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A New Actuator With Adjustable Stiffness Based on a Variable Ratio Lever
Mechanism
Article in IEEE/ASME Transactions on Mechatronics · February 2014
DOI: 10.1109/TMECH.2012.2218615
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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 1, FEBRUARY 2014
55
A New Actuator With Adjustable Stiffness Based
on a Variable Ratio Lever Mechanism
Amir Jafari, Nikos G. Tsagarakis, Irene Sardellitti, and Darwin G. Caldwell
Abstract—This paper presents the actuator with adjustable stiffness (AwAS-II), an enhanced version of the original realization
AwAS. This new variable stiffness actuator significantly differs
from its predecessor on the mechanism used for the stiffness regulation. While AwAS tunes the stiffness by regulating the position of
the compliant elements along the lever arm, AwAS-II changes the
position of the lever’s pivot point. As a result of the new principle,
AwAS-II can change the stiffness in a much broader range (from
zero to infinity) even by using softer springs and shorter lever arm,
compared to AwAS. This makes the setup of AwAS-II more compact and lighter and improves the stiffness regulation response. To
evaluate the aptitude of the fast stiffness adjustment, experiments
on reproducing the stiffness profile of the human ankle during
the stance phase of a normal walking gait are conducted. Results
indicate that AwAS-II is capable of reproducing an interpolated
stiffness profile of the ankle while providing a net positive work
and thus a sufficient amount of energy as required for the toe-off.
Index Terms—Adaptable pivot point, adjustable stiffness, variable ratio lever.
I. INTRODUCTION
N HUMAN–ROBOT interaction area, compliance has an important role [1]–[4]. Physical passive compliance decouples
the inertia of the actuator, which normally has a large value,
from inertia of the link, and it decreases the stiffness of the
joint [5]–[12]. Adaptable compliance indeed helps to improve
the performance in different circumstances. For instance, energy
efficiency can be improved by properly tuning the stiffness [13],
[14]. In the literature, different design approaches for actuation
systems that allow the stiffness regulation are proposed. These
variable compliance actuation systems typically employ two
actuator units in combination with passive elastic elements to
control, independently, the compliance and the equilibrium position of the actuated joint. These systems can be implemented
by both antagonistic and series configurations.
Antagonistic setups follow the same concept of mammalian
anatomy, i.e., a joint actuated by two muscles [15]–[19]. On
the other hand, the serial configuration is characterized by the
I
Manuscript received January 30, 2012; revised April 11, 2012; accepted July
25, 2012. Date of publication October 9, 2012; date of current version January 17, 2014. Recommended by Technical Editor H. Fujimoto. This work was
supported by the VIACTORS European Commission under Project FP7-ICT2007-3.
A. Jafari is with the Department of Mechanical and Process Engineering, BioInspired Robotic Laboratory, Swiss Federal Institute of Technology, CH-8092
Zürich, Switzerland (e-mail: ajafari@ethz.ch).
N. G. Tsagarakis, I. Sardellitti, and D. G. Caldwell are with the Department of Advanced Robotics, Istituto Italiano di Tecnologia, 16163
Genoa, Italy (e-mail: nikos.tsagarakis@iit.it; i.sardellitti@gmail.com; Darwin.
Caldwell@iit.it).
Digital Object Identifier 10.1109/TMECH.2012.2218615
mechanical series, motor–gear–compliant element and it can be
implemented both in linear and rotational designs [20]–[24].
The energy requirements for the compliance regulation depend on the implementation principle. In these systems, both
antagonistic and serial configurations, the stiffness adjustment
is obtained by tuning the pretension of the spring. As a consequence, the change of the stiffness requires a considerable
amount of energy since the actuator, which regulates the stiffness, needs to overcome the forces due to the spring deflection.
Recently a new design approach based on the variable lever
arm mechanism has been introduced. In these implementations,
the pretension of the elastic element is kept constant while the
stiffness adjustment depends on the effective ratio of the lever.
In both the actuators with adjustable stiffness (AwAS) [25] and
the hybrid dual actuator unit [26], the ratio is altered by changing the position of the springs with respect to the pivot point.
