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TRANSCRIPT
© Julien Mathieu Audet, 2020
Conception et validation expérimentale d’un système mécatronique pour la manipulation intuitive de
composantes lourdes
Mémoire
Julien Mathieu Audet
Maîtrise en génie mécanique - avec mémoire
Maître ès sciences (M. Sc.)
Québec, Canada
Conception et validation expérimentale d’un systèmemécatronique pour la manipulation intuitive de
composantes lourdes
Mémoire
Julien-Mathieu Audet
Sous la direction de:
Clément Gosselin, directeur de recherche
Résumé
Ce mémoire présente la conception et la validation expérimentale d'un système mécatronique
visant à faciliter la manipulation de composantes lourdes dans des situations industrielles
d'assemblage, par exemple l'assemblage de panneaux de fuselage d'avion.
Le principe de la redondance sous-actionnée est utilisé pour que l'interaction entre l'opérateur
humain et le robot soit sécuritaire, intuitive et réactive, tout en permettant une charge utile
relativement élevée. Ce principe consiste à utiliser un mécanisme passif à basse impédance cou-
plé à un système actif avec la charge utile à manipuler directement attachée à l'e�ecteur du
mécanisme passif. Lors du fonctionnement du dispositif, l'opérateur humain manipule directe-
ment la charge utile et induit ainsi des mouvements dans le mécanisme passif. Les variations
mesurées dans les articulations passives sont ensuite utilisées pour contrôler les articulations
actives à haute impédance du robot. Dans les travaux réalisés antérieurement, le principe a
été appliqué aux mouvements translationnels.
Le but de ce mémoire est donc d'appliquer le principe de la redondance sous-actionnée
aux mouvements rotatifs a�n d'orienter une charge utile dans l'espace tridimensionnel. Tout
d'abord, le principe est appliqué à un manipulateur plan à un degré de liberté pour évaluer la
validité du concept pour les mouvements rotatifs. Ensuite, il est appliqué à un manipulateur
spatial à deux degrés de liberté. Des contrepoids actifs sont utilisés pour équilibrer statique-
ment les deux manipulateurs. Il est à noter que le dernier mouvement rotatif n'est pas étudié
puisqu'il est facile à implémenter ; l'équilibrage statique n'étant pas requis pour la rotation
autour de l'axe vertical. Finalement, le système rotatif obtenu précédemment est combiné avec
un système translationnel existant dans le but de manipuler librement une charge utile dans
l'espace à six dimensions. Les validations expérimentales sont présentées pour montrer que le
manipulateur est intuitif, réactif et sécuritaire pour l'opérateur humain.
ii
Abstract
This Master's thesis presents the design and experimental validation of a mechatronic system
aimed at facilitating the handling of heavy components in industrial assembly situations, for
example the assembly of aircraft fuselage panels.
The principle of underactuated redundancy is used to make the interaction between the human
operator and the robot safe, intuitive and responsive, while allowing a relatively high payload.
This principle consists in using a low-impedance passive mechanism paired with an active
system with a payload directly attached to the passive mechanism's end e�ector. In the oper-
ation of the device, the human operator directly manipulates the payload and thereby induces
movements in the passive mechanism. The measured joint variables in the passive mechanism
are then used to control the high-impedance active joints of the robot. In previous works, the
principle of underactuated redundancy has been applied to translational movements.
The aim of this Master's thesis is therefore to apply the principle of underactuated redundancy
to rotations in order to rotate a payload in three-dimensional space. First, the principle is
applied to a one-degree-of-freedom planar manipulator in order to evaluate the validity of
the concept for rotational motions. Then, it is applied to a two-degree-of-freedom spatial
manipulator. Active counterweights are used to statically balance the two manipulators. It
should be noted that the last rotational motion is not studied since it is easy to implement;
static balancing is not required for the rotation around the vertical axis. Subsequently, the
rotational system obtained previously is combined with an existing translational system with
the objective of freely manipulating a payload in six-dimensional space. The experimental
validations are presented to show that the manipulator is safe, intuitive and responsive for the
human operator.
iii
Table des matières
Résumé ii
Abstract iii
Table des matières iv
Liste des tableaux v
Liste des �gures vi
Remerciements viii
Avant-propos ix
Introduction 1
1 Rotational Low Impedance Physical Human-Robot Interaction using
Underactuated Redundancy 4
1.1 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Proposed architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Alternative architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Multimedia attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.9 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Intuitive Physical Human-Robot Interaction using an Underactuated
Redundant Manipulator with Complete Rotational Capabilities 18
2.1 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Proposed mechanical architecture . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Rotational and translational motion . . . . . . . . . . . . . . . . . . . . . . 312.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.9 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
iv
Conclusion 33
Bibliographie 35
v
Liste des tableaux
2.1 Coe�cients obtained from a calibration with n con�gurations where a ± 1 de-gree error was randomly added to the reading of the passive encoders comparedto the expected coe�cients from eqs.(2.14�2.17) and eqs.(2.23�2.24) . . . . . . 28
vi
Liste des �gures
1.1 Assembly of fuselage panels using a robot and a 6-dof passive mechanism (illus-trated as a cube). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Geometric and mass parameters of the proposed gravity-balanced architecture. 81.3 Required static torque T and absolute counterweight position (αd + θ) res-
pectively, as a function of the payload position θ in order to maintain staticequilibrium ; with m1 = 5 kg, m2 = 4 kg, m3 = 5.5 kg, M = 20 kg, c = 0.4 m,r = 0.1 m, l = 0.1 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Resisting torque as a function of the payload position θ ; with m1 = 5 kg,m2 = 4 kg, m3 = 5.5 kg, M = 20 kg, c = 0.4 m, r = 0.1 m, δθ = 5◦ ; l = 0.1 mand l = 0.15 m for the �rst and second graphs respectively. . . . . . . . . . . . 10
1.5 Geometric and mass parameters of a gravity balanced architecture using anactive spring system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Geometric and mass parameters of a gravity balanced architecture using anactuated spring and a passive counterweight. . . . . . . . . . . . . . . . . . . . 12
1.7 CAD representation of the 1-dof gravity balanced tilting mechanism. . . . . . . 141.8 Experimental setup for the 1-dof gravity balanced tilting mechanism. . . . . . . 141.9 The payload's con�guration θ and the actuator's position α whose command
is αd as a function of time during an experiment ; the green circles and thered squares respectively represent the beginning and the end of the operator'sinteraction. In Fig.(a), the locking mechanism is not applied, whereas, in Fig.(b),the locking mechanism is used when there is no interaction. . . . . . . . . . . . 16
2.1 Assembly of fuselage panels using a robot and a 6-dof passive mechanism (illus-trated as a box). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Geometric and mass parameters of the proposed gravity-balanced architecture. 222.3 Absolute di�erence between the calibrated actuator position using n con�gura-
tions and the expected actuator position, that is |∆αi|, for di�erent values ofthe payload's con�guration θi where in Fig.2.3(a) i = 1, whereas in Fig.2.3(b)i = 2, for n = 4, 6, 8, 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 CAD model of the prototype of the 2-DoF rotational manipulator. . . . . . . . 292.5 Experimental setup for the 2-DoF rotational manipulator prototype. . . . . . . 302.6 Experimental setup of the 4-DoF gravity-balanced architecture. . . . . . . . . . 30
vii
An expert is a person who has
made all the mistakes that can be
made in a very narrow �eld.
Niels Bohr
viii
Remerciements
Je tiens à prendre un moment pour témoigner de ma gratitude et pour remercier tous ceux qui
ont guidé ma ré�exion à travers la réalisation de cette maîtrise. Votre support fut fortement
apprécié.
Tout d'abord, je suis grandement reconnaissant envers mon directeur de recherche, Clément
Gosselin. J'ai sincèrement apprécié sa patience, sa grande disponibilité et son écoute tout au
long de ma maîtrise. Merci de m'avoir donné cette passion immuable pour le domaine de la
robotique.
Je veux aussi remercier tous les membres du laboratoire de robotique de l'Université La-
val. Sans l'aide indispensable de Simon pour la communication temps réel avec RTLab et de
Thierry pour la conception mécanique, les prototypes de cette maîtrise n'auraient jamais vu
le jour. Merci à tous les membres du laboratoire de robotique pour les conseils techniques, les
discussions enrichissantes et bien-sûr, ces nombreuses parties inoubliables de spikeball.
Je suis également reconnaissant envers tous les membres du groupe de robotique du centre des
technologies de fabrication en aérospatiale à Montréal pour m'avoir montré le côté industriel
de la robotique. Je remercie particulièrement le chef du groupe, Bruno Monsarrat, de m'avoir
si bien accueilli dans son équipe.
Finalement, je remercie mes parents, mon frère, ma copine et mes amis de m'avoir supporté
moralement pendant ces deux ans d'étude. Merci in�niment.
ix
Avant-propos
L'ouvrage actuel est écrit sous la forme d'un mémoire par articles. Les deux premiers chapitres
présentent des articles écrits dans le cadre des travaux de maîtrise de l'étudiant à l'Univer-
sité Laval. Il est à noter que les activités de recherche de l'étudiant qui ont été e�ectuées
au centre des technologies de fabrication aérospatiale du Conseil national de recherches du
Canada (CTFA-CNRC) ne sont pas présentées dans ce mémoire pour des raisons de con�den-
tialité. Néanmoins, ce volet complémentaire n'est pas en lien avec les recherches entreprises
par l'étudiant à l'Université Laval et son absence n'a�ectera pas la présentation des résultats
dans le mémoire.
