functional electrical stimulation induced cadence control ... · neurological conditions such...
TRANSCRIPT
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Abstract—Functional electrical stimulation (FES) has become a
popular and successful form of rehabilitation for people with
neurological conditions such spinal cord injury or stroke. FES is
the direct application of electrical current across the motor
neurons of a muscle to generate artificial muscle contractions to
perform functional tasks. Arm-cycling is a beneficial
rehabilitation activity and motivation exists to use FES in arm-
cycling. It has previously been shown that closed-loop FES control
can be designed to achieve accurate, repetitive motion. In a
previous study, a robust sliding-mode controller was used to track
a desired crank cycle cadence (crank velocity) on a decoupled hand
cycle. For this thesis the hand-cycle was modified to yield an
improved tracking performance. Using a combination of FES and
volition, experimental results from three able-bodied participants
are presented for the developed control system. Two protocols
were ran, one of which activated the motor for all time and
demonstrated an average cadence tracking error of -0.04 ± 2.83 revolutions per minute (RPM) for a desired cadence of 40 RPM.
The second protocol deactivated the motor when the bicep was
stimulated and demonstrated an average cadence tracking error
of -0.02 ± 5.35 RPM.
Index Terms—Functional electrical stimulation (FES),
Lyapunov, rehabilitation robotics, switched systems, stability
I. INTRODUCTION
UNCTIONAL electrical stimulation (FES) has been used
for decades as a form of rehabilitation for individuals with
upper motor neuron lesions, caused by an event such as a stroke
or spinal cord injury (SCI) [1]. Neuromuscular electrical
stimulation (NMES) is called FES when it is used to perform
functional tasks [2]. NMES is the application of electrical
current across muscle fibers to yield artificial muscle
contractions [2]. FES has commonly been used for
rehabilitation because it helps restore voluntary muscle
function, increases bone mineral density and muscle mass,
improves cardiovascular health, and decreases spasticity [1].
This can also lead to psychological benefits such as improved
independence and self-esteem [1]. Enhancing a person’s
physiological (skeletal, cardiopulmonary and muscular
systems) and psychological well-being may enable them to
improve their ability to perform activities of daily living [3].
In research, FES applied to the lower extremities is generally
divided into three categories: cycling, standing, and walking.
FES-cycling has an advantage over standing and walking
exercises since people with paralysis can perform cycling [3].
Regular FES cycling has been shown to improve general health,
decrease the possibility of cardiovascular disease, and diminish
the effects of secondary complications of SCIs [4].
The FES stimulation intensity, in past studies, has been
controlled using proportional-derivative feedback or open-loop
control to obtain a desired cycling cadence (crank velocity) [1].
Cadence tracking was used as the primary control objective
because it is one of the most important factors in cycling
rehabilitation [1]. Furthermore, the stimulation was switched
between multiple muscle groups to achieve a predetermined
stimulation pattern throughout the crank cycle [1]. Thus, FES
cycling systems can be seen as switched control systems with
autonomous, state-dependent switching.
In addition, FES control input is commonly not applied
during kinematic dead zones in the crank cycle, where only a
small portion of the torque produced by the cyclist’s muscles is
transferred as torque about the crank axis [5]. In a previous
study, electric motors were used throughout the entire crank
cycle to help maintain the desired cadence and account for the
kinematic dead zones [4]. Recent studies have also
implemented Lyapunov-based nonlinear control techniques to
improve the performance and effectiveness of FES cycling [1].
A recent study successfully developed a controller that switches
the control input between an electric motor and FES of various
muscle groups [5]. Using the electric motor only during
kinematic dead zones when FES control input is not used, can
maximize the contribution of the cyclist’s muscles [5].
However, there are several challenges with using FES
control. One challenge is that stimulated muscles fatigue at a
higher rate than muscles that contract voluntarily [4]. The
muscles fatigue at a high rate because the stimulation only
engages small portions of each muscle, which also makes
controlling the muscles difficult [6]. Electric motors can be
used to compensate for muscle fatigue so that the person can
continue performing the exercise even while fatigued and
without stimulation [4]. Another challenge that exists relates to
the repeatability of the stimulation because the effectiveness of
the stimulation is influenced by many factors such as muscle
fatigue, body fat, body hydration, and the correlation between
stimulation and muscle reaction [6]. Electrode placement is
another factor because there is no guarantee the electrode will
be placed at the same location every time or in the best position
for stimulation [2].
