Conception et validation expérimentale d’un système mécatronique pour la manipulation intuitive de composantes lourdes

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.)

Conception et validation expérimentale d’un système

mécatronique pour la manipulation intuitive de

composantes lourdes

Mémoire

Julien-Mathieu Audet

Sous la direction de:

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'eecteur du mécanisme passif. Lors du fonctionnement du dispositif, l'opérateur humain manipule directe-ment la charge utile et induit ainsi des mouvedirecte-ments 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 an 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 mouvestatique-ment 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.

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 eector. 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.

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 . . . 4 1.2 Abstract . . . 4 1.3 Introduction. . . 5 1.4 Proposed architecture . . . 7 1.5 Alternative architectures . . . 10 1.6 Experimental validation . . . 13 1.7 Multimedia attachment . . . 15 1.8 Conclusion . . . 15 1.9 Acknowledgment . . . 17

2 Intuitive Physical Human-Robot Interaction using an Underactuated Redundant Manipulator with Complete Rotational Capabilities 18 2.1 Résumé . . . 18

2.2 Abstract . . . 18

2.3 Introduction. . . 19

2.4 Proposed mechanical architecture . . . 21

2.5 Calibration procedure . . . 24

2.6 Experimental validation . . . 29

2.7 Rotational and translational motion . . . 31

2.8 Conclusion . . . 31

Conclusion 33

Liste des tableaux

2.1 Coecients obtained from a calibration with n congurations where a ± 1 de-gree error was randomly added to the reading of the passive encoders compared

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. 8 1.3 Required static torque T and absolute counterweight position (αd + θ) res-pectively, as a function of the payload position θ in order to maintain static equilibrium ; with m1 = 5 kg, m2 = 4kg, m3 = 5.5 kg, M = 20 kg, c = 0.4 m, r = 0.1m, 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 = 5.5kg, M = 20 kg, c = 0.4 m, r = 0.1 m, δθ = 5◦; l = 0.1 m and l = 0.15 m for the rst and second graphs respectively. . . 10

1.5 Geometric and mass parameters of a gravity balanced architecture using an active spring system. . . 11

1.6 Geometric and mass parameters of a gravity balanced architecture using an actuated spring and a passive counterweight. . . 12

1.7 CAD representation of the 1-dof gravity balanced tilting mechanism. . . 14

1.8 Experimental setup for the 1-dof gravity balanced tilting mechanism. . . 14

1.9 The payload's conguration 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 respectively represent the beginning and the end of the operator's interaction. 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. 22 2.3 Absolute dierence between the calibrated actuator position using n congura-tions and the expected actuator position, that is |∆αi|, for dierent values of the payload's conguration 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. . . 29

2.5 Experimental setup for the 2-DoF rotational manipulator prototype. . . 30

An expert is a person who has made all the mistakes that can be made in a very narrow eld.

Remerciements

Je tiens à prendre un moment pour témoigner de ma gratitude et pour remercier tous ceux qui ont guidé ma réexion 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 inniment.

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 eectuées 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 conden-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'aectera pas la présentation des résultats dans le mémoire.

Le premier article présenté a été soumis à la revue scientique 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 enn, la rédaction de l'article en tant qu'auteur principal. Avant la soumission de l'article à la revue scientique, 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 scientique 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 an 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

banc d'essai pour valider le système complet, ont été réalisés par l'étudiant sous la supervision de M. Gosselin. Aucune modication n'a été apportée à l'article intégré par rapport à l'article soumis.

Introduction

Problématique

Il est parfois dicile pour l'être humain d'eectuer 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 bénéque. 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 bénécierait 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'eort. 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

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'eecteur du méca-nisme passif. Les mouvements engendrés dans les degrés de liberté (ddl) du mécaméca-nisme 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 susent 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 diciles pour un robot (plusieurs capteurs, longue programmation, peu versatile, etc) en mettant une emphase sur la manipulation intuitive de pièces aérospatiales en rotation. Les diérents 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 an 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.

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'eecteur de la composante passive du mé-canisme et doit être équilibrée statiquement. Une composante active doit être ajoutée pour contrôler la conguration de la charge utile indirectement an que le principe de la redon-dance sous-actionnée soit respecté. Diérents mécanismes sont étudiés comme candidats pour le mécanisme à 1 ddl. Par la suite, diérents types de mécanismes sériels et parallèles à 2 ddl sont explorés an 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 diérents 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 conguration de la charge utile, pour élaborer des méthodes de calibration robustes et, nalement, pour déterminer à partir des diérents 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 avec Matlab pour faciliter la visualisation des données. Entre autres, les graphiques aident à la comparaison des diérents mécanismes en illustrant certains critères comme les eorts 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.

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. Enn, une validation expéribriève-mentale 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 briey introduced. Finally, an experimental validation is provided to demonstrate the viability of the concept for rotational low impedance pHRI.

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 dicult 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 dierent 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 specic 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-eector 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-handlretain-ing 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-dofs SCARA-type motions are sucient 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 diers 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 dierent 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 briey 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.

