Development of Noise Robust State Observer for Power

Assisting Wheelchair and its Applications

Sehoon Oh∗Yoichi Hori∗

This paper suggests a state observer for a power assisting wheelchair. In order to control a power assistingwheelchair in the way that makes the user safe and comfortable, we need some information of wheelchairmotion.

Three sensors (encoder, gyroscope, and accelerometer) are employed to obtain driving velocity and in-clination angle of a wheelchair. Each sensor has inherent defect. An LQG designed observer is designedto overcome these defects. Proposed observer turns out very robust to sensor noise and to provide goodestimations.

As control applications of the observer this paper suggests three control designs for a wheelchair. Overturnprevention control, gravity compensation control on a slope, and force sensor-less assisting control.

Keywords: state observer, LQG method, power assisting wheelchair, sensor fusion, power assisting control

1. Introduction

Emerging Power Assistance Tools Nowadaysadvanced power assistance tools are drawing people’s at-tention as emerging control application. These tools areusually located near a man or attached to one’s body,and amplify human power. This operational environ-ment makes the control difficult and unique to thesetools. Though a variety of power assistance tools are be-ing developed, there is little discussion on control meth-ods for those tools.

Fig. 1. power assisting Wheelchair as an Exampleof Power Assistance Tools (YAMAHA JW II)

A power assisting wheelchair is a good example of thatkind of assistance tools. Development of controllers fora power assisting wheelchair has just started. In conven-tional power assisting wheelchairs, motors just multiplyoriginal human force to drive by up to several times.∗ Department of Electrical Engineering

University of Tokyo4-6-1 Komaba, Meguro, Tokyo, 153-8505 Japan

But, if the controller can sense some information ondriving situation, advanced power assistance controlscan be achieved. To this end, we need an observer thatcan estimate some important physical values.

Necessity of State Observer in the Con-trol of Power-assisted Wheelchair Motion of awheelchair is different from that of a car. As for thewheelchair, the motion in the pitch direction is impor-tant from the stability viewpoint (3). In order to obtainthe inclination angle of the wheelchair, the gyroscopeis adopted for the measurement. But the integrationcalculation is necessary to get the angle, and this pro-cedure makes the estimation weak to the noise in thegyroscope.

Furthermore, the accurate velocity of the wheelchairis necessary for the advanced power assisting control.With the feedback control of velocity, we can design theresponse by disturbance and human force respectively.This can realize various power assisting controls. En-coders are adopted for this measurement of the velocity.But, if the equipped encoder has low resolution, slowvelocity of the wheelchair results in noisy and incorrectvelocity estimation by the encoder.

In this paper, an observer which is robust to sensornoises is proposed in order to obtain this pitch informa-tion or inclination angle and the velocity of a wheelchair.

In section 2, two different mathematical models ofwheelchair are suggested. Based on these models, ob-servers are designed and verified by experiments in sec-tion 3, and 4. In section 5, control examples that canmake use of this observer are introduced.

2. Mathematical Modeling of a Wheelchair

Two descriptions of wheelchair motions are adoptedfor observer designs.

Inverted Pendulum Model Wheelchair systemconsists of a wheelchair and a man who rides on it.This system analogizes to a cart with an inverted pendu-lum (1). Considering this point, motion equations of θ, ϕcan be described in equation (1), (3) using the Euler-Lagrange Differential Equation. Generally, the inclina-tion angle ϕ will not change so largely that it can be as-sumed to be around 0 rad. Under this assumption, someapproximations in equation (5) can be taken. These ap-proximations linearize the equations (1), (3) into (2),(4).

τ

θ

x

ϕ

m

M

l

Jm

JM r

vm

Fig. 2. Cart with Inverted Pendulum Model

τ +dθ = {(M + m) r + JM} θ −mlrϕ cos ϕ

+mlrϕ2 sinϕ + BM θ · · · · · · · · · · · · · · · (1)

' {(M + m) r + JM} θ −mlrϕ + BM θ (2)

dϕ =(Jm + ml2

)ϕ−mlrθ cos ϕ−mgl sinϕ

+Bmϕ · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (3)

' (Jm + ml2

)ϕ−mlrθ −mglϕ + Bmϕ(4)

cos ϕ ' 1, sinϕ ' ϕ, ϕ2 ¿ 1 · · · · · · · · · · · · · (5)

Because there are front wheels in a wheelchair, themotion of ϕ is restricted in the way of equation (6). ϕ0

is the minimum value of ϕ which is restricted by frontwheels. But this restriction is not taken into considera-tion here for simplicity.

