# Design of a Lower Limb Exoskeleton: Robust Control, Simulation and Experimental Results
## 2 Motion Analysis for Acl Injury Rehabilitation
### Development of Trajectories of Motion  
The desired trajectories \(q_{d}\) or trajectories of motion were based on ACL injuries, so the trajectories proposed are for a straight leg extension routine in a sitting position, which is a combination of flexion and extension of the knee and ankle joints. The patient is in a sitting position during the rehabilitation process and undergoes a series of movements to improve the flexibility and strength of the lower limbs. To begin, the patient extends their knee joint and holds that position. Then, the patient stretches their ankle joint before returning it to its original position. The patient then moves their ankle joint into flexion before returning the knee joint to its original position, as shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**ROM of the Human**} & \multicolumn{2}{c}{**ROM of the Robot**} \\ \hline
**Joint** & **Minimum Angle** & **Maximum Angle** & **Minimum Angle** & **Maximum Angle** \\ \hline Knee & 0\({}^{\circ}\) & 134\({}^{\circ}\) & 0\({}^{\circ}\) & 110\({}^{\circ}\) \\ Ankle & 0\({}^{\circ}\) & 75.6\({}^{\circ}\) & 0\({}^{\circ}\) & 65\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Range of motion for flexion and extension of the knee and ankle joints.  
Figure 3: Straight leg extension routine in a sitting position; proposed routine for ACL injuries.  
Figure 2: Biomechanics in the sagittal plane of the lower limb; (**a**) Flexion and extension movements for the knee joint; (**b**) Flexion and extension movements for the ankle joint.  
The desired trajectory of motion was captured using the OptiTrack motion capture system at 100 FPS in a healthy person who performed the routine. This technique uses a set of markers that are fixed to the item to track the motions of the object in its six degrees of freedom (distances and turns in X, Y, and Z). The system is based on a number of high-speed cameras that are calibrated and coordinated with one another and capture the motions of markers. The markers we utilized for this project were positioned on the knee and ankle joints, as well as the toe of the foot. Then, with the graphs of the position of the points of interest, we used the inverse kinematics of the modelled system in Figure 8 to obtain the trajectory of the angle position of the joints, knee \(q_{d1}\) and ankle \(q_{d2}\). Finally, we used a mathematical process to obtain the approximation of the trajectories.  
Figure 4 shows the motion analysis laboratory where the motion trajectories were obtained.  
The trajectory for the knee joint is  
\[q_{d1}(t)\ =\ 1.55[a_{0}+a_{1}cos(tw)+b_{1}sin(tw)+a_{2}cos(2 tw)+b_{ 2}sin(2 tw)\] \[+a_{3}cos(3 tw)+b_{3}sin(3 tw)+a_{4}cos(4 tw)+b_{4}sin(4 tw)\] \[+a_{5}cos(5 tw)+b_{5}sin(5 tw)+a_{6}cos(6 tw)+b_{6}sin(6 tw)\] \[+a_{7}cos(7 tw)+b_{7}sin(7 tw)+a_{8}cos(8 tw)+b_{8}sin(8 tw)]-100,\]  
where \(a_{0}=37.42,a_{1}=-30.68,b_{1}=-8.15,a_{2}=-3.085,b_{2}=-3.583,a_{3}=3.041\), \(b_{3}=2.28,a_{4}=0.01574,b_{4}=1.125,a_{5}=-0.7442,b_{5}=-0.1605,a_{6}=-0.4798\), \(b_{6}=0.9625,a_{7}=0.2446,b_{7}=-0.2231,a_{8}=0.4532,b_{8}=-0.09683,w=0.35\), and for the ankle joint, it is  
\[q_{d2}(t)=-0.8[a_{00}+a_{01}cos(tw_{0})+b_{01}sin(tw_{0})+a_{02} cos(2 tw_{0})+b_{02}sin(2 tw_{0})\] \[+a_{03}cos(3 tw_{0})+b_{03}sin(3 tw_{0})+a_{04}cos(4 tw_{0})+b_{04} sin(4 tw_{0})\] \[+a_{05}cos(5 tw_{0})+b_{05}sin(5 tw_{0})+a_{06}cos(6 tw_{0})+b_{06} sin(6 tw_{0})\] \[+a_{07}cos(7 tw_{0})+b_{07}sin(7 tw_{0})+a_{08}cos(8 tw_{0})+b_{08} sin(8 tw_{0})]+54,\]  
Figure 4: Motion capture system to obtain the desired trajectory or adequate rehabilitation routine, through a vision system with 12 cameras.  
where \(a_{00}=70.89,a_{01}=-0.449,b_{01}=-4.386,a_{02}=-1.523,b_{02}=4.284,a_{03}=1.537,\)\(b_{03}=-7.771,a_{04}=-2.363,b_{04}=4.677,a_{05}=-0.0724,b_{05}=-2.737,\)\(a_{06}=-0.0426,b_{06}=1.991,a_{07}=-0.2973,b_{07}=-0.42,a_{08}=0.1964,b_{08}=0.1644,\)\(w_{0}=0.29\).  
The real and approximated trajectories are shown in Figure 5.