Coriolis Matrix Research Articles

On Tracking Control of Rigid-Joint Robots With Only Position Measurements

In this letter, we present tracking control with only position measurements for rigid-joint robots, by applying the canonical transformation theory for port-Hamiltonian systems. We show that besides giving the same results as presented in the literature for Euler-Lagrange systems, the canonical transformation theory also justifies a Coriolis matrix based on the desired velocities. Furthermore, we show how the initial conditions of the controller can be tuned in order to improve transient performance. Finally, we validate our results on a simple experimental setup.

Design and Real-Time Control of a 4-DOF Biped Robot

This work is focused on the design, construction and model based control of a biped robot during the walking cycle on the sagittal plane. For the analysis, the single support phase is considered to be the dominating dynamics, by assuming an instantaneous double support phase which is only described by the impact phenomenon. The joint tracking problem is analyzed by means of a model based control strategy, which incorporates a reformulation of the Coriolis matrix that allows the cancellation of non antisymmetric terms in order to formally proof the asymptotic stability of the coordinate error system representation in a local sense. Some experiments are carried out for a pre-defined reference trajectory for the walking cycle of a 4-DOF biped robot.

Recursive Composite Adaptation for Robot Manipulators

In this paper, we investigate the recursive implementation of composite adaptive control for robot manipulators. Via exploitation of the relation between the inertia matrix and the Coriolis and centrifugal matrix, we present the recursive algorithm for the derivation of the filtered manipulator model, which, to our knowledge, is the first result on this point in the literature. With this filtered model, the prediction error of the filtered torque is obtained and injected to the direct adaptation, forming the well-known composite adaptation law, with an acceptable amount of computation O(n2). A six degree-of-freedom (DOF) manipulator is employed as a simulation example to show the performance and the computational complexity of the proposed recursive algorithm.

On the Boundedness Property of the Inertia Matrix and Skew-Symmetric Property of the Coriolis Matrix for Vehicle-Manipulator Systems

This paper addresses the boundedness property of the inertia matrix and the skew-symmetric property of the Coriolis matrix for vehicle-manipulator systems. These properties are widely used in control theory and Lyapunov-based stability proofs and thus important to identify. The skew-symmetric property does not depend on the system at hand but on the parameterization of the Coriolis matrix, which is not unique. It is the authors’ experience that many researchers take this assumption for granted without taking into account that several parameterizations exist. In fact, most researchers refer to references that do not show this property for vehicle-manipulator systems but for other systems such as single rigid bodies or fixed-base manipulators. As a result, the otherwise rigorous stability proofs fall apart. In this paper, we list some relevant references and give the correct proofs for some commonly used parameterizations for future reference. Depending on the choice of state variables, the boundedness of the inertia matrix will not necessarily hold. We show that deriving the dynamics in terms of quasi-velocities leads to an inertia matrix that is bounded in its variables. To the best of our knowledge, we derive for the first time the dynamic equations of vehicle-manipulator systems with non-Euclidean joints for which both properties are true.

Synchronization Control of Planar 2-Dof Robot with Redundant Actuation

Considering the special characteristics of the redundant parallel manipulator, with emphasis on the variable of structure, relatively small workspace and the strong coupling relationship among arms，a synchronization control strategy is presented in this paper. Since in the feedforward ,the inertial and the coriolis matrix are designed constant according to relatively small workspace, position measurement of the endeffector in plane is ignored. Synchronization error and coupling error are introduced to reject the model errors of inertial and coriolis matrix as stated above. Using the method, the errors of driving arms may be reduced, as well as synchronization performance among axes improves. The stability of the controllers was proved by Lyapunov. Finally, experimental results show the feasibility.

