An Optimal Washout Filter Design for a Motion Platform ... - CiteSeerX

An Optimal Washout Filter Design for a Motion Platform with Senseless and Angular Scaling Maneuvers Sung-Hua Chen and Li-Chen Fu, Fellow, IEEE

Abstract—The motion cueing algorithms are often applied in the motion simulators. In this paper, an optimal washout filter, taking into account the limitation of the simulator’s workspace, is designed for the motion platform aiming to minimize human’ s perception error in order to provide realistic behavior. The filtering algorithm compares the human’s perception of driving simulated vehicles realized by the motion platform with that obtained based on the human vestibular model. Then, a cost function accounting for the pilot’s sensation error and the range of platform motion is being minimized in the sense that more realistic motion and more efficient usage of the limited workspace can be successfully achieved. Finally, the simulation results verify the claimed efficient utilization of platform workspace for task running and less sensation error compared to that obtained by the classical washout filter. Keywords —Motion cueing algorithm, Human perception, Washout filter, Sensation error, Senseless maneuver.

I. INTRODUCTION The motion platform is often used as a flight simulator for providing a virtual environment to operators. The motion cueing algorithm so called as washout filter is a transformation that transforms the motion from actual vehicles to simulators. The purpose of the washout filter is to provide the motion cues representing realistic human perception, besides the motion must proceed within the bounds of the simulator workspace. Its major function is to “wash out” unnecessary signals and pull the position of the simulator back to its neutral position. Some minor functions which followed after the washout filter are designed to increase the efficiency of the platform workspace. The senseless maneuver is developed to moves the platform toward its original position beyond the threshold of human perception. Many researches of washout filter have been represented in the last three decades. Classical washout filter was the first scheme that has been proposed. The scheme is composed of linear low-pass and high-pass filters. The advantages are simplicity and easy to adjust. The fixed scheme and parameters of the classical washout filter cause it inflexible and is not suitable for all circumstances of simulators. Nahon and Reid [1] suggested an adaptive washout algorithm with S. H. Chen is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]) L. C. Fu is with the Department of Electrical Engineering & Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]) This work was supported by National Science of Council under the grant NSC 98-2218-E-002-014.

the same scheme as classical washout filter and with the filter parameters self-tuning mechanism. The optimal washout algorithm [2] that takes into account models for the vestibular system was proposed by Sivan et al. This algorithm uses techniques of optimal control and contains the sensation error and platform motion in the cost function. The optimal algorithm designs an optimal structure and a set of optimal parameters subject to the assumptions of human vestibular models and platform limitation by solving the Riccati equation. Telban, Cardullo and Houck [3] formulated a linear optimal control problem similar to [2] and solved the Riccati equation in real time so that a scalar coefficient that increases control action can be tuned online. The magnitude of the scalar coefficient depends upon the platform motion. With large platform motion, the large coefficient increases and results in faster control action. Nehaoua et al. [4] applied classical, adaptive, and optimal algorithms and compared performances of these algorithms in their driving simulator. No matter what kind of platform used as the simulator, the limited workspace is an important issue in designing the motion cueing algorithm. Several works that increase the efficiency of the platform workspace have been represented. Huang and Fu [5] proposed a senseless maneuver that moves the operator with the acceleration under the threshold value of human perception to conserve the workspace. Liao et al. [6] combined the classical washout filter with an adjustable scaling filter, a yawing washout filter, a dead zone washout filter, and an adaptive washout filter in order to complete the motion planning of the simulator in restricted workspace. In this paper, an optimal washout filter is developed by applying the algorithm [2] to the human vestibular system [7]. The object is to minimize the sensation error produced from the comparison of the human vestibular signals about actual vehicle and simulator. The optimal motion cueing algorithm is then characterized by a systematic combination of linear filters that are determined through an off-line design process. In addition to the washout filter, the senseless and angular

