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A Gaussian mixture orbit determination filter is developed using a state vector that consists of equinoctial-like osculating elements. This new filter seeks to reduce the number of Gaussian mixture...

Motion-trajectory (MT) curves are used to introduce Frenet oscillations. Time-varying orientation of the motion plane that contains the absolute velocity and acceleration vectors is defined in terms of three Frenet–Euler angles; the curvature, vertical-development, and bank angles, referred to as the Frenet angles for brevity. The Frenet bank angle and the associated Frenet super-elevation of the motion plane, which measure deviation of the centrifugal inertia force from the horizontal plane, can be used to shed light on definition of the balance speed used in practice. The concept of the pre-super-elevated osculating (PSEO) plane is introduced and Rodrigues’ formula is employed to develop an orthogonal rotation matrix that provides a geometric interpretation of the PSEO plane. A new inverse-dynamics problem that utilizes experimentally or simulation recorded motion trajectories (RMT) is used to define the Frenet inertia forces and demonstrate their equivalence to the Cartesian form of the inertia forces. New expressions for the curvature vector in terms of the velocity and acceleration, limit on the magnitude of the tangential acceleration for a given forward velocity, condition required for the centrifugal force to remain horizontal, and condition of curvature singular points are derived. The Frenet bank angle can be used to prove existence of the normal vectors at the curvature singular points. It is shown that the inertia force can assume different forms, depending on the curve parameter used. The results of a simple analytical curve demonstrate Frenet oscillations and importance of distinguishing between the highway-ramp and railroad track bank angles and super-elevations, which are time-invariant, and the Frenet bank angle and super-elevation, which are motion-dependent.

At every point, a smooth plane curve can be approximated, to second order, by a circle; this circle is called osculating. One may think of the osculating circle as passing through three infinitesimally close points of the curve. A vertex of the curve is a point at which the osculating circle hyper-osculates: it approximates the curve to third order. Equivalently, a vertex is a critical point of the curvature function. Consider a (necessarily non-closed) curve, free from vertices. The classical Tait-Kneser theorem [13, 5] (see also [3, 10]), states that the osculating circles of the curve are pairwise disjoint, see Figure 1. This theorem is closely related to the four vertex theorem of S. Mukhopadhyaya [8] that a plane oval has at least 4 vertices (see again [3, 10]). Figure 1 illustrates the Tait-Kneser theorem: it shows an annulus foliated by osculating circles of a curve.

Waveriders have a high lift-to-drag ratio in hypersonic states, while the flexibility of design methods must still be improved because of some deficiencies such as unsatisfactory off-design perform...

We study geometric aspects of the imaginary Schwarzian \(S_2f\) for curves in 3-space, as introduced by Ahlfors in [1]. We show that \(S_2f\) points in the direction from the center of the osculating sphere to the point of contact with the curve. We also establish an important law of transformation of \(S_2f\) under Mobius transformations. We finally study questions of existence and uniqueness up to Mobius transformations of curves with given real and imaginary Schwarzians. We show that curves with the same generic imaginary Schwarzian are equal provided they agree to second order at one point, while prescribing in addition the real Schwarzian becomes an overdetermined problem.

We consider the spatial circular restricted three-body problem with light pressure. We obtain an analytic expression for the twice-averaged force function in the nonresonant case. We show that the integrals of the averaged equations written in osculating elements are the semi-major axis of the satellite orbit, the average value of the force function, and the Lidov–Kozai integral. We investigate stationary modes of oscillations and describe the set of equilibrium states of the averaged problem for different values of the coefficient of light pressure.

Let $X$ be a smooth complex projective variety. Using a construction devised to Gathmann, we present a recursive formula for some of the Gromov-Witten invariants of $X$. We prove that, when $X$ is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of $X$. This generalizes the classical well known pairs of inflexion (asymptotic) lines for surfaces in $\mathbb{P}^{3}$ of Salmon, as well as Darboux's $27$ osculating conics., Comment: Notation, small modification in Introduction, Prop. 4.1 and Lemma 5.2

Interpolation of parametric curves is one of the most effective methods in high-performance computer numerical control (CNC) machining. Its precision highly depends on chord errors from feedrate scheduling and contour errors from real-time machining. In order to improve machining precision, a novel feedrate optimization method is proposed in this paper. The osculating circle (OC) method can effectively estimate the chord errors, but its precision degrades if the trajectory curvature changes dramatically. Therefore, an iterative algorithm based on OC method is developed to reevaluate the chord errors, which can upgrade the estimation precision and further improve the feedrate constraints. To reduce the contour error, feature zones on the trajectory are firstly selected according to the kinestates and a specified zone length. In each feature zone, the projection of tracking error onto the zone chord-line is calculated as a new vector, based on which an indirect contour error compensation method is proposed through refining the feedrate profile. To ensure that the contour error can be reduced, the compensation performance is pre-estimated before implementation. Finally, simulation results in different scenarios are presented to validate the high precision of the proposed method.

