Optimal Wireless Aerial Sensor Node Positioning for Randomly Deployed Planar Collaborative Beamforming

Transcription

1 214 47th Hawaii International Conference on System Science Optimal Wireless Aerial Sensor Node Positioning for Randomly Deployed Planar Collaborative Beamforming Tan A. Ngo, Murali Tummala, and John C. McEachen Department of Electrical and Computer Engineering Naval Postgraduate School, 1 University Circle, Monterey, CA 93943, USA {tango, mtummala, mceachen}@nps.edu Abstract Recent advancements in low power micro sensors and wireless network technology have enabled a host of new sensor network applications. One such novel application is to perform collaborative beamforming from a wireless sensor network consisting of multirotor aerial vehicles (MAVs). Collaborative beamforming from an airborne wireless sensor network can be an effective means of communication from an unobstructed vantage point. This paper explores the feasibility of such a network. We analyze the beam pattern fluctuations due to position errors from the MAV s unsteady station keeping. We show that the effects of minor position errors are manageable. Finally, two non-linear constrained optimization techniques for sidelobe suppression utilizing the MAVs mobility and station keeping capability are introduced. We demonstrate through simulation that our techniques have similar performance while requiring lower computational cost and complexity compared to the conventional minimum variance distortionless response beamformer. 1. Introduction Collaborative beamforming is an efficient technique for wireless communication in wireless sensor networks (WSNs) [1]-[3]. It enables extended communication range through the collaborative radiating or receiving of power from a group of sensor nodes in a given direction. Recent advancements in low power micro sensors and wireless network technology have enabled the implementation of collaborative beamforming from a network of small mobile nodes. One such scenario is to form a collaborative beamforming network from a group of airborne semi-stationary vehicles such as multirotor aerial vehicles (MAVs). These systems have the unique ability to maintain a given position (station keep) in 3-D space. A collaborative beamforming WSN consisting of MAVs enables beamforming from an advantageous vantage point that ensures line-ofsight (LOS) communication at an altitude above the noisy surface environment. Multirotor systems are relatively simple machines. For instance, a quadrotor has four rotors held together with a rigid cross frame [4], [5]. Control is accomplished through the manipulation of each rotors thrust. The system is under actuated, thus the remaining degrees of freedom are handled through system dynamics. Having demonstrated their effectiveness in many applications such as 3-D environment mapping, transportation, and construction, they have become a popular research platform [6]. For more information regarding multirotor aerial vehicles see [6], [7]. This paper explores the feasibility of collaborative beamforming from a WSN consisting of multiple MAVs. Through analysis and simulation we will investigate what effects unsteady MAV nodes have on the array beam pattern. The overall concept is illustrated in Figure 1. Such a system will have a persistent LOS path to the target transmitters/receivers and can support a host of remote sensing applications beyond collaborative beamforming. Figure 1. Concept illustration of a collaborative beamforming WSN consisting of MAVs U.S. Government Work Not Protected by U.S. Copyright DOI 1.119/HICSS

2 With the effects of unsteady nodes better understood, we introduce and compare two non-linear constrained optimization techniques for sidelobe suppression that leverage the unique maneuverability and station keeping capabilities of a MAV WSN. These techniques minimize the sidelobe levels in given directions by minimizing a cost function through the reconfiguration of each nodes position. Similar optimization techniques using node positions can be found in [1]-[13]. 2. Proposed collaborative beamforming scheme In this proposed collaborative beamforming scheme there are two steps, illustrated in Figure 2. Here each MAV node consists of a MAV system and an anchor. The anchor is a ground base position estimation system that provides the MAV with highly accurate estimates of its position. Such an anchor could be realized using an outdoor VICON system [8]. The first step is the array positioning algorithm where the position of each node is configured using a non-linear constrained optimization technique. Here, each MAVs anchor is deployed in a loose linear fashion as allowed by the terrain. The operating altitude, node position bounds, and each signal direction of arrival (DOA) information is feed into the positioning algorithm. Assuming that the DOA of target signals and interfering signals are known, the positioning algorithm will generate an optimal position for each MAV in the network. These positions then are used in a standard phase shifting beamformer [15]. The goal of this step is to maximizing gain towards the target while minimizing sidelobe responses in the direction of interfering sources. The second step is where the array transmits or receives, here each node maintains its designated position derived in step one and transmits/receives collaboratively. In step two, it is important to note that any deviation in the position of each MAV node will lead to fluctuations in the beam pattern [2]. The deviations of each node can largely be attributed to position estimate errors and/or wind. Deviations from position estimate errors depend on the accuracy of the estimator and are typically obtained from GPS or VICON camera systems. Deviations due to wind are dependent on the responsiveness of the MAVs capability to station keep in the presence of wind [9]. This makes the feasibility of such a scheme dependent on the capability of each MAV node to maintain a given position i.e. station keep with relatively high accuracy. We will analyze the station keeping position errors effects on the beam pattern and determine what type of system can accommodate this error. 3. Effects of beam pattern fluctuations due to unsteady station keeping For this research we will assume that position estimates are perfect and that the only fluctuation in each node s position is due to wind. As a reference point, a GPS system can provide a few centimeters position accuracy [14] and VICON systems can deliver 5 μm accuracies indoors [6], which lends credence to this assumption, given they can be adapted for outdoor use. When a MAV recognizes it has been moved off station due to wind, its controller will move it back on station. This movement due to station keeping can be modeled as fluctuations in a node s position. With a random planar array in mind, these small positional fluctuations are modeled as a uniform random variable defined between [-b,b], with b being the max distance a MAV deviates in the x and y axes. This creates a 2-D box in the x-y plane with sides equal to 2b. This square represents the possible positions each MAV can take at any given time due to its station keeping operations. These fluctuations will be referred Normalized Gain (db) Node position bounds & operating altitude Signals DOA Anchor deployment locations Step 1: Array Positioning Algorithm Step 2: Array Tx/Rx Figure 2. Proposed two step collaborative beamforming scheme -5-1 No Position Error.5 Position Error Azimuth cut with Elevation = 9 degs Figure 3. Illustration of a phase shifting beamformer s beam pattern with and without position error. 5123

