Active Orthosis for Ankle Articulation Pathologies
Carlos André Freitas Vasconcelos
Dissertação para a obtenção do Grau de Mestre em
Engenharia Mecânica
Júri
Presidente: Doutor João Rogério Caldas Pinto
Orientador: Doutor Jorge Manuel Mateus Martins
Co-Orientador: Doutor Miguel Pedro Tavares da Silva
Vogal: Doutor José Manuel Gutierrez Sá da Costa
Vogal: Doutor Manuel Cassiano Neves
Novembro 2010
i
Resumo
Esta dissertação apresenta a análise, simulação e controlo da articulação do tornozelo durante o ciclo
de marcha, com o objectivo de desenvolver uma ortótese activa para o pé e tornozelo (Active Ankle-
Foot Orthosis, AAFO) de modo a auxiliar indivíduos sem controlo motor da articulação do tornozelo.
Para realizar este objectivo, foi necessário compreender a anatomia, fisiologia e biomecânica do
sistema locomotor para que fosse possível desenvolver correctamente um modelo biomecânico da
perna, pé e articulação do tornozelo. Deste modo, diferentes estratégias de controlo do movimento da
articulação do tornozelo podem ser aplicadas, obtendo-se resultados necessários para o posterior
projecto do sistema de actuação da ortótese activa.
Um modelo de contacto elástico do pé com o chão foi desenvolvido no software SimMechanics® com
o objectivo de simular o contacto entre o pé e o chão. Prescrevendo o movimento cinemático apenas
para a articulação do joelho e perna, deixando a articulação do tornozelo livre, o modelo tem uma
grande importância, permitindo que o movimento normal da articulação do tornozelo durante o ciclo
de marcha possa ser obtido através do controlador. O movimento do tornozelo foi controlado
utilizando três estratégias de controlo: controlo proporcional - derivativo (P-D), regulador linear
quadrático (LQR) e controlo de impedância (IC). Os dois primeiros foram implementados para
seguimento de referência com propósitos de reabilitação, enquanto o último foi implementado para
simular o controlo humano considerando a rigidez e o amortecimento do tornozelo durante o ciclo de
marcha. No controlo de impedância foram definidos diferentes valores de rigidez e amortecimento em
fases distintas do ciclo de marcha.
Um modelo computacional para uma ortótese activa foi desenvolvido com o intuito de assistir o
movimento do tornozelo durante o ciclo de marcha. As características da ortótese activa foram
baseadas nos requisitos biomecânicos para um indivíduo com uma massa corporal de 70kg. Dois
sensores foram introduzidos na base plantar da ortótese para detectar as diferentes fases do ciclo de
marcha, em particular, o contacto do calcanhar e do ante pé, enquanto um potenciómetro foi
implementado na articulação do tornozelo para medir o ângulo do tornozelo relativamente à perna.
Um actuador com elasticidade em série (SEA) proporciona um controlo activo da AAFO. O actuador
foi projectado para ser leve, compacto e com energia suficiente para providenciar o movimento
correcto do tornozelo durante ambos períodos de balanço e contacto com o chão. A ortótese activa
considerada foi projectada com o intuito de ser autónoma e auxiliar o movimento do tornozelo em
todas as fases do ciclo de marcha.
Os resultados obtidos através do modelo computacional foram essenciais para identificar a
arquitectura a adoptar no protótipo físico, assim como identificar criteriosamente quais os
componentes a adquirir e a fabricar, numa verdadeira perspectiva de projecto integrado por
computador. Durante este trabalho, os componentes para a ortótese activa desenvolvida foram
adquiridos e fabricados, procedendo-se no futuro à montagem e teste do sistema em pacientes com
patologias na articulação do tornozelo.
Palavras-chave: Ortótese activa para pé e tornozelo, Actuador com elasticidade em série, Simulação
e controlo do conjunto pé - tornozelo, Controlo de Impedância, Marcha Humana.
ii
iii
Abstract
This dissertation presents the analysis, simulation, and control of the ankle joint during gait, with the
goal of designing an active ankle-foot orthosis (AAFO) to assist individuals without motor control of
this joint. To accomplish this goal, it was firstly necessary to understand the anatomy, physiology, and
biomechanics of the locomotor unit for the correct development of a comprehensive biomechanical
model of the lower leg, foot, and ankle joint, where different control strategies, for the movement of the
ankle joint, could be applied, and the results obtained used for the later design of the actuation system
of the AAFO.
An elastic foot contact model was developed in SimMechanics® with the purpose of simulating the
foot-ground interaction. With movement kinematics prescribed only for the knee joint and leg, and not
for the foot, this model is of the most importance since it allows that the correct movement of ankle
joint, during gait cycle, could be achieved by the controller. The ankle movement was controlled using
three control strategies: proportional-derivative (P-D) control, linear quadratic regulator (LQR) control,
and impedance control (IC). The former two were used for reference tracking with rehabilitation
purpose, while the later was implemented to mimic the human control by considering an ankle
stiffness and damping during gait cycle. With impedance control, different stiffness’s and damping
values were set for the different phases of the gait cycle.
An in silico AAFO was designed for assisting the ankle movement during gait. The AAFO
characteristics were based on the biomechanical requirements of an individual with a body mass of
70kg. Two foot switches sensors were introduced in the plantar basis of the AAFO to allow the
detection of the different phases of gait, in particular heel contact and forefoot contact, while a rotary
potentiometer was included on the articulated ankle joint to measure the ankle angle relative to the
leg. A series elastic actuator (SEA) provides the active control of the AAFO. The SEA was designed to
be lightweight, compact, and with enough power to provide the correct ankle movement during both
stance and swing periods of gait. The developed AAFO is expected to be autonomous and assist the
ankle movement in all phases of the gait cycle.
The results from the computational model where essential to identify the architecture to adopt in the
physical prototype, as to identify carefully which components to acquire and manufacture, having a
true perspective of an integrated computational project. During this work, the components of the
designed AAFO have been acquired and manufactured, preceding the future assembling and testing
on patients with ankle joint pathologies.
Keywords: Active Ankle-Foot Orthosis (AAFO), Series Elastic Actuator (SEA), Ankle-Foot Complex
Simulation and Control, Impedance Control, Human Gait.
iv
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Acknowledgments
My first words of thanks go to my supervisors, Prof. Jorge Martins and Prof. Miguel Silva for
introducing me to this amazing theme and let me work with them. Their enthusiasm in this area is
contagious and helped to face several inherent difficulties. Once again, thank you!
To the people who allow me to study and accomplish my goals, my parents, always thankful!
To my brother and sisters, to my family, thank you for all the support!
To my friends, a sincere thanks!
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Contents
Resumo ......................................................................................................................................... i
Abstract ...................................................................................................................................... iii
Acknowledgments ...................................................................................................................... v
List of Tables ............................................................................................................................ xiii
Notation ..................................................................................................................................... xv
1 Introduction ...................................................................................................................... 1
1.1 Motivation ..................................................................................................................... 2
1.2 Literature Review ......................................................................................................... 2
1.3 Objectives .................................................................................................................... 4
1.4 Structure of this Dissertation ........................................................................................ 4
1.5 Contributions of this Dissertation ................................................................................. 5
2 The Ankle-Foot Complex ................................................................................................. 7
2.1 Basic Concepts of Anatomy and Physiology Related to Gait ...................................... 7
2.2 Anatomy and Physiology of Ankle-Foot Complex ....................................................... 9
2.3 Human Gait ................................................................................................................ 11
2.3.1 Kinematics of Human Gait ................................................................................ 11
2.3.2 Kinetics of Human Gait ..................................................................................... 12
2.3.3 Gait Cycle ......................................................................................................... 13
2.4 Ankle Function and Common Pathologies................................................................. 14
2.5 Solutions for Ankle Pathologies ................................................................................. 16
2.5.1 Ankle-Foot Orthosis (AFO) ............................................................................... 17
2.5.2 Active Ankle-Foot Orthosis (AAFO) .................................................................. 18
2.5.3 Functional Electrical Stimulation (FES) ............................................................ 19
3 Simulation and Control of Ankle-Foot Complex ......................................................... 21
3.1 Control System .......................................................................................................... 21
3.2 Human Gait Simulation .............................................................................................. 23
3.2.1 Gait and Anthropometric Data .......................................................................... 23
3.2.2 Simplified Ankle-Foot Model ............................................................................. 24
3.2.3 Multibody Ankle-Foot Model ............................................................................. 24
3.3 PID Control ................................................................................................................ 29
3.4 Optimal Control .......................................................................................................... 30
3.5 Impedance Control ..................................................................................................... 33
3.6 Gait States ................................................................................................................. 35
viii
3.7 Simulation with Ankle Control – Results and Discussion .......................................... 36
3.7.1 P-D and LQR .................................................................................................... 37
3.7.2 Impedance Control ........................................................................................... 40
4 Active Ankle-Foot Orthosis (AAFO) ............................................................................. 45
4.1 Kinematics and Kinetics Requirements ..................................................................... 45
4.2 AAFO Design ............................................................................................................. 46
4.3 Components of the AAFO .......................................................................................... 48
4.3.1 Ankle-Foot Orthosis (AFO) ............................................................................... 48
4.3.2 Sensors and Actuator ....................................................................................... 49
4.3.3 Input-Output devices ......................................................................................... 51
4.3.4 Power Supply .................................................................................................... 51
4.3.5 Autonomous AAFO ........................................................................................... 53
5 Actuating Unit ................................................................................................................ 55
5.1 Actuators and Coupling Mechanisms ........................................................................ 55
5.2 Series Elastic Actuator (SEA) Design ........................................................................ 57
5.2.1 Components Selection ...................................................................................... 57
5.2.2 Motor and Coupling Mechanism ....................................................................... 57
5.2.3 Structural parts ................................................................................................. 61
5.2.4 Linear Sensor ................................................................................................... 61
5.2.5 SEA Control ...................................................................................................... 62
5.2.6 Main Characteristics of SEA ............................................................................. 63
5.2.7 Maintenance and Durability of the SEA ............................................................ 64
6 Conclusions ................................................................................................................... 65
6.1 Future Work ............................................................................................................... 66
Bibliography .............................................................................................................................. 67
Appendix A ................................................................................................................................ 71
A.1 Kinematics and Kinematics Gait Data ....................................................................... 71
Appendix B ................................................................................................................................ 72
B.1 Results for P-D and LQR Controllers ........................................................................ 72
B.2 Results for Impedance Control .................................................................................. 72
ix
List of Figures
Figure 1.1: Opera Chirurgica: constructional of metal frames to fix and correct all parts of the body
and the joints (adapted from Szendrõi, 2008). ................................................................................. 2
Figure 2.1: The anatomical position, with three reference planes and six fundamental directions with
spatial coordinate system used for all data and analysis (Whittle, 2007 p. 2). ................................ 8
Figure 2.2: (a) Movements about the hip joint (above), knee joint (middle) and ankle joint (below) in
sagittal plane (Whittle, 2007 p. 4), (b) Movements about the hip joint (above), knee joint (middle)
in frontal and transverse planes (Whittle, 2007 p. 4), (c) Abduction and adduction of the foot
(Whittle, 2007 p. 4), (d) Eversion and inversion of the foot (Szendrõi, 2008 p. 24), (e) Pronation
and supination of the foot (Faller, et al., 2004 p. 184). .................................................................... 8
Figure 2.3: Functional division of the body. During walking, the locomotor unit transports the passive
passenger unit. The locomotor unit includes the pelvis and both lower limbs, where several joints
are involved (lumbosacral, hips, knees, ankles, subtalars, and metatarsophalangeal) (Perry, 1992
p. 20). ............................................................................................................................................... 9
Figure 2.4: (a) Bones and joints of the lower limbs (adapted from Whittle, 2007 p. 6), (b) Bones and
joints of the right foot – lateral view (Faller, et al., 2004 p. 183). ..................................................... 9
Figure 2.5: Foot and ankle joints with major functional significance during walking (black areas):
talocrural, subtalar, midtarsal, and metatarsophalangeal (Perry, 1992 p. 69). .............................. 10
Figure 2.6: Muscles of leg: (a) Anterior view, (b) Posterior view (Whittle, 2007 p. 13). ....................... 10
Figure 2.7: (a) Anatomical position with spatial coordinate system used for all data and analysis
(adapted from Whittle, 2007), (b) Marker location and limb and joint angles as defined using an
established convention (Winter, 2004). .......................................................................................... 11
Figure 2.8: (a) Biomechanical convention for moments of force (adapted from Winter, 1991 p. 39), (b)
Schematic of the lower leg during gait - free body diagram of the foot showing the ankle moment,
weight of the foot ( ), and ground reaction force (Rose, et al., 2006 p. 59), (c) Ankle
moment of force per body mass during a gait cycle in normal cadence (Winter, 1991 p. 41). ...... 12
Figure 2.9: Divisions of the gait cycle. Adapted from Perry, 1992 and Rose, et al., 2006. .................. 13
Figure 2.10: Ankle joint angles during a gait cycle in natural cadence (Winter, 1991 p. 29). .............. 15
Figure 2.11: (a) Foot slap due to weak dorsiflexion control, (b) Prolonged heel contact due to
excessive dorsiflexion, (c) excessive knee flexion combined with heel rise (Perry, 1992). ........... 15
Figure 2.12: Gait deviations due to excessive ankle plantar flexion: (a) toe drag during mid swing, (b)
increased hip and knee flexion to avoid toe drag during mid swing, (c) terminal swing gait
deviation, with the foot nearly parallel to the ground (Perry, 1992). .............................................. 16
Figure 2.13: Ankle-foot orthosis: (a) Conventional AFO, comprised of a metal frame with leather
straps, attached to a shoe (Orthomedics), (b) Standard Plastic AFO (Orthomedics), (c) Prolite
Carbon AFO made from injection molded carbon-reinforced polypropylene with increased rigidity
in vertical aspect (Össur, 2010), (d) Plastic Articulated AFO (Orthomedics). ................................ 17
Figure 2.14: (a) Active ankle-foot orthosis developed at Massachusetts Institute of Technology
(MIT)(Dollar, et al., 2007), (b) Powered Ankle Exoskeleton developed at The University of
Michigan (Kao, 2009), (c) Adjustable robotic tendon robot concept (Hollander, et al., 2005). ...... 18
Figure 2.15: (a) FES electrodes, (b) FES with orthosis (Phillips, 1986). .............................................. 19
Figure 3.1: Diagram of a simplified control scheme for controlling the ankle movement with the AAFO.
The control action of the AAFO is executed by an actuator (series elastic actuator), which causes
a torque about the ankle joint with the goal of assisting the ankle movement during gait. ............ 22
Figure 3.2: Main control scheme with master and slave controllers. ................................................... 23
Figure 3.3: Scheme for the multibody ankle-foot model with the representation of kinematics
prescribed at the knee joint. ........................................................................................................... 25
x
Figure 3.4: (a) Diagrammatic representation of a viscoelastic foot model (Gilchrist, et al., 1997), (b);
Contact surfaces (hiper-ellipsoid) between the foot and the ground (Barbosa, et al., 2005), (c)
Foot contact model consisting of a set of spheres (Moreira, 2009). .............................................. 25
Figure 3.5: Elastic foot contact model for the multibody ankle-foot model. .......................................... 26
Figure 3.6: Multibody biomechanical model used in the simulation (SimMechanics®). ....................... 27
Figure 3.7: Events triggers during gait cycle – Heel and Forefoot sensors activation. ........................ 28
Figure 3.8: Dynamics of gait data and model during gait cycle: (a) ankle torque, (b) horizontal shear
force, (c) vertical force. The stiffness for each virtual spring was: Horizontal spring (heel) = 5 x 104
N/m, Horizontal spring (forefoot) = 9 x 103 N/m, Vertical spring (heel) = 4.5 x 10
5 N/m, Vertical
spring (forefoot) = 1.7 x 105 N/m. ................................................................................................... 28
Figure 3.9: Block diagram of a basic PID controller. ............................................................................ 29
Figure 3.10: Block diagram of the P-D controller in closed-loop with simplified ankle-foot model. ...... 30
Figure 3.11: Block diagram of the implemented LQR. ......................................................................... 32
Figure 3.12: The desired effect of impedance control represented by the use of a rotational mass-
spring-damper system. ................................................................................................................... 33
Figure 3.13: Block diagram of implemented impedance control. ......................................................... 34
Figure 3.14: Events that cause the transition of states. Sensors enabled in red and disabled in black.
The ankle angle possibilities are represented in Figure 3.12. ....................................................... 36
Figure 3.15: Four states during a complete gait cycle. ........................................................................ 36
Figure 3.16: Ankle angle of simulated and controlled multibody ankle-foot model during gait cycle. .. 38
Figure 3.17: Ankle angle tracking error during gait cycle of controlled multibody ankle-foot mode. .... 38
Figure 3.18: Ankle torque during gait cycle of controlled multibody ankle-foot model with P-D and LQR
control strategies. ........................................................................................................................... 39
Figure 3.19: Response to a unit step input of closed-loop system for the P-D controller. ................... 39
Figure 3.20: Response to a unit step input of closed-loop system for the LQR controller. .................. 40
Figure 3.21: Ankle angle during gait cycle from gait data and controlled multibody ankle-foot model
with impedance control. Parameters of controller are presented in Table 3.3. ............................. 42
Figure 3.22: Ankle angle error between the ankle angle from gait data and from controlled model with
impedance control 2 during gait cycle. ........................................................................................... 42
Figure 3.23: Events triggers during gait cycle for the simulated model and controlled model – Heel
and Forefoot sensors activation. .................................................................................................... 43
Figure 3.24: Dynamics of simulated model and controlled model with impedance control 2 during gait
cycle: (a) ankle torque, (b) horizontal shear force, (c) vertical force. ............................................. 43
Figure 3.25: Responses to a unit step input of a closed-loop system with simplified ankle-foot model
and impedance control. .................................................................................................................. 44
Figure 4.1: Representation of different concepts for converting linear to rotational movement of the
AFO: (a) detail about scissor system with one level, (b) scissor system with one level and a
parallel spring, (c) scissor system with two levels, (d) back of the AFO used as a level arm. ....... 46
Figure 4.2: Representation of the orthosis with the respective force arm. ........................................... 46
Figure 4.3: Active ankle-foot orthosis 3D model: (a) left view, (b) right view, (c) basic ankle movement
descriptions, dorsiflexed, neutral, and plantar flexed. .................................................................... 48
Figure 4.4: Standard polypropylene AFO with aluminum joints. .......................................................... 49
Figure 4.5: Bourns 6639S-1-502 5 kΩ rotary potentiometer (RS Components Ltd, 2010, article no.
