Where am I? Sensors and Methods for Mobile Robot Positioning

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218 Appendices, References, Indexes APPENDIX A A WORD ON KALMAN FILTERS The most widely used method for sensor fusion in mobile robot applications is the Kalman filter. This filter is often used to combine all measurement data (e.g., for fusing data from different sensors) to get an optimal estimate in a statistical sense. If the system can be described with a linear model and both the system error and the sensor error can be modeled as white Gaussian noise, then the Kalman filter will provide a unique statistically optimal estimate for the fused data. This means that under certain conditions the Kalman filter is able to find the best estimates based on the “correctness” of each individual measurement. The calculation of the Kalman filter is done recursively, i.e., in each iteration, only the newest measurement and the last estimate will be used in the current calculation, so there is no need to store all the previous measurements and estimates. This characteristic of the Kalman filter makes it appropriate for use in systems that don't have large data storage capabilities and computing power. The measurements from a group of n sensors can be fused using a Kalman filter to provide both an estimate of the current state of a system and a prediction of the future state of the system. The inputs to a Kalman filter are the system measurements. The a priori information required are the system dynamics and the noise properties of the system and the sensors. The output of the Kalman filter is the estimated system state and the innovation (i.e., the difference between the predicted and observed measurement). The innovation is also a measure for the performance of the Kalman filter. At each step, the Kalman filter generates a state estimate by computing a weighted average of the predicted state (obtained from the system model) and the innovation. The weight used in the weighted average is determined by the covariance matrix, which is a direct indication of the error in state estimation. In the simplest case, when all measurements have the same accuracy and the measurements are the states to be estimated, the estimate will reduce to a simple average, i.e., a weighted average with all weights equal. Note also that the Kalman filter can be used for systems with timevariant parameters. The extended Kalman filter is used in place of the conventional Kalman filter if the system model is potentially numerically instable or if the system model is not approximately linear. The extended Kalman filter is a version of the Kalman filter that can handle non-linear dynamics or non-linear measurement equations, or both [Abidi and Gonzalez, 1992].
Appendices 219 APPENDIX B UNIT CONVERSIONS AND ABBREVIATIONS To convert from To Multiply by (Angles) degrees (E) radian (rad) 0.01745 radian (rad) degrees (E) 57.2958 milliradians (mrad) degrees (E) 0.0573 (Length) inch (in) meter (m) 0.0254 inch (in) centimeter (cm) 2.54 inch (in) millimeter (mm) 25.4 foot (ft) meter (m) 30.48 mile (mi), (U.S. statute) meter (m) 1,609 mile (mi), (international nautical) meter (m) 1,852 yard (yd) meter (m) 0.9144 (Area) inch (in ) meter (m ) 6.4516 × 10 foot (ft ) meter (m ) 9.2903 × 10 2 2 yard (yd ) 2 2 meter (m ) 0.83613 2 2 2 2 -4 2 2 2 2 -2 (Volume) foot (ft ) meter (m ) 2.8317 × 10 inch (in ) meter (m ) 1.6387 × 10 3 3 3 3 -2 3 3 3 3 -5 (Time) nanosecond (ns) second (s) 10 -9 microsecond (µs) second (s) 10 -6 millisecond (ms) second (s) 10 -3 second (s) minute (min) second (s) 60 hour (hr) second (s) 3,600 (Frequency) Hertz (Hz) 1 cycle/second (s- ) 1 Kilohertz (KHz) Hz 1,000 Megahertz (MHz) Hz 6 10 Gigahertz (GHz) Hz 9 10
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INTRODUCTION Leonard and Durrant-Wh
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Part I Sensors for Mobile Robot Pos
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