MB simulations for vehicle dynamics: reduction through parameters estimation

Transcription

1 MB simulations for vehicle dynamics: reduction through parameters estimation Gubitosa Marco The aim of this activity is to propose a methodology applicable for parameters estimation in vehicle dynamics, aiming at generating reduced models to be adopted for functional analyses and real time simulations with the focus on enabling a model conversion scheme, allowing building a communication bridge between the 1D and the 3D simulation domains. The benchmark case, i.e. the high fidelity model, is defined in the 3D multibody environment of LMS Virtual.Lab Motion, while the model is developed symbolically with the help of Maple and implemented in the block scheme oriented interface of Imagine.Lab AMESim. To estimate the physical and structural parameters of the detailed benchmark model for use in the functional model, it is important to make clear how and to what extent the response of the system depends on each parameter. Therefore sensitivity analysis and optimization loops are programmed to firstly define the most effective contribution to the behaviour of the model and secondly asses a good correlation in the dynamic performance. Reference vehicle model The use of MBS software allows the modeling and simulation of a range of vehicle subsystem representing the chassis, engine, driveline and body areas of the vehicle as shown in Figure 1, where is intended that multibody system models for each of those areas are integrated to provide a detailed representation of the complete real vehicle. Figure 1: Integration of subsystems in a full vehicle model and detail of the vehicle dynamic areas of interest Here also the modeling of road and driver are included as elements of what is considered a full vehicle system model. Restricting the discussion of full vehicle system models to a level appropriate for the vehicle dynamics, a detailed modeling of the suspension systems, anti roll bars, steering system and tires is needed as evidenced with ovals in Figure 1. Of course to

2 complete the model and give the possibility to run simulations with an acceptable realistic level of the vehicle s response, models of driver inputs as well as engine and driveline final effects should be comprised. Besides the inertia characteristics of the sprung (and un-sprung) masses have to be included. A creation of a virtual environment is than an aspect that can be considered. Set up of the MB model Figure 2: Front double wishbone and Rear 5-link suspension (example of the Acura RL and TSX 28) The double wishbone front suspension is constructed with short upper wishbones, lower transverse control arms and longitudinal rods whose front mounts absorb the dynamic rolling stiffness of the radial tires. The spring shock absorbers are supported via fork-shaped struts on the transverse control arms in order to leave space to the crank shafts and are fixed within the upper link mounts. As its name indicates, the rear suspension employs five links. The hub-carrier/spring-shockabsorber mount is located by five tubular links: a trailing link, lower link, lower control link, upper link, and upper leading link. Car manufacturers claim that this system gives even better road-holding properties, because all the various joints make the suspension almost infinitely adjustable. The linkages of the suspension parts are realized partially with bushings, representing the compliance elements, and for some of them ideal joints have been included, realizing a kinematically constrained mechanism as represented in Figures 3 and 4. The bushing element available in Virtual Lab Motion defines a six degree-of-freedom element between two bodies, producing forces along and torques about the three principal axes of the element attachments. The bushing characteristics are defined as a combination of six values of stiffness and six values of damping which are normally defined by non linear spline curves. The equation below describes the formulation for forces in the bushing. F 1 = Kz + DŜ + F K (z) + F D (Ŝ) F 2 =-F 1 where F 2 and F 1 are the force vectors applied to body 1 and 2 K and D are the stiffness and damping matrixes z and Ŝ are the relative displacement and velocity vectors between the two bodies F K (z) and F D (Ŝ) are the forces expressing stiffness and damping as functions of relative displacement and velocity in a nonlinear sense. A similar formulation is used for the calculation of torque reactions in function of relative rotation and rotation velocity between the connected bodies.

3 Table 1: Connection types (Joints and Bushings) for the front suspension Figure 3: Linkage of the Front Suspension system

4 Table 2: Connection types (Joints and Bushings) for the rear suspension Figure 4: Linkage of the Rear Suspension system

5 Assumptions and force elements considered Antiroll Bars They're also known as sway-bars or anti-sway-bars. The function of the anti-roll bars is to reduce the body roll inclination during cornering and to influence the cornering behavior in terms of under- or over-steering. The anti-roll bar is usually connected to the front, lower edge of the bottom suspension joint. It passes through two pivot points under the chassis, usually on the subframe and is attached to the same point on the opposite suspension setup. Hence, the two suspensions are not any more connected only due to the subframe and the chassis, but effectively they are joined together through the anti-roll bar. This connection clearly affects each one-sided bouncing. Figure 5: Anti-roll bar loaded by vertical forces In the model here implemented it has been considered a lumped torsional stiffness granted to a bushing element located at the mid-point connection of the two bodies representing the left and right portions of the antiroll-bar. Damper, springs and end stops Figure 6: Example of a spring dumper structure with notable elements listed The force elements included in the strut here shown are re reproduced in the MB model as non-linear splines for the damping characteristic and linear stiffnesses for the main spring and end stops.

