In terms of simulation and control holonomic constraints are well documented and thus termed standard. As non-standard constraints, we understand non-holonomic and unilateral constraints. We limit this survey to mechanical systems with a finite number of degrees of freedom. The long-term behavior of non-
					holonomic integrators as compared to structure-preserving integrators for holonomically constrained systems is briefly discussed. Some recent research regarding the treatment of unilaterally constrained systems by event-driven or time-stepping schemes for time integration and in the context of optimal control problems is outlined

Kern, Dominik

Römer, Ulrich J.

English

KITopen

Proceedings of the 8th GACM Colloquium on Computational Mechanicsfor Young Scientists from Academia and IndustryAugust 28–30, 2019 in Kassel, GermanyA Brief Survey on Non-standard Constraints: Simula-tion and Optimal ControlDominik Kern ∗ 1 and Ulrich J. Römer21TU Chemnitz, Germany2Karlsruhe Institute of Technology (KIT), GermanyAbstract. In terms of simulation and control holonomic constraints are well documented and thus termedstandard. As non-standard constraints, we understand non-holonomic and unilateral constraints. We limitthis survey to mechanical systems with a finite number of degrees of freedom. The long-term behavior of non-holonomic integrators as compared to structure-preserving integrators for holonomically constrained systemsis briefly discussed. Some recent research regarding the treatment of unilaterally constrained systems by event-driven or time-stepping schemes for time integration and in the context of optimal control problems is outlined.1 IntroductionMany technical systems are modeled as multibody systems which are comprised of rigid or elastic bodies,joints, and force elements or forcing. We restrict the discussion to discrete systems, meaning elastic bodiesare presumed to be discretized by shape functions or finite elements and the system is represented by afinite number of degrees of freedom. Joints are the typical pictorialization of constraints; for example theyconstrain a body to move only along or around a certain axis. Generally, constraints may either accountfor kinematic restrictions of a body, such as with redundant coordinates, or for interactions betweenbodies and the environment. Such interactions may be described by an equation, such as a pivot, or byan inequation, such as a non-penetration condition.Without any claim for completeness, we report about some active research related to simulation andcontrol. We adopt the common classification of constraints into: holonomic or non-holonomic (e. g.massless rigid connectors or rolling without sliding), scleronomic or rheonomic (e. g. static or movingenvironment), and unilateral or bilateral (e. g. limiters or bearings). Bilateral holonomic, particularlyholonomic-scleronomic, constraints are the simplest type of constraints and their treatment has been wellstudied and is covered in many textbooks [12, 32]. Therefore, we refer to them as standard. There aresome ongoing discussions regarding the remaining constraint types, non-holonomic and unilateral, whichis why we refer to them as non-standard.2 Non-holonomic constraintsWe need to recall the definition of a holonomic constraint to give meaning to the term non-holonomic.Having a multibody system with generalized coordinates q in mind, a holonomic-scleronomic constraintis given by an algebraic equation Ψ(q) = 0 on position level, whereas a holonomic-rheonomic constraintwould explicitly depend on time t. For further details on non-holonomic systems we refer to the classicaltextbook of Neimark & Fufaev [25] and more recently Rabier & Rheinboldt and Bloch [5, 26]. A popularformulation for the numerical solution of holonomically constrained systems was proposed by Gear, Guptaand Leimkuhler [11]∗Corresponding author: dominik.kern@mb.tu-chemnitz.de ORCID iD: 0000-0002-1958-2982MS 10: Dynamic Systems with Non–standard Constraints: Simulation, Control and Numerical Analysis211q̇ = v − GT(q)η , (1a)M(q)v̇ = f(q,v) − GT(q)λ , (1b)Ψ(q) = 0 , (1c)G(q)v = 0 , (1d)with mass matrix M(q), generalized forces f , Lagrange multipliers λ and η, and G = ∂Ψ/∂q. Itenforces the constraints on both, position and velocity level, and is also referred to as stabilized index-2formulation. A combination of differential and algebraic equations like in the example given by eq. (1) iscalled differential algebraic equation (DAE) or descriptor system.Non-holonomic constraints are usually understood as equations between differentials of dq that cannot beintegrated to an equation on position level. A general criteria for non-integrability gives the Frobenius-Theorem [14]. In mechanical systems, where most constraints appear as linear forms A(q)q̇ = 0, thereis a simple necessary condition. For a non-holonomic constraint the derivative of the corresponding rowin A must be a non-symmetric matrix.From the point of view of numerics, the inclusion of non-holonomic constraints into eq. (1) is straight-forward: they are added to the holonomic constraints on velocity level. This formulation works wellfor moderate time spans. The robustness and the excellent long-time behavior of structure-preservingintegrators for unconstrained and holonomically constrained systems [13] suggest taking advantage of thegeometrical properties of non-holonomic systems. However, non-holonomic systems do not, in general,preserve a symplectic structure. There is the open question whether energy conservation is the onlygeometric property of non-holonomic systems. Consequently, an energy-consistent scheme has alreadybeen proposed [4] to include both, holonomic and non-holonomic constraints. Recent research indicatesthat the good long-term behavior of current non-holonomic integrators is due to the special properties ofpopular benchmark systems and does not hold in general [22].Lie-group methods [8, 24] and their successful application for the description of large rotations are notconsidered further, since there is no hope to identify a Lie-group for any possible constraint.The consideration of non-holonomic constraints in optimal control problems is also more challenging thanfor holonomic ones. We restrict the discussion to feed-forward control, meaning we calculate a nominaltrajectory that is optimal in the sense of a given cost functional. These are constrained optimizations,because the equations of motion, or more generally the system