Robert M. Howe

Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrangian Equations

Conventional holonomic or nonholonomic constraints are defined as geometric constraints. The total enregy of a dynamic system can be treated as a constrained quantity for the purpose of accurate numerical simulation. In the simulation of Lagrangian equations of motion with constraint equations, the Geometric Elimination Method turns out to be more effective in controlling constraint violations than any conventional methods, including Baumgarte’s Constraint Violation Stabilization Method (CVSM). At each step, this method first goes through the numerical integration process without correction to obtain updated values of the state variables. These values are then used in a gradient-based procedure to eliminate the geometric and energy errors simultaneously before processing to the next step. For small step size, this procedure is stable and very accurate.

Hybrid Computer Systems [Guest editorial]

Todays increasing emphasis on simulation of continuous dynamic systems lends a special significance to this issue of Computer. Hybrid computers, with their tremendous computing speed resulting from the all-parallel configuration of their analog subsystem, are particularly suited for dynamic system simulation. Indeed, back in the 1950s analog computers were the only game in town when it came to real-time simulation of flight vehicles. Digital computers of that era were simply too slow in the numerical solution of differential equations.

Stability and Accuracy Analysis of Baumgarte’s Constraint Violation Stabilization Method

When Baumgartes Constraint Violation Stabilization Method (CVSM) is used in the simulation of Lagrange equations of motion with holonomic constraints, it is shown that, with suitable assumptions on the integration step size h and the eigen-values (λs) of the linearized system, the constraint variables are effectively integrated by the same algorithm as that used for the state variables. A numerical stability analysis of the constraint violations can be performed using this so-called pseudo-integration equation. A study is also made of truncation errors and their modeling in the continuous time domain. This model can be used to determine the effectiveness of various constraint controls and integration methods in reducing the errors in the solution due to truncation errors. Examples are presented to illustrate the use of a higher-order truncation error model which leads to an accurate quantitative steady-state analysis of the constraint violations.

Trigonometric Resolution in Analog Computers by Means of Multiplier Elements

A method of generating sine and cosine functions in analog computers by means of multiplier elements and integrators is discussed. Static accuracy of the method is analyzed and found to be essentially equal to the accuracy of the multipliers employed. The system accepts d?/dt as the input and generates output voltages of sin? and cos ?. Amplitude-stabilizing loops are employed to maintain sin2?+cos2? = 1. Advantages of the method include representation of unlimited range in angles, dynamic capabilities far beyond that of the multipliers alone, and possibility of employing electronic multipliers. The method has been successfully used to compute Euler angles in analog computer solutions of the three-dimensional flight equations.

Reachable domain for interception at hyperbolic speeds

Abstract The interception of an ICBM at low altitude and in a short time requires hyperbolic speeds. At the acquisition time t0, if updated information about the trajectory of the target dictates a velocity correction for the interception, then it is of interest to assess the potential for interception. The reachable surface is defined as the boundary of the set of all attainable points at a given time t, for a fuel potential, conveniently expressed in terms of the Δv available. Properties of this surface are derived and its mathematical characterization is obtained. As an application of the concept, a necessary condition for a successful interception of a target given in terms of a capture function is explicitly derived.

OPTIMAL AEROASSISTED INTERCEPT TRAJECTORIES AT HYPERBOLIC SPEEDS

This paper considers the optimization of Earth-based multistage rocket interceptors with very short flight times and long ranges. The objective is the minimization of the launch mass as a function of interceptor design variables such as stage size, engine burn times, and the angle-of-attack program. Because of the demanding target conditions, the payload reaches hyperbolic speeds and the centrifugal forces greatly exceeds the gravity force. The minimization of launch mass shows that the needed downforce on the payload is best provided by negative aerodynamic lift. The description of such negative-lift, aeroassisted optimal trajectories is a principal goal of the paper. Topics treated include the following: a model for the multistage interceptor, the formulation of the optimization problem, the mathematical derivation of a universal curve that provides a simple and accurate model for negative-lift segments of the optimal trajectories, and effective procedures for efficient numerical optimization. Results of solution studies are reported. For flight times of 6 min and a range of about 3000 miles, a five-stage interceptor requires a mass ratio of several thousand. The dependence of the optimal mass ratio on key design and target parameters is described.

A new family of real-time predictor-corrector integration algorithms

This paper describes a variation of the conventional two-pass explicit Adams- Moulton predictor-corrector integration methods which is suitable for real-time simulation. In this new method the first pass through the state equations uses an Adams-Bash forth type of predictor algo rithm to campute an estimate of the state at the n+1/2 frame instead of the n+1 frame, as is customary. This estimate is then used to compute the derivative at the n+1/2 frame which, along with derivatives at the n, n-1, n-2, ... frames is used in the final corrector pass to calculate the state at the n+1 frame. Unlike conventional two-pass Adams- Moulton methods, these new versions are compatible with real-time inputs. The paper shows that they are also superior based on dynamic accuracy measures and stability measures. A three-pass predictor-corrector integration algorithm compatible with real- time inputs is also presented and shown to yield significantly more accurate results than 3rd-order RK (Runge-Kutta) integra tion.

Generalized guidance law for collision courses

A new generalized guidance law for collision courses is presented. When the missile and target axial accelerations or decelerations are constant, there exists a rectilinear collision course. The guidance law presented, which is called the true guidance law, gives theoretical lateral acceleration commands to guide the missile on a collision course. However, since it is very difficult to realize the true guidance law on most existing tactical missiles, this paper shows a method for simply implementing the guidance law, which is called the simplified guidance law. The small perturbation equation of the true guidance law shows that the definition of an effective navigation constant is the same expression as that in the case of conventional proportional navigation. The performance of the two guidance laws presented is compared with that of proportional navigation using simulation studies of a simple model of a short range air-to-air missile. The simulation results show that the guidance laws presented can intercept the target using far smaller lateral acceleration commands than prepared for proportional navigation. The inner launch envelope shows that the guidance laws presented provide an overall performance improvement over proportional navigation.

Constraint Violation Stabilization Using Gradient Feedback in Constrained Dynamics Simulation

Conventional holonomic or nonholonomic constraints are defined as geometric constraints in this paper. If the total energy of a dynamic system can be computed from the initial energy plus the time integral of the energy input rate due to external or internal forces, then the total energy can be artificially treated as a constraint. The violation of the total energy constraint due to numerical errors during simulation can be used as information to control these errors. When geometric constraint control is combined with energy constraint control, numerical simulation of a constrained dynamic system becomes more accurate. An energy constraint control based on the gradient feedback of the energy constraint violation leads to a new method to control both geometric and energy constraint violations, so-called constraint violation stabilization using gradient feedback. A new convenient and effective method to implement energy constraint control in numerical simulation is developed based on the geometric interpretation of the relation between constraints in the phase space. Several combinations of energy constraint control with either Baumgartes constraint violation stabilization method or the new constraint violation stabilization using gradient feedback are also addressed. Finally, a new method for implementing constraint controls is developed by using the Euler method for integrating constraint control terms, even when higher-order integration methods are used for all other integrations.

Application of singular perturbation methods for three-dimensional minimum-time interception

In this paper, a feedback control law based on the singular perturbation method is developed for three-dimensional minimum-time interception. Whereas the heading and flight-path angles are considered fast variables with the same time scale, the relative position and the specific energy are considered slow variables. A zeroth-order optimal control algorithm is developed, and an extension to higher-order analysis is discussed. With the demonstrations of several numerical examples, it is shown that this time-scale separation is physically reasonable and results in a uniformly valid control law for long-, medium-, and short-range interception.