Spherical Linear Interpolation Research Articles

SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

We mimic the conventional explicit Total Variation Diminishing Runge–Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 4$$\\end{document}. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.

Advancing Global Pose Refinement: A Linear, Parameter-Free Model for Closed Circuits via Quaternion Interpolation.

Global pose refinement is a significant challenge within Simultaneous Localization and Mapping (SLAM) frameworks. For LIDAR-based SLAM systems, pose refinement is integral to correcting drift caused by the successive registration of 3D point clouds collected by the sensor. A divergence between the actual and calculated platform paths characterizes this error. In response to this challenge, we propose a linear, parameter-free model that uses a closed circuit for global trajectory corrections. Our model maps rotations to quaternions and uses Spherical Linear Interpolation (SLERP) for transitions between them. The intervals are established by the constraint set by the Least Squares (LS) method on rotation closure and are proportional to the circuit's size. Translations are globally adjusted in a distinct linear phase. Additionally, we suggest a coarse-to-fine pairwise registration method, integrating Fast Global Registration and Generalized ICP with multiscale sampling and filtering. The proposed approach is tested on three varied datasets of point clouds, including Mobile Laser Scanners and Terrestrial Laser Scanners. These diverse datasets affirm the model's effectiveness in 3D pose estimation, with substantial pose differences and efficient pose optimization in larger circuits.

An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation

A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm.In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the Bézier basis functions on the manifold SO(3).The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.

Interpolation on the unit hypersphere using the n-dimensional generalized Kuramoto model

Computer graphics, robotics, and physics are one of the many domains where interpolation on the unit sphere S n (often called a unit hypersphere or unit n-sphere) plays a crucial role. In this paper, we introduce a novel approach for achieving smooth and precise interpolation on the unit sphere S n−1 using the n-dimensional generalized Kuramoto model. The proposed algorithm finds the shortest and most direct path between two points on that non-Euclidean manifold. Our simulation results demonstrate that it achieves performance comparable to that of a Spherical Linear Interpolation algorithm. Also, the paper proposes the application of our algorithm in the interpolation of rotations that are presented in the form of four-dimensional data.

Dual-Quaternion-Based SLERP MPC Local Controller for Safe Self-Driving of Robotic Wheelchairs

In this work, the motion control of a robotic wheelchair to achieve safe and intelligent movement in an unknown scenario is proposed. The primary objective is to develop a comprehensive framework for a robotic wheelchair that combines a global path planner and a model predictive control (MPC) local controller. The A* algorithm is employed to generate a global path. To ensure safe and directional motion for the wheelchair user, an MPC local controller is implemented taking into account the via points generated by an approach combined with dual quaternions and spherical linear interpolation (SLERP). Dual quaternions are utilized for their simultaneous handling of rotation and translation, while SLERP enables smooth and continuous rotation interpolation by generating intermediate orientations between two specified orientations. The integration of these two methods optimizes navigation performance. The system is built on the Robot Operating System (ROS), with an electric wheelchair equipped with 3D-LiDAR serving as the hardware foundation. The experimental results reveal the effectiveness of the proposed method and demonstrate the ability of the robotic wheelchair to move safely from the initial position to the destination. This work contributes to the development of effective motion control for robotic wheelchairs, focusing on safety and improving the user experience when navigating in unknown environments.

Effects of model fidelity and uncertainty on a model-based attitude controller for satellites with flexible appendages

This paper investigates the effects of model fidelity and parameter uncertainty on the performance of a hybrid model-based feedback-feedforward control scheme for attitude tracking of a satellite with flexible appendages. The feedforward component is an inverse model-based term produced through a computational approach known as inverse simulation (InvSim), which works by iteratively solving a discretised reference trajectory. The hybrid controller’s feedback is proportional-derivative (PD) based, using body attitude and rate feedback to provide stability and robustness. Furthermore, to ensure that the flexible modes do not trigger instability, the PD control gains are tuned to give a closed-loop response that is significantly slower than the flexible modes. Additionally, excitation of the flexible modes is reduced by minimising jerk through polynomial rest-to-rest manoeuvres, following the shortest quaternion path using spherical–linear-interpolation (SLERP). The effects of the appendage flexing on attitude tracking are then compensated through the feedforward element of the hybrid controller, with performance being compared to a traditional PD tracking law. The effect of the model fidelity on the performance of the hybrid controller is investigated through the use of both rigid body and multiple-fidelity finite-element mathematical models. Additionally, the effect of uncertainties in the model parameters is investigated to determine the accuracy of the model required to obtain significant improvement in attitude tracking. It is found that in the absence of any model parameter uncertainty, the hybrid controller outperforms the PD tracking control law by at least one order of magnitude when the finite-element model is used. Increasing the number of finite elements was found to provide no significant improvement in performance, with one element being sufficient and favourable with its lower computational overhead. It was also found that to ensure good performance compared to the PD tracking controller, the uncertainty in the inertia tensor should be <1%. Similarly, uncertainty in the first flexible modal frequency should be <0.5 rad/s.

