Tangential Impulses Research Articles

FESD-J: Finite Elements with Switch Detection for numerical optimal control of rigid bodies with impacts and Coulomb friction

The Finite Elements with Switch Detection (FESD) is a high-accuracy method for the numerical simulation and solution of optimal control problems subject to discontinuous ODEs. In this article, we extend the FESD method (Nurkanović et al., 2022) to the dynamic equations of multiple rigid bodies that exhibit state jumps due to impacts and Coulomb friction. This new method is referred to as FESD with Jumps (FESD-J). Starting from the standard Runge–Kutta equations, we let the integration step sizes be degrees of freedom. Additional constraints are introduced to ensure exact switch detection and to remove spurious degrees of freedom if no switches occur. Moreover, at the boundaries of each integration interval (finite element), we impose the impact equations in their complementarity form, at both the position and velocity level. They compute the normal and tangential impulses in case of contact making. Otherwise, they are reduced to the continuity conditions for the velocities. FESD-J treats multiple contacts, where each contact can have a different coefficient of restitution and friction. All methods introduced in this paper are implemented in the open-source software package NOSNOC (Nurkanović and Diehl, 2022). We illustrate the use of FESD-J in both simulation and optimal control examples.

Method of SLAS's ground track manipulation based on tangential impulse thrust

Method of SLAS's ground track manipulation based on tangential impulse thrust

Reachable set analysis of practical trans-lunar orbit via a retrograde semi-analytic model

Different from the Apollo flight mode, a safer trans-lunar flight mode for the crews is preferred. The previous-arrived lunar module rendezvouses with the crew exploration vehicle at a low lunar destination orbit, and then the crews ride the lunar module to descend the lunar surface sampling destination. The lunar module, which includes the descent and ascent stages, flies from a low Earth orbit to the low lunar destination orbit with two tangential impulses. The low lunar destination orbital reachable set of the practical trans-lunar orbit limits the feasible lunar surface sampling region. Therefore, this paper addresses the low lunar destination orbital reachable set of the practical trans-lunar orbit. A retrograde semi-analytic model is proposed for rapidly computing the practical two-impulse trans-lunar orbit firstly, which refers the ephemeris table twice for more precision perilune orbital elements. The reachable set is generated using the multiple-level traversal searching approach with this retrograde semi-analytic mode. Its envelope is re-checked by the continuation theory with the high-precision orbital model. Besides, some factors that affect the reachable set are also measured. The results show that neither the Earth–Moon distance nor the trans-lunar duration affects the reachable set. However, if the trans-lunar injection inclination is smaller than the inclination of the moon’s path, the reachable set becomes smaller or even reduces into an empty set. In brief, the proposed retrograde semi-analytic model for computing the reachable set provides a helpful and fast tool for selecting an applicable lunar surface sampling site for the manned lunar mission overall design.Graphical

A graphical method for the analysis of a satellite’s in-orbit breakup through optical observations

The incessant overcrowding of the space around the Earth with new operative satellites is a critical issue in the Space Traffic Management (STM) framework, especially in last years with the beginning of the new space economy. The operative life of the active satellites is seriously endangered by the numerous uncontrolled objects, i.e., the space debris, which populate the outer space. The consequence of an impact between these objects may be catastrophic, as proved by the events in the recent past. In this context, the in-orbit explosions both of dismissed satellites and rockets stages are one of the main causes of space debris generation. For this reason, a suitable observation and detection strategy of the fragments generated by these events is necessary, e.g., by means of optical systems. In this work, the Sapienza Space Systems and Space Surveillance Laboratory (S5Lab) shows the results, by means of a graphical method, of the short-term analysis of the in-orbit explosions that involved the third stage of the Russian rocket FREGAT-SB and the fragmentation of the military Chinese satellite YUNHAI 1–02. In order to approach the study of a fragmentation event, an observation strategy and an analysis method have been developed. The research of fragments was focused on the orbital plane leading to estimation of fragments’ number and their distribution. The measurements obtained through the images were analyzed in order to understand if the objects present in the images are either included in the North American Aerospace Defense Command (NORAD) catalogue or not. The unknowns could be fragments. The next analysis follows the hypothesis of the isotropic explosion confined only on the tangential impulses. In first approximation, only semi-major axis and eccentricity change due to the impulse. Therefore, it is possible to generate a new Two-Line Elements (TLE) for each fragment leading to an association between the impulses and the affected TLEs. A new, simple, and fast analysis method was developed, specifically to improve the impulse association. It consists in a graphical method where it is possible to see the distribution of the fragments with respect to the original body using the angular distance trend between them.

