Multiphysics modeling and optimization of mechatronic multibody systems

Transcription

1 Multibody Syst Dyn DOI /s Multiphysics modeling and optimization of mechatronic multibody systems J.C. Samin O. Brüls J.F. Collard L. Sass P. Fisette Received: 9 March 2007 / Accepted: 4 June 2007 Springer Science+Business Media B.V Abstract Modeling mechatronic multibody systems requires the same type of methodology as for designing and prototyping mechatronic devices: a unified and integrated engineering approach. Various formulations are currently proposed to deal with multiphysics modeling, e.g., graph theories, equational approaches, co-simulation techniques. Recent works have pointed out their relative advantages and drawbacks, depending on the application to deal with: model size, model complexity, degree of coupling, frequency range, etc. This paper is the result of a close collaboration between three laboratories, and aims at showing that for non-academic mechatronic applications (i.e., issuing from real industrial issues), multibody dynamics formulations can be generalized to mechatronic systems, for the model generation as well as for the numerical analysis phases. Model portability being also an important aspect of the work, they must be easily interfaced with control design and optimization programs. A global demonstrator, based on an industrial case, is discussed: multiphysics modeling and mathematical optimization are carried out to illustrate the consistency and the efficiency of the proposed approaches. Keywords Multibody Multiphysics Modeling Optimization 1 Introduction The dynamical performances of modern mechatronic systems, such as robots, machinetools, vehicles or active space structures, result from the multiphysics interactions between J.C. Samin ( ) J.F. Collard P. Fisette Center for Research in Mechatronics, Université catholique de Louvain, Louvain-la-Neuve, Belgium samin@prm.ucl.ac.be O. Brüls LTAS-Vibrations et Identification des Structures, Université de Liège, Liege, Belgium L. Sass Colegio Politecnico, Universidad San Fransisco, Quito, Ecuador

2 J.C. Samin et al. mechanical, electrical, hydraulic, thermic, or electronic components. According to the system approach, the attention of the designer can focus either at the component or at the system level. At the component level, many local details can be analysed, but the interactions with the rest of the system have to be simplified. Conversely, at the system level, the interactions play a preponderant role, but local effects in the components are usually roughly approximated. As the performances of a machine are pushed ahead, additional interaction phenomena between the components become emergent at the system level. For example, a usual assumption when designing a machine or a robot is that the mechanical deformations can be neglected during the control design phase. In other words, the dynamical performances have to be limited in order to prevent any interaction between the bandwidth of the controller and the first natural frequencies of vibration. For high-speed and/or lightweight applications, this limitation should be overcome and it is necessary to account for the flexibility at the control design stage in order to achieve a simultaneous control of the motion and of the vibrations. A second example concerns electrical actuators, which can lose their ideal behavior when pushed to their limit. In this case, the interactions between the mechanical motion, the electrical network, and the control system have to be considered in a unifying framework. As a consequence, the traditional design approach, which is based on a sequential design of the components, evolves to a concurrent engineering framework, where a large range of phenomena have to be simultaneously taken into account. The complexity of this new approach represents a great challenge since experience and intuition are no more sufficient to guide the engineer to the solution. This motivates the development of multiphysics software tools for the systematic modeling, simulation and optimization of mechatronic systems. For the simulation of mechatronic systems, a first problem is to derive the nonlinear dynamical equations of the coupled system, and a second is to solve them in the time domain or in the frequency domain. For the first problem, two strategies allow to preserve the modular structure of the system in the formulation. According to the strongly coupled strategy, a global set of coupled equations is constructed by assembly of elementary contributions, so that a monolithic time-integrator can be used. According to the weakly coupled strategy, the assembly of the equations is not realized at the modeling level but the subsystems are simply treated in a sequential way with possible numerical iteration processes by the solver. For example, most commercial software relying on the block diagram language (e.g., MAT- LAB/SIMULINK) belong to this second class of methods. Weakly coupled approaches have the advantage of a better modularity: a partitioned solver with different step-sizes can be used [1, 2], and a co-simulation configuration may allow to model the different subsystems in different specialized software [3 7]. However, a consistent representation of the coupling effects, in particular in the presence of algebraic loops, can only be obtained at the cost of sophisticated algorithms and additional assumptions, which is an important drawback when compared to strongly coupled methods. The present paper essentially focuses on strongly coupled methods, for which, contrary to co-simulation approaches, the formulation of the coupled dynamical equations can be clearly distinguished from the numerical solver (time simulation, modal analysis, optimization, etc.), according to Fig. 1. Another important challenge is to find quickly and efficiently the right component parameters that achieve the best (or at least improved) system performances. The major difficulty stems from the intricate interactions and couplings between the mechatronic components, which makes difficult to understand intuitively the influence of parameter modifications. In order to achieve this task efficiently, mathematical optimization techniques are natural tools to provide a rationale methodology to solve these complex design problems. As shown in [8], the development of an optimization upper-layer over the integrated simulation would allow significant design improvements for mechatronic systems.

