3-RRR Planar Research Articles

Kinematic Modelling of a 3RRR Planar Parallel Robot Using Genetic Algorithms and Neural Networks

Kinematic modelling of parallel manipulators poses significant challenges due to the absence of analytical solutions for the Forward Kinematics (FK) problem. This study centres on a specific parallel planar robot, specifically a 3RRR configuration, and addresses the FK problem through two distinct methodologies: Genetic Algorithms (GA) and Neural Networks (NN). Utilising the Inverse Kinematic (IK) model, which is readily obtainable, both GA and NN techniques are implemented without the need for closed-loop formulations or non-systematic mathematical tools, allowing for easy extension to other robot types. A comparative analysis against an existing numerical method demonstrates that the proposed methodologies yield comparable or superior performance in terms of accuracy and time, all while reducing development costs. Despite GA’s time consumption limitations, it excels in path planning, whereas NN delivers precise results unaffected by stochastic elements. These results underscore the feasibility of using neural networks and genetic algorithms as viable alternatives for real-time kinematic modelling of robots when closed-form solutions are unavailable.

Dynamic modelling and singularity-free simulation of closed loop multibody system driven by electric motors

This paper presents the dynamic model and singularity-free simulation of electromechanical systems including closed loop multibody systems, massless gear transmission and electric motors. The dynamic model of these systems is established in matrix form and written in a Differential-Algebraic Equations form by applying the Lagrangian equation with multipliers and substructure method. Moreover, this paper deals with two difficult issues in the simulation of closed-loop multibody systems which are to overcome smoothly the singular configurations and to stabilize the constrained equations due to accumulated errors. The singularity-free simulation is solved by using null-space of Jacobian matrix to eliminate the constraint forces – Lagrangian multipliers in equations of motion. The drift in the constraint equation during simulation is restricted by a combination of Baumgarte’s stabilization and post-adjusting technique. Some numerical experiments are carried out to the planar 3RRR parallel manipulator driven by electric motors. Simulation results confirm the effectiveness of the proposed approach in overcoming the singular configurations and in stabilization of the constraint.

TOPOLOGICAL STRUCTURE OPTIMIZATION AND KINEMATIC PERFORMANCE IMPROVEMENT OF 3-RRR PLANAR PARALLEL MANIPULATOR

The typical 3-RRR planar parallel manipulator with two translations and one rotation has extensive applications such as plane location and motion transfer. But it suffers two disadvantages. One is its analytical direct kinematics is difficult to be got and another is not input-output motion decoupled. This paper focuses on its topological structure optimization and resulting kinematic performance improvementt. First, the coupling degree of this manipulator is calculated being k=1. Second, based on structure coupling-reducing principle, its coupling-reduced manipulator with zero coupling degree is designed, which not only leads to be easy to get its analytic direct kinematic solutions, but also makes input-output motion partially decoupled. Moreover, based on workspace and singularity of this coupling-reduced manipulator, comprehensive comparison of two manipulators before and after coupling-reducing showed that the main performances of structure coupling-reduced manipulator are superior than that of the typical mechanism. The work shows that structure coupling-reducing is effective method for optimization of topology structure.

Methods of workspace generation for parallel kinematic manipulators

This paper presents the geometric, discretization and Monte Carlo method to generate the workspace for Parallel Kinematic Manipulators (PKMs). The geometric method does not require the solution to the inverse kinematics as opposed to the Monte Carlo and discretization methods. Each method possesses advantages and drawbacks concerning accuracy, time and ease of implementation. A case study concerning constant orientation workspace of a planar 3RRR PKM showed that the most efficient and accurate method was the geometric method but it also possessed the most limitations. The discretization and Monte Carlo methods produced the workspace with an accuracy of 92.49% and 95.67% respectively.

Two Practical Methods for the Forward Kinematics of 3-3 Type Spatial and 3-RRR Planar Parallel Manipulators

The forward kinematics in parallel manipulators is a mathematically challenging issue, unlike serial manipulators. Kinematic constraint equations are non-linear transcendental equations that can be reduced to algebraic equations with appropriate transformations. For this reason, sophisticated and time-consuming methods such as the Bezout method, the Groebner bases method, and the like, are used. In this paper, we demonstrate that these equations can be solved by non-complicated mathematical methods for some special types of manipulators such as the 3-3 and 6-3 types of Stewart platforms, and the 3-RRR planar parallel manipulator. Our first method is an analytical approach that exploits the special structure of kinematic constraint equations and yields polynomials of 32nd and 16th order, as mentioned in the previous works. In the second method, an error function is defined. This error function is employed to find the most appropriate initial values for the non-linear equation solver which is used for solving kinematic constraint equations. Determining the initial values in this manner saves computation time and guarantees fast convergence to real solutions.