To reduce the contour error in contour-following tasks, the most common method is to design a contour controller based on the contour error model. Therefore, how to estimate the contour error in real-time and with high accuracy plays an important role in contour-following control. It is difficult to guarantee real-time performance, high estimation accuracy and strong robustness against arbitrary trajectories at the same time. In this paper, a contour error estimation method based on the third-order osculating helix is proposed. The osculating helix can fully consider the geometric characteristics of the interpolation trajectory and is closer to the desired interpolation trajectory. On this basis, PD-type tracking error sliding mode surface and PD-type contour error sliding mode surface are redesigned to fully take into account the regulating effect of the velocity tracking error and the velocity contour error. Experimental results demonstrate the effectiveness of the proposed contour control approach.

Traditional osculating axisymmetric flow (OA) theory is widely used in the inverse design of generalized shock waves. However, this theory is only applicable to uniform incoming flows. In this paper, a novel method called micro osculating axisymmetric flow (MOA) method is proposed to solve the inverse problem of the 3D generalized shock wave design under non-uniform incoming flows. This method constructs a series of micro osculating planes (MOPs) along the shock wave surface in spanwise and streamwise directions. Actual 3D flows are then approximated by 2D axisymmetric flows in each MOP. Thus, the new method breaks the restriction of no lateral velocities or pressure gradients in the traditional method. An internal conical shock wave at a 4° angle of attack and the other one in an external conical flow of a 10° cone half-angle are obtained by the novel method to validate its feasibility and applicability. Numerical simulation results of the two cases indicate that the 3D shock wave geometries completely match the target. A streamwise integration of the inward turning wavecatcher inlet and outward turning waverider forebody is also presented. This integration shows the remarkable application prospects of the MOA method in the field of air-breathing hypersonic propulsion.

The problem of the secular evolution of a thin ring around a rapidly rotating triaxial celestial body is formulated and solved. The technology for calculating secular perturbations is based on two formulas: the azimuthally averaged force field of the central body and the mutual energy $${{W}_{{{\text{mut}}}}}$$ of this body and a Gaussian ring. With $${{W}_{{{\text{mut}}}}}$$ instead of the usual perturbing function, a system of differential equations for the osculating elements of the ring is obtained. An equation is obtained that allows one to find the coefficients of the zonal harmonics of the azimuthally averaged potential of an inhomogeneous ellipsoid using a unified scheme. The method is applied to dwarf planet Haumea with refined masses of the rocky core and the ice shell and the coefficients $${{C}_{{20}}}$$ and $${{C}_{{40}}}$$ of the po-tential’s zonal harmonics. According to new data, the ring around Haumea has a slight obliquity and must precess. It was established that the period of the retrograde nodal precession of the Haumea’s ring (without regard to self-gravity) is $${{T}_{\Omega }} = 12.9 \pm 0.7$$ days and the period of the forward of the apside line precession is $${{T}_{\omega }} \approx 8.{\text{08}}\;{\text{days}}$$ . It is proven that the 3:1 orbital resonance for the particles of the Haumea’s ring is fulfilled only approximately and the averaging time of additional perturbations at a nonsharp resonance turned out to be an order of magnitude smaller than $${{T}_{\Omega }}$$ . This confirms the adequacy of the method.

Resident space object population in highly elliptical high perigee altitude (> 600 km) orbits is significantly affected by luni-solar gravity. Using regularization, an analytical orbit theory with luni-solar gravity effects as third-body perturbations in terms of Kustaanheimo-Stiefel regular elements is developed. Numerical tests with different cases resulted in good accuracy for both short- and long-term orbit propagations. It is observed that the luni-solar perturbations affect the accuracy of the analytical solution seasonally. The analytical theory is tested with the observed orbital parameters of the few objects in highly elliptical orbits. The analytical evolution of osculating perigee altitude is found to be concurrent with observed data. Solar perturbation, when compared with lunar perturbation, is established to be dominant over such orbits.

