Journal articles: 'Jacobian matrix'

1

Xu, Yong-Xian, D. Kohli, and Tzu-Chen Weng. "Direct Differential Kinematics of Hybrid-Chain Manipulators Including Singularity and Stability Analyses." Journal of Mechanical Design 116, no. 2 (June 1, 1994): 614–21. http://dx.doi.org/10.1115/1.2919422.

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A general formulation for the differential kinematics of hybrid-chain manipulators is developed based on transformation matrices. This formulation leads to velocity and acceleration analyses, as well as to the formation of Jacobians for singularity and unstable configuration analyses. A manipulator consisting of n nonsymmetrical subchains with an arbitrary arrangement of actuators in the subchain is called a hybrid-chain manipulator in this paper. The Jacobian of the manipulator (called here the system Jacobian) is a product of two matrices, namely the Jacobian of a leg and a matrix M containing the inverse of a matrix Dk, called the Jacobian of direct kinematics. The system Jacobian is singular when a leg Jacobian is singular; the resulting singularity is called the inverse kinematic singularity and it occurs at the boundary of inverse kinematic solutions. When the Dk matrix is singular, the M matrix and the system Jacobian do not exist. The singularity due to the singularity of the Dk matrix is the direct kinematic singularity and it provides positions where the manipulator as a whole loses at least one degree of freedom. Here the inputs to the manipulator become dependent on each other and are locked. While at these positions, the platform gains at least one degree of freedom, and becomes statically unstable. The system Jacobian may be used in the static force analysis. A stability index, defined in terms of the condition number of the Dk matrix, is proposed for evaluating the proximity of the configuration to the unstable configuration. Several illustrative numerical examples are presented.

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López, M., E. Castillo, G. García, and A. Bashir. "Delta robot: Inverse, direct, and intermediate Jacobians." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220, no. 1 (January 1, 2006): 103–9. http://dx.doi.org/10.1243/095440606x78263.

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In the context of a parallel manipulator, inverse and direct Jacobian matrices are known to contain information which helps us identify some of the singular configurations. In this article, we employ kinematic analysis for the Delta robot to derive the velocity of the end-effector in terms of the angular joint velocities, thus yielding the Jacobian matrices. Setting their determinants to zero, several undesirable postures of the manipulator have been extracted. The analysis of the inverse Jacobian matrix reveals that singularities are encountered when the limbs belonging to the same kinematic chain lie in a plane. Two of the possible configurations which correspond to this condition are when the robot is completely extended or contracted, indicating the boundaries of the workspace. Singularities associated with the direct Jacobian matrix, which correspond to relatively more complicated configurations of the manipulator, have also been derived and commented on. Moreover, the idea of intermediate Jacobian matrices have been introduced that are simpler to evaluate but still contain the information of the singularities mentioned earlier in addition to architectural singularities not contemplated in conventional Jacobians.

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Niitsuma, Hiroshi. "Jacobian matrix and p-basis." Banach Center Publications 26, no. 2 (1990): 185–88. http://dx.doi.org/10.4064/-26-2-185-188.

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Fang, Wang, Yang Zhen, Q. S. Kang, Shang Dong Xi, and Lin Yang Shang. "A Simulation Research on the Visual Servo Based on Pseudo-Inverse of Image Jacobian Matrix for Robot." Applied Mechanics and Materials 494-495 (February 2014): 1212–15. http://dx.doi.org/10.4028/www.scientific.net/amm.494-495.1212.

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The image Jacobian Matrix must obtain during the course of uncalibrated visual servo for classic algorithms firstly. Then the inverse of image Jacobian Matrix or pseudo-inverse of image Jacobian Matrix can be taken. But when the inverse of image Jacobian Matrix is not exist or pseudo-inverse of image Jacobian Matrix is not easy to get, the uncalibrated visual servo for robot can not realize. In this paper, a research is carried on by simulation between the classic method for uncalibrared visual servo and the strategy by computing pseudo-inverse of image Jacobian Matrix. It is conclusion that the latter not only has advantage of the performance for tracking, but also reduces computational complexity for control.

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LIU, Zhizhong. "Jacobian Matrix Normalization Based on Variable Weighting Matrix." Journal of Mechanical Engineering 50, no. 23 (2014): 29. http://dx.doi.org/10.3901/jme.2014.23.029.

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Kim, S. S., and M. J. Vanderploeg. "QR Decomposition for State Space Representation of Constrained Mechanical Dynamic Systems." Journal of Mechanisms, Transmissions, and Automation in Design 108, no. 2 (June 1, 1986): 183–88. http://dx.doi.org/10.1115/1.3260800.

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This paper presents a numerical solution method for dynamic analysis of constrained mechanical systems. This method reduces a coupled set of differential and algebraic equations to state space form. The reduction uses an independent set of velocities which lie on the tangent plane of the constraint surface. The tangent plane is defined by the nullspace of constraint Jacobian matrix. The nullspace basis is found using QR decomposition of the constraint Jacobian matrix. Because the nullspace basis is not unique, directional continuity of the nullspace is difficult to preserve each time the Jacobiar is decomposed. This paper presents an updating algorithm that is used instead oj repeated decomposition. This preserves directional continuity of the Jacobian matrix and increases efficiency. State equations are then derived in terms of independent accelerations and therefore can efficiently be integrated. Generalized velocities are integrated with constraints to obtain positions. This method has demonstrated minimal constraint violations and improved efficiency. Numerical examples with singular configurations and redundant constraints are presented to demonstrate the effectiveness of the method.

