Adam M. Fosbury

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.

Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization

We derive a continuous nonlinear control law for spacecraft attitude tracking of arbitrary continuously differentiable attitude trajectories based on rotation matrices. This formulation provides almost global stabilizability, that is, Lyapunov stability of the desired equilibrium of the error system as well as convergence from all initial states except for a subset for which the complement is open and dense. This controller thus overcomes the unwinding phenomenon associated with continuous controllers based on attitude representations, such as quaternions, that are not bijective and without resorting to discontinuous switching. The controller requires no inertiainformation,noinformation onconstant-disturbance torques,andonlyfrequencyinformation forsinusoidal disturbance torques. For slew maneuvers (that is, maneuvers with a setpoint command in the absence of disturbances), the controller specializes to a continuous, nonlinear, proportional–derivative-type, almost globally stabilizing controller, in which casethe torque inputs can be arbitrarily bounded a priori. For arbitrary maneuvers, we present an approximate saturation technique for bounding the control torques.

Relative Navigation of Air Vehicles

This paper derives the full equations of motion for relative navigation of air vehicles. An extended Kalman filter is used for estimating the relative position and attitude of two air vehicles designated leader and follower. All leader states are assumed known, whereas the relative states are estimated using line-of-sight measurements between the vehicles along with acceleration and angular rate measurements of the follower. Noise is present on all measurements, whereas biases are present only on the latter two. The global attitude is parameterized using a quaternion, whereas the local attitude error is given by a three-dimensional attitude representation. An application of the new theoretical developments is also given, which involves determining an optimal trajectory to improve the estimation accuracy of the system. A cost function is derived based upon the relative position elements of the estimator covariance. State constraints are appended to the cost function using an exponential term. Results show that minimization of this cost function yields a trajectory that improves accuracy of both the position and attitude state estimates.

Spacecraft Actuator Alignment Estimation

Spacecraft attitude control has been well studied for several decades. Dozens of algorithms using a variety of techniques have been developed. One of the factors that these algorithms have in common is that they assume accurate knowledge of the actuator alignments. As with other spacecraft parameters, this knowledge will never be perfect. Whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some alignment error will exist. Additionally, there is a possibility of launching a satellite with a sign error in at least one actuator. Whether due to a software or hardware error, this problem in and of itself can cause the spacecraft to fail. This paper develops methods for on-orbit estimation of actuator alignments. Sub-degree accuracy is demonstrated for several different scenarios. Simulation results show that alignment errors using filtered data primarily come from the finite difference approximations used to estimate the angular acceleration. Inertia knowledge errors yield alignment errors between seven and twelve degrees. While this error is significant, the rough alignment estimates provide an option for fault checking of ground-determined alignment calibrations. Overall, these approaches provide an effective means for on-orbit estimation of spacecraft actuator alignments.

Decentralized Attitude Estimation Using a Quaternion Covariance Intersection Approach

This paper derives an approach to combine estimates and covariances for decentralized attitude estimation using a quaternion parameterization. The approach is based on the covariance intersection method, which is modified to maintain quaternion normalization in the combination process. A practical simulation result is provided where local extended Kaiman filters are used on two star trackers, each running with common gyro measurements. The covariance intersection approach is shown to provide more accurate estimates than either of the local filters.

Kalman Filtering for Relative Inertial Navigation of Uninhabited Air Vehicles

An extended Kalman filter is derived for estimating the relative position and attitude of a pair of uninhabited air vehicles, designated leader and follower. All leader states are assumed known, while the relative states are estimated using line-of-sight measurements between the vehicles along with angular rate and acceleration measurements of the follower. Noise is present on all measurements, while biases are present only on the latter two. Line-of-sight measurements are generated using visual navigation beacons. The global attitude is parameterized using a quaternion, while the local attitude error is given by a three-dimensional attitude representation. The quaternion normalization constraint is maintained using a multiplicative error quaternion. Simulation results show that the relative states and measurement biases converge within their respective covariance bounds. The number of visual navigation beacons is shown to affect estimator convergence in the presence of initial condition errors.

