Localization accuracy is one of the most important parts of Unmanned Vehicle Systems, Automated Vehicles, Robotics and Navigation. The 6-DOF Inertial Measurement Unit (IMU) is a commonly used device for inertial navigation and is composed of a 3-axis accelerometer and 3-axis gyroscope. The body-fixed IMU measurements are combined with initial values to produce a position and orientation estimate in the inertial frame with every new measurement. However, IMU performance is greatly degraded by bias, scale-factor, non-orthogonality, temperature, and noise. This paper develops a sliding mode observer specifically focused on gyroscope bias estimation to improve gyro measurement results. The work presented here improves the performance of tilt sensors equipped in a commercially available smartphones with accelerometers and gyroscopes. The algorithm uses quaternions to avoid the well-known Euler angle singularities also known as gimbal lock. The observed gyro-bias can be used to reconstruct an improved estimation of the real attitude. A sliding-mode observer was constructed, and A* Matrix stability criterion were used to guarantee observer error convergence in finite time. The algorithm was verified using both a simulated IMU model and experimental tests with a custom designed rotational platform. Simulation tests used a predefined gyros-bias to ensure the algorithm-estimated results converged to the correct value. Simulation results show the observer error quickly converges to zero and the gyro-bias estimation converged to the expected values. The results also show that the proposed method is very effective for reconstructing the real attitude using the observed gyro-bias. This study presents a fast, simple gyro-bias estimation method that can help reconstruct the real attitude with a simple formulation that eliminates complicated constraints.