Diagonal Stiffness Matrix Research Articles

Cross-coupling stiffness for natural goal-directed robot motion

The capability of humans to use their natural dynamics for generating explosive motions in a highly-coordinated sequence is a feat that has yet to be reached in robotics. With the introduction of intrinsically elastic joints, great progress towards this goal has been made. However, there are still some challenges associated with this type of actuation, which limits its application. Generating goal-directed sequences has proven difficult as optimal control solutions tend to result in uncoordinated swing-up motions. This can be explained when viewing the structure of the stiffness matrix: If the elastic elements are placed in series with the motor, a diagonal stiffness matrix is generated. This in turn leads to a multitude of frequencies at which the system can oscillate. By adding off-diagonal elements, a dominant primary resonance behaviour can be achieved. Leveraging this cross-coupling stiffness, we show that robots can produce natural goal-directed oscillatory and explosive movements that closely resemble human throwing.

Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in ℝd. As a generalisation of the G. Szegö’s family in 1D (1939), the first family of multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function |x|2μe-|x|2(resp. |x|2μ) in ℝd. We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product [u,v]Hs(ℝd)= ((-Δ)s/2u, (-Δ)s/2v)ℝdassociated with the IFL of orders> 0. As an immediate consequence, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds remarkably efficient in solving PDEs with the fractional Schrödinger operator: (-Δ)s+ |x|2μwiths∈ (0,1] andμ> −1/2 in ℝdWe construct the second family of multivariate nontensorial Müntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrödinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Müntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some Schrödinger eigenvalue problems.

A Novel Geometric Design Method of Elastic Structure for 6-Axis Force/Torque Sensor

We present a novel design method of a 6-axis Force/Torque (F/T) sensor with identical stiffness in all directions. In contrast to common F/T sensor designs, the proposed method is an analytical approach to the realization of a desired diagonal stiffness matrix which implies the complete decoupling of the stiffnesses in all directions. The first step of the design is to group the column vectors of a given rank 6 diagonal stiffness matrix into in-plane type and out-of-plane type stiffnesses. Each type of stiffnesses is then synthesized by means of three line vectors and designed as the corresponding serial-chain for a limb of an F/T sensor using only circular hinges. The F/T sensor can be formed by connecting the designed two serial-chains in series. However, for a practical design with minimum structural error and better strength, each of two type stiffnesses is further divided into a set of n separate serial-chains. Finally, the n-sets of serial-chains each consisting of six circular hinges that correspond in-plane and out-of-plane type stiffnesses are connected to the moving platform in parallel to complete the design. The FEM analysis and experiments are conducted to verify the proposed design method.

Stiffness synthesis of 3-DOF planar 3RPR parallel mechanisms

SUMMARYForce control is important in robotics research for safe operation in the interaction between a manipulator and a human operator. The elasticity center is a very important characteristic for controlling the force of a manipulator, because a force acting at the elasticity center results in a pure displacement of the end-effector in the same direction as the force. Similarly, a torque acting at the elasticity center results in a pure rotation of the end-effector in the same direction as the torque. A stiffness synthesis strategy is proposed for a desired elasticity center for three-degree-of-freedom (DOF) planar parallel mechanisms (PPM) consisting of three revolute-prismatic-revolute (3RPR) links. Based on stiffness analysis, the elasticity center is derived to have a diagonal stiffness matrix in an arbitrary configuration. The stiffness synthesis is defined to determine the configuration when the elasticity center and the diagonal matrix are given. The seven nonlinear system equations are solved based on one reference input. The existence and the solvability of the nonlinear system equations were analyzed using reduced Gröbner bases. A numerical example is presented to validate the method.

6축 병렬형 순응기구를 이용한 위치/힘 동시제어

In this paper, the kinestatic control algorithm using a six-axis compliant device is presented. Unlike the traditional control methods using a force/torque sensor with very limited compliance, this method employs a compliant device to provide sufficient compliance between an industrial robot and a rigid environment. This kinestatic control method is used to simply control the position of an industrial robot with twists of compensation, which can be decomposed into twists of compliance and twists of freedom. A simple design method of a six-axis parallel-type compliant device with a diagonal stiffness matrix is presented. A compliant device prototype and kinestatic control hardware system and programming were developed. The effectiveness of the kinestatic control algorithm was verified through two kinds of kinestatic control experiments.

A model for computing the dual stiffness matrix of the human knee joint

This paper discusses an application of dual algebra to the analysis of human knee joint stiffness matrix. According to the proposed methodology, the general stiffness matrix characterizing the elasticity properties of the knee ligaments is reduced to a dual diagonal stiffness matrix. For this purpose, the similarity transform was extended to the field of dual numbers. The theoretical framework proposed seems particularly useful when comparing stiffness matrices obtained from different experimental setup and within different Cartesian coordinate systems.

