Trapezoidal Velocity Profile Research Articles

Trajectory Planning, Kinematics, and Experimental Validation of a 3D-Printed Delta Robot Manipulator

This paper presents the implementation, trajectory planning, kinematics, and experimental validation of an open-source 3D-printed delta robot manipulator. The robot’s hardware consists of three servo motors, an Arduino Uno microcontroller, and a Pulse Width Modulation (PWM) servo module. The software includes a trapezoidal velocity profile planner and an inverse and forward kinematics solver to calculate the motor angles required to achieve a desired end-effector position in task space. The 3D printed parts were obtained from an open-source Thingiverse project and assembled to form the robot’s kinematic structure. The robot’s performance was evaluated in terms of its accuracy, repeatability, and maximum speed for a pick-and-place task. Experimental results show that the robot can achieve a positioning accuracy of 2.2 mm, with a top speed of 0.4 m/s, 30 picks per minute, and load carrying capacity up to 200 g. These results are satisfactory, considering the ease of assembly and the cost-effectiveness of the robot. The comparison of the calculated motor angles with actual motor angles obtained through additional potentiometer wires soldering shows promising results. The proposed robot’s cost-effective characteristics, utilization of open-source additive manufacturing, and motion planning algorithms make it suitable for various applications, including education, research, and small-scale industrial applications.

Low-consumption stepper motor controller with real-time target position change responsiveness based on field programmable gate array.

In response to the demand for low resource consumption, parallel control, and real-time response to target position changes in precision measurement and manufacturing of multi-axis stepper motor controllers, this paper proposes a field programmable gate array-based method for generating trapezoidal velocity profiles and pulse generation, which can easily keep parallelism and independence during multi-axis control. By avoiding using multiplication and division, this controller not only reduces resource consumption but also enhances the pulse output frequency. To address the real-time responsiveness of the velocity profile generation algorithm to changes in the target position during the control process, the algorithm introduces a novel real-time comparative state transition logic for speed control, which makes it capable of adjusting the acceleration within a single clock cycle, enabling its application in scenarios that require higher levels of real-time performance. Finally, the designed controller is applied to a four-axis positioning system for performance validation.

Trapez Hız Profilinin Ayarlanması Yoluyla Hareketli Bir Kütle Altındaki Kirişin Titreşim Kontrolü

In this article, the residual vibration of a simply supported beam with a moving mass is studied. The mass moves from a starting point to an end point on the beam with a trapezoidal velocity profile having accelerating, constant velocity and decelerating time intervals. The residual vibration of the mid-point of the beam after the mass stops is analyzed. The mathematical model of the system is developed using the finite element (FE) theory. Newmark method is used for the solution of FE model having time dependent matrices because of the moving mass. The model is verified by comparing the solution results with the results given in the previous studies in the literature. It is seen that the relationship between the natural frequency of the system and the velocity profile of the moving mass has an effect on the residual vibration of the structure. If the natural frequency of the system and the inverse of the deceleration time interval of the moving mass are equal while the moving mass is at the stopping position, residual vibrations occur at a minimum level. It seen that with the right speed profile selection, the decrease in vibration levels approaches 70% during the movement and 80% after stopping.

Effects of Velocity Profile and Plate Usage on Identified Bone Strength during Instrumented Screw Insertion

Bone screws are important in orthopaedic surgery to treat fractures and secure implants. Over- or under-tightening these screws may lead to premature failure of the screw fixation, necessitating risky and costly revision surgery. Previously, a model-based method for automatically optimising bone screw insertion torque was developed. This paper expands on the prior testing of this method, to investigate how non-ideal conditions may affect the model accuracy/precision.A bone screw was inserted into pre-drilled holes in artificial bone made from PU foam. Three cases were tested: first the screw was inserted directly with a constant velocity, then it was inserted with a trapezoidal velocity profile, and lastly it was inserted with a constant velocity profile, but through a hole in a metal plate. Each case was repeated 20 times for 60 total insertions. Torque and angular displacement measurements were used with a previously-developed model to identify the foam material strength for each insertion. Summary statistics were calculated for each case and statistical tests were used to compare the means and variances for the identified values between each case.Comparing to base base case with constant velocity and no plate: we found a statistically significant (p < 0.05) difference in the mean identified strength for the trapezoidal velocity and the case with the plate. We found a statistically significant difference in the variance for the trapezoidal velocity profile (p < 0.05), but not for case with the plate present (p = 0.16).Future work should investigate these effects over a wider range of materials and screws.

