The purpose of this article is to provide a tutorial overview of information consensus in multivehicle cooperative control. Theoretical results regarding consensus-seeking under both time invariant and dynamically changing communication topologies are summarized. Several specific applications of consensus algorithms to multivehicle coordination are described

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Information consensus in multivehicle cooperative control

This paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units. The outputs of such systems are characterized by high noise levels and time varying additive biases. We formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group SO (3) driven by reconstructed attitude and angular velocity measurements. Lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error. The approach taken leads to an observer that we term the direct complementary filter. By exploiting the geometry of the special orthogonal group a related observer, termed the passive complementary filter, is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs. Both the direct and passive filters can be extended to estimate gyro bias online. The passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude. This leads to an observer on SO(3), termed the explicit complementary filter, that requires only accelerometer and gyro outputs; is suitable for implementation on embedded hardware; and provides good attitude estimates as well as estimating the gyro biases online. The performance of the observers are demonstrated with a set of experiments performed on a robotic test-bed and a radio controlled unmanned aerial vehicle.

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Nonlinear Complementary Filters on the Special Orthogonal Group

LEAST SQUARES APPROXIMATION A Curve Fitting Example Linear Batch Estimation Linear Least Squares Weighted Least Squares Constrained Least Squares Linear Sequential Estimation Nonlinear Least Squares Estimation Basis Functions Advanced Topics Matrix Decompositions in Least Squares Kronecker Factorization and Least Squares Levenberg-Marquardt Method Projections in Least Squares Summary PROBABILITY CONCEPTS IN LEAST SQUARES Minimum Variance Estimation Estimation without a Prior State Estimates Estimation with a Prior State Estimates Unbiased Estimates Maximum Likelihood Estimation Cramer-Rao Inequality Nonuniqueness of the Weight Matrix Bayesian Estimation Advanced Topics Analysis of Covariance Errors Ridge Estimation Total Least Squares Summary REVIEW OF DYNAMICAL SYSTEMS Linear System Theory The State Space Approach Homogeneous Linear Dynamical Systems Forced Linear Dynamical Systems Linear State Variable Transformations Nonlinear Dynamical Systems Parametric Differentiation Observability Discrete-Time Systems Stability of Linear and Nonlinear Systems Attitude Kinematics and Rigid Body Dynamics Attitude Kinematics Rigid Body Dynamics Spacecraft Dynamics and Orbital Mechanics Spacecraft Dynamics Orbital Mechanics Aircraft Flight Dynamics Vibration Summary PARAMETER ESTIMATION: APPLICATIONS Global Positioning System Navigation Attitude Determination Vector Measurement Models Maximum Likelihood Estimation Optimal Quaternion Solution Information Matrix Analysis Orbit Determination Aircraft Parameter Identification Eigen-system Realization Algorithm Summary SEQUENTIAL STATE ESTIMATION A Simple First-Order Filter Example Full-Order Estimators Discrete-Time Estimators The Discrete-Time Kalman Filter Kalman Filter Derivation Stability and Joseph's Form Information Filter and Sequential Processing Steady-State Kalman Filter Correlated Measurement and Process Noise Orthogonality Principle The Continuous-Time Kalman Filter Kalman Filter Derivation in Continuous Time Kalman Filter Derivation from Discrete Time Stability Steady-State Kalman Filter Correlated Measurement and Process Noise The Continuous-Discrete Kalman Filter Extended Kalman Filter Advanced Topics Factorization Methods Colored-Noise Kalman Filtering Consistency of the Kalman Filter Adaptive Filtering Error Analysis Unscented Filtering Robust Filtering Summary BATCH STATE ESTIMATION Fixed-Interval Smoothing Discrete-Time Formulation Continuous-Time Formulation Nonlinear Smoothing Fixed-Point Smoothing Discrete-Time Formulation Continuous-Time Formulation Fixed-Lag Smoothing Discrete-Time Formulation Continuous-Time Formulation Advanced Topics Estimation/Control Duality Innovations Process Summary ESTIMATION OF DYNAMIC SYSTEMS: APPLICATIONS GPS Position Estimation GPS Coordinate Transformations Extended Kalman Filter Application to GPS Attitude Estimation Multiplicative Quaternion Formulation Discrete-Time Attitude Estimation Murrell's Version Farrenkopf's Steady-State Analysis Orbit Estimation Target Tracking of Aircraft The a-b Filter The a-b-g Filter Aircraft Parameter Estimation Smoothing with the Eigen-system Realization Algorithm Summary OPTIMAL CONTROL AND ESTIMATION THEORY Calculus of Variations Optimization with Differential Equation Constraints Pontryagin's Optimal Control Necessary Conditions Discrete-Time Control Linear Regulator Problems Continuous-Time Formulation Discrete-Time Formulation Linear Quadratic-Gaussian Controllers Continuous-Time Formulation Discrete-Time Formulation Loop Transfer Recovery Spacecraft Control Design Summary APPENDIX A MATRIX PROPERTIES Basic Definitions of Matrices Vectors Matrix Norms and Definiteness Matrix Decompositions Matrix Calculus APPENDIX B BASIC PROBABILITY CONCEPTS Functions of a Single Discrete-Valued Random Variable Functions of Discrete-Valued Random Variables Functions of Continuous Random Variables Gaussian Random Variables Chi-Square Random Variables Propagation of Functions through Various Models Linear Matrix Models Nonlinear Models APPENDIX C PARAMETER OPTIMIZATION METHODS C.1 Unconstrained Extrema C.2 Equality Constrained Extrema C.3 Nonlinear Unconstrained Optimization C.3.1 Some Geometrical Insights C.3.2 Methods of Gradients C.3.3 Second-Order (Gauss-Newton) Algorithm APPENDIX D COMPUTER SOFTWARE Index

