1
Stiffness Modulation in an Elastic Articulated-Cable
Leg-Orthosis Emulator: Theory and Experiment
Aliakbar Alamdari, Reza Haghighi and Venkat Krovi, Member, IEEE
Abstract—There has been an increasing interest towards cabledriven robotic rehabilitation systems with adjustable stiffness to
provide safe and natural interactions for physical therapy. In this
paper, we investigate the effectiveness of various stiffness modulation schema and alternate attachment configurations within
a scaled planar elastic articulated-cable leg-orthosis emulator
for gait training. Elasticity, incorporated by (i) series-elastic
springs or (ii) adjustable stiffness modules connected to nonflexible cables, can add substantial robustness during forceful
interaction with uncertain environments. Other benefits include
force-sensor-free tension control i.e. without using force sensors
connected to cables and better overall tension-distribution. However, elasticity also degrades the accuracy of positioning and
make the system more disposed to external disturbances. Hence,
we examine the performance of actively modulating the effective
stiffness to smooth out any perturbations to the scaled normative
rehabilitative motion patterns and achieve natural gaits (by
simulation and experiment). Stiffness modulation can now be
achieved by varying: (i) system configuration, (ii) antagonistic
cable tension, or (iii) joint stiffness and we examine the benefits
of each. Finally, we also comparatively evaluate the functional
performance of alternate attachment configurations: (i) AnkleCable Configuration; (ii) Articulated-Cable Configuration by way
of simulation and experiments.
Index Terms—Cable-driven leg orthosis, Gait training,
Articulated-cable mechanism, Kinetostatic optimization, Stiffness
modulation
I. I NTRODUCTION
The past two decades have seen a significant increase
in the use of active exoskeletons to augment and enhance
human motor control – for both able-bodied (e.g. warfighter,
shop-floor worker) as well as motor-impaired (e.g. clinical
patient) populations. While much of the interest in ablebodied populations has been for stand-alone ungrounded
interfaces, grounded interfaces have remained the mainstay
for robotic-rehabilitation deployments (our target application
space). However, it is worth noting that ongoing development
of architectures and deployments for both able-bodied and
motor-impaired have benefitted from the enhanced insights and
cross-pollination ensuing from the convergence – see Herr et
al. [1] for a classification of the emerging diversity.
Assisted motor therapies play a critical role in reversing the
degradation of functional motor performance due to disease or
injury [2]. Systematically exploiting neurological remapping
and brain plasticity by modulating the regularity, frequency,
duration, and intensity of a rehabilitation regimen has been
shown to enhance rates of recovery [3], [4]. The success
of robotic-rehabilitation therapy (over conventional physiotherapy) ensues from the flexibility and customizability of
the training regimen deployment, as supported by numerous
studies [5], [6], [7]. In turn, this has spurred the growth of
novel robotic-rehabilitation systems targeting both upper/lower
extremities – we will, however, restrict our attention to lowerextremity systems in this paper.
Numerous lower-extremity robotic-rehabilitation systems
have emerged to fill the gaps of classical gait rehabilitation methods in restoration of normal walking patterns [8],
[9], [10], [11], [12], [13]. The Lokomat is one of the best
exemplars of a commercially-available gait training system
for lower-limb disabilities [14] facilitating adjustment of gait
speed and body weight support. The LokoHelp treadmill
gait trainer incorporates an electromechanical foot-powered
orthosis that mediates a gait motion during the training session
[15]. The robotic arms of a ReoAmbulator attach to the thigh
and ankle of the patient’s leg, and emulate the guidance
of a physiotherapist in realizing a desired stepping pattern
[15]. Added benefits of automation have included enhanced
ergonomics of physical assistance, reduction in therapy cost,
and transparency of data acquisition, measurement and recordkeeping [16], [17]. The renewed research focus is on: (i)
developing design- and control-architectures to not alter the
natural gait kinematics/dynamics; as well as (ii) achieving gradated limb-gravity compensation with the diversity of motorimpaired populations. Examples include the gravity-balancing
leg orthosis [18], the ankle-foot orthosis [19], the elastic
knee orthosis [20], Active Leg Exoskeleton (ALEX) [21], and
Lower-Extremity Powered Exoskeleton (LOPES) [22].
It is in this setting that we examine the development of
an elastic articulated-cable leg-orthosis system called RObotic Physical Exercise System (ROPES) [23], [24]. ROPES
(shown in Figure 1) is a lightweight, reconfigurable, hybrid
(articulated-multibody and cable-based) robotic rehabilitative
system, intended to act as a surrogate for a human physiotherapist for the performance of repetitive motor-therapy of
the human lower-limb. Prescribed rehabilitative exercises, built
upon normative motion patterns, are intended to be realized
using trajectory-tracking task-space impedance controllers.
Over the years, cable-driven mechanisms (where extension
and retraction of the cables enable to control the position and
orientation of a moving platform) have emerged as an important robotic-architecture. They offer many advantages over
conventional serial and parallel mechanisms including ease
of assembly/disassembly, relatively lightweight construction
and low maintenance costs. Base-fixed winches help reduce
the effective system inertia which together with high forceto-weight ratios and controllable stiffness (building upon the
redundant actuation) positively benefit the trajectory tracking
the performance of the moving platform [25], [26], [27]. In
IEEE Transactions on Robotics
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yH
xH
y0
h
C2
x0
C1
y1
x1
T ̂ , k
T ̂ , (a) Sagittal
y2
(b) Frontal
Fig. 1. An elastic articulated-cable leg-orthosis system called RObotic Physical Exercise System (ROPES). ROPES is a cable-driven robotic rehabilitation
system for Lower-Extremity such that 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.
turn, this has opened up numerous applications for haptic
devices [28], rescue operations [29], aerial cameras [30], largescale radio telescopes [31], gait training [32], [23], [24] and
upper limb rehabilitation [33], [34].
