Analytical and Graphical Solutions to the Forward Position Problem of Two Robots Manipulating a Spatial Linkage Payload

1997 ◽  
Vol 119 (3) ◽  
pp. 349-358
Author(s):  
G. R. Pennock ◽  
K. G. Mattson
Keyword(s):  
Closed Form ◽  
Orthogonal Transformation ◽  
Solution Technique ◽  
Closed Form Solutions ◽  
Transformation Matrices ◽  
Kinematic Geometry ◽  
Order Polynomial ◽  
Position Problem ◽  
Four Bar Linkage ◽  
Insight Into

This paper presents a solution to the forward position problem of two PUMA-type robots manipulating a spatial four-bar linkage payload. To simplify the kinematic analysis, the Bennett linkage, which is a special geometry spatial four-bar, will be regarded as the payload. The orientation of a specified payload link is described by a sixth-order polynomial and a specified joint displacement in the wrist subassembly of one of the robots is described by a second-order polynomial. A solution technique, based on orthogonal transformation matrices with dual number elements, is used to obtain closed-form solutions for the remaining unknown joint displacements in the wrist subassembly of each robot. An important result is that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. Repeated roots of the polynomials are shown to correspond to the stationary configurations of the system. The paper emphasizes that an understanding of the kinematic geometry of the system is essential to verify the number of possible solutions to the forward position problem. Graphical methods are also presented to provide insight into the assembly and stationary configurations. A numerical example of the two robots manipulating the Bennett linkage is included to demonstrate the importance of the polynomial and closed-form solutions.

The Forward Position Problem of Two Puma-Type Robots Manipulating a Bennett Linkage Payload

1996 ◽  
Author(s):  
Gordon R. Pennock ◽  
Keith G. Mattson
Keyword(s):  
Closed Form ◽  
Orthogonal Transformation ◽  
Solution Technique ◽  
Closed Form Solutions ◽  
Transformation Matrices ◽  
Kinematic Geometry ◽  
Order Polynomial ◽  
Position Solution ◽  
Position Problem ◽  
Insight Into

Abstract This paper presents a solution to the forward position problem of two PUMA-type robots manipulating a Bennett linkage payload. The orientation of a specified payload link is described by a sixth-order polynomial and a specified angular joint displacement in the wrist subassembly of one of the robots is described by a second-order polynomial. A solution technique, based on orthogonal transformation matrices with dual number elements, is used to obtain closed-form solutions for the remaining unknown angular joint displacements in the wrist subassembly of each robot. The paper shows that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. The polynomials provide insight into these configurations, and also reveal stationary configurations of the system. The paper emphasizes that insight into the kinematic geometry of the system is essential in developing the forward position solution. Graphical methods are presented which provide insight into the geometry, and a check of the analytical approach. For illustrative purposes, a numerical example of the two robots manipulating a Bennett linkage is included in this paper to demonstrate the importance of the polynomials and the closed-form solutions.

Symmetry transformation matrices for structures

2007 ◽  
Vol 463 (2082) ◽  
pp. 1563-1583 ◽  
Author(s):  
Markus Pagitz ◽  
Jonathan James
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Symmetry Transformation ◽  
Symmetry Groups ◽  
Vector Spherical Harmonics ◽  
Alternative Derivation ◽  
Closed Form Solutions ◽  
Transformation Matrices ◽  
Displacement Vectors ◽  
Stiffness Matrices

Many structures in nature and engineering are symmetric. Depending on the degree of symmetry, it is possible to simplify the computations considerably by block diagonalizing the stiffness matrices. Closed-form solutions of transformation matrices for such block diagonalizations can be derived using group theory for arbitrary symmetry groups. This paper presents closed-form solutions of transformation matrices based on an alternative derivation. It is shown that transformation matrices for C nv and D nh groups can be obtained from a finite Fourier series decomposition of load and displacement vectors. Furthermore, it is shown that structures with tetrahedral, octahedral and icosahedral symmetries can be block diagonalized in an elegant way using vector spherical harmonics.

Assessment of Aided-INS Performance

2011 ◽  
Vol 65 (1) ◽  
pp. 169-185 ◽  
Author(s):  
Itzik Klein ◽  
Sagi Filin ◽  
Tomer Toledo ◽  
Ilan Rusnak
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Navigation Systems ◽  
Inertial Navigation Systems ◽  
Closed Form Solutions ◽  
Land Vehicles ◽  
Error Covariance ◽  
Model Dependency ◽  
Error State ◽  
Insight Into

Aided Inertial Navigation Systems (INS) systems are commonly implemented in land vehicles for a variety of applications. Several methods have been reported in the literature for evaluating aided INS performance. Yet, the INS error-state-model dependency on time and trajectory implies that no closed-form solutions exist for such evaluation. In this paper, we derive analytical solutions to evaluate the fusion performance. We show that the derived analytical solutions manage to predict the error covariance behavior of the full aided INS error model. These solutions bring insight into the effect of the various parameters involved in the fusion of the INS and an aiding sensor.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

scholarly journals A novel-iterative simulation method for performance analysis of non-coherent FSK/ASK systems over Rice/Rayleigh channels using the wolfram language

