John E. Cochran

Effects of gravity-gradient torque on the rotational motion of A triaxial satellite in a precessing elliptic orbit

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the effects of gravity-gradient torque on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is free to precess and spin. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite to second order inn/ω0, wheren is the orbital mean motion of the center of mass andω0 is a reference value of the magnitude of the satellites rotational angular velocity. The differential equations derivable from the transformed Hamiltonian are integrable and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed and a comparison of theoretical results with observations is made.

Bank-to-Turn Guidance Law Using Lyapunov Function and Nonzero Effort Miss

Aguidancelawthatdirectly computesthepitchaccelerationandrollanglecommandsforBank-To-Turn missiles is presented. The nonzero effort miss is introduced, and a Lyapunov function is dee ned in terms of nonzero effort miss. Lyapunov’ s stability theorem is used to obtain a guidance law that completely eliminates the trigonometric polar conversion, which conversion is necessary when the guidance commands and the input to the pitch and the roll autopilots are given in different coordinate systems. When the new guidance law is used, the missile tends to maintain its acceleration command above a certain level during its e ight and thereby avoid a mathematical singularitythatariseswhenatrigonometricinversefunctionisusedtocomputetherollcommand. Arepresentative engagement scenario is used to demonstrate the effectiveness of the proposed guidance law. Numerical simulation results arecompared with previous results and with resultsobtained using a proportional navigation guidance law that uses a polar conversion.

Resonant Motion of a Spin-Stabilized Thrusting Spacecraft

The attitude instability of a spin-stabilized, thrusting upper stage spacecraft is investigated based on a two-body model consisting of a symmetric main body, representing the spacecraft, and a spherical pendulum, representing the liquefied slag pool entrapped in the aft section of the rocket motor. Exact time-varying nonlinear equations are derived and used to eliminate the drawbacks of conventional linear models. To study the stability of the spacecrafts attitude motion, both the spacecraft and pendulum are assumed to be in states of steady spin about the symmetry axis of the spacecraft and the coupled time-varying nonlinear equation of the pendulum is simplified. A quasi-stationary solution to that equation and approximate resonance conditions are determined in terms of the system parameters. The analysis shows that the pendulum is subject to a combination of parametric and external-type excitation by the main body and that energy from the excited pendulum is fed into the main body to develop the coning instability. When one of the resonance conditions and real flight data are used in the original time-varying nonlinear equations, the results match well with the observed motion before and after motor burnout of typical spin-stabilized upper stages. Some numerical examples are presented to explain the mechanism of the coning angle growth and how disturbance moments are generated.

Path Planning and State Estimation for Unmanned Aerial Vehicles in Hostile Environments

U NMANNED aerial vehicles (UAVs) are now widely used in antiterrorism activities and intelligence gathering to enhance mission performance and maximize safety. The susceptibility of these UAVs in hostile environments raises requirements for flight path planning. Path planning strategies in hostile environments are normally composed of two phases [1,2]. The first phase is a Voronoi graph search, which will generate polygonal graphs and will optimize a safety performance index. The second is to use the virtual forces emanated from the virtual field of each surveillance radar site to refine the generated Voronoi graphs. These virtual forces provide information that can be used to reduce the vertices of the Voronoi polygonal and greatly improve the UAV performance. But the curvature continuity of the refined graphs, which plays an important role in the stability of the UAVs’ turning maneuvers, often does not meet the requirements for a continuously flyable path. Many kinds of curves have been studied and designed for UAVs to accomplish their mission [3–5]. Dubins curves, first applied in robotics path planning, are curves along which the UAV can move forward. This kind of circle–line–circle curve has a jump discontinuity in curvature at the connection points between the circle and the line that will cause a robot to stop at these connection points when traveling through the whole path. Other curves, such as the Reeds– Shepp curves [6,7], also have curvature discontinuities at their joint points. An alternative choice, the composite clothoid–line–clothoid curves, can be well designed with curvature being zero at the joint points to eliminate the discontinuities. This allows one to generate a continuous-curvature path by using different kinds of simply shaped curves, although, under most circumstances, we are expecting more flexibility in the curve shape that will allow more space for change. Shanmugavel et al. [3–5]. proposed quintic Pythagorean hodograph (PH) curves for a flyable path, with ten parameters representing each curve. The PH curves are flexible in design and their curvatures are expressed in continuous polynomials. The parameter calculation of a PH curve is an iterative process in order to satisfy different constraints. Such kinds of curves can be further simplified with fewer parameters and a more efficient optimization algorithm. All the methods discussed leave room for improvement in the area of continuous-curvature path planning. The Cornu sprial (CS) [8,9], also known as a clothoid or Euler’s spiral, has wide application in highway and railroad construction, since it can be used to design gradual and smooth transitions in highway entrances or exits. Kelly and Nagy [10] used a parametric CS model to generate real-time nonholonomic trajectories for robotics to minimize the terminal posture error. Here, we consider this CSmodel for use in UAV path planning and investigate how this parametric CS curve works under different constraints. To generate a flyable and safe path with given starting and ending points for UAVs passing through areas covered, at least partially, by several radar sites, the path constraints considered here include 1) minimum accumulative exposure to all radar sites; 2) continuous curvature throughout its length, which will ensure a flyable path’ 3) maximum curvature corresponding to the maximum achievable lateral turn rate; and 4) initial and final boundary constraints. Unlike other path planning problems, including that of moving objects finding the final path, which normally result in motion planning or trajectory planning with system dynamics [11], the path considered here is in a static object environment without dynamic constraints. The work in this paper is based on the developed Voronoi graph; it refines the graph by proposing a generalized CS curve along with a simplified parameter-identification procedure. Most papers on the topic of path planning do not include the information about dynamic state variables of UAVs flying along the planned path. For control purposes, it is beneficial to estimate these state variables to construct complete information of the flight. Kalman filtering [12] has been widely used as an efficient tool in optimal filtering and prediction, especially in the field of state estimation of UAVs performing designated missions [13–17]. For example, Grillo and Vitrano [13] used an extended Kalman filter (EKF) to estimate the state variables and wind velocity for a nonlinear UAV model with Global Positioning System (GPS) measurements. Abdelkrim et al. [14] used an EKF and an H1filter to estimate the localization of UAVs for which the position, velocity, and attitude aremeasured by an inertial navigation system. Campbell andOusingsawat [15] used two different estimators to provide online state and parameter estimation for path planning in uncertain environments. The state estimations in these studies have a commonality, because the UAVs in both cases have sensors or other instruments to provide useful measurement information. If all of the Voronoi points on the initial path are fixed and expected to be followed as closely as possible, they can be assumed asmeasurement points. The generated CS curve is then treated as the reference solution so that the state variables can be estimated by the EKF. The following sections present the procedure for the pathinformation construction in three parts. The first part is the initial rough path of the Voronoi graph and the dynamic programming search algorithm. In the second part, CS curve expression and properties are introduced and different constraints and their mathematical expression are explained. This is followed by the systematicsolution nonlinear programming (NLP) solver. In the last part, the state variables are estimated based on the generated Voronoi points and the refined reference path generated in the first two parts. Simulation results are presented in each part separately.

