An Experimental Study of Resolved Acceleration Control of Robots at Singularities: Damped Least-Squares Approach

1997 ◽  
Vol 119 (1) ◽  
pp. 97-101 ◽  
Author(s):  
M. Kirc´anski ◽  
N. Kirc´anski ◽  
D. Lekovic´ ◽  
M. Vukobratovic´
Keyword(s):  
Least Squares ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Control Level ◽  
Singular Values ◽  
Control Scheme ◽  
Resolved Acceleration Control ◽  
Robot Task ◽  
Damped Least Squares ◽  
Value Decomposition

Most of the robot task space control methods based on inverse Jacobian matrix suffer from instability in singular regions of workspace. Methods based on damped least-squares algorithm (DLS) for matrix inversion have been developed but not experimentally confirmed. The application of DLS method at the kinematic control level has been reported in (Chiaverini et al., 1994). In this article, a modified DLS method combined with the resolved-acceleration control scheme, is experimentally verified on two degrees of freedom of a PUMA-560 robot. In order to decrease the position error introduced by the damping, only small singular values are damped, in contrast to the conventional damping method were all the singular values are damped. The symbolic expressions of the singular value decomposition of the Jacobian matrix were used, to decrease the computational burden.

A Task Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

2013 ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Singular Values ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Motion Approximation ◽  
Value Decomposition ◽  
Four Bar Linkage ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of planar four-bar motion approximation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the Image Space of planar displacements, we obtain a class of quadrics, called Generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using Singular Value Decomposition. The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

scholarly journals A comparison of three-dimensional rotation formalisms for least-squares and Bayesian inverse kinematics

2021 ◽  
Author(s):  
Ben Serrien ◽  
Klevis Aliaj ◽  
Todd Pataky
Keyword(s):  
Least Squares ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Estimation Error ◽  
Computational Cost ◽  
Three Dimensional ◽  
Linear Least Squares ◽  
Computationally Efficient ◽  
Rotational Degrees Of Freedom ◽  
Value Decomposition

Marker-based inverse kinematics (IK) is prone to errors arising from measurementnoise and soft-tissue artefacts. Various least-squares and Bayesian methods canbe applied to limit the estimation error to a minimum. Recently proposed meth-ods like Bayesian IK come at an increased computational cost however. In thistechnical paper, we present an overview of eight different least squares or BayesianIK methods, including their accuracy and computational load for IK problemsinvolving a single rigid body and three rotational degrees-of-freedom, whose at-titude is estimated from four noisy marker positions. The results indicate thatNon-Linear Least Squares, Variational Bayesian and full Bayesian IK are supe-rior to Singular Value Decomposition in terms of accuracy, with approximatelya two-fold error reduction. However, only Non-Linear Least Squares and Varia-tional Bayesian IK are computationally efficient enough to scale towards practicaluse in biomechanical applications, with computational durations of 1-10 ms; fullyBayesian procedures required approximately 30 s for single rotation calculations.All Python code and supplementary material can be found in this paper’s GitHubrepository: https://github.com/benserrien/pybik.

Reciprocity-Based Singular Value Decomposition for Inverse Kinematic Analysis of the Metamorphic Multifingered Hand

2012 ◽  
Vol 4 (3) ◽  
Author(s):  
Lei Cui ◽  
Jian S. Dai
Keyword(s):  
Singular Value Decomposition ◽  
Degrees Of Freedom ◽  
Singular Values ◽  
Analytical Form ◽  
Inverse Kinematic ◽  
Singular Value ◽  
Robotic Hand ◽  
Value Analysis ◽  
Constraint Equations ◽  
Value Decomposition

With a new type of multifingered hands that raise a new philosophy in the construction and study of a multifingered hand, this paper is a follow-on study of the kinematics of the metamorphic multifingered hand based on finger constraint equations. The finger constraint equations lead to a comprehensive mathematical model of the hand with a reconfigurable palm which integrates all finger motions with the additional palm motion. Singular values of the partitioned Jacobian matrix in their analytical form are derived and applied to obtaining analytical solution to inverse kinematics of a complete robotic hand. The paper for the first time solves this integrated motion and the multifingered hand model with the singular value decomposition and extra degrees of freedom are examined with the singular value analysis to avoid the singularities. The work identifies finger displacement and velocity with effect from the articulated palm and presents a new way of analyzing a multifingered robotic hand.

