Kinematic Formula Research Articles

TPTrans: Vessel Trajectory Prediction Model Based on Transformer Using AIS Data

The analysis of large amounts of vessel trajectory data can facilitate more complex traffic management and route planning, thereby reducing the risk of accidents. The application of deep learning methods in vessel trajectory prediction is becoming more and more widespread; however, due to the complexity of the marine environment, including the influence of geographical environmental factors, weather factors, and real-time traffic conditions, predicting trajectories in less constrained maritime areas is more challenging than in path network conditions. Ship trajectory prediction methods based on kinematic formulas work well in ideal conditions but struggle with real-world complexities. Machine learning methods avoid kinematic formulas but fail to fully leverage complex data due to their simple structure. Deep learning methods, which do not require preset formulas, still face challenges in achieving high-precision and long-term predictions, particularly with complex ship movements and heterogeneous data. This study introduces an innovative model based on the transformer structure to predict the trajectory of a vessel. First, by processing the raw AIS (Automatic Identification System) data, we provide the model with a more efficient input format and data that are both more representative and concise. Secondly, we combine convolutional layers with the transformer structure, using convolutional neural networks to extract local spatiotemporal features in sequences. The encoder and decoder structure of the traditional transformer structure is retained by us. The attention mechanism is used to extract the global spatiotemporal features of sequences. Finally, the model is trained and tested using publicly available AIS data. The prediction results on the field data show that the model can predict trajectories including straight lines and turns under the field data of complex terrain, and in terms of prediction accuracy, our model can reduce the mean squared error by at least 6×10−4 compared with the baseline model.

AUTOMATIC ASSEMBLY OF INTERLOCKING BUILDING CUBES BY USING VISUAL SERVOING APPROACH

Most of the robotic arms are set up for operation by the teach-and-repeat (T&R) technique, also known as coordinate-based control, using pre-recorded coordination values to generate robotic arm trajectories. This technique is inefficient because it requires teaching a robotic arm every single coordinate point to complete an operation. It is a manual, tedious and time-consuming task. Moreover, this task needs to be repeated if there are slight changes in the operation. This paper studied the visual servoing approach to replace the T&R technique in assembling different interlocking building cubes (IBC) combinations. The proposed method uses an image-based visual servo (IBVS) with a dynamic look-and-move system, and the eye-to-hand system, which requires computing the incremental joint angles of the robotic arm, a novel kinematic formula with a combination of Trigonometry formulas has been proposed in this paper. The experimental results show the feasibility of using a visual servoing approach to complete the assembly tasks without relying on the T&R technique and achieved an average success rate of 90% and an average of 40.2 seconds for task duration. Thus, this eliminates the tedious and time-consuming tasks of teaching the coordinate points to the robotic arms. Further improvement on the servo motor, camera and computer processor can be done to increase the accuracy, computation time and task duration in future.

An Efficient Kinematic Calibration Method for Parallel Robots With Compact Multi-Degrees-of-Freedom Joint Models

Abstract Forward kinematics-based modeling approaches are capable of constructing complete kinematic error models for parallel robots in a general way. The existing forward kinematics-based modeling methods replace multi-degrees-of-freedom (multi-DOF) joints with several 1DOF joints, allowing each limb of the parallel robot to be modeled like a serial robot. Nonetheless, this substitution complicates the kinematic model and results in additional computation. To overcome this limitation, an efficient kinematic calibration method adopting compact multi-DOF joint models is proposed. First, compact kinematic models for multi-DOF joints are established with the product of exponentials formula and adopted in the forward kinematic formulation of limbs. Error models of limbs are derived by simplifying the forward kinematic formulas' differentials, and the geometric error model for parallel robots is established by further concatenating and reformulating the limb error models. Next, the kinematic model is iteratively updated with the geometric parameter errors identified by the Levenberg–Marquardt algorithm. Error compensation is achieved through the inverse kinematics of the calibrated kinematic model. Finally, simulations and an experiment are implemented for validation. Compared with the existing forward kinematics-based modeling approaches, the error modeling procedures are simplified as the equivalent substitution of multi-DOF joints is avoided. The proposed approach also enhances the error compensation efficiency while maintaining high accuracy improvement.

