Paden-Kahan Sub-problems Research Articles

Inverse solution of six-degrees-of-freedom serial robot with adjacent three joint axes parallel based on screw theory

Inverse kinematics occupies a vital role in robotics and is the fundation of robot trajectory planning as well as rigid body motion, which directly affects the rapidity and accuracy of control. Compared with the traditional Denavit-Hatenberg parameter modeling, the screw method has its unique advantages. The paper establishes the kinematic model of a six-degree-of-freedom serial robot using screw theory. The Paden-Kahan sub-problem, orientation separation reduction technique and matrix transformation are used to resolve the inverse kinematics. The effectiveness and validity of the above algorithm are verified by data calculation. A new inverse solution algorithm is designed in this paper to propose an idea for the inverse kinematic solution with the second, third, and fourth joints being parallel to each other in the robot.

Formalization of the inverse kinematics of three-fingered dexterous hand

As an effective extension of human limbs, dexterous robotic hands based on cyber-physical systems (CPS) are used to perform diverse tasks in industrial production, medical operation, outer-space exploration, etc. The reliable functioning of dexterous robotic hands depends on the proper modeling and solving of inverse kinematics problems in the design phase. The traditional D-H parameter method for modeling and solving inverse kinematics problems suffers from the singularity problem, which is avoided by methods based on the screw theory. However, the manual or computer-simulation-based development of these methods could suffer from human errors and software defects. In this paper, we address this problem by using formal techniques and tools to develop a screw-theory-based method. Firstly, we formalize theories related to the Paden-Kahan sub-problem in an interactive theorem prover. Secondly, we construct a formal model for solving the inverse kinematics problem for robotic systems. Lastly, taking the three-fingered dexterous hand as a case study, we formally verified its inverse kinematics solutions. The results presented in this article help guarantee a level of safety for robotic systems that is beyond achievable by manual and computer-simulation-based development methods.

Modeling and compensation of volumetric errors for a six-axis automated fiber placement machine based on screw theory

This paper presents a novel modeling and compensation method for the volumetric errors of a six-axis gantry automated fiber placement (AFP) machine. Based on the screw theory, the forward and inverse kinematics models of the AFP machine are established. In order to improve the accuracy of the inverse kinematics solution, the Paden-Kahan sub-problem method is used to perform the inverse kinematics solution for the simplified topology of the rotary axes. Using error motion twist to establish a volumetric error transfer model for 54 geometric errors. According to the measurement data of a laser tracker and the Levenberg-Marquardt method to identify the geometric error parameters. The explicit formula of the inverse kinematics solution is used to obtain the error compensation of each motion axis, and the G code of the laying path is modified by the iterative method to realize the compensation of the volumetric errors. By comparing the positions of the tool center point before and after the error compensation, the practicability of the volumetric error modeling and iterative compensation methods are verified, and the geometric accuracy of the AFP machine can be effectively improved.

A Novel Analytical Inverse Kinematics Method for SSRMS-Type Space Manipulators Based on the POE Formula and the Paden-Kahan Subproblem

Space manipulators which have a similar symmetrical structure with seven revolute joints, such as the space station remote manipulator system (SSRMS), can be called SSRMS-type space manipulators. The analytical inverse kinematics of an SSRMS-type manipulator can be solved by locking a single joint; the locked joint (joint 1, 2, 6, or 7) can be determined by configuration analysis. Although widely used in establishing the kinematics of SSRMS-type manipulators, the Denavit-Hartenberg (DH) method has a singular problem when two adjacent joint axes are nearly parallel. To avoid this problem, this paper proposes a novel analytical inverse kinematics method for SSRMS-type manipulators based on the product of exponentials (POE) formula and the Paden-Kahan subproblem. Because of the symmetrical structure, an SSRMS-type manipulator degrades to two kinds of 6-degree-of-freedom (DOF) manipulators when locking a single joint (joint 1, 2, 6, or 7). The analytical inverse kinematics of these two kinds of 6-DOF manipulators is solved by combining the Paden-Kahan subproblems and geometric and algebraic methods, respectively. The proposed approach is not only singularity free compared with the traditional DH-based methods but also more accurate than the POE-based numerical solution. The simulation results verify the efficiency of the proposed approach.

