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A Robust H∞ Full-State Observer for Under-Tendon-Driven Prosthetic Hands

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A Robust H∞ Full-State Observer for Under-Tendon-Driven Prosthetic Hands

Abstract and Figures

Controlling different characteristics like force, speed and position is a relevant aspect in assistive robotics, because their interaction with diverse, common, everyday objects is divergent. Usual approaches to solve this issue involve the implementation of sensors; however, the unnecessary use of such devices increases the prosthetics’ prices in a significant manner. Thus, this work focuses on the design of an H∞ full-state observer to estimate the angular position and velocity of the motor’s gearhead in order to determine parameters such as the joints’ torque, fingertip force and the generalized coordinates of the digits of an under-tendon-driven system to replace the transductors. This is achieved by measuring the current demanded by the brushed DC motors operating the fingers of an open-source, 3D-printed and intrinsic prosthetic hand. Besides, the proposed method guarantees disturbance attenuation, as well as the asymptotic stability of the error estimation. In addition to that, the theoretical model was validated through its implementation on a prosthetic finger, showing successful results.
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A Robust HFull-State Observer for
Under-Tendon-Driven Prosthetic Hands
Julio Fajardo1,2, Diego Cardona1, Guillermo Maldonado1, Antonio Ribas Neto2,3and Eric Rohmer2
Abstract—Controlling different characteristics like force,
speed and position is a relevant aspect in assistive robotics,
because their interaction with diverse, common, everyday ob-
jects is divergent. Usual approaches to solve this issue involve the
implementation of sensors; however, the unnecessary use of such
devices increases the prosthetics’ prices in a significant manner.
Thus, this work focuses on the design of an Hfull-state
observer to estimate the angular position and velocity of the
motor’s gearhead in order to determine parameters such as the
joints’ torque, fingertip force and the generalized coordinates
of the digits of an under-tendon-driven system to replace
the transductors. This is achieved by measuring the current
demanded by the brushed DC motors operating the fingers of an
open-source, 3D-printed and intrinsic prosthetic hand. Besides,
the proposed method guarantees disturbance attenuation, as
well as the asymptotic stability of the error estimation. In
addition to that, the theoretical model was validated through
its implementation on a prosthetic finger, showing successful
results.
Index TermsHfiltering, linear matrix inequalities, full-
state observer, sensor-less estimation, upper-limb prosthesis.
I. INTRODUCTION
The affordability and availability of basic prosthetic care
are still limited in some parts of the world, particularly in
developing countries, since their limb-impaired inhabitants
cannot finance assistive technology worth $1000 or more and,
because their acquisition is not always guaranteed [1]–[4].
That is why the most common prosthetics in such places
are steel hooks, which have several limitations that make
them a non-competitive alternative to the more expensive and
practical bionic devices. This superiority lies in their correct
functioning in tandem with additional, diverse aspects look-
ing to ameliorate the user experience, like providing feedback
on their environment or the functionality of the device, even
if it elevates its price. [5] Thus, aiming for a low-budget,
anthropomorphic and highly functional prosthesis is relevant
to provide a solution to the accomplishing of activities of
daily living (ADLs), whilst incorporating additional, useful
features to improve the user experience.
1Author is with Turing Research Laboratory, FISICC, Galileo University,
Guatemala City, Guatemala. {juandiego.cardona,guiller}
@galileo.edu
2Author is with the Department of Computer Engineering and In-
dustrial Automation, FEEC, UNICAMP, 13083-852 Campinas, SP, Brazil.
{julioef,eric}@dca.fee.unicamp.br
3Author is with Federal Institute of Education, Science
and Technology Catarinense, 89609-000 Luzerna, SC, Brazil.
antonio.ribas@ifc.edu.br
On the other hand, the use of sensory feedback provides
the patients a more realistic substitute for their biologi-
cal counterpart, conveying information as thermal, pressure,
strain or vibrational stimuli [6,7]. This tactile feedback has
been shown to be important, since the coordination, manip-
ulation and grip selection whilst interacting with everyday
items has been demonstrated to worsen when having a
lower sensitivity [8]–[10]. However, haptics alone does not
improve the user’s interaction with common objects. This
leads to employ different kinds of transducers to close the
feedback control loops of the assistive devices to increase
their usability during ADLs. For instance, the utilization of
potentiometers and quadrature encoders, as well as the use of
force or tactile/pressure sensors, has been used to better the
functionality and the grip on items held by a prosthetic hand
by controlling the speed and the strength exerted by each
finger [11]–[14]. These approaches increase the price and, in
some cases, the size of the prosthetics themselves, leading
the patients to settle with lightweight aesthetic prostheses or
to not use any at all [15,16].
