## 2 Materials and Methods
### The Electromotor-Driven Parallel External Fixator System
#### 2.4.1 Structure Design of the Parallel External Fixator  
The developed parallel external fixator is a type of hexapod external fixator, which consists of two rings, six telescopic rods, six Hooke joints, and six spherical hinges. The telescopic rods are connected with the proximal ring and distal ring by the spherical hinges and Hooke joints. The position and orientation of the distal platform can be changed by regulating the rod length and the fixator can correct any three-dimensional malformation without modifying the frame configuration. Compared with the Ilizarov frame, it can be utilized to perform multiplane deformity simultaneous- ly with computer assistance.  
The telescopic rod is driven by lead screw transmission, and the DC (Direct Current) servo motor (Maxon DCX 12L) is connected to the lead screw through the gear reduction (\(i_{1}\) = 5:3). There is an incremental encoder at the bottom of the motor (Maxon ENX10 EASY, 1024 CPT), which can feed back the lengths of six rods to the microcontroller in real-time. The motor's weight is 50g and the total weight of six motors is 300 g. The motor will be considered to be installed in the outward direction to increase the internal space of the ring, thereby reducing the possibility of interference between the struts of the fixator and the limbs. The additional weights caused by the motor and the corresponding accessories required to install the motor are small, which has a very small impact on the patient's tolerance. Thus, it can still be effectively used clinically with the motor and control unit.  
The main parameters of the Maxon DCX 12L are as follows: the maximum rotational speed \(n=8810\) rpm, the rated torque \(M=3.88\) mN\(\cdot\)m, and the reduction ratio of the reduction gearbox (GPX12A) is \(i_{2}=62\):1. The lead screw is adopted with a single thread T thread (\(p=1\) mm), which has a good self-locking function. The minimum length between the center of the spherical joint and hook joint is 140 mm and the maximum length is 210 mm. To increase the flexibility of the structure and avoid joint abrasion during correction, rolling bearings are used at screw nuts, Hooke hinges, and spherical joints. The overall mechanical diagram of the external fixator is shown in Figure 3.  
The value of maximum output force \(F_{\text{max}}=2.52\) KN and maximum output velocity \(V_{\text{max}}=0.89\) mm/s are acquired. Thus, the motor can provide a larger force/m  
Figure 3: The mechanical diagram of the external fixator. The external fixator consists of (1) proximal ring, (2) distal ring, (3) Hooke joint, (4) spherical hinge, (5) telescopic rod, (6) DC servo motor, (7) incremental encoder, (8) Kirschner wire, (9) the holder of the telescopic rod, (10) rolling bearing mounted on the screw nut, (11) lead screw, (12) gear reduction, (13) mark number of rod, (14) motor bracket, (15) rolling bearing mounted on the spherical joint, (16) bracket of the spherical joint, (17) bush bearing, (18) end cap, (19) linkage shaft of the Hooke joint.  
to offset the larger soft tissue force required to pull the soft tissue during the deformity correction. The lower output speed allows for very slow and continuous movement, which is effective for the safety of patients. The maximum output force and velocity can be obtained by Equation (11).  
\[F_{\text{max}}\ =\ \frac{2\pi M\dot{r}\dot{\eta}}{P},\ \ \ V_{\text{max}}\ =\ \frac{\dot{n}}{\delta 0 \dot{n}}\ \cdot P \tag{11}\]