## 3 Methods  
### Human-Exoskeleton Coupling Dynamics Modeling  
The exoskeleton robot has two unique characteristics, namely physical and cognitive interaction between human limbs and the exoskeleton. The physical interaction is a force interaction, and the exoskeleton can enhance the strength of human limbs. The cognitive interaction means that the wearer can feel that he is controlling the exoskeleton. At present, in the dynamic analysis of exoskeleton system, researchers usually only pay attention to the dynamic characteristics of the exoskeleton system itself, or take the exoskeleton system and human limb as a rigid connection for dynamic analysis, and ignore the physical interaction between the exoskeleton and the human limb. It is of great significance in the study of adaptive sliding mode variable admittance control of exoskeleton robot based on human-exoskeleton physical interaction to establish a coupling dynamic model of exoskeleton and human limb. In this work, the 2-DOF connecting-rod model is adopted in coupling dynamics analysis of the exoskeleton and human limb. The 2-DOF connecting-rod model contains a trunk, thigh, and shank with the trunk locked, so the swing process is essentially the rotation of the thigh around the hip joint and the shank around the knee joint. Because the human leg is attached to the exoskeleton support, the muscle and fat in the human leg will produce soft coupling between the exoskeleton and the skeleton, which allows relative movement of the human leg skeleton and the exoskeleton. As shown in Figure 4a, this soft coupling can be modeled using linear spring with coefficient \(b_{i}\) and linear damper with coefficient \(k_{i}\). The simplified model is shown in Figure 4b. Figure 4: (**a**) The coupling model of human lower limb and exoskeleton. (**b**) The simplified model of the human lower limb (yellow) and the exoskeleton (black). Figure 3: Real-time control system of the exoskeleton. According to the Euler-Lagrange equations, the dynamic model equation of a human leg can be described by:  
\[\mathbf{M_{h}}(\mathbf{q_{h}})\ddot{\mathbf{q_{h}}}+\mathbf{C_{h}}(\mathbf{q_{h} },\dot{\mathbf{q_{h}}})\dot{\mathbf{q_{h}}}+\mathbf{G_{h}}(\mathbf{q_{h}})= \mathbf{T_{h}}, \tag{1}\]  
where \(\mathbf{q_{h}}=\left[q_{hh},q_{hk}\right]^{T}\) denotes the state vector of human joint angle, \(\dot{\mathbf{q_{h}}}=\left[\dot{q}_{hh},\dot{q}_{hk}\right]^{T}\) denotes the state vector of human joint angular velocity, \(\ddot{\mathbf{q_{h}}}=\left[\ddot{q}_{hh},\ddot{q}_{hk}\right]^{T}\) denotes the state vector of human joint angular acceleration. \(\mathbf{T_{h}}=\left[T_{hh},T_{hk}\right]^{T}\) is the state vector of torque generated by human hip joint and knee joint. \(\mathbf{M_{h}}\in\mathbf{R}^{2\times 2}\) is the inertia matrix of single human lower limb, which is a symmetric positive definite matrix. \(\mathbf{C_{h}}\in\mathbf{R}^{2}\) is the coriolis force vector. \(\mathbf{G_{h}}\in\mathbf{R}^{2}\) denotes the gravitational torque vector. Similarly, the dynamic model equation of a 2-DOF exoskeleton is given as:  
\[\mathbf{M_{e}}(\mathbf{q_{e}})\ddot{\mathbf{q_{e}}}+\mathbf{C_{e}}(\mathbf{q_ {e^{\prime}}},\dot{\mathbf{q_{e}}})\dot{\mathbf{q_{e}}}+\mathbf{G_{e}}( \mathbf{q_{e}})=\mathbf{T_{e}}, \tag{2}\]  
where \(\mathbf{q_{e}}=\left[q_{eh},q_{ek}\right]^{T}\) denotes the state vector of 2-DOF exoskeleton joint angle, \(\dot{\mathbf{q_{e}}}=\left[\dot{q}_{eh},\dot{q}_{ek}\right]^{T}\) denotes the state vector of 2-DOF exoskeleton joint angular velocity, \(\ddot{\mathbf{q_{e}}}=\left[\ddot{q}_{eh},\ddot{q}_{ek}\right]^{T}\) denotes the state vector of 2-DOF exoskeleton joint angular acceleration. \(\mathbf{T_{e}}=\left[T_{eh},T_{ek}\right]^{T}\) is the vector of torque generated by the exoskeleton. \(\mathbf{M_{e}}\in\mathbf{R}^{2\times 2}\) is the inertia matrix of exoskeleton, which is a symmetric positive definite matrix. \(\mathbf{C_{e}}\in\mathbf{R}^{2}\) is the coriolis force vector of the exoskeleton.