Fig. 1. Classical washout algorithm

scaling maneuvers are following to conserve the platform workspace by adjusting the motion gain appropriately. In Section II we shall obtain the human vestibular based optimal design for a six degree motion platform, and in Section III we shall represent the senseless and angular scaling maneuvers. After that, simulation results are provided to the performance of this algorithm and the comparison with classical washout filter in Section IV. Finally, the conclusion is made in Section V. II. OPTIMAL WASHOUT FILTER DESIGN

yA

+

uA

e

−

uS

uP yS

Fig. 2. Optimal simulator design problem

To determine the structure and parameters of the washout filter means to find a mapping from the actual vehicle input u A to simulator input uS . In Fig. 2, two paths are separated to compare the perceptions of actual vehicle and simulator. The upper path represents the actual vehicle input u A transformed by the human vestibular model, and signal y A , the sensation of operator driving actual vehicle, can be obtained. In the lower path, the vehicle input u A is transformed by washout filter and obtained the reference input of the simulator uS , then the output of the simulator u P passes through the vestibular model to get the sensation of simulator pilot. Here we discuss the ideal case the control error could be neglected and results in the output of simulator u P can approximate to the command uS . To compare the signals from two paths would obtain the sensation error e which is the object minimized in the optimal design. A. Model for the Vestibular System Here we use a linear model for the vestibular system ˆ (surge/pitch) as shown in Fig. 3 where fˆx and θɺ are the sensed specific force and the angular velocity respectively, ax and θɺ are the actual vehicle inputs, the longitudinal (surge) acceleration and the pitch angular velocity, and f x is the stimulus specific force. Fig. 4 shows the influence of pitch rotation on surge specific force. For the center of rotation at the centroid of the motion platform, the specific force is (1) f x = a x cos θ + g sin θ ≅ a x + gθ where θ is assumed to be small so that sin θ can be replaced

by θ and cos θ by 1. In Fig. 3, the sensed specific force fˆx is related to the stimulus specific force f x by the otolith model [7] shown as k (τ a s + 1) fˆx = fx (2) (τ L s + 1)(τ s s + 1) where τ a , τ L , τ s , k are computed parameters of the otolith model. Take Laplace transform of Eq. (1), and we obtain 1 f x ( s) = ax ( s) + g θɺ( s) (3) s which is substituted into Eq. (2) and results in kτ a s + k 1 fˆx = (ax ( s) + g θɺ( s)) 2 s τ Lτ s s + (τ L + τ s ) s + 1 g  gs +  τa = kτ a   τ Lτ s s 3 + (τ L + τ s ) s 2 + s  

  u τ Lτ s s 2 + (τ L + τ s ) s + 1    (4) s+

1

τa

T where u = θɺ a x  is the input of vehicle. Rearranging Eq. (4) leads to the following differential equation

ax

fˆx

fx

+ +

g 1 S

ˆ θɺ

θɺ

Fig. 3. Model of vestibular system subjected to surge linear motion and pitch angular motion

Fig. 4. Influence of rotation on specific force in longitudinal mode

ɺɺ ɺ τ Lτ s fˆx + (τ L + τ s ) fˆx + fˆx = kτ a (( g +

g ɺ 1 )θ + ( s + )ax ) τas τa

gθ 1 = kτ a ( gθɺ + + aɺ x + ax )

τa

τa

(5)

and then can be realized in state space form as xɺot = Aot xot + Bot u fˆ = C x + D u x

ot ot

(6)

ot

where xot is the state vector of the otolith model, and

  1 0  0    kτ g  0 1  , Bot =  a  τ Lτ s   kg 0 0    τ Lτ s  Cot = [1 0 0] , Dot = 0.  τL +τs − τ τ L s   1 Aot =  − τ Lτ s   0  

kτ a   τ Lτ s  k  , τ Lτ s   0  

ˆ The sensed angular velocity θɺ is related to the vehicular angular velocity θɺ by the semicircular canals model [7] as ɺ

