Inverse Jacobian Matrix Research Articles

Design and Control of a Novel 6-DOF Desktop Upper Limb Rehabilitation Robot

Abstract This paper presents the design, analysis, and development of a Novel six degrees of freedom (6-DOF) desktop upper limb rehabilitation robot. The upper limb rehabilitation robot is mainly composed of the Omnidirectional mobile platform, armrest, and 3-DOF wrist rehabilitation mechanism. The forward and inverse kinematics and Jacobian matrix of the upper limb rehabilitation robot are derived based on the kinematics of a rigid body, and its working space is analyzed based on arm kinematics as well. The forward and inverse kinematics of the arm are derived based on the D-H method. A new control strategy and control algorithm were developed based on the hardware structure of the robot system and arm model, and the control strategy and control algorithm are used to carry out the simulated rehabilitation experiment on a single joint of the arm and the simulated rehabilitation experiment on multiple joint linkages in the arm. The experimental results show that the maximum error in the simulated rehabilitation experiment on a single joint of the arm is 6.123 deg, and the maximum error in the simulated rehabilitation experiment of multiple joint linkages in the arm is 5.323 deg. Therefore, the control strategy and control algorithm can complete the corresponding rehabilitation training. This 6-DOF desktop upper limb rehabilitation robot can provide passive rehabilitation training for patients in the early stages of paralysis.

Design and analysis of a novel 3-DOF planar parallel manipulator with two kinematic chains

In this work, to enlarge the workspace, a novel 3-DOF planar parallel manipulator (PPM) with two identical RRR kinematic chains is proposed, and this manipulator has a significant advantage in orienting the moving platform (MP) across a wide range of angles. Resorting to the closed-loop vector approach, the inverse position equation and Jacobian matrix of the novel manipulator are established. Payload performance, dexterity performance, and singularity of the proposed manipulator are analyzed based on the Jacobian matrix. To investigate the distribution characteristics of the constant orientation workspace, orientation workspace, and reachable workspace of the proposed manipulator, a numerical discretization method is employed. Under the same working mode and structural parameters, constant orientation workspace and reachable workspace comparisons between the proposed manipulator and another two traditional PPMs are conducted. Focused on three kinds of performance indices, link lengths optimization of the proposed manipulator is carried out. A 4-DOF pick-and-place equipment is fabricated, and the rotational and translational capabilities are verified based on two examples.

A Kinematics-Based Optimization Design for the Leg Mechanism of a Novel Earth Rover

Abstract This paper proposes a general kinematic-based design method for optimizing the side-mounted leg mechanism of BJTUBOT, a novel multi-mission quadrupedal Earth rover. The focus issue lies in designing structural improvements that not only enhance its kinematic performance but also prevent singularity, all while meeting the demands for miniaturization and lightweight without deviating from the original leg design concept. To solve this issue, a novel 3-UPRU&PPRR mechanism is envisaged based on the original configuration. Around the unique structural features of this mechanism, its inverse kinematic solution and Jacobian matrix are calculated, and a coupled motion relation between a key limb and its moving platform (MP) is presented. In order to achieve singularity avoidance, some typical singularity configurations based on line geometry analysis are given. In accordance with this result, an initial configuration for multi-objective dimensional optimization is presented. To further enhance its kinematic performance, we introduce the use of the GCI (global conditional index) performance at extreme positions as one of the optimization criteria based on the NSGA-II (Non-dominated Sorting Genetic Algorithm) algorithm, and directly measuring the crowding distance using the position vector of the U (universal) joints on the moving platform. This optimized mechanism prototype is demonstrated in a single-leg Adams simulation, which exhibits good velocity mapping effects and displacement accuracy. Finally, a new BJTUBOT prototype was constructed based on the optimized leg, and its flexibility was tested with various classical forms of motions. The workflow in this paper significantly improves the leg performance under the current design needs.

Model-Free Intelligent Control for Space Soft Robotic Manipulators

Given the advantages of softness, lightness, low cost, and interaction safety, inverse kinematic modeling and control of soft actuators has caused a research boom. However, in realizing dexterous manipulation of space large soft manipulators, it is much more difficult to achieve precise control not only because of the greater accumulation of errors in the multiple degrees of freedom and nonlinear properties of soft materials at large scales but also because of the inability of directly solving the inverse kinematics in the cases of singular pure elongation. In this work, a model-free intelligent kinematic control strategy is proposed for these problems that exhibit a mapping relationship between the output end-effector position and the input pressure. For multiple-degree-of-freedom robots, especially pneumatic soft manipulators, traditional inverse kinematic modeling methods are complex and inverse Jacobian matrix solution often encounters geometric singularities. To address this issue, this paper proposes an inverse kinematics–multilayer perceptron (IK-MLP) method for soft manipulators. In this strategy, the trained intelligent controller can be applied to control pneumatic manipulators without establishing a traditional inverse kinematic model. The control algorithm is experimentally tested based on the ground experiment system of the space soft manipulator. Simulations and experiments are carried out to validate the given model-free intelligent controller, proving that the IK-MLP method can effectively solve the singularity of inverse kinematics.