Model Test Study on Rock Rolling Characteristics

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.

Minkowski functionals of CMB polarization intensity with pynkowski: theory and application to Planck and future data

ABSTRACT The angular power spectrum of the cosmic microwave background (CMB) anisotropies is a key tool to study the Universe. However, it is blind to the presence of non-Gaussianities and deviations from statistical isotropy, which can be detected with other statistics such as Minkowski functionals (MFs). These tools have been applied to CMB temperature and E-mode anisotropies with no detection of deviations from Gaussianity and isotropy. In this work, we extend the MF formalism to the CMB polarization intensity, P2 = Q2 + U2. We use the Gaussian kinematic formula to derive the theoretical predictions of MFs for Gaussian isotropic fields. We develop a software that computes MFs on P2healpix maps and apply it to simulations to verify the robustness of both theory and methodology. We then estimate MFs of P2 maps from Planck, both in pixel space and needlet domain, comparing them with realistic simulations that include CMB and instrumental noise residuals. We find no significant deviations from Gaussianity or isotropy in Planck CMB polarization intensity. However, MFs could play an important role in the analysis of CMB polarization measurements from upcoming experiments with improved sensitivity. Therefore, we forecast the ability of MFs applied to P2 maps to detect much fainter non-Gaussian anisotropic signals than with Planck data for two future complementary experiments: the LiteBIRD satellite and the ground-based Simons Observatory. We publicly release the software to compute MFs in arbitrary scalar healpix maps as a fully documented python package called pynkowski.

An analytical approach based on Dixon resultant for the inverse kinematics of 6R robot manipulators with offset wrists

6-revolute (6R) robot manipulators with offset wrists are widely applied in industrial scenarios. However, the autonomous and highly-precise manipulations of these robots are restricted for online industrial uses due to the lack of effective inverse kinematic methods. The inverse kinematic problem for 6R robot manipulators with offset wrists is addressed with a novel kinematic modelling and elimination approach in this study. To begin, kinematic models with non-redundant equations for joint variables are established An improved Dixon elimination approach is then introduced for inverse kinematic calculation to reduce the high computational cost and get rid of the singularity issues. The core is to efficiently extract the linear terms from variable sequences, such that the number of terms can be reduced in the polynomial equations. By further examining the characteristics of offset wrists, the fourth and fifth joint variables are eliminated from kinematic formulas, without increasing the polynomial degrees. The formulas are demonstrated to produce a polynomial with a single joint variable and a minimum degree using the elimination approach. Finally, experiments are carried out to verify the computing performance of the inverse kinematic problem. The main focus of this study is to develop a systematic and automatic procedure for solving the inverse kinematic problem for 6R robot manipulators with offset wrists and producing highly precise and efficient solutions by means of a novel modelling and elimination method.

Confidence tubes for curves on SO(3) and identification of subject-specific gait change after kneeling

Abstract In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which might be a major risk factor for the development of knee osteoarthritis, we develop confidence tubes for curves following a perturbation model on SO(3) using the Gaussian kinematic formula which are equivariant under gait similarities and have precise coverage even for small sample sizes. Applying them to gait curves from eight volunteers undergoing kneeling tasks and adjusting for different walking speeds and marker replacement at different visits, allows us to identify at which phases of the gait cycle the gait pattern changed due to kneeling.

Design and dynamics of a 2PSS+PSR+SP+RUPUR-type parallel rotor of helicopter

A novel parallel rotor of the helicopter is designed by combining a 2PSS+PSR+SP+RUPUR-type parallel manipulator with a rotor of the helicopter. Its kinematics and dynamics models are established. Firstly, the structure principle and the merit performances of the parallel rotor are explained by comparing with conventional and existing rotors of the helicopter, and the degree of freedom of the parallel manipulator is calculated. Secondly, the kinematics formulas are derived for solving the displacement, Jacobian matrix, general velocity, Hessian matrix, and acceleration of the parallel rotor and the kinematic limbs. Thirdly, a dynamics model of the parallel rotor is established based on the derived kinematics formulas. Finally, the theoretical kinematics and dynamics solutions are verified by a simulation mechanism of the parallel rotor of the helicopter.

