Constant Jerk Research Articles

Cosmological implications of the constant jerk parameter in [formula omitted] gravity theory

This study delves into modified gravity theories that are equivalent to General Relativity but involve the torsion or non-metricity scalar instead of the curvature scalar. Specifically, we focus on f(Q,T) gravity, which entails an arbitrary function of the non-metricity scalar Q non-minimally coupled to the trace of the stress–energy tensor T. We investigate the functional form f(Q,T)=f(Q)+f(T), where f(Q)=Q+αQ2 represents the Starobinsky model in f(Q) gravity and f(T)=2γT, with α and γ are constants. To obtain solutions for the Friedmann equations, we introduce the concept of a constant jerk and employ its definition to trace the evolution of other kinematic variables, including the deceleration parameter, energy density, EoS parameter, and various energy conditions. These analyses serve to validate the proposed model. We constrain our constant jerk model using the most available Pantheon set of data, the Hubble set of data, and the BAO set of data. Further, we employ Om(z) diagnostic as a means to differentiate between different theories of dark energy. Throughout our examination, all cosmological parameters under scrutiny consistently indicate an accelerating Universe.

Cosmic jerk parameter in symmetric teleparallel cosmology

In this paper, we have examined the recently proposed modified symmetric teleparallel gravity, in which gravitational Lagrangian is given by an arbitrary function of non-metricity scalar Q. We have considered a constant jerk parameter to express the Hubble rate. Moreover, we have used 31 points of OHD datasets and 1701 points of Pantheon+ datasets to constraint our model parameters by means of the Markov Chain Monte Carlo analysis. The mean values and the best fit obtained give a consistent Hubble rate and deceleration parameter compared to the observation values. In order to study the current accelerated expansion scenario of the Universe with the presence of the cosmological fluid as a perfect fluid, we have considered two forms of teleparallel gravity. We have studied the obtained field equations with the proposed forms of f(Q) models, specifically, linear $$f\left( Q\right) =\alpha Q+\beta $$ and nonlinear $$f\left( Q\right) =Q+mQ^{n}$$ models. Next, we have discussed the physical behavior of cosmological parameters such as energy density, pressure, EoS parameter, and deceleration parameter for both model. To ensure the validity of our proposed cosmological models, we have checked all energy conditions. The properties of these parameters confirm that our models describe the current acceleration of the expansion of the Universe. This result is also corroborated by the energy conditions criteria. Finally, the EoS parameter for both models indicates that the cosmological fluid behaves like a quintessence dark energy model.

Role of constant jerk parameter in f(R,T) gravity

In this study, a homogeneous and anisotropic Bianchi type-I universe model is considered taking the cosmological constant as a variable in the context of f (R, T) gravity. The presented model relies on a unique condition of constant jerk parameter as j = 1, which corresponds to a flat [Formula: see text]CDM model that leads to a variable deceleration parameter. The exact solution of the field equations is obtained under that condition, and the physical and geometrical features of the model are discussed through the paper. The model has been shown to be successful in explaining recent cosmological observations such as the universe’s expansion decelerates in the early universe and accelerates in the late universe, in agreement with their current data.

Determination of the kinematic parameters from SNe Ia and cosmic chronometers

In this work, by assuming a spatially flat Universe, we have tested 8 kinematic parametrization models with H(z) data from Cosmic Chronometers and SNe Ia from Pantheon compilation. Our aim is obtain the current values for the Hubble constant (H_0), deceleration parameter (q_0), jerk (j_0) and snap (s_0) parameters independently from a dynamical model. By using a Bayesian model comparison, three models are favoured: a model with the deceleration parameter (q) linearly dependent on the redshift, q linearly dependent on the scale factor and a model with a constant jerk. The model with constant jerk is slightly favoured by this analysis, furnishing H_0=68.8^{+3.7}_{-3.6} km/s/Mpc, q_0=-0.58pm 0.13, j_0=1.15^{+0.56}_{-0.53} and s_0=-0.25^{+0.40}_{-0.30}. The other models are compatible with the constant jerk model, except for the snap parameter, where we have found s_0=4.0^{+3.4}_{-3.0} for the model with q linearly dependent on the scale factor.

Radar Target Tracking for Unmanned Surface Vehicle Based on Square Root Sage-Husa Adaptive Robust Kalman Filter.