Based on the success of FES cycling for the lower
extremities, there is evidence that FES cycling can also be
applied to the upper extremities. Individuals with SCIs above
Functional Electrical Stimulation Induced
Cadence Control of a Hand Cycle Michael Woc, Christian Cousin, Brendon Allen, Warren E. Dixon
F
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Th1 (thoracic vertebrae 1) suffer from upper extremity
impairments and would benefit from upper limb rehabilitation
[7]. The aforementioned methods for FES cycling in the lower
extremities can be applied to FES cycling with the upper limbs.
Preliminary work such as characterizing the dynamics of an
arm-cycle system and defining the arm stimulation regions have
been performed in [6], [8]. This thesis seeks to improve the
performance (crank cycle cadence tracking) of the controller
developed in [6] by improving the cycle hardware (i.e., the
electric motor).
In this thesis, a switched-systems sliding mode controller
was developed and used on a FES arm-cycle system with
electric motor assistance. The cadence was tracked during
experiments and the cadence tracking error was used to quantify
the performance of the controller. Preliminary experiments
were conducted on three able-bodied individuals to evaluate the
performance of the controller. FES cycling rehabilitation is
intended for people with neurological conditions (NCs), but for
their safety the arm-cycle system was first tested on able-bodied
individuals. Two different protocols were implemented for the
experiments, Protocol 1 and Protocol 2. Both protocols had
each participant cycle at a specific RPM (revolutions per
minute). In Protocol 1, the motor was on during the entire crank
cycle and the participant only added volition if the cadence error
exceeded ±5 RPM. In Protocol 2, the motor was deactivated during bicep stimulation and the participant was never allowed
to add volition. In the subsequent sections the arm-cycle
dynamic model, the developed switched-systems sliding mode
controller, and the resulting stability analysis are provided.
II. MODEL
A. Arm-Cycle Dynamic Model
A person pedaling a motor assisted arm-cycle can be
modeled as single degree-of-freedom system [6], [8], which can
be expressed as:
𝑀(𝑞)�̈� + 𝑉𝑚(𝑞, �̇�)�̇� + 𝐺(𝑞) + 𝑃(𝑞, �̇�) + 𝑐𝑑�̇� + 𝑑(𝑡)
= 𝜏𝑚(𝑞, �̇�, 𝑡) + 𝜏𝑒(𝑞, �̇�, 𝑡) (1)
where 𝑞 and �̇� denote the measured crank angle and velocity of the cycle, respectively; �̈� denotes the unmeasured crank acceleration; 𝑀 is the inertia matrix; 𝑉𝑚 represents the centripetal and Coriolis effects; 𝐺 denotes the gravitational effects; 𝑃 accounts for the viscoelastic tissue forces; 𝑐𝑑 is the coefficient of viscous damping, and 𝑑 accounts for time-varying disturbances. The torque produced by the upper limb
muscle contractions of the individual is expressed as:
𝜏𝑚(𝑞, �̇�, 𝑡) = ∑ 𝐵𝑚𝑢𝑚𝑚∈ℳ
(2)
where 𝐵𝑚 denotes the uncertain, nonlinear control effectiveness term that relates the stimulation input to the torque output of a
muscle group. The term 𝑢𝑚 is the input stimulation intensity applied to each muscle group. The subscript 𝑚 indicates the muscle group and is defined as 𝑚 ∈ ℳ ≜{𝑅𝐵𝑖, 𝐿𝐵𝑖, 𝑅𝑇𝑟𝑖, 𝐿𝑇𝑅𝑖}. 𝐵𝑖 and 𝑇𝑟𝑖 represent the biceps brachii and triceps brachii muscle groups, respectively; 𝑅 and 𝐿 correspond to right and left arm, respectively. The torque
applied about the crank cycle axis by the electric motor is
represented by:
𝜏𝑒(𝑡) = 𝐵𝑒𝑢𝑒 (3) where 𝐵𝑒 is a known positive constant term that relates the motor’s applied current to the resultant torque. 𝐵𝑒 can be bounded as 0 < 𝐵𝑒 ≤ 𝑐𝑒. The current applied to the motor is represented by 𝑢𝑒.
B. Switched System Model
As previously mentioned, the stimulation needs to switch
between different muscle groups in many FES applications to
produce coordinated motion, such as cycling. Switched systems
have additional control challenges since switching between
subsystems, such as muscles and electric motors, can generate
instabilities in the overall system and it is necessary to prove
the stability of the overall switched system, even if the
subsystems are stable. The stability analysis for the arm-cycle
system used in this thesis was developed similar to that of
previous studies done on a recumbent stationary cycle [1], [5].
