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Position Estimation for Sensor Networks

Position Estimation for Sensor Networks. FRC Seminar – Dec. 19, 2007 Joseph Djugash (Speaking Qualifier Talk). Motivation. Motivation. The Problem. Accurate localization of a large network of nodes. What makes it hard?. Resource Limitation

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Position Estimation for Sensor Networks

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  1. Position Estimation for Sensor Networks FRC Seminar – Dec. 19, 2007 Joseph Djugash (Speaking Qualifier Talk)

  2. Motivation

  3. Motivation

  4. The Problem Accurate localization of a large network of nodes

  5. What makes it hard? • Resource Limitation • power, communication bandwidth, processing, cost, sensor range, etc. • Scalability • 10, 100, 1000's of sensor nodes • Robustness • maintaining accuracy under sub-optimal configurations

  6. Outline • Range-Only Estimation • Simple Optimization • Bayesian Estimation • Decentralization • Conclusion

  7. Why use range sensors? • Shortcomings of classical sensors • Line-of-sight • Practical Considerations • Environmental Constraints • Correspondence Problem • Range-only sensors • Non-Gaussian noise models • Nonlinear measurements

  8. Limitations of range Uncertainty • Highly nonlinear a measurements

  9. Outline • Range-Only Sensors • Simple Optimization • The Naïve Approach • Improved Optimization • Bayesian Estimation • Decentralization • Conclusion

  10. Problem Formulation • Inputs: • Zik: Range meas. btw. nodes i & k • Outputs: • : Node positions • Estimated node positions can be used to predict the input ranges

  11. Multi-Dimensional Scaling (MDS) Fully connected sub-graph • MDS maps the distances between the nodes into a 2D space. • Minimize, • Initial condition important • Invariant to rotation and translation • To uniquely determine a node’s relative position, it needs to belong to a clique of degree 4 or higher Observed distances btw nodes i and k Distances btw nodes within the estimate Borg1997, Moore2004, Moses2002

  12. Multi-Dimensional Scaling (MDS) Prediction #1 Prediction #2 3 out of 4 meas. needed for rigidity Ground Truth Positions Borg1997, Moore2004, Moses2002

  13. Key Problem with MDS Requires High Degree of Connectivity! Can we get around this? Borg1997, Moore2004, Moses2002

  14. Incorporating Motion • Points along the trajectory are used to increase the degree of connectivity • Motion helps resolve ambiguities in orientation and handedness Kehagias2006, Djugash2006

  15. Improved Optimization Cost for deviating from robot’s odometry Cost of errors in range measurements Cost Uncertainty in motion Uncertainty in measurements • Minimize the error in … • All range measurements • Use path history of mobile nodes to provide additional constraints • Model noise in measurements • The Cost Function: Kehagias2006, Djugash2006

  16. Shortcomings of Optimization • Increased Dimensionality • Multi-modality in the estimate is hidden Kehagias2006, Djugash2006

  17. What’s next? How can we model these ambiguities (uncertainty) in the estimate?

  18. Outline • Range-Only Sensors • Simple Optimization • Bayesian Estimation • Bayes Filter • Particle Filter • Parametric Representation • Decentralization • Conclusion

  19. Bayes Filter Motion Model Sensor Model • General Formalism • Arbitrary belief representation • Recursively computes the posterior distribution: Thrun2005

  20. Bayesian Estimation Axis Anchor Using only meas. from nodes 1 and 2 Origin Anchor Adding angle constraint for Axis Anchor Ground Truth Positions Node #2 Node #3 Node #4 Thrun2005

  21. Major Drawbacks • Requires complex belief representation • Computational costs grow with environment size • How can we reduce the computational costs? Thrun2005

  22. Particle Filtering • Represent belief using a set of samples or particles • Sequential importance sampling with re-sampling used to update the belief • Handles arbitrary motion and measurement model Ihler2004, Ing2005

  23. Particle Filtering Ihler2004, Ing2005

  24. Downside to Particle Filters Poor Scalability • Accuracy ∝ (# of Particles) ∝ Computational Cost Ihler2004, Ing2005

  25. Issue of Scalability • Consider what happens when a single additional node is added… Ihler2004, Ing2005

  26. Issue of Scalability • Exponential growth of modes • # of modes ≤ 2 *  (# of modes of observers/“parents”) • Additional particles needed to accurately represent the nonlinearity within each mode Ihler2004, Ing2005

  27. How to solve this? • Use of negative information • Ideal for certain scenarios • Difficult to determine the cause for lack of info. • Moving away from particles? Perhaps a more approximate representation of belief?

  28. Alternate Belief Representations • How to best approximate the nonlinearity in the belief? • Idea: Perhaps in a parameterized model this nonlinear distribution will become linear… • What is a good parameterization?

  29. Over-Parameterized Filter True Posterior Gaussian in [x,y] • Simple Gaussian Parameterization in [x,y] is not sufficient • Relative Over-Parameterization (ROP) • The ring-like structure can be represented in polar coordinates • range, theta, center of circle (location of unknown person) – [r, , mx, my] Djugash2008, Funiak2006

  30. ROP Representation Djugash2008, Funiak2006

  31. Multi-Modal Distributions • Standard EKF limited to unimodal Gaussian • Multiple hypothesis representation • Use multiple EKFs, one for each hypothesis • Inconsistent hypotheses are dropped (threshold on likelihood) Thrun2005, Djugash2008

  32. Example of ROP-EKF Djugash2008

  33. Drawbacks of ROP-EKF • Accuracy limited by parameterization • Singularities requires special consideration • Hypothesis count limits scalability Djugash2008, Funiak2006

  34. Outline • Range-Only Sensors • Simple Optimization • Bayesian Estimation • Decentralization • Conclusion

  35. Decentralization • How to distribute the work load without sacrificing accuracy? • Can we guarantee … • robustness? • convergence? • What, if any, information needs to be shared?

