Robust Tuning Rules for Series Elastic Actuator PID Cascade Controllers

Proceedings of the 3rd IFAC Conference on Proceedings of of the the 3rd IFAC IFAC Conference on on Proceedings Advances in Proportional-Integral-Derivative Proceedings of the 3rd 3rd IFAC Conference Conference on Control Available online at www.sciencedirect.com Advances in Proportional-Integral-Derivative Advances in Proportional-Integral-Derivative Control Proceedings of the 3rd IFAC Conference on Control Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Proceedings of the 3rd IFAC Conference on Control Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018

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IFAC PapersOnLine 51-4 (2018) 220–225

Robust Tuning Rules for Series Elastic Robust Robust Tuning Tuning Rules Rules for for Series Series Elastic Elastic Actuator PID Cascade Controllers Robust Tuning for Series Actuator PIDRules Cascade Robust Tuning Rules for Controllers Series Elastic Elastic ∗ ∗ Controllers ∗ Actuator PID Cascade Stefano Ghidini Manuel Beschi Nicola Pedrocchi ∗ ∗ Stefano GhidiniPID Manuel Beschi ∗∗ Nicola Pedrocchi ∗∗ Actuator Cascade Controllers

∗∗∗Nicola Pedrocchi ∗ Stefano Manuel Stefano Ghidini Ghidini ∗ Antonio Manuel Beschi Beschi Nicola Pedrocchi ∗ Visioli ∗∗∗ ∗∗ Visioli ∗∗ Stefano Ghidini ∗∗ Antonio Manuel Beschi Antonio Visioli Antonio Visioli ∗ Nicola Pedrocchi ∗ Stefano Ghidini Antonio Manuel Beschi Nicola Pedrocchi Visioli ∗∗ ∗ ∗∗ Automation, National of Industrial Technologies and ∗ Antonio Visioli ∗ Institute Institute of Industrial Technologies and Automation, National ∗ Institute of Industrial Technologies and Automation, National Institute of Industrial Technologies and Automation, National Research Council of Via 20133, Milan, (e-mail: ∗ Research Council of Italy, Italy, Technologies Via Corti Corti 12, 12,and 20133, Milan, Italy Italy (e-mail: Institute of Industrial Automation, National Research Council of Italy, Via Corti 12, 20133, Milan, Italy (e-mail: Research Council of Italy, Via Corti 12, 20133, Milan, Italy (e-mail: ∗ stefano.ghidini,manuel.beschi,[email protected]) Institute of Industrial Technologies and Automation, National stefano.ghidini,manuel.beschi,[email protected]) Research Council of Italy, Via Corti 12, 20133, Milan, Italy (e-mail:of stefano.ghidini,manuel.beschi,[email protected]) ∗∗ stefano.ghidini,manuel.beschi,[email protected]) Department of Mechanical and Industrial Engineering, University ∗∗ Research Council of Italy, Via Corti 12, 20133, Milan, Italy (e-mail:of ∗∗ Department of Mechanical and Industrial Engineering, University ∗∗ stefano.ghidini,manuel.beschi,[email protected]) Department of Mechanical and Industrial Engineering, University Department of Via Mechanical and Industrial Engineering, University of of Brescia, Branze 38, 25123 Brescia, Italy (e-mail: stefano.ghidini,manuel.beschi,[email protected]) ∗∗ Brescia, Via Branze 38, 25123 Brescia, Italy (e-mail: Department of Mechanical and Industrial Engineering, University of Brescia, Via Branze 38, Brescia, Italy Brescia, Via Branze 38, 25123 25123 Brescia, Italy (e-mail: (e-mail: ∗∗ [email protected]) Department of Mechanical and Industrial Engineering, University of [email protected]) Brescia, Via Branze 38, 25123 Brescia, Italy (e-mail: [email protected]) [email protected]) Brescia, Via Branze 38, 25123 Brescia, Italy (e-mail: [email protected]) [email protected]) Abstract: This paper deals with the control of a collaborative robot manipulator with series Abstract: This deals the control of collaborative robot manipulator with series Abstract: This paper paper deals with with thetuning controlrules of aaafor collaborative robot manipulator with series Abstract: This paper deals with the control of collaborative robot manipulator with series elastic actuators. In particular, robust cascade control of the joints are presented. elastic actuators. In particular, robust tuning rules for cascade control of the joints are presented. Abstract: This paper deals with the control of a collaborative robot manipulator with series elastic actuators. In particular, robust tuning rules for cascade control of the joints are presented. elastic actuators. In particular, robust tuning rules cascade of manipulator theThe joints arewith presented. Both the motor and link position control loops are control considered. proposed tuning Abstract: This velocity paper deals the control of afor collaborative robot series Both the motor velocity and link position control loops are considered. The proposed tuning elastic actuators. In particular, robust tuning rules for cascade of the joints are presented. Both allow the motor velocity and with link position control loops are control considered. The proposed tuning Both the motor velocity and link position control loops are considered. The proposed tuning rules the online computation of robust control parameters to cope with the different link elastic actuators. In particular, robust tuning rules for cascade control of the joints are presented. rules allow the online computation of robust controlloops parameters to cope cope with with the different link Both the motor velocity and link position control are considered. The proposed tuning rules allow the online computation of robust control parameters to the different link rules allow the online computation of robust control parameters to method cope with the different link reflected inertia. Experimental results show the effectiveness of the in real applications. Both the motor velocity and link position control loops are considered. The proposed tuning reflected inertia. Experimental results show the effectiveness of the method in real applications. rules allow the online computation of robust control parameters to cope with the different link reflected inertia. Experimental results show the effectiveness of the method in real applications. reflected inertia. Experimental results show the effectiveness of the method in real applications. rules allow the online computation of robust control parameters cope with therights different link reflected inertia. Experimental results show the effectiveness of the method in All real applications. © 2018, IFAC (International Federation of Automatic Control) Hosting bytoElsevier Ltd. reserved. Keywords: Collaborative robots; Series elastic actuator; Robust control; PID; Tuning rules. reflected inertia. Experimental results show the effectiveness of the method in real applications. Keywords: Collaborative robots; Series elastic actuator; Robust control; PID; Tuning rules. Keywords: Collaborative Collaborative robots; robots; Series Series elastic elastic actuator; actuator; Robust Robust control; control; PID; PID; Tuning Tuning rules. rules. Keywords: Keywords: Collaborative robots; Series elastic actuator; Robust control; PID; Tuning rules. Keywords: Collaborative robots; Series elastic actuator; Robust control; Tuning rules. 1. INTRODUCTION INTRODUCTION model mismatches: mismatches: the PID; reflected inertia strongly depends depends 1. model the inertia strongly 1. INTRODUCTION INTRODUCTION model mismatches: the reflected reflected inertia stronglyunknown) depends 1. model mismatches: the reflected inertia strongly depends on the robot configuration and on the (possibly robot configuration and on the unknown) Standard industrial industrial robots are not not typically typically designed designed to to on 1. robots INTRODUCTION model the spring reflected strongly depends on the the mismatches: robot configuration and characteristic oninertia the (possibly (possibly unknown) on the robot configuration and on the (possibly unknown) Standard are payload. Further, the can be nonnonStandardwith industrial robots are not not typically designed In to payload. 