Inertia-free spacecraft attitude trajectory tracking with internal-model-based disturbance rejection and almost global stabilization

We derive a continuous nonlinear control law for spacecraft attitude trajectory tracking of arbitrary C1 attitude trajectories based on rotation matrices. This formulation provides almost global stabilizability, that is, Lyapunov stability of the desired equilibrium of the error system as well as convergence from all initial states except for a subset whose complement is open and dense. This controller thus overcomes the unwinding phenomenon associated with continuous controllers based on attitude representations, such as quaternions, that are not bijective. The controller requires no inertia information and no information on constant disturbance torques. For slew maneuvers, that is, maneuvers with a setpoint command, in the absence of disturbances, the controller specializes to the continuous, nonlinear PD-type almost globally stabilizing controller of Chaturvedi, in which case the torque inputs can be arbitrarily bounded a priori.

Estimation with Multitemporal Measurements

This paper studies measurements that are mathematical functions of states at more than one time instance where the exact time instances may not be known. For consistency of discussion, they are referred to as multitemporal measurements. Many sensors that rely on received light waves, sound waves, or other transmitted physical phenomena can have measurements that are classified as multitemporal. Even though there is a finite time between the sending and receiving of these signals, measurement estimates are commonly written as if these actions occurred simultaneously. The application of lasers for communication across large distances has provided a system where ignoring this phenomenon results in numerically significant errors. After mathematically describing the measurements, the overall multitemporal problem is discussed in the context of time-delay measurement, out-of-sequence measurement, relative state measurement, and Global-Positioning-System problems. While some research has explored the use of Global-Positioning-System signals as a means for Global-Positioning-System satellites to estimate their own orbits, that research has made simplifying assumptions that allow the multitemporal problem to be avoided. A solution using Taylor-series approximations within an extended Kalman filter is developed and shown to provide filter convergence, whereas a filter ignoring the multitemporal problem does not.

Decentralized Relative Attitude Estimation for Three-Spacecraft Formation Flying Applications

This paper investigates the problem of relative spacecraft attitude estimation between three vehicles from a decentralized point of view. Decentralized attitude estimation is achieved through the use of local extended Kalman filters and a data fusion process known as the Covariance Intersection algorithm. Because the global attitude parameterization is the quaternion, the Covariance Intersection algorithm is modified in order to handle the quaternion norm constraint using the method of Lagrange multipliers. A multivariate Newton-Raphson iteration is developed so that state vectors containing multiple quaternions may be fused. A formation flying simulation shows that the results of state fusion via the Covariance Intersection algorithm provides substantially better results than available from any one local source.

Efficient Covariance Intersection of Attitude Estimates Using a Local-Error Representation

As the complexity of spacecraft and space systems increases, there is an ever increasing need for more accurate and robust estimators to track the system. While the demand placed on an individual spacecraft is increasing, the spacecraft itself is being pushed towards smaller, independent pieces of hardware which may have limited power and/or computational abilities. One such initiative is often refereed to as Operational Responsive Space (ORS). The primary goal of ORS systems is acceleration of the mission conception-to-launch timeline of spacecraft by means of utilizing off-the-shelf type modular components for construction of the spacecraft. With such modular systems, customized interfacing and software necessary to direct all data to a central processor may not be possible given the accelerated timeline, the unfortunate result of which would be degradation of the estimates due to missing data. Graduate Student, Department of Mechanical & Aerospace Engineering. E-mail: ckn@buffalo.edu. Student Member AIAA. Professor, Department of Mechanical & Aerospace Engineering. E-mail: johnc@buffalo.edu. Associate Fellow AIAA. Senior Professional Staff I, SEG, Space Department. Email: adam.fosbury@jhuapl.edu. Member AIAA. Assistant Professor, Department of Aerospace Engineering. Email: cheng@ae.msstate.edu, Senior Member AIAA.