Universidade de São Paulo

2013

Estimation of lower limbs angular positions

using Kalman filter and genetic algorithm. ISSNIP-IEEE Biosignals and Biorobotics Conference, 4, 2013 : Rio de Janeiro, RJhttp://www.producao.usp.br/handle/BDPI/51192

Downloaded from: Biblioteca Digital da Produção Intelectual - BDPI, Universidade de São Paulo

Biblioteca Digital da Produção Intelectual - BDPI

Departamento de Engenharia Mecânica - EESC/SEM Comunicações em Eventos - EESC/SEM

Estimation of Lower Limbs Angular Positions using Kalman Filterand Genetic Algorithm

Samuel L. Nogueira, Roberto S. Inoue, Marco H. Terra and Adriano A. G. Siqueira

Abstract—This paper presents an application of filtering inthe field of robotic rehabilitation. The proposed system is beingdeveloped to estimate the angular positions of an impedance-controlled exoskeleton for lower limbs, designed to provide motorrehabilitation of stroke and spinal cord injured people. A Kalmanfilter based on genetic algorithm is used in a sensor fusionstrategy for estimation of the angular positions, whereas Kalmanfilter fuses the data from inertial sensors and genetic algorithmtunes the weighting matrices of the filter. Also, to properly useaccelerometers in a position estimation strategy, the measuredacceleration must be close to the gravity acceleration. In thispaper, we use the three components of the three-dimensionalaccelerometers to ensure that they are measuring only the gravityvector. We compare the proposed system with our previous sensorfusion system where force sensors located in an insole system wasused for gait-phase identification, giving the periods where thefoot was in full contact with the ground and the one-dimensionalaccelerometer measurements are suitable for position estimation.Simulation results validate the effectiveness of this proposal.

I. INTRODUCTION

In the past few years, robotic devices for rehabilitation engi-neering have been developed and, among them, exoskeletonsfor upper and lower limbs have been widely exploited (see e.g.,[1], [2], [3]). However, regarding lower limb exoskeletons,some issues still need to be improved to ensure an effectivegain for the patient [4], such as force and impedance con-trols to suit the patient’s gait pattern and new strategies forenhancing balance and muscle strengthening.

As the control strategies for these robots is generally per-formed using information about absolute and relative anglesof their segments, it is necessary to use sensors which providea reliable and accurate position measurement. Thus, inertialsensors, i.e., gyroscopes and accelerometers, have been appliedsuccessfully in gait identification (see e.g., [5], [6]), or incontrol strategies to provide stability and gait-pattern adap-tation during walking. In [7], for instance, it was proposedan application using Inertial Measurement Unit (IMU) todetermine the knee joint segment angle of a leg orthosis,where the resulting measurement provide sufficient redundantinformation to control a novel active system conceived fordynamically adaptation of orthotic joint impedance.

This work is supported by grants from Coordenacao de Aperfeicoamentode Pessoal de Nıvel Superior (Capes) and Fundacao de Amparo a Pesquisado Estado de Sao Paulo (FAPESP- 2012/05552-9).

S. L. Nogueira and A. A. G. Siqueira are with the Department of Mechani-cal Engineering, University of Sao Paulo at Sao Carlos, Brazil (correspondingauthor: samlourenco@gmail.com).

R. S. Inoue and M. H. Terra are with the Department of ElectricalEngineering, University of Sao Paulo at Sao Carlos, Brazil.

M. H. Terra and A. A. G. Siqueira are with the Center for Robotics of SaoCarlos, University of Sao Paulo at Sao Carlos, Brazil.

In addition, Force Sensitive Resistors (FSRs), used hereas switch sensors, can be applied in gait-phase identificationsystems with inertial sensors for estimation of absolute angles.In [8], such a sensors are used in a device mounted in aninsole to determine the initial and final instants of the phasesin a human gait, namely heel-strike, toe-off, stance, and swingphases. According to the authors, the identification of thegait phases are important to verify where the accelerometersare measuring only the gravity acceleration, since dynamicaccelerations deteriorate the position estimation.

