## 3 Automatic Control of Biped Robot with Linear Actuators
### Control Realization for Biped Robot Using Linear Ball Sereudrivers as Actuation Devices  
The dynamic model of a single leg presented in (2) can be represented (with \(\zeta_{\alpha}:=q\) and \(\zeta_{\beta}:=\dfrac{d}{dt}q\)) as follows [17; 18]:  
\[\dfrac{d}{dt}\zeta_{\alpha}(t) = \zeta_{\beta}(t)\] \[\dfrac{d}{dt}\zeta_{\beta}(t) = f(\zeta(t),t)+g(\zeta_{\alpha}(t))\tau(t)+\Upsilon(\zeta(t),t)\] \[y(t) = \zeta_{\alpha}(t)\]  
with \(\zeta^{\top}=[\zeta_{\alpha}^{\top},\zeta_{\beta}^{\top}]\), where \(\zeta\in X\subset\mathbb{R}^{2n}\), \(X\) is an open subspace of the state space, \(\zeta_{\alpha}\in\mathbb{R}^{n}\) is a vector formed with the angles describing the angular motions related to each joint in the BRD. The vector \(\zeta_{\beta}\in\mathbb{R}^{n}\) represents the derivative of the angular motions for each joint with respect to time. The variable \(y\) represents the available output information that can be measured from the biped robot. The vector filed \(f:X\times\mathbb{R}^{+}\to\mathbb{R}^{n}\) defines the internal dynamics of the BRD. The structure of this function can be obtained using the Euler-Lagrange method. This study does not study the interactions between contacting surface and the BRD. The mathematical properties of the field \(f\) guarantee the local Lipschitz condition given here:  
\[\begin{array}{c}\|f\big{(}\zeta^{1}\big{)}-f\big{(}\zeta^{2}\big{)}\|\leq L _{f}\|\zeta^{1}-\zeta^{2}\|,\\ \zeta^{1}\in X,\zeta^{2}\in X,L_{f}\in\mathbb{R}^{+}\end{array} \tag{44}\]  
The matrix \(g:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}\) defines how the input action modifies the dynamics of the BRD. The structure of this matrix corresponds to the inverse of the inertia matrix for the BRD, which is positive definite. This matrix satisfies the following inequality (45) uniformly on \(t\geq 0\) (using the Frobenius matrix norm):  
\[\begin{array}{c}0<g^{-}\leq\|g(\zeta_{\alpha})\|_{F}\leq g^{+}<+\infty\\ g^{-}\in\mathbb{R}^{+},g^{+}\in\mathbb{R}^{+}\end{array} \tag{45}\]  
The term \(\Upsilon:X\times\mathbb{R}^{+}\to\mathbb{R}^{n}\) represents the uncertainties/perturbations, and it belongs to the following set:  
\[\Xi=\left\{\Upsilon|\sup_{t\geq 0,\ \zeta_{\in\mathbb{R}^{n}}}\|\Upsilon\|^{2} \leq\Upsilon_{0}+\Upsilon_{1}\|\zeta\|^{2}\right\} \tag{46}\]  
Because of the nature of the mechanical structure, it must satisfy the angular displacement restrictions for all the components of the state vector:  
\[-\infty<\zeta_{\alpha,i}^{-}\leq\zeta_{\alpha,i}\leq\zeta_{\alpha,i}^{+}<+ \infty,\ \ \zeta_{\alpha,i}^{-},\zeta_{\alpha,i}^{+}\in\mathbb{R} \tag{47}\]This study assumes that an upper limit exists \(\zeta^{+}\), given as follows:  
\[\frac{(\max\limits_{i}\zeta_{i}^{+})n}{\lambda_{min}\{P\}}=\zeta^{+} \tag{48}\]  
where \(\zeta_{i}\) characterizes the motion restriction for the \(i^{\text{th}}\) angular displacement and \(P\in\mathbb{R}^{n\times n}\) is a positive definite weighting matrix. Such a matrix may be used to impose some specific articulation restrictions. Therefore, \(\zeta^{+}\) provides the bounds for the BRD states according to the following relationship:  
\[\|\zeta\|_{P}^{2}\leq\zeta^{+} \tag{49}\]  
The term Y corresponds to \(\text{Y}=M^{-1}\Xi\). A biped robot actuated with linear ball screwdrivers gains postural force. Usually, an individual driver is placed at each articulation forming the configuration observed in Figure 1.