In the ROPES context, a hybrid cable-articulated multibody
system is formed when multiple cables connect a groundframe to various locations on the lower limbs. Multiple
holonomic cable-loop-closure constraints acting on a treestructured multibody system govern the relative degrees-offreedom of the system. Hence, careful coordination of the multiple cable-winches becomes critical in order to achieve the corobotic control of the overall system, to avoid development of
internal stresses and ensuring continued satisfaction of the unilateral cable-tension constraints throughout the workspace. In
particular, active stiffness control becomes more complicated
in cable-driven systems with closed-loops and unidirectional
constraints. Nonetheless, controllable stiffness is beneficial for
physical rehabilitation applications to achieve accommodation
to the effects of uncertain and unmodeled disturbances, and
provide a safe tool for human physical therapy [35].
As a precursor to the development of a full-fledged ROPES
robotic-rehabilitation system, we examine a scaled-down planar Elastic Articulated-Cable Leg-Orthosis Emulator, shown
in Figure 2, to achieve smooth and natural gait training.
Emulation has been shown to be an effective means for
quantitative evaluation of alternate hardware-configurations
and algorithms [36], [37], [38]. In [24] we examined various
alternative methods for approximate gravity-balancing (using
inline series-elastic cable-based systems) after a comprehensive comparison with other existing gravity-balancing methods
in the literature.
The main contributions of the paper come from comparative
evaluation of active stiffness modulation to achieve natural gaits within the emulator under a variety of stiffnessmodulation schema and cable-attachment to the leg-orthosis.
T ̂ , x2
C3
(a) Ankle-cable configuration
yH
xH
y0
x0
h u
C1
T1 t1 ,l1
1
C2
y1
x1 k
T2 t 2 , l 2
Fe
y 2 T3 t3 ,l3
x2
C3
(b) Articulated-cable configuration
Fig. 2. An illustration of the ankle-cable and articulated-cable configuration,
in which Ci are the position of the linear motors with respect to the fixed
frame {H}, li are the cables length, ti and tˆi are the cables’ unit vector, Ti are
the cables tension and ui denote the vectors of cable attachment point with
respect to the fixed frame for i = 1, 2, 3, and Fe is external force to the ankle,
θh is the hip joint angle, θk is the knee joint angle.
Kinematic and actuation redundancy within the system permits realization of multiple stiffness modulation schemas (by
changing the system configuration, cable tensions, or joint
stiffnesses). Similarly, varying the attachment of the cables
to the leg-orthosis results in alternate configurations such as
(i) Ankle-Cable Configuration (which offers ease of attachment/detachment); (ii) Articulated-Cable Configuration (which
allows greater co-robotic control). Note that, although the proposed methods are mainly implemented on the scaled planar
lower-leg orthosis-emulator, they can be extended for other
rehabilitation applications (e.g. upper limb [33], [34]), or more
IEEE Transactions on Robotics
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general cable-driven co-robotic manipulation applications [28],
[29], [30].
The paper is organized as follows. Section 2 presents the
stiffness analysis of ankle-cable and articulated-cable configurations and draws out the dependencies on system configuration, cable tension, and cable stiffness. Section 3 discusses the
kinetostatic optimization formulation for stiffness modulation
of ankle-cable configuration and exploitation of configuration
redundancy (with simulations and experiments). Section 4
presents the modifications for the hybrid articulated-cable configuration. Stiffness modulation of both configurations through
variation of actuation (cable-tensions) are presented in section
5 and 6, respectively. Section 7 examines the significant
benefits of the introduction of variable stiffness module (that
allows decoupling of cable-tension and cable-stiffness) for
both configurations. Finally, Section 8, presents a comparative
discussion of the overall benefits.
II. S TIFFNESS A NALYSIS OF P LANAR C ABLE -D RIVEN
A RTICULATED M ULTIBODY C ONFIGURATION
The scaled Planar Elastic Articulated-Cable Leg-Orthosis
Emulator comprises of a two-link (RR) linkage (playing the
role of lower-limb orthosis) to which actuated cables can
be attached in multiple configurations. Figure 2 depicts two
alternate configurations: (i) the Ankle-Cable configuration
with three cables attached at the end-effector of the RR
linkage (which offers ease of attachment/detachment); (ii) the
Articulated-Cable configuration with two cables attached to
the proximal-link and distal-links via cuffs (which allows
greater manipulation control). In the following, we briefly
outline the analytical stiffness formulation for these two configurations with further details available in [24].
A. Ankle-Cable Configuration
Four coordinate systems {H}, {0}, {1} and {2}, attached
to the trunk, hip, knee, and ankle, respectively, are illustrated
in Figure 2(a). The parameters include origins of the cable
with respect to the fixed-frame, Ci , cable lengths, li , cable
unit-vectors, tˆi , cable tensions, Ti , and vectors from cableattachment point to the fixed frame, ui , for i = 1, 2, 3. θh is
the hip joint angle, θk is the knee joint angle. The equilibrium
equation is written as follows:
Aank T = −Fe
(1)
where T = [T1 , T2 , T3 ]T and Aank is defined as follows:
[
]
Aank = t̂1 t̂2 t̂3
change in the cable length δ l, incremental external force δ Fe ,
and incremental tension in the cables δ T,

 δ l = −ATank δ X
δ T = Ks δ l
(3)

δ Fe = KX δ X
where Ks is the cable stiffness matrix and KX is the Cartesian
stiffness matrix. Taking the variation of equation (1), and
simplifying it using equation (3), yields
KX δ X = −δ Aank T + Aank Ks ATank δ X
Since δ Aank =
sides, gives
∂ Aank
∂ Aank
∂ Xe δ Xe + ∂ Ye δ Ye ,
KX = −
[
1
t̂i = √
(Xe − Xi )2 + (Ye −Yi )2
[
Xi − Xe
Yi −Ye
canceling δ X from both
]
∂ Aank
∂ Ye T
+ Aank Ks ATank
B. Articulated-Cable Configuration
The articulated-cable configuration is illustrated in Figure
2(b). The equilibrium equation is written as follows:
Aart T = −τe
(6)
where Aart is the Jacobian which maps cable tensions into the
joint torques, which is defined as follows:
[
]
tT1 ∂∂ uq1 tT2 ∂∂ uq2 tT3 ∂∂ uq3
1
1
1
Aart =
(7)
tT1 ∂∂ uq1 tT2 ∂∂ uq2 tT2 ∂∂ uq3
2
2
2
where q1 = 32π − θh
parameters q1 and q2
and q2 = θk . Note that the auxiliary
are defined to simplify the formulation.