θˆ =

TLTa s 2 θɺ (TL s + 1)(Ts s + 1)(Ta s + 1)

(7)

where Ta , TL , Ts are time constants. For realization purpose, Eq. (2) can be rewritten as

T3 s 2

ɺ

θˆ = where T3 =

s 3 + T2 s 2 + T1 s + T0

θɺ

(8)

TLTa T T +T T +T T , T2 = L s L a s a , TLTsTa TLTsTa

TL + Ts + Ta 1 , T0 = and can be expressed in state TLTsTa TLTsTa space form as xɺsc = Asc xsc + Bsc u (9) ɺ θˆ = Csc xsc + Dsc u T1 =

where xsc is the state vector of the semicircular canals model, and  −T2 1 0  T3 0   Asc =  −T1 0 1  , Bsc =  0 0 , Csc = [1 0 0] ,  −T0 0 0   0 0 Dsc = 0. The representations in Eqs. (6) and (9) can be integrated to form a single representation as xɺv = Av xv + Bv u (10) yˆ v = Cv xv + Dv u which represents the human vestibular model where xv is the combined states, yˆ v is the sensed responses, and Av , Bv ,

Cv , and Dv represent the vestibular models as one set of state equations: 0  A  Bsc  Csc 0  Av =  sc  , Bv =   , Cv =  , 0 A B ot    ot   0 Cot  D  Dv =  sc  .  Dot 

B. Integrated System We assume the human vestibular model can be applied to both the vehicle operator and simulator operator as shown in Fig. 2. Then the vestibular state error is defined as xe = xS − x A where xS and x A are the vestibular states for simulator and vehicle respectively. Then the pilot sensation error e can be calculated in the form xɺe = Av xe + Bv uS − Bv u A (11) e = Cv xe + Dv uS − Dv u A where u A and uS represent the simulator and vehicle inputs respectively, as shown in Fig. 2. Owing to the limited workspace of the simulator, the motion of the platform should be constrained. For this purpose, we want to incorporate the platform states into the cost function. Here the velocity, displacement, and rotation angle of the platform are considered to be accessed. The additional terms are included in the state equation: xɺc = Ac xc + Bc uS (12) T

xc =  ax dt 2 ax dt θ  represents the   additional motion platform states, and 0 1 0  0 0 Ac = 0 0 0  , Bc = 0 1  . 0 0 0  1 0  The vehicle input u A consists of filtered noise, and can be represented as xɺn = An xn + Bn w (13) u A = xn

∫∫

where

∫

where xn is the filtered white noise state, w is the white

0  − β  β1  noise, and An =  1  , Bn =   , where β1 and β 2 β 0 − 2  β2  are the first order break frequencies for each degree-of-freedom. By combining Eqs. (11), (12), and (13), we can obtain the desired system equation Xɺ = AX + Bus + Hw (14) y = CX + Dus where

T

y = e xc  ∈ ℝ5

is

the

desired

output,

T

X =  xe xc xn  ∈ ℝ11 represents the combined states, and the combined system matrices A, B, C, D and H are then given as  Av 0 − Bv   Bv  0 A =  0 Ac 0  , B =  Bv  , H =  0  ,  0 0  0   Bn  An  11×11

C C= v 0

0 I 3×3

11×2

− Dv  D  ,D =  v . 0  5×11  0  5×2

11×2

C. Derivation of Optimal Design The cost function for the optimal design is defined as

 J = E  where E {

}

∫

t1

t0

 (eT Qe + X cT Rc X c + uTs Rus )dt  

T

(15)

−1

Bv K2  K  sI − Av + Bv K1  Bv (I + K3 ) W(s) =  1   − K being sI − Ac + Bc K2   Bc K3  3 K2   Bc K1 a matrix of the optimized transfer function which links the simulator inputs uS to the vehicle inputs u A .

