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A Robust H∞Full-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 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.

Index Terms—H∞ﬁltering, 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 H∞observer

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

P≥0denoting it as non-negative deﬁnite. For a transfer

function, H(z)analytic for |z| ≥ 0,

H(z)

2and

H(z)

∞

denote the standard H2and H∞norms, 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 H∞observer 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 ft∈RLft= [ fta fte ]T, the tensile

force vector, a relationship between the joint torque vector,

τ∈RN, 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

−r−r−r(2)

Alternatively, the tensile force vector for the system can

also be deﬁned by the following equation.

ft=fb−JT

j+

τ(3)

where JT

j+

is the Moore-Penrose pseudoinverse of the

transposed Jacobian matrix, and fb∈RLis 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[l−l0−Jaθ] + 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

Jm−b

Jm

ηGrkt

Jm

0−kt

La−Ra

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 H∞FUL 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 xk∈Rn,uk∈Rp,yk∈Rq,wk∈Rsand vk∈

Rtare the states, control input, measured output, process

and measurement noise vectors, respectively. Besides, A∈

Rn×n,B1∈Rn×p,B2∈Rn×s,C∈Rq×n,D1∈Rq×t

and D2∈Rq×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 =Aˆxk+B1uk−K(yk−ˆyk)(13)

where ˆxk∈Rnis the estimated state; ˆyk∈Rnthe

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 kekk2≤γ(kwkk2+

kvkk2), with the robustness level γ∈R 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 H∞can be characterized using the Lyapunov

function, ν(xk) = xT

kPxk, as done in [34], imposing that

H(z)

∞< γ ⇔ ∃P∈Rn×nP=PT≥0(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 Z∈Rn×qand Pare the variables

of the problem [30]. In addition to that, K∈Rn×qcan be

recovered using the following expression

K=P−1Z(19)

On the other hand, to further improve this system’s ro-

bustness, a slack variable, G∈Rn×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+GT−P 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 H∞observer-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, 7.2×10−3mm.

So, this results in a root mean square error (RMSE) for θof

about 0.1394 rad and a robustness level, γ, of 2.2915×10−6.

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 H∞observer-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 (3−3.5s), 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.

REFERENCES

[1] D. Pilling, P. Barrett, and M. Floyd, “Disabled people and the internet:

Experiences, barriers and opportunities,” 2004.

[2] W. H. Organization et al., “World report on disability: World health

organization; 2011.”

[3] D. Cummings, “Prosthetics in the developing world: a review of the

literature,” Prosthetics and orthotics international, vol. 20, no. 1, pp.

51–60, 1996.

[4] J. Ten Kate, G. Smit, and P. Breedveld, “3d-printed upper limb pros-

theses: a review,” Disability and Rehabilitation: Assistive Technology,

vol. 12, no. 3, pp. 300–314, 2017.

[5] C. A. Caspers, “Pressure/temperature monitoring device for prosthet-

ics,” Dec. 19 2006, uS Patent 7,150,762.

[6] A. Chortos, J. Liu, and Z. Bao, “Pursuing prosthetic electronic skin,”

Nature materials, vol. 15, no. 9, p. 937, 2016.

[7] M. C. Jimenez and J. A. Fishel, “Evaluation of force, vibration and

thermal tactile feedback in prosthetic limbs,” in 2014 IEEE Haptics

Symposium (HAPTICS). IEEE, 2014, pp. 437–441.

[8] J. Rothwell, M. Traub, B. Day, J. Obeso, P. Thomas, and C. Marsden,

“Manual motor performance in a deafferented man,” Brain, vol. 105,

no. 3, pp. 515–542, 1982.

[9] R. S. Johansson and G. Westling, “Roles of glabrous skin receptors

and sensorimotor memory in automatic control of precision grip when

lifting rougher or more slippery objects,” Experimental brain research,

vol. 56, no. 3, pp. 550–564, 1984.

[10] J. Monz ´

ee, Y. Lamarre, and A. M. Smith, “The effects of digital

anesthesia on force control using a precision grip,” Journal of neu-

rophysiology, vol. 89, no. 2, pp. 672–683, 2003.

[11] A. Cranny, D. Cotton, P. Chappell, S. Beeby, and N. White, “Thick-ﬁlm

force, slip and temperature sensors for a prosthetic hand,” Measurement

Science and Technology, vol. 16, no. 4, p. 931, 2005.

[12] C. Cipriani, M. Controzzi, and M. C. Carrozza, “The smarthand

transradial prosthesis,” Journal of neuroengineering and rehabilitation,

vol. 8, no. 1, p. 29, 2011.

[13] A. Akhtar, K. Y. Choi, M. Fatina, J. Cornman, E. Wu, J. Sombeck,

C. Yim, P. Slade, J. Lee, J. Moore et al., “A low-cost, open-source,

compliant hand for enabling sensorimotor control for people with tran-

sradial amputations,” in 2016 38th Annual International Conference

of the IEEE Engineering in Medicine and Biology Society (EMBC).

IEEE, 2016, pp. 4642–4645.

[14] L. Jiang, B. Zeng, S. Fan, K. Sun, T. Zhang, and H. Liu, “A modular

multisensory prosthetic hand,” in 2014 IEEE International Conference

on Information and Automation (ICIA). IEEE, 2014, pp. 648–653.

