## 3 Results
### Trajectory Generating  
The parameters of the deformed bone and external fixator are designed as follows: the deformed bone has 8\({}^{\circ}\) and 13\({}^{\circ}\) angulation deformities around the \(x\) and \(y\)-axes, respectively, 8 mm and 5 mm translation deformities along the \(x\) and \(y\)-axes, respectively, 10\({}^{\circ}\) rotation deformity around the \(z\)-axis, and 20 mm shortened deformity along the \(z\)-axis in the frame \(O\)-\(xyz\). The diameter of both the proximal and distal ring is 160 mm. For the above parameters of deformed bone, the translation deformity along the \(z\)-axis is the largest (20 mm). Therefore, it is assumed that the correction time is 20 days to ensure that the optimal correction rate is 1 mm/day along the \(z\)-axis. Meanwhile, the parameters can also ensure that the correction rate in the other five directions does not exceed 1 mm/day. The patient needs to wear the external fixator all the time and needs to maintain the fixator's stability. When the new bone tissue is completely calcified, the external fixator can be removed [18]. In reference to [6], it is assumed that the growth interface of bone is a circle centered on the central point of the bone end and that the new bone tissue fills the volume swept by the growth interface. A discrete set of target positions are fitted by MATLAB (R2011a, The Math Works, Inc., Novi, MI, USA) to generate the correction trajectory. The resulting bone shape, the volume swept by the growth interface, is then generated by fitting a series of growth interfaces. Due to the small magnitude of deformity parameters, a circle with a diameter of 8 mm is adopted to indicate the growth interface to visually and clearly express the characteristics of the new bone shape. The correction trajectory and bone shape under the scheme of equal joint adjustment are generated through the numerical method, as shown in Figure 7. The generated trajectory is a smooth spatial curve, and the bone shape has a smooth curvature. The angulation deformities and translation deformities at the target position are completely corrected and the proximal and distal bone axes are aligned along the anatomical axis of the tibia (the axes axi1 and axi2 are aligned in the vertical direction). The advantage of this strategy is that it avoids the complicated process of calculating the correction prescription. There is a nonlinear mapping relationship between the joint and operating space of the frame, which causes the possibility that the adjustment value of the bone position may be too high or too small during the correction, and it also causes the orientation of the growth interface to change unevenly. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **The Maximum Correction Capability** & \multicolumn{2}{c}{**The Allowable**} \\  & & & **Strokes of the Rod** \\ \hline Translation along & \(-\)92\(\sim\)+92 mm & Rotation around & \(-\)18\({}^{\circ}\)\(\sim\)+18\({}^{\circ}\) & \\ the \(x\)-axis & & the \(x\)-axis & \\ Translation along & & Rotation around & 140\(\sim\)210 mm \\ the \(y\)-axis & & the \(y\)-axis & \(-\)20\({}^{\circ}\)\(\sim\)+20\({}^{\circ}\) & \\ Translation along & & Rotation around & \\ the \(z\)-axis & & the \(z\)-axis & \(-\)34\({}^{\circ}\)\(\sim\)+34\({}^{\circ}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The maximum correction capability in space-reachable position. The correction trajectory and bone shape of the given case are generated according to the strategy of joint adjustment for equal bone distraction, which can be found in Figure 8. The bone trajectory is a smooth and uniformly spaced linear trajectory and the shape of the new bone tissue is straight. The axis of distal bone is aligned with the anatomical axis at the desired position (the axes axi1 and axi2 are aligned in the vertical direction). The orientation of the bone end-plane changes uniformly, and there is no bone end interference between adjacent growth interfaces until the correction is completed.