In this paper, we consider the plane restricted four-body problem taking into account the light pressure forces when the Earth’s orbit is a Keplerian ellipse with a focus at the Sun, the Moon’s orbit is a Keplerian ellipse with a focus at the Earth, and the balloon satellite is a passively gravitating body. The averaged perturbing function of the problem in osculating elements is obtained in the nonresonant case, when the unperturbed orbit of the Earth satellite lies in the outer sphere of the Earth’s gravitational influence, located beyond the lunar Hill sphere. It is found that the integrals of averaged equations in the osculating elements are the semimajor axis of the satellite orbit as well as the average value of the force function. The stationary modes of oscillations and their bifurcation depending on the light pressure coefficient and the semimajor axis are investigated. Phase oscillation portraits are constructed for different values of the light pressure coefficient. The librational and rotational orbits of the satellite in the ecliptic plane are calculated.

The motion of a point with zero mass under the action of attraction to the central body $$\mathcal{S}$$ and perturbing acceleration $${\mathbf{P}}{\kern 1pt} ' = {\mathbf{P}}{\text{/}}{{r}^{2}}$$ , inversely proportional to the square of the distance to $$\mathcal{S}$$ , is considered. It is assumed that $${\mathbf{P}}{\kern 1pt} '$$ is small in absolute value compared to the main acceleration, caused by the attraction of the central body. Further, the vector $${\mathbf{P}}$$ components are constant in the reference frame used in astronomy, with the origin in the central body and the axes directed along the radius vector, the transversal (perpendicular to the radius vector in the osculating plane in the direction of motion), and the binormal (directed along the area vector). Earlier, we performed an averaging transformation of Euler-type equations of motion in osculating elements and obtained mean element evolutionary differential equations of motion in the first approximation in a small parameter. This article is devoted to solving the averaged equations, which are integrated completely. Moreover, the quadratures were expressed via elementary functions. The solution found has singularities at zero eccentricity and in the absence of the transverse acceleration. These and some other special cases are considered separately. There are at least two applications of the problem considered which are: an asteroid’s motion with allowance for the Yarkovsky–Radzievsky effect and a spacecraft’s motion with a solar sail. In both cases, the perturbation is inversely proportional to the squared distance from the Sun.

Multilevel secret sharing schemes are used in cryptography when a responsibility has to be delegated to qualified groups of participants rather than to a single one. In the present paper we propose a mathematical model for such a scheme based on projective geometry over a finite field. The main ingredients are normal rational curves and their osculating subspaces.

When considering the practical engineering application of a waverider, the on-design and off-design aerodynamic characteristics of the design conditions, especially the lift-to-drag ratio and the stability, deserve attention. According to recently studies, the planform and rear sight shape of a waverider are closely related to the above aerodynamic performance. Thus, the planform leading-edge profile curve used to design the planform shape of a vehicle is applied to designing an osculating cone waverider. Two key parameters concerned in planform and rear sight shape, namely the plan view sweep angle of the leading edge and the dihedral angle of the underside are introduced to the waverider design process. Each parameter is inserted in the control curve equation. Especially, a parameterization scheme is put forward for the free adjustment of the sweep angle along the leading edge. Finally, three examples are generated for verification and investigation. After the verification process based on the inviscid flow field of one case, the influences of the sweep and dihedral angles on the lift-to-drag ratio and the lateral static stability are evaluated, and meaningful results are obtained. Based on these results, we can conclude that, considering the maximum lift-to-drag ratio, the sweep angle plays a role on the lift-to-drag ratio only at subsonic and trans/supersonic speed as a negligible effect is observed at hypersonic speeds, whereas the dihedral angle is seem to produce a relevant difference at hypersonic speeds. Considering the lateral static stability, the dihedral angles have more influence on the waverider than the sweep angles.

