CONTROLLER DESIGN FOR A STANCE-CONTROL

KNEE-ANKLE-FOOT ORTHOSIS BASED ON OPTIMIZATION

TECHNIQUES

S. H. HosseinNia

, F. Romero

, B. Vinagre

, F. J. Alonso

and I. Tejado

Dept. of Electrical, Electronic and Automation Engineering, University of Extremadura,

Avda de Elvas S/N, Badajoz, Spain

Dept. of Mechanical Engineering, Energetics and Materials, University of Extremadura,

Avda de Elvas S/N, Badajoz, Spain

Keywords:

Active Orthosis, Biomechanics, Muscle Modeling, Optimization, Optimal Control.

Abstract:

Design of active orthosis is a challenging problem from both the dynamic simulation and control points of

view. The redundancy problem of the simultaneous human-orthosis actuation is an interesting exercise to

solve concerning the analytical and computational cost effectiveness. The physiological static optimization

approach tries to solve the actuation sharing problem. Its objective is to quantify the contributions of muscles

and active orthosis to the net joint torques in order to select the proper actuator for the joint. Depending on

the disability of each patient, different controllers can be designed. As a matter of fact, the duration of the

gait cycle for each patient should be different. In this paper, a PI controller is designed whose parameters are

tuned by optimizing a cost function which takes into account the patients muscle power and the error of the

knee angle with the reference value. Moreover, the ﬁnal time is obtained by minimizing the mean of integral

squared errors. The performance of the method is shown by designing the controller for three types of patients,

ordered from low to high disability. The objective of this work is to use optimal control techniques based on

physiological static optimization approach to the design of active orthosis and its control.

1 INTRODUCTION

Spinal cord injuries (SCI) cause paralysis of the lower

extremities because of the break of the connection be-

tween nervous central system and muscular units of

the lower body. According to the standard neurolog-

ical classiﬁcation of the American Spinal Injury As-

sociation (ASIA), there are different SCI levels de-

pending on motor and sensory function to be pre-

served. The ASIA Impairment SCALE (AIS) range

them from A (complete SCI) to E (normal and sen-

sory function). This work focuses in the assistance

of incomplete SCI subjects with AIS level C or D.

Those patients have partially preserved motor func-

tion in the key lower limb muscle groups, and can

perform a low-speed and high-cost pathological gait

by using walking aids. The energy cost and aesthet-

ics of this walk can be performed by means of active

orthosis, requiring external actuation mechanisms to

assist the motion of the lower limb joints during gait

cycle. Considerable efforts have been focused on the

design and application of passive and active orthoses

to assist standing and walking of SCI individuals.

There is a great evolution between the ﬁrst con-

trollable active, a patent by Filippi in 1942 (Filippi,

1942) of a hydraulically-actuated device for adding

power at the hip and knee joints, and the actual or-

thotic devices. Concerning the ﬁrst, developed at the

University of Belgrade in the 60’s and 70’s by Vuko-

bratovich et al. (Vukobratovic et al., 1972), these

early devices to aid people with paraplegia resulting

from spinal cord injury were limited to predeﬁned

motions and had limited success. Nowadays, orthotic

systems use predeﬁned patterns of joint motions and

torques together with classical control techniques or

EMG-based control, with the aim of integrating the

human musculoskeletal system and the assisting de-

vice. There are different designs in the literature, see

for example the review of Dollar (Dollar and Herr,

2008). Nevertheless, few studies (Silva et al., 2010;

Kao et al., 2010) examine the moment joint patterns

of combined patient-orthosis systems. Moreover, the

number of studies testing these systems on handi-

capped subjects is paradoxically low when comparing

H. HosseinNia S., Romero F., Vinagre B., J. Alonso F. and Tejado I..

CONTROLLER DESIGN FOR A STANCE-CONTROL KNEE-ANKLE-FOOT ORTHOSIS BASED ON OPTIMIZATION TECHNIQUES.

DOI: 10.5220/0003783900590064

In Proceedings of the International Conference on Biomedical Electronics and Devices (BIODEVICES-2012), pages 59-64

ISBN: 978-989-8425-91-1

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

with the studies on able-bodied subjects wearing the

orthosis.