Optimization of Fixture Layout Based on Error Amplification Factors

2018 ◽  
Vol 18 (4) ◽  
Author(s):  
Xiao-Jin Wan ◽  
Junqiang Yang ◽  
Hanjie Zhang ◽  
Zhi-Yong Feng ◽  
Zhigang Xu
Keyword(s):  
Degrees Of Freedom ◽  
Cutting Tool ◽  
Amplification Factor ◽  
Singular Values ◽  
Singular Value ◽  
Machining Accuracy ◽  
Nsga Ii ◽  
Amplification Factors ◽  
Error Amplification ◽  
Value Decomposition

Fixture locators are used to precisely locate and stably support a workpiece so that the desired position and orientation (pose) of the workpiece relative to the cutting tool can be maintained during machining or inspection process. It is believed that manufacturing errors of locators and locating datum surfaces are key factors for the pose error between the workpiece and the cutting tool. Optimizing the layout of locators is helpful to reduce the pose error so as to improve machining accuracy of the workpiece. In order to minimize the pose error, we introduced, for the first time, a singular value decomposition (SVD) technique for the location matrix to derive error amplification factors (EAF) in six degrees-of-freedom of the workpiece. The EAF principle defines the maximal singular value, the condition number, the product of all singular values and the manipulability as the maximal error amplification factor, the relative error amplification factor, the error ellipsoid volume and the location stability, respectively. The four defined indices taken as objective functions are optimized, by a nondominated sort genetic algorithm (NSGA-II), so that an optimal layout of locators is obtained due to the minimization of the pose error. Also, the feasibility of the proposed method was comprehensively validated by simulation and machining experiments.

Structure Parameter Identification of Articulated Arm Coordinate Measuring Machines Based on the Least Squares Method

2013 ◽  
Vol 303-306 ◽  
pp. 638-641
Author(s):  
Jian Lu ◽  
Guan Bin Gao ◽  
Hui Ping Yang
Keyword(s):  
Least Squares ◽  
Jacobian Matrix ◽  
Least Squares Method ◽  
Structural Parameters ◽  
Structure Parameters ◽  
Measuring Accuracy ◽  
Total Differential ◽  
Cartesian System ◽  
Main Factors ◽  
Value Decomposition

The AACMM is a kind of new coordinate measuring instrument based on non-Cartesian system. The errors of structure parameters are the main factors which affect the measuring accuracy, and the structural parameters identification can effectively improve the measuring accuracy of AACMM. The measuring model of an AACMM was established using the D-H method. Then the error model of the AACMM was established with total differential transforming method, from which the Jacobian matrix was obtained. The structure parameters’ errors which need to be identified through singular value decomposition of Jacobian matrix and the decomposition of orthogonal matrix elementary row transform can be determined. Finally, the least squares method was used in identifying the structure parameters’ errors of the AACMM, the results shows that the method proposed in this paper can identify the structure parameters efficiently.

Cartesian workspace optimization of Tricept parallel manipulator with machining application

Robotica ◽  
2014 ◽  
Vol 33 (9) ◽  
pp. 1948-1957 ◽  
Author(s):  
M. A. Hosseini ◽  
H. M. Daniali
Keyword(s):  
Velocity Vector ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Research Work ◽  
Singular Values ◽  
Maximum Size ◽  
Space Velocity ◽  
Parallel Robot ◽  
Joint Space ◽  
Work Piece

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.

Applications of Damped Least-Squares Methods to Resolved-Rate and Resolved-Acceleration Control of Manipulators

1988 ◽  
Vol 110 (1) ◽  
pp. 31-38 ◽  
Author(s):  
C. W. Wampler ◽  
L. J. Leifer
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Sensory Feedback ◽  
Inverse Kinematic ◽  
Redundant Manipulators ◽  
Approximate Inverse ◽  
Singular Configurations ◽  
Large Joint ◽  
Resolved Acceleration Control ◽  
Damped Least Squares

Resolved-rate and resolved-acceleration controllers have been proposed for manipulators whose trajectories are determined by real-time sensory feedback. For redundant manipulators, these controllers have been generalized using the pseudoinverse of the manipulator Jacobian. However, near singular configurations, these controllers fail in that they require infeasibly large joint speeds. A damped least-squares reformation of the problem gives approximate inverse kinematic solutions that are free of singularities. Away from singularities the new controllers closely approximate their conventional counterparts; near singular configurations the new controllers remain well-behaved, although the rate of convergence decreases. This paper defines the new controllers and proves their stability. Some aspects of the behavior of the new resolved-rate controller are illustrated in simulations.

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