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In some cases, tendon-sheath transmission systems were used to transmit power into work places through complex paths, such as the small narrow crevices possibly existed in the ruins of disasters or complex pipelines. Thus the tendon-sheath systems might work in arbitrarily curved shape. Based on the results in constant curvature, the transmission characteristics of tendon-sheath in arbitrarily curved shape were investigated in this paper. It was showed that the transmission characteristics were independent of the tendon torsion. By considering static balance of a differential tendon element with Coulomb friction, the transmission characteristics were formulated in terms of total curvature and friction coefficient. The experiment results showed that the formulas of the transmission characteristics were only correct when the change of the input force was slow enough, which was in consistent with the static balance assumption during formulation. A robot with a tendon sheath transmission system was designed, which could finish some dexterous actions in the narrow spaces.