the same set of discrete-time equations as in (69) and (70) at $q_i$, replacing all quantities indexed by $i+1$ by $i+1/2$ for the compression phase, ie, one writes $M(\dot{q}_{i+1/2}-\dot{q}_i) = \nabla\phi(q_i)N_{i+1/2}+D(q_i)\beta_{i+1/2}$. For the expansion phase, one replaces $\dot{q}_i$ by $\dot{q}_{i+1/2}$ (ie, one writes $M(\dot{q}_{i+1}-\dot{q}_{i+1/2})$) and the right-hand-side of the Lagrange dynamics becomes $\nabla\phi(q_i)N_{i+1}+D(q_i)\beta_{i+1}+e_p(\nabla\phi(q_i)N_{i+1/2}+D(q_i)\beta_{i+1/2})$. Finally, $q_{i+1}=q_i+h\dot{q}_{i+1/2}$ after these two half-steps. Let us note that the scheme in Eq. (55) incorporates the term $hM_{m,i}^{-1}Q_{m,i}$ in the calculation of the postimpact velocity, see Eq. (56). This is not the case for the algorithms in [84,86]. The term $hQ(q_i, \dot{q}_i)$ is however present in the right-hand-side for the compression phase calculation in [87]. Actually adding this term permits to better handle the problem posed by the capture of a constraint after an infinite series of rebounds (like the classical ball that rebounds on the ground), which is not the case if one applies an impact rule which neglects all non-impulsive terms. This can be understood with the 1D bouncing ball. In this case, one has $v_{i+1}=\text{prox}[V(q_{m,i}), v_i - h/2g]$. When $v_i$ becomes very small, then $h/2g$ dominates $|v_i|$ and there is no numerical problem in continuing the calculations. Since the term $-h/2g$ is $<0$, $v_{i+1}=0$ in the subsequent steps. So the threshold parameter $\gamma$ introduced in Section 6.3 is directly incorporated via $h$. On the other hand this numerical trick hampers the simulation of reversed accumulations of impacts [49,106] (a situation that might occur with a particle at rest on a table submitted to some excitation). However, such detachment conditions are met much less often than capture in practice.

There is, therefore, a significant difference between the schemes in [33] and [86]. Indeed in the second case, one integrates the motion and applies the impact rule when a collision has been detected, ie, $\phi(q_i)<0$. Then the algorithm computes $q_{i+1}=q_i+h\dot{q}_{i+1/2}$ after the two steps of the collision rule. When applying the maximal dissipation at $i$, the algorithm computes the quantities at $i+1$ by modifying abruptly the velocity direction if needed, but the forces $Q_i(q_i, \dot{q}_i)$ are part of the calculation (as in Moreau's scheme).

**Remark 16:**

* Similar backward Euler methods are used for the simulation of complex electrical circuits [160,221,222]. Notice that event-driven algorithms have also been used in this framework [223,224]. The discretization of so-called Linear Complementarity Systems [18,39,40] with an initial state jump as in Eq. (46) is studied in [159]. Sufficient conditions for consistency (ie, convergence of the discrete-time solution towards a solution of the continuous-time system) are given.

* Concerning multiple shocks with friction, it seems that both the algorithms in this section and Section VII A yield similar results, in the sense that the outcomes they provide are rather unpredictable: although they are formulated as a generalization of the frictionless plastic impact in Eq. (12), they may yield rebound depending on the parameters and data. The extension of the maximum dissipation principle and the generalization of Gauss' principle towards multiple collisions with friction, is far from being a trivial matter.

Several ways to formulate Eqs. (69) and (70) have been proposed by this group of authors. In [86], $M(q) \equiv M$ and $\nabla\phi(q_{i+1})\dot{q}_{i+1}$ is used. In [33], $\nabla\phi(q_i)\dot{q}_{i+1}$ and $M(q_{i+1})$ are used (which in practice leads to solving a NCP), but the work is essentially aimed at convergence proofs. In [85], the normal constraint is formulated as $\nabla\phi(q_i+h\dot{q}_i)q_{i+1} \ge \alpha$, which assures the respect of the constraints for the linearized dynamics. It is argued that the various quantities (mass matrix, $\nabla\phi$, matrix $D(q)$) can be calculated with $q_i+h\dot{q}_i$ or $q_{i+1}$ or $q_i+q_{i+1}/2$. But the last two approaches yield NCPs, no longer LCPs, consequently more difficult to implement, although perhaps more accurate. These variations are not gratuitous, as they may have strong consequences on:

* The implementation of the algorithms and their speed, accuracy, robustness, etc.

