Symbolic Computation of Inverse Kinematics for General 6R Manipulators Based on Raghavan and Roth’s Solution

2020 ◽  
Author(s):  
Keisuke Arikawa
Keyword(s):  
Characteristic Polynomial ◽  
Symbolic Computation ◽  
Inverse Kinematics ◽  
Structural Parameters ◽  
Efficient Computation ◽  
Joint Angles ◽  
Arbitrary Geometries ◽  
The Matrix ◽  
Rational Polynomials ◽  
Symbolic Constraints

Abstract We discuss the symbolic computation of inverse kinematics for serial 6R manipulators with arbitrary geometries (general 6R manipulators) based on Raghavan and Roth’s solution. The elements of the matrices required in the solution were symbolically calculated. In the symbolic computation, an algorithm for simplifying polynomials upon considering the symbolic constraints (constraints of the trigonometric functions and those of the rotation matrix), a method for symbolic elimination of the joint variables, and an efficient computation of the rational polynomials are presented. The elements of the matrix whose determinant produces a 16th-order single variable polynomial (characteristic polynomial) were symbolically calculated by using structural parameters (parameters that define the geometry of the manipulator) and hand configuration parameters (parameters that define the hand configuration). The symbolic determinant of the matrix consists of huge number of terms even when each element is replaced by a single symbol. Instead of expressing the coefficients in a characteristic polynomial by structural parameters and hand configuration parameters, we substituted appropriate rational numbers that strictly satisfy the constraints of the symbols for the elements of the matrix and calculated the determinant (numerical error free calculation). By numerically calculating the real roots of the rational characteristic polynomial and the joint angles for each root, we verified the formulation for the symbolic computation.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

A Model of the Human Spine: Forward and Inverse Kinematics

1992 ◽  
Author(s):  
Sunil Kumar Agrawal ◽  
Siyan Li ◽  
Glen Desmier
Keyword(s):  
Range Of Motion ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Kinematic Model ◽  
Joint Angles ◽  
Human Spine ◽  
Forward And Inverse Kinematics ◽  
Position And Orientation ◽  
Three Degrees Of Freedom ◽  
Adjacent Vertebra

Abstract The human spine is a sophisticated mechanism consisting of 24 vertebrae which are arranged in a series-chain between the pelvis and the skull. By careful articulation of these vertebrae, a human being achieves fine motion of the skull. The spine can be modeled as a series-chain with 24 rigid links, the vertebrae, where each vertebra has three degrees-of-freedom relative to an adjacent vertebra. From the studies in the literature, the vertebral geometry and the range of motion between adjacent vertebrae are well-known. The objectives of this paper are to present a kinematic model of the spine using the available data in the literature and an algorithm to compute the inter vertebral joint angles given the position and orientation of the skull. This algorithm is based on the observation that the backbone can be described analytically by a space curve which is used to find the joint solutions..

A Rapid Approach for Calculating the Damped Eigenvalues of a Gas Turbine on a Minicomputer: Theory

1984 ◽  
Vol 106 (2) ◽  
pp. 239-249 ◽  
Author(s):  
E. J. Gunter ◽  
R. R. Humphris ◽  
H. Springer
Keyword(s):  
Gas Turbine ◽  
Characteristic Polynomial ◽  
Normal Modes ◽  
Complex Matrix ◽  
Large Computer ◽  
Mainframe Computer ◽  
The Matrix ◽  
Rigid Body Modes ◽  
Transfer Method ◽  
Matrix Transfer Method

The calculation of the damped eigenvalues of a large multistation gas turbine by the complex matrix transfer procedure may encounter numerical difficulties, even on a large computer due to numerical round-off errors. In this paper, a procedure is presented in which the damped eigenvalues may be rapidly and accurately calculated on a minicomputer with accuracy which rivals that of a mainframe computer using the matrix transfer method. The method presented in this paper is based upon the use of constrained normal modes plus the rigid body modes in order to generate the characteristic polynomial of the system. The constrained undamped modes, using the matrix transfer process with scaling, may be very accurately calculated for a multistation turbine on a minicomputer. In this paper, a five station rotor is evaluated to demonstrate the procedure. A method is presented in which the characteristic polynomial may be automatically generated by Leverrier’s algorithm. The characteristic polynomial may be directly solved or the coefficients of the polynomial may be examined by the Routh criteria to determine stability. The method is accurate and easy to implement on a 16 bit minicomputer.