3 Multiphysics modeling and optimization of mechatronic multibody Fig. 1 Modeling versus analysis of mechatronic systems The paper is organized as follows. Section 2 focuses on multiphysics formalisms for the generation of the dynamical equations (see Fig. 1, left): after a short review of unifying approaches in Sect. 2.1, two formalisms issuing from the field of multibody dynamics will be presented in Sect Time integration (see Fig. 1, right) will be shortly presented in Sect. 3, considering the two previous modeling approaches. Then, Sect. 4 presents an industrial application for which multiphysics modeling and system optimization have been commonly tackled to demonstrate the feasibility of a global mechatronic approach, in the field of virtual engineering. 2 Modeling of multiphysics systems A modeling software aims at formulating automatically the coupled dynamical equations from a high-level description. Several approaches based on unified theories are reported in the literature; Sect. 2.1 shortly compares linear graphs, bond graphs and an equational approach based on a Lagrange formulation. Methods based on modeling languages, such as Dymola or Modelica [9, 10], will not be extensively discussed here. Alternatives approaches, based on multibody formalisms are also proposed in this paper. Indeed, any classical formulation in multibody dynamics aims at the modular and reliable representation of complex mechanical systems, composed of interconnected mechanical components (bodies, joints, etc.). They can thus be extended with a minimal effort and guaranteed consistency to mechatronic systems composed of mechanical and nonmechanical components. The most fundamental distinction among multibody formalisms concerns the assembly procedure of the coupled equations of motion, which can be implemented either symbolically or numerically. The symbolic format has the advantages of portability and efficiency, and it provides interesting insights in the analytical structure of the equations. However, numerical methods are able to deal with a more general class of problems, and they are especially suitable to model the dynamics of complex flexible mechanisms in a systematic way. Section 2.2 presents two original multiphysics modeling approaches, which extend classical multibody formalisms and lead to strongly coupled dynamical equations. Using the symbolic approach, we describe a method to model rigid multibody systems with electromechanical components or electrical circuits. Using the numerical approach, we present a general method to model flexible multibody systems with any kind of control system. 2.1 Methods based on unifying theories Unified modeling of mechatronic applications with strongly coupled mechanical, electrical, hydraulic, or thermic effects can be achieved using different approaches, among which: the

4 J.C. Samin et al. linear graph theory, the bond graph theory and a so-called equational formulation. A short review of these methods is proposed in this section Linear graph approach The linear graph theory is a branch of mathematics devoted to the study of systems topology. It was invented by L. Euler in the 1700s to study problems of connectivity and was extended in the 1900s [11] to the modeling of physical systems: individual components are identified and their constitutive equations are determined (theoretically and/or experimentally). In these equations, one distinguishes the through variables (measured by an instrument in series with the component) from the across variables obtained by an instrument in parallel. Figure 2 illustrates an example of linear graph (right) representing the electromechanical model of a loudspeaker (left), made of a variable capacitor connected in series with a resistor, an inductor and a voltage source (electrical part) and linked, on the other side, with a springdamper system (mechanical part). Without going into details, linear graph components are of two types: storage if they are able to store energy or dissipation if they dissipate energy (see Table 1). Each component is represented by a line (or edge). The nodes (or vertices) represent the physical connections between components. A constitutive equation is associated with each node and is expressed in terms of the through and across variables, their product being the power provided or dissipated by the component (these variables are commonly denoted as power variables). For instance, considering the constitutive equation of the damper in Fig. 2 A loudspeaker: electromechanical model and its linear graph representation Table 1 Linear graph analogies between domains (source [12]) Power variables Storage elements Dissipation Across variable Through variable A-type T-type elements Mechanics Velocity Force Mass (inertia) Spring Damper Electricity Voltage drop Current Capacitor Inductor Resistor Hydraulics Pressure Flow rate Capacitance Inertance Resistance Thermics Temperature Heat flow Capacitance Resistance

5 Multiphysics modeling and optimization of mechatronic multibody the loudspeaker example, F 6d = d 6 v 6 (d 6 being the damping coefficient), v 6, the relative velocity between the damper connections, is an across variable. The current i 3 in the inductor is typically a through variable which appears in the differential equation u 3 = L 3 di 3 /dt, u 3 being the associated voltage drop, and thus an across variable. To couple the different disciplines (in the example: the mechanical and electrical parts), specific graph elements called transformers are introduced and involve multiphysics equations in terms of the power variables. In the example, since the capacitor affects the physics of both domains, there is an edge for capacitor C 2 in the electrical and in the mechanical linear graph (see the dotted rectangle in Fig. 2). Once established, the global linear graph is at the root of the linear graph theory (see [11, 12] for more details) whose rules allow us to generate the minimal set of state equations that represent the dynamics of the whole system. Let us mention the advantages and drawbacks of the approach, with respect to the applications we want to deal with: Advantages: Linear graphs are intrinsically well-suited to multiphysics problems, being based on fundamental notions shared by the involved disciplines: type of variables, delivered and dissipated power. A linear graph bears a striking resemblance to the physical system, which is a real appeal when dealing with large and/or complex systems. For multibody systems (mechanical part), it is straightforward to switch from a relative coordinate to an absolute coordinate formulation. Drawbacks (mainly associated with the modeling of 3D multibody systems): Beside kinematic loop constraints which can be directly deduced from the graph, other more complex mechanical constraints (e.g. a wheel on a rail, a rolling without slip constraint) require an a posteriori introduction of the associated equations. The intrinsic efficiency of the dynamical models (i.e., equation compactness) generated by linear graph theory (for 3D mechanical systems) is rather low when compared to that obtained by recursive multibody formulations. A post-process can of course be achieved but this denatures the approach for the generation step Bond graph approach Bond graphs are more recent. They have been introduced by H.M. Paynter in 1959 and published in 1961 [13]. Bond graphs are based on the notion of energy whose exchange between components and with the environment is done via interactive ports which involve two scalar variables: a flow-type and an effort-type. Their product represents the instantaneous transmitted power. Table 2 shows the bond graph analogies between the elements (variables and components) used in the different fields. Table 2 Bond graph analogies between domains (source [12]) Power variables Storage elements Dissipation Effort variable Flow variable Capacitor Inertia elements Mechanics Force Velocity Spring Mass (inertia) Damper Electricity Voltage drop Current Capacitor Inductor Resistor Hydraulics Pressure Flow rate Capacitance Inertance Resistance Thermics Temperature Heat flow Capacitance Resistance