dynamics, are imposed as constraints onthe optimal control problem. For a general discussion, we refer to the comprehensive textbook [12] andtake a glance at concepts motivated by structure preservation. Both, the discretization-based approachDiscrete Mechanics and Optimal Control of Constrained systems (DMOCC) [19] and the functional-based approach [15], are designed for holonomic constraints. Nevertheless, both allow different timediscretizations than variational integrators. Consequently, it will be interesting to find out how anenergy-consistent time discretization like the one proposed by Betsch [4] performs in this context.3 Unilateral constraintsWhile bilateral constraints describe relationships between some or all state variables via equations, unilat-eral constraints introduce one-sided limits on the admissible domain which result in inequations φ(q) ≥ 0or inclusions. Consequently, solutions may be non-smooth or even discontinuous. In mechanics, typicalorigins of unilateral constraints are collisions and friction. Several approaches exist to describe such sys-tems, we refer to [6, 17] for further details. Numerical methods for approximate solutions have to dealwith the (piecewise) non-smooth dynamics and follow one of two basic philosophies: event-driven or time-stepping, cf. figure 1. Event-driven schemes rely on algorithms for ODEs or DAEs to solve the smoothFigure 1. Numerical approximation of time behavior of state qj : left smooth solution (variable time step); middlenon-smooth solution with event-driven approach (variable time step); right non-smooth solution withtime-stepping scheme (fixed time step).MS 10: Dynamic Systems with Non–standard Constraints: Simulation, Control and Numerical Analysis212parts of the solution trajectory [3], discrete events where inequalities become active are determined itera-tively [29], a discrete state update is evaluated and the smooth solver is restarted. Time-stepping schemesapproximate the solution trajectory at fixed time steps where constraints are satisfied exactly (subjectonly to numerical precision), but might be (slightly) violated in between [1, 6]. Event-driven schemes arestandard in most commercial software tools, since high-order methods can be used for the smooth partsof the solution and good performance is achieved if there is a moderate number of events. They performbadly if there is a high number of events or even accumulation points (Zeno behavior), since this leads tovery small time steps and the algorithm gets stuck. Furthermore, existence and uniqueness of a solutionof the ODE or DAE after an event is not guaranteed (cf. the Painlevé paradox) [31]. Time-steppingschemes overcome these problems by “averaging” over the influence of all events during fixed time steps[23]. Velocity jumps due to collisions are treated by a coefficient of restitution which acts as though allcollisions within a time step occur simultaneously [30]. They can be formulated as implicit time integra-tion methods which require the solution of a linear or nonlinear complementary problem at every timestep [2].In terms of structure-preservation, event-driven collision integrators follow naturally from Hamilton’sprinciple, when the collision time is an additional variable [9]. Note, that this use of Hamilton’s principlein equality form [9] does not allow trajectories to approach or to exit motions along the admissibleset boundary, i.e. contacts being opened or closed. In addition, the adjusted time steps may slightlyviolate the conservation properties of variational integrators and thus accumulate errors. The extension ofclassical analytical mechanics to perfect unilateral constraints, with possible persistent contact, requires areformulation of the variational principles of mechanics in terms of variational inequalities [18]. It is thusnatural to consider a nonsmooth, discrete Variational Principle for inequality constrained systems [16].Since variational integrators are formulated in terms of momenta and avoid direct computation of forcesthey are a convenient choice for time-stepping schemes. There is ongoing research in the developmentof higher order methods to increase time step size and performance [28]. In the context of transientsimulations, the competition between event-driven and time-stepping schemes is not decided yet andmaybe never will.For the optimal control problem (feed-forward), we distinguish two cases: firstly one only requires com-pliance with the unilateral constraints, secondly there are collisions with the admissible boundary set.The first case, pure compliance with unilateral constraints such as actuator saturation, is comparativelysimple and commonly accounted for by the Karush-Kuhn-Tucker conditions [7]. The second case, whenthe events of interaction with the boundary of the admissible domain become subject of the optimization,requires more sophistication since then the related dynamics are non-smooth and the events, their numberand their states, are additional unknown variables. As an example, consider a pendulum-on-cart-systemwhere the cart moves between two limiters and the collisions with these boundaries are to be utilizedfor a swing-up of the pendulum. Since classical control theory presumes smooth dynamics, the directapproach, not to say brute-force, splits this problem into two layers. The inner layer finds the optimalsolution between two collisions. The earlier collision is given by a fixed post-collision state and the laterone by a fixed pre-collision state, defining already the subsequent post-collision state. The outer layer isused to find the optimal collision times and states [10]. A further outer layer may optimize the contactmode sequence, which is an integer programming problem growing exponentially with the number ofcontact constraints. Alternatively, time transformations [27] or contact-implicit trajectory optimization[21] can be used to avoid this mixed integer optimal control problem. We further mention that there isa close relation to the theory of switched systems [20].References[1] V. Acary and B. Brogliato. Numerical methods for nonsmooth dynamical systems: applications inmechanics and electronics. 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A Brief Survey on Non-standard Constraints: Simulation and Optimal Control

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A Brief Survey on Non-standard Constraints: Simulation and Optimal Control

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