Alternatively in the vsaUT, proposed by [27], the ratio is set by
moving the load point on the lever. Theoretically, by varying the
position of the springs, the stiffness can be adjusted from zero
to a maximum value, which depends on the spring’s rate and
length of the lever. Alternatively, by shifting the load point the
stiffness range is defined from a minimum value, which depends
on the spring’s rate, up to the infinity.
This paper proposes the second version of the AwAS-II. The
novelty with respect to other variable lever mechanisms consists
in the principle implemented for adjusting the stiffness. In the
AwAS-II, the position of the pivot point changes while both the
position of the springs and the load point are kept constant. The
stiffness range does not depend on the spring’s rate and length
of the lever, and it can be adjusted from zero to completely
rigid. This allows for the introduction of shorter lever compared
to AwAS which permits AwAS-II to be a lighter and more
compact setup. The reduced lever length also allows for a faster
regulation of the stiffness level since the total distance that the
pivot needs to travel is reduced. Even though using softer springs
can also help to reduce the size and weight of the system, energy
storing capacity in the AwAS-II will decrease if softer springs
are used.
Currently, the regulation of the compliance in assistive systems, such as prosthetic devices and exoskeleton is gaining interest due to the energy efficiency [28]–[32]. In addition, since
these devices require a significantly fast stiffness adjustment
according to the human performance, the speed of stiffness
adjustment is vital. For instance, during normal walking, the
ankle stiffness rapidly changes over the stance phase in order to
provide the net positive work which is required for the toeoff [33]. However, most of the commercially available prosthetic feet have not the capability to change the stiffness [29].
Therefore, they are not widely accepted by the users since an
1083-4435 © 2012 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 1, FEBRUARY 2014
Fig. 1. Schematic of the several adjustable stiffness concepts based on the lever mechanism. (a) Stiffness is altered by changing the effective arm length through
variation of the spring location (AwAS [24], hybrid dual actuator unit [25]). (b) Stiffness is adjusted by changing the point where the force is applied on the lever
(vsaUT [26]); (c) Stiffness is changed through the variation of the lever ratio by moving the location of the pivot position.
inappropriate stiffness profile of the ankle prosthetic may lead
to increase the muscle activities that contribute to body forward
propulsion. Some other prosthetic systems have a prescribed
stiffness variation profile. The effect of different preset stiffness
profiles in energy storage and return prosthetic ankle on muscle activity has been studied in [32]. However, since the stiffness profiles are predefined, these kinds of prosthesis are only
suitable for particular conditions, for instance certain walking
speed. To realize an ankle prosthetic device applicable to different situations, variable stiffness actuators are expected to be a
promising approach. However, when a variable stiffness actuator
is introduced to drive the ankle prosthetic, it needs to guarantee a reasonably fast stiffness adjustment as well as the energy
storage capacity required for the toe-off. To this purpose, experiments are conducted to verify the effectiveness of AwAS-II
in reproducing the ankle stiffness profile behavior during the
stance phase.
This paper is organized as follows. In Section II, the concept of AwAS-II is explained. The mechanical realization of
AwAS-II prototype is presented in Section III. Section IV discusses the modeling of AwA-II. The performance of AwAS-II
prototype in tuning the stiffness is analyzed in Section V. Experimental results are discussed in Section VI. Finally, the conclusion and future works are presented in Section VII.
II. CONCEPT OF THE MECHANISM IMPLEMENTED IN AWAS-II
To explain the advantages of the AwAS-II concept with respect to the other variable stiffness actuators which employ a
lever mechanism, a comparison is presented in this section.
The mechanism for adjusting the stiffness applied to AwAS
[24] and to the hybrid dual actuator unit [25] is based on a variable lever arm mechanism. Assuming a lever which is rotating
around its pivot [see Fig. 1(a)], with two springs antagonistically
attached and able to move with respect to the pivot, the lever
arm is defined as the distance between the pivot and the springs.
Therefore, the apparent stiffness at the joint can be tuned by
changing the length of the lever arm. In detail, the joint is stiffer
when the length of the lever arm increases. The range of the
stiffness is defined between zero and a maximum value which
depends on the total length of the lever and also on the spring’s
rate.
