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The design of strapdown inertial navigation system (INS) algorithms based on dual quaternions is addressed. Dual quaternion is a most concise and efficient mathematical tool to represent rotation and translation simultaneously, i.e., the general displacement of a rigid body. The principle of strapdown inertial navigation is represented using the tool of dual quaternion. It is shown that the principle can be expressed by three continuous kinematic equations in dual quaternion. These equations take the same form as the attitude quaternion rate equation. Subsequently, one new numerical integration algorithm is structured to solve the three kinematic equations, utilizing the traditional two-speed approach originally developed in attitude integration. The duality between the coning and sculling corrections, raised in the recent literature, can be essentially explained by splitting the new algorithm into the corresponding rotational and translational parts. The superiority of the new algorithm over conventional ones in accuracy is analytically derived. A variety of simulations are carried out to support the analytic results. The numerical results agree well with the analyses. The new algorithm turns out to be a better choice than any conventional algorithm for high-precision navigation systems and high-maneuver applications. Several guidelines in choosing a suitable navigation algorithm are also provided.