Model-based predictive motion cueing strategy for vehicle driving simulators

ARTICLE IN PRESS Control Engineering Practice 17 (2009) 995–1003

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Model-based predictive motion cueing strategy for vehicle driving simulators Mehmet Dagdelen a,, Gilles Reymond a, Andras Kemeny a, Marc Bordier b, Nadia Maı¨zi b a b

Centre Technique de Simulation, Renault TCR AVA 0 13, 1 avenue du Golf, 78288 Guyancourt, France Centre de Mathe´matiques Applique´es, 2004 Route des Lucioles BP 93, 06902 Sophia Antipolis, France

a r t i c l e in fo

abstract

Article history: Received 20 December 2007 Accepted 11 March 2009 Available online 16 April 2009

Motion-based driving simulators are designed to render accelerations perceived as realistic, while keeping the motion system within its physical limits. These objectives are generally contradictory, and consequently motion control strategies are difficult to customize for different simulator configurations. In this paper, a novel approach is presented for the design of motion rendering strategies, using the model-based predictive control theory. Compared to traditional cueing techniques, actuator constraints are always respected, and the use of motion workspace is maximized during simulations. Models of human motion perception can be integrated to reduce sensory conflicts. A practical implementation on a high-performance automotive driving simulator is presented. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Human factors Human-centered design Predictive control Robot control Vehicle simulators

1. Introduction Several studies show that in driving simulators, the reproduction of inertial cues favors a realistic driver behavior (Hosman & van der Vaart, 1981; Kemeny & Panerai, 2002; Reymond, Kemeny, Droulez, & Berthoz, 2001; Wierwille, Casali, & Repa, 1983). Yet, the reproduction of vehicle accelerations is only possible within the performance limits of the motion system actuators, in terms of displacement, velocity and acceleration. The transformation of the vehicle accelerations into admissible motion commands is performed by a motion cueing strategy (MCS), also called washout filter or motion drive algorithm. The ‘‘classical’’ MCS, initially developed for flight simulators (Schmidt & Conrad, 1970) computes the acceleration of the motion platform along any degree of freedom, aðtÞ by high-pass filtering the corresponding vehicle acceleration rðtÞ (Fig. 1). By thus removing the low-frequency component of rðtÞ, platform excursions are reduced. The filter parameters are tuned to maintain the platform displacement within its physical limits. However, removing the low-frequency component also reduces the rendered acceleration. Since the classical MCS is composed of linear filters, the trade-off between optimizing acceleration rendering and keeping the motion system within its physical limits is typically handled beforehand by considering a ‘‘worst case’’ driving situation in terms of acceleration amplitude and duration. As a consequence, in usual driving conditions, only a fraction of the platform workspace is actually used. To overcome this situation, Parrish, Dieudonne, and Martin (1975) proposed an ‘‘adaptive’’ strategy, further developed by Ariel

and Sivan (1984), Reid and Nahon (1988) and Zong, Gao, and Mai (2000). At each computational step k, the gains and cut-off frequencies of the classical MCS filters are derived from the minimization of a cost function V k , of the form: V k ¼ ðr k  ak Þ2 þ w1  v2k þ w2  p2k

(1)

The first term of V k expresses the discrepancy between the vehicle acceleration r k and the platform acceleration ak . The two other terms, related to the platform velocity vk and position pk , are introduced to restrain the platform excursion. Compared to the linear classical MCS, this adaptive strategy increases the driving fidelity when the simulator is near to its neutral position. The weights w1 and w2 define the trade-off between the acceleration rendering and the limitation of the platform motion: an increase of w1 and w2 penalizes large displacement and velocity whereas a decrease of w1 and w2 favors the minimization of the acceleration tracking error. Nevertheless, this trade-off is still tuned off-line, considering a specific driving case situation and a specific vehicle acceleration profile. More recent strategies are based on the optimal control formalism (Chang, Liao, & Chieng, 2009; Friedland, Ling, & Hutton, 1973; Ish-shalom, 1982; Sturgeon, 1978; Sullivan, 1985; Telban & Cardullo, 2002). In particular, Sivan, Ish shalom, and Huang (1982) determined an optimal MCS that takes into account the linear motion perception model proposed by Zacharias (1978) and by Hosman and van der Vaart (1978), as well as a linear model of the motion system dynamics. This MCS minimizes a global cost function V defined as Z 1 ½ðr^ ðtÞ  a^ ðtÞÞ2 þ w1  v2 ðtÞ þ w2  p2 ðtÞ dt V½uðtÞ ¼ 0