164-2661). ...................................................................................................................................... 49
Figure 4.6: Flexible event switch sensor (Motion Lab Systems, 2009). ............................................... 50
Figure 4.7: Foot pressure versus time, where the hot colors represent high pressure areas (Tekscan,
2009). ............................................................................................................................................. 50
Figure 4.8: Series elastic actuator designed in this work. .................................................................... 51
xi
Figure 4.9: National Instruments USB-6009, 14-Bit, 48kS/s Low-Cost Multifunction DAQ (National
Instruments, 2010). ........................................................................................................................ 51
Figure 4.10: Kokam lithium polymer battery pack (model: H5 3600mAh, 18.5V, 30C). ...................... 52
Figure 4.11: Autonomous AAFO with the individual carrying the batteries and processing unit in a
backpack. ....................................................................................................................................... 53
Figure 5.1: A typical actuating unit (Bishop, 2008). .............................................................................. 55
Figure 5.2: (a) BS23 High Performance rotary motor (Moog, 2010), (b) Linear ET032 actuator (Parker,
2009), (c) Series Elastic Actuator (Robinson, 2000). ..................................................................... 56
Figure 5.3: (a) Linear actuator with rotary motor coupled to a screw shaft by a gear drive (Parker,
2009), (b) SKF lead screw (SKF), (c) Linear actuator with rotary motor coupled to a screw shaft by
a belt drive (Parker, 2009). ............................................................................................................. 56
Figure 5.4: Series Elastic Actuator three dimensional model. .............................................................. 57
Figure 5.5: (a) BS23 High Performance rotary motor (Moog, 2010), BN23-23IP-03 rotary motor
(Moog, 2010), (c) maxon EC–4pole 30-200W (maxon motor ag, 2010, article no. 305015). ........ 58
Figure 5.6: AFO endowed with the three considered rotary motors. (a) SEA with BN23-23IP-03 rotary
motor, (b) SEA with BS23 High Performance rotary motor, (c) SEA with maxon EC–4pole 30-
200W. ............................................................................................................................................. 58
Figure 5.7: SKF nut and ball screw shaft (model: BD 10x4 R). ............................................................ 60
Figure 5.8: Maxon motor control’s EPOS2 50/5 (maxon motor ag, 2010, article no. 347717). ........... 60
Figure 5.9: Set of springs with different stiffness’s. .............................................................................. 61
Figure 5.10: Linear potentiometer (RS Components Ltd, 2010). ......................................................... 61
Figure 5.11: Electro-magnetic series elastic actuator model. The lumped mass has a driving force and
a viscous friction. The controller drives the lumped mass to compress the spring which gives the
desired force output. Adapted from Robinson, 2000 p. 92. ........................................................... 62
Figure 5.12: Unit step response of the closed-loop transfer function of SEA. ..................................... 64
Figure A.1: Kinematics and kinetics of human gait during gait cycle, for three different cadences: (a)
joint ankle angle, (b) ankle angular velocity, (c) ankle angular acceleration, (d) ankle moment of
force (torque) per unit of body mass, (e) ankle power per unit of body mass (Winter, 1991). ....... 71
Figure B.1: Ankle angle tracking during gait cycle of controlled simplified ankle-foot model. ............. 72
Figure B.2: Ankle angle tracking error during gait cycle of controlled simplified ankle-foot model. ..... 72
Figure B.3: (a) Ankle angle during gait cycle (Winter, 2004), (b) Ankle angle during swing period (Gait
Data) and second-order model for approximation of gait data (step response with negative offset).
........................................................................................................................................................ 73
Figure B.4: Dynamics from simulated model and controlled model with impedance control 1 during
gait cycle: (a) ankle torque, (b) horizontal shear force, (c) vertical force. ...................................... 74
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List of Tables
Table 3.1: Parameters of P-D controller for reference tracking of simplified and multibody ankle-foot
models. ........................................................................................................................................... 37
Table 3.2: Parameters of LQR controller for reference tracking of simplified and multibody ankle-foot
models. ........................................................................................................................................... 37
Table 3.3: Parameters and characteristics of implemented impedance control of multibody ankle-foot
model during GC. ........................................................................................................................... 41
Table 4.1: Ankle maximum negative and positive values for different quantities of gait data, in normal
cadence, of an individual with a total mass of 70kg (Winter, 1991). .............................................. 46
Table 4.2: Linear quantities required for an actuator to assist the ankle movement, based on data from
Table 4.1. ....................................................................................................................................... 47
Table 4.3: Technical description of Kokam lithium polymer batteries packs (model: H5 3600mAh,
18.5V, 30C). ................................................................................................................................... 52
Table 5.1: Brushless DC motors specifications with best overall characteristics. ................................ 58
Table 5.2: Nut and Ball screws specifications provided by SKF® Ball screws catalogue (SKF). Peak
torque and peak angular velocity are obtain according to the requirements presented in Table 4.2.
........................................................................................................................................................ 59
Table 5.3: Properties of the modeled actuator used in the simulation control. Some of the values are
calculated from motor, and screw literature. .................................................................................. 63
xiv
xv
Notation
Acronyms
AAFO Ankle Foot Orthosis
AAFOs Ankle Foot Orthoses
AC Alternating Current
AFO Ankle Foot Orthosis
AFOs Ankle Foot Orthoses
CAD Computer Aided Design
CAN Control Area Network
CG Center of Gravity
COM Center Of Mass
COP Center Of Pressure
DACHOR Multibody Dynamic and Control of Hybrid Active Orthoses
DC Direct Current
DOF Degree Of Freedom
FES Functional Electrical Stimulation
GC Gait Cycle
GRF Ground Reaction Force
HAT Head-Arms-Trunk
IC Impedance Controller
LiPo Lithium Polymer
LQR Linear Quadratic Regulator
LQR Linear Quadratic Regulator
PID Proportional Integrative Derivative
RS232 Recommended Standard 232
SEA Series Elastic Actuator
USB Universal Serial Bus
DAQ Data Acquisition
xvi
Nomenclature
Typeface
italic – scalar variables
bold – vector or matrix variables
Variables Description
Symbol Unit Definition
rad/s2 Angular acceleration
rad/s2 Ankle angular acceleration
m Foot contact sensor penetration
None Damping ratio
None SEA damping ratio
rad Reference target ankle angle
rad/s Reference target ankle angular velocity
rad Ankle angle
rad/s Ankle angular velocity
rad/s2 Ankle angular acceleration
rad Reference ankle angle
m/s Linear velocity
m/s Nut linear velocity
N.m Torque
N.m Ankle moment of force / torque
N.m Motor torque
N.m Screw shaft torque
rad/s Angular velocity
rad/s SEA operational frequency
rad/s Ankle angular velocity
rad/s SEA natural frequency
rad/s Natural frequency
rad/s Screw shaft angular velocity
m/s2 Linear acceleration
none Dynamic matrix
Nms/rad Rotational damping
xvii
none Input matrix
N.s/m SEA damping
none Output matrix
J Energy
N Actuator force
N Desired force
N Load force
N Screw shaft nut force
N External ground reaction force applied on the foot
none Ankle-foot transfer function
none Controller transfer function
none Closed-loop transfer function
rad/m Gear reduction between motor and nut
none Gear reduction between motor and screw shaft
rad/m Gear reduction between screw shaft and nut
m Height
A Electrical current
kg.m2 Rotational inertia
kg.m2 Ankle rotational inertia due to foot mass
kg.m2 Rotor rotational inertia
kg.m2 Screw shaft rotational inertia
none Index performance
Nm/rad Rotational stiffness
none Derivative gain of PID controller
none Integrative gain of PID controller
none Proportional gain of PID controller
N/m Total spring stiffness
m Lever arm between ankle joint and SEA force application
m Screw shaft lead
kg Mass
N.m Ankle moment of force
kg Lumped mass (total dynamic mass of rotor and screw shaft)
kg Dynamic mass of motor rotor
kg Dynamic mass of screw shaft
none Maximum overshoot
xviii
none Equation system order
W Power
W Ankle power
W Actuator power
none Positive-definite Hermitian matrix
none Positive-definite Hermitian matrix
M Position vector of COM relative to ankle joint center
M Position vector of COP relative to ankle joint center
ºC Celsius degrees
S Peak time
V Electrical potential difference (Voltage)
none Control input vector
N Weight of foot
none State vector
m Spring deflection
m Actuator linear displacement
m/s Actuator linear velocity
1
Chapter 1
1 Introduction
“We see others walking, see what they can do with their walking, what
we cannot do, and we want to do it too”, (Winter, 1991).
Standing or walking is something people take it for granted, however, every year thousands of people
are prevented from doing it. In a physical and psychological perspective, gait disabilities have a
negative impact on people’s life, compromising their ability to work, engage in social and leisure
activities, or in the worst cases, participate in activities associated with an independent lifestyle (Rose,
et al., 2006 p. 209). Thus, it is expectable that affected people will tend to become sedentary, fact that
will further affect their health condition. Gait disabilities are a serious problem that affects millions of
people around the world, and although the existence of many walking aids that directly benefit the
affected person, it is clear that with current technology greater benefits are possible to achieve in
some pathologies. The great majority of current assistive methods rely on passive orthotic systems,
which can provide enough support when limbs have some functional pathology, but fail to assist when
external power output is required (Rose, et al., 2006).
Facing various gait disabilities, several approaches of active systems, such as assist people actively
with robotic solutions, have been made with the purpose of improve patient’s quality of life. It is in this
context that the concept of wearable robots has emerged, where the robotic counterparts of current
orthoses are referred to as robotic exoskeletons (Pons, 2008). The function of the exoskeleton has a
wide range of applicability, since it can be applied not only to restore and rehabilitate handicapped
functions of the body, but also on the improvement and enhancement of normal human body
performance. Apply exoskeletons to restore the full walking capacity of a spinal cord injury (SCI)
patient is one of the most ambitious objectives that researchers try to achieve. The estimates indicate
that in the United States only, 250,000-400,000 individuals live with SCI (NSCIA, 2010). The direct
costs alone are estimated at $10 billion, neglecting the indirect income potential. Even more difficult to
measure is the psychological impact on patients and their families (Cooper, 2006 p. 59). Restore the
movement of the locomotor unit is a complex task, but the approach to such goal can be simplified by
dividing the locomotor unit in several sub-systems and studying and developing assistance methods
separately for each sub-system, i.e., developing a system to each locomotor unit joint separately,
starting from the ankle and moving upwards to the knee and hip.
As a part of the locomotor unit, the ankle and foot are the final segments that provide support to the
body by distributing gravitational and inertial loads. Thus, this dissertation investigates in particular the
disabilities of the ankle joint, considering as design requirement the worst case scenario the active
support to an individual that has very low or no motor control of the ankle joint. Assist this pathology,
which can occur due to weakness in dorsiflexor and plantar flexor muscles, allows the recover of full
functioning of the ankle joint and faces the challenge of interrelating biomechanical principles with
engineering concepts.
2
1.1 Motivation
Throughout ages Humans have developed means of transportation that allow them to travel safely
through land, water, air and in space, enabling people to move from one point to another with celerity
and commodity. However, Humans have not been able to develop systems capable of restoring basic
human movements like autonomous walking with commodity and safety. This is the major motivation
for this dissertation.
This task can be achieved by designing an active system that assists the patient during walking. This
system, which can be characterized by an exoskeleton or wearable robot, would be fitted on the
individual anatomy, detecting the individual intentions and restoring actively the movement.
Considering the ankle joint, the main objective is to develop a system that assists all ankle pathologies
due to functional errors in the neuromuscular system. With this, a part of the locomotor unit would
have autonomy to later, integrated with other assisting devices on lower limb joints, provide the full
movement of the locomotor unit.
Overall, people’s health is the major motivation!
1.2 Literature Review
People are subjected everyday to many situations that can lead to functional disabilities in
musculoskeletal and neuromuscular systems. Correcting this problematic requires the development of
prosthetic an orthotic devices. In the Middle Age, similarly to surgery, the treatment of the
musculoskeletal diseases was the task of healers and blacksmiths (Szendrõi, 2008), proposing
devices like the presented in Figure 1.1. Nowadays, the treatment of these disabilities involves several
areas so that improvements can be achieved in assisting several different pathologies.
Figure 1.1: Opera Chirurgica: constructional of metal frames to fix and correct all parts of the body and the joints (adapted from Szendrõi, 2008).
As a part of the locomotor unit, the ankle-foot complex provides the support and propels the body
forward during gait (Perry, 1992). In the presence of functional disabilities on the neuromuscular
system that affect the ankle-foot complex on the overall movements of the body, the gait becomes
pathological (Whittle, 2007). Several pathologies can affect the ankle, but these can all be classified
as excessive dorsiflexion or plantar flexion (Perry, 1992).
Several techniques on assisting ankle pathologies have been developed for the different pathologies
(Rose, et al., 2006), however, ankle-foot orthoses (AFOs) are the most used systems. AFO is usually
an orthosis that covers the foot, spans the ankle joint and covers the lower leg. This passive lower
limb orthotic device can be divided into categories of metal, plastic and more recently carbon fiber
3
(Cooper, 2006). In the case of articulated AFOs, mechanical joints are used to interconnect the upper
and lower elements. These joints are usually set to allow as much motion as possible, while blocking
unwanted movements (Össur, 2010). AFOs have been shown to improve walking (Harris, et al.,
2008), presenting different characteristics depending on the pathology (Jamshidi, et al., 2009), but are
limited in these action due to the absence of an active behavior. However, AFOs are in many cases
the cheapest solution to solve ankle pathologies. With the necessity of assisting ankle pathologies
more actively, the AFOs have been endowed of active systems, generally denominated of active
ankle-foot orthoses (AAFOs). Several active ankle-foot orthosis have been developed in the last
decade for rehabilitation and gait assist purposes. A prototype to assist drop foot (dorsiflexor muscles
weakness) was developed at the MIT Biomechatronics Lab (Blaya, 2003). The prototype consists of a
modified AFO, with pressure sensors on the plantar surface, rotary sensor at the ankle joint, and a
series elastic actuator (SEA) on its back. The SEA allowed a variation in the impedance for
dorsiflexion and plantar flexion, recurring to adaptive control to change the ankle stiffness during
walking. It was shown with this prototype that foot slap could be reduce during gait, however, the
system portability and weight needed improvement. The SEA used in the prototype has the advantage
of having the motor isolated from shock loads, and the effects of backlash, torque ripple, and friction
are filtered by the spring (Robinson, 2000). With rehabilitation purposes, several lightweight AFOs
endowed of artificial pneumatic muscles were developed by the University of Michigan (Kao, 2009),
allowing dorsiflexion and plantar flexion movement control. In the former prototypes, the pneumatic
muscles activation was made by user selection through commands (Kao, 2009), but later a
myoelectric control was implemented (Ferris, et al., 2006), i.e., by electromyography reading of
muscles reactions, the artificial muscles were activated. A promising system with an adjustable robotic
tendon was proposed by Hollander, et al., 2005, where a lightweight system with reduced peak power
of the motor was the major highlight. The system uses as an actuator basis a Jack Spring™, which
combines compliance and energy storage for its actuation tasks. Another technique in development is
the functional electric stimulation of affected muscles (FES), which can generate useful movements of
the extremities through electrical activation of paralyzed or spastic muscles (Rose, et al., 2006 p. 209).
In the framework of FCT DACHOR project which this work also develops, the main goal is to develop
a hybrid solution where an AAFO and a FES system are integrated. Since the integration of FES with
orthotic devices has proven to be a good solution (Phillips, 1986), a control architecture for the
musculoskeletal system of the ankle joint was developed by Malcata, (Malcata, 2009), also in the
framework of the FCT DACHOR project.
When developing systems to assist human movements (wearable robots), it is necessary to
understand the biomechanics involved in those movements. The ankle function during gait (Perry,
1992) and its biomechanics are the basis for the development of the present AAFO, allowing
simulating the ankle movement and creating biomechanical models. Biomechanical models are
fundamental when developing control strategies for assisting human limbs, and thus, several dynamic
models with different inputs and outputs for the simulation of human movement have been developed.
Some simulations take in account the entire locomotor system (Jamshidi, et al., 2009), while others
focus on the particular system in study. Several approaches for modeling the foot have been proposed
in order to achieve a model of the ankle-foot system. A currently accepted approach to quantify foot
and ankle kinematics during gait is to represent the entire foot as a single rigid body with a revolute
ankle joint, generally for sagittal plane studies (Harris, et al., 2008). Focusing the foot contact with the
ground, Girlchrist and Winter (Gilchrist, et al., 1997) developed a foot model for the stance period of
gait, where the foot ground contact is described by means of viscoelastic elements. Barbosa, et al.
(Barbosa, et al., 2005) represented the contact between the ground and the foot by approaching the
contact surfaces to hiper-ellipsoids with certain material and geometric characteristics, restrained to
the most important areas on the foot during gait. Inserted in the FCT DACHOR project, Moreira
(Moreira, 2009) developed a foot contact model consisting of a set of spheres located under the
plantar surface to model and check the contact foot-ground.
4
Controlling the AAFO is the key concept to achieve an effective system capable of mimic the natural
ankle behavior. When developing systems that interact with humans, the control strategy generally
relies on impedance control (Ikeura, et al., 1995). Studies have shown that ankle stiffness varies with
gait speed (Palmer, 2002), and thus the system needs to adapt to the different gait speeds, requiring
the adaptive inclusion of the feature. In the prototype presented by the MIT Biomechatronics Lab
(Blaya, 2003), the ankle stiffness was adapted during gait according to the number of foot slaps and
gait speed. A different system was presented by Veneva (Veneva, 2009), where using a position
controller, the actuator joint torque was automatically modulated in order to optimize the heel-to-
forefoot transition during the stance or the swing phase of walking.
1.3 Objectives
The objective of this work is to develop a lightweight and autonomous computational system capable
of providing ankle movement during gait cycle.
The system will be able to assist neuromuscular ankle pathology, referred as ankle palsy, where the
individual has very low or no control of the ankle joint due to weakness in dorsiflexor and plantar flexor
muscles. Other neuromuscular disabilities will be covered with this extreme pathology.
A passive ankle-foot orthosis will be the basis of the system, which should be endowed of sensing
devices that identify the different phases of the gait, and of an actuator with enough power to provide
the ankle movement during the gait cycle. Several control strategies will be developed to the system in
order to control the angular position of the ankle joint during the gait cycle.
This work is inserted in the FCT DACHOR project – Multibody Dynamics and Control of Hybrid
Orthoses (MIT-Pt/BS-HHMS/0042/2008). The main goal of this project is to develop a hybrid ankle-
foot orthosis (AFO) to assist patients with gait disorders. The hybrid system consists of the AAFO and
a functional electrical system (FES) to use the residual muscles strength.
1.4 Structure of this Dissertation
This dissertation is structured so the reader follows the steps taken in the design and development of
an active ankle-foot orthosis.
Chapter 2: biomechanical aspects involved in the ankle joint movement, addressing
pathologies and current solutions.
Chapter 3: modeling, control, and simulation of the ankle-foot system with different control
strategies during the gait cycle.
Chapter 4: steps taken in the design of an active ankle-foot orthosis (AAFO), fulfilling the
biomechanical requirements.
Chapter 5: steps taken in the design of a series elastic actuator (SEA) to be used in the
AAFO.
Chapter 6: discussion of results and conclusion exposition of this work with some suggestions
for future work.
5
1.5 Contributions of this Dissertation
The main contributions of this dissertation are the following:
development of multibody foot contact model in SimMechanics that allows the implementation
of different control strategies to control the ankle joint during the gait cycle;
development of a control architecture that identifies the different phases of gait cycle;
implementation of two control strategies of reference following to control the ankle joint during
the gait cycle (proportional-integral-derivative and linear quadratic regulator controllers);
implementation of a control strategy that mimics the movement of the ankle joint during the
gait cycle by varying the ankle stiffness and damping (impedance control);
design of an active ankle-foot orthosis that assists ankle joint movement during all phases of
gait cycle with power autonomy;
improvement of existing series elastic actuators to a compact, lighter, and powerful unit;
In the overall, this dissertation presents the design, step by step, of an active ankle-foot orthosis
prototype that will be built in a nearby future.
6
7
Chapter 2
2 The Ankle-Foot Complex
Biomechanics of human movement can be defined as the interdiscipline which describes, analyzes
and assesses human movement (Winter, 2004 p. 1). Understanding human movement is essential
when developing systems capable of assisting the human body, and such requires a close study of its
anatomy and physiology. The study of this theme provides essential labels for musculoskeletal
structures, joint motions, and function of the different structures (Knudson, 2007 p. 40).