Deep generative design of porous organic cages via a variational autoencoder.

Porous organic cages (POCs) are a class of porous molecular materials characterised by their tunable, intrinsic porosity; this functional property makes them candidates for applications including guest storage and separation. Typically formed via dynamic covalent chemistry reactions from multifunctionalised molecular precursors, there is an enormous potential chemical space for POCs due to the fact they can be formed by combining two relatively small organic molecules, which themselves have an enormous chemical space. However, identifying suitable molecular precursors for POC formation is challenging, as POCs often lack shape persistence (the cage collapses upon solvent removal with loss of its cavity), thus losing a key functional property (porosity). Generative machine learning models have potential for targeted computational design of large functional molecular systems such as POCs. Here, we present a deep-learning-enabled generative model, Cage-VAE, for the targeted generation of shape-persistent POCs. We demonstrate the capacity of Cage-VAE to propose novel, shape-persistent POCs, via integration with multiple efficient sampling methods, including Bayesian optimisation and spherical linear interpolation.

Facilitating Large-Amplitude Motions of Wave Energy Converters in OpenFOAM by a modified Mesh Morphing Approach

High-fidelity simulations using computational fluid dynamics (CFD) for wave-body interaction are becoming increasingly common and important for wave energy converter (WEC) design. The open source finite volume toolbox OpenFOAM is one of the most frequently used platforms for wave energy. There are currently two ways to account for moving bodies in OpenFOAM: (i) mesh morphing, where the mesh deforms around the body; and (ii) an overset mesh method where a separate body mesh moves on top of a background mesh. Mesh morphing is computationally efficient but may introduce highly deformed cells for combinations of large translational and rotational motions. The overset method allows for arbitrarily large body motions and retains the quality of the mesh. However, it comes with a substantial increase in computational cost and possible loss of energy conservation due to the interpolation. In this paper we present a straightforward extension of the spherical linear interpolation (SLERP) based mesh morphing algorithm that increase the stability range of the method. The mesh deformation is allowed to be interpolated independently for different modes of motion, which facilitates tailored mesh motion simulations. The paper details the implementation of the method and evaluates its performance with computational examples of a cylinder with a moonpool. The examples show that the modified mesh morphing approach handles large motions well and provides a cost effective alternative to overset mesh for survival conditions.

A sensor fusion approach to MARG module orientation estimation for a real-time hand tracking application

This paper introduces a new algorithm (Gravity & Magnetic North Vector correction — Double SLERP, or “GMV-D”) to estimate the orientation of a MEMS Magnetic/Angular-Rate/Gravity (MARG) sensor module using sensor fusion in the context of a real-time hand tracking application, for human–computer interaction purposes.Integrated MEMS MARG modules are affordable, small, light and consume minimal power. As such, there is interest in using them for monitoring the orientation of various body segments, to which they can be attached (e.g., the finger segments of a gloved hand). However, each of the 3 types of signals they provide has proven insufficient to yield robust orientation estimates, particularly in regions of space where the geomagnetic field is distorted. The significance (main contribution) of the approach we present is the computation of a final orientation estimate that uses all the signals generated by inexpensive (e.g., less than 20 USD, in large quantities) integrated MEMS MARG modules but weighs their contributions with simple, real-time-updatable parameters that prevent erroneous corrections when the pre-conditions for their valid use are not met. This will enable the use of inexpensive integrated MEMS MARG modules for hand tracking applications in human–computer interaction and other areas of work where tracking the orientation of body segments in real-time is important.In each iteration, GMV-D defines an initial orientation estimate from integration of gyroscopic signals (“dead reckoning”), and also calculates accelerometer-based and magnetometer-based corrections. These corrections are defined on the assumptions that the module is near static and affected by an undistorted geomagnetic field.Because these assumptions are seldom fully met simultaneously, the information fusion challenge is to apply each correction only to the extent that its corresponding pre-conditions are met, as inappropriate corrections will introduce significant error in the future orientation estimates. To achieve this, GMV-D develops an accelerometer correction trustworthiness parameter, 0 ≤α≤ 1, and a magnetometer correction trustworthiness parameter, 0 ≤μ≤ 1, both of which are updated on a sample-by-sample basis and are available at each iteration of the algorithm. The information fusion phase of the algorithm implements the corrections in a two-tiered application of Spherical Linear Interpolation (SLERP) of the quaternions representing the initial dead reckoning estimate and the available corrections, scaled according to their corresponding levels of trustworthiness.GMV-D was evaluated in comparison to 2 other orientation correction approaches (Kalman Filtering and GMV-S) and contrasted with 2 contemporary complementary filter approaches (Madgwick , Mahony). The results confirm that GMV-D displayed better orientation estimation performance when the algorithms operated in an area with known distortion of the geomagnetic field.