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 conguration of the link and payload.

In principle, if the counterweight is perfectly adjusted, the mechanism is statically balanced around the xed pivot and all congurations are equilibrium congurations. 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 conguration. 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 conguration will exist. The counterweight is chosen such that one has the static balancing condition in the nominal condition, namely

M r = 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

m

M

θ

α

F

g

r

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 congurations 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 coecient, 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 = −

π

2 − 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 congurations. In this state, energy expenditure is kept at a minimum since the torque (T ) used to raise the counterweight is

T = M gl 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 coecient C may not be exactly equal to zero. Consequently, the actuator has to rotate the second link out of its resting vertical conguration 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 coecient C appearing in eq.(1.3), namely

C = cos(α + θ)

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 for m1= 5kg can also be calibrated for new payloads m2 = 4kg

and m3= 5.5kg (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 [N m ] 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; with m1= 5 kg, m2 = 4 kg, m3 = 5.5kg, 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 conguration (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 conguration. 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

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 satised, the resisting torque R is constant for every conguration and is proportional to the length l . Mathematically, the resisting torque R in the nominal condition can be expressed as

R = −M gl cos(δθ −π 2). (1.9) −180 −90 0 90 180 −4 −3 −2 −1 θ [◦] R [N m ] Error on mass 0% -20% +10% −180 −90 0 90 180 −4 −3 −2 −1 θ [◦] R [N m ]

Figure 1.4  Resisting torque as a function of the payload position θ; with m1= 5kg, m2 = 4

kg, m3 = 5.5kg, M = 20 kg, c = 0.4 m, r = 0.1 m, δθ = 5◦ ; l = 0.1 m and l = 0.15 m for

the rst and second graphs respectively.

However, if the nominal condition is not satised, 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,

m

θ

α

k

g

l

c

h

s

Figure 1.5  Geometric and mass parameters of a gravity balanced architecture using an active 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 stiness 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

khl cos 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.

m

θ

α

k

M

g

r

l

_c

h

s

Figure 1.6  Geometric and mass parameters of a gravity balanced architecture using an actuated 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

M r = 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 stiness 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 − M r)g sin θ + 1

2k(s − s0)

2_{, (1.14)}

where s2 _{is described by eq.(1.11) and s}

0 = 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 − M r)g

khl cos θ, (1.15)

from which the two solutions for the desired αd position can be calculated. The operation of

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 satised, 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 = (M gr − mgc) cos(θ + δθ) + khl sin αd− (θ + δθ)
. (1.17)

If the nominal condition of eq.(1.13) is satised, the resisting torque R is constant for every payload conguration 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 countercounter-weight from section 1.4 is chosen as a prototype because of its simplicity and eectiveness.

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 congurations. 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 eects, 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

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 signicantly 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,

smaller exible electric cables could have been used, passing them through the passive pivot to substantially reduce their undesirable eect 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 congurations 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 benet of locking the active counterweight while there is no interaction is shown since it forces the payload to have a single stable conguration even under the eects 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 eectiveness 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 aect 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). Conicts 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

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 conguration 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 respectively represent the beginning and the end of the operator's interaction. In Fig.(a), the locking

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.

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 an de fournir la conguration requise des contrepoids pour avoir un mécanisme équilibré statiquement. Une méthode de calibration du mécanisme, qui établit les coecients des équations d'équilibre statique, est également présentée. An 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 conguration of the counterweights for a statically balanced mecha-nism. A method for calibrating the mechanism, which establishes the coecients of the static equilibrium equations, is also presented. In order to both translate and rotate the payload

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 beneting 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, dierent 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 specic 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

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, sucient 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 signicant 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-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 veried 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 dened 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 oset 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 conguration. Therefore, the payload must be gravity balanced in all congurations. 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

M

θ

α

g

r

l

α

θ

c

F

m

θ

M

θ

M

l

r

m

α

c

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 conguration 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 conguration where the system is statically balanced. In other words, the equilibrium conguration is constantly adjusted according to the user input. Also, around the equilibrium conguration, the payload can be moved with very little eort 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 conguration (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 specied conguration (θ1, θ2) by selecting the appropriate actuated

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

M1l1cos(α1+ θ1) + M2r2sin θ2sin θ1

− M₂l2sin(α2+ θ2) sin θ1−

mc2

cos θ0

sin(θ2+ θ0) sin θ1

+ mc1cos θ1− M1r1cos θ1 = 0. (2.3)

Expanding sin(θ2+ θ0) yields

sin(θ2+ θ0) = sin θ2cos θ0+ cos θ2sin θ0. (2.4)

Substituting eq.(2.4) into eq.(2.3) and simplifying, we then obtain M1l1cos(α1+ θ1) + M2r2sin θ2sin θ1

