Antonio F. Bertachini A. Prado

Maria Cecilia Zanardi

Tadashi Yokoyama

Silvia Maria Giuliatti Winter

Space Dynamics

The attitude control systems of satellites with rigid and flexible components are demanding more and more better performance resulting in the development of several methods control. For that reason, control design methods presently available, including parameters and states estimation, robust and adaptive control, as well as linear and nonlinear theory, need more investigation to know their capability and limitations. In this paper the investigated technique is H-Infinity method in the performance of the Attitude Control System of a Rigid-Flexible Satellite.

Ximena Celia Méndez Cubillos

Luiz Carlos Gadelha de Souza

Using of H-Infinity Control Method in Attitude Control System of Rigid-Flexible Satellite

It is well known that lunar satellites in polar orbits suffer a high increase on the eccentricity due to the gravitational perturbation of the Earth. The final fate of such satellites is the collision with the Moon. Therefore, the control of the orbital eccentricity leads to the control of the satellite's lifetime. In the present work we study this problem and introduce an approach in order to keep the orbital eccentricity of the satellite at low values. The whole work was made considering two systems: the 3-body problem, Moon-Earth-satellite, and the 4-body problem, Moon-Earth-Sun-satellite. First, we simulated the systems considering a satellite with initial eccentricity equals to 0.0001 and a range of initial altitudes between 100 km and 5000 km. In such simulations we followed the evolution of the satellite's eccentricity. We also obtained an empirical expression for the length of time needed to occur the collision with the Moon as a function of the initial altitude. The results found for the 3-body model were not significantly different from those found for the 4-body model. Secondly, using low-thrust propulsion, we introduced a correction of the eccentricity every time it reached the value 0.05.

O. C. Winter

D. C. Mourão

C. F. de Melo

E. N. Macau

J. L. Ferreira

J. P. S. Carvalho

Controlling the Eccentricity of Polar Lunar Orbits with Low-Thrust Propulsion

Herein, we consider the problem of a lunar artificial satellite perturbed by the nonuniform distribution of mass of the Moon taking into account the oblateness (<svg style="vertical-align:-3.27605pt;width:14.225px;" id="M1" height="15.25" version="1.1" viewBox="0 0 14.225 15.25" width="14.225" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D43D" d="M434 650l-5 -28q-60 -8 -75 -20.5t-25 -69.5l-71 -395q-20 -107 -57.5 -168t-106.5 -94q-40 -18 -61 -20l-10 32q77 23 113 108q21 50 41 160l68 377q10 57 -1.5 69.5t-76.5 20.5l6 28h261z" /></g><g transform="matrix(.012,-0,0,-.012,7.825,15.187)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg>) and the equatorial ellipticity (sectorial term <svg style="vertical-align:-3.27605pt;width:24.15px;" id="M2" height="15.5375" version="1.1" viewBox="0 0 24.15 15.5375" width="24.15" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.012,-0,0,-.012,12.05,15.475)"><use xlink:href="#x32" /></g><g transform="matrix(.012,-0,0,-.012,17.761,15.475)"><use xlink:href="#x32" /></g></svg>). Using Lie-Hori method up to the second order short-period terms of the Hamiltonian are eliminated. A study is done for the critical inclination in first and second order of the disturbing potential. Coupling terms due to the nonuniform distribution of mass of the Moon are analyzed. Numerical simulations are presented with the disturbing potential of first and second order is. It an approach for the behavior of the longitude of the ascending node of a near Sun-synchronous polar lunar orbit is presented.

Jean Paulo dos Santos Carvalho

Rodolpho Vilhena de Moraes

Antônio Fernando Bertachini de Almeida Prado

Nonsphericity of the Moon and Near Sun-Synchronous Polar Lunar Orbits

A simple mathematical model for a constrained robotic manipulator is investigated. Besides the fact that this model is relatively simple, all the features present in more complex problems are similar to the ones analyzed here. The fully plastic impact is considered in this paper. Expressions for the velocities of the colliding bodies after impact are developed. These expressions are important in the numerical integration of the governing equations of motion when one must exchange the set of unconstrained equations for the set of constrained equation. The theory presented in this work can be applied to problems in which robots have to follow some prescribed patterns or trajectories when in contact with the environment. It can also de applied to problems in which robotic manipulators must handle payloads.

