# Dynamic Parameter Identification of a Lower Extremity Exoskeleton Using RLS-PSO
## 2 Methods
### Parametric Dynamics Equation of Exoskeleton  
The RLS-PSO was verified on the active power-assist exoskeleton (APAL) [36] as shown in Figure 1a, which consists of trunk, thigh, shank, and foot modules. As shown in Figure 1b, the trunk and left lower extremity were fixed on the pedestal during the experiment. The right lower extremity was suspended. The hip and knee joints can only swing in the sagittal plane. The forefoot of the exoskeleton was attached to the shank by the fastening band in Figure 1b. The ankle has zero degrees of freedom. The hip and knee joints are driven by hydraulic cylinders, and the exoskeleton is driven only by torque and gravity. The hydraulic actuator of hip joint A is fixed in the trunk and will not affect the dynamic parameters which are related to the thigh and hip joint. Hydraulic cylinder CD drives knee joint B and is fixed on AB. Through the four-linkage mechanism, ABCD, the linear motion of the CD is converted to the rotation of B. The stretching and swinging of the CD cause changes in the position of lower extremity's center of mass (COM) [36]. Therefore, the dynamic parameters of the thigh and shank fluctuate. The fluctuation was used for defining the search space of PSO. In the sagittal plane, the right leg of the APAL exoskeleton was simplified into a two-linkage model [37], as shown in Figure 2.  
Equation (1) is the dynamic equation of APAL's lower extremity. Equation (1) was derived by the Lagrange method. \[ \begin{array}{l}T_{h}=[I_{s}+m_{s}((L_{s}-L_{Gs})^{2}-h_{Gs}^{2})+I_{t}+m_{t} ((L_{t}-L_{Gt})^{2}-h_{Gt}^{2})+m_{s}L_{t}^{2}+2L_{t}(m_{s}(L_{s}-L_{Gs})\cos q _{2}-\\ m_{s}h_{Gs}\sin q_{2})]\tilde{q}_{1}+[I_{s}+m_{s}((L_{s}-L_{Gs})^{2}-h_{Gs}^{2} )+L_{t}(m_{s}(L_{s}-L_{Gs})\cos q_{2}-m_{s}h_{Gs}\sin q_{2})]\tilde{q}_{2}+\\ L_{t}(m_{s}(L_{s}-L_{Gs})\sin q_{2}+m_{s}h_{Gs}\cos q_{2})(\tilde{q}_{1}^{2}-( \hat{q}_{1}+\hat{q}_{2})^{2})+g[(m_{t}(L_{t}-L_{Gt})+m_{s}L_{t})\sin q_{1}+\\ m_{t}h_{Gt}\cos q_{1}+m_{s}(L_{s}-L_{Gs})\sin(\hat{q}_{1}+q_{2})+m_{s}h_{Gs} \cos(q_{1}+q_{2})]\\ T_{k}=[I_{k}+m_{s}((L_{s}-L_{Gs})^{2}-h_{Gs}^{2})+L_{t}(m_{s}(L_{s}-L_{Gs}) \cos q_{2}-m_{s}h_{Gs}\sin q_{2})]\tilde{q}_{1}+[I_{s}+m_{s}((L_{s}-\\ L_{Gs})^{2}-h_{Gs}^{2})]\tilde{q}_{2}+L_{t}(m_{s}(L_{s}-L_{Gs})\sin q_{2}+m_{s}h _{Gs}\cos q_{2})\tilde{q}_{1}^{2}+g[m_{s}(L_{s}-L_{Gs})\sin(q_{1}+q_{2})+\\ m_{s}h_{Gs}\cos(q_{1}+q_{2})]\end{array}  \tag{1}\]  
Figure 1: Active power-assist exoskeleton (APAL): (**a**) APAL; (**b**) Schematic diagram of APAL lower extremity mechanical structure. where \(\dot{q}_{1}\) and \(\dot{q}_{2}\) represent the angular velocity of hip and knee joint, respectively, and \(\ddot{q}_{1}\) and \(\ddot{q}_{2}\) represent the angular acceleration of hip and knee joint, respectively. \(T_{h}\) and \(T_{k}\) represent the hip and knee drive torque required by APAL to complete the trajectory of \(q_{1}\) and \(q_{2}\), respectively. When the lower extremity of APAL is vertically downward, \(q_{1}\) and \(q_{2}\) are zero. The counterclockwise rotation is positive. Considering the friction on the hip and knee joint, friction torques \(T_{f1}\) and \(T_{f2}\) were introduced. Equation (2) is the real joint torque \(T_{1}\) and \(T_{2}\). \[ \begin{array}{l}T_{1}=T_{h}+T_{f1}\\ T_{2}=T_{k}+T_{f2}\end{array}  \tag{2}\]  
where  
\[ \begin{array}{l}T_{f1}=f_{c1}\text{sgn}(\dot{q}_{1})+f_{c1}\dot{q}_{1}\\ T_{f2}=f_{c2}\text{sgn}(\dot{q}_{2})+f_{c2}\dot{q}_{2}\end{array} \]  
\(f_{c1}\) and \(f_{c1}\) represent the friction parameters of the hip joint, \(f_{c2}\) and \(f_{c2}\) represent the friction parameter of the knee joint. Since there are couplings between the physical parameters \(I_{t}\), \(I_{s}\), \(m_{t}\), \(m_{s}\), \(L_{t}\), \(L_{s}\), \(L_{Gt}\), \(L_{Gs}\), \(h_{Gt}\), \(h_{Gs}\) in Equation (2), Equation (2) is nonlinear to the above physical parameters. In order to meet the linearization requirements of RLS [38; 39], Equation (2) was parameterized to Equation (3).