Alternatively, in the stiffness adjustment mechanism applied
to the vsaUT [26], as it is shown in Fig. 1(b), the springs are
fixed and the load point varies on the lever. The length of the
effective lever arm, in this case is the distance between the pivot
and the load point. As a consequence, also in this case, the
stiffness of the lever can be tuned by changing the length of
the effective lever arm. However, the shorter is the length of
the lever arm, the higher is the apparent stiffness. The range of
the stiffness is ultimately defined between a minimum value,
which depends on the spring’s rate, up to infinity. Important is
to note that this mechanism is only applicable to prismatic joint
(where the stiffness is defined as the force required for one unit
of linear displacement). Whereas, in the revolute joints (where
the stiffness is defined as the torque required for the unit of
angular displacement) changing the location of the load point,
also implies a variation of the torque, thus the apparent stiffness
at the output link remains unchanged.
Differently from the principles described earlier, the concept
of AwAS-II is based on a lever mechanism with variable ratio.
In this case, as it is shown in Fig. 1(c), to tune the stiffness
the position of the pivot is changed while the location of both
the force application and the springs is kept constant. When the
pivot point moves, the ratio of the lever changes. This ratio is
defined as L1 /L2 , where L1 is the distance between the pivot
and the point where the springs are connected and L2 is the
distance between the pivot and the load point. When the pivot is
aligned with the springs (L1 = 0), the ratio becomes zero thus
the lever stiffness is zero. On the other hand, when the pivot is
aligned with the load point (L2 = 0) the stiffness increases up
to infinity and the link becomes completely rigid.
III. MECHANICAL REALIZATION OF AWAS-II
The concept of the lever with variable pivot point is implemented in AwAS-II as it is shown in Fig. 2(a).
The lever is connected to the springs from one end and from
the other end it is connected to the output link through a rotary joint which allows transmit of force between the lever and
the output link while the lever can rotate around this rotary
joint when the output link is loaded. Therefore, this rotary joint
corresponds to the force point in Fig. 1(c). The CAD view of
AwAS-II, as it is shown in Fig. 2(b), consists of a motor M1
devoted to set the link position and a motor M2 inserted to tune
the stiffness. M1 is a brushless frameless dc motor [Emoteq:HT02300] with a peak torque of 2.35 N·m connected to a harmonic
drive gearbox with a ratio of 50:1. The output of the harmonic
drive is then connected to the intermediate link. The motor
M2 mounted on the intermediate link is a miniature brushed
JAFARI et al.: NEW ACTUATOR WITH ADJUSTABLE STIFFNESS BASED ON A VARIABLE RATIO LEVER MECHANISM
57
Fig. 2. (a) Implementation of the lever with adjustable pivot point in AwAS-II, (b) cross section of the CAD design while the output link is unloaded, (b) stiffness
regulation system and the prototype of AwAS-II (d).
frameless torque motor [Kollmorgen:QT-0707] with a peak
torque of 0.05 N·m. It is connected to a miniature harmonic
drive gearbox with ratio of 100:1. The output of the miniature
harmonic drive is connected to a ballscrew drive. The pivot
is a cam follower [McGill:MCFR] which has a diameter of
13 mm mounted on the ballscrew’s nut and can be linearly
moved by motor M2 [see Fig. 2(c)]. A linear guide, parallel to
the ballscrew, while preventing the nut to rotate, constraints the
pivot to move along the lever. The linear guide also facilitates for
the lateral loads applied to the ballscrew. In the AwAS-II prototype [see Fig. 2(d)], two torsion springs with rate of 10 N·m/rad
are antagonistically placed with a preangular deflection equal
to half of their maximum allowable deflection. This guarantees
the continuous engagement of the springs between the lever and
the output link for different angular deflections of the joint. The
angular difference between the intermediate link and the output
link is the angular deflection. A mechanical lock constrains this
angular deflection to be in the range of [–0.3, 0.3] rad. When
the link deviates from its equilibrium position, the end of the
lever, which is connected to the springs, slides along the spring’s
legs. To have a frictionless sliding motion, two cam followers
are placed between the lever and each spring [see Fig. 2(b)].