1. robots INTRODUCTION model the spring reflected strongly depends Further, the can be Standard industrial are typically designed to co-exist humans in the the same same working environment. on the mismatches: robot configuration and characteristic oninertia theand (possibly payload. Further, the spring characteristic canunknown) be static nonpayload. Further, the spring characteristic can be nonco-exist with humans in working environment. In linear, aging can increase hysteresis backlash, Standard industrial robots are not typically designed to linear, co-exist with humans in the same working environment. In on the robot configuration and characteristic on theand (possibly unknown) aging can increase hysteresis backlash, static co-exist with humans in the same working environment. In fact, industrial robots are usually employed in several tasks payload. Further, the spring can be nonlinear, aging can increase hysteresis and backlash, static Standard industrial robots are not typically designed to aging can increase hysteresis and backlash, fact, industrial robots are usually in several tasks and viscous friction terms strongly depend on can temperature co-exist with humans same employed working In linear, fact, industrial robots in arethe usually employed in fences. several They tasks payload. Further, the spring characteristic be static nonfriction terms strongly depend on fact, robots are usually employed in several tasks in anindustrial isolated workspace by means of walls wallsenvironment. or linear, aging can increase hysteresis andassumption backlash, static and viscous viscous friction terms strongly depend on temperature temperature co-exist with humans in the same employed working environment. In and and viscous friction terms strongly depend on temperature in an isolated workspace by means of or fences. They (Simoni et al., 2017), and the rigid body could fact, industrial robots are usually in several tasks in an isolated workspace by means of walls or fences. They linear, aging can increase hysteresis and backlash, static (Simoni et al., 2017), and the rigid body assumption could in an isolated workspace by means of walls or fences. They are designed to provide high accuracy and precision, but and viscous friction terms strongly depend on temperature (Simoni et al., 2017), and the rigid body assumption could fact, industrial robots arehigh usually employed inprecision, several They tasks et accurate al., 2017), and the rigid body assumption are provide accuracy and but not be an hypothesis for lightweight lightweight links. could in isolated to workspace by means of walls fences. areandesigned designed to provide high accuracy andor precision, but (Simoni and viscous friction terms strongly depend on temperature not an hypothesis for links. are designed to provide accuracy and precision, but the achievement of such suchhigh a high high level performance makes (Simoni et accurate al., 2017), and the rigid body assumption could not be be an accurate hypothesis for lightweight lightweight links. in an isolated workspace by means of walls or fences. They not be an accurate hypothesis for links. the achievement of a level performance makes Those model mismatches can imply significant detriment are to provide high accuracy and precision, but Those the designed achievement of by such high level performance makes (Simoni et accurate al.,mismatches 2017), and the rigid body assumption could model can imply significant detriment the achievement of such aa high level performance makes them characterized a rigid and heavy structure. These not be an hypothesis for lightweight links. Those model mismatches can imply significant detriment are designed to provide high accuracy and precision, but model mismatches can imply significant detriment them characterized aa rigid and heavy structure. These in model-based control strategies which do not notlinks. take them the of by such a high level performance makes themachievement characterized by rigid and heavy structure. These Those not be an accurate hypothesis for lightweight in model-based control strategies which do take them them characterized by a rigid and heavy structure. These features do not allow robots to reach the safety conditions Those model mismatches canitimply significant detriment in model-based model-based control strategies which do not not to take them the achievement of by such a high level performance makes in control strategies which do take them features do not allow robots to reach the safety conditions into account. In this case, is reasonable employ them characterized a rigid and heavy structure. These features do not allow robots to reach the safety conditions Those model mismatches can imply significant detriment account. In this case, it is reasonable to employ features do not allow robots to reach theenvironment safety conditions that are necessary to share a and working with into in model-based control strategies do not take them into account. In thiscontrol case, itarchitecture iswhich reasonable to employ them characterized by a rigid heavy structure. These into account. In this case, it is reasonable to employ that are necessary to share a working environment with a robust PID-based instead of an features do not allow to reach theenvironment safetyskin. conditions that are are without necessary torobots sharetools working environment with ain model-based control strategies which do not take them PID-based instead of an that necessary to share aa working with humans expensive like sensitive sensitive into account. In thiscontrol case, itarchitecture isOne reasonable to employ a robust robust PID-based control architecture instead of an features do not allow robots to reach the safety conditions a robust PID-based control architecture instead of an humans without expensive tools like skin. adaptive model-based algorithm. limit of this solution that are necessary to share a working environment with humans without expensive tools like sensitive sensitive skin. skin. into account. In thiscontrol case, itarchitecture isOne reasonable to solution employ adaptive model-based algorithm. limit of this humans without expensive tools like Due to the increased demand of human-robot interaction, a robust PID-based instead of an adaptive model-based algorithm. One limit of this solution that are necessary to share a working environment with adaptive model-based algorithm. Onead-hoc limit of this solution Due to increased demand of interaction, is the computational computational burden of an robust tuning humans without tools like sensitivedesign skin. Due to the the increased demand of human-robot human-robot interaction, aadaptive PID-based burden control architecture instead of an the of an robust tuning Due to the increased demand of human-robot interaction, paradigm shift expensive is changing changing the structural of the the is model-based algorithm. Onead-hoc limit modular of this solution is robust the computational computational burden of an ad-hoc robust tuning without expensive tools like sensitivedesign skin. is the burden of an ad-hoc robust tuning aahumans paradigm shift is the structural of optimization, which is not compatible with design Due to the increased demand of human-robot interaction, a paradigm shift is changing the structural design of the adaptive model-based algorithm. One limit of this solution optimization, which is not compatible with modular design arobot paradigm shift is changing the structural design of the to reduce the impact forces during possible collision. is the computational burden of an ad-hoc robust tuning optimization, which is not compatible with modular design Due toto increased demand of human-robot interaction, is burden not compatible with modular design robot reduce impact forces during possible collision. (Maurtua et which al., 2016). 