6 Figure 7: Setting the damping curves Front Suspension Coil Spring Stiffness 48 N/m Preload 58 N AntiRoll Bar Torsional Stiffness 2 Nm/rad Bump Stop Stiffness 35 N/m Clearance 4 mm Rebound Stop Stiffness 68 N/m Clearance 5 mm Rear Suspension Coil Spring Stiffness 36 N/m Preload 35 N AntiRoll Bar Torsional Stiffness 5 Nm/rad Bump Stop Stiffness 35 N/m Clearance 26 mm Rebound Stop Stiffness 7 N/m Clearance 75 mm Table 3: Characteristics of the Force Elements Steering system The simplest and also most common steering system to be created is the rack and pinion steering system. Firstly the rotations of the steering wheel are transformed by the steering box to the rack travel which is travels along a straight rail activated by the rotations of a pinion. At the extremities of the rack two tie rods permit the transformation of this translational movement to the rotation around the steering axis of the suspension. Hence the overall steering ratio depends on the ratio of the steering box and the kinematics of the steering linkage.

7 Table 4: Connection types for the steering mechanism Figure 7: Representation of the joints and configuration of the mechanism of the steer In Figure 7 the hierarchical organization of the joints is shown. Here is possible to see (in the block scheme) two green arrows indicating the revolute joint of the steer on the chassis and the translational joint of the rack. This means that there is a correlation between the two (set by a relative driver) which is programmed by the steering ratio. Tires modelling An accurate modelling of the tire force elements is achieved by including the so called TNO- MF tire (version 6.), which is based on the renowned Magic Formula tire model of Pacejka. The model takes as input a series of parameters (i.e. a vector with more than 1 elements) for each calculation to be performed, which are empirically determined coefficients that address the complexity of the model. Equations of motion Virtual.Lab Motion is based on a Cartesian coordinates approach for the assembly of the equations of motion. The solver uses Euler parameters to represent the rotational degrees of freedom (avoiding therefore the intrinsic singularity of the angular notation) and Lagrangian formulation for the assembly and generation of equations of motion. The joints between bodies are expressed in a set of algebraic equations, subsequently assembled in a second derivative structure, obtaining finally a set of Differential Algebraic Equations (DAEs) packable in the following form: M Φ q ( q) Φ( q) ( ) T q&& Q q = λ γ (,&, q t)

8 Here M is the mass matrix q is the vector of the generalized coordinates Q is the vector of the generalized forces applied to the rigid bodies λ is the vector of the Lagrange multipliers Φ is the Jacobian of the constraint forces γ the right-hand-side of the second derivative the constraint equations This model includes 52 bodies; therefore a total of 52 x 7 = 364 configuration parameters are used by the pre-processor to build the set of equations of motion. For the settling configuration, in which the vehicle is just let rest on the ground with null initial conditions, joints and drivers are for a total of 234, therefore leaving the system with 13 degrees of freedom. While setting up a manoeuvre, instead, additional constraint is added to the system in terms of position driver on the steering wheel, commanded in open loop, and forces are acting on the wheel s revolute joints to represent the driving torque. Moreover, non-zero initial conditions at velocity and position level are added to every body. Between the different solvers solution proposed in the Virtual.Lab Motion (here below reported), the BDF has been selected, granting a good stable behaviour for such a stiff system. Acronym Name Code based on Type Strengths PECE Predict -Evaluate- Correct-Evaluate Adams-Bashforth- Moulton Method Shampine- Gordon s DE Explicit, Multistep Discontinuous Systems and Non-stiff systems BDF Backwards Difference Formulation DASSL Implicit, Multistep Smooth, stiff, Systems RK Runge-Kutta DOPRI5 Explicit, Singlestep Extremely discontinuous systems Table 5: Solvers available in Virtual.Lab Motion

