## 6 Experimental Validation
### Key Performance Indicators  
The Key Performance Indicators (KPIs) are a series of indicators developed in the European Project KoroiBot ([[http://www.koroibot.eu](http://www.koroibot.eu)]([http://www.koroibot.eu](http://www.koroibot.eu))) designed to compare the performances of different methods applied on humanoid robots. These indicators are meant to compare the walking capabilities of the same humanoid robot with different control approaches and/or methodologies, and not to compare different robots, as it is highly difficult to compare robots that are very different in size and mechanical structure. For the HeiCub, a subset of the KPIs has been defined in our previous works [8; 23]. In the following, we give a brief explanation of the measured indicators, which were chosen based on quantities that are measurable from the robot sensors:  
* Walking velocity \(v_{max}\): The maximum achieved walking velocity; in the case of motions generated with optimal control, this corresponds to the velocity of the resulting sequence. * Walking timings \(t^{ss}/t^{ds}\) and step period: Single- and double-support times of a single step, the whole duration of which is indicated as the step period. In the case of optimal control, we consider the timings of the periodic step. * Cost of transport: \[\mathbf{E}_{CT}=\frac{\sum_{m=1}^{M}\int_{t_{0}}^{t_{f}}I_{m}(t)V_{m}(t)dt}{m_ {robot}g\cdot d}\] (19) The cost of transport is defined as a unitless quantity, where \(M\) is the total number of motors, \(I_{m}\) and \(V_{m}\) are the current and voltage measurements of the motor \(m\), \(m_{robot}\) is the mass of the robot and \(d\) is the traveled distance. * Froude number: \[Fr=v_{max}/\sqrt{g\cdot h},\text{ for }h=l_{leg}\] (20) where \(l_{leg}\) is the robot leg length. The Froude number is a dimensionless number used in fluid mechanics to characterize the resistance of an object moving through water. [PERSON] used it to characterize animal locomotion, given the that also legged locomotion is a dynamic motion in gravity [39]. A given Froude number can be assigned to a certain walking style. Running, for example, starts at a Froude number of approximately 1.0  
Figure 10: [Hardware experiment result.] Center of mass trajectories obtained on the robot (in red) with all four combinations of objective functions as in Table 2.  
* Precision of task execution: Expressed in terms of tracking errors. The root mean square error is computed by summing the squared difference between the measurement and the desired position over all points of the whole trajectory. The KPIs computed for each of the combinations of objective functions as in Table 2 are reported in Table 3 for the whole sequence, while in Tables 4-6, a few KPIs referred to the three steps of the optimized sequence: starting step, periodic step and ending step. In these tables, we report also the ones obtained from previous works [8; 23] for comparison purposes. As we can see from these tables, the cost of transport could be reduced with the minimization of torques, while the minimization of mechanical work does not show significant improvements in these indicators with respect to the reduced models. In particular, when minimizing work, the cost of transport is higher with respect to the LIPM model, which seems to contradict the  
When stabilizing the torso orientation, the motion generated using optimal control outperforms the reduced model in walking speed as the double-support time is much shorter; as a matter of fact, also the cost of transport and Froude number are better in this case with respect to the other cases. A higher Froude number indicates basically faster walking both in terms of speed and covered distance; in fact, we can observe that in the case of minimizing torque and introducing torso orientation minimization, the Froude number is the highest, while in the case of minimum work only, it is quite low due to the longer step timings.