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Design and Analysis of a Cable-Driven
Articulated Rehabilitation System for Gait
Training
Aliakbar Alamdari ∗
Graduate Research Assistant, Student Member of ASME
Mechanical and Aerospace Engineering
SUNY at Buffalo
Buffalo, NY, 14260
aalamdar@buffalo.edu
Venkat Krovi
Professor, ASME Fellow
Mechanical and Aerospace Engineering
SUNY at Buffalo
Buffalo, NY, 14260
vkrovi@buffalo.edu
Assisted motor therapies play a critical role in enhancing
functional musculoskeletal recovery and neurological reha-
bilitation. Our long term goal is to assist and automate
the performance of repetitive motor-therapy of the human
lower limbs. Hence, in this paper, we examine the viabil-
ity of a light-weight and reconfigurable hybrid (articulated-
multibody and cable) robotic system for assisting lower-
extremity rehabilitation and analyze its performance. A hy-
brid cable-actuated articulated multibody system is formed
when multiple cables are attached from a ground-frame to
various locations on an articulated-linkage based orthosis.
Our efforts initially focus on developing an analysis and
simulation framework for the kinematics and dynamics of
the cable-driven lower limb orthosis. A Monte Carlo ap-
proach is employed to select configuration parameters in-
cluding cuff sizes, cuff locations, and the position of fixed
winches. The desired motions for the rehabilitative exer-
cises are prescribed based upon motion patterns from a nor-
mative subject cohort. We examine the viability of using
two controllers —a joint-space feedback linearized PD con-
troller and a task-space force-control strategy —to realize
trajectory- and path- tracking of the desired motions within
a simulation environment. In particular, we examine per-
formance in terms of (i) coordinated control of the redun-
dant system; (ii) reducing internal stresses within the lower-
extremity joints; and (iii) continued satisfaction of the uni-
lateral cable-tension constraints throughout the workspace.
∗Corresponding author.
1 INTRODUCTION
Several neurological disorders including stroke, spinal
cord injury, cerebellar disorders, and neuromuscular diseases
manifest themselves via generation of abnormal patterns of
lower limb motion. Numerous studies have noted that sys-
tematic deployment of rehabilitation regimen (of adequate
intensity, duration, and consistency) can help restore mo-
tor functionalities in such patients [1]. However, significant
challenges exist for realization of an automated rehabilita-
tion system that can function in close coordination with a
patient’s musculoskeletal system.
Traditionally, lower limbs motor therapy is carried out
manually, requiring multiple (often 3 or more) physiothera-
pists. The difficulty and inconsistency in therapy from one
session to the next motivated researchers to develop gait
training treadmills with body weight support to provide con-
sistent gait motion [2–4]. Our interest is in rehabilitation sys-
tems which allow for significant greater flexibility of thera-
peutic motions/forces as well as customization of the train-
ing regimen. Volpe et al. [5] review the successes noted
from several groups in reducing impairment and enhanc-
ing human motor control with task specific exercises de-
livered by robotic devices. It is worth noting that several
exoskeleton systems such as BLEEX [6], MIT exoskeleton
[7], HAL-3 [8] have been developed (in non-rehabilitation
contexts) to augment human walking. Additionally, many
active/passive orthoses such as gravity-balancing leg ortho-
sis [9], pneumatic-muscle powered ankle-foot orthosis [10],
elastic knee orthosis [11], ALEX [12], and LOPES [13] have
been developed.
Alamdari 1 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
1
2
3
4
6
75
Fig. 1. RObotic Physical Exercise and System (ROPES): A Cable-
Driven Robotic Rehabilitation System for Lower-Extremity. Motors 1,
2, 3 and 4 are placed in appropriate positions to generate positive
cable tensions to move lower limbs in the sagittal plane along the
desired trajectory, and likewise motors 5, 6 and 7 are placed in frontal
plane to generate positive cable tensions based upon the prescribed
lateral exercises for lower limbs.
Several of these architectures are now being evaluated
for rehabilitation application potential to help patients to
overcome muscle-weakness and remedy abnormal motor
control etc. The challenge arises both for design of robotic
system with adequate degrees of freedom to not hinder natu-
ral gait kinematics/dynamics as well as coordinated control
to achieve the normal physiological gait patterns. Among
clinically-deployed rehabilitation systems, Lokomat [14] is
an exoskeletal orthosis which is attached to a patient’s legs
to assist the person to walk on the treadmill. The Haptic
Walker [15] is a multidegree-of-freedom system intended to
generates foot pedal motions to simulate regular and stair
walking. However, in addition to performance limitations,
these devices are expensive and only available in some clini-
cal or rehabilitation centers.
Hence, in this paper, we examine the feasibility of a
light-weight and reconfigurable hybrid (articulated multi-
body and cable) robotic system for assisting lower-extremity
rehabilitation and analyze its performance. Many of the com-
monly prescribed rehabilitation exercises engender closed-
loop ankle trajectories which can be realized by an appro-
priate cable-driven rehabilitation device. Such closed-loop
ankle trajectories can serve to guide the other leg-members
(shank/thigh) via the natural kinematics and dynamics of the
limbs. Cable robots are well-known for their low inertia, rel-
atively large workspace, low fabrication costs and reconfig-
urability. The use of cables allows the relatively heavy mo-
tors and gearboxes to be moved from the joints to the base.
This reduces the mass and inertia of the moving bodies and
can allow the robot to be designed with smaller, less costly
motors.
Several recent exoskeleton designs have emerged to take
advantage of the cable-based architecture. In [16] a cable-
driven robotic gait training system called CaLT was designed
for gait training of spinal cord injury patients, and it deliv-
ered a promising and acceptable experimental results. In this
end-effector type apparatus cables are directly connected to
the subject’s ankle, and only apply force at the ankle. In
the C-ALEX [17] design, no rigid or joints are employed
within the exoskeleton. Instead, three cuffs are connected
to the waist, thigh, and shank, and four cables are routed
through the cuffs to actuate two degrees-of-freedom of the
human user lower limbs. The exoskeleton is controlled in
force mode using assist-as-needed control paradigm. Our
proposed design differs in seeking to take advantage of the
hybrid cable-articulated architecture.
The main challenge in any cable-robot control scheme
is satisfaction of the tensionability conditions, i.e., assuring
that all cables are always in tension. The manipulation is
performed via increasing and decreasing the lengths of the
cables connected to the end-effector. One can increase the
number of redundant cables to satisfy the tensionability of
the cables. However, this increases the interference of the
cables in the working space. The minimum sufficient number
of cables in mulibody systems that guarantees all cables are
in tension have been investigated in [18].
In traditional cable robots, multiple cables are at-
tached to single-payload/platform from multiple points on
the ground. In contrast, in the proposed articulated multi-
body cable-robot system, cables are connected to different
links of the multibody system from multiple points on the
ground as illustrated in Fig. 1. In this work, we assume a
simplified model wherein each human lower limb segment
(i.e. foot, shank and thigh) and its corresponding lorthosis-
link are considered as one part. Thus, each human lower
limbs maybe considered as a serial multibody system driven
by the cables attached to them.
