John L. Crassidis

Survey of Nonlinear Attitude Estimation Methods

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.

Unscented Filtering for Spacecraft Attitude Estimation

A new spacecraft attitude estimation approach based on the unscented filter is derived. For nonlinear systems the unscented filter uses a carefully selected set of sample points to map the probability distribution more accurately than the linearization of the standard extended Kalman filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The filter formulation is based on standard attitude-vector measurements using a gyro-based model for attitude propagation. The global attitude parameterization is given by a quaternion, whereas a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is derived from the local attitude error, which guarantees that quaternion normalization is maintained in the filter. Simulation results indicate that the unscented filter is more robust than the extended Kalman filter under realistic initial attitude-error conditions.

Sigma-point Kalman filtering for integrated GPS and inertial navigation

A sigma-point Kalman filter is derived for integrating GPS measurements with inertial measurements from gyros and accelerometers to determine both the position and the attitude of a moving vehicle. Sigma-point filters use a carefully selected set of sample points to more accurately map the probability distribution than the linearization of the standard extended Kalman filter (KKF), leading to faster convergence from inaccurate initial conditions in position/attitude estimation problems. The filter formulation is based on standard inertial navigation equations. The global attitude parameterization is given by a quaternion, while a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is used to guarantee that quaternion normalization is maintained in the filter. Simulation and experimental results are shown to compare the performance of the sigma-point filter with a standard EKF approach.

Kalman filtering for relative spacecraft attitude and position estimation

In this paper a novel approach is developed for relative navigation and attitude estimation of spacecraft flying in formation. The approach uses information from an optical sensor, which employs relatively simple electronic circuits with modest digital signal processing requirements, to provide multiple line-of-sight vectors from spacecraft to another. The sensor mechanism is well suited for both near-Earth and deep space applications since it is fully independent of any external systems. The line-of-sight measurements are coupled with gyro measurements and dynamical models in an extended Kalman filter to determine relative attitude, position and gyro biases. The quaternion is used to describe the relative kinematics and general relative orbital equations are used to describe the positional dynamics. Simulation results indicate that the combined sensor/estimator approach provides accurate relative position and attitude estimates.

Sliding mode control using modified Rodrigues parameters

Introduction Spacecraft pointing poses a complex problem involving nonlinear dynamics with either linear and/or nonlinear control laws. Primary control actuators usually include thrusters for rapid and coarse attitude maneuvers, and reaction wheels for slow and precise attitude maneuvers. Other types of control mechanisms include gravity-gradient stabilization and magnetic torquer assemblies. Control algorithms can be divided into open-loop systems and closed-loop (feedback) systems. Open-loop systems usually require a pre-determined pointing maneuver, and are typically determined using optimal control techniques which involve the solution of a twopoint-boundary-problem. An example of open-loop control is the time-optimal attitude maneuver (e.g., see the excellent survey paper by Scrivener and Thompson [1]). Closed-loop systems can provide robustness with respect to spacecraft modeling uncertainties and unexpected disturbances.

Real-Time Attitude-Independent Three-Axis Magnetometer Calibration

In this paper new real-time approaches for three-axis magnetometer sensor calibration are derived. These approaches rely on a conversion of the magnetometer-body and geomagnetic-reference vectors into an attitude independent observation by using scalar checking. The goal of the full calibration problem involves the determination of the magnetometer bias vector, scale factors and non-orthogonality corrections. Although the actual solution to this full calibration problem involves the minimization of a quartic loss function, the problem can be converted into a quadratic loss function by a centering approximation. This leads to a simple batch linear least squares solution. In this paper we develop alternative real-time algorithms based on both the extended Kalman filter and Unscented filter. With these real-time algorithms, a full magnetometer calibration can now be performed on-orbit during typical spacecraft mission-mode operations. Simulation results indicate that both algorithms provide accurate integer resolution in real time, but the Unscented filter is more robust to large initial condition errors than the extended Kalman filter. The algorithms are also tested using actual data from the Transition Region and Coronal Explorer (TRACE).

