Jeng-Shing Chern

Ideal Proportional Navigation

Proportional navigation has been proved to be a useful guidance technique in several surface-to-air and air-to-air homing systems for interception of airborne targets. Besides the familiar pure, true, and generalized proportional navigation guidance laws, a new guidance scheme, called ideal proportional navigation, with commanded acceleration applied in the direction normal to the relative velocity between interceptor and target, is presented. In this study the closed-form solutions of ideal proportional navigation are completely derived for maneuvering and nonmaneuvering targets, and some important characteristics related to the system performance are introduced. Under this scheme the capture criterion is related to the effective proportional navigation constant only, no matter where the initial condition and target maneuver are. With some more energy consumption, this new guidance scheme has a larger capture area and is much more effective than the other schemes.

Analytic Study of Biased Proportional Navigation

Proportional navigation has proved to be a useful guidance technique in several surface-to-air and air-to-air missile systems for interception of airborne targets. An analytic study of the biased proportional navigation with varying closing speed and maneuvering target is presented in this article. A specific target maneuver is considered that is proportional to the closing rate for effective escape during intercept period. The closed-form solutions are completely derived, and some important characteristics related to the system performance are discussed. The effects of the bias factor and target escape factor on the capture criterion and cumulative velocity increment are investigated in detail. In this scheme, the line-of-sight rate approaches a nonzero value, which is beyond the dead-zone of the system. But intercept still can be achieved with a cost of additional energy, and the capture area will be decreased due to the effect of bias.

Minimum-time aerobraking maneuver at constant altitude

Abstract The minimum-time aerobraking maneuver at constant altitude is studied. The space vehicle has lift capability. Its initial position is specified on the longitude-latitude surface at a certain high altitude with initial velocity vector given. A very strong final condition with specified final position and final velocity vector is imposed. The vehicle then performs aerodynamic maneuver at constant altitude to reach the final condition within the shortest time of flight. In order to minimize the time of flight, the vehicle must try to approach the final position as soon as possible. And then at a certain distance from the final position, it performs chattering maneuver to decelerate as quickly as possible so that the final speed can be satisfied. This is a theoretical work to illustrate the existence of chattering arc in an optimal trajectory. The constant altitude restriction is necessary to obtain an exact solution for the control. Dimensionless equations of motion are derived. For one-dimensional flight, we have analytic solutions for both coasting arc and chattering arc. This means that we can obtain an analytic solution for the whole trajectory of one-dimensional flight. For two-dimensional flight, variational formulation is required. The coasting arc proceeds to the final point in some trajectories, or, in many other trajectories, switches to the chattering arc which then leads to the final point. The allowable regions for different cases are constructed.

Optimal trajectories for maximum endurance gliding in a horizontal plane

Nomenclature C0,C/,C2,Cj = constants of integration CD = drag coefficient CDO =zero lift drag coefficient CL = lift coefficient CL* = lift coefficient for maximum lift-to-drag ratio D = drag force E* = maximum lift-to-drag ratio h = altitude H =Hamiltonian function kj,k2,k3 =constants, =C//C0, C2/C0, C3/C0, respectively K = induced drag factor L =lift force m =mass of the vehicle Px>Py>Pu>Pt>Pe = adjoint variables associated with state variables r — radius of penetration 5 = reference area / =time u = dimensionless speed V — speed of vehicle W — weight of vehicle x,y = dimensionless coordinates X,Y = position coordinates of vehicle A =tan/>t 0 = dimensionless time X = normalized lift coefficient jn =bank angle p = density of atmosphere T = normalized time 0 = velocity yaw angle co = dimensionless wing loading

Optimal lift and bank modulations for three-dimensional reentry trajectories with heat constraint

Abstract For hypersonic reentry flight, the heat problem is usually the most severe problem. Therefore, it is of necessity and interest to consider the heat constraint in solving optimal reentry trajectories. This paper, under the facilities of the continuation method and the multiple shooting method, investigates the optimal lift and bank modulations for three-dimensional reentry trajectories with heating rate constraint. The modified Newton method is used to induce and accelerate convergence. From the variational formulation, the optimal lift and bank control laws and the transversality conditions are derived. The non-constrained optimal trajectories leading to the boundary of the maximum reachable domain of a typical lifting reentry vehicle are solved at first. It is a three-parameter two-point boundary-value problem. Then the heating rate constraint is imposed and the constrained maximum reachable domain is constructed finally. Because the equilibrium glide condition is eliminated in this paper, the maximum reachable domain obtained is larger than the one obtained under the equilibrium glide assumption. Besides, both optimal lift and optimal bank control histories are presented and discussed.