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Leung, A. Y. T., and T. Ge. "Toeplitz Jacobian Matrix for Nonlinear Periodic Vibration." Journal of Applied Mechanics 62, no. 3 (September 1, 1995): 709–17. http://dx.doi.org/10.1115/1.2897004.

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The main difference between a linear system and a nonlinear system is in the non-uniqueness of solutions manifested by the singular Jacobian matrix. It is important to be able to express the Jacobian accurately, completely, and efficiently in an algorithm to analyze a nonlinear system. For periodic response, the incremental harmonic balance (IHB) method is widely used. The existing IHB methods, however, requiring double summations to form the Jacobian matrix, are often extremely time-consuming when higher order harmonic terms are retained to fulfill the completeness requirement. A new algorithm to compute the Jacobian is to be introduced with the application of fast Fourier transforms (FFT) and Toeplitz formulation. The resulting Jacobian matrix is constructed explicitly by three vectors in terms of the current Fourier coefficients of response, depending respectively on the synchronizing mass, damping, and stiffness functions. The part of the Jacobian matrix depending on the nonlinear stiffness is actually a Toeplitz matrix. A Toeplitz matrix is a matrix whose k, r position depends only on their difference k-r. The other parts of the Jacobian matrix depending on the nonlinear mass and damping are Toeplitz matrices modified by diagonal matrices. If the synchronizing mass is normalized in the beginning, we need only two real vectors to construct the Toeplitz Jacobian matrix (TJM), which can be treated in one complex fast Fourier transforms. The present method of TJM is found to be superior in both computation time and storage than all existing IHB methods due to the simplified explicit analytical form and the use of FFT.

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van den Essen, Arno, and Engelbert Hubbers. "Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture." Linear Algebra and its Applications 247 (November 1996): 121–32. http://dx.doi.org/10.1016/0024-3795(95)00095-x.

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9

Reutenauer, Christophe. "Applications of a noncommutative jacobian matrix." Journal of Pure and Applied Algebra 77, no. 2 (February 1992): 169–81. http://dx.doi.org/10.1016/0022-4049(92)90083-r.

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Chèze, G., and S. Najib. "Indecomposability of polynomials via Jacobian matrix." Journal of Algebra 324, no. 1 (July 2010): 1–11. http://dx.doi.org/10.1016/j.jalgebra.2010.01.007.

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11

Lu, Guangda, Aimei Zhang, Jing Zhou, Shigang Cui, and Li Zhao. "The Statics Analysis and Verification of 3-DOF Parallel Mechanism Based on Two Methods." International Journal of Automation Technology 7, no. 2 (March 5, 2013): 237–44. http://dx.doi.org/10.20965/ijat.2013.p0237.

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Statics of the 3-RSS/S parallel ankle-rehabilitation robot is analyzed in this paper using two methods, i.e. the component vector method and the principle of virtual work. Static equilibrium equations based on component vector theory were established on a moving platform, and cranks of 3-RSS/S parallel Ankle-rehabilitation Robot, using this method, to obtain mathematical relationships between the external torque of moving platform and the output torque of three cranks. The velocity Jacobian matrix of the robot is calculated firstly using the principle of virtual work method, then the force Jacobian matrix is obtained based on the relationship between velocity Jacobian matrix and force Jacobian matrix. The results of the two methods are verified and found to be consistent by calculation, and the force Jacobian matrix of the robot is the basis of the force feedback control for the Ankle-rehabilitation Robot.

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Alwan, Hassan Mohammed, and Riyadh Ahmed Sarhan. "Design Blocks in Simulink to Detection Singularity in the Workspace of Gough-Stewart Robot Manipulator." Journal of University of Babylon for Engineering Sciences 27, no. 1 (January 28, 2019): 1–15. http://dx.doi.org/10.29196/jubes.v27i1.1967.

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This work deals with Gough-Stewart robot manipulator, which has six degrees of freedom, six actuators, fixed base, and moving platforms. Here, the Jacobian matrix derived to detect the singular point in the workspace for manipulator at determinant of Jacobian matrix equal to zero, then derived the equation of motion from the dynamic analysis by Lagrange method to verify the singular points with Jacobian where the forces increase rapidly at this point. Finally, design blocks in Simulink include the Jacobian matrix and the equations of motion to detection the singularities at any time for current input parameters (X, Y, Z, α, β, γ), where the determinant of the Jacobian equal to zero at maximum forces.

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Shi, Yan, Hong Xin Yue, Yi Lu, and Lian He Guo. "Singularity Analysis of a Plane-Symmetry 3-RPS Parallel Robot Based on Translational/Rotational Jacobian Matrices." Applied Mechanics and Materials 121-126 (October 2011): 1590–94. http://dx.doi.org/10.4028/www.scientific.net/amm.121-126.1590.