Let τe be the applied torque on the hip and knee joints such
that τe = JeT Fe where Je is the conventional Jacobian matrix
which maps the external force on the ankle (Fe ) into the torque
in the joints (τe ). The incremental angular displacement δ q
and the incremental torque δ τe are related by the stiffness
matrix KQ as δ τe = KQ δ q. Similarly, the incremental cable
displacement δ l and the incremental force in cables δ T are
related by the diagonal cable stiffness matrix Ks as δ T = Ks δ l,
where δ l = −ATart δ q. Differentiating equation Aart T = −τe
and substituting δ τe , δ T and δ l = −ATart δ q in the resulting
equation yields:
KQ δ q = −δ Aart T + Aart Ks ATart δ q
(2)
]
such that (Xi ,Yi ) are the pulleys’ position Ci and (Xe ,Ye ) is the
ankle’s position. Let Fe be the applied force on the ankle, the
following equations hold between the incremental displacement of the ankle’s position δ X = (δ Xe , δ Ye ), incremental
(5)
Note that the Cartesian task-space stiffness matrix, KX , can
vary not only due to change in cable stiffness Ks , but also the
cable tension forces T, as well as system configuration Aank .
This offers significant opportunities to modulate the task-space
stiffness which we exploit.
m
Since δ Aart = ∑
j=1
where
∂ Aank
∂ Xe T
(4)
yields
KQ = −
[
∂ Aart
∂ q j δ q j,
∂ Aart
∂ q1 T
(8)
canceling δ q from both sides
∂ Aart
∂ q2 T
]
+ Aart Ks ATart
(9)
Subsequently, from equations (5) and (9) the stiffness control of the cable-driven configurations can be realized in 3
ways (see Figure 3), (i) changing the system configuration
IEEE Transactions on Robotics
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Stiffness Modulation
of Leg Orthosis
Task-Space
Joint-Space
Ankle-Cable Mechanism
Articulated-Cable Mechanism
Changing
System
Configuration
Changing
Cable
Tension
Changing
Joint
Stiffness
Changing
System
Configuration
Changing
Cable
Tension
Changing
Joint
Stiffness
Fig. 3. Classification of stiffness modulation in joint space and task space with ankle-cable and articulated-cable configurations. Considering equations (5)
ank
art
and (9) the stiffness control of the cable-driven configurations can be realized in 3 ways, (i) changing the system configuration ∂ A∂ X
and ∂ A
∂ q , (ii) changing
T
T
antagonistic cable tensions T to alter the stiffness [39], and finally (iii) changing joint stiffness matrix Aank Ks Aank and Aart Ks Aart using an adjustable
stiffness module.
∂ Aank
∂X
art
and ∂ A
∂ q , (ii) changing antagonistic cable tensions T to
alter the stiffness [39], and finally (iii) changing joint stiffness
matrix Aank Ks ATank and Aart Ks Aart T using an adjustable
stiffness module.
Fig. 4. Ankle-cable configuration with configuration redundancy. A fixed
coordinate frame {F} is located at the center of triangular base, and the local
coordinate frames {Oi } are placed at the center of each side of triangle.
However, the positivity of the tension of the cables must be
verified. Hence, the wrench closure-workspace 1 and wrenchfeasible workspace 2 of such configurations are required to
be evaluated before setting up the experiment to find suitable
position for point (Xi ,Yi ). Figure 5 depicts the wrench-closure
workspace and the tension factor 3 [24] of the experimental
setup with bases fixed at the points (Xi ,Yi ). The tension factor
is close to 1 around the origin of the platform (due to the
symmetric configuration around the center) but progressively
reduces as the ankle point moves towards the point (Xi ,Yi ). The
significantly reduced wrench-feasible workspace is depicted in
Figure 6 when bounds 0 < T ≤ 5 are placed on the tension
of the cables. Further, Figure 7, depicts the reduction of the
wrench-feasible workspace as the external forces placed at the
ankle point are increased.
1 A wrench-closure workspace (WCW) refers to a workspace that corresponds to the set of static poses of the platform where the mechanism is fully
constrained by the cables.
2 The wrench-feasible workspace (WFW) is the set of poses of the platform
where the cables can balance any wrench in a specified set of wrenches, such
that the tension of the cables remain within a prescribed range.
3 The tension factor is defined as the ratio of the minimum cable tension to
the maximum cable tension.