To assist the proper design of active orthoses for

incomplete SCI, it is necessary to quantify the simul-

taneous contributions of muscles and active orthosis

to the net joint torques of the human-orthosis system.

Simulation of walking in individuals with incomplete

SCI wearing an active orthosis is a challenging prob-

lem from both the analytical and the computational

points of view, due to the redundant nature of the

simultaneous actuation of the two systems. In this

work, the functional innervated muscles of SCI pa-

tients will be modeled as Hill-type actuators, while

the idle muscles will be represented by stiff and dis-

sipative elements that increment the passive moments

of the inactive joints. The orthosis will be consid-

ered as a set of external torques added to the an-

kles, knees and hips to obtain net joint torque patterns

similar to those of normal unassisted walking. Kao

(Kao et al., 2010) suggests that able-bodied subjects

aim for similar joint moment patterns when walking

with and without robotic assistance rather than sim-

ilar kinematic patterns. This is the fundamental hy-

pothesis of this approach to obtain muscular forces:

the combined actuation of the musculoskeletal system

of the SCI subject and the active orthosis produce net

joint moment patterns similar to those of normal unas-

sisted walking. The muscle-orthosis redundant actua-

tor problem was solved through a physiological static

optimization approach (Alonso et al., 2011). A com-

parison between cost functions and various sets of in-

nervated muscles can be found in this work. Based

on these results, as Font et al. explained in (Font-

Llagunes et al., 2011), the proper actuation can be se-

lected, but control techniques are required to achieve

a suitable gait.

The objective of this work is to design an opti-

mal controller based on the minimization of a cost

function that takes into account the patients muscle

power and the tracking error of the knee angle. For

the patients with less capability, the weigh of the mus-

cle power in the cost function will be chosen bigger

whereas for patients with more capability, this weigh

will be chosen smaller. Therefore, this controller will

consider a trade off between accuracy of knee move-

ment regarding to healthy human waking and muscle

power of each patient.

The rest of the paper is organized as follows. In

Section 2, musculoskeletal modeling is stated. In ad-

dition, in order to obtain the muscular power devel-

oped by each muscle during gait cycle, the optimiza-

tion approach proposed in Alonso et al. (Alonso et al.,

2011) is applied. Section 3 addresses the design of

the optimal controller. Finally, Section 4 includes the

main conclusions of this work.

2 MUSCULOSKELETAL

MODELING

In this section, the biomechanical model adopted to

obtain net joint torques for normal walking is pre-

sented, as well as the muscle models for the functional

(innervated) and partially denervated muscles of the

spinal cord injured subject.

2.1 Biomechanical Multibody Model

The biomechanical model used has 12 degrees of free-

dom. It consists of twelve rigid bodies linked with

revolute joints (see Fig. 1), and is constrained to move

in the sagittal plane. Each rigid body is characterized

by mass, length, moment of inertia about the center

of mass, and distance from the center of mass to the

proximal joint. The equations of motion of the biome-

chanical multi-body system can be written as:

M ¨q + Φ

λ = Q, (1)

where M is the global (human-orthosis) mass ma-

trix, Φ

is the Jacobian matrix of the constraint equa-

tions, ¨q is the acceleration vector, Q is the generalized

force vector and λ are the Lagrange multipliers. Us-

ing kinematic and anthropometric data in (1), the net

joint torques during a physical activity or motion and

the resultant force and moment due to body-ground

contact can be calculated.

Figure 1: (a) Biomechanical model of the human orthosis

system. (b) Muscle groups of the lower limbs: 1 - Iliopsoas,

2 - Rectus Femoris, 3 - Glutei, 4 - Hamstrings, 5 - Vasti, 6 -

Gastrocnemius, 7 - Tibialis Anterior, 8 - Soleus.

2.2 Muscle Modeling: Innervated and

Denervated Muscles

According to AIS, it is possible to deﬁne several lev-

els which indicate the severity of the injury from A

BIODEVICES 2012 - International Conference on Biomedical Electronics and Devices

(complete) to E (normal motor and sensory function).