IBOA-Based Optimization of Cross-Sectional Dimension of Rods for a 3-RRR PPM to Minimize Energy Consumption

For obtaining optimal cross-sectional dimensions of rods for a 3-RRR planar parallel manipulator (PPM) to minimize energy consumption, the inverse dynamics of the manipulator is modeled based on the Newton–Euler method, after which the coefficient matrix of the inverse dynamics equation is decomposed based on matrix theory. Hence, the objective function, that is, the logical relationship between the energy consumption of the manipulator and the cross-sectional dimension of each rod, is established. However, in solving the multidimensional constrained single-object optimization problem, there are difficulties such as the penalty function’s sensitivity to the penalty factors if the problem is transformed into the one of unconstrained multiobjective optimization. Therefore, to properly handle the constraints, an improved butterfly optimization algorithm (IBOA) is presented to ensure that the new iterated point always falls into the feasible region according to the butterfly optimization algorithm (BOA). Finally, the comparisons among the IBOA, particle swarm optimization (PSO), and BOA and further experiments of the physical prototype are implemented to validate the effectiveness of the proposed theoretical model and numerical algorithm. Results indicate that the proposed IBOA is more suitable for solving the constrained single-object optimization problem with better convergence speed and accuracy.

Evaluating the Maximum Directional Kinematic Capability of a Redundant Manipulator Based on Allowable Velocity and Force

In this study, a new concept of allowable velocity and force is proposed to precisely evaluate the maximum directional kinematic capability of a redundant manipulator. For a general redundant manipulator, an optimization problem is formulated to determine the maximum achievable velocity and force projected along the base direction at any target position in the workspace. This provides quantitative information on allowable (i.e., maximum directional) velocity and force to be precisely visualized in 2D and 3D complicated shapes, which conventional manipulability ellipsoid cannot provide. As application examples, allowable velocity and force are evaluated for a distributed actuation mechanism (DAM)-based three-link planar manipulator, 3RRR planar parallel manipulator, and the UR5 robot (a spatial manipulator with 6 degrees of freedom). The simulation and experimental results validate that the proposed method can precisely determine allowable velocity and force, thereby contributing to planning the optimal operation for a given task.

Optimum transmission performance of 3-RRR planar parallel manipulators and sensitivity analysis

Optimum transmission performance of 3-RRR planar parallel manipulators and sensitivity analysis

Radial basis function neural network vibration control of a flexible planar parallel manipulator based on acceleration feedback

The self-excited vibration of flexible planar 3-RRR parallel manipulators is converted from the residual vibration after high-speed motion and is a resonance of the strongly coupled and nonlinear electromechanical system. This makes the active vibration control quite a challenging task. In this study, we attempt to adopt the radial basis function neural network control algorithm based on acceleration feedback for suppressing the self-excited vibration and guarantee its position accuracy. The stability of the controlled system is proved by the Lyapunov concept. Self-excited vibration control experiments are conducted near the singular region. Experimental results demonstrate the effectiveness of our adopted controller.

Control of a 3-RRR Planar Parallel Robot Using Fractional Order PID Controller

3-RRR planar parallel robots are utilized for solving precise material-handling problems in industrial automation applications. Thus, robust and stable control is required to deliver high accuracy in comparison to the state of the art. The operation of the mechanism is achieved based on three revolute (3-RRR) joints which are geometrically designed using an open-loop spatial robotic platform. The inverse kinematic model of the system is derived and analyzed by using the geometric structure with three revolute joints. The main variables in our design are the platform base positions, the geometry of the joint angles, and links of the 3-RRR planar parallel robot. These variables are calculated based on Cayley-Menger determinants and bilateration to determine the final position of the platform when moving and placing objects. Additionally, a proposed fractional order proportional integral derivative (FOPID) is optimized using the bat optimization algorithm to control the path tracking of the center of the 3-RRR planar parallel robot. The design is compared with the state of the art and simulated using the Matlab environment to validate the effectiveness of the proposed controller. Furthermore, real-time implementation has been tested to prove that the design performance is practical.

Time-dependent system kinematic reliability analysis for planar parallel manipulators

Time-dependent system kinematic reliability of the planar parallel manipulator refers to the probability of the pose error falling into the allowable safe region over the whole specified trajectory, which is essential for its work performance. However, works regarding this issue are quite limited. Consequently, this study conducts time-dependent kinematic reliability analysis for planar parallel manipulators considering joint clearance, input uncertainty, and manufacturing imperfection based on the first-passage method. First, the limit-state pose error function is established by means of the Baker–Campbell–Hausdorff formula and the theory of the Lie group and Lie algebra. Then, the analytical solution to the outcrossing rate of the non-stationary vector stochastic kinematic process is derived by employing the expectation propagation and closed-form properties of conditional and marginal statistical moments (mean and covariance) of the multivariate Gaussian. On this basis, the time-dependent system kinematic reliability is calculated upon the assumption that the outcrossing events are independent. Finally, the planar 3-RRR parallel manipulator is used to demonstrate the proposed method, and its effectiveness is validated by comparison with the Monte Carlo simulation method.

