Variable Stiffness Mechanism for suppressing unintended forces in
physical Human Robot
Interaction.
University at Buffalo, SUNY Buffalo, New York 14260
Email:
[email protected]
University at Buffalo, SUNY Buffalo, New York 14260
Email:
[email protected]
University at Buffalo, SUNY Buffalo, New York 14260
Email:
[email protected]
University at Buffalo, SUNY Buffalo, New York 14260
Email:
[email protected]
ABSTRACT This paper presents the design of a two-degree-of-freedom
variable stiffness mechanism and demonstrates how
its adjustable compliance can enhance the robustness of physical
human-robot interaction. Compliance on the grasp handle is achieved
by suspending it in between magnets in pre-loaded repelling
configuration to act as nonlinear springs. By adjusting the air
gaps between the outer magnets, the stiffness of the mechanism in
each direction can be adjusted independently. Moreover, the
capability of the proposed design in suppressing unintended
interaction forces is evaluated in two different experiments. In
the first experiment, improper admittance controller gain leads to
unstable interaction, whereas in the second case, high-frequency
involuntary forces are caused by the tremor.
1 INTRODUCTION In recent years, adjustable compliance actuators or
variable stiffness actuators (VSA) have received much attention
in
wearable robotics, rehabilitation robotics, and home-based
interactive robots [1, 2]. In comparison with stiff actuators, they
can improve stability during physical Human-Robot Interaction
(pHRI). The interaction safety can also be improved by minimizing
large external forces due to shocks [1, 2]. Placing an elastic
element at the port of interaction provides two main benefits:
first, it can store and release energy when required and second, it
can act as a low-pass filter for attenuating high- frequency
involuntary forces. Moreover, compliant actuators reduce the
reflected inertia during human-robot interaction [3, 4].
Complaint actuators can be classified into two major categories,
active and passive depending upon their adaptability and energy
storing capability [1]. Passive compliant actuators consist of an
elastic element to store energy, whereas active compliant actuators
use control strategies to mimic the spring-like behavior [5].
Therefore, the adjustable behavior is an inherent hardware property
[6] of passive actuators, whereas it’s a control strategy in the
active actuators.
During a pHRI task, the robot senses the input force or motion from
a human and produces corresponding reactions. Several control
strategies are proposed for effective pHRI, out of which two
popular control strategies are the impedance control [7] and the
admittance control [8,9]. They both deal with the modulation of
inertia, stiffness, and damping parameters of the apparent dynamics
simulated by the robot for efficient interaction. Hogan [7]
suggested the use of impedance control for manipulators by
describing the environment as admittance and the manipulator as
impedance. However, the bandwidth of achievable forces is highly
limited due to the possible non-collocation of the force sensor and
the actuator [10]. On the other hand, the admittance control
strategy can achieve a wider bandwidth of apparent dynamics [11]
and thus are more effective in human-robot cooperation [12].
However, designing the admittance controller for a low-effort task
by selecting the desired
∗Address all correspondence to this author.
ESFAHANI JMR-18-1384 1
inertia below the end-effector’s physical inertia can lead to
instability [12]. The interaction stability in such situations is
greatly affected by the human limb stiffness [13] Attaching a
passive compliant element to the end-effector and controlling the
endpoint stiffness can improve the interaction stability in these
cases.
Passive compliant elements are also used as low-pass filters in
some health-care robotic applications for suppressing tremor [14,
15]. Tremor is a rhythmic and involuntary movement caused due to
the oscillatory innervation of antagonistic muscle fibers [16]. In
particular, physiological tremor has small amplitudes in the
bandwidth of 8-16 Hz, and affects the limbs and causes significant
difference in the precision of hand movements. Some of the tremor
suppression strategies include biomechanical loading of the limb
[17, 18], passive devices such as dynamic vibration absorbers [15]
and active control strategies based on jerk minimization
[19].