9 Simplified modelling approach ψ, ψ&, & ψ Yaw angle, yaw velocity and yaw acceleration ϕ, &, ϕ & ϕ Roll angle, roll velocity and roll acceleration β, & β Car-body sideslip angle, velocity at center of gravity v Absolute car-body velocity at center of gravity δ 1 δ 2 Steering angle of the front wheels and rear λ 1, λ 2 Coefficient for camber angle induced by roll (front and rear) a 1, a 2 Front and rear wheelbases b 1, b 2 Front and rear half tracks h relative position of roll center with respect to car-body CG M Total mass of the vehicle I xx I zz K φ b φ C 1, C 2 g δ axle elas + δ tire δ 1 axle kin δ 1 axle elas δ 2 axle kin δ2 axle elas axle axle axle Roll inertia of the vehicle Yaw inertia of the vehicle Total anti-roll stiffness (Kr1 + Kr2) Total roll damper rate Camber stiffness, resp. for front and rear axle Constant of gravity (defaulted to m/s Total front and rear sideslip (axle + tire) Steering angle of the wheel due to axle kinematics - Front axle Toe angle of the wheel due to axle elasto-kinematics - Front axle Steering angle of the wheel due to axle kinematics - Rear axle Toe angle of the wheel due to axle elasto-kinematics - Rear axle M, k, b Mass, stiffness and damping of the axle in lateral direction v y1 Lateral deformation velocity of the axle (front and rear) Table 6: List of symbols for the model The domain of lateral vehicle dynamics is here investigated. As mentioned before a range of possible approaches has been reported to model the dynamics of a vehicle. Depending on the field of study and the accuracy required, the details to be included vary considerably. The solution for this dilemma, and a trustworthy help to the vehicle dynamics engineer comes from the adoption of a modular modeling approach. The model for lateral dynamics studies proposed in the following is a four wheels chassis model with medium wheel approach for front and rear axles. The structure of this model, well known in the literature, is meant for handling modeling and lateral dynamics studies and has 3 DOF: yaw velocity (ψ& ), carbody sideslip angle at center of gravity (β ) and roll angle (φ). The equation of motion are obtained cascading the overall dynamics to the following set of differential equations, expressed in what are normally called quasicoordinates and generated by forces and moments balance of the Newton-Euler approach. Moreover, linearization in the McLaurin series (assuming to be in steady state conditions and close enough to the equilibrium position) brings to the condensed formulation: [ ( ψ& + & β ) h&& ϕ] M v I && ψ = a F zz 1 y tire1 2 ( I + Mh )&& ϕ Mh[ v( ψ& + & β ) h && ϕ] xx = a F F y tires 2 y tire 2 & ϕ ϕ + b ϕ + k ϕ = h F y tires

10 Hence a global motion is allowed with respect to the ground, including also the car-body roll effect on the generalized sideslip and yaw velocity (due to roll center heights and axle kinematics). In the state equations, the relative position of roll center with respect to the carbody center of gravity can be computed with relative height of roll center above front and rear axle: a 2 a1 h = h = + G h hg hr1 hr 2 a1 + a2 a1 + a2 In addition the load transfer between left and right is included (effect while negotiating a turn) which brings in the variation of lateral force available at the tires, computed via: F z F y tire = a δ tire a 3 sin 2 arctan 4 The total axle sideslip angle gives an extra contribution to the lateral force acting on the axle due to the synthesis parameters of camber stiffness (C 1 λ 1, C 2 λ 2 ) and lateral stiffness and damping. Therefore the combination of axle kinematic steering angle (axle kin) and axle compliance contribution (axle elas) is considered; the tire slip angle can be hence written as: a1ψ& vy1 ( δtire ) = β + δ front 1 axle kin δ1 axle elas v 1 δ 1 axle elas = Fy tire ( M axle, kaxle, baxle,δ tire ) 2D1 In addition the relaxation length is considered as a simple first order filter with fixed time constant. Figure 8: Representation of the vehicle model in Imagine.Lab AMESim for studying lateral dynamics The set of equations therefore obtained is presented in the form of an ODE system, but with an implicit loop for the computation of the lateral force on the tire: infact it depends on the lateral slip that is computed from the lateral force again. In this condition AMESim automatically selects a solver based on DASSL (Differential Algebraic System Solver), therefore a sort of BDF formula is used. Whether a full ODE system was provided the solver would have been selected between the ADAMS or Gear s method, actually both included in the same algorithm (LSODA) which switches between the two based on an index to identify the stiffness of the problem.

11 Approach for the Estimation scheme The systematic approach proposed is distinguished then in two steps: 1. in the first step, the Assured and Calculated parameters are provided as input to the models and considered as fixed 2. in the second step optimization algorithms run to determine the parameters in a loop that aims to minimize multiple objectives. Assured Calculated Obtained from the high fidelity model (the real vehicle) without experiments Can be computed a priori from known parameters or by basic measurements These include synthesis parameters and reduced model topological definition classified based on LH-DOE Case study of Lateral Dynamics Table 7: Phenomenological classification of system s parameters In general the motion equations governing a mechanical system are in the form of second order differential equations. Three types of solutions can be computed from this mathematical formulation, corresponding to three types of driving circumstances: the steady-state, the stability solution and the frequency response. To be able to explore those three domains, different maneuvers have been selected as comparison between the high fidelity model and the model: slalom, step steer and a form of open loop double lane change. Since it is commonly accepted that yaw rate relates mainly to what a driver sees and lateral acceleration relates to the human feelings, both aspects will be considered as output parameters. The identification method illustrated here is then based on the error minimization calculated in three different time windows, as shown here below, considering basically the equilibrium starting condition in error 1, transient behavior in error 2 and steady state after distortion (steering input) in error 3. yaw rate [degree/s] lateral acceleration [G] Figure 9: Distinction of the errors domain