[15] E. National Academies of Sciences, Medicine et al.,The promise

of assistive technology to enhance activity and work participation.

National Academies Press, 2017.

[16] S.-K. Sul, Control of electric machine drive systems. John Wiley &

Sons, 2011, vol. 88.

[17] E. Vazquez-Sanchez, J. Sottile, and J. Gomez-Gil, “A novel method

for sensorless speed detection of brushed dc motors,” Applied Sciences,

vol. 7, no. 1, p. 14, 2017.

[18] G. C. Sincero, J. Cros, and P. Viarouge, “Arc models for simulation of

brush motor commutations,” IEEE transactions on magnetics, vol. 44,

no. 6, pp. 1518–1521, 2008.

[19] T. Figarella and M. Jansen, “Brush wear detection by continuous

wavelet transform,” Mechanical systems and signal processing, vol. 21,

no. 3, pp. 1212–1222, 2007.

[20] P. Chevrel and S. Siala, “Robust dc-motor speed control without any

mechanical sensor,” in Proceedings of the 1997 IEEE international

conference on control applications. IEEE, 1997, pp. 244–246.

[21] S. Yachiangkam, C. Prapanavarat, U. Yungyuen, and S. Po-ngam,

“Speed-sensorless separately excited dc motor drive with an adaptive

observer,” in 2004 IEEE Region 10 Conference TENCON 2004., vol.

500. IEEE, 2004, pp. 163–166.

[22] S. R. Bowes, A. Sevinc, and D. Holliday, “New natural observer

applied to speed-sensorless dc servo and induction motors,” IEEE

transactions on industrial electronics, vol. 51, no. 5, pp. 1025–1032,

2004.

[23] C. E. Castaneda, A. G. Loukianov, E. N. Sanchez, and B. Castillo-

Toledo, “Discrete-time neural sliding-mode block control for a dc mo-

tor with controlled ﬂux,” IEEE Transactions on Industrial Electronics,

vol. 59, no. 2, pp. 1194–1207, 2011.

[24] Z. Z. Liu, F. L. Luo, and M. H. Rashid, “Speed nonlinear control of

dc motor drive with ﬁeld weakening,” IEEE Transactions on Industry

Applications, vol. 39, no. 2, pp. 417–423, 2003.

[25] F. Farkas, S. Hal´

asz, and I. K´

ad´

ar, “Speed sensorless neuro-fuzzy

controller for brush type dc machines,” in Proceedings of the 5th

International Symposium of Hungarian Researchers on Computational

Intelligence, Budapest, Hungray, 2004, pp. 11–12.

[26] S. Weerasooriya and M. A. El-Sharkawi, “Identiﬁcation and control

of a dc motor using back-propagation neural networks,” IEEE trans-

actions on Energy Conversion, vol. 6, no. 4, pp. 663–669, 1991.

[27] S. Praesomboon, S. Athaphaisal, S. Yimman, R. Boontawan, and

K. Dejhan, “Sensorless speed control of dc servo motor using kalman

ﬁlter,” in 2009 7th International Conference on Information, Commu-

nications and Signal Processing (ICICS). IEEE, 2009, pp. 1–5.

[28] A. Khalid and A. Nawaz, “Sensor less control of dc motor using

kalman ﬁlter for low cost cnc machine,” in 2014 International Con-

ference on Robotics and Emerging Allied Technologies in Engineering

(iCREATE). IEEE, 2014, pp. 180–185.

[29] O. Aydogmus and M. F. Talu, “Comparison of extended-kalman-and

particle-ﬁlter-based sensorless speed control,” IEEE Transactions on

Instrumentation and Measurement, vol. 61, no. 2, pp. 402–410, 2011.

[30] M. C. De Oliveira, J. C. Geromel, and J. Bernussou, “Extended H2

and H∞norm characterizations and controller parametrizations for

discrete-time systems,” International Journal of Control, vol. 75, no. 9,

pp. 666–679, 2002.

[31] J. Fajardo, V. Ferman, A. Lemus, and E. Rohmer, “An affordable open-

source multifunctional upper-limb prosthesis with intrinsic actuation,”

in 2017 IEEE Workshop on Advanced Robotics and its Social Impacts

(ARSO). IEEE, 2017, pp. 1–6.

[32] J. Fajardo, V. Ferman, D. Cardona, G. Maldonado, A. Lemus, and

E. Rohmer, “Galileo hand: An anthropomorphic and affordable upper-

limb prosthesis,” IEEE Access, vol. 8, pp. 81365–81 377, 2020.

[33] R. Ozawa, K. Hashirii, and H. Kobayashi, “Design and control of

underactuated tendon-driven mechanisms,” in 2009 IEEE International

Conference on Robotics and Automation. IEEE, 2009, pp. 1522–1527.

[34] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix

inequalities in system and control theory. Siam, 1994, vol. 15.

[35] M. ApS, “The MOSEK optimization toolbox for MATLAB,” User’s

Guide and Reference Manual, version, vol. 4, 2019.

[36] J. Lofberg, “YALMIP: A toolbox for modeling and optimization in

MATLAB,” in 2004 IEEE international conference on robotics and

automation (IEEE Cat. No. 04CH37508). IEEE, 2004, pp. 284–289.