6 J.C. Samin et al. Fig. 3 A loudspeaker: electromechanical model and bond graph representation A bond graph is obtained by the interconnection of junctions and elements by means of bonds [14], according to the system topology. Contrary to linear graphs, a bond graph does not bear much resemblance to the physical system (see Fig. 3) but actually indicates the structure through which energy is exchanged. This can be of interest to visualize on-line the instantaneous transmitted power between components and between domains. The rules to produce the state equations are of course directly inspired from the graph and the type of elements and variables (see [12, 14] for more details). An interesting feature of bond graphs which must be pointed out is that they are naturally acausal, i.e. the bond graph structure is independent of the equation s causality. In other words, the same bond graph can be used to obtain different forms of models, for instance the direct dynamics or the inverse dynamics of a mechanical system: this feature is at the root of some multiphysics software based on the bond graph approach. According to our applications, advantages and drawbacks of bond graphs can be summarized as follows: Advantages: Acausal formulation. Straightforward energetical interpretation. Efficient detection of constraints between variables via the causality analysis [14]. Drawbacks (once again, they mainly concern 3D mechanics): Bond graphs do not resemble to the actual system: this can be really problematic for complex mechanical structures (see for instance Fig. 4 which shows, for a purely illustrative purpose, the bond graph of a double four-bar planar mechanism). Being one-dimensional, their extension to 3D multibody systems is complex and requires the development of vectorial bond graphs (or multi-bond graphs ) [15, 16]. In addition to a certain graphical heaviness (see Fig. 4), kinematic constraints are difficult to implement and second-order tensors are frame-dependent, which makes the 3D modeling of mechanical systems quite tedious [12] Graph theories and multibody dynamics Several authors have extended the concept of bond graph and linear graph for multibody systems, see [17, 18] and[19 22], respectively. According to the comparison study by Sass [12], the linear graph is more appropriate in that case, and it is noticeable that the

7 Multiphysics modeling and optimization of mechatronic multibody Fig. 4 Bond graph of a simple multibody system: a double four-bar

8 J.C. Samin et al. topological analysis automatically solves the kinematic problem. The extension of the linear graph method for flexible multibody systems was presented in [23, 24], relying on an a priori spatial discretization with assumed-modes. McPhee et al. [22, 25] also demonstrated the relevance of the linear graph method in electromechanics. However, as briefly discussed above, the main interest of linear and bond graph methods consists in using analogies between sub-domains via the use of power variables (through/across and flow/effort respectively) to express the energy exchange between elements and with the environment. Once the graph is built, rules can be applied to generate the state equations but in case of systems involving complex mechanics (see Sect. 4 for instance), the formulation of the equations is far from being optimal in terms of efficiency, unless a post-process is used to simplify them via the resort to recursive techniques, symbolic manipulations, etc Equational approach Several authors [26, 27] applied the virtual work principle for the modeling of electromechanical systems. Its scalar formulation is a definite advantage to mix various contributions from electrical and mechanical subsystems. The method relies on the selection of a set of independent generalized coordinates, able to describe the electromechanical configuration of the system. For rigid mechanisms, this principle leads to the standard Newton Euler equations in terms of independent joint coordinates. Hadwitch and Pfeiffer [26] applied the virtual work principle to rigid multibody systems interacting with electrical networks. Variational principles are also very useful for the discretization of coupled distributed problems (e.g. coupling of electrostatic and elastic deformation fields [28, 29]). Let us for instance consider the field of electromechanical (multibody) systems. Using the Lagrange formulation (which, in fact, can be deduced from the virtual work principle), one may define (see [12, 26]): for the mechanical part, the mechanical Lagrangian: L m = T U with T, the kinetic energy and U, the potential energy, for the electrical part, the electrical Lagrangian: L e = Wm W e with Wm, the magnetic co-energy and W e, the electrical energy of the circuit. It is shown in [12] that the Lagrange equations of the electromechanical system reads: where: d dt ( Lem ṡ ) L em Γ = 0 (1) s L em = L m + L e represents the electromechanical Lagrangian, s = (s m,s e ) represents the mechanical and electrical generalized coordinates, Γ are the mechanical and electrical applied generalized forces. The analytical developments to end up with the above formulation are rather long. 1 As for the previous methods, let us mention some advantages and drawbacks of this equational approach [12]: Advantages: Unified electromechanical modeling able to deal with 3D mechanical applications. 1 More than ten pages for the mechanical part for unconstrained multibody systems.