* The mathematical analysis needed to prove their properties (consistency, stability, etc).

These influences may have opposite effects! Some of these properties are discussed in the related papers. With respect to Löstedt's schemes, Stewart et al have improved the algorithm in several directions:

* **Friction:** 3-dimensional and 2-dimensional friction (Painlevé paradoxes for one contact are treated without resorting to any numerical trick).

* **Dissipativeness of the numerical scheme:** Stewart [33,88] proves that the algorithm in Eq. (70) with the maximal dissipation principle, is dissipative if $M(q)$ and $Q(q, \dot{q})$ are constant (linearized dynamics, or affine potential energy). The same is proved for the scheme in [85] provided $K$ is convex and $\phi(q_i) \ge 0$ for all $i$ (no violation of the constraints). Anistescu and Potra [86] prove a similar result when $M(q) \equiv M$. However Stewart proves that the limit of his scheme is dissipative (which makes sense since otherwise it could not be a solution of the original problem).

* **Convergence:** Convergence (but not uniqueness) of the solutions of Eqs. (69) and (70) towards a solution of Eq. (2) has been proved only for the one frictional contact case [33,34,87], encompassing Painlevé paradoxes. In the multicontact case, it seems difficult to prove the convergence of the piecewise constant solutions of Eq. (70) towards a solution of Eq. (2) that satisfies Coulomb friction law. It is not clear whether this problem comes from the formulation of Coulomb friction itself (through the generalized cone $C$ and the maximum dissipation), or from the interaction between the discretization and this formulation. In any case the problem of multiple contacts with friction still requires investigations. No convergence proof is available for the schemes in [84,86].

Some questions need to be still investigated: why solving LCPs at each step when the constraints are independent? What happens when systems are hyperstatic? Which problems does the multiplication of LCPs for solving Coulomb friction (polyhedral cone) create for real-time applications? Is the formulation of generalized friction for multiple contacts equivalent to Coulomb friction at each contact? Is it possible to recover sticking and sliding contacts from the
multicontact generalized formulation, in all cases? Solutions proposed for treating Coulomb with the revolution cone and new methods (homotopy etc) to solve NCPs need to be compared carefully with Moreau's Gauss-Seidel method. More generally it seems that there is a strong need for clarifying the domains of application and the performance of the various methods that allow to solve LCPs or NCPs (Lemke, NCP or LCP-functions and homotopy, nonsmooth Newton algorithms, QPs, interior point). See comments above and in Remark 9. From a general point of view, there is a lack of available numerical studies concerning the time-stepping schemes in Section 7.4 and comparisons with other methods.

## 7.5 Convergence studies

A very important mathematical study is the proof of convergence of the approximate (piecewise constant) discrete solution, towards a continuous function that is the solution of the continuous dynamics. These are the results of Monteiro-Marques, Stewart, Mabrouk and Paoli and Schatzman that we have outlined above. Let us define the approximate solution $q_h$ by

$$q_h(t) = q_i + (t - t_i) \frac{q_{i+1} - q_i}{h} \quad \text{if } t \in [t_i, t_{i+1}).$$

The goal is to prove that one can extract a subsequence, still denoted $q_h$, which converges uniformly to a solution $q$ of the Cauchy's problem. One shows first that the sequence $(\dot{q}_h)_h$ is bounded independently of $h$. Hence, Ascoli's theorem implies that there is a subsequence of $\{q_h\}_h$ which converges uniformly. Moreover, one establishes that the total variation of $\dot{q}_h$ is also bounded independently of $h$ and with Helly's theorem it follows that the sequence $(\dot{q}_h)_h$ converges pointwise (except perhaps on a countable set of points) to $\dot{q}$, and $(\ddot{q}_h)_h$ converges weakly to $\ddot{q}$ in the space of vector valued measures.

**Remark 17:** This notion of convergence is needed for the acceleration, because it allows one to get convergence of functions towards (singular) measures like the Dirac measure. This is not possible with other convergence notions. This is clearly explained in [87] in a way accessible to non-mathematicians.