IEEE Transactions on Robotics
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TF
Wrench closure workspace of ankle−cable robot with fixed bases
Wrench Feasible Workspace of Ankle-Cable Robot with Variant External Forces
600
500
F =1
400
0.9
500
300
0.8
400
Fe=3
300
F =4
e
F =2
e
0.7
100
0.6
0
0.5
−100
0.4
e
F =5
e
200
F =6
e
Y (mm)
Y(mm)
200
100
0
-100
−200
0.3
−300
0.2
−400
0.1
−500
−600
−400
−200
0
X (mm)
200
400
600
-200
-300
-400
-500
-600
-400
-200
0
200
400
600
X (mm)
Fig. 5. Wrench closure workspace and Tension Factor (TF) of ankle-cable
configuration with fixed bases. The tension factor is close to 1 around the
origin of the platform (due to the symmetric configuration around the center)
but progressively reduces as the ankle point moves towards the point (Xi ,Yi ).
Fig. 7. Wrench feasible workspace of ankle-cable configuration with variant
external forces. It is illustrated that the wrench feasible workspace reduces as
the external forces placed at the ankle point increase.
Wrench feasible workspace of ankle−cable robot with fixed bases 0<T<5N
500
400
300
200
A1 ,(X1 , Y1 )
{H}
Y(mm)
100
S1
0
o1
−100
−200
{F}
S′1′
A1′ ,(X1′ , Y1′)
sy
−300
{E}
−400
−500
−600
{O1 }
Rail 3
−400
−200
0
X (mm)
200
400
(Xe ,Ye )
600
Fig. 6. Wrench feasible workspace of ankle-cable configuration with fixed
bases and bounded cable tension 0 < T ≤ 5 N. It is illustrated that the wrenchfeasible workspace is significantly reduced.
III. S TIFFNESS M ODULATION OF L EG O RTHOSIS IN THE
A NKLE -C ABLE C ONFIGURATION E XPLOITING
C ONFIGURATION R EDUNDANCY
The mobility of base-pulleys creates kinematic configuration
redundancy and can be exploited to enhance the stiffnessmodulation capabilities (over conventional cable-driven mechanisms with fixed base pulleys) [40], [41], [42], [43]. The
schematic of the ankle-cable configuration with mobile basepulleys to realize configuration redundancy is depicted in
Figure 4. As illustrated in Figure 4, the base has a triangular
shape with two linear sliders on each side. A fixed coordinate
frame {F} is located at the center of the base, and the local
coordinate frames {Oi } are placed at the center of each side
of the triangle. The linear displacement of each slider with
respect to the coordinate frame {Oi } is identified by Si and
S′ i .
A. Kinetostatic formulation
Figure 8 is a schematic of one side of the ankle-cable configuration with kinematic configuration redundancy provided
by mobile base-pulleys. A linear spring is connected in series
Fig. 8. Parameters defined for the motion analysis of the ankle-cable
configuration with configuration redundancy.
to cable and is utilized to model the tension in cable-spring
with stiffness Ksi expressed as follows:
(
)
Ti = Ksi li + Si′ + Si − L0i
(10)
where parameters L0i are the length of cables together with
free-length spring. The relative motion of the two sliders can
be adjusted to achieve the following two cases: (i) Fixed total
cable length (which includes the length of cable and spring)
while the position of pulley (Xi ,Yi ) changes, (ii) Varying
total cable length (which leads to the change in the spring
length and the cable tension). Given the sliders’ position, the
ankle position now depends on the static equilibrium equations
expressed by (1) and the resulting ankle position can be
numerically calculated. However, for a given ankle position,
infinite sliders positions are possible due to the configuration
redundancy. This kinematic redundancy can be exploited to
optimize the configuration based on chosen stiffness modulation objective function.
ank
T+
The Cartesian stiffness is derived as KX = − ∂ A∂ X
T
Aank Ks Aank and a stiffness optimization helps to resolve
the configuration redundancy. Let S = [S1 , S1′ , S2 , S2′ , S3 , S3′ ]T be
the vector of sliders’ position with respect to its corresponding
origin Oi at each side of the triangular base, and Simin , Si′min and
Simax , Si′max are minimum and maximum distance of each slider
IEEE Transactions on Robotics
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with respect to the origin Oi . The problem may be generally
formulated as follows:
Minimize:
C(S)
(11)
Subject to:
[
t̂1
t̂2


′
] Ks1 (l1 + S1′ + S1 − L01 )
t̂3 Ks2 (l2 + S2 + S2 − L02 ) = −Fe
Ks3 (l3 + S3′ + S3 − L03 )
[
]
Si ∈ [Simin , Simax ] , Si′ ∈ Si′min , Si′max
(
)
li + Si′ + Si − L0i ∈ [0, ∆Simax ]
B. Simulation results for trajectory tracking with stiffness
modulation
In this section, we consider the stiffness modulation in
trajectory tracking for a normalized gait trajectory starting
from [−194.6, −102.9] mm. We used the cost function (14)
to achieve maximum stiffness in both directions of the AnkleCable configuration along the desired trajectory. The resulting
cable tension forces are depicted in Figure 9.
|Ṡi′ | ≤ Ṡi′max , |Ṡi | ≤ Ṡimax
where C(S) is a cost function (defined later based on the
desired performance) that is a function of the Cartesian stiffness matrix. The common constraints are force closure, slider
limits, maximum extension of the springs ∆Simax , and velocity
limits of the sliders i.e. Ṡi′max , Ṡimax .
Velocity limits for the sliders prevent jerky motions emerging from (i) multiple optima; (ii) delays in re-positioning of
the sliders to the optimal configuration. While the velocity
constraints may limit the results to local optima, this is
acceptable in practical applications.
1) Isotropic stiffness: To achieve equal stiffness in both
X and Y directions which provides equal resistance in both
directions, the eigenvectors of the stiffness matrix KX should
form an orthogonal basis. Hence, the stiffness matrix KX
should have an eigenvalue with multiplicity two. This can be
realized by the following cost function:
Fig. 9. Cable tension forces in ankle-cable configuration for task space
stiffness modulation through configuration redundancy.