Kinematic and dynamic analysis of a 4-DOF over-constraint parallel driving mechanism with planar sub-closed chains

AbstractIn this paper, a new over-constrained parallel driving mechanism (PDM) with planar sub-closed chains is proposed. First, the number of over-constraints on the PDM is calculated. Then, an analysis is conducted as to the kinematics of the hybrid manipulator, including positions, velocities, and accelerations of all bodies. Furthermore, the Newton–Euler approach is taken to deduce the kinematic formula of each link and the formula of inertial force at the center of mass. However, it remains difficult to solve the equation since the number of equations is smaller than that of unknown variables. To solve this problem, the screw theory is applied in the present study to analyze the cause of over-constraints, with the link’s elastic deformation introduced as the supplement of deformation compatibility equations. Moreover, the actuation forces and constrained forces/moments are calculated simultaneously. Finally, the dynamic model is verified through simulation and experimentation. The proposed modeling approach provides a fundamental basis for the structural optimization and friction force computation of the over-constrained PDM.

Development and dynamics of a novel couple-constrained parallel wrist with three measuring force flexible fingers

AbstractA novel couple-constrained parallel wrist with three measuring force flexible fingers is designed for grabbing heavy objects and measuring grabbed forces. Its prototype is developed, its dynamics model is established, and its grabbing forces are measured. First, using the extended formulas of the skew-symmetric matrix, the kinematic formulas are derived for solving the Jacobian/Hessian matrices and the general velocity/acceleration of the moving links in the couple-constrained parallel wrist. Second, a dynamics model is established for solving the dynamic actuation forces, the couple-constrained forces, and the torque in the couple-constrained parallel wrist. Third, the theoretical solutions of the kinematics/dynamics of the couple-constrained parallel wrist are verified using a simulation mechanism. Finally, the grabbing forces of the three flexible fingers are measured and analyzed.

Inverse kinematics and singularity analysis of a redundant parallel robot

The 3C (computer, communications, consumer electronics) industry belongs to labor-intensive and the requirements of the 3C industry are precisely high. The parallel robot is a good choice to replace human labor to manufacture these products. This paper presents a new 3- redundant parallel mechanism for 3C manufacturing. Firstly, some advantages of redundant parallel robots in the current industrial demand are discussed. Then, the inverse kinematics of the redundant parallel manipulator is analyzed according to the closed-loop equation, the inverse displacement problem is solved. The machine is simple and low cost, and as a less freedom robot, it has a higher precision than the traditional robot and has a broad prospect for 3C manufacturing. Finally, the Jacobian matrix of the redundant parallel manipulator was calculated and established by the inverse kinematics formula, and three kinds of the singularity of the mechanism were analyzed by the Jacobian matrix, which are the forward singularity, reverse singularity, mixed singularity respectively.

Higher order kinematic formulas and its numerical computation employing dual numbers

In the analysis of robots and mechanisms, the computation of velocities and accelerations is frequently needed. In other cases, higher-order kinematic formulas such as the jerk and the jounce/snap—the third and fourth-order time derivative of the position vector, respectively—are also needed. While procedures for computing velocity and accelerations can be found elsewhere in the literature, the computations of the jerk and especially of the jounce/snap are not so common. In this study, a novel formulation to compute all kinematic quantities up to the fourth order, based on the dual number is presented. In dual number formalism, kinematic analysis equations are computed automatically once the position vector is constructed. This provides a concise and efficient method to conduct kinematic analysis. To prove the validity of the implemented formulation, we compute the velocity, acceleration, jerk, and jounce/snap for the end effector of the RC robot manipulator. In addition, we compute the jerk and the jounce/snap for the coupler point of the spherical 4 R mechanism.

