Kinematics of Articulated Planar Linkages

This paper proposes a kinematics algorithm in screw coordinates for articulated linkages. As the screw consists of velocity and position variables of a joint, the solutions of the forward and inverse velocities are the functions of position coordinates and their time derivatives. The most prominent merit of this kinematic algorithm is that we only need the first order numerical differential interpolation for computing the acceleration. To calculate the displacement, we also only need the first order numerical integral of the velocity. This benefit stems from the screw the coordinates of which are velocity components. Both the forward and the inverse kinematics have the similar calculation process in this method. Through examples of planar open-chain linkage, single closed-chain linkage and multiple closed-chain linkage, the kinematics algorithm is validated. It is particularly fit for developing numerical programmers for forward and inverse kinematics in the same procedures, including the velocity, displacement and acceleration which provide the fundamental information for dynamics of the linkage.

A Constraint-Based Approach to Manipulator Kinematics and Singularities

<p>In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories. The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family. At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4]. Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators. The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.</p>

A Constraint-Based Approach to Manipulator Kinematics and Singularities

<p>In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories. The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family. At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4]. Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators. The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.</p>

Manipulator Kinematics and Dynamics On Differentiable Manifolds: Part II Dynamics

The manipulator differentiable manifold formulation presented in Part I of this paper is used to create algorithms for forward and inverse kinematics on maximal, singularity free, path connected manifold components. Existence of forward and inverse configuration mappings in manifold components is extended to obtain forward and inverse velocity mappings. Computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of the four categories of manipulator treated. Manifold parameterizations derived in Part I are used to transform variational equations of motion in Cartesian generalized coordinates to second order ordinary differential equations of manipulator dynamics, in terms of both input and output coordinates, avoiding ad-hoc derivation of equations of motion. This process is illustrated in evaluating terms required for equations of motion of the four model manipulators analyzed in Part I. It is shown that computation involved in forward and inverse kinematics and in evaluation of equations of manipulator dynamics can be carried out in real-time on modern microprocessors, supporting on-line implementation of modern methods of manipulator control.

Kinematic Modeling of a Novel RR-RP Hybrid Serial-Parallel Mechanism With Variable Topology

This paper introduces a new mechanism that will be classified as an RR-RP hybrid serial-parallel mechanism with variable topology. A mechanism with variable topology is a mechanism that can change its topology due to a change it its joints constraint geometric profile. The RR-RP is unique in that it combines the functionality of both an RR and RP serial manipulator without the need for an additional actuator, leading to a lower weight, lower cost, and more efficient mechanism. The new mechanism and its topology are presented, followed by a workspace analysis, derivation of the forward and inverse kinematics, and velocity analysis of the new mechanism.

Kinematics of a planar slider-crank linkage in screw form

This paper investigates the forward and inverse kinematics in screw coordinates for a planar slider-crank linkage. According to the definition of a screw, both the angular velocity of a rigid body and the linear velocity of a point on it are expressed in screw components. Through numerical integration on the velocity solution, we get the displacement. Through numerical differential interpolation of velocity, we gain the acceleration of any joint. Traditionally, position and angular parameters are usually the only variables for establishing the displacement equations of a mechanism. For a series mechanism, the forward kinematics can be expressed explicitly in the variable of position parameters while the inverse kinematics will have to resort to numerical algorithms because of the multiplicity of solution. For a parallel mechanism, the inverse kinematics can be expressed explicitly in the variable of position parameters of the end effector while the forward kinematics will have to resort to numerical algorithms because of the nonlinearity of system. Therefore this will surely lead to second order numerical differential interpolation for the calculation of accelerations. The most prominent merit of this kinematic algorithm is that we only need the first order numerical differential interpolation for computing the acceleration. To calculate the displacement, we also only need the first order numerical integral of the velocity. This benefit stems from the screw the coordinates of which are velocity components. The example of planar four-bar and five-bar slider-crank linkages validate this algorithm. It is especially suited to developing numerical algorithms for forward and inverse velocity, displacement and acceleration of a linkage.

Forward and Inverse Kinematics Demonstration using RoboDK and C#

Purpose: Robot researchers need a simulator to understand better the algorithm on path planning, arm movement, and many more. They need a good simulator. RoboDK is an excellent simulator to fulfill the research work. It has calibration facilities, so it is industrial-grade software. Its forward and inverse kinematics accuracy is better than any competing software. The main advantage is all robots under one IDE. When we use an industrial robot, and we must use their software environment to operate the robot. But the RoboDK covers most of the robots and runs under one roof. And we need to learn only one IDE. The RoboDK online library is full of the standard robot. And all robot’s operation procedure is the same. So, the learning curve of new robots is easy. It is easy to simulate, and it can connect with a practical robot to execute the task. Using this software, we can quickly create digital twins for the industry. Now we think about control the robot from our application. When we use to control the robot from an external environment or remote software, we need the use the API to control the robot. Here we will see how easily we can operate the robot from our custom application. We adopted RoboDK C# API and integrated it into Visual studio using a User interface to control the robot movement. Keeping this research as a reference, the robotic arm researcher can add value to their research. Our primary purpose is to shorten the learning curve to integrate the RoboDK with their custom application. Design/Methodology/Approach: Taking the RoboDK C# API they provided, we customized it according to our purpose with minimal components. After developing a graphical user interface, we interact through API. Then, opening both RoboDK IDE and C# application, we can send the End effector position using the sliding movement. Findings/Result: After our research, we found that RoboDK is a good IDE for our research on the robotics arm. We can easily integrate the C# API they provided with our custom application for research purposes. Originality/Value: If we want to test robotic arm movement in the simulator, we need an excellent simulator like RoboDK. Integrating the RoboDK C# API is a little bit time-consuming. Using our approach, the researcher can continue their research in a minimal period. And find adequate information here to integrate easily into their project. Paper Type: Simulation-based Research.

Kinematics Analysis of 6-DoF Articulated Robot with Spherical Wrist

The aim of the paper is to study the kinematics of the manipulator. The articulated robot with a spherical wrist has been used for this purpose. The Comau NM45 Manipulator has been chosen for the kinematic model study. The manipulator contains six revolution joints. Pieper’s approach has been employed to study the kinematics (inverse) of the robot manipulator. Using this approach, the inverse kinematic problem is divided into two small less complex problems. This reduces the time of analysing the manipulator kinematically. The forward and inverse kinematics has been performed, and mathematical solutions are detailed based on D-H (Denavit–Hartenberg) parameters. The kinematics solution has been verified by solving the manipulator’s motion. It has been observed that the model is accurate as the motion trajectory was smoothly followed by the manipulator.