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Convergence and robustness of the Hopf oscillator
applied to an ABLE exoskeleton: reachability analysis
and experimentation
Abdelwaheb Hafs1,2, Dorian Verdel1,2, Jawher Jerray3, Olivier Bruneau4, Nicolas
Vignais1,2, Bastien Berret1,2,5and Laurent Fribourg6
1CIAMS, Universit´
e Paris-Saclay, Bˆ
at. 335, Bures-sur-Yvette, 91405 Orsay cedex, France
{abdelwaheb.hafs, dorian.verdel, nicolas.vignais,
bastien.berret}@universite-paris-saclay.fr
2CIAMS, Universit´
e d’Orl´
eans, Orl´
eans, France
3LTCI, T´
el´
ecom Paris, Institut Polytechnique de Paris, Sophia-Antipolis, France
jawher.jerray@telecom-paris.fr
4LURPA, ENS Paris-Saclay, 4 Av. des Sciences, Gif-sur-Yvette, 91190, France
olivier.bruneau@ens-paris-saclay.fr
5Institut Universitaire de France, Paris, France
6University Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, F-91190 Gif-Sur-Yvette, France
fribourg@lsv.fr
Abstract
In this paper, we consider an industrial exoskeleton called ABLE, which we controlled using an
adaptive oscillator. In the context of periodic human movements around the elbow axis we show
that the exoskeleton anticipates smoothly the human movement, converges towards a limit cycle, and
is “robust” against non-deterministic bounded perturbations. In particular, the properties of stability
and robustness are formally proven, at the model level, using a recent development of the method of
“reachability analysis”. These properties are then confronted to preliminary experimental data.
1 Introduction
Active exoskeletons are wearable robotic devices, developed to assist human movement [1,13]. Their
versatility makes it possible to consider numerous applications, in particular to improve rehabilitation
protocols [9,10,19,21,23] and to prevent musculoskeletal disorders in workers [5,6,22,29]. The imple-
mentation of exoskeletons for these different applications is nevertheless constrained by the impossibil-
ity of completely predicting human movement intention [1,14]. Several approaches have been proposed
in the literature to solve this problem. The first approach relies on controllers based on bioelectric sig-
nals such as electromyography [30,31] or electroencephalography [7]. Other biological signals such as
gaze were also included in the control strategies to detect human intention [20]. All these strategies rely
on the appearance of the signals considered before the appearance of the movement kinematics, which
is called electromechanical delay in the case of myoelectrical signals. Another proposed approach was
to learn the human movement [14], which led to a significant reduction of unwanted interaction efforts
in reaching movements. In the case of periodical movements, approaches based on adaptive oscilla-
tors have been proposed and successfully experimented [24–27]. At the theoretical level, the authors
of [25] proved the stability of the model of one of these oscillators: the Hopf oscillator. The proof
uses the classical “perturbation method” that consists in introducing an infinitesimal perturbation εin
the model, expressing the solution as a power series in ε, and showing the positiveness of the radius
of convergence of the series. The Hopf oscillator is also considered here, but the stability is proven
by introducing a bounded perturbation w (instead of an infinitesimal perturbation ε) taking its values
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
non-deterministically in a given set Wof the form [−c;c]with c>0. In order to show the stability of
the system in presence of such perturbations, we make use of a recent development of the method of
“reachability analysis” (see [2] for a survey). Given an initial set of states S0, this method constructs
iteratively by “set-based integration” the set S1of the states reached at the end of one time step, then
S2at the end of a new time step, etc. These sets S1;S2;... are over-approximations of the exact sets of
solutions of the system, for any initial state in S0and for any admissible perturbation taking its values
in W. When one succeeds in generating a finite representation of the set I:=∪∞
i=0Si, one obtains a
description of a superset of all the states that can be reached from S0. The set Iis an invariant set of the
system: the trajectory from any point of Iremains in Iindefinitely (see [4]). We explain here how to
successfully generate such a set Ifor the Hopf oscillator. In the present paper, this previously defined
oscillator [27] is applied to an upper-limb exoskeleton to design a transparent control. Transparency is
defined as the interaction between the human and the exoskeleton with minimum efforts. Our primary
contributions lie in formally proving the convergence and stability of the oscillator and validating this
proof through experimental results. The theoretical development allows a priori evaluation of whether
the system converges to a limit cycle and is robust to non-deterministic bounded amplitude disturbances.