On the Recursive Adaptive Control for Free-floating Space Manipulators

This paper is devoted to investigating the recursive implementation schemes of adaptive control for free-floating space manipulators. Using spatial vector tool and some physical properties that free-floating space manipulators enjoy, we establish a general framework on the seeking of the centripetal and Coriolis matrix that satisfies the skew symmetry requirement. Under this general framework, we propose a recursive adaptive algorithm for free-floating manipulators, which is composed of two parts: the first part is the recursive derivation of the required manipulator control torques, and the second part is the recursive updating of the spacecraft reference velocity and acceleration. To guarantee the uniform positive definiteness of the estimated spacecraft inertia, we present a parameter projection algorithm to project the estimated parameters into some pre-specified parameter region. In the next, we extend the proposed recursive adaptive algorithm to task-space control of free-floating space manipulators. We examine the performance of the proposed recursive adaptive algorithms via numerical simulation on a six-DOF space manipulator.

Singularity-free dynamic equations of spacecraft-manipulator systems

In this paper we derive the singularity-free dynamic equations of spacecraft-manipulator systems using a minimal representation. Spacecraft are normally modeled using Euler angles, which leads to singularities, or Euler parameters, which is not a minimal representation and thus not suited for Lagrange's equations. We circumvent these issues by introducing quasi-coordinates which allows us to derive the dynamics using minimal and globally valid non-Euclidean configuration coordinates. This is a great advantage as the configuration space of a spacecraft is non-Euclidean. We thus obtain a computationally efficient and singularity-free formulation of the dynamic equations with the same complexity as the conventional Lagrangian approach. The closed form formulation makes the proposed approach well suited for system analysis and model-based control. This paper focuses on the dynamic properties of free-floating and free-flying spacecraft-manipulator systems and we show how to calculate the inertia and Coriolis matrices in such a way that this can be implemented for simulation and control purposes without extensive knowledge of the mathematical background. This paper represents the first detailed study of modeling of spacecraft-manipulator systems with a focus on a singularity free formulation using the proposed framework.

Adaptive output-feedback control for trajectory tracking of electrically driven non-holonomic mobile robots

In this study, the authors propose an adaptive output-feedback controller for trajectory tracking of electrically driven non-holonomic mobile robots in the presence of parametric uncertainties. A new adaptive observer using the transformation matrices is developed to estimate the unmeasured velocities of the mobile robot. By using the transformation matrices, the designed adaptive observer can deal with quadratic velocity terms caused by the Coriolis matrix in the mobile robot dynamics as well as uncertain parameters in quadratic velocity terms. Based on the designed adaptive observer, a simple tracking controller at the actuator level is induced from the dynamic surface design methodology. Using the Lyapunov stability theory, the authors prove that all errors in a closed-loop system are uniformly ultimately bounded and converge to an adjustable neighbourhood of the origin.

Coulomb excitation of aAm242isomeric target:E2andE3strengths, rotational alignment, and collective enhancement