It should be mentioned here that when the pivot is aligned
with the rotary joint between the lever and the output link [see
L2 = 0 in Fig. 2(a)], the force which is transmitted between
the lever and the output link cannot produce any torque around
that rotary joint to rotate the lever and is directly transferred
from the output link to the intermediate link (without involving
the springs). In this case, the AwAS-II joint is completely rigid.
On the other side, when the pivot is aligned with the center
of rotation of the AwAS-II joint (L1 = 0 in Fig. 2(a), which
is also where springs are attached to the lever), by loading the
Fig. 3.
Connections between essential elements of AwAS-II.
output link, the lever (which always rotates around the pivot),
will rotate around the center of rotation of the AwAS-II joint
together with the output link and so there would be no rotation
between the lever and the output link. Therefore, springs will
not be deflected and so there would be no elastic torque and thus
the stiffness is zero.
Fig. 3 shows the block diagram of the connections between
essential elements of AwAS-II with a partial section on the motor
M1 components. As it is clear from this graph, the stator of M1
is connected to the base frame and the rotor M1 is connected
to the intermediate link (through the harmonic gear). The pivot
is sliding along the intermediate link. The lever is connected to
the output link through a rotary joint and from the other end it
is attached to the springs. Springs are located between the lever
and the output link. The output link is also connected to the base
frame through another rotary joint (see Fig. 2) which allows the
rotation of the output link around the same axis of the M1.
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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 1, FEBRUARY 2014
TABLE I
GENERAL SPECIFICATIONS OF AWAS-II
TABLE II
PARAMETERS FOR AWAS-II
Fig. 5.
spring.
Fig. 4.
Schematic of AwAS-II.
The sensing system of AwAS-II includes four position sensors
and one torque sensor. In detail, an optical encoder measures
the angular position of the motor M1, two absolute magnetic
encoders measure the position of the output link and the intermediate link and an incremental encoder measures the rotation
of the ball screw. In addition, a torque sensor has been installed
to measure the torque applied to the intermediate link and it is
based on a custom load cell element equipped with semiconductor strain gauges.
A comparison between the specifications of the original
AwAS [24] and AwAS-II is shown in Table I.
AwAS-II design achieves a larger range of motion and stiffness regulation. In addition, it can rapidly adjust the stiffness
level due to the reduced length of both the ballscrew and the
lever. AwAS-II is capable of storing more energy even though
it has a significantly reduced size.
IV. MODELING OF AWAS-II
The schematic of AwAS-II model is illustrated in Fig. 4.
The dynamics of this system, neglecting the gravity contribution, can be described as
are the external, elastic, and resistant torques, respectively. The
other parameters in (1) are described in Table II.
The joint stiffness K = ∂τE /∂φ, is defined as the derivative
of the elastic torque τE with respect to the angular deflection ϕ =
q − θ1 where the elastic torque τE = ∂E/∂φ is the derivative
of the elastic energy E with respect to the angular deflection
ϕ. Therefore, to formulate the joint stiffness, the elastic torque
has to be obtained which can be determined from the elastic
energy. To derive the elastic energy of the AwAS-II, the torsion
springs are initially replaced by equivalent virtual compression
springs. In this case, the rate of the compression springs has to
be expressed in terms of torsion spring’s rate. As it is shown in
the Fig. 5, the force Ft is applied to the torsion spring at the
distance rt from the center of rotation of the torsion spring.
To simplify the model, it is assumed that the force Ft is always
applied in alignment with the axis of the compression spring.
The force Ft deflects the torsion spring by δγ
x
(2)
δγ ≈
rt
where x is the linear displacement of the torsion spring’s arm
as it is shown in the Fig. 5. The resultant torque applied to the
torsion spring is then given by
Tt = Ft rt .
B1 θ̈1 + ϕ1 θ̇1 − τE = τM 1
(1)
where θi and τM i are the position (after the gear reduction)
and the torque associated with the motor Mi with iε [1], [2],
respectively; q is the position of the output link, τext , τE , and τr
(3)
Using (2) and (3), the rate of the torsion spring can be obtained
such as
Kt = Ks rt2
I q̈ + N q̇ + τE = τext
B2 θ̈2 + ϕ2 θ̇2 + τr = τM 2
Replacing the torsion spring by an equivalent virtual compression
(4)
where the term Ks represents the rate of virtual compression
spring. Since the torsion springs are assembled with a predeflection, the virtual compression springs are accordingly assumed
to have a precompression at the no load equilibrium position.
