Trajectory Interpolation Research Articles

Acceleration-based FIR filters for trajectory interpolation with different kinematic constraints and target velocities for NC systems

Motion planning is a critical issue in CAD/CAM manufacturing systems. Many interpolation algorithms based on velocity-based finite-impulse-response (FIR) filters have been proposed to interpolate fast and smooth trajectories, in which the reference trajectories are usually composed of a series of linear segments. Those methods, however, possess two main limitations: first, they can only interpolate trajectories that maintain the same target velocity, i.e., the corner error equations will fail if the target velocities are different around different corners. Second, many systems have different kinematic constraints around different corners, but all of the methods can only process fixed and conservative kinematic constraints. In order to overcome these two limitations, an interpolation algorithm based on acceleration-based FIR filters is proposed. By applying FIR filters to the acceleration base, the motion planning process will comprise several individual convolution processes. The filters and constraints in each convolution process can be designed differently. Therefore, the interpolation algorithm can generate trajectories with different target velocities and kinematic constraints because the taps and types of FIR filters can be changed accordingly during the interpolation process. The experiment results show that, by applying the proposed method to a non-linear kinematic system, more than 10% of motion time is saved when compared to a velocity-based method, the performance of the system can be fully utilized, and efficiency can be markedly improved.

Trajectory planning of a 4-RR(SS)2 high-speed parallel robot

AbstractThe trajectory-planning method for a novel 4-degree-of-freedom high-speed parallel robot is studied herein. The robot’s motion mechanism adopts RR(SS)2 as branch chains and has a single moving platform structure. Compared with a double moving platform structure, the proposed parallel robot has better acceleration and deceleration performance since the mass of its moving platform is lighter. An inverse kinematics model of the mechanism is established, and the corresponding relationship between the motion parameters of the end-moving platform and the active arm with three end-motion laws is obtained, followed by the optimization of the motion laws by considering the motion laws’ duration and stability. A Lamé curve is used to transition the right-angled part of the traditional gate trajectory, and the parameters of the Lamé curve are optimized to achieve the shortest movement time and minimum acceleration peak. A method for solving Lamé curve trajectory interpolation points based on deduplication optimization is proposed, and a grasping frequency experiment is conducted on a robot prototype. Results show that the grasping frequency of the optimized Lamé curve prototype can be increased to 147 times/min, and its work efficiency is 54.7% higher than that obtained using the traditional Adept gate-shaped trajectory.

Interpolation on the special orthogonal group with high-dimensional Kuramoto model

The construction of smooth interpolation trajectories in different non-Euclidean spaces finds application in robotics, computer graphics, and many other engineering fields. This paper proposes a method for generating interpolation trajectories on the special orthogonal group SO(3), called the rotation group. Our method is based on a high-dimensional generalization of the Kuramoto model which is a well-known mathematical description of self-organization in large populations of coupled oscillators. We present the method through several simulations and visualize each simulation as trajectories on unit spheres S2. In addition, we applied our method to the specific problem of object rotation interpolation.

Experimental and analytical evaluation of tool path error using computer integrated nonlinear kinematical modeling for a 4DOF parallel milling machine

ABSTRACT Machining errors in parallel kinematic machines primarily depend on their extent of kinematic nonlinearity. In this paper, a computer integrated model of the kinematic nonlinearity and trajectory interpolation method for a 4DOF parallel milling machine is developed. The nonlinear errors prompted during various trajectory modes are analyzed. It is proved that in order to avoid significant non-linearity, the actuator and the end-effector space must possess identical order of trajectory. The effects of tool path length, its spatial location, and the Jacobian matrix properties on the kinematic nonlinearity error, are studied. Results showed that the path length is the governing factor for the nonlinear behavior of the mechanism. Moreover, the kinematic error is illustrated to have a reverse relationship with maximum singular values of the inverse Jacobian matrix. Experiments are conducted using digital dial indicators. Experimental results verified the accuracy of the proposed mathematical approach and confirmed its applicability in actual machining processes. Moreover, the proposed interpolation algorithm successfully limits the kinematic error under the machining tolerance by minimal segmentation. Finally, it is demonstrated that the proposed method is superior in terms of accuracy, to the median osculating circle (MOC) method.