An Enhanced Navigation Algorithm with an Adaptive Controller for Wheeled Mobile Robot Based on Bidirectional RRT

A navigation algorithm providing motion planning for two-wheeled mobile robots is proposed in this paper. The motion planning integrates path planning, velocity planning and controller design. Bidirectional rapidly-exploring random trees algorithms (RRT) with path pruning and smoothing mechanism are firstly used to obtain a collision-free path to the robot destination with directional continuity. Secondly, velocity planning based on trapezoidal velocity profile is used in both linear and angular velocities, but the position error of the endpoint of the curve appears due to the coupling problem of the nonlinear system. To reduce the error, an approximation method is used to gradually modify several parts of time length of the trapezoidal velocity profile, so the continuity of the path can still be maintained. Thirdly, the controller keeping the robot on the planned path and velocity is designed based on the dynamic model of the robot. The parameters of this controller are estimated by the adaptive low, and the gain of controller is dynamic adjusted by fuzzy logic control to avoid the case that the control value is saturated. The controller stability and the convergence of tracking error is guaranteed by Lyapunov theory. Simulation results are presented to illustrate the effectiveness and efficiency.

Real - Time Trajectory and Velocity Planning for Autonomous Vehicles

Path planning algorithm integrated with a velocity profile generation-based navigation system is one of the most important aspects of an autonomous driving system. In this paper, a real-time path planning solution to obtain a feasible and collision-free trajectory is proposed for navigating an autonomous car on a virtual highway. This is achieved by designing the navigation algorithm to incorporate a path planner for finding the optimal path, and a velocity planning algorithm for ensuring a safe and comfortable motion along the obtained path. The navigation algorithm was validated on the Unity 3D Highway-Simulated Environment for practical driving while maintaining velocity and acceleration constraints. The autonomous vehicle drives at the maximum specified velocity until interrupted by vehicular traffic, whereas then, the path planner, based on the various constraints provided by the simulator using µWebSockets, decides to either decelerate the vehicle or shift to a more secure lane. Subsequently, a splinebased trajectory generation for this path results in continuous and smooth trajectories. The velocity planner employs an analytical method based on trapezoidal velocity profile to generate velocities for the vehicle traveling along the precomputed path. To provide smooth control, an s-like trapezoidal profile is considered that uses a cubic spline for generating velocities for the ramp-up and ramp-down portions of the curve. The acceleration and velocity constraints, which are derived from road limitations and physical systems, are explicitly considered. Depending upon these constraints and higher module requirements (e.g., maintaining velocity, and stopping), an appropriate segment of the velocity profile is deployed. The motion profiles for all the use-cases are generated and verified graphically.

FPGA-Based Hybrid Stepper Motor Drive System Design by Variable Structure Control

A stepper motor is usually operated in position open-loop control for simplicity. However, in order to improve the transient and steady-state performances of the stepper motor-based drive system, a high performance stepper motor drive system is usually designed to feed the encoder signal back to form a closed-loop system such as a general servo motor drive, and high-performance position and speed loop controller can then be used to overcome the nonlinear characteristics of the motor, such as the cogging force and impacts from external load. On the other hand, the existed cogging force of the hybrid stepper motor must be solved to increase the positioning precision. The variable structure control (VSC) is insensitive to the bounded uncertainty and load disturbance, and has been known as a high-performance controller. A predefined sliding surface is used to shape the system performances, and incorporate with switching control to achieve the robustness property. Thus, we applied the VSC to implement the stepper motor drive system about the position and speed control, and the switching control is used to overcome the cogging force. The field programmable gate array (FPGA) is a good alternative to be used to realize a motor drive system by considering its programmable ability and diverse designing environment. It is easily developed as an intellectual property (IP) for future use or combined as a part of a large control system. This paper showed the procedures to develop the hardware circuits for the variable structure controller, and applied it to stepper motor position and velocity control. Functions such as PI controllers, dq-axis transformation and two-phase space vector space vector modulation (SVPWM) designed for the stepper motor drives are also shown in the paper. The system simulation and hardware circuit realization are based on MATLAB/Simulink, and realized on Altera FPGA. Simulations on MATLAB/Simulink with trapezoidal velocity profile command, and experiments with and without the load added are shown to demonstrate the hardware performances and correctness.