Predictive Filtering for Attitude Estimation Without Rate Sensors

A real-time predictive e lter is derived for spacecraft attitude estimation without the utilization of angular rate measurements from gyros. The formulation is shown using only attitude sensors (three-axis magnetometers, sun sensors,startrackers, etc. ).Thenew real-timenonlineare lter predicts the requiredtorquemodeling errorinput in ordertopropagatethespacecraftdynamicmodel,sothatmodelresponsesmatchthemeasuredvectorobservations. The real-time predictive e lter is used to estimate the attitude of the Solar, Anomalous, Magnetospheric Particle Explorer (SAMPEX) spacecraft. Results using this new algorithm indicate that the real-time predictive e lter accurately estimates the attitude of an actual spacecraft with the sole use of magnetometer sensor measurements.

Predictive Filtering for Nonlinear Systems

A real-time predictive e lter is derived for nonlinear systems. The major advantage of this new e lter over conventional e lters is that it providesa method of determining optimalstate estimatesin the presenceof signie cant error in the assumed (nominal)model. The new real-time nonlinear e lter determines (predicts)the optimal model errortrajectorysothatthemeasurement-minus-estimatecovariancestatisticallymatchestheknownmeasurement- minus-truth covariance. The optimal model error is found by using a one-time step ahead control approach. Also, because the continuous model is used to determine state estimates, the e lter avoids discrete state jumps. The predictive e lter is used to estimate the position and velocity of nonlinear mass-damper-spring system. Results using this new algorithm indicate that the real-time predictive e lter provides accurate estimates in the presence of highly nonlinear dynamics and signie cant errors in the model parameters. ONVENTIONAL e lter methods, such as the Kalman e lter, 1 have proven to be extremely useful in a wide range of appli- cations, including noise reduction of signals, trajectory tracking of moving objects, and control of linear or nonlinear systems. The es- sential feature of the Kalman e lter is the utilization of state-space formulations for the system model. Errors in the dynamics system can be separated into process noise errors or modeling errors. Pro- cess noise errors are usually represented by a zero-mean Gaussian errorprocesswithknowncovariance (e.g.,agyro-errormodelcanbe represented by a random walk process ). Modeling errors are usu- ally not known explicitly, because system models are not usually improved or updated during the estimation process. The theoretical derivation of the expression for the estimate error covariance in the Kalman e lter is only available if one makes assumptions about the model error. The most common assumptions about the model error are that it is also a zero-mean Gaussian noise process. Therefore, in the e lter-type literature, most often process noise and model error are treated equally. The Kalman e lter satise es an optimality criterion, which min- imizes the trace of the covariance of the estimate error between the system model responses and actual measurements. Statistical properties of the process noise and measurement error are used to determine an optimal e lter design. Therefore, model characteristics are combined with sequential measurements in order to obtain state estimates that are more accurate than both the measurements and model responses. As already stated, errors in the system model of the Kalman e l- ter are usually assumed to be represented by a zero-mean Gaussian noise process with known covariance. In actual practice the noise covariance is usually determined by an ad hoc and/or heuristic es- timation approach, which may result in suboptimal e lter designs. Otherapplicationsalsodetermineasteady-stategaindirectly,which may even produce unstable e lterdesigns. 2 Also, in many cases such as nonlinearities in the actual system responses or nonstationary processes, the assumption of a Gaussian model error process can lead to severely degraded state estimates.

Nonlinear Attitude Filtering Methods

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, the backwards-smoothing extended Kalman filter, and the interlaced 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 two-step 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, including particle filters and a Bayesian filter based on a non-Gaussian, finite-parameter probability density function on SO(3). Finally, the predictive filter, nonlinear observers and adaptive approaches are shown. The strengths and weaknesses of the various approaches are discussed.

Optimal Variable-Structure Control Tracking of Spacecraft Maneuvers

An optimal control approach using variable-structure (sliding-mode) tracking for large angle spacecraft maneuvers is presented. The approach expands upon a previously derived regulation result using a quaternion parameterization for the kinematic equations of motion. This parameterization is used since it is free of singularities. The main contribution of this paper is the utilization of a simple term in the control law that produces a maneuver to the reference attitude trajectory in the shortest distance. Also, a multiplicative error quaternion between the desired and actual attitude is used to derive the control law. Sliding-mode switching surfaces are derived using an optimal-control analysis. Control laws are given using either external torque commands or reaction wheel commands. Global asymptotic stability is shown for both cases using a Lyapunov analysis. Simulation results are shown which use the new control strategy to stabilize the motion of the Microwave Anisotropy Probe spacecraft.