  36. Belief Propagation Normalization Constant • An Inference method on graphs • The set of sensor nodes are the graphical model • Combine the observations from all nodes via message-passing operations • Belief Computation Observations of node “s” Messages from neighbors Belief =  of all inputs into node “s” Sudderth2003

  37. Belief Propagation • Message Computation • Message Product: • Belief based on all nodes except node “s” • Message Propagation: • Marginalize over node “t” to compute belief of node “s” Message Product Message Propagation Sudderth2003

  38. Properties of BP • Produces exact conditional marginals for tree-like graphs • Excellent empirical performance • Nonparametric BP – Ihler2004 • Non-Gaussian and continuous distributions • Transmit samples of the message distribution Sudderth2003, Ihler2004

  39. Outline • Range-Only Sensors • Simple Optimization • Bayesian Estimation • Decentralization • Conclusion

  40. Comparison

  41. Complexity vs. Accuracy • Striking a Good Compromise Requires • Improved Representation! • Distributable Computation!

  42. References • Borg1997: I. Borg and P. Groenen, “Modern multidimensional scaling: theory and applications,” New York: Springer, 1997. • Moore2004: D. Moore, J. Leonard, D. Rus, and S. Teller, “Robust distributednetwork localization with noisy range measurements,” in in Sen-Sys’04: Proc 2nd international conference on Embedded networked sensor systems. New York: ACM Press, 2004, pp. 50–61. • Moses2002: R. Moses and R. Patterson, “Self-calibration of sensor networks,” Unattended Ground Sensor Technologies and Applications IV, vol. 4743 in SPIE, 2002. • Kehagias2006: A. Kehagias, J. Djugash, and S. Singh, “Range-only slam with interpolated range data,” tech. report CMU-RI-TR-06-26, Robotics Institute, Carnegie Mellon University, May, 2006, Tech. Rep. • Djugash2006: J. Djugash, S. Singh, G. Kantor, and W. Zhang, “Range-only slam for robots operating cooperatively with sensor networks,” in IEEE Int’l Conf. on Robotics and Automation (ICRA ‘06), 2006. • Thrun2005: S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics. Cambridge, MA: MIT Press, 2005.

  43. References • Ihler2004: A. T. Ihler, J. W. Fisher III, R. L. Moses, and A. S. Willsky, “Nonparametric belief propagation for self-calibration in sensor networks,” in Information Processing in Sensor Networks, 2004. • Ing2005: G.Ing, M.J.Coates, "Parallel particle filters for tracking in wireless sensor networks," Signal Processing Advances in Wireless Communications, 2005 IEEE 6th Workshop on , vol., no., pp. 935-939, 5-8 June 2005 • Funiak2006: S. Funiak, C. E. Guestrin, R. Sukthankar, and M. Paskin, “Distributed localization of networked cameras,” in Fifth International Conference on Information Processing in Sensor Networks (IPSN’06), April 2006, pp. 34 – 42. • Stump2006: E. Stump, B. Grocholsky, and V. Kumar, “Extensive representations and algorithms for nonlinear filtering and estimation,” in The Seventh International Workshop on the Algorithmic Foundations of Robotics, July 2006. • Djugash2008: J. Djugash, B. Grocholsky, and S. Singh, “Decentralized Mapping of Robot-Aided Sensor Networks,” in IEEE Int’l Conf. on Robotics and Automation (ICRA ‘08), 2008.

  44. References • Sudderth2003: E. Sudderth, A. Ihler, W. Freeman, and A. Willsk, “Nonparametric Belief Propagation,” Computer Vision and Pattern Recognition (CVPR), June 2003. • Olfati-Saber2005: R.Olfati-Saber, J.S.Shamma, "Consensus Filters for Sensor Networks and Distributed Sensor Fusion," Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on , vol., no., pp. 6698-6703, 12-15 Dec. 2005 • Paskin2005: M. Paskin, C. Guestrin, and J. McFadden. “A robust architecture for inference in sensor networks,” In Proc. IPSN, 2005.

  45. Thank You Advisor: Sanjiv Singh Committee Members Brett Browning Paul Rybski Nathaniel Fairfield

  46. Conclusion • Motion helps with sparse connectivity • Modeling of uncertainty is necessary • Parametric belief representations • Preserve scalability and robustness • Little loss in accuracy • Decentralization improves scalability

  47. Belief Propagation with ROP-EKF Djugash2008

  48. Exploiting Negative Information

  49. Coordinate System + Handedness • In the absence of anchor nodes… • Arbitrarily assign a node to the origin • A second node (observable from the origin node) determines one of the axis • The other axis is left ambiguous • Unless handedness is resolved, the flip solution offers another equally likely solution in most cases Z = range btw node One Solution Flip Solution Z Z Z Origin Anchor Axis Anchor Global Coordinate Estimate Coordinate Estimate Coordinate

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