## 5 Results
### Numerical Results  
We generated different motions with combinations of different optimality criteria. In particular, we have used the following combinations, which are summarized in Table 2:  
1. Minimization of joint torques, with a small weight on minimization of joint accelerations. 2. Minimization of joint torques and torso orientation minimization, with a small weight on minimization of joint accelerations. 3. Minimization of absolute work, with a small weight on minimization of joint torques and joint accelerations. 4. Minimization of absolute work and torso orientation, with a small weight on minimization of joint torques and joint accelerations. The computations have been carried out using MUSCOD-II and RBDL on a standard desktop PC with the i7 CPU running at 3.60 GHz with eight cores. For the whole sequence to be computed, the average time is about 12-15 h, due to the chosen high number of intervals. The box constraints on states and controls are set as per the hardware limits of the robot as in Table 1. The initial guess is generated from a sequence of walking motion obtained with the reduced models from our previous works and set as the same for all combinations of objectives considered, as setting the initial guesses all to zeros might lead to longer computation times in such a large and complex nonlinear optimization problem. We cannot ensure that the obtained results are completely independent of the initial guess, but the outcomes have shown that the resulting motions show differences with respect to the ones generated with reduced models. This can be seen in a more intuitive way from the example of a motion sequence of a single step shown in Figure 6. The first sequence is a motion generated with the 3D Linear Inverted Pendulum Model (LIPM), in this case, the whole-body motion was obtained applying inverse kinematics to the given center of mass (CoM) and feet trajectories from a pattern generator. In order to generate feasible trajectories, the orientations of feet and torso have been constrained with respect to the world reference frame. In contrast to the first sequence, the second sequence, which is generated with optimal control with minimization of torques (Case 1 in Table 2), shows a clear change in the orientation of both torso and swing foot. As also discussed in Section 4, the stabilization of the torso might be a desired feature; therefore, in the third sequence, it is possible to see how the torso is stabilized using the set of objectives as per Case 2 in Table 2.  
In order to show the results in a more consistent way, the intermediate periodic step has been concatenated into nine steps, resulting in a sequence of a total of 11 steps for each of the cases as listed in Table 2. The resulting CoM trajectories are shown in Figure 7, and the whole-body joint trajectories are shown in Figure 8. The effect of the torso orientation minimization can be seen in a clear way in \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Objectives** & \(\mathbf{\Phi_{\tau}(\cdot)}\) & \(\mathbf{\Phi_{work}(\cdot)}\) & \(\mathbf{\Phi_{\bar{q}}(\cdot)}\) & \(\mathbf{\Phi_{torso}(\cdot)}\) \\ \hline
2 & 1 & 0 & \(10^{-4}\) & 10 \\
3 & \(10^{-4}\) & 1 & \(10^{-4}\) & 0 \\
4 & \(10^{-4}\) & 1 & \(10^{-4}\) & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Combination of objective functions. Minimization of torques and joint accelerations is always included with small weighting factors to ensure smooth trajectories. Figure 8: Joint angle trajectories of all the 15 DOF of the robot, obtained for the four combinations of objective functions. Time is normalized for all trajectories for comparison purposes. It is to be noted that torso orientation minimization does not mean reduction of the movements of the torso joints, but the orientation w.r.t. the world reference.