Imagery Synthesis for Drone Celestial Navigation Simulation

Simulation plays a critical role in the development of UAV navigation systems. In the context of celestial navigation, the ability to simulate celestial imagery is particularly important, due to the logistical and legal constraints of conducting UAV flight trials after dusk. We present a method for simulating night-sky star field imagery captured from a rigidly mounted ‘strapdown’ UAV camera system, with reference to a single static reference image captured on the ground. Using fast attitude updates and spherical linear interpolation, images are superimposed to produce a finite-exposure image that accurately captures motion blur due to aircraft actuation and aerodynamic turbulence. The simulation images are validated against a real data set, showing similarity in both star trail path and magnitude. The outcomes of this work provide a simulation test environment for the development of celestial navigation algorithms.

Cartesian space track planning for welding robot with inverse solution multi-objective optimization

Aiming at the problem that the traditional inverse solution optimization method is not comprehensive and does not consider the robot structure and actual working conditions, an inverse solution multi-objective optimization method is proposed. This method comprehensively considers the structural size and working state of the welding robot, establishes the stiffness performance evaluation index, optimizes the performance index of the subsequent connecting rod movement area caused by joint rotation, and selects the optimal inverse solution combined with the principle of “minimum joint displacement.” Compared with the traditional method, this method is more comprehensive. It makes the variation range of the rear three small joints of the welding robot larger than the front three large joints, which reduces the power consumption. In addition, it also improves the stiffness of the welding robot at the trajectory point, which ensures the reliability of the welding robot. On this basis, for linear and arc welds, the position interpolation of cartesian space line and arc trajectory based on the S-shaped acceleration and deceleration curve and the posture interpolation of spherical linear interpolation based on unit quaternion are realized. MATLAB simulation results show that the combination of the inverse optimization method and the interpolation method makes the end trajectory, velocity, acceleration, posture curve, and joint displacement curve of the welding robot continuous, smooth, and without mutation.

Combined Predictive and Feedback Contour Error Control With Dynamic Contour Error Estimation for Industrial Five-Axis Machine Tools

This article proposes a real-time method to control the contour error for industrial five-axis machine tools by combining the generalized predictive control (GPC) and the feedback correction (FBC) with dynamic contour error estimation (CEE). The CEE is developed by well considering the curve’s curvature and torsion based on Taylor series expansion and Frenet frame theory. By utilizing the spherical linear interpolation, the tool orientation and the tool tip position are synchronized with respect to the curve length. A novel dynamic foot point searching procedure is established to weaken the influences of the tracking errors’ magnitude on the CEE precision. To tackle the transmission effect, the GPC is adopted to predict the servo systems’ outputs, and then, the induced tool pose deviations, which are subsequently utilized to counteract the contour errors, are predicted by constructing Jacobian matrixes. Especially, the FBC loops are constructed to suppress the influences of disturbances and further to reduce the magnitudes of contour errors. Simulations and experiments are conducted to verify the effectiveness of the proposed methods.

