Gabriel Hugh Elkaim

A Geometric Approach to Strapdown Magnetometer Calibration in Sensor Frame

Abstract In this work a new algorithm is derived for the onboard calibration of three-axis strapdown magnetometers. The proposed calibration method is written in the sensor frame, and compensates for the combined effect of all linear time-invariant distortions, namely soft iron, hard iron, sensor non-orthogonality, bias, among others. A Maximum Likelihood Estimator (MLE) is formulated to iteratively find the optimal calibration parameters that best fit to the onboard sensor readings, without requiring external attitude references. It is shown that the proposed calibration technique is equivalent to the estimation of an ellipsoidal surface, and that the sensor alignment matrix is given by the solution of the orthogonal Procrustes problem. Good initial conditions for the iterative algorithm are obtained by a suboptimal batch least squares computation. Simulation and experimental results with low-cost sensors data are presented, supporting the application of the algorithm to autonomous vehicles and other robotic platforms.

A gyro-free quaternion-based attitude determination system suitable for implementation using low cost sensors

Attitude determination systems that use inexpensive sensors and are based on computationally efficient and robust algorithms are indispensable for real-time vehicle navigation, guidance and control applications. This paper describes an attitude determination system that is based on two vector measurements of non-zero, non-colinear vectors. The algorithm is based on a quaternion formulation of Wahbas (1966) problem, whereby the error quaternion (q/sub e/) becomes the observed state and can be cast into a standard linear measurement equation. Using the Earths magnetic field and gravity as the two measured quantities, a low-cost attitude determination system is proposed. An iterated least-squares solution to the attitude determination problem is tested on simulated static cases, and shown to be globally convergent. A time-varying Kalman filter implementation of the same formulation is tested on simulated data and experimental data from a maneuvering aircraft. The time-varying Kalman filter implementation of this algorithm is exercised on simulated and real data collected from an inexpensive triad of accelerometers and magnetometers. The accelerometers in conjunction with the derivative of GPS velocity provided a measure of the gravitation field vector and the magnetometers measured the Earths magnetic field vector. Tracking errors on experimental data are shown to be less than 1 degree mean and standard deviation of approximately 11 degrees in yaw, and 3 degrees in pitch and roll. Best case performance of the system during maneuvering is shown to improve standard deviations to approximately 3 degrees in yaw, and 1.5 degrees in pitch and roll.

Extension of a two-step calibration methodology to include nonorthogonal sensor axes

We present an extension of the nonlinear two-step estimation algorithm originally developed for the calibration of solid-state strapdown magnetometers. We expand the algorithm to include nonorthogonality within a sensor set for both two- and three-axis sensors. Nonorthogonality can result from manufacturing issues, installation geometry, and in the case of magnetometers, from soft iron bias errors. Simulation studies for both two- and three-axis sensors show convergence of the improved algorithm to the true values, even in the presence of realistic measurement noise. Finally the algorithm is experimentally validated on a low-cost solid-state three-axis magnetometer set, which shows definite improvement postcalibration. We note that the algorithm is general and can be applied to any two- or three-axis sensor set (such as accelerometers) with an error model consisting of scale, offset, and nonorthogonality errors.

Geometric Approach to Strapdown Magnetometer Calibration in Sensor Frame

In this work a new algorithm is derived for the onboard calibration of three-axis strapdown magnetometers. The proposed calibration method is written in the sensor frame, and compensates for the combined effect of all linear time-invariant distortions, namely soft iron, hard iron, sensor nonorthogonality, and bias, among others. A maximum likelihood estimator (MLE) is formulated to iteratively find the optimal calibration parameters that best fit to the onboard sensor readings, without requiring external attitude references. It is shown that the proposed calibration technique is equivalent to the estimation of a rotation, scaling and translation transformation, and that the sensor alignment matrix is given by the solution of the orthogonal Procrustes problem. Good initial conditions for the iterative algorithm are obtained by a suboptimal batch least squares computation. Simulation and experimental results with low-cost sensors data are presented and discussed, supporting the application of the algorithm to autonomous vehicles and other robotic platforms.

Path Planning Based on Bézier Curve for Autonomous Ground Vehicles

In this paper we present two path planning algorithms based on Bezier curves for autonomous vehicles with way points and corridor constraints. Bezier curves have useful properties for the path generation problem. The paper describes how the algorithms apply these properties to generate the reference trajectory for vehicles to satisfy the path constraints. Both algorithms join cubic Bezier curve segments smoothly to generate the path. Additionally, we discuss the constrained optimization problem that optimizes the resulting path for a user-defined cost function. The simulation shows the generation of successful routes for autonomous vehicles using these algorithms as well as control results for a simple kinematic vehicle. Extensions of these algorithms towards navigating through an unstructured environment with limited sensor range are discussed.