Dynamic information such as the position and velocity of the target detected by marine radar is frequently susceptible to external measurement white noise generated by the oscillations of an unmanned surface vehicle (USV) and target. Although the Sage–Husa adaptive Kalman filter (SHAKF) has been applied to the target tracking field, the precision and stability of SHAKF remain to be improved. In this paper, a square root Sage–Husa adaptive robust Kalman filter (SR-SHARKF) algorithm together with the constant jerk model is proposed, which can not only solve the problem of filtering divergence triggered by numerical rounding errors, inaccurate system mathematics, and noise statistical models, but also improve the filtering accuracy. First, a novel square root decomposition method is proposed in the SR-SHARKF algorithm for decomposing the covariance matrix of SHAKF to assure its non-negative definiteness. After that, a three-segment approach is adopted to balance the observed and predicted states by evaluating the adaptive scale factor. Finally, the unbiased and the biased noise estimators are integrated while the interval scope of the measurement noise is constrained to jointly evaluate the measurement and observation noise for better adaptability and reliability. Simulation and experimental results demonstrate the effectiveness of the proposed algorithm in eliminating white noise triggered by the USV and target oscillations.

Kinematic Tool-Path Smoothing for 6-Axis Industrial Machining Robots

The demands for machining by industrial robots have been increasing owing to their low installation cost and high flexibility. A novel trajectory generation algorithm for high-speed and high-accuracy machining by industrial robots is proposed in this paper. Linear interpolation in the workspace and smooth trajectory generation at the corners are important in industrial machining robots. Because industrial robots are composed of rotational joints, the joint space has a nonlinear relationship with the workspace. Therefore, linear interpolation in the joint space, which has been widely used in conventional machine tools, does not guarantee linear interpolation in the actual machining workspace. This results in the degradation of the machining surface. The proposed trajectory generation algorithm based on the decoupled approach can achieve linear interpolation in the workspace by separating the position commands into Cartesian coordinates and the orientation commands into spherical coordinates. In addition, a novel corner smoothing method that generates a smooth and continuous trajectory from discrete commands is proposed in this paper. The proposed kinematic local corner smoothing generates a smooth trajectory by using a 3-segmented constant jerk profile at the corners in the joint space. The sharp corners can thereby be replaced by smooth curves. The resulting cornering error is controlled by varying the cornering duration. The simulation results demonstrate the effectiveness of the proposed kinematic smoothing algorithm in achieving linear tool motion in straight sections and in generating smooth trajectories at corner sections within the user-defined tolerance.

A study of the cosmologically reconstructed f(T, B) gravity from the cosmological jerk parameter

The f(T, B) gravity which can also be considered as a natural modification of teleparallel gravity has been successfully used to substitute DE along with providing an explanation for the long-standing and unresolved cosmological puzzles. The present paper attempts to assess the cosmological significance of the reconstructed f(T, B) gravity on the basis of well-known choices of jerk parameter j. To achieve this goal, we consider a spatially flat Friedmann–Robertson–Walker geometry having a homogeneous and isotropic matter distribution. We study three ansatzes of the jerk parameter: Initially, we reconstruct a specific model of f(T, B) from a $$\Lambda $$ CDM inspired constant jerk parameter, secondly as a function of Hubble parameter H and lastly as a gradually changing function of the redshift parameter. Moreover, to explore the cosmic consequences of these models we determine the disposition of the most critical cosmic parameters including deceleration parameter q and equation of state $$\omega _{TB}$$ under the effect of three choices of matter equation of state parameter. Furthermore, we also study the null energy condition and stability of the aforementioned models against the linear, isotropic and homogeneous perturbations.

Raychaudhuri-based reconstruction of anisotropic Einstein\u2013Maxwell equation in 1+3 covariant formalism of f(R)-gravity

Recently, a new strategy to the reconstruction of f(R)-gravity models based on the Raychaudhuri equation has been suggested by Choudhury et al. (MNRAS 485:5693, 2019). In this paper, utilizing this method, the reconstruction of anisotropic Einstein–Maxwell equation in the 1+3 covariant formalism of f(R)-gravity is investigated in four modes: (i) Reconstruction from a negative constant deceleration parameter refereeing to an ever-accelerating universe; (ii) Reconstruction from a constant jerk parameter j=1 which recovers celebrated varLambda text {CDM} mode of evolution; (iii) Reconstruction from a variable jerk parameter j=Q(t); and (iv) Reconstruction from a slowly varying jerk parameter. Furthermore, two suggestions for enhancing the method are proposed.

