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In this paper, we present a new factorized quaternion approach for determining the arm limbs' orientation using triaxial accelerometers with consideration of anatomical and sensor constraints. Typical use of the quaternion method determines the angle and axis of rotation represented by a single angle-axis quaternion. Different from the conventional approach, we propose using the factorized quaternion approach for the determination of arm motions. This approach allows the implementation of anatomical arm constraints which match the range of motion of the human arm and also reduces the ambiguity in solutions. In addition, the singularities arising from the use of triaxial accelerometers can be detected and resolved for a transient state. Measurement of the upper arm motion is demonstrated along a vertical plane and extended along a tilted plane for the forearm. Experiments have been conducted using a wireless sensor network equipped with triaxial accelerometers attached to the arm. The results have been benchmarked with a commercial inertial measurement unit to validate the feasibility and advantages of this new approach. Comparable performance in terms of accuracy has been obtained at a much reduced cost and power consumption.