2016). In fact, fact, the robot presents arobot paradigm shiftthe changing the structural design of has the optimization, tothe reduce the impact forces during possible collision. is the computational of an the ad-hoc robust tuning (Maurtua et al., In robot presents aaa robot to reduce the impact forces during possible collision. In recent years, a is class of lightweight lightweight industrial robot optimization, is its notoverall compatible with modular design (Maurtua et which al.,thus 2016). In fact, fact, the robot presents a paradigm shift is changing the structural design of the (Maurtua et al., 2016). In the robot presents a In recent years, a class of industrial robot has modular design, inertia may change based robot to reduce the impact forces during possible collision. In recent recent years, (Guiochet class of of lightweight lightweight industrial robot has modular optimization, is its notoverall compatible with modular design design, thus inertia may change based In years, aa class industrial robot has been presented et al., during 2017) where the impact (Maurtua et which al., 2016). In fact, the robot presents a modular design, thus its overall inertia may change based robot to reduce the impact forces possible collision. modular design, thus its overall inertia may change based been presented (Guiochet et al., 2017) where the impact on the setup of the mechanical structure. Moreover, in In recent years, a class of lightweight industrial robot has been presented (Guiochet et al., 2017) where the impact (Maurtua et al., 2016). In fact, the robot presents a on the setup of the mechanical structure. Moreover, in been presented (Guiochet et al., the 2017) where impact forces areyears, reduced by of limiting mass, andthe therefore modular design, thus its overall inertia may change based on the setup of the mechanical structure. Moreover, in In recent a class lightweight industrial robot has on the setup of the mechanical structure. Moreover, in forces are reduced by limiting the mass, and therefore many applications, the usage of a single controller for the been presented (Guiochet et al., 2017) where the impact forces are reduced by limiting the mass, and therefore modular design, thus its overall inertia may change based usage of aa structure. single controller for forces are reduced by robot, limiting the mass, andthetherefore the momentum of the the and the maximum exerted on theapplications, setupworkspace of thethe Moreover, in many applications, themechanical usage of to single controller for the the been presented (Guiochet et al., 2017) where impact many many usage of a structure. single controller for the the of and the maximum exerted entire robot can lead very poor poor performance, forces are reduced by robot, limiting the mass, and therefore the momentum momentum ofmany the robot, and the maximum exerted on theapplications, setupworkspace of thethe mechanical Moreover, in entire robot can lead to very performance, the momentum of the robot, and the maximum exerted forces. However, tasks do not require the precision many applications, the usage of a single controller for the entire robot workspace can lead to very poor performance, forces are reduced by limiting the mass, and therefore entire robot workspace can very poor performance, forces. However, tasks do not require the requiring the retuning of lead theof to controllers. In thisforcase, case, the momentum ofmany the robot, and maximum exerted forces. However, many tasks do notthe require the precision precision many applications, the usage a single controller the requiring the retuning of the controllers. In this forces. However, many tasks do not require the precision provided by an industrial robot and therefore impact forces entire robot workspace can to very to poor performance, requiring the retuning of lead the controllers. In this case, the momentum ofmany the robot, and maximum exerted the retuning of the controllers. case, provided by an industrial robot and therefore impact forces gain-scheduling approach is proven proven beIn anthis effective forces. However, tasks notthe require the precision provided by anbe industrial robot and therefore impact forces requiring robot workspace can to very to poor performance, aaentire approach is be an effective provided by an industrial robot and therefore impact forces reduction can obtained by do introducing elastic elements requiring the retuning of lead the controllers. case, a gain-scheduling gain-scheduling approach is proven proven to beIn anthis effective forces. However, many tasks do not require the precision a gain-scheduling approach is to be an effective reduction can be obtained by introducing elastic elements solution. Also in this case, effective robust tuning rules provided by an industrial robot andal., therefore impact forces solution. reduction can be obtained by introducing introducing elastic elements requiring the retuning of the controllers. In this case, Also in this case, effective robust tuning rules reduction can be obtained by elastic elements in the robot robot chain (Verstraten et 2016). Series Elastic asolution. gain-scheduling approach is proven to be an effective Also in this case, effective robust tuning rules provided by an industrial robot andal., therefore impact forces solution. Also this effective robust tuning rules in the chain (Verstraten et 2016). Series Elastic can be used used to in tune thecase, controller without the need of an reduction can be obtained by introducing elastic elements in the robot chain (Verstraten et al., 2016). Series Elastic a gain-scheduling approach is proven to be an effective can be to tune the controller without the need of an in the robot chain (Verstraten et al., 2016). Series Elastic Actuators (SEAs) are explicitly explicitly designedelastic to pursue pursue this can solution. Also this effective robust tuning rules can be be used used to in tune thecase, controller without the need of of an reduction can be obtained by introducing elements to tune the controller without the need an Actuators (SEAs) are designed to this external dedicated computer (Misgeld et al., 2017). in the robot chain (Verstraten et al., 2016). Series Elastic Actuators (SEAs) are explicitly designed to pursue this solution. Also in this case, effective robust tuning rules dedicated computer (Misgeld et al., 2017). Actuators (SEAs) are explicitly toSeries pursue idea. Inrobot fact, an elastic elastic elementetis isdesigned inserted in series to this the external can be used to tune the controller without the need of an external dedicated computer (Misgeld et al., 2017). in the chain (Verstraten al., 2016). Elastic external dedicated computer (Misgeld et al., 2017). idea. In fact, an element inserted in series to the The approximation of robust problem solutions withoftuntunActuators (SEAs) are explicitly tothat pursue idea. In In fact, fact, an elastic element isofdesigned inserted in series to this the The can be used to tunecomputer the controller without the need an approximation of robust problem solutions with idea. an elastic element inserted in series to the actuator thus reducing the part part is the mass mass directly external dedicated (Misgeld et al., 2017). Therules approximation of robust problem solutions with tunActuators (SEAs) are explicitly designed tothat pursue this ˚str¨ The approximation of robust problem solutions with tunactuator thus reducing the of the directly ing is a common strategy in process control ( A o m idea. In fact, an elastic element is inserted in series to the actuator thus reducing the part of the mass that directly ˚ external dedicated computer (Misgeld et al., 2017). ing rules is a common strategy in process control ( A str¨ o actuator thus reducing the part isofthe theoverall mass that directly ˚ impacts the human, andelement therefore impact force. The approximation of strategy robust problem solutions with tuning rules rules is aa common common strategy in are process controlrules (˚ Astr¨ str¨ om m idea. In fact, an elastic inserted in series to the ing is in process control ( A o m impacts the human, and therefore the overall impact force. and H¨ a gglund, 2004), but there no tuning that actuator thus reducing part ofthe theoverall massexerted that directly impacts the the human, andthe therefore the overall impact force. and The approximation of strategy robust problem solutions with tunH¨ aagglund, 2004), but there are no tuning rules that impacts human, and therefore impact force. ˚ Moreover, springs allow estimating the forces ing rules is a common in process control ( A str¨ o m and H¨ gglund, 2004), but there are no tuning rules that actuator thus reducing the part of the mass that directly H¨agglund, 2004), but there no tuning that Moreover, springs allow estimating the exerted address the control of elastic