A $98%$ pure ${}^{242\mathrm{m}}$Am ($K={5}^{\ensuremath{-}}$, ${t}_{1/2}=141$ years) isomeric target was Coulomb excited with a $170.5$-MeV $^{40}\mathrm{Ar}$ beam. The selectivity of Coulomb excitation, coupled with the sensitivity of Gammasphere plus CHICO, was sufficient to identify $46$ new states up to spin $18\ensuremath{\hbar}$ in at least four rotational bands; $11$ of these new states lie in the isomer band, $13$ in a previously unknown yrast ${K}^{\ensuremath{\pi}}={6}^{\ensuremath{-}}$ rotational band, and $13$ in a band tentatively identified as the predicted yrast ${K}^{\ensuremath{\pi}}={5}^{+}$ band. The rotational bands based on the ${K}^{\ensuremath{\pi}}={5}^{\ensuremath{-}}$ isomer and the ${6}^{\ensuremath{-}}$ bandhead were populated by Coulomb excitation with unexpectedly equal cross sections. The $\ensuremath{\gamma}$-ray yields are reproduced by Coulomb excitation calculations using a two-particle plus rotor model (PRM), implying nearly complete $\ensuremath{\Delta}K=1$ mixing of the two almost-degenerate rotational bands, but recovering the Alaga rule for the unperturbed states. The degeneracy of the ${5}^{\ensuremath{-}}$ and ${6}^{\ensuremath{-}}$ bands allows for precise determination of the mixing interaction strength $V$, which approaches the strong-mixing limit; this agrees with the $50%$ attenuation of the Coriolis matrix element assumed in the model calculations. The fractional admixture of the ${I}_{K}^{\ensuremath{\pi}}={6}_{6}^{\ensuremath{-}}$ state in the nominal ${6}_{5}^{\ensuremath{-}}$ isomer band state is measured within the PRM as $45.{6}_{\ensuremath{-}1.1}^{+0.3}%$. The $E$2 and $M$$1$ strengths coupling the ${5}^{\ensuremath{-}}$ and ${6}^{\ensuremath{-}}$ bands are enhanced significantly by the mixing, while $E1$ and $E2$ couplings to other low-$K$ bands are not measurably enhanced. The yields of the ${5}^{+}$ band are reproduced by an $E3$ strength of $\ensuremath{\approx}$15 W.u., competitive with the interband $E2$ strength. Alignments of the identified two-particle Nilsson states in $^{242}\mathrm{Am}$ are compared with the single-particle alignments in $^{241}\mathrm{Am}$.

Singularity-free dynamic equations of vehicle–manipulator systems

In this paper we derive the singularity-free dynamic equations of vehicle–manipulator systems using a minimal representation. These systems are normally modeled using Euler angles, which leads to singularities, or Euler parameters, which is not a minimal representation and thus not suited for Lagrange’s equations. We circumvent these issues by introducing quasi-coordinates which allows us to derive the dynamics using minimal and globally valid non-Euclidean configuration coordinates. This is a great advantage as the configuration space of the vehicle in general is non-Euclidean. We thus obtain a computationally efficient and singularity-free formulation of the dynamic equations with the same complexity as the conventional Lagrangian approach. The closed form formulation makes the proposed approach well suited for system analysis and model-based control. This paper focuses on the dynamic properties of vehicle–manipulator systems and we present the explicit matrices needed for implementation together with several mathematical relations that can be used to speed up the algorithms. We also show how to calculate the inertia and Coriolis matrices and present these for several different vehicle–manipulator systems in such a way that this can be implemented for simulation and control purposes without extensive knowledge of the mathematical background. By presenting the explicit equations needed for implementation, the approach presented becomes more accessible and should reach a wider audience, including engineers and programmers.

On the Boundedness and Skew-Symmetric Properties of the Inertia and Coriolis Matrices for Vehicle-Manipulator Systems

This paper addresses the boundedness property of the inertia matrix and the skew-symmetric property of the Coriolis matrix for vehicle-manipulator systems. These properties are widely used in Lyapunov-based stability proofs and are therefore important to identify. For example, the skew-symmetric property does not depend on the system at hand, but on the choice of parameterisation of the Coriolis matrix, which is not unique. It is the authors' experience that many researchers take this assumption for granted without taking into account that there exist several parameterisations for which this is not true. In fact, most researchers refer to references that do not show this property for vehicle-manipulator systems, but for other systems such as single rigid bodies or manipulators on a fixed base. As a result, the otherwise rigorous stability proofs fall apart. In this paper we point out several references that are widely used, but that do not show this property and we refer to the correct references. As most references on this topics are not easily accessible, we also give the correct proofs for commonly used parameterisations of the Coriolis matrix and thus provide a proof for future reference.The same is the case for the boundedness property of the inertia matrix which for a bad choice of state variables will not necessarily hold. This can be solved by deriving the dynamics for vehicle-manipulator systems in terms of quasi-velocities, which allows us to describe the dynamics without the presence of the Euler angle singularities that normally arise in vehicle-manipulator dynamics. To the authors' best knowledge we derive for the first time the dynamic equations with both the skew-symmetric property of the Coriolis matrix and the boundedness property of the inertia matrix for vehicle-manipulator systems with non-Euclidean joints.

