# Natural Motion for Energy Saving in Robotic and Mechatronic Systems
## 3 Design of Desired Natural Dynamics  
The concept of natural motion can be exploited to perform tasks that are compatible with the natural dynamics of a mechanical systems, that is, cycle tasks. For this reason, such a concept is mainly adopted for locomotion of legged robots or, in the industrial field, for pick-and-place and palletizing tasks.  
A pick-and-place (or palletizing) task has strict requirements on the positions where objects are located and the corresponding velocities. Specifications on the task time and the path could potentially be relaxed constraints.  
Conversely, walking robots do not need strict requirements for the gait. In this case, if the robotic system is already equipped with elastic elements, the desired task (the robot gait) can be adapted to the system characteristics, that is, The task can be performed by adopting the trajectory naturally generated by the system. Such an approach can be referred to as _natural dynamic exploitation_[96]. Generally speaking, when there are requirements on the task to execute, it is always necessary to modify the system to fulfill them. The adaptation of the system to perform a given periodic task can be indicated as _natural dynamic modification_[73].  
As discussed in Section 1, the modifications of a robotic system to exploit the natural motion rely on spring additions. Therefore, the problem can be reformulated in this way: how should the spring parameters be determined in order to fulfill the task requirements exploiting system dynamics? Such a question has not a unique and straightforward answer. Requirements on task are typically given in terms of positions, velocities and time. Dynamic models of robotic systems depend non-linearly on the first two. Additionally, to find an analytical solution for the motion equations is very difficult, if not impossible, and hence also the setting on the time requirement is impossible, unless simplifications (such as linearization) are adopted or assumptions on the system response law are made. Therefore, since the optimal spring depends on the trajectory and _vice versa_, we classify the methods concerning with natural motion according to the trajectory followed by the system:  
* Defined trajectory. A feasible trajectory for the desired task (typically harmonic) is imposed and the spring parameters are optimized to minimize a given objective function related to energy consumption;
* Optimized trajectory. The spring parameters and the system trajectories are concurrently optimized thanks to a parametric representation of both or by adopting the optimal control theory;
* Free-vibration response. The trajectory is not imposed. The optimal spring parameters are identified so that the free response of the system fulfills task requirements. Such a result can be obtained by means of linearized dynamic models or multibody simulators;
* Periodic trajectory learning. The robotic system is not modified. The forced response at resonance of the system is learned by means of proper tools and used as reference trajectory.  
These four categories are analyzed and discussed in the following.  
\begin{table}
\begin{tabular}{l l l l l} \hline
**Ref.** & **1 st Author** & **Year** & **Path and Trajectory Planning** & **Compliance of Elastic Elements** \\ \hline \multicolumn{4}{l}{**Free-vibration response**} \\
[103] & [PERSON] & 2017 & defined by the multibody simulator & linear parallel \\
[81] & [PERSON] & 2018 & defined by the multibody simulator & linear parallel \\
[104] & [PERSON] & 2017 & periodic motion defined by the natural & variable serial \\ \hline \multicolumn{4}{l}{**Periodic trajectory learning**} \\
[105] & [PERSON] & 2017 & harmonic trajectory & linear/variable parallel \\
[83] & [PERSON] & 2016 & sinusoidal, polynomial and sign basis & linear parallel \\
[106] & [PERSON] & 2013 & motion defined by the pendulum & linear, applied at pendulum tip \\ \hline \end{tabular}
\end{table}
Table 1: Cont.