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E-mail address: [email protected] (M. Dagdelen). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.03.002

_ ¼ AxðtÞ þ BuðtÞ subject to xðtÞ p v a a^ T with x ¼ ½

(2)

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Linear Acceleration

Vehicle Dynamics

Angular Acceleration

r(t)

Low-Pass Filter

L

Anti-backlash Filter

High-Pass Filter

L

1

Linear Displacement

s²

Motion System

Rate Limiter

Normalisation

High-Pass Filter

a(t)

1

Anti-backlash Filter

+

s²

Angular Displacement

Fig. 1. Flow chart of the classical strategy. L is the rotation matrix that transforms vector components from the simulator to the inertial reference frame and s, the Laplace variable.

where ðA; BÞ represent the model of the motion system and of the motion perception, a^ denotes the acceleration perceived in the simulator, and uðtÞ the motion system control input. Computing the minimum of this cost function requires some knowledge of the vehicle acceleration rðtÞ over the time period ½0; 1. When rðtÞ is assumed being a filtered white noise, the authors showed that the resulting strategy is a combination of linear high-order high-pass and low-pass filters. Here again, the tuning of the weighting coefficients is done off-line, according to an acceleration profile (white noise) that may not correspond to a typical driving situation. Thus, existing MCS are designed to enhance the sensation of self-motion in driving simulator by extracting a specific bandwidth of the vehicle acceleration signal. However, this frequential rendering may introduce phase shifts and lead to situations where the simulator acceleration is opposite to the vehicle acceleration, creating ‘‘conflicts’’ between visual and inertial motions. To reduce the impact of these conflicts on the self-motion perception and the drivers’ behavior, these filters are tuned to minimize the magnitudes of these platform accelerations. As, the filter parameters (gains, cut-off frequencies or weighting coefficients) are not directly related to the physical and perceptive constraints, the tuning of these MCS are not intuitive for inexperienced simulator users. This paper presents a completely new strategy based on model-based predictive control (MPC) theory that follows closely the reference signal as long as the motion system can and to handle ‘‘smooth’’ deceleration near its physical limits, avoiding false motion cues. Over the last decades, the MPC theory has received increasing attention (Qin & Badgwell, 2003), mainly due to its ability to handle, simply and effectively, hard constraints on control and states (Mayne, 2000). Given the plant dynamic model and the constraints on control and state, the current control action corresponds to the first element of an optimal control sequence that solves a finite-horizon open-loop control problem in which the initial state is the current state of the plant. The whole optimization cycle is repeated at the next computational step over a shifted horizon. Numerical routines can be used to solve the constrained optimization problem defined at each step (Bonnans, Gilbert, Lemare´chal, & Sagastizabal, 2002). The proposed predictive MCS computes the control input to minimize the local difference between the acceleration perceived in the simulator and the acceleration perceived in the vehicle, while maintaining

the motion system trajectory within its bounds. Its parameters correspond explicitly to the motion system physical limits and motion detection thresholds. Therefore, it can be tuned by inexperienced simulator users. This paper is organized as follows. Section 2 presents the proposed predictive MCS. Experimental and simulation results are presented in Section 3. Section 4 discusses some properties and evolution of this strategy and Section 5 draws the conclusions from this work.

2. Predictive MCS 2.1. Motion system dynamics The motion systems used in dynamic driving simulators consist of mechanical systems (generally parallel manipulators such as Gough–Stewart platforms) equipped with actuators moved by specific controllers (Fig. 3). The dynamics of mechanical systems can be written in the general form: € þ F½qðtÞ; qðtÞ _ M½qðtÞqðtÞ ¼ sðtÞ n