is the mathematical mean of statistical variable,

Q and Rc are positive semi-definite matrices, and R is a positive definite matrix. Three variables are to be constrained in the cost function, namely, the sensation error e along with the additional terms xc and the input uS which contains the linear and angular motion of the platform. From the cost function, it can be seen that the design considers reducing the sensation error and the limited workspace of the platform. The system equation and cost function can be transformed to comply with the optimal control formulation as: Xɺ = A ' X + Bu '+ Hw (16)  t1  J ' = E  ( X T R1 ' X + u 'T R2 u ')dt   t0  where

∫

T T A ' = A − BR2−1 R12 , u ' = us + R2−1 R12 X T R1 ' = R1 − R12 R2−1 R12 , R1 = C T GC , R12 = R + C T GD, .

R2 = R + DT GD, G = diag Q Rc  The cost function is minimized when (17) u ' = − R2−1 BT PX where P is the solution of the algebraic Riccati equation (18) R1 '− PBR2−1 BT P + A 'T P + PA ' = 0 . Solve for uS , and we can obtain

uS = − KX

(19)

T where K = − R2−1 ( BT P + R12 ) . Let K be partitioned to three parts according the partition of X in Eq. (14), then we have:  Xe  uS = − [ K1 K 2 K 3 ]  X c  . (20)  X n 

Since xn = u A , we remove the state corresponding to xn in Eq. (20) so that :  xe   xɺe   Av 0 − Bv     Bv  = x + u (21)  xɺ   0 A 0   c   Bc  S c  c  u A  Substitution of (20) into (21) then leads to

xɺe   Av − Bv K1 −Bv K2 xe  −Bv (I + K3 ) xɺ  =  −B K   +   uA (22)  c   c 1 Ac − Bc K2 xc   −Bc K3  After observing the state space forms respectively described by (22) and (11), the following equation can readily be derived in the Laplace domain: u S ( s ) = W ( s )u A ( s ) (23) with

III. SENSELESS AND ANGULAR SCALING MANEUVERS Two additional processes are integrated into the whole loop. In order to increase the efficiency of the limited workspace of the platform, the senseless maneuver is applied. The operation pulling the cockpit back to origin can be manipulated with the linear acceleration under human sensed threshold. Another operation is to adjust the angular scale in the outer range of cockpit workspace. When proceeding large angular motion, the scale of angular motion decreases appropriately with the distance to the edge of the workspace. This maneuver is designed to prevent the collision of the platform in restricted workspace. A. Senseless Maneuver The acceleration in the section from 0.17m/s2 to 0.28m/s2 could not be sensed by vestibular system. We don’t proceed to extra operation to the acceleration command when the acceleration exceeds this section. While the acceleration is under the senseless range, appropriate adjustments can conserve the platform workspace and do not change the human perception in the same time. The algorithm is as follows. sign(vi ) = −sign(di ) ⇒ ai = ath If aw,i ≤ ath   sign(vi ) = sign(di ) ⇒ ai = 0 If aw,i > ath ⇒ ai = aw,i where ai (i=x,y, and z) is the acceleration command given to the platform, aw,i is the output from the washout filter, ath is the threshold acceleration, vi and di are the platform velocity and displacement respectively. In the restoration period (the direction of the velocity is the same with the displacement), the platform acceleration is set to the threshold acceleration to increase the speed of returning home position. If it is not restoring, the acceleration is set to zero to prevent wasting of simulator workspace. This operation would be carried out under the range without perception. So this is called as senseless maneuver. B. Angular Scaling Maneuver Owing to the limited workspace of the simulator, the 3-axis rotation angles are also adjusted appropriately. The adjustments are as follows. If φi ≤ φcr ⇒ φi = φw,i

If φi > φcr ⇒ φi = ρ (φw,i − φcr ) + φcr −α (φw ,i −φcr )

,

ρ =e where φi (i=x,y, and z) is the command of rotation angle given to the platform, φw,i is the output from the washout

