This paper presents the development of an optimization guidance algorithm capable of generating three-dimensional trajectories enabling a small free-flyer robotic spacecraft to maneuver in close proximity to translating and tumbling satellites. Specifically, the proximity guidance law parameterizes the velocity trajectory using Legendre polynomials and optimizes their coefficients in a two-step fashion. In the first step, a sub-optimal solution that satisfies the boundary conditions, dynamics constraints, and performance limitations is obtained. In the second step, time-permitting, the optimal solution which also minimizes the path length is obtained. The performance of the guidance law is evaluated in simulation results for a fly-around scenario of both a non-cooperative stationary and tumbling target satellite.
The measurement of the acceleration due to gravity by means of the compound pendulum is a very common laboratory experiment; but there are certain simple principles involved in the theory of it which are not, as it seems to me, sufficiently emphasized by teachers, and which are not stated (at all events explicitly) in any text-book known to me. It may be of use to some to enunciate them here, and to shew their importance both to the instrument-maker and the student.
We may now turn our attention to the questions which arise specially in connection with a reversible pendulum, such as Kater’s. Of course the object in view is so to place two knife-edges on opposite sides of the C.G. that the period shall be the same from either, the two positions being selected according to the rule at the end of § 4. But the usual procedure will be to decide on the distance between the knife-edges at the start, say one metre; then to clamp them to the more or less uniform bar of the pendulum, using a standard distance-piece to secure the proper interval between them, and finally to adjust a sliding weight in such a position as to make the period the same for both. We have seen in § 2 how the period, or rather the length of the equivalent S.P., varies with the position of the sliding weight, but it will be convenient now to change the notation. Let I1, I2, ... denote the moments of inertia of the various masses making up the pendulum, each about its own C.G., M1, M2.. the masses; x1, x2... the co-ordinates of the centres of gravity, referred to one of the knife-edges as origin; let the co-ordinate of the other knife-edge be d, and let the letters I, M, x without suffixes refer to a moveable mass which is being adjusted. We shall at once see how much simpler it is to discuss the graph whose ordinate is the length of the S.P., than the one where the period is used, for the position which the moveable mass should occupy of course corresponds to the intersection of the two graphs relating respectively to the two knife-edges: and it is surely better to have to deal with the intersections of conics than of cubics.
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