Then one has to check that the limit $(q, \dot{q})$ satisfies the constraints (ie, $\phi(q(t)) \ge 0$ for all $t$) and the impact and friction laws. Convergence ensures that a scheme gives a good approximation of the continuous motion even when there is an accumulation point in the set of impact times. This is a great advantage of the time-stepping methods presented in Sections 7.2, 7.3 and 7.4 over all other methods, especially over the event-driven algorithms based on a systematic detection of impacts which may fail whenever there is an infinite converging sequence of impacts, without special attention to such phenomena in the schemes.

## 7.6 General comments

We have seen that Moreau and the NSCD method use a first-order discretization, the works in this section and in Section 7.3 as well, whereas Lötstedt chose multistep methods. Actually, if the goal is to build a LCP($\lambda$) at each step,

this is quite understandable since it may simplify the construction of this LCP. However there is another reason (that is also known in the DAE literature for problems with inconsistent initial data [29]): after each collision the order decreases to one. So if the system undergoes many collisions, why using higher order and/or multistep algorithms? Here comes into play the nature of the system to be simulated. It is argued in [86] that Runge-Kutta methods could be used, without further argumentation however. In [26] (page 264), it is indicated that multistep BDF methods result in a severe reduction of the step size $h_i$ when a discontinuity in the state occurs, because it tries to fit a polynomial through this jump. Reinitialization techniques must be used. This, however, may apply to systems that consist of switching DAEs, but cannot reasonably constitute a general method for MDIs simulation.

The problem of drift away from the constraints is seldom discussed in these works. In [85], (see also [27]) it is pointed out that the normal complementarity conditions could be stated as $\phi(q_i) \ge 0$, $N_i \ge 0$, $\phi(q_i)^T N_i = 0$. However this yields a NCP. It can be solved as a series of LCPs, but this may drastically increase the number of LCPs to be solved at each step (if one considers the ones coming from approximated friction). In [91], it is pointed out that using the conditions $0 \le \phi(q_{i+1}) \perp N_{i+1} \ge 0$ does not work because the behavior is that of a random impact when a shock occurs. Projection of $q_{i+1}$ back on the constraints is also alluded to as a remedy to drift. Lötstedt [27] points out that the velocity complementary slackness formulation in Eq. (49) may yield drift, and proposes to use Baumgarte's method as a remedy.

The integration step can be chosen as time-varying. Actually, the scheme in Eqs. (69) and (70) is of order 1 therefore not very accurate unless $h$ is decreased a lot. Similarly as Moreau's scheme, it should therefore be preferred for systems with a lot of events only.

Notice that it may however be argued that such modifications of time-stepping schemes may make them lose their basic interest, which is to remain simple enough but robust. In other words, they should be able to detect the main characteristics of a system (strange attractors, periodic orbits etc), without necessarily providing very accurate results (for instance if the solutions are very sensitive to initial data, it may be hopeless to get accurate numerical results).

However the reader should keep in mind that some of the presented time-stepping schemes provide a new value of the state and contact forces, at each step $i$. In many cases, the resolution of LCPs or NCPs passes through a fixed-point calculation (Newton's like, smooth or nonsmooth), similarly to the proximations or projections operations. Except when there is an abrupt change in the system topology (deletion or activation of a constraint, which implies a change in the indices sets and consequently in the LCP dimension), the root at step $i$ should not be too far from that at step $i-1$. Hence the apparent complexity of having to solve LCPs or NCPs at each step may be greatly simplified and accelerated in practice, provided $h$ is taken small enough (and provided the algorithm used permits to fix the initial conditions at will, which is for instance not the case for Lemke's). More generally it is of interest to reuse the data of the previous steps to
decrease the computational efforts in all the modules of the software, like contact force calculation [185] and collision detection [225]. The combined BDF-Newton Raphson algorithm proposed in [67] for systems as in Eq. (7) uses iterations to calculate $\mu_i$, $x_{1,i}$ and $x_{2,i}$ that satisfy the two equality constraints. Interestingly enough, it is pointed out in [67] that one iteration of the Newton-Raphson seems sufficient to get stability, from numerical experiments.