Le premier article présenté a été soumis à la revue scienti�que Journal of Mechanisms and
Robotics de l'American Society of Mechanical Engineers (ASME) le 30 mars 2020, une re-
vue dont la portée internationale exige une rédaction en anglais. Les activités de recherche
réalisées par l'étudiant, sous la supervision de son directeur de recherche Clément Gosselin,
sont principalement la conception du mécanisme, l'élaboration de l'algorithme de contrôle et
en�n, la rédaction de l'article en tant qu'auteur principal. Avant la soumission de l'article à
la revue scienti�que, le directeur de recherche a révisé l'article. Il est à noter que la �gure
1.9 du chapitre 1 du mémoire est une combinaison de deux �gures de l'article soumis, soit les
�gures 9 et 10, dans le but de condenser la description des deux �gures en une seule pour une
meilleure lecture.
Le deuxième article présenté a été soumis à la revue scienti�que IEEE/ASME Transactions
on Mechatronics le 4 mai 2020. Le caractère international de la revue exige également une
rédaction en anglais. Sous la supervision de son directeur de recherche (et co-auteur) Clément
Gosselin, l'étudiant a rédigé en grande partie l'article, ce qui fait de lui l'auteur principal.
De son côté, M. Gosselin a révisé l'article et a apporté des corrections. L'article reprend les
concepts abordés dans le premier article a�n d'élargir l'utilisation du principe de la redondance
sous-actionnée. Appliqué à un manipulateur plan à un seul degré de liberté jusqu'à maintenant,
le principe de la redondance sous-actionnée est utilisé pour concevoir un manipulateur ayant
toutes les capacités rotationnelles et translationnelles. Les préparations pour l'article, soit la
conception du manipulateur rotatif, l'élaboration d'un système de contrôle, la conception d'un
module pour lier la partie rotative à la partie translationnelle du système et l'élaboration d'un
x
banc d'essai pour valider le système complet, ont été réalisés par l'étudiant sous la supervision
de M. Gosselin. Aucune modi�cation n'a été apportée à l'article intégré par rapport à l'article
soumis.
xi
Introduction
Problématique
Il est parfois di�cile pour l'être humain d'e�ectuer certaines tâches par lui-même. C'est pour-
quoi, dans les derniers siècles, la société n'a cessé d'inventer et d'innover pour faciliter la vie
des hommes et des femmes. On pense à l'invention de la roue, de la machine à vapeur, des
moteurs à combustion, etc. Dans la dernière décennie, l'arrivée de l'automatisation a même
détrôné l'être humain dans certaines industries comme l'industrie manufacturière, où la main-
d'oeuvre est dominée par la machinerie automatisée. Là-bas, les robots industriels réalisent
des tâches jugées simples, répétitives et dangereuses pour l'humain. Ils sont isolés dans des
cellules où l'interaction avec le vivant est inexistante, compte tenu des vitesses élevées que
peuvent atteindre ces robots.
Or, il existe certaines tâches où cette interaction serait béné�que. D'un côté, la robotique assure
une haute charge utile, une meilleure précision et une �abilité supérieure. D'un autre côté, les
humains possèdent des habilités que les robots n'ont pas, comme l'intuition, la polyvalence et
la conscience. En unissant les compétences des humains et des robots, on obtient le meilleur
des deux mondes. Une tâche qui béné�cierait de cette alliance � et qui est l'objet de cette
maîtrise � est l'assemblage de pièces aérospatiales lourdes ; plus précisément, les panneaux
de fuselage d'avion. L'utilisation d'un robot comme compensateur de charge permettrait à
l'opérateur humain de manipuler une pièce relativement lourde dans l'espace avec peu d'e�ort.
En revanche, comment est-il possible de s'assurer de la sécurité de l'opérateur pendant la
tâche ?
Dans les dernières années, des nouveaux robots collaboratifs, surnommés les cobots, ont été
introduits. Ces nouvelles avancées en robotique permettent aux humains d'interagir avec le
cobot tout en respectant les normes de sécurité. Par contre, ces robots collaboratifs ont une
limite de charge utile assez contraignante. De plus, l'aspect sécuritaire de ces robots est à
questionner [3]. Il est donc intéressant d'élaborer un nouveau type de cobot industriel pour
contrer ces désavantages. Ceci fut réalisé antérieurement pour les mouvements translationnels
en appliquant le principe de la redondance sous-actionnée [13] dans le but d'assister un opéra-
teur pour l'assemblage de portes de voitures (avec la possibilité de rotation par rapport à l'axe
1
vertical). Ce principe consiste à utiliser un mécanisme passif ainsi qu'un système actif (robot
conventionnel, système portique, etc). Dans un tel système, le mécanisme passif � qui, par
sa passivité, est de faible impédance, et donc sécuritaire et intuitif � agit comme interface de
manipulation pour l'opérateur. La charge utile à manipuler est placée sur l'e�ecteur du méca-
nisme passif. Les mouvements engendrés dans les degrés de liberté (ddl) du mécanisme passif,
produits par l'opérateur qui manipule la charge, servent de commande pour l'actionnement
des ddl correspondants du système actif. En séparant l'opérateur de la partie active, la haute
impédance du système actif devient isolée et la sécurité de l'opérateur est assurée. Un tel sys-
tème permet une haute charge utile ; celle-ci est seulement limitée par l'intégrité structurelle
du mécanisme passif et par la puissance du système actif. De plus, il s'agit d'un dispositif de
manipulation intuitif, réactif et sécuritaire. Il est donc intéressant d'appliquer le principe de la
redondance sous-actionnée aux rotations, puisque les mouvements de type SCARA, quoique
acceptables pour la plupart des tâches industrielles, ne su�sent pas pour la tâche d'assemblage
décrite plus haut. Pour réaliser cette opération, une manipulation complète de la charge dans
l'espace est essentielle.
Objectifs de la recherche
Le but de ce mémoire est d'explorer la collaboration humain-robot pour des tâches industrielles
jugées impossibles ou non-ergonomiques pour l'être humain (charge très lourde) et di�ciles
pour un robot (plusieurs capteurs, longue programmation, peu versatile, etc) en mettant une
emphase sur la manipulation intuitive de pièces aérospatiales en rotation. Les di�érents ob-
jectifs de ce mémoire sont énumérés ci-dessous dans le but d'illustrer ce qui a été traité par
les deux articles de ce mémoire :
1. Élaborer un mécanisme rotatif, équilibré statiquement, qui utilise le principe de la re-
dondance sous-actionnée.
2. Concevoir un premier prototype à 1 ddl et à petite échelle a�n de valider expérimenta-
lement le principe de la redondance sous-actionnée pour les rotations.
3. Élaborer un mécanisme rotatif à 2 ddl et concevoir un second prototype à moyenne échelle
et puis, valider expérimentalement l'intuitivité et la réactivité quant à la manipulation
de la charge utile par l'opérateur humain.
4. Associer le système rotatif obtenu au point 3) à un système translationnel existant pour
obtenir un système ayant la capacité de manipuler la charge utile dans l'espace.
5. Réaliser une tâche simple d'insertion de la charge utile en utilisant le système obtenu
dans le point 4) pour démontrer les avantages d'un tel système.
Les points 1 à 2 sont abordés dans le chapitre 1 tandis que les points 3 à 5 sont abordés dans
le chapitre 2.
2
Méthodologie de recherche
Puisque les articles survolent la méthodologie utilisée pendant la réalisation de cette maîtrise,
une section dédiée à la méthodologie est décrite.
Au tout début, une étude est réalisée pour trouver un mécanisme équilibré statiquement qui
respecte les exigences et qui fera partie du système rotatif. Le mouvement d'inclinaison de
la charge utile est étudié en premier. Les exigences du mécanisme sont décrites ci-dessous.
La charge utile à manipuler doit être placée sur l'e�ecteur de la composante passive du mé-
canisme et doit être équilibrée statiquement. Une composante active doit être ajoutée pour
contrôler la con�guration de la charge utile indirectement a�n que le principe de la redon-
dance sous-actionnée soit respecté. Di�érents mécanismes sont étudiés comme candidats pour
le mécanisme à 1 ddl. Par la suite, di�érents types de mécanismes sériels et parallèles à 2 ddl
sont explorés a�n d'ajouter un mouvement de rotation additionnel à la charge utile, soit le
mouvement de roulis. Le dernier mouvement de rotation possible de la charge utile, soit le
mouvement de pivot, n'est pas étudié dans cette maîtrise puisqu'il est facile à implémenter ;
l'équilibrage statique n'étant pas requis pour cette rotation.
Pour évaluer ces di�érents mécanismes, le logiciel Matlab est privilégié. Il est utilisé pour
résoudre les équations d'équilibre statique, pour établir les couples requis des actionneurs
selon la con�guration de la charge utile, pour élaborer des méthodes de calibration robustes
et, �nalement, pour déterminer à partir des di�érents mécanismes étudiés le mécanisme le plus
approprié pour le manipulateur rotatif à 1 ddl et, ensuite, pour celui à 2 ddl. Des graphiques
sont générés avecMatlab pour faciliter la visualisation des données. Entre autres, les graphiques
aident à la comparaison des di�érents mécanismes en illustrant certains critères comme les
e�orts requis au niveau des actionneurs.
En ce qui concerne les prototypes, le logiciel de modélisation 3D Creo est utilisé pour la
modélisation des mécanismes. Par la suite, les pièces sont usinées par l'équipe de l'atelier
d'usinage de l'Université Laval. L'assemblage des prototypes et la préparation des validations
expérimentales sont réalisés par l'étudiant.