Solar Sail Augmented Hohmann Transfer

The aim of this Correspondence is to analyze the performance of a solar sail-based spacecraft in a two-impulse orbit transfer between circular and coplanar heliocentric orbits of assigned radii. In particular, assuming a transfer trajectory with a sweep angle and a flight time equal to those of a classical Hohmann orbit, the sail propulsive acceleration is used to minimize the total velocity variation required by the two tangential impulses. In this sense the work quantifies the optimal performance of a solar sail augmented Hohmann transfer as a function of the sail characteristics.

Analysis of normal and tangential restitution coefficients in car collisions based on finite element method

In momentum-based impact models used in accident reconstructions, the restitution coefficient is a key role to determine impulses and delta-V accurately in car-to-car collisions. The restitution coefficients can be defined as the normal and the tangential direction in the contact plane, and the same value of the normal and tangential coefficients is employed in a two-dimensional PC-Crash model. Ishikawa (1993) proposed an impact model with the normal and tangential restitution coefficient, however, to the best of the author's knowledge, few studies have further investigated tangential restitution coefficients in detail. In this study, the normal and tangential restitution coefficients of the impact model were investigated based on the finite element (FE) analysis. FE simulations of angled barrier crashes with varying barrier angles as well as 16 car-to-car collisions were conducted. As the contact plane slides, the ratio of the tangential impulse to the normal impulse becomes the same value with the maximum friction coefficient. The tangential restitution coefficient is related to whether sliding occurs on the contact plane, and can be expressed as a function of the impulse ratio. The tangential restitution coefficient decreased with increasing the impulse ratio and is positive in non-sliding collisions and negative in sliding collisions. An approximate expression for the tangential restitution coefficient was formulated as a function of the generalized impulse ratio. The calculated impulses of the impact model with normal and tangential restitution coefficients were closer to the FE analysis than the PC-Crash model which uses a constant restitution coefficient of 0.1.

Transversal mass transfer and shear stress formation during rapid gravity flow of a granular medium

The micro structural models for shear stress generation during rapid gravity flow of granular materials on a rough chute are discussed. The mechanism of the shear stress formation, taking into account the tangential impulse formed under transversal mass transfer of particles, is suggested. The analogy between granular media during rapid shear deformation and dense gases is used to develop the suggested mechanism on the basis of kinetic theory. The total shear stress is determined as the sum of the stress components induced by collisions, transversal mass transfer and contact interactions of uniform cohesionless inelastic spherical particles. The mathematical models describing the components of shear kinetic stresses are developed as the functions of particle properties, structural and kinematical gravity flow characteristics. The equations of impulse and energy conservation in the course of rapid gravity flow of uniform cohesionless particles are formulated. A variant of the formulation of boundary conditions at the flow bottom is proposed for mathematical modeling of the dynamics of rapid gravity flows of granular materials on a rough chute. The variant assumes the displacement of the area with the most intense shear rate inside the flow and into its layers adjacent to the rough chute surface.