Optimal Estimation of Dynamic Systems

This paper provides a survey of modern nonlinear filtering methods for attitude estimation. Early applications relied mostly on the extended Kalman filter for attitude estimation. Since these applications, several new approaches have been developed that have proven to be superior to the extended Kalman filter. Several of these approaches maintain the basic structure of the extended Kalman filter, but employ various modifications in order to provide better convergence or improve other performance characteristics. Examples of such approaches include: filter QUEST, extended QUEST and the backwards-smoothing extended Kalman filter. Filters that propagate and update a discrete set of sigma points rather than using linearized equations for the mean and covariance are also reviewed. A twostep approach is discussed with a first-step state that linearizes the measurement model and an iterative second step to recover the desired attitude states. These approaches are all based on the Gaussian assumption that the probability density function is adequately specified by its mean and covariance. Other approaches that do not require this assumption are reviewed, Associate Professor, Department of Mechanical & Aerospace Engineering. Email: johnc@eng.buffalo.edu. Associate Fellow AIAA. Aerospace Engineer, Guidance, Navigation and Control Systems Engineering Branch. Email: Landis.Markley@nasa.gov. Fellow AIAA. Postdoctoral Research Fellow, Department of Mechanical & Aerospace Engineering. Email: cheng3@eng.buffalo.edu. Member AIAA.

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Survey of nonlinear attitude estimation methods

Probabilistic inference is the problem of estimating the hidden variables (states or parameters) of a system in an optimal and consistent fashion as a set of noisy or incomplete observations of the system becomes available online. The optimal solution to this problem is given by the recursive Bayesian estimation algorithm which recursively updates the posterior density of the system state as new observations arrive. This posterior density constitutes the complete solution to the probabilistic inference problem, and allows us to calculate any “optimal” estimate of the state. Unfortunately, for most real-world problems, the optimal Bayesian recursion is intractable and approximate solutions must be used. Within the space of approximate solutions, the extended Kalman filter (EKF) has become one of the most widely used algorithms with applications in state, parameter and dual estimation. Unfortunately, the EKF is based on a sub-optimal implementation of the recursive Bayesian estimation framework applied to Gaussian random variables. This can seriously affect the accuracy or even lead to divergence of any inference system that is based on the EKF or that uses the EKF as a component part. Recently a number of related novel, more accurate and theoretically better motivated algorithmic alternatives to the EKF have surfaced in the literature, with specific application to state estimation for automatic control. We have extended these algorithms, all based on derivativeless deterministic sampling based approximations of the relevant Gaussian statistics, to a family of algorithms called Sigma-Point Kalman Filters (SPKF). Furthermore, we successfully expanded the use of this group of algorithms (SPKFs) within the general field of probabilistic inference and machine learning, both as stand-alone filters and as subcomponents of more powerful sequential Monte Carlo methods (particle filters). We have consistently shown that there are large performance benefits to be gained by applying Sigma-Point Kalman filters to areas where EKFs have been used as the de facto standard in the past, as well as in new areas where the use of the EKF is impossible.

