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In this paper, a motion-sensing based management system for smart context-awareness rehabilitation healthcare including various balance exercise is built by the integration of the physiological sensing and feedback coaching. The home-end system can not only provide the exercise coaching instruction, the balance stability analysis, and the motion similarity analysis in real-time, but also simultaneously transmit the user image, exercise skeleton streaming, center of pressure (COP), center of gravity (COG) and physiological information to the telecare-end center. According to the combination of the home-end and the telecare-end as well as the real-time care management of one-to-multiple personal balance exercise monitor, this system can provide user various personalized balance exercise prescription and cardiac rehabilitation coaching in an effectiveness rehabilitation exercise environment. Therefore, via this tele-system, the spinocerebellar ataxia (SCA) patients in balance rehabilitation stage not only can be monitored execution status of the rehabilitation exercise prescription, but also can be long-term monitored and evaluated the predicted goal of the rehabilitation exercise balance stability in order to improve patient's compliance.

One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4x4 homogeneous transformation matrix. In previous work we applied this matrix to the optical domains of flat and spherical surfaces for: (1) skew ray tracing to determine the paths of skew rays being reflected/refracted; (2) sensitivity analysis to determine by direct mathematical analysis the differential change of incident point and reflected/refracted vector with respect to change in incident light source. The present work extends our previous work to include the case of parabaloidal boundary surfaces.

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