## 6 Preliminary Results for the ASLB Platform  
This section presents the simulation of the algorithm for the proposed method. The simulation is not executed in a physics environment, and trajectories illustrated in this section are estimated trajectories only. Solutions are obtained on Intel(R) Core i7-4720 HQ CPU @2.60 GHz 16 GB RAM PC with MATLAB 2019b software. The sampling time \(\Delta t\) for the discrete model is selected to be \(\Delta t\ =\ 0.05\)s. Nominal values for \(N_{u}\) and \(N_{p}\) are selected as \(N_{n}\ =\ 20\) and \(N_{p}\ =\ 4\), respectively. \(\mathbf{Q}_{s}\) and \(\mathbf{Q}_{p}\) for contact phase optimization is selected to be \(\mathbf{Q}_{s}\ =\ 5\) and \(\mathbf{Q}_{p}\ =\ 2\). \(\mathbf{Q}_{0}\), \(\mathbf{Q}_{f}\), and \(\mathbf{Q}_{t}\) for swing phase optimization is selected to be \(\mathbf{Q}_{0}\ =\ 12.5\), \(\mathbf{Q}_{f}\ =\ 12.5\), and \(\mathbf{Q}_{t}\ =\ 30\). Finally, \(\mathbf{Q}_{u}\) and \(\mathbf{Q}_{w}\) for cooperative force optimization are selected to be \(\mathbf{Q}_{u}\ =\ 100\) and \(\mathbf{Q}_{w}\ =\ 5\). Based on these settings, and from the initial conditions of \(\mathbf{u}_{B}(t=0)\ =\ \left[ \begin{array}{cccc}0&0&0\end{array} \right]^{T}\), \(\mathbf{r}_{2}(t=0)\ =\ \left[ \begin{array}{cccc}0.01&-0.344&0.063\end{array} \right]^{T}\), and \(\mathbf{r}_{3}(t=0)\ =\ \left[ \begin{array}{cccc}0.045&-0.344&-0.239\end{array} \right]^{T}\), ASLB is asked to move forward by \(\mathbf{u}_{B,ref}\ =\ \left[ \begin{array}{cccc}0.1&0&0\end{array} \right]^{T}\). The results are provided in the following figures. Figure 10 illustrates the calculated swing and contact trajectories. \(N_{n}\) for the first optimization is minimized to \(N_{n}\ =\ 6\) from a nominal 20 as the contact initial position for \({}^{p=1}\mathbf{r}_{c,1}\) is already provided by the sensor information. For a feasible finite horizon, \(N_{n}=6\), and contact leg, which is the right leg or **Agent 2**, is used to calculate an approximate trajectory for \(\mathbf{r}_{c}\) starting from \(\mathbf{r}_{2}(t=0)\) and diverge from the CoM toward the (+) x and (+) z directions. Note that this trajectory is calculated with respect to \(\mathcal{F}_{B}\). The swing phase optimization connects \(\mathbf{r}_{3}(t=0)\) to the \({}^{p=2}\mathbf{r}_{c,1}\) without violating the kinematic bounds. Figure 11 illustrates the nominal force trajectory for the contact leg as it interacts with the ground. \({}_{0}\)F\({}_{\text{C}}\) is the initial trajectory that is calculated by substituting \(\mathbf{q}_{a,i}\) and \(\mathbf{x}_{i}\) into the agent dynamics. \({}_{opt}\)F\({}_{\text{C}}\) is the resultant contact force trajectory. It is visible from the figure that there is not a significant change in the y-axis. However, \({}_{opt}\)F\({}_{\text{C}}\) is shifted in the (+) x direction by approximately 0.4 N.  
Figure 10: Calculated swing and contact leg trajectories for first step. Figure 12 shows the contact and swing trajectories for the second step, where \(N_{n}\) is minimized to \(N_{n}\ =\ 16\). Similar to the first step, \({}^{p=1}\mathbf{r}_{c,1}\) for contact phase optimization is the initial position of the current contact leg, which is **Agent 3**. \(\mathbf{r}_{c}\) in Figure 12 converges to the CoM in the beginning of this phase and then pushes away toward the (-) x and (-) z directions. A smaller correction occurs in the contact force as \({}_{0}\mathbf{F}_{\mathrm{C}}\) and \({}_{opt}\mathbf{F}_{\mathrm{C}}\) have slight differences in all three directions as seen in Figure 13.  
Figure 14 provides the \(\mathbf{u}_{B}\) trajectory throughout the walking simulation along with the \(\mathbf{r}_{c,i}\) for \(i=2,3\). This figure illustrates the characteristic difference in the method, which calculates the contact positions and forces them to track a reference velocity \(\mathbf{u}_{B,ref}\). The black circle in Figure 14 indicates the CoM and decision variables are defined with respect to that. The trajectory of \(\mathbf{u}_{B}\), which is given with the blue line, settles in a cyclic pattern.