Polar Frame Research Articles

A Lagrangian Approach for Trajectory Tracking of Underactuated Marine Vessels

This paper attempts to explore the trajectory tracking problem for underactuated marine vessels from a Lagrangian perspective. First, we derive the vessel’s dynamics from Euler-Lagrangian equations and its kinematics from the mechanical constraint, both in the polar frame. The benefit of using polar frame is that we can transform the two-input-three-output underactauted system into a two-input-two-output form such that it becomes easy for us to solve the control problem using general backstepping method. In this paper, the main focus is taken on that how to improve the control performance by fully utilizing the features of the vessel’s nonlinear dynamics from the Lagrangian perspective. Some numerical studies are also carried out to investigate and analyse the effectiveness of the proposed scheme.

A Tether System at the L1, L2 Collinear Libration Points of the Mars–Phobos System: Analytical Solutions

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

Asymmetries in the discrimination of motion direction around the visual field.

The discriminability of motion direction is asymmetric, with some motion directions that are better discriminated than others. For example, discrimination of directions near the cardinal axes (upward/downward/leftward/rightward) tends to be better than oblique directions. Here, we tested discriminability for multiple motion directions at multiple polar angle locations. We found three systematic asymmetries. First, we found a large cardinal advantage in a cartesian reference frame - better discriminability for motion near cardinal reference directions than oblique directions. Second, we found a moderate cardinal advantage in a polar reference frame - better discriminability for motion near radial (inward/outward) and tangential (clockwise/counterclockwise) reference directions than other directions. Third, we found a small advantage for discriminating motion near radial compared to tangential reference directions. The three advantages combine in an approximately linear manner, and together predict variation in motion discrimination as a function of both motion direction and location around the visual field. For example, best performance is found for radial motion on the horizontal and vertical meridians, as these directions encompass all three advantages, whereas poorest performance is found for oblique motion stimuli located on the horizontal and vertical meridians, as these directions encompass all three disadvantages. Our results constrain models of motion perception and suggest that reference frames at multiple stages of the visual processing hierarchy limit performance.

Novel Fiducial Fiber Scanning Measurement Method Based on a Polar Coordinate System for LAMOST

The multiobject fiber-fed spectrograph is the core scientific instrument for large-scale spectroscopic surveys. For closed-loop control of fiber positioning, fiber metrology systems are implemented in numerous fiber-fed spectrographs. The position accuracy of the fiducial fiber in the focal plate directly affects the performance of the fiber metrology system. However, there are currently no suitable methods and devices for measuring the fiducial fibers with high accuracy in the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST). To solve this problem, this study proposed a novel online scanning measurement method for fiducial fibers in which a scanning camera was set up in front of the focal plate, and the rotation and translation movements of the focal plate were combined to set a polar coordinate measurement system. First, the pole and polar axis of the polar coordinate frame were determined, and the compensation values of the polar radius and angle of the polar coordinate of the fiducial fibers were solved in the field of view of the scanning camera. A prototype measurement platform was set up to verify the feasibility and scientific validity of the method. Experiment results show that the polar radius accuracy of the proposed method met the measurement requirements. The X–Y measurement accuracy can be further improved when a higher-precision rotary stage is adopted. Thus, the difficulties in online accurate measurement for fiducial fibers can be tackled by the proposed method with good operability in LAMOST.

Trajectory Tracking Performance Transition Analysis from Polar to Cartesian Coordinates

Trajectory tracking of underactuated marine vessels has become an intense research area in the nonlinear control community over the past two decades. In the authors’ previous work, an asymptotic tracking scheme was proposed by introducing two polar coordinate transformations, which can transform two-input-three-output underactuated system model into a certain two-input-two-output one. Therefore, it becomes possible to solve the trajectory tracking problem of these kinds of underactuated system using general backstepping method. However, since the polar coordinate transformation does not satisfy to be a diffeomorphism, the proposed tracking scheme in the polar frame cannot guarantee the same tracking performance in the Cartesian frame. With this consideration, in this paper the authors aim to analyze and establish a generalized condition under which the proposed tracking scheme in the previous work can guarantee the same tracking property in the Cartesian frame. The analysis results are verified through numerical studies as well.