In both studies, the stimulation was not applied during
kinematic dead zones. In [5], it was shown that coupling a
motor to the cycle increased the controllability of the dead
zones. As a result, the arm-cycle system was analyzed as a
switched system with control input switching between
stimulation of the biceps and triceps muscle groups and the
electric motor.
The stimulation regions for each muscle were defined by a
range of crank angles in the crank cycle that allow for that
muscle to produce positive torque about the crank axis. The
motor-controlled regions are the kinematic dead zones or
kinematically inefficient regions for any of the muscles to
produce positive torque. Let the set of crank angles (ℚ ) vary from [0, 2π), the crank cycle regions where stimulation is
applied to each muscle be represented by ℚ𝑚 ⊂ ℚ, and the regions of the crank cycle where the electric motor is used be
ℚ𝑒 ⊂ ℚ. The union of crank regions where FES is applied is denotes as 𝑄𝐹𝐸𝑆 is defined as:
ℚ𝐹𝐸𝑆 ≜ ⋃ ℚ𝑚𝑚∈ℳ
(4)
The 𝑄𝑒 regions can be expressed as: ℚ𝑒 ≜ ℚ\QFES (5) A graphical representation of the kinematic dead zones (also
known as ℚ𝑒 regions in this thesis) and the stimulation regions, ℚ𝑚, using crank angles over the crank cycle is in Fig. 1.
Fig. 1. Sample schematic showing the arm muscle stimulation regions for the biceps (red region) and the triceps (green region). The kinematic dead zones
(KDZ) are also displayed with the gray areas [6].
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The control inputs for the muscle stimulation (𝑢𝑚) and current input (𝑢𝑒) for the electric motor are represented as:
𝑢𝑚 ≜ 𝑘𝑚𝜎𝑚𝑢𝑠 (6)
𝑢𝑒 ≜ 𝑘𝑒𝜎𝑒𝑢𝑐 (7)
where 𝑢𝑠, 𝑢𝑐 are subsequently designed control inputs and 𝑘𝑚, 𝑘𝑒 are positive, constant control gains. A switching signal for the muscle groups (𝜎𝑚) and the motor (𝜎𝑒) are defined as:
𝜎𝑚 ≜ {
1,0,
𝑖𝑓 𝑞 ∈ ℚ𝑚 𝑖𝑓 𝑞 ∉ ℚ𝑚
(8)
𝜎𝑒 ≜ {
1,0,
𝑖𝑓 𝑞 ∈ ℚ𝑒 𝑖𝑓 𝑞 ∉ ℚ𝑒
(9)
Switched control effectiveness terms 𝐵𝑀 and 𝐵𝐸 are introduced to rewrite the dynamics of the overall arm-cycle
system. 𝐵𝑀 is defined as:
𝐵𝑀 = ∑ 𝐵𝑚𝑘𝑚𝜎𝑚𝑚∈ℳ
(10)
and 𝐵𝐸 is defined as:
𝐵𝐸 = ∑ 𝐵𝑒𝑘𝑒𝜎𝑒𝑚∈ℳ
(11)
The arm-cycle system dynamics can be rewritten, accounting
for the switching signals, as:
𝑀�̈� + 𝑉𝑚�̇� + 𝐺 + 𝑃 + 𝑐𝑑�̇� + 𝑑 = 𝐵𝑀𝑢𝑠 + 𝐵𝐸𝑢𝑐 (12)
Finally, the overall switched system adheres to the following
properties:
Property 1: The inertia matrix (𝑀) is positive-definite and can be bounded as 𝑐𝑚 ≤ 𝑀 ≤ 𝑐𝑀, where 𝑐𝑚 and 𝑐𝑀 are known constants.
Property 2: The centripetal and Coriolis matrix (𝑉𝑚) can be upper bounded as |𝑉𝑚| ≤ 𝑐𝑣|�̇�|, where 𝑐𝑣 is a known constant.
Property 3: The gravitational effects matrix (𝐺) can be upper bounded as |𝐺| ≤ 𝑐𝐺, where 𝑐𝐺 is a known constant.
Property 4: The passive and viscoelastic tissue effects (𝑃) can be bounded as |𝑃| ≤ 𝑐𝑃1 + 𝑐𝑃2|�̇�|, where 𝑐𝑃1 and 𝑐𝑃2 are known constants.