However, the gait-phase identification does not provide agood indication of accelerometer reliability for all segmentsin a lower limbs. Since the dynamics are different for eachsegment during walking, the switch sensor approach providesa good result only for the foot segment. Hence, we have agood accuracy of the foot measurement, but a lower qualityof measurement when we move in the direction of the hip.

In order to overcome this problem, we propose to use thethree components of the accelerometers instead of identify gaitphases. According to [9], the norm of the three-dimensionalaccelerometer measurements of a given segment, regardingconstant speed, must be approximately the gravity vector.Then, if the norm is close to the gravity acceleration, thesegment is not being accelerated.

The main objective of this work is to estimate the absoluteand relative angles of all segments of an exoskeleton forlower limbs, using only information from gyroscopes andthree-dimensional accelerometers. The proposed sensor fusionstrategy uses Kalman filters, where the weighting matrices areadjusted by an automatic procedure based on Genetic algo-rithms. We compare the proposed system with our previoussensor fusion system [10], where force sensors located in aninsole system was used for gait-phase identification, giving theperiods where the foot was in full contact with the ground andthe one-dimensional accelerometer measurements are suitablefor position estimation. Simulation results performed in theOpenSim 2.4 software validate the effectiveness of this pro-posal.

This paper is organized as follows: Section II presentsa brief description of the exoskeleton for lower limbs, thesimulation environment, the lower limbs angular positionestimation problem and the solution using Kalman filter andGenetic algorithm; Section III presents the simulation resultsfor the proposed sensor fusion strategy, with a comparisonwith a previously reported force sensor-based strategy .

II. MATERIAL AND METHODSIn this section, we present the sensor fusion strategy based

on Kalman filter and Genetic algorithm designed to estimatethe absolute and relative angular positions of an exoskeletonfor lower limbs. The exoskeleton, named Exo-Kanguera, canbe used to evaluate control strategies and rehabilitation proto-cols that consider the natural characteristics of human walk-ing. The proposed robotic system is driven by series elasticactuators that allow the implementation of impedance control,ensuring thus patient safety and the ability to specify differentinteraction behaviors. Figure 1 shows a user performing a stepwith the Exo-Kanguera (end of the swing phase).

Fig. 1. User performing a step with the Exo-Kanguera.

Using simulation data, this paper shows the effectivenessof using KF and GA, with only IMU sensors, to estimateabsolute angular positions of lower limbs, Figure 2. First weuse the software OpenSim 2.4, see Section II-A, to generatekinematics (positions, velocities and accelerations) of eachsegment.

x

y

θ4  

θ1  

θ5  

θ6  

θ2  

θ3  

Fig. 2. Skeleton with the arrangement of IMU sensors

With these data, we use the model of the gyroscope andaccelerometer presented in section II-B to create the simulation

signals of Inertial Measurement Units (IMU) present in eachsegment of legs.

An approach based on a sensor fusion strategy using gyro-scopes and accelerometers is then used. Since this approach ismodelled considering that the angles are obtained directly fromthe gyroscopes, the accelerometers-based positions are usedas redundant measurements to correct the gyroscopes intrinsicerrors, see section II-D.

As this work is intend to compare two very similar ap-proaches, we used a GA optimizer method, based on elitistselection, to tune the weighting matrices R and Q of the KF.

The Figure 3 illustrates the proposed sensor fusion systembased on Kalman filter and GA. In the sequence we show themain tools that are used in this work.

Filter Observation Prediction

1-axial gyro

3-axial acel. Update

Atual

xk =Δθ(k)Δb(k)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

θ(k) = θg (k)+ Δθ(k)

b(k) = bg (k)+ Δb(k)

Genetic Algorithm

Fig. 3. Kalman filter, whereas θ(k) and b(k) are the angle and offset of thegyroscope in the time k.

A. OpenSim 2.4

OpenSim is a freely available, user extensible softwaresystem that lets users develop models of musculoskeletal struc-tures and create dynamic simulations of movement, Figure 4,for more information see [11]. Moreover in this software youcan use open models like [12] or [13], which load marketset files containing motion capture of legs, and calculate theinverse Kinematics to get angular positions, velocities andaccelerations. See the Figure 4. These kinematics are usedin Section II-B to generate simulated sensor signals.