In particular, the experiments have confirmed the invariance property of the set I: once a first experi-
mental datum relative to the arm position lies in I, all the subsequent data also lie in I. In the present
study, the transparent control mode is chosen to experimentally evaluate the stability of the oscillator,
independently from model errors that can appear when using assistance control modes and that can lead
to instability [27]. In this control mode, the human movement should be impacted as little as possible
by the exoskeleton [1,3,14,32]. The experiments are conducted on elbow flexion/extension with an
upper-limb exoskeleton called ABLE [11,29] (see Section 2.2 for details).
2 Material and methods
2.1 Adaptive oscillators
As mentioned in the introduction, human intention prediction may be achieved by adaptive oscillators
for periodic movements. These oscillators allow to estimate the fundamental parameters (amplitude and
frequency) of arbitrary rhythmic signals in a supervised learning framework. Among different types of
adaptive oscillators, we focus here on the Hopf oscillator, which is one of the most widely used types in
the field of robotics and assistive technology [27] [25] [34] [37].
The Hopf oscillator is defined by a system of two differential equations (Eq. (1)).
˙x(t) = γ(µ−r2)x(t)−ω(t)y(t) + νe(t)
˙y(t) = γ(µ−r2)y(t) + ω(t)x(t)(1)
With r=px2+y2, where x(t)and y(t)are the orthogonal coordinates of the oscillator, eis the
oscillator’s external force, νthe coupling gain,µand γare the amplitude and attractiveness of the os-
cillator respectively [26]. For this study γ=1 (as in [25–27]), and µ=1 (so that the intrinsic amplitude
of the oscillator is equal to 1).
The input of the oscillator is the real position θ. In this case the external force of the oscillator e
represents the difference between the real position θof the human elbow and the estimated position ˆ
θ.
Righetti and Ijspeert [24] augmented this oscillator to learn the frequency ω, amplitude α1and offset
2
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
Figure 1: Example of oscillator’s adaptation. Top panel: the oscillator’s input. Bottom panel: evolution
of the learned frequency
α0of the input using integrators. The complete estimation algorithm is presented in Eq. (2)
˙
ω(t) = −νe(t)sin(φ(t))
˙
φ(t) = ω(t)−νesin(φ(t))
˙
α1(t) = ηcos(φ(t))e(t)
˙
α0(t) = ηe(t)
ˆ
θ(t) = α0(t) + α1(t)cos(φ(t))
(2)
Where ηis the integrator gain and φthe output phase. Finally the estimated position ˆ
θis used to
compute the torque generated by the exoskeleton to assist the human movement (see Section 2.2 for
details).
2.2 Exoskeleton modeling
ABLE is an upper limb exoskeleton. This exoskeleton is based on a screw and cable transmission
allowing high levels of reversibility and transparency [11,12]. It has four active degrees of freedom (see
Fig. 2). The first three correspond to rotations of the human shoulder, the fourth correspond to rotation
of the elbow (flexion/extension). The physical interfaces between the exoskeleton and the user include
passive rotations and translation to minimize undesired efforts due to the connection hyperstatism [15,
28]. Additional settings are available to adapt the exoskeleton to the user. The robot is mounted on a
fixed frame with a winch to adapt its height.
The proposed method consists of using the joint position and velocity estimated by the oscillator
instead of the measured position and velocity. As we focus on the last axis (elbow), the inverse dynamic
model of the robots elbow axis is computed upon the basis of classical robot dynamics [32,33]. and
represented in Eq. (3).
ˆ
τm= (g(−xmcos(ˆ
θ)−ymsin(ˆ
θ))
+˙
ˆ
θ µv+sign(˙
ˆ
θ)µc)1
r
(3)
3
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
Figure 2: Exoskeleton ABLE
Here g=9.81 m/s2the gravity acceleration, (xm,ym)the exoskeleton’s forearm centre of gravity
coordinates, µvthe friction coefficient, µcthe Coulomb dry friction coefficient, ˆ
τmthe estimated motor
torque, and rthe torque reduction ratio.
Finally Fig. 3 represents the exoskeleton control scheme. The position θis measured by exoskeleton
internal sensors. As shown, the oscillator’s input depends on both the control law and the modulation
performed by the human, where the human central pattern is represented by the transfer function on the
feedback loop.
