Guest Editorial

J. Comput. Nonlinear Dynam. 2017;12(2):020301-020301-2. doi:10.1115/1.4035538.

Multibody system (MBS) dynamics is a branch of computational mechanics dealing with modeling principles and computational methods for dynamic analysis, simulation, and control of complex mechanical systems. From its outset as engineering research field during 1970s, it focuses on mechanical models, mathematical formulations, and computational methods with the aim to establish efficient and reliable numerical simulation methods that allow for studying the dynamics of complex MBS. However, although many advanced formulations and algorithms have been proposed over the years, the steady growth of technological challenges requires ever more detailed high-fidelity models and the corresponding simulation procedures. After three decades of intensive research, a variety of reliable and computationally efficient numerical tools exist and have become an integral part of the development cycle in various industry sectors. At the same time, there is a growing demand for more robust and efficient numerical codes as well as for a better understanding of their mathematical backgrounds. This is where the geometric point of view finds its full justification.

Commentary by Dr. Valentin Fuster

Research Papers

J. Comput. Nonlinear Dynam. 2016;12(2):021001-021001-14. doi:10.1115/1.4034390.

We present a formulation of nonpenetration constraint between pairs of polytopes which accounts for all possible combinations of active contact between geometric features. This is the first formulation that exactly models the body geometries near points of potential contact, preventing interpenetration while not overconstraining body motions. Unlike many popular methods, ours does not wait for penetrations to occur as a way to identify which contact constraints to enforce. Nor do we overconstrain by representing the free space between pairs of bodies as convex, when it is in fact nonconvex. Instead, each contact constraint incorporates all feasible potential contacts in a way that represents the true geometry of the bodies. This ensures penetration-free, physically correct configurations at the end of each time step while allowing bodies to accurately traverse the free space surrounding other bodies. The new formulation improves accuracy, dramatically reduces the need for ad hoc corrections of constraint violations, and avoids many of the inevitable instabilities consequent of other contact models. Although the dynamics problem at each time step is larger, the inherent stability of our method means that much larger time steps can be taken without loss of physical fidelity. As will be seen, the results obtained with our method demonstrate the effective elimination of interpenetration, and as a result, correction-induced instabilities, in multibody simulations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021002-021002-8. doi:10.1115/1.4033441.

Configuration spaces with Lie group structure display kinematical nonlinearities of mechanical systems. In Lie group time integration, this nonlinear structure is also considered at the time-discrete level using nonlinear updates of the configuration variables. For practical implementation purposes, these update formulae have to be adapted to each specific Lie group setting that may be characterized from the algorithmic viewpoint by group operation, exponential map, tilde, and tangent operator. In this paper, we discuss these practical aspects for the time integration of a geometrically exact Cosserat rod model with rotational degrees-of-freedom being represented by unit quaternions. Shearing and longitudinal extension of the Cosserat rod may be neglected using suitable constraints that result in a differential-algebraic equation (DAE) formulation of the beam structure. The specific structure of unconstrained systems and constrained systems is exploited by tailored algorithms for the corrector iteration of the generalized-α Lie group integrator.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021003-021003-7. doi:10.1115/1.4034727.

Inspired by the designs of underwater gliders, hybrid autonomous underwater vehicles (AUVs) have emerged recently, which use internal actuators instead of control surfaces to control the heading angle and depth of the vehicles. In this paper, we focus on controlling the heading angle of a REMUS AUV by using an internal moving mass. We derive a nonlinear dynamical model of the vehicle with hydrodynamic forces and coupling between the vehicle and the internal moving mass. The model is used to study the stability of the horizontal-plane motion of the vehicle and to design a linear feedback law to stabilize its heading angle around a desired direction. Simulation results demonstrate that a controlled internal moving mass is able to fulfill the purpose of heading control.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021004-021004-18. doi:10.1115/1.4033520.

In this paper, nonsmooth contact dynamics of articulated rigid multibody systems is formulated as a complementarity problem. Minimal coordinate (MC) formulation is used to derive the dynamic equations of motion as it provides significant computational cost benefits and leads to a smaller-sized complementarity problem when compared with the frequently used redundant coordinate (RC) formulation. Additionally, an operational space (OS) formulation is employed to take advantage of the low-order structure-based recursive algorithms that do not require mass matrix inversion, leading to a further reduction in these computational costs. Based on the accuracy with which Coulomb's friction cone is modeled, the complementarity problem can be posed either as a linear complementarity problem (LCP), where the friction cone is approximated using a polygon, or as a nonlinear complementarity problem (NCP), where the friction cone is modeled exactly. Both formulations are studied in this paper. These complementarity problems are further recast as nonsmooth unconstrained optimization problems, which are solved by employing a class of Levenberg–Marquardt (LM) algorithms. The necessary theory detailing these techniques is discussed and five solvers are implemented to solve contact dynamics problems. A simple test case of a sphere moving on a plane surface is used to validate these solvers for a single contact, whereas a 12-link complex pendulum example is chosen to compare the accuracy of the solvers for the case of multiple simultaneous contacts. The simulation results validate the MC-based NCP formulations developed in this paper. Moreover, we observe that the LCP solvers deliver accuracy comparable to that of the NCP solvers when the friction cone direction vectors in the contact tangent plane are aligned with the sliding contact velocity at each time step. The theory and simulation results show that the NCP approach can be seamlessly recast into an MC OS formulation, thus allowing for accurate modeling of frictional contacts, while at the same time reducing overall computational costs associated with contact and collision dynamics problems in articulated rigid body systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021005-021005-10. doi:10.1115/1.4034728.