To mitigate these issues, most typical solutions rely on
the use of sensor-less observers, which estimate the full state
of the system depending on the current or voltage measure-
ments. These methods not only reduce the cost, weight and
size of the prosthesis itself, but also offer other advantages,
such as easy maintenance and repairability, since the system
is considerably simplified [17]. Sensor-less observers are
typically divided into two groups; the first one estimates the
angular speed of the shaft based on the ripple component of
the measured signal, which results from the electromotive
force induced in each coil or, when the brushes in the
commutator short adjacent segments [18,19]; the second is
built upon the dynamic linear model of brushed DC motors
and is able to estimate its own states [20]–[22].
Other approaches include the use of the brushed DC
motors’ non-linear model [23,24], as well as the utilization
of neural networks to obtain an approximation for the re-
sulting non-linear system, leading to more complex systems
with high computational costs [25,26]. Furthermore, alternate
versions consider the implementation of more specialized
methods in order to improve the estimation of the states under
a stochastic dynamical system. In this manner, methods such
as the Kalman (KF), extended Kalman (EKF) and the particle
(PF) filters provide robustness to exogenous disturbances
surging from both, the process and the sensor [27]–[29].
However, these errors need to be modeled as Gaussian noise,
which causes issues in real applications, especially due to
the manual noise covariance tuning parameters. Similarly,
H-based observers can also handle with such uncertainty,
but only require to be energy-bounded, thus no assumptions
regarding the noise are needed. In addition to that, this
methodology ensures that the energy gain from the noise
inputs to the estimation error ratio is limited by an upper-
bound limit, which guarantees the convergence of its solution.
This work proposes a method to obtain the Hobserver
gain matrix through the use of linear matrix inequalities
(LMIs) methodologies [30] in order to estimate the full state
of the discrete-time model of a brushed DC motor actuating
the fingers of an assistive device for transradial amputees,
in this case, the Galileo Hand, an intrinsic, under-tendon-
driven (UTD), upper-limb prosthesis [31,32]. In addition
to that, the position and velocity of the fingers can also
be estimated by measuring the current demanded by the
actuators operating each finger on the artificial hand.
The notation used throughout this work is as follows: capi-
tal and lower-case bold letters stand for matrices and vectors,
respectively; the rest denote scalars. For symmetric matrices,
P>0indicates that Pis positive definite; similarly with
P0denoting it as non-negative definite. For a transfer
function, H(z)analytic for |z| ≥ 0,
H(z)
2and
H(z)
denote the standard H2and Hnorms, correspondingly.
Furthermore, for the sake of easing the notation of partitioned
symmetric matrices, the symbol ?indicates, generically, each
of its symmetric blocks.
The rest of this paper is structured as follows: Section II
elaborates on the UTD system used in the upper-limb pros-
thesis and its implications, Section III states the issues of
designing an observer for the prosthetic hand described in
the previous section, whilst Section IV proposes a discrete-
time Hobserver to handle with unknown measurements
and process noises, as well as a method to find its gain
through the utilization of LMI methods. Finally, experimental
results and conclusions are presented in Sections V and VI,
accordingly.
II. TH E UND ER -TE ND ON -DRIVEN MACH IN E
The Galileo Hand is an affordable, open-source, anthro-
pomorphic and UTD myoelectric upper-limb prosthesis for
transradial amputees, whose intrinsic design allows for indi-
vidual finger control [31,32]. These digits are conformed by
three phalanges: distal, proximal and middle; as well as three
joints: distal and proximal interphalangeal (DIP and PIP) and
the metacarpophalangeal (MCP) one (illustrated in Fig. 1).
Thus, each finger possesses 3degrees of freedom (DOF); but,
since each one is operated by a single motor, only one degree
of actuation (DOA). Such a system permits the extension
and flexion of each member by operating its two tendons; an
active and a passive one, which run along the internal canals
inside the finger (the blue sections in Fig. 1). The first one
consists of a waxed nylon cord extended along the finger’s
dorsal side, which is actuated by a brushed DC motor with a
gear ratio of 250:1; thus, generating a positive tensile force,
fta, that flexes the finger. The second one is composed by a
round, surgical-grade elastic going through the duct inside the
digit’s volar face; this results in a passive tensile force, fte,
opposing itself to the actuator’s drive that depends uniquely
on the joints’ deflection, resulting in springing the limb back
open [33].