(3) m

where qðtÞ 2 R is the position vector, sðtÞ 2 R is the vector of actuation forces and torques, F : Rn  Rn /Rn is a given vector-value function (the matrix of Coriolis, centrifugal and gravitational terms), and M : Rn /Rnn is a given function that corresponds to the inertia matrix (Merlet, 2006; Spong & Vidyasagar, 2004). The actuator controllers are usually based on a ‘‘computed torque’’ control law to handle the nonlinearities of the mechanical system (Slotine & Li, 1995; Spong & Vidyasagar, 1989). Indeed, under the assumption of a perfect knowledge of M and F, a computed torque controller generates the input command: _ sðtÞ ¼ M½qðtÞvðtÞ þ F½qðtÞ; qðtÞ

(4)

where v is the new control input, leading to the uncoupled double integrator system: € qðtÞ ¼ vðtÞ

(5)

When the control input vðtÞ is chosen as _~  l2 qðtÞ ~ vðtÞ ¼ q€ d ðtÞ  2lqðtÞ

(6)

ARTICLE IN PRESS M. Dagdelen et al. / Control Engineering Practice 17 (2009) 995–1003

997

Linear System Nonlinear System ud xd

Reference FeedForward Compensator (Motion Cueing Strategy)

Feedback Compensator (Feedback Linearization)

u

Mechanical System

Output

x Fig. 2. Two-degree-of-freedom control design. The feedforward compensator, representing the motion cueing strategy, generates the nominal input ud required to track a given reference trajectory. The feedback compensator based on feedback linearization techniques corrects for errors between the desired ðxd Þ and the actual ðxÞ motion system trajectories.

Fig. 3. The ULTIMATE simulator at Renault, Technical Center for Simulation. The cockpit is on the upper platform of a hexapod motion system that moves on perpendicular XY rails.

with l40 and q~ ¼ q  qd where qd is the desired position vector, the closed-loop system is exponentially stable: _~ þ l2 qðtÞ €~ þ 2lqðtÞ ~ qðtÞ ¼0

(7)

Thus, the global simulator motion control scheme appears to be composed of a feedback compensator (the actuator controller) that handles the nonlinearities of the mechanical system dynamics, and of a feedforward compensator (the motion cueing algorithm) that defines the trajectory to follow (Fig. 2). Based on these assumptions, the motion system dynamics (parallel manipulator and actuator controllers) can be assumed to be linear and uncoupled during the design of a MCS. In the following, without loss of generality, the dynamics of the motion system along one axis of motion are assumed to be x_ ðtÞ ¼ AxðtÞ þ BuðtÞ (8) n

where uðtÞ 2 R is the acceleration control input, xðtÞ 2 R is the current state vector, aðtÞ is the acceleration of the motion system along the chosen axis of motion, and A, B, C, D are matrices of the proper dimension. Due to the physical limits, this system is constrained both on the control u and the state x: uðtÞ 2 U  R

where X and U are assumed to be convex sets.

Located in the inner ear, the vestibular system is the main sensory organ for self-motion acceleration perception. It is composed of semicircular canals, sensitive to angular accelerations, and of otoliths, sensitive to linear accelerations and gravity, i.e. specific force. The responses of the vestibular organs to selfmotion accelerations can be considered as uncoupled linear filters (Berthoz & Droulez, 1982). Let aðtÞ be the longitudinal acceleration of the driver’s head, the longitudinal perceived acceleration a^ ðtÞ can be expressed as ^ xðtÞ ^ ^ þ BaðtÞ x_^ ðtÞ ¼ A ^ ^aðtÞ ¼ C^ x^ ðtÞ þ DaðtÞ

(9)

(10)

^ B, ^ D ^ C, ^ are matrices modeling the motion where x^ ðtÞ 2 R , and A, perception filter. Benson (1990) observed that subjects linearly accelerated in the dark below 0:05 m=s2 perceive stationarity, suggesting the existence of an acceleration detection threshold. Gianna, Heimbrand, and Gresty (1996) reported that, in the dark, this threshold is higher when the acceleration profile is linear or parabolic than when it is a step. Berthoz, Pavard, and Young (1975) showed that the motion detection error is higher when the visual stimuli are not coherent with the inertial stimuli. Pavard and Berthoz (1977) also noticed that the motion detection time of a visual target is higher when the subject is accelerated than when at rest. The m

aðtÞ ¼ CxðtÞ þ DuðtÞ

xðtÞ 2 X  Rn ;