{ϕ = 0 if ϕ ≤ ϕ0 and ϕ ≤ 0ϕ moves according to equation (1),(3) if ϕ > ϕ0

(6)

Simple Two Bodies Model This model uses thesame 2 as a model of a wheelchair, but does not considerthe detail relationship between a wheelchair and a rider.

The rotating angle of wheelchair wheel (θ) is propelledby external torque τ and disturbance dθ. The inclina-tion angle of a rider (ϕ) is propelled by disturbance dϕ

and not related to the inertia force of the wheel (θ) mo-tion. It is certain that the motions of two variables arerelated but it is the strategy of this model to sum thoserelationships into the disturbance and make the observersimple. The dynamics of each variable are described inequation (7).

Jwθ = −Bwθ+τ+dθ, Jbϕ = −Bbϕ+dϕ, x = rθ(7)

Ignorance of the connection between wheels and arider will make the observation incorrect; neverthelessthe disturbance inputs will compensate this ignorance.

3. Observer Design Using LQG Method

3.1 Output Equation of Multisensor As saidin the introduction, we need to obtain correct informa-tion on driving condition of a wheelchair in real-time. Tothis end, here three sensors are used for measurements:encoder, gyroscope and accelerometer. Each sensor hasweak point when it is used alone. To overcome theseweak points, an observer that will estimate the incli-nation angle (ϕ) and the driving speed(x) is designed.This can be a kind of sensor fusion.

In order to design this observer, states in equation (8)are adopted.

x =(

θ ϕ θ ϕ dθ dϕ

)T · · · · · · · · · · · · · (8)

Velocity of a wheelchair x is calculated by x = rθ ig-noring slips on wheels. Based on these states, outputequations of three sensors are described in Table (1).

Encoder yenc = θ

Gyroscope ygyro = ϕ

Accelerometer yaccx (ax) = rθ cos ϕ + g sin ϕ

yaccy (ay) = g cos ϕ− rθ sin ϕ

Table 1. Output Equation of Each Sensor

The measurements ax, ay in the accelerometer are an-alyzed in figure(3). ax is linearized on the assumptionof equation (5). This results in the output equation forthe observer shown in equation (9) and (10).

ax

g ay

ϕx

Fig. 3. Accelerations Measured by Accelerometer

y =

ϕθax

=

0 1 0 0 0 00 0 1 0 0 00 −Bwr

Jwg 0 0 r

Jw

x +

00r

Jw

u(9)

Equation (9) is the output equation based on the sim-ple two rotating bodies model, and equation (10) is theoutput equation based on the inverted pendulum model.

In this research, the observer gain is decided using theLQG (Linear Quadratic Gaussian) method, for this ob-server uses the multisensor. The noise covariance datafor the determination of the observer gain are like below.

Qx = diag(Qθ, Qθ, Qϕ, Qϕ, Qdθ

, Qdϕ

) · · · · · · (11)Ry = diag (Rgyro, Renc, Racc) · · · · · · · · · · · · · · · (12)

As Qθ, Qθ become smaller, the noise in the estimated

y =

(0 1 0 0 0 0

0 0 1 0 0 0

− rD

(Jm + ml2

)BM

rD mlrBm 0 − r

D m2rgl2 + g rD

(Jm + ml2

)− r

D mlr

)x +

(0

0rD

(Jm + ml2

))

u · · · · · · · · · · · · (10)

θ becomes smaller. The same is for Qϕ, Qϕ, and thesmaller they are, the less noise will ϕ have. For goodcontrol performance, ˆ

θ should have small noise. LargeRacc will make the estimated states noiseless.

4. Experimental Results

Due to the limitation of sensors, the measurements ofx and ϕ can be incorrect. In order to get vf information,we should differentiate the encoder output discretely. Ifthe resolution of the encoder is too low or the angularvelocity of the wheel is too low, the discretely differ-entiated velocity will be very noisy and incorrect. Toovercome this, a low pass filter is utilized, but it willmake the estimation slow.

And for the measurement of ϕ, the value of a gyro-scope is integrated. If there is some noise in the outputof the gyroscope, the drift phenomenon will occur.

