Analytical derivation and application of the Jacobian matrix of parallel kinematic manipulators

Taking the 3DOF parallel mechanism within the Tricept robot as an example, this paper presents an analytical approach for the stiffness modeling of parallel kinematic machines having a properly constrained passive limb. The stiffness model is formulated using the 6×6overall Jacobian. It takes particular interest in the precise formulation of the bending stiffness matrix of the properly constrained passive limb by considering the compatibility conditions of the system. Stiffness evaluation of a sample Tricept robot is carried out using two global indices obtained from singular value decomposition of the compliance matrix.

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Error modeling and sensitivity analysis of a novel 5-degree-of-freedom parallel kinematic machine tool

Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture ◽

10.1177/0954405418782283 ◽

2018 ◽

Vol 233 (6) ◽

pp. 1637-1652 ◽

Cited By ~ 6

Author(s):

Xuan Luo ◽

Fugui Xie ◽

Xin-Jun Liu ◽

Jie Li

Keyword(s):

Sensitivity Analysis ◽

Machine Tool ◽

Jacobian Matrix ◽

Error Modeling ◽

Degree Of Freedom ◽

Geometric Errors ◽

Parallel Kinematic Machine ◽

Parallel Kinematic ◽

Position And Orientation ◽

Five Axis

5-Degree-of-freedom parallel kinematic machine tools are always attractive in manufacturing industry due to the ability of five-axis machining with high stiffness/mass ratio and flexibility. In this article, error modeling and sensitivity analysis of a novel 5-degree-of-freedom parallel kinematic machine tool are discussed for its accuracy issues. An error modeling method based on screw theory is applied to each limb, and then the error model of the parallel kinematic machine tool is established and the error mapping Jacobian matrix of 53 geometric errors is derived. Considering that geometric errors exert both impacts on value and direction of the end-effector’s pose error, a set of sensitivity indices and an easy routine for sensitivity analysis are proposed according to the error mapping Jacobian matrix. On this basis, 10 vital errors and 10 trivial errors are identified over the prescribed workspace. To validate the effects of sensitivity analysis, several numerical simulations of accuracy design are conducted, and three-dimensional model assemblies with relevant geometric errors are established as well. The simulations exhibit maximal −0.10% and 0.34% improvements of the position and orientation errors, respectively, after modifying 10 trivial errors, while minimal 65.56% and 55.17% improvements of the position and orientation errors, respectively, after modifying 10 vital errors. Besides the assembly reveals an output pose error of (0.0134 mm, 0.0020 rad) with only trivial errors, while (2.0338 mm, 0.0048 rad) with only vital errors. In consequence, both results of simulations and assemblies validate the correctness of the sensitivity analysis. Moreover, this procedure can be extended to any other parallel kinematic mechanisms easily.

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On the Structural Constraint and Motion of 3-PRS Parallel Kinematic Machines

10.1115/detc2021-70160 ◽

2021 ◽

Author(s):

Hassen Nigatu ◽

Yun Ho Choi ◽

Doik Kim

Keyword(s):

Jacobian Matrix ◽

Constraint Equation ◽

Parallel Manipulators ◽

Forward Rate ◽

Structural Constraint ◽

Usual Practice ◽

Parasitic Motion ◽

Parallel Kinematic ◽

Joint Velocity ◽

Velocity Level

Abstract This paper presents a consistent analytic kinematic formulation of the 3-PRS parallel manipulator (PM) with a parasitic motion by embedding the velocity level structural constraint equation into the motion expression. Inverse rate kinematics (IRK) is solved with a simple constraint compatible velocity profile, which is obtained by projecting the instantaneous restriction space onto the motion space. Moreover, the systematic method to reveal the parasitic motion is introduced. Thus, the parasitic terms are automatically identified from the main motions. Unlike the usual approach, this study does not consider any explicit parasitic motion expression. Consequently, the derivation of constraint compatible input velocity, which comprises the parasitic term, is simplified. To incorporate the parasitic motion into the task velocity, constraint Jacobian of the manipulator is analytically obtained first. The manipulator Jacobian is extended to incorporate the passive joint’s information apart from the active joints and structural constraint. Hence, the dimension of the Jacobian matrix used to solve IRK is 9 × 6. The validity of the IRK is proved by the Bordered Gramian based forward rate kinematics (FRK). Then, an accurate numerical integration, RK4, is applied to the joint velocity of IRK to obtain the manipulator’s joint values. Consequently, the moving plate’s pose is obtained via forward position kinematics computed using integrated active and passive joint values for validation. The projection matrix used to get compatible constraint motion adjusts our input velocity and makes it compatible with the structural constraint policy, and the parasitic motion is embedded easily. Thus, an explicit formulation of the parasitic motion equation is not required, as the usual practice. Finally, the study presented numerical simulations to show the validity of the outlined resolutions. This paper’s result and analysis can be uniformly applied to other parallel manipulators with less than 6 DoFs.