What is it that leads the authors to use one discretization procedure or another one? We already provided the reader with some elements of an answer, concerning Moreau's midpoint scheme. This may be the ability of constructing a LCP($\lambda_i$) or NCP($\lambda_i$) at each step (Lötstedt, Stewart et al), the combination of a second-order ODE discretization with shock dynamics (Paoli and Schatzman, Eqs. (65) and (64)). Recall also that the four classes of time-stepping schemes presented above, discretize different models. About DAEs, Petzold said that “BDF is so beautiful that it is hard to imagine something else could be better” [25], page 481. BDF combined with the reduction index in Eq. (7) has been applied to bilaterally constrained mechanical systems in [67] and has been shown to converge. This also may have been a motivation for the choice in Eqs. (49) and (50). There may also be significant discrepancies between the schemes constructed for mathematical convergence proofs, and the ones with more practical goals (compare Eqs. (57) and (55)). Moreau [28] (page 33) noted that multiple step methods seem a priori inadequate since one looks for algorithms allowing to take impacts into account. Consequently it seems that single step methods should be preferred to multistep ones. Then the advantage or drawbacks of Euler versus RK algorithms remains an open issue.

# 8 MATHEMATICAL ISSUES

Even for simple ODEs it is known that numerical methods that converge do not necessarily yield stable and robust results (see eg, [218], Chs 3 and 5). For instance an ODE as simple as $\dot{x} = -ax$ must be simulated with care when $a$ becomes large [218] (page 238) ie, when the problem becomes stiff. Then implicit methods are known to provide much better results [25]. For ODEs analysis of round-off errors influence, problems of conditional stability, are understood. For MDIs, things complicate so drastically that mere convergence is in general very hard to prove [32,33,54,55]. If one wants ultimately to avoid ad hoc tricks in the course of the simulation, schemes with strong mathematical foundations seem mandatory. Additional problems arise with the non-uniqueness of solutions: uniqueness is assured only if the data (constraints, external forces and mass matrix) satisfy some stringent conditions like analyticity [44,45,49–52,106]. Coulomb friction may also result in non-uniqueness of solutions [59,200]. We reiterate that such phenomena, although they are due to the rigid body assumption and/or the friction model, are not to be considered as artifacts. Rather, these simplified models allow one to highlight physical phenomena which otherwise would have remained hidden. For instance, the unbounded force/bounded impulse phenomenon of Painlevé's problem (see Section 8.1) can hardly be

guessed if a penalized scheme is adopted. But in the course of a simulation with a large stiffness, the contact forces may become very large as well and create subsequent numerical inaccuracies. The rigid body analysis allows one to better understand such a phenomenon, and consequently to better circumvent it.

## 8.1 Is Painlevé a real obstacle?

One should not think that Painlevé paradoxes (ie, non-existence or non-uniqueness of solutions to the LCP($\lambda$) for frictional contacts) occur only for unrealistic friction coefficients. This depends a lot on the contact geometry [201]. From the results of [33] and [200], the Painlevé paradoxes are better understood, at least in the simplest case of one contact. Time-stepping impulse-velocity schemes do not face Painlevé paradoxes since impulse remains bounded in the vicinity of critical points. Moreover the maximal dissipation principle allows one to impose a velocity jump (the so-called impacts without collisions) that prevents the system from penetrating into the zones of non-existence of a solution to the LCP(N), even if force-acceleration schemes are used. Notice that a penalized problem with high stiffness will necessarily yield the same problem, ie, computation of very large interaction forces. Once again the study of the rigid body case allows one to highlight some crucial properties of the models which have an important consequence for numerical applications. In other words, a compliant contact model with Coulomb (or any variant of Coulomb) friction will show instability phenomena for certain configurations and large enough coefficient of friction, see eg, [179]. However for finite stiffnesses the contact force always remains bounded. It is only the study of the limit case that reveals the underlying structure and the fundamental nature of the problem. A comparative study of compliant models behavior when the stiffness is large (via singular perturbations analysis) and the LCP(N) has been made in [125]. Since in many practical applications the contact stiffnesses are finite but quite large (gears commonly possess contact stiffnesses $k = 10^{10}$ N/m or higher) such studies are far from being of pure theoretical interest. Results in [226] and [227] show that the impacts without collisions are the limit behavior of solutions of penalized problems (with finite but high stiffness): there are configurations in which the velocity varies very rapidly when $k < +\infty$ and jumps in the rigid body limit as $k \to +\infty$. Though the results in [33,200] are for the moment essentially limited to the case $m=1$, they look quite promising. Some numerical results can be found in [227] for the scheme in Eqs. (69) and (70). As we pointed out at other places of this paper, one big challenge in multibody dynamics is a better modeling and understanding of dry friction in the multi-contact case. Painlevé-like phenomena require future investigations (such as the influence of the friction models, extension to multi frictional contacts).

