Efficient response optimization of realistic vehicle models using an automatically-differentiated semi-recursive formulation Alfonso Callejo1 , Javier García de Jalón1 , Pablo Luque2 , Daniel Álvarez2 1 2
Instituto Universitario de Investigación del automóvil (INSIA), Universidad Politécnica de Madrid (UPM), Ctra. de Valencia km 7, 28031 Madrid, {a.callejo,javier.garciadejalon}@upm.es Área de Ingeniería e Infraestructura de los Transportes, Universidad de Oviedo, Campus de Viesques s/n, 33204 Gijón, {luque,mantaras}@uniovi.es
Abstract This article deals with the dynamic response optimization of mechanical systems. Specifically, the dynamic behavior of a coach is analyzed in detail, so as to improve its response in terms of handling and comfort characteristics. To that end, the coach is modeled as a 15-DOF multibody system by means of an efficient double-step semi-recursive formulation based on Maggi’s equations. Then, the effect of 19 design parameters on the dynamic response is assessed by performing a full state-of-the-art sensitivity analysis based on the direct differentiation method and the automatic differentiation of the motion differential equations. Finally, handling and comfort objective functions are defined and a multiobjective algorithm is run, improving the vehicle response in an effective yet automatic way. Keywords: response optimization, multibody dynamics, automatic differentiation, handling, comfort
1
Introduction
The design optimization of vehicles in the context of multibody dynamics has been widely studied over the last thirty years. The multibody dynamics approach enables a meaningful optimization of vehicles, provided both a rigorous model and a manageable optimization algorithm are available. Not infrequently, one of these requirements (or both) is disregarded, which results in either too simple academic examples or excessively cumbersome formulations. The latter is especially frequent when approaching the problem of computing sensitivities. Sensitivities are a numerical measure of how the design parameters affect the dynamic response of the system, and are often part of the optimization problem. This article deals with the optimization of the dynamic behavior of a real-life 18-DOF coach from a comprehensive point of view. All steps from the sensitivity analysis of the coach to the comfort and handling optimization are tackled in depth. The multibody formulation employed in this paper is based on a semi-recursive formulation proven to be both accurate and efficient: the double-step Maggi’s formulation developed by García de Jalón et al. [1]. According to this formulation, the motion differential equations of the multibody system can be written concisely as: ˆ z, z˙ ) − P(t, ˆ ˆ z, z˙ ) M(z)¨ zi (t) = Q(t,
(1)
ˆ is the vector of external forces, Pˆ is the vector of velocity-dependent inertia forces, z ˆ is the inertia matrix, Q where M and z˙ are the vectors of dependent relative positions and dependent relative velocities, and z¨ i is the vector of independent relative accelerations. Equation (1) can be integrated over time so as to compute the forward dynamics of the multibody system. In this paper, it will be used to evaluate the dynamic response of realistic vehicles. Once the dynamic response is known, optimization procedures can be used to improve it. The basic objective of any optimization procedure is to improve an objective function by varying the value of certain design parameters. Let us assume a generic dynamic response ψ of the multibody system depends on a set of design parameters b, such as inertias, masses, spring stiffnesses, damping coefficients, distances, angles, or any other model parameter. ψ = ψ (t, z(b), z˙ (b), z¨ (b), b)
(2)
A first step towards optimizing the model would be to quantify the influence of the design parameters on the system response. The problem of computing global sensitivities consists of differentiating such response with respect to b: ∂ ψ dz ∂ ψ d z˙ ∂ ψ d z¨ ∂ ψ dψ = + + + db ∂ z db ∂ z˙ db ∂ z¨ db ∂ b
(3)
where operator d(.) d(.) denotes total derivatives, and operator ∂ (.) ∂ (.) denotes partial derivatives. In this process, another level of derivatives is required: state sensitivities zb , z˙ b and z¨ b . Traditionally, three approaches have been used to compute state sensitivities in the context of multibody systems: numerical differentiation (ND), the direct differentiation method (DDM) and the adjoint variable method (AVM) [2]. The only straightforward technique of the three is ND, since the other two require a manual (analytic) differentiation of the motion differential equations. Nevertheless, ND is also the one with the largest error. Instead, in this article this equations are differentiated using automatic differentiation (AD) [3]. AD is a computational-mathematical technique that provides machine-precision derivatives of arbitrary computer functions with reasonable computational efficiency.
Figure 1. Realistic multibody model of the coach.
