The gaits used in turning motion of a quadruped are studied. Depending on the location of the turning center, turning gaits, are divided into two types: spinning gait and circling gait. The former is applied when the turning center is at or close to the gravity center, whereas the latter is suitable for a more distant turning center. The spinning gait is developed on the basis of the z-type creeping gait and is optimized for stability. The optimization procedure issues in two special gaits-the wave-spinning gait and the equal-phase-spinning gait. The circling gaits are developed based on the wave-crab gaits and the gaits are called wave-circling gaits. The stability margin curve of a wave-circling gait is found to be similar to a wave-crab gait when the turning radius is large. In order to assist the selection of a turning gait, the stable areas of turning center of both spinning and circling gaits are generated by numerical method.>

The condition under which closed-form solutions of forward kinematics of parallel platforms are obtainable is explored. It is found that forward position analysis has closed-form solutions if one rotational degree of freedom (DOF) of a parallel platform is decoupled from the other five DOFs. Geometrically, this condition is satisfied when five end points at the platform or at the base are on the same line. A general case in which these five points do not coincide with each other is studied first and is shown to have 16 possible closed-form solutions. The computation of these solutions is very efficient compared to that of Stewart platforms. The different geometries of parallel platforms with closed-form solutions are discussed.>