Property 5: The time-varying disturbance term (𝑑) can be upper bounded as |𝑑| ≤ 𝑐𝑑, where 𝑐𝑑 is a known constant.
Property 6: The lumped muscle control effectiveness term
(𝐵𝑀) can be upper and lower bounded as 𝑐𝐵1 ≤ 𝐵𝑀 ≤ 𝑐𝐵2, where 𝑐𝐵1 and 𝑐𝐵2 are known constants.
Property 7: The motor control effectiveness term (𝐵𝐸) can be upper and lower bounded as 𝑐𝐸1 ≤ 𝐵𝐸 ≤ 𝑐𝐸2, where 𝑐𝐸1 and 𝑐𝐸2 are known constants.
Property 8: The inertia and centripetal and Coriolis matrices
follow the skew-symmetry relationship: 1
2�̇� − 𝑉𝑚 = 0.
III. CONTROL DEVELOPMENT
The control objective of the arm-cycle system is to track a
desired crank cadence. Two tracking errors are measured and
the error signals are expressed as:
𝑒1 ≜ 𝑞𝑑 − 𝑞 (13)
𝑒2 ≜ �̇�1 + 𝛼𝑒1 (14)
where 𝑒1 is the position tracking error, 𝑒2 is an auxiliary filtered tracking error that considers errors in position and cadence
tracking, and 𝛼 is a user-defined positive gain. The desired crank position (𝑞𝑑) is designed so that its derivatives exist and are bounded as �̇�𝑑, �̈�𝑑 ∈ ℒ∞ . Additionally, the tracking errors are combined to generate an error vector (𝑧):
𝑧 ≜ [𝑒1 𝑒2]𝑇 (15)
The open-loop error system is calculated by taking the time
derivative of (14), multiplying the derivative by the inertia
matrix (𝑀), and using (12)-(14) to yield:
𝑀�̇�2 = 𝜒 − 𝑒1 − 𝑉𝑚𝑒2 − 𝐵𝑀𝑢𝑠 − 𝐵𝐸𝑢𝑐 (16)
where 𝜒 represents a group of constant and state-dependent terms that can be bounded using Properties 1-8 by a known
function of the states. Equation 16 and the Lyapunov-based
stability analysis from the subsequent section are used to design
the robust sliding mode controllers for the stimulation (𝑢𝑠):
𝑢𝑠 ≜ 𝑘1𝑒2 + (𝑘2 + 𝑘3‖𝑧‖ + 𝑘4(‖𝑧‖)2)𝑠𝑔𝑛(𝑒2) (17)
and the current supplied to the motor (𝑢𝑐):
𝑢𝑐 ≜ 𝑘5𝑒2 + (𝑘6 + 𝑘7‖𝑧‖ + 𝑘8(‖𝑧‖)2)𝑠𝑔𝑛(𝑒2) (18)
where 𝑘1, 𝑘2, 𝑘3, 𝑘4, 𝑘5, 𝑘6, 𝑘7, and 𝑘8 are selectable positive control gains and sgn( ̇ ∙) is the signum function. The control inputs in (17)-(18) are substituted into the open-loop error
system in (16) to generate the closed-loop system dynamics:
𝑀�̇�2 = 𝜒 − 𝑒1 − 𝑉𝑚𝑒2 − 𝐵𝑀[𝑘1𝑒2 + (𝑘2 + 𝑘3‖𝑧‖
+𝑘4(‖𝑧‖)2)𝑠𝑔𝑛(𝑒2)]- 𝐵𝐸[𝑘5𝑒2 + (𝑘6 + 𝑘7‖𝑧‖ +
𝑘8(‖𝑧‖)2)𝑠𝑔𝑛(𝑒2)]
(19)
IV. STABILITY ANALYSIS
The stability analysis begins by defining 𝑉𝐿, a positive-definite, continuously differentiable common Lyapunov
function candidate, denoted as:
𝑉𝐿 ≜
1
2𝑀𝑒2
2 +1
2𝑒1
2 (20)
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where 𝑉𝐿 can be bounded as:
𝜆1(‖𝑧‖)2 ≤ 𝑉𝐿 ≤ 𝜆2(‖𝑧‖)
2 (21)
where 𝜆1 and 𝜆2 are known constants. Taking the derivative of (20) and substituting in (14) and (16) results in:
�̇�𝐿 = 𝑒2(𝜒 − 𝑒1 − 𝑉𝑚𝑒2 − 𝐵𝑀𝑢𝑠 − 𝐵𝐸𝑢𝑐)
+1
2�̇�𝑒2
2 + 𝑒1(𝑒2 − 𝛼𝑒1) (22)
For the case when 𝜎𝑚 = 1 and 𝜎𝑒 = 0, (22) can be simplified by using Properties 6-8, (17)-(18), cancelling terms, upper
bounding, and assuming certain gain conditions are met to
produce:
�̇�𝐿 ≤ −𝑐𝐵1𝑘1𝑒22 − 𝛼𝑒1
2 (23)
For the case when 𝜎𝑚 = 0 and 𝜎𝑒 = 1, (22) can be simplified by using Properties 6 and 8, (17)-(18), cancelling terms, upper
bounding, and assuming certain gain conditions are met to
produce:
�̇�𝐿 ≤ −𝑐𝐸1𝑘5𝑒22 − 𝛼𝑒1
2 (24)
The bounds in (23) and (24) can be expressed in terms of the
error vector (𝑧) and a known constant (𝜆3) for all time as:
�̇�𝐿 ≤ −𝜆3(‖𝑧‖)2 (25)
