A HYBRID METHOD BASED ON FUZZY INFERENCE AND

NON-LINEAR OSCILLATORS FOR REAL-TIME

CONTROL OF GAIT

J. C. Moreno

1,2

, J. L. Pons, E. Rocon

Bioengineering Group, Industrial Automation Institute, the Spanish National Research Council (CSIC), 28500 Madrid, Spain

Technaid S.L., 28500 Madrid, Spain

Y. Demiris

Intelligent Systems and Networks Group, Imperial College London, SW7 2BT London, UK

Keywords:

Fuzzy inference systems, non-linear oscillators, inertial sensing, gait compensation.

Abstract:

Robust generation of motor commands for real-time control of locomotion with artiﬁcial means is crucial for

human safety. This paper addresses the combination of fuzzy inference for determination of rules with a non

linear oscillator system, as generators of motor commands for the control of human leg joints during walking,

by means of external gait compensators, e.g. exoskeletons, functional electrical stimulation or hybrid systems.

The response of the proposed method is evaluated for variations in stride frequency and step length. The

testing during gait conditions is performed considering inertial sensing as feedback in a simulation study. The

reference data considered is obtained in multiple experiments with healthy subjects walking with a control-

lable exoskeleton designed to compensate quadriceps weakness. A model of the operation of the knee joint

compensation provided by the exoskeleton is obtained as reference to evaluate the method based on real data.

The results demonstrate the beneﬁts of both incorporating a) the fuzzy inference system in cyclical decision

making for generation of motor commands and b) the dynamic adaptation of the timing parameters of the

external compensator provided by the van der Pol oscillator.

1 INTRODUCTION

Robust generation of motor commands for real-time

control of locomotion with artiﬁcial means is cru-

cial for human safety. Broadly, current active ex-

ternal compensators of pathological gait under re-

search can be conﬁgured as functional electrical stim-

ulators (FES), (Popovic et al., 1999), (Skelly and

Chizeck, 2001) controllable leg exoskeletons or or-

thoses, (Blaya and Herr, 2004), (Irby et al., 1999),

(Moreno et al., 2005) or as a combination of both,

known as hybrid systems, (Gharooni et al., 2000),

(Goldfarb and Durfee, 1996). From the control point

of view, the design of robust controller of locomo-

tion with such devices, towards real life application,

must be easy to customise, adapt dynamically to typ-

ical variations in gait pase and preferably should in-

corporate a coordinated development with the user.

1.1 Gait Compensation

A wide range of external gait compensators, e.g. ex-

oskeletons, functional electrical stimulation or hybrid

systems, have been considered to restore human gait.

In particular leg exoskeletons or orthoses, can be pre-

scribed for cerebrovascular accident, polyo myelitis

or cerebral palsy patients with leg muscle weakness,

in order to provide knee stability, reducing falling risk

and enabling a certain degree of mobility.

In order to control an exoskeleton, it is not clear

the hypothesis that instantaneous control of trajec-

tory of the joint angle is essential for the lower limb

system, since the reduced mechanical output —joint

torque— limits its transitory response, in relation with

the inertial properties of the musculoskeletal system

(Franken, 1995). Velocity or position control loops

are more appropriate and safe in training and reha-

bilitation applications where controlled generation of

joint trajectories is required, with application of oscil-

latory signals and modulation techniques during gait

C. Moreno J., L. Pons J., Rocon E. and Demiris Y. (2008).

A HYBRID METHOD BASED ON FUZZY INFERENCE AND NON-LINEAR OSCILLATORS FOR REAL-TIME CONTROL OF GAIT.

In Proceedings of the First International Conference on Bio-inspired Systems and Signal Processing, pages 44-51

DOI: 10.5220/0001060800440051

 SciTePress

Swing

Stance

Figure 1: Mechanical adaptation for gait compensation dur-

ing one gait cycle at the knee joint.

cycles for training subjects following neural/motor in-

juries.