Several key concepts are related with human gait, which is the most common of all human
movements (Winter, 1991 p. 1). This chapter gives a general overview of the biomechanics involved in
human gait, starting in section 2.1 with basic concepts of human anatomy and physiology, and
focusing this theme for the ankle-foot complex in section 2.2. Human gait is characterized in section
2.3, where its kinematics and kinetics are firstly presented. These important variables are fundamental
in the description of the gait cycle. To assist pathological gait, it is first necessary to understand its
normal functioning. For that reason, section 2.4 presents the ankle function during gait and its most
common pathologies that the devices presented in section 2.5 try to assist.
2.1 Basic Concepts of Anatomy and Physiology Related to
Gait
Knowledge of biomechanics must be combined with anatomy to accurately determine the
musculoskeletal causes or how human movement is created. Anatomy familiarity also provides a
common “language” of human body when communicating with kinesiology and medical professionals
(Knudson, 2007 p. 41).
For a correct understanding and later description of human gait, it is necessary to define a reference
anatomical position with respect to which reference planes, axis, and other major body movements.
The anatomical terms describing the relationships between different parts of the body are based on
this anatomical position, in which a person is standing upright, with the feet together and the arms by
the side of the body, with the hand palms facing forward. This position, together with the three
reference planes (sagittal, frontal and transverse), axis, and six fundamental directions is illustrated in
Figure 2.1.
8
Figure 2.1: The anatomical position, with three reference planes and six fundamental directions with spatial coordinate system used for all data and analysis (Whittle, 2007 p. 2).
Specific terminology is also used to describe the major rotations of bones at the joints. Most joints can
only move in one or two of the three considered planes. Focusing the lower limb system, several
movements are possible. Movements in the sagittal plane are denominated by flexion and extension,
with the exception for the ankle, where these are denominated as dorsiflexion and plantar flexion,
respectively (Figure 2.2a). Movements about frontal plane are called abduction and adduction, while
internal and external rotation take place in the transverse plane (Figure 2.2b,c) (Whittle, 2007 p. 3).
Other specific movements on the foot are eversion and inversion (Figure 2.2d), and pronation and
supination (Figure 2.2e).
Figure 2.2: (a) Movements about the hip joint (above), knee joint (middle) and ankle joint (below) in sagittal plane (Whittle, 2007 p. 4), (b) Movements about the hip joint (above), knee joint (middle) in frontal and transverse
planes (Whittle, 2007 p. 4), (c) Abduction and adduction of the foot (Whittle, 2007 p. 4), (d) Eversion and inversion of the foot (Szendrõi, 2008 p. 24), (e) Pronation and supination of the foot (Faller, et al., 2004 p. 184).
During gait, human body can be approximated by two functional units: passenger and locomotor
(Figure 2.3). In this approximation, the passenger unit is responsible for its own postural integrity,
while the locomotor unit carries the body to the desired position. Focusing the locomotor unit during
gait, several articulations (joints) are involved in the motion: lumbosacral, bilateral hip, knee, ankle,
subtalar, and metatarsophalangeal (Perry, 1992 p. 50).
9
Figure 2.3: Functional division of the body. During walking, the locomotor unit transports the passive passenger unit. The locomotor unit includes the pelvis and both lower limbs, where several joints are involved (lumbosacral,
hips, knees, ankles, subtalars, and metatarsophalangeal) (Perry, 1992 p. 20).
2.2 Anatomy and Physiology of Ankle-Foot Complex
Leg and foot are part of the locomotor unit, more properly, the free lower limb. The free lower limb is
connected to the pelvic girdle by the hip joint and consists of the thigh bone (femur), the leg (crus)
including tibia and fibula, and the foot (pes), which includes the ankle (tarsus), metatarsals, and toes
(digits) (Figure 2.4a).
The foot and ankle joint is a complex structure, formed by several joints with different characteristics,
that are involved in the motion occurring between the foot and the lower leg. This structure supports
and propels the body forward and absorbs the forces of a step, while providing rotation for adaptations
on uneven terrains. Foot can also be classified in three elements: forefoot (five metatarsal bones with
the phalanges), midfoot (navicular, cuboid and three cuneiform), and hindfoot (calcaneus and talus
bones).
Figure 2.4: (a) Bones and joints of the lower limbs (adapted from Whittle, 2007 p. 6), (b) Bones and joints of the right foot – lateral view (Faller, et al., 2004 p. 183).
Four major joints compose the foot and ankle joint complex (Figure 2.5): talocrural (true ankle joint),
talocalcaneal (subtalar), transverse tarsal (midtarsal), and metatarsophalangeal (Harris, et al., 2008 p.
23). The ankle joint is formed by the tibia, fibula and talus, and the movements at this joint are called
dorsiflexion and plantar flexion. In the intertarsal joint, the talus articulates with the calcaneus and the
10
navicular bone. The In the subtalar joint, the talus articulates with the calcaneus. In the midtarsal joint,
the ball-shaped head of the talus articulates with the calcaneus and the navicular bone. The
movements in midtarsal joint are lateral movements, called supination and pronation (Faller, et al.,
2004 p. 182).
Figure 2.5: Foot and ankle joints with major functional significance during walking (black areas): talocrural, subtalar, midtarsal, and metatarsophalangeal (Perry, 1992 p. 69).
Aiming for a simpler computational model of the locomotor unit, it was considered an accepted
approach of quantifying foot and ankle kinematics during gait as the representation of the entire foot
as a single rigid body, with a revolute ankle joint in the sagittal plane (Harris, et al., 2008 p. 384). For
the same reason, the lumbosacral articulation was considered to be fixed. This way, only movements
taken place in sagittal plane are considered, disregarding the movements in the others planes (Figure
2.2a).
Muscle forces are the main internal motors and brakes for human movement. The torques created by
skeletal muscles are coordinated with torques from external forces, providing the human motion of
interest (Knudson, 2007 p. 49).
According to the theme of this dissertation, the muscles of major interest are the ones which provide
movement through the ankle joint (Figure 2.6). As the ankle joint moves in the sagittal plane, all the
controlling muscles function are classified either as dorsiflexors or plantar flexors (Perry, 1992 p. 55).
Figure 2.6: Muscles of leg: (a) Anterior view, (b) Posterior view (Whittle, 2007 p. 13).
The anterior tibial group is responsible for the dorsiflexion and supination of the foot, being composed
of four muscles: extensor hallucis longus, extensor digitorum longus, peroneus tertius, and tibialis
anterior. The two former muscles are inserted into the toes, which they extend. Peroneus tertius
11
muscle is inserted into the tarsal bones, raising the foot on the lateral side. Tibialis anterior muscle is
inserted into the tarsal bones, being the main dorsiflexor muscle and supinator of the foot, while the
others muscles are weak dorsiflexors.
Foot plantar flexion is generally acted by a group of seven to eight muscles: soleus, gastrocnemius,
tibialis posterior, flexor hallucis longus, flexor digitorum longus, peroneus longus, peroneus brevis, and
sometimes the plantaris.
The strongest muscle group in plantar flexion, triceps surae, consists on superficial muscles from the
calf, usually composed of two or three muscles, the soleus, the gastrocnemius, and sometimes the
plantaris. They join to form the Achilles tendon, which is inserted into the calcaneal tubercle (Faller, et
al., 2004 p. 182). Five other muscles act as weak plantar flexors.The muscles on the lateral
compartment of the calf, peronei longus and brevis, are primarily pronators. Flexor hallucis longus,
flexor digitorum longus, and tibialis posterior are posterior calf muscles, assisting primarily as
supinators (Whittle, 2007 p. 14).
2.3 Human Gait
2.3.1 Kinematics of Human Gait
Kinematics is the term used in the description of human movement, disregarding the forces, either
internal or external, that cause the movement. The evaluated variables are usually linear and angular
displacements, velocities, and accelerations (Winter, 2004 p. 9).
A complete description of spatial coordinate system and its conventions is fundamental when
discussing kinematic variables (Figure 2.7a). Limb angles in the spatial reference system are defined
using counterclockwise from the horizontal as positive (Figure 2.7b). Thus angular velocities and
accelerations are also positive in a counterclockwise direction in the plane of movement, which is
essential for consistent use in subsequent kinetic analyses. Convention for joint angles (which are
relative) is subject to wide variation among researchers, having the necessity to clarify it (Winter, 2004
p. 54). Focusing the movements through the ankle joint, dorsiflexion causes positive ankle joint
angles, while plantar flexion causes negative ankle joint angles (Figure 2.10).
Figure 2.7: (a) Anatomical position with spatial coordinate system used for all data and analysis (adapted from Whittle, 2007), (b) Marker location and limb and joint angles as defined using an established convention (Winter,
2004).
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2.3.2 Kinetics of Human Gait
Kinetics in human gait represents the forces and torques that cause the motion of the body (Harris, et
al., 2008 p. 133), where both internal and external forces are included. Internal forces come from
muscle activity, ligaments or friction in the muscles and forces, while external forces come from the
ground or from external loads (Winter, 2004 p. 10).
The future of biomechanics lies in kinetic analyses, since the information acquired allows making
definitive assessments and interpretations (Winter, 2004 p. 10), which are important for the
development of systems and methods for assisting pathological gaits.
Comparatively to kinematics, kinetics is more difficult to evaluate, since there are many combinations
of muscle forces that can result in the same movement pattern (Winter, 1991 p. 35), and the
measurements cannot be directly observed (Rose, et al., 2006 p. 53). The calculation of different
variables such as intersegmental moments, work, mechanical energy, and power, implies the use of
Newton’s laws (Tipler, et al., 2004 p. 85) and the law of conservation of energy (Tipler, et al., 2004 p.
183), in order to interpret what is happening at each phase of the gait (Winter, 1991 p. 35).
In this work, the main focus about kinetic data regards to joint moments. The standard convention for
moments of force in the plane of progression is shown in Figure 2.8a. Counterclockwise moments are
positive, calculated at the proximal end of each segment, while clockwise moments are negative.
Thus, all moments of force in this work are reported in this basis.
Figure 2.8: (a) Biomechanical convention for moments of force (adapted from Winter, 1991 p. 39), (b) Schematic
of the lower leg during gait - free body diagram of the foot showing the ankle moment, weight of the foot ( ),
and ground reaction force (Rose, et al., 2006 p. 59), (c) Ankle moment of force per body mass during a gait cycle in normal cadence (Winter, 1991 p. 41).
Focusing the ankle moment of force, dorsiflexion causes negative moments at the foot, while plantar
flexion causes positive moments. In normal gait, the joint angles do not reach their extreme limits,
resulting that the net moment is the result of muscle forces only. Focusing the kinetics through the
ankle joint, the free body diagram presented in Figure 2.8b exhibits the considered forces applied to
the foot during the gait cycle. Thus, the moment of force acting through ankle joint, , becomes
(2.1)
where is the ankle rotational inertia due to the mass of the foot, the ankle angular acceleration,
and the position vectors of the center of mass (COM) and center of pressure (COP) relative
to ankle joint center, respectively. Generally, the ground reaction forces and the center of pressure,
which varies in time, are measured using a force transducer.
13
2.3.3 Gait Cycle
Walking uses a repetitious sequence of limb motions to move the body forward while simultaneously
maintaining stance stability. As the body moves forward, one limb serves as a mobile source of
support while the other limb advances itself to a new support site. Then the limbs reverse their roles.
For the transfer of body weight from one limb to the other, both feet are in contact with the ground.
This series of events is repeated by each limb with reciprocal timing until the person’s destination is
reached. A single sequence of these functions by one limb is called a gait cycle (GC) (Perry, 1992 pp.
3-4).
In gait analysis, the basic unit is the interval between identical positions, i.e. a gait cycle. A step is
taken to mean the period of motion of the limb from one initial contact till the next initial contact of the
same foot. The gait cycle is characterized by timing and length (Szendrõi, 2008 p. 28). Unfortunately,
the nomenclature used to describe the gait cycle varies considerably from one publication to another.
The present work attempts to use terms which will be understood by most people working in the field
(Figure 2.9).
Figure 2.9: Divisions of the gait cycle. Adapted from Perry, 1992 and Rose, et al., 2006.
The gait cycle can be divided into two periods. First, the stance period, occurs when the foot is in
contact with the ground, and the second, the swing period, occurs when the foot is in the air(Rose, et
al., 2006 p. 224). During walking at a comfortable pace, 60% of the cycle is in the stance period and
40% in the swing period (Szendrõi, 2008 p. 29). However, this varies with the walking speed, where
the swing period becomes proportionately longer and the stance period shorter as speed increases.
The final disappearance of the double support marks the transition from walking to running (Whittle,
2007 p. 54).
Stance period can be subdivided into five phases known as: (1) initial contact, (2) loading response,
(3) mid stance, (4) terminal stance, and (5) pre swing. In addition, swing period into three phases: (6)
initial swing, (7) mid swing, and (8) terminal swing.
14
Considering the five phases of stance period, the first phase, initial contact, begins when the foot just
touches the ground. Another term for the onset of stance customarily has been called heel strike. Yet,
the heel of a paralytic patient may never contact the ground or do so much later in the gait cycle
(Perry, 1992 p. 10). The second phase, loading response, is a time of double limb support when both
feet are on the ground. During this phase, the limb accepts the weight of the body. Loading response
ends when the opposite foot is lifted for swing.
Mid stance is the phase when the center of gravity moves over the foot and the limb fully supports the
weight of the body. The fourth phase, terminal stance, is characterized by heel rise and continuous
dorsiflexion, while the other foot strikes the ground. The last phase of stance is pre swing, another
time of double limb support, when the foot is about to become airborne and the opposite limb
progressively accepts more weight.
The last three phases in swing period starts with the initial swing phase. This phase begins when the
foot leaves the ground and continues as the knee flexes. Mid swing phase begins with the knee in
maximum flexion and ends when the leg is perpendicular to the ground. The last phase, terminal
swing, begins with the leg perpendicular to the ground and ends when the foot contacts the ground
again (Rose, et al., 2006 p. 225).
2.4 Ankle Function and Common Pathologies
During a gait cycle, the foot and ankle joint are subject to several forces and movements, so the body
movement occurs. The ankle movement is crucial to provide correct absorption of forces and limb
advancement. Aiming for a system capable of assisting patients with pathological gait, it is first
necessary to understand normal gait, since this provides the standard against which the gait of a
patient can judged.
At the time of initial contact, the ankle is usually close to its neutral position. With tibia sloping
backwards and the foot sloping upward, only the heel contacts the ground. Activated since swing,
tibialis anterior maintains the foot dorsiflexed, in preparation for the controlled plantar flexion after
initial contact. Loading response phase involves plantar flexion at the ankle caused by external forces
at the heel, which is controlled by the tibialis anterior muscle. This movement is accompanied by
internal rotation of the tibia, rolling the body weight forward on the heel, and bringing the forefoot onto
the ground. In mid stance, the direction of ankle motion changes to dorsiflexion, as the tibia moves
over the stationary foot. Tibialis anterior ceases to contract, while the contraction of the triceps surae
begins. Terminal stance is characterized by forward rotation of the tibia about the ankle joint, with the
forefoot remaining flat on the ground and the heel rise. Peak ankle dorsiflexion is reached around the
final of terminal stance, with triceps surae initially maintaining the ankle angle, while the knee flexes, to
later move into plantar flexion. During pre swing, the ankle moves into plantar flexion due to concentric
contraction of the triceps surae. Peak ankle angle for plantar flexion occurs just after initial swing.
Triceps surae ceases and tibialis anterior begins to contract, with the ankle movement from plantar
flexion to dorsiflexion continuing during mid swing. Nevertheless, this contraction is much smaller than
the required at the foot lowering after initial contact. In the last phase, terminal swing, ankle position
turns close to neutral position. Tibialis anterior continues to contract, holding the ankle in position, but
its activity usually increases prior to initial contact, anticipating the greater contraction which will be
needed during the loading response (Whittle, 2007 pp. 64-79). A normal ankle movement during a gait
cycle is presented in Figure 2.10.
15
Figure 2.10: Ankle joint angles during a gait cycle in natural cadence (Winter, 1991 p. 29).
A large number of diseases or accidents affect the neuromuscular and musculoskeletal systems,
which may lead to disorders in the ankle. Among the most important are: Cerebral palsy,
Parkinsonism, Muscular dystrophy, Osteoarthritis, Rheumatoid arthritis, Stroke, Head injury, Spinal
cord injury, and Multiple sclerosis (Whittle, 2007 p. 187). Even with the existence of many walking aids
to benefit a person affected by any of these conditions, it is clear that greater benefits are possible in
some pathologies than others.
A possible classification of all functional errors at the ankle can be by either considering the existence
of excessive dorsiflexion (inadequate plantar flexion), or excessive plantar flexion (inadequate
dorsiflexion) (Perry, 1992 p. 186). In this work the focus will be placed in patients who have both
functional errors, usually patients without motor control at the ankle joint, i.e., ankle palsy. This
pathology combines different functional errors presented in both excessive dorsiflexion and excessive
plantar flexion, having different impacts during gait cycle. While excessive dorsiflexion has more
functional significance in stance than in swing, excessive plantar flexion has functional significance in
both periods. Generally, excessive dorsiflexion is caused by triceps surae weakness, while excessive
plantar flexion is caused by pretibial muscles weakness.
In a pathological gait cycle, the normal event of heel strike as initial contact can be replaced by
abbreviated heel strike or forefoot contacts (Rose, et al., 2006 p. 166) due to excessive plantar flexion.
The action that accompanies loading the limb varies with the mode of initial contact. Abbreviated heel
strike can be followed by instantaneous foot drop due to weak dorsiflexion control (Figure 2.11a), and
in case of forefoot contact, foot rapidly drops onto the heel while the tibia stays vertical (Perry, 1992 p.
187). Mid stance with excessive dorsiflexion causes an accelerated rate of ankle dorsiflexion from its
initial position of plantar flexion, leading to instability at the onset of single limb support (Perry, 1992 p.
197). The lack of plantar flexion allows unrestrained tibial advancement during terminal stance,
resulting in flexed knee and a possible loss of heel rise, leading to a prolonged heel contact in pre
swing (Figure 2.11b,c). Generally, heel rise begins when tibia moves forward as result of reaching the
ankle’s passive range (Perry, 1992 p. 198).
Figure 2.11: (a) Foot slap due to weak dorsiflexion control, (b) Prolonged heel contact due to excessive dorsiflexion, (c) excessive knee flexion combined with heel rise (Perry, 1992).
16
During swing, the most compromising functional error is excessive plantar flexion. The inability to
dorsiflex the foot during the swing phase causes a functional leg length discrepancy, and toe drag is
observed when the subject fails to compensate (Whittle, 2007 p. 117). Unless it is extreme, excessive
ankle plantar flexion in initial swing has no clinical influence, since the trailing posture of the tibia tends
to minimize the effect increased ankle plantar flexion has on toe drag (Perry, 1992 p. 190). Toe drag in
mid swing due excessive plantar flexion inhibits limb advancement (Figure 2.12a). As a result, swing is
prematurely terminated unless there is adequate substitution to preserve floor clearance. The most
direct substitution for lack of adequate ankle dorsiflexion in swing is increased hip flexion to lift the limb
and, hence, the foot Figure 2.12b) (Perry, 1992 p. 191). In terminal swing, excessive plantar flexion
influences the way initial contact occurs, compromising the following phases.