9 Multiphysics modeling and optimization of mechatronic multibody Solution of algebraic and kinematic constraints (even nonlinear) using robust methods, well-proven in multibody dynamics straightforward extension to electrical constraints. Drawbacks: Introduction of electrical variables which have no physical sense (e.g. electrical charge in inductor). Equation formulation (Lagrange or virtual work principle) not well-suited to recursive formulation, at the root of efficient multibody formalisms. No physical interpretation in terms of power and energy, without resorting to additional post-process. 2.2 Alternative methods The main goal of the unifying approaches presented in the previous section is the generation of the strongly coupled equations describing the system dynamics, i.e., without resorting to co-simulation techniques. However, those methods have some limitations when considering the two following important specifications. 1. The nonacademic nature of our applications (e.g., articulated railway boogies equipped with induction motors, cars with electro-hydraulic semi-active suspensions, lightweight manipulators with active vibration control) requires the introduction of very diversified and specific models related, for instance, to wheel rail contact, three-phase induction motor circuits, valve dynamics or control systems. 2. In recent virtual engineering applications, dynamical models are not only used for timedomain simulation, but also for control design and optimization. Therefore, the model should be easily portable towards these disciplines; but moreover, it must be very efficient from a computational point of view since optimization methods may require a relatively high number of full simulations (e.g., one simulation for each evaluation of the cost function). These preliminary comments justify our choice to tackle multiphysics applications by generalizing the modeling capabilities of well-established multibody formalisms, rather than starting from graph theories, in spite of their intrinsic multiphysics nature. Before developing two approaches (a symbolic-based in Sect and a numerical-based in Sect ), a common multiphysics formulation can be written. The equations of motion of a rigid or flexible multibody system have the general form: M(q) q = g(q, q,t) J T λ, (2) ϕ(q,t) = 0 (3) where q is the n 1 vector of mechanical coordinates λ is the m 1 vector of Lagrange multipliers associated with the m constraints ϕ. M is the configuration-dependent mass matrix, g gathers internal, external and complementary inertia forces, and J is the Jacobian matrix associated with the constraints ϕ(q,t) = 0. Nonholonomic constraints could also be included, but they are not considered here for conciseness of notations. In a mechatronic context, additional equations are necessary to account for the nonmechanical components. They can usually be given as first-order state equations in descriptor form: Eẋ = f(x,q, q,t) (4)

10 J.C. Samin et al. where x is the state vector and E is the descriptor matrix (for an explicit state-space system, E would simply be the identity). In order to describe the nonmechanical components in a modular way, we will see that algebraic variables y should be introduced in order to represent the interactions between subsystems: 0 = h (y,x,q, q,t). (5) The actuators lead to an additional force g which is added to g in the mechanical equation. We thus define: g (q, q,x,y,t) = g(q, q,t)+ g (q, q,x,y,t). (6) The mechanical and nonmechanical variables are thus strongly coupled in the global dynamical equations. The two formalisms presented in the following sections fit into this general framework. However, the internal structure of the equations and the interpretation of the variables x and y are quite different in both cases Symbolic modeling of multiphysics systems Symbolic generation of multibody models manipulates arithmetical operators (+,,,/, etc.) and strings of alphanumeric characters (mi, qdi, dij,...) to generate in a set of files the analytical form of the system kinematics and dynamics using a desired syntax (e.g., C, Fortran, MATLAB, SIMULINK). For a given multibody application, this symbolic generation is performed only once, exactly as in manual generation for very small systems. For rigid multibody applications, symbolic generators exhibit the same level of generality as their numerical competitors in the sense that they can handle large systems with any topology and containing up to 100 degrees of freedom. However, they allow drastic simplifications, from the most trivial (addition/multiplication by zero) to the most complex ones (simplification of trigonometric expressions, elimination of unnecessary equations). Comparison tests [30] showed that, all other things being equal, a symbolic multibody model performs a time simulation between five and ten times faster than a purely numerical one, which is not at all negligible. There do exist commercial general purpose symbolic computation packages. Why do we not use these? There are two reasons for this. The first and most important one relates to the amount of computer memory they require for generating medium-sized and large models symbolically, which can be incompatible with the current computer technology. The second reason concerns the possible simplifications of the symbolic expressions. Although the simplification capabilities of commercial packages are powerful, the condensation of multibody equations relies on specific rules which can be applied more easily by developing a dedicated symbolic program;thiswasthe main motivation to develop the multibody program ROBOTRAN [31]. More recently, another interesting aspect of the symbolic approach has emerged in the framework of the analysis of hybrid mechatronic systems involving enlarged multibody models (i.e. including electrical actuators, hydraulic devices,...) and requiring enlarged analysis (control, optimization,...). Indeed, since symbolic multibody programs clearly separate the modeling part from the analysis part (see Fig. 1), it is quite straightforward to extract from them the symbolic model in order to export it towards another environment (e.g., for control or optimization purpose). In other words, symbolic multibody models are portable, and this feature is very attractive for the analysis of mechatronic applications. From the modeling point of view, the extension of the symbolic approach to other fields than mechanics has been achieved by Sass [12] for electromechanical systems. The main steps of the approach are summarized here below.