Our previous work has consisted in the implemen-

tation of intermittent control of resistance of the knee

joint with an unilateral exoskeleton applying selec-

tively different constant stiffnesses depending on gait

phase, to approach more natural proﬁles and avoid

collapsing of the knee and risk of falling, see ﬁg. 2.

Linearsolenoid

Sensorset

Anklepassive

actuatorand

carbonfiberinsole

Controller

Kneeactuator

Figure 2: Controllable ambulatory exoskeleton.

Under this approach, a knee actuator is controlled

to apply a given impedance K

in the stance phase,

during a period of time ensuring the joint stability

and shift during swing phase releasing the joint for a

free swing while applying K

>> K

), for smooth

transition and storage/recover of energy to assist the

leg extension, see ﬁgure 1.

1.2 Rule based Control

Reliability of control in such a wearable solution for

pathological cases is a critical issue that has an impact

in human safety.

Figure 3: Typical normal gait pattern of foot and shank seg-

ments rotations and rotational velocities (sagittal plane) dur-

ing a walking task at 34 m/min speed, with the cable-driven

exoskeleton, after training of the subject. A system with a

cable triggers the knee mechanism (onset) depending on a

ﬁxed degree of dorsiﬂexion.

The output of the controller is the motor com-

mand for the actuator, characterized by two param-

eters: activation onset and period. The activation on-

sets during each stride are calculated by rule-based

conditions, evaluated according to segments orienta-

tion or rate velocity (See pattern during stance phase,

ﬁgure 3). The system is a reactive controller perform-

ing according to the motion of the leg. The criteria to

cyclically adapt the activation period (pulse width) of

the actuator is deﬁned considering temporal parame-

ters relative to stance phase of current S(k) and past

S(k-1) strides, and initial conditions S(0), given by

average expected values.

Rule-based

condition

Intermittent

mechanism

Leg

dynamics

Linearsolenoid

excitation

b(k)

f s

Figure 4: Control scheme for walking.

Experimental trials have demonstrated short-term

adaptation of human motor system when apply-

ing functional compensations with customized tun-

ning of the discrete rule-based controllers ,(Moreno,

A HYBRID METHOD BASED ON FUZZY INFERENCE AND NON-LINEAR OSCILLATORS FOR REAL-TIME

CONTROL OF GAIT

2006). The adaptation of cyclical activation, has

demonstrated proper results at self-preferred constant

speeds. The next proposed method is an improvement

intended to provide the required dynamicaladaptation

to changes of step frequency/length by the user.

1.3 Bipedal walking with Central

Pattern Generators (CPGs):

Simulation

It has been demonstrated previously how the use of

the dynamical systems paradigm can realize a walk-

ing behavior in robotic walking platforms (Veskos

and Demiris, 2006). The neural architecture has

demonstrated successful operation in swinging and

planar walking in a bipedal platform, incorporating

van der Pol oscilators as generators of motor com-

mands.

Medium and short term application of a walking

real-time controller for the mentioned application sce-

narios, ought include mechanisms that provide adapt-

ability and stable response to variations of frequency

in the feedback signals, can led to an approach of co-

operative development with the user/environment. In

the following, the analysis of the response of the pro-

posed hybrid controller to variations in gait frequency

is evaluated with real data measured with the orthotic

walking platform.

