The preoperative procedure for treating peripheral arterial disease requires 3D mapping of the blood vessel of interest. Because the available technologies are costly and invasive, and have an iodizing effect, new 3D imaging systems are being developed from ultrasound scanning technology using a robot as the probe manipulator. The authors of this paper have designed a new parallel robot along these lines. In response to the great concern for safety generated by the use of robots in medicine, we present a new approach for static balancing to enhance the safety of the proposed robot. Because total balancing is not practical for this device, the approach we have chosen is an optimization based on the addition of torsion springs on the actuated and the passive revolute joints. The optimization consists of a sequence of objectives, which are met using a linear programming technique, since the equations of torques and forces are linear with respect to the unknown variables.