The rigid-body attitude tracking using vector and biased gyro measurements with unknown inertia matrix is studied in this note. First, a gyro-bias observer with global exponential stability is designed. Then, an attitude tracking controller based on this observer is devised, ensuring almost global asymptotic stability and almost semiglobal exponential stability. A separation property of the combined observer-controller is proved. Lastly, an adaptive attitude tracking controller relying on a modified gyro-bias observer and with no over-parametrization is developed to deal with the unknown inertia matrix. The proposed control schemes require neither an explicit attitude representation nor any attitude estimation, but only the measurement of at least two non-collinear known inertial reference vectors and biased gyro rate, which can be obtained by common low-cost IMU sensors. Simulations are included to illustrate the proposed adaptive controller under noisy measurements.

This paper presents a novel Lagrangian approach to attitude tracking for rigid spacecraft using unit quaternions, where the motion equations of a spacecraft are described by a four degrees of freedom Lagrangian dynamics subject to a holonomic constraint imposed by the norm of a unit quaternion. The basic energy-conservation property as well as some additional useful properties of the Lagrangian dynamics are explored, enabling to develop quaternion-based attitude tracking controllers by taking full advantage of a broad class of tracking control designs for mechanical systems based on energy-shaping methodology. Global tracking of a desired attitude on the unit sphere is achieved by designing control laws that render the tracking error on the four-dimensional Euclidean space to converge to the origin. The topological constraints for globally exponentially tracking by a quaternion-based continuous controller and singularities in controller designs based on any three-parameter representation of the attitude are then avoided. Using this approach, a full-state feedback controller is first developed, and then several important issues, such as robustness to noise in quaternion measurements, unknown on-orbit torque disturbances, uncertainty in the inertial matrix, and lack of angular-velocity measurements are addressed progressively, by designing a hybrid statefeedback controller, an adaptive hybrid state-feedback controller, and an adaptive hybrid attitude-feedback controller. Global asymptotic stability is established for each controller. Simulations are included to illustrate the theoretical results.