## 3 Dynamics of the Knee Exoskeleton and Problem Formulation
### Dynamic Model of the Knee-Joint Exoskeleton  
Using a pendulum model to represent the exoskeleton robot, we can derive its dynamic model using the Lagrangian mechanical system, resulting in the following equations of motion:  
\[\frac{d}{dt}\bigg{(}\frac{\partial\mathcal{L}}{\partial\dot{ \theta}}\bigg{)}-\frac{\partial\mathcal{L}}{\partial\theta}=\mathcal{T} \tag{2}\]  
In this context, \(T\), representing the total torques acting on the robot, is defined by the following equation:  
\[\mathcal{T}=-\mathcal{T}_{\mathrm{s}}-\mathcal{T}_{\mathrm{p}}- \delta_{t}+u \tag{3}\]  
where \(\mathcal{T}_{\mathrm{p}}=f_{\mathrm{s}}\dot{\theta}\) represents the torque due to viscous friction, while \(\mathcal{T}_{\mathrm{s}}=f_{\mathrm{s}}\mathrm{sgn}(\dot{\theta})\) accounts for solid friction. Additionally, \(\delta_{t}\) denotes the load torque, and \(u\) signifies the control torque applied to the system. In Equation (2), \(\mathcal{L}\) refers to the Lagrangian function, described as follows:  
\[\mathcal{L}=-\mathcal{G}_{e}+\mathcal{K}_{e} \tag{4}\]  
where \(\mathcal{G}_{e}\) denotes the gravitational potential energy, and \(\mathcal{K}_{e}\) is the kinetic energy. By inserting Equations (3) and (4) into the Lagrangian Equation (2), we derive the dynamics of the knee exoskeleton depicted in Figure 1, considering uncertainties including external disturbances, unmodeled dynamics, and parameter variations:  
\[\
u\ddot{\theta}=-f_{\mathrm{v}}\dot{\theta}+u+a\cos(\theta)-f_{ \mathrm{s}}\mathrm{sgn}(\dot{\theta})-\delta_{t} \tag{5}\]  
In the context of the knee exoskeleton, the symbol \(\
u\) represents the inertia of the system, the viscous friction coefficient is denoted by \(f_{\mathrm{v}}\), and \(f_{\mathrm{s}}\) corresponds to the solid  
Figure 1: Visual depiction of the knee exoskeleton’s geometry [30], adopted from [51]. friction coefficient. The scalar \(a=mgl\), where \(m\) is the mass of the system, \(g\) denotes the acceleration due to gravity, and \(l\) represents a characteristic length. In the model (5), the applied torque is denoted by \(u\). Furthermore, \(\delta_{t}\) incorporates the effects of external forces as well as any uncertainties that may arise. Indeed, it is crucial to acknowledge that precisely determining parameters for a dynamic model is often unattainable in advance, leading to system uncertainty arising from parameter fluctuations and unmodeled dynamics. Consequently, the dynamics are partitioned into a nominal component and an uncertain component, with the system's parameters expressed as follows:  
\[a=a_{\star}+\Delta a,\qquad\
u=\
u_{\star}+\Delta\
u,\qquad f_{v}=f_{v}^{ \star}+\Delta f_{v},\qquad\text{ and }\qquad f_{s}=f_{s}^{\star}+\Delta f_{s} \tag{6}\]  
By substituting the previously defined expressions from (6) into Equation (5), the subsequent result is derived:  
\[\
u_{\star}\ddot{\theta}=-f_{v}^{\star}\dot{\theta}+u+\Delta_{t}-f_{s}^{\star} \text{sgn}(\dot{\theta})+a_{\star}\cos(\theta) \tag{7}\]  
The equation representing the lumped or total disturbance, denoted as \(\Delta_{t}\), is as follows:  
\[\Delta_{t}=-\Delta f\ddot{\theta}-\Delta f_{v}\dot{\theta}+\Delta a\cos(\theta )-\Delta f_{s}\text{sgn}(\dot{\theta})-\delta(t) \tag{8}\]  
Here, we chose to represent the uncertainties through a single lumped term, \(\Delta_{t}\), to simplify the analysis and facilitate the design of the control framework. This approach ensures that the aggregated effects of all uncertainties are bounded by a uniform term, \(|\Delta_{t}|\leq\bar{\delta}\), which is particularly advantageous when applying robust control methods such as the LMI framework. Similar strategies have been used effectively in rehabilitation robotics, as seen in [56], where lumped uncertainties simplified robust control design, and in [57], where a high-order observer efficiently handled fast time-varying disturbances. Additionally, [58] demonstrated the benefits of addressing bounded uncertainties through lumped modeling in exoskeleton systems.