Pour ce qui est de la partie électronique, le contrôle des actionneurs et la lecture des encodeurs
et des signaux de l'interrupteur sont réalisés à partir de deux logiciels, soit Simulink pour la
programmation des algorithmes du système et RTLab pour la communication et le contrôle
en temps réel des éléments du système.
3
Chapitre 1
Rotational Low Impedance Physical
Human-Robot Interaction using
Underactuated Redundancy
1.1 Résumé
Cet article vise à appliquer le principe de la redondance sous-actionnée pour de l'interac-
tion physique humain-robot à un contexte d'assemblage industriel en introduisant un nouveau
manipulateur rotatif équilibré statiquement à 1 degré de liberté. L'architecture proposée est
composée d'un contrepoids actif en rotation et d'un pivot passif équipé d'un encodeur. L'archi-
tecture proposée est d'abord présentée et les conditions d'équilibre statique sont utilisées pour
décrire le fonctionnement du mécanisme. Ensuite, des architectures alternatives sont briève-
ment présentées. En�n, une validation expérimentale est fournie pour démontrer la viabilité
du concept pour de l'interaction physique humain-robot rotatif à faible impédance.
1.2 Abstract
This paper extends the concept of underactuated redundancy for physical human-robot in-
teraction (pHRI) in a context of industrial assembly by introducing a novel 1-dof gravity
balanced rotational manipulator. The proposed architecture consists of a rotational active
counterweight with a passive joint equipped with an encoder. The proposed architecture is
�rst described and the static equilibrium conditions are used to describe the operation of the
mechanism. Then, alternative architectures are brie�y introduced. Finally, an experimental
validation is provided to demonstrate the viability of the concept for rotational low impedance
pHRI.
4
1.3 Introduction
Physical human-robot interaction (pHRI) is becoming more common in various industrial
applications, such as manufacturing [1], [2]. Indeed, while robots have better precision and can
carry higher payloads, humans possess abilities that robots lack like intuitiveness, versatility
and awareness. Unfortunately, most robots cannot interact with humans without proper safety
precautions because of their high payloads and speed (such as conventional industrial robots).
Those that can, i.e., commercial collaborative robots, have a relatively limited payload and
yet, their safety features are far from �awless [3]. In an industrial assembly environment,
high payload-handling abilities and precise manipulation together with stringent safety are
required. Therefore, pHRI is di�cult to integrate in such an environment. Multiple strategies
were conceived to allow humans and robots to share a common workspace and collaborate in
the performance of tasks.
In order to reduce the perceived combined inertia of the payload and the robot, the prevalent
approach in pHRI applications has been the use of force/torque sensors. Paired with an admit-
tance controller, this approach can be used to emulate di�erent impedances [4], [5]. In some
instances, a proportional-integral (PI) controller [6], or even lead and lag compensators [7] are
used. However, such techniques are limited in their abilities to reduce the apparent impedance
due to hardware dynamics [8] and leads to unstable behaviours if used to go below a certain
proportion of the intrinsic inertia [9]. Based on techniques used in [4], [5] and [7], reduction
ratios of �ve to seven times the inertia were achievable. Another approach can mechanically
�lter the high-frequency interactions using force sensors paired with compliant material [10].
Nonetheless, physical contacts remain limited to speci�c ranges of environment dynamics, con-
sidering that these large inertia reduction ratios are obtained only by overstepping the concept
of passivity [11], [12].
One potential avenue for intuitive pHRI is the decoupling of the human and robot dynamics,
which has been successfully studied in previous works. In this approach, the human operator
is working in the manipulative space while the robot provides forces and moments in the con-
strained space while allowing the large amplitude motions in the manipulative space [13]-[15].
This is obtained by using a low-impedance passive manipulator (in the manipulative space) and
a high-impedance active robot. The input driving the robot is the end-e�ector displacement
of the passive mechanism, with which the operator interacts using his/her own mechanical
impedance. Therefore, high bandwidth interaction can be achieved. Underactuated redun-
dant mechanisms provide lower apparent impedance than any actuated mechanisms, ensuring
safety standards and allowing precise and intuitive manipulation by the operator while retain-
ing the same high payload-handling abilities. This is ideal for pHRI applications.
However, in previous work [13]-[15], only the translational degrees of freedom (dofs) (and
possibly a rotation about a vertical axis) were included in the manipulative space. Although 4-
5
dofs SCARA-type motions are su�cient for many industrial assembly tasks, in some instances
tilting rotations are necessary (i.e., the two rotations that are constrained in the SCARA
motions). For instance, aerospace components like fuselage panels need accurate adjustments
in 6 dofs to assemble correctly, which is why rotational dofs are required for advanced assembly
tasks. This is illustrated schematically in Fig. 1.1, where a fuselage panel is supported by
a robot (where all rotations are constrained) and manipulated by an operator, using a 6-dof
underactuated redundant mechanism (illustrated as a cube in Fig. 1.1) for �ne and intuitive
assembly.
Figure 1.1 � Assembly of fuselage panels using a robot and a 6-dof passive mechanism(illustrated as a cube).
As shown in [13], [14], only the vertical dof (Z motion) requires gravity balancing since both
horizontal dofs (X and Y motions) are perpendicular to the direction of gravity. This di�ers
from the rotational dofs where the tilting dof and the rolling dof both need gravity balancing
since vertical motions of the centre of mass of the payload can be present in these dofs. In
a di�erent context, Kawamura et al. explored the principle of underactuated redundancy in
order to obtain an agile low-power robotic arm using a movable counterweight [16] to indirectly
drive a passive joint. Similarly, a rotational counterweight [17] was also developed, improving
the limited workspace and torque.
In this paper, we propose to extend the principle of underactuated redundant robot presented
in [13], [14] to a tilting rotational dof using a redundant active counterweight as a �rst step
toward a 6-dof low-impedance manipulator for industrial applications. The paper is structured
as follows. Section 1.4 explores the architecture used for the 1-dof underactuated redundant
tilting mechanism. Section 1.5 then brie�y describes viable alternative architectures. Section
1.6 presents the experimental validation and section 1.7 describes the multimedia attachment.
In section 1.8, conclusions are drawn.
6
1.4 Proposed architecture
The proposed architecture is illustrated schematically in Fig. 1.2. It consists of a link that
can be tilted in a vertical plane and which is used to manipulate a payload. The payload is
rigidly attached to the link and together they have a mass m with a centre of mass (CoM)
located at a distance c from the passive pivot. The passive pivot is equipped with an encoder
that measures the angle of the link, noted θ, with respect to a horizontal reference. The
objective is to allow a human operator to freely manipulate the payload that is attached to
the link and tilt it around the �xed pivot. When the operator lets go of the payload, the
payload should remain stationary in its current orientation. In order to balance the payload
and link, a counterweight of mass M is mounted on a second link of length ` that is attached
to the main link by an actuated pivot, located at a distance r from the �xed pivot. Angle α
represents the angle between the two links, associated with the motion of the actuator. Using
the actuator to control angle α, it is possible to control the equilibrium con�guration of the
link and payload.
In principle, if the counterweight is perfectly adjusted, the mechanism is statically balanced
around the �xed pivot and all con�gurations are equilibrium con�gurations. In such a situ-
ation, it is not necessary to actuate the joint between the two links in order to balance the
mechanism in a certain con�guration. However, the mass and location of the CoM of the link
and payload are not known precisely in practice. Therefore, if the pivot on which the payload
is mounted is not actuated, it is very likely that the mechanism will not be perfectly balanced
and that only one equilibrium con�guration will exist. The counterweight is chosen such that
one has the static balancing condition in the nominal condition, namely
Mr = mc. (1.1)
To obtain the desired behaviour (active-passive decoupling), the active pivot is used. In the
operation of the device, the human user applies an interaction force (Fi) on the payload to
move it in the desired position. At the same time, the rotation of the passive pivot (angle θ)
is detected by the encoder and the actuator then rotates the counterweight link in order to
maintain static equilibrium in the desired θ position.
For the proposed architecture to be statically balanced, the CoM of the entire system must
be located on the vertical line passing through the passive pivot, which yields
−M(r cos θ − l cos(α+ θ)) +mc cos θ = 0. (1.2)
Solving eq.(1.2) for α, we obtain
αd = ± arccos(C cos θ)− θ, (1.3)
where αd is the desired α position for static equilibrium and
C =Mr −mc
Ml. (1.4)
7
m
M
θ
α
Fi
gr
c
l
Figure 1.2 � Geometric and mass parameters of the proposed gravity-balanced architecture.