We study curves whose position vector always lies in the planes spanned by Darboux frame along the curves on the surface. For $$\gamma$$ curves on the surface, we define $$\gamma$$ as the osculating Darboux curve when the position vector of $$\gamma$$ always lies in the tangent plane of the surface and define $$\gamma$$ as the rectifying Darboux curve when the position vector of $$\gamma$$ always lies in the Darboux rectifying plane. In this study, we find the geometric properties of the osculating Darboux curve and rectifying Darboux curve, classify these curves into two cases by using special invariants, and we propose approaches to construct examples of these curves by using normalized osculating Darboux vectors and normalized rectifying Darboux vectors. Moreover, we study the classification of osculating Darboux developables and normal developable surfaces under the condition of the existence of osculating Darboux curve and rectifying Darboux curve as a line of curvature and clad helices.

To expand the selection of preassigned shock surfaces, a new inverse design method called the local-turning osculating cones method has been proposed for waverider design. The new method discretize...

As a type of spatial transmission mechanism, noncircular bevel gears can transfer power and motion between two intersecting axes with variable transmission. In this paper, the relation of arc length is expressed by Angle in spatial polar coordinate system, utilizing the spherical triangle theorem, the parametric equations of base curves are established by applying arc length relation. Further, an example is given to verify whether pitch curves is concave on osculating circle of noncircular bevel gear section cone. Similarly, this example illustrates the cause of the mutation for the base curves.

This study is devoted to searching for lunar orbits that are accessible during departure along an unstable invariant manifold of halo orbits around lunar libration points L1 and L2. The spacecraft is assumed to be equipped with a low-thrust engine. Two levels of jet acceleration from the typical range of small-size spacecraft with low-thrust engines are considered. The study consists of two parts: (1) searching for osculating lunar orbits obtained during passive motion from halo orbits and (2) stabilization of orbits by the thrust force, directed against the spacecraft velocity. Many halo-orbits were found, the descent from which along the unstable manifold makes it possible to reach polar and near-polar orbits. The problem is solved within the model of a circular, restricted three-body problem.

The averaging method is widely used in celestial mechanics, in which a mean orbit is introduced and slightly deviates from an osculating one, as long as disturbing forces are small. The difference $$\delta {\mathbf{r}}$$ in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm $$\left\| {\delta {\mathbf{r}}} \right\|$$ for deviation is interesting to note. Earlier, the exact expression of the mean-square norm for one problem of celestial mechanics was obtained: a zero-mass point moves under the gravitation of a central body and a small perturbing acceleration $${\mathbf{F}}$$. The vector $${\mathbf{F}}$$ is taken to be constant in a co-moving coordinate system with axes directed along the radius vector, the transversal, and the angular momentum vector. Here, we solved a similar problem, assuming the vector $${\mathbf{F}}$$ to be constant in the reference frame with axes directed along the tangent, the principal normal, and the angular momentum vector. It turned out that $${{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}$$ is proportional to $${{a}^{6}}$$, where $$a$$ is the semi-major axis. The value $${{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}{{a}^{{ - 6}}}$$ is the weighted sum of the component squares of $${\mathbf{F}}$$. The quadratic form coefficients depend only on the eccentricity and are represented by the Maclaurin series in even powers of $$e$$ that converge, at least for $$e < 1$$. The series coefficients are calculated up to $${{e}^{4}}$$ inclusive, so that the correction terms are of order $${{e}^{6}}$$.

The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

The traditional osculating cones’ (OCs’) waverider design method is widely used in hypersonic waverider airframe design. However, it becomes ineffective when the shock strength in each osculating p...

The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, $-F \subset S_{\text{EE}}$. For a disk region, $F|_{\rm disk}\equiv F_0$ coincides with the free energy on $\mathbb{S}^3$ and provides an RG-monotone for general theories. As opposed to the analogous quantity in four dimensions, the value of $F$ generally depends in a complicated (and non-local) way on the geometry of the region and the theory under consideration. For small geometric deformations of the disk in general CFTs as well as for arbitrary regions in holographic theories, it has been argued that $F$ is precisely minimized by disks. Here, we argue that $F$ is globally minimized by disks with respect to arbitrary regions and for general theories. The proof makes use of the strong subadditivity of entanglement entropy and the geometric fact that one can always place an osculating circle within a given smooth entangling region. For topologically non-trivial entangling regions with $n_B$ boundaries, the general bound can be improved to $F \geq n_B F_0$. In addition, we provide accurate approximations to $F$ valid for general CFTs in the case of elliptic regions for arbitrary values of the eccentricity which we check against lattice calculations for free fields. We also evaluate $F$ numerically for more general shapes in the so-called "Extensive Mutual Information model", verifying the general bound., Comment: 38 pages, 10 figures. v2: some new references, comments and improved discussions added; matches version to appear in JHEP

