# Design of Hybrid Phase Sliding Mode Control Scheme for Lower Extremity Exoskeleton
## 2 Methods  
### Dynamic Model of Lower Extremity Exoskeleton  
As shown in Figure 2, a numerical characterization method [26; 27] of the lower limbs is adopted to obtain the switching model in the sagittal plane. In this method, it is assumed the lower limbs of the human body usually perform the periodic motion, and the human legs are symmetrical. Therefore, the simplified model can be seen as a double link, which is towed by pneumatic muscles. In the swing phase, the oscillating leg rotates around the hip joint \(P_{0}(x_{0},y_{0})\); During the stance phase, the thigh and upper torso are approximated as the fixed axis motion rotating around the foot, the axis is between the foot and the ground \(P_{0}(x_{0},y_{0})\), and \(P_{16}\) represents the center of mass of the torso. The schematic diagram of the lower extremity exoskeleton is shown in Figure 2, and \(P_{11}P_{14}\), \(P_{1}P_{15}\), \(P_{10}P_{8}\) are corresponding to three pneumatic muscles. The meaning of the symbols in Figure 2 is shown as follows:  
* \(P_{4}\) : The knee joint
* \(P_{0}\) : The hip joint/The foothold point
* \(P_{5}\) : The end of the pole fixed on the calf
* \(P_{8}\) : The connection point between pneumatic muscle 3 and calf
* \(P_{1},P_{3}\) : The connection point between pneumatic muscle 1,2 and thigh
* \(P_{10},P_{11}\) : The connection point between pneumatic muscle 1,3 and waist
* \(m_{1},m_{2}\) : The quality of thigh and calf
* \(m_{6}\) : The quality of the human upper body
* \(m_{3},m_{4},m_{5}\) : The quality of pneumatic muscle 1-3
* \(r_{1}\) : The length between \(P_{1}\) and \(P_{0}\)
* \(r_{2}\) : The length between \(P_{3}\) and \(P_{0}\)
* \(r_{3}\) : The length between \(P_{4}\) and \(P_{6}\)  
Figure 2: Swing phase model (**Left**) and Stance phase model (**Right**). * \(r_{4}\) : The length between \(P_{4}\) and \(P_{8}\)
* \(l_{1},l_{2}\) : The length between \(P_{10},P_{11}\) and \(P_{0}\)
* \(q_{1},q_{2}\) : The angle of hip and knee  
Based on the mention above, the model of the single-leg during the swing and stance phase are respectively accomplished by using the Lagrange dynamics method, the general hybrid phase dynamics of the lower extremity exoskeleton system can be written as follows (See Appendix A for details):  
\[F_{i}\left(q\right)\ddot{q}+C_{i}\left(q,\dot{q}\right)\dot{q}+G_{i}\left(q \right)=\tau_{act}+\tau_{hm} \tag{1}\]  
where \(q\) is the angle vector of the joints, \(F(q)\)is a inertia matrix and a function of \(q\), \(C(q,\dot{q})\) a centripetal and Coriolis matrix, \(G(q)\) is a vector of gravitational torque. \(\tau_{act}\) is the actuator torque vector and \(\tau_{hm}\) may contain two part: \(\tau_{1}=\int^{T}F_{1}\) and \(\tau_{2}=\int^{T}F_{2}\). \(F_{1}\) is the interactive force of the human body acting on the exoskeleton and \(F_{2}\) is the contact force provided by the ground only in the stance phase. They can be transformed into control torque on the joints through the Jacobian matrix \(J\). \(\tau_{1},\tau_{2}\) can be seen as the extern interference for exoskeleton system. The value of subscript can be selected as 1 or 2, which represent the swing phase and the stance phase respectively. Equation (1) can be deduced to:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-b(C\dot{q}+G)+bu-bd\\ y=x_{1}\end{array} \right. \tag{2}\]  
where \(x_{1}=q=\left[ \begin{array}{cc}q_{1}&q_{2}\end{array} \right]^{\mathbf{T}}\), \(x_{2}=\dot{q}=\left[ \begin{array}{cc}q_{1}&q_{2}\end{array} \right]^{\mathbf{ T}}\), the output of exoskeleton system is denoted as \(y\in R^{2\times 1}\). \(q,\dot{q},\ddot{q}\in R^{2\times 1}\) denote the joint angle, angular velocity and angular acceleration of the lower extremity exoskeleton. \(b=F^{-1}\) is a nonlinear smoothing function and the system disturbances are denoted by \(d\). The input torque of the hip and knee are \(u=\tau\in R^{2\times 1}\). Hence, the system can be rewritten as multiple-input multiple-output (MIMO) system.