For real solutions, C ∈ [−1, 1]. The plus-minus sign in eq.(1.3) corresponds to the fact that
there exist two values of angle α that correspond to a given value of θ, for a given set of
masses and lengths. The two con�gurations are analagous to an inverted pendulum and a
conventional pendulum. In practice, the position with the lower CoM is chosen for stability
purposes and in order to reduce the required actuator torques. Substituting eq.(1.1) into
eq.(1.4), we �nd that the coe�cient, noted C, is equal to zero in the nominal condition chosen
earlier. Eq.(1.3) can then be rewritten as
αd = − arccos(0 cos θ)− θ = −π2− θ. (1.5)
Therefore, if the nominal condition is respected as dictated by eq.(1.1), the second link of
length l must be parallel to the direction of gravity in order to be statically balanced in all
con�gurations. In this state, energy expenditure is kept at a minimum since the torque (T )
used to raise the counterweight is
T = Mgl cos(α+ θ), (1.6)
which amounts to zero using eq.(1.5). However, if the mass and the payload's CoM are
not known precisely, the coe�cient C may not be exactly equal to zero. Consequently, the
actuator has to rotate the second link out of its resting vertical con�guration to indirectly
increase or decrease the torque acting on the passive pivot. Experimentally, the architecture
can be calibrated using a single set of angles (θ and α) to calculate the coe�cient C appearing
in eq.(1.3), namely
C =cos(α+ θ)
cos(θ). (1.7)
8
Therefore, the same counterweight M can be used for similar payloads with varied CoM and
mass using the actuator to generate torque indirectly. As shown in Fig. 1.3, a mechanism ini-
tially designed and calibrated form1 = 5 kg can also be calibrated for new payloadsm2 = 4 kg
and m3 = 5.5 kg (which is 20% less and 10% more mass than anticipated respectively). These
relatively high corrections are used only to better illustrate the principle. In practice, mass
would be added to or removed from the counterweight and the actuator would be used only for
correcting the remaining unbalance in order to avoid energy expenditure during interaction.
−180 −90 0 90 180−10
−5
0
5
10
θ [◦]
T[Nm
]
Error on mass
0%-20%+10%
−180 −90 0 90 180−110
−100
−90
−80
−70
θ [◦]
αd
+θ
[◦]
Figure 1.3 � Required static torque T and absolute counterweight position (αd + θ) respec-tively, as a function of the payload position θ in order to maintain static equilibrium; withm1 = 5 kg, m2 = 4 kg, m3 = 5.5 kg, M = 20 kg, c = 0.4 m, r = 0.1 m, l = 0.1 m.
On another note, it was shown in a dynamic simulation that small unaccounted external
forces (such as tension from the actuator electric cables) can rotate the payload out of the
desired orientation even if there is no interaction with the operator. Additionally, imperfect
initialization of the encoders can result in both links slowly moving toward a low energy
con�guration (links vertical). To avoid such situations, a load cell is mounted on the payload's
link in order to measure the interaction force. If the interaction force is below a certain
threshold or uncharacteristic of a human interaction, the actuated joint of the mechanism
is locked in order to constrain the system to a single equilibrium con�guration. A non-
backdrivable (self-locking) actuated joint is used to avoid energy expenditure of the actuator
during lockdown.
It is also worth noting that the apparent impedance, or resistance to motion due to gravity,
can be adjusted using the second link length (l). This impedance can be evaluated by rotating
the �rst link by a certain angle δθ while locking the second link which simulates the system's
response rate and then calculating the resisting torque R to the movement neglecting inertia
(assuming low velocity), that is the sum of moments around the passive pivot due to gravity,
which yields
R = (Mgr −mgc) cos(θ + δθ)−Mgl cos(αd + (θ + δθ)
). (1.8)
9
The resisting torque R as a function of the payload position θ is depicted in Fig. 1.4 using
the same parameters as before but with l = 0.1 m and l = 0.15 m for the �rst and second
graphs. It can be observed in Fig. 1.4 that if the nominal condition of eq.(1.1) is satis�ed,
the resisting torque R is constant for every con�guration and is proportional to the length l .
Mathematically, the resisting torque R in the nominal condition can be expressed as
R = −Mgl cos(δθ − π
2). (1.9)
−180 −90 0 90 180−4
−3
−2
−1
θ [◦]
R[Nm
]
Error on mass
0%-20%+10%
−180 −90 0 90 180−4
−3
−2
−1
θ [◦]
R[Nm
]
Figure 1.4 � Resisting torque as a function of the payload position θ; withm1 = 5 kg, m2 = 4kg, m3 = 5.5 kg, M = 20 kg, c = 0.4 m, r = 0.1 m, δθ = 5◦ ; l = 0.1 m and l = 0.15 m forthe �rst and second graphs respectively.
However, if the nominal condition is not satis�ed, the resisting torque varies according to
θ, which can be calculated with eq.(1.8). Since the calibration process is used for small
corrections, only slight �uctuations will be felt by the operator. In this architecture, the
counterweight can also be used to increase or decrease the apparent impedance. However,
parameter M is linked to the static balancing condition of eq.(1.1), which makes it harder to
use.
Hence, a 1-dof underactuated redundant tilting mechanism is obtained with the passive motion
of the system preserved at all times. To validate the system experimentally, a prototype was
built and is presented in section 1.6.
1.5 Alternative architectures
Several alternative architectures were studied for the gravity balanced tilting mechanism,
although, an active counterweight system was shown to be the simplest system for a prototype.
As an example, a redundant active spring system could have been used as shown in Fig. 1.5.
Similarly to the proposed architecture, the payload is rigidly �xed on a link of length l.
Together, they have a mass m with a CoM located at a distance c from the passive pivot,
10
m
θ
α
k
g
lc
h
s
Figure 1.5 � Geometric and mass parameters of a gravity balanced architecture using anactive spring system.
which the link is attached to. A second link of length h is mounted to the same pivot, but is
actuated to incorporate the principle of underactuated redundancy. The �rst and second links
are connected by a zero-free-length spring of sti�ness k and extended length s, attached to
the end of both links and their rotations are described by θ and α with respect to a horizontal
reference. To reduce the size of the system, a torsional spring could alternatively be used.
The potential energy of the redundant active spring system can be written as
E = mgc sin θ +1
2k(s− s0)2, (1.10)
where
s2 = h2 + l2 − 2hl cos(α− θ) (1.11)
and s0 = 0 for zero-free-length springs. For the static balancing condition of this architec-
ture to be met, the partial derivative of eq.(1.10) with respect to θ must be equal to zero.
Substituting eq.(1.11) into eq.(1.10) (with s0 = 0) and taking the derivative then yields
sin(αd − θ) =mgc
khlcos θ. (1.12)
Using eq.(1.12), the two solutions for the desired αd position can be calculated. During the
operation of the device, the rotation of the passive pivot θ, induced by the operator, is detected
by the encoder. The actuator then rotates the second link to the αd position calculated with
eq.(1.12) using θ. Therefore, the static equilibrium of the system is preserved at all times. In
this architecture, the actuator must generally apply large torques compared to the proposed
active counterweight architecture.
11
m
θ
αk
M
g
r
l c
h
s
Figure 1.6 � Geometric and mass parameters of a gravity balanced architecture using anactuated spring and a passive counterweight.
An alternative architecture, using both a counterweight and a spring, was also studied as
illustrated in Fig. 1.6. In this hybrid version, the payload is rigidly �xed on a link connected
to a passive pivot. Together, they have a mass m with a CoM at a distance c from the passive
pivot. A counterweight M is mounted on the other end of the same link located at a distance
of r from the passive pivot, aligned with the payload. Like in the proposed architecture, the
counterweight is chosen to satisfy the condition for static balancing, that is
Mr = mc. (1.13)
To incorporate the principle of underactuated redundancy, a second link of length h is con-
nected to the �xed pivot and is actuated. A zero-free-length spring of sti�ness k and extended
length s is attached from one end to the tip of the second link and from the other end to a
location on the �rst link at a distance l from the passive pivot. Angle α represents the rotation
of the second link while angle θ is the tilting rotation of the payload, both with respect to the
horizontal reference. The potential energy of this hybrid architecture can be written as
E = (mc−Mr)g sin θ +1
2k(s− s0)2, (1.14)
where s2 is described by eq.(1.11) and s0 = 0 for zero-free-length springs. For equilibrium,
the partial derivative of eq.(1.14) with respect to θ must be zero. Substituting eq.(1.11) into
eq.(1.14), taking the derivative and equating it to zero then yields
sin(αd − θ) =(mc−Mr)g
khlcos θ, (1.15)
from which the two solutions for the desired αd position can be calculated. The operation of
the device is the same as for the previous architectures, using the passive pivot rotation θ in
12
order to calculate αd which is then used for the actuator's input in order to maintain static
equilibrium. If the nominal condition of eq.(1.13) is satis�ed, the second link must be aligned
with the �rst link according to eq.(1.15), that is
αd = θ or αd = θ + π. (1.16)
Since the counterweight directly balances the payload, there is minimal strain on the actuator
in the nominal condition contrary to the previous architecture of Fig. 1.5. Likewise, the
use of torsional springs instead of extension springs would result in a more compact system.
Additionally, the parameters k, h and l can be selected according to the impedance that
the operator is comfortable with during interaction so that it feels intuitive. Employing the
same method used previously to obtain eq.(1.8), the resisting torque R to movement can be
calculated, that is
R = (Mgr −mgc) cos(θ + δθ) + khl sin(αd − (θ + δθ)
). (1.17)
If the nominal condition of eq.(1.13) is satis�ed, the resisting torque R is constant for every
payload con�guration and is proportional to parameters k, h and l.
Other possible architectures include the movable counterweight [16] and rotational counter-
weight [17]. For the initial experimental validation, the active counterweight from section 1.4
is chosen as a prototype because of its simplicity and e�ectiveness.
1.6 Experimental validation
To demonstrate the validity of the proposed architecture for low impedance pHRI, an experi-
ment is conducted with a redundant active counterweight prototype. The prototype is based
on the design shown in Fig. 1.7. The geometric and mass parameters of the prototype were
chosen according to eq.(1.1). Small brass weights are used for the payload m and the counter-
weight M (0.2 kg and 1.2 kg respectively). The actuator is moved away from the active pivot
using two sprockets and a timing belt (not illustrated) in order to allow a full rotation. The
actuator and the passive joint are equipped with encoders since the values of θ and α must
both be known. A simple PD controller is used with the angle α whose command is the value
calculated by eq.(1.3). The experimental setup with the prototype is shown in Fig. 1.8.