In this paper, we define some new vector fields along a space curve with nonvanishing curvatures in Euclidean 4-space. By using these vector fields we determine some new planes, curves, and ruled hypersurfaces. We show that the determined new planes play the role of the Darboux vector. We also show that, contrary to their definitions, osculating curves of the first kind and rectifying curves in Euclidean 4-space can be considered as space curves whose position vectors always lie in a two-dimensional subspace. Furthermore, we construct developable and nondevelopable ruled hypersurfaces associated with the new vector fields in which the base curve is always a geodesic on the developable one.

Track geometry is an important parameter, used by railways, for routine track maintenance. Laser based Track Recording Coaches (LTRCs) are widely availed by railways, for recording track geometry defects, such as: vertical profile, alignment, gauge etc. In this paper, a novel method for monitoring vertical profile irregularities by in-service rail vehicle dynamics measurements has been proposed. The procedure takes as input, data from bogie installed inertial measurement unit (IMU) measuring vertical acceleration and pitch-rate, and encoder measuring longitudinal velocity. Extended Kalman Filter (EKF) with Rauch-Tung-Striebel (RTS) smoothing and Extended Kalman Particle Filter (EPF) are reviewed, for detecting vertical profile irregularities. Process model of nonlinear state estimation filters, EKF and EPF, has been developed by applying a novel analytical technique that approximates osculating circle of a point on the vertical profile curve. Further, measurement model has been devised by a novel procedure estimating curvatures from circles approximating trajectory sensed by vertical acceleration and pitch-rate curvature. Proposed method has been field-tested by comparing its output with LTRC’s recorded data.

A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The body masses vary isotropically at different rates. Equations of motion of the satellite in a relative system of coordinates are derived. The problem is studied by the methods of perturbation theory. Equations of secular perturbations of the translational-rotational motion of the satellite in analogues of Delaunay–Andoyer osculating elements are deduced. All necessary symbolic computations are performed using the Wolfram Mathematica computer algebra system.

This article is related to the problem of the construction of planetary motion theory. We have expanded the Hamiltonian of the four-planetary problem into the Poisson series in osculating elements of the second Poincare system. The series expansion is constructed up to the third degree of the small parameter. The averaging procedure of the Hamiltonian is performed by the Hori–Deprit method. It allows to eliminate short-periodic perturbations and sufficiently increase time step of the integration of the equations of motion. This method is based on Lie transformation theory. The equations of motion in averaged elements are constructed as the Poisson brackets of the averaged Hamiltonian and corresponding orbital element. The transformation between averaged and osculating elements is given by the change-variable functions, which are obtained in the second approximation of the Hori–Deprit method. We used computer algebra system Piranha for the implementation of the Hori–Deprit method. Piranha is an echeloned Poisson series processor authored by F. Biscani. The properties of the obtained series are discussed. The numerical integration of the equations of motion is performed by Everhart method for the Solar system’s giant planets.

The classical problem of three bodies of variable masses is considered in the case when two of the bodies are protoplanets and all the masses vary non-isotropically at different rates. The problem is analyzed in the framework of the planetary perturbation theory in terms of the osculating elements of aperiodic motion on quasi-conic sections. An algorithm for symbolic computation of the disturbing function and its expansion into power series in terms of the eccentricities and inclinations is discussed in detail. Differential equations describing the long-term evolution of the orbital parameters are derived in the form of Lagrange’s planetary equations. All the relevant calculations are done with the computer algebra system Wolfram Mathematica.

Low-energy trajectories take advantage of the mutual action of multiple celestial bodies on the spacecraft, and can conclude with ballistic capture about the arrival body, thus allowing significant savings in terms of propellant consumption, if compared to more traditional transfers. Because of the chaotic nature of multibody environments, the design of low-energy trajectories with given constraints can be complex and it is often obtained after a long, iterative, and eventually computationally expensive process. This work is aimed at identifying a limited set of characteristic parameters related both to the time behavior of three-dimensional ballistic capture orbits and to some osculating orbit elements (i.e., inclination, semimajor axis, and eccentricity), relative either to the departure or to the arrival body. The analysis is performed using the linear expansion of the Hamiltonian equations of motion in the equilibrium region around the collinear libration point L1 (or L2), in the dynamical framework of the 3 dimensional circular restricted 3-body problem. A correlation among some Hamiltonian parameters in the equilibrium region and the trajectory osculating orbital elements at capture is established. This result is used to design missions with ballistic capture having required orbital parameters at the arrival planet and it provides a strategy to control the target orbital elements at capture by small thrust maneuvers at the equilibrium region. Because of the long flight time, the solar perturbation is considered in the analysis, and suitable launch dates for the ballistically captured missions are determined.