C. Experimental setup and results
The overall experimental setup (for all experiments reported
in this paper) is depicted in Figure 10, with further details
available from [24]. Ground-truth results are obtained from
OptiTrack cameras mounted to rigid frame and track markers
associated with each rigid body (3 per planar moving body).
λmin
(12)
λmax
where λmin and λmax are the minimum and maximum eigenvalues of the stiffness matrix KX , respectively.
2) Directional stiffness: Significant stiffness in the desired
direction can be achieved by aligning the eigenvector associated with its maximum eigenvalue of the stiffness matrix KX
to be parallel to the desired direction. The cost-function to
realize such as case can be formulated as:
C(S) = 1 −
C(S) = |vmax ||u| − |vTmax u|
(13)
where vmax is the eigenvector associated with the maximum
eigenvalue of the stiffness matrix KX , and u is the desired unit
direction vector. This case can be utilized in the trajectory
tracking problem where the eigenvector associated with the
maximum eigenvalue of the stiffness matrix is reoriented to
be perpendicular to the tangent of the trajectory to reject the
lateral disturbances.
3) Maximum stiffness: Inspired by [44], we present the following cost function to maximize stiffness in both directions:
C(S) =
2 λ2
λmin
max
2
2
λmin + λmax
(14)
In order to eliminate the effect of unknown disturbances [35],
while moving along the desired trajectory, the cost function
(14) can be used to obtain the maximum stiffness along the
trajectory in both directions of eigenvectors.
Fig. 10. Experimental setup and OptiTrack system. OptiTrack cameras
mounted on the frame are used to provide ground truth.
The setup for exploiting the configuration redundancy for
ankle-cable configuration is depicted in Figure 11. There
are two slides on each side of the triangle. The sliders can
travel within the range [60, 320] mm. The total cables’ length
considering the springs in their rest states is [860, 670, 675]
mm. Using the linear fitting, the stiffness of the springs are
obtained as [61, 69, 69] N/m. The maximum springs extension
are set to 100 mm for safety purposes. The numerical optimization is performed point-to-point, and we used the interiorpoint algorithm which is available in MATLAB. The video
of the experiment is provided in supplementary materials,
IEEE Transactions on Robotics
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{H}
{O2 }
{O1 }
{F}
Fig. 11. Experimental setup for stiffness modulation of ankle-cable configuration with configuration redundancy where cables are connected to a single
point representing the ankle.
which demonstrates the real-time performance of the proposed
method.
In the experiment, we seek to maximize stiffness during the
gait cycle. Note that, here we have considered a quasi-static
motion for the assistance of paralyzed patients in which gait
cycle period is longer than average stride period. The experimentally obtained sliders’ positions are depicted in Figure 12,
which indicate that sliders’ positions remain in the predefined
range of [60, 320] mm. The values of eigenvalues λmax and
λmin are reported as λmax = 0.1418 and λmin = 0.07333. The
results also illustrate smoothness of slider motions achieved
by placing velocity limits. Both sliders on each side of the
base (S1 and Sp1 ) follow a similar motion but with a constant
offset from each other. Note that the cable tension becomes
larger as the offset between Si and Spi increases. For instance,
since the offset between S3 and Sp3 is more than the offset
between other sliders in the time interval 1s to 2s, the cable
tension T3 has larger value with respect to T1 and T2 during
that interval as depicted in Figure 9.
{O3 }
Fig. 13. Articulated-cable configuration with configuration redundancy. A
fixed coordinate frame {F} is located at the center of the triangular base, and
the local coordinate frames {Oi } are placed at the center of each side of the
triangle.
{H}
A1 ,(X1 , Y1 )
‫ݑ‬ଵ
݇ଵ
o1
S1
{O1 }
S1′
{F}
A1′ ,(X1′ , Y1′)
sy
{E}
(Xe ,Ye )
Rail 3
Fig. 14. Parameters defined for the motion analysis of the articulated-cable
configuration with configuration redundancy.
Considering the kinematics and statics of the configuration
depicted in the Figure 14, the optimization problem is formulated as follows:
Maximize:
λ2 λ2
C(S) = 2 max min2
(15)
λmax + λmin
Fig. 12. Sliders’ positions in the ankle-cable configuration for task space
stiffness modulation through configuration redundancy.
Subject to:
[
tT1 ∂∂ uq1
1
IV. S TIFFNESS M ODULATION OF L EG O RTHOSIS IN THE
A RTICULATED -C ABLE C ONFIGURATION E XPLOITING
C ONFIGURATION R EDUNDANCY
Joint stiffness relates the angular displacement of the hip
and knee joints in the articulated leg orthosis to the external
torques exerted at these joints. The attached cables can apply
forces to the articulated orthosis provided that they are always
in tension. Here, as shown in Figure 13 and Figure 14, we
attempt to control the stiffness of the hip and knee by changing
system configuration such that equilibrium equation Aart T =
−τe is satisfied.
tT1 ∂∂ uq1
2

]
Ks1 (l1 + S′ 1 + S1 − L01 )
1
1
 Ks2 (l2 + S′ 2 + S2 − L02 )  = −τe
tT2 ∂∂ uq2 tT2 ∂∂ uq3
Ks3 (l3 + S′ 3 + S3 − L03 )
2
2
[
]
Si ∈ [Simin , Simax ] , Si′ ∈ Si′min , Si′max
(
)
li + Si′ + Si − L0i ∈ [0, ∆Simax ]
tT2 ∂∂ uq2
tT3 ∂∂ uq3
|Ṡi′ | ≤ Ṡi′max , |Ṡi | ≤ Ṡimax
The experimental setup for exploiting the configuration
redundancy using articulated-cable configuration is demonstrated in Figure 15 with further details available in [24].