Thus, letting Lbe the number of tendons; Nthe amount
of joints; and ftRLft= [ fta fte ]T, the tensile
force vector, a relationship between the joint torque vector,
τRN, can be given by
τ=JT
jft(1)
where Jj=Jja Jj e Tis the Jacobian matrix for the
active and passive tendons.
Furthermore, considering ris the radius of the joint’s
pulleys and, taking into account the tendon-driven machine
described before (resulting in L= 2 and N= 3), the
following is true for each finger
Jj=r r r
rrr(2)
Alternatively, the tensile force vector for the system can
also be defined by the following equation.
ft=fbJT
j+
τ(3)
where JT
j+
is the Moore-Penrose pseudoinverse of the
transposed Jacobian matrix, and fbRLis a bias force
vector that prevents the tendons from loosening and does not
have an impact on τ, which is defined as follows
fb=Aξ,A=hIL(JT
j)+JT
ji(4)
Fig. 1: Mechanical design for the fingers, where ris the
pulley’s radius; and θ, the gearhead shaft’s angular position.
such that ξis a compatible dimensional vector with Aand
ILis the identity matrix of size L.
Considering the previous equations, the relationship be-
tween the generalized coordinates, q, and the motor angle
vector, θ, can be defined as the following
q=J+
j[ll0Jaθ] + q0(5)
where l= [ lale]Tis the deflection of the tendons,
such that laand leare the expansion of active and passive
ones, respectively; l0= [ 0 le0]T, an initial expansion of
the tendons to prevent them from loosening; q0, an initial
angular displacement of the joints; and Ja, the Jacobian
matrix related to the actuator.
Therefore, since a positive initial expansion of the passive
tendon le0is considered for each finger, it is evident that
the bias force fb>0, resulting in a tendon-driven machine
and, moreover, since rank(Jj) = 1 <N, the system is,
additionally, a UTD mechanism.
Furthermore, the dynamic equations of the tendon-driven
system are given by the following equations
M¨
q+1
2˙
M+S+B0˙
q+Ggq=τ(6)
Jm¨
θ+b˙
θ+rpfta =τm(7)
where Mand B0are the inertia and damping matrices
of the finger, accordingly, Sis a skew-symmetric matrix and
Ggis the gravity load matrix. Additionally, Jmand bare the
gearhead’s moment of inertia and friction coefficient, corre-
spondingly; τm, the torque exerted by the motor gearhead’s
shaft; and rp, the radius of the pulley mounted on it [33].
III. PROB LE M STATEM EN T
Since the dynamic behavior of the finger is non-linear,
particularly due to the inertia matrix and the centripetal
and Coriolis terms expressed in Eq. (6), one cannot simply
estimate the full state of the coupled system of differential
equations, (6)-(7). Thus, an approximated linear model was
created instead, which considers the dynamic equations of the
finger as a mass-spring system, whose behavior is similar to
that of a UTD machine (as the passive tendon opposes itself
to the flexion movement, but favors the extension one). In
addition, this also simplifies the computational load, since
it is not necessary to linearize the model on each operating
point, permitting its implementation on the microcontroller
unit (MCU) used on the prosthetic device.
Furthermore, the mechanism that drives the fingers does
not have a mechanical limit to cease the extension movement,
causing the motor to continue actuating the digit and flexing it
again (as the pulley coils the string in the opposite direction).
Therefore, the purpose of implementing such an observer is
to determine the state of the fingers (opened or closed) using
the estimation of the angular displacement of the gearhead’s
shaft only, leading to not requiring an exact result for the
generalized coordinates q. However, an approximation for it
can still be determined from Eq. (5); similarly with the joints’
torque, τfrom Eq. (6).
