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Even though classical mechanics states that it is not possible to find an inertial reference frame, relative measurements are used to determine the mechanical behaviour of celestial bodies with very small dimensions compared to the distances between them. The present work aims to demonstrate that the two fundamental theorems of mechanics, the momentum theorem and the angular momentum theorem, remain valid if we use the relative distances and velocities between these celestial bodies. The starting point for developing such a result is that the basic assumptions of Newtonian mechanics are considered to be valid and are used with the assumption that an inertial reference frame can exist, even if only hypothetically. Relationships are established between the relative accelerations and the global forces acting on each body in a two-body ensemble, which are quite similar to those stated in an inertial reference frame.

Several researchers are considering the plausibility of being able to rapidly launch a mission to an asteroid, which would fly in close proximity of the asteroid to deliver an impulse in a particular direction so as to deflect the asteroid from its current orbit. Planetary motion, in general, and the motion of asteroids, in particular, are subject to planetary influences that are characterised by a kind of natural symmetry, which results in an asteroid orbiting in a stable and periodic or almost periodic orbit exhibiting a number of natural orbital symmetries. Tracking and following an asteroid, in close proximity, is the subject of this paper. In this paper, the problem of synthesizing an optimal trajectory to a NEO such as an asteroid is considered. A particular strategy involving the optimization of a co-planar trajectory segment that permits the satellite to approach and fly alongside the asteroid is chosen. Two different state space representations of the Hill–Clohessy–Wiltshire (HCW) linearized equations of relative motion are used to obtain optimal trajectories for a spacecraft approaching an asteroid. It is shown that by using a state space representation of HCW equations where the secular states are explicitly represented, the optimal trajectories are not only synthesized rapidly but also result in lower magnitudes of control inputs which must be applied continuously over extended periods of time. Thus, the solutions obtained are particularly suitable for low thrust control of the satellites orbit which can be realized by electric thrusters.

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the J 2 effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the J 2 perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a J 2 -perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the J 2 perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a J 2 -perturbed reference orbit is also introduced.

The work concerns dynamic similarity criteria of various phenomena occurring in hydraulics and fluid dynamics originally derived from ratios of forces and forces moments affecting these phenomena. The base of dynamic similarity criteria formulations and considerations is A. Flaga’s method and procedure for determining dynamic similarity criteria in different issues of fluid–solid interactions i.e. at different fluid–solid relative motions. The paper concerns the determination and analysis of dynamic similarity criteria for various practical problems encountered mainly in hydraulics and fluid dynamics at steady, smooth fluid onflow in front of a solid. Moreover, the cases of mechanically induced vibrations of a body in a stationary fluid moving with constant velocity in front of the body have been presented. Assuming authorial method and procedure for determining dynamic similarity criteria, its have been presented and analysed in the paper both well known similarity numbers obtained in another way (e.g. from dimensional analysis or differential equations for particular problems – as Reynolds, Froude, Euler, Cauchy, Strouhal, Mach numbers) – as well as several new similarity numbers encountered in different fluid solid interaction problems (e.g. new forces and moments coefficients encountered in problems of vibrating solid bodies in fluids).

The Flying Bluebottle Illusion is a compelling example of how the perceived trajectory of moving objects can be greatly influenced by other motion sources in the visual scene. In this article, we present a series of simplified variants of the Flying Bluebottle Illusion in which the true motion of a target is a circular orbit around a central point. However, when a similar but offset orbiting motion trajectory is added to a set of surrounding inducers, the perceived trajectory of the target is drastically altered in both extent and direction. In other words, the perceived orbiting motion of the target is “pulled” and distorted by the orbiting motion of the inducers. For simplicity's sake, we refer to the illusory effect revealed by these dueling orbits as the Dueling Orbit Illusion. These simplified variants lend themselves to empirical study with resultant effects that can be readily modeled. Here, we present a series of case examples for how the parameters of the stimuli may be varied to yield predictable effects, describe a straightforward computational model for quantifying the magnitude of the contextual influence, and discuss how the model may be leveraged to gain insight into the phenomenon of induced motion across a range of within and between observer domains.

In flying Ad hoc networks (FANETs), traditional geographic routing protocols such as greedy perimeter stateless routing (GPSR) protocol, usually adopts a periodic Hello mechanism and a greedy forwarding mechanism based on maximum advanced distance. However, the traditional periodic Hello mechanism ignores node mobility, network connectivity, and traffic type, thus causing temporary communication blindness (TCB). Additionally, the traditionally geographic routing protocols choose the next-hop node based on the maximum advanced distance. The greedy forwarding mechanism based on maximum advanced distance towards the destination ignores the impact of network dynamics on links, leading to short link lifetime and frequent link breakage. To address these problems, a novel airborne greedy geographic routing (AGGR) protocol for FANETs is proposed in this paper. It includes an adaptive Hello mechanism and a greedy forwarding mechanism based on the relative motion of nodes. In the adaptive Hello mechanism, the Hello period of working nodes is calculated according to the real-time relative characteristic values between the node and its upstream node, and the Hello period of idle nodes adopts a fixed value according to the movement characteristics relative to all neighbor nodes. In the greedy forwarding mechanism based on the relative motion of nodes, in order to adapt to the high dynamic changes of network topology, the time to enter the communication range of the destination is taken as routing criteria to choose the next-hop node. Moreover, the AGGR protocol is compared with the original GPSR in simulation. The results show that it significantly mitigates the TCB problem and gains significant improvement in network performance.