Non-destructive Testing Assisted by Six-Axis Manipulator Based on POE Formula

At present, aluminum plates are widely used in aerospace, machinery, and other fields because of their light weight, good manufacturability, and strong corrosion resistance. However, during long-term use, defects that will affect the structural strength such as cracks are easy to appear. Therefore, it is necessary to conduct automatic non-destructive testing of large metal plates to determine the location, size, and structural damage to their defects. Compared with manual detection, automated detection has many advantages such as high detection efficiency, high repeatability, strong flexibility, and low labor intensity. Based on the spinor theory, this paper uses the product of exponentials (POE) to model the kinematics of the Aubo-i5 manipulator and obtain the positive kinematics equations, which are compiled by analytic geometry and algebraic methods combined with Paden-Kahan subproblems and matrix theory. The inverse kinematics algorithm of the 6R manipulator is based on this model. An automated non-destructive testing system including a robotic arm and ultrasonic phased array testing equipment was built. At the same time, a robotic arm simulation platform was built based on ROS, and the inverse kinematics algorithm was used to control the robotic arm through the MoveIt!. The angle error can be controlled within 2.5°.

A novel solution of inverse kinematic for 6R robot manipulator with offset joint based on screw theory

This article proposes a novel inverse kinematic approach with translation transformation matrix based on screw theory to solve the inverse kinematic problem for 6R robot manipulator with offset joint. The translation transformation matrix is introduced to convert the 6R robot manipulator with offset joint to a new configuration with intersecting axes, and the mapping relationship from the end effector to the joint angle is established along with the Paden–Kahan subproblems. The eight closed solutions of the specific configuration are deduced, which automatically eliminate the singularity solutions. Moreover, the precision and efficiency of the proposed method are verified through a numerical example. Unlike other approaches, the presented algorithm not only inherits the superior accuracy of the other geometric approaches but also exhibits an outperform efficiency. Finally, the method is generalized to other 6R robots, which has closed-form solutions to further verify its versatility. The presented study provides some basis for further investigations, such as trajectory planning and motion control, which provides a new tool on the analysis and application of this kind of robot manipulator.

Inverse displacement analysis of a novel hybrid humanoid robotic arm

The application of hybrid mechanisms in the field of humanoid robots has attracted significant attention. A novel eight-degree-of-freedom hybrid manipulator is proposed to realize a kinematic function similar to that of the human arm. A general method for solving the inverse displacement problem of hybrid mechanisms is given, and the proposed humanoid robotic arm is taken as an example to demonstrate the solution process of this method. Furthermore, a closed-form solution for the inverse displacement problem of the hybrid humanoid robotic arm is derived by using the given method based on screw theory, exponential product formula, and Paden–Kahan subproblems. In addition, the problem of verifying and selecting the appropriate solutions according to the starting postures is also illustrated in a series of simulation experiments. Simulation experiment results show that there are (at most) 32 sets of solutions for the proposed humanoid robotic arm according to the same target position-orientation matrix and the given redundant input variables, and the accuracy of the proposed method for solving the inverse displacement problem is verified.

Optimizing method of mechanism angle of upper limb rehabilitation robot at glenohumeral joint

At present, there are seventy million stroke patients in China and annual death toll of stroke is 1 million 650 thousand people. The survivors about 75% become disabled persons and lose the ability to move. To address this issue, a kind of rehabilitation exoskeleton robot called YANARM is proposed which for training of shoulder complex. This paper, a kind of method for optimizing the angle parameters of series dynamic axes under given workspace conditions is presented. The forward and inverse kinematics solutions of glenohumeral mechanism are solved based on the exponential product formula (POE) and the Paden-Kahan sub-problem. The range of joint rotation angle can be inversely solved according to the end of the arm position at the borderline of the workspace. A curve between the angle and the CP which equal mean variance of joint rotation range plus sum of all joint rotation ranges is solved. The mechanism angle of glenohumeral joint is optimized by this method.

Analytical Solution for Inverse Kinematics Using Dual Quaternions

This paper presents a new solution for inverse kinematics (IK) by using dual quaternions (DQ) as operators and combining them with the Paden–Kahan subproblem to determine the analytical solution. Kinematics under the structure of screw theory makes the process mathematically efficient and gives geometric meaning. Both derived results and simulation outgrowths, when compared with the generally used numerical solution and matrix-based analytical solution, show that our method is faster and does not suffer from the numerical instability caused by being near to a singular configuration. The method has been proven to have the advantages of reducing the computational load and promoting precision, and it can provide multiple choices for IK.