Envelopes of Spacecraft Trajectories with a Single Impulse

This paper analyzes reachable domains of spacecraft using a single impulsive maneuver. In particular, compact expressions are obtained of the envelopes of spacecraft trajectories in closed form, in both cases of either radial or tangential impulse. Suitable bounds are enforced on the magnitude of the velocity variation to obtain an elliptic transfer trajectory after the maneuver with a pericenter radius greater than the primary body’s radius. Three different cases are investigated: (1) the impulse point is fixed and the velocity variation may be varied; (2) a fixed impulse magnitude and free maneuver point; (3) both free impulse point and magnitude. Finally, some mission scenarios are analyzed to show the effectiveness of the proposed analytical method.

High-speed helmeted head impacts in motorcycling: A computational study

The motorcyclist is exposed to the risk of falling and impacting ground head-first at a wide range of travelling speeds – from a speed limit of less than 50 km/h on the urban road to the race circuit where speed can reach well above 200 km/h. However, motorcycle helmets today are tested at a single and much lower impact speed, i.e. 30 km/h. There is a knowledge gap in understanding the dynamics and head impact responses at high travelling speeds due to the limitation of existing laboratory rigs.This study used a finite element head model coupled with a motorcycle helmet model to simulate head-first falls at travelling speed (or tangential velocity at impact) from 0 to 216 km/h. The effect of different falling heights (1.6 m and 0.25 m) and coefficient of frictions (0.20 and 0.45) between the helmet outer shell and ground were also examined. The simulation results were analysed together with the analytical model to better comprehend rolling and/or sliding phenomena that are often observed in helmet oblique impacts.Three types of helmet-to-ground interactions are found when the helmet impacts ground from low to high tangential velocities: (1) helmet rolling without slipping; (2) a combination of sliding and rolling; and (3) continuous sliding. The tangential impulse transmitted to the head-helmet system, peak angular head kinematics and brain strain increase almost linearly with the tangential velocity when the helmet rolls but plateaus when the helmet slides. The critical tangential velocity at which the motion transit from the rolling regime to the sliding regime depends on both the falling height and friction coefficient. Typically, for a fall height of 1.63 m and a friction coefficient of 0.45, the rolling/sliding transition occurs at a tangential velocity of 10.8 m/s (38.9 km/h).Low sliding resistance in helmet design, i.e. by the means of a lower friction coefficient between the helmet outer shell and ground, has shown a higher reduction of brain tissue strain in the sliding regime than in the rolling regime. This study uncovers the underlying dynamics of rolling and sliding phenomena in high-speed oblique impacts, which largely affect head impact biomechanics. Besides, the study highlights the importance of testing helmets at speeds covering both the rolling and sliding regime since potential designs for improved head protection at high-speed impacts can be more distinguishable in the sliding regime than in the rolling regime.

On state and inertial parameter estimation of free-falling planar rigid bodies subject to unscheduled frictional impacts

This paper addresses the problem of simultaneous state estimation and inertial and frictional parameter identification for planar rigid-bodies subject to unscheduled frictional impacts. The aim is to evaluate to what level of accuracy, given noisy captured poses of an object free-falling under gravity and impacting the surrounding environment, it is conceivable to reconstruct its states, the sequence of normal and tangential impulses and, concurrently, estimate its inertial properties along with Coulomb’s coefficient of friction at contacts.To this aim we set up a constrained nonlinear optimization problem, where the unscheduled impacts are handled via a complementarity formulation. To assess the validity of the proposed approach we test the identification results both (i) with respect to ground truth values produced with a simulator, and (ii) with respect to real experimental data. In both cases, we are able to provide accurate/realistic estimates of the inertia-to-mass ratio and friction coefficient along with a satisfactory reconstruction of systems states and contact impulses.

