International Conference on Engineering Vibration Ljubljana, Slovenia, 7-10 September 2015

ACTIVE SUSPENSION LQ CONTROL FOR IMPROVING RIDING COMFORT

Francesco Cosco1, 2, Ion Borozan*2, Stefano Candreva3, Domenico Mundo3

1 Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300B, 3001 Leuven, Belgium francesco.cosco@kuleuven.be

2 G&G Design and Engineering SRL, Via G. Barrio, SN 87100 Cosenza, Italy {francesco.cosco, ionutz_borozan}@gegde.com

3University of Calabria, Cosenza, Italy {stefano.candreva, domenico.mundo}@unical.it

Keywords: quarter-car, driver model, active suspension, optimal control, comfort

Abstract. The purpose of this research is by means of a properly tuned optimally controlled active suspension, exploiting conceptual human body models as a tool for virtually sense human body vibrations. Canonical approaches achieve comfort performance by reducing vehicle body accelerations. Complimentary, our ap-proach aims to reduce also human body vibrations, taking into account their estimation by means of a simplified human body model. In order to assess whether and to what extent the proposed approach improves comfort, we defined a case study where just vertical vibrations are considered. In this scenario, several conceptual human body models are available in literature, specifically designed for capturing the vertical vibrational behaviour of the driver while in the seated position. By coupling such a two degrees-of-freedom (DOFs) human body model with a standard two DOFs quarter car model we get a four DOFs driver-car system on which the linear quadratic optimal control design was performed to obtain the optimally controlled behaviour. For validating the ap-proach, we derived also the optimal control of the active suspension in a canonical sense (i.e. by aiming at reducing vehicle body vibrations) which we adopted as the reference model. By use of numerical simulations, we finally compared the proposed active suspension optimal con-trol highlighting the main differences with respect to the reference model in terms of head vi-brations (vertical accelerations), vehicle body vibrations, suspension travel, and tire deflection. In particular, we considered two different scenarios: one where the vehicle travels just over the planar straight road roughness at a constant speed of 72 km/h, and the other one where the vehicle passes over a sinusoidal bump. Numerical results demonstrate that the proposed con-troller is able to reduce the driveapproach, leading this way to an improvement of comfort.

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

1 INTRODUCTION

A vehicle's suspension system typically consists of springs and shock absorbers that help to isolate the vehicle chassis and occupants from sudden vertical displacements of the wheel as-semblies when driving. A well-tuned suspension system is important for the comfort and safety of the vehicle occupants as well as the long-term durability of the vehicle's electronic and me-

and thus the passengers, from the road inputs. Occupant comfort requires the minimization of sprung mass accelerations [1], as well as the

optimization of the seating design [2-4]. Because of its response to varying road conditions, active suspensions offer superior handling, road feel, responsiveness and safety compared to passive suspension systems. For instance, in a passive suspension system each improvement comes at the expense of performance in another area. In general, passive suspensions are the best solution when considering a single operating point, situation not common in real life. While adding complexity, active suspensions have shown an ability to minimize vehicle accelerations, rattle space utilization, or to improve road holding to various degrees over certain frequency ranges.

One common control technique that is based on full state feedback relates with the Linear Quadratic Regulator (LQR). Many works are available in the literature regarding LQR optimal control design of active suspension based on quarter car, half car and also full car models. In our work, we take into account a quarter car model coupled with a two DOFs lumped parame-ters human body model, with the aim of designing the active suspension also from the driver point of view, i.e. taking into account body and head accelerations.

In the following sections the driver-car system model is first presented (Section 2), then exploited for the optimal design of the LQR (Section 3). Numerical simulation results of the obtained controlled suspension are presented for two different scenarios: random road rough-ness, and a speed bumper, both engaged at constant speed. For each scenario, a comparison between the passive solution, the reference controller (i.e. a controller obtained optimizing the performance in a canonical sense), and the proposed controller are presented and discussed in Section 4. Numerical results demonstrate that the proposed controller is able to reduce the driver

ons in both situations more than the standard approach, leading this way to an improvement of comfort.

2 DRIVER-CAR SYSTEM MODEL

Several authors already addressed the problem of capturing the vertical dynamics of a car by means of lumped parameters models [5]. The use of quarter-car, half-car and even full car mod-els is a well-established technique with widespread application in the field of vehicle dynamics and control. In the particular case of improving ride quality, Wakeham and Rideout highlighted that using two quarter-car-based LQ controllers better performance are achieved compared to half-car-based LQ controller[1], unless pitch control is of great importance.

As a consequence, the - r to add a physical and compact description of the vertical vibrational behaviour of the driver. To this end, several lumped-parameters models were proposed in literature [6-12], all considering the human body as several concentrated masses, interconnected by springs and dampers. As reported in a recent survey [10], complexity of these models ranges from simple one DOF, to increasing levels of detail alternatives, like the dedicated eleven DOFs model proposed by Quassem et al. to capture the effect of pregnant women [11].

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

Figure 1. System lumped-parameters model

After considering all alternatives, and taking into account the goal of this research, i.e. the

synthesis of a LQR, we choose the two DOF model representation developed by Allen in [12]. As depicted in Figure 1, the driver body is composed of two masses, and , corresponding to the head and the rest of the body respectively, which are inter-connected by a spring-damper element ( and ) representing the neck articulation. The driver model is attached to the car model by means of a soft connection ( and ).

This connection was modelled assuming that the seat mass foreseen by the Allen model is part of the sprung mass, , of the quarter car system. The remaining mass, , completes the system model, by concentrating the inertia of the unsprung elements of the suspension; with the passive stiffness and damping of the suspension being concentrated in and . Whereas, the stiffness and damping of the tyre is modelled by and Values of the lumped parame-ters used in this work are listed in table 1.