Figure 2.12: Gait deviations due to excessive ankle plantar flexion: (a) toe drag during mid swing, (b) increased hip and knee flexion to avoid toe drag during mid swing, (c) terminal swing gait deviation, with the foot nearly
parallel to the ground (Perry, 1992).
Some patients compensate the effects of dropped foot by altering the way that they walk. In time, this
can lead to further problems such as pain in the hips. Compensatory strategies depend on residual
muscle strength, motor control, joint mobility and sensory capabilities (Rose, et al., 2006 p. 165). A
good diagnosis of the functional error is essential to improve the gait, and therefore patient’s health.
2.5 Solutions for Ankle Pathologies
Different ankle pathologies demand different solutions. This way, an essential first step in the clinical
application of gait analysis is identification of the patient's medical diagnosis as means of classifying
the primary impairments, which can fall into two broad practice patterns: neuromuscular and
musculoskeletal (Rose, et al., 2006 p. 168). According to the patient’s deficit, an orthosis can be
prescribed. An orthosis consists on a structure that lies on to the anatomy of the human limb. Its
purpose is to restore lost or weak functions, usually following a disease or a neurological condition, to
their natural levels (Pons, 2008).
In the past few decades there have been significant changes as regards the materials and technology
used to develop and manufacture new orthosis. Orthosis for ankle pathologies can go from exterior
structures with light metal and plastic to systems that interact directly with the nervous system. Even
so, for the different orthosis there are several goals which they are expected to achieve:
Restore the expected function
Cause the least possible inconveniences
Be safe, and not cause injury (e.g. they must not break)
Be as esthetic as possible
Be easy to use and maintain
Be lasting (adaptation to a new orthosis can cause inconveniences
Be inexpensive
17
It is obvious that it is unlikely that all of these expectations will be completely satisfied. It is therefore
especially important to verify if the prescribed orthosis meets the functional aims (Szendrõi, 2008 p.
74).
2.5.1 Ankle-Foot Orthosis (AFO)
An ankle-foot orthosis is usually an orthosis that covers the foot, spans the ankle joint and covers the
lower leg (Figure 2.13). This lower limb orthotic device can be divided into categories of metal, plastic,
and more recently carbon.
Metal AFOs are still used for several indications, including the insensate foot, the foot with fluctuating
edema, or when the need for adjustability or progressive changes in the device are indicated (Figure
2.13a).
The plastic design is used more often and can be fabricated from a cast or molding of the patient’s
limb (Cooper, 2006 p. 108). Some off-the-shelf designs may be suitable for short term use, but custom
designs are better for long term use, allowing choosing the plastic type, color, and thickness. These
general features of plastic AFOs affect the degree of rigidity, degrees of dorsiflexion and foot plate
design (Figure 2.13b,d).
Carbon fiber AFOs have been widely used throughout the past decade. Carbon fiber is extremely light
weighted, and durable. This style of AFO is best used for isolated drop foot, and when other
instabilities are present, a custom version can be made to accommodate any need (Figure 2.13c).
Figure 2.13: Ankle-foot orthosis: (a) Conventional AFO, comprised of a metal frame with leather straps, attached to a shoe (Orthomedics), (b) Standard Plastic AFO (Orthomedics), (c) Prolite Carbon AFO made from injection
molded carbon-reinforced polypropylene with increased rigidity in vertical aspect (Össur, 2010), (d) Plastic Articulated AFO (Orthomedics).
AFOs can also incorporate several special features:
Hinged joints allow some dorsiflexion and limited plantar flexion, or more adjustable metal
AFO joints.
Footplate designs can incorporate three-quarter length, stopping just before the metatarsal
heads for easier access into shoes, or a full length footplate with padding, which is generally
used for the most spastic or most vulnerable foot.
Inversion control features include a high medial wall on the footplate and a large lateral
phalange at the fibula to prevent inversion positioning of the foot in the brace.
Most of the AFOs used today incorporate an ankle joint of some kind to allow ankle motion. Main goal
is to allow as much motion as possible while blocking unwanted movement. Selection of an ankle joint
depends entirely on what the AFO is designed to do, and how it will be used. Usually, standard AFOs
(Figure 2.13b) are the basis for applying joints, becoming articulated AFOs (Figure 2.13d). These
joints can have several features like locking the ankle in any selected position (Cooper, 2006), provide
18
control both in dorsiflexion and plantar flexion directions (Cooper, 2006). More recent technology is the
Tamarack Flexure Joint™, which is a self-aligning flexure joint technology without unwanted joint
elongation.
AFOs have been shown to improve ankle kinematics during stance phase, increase step and stride
length, decrease cadence, and decrease energy costs in walking, while improving walking, running,
and jumping skills (Harris, et al., 2008 p. 177), being many times the cheapest solution to solve ankle
pathologies.
2.5.2 Active Ankle-Foot Orthosis (AAFO)
An Active ankle-foot orthosis (AAFO) consists on a generic AFO (passive system) endowed of an
electromechanical device (active system) with the aim of controlling the ankle movements. Many of
these systems are still in development, emerging more and more prototypes. Unlike passive orthoses,
active orthoses have the potential of controlling the ankle joint, providing the necessary torque during
the gait cycle. These systems assist patients to walk, allowing them to walk more naturally or enabling
walking in patients who could not with passive orthoses.
A powered AFO was developed in the MIT Biomechatronics Lab to assist drop foot gait (Figure 2.14a).
Using a biomimetic variable-impedance control algorithm for this pathological gait, the system consists
of a modified AFO with a series elastic actuator (SEA), that is used to adjust the impedance of
dorsiflexion or plantar flexion, according to the ankle motion monitored by the ankle angle sensor and
force sensors at the sole of the AFO. With approximately 1.82 kg of mass (AFO, SEA, and sensors),
this system provided a functional work platform for later developments, such as portability and a lighter
system (Blaya, 2003).
Figure 2.14: (a) Active ankle-foot orthosis developed at Massachusetts Institute of Technology (MIT)(Dollar, et al., 2007), (b) Powered Ankle Exoskeleton developed at The University of Michigan (Kao, 2009), (c) Adjustable
robotic tendon robot concept (Hollander, et al., 2005).
Artificial pneumatic muscles endowing a carbon fiber shank section with a polypropylene foot section
of movement (Figure 2.14b) was developed by the University of Michigan. By increasing air pressure,
the pneumatic muscles (McKibbon muscles) start developing tension and become shortened, allowing
the powered exoskeleton to provide plantar flexor movement, i.e., artificial soleus muscle activation.
Designed for rehabilitation purpose, this system has a mass of approximately 1.1 kg (Kao, 2009).
Other identical system was developed by the same laboratory, which provided both dorsiflexion and
plantar flexion control of the ankle joint (Ferris, et al., 2006).
19
An adjustable robotic tendon was proposed by Hollander, et al., 2005, which can be used in a gait
assistance device (Figure 2.14b), where a spring based linear actuator tries to mimic human ankle
motion. In the design, the spring stiffness is varied by changing the length of a linear coil spring. A
power amplification factor of 3 times was reported, reducing the peak power and energy requirements
for its motor. The proposed system is expected to provide 100% of the power and energy necessary
for ankle gait with a total mass of 0.84 kg.
2.5.3 Functional Electrical Stimulation (FES)
Functional electrical stimulation is a technique that uses electrical current to contract muscles. This
developing assistive technology can generate useful movements of the extremities through such
electrical activation of paralyzed or spastic muscles (Rose, et al., 2006 p. 209). Electric impulses are
generated by electrodes, which can be implanted or placed on the surface of the skin.
Figure 2.15: (a) FES electrodes1, (b) FES with orthosis (Phillips, 1986).
Early FES devices were based upon pure analog designs which rendered the usage and accuracy of
usage of the FES device an unpredictable one (Karthikeyan, et al., 2008). In 1982, computer
controlled standing and walking was introduced by researchers at Wright State University (Dayton,
Ohio), using a closed loop-control (Phillips, 1986). Recent developments in digital technologies led to
FES systems as a reliable method of rehabilitation (Karthikeyan, et al., 2008). To have a proper use of
FES, stability, repeatability, and regulation of muscle properties are required under several different
conditions, fatigue and muscle length. This way, prior to the development of FES patterns, stimulation
levels for each muscle must be determined individually to provide optimal exercise and walking
programs (Johnston, et al., 2003).
Closed-loop control with FES requires that data obtained from sensors modifies the output signal from
the controller, in order to help coordinating walking. Sensors are placed on the hips, knees, or ankles
of the paralyzed patients, providing positional data for the computer controller (Phillips, 1986). The
integration of FES with orthotic devices has proved to be a good solution, decreasing muscle fatigue
and the number of joints needing control (Phillips, 1986). Coordinating actions of weak, paralyzed, or
uncontrollable muscles with each other can be achieved in patients with spinal cord injuries, making
standing and walking possible (Rose, et al., 2006 p. 209).
1 http://www.mstrust.org.uk/professionals/information/wayahead/articles/13032009_03.jsp
20
21
Chapter 3
3 Simulation and Control of Ankle-Foot Complex
When a passive system is endowed with active components, it is necessary to provide a control
strategy that assumes a desired behavior. When endowing an ankle-foot orthosis (AFO) with an
actuator to assist ankle pathologies, several control strategies can be implemented.
When designing a control system, it is necessary to devise a control scheme where the manipulated
and controlled variables are set. Section 3.1 presents the control scheme to implement, describing the
manipulated and controlled variables. Setting the control scheme, it is necessary to know the dynamic
behavior of physical phenomena’s, which can be understood by studying their mathematical
descriptions. Mathematical equations, typically differential or difference equations are used to describe
the behavior of processes and predict their response to certain inputs (Antsaklis, et al., 2006). For the
active ankle-foot orthosis (AAFO) control, it is important to have an ankle-foot model, where the
movement, forces, and moment of forces acting through the ankle-foot system can be obtained. For
this reason, section 3.2 presents the simulation of human gait, where simplified and multibody ankle-
foot models are proposed.
Three control strategies were proposed in this work. With rehabilitation purposes in mind, a variation
of a Proportional-Integral-Derivative (PID) controller and a Linear Quadratic Regulator (LQR) are
presented in sections 3.3 and 3.4, controlling the system by reference tracking. With a control
behavior similar to the human control, Impedance Control (IC) is presented in section 3.5, which is the
control strategy for the AAFO during free walking where the controller tries to mimic the human
control. Since the ankle-foot system is a nonlinear system, with large changes in the dynamic
characteristics during gait cycle (GC), section 3.6 presents a set of triggers and states for changing
the control system parameters during the cycle. The results for the controlled systems with the
different strategies are presented in section 3.7, with the respective analyzes and contextualization in
case of a real system.
3.1 Control System
A control system consists of subsystems and processes (or plants) assembled for the purpose of
controlling the output of the processes (Nise, 2004 p. 2). These processes are characterized by its
inputs and outputs. The system input is the manipulated variable, which is the condition that is varied
by the controller to affect the controlled variable. The controlled variable is the system output, which is
the condition that is measured and controlled (Ogata, 2002 p. 2). Manipulated and controlled variables
are the key concept for the entire designing, since a bad choice of variables can affect the possibility
of controlling the system.
The control scheme presented in Figure 3.1 is a closed-loop control system, where the difference in
the desired and actual condition creates a correction control command to remove the error (Bishop,
22
2008 pp. 3-8). The controlled variable involved in this control scheme is the ankle angle sensor output,
while the manipulated variable is the ankle torque that causes the rotation of the ankle joint. In fact,
the ankle torque results from the output force of a series elastic actuator (SEA), this is a linear actuator
in series with a spring that causes a torque through the ankle joint due to the offset of the application
point with respect to the joint center. The existence of a spring in the actuator provides some
mechanical compliance, acting also as an indirect force sensor by measuring the deflection of the
spring. It also makes the system more robust to the application of sudden external forces and more
close to its biological counterpart, improving the overall response of the system in the correction of
pathological gait.
Figure 3.1: Diagram of a simplified control scheme for controlling the ankle movement with the AAFO. The control action of the AAFO is executed by an actuator (series elastic actuator), which causes a torque about the
ankle joint with the goal of assisting the ankle movement during gait.
In this work, three control strategies were proposed with the main goal of assisting the ankle
movement. With rehabilitation purposes in mind, proportional-integral-derivative (PID) and linear
quadratic regulator (LQR) are proposed for assisting the ankle movement during the gait cycle. During
rehabilitation, these two control strategies act in reference tracking, imposing the individual who wears
the device a certain movement during gait. It is expected that the rehabilitation environment provides
safe walking, minimizing the unexpected forces on the foot. With the attempt of mimicking the human
control of limbs, impedance control (IC) was also proposed. This control strategy tries to mimic the
behavior of biological ankle movement by varying the stiffness and damping of the ankle joint during
movement.
The main control scheme for controlling the system is presented in Figure 3.2, where the slave
controller is responsible of controlling the torque output from the actuator and the master controller is
responsible of controlling the ankle joint angle by manipulating the torque to apply on the ankle joint.
So that the dynamics of the actuator does not influence the control of the ankle joint, the slave
controller bandwidth has to be larger than the one from the master controller. Thus, the closed-loop
model of the SEA can be disregarded in the closed-loop transfer function of the system with the
simplified ankle-foot model, but implemented when simulating the system control. The control of the
SEA was made separately, being its modeling and control described in detail in section 5.2.5.
23
Figure 3.2: Main control scheme with master and slave controllers.
Overall, three linear control strategies are proposed for the control of this nonlinear system, which is
subject to external forces and to a large range of movement in terms of ankle angle.
3.2 Human Gait Simulation
The design and control of robotic systems requires the use of mathematical models as an attempt of
reproducing the reality. A model is generally a simplification of the system intended to promote
understanding, allowing that complex or long actions can be executed in a safer, economic, and faster
way.
Since it was opted to control the system with a force based actuator, it was deemed necessary to
develop a dynamic model of the ankle-foot complex that provided the relations between torques and
forces of the actuator, and the positions, velocities, and accelerations of the joints. Subsection 3.2.1
provides the information relative to the biomechanical data necessary for the simulation models.
During the gait cycle, the ankle-foot complex is subjected to various states, where a wide range of
ankle movement occurs and external forces are applied to the foot, characterizing the system as
nonlinear. A simplified model is presented in subsection 3.2.2, where the ankle-foot complex is
approximated by a revolute joint with inertia. This model is acceptable in the swing period, but very
poor in the stance period. Stance period is a much more complex state, since it has external forces
acting on the foot. Requiring a more realistic model, subsection 3.2.3 presents a basic multibody
ankle-foot model. This biomechanical model considers inertial and ground reaction forces during the
gait cycle, where the influence of abnormal motor patterns can be predicted and simulated.
3.2.1 Gait and Anthropometric Data
The gait data used in the modeling and control of the computational ankle-foot models was obtained
from tables in(Winter, 2004). The gait data was from an individual with approximately 1.6 m high, 56.7
kg of body mass, walking at a speed of 1.42 m/s.
The gait data includes ground force and motion capture data for approximately 1.51 seconds of
walking, at a frame rate of 70 frames per second. The markers where located over the edges of the
locomotor system, toe, fifth metatarsal, heel, lateral malleolus (ankle), head of fibula, lateral
epicondyle of thigh (knee), greater trochanter (hip), and base of rib cage. These tables present the raw
coordinate data (before filtering), filtered marker kinematics, linear and angular kinematics and kinetics
of the lower limbs. Corresponding potential and total energies of the lower limbs are also available.
The coordinate system of the gait data is the system presented in Figure 2.7a.
When developing biomechanical models, it is important to consider the masses of the limb segments,
location of the COM, segment lengths, centers of rotation, and moments of inertia. A realistic model
depends as much on the kinematics and kinetics as on the quality and completeness of the
anthropometric measurements (Winter, 2004 p. 9). For the biomechanical models presented in the
24
following subsections, the parameters were obtained from the anthropometric data table in (Winter,
2004), p. 63. In particular, the average ankle moment of inertia, , calculated trough the
anthropometric data was 0.012 kg m2.
3.2.2 Simplified Ankle-Foot Model
In general, when solving a new problem it is desirable to build a simplified model that allows having a
general feeling of the solution. Depending on the particular system and the particular circumstances, a
specific simplified model may be better suited than other more comprehensive models (Ogata, 2002 p.
53). The knowledge of the mathematical formulation of the system allows a theoretical approach when
designing the system controller.
The approach that was taken when obtaining the simplified ankle-foot model resided in a compromise
between the simplicity of the model and the accuracy of the results, considering that it was necessary
to ignore some of the nonlinearities of the system by using a simple model with easy implementation
than a realistic model difficult to achieve. The main goal was to obtain a model that could replicate the
fundamental behavior of the physical system. Thus, the approximation for the simplified ankle-foot
model is obtained by the application of Newton’s second law for rotational motion (Tipler, et al., 2004
p. 287), with the ankle torque, , as:
(3.1)
where is the ankle rotational inertia due to foot mass, and the ankle angular acceleration. This
equation can be written as a transfer function using the Laplace transformation, resulting:
(3.2)
The resulting transfer function represents a second order system, unstable, characterized by the two
poles at the origin. For simplicity, this model excludes the ground reaction forces, which can be
compensated in a feed forward manner.
3.2.3 Multibody Ankle-Foot Model
Instead of deriving and programming equations, biomechanical models can be developed in multibody
simulation tools. Multibody systems are used to model the dynamic behavior of interconnected rigid or
flexible bodies that have their relative motion constrained by kinematic joints and that are actuated by
forces. When the information of the kinematic description of the movement, the anthropometric
measures of the subject, and the external force measures are available, it is possible to calculate joint
reaction forces and moments at the joints of the model (Winter, 2004 p. 86), i.e., the so-called inverse
dynamic analysis. The inverse scenario is also possible, i.e., calculating the resulting kinematics due
to forces and moments applied on the model in an analysis that is usually referred as forward
dynamics analysis.
Several dynamic models with different inputs and outputs for the simulation of human movement have
been developed. Some simulations take into account the entire locomotor unit, trying to create the
model of a bipedal robotic system, while others focus on a part of the locomotor unit in particular. In
this work, the focus was made on the movement of the limbs below the knee joint. The goal was to
create a biomechanical model where the knee and leg kinematics were fully prescribed, leaving the
ankle joint free to rotate due to foot weight and torque caused by the interaction forces between the
foot and the ground. Thus, it was necessary the existence of a foot contact model, i.e., a model that
25
considered the external forces caused by the contact with the ground (ground reaction forces (GRF)).
A representation of the multibody ankle-foot model is presented in Figure 3.3.
Figure 3.3: Scheme for the multibody ankle-foot model with the representation of kinematics prescribed at the knee joint.
Several foot models have been widely proposed as an attempt to calculate the ground reaction forces
on the foot or the torque through the ankle joint.
A foot contact model with viscoelastic elements was developed by Girlchrist and Winter (Gilchrist, et
al., 1997). Focusing on the stance phase, the presented model in Figure 3.4a shows a set of vertical
spring-damper elements were located along the midline of the foot, combining the viscoelastic
behavior of foot, shoe, and ground. Two orthogonally horizontal dampers were also associated to
each vertical element, in order to account for the shear forces.
Barbosa, et al. (Barbosa, et al., 2005) represented the contact between the ground and the foot by
approximating the contact surfaces by hiper-ellipsoids with certain material and geometric
characteristics, restrained to the most important areas on the foot during gait. Acting like a spring-
damper element, the ground reaction force was calculated based on the penetration of the ellipsoid
into the ground and the equivalent stiffness and damping properties of the ellipsoid. A representation
of the model is presented in Figure 3.4b.
