Time integration of rigid bodies modelled with three rotation parameters

Holzinger, Stefan; Gerstmayr, Johannes

doi:10.1007/s11044-021-09778-w

Cited by 13 publications

(13 citation statements)

References 45 publications

(108 reference statements)

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“…In Eq. (11), ω (i) , v (i) ρ ∈ R 3 represents the angular velocity vector expressed in the body-attached frame, m (i) is the mass of the rigid body,…”

Section: Classical Formulationmentioning

confidence: 99%

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

Holzinger

Arnold

Gerstmayr

2023

Preprint

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As commonly known, standard time integration of the kinematic equations of rigid bodies modelled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using several typical rigid multibody systems. It is shown, that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turned out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the average number of Newton iterations required per time step is most often smaller for the Lie group integration method. It also turns out, that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly.

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“…In Eq. (11), ω (i) , v (i) ρ ∈ R 3 represents the angular velocity vector expressed in the body-attached frame, m (i) is the mass of the rigid body,…”

Section: Classical Formulationmentioning

confidence: 99%

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

Holzinger

Arnold

Gerstmayr

2023

Preprint

Self Cite

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“…( 2 ), the well-established approach [ 31 , 32 ] is applied, with the terms incremental rotation vector , see Eq. ( 6 ), and Euler–Rodrigues formula [ 33 ] with the cardinal sine function [ 34 ] The Euler–Rodrigues formula, see Eq. ( 4 ), is a common approach to compute the exponential map, thus mapping elements of into SO(3).…”

Section: Motion Reconstructionmentioning

confidence: 99%

A novel motion-reconstruction method for inertial sensors with constraints

Neurauter

Gerstmayr

2022

Multibody Syst Dyn

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Motion reconstruction for rigid bodies and rigid-body frames using data from inertial measurement units (IMUs) is a challenging task. Position and orientation determination by means of IMUs is erroneous, as deterministic and stochastic errors accumulate over time. The former of which errors can be minimized by standard calibration approaches, however, sensor calibration with respect to a common reference coordinate system to correct misalignment, has not been fully addressed yet. The latter stochastic errors are mostly reduced using sensor fusion. In this paper, we present a novel motion-reconstruction method utilizing optimization to correct measured IMU data by means of correction polynomials to minimize the deviation of motion constraints. In addition, we perform gyrometer and accelerometer calibration with an industrial manipulator to address deterministic IMU errors, especially misalignment. To evaluate the performance of the novel methods, two types of experiments, one at constant orientation and another with simultaneous translation and rotation, were conducted utilizing the manipulator. The experiments were repeated for five individual IMUs successively. Application of the calibration and optimization methods yielded an average decrease of 95% in the maximum position error compared to the results of common motion reconstruction. Moreover, the average position error over the measurement duration decreased by nearly 90%. The proposed method is applicable to velocity, position, and orientation constraints for every experiment that starts and ends at standstill.

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“…Accurate and efficient simulation of rigid body rotation is of great importance in multibody dynamics [1][2][3][4][5][6][7][8][9][10], aerospace [11][12][13], robotic systems [14,15], and so on, which closely depends on the selection of parametrization of spatial rotation. Euler parameters [16] that a special case of quaternions have been widely employed in the description of rigid body rotation, and the corresponding motion equations are singularity-free and do not contain trigonometric functions.…”

Section: Introductionmentioning

confidence: 99%

A three-sub-step composite method for the analysis of rigid body rotation with Euler parameters

Xing

2022

Preprint

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This paper proposes a composite method for the analysis of rigid body rotation based on Euler parameters. The proposed method contains three sub-steps, wherein for keeping as much low-frequency information as possible the first two sub-steps adopt the trapezoidal rule, and the four-point backward interpolation formula is used in the last sub-step to flexibly control the amount of high-frequency dissipation. On this basis, in terms of the relation between Euler parameters and angular velocity, the stepping formulations of the proposed method are further modified for maximizing the accuracy of the angular velocity. For the analysis of rigid body rotation, the accuracy of the proposed method can converge to second-order, and the amount of its high-frequency dissipation can smoothly range from one (conservative scheme) to zero (annihilating scheme). Additionally, in the proposed method, the constraints at the displacement and velocity levels are strictly satisfied, and the numerical drifts at the acceleration level can be effectively eliminated. Several benchmark rigid body rotation problems show the advantages of the proposed method in stability, accuracy, dissipation, efficiency, and energy conservation.

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Time integration of rigid bodies modelled with three rotation parameters

Cited by 13 publications

References 45 publications

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

A novel motion-reconstruction method for inertial sensors with constraints

A three-sub-step composite method for the analysis of rigid body rotation with Euler parameters

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