Knowing that 𝑉𝐿 is bounded by (21), then �̇�𝐿 can be expressed as a first order differential equation:
�̇�𝐿 ≤ −
𝜆3𝜆2
𝑉𝐿 (26)
The bound in (26) was solved for 𝑉𝐿 in terms of the initial condition of 𝑉𝐿 (𝑉𝐿,0), 𝜆2, 𝜆3, and 𝑡 to yield:
𝑉𝐿 ≤ 𝑉𝐿,0exp (−𝜆3(𝑡−𝑡0)
𝜆2) (27)
Using (21) and the initial condition of the error vector (𝑧0), (27) can be rewritten as:
‖𝑧‖ ≤ √𝜆2𝜆1
‖𝑧0‖exp (−𝜆3(𝑡 − 𝑡0)
𝜆2) (28)
Hence, the error system is bounded by an exponentially
decaying envelope. As a result, the stability analysis shows that
the closed-loop error system in (19) is globally, exponentially
stable for all 𝑡 ∈ [𝑡0, ∞), where 𝑡0 is initial time.
V. EXPERIMENTS
For simplicity, the performance of the controllers in (17)-(18)
were assessed using preliminary experiments on the right arm.
The goal of the experiments was to track a desired crank
velocity of 40 RPM. In the experiments, only the biceps and
triceps muscles of the right arm were stimulated. Experiments
were performed on three able-bodied individuals (three male)
23-27 years old. Prior to the experiment, each individual gave
written informed consent approved by the University of Florida
Institutional Review Board.
A. Arm-Cycle Testbed Setup
The arm-cycle system, shown in Fig. 2, consists of two
independently controlled arm-cycles. Each arm-cycle has its
own electric motor, shaft, handle, torque sensor, and encoder.
The motor is a 250 Watt, brushed, 24 VDC electric motor
(Unite Motor Co. Ltd. MY1016Z) mounted to the system frame
and coupled to the handle. The torque sensor is located on the
same shaft as the handle. The encoder (US Digital H1) is used
to provide crank position feedback. The handle has straps to
ensure the user’s arm remains fixed to the cycle. There is also
an emergency stop button that enables the user to stop the
experiment immediately, if necessary.
Fig. 2. Right arm-cycle testbed setup. (A) Electric motor. (B) Handle. (C)
Encoder. (D) Torque Sensor.
The current supplied to the motor was controlled with data
acquisition hardware (Quanser QPIDe) and an Advanced
Motion Controls motor driver. A desktop computer was used to
run real-time control software (QUARC 2.5,
MATLAB/Simulink, Windows 10) at a sampling rate of 500
HZ to implement the controller. A current-controlled stimulator
(Hasomed RehaStim) and PALS electrodes were used to deliver
stimulation to the biceps and triceps; the same computer and
control software were used to control the stimulation.
B. Experimental Setup
Each participant was seated in a chair in front of the arm-
cycle testbed at a distance that allowed the entire crank cycle to
be completed without fully extending the right arm. Electrodes
were placed on the individual’s right biceps and triceps, as seen
in Fig. 3. The top biceps and triceps electrodes were placed high
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enough on the arm to ensure there was enough space for the
bottom electrodes, but low enough that the shoulder would not
be stimulated. The individuals were instructed to remain
passive throughout the entire exercise, unless otherwise
instructed such as in Protocol 1, and to stay fixed in the chair;
this was done to minimize the volitional effort in the
experiment.
