By Ye Yan, Xu Huang, Yueneng Yang

This booklet develops a dynamical version of the orbital movement of Lorentz spacecraft in either unperturbed and J2-perturbed environments. It explicitly discusses 3 sorts of ordinary area missions related to relative orbital keep an eye on: spacecraft soaring, rendezvous, and formation flying. as a result, it places ahead designs for either open-loop and closed-loop regulate schemes propelled or augmented by way of the geomagnetic Lorentz strength. those regulate schemes are completely novel and symbolize a considerably departure from past approaches.

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**Sample text**

N ð3:21Þ where N is the number of the beacons, fd is the known focal length, and Ajk is the unknown elements of the attitude matrix AðqÞ. Rewrite the ideal observation equation in a unit-vector form as [3] bi ¼ Ari i ¼ 1; 2; . ; N ð3:22Þ where the unit vectors bi and ri are, respectively, given by 2 3 Àvi 1 4 Àci 5 bi ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fd2 þ v2i þ c2i fd ð3:23Þ 2 3 Xi À x 1 ri ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 Yi À y 5 ðXi À xÞ2 þ ðYi À yÞ2 þ ðZi À zÞ2 Zi À z ð3:24Þ Due to the existence of observation noises, the actual observation model is [3] ~bi ¼ Ari þ gi ð3:25Þ where ~ bi is the ith observation vector, and gi is the ith noise vector satisfying E½gi ¼ 03Â1 ; E½gi gTi ¼ r2i I 3Â3 ð3:26Þ where E½ denotes the expectation.

43). In the presence of J2 perturbation, the velocity of the Lorentz spacecraft with respect to the local magnetic ﬁeld is revised as V rel;J2 ¼ V J2 À xE Â RL ¼ ½ Vx;J2 Vy;J2 Vz;J2 T 3 2 R_ T þ x_ À yðxz À xE cos iT Þ À zxE sin iT cos uT 7 6 ¼ 4 y_ þ ðRT þ xÞðxz À xE cos iT Þ À zðxx À xE sin iT sin uT Þ 5 ð2:74Þ z_ þ ðRT þ xÞxE sin iT cos uT þ yðxx À xE sin iT sin uT Þ Furthermore, the gravitational potential (per unit mass) of the Lorentz spacecraft, including J2 perturbation, is [5] Ug;J2 ¼ À l kJ À 3 ð1 À 3 cos2 hÞ RL 3RL ð2:75Þ with cos h ¼ RLz =RL ð2:76Þ where RLz is the projection of RL on the ZI axis of the ECI frame, given by RLz ¼ ðRT þ xÞ sin iT sin uT þ y sin iT cos uT þ z cos iT ð2:77Þ Substitution of Eqs.

65). Heretofore, the dynamical model that describes the relative orbital motion of a Lorentz spacecraft about a J2-perturbed reference orbit has been derived as Eqs. 91). 2 Numerical Simulations A typical scenario in LEO is simulated to evaluate effect of J2 perturbation on the Lorentz-augmented relative motion. 1. Also, other simulation parameters are chosen the same as those given in Sect. 2. The only difference is the inclusion of J2 perturbation here. The exact trajectories of the relative position are generated by numerical integrations of the nonlinear equations of J2-perturbed relative motion from Eqs.