Inserted in the FCT DACHOR project, Moreira (Moreira, 2009) developed a foot model consisting of a
set of spheres located under the plantar surface to model foot-ground contact. Constitutive laws for
the contact phenomena were applied to simulate the interaction between the foot and ground
surfaces, taking in account vertical reaction force and Coulomb and viscous friction. A representation
of the model is presented in Figure 3.4c.
Figure 3.4: (a) Diagrammatic representation of a viscoelastic foot model (Gilchrist, et al., 1997), (b); Contact surfaces (hiper-ellipsoid) between the foot and the ground (Barbosa, et al., 2005), (c) Foot contact model
consisting of a set of spheres (Moreira, 2009).
In the modeling of the ankle-foot complex, a currently accepted approach to quantify foot and ankle
kinematics during gait is to represent the entire foot as a single rigid body with a revolute ankle joint,
26
generally for sagittal plane studies (Harris, et al., 2008 p. 384). Thus, a simple foot contact model was
considered for the modeling, adding to the foot (rigid body) elastic contacts.
The developed foot model, presented in Figure 3.5, has two elastic contact points, considering for
each vertical and horizontal components of the GRF. The contact forces, , are defined by Hertz
law (Tipler, et al., 2004)
(3.3)
where represents the foot penetration on the ground, and the relative stiffness at the contact
point. The foot penetration is the displacement of the contact point below the ground (Y=0), i.e., when
contact point is at negative values for the vertical coordinate (Y) the foot is actuated by the ground
reaction forces.
The software used in the development of the multibody biomechanical model was SimMechanics®.
SimMechanics® is a module of Simulink
®, which is a simulation tool from MATLAB
® (MathWorks)
software. With SimMechanics® software, it is possible to build models composed of bodies (links or
segments), joints, constraints, and force elements that reflect the structure of the system by the use of
Newtonian dynamics of forces and torques. This software incorporate an automatically generated 3-D
animation, being possible to visualize the system dynamics, and import complete models with mass,
inertia, constraints, and 3-D geometry from several CAD systems.
Figure 3.5: Elastic foot contact model for the multibody ankle-foot model.
For the configuration of the ankle-foot model, the development of the multibody system was realized in
two phases. First phase was characterized by all joints having their kinematics prescribed, being the
ankle torque obtained through inverse dynamics. In the second phase, the ankle joint ceased to have
prescribed kinematics, setting the ankle joint as a forward dynamics model.
The first phase of the multibody system development can be considered as the creation of an inverse
dynamic model. Even so, to obtain a valid model, it was necessary to make some assumptions with
respect to the model (Winter, 2004 p. 87):
Each segment has a fixed mass, located as a point mass at its center of mass (COM).
The location of each segment’s COM remains fixed during the movement.
The joints are considered to be hinge joints.
The moment of inertia of each segment about its COM is constant during the movement.
The length of each segment remains constant during the movement.
The second phase of the multibody development can be considered as the creation of an inverse and
forward dynamics model, where the movement of the leg is prescribed and the movement of the ankle
joint is dictated by external forces on the foot and torque inputs on the ankle joint. Maintaining the
assumptions for the inverse dynamic model, some additional assumptions were necessary for the
forward dynamic model (Winter, 2004 p. 157):
There must be no kinematic driving constraints whatsoever associated with the ankle joint.
27
The initial conditions must include the position and velocity of every segment.
The only inputs to the model are externally applied forces and internally generated muscle
forces or moments.
The model must include all important degrees of freedom (DOF) and constraints.
External reaction forces must be calculated by the forward dynamic analysis procedure.
The multibody model developed in this work and presented in Figure 3.6, was composed of two sub-
models: the principal and secondary multibody models. The principal multibody model, composed by
the foot (body), ankle (joint), leg (body), and knee (joint) was used to calculate the ground reaction
forces on the foot, and thus, the ankle torque. The secondary multibody model, composed by the
ground (body), thigh (body), hip (joint), and head-arm-trunk (HAT) (body) was created for visualization
purposes only. Each body was characterized by its mass, moment of inertia tensor, center of mass
(COM), and dimensions, while the joints by the DOF and constraints.
To perform the model analysis in the first phase, it was necessary to prescribe the kinematics of each
joint. Translational movement uses (X, Y) coordinates, while rotational movement occurs around the Z
axis, in the sagittal plane (see spatial coordinates in Figure 2.7a). In the principal multibody model,
translational and rotational movement was prescribed to the knee joint, where the leg, ankle joint, and
foot elements were connected, respectively. The rotational movement of the foot through the ankle
joint was prescribed relatively to the position of the leg, where the neutral position (0º) of the foot
occurs when the foot and leg are perpendicular. The secondary multibody model had its kinematic
prescribed through the hip joint, except the ground body, which was set fixed at the origin of the
coordinate system. The hip joint has translational and rotational movement, noticing that the rotational
movement only affects the thigh, maintaining the HAT always vertical.
Figure 3.6: Multibody biomechanical model used in the simulation (SimMechanics®).
With all joints kinematics prescribed, it was necessary to set the parameters of the ankle-foot model to
use. The main goal was to assign a position and stiffness to each contact point on the foot, in order to
replicate the ankle torque during gait cycle on the ankle joint due to reaction forces on the contact
points.
In an iterative procedure, the positions of the contact points on the foot were first chosen in an attempt
to match some particular events during the stance period with the intersection of the ground with the
contact points. The events used in this positioning procedure were heel contact (initial contact), toe
contact, heel off, and toe off (Figure 3.7).
28
Figure 3.7: Events triggers during gait cycle – Heel and Forefoot sensors activation.
With the position of the contact points defined, the following step was to set to each contact point a
determined stiffness. The method in this procedure was to assign stiffness’s to each contact point and
compare the ankle torque from the model with the ankle torque from the gait tables, as an attempt of
having similar ankle torques on the model (Figure 3.8a). The same procedure was applied to the
ground reaction forces, where the vertical and horizontal ground reaction forces (GRF) in each contact
point were compared with the respective values from the gait tables (Figure 3.8b,c), however, the
results were very poor. Therefore, it was given more focus on matching the ankle torques during the
gait cycle.
Figure 3.8: Dynamics of gait data and model during gait cycle: (a) ankle torque, (b) horizontal shear force, (c) vertical force. The stiffness for each virtual spring was: Horizontal spring (heel) = 5 x 10
4 N/m, Horizontal spring
(forefoot) = 9 x 103 N/m, Vertical spring (heel) = 4.5 x 10
5 N/m, Vertical spring (forefoot) = 1.7 x 10
5 N/m.
Achieving an acceptable ankle-foot model for the gait cycle, the ankle joint ceased to have prescribed
kinematics. In this situation, the ankle-foot model relies on the functional error for which this work has
considered to assist, i.e., excessive dorsiflexion and plantar flexion. Thus, this model can be
connected to a controller in order to assist the movement of the ankle joint during gait cycle.
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
Gait Cycle (%)
Conta
ct
1=
yes 0
=no
Heel Sensor
Forefoot Sensor
29
3.3 PID Control
The PID controller is the most commonly used controller (Levine, W., 2000 p. 216). The basic form of
PID consists of three ways of acting through the error between the reference signal and the system
output. The first is the proportional term, P, representing the control action of the controller
proportionally to the error, i.e., the larger the error, the larger the correction. The second is the integral
term, I, representing the integral of the error over the time, where the correction considers the time
that the error has been present, i.e., the longer the error persists, the bigger the correction. The third is
the derivative term, D, where the corrective action is related with the derivative of the error over the
time, i.e., the faster the error changes, the larger the correction (Bishop, 2008 pp. 3-8). Figure 3.9
presents the block diagram of the basic PID controller, indicating the controller terms, P, I, D, the
system input, R(s), the system output, Y(s), the system error, E(s), and the control action, U(s).
Figure 3.9: Block diagram of a basic PID controller.
The goal for the controller presented in this section is that the output of the system follows the input
reference which is time varying, and in this case equal to the angle of the ankle joint during gait cycle.
For the modeled system, it is expected that a satisfactory performance can be achieved by following
the reference input.
Several control strategies based on PID controller have been widely developed (Ogata, 2002 p. 700),
where the application of the different strategies generally depends on the system or on the
characteristics the user desires from it. When choosing the PID strategy to apply on the control of the
ankle joint, it is necessary to take into account that gait cycle is a periodic motion and subject to noise.
Thus, with the intent of having a stable and fast controller, the integral term was disregarded, resulting
in a PD structure.
In the basic PID control system (Figure 3.9), when the reference input is a step function, the presence
of the derivative term in the control action, the manipulated variable, , involves an impulse
function known as derivative kick (Ogata, 2002 p. 700). The derivative term is also very sensitive to
the noise (as it amplifies the high frequency noise), which may also lead to the derivative kick.
Derivative kick is a phenomenon that generally leads to the instability of the system, and also to the
damage of its physical components. To avoid the derivative kick phenomenon, it is necessary to
operate the derivative action only in the feedback path so that differentiation occurs only on the
feedback signal and not on the reference signal. The control scheme arranged in this way is called P-
D control and was implemented in this work. Figure 3.10 shows the implemented P-D controller
system, where the system input is the reference ankle angle, , the system output is the ankle
angle, , the control action is the ankle torque, , the proportional term is , and the derivative
term is .
30
Figure 3.10: Block diagram of the P-D controller in closed-loop with simplified ankle-foot model.
Considering the transfer function of the system plant (simplified ankle-foot model) that was presented
in subsection 3.2.2,
results the following transfer function of the closed-loop system with a P-D controller,
(3.4)
The closed-loop transfer function of the P-D controller can be compared with the typical transfer
function of a pure second-order system
(3.5)
where is the system natural frequency and is the damping ratio. With a suitable choice of gains, it
is possible to obtain any value of the natural frequency and damping ratio. Hence, if and are
given as design specifications, the following relations can be found
(3.6)
(3.7)
The process of selecting the controller parameters to meet a given performance specifications is
known as controller tuning, which is based on the system characteristics or by the user know-how.
The main goal of P-D control in this work is to provide an effective reference tracking of the ankle
angle during gait. This control strategy was implemented with the simplified and multibody ankle-foot
models, with the results presented in section 3.7.1. Although the dynamics of the series elastic
actuator (SEA) has not been considered in the closed-loop transfer function of this section, the
corresponding transfer function of the SEA was integrated in the simulation models. This additional
dynamics can be considered as non-modeled dynamics, that the controller has to be capable of
controlling. Section 5.2.5.1 presents the dynamics of the SEA as a force actuator.
3.4 Optimal Control
An optimal control system is a system whose design “optimizes” the value of a function chosen as the
performance index (Ogata, 1995 p. 566), e.g., the application of mathematical optimization methods
with control purposes. Thus, given a system, the optimal control objective consists on finding a control
law by recurring to a certain optimality criterion.
31
Optimal control is generally implemented in a state space representation, which is a method that
simplifies the complexity of mathematical expressions, by describing the equations system in terms of
first order differential equations, which may be combined into a first order vector matrix differential
equation (Ogata, 2002 p. 752). The system description in state space can be given as:
(3.8)
(3.9)
where is the state vector, is the control input vector, and is the output vector. The matrix with
dimension is the dynamic matrix, the matrix is the input matrix, and the matrix is the
output matrix.
Several methods can be considered in optimal control, where the most common are pole placement
and linear quadratic regulator (LQR). In this work, the LQR was considered since it has the advantage
of providing a systematic way of computing the state feedback control gain matrix. When designing an
optimal control system, it is required the definition of a control decision method, subjected to certain
constraints, so as to minimize some measure of the deviation from ideal behavior (Ogata, 1995 p.
566).
In order to design a controller recurring to the LQR, the performance index, , is given by the
following expression:
(3.10)
where is a positive-definite (or positive-semi definite) Hermitian matrix and is a positive-definite
Hermitian matrix. The matrices and determine the relative importance of the error and the
expenditure of energy (Ogata, 2002 p. 897).
In the LQR problem, given the system equation
the gain matrix , of the optimal control vector
(3.11)
may be determined so as to minimize the performance index . To find the values for the gain matrix
in the control input, it is necessary to solve the algebraic Riccati equation for the symmetric positive
definite Hermitian matrix :
(3.12)
Finally, the gain matrix can be obtained
(3.13)
Considering the transfer function of the system plant (simplified ankle-foot model) that was presented
in subsection 3.2.2, , and the following variables correspondence
results
32
(3.14)
With this, the corresponding model in state space is
(3.15)
(3.16)
where is the ankle rotational inertia, and the state variables and are the ankle angle and ankle
angular velocity, respectively. In Figure 3.11 it is presented the block diagram of the LQR control
scheme used in this work.
Figure 3.11: Block diagram of the implemented LQR.
where is the ankle angle reference introduced in the state vector as
, and are
the control gains for state variables and respectively. As we can see, this is the same control
architecture as that presented in Figure 3.10. However, now we have a control systematic (optimal)
procedure for determining the control gains.
Considering the presented block diagram, the control actions results:
(3.17)
Thus, the state equation for the designed system is
(3.18)
and the output equation is .
Solving the Riccati equation, the Hermitian matrices and have the following form:
where and are the weight for the state variables and respectively, and the weight for
the control action. Thus, it follows the gain matrix,
The tuning procedure for this controller is made by choosing the weights according to the desired
performance for the system.
33
As in P-D control, the main goal of LQR control in this work is to provide an effective stabilization of
the ankle angle during gait. In an identical procedure, LQR control was implemented with the
simplified and multibody ankle-foot models, with the results presented in section 3.7.1, in a
comparison with results from P-D control. Similar to the P-D controller, the dynamics of the SEA have
not been included here.
3.5 Impedance Control
In wearable robotics, the most common approach to control the interaction forces between the robot
and the human is impedance control (IC) (Vukobratovic, et al., 2009). During the interaction of a
robotic system and the environment, the environment imposes constraints on the trajectories that the
robotic system can follow. Therefore, the use of position controls like those described in sections 3.3
and 3.4 are not recommended, unless the trajectory of the robotic system can be planned extremely
accurately and the control ensures a perfect monitoring of this trajectory. For such, accurate models of
the robotic system and environment (geometry and mechanical characteristics) are necessary
(Dombre, et al., 2007 p. 257), but in the absence of these environmental models, it is therefore
necessary to implement controls that are not “pure” controls in position. An alternative control strategy
is the impedance control, where the main goal is to realize a reference target model by specifying the
interaction between the robotic system and the environment. The main impedance control
performance specification is given by the capability to achieve the target model, while conventional
control systems are analyzed for its ability to track standard input signals in time (Vukobratovic, et al.,
2009 p. 256). Impedance control acts more like a constrained motion than a concrete control scheme,
where the basic idea is to have a closed-loop control system whose dynamics can be mathematically
described by the following equation (for a rotational system):
(3.19)
where and represent the rotational inertia, damping and stiffness of the interactive system
respectively, the system joint torque, and the generalized coordinates in which the coordinates
with subscript, , represents the target reference. The parameters for the interactive system can be
selected in order to correspond to various objectives of a given task. High value of stiffness parameter,
, is selected in the directions where the environment is compliant and positioning accuracy is
important, while low value of stiffness is selected in the directions where small interaction forces have
to be maintained. The damping parameter, , has a positive proportionality with energy dissipation,
while the inertia parameter, , is used to provide a smooth transient behavior in the system response
during the contact (Vukobratovic, et al., 2009 pp. 19-20). For our orthotic device, a representation of
the impedance control system is presented in Figure 3.12.
Figure 3.12: The desired effect of impedance control represented by the use of a rotational mass-spring-damper system.
34
IC is therefore a control strategy that fits the AAFO in order to mimic the ankle movement during gait.
It can be considered that the AAFO is in contact with two different environments during gait: a
permanent contact with human body and a periodic contact with the ground (stance period).
There are different types of approach to obtain the behavior presented in equation 3.19, varying
mainly in the fact of requiring or not the measurement of external forces. For this control strategy, it
was considered an implicit force control that does not require external force measurement and does
not take into account the dynamic model of the robotic system (Dombre, et al., 2007 p. 257). Thus, the
control law was approximated by a linear rotary spring-damper system, with variable stiffness and
damping during different states of system which were presented in section 3.5. The linear rotary
spring-damper system is described by the following equation:
(3.20)
where is the ankle torque, the target ankle angular displacement, and the ankle angular
displacement. Using the Laplace transformation, this function can be written as a transfer function,
assuming that the difference between ankle angular displacements, , is the input and the
ankle torque, , is the output.
(3.21)
This control law can be related to a proportional and derivative control in the operational space, which
was presented in section 3.3. The performance differences between the two approaches reside in how
the controllers are used. In the P-D controller, joint motion is performed by varying in time, while
keeping high values of natural frequency with critical damping. On the other hand, in the impedance
controller, joint motion results from using low controller gains while keeping constant.
The implemented control scheme is presented in Figure 3.13, in the form of block diagram.
Figure 3.13: Block diagram of implemented impedance control.
For a simulation purposes without reaction forces, it is possible to calculate the closed-loop transfer
function by considering the simplified model of the ankle-foot system presented in subsection 3.2.2,
where is the ankle rotational inertia. Thus, the closed-loop transfer function of the system with
impedance controller is given by
(3.22)
35
The closed-loop transfer function can be compared with the typical transfer function of a second-order
system,
(3.23)
Arranging equation 3.22, follows
(3.24)
where the system natural frequency, , and damping ratio, , are related with the system rotational
stiffness and damping terms by the following equations:
(3.25)
(3.26)
It is also possible to notice that the rotational damping term introduces a zero in the system, in
, due to the consideration of the ankle angular velocity as target reference. In fact, the
impedance control structure presented in this section differs from the P-D structure presented in
section 3.3 by this additional zero.
With the main goal of mimicking human control of the ankle joint, the parameters tuning of the
impedance controllers are based on setting the ankle stiffness and damping , which are related
with the target references for the ankle angle and ankle angular velocity . Selecting good
impedance parameters to ensure a satisfactory behavior requires a hard iterative procedure, with the
dynamics of the closed-loop system acting in different modes when in free space or during interaction.
Since the goal of this control strategy was based on the interaction with the ground, this was only
implemented in the control of the multibody ankle-foot model, considering also the closed-loop transfer
function of the series elastic actuator (SEA). The additional dynamic due to the SEA can be
considered as non-modeled dynamic, which the controller has to be capable of controlling.
3.6 Gait States
During the gait cycle, the ankle joint is subjected to a large movement and torque, which characterizes
the system as nonlinear. Therefore, it is preferable to linearize the system by defining different states
for the gait cycle, allowing the controllers to have specific parameters for each state.
A total of four states were defined for the gait cycle: state 1 as loading response, state 2 as stance,
state 3 as pre swing, and state 4 as swing. The division into the four states was based on the foot
contacts and extreme positions. State 1, loading response, starts with the heel contact, accepts the
forefoot contact and ends with the heel rise. State 2, stance, starts with the heel rise and ends when a
defined maximum dorsiflexion ankle angle is reached. The definition of the maximum dorsiflexion
ankle angle depends on the velocity of the gait, which in the used gait data was 6.9 degrees. State 3,
pre swing, starts with the maximum dorsiflexion ankle angle and forefoot contact, ending when the
forefoot is no longer in contact. Since state 2 and state 3 both consider the forefoot contact, state 3 is
only triggered after the occurrence of state 2. State 4, swing, is characterized by not having contact in
both foot sensors. In Figure 3.14 is presented a schematic representation of the different events that
cause the states transition.