Limited literature has examined multi-body articulated
cable-drive systems in the past. For example, the wrench
closure workspace of multibody cable-driven mechanisms is
determined based on Lagrange’s approach [18] and recipro-
cal screw theory [19]. The concept of generalized forces and
Lagrange’s method have been employed to eliminate forces
and moments from the equilibrium equations. It is notewor-
thy that the operational workspace of multibody cable-driven
systems is largely impacted by the choice of cable placement
and routing. Bryson and Agrawal [20] identified and ana-
lyzed cable configurations for serial robot driven by cables.
Yan et al. [21] designed a 7-DOF humanoid arm driven with
14 cables. They employed the force closure method to evalu-
ate the workspace of multi-finger grasping. In [22, 23], a hy-
brid articulated-cable parallel mechanism named PACER de-
veloped by authors for upper limb rehabilitation in 3D space
in which with appropriate design, and proper selection of the
position of cable winches and cable attachment points to the
multibody system, the positive tension in the cables assured.
The key contribution of this paper is the development
and feasibility/performance evaluation of a reconfigurable
hybrid cable-articulated architecture that will work closely
with the human musculoskeletal system to provide motor
therapies. This system is an extension of traditional single-
body cable-driven to multibody cable-driven system. The
combined system features multiple holonomic cable-loop-
closure constraints acting on a tree-structured multibody
Alamdari 2 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
system. The wrench feasible workspace [24] of RObotic
Physical Exercise and System (ROPES) is determined and
improved by adding linear tensional and torsional springs
to articulated-multibody system to cover whole operational
workspace of the prescribed ankle trajectory. Such springs
help in keeping cables taut, and result in larger workspace.
There remains significant design-freedom in determining the
location of base spooling motors as well as the sizing of
the cable-attachment (cuffs) to the articulated-leg-frame. We
highlight the opportunity to exploit this freedom by design-
optimization to enhance the functional performance. Specif-
ically, the focus of this paper is to assist the performance
of repetitive therapy of human lower limbs in sagittal plane.
The desired motion is prescribed based upon normative sub-
jects’ motion patterns. The appropriate coordination of ca-
ble forces to each segment of the lower limbs is realized by
an impedance/force-field and a feedback linearized PD con-
trollers to (i) assure the tensionability in cables, and (ii) avoid
increasing the internal forces at the lower extremity joints.
The remainder of the paper is organized as follows: In
Section 2, the kinematic analysis of ROPES is presented.
The dynamic equations of motion of ROPES using the
Newton-Euler, and Lagrangian approach are derived in both
joint-space and task-space in Section 3. In Section 4, the
optimal cable configuration analysis of ROPES is identified,
and then the tension distribution in the resulting optimal con-
figuration is examined in Section 5, and Section 6 and 7 are
devoted to designing feedback-linearized PD controller and
impedance/force-field controller for ROPES, and finally Sec-
tion 8 concludes the paper.
2 KINEMATIC ANALYSIS OF ROPES
In order to describe the motion of the lower limbs driven
with cables, the coordinate systems {F},{0},{1},{2}and
{3}are attached to the trunk, hip, knee, ankle and the end
of the foot, respectively, as illustrated in Fig. 2. The cable
length li, cable unit vector tiand Jacobian matrix JTcan be
found by writing the loop-closure equations for each cable.
In these relations widenotes the winch position respect to the
fixed frame {F}, and uidenotes the position vector of cable
attachment point to orthosis respect to the fixed frame {F}.
ui−wi+liti=0,i={1,2,3,4}(1)
li=q(wi−ui)T(wi−ui)(2)
ti= (wi−ui)/li(3)
By taking the derivative of both sides of Eqn. (1) with
respect to time, and multiplying both sides of resulting equa-
tion to tT
i, it gives the Jacobian matrix as follows
˙
li=−tT
i
∂ui
∂q˙
q=−JT˙
q(4)
where ˙
q= [ ˙q1,˙q2,˙q3]T(note that q1=3π
2−θh,q2=θkand
q3=3π
2−θawhich θh,θk,θaare hip, knee and ankle joint an-
0
x
0
y
1
y
1
x
2
y
2
x
h
q
k
q
3
y
3
x
a
2
p+ q
2
y
2
x
a
2
p
+ q
F
x
F
y
4
w
4
u
t1
K
t2
K
4 4
t ,l
4
T
3 3
t ,l
3
T
2 2
t ,l
2
T
1 1
t ,l
1
T
4y
d
3y
d
4x
d
3x
d
1y
d
2y
d
R
0
C
1
C
2
C
3
C
4
C
3
q
Fig. 2. A cable-driven robotic rehabilitation system, in which Ti,ti
are cable tension and cable unit vector, respectively; diy and dix are
cuff size and its position in the local frame; Kti is tensional spring for
increasing the wrench feasible workspace.
gles as shown in Fig. 2), and JT=tT
i
∂ui
∂qis a Jacobian matrix
which maps joint angular velocities to the cable velocities.
The angular velocity and acceleration of each segment
i.e., thigh, shank and foot in sagittal plane can be found such
that for thigh is, ωt=˙q1,˙
ωt=¨q1, and linear acceleration
of the mass center of thigh is ac
t=˙
ωt×lc
t+ωt×(ωt×lc
t),
where lc
t=FR11lc
tis the thigh mass center vector expressed
in fixed frame; 1lc
tis the thigh mass center vector expressed
in frame {1}(position vector from the origin of frame {0}
to the mass center of thigh), and FRiis rotation matrix from
fixed frame to ith frame of reference.
Similarly, the angular velocity and acceleration for the
shank can be expressed by, ωs=˙q1+˙q2,˙
ωs=¨q1+¨q2, and
linear acceleration of the mass center of shank is ac
s=at+
˙
ωs×lc
s+ωs×(ωs×lc
s), where lc
s=FR22lc
sis the link (shank)
mass center vector expressed in fixed frame; 2lc
sis the shank
mass center vector expressed in frame {2}(position vector
from the origin of frame {1}to the mass center of shank),
and linear acceleration atcan be found by substitution of lt
instead of lc
tin equation of ac
t.
Finally for the foot, the angular velocity and accelera-
tion can be written as, ωf=˙q1+˙q2+˙q3,˙
ωf=¨q1+¨q2+¨q3,
and linear acceleration is ac
f=as+˙
ωf×lc
f+ωf×(ωf×lc
f),
where lc
f=FR33lc
fis the foot mass center vector expressed in
fixed frame, and 3lc
fis the foot mass center vector expressed
in frame {3}(position vector from the origin of frame {2}to
the mass center of the foot), and linear acceleration ascan be
found by substitution of lsinstead of lc
sin equation of ac
s.