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

time (sec)

velocity

(rad/sec

)

Angular Velocity

ObservedDifferentiated

0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

time (sec)

angle (r

ad)

CoG angle

ObservedIntegrated

(a) on level ground

0 5 10 15 20−3

−2

−1

0

1

2

3

time (sec)

velocity

(rad/sec

)

Angular Velocity

ObservedDifferentiated

0 5 10 15 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time (sec)

angle (r

ad)

CoG angle

ObservedIntegrated

(b) with a wheelie

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

time (sec)

velocity

(rad/sec

)

Angular Velocity

ObservedDifferentiated

0 5 10 15 20−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time (sec)

angle (r

ad)

CoG angle

ObservedIntegrated

(c) on a slope

Fig. 4. Estimated States : angular velocity (ˆθ), in-

clination angle(ϕ) (Using Two Isonlated RotatingBodies Model)

Proposed observer can overcome these two problems.Experimental results shown in figure 4 explain thispoint. Each figure shows θ and ϕ respectively. In (a),the wheelchair runs straightly without a wheelie, whilein (b) there is a wheelie around 4 sec. Red line in theupper figure of each experiment, which shows θ, is thedifferential of the encoder output. This value is verynoisy and delayed, while the estimation of the proposedobserver (blue line) is fast and not noisy. To investi-gate the robustness of proposed observer, we added anoise to the gyroscope output at 10 sec. Compared tothe integration of the gyroscope output (red line in thebelow figure of each experiment), the proposed observerestimation (blue line) shows robust observation results.

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

time (sec)

velocity

(rad/sec

)

Angular Velocity

ObservedDifferentiated

0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

time (sec)

angle (r

ad)

CoG angle

ObservedIntegrated

(a) on level ground

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

time (sec)

velocity

(rad/sec

)

Angular Velocity

ObservedDifferentiated

0 5 10 15 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time (sec)

angle (r

ad)

CoG angle

ObservedIntegrated

(b) with a wheelie

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

time (sec)

velocity

(rad/sec

)

Angular Velocity

ObservedDifferentiated

0 5 10 15 20−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time (sec)

angle (r

ad)

CoG angle

ObservedIntegrated

(c) on a slope

Fig. 5. Estimated States : angular velocity (ˆθ),

inclination angle(ϕ) (Using Inverted PendulumModel)

Figure 4 shows the experimental results done based

on the inverted pendulum model with the same covari-ance data Qx and Ry. This shows similar results withthe simple model observer, which ascertains that thissimple model is useful enough in the observer designof a wheelchair. From this experimental result, we canconclude that using the proposed observer, the drift phe-nomenon can be avoided and better velocity informationwill be obtained.

Figure 4 shows different observation results whenchoosing different Qx and Ry. This suggests that theselection of covariance matrix be adjusted according tothe specification of each sensor.

0 5 10 15 20−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time(sec)

angle

(rad)

Inclination Angle

0 5 10 15 20−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time(sec)

angle

(rad)

Fig. 6. Different Observation Results by DifferentObserver Gain

5. Application for the Advanced Controlof Power Assisting Wheelchair

In this section, control designs that are related withthe proposed observer are introduced. Variable assistratio controller (3), flexible gravity compensation con-troller (4), and force(torque) sensor-less power assistingcontrol are the control designs.

5.1 Prevention of Overturn The position ofthe inclination angle ϕ in the phase plane can denote thestability of the wheelchair in the pitch direction. Figure7 shows the phase plane of ϕ. It is divided into three re-gions depending on the level of danger; A) proper safetyzone (ϕ < 0 and below the negative slope asymptote),B) semi-safety zone (ϕ < 0, ϕ > 0 and below the nega-tive slope asymptote), and C) dangerous zone (above thenegative slope asymptote), and the location of ϕ in thesethree zones indicates the stability of the wheelchair.

Power assistance with fixed assist ratio (which meansthe extent of increase) sometimes provides excessive as-sisting power, and the excessive power results in theoverturn of the wheelchair especially on slopes. The ra-tio should be changed according to of ϕ.

Using the estimation of ϕ, the assist ratio is changedas shown in the equation (13) and the figure 8, andthis change stabilizes the wheelchair and prevent it fromoverturning on hills.

α = αmax exp(βωp

θp

) · · · · · · · · · · · · · · · · · · · · · · · · (13)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

B

C

A

ϕ[rad]

ϕ[rad/s]

P

Fig. 7. Man-wheelchair Phase Plane.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

ϕ [rad]ϕ

[ra

d/s

]

B C

A

CoG Movements on the Phase

Plane

time[sec]

[rad], A

ssis

tance-r

atio

ϕ^

assistance-ratio

ϕ

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7 8 9 10

Assis-ratio and ϕ Changes

Fig. 8. Analysis Using Phase Plane of ϕ and itsApplication to Variable Assist Ratio Control

5.2 Gravity Compensation Control We sug-gested the effect of gravity on a wheelchair be removedusing two degree of freedom controller. This controlleruses the velocity(θ) of a wheelchair, and feedbacks veloc-ity errors. Figure 9 shows the structure of this gravitycompensation controller. If the velocity information isobtained by the differential calculation of encoder read-ings, it will be very noisy especially at low speed. Thisnoise in the velocity information in the feedback loopworsens comfortability. Using the proposed observer,this noise problem can be solved.