2016 ◽  
Vol 13 (2) ◽  
pp. 157-174 ◽  
Author(s):  
Vladimir Mladenovic ◽  
Danijela Milosevic
Keyword(s):  
Closed Form ◽  
Communication Systems ◽  
Rayleigh Fading ◽  
Simulation Method ◽  
Rician Fading ◽  
New Approach ◽  
Closed Form Solutions ◽  
Numerical Tools ◽  
The Impact ◽  
Insight Into

In this paper, a new approach in solving and analysing the performances of the digital telecommunication non-coherent FSK/ASK system in the presence of noise is derived, by using a computer algebra system. So far, most previous solutions cannot be obtained in closed form, which can be a problem for detailed analysis of complex communication systems. In this case, there is no insight into the influence of certain parameters on the performance of the system. The analysis, modelling and design can be time-consuming. One of the main reasons is that these solutions are obtained by utilising traditional numerical tools in the shape of closed-form expressions. Our results were obtained in closed-form solutions. They are resolved by the introduction of an iteration-based simulation method. The Wolfram language is used for describing applied symbolic tools, and SchematicSolver application package has been used for designing. In a new way, the probability density function and the impact of the newly introduced parameter of iteration are performed when errors are calculated. Analyses of the new method are applied to several scenarios: without fading, in the presence of Rayleigh fading, Rician fading, and in cases when the signals are correlated and uncorrelated.

Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. G1-G13 ◽  
Author(s):  
Zhengyong Ren ◽  
Yiyuan Zhong ◽  
Chaojian Chen ◽  
Jingtian Tang ◽  
Kejia Pan
Keyword(s):  
Closed Form ◽  
Gaussian Quadrature ◽  
Analytical Formula ◽  
Gravity Anomalies ◽  
Surface Integral ◽  
Third Order ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Relative Errors ◽  
Analytical Expressions

A new singularity-free analytical formula has been developed for the gravity field of arbitrary 3D polyhedral mass bodies with horizontally and vertically varying density contrast using third-order polynomial functions. First, the observation sites are moved to the origin of the coordinate system. Then, the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies. Furthermore, singularity-free closed-form solutions are derived for these line integrals over the edges. Thus, the observation sites can be located inside, on, or outside the 3D distributions. A synthetic prismatic mass body is adopted to verify the accuracy and singularity-free property of our newly developed analytical expressions. Excellent agreements are obtained between our solutions and other published closed-form solutions with relative errors in the order of [Formula: see text] to [Formula: see text]. In addition, an octahedral model and a near-Earth asteroid model are used to verify the accuracy of the presented method for complicated target structures by comparing the results with those from a high-order Gaussian quadrature approach.

Forward Position Analysis of Two 3-R Robots Manipulating a Planar Linkage Payload

1995 ◽  
Vol 117 (2A) ◽  
pp. 292-297 ◽  
Author(s):  
G. R. Pennock ◽  
K. G. Mattson
Keyword(s):  
Closed Loop ◽  
Degree Of Freedom ◽  
Sixth Order ◽  
Position Analysis ◽  
Revolute Joints ◽  
Order Polynomial ◽  
Cooperating Robots ◽  
Position Problem ◽  
Three Degree Of Freedom ◽  
Insight Into

This paper presents the forward position analysis of two planar three degree-of-freedom robots, with all revolute joints, manipulating a single degree-of-freedom closed-loop linkage payload. Kinematic constraint relations are developed which provide geometric insight into the cooperating robot-payload system and are important in the control of the two robots. For illustrative purposes, the payload that is considered here is a planar four-bar linkage. The paper shows that the orientation of a specified link in the payload can be described by a sixth-order polynomial. This polynomial is an important contribution, not only to the kinematics of the cooperating robots, but to the multiple-input closed-loop nine-bar linkage formed by the two robots and the payload. The polynomial contains important information regarding the assembly configurations and the stationary configurations of the system. The paper shows that zero, two, four, or six assembly configurations are possible and that each configuration corresponds to a different circuit of the system. Graphical methods are utilized to provide geometric insight into the assembly and stationary configurations and to check the results obtained from the sixth-order polynomial. A numerical example is included which demonstrates the importance of the polynomial in solving the forward position problem, and in determining the number of assembly configurations.

Generalized Reference Basis Model Updating in Structural Dynamics

1997 ◽  
Author(s):  
Rachel Kenigsbuch ◽  
Yoram Halevi
Keyword(s):  
Experimental Data ◽  
Closed Form ◽  
Prior Knowledge ◽  
Structural Dynamics ◽  
Optimization Problem ◽  
Model Updating ◽  
Optimization Criterion ◽  
Constrained Optimization Problem ◽  
Closed Form Solutions ◽  
Insight Into

Abstract The paper considers the problem of updating an analytical model from experimental data. The approach that is taken is the reference basis, where some of the parameters are considered to be completely accurate while the others are updated by solving a constrained optimization problem. The main results of this paper are closed form solutions to these problems with general weighting matrices in the optimization criterion. These are generalizations of several model reference updating problems that were solved and reported in the literature. The importance of this generalization is the ability to incorporate prior knowledge regarding the accuracy of the model in specified areas into the method. Another aspect of this work is the investigation of geometrical interpretation of the results which provides insight into the mechanism of the updating process. The advantages of the new updating schemes are demonstrated by means of examples.

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