The dynamics of parallel Schönflies motion generators: The case of a two-limb system

The formulation of the mathematical model governing the dynamics of parallel Schönflies motion generators (SMGs) is the subject of this paper. These are robotic systems intended to produce motions that entail displacements of the Schönflies subgroup of rigid-body displacements. Such motions are representative of those produced by the serial robots termed SCARA, which involve three independent translations and one rotation about an axis of fixed direction. The main features of SMGs are illustrated with the aid of the McGill Schönflies motion generator. This robot is composed of two limbs, each being a four-joint RΠΠR - R representing a revolute, Π a Π joint, or a parallelogram linkage - kinematic chain, with only two joints actuated. One important feature of the McGill SMG is its mass distribution, as its moving parts account for about 10 per cent of the mass of its drive units, which are (a) fixed to the robot frame and (b) the only steel parts of the whole robot. Moreover, its joints account for roughly 10 per cent of the total mass of its moving parts, fabricated from aluminium, which justifies neglecting the joint inertia in the mathematical model. This feature calls for a formulation of the model in question in terms of the motor displacements, rather than the joint displacements, as the generalized coordinates of the model. Furthermore, in order to derive the model, the robot is decomposed into two subsystems, the drive and the linkage, the drive being decomposed, in turn, into two subsubsystems, the epicyclic gear train and the right-angled gearbox. The robot kinematics is first derived and then the dynamics model is formulated by means of the natural orthogonal complement. In the framework of this methodology, the inertia and what are called the Coriolis matrices of the mathematical model are additive, in the sense that they can be computed as the sum of the contributions of the different subsystems and subsubsystems of a given mechanical system. The contributions of the subsystems and subsubsystems, in turn, can be computed as the sum of the contributions of the individual moving parts of these. Some general results applicable to all SMGs are derived, which leads to the simplification of the mathematical model of such systems.

Nonstationary Vibration of a Fully Flexible Parallel Kinematic Machine

This paper studies the problem of the nonstationary vibration of a fully flexible parallel kinematic machine (PKM) that has flexibilities both in links and in joints. In the stationary case, the PKM was treated as a varying structure and the natural frequencies and mode shapes changed with the changes in the PKM configuration, without consideration of the PKM nominal motion. In the nonstationary case as studied in this paper, the nominal motion is included to investigate how it would affect the natural frequencies and mode shapes. To do so, a nonstationary model is developed using the elasto-dynamics method. First, a kinematic model is built based on rigid links and ideal joints, which is used to solve the PKM nominal motion. Second, the kinetic model is developed considering the flexibilities in the links and joints. In this case, the vibration equations would contain the Coriolis and gyroscopic damping matrix and the tangential and normal stiffening matrix, which are the terms resulting from the nominal motion. The instantaneous eigensolutions are obtained from the nonstationary eigenequations. The results show that (i) the slider velocity affects the instantaneous natural frequencies more than the slider acceleration; and (ii) the nominal motion has an effect on the system eigencharacteristics (e.g., the nonstationary frequencies can be higher or lower than the stationary ones) but the effect is small in an absolute amount (within 2.1Hz in natural frequencies presented at set nominal motions of the studied PKM prototype). This is because the extra inertial force from the nominal motion is always much smaller than the stiffness force in the system bodies as long as the bodies are made of hard material. The method presented is more convenient to use for the multibody system with flexible joints than other methods.