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Firstly, 3-DOF parallel robots were classified into different types from the view of moving form. A new method of analyzing the singularity of 3-DOF parallel robots was introduced, which is based on translational Jacobian matrix and rotational Jacobian matrix. The singularity of parallel robots with pure translational form and pure rotational form was introduced summarily. Secondly, the process of solving the plane-symmetry 3-RPS parallel robot with combined moving forms was focused on, through which translational Jacobian matrix and rotational Jacobian matrix were adopted. Finally, the solving results were compared with the axis-symmetry 3-RPS parallel robot, which showed more general singularity can be solved through the new method.

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Butuk, N., and J. P. Pemba. "Computing CHEMKIN Sensitivities Using Complex Variables." Journal of Engineering for Gas Turbines and Power 125, no. 3 (July 1, 2003): 854–58. http://dx.doi.org/10.1115/1.1469006.

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This paper discusses an accurate numerical approach based on complex variables for the computation of the Jacobian matrix of complex chemical reaction mechanisms. The Jacobian matrix is required in the calculation of low dimensional manifolds during kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large-scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a nonlinear reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size h.

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Lu, Yi, Yan Shi, and Jianping Yu. "Determination of singularities of some 4-DOF parallel manipulators by translational/rotational Jacobian matrices." Robotica 28, no. 6 (September 21, 2009): 811–19. http://dx.doi.org/10.1017/s0263574709990518.

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SUMMARYA novel analytic approach is proposed for determining the singularities of some four degree of freedom (DOF) parallel manipulators (PMs). First, the constraint and displacement of a general 4-DOF PM are analyzed. Second, a common 3 × 4 translational Jacobian matrix Jν and a common 3 × 4 rotational Jacobian matrix Jω are derived, and a 4 × 4 general Jacobian matrix J of the 4-DOF PMs is derived from Jν and Jω. Since a complicated process to determine singularities from the 4 × 6 Jacobian matrix is transformed into a simple process to determine singularity from J, the singularities of the some 4-DOF PMs with 3 translations and 1 rotation, or with 3 rotations and 1 translation, or with combined translation–rotations are analyzed and determined easily by this approach.

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Nurahmi, Latifah, and Stéphane Caro. "Dimensionally Homogeneous Jacobian and Condition Number." Applied Mechanics and Materials 836 (June 2016): 42–47. http://dx.doi.org/10.4028/www.scientific.net/amm.836.42.

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This paper deals with the formulation of the dimensionally homogeneous extended Jacobian matrix, which is an important issue for the performance analysis of f degrees-of-freedom (f ≤6) parallel manipulators having coupled rotational and translational motions. By using the f independent coordinates to define the permitted motions and (6-f) independent coordinates to define the restricted motions of the moving platform, the 6×6 dimensionally homogeneous extended Jacobian matrix is derived for non-redundant parallel manipulators. The conditioning number of the parallel manipulators is computed to evaluate the homogeneous extended Jacobian matrix, the homogeneous actuation wrench matrix, and the homogeneous constraint wrench matrix to evaluate the performance of the parallel manipulators. By using these indices, the closeness of a pose to different singularities can be detected. An illustrative example with the 3-RPS parallel manipulator is provided to highlight the effectiveness of the approach and the proposed indices.

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Wen, Ke Fei, and Jeh Won Lee. "Statics, Instantaneous Kinematics and Singularity Analysis of Planar Parallel Manipulators via Grassmann-Cayley Algebra." Applied Mechanics and Materials 532 (February 2014): 378–81. http://dx.doi.org/10.4028/www.scientific.net/amm.532.378.

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The wrench Jacobian matrix plays an important role in statics and singularity analysis of planar parallel manipulators (PPMs). It is easy to obtain this matrix based on plücker coordinate method. In this paper, a new approach is proposed to the analysis of the forward and inverse wrench Jacobian matrix used by Grassmann-Cayley algebra (GCA). A symbolic formula for the inverse statics and a coordinate free formula for the singularity analysis are obtained based on this Jacobian. As an example, this approach is implemented for the 3-RPR PPMs.

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Liu, Bing, Ying Wang, Zhen Yang, Yun Wei Li, Hua Zhi Xie, Yan Zhang, Wen Chao Zhang, Xiang Yang Deng, and Cheng Ming He. "Evaluation Method of Voltage Stability Based on Minimum Singular Value." Applied Mechanics and Materials 511-512 (February 2014): 1128–32. http://dx.doi.org/10.4028/www.scientific.net/amm.511-512.1128.

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When the system reaches the static voltage stability limit point, Jacobian matrix is singular. By checking if minimum singular value is zero, its very easy to determine whether or not Jacobian matrix is singular. So the minimum singular value of Jacobian matrix can reflects the degree of system voltage stability effectively. Firstly this paper introduces singular value decomposition, analyses load characteristic and excitation limits involved in this method and examples on PSD-FDS are demonstrated. At last, suggestion on voltage stability monitoring by minimum singular value is proposed.

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Duleba, Ignacy, and Jerzy Z. Sasiadek. "Modified Jacobian method of transversal passing through the smallest deficiency singularities for robot manipulators." Robotica 20, no. 4 (June 24, 2002): 405–15. http://dx.doi.org/10.1017/s0263574702004095.