elastic mechanical systems, as far as ˚ impacts the human, and therefore the overall impact forces force. and Moreover, springs allow estimating the sensors. exerted forces ing rules is acontrol common strategy in are process controlrules (as Astr¨ oas m the of mechanical systems, far Moreover, allow estimating the exerted without thesprings use of of expensive expensive force/torque and H¨agglund, 2004), but there are no tuning rules that address the control of elastic elastic mechanical systems, as far as impacts the human, and therefore the overall impact forces force. address address the control of mechanical systems, as far as without use force/torque sensors. the authors know. The aim of this paper is to cope with Moreover, springs allow estimating the exerted forces without the use of expensive force/torque sensors. and H¨ a gglund, 2004), but there are no tuning rules that the authors know. The aim of this paper is to cope with without the use of expensive force/torque sensors. However, this type of structure makes the control of the address the control of elastic mechanical systems, as far as the authors know. The aim of this paper is to cope with Moreover, springs allow estimating the exerted forces authors know. The aim of this paper is to cope with However, this type of structure makes the control of the this issuethe bycontrol proposing robust tuning rulessystems, for series elastic without the use of expensive force/torque sensors. However, this type of structure makeset the control of the the address of elastic mechanical aselastic far as this issue proposing robust tuning rules for series However, this type of structure makes the control the link position more difficult (Maleki al., 2016; of Paine the know. The aim of this paper is to cope with this authors issue by by proposing robust tuning rules for series elastic without the use of expensive force/torque sensors. this issue by proposing robust tuning rules for series elastic link position more difficult (Maleki et al., 2016; Paine actuators. In this way, it is possible to guarantee a robust However, this type of structure makes the control of the link position more difficult (Maleki et al., 2016; Paine the authors know. The aim of this paper is to cope with actuators. In this way, it is possible to guarantee a robust link position more difficult (Maleki et al., 2016; Paine et al., 2014; 2014; Ragonesi et al., al., 2011; 2011; Calanca and Fiorini, Fiorini, this issue by tuningto rules for series elastic actuators. Inproposing thislimiting way, robust it the is possible possible to guarantee robust However, thisRagonesi type of structure makes control of the actuators. In this way, it is guarantee aa in robust et et Calanca and performance by maximum sensitivity link link more etthe al., 2016;Fiorini, Paine et al., al.,position 2014; Ragonesi et al., al., 2011; Calanca and Fiorini, this issue by proposing robust tuningto rules for series elastic performance by limiting the maximum sensitivity in link et al., 2014; Ragonesi et 2011; and 2017; Calanca et al., al., difficult 2017; Dos(Maleki SantosCalanca and Siqueira, 2014). actuators. In this way, it is possible guarantee a robust performance by limiting the maximum sensitivity in link link position more difficult (Maleki et al., 2016; Paine performance by limiting the maximum sensitivity in link 2017; Calanca et 2017; Dos Santos and Siqueira, 2014). position and motor velocity controllers for different values et al., 2014; Ragonesi et al., 2011; Calanca and Fiorini, 2017; Calanca et al., 2017; Dos Santos and Siqueira, 2014). actuators. In this way, it is possible to guarantee a robust and velocity controllers for different values 2017; et al., position 2017; Dosof2011; Santos and isSiqueira, 2014). Indeed, the actual actual the Calanca link given by the position performance byinertia limiting thedifferent maximum sensitivity link position and motor motor velocity controllers for different values et al.,Calanca 2014; Ragonesi et al., and Fiorini, and motor velocity controllers for different values Indeed, the the link is given by the of the reflected reflected and damping levels.in 2017; Calanca et al., position 2017; Dosof andthe 2014). Indeed, the actual position ofSantos the plus link isSiqueira, given by the position performance byinertia limiting thedifferent maximum sensitivity in link of the and damping levels. Indeed, the actual position of the link is given by the position of the gearbox output shaft spring deforposition and motor velocity controllers for different values of the reflected inertia and different damping levels. 2017; Calanca et al., position 2017; DosofSantos andthe Siqueira, 2014). of the reflected inertia and different damping position of the gearbox output shaft plus spring deforThe paper is organized organized as follows: follows: in Section Section 2 levels. is described described Indeed, the actual the link is given by the position of the gearbox output shaft plus the spring deforposition and motor velocity controllers for different values The paper is as in 2 is position of the gearbox output shaft plus the spring deformation. The elasticity impliesofthe the presence ofgiven resonance of the reflected inertia different damping The paper is organized organized as follows: in Section Section is described described Indeed, the actual position the link the is of by the The paper is as in 22 levels. is mation. The elasticity implies presence aaa resonance the dynamic model of and thefollows: the SEA. SEA. In Section Section is proproposition of the gearbox outputin shaft plus deformation. The elasticity implies the presence ofspring resonance of the reflected inertia and different damping levels. the dynamic model of the the In 333 is mation. The elasticity implies the presence of a resonance and antiresonance behavior the frequency response, The paper is organized as follows: in Section 2 is described the dynamic model of the the SEA. In Section is proproposition of the gearbox output shaft plus the spring deforthe dynamic model of the the SEA. In Section 3 is and antiresonance behavior in the frequency response, posed the optimization problem performed to obtain spemation. The elasticity implies the presence of a resonance and antiresonance behavior in the frequency response, The paper is organized as follows: in Section 2 is described the problem performed to obtain speand antiresonance behavior the frequency response, coupled with elasticity the typical typical smallin damping factor ofresonance metallic posed the dynamic modelable of the the SEA. In Section 3 is proposed the optimization optimization problem performed toperformance. obtain spemation. The implies the presence of a posed the optimization problem performed to obtain specoupled with the small damping factor of metallic cific tuning rules, to guarantee robust and antiresonance behavior in the frequency response, coupled with the typical small damping factor of metallic the dynamic model of the the SEA. In Section 3 is procific tuning rules, able to guarantee robust performance. coupled with the Moreover, typical small damping factor of metallic spring elements. there are multiple sources of posed the optimization problem performed to obtain specific tuning rules, able to guarantee robust performance. and antiresonance behavior in the frequency response, tuning rules, able to guarantee robusttoperformance. spring there are sources of coupled with the Moreover, typical small damping factor of metallic spring elements. elements. Moreover, there are multiple multiple sources of cific posed the optimization problem performed obtain spespring elements. there are multiple sources of cific tuning rules, able to guarantee robust performance. coupled with the Moreover, typical small damping factor of metallic spring elements. Moreover, there are multiple sources of cific tuning rules, able to guarantee 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. robust performance. spring elements. Moreover,of there are multiple sources of220 Copyright © 2018 IFAC Copyright 2018 responsibility IFAC 220Control. Peer review©under International Federation of Automatic