Dynamic Behaviour of Spindle on Large EDM Machine Induced by High-Speed Jump Motion with Different Control Strategies

High-speed jump motion in electrical discharge machining (EDM) has been proven to be an important reason for flexible impact and vibration on EDM machine tools. Especially for large EDM machines, the moving parts possess large mass, and different jump control strategies cause different vibration characteristics. However, little attention has been paid to the kinematic comparison between different strategies, and the effect of jump motion on EDM spindle dynamics behaviour is also neglected. In this paper, the dynamic behaviour of spindle on a large EDM machine induced by high-speed jump motion has been investigated. Control strategies including trapezoidal velocity profile, constant jerk profile and sinusoidal jerk profile are considered. The kinematic characteristics of these jump control strategies and corresponding dynamic behaviour of EDM spindle were studied. As a result, under the same maximum velocity of 80 mm/s and maximum jerk of 128 m/s3, the jump period and height of sinusoidal jerk profile are about 0.05 s longer and 1.2 mm larger than that of the other two strategies, respectively. Trapezoidal control strategy causes the largest vibration amplitude of 48.7 μm on spindle during jump motion and thus the strongest forced vibration. The experimental measurement platform for vibration displacement was established. Comparing simulation and experiment data, the relative error is lower than 14%, which verifies the simulation model. Moreover, it can be found that large jump velocity increases vibration amplitude of spindle significantly during machining. When the velocity is 80 mm/s, the maximum vibration amplitude is 23.0 μm. And large velocity of jump motion makes spindle amplitude attenuate faster.

Recovering $$\Lambda $$CDM model from a cosmographic study

Using the mathematical definitions of deceleration and jerk parameters we obtain a general differential equation for squared Hubble parameter. For a constant jerk, this differential equation leads to an exact function for Hubble parameter. By the aid of this exact Hubble function we can exactly reconstruct any other cosmographic parameters. We also obtained a general function for transition redshift as well as spacetime curvature. Our derived functions clearly impose a lower limit on the jerk parameter which is $j_{min}\geq-0.125$. Moreover, we found that the jerk parameter indicates the geometry of the spacetime i.e any deviation from $j=1$ imply to a non-flat spacetime. In other word $j\neq 1$ reefers to a dynamical, time varying, dark energy. From obtained Hubble function we recover the analogue of $\Lambda$CDM model. To constrain cosmographic parameters as well as transition redshift and spacetime curvature of the recovered $\Lambda$CDM model, we used Metropolis-Hasting algorithm to perform Monte Carlo Markov Chain analysis by using observational Hubble data obtained from cosmic chronometric (CC) technique, BAO data, Pantheon compilation of Supernovae type Ia, and their joint combination. The only free parameters are $H$, $A(\Omega_{m})$ and $j$. From joint analysis we obtained $H_{0}=69.9\pm 1.7$, $A(\sim\Omega_{0m})=0.279^{+0.013}_{-0.017}$, $B(\sim\Omega_{0X})=0.721^{+0.017}_{-0.013}$, $j_{0}=1.038^{+0.061}_{-0.023}$ and $z_{t}=0.706^{+0.031}_{-0.034}$.

Lie symmetries of nonrelativistic and relativistic motions

We study the Lie symmetries of non-relativistic and relativistic higher order constant motions, in $d$ spatial dimensions, like constant acceleration, constant rate-of-change -of-acceleration (constant jerk), and so on. In the non-relativistic case, these symmetries contain the $z=\frac 2N$ Galilean conformal transformations, where $N$ is the order of the differential equation that defines the constant motion. The dimension of this group grows with $N$. In the relativistic case the vanishing of the ($d+1$)-dimensional space-time relativistic acceleration, jerk, snap, ... , is equivalent, in each case, to the vanishing of a $d$-dimensional spatial vector. These vectors are the $d$-dimensional non-relativistic ones plus additional terms that guarantee the relativistic transformation properties of the corresponding $d+1$ dimensional vectors. In the case of acceleration there are no corrections, which implies that the Lie symmetries of zero acceleration motions are the same in the non-relativistic and relativistic cases. The number of Lie symmetries that are obtained in the relativistic case does not increase from the four-derivative order (zero relativistic snap) onwards. We also deduce a recurrence relation for the spatial vectors that in the relativistic case characterize the constant motions.