During the experimental validation, the static balancing of the mechanism was evaluated in
various con�gurations. The payload was �rst manipulated without the locking mechanism,
which locks down the active joint when no interaction is detected. As expected, the power
cable of the actuator was inconvenient since it causes an undesirable force that the system
cannot distinguish from the operator's force. In order to diminish the e�ects, the cable is
held during the demonstration. Even then, there is a slight deviation of the payload's desired
orientation when the payload is let go which can be seen in Fig. 1.9 a) (see the variations of
13
M
m
Passive pivot
Active pivot
Actuator with encoder
Passive encoder (hidden)
Figure 1.7 � CAD representation of the 1-dof gravity balanced tilting mechanism.
angle θ between the human interactions). Afterwards, we introduced the locking mechanism
so that the active counterweight was locked when the operator was not interacting with the
payload. Doing so signi�cantly reduced the error caused by an imperfect initialization of
the encoders and by continuous external forces. The interaction was then shown to be quite
intuitive as shown in Fig. 1.9 b). After adding mass to the payload (a small nut of 16.5 grams
which is approximately 8% of the initial payload weight), we proceeded with the recalibration
process, which was successful using eq.(1.7). In view of what was observed, the principle of
underactuated redundancy for low impedance rotational interaction was validated during the
demonstration.
Passive pivot
Actuator with encoder
m
M
Active pivot
Passive encoder (hidden)
Figure 1.8 � Experimental setup for the 1-dof gravity balanced tilting mechanism.
In order to improve the current prototype, some strategies could be devised. For example,
14
smaller �exible electric cables could have been used, passing them through the passive pivot
to substantially reduce their undesirable e�ect on the pivot. Fixing the actuator on the frame
and moving the second link with a 5-bar linkage would also be an option to eliminate the
torsion completely, but such an arrangement was deemed too complex for the simple validation
conducted here.
1.7 Multimedia attachment
A video accompanies this paper. It is available at https://youtu.be/drFbH3sFtpg. The video
shows the operator manipulating the payload and evaluating the static balancing of the system
in various con�gurations without the locking mechanism at �rst. The actuator cables are held
during the demonstration to reduce the resulting forces on the system. The interaction is
shown to be intuitive aside from the fact that the payload slightly moves out of its desired
position because of external disturbances (from the cables mainly). Afterwards, the bene�t
of locking the active counterweight while there is no interaction is shown since it forces the
payload to have a single stable con�guration even under the e�ects of external disturbances.
To keep the prototype simple, the active counterweight is blocked manually without the use
of a load-cell. Subsequent tests will be conducted to include the force sensor. Finally, the
calibration using eq.(1.7) is successfully tested using a heavier payload (adding a small nut of
16.5 g which is approximately 8% of the initial payload weight).
1.8 Conclusion
A 1-dof tilting manipulator with low impedance was developed using the principle of active-
passive dynamics decoupling. The proposed architecture, which uses an active rotational
counterweight, was explained and alternative architectures were explored. The experimental
validation showed that the interaction between the system and the operator was easy and in-
tuitive. The e�ectiveness of locking the actuator when the operator is not interacting with the
system was also proven since relatively small external disturbances or even imperfect initial-
ization of the encoders can a�ect the balancing substantially. Therefore, future experiments
with a load cell mounted on the payload's link will be conducted to evaluate the practicability
of this solution.
This 1-dof tilting mechanism can be used with the 3-dof translational uMan system [13]
mentioned earlier as a serial 4-dof mechanism for precise assembly tasks (with the possibility
of adding a rotation around the vertical axis). Con�icts between dofs could arise (a vertical
interaction force induces both a vertical translation and a tilting motion for example), therefore
intuitive control will be implemented. Another problem that needs to be addressed is the
unloading of the payload after assembly. Based on the positive results, a larger prototype
15
0 2 4 6 8 10 12 14
Time [s]
-400
-350
-300
-250
-200
-150
-100
[°]
Beginning of interaction
End of interaction
0 2 4 6 8 10 12 14
Time [s]
0
50
100
150
200
250
300
[°]
d
(a)
0 2 4 6 8 10 12 14 16
Time [s]
-40
-20
0
20
40
[°]
Beginning of interaction
End of interaction
0 2 4 6 8 10 12 14 16
Time [s]
-140
-120
-100
-80
-60
[°]
d
(b)
Figure 1.9 � The payload's con�guration θ and the actuator's position α whose command is αd
as a function of time during an experiment; the green circles and the red squares respectivelyrepresent the beginning and the end of the operator's interaction. In Fig.(a), the lockingmechanism is not applied, whereas, in Fig.(b), the locking mechanism is used when there isno interaction.
16
is currently under development, adding another dof (the rolling motion) to the mechanism,
making it a 2-dof underactuated redundant rotational manipulator.
1.9 Acknowledgment
This study was supported by the Natural Sciences and Engineering Research Council of
Canada (NSERC) and by the Fonds de recherche du Québec - Nature et Technologies (FRQNT).
The authors would like to acknowledge the help of Thierry Laliberté and Simon Foucault with
the experimental validation of the prototype.
17
Chapitre 2
Intuitive Physical Human-Robot
Interaction using an Underactuated
Redundant Manipulator with
Complete Rotational Capabilities
2.1 Résumé
Dans cet article, le principe de la redondance sous-actionnée est présenté à l'aide d'un nouveau
manipulateur rotatif à deux degrés de liberté (2 ddl) équilibré statiquement, composé de
contrepoids mobiles. Les équations d'équilibre statique de l'architecture à 2 ddl sont d'abord
obtenues a�n de fournir la con�guration requise des contrepoids pour avoir un mécanisme
équilibré statiquement. Une méthode de calibration du mécanisme, qui établit les coe�cients
des équations d'équilibre statique, est également présentée. A�n de déplacer et d'orienter
la charge utile pendant l'interaction, le manipulateur rotatif est monté sur un manipulateur
translationnel existant. Des validations expérimentales des deux systèmes sont présentées pour
démontrer le comportement intuitif et réactif des manipulateurs lors des interactions physiques
humain-robot.
2.2 Abstract
In this paper, the concept of underactuated redundancy is presented using a novel two-degree-
of-freedom (2-DoF) gravity balanced rotational manipulator, composed of movable counter-
weights. The static equilibrium equations of the 2-DoF architecture are �rst described in order
to provide the required con�guration of the counterweights for a statically balanced mecha-
nism. A method for calibrating the mechanism, which establishes the coe�cients of the static
equilibrium equations, is also presented. In order to both translate and rotate the payload
18
during manipulation, the rotational manipulator is mounted on an existing translational ma-
nipulator. Experimental validations of both systems are presented to demonstrate the intuitive
and responsive behaviour of the manipulators during physical human-robot interactions.
2.3 Introduction
Fully manual systems are becoming less common in industrial applications since the intro-
duction of physical human-robot interactions (pHRI) [1], [2]. The concept of pHRI allows
industries to incorporate the adaptability and intuitiveness of human operators into the in-
dustrial process, while bene�ting from the high payload capabilities and precise manipulation
control of robots, as well as reducing potential ergonomic injuries for the operators. A typical
example of intuitive pHRI is a human operator guiding and assembling a heavy part using
his/her own impedance through direct physical contacts, while the robot � used as a gravity
compensator � bears the brunt of the payload. Unfortunately, challenges remain in the imple-
mentation of pHRI in industrial applications, especially in the area of safety. Without proper
safety measures, humans cannot interact and share a common workspace with most conven-
tional industrial robots, since their payload and speed are relatively high. Even commercial
collaborative robots, which are designed for pHRI and are limited in both payload and speed,
can raise safety issues [3].
Active research in pHRI has yielded multiple approaches which attempt to satisfy the strict
safety measures in order to allow robots and humans to perform safe and intuitive collaborative
tasks. The usage of force/torque sensors has been the prevalent approach for reducing the per-
ceived combined inertia of the payload and robot since they can be used to sense and regulate
the interaction between the human user and the mechanical system. Paired with an admittance
controller, di�erent impedances can be emulated [4], [5]. In rarer cases, a proportional-integral
(PI) controller [6], or even lead and lag compensators [7] are used with the force/torque sen-
sors. Due to hardware dynamics, the reduction of the perceived inertia is limited using such
techniques [8]. Additionally, unstable behaviours are observed if the apparent impedance is
reduced below a certain fraction of the intrinsic inertia [9]. In [4], [5] and [7], it was demons-
trated that reduction ratios of �ve to seven of the intrinsic inertia were attainable. In another
approach, compliant material is used with force sensors with the purpose of mechanically �lte-
ring the high-frequency interactions [10]. Regardless, large inertia reduction ratios cannot be
obtained without overstepping the concept of passivity [11], [12]. Therefore, physical contacts
must remain limited to speci�c ranges of environment dynamics.
An interesting approach is to employ the principle of underactuated redundancy, which was
successfully implemented for translational collaborative tasks using a serial architecture [13]
and using a parallel architecture [14]. In pHRI, safety issues mainly originate from the inherent
high impedance of the robot. Using underactuated redundancy, it is possible to decouple
19
the human and robot dynamics, therefore segregating the human operator apart from the
robot's high inertia during collaborative tasks. This is done by using low-impedance passive
joints, that the operator interacts with, whose measured joint variables are used to control the
high-impedance active joints of the robot. Furthermore, the low-inertia joints enable intuitive
interactions which provide a higher bandwidth than any force controlled methods.