Therefore, the elastic energy stored at the equivalent compression springs due to the angular deflection of the output link is
JAFARI et al.: NEW ACTUATOR WITH ADJUSTABLE STIFFNESS BASED ON A VARIABLE RATIO LEVER MECHANISM
Fig. 6.
59
Schematic of the stiffness adjustment principle.
given as
E = Ks x2
(5)
where
x = P Ā sin α
(6)
is the displacement of the virtual spring, P Ā is the distance of
the pivot with respect to the springs, and α is the rotation angle
of the lever with respect to the intermediate link, as shown in
Fig. 6. This can be formulated such as
α = sin−1 [(1 + r) sin φ]
(7)
where
r=
P Ā
L − P Ā
(8)
is the ratio of the lever and L is the length of the lever. The
location of the pivot is adjusted by rotating the ballscrew by the
motor M2, therefore,
P Ā = nθ2
(9)
where n (0.0025/2π m/rad) is the transmission ratio between
the motor M2 and the ballscrew.
By considering (2)–(9), the elastic energy can be derived such
as
2
nθ2
Kt
E = 2 L2
sin2 φ.
(10)
rt
L − nθ2
Therefore, the elastic torque can be formulated such as
2
nθ2
Kt
τE = 2 L2
sin(2φ)
(11)
rt
L − nθ2
and thus the stiffness can be described as
2
nθ2
Kt 2
K=2 2L
cos(2φ).
rt
L − nθ2
(12)
From (12), it can be seen that the joint stiffness K depends
on the lever ratio r, the rate of the springs Kt , the length of
the torsion spring’s arm rt , the total length of the lever L, and
the angular deflection ϕ of the output link with respect to the
intermediate link. However, the range of the stiffness, which is
between zero to infinity, is due to the range of the lever ratio r.
The other parameters in (12) only affect the nonlinearity of the
stiffness curve.
Fig. 7. Stiffness regulation for the AwAS-II prototype based on the traveling
distance of the pivot inside the lever.
To adjust the stiffness, the motor M2 needs to overcome a
resistant torque τr in addition to the friction and inertia components related to the motor M2 and the ballscrew. The energy
spent by the motor M2 to overcome the resistant torque is the
same as the one stored into the springs. Therefore, the resistant
torque τr = ∂E/∂θ2 defined as the derivative of the elastic energy E with respect to the position of the second motor θ2 can
be expressed as
3
L
Kt
τr = 2 2
n2 θ2 sin2 φ.
(13)
rt L − nθ2
Therefore, on the basis of (13) the resistant torque τr is zero
when there is no angular deflection (ϕ = 0). As a consequence,
in this case, to set the stiffness the motor M2 only needs to
overcome the friction and accelerate the inertia of the motor M2
and the ballscrew. This is an important advantage of variable
stiffness actuators which regulates the stiffness using the lever
mechanism. Other types, both in antagonistic and serial configurations, which tune the stiffness through variation of the spring’s
pretension, require more energy since they have to work against
the force due to the spring’s deflection.
V. PERFORMANCE ANALYSIS
To show the variation of the joint stiffness with respect to
the movement of the pivot, a parameter λ = PA/L is introduced.
This parameter is the normalized traveling distance of the pivot
and using (8) can be reformulated such as
λ=
r
.
1+r
(14)
Fig. 7 shows the stiffness variation of the AwAS-II prototype
(12) as a function of the pivot normalized traveling distance λ.
As this graph shows, for λ < 0.6, the stiffness increases slightly
and the range of the pivot traveling distance which effectively
tunes the stiffness is about 40% of the total length of the lever.
Fig. 8 shows the required energy to tune the stiffness which
has to be provided by the motor M2 as a function of angular
deflection ϕ and ratio r calculated on the basis of (9).
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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 1, FEBRUARY 2014
Fig. 10.
Fig. 8.
Energy required to change the stiffness.