Reduction of finite element equations for a rotor model on non-isotropic spring support in a rotating frame

A simple method of reduction is presented for finite element model of non-axisymmetric rotors on non-isotropic spring support in a rotating frame. The frame is rotating about the undeformed centreline of the bearings with a speed equal to the shaft spin speed. In this frame the stiffness matrix, mass matrix and Coriolis matrix for the non-axisymmetric rotor (rotor with rectangular cross-section, cracked rotor, etc.) is independent of time but the support forces become periodic function of time. Therefore, in a rotating frame, it becomes necessary to deal with a large set of linear ordinary differential equations with periodic coefficients at support degrees of freedom, which requires substantial computational effort. To effectively handle this large system it needs to be reduced keeping the essential information almost intact.

Interpretation and deperturbation of the Λ-type doubling in the a 3Π state of GaH

The unusual J dependence of the Λ doubling splitting that separates e and f levels in the a 3Π state of GaH has been interpreted using a deperturbation study carried out in an effective Hamiltonian point of view and based on a case (c) partition of the Hamiltonian: rather than limiting the zeroth-order part of the Hamiltonian to electronic and vibrational motions, spin-orbit interaction is also included, leaving outside just those effects linked with nuclear rotational motion. Thus the reasons for the peculiar behavior of the Λ splitting come out in a simple and appealing way. Essentially, with such a procedure, the variation with increasing rotation of the Λ-doubling splitting, i.e., the variations of the energy differences between e and f levels, are directly connected with fairly large differences between the rotational parameters (including mandatorily first and second order centrifugal distorsion) occurring in the off-diagonal Coriolis matrix elements within the 3Π block. To a lesser extent, smaller differences between the e and f rotational constants in the diagonal elements also contribute to the pattern of splittings. The consistency of the resulting values of the main fitted difference parameters (3Π0 e/f splitting at null rotation and first-order diagonal and off-diagonal rotational constant differences) is examined on the basis of experimental data on the one hand and new ab initio calculations on the other. Correct orders of magnitude are obtained although rotational constant differences but one are experimentally larger than expected theoretically from a model restricted to the four electronic states dissociating into ground state products.

Head and Tail Deformations, Torsional Coriolis Coupling, and E1d-E2d Vibrational Mixing in Ethane-Like Molecules

The mechanism of torsional Coriolis interaction of E1d and E2d vibrational modes in ethane-like molecules is investigated, and it is shown that this coupling can drastically affect the torsional splitting in the degenerate vibrational states. A basic point of our treatment is that the sets of coordinates of head and tail which combine with the + sign to generate E1d normal coordinates are in general different from those which combine with the − sign to generate E2d normal coordinates. It is shown that the ζγ torsional Coriolis coefficients calculated by the usual methods of normal mode analysis are related to the vibrational angular momenta within head and tail referred to the internal rotor axis systems. With knowledge of the L and L−1 matrices it is possible to transform these coefficients for reference to the molecule-fixed frame. It is peculiar that torsional Coriolis matrix elements occur between E1d and E2d vibrational components with the same x or y orientation in the molecule-fixed frame. The matrix elements of the torsional Coriolis operator and other operators responsible for the end-to-end coupling are determined, and a method for calculating vibration–torsion energies, and then torsional splittings, in degenerate vibrational states is outlined. Detailed calculations require a global model, involving all the degenerate vibrational basis states in a complex mechanism of interactions, but it is shown that useful information can be obtained by means of simplified models. Our semiempirical rule that degenerate vibrational states with a large negative value of the diagonal vibration–rotation Coriolis coefficient are likely to deviate much from the behavior of E1d or E2d vibrational states, with a sensible decrease of the torsional splittings, is confirmed.