Transversal vibrations of a toothed belt in linear drives during operation

AbstractVibrations of the toothed belt of a linear drive can affect the dynamic positioning accuracy and smoothness which is important for practical applications. An analytical, one‐dimensional model of a belt section is derived, taking into account the preload and flexural rigidity, as well as the non‐linear cable effect and the acceleration forces due to the belt mass. In addition to the case of constant belt length, the temporal change of the length as a result of the axial motion is an essential part of the modeling. The solution of the equation of motion is limited to the dominant transverse deflection of the belt. For the longitudinal extension, the assumption of a quasi‐static solution is taken. A FEM program is implemented to calculate the solution for given operation conditions. The evaluations are made for selected motions according to a path planning with trapezoidal velocity profiles. A modification of individual parameters, based on a practical parameter set, enables the analysis of the respective effects.

Energy‐optimal transport trajectory planning and online trajectory modification for holonomic robots

AbstractTo maximize energy efficiency, we suggest minimum‐energy trajectory planning and online trajectory modification algorithms for three‐wheeled omni‐directional mobile robots (TOMRs). First, minimum‐energy multisection trajectory planning for various constraints is performed using Pontryagin's minimum principle. Two strategies are established to formulate an efficient minimum‐energy, multisection trajectory generation algorithm: an online trajectory modification and a control algorithm that maintains energy efficiency are investigated using novel heuristic functions. Compared with the conventional trajectory using the trapezoidal velocity profile, the generated transport trajectory can save energy.

A fast processing method to perform transient analysis for vibration control

Transient vibration analyses of structures with complex shapes are performed by finite element (FE) methods. Transient vibration analyses take long processing times for the systems having large number of degrees of freedom. A method performing the transient analyses very fast using the Fast Fourier Transform (FFT) is introduced in this work, and it is applied to a curved manipulator. Various analyses are required to determine the input parameters for vibration control, and each analysis takes long FE solution times.In this work, an FFT method is developed where the samples of the impulse response of the system under study is obtained by the FE software ANSYS first. The samples of the transfer function are then obtained by FFT, and transient responses are found for various vibration control parameters by using FFT. Transient response results for the complex system considered in this study are obtained approximately in 1 s with FFT method, while it takes 36 h with ANSYS.A one degree of freedom system is considered to verify the results of FFT method. Newmark's numerical, ANSYS, and FFT method results are compared, and it is observed that the results are in good agreement. A curved non-uniform steel manipulator with a complex shape is considered after the verification. The samples of the impulse response of the manipulator is found by ANSYS. Then, FFT method is used to obtain transient responses. A trapezoidal velocity profile is considered to analyze the residual vibration. The effect of the deceleration time on the root-mean-square (rms) values of the residual vibration is studied experimentally and using FFT method. It is observed that the results are in good agreement, and it is concluded that FFT method introduced in this study is very effective to study transient vibration problems in complex systems.

Dynamic Behaviour of Spindle on Large EDM Machine Induced by High-Speed Jump Motion with Different Control Strategies

High-speed jump motion in electrical discharge machining (EDM) has been proven to be an important reason for flexible impact and vibration on EDM machine tools. Especially for large EDM machines, the moving parts possess large mass, and different jump control strategies cause different vibration characteristics. However, little attention has been paid to the kinematic comparison between different strategies, and the effect of jump motion on EDM spindle dynamics behaviour is also neglected. In this paper, the dynamic behaviour of spindle on a large EDM machine induced by high-speed jump motion has been investigated. Control strategies including trapezoidal velocity profile, constant jerk profile and sinusoidal jerk profile are considered. The kinematic characteristics of these jump control strategies and corresponding dynamic behaviour of EDM spindle were studied. As a result, under the same maximum velocity of 80 mm/s and maximum jerk of 128 m/s3, the jump period and height of sinusoidal jerk profile are about 0.05 s longer and 1.2 mm larger than that of the other two strategies, respectively. Trapezoidal control strategy causes the largest vibration amplitude of 48.7 μm on spindle during jump motion and thus the strongest forced vibration. The experimental measurement platform for vibration displacement was established. Comparing simulation and experiment data, the relative error is lower than 14%, which verifies the simulation model. Moreover, it can be found that large jump velocity increases vibration amplitude of spindle significantly during machining. When the velocity is 80 mm/s, the maximum vibration amplitude is 23.0 μm. And large velocity of jump motion makes spindle amplitude attenuate faster.