Once the sensitivity information is available, the optimization procedure can be carried out. Two basic qualities are usually considered in order to assess the quality of the dynamic response of vehicles: handling and comfort. Handling is related to lateral and directional stability, whereas comfort has to do with the subjective experience of vibration. Both concepts are tackled in this paper and improved in the coach model case by means of mathematical optimization. Then, due to the fact that improving one of them often makes the other worse, both objectives are considered simultaneously in the mathematical optimization process by running a multiobjective optimization algorithm.
2
Realistic modeling of vehicle dynamics
One of the key aspects of dynamic response optimization is to develop a realistic model of the system under study, so that the optimization results are not only feasible mathematical solutions, but also physically valid configurations. In this section, the main vehicle model characteristics and components are briefly explained. Fs f0
l0
c2
Fd
P = P loaded
c1
k
v 22
l
c4
v 11
c3
Compression
v
Rebound
Figure 2. Air spring and damper forces.
The coach under study is a Noge Touring 345 vehicle with frame from Mercedes-Benz. A coordinate-measuring machine has been used on the unloaded coach to obtain global dimensions and the position of key suspension points and joints. A general view of the real coach is shown in Figure 1. The coach has two axles: the front one has two wheels and the rear one four (assembled as two sets of dual wheels). The total mass of the coach is 13,498 kg when unloaded, and 17,048 kg when loaded. Throughout this paper, it is considered loaded for the purpose of considering the most critical conditions. The front suspension system is independent, while the rear one is a rigid axle. Suspension elasticity is provided through six air springs and two stabilizer bars, and damping through six regular dampers. Figure 2a shows the value of
the air spring force, which is considered linear w.r.t. the elongation of the spring. Figure 2b shows the damping force, which is modeled as a piecewise linear function w.r.t. the relative velocity of the ends. On the other hand, stabilizer bars are modeled as an angular spring between both bar ends, neglecting bending stiffness. All vehicle parameters have been measured when possible, and estimated otherwise. Regarding the torsion stiffness of the bodywork, it is considered by dividing the bodywork in two separate bodies linked by a revolute joint with a torsion spring acting along X-direction. The steering coordinate δ corresponds to the rotation of the steering actuator, which acts on the steering rods through the steering mechanism. In dynamic simulations, coordinate δ must be kinematically driven, and thus set by the user (usually as a function of time) in every step of the simulation. As far as tire forces, Pacejka’s Magic Formula is used to compute the contact point and the six contact forces (three linear forces and three torques). Also, a speed control is implemented so that the vehicle speed is constant during the two maneuvers presented later on. The resulting model is a 18-DOF multibody system.
3
Sensitivity analysis via automatic differentiation
Among the different ways of computing sensitivities presented in the Introduction, only DDM is considered because its efficiency is higher for problems with multiple outputs and because it enables a straightforward implementation in complex computer codes [4]. This approach requires the derivatives of the positions, velocities, accelerations (and Lagrange multipliers, if included in the formulation) with respect to the design parameters [5]. To that end, AD is implemented. 3.1
Direct differentiation approach
Based on the direct differentiation technique presented by Tomovic, Krishnaswami and Bhatti [6] and Chang and Nikravesh [7] presented the direct differentiation approach as a conceptually simpler alternative to the adjoint variable method. The method is based on the linearization of the objective function and the constraint equations presented in Eq. (2). In these expressions, as long as the vectors of state sensitivities zb , z˙ b and z¨ b are known, global sensitivities ψb can be calculated by integrating them over the time interval. Thus, in the direct differentiation approach the quid is to compute state sensitivities rather than global sensitivities. State sensitivities can be computed by applying the chain rule to the equations of motion in relative coordinates coming from Maggi’s approach, which have been previously presented in Eq. (1). ˆ z zb + Q ˆ z˙ z˙ b + Q ˆ b ) − (Pˆ z zb + Pˆ z˙ z˙ b + Pˆ b ) ˆ zi )z zb + (M¨ ˆ zi )b = (Q (M¨
(4)
where subscripts denote partial differentiation. Expanding the derivative terms and rearranging terms: ˆ b +Q ˆ q qb + Q ˆ q˙ q˙ b − Pˆ b − Pˆ q qb − Pˆ q˙ q˙ b − M ˆ zib = Q ˆ b z¨ i − (M¨ ˆ zi )q qb M¨
(5)