2.1 Cable attachments in ROPES
The combined human leg and orthosis is controlled in
sagittal plane using four cables. More cables may unnec-
Alamdari 3 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
essarily complicate the workspace design and analysis, and
fewer than four cables makes it impossible for unilateral
cable-tension constraints to be satisfied. The cables are con-
nected to each link of orthosis using cuffs. The cuff for the
thigh link is of radius d4yand is positioned a distance d4x
from the origin of frame {0}. The cuff for the shank link
is of radius d3yand is placed a distance d3xfrom the ori-
gin of frame {1}. Similarly, the cuffs for the foot link are
of radius d2yand d1y, respectively, and they are positioned a
distance d2xand d1xfrom the origin of frame {2}. It is as-
sumed each cable attachment point, Ci, as shown in Fig. 2,
is placed on a circle with center of C0, of radius R, and angle
θi,i={1,2,3,4}respect to the fixed coordinate frame.
Table 1. Design parameter ranges of independent parameters of
ROPES.
Parameter Range Units Parameter Range Units
d1y[45,85]mm d2y[50,85]mm
d3y[55,95]mm d4y[75,125]mm
d1x[70,220]mm d2x[70,220]mm
d3x[110,440]mm d4x[150,450]mm
θ1[170,195]deg θ2[260,330]deg
θ3[−10,80]deg θ4[110,165]deg
The cable configuration parameters, as described in Fig.
2, are allowed to vary within the ranges given in Table. 1. For
example, the radius of cable cuff attached to the thigh, d4y, is
allowed to vary from the 75mm to 125mm in increments of
5mm, and its position, d4x, is allowed to range from 150mm
to 450mm in increments of 5mm.
3 DYNAMIC EQUATIONS OF MOTION OF LOWER
LIMBS DRIVEN WITH CABLES
Guaranteeing positive cable tensions for the cable-
driven articulated orthosis (within an adequately large work-
region) remains a critical issue. Hence the dynamic equa-
tions of human lower limb (orthosis + human leg) were de-
veloped to facilitate model-based control of the tension dur-
ing the rehabilitation exercises. The exercises include normal
walking in the sagittal plane and lateral leg-lifting in frontal
plane. For this purpose, two dimensional equations of mo-
tion for each segment of the human leg was formulated sep-
arately in the sagittal and frontal planes. In sagittal plane,
the hip and knee flexion/extension motion and the dorsiflex-
ion/plantar flexion motion of the ankle are considered. Sim-
ilarly, the equation of motion for lateral leg-lifting exercise
is derived in the frontal plane while considering additional
degrees-of-freedom at the hip (abduction/adduction) and an-
kle (inversion/eversion) joints.
As shown in [25], adding springs between ground and
multibody or between links can improve wrench feasible
workspace of cable-driven system. Exploiting this idea
within the cable-articulated orthosis, one linear tensional
spring is attached between the fixed ground and the thigh,
Kt1, and another tensional spring is connected the thigh to
the shank, Kt2. Such springs help in keeping cables taut, and
improve the wrench feasible workspace without adding re-
dundant cables to cover whole operational workspace of the
predefined ankle trajectory. Without loss of generality, the
springs are modeled as linear springs with stiffness constant
K and zero free-length. The generated force will then be-
come Ktilt i which lti is the vector of zero-free length springs.
Many designers of orthoses and prostheses have sized
their devices based on the average kinetic and kinematic
data of human [26, 27]. The compliance of lower extrem-
ity joints during locomotion can be investigated by the con-
cept of quasi-stiffness. This term can be distinguished from
the passive and active stiffness of a joint typically used to
describe the local tangent to the moment-angle curve shown
for given joint at a specific angle [28]. As illustrated in Fig.
3, the quasi-stiffness of a joint is defined more globally, as
the slope of the best linear fit on the moment-angle curve of
a joint over a whole stride or specific phase of a stride [29].
A series of empirical studies [29–31] have been conducted to
characterize the quasi-stiffness and linear behavior of lower
extremity joints during walking for adult humans as function
of body size (height and weight). This forms the basis of
a normative anthropometric statistical model which we will
employ to predict the hip, knee and ankle quasi-stiffness for
adults walking on level ground. The mean value of quasi-
stiffness of each joint is enumerated in Table 2.
Table 2. Quasi-stiffness of lower extremity joints and stiffness of tor-
sional springs placed on the orthosis with units of Nm/rad [29–31].
Parameter Value
Hip quasi-stiffness in extension phase KHe 320
Hip quasi-stiffness in flexion phase KH f 335
Knee quasi-stiffness in extension phase KKe 263
Knee quasi-stiffness in flexion phase KK f 304
Ankle quasi-stiffness in plantar-flexion phase KA p 202
Ankle quasi-stiffness in dorsi-flexion phase KAd 246
Hip stiffness on the orthosis KOH 327
Knee stiffness on the orthosis KOK 283
Ankle stiffness on the orthosis KOA 224
Moreover, additional torsional springs are placed on the
orthosis at hip (KOH ), knee (KOK), and ankle (KOA) joints,
and their values are defined in Table 2.
3.1 Newton-Euler dynamic formulation in sagittal
plane
In this section, a Newton-Euler formulation is used to
derive dynamics equations of the cable-driven articulated or-
thosis (Fig. 2) in the sagittal plane. Such a formulation is
Alamdari 4 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
0 0.4 0.8 1.2
0
20
40
60
80
100
120
140
160
Ankle Moment (N.m)
Ankle Angle (rad)
0 0.4 0.8 1.2
-30
-20
-10
0
10
20
30
40
50
60
70
Knee Moment (N.m)
Knee Angle (rad)
-0.3 -0.1 0 .1 0.3 0.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
Hip Moment (N.m)
Hip Angle (rad)
Hf
K
He
K
Kf
K
Ap
K
Ad
K
Fig. 3. Ankle, knee, and hip moment vs. angle curve for repre-
sentative subject walking at 1.25 m/s. Quasi-stiffness is claculated
based on the slope of the best-line fit to the moment-angle curve for
ankle plantar-flexion (KAp), ankle dorsi-flexion (KAd ), knee flexion
(KK f ), knee extension (KKe), and hip extension (KHe ) and flexion
(KH f ) [29–31].