2.2. Motion perception dynamics

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acceleration detection threshold is therefore somewhat contextdependent, but in the following, it is assumed it has a fixed absolute value, denote Athrld for each axis of movement. Future developments of this predictive MCS will include these more detailed models of acceleration perception. It is assumed here that in a driving simulator, the negative consequences of motion conflicts on drivers’ self-motion perception and behavior are reduced when the platform acceleration is below the acceleration detection threshold. 2.3. Predictive MCS The predictive MCS is designed to be implemented in a digital controller. Therefore, the models of the motion system dynamics and the motion perception are discretized with a sampling period T. In order to simplify the notation, the discrete-time models are denoted by the same matrix as the continuous-time models. An augmented state, denoted y ¼ ½ x x^ T is defined: ykþ1 ¼ Fyk þ Guk a^ k ¼ Myk þ Nuk where "

(11)

(12)

2.3.1. Control algorithm At computational step k, the predictive MCS computes a feasible control sequence fukþj gN1 j¼0 and a feasible state sequence fykþj gN j¼1 over an horizon N such that:

 the control input uk minimizes the quadratic difference 

Remark 2. As the cost function is convex and the linear constraints are convex, Pðy; r^ Þ is a convex optimization problem, and therefore if there is a solution to the problem, the solution is unique. 2.3.2. Reduction of the computational burden Let SN denotes the set of all initial states from which it takes N time units to reach an equilibrium point with an acceleration below the detection threshold, with controls in U and passing via states all in Y: 8 y0 2 Y > > > > > y for j ¼ 0 . . . N  1 > jþ1 ¼ Fyj þ Guj > > > > ^ > a ¼ My þ Nu for j ¼ 0 . . . N  1 j j > < jþ1 ^ jpA for j ¼ 1 . . . N j a y0 2 SN 3 (14) j thld > > > 2 Y for j ¼ 1 . . . N y > j > > > > for j ¼ 0 . . . N  1 uj 2 U > > > > : y 2 Xf  Rm N

# " # 0 B F¼ ; G¼ ^ ^ ^ BC A BD h i ^ ^ M ¼ DC C^ ; N ¼ DD A

yk 2 Y ¼ X  Rm

equilibrium point over the horizon N. It aims at reducing the impact of motion conflicts that might occur by maintaining the simulator acceleration magnitude below the acceleration detection threshold.

between the acceleration perceived in the simulator and the acceleration perceived in the vehicle; the control sequence fukþj gN1 j¼1 brings the motion system to a stop within its bounds: xkþN 2 Xf , where Xf  X is the convex hull of the equilibrium points of ðA; BÞ.

Only the first element of this control sequence is applied to the system. The procedure is repeated at the each time step. Thus, at each step, the following optimization problem Pðyk ; r^ k Þ is solved: 8 min ðr^  a^ 1 Þ2 > > > fuj gN1 ;fyj gN > 0 1 > > > > > subject to yjþ1 ¼ Fyj þ Guj for j ¼ 0 . . . N  1 > > > > > a^ jþ1 ¼ Myj þ Nuj for j ¼ 0 . . . N  1 > > < ja^ jþ2 jpAthld for j ¼ 0 . . . N  2 Pðy; r^ Þ : (13) > > > 2 Y for j ¼ 0 . . . N  1 y jþ1 > > > > > uj 2 U for j ¼ 0 . . . N  1 > > > m > > y > N 2 Xf  R > > : y ¼y

The complexity of the optimization problem Pðy; r^ Þ can be reduced if SN1 is computed off-line. The N  2 dimensional constrained optimization problem Pðy; r^ Þ can then be transformed into the 1  2 dimensional constrained optimization problem PSðy; r^ Þ: 8 min ðr^  a^ 1 Þ2 > > > u0 ;y1 > > > > > subject to y1 ¼ Fy0 þ Gu0 > > < a^ 1 ¼ My0 þ Nu0 PSðy; r^ Þ : (15) > > y1 2 SN1 > > > > > u0 2 U > > > : y ¼y 0

