Search Results (12)

Let $K\subset {\mathbb{R}}^n$ be a bounded convex body. We prove a lattice-averaging formula that gives upper bounds for the number of unit balls required to cover K. If the Voronoi cell P of a lattice is contained in the Euclidean unit ball, then some translate and rotation of the lattice produces a covering whose size is at most a linear combination of the intrinsic volumes of K; the coefficients are determined by the intrinsic volumes of P. The proof averages the number of Voronoi cells meeting K over one fundamental cell and over $SO\left(n\right)$. For the regular hexagonal lattice in ${\mathbb{R}}^2$, the formula reproduces the classical planar Blaschke bound. For the cubic lattice, it gives a closed-form estimate in every dimension $n\ge 2$. In ${\mathbb{R}}^3$, explicit computations for the cubic, face-centered cubic, and body-centered cubic Voronoi cells show that the body-centered cubic lattice has the smallest coefficientwise bound among these three lattices. In ${\mathbb{R}}^4$, the intrinsic volumes of the $A_4^{\ast }$ permutohedron are computed from its graphical-zonotope representation, leading to a sharper bound than for the cubic lattice. Full article

This paper proposes a superelliptic dual quaternion framework that extends classical dual quaternion kinematics by replacing the Euclidean metric structure with a Gielis-formula-induced superelliptic inner product and its associated vector product. Within the resulting space ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^3$, superelliptic dual numbers, dual vectors, and an E-Study-type correspondence between unit dual vectors and directed superelliptic lines are established, yielding an algebraic model adapted to non-Euclidean geometric profiles. In contrast to the standard Euclidean dual quaternion formalism, where rotations, translations, and screw motions are governed by the ordinary inner product of ${\mathbb{R}}^3$, the present formulation encodes these motions relative to a parameter-dependent superelliptic geometry determined by Gielis’ superformula. This distinction enables the kinematic description of motions associated with superelliptic axes and trajectories that cannot be represented naturally within the classical Euclidean setting. A superelliptic screw motion theorem is obtained, showing that a unit superelliptic dual quaternion generates simultaneous rotation about and translation along a common superelliptic axis. The framework offers a compact mathematical basis for advanced rigid-body modeling in robotics and geometric design. The proposed framework represents rotation, translation, and screw displacement by a single unit superelliptic dual quaternion, providing a compact basis for shape-dependent rigid-body modeling in robotics and geometric design. Full article

To address the challenges of the lack of specialized machinery adapted to traditional agronomic requirements, high labor intensity, and low efficiency in the planting of Ligusticum chuanxiong stalk segments (commonly known as Chuanxiong seed stalk or Lingzhong), a planting mechanism for the cutting of Chuanxiong seed stalk was developed in accordance with traditional agronomic requirements. A kinematic model of the gripping point was established, from which a plant spacing formula was derived. Based on the zero-speed planting principle, a cuttage planting scheme for Chuanxiong seed stalks was proposed, in which the gripper trajectory as well as the forward-tilt x_t and correction x_c were defined, and the decisive role of installation height on planting depth and the influence of driven-sprocket motion parameters on planting uprightness were elucidated. A 3D model and a DEM-MBD coupled simulation model were constructed to analyze planter–soil–seed interaction. A three-factor, three-level Box–Behnken experiment was conducted, and a response surface model was built and optimized using ‘Design-Expert’ software. The optimal parameters were a driven sprocket angular velocity of 0.654 rad/s, a rotation radius of 100.787 mm, and a release angle of 90.647°, yielding an average planting uprightness of 85.264°, with the corresponding x_t and x_c of 5.18 mm and 2.69 mm, respectively; the factor influence ranked as angular velocity > rotation radius > release angle. Seed–soil interaction analysis verified the mechanism’s feasibility and the accuracy of the theoretical models. Field tests showed average qualification rates of 87.13% for plant spacing, 96.01% for planting depth, and 90.41% for uprightness, with corresponding coefficients of variation of 4.37%, 2.95%, and 3.73%, indicating stable and reliable field performance. Full article

In this study, equidistant ruled surfaces generated by the Darboux vector, which has significant kinematic importance and characterizes the instantaneous rotation of a moving frame, are investigated specifically for the Frenet frame. By establishing a structural relationship between a surface and its equidistant ruled surface, transition formulas are provided for shape operators, Gaussian and mean curvatures, and fundamental forms, revealing that the equidistant surface is a scaled transformation of the original one. The obtained results demonstrate that both surfaces are developable and that the geometric properties of the equidistant ruled surfaces can be expressed dependently on each other. Furthermore, it is shown that the geometric character of the equidistant surface, including the invariance of asymptotic lines and the preservation of umbilical points under constant angle conditions, is determined by the rotational dynamics of the base curve. These findings constitute a theoretical foundation for cases involving the use of Darboux axes of different frames in higher dimensions or the investigation of similar structures in different geometric spaces. The geometric interpretation of this theoretical framework is elucidated through the fundamental properties of the surfaces. Finally, a concrete example is presented, where the symmetry of the central planes of the equidistant ruled surfaces at appropriate points is visualized using Maple 2017 software. Full article