36
Figure 3.14: Events that cause the transition of states. Sensors enabled in red and disabled in black. The ankle angle possibilities are represented in Figure 3.12.
A safety state was considered in case of ankle angle error or sensors error. However, it is not
expected the occurrence of this type of errors during simulation. Figure 3.15 presents the evolution of
states during gait cycle in parallel with the evolution of phases.
Figure 3.15: Four states during a complete gait cycle.
It is important to notice that the corresponding gait cycle indicated in Figure 3.15 is estimated, since a
state transition can occur due to some functional error.
3.7 Simulation with Ankle Control – Results and Discussion
The simulation and control of ankle-foot models presented previously was implemented in Simulink®,
which is a simulation tool from MATLAB® (MathWorks). The multibody ankle-foot model presented in
subsection 3.2.3 also used the SimMechanics® software, which is a module of Simulink
®. Here, the
controller used the input and output variables from the multibody model.
37
3.7.1 P-D and LQR
The P-D and LQR controls were implemented in the control of the simplified and of the multibody
ankle-foot models with the goal of performing reference following, where the reference consists on the
ankle angle during the gait cycle. Both models were considered with rehabilitation purposes, where
the simplified model simulates the rehabilitation without ground contact and the multibody model
simulates the rehabilitation with ground contact, having included the SEA dynamics. The desired
performance specification in the control of both models was to have a following error inferior to three
degrees.
The design of the P-D controller requires the tuning of two parameters: damping ratio and natural
frequency. Therefore, the damping ratio was fixed to 0.7, allowing for a transient response that is
sufficiently fast and sufficiently damped. With this damping ratio, the underdamped system gets close
to the final value more rapidly than a critically damped or overdamped system. Applying the equations
presented in section 3.3 for P-D controller, with a damping ratio of 0.7 and varying the natural
frequency, it was possible to achieve a controller for reference following with an ankle angle error
inferior to three degrees. The parameters for both simplified and multibody ankle-foot models are
presented in Table 3.1. It is important to notice that in the control of the multibody ankle-foot model, it
was preferable to have different parameters depending on the system state (presented in section 3.6).
This requirement is justified by the non linearity of the system.
Table 3.1: Parameters of P-D controller for reference tracking of simplified and multibody ankle-foot models.
Parameters Simplified Model Multibody Model
(states 1 and 4)
Multibody Model
(states 2 and 3)
Damping ratio - 0.7 0.7 0.7
Natural frequency - (rad/s) 125 160 360
Proportional gain – 187.5 307.2 1555
Derivative gain – 2.1 2.688 6.048
The LQR design is based on the weighting of three parameters, the two state vectors and control
action. Considering the equations presented in section 3.4 and an iterative procedure, several values
for the parameters were tested, noting that for small ankle angle error, the parameter should be
sufficiently large compared to and . The parameters that led to an ankle angle error inferior to
three degrees are presented in Table 3.2. As in P-D controller, the LQR also has different parameters
depending on the system state when considering the multibody model.
Table 3.2: Parameters of LQR controller for reference tracking of simplified and multibody ankle-foot models.
Parameters Simplified Model Multibody Model
(states 1 and 4)
Multibody Model
(states 2 and 3)
State variable x1 weight - 6 x 107
8 x 107 1 x 10
10
State variable x2 weight - 5 x 102 5 x 10
2 5 x 10
2
Control action weight - 6 x 102 3 x 10
2 3 x 10
3
Damping ratio - 0.745 0.753 0.708
Natural frequency - (rad/s) 162 207 390
38
It is possible to observe in both previous tables that the parameters of both controllers present a
higher natural frequency for the multibody ankle-foot model. This fact can be justified by the higher
control effort required for controlling the multibody model, since it is a model that considers more
dynamics, including the contact of the foot with the ground.
Figure 3.16 presents the ankle angle during the GC when controlling the multibody ankle-foot model
with both P-D and LQR controllers. It should be noted that the ankle angle from the simulated model
corresponds to the reference angle.
Figure 3.16: Ankle angle of simulated and controlled multibody ankle-foot model during gait cycle.
In general, the reference tracking was acceptable, not showing abrupt changes during the GC, even
with the changing of parameters during the system states.
The peaks in the tracking error occur around 54% and 68% of GC, with values near to the three
degrees (Figure 3.17). In terms of biomechanical movements, these peaks are acceptable since it fits
in the variability of ankle angle during gait.
Figure 3.17: Ankle angle tracking error during gait cycle of controlled multibody ankle-foot mode.
Although the value of the angle error can have biomechanical acceptance, when analyzing the ankle
torque of the controlled models, it is visible that the peak ankle torque was not achieved (Figure 3.18).
The ankle torque deficit is visible in the interval from 35% to 55% of GC, which includes the peak
torque. In a realistic situation, this ankle torque deficit could cause some unrestrained tibial
advancement and failure in propelling the body forward. Little discontinuities are also visible around
22% and 60% of the GC in the ankle torque, which are caused by the changing in the parameters of
the controllers. The use of same parameters for the entire GC was tested, however, the control action
at states 1 and 4 was very oscillatory.
0 10 20 30 40 50 60 70 80 90 100-25
-20
-15
-10
-5
0
5
10
Gait Cycle (%)
Ankle
Angle
(degre
es)
Simulated Model
P-D Controller
LQR Controller
0 10 20 30 40 50 60 70 80 90 100-5
0
5
Gait Cycle (%)
Ankle
Angle
Err
or
(degre
es)
P-D Controller
LQR Controller
39
Figure 3.18: Ankle torque during gait cycle of controlled multibody ankle-foot model with P-D and LQR control strategies.
The results for the control of the simplified ankle-foot model, presented in Appendix B.1, show that the
reference following could be achieved without changing the parameters during the gait cycle. The
controller presented a following error inferior to three degrees, presenting lower parameters when
comparing to the parameters used in the control of multibody ankle-foot model.
The response to a unit step of a close-loop system allows characterizing the transitory and stable
phases of that system. In this particular situation, the input of the system was a unitary change in the
ankle angle, for which the controller needs to provide a control action so the output of the system, the
ankle angle, reaches the desired value of the input.
The unit step response of closed-loop system using the P-D controller is presented in Figure 3.19. As
expected, the simplified model presents the slower and less oscillatory response, since it controls a
less demanding model. In the multibody model, states 1 and 4 require much less effort from the
controller than states 2 and 3. Thus, the unit step response of system when in states 1 or 4 is slower
and less oscillatory. Requiring a faster response for states 2 and 3, the natural frequency of the
system was increased. However, lowering the rise time leads to an increase of overshoot, which in the
system causes some instability on the control action. In fact, this was the reason for considering
different parameters for the controller during the different states, which depended on the control effort.
Figure 3.19: Response to a unit step input of closed-loop system for the P-D controller.
The unit step response of closed-loop using LQR controller is presented in Figure 3.20. Since this
controller uses two state vectors for controlling the ankle angle, it is common to present the responses
for the state vectors. Identical analysis can be made for the response of state vector x1 as the ones
made for the P-D controller, since it corresponds to the ankle angle. It is visible that the response
speed and oscillations increase with the control effort. Analyzing the response for the state vector x2,
0 10 20 30 40 50 60 70 80 90 100-20
0
20
40
60
80
100
Gait Cycle (%)
Ankle
Torq
ue (
Nm
)
Simulated Model
P-D Controller
LQR Controller
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plit
ude
mathematical model
multibody model - states 1,4
multibody model - states 2,3
40
it is visible that the closed-loop with parameters for controlling the states 2 and 3 reaches almost the
double peak velocity than with the others parameters.
In a physical implementation of these controllers, the chosen parameters may not lead to the same
type of responses, since it depends on the characteristics of the physical actuator in use.
Figure 3.20: Response to a unit step input of closed-loop system for the LQR controller.
Overall, both controllers presented acceptable reference following, which for rehabilitation purposes is
acceptable. In the control of the multibody model, the discrepancy in the ankle torque is related with
the multibody ankle-foot model. A small lag is enough for the foot to reach certain ankle angles without
the necessary ankle torque, since the knee has the kinematics prescribed and is moving forward.
Thus, it is not necessary to provide the correct ankle torque to move the body forward.
As expected, both controllers presented similar behavior since LQR controller acts like a P-D
controller. The controllers proved to be capable of following the ankle angle during gait cycle for both
simplified and multibody ankle-foot models. However, by assuming a certain following error, the ankle
torque presented a large deficit during the stance period. In a realistic situation, this ankle torque
deficit could cause unrestrained tibial advancement or lack of body forward propelling. In the case of
rehabilitation without ground contact, these controllers are a good solution.
3.7.2 Impedance Control
Impedance control was implemented in the multibody ankle-foot model with the goal of assisting the
ankle movement during gait by mimicking the human control of the ankle joint. Impedance control
uses a target reference in the control law in which the system stiffness and damping are related. Since
there is no reference following, this controller requires a trigger in order to update the target reference,
considering that these triggers cannot rely solely on the ankle angle. Therefore, the impedance control
was only implemented with the multibody ankle-foot model, where the contact with the ground could
provide the triggers for the different states of the gait cycle (GC).
In the implementation of impedance control, it was necessary the tuning of four parameters: ankle
angle target, ankle angular velocity target, ankle stiffness, and ankle damping. Considering the states
triggers presented in section 3.6, the ankle angle target was set by the desired ankle angle in the end
of the considered state, i.e., if in the end of state 2 (weight acceptance) the ankle is expected to be the
neutral position, then zero degrees will be the ankle angle target. The same procedure was applied to
the ankle angular velocity target. The other two parameters were set by trial and error, in order to
achieve the following goals:
State 1: Controlled plantar flexion to avoid foot slap;
State 2: Avoid unrestrained tibial advancement and initiate the propulsion;
State 3: Provide final propulsion, moving to peak angle plantar flexion;
0 0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plit
ude
State x1 (mathematical model)
State x1
(multibody model - states 1,4)
State x1
(multibody model - states 2,3)
0 0.01 0.02 0.03 0.04 0.05-50
0
50
100
150
200
Time (s)
Am
plit
ude
State x2 (mathematical model)
State x2
(multibody model - states 1,4)
State x2
(multibody model - states 2,3)
41
State 4: Provide toe clearance to avoid toe drag and finalize the swing period with the ankle
smoothly dorsiflexed;
An alternative method for determining the controller parameters during swing state is presented in the
Appendix B.1. The method consists of modeling the ankle angle as a linear rotary, second-order
model, under-damped system with an initial position offset, by extracting the characteristics from the
ankle angle trajectory during the swing. This method provided acceptable parameters for controlling
the swing period in the simplified ankle-foot model. However, different parameters were necessary for
controlling the multibody ankle-foot model.
Using an iterative procedure to find the different parameters for each state, it was achieved a set of
parameters that meet the generality of goals. The parameters used in the impedance control are
presented in Table 3.3.
Table 3.3: Parameters and characteristics of implemented impedance control of multibody ankle-foot model during GC.
States Loading Response
( 1 )
Stance
( 2 )
Pre Swing
( 3 )
Swing
( 4 )
Rotational Damping - B (Nms/rad) 1.2 10 3 1.5
Rotational Stiffness - K (Nm/rad) 60 320 180 30
Target ankle angle - (deg) 0 0 -20 2
Target ankle angular velocity - (deg/s) 50 0 -100 0
Natural frequency - (rad/s) 70.7 163.3 122.5 50
Damping ratio - 0.70 2.55 1.02 1.25
Being the GC a cyclic process, it is expected that the final ankle angle of the GC approximates to the
initial ankle angle of the proceeding GC. When an individual starts to walk, the initial ankle angle of
GC is different than the remaining and yet, there is variability in the ankle angle during continuous
walking. For that reason, Figure 3.21 represents the ankle angle from gait data and from controlled
multibody ankle-foot model during an entire GC (0-100%) and the initial 20% of another GC. For the
simulation initialization, initial conditions were set on the model, expecting the controller to deal with
two different conditions at heel contact time. Observing the results, the both controllers were capable
of maintained the cyclic behavior by presenting a similar output after heel contact. Thus, the following
results will only refer to a normal GC (0-100%).
Since the controller parameters vary during the GC according to the different states, the transition
between states causes discontinuities. These discontinuities cause peak control actions, which may
lead to the system instability and also to the damage of its physical components. For that reason, two
impedance control methods were implemented, differing in the way parameters change between
states.
In the first method, impedance control 1, the controller parameters change instantly to the parameters
of the next state, while in the second, impedance control 2, the controller parameters change in a
continuous way. The continuous transition is made by increasing or decreasing the starting parameter
value by 2 to 5 % over time until reaching the target parameter value. Since impedance control 2
presents better results, more focus was given to these results.
42
Figure 3.21: Ankle angle during gait cycle from gait data and controlled multibody ankle-foot model with impedance control. Parameters of controller are presented in Table 3.3.
Observing the ankle angle from the model controlled with impedance control 1 in Figure 3.21, it is
clearly visible the states transitions, where abrupt changes in the ankle angle occur. The transition
from the second to the third states is the most problematic, where the large change in the parameters
causes an instantaneous increase in the ankle torque of around 40 Nm. The ground reaction forces
and ankle torque during GC with impedance control 1 are presented in Appendix B.2, Figure B.4.
A smooth transition between states was achieved in general with the impedance control 2 (Figure
3.21), since the controller parameters vary smoothly from one state to another.
After initial contact, the system presents a small lag when comparing with the ankle angle from data,
with the large ankle angle error occurring around the 4% GC (Figure 3.22), which reduces to zero
around 8% of the GC. The lag is justified by the position of the heel contact sensor, causing a
posterior heel contact of the multibody ankle-foot model when comparing with the gait data (contact on
the heel sensor (Figure 3.23)). This lag causes a small ankle peak torque at the beginning of GC
(Figure 3.24).
Figure 3.22: Ankle angle error between the ankle angle from gait data and from controlled model with impedance control 2 during gait cycle.
The transition from state 1 to state 2 occurs with the heel rise (no contact in the heel sensor (Figure
3.23)), which occurs sooner than in the simulated model. In this transition, it is possible to observe a
continuous error in the ankle angle due to higher ankle torques when comparing with the simulated
model. During this state the ankle stiffness has a large increase, since it requires large ankle torque
per small ankle angle. This requirement causes the excessive ankle torque in the first two thirds of the
time of this state, when compared to the simulated model. The fact of not having an impedance control
with external force measurement results in having ankle torques superior to the ones expected.
0 10 20 30 40 50 60 70 80 90 100-10
-5
0
5
Gait Cycle (%)
Ankle
Angle
Err
or
(degre
es)
43
Figure 3.23: Events triggers during gait cycle for the simulated model and controlled model – Heel and Forefoot sensors activation.
With forefoot contact and the ankle angle equal to seven degrees, it starts the transition from state 2 to
state 3. This transition and the associate state 3, present the biggest discrepancy with respect to the
ankle torque. The large parameters transition causes an instantaneous increase in the ankle torque of
around 20 Nm (Figure 3.24), leading to some instability of the ankle movement. Considering a realistic
situation, this instability could restrict the rotation of the tibia about the ankle joint, since it has a
prolonged dorsiflexion. The peak ankle torque during GC was also not achieved, which in a realistic
situation could cause some unrestrained tibial advancement and affect the propelling of body forward.
With the foot rotation for plantar flexion in the time of double limb support, the ankle torque and angle
matches, with the toe off occurring at the same time (no contact in the forefoot sensor (Figure 3.23)).
In the final state the foot becomes airborne, leading to an abrupt change in the controller parameters.
Starting with a negative peak torque in order to cancel the clockwise movement of the ankle at that
instant, the ankle error reaches in this state its maximum value during GC (Figure 3.22), due to the
fast dorsiflexion movement. However, the toe clearance was achieved and in the final of swing state,
the ankle was slightly dorsiflexed.
Figure 3.24: Dynamics of simulated model and controlled model with impedance control 2 during gait cycle: (a) ankle torque, (b) horizontal shear force, (c) vertical force.
0 10 20 30 40 50 60 70 80 90 100
0
0.5
1
Gait Cycle (%)
Conta
ct
1=
yes 0
=no
Heel Sensor (Sim. Model)
Forefoot Sensor (Sim. Model)
Heel Sensor (IC Model)
Forefoot Sensor (IC Model)
44
Considering a second-order model approach, the parameters used in the multibody ankle-foot model
were applied on the closed-loop system with the simplified ankle-foot model. Thus, for the different
state parameters presented in Table 3.3, the closed-loop transfer function was subjected to a unit step
input, with the responses presented in Figure 3.25.
Figure 3.25: Responses to a unit step input of a closed-loop system with simplified ankle-foot model and impedance control.
In a basic second-order system, a damping ratio superior to the unity corresponds to an overdamped
system, verifying no overshoot on the step response. However, in this particular system, the controller
adds a zero to the system which causes the overshoot in the step response (Figure 3.25), even for
damping ratios superior to the unity. This event, that is present when the system is in the second,
third, and fourth states, allows a decrease in either the rise time and in the overshoot, increasing the
control possibilities.
As expected, smaller rising times correspond to the states where the stiffness parameter is higher.
State 2 corresponds to the state with large control effort, presenting the lower rise time and overshoot,
followed by the state 3. Lower control effort is required by states 1 and 4. It was expected that state 4
had the lower control effort since it has the ankle joint rotating without ground contact, however, the
parameters found for state 1 required less control effort. Thus, state 1 response exhibits a large
overshoot when compared to the response of state 4, since it has a large stiffness and lower damping.
In a physical implementation of this controller, the chosen parameters may not lead to the same type
of responses, since it depends on the characteristics of the physical actuator in use. In a general
comparison between the gains of the three control strategies, the impedance control strategy required
less control effort. However, the ankle angle error in impedance control was substantially larger than in
the other two control strategies.
Overall, the impedance control strategy presented good results in the control of multibody ankle-foot
model. Disregarding some discontinuities in the states transitions, the controller proved to be capable
of avoid foot slap, provide a controlled tibial advancement and propel the body forward. Although a
large ankle angle error occurred in the initial swing phase, the toe drag was avoided and the swing
period ended with the foot slightly dorsiflexed.
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plit
ude
State 1
State 2
State 3
State 4
45
Chapter 4
4 Active Ankle-Foot Orthosis (AAFO)
A full assistance during the different gait phases cannot be provided by a common ankle-foot orthosis
(AFO). These common orthoses are passive systems, acting like energy storing systems, which
generally take the foot back to its neutral position. Aiming for a system that is capable of assisting the
ankle movement during the gait cycle led to the development of an active ankle-foot orthosis (AAFO).
This chapter presents the several design assumptions that were necessary in the development of an
AAFO for assisting the ankle movement during the different phases of the gait cycle. A good
understanding of ankle dynamics is crucial for a correct development of the AAFO. Therefore, overall
ankle kinematics and kinetics requirements are presented in section 4.1. Section 4.2 presents several
configurations evaluated for the AAFO. The considered AAFO configuration is presented in section
4.3, where the different components are described.
4.1 Kinematics and Kinetics Requirements
When choosing the different components for an active ankle-foot orthosis (AAFO), it is important to
have a full knowledge of the kinematics and kinetics involved in the ankle movement.
As project requirements, it was considered that the AAFO would assist the ankle movement in all
phases of the gait cycle of an individual with a total mass of 70 kg. Therefore, the actuating unit needs
to have enough power to assist the patient in all the different phases of the gait.