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Stefano Ghidini et al. / IFAC PapersOnLine 51-4 (2018) 220–225

221

. qt

Motion planner

+ qt

+

Link position control − q

. Θt

+ −

Motor velocity control

τ

SEA actuator

. Θ

Fig. 2. Cascade control scheme. Fig. 1. Illustrative scheme of the SEA model. The experimental results are described in Section 4 , they have been obtained by means of two trajectories on a three degree-of-freedom series elastic actuated robot. 2. SERIES ELASTIC ACTUATOR MODEL A series elastic actuator manipulator can be modeled as an open kinematic chain consisting of a rigid link and lumped elastic part coaxial with the joint rotational axis. The proposed approach considers a decentralized model, where the coupling effects are considered as model mismatches. This model takes into account two inertias: the first one represents the motor and the gearbox, while the second one represents the inertia of the link and of the downstream part of the kinematic chain for the actual joint configuration. Thus, the state variables are the motor ˙ the spring deflection δ and position θ, the motor velocity θ, ˙ the spring velocity δ. Other interesting variables are the ˙ which link position q = θ + δ and its velocity q˙ = θ˙ + δ, represent the real motion of the link. A joint model can be derived by dynamic laws as:  Jm θ¨ = hδ˙ + kδ − fm θ˙ + τ − Cm sign θ˙ (1) ¨ = −hδ˙ − kδ Jl (θ¨ + δ) where Jm is the motor inertia, Jl is the link (and the downstream part of the kinematic chain) inertia, τ is the motor torque, fm is the motor viscous friction (between motor and stator), Cm is the motor Coulomb friction (between motor and stator), k is the elasticity (between link and motor), and h is the viscosity (between link and motor). An illustrative scheme of the model is shown in Figure 1. Equation (1) can be linearized by neglecting the Coulomb friction terms, so that the resulting transfer function between the motor torque and the motor velocity is: Pθ,τ ˙ (s) =

˙ θ(s) τ (s)

=

Jl s2 +hs+k Jl Jm s3 +(Jl fm +(Jm +Jl )h)s2 +((Jl +Jm )k+fm h)s+fm k

(2)

while the transfer function between the motor torque and the link position is: q(s) = Pq,τ (s) = τ (s) (3) hs+k (Jl Jm s3 +(Jl fm +(Jm +Jl )h)s2 +((Jl +Jm )k+fm h)s+fm k)s

It is worth stressing that the link inertia Jl strongly depends on the robot configuration and on the (possibly unknown) payload, while several possible model mismatches can occur in real setup. For example, the spring characteristic can be nonlinear, and aging can increase hysteresis and backlash. Finally, static and viscous friction terms might strongly depend on temperature (Simoni et al., 2017). Those model mismatches can imply significant detriment in model-based control strategies which do not take them into account. 221

A robust cascade controller is therefore used to cope with mismatches without complex model adaptivity algorithms. The primary loop regulates the link position by acting on the motor velocity set-point θ˙sp , that is the input of the secondary loop, which computes the desired torque. The current loop is considered fast enough to be neglected. Figure 2 shows the overall scheme. 2.1 Reduction of model parameters In order to simplify the robust tuning rules expression, a model similarity is used to reduce the number of model parameters. Rewriting (1) in the Laplace domain it is possible to obtain: Jm s2 θ = hsδ + kδ − fm sθ + τ (4) 2 Jl s (θ + δ) = −hsδ − kδ (5) Then, by defining the following quantities   k Jm s, sˆ ⇔ sˆ = s= Jm k J l , θˆ = kθ δˆ = kδ, Jˆl = Jm (6)   fm k k ˆ=h fˆm = , h k Jm k Jm it is possible to rewrite (4) as ˆ sδˆ + δˆ − fˆm sˆθˆ + τ sˆ2 θˆ = hˆ

(7)

and (5) as

ˆ = −hˆ ˆ sδˆ − δˆ (8) Jˆl sˆ2 (θˆ + δ) The ratio Jˆl is the most important parameter, in fact, it causes a decrement of the resonance and antiresonance frequencies as shown in Figure 3 and, at the same time, it ˆ shown in Figure 4, is increases their peaks. The effect of h, mainly on resonance and antiresonance peaks, while their frequencies present a slight shift. Finally, the motor viscous friction parameter changes the low frequency behavior and it reduces the resonance peak. ˆ has normally a small value It is worth stressing that h for mechanical elastic elements (like torsional, coned-disc and coil springs) without dampers, while fˆm has not a significant impact on the crossover frequency range. Finally, it is important to note that the motor velocity frequency response is less sensible to parameter changes for frequencies higher than the resonance frequency. 3. ROBUST OPTIMAL TUNING The tuning procedure starts by considering the inner loop and continues with the outer one. The process of the inner ˙ Once loop has a transfer function defined between τ and θ. the tuning parameters of this loop have been detected, the velocity closed loop obtained will be seen as the process of the outer loop, which has a transfer function defined between θ˙ and q.

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40

40

30

30

20

20

10

10

0

0

-10

-10

-20

-20

-30

-30

-40 10 -2

10 -1

10 0

10 1

-40 10 -2

10 2

Normalized angular frequency [rad/s] 100

50

50

0

0

-50

-50

-100

-100

-150 10 -2

10 -1

10 0

10 -1

10 0

10 1

10 2

10 1

10 2

Normalized angular frequency [rad/s]

100

10 1

-150 10 -2

10 2

10 -1

10 0

Normalized angular frequency [rad/s]

Normalized angular frequency [rad/s]

Fig. 3. Effect of the parameter Jˆl on frequency responses Pθ,τ ˆ˙ (s) (upper plot), Pqˆ,τ (s) (lower plot). Blue line: ˆ = 0.2, fˆm = 0. Magenta line: Jˆl = 1, h ˆ= Jˆl = 0.5, h ˆ ˆ ˆ ˆ 0.2, fm = 0. Ocher line: Jl = 2, h = 0.2, fm = 0. ˆ = 0.2, fˆm = 0. Violet line: Jˆl = 4, h

Fig. 5. Effect of the parameter fˆm on frequency responses Pθ,τ ˆ˙ (s) (upper plot), Pqˆ,τ (s) (lower plot). Blue line: ˆ = 0.3, fˆm = 0. Magenta line: Jˆl = 1, h ˆ = Jˆl = 1, h ˆ ˆ ˆ ˆ 0.3, fm = 0.1. Ocher line: Jl = 1, h = 0.3, fm = 0.2. ˆ = 0.3, fˆm = 0.3. Violet line: Jˆl = 1, h  k/Jm ˆ p,v  k , Kp,v = K , k k/Jm (10) 1 ˆ ˆ Ki,v = Ki,v k, Ki,v = Ki,v , k The controller parameters are then computed by solving the following optimization problem:

40

ˆ p,v = Kp,v K

30 20 10 0 -10 -20 -30 -40 10 -2

2

10 -1

10 0

10 1

min(ˆ ωc,v − ω ˆ d,v ) − λv φm  φ ≥ φm,min   m Ms ≤ Ms,max subject to: ω )| ≤ Lh,v,max , if ω ˆ≥ω ˆh   |Lv (ˆ |Lv (ˆ ω )| ≥ Ll,v,min , if ω ˆ≤ω ˆl

10 2

Normalized angular frequency [rad/s] 100

50

0

-50

-100

-150 10 -2

10 -1

10 0

10 1

10 2

Normalized angular frequency [rad/s]

ˆ on frequency responses Fig. 4. Effect of the parameter h Pθ,τ ˆ˙ (s) (upper plot), Pqˆ,τ (s) (lower plot). Blue line: ˆ = 0.1, fˆm = 0. Magenta line: Jˆl = 1, h ˆ = ˆ Jl = 1, h ˆ ˆ ˆ ˆ 0.2, fm = 0. Ocher line: Jl = 1, h = 0.3, fm = 0. ˆ = 0.4, fˆm = 0. Violet line: Jˆl = 1, h 3.1 Motor velocity control A PI controller regulates the motor torque to reduce the speed trajectory error. The controller is described by the following transfer function: Ki,v (9) s where Kp,v is the proportional gain and Ki,v is the integral gain. The controller parameters can be rewritten by following (6) as Cv (s) = Kp,v +

222

(11)

where ω ˆ is the normalized angular frequency, ω ˆ c,v and ω ˆ d,v are, respectively, the cutoff frequency and its desired value, λv > 0 is a weighting factor, φm and Ms are the phase margin and the maximum sensitivity, Lv (ˆ ω) = Cv (ˆ ω )Pθ,τ ω ) is the loop transfer function, Lh,v,max is the ˙ˆ (ˆ maximum allowed value of |L(ˆ ω )| for frequencies greater than ω ˆ h , and Ll,v,min is the minimum allowed value for frequencies lower than ω ˆ l . In the tuning procedure, those values has been set equal to: ˆ h = 100ˆ ωd , Lh,v,max = −40dB, ω Ll,v,min = 40dB, ω ˆ h = 0.01ˆ ωd , Ms,max = 1.4, φm,min = 60◦ λv = 10−2

(12)

It is worth stressing that the presence of the antiresonance zeros can cause multiple crossing of the 0dB axis, and therefore the system has more than one phase margin. In this case, the most critical value is taken into account in the optimization algorithm. Due to these multiple cutoff frequencies, the maximum sensitivity Ms is therefore of utmost importance to guarantee the required level of robustness. The term −λφm of the cost function discerns between solutions with the same cutoff frequency by selecting the one with the biggest phase margin.

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3.2 Link position control A PD controller addresses the link position control by computing the target motor speed signal θ˙t . The controller is described by the following transfer function: Kd,p s Cp (s) = Kp,p + (13) Tf,p s + 1 where Kp,p is the proportional gain, Kd,p is the derivative gain, and Tf,p is the filter time constant. Controller Cp (s) acts on the process made by the secondary loop Pθ˙t ,q (ˆ ω) ˙ as shown in Figure 2. The target motor speed signal θt is given by the controller output plus a feedforward action equal to the velocity profile provided by the link motion planner. Integral action is not employed since Pθ˙t ,q (ˆ ω ) has a pole at the origin and external disturbances are managed by the secondary loop. The controller parameters can be rewritten by following (6) as   Jm k ˆ ˆ , Kp,p = Kp,v , Kp,p = Kp,p k Jm ˆ d,p , ˆ d,p = Kd,p , (14) Kd,p = K K   k Jm , , Tf,v = Tˆf,p Tˆf,p = Tf,p Jm k As in Section 3.1, the controller parameters are computed by solving the following optimization problem: 2

min(ˆ ωc,p − ω ˆ d,p ) − λp φm  φ ≥ φm,min   m (15) Ms ≤ Ms,max subject to: |L (ˆ ω )| ≤ L , if ω ˆ ≥ ω ˆ  p h,p,max h  |Lp (ˆ ω )| ≥ Ll,p,min , if ω ˆ≤ω ˆl where ω ˆ c,v and ω ˆ d,v are, respectively, the cutoff frequency and its desired value, Lp (ˆ ω ) = Cv (ˆ ω )Pθˆ˙ ,q (ˆ ω )) is the loop t transfer function, Lh,p,max is the maximum allowed value of |L(ˆ ω )| for frequencies greater than ω ˆ h , while Ll,p,min is the minimum allowed value for frequencies lower than ω ˆl and λv > 0 is a weighting factor. In the tuning procedure, those values have been set equal to: Lh,p,max = 40dB, ω ˆ h = 100ˆ ωd , Ll,p,min = 20dB, ω ˆ h = 0.01ˆ ωd , (16) Ms,max = 1.4, φm,min = 60◦ −2 λp = 10

Table 1. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 3 MK ˆp

MK ˆ

i

2.531878 0.065366 0.000677 0.001504 0.051657 0.010598

During the robot working activities, the link inertia can significantly change, and therefore a gain scheduling approach is highly recommended. However, solving (11) and (15) online is not possible due to computational burdens. For this reason, the development of an approximate solution by means of tuning rules has been performed, in order to allow the online computation of the robust controller for a given value of Jl . The motor velocity controller tuning is performed by imposing the desired cutoff frequency to be part of the set ω ˆ d,v ∈ {3, 4, 5, 6, 7, 8, 9, 10}. In order to find the tuning rules, the optimization problem (11) has been ˆ in the interval [0.01, 0.5], repeated 2000 times by varying h ˆ fm in the interval [0.0, 0.5], and Jˆm in the interval [0.5, 2] for each value of cutoff frequencies. The resulting controllers parameters are approximated by means of the following tuning rules: 223

