This paper addresses the problem of estimating the attitude of a robotic platform using biased measurements of: (i) the direction of the gravitational field and (ii) angular velocity obtained from a set of high-grade gyroscopes sensitive to the Earth's rotation. A cascade solution is proposed that features a Kalman filter (KF) tied to a rotation matrix observer built on the special orthogonal group of order 3. The KF, whose model stems from a uniformly observable linear time-varying system, yields estimates of: (i) the Earth's total rotational rate; (ii) two sensor biases associated with the aforementioned measurements; and, (iii) noise-filtered and bias-corrected accelerometer data. All estimates are expressed in the platform's body-fixed frame. In turn, the attitude observer put forward is shown to be almost globally asymptotically stable, in particular locally input-to-state stable with respect to the KF errors. Experimental results are showcased that successfully demonstrate the efficiency of the proposed attitude estimation solution.