where the notation of a bar under a term ( ) indicates that it is treated as constant in the partial differentiation. Thus, the direct differentiation approach poses one system of ODEs per design parameter (see Eq. (5)). These sets of ODEs can (and must [8]) be integrated together with the set of ODEs in Eq. (1) for the independent sensitivities zib , as outlined by ˆ and [9]. One of the advantages of the direct differentiation method is that Eqs. (1) and (5) share the system matrix M, ˆ thus the factorization of M can be reused. 3.2
Implementation of automatic differentiation
Analytical differentiation is not a straightforward way of solving direct differentiation equations (5). It is by nature a time-consuming and error-prone approach. Automatic differentiation (AD) is an alternative computational-mathematical technique for the differentiation of computer functions. It is based on the decomposition of computer functions in elementary arithmetic operations (addition, subtraction, product, division) and calls to library functions (sine, cosine, exponential, etc.), and on the systematic application of the chain rule of differentiation [3]. Using AD, any mathematical code, no matter how complex, can be differentiated automatically. In this section, instead of applying Eq. (5), AD is used to differentiate Eq. (1) and obtain acceleration sensitivities z¨ ib . There are two main ways of implementing AD: using operator overloading tools and source transformation tools. Both have the same theoretical foundation. Source transformation tools generate a new code with the explicit expressions of the derivatives. This new code has to be called by the program where the derivatives are needed and conveniently compiled. On the other hand, operator overloading tools transform each variable of the function into a structure whose member variables are the value of the variable and the value of the derivatives with respect to the independent variables. Then, arithmetic operations and library functions are overloaded so that they handle those structures and compute both
Figure 3. Coach riding over speed bumps.
the result of the operation and the corresponding derivative, following the rules of differentiation. The original program remains almost unchanged, because all those substitutions of variables, operators and functions are carried out at compile time. This way, the implementation of AD is very versatile and valid for nearly all functions. However, the execution is rather slow because of the indirect costs of overloading operators and functions. In the present work, the C/C++ language operator overloading tool ADOL-C has been implemented [10]. Regarding the mode of differentiation (see [3]), as the number of inputs (i.e., the number of parameters) is smaller than the number of outputs, the forward mode is used. The code under differentiation, in terms of the state vector used in the time integration algorithm, is function y˙ = y˙ (y, b,t): z y≡ (6) z˙ i z˙ y˙ = y˙ (t, y, b) = (7) ˆ z, z˙ , b) − P(t, ˆ ˆ z, z˙ , b)) M(z, b)−1 (Q(t, The first time function y˙ (t, y, b) is executed, the computational graph (that is, the sequence of basic operations and calls to library functions) is recorded by the AD tool. Later on, this registry can be used to execute the function again or to compute the derivatives of the function with respect to the inputs. In this case, those derivatives are equivalent to the Jacobian matrix of y˙ with respect to b: zb yb ≡ (8) z˙ ib y˙ b = y˙ b (t, yb , b)
(9)
This way, motion differential equations (6)-(7) and sensitivity equations (8)-(9) can be integrated jointly using the same time integration scheme and without having to differentiate the equations of motion by hand. This approach, although not totally “automatic”, certainly allows for a rather automated way of computing derivatives, which can result in a versatile and easy-to-modify implementation of the sensitivity analysis, and therefore of the optimization process. 3.3
Selection of design parameters
One of the basic uses of sensitivity analyses is the identification of the most meaningful design parameters in a particular dynamic simulation. Without this kind of analysis, the designer might be disoriented about the parameter choice until the optimization is run. Another useful application is to run a sensitivity analysis in the optimal solution, or even in the set of Pareto-optimal solutions (in case a multiobjective optimization approach is followed). This would increase the knowledge about the sensitivity of the solutions. See [11] for more information on this issue. A full sensitivity analysis has been run prior to the optimization of the dynamic response. In it, 19 design parameters have been considered (see Table 1) and 2 maneuvers (corresponding to handling and comfort) have been run. About the dynamic maneuvers, more details will be given in the next section, as they will be used in the optimization procedure. Table 1 shows a list of the chosen parameters, their values and their units. Obviously, the sensitivity analysis returns a great deal of data, that is, the derivatives of all state variables (positions, velocities, accelerations) with respect to all parameters
Table 1. Initial choice of design parameters.