Stress and Vibration Analysis of a PDC (Pulse Detonation Chamber)

Abstract Pulse detonation engines work similar to pulsejets, with the major difference taking place in the combustion mechanism. Having detonation, instead of conventional combustion comes with certain benefits, but also, a couple of major issues which will be addressed in this paper. These changes come as a result of the increased amount of energy released by the explosive process, with much higher fluid velocities and temperatures that easily surpass those of conventional engines. These high values of the thermodynamic parameters prove to be a tough technological problem when it comes to producing this type of engines, so this paper aims at analyzing the evolution of these parameters throughout a working cycle of the Tangential Impulse Detonation Engine (TIDE) and how they affect the internal structure of the detonation chamber. The non-steady character of the thermodynamic parameters from a Computational Fluid Dynamics (CFD) simulation will be used a as a starting point for the Finite Element Analysis (FEA), which will help determine whether or not, the structure can withstand the immense load that comes with such violent processes. The final part of this article will provide optimization criteria to decrease the loads generated by the pulsed detonations.

Optimal transfer between elliptic orbits with three tangential impulses

This paper introduces a mathematical model that can be used to evaluate the total velocity variation required to accomplish a given two-dimensional orbit transfer, using up to three tangential impulsive maneuvers. The problem is addressed in an optimal framework, by looking for the transfer trajectory that minimizes the total velocity variation. In particular, by exploiting the boundary nonlinear constraint equations, the total velocity variation can be calculated as a function only of the spacecraft angular position at which the impulses are applied. The small number of control variables involved in the algorithm allows the optimization problem to be solved in a simple and robust way, with a small computational effort. The algorithm is able to find the optimal transfer strategy in a generic ellipse-to-ellipse, two-dimensional, mission scenario.

Joint kinematics and ground reaction forces in overground versus treadmill graded running

BackgroundTreadmills are often used to assess running biomechanics, however the validity of applying results from treadmill graded running to overground graded running is currently unknown. Research questionThe purpose of this study was to investigate whether treadmill and overground graded running have comparable kinematics and ground reaction force parameters. MethodsEleven healthy male adults ran overground and on an instrumented treadmill as motion capture and force platform data were collected for the following conditions: downhill running at a slope of −8° at 10, 13 and 16 km⋅h−1; level running at 10 and 13 km⋅h−1; uphill running at a slope of +8° at 8, 10 and 13 km⋅h−1. Sagittal joint angles at heel strike, mid-stance, and toe-off were computed for the ankle, knee and hip. Ground reaction force parameters including peak average and instantaneous normal loading rate, peak impact and active normal force, peak tangential (braking and propulsive) forces, and normal and tangential impulses were also calculated. ResultsJoint kinematics and ground reaction forces for level running were generally similar between overground and treadmill conditions. The following variables were significantly higher during overground uphill running (mean difference ± SD): average normal loading rate (14.4 ± 7.1 BW⋅s−1), normal impulse (0.04 ± 0.02 BW⋅s), propulsive impulse (0.04 ± 0.02 BW⋅s), and vertical center of mass excursion (0.092 ± 0.031 m). The following variables were significantly higher during overground downhill running (mean difference ± SD): ankle plantarflexion at toe-off (−5.39 ± 6.19°) and vertical center of mass excursion (0.046 ± 0.039 m). SignificanceThese findings suggest that subtle differences in kinematics and ground reaction forces exist between overground and treadmill graded running. These differences aside, we believe that overground kinematics and ground reaction forces in graded running are reasonably replicated on a treadmill.

Transformation of sliding motion to rolling during spheres collision

In this research, we have investigated the three-dimensional elastic collision of two balls, based on friction in the tangential plane. Our aim is to offer analytical closed form relations for post collision parameters such as linear and angular velocities, collision time and tangential and normal impulse in three dimensions. To simplify the problem, stick regime is not considered. In other words, balls have a low tangential coefficient of restitution. Sliding, sliding then rolling, and rolling at the beginning of contact are three cases that can occur during impact which have been considered in our research. The normal interaction force is described by the Hertz contact force and dimensionless analysis is used for investigating normal interaction force; furthermore, Coulomb friction is considered during sliding. Experimental data for collisions show when sliding exists through the impact, tangential impulses can be taken as frictional impulses using the Coulomb law if the dynamic regime is not stick regime. To identify transformation of sliding motion to rolling or sticking during the impact process, linear and trigonometric functions are considered as an approximation for the normal interaction force. Afterwards, we have obtained the condition for the possibility of this transformation; moreover, we can estimate the duration of sliding and rolling or sticking. We have obtained an analytical solution for maximum force and deformation, collision time, impulses and post-collision linear and angular velocities in three dimensions.