very useful in a design/simulation setting in helping analyze
force-profiles (both internal forces/moments within as well
as the external forces needed to drive) the cable-articulated
orthosis. Each segment’s dynamic equations, subjected to
cable-and constraint-forces, is derived separately. Note that
each segment is subjected to cable forces as well as con-
straint forces and moments. Since the weight and moment of
inertia of the orthosis segments and cuffs are much smaller
than lower limbs segments, their weight and moment of in-
ertia are ignored in this simulation . The recursive Newton-
Euler formulation in the sagittal plane is written for the foot
segment in matrix form as follows (notice that boldface sym-
bols have been used for vectors and matrices),
t1t2ts f 0
uf1×t1uf2×t2uf3×ts f ms f
T1
T2
Fs f
Ms f
(5)
=mfaf−Fext −mfg
If˙
ωf+ [ωf×]Ifωf−Mext −uf4×Fext +Ja
where Ja=KAP/Ad(q3−3π
2)+KOA (q3−Φ3);Φ3is the initial
angular position of the torsion spring installed at the ankle
joint on the orthosis, and ti,i={1,2,3,4}is a unit vector
along the cable expressed in fixed frame and Tiis the mag-
nitude of cable forces; ts f and ms f are unit vectors of force
Fs f and moment Ms f exerted from shank to the foot in sagit-
tal plane; Fext and Mext are given external forces and mo-
ments from the ground to the foot; mfis the mass of the
foot; uf i ,i={1,2,3}is the position vector from the mass
center of the foot to the point of application of the applied
forces expressed in the fixed frame. Notice that the iner-
tia tensor Ifof the foot is expressed in fixed frame of refer-
ence i.e., If=FR33IfFRT
3;KAP/Ad is quasi-stiffness of ankle
which depends on the stance phase during walking, for ex-
ample, KAP/Ad =KAP in plantar-flexion phase, KAP/Ad =KAd
in dorsi-flexion phase; Eq. (5) concisely can be written as
JfTf=Ff. It is worth of mentioning that at each instant,
Eqn. (5) is a linear equation in terms of Tf.
Similarly, the formulation can be constructed for shank
in sagittal plane as
t3−ts f tts 0 0
us3×t3−us2×ts f us1×tts −ms f mts
T3
Fs f
F
ts
Ms f
Mts
(6)
=msas−msg
Is˙
ωs+ [ωs×]Isωs+Jk−Ja
where Jk= (KKe/K f +KOK )q2−st2×Kt2lt2,Is=
FR22IsFRT
2;t3is the unit vector along the cable with
magnitude T3;tt s and mts are the unit vectors of force F
ts
and moment Ms f exerted from thigh to the shank; msis the
mass of the shank; usi ,i={1,2,3}are the position vector
from the mass center of the shank to the point of application
of the applied force expressed in the fixed frame; KKe/K f is
quasi-stiffness of knee which depends on the stance phase
during walking, for example, KKe/K f =KK f in flexion
phase, KKe/K f =KKe in extension phase; both inertia tensor
Isand the angular velocity ωsof the foot are expressed in
fixed frame of reference, and finally Kt2and lt2are zero-free
length tensional spring stiffness and vector along the spring,
respectively; st2is the position vector from the mass center
of the shank to the spring attachment point on the orthosis;
the generated force by the linear axial springs is equal to
Kt2lt2which lt2is the vector of zero-free length spring. This
spring provides a tensile force proportional to its length.
Equation (6) can be written compactly as JsTs=Fs.
Newton-Euler formulation for thigh can also be expressed
by JtTt=Ft, where
Jt=t4−tts tbt 0 0
ut4×t4−ut2×tts ut3×tbt −mts mbt(7)
Tt=T4F
ts Fbt Mt s MbtT
Ft=mtat−mtg
It˙
ωt+ [ωt×]Itωt−Jk+Jh
where Jh=KHe/H f (q1−3π
2) + KOH (q1−Φ1)−st1×Kt1lt1,
It=FR11ItFRT
1;mtis the mass of the thigh; t4is the unit
vector along the cable with magnitude T4;tbt and mbt are
the unit vectors of force Fbt and moment Mbt exerted from
the human body to the thigh; uti ,i={2,3,4}is the position
vector from the mass center of the thigh to the point of ap-
plication of the applied force expressed in the fixed frame;
KHe/H f is quasi-stiffness of hip which depends on the stance
phase during walking, for example, KHe/H f =KH f in flexion
phase, KHe/H f =KHe in extension phase; Φ1is the initial an-
gular position of the torsion spring installed at the hip joint;
st1is the position vector from the mass center of the thigh to
Alamdari 5 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
the spring attachment point on the orthosis, and Kt1and lt1
are zero-free length tensional spring stiffness constant and
vector along the spring, respectively.
By assembling Eqns. (5), (6) and (7) the number of in-
dependent action-reaction forces and moments at the joint
can be reduced, and the compact form dynamic equations of
lower limb motion with cable-driven system in sagittal plane
can be obtained as
J9×10T10×1=F9×1(8)
where
J=
J1
fJ2
f0 0 J3
f0 0 J4
f0 0
0 0 J1
s0 J2
sJ3
s0 J4
sJ5
s0
0 0 0 J1
t0 J2
tJ3
t0 J4
tJ5
t
T=T1T2T3T4Fs f F
ts Fbt Ms f Mts Mbt T
F=FT
fFT
sFT
tT
where Ji
f,Ji
sand Ji
tare the ith column of the Jf,Jsand Jt,
respectively, defined in Eqns. (5), (6) and (7). Similarly,
dynamic equations of each segment of human leg can be
derived separately for the orthosis in the frontal plane (but
omitted here for brevity [32]).
3.2 Closed-form dynamic formulation in joint-space
The Lagrangian formulation is developed to reduce the
size of structure matrix Jin Eqn. (8). The general form of
dynamic equations for the cable-driven system in joint-space
can be written as
M(q)¨
q+V(˙
q,q) + G(q) = Q(9)
where vector q= [q1,q2,q3]Tare the generalized coordi-
nates; Mis the inertial matrix which is positive-definite,
symmetric and hence invertible; V= [V1,V2,V3]Tis the
velocity coupling vector which includes velocity-squared
terms (centrifugal forces) and velocity product terms (Cori-
olis forces), and G= [G1,G2,G3]Tis combination of gravi-
tational forces and spring forces.
Except for gravitational and inertial forces, the gener-
alized forces Qaccount for all other forces acting on lower
limb. So, the contribution of cable forces to the dynamics of
multibody system is modeled as point forces applied to the
links. By the principle of virtual work
QTδq=FT
eδP+
4
∑
i=1
TitT
i(δXTi )(10)
=FT
e(Jeδq) +
4
∑
i=1
TitT
i∂ui
∂qδq
where Jeis the conventional Jacobian matrix which maps
end-effector output force into n-dimensional joint torques
and JTis a Jacobian which maps cable tensions into the joint
torques (Eqn. (4)); δPand δXT i denote the virtual displace-
ment vector of the end-effector and cable attachment point
on each segment of lower limb, then they are substituted
by the relations δP=Jeδqand δXTi =∂ui
∂qδq, respectively;
the vector uidenotes the position vector of cable attachment
points with respect to fixed frame of reference.
One can find the generalized coordinates as
Q=JT
TT+JT
eFe(11)
where
JT
T=
tT
1
∂u1
∂q1tT
2
∂u2
∂q1tT
3
∂u3
∂q1tT
4
∂u4
∂q1
tT
1
∂u1
∂q2tT
2
∂u2
∂q2tT
3
∂u3
∂q2tT
4
∂u4
∂q2
tT
1
∂u1
∂q3tT
2
∂u2
∂q3tT
3
∂u3
∂q3tT
4
∂u4
∂q3
where JT
TTis the part of generalized force related to cable
forces and JT
eFeincludes generalized external forces and mo-
ments (Fe= [Fext,Mext ]T).