The determination of maximum singularity-free space is critical to structural design and motion control strategy in the Stewart platform. Nevertheless, in practical applications, there exist several limitations such as computational efficiency, calculation precision, and the reliability of computational results. To overcome those shortcomings, this work proposes an efficient and high-precision method for computing the maximum singularity-free space within the Stewart platform. Firstly, apply K-Means clustering to group the variables, including the range, mean, and standard deviation of driving rod lengths, and the clustering centroids and extreme rod lengths collectively form a set of scenarios to avoid large-scale searching. An additional sorting methodology with a specific parameter is proposed for sorting the aforementioned scenarios in descending order and detecting singular-prone cases. Secondly, compute the initial solution for maximum singularity-free length without gimbal lock through an analytical solution formula, enabling reduction in the search scope. Thirdly, introduce a novel scaling factor to resolve the problem of dimensional inconsistency between rotation and translation within the Jacobian matrix using dual quaternions, and determine the singularity based on the determinant of the newly proposed Jacobian matrix. Finally, employ a CNN-LSTM-Attention model for a secondary verification procedure, specifically targeting the challenge of singularities encountered when solving the forward kinematics of the Stewart platform using zero-position values. The experiments demonstrate that the accelerated discretization method for maximum singularity-free joint space and workspace is applicable to devices with diverse geometric configurations. For two practical Stewart platforms, compared with two conventional methods, this method improves computational efficiency and precision significantly. The computation time of the first platform is reduced by 97.54% and 98.07% respectively, while that of the second platform is cut by 80.84% and 81.80% respectively. In terms of precision, the first platform demonstrates 95.83% and 78% improvement respectively, and the second platform attains 99.99% improvement over two conventional methods. Full article

The present challenges in the automotive industry require the development and practical implication of novel machining procedures, which will provide appropriate solutions. These procedures should still meet the requirements of productivity, surface quality and energy efficiency. The further development of novel machining procedures introduces new problems that did not occur (or occurred to a lesser extent) with traditionally applied procedures. Rotational turning has come to the attention of production engineers in the previous decade since it can be used to machine ground-like surfaces in an ecologically friendly and highly productive manner. However, the chip removal characteristic is slightly different from traditional turning due to the applied special kinematic relation and complex tool edge geometry. The run-in phase will take longer, which is the time period between the first contact of the tool and the formation of a constant chip cross-sectional area. The clarification of the chip formation is important in any machining procedure. To achieve this goal, the geometric parameters of the chip must be determined. Since the start of the chip removal is a crucial stage in rotational turning due to its length, the chip height, chip width and the cross-sectional area of the chip should be separately defined in the initial stage. Therefore, in this paper, the initial phase of chip removal in rotational turning is studied. The increasing cross-sectional area of the chip is determined analytically by the application of the previously elaborated equation of the cut surface. Calculating formulas are defined for the different stages of the start of the chip removal, which could be used in the forthcoming studies to analyze the chip formation. The effects of different determining parameters are analyzed theoretically by the deduced formulas of the run-in phase and practical experiments are also carried out. The analytical and experimental analyses showed that increasing feed also increases the dynamic load on the cutting edge, while the depth of cut lowers the growth of the characteristic parameters of the chip, which results in a lower dynamic load on the tool. Full article

In this study, a dual-mode reconfigurable parallel ankle joint rehabilitation mechanism is proposed to meet the needs of patients in different ankle rehabilitation stages. This mechanism can switch between the 1T2R (where R represents rotation and T represents translation) and 2T1R motion modes. The screw theory and the modified G-K formula were used to analyze and verify the degree of freedom of the mechanism. The non-parasitic motion characteristics were analyzed by examining the topological structure of the mechanism. An inverse kinematics model was established using the closed-loop vector method, and the mechanism’s singularity was analyzed based on the Jacobian matrix. The Jacobian matrix and the numerical method were used to compare and analyze the workspace index, the rotational dexterity index and the load capacity performance index before and after the introduction of branched chains with actuation redundancy. A particle swarm optimization algorithm was used to optimize the geometric dimensional parameters of the mechanism. The results show that the mechanism exhibits the characteristics of a parallel mechanism without parasitic motion in the two motion modes. Using branched chains with actuation redundancy can significantly improve the rotational dexterity and load capacity performance index, without affecting the workspace index. Compared to the original mechanism, the kinematic performance of the optimized mechanism is significantly improved. It is concluded that the proposed mechanism can meet the needs of ankle joint activity training in the 1T2R motion mode and the needs of ankle joint proprioception training in the 2T1R motion mode, which can better meet the needs of patients in different rehabilitation stages. Full article