B. IMU Modeling

This section describes how to model gyroscope and ac-celerometer signals. The angular velocity obtained from thegyroscope, θ(t), at the instant t, can be modeled as

θ(t) = θg(t) + bg(t) + ηg(t), (1)

bg(t) = − 1

τgb(t) + ηbg (t), (2)

where θg(t) is the angular velocity measured in the Z axis,ηg(t) is the white Gaussian noise with variance σ2

g , bg(t) isthe gyroscope bias, τg is the Markov process correlation time,and and ηbg(t) is the white Gaussian noise of gyroscope biaswith σ2

bg.

Fig. 4. OpenSim 2.4 using a lower limb model.

In an analogous manner the acceleration obtained from theaccelerometer, a(t), at the instant t, can be modeled as

a(t) = aa(t) + ba(t) + ηa(t), (3)

b(t) = − 1

τab(t) + ηba(t), (4)

where aa(t) is the acceleration measured in local coordinatesystem from IMU, in the X axis direction, ηa(t) is the whiteGaussian noise of the accelerometer with variance σ2

a , ba(t)is the accelerometer bias, τa is the Markov process correlationtime, and ηba(t) is the white Gaussian noise of accelerometerbias with variance σ2

ba.

With these, and since that the variances values σ2g , σ2

bg,

σ2a , σ2

baare know, through (1) and (3) we can generate the

simulated signals if we consider that θg and aa are respectivelythe velocity and acceleration of each segment of legs.

C. State-Space System Modeling

Considering (1) and (2), the estimation of the angularvelocity and of the gyroscope bias is given by

˙θ(t) = θg(t) + bg(t), (5)

˙bg(t) = − 1

τgbg(t). (6)

Hence, making

∆θ(t) = θ(t)− ˙θ(t)

= bg(t)− bg(t) + ηbg(t)

= ∆bg(t) + ηbg(t), (7)

and

∆bg(t) = bg(t)− ˙bg(t)

= − 1

τg(bg(t)− bg(t)) + ηbg(t)

= − 1

τg∆bg(t) + ηbg(t), (8)

we obtain the system dynamic equation:[∆θ(t)

∆bg(t)

]=

[0 10 − 1

τg

] [∆θ(t)

∆bg(t)

]+

[1 00 1

] [ηg(t)ηb(t)

],

(9)where ∆θ(t) and ∆bg(t) are the state variables, the absoluteposition error and the gyroscope offset error, respectively.

The position obtained from the accelerometer is used in thesensor fusion strategy as a redundant measurement with thepurpose of correcting the gyroscope-based position. As it isknown the accelerometer-based position is given by:

θa(t) = arcsin

(aa

g

)= θ(t) + ηa(t), (10)

where g = 9.81m/s2 is the gravitational acceleration, andηa(t) is the noise.

To obtain the output equation of the state-space model,which is represented by the difference between theaccelerometer-based position and the gyroscope-based posi-tion, we make:

∆z(t) = θa(t)− θg(t)= θ(t) + ηa(t)− θg(t)= ∆θ(t) + ηa(t). (11)

Consequently:

∆z(t) =[

1 0] [ ∆θ(t)

∆bg(t)

]+ ηa(t), (12)

where ∆z(t) corresponds to the position measurement whichprovides feedback to the filter.

After the estimation of each pair of states, the positionmeasurement and the gyroscope offset are corrected by:

θ(t) = θg(t) + ∆θ(t), (13)

andbg(t) = bg(t) + ∆bg(t). (14)

D. Kalman Filter

Considering now the discrete time state-space model of (9)and (12),

xk+1 = Φkxk +Gkηk, (15)

zk = Hkxk + vk. (16)

Using the methodology presented in [14], the matrices ofthe system represented in (15) and (16) in discrete time, aregiven by

Φ =

[1 T

0τgi−Tτgi

]G =

[T

12 0

0 T12

]

H =[

1 0]

where T is the time sample and τgi , i = 1, ..., 6 is a timeconstant of a random process, for each segment of the legs.