For the classical Painlevé example (a slender rod that slides on a rigid ground, $m=1$ and $n=3$), the last problem that remains to be solved is uniqueness of the solutions. In particular as shown in [200] there exists in the phase plane a critical point that some trajectories may cross with un-
bounded velocities (but bounded impulses). However, just after this point, two solutions are possible: additional *rules* are necessary to decide which one has to be chosen. It is possible that studying penalized problems is going to provide us with such rules (which could hinge for instance on a certain relationship between the friction coefficient and the tangential and normal stiffnesses). This is reinforced by the fact that mathematical works [44,45,49,51,52] conclude that uniqueness holds only under very restrictive conditions on $\phi_j(q)$ and analyticity of all data. Since analyticity is much too stringent for practical applications, adding information into the model (but keeping the rigid body approach) seems mandatory. This should be done in a way similar to what has been done for collision rules [107,108]: use a better understanding of the physical process and lump these informations in some model parameters.

## 8.2 Is the discontinuity with respect to initial data a real obstacle?

Such phenomenon seems unavoidable and part of the dynamics. From a mathematical point of view (well-posedness), this does not necessarily preclude to get existence and uniqueness of solutions, see [40]. For stability of trajectories this may be an obstacle (it is known that Lyapunov stability is equivalent to continuity with respect to initial data [103] (page 124) uniformly in $t$ over $\mathbb{R}^+$). This is the reason why the available results on control of systems as in Eq. (2) remain until now restricted to codimension one or to orthogonal constraints, see [18–20]. New notions of stability have to be studied. For numerical simulations, there is no way to strike right at the singularity of $\partial K$ due to the finite accuracy of the calculations, so one can always apply a sequence of simple impacts and treat possible accumulations as usual. Evidently when getting close to the singularity the outcome becomes random [74,75,126]. This point added to the fact that even low-dimensional systems may possess a chaotic behavior [228,229], raises the question on whether it is useful to focus on the accuracy of the numerical algorithms. In this setting, the numerical computation of Lyapunov exponents is of primary importance in many studies, since they are used to detect chaos. As shown in [155], the discretization procedure has a strong influence on the numerical result.

## 9 COMMERCIAL SOFTWARE PACKAGES

Commercial softwares can be classified in two categories as a function of the frictional contact model they have adopted:

* Software packages with a penalized contact model.

* Software packages with unilateral contact model.

In this section, we will briefly discuss these two categories (and the contact models).

The penalized contact models are the most widespread mainly because these contact models are very easy to implement. Contact forces (normal and tangential) are considered as external forces, so they are just added to the right hand side of the dynamic equations. The normal contact force is usually modelled by a non-linear spring (ie, the elastic part)

and a damper (ie, the dissipative part) that is added between the two contacting bodies, in order to model a visco-elastic behavior of the materials. The normal contact force is then expressed in the following form:

$$F_n = F_{\text{elastic}} + F_{\text{damper}}, \\ \text{with } \begin{cases} F_{\text{elastic}} = K y^{\alpha} \\ F_{\text{damper}} = C \dot{y} \end{cases} \quad \text{if } y \ge 0. \quad (73)$$

Here $y$ denotes the penetration between the contacting bodies (positive means in contact), and $\dot{y}$ its first derivative as a function of time. The parameters $K$ and $\alpha$ can be estimated using Hertz theory (see, eg, [5]). It must be remarked that this theory is only valid under the following conditions:

* Central impact with the gravity centers of the contacting bodies and the contact point on the same line.

* Quasistatic phenomenon.

* Elastic impact.