홀로노믹 물류로봇의 다 구간 최소 에너지 수송경로 생성

The minimum-energy multi-section trajectory was generated for holonomic robots. First, minimum-energy trajectory planning on a section for various constraints was performed using the optimal control theory, dynamic models, and a cost function, which represent the energy drawn from the battery. Then, two novel strategies were established to formulate a minimum-energy, multi-section trajectory generation algorithm. Through these strategies on loop arrangement considering the computational load of each trajectory generation algorithm on a section, unknown variables for a multi-section could be found within a practical execution time. The generated trajectory for logistics robots saved 7.18% energy compared with the conventional trajectory using the trapezoidal velocity profile. Further, the actual performance was shown using the resolved-acceleration controller, which was implemented in the preliminary research.

Field Programmable Gate Array-Based Linear Shaft Motor Drive System Design in Terms of the Trapezoidal Velocity Profile Consideration

Owing to the benefits of programmable and parallel processing of the field programmable gate arrays (FPGAs), they have been widely used to the realization of digital controllers and motor drive systems. In this study, we adopt the FPGA chip to realize the Linear Shaft Motor (LSM) drive system which includes the position, speed and current vector controls. Linear shaft motor is a special motor which has the magnet as the shaft and stator coils are on the forcer. Thus, it reveals a small mechanical time constant. In addition, the electrical side can be looked as a general three-phase Alternating Current (AC) system which can be powered by inverter and vector control technique can be applied to the system. The designed system needs high performance calculation ability about position/speed control and vector current control loops. The mathematical model of linear shaft motor drive system is first built and simulated by MATLAB/Simulink and the accuracy about the effect from the speed estimation method is proposed. The resulting digital model of the drive system is stored into Verilog Hardware Description Language (Verilog HDL) codes and realized by FPGA. At last, the hardware circuits as well as the power module are used to test the performance of the developed hardware system in terms of the trapezoidal velocity profile. The experimental results show that the designed system realized by FPGA has attained the desired performance.

Optimal Constant Acceleration Motion Primitives

This paper proposes new motion primitives that are time optimal and feasible for a vehicle (wheeled-mobile robot). They are parameterized by a constant acceleration and a constant deceleration in an obstacle-free environment between an initial and a final configuration with given poses and velocities. The derived compact parametric solution has an implicit optimal velocity profile, is continuous in C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , considers driving constraints, and is computationally efficient. The path is derived analytically considering constraints on maximal allowable driving velocity and accelerations. The obtained motion primitives have continuous transition of curvature and are, thus, easily drivable by the vehicle. The proposed motion primitives are evaluated on several path-planning examples, and the obtained solutions are compared to a numerically expensive planner using Bernstein-Bézier curve with path shape and velocity profile optimization. Numerous experiments have been conducted not only in the simulation environment, but also during testing and comparisons on the actual mobile robot platforms. It is shown that the proposed solution is computationally efficient, time optimal under given constraints and trapezoidal velocity profile assumptions, and is close to globally optimal solution with an arbitrary velocity profile.

Hybrid vibration control of a flexible composite box cross-sectional manipulator with piezoelectric actuators

Robotic arms can be produced from composite materials with different cross-sections due to their lightness and high strength properties. The flexibility of robotic arms affects the vibration behavior of robots in engineering applications. The suppression of vibrations increases the dynamic performance of flexible robots. In this study, hybrid vibration control of a smart composite box manipulator (SCBM) is investigated numerically and experimentally. Hybrid vibration control is achieved using passive control (PC) and active control (AC) techniques. The single-axis epoxy glass SCBM has a layer orientation of [0/90/0/90]. The PC is achieved with a trapezoidal velocity profile. Experiments are conducted to verify the simulation results. For AC, the PID control with displacement and strain feedbacks is applied to the SCBM, which is driven with the cases in the PC for further reducing residual vibrations. Simulations are performed by embedding the hybrid control method into the finite element solutions in ANSYS. Controller gains are initially found by the tuning methods and then optimized according to the maximum actuation voltage. The residual vibrations of the composite manipulator have been successfully suppressed for different payloads. The effectiveness of the hybrid control technique on residual vibration results is investigated.

A New Seven-Segment Profile Algorithm for an Open Source Architecture in a Hybrid Electronic Platform

The velocity profiles are used in the design of trajectories in motion control systems. It is necessary to design smoother movements to avoid high stress in the motor. In this paper, the rate of change in acceleration value is used to develop an S-curve velocity profile which presents an acceleration and deceleration stage smoother than the trapezoidal velocity profile reducing the error at the end of the duty-cycle pre-established in one degree of freedom (DoF) application. Furthermore, a new methodology is developed to generate a seven-segment profile that works with negative velocity and displacement constraints applying an open source architecture in a hybrid electronic platform compounded by a system on a chip (SoC) Raspberry Pi 3 and a field programmable gate array (FPGA). The performance of the motion controller is measured through the comparison of the error obtained in real-time application with a trapezoidal velocity profile. As a result, a low-cost platform and an open architecture system are achieved.