− M2l2sin(α2+ θ2) sin θ1− mc2(sin θ2+ tan θ0cos θ2) sin θ1

+ mc1cos θ1− M1r1cos 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(l2sin(α2+ θ2) − r2sin θ2) + mc2(sin θ2+ tan θ0cos θ2)

 sin θ1

M1l1

#

− θ₁. (2.6) Considering now the second passive pivot yields

− M₂r2cos θ2+ M2l2cos(α2+ θ2) − mc2tan θ0sin θ2+ mc2cos 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+ mc2tan θ0sin 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= −

π

2, (2.9) α2= −

π

In this conguration, 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 = M1gl1cos(α1+ θ1) (2.11)

and

T2 = M2gl2cos(α2+ θ2) cos(θ1), (2.12)

which are both equal to zero in the nominal conguration 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 conguration since the mechanism is always in an equilibrium conguration if both eq.(2.6) and eq.(2.8) are satised and since friction in the passive pivots is low. In order to dierentiate 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 parapara-meters is usually not known exactly, a calibration procedure can be used to determine the coecients to be included in the equations. The calibration procedure can also be used to calibrate the system for a dierent 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 coecients to be determined in eq.(2.5) and rearranging, we obtain

with C11= M1r1− mc1 M1l1 , (2.14) C12= mc2− M2r2 M1l1 , (2.15) C13= M2l2 M1l1 , (2.16) C14= mc2tan θ0 M1l1 . (2.17)

If the parameters are chosen for the system to be statically balanced in the nominal condition, that is,

M r = mc,

it follows from eqs.(2.142.15) that coecients 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 coecients close to zero (C13and C14can only be minimised). Equation (13)

is then written for four dierent congurations, 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 θ1Asin θ2A sin θ1Asin(α2A+ θ2A) sin θ1Acos θ2A

cos θ1B sin θ1Bsin θ2B sin θ1Bsin(α2B+ θ2B) sin θ1Bcos θ2B

cos θ1C sin θ1Csin θ2C sin θ1Csin(αC2+ θ2C) sin θ1Ccos θ2C

cos θ1D sin θ1Dsin θ2D sin θ1Dsin(α2D+ θ2D) sin θ1Dcos 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 conguration A (similarly for

congu-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 coecients C1n can be found and the rst

2.5.2 Second joint

Replacing mass and length parameters by coecients into eq.(2.7) and rearranging in the same way as with the rst joint, we obtain

cos(θ2+ α2) = C21cos θ2+ C22sin θ2, (2.22)

where C21= M2r2− mc2 M2l2 , (2.23) C22= mc2tan θ0 M2l2 . (2.24)

If the system is statically balanced in the nominal conguration, coecient C21 is equal to

zero. Writing eq.(2.22) for two dierent congurations 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 coecients 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 θ2Asin θ2B− sin θ2Acos 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 coecients 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 congurations was selected for the cali-bration procedure. Although this procedure is theoretically correct, in practice it may not

yield the best results due to measurement errors. In order to improve the robustness of the calibration procedure, a number of congurations 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+= (AiTAi)−1AiT.

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 θ1Asin θ2A sin θ1Asin(α2A+ θ2A) sin θ1Acos θ2A

cos θ1B sin θ1Bsin θ2B sin θ1Bsin(α2B+ θ2B) sin θ1Bcos θ2B

cos θ1C sin θ1Csin θ2C sin θ1Csin(α2C + θ2C) sin θ1Ccos θ2C

cos θ1D sin θ1Dsin θ2D sin θ1Dsin(α2D+ θ2D) sin θ1Dcos θ2D

... ... ... ...

cos θ1n sin θ1nsin θ2n sin θ1nsin(α2n+ θ2n) sin θ1ncos 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 # , _b 2 =       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 encoders1_{, Table 2.1 shows the expected calibration}

1. Inaccuracies that would originate from the incremental nature of the encoder, from the joint friction and from the actuators' electric cables (interfering with the static balancing equations).

coecients, derived from eqs.(2.142.17) and eqs.(2.232.24), and the calibrated coecients obtained with four to ten equations.

Table 2.1  Coecients obtained from a calibration with n congurations where a ± 1 degree error was randomly added to the reading of the passive encoders compared to the expected coecients from eqs.(2.142.17) and eqs.(2.232.24)

Calibration coecients n C11 C12 C13 C14 C21 C22 - 0 0 0.1800 -0.0804 0 -0.4466 4 0.0214 0.2777 2.2882 2.1743 0.0115 -0.4120 6 -0.0001 0.1812 1.1928 1.0027 -0.0017 -0.4251 8 -0.0130 0.0032 0.0980 -0.1729 0.0061 -0.4344 10 -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 dierence between the calibrated actuator position using n congura-tions and the expected actuator position, that is |∆αi|, for dierent values of the payload's

conguration θi where in Fig.2.3(a) i = 1, whereas in Fig.2.3(b) i = 2, for n = 4, 6, 8, 10.

The coecients 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 congurations of the payload before theoretically calibrating. Alternatively, the absolute dierence between the calibrated actuator position using n congurations and the expected actuator position for dierent values of the payload's conguration, that is |∆αi|as a function

of 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 congurations provides an error of less than 1 degree. Therefore, it is assumed that using eight congurations should be sucient for the calibration.