+

FF Cont.(Assist)

HumanTorque

Disturbance(d)

Velocity++

+-

-

Feedback Controller

r

PLANT

(Wheelchair)

Fig. 9. Structure of Gravity Attenuation Con-troller in a Power Assisting Wheelchair

Two kinds experiment were done. One is done onlevel ground and the other is done on a hill. The re-sult is shown in figure 10. In contrast to the drive onlevel ground, on a hill the controller produces a certainamount of motor torque while there is no human torqueinput. Almost same torque is produced even when thewheelchair descend the hill.

The amount of produced torque can be adjusted bythe Bd parameter, and the parameter Jd will adjust the

response time against gravity.

-8

-6

-4

-2

0

2

4

6

8

0 2 4 6 8 10 12 14 16 18 20

human torqueangle

motor torque

-2

-1

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20

human torqueangle

motor torque

Fig. 10. Experimental Results (Left: drive on levelground, Right: drive on hill))

5.3 Force Sensor-less Power Assisting ControlPower assisting controller usually employs force sensorsto measure the force to be assisted. This limits theassistance and only the force applied to the certain re-gion is assisted. The proposed observer also estimatesthe disturbance(dθ) which is working on the wheelchair.This observed disturbance can indicate the exerted hu-man torque roughly. Using this observed disturbancedθ, torque sensor-less power assisting controller for awheelchair is designed like figure 11.

Js + B

1

A

Human force

Velocity

Jns + Bn

Tas + 1

1

Tas + 1

1

JMs + BM

+

+

+

-

-

Feedforward

Motor Torque

+

+

PLANT

(Wheelchair)

Fig. 11. Block Diagram of Proposed ControlDesign

The purpose of this controller is not the increase in thepower, but the decrease in the inertia of the wheelchair.With this control system, the transfer function from hu-man force to velocity is:

TC2(s) =1

Js + B + A

(JMs + BM + A

JMs + BM

)· · (14)

JM , BM are parameters of model dynamics and canbe chosen arbitrarily. A is a feedback gain for velocitytracking. Appropriately chosen JM , BM (smaller thanJ,B ) will make a system sensitive to a proper extentand provide good assistance. The experimental resultsare presented in figure 12.

In this experiment, two types of wheelchairs are pro-vided to a user, and the user rides wheelchairs and pro-pels the wheelchair with almost the same torque in bothcases. Figure (a) shows the observed disturbances in-cluding human force. These disturbances almost cor-respond to the imposed human torque, and the rangesof the observed disturbances in (a) are similar in both

0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time(sec)

Obse

rved D

isturba

nce

Without ControlWith Assisting Control

(a) observed disturbance (including human force)

0 5 10 15−10

−8

−6

−4

−2

0

2

4

6

8

time(sec)

Angu

lar Ve

locity(

rad/se

c)

Without ControlWith Assisting Control

(b) velocities of the robot arm (with and without control)

Fig. 12. Experimental Results

cases, especially in the first stroke. Figure (b) showsthe velocities of the wheelchair. Velocity with proposedcontrol is bigger than the one without control. The pro-posed control works as power assisting control.

6. Conclusion

In this paper, we proposed an observer design whichis useful in wheelchair controls, and showed that theLQG designed observer using multisensor is very robustto sensor noise and provides good estimations. Threecontrollers that are related with this observer are intro-duced. The selection of the covariance matrix for theLQG observer design are left for further discussion.

This observer design is a necessary technology in thehuman-friendly motion control (4).

References

( 1 ) Yoshihiko Takahashi, Tsuyoshi Takagaki, Jun Kishi, and

Yohei Ishii: ”Back and forward moving scheme of front wheel

raising for inverse pendulum control wheelchair robot”, Proc.

of the 2001 IEEE Int. Conf. on Robotics and Automa-

tion,pp. 3189-3194, Korea, May 21-26, 2001.

( 2 ) Y.Hori: ”Design of Inertia Imulator using Disturbance Torque

Observer”, National Convention Record IEEJ - Industry Ap-

plications Society, pp.91-94, 1987.

( 3 ) Naoki Hata, Yoichi Hori: ”Backward Tumbling Control for

Power Assisted Wheel-chair based on Phase Plane Analysis”,

Medicine and Biology, 2003.9, Mexico

( 4 ) Sehoon Oh, Naoki Hata, Yoichi Hori: ”Developments of var-

ious observers for the intelligent power assisted control of

wheelchair”, SICE Annual Conference 2004, pp.2214-2219,

2004