Fig. 3. Arm-cycle testbed with an individual. (A) PALS electrodes for Biceps.
(B) PALS electrodes for Triceps.
Two different protocols were performed on the arm-cycle
testbed, identified as Protocol 1 and Protocol 2. In Protocol 1,
the motor was always on and muscle stimulation was applied in
the muscle stimulation regions. Individuals were shown a
monitor with the actual and desired cadence; they were also
instructed to briefly apply volitional effort if the cadence error
began to exceed ±5 RPM. Protocol 1 was conducted to evaluate the performance of the modified arm-cycle testbed. In
Protocol 2, the motor was on in all regions except the bicep
stimulation region. Protocol 2 was conducted without the motor
during the bicep stimulation region to explore the possibility of
only having the motor on during kinematic dead zones.
Each experiment was designed to last 180 seconds. The
experiment began by using the motor to bring the arm-cycle to
40 RPM before starting the switching between muscle
stimulation and motor. The arm cycle speed was increased
exponentially to 40 RPM before leveling out. The switching
signal was designed to start at 20 seconds and from that point
on the participant received stimulation until the end of the
experiment when in FES regions of the crank cycle.
Furthermore, the motor had a current offset of 0.05 A to
ensure smooth cycling in the muscle stimulation regions, since
the motor was never completely turned off in Protocol 1. The
stimulation regions were found experimentally and identified
as:
5.06 𝑟𝑎𝑑 < 𝑄𝑏𝑖 < 5.76 𝑟𝑎𝑑 (28)
1.75 𝑟𝑎𝑑 < 𝑄𝑡𝑟𝑖 < 2.44 𝑟𝑎𝑑 (29)
where 𝑄𝑏𝑖 and 𝑄𝑡𝑟𝑖 represent the regions where the biceps and triceps were stimulated, respectively. Multiple experiments
were performed to adjust the gains in (14) and (17)-(18) to
ensure that the controller tracked the desired cadence
effectively. Once the controller was tuned, a final experiment
was performed to collect data.
C. Results
Protocol 1
The cadence, cadence error, motor current input, and pulse
width stimulation for each subject for Protocol 1 are seen in
Fig. 4-9.
Fig. 4. Plots of Subject 1 (S1) results. The top graph compares the actual
cadence (blue) to the desired cadence (red). The bottom graph shows the
cadence and position errors, where 𝑒1̇ (blue) is the cadence error in RPM and 𝑒1 (red) is the position error in rad. The plots start at 20 seconds since the rider starts at rest and ramps up to the desired cadence (40 RPM) in 20 seconds, after
which the controllers in (17)-(18) turn on.
Fig. 5. Plots of Subject 1 (S1) results. The top graph shows the amount of
current sent to the motor. The bottom graph shows the pulse width stimulation
over the course of the experiment. The vertical black line indicates when the controllers in (17)-(18) were activated.
Fig. 6. Plots of Subject 2 (S2) results. The top graph compares the actual
cadence (blue) to the desired cadence (red). The bottom graph shows the
cadence and position errors, where 𝑒1̇ (blue) is the cadence error in RPM and 𝑒1 (red) is the position error in rad. The plots start at 20 seconds since the rider starts at rest and ramps up to the desired cadence (40 RPM) in 20 seconds, after which the controllers in (17)-(18) turn on.
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Fig. 7. Plots of Subject 2 (S2) results. The top graph shows the amount of current sent to the motor. The bottom graph shows the pulse width stimulation
over the course of the experiment. The vertical black line indicates when the
controllers in (17)-(18) were activated.
Fig. 8. Plots of Subject 3 (S3) results. The top graph compares the actual
cadence (blue) to the desired cadence (red). The bottom graph shows the
cadence and position errors, where 𝑒1̇ (blue) is the cadence error in RPM and 𝑒1 (red) is the position error in rad. The plots start at 20 seconds since the rider starts at rest and ramps up to the desired cadence (40 RPM) in 20 seconds, after
which the controllers in (17)-(18) turn on.
Fig. 9. Plots of Subject 3 (S3) results. The top graph shows the amount of
current sent to the motor. The bottom graph shows the pulse width stimulation
over the course of the experiment. The vertical black line indicates when the controllers in (17)-(18) were activated.
