An application of symplectic integration for general relativistic planetary orbitography subject to non-gravitational forces

Spacecraft propagation tools describe the motion of near-Earth objects and interplanetary probes using Newton’s theory of gravity supplemented with the approximate general relativistic n-body Einstein–Infeld–Hoffmann equations of motion. With respect to the general theory of relativity and the long-standing recommendations of the International Astronomical Union for astrometry, celestial mechanics and metrology, we believe modern orbitography software is now reaching its limits in terms of complexity. In this paper, we present the first results of a prototype software titled General Relativistic Accelerometer-based Propagation Environment (GRAPE). We describe the motion of interplanetary probes and spacecraft using extended general relativistic equations of motion which account for non-gravitational forces using end-user supplied accelerometer data or approximate dynamical models. We exploit the unique general relativistic quadratic invariant associated with the orthogonality between four-velocity and acceleration and simulate the perturbed orbits for Molniya, Parker Solar Probe and Mercury Planetary Orbiter-like test particles subject to a radiation-like four-force. The accuracy of the numerical procedure is maintained using a 5-stage, $$10^\mathrm{th}$$ 10 th -order structure-preserving Gauss collocation symplectic integration scheme. GRAPE preserves the norm of the tangent vector to the test particle worldline at the order of $$10^{-32}$$ 10 - 32 .

Symplectic Integration for Multivariate Dynamic Spline-Based Model of Deformable Linear Objects

Deformable Linear Objects (DLOs) such as ropes, cables, and surgical sutures have a wide variety of uses in automotive engineering, surgery, and electromechanical industries. Therefore, modeling of DLOs as well as a computationally efficient way to predict the DLO behavior are of great importance, in particular to enable robotic manipulation of DLOs. The main motivation of this work is to enable efficient pre- diction of the DLO behavior during robotic manipulation. In this paper, the DLO is modeled by a multivariate dynamic spline, while a symplectic integration method is used to solve the model iteratively by interpolating the DLO shape during the manipulation process. Comparisons between the symplectic, Runge-Kutta and Zhai integrators are reported. The presented results show the capabilities of the symplectic integrator to overcome other integration methods in predicting the DLO behavior. Moreover, the results obtained with different sets of model parameters integrated by means of the symplectic method are reported to show how they influence the DLO behavior estimation.