Beyond-the-static-workspace point-to-point trajectory planning of a 6-DoF cable-suspended mechanism using oscillating SLERP

This paper presents a general framework for the planning of point-to-point motions that extend beyond the static workspace of six-degree-of-freedom cable-suspended parallel mechanisms. The proposed translational trajectories are based on a generalization of the hypocycloidal motion previously introduced for cable-suspended robots. Also, the concept of ideal kinematic state is used to maximize the chances of obtaining feasible trajectories, i.e., trajectories in which tension is maintained in the cables. The rotational component of the trajectories is based on the use of spherical linear interpolation. A novel formulation is proposed to connect two arbitrary orientations through oscillations going through the reference orientation. It is shown that the impact of the translational trajectories on the tension constraints is largely dominant, compared to that of the rotational trajectories. A procedure is presented for the determination of the trajectory parameters in order to ensure the continuity of the trajectories. Trajectory feasibility is verified, including mechanical interferences and singularity detection. Finally, a natural cylindrical coordinate frame is proposed that yields a very intuitive description of the trajectories and simulation results are given to illustrate the effectiveness of the proposed approach.

A contour error definition, estimation approach and control structure for six-dimensional robotic machining tasks

Robots are increasingly employed in machining applications, such as milling, grinding and polishing, etc, and contour error can be used to assess the robotic machining accuracy quantitatively. For the robotic machining tasks, such as the grinding of complex surface parts, the reduction of contour error needs to consider the six dimensions of the robotic task space. However, at present, there is no suitable research work directly related to the contour error definition and estimation for the six-dimensional (6D) robotic machining tasks, and the corresponding control structure is rarely involved. For resolving the above problems, this article presents a contour error definition, estimation approach and control structure for the 6D robotic machining tasks. To be specific, the definition of robotic contour error in the 6D machining is first introduced. Then, based on the bidirectional search, Fibonacci search and spherical linear interpolation algorithms, a locally iterative robotic contour error estimation approach with high accuracy and high efficiency is provided. Finally, by compensating the weighted contour error components to the velocity commands in the robotic task space, a simple as well as effective components-based contouring control structure is designed. The real experiments are performed on a six-degrees-of-freedom robot. The results reveal that the definition of robotic contour error given is intuitive, and the discrepancies between the position and orientation contour error estimates and the corresponding true values are very small within 0.01062 µm and 0.8959 µrad, respectively. The designed contouring control structure can reduce effectively the contour errors, and compared with the tracking control only, the mean position and orientation contour errors are reduced by about 30% and 28%, respectively.

Optimized Bézier-curve-based command generation and robust inverse optimal control for attitude tracking of spacecraft

In this paper we address the optimized attitude command generation and robust inverse optimal control design for the spacecraft attitude tracking with large angle slews. Firstly, with the reference attitude command described in a Bézier curve through the spherical linear interpolation, an optimized reference command is generated to meet the principles of small energy consumption and short turning time. The applied optimization combines the particle swarm optimization algorithm and genetic algorithm based on information entropy to guarantee the diversity of the population and improve the global search capabilities. Next, a robust inverse optimal control is developed for the attitude tracking of the obtained optimized reference attitude command, consisting of an inverse optimal Lyapunov approach for performance guarantee and an arctangent-based integral sliding mode control for the disturbance attenuation. A stronger generality of the control Lyapunov function is obtained for the inverse optimality. Furthermore, the developed control efforts are of smaller amplitudes due to introducing the auxiliary angular velocity for the smooth transition to the reference attitude command. Finally, simulation results are included to show the performance of the proposed scheme.

Efficient 3D Object Detection of Indoor Scenes Based on RGB-D Video Stream

<p indent=0mm>For indoor object detection, the input complex scenes often have some defects such as incomplete RGB-D scanning data or mutual occlusion of its objects. Meanwhile, due to the limitations of single RGB-D data or point cloud data input of indoor scenes, it is always difficult to detect all of 3D objects simultaneously. In order to overcome this issue and also alleviate its low efficiency for indoor object detection, an efficient 3D object detection method is proposed which takes RGB-D video streams as input. First, the RGB-D video stream of different indoor environments can be obtained using Kinect camera, and also captured its continuous RGB frames and corresponding point cloud data. Secondly, the Hash function is adopted to extract the content-sensitive key frames from the continuous RGB frames, and the objects semantic relationship can also be constructed according to the type/number of 3D objects contained in adjacent key frames for ensuring that different objects will appear in each key frame. Then, 3D objects of the extracted key frames can be detected by using VoteNet, and the detection results of other frames can be estimated owing to relative posture relationship between adjacent frames by using the quaternion spherical linear interpolation algorithm. Finally, it can achieve efficient 3D object detection for each frame in the RGB-D video stream. Using SUN RGB-D dataset to train the object detection network of key frame, the detection result of proposed method is accurate, and the overall detection time is greatly reduced if comparing with the VoteNet based frame-by-frame detection scheme. Experimental results demonstrate that proposed method is effective and efficient.