β1 = 0.01 rad/s, β 2 = 0.025 rad/s 0 0  0.02 0  17 0   2.4  Q= 0.005 0  .  , R =  0 0.01 , Rc =  0 0 5      0 0 0.03 The washout filter W ( s) is obtained by Eq. (23) and its elements will be denoted as W ( s) W12 ( s)  W ( s) =  11 (24) . W21 ( s) W22 ( s)  The elements Wij ( s ), i, j = 1, 2 turn out to be transfer function of dimension nine, which corresponds to the dimension of the matrix A. This transfer function maps the actual motion to simulator motion which are 2x2 signals. The selection of the parameters Q, R, Rc depends on the emphasis on which part, and determines the coefficients of the washout filter. Fig. 5 shows the Bode plots of each elements of transfer function W ( s) . Table 2 Human Vestibular Model Parameter [7] Specific Force Sensation Model Parameter Parameter

τL τs τa

Surge

Sway

Heave

5.33

5.33

5.33

0.66

0.66

0.66

13.2

13.2

13.2

k

0.4

0.4

0.4

dTH

0.17

0.17

0.28

Bode Diagram

200 W11

0 M a g n it u d e ( d B )

The results of simulations present linear acceleration in surge and angular velocity in pitch. The parameters of human perception model are shown by Table 2. The numerical values of other parameters are shown as follows:

W12

-200

W22

-400

W21

-600 720 W22

540 P h a s e (d e g )

IV. SIMULATIONS

In the simulations, we compare the sensation errors of the classical washout filter and the optimal design. First case shows the actual motion that accelerated forward at 2 m/s2 for 10 sec and then held the constant speed for 25 sec. The results of the simulation are plotted in Fig. 6~9. Fig. 6(a) and (b) show the comparison of the actual and simulator inputs for linear acceleration and angular velocity respectively. The optimal washout filter transforms the actual input to simulator input. In Fig. 6(a), the transformed acceleration decreases rapidly and vibrates substantially in the change of actual acceleration. The transformation of washout filter conserves the high frequency motion and filters others to keep the platform moving in the workspace. In Fig. 6(b), although the actual angular velocity holds on zero, the simulator input produces motions to present the low frequency linear acceleration motion. This phenomenon can be revealed by comparing Fig. 6(a) and (b).

W21

360 180

W12

0

W11

-180 -20 10

-15

-10

10

-5

10

0

10

10

Frequency (rad/sec)

Fig. 5. Bode plots for each elements of transfer function W(s) 2.5 Simulator input Actual input

2 1.5

A c c e le ra tio n in p u t (m /s2 )

filter, φcr is the critical angle, ρ and α are the angular scaling parameters. A critical angle is set to determine the switch of the angular scaling maneuver. There is no adjustment when the command is smaller than critical angle. If the command exceeds the range of critical angle, the exceeding part is multiplied by the gain ρ which is an exponential function of the angle. The critical angle is chosen to approach the fringe of the workspace for not to affect the human perception.

1 0.5 0 -0.5 -1 -1.5 -2

0

5

10

15

20

25

0.8 Actual input

Pitch

Yaw

TL

6.1

5.3

10.2

Ts

0.1

0.1

0.1

Ta

30

30

30

δTH

3.0

3.6

2.6

Simulator input

A n g u la r v e lo c ity in p u t (ra d /s )

Roll

35

(a)

0.6

Rotational Motion Sensation Model Parameter

30

Time (sec)

0.4

0.2 0 -0.2

-0.4 -0.6 -0.8

0

5

10

15

20

25

30

35

Time (sec)

(b) Fig. 6. Actual and Simulator inputs (a) Acceleration (b) Angular velocity

V. CONCLUSION

2 Actual Sensation Optimal Design Classical Design

S e n s e d a c c e l e ra tio n (m /s2 )

1.5

1

0.5

0

-0.5

-1

0

5

10

15

20

25

30

35

Time (sec)