In the design cases C and D, the motor function is

preserved below the neurological level (lowest seg-

ment where motor and sensory functions are normal),

being the difference between them the muscle activ-

ity grade of the key muscles. This grade ranges from

0 (total paralysis) to 5 (active movement, full range

of motion, normal resistance). As Alonso et al. pro-

posed in (Alonso et al., 2011), the weakness of the

denervated muscles is modeled through a weakness

factor p ∈ [0, 1] that limits the maximum activation of

this muscles.

Both innervated and denervated are modeled as

Hill-type actuators. The Hill-type muscle-tendon

model (Zajac, 1989; Winter, 1991) is shown in Fig. 2

(a) and 2 (b). It consists of a contractile element (CE)

that generates the force, a nonlinear parallel elastic

element (PE), representing the stiffness of the struc-

tures in parallel with muscle ﬁbers, and a nonlinear

series elastic (SE) element that represents the stiffness

of the tendon which is serially attached to the muscle

and completes the muscletendon unit. The two dif-

ferential equations that govern the muscle dynamics

are:

˙a = h(u, a), (2)

= g(a, f

). (3)

Equation (2) refers to the activation dynamics,

which relates the muscle excitation u from the cen-

tral nervous system (CNS) and the muscle activation

a ∈ [0,1]. On the other hand, equation (3) deﬁnes the

force-generation properties as a function of the mus-

cle tendon length l

and velocity v

. Activation dy-

namics is not considered for the purpose of this work.

If the pennation angle α is constant, in accordance

with Fig. 2 (b):

= l

+ l

cos(α), (4)

= f

= ( f

+ f

)cos(α) ≈ f

cos(α), (5)

where the force of the parallel elastic element is set to

zero (Ackermann, 2007; Ackermann and Schiehlen,

2006). The tendon (SE) can be modeled by a sim-

ple quadratic force-strain curve depending on tendon

stiffness as follows:



0 if l

< l

− l

)

if l

> l

, (6)

where l

is the tendon slack length and k

is the SE

stiffness, which is given by:

(ε

, (7)

being ε

(3% to 5%) the strain occurring at the max-

imal isometric muscle force f

(Ackermann, 2007).

The force generated by the CE f

is a function of the

activation a, its length l

, and its contraction veloc-

ity v

. The expression for the concentric contraction

< 0) reads as:

= a

( f

iso

+ A

) − A

−

˜v

)

−

˜v

, (8)

where ˜v

= v

opt

, A

= 0.41, B

= 0.52 and f

iso

(w,l

opt

), which corresponds to the muscle iso-

metric force relative to the maximal isometric muscle

force f

and f

= min(1,3.33a).

The expression for the eccentric contraction (v

0) depends on ˜v

and f

iso

. The force-length-velocity

relationship is shown in Fig. 2 (c).

Figure 2: Muscle model: (a) Conceptual scheme. (b) Hill

model (Zajac, 1989). (c) Force-length-velocity model.

In order to quantify the muscle weakness, the

muscle activation will be multiplied by the men-

tioned weakness factor p, where p = 1 for innervated

muscles, 0 < p < 1 for partially denervated muscles

and p = 0 for totally denervated muscles (no activ-

ity). The atrophy of denervated muscles, as exposed

by Thomas and Grumbles (Thomas and Grumbles,

2005), depends on the elapsed time from the injury.

This atrophy increases the passive torques at the joint.

Several studies (Edrich et al., 2000; Lebiedowska and

Fisk, 1999; Amankwah et al., 2004) show that pas-

sive torques tend to be larger in pathological than

in healthy individuals. To take this fact into ac-

count stiff and dissipative elements are included into

the model using the deﬁnitions given in (Amankwah

et al., 2004). Fig. 3 shows the increment of the knee

torque due to pathological passive torque compared

with the torques in normal gait (obtained through in-

verse dynamics analysis from the 2D walking kine-

matic benchmark from Winter (Winter, 1991)). As

CONTROLLER DESIGN FOR A STANCE-CONTROL KNEE-ANKLE-FOOT ORTHOSIS BASED ON

OPTIMIZATION TECHNIQUES

can be observed, only very slight differences can be

found between both torques.

Figure 3: Knee torques during normal gait (solid blue line)

and pathological gait (dashed red line) for the right leg.