SN1 can be pre-computed using the algorithm proposed by Gutman and Cwikel (1987): starting with S0 ¼ Xf  Rm , S1 is constructed as the set of points z 2 Y such that z ¼ F 1 y  F 1 Gu with y 2 S1 , u 2 U and Mz þ Nu 2 ½Athld ; Athld , and the procedure is repeated recursively. Remark 3. If there exists N such that SN ¼ SN1 ð¼ Smax Þ, then SN is the set of all initial states from which it takes a finite time to reach an equilibrium point with an acceleration below the detection threshold, with controls in U and passing via states all in Y. The integration of Smax instead of SN1 in PSðy; r^ Þ (i.e. y1 2 Smax instead of y1 2 SN1 ) reduces the number of parameters to tune. 2.3.3. Properties and stability Provided the initial state is inside SN1 then:

 SN1 is a positively invariant set: for all y 2 SN1 , the states

0

where r^ k is the longitudinal acceleration perceived in the vehicle1 and Xf  X is the convex hull of the equilibrium points of ðA; BÞ. Remark 1. The constraint ja^ jþ2 jpAthld requires the motion system acceleration profile to stay, from two time step, below the driver’s self-motion detection threshold, when it slows down to reach an 1 The acceleration perceived in the vehicle is assumed to be the acceleration of ^ B; ^ DÞ. ^ C; ^ As the predictive strategy is based on a the vehicle filtered by the ðA; discrete control scheme, one time unit delay is introduced.



reached by the closed-loop system remain in SN1 . This property indicates that the motion system trajectory, planned by the predictive cueing strategy is maintained in a set characterized by the horizon N and consistent with the constraints on states and control. The closed-loop system is Lyapunov-stable. Indeed, if the closed loop system is in SN1 at step k ¼ 0, then from the invariance of SN1 , there always exists a feasible control input sequence that brings the closed-loop system to the neutral position.

These properties are also valid for a more general class of MPC scheme: receding horizon control of linear discrete-time system

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subject to convex constraints on control and states, with a terminal equality constraint and a convex cost function (Goodwin, Seron, & De Dona´, 2004). Practically, the initial state is generally the origin, which belong to SN1 and these properties are ensured.

3. Results

(x-axis) acceleration [m/s²]

This section presents the application of the proposed strategy to a high performance vehicle driving simulator (Fig. 3). The ULTIMATE simulator was built by RENAULT Technical Center for Simulation, in the framework of an European research project (Eureka #1493), for the design of advanced automotive R&D applications and to study driver behavior (Reymond, 2003). It is composed of a generic car mock-up equipped with modular cockpit instruments, a panoramic on-board screen (SEOS, UK) and a hexapod that can move on XY rails (Rexroth-Hydraudyne, The Netherlands). The longitudinal and lateral accelerations of the vehicle are rendered by the displacement of the hexapod and the rails, whereas the vertical and the angular accelerations of the vehicle are rendered by the hexapod only. The RENAULT simulation software SCANeR&II (Heidet, Warusfel, Vandernoot, Saint-Loubry, & Kemeny, 2001) is used to manage all the features of the simulator: motion, sound and visual rendering, scenario management, intelligent traffic generation, etc. The predictive MCS was originally implemented in SCANeR&II with a sampling period of 10 ms and could render in real-time (Windows 2000 PC host, Pentium IV, 3 GHz processor) the vehicle acceleration feedback along all degrees of freedom (heave, surge, sway, roll, pitch and yaw). In its initial configuration, the motion perception model was assumed to be a unity gain (i.e. ak ¼ a^ k , and ^ ¼ 0, B ^ ¼ 0, C^ ¼ 0 and D ^ ¼ 1 in (8)). The maximum values of then A displacement, velocity and acceleration over the rails are chosen, respectively, as: 3:0 m, 2:0 m=s and 5:0 m=s2 to fit within the motion system capabilities. The set Smax is computed off-line using the algorithm proposed by Gutman and Cwikel (1987). The