Spectroscopy and dynamics of rare gas-spherical top complexes. III. The infrared spectrum of the ν3 band of Ne–SiH4 ( j=1←1 and j=2←1 transitions)

The infrared spectrum of the rare gas-spherical top complex Ne–SiH4 has been recorded in a supersonic jet in the region of the SiH4 ν3 triply degenerate stretching vibration at ∼2189 cm−1. The Ne–SiH4 spectrum is complex with no obvious regular band structure; a previous paper has reported the assignment and analysis of bands correlating with the SiH4 R(0) (K=0←0, K=1←0) and P(1) (K=0←0, K=0←1) transitions. This paper concludes the study of Ne–SiH4, with the assignment and analysis of bands correlating with the SiH4 Q(1) (K=0←0, K=1←0, K=0←1, K=1←1) and R(1) (K=0←0, K=1←0, K=1←1, K=2←1) transitions. This was facilitated by a twofold approach: The bands were identified by the use of a theoretical intermolecular potential, with potential parameters determined by reference to the two bands previously fitted; and the relative magnitudes of the Coriolis interactions between the K sublevels within the bands determined by explicit calculation of the Coriolis matrix elements. Having fitted the bands to appropriate energy-level expressions, an effective anisotropic potential for the internal rotation of the SiH4 molecule within the complex was fitted to the total of 15 observed band origins. With a fixed R approximation, the V3 anisotropic term was found to be ∼30 cm−1, approximately one-third that of Ar–SiH4, reflecting the freer rotation of the SiH4 molecule when complexed with the less polarizable Ne atom. However, the overall quality of the fit was an order of magnitude worse than that for Ar–SiH4, indicating increased importance of the radial dependence of the Ne–SiH4 potential compared to that of Ar–SiH4.

Rigorous Theory of Vibration Analysis of Rotating Blade Based on Transfer Matrix Method. 3rd Report. Symmetry of Inertia, Stiffness and Centrifugal Force Matrix, Antisymmetry of Coriolis Matrix, and Positive Definiteness of Intertia and Total Stiffness Matrix.

In order to show the features of the method of analyzing blade natural vibration described in the first and second reports, it must be demonstrated that the inertia, stiffness, centrifugal force and Coriolis matrices, which are equivalent to the transfer matrix of the second report, are symmetric, symmetric, symmetric and antisymmetric matrices, respectively, and that the inertia matrix and the total stiffness matrix-the latter is the sum of the stiffness and centrifugal force matrices-are positive definite matrices. It also means that natural frequencies are positive real number, and that each natural mode shape is orthogonal to the others. In addition, the result is effective in ascertaining that the selection of state variables and the treatment of centrifugal tension resistance are appropriate. In the process of the above proof, an equivalent method for symmetry verification is proposed and used, in order to make symmetry verification easier.

Rigorous Theory of Vibration Analysis of Rotating Blade Based on Transfer Matrix Method. 2nd Report. Blade Model and Transfer Matrix.

This is a research on rigorous theory for the natural vibration analysis of blade, a geometrically and environmentally complicated beam. The reason why the term "rigorous" is emphasized here is that (1) all the factors related to beam deformation are taken into consideration, and (2) those factors are combined to form an organically unified theory of non-contradiction, which is fully confirmed. In the 1st report, such a unified theory has been formulated in the form of a set of state equations. In this 2nd report, they are converted into a set of transfer matrices, assuming lumped inertia. The reason of acceptability of this assumption is that the order of major blade vibrations of engineering importance is much lower than the number of lumped inertia. The reason of beneficialness of this assumption is that it separates inertia, Coriolis and centrifugal-force matrices from stiffness matrix to enable theoritical development of good perspective.

A Symbolic Formulation of Dynamic Equations For a Manipulator With Rigid and Flexible Links

The objective of this article is to present an efficient proce dure for computer generation of the dynamic equations for a planar robot manipulator with arbitrarily assigned rigid or flexible links using any desired flexible mode shape functions. The dynamic equations for the serial link manipulator are derived using Lagrange's formulation and elastic deflection with the assumed-mode method. Fewer approximations are made than in other approaches, resulting in greater accuracy. A method to determine the centrifugal and Coriolis matrix is presented that yields an important structural property. The approach is systematic and allows a symbolic program to be written in Mathematica using a system of several groups and a constructed database. Four examples are illustrated to verify the dynamic equations. The stability of the zero dynamics is compared for different mode shape functions.