Swing phase occurs from 0 to 0.4 in normalized time and

stance phase from 0.4 to 1.

2.3 Optimization Approach: Muscle

and Orthosis Actuation

In order to solve the load sharing problem in biome-

chanics, optimization procedures are used next. There

are several optimization methods (static and dynamic

optimization, large-scale optimization) and optimiza-

tion criteria (minimization of the metabolic cost of

transport, minimization of muscle stresses) in the lit-

erature (Menegaldo et al., 2006; Yamaguchi et al.,

1995). In order to obtain the forces that will be used in

the design of the controller, we use the physiological

static optimization approach (Alonso et al., 2011).

This modiﬁed version of the classical static opti-

mization approach considers muscle contraction dy-

namics, ensuring the physiological consistency of the

solution. This approach comprises two steps. In the

ﬁrst one, the inversion of the contraction dynamics

is solved assuming that muscle activation are max-

ima. The length (l

) and velocity (v

) of each ten-

don unit involved in the process are obtained from

the generalized coordinates of the multi-body model

and the maximum muscle-tendon length (Gerritsen

et al., 1998). Then, the maximum muscle force histo-

ries f

∗

(t) compatible with contraction dynamics are

calculated assuming the muscle activation is maxima

at each instant, i.e. A

= [a

,. .. , a

]

= [1,...,1]

Brieﬂy, for each muscle, the contraction dynamics

differential equation is integrated as:

d f

∗

= g((a = 1) · p, f

∗

). (9)

In the second step, the muscle activations and or-

thosis actuation is calculated by solving the optimiza-

tion problem:

Min J(F

) = ω

∑

j=1

(− f

ce, j

)

+ ω

∑

k=1

o,k

)

s.t. R · (AF

∗

) = T

0 ≤ a

≤ 1, j = 1, . .., N = 8

−1 ≤ o

≤ 1, k = 1, 2,3

, (10)

where AF

∗

= [a

· p

· f

∗

mt,1

,. .. , a

· p

· f

∗

mt,8

∗

o,1

,. .. , o

· T

∗

o,3

With this approach, muscle forces and orthosis ac-

tuation are calculated for a gait cycle in order to op-

timize the cost function and obtain the parameters of

the PI controller proposed. The 2D walking kinematic

benchmark data from Winter (Winter, 1991) was used

to perform an inverse dynamic analysis. This move-

ment corresponds to a healthy female subject with

57.75 kg of weight with normal gait. Once joint

torques have been calculated, the optimization prob-

lem is solved by using MATLAB routine fmincon im-

plemented in the optimization toolbox that uses a Se-

quential Quadratic Programming (SQP) method. The

simulated muscle-orthosis actuation was performed

for an AIS C subject: motor function partially pre-

served below the neurological level and more than

half of the key muscles below the neurological level

have a muscle grade less than 3. To simulate this

kind of injury, we have deﬁned the following vector

of weakness factor:

p = [1,0.2,1,0.2,0.2,0.2, 0.2, 0.2, ]

The orthosis actuation prevents stance phase knee

ﬂexion due to quadriceps and assists swing-phase

ﬂexion depending on the ability of each patient, as

shown in the simulation results for the knee in Fig.

3 CONTROLLER DESIGN

In order to control the orthosis, a mathematical model

of the motor is needed. In particular, the following

second order transfer function is used (see (Hossein-

Nia et al., 2011)):

P(s) =

3.58

s(0.01s + 1)

. (11)

To control the orthosis, a classic PI controller is

considered as:

C(s) = k

. (12)

The aim is to tune the parameters k

and k

in order

to optimize the following cost function,

J =

β(− f

)

+ (1 − β)e

, (13)

with e = θ

re f

− θ

. This cost function consists of two

parts. The ﬁrst one corresponds to the muscle power

BIODEVICES 2012 - International Conference on Biomedical Electronics and Devices

Table 1: The optimized controller parameters.

β k

0.1 40 45

0.5 15 20

0.9 1 6

where f

is the muscular forces obtained by opti-

mization and v

the muscular velocities. The sec-

ond part refers to the error between the knee angle

and the reference knee angle θ

re f

, respectively.