A triaxial linkage algorithm for the irregular surface machining of backplane using turn-milling compound method

The manufacture of electronic communication equipment has become an important topic in manufacturing industry as they are used more and more widely in daily life. The fine surface machining of electronic communication backplane is still using milling method, and the production efficiency is so low that the production capacity is seriously affected. In this paper, a new method of turn-milling compound machining is proposed, which is applied to the noncircular surface machining of communication backplane, and the algorithm of triaxial linkage turn-milling is studied emphatically. This method is to add a high-speed turntable to the platform of the three-axis machining center as the fourth axis and proposed trajectory fitting and interpolation algorithm which combined UG point cloud information with spline curve fitting; the interpolation and velocity planning algorithm is based on the principle of constant linear velocity of machining tool tip and can improve the machining speed and acceleration smoothing performance perfectly. Through machining verification, the machining time of the same workpiece is decreased from 30 min using traditional milling to 4 min using this method, the surface roughness is less than 1 μm, and the precision is less than ±0.05 mm, which all fully meet the requirements of product.

Vehicle Trajectory Interpolation Based on Ensemble Transfer Regression

Vehicle trajectory collection usually faces challenges such as inaccurate and incomplete trajectory data, mainly due to missing trajectories caused by Global Navigation Satellite System (GNSS) outages. In this paper, a novel ensemble transfer regression framework is proposed for urban environments with transfer learning as the primary solution for constructing a fine-grained trajectory dataset during GNSS outages. First, GNSS and motion information are fused for the training process. Then, a regression-to-classification (R2C) process is employed to implement incremental training to adapt to dynamically changing environments. Third, to account for GNSS outages, transfer learning is integrated to construct a data filtering strategy that minimizes negative sample weights during the current scenario. Finally, a more accurate classification-type loss function for ensemble learning is designed to obtain the ensemble transfer regression model. We utilize real-world datasets to verify the accuracy of the comparative methods and the proposed framework in trajectory interpolation prediction. The experimental results show that our framework is significantly superior to the comparative methods.

Control system of 4-DOF palletizing robot based on improved R control multi-objective trajectory planning

To realize efficient palletizing robot trajectory planning and ensure ultimate robot control system universality and extensibility, the B-spline trajectory planning algorithm is used to establish a palletizing robot control system and the system is tested and analyzed. Simultaneously, to improve trajectory planning speeds, R control trajectory planning is used. Through improved algorithm design, a trajectory interpolation algorithm is established. The robot control system is based on R-dominated multi-objective trajectory planning. System stack function testing and system accuracy testing are conducted in a production environment. During palletizing function testing, the system’s single-step code packet time is stable at approximately 5.8 s and the average evolutionary algebra for each layer ranges between 32.49 and 45.66, which can save trajectory planning time. During system accuracy testing, the palletizing robot system’s repeated positioning accuracy is tested. The repeated positioning accuracy error is currently 10−1 mm and is mainly caused by friction and the machining process. By studying the control system of a four-degrees-of-freedom (4-DOF) palletizing robot based on the trajectory planning algorithm, the design predictions and effects are realized, thus providing a reference for more efficient future palletizing robot design. Although the working process still has some shortcomings, the research has major practical significance.

Research on trajectory planning of dual-arm cooperative robot

To meet the requirements of the market for complex and diverse operation tasks, the reasonable trajectory planning of the manipulator plays an important role in the completion of the task, which affects the motion accuracy and operation efficiency of the manipulator, the trajectory planning of the manipulator needs to meet the smooth change of motion parameters and no sudden change to ensure the smooth operation of the system. In this paper, the method of middle point interpolation is used to calculate and analyze the linear interpolation trajectory planning and arc interpolation trajectory planning of 7-DOF(degrees of freedom) redundant manipulator, and to realize the accurate trajectory planning of the end position of the dual-arm cooperative robot.