2 METHODS

2.1 Gait Patterns with Knee Joint

Compensator

Subjects wearing an exoskeleton, need to adapt their

walking strategy to drive the system to successfully

switch between two knee spring damper conﬁgura-

tions. During the entrainment of the subject with the

controllable exoskeleton it is necessary to reach a cer-

tain ankle dorsiﬂexion angle which is variable during

normal gait. Although this angle is adjustable, sub-

jects change their gait pattern until they learn to use

the exoskeleton. The learning process (which can be

seen as an adaptation) in the use of the controllable

exoskeleton has been previously studied in (Forner-

Cordero et al., 2006). In order to obtain sampled

data of different gait speeds, experimental trials with

a healthy subject have been conducted after the adap-

tation process, consisting in walking back and forth

along a 10 meter path, with deﬁnition of the step

length with marks on the ﬂoor and the gait speed by

means of a metronome, and systematic adjustments

of the cable mechanism to provide a comfortable gait

pattern (see table 1). The gait velocity and step length

variations were deﬁned according to average values

taken from Perry, (Perry, 1999), consisting in feasi-

ble combinations of 100%, 70%, 60% and 50%. Rate

gyroscopes ﬁxed at the shank and leg segments of the

external device were used to measure rotational veloc-

ities along the sagittal plane. Motions of interest oc-

cur at normal (2.6 km/h) and low (2 km/h) gait speeds,

and therefore, signals outside the band frequency re-

lated to gait kinematics (0.3–20 Hz), are rejected from

the sensor outputs with -3 dB low pass ﬁlters, refer

(Moreno et al., 2006) to for details. A precision angu-

lar position sensor was ﬁxed at the knee joint to track

the knee joint angle in the sagittal plane. A resistive

pressure sensor (5 mm in diameter active area, 0.30

mm thickness) is used to monitor the activation status

of the knee actuator.

Collection of input/ouput data is utilized to gener-

ate training and checking data sets, of both multiple

speed trials, and constant speed separated trials.

2.2 Validation Model

A robust Model describing the dynamics of the knee-

orthotic hinge system during cyclic walking condi-

tions can be used as a reference to analyze the per-

formance of the advanced control system. We pro-

pose the identiﬁcation of the model the activation pat-

terns provided by the cable driven exoskeleton, with

time-series of kinematic data. A broadly used signal

processing paradigm is the state-space model. De-

ﬁned by two equations, the state-space model has

been broadly applied in signal processing (Smith and

Brown, 2003). A ﬁrst equation describes how the hid-

den state or latent process is observed and a second

(state) equation that deﬁnes the evolution of the pro-

cess through time. Based on the formulation given by

(Haverkampet al., 1996), we proposeidentiﬁcation of

a multiple-input single-output continuous-time model

from the experimentally collected input and output

data.

Considering the state-space model in the innova-

tions form

dx(t)

= Ax(t) + Bu(t) (1)

y(t) = Cx(t) + Du(t) (2)

where u(t) denotes the sampled inputs, being the

foot and shank rotations in the sagittal plane dur-

ing walking, for continuous measurements at 100 Hz

sampling frequency, with transitions from low to high

speed, and progressive variations in step length and

given the measured output reference; y(t), as the en-

trained knee joint status (actuator activation period)

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

Table 1: Systematic variations of healthy subject walking with the cable driven prototype (* Not feasible combinations).

Percentage Step length[m] Stride length[m] Speed[m/s]

100 0.73 1.46 1.35 0.94 0.81 0.67*

70 0.51 1.02 0.94* 0.66 0.56 0.47

60 0.44 0.88 0.81* 0.56 0.48 0.40

50 0.37 0.73 0.67* 0.47 0.40 0.33

Cadence (step/min) 111 78 67 56

Metronome (bpm) 1.85 1.30 1.11 0.93

for normal walking, x(t) is the internal state of the

system and [A, B,C, D] are the deterministic system

matrices.

The reference sampled input and output data u(t)

and y(t) is obtained from experiments with healthy

subjects wearing a orthotic walking platform, manu-

ally adjusted at each velocity to trigger the knee actu-

ator based on the ankle dorsiﬂexion.

The goal of the state-space model identiﬁcation

process implemented in MATLAB is to ﬁnd the sys-

tem matrices [A, B,C, D] according to the model struc-

ture. This resulted in a second order model as the best

to the input-output behavior of the system, selected

upon the analysis of the singular values (1st order,

53.23; 2nd order, 3.77; 3rd order, 0.34; 4th, 0.30).