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Sigma-point kalman filters for probabilistic inference in dynamic state-space models

A survey of attitude representation

Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations. The first of these, the TRIAD algorithm, provides a deterministic (i.e., nonoptimal) solution for the attitude based on two vector observations. The second, the QUEST algorithm, is an optimal algorithm which determines the attitude that achieves the best weighted overlap of an arbitrary number of reference and observation vectors. Analytical expressions are given for the covariance matrices for the two algorithms using a fairly realistic model for the measurement errors. The mathematical relationship of the two algorithms and their relative merits are discussed and numerical examples are given. The advantage of computing the covariance matrix in the body frame rather than in the inertial frame (e.g., in terms of Euler angles) is emphasized. These results are valuable when a single-frame attitude must be computed frequently. They will also be useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

Three-axis attitude determination from vector observations

HIS report reviews the methods of Kalman filtering in attitude estimation and their development over the last two decades. This review is not intended to be complete but is limited to algorithms suitable for spacecraft equipped with three-axis gyros as well as attitude sensors. These are the systems to which we feel that Kalman filtering is most ap- plicable. The Kalman filter uses a dynamical model for the time development of the system and a model of the sensor measurements to obtain the most accurate estimate possible of the system state using a linear estimator based on present and past measurements. It is, thus, ideally suited to both ground-based and on-board attitude determination. However, the applicability of the Kalman filtering technique rests on the availability of an accurate dynamical model. The dynamic equations for the spacecraft attitude pose many difficulties in the filter modeling. In particular, the external torques and the distribution of momentum internally due to the use of rotating or rastering instruments lead to significant uncertainties in the modeling. For autonomous spacecraft the use of inertial reference units as a model replacement permits the circumvention of these problems. In this representation the angular velocity of the spacecraft is obtained from the gyro data. The kinematic equations are used to obtain the attitude state and this is augmented by means of additional state-vector components for the gyro biases. Thus, gyro data are not treated as observations and the gyro noise appears as state noise rather than as observation noise. It is theoretically possible that a spacecraft is three-axis stabilized with such rigidity that the time development of the system can be described accurately without gyro information, or that it is one-axis stabilized so that only a single gyro is needed to provide information on the time history of the system. The modification of the algorithms presented here in order to apply to those cases is slight. However, this is of little practical importance because a control system capable of such

Kalman Filtering for Spacecraft Attitude Estimation

The TWOSTEP algorithm, which has been applied successfully to the attitude independent estimation of magnetometer biases is extended to estimate scale factors and nonorthogonality corrections. For the case of a spinning spacecraft, the algorithm performs well. This goes against the common wisdom that not much more than biases can be determined without knowledge of the attitude.

Complete Linear Attitude-Independent Magnetometer Calibration

A fast robust algorithm is developed for the inflight estimation of magnetometer biases when the attitude is not known. This algorithm combines the convergence in a single step of an heuristic algorithm currently in use with the correct treatment of the statistics of the measurement, and does this without discarding data. The new algorithm works well even when the magnetometer bias is comparable in magnitude to the ambient magnetic field. The algorithm performance is examined using simulated data for both spinning and inertially stabilized spacecraft.

Malcolm D. Shuster

Papers

A survey of attitude representation

Three-axis attitude determination from vector observations

Kalman Filtering for Spacecraft Attitude Estimation

Complete Linear Attitude-Independent Magnetometer Calibration

TWOSTEP: A Fast Robust Algorithm for Attitude-Independent Magnetometer-Bias Determination