Hybrid Impedance and Nonlinear Adaptive Control for a 7-DoF Upper

Limb Rehabilitation Robot: Formulation and Stability Analysis

Andres Guatibonza

, Leonardo Solaque

, Alexandra Velasco

and Lina Pe

nuela

Militar Nueva Granada University, Bogot

a, Colombia

Keywords:

Hybrid Impedance Control, Nonlinear Adaptive Formulation, Rehabilitation Robotics, Upper Limb.

Abstract:

Physical rehabilitation aims to improve the condition of people with any musculoskeletal disorder. Different

assistive technologies have been developed to provide support to this process. In this context, human-machine

interaction has progressively improved to avoid abrupt movements and vibrations, to obtain a more natural

interaction, where control strategies play a key role. In this work, a control technique based on the combina-

tion of nonlinear adaptive theory with a hybrid impedance control applied to a 7-DoF upper limb assistance

robotic device is proposed. Additionally, we include the stability analysis using Lyapunov functions. Then,

we validate the strategies through simulations for one rehabilitation routine test. The articular and cartesian

obtained results demonstrate the effectiveness of the control to follow trajectories. The control stabilizes the

trajectories in 0.9 seconds even when the initial conditions start far from the desired trajectories, without pro-

ducing vibrations or overshoots, which is the desired behavior in rehabilitation applications like the one we

propose.

1 INTRODUCTION

People who suffer any trauma or musculoskeletal dis-

order require a rehabilitation process that focuses on

improving the functional capabilities and allows the

patient to recover socially, physically, and occupa-

tionally (Ritchie, 2003; hil, 2012; reh, 1950). To re-

gain the limb functionality, the patients undergo treat-

ments that include exposing the muscular tissues to

stress in a progressive and appropriate manner, in-

creasing the range of mobility and muscle strength

progressively (McHugh et al., 2013; Wattchow et al.,

2018; Bruder et al., 2017; Milicin and S

ırbu, 2018;

Gates et al., 2015; Ritchie, 2003).

Assistive technologies are oriented to support

physical rehabilitation processes, adapting to the pa-

tients’ condition, according to their disability (Linda

et al., 2018; Olanrewaju et al., 2015). The use of these

technologies has increased in the last few years due to

the effectiveness of the therapy (Ballantyne and Rea,

2019; Molteni et al., 2018). Moreover, assistive tech-

nologies allow the acquisition of accurate measure-

https://orcid.org/0000-0001-6102-563X

https://orcid.org/0000-0002-2773-1028

https://orcid.org/0000-0001-7786-880X

https://orcid.org/0000-0002-1925-9296

ments through sensors or smart devices to evaluate the

progress of the patients (Munih and Bajd, 2011). Re-

habilitation robots may be used for recovery or com-

pensatory purposes to improve the rehabilitation pro-

cesses and the patients’ quality of life in the shortest

possible time (reh, 1950).

Rehabilitation robots are designed to adapt to

the conditions of motion and strength assistance, ac-

cording to the level of intervention that the patient

requires. According to (Mancisidor et al., 2019a;

Akdo

gan et al., 2018) we deﬁne ﬁve levels of assis-

tance, namely (i) passive, that requires total robot in-

tervention, (ii) assistive, that requires partial robot in-

tervention, (iii) isotonic, which means no robot inter-

vention (Munih and Bajd, 2011; Trochimczuk et al.,

2018; Akdo

gan et al., 2018), (iv) isometric, where

there is a robotic-supplied static muscle level con-

traction, and (v) resistive, where there is a robotic-

supplied dynamic muscle strengthening (Munih and

Bajd, 2011). The assistance modes allow the parame-

terization of the exercises according to the patients’

condition. Here is where a proper control strategy

comes into play (Meng et al., 2015).

Rehabilitation robots are systems based on

human-machine interaction at the clinical level. In

this interaction, an adequate control strategy is neces-

Guatibonza, A., Solaque, L., Velasco, A. and Peñuela, L.

Hybrid Impedance and Nonlinear Adaptive Control for a 7-DoF Upper Limb Rehabilitation Robot: Formulation and Stability Analysis.

DOI: 10.5220/0010579206850692

In Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2021), pages 685-692

ISBN: 978-989-758-522-7

685

Figure 1: General scheme of proposed control.

sary to prevent further harm to the user. For this, the

controller have to accurately track trajectories within

the rehabilitation routines, and compensate the sys-

tem against disturbances and unwanted loads (phy,

1932).