The continuous-time model describes the relation

between the foot and shank segments angular veloci-

ties and the output activation at the knee joint actuator

for the range of tested speeds, by the state differential

equation 1 and the output equation 2, where

A =



0.994 −0.063

−0.003 0.933



; (3)

B =



−3.05e

−6

−8.28e

−6

−2e

−5

−3.39e

−5



; (4)

C =



14.55 −0.009



; (5)

Assuming the initial state as zero, from the evalu-

ation of the transient (impulse) response of the second

order system, it can be concluded a stable system with

= 0.5 s, as the time to reach the peak value, and a

settling time t

of approximately 10 s, after persistent

excitation.

Evaluation of the response of the model compared

against the external compensator operation is then

performed, with the checking data set corresponding

to multiple speeds. The crossing zeros (time inter-

polation) of the oscillatory output signal during the

steady state are detected as equivalent onset and offset

timings of the measured events. The correlation coef-

ﬁcient r

, calculated for the modeled and measured

outputs is 0.999.

2.3 Architecture

The control scheme consists of different modules (see

ﬁgure 6). A fuzzy inference system with two inputs

and a single output node is identiﬁed and trained to

map the inputs and trigger the actuator. The crisp out-

put of the fuzzy inference system during each cycle

is critical in providing transition between restrained

knee ﬂexion in stance to a free swinging leg. The

activation period of the knee actuator (pulse width)

during the swing phase is cyclically adapted by a

second module composed by an nonlinear oscilla-

tor. This nonlinear system incorporates real-time es-

timated gait temporal parameters as feedback in the

generation of an oscillatory signal which adapts the

duty cycle of an external compensator.

2.3.1 Fuzzy Inference System

Conventional PID controllers have been applied in the

control of cyclical movements in legs of paraplegic

subjects (Franken, 1995). Introduction of dynamical

adaptation of the rules commanding FES systems has

been investigated, in order to cover a wider range of

unsafe and uncertain situations in application of stim-

ulation . A Sugeno system is suited for modeling non-

linear systems. A training scheme with a fuzzy mod-

eling network structure has been combined to develop

a gait synthesis learning scheme, (Horikawa et al.,

1990).

Obtaining a fuzzy system corresponds with ap-

proximated reasoning, which refers to methodologies

to describe physical systems which include complex-

ity due to nonlinearities and uncertainties. Let us sup-

pose that our unknown system is a black box only ca-

pable of measuring a set of inputs x

,...,x

and outputs

,... ,y

. A fuzzy system with a crisp output and the

following type of rules is to be obtained

: IF x

es S

and...and x

es S

, THEN and es c

(6)

The fuzzy inference system is generated by means

of the grid partition method. For the identiﬁcation a

A HYBRID METHOD BASED ON FUZZY INFERENCE AND NON-LINEAR OSCILLATORS FOR REAL-TIME

CONTROL OF GAIT

Angle [°]

−400

−200

200

400

Ang. Velocity [°/s]

−200

200

Ang. Velocity [°/s]

10 15 20 25 30 35 40 45 50

off

Time [s]

Error

Time[s]

0.16(+)

0.73(−)

Knee status

1.35

0.94 0.81 0.66

0.56 0.47 0.56 0.48 0.4 0.47 0.4 0.33 Gait Speed

Knee

Foot

Shank

1.85

1.30

1.30 1.30

1.11

1.11 1.11 1.11 0.93

0.93 0.93 Stride length

Figure 5: Data of measured input/ouputs of healthy subject with the compensator, after adaptation, at multiple gait speeds

and space-state model performance. (a) Knee joint measured angle with external compensation, checking data set of (b) Foot

angular velocity and (c) Shank angular velocity; (d) the measured activation status of the actuator (dotted line), model ouput

(dashed line) onset and offset timings given by the model (circles), and time difference per cycle.