Adelstein [17] modeled the human musculoskeletal system as a second
order dynamical system whose frequency re- sponse is characterized
by a low-pass filter with cut-off frequency subject to the
stiffness, inertia and damping parameters of the limb. This concept
can be used to design an elastic element that can act as a low-pass
filter to suppress the effect of invol- untary forces. By placing
this elastic element at the end-effector of a lightweight robotic
arm, the high-frequency impedance of the actuator is limited to the
low-frequency stiffness of the elastic element [20]. This can
prevent the transmission of high-frequency forces and displacements
to the system being operated. In this context, a VSA can be used in
suppressing a particular bandwidth of frequency by adjusting the
stiffness of the end-effector without affecting the inherent
dynamics of the robot.
In this regard, this paper presents a two degree of freedom (DoF)
variable stiffness mechanism (VSM) that can be used to improve the
robustness of interaction in pHRI. The proposed design is based on
permanent magnets in a pre-loaded repelling configuration. A
data-driven model is used to estimate the stiffness of the
mechanism based on the air gap between actuated magnets. Moreover,
frequency responses are used to study the behavior of the system
around the characteristic frequency. Fi- nally, the effectiveness
of this mechanism in suppressing high-frequency forces caused due
to tremor or admittance controller is demonstrated and
discussed.
2 MODELING AND DESIGN OF VSM The design concept of the proposed
variable stiffness mechanism is illustrated in Fig.1. The handle is
attached to a
passive magnet that is suspended by springs in between two actuated
magnets in an antagonistic configuration. The nonlinear springs
between the blocks provide the capability of adjusting the
stiffness of the system by controlling the distance between the
actuated blocks [1]. We previously used permanent magnets in a
pre-loaded repelling configuration as nonlinear springs to develop
a variable stiffness gripper [21] that can be used for safe object
handling during dynamic manipulation [22]. Motivated from this
design, permanent magnets in repelling antagonistic configuration
are used in two orthogonal directions to allow stiffness adjustment
in planar motions. Magnets are selected in the design of the VSM as
the interaction forces between them are non-contact and thus can
allow zero separation, unlike springs that impose minimum and
maximum separation constraints. For convenience, the external
magnets are referred to as ‘active magnets’ and the suspended
magnet as the ‘passive magnet’ in this paper. A pulley-belt
assembly attached to a servo motor with position control is used to
control the position of active magnets. The active magnets can be
displaced in an equal and opposite direction as shown by similar
color arrows in Fig.1C. Cylindrical permanent magnets of diameter
15 mm, height 5 mm, and uniform magnetization 6.05x105 A/m in the
pre-loaded repelling configuration are chosen to serve as nonlinear
springs. The selection of these magnets is based on the size
(compact design) and the required bandwidth of repulsive forces
(maximum of 25 N for a low-effort HRI) in an antagonistic
configuration.
2.1 MECHANICAL DESIGN Fig.2 displays the two DoF variable stiffness
mechanism and its components. The movement of all magnet blocks
are
linearly constrained with the help of low friction linear guides
and the displacement of passive magnets from the equilibrium
position are measured using sliding potentiometers as shown in
Fig.2B.
Active magnets are controlled by a pulley belt sub-assembly mounted
on the shaft of a servo motor with position encoder. The
interaction between each magnet pair is non-contact as long as the
applied force is within permissible limits. If the applied force
exceeds the opposing magnetic forces, the passive block makes a
contact with the active block in the direction of applied force and
the stiffness can no longer play a role.
2.2 MAGNET FORCE MODEL A data-driven model has shown to best
represent the behavior of the magnets within the feasible range of
motion [22].
For this purpose, we conduct a series of experiments in which a
force sensor (PCB Piezotronics 208C03) is attached to the base of
VSM to measure the interaction forces between each magnet pair. The
handle attached to the passive magnet is then displaced using a
low-frequency sine sweep (0.5 Hz) and the force data is recorded
for five different air-gaps. An expo- nential model, Fm(x) =
C1e−C2(x), is used to fit the experimental data for different
magnet air-gaps. The force-displacement
ESFAHANI JMR-18-1384 2
Fig. 1: A) Single DoF spring model B) Equivalent magnet model C)
Interaction forces in magnet model.
relationship of the antagonistic configuration can be described by
Eq.1.