Control strategies for rehabilitation applications

are designed to improve the system’s performance

and compensate undesired behaviors that may com-

promise the integrity of the person (Marchal-Crespo

and Reinkensmeyer, 2009; Meng et al., 2015). In the

literature, control strategies are designed to improve

the interaction capabilities of robotic systems with pa-

tients. This implies that both the rehabilitation sys-

tems and the control strategies have a certain level of

complexity in the design, especially if they are used

in robotic systems of more than 6-DoF.

In this paper we propose a control strategy that

combines hybrid impedance and nonlinear adaptive

control for assistive rehabilitation robotic systems.

In the literature, we ﬁnd recent works that propose

control strategies applied to assistive robotic systems,

focused on performing more natural motions. We

have evidenced that one of the most used controls is

impedance control. This strategy tries to imitate natu-

ral movements taking as a reference a desired damper-

spring-mass model. The controller can be force-based

or position-based (Jutinico et al., 2017; Song et al.,

2017). In this type of control, a concept called As-

sist as needed (AAN) has been implemented. This is

an assistance strategy that relates directly the levels of

intervention that the robot provides to the patient. For

example, in (Wu and Wu, 2018) an impedance con-

trol, a.k.a multimodal control is applied to a therapeu-

tic exoskeleton for upper limb rehabilitation. In (Asl

et al., 2020) a control strategy is developed to max-

imize the patient’s participation in the rehabilitation

process, using the AAN strategy with an impedance

controller for speed tracking, and a velocity tracking

controller, which adjusts its contribution in an AAN

way, by monitoring the tracking error. In the same

way, in (Chen et al., 2016) a control strategy based

on ANN is applied to a 7-degrees-of-freedom (DoF)

upper limb rehabilitation system. Moreover, in (Man-

cisidor et al., 2019b), an inclusive control based on

adaptive assist modes is applied for upper limb re-

habilitation using transition impedance control based

on position and strength. It is worth to remark that

the stability of impedance controllers is usually ana-

lyzed by theoretical methods such as Lyapunov equa-

tions, which are constrained to design conditions and

workspaces according to the application. In general,

to the best of our knowledge we have not found in

the literature an impedance controller that can adapt

to the dynamics of a system, and to other dynamics

and uncertainties. In this work, we propose the adapt-

ability that is required.

On the other hand, hybrid strategies have been

proposed to enhance the capabilities of impedance

controllers, see for instance, (Akdo

gan et al., 2018).

In the same way, an adaptive impedance control based

on backstepping theory and fuzzy logic has been pre-

sented in (Bai et al., 2019). In (Li et al., 2017), an

adaptive impedance control is presented using elec-

tromyography signals (sEMG) that are used to de-

sign the optimal reference impedance model. Fur-

thermore, an adaptive neural network control with a

high-gain observer is developed to approximate the

effect of the dead zone and the robot’s dynamics. Like

impedance control, stability in this type of controls

plays a very important role and must to be analyzed.

In the same way, alternatives of controls have been

proposed, some that use EMG as the main basis such

as (Lee et al., 2017) where one proposes a novel con-

trol method to minimize muscle energy for robotic

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

686

systems that support the movements of a user under

unknown external disturbances, using sEMG. Oth-

ers that use alternatives for the integration of con-

trol methods such as (Brahmi et al., 2018) where it

proposes an adaptive control through rehabilitation

tasks using integrated backstepping theory with time-

delay estimation, or as in (Miao et al., 2020) where a

new strategy of control for a bilateral upper limb sys-

tem using regulation based on human-robot interac-

tion force measured using compliance control based

on position controller (low level) and admittance con-

troller (high level). Also in (Huang et al., 2018),

an adaptive control based on force is proposed un-

der the AAN concept. Or also as an alternative in

(Wu et al., 2018), where a neural-diffuse adaptive

controller (NFAC) based on a radial basis function

network (RBFN) is proposed to guarantee the preci-

sion of the path tracking with parametric uncertain-

ties and environmental perturbations of an upper limb

exoskeleton. All these proposed techniques generate

new alternatives to improve or propose alternative or

complementary control strategies on this type of ap-

plications in physical rehabilitation.

