The problem of determining the attitude and gyro bias for rigid bodies is addressed by utilizing biased angular velocity and multiple body-frame vector measurements. We first formulate the gyro bias estimation problem as a parameter estimation problem. Then, a parameter estimation-based observer is proposed for gyro bias estimation. Thereafter, we propose a nonlinear geometric attitude observer on Lie group \(\mathcal{S}\mathcal{O}(3)\) , based on the estimated gyro bias and vector measurements. The overall estimation system is shown to be almost globally asymptotically stable.Numerical examples are presented to demonstrate the effectiveness of the proposed estimation scheme.