This form is well suited for trajectory tracking control
applications where the desired end-effector trajectory is pre-
sented in terms of joint angles, velocities, and accelerations
for which a feedback linearization controller is developed.
3.3 Closed-form dynamic formulation in task-space
However, in other circumstances (e.g. development of
an impedance controller) we may wish to move the ankle
on a trajectory in Cartesian space without converting task-
space variables. In such cases, the joint-space dynamics can
be projected into the task-space to realize the (often lower-
order) task-space dynamics equations as follows
˜
M¨
P+˜
V+˜
G=˜
Q(12)
where ˜
M=J−T
eMJ−1
e,˜
V=J−T
eV−MJ−1
e˙
Je˙
q,˜
G=
J−T
eG, and ˜
Q=J−T
eQ.
It is noteworthy that although the equations are ex-
pressed in task-space, still some terms such as ˜
V,˜
G, and ˜
Q
are written as function of joint variables q. Due to nonlinear-
ity of inverse kinematics, it is impossible to write everything
in terms of task-space variables, P.
4 PRELIMINARY CONFIGURATION ANALYSIS IN
ROPES VIA MONTE CARLO APPROACH
In order to account for the positive tension condition
present in cable-driven systems, more useful workspaces
have been proposed. Two of the more commonly utilized
ones are introduced here. The wrench-closure workspace
(WCW) [33] (also called the controllable workspace [34],
or force-closure workspace [35, 36]), is defined as the set
of end effector poses for which any arbitrary wrench can be
resisted/exerted by the platform while maintaining positive
Alamdari 6 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
cable tensions. This workspace depends only on the geo-
metric parameters of the system i.e. the locations of the
cable attachment points on the base and platform and the
pose of the platform. Several algorithms have been proposed
for efficiently computing the boundary of the wrench-closure
workspace.
If JT
Tis full rank, then Eqn. (11) is under-determined and
has many tension solutions for a given pose, q, and desired
set of Q−JT
eFe. In general, these tension solutions may not
be strictly positive, but only positive cable tensions are valid
for controlling ROPES. Solving Eqn. (11) for the tension
vector, T, yields the following Eqn. (13);
T= (JT
T)#(Q−JT
eFe) + ηJT
Tλ(13)
where (JT
T)#is the Moore-Penrose pseudo-inverse of matrix
JT
T,λis an arbitrary constant, and ηJT
Tis a basis for the null
space of JT
Tsuch that JT
TηJT
T=0.
For this case, since JT
Tis 3 ×4, then ηJT
Tis 4 ×1 vector.
The λ=0 case of Eqn. (13) is the minimum norm solution
but the tension vector components are not guaranteed to be
non-negative.
If all elements of the null vector, ηJT
T, have the same
sign, then it can be seen that regardless of minimum norm
solution values a λcan be chosen such that the ηJT
Tterm
be more than particular solution and gives a tension solution
which all components are positive. Here, we calculate the
wrench-closure workspace by exploiting this characteristic
(i.e., if all elements of the null vector are the same sign, then
the point belongs to the wrench-closure workspace). Since
the conditions of the wrench-closure workspace are quite
strict, and set of wrenches that the end-effector will have
to resist/exert are known, a more useful workspace called
wrench-feasible workspace (WFW) [24] can be used such
that cable tensions are greater than some prescribed mini-
mum and less than some prescribed maximum values. This
workspace depends not only on geometric parameters, but
also on the allowable tension ranges, gravitational effects,
and the required wrench set.
The choice of cable attachment both on fixed frame and
cuffs have significant effects on the operational workspace
of the ROPES. Inappropriate selections in cable attachments
can mitigate or restrain the ability of ROPES to perform
the prescribed task, while a proper configuration might pro-
vide additional capabilities and enlarge the wrench-feasible
workspace.
4.1 Optimization of configuration parameters
Table 3 shows parameter values for subject’s thigh,
shank, and foot (length, weight, moment of inertia, and
mass-center location) taken from an anthropometric database
[37]. From the a typical walking gait in task-space, the
desired task in joint-space is generated and shown Fig. 4.
The desired workspace in joint-space is also defined as θh=
−10◦to 50◦(or q1=220◦to 280◦), θk=q2=0◦to 60◦, and
θa=−10◦to 25◦(or q3=245◦to 280◦).
The total number of independent parameters is 12 (eight
cuff parameters dix and diy which i={1,2,3,4}, and four
angles for the position of cable attachment points to the fixed
frame, θi). Systematically determining each configuration in
the range defined in Table 1 would be extremely time con-
suming. Thus, alternatively, a Monte Carlo approach is em-
ployed to rapidly explore configurations within quite large
independent configurations.
0 1 2 3
−20
−10
0
10
20
30
Time (sec)
Ankle range of motion (deg)
0.5 m/s 1 m/s 1.5 m/s
0 1 2 3
0
20
40
60
Time (sec)
Knee range of motion (deg)
0.5 m/s
1 m/s
1.5 m/s
0 1 2 3
−20
0
20
40
60
Time (sec)
Hip range of motion (deg)
0.5 m/s
1 m/s
1.5 m/s
0 1 2 3
0
0.5
1
1.5
Time (sec)
Normalized Ground Reaction Force
0.5 m/s
1 m/s
1.5 m/s
Fig. 4. Hip, knee, ankle and normalized ground reaction forces of
healthy subject during walking with different speeds. These values
are considered as desired angles and forces, in trajectory tracking
problem [38].
To accomplish this analysis, a random value for each of
these 12 independent parameters in a given range, shown in
Table 1, is selected. Then, the subsequent randomly selected
configuration is checked for wrench feasible workspace.
Configurations which satisfy the wrench feasibility con-
straints are accumulated in database of desired configura-
tions.
To visualize the trends of these high-dimensional data-
set, each possible parameter value is shown in separate x-axis
on the plot, in a range defined for parameter values in Table 1,
and the number of wrench feasible configurations including
each possible parameter value is drawn on y-axis (see Fig. 5,
6, and 7).
Figures 5, 6, and 7 show configuration analysis plot
which all configuration parameters were randomly selected
with uniform distribution. These plots can provide in-
sight into the parameter value trends which result in high-
performance (higher WFW). If higher WFW be achieved in a
range defined for parameter values shown in Table 1, ROPES
would be more robust to errors in setup and assembly.
As shown in Fig. 5, the trend for d1xsuggests that pa-
rameter should be as close as possible to ankle joint, and
trend for d2xsuggests that the parameter should be as far as
possible from the ankle joint. The trends for d3xand d4xex-
hibit that it is preferable to select a value in range of 220◦
Alamdari 7 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
Table 3. Anthropometry and mass distribution for human body [37]
Parameter Value Units Parameter Value Units
lt472 mm ls466 mm
lf243.3mm mt11.8kg
ms4.5kg m f1.1kg
(It)z1157.9kg.cm2(Is)z392.8kg.cm2
(If)z30.4kg.cm2uf3117 mm
us1183 mm us2283 mm
ut2262 mm ut3210 mm
to 260◦. As illustrated in Fig. 6, the trend for d1yindicates
an evidence that the cuff radius is better to be chosen with
smaller size, here 56mm is selected for d1y. The trend for d2y
suggest bigger size for second cuff radius. Accordingly, the
trends for cuffs radius, d3yand d4y, suggests that there are
better to be selected around 65mm and 115mm, respectively.