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Name Front spring stiffness k Front spring relaxed length l0 Front damper c1 coefficient Front damper c2 coefficient Front damper c3 coefficient Front damper c4 coefficient Front damper v11 coefficient Front damper v22 coefficient Rear spring stiffness k Rear spring relaxed length l0 Rear damper c1 coefficient Rear damper c2 coefficient Rear damper c3 coefficient Rear damper c4 coefficient Rear damper v11 coefficient Rear damper v22 coefficient Front stabilizer bar stiffness Rear stabilizer bar stiffness Bodywork torsion stiffness
Initial value 555555 0.2344 19830 9915 9915 4957 0.01 -0.01 555555 0.2344 19830 9915 9915 4957 0.01 -0.01 216000 200000 300000
Units N/m m Ns/m Ns/m Ns/m Ns/m m/s m/s N/m m Ns/m Ns/m Ns/m Ns/m m/s m/s N/rad N/rad N/rad
Table 2. Parameters sorted by descending maximum value of position sensitivities, and corresponding DOF.
# 2 1 10 17 9 18 3 13 11 5 15 16 7 8 12 14 6 4 19
Value 36.274482 8.182579 4.041814 3.518830 2.300821 0.915893 0.008292 0.004751 0.004557 0.004326 0.001761 0.001453 0.001078 0.001068 0.000853 0.000725 0.000561 0.000214 0.000000
Handling DOF Bodywork X-coordinate Bodywork X-coordinate Bodywork X-coordinate Bodywork X-coordinate Bodywork X-coordinate Front right wheel Bodywork X-coordinate Bodywork X-coordinate Front right wheel Bodywork X-coordinate Front right wheel Bodywork X-coordinate Bodywork X-coordinate Bodywork X-coordinate Front right wheel Bodywork X-coordinate Bodywork X-coordinate Bodywork X-coordinate -
# 2 10 9 1 14 12 4 6 18 11 13 17 15 16 3 5 7 8 19
Value 1.055710 0.970520 0.687086 0.257599 0.248523 0.108923 0.041066 0.026745 0.005639 0.004510 0.003946 0.003725 0.002232 0.001982 0.001716 0.000956 0.000819 0.000443 0.000000
Comfort DOF Upper left triangle Rear left stabilizer arm Exterior rear right wheel Upper left triangle Exterior rear right wheel Rear left stabilizer arm Upper right triangle Upper left triangle Bodywork Y-coordinate Rear left stabilizer arm Exterior rear right wheel Upper left triangle Rear left stabilizer arm Exterior rear right wheel Upper right triangle Upper right triangle Upper right triangle Upper right triangle -
over time. In order to visualize the importance of each parameter, the parameters have been sorted in descending order of the maximum absolute value of position sensitivities, as Table 2 shows.
dz (m/ m) dl0f
0.2 0.15 0.1 0.05 0
0
0.5
1
1.5
2
t (s)
2.5
3
3.5
4
Figure 4. Sensitivity of the Z-position w.r.t. the relaxed length of the front springs.
dz (m 2 / N) dk f
0.04 0.03 0.02 0.01 0
0
0.5
1
1.5
2
2.5
3
3.5
4
t (s)
Figure 5. Sensitivity of the Z-position w.r.t. the stiffness of the front springs.
From these tables, one can identify which parameters have influence on the response and, most importantly, which do not. Also, one position sensitivity has been plotted for each maneuver, and they are presented in Figures 4 and 5. Although not done here because of space constraints, it can be proven that the error when using numerical differentiation is noticeable and difficult to control, whereas automatic differentiation yields machine-precision sensitivities.
4
Dynamic response optimization
Once the sensitivity information is available, it is provided to the actual optimization algorithm, which is in charge of improving the dynamic response. To that end, objective functions have to be defined so as to monitor and assess the quality of suspension setups. Particularly, handling and comfort objectives are defined. When optimized separately, the results might not comply with the requirements of a mainstream coach, and thus a multiobjective approach is proposed. 4.1
Handling
The way dynamic responses are graded for a particular objective is always subjective, and depends on the specific vehicle and on the purpose of the optimization. The reason for this is that results are usually not general, and thus cannot be applied to other vehicles or situations. There are, however, a few guidelines available in the literature, which can help identify desirable handling characteristics. For instance, the vehicle behavior can be monitored when undertaking the following standard maneuvers: step-steer test (ISO 7401), constant-speed test (ISO 4138), double lane-change maneuver (ISO 3888-1), obstacle-avoidance maneuver (ISO 3888-2), severe braking and severe acceleration. The main purpose of this paper is to propose an optimization methodology, and thus only one maneuver is considered for the improvement of handling: the step-steer test. The maneuver consists of applying a severe step input on the steering mechanism, which makes the vehicle turn. Figure 6 shows the steering input over time and the corresponding lateral acceleration response during the step-steer test. Among all dynamic variables, the lateral acceleration is chosen as the monitored magnitude. The sooner the vehicle responds to the steering input, the better the vehicle is going to follow the driver’s commands. In terms of the lateral acceleration, the sooner the lateral acceleration starts to change, the better. One possible way of formulating this idea over time is to compute the definite integral of the lateral acceleration during the transient steering interval. Of course other
ay
−ψ h t
δ
t0
t
t1
Figure 6. Steering wheel input and lateral acceleration in the step steer test.