11 Multiphysics modeling and optimization of mechatronic multibody Symbolic modeling of multibody systems To describe the dynamics of a multibody system composed of N rigid bodies, one can write the virtual work principle (from which the Lagrange equations can be easily obtained) in the form: N ( δr T i (m i r i F i ) + δψi T i=1 ( H i L i )) = 0 (7) in which, r i and ψ i respectively denote the absolute position and (3) orientation parameters describing body i configuration. r i is the absolute acceleration of the center of mass of body i. H i is the time derivative of the angular momentum of body i with respect to its center of mass. F i and L i are respectively the resultant force applied to body i, and the resultant moment about its center of mass. From the body kinematics description, the virtual quantities δr i and δψ i can be expressed in terms of generalized virtual coordinates δq. Equation (7) can thus be rewritten in terms of the generalized coordinates q, q, q and δq, leading to: Φ T (q, q, q)δq = 0 (8) which, being linear in terms of the accelerations q, can be rewritten in the form (2) (without considering the constraint forces term). From the mechanical point of view, Lagrange equations and the virtual power principle formulation (8) have a theoretical interest, especially to prove in (2) the symmetry and the positive definite character of the mass matrix M and to justify in(2) the constraint force term J T λ using the Lagrange multipliers technique [32]. Computer implementation the problem of model efficiency is not solved because the complexity of the multibody equations (in relative coordinates) produced by the virtual work principle (8) is comparable to that obtained from the Lagrange formulation, unless a recursive formulation is achieved. 2 From a computational point of view, recursive multibody formalisms (e.g., Newton Euler recursive scheme, Order-N scheme) are known to be very efficient when dealing with large systems in relative coordinates (e.g. the car of Sect. 4). The virtual power principle and the recursive Newton Euler formulations for unconstrained systems being fully equivalent [32], recursive methods can be advantageously used to compute system (2) in order to get a maximal efficiency. Moreover, since symbolic manipulations can highly profit from that recursivity [32], it is possible to generate multibody models with the ROBOTRAN symbolic program [31] in a very efficient manner, compared to other formalisms and to pure numerical approaches. Symbolic modeling of electrical systems For electrical systems (actuators,...), Lagrangetype equations can be obtained (see Sect ), or, equivalently, a virtual work principle can also be written, starting from the electrical energy of an electrical circuit defined by E = u T idt = u T idq ch = i T dφ in which u and i are the column matrices of the voltage 2 Which is not obvious with such formalisms.

12 J.C. Samin et al. drops across the elements of the circuit, and the currents flowing through them, respectively. The amount of charge accumulated in the elements constituting the circuit is represented by q ch = idt, while φ = udt represents the flux passing through the elements. These two ways of calculating the electrical energy comprise the two main formulations for the principle of virtual work: Formulation in terms of charge variations leading to charges as generalized variables: u T δq ch = 0. (9) Formulation in terms of flux variations leading to flux as generalized variables: i T δφ = 0. (10) Computer implementation: as for the multibody model, a particular attention has been paid by L. Sass [12] to the computational efficiency of the model. This is the purpose of the ELECTRAN symbolic program. 3 In ELECTRAN, a circuit-based formalism different from classical graph representations of Sect. 2.1 is proposed and is based on topological concepts inspired from multibody representation, allowing to deduce electrical dynamics and constraints in a fully symbolical form, in a far more efficient way than when starting from the general formulation (9). The formalism is summarized here below. Based on the same starting equations as for the virtual work principle, Kirchoff s laws and constitutive equations, the proposed circuit theory formalism leads to similar equations expressed in terms of voltages and currents, instead of charges of fluxes that suffer from physical insights (see Sect. 2.4). Each element k of the circuit is characterized by the current i k flowing through it, the voltage drop u kk between its starting and end nodes and a constitutive equation relating these two variables. Among these variables, the current through the inductors i l and the voltage drops across the capacitors u c will be the state variables: x T = ( i lt u ct ). (11) The other variables, denoted y, arethealgebraic variables, appearing only in purely algebraic relations: y T = ( ) i st i rt u st u rt u lt i ct (12) in which r, s stand for resistor and source elements respectively. Together with Kirchoff s laws, resistances, current and voltage sources constitutive equations form a set of pure algebraic equations, involving x and y: Ay = Bx + Cz (13) in which z k represents the value of the (current or voltage) source element k. Considering the l + c differential equations of inductors (u l = f(i l )) and capacitors (i c = f(u c )), we can write the dynamical equations of the electrical system: in which d(m e x) dt ( u l = i c ) + d dt ( ) φ p 0 (14) 3 ELECTRAN is the ROBOTRAN s companion program for electrical dynamics.

13 Multiphysics modeling and optimization of mechatronic multibody M e denotes the electrical mass matrix and contains the inductances and capacitances of the circuit. φ p represents possible permanent magnets fluxes. Since (u lt i ct ) contains the subset of algebraic variables y, one can write: ( ) u l = Dy and (14) then reads: i c M e ẋ = dme x + Dy + d dt dt ( φ p 0 ). (15) For electromechanical converters, matrix M e becomes function of mechanical variables varying with time and thus the term dme must be taken into account in (15). dt When matrix A is full rank, it is possible to solve the algebraic system (13) with respect to y. The corresponding solution y = A 1 (Bx + Cz) (16) can be introduced in (15) which then becomes a pure set of ODEs in the state variables x. Symbolic modeling of electromechanical multibody systems When considering electromechanical systems, 4 (2), (13) and(15) apply. Let us assume that only mechanical positions q influence the electrical parameters, except for sources that may also be influenced by velocities q. The electrical equations become: A(q)y = Bx + Cz(q, q), (17) ) ( ) M e (q)ẋ = ( dme φ p (q, q) x + Dy +. (18) dt 0 In the above equations, B and D only concern Kirchoff s equations and are not influenced by mechanical variables. In the mechanical equations, supplementary generalized forces of electrical origin (cf. (6)) will appear: g em = W m q act W e q act (19) where W m, W e have been defined in Sect and q act corresponds in this particular case to the actuated joint in the multibody system. Equations (2), (17, 18) and(19) represent the global dynamical model of the multibody electromechanical system. They can be computed within the same numerical step of a numerical analysis (for instance, a time integrator function evaluation), which is a key point when dealing with multiphysics modeling. 4 Momentarily unconstrained.