AwAS-II torques/angular deflection as a function of the ratio r.
uation of link oscillation. The feed forward term g(q) is added
for gravity compensation.
The controller of the motor M2 is given by
τM 2 = kM 2p (θ2d − θ2 ) + kM 2d (θ̇2d − θ̇2 )
Fig. 9.
where the reference position θ2d for the motor M2, which adjusts
the pivot position for a given desired stiffness Kd is obtained
from (12) such as
⎛ ⎞
Kd
2
K t L cos(2φ)
L
⎠.
(18)
θ2d = ⎝ √
Kd
2
n
+
Maximum angular deflection as a function of joint stiffness.
rt
As it is clear from the graph when the angular deflection
is small, a little amount of energy is required to change the
stiffness.
In AwAS-II, the maximum angular deflection ϕm ax is constrained to be 0.3 rad using a mechanical lock. However, as it is
shown in Fig. 9, after a certain level of the stiffness and while the
output link is loaded, the springs become fully deflected. In this
case, the potential energy of the output link reaches the maximum value of 5.8 J, as it was mentioned in Table I. Therefore,
if the stiffness is further increased, this results in a reduction
of the maximum angular deflection while the maximum energy
stored at the output link remains constant.
VI. EXPERIMENTAL RESULTS
In order to verify the performance of the AwAS-II, both motors M1 and M2 were controlled through position control strategy running at 1 kHz. The reference position θ1d for the motor
M1 was formulated such as
θ1d = qd + φ
(17)
(15)
where qd is the reference link position and ϕ = τE /K is the
passive deflection computed through the measured joint elastic
torque τE and the model-based joint stiffness K from (12).
The controller of the motor M1 is
τM 1 = kM 1p (θ1d − θ1 ) + kM 1d (θ̇1d − θ̇1 ) + kld q̇ + g(q).
(16)
The controller for the motor M1 positioning in (16) contains
an additional active damping term which is used for the atten-
K t L 2 cos(2φ)
A. Stiffness Tuning
To evaluate the ability of AwAS-II in changing the stiffness,
the intermediate link position is fixed by setting the M1 reference to a given value while the output link is manually moved
from its no load equilibrium position. The angular deflection
between the output link and the intermediate link was computed
from the encoder measurements and the elastic joint torque was
monitored by the torque sensor. The experiment was repeated
for different sets of the ratio r. The experimental results (solid
lines) are plotted in Fig. 10 together with the theoretical results
(dotted lines). As it is clear from the graph, the experimental
data match the predicted values.
The slope of each curve represents the joint stiffness K for
several values of the lever ratio r. The arrow in Fig. 10 is toward
increasing the ratio. It is clear that by changing the ratio, the
stiffness can be tuned in a wide range. When the ratio is 0.05 the
joint stiffness is almost zero. However, when the ratio becomes
bigger, the slop of the curve increases.
B. Tracking Simple Periodic Trajectories
Initially to examine the ability of AwAS-II to control position
and stiffness independently, both motors M1 for position and M2
for stiffness were simultaneously controlled to follow sinusoidal
position and stiffness trajectories of different frequencies. To
follow a sinusoidal position, the stiffness motor has to vary
according to (18).
Fig. 11 presents the motor positions and stiffness trajectories
against the reference ones revealing the capability of the actuator
to control both variables independently with good fidelity.
JAFARI et al.: NEW ACTUATOR WITH ADJUSTABLE STIFFNESS BASED ON A VARIABLE RATIO LEVER MECHANISM
61
Fig. 13. Stiffness variation during the stance phase based on the typical ankle
torque/angle profile.
Fig. 11. Tracking a sine wave trajectory for the position of motor M1, trajectory of the motor M2 in order to achieve desired sine wave trajectory for the
stiffness.
Fig. 12. Typical and interpolated ankle torque/angle profile of a 75 kg person
during normal walking [33].
C. Reproducing Position and Stiffness of Human Ankle
The stiffness adjustment has an important role in human ankle
over the stance phase of a gait cycle [31]. In this experiment, we
verified the performance of the AwAS-II as a variable stiffness
actuator used for driving an orthosis for the human ankle.