Driver model(M = 56.8 kg) [10, 12] Quarter car model parameters [5]

MH [Kg] 5.5 MS [Kg] 240

MB [Kg] 51.3 MU [Kg] 36

KN [N/m] 41000 Ks [N/m] 16000

CN [N s/m] 318 Cs [N s/m] 980

KC [N/m] 74300 KT [N/m] 160000

CC [N s/m] 2807 CT [N s/m] 0

Table 1: List of the lumped parameters used for the Driver-Car system model.

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

3 LQ CONTROLLER DESIGN

For the design of the regulator gains, we extended the approach suggested by Wakeham and Rideout [1], by adopting the extended four DOFs quarter car model (as described in the previ-ous section). The corresponding general linear system expressed in the state-space form is de-scribed as follows:

(1)

where the system states vector,

(2) is defined considering the head, body, sprung and unsprung velocities, followed by neck, cush-ion, suspensions and tire deflections. Moreover, vectors u and d represent the control action (force actuation) and the external disturbance (time derivative of the road profile variations) respectively. Given the state variables, the state transition matrix A, as well as the control matrix, B, and the disturbance matrix C, are derived as follows:

(3)

(4)

(5) According to the LQR method, a parametric performance index needs to be defined, in order

to retrieve the corresponding optimal gains. The proposed performance index is derived as an extension of [1]:

(6) where two extra terms (last two terms in eq. 6) were added to allow the regulator minimizing

the performance also from the perspective of the human body vibrations.

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

4 SIMULATIONS RESULTS

We performed a set of time-domain simulations, which highlight the performances of the proposed approach with respect to the reference regulator obtained in [1], in two different sce-narios: a sinusoidal bump and a random road profile, according to the ISO 8606 specifica-tion.Weights tuned for the design of the optimal LQ regulators are summarized in table 2.

2DOF-LQR

(Reference) 4DOF-LQR

0.4 0.4

0.16 0.16

0.4 0.4

0.16 0.16

- 100

- 100

Table 2: LQR weights for the design of the reference 2 DOFs regulator, and the proposed 4DOFs regulator.

For both situations, the following discretized form of equation (1) is first derived, by con-sidering an exponential integrator scheme, with a fixed time-step : (7)

with: (8) (9) (10)

The optimal gains, , obtained from the LQR design are used online to get the full-state control feedback: (11)

4.1 Sinusoidal bump

For the discrete bump scenario, the input disturbance (i.e. road profile height, as showed in Figure 1) was defined according to the following continuous function:

(12)

The input disturbance series, , for simulating the scenario was then computed by differenti-ating the given formula for a speed, V, of 50 km/h; a bump height, a, of 0.1 m; and a bump length, L of 2 m.

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

As depicted in Figure 2, both the optimal regulators outperforms the passive solutions in term of ride quality. In particular the 4DOFs solution tends to allow slightly larger vibrations of the sprung mass,of ride quality. In particular the 4DOFs solution tends to allow slightly larger vibrations of the sprung mass, but obtains a substantial reduction (about two order of magni-tudes) of the vibration transmitted to the driver head (Figure 3). This results are achieved with-out compromising neither the force demanded by the actuators (Figure 4), nor the road-holding performance estimated by considering the tire deflections (Figure 5).

Figure 3. Sinusoidal bump: driver head acceleration (Objective ride quality)

Figure 2. Sinusoidal bump: sprung mass acceleration (Conventional-Ride quality)

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

4.2 Random road profile

Following the approach described in [13], a class E rough road profile (ISO 8606), corre-sponding to a road with a rating of very poor quality, was generated. For this test, the velocity of the car is assumed to remain constant at 72 km/h (20 m/s). The corresponding disturbance signal in time domain is showed in Figure 6.

Simulation results of the random input are qualitatively identical to those related to the si-nusoidal bump scenario. As depicted in Figures 7 to 10, both the controllers outperform the passive suspension, with the 4DOFs presenting again a small performance reduction in terms of sprung mass acceleration (Figure 7), force actuation (Figure 9) and road holding (Figure 10), but with great improvements (Figure 8: the rms of the head vibrations scales from 0.2 to 0.002

) in term of head and body vibrations.

Figure 4. Sinusoidal bump: force demanded by the controllers

Figure 5. Sinusoidal bump: tire deflections (Road holding performance)

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

Figure 8. Random road profile: driver head acceleration (Objective ride quality)

Figure 6. Random road profile, ISO 8606 class E engaged at 72 km/h

Figure 7. Random road profile: sprung mass acceleration (Conventional-Ride quality)

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

Figure 9. Random road profile: force demanded by the controllers

Figure 10. Random road profile: tire deflections (Road holding performance)

5 CONCLUSIONS

A novel approach for designing LQR optimal control of an active suspension system was presented, whose peculiarity lies in expanding the well-known quarter car model by adding the inertia, stiffness and damping elements required to capture human vibrational behaviour. After dealing with the mathematical background of the used models, and the derivation of a suitable performance index, the LQR was applied to derive the controlled suspension.

Numerical results demonstrate that the proposed controller is able tvibrations in the two driving scenarios analysed in the present work more than the standard approach, leading this way to a deep improvement of comfort, which in both of our test cases was in the range of two orders of magnitude.

AKNOWLEDGEMNENTS

The research leading to these results has received funding from the People Programme (Ma-rie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007-2013/

Francesco Cosco, Ion Borozan, Stefano Candreva, Domenico Mundo

under REA grant agreement n° [285808] INTERACTIVE: Innovative Concept Modelling Techniques for Multi-Attribute Optimization of Active Vehicles (www.fp7interactive.eu).

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