Optimal Adjustment Procedure of the External Fixators to Minimize the Soft Tissue Injury

A typical fixator of tibia consists of an axial rotary joint, 4 revolute joints and 2 prismatic joints in the ends providing a total of 7 degrees of freedom for its maneuverability to reduce the bone fracture in the 3-D space. The purpose of the present study was to calculate the final configuration of the fixator joints to treat a general fracture and to optimize the path to this configuration. To obtain the final configuration, the known space orientation 4×4 matrix of the assumed healed bone was set equal to the orientation matrix of the fixator and the values for the seven joints were calculated assuming a seventh equation stating the final values of the two prismatic joints to be equal. In the second part of the study, the optimal path of the adjustment procedure to the final configuration of fixator was investigated. The optimization criterion was defined as the length of the path of the bone tips throughout the procedure, so that the connective soft tissues are minimally injured. The integral of path length with respect to time was calculated, then the Lagrange equations specifying equalities between joint values and their first and second derivatives were derived. The resulting set of 2^nd- order differential equations were transformed into a set of 14 1^st- order differential equations, and solved using MATLAB. The significance of this approach was examined considering a simple 2 link planar mechanism, going from a specified starting position to a final configuration. It was found that implication of the optimization procedure reduces the path length by 14.3% in comparison with when the joint angles are change linearly.