Inverse kinematic solution of 6R robot manipulators based on screw theory and the Paden–Kahan subproblem

The traditional Denavit–Hatenberg method is a relatively mature method for modeling the kinematics of robots. However, it has an obvious drawback, in that the parameters of the Denavit–Hatenberg model are discontinuous, resulting in singularity when the adjacent joint axes are parallel or close to parallel. As a result, this model is not suitable for kinematic calibration. In this article, to avoid the problem of singularity, the product of exponentials method based on screw theory is employed for kinematics modeling. In addition, the inverse kinematics of the 6R robot manipulator is solved by adopting analytical, geometric, and algebraic methods combined with the Paden–Kahan subproblem as well as matrix theory. Moreover, the kinematic parameters of the Denavit–Hatenberg and the product of exponentials-based models are analyzed, and the singularity of the two models is illustrated. Finally, eight solutions of inverse kinematics are obtained, and the correctness and high level of accuracy of the algorithm proposed in this article are verified. This algorithm provides a reference for the inverse kinematics of robots with three adjacent parallel joints.

General inverse solution of six-degrees-of-freedom serial robots based on the product of exponentials model

PurposeDeveloping general closed-form solutions for six-degrees-of-freedom (DOF) serial robots is a significant challenge. This paper thus aims to present a general solution for six-DOF robots based on the product of exponentials model, which adapts to a class of robots satisfying the Pieper criterion with two parallel or intersecting axes among its first three axes.Design/methodology/approachThe proposed solution can be represented as uniform expressions by using geometrical properties and a modified Paden–Kahan sub-problem, which mainly adopts the screw theory.FindingsA simulation and experiments validated the correctness and effectiveness of the proposed method (general resolution for six-DOF robots based on the product of exponentials model).Originality/valueThe Rodrigues rotation formula is additionally used to turn the complex problem into a solvable trigonometric function and uniformly express six solutions using two formulas.

General frame for arbitrary 3R subproblems based on the POE model

Inverse kinematic (IK) problems based on the product of exponentials (POE) model are often transformed to a series of Paden–Kahan subproblems, which are based on geometric methods with geometric constraints. The focus of subproblem is to solve the 3rd order subproblems, among which the 3R-type subproblem is the most prevalent. In this paper, a general frame for the inverse solution of arbitrary 3R types, without geometric constraints, is presented. It can be applied in different 3R types of practical cases, including those with parallel, intersecting, and skewed relationships. This paper mainly focuses on: (1) developing a real algebraic geometric (RAG) method based on the properties of the screw theory and the Rodrigues’ rotation formula; (2) obtaining the closed-form solutions for arbitrary 3R subproblems, and ensuring the accuracies of these solutions; (3) expanding the Paden–Kahan subproblems and meeting the demands of online real-time applications; and finally, (4) verifying the effectiveness of the RAG method, through comparisons with the geometric method, using simulations and real experiments. The proposed frame can be widely applied in series, reconfigurable, and other types of robots.

A novel second subproblem for two arbitrary axes of robots

The Paden–kahan subproblem is a simple and flexible method to solve the closed-form inverse resolution but limited by the geometrical structure of robots, which is very difficult to be kept because...

Algorithmic approach to geometric solution of generalized Paden–Kahan subproblem and its extension

Kinematics as a science of geometry of motion describes motion by means of position, orientation, and their time derivatives. The focus of this article aims screw theory approach for the solution of inverse kinematics problem. The kinematic elements are mathematically assembled through screw theory by using only the base, tool, and workpiece coordinate systems—opposite to conventional Denavit–Hartenberg approach, where at least n + 1 coordinate frames are needed for a robot manipulator with n joints. The inverse kinematics solution in Denavit–Hartenberg convention is implicit. Instead, explicit solutions to inverse kinematics using the Paden–Kahan subproblems could be expressed. This article gives step-by-step application of geometric algorithm for the solution of all the cases of Paden–Kahan subproblem 2 and some extension of that subproblem based on subproblem 2. The algorithm described here covers all of the cases that can appear in the generalized subproblem 2 definition, which makes it applicable for multiple movement configurations. The extended subproblem is used to solve inverse kinematics of a manipulator that cannot be solved using only three basic Paden–Kahan subproblems, as they are originally formulated. Instead, here is provided solution for the case of three subsequent rotations, where last two axes are parallel and the first one does not lie in the same plane with neither of the other axes. Since the inverse kinematics problem may have no solution, unique solution, or many solutions, this article gives a thorough discussion about the necessary conditions for the existence and number of solutions.