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This paper proposes a method for transversal passing through singularities of corank 1, both for nonredundant and redundant robotic manipulators. The method modifies the Jacobian matrix of manipulator's forward kinematics to retrieve its full rank at singularities. Natural candidates for the Jacobian matrix modification are derivatives of determinants of full size sub-matrices of the Jacobian matrix. The method is illustrated with examples, including a PUMA manipulator and 2-link and 3-link planar manipulators. Some restrictions on the applicability of the method for nonredundant manipulators are also discussed.

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CAI, Q. D. "CONTINUOUS NEWTON METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATION." Modern Physics Letters B 24, no. 13 (May 30, 2010): 1303–6. http://dx.doi.org/10.1142/s0217984910023487.

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Newton method is a widely used iteration method in solving nonlinear algebraic equations. In this method, a linear algebraic equations need to be solved in every step. The coefficient matrix of the algebraic equations is so-called Jacobian matrix, which needs to be determined at every step. For a complex non-linear system, usually no explicit form of Jacobian matrix can be found. Several methods are introduced to obtain an approximated matrix, which are classified as Jacobian-free method. The finite difference method is used to approximate the derivatives in Jacobian matrix, and a small parameter is needed in this process. Some problems may arise because of the interaction of this parameter and round-off errors. In the present work, we show that this kind of Newton method may encounter difficulties in solving non-linear partial differential equation (PDE) on fine mesh. To avoid this problem, the continuous Newton method is presented, which is a modification of classical Newton method for non-linear PDE.

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Rivals, Isabelle, and Léon Personnaz. "Jacobian Conditioning Analysis for Model Validation." Neural Computation 16, no. 2 (February 1, 2004): 401–18. http://dx.doi.org/10.1162/089976604322742083.

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Our aim is to stress the importance of Jacobian matrix conditioning for model validation. We also comment on Monari and Dreyfus (2002), where, following Rivals and Personnaz (2000), it is proposed to discard neural candidates that are likely to overfit and/or for which quantities of interest such as confidence intervals cannot be computed accurately. In Rivals and Personnaz (2000), we argued that such models are to be discarded on the basis of the condition number of their Jacobian matrix. But Monari and Dreyfus (2002) suggest making the decision on the basis of the computed values of the leverages, the diagonal elements of the projection matrix on the range of the Jacobian, or “hat” matrix: they propose to discard a model if computed leverages are outside some theoretical bounds, pretending that it is the symptom of the Jacobian rank deficiency. We question this proposition because, theoretically, the hat matrix is defined whatever the rank of the Jacobian and because, in practice, the computed leverages of very ill-conditioned networks may respect their theoretical bounds while confidence intervals cannot be estimated accurately enough, two facts that have escaped Monari and Dreyfus's attention. We also recall the most accurate way to estimate the leverages and the properties of these estimations. Finally, we make an additional comment concerning the performance estimation in Monari and Dreyfus (2002).

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Yoshimura, H. "A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations." Mechanical Sciences 4, no. 1 (June 6, 2013): 243–50. http://dx.doi.org/10.5194/ms-4-243-2013.

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Abstract. In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

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Xu, Shu Ai, Shao Hui Cui, Yuan Zhou, Zhen Bin Tang, and Wen Jie Zhu. "The Application of Modified Covariance EKF Algorithm to Target-Tracking Modeling." Applied Mechanics and Materials 427-429 (September 2013): 953–56. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.953.

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In traditional EKF algorithm, biased prediction is used to calculate the jacobian matrix, which leads to get an inaccurate covariance matrix and influence the estimated performance. This paper applies Modified Covariance EKF to target-tracking modeling to solve the problem. In this algorithm, jacobian matrix is calculated again with state estimation. Through this way, measured values are used to modify the covariance matrix, which makes it more accurate. Consequently, estimated performance is improved.

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Dulęba, Ignacy, and Michał Opałka. "A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators." International Journal of Applied Mathematics and Computer Science 23, no. 2 (June 1, 2013): 373–82. http://dx.doi.org/10.2478/amcs-2013-0028.

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The objective of this paper is to present and make a comparative study of several inverse kinematics methods for serial manipulators, based on the Jacobian matrix. Besides the well-known Jacobian transpose and Jacobian pseudo-inverse methods, three others, borrowed from numerical analysis, are presented. Among them, two approximation methods avoid the explicit manipulability matrix inversion, while the third one is a slightly modified version of the Levenberg-Marquardt method (mLM). Their comparison is based on the evaluation of a short distance approaching the goal point and on their computational complexity. As the reference method, the Jacobian pseudo-inverse is utilized. Simulation results reveal that the modified Levenberg-Marquardt method is promising, while the first order approximation method is reliable and requires mild computational costs. Some hints are formulated concerning the application of Jacobian-based methods in practice.

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Li, Maokun, Aria Abubakar, Jianguo Liu, Guangdong Pan, and Tarek M. Habashy. "A compressed implicit Jacobian scheme for 3D electromagnetic data inversion." GEOPHYSICS 76, no. 3 (May 2011): F173—F183. http://dx.doi.org/10.1190/1.3569482.