Even if Hertz theory is not generally valid (due to the preceding hypothesis), it can give a rough idea of the values of parameters $K$ and $\alpha$. $K$ is sometime considered as a constant parameter (see for example Adams, MDI software) or is calculated as a function of the geometry of the contact bodies (see for example Mechanica Motion, PTC software). The normal elastic contact force can be considered to be as a good approximation. Things are much more complicated for the normal damping contact force. In fact, each software with a penalized contact model proposes its own formula for the damping factor $C$. We report here some expressions given in some widespread simulation tools that are valid for $y \ge 0$.

$$\left\{ \begin{aligned} F_{\text{damper}} &= \text{regul}(d, C_1)\dot{y} \\ F_{\text{damper}} &= C_2 K y^{3/2}\dot{y} \\ F_{\text{damper}} &= C_3 y^{1/2}\dot{y} \\ F_{\text{damper}} &= 2e_r m \sqrt{\frac{K}{m}}\dot{y}, \quad \text{with } \bar{K} = \frac{3}{2}\sqrt{mgK^2}. \end{aligned} \right. \quad (74)$$

The function regul is plotted in Fig. 4, $C_1$, $C_2$ and $C_3$ are damping factors, $e_r$ a parameter homogeneous to a restitution coefficient, and $m$ the equivalent mass of the contacting bodies.

The expressions in Eq. (74) are used respectively in Adams (MDI), Mechanica Motion (PTC), SDS (Solid Dynamics) and Dads (LMS). All these expressions verify $C=0$ for a zero penetration $y=0$. Moreover, the normal contact force is continuous as a function of the penetration $y$. The expressions in Eq. (74) can lead in some cases to energy gains. For example, in case of the simple example of a ball bouncing on a plane, the dynamic equation in the direction normal to the plane can be written (using Hertz theory for the elastic component):

$$m\ddot{y} = -F_n + mg, \quad (75)$$

where $F_n = Ky^{3/2} + C\dot{y}$ is the normal contact force. When $C$ equals a constant value or $C = C_3y^{1/2}$, the contact force $F_n$
<figure><img src="image_5.png" /><figcaption>Fig. 4 The function regul</figcaption></figure>

can be negative, that is to say attractive. As a consequence, the formulation of the normal contact force is changed, for example according to:

$$ 
\begin{aligned}
F_n &= \text{abs}(Ky^{3/2} + Cy'), \quad \text{or} \\
F_n &= \max(0, Ky^{3/2} + Cy').
\end{aligned} 
\qquad (76) 
$$

It is remarked that already in the case of the bouncing ball, for the same ball movement (ie, same velocity before and after the shock) the normal contact forces predicted by the preceding models are very different. So the impact forces must be considered with care and cannot be used as a precise estimation for structural analysis. The determination of impact forces is still an active research area. To end this discussion, let us state that the parameters K, α, and C are difficult to predict and cannot be measured easily.

For the tangential contact force, penalized softwares usually adopt a regularized Coulomb law (see Fig. 5a) or the sign function (see Fig. 5b) In Fig. 5, $F_t$ is the tangential contact force, $v_t$ the relative tangential velocity of the contacting bodies, $\mu$ the friction coefficient, and $veltol$ a parameter specified by the user.

Some recent softwares have started to implement unilateral contact models as an alternative to the penalized contact model. It is not easy to get reliable and precise informations on these contact models because vendors do not, in general, share implementation details with users. Consequently, we report here some very general informations concerning some of these softwares.

Probably the most famous software is Working Model (MSC). One of this software's particularities is its automatic

contact detection (in 2D and in 3D). It uses a Newton restitution coefficient in order to model dissipation during impacts (which may cause in certain cases energy gain), together with a fifth-order Runge-Kutta integration scheme (Kutta Merson). We assume that this program is an event-driven scheme, but are not sure that each event (like impact, lift-off or stick-slip transition) is accurately located. No precise informations could be obtained on the formulation of the frictional contact problem and on the way it is solved.

More recently, SDS (Solid Dynamics) has also adopted a 2D frictional contact model based on unilateral contact theory. Unlike Working Model, the contact model used in SDS is fairly well documented. Its general description can be found in [5]. It uses a Poisson restitution coefficient with a fifth-order Runge-Kutta integration scheme (Dormand and Prince). Each event is located accurately using a dichotomy procedure. The problem is written on the form of two coupled quadratic programming problems and is solved using the famous Moreau relaxation method (ie, Gauss-Seidel with a projection method).

