Sigrid Leyendecker

Variational integrators for constrained dynamical systems

A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps. Furthermore, the discrete null space method which has been introduced in the framework of energy-momentum conserving integration of constrained systems is adapted to the framework of variational integrators. It eliminates the constraint forces (including the Lagrange multipliers) from the timestepping scheme and subsequently reduces its dimension to the minimal possible number. While retaining the structure preserving properties of the specific integrator, the solution of the smaller dimensional system saves computational costs and does not suffer from conditioning problems. The performance of the variational discrete null space method is illustrated by numerical examples dealing with mass point systems, a closed kinematic chain of rigid bodies and flexible multibody dynamics and the solutions are compared to those obtained by an energy-momentum scheme.

A Variational Approach to Multirate Integration for Constrained Systems

The simulation of systems with dynamics on strongly varying time scales is quite challenging and demanding with regard to possible numerical methods. A rather naive approach is to use the smallest necessary time step to guarantee a stable integration of the fast frequencies. However, this typically leads to unacceptable computational loads. Alternatively, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. In this work, a multirate integrator for constrained dynamical systems is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Being based on a discrete version of Hamilton’s principle, the resulting variational multirate integrator is a symplectic and momentum preserving integration scheme and also exhibits good energy behaviour. Depending on the discrete approximations for the Lagrangian function, one obtains different integrators, e.g. purely implicit or purely explicit schemes, or methods that treat the fast and slow parts in different ways. The performance of the multirate integrator is demonstrated by means of several examples.

Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics

This paper formulates the dynamical equations of mechanics subject to holonomic constraints in terms of the states and controls using a constrained version of the Lagrange-d’Alembert principle. Based on a discrete version of this principle, a structure preserving time-stepping scheme is derived. It is shown that this respect for the mechanical structure (such as a reliable computation of the energy and momentum budget, without numerical dissipation) is retained when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced time-stepping equations serve as nonlinear equality constraints for the minimisation of a given cost functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. The resulting discrete optimal control algorithm is shown to have excellent energy and momentum properties, which are illustrated by two specific examples, namely reorientation and repositioning of a rigid body subject to external forces and the reorientation of a rigid body with internal momentum wheels.Copyright

Discrete variational Lie group formulation of geometrically exact beam dynamics

The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almost-perfect energy conservation. Comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically.

Γ-convergence of Variational Integrators for Constrained Systems

For a physical system described by a motion in an energy landscape under holonomic constraints, we study the Γ-convergence of variational integrators to the corresponding continuum action functional and the convergence properties of solutions of the discrete Euler–Lagrange equations to stationary points of the continuum problem. This extends the results in Müller and Ortiz (J. Nonlinear Sci. 14:279–296, 2004) to constrained systems. The convergence result is illustrated with examples of mass point systems and flexible multibody dynamics.

On the relevance of structure preservation to simulations of muscle actuated movements.

In this work, we implement a typical nonlinear Hill-type muscle model in a structure-preserving simulation framework and investigate the differences to standard simulations of muscle-actuated movements with MATLAB/Simulink. The latter is a common tool to solve dynamical problems, in particular, in biomechanic investigations. Despite the simplicity of the examples used for comparison, it becomes obvious that the MATLAB/Simulink integrators artificially loose or gain energy and angular momentum during dynamic simulations. The relative energy error of the MATLAB/Simulink integrators related to a very low actual muscle work can naturally reach large values, even higher than 100%. But also during periods with large muscle work, the relative energy error reaches up to 2%. Even in simulations with very small time steps, energy and angular momentum errors are still present using MATLAB/Simulink and can (at least partially) be responsible for phase errors in long-term simulations. This typical behaviour of commercial integrators is known to increase for more complex models or for computations with larger time steps, whose use is crucial for efficiency, especially in the context of optimal control simulations. In contrast to that, time-stepping schemes being derived from a discrete variational principle yield discrete analogues of the Euler–Lagrange equations and Noethers theorem. This ensures that the structure of the system is preserved, i.e. the simulation results are symplectic and momentum consistent and exhibit a good energy behaviour (no drift).