Protocol 2
The cadence, cadence error, motor current input, and
pulse width stimulation for each subject for Protocol 2 are
seen in Fig. 10-15.
Fig. 10. Plots of Subject 1 (S1) results. The top graph compares the actual cadence (blue) to the desired cadence (red). The bottom graph shows the
cadence and position errors, where 𝑒1̇ (blue) is the cadence error in RPM and 𝑒1 (red) is the position error in rad. The plots start at 20 seconds since the rider starts at rest and ramps up to the desired cadence (40 RPM) in 20 seconds, after
which the controllers in (17)-(18) turn on.
Fig. 11. Plots of Subject 1 (S1) results. The top graph shows the amount of current sent to the motor. The bottom graph shows the pulse width stimulation
over the course of the experiment. The vertical black line indicates when the
controllers in (17)-(18) were activated.
Fig. 12. Plots of Subject 2 (S2) results. The top graph compares the actual
cadence (blue) to the desired cadence (red). The bottom graph shows the
cadence and position errors, where 𝑒1̇ (blue) is the cadence error in RPM and 𝑒1 (red) is the position error in rad. The plots start at 20 seconds since the rider starts at rest and ramps up to the desired cadence (40 RPM) in 20 seconds, after
which the controllers in (17)-(18) turn on.
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Fig. 13. Plots of Subject 2 (S2) results. The top graph shows the amount of current sent to the motor. The bottom graph shows the pulse width stimulation
over the course of the experiment. The vertical black line indicates when the
controllers in (17)-(18) were activated.
Fig. 14. Plots of Subject 3 (S3) results. The top graph compares the actual cadence (blue) to the desired cadence (red). The bottom graph shows the
cadence and position errors, where 𝑒1̇ (blue) is the cadence error in RPM and 𝑒1 (red) is the position error in rad. The plots start at 20 seconds since the rider starts at rest and ramps up to the desired cadence (40 RPM) in 20 seconds, after
which the controllers in (17)-(18) turn on.
Fig. 15. Plots of Subject 3 (S3) results. The top graph shows the amount of current sent to the motor. The bottom graph shows the pulse width stimulation
over the course of the experiment. The vertical black line indicates when the
controllers in (17)-(18) were activated.
The mean and standard deviation of the cadence error (𝑒1̇) and cadence were calculated for each subject in Protocols 1-2 and
the results are shown in Table I-II.
TABLE I
PROTOCOL 1 ANALYSIS
Subject 1 Subject 2 Subject 3 Averages
𝑒1̇ Mean (RPM) -0.04 -0.06 -0.04 -0.04 𝑒1̇ STD (RPM) 2.69 3.43
2.35 2.83
Cadence Mean (RPM)
40.04 40.06 40.04 40.04
Cadence STD
(RPM) 2.69 3.43 2.35 2.83
TABLE II
PROTOCOL 2 ANALYSIS
Subject 1 Subject 2 Subject 3 Averages
𝑒1̇ Mean (RPM) -0.02 -0.03 -0.04 -0.03 𝑒1̇ STD (RPM) 4.74 5.37
5.94 5.35
Cadence Mean
(RPM) 40.02 40.03 40.04 40.03
Cadence STD
(RPM) 4.74 5.37 5.94 5.35
VI. DISCUSSION
A. Experimental Results
The experimental results from Protocol 1 demonstrate the
ability of the controllers in (17)-(18) to distribute current to the
electric motor and apply FES to the rider’s muscles on the
modified arm-cycle. The experiment in [6] had the same
protocol as Protocol 1 except for the cadence tracked. In [6], a
cadence of 65 RPM was tracked, while the experiments
performed in this thesis had a cadence of 40 RPM. A cadence
of 65 RPM was deemed too fast to be an effective arm cycling
cadence and was reduced to 40 RPM. In addition, participants
were not allowed to apply volition in [6] and Protocol 2 was
also not performed in the previous study [6]. The experiment in
[6] had a cadence error of -0.06 ± 7.96 RPM and the experiments in this thesis had an average cadence error of -0.04
± 2.83 RPM. The standard deviations of the cadence error (𝑒1̇) and cadence
for each subject in Protocol 1 were all under 4 RPM. However,
the standard deviations approximately double in Protocol 2
compared to Protocol 1. The average cadence error in Protocol
2 was -0.03 ± 5.35 RPM, which is also smaller than the error in [6]. Thus, these results display the arm-cycle testbed’s
potential for future experiments.