The two preceding softwares are widely used in the industry. Of course other programs exist that support unilateral contact theory, like for example Simpack [192], but it seems that it is less complete than Working Model or SDS. For Simpack, only one reference could be traced on this software that only describes frictionless contacts. The problem is formulated in the form of a DAE and it is solved using a trial and error method (which may in practice be untractable).

All the commercial software packages are using an acceleration formulation (ie, event-driven formulation, with event detection). This kind of method is fast and reliable for problems with less than, say, a hundred of frictional contacts. If one plans to treat problems with thousands of contacts, one should use a time-stepping scheme (no event detection) to get an idea of the behavior of the system. (Even a rough idea because these time-stepping formulations are usually based on a first-order integration scheme.) But time-stepping schemes are research codes like LMGC of Moreau that has been extensively used for 15 years for the simulation of granular materials [13].

In conclusion, the penalized and the unilateral contact models have some advantages and some drawbacks, and the user has to be aware of them in order to choose the software

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5 Examples of tangential contact force approximations</figcaption></figure>
<table><caption>Table 1. Advantages and drawbacks of a number of softwares</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Penalized contact model</th><th colspan="2">Unilateral contact model</th></tr><tr><th>Event-driven schemes</th><th>Time-stepping schemes</th></tr></thead><tbody><tr><td>Most famous softwares</td><td>• Adams (MDI)<br>• Dads (LMS)<br>• Mechanica Motion (PTC)</td><td>• Working Model 2D & 3D (MSC)<br>• SDS (Solid Dynamics)</td><td>• LMGC (Moreau)</td></tr><tr><td>Advantages</td><td>• Model easy to implement<br>• No problem of redundancies<br>• No problem of impulse propagation (see also Newton cradle example)</td><td>• Restitution coefficient can be estimated<br>• Sticking effect taken into account (real Coulomb law)<br>• No oscillations of the contact forces<br>• Effective for less than 100 of frictional contacts (good CPU time)<br>• Very accurate</td><td>• Usable for thousands of frictional contacts (huge CPU time)<br>• Can give a rough idea of the results</td></tr><tr><td>Drawbacks</td><td>• Contact parameters unknown (K, α, C)<br>• Sticking effect not modeled<br>• Stiff differential systems<br>• Oscillations of the contact forces</td><td>• Redundancy varies during simulation<br>• No impulse propagation (see Newton cradle example)</td><td></td></tr></tbody></table>

to solve the problem at hand. To close this section on commercial softwares, let us summarize these advantages and drawbacks in Table I. This has to be considered as a complement of informations with respect to the material in Section 4.

**Remark 18 (Vehicle crash dynamics and simulation):**

The general problem is quite different: find how to reconstitute the motion of accidents from some data (physical parameters, estimation of dissipated energies etc.). Details and references can be found in [93] and [230].

## 10 GENERAL CONCLUSIONS

“...unilateral problems are slowly noticed by the scientific community, but are migrating fast into industrial applications” [231]. It is clear that the development of a reliable and efficient software for nonsmooth multibody systems is the result of assembling various modules. We have chosen not to incorporate numerical examples in this paper. The main reason is that it is not our aim here to classify the approaches (if this is possible), but to introduce the reader to the field of numerical analysis and simulation of nonsmooth multibody systems. Despite the many open problems that remain to be solved before getting a satisfactory software, the rigid body approach is quite interesting. One of the reasons is that compliant contact models may really fail in providing reliable schemes.

In this article, the state of the art about numerical simulation of unilaterally constrained mechanical systems has been described. The general message is that the analysis of such systems (mathematical, numerical, system theoretical) is by far not a simple extension of that of DAEs. Also the interconnections between modeling and these topics is an important point. It is not possible to decouple the mathematical, modeling and numerical problems. For instance multiple impacts create deep modeling problems. Even the frictionless case has not yet been solved in its generality, although the work in [108] seems quite promising. As pointed out above in this article, some authors argue that statistical modeling should be investigated. In regard to this point, it could be that