-0.187629 0.590771 -0.295008 3.001502 -0.256998 -0.028327

0.766849 -1.589730 0.763627 -0.006152 0.367284 0.025340

0.176141 0.125980 -0.124333 -0.000658 -0.000003 0.000178

2.369994 -2.663157 0.946668 -0.009293 0.009802 -0.003194

Table 2. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 4 MK ˆp

MK ˆ

i

3.708948 0.011955 0.003181 -0.003161 0.127712 0.015017

-0.108430 0.260642 -0.116833 4.029311 -0.628493 -0.008518

0.377811 -0.653636 0.292964 -0.046981 0.890955 -0.026994

0.037026 0.130305 -0.071785 -0.000382 -0.000331 0.000229

1.437171 -1.496274 0.498541 -0.009993 0.009936 -0.003068

Table 3. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 5 MK ˆp

MK ˆ

i

4.791588 -0.003609 0.003717 -0.009775 0.249213 0.014515

-0.113563 0.189006 -0.074892 5.061989 -1.201760 0.064108

0.302272 -0.436630 0.180993 -0.088155 1.679001 -0.166778

0.032533 0.068670 -0.036597 -0.000461 -0.000247 0.000201

0.976789 -0.936631 0.298819 -0.010757 0.010181 -0.003044

Table 4. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 6 MK ˆp

MK ˆ

i

5.850311 -0.025962 0.010387 -0.015671 0.418010 0.007221

-0.174389 0.249436 -0.094186 6.076707 -1.954373 0.196555

0.379250 -0.541724 0.217908 -0.089640 2.674216 -0.397502

0.000514 0.102136 -0.049836 -0.000315 -0.000646 0.000421

0.796468 -0.787528 0.261223 -0.012242 0.011784 -0.003593

Table 5. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 7 MK ˆp

MK ˆ

i

6.897957 -0.054282 0.020995 -0.017312 0.631822 -0.007112

ˆp K

3.3 Tuning rules

223

ˆi K

-0.263959 0.386998 -0.150384 7.046733 -2.839686 0.383480

0.537100 -0.835283 0.344335 -0.007167 3.784653 -0.702280

-0.048779 0.186529 -0.088107 0.000191 -0.001803 0.001026

 1 2 ˆ ˆ = 1 fˆm fˆm h h MKˆ p  Jˆl  ˆ2  Jl  1   2 ˆ ˆ2 M  Jˆl  = 1 fˆm fˆm h h ˆ Ki Jˆl2 

 2

0.747911 -0.833085 0.301129 -0.014580 0.015161 -0.004937



(17)

The resulting values for each ω ˆ d,v are in Tables 1-8. For all the values of ω ˆ d,v the maximum sensitivity is much smaller than 1.4, as shown in Figure 6, which demonstrates the effectiveness of the tuning rules to ensure the required robustness. The link position controller tuning rules are also computed by solving 2000 times the optimization problem (15) by using the motor velocity tuning rules with ω ˆ d,v = 3, even if it can be used also with faster secondary controlled systems. The primary loop cutoff frequency is set to ω ˆ d,p = 1 but, differently to the secondary loop case, the constraint Ms = 1.4 limits the bandwidth in the range [0.3,

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Table 9. Matrices coeffients of the link position tuning rules when ω ˆ d,p = 1

Table 6. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 8 MK ˆp

MK ˆ

i

7.946372 -0.101432 0.040938 -0.011250 0.886549 -0.027762

-0.410546 0.656619 -0.269486 7.948294 -3.804287 0.613100

0.836845 -1.444846 0.621562 0.195722 4.914911 -1.055465

-0.142072 0.373792 -0.175203 0.001579 -0.004912 0.002601

MK ˆp

0.832658 -1.122394 0.448979 -0.018751 0.022654 -0.008212

MK ˆ

d

0.063902 0.035825 -0.019325 0.026688 -0.042239 0.026225

0.152822 -0.212446 0.069888 -0.275392 0.780534 -0.414427

-0.281618 0.385927 -0.132506 0.682902 -1.706131 0.850398

-0.057527 0.071206 0.059622 -0.568180 0.081026 0.176679

2.330662 -2.876210 0.735524 2.708334 -1.507766 0.063388

Table 7. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 9 MK ˆp

MK ˆ

i

8.996517 -0.163220 0.068099 0.005449 1.177551 -0.053453

-0.605634 1.035979 -0.440535 8.763461 -4.797556 0.871060

1.256875 -2.316475 1.022770 0.543840 5.980280 -1.428947

-0.259264 0.619581 -0.291915 0.003918 -0.010142 0.005280

0.988444 -1.544295 0.659486 -0.024746 0.034360 -0.013569

Table 8. Matrices coeffients of the motor velocity tuning rules when ω ˆ d,v = 10 MK ˆp

MK ˆ

i

10.058496 -0.257929 0.111149 0.034812 1.500984 -0.082965

-0.854067 1.543944 -0.676386 9.481141 -5.778026 1.143977

1.833301 -3.549797 1.604185 1.048431 6.918657 -1.799310

-0.489032 1.106527 -0.522677 0.009575 -0.022352 0.011330

1.374578 -2.438658 1.091335 -0.037039 0.059535 -0.025308

Fig. 7. Nyquist plot of the link position open loop transfer function (ˆ ωd,v = 10 and ω ˆ d,p = 1). The dashed circle indicates the value Ms = 1.4. Fig. 8. FourByThree Series Elastic Actuated robot Parameter Jm k h fm Jl (no payload)

Value 0.883024 [kgm2 ] 735.343764 [Nm/rad] 7.8928 [Nm/(rad/s)] 6.966257 [Nm/(rad/s)] 0.650 [kgm2 ]

Reduced value [-] [-] 0.301 0.27 0.736

Table 10. Dynamics parameters of Joint 1 4. EXPERIMENTAL RESULTS

Fig. 6. Nyquist plot of the motor velocity open loop transfer function (ˆ ωd,v = 10). The dashed circle indicates the value Ms = 1.4. ˆ fˆm , and Jˆl . Thus, the 0.6] depending on the values of h, proposed tuning rules provide the fastest primary loop. The tuning rules are expressed by means of the following equation:   1   ˆ p = 1 fˆm fˆ2 h ˆ h ˆ 2 M ˆ  Jˆl  K m Kp ˆ2  Jl  1 (18)   ˆ d = 1 fˆm fˆ2 h ˆ h ˆ 2 M ˆ  Jˆl  K m Kd Jˆl2 1 Tˆf = 5 The resulting values are in Table 9. Figure 7 shows that the maximum sensitivity is limited to 1.4 demonstrating the effectiveness of the tuning rules to ensure the required robustness. 224

The proposed tuning rules are tested on a three degree-offreedom series elastic actuated robot, designed as a part of the FourByThree European project (FourByThree, 2018). The actuators are composed by a brushless DC motor, an harmonic driver and an elastic element made by a series of coil springs. The actuator has two encoders: one measures the motor position after the harmonic drive while the second measures the angle after the spring element. The current loop is implemented inside a FPGA board and it is considered as a unitary gain for the position and velocity controllers. The torque reference signal is sent to the FPGA board by using the ROS operating system framework (ROS, 2018). Joint 1 has been used to show the effectiveness of the proposed method during a trajectory tracking task. The dynamics parameters of the robot are in Table 10. Moreover, there is a static friction of Cm = 2.3 [Nm] and a backlash of 0.015 [rad] (around 1 [deg]) which are considered as model mismatches, as well as the coupling effect caused by the other links. The control performance are tested by means of two trajectories. In particular, a trapezoidal motion profile from position 0 [rad] to 1.5708 [rad] followed by another trapezoidal motion profile from 1.5708 [rad] to 0 [rad]