12 The following table summarizes the parameter classification adopted for the lateral dynamics model. Sub model Parameters Definition 3 DOF Vehicle Model Axis compliances Tire Models mass of vehicle yaw inertia roll inertia front wheelbase rear wheelbase front half track rear half track height of centre of gravity height of front roll centre (absolute) height of rear roll centre (absolute) front anti-roll stiffness rear anti-roll stiffness roll damper rating steer angle ratio toe coefficient induced by roll - front & rear camber coefficient induced by roll - front & rear front castor offset compliance gain - front & rear lumped mass axis - front & rear compliance spring - front & rear compliance damper - front & rear slope at the origin - maximum cornering stiffness slope at the origin - vertical load for maximum cornering stiffness camber stiffness Assured Calculated Calculated Assured Assured Assured Assured Calculated Calculated Calculated Calculated Steering Mechanism radius of the pinion Calculated Table 8: Parameters classification for the lateral dynamics case study Before the optimization, an accurate sensitivity analysis is run. The number of sampling points in the Latin Hypercube-DOE is set to 3. Since the number of parameters is high, the importance of selection the appropriate excitation for the target parameters is a crucial process since one must avoid ending up with an ill-posed inverse problem. The objective functions of each stage are hence defined by results of the sensitivity analysis. The subsequent optimization is divided into three stages, cascading from the highest (most contributing) to lower sensitive parameters with respect to the selected cost functions. The combinations of Design Variables and Object functions are summarized in the below Table, where part 1, 2 and 3 refer to the error classification scheme previously defined.

13 Stages Design Variables Objective Functions Stage1 steer angle ratio part3_error_yaw_rate roll damper rating front anti-roll stiffness rear anti-roll stiffness Slalom part2_error_lateral_acceleration part3_error_lateral_acceleration Stage2 front anti-roll stiffness rear anti-roll stiffness compliance gain - front compliance gain - rear Double Lane Change part1_error_yaw_rate part3_error_yaw_rate toe coefficient induced by roll - front & rear lumped mass axis - front & rear Slalom part3_error_yaw_rate part1_error_lateral_acceleration height of rear roll centre part1_error_yaw_rate height of front roll centre Stage3 camber coefficient induced by roll - front & rear compliance spring - front Double Lane Change part3_error_yaw_rate compliance spring - rear front castor offset part3_error_lateral_acceleration compliance damper - front compliance damper - rear Step steer part2_error_yaw_rate part2_error_yaw_rate Table 9: Combination of design variables and object functions for each stage As the multi-objective optimization of the third stage reaches the stopping criteria, the differential evolution algorithm provides the Pareto set (collection of optimal solutions) that minimizes the concurrent objective functions. As clarified by the objective contribution plot (Figure 1) a parameter set that happens to minimize one cost function, is instead acting negatively for another objective. The optimal point is selected afterwards based on a trade-off between the single contributions. The results of the final comparison between the high fidelity model and lateral dynamic model are shown in Figure 11 a, b, c, d. In particular Figure 11 d proposes a cross validation of the model by applying to it a manoeuvre for which parameters estimation has not been performed (i.e. random steering action in time). Figure 1: Objective contribution plot of optimization of stage 3 with respect to the 7 objective functions of the different maneuvers of slalom (SLALOM), double lane change (LANE), step steer (CSA)

14 yaw rate [degree/s] lateral acceleration [G] yaw rate [degree/s] lateral acceleration [G] (a) (c) yaw rate [degree/s] lateral acceleration [G] lateral acceleration [G] yaw rate [degree/s] (b) (d) Figure 11: Comparison of the behavior of and model for a) slalom maneuver, b) double lane change, c) step steer, d) polynomial steering angle Conclusion An observation can be made regarding the model validity. Simplified vehicle models are often used by control engineers for control design and online implementation of on-board safety systems. Typically, these models are intensively used for efficient cycle computation within a limited validity range: several physical phenomena and dynamics effects are not included. This means that from the analysis of these models, the controls engineer does not learn about possible interactions of the neglected dynamics with the control law and furthermore, the effect of uncertainty in the input parameters on the controlled system performance is not assessed. An example is indeed in the notable gap between the prediction of the model and the high fidelity multibody simulation when large lateral acceleration is required.

15 yaw rate [degree/s] lateral acceleration [G] Figure 1: Step steer manoeuvre with large steering input (i.e. high lateral acceleration) For a proper insight in the physical performance of the end product (i.e. the active vehicle on the road), an improved engineering process is needed to guarantee the vehicle and the controller performance even in the presence of unmodelled physical effects and uncertainty in the input parameters. A work in progress is indeed in this direction, aiming at achieving higher model accuracy while keeping them as simple as possible.