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We developed a compressed implicit Jacobian scheme for the regularized Gauss-Newton inversion algorithm for reconstructing 3D conductivity distributions from electromagnetic data. In this algorithm, the Jacobian matrix, whose storage usually requires a large amount of memory, is decomposed in terms of electric fields excited by sources located and oriented identically to the physical sources and receivers. As a result, the memory usage for the Jacobian matrix reduces from O(NFNSNRNP) to O[NF(NS + NR)NP], where NF is the number of frequencies, NS is the number of sources, NR is the number of receivers, and NP is the number of conductivity cells to be inverted. When solving the Gauss-Newton linear system of equations using iterative solvers, the multiplication of the Jacobian matrix with a vector is converted to matrix-vector operations between the matrices of the electric fields and the vector. In order to mitigate the additional computational overhead of this scheme, these fields are further compressed using the adaptive cross approximation (ACA) method. The compressed implicit Jacobian scheme provides a good balance between memory usage and computational time and renders the Gauss-Newton algorithm more efficient. We demonstrated the benefits of this scheme using numerical examples including both synthetic and field data for both crosswell and controlled-source electromagnetic (CSEM) applications.

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Gou, B. "Jacobian Matrix-Based Observability Analysis for State Estimation." IEEE Transactions on Power Systems 21, no. 1 (February 2006): 348–56. http://dx.doi.org/10.1109/tpwrs.2005.860934.

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Slutsker, Ilya W., and Jon M. Scudder. "Network Observability Analysis Through Measurement Jacobian Matrix Reduction." IEEE Power Engineering Review PER-7, no. 5 (May 1987): 40–41. http://dx.doi.org/10.1109/mper.1987.5527245.

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Slutsker, Ilya W., and Jon M. Scudder. "Network Observability Analysis through Measurement Jacobian Matrix Reduction." IEEE Transactions on Power Systems 2, no. 2 (1987): 331–36. http://dx.doi.org/10.1109/tpwrs.1987.4335128.

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Gupta, K. C., and R. Ma. "Formulation of manipulator Jacobian using the velocity similarity principle." Robotica 8, no. 1 (January 1990): 81–84. http://dx.doi.org/10.1017/s0263574700007359.

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SUMMARYVelocity similarity principle V(θ., ṡ, ub, Qb) = DabV (θ., ṡ, ua, Qa)D ab–1 is presented and used to derive several useful forms of the Jacobian matrix for the manipulator from its basic kinematic equations in 4 X 4 matrix form. The zero reference position representation is used and, therefore, the base system is the only coordinate system utilized in the derivations. For manipulators with a spherical wrist, a modified form of the Jacobian is presented in which the Jacobian columns corresponding to the regional structure are completely decoupled from those corresponding to the wrist structure.

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Zhao, Heng Hua, and Chuan Qiang Wang. "Research on the Singularity and Stationarity of 3-TPT Parallel Machine Tool." Advanced Materials Research 383-390 (November 2011): 190–95. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.190.

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This paper takes a new-type 3-TPT 3-DOF parallel machine tool as the object of study, established the positive and negative solution of kinematic equations of the machine, and derived the Jacobian matrix and Jacobian inverse matrix of the machine tool. On this basis, the singularity and stationarity of the machine tool were researched by using Matlab and got the expression of the absolute value of Jacobian matrix determinant of the machine tool. The velocity distribution curves of telescopic driving rods of the machine tool were simulated, too. The results show that the machine tool has advantages such as good stationarity and no singularity and so on. The study provides certain theoretical basis for the structural optimum design and performance analysis of parallel machine tool.

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Lin, Gui Hua, Yan Jun Zhang, Tao Wang, and Yu Ying Wang. "State Estimation of Equivalent Current Measurement Transformation Based on Generalized Tellegen's Theorem." Advanced Materials Research 732-733 (August 2013): 941–47. http://dx.doi.org/10.4028/www.scientific.net/amr.732-733.941.

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One of the most important ways to enhance the speed of state estimation is to establish the constant matrix Jacobian. This essay puts forward the state estimation method of the equivalent current transformation based on the Generalized Tellegen’s Theorem. This estimation method establishes the constant Jacobian matrix without neglecting the secondary factor making use of the Generalized Tellegen’s Theorem, solves the numerical stability problem caused by the establishment of the constant Jacobian matrix in the current state estimation, and has the advantages of a relatively rapid computing rate and an unparalleled astringency. The method put forward in this essay has been verified through IEEE-30 Node System, and the efficiency of it has been fully proved by the example results.

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Liauh, C. T., and R. B. Roemer. "A Semilinear State and Parameter Estimation Algorithm for Inverse Hyperthermia Problems." Journal of Biomechanical Engineering 115, no. 3 (August 1, 1993): 257–61. http://dx.doi.org/10.1115/1.2895484.

