MR compatible apparatus is essential to design new tasks for fMRI study. This paper described the development of an MR compatible manipulandum actuated by the ultrasonic motors, which was able to work within MRI scanner and to perform fMRI task continuously during finger movements. The prototype of maniulandum was able to produce two discriminable force fields: position dependent and velocity dependent force field. The control law was derived by the characteristic of velocity-phase difference-torque relationship of the ultrasonic motor. After MR compatibility was verified with signal noise ratio of the images, it was shown that the proposed prototype was able to use for fMRI study.

Derivative-free Abel inversions named here as 'Indirect' method is provedto be far superior to conventional 'Direct' method involving derivative term, by numerical experiments. With powerful least-square fitting method named SALS developed at computer center of Tokyo University, and simple power-series expansions of emission coefficient e(r) is sufficient without any iterations nor deconvolutions. 10-test functions often used in literatures are treated and the result of numerical experiments show reconstructions reached to the ultimate of computer precisions.

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