Considering Gras the gear ratio, ktas the motor’s
constant, iaas the current it demands, and ηas the gearhead’s
efficiency, τmcan be obtained with the following expression:
τm=ηGrktia(8)
In this way, the continuous-time model for a brushed DC
motor in the space state results in:
˙
x=
0 1 0
ker2
p
Jmb
Jm
ηGrkt
Jm
0kt
LaRa
La
x+
0
0
1
La
u(9)
y= [0 0 1]x(10)
where x=hθ˙
θ iaiT
, with θand ˙
θbeing the gearhead’s
angular position and velocity, respectively; keis the elastic
constant of the passive tendon; Lais the motor’s inductance;
Raand uare the armature’s resistance and voltage, accord-
ingly; and yis the measured output.
IV. DISCRETE-TIME HFUL L- STATE OBSERVER
For designing the observer, a discretization of the afore-
mentioned system is required. Considering the noise compo-
nents and a sampling time k, it results in the following:
xk+1 =Axk+B1uk+B2wk(11)
yk=Cxk+D1vk+D2wk(12)
where xkRn,ukRp,ykRq,wkRsand vk
Rtare the states, control input, measured output, process
and measurement noise vectors, respectively. Besides, A
Rn×n,B1Rn×p,B2Rn×s,CRq×n,D1Rq×t
and D2Rq×sare the process, input control and input
process noise, measured output, as well as the output process
and output sensor noise matrices, correspondingly. Then, by
defining a general noise vector, ˜wk= [wkvk]T, an observer-
based filter can be described by
ˆxk+1 =xk+B1ukK(ykˆyk)(13)
where ˆxkRnis the estimated state; ˆykRnthe
estimated output; and K, the observer gain.
Since the initial conditions of the estimated state, ˆx0, are
equal to those of the initial state, x0= [0 0 0]T, one can
determine the filtering error dynamic, from the expressions
(11)-(13), with the following augmented system:
ek+1 =˜
Aek+˜
B ˜wk(14)
˜yk=˜
Cek+˜
D ˜wk(15)
with
˜
A=A+KC,˜
B= [B2+KD2KD1]
˜
C=C,˜
D= [D2D1]
The main goal is to find an optimal robust observer-based
filter for the system composed by (11) and (12), where the
error filtering, ek, has to satisfy that kekk2γ(kwkk2+
kvkk2), with the robustness level γR  γ > 0. Therefore,
from the bounded-real lemma and given the transfer function
H(z)in the complex frequency-domain for the system (14-
15), the norm Hcan be characterized using the Lyapunov
function, ν(xk) = xT
kPxk, as done in [34], imposing that
H(z)
< γ ⇔ ∃PRn×nP=PT0(16)
Hence, an observer meeting the aforementioned require-
ments can be successfully established if a solution to the
following convex optimization problem can be found
min
Z,P=PT>0
γ(17)
which is subject to the following LMI
P ATP+CTZT0n×s0n×sCT
?P PB2+ZD2ZD10n×q
? ? Is0s×sDT
2
? ? ? IsDT
1
? ? ? ? γ2Iq
>0
(18)
where the matrices ZRn×qand Pare the variables
of the problem [30]. In addition to that, KRn×qcan be
recovered using the following expression
K=P1Z(19)
On the other hand, to further improve this system’s ro-
bustness, a slack variable, GRn×n, can be incorporated
so that the optimization problem is now
min
Z,G,P=PT>0
γ(20)
subjected to the following LMI
P ATG+CTZT0n×s0n×sCT
?G+GTP GTB2+ZD2ZD10n×q
? ? Is0s×sDT
2
? ? ? IsDT
1
? ? ? ? γ2Iq
>0
(21)
Moreover, since G+GT>P>0, this implies that Gis
non-singular [30], resulting in Kbeing able to be recovered
by evaluating the equation mentioned underneath.
K= (GT)1Z(22)
V. RE SU LTS
The experiments to test and validate the methods proposed
in Sections III and IV were carried out using the index finger
of the Galileo Hand, which is controlled by a customized
board located on the inside of the palm of the artificial hand,
with its volar side in a supine position [31,32]. Additionally,
to design the robust Hobserver-based filter and to solve
the convex optimization problems subjected to the LMIs de-
scribed in Eqs. (17)-(22), MATLAB, YALMIP and MOSEK
were employed [35,36]. Later, with the resulting gain, the
observer described in Eq. (13) was implemented on the MCU
(ARM Cortex-M4 architecture) actuating the fingers of the
assistive device.
