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Position error e ¼ rðqÞ À r d ðtÞ for the case study examples A (top row) and D (bottom row). Methods VF and AF are shown in the left and the middle column with the almost
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For both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That...
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Context 1
... parameter values were a 0 ¼ 1 m, a 1 ¼ 1 m, x ¼ 1 rad=s and c ¼ 0:2 m. The simulations were performed with an initial error as it is shown in the top panels of Fig 5. The gain parameters and the time-step were j ¼ 19 1=s, j P ¼ 250 1=s 2 , j D ¼ 8 1=s and h ¼ 0:05 s respectively for the simulations that are carried out in the inner vicinity of the stability border shown in Fig. 4 (close to the marginal stability). A parameter set deep inside the stable region was j ¼ 5 1=s and j P ¼ 50 1=s without changing any other parameter. ...
Context 2
... other parameter is the same as in the nonredundant case example A. The initial conditions were q 0 ¼ ½1; 1 | and _ q 0 ¼ 0. The simulation results are shown in the bottom panels of Fig 5. As we can see in the graphs for VF, and AF methods, the almost unstable parameter sets provide stability; however the error decays very slowly with high oscillations. The convergence speeds up when choosing the parameters from deep inside the stable range. ...
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Citations
... In order to update the list of references provided in [3] with the most recent articles taking advantage of the Moore-Penrose inverse, let us mention that: in [16] the inverse of a stiffness matrix was used to solve the characteristic finite-element equations in considerations dealing with modeling of elastic deformation and non-local brittle fracture of solids, in [17] the inverse of a correlation matrix was used in a study of correlated pulsar-pairs, in [18] the inverse of operator matrices in a Hilbert space framework was used to derive solutions of constrained linear equations using variational principles in considerations of a class of problems (called Z-problems) that arise in effective media theory, in [19] the inverse was used to derive numerical solutions to particular nonlinear ordinary differential equations and differential algebraic equations involved in neural networks analysis, in [20] the inverse of a key matrix was used to derive a novel image encryption algorithm, in [21] the inverse was used to construct an image fusion algorithm based on a matrix factorization, which aims at optimizing computational workload, in [22] the inverse was used to solve a set of linear equations resulting from a novel variant of the multiconfiguration perturbation theory proposed in order to ensure its pivot invariance, and in [23] the inverse was used to specify new mode time coefficient (equivalent to the best-fit solution in the least squares sense) for dynamic mode decomposition utilized to extract dominant modes of unsteady flow fields. Additionally, in [24] the Moore-Penrose inverse was used in comparison of alternative models of velocity and acceleration level inverse kinematic for redundant manipulators, with the inverse of a Jacobian matrix involved in each model; orthogonal projectors onto the fundamental subspaces associated with the Jacobian matrix were utilized in the analysis as well. Redundant manipulators were also referred to in [25], where two algorithms for a dynamic Moore-Penrose inversion were introduced and implemented. ...
The paper aims to play an expository role, providing a tailored introduction to the theory of matrix generalized inverses determined by the so-called Penrose equations, with the Moore–Penrose inverse as the jewel in the crown. The tailoring is made taking into account applicability of the inverses to solvability of a system of linear equations, which covers, inter alia, the least squares method. The method is introduced in a formal, though compendious way, with an intention to support researchers who want to consciously utilize it in their investigations. Additionally, the paper points out various links between the generalized inverses and theory of projectors, indicating issues which are relevant from the perspective of physics. The article can be viewed as a sequel of [O.M. Baksalary and G. Trenkler, ‘The Moore–Penrose inverse—a hundred years on a frontline of physics research,’ Eur. Phys. J. H 46, 9 (2021)], the paper prepared to celebrate the 100th anniversary of the first definition of the Moore–Penrose inverse, which shades a spotlight on the role the inverse plays in physics.
Specific studies on Kinematics date back to the XVIII Century. Hence, it is rather complicate to provide any generally agreed framework where a pertinent paper can be classified. This task has been herein bravely attempted for the papers published in Meccanica. Particularly, six kinematic-related categories have been identified and 84 papers have been classified accordingly. A further division was made between two equally numerous groups of contributions from either Italian or Foreign Authors.KeywordsKinematicsMechanisms topologyCams and transmissionsMechanism design and analysisCompliant mechanisms
