Attitude estimation is one of the basic processes of the attitude and orbit control system hosted by every satellite. The spacecraft should be orientated properly during the orbit, according to the mission objectives. The estimation is based on a plethora of sensors, like rate gyroscopes, star trackers, sun and earth sensors, magnetometers (for low earth orbits). However, during the orbit the spacecraft operates according to some protocol, which is characterized from its different modes. During safe mode conditions, gyroscopes and/or star trackers might be turned off for energy saving reasons. Then the satellite senses only the geomagnetic field's direction and directions of other celestial bodies. In addition, the motion of a rigid body is described from well-known dynamic equations up to model uncertainties. The goal is to estimate how the satellite faces the Earth at each instant of time, by utilizing the two aforementioned sources of information. Limited research has been undertaken for this problem; on the contrary, the vast majority of published works provide solutions to the problem at the presence of rate sensors. Within a general non-linear framework, the problem of orientation and rate estimation can be set under the Bayesian formulation of estimation. The available tools provided by this approach, are the widely-known Kalman-Based filters. An alternative compelling approach considers the dual optimal control problem thus being more compatible with the geometric nature of the problem. This approach is also more promising, since it overcomes some critical difficulties of the former one and it is applicable in a coordinate free fashion. In this thesis, the aforementioned observation is emphasized and looked into. Therefore, two specific filters are studied: the second-order-optimal minimum energy filter on the special orthogonal group; and the predictive filter on, which is derived on this work. The two filters perform very well with the predictive filter to be less tolerant in measurement noise. Both filters are analyzed and simulated in various scenarios and a comparison is demonstrated, given particular emphasis to their structure. Finally, the predictive filter on is extended and used in an attempt to provide an alternative approach for the problem of the orbital position estimation by utilizing a kinematic model for the Earth's geomagnetic field. Towards this direction it is proved that under some realistic assumptions this later problem is equivalent to two separate problems and therefore the resulting filter can be decomposed in two distinct filters interconnected in series. The optimal correction term which is determined through the estimation process provides information regarding the orbital rotation rate.