3 to as position error throughout this paper. For this analysis, the following is assumed: MAVs are connected as synchronized nodes in a WSN. All MAV nodes carry isotropic antennas with matched polarization. DOA data for all signals intended and interfering are known and static. From [15], the beam pattern of a random planar phase shifting array is expressed as Mean of Beam Pattern Error Variance of Beam Pattern Error (, ) = Standard Deviation of Position Error (wavelengths) Figure 4. Mean and variance of the beam pattern error for a 1-node random planar phase shifting array versus position error. Data points are averages taken over 5 trials. where is the number of nodes and (, )is the angle of reception (elevation, azimuth). α is the phase weigths and ξ is the current signal phase. They are defined as =[ sin cos + sin sin ] (2) =[ sin cos + sin sin ] (3) where =2/ with being the wavelength, (, ) is the ith nodes position, and (, ) is the steering angle. Equation (1) with the inclusion of a uniform random position error, can be expressed as where is defined as (,, ) =. (4) = sin cos + sin sin (5) In (4) we can see that introduces a phase angle error which is dependent on the random variables, and the angle of reception (, ). In the direction of the main beam, (, ) and (, ) are equal and the main beam gain becomes (, ) =. (6) Since we represent the error as a set of uniform random variables defined between [, ] we can represent as a set of independent random variables, with mean of the ith being = ( ) ) (. (7) The mean and variance of the main beam gain with uniform position errors is then derived as =, (8) =1, (9) The derivation of (9) is shown in appendix A. Intuitively, from (5) we see that if the phase angle error is small the effect on each node s phasing is also small, thus resulting in minor fluctuations in the beam pattern. In order to quantify this fluctuation we use the beam pattern error function ψ = ((, ) (,) ), to calculate the difference between a normalized beam pattern with no position errors (, ) and with position errors (, ). From these results we can infer that minor position errors induce a corresponding minor phase angle error at each node. If these phase angle errors are small enough, the overall cumulative Max Allowable Position Deviation (m) Operating Frequency (MHz) Figure 5. Max Allowable Position Deviation (station keeping requirement) versus Operating Frequency for Beam pattern Error of