Mechanics of the Segmentation of an Embedded Fiber, Part I: Experimental Investigations

1995 ◽  
Vol 62 (1) ◽  
pp. 87-97 ◽  
Author(s):  
A. ten Busschen ◽  
A. P. S. Selvadurai
Keyword(s):  
Structural Parameters ◽  
Matrix Material ◽  
Micromechanical Modeling ◽  
Experimental Investigations ◽  
Shear Stresses ◽  
Research Attention ◽  
Fragmentation Test ◽  
The Matrix ◽  
Matrix Region ◽  
Interface Bond

Micromechanical modeling is an important aspect in the study of fiber-reinforced composites. In such studies, an important class of structural parameters is formed by the interaction between the matrix and the embedded fibers. These interactive processes can be investigated by an appeal to a test which involves the segmentation of an embedded fiber. This test is referred to as a “fragmentation test.” During a fragmentation test, two distinct fracture phenomena are observed. These phenomena are directly related to the integrity of bond between the embedded fiber and the matrix. The first phenomenon involves situations where the interface bond is weaker than the matrix material. In this case the fiber fragment ends will slip and in this region shear stresses are transmitted by friction and/or interlocking mechanical actions. In contrast, when the interface bond has stronger properties than the matrix material, cracking will occur in the matrix region. Here, a crack initiated in the fiber will propagate into the matrix region typically forming conoidal cracks, or combinations of conoidal and flat cracks. This paper describes the background of the fragmentation test and the associated experimental research. Attention is focused on the experimental evaluation of matrix fracture topographies encountered in the fragmentation test.

Efficient computation of one-electron two-centre exchange integrals in ion–atom collisions

1976 ◽  
Vol 54 (9) ◽  
pp. 944-949 ◽  
Author(s):  
Alfred Msezane
Keyword(s):  
Principal Quantum Number ◽  
Computing Time ◽  
Translational Motion ◽  
Matrix Elements ◽  
Efficient Computation ◽  
Close Coupling ◽  
One Dimensional ◽  
State Approximation ◽  
The Matrix ◽  
Exchange Matrix

A scheme is presented for the reduction to one-dimensional integrals of any one-electron two-centre exchange matrix elements which incorporate the momentum associated with the translational motion of the electron. These elements are of the types occurring in close coupling-based treatments of ion–atom collisions. It is shown in a six state approximation, by coupling both eigenstates and pseudostates for the asymmetric He2+–H collision process, that computing time for the evaluation of the matrix elements is determined mainly by the number of different exponents in the matrix elements. The coupling of additional states with the same principal quantum number as the already coupled ones alters computing time insignificantly.

Velocity and Force Observers for the Control of Robot Manipulators

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Marco A. Arteaga–Pérez ◽  
Juan C. Rivera–Dueñas ◽  
Alejandro Gutiérrez–Giles
Keyword(s):  
Dynamic Model ◽  
Tracking Control ◽  
Inverse Kinematics ◽  
Robot Manipulator ◽  
Robot Manipulators ◽  
Asymptotical Stability ◽  
Joint Angles ◽  
Force Tracking

In this paper, position/force tracking control for rigid robot manipulators interacting with its environment is considered. It is assumed that only joint angles are available for feedback, so that velocity and force observers are designed. The principle of orthogonalization is employed for this particular purpose and some of its main properties are fully exploited to guarantee local asymptotical stability. Only the force observer requires the dynamic model of the robot manipulator for implementation, and the scheme is developed directly in workspace coordinates, so that no inverse kinematics is required. The proposed approach is tested experimentally and compared with a well–known algorithm.