999

optimization is carried out numerically using sequential quadratic programming routine (Bonnans et al., 2002) which generally converges in less than four iterations due to simplicity of the cost function. Fig. 4 shows the forward (x-axis) acceleration of the motion system planned by the predictive MCS when the simulator driver accelerates from 0 to 70 km/h using the 1st, 2nd and 3rd gears. The motion system acceleration profile (see Fig. 4) is composed of phases where the vehicle accelerations are rendered with unity gain, followed by phases where the motion system accelerations are opposite to that of the vehicle (actuator limit approach phase in Fig. 4), creating motion conflicts. Such conflicts are inevitable since the motion system has eventually to slow down not to hit its bounds. However, they are controlled by the predictive MCS by keeping the acceleration below an acceleration detection threshold. The transition from one phase to another results from the constrained optimization problem solved at each computational step. As seen in Section 2.3, there is always one feasible solution: that maintaining the motion system within the physical and perceptive constraints over the receding horizon. This solution is optimal when there is only one feasible solution. Roughly, when the motion system acceleration does not follow one to one the vehicle acceleration, the predictive MCS plans the only one feasible solution: the one that stops the motion system within its limits and the perceptive constraints. The rendering resumes when the vehicle acceleration changes in sign. During test driving sessions, the drivers reported they would prefer a ‘‘smoother’’ transition between acceleration rendering and actuator approach. At the best of the authors knowledge, the impact of this transition profile on the driving fidelity is not completely known yet. Therefore, the following constraint was added to smoothen the transition profile: at step k, the acceleration of the motion system is forced to decrease according to the linear law: akþiþ1 ¼ akþi M  iM

for i ¼ 0 . . . M

4

(16)

motion system vehicle

3

Magnitude of the acceleration is below the perception threshold

2 1 0 2

4

8

6

10

12

14

(x-axis) acceleration [m/s²]

time [s] 4

Acceleration rendering phase

Acceleration rendering phase

2 Actuator limit approach phase

-0.05 1

1.5

2

2.5 time [s]

3

3.5

Fig. 4. Response of the predictive strategy, when motion perception dynamics are neglected.

4

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4.2. Comparison with a conventional strategy

1.5 actual accel. of the motion system accel. requested by the predictive strategy accel. of the vehicle

(x-axis) acceleration [m/s²]

1 0.5 0 -0.5

Acceleration rendering phase

-1

Actuator limit approach phase

-1.5 -2 -2.5 1

2

3

4

5

6 7 time [s]

8

9

10

11

Fig. 5. Response of the predictive strategy and of the actual motion system, using end stop smooth approach transition profiles.

where M is a transition horizon that was estimated by trial-anderror ðM ¼ 20Þ. Fig. 5 shows the longitudinal (x-axis) acceleration of the motion system planned by the resulting strategy and the actual platform acceleration, during a driving session that generates accelerations with a high positive and negative magnitudes (stop-and-go driving session). The motion system actual acceleration was measured by a AHRS-DHX inertial sensor (Crossbow, USA), with a sampling frequency of 133 Hz and low pass filtered at 7 Hz. One can observe that the requested motion system acceleration is smoother than in the preceding configuration (Fig. 4) and that the feedback compensator (Fig. 2) provides an acceptable regulation performance.

4. Discussion 4.1. Tuning parameters The parameters of the predictive MCS can be tuned by inexperienced simulator users based on the following guidelines:

 The physical limits of the motion system are usually known, 





but one can set lower values to study for instance the impact of the motion system limits on the simulation fidelity. The control horizon N defines the set of states that can be reached by the closed-loop system, i.e. SN . The higher is N, the better is the tracking performance of the closed-loop system. However, high N values increase the complexity of the optimization problem. When Smax can be defined explicitly (as it is the case when the system results from the discretization of a time-continuous linear system), this parameter is not needed. The transition horizon M is introduced to smoothen the decrease of the motion system acceleration, and is directly related to the time period between a perfect tracking phase and a deceleration phase. However, the higher M, the shorter is the perfect tracking phase. The acceleration detection threshold can be defined from psychophysical data or from specific experimentations. The higher the threshold, the longer is the tracking phase but the ‘‘stronger’’ is the motion conflict during the braking phase.