In this paper, we design a control technique based

on the combination of nonlinear adaptive theory and

hybrid impedance control (position-based and force-

based strategy) applied to a 7-DoF upper limb assis-

tive robotic system. This is a interesting strategy that

exploits the characteristics of both techniques. We

carry out a stability analysis using Lyapunov func-

tions to prove the performance of the control, con-

sidering the stabilization time, the response to distur-

bances, and the error tracking. This work is a previ-

ous step to the implementation in the physical assis-

tive system. The control model that we propose aims

to improve and widen the application of (impedance)

nonlinear adaptive control strategies to assistive re-

habilitation systems. The proposed control model is

shown in Fig. 1. It consists in an adaptive nonlinear

control using an impedance control strategy, with the

option of switching between position-based or force-

based control when required, for a 7-DoF assistive

rehabilitation device. The results obtained from the

simulation of three common routines are the positions

of the trajectories in the joint space and the positions

and orientations of the end-effector trajectories. The

controller tracks with a maximum error of 2% the de-

sired trajectories. These results show the effectiveness

of the strategy proposed.

This paper is organized as follows: section II

presents the proposed assistive robotic system and the

dynamic formulation. In Section III we describe the

proposed control strategy and the stability analysis

performed using Lyapunov functions. The set up for

the validation and simulation results of common rou-

tine trajectories in upper limb rehabilitation are pre-

sented in Section IV. Finally we give some conclu-

sions and future work perspectives in Section V.

2 ASSISTIVE ROBOTIC

DYNAMIC MODEL

We have designed a 7-Dof assistive robotic system

for the rehabilitation of upper limb tendinopathies.

This system allows the main movements of each

joint of the upper limb, which are: (i) scapular

band, protraction/retraction movements. (ii) shoul-

der, ﬂexion/extension, abduction/adduction and inter-

nal/external rotation movements. (iii) elbow, ﬂex-

ion/extension movements. and (iv) wrist, ﬂex-

ion/extension, and pronation/supination movements.

Fig. 2 shows the concept of the robotic system and the

simpliﬁed diagram for the calculation of the transfor-

mation matrices, using the Denavit-Hartenberg con-

vention. Table 1 shows the corresponding parameters

to obtain the transformation matrices and the Jaco-

bian J(q) for the robot’s dynamic model and control

formulation.

Table 1: Denavit-Hartenberg parameters.

Joint

1 θ

∗

2 θ

∗

− π/2 L

0 −π/2

3 θ

∗

− π/2 L

0 −π/2

4 θ

∗

0 −L

π/2

5 θ

∗

0 0 π/2

6 θ

∗

+ π/2 L

0 π/2

7 θ

∗

+ π/2 0 L

Let us write the general dynamics model of the

assistive device in the joint space as,

M(q) ¨q +C(q, ˙q) ˙q + G(q) = τ

total

(1)

Where q is the vector R

7x1

of generalized joint co-

ordinates, M(q) is the inertia matrix R

7x7

, C(q, ˙q) is

the Coriolis matrix R

7x7

, G(q) is the vector of gravity

forces R

7x1

(due to the weight of each link) and τ :

is the vector of generalized (non-conservative) forces

7x1

Through the Euler-Lagrange formulation, the La-

grangian operator is

L(q, ˙q) = K(q, ˙q) − P(q) (2)

Where K(q, ˙q) is the kinetic energy and P(q) is the

potential energy of the system, then

K(q, ˙q) =

˙q

M(q) ˙q (3)

Hybrid Impedance and Nonlinear Adaptive Control for a 7-DoF Upper Limb Rehabilitation Robot: Formulation and Stability Analysis

687

Figure 2: Concept design and simpliﬁed diagram of the robotic system. Movements performed: 1. protraction/retraction. 2.

internal/external rotation. 3. ﬂexion/extension. 4. abduction/adduction. 5. ﬂexion/extension. 6. pronation/supination and 7.

ﬂexion/extension.

and

M(q) =

∑

i=1

+ J

) (4)

Where m

is the mass of link i, v

is the velocity of

the center mass of link i, ω

is the angular velocity of

the center of mass of link i

is the constant inertia

tensor, J

(q) is the lineal velocity component of the

Jacobian matrix of link i and J

(q) is is the angular

velocity component of the Jacobian matrix of link i.

Then, the potential energy is deﬁned as

P(q) = −

∑

i=1

(5)

Where m

is the mass of link i, g

is the gravity vector

and P

is the position of the center of mass of link i.