The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface XB of degree 7 in P5 with its osculating spaces of order 2 which in every point of XB have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfaces.

Generally speaking, a spiral is a 2D curve which winds about a fixed point. Now, we present a new, alternative, and easy way to describe and generate spirals by means of the use of the Slope Chain Code (SCC) [E. Bribiesca, A measure of tortuosity based on chain coding, Pattern Recognition 46 (2013) 716-724]. Thus, each spiral is represented by only one chain. The chain elements produce a finite alphabet which allows us to use grammatical techniques for spiral classification. Spirals are composed of constant straight-line segments and their chain elements are obtained by calculating the slope changes between contiguous straight-line segments (angle of contingence) scaled to a continuous range from − 1 ( − 180 ∘ ) to 1 (180 ∘ ). The SCC notation is invariant under translation, rotation, optionally under scaling, and it does not use a grid. Other interesting properties can be derived from this notation, such as: the mirror symmetry and inverse spirals, the accumulated slope, the slope change mean, and tortuosity for spirals. We introduce new concepts of projective polygonal paths and osculating polygons. We present a new spiral called the SCC polygonal spiral and its chain which is described by the numerical sequence 2 n for n ≥ 3, to the best of our knowledge this is the first time that this spiral and its chain are presented. The importance of this spiral and its chain is that this chain is covering all the slope changes of all the regular polygons composed of n edges (n-gons). Also, we describe the chain which generates the spiral of Archimedes. Finally, we present some results of different kind of spirals from the real world, including spiral patterns in shells.

This study investigates the influence of surface distribution of the basic flowfield on the shapes and performances of the osculating axisymmetric waveriders. First, the design method of osculating...

Purpose The purpose of this study is to obtain the differential geometric analysis of autonomous wheel-legged robots and their trajectories on the terrain. Design/methodology/approach The author uses a wheel using the osculating sphere of the curve on rough terrain. Additionally, the author expresses a triple osculating sphere wheel by taking advantage of differential geometry. Moreover, the author examined the consecutive wheel center-curves to obtain the optimum posture of a micro-hydraulic toolkit (MHT) robot. Findings The author examined the terrain path, which is crucial for trajectory planning in terms of the geometric perspective. The author designed the triple MHT wheel using the osculating sphere of the MHT robot trajectory by taking advantage of local differential geometric properties of this curve on the terrain. The consecutive wheel center-curves were expressed and studied based on differential geometry. Originality/value The author provides a novel approach for the optimum posture of an MHT robot using consecutive wheel-center curves and provides an original perspective to MHT robot and its trajectory by using differential geometry.

In order to guarantee the good aerodynamic performance of the waverider vehicle in the wide-speed range, the variable Mach number osculating flowfield waverider design theory based on conical basic flowfield is put forward. The shape and aerodynamic characteristics of the variable Mach number osculating flowfield waverider and osculating cone waverider are compared and analyzed. And the inviscid numerical simulation method is used to verify the correctness of the theory and the validity of the design method. The obtained results show that the osculating flowfield waverider has good waverider properties under the conditions of wide-ranged flight Mach numbers. Furthermore, the comparison of osculating flowfield waverider and osculating cone waveriders indicates that the variable Mach number osculating flowfield waverider has balanced internal loading capacity and aerodynamic performance, which is more suitable for wide-speed flight.

For a real algebraic link in $${{\mathbb {RP}}}^3$$, we prove that its encomplexed writhe (an invariant introduced by Viro) is maximal for a given degree and genus if and only if its self-linking number with respect to the framing by the osculating planes is maximal for a given degree.

