Ming Xin

High-degree cubature Kalman filter

The cubature Kalman filter (CKF), which is based on the third degree spherical-radial cubature rule, is numerically more stable than the unscented Kalman filter (UKF) but less accurate than the Gauss-Hermite quadrature filter (GHQF). To improve the performance of the CKF, a new class of CKFs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed. The third-degree CKF is a special case of the class. The high-degree CKFs of the class can achieve the accuracy and stability performances close to those of the GHQF but at lower computational cost. A numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high-degree cubature rules to improve the performance. The target tracking simulation shows that the fifth-degree CKF can achieve higher accuracy than the extended Kalman filter, the UKF, the third-degree CKF, and the particle filter, and is computationally much more efficient than the GHQF.

Sparse-grid quadrature nonlinear filtering

In this paper, a novel nonlinear filter named Sparse-grid Quadrature Filter (SGQF) is proposed. The filter utilizes weighted sparse-grid quadrature points to approximate the multi-dimensional integrals in the nonlinear Bayesian estimation algorithm. The locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method. Then the univariate quadrature point sets are extended to form a multi-dimensional grid using the sparse-grid theory. Compared with the conventional point-based methods, the estimation accuracy level of the SGQF can be flexibly controlled and the number of sparse-grid quadrature points for the SGQF is a polynomial of the dimension of the system, which alleviates the curse of dimensionality for high dimensional problems. The Unscented Kalman Filter (UKF) is proven to be a subset of the SGQF at the level-2 accuracy. The performance of this filter is demonstrated by an orbit estimation problem. The simulation results show that the SGQF achieves higher accuracy than the Extended Kalman Filter (EKF), the UKF, and the Cubature Kalman Filter (CKF). In addition, the SGQF is computationally much more efficient than the multi-dimensional Gauss-Hermite Quadrature Filter (GHQF) with the same performance.

Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation

A novel sparse Gauss―Hermite quadrature filter is proposed using a sparse-grid method for multidimensional numerical integration in the Bayesian estimation framework. The conventional Gauss―Hermite quadrature filter is computationally expensive for multidimensional problems, because the number of Gauss―Hermite quadrature points increases exponentially with the dimension. The number of sparse-grid points of the computationally efficient sparse Gauss―Hermite quadrature filter, however, increases only polynomially with the dimension. In addition, it is proven in this paper that the unscented Kalman filter using the suggested optimal parameter is a subset of the sparse Gauss―Hermite quadrature filter. The sparse Gauss-Hermite quadrature filter is therefore more flexible to use than the unscented Kalman filter in terms of the number of points and accuracy level, and it is more efficient than the conventional Gauss―Hermite quadrature filter. The application to the spacecraft attitude estimation problem demonstrates better performance of the sparse Gauss―Hermite quadrature filter in comparison with the extended Kalman filter, the cubature Kalman filter, and the unscented Kalman filter.

A new method for suboptimal control of a class of nonlinear systems

In this paper, a new nonlinear control synthesis technique (/spl theta/ - D approximation) is presented. This approach achieves suboptimal solutions to nonlinear optimal control problems in the sense that it solves the Hamilton-Jacobi-Bellman (HJB) equation approximately by adding perturbations to the cost function. By manipulating the perturbation terms both semi-globally asymptotic stability and suboptimality properties can be obtained. The convergence and stability proofs are given. This method overcomes the large control for large initial states problem that occurs in some other Taylor expansion based methods. It does not need time-consuming online computations like the state dependent Riccati equation (SDRE) technique. A vector problem is investigated to demonstrate the effectiveness of this new technique.

New Nonlinear Control Technique for Ascent Phase of Reusable Launch Vehicles

Current flight control of reusable launch vehicles is based on table look-up gains for specific flight conditions. A new suboptimal nonlinear control technique for the complete ascent phase of reusable launch vehicles is presented. The new technique, called the θ-D method, is synthesized by adding perturbations to a typical optimal control formulation with a quadratic cost function. The controller expressions are obtained by getting an approximate closed-form solution to the Hamilton-Jacobi-Bellman equation. The θ-D method avoids iterative online solutions. A controller using this new method has been designed for the ascent phase of a reusable launch vehicle and implemented in a six-degrees-of-freedom high-fidelity simulator being run at the NASA Marshall Space Flight Center. Simulation results show that the 0-D controller achieves accurate tracking for the ascent phase of the reusable launch vehicle while being robust to external disturbances and plant uncertainties.