Low-frequency ambient ocean-bottom node surface-wave seismology: A Gulf of Mexico case history

Long-time marine seismic recordings are becoming more common with the increased use of ocean-bottom nodes (OBNs), which can measure ambient seismic energy at frequencies lower than the typical minimum values in active-source compressed air-gun surveys. Interferometric processing on long-time ambient multicomponent data allows for the extraction of low-frequency (sub-2.0 Hz) responses in virtual source gathers (VSGs). Using 40 days of continuous OBN recordings acquired on a large dense array during a field experiment in the Gulf of Mexico, we find that sub-2.0 Hz surface-wave energy in the computed VSGs is strongly coherent and exhibits an identifiable spatially varying character. In particular, after rotating the data components from a Cartesian geographic into a polar wave-vector reference frame, we find that radial VSGs (i.e., oriented along the vector connecting the virtual source and receiver) clearly indicate that surface-wave propagation is influenced by salt bodies as identified in a colocated active-source survey situated at a minimum of 0.7 km depth below the seafloor, an observation consistent with calculated 0.25–0.50 Hz surface-wave sensitivity kernels. This suggests that low-frequency ambient OBN surface-wave seismology could be important for estimating the long-wavelength elastic material properties (particularly S-wave velocity) and identifying the lateral boundaries of salt bodies without any prior knowledge of subsurface geology.

Risley prisms as a conformal transformation device. I. Complex and graphic analyses of mapping images conformality.

Mapping conformality and the singularities in the control law of the Risley prisms for target tracking are two internally related topics well suited for an investigation in one study. This is because an intrinsic ability of the Risley prisms is to produce 2D scan patterns in a polar frame by means of a combination of two 1D motions of the two prisms when their orientations are specified by two independent variables expressible as the x and y coordinates of a single point in a rectangular frame. In transformation of points between the rectangular and the polar frames, a singularity occurs inevitably at the pole where r=0 in the polar frame. For a better understanding of the two topics and their relationship, this study presents an in-depth analysis, starting from the first-order approximation, and then the far-field condition is considered and finally is the rigorous theory developed for the Risley-prism-based beam steering system under the basis of a consideration of the effect of the thickness of the Risley prisms on the accuracy of beam steering. The rigorous theory provides correct results of mapping conformality in both the near and the far zones of the beam steering field.

Risley prisms as a conformal transformation device: II. Derivatives of the absolute functions and an attributive analysis of the control singularities.

Risley prisms have quite a few advantages over their competitive mirror scanning systems, a highly commendable one being the conformal transformation of the angular motions of the two prisms into a 2D pattern in the polar frame. However, in such a transformation, knotty problems relating to the so-called control singularities unpreventably occur when the two prisms are either in parallel with their apexes aligned or in opposition to each other. The aim of Part II of this study is an extension of the theories and their applications presented in Part I of this study [J. Opt. Soc. Am. A39, 1540 (2022)JOAOD60740-323210.1364/JOSAA.463724] to a parametric analysis of the singularities and their describing functions for a better understanding of how the scanning geometry affects the attributes of the singularities and how we can reduce their unfavorable influence by properly selecting the parameters of this two-prism system.

A Nonlinear Observability Analysis of Ambient Wind Estimation with Uncalibrated Sensors, Inspired by Insect Neural Encoding.

Estimating the direction of ambient fluid flow is key for many flying or swimming animals and robots, but can only be accomplished through indirect measurements and active control. Recent work with tethered flying insects indicates that their sensory representation of orientation, apparent wind, direction of movement, and control is represented by a 2-dimensional angular encoding in the central brain. This representation simplifies sensory integration by projecting the direction (but not scale) of measurements with different units onto a universal polar coordinate frame. To align these angular measurements with one another and the motor system does, however, require a calibration of angular gain and offset for each sensor. This calibration could change with time due to changes in the environment or physical structure. The circumstances under which small robots and animals with angular sensors and changing calibrations could self-calibrate and estimate the direction of ambient fluid flow while moving remains an open question. Here, a methodical nonlinear observability analysis is presented to address this. The analysis shows that it is mathematically feasible to continuously estimate flow direction and perform self-calibrations by adopting frequent changes in course (or active prevention thereof) and orientation, and requires fusion and temporal differentiation of three sensory measurements: apparent flow, orientation (or its derivative), and direction of motion (or its derivative). These conclusions are consistent with the zigzagging trajectories exhibited by many plume tracking organisms, suggesting that perhaps flow estimation is a secondary driver of their trajectory structure.