14 J.C. Samin et al. Fig. 5 Electrical constraints: loop of capacitors, cutset of inductors Constrained electromechanical systems As regards constraints, which can be of mechanical or electrical nature, the coordinate partitioning method [33] can be used for both domains. The method was originally proposed for multibody systems in which nonlinear constraints may exist, and can be easily extended to electrical systems according to the previous circuit formulation. The method exhibits various advantages: 1. The selection of an optimal 5 subset of independent electromechanical coordinates. 2. For electrical circuits, the automatic detection of (Fig. 5): Loops of capacitors and voltage sources Cutsets of inductors and current sources which, as they only involve state variables and source values, make A singular in expression (13). A Gauss triangularization with pivoting allows us to solve the problem and additionally to compute, as for mechanical systems, the electrical number of degrees of freedom. 3. For mechanical systems, the exact solution of nonlinear constraints at position, velocity and acceleration levels at each time step, within the model computation (and not via the time integrator process). 4. The full reduction of the differential/algebraic system (2) to a pure ODE system (size = number of electromechanical degrees of freedom), compatible with standard integration schemes (see Sect. 3). The main drawback of the method is the necessity of a on-line re-partitioning in case of system re-configuration (mechanical part) or when the morphology changes; this is the case for instance with intermittent contact in mechanics, or with perfect diods and/or transistors in electrical circuits (see [34, 35] for instance) Numerical modeling of multiphysics systems Symbolic approaches allow to generate compact models for rigid (electromechanical) multibody systems, which can be easily exported to other software for simulation, control or optimization. However, when flexible effects arise in the mechanical subsystem, more general analysis and design tools have to be considered. The nonlinear finite element method is certainly the most general, consistent and systematic approach in flexible multibody dynamics, see Géradin and Cardona [36] for a detailed description. Each flexible body is divided into smaller elements, whose displacement fields are parameterized using nodal coordinates, namely translations and rotations with respect to an inertial frame. This parameterization leads to sparse equations of motion in DAE form. 5 Numerically speaking.

15 Multiphysics modeling and optimization of mechatronic multibody Fig. 6 Model of a mechatronic system. Using the block diagram language, the control system is described as a set of interconnected subsystems This section presents some extensions of the finite element method for the modeling and simulation of mechatronic systems. The description of nonmechanical elements within a finite element context is not an innovative idea, and for instance, user elements are available in most industrial software for multibody systems. However, the implementation of a user element is an intricate, time-consuming and sometimes unreliable process, which motivates a more systematic and theoretically founded approach. Hence, we propose to integrate the block diagram language within the finite element formulation. Relying on a nonlinear state space description, this approach presents the advantages of modularity, generality, with a language very familiar to control engineers. Moreover, the finite element assembly procedure allows the construction of the strongly coupled dynamical equations in a systematic way. Figure 6 illustrates the model of a mechatronic system, where the block diagram language is used to describe the control system (controller, actuators and sensors). Mathematically, the global interactions between the mechanical and control systems are represented by two vectors: the n w 1 vector w contains the sensor measurements exploited by the control system and any other mechanical variables that may influence its dynamical behavior. Therefore, w is associated with the displacements and velocities: w = L wq q + L w q q (20) where L wq and L w q are sensor localization matrices. The n a 1 vector g a represents the generalized forces exerted by the actuators on the mechanism: g = L qa g a (21) where L qa is the actuator localization matrix. Each block or subsystem (k) is modeled by explicit state equations with respects to its own inputs u (k) and outputs y (k) : ẋ (k) = f (k)( u (k),x (k),t ), y (k) = h (k)( u (k),x (k),t ). (22) Hence, one defines the global vectors of input, state and output variables: u (1) x (1) y (1) u =., x =., y =. (23) u (n block) x (n block) y (n block)

