In this work, a Kalman filtering algorithm is proposed that estimates the spacecraft attitude and attitude parameters without gyroscope measurements for nanosatellites. The attitude parameters include sensor and actuator alignment, spacecraft body moment of inertia, reaction wheel moment of inertia, reaction wheel speed, and the dipole moment of the spacecraft. The new filtering formulation is based on the differential form of the rigid-body rotational dynamics, and so the body rate and the other attitude parameters can be updated directly by attitude measurements such that the gyroscope reading is not required. The new filter is derived in a closed form for implementation, and physical and mathematical approaches toward achieving convergence and stability with this filter are discussed. A detailed simulation is presented that demonstrates the utility of the proposed algorithm for three different types of unmodeled disturbance torques.