Fmag = Fm(s− x)−Fm(s+ x) = 2C1e−C2s sinh (C2x) (1)
Where s is the air-gap between external magnets and x is the
displacement of the passive magnet from the equilibrium posi- tion.
A nonlinear least squares approach with Levenberg-Marquardt
algorithm is used to calculate the modeling parameters as C1=28.41
and C2=206.35.
The experimental data and the corresponding fitted model are shown
in Fig.3. Negative displacement represents move- ments towards the
left and positive towards the right. The resultant stiffness can be
obtained by differentiating Eq.1 with respect to x and thus the
slope of the force-displacement curve represents stiffness which
also decreases as the air-gap between active magnets
increases.
The forces between each magnet pair diminish significantly when the
air-gap is greater than 0.02 m. Therefore, the model is only
effective if the air-gap on each side of the passive magnets is
within a 0.02 m limit. It is also observed that the resultant force
and stiffness at the equilibrium position is zero and exponentially
increases towards both the ends. Any external force Fe is
compensated by the repelling forces due to magnets as shown in
Fig.1C. More details on the bandwidth of stiffness and its
dependency on air-gap are discussed in [23].
2.3 MODEL EVALUATION - SIMULATIONS Using the fitted model for
magnetic repulsive forces, one can study the change in frequency
response of the system
with respect to its resultant stiffness. Two main factors affecting
the frequency response of the system are the air-gap and magnitude
of the external force. For the purpose of verification, a single
DoF sub-assembly is considered for simulation. In the presence of
an external force Fe(t) applied on the passive magnet, the equation
of motion for the single DoF-VSM about the equilibrium position is
described by Eq.2.
m Üx+ c Ûx+Fm(s, x) = Fe(t) (2)
Fig. 2: Mechanical Design of two DoF variable stiffness in A)
Isometric view and B) Top View
ESFAHANI JMR-18-1384 3
Where m is the mass attached to the passive magnet, F(s− x) and
F(s+ x) are the repelling forces between magnet pairs as a function
of the air-gap (s− x) and (s+ x) respectively, and c Ûx represents
the viscous damping term (to compensate for the dynamic friction
due to the linear guide). In our simulation study, the external
force is considered to be a sinusoidal function of Fe(t) = Aesin(2π
f t). Thus, Eq.2 can be simplified as Eq.3.
m Üx+ c Ûx+2C1e−C2s sinhC2x = Aesin(2π f t) (3)
The frequency response of the system is characterized by the
resultant stiffness which in turn depends on the air-gap between
magnets and the displacement of the handle (Fig.3). To understand
the influence of air-gap, magnitude of force, and frequency on the
dynamic behavior of the single DoF system, the differential
equation (Eq.2) is evaluated using an explicit 4th order
Runge-Kutta method with variable time step for the force range of
0.4–1.2 N, frequency of 2–20 Hz and the air-gap between active
magnets 0.04–0.01 m. In all the simulation cases, a damping
constant of 0.1 N-s/m and attached mass of 0.18 kg is
considered.
A bandwidth of force and frequency are considered to evaluate the
mechanism for tremor suppression, because tremor is known to exist
at frequencies greater than that of voluntary movement (i.e. in the
range of 4-16 Hz) and the amplitude of tremor is significantly
lower than voluntary movement. However, the intensity of tremor can
vary from task to task and individual to individual. Therefore, the
response of the handle for different force and frequency is studied
in the first set of simulations. Force is varied (0.4-1.2 N) for a
constant air-gap (10 mm) and the results for this case are
presented in Fig.4A.
It is observed that the peak shifts towards the right with an
increase in the magnitude of applied force. This is obvious from
the fact that large force amplitudes cause larger deviations (x) of
the handle from its equilibrium position and thus experiences
higher stiffness (displacement from equilibrium position increases
slope). Therefore, the magnetic air gap should be properly selected
to suppress a particular bandwidth of frequency.