Towards Fast and Robust Simulation in Integrated Electricity and Gas System: A Sequential United Method

The decoupling method, in which different energy vectors are solved alternatively, is widely adopted in the integrated electricity and gas systems (IEGS) analysis. Nevertheless, the bidirectional interaction created by power to gas (P2G) and gas turbines (GT) requires iterative calculations between the electric power system (EPS) and the natural gas network (NGN). Without proper solutions, the iterations will raise predictable concerns about simulation efficiency and convergence. Therefore, this paper investigates the fast and robust simulation in IEGS considering the nonlinearity and dynamics from both the network and device sides. To address the bidirectional interaction, a state decoupling method (SDM) is first proposed as an efficient solver for the dynamic NGN model. Based on this, a sequential united method (SUM) is developed to reduce the computational cost and improve the solver robustness, which decouples the large-scale Jacobian matrix inversion into the sequential solution of multiple small sub-matrixes. Case studies in two systems demonstrate the superiority of the proposed method compared with the decoupling method, indicating a significant improvement of SUM in simulation efficiency and robustness.

A Novel Integrated Design Method of Parallel Mechanisms Based on Performance Requirements

Abstract Due to the lack of connection between configuration synthesis and performance indices, many configurations obtained cannot meet the performance requirements, increasing the difficulty of configuration selection and prolonging the design cycle of the parallel mechanism (PM). In order to solve this problem, this paper proposes an inverse Jacobian matrix construction method based on performance indices. The method is realized by constructing singular values and singular vectors directly related to the performance indices. Furthermore, based on the screw expression form of the inverse Jacobian matrix, a new integrated design method that can directly meet the performance requirements is proposed. Finally, a novel ankle rehabilitation mechanism is presented using this method, and the correctness and effectiveness of the integrated design method are verified by theoretical analysis. Meanwhile, the analysis results show that the proposed method can effectively shorten PM’s design time and simplify PM’s design process, which has a good application value.

Design of a 2T1R-Type Parallel Mechanism: Performance Analysis and Size Optimization

The planar three-degree-of-freedom (DoF) parallel mechanism plays a significant role in the fields of automobiles and electronics due to its advantages such as high stiffness, large carrying capability, and stable operation. In this paper, a planar 3-DoF (2T1R) parallel mechanism is derived by structural evolution from the parallelogram by means of Grassmann line geometry and the Atlas method. First, the position equation of the mechanism is established and the inverse kinematics and Jacobian matrix are investigated. The maximum deviation between the theory and simulation results of the inverse kinematics and Jacobian matrix is 0.48%, which verifies the accuracy of the theoretical model. Then, the constraint conditions of the mechanism are defined. Based on the inverse position solution, the numerical method is used to solve the workspace of the mechanism. It is concluded that the ratio of the workspace to the entire triangular space is about 15% higher than that of the 3-PRR parallel mechanism. Next, based on the Jacobian matrix, the performance indexes are established to evaluate stiffness and dexterity. Finally, the size of the mechanism is optimized by means of the genetic algorithm to improve the comprehensive performance.

Symbolic and Numerical Mathematical Modeling of the Jacobian of the Yaskawa Motoman - Gp7 Robot using Matlab©

Robotic manipulators have besides static positioning problems, but besides static we have in the robot its speeds and forces applied in any kind of movement. one of the show important activities of robot motion control is the definition of its movements in the workspace. to perform these moves its joints need I ‘m moving and to define these positions in the spaces we need to calculate the direct kinematics and the opposite we call the inverse kinematics. The main objective of this paper is to present a study on velocity analysis, known as the Jacobian matrix. The methodology employed in this exploratory scientific research will ok developed from experimental tests, bibliographic references and case studies applied at the Advanced robotics Institute (IAR). the robot under study is the YASKAWA-MOTOMAN-GP7 present at the robotics Laboratory of the IAR. The work brings as contribution the implementation of the Denavit-Hartenberg notation and the validation of the Jacobian matrix of the robot. The results are the determination of the equations of the inverse kinematics using the method based on the inverse Jacobian matrix. The code was implemented in MATLAB© software capable of proving the developed mathematical model, which is applicable in industrial robots.