So the Kalman Filter is given by, see [15]

Predict

xk+1|k = Φxk|k

Pk+1|k = ΦPk|kΦT +GQGT (17)

Update

yk+1 = zk+1 −Hxk+1|k

Kk+1 = Pk|k−1HT (HPk|k−1H

T +R)−1

xk+1|k+1 = xk+1|k +Kk+1yk+1

Pk+1|k+1 = (I −Kk+1H)Pk+1|k (18)

where Q and R are considered in this problem as timeinvariant, given by

Q =

[σ2

g 00 σ2

bg

](19)

andR =

[σ2

a

]. (20)

Usually the variances values of gyroscopes (σ2g and σ2

bg)

and accelerometers (σ2a and σ2

ba) are not available in the

datasheet of low cost sensors. In these cases we can usegenetic algorithm to estimate this values or another methodof optimization.

E. Genetic Algorithm

A genetic algorithm (GA) can be defined as search heuristicthat mimics the process of natural evolution. This process isnormally used to generate solutions to optimization and searchproblems, see [16]. In this work the topology of the appliedGA is represented in Figure 5.

Fitness Function

Define Initial Population

Selection

Crossover

Mutation

Returns the best

Termina(on  No

Yes

Elitism

Update

Fig. 5. Genetic Algorithm

The goal of the proposed GA is to optimize the diagonalelements of matrices Q and R, with this the performance ofthe estimation is calculated using Fitness Function block showin Figure 5, this evaluates the sum error between the referenceθref (t), obtained in Section II-A, and the estimated angles θ(t)by (13), as follows

JGA =1

T

T∑i=0

|θref (i)− θ(i)| (21)

F. Required Conditions to Apply the Filter

To use the Kalman filter correctly, the feedback signal ofthe filter must be extremely reliable. Thus, it is necessary toestablish conditions to reach this requirement. The standardsolution for this kind of problem is proposed in Case 1,whereas in Case 2 is proposed the same solution used inUnmanned Aerial Vehicles (UAV), [9].

Case 1: This condition concerns to the intervals in whichthe position measurement obtained from the accelerometerpresents the least possible variance. These intervals are ob-tained using the three FSRs placed in the sole of shoe. It wasdefined that the accelerometer signal is not take into accountwhen the heel hits the ground because, around this event, high-magnitude accelerations are sensed. However, when the gait isin transition between stance phase and swing phase, minimalvariances are observed. Thus, Table I summarizes the secondcondition.

TABLE IGAIT INTERVALS TO ENABLE THE FILTER.

FSRheel FSRmiddle FSRfront

Interval 1 Low High or Low HighInterval 2 Low Low Low

Case 2: Using only the signals obtained from the triaxialaccelerometer we use a variation of the validation presentedin [9]. This validation verify when the signal obtained fromaccelerometers is reliable, and in these moments their signalsare take into account to feed the filter. This checking is givenbelow ∣∣∣‖ [ax ay az]

T ‖ − g∣∣∣ ≤ ζ, (22)

where ax, ay and ax are the values of the 3-axis accelerometer.So when the norm of the three components of the acceler-

ation, ζ, is lower than the norm of gravitational vector thenthe signal is considered reliable. However, due to the biasdeviation, this value needs to be tuned. and it was added tobe tuned by the GA, present in Section II-E.

Regarding these two aforementioned cases, it was createda switching logic to let the Kalman filter algorithm to correctthe absolute angles in a reliable way, for each segment ofthe exoskeleton. In the following Section III, we make acomparison between these two cases.

III. RESULTS

The diagonal values of Qi and Ri, in (19) and (20), fori = 1, ..., 6 are tuned by GA present in Section II-E. Thisweighting values are shown in the tables II and III.