In the second set of simulations, the frequency response is studied
for varying air-gaps and constant input force (0.45 N). Fig.4B
illustrates the shift in peak corresponding to the natural
frequency towards a lower frequency by increasing the air-gap, this
shows the significance of the air-gap in decreasing the stiffness
of the system.
2.4 MODEL EVALUATION - EXPERIMENTS The single DoF mechanism is
attached to the base of a shaker table in the horizontal direction
to avoid the effect of
gravity on the magnets (Fig.5). The linear actuator is connected to
the handle by aligning it’s axis to the allowable motion of the
passive magnet. A force sensor is attached at the interface
connecting the handle and the actuator to monitor and ensure
constant input force throughout the experiment. The position of
active magnets is controlled using a servo motor constantly powered
at 11 Volts.
A constant amplitude sine wave is used for actuating the shaker
with the oscillation frequency increasing at the rate of 1
octave/min in the 2-20 Hz range. A high precision laser vibrometer
is used to record the displacement and velocity of the
handle.
Fig. 3: Force-Displacement characteristics of the antagonistic
configuration measured for different air-gaps between the magnets.
The solid lines are the average of the loading and unloading cycle
(dotted lines) for each condition. The dashed line corresponds to
the equilibrium point.
ESFAHANI JMR-18-1384 4
Fig. 4: Frequency response of simulation data A) Constant air-gap
(10 mm) and varying force input, B) Constant force input (0.45 N)
and varying air-gap
The first set of experiments are performed by keeping the air-gap
constant (10 mm) and changing the force (0.4-1.2 N). The
displacement transfer function for each amplitude is then plotted
against frequency in Fig.6A. It can be seen that with the increase
in the amplitude of input force, the peak shifts towards the right
(similar to simulation results in Fig.4A). When the air-gap is
decreased, higher forces are encountered by the passive magnet on
each side which in turn increases the stiffness and natural
frequency.
In the second set of experiments, response of the system for a
constant input force (0.45 N) and varied air-gap from 3 mm (high
stiffness) to 19 mm (low stiffness) are demonstrated in Fig.6B.
Experimental results are similar to the simulation results (shown
in Fig.4B) i.e. the air gap affects the natural frequency of the
system. The frequency bandwidth of the VSM observed in these
experiments is in the range of 3-20 Hz due to the minimum and
maximum separation constraints. The slight difference between the
natural frequency of the system in experimental and simulation
results is due to inadequate information to model friction during
simulations.
3 EXPERIMENTAL VALIDATION 3.1 EXPERIMENT SETUP
An experiment is designed to demonstrate the effectiveness of VSM
for stabilizing and suppressing high-frequency forces during
physical human-robot interaction. The two DoF VSM is mounted on top
of a 6-axis force/torque sensor attached to the end effector of a
light-weight 6-DoF robot arm (SCHUNK PowerBall-LWA). Furthermore, a
cartesian space- based admittance controller is implemented on the
robot to mimic the behavior of a second order dynamic system with
null
Fig. 5: Test rig for experimentation.
ESFAHANI JMR-18-1384 5
Fig. 6: Frequency response of experimental data A) Constant air-gap
(10 mm) and varying force input, B) Constant force input (0.45 N)
and varying air-gap
stiffness. This controller consists of two feedback loops, the
first one to convert the input force from human to a desired
end-effector position (Eq.4) and the second to track the desired
joint velocities of the robot (Eq.5).
Eq.4 represents a virtual second order dynamical system of inertia,
M ε R6X6 and damping, C ε R6X6 that converts the human input force
(Fh ε R
6) to the end-effector velocity (vd ε R6).
M Ûvd +Cvd = Fh (4)
The end-effector velocity is mapped to the joint space velocity
using the inverse of Jacobian (J) as represented in Eq.5. The input
force is sampled at 256 Hz and the desired joint space velocity is
attained with the help of a closed loop feedback control.