A simple algorithm for diffuse optical tomography without Jacobian inversion

A computationally simpler algorithm to reconstruct the optical property distribution of turbid media using diffuse optical tomographic principles is presented. The proposed algorithm eliminates the requirement of large Jacobian matrix inversion which otherwise is essential for tomographic imaging. The most significant Jacobians are identified based on proper thresholding of the measurement and the intersection of these Jacobians gives the approximate spatial location of the inhomogeneity. The algorithm is tested and optimized using simulations and further validated using tissue-mimicking phantom-based experiments and in-vivo small-animal experiments.

Probabilistic three-phase power flow in a distribution system applying the pseudo-inverse and cumulant method

Abstract A new, analytical approach using the cumulant method is proposed for the three-phase probabilistic power-flow (PPF) analysis. The approach to forming the sensitivity matrix is based on quantifying the pseudo-inverse instead of the inverse jacobian matrix, since it is commonly singular in a distribution power network. The results are compared with those obtained using the point-estimate method (PEM) and the Monte Carlo (MC) method, which is a commonly used reference method for the PPF analysis in a distribution power network.

케이블 구동형 다자유도 유연 로봇 암 개발

This report presents the development of a cable-driven hyper redundant flexible robot arm that has 10 joints with 20 degrees of freedom. The robot arm has the cable-driven mechanism for actuating each joint of the robot; instead of attaching the motors directly at each joints of the robot, the motors are deployed outside of the robot and the actuation force from the motors are transmitted by the flexible cables to each joint. The cable-driven mechanism makes the motion of the robot more flexible and dynamic by reducing the loads of motors to the robot. The control algorithm is designed based on the inverse kinematics using Pseudo inverse of Jacobian matrix. Detailed performances of this 10-joint robot arm and experimental results are outlined in further detail throughout this report.

A modified firefly algorithm for the inverse kinematics solutions of robotic manipulators

The inverse kinematics of robotic manipulators consists of finding a joint configuration to reach a desired end-effector pose. Since inverse kinematics is a complex non-linear problem with redundant solutions, sophisticated optimization techniques are often required to solve this problem; a possible solution can be found in metaheuristic algorithms. In this work, a modified version of the firefly algorithm for multimodal optimization is proposed to solve the inverse kinematics. This modified version can provide multiple joint configurations leading to the same end-effector pose, improving the classic firefly algorithm performance. Moreover, the proposed approach avoids singularities because it does not require any Jacobian matrix inversion, which is the main problem of conventional approaches. The proposed approach can be implemented in robotic manipulators composed of revolute or prismatic joints of n degrees of freedom considering joint limits constrains. Simulations with different robotic manipulators show the accuracy and robustness of the proposed approach. Additionally, non-parametric statistical tests are included to show that the proposed method has a statistically significant improvement over other multimodal optimization algorithms. Finally, real-time experiments on five degrees of freedom robotic manipulator illustrate the applicability of this approach.

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.

Application of parametric function in construction of particle shape and discrete element simulation

In previous studies, particles, such as ellipsoids and super-ellipsoids, are mainly described by implicit functions. However, parametric functions can define more surfaces that can be used to represent a wider variety of particle shapes. In this study, parametric functions are used to construct particles, and the algorithm to determine the multi-point contact between concave particles is also given. Furthermore, the geometric parameters at the contact point, for instance, normal vectors, curvatures, and overlapping, are formulated by the parameters in the parametric function. To verify the proposed method, we use the discrete element method to simulate the systems of torus-shaped particles nested within each other, and analyze the momentum and kinetic energy changes with time. The equilibrium state with approximately zero energy is obtained, which means that the algorithm for multi-point contact of concave particles is suitable and stable. In addition to the torus-shaped particle, another two concave particles defined by parametric function are modeled. The simulation results indicate that the method is universal. Any plane curve can be used to construct surfaces by using the method provided in this work, and computational efficiency of simulations of particles defined by parametric function is higher because calculations of the inverse Jacobian matrix in the Newton-Raphson method are unnecessary. The parametric function method extends the scope of previous studies on particle shape.

Design and Analysis of a Novel 2T2R Parallel Mechanism with the Closed-loop Limbs

This paper presents a novel four degrees of freedom (DOF) parallel mechanism with the closed-loop limbs, which includes two translational (2T) DOF and two rotational (2R) DOF. By connecting the proposed parallel mechanism with the guide rail in series, the 5-DOF hybrid robot system is obtained, which can be applied for the composite material tape laying in aerospace industry. The analysis in this paper mainly focuses on the parallel module of the hybrid robot system. First, the freedom of the proposed parallel mechanism is calculated based on the screw theory. Then, according to the closed-loop vector equation, the inverse kinematics and Jacobian matrix of the parallel mechanism are carried out. Next, the workspace stiffness and dexterity analysis of the parallel mechanism are investigated based on the constraint equations, static stiffness matrix and Jacobian condition number. Finally, the correctness of the inverse kinematics and the high stiffness of the parallel mechanism are verified by the kinematics and stiffness simulation analysis, which lays a foundation for the automatic composite material tape laying.