objectives could be conveniently weighted and added to this equation so as to capture the response more accurately. ψh = −
Z t1 t0
ay (t, b) dt
(10)
Differentiating this expression with respect to the design parameters, the gradient of the objective function can be written inp terms of the state sensitivities. Considering only one design parameter b for the purpose of clarity and assuming that ay ' x¨2 + y¨2 because longitudinal velocity is horizontal and almost constant, it follows: Z t1 Z t1 day dψh d x¨ d y¨ 1 p =− dt = − 2x¨ + 2y¨ dt (11) db db db t0 db t0 2 x¨2 + y¨2 Finally, a meaningful mathematical optimization requires the definition of constraint equations, especially in the case of real-life systems like the present one. Optimization constraints ensure that the results are not only mathematically feasible, but also physically reasonable. In the case of handling optimization, two basic types of constraints have been defined. The first kind enforces tire grip during the whole simulation, so that the vehicle does not lose lateral stability. The second one prevents the tires from losing contact with the ground. Both situations are clearly not desirable for coaches, and thus are banned in the optimization process. 4.2
Comfort
Comfort is usually considered as opposed to handling, i.e., the improvement of the handling response usually brings a worsening of comfort characteristics. Therefore, the design of regular car suspensions requires a tradeoff between handling and comfort responses, as does the design of coach suspensions. The analysis of the vehicle response with respect to comfort is also a very active research topic. Not only the methods for measuring vibrations can be complex, but also the effect of vibration on the comfort and health of the human body is difficult to assess [12]. In spite of this, some criteria have been gathered in international standards, most importantly in standard ISO 2631. Also, authors agree that vibration is most accurately evaluated looking at acceleration effects. As an example of how comfort can be evaluated, the coach is ridden over a set of five pairs of speed bumps. In order to cause a pitch movement on the bus, each pair of left and right bumps have the same X-coordinate, and thus are reached simultaneously by left and right wheels. See Figure 3 for a screenshot of the bump test. Regarding the formulation of the objective function, there are several ways of adding up accelerations over time: mean value, root mean square value, vibration dose value, etc. (see [12] for more details). In all of them, the objective is to obtain a single scalar that quantifies the amount of acceleration during the maneuver. In the present approach, the RMS value of the Z-acceleration of the bodywork is chosen for its simplicity. s ψc =
Z tend
z¨(t, b)2 dt
(12)
tini
Accordingly, the expression of the gradient in terms of the state sensitivities and a single parameter b can be obtained by applying the chain rule as follows: Z tend dψc 1 d z¨ = (13) 2¨z dt db 2ψc tini db Both the objective function and its gradient with respect to the design parameters can now be computed using these expressions together with the state sensitivities.
4.3
Multiobjective approach
As is known, optimization problems with more than one objective function are not straightforward to solve. In them, the designer plays a more important role than in standard nonlinear programming problems, mainly because multiobjective optimization problems do not have a unique solution. Very often the objectives are weighted at the designer’s discretion and added into a single objective function, which can then be optimized using standard methods. Moreover, multiobjective approaches are not always easy to implement and apply. Due to either of these reasons, multiobjective mechanical engineering problems are often not solved from a proper angle [11]. To the authors’ knowledge, the optimization of dynamic response of vehicles is not an exception. 3
1
2.9
0.8
2.7
0.6
ψc
b2
2.8
0.4
2.6 2.5 2.4
0.2
2.3
0
2.2
0
0.2
0.4
b1
0.6
0.8
1
−0.8
−0.7
−0.6
ψh
−0.5
−0.4
Figure 7. Design variable space and objective function space.
