# Adaptive Decentralized Control of Mobile Underwater Sensor Networks and Robots for Modeling Underwater Phenomena

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## Abstract

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## 1. Introduction

## 2. Underwater Sensor Network and Robot Platform

## 3. Modeling Underwater Phenomena

## 4. Decentralized Control Algorithm

#### 4.1. Problem Formulation and Intuition

**Figure 3.**Dashed lines are the motion constraints on the AquaNode motion and green circles are the points of sensor interest. Solid lines and the labels indicate the covariance between the point of interest and the indicated sensor.

#### 4.2. Objective Function

#### 4.3. General Decentralized Controller

#### 4.4. Gaussian Sensing Function

#### 4.5. Gaussian-Based Decentralized Controller

#### 4.6. Controller Convergence

- $\mathcal{H}$ must be differentiable;
- $\frac{\partial \mathcal{H}}{\partial {z}_{i}}$ must be locally Lipschitz;
- $\mathcal{H}$ must have a lower bound;
- $\mathcal{H}$ must be radially unbounded or the trajectories of the system must be bounded.

**Theorem 1.**The controller $-k\frac{\partial \mathcal{H}}{\partial {z}_{i}}$ converges to a critical point of $\mathcal{H}$. In other words, as time t progresses, the output of the controller will go to zero:

**Proof.**We show that the objective function satisfies the conditions outlined above. In Section 4.5 we determined the gradient of $\mathcal{H}$, satisfying condition 1. $\frac{\partial \mathcal{H}}{\partial {z}_{i}}$ has a locally bounded slope, meaning it is locally Lipschitz and satisfies condition 2.

## 5. Collaborating with Underwater Robots

#### 5.1. Problem Formulation

#### 5.2. Robot-Sensor Node Decentralized Controller

## 6. Simulation Experiments

#### 6.1. Implementation

`F(p_i,x,y,z)`and

`FDz(p_i,x,y,z)`. These functions take the sensor location ${p}_{i}$ and the point $[x,y,z]$ that we want to cover. The first function,

`F(p_i,x,y,z)`, computes the covariance between the sensor location and the point of interest. The second function,

`FDz(p_i,x,y,z)`, computes the gradient of the covariance function with respect to z at the same pair of points.

`changeDepth`puts a hard limit on the range of the nodes, preventing them from moving out of the water column. We found that this had little impact on the results.

Algorithm 1 Decentralized Depth Controller |

1: |

#### 6.2. Covariance Model

Algorithm 2 Decentralized Depth Controller with Robot Sensing |

1: |

**Figure 4.**(

**Bottom**) Model of the CDOM concentration in the Neponset river when tide caused a river depth of on average 2 m (a) and 3 m (b). (

**Top**) The corresponding numerically computed covariance.

`newrb`function. Figure 5 shows the result of the basis function fit. The error in the fit depends on the number of Gaussians used. For this plot with 6 elements the error is 1.88%, whereas using 10 elements gives an error of 0.54%. Using a basis function gives us a compact representation of the covariance function that a sensor node can easily store and compute.

#### 6.3. Simulation Parameters

**Figure 6.**The final positions after the distributed controller converges for a (

**a**) 2D and (

**b**) 3D setup.

#### 6.3.1. Changing k

**Figure 7.**The objective value and posterior error found when different “k” values are used in a system with 20 nodes. Inset (

**a**) shows the full range of values explored. Inset (

**b**) shows a zoomed in section of (a) to highlight the area where values of “k” can produce some instabilities.

#### 6.3.2. Changing Neighborhood Size

**Figure 8.**The (

**a**) objective value and (

**b**) runtime for a 15 node network when changing the size of the neighborhood over which the integration occurs.

#### 6.3.3. Changing Grid Size

**Figure 9.**Changing the grid size. (

**a**) Objective value and (

**b**) total search time as the step size changes.

#### 6.3.4. Changing Start Configurations

**Figure 10.**(

**a,b**) The results of the running the depth adjustment algorithm on various node start positions (circles) and (

**c**) the resultant objective value and posterior error.

#### 6.3.5. Random Placement Error

**Figure 11.**Nodes deployed with random error in x-axis placement. (

**a**) Plot of 100 runs with 6 m error. (

**b**) Average over many runs and positions.

#### 6.4. Data Reconstruction

`griddata`. The sum of squared error values are shown in the right of Figure 12. The dynamic depth adjustment algorithm outperforms the three manually chosen configurations.

**Figure 12.**From top: model data, reconstructed data for three manual configurations, and for algorithm positioning.

#### 6.5. Comparison with Posterior Error

#### 6.6. Mobile Robot

**Figure 14.**Example showing how the (

**a**) sensor nodes move to accommodate the robot and (

**b**) the corresponding objective functions (normalized).

## 7. Hardware Experiments

`FDz(p_i,qx,qy,qz)`(line 10) we use the numerical gradient of

`F`at that position. We use this to avoid deriving the gradient for every covariance function. The two different covariance models require different implementations of the function

`F`(line 8).

`F`as:

withexp(-(((px-qx)*(px-qx)+(py-qy)*(py-qy))/(2.0*SIGMA_SURF*SIGMA_SURF)+((pz-qz)*(pz-qz))/(2.0*SIGMA_DEPTH*SIGMA_DEPTH)));

`SIGMA_DEPTH = 4.0`and

`SIGMA_SURF = 10.0`. Some additional optimizations were made to limit duplicate computations. For the algorithm, the node locations were scaled to be 15 m apart along the x-axis, the neighborhood size was ±20 m along the x-axis, the virtual depth ranged in z from 0 to 30 m, and a step size of 1 m was used.