Decentralized targeting control of collinear agents

AbstractIn this paper, we deal with the decentralized targeting control problem of collinear agents in three‐dimensional space. Here the direction half‐lines associated with these agents are not coplanar on the baseplane determined by the agents and the target at the very beginning. The two direction angles of each direction half‐line in a local polar coordinate frame associated with each agent are adopted to denote the direction of the corresponding agent. The two agents at the two ends are assumed to already point at the target and stay there. Only through local information exchanges with the left and right neighbors, the remaining agents regulate their own direction angles locally so that their direction half‐lines eventually point to the same target. All the agents and the target are stationary in the sense that their locations do not change. The crucial idea of our designed decentralized targeting control algorithm is to exploit the geometric relations of the direction angles and the relative distances among three neighboring agents. The regulation of the projection lines of the direction half‐lines is the critical step to the present three‐dimensional case from the two‐dimensional one. Finally, the decentralized targeting control problem is converted into a Lyapunov stability problem and resolved. The validity of the obtained theoretical results is illustrated through numerical simulations.

Rotor Balancing for Magnetically Levitated TMPs Integrated With Vibration Self-Sensing of Magnetic Bearings

Due to the rotor unbalance of the magnetically levitated turbo molecular pump (TMP), this article presents a novel high-precision field balancing method based on regular control mode without trial weight. First, the static/dynamic unbalance vector expressions of the magnetically levitated rotor are analyzed and derived in polar coordinates frame. Second, according to the synchronous currents and displacement responses of the rotor at a fixed speed, the correction weight can be solved by the double-plane correction method based on equivalence principle of the magnetic forces. Then, in order to reduce the deviation between theoretical and actual value of the magnetic bearing parameters, this article proposes a second correction method by correcting the conversion coefficient matrix from online identification procedures. The proposed method, which can eliminate the errors introduced by the oversimplification of dynamic model in regular control, is especially suitable for the magnetically levitated rotor system with strong gyroscopic effects and asymmetric structure. Finally, the experimental results on a magnetically levitated TMP prototype validate the effectiveness of the proposed method. Compared with the original unbalance of rotor, the synchronous current amplitude declined by 99.0%, and the synchronous displacement amplitude also declined by 96.1% with the proposed method. A high-precision field balancing has been achieved.

General 3-D Type-II Fuzzy Logic Systems in the Polar Frame: Concept and Practice

This paper deals with the concept and application of general three-dimensional (3-D) type-2 fuzzy logic systems in the polar frame (G3DT2FLS). 1 1 Because many acronyms are used in this paper, they are summarized in Table I for the convenience of the reader. We focused on the automatic membership function (MF) Generator, general type-2 polar fuzzy membership, new geometric operators in polar frame, inference consisting of fuzzy 3-D polar rules and antecedents/consequents $\theta$ -slice and $\alpha$ -planes. The cubic smoothing spline is introduced to generate the upper and lower MFs according to the information theory. Three indices of compactness, smoothness, and entropy are employed for tuning of MFs. A measure of ultrafuzziness, min, max, equal, and reduced ultrafuzziness are suggested for the polar method. Additionally, the set theoretic operations of 3-D type-2 fuzzy sets in the polar frame are discussed. We also prove the join, meet, centroid, and type-reduction operations in the polar frame. Several rule sets are given to show the usefulness and complexity of the proposed method. The performance of the partial general 3D polar type-2 fuzzy logic system showed linear growth as the number of rules was increased. Computation time tests showed that the algorithm reduces the computation time by a maximum of 67% compared with discrete MF and is shown by an extreme 98% for the geometric procedure. These results indicate significant improvements in computation time for the spline interpolation over the existing methods.