a pure *continuous* discretization of the dynamics in Eqs. (2) and (3) is not sufficient in general. A possible path is to resort to *hybrid* simulators which incorporate not only the classical event-driven or time-stepping algorithms, but also some *high-level rules* that would guide the user in the choice of future events (detachment, impact, sticking etc). This can be done via the construction of a tableau containing possible choices and their probability to occur. Such data may be obtained from experiments. This may constitute a way to connect experiments, modeling and numerics via some logic included in the software [65] (page 175) The determination of the qualitative properties of a system may be sufficient in certain applications (ie, one requires only to detect some dynamical invariants in the system, like attractors). Thus those schemes with less accuracy but high robustness and consistency results may prove quite useful (like time-stepping schemes). But qualitative properties may not be sufficient: quantitative ones can be crucial in industrial applications (eg, aeronautics where long-run simulations may be needed, sometimes one wants to observe the evolution of a system on several hours or days). Notice that the length of the simulation has to be modulated by the number of events occurring during the simulation: some very short motions (like in circuit breakers) can involve a great number of events on a very small time interval. Hybrid schemes that switch between various integrators depending on the phase of motion [232] may also represent a promising path for simulation of complex multibody systems.

Let us end this article by citing some of the important problems which still deserve deep research study (this list does not pretend to be exhaustive):

* Comparison and determination of the domains of applications of complementarity problems solvers.

* More generally, determination of the domains of applications of the algorithms on suitable benchmarks allowing one to detect their capabilities with respect to various dynamical situations (periodic/non-periodic motion, sensitivity to initial data or not, large or small N, etc)

* Better understanding of Painlevé-like phenomena in higher dimensions and multiple contacts.

* Modeling of multiple impacts with or without friction.
* Incorporation of more sophisticated dry friction models that satisfy the principle of maximum dissipation, in a mathematical programming framework.

## ACKNOWLEDGMENT

Discussions with David Stewart and Christoph Glocker on contacts with friction have been very useful to complete some parts of the article. This article was written while the first author was with the Laboratoire d'Automatique de Grenoble, France.

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<figure><img src="image_8.png" /></figure>

Bernard Brogliato graduated from the Ecole Normale Supérieure de Cachan in 1987, and got his PhD degree from the Institut National Polytechnique de Grenoble in 1991. He was a CNRS investigator from September 1991 to September 2001 and has been at INRIA Rhone-Alpes since September 2001. His main research interests are in dissipative systems and in nonsmooth dynamical systems analysis and control.

<figure><img src="image_9.png" /></figure>

AA (Tonny) ten Dam obtained his masters degree in Mathematics (specializing in Systems and Control) in 1986 and his doctoral degree in Mathematics in 1997, both from the University Groningen. In 1988, he joined the National Aerospace Laboratory NLR, Amsterdam, The Netherlands. He acts as consultant for simulation solutions and is involved in NLR's real-time high-performance simulation activities. His main research interests are representation, simulation, and control issues of constrained dynamical systems, with emphasis on aerospace applications.
<figure><img src="image_10.png" /></figure>

**Laetitia Paoli** obtained a civil engineer degree from ENTPE (Ecole Nationale des Travaux Publics de l'Etat, Vaulx-en-Velin, France) in 1989 and got her PhD degree from the University of Lyon 1 in 1993 in Applied Mathematics. She is an Associate Professor in the University Jean Monnet-Saint Etienne, and her main research interests are in the mathematical and numerical analysis of nonsmooth dynamical systems.

<figure><img src="image_11.png" /></figure>

**Frank Génot** graduated from ENSIMAG (Ecole Nationale Supérieure d'Informatique et de Mathématiques Appliquées de Grenoble) in 1993 and received his PhD degree from the INPG (Institut National Polytechnique de Grenoble) in 1998. He spent one year at the Mechanical Engineering Dept, MIT, and then joined the INRIA Rocquencourt, Paris. His main research interests are in modeling and control of non-smooth mechanical systems and structural control.

<figure><img src="image_12.png" /></figure>

**Michel Abadie** graduated from the Ecole Nationale Supérieure de Micro-Mécanique (ENSMM, Besançon, France) in 1993, and obtained his PhD degree from the University of Montpellier in 1998. Since 1995, he has been with Schneider Electric, working on the development of Mechanism Simulation. He is now in charge of the Mechanical Simulation group for the Low Voltage activity.