Orthodontic Treatment Based on Wearable Mirror-Type Oral Prosthetic Tongue Flap without Bracket Correction.

White teeth can make people full of confidence and satisfy the concept of modern life from the love of beauty. Due to the fusion of computer-aided design and teeth, invisible orthodontics has become the focus of research. Invisible orthodontic treatment technology can predict the results of orthodontics. How to automatically calculate the position and posture of the teeth in the middle stage of orthodontics is the key point of the treatment technology. In order to solve this problem, this article is divided into two parts to start research. Aiming at the problem of tooth orthodontic path planning, quaternion is used to define the tooth posture, combined with the initial posture and target posture of the tooth. A two-stage method is given to plan a collision-free path for the orthodontic tooth. In the first stage, the quaternion spherical linear interpolation and position linear interpolation are used to obtain the intermediate posture of the tooth during orthodontics, and the initial value of the orthodontic stage is obtained, and the obtained intermediate posture is used as a sampling node to apply to the next stage. In the second phase, considering the problem of orthodontic collision and interference, a scheme for calculating the priority of orthodontics is proposed, and the random node expansion part in the RRT (Rapid-exploration Random Tree) algorithm is improved. The initial value of the orthodontic phase is used to calculate the initial value of the iteration. Finally, a path with no collision and the least number of orthodontic stages is searched from the random tree of each tooth node. The experimental results and analysis show that this method can quickly and effectively solve the orthodontic path of teeth, and it is used clinically. The clasp-free invisible correction technology pushes the molars far away to leave gaps for treating patients with mild to moderate overcrowding. The treatment time should be reduced by at least 30%; the stability of the gaps and the long-term healing effect of the treatment provide a reference.

Manipulation Planning for Object Re-Orientation Based on Semantic Segmentation Keypoint Detection.

In this paper, a manipulation planning method for object re-orientation based on semantic segmentation keypoint detection is proposed for robot manipulator which is able to detect and re-orientate the randomly placed objects to a specified position and pose. There are two main parts: (1) 3D keypoint detection system; and (2) manipulation planning system for object re-orientation. In the 3D keypoint detection system, an RGB-D camera is used to obtain the information of the environment and can generate 3D keypoints of the target object as inputs to represent its corresponding position and pose. This process simplifies the 3D model representation so that the manipulation planning for object re-orientation can be executed in a category-level manner by adding various training data of the object in the training phase. In addition, 3D suction points in both the object’s current and expected poses are also generated as the inputs of the next operation stage. During the next stage, Mask Region-Convolutional Neural Network (Mask R-CNN) algorithm is used for preliminary object detection and object image. The highest confidence index image is selected as the input of the semantic segmentation system in order to classify each pixel in the picture for the corresponding pack unit of the object. In addition, after using a convolutional neural network for semantic segmentation, the Conditional Random Fields (CRFs) method is used to perform several iterations to obtain a more accurate result of object recognition. When the target object is segmented into the pack units of image process, the center position of each pack unit can be obtained. Then, a normal vector of each pack unit’s center points is generated by the depth image information and pose of the object, which can be obtained by connecting the center points of each pack unit. In the manipulation planning system for object re-orientation, the pose of the object and the normal vector of each pack unit are first converted into the working coordinate system of the robot manipulator. Then, according to the current and expected pose of the object, the spherical linear interpolation (Slerp) algorithm is used to generate a series of movements in the workspace for object re-orientation on the robot manipulator. In addition, the pose of the object is adjusted on the z-axis of the object’s geodetic coordinate system based on the image features on the surface of the object, so that the pose of the placed object can approach the desired pose. Finally, a robot manipulator and a vacuum suction cup made by the laboratory are used to verify that the proposed system can indeed complete the planned task of object re-orientation.

Study on the Cable Traversing Robot on Spatially Structured Cableway

This paper describes the kinematics, motion planning and mechanism optimization of the dual-arm type cable traversing robot on spatially-structured cable way. Since the inverse kinematics cannot be perfectly defined due to one of the orientation parameters is determined by static condition, its inverse kinematics algorithm utilizing both the kinematics of 3R serial spherical mechanism and the static equilibrium condition of the whole system was developed. The motion planning method to traverse cable node is based on the kinematics and Spherical Linear Interpolation using quaternion. Furthermore, mechanism optimization for avoidance of collision with cables and improvement of characteristics is also conducted.