During walking, the ankle has to provide a net positive work
over the stance phase in order to supply sufficient energy for
the toe-off. The stance phase can be divided into two steps;
loading and unloading [32]. During the loading operation, the
ankle stores energy and over the unloading operation it releases
the energy. However, in order to generate a net positive work
over the stance phase, the amount of released energy has to be
more than the amount of energy stored. In a human ankle this is
performed through the stiffness adjustment. Fig. 12 shows the
ankle/position profile during a gait cycle for a healthy person of
75 Kg [33]. It is important to note that Fig. 12 is only a typical
graph since each person has a different gait pattern. In this
graph, the network is positive since the profile is a CCW closed
loop chain (abcdefgha). The area inside the loop represents the
amount of the work.
The stance phase is from a to h and the swing phase of the
gait cycle is from h to a. From a (Heel Strike) to b (Foot Flat),
is controlled plantarflexion period in which ankle behaves like
a linear spring and so does not contribute on the mechanical
work [35]. However, the ankle produces the net positive work
by changing the stiffness during the stance phase from b to h.
The stiffness profile associated with the torque/position characteristic of the stance phase after the foot flat (b to h) is shown
in Fig. 12. In a normal speed of walking, this period is about
90% of the stance phase interval [33]. As it is clear, stiffness
variation in the human ankle is very fast.
The goal of this experiment is to generate the same net positive
work as the human ankle does by regulating the stiffness and
not tracking the stiffness profile accurately. The tracked stiffness profile should be as close as possible to that human ankle
with minimum stiffness variation since changing the stiffness
requires energy consumption. Therefore, in Fig. 12 the data
are initially divided into several groups and then interpolated
through linear least square method. This pricewise interpolation
of torque/angle generates almost the same net positive work but
it requires less stiffness variation compare to human ankle.
It is important to note that the ankle position in Fig. 12 is given
with respect to the equilibrium position in which the shank is
perpendicular to the foot. In addition, it should be clarified that
the position calculated for the ankle assumes a single stationary
rotation axis. However, this assumption is often introduced in
prosthetic ankle studies [32], [34]. As a consequence, to experimentally reproduce the ankle stiffness and position profile, the
output link of AwAS-II is locked in a given position while the
intermediate link (motor M1) is free to rotate around its axis
(see Fig. 14). Therefore, the reference position sent to the motor
M1 corresponds to the ankle position during the stance phase.
Fig. 15 shows the result of the stiffness and position tracking
over the stance phase. In this graph the interpolated stiffness is
tracked, however the goal of this experiment is to generate the
same amount of positive work over the stance phase by tuning
the stiffness and not to accurately track the stiffness profile in
Fig. 13. This graph demonstrates that AwAS-II is capable of
reproducing position and interpolated stiffness of the human
ankle over the stance phase in order to provide the same amount
of net positive work required for the push-off in human ankle.
62
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 1, FEBRUARY 2014
of the ankle stiffness profile. It is important to mention here
that the capability of AwAS-II (and in general any other actuator) in generating the net positive work depends on the torque
and speed characteristics of its main motor and existence of the
spring in compliant actuators helps to reduce the required power
by the main motor. The capability of AwAS-II in this optimal
scenario is constrained due to its limited energy storage capability. Therefore, AwAS-II may need to exert additional torque
through its main motor for generating larger net positive works.
Fig. 14. Experimental setup for reproducing the ankle position and stiffness
during the stance phase; the output link if fixed horizontally while the intermediate link (motor M1) can rotate around its axis.
Fig. 15. Reproducing the human ankle during the stance phase: (a) interpolated stiffness and (b) position.
VII. CONCLUSION AND FUTURE WORK
In this paper, a new actuator with adjustable stiffness
AwAS-II was presented, which is an improved version of the
original actuator AwAS [24].
The proposed actuator has a wider range of stiffness (from
zero to infinity) which is achieved through a variable ratio lever
mechanism. The stiffness is mostly affected by the lever ratio
with a minor contribution coming from the load. Compared to
the original prototype of AwAS, the new actuator can adjust the
stiffness faster due to the reduced pivot travel distance. When
the link is not loaded, the force due to the spring’s predeflection
is perpendicular to the pivot displacement, thus the requested
energy to adjust the stiffness is minimum. In addition, the new
version results in a lighter and more compact structure with an
increased energy storage capacity.