Similarly, the preferred values for winch positions are shown
in Fig. 7.
These initial trends indicate that there is an opportunity
to further optimize the performance of the ROPES system.
For example, the d2ytrend indicates there might be value in
extending that particular parameter beyond 90 mm. We high-
light the opportunity and intend to pursue this as one aspect
of future work.
For the subsequent control analysis, one particular con-
figuration (from among the numerous high-performance con-
figurations) was selected (as shown in Figs. 5, 6 and 7 with
red arrows).
60 100 140 180 220 260 300 340 380 420 460
0
100
200
300
400
500
600
700
dix (mm)
Number of configurations
180 220 260 300 340 380 420
0
50
100
d1x
d2x
d3x
d4x
Fig. 5. The distance of each cuff from the local frame origin as
shown in Fig. 2.
5 TENSION DISTRIBUTION IN ROPES
A variety of tension distribution algorithms have been
proposed for resolving the actuation redundancy present in
most cable-driven robots. Each of these approaches have
different characteristics and varying computational cost [39].
40 50 60 70 80 90 100 110 120 130
160
180
200
220
240
260
280
300
320
340
diy (mm)
Number of configurations
d1y
d2y
d3y
d4y
Fig. 6. The trend of each cuff radius.
−10 15 40 65 90 115 140 165 190 215 240 265 290 315 340
0
100
200
300
400
500
600
θi (deg)
Number of configurations
θ1
θ2
θ3
θ4
165 190
0
200
400
Fig. 7. The trend of cable placements on the fixed frame.
Perhaps the most commonly implemented method in the lit-
erature relies upon the minimization (or maximization) of
some function of the cable tensions. In [34], it was shown
that while the L∞-norm provides optimal solutions, they may
be discontinuous along a given trajectory. The optimal solu-
tion can, however, be approximated using a p-norm (1 <p<
∞), and the resulting minimum-norm solution is proven to be
unique and continuous except at singular configurations.
In [40], a discussion on the potential limitations of the
more traditional L1and L2norms is provided in the context
of tension distribution. Use of the L1-norm results in a linear
programming problem which seeks to minimize (or maxi-
mize) the sum of the tensions. The primary drawback of
this approach is that it is susceptible to discontinuities, as
the optimal operating point may jump from one vertex of the
feasible polyhedron to another between successive computa-
tions. This can potentially excite high-frequency modes and
degrade the stability of the system. The L2-norm, which re-
sults in a quadratic programming problem and seeks to min-
imize (or maximize) the sum of the squared tensions, im-
proves upon this limitation by providing a smooth objective
function.
Both methods, however, result in tensions that fre-
quently lie on the lower or upper tension limits. Opera-
tion at the lower tension limits increases the risk of a cable
Alamdari 8 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
becoming slack, and can result in low stiffness properties.
Operation at the upper tension limits results in excessively
high torque requirements. Thus, the alternative methods pre-
sented by these authors allow for the cable tensions to be
steered towards a desired region of operation. Despite the
potential limitations of utilizing the L1and L2norms for the
purposes of cable tension distribution, efficient algorithms
for these problems are readily found in general optimization
packages.
The tensionability of the multibody system, driven by
cables, can be evaluated by the analysis of rank and null
space of the structure matrix JT
Tof the multibody system.
This equation at each instant of time is a set of linear equa-
tions in terms of T. The size of Jacobian matrix JT
Tis 3 ×4
which indicates that the number of unknowns is one more
than the the number of equations. If the number of unknowns
was equal to the number of equations, there would be one and
only one solution for each cable.
While a variety of redundancy resolution techniques
have been proposed, one approach commonly used is based
on minimizing the norm of the cable tensions. This has
the beneficial effect of minimizing the energy and torque re-
quirements of the system. The optimization problem can be
formulated as:
minkTkp(14)
subject to: JT
TT=Q−JT
eFeand Tmin ≤Ti≤Tmax ,i=
{1,2,3,4}. Thus, the optimized cable tensions must satisfy
the dynamic equilibrium equations and remain within some
specified upper and lower bounds. In general, the lower limit
corresponds to the amount of tension required to keep the
cables taut, while the upper bound depends on the torque ca-
pacity of the motors and/or the failure point of the cables.
6 TRAJECTORY TRACKING CONTROLLER DE-
SIGN FOR THE MULTIBODY CABLE-DRIVEN
SYSTEM
Appropriate controller design is a critical aspect of de-
velopment of rehabilitative robots and motor therapy. In
[41], control strategies have been categorized in four groups.
The first group is assisting controllers which move the pa-
tients injured limbs in a predefined trajectory to stretch the
limb muscles and rebuild the human motor control system.
Effort is thought to be essential for provoking motor plas-
ticity [42], and stretching can help prevent stiffening of
soft tissue and reduce spasticity, at least temporarily [43].
Impedance control is the main approach in assisting control
paradigm which helps the patient to follow the desired tra-
jectory with some deviation, depending on the impedance
gains [44]. More recent controllers have used more sophisti-
cated forms of mechanical impedance than stiffness, includ-
ing for example viscous force fields [45], and creating virtual
objects that assist in achieving the desired movement [46].
The second group is challenge-based controllers which
try to strengthen the muscles by providing resistance against
the movement, hence it increases the error [47,48]. Simulat-
ing everyday normal activities by using haptic interfaces is
the third group of controllers [49]. Finally, the last group is
robot for encouraging patients for performing exercises [50].
All these controllers are always implemented using position,
force or impedance control.
Here, the trajectory tracking controller tracks the desired
normal cycle using a feedback linearized PD controller. The
desired trajectory was obtained from recorded data of healthy
subject during walking with attached markers to human leg
as shown in Fig. 4 [38].
In this controller, the desired trajectory in terms of gen-
eralized coordinates is defined as a function of time qd=
qd(t). The block diagram of the proposed control law for
trajectory tracking system is shown in Fig. 8. Hence, from
closed-form dynamic equations (Eqn. 9) one can write the
virtual control law as
τ1=M(q)(¨
qd+Kd˙
qe+Kpqe) + V(˙
q,q) + G(q)(15)
This law linearizes the equations to an exponentially
stable system, ¨
qe+Kd˙
qe+Kpqe=0, where qe=qd−q,
Kp=diag{Kp1,Kp2,Kp3}, and Kd=diag{Kd1,Kd2,Kd3}
are positive matrices. In this simulation the controller gain
coefficients are selected to be Kpi =125 and Kdi =15, and
user-determined allowable minimum and maximum cable
tensions are chosen to be 2 and 80N, respectively.