4.2.1. Classical strategy The classical MCS is the most widely used strategy in commercial simulators (Fig. 1). The linear and angular accelerations of the vehicle are filtered using second-order high-pass filters to maintain the motion system in its workspace, and a firstorder low-pass filter is applied to return (washout) the motion base to its neutral position (Reid & Nahon, 1988). An ‘‘antibacklash’’ filter can be added to reduce certain artifacts that are induced by the high-pass filters (Reymond & Kemeny, 2000). The longitudinal and lateral accelerations of the vehicle are low-pass filtered, scaled and passed through rate limiters to produce additional pitch and roll tilt angles. This ‘‘tilt-coordination’’ aims at creating an illusion of sustained acceleration through slow platform tilting. As the driver is rotated by an angle y while the visual environment is kept stable relative to him/her, a fraction of the gravity vector can be perceived as a linear selfmotion acceleration of magnitude g  sinðyÞ, where g is the gravity constant (Groen & Bles, 2004; van der Steen, 1998). This technique, efficient only for very slow variations of the vehicle acceleration, is not considered in the following comparison presented below. As the motion system acceleration planned by this strategy depends linearly on the acceleration of the vehicle, the secondorder high-pass filter is tuned considering a ‘‘worst case’’ driving situation in terms of acceleration amplitude and duration, chosen as a step of 5 m=s2 (the resulting cut-off frequencies are 1.8 and 0.02 Hz). 4.2.2. Comparison results According to a first experimental tests on the simulator, the Renault expert drivers subjectively preferred simulation sessions with the predictive strategy to simulation sessions with the classical strategy. Some aspects of the psychophysical experimentations planed to provide more proper evidence are discussed by Dagdelen, Berlioux, Panerai, Reymond, and Kemeny (2006) and Filliard, Vailleau, Reymond, and Kemeny (2009). From a quantitative point of view, the predictive MCS provides better handling of the conflict phases, and a better usage of the available workspace of the motion system. Fig. 6 shows the motion system longitudinal acceleration planned by the classical and the predictive strategies, for a typical 0–100 km/h driving session. The intervals where the classical strategy renders the vehicle accelerations with a unity gain are shorter than those defined by the predictive strategy (see intervals A and D). Over the interval B, the two strategies decelerate the platform motion. Over the interval C, the accelerations rendered by the two algorithms do not reproduce the vehicle accelerations. The jerkiness of the acceleration rendered by the classical algorithm seems to correspond to the vehicle jerks. However, it leads to false motion cues: the platform acceleration is opposite to the vehicle acceleration with a magnitude higher than the detection threshold. The predictive motion cueing algorithm avoids these false cues and provides in these intervals a controlled motion conflict situation: the platform acceleration is below the acceleration detection threshold, an explicit parameter of the algorithm. The predictive and the classical algorithms make a different use of the platform workspace. Fig. 7 shows the state-space trajectories ðp; vÞ computed by the two MCS, for a complete driving session of 5 min. The virtual road is a typical test track with 10 curves of different radii of curvature. The motion system trajectory defined by the classical strategy covers an area A2 smaller than the area A1 defined by the predictive strategy, which

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4.5

Vehicle motion Predictive strategy Ciassical strategy

accel. of the motion system accel. perceived in the simulator accel. of the vehicle accel. perceived in the vehicle

4

3

[x-axis] acceleration [m/s2]

(x-axis) acceleration [m/s2]

4

1001

2 Magnitude of the acceleration is higher than the perception threshold

1

0

3.5 3 2.5 2 1.5 1 0.5

-1

A

0

D

C

B

0.5 1

1.5

2

2.5 time [s]

3

3.5

4

1

Fig. 6. Results of the classical and the predictive strategies at the beginning of the simulation (longitudinal axis). See text for interpretation of A–D intervals.

Area A1

X-Position [m]

Classical Strategy Predictive Strategy

1 Area A2

0 -1 -2

Velocity limit Position limit

-3 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2.5

3 time [s]

3.5

4

4.5

5

Fig. 8. Response of the predictive strategy when the driver starts: integration of the self-motion perception filter.

2

X-Velocity [m/s] Fig. 7. State space trajectories defined by the classical and the predictive strategies during a 5 min driving session on a test track.

corresponds in fact to Smax . This confirms the ability of the predictive scheme to better use the available actuator space to track the reference acceleration. 4.3. Extensions of the MPC control scheme 4.3.1. Integration of a motion perception model In the following, the results obtained in the first implementation are compared with those from a version integrating a linear model of self-motion perception. By considering that the otolith organs behave as kinetic and position transducers, Meiry (1965) proposed a second-order lowpass filter that fits several self-motion perception data obtained in psychophysical experiments in the dark. In order to model the steady-state perception of body tilt, a lead term was added by Young, Meiry, and Li (1968), resulting in the following transfer function: ^ AðsÞ 1:5ðs þ 0:076Þ ¼ AðsÞ ðs þ 0:19Þðs þ 1:5Þ

