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Bd TEXT ( k ) ⎣ k⎦ (11) To achieve the fast non-oscillatory response on the external force, we assigned the eigenvalues λ1 and λ2 of matrix A as real and unequal λ1≠λ2. By using Cayley-Hamilton method for matrix exponential determination, we have: Ad = e AT λ2T λ1T 1 ⎡ e λ1 − e λ2 ⎢ = λ1 − λ2 ⎢⎣ −b eλ1T − eλ2T ( Bd = ( Ad − I ) A−1 B = − ) eλ1T − eλ2T ⎤ ⎥, eλ2T ( λ1 + a ) − eλ1T ( λ2 + a ) ⎥⎦ (12) ⎡eλ2T λ1 − eλ1T λ2 − ( λ1 − λ2 ) ⎤ c ⎢ ⎥, λ1T λ2T b ( λ1 − λ2 ) ⎢ b e e − − ⎥⎦ ⎣ ( (13) ) where T is the sampling time; coefficients a, b, and c equal to Dd/Md, Kd/Md, and 1/Md, respectively; I is the identity matrix.

Introduction Recently, various kinds of tactile sensors have been investigated and reported for tactile applications with robot fingertips. Typical specifications of human fingertips are known as follows; spatial resolution of human fingertip is around 1 mm, time resolution is below 1 msec (1 kHz), and the minimum force resolution is around 1-10 mN. Also, human fingertip can recognize the three-dimensional (3-D) shape of touching object using flexible deformation in the convex shape of fingertip skin.

An optical 6-axis force sensor for brain function analysis using fMRI, Proceedings of 2nd IEEE Int. Conf. on Sensors, Vol. 1, pp. 253-258, ISBN 0-7803-8133-5, Toronto, October 2003, IEEE Press. ; Kawakami, N. & Tachi. S. (2007). Towards safe human-robot interaction: joint impedance control of a new teleoperated robot 36 Sensors, Focus on Tactile, Force and Stress Sensors arm. Proceedings of IEEE Int. Symposium on Robot and Human Interactive Communication (RO-MAN), pp. 860-865, ISBN 1-4244-1635-6, Jeju island, August 2007, IEEE Press.

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