Andr&#233; Fenili

Bernd Sch&#228;fer

The Determination of the Velocities after Impact for the Constrained Bar Problem

Swing-by techniques are extensively used in interplanetary missions to minimize fuel consumption and to raise payloads of spacecrafts. The effectiveness of this type of maneuver has been proven since the beginning of space exploration. According to this premise, we have explored the existence of a natural and direct link between low Earth orbits and the lunar sphere of influence to get low-energy transfer trajectories to the Near Earth Objects (NEOs) 99942 Apophis, 1994 WR12, and 2007 UW1 through swing-bys with the Moon. The existence of this link is related to a family of retrograde periodic orbits around the Lagrangian equilibrium point L1 predicted for the circular, planar, restricted three-body Earth-Moon-particle problem. The trajectories in this link are sensitive to small disturbances. This enables them to be conveniently diverted reducing so the cost of the swing-by maneuver. These maneuvers allow a gain in energy sufficient for the trajectories to escape from the Earth-Moon system and to stabilize in heliocentric orbits between the Earth and Venus or Earth and Mars. Therefore, the trajectories have sufficient reach to intercept the NEOs' orbits.

E. E. N. Macau

Alternative Transfers to the NEOs 99942 Apophis, 1994 WR12, and 2007 UW1 via Derived Trajectories from Periodic Orbits of Family G

Several irregular satellites of the giant planets were found in the last years. Their orbital configuration suggests that these
satellites were asteroids captured by the planets. The restricted three-body problem can explain the dynamics of the capture, but
the capture is temporary. It is necessary some kind of dissipative effect to turn the temporary capture into a permanent one. In this
work we study an asteroid suffering a gas drag at an extended atmosphere of a planet to turn a temporary capture into a
permanent one. In the primordial Solar System, gas envelopes were created around the planet. An asteroid that was gravitationally
captured by the planet got its velocity reduced and could been trapped as an irregular satellite. It is well known that, depending on the time scale of the gas envelope, an asteroid will spiral and collide with the planet. So, we simulate the passage of the asteroid in the gas envelope with its density decreasing along the time. Using this approach, we found effective captures, and have a better understanding of the whole process. Finally, we conclude that the origin of the irregular satellites cannot be attributed to the gas drag capture mechanism alone.

E. Vieira Neto

Gravitational Capture of Asteroids by Gas Drag

The spatial equilateral restricted four-body problem (ERFBP) is a four
body problem where a mass point of negligible mass is moving under the
Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass
fixed at the origin of the coordinate system such that their configuration is
always an equilateral triangle. Since fourth mass is small, it does not affect
the motion of the three primaries. In our model we assume
that the two masses of the primaries <svg style="vertical-align:-3.27605pt;width:19.875px;" id="M1" height="11.825" version="1.1" viewBox="0 0 19.875 11.825" width="19.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" /></g><g transform="matrix(.012,-0,0,-.012,13.475,11.762)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg> and <svg style="vertical-align:-3.3907pt;width:19.875px;" id="M2" height="11.9625" version="1.1" viewBox="0 0 19.875 11.9625" width="19.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45A" /></g><g transform="matrix(.012,-0,0,-.012,13.475,11.762)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g></svg> are equal to <svg style="vertical-align:-3.56265pt;width:9.5625px;" id="M3" height="12.175" version="1.1" viewBox="0 0 9.5625 12.175" width="9.5625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D707" d="M538 96q-38 -45 -84 -76.5t-71 -31.5q-44 0 -15 117q12 49 18 89h-2q-99 -128 -163 -178q-36 -28 -70 -28q-26 0 -44 38h-2q-16 -69 -16 -129q0 -43 11 -75.5t24 -42.5l1 -6q-7 -12 -24.5 -23t-37.5 -11q-40 0 -40 76q0 52 35 202l94 407l78 24l5 -5q-32 -122 -72 -310&#xA;q-7 -33 0 -55t25 -22q36 0 105.5 75t107.5 145l32 146l75 26h10q-50 -197 -78 -347q-7 -38 6 -38q7 0 31 17t47 39z" /></g></svg> and the mass <svg style="vertical-align:-3.27605pt;width:19.875px;" id="M4" height="11.825" version="1.1" viewBox="0 0 19.875 11.825" width="19.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45A" /></g><g transform="matrix(.012,-0,0,-.012,13.475,11.762)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g></svg> is <svg style="vertical-align:-3.56265pt;width:43.412498px;" id="M5" height="15.35" version="1.1" viewBox="0 0 43.412498 15.35" width="43.412498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,11.996,10.862)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,25.748,10.862)"><use xlink:href="#x32" /></g><g transform="matrix(.017,-0,0,-.017,33.907,10.862)"><use xlink:href="#x1D707" /></g></svg>. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.