Considering J

(q) =

∂P

∂q

, the general vector of gravity

forces G(q) is

G(q) = −

∑

i=1

(q)m

(6)

(q) is the moment of joint i due to the gravity

(weight). Then, the coriolis matrix C(q, ˙q), can be

deﬁned by means of Euler-Lagrange formulation, us-

ing the Christoffel symbols of the ﬁrst kind. Then, we

have

i j

∑

k=1

i jk

˙q

(7)

i jk



∂m

i j

∂q

∂m

∂q

−

∂m

∂q



i jk

= c

ik j

Where n is the number of DoF, m

is the moment of

inertia at the i − th joint, when the other joints do

not move. m

i j

is the i − th joint acceleration effect

at the joint j (coupling effect), c

i j j

˙q

is the centrifu-

gal force at joint i due to the j −th joint velocity and

i jk

˙q

is the Coriolis effect on i − th joint, due to

j − th and k −th. joint velocity

Let us deﬁne the components of the generalized

forces τ

total

as:

total

= τ − f

ext

− f (q, ˙q) (8)

Where τ is the torque applied (by the actuators)

to the joints, f

ext

is the torque due to external

forces/moments and f (q, ˙q) is the torque due to fric-

tion in the joints deﬁned as

f (q, ˙q) = F

sgn( ˙q) +F

˙q (9)

Where F

is the static friction matrix and F

is the vis-

cous friction matrix (Akdo

gan et al., 2018; Brahmi

et al., 2018).

3 CONTROL STRATEGY:

HYBRID IMPEDANCE AND

NONLINEAR ADAPTIVE

CONTROL

Impedance control method is widely used in rehabil-

itation systems. In this paper, hybrid impedance con-

trol and nonlinear adaptive control are combined into

a single structure for a 7-DoF upper limb rehabilita-

tion system. We propose a control scheme as shown in

Fig. 1. Therefore, the desired mechanical impedance

of the robot’s end-effector can be adjusted while the

robot follows the trajectory of the desired force or po-

sition. This behavior aims to reproduce the actions

that the specialist performs during physiotherapy, re-

garding strength and position. In our case, a switching

matrix allows the transition from the position-based

control to the force-based control. This change and

the level of intervention of the system depends on

the therapist’s decision or on the system’s position or

force error. Physical exercises are modeled using the

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

688

hybrid impedance control approach and are classiﬁed

considering the level of intervention. According to

(Akdo

gan et al., 2018), the desired hybrid impedance

can be stated as

−J(q)

ext

= M

( ¨x − S ¨x

) + B

( ˙x − S ˙x

) + SK

− Sx

) + (I − S)F

(10)

Where M

is the desired inertia matrix R

6x6

, B

the desired damping matrix R

6x6

, K

is the matrix of

desired stiffness R

6x6

, x is the position and orientation

vector of the end effector R

6x1

, x

is the desired posi-

tion and orientation vector of the end effector R

6x1

, F

is the desired force vector R

6x1

and S is a switching

matrix between position-based and force-based con-

trols, for more information about this control equa-

tion, we refer to (Akdo

gan et al., 2018). Merging (1)

and (10), we can write the hybrid impedance control

law as

τ = M(q)J(q)

(S ¨x

+ M

−1

[(I − S)F

− B

( ˙x − S ˙x

)

− K

(x − Sx

) − J(q)

ext

] −

J(q) ˙q) +C(q, ˙q) + G(q)

+ f (q, ˙q) − f

ext

(11)

Where the velocity ˙x and acceleration ¨x of the end

effector are

˙x = J(q) ˙q ¨x = J(q) ¨q +

J(q) ˙q

Then, on the basis of the hybrid control deﬁnition,

we reformulate and propose our control strategy. This

formulation intends to implement hybrid impedance

control with the demonstration of stability using non-

linear adaptive control theory by applying Lyapunov

candidates, which guarantee stability, if there exists a

function V (x) that fulﬁlls the following conditions:

• V (x) positive deﬁnite for x 6= 0V (x) > 0

•

V (x) negative deﬁnite for x 6= 0

V (x) < 0

• V (x) → ∞ for kxk → ∞

• V (0) = 0

Hybrid impedance control allows both position-

based trajectory control when the patient needs to re-

gain mobility, and force-based control when the pa-

tient needs to regain muscle strength. The disadvan-

tage of using only this control is that there is no way to

ensure stability when the position-force control tran-

sition is made. In this case, the nonlinear adaptive the-

ory, with the Lyapunov candidates solves this draw-

back.