Aircraft planar trajectories in crosswind navigation: some hypergeometric solutions

We study the planar motion of a self-propelled object against a withstanding flow, such as wind or sea current, whose speed is capable of perturbing the trajectory of the object, heading it off its path. Seven cases are dealt with, corresponding to fairly general wind types and all leading to nonlinear ordinary differential equations for the object motion. The first two cases concern motions described in Cartesian coordinates. They are followed by motions under a cyclonic wind, analysed in a polar reference frame; the wind is modelled as a logarithmic or hyperbolic spiral and closed-form solutions are obtained by means of the Gauss hypergeometric function and the Lerch transcendent. Further wind instances, treated in a Cartesian or in a polar reference system, require some special functions, such as the Lambert function, while finding the motion time law leads to a trinomial equation, in the most interesting situation of a mobile object that is faster than the wind. In the last case of a horizontal hyperbolic wind, time law and trajectory are computed in a polar reference frame by means of the Lambert function.

A New Rotation-Free Shell Formulation Using Exact Corotational Frame for Dynamic Analysis and Applications

Rotation-free shell formulations were proved to be an effective approach to speed up solving large-scaled problems. It reduces systems' degrees-of-freedom (DOF) and avoids shortages of using rotational DOF, such as singular problem and rotational interpolation. The rotation-free element can be extended for solving geometrically nonlinear problems using a corotational (CR) frame. However, its accuracy may be lost if the approach is used directly. Therefore, a new nonlinear rotation-free shell element is formulated to improve the accuracy of the local bending strain energy using a CR frame. The linear strain for bending is obtained by combining two re-derived elements, while the nonlinear part is deduced with the side rotation concept. Furthermore, a local frame is presented to correct the conventional local CR frame. An explicit tangential stiffness matrix is derived based on plane polar decomposition local frame. Simple elemental rotation tests show that the stiffness matrix and the proposed local frame are both correct. Several numerical examples and the application of drape simulations are given to verify the accuracy of nonlinear behavior of the presented element, and some of the results show that the presented method only requires few elements to obtain an accurate solution to the problem studied.

A comparative study on the smart damping capabilities of cylindrically orthotropic piezoelectric fiber–reinforced composite actuators in vibration control of simply supported/fully clamped isotropic annular plate

For active control of vibration of plane structures of revolution, a cylindrically orthotropic piezoelectric fiber–reinforced composite actuator is addressed in a recent study. Some other available piezoelectric composites, which are originally constructed in Cartesian coordinate frame, can also be reconstructed in polar coordinate frame for similar application. Making use of all these piezoelectric composites in polar coordinate frame of an annular plate, a comparative study on their control capabilities is performed at present. All the piezoelectric composites are used in the form of patches which are optimally configured over the host plate surface and utilized as smart dampers. The geometrical and material properties of the piezoelectric composites are taken in uniform manner and the corresponding electric fields in function of applied voltage are evaluated for derivation of a closed-loop finite element model of the smart annular plate. First, the patches of every piezoelectric composite are optimally configured through the proposition of a new methodology. Next, the control/damping capabilities of the piezoelectric composites are evaluated for fundamental symmetric/asymmetric mode of vibration of the plate. The numerical results first illustrate the implementation and suitability of the present methodology for optimal size and locations of actuator patches. Subsequently, the results demonstrate damping capabilities of all the piezoelectric composites to label potential ones.

Rahbin: A quadcopter unmanned aerial vehicle based on a systematic image processing approach toward an automated asphalt pavement inspection

Automatic inspection of pavement cracking is a critical issue in pavement management systems. In this study, a quadcopter-based digital imaging system is introduced for collecting pavement surface data over a distressed area for visual conditions interpretation. An aerial evaluation was carried out using a quadcopter unmanned aerial vehicle (QUAV), which is a device equipped with a set of automatic systems. Since QUAV flies autonomously and has high maneuverability, it is potentially useful in a variety of conditions particularly the positions dangerous for surveillance and reconnaissance. The main purpose of this work is to design a multi-stage system for QUAV image analysis consisting of image processing, threshold selection, and classification stages. The images are transformed into a new domain; then, an adaptive thresholding is applied to build the pattern of transformed cracks; and finally, the polar support vector machine (the PSVM) is applied for interpretation of crack distress. The PSVM is an automation procedure based on the support vector machine (SVM) classifier defined in the polar coordinate frame. A Mixture of Wavelet modulus and three-dimensional polar Radon transform (3DPRT) are used for feature generation. We show that the PSVM method can be successfully applied to classify the crack and is capable of providing new features about cracking distress, threshold selection and classification. In order to show the applicability and efficiency of the proposed system and method, a test was conducted applying a variety of pavement distresses. The experimental results demonstrate that the applied system provides reliable output. In addition, the comparison of the derived information with the on-site manual quantifications revealed the potentiality of the QUAV and multi-stage system for future practice.