Invariant kinematic formulas and inequalities for chord power integrals in integral geometry

In this paper, we first introduce the history, developing process, recent progress, andworks contributed by Chinese researchers in integral and convex geometry.Then we show our recent works in invariant kinematicformulas and chord power integrals. The affine isoperimetric inequalities on the set of convex bodies, closely related to integral geometry, are also addressed.

Inverse kinematic formula for a new class of 6R robotic arms with simple constraints

A systematic and unified kinematic modelling method for a new class of 6R robotic arms is investigated, which separates the coupled joint variables without raising degrees. To begin, successive products of the tangent-form DCMs and quaternions yield the non-redundant kinematic equations. By providing the derived position vectors, new inverse kinematic formulas are then invented for general 6R robotic arms. Based on this, a degree-reduced condition is found out to ideally reduce the kinematic formulas of five coupling variables to three while the degrees remain constant, which benefits for a light computational resource. A new class of robotic arms satisfying the degree-reduced condition is proposed, including simple kinematic constraints of two adjacent parallel or orthogonal. The inverse kinematic software testing proves the efficiency of the method: the running time is less than 2 milliseconds; the relative position and orientation variations are less than 10−11. This study can be taken as a theoretical basis for the inverse kinematic research of 6R robotic arms, which provides a brand-new way to establish the kinematic formulas and exhibits a relatively simple kinematic structure for precise industrial use. Moreover, the proposed method is expected to be used to reduce the degrees of kinematic modelling polynomials of higher-DOF robotic arms.

The Wearable Lower Limb Rehabilitation Exoskeleton Kinematic Analysis and Simulation.

In recent years, due to the increase in the incidence of traffic accidents, the number of people with limb injuries has also increased. At the same time, among the aging population, neurological diseases or cardiovascular and cerebrovascular diseases have caused many people to have limb hemiplegia. It has been clinically proven that the use of rehabilitation equipment can help patients with limb injuries to restore limb motor function. This paper takes wearable lower limb rehabilitation exoskeleton as the research object, and its main contents are mechanical structure design, kinematics analysis, gait planning, virtual prototype simulation, and experimental verification and analysis. Based on the physiological characteristics of human body and the principle of comfortable and reliable wearing, this paper designs wearable exoskeleton for lower limb rehabilitation. Firstly, the physiological structure characteristics and movement mechanism of human lower limbs were studied and analyzed. By referring to the rotation range and height and size of each joint of human lower limbs, the overall scheme of wearable lower limb rehabilitation exoskeleton was designed and the degree of freedom was allocated. At the same time, Solidworks was used to establish a three-dimensional model. On the basis of a 3D model, a kinematics model was established, and the forward kinematics solution was obtained by using homogeneous coordinate transformation. Since the inverse kinematics solution was relatively complicated, the inverse kinematics solution was conducted in this paper according to the geometric relations of the joints of the lower limbs. Kinematics analysis of the exoskeleton structure of wearable lower limb rehabilitation was carried out to lay a theoretical foundation for gait planning. The off-line gait planning was carried out by using the method based on ZMP stability criterion, and the gait planning was divided into five stages: squat, start, middle step, stop and rise, and the motion trajectory of the center of mass and ankle joint was planned. Based on the inverse kinematics formula, the function of the change of joint angle with time in walking process is derived. The virtual prototype is established in ADAMS, and the simulation of virtual prototype is carried out by using the function of gait planning's joint angle and time. The correctness of structural design and gait planning was verified by measuring the trajectories of the centroid and ankle joints in each gait stage and the functional relationship between the rotation angle of each joint and time. Then, using a 3D dynamic capture system to capture the human lower limb motion trajectory of each joint, each joint trajectory data output, using the MATLAB software to output data, gets the joint trajectory change over time function curve and is used to verify feasibility and applicability of human gait planning. Through the research and analysis of the joints of the lower limbs of the human body, it can be concluded that the hip joint and the knee joint have 3 degrees of freedom, respectively, and the knee joint has 1 degree of freedom.