Using a Cable-Driven Parallel Robot with Applications in 3D Concrete Printing

In construction, a large-scale 3D printing method for construction is used to build houses quickly, based on Computerized Aid Design. Currently, the construction industry is beginning to apply quite a lot of 3D printing technologies to create buildings that require a quick construction time and complex structures that classical methods cannot implement. In this paper, a Cable-Driven Parallel Robot (CDPR) is described for the 3D printing of concrete for building a house. The CDPR structures are designed to be suitable for 3D printing in a large workspace. A linear programming algorithm was used to quickly calculate the inverse kinematic problem with the force equilibrium condition for the moving platform; this method is suitable for the flexible configuration of a CDPR corresponding to the various spaces. Cable sagging was also analyzed by the Trust-Region-Dogleg algorithm to increase the accuracy of the inverse kinematic problem for controlling the robot to perform basic trajectory interpolation movements. The paper also covers the design and analysis of a concrete extruder for the 3D printing method. The analytical results are experimented with based on a prototype of the CDPR to evaluate the work ability and suitability of this design. The results show that this design is suitable for 3D printing in construction, with high precision and a stable trajectory printing. The robot configuration can be easily adjusted and calculated to suit the construction space, while maintaining rigidity as well as an adequate operating space. The actuators are compact, easy to disassemble and move, and capable of accommodating a wide variety of dimensions.

Investigation of the class of algebraical functions for interpolation of boundary trajectories of short-focus electron beams

The method for interpolating the dependence of the electron beam radius on the cutting plane position by the longitudinal coordinate, based on using the special class of interpolation functions, was investigated in this article. The roots of n-th order from the polynomials of the same order are used as the class of interpolation functions, which correspond to the ravine dependence of the radius of a short-focus electron beam on the distance of propagation. The results of testing have shown that, depending on choosing the set of basic points, including its position relatively to the minimum and their quantity, the relative interpolation error is between 0,7% and 8%. Obtained results of mathematical simulation of focal parameters of electron beams have the significant practical importance for the specialists, involved in the development and introduction into industrial production of modern highly effective electron-beam technological equipment.

Real-time quintic Hermite interpolation for robot trajectory execution.

This paper presents a real-time joint trajectory interpolation system for the purpose of frequency scaling the low cycle time of a robot controller, allowing a Python application to real-time control the robot at a moderate cycle time. Interpolation is based on quintic Hermite piece-wise splines. The splines are calculated in real-time, in a piecewise manner between the high-level, long cycle time trajectory points, while sampling of these splines at an appropriate, shorter cycle time for the real-time requirement of the lower-level system. The principle is usable in general, and the specific implementation presented is for control of the Panda robot from Franka Emika. Tracking delay analysis is presented based on a cosine trajectory. A simple test application has been implemented, demonstrating real-time feeding of a pre-calculated trajectory for cutting with a knife. Estimated forces on the robot wrist are recorded during cutting and presented in the paper.

Online local modeling and prediction of batch process trajectories using just-in-time learning and LSTM neural network

This paper deals with the online sample trajectory prediction problem of batch processes considering complex data characteristics and batch-to-batch variations. Although some methods have been proposed to implement the trajectory interpolation problem for quality prediction and monitoring applications, the accuracy and reliability are not ensured due to data nonlinearity, dynamics and other complicated feature. To improve the data interpolation performance, an improved JITL-LSTM approach is designed in this work. Firstly, an improved trajectory-based JITL strategy is developed to extract similar local trajectories. Then the LSTM neural network is used on the basis of the extracted trajectories with a modified network structure. Therefore, trajectory prediction and interpolation can be achieved according to the local JITL-LSTM model at each time index. A simulated fed-batch reactor process is presented to demonstrate the effectiveness of the proposed method.