The approach followed hereby is not based on the scalarization of the objective function, but rather on the concept of Pareto optimality [11]. In order to visualize the concepts described here, parameters 2 and 10 are selected from the list of design parameters, and handling and comfort are considered as objective functions. Also, parameters are normalized as: bi =
bcurrent − blower bupper − blower
(14)
where bcurrent is the current value of the parameter, blower is the lower bound and bupper is the upper bound. Lower and upper bounds are set as blower = 0.5binitial and bupper = 1.5binitial , respectively. By defining a uniformly-distributed sequence of design parameters, the design space can be plotted in a 2D chart (see Figure 7a). The corresponding objective function space can also be computed and plotted (see Figure 7b). In these figures, blue dots correspond to parameter configurations that fulfill optimization constraints (and thus are feasible points), and red dots to configurations that do not. This way of computing the objective function space is an inefficient but effective way of assessing the optimality of the different parameter configurations. Strictly speaking, multiobjective problems have infinite Pareto-optimal solutions, all of them belonging to the Pareto front. In such front, one cannot improve an objective function without worsening the other one. A decision has to be made on what point of the Pareto front is chosen. If none of the objective functions is to be favoured, or the information on the importance of each objective is not available, the intersection of the first quadrant bisector with the Pareto front can be chosen as objective point. This is the approach followed here. Mathematically, this problem would be equivalent to the following one: min max ψi (t, z, z˙ , z¨ , b) (15) i
b
where ψi are the objective functions. This expression is equivalent to the following optimization problem: minimize γ ψh ≤ γ s.t. ψc ≤ γ
(16)
Regarding the software implementation, MATLAB’s minimax function has been used to solve the multiobjective optimization problem. In order to run the objective functions efficiently, they have been programmed in C/C++ and encapsulated in MEX-functions, which can then be called and controlled from MATLAB. The results of the multiobjective optimization problem are shown in Figures 9-8. The history of iterations and design parameter values can be seen in Figure 8. The difference between the optimized setup and the original one is clear in the plot of lateral acceleration during the handling maneuver(Figure 9a) and in the plot of vertical acceleration during the comfort maneuver (Figure 9b). Although the objective function reduction is not very large, the methodology is effective and the objective functions are improved. It is also interesting to see how the optimal point makes sense when comparing it with the objective function space in Figure 7. A remaining task would be to test the robustness of the method by moving away the parameters from the values of the real coach, and see if the method converges to the same results. Also, global methods could be used together with gradient-based methods to look for the optimum. 0.5
Normalized parameter
0.45
0.4
b1 (parameter #2) b2 (parameter #10)
0.35
0.3 0
0.5
1
1.5
2
2.5 Iteration
3
3.5
4
4.5
5
Figure 8. History of parameter values during the optimization process.
0.4
0.35 0.3
0.2
0.2
z (g)
y
a (g)
0.25
0.15 Initial Optimized
0.1 0.05 0 3
3.5
t (s)
4
4.5
0 −0.2 −0.4 −0.6
3.5
4 t (s)
4.5
Figure 9. Lateral acceleration during the step-steer maneuver and vertical acceleration on the speed bump maneuver.
5
Conclusions
A simple but powerful approach has been used to perform dynamic response optimization of a realistic coach vehicle. The coach has been modeled using an efficient semi-recursive multibody formulation implemented in C/C++, with which vehicle dynamics can be evaluated effectively. Then, a full sensitivity analysis of 19 design parameters has been carried out by using a state-of-the-art automatic differentiation tool called ADOL-C, which provides some insight about the relevance of each design parameter. With this approach, sensitivities are computed with a very limited amount of effort from the designer, who does not have to calculate analytic derivatives, and with a fast evaluation of the design changes. Finally,
the dynamic response of the vehicle has been optimized in terms of the handling behavior and the comfort characteristics. To that end, a multiobjective method has been applied. Overall, an all-inclusive optimization of the dynamic response of a realistic coach vehicle has been carried out. Even though the presented multiobjective problem can be considered as a “toy” problem, both the multibody model and the sensitivity analysis are considered in depth, and the approach is globally an effective dynamic response methodology.
Acknowledgments The authors acknowledge the support of the Ministry of Science and Innovation of Spain under Research Project TRA200914513-C02-01 (OPTIVIRTEST), and of the Education Department of the Government of Navarre, Spain.
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