A tracking algorithm of hypersonic target based on dynamic model and square-root cubature Kalman filter

Because of the complexities of target motion features, the tracking of hypersonic target in near space is difficult. In this paper, a new adaptive tracking algorithm based on dynamic model and improved square-root cubature Kalman filter(SRCKF) is proposed. According to the target movement characteristics, the improved adaptive piecewise constant Jerk model is proposed based on the dynamic model. The improved square-root cubature Kalman filter which the model mismatch detection function is introduced to reduce the state estimation error. The simulation results indicate that the proposed method which combines the model and the estimation algorithm has higher accuracy and stability on maneuvering target tracking than other existing methods. The research provides a feasible solution to the further improvement of the real time tracking accuracy of near-space hypersonic target.

Research of Clock Pulse Interpolation Technology Based on Constant Acceleration Model

Clock pulse interpolation method which relies on the software implementation is widely used in the measurement occasions when grating is used. For the displacement less than a grating pitch, common method is to estimate it using adjacent cycle information based on constant speed model or linear model. When the measured object moved with constant speed, these two methods would not introduce errors. However, when the measured object moved with constant acceleration or constant jerk, principle errors would be produced in these two methods. For this problem, this paper developed a method to measure the displacement less than one cycle in clock pulse interpolation method’s applications based on constant acceleration model. Simulation result demonstrated that this method will not introduce error when the measured object moves with constant acceleration, and its overall effect is better than the estimation effect using constant speed model or linear model when the object moves with constant jerk.

In search of the dark matter dark energy interaction: a kinematic approach

The present work deals with a kinematic approach to modelling the late time dynamics of the universe. This approach is based upon the assumption of constant value of cosmological jerk parameter, which is the dimensionless representation of the third order time derivative of the scale factor. For the CDM model, the value of jerk parameter is −1 throughout the evolution history. Now any model dependent estimation of the value of the jerk parameter would indicate the deviation of the model from the cosmological constant. In the present work, it has also been shown that for a constant jerk parameter model, any deviation of its value from −1 would not allow the dark matter to have an independent conservation, thus indicating an interaction between dark matter and dark energy. Statistical analysis with different observational data sets (namely the observational Hubble parameter data (OHD), the type Ia supernova data (SNe), and the baryon acoustic oscillation data (BAO)) lead to well constrained values of the jerk parameter and the model remains at a very close proximity of the CDM. The possibility of interaction is found to be more likely at high redshift rather than at the present epoch.

On an Optimized Fuzzy Supervized Multiphase Guidance Law

AbstractProportional navigation (PN) is an effective guidance law in chasing a constant speed target. However, intercepting a maneuvering target requires extra provisions to contain target acceleration. This is observed in the next generation of guidance laws, namely augmented proportional navigation (APN) and the optimal guidance law (OGL). The other advanced guidance method is integral sliding mode (ISM), which exhibits superb low miss distance (MD), but unfortunately at the cost of disappointing effective lateral acceleration (Ceff), higher than the modest level that APN and OGL demand. Reducing both MD and Ceff can be achieved using a multiphase algorithm. A setup of APN, OGL and ISM is proposed to integrate the strength of each and overcome their weaknesses. Hybrid algorithm smooth management is conducted by an optimally tuned fuzzy supervisory controller. The results indicate that in facing a constant acceleration target, an APN‐ISM twin yields the best results, while in case of a constant jerk target an APN‐OGL‐ISM triplet renders excellent interception. Miss distance as low as 0.012m can be achieved with significantly lower control effort than required by ISM alone. The simulations results confirm the conclusions and illustrate the capacity of the algorithm in combating maneuvering targets.