FuzzyInference

System

x: w w

f s

Knee

Actuator

Lowerlimb

dynamics

MotorCommand

Generator

NonlinearPattern

Generator

w w

f s

du/dt

onset

y (k+ )

f:

Pulsemod.

width

Figure 6: Hybrid architecture for control of gait external compensation at knee level, based on inertial sensing data. First

module contains the fuzzy inference system with a crisp output The second module contains a nonlinear system predicting

the activation period of the knee actuator as function of gait frequency (forced oscillator), with proprioceptive feedback.

training data set is generated from the experimenta-

tion. The identiﬁcation method consists in the appli-

cation of the adaptive network fuzzy inference system

(ANFIS) proposed by Jang, (Jang, 1993), in order to

build the fuzzy rules with membership functions to

generate input/output data pairs. Iteratively, input pa-

rameters of the membership functions are learnt by

means of back-propagation in an adaptive network

and while the parameters of output functions are opti-

mized by the least squares ﬁtting method. The adap-

tive network is a feedforward multilayered network,

with a supervised learning scheme. The functions of

given nodes in a layer are similar. For means of sim-

plicity, we consider a ﬁrst order Sugeno type model,

as the inference system. Having the kinematic inputs,

the output E(t) and n fuzzy rules:

: IF

is A

, AND

is B

THEN E = p

+ t

(7)

Gaussian membership functions have been se-

lected for smooth transition. A total of 4 Sugeno

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

type fuzzy rules were deﬁned, with a network with

21 nodes. These rules were of AND (minimum) type

antecedent. The defuzziﬁcation method, calculating

the output, is performed by the centroid method. The

clustering radius r = 0.2 was adjusted for tunning.

The optimization process spanned 13 epochs, with the

training data set. The ﬁgure depicts the output surface

of the ﬁnal identiﬁed system given the two inputs.

−200

−100

100

200

−200

−150

−100

−50

100

−10

−8

−6

−4

−2

Foot angular velocity [°/s]

Shank angular velocity [°/s]

Output

Figure 7: Inputs (foot and shank angular velocities) output

(knee actuator activation) surface of the fuzzy system.

2.3.2 Forced Nonlinear Oscillator

The dynamic robustness of a pattern generator to

noise and other external disturbances can be improved

by incorporating nonlinearities to the system. A van

der Pol oscillator, requires a reduced number of pa-

rameters, and has the advantages of robustness and

ease of computational implementation. Such nonlin-

ear system can be applied as an adaptive oscillator

during the swing phase to determine the time of acti-

vation of the external compensator. To unlock the fre-

quency of the oscillator and provide it with adaptabil-

ity to the leg motion, the nonlinear system is forced to

oscillate at a frequency, which depends on the spatio-

temporal behavior of gait. Let us consider the forced

nonlinear oscillator

[!top] ˙x = y (8)

˙y = − µ(x

− 1)y− ωx+ Acosθt (9)

with ω as the natural frequency µ as the damping

parameter, θ as the forcing frequency and A as the

amplitude of the forcing function. An approximate

solution of the non-linear system, satisfying the initial

conditions x = 0, y = 0 is calculated during each cycle

i with

(10)

where T

is the stance phase period in cycle i,

and R, a frequency scaling factor. T

is estimated

from consecutivelocal minima (peak) values from the

foot rotational velocity, as described in (Moreno et al.,

2006).

2.4 Hybrid Controller

Local minima values are detected from the output of

the fuzzy system, upon numerically integration. The

sensitivity of the local minima detector is given by

δ, which corresponds to the minimum difference in

amplitude with the neighbor samples. With the cali-

brated gyroscopes raw data, a δ = 40 was satisfactory

for all conditions. Thus, cyclically the fuzzy system

provides the activation onset, the controller incorpo-

rates the output of the nonlinear oscillator to predict

the width τ or duration period for the knee external

compensator, with

τ =

2ω

(11)

where D determines the duty cycle percentage. In-

corporating the prediction given by the forced oscilla-

tor, D = 0.8 was deﬁned and remained constant in all

further experiments.