Figure 3: Exoskeleton Control scheme
4
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
3 Reachability Analysis
We focus here on sinusoidal inputs (corresponding to sinusoidal movements of the users) with bounded
uncertainty.
The stability of the Hopf oscillator was proved in [25] using the “perturbation method”. The per-
turbation method was developed in the 19th century by Laplace and others to show, for example, the
orbital stability of the moon around the Earth despite the influence of the sun. Roughly speaking, the
equation of the ideal motion (without perturbation) is modified by injecting an infinitesimal value ε, and
the solution of the modified equation is expressed as a power series in ε. The stability is demonstrated
by showing that the radius of convergence of the series is non-zero. We prove here the stability of the
Hopf oscillator using the method of “reachability analysis”. Reachability analysis has been developed
since a couple of decades in the community of formal methods and model checking (see [2] for a re-
cent review). In this method, one is given a set S0of initial points and a perturbation w, which is not
infinitesimal as in the perturbation method, but of bounded amplitude (typically w∈W= [−c,c]with
c>0). Reachability analysis then constructs by “set-based integration” first the set S1of the states
reached at the end of one time step, then S2at the end of a new time step, etc. These sets S1,S2, ...
are over-approximations of the exact sets of solutions of the system, for any initial state in S0and for
any admissible perturbation taking its values non-deterministically in W. The set SK
k=0Skcharacterises
a superset of all the states on the time interval [0,Kh], where his the time-step size. This set is often
referred to as the “reachability tube”.
In the case where we can detect that at a time t=Lh for some L∈N, the set of states SLis contained
in the tube currently generated (SL⊂SL−1
k=0Sk), we obtain a finite representation of a superset of all the
states that can be reached from S0. The set I=SL
k=0Skis an invariant set of the system: the trajectory
from any point of Iremains in Iindefinitely (see [4]).
For an oscillatory system, this invariant set is shaped like a tube rounding on itself (a “doughnut” in
3D). In [8], we represented the set Skas a ball B(yk,δ(k)) of centre ykand radius δ(k). The centre ykof
the ball B(yk,δ(k)) is the value computed by Euler’s explicit method at the k-th step. The radius δ(k)is
determined by an analytical formula giving an upper bound on the error introduced by Euler’s method
(see [8]). The detection of the inclusion SL⊆SL−1
k=0Sk, is done by finding a value Ksuch that
B(yL,δ(L)) ⊆B(yK,δ(K))
with L=K+ℓhfor some integer ℓ. The value T=ℓhis the estimated value of the system period
(see [18] for details). The invariant set I=SL
k=0Skis here a looping tube, centred on a set of values
y0,y1,...,yLcorresponding to the “limit cycle” Cof the system.
The main advantage of the reachability analysis over the perturbation method is its ability to take
into account perturbations of a given amplitude, and to construct an invariant set Iin which the state
of the system is guaranteed to be confined. This allows us to characterize the “robustness” of the system
against bounded perturbations.
Our method of reachability analysis has been implemented in Python in a software called ORBITA-
DOR. The method has been successfully applied to various systems, for example we have shown the
stability of a parametric Van der Pol system in [17] and a biochemical process model in [18]. We also
studied the stability of hybrid systems such as the passive biped model in [16]. Here ORBITADOR is
applied to the system Eq. (2) for W= [ −2π
4000 ,2π
4000 ](which corresponds to the true sensitivity of the optical
sensors embedded in the exoskeleton), h=10−3(which corresponds to the real time sampling frequency
of exoskeleton control), δ0=0.084, and the initial condition x0= (α1(0),α0(0),φ(0),ω(0),ˆ
θ(0)) =
(0.059 rad,0.059 rad,0.123 rad,0.062 rad/s,0.117 rad). It automatically finds that, for T=5.168 s,
B(yL,δ(L)) ⊆B(yK,δ(K)),
5
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
Figure 4: 2D projections of the limit cycle C.
with L=10336 and K=5168. The associated limit cycle Cand controlled invariant set Iare depicted
on Fig. 4 and Fig. 5 respectively, according to various 2D projections. In Fig. 5, the ball B(yK,δ(K)) is
represented in orange, and the ball B(yL,δ(L)) in green (we see B(yL,δ(L)) ⊂B(yK,δ(K))).
Figure 5: Evolution of the size δ(t)of the controlled invariant set I(top), followed by 2D projections
of Ifor W= [ −2π
4000 ,2π
4000 ].