In this way, a current on-off controller was used to de-
termine when the finger is closed or grabbing some object,
whereas, based on the encoder measurements (the ground
truth), a PID position controller was implemented to fully
open the finger. This leads to what is shown in Fig. 2,
which illustrates the flexion and extension processes of the
finger. On the upper graph, the estimation of the angular
displacement, ˆ
θ, is juxtaposed to its ground truth alternative,
θ, whilst the lower one represents the current measured
on the Shunt resistor installed on the motor driver. The
aforementioned estimation was established based on the data
gathered by a 100 Hz reading of a quadrature encoder
and the on-chip ADC, respectively. Furthermore, the angular
displacement of the motor gearhead’s shaft when the finger
is completely flexed, is about 4.5971 rad; while its estimated
value, of 4.6775 rad. This implies that the active tendon
was coiled around 16.5mm, as opposed to the 16.8mm
estimation. A similar discrepance occurs on the extension
process, where that error is, approximately, 7.2×103mm.
So, this results in a root mean square error (RMSE) for θof
about 0.1394 rad and a robustness level, γ, of 2.2915×106.
Besides, a comparison at different points in time between
the experimental and estimated results for qwas established
(shown in Fig. 3), considering it can be calculated by Eq. (5).
Moreover, utilizing the aforementioned results for ˆq, one
can determine the resulting torque on each of the joints’ axes
using the Eq. (6). This can be visualized in Fig. 4, where the
torques exerted on the MCP, PIP and DIP joints correspond
to τ1,τ2and τ3, accordingly.
VI. CONCLUSIONS
A simplified dynamic model of the finger, together with
the design of an Hobserver-based filter (without making
any assumptions regarding the effects of noise) has proven
to be a successful alternative to installing sensors for the
estimation of the angular position of the motor gearhead’s
shaft from an under-tendon-driven prosthesis for transradial
amputees, as can be seen in Fig. 2. Since the purpose of an
artificial hand is to determine whether the fingers are fully
closed, opened or grasping an object, rather than a precise
position and orientation of the fingertips, the estimation error
obtained is sufficient for the apt fulfillment of ADLs.
Fig. 2: (a) Motor gearhead shaft’s angular displacement, θ. The dotted red line represents the estimation ˆ
θ; while the solid
blue line, the ground truth measured with a quadrature encoder. (b) Current measured on the motor’s armature, ia.
Fig. 3: Finger movement processes: the ground truth and
its estimation, from left to right, respectively, where the
movement starts in q0.
Additionally, this data can be used to determine the kine-
matics and dynamics of each finger of the assistive device by
estimating its generalized coordinates, as shown in Figs. 3
and 4, and employ this information in robust torque and
impedance controllers. Moreover, such a model enables its
implementation in an MCU, allowing for a more compact
and affordable option to install on prosthetics.
Furthermore, observing the comportment of the torques’
behaviour, shown in Fig. 4, one can trace the finger’s move-
ments as it flexes and extends. The first main peak indicates
when the motor starts to coil the string, leading the motor, the
elastic and the joints to have to overcome the static friction
coefficient to start mobilizing; therefore a higher tensile
force (and torque) has to be exerted (Fig. 5). Additionally,
the second peak is a negative one, as the actuator breaks,
which causes it to spin in the opposite direction; similarly,
with the extension process. Other relevant aspects to note
on Fig. 4 are the offsets in torque and the fact that their
Fig. 4: Torque τapplied on the MCP, PIP and DIP joints’
axes (τ1, τ2and τ3, correspondingly).
derivatives and peaks increase in magnitude as they are closer
to the metacarpus. The first one depends on the gravitational
energy impacting each joint, while the latter is consequence
of moving a larger lever as the joints are farther from the
fingertip. In addition to that, a change in the direction of the
torque is also palpable in the DIP and PIP joints, because
their coordinate systems are modified as the proximal and
middle phalanges rotate. Analyzing the graph permits to
corroborate the functionality of the prosthesis and facilitates
the pairing with a robust controller to properly regulate the
prosthetic hand’s overall torque to improve its grips.
Fig. 5: Active tensile force fta exerted during the process.
Finally, despite the disturbances presented in the current
measurement, as shown in Fig. 2 (33.5s), the methods
proposed in this work behaves as expected, reducing the
effects of noise on estimation. This can be improved by
designing a robust, full-order filter, also based on LMI
methods, guaranteeing a lower robustness level. On the other
hand, a better approximation of the model employing the
Takagi-Sugeno technique could handle the uncertainties in a
better way, both for the robust observer and the controller.
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