Ûqd = J(q)−1vd (5)
Input force to the admittance controller (Fi) is modulated by the
VSM that is used to couple the human and robot at the interaction
port, the equivalent block diagram representation of the complete
setup is shown in Fig.7.
3.2 HUMAN SUBJECT STUDY A single subject study with eight different
stiffness settings (randomized during each session) is conducted
after ob-
taining an approval from the Institutional Review Board (IRB
no.030-801361). The experiment comprises of four different
interaction configurations to assess the performance of the
variable stiffness mechanism in suppressing unintended forces in
pHRI. These cases are shown in Fig.8 and are as follow:
Case-1: Stiff handle which is used as the control case. Case-2: 2
Dof variable stiffness mechanism. Case-3: Stiff handle in presence
of simulated tremor Case-4: VSM in presence of simulated
tremor
To simulate the tremor in the last two cases, the subject is asked
to wear a glove on which a DC motor with an offset mass mounted
(Fig.8C). The frequency of oscillation is tuned to be around ∼ 12
Hz by changing the input to the DC motor.
In all cases, the subject is asked to perform a visuomotor task
which includes moving the handle inside the boundaries of a maze
(Fig.7B) consisting of a fine and a gross motor region. For the
purpose of this experiment, the end-effector movement is restricted
to planar motion (X-Y plane) and the corresponding velocities are
transformed to position and displayed on a computer screen as
visual feedback to the subject. A total of 18 levels were selected,
1 each in case-1 and case-3, and 8 each
ESFAHANI JMR-18-1384 6
in case-2 and case-4 (8 different magnet separations). The magnet
separation can be varied continuously from 0-30 mm on each side of
the passive magnet but for the subject study, 8 separations equally
spaced between 3-17 mm were considered. Prior to performing the
experiment, the subject has practiced the levels several times and
is accustomed to the settings of the VSM and the robot. Each level
requires the subject to perform two trials of the visuomotor task
during which the input force, the end-effector’s velocity and the
displacement in X and Y directions are recorded.
4 RESULTS AND DISCUSSION 4.1 STABILITY DURING INTERACTION
The selected inertia and damping values for the admittance
controller of the robot are 3 kg and 25 Ns/m respectively. Here,
the inertia of the apparent dynamics simulated by the admittance
controller is selected below the physical inertia of the
end-effector. These admittance parameters are chosen such that the
task requires low-effort (low inertia and damping) during an
interaction, but at the expense of interaction stability caused due
to the stiffness variation in human limb [12].
Comparing the outcome of case-1 and case-2, one can examine if
adjusting the stiffness in VSM would, in fact, improve the
stability of interaction. Furthermore, the comparison of case-3 and
case-4 can demonstrate the effectiveness of VSM in suppressing the
high-frequency unintended forces during pHRI.
The mechanism proposed in this work creates an interface to
stabilize the cooperative task by varying the stiffness of the
mechanism and thus affecting the resultant stiffness of the
combined setup (human and mechanism) at the interaction port. Note
that the VSM modulates the components of the human input force (Fh)
to the admittance controller Fi by varying the end-point
stiffness.
To better understand the components of force (Fi) causing
instability in an admittance controller, a frequency domain
analysis is performed using force and velocity signals recorded
during the experiment [12]. The Power Spectrum Density (PSD) of the
force and velocity signals from each trial are obtained using the
Welch method with a two second Hanning window and 50% overlap. Post
experiment completion, the resultant PSD’s of force and velocity
are averaged across trials and is shown in Fig.9A.