For the nonlinear adaptive formulation, let us de-

ﬁne η

= q,

= ˙q, η

= ˙q and

= ¨q and rewrite

this terms into the general dynamic model equation

(1). Merging (8) into (1) yields,

M(q)

+C(q, ˙q)η

+G(q)+ f (q, ˙q)+ f

ext

= τ (12)

=η

=M(q)

−1

[τ −C(q, ˙q)η

− G(q) − f (q, ˙q) − f

ext

]

= U (t) − F(t)

(13)

Where: U(t) = M(q)

−1

τ and F(t) =

M(q)

−1

[C(q, ˙q)η

+ G(q) + f (q, ˙q) + f

ext

]

Then by non-adaptive linear control methods, the

control law can be stated as

U(t) = F(t)+

ξ − K

− e

= M(q)

−1

[C(q, ˙q)η

+ G(q) + f (q, ˙q) + f

ext

] +

− K

− e

(14)

Where ξ is a virtual control input to e

= η

−

, η

is the desired position and e

= η

− ξ. And

consider that ξ(t) =

− K

, where K

is a gain

matrix of R

7x7

, and K

is a gain matrix of R

7x7

Then, the same procedure is applied to (10) of hy-

brid impedance rewriting with e

= (x − Sx

), where

is the selective error that allows to switch be-

tween the expressions of position-based and force-

based impedance control.

Now, let us introduce a variable as λ

= e

, and

the derivatives

= ˙e

, λ

= ˙e

and

= ¨e

. Then we

obtain the control law as

(t) = F

(t) − S ¨x

− K

− e

(15)

Where U

(t) = − f

ext

− (I − S)F

y F

(t) =

−1

+ SK

], ξ

(t) = S ˙x

− K

is a second

virtual control input, e

= λ

, e

= λ

+S ˙x

−ξ

is a gain matrix of R

6x6

and K

is a gain matrix of

6x6

Finally, the control law that relates the system

dynamics with the desired dynamics yields

U(t) = U

(t) + [F(t) − F

(t)] + [

ξ −

+ S ¨x

− K

] + [e

− e

]

(16)

Notice that (16) deﬁnes the control system states

as state errors. The errors compare the assistive sys-

tem dynamics with the impedance desired dynam-

ics. This control law requires the joint space errors,

end-effector cartesian space errors and force errors.

Therefore, we need to compute the kinematics and the

Jacobian matrix to move from one space to the other

during the simulation.

In addition, merging (14) into

, and merging

(16) into

, we guarantee stability by proving Lya-

punov functions are negative,

= −e

− e

= −e

(17)

Regardless the error, for (K

) > 0, stability

conditions of

and

will be satisﬁed.

Hybrid Impedance and Nonlinear Adaptive Control for a 7-DoF Upper Limb Rehabilitation Robot: Formulation and Stability Analysis

689

4 CONFIGURATION AND

SIMULATION RESULTS

For the validation of the control law in (16), we car-

ried out three different tests where we deﬁne com-

mon rehabilitation routines. The tests consists in in-

dividual movements of each joint and combined joint

movements of the whole upper limb. We deﬁned

the required joint trajectories for the following rou-

tines: elbow ﬂexion/extension, joint ﬂexion/extension

of elbow and shoulder, and shoulder ﬂexion/extension

with extended forearm. These are basic exercises that

are usually performed when an elbow injury occurs

for example. Here we present and analyze the results

of the shoulder ﬂexion/extension with extended fore-

arm.