Two weights 0 < β < 1 and (1 − β) can be chosen

regarding to the muscle power of the patients. The

idea is to design an optimal controller based on the

patients muscle power and the tracking of a reference

signal, where, for the patients with less muscular ca-

pacity, β will be chosen bigger in order to minimize

power and perform the movement and, for patients

with more capacity, the value of β will be minor, so

the cost function prioritizes the minimization of the

tracking error and the movement is going to be made

in less time. Therefore, this controller will consider a

trade off between accuracy of knee movement (con-

cerning healthy human walking) and muscle power

of each patient, taking into account that patients with

less capability need more time to perform the same

movement.

In order to show the performance of the proposed

method, the controller will be designed regarding to

the following three weight options:

• β = 0.1 for the patients with low disability,

• β = 0.5 for the patients with fair disability,

• β = 0.9 for the patients with high disability.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−20

−10

knee

(Nm)

(β=0.1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−20

−10

knee

(Nm)

Gate cycle

(β=0.9)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−20

−10

knee

(Nm)

(β=0.5)

orthosis

muscle

total

Figure 4: Comparsion of orthosis and muscle torque ac-

cording to the patients ability.

Table 2: Final time.

β Final time

0.1 0.75

0.5 1.4

0.9 6.1

Fig. 4 shows the orthosis torque, muscle torque

and total torque corresponding to each controlled sys-

tem (the controller parameters ara given in Table 2).

As can be seen, prioritizing the muscular power with

a big value of β, we consider that patients have a ma-

jor disability and the assistance torque provided by

the orthosis should be higher to compensate the de-

ﬁciency. On the other hand, prioritizing the tracking

error with a low value of β means that patients have

more ability to perform the movement, so the torque

provided is lower and the movement is achieved in

less time with more accuracy. The ﬁnal times corre-

sponding to each case are calculated based on mini-

mizing the following mean of integral squared errors

(MISE):

MISE = E



− θ

re f



= E



(t)− θ

re f

(t)



dt,

(14)

where t

denotes the ﬁnal time in a gate cycle and

E denotes the expected value with respect to that sam-

ple. Minimizing MISEfor each controller designed,

the ﬁnal time is obtained based on an optimization

program to satisfy MISE < 0.01):

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.5

0.5

1.5

(rad)

β=0.1

0 0.5 1 1.5

−0.5

0.5

1.5

(rad)

β=0.5

0 1 2 3 4 5 6 7

−0.5

0.5

1.5

Time (s)

(rad)

β=0.9

Knee angle

Reference knee angle

Figure 5: Effect of disability in ﬁnal time.

Fig. 5 shows the effect of the disability of the pa-

tients in ﬁnal times. As can be observed, patients with

higher β (considered with disability) need more time

to perform the complete gait cycle than patients with

CONTROLLER DESIGN FOR A STANCE-CONTROL KNEE-ANKLE-FOOT ORTHOSIS BASED ON

OPTIMIZATION TECHNIQUES

low value of β, corresponding to low disability. In the

same way, high disability correspond to higher track-

ing error, and lower disability correspond to a better

accuracy in the performance of the movement com-

pared with the healthy subject.

4 CONCLUSIONS

In this paper, in order to control an orthosis, an op-

timal approach is proposed to design a PI controller

according to disability of the patients. This disabil-

ity is simulated by means of physiological static opti-

mization approach where the muscular forces of SCI

are obtained in a process that combines the actuation

of the muscles and the external actuation provided by

the orthosis. Those forces are used to design a proper

controller for the external actuation. Considering pa-

tients with a high disability, the controller is tuned to

perform the movement so as to allow the patient to

achieve the movement but in a longer cycle compared

with patients with less disability, where the controller

is tuned giving priority to the accuracy of the move-

ment. Patients with less power in his muscle –high

disability–, need more time in a gate cycle to walk,

whereas patients with low disability need less time.

This idea is shown through some three types of dis-

ability, i.e. high disability, fair disability and low dis-

ability. The simulation results show the efﬁciency of

the proposed method.

ACKNOWLEDGEMENTS

This work was supported by the Spanish Ministry of

Science and Innovation under the project DPI2009-

13438-C03. The support is gratefully acknowledged.

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