## 3 Automatic Control of Biped Robot with Linear Actuators
### Output Feedback Control of Biped Robot  
This section details all the elements in the automatic control design, including the implementation of the output feedback form, the implementation strategy as well as a brief sketch of the stability proof for the close loop dynamics. The aim of this study is to develop an output feedback control \(u=u(t)\) for a biped robot such that  
\[\|\zeta_{a}(t)-\zeta_{a}^{*}(t)\|\leq\beta_{0}\quad,\quad\forall t\geq T>0 \tag{50}\]  
Here, \(T\) is a given time, and the positive scalar \(\beta_{0}\) defines the quality of the tracking trajectory for the reference trajectory \(\zeta_{a}^{*}\), which satisfies the following ordinary differential equation:  
\[ \begin{split}\frac{d}{dt}\zeta_{a}^{*}(t)& =\zeta_{\beta}^{*}(t)\\ \frac{d}{dt}\zeta_{\beta}^{*}(t)&=\!h\!\left(\zeta_ {a}^{*}(t),\zeta_{\beta}^{*}(t),t\right)\end{split}  \tag{51}\]  
Here the vector field \(h:X\times X\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{n}\) is a locally Lipschitz function, which characterizes the reference trajectories. The reference trajectories are proposed to satisfy the same bounds that are admitted for the angular displacements of each joint, that is  
\[-\infty<\zeta_{a,i}^{-}\leq\zeta_{a,i}^{*}\leq\zeta_{a,i}^{+}<+\infty,\ \ \zeta_{a,i}^{-},\zeta_{a,i}^{+}\in\mathbb{R} \tag{52}\]  
Defining the trajectory tracking error \(\delta(t)\in\mathbb{R}^{2n}\), which is given by \(\delta=\zeta-\zeta^{*}\), where \(\zeta^{*}=[(\zeta_{a}^{*})^{\top},(\zeta_{\beta}^{*})^{\top}]^{\top}\). Hence, the dynamics of \(\delta(t)\) are given by:  
\[ \begin{split}\frac{d}{dt}\delta(t)&=\!A\delta(t)+B (f(\zeta(t),t)+g(\zeta_{a}(t))\tau(t))\\ +& B\!\left(\text{Y}(\zeta(t),t)-h\!\left(\zeta_{a}^ {*}(t),\zeta_{\beta}^{*}(t),t\right)\right)\\ A&=\left[\begin{array}{cc}0_{n}&I_{n}\\ 0_{n}&0_{n}\end{array}\right]\ B=\left[\begin{array}{c}0_{n}\\ I_{n}\end{array}\right]\end{split}  \tag{53}\]  
The matrices \(A\) and \(B\) satisfy a companion controllable form with the correct dimensions. The controller design considered in this study assumes that the control form satisfies the following linear form with a state-dependent gain  
\[\tau(t)=g^{-1}(\zeta_{a}(t))\left(K(t)\delta(t)-h\!\left(\zeta_{a}^{*}(t), \zeta_{\beta}^{*}(t),t\right)-f(\zeta(t),t)\right) \tag{54}\]  
Notice that this formulation appears to be a class of adaptive state-dependent feedback for the class of biped robot. In the controller form, \(\delta\) is the tracking error using the estimates of \(\zeta_{\xi}\). The gain of the suggested controller is defined as \(K(t)=[K_{P}(t)\;\;K_{D}(t)]\) with \(K_{P}\in\mathbb{R}^{n\times n}\) and \(K_{D}\in\mathbb{R}^{n\times n}\). The linear control form has well-known efficient tracking characteristics, and its implementation requires measuring \(\dfrac{d}{dt}y(t)\). The possibility of obtaining the estimation of such derivatives online usually implies important technical resources such as additional sensors. A feasible alternative is instrumenting a class of RED, which can yield an accurate estimation of the time derivative of \(\dfrac{d}{dt}y(t)\)[19]. This study uses the significant benefits of the STA method, which has proven to be one of the more efficient alternatives to obtain an accurate estimation of the time derivative of the output information, while it can be used as part of output feedback control forms [20]. For a scalar variable, the STA operating as a RED can be detailed as follows. Consider that \(w_{1}(t)=r(t)\), with \(r(t)\in\mathbb{R}\) being the signal that must be differentiated, and \(w_{2}(t)=\dfrac{d}{dt}r(t)\) describes its time-derivative. Assuming that \(\left|\dfrac{d^{2}}{dt^{2}}r(t)\right|\leq r^{+}\), it is feasible to propose the following auxiliary dynamic representation  
\[\dfrac{d}{dt}w_{1}(t)=w_{2}(t),\;\;\;\;\;\dfrac{d}{dt}w_{2}(t)=\dfrac{d^{2}}{ dt^{2}}r(t) \tag{55}\]  
The set of differential equations (55) consists of the state-space description for \(r(t)\).