# The Energy Consumption and Robust Case Torque Control of a Rehabilitation Hip Exoskeleton
## 2 Theory
### The Linear Quadratic Regulator  
Trajectory tracking with predefined movements is commonly used in rehabilitation exoskeletons for the lower extremities because patients can be trained passively to follow desired reference motions [8; 10]. Therefore, an admittance controller based on the torque of a healthy hip joint \(\mathsf{\mathsf{\mathsf{\mathsf{T}}}}_{\rm HH}\) was used for the rehabilitation exoskeleton to train the patient's weak muscles. The controller was used with an SEA and LQR that were chosen for the reasons set out in Section 1. The SEA was used to generate the proper current for the desired torque \(\mathsf{\mathsf{\mathsf{\mathsf{T}}}}_{\rm reference}\) and applications. Thus, an LQR with a Luenberger observer was used, but a mathematical model expressing the rehabilitation hip exoskeleton was needed, as illustrated in [8] and depicted based on the literature [28; 36; 45].  
To use the LQR, the state space model of the exoskeleton was represented in the following form  
\[\dot{x} =Ax+Bu \tag{1}\] \[y =Cx+Du,\]  
where \(\dot{x}\) is a vector matrix representing the system states; \(A\), \(B\), \(C\) and \(D\) are the system's dynamic matrices; \(u\) is the system input signal; and \(y\) is the output signal. However, the states of the system are not always accessible for sensors; thus, an observer is commonly used. Many observers are used through control systems, but the one that can increase the control system's bandwidth and improve the disturbance response indirectly is the Luenberger observer [46], which estimates the states \(\dot{x}\) by using the inputs \(u\) and the outputs \(y\) of the system. Finally, it subtracts the output of the observer, \(\dot{y}\), from the output of the system, \(y\).  
Figure 1: The hip torque during STS activity [8; 44].  
Figure 2: The hip torque during walking activity [8; 44].  
On the LQR controller side, the optimal placement of the system poles was based on the weights of the \(Q\) and \(R\) matrices by minimising the quadratic cost function \(J\) and finding the minimum steady-state error and the best feedback gain [47].  
\[J=E\left\{\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\Bigl{[}x^{T}Qx+u^{T}Ru\Bigr{]} dt\right\}, \tag{2}\]  
where \(Q\) and \(R\) are the semi-definite and positively definite design matrices, respectively. By tuning \(Q\) and \(R\), the placement of the poles was changed, whereas the feedback gain, \(K\), was determined from the Hamilton-Jacobi-Bellman method and solved by using the Riccati equation [8; 48] that ends with  
\[u=-Kx. \tag{3}\]  
The feedback gain, \(K\), increased or decreased depending on the values selected for \(Q\) and \(R\). A larger weight value in \(R\) set a higher penalty on the input signal, \(u\), which decreased the feedback gain, \(K\) and slowed the steady state and the energy needed. Conversely, the large weight values of \(Q\) meant a higher penalty on the state of the system, \(x\), when finding the minimum cost function, \(J\), according to (2), which increased the value of the feedback gain, \(K\), in (3) and accelerated the system's response accordingly.  
The system states, \(\hat{x}\), were necessary for the LQR algorithm, which can be inaccessible in many real systems or expensive to measure with certain sensors; thus, an LO was a suitable solution for deriving the system states [8].  
The system states, \(\hat{x}\), can be estimated by putting the input, \(u\), and the system output, \(y\), into LO. The procedures for obtaining the states and the LO diagram are discussed in more detail elsewhere [8]. Although the states are critical to finding the mathematical model of the system, which needs to be optimally controlled using the LQR, the model of the system will nevertheless include errors and imperfect model representation is more common compared to real systems. Therefore, the system robustness for the various \(Q\) and \(R\) is analysed and the energy consumption was studied regarding the sensitivity of the system to the model uncertainty.