## 4 Aslb--A Bipedal Robot for Dynamics Locomotion
### Cooperative Force Optimization
#### 5.3.3 Cooperative Force Optimization: Force Cone Constraint  
The coooperative force optimization phase is designed as a quadratic problem; therefore, constraints have to be set accordingly. Contact constraints are dedicated to keep tangential forces small so that no slipping occurs. To do so, a friction pyramid is created inside a friction cone. The friction cone is a geometric interpretation of the magnitude of the allowable tangential force that can be applied on the ground. The allowable limit is calculated by simply multiplying the normal component of the applied force by the contact point with the friction coefficient. The nrmal component of the force is denoted as \({}_{B}f_{c,n}\) and the tangential components of the applied force are denoted as \({}_{B}f_{c,t}\), \({}_{B}f_{c,s}\), respectively. It should be noted that the applied force is defined with respect to \({\cal F}_{B}\). The friction coefficient is denoted as \(\mu\). The radius of the friction cone is defined as \(\mathbf{r}_{fc}\ =\ \mu_{B}f_{c,n}\). The friction pyramid is defined such that it is always upper-bounded by the \(\mathbf{r}_{fc}\), and this is achieved by setting linear bounds that are denoted as \(r_{fc,s}\) and \(r_{fc,t}\). These bounds are calculated such that \(|r_{fc,s}|\ \approx\ 0.707\mathbf{r}_{fc}\) and \(|r_{fc,s}|\ \approx\ 0.707\mathbf{r}_{fc}\). It should be noted that the bounding \(\mathbf{r}_{fc}\) creates a non-linear relationship and makes optimization a non-linear problem; however, bounds that are defined for tangential components can be implemented in a linear fashion. The geometric interpretation of the friction cone (red solid line) and pyramid (blue solid line) is provided in Figure 9.  
Figure 9: Geometric interpretation of the friction pyramid.  
Based on the linear and conservative bounds, the following linear constraints are defined for the tangential forces.  
\[\begin{split}-0.707\mu_{B}f_{c,n}&\leq\ b_{B}f_{c,t} \leq\ 0.707\mu_{B}f_{c,n}\\ -0.707\mu_{B}f_{c,n}&\leq\ b_{B}f_{c,s}\leq\ 0.707\mu_{B}f_{c,n}\\ -2\times 0.707\mu_{B}f_{c,n}&\leq\ b_{B}f_{c,t}+\ b_{B }f_{c,s}\leq\ 2\times 0.707\mu_{B}f_{c,n}\end{split} \tag{54}\]  
The relationship given in Equation (54) is written compactly as provided in Equation (55), where \({}_{B}F_{\mathbb{C},k}\) is the vector containing the decision variables. \({}_{B}\mathbf{n}\), \({}_{B}\mathbf{t}\) and \({}_{B}\mathbf{s}\) are the unit vectors attached on the contact points and defined in \(\mathcal{F}_{B}\). As a practical note, calculating these unit vectors with respect to \(\mathcal{F}_{B}\) is a straightforward calculation when there are passive joints at the ankle of the contact legs. Depending on the kinematic structure of the leg, certain unit vectors can be assumed to be in the same direction with the axes of the body frame.  
\[\begin{split}\left(-2\times 0.707\mu_{B}\mathbf{n}^{T}-{}_{B}\mathbf{t}^ {T}-{}_{B}\mathbf{s}^{T}\right){}_{B}\mathbf{F}_{\mathbb{C},k}&\leq\ \mathbf{0}\\ \left(\ 2\times 0.707\mu_{B}\mathbf{n}^{T}+{}_{B}\mathbf{t}^{T}+{}_{B} \mathbf{s}^{T}\right){}_{B}\mathbf{F}_{\mathbb{C},k}&\leq\ \mathbf{0}\end{split} \tag{55}\]  
Within the current work, \({}_{B}\mathbf{n}\), \({}_{B}\mathbf{t}\) and \({}_{B}\mathbf{s}\) are assumed to be constant and defined as \({}_{B}\mathbf{n}^{T}=\left[\begin{array}{cccc}0&1&0&0\end{array}\right]^{T}\), \({}_{B}\mathbf{t}=\left[\begin{array}{cccc}1&0&0\end{array}\right]^{T}\) and \({}_{B}\mathbf{s}=\left[\begin{array}{cccc}0&0&1\end{array}\right]^{T}\).