## 3 Results  
### ROM of the External Fixator  
The range of motion (ROM) is a crucial indicator to assess the correction capability of the fixator and can be described as the size of the motion volume, a set of arbitrary reachable postures, of the frame end. The external fixator must have a sufficient ROM to accommodate the correction requirement of different indications; otherwise, the correction plan will not be carried out. The ROM can be described in detail as: giving a searching range in the three-dimensional space, and then detecting whether each posture (corresponding to a set of points) of the frame end is within the searching range. The points that meet the constraints are recorded; otherwise, they will be discarded. Finally, when all the points that meet the conditions are obtained, the reachable area of the frame end can be plotted, which is represented as a closed motion volume. The diameter of the proximal ring and the distal ring are both 220 mm and the distance of \(A_{i}A_{j}\) (\(ij\) = 16, 23, 45) is 32 mm. The constraint condition is the ROM of six telescopic rods; specifically, the minimum and maximum allowable strokes are set to 140 mm and 210 mm, respectively. With the assistance of MATLAB (R2011a, The Math Works, Inc., Novi, MI, USA), the numerical searching method is adopted to simulate the ROM of the mechanism on basis of the inverse position solution. The simulation results of the position ROM and orientation ROM are shown in Figures 5 and 6. Figure 5: The position range of motion (ROM) of the external fixator. (**a**) The position ROM of the fixator when the orientation is (0,0,0). (**b**) The position ROM of the fixator when the orientation is (0,0,20). (**c**) The position ROM of the fixator when the orientation is (0,13,0). (**d**) The position ROM of the fixator when the orientation is (13,0,0). Four different points are set up in this part to view the ROM in the different orientations of the external fixator. The comparison results show that when the orientation is (0,0,0), the fixator has the largest position ROM, as shown in Figure 5a. In this particular configuration, it has a ROM with \(-92\) to \(+92\) mm of translation along the \(x\)-axis, \(-82\) to \(+82\) mm of translation along the \(y\)-axis, and \(+122\) to \(+196\) mm of translation along the \(z\)-axis. It has also been found that the value of \(\theta_{2}\) has a greater impact on the displacement along the \(y\)-axis and the value of \(\theta_{1}\) has more constraints on the displacement along the \(X\)-axis, and the value of \(\theta_{3}\) will cause the entire ROM to shrink, as shown in Figure 5b-d.  
To investigate the orientation ROM of the external fixator, four different points are set up in different positions. The simulation result shows that when the position is (0,0,159), the middle position among the reachable position along the z-axis of the external fixator, the fixator has the largest position ROM, as shown in Figure 6a. In this particular configuration, it has a ROM with \(-18^{\circ}\) to \(+18^{\circ}\) of rotation around the \(x\)-axis, \(-20^{\circ}\) to \(+20^{\circ}\) of rotation around the y-axis, and \(-34^{\circ}\) to \(+34^{\circ}\) of rotation around the z-axis. It has also been found that the value of \(Y_{G1}\) has a greater impact on the rotation around the y-axis and the value of \(X_{G1}\) has more constraints on the rotation around the \(x\)-axis, and the value of \(Z_{G1}\) will cause the entire ROM to shrink, as shown in Figure 6b-d. The correction capability of the orientation of the fixator in any position can be determined, represented intuitively in the form of the volume of space-reachable position, and the position operation capability in any orientation of the fixator can also be determined through this method. Meanwhile, this method can also be used to determine whether any given position and orientation are within the correction capability of the external fixator.