Improving eigenpairs of automated multilevel substructuring with subspace iterations

This paper improves the eigenpair approximations obtained from the automated multilevel substructuring (AMLS) method by subspace iterations. Two variants of AMLS hybrid Subspace Iteration Method (AMLS-SIMa and AMLS-SIMb) are proposed. AMLS-SIMa is a derivative of the basic subspace iteration by utilizing the AMLS approximations as initial vectors. AMLS-SIMb further takes advantage of the AMLS transformed block diagonal stiffness matrix to avoid factorization of the original stiffness matrix. Numerical experiments show that: (a) the error of AMLS approximate eigenpairs can be significantly reduced with just a few iteration steps; (b) AMLS-SIMb is more efficient than AMLS-SIMa with less execution time.

Time-Varying Total Stiffness Matrix of a Rigid Machine Spindle-Angular Contact Ball Bearings Assembly: Theory and Analytical/Experimental Verifications

A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent.

Time-Varying Total Stiffness Matrix of a Rigid Machine Spindle-Angular Contact Ball Bearings Assembly: Theory and Analytical/Experimental Verifications

A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent.

Modeling of a light elastic beam by a system of rigid bodies

This paper has shown that a light elastic beam, in the case of small elastic deformations, can be modeled by a kinematic chain without branching composed of rigid bodies which are connected by passive revolute or prismatic joints with corresponding springs in them. Elastic properties of the beam are modeled by the springs introduced. The potential energy of the elastic beam is expressed as a function of components of the vector of elastic displacement and the vector of elastic rotation calculated for the elastic centre of the beam, which results in the diagonal stiffness matrix of the beam. As the potential energy of the introduced system of bodies with springs is expressed in the function of relative joint displacements, the diagonal stiffness matrix is obtained. In addition, these two stiffness matrices are equal. The modeling process has been demonstrated on the example of an elastic beam rotating about a fixed vertical axis, with a rigid body whose mass is considerably larger than the beam mass fixed to its free end. Differential equations of motion have been formed for this mechanical system. The modeling technique described here aims at expanding of usage of well developed methods of dynamics of systems of rigid bodies to the analysis of systems with elastic bodies. .

A compliance control strategy for robot manipulators under unknown environment

In this paper, a compliance control strategy for robot manipulators that employs a self-adjusting stiffness function is proposed. Based on the contact force, each entry of the diagonal stiffness matrix corresponding to a task coordinate in the operational space is adaptively adjusted during contact along the corresponding axis. The proposed method can be used for both the unconstrained and constrained motions without any switching mechanism which often causes undesirable instability and/or vibrational motion of the end-effector. The experimental results involving a two-link direct drive manipulator interacting with an unknown environment demonstrates the effectiveness of the proposed method.

Solving plate bending problems using finite strips on networked workstations

In this paper, we present two parallelization schemes for solving a particular class of plate-bending problems using the finite strip method on a network of workstations equipped with the well-known software package PVM (parallel virtual machine). The first scheme is based on the master-slave model and the second is a multilevel master-slave approach. The finite strip method, first developed in the context of thin plate bending analysis, is a semi-analytical finite element process. It takes advantage of the orthogonality properties of harmonic functions in the stiffness matrix formulation to yield a block diagonal stiffness matrix and, thus, decomposes a single problem in two dimensions into several independent subproblems, each corresponding to a problem in one dimension. The method not only reduces the bandwidth of the system stiffness matrix, but offers obvious advantages for solving the problem on either shared-memory machines or distributed-memory multiprocessors. Our parallelization strategies are implemented on a network of SUN workstations interconnected by the Ethernet. Numerical experiments and the performance of the implementations are presented. Our results clearly indicate that the multilevel master-slave model is better than the master-slave scheme on the networked system using PVM.

Isoelastic behavior of parallel robots

SUMMARYThis paper presents a particular architecture of a parallel robot which is characterized by a diagonal stiffness matrix. This result suggests the use of a parallel robot in the final phase of insertion as a passive compliance device. The stiffness rate of this device is controlled by the gain of the feedback loop. As the correction of small angular misalignments due to contact forces do not generate lateral errors and vice versa, we have the equivalent of an isoelastic swivel.

Thermodynamic expansion processes for argon plasma in a convergent-divergent nozzle.

which are evaluated on r — a for the structural interface and on 6 = 77/2 for the free surface. Solution of the present problem in terms of uncoupled spherical harmonics is possible only as a result of the idealized structural law, P = KUr. Mathematical representation of the container structure in terms of a shell theory would complicate the solution in that the modes would be linear combinations of the functions, i[fMN for 0 . Although the exact solution of the present problem results from an unrealistic structural representation, it is useful as a reference solution in the evaluation of numerical hydroelastic methods (Note: A lumped parameter representation of the idealized container structure consists of a diagonal stiffness matrix with element values proportional to the elemental surface areas.)