The data relative to ankle kinematics and kinetics were obtained from gait tables (Winter, 1991). The
ankle data include angular displacement, angular velocities, angular accelerations, moments of force,
and power requirements of ankle joint during gait cycle. Kinematic data is based on the convention
presented in Figure 2.7b, while kinetic data is based on the convention presented in Figure 2.8a. The
considered data consists on averaged inter-subject joint values obtained for the referred quantities,
with all results clustered into three cadence groups: slow, natural, and fast. Data relative to joints
moments of force and power patterns are also normalized by body mass (Winter, 1991). The
respective charts of these quantities are available in appendix A.1.
From the analysis of these data, it was possible to achieve a list of requirements to fulfill, considering
that the device will be assisting natural cadence gait. Table 4.1 presents the maximum amplitudes
registered for the different quantities of the ankle gait data, in normal cadence, for the referred
individual. It is important to notice that peak ankle power does not occur either at peak angular velocity
or at peak ankle torque.
46
Table 4.1: Ankle maximum negative and positive values for different quantities of gait data, in normal cadence, of an individual with a total mass of 70kg (Winter, 1991).
Quantity Maximum negative Maximum positive
Ankle Angle - (deg) -19.77 9.62
Ankle Angular Velocity - (rad.s-1
) -3.75 2.46
Ankle Angular Acceleration - (rad.s-2
) -33.08 66.07
Ankle Torque - (Nm) -4.48 113.96
Ankle Power - (W) -35.43 228.51
4.2 AAFO Design
As design requirement, it was defined that the AFO would be actuated by a system based on linear
motion. There are several mechanisms that convert linear into rotational motion with a certain gear
ratio. Figure 4.1 presents different mechanisms that were considered for converting linear into
rotational motion of the AFO.
Figure 4.1: Representation of different concepts for converting linear to rotational movement of the AFO: (a) detail about scissor system with one level, (b) scissor system with one level and a parallel spring, (c) scissor
system with two levels, (d) back of the AFO used as a level arm.
From the different possibilities presented in Figure 4.1, the solution that uses the back of the AFO as a
level arm was adopted. This solution requires as represented in Figure 4.1d less moving parts and has
the advantage of being in the back of the leg, enjoying the space provided by the shape of the leg. It
also decreases the possibility of damaging the system, since the external side of the leg is generally
more subjected to collisions. Therefore, it was necessary to determine the corresponding linear forces,
displacements, velocities, and accelerations in order to know the requirements the actuator should
fulfill.
Figure 4.2: Representation of the orthosis with the respective force arm.
47
Considering the actuator on the back of the leg, positive forces correspond to plantar flexion
movement, while negative forces to dorsiflexion movement, i.e., positive force will cause a negative
moment, in order to equal the positive moment due to floor reaction. It is also visible that plantar
flexion movements (negative ankle angles) will correspond to positive linear displacements.
It was also taken into account the fact that the arm does not have a constant value, because the line
of action of the actuator is practically vertical and the point of force application rotates about the ankle
axis. The corresponding linear quantities were calculated considering that the arm between the ankle
axis and point of force application, , had 0.08m of length, with the results presented in Table 4.2.
Table 4.2: Linear quantities required for an actuator to assist the ankle movement, based on data from Table 4.1.
Quantity Maximum negative Maximum positive
Linear Displacement - (mm) -13.37 27.06
Linear Velocity - (mm.s-1
) -192.82 296.68
Linear Acceleration - (mm.s-2
) -4954.82 2622.16
Linear Force - (N) -56.13 1441.08
Power - (W) -35.43 228.51
The required kinematic for the actuator to fulfill were obtained by trigonometric relations. The required
actuator displacement is
(4.1)
with corresponding linear velocity
(4.2)
and linear acceleration
(4.3)
Kinetic quantities were also obtained through trigonometric relations, where the required actuator force
given by
(4.4)
The actuator power can also be given by relating linear force and linear velocity, which must match we
the power obtained by relating ankle torque and angular velocity,
(4.5)
Finally, the relation between the screw shaft lead, , i.e., the distance the nut moves parallel to the
screw axis when the screw is given one turn (Shigley, et al., 2004 p. 396), screw shaft angular
velocity, , and actuator linear velocity is given by
(4.6)
Thus, with equations 4.6 and 4.2, it is possible to make the comparison of linear velocity required for
the ankle movement and the linear velocity that linear actuator needs to provide, considering the
screw shaft angular velocity. Further details about the actuator gear ratios are presented in subsection
5.2.2.2.
48
4.3 Components of the AAFO
Having defined in the previous section how the AAFO would be actuated, this section exposes the
chosen physical components that allow the control and autonomy of the AAFO.
The basic component of the AAFO is the AFO, where all components are coupled. It is expected from
the AFO that the dynamics of the system is not amended. For such, the AFO must be sufficiently rigid
for not deforming in the areas where the actuator is fixed, and allow a free motion of the ankle joint
inside the normal range of motion, which led to the choice of a standard AFO. For the measurement of
ankle angle it was chosen a rotary potentiometer, while foot switches were the most suitable choice as
heel and forefoot contact sensors. To provide the motion, it was chosen a series elastic actuator
(SEA), where the compliance of the springs allows an indirect force measurement.
Since this device is a prototype, the power supply and controlling system are not integrated with the
AFO. For enjoying autonomy, the power supply and controlling system are to be placed in a backpack
for easier transportation of the individual wearing the AAFO.
The various components of AAFO are shown in Figure 4.3, disregarding the power supply and the
controlling system.
Figure 4.3: Active ankle-foot orthosis 3D model: (a) left view, (b) right view, (c) basic ankle movement descriptions, dorsiflexed, neutral, and plantar flexed.
Following subsections provide a detailed description of the several components that compose the
AAFO, which has an expected total mass of 1,3kg.
4.3.1 Ankle-Foot Orthosis (AFO)
The AFO chosen for the system was a standard polypropylene AFO, with approximately 5 millimeters
of thickness, and articulated aluminum alloy joints as represented in Figure 4.4. These joints allow
motion in the sagittal plane and restrict the motion in the others planes. The motion in the sagittal
plane is limited to the ankle angle range in that plane for normal gait, acting as a safety locking device
in other situations.
49
Figure 4.4: Standard polypropylene AFO with aluminum joints.
Some modifications on the AFO are required for the adaptation of the ankle angle sensor and fixation
of the SEA, like presented in Figure 4.3. The total mass of the AFO is approximately 490g.
4.3.2 Sensors and Actuator
4.3.2.1 Ankle Angle Sensor
The most common method of measuring the range of motion in human joints is the goniometry
(Montgomery, et al., 2003 p. 179). These measures can be performed by using an electrogoniometer,
which consists of a rotary potentiometer with arms fixed to the shaft and base for attachment to the
body segments juxtaposed to the joint of interest (Peterson, et al., 2008 pp. 5-4). A rotary
potentiometer is a variable resistor, in which turning the central spindle produces a change in electrical
resistance to be measured by an external circuit. These devices offer the advantages of real-time
display and rapid collection of single joint information. The selected rotary potentiometer was a Bourns
6639S-1-502 5 kΩ (Figure 4.5).
Figure 4.5: Bourns 6639S-1-502 5 kΩ rotary potentiometer (RS Components Ltd, 2010, article no. 164-2661).
To ensure accurate measures of the ankle angle between the leg and the foot (ankle angle), the
potentiometer rotation axis must agree with the AFO joint rotation axis (Figure 4.3). The mass of this
sensor is approximately 20g.
4.3.2.2 Footswitches
Footswitches, often called by event switches, are generally used for acquiring the timing of gait. The
data from footswitches allows determining the time duration of the stance period, the stride period, and
also the transition between phases. The event switch sensor presented in Figure 4.6 has only 1mm
thick and a negligible weight of 1g, which do not cause any disturbance to the gait, which allows the
use of several sensors.
50
Figure 4.6: Flexible event switch sensor (Motion Lab Systems, 2009).
Generally, these sensors have short life time, but by being relatively inexpensive make’s them a
suitable choice.
A simple footswitch can be made from two layers of metal mesh, separated by a thin sheet of plastic
foam with a hole in it. When pressure is applied, the sheets of mesh contact each other through the
hole and complete an electrical circuit (Whittle, 2007 p. 147).
The placement of the foot sensors is crucial for detecting correctly the desired states transition. In the
modeling of the ankle-foot complex (subsection 3.2.3), the placement of the heel and forefoot contact
points was made with the goal of detecting major events as heel strike, heel rise, forefoot contact, and
toe off. It is visible in Figure 4.7 that heel, forefoot and in particular, the toe, are the areas with more
functional relevance during stance, justifying the position for the foot contact sensors.
Figure 4.7: Foot pressure versus time, where the hot colors represent high pressure areas (Tekscan, 2009).
4.3.2.3 Series Elastic Actuator (SEA)
Assisting the ankle movement during all phases of gait requires the selection of an actuator with high
output power, where size and weight cannot be neglected. An additional requirement for the
equipment is the capacity of dealing with shock loads and work in parallel with a human. Series elastic
actuator fulfills all these requirements by having a high power density, low weight, and capacity of
absorbing some external forces without damaging the motor due to its low impedance. This actuator,
presented in Figure 4.8, consists of a brushless DC motor coupled laterally to a ball screw shaft by a
gear drive. The nut of the ball screw shaft is connected to a set of springs placed in series with carbon
bars that transmit the forces to the output. These springs are responsible for the low impedance of the
actuator. The mass of the SEA is approximately 0.75kg, and a detailed description of this system is
presented in chapter 5.
51
Figure 4.8: Series elastic actuator designed in this work.
4.3.3 Input-Output devices
Communicating with the sensors and controlling the actuator in a physical prototype of the AAFO
requires the use of a data acquisition unit (DAQ) and a controller for the rotary motor. The DAQ unit
establishes the communication between the sensors, actuator, and the overall processing unit, the
laptop. The motor controller, which is detailed in subsection 5.2.2.4, page 60, is a dedicate controller
for the rotary DC motor and also communicates with the main controller by the DAQ unit.
The chosen DAQ unit for this purpose is presented in Figure 4.9, which is a basic DAQ unit with 8
analog inputs, 2 analog outputs, 12 digital I/O, a 32-bit counter, and a USB connection. The mass of
this device is approximately 80g.
Figure 4.9: National Instruments USB-6009, 14-Bit, 48kS/s Low-Cost Multifunction DAQ (National Instruments, 2010).
4.3.4 Power Supply
Aiming for an autonomous system, the energy to provide the system cannot be external. Two
elements in the system require the majority of the energy: the laptop and the actuator. As laptop has
its own battery, it was necessary to choose a battery to provide energy to the actuator. The
considerations which were taken in account while choosing the battery were voltage, discharge
current, energy capacity, size, and weight.
52
First two parameters are imposed by the DC motor actuator, a maxon EC-305015 as presented in
subsection 5.2.2.1, requiring a minimum supply voltage of 48 volts (V) and a minimum current of 10
amperes (A) in order to achieve motor’s peak power (Table 5.1).
A published study (Tudor-Locke, et al., 2008) presented a zone-based hierarchy about daily steps in
adults, where adults were considered active when performing an average of daily steps between
10,000 and 12,499. Considering that the AAFO is used on a daily basis, a regular individual with a
normal cadence of 105.3 steps per minute (Winter, 1991), takes near two hours to perform 12,000
steps. For the same cadence, and for an individual mass of 70kg, the energy requirements during gait
cycle for the ankle joint have been computed to be approximately 16.5J, by integrating the power
curve shown in Figure A.1e, of appendix A. This way, 12,000 steps are expected to require an energy
total of 198kJ.
Current advances in the field of batteries have provided lighter and smaller batteries, not forgetting
their energy density. For this application the choice fell on a lithium polymer (LiPo) battery, Figure
4.10, a type of rechargeable battery with high energy density. It has many applications in radio-
controlled cars and aircrafts due to its high performance.
Figure 4.10: Kokam lithium polymer battery pack (model: H5 3600mAh, 18.5V, 30C).
Since the great majority of LiPo batteries available in market have voltages lower than 20V, it was
necessary to consider the connection of batteries packs in serial connection. In the selected battery
each pack contains five cells. Its technical description is presented in Table 4.3.
Table 4.3: Technical description of Kokam lithium polymer batteries packs (model: H5 3600mAh, 18.5V, 30C).
Technical Description Single pack Three packs in
serial connection
Voltage (V) 18.5 55.5
Form Pack of 5 cells 3 Packs of 5 cells
Mass (kg) 0.528 1.584
Maximum discharging current (A) 108 108
Energy Capacity (kJ) 207.36
622.08
Length x Height x Width (mm) 135 x 42 x 44 135 x 126 x 44
Mass (kg) 0.528 1.584
With the three pack configuration, the total energy capacity, 622.08kJ, has around three times the
required energy for 12,000 steps in normal cadence. This extra available energy can cover the time
when the patient still moves the foot but doesn’t walk, or energy losses due to cable heating. Another
reason for selecting batteries with high energy capacity was due to the fact that these batteries lose
some of their power supply capacity with the available energy, and so, it is ensured in the prototype
phase that batteries will not compromise the AAFO performance.
53
4.3.5 Autonomous AAFO
The main goal of this work is presented in Figure 4.11, an autonomous AAFO capable of assisting an
individual during gait.
As an in silico prototype, the proposed AAFO has the sensor-actuating unit in the leg-ankle-foot, and
the controlling unit, with the respective power supply in a backpack. It is expected that the sensor-
actuating unit of the AAFO will have approximately 1.3kg of mass. The rest of the unit includes wires,
batteries, controller, DAQ, and processing unit. This unit is expected to have approximately 6kg, by
considering a processing unit (laptop) with 2.5kg of mass, wires with a total mass of 1.5kg, batteries
with a total mass of 1.6kg, controller with 0,25kg of mass, and the DAQ with 0,08kg of mass.
It is important to notice that three quarters of the weight in the backpack is due to batteries, laptop
and cables. In a future system, a dedicated control unit for the entire system will be used, allowing that
the all control can systems can be fitted to the AFO, including the batteries.
Figure 4.11: Autonomous AAFO with the individual carrying the batteries and processing unit in a backpack.
54
55
Chapter 5
5 Actuating Unit
The demanding task of providing power to the ankle joint during gait requires an actuating unit with
high power density. The fact of being coupled to an ankle-foot orthosis (AFO) and thus to the human
leg, implies the actuating unit to be the smallest and the lightest possible. The main idea is to have a
system capable of supplying enough energy to provide the ankle movement during gait, without
neglecting the size and weight.
A good knowledge of the ankle requirements is essential for a successful choice of the actuating
system, and therefore, the execution of a task. For this reason, this chapter presents the guidelines for
choosing an adequate actuating unit to assist the ankle movement. Section 4.1 of previous chapter
presented the kinematics and kinetics requirements which the actuating units considered in section
5.1 have to fulfill. Choosing a series elastic actuator (SEA) for endowing the AFO, required the
redesign of the considered SEA with the goal of a more compact and powerful SEA. All the redesign
process is described in section 5.2.
5.1 Actuators and Coupling Mechanisms
An actuator is a system that establishes a flow of energy between an input port and an output port.
Generally, input port is electrical and the output port is mechanical, transducing some sort of input
power into mechanical power (Pons, 2005 p. 2).
Actuators are essentially of electrical, electromechanical, electromagnetic, pneumatic, or hydraulic
type. Normally, these are used with a power supply and a coupling mechanism (Figure 5.1). The
power unit provides either AC or DC power at the rated voltage and current, while the coupling system
acts as the interface between the actuator and the physical system. These interface systems include
belt drive, lead screw and nut, rack and pinion, gear drive, piston, and linkages (Bishop, 2008 pp. 17-
11).
Figure 5.1: A typical actuating unit (Bishop, 2008).
In the selection of the actuator to assist the AFO, two options were considered: choose an existing
actuator in the market or develop an actuator for this purpose. Since the actuator’s market has a huge
variety of different types of actuators, the first option was initially chosen, searching for rotary,
hydraulic, pneumatic and linear actuators.
56
Rotary actuators (Figure 5.2a) are the most common actuators, being frequently the basis of other
types of actuators when coupled with different mechanisms. The purpose of coupling mechanisms is
to increase the force and power density of the actuator, or also to convert rotation to longitudinal
movement. While mathematically the coupling mechanism works wonderfully, physically it presents
problems such as backlash, increased dynamic mass, and increased output impedance (Robinson,
2000 p. 34).
Linear actuators consist of rotary motors coupled to a screw shaft, so that rotation can be converted
into longitudinal movement. The coupling of the two elements is generally held by gear drives, belts or
joints coupling when the axis of rotation matches. Figure 5.2b presents a common linear actuator
where the rotary motor is coupled in parallel to the screw shaft. A more advanced linear actuator is
presented in Figure 5.2c, the series elastic actuator (SEA), which is a linear actuator in series with a
set of springs that offers the system low impedance. In the case of SEA, the screw shaft acts also as
motor rotor, avoiding the use of coupling mechanisms.
Figure 5.2: (a) BS23 High Performance rotary motor (Moog, 2010), (b) Linear ET032 actuator (Parker, 2009), (c) Series Elastic Actuator (Robinson, 2000).
When comparing the performance of the linear ET 032 actuator, with the SEA, it is clearly the high
performance that the SEA offers. The ET 032 actuator presents a maximum velocity of 396mm/s,
maximum force of 600N, and a mass of 1.3kg, while the SEA presents a maximum velocity of
270mm/s, maximum force of 1300N, and a mass of 1.1kg. The SEA also has the advantage of having
low impedance, which protects the motor from shock loads, and presents a more stable behavior
when used in parallel with humans.
Some of the most common rotational power transmission systems are belt, cable, and drive chain
mechanisms. Although being relatively inexpensive, some of these systems offer generally low
precision, repeatability, or safety. Rack and pinion gear systems can provide power transmission with
high accuracy (Figure 5.3a). However, the accuracy decreases with the wearing, being these systems
generally subjected to stiction and backlash. The most common of linear coupling systems are lead
and ball screws (Figure 5.3b). Lead screws are generally more inexpensive, however, more heavy and
more often used for tasks of high forces. Ball screws provide high precision and low rotational inertia
and friction. To have accurate and low losses in power transmission, it is required some maintenance
of the system, mainly for tuning (Figure 5.3c).
Figure 5.3: (a) Linear actuator with rotary motor coupled to a screw shaft by a gear drive (Parker, 2009), (b) SKF lead screw (SKF), (c) Linear actuator with rotary motor coupled to a screw shaft by a belt drive (Parker, 2009).
57
Hydraulic and pneumatic actuators are differentiated in the type of fluid, as air is generally found on
pneumatic devices and oil on the hydraulic ones (Ogata, 2002 p. 158). These systems are generally
associated to high forces and large motion at the output. However, the weight and safety of these
systems is the great disadvantage for using them in the AFO. The lack of safety is justified by the
possibility of rupture in the fluid line. Even so, pneumatic actuators have been used in AFO for
rehabilitation purposes, limiting the use of the system to the indoor due to the compressed air supply.
An example is the powered ankle exoskeleton developed at The University of Michigan (Kao, 2009),
which had a total mass of 1.6 kg (Figure 2.14, page 18)
Taking in account the performed survey, the SEA was considered as the device more suitable for
assisting the movement of the ankle joint. However, the SEA was still heavier than the desired, taking
in account that it did not met the kinetic and kinematic requirements. For this reason, it was decided to
improve the SEA concept, by redesigning it with the goal of fitting the AAFO.