A challenge associated with decoupled cycling, such as the
arm-cycle in this thesis, is that for a portion of the crank cycle
the participant must fight against gravity. With a coupled cycle,
when one side is being resisted by gravity, the other is being
aided to assist in offsetting the gravitational effect. However,
this is not the case with a decoupled cycle. This results in a
portion of the cycle having an increased difficulty, which is
partially responsible for the worse cadence error when the
participant provides no voluntary input. To improve this result,
a more advanced controller will be necessary, such as an
adaptive controller to learn these periodic dynamics that occur
throughout the crank cycle.
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B. Arm-cycle Testbed Design
The arm-cycle testbed used for this thesis is similar to the one
used in [6]; Fig. 16 shows the left arm-cycle testbed used in [6].
The arm-cycle testbed in this thesis used the same arm-cycle
frame, handle, shaft couplers, encoder, and torque sensor. The
direct drive electric motor was replaced with the chain driven
electric motor mentioned before. The existing direct drive
motor lacked resolution resulting in an inability to effectively
control the system using a robust sliding-mode controller.
Fig. 16 Original left arm-cycle testbed.
To accommodate the new motor, a motor mount was designed
and machined so that the motor was mounted to the existing
arm-cycle testbed structure. A new shaft was machined to
couple the handle, sprocket driven by the motor, and the gear
connected to the encoder. The chain used to drive the sprocket
was fed through a non-spring-loaded chain tensioner to
maintain tension in the chain.
Furthermore, a sleeve bearing was mounted near the encoder
to provide stability at the end of the shaft and to ensure accurate
measurements were recorded by the encoder. The previous
sleeve bearing near the handle was replaced with one that only
had a 5° shaft misalignment capability; the existing bearing had
a 23° misalignment capability. This sleeve bearing was replaced
to reduce disturbances seen by the system during cycling.
C. Future Design Improvements
The arm-cycle testbed can be improved for future
experiments by increasing the rigidity of the system. The
system currently flexes a visible amount along the coupling
shaft. Flexible couplers were initially chosen to help account
for misalignment but have proven to be detrimental. The
flexible couplers can be replaced with rigid couplers. In
addition, an aluminum shaft was used to couple the sprocket
and encoder gear which can be replaced with a steel shaft.
Lastly, the coupling shaft portion of the system can be
shortened.
VII. CONCLUSION
This thesis discussed the challenges and benefits of
controlling FES systems, particularly for rehabilitation
applications. The application discussed in this thesis was for an
arm-cycle system and the thesis sought to improve the
performance (cadence tracking) of the arm-cycle system in a
previous study. The changes included implementing new
hardware (i.e. electric motor) and software (i.e. modified robust
sliding-mode controller). The results from the preliminary
experiments indicate the controllers’ ability to sufficiently track
a desired cadence; the results also showed that the new system
is an improvement over the previous system.
The improved arm-cycle testbed will enable future research
in FES rehabilitation for upper extremities. There is potential
for restoring muscle function in individuals with NCs using
arm-cycling; however, experimental results from experiments
conducted on able-bodied individuals do not represent how
individuals with NCs would perform. Moreover, additional
experiments, especially on individuals with NCs, will need to
be performed to further refine the controller and investigate
how this system would affect people with NCs. Future research
will involve characterizing the stimulation regions analytically
using upper arm kinematics and implementing adaptive control
schemes to account for unknown disturbances. Finally, future
experiments will involve simultaneous, independent cycling of
the left and right arms.
ACKNOWLEDGMENT
I would like to acknowledge my advisor, Dr. Warren E.
Dixon for allowing me to work in his lab the last three years,
my committee members and colleagues at the Nonlinear
Controls and Robotics Laboratory. I would also like to
acknowledge my mentors: Christian Cousin, Courtney Rouse,
and Brendon Allen for all the help and knowledge they have
provided me throughout the years. I would not be the engineer
I am today without everyone’s help and for that I will be
eternally grateful.
REFERENCES
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and Rehabilitation Engineering, vol. 24, no. 12, pp. 1373-1383 (2016).
[2] W. E. Dixon and M. J. Bellman, "Cycling Induced by Functional Electrical Stimulation: A Control Systems Perspective," ASME Dynamic
Systems & Control Magazine, vol. 4, no. 3, pp. 3-7 (2016).
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[7] G. J. Snoek, M. J. Ijzerman, H. J. Hermens, D. Maxwell, and F. Biering-Sorensen, “Survey of the needs of patients with spinal cord injury: impact
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