# Petri Net Toolbox for Multi-Robot Planning under Uncertainty
## 3 Materials and Methods
### Application-Solarfarm Inspection
It also gives a place reward of \(-1\) per unit of time for every small UGV that has low battery. This discourages the system from keeping robots with low battery for long periods of time. We used a discount factor of 0.99 and a convergence error of 0.01. The GSPNR model parameters were estimated by running every action once and using the action duration to compute the timed transitions rates. We obtained the optimal policy using the toolbox _policy synthesis module_, that we shall call GSPNR policy. We then executed and compared the policy obtained with the toolbox against a policy synthesized from a marked ordinary Petri net model. We also compared it against two baselines: a random policy and an handcrafted policy. The random policy consists of randomly choosing an action every time the system has to make a decision. The handcrafted policy was carefully designed and follows a greedy policy. As a result, the small UGVs choose to inspect the closest available panel, while the large UGV stays stationary in the last node visited until a small UGV discharges. When this occurs the large UGV moves to the closest discharged robot and initiates the charging procedure. The main difference between the MOPN model and the GSPNR is that the former does not capture time uncertainty since it only has immediate transitions. So the MOPN model is identical to the GSPNR model described above, but the timed transitions were replaced by immediate transitions. The immediate transitions, that replace the timed transitions, have weights assigned that are proportional to the time that it takes to execute that action. So if a particular action takes less time, then the probability of the corresponding transition firing is higher. Moreover, the place rewards disappear in this model since there is no notion of continuous time. So we only provide a positive transition reward, that is given when an inspection action is selected. Each policy was executed in the simulated Gazebo environment for 8 runs of 1 h each. The number of solar panels inspected over time is depicted in Figure 15. The handcrafted policy is substantially better than the random policy showing that it is highly efficient. Nevertheless, both policies obtained from the GSPNR and the MOPN models outperform the handcrafted policy. This shows that formal methods are able to produce more efficient behaviors than carefully engineered policies. Longer runs suggest that as time increases the GSPNR policy diverges further from the handcrafted policy. The designed MOPN model does not capture time uncertainty, but it is able to model uncertainty in the outcomes of actions. As a result, the MOPN is able to distinguish between longer actions and shorter actions as the latter occur more often. The GSPNR captures time uncertainty as an exponential distribution, that, in some cases, does not correspond to the action duration true distribution. Nevertheless, the GSPNR policy outperforms the MOPN policy providing evidence that taking action duration uncertainty into account can improve the quality of the yielded policy. The standard deviation of the GSPNR policy is larger than of the MOPN policy. This occurs since the GSPNR model forces the system to always select an action, unlike in the MOPN. The consequence is that, for example the large UGV moves around significantly more, and crosses in front of the small UGVs more often. Thus the path planning algorithm introduces an higher variability in the time it takes to move between locations. For this reason, around second 1750 the MOPN policy  
Figure 14: Solar farm full GSPNR model with N battery levels and recharge action. Blue color—inspection model; yellow color—navigation model; red color—discharging model; green color—recharging model. has more solar panels inspected, on average, than the GSPNR policy. Finally, the MOPN model state space grows as fast as the GSPNR model, existing no difference in the time it takes to synthesize the optimal policy.