5.2 Series Elastic Actuator (SEA) Design
A linear series elastic actuator (SEA) is an actuator that has an elastic element in series with the motor
and the ball screw. A sensor measures the displacement of the elastic element and force is generated
by Hooke’s law. By placing a spring in series with the output of an electric motor, the force control
performance is improved. The motor is isolated from shock loads, and the effects of torque ripple,
friction, and backlash are filtered by the elastic element (Williamson, 1995 p. 77).
5.2.1 Components Selection
The design possibilities for a SEA are very large. Besides topology and geometry, there are six major
components to take in account for the design: motor, amplifier, transmission, stiffness, sensor, and
controller.
Figure 5.4: Series Elastic Actuator three dimensional model.
5.2.2 Motor and Coupling Mechanism
5.2.2.1 DC Motor and Encoder
58
Brushless rotary motors, with electronic commutation, have shown to be the most suitable motors to
develop a compact linear actuator. These have high efficiency, fast response, long life, high reliability,
no maintenance, low radio frequency interference and noise production (Bishop, 2008 pp. 17-12). The
considerations which were taken into account while choosing the motor were the power of the motor,
output speed under full load, maximum allowable torque, weight, and size. Motors from different
manufacturers were considered as depicted in Figure 5.5, being the specifications of the three motors
with best overall characteristics show in Table 5.1.
Figure 5.5: (a) BS23 High Performance rotary motor (Moog, 2010), BN23-23IP-03 rotary motor (Moog, 2010), (c) maxon EC–4pole 30-200W (maxon motor ag, 2010, article no. 305015).
Figure 5.6 presents an overall view of the implementation of a SEA with all three rotary motors. It is
important to notice that the BN23-23IP-03 rotary motor is the same motor that fits the SEA presented
in Figure 5.2c.
Figure 5.6: AFO endowed with the three considered rotary motors. (a) SEA with BN23-23IP-03 rotary motor, (b) SEA with BS23 High Performance rotary motor, (c) SEA with maxon EC–4pole 30-200W.
Table 5.1: Brushless DC motors specifications with best overall characteristics.
Motor BS23-23-HP-03 (Moog, 2010)
BN23-23IP-03 (Moog, 2010)
Maxon EC–4pole 30-200W (maxon motor ag, 2010,
article no. 305015)
Voltage (V) 48 48 48
Rated current (A) 3.3 5 4.7
Rated Power (W) 137 193 200
Rated Speed (rpm) 5151 6250 15800
Peak torque (Nm) 0.791 0.678 0.276
Torque constant (Nm/A) 0.083 0.055 0.0276
Rotor Inertia - (kg.m2) 106 x 10
-7 120 x 10
-7 33 x 10
-7
Length (mm) 61.2 87.1 64
Diameter (mm) 66 57.2 30
Mass (g) 596 542 270
Taking in account the motors specifications, all three motors can provide enough power to assist the
ankle movement during gait. This way, the choice was mainly based on the weight and size of the
59
motor, where the maxon EC-4pole motor have the best characteristics. Therefore, the motor choice
fell on the maxon EC-4pole 30-200W, due to its compact size and low weight. With the possibility of
coupling the encoder MR-Type ML 128-1000 (maxon motor ag, 2010, article no. 225780) on its back,
the position of motor output shaft can be measured accurately. Since the increase in weight and
length was low, this option was considered, with the aggregation performing a total length of 76.2mm
and 310g of mass (maxon motor ag, 2010, article no. 314176). This rotary motor can develop a peak
power of near 400W, satisfying the peak power occurred during gait cycle.
5.2.2.2 Ball Screw
A ball screw is an element used to convert rotation to longitudinal motion. It consists on a threaded rod
linked to a threaded nut by ball bearings, constrained to roll in the space formed by the threads,
reducing friction (McGraw-Hill, 2003 p. 49). This way, rotational power from motor can be converted
into linear power.
This system choice relies on its combination of speed, accuracy, efficiency, repeatability, lubrication
retention, load capacity, and compactness.
Table 5.2: Nut and Ball screws specifications provided by SKF® Ball screws catalogue (SKF). Peak torque and
peak angular velocity are obtain according to the requirements presented in Table 4.2.
Nut designation SH 6x2 R BD 8x2.5 R BD 10x2 R SH 10x3 R BD 10x4 R
Nominal screw diameter (mm) 6 8 10 10 10
Lead - (mm) 2 2.5 2 3 4
Nut mass (kg) 0.025 0.025 0.030 0.050 0.040
Screw mass (kg) 0.022 0.038 0.061 0.060 0.052
Screw inertia - (kg.m2) 8.4 x 10
-8 2.5 x 10
-7 6.24 x 10
-7 6.1 x 10
-7 4.7 x 10
-7
Peak torque (Nm) 0.46 0.57 0.46 0.69 0.92
Peak angular velocity (rpm) 8900 7120 8900 5934 4450
The relation between the screw shaft angular velocity, and the nut linear velocity, , is described
by the gear reduction between screw shaft and nut ( ):
(5.1)
where is the screw lead, i.e., the distance the nut moves parallel to the screw axis when the screw is
given one turn (Shigley, et al., 2004 p. 396).
The gear reduction parameter allows an easier mathematical relation between rotational and linear
domains. Given a desired force on the nut, , it is necessary to calculate the corresponding torque to
provide to the screw shaft, :
(5.2)
In torque calculations, it was considered that there were no losses when converting rotational to linear
motion. When choosing the ball screw, the choice was made on the ball screw with the overall low
peak angular velocity and low rotational inertia, leading to the ball screw with the nut BD 10x4 R (Figure
5.7).
60
Figure 5.7: SKF nut and ball screw shaft (model: BD 10x4 R).
5.2.2.3 Gears
To have a compact actuator, the motor was coupled laterally to the screw, requiring the use of gears
to transmit the power from the motor to the screw. This choice relies on the commitment between the
actuator efficiency and its compactness.
Considering that there were no power losses in the gear drives, the gear reduction was found by
considering the peak force during gait and the peak torque the motor can provide, which follows the
expression
(5.3)
where is the gear reduction from the motor to the nut.
Therefore, two gears were selected, one with 56mm and the other with 16mm of diameter, which are
presented in Figure 5.4.
5.2.2.4 DC Motor Controller – Amplifier
A motor controller is a device that serves to govern the operation of an electric motor. Several aspects
of the motor can be operated depending on the type of motor and also the type of controller, such as
selecting forward or reverse rotation, selecting and regulating the speed, regulating or limiting the
torque, and protecting against overloads and faults.
To ensure compatibility between the motor and the controller, only controllers recommended by the
motor manufacturer (maxon motor ag, 2010) were considered. With the interest of driving the system
in current control, the controller choice fell on the EPOS2 50/5 Positioning Controller (Figure 5.8).
Figure 5.8: Maxon motor control’s EPOS2 50/5 (maxon motor ag, 2010, article no. 347717).
Maxon motor controller EPOS2 50/5 is a small-sized, full digital smart motion control unit. The
controller can drive the EC motor due to its flexible and high efficient power stage. The sinusoidal
61
current commutation by space vector control allows driving the brushless motor with minimal torque
ripple and low noise. It features position, velocity and current control functionalities, and can be
operated through CANopen network, USB or RS232 interfaces. This device has approximately 240g
of mass.
5.2.2.5 Springs
Choosing proper spring stiffness and designing an actuator that houses the elasticity can prove
challenging since the spring travel is so large (Robinson, 2000 p. 17). The spring stiffness is related
with the mechanical impedance of the SEA, as also with the desired force output. Facing this, three
different springs were chosen, differing on the stiffness.
Figure 5.9: Set of springs with different stiffness’s.
The spring stiffness is also defined by the control parameters of the SEA, which are discussed in
subsection 5.2.5. Each of the selected spring has approximately 13g of mass.
5.2.3 Structural parts
The structural parts of the SEA consist of aluminum parts and carbon bars. The aluminum parts were
designed in Solidworks® software. The finite-element study was also made in the Solidworks
® through
the module COSMOSWorks®. The aluminum parts have all together approximately 160g of mass.
For transmitting the power and connect the aluminum parts it were used carbon bars. Extremely rigid,
light and with low expansion coefficient, these carbon bars are a good solution as structural elements,
presenting approximately 35g of mass.
5.2.4 Linear Sensor
For measuring the spring displacement, a linear potentiometer was chosen. This sensor has
approximately 10g of mass.
Figure 5.10: Linear potentiometer (RS Components Ltd, 2010).
62
5.2.5 SEA Control
The control scheme for this actuator was mainly based on the control scheme proposed in the Electro-
Magnetic SEA developed in MIT (Robinson, 2000). The main idea in series elasticity is to take a
standard actuation method with high output impedance, to add compliance at the end point and to use
feedback control to modulate the force acting through the spring. This way, the major difference
between series elasticity and other actuators with closed-loop force control lies in stiffness of the
spring.
5.2.5.1 Actuator Model
The considered actuator model can be divided into signal and power domains, being this division
indicated by a shaded line in the model representation (Figure 5.11). Signal domain mainly consists on
a deflection sensor (linear potentiometer), controller, and input signal (desired force), while power
domain consists on a motor amplifier, lumped mass and a viscous friction element (Robinson, 2000).
Figure 5.11 presents the control scheme for SEA, where the actuator model is controlled by a PD
controller.
Figure 5.11: Electro-magnetic series elastic actuator model. The lumped mass has a driving force and a viscous friction. The controller drives the lumped mass to compress the spring which gives the desired force output.
Adapted from Robinson, 2000 p. 92.
The elasticity on the actuator is provided by a linear spring, with a spring constant . Depending on
the spring stiffness, the deflection can be significant in order to cover the full force output range of the
actuator. The spring deflection is measured by a sensor which is a representation of the force, ,
acting through the spring. It is assumed that there is no hysteresis in the spring and therefore the
relationship between force and spring deflection is defined by Hooke’s law (Tipler, et al., 2004 p. 93):
(5.4)
where is the spring deflection.
The close-loop transfer function for the control of the SEA, with the model presented in Figure 5.11 is
given by
(5.5)
where is the proportional gain, is the derivative gain, is the damping term, is the lumped
mass, and is the total spring stiffness.
The lumped mass, , presented in the actuator model consists on a dynamic mass, i.e., an
equivalent linear mass, caused by the rotational inertia of screw shaft and motor rotor, where an
analogy about input and output power can be made for rotational and linear motion. Power due to
rotational motion, , governs motor and screw shaft motion, while power due to linear motion,
63
, governs nut linear motion (Tipler, et al., 2004 p. 287). Taking in account that
, and
, the dynamic mass can be attained:
(5.6)
where
is the gear reduction between the rotational and linear elements, and the rotational inertia
of the rotational element. This way, the dynamic mass contributions from the rotor, , and screw
shaft, , can be obtained:
(5.7)
(5.8)
(5.9)
Choosing the spring stiffness requires some iterative procedure. Some guidelines were proposed by
Robinson (Robinson, 2000), which were taken in consideration during this work.
5.2.6 Main Characteristics of SEA
The main characteristics of the SEA discussed in this chapter are presented in Table 5.3.
Table 5.3: Properties of the modeled actuator used in the simulation control. Some of the values are calculated from motor, and screw literature.
Parameter Value Units
Maximum Force 1515 N
Maximum Speed 0.31 m/s
Intermittent Power 476 W
Actuator Mass 0.75 kg
Dynamic Mass - 100.7 kg
Spring Constant - 400 kN / m
Damping - 2500 Ns / m
Operational bandwidth - 9.6 Hz
Natural frequency - 95.7 Hz
Damping ratio - 0.7 No units
Nominal Voltage – U 48 V
Maximum current (peak) 10 A
Gear reduction screw-nut - 1570 No units
Gear reduction motor-nut - 5495 No units
With the parameters present in Table 5.3, the closed-loop transfer function of the SEA was subjected
to a unitary step, with the response presented in Figure 5.12. As expected, the step response presents
a considerable overshoot, representing the low impedance of the system.
64
Figure 5.12: Unit step response of the closed-loop transfer function of SEA.
The implementation of the SEA in the control of the ankle joint did not affect those systems. However,
the selection of parameters for the SEA controller can be improved so the effect of the SEA in the
human body control can present better results.
The proposed design for the SEA is expected to have a total mass of 0.75kg, provide a maximum
force of 1515 N, and a maximum velocity of 0.31m/s. Although the losses in gear reductions have
been taken in account, it is expected a good performance from the designed SEA.
5.2.7 Maintenance and Durability of the SEA
To extend the durability of the SEA it is important to make a proper maintenance of the different parts
of the system.
Good lubrication of the screw is essential for the proper functioning and for its long reliability. Since the
oil lubrication is not practicable, it can be used grease. Lithium base greases, for standard
applications, are generally suitable for use from -30ºC to +110ºC. After a few full strokes, the grease
will be spread evenly over the useful threaded length of the ball screw, which will also help to protect
the screw against corrosion (SKF).
0 0.002 0.004 0.006 0.008 0.01 0.0120
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plit
ude
65
Chapter 6
6 Conclusions
In this work, an in silico active ankle-foot orthosis (AAFO) was developed for assisting individuals with
ankle pathologies.
In the development of the AAFO, a simplified and a multibody model for the ankle-foot complex was
developed with the goal of implementing control strategies to assist the ankle movement during gait.
The simplified model, by only considering inertial forces, did not provided realistic results. However,
the approximation to a second-order system allowed a mathematical characterization of the close-loop
transfer function. The multibody ankle-foot model presented a close to reality behavior, due to the use
of an elastic foot contact model, which allowed the application of the external forces due to ground
contact. However, since the model had only two contacts with the ground and the foot was a rigid
body, a big discrepancy in the horizontal shear force and vertical force was visible. Therefore, the
multibody model was approximated to the reality in terms of the ankle torque.
Three control strategies were implemented in the control of the ankle joint. P-D and LQR presented
similar results for both simplified and multibody ankle-foot models, although in the multibody model it
was deemed necessary a change in the parameters between the different gait states. The P-D and
LQR proved to be capable of following the ankle angle during the gait cycle for both simplified and
multibody ankle-foot models. However, by assuming a following error, the ankle torque presented a
large deficit during the stance period. In a realistic situation, this ankle torque deficit could cause
unrestrained tibial advancement or lack of body forward propelling. About impedance control (IC), the
control strategy was only implemented in the multibody model. Overall, impedance control strategy
presented good results in the control of the multibody ankle-foot model. Disregarding some
discontinuities in the states transitions, the controller proved to be capable of avoiding foot slap,
provide a controlled tibial advancement and propel the body forward. Although a large ankle angle
error occurred in the initial swing phase, the toe drag was avoided and the swing period ended with
the foot slightly dorsiflexed.
In a general comparison between the controller’s gains, impedance control strategy required less
control effort. However, the ankle angle error in impedance control was substantially larger than in the
other two control strategies. Overall, the control strategies presented high parameters values, which in
a physical implementation may not be possible to achieve.
The designed AAFO and SEA are expected to provide a full assistance during gait in an autonomous
way. With the selected parts for measuring the gait and provide energy, as the redesign of the SEA
that made it more compact, light and with more power output, the construction of the prototype is the
next step.
66
6.1 Future Work
The areas for future work fall into three main categories: simulation and control, AAFO design, and
SEA design.
In chapter 3 the simulation and control of ankle-foot complex was presented. It would be good to use a
more realistic foot contact model so that the dynamics through the ankle joint could be more close to
the gait tables. Thus, with full modeling and control of the locomotor unit, it would be possible to study
the functional errors of the ankle joint that affect the gait. For the ankle joint control, an impedance
controller with external force measurement would avoid having ankle torques superior to the expected
in gait, and thus, safeguard the human body.
The design of an AAFO was presented in chapter 4, where it was visible that the weight of the AAFO
is intrinsic to the actual technology. A further study in systems capable of absorbing external forces to
output later during the peak torque would allow having smaller motor and thus, smaller battery. For
instance, coupling parallel rotational springs in the ankle joint could absorb the energy during loading
response phase and output the energy during the mid stance phase.
The series elastic actuator (SEA) presented in chapter 5 can be considered as the main working point
for weight saving. A good improvement would be merging as much as possible the actuator and the
ankle-foot orthosis (AFO), with the goal of having a compact system. The springs in the actuator are
responsible for big part of the volume. An alternative solution for reading the actuator force would pass
by connecting the ankle joint and the actuator application force point with a beam. By setting gages in
the beam, the deflection of the beam could be obtained and thus calculate the force applied by the
actuator.
67
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71
Appendix A
Biomechanical Data
A.1 Kinematics and Kinematics Gait Data
Figure A.1: Kinematics and kinetics of human gait during gait cycle, for three different cadences: (a) joint ankle angle, (b) ankle angular velocity, (c) ankle angular acceleration, (d) ankle moment of force (torque) per unit of
body mass, (e) ankle power per unit of body mass (Winter, 1991).
72
Appendix B
Simulation and Control
B.1 Results for P-D and LQR Controllers
Figure B.1: Ankle angle tracking during gait cycle of controlled simplified ankle-foot model.
Figure B.2: Ankle angle tracking error during gait cycle of controlled simplified ankle-foot model.
B.2 Results for Impedance Control
During swing period, the ankle-foot system is not subjected to external forces due to ground contact.
Thus, the ankle angle was modeled as a linear rotary, second-order model, under-damped system
with an initial position offset. Figure B.3a presents the ankle angle during GC, where the swing period
initializes around 62% GC. Analyzing the ankle angle data in that period (Figure B.3b), it is possible to
extract some second-order models characteristics such as peak time, , and maximum overshoot,
. Peak time is the time required for the response to reach the first peak of the overshoot, i.e., the
maximum (percent) overshoot, which is the maximum peak value of the response curve measured
from unity (Ogata, 2002 p. 219). Thus it is possible to calculate the corresponding damping ratio, ,
and natural frequency, , with the following expressions:
(6.1)
(6.2)
0 10 20 30 40 50 60 70 80 90 100-25
-20
-15
-10
-5
0
5
10
Gait Cycle (%)
Ankle
Angle
(degre
es)
Simulated Model
P-D Controller
LQR Controller
0 10 20 30 40 50 60 70 80 90 100-5
0
5
Gait Cycle (%)
Ankle
Angle
Err
or
(degre
es)
P-D Controller
LQR Controller
73
It is possible to observe in the gait data of Figure B.3b that after the time 0.23 seconds there is
another increase in the ankle angle, which invalids the second-order model approximation. Thus, it
was disregard the data after the time 0.23 seconds, considering that the peak time was reached at 0.2
seconds, with a maximum overshoot of 6.34%.
Figure B.3: (a) Ankle angle during gait cycle (Winter, 2004), (b) Ankle angle during swing period (Gait Data) and second-order model for approximation of gait data (step response with negative offset).
The parameters of the determined model are presented in Table B.1, with step response with negative
offset in Figure B.3b. For that, the ankle stiffness and damping were determined by the equations
described in section 3.5, with a rotational ankle inertia, , of . With the determined
parameters, the step response with negative offset shows a very similar behavior until 0.23 seconds,
concluding that the approximation is acceptable for that region using the simplified ankle-foot model.
Table B.1: Calculated ankle stiffness and damping from second-order model.
Parameters Swing
Maximum overshoot - (%) 6.34
Peak time - (s) 0.2
Damping ratio - 0.659
Natural frequency - (rad/s) 20.9
Rotational Damping - B (Nms/rad) 0.331
Rotational Stiffness - K (Nm/rad) 5.243
74
Figure B.4: Dynamics from simulated model and controlled model with impedance control 1 during gait cycle: (a) ankle torque, (b) horizontal shear force, (c) vertical force.
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