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Kinematic analysis of limited-dof parallel manipulators based on translational/rotational Jacobian and Hessian matrices

Robotica ◽

10.1017/s0263574709005396 ◽

2009 ◽

Vol 27 (7) ◽

pp. 971-980 ◽

Cited By ~ 6

Author(s):

Yi Lu ◽

Yan Shi ◽

Jianping Yu

Keyword(s):

Kinematic Analysis ◽

Jacobian Matrix ◽

Parallel Manipulators ◽

Parallel Kinematic Machines ◽

Parallel Kinematic

SUMMARYThis paper proposes an approach for solving the velocity and acceleration of the limited-dof (dof n < 6) parallel kinematic machines with linear active legs by means of translational/rotational Jacobian and Hessian matrices. First, based on the established or derived constraint and displacement equations, the translational/rotational Jacobian and Hessian matrices are derived. Second, the formulae for solving inverse/forward velocities and accelerations are derived from translational and rotational Jacobian/Hessian matrices. Third, a 2SPR + UPU PKM and a 2SPS + RPRR PKM are illustrated for explaining how to use this method. This approach is simple because it needs neither to eliminate 6-n rows of an n × 6 Jacobian matrix nor to determine the screw or pose of the constrained wrench.

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An Application of Jacobian Matrix in Kinematic of Multi-loop Spatial Mechanism

International Journal of Innovative Technology and Exploring Engineering - Special Issue ◽

10.35940/ijitee.l1166.10812s19 ◽

2019 ◽

Vol 8 (12S) ◽

pp. 701-704

Keyword(s):

Review Paper ◽

Jacobian Matrix ◽

Robot Manipulator ◽

Industrial Applications ◽

Design Synthesis ◽

Spatial Mechanism ◽

Kinematic Geometry ◽

Parallel Kinematic ◽

Input Motion ◽

Kinematic Singularities

Mechanism is an intended to turn input motion into a required set of output motion and forces. Now a days the spatial mechanism is widely used in various industrial applications like parallel kinematic machines as a robot manipulator. The main aim of the review paper is to use the Jacobian matrix into the kinematic analysis. From the review papers analysis, it has been concluded that the Kinematic analysiswith help of Jacobian matrix performed various Multi-loop Spatial Mechanism settings for Kinematic geometry, its topology and to carried out displacement modelling. It also used to Kinematic Design, synthesis and optimization. Finally, Comparison of existing methods for Kinematic Singularities with Jacobian in Parallel Manipulator.

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Workspace Decomposition Based Dimensional Synthesis of a Novel Hybrid Reconfigurable Robot

Journal of Mechanisms and Robotics ◽

10.1115/1.4001781 ◽

2010 ◽

Vol 2 (3) ◽

Cited By ~ 38

Author(s):

Tao Sun ◽

Yimin Song ◽

Yonggang Li ◽

Jun Zhang

Keyword(s):

Jacobian Matrix ◽

Screw Theory ◽

Dimensional Synthesis ◽

End Effector ◽

Global View ◽

Hybrid Module ◽

Reconfigurable Robot ◽

Parallel Kinematic ◽

Kinematic Performance ◽

Position Problem

A novel 5-axis hybrid reconfigurable robot named Tricept-IV, including one 4-degree-of-freedom (4DOF) hybrid module and one 2DOF end effector, is investigated. Compared with extensive research that has been pursued for the parallel kinematic machines such as Tricept, Sprint Z3 Head, and so on, the hybrid kinematic machines have not adequately been studied. This paper demonstrates a method of workspace decomposition applied in the dimensional synthesis of a 4DOF hybrid module, which is the underlying architecture of the newly invented robot. This paper starts with an introduction of the Tricept-IV robot. After dividing the 4DOF hybrid module into one position-orientation coupling (POC) subsystem and one position-feed (PF) subsystem, its workspace is decomposed into POC and PF subspaces accordingly, and then the inverse position problem may be solved by means of one prejudgment method. Furthermore, the Jacobian matrix of the POC subsystem is obtained by the screw theory so as to formulate its kinematic performance index. Finally, the dimensional synthesis based on workspace decomposition of the hybrid module is carried out by taking a global view of the dimensional synthesis of these two subsystems.

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Improved Cubature Kalman Filtering-Based Estimation of the Jacobian Matrix for Environment Mutation

2019 Chinese Control Conference (CCC) ◽

10.23919/chicc.2019.8865132 ◽

2019 ◽

Author(s):

Xiuqian Jia ◽

Haixia Wang ◽

Chunyang Sheng ◽

Zhiguo Zhang ◽

Wei Cui ◽

...

Keyword(s):

Kalman Filtering ◽

Jacobian Matrix

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The dynamics of a Leslie type predator–prey model with fear and Allee effect

Advances in Difference Equations ◽

10.1186/s13662-021-03490-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

S. Vinoth ◽

R. Sivasamy ◽

K. Sathiyanathan ◽

Bundit Unyong ◽

Grienggrai Rajchakit ◽

...

Keyword(s):

Local Stability ◽

Allee Effect ◽

Jacobian Matrix ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Lyapunov Coefficient ◽

Points Of View ◽

Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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