Existing aerospace flight trajectory programs simulate the motion of aerospace vehicles by modeling external forces and moments acting on each body, but lack provisions for determining reaction forces and moments exerted by one body on another through a connecting joint. These reaction forces and moments are also known as constraint forces and moments because they permit specified motion of one body relative to another, and, at the same time, prohibit all other relative motions. In other words, a joint imposes certain constraints on relative motion.

Previous techniques for simulating the motion of aerospace vehicles having multiple bodies connected by joints fall into one of two categories. The first category consists of simulations that are created for specific vehicle configurations. They are often performed in a single simulation environment, highly customized for a particular vehicle configuration, and require significant effort to develop. Such specific, customized simulations cannot be easily modified for different applications. Techniques that fall within this category usually approximate the constraint forces and moments with spring/damper systems to simulate constrained motion in a launch vehicle staging environment.

A second category involves approaches that subdivide the trajectory simulation problem into distinct phases, each of which is handled by a different software program. Typically, a conventional trajectory simulation program is used to model the unconstrained motion, and a specialized mechanical analysis and design program solves the constrained-motion (separation) segment of flight. Performing end-to-end mission simulation with this approach involves linking of multiple codes and ensuring proper input/output interfaces.

This innovation, known as the Constraint Force Equation (CFE) Solver, is a physics-based approach for simulating the constrained motion of aerospace vehicles or vehicle segments that are connected by simple joints. The CFE Solver can be easily implemented in a conventional trajectory simulation software program used to simulate unconstrained motion, thus providing a means for these existing, widely used tools to accurately simulate vehicle motion that is constrained by joints.

The CFE Solver computes the internal joint loads (constraint forces/moments) acting at the joints connecting multiple bodies in motion, and applies them as additional external forces/moments. The CFE solver enables users to simulate the dynamics of systems of multiple rigid bodies connected by multiple joints that permit certain relative motion between the bodies. The CFE method is not new, but this approach of using CFE Solver to compute internal joint loads and apply them as external loads is new. This new approach enables users to solve equations governing complex multi-body dynamics in a simple manner, and also enables implementation of CFE solver in a trajectory optimization environment like POST2 (Program to Optimize Simulated Trajectories II) without major modification to the basic trajectory optimization code. The CFE Solver implemented in POST2 enables users to perform trajectory optimization and end-to-end simulation of launch vehicle trajectories, including multi-body stage separation in a seamless and efficient manner.

The CFE Solver is based on Newton’s laws of motion. In the literature, it is also known as the Lagrange multiplier method for treating the constrained motion of multiple bodies. The CFE Solver consists of an algorithm that is divided into several computer programs. These algorithms are designed so that the user can select the type of the joint(s) connecting multiple bodies, and selectively turn on or turn off the applicable degrees of translational or rotational freedom.

Once the user specifies all of the inputs to the CFE algorithm, the CFE code is compiled and ready for execution. If the user wants to run it in a standalone mode, they can do that. However, if the user wants to run it in a trajectory optimization/simulation environment like POST2, then he or she compiles/links the CFE Solver with POST2 software and runs POST2 in the usual fashion.

This work was done by Bandu Pamadi, Paul Tartabini, Matthew Toniolo, Carlos Roithmayr, Chris Karlgaard, and Cindy Albertson of Langley Research Center. For more information, contact Langley Research Center at This email address is being protected from spambots. You need JavaScript enabled to view it.. Refer to LAR-18349-1.

NASA Tech Briefs Magazine

This article first appeared in the April, 2016 issue of NASA Tech Briefs Magazine.

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