k = 1, . . . , K − 1. Here u1 (k ) ∈ R and u2 (k ) ∈ R are the (nonnegative) thruster force magnitudes, θ1 and θ2 are the
directions of the thrust forces, and g = 10 is the constant acceleration due to gravity.
The total fuel use is K −1 (u1 (k ) + u2 (k )) . F=
k=1 (Recall that u1 (k ) ≥ 0, u2 (k ) ≥...
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- Fall '13
- The Aeneid