We present a new method for offset surface trimming that eliminates redundant parts of an offset surface that are closer than the offset distance to the original surface. The proposed approach deals with numerical instability around near-singular regions of an offset surface using the concept of offset trimming regions in the parameter space and carrying out numerical computations based on the regularity and intrinsic properties of the given input surface. In particular, we replace the self-intersection of an offset surface (which can be unstable around near-singular regions) by computation on the original input surface (and its derivatives) only, and also by the intersection of osculating tori that can be constructed in a highly stable way by offsetting the osculating tori of the given input regular surface. We demonstrate the effectiveness of our approach using non-trivial test examples of offset surface trimming, including some examples from the previous publications for the purpose of comparison.

A novel design methodology for waverider is proposed based on the conical flow field, named volume-improved osculating-cone waveriders. The new waverider is generated through three given curves, including flow capture curve, inlet capture curve, and curve of the center of exit shock. To meet more design requirements, the curve of the center of exit shock, which can be flexibly designed, is newly introduced to generate configurations with different properties and performances. The feasibility and effectiveness of the method have been verified by computational fluid dynamics (CFD) simulation. The discrepancies between the volume-improved osculating-cone waverider and conventional waveriders are numerically analyzed in detail. The results demonstrate that the new waverider takes the advantage of the volumetric efficiency with little loss of performance compared to the conventional waveriders in design conditions. Furthermore, the results of the off-design conditions show excellent aerodynamic performance as the conventional waverider. Moreover, for bluntness with a radius of 10 mm, the new waverider synthetically owns higher viscous lift-to-drag ratio and greater volumetric efficiency than conventional configurations at 0° angle of attack. Therefore, the novel approach is useful to design the hypersonic waverider vehicle.

The wide-speed range characteristics of aerodynamic configuration for aircraft are critical to the design of the hypersonic vehicle, and many novel strategies have been proposed. As one of the valid schemes of aircraft with a wide-speed range, the vortex lift waverider with a wide-speed range has attracted an increasing attention. In the current study, the low speed aerodynamic performance of vortex lift waveriders with a wide-speed range has been investigated numerically by means of the three-dimensional implicit Reynolds-averaged Navier-Stokes (RANS) equations coupled with the SST k-ω turbulence model. At the same time, the numerical approaches employed have been validated against the available experimental data in the open literature. The obtained results show that the low speed lift-to-drag ratio of the cuspidal waverider is higher than that of the general osculating cone waverider, while the low speed lift-to-drag ratio of the delta-winged waverider is lower than that of the general osculating cone waverider except for the small and negative angles of attack. At the same time, the low speed vortex lift characteristics of the three types of osculating cone waveriders have been compares and analyzes, and some valuable conclusions have been obtained.

With the use of quaternions and the maximum principle, we solve the optimal orbit transfer problem for a variable-mass spacecraft to a given plane in a nonlinear setting. The motion control of the spacecraft is carried out with the help of a jet thrust, bounded in absolute value and orthogonal to the plane of the osculating spacecraft orbit. We take into account the change in mass of the spacecraft due to the consumption of the working fluid in the control process. The functional that determines the quality of the control process is a linear convolution with weight factors for two criteria: time and total thrust impulse spent on the control process. We provide an exposition of the theory of the problem’s solution. We show results of optimal control calculations for cases when both criteria are simultaneously taken into account in the minimized combined quality functional of the control process, and for cases when only the total thrust impulse is minimized. We obtain examples of optimal control with up to 192 passive and active stages. We also establish optimal control laws for the rotation of the spacecraft’s orbital plane.

Polynomial interpolation is widely used in channel modeling for aerospace tracking, telemetry, and command channel simulators, to accurately regenerate the satellite motion parameters with a high sampling rate from the available low sampling rate motion data. In this paper, a novel piecewise osculating interpolation approach is proposed to improve modeling precision. It uses the available distance and its higher-order derivatives at each pair of adjacent knots to construct the piecewise interpolation polynomial, thus achieving a better precision and continuity of the resulting motion parameters. Moreover, it is efficiently implemented with an interpolation filter based on a modified Farrow structure, and the designed filter can interpolate the motion data obtained both through uniform or nonuniform sampling. Additionally, a sampling optimization algorithm is proposed. It optimizes the motion data sampling by minimizing the maximum interpolation error given the average sampling interval and interpolation polynomial order, and therefore, improves the precision further without increasing the costs. The validity of the proposed approach is verified by numerical simulation, and its advantages are demonstrated by comparison with existing methods.