The Kinematics Analysis of a Novel Self-Reconfigurable Modular Robot Based on Screw Theory

This paper introduces our newly designed self-reconfigurable modular robot named Larva-Bot, which has two DOFs with intersecting rotation axis normal to each other. Each module has autonomous mobility, and can assemble with other modules to form various configurations. In this study, a typical 6-DOF structure, composed of Larva-Bots, is proposed to analyze the kinematic characteristics of the modular robot. The common kinematics model of the robot is based on the Denavit-Hartenberg (D-H) parameter method, while its inverse kinematics has inefficient calculation and complicated solutions. To solve the problems, the screw theory method is applied in this paper. The forward kinematics model is constructed by using the product of exponentials (POEs) formula, while the inverse kinematics analysis is based on Paden-Kahan Sub-problems. Finally, the reliability of the method has been proved by computations and simulation.

Improvement of the Algorithm of the Inverse Kinematics Calculation for 6R Series Robots Based on One Novel Paden-Kahan Sub-problem

Improvement of the Algorithm of the Inverse Kinematics Calculation for 6R Series Robots Based on One Novel Paden-Kahan Sub-problem

Improved Inverse Kinematics Algorithm Using Screw Theory for a Six-DOF Robot Manipulator

Based on screw theory, a novel improved inverse-kinematics approach for a type of six-DOF serial robot, “Qianjiang I”, is proposed in this paper. The common kinematics model of the robot is based on the Denavit-Hartenberg (D-H) notation method while its inverse kinematics has inefficient calculation and complicated solution, which cannot meet the demands of online real-time application. To solve this problem, this paper presents a new method to improve the efficiency of the inverse kinematics solution by introducing the screw theory. Unlike other methods, the proposed method only establishes two coordinates, namely the inertial coordinate and the tool coordinate; the screw motion of each link is carried out based on the inertial coordinate, ensuring definite geometric meaning. Furthermore, we adopt a new inverse kinematics algorithm, developing an improved sub-problem method along with Paden-Kahan sub-problems. This method has high efficiency and can be applied in real-time industrial operation. It is convenient to select the desired solutions directly from among multiple solutions by examining clear geometric meaning. Finally, the effectiveness and reliability performance of the new algorithm are analysed and verified in comparative experiments carried out on the six-DOF serial robot “Qianjiang I”.

A new formulation method for solving kinematic problems of multiarm robot systems using quaternion algebra in the screw theory framework

We present a new formulation method to solve the kinematic problem of multiarm robot systems. Our major aims were to formulize the kinematic problem in a compact closed form and avoid singularity problems in the inverse kinematic solution. The new formulation method is based on screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of a rigid body and avoids singularities due to the use of the local coordinates. The dual quaternion, the most compact and efficient dual operator to express screw displacement, was used as a screw motion operator to obtain the formulation in a compact closed form. Inverse kinematic solutions were obtained using Paden-Kahan subproblems. This new formulation method was implemented into the cooperative working of 2 St\"{a}ubli RX160 industrial robot-arm manipulators. Simulation and experimental results were derived.

Kinematic Analysis of Humanoid Based on Screw Theory

It is more difficult and sophisticated to analyze the humanoid kinematics than the common industrial serial manipulator’s because of its high number of DOF, complex model, and balance constraints. Taking the advantages of the characteristics in the configuration of leg’s joints of humanoid, based on the screw theory, according to the trajectory planning of the center of mass and the feet, a method is proposed to solve the kinematics of humanoid’s legs using Paden-Kahan sub-problems approach. The solving process indicates that the mathematical modeling is of simpleness and briefness, while it is of convenience to solve the inverse kinematics

A Comparative Study of Three Inverse Kinematic Methods of Serial Industrial Robot Manipulators in the Screw Theory Framework

In this paper, we compare three inverse kinematic formulation methods for the serial industrial robot manipulators. All formulation methods are based on screw theory. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. In these three formulation methods, the first one is based on quaternion algebra, the second one is based on dual-quaternions, and the last one that is called exponential mapping method is based on matrix algebra. Compared with the matrix algebra, quaternion algebra based solutions are more computationally efficient and they need less storage area. The method which is based on dual-quaternion gives the most compact and computationally efficient solution. Paden-Kahan sub-problems are used to derive inverse kinematic solutions. 6-DOF industrial robot manipulator's forward and inverse kinematic equations are derived using these formulation methods. Simulation and experimental results are given.