bv_cvxbook_extra_exercises

E in the p1 p2 plane verify that it passes through the

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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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