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An improved state and parameter estimation algorithm has been developed which decreases the total computational time required to accurately reconstruct complete hyperthermia temperature fields. Within this improved iterative estimation algorithm, if the change in the unknown perfusion parameters is small a linear approximation scheme is implemented in which the old Jacobian matrix (the sensitivity matrix) is used, instead of recalculating the new Jacobian matrix for the next iteration. In the hyperthermia temperature estimation problem the relationship between the temperature and the blood perfusion based on the bioheat transfer equation is generally nonlinear. However, the temperature can be approximated as a linear function of the blood perfusion over a certain range thus allowing this improved approach to work. Results show that if the temperature is approximated as a linear (or quasi-linear) function of the blood perfusion, the linearizing approach considerably reduces the CPU time required to accurately reconstruct the temperature field. The limiting case of implementing this approach is to calculate the Jacobian matrix for each iteration, which is identical to the approach used in the original nonlinear algorithm. Critical values of determining whether or not there is a need to recalculate the new Jacobian matrix during the iterations are presented for several inverse hyperthermia temperature estimation problems.

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Dharssi, I., B. Candy, and P. Steinle. "Analysis of the linearised observation operator in a soil moisture and temperature analysis scheme." SOIL Discussions 2, no. 1 (June 1, 2015): 505–35. http://dx.doi.org/10.5194/soild-2-505-2015.

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Abstract. Several weather forecasting agencies have developed advanced land data assimilation systems that, in principle, can analyse any model land variable. Such systems can make use of a wide variety of observation types, such as screen level (2 m above the surface) observations and satellite based measurements of surface soil moisture and skin temperature. Indirect measurements can be used and information propagated from the surface into the deeper soil layers. A key component of the system is the calculation of the linearised observation operator matrix (Jacobian matrix) which describes the link between the observations and the land surface model variables. The elements of the Jacobian matrix (Jacobians) are estimated using finite difference by performing short model forecasts with perturbed initial conditions. The calculated Jacobians show that there can be strong coupling between the screen level and the soil. The coupling between the screen level and surface soil moisture is found to be due to a number of processes including bare soil evaporation, soil thermal conductivity as well as transpiration by plants. Therefore, there is significant coupling both during the day and at night. The coupling between the screen level and root-zone soil moisture is primarily through transpiration by plants. Therefore the coupling is only significant during the day and the vertical variation of the coupling is modulated by the vegetation root depths. The calculated Jacobians that link screen level temperature to model soil temperature are found to be largest for the topmost model soil layer and become very small for the lower soil layers. These values are largest during the night and generally positive in value. It is found that the Jacobians that link observations of surface soil moisture to model soil moisture are strongly affected by the soil hydraulic conductivity. Generally, for the Joint UK Land Environment Simulator (JULES) land surface model, the coupling between the surface and root zone soil moisture is weak. Finally, the Jacobians linking observations of skin temperature to model soil temperature and moisture are calculated. These Jacobians are found to have a similar spatial pattern to the Jacobians for observations of screen level temperature. Analysis is also performed of the sensitivity of the calculated Jacobians to the magnitude of the perturbations used.

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Nesser, Hannah, Daniel J. Jacob, Joannes D. Maasakkers, Tia R. Scarpelli, Melissa P. Sulprizio, Yuzhong Zhang, and Chris H. Rycroft. "Reduced-cost construction of Jacobian matrices for high-resolution inversions of satellite observations of atmospheric composition." Atmospheric Measurement Techniques 14, no. 8 (August 12, 2021): 5521–34. http://dx.doi.org/10.5194/amt-14-5521-2021.

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Abstract. Global high-resolution observations of atmospheric composition from satellites can greatly improve our understanding of surface emissions through inverse analyses. Variational inverse methods can optimize surface emissions at any resolution but do not readily quantify the error and information content of the posterior solution. The information content of satellite data may be much lower than its coverage would suggest because of failed retrievals, instrument noise, and error correlations that propagate through the inversion. Analytical solution of the inverse problem provides closed-form characterization of posterior error statistics and information content but requires the construction of the Jacobian matrix that relates emissions to atmospheric concentrations. Building the Jacobian matrix is computationally expensive at high resolution because it involves perturbing each emission element, typically individual grid cells, in the atmospheric transport model used as the forward model for the inversion. We propose and analyze two methods, reduced dimension and reduced rank, to construct the Jacobian matrix at greatly decreased computational cost while retaining information content. Both methods are two-step iterative procedures that begin from an initial native-resolution estimate of the Jacobian matrix constructed at no computational cost by assuming that atmospheric concentrations are most sensitive to local emissions. The reduced-dimension method uses this estimate to construct a Jacobian matrix on a multiscale grid that maintains a high resolution in areas with high information content and aggregates grid cells elsewhere. The reduced-rank method constructs the Jacobian matrix at native resolution by perturbing the leading patterns of information content given by the initial estimate. We demonstrate both methods in an analytical Bayesian inversion of Greenhouse Gases Observing Satellite (GOSAT) methane data with augmented information content over North America in July 2009. We show that both methods reproduce the results of the native-resolution inversion while achieving a factor of 4 improvement in computational performance. The reduced-dimension method produces an exact solution at a lower spatial resolution, while the reduced-rank method solves the inversion at native resolution in areas of high information content and defaults to the prior estimate elsewhere.