A lightweight formation control methodology for a swarm of non-holonomic vehicles

Multi-vehicle swarms offer the potential for increased performance and robustness in several key robotic and autonomous applications. Emergent swarm behavior demonstrated in biological systems show performance that far outstrips the abilities of the individual members. This paper discusses a lightweight formation control methodology using conservative potential functions to ensure group cohesion, yet requiring very modest communication and control requirements for each individual node. Previous efforts have demonstrated distributed methods to navigate a vehicle swarm through a complex obstacle environment while remaining computationally simple and having low bandwidth requirements. It is shown that arbitrary formation can be held and morphed within the lightweight framework. Simulations of the lightweight framework applied to realistic non-holonomic tricycle vehicles highlight the swarms ability to form arbitrary formations from random initial vehicle distributions and formation morphing capabilities, as well as navigate complex obstacle fields while maintaining formation. The non-holonomic constraints are used to implement realistic controls

MAV attitude determination by vector matching

An attitude determination algorithm suitable for micro aerial vehicle (MAV) applications is developed. The algorithm uses Earths magnetic and gravity field vectors as observations. The magnetic field vector measurements are obtained from a magnetometer triad. The gravity field vector is measured by fusing information from an accelerometer triad with GPS/WAAS (wide area augmentation system) velocity measurements. Two linearization and estimator designs for implementing the algorithm are discussed. Simulation and experimental flight test results validating the algorithm are presented.

Analysis of a Spline Based, Obstacle Avoiding Path Planning Algorithm

The Overbot is one of the original DARPA Grand Challenge vehicles now being used as a platform for autonomous vehicle research. The vehicle, equipped with a complete actuator and sensor suite, provides for an extremely capable robotic platform with computing infrastructure and software framework already in place to create a reconfigurable testbed. For point to point navigation, calculating suitable paths is computationally difficult. Maneuvering an autonomous vehicle safely around obstacles is essential, and the ability to generate safe paths in a real time environment is crucial for vehicle viability. We previously presented a method for developing feasible paths through complicated environments using a baseline smooth path based on cubic splines. This method is able to iteratively refine the path to more directly compute a feasible path and thus find an efficient, collision free path in real time through an unstructured environment. This method, when implemented in a receding horizon fashion, becomes the basis for high level control. In this work we perform Monte Carlo simulations to validate algorithm performance. The algorithm demonstrates a high success rate for all but the toughest of environments.

Spatially deconflicted path generation for multiple UAVs in a bounded airspace

This paper presents a preliminary framework for generating spatially deconflicted paths for multiple UAVs using Bezier curves. The critical issue addressed is that of guaranteeing that all the paths lie inside a pre-defined airspace volume. Its is shown that Bezier curves reperesent a natural tool for meeting this requirement. The paper reviews the essential properties of the Bezier curves that are used to guarantee spatial deconfliction between the UAV paths as well as airspace volume contsraints. The generated curves are not only non-overlapping but separated by a minimum distance chosen prior to flight. It is then shown that the path generation problem can be formulated as a constrained optimization problem over a finite optimization set and solved using standard MATLAB optimization tools. Simulation results are presented along with its discussion. The paper includes an analysis of numerical solutions obtained as well as discussion of future work.

Comparison of low-cost GPS/INS sensors for Autonomous Vehicle applications

Autonomous Vehicle applications (Unmanned Ground Vehicles, Micro-Air Vehicles, UAVpsilas, and Marine Surface Vehicles) all require accurate position and attitude to be effective. Commercial units range in both cost and accuracy, as well as power, size, and weight. With the advent of low-cost blended GPS/INS solutions, several new options are available to accomplish the positioning task. In this work, we experimentally compare three commercially available, off-the-shelf units insitu, in terms of both position, and attitude. The compared units are a Microbotics MIDG-II, a Tokimec VSAS-2GM, along with a KVH Fiber Optic Gyro. The position truth measure is from a Trimble Ag122 DGPS receiver, and the attitude truth is from the KVH in yaw. Care is taken to make sure that all measurements are taken simultaneously, and that the sensors are all mounted rigidly to the vehicle chassis. A series of measurement trials are performed, including light driving on coastal roads and highway speeds, static bench testing, and flight data taken in a light aircraft both flying up the coast as well as aggressively maneuvering. Allan Variance analysis performed on all of the sensors, and their noise characteristics are compared directly. A table is included with the final consistent models for these sensors, and a methodology for creating such models for any additional sensors as they are made available. The Microbotics MIDG-II demonstrates performance that is superior to the Tokimec VSAS-2GM, both in terms of raw positioning data, as well as attitude data. While both perform quite well during flight, the MIDG is much better during driving tests. This is due to the MIDG internal tightly-coupled architecture, which is able to better fuse the GPS information with the noisy inertial sensor measurements.