(a) 0.4 Actual sensation Optimal design Classical design

S e n s e d a n g u la r v e l o c ity (ra d /s )

0.3 0.2

In this paper, an optimal design has been applied to minimize the sensation error and conserve the workspace. By building the human vestibular system, we can obtain the sensed acceleration and angular velocities which are defined as the human perception. For our designing purpose, the cost function contains the pilot’s sensation error, the motion of platform and the input. The senseless maneuver operates the platform outside the range of human perception, and the angular scaling maneuver decrease the rotation angle in the outer part of the cockpit workspace. These two maneuvers can help increasing the efficiency of the limited workspace without affecting the human perception. In the simulations, the sensation errors of classical washout filter and the optimal design are compared to show the performance of this algorithm. REFERENCES

0.1

[1]

0

-0.1

[2]

-0.2 -0.3

[3] -0.4

0

5

10

15

20

25

30

35

Time (sec)

(b) Fig. 7. Sensation of vehicle and simulator operators (a) Sensed acceleration (b) Sensed angular velocity

[5]

0.5 0.4

S e n s a tio n e rro r f a c c e le ra ti o n (m /s2 )

[4]

Optimal Classical

0.3 0.2

[6]

0.1 0

[7]

-0.1 -0.2 -0.3 -0.4 -0.5

0

5

10

15

20

25

30

35

Time (sec)

Fig. 8. Sensation errors of classical and optimal washout filters

Fig. 7 shows the sensation of actual vehicle operator y A comparing to the sensation of simulator operator yS referred to Fig. 2. The sensed accelerations of optimal and classical washout filters are both shown in Fig. 7(a). Obviously, the trajectory of optimal design is more close to actual sensation than classical algorithm. Fig. 7(b) also shows the optimal design has better performance in angular velocity. Fig. 8 shows the sensation error e of the classical washout filter and the optimal washout filter. The RMS of sensation error for the acceleration of optimal design is 0.0237 which is smaller than that of classical design 0.1107. The results reveals optimal design can perform more realistic human perception signals than classical design.

M. A. Nahon, L. D. Reid, and J. Kirdeikis, “Adaptive Simulator Motion Software with Supervisory Control,” Journal of Guidance, Control, and Dynamics, vol. 15, no. 2. March-April 1992. R. Sivan, J. Ish-Shalom, J. K. Huang, “An optimal Control Approach to the Design of Moving Flight Simulators,” IEEE Trans. On Systems, Man, and cybernetics, vol. SMC-12, no. 6, November/December 1982. R. J. Telban, F. M. Cardullo, and J. A. Houck, “A Nonlinear, Human-Centered Approach to Motion Cueing with a Neurocomputing Solver,” AIAA Modeling and Simulation Technologies Conference and Exhibit, Monterey, California, Aug. 5-8, 2002. L. Nehaoua, H. Mohellebi, A. Amouri, and H. Arioui, S. Espié, and A. Kheddar, “Design and Control of a Small-Clearance Driving Simulator,” IEEE Transactions on Vehicular Technology, vol. 57, no. 2, March 2008. Chin-I Huang, and Li-Chen Fu, “Human Vestibular Based (HVB) Senseless Maneuver Optimal Washout Filter Design for VR-based Motion Simulator,” Systems, Man and Cybernetics, SMC '06. IEEE International Conference on Volume 5, 8-11 Oct. 2006 Page(s):4451 – 4458. C. S. Liao, C. F. Huang, and W. H. Chieng, “A Novel Washout Filter Design for a Six Degree-of-Freedom Motion Simulator,” JSME International Journal, Series C, vol. 47, no. 2, 2004. M. K. Park, M. C. Lee, K. S. Yoo, K. Son, W. S. Yoo, and M. C. Han, “Development of the PNU Vehicle Driving Simulator and Its Performance Evaluation,” Proceeding of IEEE International Conference on Robotics and Automation, pp. 2325-2330, 2001.