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Park, Jaeheung. "The relationship between controlled joint torque and end-effector force in underactuated robotic systems." Robotica 29, no. 4 (August 16, 2010): 581–84. http://dx.doi.org/10.1017/s0263574710000391.

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SUMMARYThe generalized Jacobian matrix was introduced for dealing with end-effector control in space robots. One of the applications of this Jacobian is to be used in Jacobian transpose control to generate joint torques given end-effector position error. It would be misleading, however, to consider the transpose of this Jacobian as a mapping from end-effector force/moment to controlled joint torques for underactuated systems or floating base robots. This paper explains why it does not represent the mapping and provides a simple example. Later, the correct mapping is provided using the dynamically consistent Jacobian inverse and then a method to compute the actuated-joint torques is explained given the desired end-effector force. Finally, the effect of using the generalized Jacobian in the Jacobian transpose control is analyzed.

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Lin, Gui Hua, Tao Wang, Yu Ying Wang, and Li Guo Zheng. "Mixed Measurement State Estimation of the Equivalent Current Transformation Based on Generalized Tellegen's Theorem." Applied Mechanics and Materials 373-375 (August 2013): 970–75. http://dx.doi.org/10.4028/www.scientific.net/amm.373-375.970.

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One of the most important ways to enhance the speed of state estimation is to establish the constant matrix Jacobian. This essay puts forward the state estimation method of the equivalent current transformation based on the Generalized Tellegens Theorem. This estimation method establishes the constant Jacobian matrix without neglecting the secondary factor making use of the Generalized Tellegens Theorem, solves the numerical stability problem caused by the establishment of the constant Jacobian matrix in the current state estimation, and has the advantages of a relatively rapid computing rate, making advantage of measuremnt of WAMS and SCADA system, and an unparalleled astringency. The method put forward in this essay has been verified through IEEE-30 Node System, and the efficiency of it has been fully proved by the example results.

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Xia, Youshen, and Gang Feng. "On Convergence Conditions of an Extended Projection Neural Network." Neural Computation 17, no. 3 (March 1, 2005): 515–25. http://dx.doi.org/10.1162/0899766053019926.

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The output trajectory convergence of an extended projection neural network was developed under the positive definiteness condition of the Jacobian matrix of nonlinear mapping. This note offers several new convergence results. The state trajectory convergence and the output trajectory convergence of the extended projection neural network are obtained under the positive semidefiniteness condition of the Jacobian matrix. Comparison and illustrative examples demonstrate applied significance of these new results.

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Usman Moyi, Aliyu. "Chord Newton’s Method for Solving Fuzzy Nonlinear Equations." International Journal of Advanced Mathematical Sciences 7, no. 1 (December 11, 2019): 16. http://dx.doi.org/10.14419/ijams.v7i1.30098.

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In this paper, we present a new approach for solving fuzzy nonlinear equations. Our approach requires to compute the Jacobian matrix once throughout the iterations unlike some Newton’s-like methods which needs to compute the Jacobian matrix in every iterations. The fuzzy coefficients are presented in parametric form. Numerical results on well-known benchmarks fuzzy nonlinear equations are reported to authenticate the effectiveness and efficiency of the approach.

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Hosseini, M. A., and H. M. Daniali. "Cartesian workspace optimization of Tricept parallel manipulator with machining application." Robotica 33, no. 9 (May 14, 2014): 1948–57. http://dx.doi.org/10.1017/s0263574714000861.

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SUMMARYIn this research work, the maximum Cartesian workspace of a Tricept parallel robot with two rotational and one translational degrees of freedom was investigated. Generally, the Cartesian workspace identifies the maximum size of a work-piece, specifying its cubicx,yandzdimensions, on which the milling machine could perform operations. However, the workspace of a robot can be considered in its task space, such as ψ × θ ×zfor the Tricept Parallel Kinematic Mechanism (PKM). A novel homogeneous Jacobian matrix which transforms joint space velocity vector into end-effector Cartesian velocity vector has been generated named as a Cartesian Jacobian matrix. Using the indices derived from the homogeneous Cartesian Jacobian matrix, i.e. the maximum singular values and local conditioning indices, the manipulator is designed to reach the Cartesian workspace with rapid positioning rates as well as with singularity avoidance.

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Simaan, N., and M. Shoham. "Geometric Interpretation of the Derivatives of Parallel Robots’ Jacobian Matrix With Application to Stiffness Control." Journal of Mechanical Design 125, no. 1 (March 1, 2003): 33–42. http://dx.doi.org/10.1115/1.1539514.

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This paper presents a closed-form formulation and geometrical interpretation of the derivatives of the Jacobian matrix of fully parallel robots with respect to the moving platforms’ position/orientation variables. Similar to the Jacobian matrix, these derivatives are proven to be also groups of lines that together with the lines of the instantaneous direct kinematics matrix govern the singularities of the active stiffness control. This geometric interpretation is utilized in an example of a planar 3 degrees-of-freedom redundant robot to determine its active stiffness control singularity.

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Hosseinzadegan, Samar, Andreas Fhager, Mikael Persson, Shireen Geimer, and Paul Meaney. "Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix." Sensors 21, no. 3 (January 22, 2021): 729. http://dx.doi.org/10.3390/s21030729.