Sliding mode control with third-order contour error estimation for free-form contour following

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

Optimal Motion Planning for Differential Drive Mobile Robots based on Multiple-Interval Chebyshev Pseudospectral Methods

SUMMARYThis paper presents a Chebyshev Pseudospectral (PS) method for solving the motion planning problem of nonholonomic mobile robots with kinematic and dynamic constraints. The state and control variables are expanded in the Chebyshev polynomial of orderN, and Chebyshev–Gauss–Lobatto (CGL) nodes are provided for approximating the system dynamics, boundary conditions, and performance index. For the lack of enough nodes nearby the obstacles, the interpolation of trajectory may violate the obstacles and the multiple-interval strategy is proposed to deal with the violation. Numerical examples demonstrate that multiple-interval strategy yields more accurate results than the single-interval Chebyshev PS method.

COMPORATIVE ANALIZE OF THE METODS OF INTERPOLATION AND APPROXIMATION OF THE BOUNDARY TRAJECTORY OF SHORT-FOCUS ELECTRON BEAM IN CONDITIONS OF ITS PROPAGATION IN THE IONIZED GAS

Possibilities and particularities of using the functions, which are formed as the roots of corresponded power from the polynomials of same power, for interpolation and approximation of the boundary trajectory of electron beam during its propagation in the ionized gas in the conditions of compensation of the intrinsic space charge of electrons by the positive ions of residual gas, are considered in the article. It’s have been shown, that the nature of considered functional dependences is strongly corresponded to adequate describing of the boundary trajectories of electron beams for the defined physical conditions. The mathematical functions, which are set as the roots of corresponded power from the polynomials of same power, always have the one global minimum, and outside the minimum area its nature is mostly seems to linear. Namely such functions in the case of correct choosing of polynomial coefficients most accurately, as well as adequate, described the boundary trajectories of short-focus electron beams in the case of its guiding in the ionized gas. It is reasoned, that for finding the optimal values of polynomial coefficients during the solving the approximation task is suitable to use the symbolic processor and the library functions of the system of scientific and technical calculations MatLab. The test examples of such approximation are also considered. It is shown, that generally the error of approximation is smaller, than 10%.

Robot Time-Optimal Trajectory Planning Based on Improved Cuckoo Search Algorithm

The trajectory planning time is uncertain when there is no velocity constraint in the joint space of an operating robot. This paper proposes a 3-5-3 polynomial interpolation trajectory planning algorithm based on improved cuckoo search algorithm (ICS) which functions under a velocity constraint. Kinematics analysis of the robot is conducted and the 3-5-3 polynomial interpolation function is established as the foundation for trajectory planning. The size of the step size control factor is set as a variant which varies with the number of iterations, which accelerates the algorithm’s convergence and prevents premature falling into the local optimal solution, both problems in the traditional cuckoo search algorithm where the step size control factor is fixed. The proposed algorithm was used to optimize the trajectory of a test robot with interpolation time as the search space for optimization. Any interpolation time that did not meet the speed constraint was eliminated to secure the shortest possible robot running time. The UR robot was taken as a research object to simulate the joint motion trajectory planning in MATLAB. The proposed algorithm performed well and better realized the time-optimal trajectory under the constraint of speed compared to the traditional cuckoo search algorithm, particle swarm optimization algorithm, and genetic algorithm.