Analysis and Computation of Two Body Impact in Three Dimensions

A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

An Impact Dynamics Modeling Method with Tangential Impulse in Three-Dimensional Space

This paper studies the impact of two rigid bodies considering tangential friction in three dimensional space. It extends Stronge’s spring-based planar contact structure to three dimensions by introducing three orthogonal virtual springs. Impact analysis is carried out using normal impulse rather than time as the only independent variable, unlike previous work on tangential impulse. This makes the algorithm more compact. At last, collision is governed by a system of differential equations. Modularity of the impact model makes it easy to be integrated into a multi-body system.

Impact behavior of a superball

When a superball is incident with backspin on a horizontal surface, at an angle near the normal, it bounces backwards and with a reversal in the spin direction. The effect is less noticeable with other balls, which may bounce with reduced spin and without a reversal in the spin direction. The outcome depends on the friction force acting on the ball. Measurements with several different rubber balls are presented showing that the time history of the friction force, the resulting tangential impulse, and the resulting tangential coefficient of restitution all depend on both the coefficient of sliding friction and the ratio of the tangential vibration period to the impact duration of the ball. Grip and slip phases of the bounce are identified visually by allowing the ball to impact on chalk lines drawn on a blackboard, and by video-recording the bounce of a rectangular block of rubber.

Generalised multi-impulsive manoeuvres for optimum spacecraft rendezvous in near-circular orbit

This work describes the design of an impulsive manoeuvres' planner meant for onboard autonomous optimum formation flying reconfigurations in near-circular orbit. The whole variation of the relative orbit is stepwise achieved through intermediate configurations, so that passive safety and delta-v consumption minimisation are pursued. The description of the relative motion is accomplished in terms of relative orbital elements and the reconfiguration plan takes into account mean effects due to the Earth oblateness coefficient and differential drag. Manoeuvres consist of sets of triple tangential impulses and a single out-of-plane burn to establish each intermediate configuration. They are scheduled in time intervals compliant with the user-defined permissible time control windows.

Three-dimensional impact: energy-based modeling of tangential compliance

Impact is indispensable in robotic manipulation tasks in which objects and/or manipulators move at high speeds. Applied research using impact has been hindered by underdeveloped computational foundations for rigid-body collision. This paper studies the computation of tangential impulse as two rigid bodies in the space collide at a point with both tangential compliance and friction. It extends Stronge’s spring-based planar contact structure to three dimensions by modeling the contact point as a massless particle able to move tangentially on one body while connected to an infinitesimal region on the other body via three orthogonal springs. Slip or stick is indicated by whether the particle is still or moving. Impact analysis is carried out using normal impulse rather than time as the only independent variable, unlike in previous work on tangential compliance. This is due to the ability to update the energies stored in the three springs. Collision is governed by a system of differential equations that are solvable numerically. Modularity of the impact model makes it easy to be integrated into a multibody system, with one copy at each contact, in combination with a model for multiple impacts that governs normal impulses at different contacts.

Analytic solution of the problem of rotation of a sphere in a rarefied molecular gas with allowance for the accommodation coefficients

In the case of the slip flow regime expressions both for the mass velocity of a rarefied molecular gas entrained by a rotating sphere and for the moment of the friction forces exerted on the sphere by the gas are obtained with allowance for the second-order correction with respect to the Knudsen number. It is demonstrated that these quantities depend on the Prandtl number and the accommodation coefficients of the tangential impulse of the gas molecules and its flux toward the sphere surface. The results are compared with analogous results obtained in the case of diffuse reflection of the gas molecules by the sphere surface.