The underactuated redundant robots proposed in [13], [14] allowed only translations (with the
possibility of adding a rotation around a vertical axis). Although SCARA-type motions are, in
many instances, su�cient for industrial applications, there are several cases in which additional
degrees of freedom are needed. For example, airplane fuselage panel assembly requires 6-DoF
adjustments in order to satisfy the strict geometric tolerances. This is illustrated schematically
in Fig. 2.1, where a fuselage panel is supported by a 6-DoF robot and manipulated by an
operator. A 6-DoF passive mechanism, illustrated schematically as a box in Fig. 2.1, is used
to connect the payload to the robot. Hence, to fully manipulate a payload with six degrees
of freedom using the concept of underactuated redundancy, a passive mechanism that allows
translations and rotations must be developed. Compared to the passive mechanisms proposed
in [13] and [14], the development of a passive mechanism that can handle rotations while
preserving static balancing is a signi�cant challenge.
Figure 2.1 � Assembly of fuselage panels using a robot and a 6-dof passive mechanism(illustrated as a box).
An underactuated redundant manipulator that can handle rotations of the payload was pro-
posed in [18] using a passive joint to tilt the payload and a movable counterweight to control
the equilibrium position of the payload. However, this architecture allows only one rotational
degree of freedom and is therefore mainly relevant for planar tasks. In order to allow a payload
to freely rotate in three-dimensional space, a manipulator with 3-DoF rotational capabilities is
needed. However, it should be pointed out that, for the payload to be statically balanced, gra-
20
vity compensation is required in only two rotational DoFs, i.e., the pitch and the roll motions.
Gravity cannot act on the third rotational DoF, the yaw motion, considering that the axis of
rotation is parallel to the line of action of gravity. Therefore, in this paper, an underactuated
redundant 2-DoF manipulator is proposed, with the possibility of adding a passive third DoF.
A translational unit, such as the one presented in [13], can also be included, to yield a 6-DoF
underactuated redundant manipulator.
This paper is structured as follows. Section 2.4 extends the principle of underactuated redun-
dancy of a 1-DoF gravity-balanced rotational manipulator to a spatial 2-DoF manipulator.
Section 2.5 describes a method for calibrating the proposed manipulator. In Section 2.6, a
prototype is described which has been built and validated experimentally. Section 2.7 explores
the combination of the proposed mechanism with a translational unit in order to freely mani-
pulate a payload in six-dimensional space. The intuitive and responsive behaviour of the whole
system is veri�ed experimentally using a simple insertion task. In Section 2.8, conclusions are
drawn.
2.4 Proposed mechanical architecture
The proposed 2-DoF architecture is represented schematically in Fig. 2.2. It consists of a �rst
link, mounted on a �xed joint, that can be tilted in a vertical plane. Angle θ1 represents
the angle between the �rst link and a �xed horizontal reference, associated with this tilting
motion. A second axis of rotation is de�ned along the link and corresponds to a second revolute
joint, to which a second moving link is attached. This second moving link is represented as
a shaft mounted along the �rst link in Fig. 2.2. Angle θ2 represents the rotation around this
axis, associated with the rolling motion. The payload, of mass m, is rigidly attached to the
second moving link and its centre of mass (CoM) is located at a distance c1 from the �xed
pivot, measured in the direction of the �rst link and at a distance c2 from the second axis
of rotation. Since the actual location of the payload's CoM is usually not known precisely,
an o�set angle θ0 is included in the model. The objective of the proposed architecture is to
allow a human operator to freely manipulate the payload along both the tilting motion and
the rolling motion, without having to support the weight of the payload in any con�guration.
Therefore, the payload must be gravity balanced in all con�gurations. Counterweights are used
to statically balance the mechanism. A �rst counterweight of mass M1 is mounted on a link of
length l1 that is attached to the �rst moving link by an actuated pivot, located at a distance
r1 from the �xed pivot. A second counterweight of mass M2 is mounted similarly on a link of
length l2 that is attached to the second moving link (shaft) by another actuated pivot, located
at a distance r2 from the second revolute joint. Angle αi, i = 1, 2 represents the angle between
the axis of link i and the direction of the link supporting counterweight Mi. This angle is
associated with the motion of the ith counterweight actuator. In summary, the two pivots
associated with the motion of the links are unactuated while the two pivots associated with
21
M1
θ1
α1
g
r1
l1
α2
θ2
c1 Fi
mView
View
θ2
M2
θ0
M2
l2
r2
m
α2
Payload
Payload
c2
Figure 2.2 � Geometric and mass parameters of the proposed gravity-balanced architecture.
the motion of the counterweights are actuated. Four encoders are used to measure the rotation
of all four joints. Using the two actuators to control angles α1 and α2, it is possible to control
the equilibrium con�guration of the payload (angles θ1 and θ2). During the performance of a
task, the human user applies an interaction force Fi on the payload in order to guide it to the
desired orientation. Simultaneously, the encoders detect both passive rotations which inform
the actuators to rotate the counterweight links in a con�guration where the system is statically
balanced. In other words, the equilibrium con�guration is constantly adjusted according to
the user input. Also, around the equilibrium con�guration, the payload can be moved with
very little e�ort from the user which means that the interaction forces remain very low.
The counterweight parameters r1, r2 and M1, M2 are chosen for the system to be statically
balanced in the nominal con�guration (θ1 = θ2 = 0) for a nominal payload m, that is,
M1r1 = mc1 (2.1)
and
M2r2 = mc2. (2.2)
Writing the mechanism's equilibrium equations, the relation between the passive rotations and
the active rotations is obtained and it is possible for the proposed architecture to be statically
balanced for any given speci�ed con�guration (θ1, θ2) by selecting the appropriate actuated
22
angles α1 and α2. The requirement for static equilibrium is, for both joints, to have the CoM
of the rotational system located on the line of action of gravity passing through the base pivot.
For stability purposes, the CoM should be lower than the pivot. Considering the �rst pivot
yields
M1l1 cos(α1 + θ1) +M2r2 sin θ2 sin θ1
−M2l2 sin(α2 + θ2) sin θ1 −mc2
cos θ0sin(θ2 + θ0) sin θ1
+mc1 cos θ1 −M1r1 cos θ1 = 0. (2.3)
Expanding sin(θ2 + θ0) yields
sin(θ2 + θ0) = sin θ2 cos θ0 + cos θ2 sin θ0. (2.4)
Substituting eq.(2.4) into eq.(2.3) and simplifying, we then obtain
M1l1 cos(α1 + θ1) +M2r2 sin θ2 sin θ1
−M2l2 sin(α2 + θ2) sin θ1 −mc2(sin θ2 + tan θ0 cos θ2) sin θ1
+mc1 cos θ1 −M1r1 cos θ1 = 0. (2.5)
Solving eq.(2.5) for α1, we �nd that the equilibrium is obtained if
α1 = ± arccos
[(M1r1 −mc1) cos θ1
M1l1+(
M2(l2 sin(α2 + θ2)− r2 sin θ2) +mc2(sin θ2 + tan θ0 cos θ2))
sin θ1
M1l1
]− θ1. (2.6)
Considering now the second passive pivot yields
−M2r2 cos θ2 +M2l2 cos(α2 + θ2)−mc2 tan θ0 sin θ2 +mc2 cos θ2 = 0, (2.7)
which can be similarly solved for α2. The expression of the second actuated joint coordinate
that leads to equilibrium is then obtained as
α2 = ± arccos
[(M2r2 −mc2) cos θ2 +mc2 tan θ0 sin θ2
M2l2
]− θ2. (2.8)
The solutions for α1 and α2 in eq.(2.6) and eq.(2.8) that lead to the lowest position of the
CoM are chosen for stability. The workspace of this architecture is only limited by mechanical
interferences and by the torque limits of its actuators. Substituting eq.(2.1) and θ1 = 0 into
eq.(2.6) and substituting eq.(2.2) and θ2 = 0 into eq.(2.8), we �nd that in this case, one has
α1 = −π2, (2.9) α2 = −π
2. (2.10)
23
In this con�guration, the torque required at the actuators is minimum. Indeed, the torques
(T1 and T2) required to raise the �rst and second counterweights can be respectively written
as
T1 = M1gl1 cos(α1 + θ1) (2.11)
and
T2 = M2gl2 cos(α2 + θ2) cos(θ1), (2.12)
which are both equal to zero in the nominal con�guration of the mechanism (θ1 = 0 and
θ2 = 0) using the nominal conditions.
Relatively small unaccounted for external forces can easily rotate the payload away from its
current con�guration since the mechanism is always in an equilibrium con�guration if both
eq.(2.6) and eq.(2.8) are satis�ed and since friction in the passive pivots is low. In order to
di�erentiate the operator's intentions from external disturbances, a device can be used to
lock the counterweight joints when there is no interaction between the human user and the
robot. This can be simply implemented by using a mechanical switch on the device to turn on
interactions when pressed, similarly to a dead man's switch. To avoid any energy expenditure
during lockdown, non-backdrivable (self-locking) actuated joints are used. It can also be noted
that, theoretically, the load range of the mechanism is relatively high since an extremely
heavy payload can be balanced by similarly heavy counterweights. The true limitations lie in
the structural strength of the architecture and in the torque limits of the actuators (which
constrain the workspace).