Energy Consumption Prediction for 3-RRR PPM through Combining LSTM Neural Network with Whale Optimization Algorithm

In the process of minimizing the energy consumption of a 3-RRR planar parallel manipulator (3-RRR PPM) and even general parallel kinematic manipulators, obtaining optimal results usually depends on particular functional relation between the instantaneous position of the moving platform and the kinetic time, which is called a displacement model (DM). Nevertheless, it is likely that although the movement time and path of a moving platform are the same, different amounts of energy are consumed for different DMs of the moving platform. To address this, a method of using long short-term memory neural network (LSTM-NN) instead of a complex theoretical model to predict the energy consumption of a 3-RRR PPM was presented. Subsequently, inverse dynamic equations of 3-RRR PPM were established based on the Newton–Euler method and solved using QR decomposition. Meanwhile, energy consumption between any two points in workspace of the 3-RRR PPM was programmed to provide the LSTM-NN with abundant precise training data. In view of time-varying characteristics of energy consumption prediction, the network architecture was developed based on the principle of LSTM-NN, and root-mean-square error (RMSE) was taken as the loss function. After acquiring training data, the RMSE of the LSTM-NN reached 0.00041 using whale optimization algorithm (WOA) with no need for the gradient of the loss function, so the lack of solving precision in training LSTM-NN was effectively improved. Finally, two different DMs of a moving platform with the same path and movement time were chosen to compare the total energy consumption of the 3-RRR PPM from the simulations, predictions, and experiments. The results showed that the relative error between predicted and experimental data was less than 2.50%. Therefore, the energy consumption prediction based on the LSTM-NN will be useful for achieving the intelligent application of 3-RRR PPMs.

Optimum configuration design and sensitivity analysis of the 3-RRR PPMs with a general kinematic model

In this work, a parameterized model of 3-RRR planar parallel manipulator is proposed, aiming to optimize the overall performance with optimum shapes and sizes of the base and mobile platforms. In the model, the inverse and direct kinematics are addressed, upon which the global kinematic performance in terms of dexterity index is calculated and analyzed. A shape and dimensional optimization is accomplished through the atlas-based method. Optimum configurations are obtained from the global performance atlases. Moreover, sensitivity analysis of overall kinematic performance with respect to each configuration parameter of planar 3-RRR parallel manipulator is performed to reveal the influences of the design parameters. A case study is included to present a more detailed design of 3-RRR manipulator.

An Alternative Method for Shaking Force Balancing of the 3RRR PPM through Acceleration Control of the Center of Mass

The problem of shaking force balancing of robotic manipulators, which allows the elimination or substantial reduction of the variable force transmitted to the fixed frame, has been traditionally solved by optimal mass redistribution of the moving links. The resulting configurations have been achieved by adding counterweights, by adding auxiliary structures or, by modifying the form of the links from the early design phase. This leads to an increase in the mass of the elements of the mechanism, which in turn leads to an increment of the torque transmitted to the base (the shaking moment) and of the driving torque. Thus, a balancing method that avoids the increment in mass is very desirable. In this article, the reduction of the shaking force of robotic manipulators is proposed by the optimal trajectory planning of the common center of mass of the system, which is carried out by “bang-bang” profile. This allows a considerable reduction in shaking forces without requiring counterweights, additional structures, or changes in form. The method, already presented in the literature, is resumed in this case using a direct and easy to automate modeling technique based on fully Cartesian coordinates. This permits to express the common center of mass, the shaking force, and the shaking moment of the manipulator as simple analytic expressions. The suggested modeling procedure and balancing technique are illustrated through the balancing of the 3RRR planar parallel manipulator (PPM). Results from computer simulations are reported.

Dynamic Filtered Path Tracking Control for a 3RRR Robot Using Optimal Recursive Path Planning and Vision-Based Pose Estimation

This study proposes a dynamic filtered path tracking control schema for a 3RRR planar parallel robot to reduce the positioning errors caused by various sources such as backlash in the mechanical system, system nonlinearities, uncertainties, and unknown disturbances. The optimal recursive algorithm is implemented absolutely in path planning based on the optimization problem. Compared to other methods, this algorithm not only helps the robot avoid singularities but also ensures the shortest movement distance of links thanks to the optimization in path planning. The vision system using the 3D Intel RealSense camera D435i is proposed to provide the visual measurement as feedback for the pose estimation of the 3RRR planar parallel robot. The unscented Kalman filter is designed to reduce the errors of the pose estimation due to the camera’s intrinsic parameters, the vibration of the mechanical system, and unknown noises. The establishment of important equations and parameters of the unscented Kalman filter is detailed in this study. The simulation, experiment, and comparison are conducted to verify the effectiveness and feasibility of the proposed approaches.