In control simulation, the gain matrices K

, K

are deﬁned as

= 200 ∗diag[1,1, 1,1,1, 1,1]

= 200 ∗diag[1,1, 1,1,2, 1,1]

= 0.2 ∗ diag[1,1, 1,1,1, 1]

The values were ﬁne-tuned according to the desired

results obtained. For the gains of the desired inertia,

stiffness and damping matrices, the following values

were used:

= 0.01 ∗ diag[1,1, 1,1,1, 1]

= 0.1 ∗ diag[1,1, 1,1,1, 1]

Desired trajectories are periodic, proposing a fre-

quency of f = 0.1 Hz to obtain slow movements as a

regular routine. Disturbances between 0.1 to 0.5 Nm

are included to evaluate the behavior of the system

when external forces appear, for instance, involuntary

movements of the patient due to pain. Three differ-

ent tests were carried out in simulation. Here, we

present results of one routine that consists in shoulder

ﬂexion/extension. The simulation results are shown

in Fig.3. The resulting joint space and end-effector

trajectories are shown in Fig.3a. Joint errors and

torque control switching for desired forces are shown

in Fig.3b. In all cases the control stabilizes at 0.9

seconds approximately without presenting overshoots

despite the quick stabilization. Notice in Fig.3a there

is no perceptible vibrations and errors throughout the

trajectory. However, in Fig.3b (left), for constant val-

ues in the curves there are vibrations at maximum am-

plitudes of ±0.015 (radians) for both joint curves and

end effector curves. Moreover, the response obtained

is free of noise despite the disturbances.

In all the tests carried out, the error is under 2%.

The matrix of position-based to force-based control

switched in the middle of the routine. Fig.3b (right)

shows the moment when desired forces are com-

muted. Notice that the performance of the trajecto-

ries was preserved. For stability validation, the type

of control was changed from position-based to force-

based by switching the matrix S starting with desired

forces of value Fd = 1Nm and then, increased by an

average of 6% from second t = 50. The trajectory

does not diverge and the properties of the signal are

still preserved. That means, the control is working at a

different desired force without deviating from the de-

sired trajectory of the routine. The other two routines

had the same performance, the properties of the tra-

jectory obtained were still being preserved throughout

the simulation.

The obtained behavior is what is expected to be

obtained in robotic rehabilitation systems due to the

adaptive properties of the control to changes in po-

sition and force as in a conventional routine. These

changes can also be a consequence of sudden move-

ments that are usually produced by pain, but also oc-

cur when forces are applied progressively to the pa-

tient to increase muscle strength and on the other

hand, there is the option of adaptation to any therapy

under the AAN concept.

5 CONCLUSIONS

In this work, a control technique based on the combi-

nation of nonlinear adaptive control theory and hybrid

impedance (position-based and force-based control)

applied to a 7-DoF upper limb assistive robotic sys-

tem is proposed. Hybrid impedance control is respon-

sible for providing the levels of intervention of the

assistive system to the patient for mobility recovery

(position-based control) and muscular strength (force-

based control).

Through nonlinear adaptive theory, we ensure the

stability when changing the type of control. We per-

formed a stability analysis using Lyapunov functions

and we validated through simulations. The approach

of the Lyapunov functions helps to develop the con-

trol laws, guarantee stability, and also provides adapt-

ability to non-linearities and disturbances. Three tests

were considered as part of the validation of the pro-

posed control technique. Here we analyze the be-

havior of the system when performing the shoulder

ﬂexion/extension routine. The results show that the

controller tracks with (almost) zero error (2%) the de-

sired trajectories, demonstrating the effectiveness of

the proposed strategy. Therefore, this control gives an

insight into new opportunities for the improvement of

existing or development of new strategies applied to

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

690

(a) Shoulder ﬂexion/extension routine: joint space trajectory (left) with desired joint positions (continuous line) Q

d in

radians and obtained joint position (dashed line) Q

in radians for i = 1..7 and cartesian space (right) with desired cartesian

positions (continuous line) X

in meters and desired orientations (continuous line) beta

,al pha

,rho

in radians,

and obtained cartesian positions (dashed line) X,Y , Z in meters and obtained orientations (dashed line) beta, al pha,rho in

radians.

(b) Joint errors (left) in radians and desired forces conmutation (right) from position-based to force-based control

Figure 3: Shoulder ﬂexion/extension routine: trajectory in articular and Cartesian end-effector space, joint errors and desired

forces conmutation from position-based to force-based control.

physical rehabilitation systems in order to guarantee

safety in the interaction with patients. Future work

will be focused on implementation and validation of

the control in the physical rehabilitation device.

ACKNOWLEDGEMENTS

This work is supported by Universidad Militar Nueva

Granada- Vicerrectoria de Investigaciones, under re-

search project IMP-ING-3127, entitled ’Dise

no e im-

plementaci

on de un sistema rob

otico asistencial para

apoyo al diagn

ostico y rehabilitaci

on de tendinopat

ıas

del codo’.

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