2

^ where AðsÞ is the Laplace transform of the perceived acceleration a^ ðtÞ, and AðsÞ is the Laplace transform of the head acceleration aðtÞ. The integration of this model requires a discretization of (17), ^ B, ^ C^ and D ^ as defined in (8). Fig. 8 leading to the matrices: A, shows the responses of the predictive strategy based on this discretized model. The acceleration perceived in the simulator now tracks the acceleration perceived in the vehicle. In the time period [3–4 s], the motion system acceleration remains constant, whereas it followed the vehicle acceleration in the first implementation (Fig. 4). Compared to configuration showed by Fig. 6, the motion system acceleration remains constant over the time period [3–4 s] to keep the magnitude of the perceived acceleration below the detection threshold, instead of tracking the vehicle acceleration itself.

3 2

1.5

(17)

4.3.2. Integration of the washout and the tilt coordination in the predictive strategy In existing motion cueing strategies, the washout is an additional process that reduces the magnitude of the platform acceleration. In the classical scheme, it results generally from the increase of the filters order; in the adaptive and optimal strategies, it is realized thanks to the integration of the motion system position and velocity in the cost function. In the proposed approach, the washout is integrated as a process that runs in parallel to the predictive strategy as defined before (Fig. 9). The switching criterion is based on the variation of the vehicle acceleration, corresponding to the variation of drivers actions. To reduce sensory conflicts, the washout is performed with an acceleration below the detection threshold, using a minimum-time ‘‘bang-bang’’ motion control scheme (Dagdelen, Reymond, & Kemeny, 2005). The tilt-coordination uses a rotational degree of freedom to render an illusion of linear acceleration. Numerous tilt coordination schemes can be proposed. The chosen tilting scheme computes first the tilt coordination angles using the predictive strategy proposed in the preceding section, then the linear excursion of the platform to render the part of the acceleration not rendered by tilt coordination. However, the validity of tiltcoordination as applied to car driving simulations is questionable in general, considering the conflict between the normal rates of

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Activation Law

Vehicle

Washout Linear Displacement

Dynamics Main Strategy

Linear Acceleration

Motion Tilt Coordination

System

Angular Acceleration Activation Law Main Strategy +

Angular Displacement

Washout Fig. 9. Integration of the washout and tilt-coordination in the predictive strategy.

change of a vehicle’s acceleration and the human tilt detection threshold (typically 3 =s, see Groen & Bles, 2004).

5. Conclusion The purpose of MCS is to enhance the sensation of self-motion in a driving simulator by providing inertial cues consistent with the motion system physical limits and the strong visual cues. By rendering specific bandwidths of the vehicle acceleration, existing approaches produce problematic motion artifacts. The new predictive MCS plans a trajectory for the motion system that renders the perceived vehicle acceleration as long as the motion system can do it and a smooth deceleration when it approaches its limits. By handling explicitly the models of the motion system and the driver’s perception as hard constraints into a MPC scheme, it avoids frequency specific motion artifacts, and facilitates the tuning. The performance of this predictive strategy is compared to the one of the strategy widely used dynamic simulators (classical MCS). Its ability to maximize the motion system envelope and to control the sensory conflicts is observed in typical driving situations, and confirmed in the real-time simulator implementation. According to a first experimental test, drivers subjectively preferred simulation sessions with the predictive strategy to simulation sessions with the classical strategy. However, a psychophysical experiment is planned to compare the impact of these two algorithms on the drivers’ behaviors throughout an analysis of the scaling effects and the motion perception models. The integration of a washout and a tilt coordination process is demonstrated as extensions of the basic algorithm. The MPC theory framework also allows to consider nonlinear systems (for instance for hexapods) and nonlinear effects of motion perception. If some predictions of the vehicle acceleration are available, the cost function of the constrained optimization problem can be updated to integrate the quadratic difference between the vehicle and the motion system acceleration over the receding horizon. As a consequence, the control sequence that solves the resulting optimization problem would stabilize the platform, with respect to its origin, maximizing the range of motion for the predicted vehicle acceleration.

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