Robust state dependent Riccati equation based robot manipulator control

We present a new optimal control approach to robust control of robot manipulators in the framework of state dependent Riccati equation (SDRE) technique. To treat this highly nonlinear control system, we formulate it as a nonlinear optimal regulator problem. SDRE technique was used to synthesize an optimal controller to this class of robot control problem. We also synthesize a neural network based extra controller to achieve the robustness in the presence of the parameter uncertainties. A typical two-link robot position control problem was studied to show the effectiveness of SDRE approach and robust extra control design to robotic application.

Relations Between Sparse-Grid Quadrature Rule and Spherical-Radial Cubature Rule in Nonlinear Gaussian Estimation

The recently emerging cubature Kalman filter and the sparse-grid quadrature filter approximate the numerical integrations for the mean and covariance in Gaussian filters using the spherical-radial cubature rule and the sparse-grid quadrature rule, respectively. This technical note reveals that 1) spherical rules can be obtained by the projection of sparse-grid quadrature rules; 2) the third- and some of the fifth-degree spherical-radial cubature rules can be directly constructed from sparse-grid quadrature rules.

Sparse Gauss-Hermite Quadrature filter for spacecraft attitude estimation

In this paper, a new nonlinear filter based on Sparse Gauss-Hermite Quadrature (SGHQ) is proposed for spacecraft attitude estimation. Gauss-Hermite Quadrature (GHQ) has been widely used in numerical integration and nonlinear filtering. However, for multi-dimensional problems, the conventional GHQ based filter using product operations is difficult to implement because the number of points increases exponentially with dimensions. To solve this problem, the Smolyaks product rule has been used to extend GHQ rule to high dimensional problems. The contribution of this work is to design a new sparse-grid GHQ filter using Smolyaks product rule to alleviate the curse-of-dimensionality problem of the conventional GHQ filter. The number of SGHQ points needed for high dimensional problems is considerably smaller than the original GHQ method. Hence, the efficiency of using GHQ can be significantly improved. The performance of this new filter is demonstrated by the application to the spacecraft attitude estimation problem, which shows better results than the Extended Kalman Filter (EKF).

Position and Attitude Control of Deep-Space Spacecraft Formation Flying Via Virtual Structure and Theta-D Technique

Control of deep-space spacecraft formation flying is investigated in this paper using the virtual structure approach and the − θ D suboptimal control technique. The circular restricted three-body problem with the Sun and the Earth as the two primaries is utilized as a framework for study and a two-satellite formation flying scheme is considered. The virtual structure is stationkept in a nominal orbit around the 2 L libration point. A maneuver mode of formation flying is then considered. Each spacecraft is required to maneuver to a new position and the formation line-of-sight is required to rotate to a desired orientation to acquire new science targets. During the rotation, the formation needs to be maintained and each spacecraft’s attitude must align with the rotating formation orientation. The basic strategy is used on a ‘Virtual structure’ approach. A nonlinear model is developed that describes the relative formation dynamics. This highly nonlinear position and attitude control problem is solved by employing a recently developed nonlinear control approach, called the -D technique. This method is based on solution to the Hamilton-JacobiBellman equation and yields a closed-form suboptimal feedback solution. The controller is designed such that the relative position error of the formation is maintained within one centimeter accuracy.

Libration point stationkeeping using the θ-D technique

Unstable orbits about the collinear libration points require stationkeeping maneuvers to maintain the nominal path. A new method for stationkeeping such unstable orbits is proposed here using continuous thrust. The stationkeeping challenge is formulated as a nonlinear optimal control problem in the framework of the circular restricted three-body problem with the Sun and Earth/Moon center of mass as the two primaries. A recently developed control technique known as the “θ-D controller” is employed to provide a closed-form suboptimal feedback solution to this nonlinear control problem. In this approach an approximate solution to the Hamiltonian-Jacobi-Bellman (HJB) equation is found by adding perturbation terms to the cost function. The controller is designed such that the actual (numerically integrated) trajectory tracks a predetermined Lissajous reference orbit with good accuracy. Numerical results employing this method demonstrate the potential of this approach with stationkeeping costs varying between 0.52 ∼ 0.68 m/s per year (the range depending on the particular simulation parameters used), which is of the same order of magnitude as other methods using discrete maneuvers with halo orbits. The costs are modest and the method provides flexibility in selecting the “tightness” of the control versus fuel consumption. The algorithm is well-suited for integration with onboard flight software, as the nonlinear optimal control law is solved in closed-form and the Riccati equation must only be solved once, resulting in a computationally efficient controller.