`F`based on the Gaussian basis function. Since we used MATLAB’s

`newrb`function to compute the basis function, we used their documentation to determine the reconstruction of

`F`:

val = 0.0; for(i = 0; i < NUM_BASIS; i++){ val += netLWX[i]*exp(-pow(((fabs(px-qx)-netIWX[i])*netb1X[i]),2)); val += netLWZ[i]*exp(-pow(((fabs(px-qx)-netIWZ[i])*netb1Z[i]),2));}val += netb2X + netb2Z; Yme = Yme + net.b{2};

`netLW`is the amplitude and

`netIW`is the center of the Gaussian as reported by MATLAB. The factor

`netb1`is the inverse of the variance of the Gaussian. The actual values of these variables are dependent on the model data. As discussed in Section 6.2 and shown in Figure 5, a Gaussian basis function produces a good fit as compared with the actual data for the Neponset River CDOM covariance. For this setup, the node locations were scaled to 500 m spacing along the x-axis, the neighborhood size was ±800 m, the depth ranged from 0 to 3 m, the step size was 40 m along the x-axis, and a step size of 0.1 m in depth was used.

#### 7.1. Lab and Pool

#### 7.1.1. Results

**Figure 17.**(

**a**) The value of $\frac{\partial \mathcal{H}}{\partial {z}_{i}}$. (

**b**) The depths over the course of an experiment.

Node0 | Node1 | Node2 | Node3 | |
---|---|---|---|---|

Bucket 1 Start | 10.0 m | 10.0 m | 10.0 m | 10.0 m |

Bucket 1 Final | 10.3 m | 24.1 m | 5.9 m | 19.7 m |

Bucket 2 Start | 20.0 m | 20.0 m | 20.0 m | 20.0 m |

Bucket 2 End | 19.8 m | 5.9 m | 23.8 m | 10.2 m |

Bucket 3 Start | 3.7 m | 7.8 m | 12.2 m | 15.9 m |

Bucket 3 End | 9.5 m | 22.9 m | 23.9 m | 9.6 m |

Pool 1 Start | 10.2 m | 9.9 m | 10.1 m | 9.8 m |

Pool 1 End | 20.6 m | 6.9 m | 24.1 m | 10.2 m |

Pool 2 Start | 20.0 m | 20.1 m | 20.3 m | 20.1 m |

Pool 2 End | 9.5 m | 23.9 m | 5.6 m | 18.8 m |

Pool 3 Start | 20.2 m | 19.9 m | 20.3 m | 20.1 m |

Pool 3 End | 9.6 m | 24.0 m | 5.8 m | 19.7 m |

Node0 | Node1 | Node2 | Node3 | |
---|---|---|---|---|

Bucket 1 Start | 1.0 m | 1.0 m | 1.0 m | 1.0 m |

Bucket 1 Final | 1.6 m | 0.7 m | 0.6 m | 2.4 m |

Bucket 2 Start | 1.0 m | 1.0 m | 1.1 m | 1.0 m |

Bucket 2 End | 2.5 m | 1.5 m | 1.7 m | 0.5 m |

Bucket 3 Start | 1.0 m | 2.0 m | 1.0 m | 2.0 m |

Bucket 3 End | 0.8 m | 2.4 m | 0.6 m | 2.1 m |

Pool 1 Start | 1.0 m | 1.0 m | 1.0 m | 1.0 m |

Pool 1 End | 2.4 m | 1.5 m | 0.5 m | 2.4 m |

Pool 2 Start | 2.0 m | 2.0 m | 2.0 m | 2.0 m |

Pool 2 End | 0.7 m | 2.3 m | 0.7 m | 2.2 m |

Pool 3 Start | 2.0 m | 2.1 m | 2.0 m | 2.0 m |

Pool 3 End | 0.7 m | 2.8 m | 0.8 m | 2.8 m |

#### 7.1.2. Communication Performance

**Figure 18.**(

**a**) Number of neighbors used to calculate $\frac{\partial \mathcal{H}}{\partial {z}_{i}}$. (

**b**) One node’s estimate of the others’ depths versus actual.

#### 7.2. Charles River Hardware With Changing Covariance

#### 7.3. Neponset River Experiment

Node0 | Node1 | Node2 | Node3 | |
---|---|---|---|---|

Node 0 | - | 8.3 | 8.1 | 200.7 |

Node 1 | 4.6 | - | 6.6 | 238.7 |

Node 2 | 39.0 | 59.0 | - | 46.1 |

Node 3 | 248.6 | 248.6 | 51.2 | - |

## 8. Related Work

## 9. Conclusions and Future Work

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Detweiler, C.; Banerjee, S.; Doniec, M.; Jiang, M.; Peri, F.; Chen, R.F.; Rus, D.
Adaptive Decentralized Control of Mobile Underwater Sensor Networks and Robots for Modeling Underwater Phenomena. *J. Sens. Actuator Netw.* **2014**, *3*, 113-149.
https://doi.org/10.3390/jsan3020113

**AMA Style**

Detweiler C, Banerjee S, Doniec M, Jiang M, Peri F, Chen RF, Rus D.
Adaptive Decentralized Control of Mobile Underwater Sensor Networks and Robots for Modeling Underwater Phenomena. *Journal of Sensor and Actuator Networks*. 2014; 3(2):113-149.
https://doi.org/10.3390/jsan3020113

**Chicago/Turabian Style**

Detweiler, Carrick, Sreeja Banerjee, Marek Doniec, Mingshun Jiang, Francesco Peri, Robert F. Chen, and Daniela Rus.
2014. "Adaptive Decentralized Control of Mobile Underwater Sensor Networks and Robots for Modeling Underwater Phenomena" *Journal of Sensor and Actuator Networks* 3, no. 2: 113-149.
https://doi.org/10.3390/jsan3020113