As previously noted, there are more cables than the DOF
of orthosis, therefore there are many solutions for cable ten-
sions. As we discussed in cable tension distribution section,
it is desirable to find the set of cable tensions with smaller
positive values. This can be solved by quadratic program-
ming approach. Then, the cable force distribution can be
obtained from Eqn. (14) and JT
TT=τ1−JT
eFe=τ2.
Lower Limbs
+ Orthosis
p
min T
Compensation
V(q, q),G (q)
M(q)
PD
qd
+
-
+
+
1
t
q
T
IMU
sensors
-
+
T
e
J (q)
e
F
2
t
d
q
+
+
Fig. 8. Block diagram of trajectory tracking controller for human user
lower limbs.
With this controller and quadratic programming, the
ROPES is able to follow the desired trajectory as shown
in Fig. 9. The hip, knee and ankle angles corresponding
to the closed-loop ankle trajectory, and cable length varia-
tions are shown in Fig. 9, and tensile cable forces are illus-
trated in Fig. 10. Moreover, the Fig. 11 shows the internal
forces/moments at the lower extremity joints obtained from
Eqns. (5), (6) and (7). These results exhibit and ensure that
the internal stresses at these joints never exceed the corre-
sponding forces/moments during normal walking.
Alamdari 9 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
0 0.5 1 1.5 2 2.5 3
−20
0
20
40
Hip angle(deg)
Hip, knee and ankle joint angles with ROPES
0 0.5 1 1.5 2 2.5 3
0
20
40
60
Knee angle(deg)
0 0.5 1 1.5 2 2.5 3
−20
0
20
40
Time(sec)
Ankle angle(deg)
Actual
Desired
0 0.5 1 1.5 2 2.5 3
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec)
Cable Length (m)
Cable Length Variations with ROPES
L1L2L3L4
Fig. 9. Hip, knee, ankle, and cable length variations during a gait
cycle.
0 0.5 1 1.5 2 2.5 3
0
10
20
30
40
50
60
70
80
Time (sec)
Cable Tension Forces (N)
T1T2T3T4
Fig. 10. Cable tension forces in ROPES.
0 1 2 3
0
20
40
Reaction forces/moments at the hip, knee and ankle joints
Forces at the Ankle (N)
Anterior and posterior forces
Superior and inferior forces
0 1 2 3
−40
−20
0
20
40
60
Forces at the Knee (N)
0 1 2 3
−100
0
100
200
Forces at the Hip (N)
0 1 2 3
−40
−20
0
20
Joint Moment (N.m)
Time (sec)
Ankle Knee Hip
Fig. 11. Internal forces/moments at the lower extremity joints due to
cable tensions.
7 FORCE-FIELD AND POSITION-BASED
IMPEDANCE CONTROLLER DESIGN
Following the discussion in Section 6 about different
control strategies in development of rehabilitative robots,
research has exhibited some evidence that force-control
based strategies can be more effective for rehabilitation than
position-based control alone [41]. Impedance [44,51], force-
field [12, 17, 51], and admittance control [52] have been
implemented before for rehabilitation purposes, and have
shown their capabilities for providing compliant interaction
with the human limbs. Here, we present the development of
impedance and force-field control for human lower limbs.
7.1 Impedance control
The goal of the position-based impedance controller is
to create a virtual force to the leg while it is moving along a
target path. Let Pbe the current position of ankle in a Carte-
sian reference frame attached to subject’s trunk. The more
distance between the current position and desired position,
the more forces will be applied to thigh and shank to bring
back ankle point toward the target path in sagittal plane. The
impedance controller tries to control the lower limb such that
against an external force it acts as a mass-spring-damper sys-
tem. So,
Fd=Ka¨
Pd−¨
P+Kv˙
Pd−˙
P+Kx(Pd−P)(16)
where Ka=diag{Ka1,Ka2},Kv=diag{Kv1,Kv2}, and Kx=
diag{Kx1,Kx2}are impedance gains matrices, Pis the cur-
rent position of the ankle, and Pdrepresents the desired po-
sition.
Similarly, if foot angle deviates from the desired foot
angle trajectory q3d, a torque τdat the ankle joint will bring
the foot towards the desired trajectory, so one can define τd
as
τd=Kfα(¨
q3d−¨
q3) + Kfω(˙
q3d−˙
q3) + Kfθ(q3d−q3)
(17)
where Kfα,Kfω, and Kfθare impedance gains, q3is the cur-
rent foot angle, and q3drepresents the desired foot angle. In
this simulation, the parameters are defined in Table 4.
As illustrated in the block diagram Fig. 12, the haptic in-
terface in impedance controller design consists of two major
components, hardware and software. The hardware consists
of motors, load cells, orthosis and IMU sensors, and is simu-
lated for purposes of the current study. The software compo-
nent consists of forward kinematics, gravity compensation,
and impedance controller blocks which are implemented in
Matlab/Simulink. The ankle point position (or foot angle)
sent from Simulink to virtual reality will be compared to
desired target Pd(or desired angle q3d), and then the de-
sired force Fd(or torque τd) will be generated. Impedance
controller based on the current ankle position/foot angle and
desired target reflects forces to cable-driven system. Cable-
based impedance controller utilizes the principle of virtual
work to create the forces in cable system.
7.2 Force-field control
The goal of force-field controller is to assist the indi-
vidual to move ankle point along target path, also help the
individual in plantar- and dorsi-flexion motion of the foot
during normal walking, for those who suffer from a signif-
icant weakness of ankle muscles [53]. Force-field control
constructs a virtual tunnel like force-field around the ankle
Alamdari 10 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
Impedance /
Force-Field
Forward
Kinematics
Lower limbs +
Orthosis
P
IMU Sensors
q
T
e
J (q)
T
SoftwareHardware
d
P
Gravity
Compensation
+
+
p
min T
d
F
PD
Motors
Load Cell
+
-
d
T
Fig. 12. A block diagram for impedance control of human user lower
limbs by creating a virtual force to the ankle point to move it along the
target path.
Table 4. Parameters/gains of impedance and force-field controller
Parameter Value Parameter Value Parameter Value
Kai 1Kvi 10 Kxi 25
Kfα0.5Kfω4Kfθ8
δn8δt3rn8
rt2Kn20 Kt10
ζn20 ζt10
point target path, and along the desired foot angle trajectory.
If the ankle point (or foot angle) deviate from the target path
(or foot desired angle trajectory), the controller acts as spring
and brings them back to target.
The high level force-field controller generates: (i) a
force Fdat the ankle point that has a normal (Fn) and tan-
gential (Ft) components, i.e., Fd=Fn+Ft, (ii) a torque vec-
tor τdwhich similarly has a normal (τn) and tangential (τt)
components. The normal components i.e. Fnand τnare re-
sponsible for pulling ankle point (or foot angle) towards the
target path (or desired joint angle trajectory), and tangential
components i.e. Ftand τtare tangential to target path/desired
joint angle trajectory, and assist in tracking the target path (or
desired joint angle trajectory) [17]. So,
kFnk=Kn1−e−(2dmin
rn)2,kFtk=Kte−(2dmin
rt)2(18)
kτnk=ζn1−e−(2qmin
δn)2,kτtk=ζte−(2qmin
δt)2(19)
where Knand ζnare gain vectors for normal force-field, and
Ktand ζtare gain vectors for tangential force-field, respec-
tively; dmin is minimum distance from the ankle point to a
point on the target path, and qmin is the distance between the
current foot angle and desired joint angle.