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This paper focuses on the construction of the Jacobian matrix required in tomographic reconstruction algorithms. In microwave tomography, computing the forward solutions during the iterative reconstruction process impacts the accuracy and computational efficiency. Towards this end, we have applied the discrete dipole approximation for the forward solutions with significant time savings. However, while we have discovered that the imaging problem configuration can dramatically impact the computation time required for the forward solver, it can be equally beneficial in constructing the Jacobian matrix calculated in iterative image reconstruction algorithms. Key to this implementation, we propose to use the same simulation grid for both the forward and imaging domain discretizations for the discrete dipole approximation solutions and report in detail the theoretical aspects for this localization. In this way, the computational cost of the nodal adjoint method decreases by several orders of magnitude. Our investigations show that this expansion is a significant enhancement compared to previous implementations and results in a rapid calculation of the Jacobian matrix with a high level of accuracy. The discrete dipole approximation and the newly efficient Jacobian matrices are effectively implemented to produce quantitative images of the simplified breast phantom from the microwave imaging system.

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Ji, Zhiming, and Ming C. Leu. "Mapping of Kinematic and Dynamic Parameters for Coupled Manipulators." Journal of Mechanical Design 115, no. 2 (June 1, 1993): 283–88. http://dx.doi.org/10.1115/1.2919189.

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Industrial manipulators use various types of transmissions, such as gears, belts, chains, and parallel mechanisms, to transmit driving power to the links. By arrangement many such transmissions cause coupled joint motions. Manipulators having coupled joint motions are referred to as coupled manipulators. Conventional methods for constructing Jacobian matrix and compliance matrix are not directly applicable to coupled manipulators. The concept of the mapping matrix is used in this paper for establishing relationships of torque, speed, Jacobian and compliance of these manipulators with those obtained with conventional methods. A method of constructing the mapping matrix systematically is discussed. Two examples show that the proposed method is easy to implement.

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Yan, Yu, and Gong Ping Wu. "The Jacobian Matrix Structure of the Barrier Arm of Inspection Robot." Advanced Materials Research 706-708 (June 2013): 1183–86. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.1183.

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The Jacobian matrix represents the relationship of the linear mapping between the space velocity of robot operation and joint space velocity, which is the important link in the process of robot control. According to analyzing the gesture character of the inspection robot when it gets over the obstacles. This paper sets up the dynamitic model of robot by utilizing D-H. Based on that, it builds the Jacobian matrix by adopting differential transformation method, which establishes the foundation of the robots motion planning and real-time control.

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Kosmopoulos, D. I. "Robust Jacobian matrix estimation for image-based visual servoing." Robotics and Computer-Integrated Manufacturing 27, no. 1 (February 2011): 82–87. http://dx.doi.org/10.1016/j.rcim.2010.06.013.

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Campbell, L. Andrew. "Characteristic values of the Jacobian matrix and global invertibility." Annales Polonici Mathematici 76, no. 1-2 (2001): 11–20. http://dx.doi.org/10.4064/ap76-1-2.

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Waldron, K. J., Shih-Liang Wang, and S. J. Bolin. "A Study of the Jacobian Matrix of Serial Manipulators." Journal of Mechanisms, Transmissions, and Automation in Design 107, no. 2 (June 1, 1985): 230–37. http://dx.doi.org/10.1115/1.3258714.

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Inversion of the Jacobian matrix is the critical step in rate decomposition which is used to solve the so-called “inverse kinematics” problem of robotics. This is the problem of achieving a coordinated motion relative to the fixed reference frame. In this paper a general methodology is presented for formulation and manipulation of the Jacobian matrix. The formulation is closely tied to the geometry of the system and lends itself to simplification using appropriate coordinate transformations. This is of great importance since it gives a systematic approach to the derivation of efficient, analytical inverses. The method is also applied to the examination of geometrically singular positions. Several important general results relating to the structure of the singularity field are deducible from the structure of the algebraic system.

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Chen, Yu Christine, Jianhui Wang, Alejandro D. Dominguez-Garcia, and Peter W. Sauer. "Measurement-Based Estimation of the Power Flow Jacobian Matrix." IEEE Transactions on Smart Grid 7, no. 5 (September 2016): 2507–15. http://dx.doi.org/10.1109/tsg.2015.2502484.

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Parthasarathy, T., and G. Ravindran. "The Jacobian Matrix, Global Univalence and Completely Mixed Games." Mathematics of Operations Research 11, no. 4 (November 1986): 663–71. http://dx.doi.org/10.1287/moor.11.4.663.

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Hossain, A. K. M. Shahadat, and Trond Steihaug. "Computing a sparse Jacobian matrix by rows and columns." Optimization Methods and Software 10, no. 1 (January 1998): 33–48. http://dx.doi.org/10.1080/10556789808805700.

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Lee, Hung-I., Chen-Yo Han, and James Chien-Mo Li. "A Multicircuit Simulator Based on Inverse Jacobian Matrix Reuse." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 35, no. 7 (July 2016): 1130–37. http://dx.doi.org/10.1109/tcad.2015.2501308.

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Journal articles on the topic 'Jacobian matrix'

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