2. Localization Algorithm based on Spring Model (LASM)
- The sensor nodes can be deployed in a two or three-dimensional space. To simplify the explanation, we assume that they are deployed in a two-dimensional space in the rest of the paper.
- The wireless sensor network is a dynamic network, which means that the sensor nodes can add in or leave from the network at any time. In addition, the sensor nodes can fail at any moment.
- There are at least 3 anchor nodes that know their accurate positions in the wireless sensor network. Other nodes are blind nodes that do not know their positions. The anchor nodes can be realized by GPS or by manual placing in specific positions.
- The sensor nodes are able to communicate with neighbor nodes that are in a range of radio range R. They can also distinguish their neighbor nodes by their IDs.
- The distance estimates of neighbor nodes can be obtained. It can be realized by RSSI, ToA, TDoA etc as discussed before.
2.1. System model
2.1.1 Static and dynamic equations
2.1.2 Localization process
2.2. The basic LASM (LASM(B))
- Step 1:
- InitializationAssign its virtual position at (xi, yi) randomly and its velocity as zero.
- Step 2:
- CommunicationCommunicate with its neighbor particles, obtaining the distances between its neighbor particles and itself (the distances can be calculated by RSSI, TDoA, AoA etc.) and the virtual positions of its neighbors.
- Step 3:
- CalculationCalculate the total force exerted by its neighbors, the acceleration a, and then the next virtual position and velocity using (2)(3)(5). It can be noticed that the next step of position is related with ΔT. The larger the ΔT is, the more rapidly the position changes. Therefore, in the beginning, we set the ΔT larger to let the particle run to the right position more rapidly. After that, we set the ΔT smaller to let the particle adjust its position lightly. Here, we set ΔT be:
- Step 4:
- IterationRepeat step 2,3 until the total force exerted by neighbors is smaller than a threshold Tforce or l is larger than lstep.
- Step 1:
- each sensor node sends the distances between itself and its neighbors to the sink node. The anchor nodes also send their absolute positions to the sink node.
- Step 2:
- the sink node calculates the positions and velocities for all nodes, using the formula similar with the step3 in the distributed scheme described before.
- Step 3:
- repeat step 2 until the total forces of all particles exerted by neighbors are smaller than a threshold Tforce or l is larger than lstep.
2.2.1 Convergence analysis
2.2.2 Error analysis
2.3. Patches for the basic localization algorithm (LASM(P))
2.3.1 Patch A for particles at needless point
2.3.2 Patch B for bad nodes
- The node cannot be localized;
- The node is localized at needless points.
- Step 1
- for each particle, if it is at needless point, the trust of this particle is set to be 0.5; if it only has one or two neighbor particles, the trust of this particle is set to be 0. The trusts of other particles are set to be 1.
- Step 2
- for each particle, if its trust is not equal to 1, cut off all the springs connected with it. Therefore, it will not be the neighbor particles of others. And then tells its former neighbors to delete itself from the neighbors' neighbor list.
- Step 3
- repeat steps 1&2, until no particles need to change their trust.
2.3.3 Patch C for node variation
3. Simulation Results
3.1. Performance of LASM(B) and LASM(P)
3.2. Comparisons with MDS-MAP
3.3. Initial experiment
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- wireless sensor network
- Spring model.
- Drawing blind particles in random positions
- Blind particles go back to their stable positions
|Distributed LASM(B)||Distributed LASM(P)||MDS-MAP(P)|
|Communication Cost||O(1)*n||O(1)*n||O(n log n)|
|Centralized LASM(B)||Centralized LASM(P)||MDS-MAP(C)|
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