Estimation and filtering are important tasks in most modern control systems. These methods rely on accurate discrete-time approximations of the system dynamics. We present filtering algorithms that are based on discrete mechanics techniques (variational integrators), which are known to preserve system structures (momentum, symplecticity, and constraints, for instance) and have stable long-term energy behavior. These filtering methods show increased performance in simulations and experiments on a real digital control system. The particle filter as well as the extended Kalman filter benefits from the statistics-preserving properties of a variational integrator discretization, especially in low bandwidth applications. Moreover, it is shown how the optimality of the Kalman filter can be preserved through discretization by means of modified discrete-time Riccati equations for the covariance updates. This leads to further improvement in filter accuracy, even in a simple test example.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021006-021006-12. doi:10.1115/1.4035213.

In this work, we optimally control the upright gait of a three-dimensional symmetric bipedal walking model with flat feet. The whole walking cycle is assumed to occur during a fixed time span while the time span for each of the cycle phases is variable and part of the optimization. The implemented flat foot model allows to distinguish forefoot and heel contact such that a half walking cycle consists of five different phases. A fixed number of discrete time nodes in combination with a variable time interval length assure that the discretized problem is differentiable even though the particular time of establishing or releasing the contact between the foot and the ground is variable. Moreover, the perfectly plastic contact model prevents penetration of the ground. The optimal control problem is solved by our structure preserving discrete mechanics and optimal control for constrained systems (DMOCC) approach where the considered cost function is physiologically motivated and the obtained results are analyzed with regard to the gait of humans walking on a horizontal and an inclined plane.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021007-021007-14. doi:10.1115/1.4034729.

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021008-021008-7. doi:10.1115/1.4034862.

A mathematical model that invokes the Kutta condition to account for vortex shedding from the trailing edge of a free hydrofoil in a planar ideal fluid is compared with a canonical model for the dynamics of a terrestrial vehicle subject to a nonintegrable velocity constraint. The Kutta condition is shown to be nonintegrable in a sense that parallels that in which the constraint on the terrestrial vehicle is nonintegrable. Simulations of the two systems' dynamics reinforce the analogy between the two.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021009-021009-11. doi:10.1115/1.4034398.

This paper proposes an approach to formulation and integration of the governing equations for aircraft flight simulation that is based on a Lie group setting, and leads to a nonsingular coordinate-free numerical integration. Dynamical model of an aircraft is formulated in Lie group state space form and integrated by ordinary-differential-equation (ODE)-on-Lie groups Munthe-Kaas (MK) type of integrator. By following such an approach, it is assured that kinematic singularities, which are unavoidable if a three-angles-based rotation parameterization is applied for the whole 3D rotation domain, do not occur in the proposed noncoordinate formulation form. Moreover, in contrast to the quaternion rotation parameterization that imposes additional algebraic constraint and leads to integration of differential-algebraic equations (DAEs) (with necessary algebraic-equation-violation stabilization step), the proposed formulation leads to a nonredundant ODE integration in minimal form. To this end, this approach combines benefits of both traditional approaches to aircraft simulation (i.e., three angles parameterization and quaternions), while at the same time it avoids related drawbacks of the classical models. Besides solving kinematic singularity problem without introducing DAEs, the proposed formulation also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation and when aircraft motion pattern comprises steady rotational component of its 3D motion. This is due to the fact that a Lie group setting and applied MK integrator determine vehicle orientation on the basis of integration of local (tangent, nonlinear) kinematical differential equations (KDEs) that model process of 3D rotations (i.e., vehicle attitude reconstruction on nonlinear manifold SO(3)) more accurately than “global” KDEs of the classical formulations (that are linear in differential equations part in the case of standard quaternion models).

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(2):021010-021010-10. doi:10.1115/1.4034730.

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

Topics: Rotation , Screws
Commentary by Dr. Valentin Fuster

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