Performance analysis of α- β- γtracking filters using position and velocity measurements

This paper examines the performance of two position-velocity-measured (PVM) α- β- γ tracking filters. The first estimates the target acceleration using the measured velocity, and the second, which is proposed for the first time in this paper, estimates acceleration using the measured position. To quantify the performance of these PVM α- β- γ filters, we analytically derive steady-state errors that assume that the target is moving with constant acceleration or jerk. With these performance indices, the optimal gains of the PVM α- β- γ filters are determined using a minimum-variance filter criterion. The performance of each filter under these optimal gains is then analyzed and compared. Numerical analyses clarify the performance of the PVM α- β- γ filters and verify that their accuracy is better than that of the general position-only-measured α- β- γ filter, even when the variance in velocity measurement noise is comparatively large. We identify the conditions under which the proposed PVM α- β- γ filter outperforms the general α- β- γ filter for different ratios of noise variance in the velocity and position measurements. Finally, numerical simulations verify the effectiveness of the PVM α- β- γ filters for a realistic maneuvering target.

Optimal Trajectories for Vibration Reduction Based on Exponential Filters

In this paper, a new type of trajectory, based on an exponential jerk, is presented along with filters for their online generation. The goal is to generalize constant jerk trajectories, widely used in industrial applications, in order to reduce vibrations of motion systems. As a matter of fact, constant jerk trajectories do not assure a complete vibration suppression when the damping of the resonant modes is not negligible. The values of the parameters (decay rate and duration) of the jerk impulses that allow residual vibration cancellation are derived in an analytical way as a function of the dynamic characteristics of the plant. Comparisons with the well-known input shaping techniques and with system-inversion-based filters show the advantages of the proposed method in terms of robustness with respect to modeling errors, smoothness of the resulting trajectory, and time duration of the motion under velocity and acceleration constraints.

Bio-inspired Trajectory Planning for Robot Catching Movements Based on Tau Theory

A bio-inspired trajectory planning method for robot reaching movement to catch an object is presented. The method is based on the Tau theory which has been established by cognitive science researchers from studying the natural motion patterns when animals(including humans) approaching and catching a target object. Based on the Tau theory, two fundamental motion strategies, namely, the Tau coupling strategy and the intrinsic Tau-g guidance strategy have been analyzed. In order to solve the problem that the initial acceleration value is nonzero in the intrinsic Tau-g guidance strategy, a new strategy, namely, the intrinsic constant jerk Tau guidance strategy is presented. The trajectory planning formulae based on the Tau theory are derived. Trajectory planning, kinetics analysis, dynamic analysis, computer simulation and experiments are performed on a 3-DOF planar manipulator. The results show that, by using the proposed method, the end-effector can achieve reaching the target position and attitude at the same definite time with good continuity of joint velocity and acceleration, and the trajectory in Cartesian space is a straight path with a smooth bell-shaped velocity profile.

Four State Filter with Position Measurements and also with Position and Doppler Measurements

A one-dimensional, four state Kalman tracking filter is described for two cases: In case 1, the filter is described for position measurements only and in case 2, the filter is described for both position and Doppler measurements. The filter estimates the range, the range-rate, the rangeacceleration and the range-jerk of a moving target such as an aircraft moving with constant jerk perturbed by a plant noise of zero mean and constant variance which accounts for maneuvers and/or other random factors. In case 1, the range coordinate of the vehicle is assumed to be measured by a track-while-scan radar sensor and incase 2, the rangerate (Doppler) measurements are obtained by a trackwhile-scan radar sensor which employs pulsed Doppler processing such as a moving target detector providing unambiguous Doppler data [1,2]. The measurements are obtained at uniform sampling intervals of time T seconds and all measurements are noisy. The filter structures have been defined and the steady state results are given for both cases.

DARK ENERGY FROM A POSITIVE JERK PARAMETER

In this paper, we demonstrate that the variation of acceleration, namely the jerk parameter j, could give hints for determining the dark energy equation of state (EoS). In particular, it is possible to show that a viable cosmological model is compatible with a constant jerk, here conventionally rewritten as j = 1+ϵ, with ϵ > 0 representing a departure from the ΛCDM model. This suggests that the cosmological constant could be seen as a limiting case of a more general dark energy model. We use the most recent union 2.1 compilation of supernovae Ia, showing at 1σ confidence level, that j is compatible with the condition j ≥ 1. In doing so, we infer a corresponding cosmological model, viable with a negative acceleration parameter, in the observed range -1 < q0 < 0.