IFAC PID 2018 Ghent, Belgium, May 9-11, 2018

Parameter Jl (no payload) Jl (m= 0.5 [kg]) Jl (m= 1 [kg]) Jl (m= 2 [kg]) Jl (m= 5 [kg])

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Value 0.650 [kgm2 ] 0.740 [kgm2 ] 0.786 [kgm2 ] 0.924 [kgm2 ] 1.514 [kgm2 ]

REFERENCES

Reduced value 0.736 0.8385 0.890 1.047 1.714

Table 11. Link inertia values with disk payloads. |¯ e|p 0.0014 [rad] 0.0019 [rad] 0.0018 [rad] 0.0012 [rad] 0.0020 [rad]

Payload Jl (no payload) Jl (m= 0.5 [kg]) Jl (m= 1 [kg]) Jl (m= 2 [kg]) Jl (m= 5 [kg])

|¯ e |v 0.0548 [rad/s] 0.0471 [rad/s] 0.0720 [rad/s] 0.0498 [rad/s] 0.0648 [rad/s]

ce 0.3055 [Nm] 0.3073 [Nm] 0.4243 [Nm] 0.3466 [Nm] 0.3902 [Nm]

Table 12. performance indexes for the different Jl values. 1.6 1.4

Link position [rad]

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

Time [s]

0.015

Link error [rad]

0.01

0.005

0

-0.005

-0.01

-0.015 0

0.5

1

1.5

2

2.5

225

Time [s]

Fig. 9. Link position and error trends with different Jl values. Blue line: no payload. Magenta line: 0.5 [kg] payload. Ocher line: 1 [kg] payload. Violet line: 2 [kg] payload. Green line: 5 [kg] payload. has been considered. The motor velocity cutoff frequency has been chosen equal to ω ˆ d,p = 3, which corresponds to ωd,p = 86.5735 [rad/s]. In order to demonstrate the robustness of the controller, four different disk payloads have been attached to the robot flange, changing Jl as shown in Table 11. For each test, the performance are evaluated by computing the link position and the motor velocity mean absolute errors (respectively, |¯ e|p and |¯ e|v ) and the control effort ce defined as  1 T ce = |∆τ |dt T t=0 The obtained values are in Table 12, while Figure 9 shows the position and the torque trends for each Jl values. It is possible to note the effectiveness of the proposed method to follow the required set-point signal in presence of an unknown payload. In fact, the mean absolute errors are limited also in the most critical situation (namely, the 5 [kg] payload case). 5. CONCLUSIONS Robust tuning rules for Series Elastic Actuator control have been presented. The tuning rules are designed for a PD-PI cascade control and allow the online computation of the robust control parameters for each value of the link inertia. Experimental results show the effectiveness of the methodology on a three degree-of-freedom series elastic actuated robot. ACKNOWLEDGEMENTS The work is partially supported by FourByThree Project H2020-FoF-06-2014-737095. 225

˚ Astr¨om, K.J. and H¨agglund, T. (2004). Revisiting the Ziegler-Nichols step response method for PID control. Journal of Process Control, 14(6), 635–650. doi: 10.1016/j.jprocont.2004.01.002. Calanca, A. and Fiorini, P. (2017). Impedance control of series elastic actuators based on well-defined force dynamics. Robotics and Autonomous Systems, 96, 81– 92. doi:10.1016/j.robot.2017.06.013. Calanca, A., Muradore, R., and Fiorini, P. (2017). Impedance control of series elastic actuators: Passivity and acceleration-based control. Mechatronics, 47, 37–48. doi:10.1016/j.mechatronics.2017.08.010. Dos Santos, W.M. and Siqueira, A.A.G. (2014). Impedance control of a rotary series elastic actuator for knee rehabilitation, volume 19. IFAC. doi: 10.1109/ICAR.2013.6766567. FourByThree (2018). Fourbythree project homepage. http://http://fourbythree.eu/. Accessed: 2018-0131. Guiochet, J., Machin, M., and Waeselynck, H. (2017). Safety-critical advanced robots: A survey. Robotics and Autonomous Systems, 94, 43–52. doi: https://doi.org/10.1016/j.robot.2017.04.004. Maleki, S., Parsa, A., and Ahmadabadi, M.N. (2016). Feed-forward learning with frequency adaptation towards the control of series elastic actuators. In 2016 4th International Conference on Robotics and Mechatronics (ICROM), 196–201. doi:10.1109/ICRoM.2016.7886846. Maurtua, I., Pedrocchi, N., Orlandini, A., d. G. Fernndez, J., Vogel, C., Geenen, A., Althoefer, K., and Shafti, A. (2016). Fourbythree: Imagine humans and robots working hand in hand. In 2016 IEEE 21st International Conference on Emerging Technologies and Factory Automation (ETFA), 1–8. doi:10.1109/ETFA.2016.7733583. Misgeld, B.J., Hewing, L., Liu, L., and Leonhardt, S. (2017). Robust gain-scheduled control of variable stiffness actuators. IFAC-PapersOnLine, 50(1), 8804–8809. doi:10.1016/j.ifacol.2017.08.1535. Paine, N., Oh, S., and Sentis, L. (2014). Design and control considerations for high-performance series elastic actuators. IEEE/ASME Transactions on Mechatronics, 19(3), 1080–1091. doi:10.1109/TMECH.2013.2270435. Ragonesi, D., Agrawal, S., Sample, W., and Rahman, T. (2011). Series elastic actuator control of a powered exoskeleton. In 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 3515–3518. doi:10.1109/IEMBS.2011.6090583. ROS (2018). ROS homepage. http://wwww.ros.org. Accessed: 2018-01-31. Simoni, L., Beschi, M., Legnani, G., and Visioli, A. (2017). Modelling the temperature in joint friction of industrial manipulators. Robotica, 122. doi: 10.1017/S0263574717000509. Verstraten, T., Beckerle, P., Furn´emont, R., Mathijssen, G., Vanderborght, B., and Lefeber, D. (2016). Series and Parallel Elastic Actuation: Impact of natural dynamics on power and energy consumption. Mechanism and Machine Theory, 102, 232–246. doi: https://doi.org/10.1016/j.mechmachtheory.2016.04.004.