The video of the experiment is provided in supplementary
materials.
IEEE Transactions on Robotics
8
Subject to: T > 0, where λ () refers to eigenvalues of a
matrix.
A. Kinetostatic formulation
A1 ,(X1′ ,Y1′)
b1
C1 ,(X1 , Y1 )
P1
c1
l1
l1
l′1
h1
{F}
k1
S1
o1 {O1 }
S1
S′1′
{O1 }
sy
V. C ARTESIAN S TIFFNESS M ODULATION OF L EG
O RTHOSIS IN THE A NKLE -C ABLE C ONFIGURATION U SING
ACTUATION R EDUNDANCY
Rail 3
{E}
C1 ,(X1 ,Y1 ) A1 ,(X1′ , Y1′)
k1
{H}
Fig. 15. Experimental setup for stiffness modulation of articulated-cable
configuration with configuration redundancy.
Active stiffness control problem can be addressed using
actuation redundancy by employing surplus cables [45]. However, this method cannot provide a satisfactory result when the
number of actuators is less than the independent components
of desired stiffness matrix. Hence, a feasible solution may
not always exist, which means the desired stiffness cannot
be achieved. Inspired by [46], [47], we propose a method
based on the smallest eigenvalue control to guarantee the
lower bound of the stiffness. Note that such a bound improves
the performance of the system in dealing with the trajectory
tracking and disturbance rejection problems.
l′1
S′1′
(Xe ,Ye )
Fig. 17. Parameters defined for the motion analysis of the ankle-cable
configuration with actuation redundancy.
The cable elasticity couples the kinematics problem with
statics problems. The tension in the cable is modeled as linear
spring:
(
)
Ti = Ksi li + 2Si′ + Si + bi + li − L0i
(17)
where bi is the distance between the point (Xi′ ,Yi′ ) and (Xi ,Yi )
as shown in Figure 17. Resolving actuation redundancy is
achieved through stiffness optimization. The problem may be
generally formulated as follows:
Minimize:
)2
)
( (
∂ Aank
d
(18)
λ −
T + Aank Ks ATank − λmin
∂X
Subject to:
{H}
{O2 }
{F}
{O1 }
{O3 }
Fig. 16. Ankle-cable Configuration with actuation redundancy and series
elastic cables
As mentioned in [47], one way for resolving redundancy
resolution is adding desired task stiffness KdX to the system,
but since the strict desired stiffness is not achievable, we
relax this criterion with defining a lower bound stiffness to
resolve the redundancy problem. In general, the matrix KX is
not a symmetric matrix, hence, it should be assured that the
smallest eigenvalue of task stiffness matrix, λmin , is greater
d , at each desired point in
than a predefined lower bound, λmin
the space. Therefore, the objective function can be written as:
Minimize:
)2
)
( (
∂ Aank
d
(16)
λ −
T + Aank Ks ATank − λmin
∂X
[
t̂1
t̂2


′
′
] Ks1 (l1 + 2S1′ + S1 + b1 + l1′ − L01 )
t̂3 Ks1 (l2 + 2S2 + S2 + b2 + l2 − L02 ) = −Fe
Ks1 (l3 + 2S3′ + S3 + b3 + l3′ − L03 )
]
[
Si ∈ [Simin , Simax ] , Si′ ∈ Si′min , Si′max
(
)
li + 2Si′ + Si + bi + li′ − L0i ∈ [0, ∆Simax ]
|Ṡi′ | ≤ Ṡi′max , |Ṡi | ≤ Ṡimax
In both ankle- and articulated-cable configurations with configuration redundancy, the base mobility significantly improves
the wrench-closure workspace (WCW) [48] and wrenchfeasible workspace (WFW) [49], [50] compared to the setup
with fixed bases. Therefore, the positivity of cables tension for
the given end-effector trajectory in such configurations with
the mobile bases (Xi ,Yi ) is guaranteed. Figure 18 illustrates
the cable tension forces obtained by simulation.
B. Experimental setup and results
The experimental setup is shown in Figure 19 with further
details available in [24]. In the experiment, the stiffness is
maximized during tracking the normal gait. The value of lower
d = 0.079942. The sliders’ position
bound stiffness is set as λmin
IEEE Transactions on Robotics
9
{H}
{O1 }
{O 2 }
{F}
Fig. 18. Cable tension forces in the ankle-cable configuration for task space
stiffness modulation through actuation redundancy.
{O 3 }
are depicted in Figure 20. The video of the experiment is
provided in supplementary materials.
Comparing Figure 12 and Figure 20, one can observe the
range of sliders’ position based on configuration redundancy
is larger than the range of sliders’ position based on actuation
redundancy. Further, since maximizing the Cartesian stiffness
in all direction requires an increase in the internal tension of
the system, the cable tension in Figure 18 is much higher than
the cable tension in Figure 9.
Fig. 21. Stiffness modulation of articulated-cable configuration with the
variation of actuation forces.
A1 ,(X1′ , Y1′)
b1
C1 ,(X1 , Y1 )
P1
c1
l′1
l1
l1
{H}
l′1
h1
{F}
k1
S1
o1 {O1 }
S1
S′1′
{O1 }
sy
Rail 3
{E}
C1 ,(X1 , Y1 ) A1 ,(X1′ , Y1′)
k1
S′1′
(Xe , Ye )
Fig. 22. Variables and parameters of articulated-cable configuration with the
variation of actuation forces.
Fig. 19. Experimental setup for stiffness modulation of ankle-cable configuration with actuation redundancy
redundancy such that equilibrium equation Aart T = −τe to be
satisfied.