4 effect on the beam pattern will be minimal. For reference, Figure 3 shows the beam pattern of a random planar phase shifting array with a beam pattern error of.46. To confirm these findings two simulations were evaluated. In the first simulation, a 1-node random planar phase shifting array with position error implemented as a uniform random variable was completed. Figure 4 shows the mean and variance of the beam pattern error increases with position error (each data point is taken over 5 trails). Here, we see that a position error of.5 standard deviation has a corresponding beam pattern error of.46. This implies that systems with the capability to contain position errors under.5 standard deviation will have minimal beam pattern fluctuations. Figure 5 shows the station keeping requirement at different operating frequencies to maintain a position error of.46. As a reference point, to maintain this level of error a system operating at a frequency of 1 MHz needs each node to station keep within a square with sides equal to.5 meters. For systems concerned with only the main beam gain, there is a higher tolerance for position error as indicated by Figure 6, (8) and (9). This concept was tested in the second simulation which evaluated the main beam s response to position errors. It evaluated a random planar phase shifting array s main beam response at increasing position errors over 1 trails. Figure 6 and 7 show the conformance of the results with (8) and (9), and that the overall effect of the minor position errors on the main beam gain is manageable. 4. Array positioning algorithm using nonlinear constrained optimization Since we have shown that the effect of minor position errors on the beam pattern is manageable. We can now introduce two non-linear constrained optimization techniques that utilize the MAVs unique ability to maintain a given position in 3-D space. With every given phase shifting array configuration, there is a corresponding sidelobe response. These two techniques take advantage of this relationship by finding a position for each MAV node. These positions are derived in a manner that yields an optimal sidelobe response. Since each MAV unit must stay near its anchor to receive position estimates, their positions are constrained to the vicinity of their anchors. It is important to note that since these techniques are non-convex, they cannot guarantee a global solution and are sensitive to their initialization points. Both techniques will produce the node positions for a random planar phase shifting array. The first technique Min ((, ), ).. (, ),(, ) <, < =1,2,, = 1,2,,, uses a set of non-linear constraint functions to ensure an upper bounded response in each interfering signals direction. Here, (, ), (, ), (, ), and are the steering angle, angles of interfering sources, position of the ith node, and the number of interfering sources, respectively. and are the lower and upper bounds on each node's position on the (, ) plane and represents the positional constraints imposed by their anchors. This technique has the benefit of controlling the upper bound on the beam pattern's response in given directions. However, there is a potential drawback. If the upper bounds are set to low and/or the number of interfering signals is too high, the feasible solution set goes to the null set. In the second technique Min (, ),(, ) + (, ),(, ) + +(, ),(, ).. <, < =1,2,, =1,2,,, all the constraints except the position bounds replace the objective function from the first technique. This gives the solver more latitude in reducing the sum of all the selected sidelobe responses. As a result, it is guaranteed to produce a solution as opposed to the first technique. One drawback is that there is no guaranteed upper bound on each selected sidelobe response as in the first technique. Figure 8-1 show the resulting beam pattern of the first and second techniques compared to a conventional minimum variance distortionless response beamformer (MVDR) [17], [18]. Here, a 5-node random planar phase shifting array was created with the steering angle at (θ, ϕ) =(45,9 ), and four interfering sources at (θ, ϕ) =(1,9 ), (9,9 ), (135,9 ), and (175,9 ). In both cases the solver was initialized with each node given a random initial position. Each node position was then bounded to a square area with sides equal to two wavelengths. In technique one, each constraint upper bound was set to zero. The minimization problems were solved using the interior point algorithm in the MATLAB optimization toolbox. Once the optimal positions were derived 5125

5 Mean Main Beam Gain Simulated Theoretical Standard Deviation of Position Error (wavelengths) Figure 6. Mean main beam gain versus position error, theoretical and simulated. 1 node random planar phase shifting array. 2 MVDR Technique 1 Variance of Main Beam Gain Simulated Theoretical Standard Deviation of Position Error (wavelengths) Figure 7. Variance of main beam gain versus position error, theoretical and simulated. 1 node random planar phase shifting array. 2 MVDR Technique 2 Normalized Power (db) Normalized Power (db) Azimuth Angle (degrees) Figure 8. Beam pattern (Azimuth cut at o Elevation) of a 5-node random planar phase shifting beamformer derived using technique one versus MVDR beamformer. The main beam is at (45 o, 9 o )and interfering signals all at 9 o El and Az at 1 o,9 o,135 o, and 175 o. Figure 1. 3D beam pattern of 5-node random planar phase shifting beamformer. Technique one (left). Technique two (right). using technique one and two, they were used to implement a standard phase shifting beamformer, and a MVDR beamformer. In the phase shifting beamformer the complex weights of the array are derived from the calculated node positions using (1), whereas in the Azimuth Angle (degrees) Figure 9. Beam pattern (Azimuth cut at o Elevation) of a 5-node random planar phase shifting beamformer derived using technique two versus MVDR beamformer. The main beam is at (45 o, 9 o )and interfering signals all at 9 o El and Az at 1 o,9 o,135 o, and 175 o. MVDR they are derived optimally to minimize the output variance [17]. As indicated in Figure 8-9, both techniques yield similar beam patterns and performance to that of the MVDR beamformer. Both techniques have the added benefit of a lower computational cost than the MVDR beamformer since there is no matrix inversion operations required. They also do not require knowledge of the interference/noise correlation matrix. This allows for less complex implementation as the node positions can be calculated prior to deployment and implemented via look-up tables. 5. Conclusion The effect of node position errors on the beam pattern of a collaborative beamforming MAV WSN 5126