Variational collision integrator for polymer chains

The numerical simulation of many-particle systems (e.g. in molecular dynamics) often involves constraints of various forms. We present a symplectic integrator for mechanical systems with holonomic (bilateral) and unilateral contact constraints, the latter being in the form of a non-penetration condition. The scheme is based on a discrete variant of Hamiltons principle in which both the discrete trajectory and the unknown collision time are varied (cf. R. Fetecau, J. Marsden, M. Ortiz, M. West, Nonsmooth Lagrangian mechanics and variational collision integrators, SIAM J. Appl. Dyn. Syst. 2 (2003) 381-416]). As a consequence, the collision event enters the discrete equations of motion as an unknown that has to be computed on-the-fly whenever a collision is imminent. The additional bilateral constraints are efficiently dealt with employing a discrete null space reduction (including a projection and a local reparametrisation step) which considerably reduces the number of unknowns and improves the condition number during each time-step as compared to a standard treatment with Lagrange multipliers. We illustrate the numerical scheme with a simple example from polymer dynamics, a linear chain of beads, and test it against other standard numerical schemes for collision problems.

Optimal Control Strategies for Robust Certification

We present an optimal control methodology, which we refer to as concentration-of-measure optimal control (COMOC), that seeks to minimize a concentration-of-measure upper bound on the probability of failure of a system. The systems under consideration are characterized by a single performance measure that depends on random inputs through a known response function. For these systems, concentration-of-measure upper bound on the probability of failure of a system can be formulated in terms of the mean performance measure and a system diameter that measures the uncertainty in the operation of the system. COMOC then seeks to determine the optimal controls that maximize the confidence in the safe operation of the system, defined as the ratio of the design margin, which is measured by the difference between the mean performance and the design threshold, to the system uncertainty, which is measured by the system diameter. This strategy has been assessed in the case of a robot-arm maneuver for which the performance measure of interest is assumed to be the placement accuracy of the arm tip. The ability of COMOC to significantly increase the design confidence in that particular example of application is demonstrated.

Biomechanical optimal control of human arm motion

As both ordinary and well-trained human motion is mostly planned and controlled unconsciously by the central nervous system (CNS), human control mechanisms remain relatively obscure. Despite, they are an interesting topic, for example, with regard to improve protheses or athletic motion. To learn and understand more about the control of human motion, we use rigid multibody systems to represent bones and joints and formulate an optimal control problem (OCP) with the goal to minimise a physiologically motivated cost function, while the equations of motion and further nonlinear constraints have to be fulfilled. The investigated biomechanical movements are induced either via joint torques or via Hill-type muscle forces. We compare several cost functions known from literature to another one concerning the impact on the joints by involving the constraint forces. A direct transcription method called DMOCC (discrete mechanics and optimal control for constraint systems) is used to solve the OCP, whereby we benefit from its structure preserving formulation, as the resulting optimal discrete trajectories are symplectic-momentum preserving.

Frustration‐guided motion planning reveals conformational transitions in proteins

Proteins exist as conformational ensembles, exchanging between substates to perform their function. Advances in experimental techniques yield unprecedented access to structural snapshots of their conformational landscape. However, computationally modeling how proteins use collective motions to transition between substates is challenging owing to a rugged landscape and large energy barriers. Here, we present a new, robotics‐inspired motion planning procedure called dCC‐RRT that navigates the rugged landscape between substates by introducing dynamic, interatomic constraints to modulate frustration. The constraints balance non‐native contacts and flexibility, and instantaneously redirect the motion towards sterically favorable conformations. On a test set of eight proteins determined in two conformations separated by, on average, 7.5 Å root mean square deviation (RMSD), our pathways reduced the Cα atom RMSD to the goal conformation by 78%, outperforming peer methods. We then applied dCC‐RRT to examine how collective, small‐scale motions of four side‐chains in the active site of cyclophilin A propagate through the protein. dCC‐RRT uncovered a spatially contiguous network of residues linked by steric interactions and collective motion connecting the active site to a recently proposed, non‐canonical capsid binding site 25 Å away, rationalizing NMR and multi‐temperature crystallography experiments. In all, dCC‐RRT can reveal detailed, all‐atom molecular mechanisms for small and large amplitude motions. Source code and binaries are freely available at https://github.com/ExcitedStates/KGS/.