Hence, a 2-DoF rotational manipulator which uses underactuated redundancy is obtained.
2.5 Calibration procedure
In order to establish the equilibrium equations of the system, several mass and geometric para-
meters must be known, as shown in eq.(2.6) and eq.(2.8). Since the value of these parameters
is usually not known exactly, a calibration procedure can be used to determine the coe�cients
to be included in the equations. The calibration procedure can also be used to calibrate the
system for a di�erent payload (as a general rule, the system should be designed to be stati-
cally balanced in the nominal conditions to reduce the torque demands on the actuators). The
calibration procedure of the proposed architecture is presented in this section.
2.5.1 First joint
Replacing mass and geometric parameters by coe�cients to be determined in eq.(2.5) and
rearranging, we obtain
cos(θ1 +α1) = C11 cos θ1 +C12 sin θ1 sin θ2 +C13 sin θ1 sin(α2 + θ2) +C14 sin θ1 cos θ2, (2.13)
24
with
C11 =M1r1 −mc1
M1l1, (2.14) C12 =
mc2 −M2r2M1l1
, (2.15)
C13 =M2l2M1l1
, (2.16) C14 =mc2 tan θ0M1l1
. (2.17)
If the parameters are chosen for the system to be statically balanced in the nominal condition,
that is,
Mr = mc,
it follows from eqs.(2.14�2.15) that coe�cients C11 and C12, are equal to zero. In order to
reduce the required torque, computed with eq.(2.11), the right-hand side of eq.(2.13) must
be close to zero. This is why the prototype presented in an upcoming section of the paper is
designed with the coe�cients close to zero (C13 and C14 can only be minimised). Equation (13)
is then written for four di�erent con�gurations, noted A, B, C and D and the four equations
obtained are written as a system of linear equations in matrix form. We obtain
A1x1 = b1 (2.18)
with
A1 =
cos θ1A sin θ1A sin θ2A sin θ1A sin(α2A + θ2A) sin θ1A cos θ2A
cos θ1B sin θ1B sin θ2B sin θ1B sin(α2B + θ2B) sin θ1B cos θ2B
cos θ1C sin θ1C sin θ2C sin θ1C sin(αC2 + θ2C) sin θ1C cos θ2C
cos θ1D sin θ1D sin θ2D sin θ1D sin(α2D + θ2D) sin θ1D cos θ2D
, (2.19)
x1 =
C11
C12
C13
C14
, (2.20) b1 =
cos(θ1A + α1A)
cos(θ1B + α1B)
cos(θ1C + α1C)
cos(θ1D + α1D)
, (2.21)
where θiA and αiA stand for the values of θi and αi in con�guration A (similarly for con�gu-
rations B, C and D). To solve the linear equations, A1 must be invertible, which leads to the
condition that the det(A1) must not be equal to zero. The sets of passive and active rotations
must therefore be independent from one another to avoid singularity.
Solving for x1 and using four independent sets of both passive and active rotations, that is
θ1n, θ2n, α1n and α2n for n = A,B,C,D, the four coe�cients C1n can be found and the �rst
joint is calibrated.
25
2.5.2 Second joint
Replacing mass and length parameters by coe�cients into eq.(2.7) and rearranging in the same
way as with the �rst joint, we obtain
cos(θ2 + α2) = C21 cos θ2 + C22 sin θ2, (2.22)
where
C21 =M2r2 −mc2
M2l2, (2.23) C22 =
mc2 tan θ0M2l2
. (2.24)
If the system is statically balanced in the nominal con�guration, coe�cient C21 is equal to
zero. Writing eq.(2.22) for two di�erent con�gurations noted A and B and assembling the
linear equations in matrix form, we obtain
A2x2 = b2 (2.25)
with
A2 =
[cos θ2A sin θ2A
cos θ2B sin θ2B
], (2.26)
x2 =
[C21
C22
], (2.27) b2 =
[cos(θ2A + α2A)
cos(θ2B + α2B)
], (2.28)
where a notation similar to the one used for the �rst joint is employed here. The condition
under which the coe�cients cannot be solved from the above linear system can be obtained
by setting the determinant of matrix A2 to zero, that is
det(A2) = cos θ2A sin θ2B − sin θ2A cos θ2B = 0,
which yields
θ2A = θ2B + πi i ∈ Z.
Using two independent sets of the second joint passive and active rotations, that is θ2n and
α2n for n = A,B, the two coe�cients can be found and the second joint is calibrated. In this
case, the �rst joint requires four sets of angles, therefore, the same values will be used for the
second joint.
2.5.3 Robustness of the calibration procedure
In the above subsections, the minimum number of con�gurations was selected for the cali-
bration procedure. Although this procedure is theoretically correct, in practice it may not
26
yield the best results due to measurement errors. In order to improve the robustness of the
calibration procedure, a number of con�gurations larger than the minimum required can be
used, leading to an overdetermined system of linear equations. The overdetermined system of
equations can then be solved using a least squares approach, namely
xi = Ai+bi with i = 1, 2,
where
Ai+ = (Ai
TAi)−1Ai
T .
For the �rst joint, a minimum of four equations (A, B, C and D) is required for calibration,
which gives us
A1 =
cos θ1A sin θ1A sin θ2A sin θ1A sin(α2A + θ2A) sin θ1A cos θ2A
cos θ1B sin θ1B sin θ2B sin θ1B sin(α2B + θ2B) sin θ1B cos θ2B
cos θ1C sin θ1C sin θ2C sin θ1C sin(α2C + θ2C) sin θ1C cos θ2C
cos θ1D sin θ1D sin θ2D sin θ1D sin(α2D + θ2D) sin θ1D cos θ2D
... ... ... ...
cos θ1n sin θ1n sin θ2n sin θ1n sin(α2n + θ2n) sin θ1n cos θ2n
,
x1 =
C11
C12
C13
C14
, b1 =
cos(θ1A + α1A)
cos(θ1B + α1B)
cos(θ1C + α1C)
cos(θ1D + α1D)
...
cos(θ1n + α1n)
.
For the second joint, a minimum of two equations is required for calibration, which can be
represented by the following equations
A2 =
cos θ2A sin θ2A
cos θ2B sin θ2B
... ...
cos θ2n sin θ2n
, x2 =
[C21
C22
], b2 =
cos(θ2A + α2A)
cos(θ2B + α2B)
...
cos(θ2n + α2n)
.
In order to adequately calibrate the system, the number of equilibrium equations needed for
the calibration is investigated. With the assumption that an error of ± 1 degree is randomly
applied to the reading of the passive encoders 1, Table 2.1 shows the expected calibration
1. Inaccuracies that would originate from the incremental nature of the encoder, from the joint friction andfrom the actuators' electric cables (interfering with the static balancing equations).
27
coe�cients, derived from eqs.(2.14�2.17) and eqs.(2.23�2.24), and the calibrated coe�cients
obtained with four to ten equations.
Table 2.1 � Coe�cients obtained from a calibration with n con�gurations where a ± 1 degreeerror was randomly added to the reading of the passive encoders compared to the expectedcoe�cients from eqs.(2.14�2.17) and eqs.(2.23�2.24)
Calibration coe�cients
n C11 C12 C13 C14 C21 C22
- 0 0 0.1800 -0.0804 0 -0.44664 0.0214 0.2777 2.2882 2.1743 0.0115 -0.41206 -0.0001 0.1812 1.1928 1.0027 -0.0017 -0.42518 -0.0130 0.0032 0.0980 -0.1729 0.0061 -0.434410 -0.0105 0.0185 0.3005 0.0494 0.0011 -0.4380
-40 -30 -20 -10 0 10 20 30 40
1 [°]
0
1
2
3
4
5
6
7
|1| [°
]
n = 4
n = 6
n = 8
n = 10
(a)
-40 -30 -20 -10 0 10 20 30 40
2 [°]
0
0.5
1
1.5
2|
2| [°
]
n = 4
n = 6
n = 8
n = 10
(b)
Figure 2.3 � Absolute di�erence between the calibrated actuator position using n con�gura-tions and the expected actuator position, that is |∆αi|, for di�erent values of the payload'scon�guration θi where in Fig.2.3(a) i = 1, whereas in Fig.2.3(b) i = 2, for n = 4, 6, 8, 10.
The coe�cients produced by the calibration procedure are still far from the expected coef-
�cients according to the data in Table 2.1 : this is expected since an error was added to
the con�gurations of the payload before theoretically calibrating. Alternatively, the absolute
di�erence between the calibrated actuator position using n con�gurations and the expected
actuator position for di�erent values of the payload's con�guration, that is |∆αi| as a functionof θi for i = 1, 2, can be examined as exposed in Fig.2.3. According to the results in Fig.2.3,
it is observed that using at least eight con�gurations provides an error of less than 1 degree.
Therefore, it is assumed that using eight con�gurations should be su�cient for the calibration.
28
2.6 Experimental validation
A small-scale prototype based on the 2-DoF rotational manipulator was built according to
the design illustrated in Fig. 2.4. Both active joints are actuated with harmonic drive motors
for a higher gear ratio. The actuators are placed at the end of each of the primary links with
the aim of reducing the size of the counterweights. The payload has a mass of 2.27 kg. The
counterweights for the pitch motion and the roll motion respectively have a mass of 6.80 kg and
2.27 kg. The counterweights are chosen using the nominal conditions of eq.(2.1) and eq.(2.2).
The actuators are driven by a simple PD controller with eq.(2.6) and eq.(2.8) providing the
command values. All joints are equipped with encoders.