The black solid line represents the PSD corresponding to the
experiment performed without the mechanism (case-1), and the blue
and red dashed lines represents the PSD corresponding to maximum
and minimum stiffness respectively (case-2). From the PSD values of
force, it is clear that the oscillations due to an unstable motion
for the designed admittance controller are more predominant around
the ∼ 5 Hz region (represented by a rectangular patch in the
figure). It is also observed during the experiments that the power
of the peak decreased significantly (from blue to red) by
decreasing the stiffness of the mechanism. The bounded grey band
represents the bandwidth of PSD’s achievable for different levels
of stiffness (from high
Fig. 7: A) Human-robot coupling at the interaction port using the 2
DoF VSM. Fh and Vh represents the input force and output velocity
respectively at the interface. Fi and Vo represents the
input-output pair of robot’s admittance control. B) Visuomotor task
(Fine and gross region)
ESFAHANI JMR-18-1384 7
Fig. 8: Representation of three different configuration: A) Stiff
handle, B) Mechanism attached, C) Motor with offset mass attached
to human limb to simulate tremor
Fig. 9: PSD of force and velocity for A) case-1 and case-2, B)
case-3 and case-4. Grey band corresponds to regions of interest,
3–6.5 Hz in A) and both 3–6.5 Hz and 9–16 Hz bands in B).
and low). Based on these observations, the contribution of unstable
force components can be decreased by decreasing the stiffness
of the mechanism during the interaction.
4.2 HIGH FREQUENCY FORCE SUPPRESSION To assess the effectiveness of
VSM in suppressing unintended high-frequency forces, we present and
discuss the results
of case-3 and case-4 in the presence of tremor. Haptic stability
observer (Eq.6) [24] is often used to quantify the level of
stability in pHRI with an admittance control robot [12, 25].
I = ∑ωs/2
ω=ωo Pf (ω)
(6)
A modified version of this index (Eq.7) is used to differentiate
tremor from the controller instability.
IR =
(7)
Where ωs is the sampling rate (ωs/2 using Nyquist criteria), ω0 is
the lowest frequency corresponding to the PSD (Pf ). ωl and ωu are
the lower and upper bounds of the frequency band of interest. In
the present experiment, two frequency bands are considered, the
first band corresponds to controller instability (3–6.5 Hz) as
mentioned in the previous sub-section and the second band
corresponds to the frequency caused due to tremor (9–16 Hz) as
shown in Fig.9B. The effect of these high-frequency forces on the
end-effector displacement for four different settings of the VSM is
shown in Fig.10.
ESFAHANI JMR-18-1384 8
Fig. 10: Displacement of the end-effector in the fine motor region
A) Case-3, B–D) Case-4 with maximum, optimum and minimum stiffness,
respectively.
Table 1: Instability indices corresponding to the two frequency
bands for different levels of stiffness.
Air-gap Stiffness IR (3-6.5Hz) IR (9-16Hz)
X (Handle) High .356 0.072
3 mm 6937 N/m .230 0.019
5 mm 5392 N/m .097 0.019
7 mm 4477 N/m .028 0.019
9 mm 3918 N/m .012 0.013
11 mm 3568 N/m .011 0.012
13 mm 3346 N/m .005 0.016
15 mm 3202 N/m .004 0.026
17 mm 3109 N/m .007 0.020
The modified instability index is calculated for these two
frequency bands using the PSD of force signal without (case-3) and
with (case-4) the mechanism and the results are tabulated in Table
1. The range of IR lies in-between 0 and 1, where values close to 0
correspond to higher attenuation of forces and values close to 1
correspond to lower attenuation of forces in that frequency
band.
The instability index values for eight different magnet separations
are represented alongside the index for the stiff handle in the two
frequency bands using a bar plot (Fig.11). It is observed that
decreasing the stiffness leads to higher stability in the first
frequency band whereas the attenuation increases and then decreases
with the continuous decrease in stiffness in the second frequency
band. Such kind of behavior is observed when the tremor frequency
is close to the natural frequency of the system for that particular
level of stiffness. Therefore in such situations, a choice of
optimum stiffness level has to be made based on the instability
index values to suppress most part of the involuntary forces in
both the frequency bands. It is also observed that when the
mechanism is attached (case-4), index values in both the frequency
bands (for all stiffness levels) are lower compared to the results
in case-3. This proves the effectiveness of the mechanism in
stabilizing the controller during stiff interaction and also in the
case of the tremor.
The index values presented in this paper are obtained from the
experimental data post-trial completion. These index values act as
a reference to obtain an optimum level of stiffness for a
subject.