Motion planning and numerical simulation of material deposition at corners in extrusion additive manufacturing

The material deposition along a toolpath with a sharp corner is simulated with a computational fluid dynamics model. We investigate the effects of smoothing the toolpath and material over-extrusion on the corner rounding and the corner swelling, for 90° and 30° turns. The toolpath motion is controlled with trapezoidal velocity profiles constrained by a maximal acceleration. The toolpath smoothing of the corner is parametrized by a blending acceleration factor. Analytical solutions for the deviation of the smoothed toolpath from the trajectory of the sharp corner, as well as the additional printing time required by the deceleration and acceleration phases in the vicinity of the turn are provided. Moreover, several scenarios with different blending acceleration factors are simulated, for the cases of a constant extrusion rate and an extrusion rate proportional to the printing head speed. The constant extrusion rate causes material over-extrusion during the deceleration and acceleration phases of the printing head. However, the toolpath smoothing reduces the corner swelling. The amount of underfill and overfill at the inside and outside of the corner are quantified, as compared to an ideal case where the printing head would instantaneously change its speed direction at the corner. The numerical results show that there is an optimal amount of toolpath smoothing where the over-extrusion compensates for the corner rounding; hence improving the quality of the corner. A uniform road width is obtained with the proportional extrusion rate.

A Trapezoidal Velocity Profile Generator for Position Control Using a Feedback Strategy

Position control is usually achieved using a position controller and a profile generator. The profile generator produces a desired position trajectory from a position reference and predefined profiles. The position controller forces the actual position to trace the generated position trajectory. A time-based profile generator is the most famous profile generator due to its capability of generating various profiles. However, time base difference in analysis and implementation causes a steady-state error. In order to remove the steady-state error, this paper proposes a novel profile generator for a trapezoidal velocity profile generation. The proposed generator is based on a cascaded P-PI position controller which is designed to trace the position reference. A dynamic range limiter is adopted to provide the acceleration and velocity restrictions which are basic functions for generating the trapezoidal profile. In spite of these restrictions, it cannot make a desired velocity profile only using the limiter because deceleration point is inaccurate. To adjust the deceleration point, a feedback compensator is designed which requires the velocity of the deceleration point. The velocity of the deceleration point is estimated from the initial position error. The compensator moves the deceleration point to the appropriate point which can generate the desired velocity profile. The proposed profile generator can remove the steady-state error, and the position response can be easily adjusted to be either overdamped or underdamped by selecting the two gains appropriately. Several experimental results are presented to verify the usefulness of the proposed generator.

Online near time-optimal trajectory planning for industrial robots

Abstract A novel trajectory planning approach is presented suitable for online industrial robot applications along short path segments such as spot-welding. The proposed method generates trajectories that are computationally efficient, dynamically near time-optimal, and maintain path-following integrity in high-frequency planning-and-control cycles. The method is based on the well-known path constrained time-optimal motion. We show that this trajectory can be quickly approximated with trapezoidal velocity profiles, resulting in near time-optimal trajectories, requiring only four robot dynamics computations per path segment. For continuous motions, a method to safely transit between adjacent optimal path segments within geometric bounds is also presented. We then show how the generated second order trajectory can be successfully used with a generic control loop by adopting feedforward control based on an elastic model. A real-world experiment with a 6DOF industrial robot validates our approach.

Trapezoidal Motion Profile to Suppress Residual Vibration of Flexible Object Moved by Robot

The residual vibration when a robot manipulator is operated at high speed needs to be suppressed. These vibrations are generated by the resonance of a flexible object being moved by the robot, and research on control algorithms and motion profiles is ongoing to reduce them. In this paper, we propose a method to reduce the residual vibration of an object moved by a robot manipulator by optimizing the acceleration/deceleration time calculated using the object’s natural frequency. The relationship between acceleration/deceleration time and the residual vibration in a trapezoidal velocity profile is considered by analyzing the scenario when the jerking motion characteristic of such vibrations occurs. The results of experiments using a commercial robot show that residual vibrations can be reduced by the proposed method without the need for an additional feedback control algorithm while transferring a flexible object over small and large distances.