The redundancy resolution problem for the ArticulatedCable Configuration with Actuation Redundancy, as shown in
the Figure 22, may be formulated as:
Minimize:
( (
)
)2
∂ Aart
d
(19)
λ −
T + Aart Ks Aart T − λmin
∂X
Subject to:
Fig. 20. Sliders’ position in the ankle-cable configuration for task space
stiffness modulation through configuration redundancy
VI. C ARTESIAN S TIFFNESS M ODULATION OF
A RTICULATED -C ABLE C ONFIGURATION U SING
ACTUATION R EDUNDANCY
Joint stiffness KQ relates the angular displacement of the
hip and knee joint in the articulated leg orthosis to the external
torques exerted at these joints. Cable mechanisms can apply
forces to the articulated orthosis provided that they are always
in tension. Here, as shown in Figure 21, 22, we attempt to
control the joint stiffness of the hip and knee by actuation
[
tT1 ∂∂ uq1
1
tT2 ∂∂ uq2
1
tT3 ∂∂ uq3
2
2
2
tT1 ∂∂ uq1
tT2 ∂∂ uq2
1
tT2 ∂∂ uq3

]
Ks1 (l1 + 2S1′ + S1 + b1 + l1′ − L01 )
 Ks2 (l2 + 2S2′ + S2 + b2 + l2′ − L02 )
Ks3 (l3 + 2S3′ + S3 + b3 + l3′ − L03 )
= −τe
[
]
Si ∈ [Simin , Simax ] , Si′ ∈ Si′min , Si′max
(
)
li + 2Si′ + Si + bi + li′ − L0i ∈ [0, ∆Simax ]
|Ṡi′ | ≤ Ṡi′max , |Ṡi | ≤ Ṡimax
The experimental setup is shown in Figure 23. The video
of the experiment is provided in supplementary materials.
IEEE Transactions on Robotics
10
A. Variable stiffness module
Figure 24 illustrates the overall leg-orthosis emulator in
the ankle-cable configuration with the active variable-stiffness
module attached. The cables are represented by a thick blue
line which is routed through fixed pulleys as depicted with
the red circles. Pulleys Pi as depicted with green circles are
connected to the sliders {Si } by linear springs. Figure 25
presents a detailed view of one cable with the active variable
stiffness module and depicts the pertinent parameters.
A1 ,(X1′ , Y1′)
Fig. 23. Experimental setup for stiffness modulation of articulated-cable
configuration with actuation redundancy.
α1
C1 ,(X1 ,Y1 )
a1
P1
c1
c1
l1
α1
l1
o1 {O1 }
Ŝ1
S′1′
S1
sy
{E}
Rail 3
ĉ1
S1
h
{F}
â1
k1
{H}
VII. A NKLE -C ABLE C ONFIGURATION WITH VARIABLE
S TIFFNESS M ODULES
Another way of modulating the stiffness in the ankleand articulated-cable configurations is to introduce variable
stiffness module for each individual cable. Tonietti et. al [51]
developed an electromechanical variable stiffness actuation
motor for physical human-robot interaction. Azadi et. al [25]
presented the concept of variable stiffness elements based
on antagonistic forces in the cable-driven mechanisms. Yeo
et. al [44] developed a passive variable stiffness module to
achieve variable stiffness for the cable-driven mechanisms.
Such variable stiffness modules increase robustness in the
cable-driven mechanisms during interactions with unknown
environments. However, the main drawback with the passive
variable stiffness modules is the stiffness varies proportionally
to the internal tension. Hence, to achieve a higher stiffness,
motors are required to draw more current, thereby increasing
overall energy consumption.
In [52], Zhou et. al designed active variable stiffness modules to independently adjust the stiffness and internal tension
in planar cable mechanisms. The method employs a second
motor to adjust the spring attachment point – effectively
adjusting the equilibrium configuration – while keeping cable
tension relatively small. In this section, we adopt this active
variable stiffness module for the leg-orthosis emulator and
demonstrate that this approach is simpler and performs better
compared to the configuration redundancy approach.
b1
B1
{O1 }
(Xe ,Ye )
Fig. 25. Variables and parameters of one side of leg orthosis-like mechanism
with variable stiffness modules.
In the side 1 of triangular base as depicted in Figure 25, let
|A1C1 | = b, |P1 B1 | = a1 , |P1 O1 | = l1′′ , and ∠A1 P1 B1 = α1 . By
the principle of virtual work:
T δ L + Fs δ l1′′ = 0
(20)
′′ ) such that l ′′ is free length of spring,
where Fs = Ks1 (l1′′ − l01
01
and l1′′ = l1′ − S1 − a1 . Therefore,
Ks1
(21)
4 cos2 α1
where KL1 is output cable stiffness of cable 1 which is related
to the constant spring stiffness Ks1 . Thus, the cable tension
becomes:
KL1 =
T=
Fs
2 cos α1
(22)
The linear slider {Si } moves the spring attachment point to
adjust the equilibrium position of the linear spring to adjust
the stiffness. The output cable stiffness is related to angle
α1 – higher stiffness can be achieved by increasing angle α1
even with a lower internal cable tension. This leads to smaller
actuators, lower power consumption, and increasing safety in
interaction with humans.
{H}
{O1 }
{O2 }
{F}
{O3 }
Fig. 24. Leg orthosis-like configuration with variable stiffness modules.