Prospective, multicenter, consecutive case series INTRODUCTION: There are 3 categories of relative motion orthoses; protective, exercise and adaptive, with only 2 unpublished studies that prescribed for exercise. These orthoses are of 2 types: relative motion extension (RME) orthoses and relative motion flexion (RMF) orthoses. To describe prescription of relative motion (RME and RMF) exercise orthoses when used to assist recovery of proximal interphalangeal joint (PIPJ) movement after injury or surgery. Therapists enrolled patients who had limited PIPJ movement after injury or surgery and demonstrated greater passive than active isolated PIPJ movement. Relative motion exercise orthoses and usual hand therapy treatments were implemented for 6 weeks. Measures of PIPJ motion, pain, and patient-report of orthotic wear time and perceived benefit were recorded at the time of orthotic intervention, at 3 weeks and at 6 weeks. Eight therapists from 4 private hand therapy clinics implemented RM exercise orthoses in 14 patients with limited PIPJ flexion (RME orthoses) and 6 patients with limited PIPJ extension (RMF orthoses). One participant prescribed a RMF orthosis failed to complete the study. Isolated PIPJ active flexion improved for those prescribed RME orthoses (n = 14/14) and isolated PIPJ active extension improved for those prescribe a RMF orthosis (n = 2/5). Most patient-reports were positive about the relative motion experience. Although diagnoses and prescription times differed, the outcomes of this patient series prescribed relative motion exercise orthoses agree with those of 2 unpublished case series; all in support relative motion exercise orthoses to improve limited PIPJ movement. Future studies implementing relative motion exercise orthoses to recover limited PIPJ movement after injury or surgery may be worthwhile.

The rupture of the central slip of the extensor tendon of the finger causes a deformity, characterized by flexion at the proximal interphalangeal (PIP) joint and hyperextension at the distal interphalangeal joint (the boutonnière deformity). Various treatments are used in this condition, including conservative and operative methods, however, there is no standard treatment guideline. The idea the relative motion flexion splint method is based on an importance of keeping the metacarpo-phalangeal joint of the involved finger relatively flexed compared with these joints of the adjacent fingers. This is done with a relative motion flexion splint. Flexion in the metacarpo-phalangeal joint allows the lateral slips of the extrinsic extensor tendon to pull the lateral bands dorsal to the axis of the PIP joint. It allows also the relaxation of the intrinsic muscles and the lateral bands so they can migrate dorsal to the axis of the joint. This article presents the method and provides literature review about outcomes of treatment of the condition.

•Prescription of exercise relative motion orthoses is common practice with limited evidence.•Exercise relative motion orthosis (RM) orthosis were implemented for 6 weeks to improve limited proximal interphalangeal joint motion.•Active and passive proximal interphalangeal joint movement improved, agreeing with unpublished results.•Patient comments were mostly positive about exercise RM orthosis wear.•Orthotic wear instruction for exercise RM orthoses differs from protective RM orthoses. Prospective, multicenter, consecutive case series There are 3 categories of relative motion orthoses; protective, exercise and adaptive, with only 2 unpublished studies that prescribed for exercise. These orthoses are of 2 types: relative motion extension (RME) orthoses and relative motion flexion (RMF) orthoses. To describe prescription of relative motion (RME and RMF) exercise orthoses when used to assist recovery of proximal interphalangeal joint (PIPJ) movement after injury or surgery. Therapists enrolled patients who had limited PIPJ movement after injury or surgery and demonstrated greater passive than active isolated PIPJ movement. Relative motion exercise orthoses and usual hand therapy treatments were implemented for 6 weeks. Measures of PIPJ motion, pain, and patient-report of orthotic wear time and perceived benefit were recorded at the time of orthotic intervention, at 3 weeks and at 6 weeks. Eight therapists from 4 private hand therapy clinics implemented RM exercise orthoses in 14 patients with limited PIPJ flexion (RME orthoses) and 6 patients with limited PIPJ extension (RMF orthoses). One participant prescribed a RMF orthosis failed to complete the study. Isolated PIPJ active flexion improved for those prescribed RME orthoses (n = 14/14) and isolated PIPJ active extension improved for those prescribe a RMF orthosis (n = 2/5). Most patient-reports were positive about the relative motion experience. Although diagnoses and prescription times differed, the outcomes of this patient series prescribed relative motion exercise orthoses agree with those of 2 unpublished case series; all in support relative motion exercise orthoses to improve limited PIPJ movement. Future studies implementing relative motion exercise orthoses to recover limited PIPJ movement after injury or surgery may be worthwhile.

In order to address the issue of quantitatively solving the impact of perturbation forces and initial errors on rendezvous accuracy, traditional analytical methods are not suitable, and the calculation steps for statistical methods based on Monte Carlo simulation analysis are complex and computationally time-consuming. To overcome these challenges, this paper proposes a method for analysing rendezvous accuracy based on an improved relative motion dynamics model. Describing the relative motion between a sub-spacecraft and a target spacecraft by an improved Clohessy-Wiltshire (C-W) equation. By employing the derived state transition matrix from the model, it is possible to calculate the error propagation during the rendezvous process and express the rendezvous accuracy analytically. Through simulation examples, the proposed method proves to be both accurate and efficient in analysing the influence of perturbation forces and random errors such as initial position and velocity on rendezvous accuracy. This study verifies the correctness and effectiveness of the approach.