Design and dynamics of a novel parallel coaxial twin rotor of helicopter

In order to simplify a conventional coaxial rotor of the helicopter, to improve the dynamic performances of the coaxial rotor, to reduce the weight of the helicopter and to increase its safety, a novel parallel coaxial twin rotor (PCTR) of the helicopter is designed by combining a 2PSS + PSR + SP parallel mechanism with a coaxial twin rotor. Its kinematics and dynamics models are established. Firstly, the structure principle and the performances of the PCTR are explained, and the degree of freedom of the 2PSS + PSR + SP parallel mechanism is calculated. Secondly, the kinematics formulas are derived for solving the displacement, Jacobian matrix, general velocity, Hessian matrix, and acceleration of the PCTR and its moving limbs. Thirdly, a dynamics model of the PCTR is established for solving the dynamic actuation forces and the dynamic constrained forces. Finally, the theoretical solutions of the kinematics and dynamics of the PCTR of the helicopter are verified to be correct by its simulation mechanism.

Dynamic Modeling and Simulation of a Hybrid Robot

Abstract The unique structure of hybrid robot makes its dynamic characteristic different from that of the traditional machine tools. Therefore, the dynamic model is crucial to both designing and application of hybrid robot. In this paper, a new type of five-degrees-of-freedom (5DoF) hybrid robot is introduced, and its dynamic model is established. First, the kinematic formulas are derived for all the component, and then, the inertia forces or moments are calculated. Second, the active forces or moments in the joints are assumed as variables and the number of variables is reduced by analyzing joint types. Then, an equation set of 36 equilibrium equations with 38 variables is obtained using D'Alembert's principle. Based on the spatial deformation compatibility analysis of two branches, two supplementary equations are derived to determine the solution of dynamic model of the hybrid robot with redundant constraints in its parallel mechanism. Several cases are studied by comparing with ADAMS simulation. The result shows the good accuracy of the proposed dynamic model, which provides a practical method to calculate the reaction force or moment in any joint at any instant for the hybrid robot and thus facilitates dimensional synthesis, trajectory optimization, and smoothing control.

Principal kinematic formulas for germs of closed definable sets

We prove two principal kinematic formulas for germs of closed definable sets in Rn, that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula due to the author. In this setting, we do not integrate on the space of euclidean motions SO(n)⋉Rn, but on the manifold SO(n)×Sn−1.

Kac-Rice formula for transverse intersections

We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere.

Dynamics of moving-object grasped by a hybrid hand

When a heavy object is cooperatively grasped to move by several fingers of the robot hybrid hand, the inertial properties and the mass distribution of the object must influence largely on the operation precision, grasping stability, and the safety of both the hybrid hand and the object. Hence, it is an important and significant issue to establish and analyze the dynamics model of the moving-object cooperatively grasped by the hybrid hand in order to ensure the safety and grasping stability of the hybrid hand and the object. However, this research has not been conducted. In this paper, a dynamics model of the moving-object grasped by the hybrid hand is established, and its dynamics is studied and analyzed. First, a three-dimensional model of a hybrid hand formed by a novel parallel manipulator and three fingers is designed for cooperatively grasping object. Second, the kinematic formulas for solving the Jacobian matrices, the Hessian matrices, the general velocity/acceleration of the moving platform, and four active limbs of the parallel manipulator are derived. Third, the composite Jacobian matrix and the composite Hessian matrix of the hybrid hand are derived, and the general velocity/acceleration of the moving-object grasped by the hybrid hand is derived. Fourth, dynamics model of the hybrid hand is established, the formulas for solving the dynamic actuation forces of the three fingers and the dynamic actuation forces/torque and constrained forces of the parallel manipulator are derived. Finally, the theoretical solutions of the dynamics model of the moving-object grasped by the hybrid hand are verified by its simulation mechanism.