Experiments results verified the performance of the AwAS-II
in adjusting the stiffness. Initially the tracking of primitive profiles such as sine waves for both the position and the stiffness
of the link were tested, showing good performance. Then, experiments were carried out to verify the ability of AwAS-II in
reproducing the stiffness variation of the human ankle over the
stance phase. For this purpose, a profile of human ankle stiffness was considered and then sent as input reference to the
stiffness motor. Results indicated that the AwAS-II changed the
stiffness rapidly and provided a sufficient amount of net positive work required for the push-off. Future work consists in the
implementation of AwAS-II for driving the ankle joint of a leg
exoskeleton.
REFERENCES
Fig. 16.
phase.
Power generated by the human ankle and AwAS-II over the stance
To verify the positive work, the power generated at the human ankle is plotted in Fig. 16 together with the power at the
AwAS-II joint by using the torque (through the stiffness and position) and the velocity (time derivative of the position) profiles.
This graph shows that AwAS-II was able to generate almost
the same amount of positive work (5.1 J) as the human ankle
(5.8 J) over the stance phase by tracking the linear interpolation
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Amir Jafari received the B.S. degree in industrial
engineering and the M.S. degree in mechanical engineering from Isfahan University of Technology,
Khomeynīshahr, Iran, in 2004 and 2006, respectively,
and the Ph.D. degree in robotics from the Italian Institute of Technology, Genoa, Italy, in April 2011.
He is currently a Postdoctoral Researcher at the
Bio-Inspired Robotic Laboratory, Swiss Federal Institute of Technology, Zurich, Switzerland. His research interests include design and control of variable stiffness actuators, rehabilitation robotics, and
exoskeletons.
Nikos G. Tsagarakis received the D.Eng. degree in
electrical and computer science engineering from the
Polytechnic School of Aristotle University, Greece,
in 1995, and the M.Sc degree in control engineering
and the Ph.D. degree in robotics from the Univeristy
of Salford, U.K., in 1997 and 2000, respectively.
He is currently a Senior Researcher at the Italian
Institute of Technology (IIT) (Head of Humanoids
Lab, ADVR), Genoa, Italy, with overall responsibility
for humanoid design (iCub and cCub). Before joining
IIT, he was a Research Fellow and then a Senior
Research Fellow in the Centre for Robotics and Automation, University of
Salford, where he worked on haptic systems, wearable exoskeletons, humanoid
robots, mechanism design, and actuation systems. He is the author of more
than 90 papers published in research journals and presented at international
conferences.
Irene Sardellitti received the M.S. degree in biomedical engineering (cum laude) from Campus BioMedico University of Rome, Rome, Italy, in 2004,
and the Ph.D. degree in bioengineering from the
Scuola Superiore Sant’ Anna of Pisa, Pisa, Italy, in
May 2008.
In January 2005, she joined the Advanced Robotic
Technology and System Laboratories, Scuola Superiore Sant’Anna. In June 2006, she joined, as a visiting
student, the Stanford Robotics Laboratory where she
pursued her research for two years. Since July 2008,
she holds a Postdoctoral position in the Advanced Robotics Department, Italian
Institute of Technology, Genoa, Italy. Her research interests include humanfriendly robotics, safety analysis, and control strategies for compliant actuation.
Darwin G. Caldwell received the B.Sc. and Ph.D degrees in robotics from the University of Hull, U.K.,
in 1986 and 1990, respectively. In 1994, he received
the M.Sc. degree in management from the University
of Salford, U.K.
He is currently the Director of Robotics at the
Italian Institute of Technology, Genoa, Italy. He is a
Visiting/Honorary/Emeritus Professor at the Universities of Sheffield, Manchester, and Wales, Bangor.
His research interests include innovative actuators
and sensors, haptic feedback, force augmentation exoskeletons, dexterous manipulators, humanoid robotics, biomimetic systems,
rehabilitation robotics, telepresence and teleoperation procedures, and robotics
and automation systems for the food industry.