6 was discussed in this paper. Analysis and simulation showed that the effects of minor position errors on the beam pattern are manageable. Consequently, two constrained optimization techniques were introduced that leverage the MAV s station keeping capability. By leveraging the optimal position of each node, these techniques were able to show a significant reduction in the sidelobe responses at given angles. Finally, their performance was shown to be comparable to that of a MVDR beamformer while providing computational advantages. 6. Appendix A: Derivation of (9) Using the equation for variance from [16] as = ({ }), the second moment of a given node is written as = ( ) =1. Substituting (14) and (8) into (13) we get =1. From [16] we see that the variance of a sum of independent random variables is simply the sum of their variances. Since the set of random variables are independent, the variance of is the summation of each variances which is shown in (9). 7. References [1] Mudumbai, R.; Barriac, G.; Madhow, U., "On the Feasibility of Distributed Beamforming in Wireless Networks," Wireless Communications, IEEE Transactions on, vol.6, no.5, pp.1754,1763, May 27 [2] Ochiai, H.; Mitran, P.; Poor, H.V.; Tarokh, Vahid, "Collaborative beamforming for distributed wireless ad hoc sensor networks," Signal Processing, IEEE Transactions on, vol.53, no.11, pp.411,4124, Nov. 25 [3] Ahmed, M.F.A.; Vorobyov, S.A., "Collaborative beamforming for wireless sensor networks with Gaussian distributed sensor nodes," Wireless Communications, IEEE Transactions on, vol.8, no.2, pp.638,643, Feb. 29 [4] Hoffmann, G.; Rajnarayan, D.G.; Waslander, S.L.; Dostal, D.; Jung Soon Jang; Tomlin, C.J., "The Stanford testbed of autonomous rotorcraft for multi agent control (STARMAC)," Digital Avionics Systems Conference, 24. DASC 4. The 23rd, vol.2, no., pp.12.e.4,121-1 Vol.2, Oct. 24 [5] Meier, L.; Tanskanen, P.; Fraundorfer, F.; Pollefeys, M., "PIXHAWK: A system for autonomous flight using onboard computer vision," Robotics and Automation (ICRA), 211 IEEE International Conference on, vol., no., pp.2992,2997, 9-13 May 211 [6] Mahony, R.; Kumar, V.; Corke, P., "Multirotor Aerial Vehicles: Modeling, Estimation, and Control of Quadrotor," Robotics & Automation Magazine, IEEE, vol.19, no.3, pp.2,32, Sept. 212 [7] Hyon Lim; Jaemann Park; Daewon Lee; Kim, H. J., "Build Your Own Quadrotor: Open-Source Projects on Unmanned Aerial Vehicles," Robotics & Automation Magazine, IEEE, vol.19, no.3, pp.33,45, Sept. 212 [8] Vicon MX Systems [Online]. Available: [9] Madani, T.; Benallegue, A., "Sliding Mode Observer and Backstepping Control for a Quadrotor Unmanned Aerial Vehicles," American Control Conference, 27. ACC '7, vol., no., pp.5887,5892, 9-13 July 27 [1] Khodier, M.M.; Christodoulou, C.G., "Linear Array Geometry Synthesis With Minimum Sidelobe Level and Null Control Using Particle Swarm Optimization," Antennas and Propagation, IEEE Transactions on, vol.53, no.8, pp.2674,2679, Aug. 25 [11] Leahy, R.M.; Jeffs, B.D., "On the design of maximally sparse beamforming arrays," Antennas and Propagation, IEEE Transactions on, vol.39, no.8, pp.1178,1187, Aug 1991 [12] Skolnik, M.I.; Nemhauser, G.; Sherman, J., III, "Dynamic programming applied to unequally spaced arrays," Antennas and Propagation, IEEE Transactions on, vol.12, no.1, pp.35,43, Jan 1964 [13] Skolnik, M.I.; Sherman, J.W., "Planar arrays with unequally spaced elements," Radio and Electronic Engineer, vol.28, no.3, pp.173,184, September 1964 [14] Haomiao Huang; Hoffmann, G.M.; Waslander, S.L.; Tomlin, C.J., "Aerodynamics and control of autonomous quadrotor helicopters in aggressive maneuvering," Robotics and Automation, 29. ICRA '9. IEEE International Conference on, vol., no., pp.3277,3282, May 29 [15] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd Edition, John Wiley & Sons, Inc., [16] C. Therrien and M. Tummala, Probability and Random Processes for Electrical and Computerengineers, 2nd Edition, CRC Press,

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