16 J.C. Samin et al. with the connectivity relations (see Fig. 6): u = L uw w + L uy y, g a = L ay y. (24) The coupled equations of motion are then derived: M(q) q = g(q, q,t) J T λ + L qy y, 0 = ϕ(q,t), ẋ = f(u,x,t), 0 = y h(u, x, t) (25) where L qy = L qa L ay. The formulation could be extended to account for acceleration measurements. The vector u does not contain any independent variables; it is simply a convenient notation for u = L uq q + L u q q + L uy y. (26) This equation is obtained from (20)and(24), with L uq = L uw L wq and L u q = L uw L w q. 3 Time integration The coupled dynamical equations have a DAE structure: the multipliers and the interconnection variables y are algebraic in nature. It is well known that the differentiation index associated with the mechanical subsystem is three, see Brenan et al. [37]or Hairer and Wanner [38]. In contrast, the nonmechanical subsystem is represented by a set of semi-explicit index-1 DAEs as shown in [39]. It is possible to solve those equations directly using a DAE solver, but it is often useful to reformulate the equations of motion before integration. We shall briefly review both approaches. The coupled dynamical equations can be solved using a monolithic time integration algorithm. The classical approach relies on the reformulation of the mechanical equations as first-order DAEs, by the introduction of the velocity vector v: q = v, M(q) v = g (q,v,x,y,t) J T λ, 0 = ϕ(q,t), Eẋ = f(x,q,v,t), 0 = h (y,x,q,v,t). (27) Those equations can then be solved using a solver for first-order DAEs, such as RADAU5 [38]. A dual approach is to reformulate the state equations as second-order DAEs, by the introduction of the integral state vector z: z(t) = t 0 x(τ)dτ (28)

17 Multiphysics modeling and optimization of mechatronic multibody leading to the equations: M(q) q = g (q, q,x,y,t) J T λ, 0 = ϕ(q,t), E z = f(ż, q,v,t), 0 = h (y, ż, q, q,t). (29) We note that z is only meaningful at the velocity level (ż = x) or at the acceleration level ( z =ẋ). Those equations can be solved using a solver for second-order DAEs, such as the generalized-α scheme [39]. This latest allows to adjust the numerical damping at high frequencies, which is very useful for finite element models with unresolved modes in the highfrequency range. The time integration of high-index DAEs leads to important numerical difficulties. Therefore, it is often recommended to make an index reduction before time integration. Indexreduction techniques are classical in multibody dynamics, and we do not intend to give here an overview on this topic. Let us mention constraint differentiation methods (with a possible projection step), Baumgarte stabilization [40], the Gear Gupta Leimkhuler stabilization [41], the method based on overdetermined DAEs [42], the coordinate partitioning [33], and the coordinate splitting [44]. As regards the coordinate partitioning method [33], which is a well-known technique to transform the DAE multibody equations of constrained multibody systems into a reduced ODE system, it can be easily extended to mechatronic systems, as used by the symbolic approach of Sect For a purely mechanical multibody system with m constraints, 6 let us recall that the method splits the n coordinates into n m independent coordinates q I and m dependent coordinates q D. ( q I ) q = q D. (30) The dependent coordinates q D can be eliminated within each function evaluation (of any numerical algorithm) from the equations of motion by solving the constraints ϕ(q,t) at position, 7 velocity and acceleration levels. After elimination of the Lagrange multipliers, a reduced unconstrained system is obtained in terms of the independent coordinates: M ( q I) q I + g ( q I, q I,t ) = 0. (31) This system can then be solved easily using standard ODE integrator algorithms. Of course depending on the nonlinear and/or stiff character of (31), care must be taken in choosing an adapted scheme. With respect to equation numerical stiffness, it was shown in [43] thatimplicit integrators are not incompatible with the above coordinate partitioning ODE reduction: equations in (31) can be re-formulated as an inverse dynamic problem additionally leading to an advantageous order-n formulation which can be used as the implicit integrator residual. The iterative tangent matrices needed to make the time-step solution progress (ex. in a Newmark-type implicit scheme) can be computed numerically or even symbolically, without any constraints violation (see [43]). 6 Which are generally non linear and can be holonomic or not, rheonomic or scleronomic. 7 For holonomic constraints.

18 J.C. Samin et al. Among the drawbacks of the coordinate partitioning method, let use mention that, due to mechanical singularities of closed-loop systems, a given set of independent coordinates q I does not allow a description of the motion in the whole configuration space. Therefore, a re-partitioning strategy must be implemented to overcome this problem: this is certainly the main difficulty of the approach but this can be solved easily using modern computer programming environment. The extension of the coordinate partitioning approach to mechatronics is rather straightforward. Indeed, if the descriptor matrix E in (4) is regular, the state variables x are independent, whereas the variables y (see (5)) are dependent. Conversely, in case of a singular matrix E, the state variables simply need to be partitioned into dependent and independent variables, leading to define the nonmechanical degrees of freedom. This approach has been followed in [45]. 4 Demonstrator: semi-active suspension of an Audi A6 The system under consideration is an Audi A6 car (see Fig. 7) equipped with four semiactive hydraulic dampers, whose force-velocity characteristics can be modified and controlled via an electro-valve. As semi-active hydraulic actuators with current controlled valves are considered, the suspension system is moved from a purely mechanical system to a mechatronic design. Figure 7 shows also a test-vehicle on a test-rig with four hydraulic shakers which are capable of independently exciting the four car-wheels with a desired road profile. The mechatronic model of the car has been derived using both formalisms of Sects and Itinvolves: A mechanical sub-model: the vehicle (chassis suspensions wheels) for which a multibody approach is used. An electro-hydraulic sub-model: to describe the behavior of the semi-active shock absorbers, oil pressures and flows in the damper chambers are computed taking into account oil compressibility, active and passive valves characteristics and the accumulator behavior. A control sub-model: the latter has been developed by Lauwerys et al. [48] and tailored to satisfy some comfort and/or ride criteria (input: car body acceleration and damper velocity output: electro-valve current). This model, whatever the formalism which produced it, will be in the heart of the optimization cost function and thus must be efficient. Fig. 7 Audi A6 (left) and the same vehicle on 4 poster rig for testing and its instrumentation with corner accelerometers (right)