An example of the hybrid controller for cyclic gait

at 0.94 m/s (stride length, 1.46 m) is depicted in ﬁg. 8

3 RESULTS

The performance of the hybrid controller is compared

with the validation model and the testing data set. The

mean errors and standard deviations are calculated,

considering 4 continuous gait cycles per each con-

dition, for the output of the fuzzy inference system

module and the nonlinear oscillator module (see Table

2). A negative error (in seconds) means anticipation

with respect to the reference. For the tested condi-

tions, the maximum average error for the fuzzy rule-

based detection was 0.19 s demonstrating the robust-

ness of a single fuzzy model to drastic variations in

stride frequency. The discrete rule-basedmethod, pre-

vious tests showed signiﬁcantly better performance

for the application of thresholds, during slow gait ve-

locities in comparison with the results with higher

velocities. The response with the fuzzy rule-based

method can be regarded as uniform for the tested con-

ditions. The maximum average error for the oscillator

was 0.32 s and therefore, the robustness to the varia-

tions in the timing of the generated motor commands

was observed. The evaluation with the continuous

data set provide a good indication of the accuracy and

robustness of the hybrid method.

A HYBRID METHOD BASED ON FUZZY INFERENCE AND NON-LINEAR OSCILLATORS FOR REAL-TIME

CONTROL OF GAIT

Knee sagital (continuous) − Validation model output (dashed)

Knee angle [°]

−400

−200

200

400

Shank segment (continuous) − Foot (dashed)

Angular velocity [°/s]

0 100 200 300 400 500 600

−200

200

Fuzzy Inference System Output (continuous) − Forced oscillator y ouput (dashed)

Sample

Amplitude

1/2ω

Figure 8: Example of the simulation results of the hybrid system at a relative fast speed. Measured knee angle and reference

output given by validation model (top); Inputs (middle); Fuzzy inference system and forced linear oscillator outputs. The

hybrid system generates the triggers (continuous vertical lines) and activation periods (dashed vertical lines).

Table 2: Results of the hybrid controller for the 12 testing conditions. Mean errors and standard deviations with respect to the

evaluation model output are calculated taking 4 continuous gait cycles per each condition.

Fuzzy System Forced oscillator

Step length[m] Speed [m/s] Mean error [s] SD Mean error [s] SD

1.46 1.3505 0.0995 0.0123 0.005 0.0451

1.46 0.94535 0.041 0.0744 -0.025 0.0719

1.46 0.8103 0.154 0.1847 -0.245 0.0806

1.022 0.6617 0.0685 0.0296 -0.06 0.051

1.022 0.5672 -0.042 0.0238 -0.17 0.0497

1.022 0.4726 -0.13 0.1238 -0.1775 0.1072

0.876 0.56721 0.0335 0.023 -0.1275 0.0629

0.876 0.48618 -0.0125 0.0728 -0.1725 0.083

0.876 0.4051 -0.1075 0.12 -0.32 0.0668

0.73 0.47267 0.0025 0.031 -0.16 0.0462

0.73 0.4051 -0.1205 0.0689 -0.0625 0.0998

0.73 0.3376 -0.1915 0.1003 0.15 0.2149

4 CONCLUSIONS

The evaluation with the continuous data set provide a

good indication of the accuracy and robustness of the

hybrid method. For the tested conditions, the results

demonstrate a proper means to combine a learning

method which incorporates fuzziness with the adap-

tive nature of a non lineal oscillator, to generate mo-

tor commands to control gait. A validation model

has been used in order to simulate the real mechani-

cal system (human leg and exoskeleton) in this study.

Further work includes a simulation study of the re-

sponse of the methos to external perturbations (foot

contact with the ground during the swing and obsta-

cles) and testing with subjects of the embedded appli-

cation.

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

ACKNOWLEDGEMENTS

The authors would like to thank Paschalis Veskos.

The work presented in this paper has been partially

founded by I3P programme of the Spanish Scientiﬁc

Research Council.

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A HYBRID METHOD BASED ON FUZZY INFERENCE AND NON-LINEAR OSCILLATORS FOR REAL-TIME

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