4 Experimental results
4.1 Participants
One healthy right-handed adult took part in the experiments. A written consent was signed by the
participant, and obtained as required by the Helsinki declaration [35]. The study was validated by a
research ethics committee of Paris-Saclay (Universit´
e Paris-Saclay, 2021-303).
4.2 Kinematics
The kinematics of the human movement are measured by the opto-electronic motion capture device
called ”Qualysis”. This system is composed of 10 cameras to capture the human movement at 179 Hz.
There are 7 reflecting markers placed on the participant forearm, which allow the construction of plans
based on the recommendations of Ge Wu et al [36]. The identification of the 7 markers is carried out
using an AIM (Automatic Idenfication Markers) model on the Qualysis Track Manager software. In
6
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
order to make the participant do oscillatory movements, a sinusoidally moving target is projected on a
screen in front of the subject.
4.3 Data acquisition
The code which controls the exoskeleton, also allows us to obtain its position as well as the parameters
of Hopf oscillator. This allows to analyze the convergence of the oscillator. The data is first cleaned up,
keeping only the part where the experiment starts.
4.4 Motor task
The participant is first placed in the robot at a distance from the targets corresponding to 2 times the
size of his arm. The first three axis of the exoskeleton were mechanically blocked to avoid unwanted
movement. Then a moving target is projected in a big screen in front of the participant. The target moves
vertically with a varying sinusoidal trajectory. The participant is asked to follow the target by performing
only the flexion/extension of the elbow. The pointing position is computed as the intersection between
the line of the index and the plan of the screen, it is then projected in real time in the screen as a visual
feedback for the participant. The subject starts with the forearm pointed at the fixed target in the middle
(starting target) and then starts the experiment when the target starts to move as shown in Fig. 6.
The target follows the following sinusoidal trajectory :
x(t) = Asin(ωt),
A=0.3rad,ω=5 rad.s−1.(4)
Figure 6: Description of the experimental setup. A: Markers on the forearms (human and exoskeleton).
B: Target tracking example.
4.5 Experimental data
This section presents the behaviour of the set of states obtained in the experiment where we used the
control scheme presented in Section 2.2.Fig. 7 represents the real position θ(also the oscillator input)
7
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
and the estimated position ˆ
θcomputed by the oscillator. We can see that the estimation follows the
desired position θ. We now compare the location of the experimental data with the invariant set I
Figure 7: Evolution of the estimated position
Figure 8: Evolution of the experimental data (black at t<4.918 s then blue at t≥4.918 s) in the
controlled invariant (red)
where W= [ −2π
4000 ,2π
4000 ]constructed by reachability analysis in Section 3.Fig. 8 represents the set I
together with 5000 points chosen randomly within the set of experimental data. At the beginning of
the movement at t<4.918 s, we see on Fig. 8 that the experimental points (black) are outside of the
invariant set I; then at t≥4.918 s the points (blue) are all located inside I.
More precisely, there are in total 10336 experimental data points: 4918 points for t<4.918 s and
5418 points for t≥4.918 s. All the 5418 points at t≥4.918 s are located inside the invariant set I. For
t<4.918 s, the average of the distance between the experimental data and the limit cycle Cis 0.0493
and the standard deviation 0.0255; for t≥4.918 s, the distance average is 0.0028 and the standard
deviation 0.0023.
Fig. 9 shows the evolution of the distance between the experimental data and the limit cycle.
8
Convergence and robustness of the Hopf oscillator applied to an ABLE exoskeleton A. Hafs et al.
Figure 9: Evolution of the distance between the experimental data and the limit cycle.
As a conclusion, the experimental results allowed us to verify the efficiency of the Hopf oscillator
on a complex robot such as ABLE.
5 Conclusion
We have described a methodology for
1. formally guaranteeing the convergence and robustness of the controlled system, at the model level,
using a recent method of reachability analysis of periodic movements [18], and
2. verifying all these properties at the experimental level, via a protocol on human subjects.
In this pilot study we have verified that all the data of the experiments (after a possible transient phase)
are located within the controlled invariant set Ifound, at the theoretical level, by reachability analysis
(see Section 3). We are presently enriching the model by taking into account additional axes of move-
ment, and considering more sophisticated adaptive oscillators than the Hopf oscillator. We think that
the methodology presented here (relying on control with adaptive oscillators, and formal verification via
reachability analysis) is still well-suited to the validation of such extensions.
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