Path planning for multiple unmanned aerial vehicles by parameterized cornu-spirals

In this paper, a group of cooperative planning paths for simultaneous starting and arriving Unmanned Aerial Vehicles (UAVs) are generated by parameterized Cornu-Spirals (CSs). The continuity and smoothness requirements for the designed flyable paths are achieved by the continuous curvature characteristics of CSs. The final curves are minimized in length with the least number of parameters representing the polynomial expression of the path curvature, while satisfying the maximum curvature constraints, equal length constraints, and collision avoidance constraints. The paths are integrated from initial points to final points by a trapezoidal integration algorithm. A nonlinear programming solver is used to calculate the optimized parameters. Simulation results for four simultaneous UAV paths are presented with designated initial and final positions and attitudes.

Stability criteria of slosh motion with periodicity in a spinning spacecraft

a = transverse component of the system angular momentum C1, C = centers of mass of the main body and the system F = thrust vector (0 0 F3) along the symmetry axis passing through C1 H = angular momentum vector of the system I1, I2, I3 = moments of inertia of the main body about the centroidal principal axes x , y, and z m1, m2 = masses of spacecraft main body and pendulum bob O = pivot point of the pendulum r = length of massless rod connecting the pendulum bob r, rO , rp = position vectors from O to m2, C1 to O, and C1 to m2 r1, r2 = position vectors from C to C1 and m2, respectively xyz = body-fixed frame having its origin at C1 β, βd = resonant and detuned frequencies e = nondimensional small parameter θ , ψ = generalized coordinates of the pendulum θs , θ = stationary point of θ and variation from that point μ = ratio of m2 to the total mass, m1 + m2 σ = parameter describing the nearness of the detuned frequency to the resonance frequency τ = scaled time = angle of rotation of the main body ω = angular velocity of the coordinate system xyz

Predicting the orbital lifetime of tethered satellite systems

Abstract Two models of uncontrolled tethered systems—a lumped-mass model and an orbital element propagation model—are described. Feed-forward artificial neural network architecture and training are discussed. A network training algorithm and hybrid training approach are described and used to train two lifetime prediction networks: one for free tethers and one for upward-deployed tether-trailing satellites. The accuracy of network predictions is demonstrated.

Existence of periodic motions of a tether trailing satellite

Tethered satellite systems are becoming more widely used in space explorations. In this analytical study, the dynamics of a satellite trailing a tether are addressed. More precisely, we derive the conditions for existence of periodic motions about the relative equilibrium states of the tethered system. The system is considered to be affected by the atmospheric drag and nonspherical Earth. The mathematical proof of the existence result is based on the Leray-Schauder degree theory. Main conclusions can be easily generalized for any bounded forcing and/or forces with linear growth.

Simplified Model for Calculation of Backflow Contamination from Rocket Exhausts in Vacuum

A computationally simple procedure for estimating the backflow from a plume expanding into a vacuum has been developed. The continuum flow is modeled as in the Simons method, with straight streamlines radiating from a point on the plume axis of symmetry, constant velocity, and density varying inversely as the square of the radius. The continuous transition from continuum to free-molecular flow is replaced by a suitably defined discontinuity surf ace. Particle number densities outside the discontinuity surface are calculated by assuming that a Maxwellian velocity distribution exists for particles located within small volumes adjacent to that surface. The boundary layer at the nozzle exit is accounted for and may be large compared with the freestream inviscid flow region. Calculated results are compared with existing backflow data for nozzles using CCh as a propellant.

Constrained Initial Guidance Algorithm

An algorithm is presented for determining two-body trajectories that originate on a given two-body orbit, intercept a moving target at a free final time, and are generated by an impulsive velocity change of fixed magnitude. By using a coordinate system defined by the initial and final position vectors of the interceptor, the problem is first reduced to one of solving for the real root of a quartic equation that corresponds to minimum flight time between two specified points subject to the velocity change constraint. An iterative procedure then is used to correct the targeted positon until convergence is obtained with respect to the position of the target. Numerical simulation results presented indicate excellent algorithm performance. Typically, less than 20 iterations are needed io reduce miss distance to less than 1 m if perfect target position information is available. The algorithm provides a rapid method of determining the performance boundaries for systems with specified maximum impulse capability.