Figures 6 to 11 show the estimative of absolute angles inthe feet is good for both approach, figures 6 and 9. Howeverwhen the segment moves away from the feet, the estimativebecomes inaccurate, figure 8 and 11. The great advantage of

TABLE IIKALMAN FILTER GAINS - RIGHT LEG

Limb segment ζ σ2g σ2

bgσ2a τg

R. Thigh 0.01 8100.6 465.5 3411.1 9694.7R. Shin 0.9 9507.4 8892.3 2144 2980.1R. Foot 0.59 5098.6 47.6 3241.2 862.9

R. Thigh with FSR - 85.6 627 8810.3 5931.8R. Shin with FSR - 157.9 9411 2728.7 6694.4R. Foot with FSR - 19.4 1149.8 9937.7 6346.3

TABLE IIIKALMAN FILTER GAINS - LEF LEG

Limb segment ζ σ2g σ2

bgσ2a τg

L. Thigh 0 3045.5 5244 1530 8556.2L. Shin 0 655.4 9776.3 7047.6 9288.7L. Foot 0 4364.8 6646.9 6481.5 4823.8

L. Thigh with FSR - 3660.8 1082.2 830.4 8796.4L. Shin with FSR - 9010.1 1288.5 3014.9 8455.1L. Foot with FSR - 12.6 4724.3 2178.8 5582.6

using Case 2 to verify when the signal is reliable is that thereare more scattered points, while in Case 1 there are pointsconcentrated in a small region; a good example of this can beseen in figures 10 and 11.

IV. CONCLUSION

In this work simulation results comproved that the ap-plication of inertial sensors, to switch logic the KF on theestimation of absolute angles for lower limb, is more efficientthan using FSR in an insole. Moreover, we show that a GAcan be used to optimize the weighting matrices of the KF forlower limb angular positions estimation.

0.5 1 1.5 2 2.5 3 3.5 4

−1.2

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0

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Ab

so

lute

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d)

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(a) Left Foot with FSR

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(b) Left Foot with 3-axial acel.

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Absolu

te a

ngle

(rad)

FSR

Filtered

Kalman

(c) Correction with FSR

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Ab

so

lute

an

gle

(ra

d)

3−Accel

Filtered

Kalman

(d) Correction with 3-axial acel.

Fig. 6. Left foot angle

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−1

−0.5

0

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Ab

so

lute

an

gle

(ra

d)

FSR

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Refer.

(a) Left Shin with FSR

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−0.5

0

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Ab

so

lute

an

gle

(ra

d)

3−Accel

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Refer.

(b) Left Shin with 3-axial acel.

0.5 1 1.5 2 2.5 3 3.5 4

−1

−0.5

0

0.5

Time(s)

Absolu

te a

ngle

(rad)

FSR

Filtered

Kalman

(c) Correction with FSR

0.5 1 1.5 2 2.5 3 3.5 4

−1

−0.5

0

0.5

Time(s)

Ab

so

lute

an

gle

(ra

d)

3−Accel

Filtered

Kalman

(d) Correction with 3-axial acel.

Fig. 7. Left shin angle

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so

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d)

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(a) Left thigh with FSR

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(b) Left thigh with 3-axial acel.

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ngle

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FSR

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Kalman

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gle

(ra

d)

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Kalman

(d) Correction with 3-axial acel.

Fig. 8. Left thigh angle

V. ACKNOWLEDGMENT

This work is supported by grants from Coordenacao deAperfeicoamento de Pessoal de Nıvel Superior (Capes) andFundacao de Amparo a Pesquisa do Estado de Sao Paulo(FAPESP-2012/05552-9).

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(a) Right Foot with FSR

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Ab

so

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Ab

so

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an

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Kalman

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Ab

so

lute

an

gle

(ra

d)

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Kalman

(d) Correction with 3-axial acel.

Fig. 9. Right foot angle

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−1

−0.5

0

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Time(s)

Ab

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gle

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Refer.

(a) Right Shin with FSR

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−1

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0

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Time(s)

Ab

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3−Accel

Filtered

Refer.

(b) Right Shin with 3-axial acel.

0.5 1 1.5 2 2.5 3 3.5 4

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−0.5

0

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(c) Correction with FSR

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−1

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0

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Time(s)

Ab

so

lute

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gle

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d)

3−Accel

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Kalman

(d) Correction with 3-axial acel.

Fig. 10. Right shin angle

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0.5 1 1.5 2 2.5 3 3.5 4−0.4

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(a) Right thigh with FSR

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(b) Right thigh with 3-axial acel.

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Fig. 11. Right thigh angle

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