VTDP: Privately Sanitizing Fine-grained Vehicle Trajectory Data with Boosted Utility

With the rapidly growing deployment of intelligent transportation systems (ITS) and smart traffic applications, vehicle trajectory data are ubiquitously generated, e.g., from GPS navigation systems, mobile applications, and urban traffic cameras. Analyzing such fine-grained data would greatly benefit the development of ITS and smart cities, yet pose severe privacy risks due to the recorded drivers’ visited locations, routes, and driving habits. Recently, some privacy enhancing techniques were proposed to sanitize such data. However, such schemes have some major limitations–they either lack formal privacy notions to quantify and bound the privacy risks, or result in very limited utility, e.g., only a sequence of locations or aggregated information can be released (without retaining the speeds, accelerations and the timestamps of vehicles). In this article, we propose a novel framework to sanitize the fine-grained vehicle trajectories with differential privacy (VTDP), which provides rigorous privacy protection against adversaries who possess arbitrary background knowledge. Our VTDP technique involves three phases of differentially private sampling, which sequentially generate all the three categories of data (besides a pseudo identity for each vehicle)– position, moving, and timestamps . It also includes a vehicle trajectory interpolation procedure to further improve the output utility with the properties of fine-grained vehicle trajectory data. We conducted experiments on real vehicle trajectory datasets to validate the performance of our approach.

Batting an in-flight object to the target

Striking a flying object such as a ball to some target location is a highly skillful maneuver that a human being has to learn through a great deal of practice. In robotic manipulation, precision batting remains one of the most challenging tasks in which computer vision, modeling, planning, control, and action must be tightly coordinated in a split second. This paper investigates the problem of a two-degree-of-freedom robotic arm intercepting an object in free flight and redirecting it to some target with a single strike, assuming all the movements take place in one vertical plane. Two-dimensional impact is solved under Coulomb friction and energy-based restitution with a proof of termination. Planning combines impact dynamics and projectile flight mechanics with manipulator kinematics and image-based motion estimation. As the object is on the incoming flight, the post-impact task constraint of reaching the target is propagated backward in time, while the arm’s kinematic constraints are propagated forward (via joint trajectory interpolation), all to the pre-impact instant when they will meet constraints that allow batting to happen. All the constraints (16 in total) are then exerted on the arm’s pre-impact joint angles and velocities, which are repeatedly planned based on updated estimates of the object’s motion captured by a high-speed camera. The arm keeps adjusting its motion in sync with planning until batting takes place. Experiments have demonstrated a better batting performance by a Barrett Technology WAM Arm than by a human being without training.

A helical interpolation precision truing and error compensation for arc-shaped diamond grinding wheel

In the grinding of aspherical optical surface, an arc-shaped diamond grinding wheel is generally employed, whereas its profile errors greatly influence the form accuracy of ground workpiece. A high efficiency helical interpolation precision truing method for the arc-shaped diamond grinding wheel is proposed in this paper. Helical interpolation trajectory and decreasing interpolation radius with the wear of a rotary truing wheel ensure a constant contact length and a uniform wear of the diamond grinding wheel and truer. The mathematical models of tool setting errors and measurement error of truer diameter in the truing process were established firstly. The prediction models considering the wear of truer are coincided well with the truing experiment results. Based on the mathematical error models, an error compensation truing process was proposed. The profile accuracy of an arc-shaped diamond grinding wheel was improved to 5 μm (PV) effectively by the error compensation truing process. An ellipsoidal surface of a fused silica workpiece was ground by the well-trued diamond grinding wheel. A form error (in PV) below 4.5 μm was obtained after precision grinding.

A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve

Optical polishing can accurately correct the surface error through controlling the dwell time of the polishing tool on the element surface. Thus, the precision of the trajectory and the dwell time (the runtime of the trajectory) are important factors affecting the polishing quality. This study introduces a systematic interpolation method for optical polishing using a non-uniform rational B-spline (NURBS). A numerical method for solving all the control points of NURBS was proposed with the help of a successive over relaxation (SOR) iterative theory, to overcome the problem of large computation. Then, an optimisation algorithm was applied to smooth the NURBS by taking the shear jerk as the evaluation index. Finally, a trajectory interpolation scheme was investigated for guaranteeing the precision of the trajectory runtime. The experiments on a prototype showed that, compared to the linear interpolation method, there was an order of magnitude improvement in interpolation, and runtime, errors. Correspondingly, the convergence rate of the surface error of elements improved from 37.59% to 44.44%.