Deceleration and heating constrained footprint of shuttle vehicles

With increasing frequency in shuttle operation, it is of interest to have more than one or two landing fields within the boundary of the reachable area of the reentry vehicle. This boundary, called the footprint, depends on the aerodynamic characteristics of the vehicle and is severely restricted by the deceleration and heating constraints imposed upon the atmospheric reentry trajectory. This paper gives a general assessment of the footprint as a function of various deceleration and heating constraints. The difficulties in the computation of the three-dimensional reentry trajectories with optimal modulation in both the angle-of-attack and the bank angle are alleviated by the following devices: (a) nondimensionalizing of the equations of motion and use of the density as the altitude variable; (b) use of the classical integrals of the motion; (c) transformation of the adjoint variables into physical variables; and (d) spherical rotation of the coordinates.

Some aspects of ROCSAT-2 system engineering

Abstract The ROCSAT-2 is designated to the second satellite of the Republic of China. It is expected to be launched in 2003, and is designed primarily to image the region of Taiwan Island. The satellite will be operated in a circular sun-synchronous orbit with exactly 14 revolutions per day. By virtue of the nearly global coverage of the orbit, the images taken by ROCSAT-2 can also be acquired through international cooperation if the foreign ground stations are capable of receiving the downlink data. The satellite will provide high tasking agility for along-track or cross-track imaging with 45° field of regard. The remote sensing instrument (RSI) has four Landsat-like multi-spectral visible bands. It will provide images for 2 m ground sampling distance (GSD) in panchromatic band and 8 m GSD in multi-spectral bands over 24 km swath in the nadir direction. The secondary payload of ROCSAT-2 is a scientific instrument called the imager of sprite, the upper atmospheric lightning (ISUAL). It will be the first payload to observe the upper atmospheric lightning from space. The objective of this experiment is to study the nature of the electrodynamic coupling between thundercloud and upper atmosphere. The system engineering analysis for ROCSAT-2 is presented in this paper.

Approximate chattering arc for minimum time flight

Abstract The purpose of this paper is to investigate the G-constrained approximate chattering arc for the minimum-time aerobraking maneuver of the shuttle-type space vehicle at constant altitude. Theoretically, in a chattering arc of the first kind, the control chatters between its maximum and minimum values at an infinite rate. As an example, for flight at constant altitude, the bank angle switches between its positive and negative maximum values at an infinite rate to generate maximum drag. The resulting flight path is along the arc of a large circle and is one-dimensional. There is a complete analytical solution for this theoretical chattering arc. For practical application, switching of the bank control at an infinite rate is not possible. In the approximate chattering arc, the bank control switches at a finite rate. The resulting flight path is two-dimensional and there is the penalty of a shorter longitudinal range. If we allow the vehicle to coast for a short distance and then change to an approximate chattering arc, the longitudinal range is satisfied and longer flight time becomes the penalty. The penalty of longer flight time is minimized by increasing the number of control switchings and, at the same time, selecting the optimal instants for the switchings. It is found that when the number of control switchings is five, the resulting optimal trajectory is good enough. With more times of control switchings, too much numerical computation must be made while the improvement in the performance index is small. The G constraint has a significant effect on the trajectory.

Optimal three-dimensional glide for maximum reachable domain

A problem of optimal three-dimensional glide for maximum reachable domain is solved by using both the second-order gradient method (GM) and the singular perturbation method (SPM). This problem is essentially an extension of the maximum-range glide problem, in which the flight is constrained in a vertical plane. A set of closed-loop guidance laws which are implementable on an on-board computer are derived by using the SPM. With a numerical example, the trajectory produced by the closed-loop guidance laws is shown to have the same characteristics with the one computed by using the GM. Also, the maximum reachable domain computed by using the SPM is shown to be smaller than the one computed by using the GM.

Optimal vertical ascent to GEO with thrust acceleration and dynamic pressure constraints

Abstract The optimal trajectory for vertical ascent to the geosynchronous earth orbit (GEO) with both dynamic pressure and thrust acceleration constraints will be solved by using the parameter optimization method. The performance index is to maximize the final mass. In other words, the propellant consumption is to be minimized. The time derivative of the velocity magnitude of the vehicle is assumed to be a polynomial function of the flight time, with the coefficients and the flight time as free parameters to be selected. When the thrust control is interior, the required thrust magnitude and angle are derived as functions of the state variables and the polynomial. The acceleration due to the thrust is limited to 2.5 times the gravitational acceleration at the Earths surface. The dynamic pressure is limited to the maximum allowable level for a Space Shuttle ascending flight. For each constraint, the thrust control on the constraint boundary is derived, respectively. A first order polynomial function is adopted for numerical computation. The flight time and the two coefficients are selected such that the final condition for GEO insertion is satisfied and the final mass is maximized. The two constraints are considered separately at first, and then considered together. A laser propulsion system is used with different specific impulse values of 500, 1000, 1500, 2000 and 2500s, respectively. With both constraints, it is found that the final mass remaining is 0.1130, 3.371, 10.42, 18.34 and 25.74%, respectively. The ascending flight time is 1.992 h. The penalty on the performance index incurred by the constraints is less than 0.1%. For a vertical ascending trajectory, the relative speed of the vehicle with respect to the atmosphere is the vertical component of the inertial vehicle velocity.