Two virtual force-field tunnels with diameters rnand rt
are created around the target path in task-space, and sim-
ilarly two force-field tunnels with diameters δnand δtare
created around the desired joint angle trajectory in the joint
angle space. And also, two upper and lower bounds are cre-
ated as joint angle limits, qup and ql ow for the safety of foot
to avoid increasing the absolute value of the foot angle (see
Fig. 14). To understand the force field tunnels concept, a
force field around the ankle target path (thick red line) is il-
lustrated in Fig. 13. For the normal force Fn, outside the
tunnel rn, the magnitude almost equals to Knand apply force
along the stream line (blue lines) and normal to target path,
and the magnitude close to the target path equals to zero.
Likewise, for the tangential force Ft, outside the tunnel rt,
the magnitude roughly equals to zero and gradually increase
to Ktinside the tunnel.
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
−1
−0.95
−0.9
−0.85
−0.8
−0.75
X(m)
Y(m)
Magnitude of force−field around the desired path
Fig. 13. The magnitude of force-field (black arrows) around the an-
kle path (red line), and the stream of forces around the path (blue
curve lines).
−0.5
−0.4
−0.3
−0.2
−0.1
00.10.2
−1
−0.95
−0.9
−0.85
−0.8
−0.75
−10
0
10
20
X(m)
Foot angle variation along the traget path
Y(m)
Foot angle (deg)
Foot angle magnitude
Upper bound
Lower bound
Foot angle path
Target path
Fig. 14. The magnitude of foot angle (cyan line) on the target path
(red line), and upper and lower bound around the foot angle (blue
curve lines).
8 Discussion
In this paper we examined various aspects of the mod-
eling, analysis and simulation of an articulated-multibody
cable-driven system for rehabilitative exercises on lower ex-
tremity. The sagittal-plane dynamic model of lower limbs
with 3-DOF was formulated based on the both Newton-Euler
and Lagrangian formulations to support the design/control
Alamdari 11 JMR-15-1266, 2015. doi:10.1115/1.4032274
Journal of Mechanisms and Robotics Accepted manuscript posted December 17, 2015. doi:10.1115/1.4032274
efforts. The Newton-Euler approach allows for monitoring
of the internal forces (both within the orthosis as well as
the human) which is critical from a design perspective. The
Lagrangian formulation aids development of controllers for
simulation- or hardware-testing) after elimination of inter-
nal constraint forces. We highlighted how the wrench feasi-
ble workspace of ROPES depends on the selection of cable
placement both on fixed frame and mobile frames (attached
to the orthosis). We identified and analyzed the design-space
for cable configuration including cuff sizes, distance of cuffs
from local frames, and cable attachment points to the ground
frame. This analysis helped to identify not only the most-
sensitive configurations of ROPES but also the most-robust
(with respect to assembly- and set-up errors). We highlight
the opportunity and intend to pursue this as one aspect of
future work.
For the subsequent control analysis, we down-selected
one particular configuration (from among the numerous
high-performance configurations). Two types of controllers
were implemented for Lagrangian model, trajectory tracking
PD controller in joint-space, and force-control strategies in
task-space. Simulation results for both controllers were pre-
sented to show how model-based controller can apply forces
through cable-driven system. Finally, using the cable tension
results, and splitting the Jacobian matrix derived in Newton-
Euler formulation, we were able to calculate the internal
forces and moments at the hip, knee and ankle joints due
to forces through cable-driven systems to avoid increasing of
internal stresses at lower extremity joints.
For future work, there is an opportunity to include mo-
bility in fixed bases and add springs in series with cables to
(i) generate appropriate pretension in cables without becom-
ing slack, (ii) be able to change the ROPES stiffness, and
(iii) increase the safety of mechanism as well. Last but not
least is the need for experimental validation of this overall
framework.
Acknowledgment
This work was partially supported by the National Sci-
ence Foundation awards CNS-1314484 and IIS-1319084.
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List of Figures
1 RObotic Physical Exercise and System (ROPES): A Cable-Driven Robotic Rehabilitation System for Lower-Extremity.
Motors 1, 2, 3 and 4 are placed in appropriate positions to generate positive cable tensions to move lower limbs in the
sagittal plane along the desired trajectory, and likewise motors 5, 6 and 7 are placed in frontal plane to generate positive
cable tensions based upon the prescribed lateral exercises for lower limbs. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 A cable-driven robotic rehabilitation system, in which Ti,tiare cable tension and cable unit vector, respectively; diy and
dix are cuff size and its position in the local frame; Kti is tensional spring for increasing the wrench feasible workspace. . . 3
3 Ankle, knee, and hip moment vs. angle curve for representative subject walking at 1.25 m/s. Quasi-stiffness is claculated
based on the slope of the best-line fit to the moment-angle curve for ankle plantar-flexion (KAp), ankle dorsi-flexion (KAd ),
knee flexion (KK f ), knee extension (KKe), and hip extension (KHe) and flexion (KH f )[29–31]. ............... 5
4 Hip, knee, ankle and normalized ground reaction forces of healthy subject during walking with different speeds. These
values are considered as desired angles and forces, in trajectory tracking problem [38]. . . . . . . . . . . . . . . . . . . . 7
5 The distance of each cuff from the local frame origin as shown in Fig. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Thetrendofeachcuffradius................................................... 8
7 The trend of cable placements on the fixed frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
8 Block diagram of trajectory tracking controller for human user lower limbs. . . . . . . . . . . . . . . . . . . . . . . . . . 9
9 Hip, knee, ankle, and cable length variations during a gait cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
10 CabletensionforcesinROPES. ................................................ 10
11 Internal forces/moments at the lower extremity joints due to cable tensions. . . . . . . . . . . . . . . . . . . . . . . . . . 10
12 A block diagram for impedance control of human user lower limbs by creating a virtual force to the ankle point to move it
alongthetargetpath. ...................................................... 11
13 The magnitude of force-field (black arrows) around the ankle path (red line), and the stream of forces around the path (blue
curvelines)............................................................ 11
14 The magnitude of foot angle (cyan line) on the target path (red line), and upper and lower bound around the foot angle
(bluecurvelines)......................................................... 11
List of Tables
1 Design parameter ranges of independent parameters of ROPES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Quasi-stiffness of lower extremity joints and stiffness of torsional springs placed on the orthosis with units of Nm/rad [29–31]. 4
3 Anthropometry and mass distribution for human body [37] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Parameters/gains of impedance and force-field controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Alamdari 15 JMR-15-1266, 2015. doi:10.1115/1.4032274
