## 7 Discussion and Conclusions  
In this paper, we have presented the first results of optimal control methods applied to whole-body models of an iCub robot for generating walking motions using a direct multiple shooting method. We have also described the underlying framework and model in much detail. In contrast to other approaches to motion generation, optimization is an all-at-once approach that determines all characteristics of a motion simultaneously to optimize a chosen performance criterion called the objective or cost function and to satisfy all important constraints related to the robot and the task description. For this research, we have focused on the HeiCub robot, which is a reduced version of the iCub with no head and no arms, but the same approach could be applied to the full iCub by just using the corresponding model, which is also available. In our previous research, the HeiCub had already achieved walking, using methods based on reduced models such as the LIPM and the table cart, and these approaches also have been transferred to the full iCub. These experiences also allowed us to identify parameters and physical constraints that were now included in the optimal control problem formulations. Even though the simplified approaches allowed one to generate a variety of walking motions on level ground in different directions (forward, backward, sidewards) and up and down stairs and slopes, the motions showed some of the typical characteristics of walking based on simplified models, e.g., keeping the pelvis at (nearly) constant height, resulting in a less dynamic appearance of the gait. In addition, the weight distribution of the HeiCub robot violates some of the center of mass assumptions taken for the simple models (see Section 2), because its legs are comparatively heavy. Walking generation methods taking into account the precise mass distribution of the robot therefore seemed to be very promising to generate more suitable motions on the robot, especially since we already had positive experiences with such methods with generating squatting motions and simple push recovery steps for the same robot [40; 41], as well as complex walking motions for other humanoid robots, such as HRP-2 [15]. The results of this first optimization study are very encouraging. We have shown that  
* It is possible to formulate optimal control problems for the iCub/HeiCub robots that allow one to simultaneously generate periodic walking motions, as well as the necessary starting and stopping \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**KPIs**} & \multirow{2}{*}{**Cart-Table**} &
\begin{tabular}{c} **NMPC** \\ **(LIPM)** \\ \end{tabular} & \multirow{2}{*}{**Min Torques**} & \multirow{2}{*}{**Min Torques**} & \multirow{2}{*}{**Min Torques**} & \multirow{2}{*}{**Min Work**} & \multirow{2}{*}{**Min Work**} \\  & & & & & & & \& **Torso ort.** \\ \hline Cost of Transport & 4.05 & 2.64 & 2.54 & 2.20 & 2.61 & 2.62 \\ Joint error (deg) & 1.35 & 1.15 & 1.20 & 1.20 & 1.08 & 0.85 \\ CoM error (cm) & 0.61 & 0.43 & 0.26 & 0.30 & 0.38 & 0.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: [Hardware experiment result.] Key Performance Indicators (KPIs) for the ending step. steps that take the robot from standing position to the periodic cycle, as well as from back to standing position. So far, only single starting and stopping steps have been included in the formulation, but it is straightforward to extend this to multiple steps, which may be required for faster walking. * Different objective functions result in visibly different walking styles for the robot. In particular we have compared a minimization of torques squared and of absolute mechanical work, in both cases with and without a combined term on torso stabilization. A very small term on joint accelerations was present in all objective functions. Further objective functions are the subject of current research.