Martha Álvarez-Ramírez

Claudio Vidal

Dynamical Aspects of an Equilateral Restricted Four-Body Problem

Rolling-element bearings are simple machine elements of great
utility used both in simple commercial devices as in complex
engineering mechanisms. Because of being a very popular machine
element, there is a lot of literature on the subject. With regard
to the behavior of internal loading distribution, elastic
deformations at point or line contacts, and geometric parameters
under loading, although there are many works describing the
parameters variation models, few works show such variations in
practice, even under simple static loadings. In an attempt to
cover this gap some studies are being developed in parallel.
Particularly in this work, a <i>new</i>, iterative computational
procedure is introduced which calculates internal normal ball
loads in statically loaded single-row, angular-contact ball
bearings, subjected to a known thrust load which is applied to a
variable distance (lever arm or eccentricity) from the geometric
bearing center line. Numerical examples results for a 218
angular-contact ball bearing have been compared with those from
the literature. Fifty figures are presented showing geometrical
features and the following parameters variations as functions of
the thrust load and eccentricity: contact angle, contact ellipse
parameters, normal ball loads, distances between groove curvature
centers, normal and axial deflections, and loading zones.

Mário César Ricci

Internal Loading Distribution in Statically Loaded Ball Bearings Subjected to an Eccentric Thrust Load

Herein, the purpose is to present a Kalman filter based on the sigma point unscented transformation development, aiming at real-time satellite orbit determination using GPS measurements. First, a brief review of the extended Kalman filter will be done. After, the sigma point Kalman filter will be introduced as well as the basic idea of the unscented transformation, in which this filter is based. Following, the unscented Kalman filter applied to orbit determination will be explained. Such explanation encloses formulations about the orbit determination through GPS; the dynamic model; the observation model; the unmodeled acceleration estimation;  also an application of this new filter approaches on orbit determination using GPS measurements discussion.

Paula Cristiane Pinto Mesquita Pardal

Helio Koiti Kuga

A Discussion Related to Orbit Determination Using Nonlinear Sigma Point Kalman Filter

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Mathematical Problems in Engineering

Space Dynamics

Space Dynamics

Articles

Space Dynamics

Using of H-Infinity Control Method in Attitude Control System of Rigid-Flexible Satellite

Controlling the Eccentricity of Polar Lunar Orbits with Low-Thrust Propulsion

Nonsphericity of the Moon and Near Sun-Synchronous Polar Lunar Orbits

The Determination of the Velocities after Impact for the Constrained Bar Problem

Alternative Transfers to the NEOs 99942 Apophis, 1994 WR12, and 2007 UW1 via Derived Trajectories from Periodic Orbits of Family G

Gravitational Capture of Asteroids by Gas Drag

Dynamical Aspects of an Equilateral Restricted Four-Body Problem

Internal Loading Distribution in Statically Loaded Ball Bearings Subjected to an Eccentric Thrust Load

A Discussion Related to Orbit Determination Using Nonlinear Sigma Point Kalman Filter