B. Kinetostatics formulation
The active-module stiffness and its corresponding slider
positions may be obtained by solving the following equation
(23), with desired Cartesian stiffness KdX , desired tension Td ,
end-effector location Xe and Ye the corresponding structure
matrix all known. Note that, because of the symmetry of
stiffness matrix, this results in a set of three linear equations
with three module stiffness as unknowns are straightforward
to solve. However, due to real-world constraints such as slider
travel limits and linear spring extension limits, it may not be
Subject to:
∂ A∗ d
=−
T + A∗ Ks A∗ T
∂X
−A+ Fext + β Ker(A∗ ) > 0
[
]
Si ∈ [Simin , Simax ] , Si′ ∈ Si′min , Si′max
KdX
li′ − Si − ai − L0i ∈ [0, ∆Simax ]
1245
124
1215
1
01215
0124
01245
01023123
7
4
7
125
1
69
easy to prescribe a desired tension that will work for all cases
as shown in Figure 26. Therefore it is more convenient to pose
it as the following optimization problem with the objective of
minimal tension:
Minimize:
(23)
β2
11
69
IEEE Transactions on Robotics
0125
0124
1
679
124
123
004425 04 0125 1 125 4
679
425
Fig. 27. Changing Cartesian stiffness with (a) increasing internal tension, (b)
adjusting the module stiffness. Note that, these figures compare the Cartesian
stiffness variation by increasing internal tension as depicted in sub-figure (a)
and module stiffness as depicted in sub-figure (b). The direction of increasing
module stiffness is shown with a red arrow in each sub-figure.
|Ṡi′ | ≤ Ṡi′max , |Ṡi | ≤ Ṡimax
C. Experimental setup and results
where T = −A+
∗ Fext + β Ker(A∗ ) such that A∗ is Aank for
ankle-cable configurations and Aart for articulated-cable configurations, A+
∗ is the Moore-Penrose pseudoinverse, and Fext
is Fe for ankle-cable configurations and τe for articulated-cable
configurations. To avoid getting stuck in local optima, the
interior point solver is used which meet the constraints within
step time limits. Since sliders have limited speed and may not
be able to reach the desired speed and position instantaneously,
the desired Cartesian stiffness may not be achieved in entire
trajectory. In this case, we can relax the equality constraints
on stiffness, i.e. instead of specifying the desired stiffness
matrix, we can optimize a secondary objective function such
as directional stiffness, as shown in equation 13.
Fig. 26. Cable tension in ankle-cable configuration with variable stiffness
module. It is illustrated that the cable tension is remained constant for more
than half of the gait cycle, while the stiffness modulation of the ankle-cable
mechanism is satisfied.
Unlike changing the internal cable tension, changing the
joint stiffness Ksi , enable us to have better control over
the task space stiffness. Assuming a payload is located at
the center of platform, the initial stiffness is set as Ksi =
diag([0.07, 0.07, 0.07]) N/mm, and cable tension T = [1, 1, 1]T
N, which satisfies the static equation with zero external
wrench. According to Eq. (5), the Cartesian stiffness matrix
Kx can be adjusted by variation of (i) system configuration
A∗ , (ii) cable tension T and (iii) module stiffness Ks . Figure
27(a) depicts the variation in Cartesian stiffness by increasing
internal tension ten-fold. This is at least an order of magnitude
less than the Cartesian stiffness variation achieved by a tenfold change in joint-stiffness using the active variable-stiffness
module, as seen in Figure 27(b).
The experimental setup is shown in Figure 28 where the
stiffness is maximized during the normalized gait trajectory.
The sliders’ position are depicted in Figure 29. The video of
the experiment is provided in supplementary materials.
As seen in Figure 26, the cable tension is remained constant
more than half of the gait cycle, while the stiffness modulation
of the ankle-cable configuration is satisfied. In this case, the
stiffness modulation is mainly accomplished by variation of
angle α1 which is adjusted by changing the slider Si position.
As illustrated in Figure 29, the slider Si position variation is
significantly reduced in comparison with other cases as well.
Fig. 28. Experimental setup for stiffness modulation of ankle-cable configuration with variable stiffness module.
Fig. 29. Sliders’ positions in the ankle-cable configuration with variable
stiffness modules. As illustrated, the sliders’ positions on each side are the
same to keep the internal cable tension constant, which lead to a significant
reduction in the sliders’ position variation.
IEEE Transactions on Robotics
VIII. D ISCUSSION
In this paper, we presented and evaluated alternate methods
for stiffness modulation of alternate cable-attachment configurations to achieve smooth and natural gait rehabilitation. The
three proposed methods for stiffness modulation are based
on (i) exploiting configuration redundancy introduced by way
of adding mobility into the bases; (ii) actively changing
the cable tension; and (iii) employing an inline variablestiffness module to vary the stiffness of each individual cable.
These three methods were evaluated with two alternate cableattachment configurations: (i) the ankle-cable configuration
and (ii) articulated-cable configuration. Experimental results
show that among the three proposed methods, the variable
active-stiffness technique was more effective in modulating the
overall system stiffness. This method permits the independent
modulation of the perceived stiffness with controlling the
structural parameters.
Determining a suitable location of cuffs on the scaledleg with respect to the corresponding joints (hip and knee
joints), and finding the proper diameters for the cuffs is
critical. Similarly, the fixed location of pulley Ci (Xi ,Yi ) in
the following configurations (i) changing cable tension and (ii)
changing joint stiffness is critical. In both scenarios, we used
a Monte-Carlo approach within the parameter-space to rapidly explore configurations and downselect feasible parameter
ranges based on satisfaction of wrench-feasible constraints.
The small differences between theoretical and experimental
results are attributed to uncertainties introduced by unmodeled
(friction at joints, friction between pulley/cable), imperfectly
modeled (linear least-squares fit for spring elasticity, inelastic
cables) or calibration (motion-capture cameras).
Numerous additional issues need to be taken into account
in realizing a full-scale gait training system principal of
these would be ensuring safety. We focused on passive mode
rehabilitation i.e. when there is no muscle activity. We will
look forward to exploring the other modes (resistive- and
active-mode rehabilitation) where personalized human-user
muscle activation capabilities (and ensuing bilateral physicalinteractions) must be considered.
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