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Fig. 8 Tracking multiple objects; target trajectories and mobile sensor trajectories.
An approach is proposed to joint activation of stationary sensor nodes and design of mobile sensor trajectories in a hybrid sensor network collecting measurements for parameter estimation of a process described by a partial differential equation.
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Definition 5. Minimum max-min trajectory schedule: Among the feasible monitoring schedules of k mobile sensors, each of which can be represented as a set of the k mobile sensors' trajectories, minimum max-min trajectory schedule is the one with minimum 2MDkT.
This is mainly because in every step of SSR algorithm, when choosing the next POI to join one of the mobile sensors' trajectory, we always guarantee that the 2MDkT of current schedule is always minimized after this POI is added.
C = { c 1, c 2, …, c k }, c i is the i th mobile sensor's trajectory.
Repeat steps (3) to (4) until the trajectories in C contain all vertexes of V. Check the trajectory of each mobile sensor, if the trajectory intersects with other trajectories and the distance from start position to the point of intersection is the same, there will be collision between mobile sensors.
(5) Repeat steps (3) to (4) until the trajectories in C contain all vertexes of V. (6) Check the trajectory of each mobile sensor, if the trajectory intersects with other trajectories and the distance from start position to the point of intersection is the same, there will be collision between mobile sensors.
One is constructing a traveling salesman problem (TSP) ring using the polynomial-time approximation scheme (PTAS) for the TSP [3], then divide the ring into k trajectories and let every mobile sensor move along with a trajectory, respectively.
Finally, each mobile sensor moves along one trajectory back and forth.
Secondly, after tentatively adding each POI to all the trajectories, the calculation of length difference between the maximal and minimal trajectories in current mobile sensor network costs O k|V|) time.
Then add t to the trajectory of mobile sensor s.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com