## 2 Model-Based Robust Control Design
### Mathematical Model Using Bond Graph  
A mathematical model of the three-link exoskeleton manipulator, in the form of a transfer function and state space equations, was obtained using the bond graph modeling technique as given in (3) and (4). A schematic diagram of the exoskeleton and the bond graph model of the system generated using the 20-sim software are presented in Figures 2 and 3, respectively. For the control of link_1, associated with the hip joint, a composite model is required because, during the locomotion assistance and gait training, the motor at the hip joint (mhi), including the inertial mass, stiffness, and viscous friction associated with link_1, experiences the effect of the remaining structure, i.e., two motors (the motor at the knee joint (mkj) and the motor at the ankle joint (maj), two links, and the human leg except the thigh, as a mass-spring-damper load [13; 46]. Similarly, the actuator at the knee joint is influenced by the third motor and third link in the form of a mass-spring-damper load. Hence, for the velocity control of the first link, a composite model can be realized with the combined masses of the remaining two actuators (motor_2 and motor_3), links (link_2 and link_3), and the mass of the remaining human leg, considered at link_1 (as \(I:Mass\_lift\_lift\_dot\)). Similarly, combined damping, stiffness, and gravitational force associated with link_2 and link_3 are considered at link_1 (as \(R:Fv\_lift\_lift\_to\dot{\mathit{t}}\mathit{of}\mathit{of}\mathit{of}\mathit{of} \mathit{of}\mathit{of}\mathit{d}\), \(C:Ks\_ffif\_to\dot{\mathit{t}}\mathit{of}\mathit{of}\mathit{of}\mathit{d}\), and \(Se:g\_Mass\_lift\_lift\_dot\)), respectively, as shown in Figure 4. A composite model obtained as the mathematical model to control the position of link_1 is presented in Figure 4. The description of variables of the bond graph [13] of the composite model, and the transfer function of the composite model defining a relationship between the applied input voltage (\(E\)) and the link velocity (\(V\)) of link_1 with the combined effect at the output, are presented in Table 1 and Equation (3), respectively. \[G_{P}(s)=\frac{V(s)}{E(s)}=\frac{5e^{5}s+1250}{s^{5}+100s^{4}+1.461e^{4}s^{3}+2.1e^{5}s^{2}+1.256e^{6}s+3125} \tag{3}\]  
Figure 1: Research flow. Figure 4: Composite model–bond graph using 20-sim. Figure 3: Bond graph model of lower limb rehabilitation exoskeleton using 20-sim. Figure 2: Schematic diagram of lower limb rehabilitation exoskeleton. **Remark 1**. : _The \(H_{\infty}\) controller ensures._  
* _Internal stability of the closed loop system_
* \(\|T_{2\omega}(s)\|<\gamma\) _where,_ \(\gamma\) _is a specified number_ \(<0\)__  
**Remark 2**. : _The compulsory assumptions for the controller design are checked. These include:_  
* \((A,\ B_{2})\) _is stabilizable and_ \((A,\ C_{2})\) _is detectable_
* _Rank of matrix_ \(\left[ \begin{array}{cc}A-\omega jI&B_{2}\\ C_{1}&D_{12}\end{array} \right]\) _should be full column rank and rank of matrix_ \(\left[ \begin{array}{cc}A-\omega jI&B_{1}\\ C_{2}&D_{21}\end{array} \right]\) _should be full row rank_
* _For proper and realizable controller: Rank of_ \(D_{12}\) _is full column rank and rank of_ \(D_{21}\) _is full row rank._  
**Remark 3**. : _The algorithm steps for the controller design followed are:_  
* _The system is realized in the form of state space and the status of assumptions are checked to be satisfactory._
* _A large positive value for_ \(\gamma\) _is selected and the controller_ \(K(s)\) _is found after solving the Algebraic Riccati Equations (AREs) until a satisfactory solution is obtained, i.e., subject to the fulfilment of the following condition with lowering_ \(\gamma\)_:_ \[\rho(X_{\infty},\ Y_{\infty})<\gamma^{2}\] _where_ \(X_{\infty}\) _and_ \(Y_{\infty}\) _are the solutions of the AREs._