19 Multiphysics modeling and optimization of mechatronic multibody 4.1 Mechanical model Vehicle The chassis and the suspension system of the Audi A6 are modeled as a multibody system (see Fig. 8, left). It includes the car-body and chassis, rear and front multi-link suspension mechanisms (including the passive springs) (see Fig. 8, right), the slider crank direction mechanism for the front wheels and the tyre model. From the multibody point of view, this 3D model is rather complex since it involves: around 50 revolute or prismatic joints, 18 closed-loops, 4 wheel ground contact models with a vertical compliant force and a lateral slip model. The current model could be extended to include the stiffness of the suspension bushings, the flexibility of the chassis, and coupled longitudinal-lateral model for the wheels. However, in the framework of this simplified study, the current model is sufficient. Both approaches described earlier have been used to develop the multibody system model: the symbolic approach (Sect ) has been carried out with the ROBOTRAN software [31], whereas the numerical approach (Sect ) is implemented in the OOFELIE software [46] Semi-active shock absorbers The semi-active shock absorber hardware (see Fig. 9 left) corresponds to that of a passive shock absorber in which the piston and base valve are each replaced by a check valve. A current controlled CVSA valve (continually variable semi-active valve) connects the two damper chambers. The current to this valve is limited between i = 0.3A and i + = 1.6A, which corresponds to the least and most restrictive positions of the valve (i.e., open and closed), respectively. When the rod moves up (positive rattle velocity), the piston checkvalve closes and oil flows through the CVSA valve. Because the volume of the rod inside the cylinder reduces, oil is forced from the accumulator into the cylinder through the base check-valve. When the rod moves down (negative rattle velocity), the piston check-valve opens. Because the volume of the rod inside the cylinder increases, the base check-valve closes and oil flows from the cylinder into the accumulator through the CVSA valve. The dynamical behavior is quite complex, and a detailed nonlinear model [47], available as C-functions, has been calibrated by the manufacturer of the shock absorber, Tenneco Automotive Company (Saint Trond, Belgium). This model can be presented in nonlinear Fig. 8 Multibody model of the car chassis (left) and the multi-link rear suspension

20 J.C. Samin et al. Fig. 9 Semi-active damper (left) and(right) fitted curves of the shock absorber damping force as a function of rattle velocity for different CVSA valve current. A low/high current to the CVSA valve corresponds to a small/large restriction yielding a low/high damping ratio state-space format, defining the inputs u (damp) =[l r dl r /dt i v ] T (l r, the rattle extension, i v, the electrical current in the CVSA valve), the state variables x (damp) =[p reb p comp ] T (p reb and p comp the pressures in the rebond and compression chambers) and the output y (damp) =[g a ] T (g a the force exerted by the damper): ẋ (damp) = f (damp)( u (damp),x (damp)), y (damp) = h (damp)( u (damp),x (damp)). (32) However, when coupled with the multibody dynamics model, model (32), assuming oil compressibility, results in a set of a stiff differential-algebraic system. Therefore, a fitted set of curves (see Fig. 9, right) has been realized on the basis of experimental data. The fitted relationships provide quasi-static damper force with respect to the CVSA electrical current, and the rattle velocity: g a = g a (i v,v r ). (33) They can provide a satisfactory model of the shock absorber in the design perspective. A flexible model-free control structure has been developed by Lauwerys et al. [48]andthey have implemented it on a real passenger car equipped with a semi-active suspension. Part of this control structure is adopted here and tuned by simulation and optimization methods. The control structure consists of three stages (see Fig. 10): a feedback linearization (inverse actuator models), a transformation into modal space (coupling and decoupling operations), and a linear integral control, which corresponds to the well-known skyhook controller. Here, the car and the dampers are represented as a black box, whose inputs are the four CVSA electrical currents i v and the road excitation (uncontrollable inputs), and whose outputs are the rattle velocities v r = dl r /dt, and the accelerations measured at the four corners of the car a b. 4.2 Simulation According to Sect. 2.2, two approaches are used to carry out the integrated simulation of the mechatronic system. As regards the symbolic approach, which enables to generate a standalone dynamical model, the full mechatronic model is exported towards the SIMULINK

21 Multiphysics modeling and optimization of mechatronic multibody Fig. 10 Controller of the semi-active suspension. vr stands for the rattle velocities, ab for the car body accelerations, and iv for the valve currents (from Lauwerys et al. [48]) Fig. 11 Lane change maneuver (standard qualification test) environment via S-functions (in compiled C language), ready-to-use for fast-running simulations or for the implementation of a controller. Let us emphasize that the optimization algorithms (see Sect. 4) will need to iterate on a full SIMULINK time integration of the model 8 whose very fast computation allows us to produce the optimization results in a reasonable CPU time (a couple of hours). Conversely, when using the FEM approach for the MBS modeling, the integrated simulation consists in implementing the hydraulic and the controller models within the MBS simulation package using the block diagram language explained in Sect A lane change maneuver has been simulated (Fig. 11). The car has a 10 m/s initial velocity, and a driver applies an open-loop steering command, without any real-time correction (blind driver assumption). The motor and the brakes do not produce any torque on 8 Which consists of the 3D multibody structure with four wheel/road contacts, the coupled hydraulic model of the shock absorbers and the associated controller.