5 CONCLUSION This paper presents the design and validation of a
two-DoF passive variable stiffness mechanism using permanent
magnets. The interaction forces between magnets in an antagonistic
configuration are modeled as a function of air-gap between them and
the displacement of the center magnet from the equilibrium
position. It is shown that the mechanism endpoint stiffness can be
varied by changing the air-gap between magnets. The design concept
is validated using simulations and experimental results by
characterizing the stiffness variation as a change in the behavior
of its frequency response. These results show that the stiffness at
the interaction port is effected by air-gap between active magnets
as well as the magnitude
ESFAHANI JMR-18-1384 9
Fig. 11: Instability index of handle (black) and VSM with 8
different stiffness (green) in two frequency bands: A) 3-6.5Hz band
(instability due to controller), B) 9-16Hz band (tremor)
of the input force. Moreover, a human subject study is conducted
with the mechanism attached to the robot with an admittance
controller
and validated for a motor-task. It is observed that by changing the
stiffness of the mechanism at the port of interaction, the
instability caused due to the stiff environment could be
significantly reduced. The above experiment is further extended to
verify the effect of stiffness in suppressing the high-frequency
involuntary forces caused due to tremor. Choosing a proper level of
stiffness can suppress the high-frequency forces to some extent
indicating the effectiveness of the proposed mechanism in reducing
the backlash of involuntary components during pHRI.
Future work should address a real-time control strategy to extract
the involuntary movements and reduce its effect by controlling the
position of magnets. Moreover, because of the individual
differences in muscle stiffness, the optimum stiffness of the
proposed mechanism in pHRI is likely to be subject dependent which
should be investigated further in future research.
Acknowledgements This work was supported by the National Science
Foundation under Grant No.1502287.
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ESFAHANI JMR-18-1384 11
List of Tables 1 Instability indices corresponding to the two
frequency bands for different levels of stiffness. . . . . . . . .
. 9
List of Figures 1 A) Single DoF spring model B) Equivalent magnet
model C) Interaction forces in magnet model. . . . . . . 3 2
Mechanical Design of two DoF variable stiffness in A) Isometric
view and B) Top View . . . . . . . . . . . 3 3 Force-Displacement
characteristics of the antagonistic configuration measured for
different air-gaps between
the magnets. The solid lines are the average of the loading and
unloading cycle (dotted lines) for each condition. The dashed line
corresponds to the equilibrium point. . . . . . . . . . . . . . . .
. . . . . . . . 4
4 Frequency response of simulation data A) Constant air-gap (10 mm)
and varying force input, B) Constant force input (0.45 N) and
varying air-gap . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 5
5 Test rig for experimentation. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 5 6 Frequency
response of experimental data A) Constant air-gap (10 mm) and
varying force input, B) Constant
force input (0.45 N) and varying air-gap . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 6 7 A) Human-robot
coupling at the interaction port using the 2 DoF VSM. Fh and Vh
represents the input
force and output velocity respectively at the interface. Fi and Vo
represents the input-output pair of robot’s admittance control. B)
Visuomotor task (Fine and gross region) . . . . . . . . . . . . . .
. . . . . . . . . 7
8 Representation of three different configuration: A) Stiff handle,
B) Mechanism attached, C) Motor with offset mass attached to human
limb to simulate tremor . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 8
9 PSD of force and velocity for A) case-1 and case-2, B) case-3 and
case-4. Grey band corresponds to regions of interest, 3–6.5 Hz in
A) and both 3–6.5 Hz and 9–16 Hz bands in B). . . . . . . . . . . .
. . . . . . . . 8
10 Displacement of the end-effector in the fine motor region A)
Case-3, B–D) Case-4 with maximum, optimum and minimum stiffness,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 9
11 Instability index of handle (black) and VSM with 8 different
stiffness (green) in two frequency bands: A) 3-6.5Hz band
(instability due to controller), B) 9-16Hz band (tremor) . . . . .
. . . . . . . . . . . . . . . 10
ESFAHANI JMR-18-1384 12