M2
M1
m
α2
α1
θ1
θ2
Figure 2.4 � CAD model of the prototype of the 2-DoF rotational manipulator.
The prototype is mounted in an experimental setup, as shown in Fig. 2.5, in order to validate
both the calibration procedure and the manipulability/intuitiveness of the mechanism. In
the �rst video accompanying this paper, available at https ://youtu.be/�zElABZHdw, the
intuitive manipulation of the robot and the calibration procedure are demonstrated. First,
the intuitiveness of the operation of the system is evaluated qualitatively. Using the static
balancing equations with the expected coe�cients from eqs.(2.14�2.17) and eqs.(2.23�2.24) to
drive the actuators, the manipulation is shown to be intuitive and reactive to the operator, as
demonstrated in the video.
Then, the calibration procedure is evaluated experimentally with the aim of verifying if the
equilibrium coe�cients are identi�ed correctly. To show that the system can be easily recali-
brated for a higher or lower than anticipated payload, the payload is increased by adding a 0.2
kg mass to the payload (an increase of ≈ 8.8% of the initial payload). Di�erent con�gurations
are used during the calibration procedure and both the active and passive joint angles are
measured. Solving the systems of linear equations of eq.(2.18) and eq.(2.25) using the least-
29
M2
M1
Payload mActuatorswith encoders
Passive pivotswith encoders
Dead man’sswitchCounterweights
Figure 2.5 � Experimental setup for the 2-DoF rotational manipulator prototype.
squares approach, the calibrated equilibrium coe�cients are found. The modi�ed payload is
then manipulated to demonstrate the e�ectiveness of the calibration.
3-DoF Gantry
Rotationalsystem
systemTranslational
Payload
Figure 2.6 � Experimental setup of the 4-DoF gravity-balanced architecture.
30
2.7 Rotational and translational motion
In order to freely manipulate a payload with 6 DoFs, a translational manipulator can be com-
bined with the rotational manipulator presented in the previous sections. The robot proposed
in [13] is ideal for the translational component of the 6-DoF serial architecture since it operates
using the same principle of underactuated redundancy, leading to an intuitive and reactive be-
haviour for all DoFs. The translational manipulator consists of a 3-DoF active gantry system
coupled with a passive translational manipulator equipped with encoders. The rotational ma-
nipulator is simply mounted on the end of the translational manipulator. In order to simplify
the experiments, only two of the translational DoFs of the manipulator presented in [13] (hori-
zontal motion) are included in this work, leading to a 4-DoF system allowing two translational
DoFs (horizontal motion) and two rotational degrees of freedom. The combined prototype is
shown in Fig. 2.6.
Considering that the control systems of both manipulators are mainly driven by the relative
positions of their passive components, no major changes are to be made to the control systems
when the rotational and the translational manipulators are combined. Statically, both manipu-
lators are independent from one another. Nevertheless, they do a�ect each other dynamically.
However, the low dynamic responses produced by their interactions during pHRI � conside-
red quasi-static because of the low speed/acceleration � are trivial to the equilibrium of the
whole system since it is mechanically damped by the human operator during interactions.
The serial architecture was validated experimentally in order to con�rm its intuitive behaviour
for pHRI. This is demonstrated in the second video accompanying this paper, available at
https ://youtu.be/z8nbQrFxoQM, where a human operator manipulating the device performs
a 4-DoF insertion task. First, the translational movements are shown. Then, the rotational
capabilities are presented. In order to demonstrate the intuitiveness of the device for industrial
applications such as aerospace panel assembly, two pins are added to the payload, that need
to be assembled on a plate with two corresponding sockets. The plate and the manipulator
are placed in a con�guration where 4 DoFs are needed to complete the task. From the video,
it can be observed that the operator can easily insert the two pins in the sockets, thereby
demonstrating that the whole system is stable, responsive and intuitive for the operator.
2.8 Conclusion
In this paper, a 2-DoF rotational manipulator, composed of movable counterweights, was
proposed using the principle of underactuated redundancy. A third rotational DoF, the yaw
motion, is not included since static balancing is not required for this DoF. In order to control
the payload's con�guration, the static equilibrium equations and actuators' torque equations
were developed. A calibration procedure was presented with the purpose of adjusting the
static equilibrium equations on the assumption that the mass and geometric parameters can
31
be either imprecise or unknown and that a di�erent payload can be used (without changing
the counterweights). A small-scale prototype was developed to validate both the intuitiveness
and the calibration of the manipulator. It was found that the system was both intuitive and
responsive for the operator.
In order to achieve intuitive pHRI for applications requiring 6-DoF manipulation like aerospace
panel assembly, the 2-DoF rotational manipulator is attached to an existing translational
manipulator from a previous work [13]. A simple 4-DoF insertion task is used to demonstrate
the viability of the system, where high-bandwidth low-impedance interactions are achieved
between the operator and the system.
2.9 Acknowledgement
This study was supported by the Natural Sciences and Engineering Research Council of Ca-
nada (NSERC) and by the Fonds de Recherche du Québec - Nature et Technologies (FRQNT).
The authors would like to acknowledge the help of Thierry Laliberté and Simon Foucault with
the experimental validation of the prototype.
32
Conclusion
La problématique abordée dans ce mémoire est la manipulation de pièces aérospatiales lourdes,
soit l'assemblage de panneaux de fuselage d'avion, situation industrielle où l'interaction humain-
robot est sans doute béné�que. On pense à une réduction des blessures de nature ergonomique,
à une augmentation de la productivité et à une réduction des coûts d'opération ; une véritable
synergie entre l'humain et le robot. Pour soutenir les opérateurs humains lors de ces tâches
d'assemblage, un mécanisme capable de supporter et déplacer une charge utile élevée, en plus
d'avoir la capacité de l'orienter dans l'espace tridimensionnel, est investigué. Ce mécanisme
doit être intuitif, réactif et sécuritaire.
Les travaux de recherche présentés dans le premier chapitre ont tout d'abord permis d'ap-
pliquer le principe de la redondance sous-actionnée aux mouvements rotatifs, principe qui,
jusqu'à maintenant, n'avait été appliqué qu'aux mouvements translationnels. Le principe de
la redondance sous-actionnée est utilisé parce qu'il permet une interaction intuitive, réactive
et sécuritaire entre l'opérateur humain et le robot. Di�érents mécanismes rotatifs, équilibrés
statiquement, sont étudiés comme candidats pour le premier manipulateur à 1 degré de liberté.
Le mécanisme à contrepoids actif est sélectionné et un prototype est construit. Le mécanisme
est ensuite validé expérimentalement.
Appuyé par la preuve de concept du premier chapitre, le deuxième chapitre introduit un mé-
canisme ayant la capacité d'orienter une charge utile dans l'espace tridimensionnel, toujours
en utilisant le principe de la redondance sous-actionnée. Ce manipulateur rotatif est ensuite
jumelé à un manipulateur translationnel existant dans le but d'ajouter au système la capacité
de positionnement de la charge utile. Le manipulateur rotatif, ainsi que sa combinaison avec
le manipulateur translationnel, sont validés expérimentalement. Les résultats sont très satis-
faisants : l'opérateur humain manipule la charge utile facilement dans l'espace et la passivité
du dispositif assure un milieu sécuritaire pour l'opérateur. Ceci est d'ailleurs démontré par
une tâche d'insertion à quatre degrés de liberté.
En se basant sur les résultats présentés dans les deux chapitres de ce mémoire, on peut conclure
qu'un manipulateur intuitif, réactif et sécuritaire, basé sur le principe de la redondance sous-
actionnée et ayant les capacités d'orienter et de positionner une charge utile dans toutes les
dimensions, est obtenu. Pour intégrer le manipulateur dans un contexte industriel, il faudrait
33
au préalable appliquer quelques modi�cations. Premièrement, l'architecture complète com-
binant les translations et les rotations doit être réduite en taille pour en faire une variante
compacte plus facile à utiliser. Pour la partie translationnelle du manipulateur, l'architecture
utilisée jusqu'à maintenant [13] pourrait être remplacée par son homologue parallèle [14], éli-
minant ainsi la longue chaîne passive en série. Pour la partie rotative du manipulateur, il est
possible de réduire la taille de celle-ci en remplaçant les contrepoids rotatifs, utilisés pour
l'équilibrage statique, par un système de ressorts industriels compacts. Deuxièmement, une
méthode intuitive pour le chargement et le déchargement de la charge utile doit être élaborée.
En ce moment, le manipulateur rotatif est équilibré statiquement lorsque la charge utile est
montée sur l'e�ecteur du robot. Une stratégie temporaire, qui peut facilement être implémen-
tée, est de bloquer les articulations passives du manipulateur rotatif lorsque la charge utile à
manipuler est absente. Le désavantage de cette stratégie est bien évidemment la perte de la
passivité du système pendant un certain moment. Dans la même optique, il faudrait analyser
les dangers liés à la défaillance d'un moteur puisque l'équilibre statique ne serait plus assuré
suite à un tel événement. Finalement, il serait béné�que de diminuer ou d'éliminer l'in�uence
des �ls électriques sur les composantes passives de la partie rotative du manipulateur a�n
d'améliorer l'équilibrage statique de celle-ci. Par exemple, l'utilisation de bagues collectrices
pourrait être étudiée pour remplacer les connexions électriques actuelles du manipulateur ro-
tatif (particulièrement celles des moteurs). Cela permettrait de réduire grandement le moment
généré par la torsion des �ls électriques lors du fonctionnement de l'appareil.
34
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