Increasing the effective workspace of the planar 3-RRR mechanism by using kinematic redundancy

A significant issue of parallel mechanisms is the existence of singularities where the end-effector can lose their mobility or cannot be controlled. The closeness to singularities affects the performance of the mechanism, for example, the actuation efforts increase, thus reducing the effective workspace of the mechanism. Kinematic redundancy can be used for avoiding the singularity and increase the effective workspace. The approach is considered on the example of a planar 3-RRR mechanism.

Comparative Kinematic Analysis and Design Optimization of Redundant and Nonredundant Planar Parallel Manipulators Intended for Haptic Use

SUMMARYThis paper investigates a comparative kinematic analysis between nonredundant and redundant 2-Degree Of Freedom parallel manipulators. The nonredundant manipulator is based on the Five-Bar mechanism, and the redundant one is a 3-RRR planar parallel manipulator. This study is aimed to select the best structure for a haptic application. This latter requires a mechanism with a desired workspace of 10 cm × 10 cm and an admissible force of 5 N in all directions. The analysis criteria are the accuracy of the forward kinematic model and the required actuator torques. Thereby, the geometric parameters of the two structures are optimized in order to satisfy the required workspace such that parallel singularities are overcome. The analysis showed that the nonredundant optimally designed manipulator is more suitable for the haptic application.

Small-angle perturbation method for moving platform orientation to avoid singularity of asymmetrical 3-RRR planner parallel manipulator

To overcome the weakness of the present methods of singularity avoidance for 3-RRR planar parallel manipulator (PPM), a new method of avoiding singular points on planed paths in workspace via orientation perturbation to moving platforms is proposed. First, an inverse kinematics model of 3-RRR PPM is built. A discriminant matrix of type II singularity is specified and the mapping between determinant values of the matrix and points in workspace of 3-RRR PPM is defined as a singularity surface. Then, the planned path is regarded as a directrix and intersecting line between cylindrical surfaces (its generating line is parallel to the z-axis) and the singularity surface is defined as a singularity curve corresponding to a planned path. By determining the critical perturbation value of the moving platform’s orientation angle, the maximum value of the singularity curve corresponding to the planned path is no larger than 0 or its minimum value is no smaller than 0. Therefore, singular points on the planned path are avoided by changing the moving platform’s orientation angle without changing the planned path. Moreover, a numerical realization method is given; example and experiment verification indicates that singular points in the planned path can be avoided when perturbation to the moving platform’s orientation angle is no larger than 0.1745 rad. Therefore, the proposed method lays a solid foundation for singularity-free kinematics control based on inverse solutions.

Dynamic modeling of parallel manipulators based on Lagrange–D’Alembert formulation and Jacobian/Hessian matrices

This paper describes the development of a dynamic model for parallel manipulators based on the Lagrange–D’Alembert equation, the Hessian matrix of the kinetic energy of the manipulator, and the Jacobian/Hessian matrices of the constraint equations. The model generates the forward and inverse dynamics formulations of parallel manipulators. First, the kinematic relations between the input actuated variables and the output variables are defined in terms of the Jacobian and Hessian matrices of the constraint equations. Then, a new form of Lagrange–D’Alembert equation is introduced and used to derive the dynamic model of parallel manipulators. In the present model, the inertia, centrifugal, and Coriolis generalized forces/torques are obtained directly from Hessian matrix of the kinetic energy of the manipulator. The developed model is simple, straightforward, and appropriate for parallel processing techniques as the elements of the Jacobian and Hessian matrices can be calculated simultaneously. The developed model is validated using the conservation of work and energy principle. A trajectory tracking control of a simple 3RRR planar parallel manipulator is used to illustrate the developed model.

A Visual Servoing Strategy Under Limited Frame Rates for Planar Parallel Kinematic Machines

In this manuscript, an affordable alternative to control parallel kinematic machines is proposed based in a high-authority/low-authority control strategy. The high-authority loop is implemented using a model-based Cartesian control strategy. This feedback loop exploits a Pose-Based Visual Servo (PBVS) using an eye-to-hand limited frame rate camera. The low-authority loop is implemented using an independent space joint space control strategy. This proposal is experimentally compared with the Cartesian space computed torque control and with the independent joint space control strategies. The experimental campaign is carried out for tracking-trajectory problems using a planar 3R RR parallel manipulator. The experimental results demonstrate that the proposal can achieve results that are more accurate without increasing the manipulator’s energy consumption when compared to the use of the independent joint space control strategy. For a predefined square trajectory, a reduction of 4.9% and 24.5% was achieved in the root mean square error of the linear and angular positions, respectively.