Joint angle variations analyses of the two link planar manipulator in welding by using inverse kinematics

Robotica ◽  
2005 ◽  
Vol 24 (3) ◽  
pp. 355-363 ◽  
Author(s):  
S. Bulut ◽  
M. B. Terzioǧlu
Keyword(s):  
Angular Velocity ◽  
Data Collection ◽  
Inverse Kinematics ◽  
Joint Angle ◽  
Joint Angles ◽  
End Effector ◽  
Fortran 90

In this paper, the joint angles of a two link planar manipulator are calculated by using inverse kinematics equations together with some geometric equalities. For a given position of the end-effector the joint angle and angular velocity of the links are derived. The analyses contains many equations which have to be solved. However, the solutions are rather cumbersome and complicated, therefore a program is written in Fortran 90 in order to do, the whole calculation and data collection. The results are given at the end of this paper.

scholarly journals Structural Parameters and Working Range Estimation of Excavator Backhoe Mechanism

2019 ◽  
Vol 8 (2S10) ◽  
pp. 867-872
Keyword(s):  
Structural Parameters ◽  
Kinematic Model ◽  
Computer Algorithm ◽  
Joint Angles ◽  
Range Estimation ◽  
Kinematical Model ◽  
Physical Prototype ◽  
Solid Foundation ◽  
Forward Kinematic ◽  
Earthmoving Machines

Earthmoving machines like excavators and loaders characteristics such as productivity, weight, reliability depend on their backhoe mechanism. For that, the backhoe mechanism has to deliver the desired working range, digging forces and stability which are dependent on structural parameters like components length and joint angles. This paper describes the method of developing a backhoe mechanism for the desired working range which constitutes cutting heights and reaches by using structural parameters. This requires to develop forward kinematical model by considering the backhoe mechanism as a mechanical manipulator. A computer algorithm was developed, that uses the forward kinematic model, to estimate the working range. Also, a relationship is established between joint angles and cylinder lengths. Results of Virtual prototype, modeled and simulated in MSC ADAMS along with the testing results of BEML designed Physical prototype were used to validate the working range and structural parameters. This research provides a solid foundation for analyzing the effect of structural parameters on digging forces and stability.

scholarly journals The importance of a consistent workflow to estimate muscle-tendon lengths based on joint angles from the conventional gait model

2021 ◽  
Author(s):  
Hans Kainz ◽  
Michael H Schwartz
Keyword(s):  
Cerebral Palsy ◽  
Inverse Kinematics ◽  
Clinical Decision Making ◽  
Joint Kinematics ◽  
Joint Angles ◽  
Modified Model ◽  
Model Driven ◽  
Tendon Length ◽  
Musculoskeletal Models ◽  
Children With Cerebral Palsy

AbstractBackgroundMusculoskeletal models enable us to estimate muscle-tendon length, which has been shown to improve clinical decision-making and outcomes in children with cerebral palsy. Most clinical gait analysis services, however, do not include muscle-tendon length estimation in their clinical routine. This is due, in part, to a lack of knowledge and trust in the musculoskeletal models, and to the complexity involved in the workflow to obtain the muscle-tendon length.Research questionCan the joint angles obtained with the conventional gait model (CGM) be used to generate accurate muscle-tendon length estimates?MethodsThree-dimensional motion capture data of 15 children with cerebral palsy and 15 typically developing children were retrospectively analyzed and used to estimate muscle-tendon length with the following four modelling frameworks: (1) 2392-OSM-IK-angles: standard OpenSim workflow including scaling, inverse kinematics and muscle analysis; (2) 2392-OSM-CGM-angle: generic 2392-OpenSim model driven with joint angles from the CGM; (3) modif-OSM-IK-angles: standard OpenSim workflow including inverse kinematics and a modified model with segment coordinate systems and joint degrees-of-freedom similar to the CGM; (4) modif-OSM-CGM-angles: modified model driven with joint angles from the CGM. Joint kinematics and muscle-tendon length were compared between the different modelling frameworks.ResultsLarge differences in hip joint kinematics were observed between the CGM and the 2392-OpenSim model. The modif-OSM showed similar kinematics as the CGM. Muscle-tendon length obtained with modif-OSM-IK-angles and modif-OSM-CGM-angles were similar, whereas large differences in some muscle-tendon length were observed between 2392-OSM-IK-angles and 2392-OSM-CGM-angles.SignificanceThe modif-OSM-CGM-angles framework enabled us to estimate muscle-tendon lengths without the need for scaling a musculoskeletal model and running inverse kinematics. Hence, muscle-tendon length estimates can be obtained simply, without the need for the complexity, knowledge and time required for musculoskeletal modeling and associated software. An instruction showing how the framework can be used in a clinical setting is provided on https://github.com/HansUniVie/MuscleLength.

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