Olivier Bruls

On the Use of Lie Group Time Integrators in Multibody Dynamics

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-a method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R 3 (the Euclidean space) and the group of rotations SO(3 ) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3 ) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3 ) offers the advantage of an almost constant iteration matrix.Copyright

The Generalized-α Scheme as a Linear Multistep Integrator: Toward a General Mechatronic Simulator

This paper presents a consistent formulation of the generalized-a time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller dynamics, and kinematic constraints. The theoretical background relies on the analogy with linear multistep formulas, which leads to elegant results related to consistency, order conditions for constant and variable step-size methods, as well as global convergence. Those results are illustrated for a controlled spring-mass system, and the method is also applied for the simulation of a vehicle semi-active suspension.

Error analysis of generalized- α Lie group time integration methods for constrained mechanical systems

Generalized-

Validated extraction of gait events from 3D accelerometer recordings

\alpha

Modeling joints with clearance and friction in multibody dynamic simulation of automotive differentials

α methods are very popular in structural dynamics. They are methods of Newmark type and combine favourable stability properties with second order convergence for unconstrained second order systems in linear spaces. Recently, they were extended to constrained systems in flexible multibody dynamics that have a configuration space with Lie group structure. In the present paper, the convergence of these Lie group methods is analysed by a coupled one-step error recursion for differential and algebraic solution components. It is shown that spurious oscillations in the transient phase result from order reduction that may be avoided by a perturbation of starting values or by index reduction. Numerical tests for a benchmark problem from the literature illustrate the results of the theoretical investigations.

SIMULATION OF DIFFERENTIALS IN FOUR-WHEEL DRIVE VEHICLES USING MULTIBODY DYNAMICS

BACKGROUND The contralateral shoulder is often used as a reference when evaluating a pathologic shoulder. However, the literature provides contradictory results regarding the symmetry of the scapular pattern in a healthy population. We assume that several factors including gender and type of motion may influence the bilateral symmetry of the scapulae. MATERIALS AND METHODS The dominant and nondominant shoulders of 2 populations of men and women comprising 11 subjects each were evaluated for 3 distinct motions: flexion in the sagittal plane, abduction in the frontal plane, and glenohumeral internal/external rotation with the arm abducted at 90°. Posture, kinematics, and range of motion were studied separately. RESULTS Asymmetries are observed for motions performed in the frontal and sagittal plane but not for internal/external rotation with the arm abducted at 90°. For both male and female populations, multiplanar asymmetries are observed and the dominant scapula has a larger upward rotation. The asymmetries mainly originate in the scapulas kinematics and not in its original posture. CONCLUSION Small but significant asymmetries exist between the dominant and nondominant shoulders in terms of kinematics. One should be aware of these differences when using the contralateral shoulder as a reference. LEVEL OF EVIDENCE Basic science study, kinematics

Modelling, Simulation and Control of Flexible Multibody Systems

This work is part of a project that deals with the three-dimensional (3D) analysis of normal and pathological gaits based on a newly developed system for clinical applications, using low-cost wireless accelerometers and a signal processing algorithm. This system automatically extracts relevant gait events such as the heel strikes (HS) and the toe-offs (TO), which characterize the stance and the swing phases of walking. The performances of the low-cost accelerometer hardware and related algorithm have been compared to those obtained by a kinematic 3D analysis system and a force plate, used as gold standard methods. The HS and TO times obtained from the gait data of 7 healthy volunteers (147 trials) have been found to be (mean ± standard deviation) 0.42±7.92 ms and 3.11±10.08 ms later than those determined by the force plate, respectively. The experimental results demonstrate that the new hardware and associated algorithm constitute an effective low-cost gait analysis system, which could thus be used for the assessment of mobility in routine clinical practice.