Nonlinear differential geometric guidance for maneuvering target

Based on the idea of zeroing the line of sight rate（LOSR）,a novel nonlinear differential geometric（DG） law for intercepting the agile target is proposed.In the first part,the DG formulations are utilized to describe the relatively kinematics model of missile and target,and the nonlinear DG guidance（DGG） law is proposed based on the nonlinear control theory to eliminate the influence brought by target.Further,the missile guidance commands are derived to overcome the information loss caused by decoupling condition,the new necessary initial condition is developed to guarantee capture the agile target.Then,the designed nonlinear DGG commands are transformed from an arc-length system to the time domain.A desirable aspect of the designed guidance law is that it does not require rigorous information about target acceleration.Representative numerical results show that the designed guidance law obtain a better performance than the traditional DGG law for agile target.

Extended differential geometric guidance law for intercepting maneuvering targets

Without assumptions made on motion states of missile and target, an extended differential geometric guidance law is derived. Through introducing a line of sight rotation coordinate system, the derivation is simplified and has more explicit physical significances. The extended law is theoretically applicable to any engagement scenarios. Then, on basis of the extended law, a modified one is designed without the requirement of target acceleration and an approach is proposed to determining the applied direction of commanded missile acceleration. Qualitative analysis is carried out to study the capture performance and a criterion for capture is given. Simulation results indicate the two laws are effective and make up the deficiency that pure proportional navigation suitable for endoatmospheric interceptions cannot deal with high-speed maneuvering targets. Furthermore, the correctness of the criterion is validated.

DIFFERENTIAL GEOMETRICAL METHOD IN ELASTICCOMPOSITE WITH IMPERFECT INTERFACES

A differential geometrical method is for the first time used to calculate the effective moduli of a two-phase elastic composite materials with imperfect interface which the inclusions are assumed to be ellipsoidal of revolutions. All of the interface integral items participating in forming the potential and complementary energy functionals of the composite materials are expressed in terms of intrinsic quantities of the ellipsoidal of revolutions. Based on this, the upper and the lower bound for the effective elastic moduli of the composite materials with inclusions described above have been derived. Under three limiting conditions of sphere, disk and needle shaped inclusions, the results of this paper will return to the bounds obtained by Hashin([6]) (1992).

Orbital rendezvous performance comparison of differential geometric and ZEM/ZEV feedback guidance algorithms

In this paper,the performance of two distinct classes of feedback guidance algorithms is evaluated for a spacecraft rendezvous problem utilizing a continuous low-thrust propulsion system.They are the DG(Differential Geometric)and ZEM/ZEV(Zero-Effort-Miss/Zero-Effort-Velocity)feedback guidance algorithms.Even though these two guidance algorithms do not attempt to minimize the onboard fuel consumption orΔV directly,theΔV requirement is used as a measure of their orbital rendezvous performance for various initial conditions and a wide range of the rendezvous time(within less than one orbital period of the target vehicle).For the DG guidance,the effects of its guidance parameter and terminal time on the closed-loop performance are evaluated by numerical simulations.For the ZEM/ZEV guidance,its nearfuel-optimality is further demonstrated for a rapid,short-range orbital rendezvous,in comparison with the corresponding open-loop optimal solutions.Furthermore,the poorΔV performance of the ZEM/ZEV guidance for a slow,long-range orbital rendezvous is remedied by simply adding an initial drift phase.The ZEM/ZEV feedback guidance algorithm and its appropriate variants are then shown to be a simple practical solution to a non-impulsive rendezvous problem,in comparison with the DG guidance as well as the open-loop optimal guidance.

Performance analysis of three-dimensional differential geometric guidance law against low-speed maneuvering targets

The performance of the three-dimensional differential geometric guidance law with proportional navigation formation against a target maneuvering arbitrarily with time-varying normal acceleration is thoroughly analyzed using the Lyapunov-like approach.The validation of this guidance law is firstly proved,and then the performance issues such as capturability,heading error control efficiency,line of sight rate convergence,and commanded acceleration requirement are analyzed,under the condition that the missile is initially flying toward the target with a speed advantage.It is proved that an intercept can occur and the line of sight rate and missile commanded acceleration can be limited in certain ranges,if the initial heading error is small and the navigation gain is sufficiently large.The nonlinear relative dynamics between the missile and the target is taken into full account,and the analysis process is simple and intuitive,due to the use of a convenient line of sight rotating coordinate system.Finally,the new theoretical findings are validated by numerical simulations.

CONSTRUCTION OF GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS FOR LEVEL SETS

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.

Higher-Order Level-Set Method and Its Application in Biomolecular Surfaces Construction

We present a general framework for a higher-order spline level-set （HLS） method and apply this to biomolecule surfaces construction. Starting from a first order energy functional, we obtain a general level set formulation of geometric partial differential equation, and provide an efficient approach to solving this partial differential equation using a C2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. One example of our HLS method is demonstrated their individual atomic coordinates which is the construction of biomolecule and solvated radii as prerequisites. surfaces （an implicit solvation interface） with

A Neural Paradigm forTime-Varying Motion Segmentation

This paper proposes a new neural algorithm to perform the segmentation of an observed scene into regions corresponding to different moving objects byanalyzing a time-varying images sequence. The method consists of a classificationstep, where the motion of small patches is characterized through an optimizationapproach, and a segmentation step merging neighboring patches characterized bythe same motion. Classification of motion is performed without optical flow computation, but considering only the spatial and temporal image gradients into anappropriate energy function minimized with a Hopfield-like neural network givingas output directly the 3D motion parameter estimates. Network convergence is accelerated by integrating the quantitative estimation of motion parameters with aqualitative estimate of dominant motion using the geometric theory of differentialequations.

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.