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Variable impedance control has been required to perform a variety of interactive tasks in contact with environments. In some cases, the time-varying stiffness matrix of the impedance model can be used to achieve high performance for uneven contact tasks. In the paper, two sufficient conditions are proposed to ensure the input-to-state stability (ISS) irrespective of time-varying stiffness. Furthermore, the update rule of the stiffness is also suggested in such a way that the asymptotic stability is guaranteed under certain region conditions. Even when the update rule is not applied, the ISS is at least assured. In other words, the error is always bounded only if the external force/torque is bounded. In detail, two sufficient conditions offer the lower bound of stiffness and the upper bound of its time derivative. Simulation results show that the ISS of variable impedance control is achieved if the proposed sufficient conditions are satisfied. Also, we can confirm the asymptotic behavior in the simulation when the stiffness is updated according to the given rule.

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Our purpose was to compare the contributions of these two systems to assess PIP joint extension in fresh cadaver models. Nine middle fingers of fresh cadavers were used. The PIP joint angle was measured while an extension load was applied on the extensor tendons. Specimens on which extension load was applied on the extrinsic extensors were classified as the extrinsic group, and those on which extension load was applied on the intrinsic extensors were classified as the intrinsic group. Linear regression analyses were performed to obtain regression equation and the extension load-PIP joint angle curve. The mean of slope of the curve was compared between the two groups using paired t-test. The same experiments were done for the metacarpophalangeal (MP) joint in 0° and 60° flexion to evaluate the effect of MP joint flexion on PIP joint extension. The mean slope of the extension load-PIP joint angle curve of the extrinsic group was significantly greater than that of the intrinsic group. With the MP joint in 0° flexion, the mean slope of the extrinsic and intrinsic groups was -0.148 and -0.117, respectively (greater absolute value means greater slope, p=0.01). With the MP joint in 60° flexion, the mean slopes were -0.147 and -0.104, respectively (p=0.015). The contribution of the intrinsic extensor for PIP joint extension shows decreasing trends with MP joint flexion. The extrinsic extensors have greater contribution for PIP joint extension compared with the intrinsic extensors.

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