Point Stabilization for Wheeled Mobile Robots Using Model Predictive Control

A model predictive control method is proposed for the point stabilization of wheeled mobile robots (WMRs) subject to nonholonomic constraints. The problem is simplified by considering only the steering system and neglecting the vehicle dynamics. A linearized error model is then formulated by transforming the robot position into polar frame. The feedback control policy is obtained by minimizing a quadratic cost function which penalizes the predicted errors and control variables in each sampling time over a finite horizon. The proposed control law is proven to guarantee the exponential stability of the robot system by considering additive inequality constraints in the optimization process. The performance of the stabilization algorithm is verified through computer simulations showing that the proposed method has a good regulation performance and convergence.

Radially accelerated trajectories for variable mass spacecraft

This paper proposes a new approach for mission design of a spacecraft with a time-variable mass and subjected to a continuous propulsive thrust in the radial direction. Two different mission scenarios are discussed according to whether the thrust remains constant along the whole mission or its modulus varies inversely proportional to the distance between the spacecraft and the primary body. In the first case a graphical solution is discussed for evaluating the spacecraft performance during an escape mission scenario. In the second case a closed-form solution, which makes use of the trigonometric integral functions, is found for describing the spacecraft trajectory in a polar reference frame. Some examples are discussed where the methodology is applied to a spacecraft with an electric thruster or a mini-magnetospheric propulsion system.

Task Polar Coordinate Frame-Based Contouring Control of Biaxial Systems

Contouring control is crucial in high-speed and high-precision manufacturing. In this paper, a novel task polar coordinate frame (TPCF), moving along the desired contour, is proposed to naturally calculate and control the estimated contouring error by the circular approximation, a second-order approximation. The dynamics in the world Cartesian coordinate frame is transformed into radial and angular dynamics in the local polar coordinate frame. By the feedback linearization technique and an input feedforward compensation, the closed-loop dynamics are decoupled in terms of the estimated contouring error and the angular error, respectively. Proportional-plus-derivative controllers can be assigned to stabilize the individual axis dynamics in the TPCF. By tuning the control parameters, different strengthening on estimated contouring error and angular error can be imposed explicitly and directly. Various experiments on an XY-stage biaxial system with typical contours, a circle and a figure-“8,” were conducted. Comparative studies are carried out for the TPCF- and traditional Frenet frame-based controls. The contouring errors were drastically reduced by the proposed approach, particularly in high-speed and large-curvature contouring cases.

Design of target state estimation algorithm with hybrid measurements

ÖZETBazı güdümlü füzelerde, hedefe dair bilgiler (pozisyon/hız/ivme), arayıcının henüz hedefe kilitlenmediği durumda çalıştırılan arasafha güdüm algoritmalarında kullanılmak üzere, yer sisteminde bulunan radar tarafından ölçülerek/hesaplanarak veribağı yoluyla füzeye iletilir. Veribağı iletiminin terminal fazda sürmesi durumunda, bu bilgiler aynı zamanda terminal fazda tanımlı ileri güdüm yöntemlerinin uygulanması ve güdüm performansının arttırılması için kullanılabilir. Bu çalışmada, arayıcı ve radar tarafından sağlanan farklı frekanslardaki ölçümler, Extended Kalman Filtresi tabanlı kestirim yöntemiyle tümleştirilmiştir. 3 boyutlu füzehedef kinematiğini yansıtan küresel koordinat sisteminde tanımlı sistem modelinde, hedefin ivmelenmediği varsayılmıştır. Bu çalışmanın sonucunda, hedefe kalan mesafe ve yaklaşma hızı kestirimi elde edilmiştir.