In order to study the influence of falling rock shapes on their rolling characteristics and to determine the optimization of falling rock protection design, a series of research experiments were conducted. Model experiments were designed to explore the rolling characteristics of rockfalls with different shapes. Based on the experimental results, it was found that the slenderness ratio, center of gravity, and rotational inertia of the rockfalls can affect their rolling characteristics, leading to swaying and changing the rolling axis during the rolling process, thereby affecting their rolling speed. Building upon these findings, an analysis of the formation mechanism of rolling resistance was conducted. It was determined that the primary cause of energy loss was the rolling resistance arm formed with the rolling surface during rockfall motion. A shape parameter was proposed to quantify the rolling resistance. These parameters were incorporated into a kinematic formula that considered the influence of rockfall shape, slope, and slope roughness on the rolling speed. Combined with the offset and initial position of the rockfall, the formula could be used to calculate the rolling speed and impact energy in the rolling region at any position in the region. The calculation formula was validated using model experimental data, and the results showed that the error between the experimental and calculated values was small. The error was corrected based on the experimental data. After on-site testing and verification, it could provide reference for the management of rockfall disasters. Full article

In this paper, a spatial cable-driven parallel mechanism in a V-shaped cable arrangement is proposed. It is further simplified as a planar hybrid cable-driven parallel mechanism to analyze its kinematics, which consists of two identical active cable chains and a passive cross-slide mechanism. In order to investigate the degrees of freedom (DoFs) of the hybrid mechanism using screw theory, cable chains are represented as rotational–prismatic–rotational (RPR) chains. The motion pairs of all the chains are denoted according to screw theory. Firstly, the number and the types of DoFs of each chain are determined. Then, the number and the types of DoFs for the hybrid mechanism are calculated. Furthermore, the theoretical result is verified using the modified Grübler–Kutzbach (G-K) formula. It shows that the unique DoF of the equivalent mechanism is a rotation with a continuously changing axis, which is consistent with the V-type cable-driven mechanism with elliptical trajectories. Finally, the kinematics analysis of the cross-slider mechanism driven by two cables is carried out. The length, velocity and acceleration of the cables are obtained from numerical calculation in MATLAB, and the results are demonstrated using ADAMS simulation. Full article

This paper introduces a new method for kinematic modeling of the robot arm by deriving a new elegant mathematical formula based on the axis vector with the tangent of the rotation angle. For this purpose, an innovative analytical quaternion is introduced through integration between Axis-Invariants and unit quaternion features named Ju-Gibbs quaternion, which expresses the body rotation with non-redundant parameters compared with the quaternions in literature. Two theorems based on the new form of the quaternion are developed and proved for the kinematic modeling of the robot arm. The first is attitude alignment, which is essential in multiaxial rotation systems. The second theorem for the wrist inverse kinematic (IK) solution is utilized to obtain the joint variables for the last joints of the end effector. In order to verify the effectiveness and accuracy of the proposed method, a numerical example and simulation of different structural configurations of robot and human arms are intensively studied. The novel quaternion provides a new tool for kinematic analysis and reduces the computational complexity of the kinematic solutions of the Robot-Arms wrist. Furthermore, the method laid a new foundation for the IKs of multi-axis systems based on Axis-Invariant and tangent quaternion. Full article

Teaching kinematic rotations is a daunting task for even some of the most advanced mathematical minds. However, changing the paradigm can highly simplify envisioning and explaining the three-dimensional rotations. This paradigm change allows a high school student with an understanding of geometry to develop the matrix and explain the rotations at a collegiate level. The proposed method includes the assumption of a point (P) within the initial three-dimensional frame with axes (${\widehat{x}}_i$, ${\widehat{y}}_i$, ${\widehat{z}}_i$). The method then utilizes a two-dimensional rotation view (2DRV) to measure how the coordinates of point P translate after a rotation around the initial axis. The equations are used in matrix notation to develop a rotation matrix for follow-on direction cosine matrixes. The method removes the requirement to use Euler’s formula, ultimately, providing a high school student with an elementary and repeatable process to compose and explain kinematic rotations, which are critical to attitude direction control systems commonly found in vehicles. Full article

With the widespread application of robots in aerospace, medicine, automation, and other fields, their motion safety is essential for the well-being of humans and the accomplishment of vital socially beneficial programs. Conventional robot hardware and software designs mainly rely on experiential knowledge and manual testing to ensure safety, but this fails to cover all possible testing paths and adds risks. Alternatively, formal, mathematically rigorous verifications can provide predictable and reliable guarantees of robot motion safety. To demonstrate the feasibility of this approach, we formalize the mathematical coordinate transformation of a robot’s rigid-body kinematics using the Coq Proof Assistant to verify the correctness of its theoretical design. First, based on record-type matrix formalization, we define and verify a robot’s spatial geometry by constructing formal expressions of the matrix’ Frobenius norm, trace, and inner product. Second, we divide rotary motion into revolution and rotation construct and provide their formal definitions. Next, we formally verify the rotational matrices of angle conventions (e.g., roll–pitch–yaw and Euler), and we complete the formal verification of the Rodriguez formula to formally verify the correctness of the motion theory in specific rotating kinematics problems. The formal work of this paper has a variety of essential applications and provides a generalizable kinematics analysis framework for robot control system verification. Moreover, it paves the way for automatic programming capabilities. Full article