Power flow solution using a novel generalized linear Hopfield network based on Moore–Penrose pseudoinverse

This paper proposes a novel generalized linear Hopfield neural network-based power flow analysis technique using Moore–Penrose Inverse (MPI) to solve the nonlinear power flow equations (PFEs). The Hopfield neural network (HNN) with linear activation function augmented by a feed forward layer is used to compute the MPI. In this work, the inverse of Jacobian matrix in solving the PFEs is determined by including feed forward network along with feedback network. The developed power flow technique is coded in MATLAB, and its effectiveness is tested on well-conditioned IEEE bus systems (9-bus, 14-bus, 30-bus, and 118-bus), naturally ill-conditioned systems (11-bus and 13-bus), and real-time Malaysian 87-bus system. The results of voltage magnitude and phase angle obtained are compared with standard Newton–Raphson method in case of well-conditioned system. Further, the sensitivity analysis of this approach is carried out against change in initial conditions, line outage, and increase in power generation to validate its robustness. The computational cost of convergence time is compared with well-known power flow techniques of Modified HNN, fourth-order Runge–Kutta (RK4), Iwamoto, and Euler method. The convergence of solution obtained from proposed technique is ensured by Lyapunov notion of stability.

Adaptive control of parallel robots with uncertain kinematics and dynamics

One of the most challenging issues in adaptive control of robot manipulators with kinematic uncertainties is the requirement of inverse Jacobian matrix in regressor form. This requirement is inevitable in the case of the control of parallel robots, whose dynamics formulation are derived in the task space. In this paper, an adaptive controller is proposed for parallel robots based on representation of Jacobian matrix in regressor form with asymptotic trajectory tracking. The main idea of this paper is separation of determinant and adjugate of Jacobian matrix in order to represent them into a new regressor forms. Simulation and experimental results on a 2-DOF RPR and a 3-DOF redundant cable driven robot, respectively, verify promising performance of the proposed method in practice.

Industrial robot accurate trajectory generation by nested loop iterative learning control

Common error sources of industrial robot manipulators include joint servoing error, imprecise kinematics, mechanical compliance, and transmission error. In this work we present a nested loop iterative learning control (ILC) feedforward structure: an inner loop that compensates for motor dynamics, and an outer loop that corrects the deviation along the path tracked, that features practically efficient implementation. Taking advantage of industrial robot’s speed reduction transmission, single-input-single-output method is demonstrated effective for the nonlinear coupled robot dynamics. Data-based inversion technique that incorporates motion constraint is used for fast inner loop convergence. The outer loop utilizes inverse Jacobian matrix for joint reference modification. For nonlinear static friction that is difficult to be compensated for with only joint command, notch filtering is utilized in the learning process to avoid exciting vibration inherently exists in the robot. The proposed nested loop ILC requires only the nominal kinematic parameters from the robot manufacturer, and can be readily implemented without modifying the existing robot controllers. The effectiveness of the proposed method is experimentally verified on a six degree-of-freedom robot manipulator.

Kinematics Analysis of A Parallel Robot with 3 DOF and 4 Segments in Pure Translation.(Dept.M)

In this paper we present the direct and inverse analysis of a geometrical model of an UPS manipulator with three degrees of freedom added to a PPP passive central segment. This structure provides a pure translation motion. We will also determine the relations between generalized and articular velocities by using the inverse Jacobian matrix, Further, we determine the reciprocal relations between cartesian and angular velocities of the end effector via articular velocities by direct re-inversion of the Jacobian matrix. This study aims of implementing the control of a parallel robot manipulator. A prototype of a parallel robot has been built up in our laboratory in order to validate the proposed models.

Multi-agent reinforcement learning for redundant robot control in task-space

Task-space control needs the inverse kinematics solution or Jacobian matrix for the transformation from task space to joint space. However, they are not always available for redundant robots because there are more joint degrees-of-freedom than Cartesian degrees-of-freedom. Intelligent learning methods, such as neural networks (NN) and reinforcement learning (RL) can learn the inverse kinematics solution. However, NN needs big data and classical RL is not suitable for multi-link robots controlled in task space. In this paper, we propose a fully cooperative multi-agent reinforcement learning (MARL) to solve the kinematic problem of redundant robots. Each joint of the robot is regarded as one agent. The fully cooperative MARL uses a kinematic learning to avoid function approximators and large learning space. The convergence property of the proposed MARL is analyzed. The experimental results show that our MARL is much more better compared with the classic methods such as Jacobian-based methods and neural networks.