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Control Engineering Practice 115 (2021) 104887
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Contents lists available at ScienceDirect
Control Engineering Practice
journal homepage: www.elsevier.com/locate/conengprac
isturbance observer-based LPV feedback control of aπ-DoF roboticanipulator including compliance through gain shifting
lberto San-Miguel β, VicenΓ§ Puig, Guillem AlenyΓ nstitut de RobΓ²tica i InformΓ tica Industrial (CSIC-UPC), Llorens i Artigas, 4-6, 08028 Barcelona, Spain
R T I C L E I N F O
eywords:obotic systeminear parameter varyingnknown-input observerhifting paradigmobustnesservice robot
A B S T R A C T
This paper proposes a control scheme for a π-DoF robotic manipulator in a joint-regulation motion problem,dealing with disturbances (as e.g. exogenous forces, unmodelled dynamics) that hinder task fulfilment, and alsoconsidering that not all the required states are available online. Existing literature tackles this problem throughDisturbance Observer (DO) strategies which imply complex analysis and design methods or introducing strongassumptions. Conversely, we propose to formulate the system as a Linear Parameter Varying (LPV) model,which allows a straightforward application of the existing linear control structures but without neglectingits non-linear behaviour. We make use of the Robust Unknown Input Observer (RUIO) to obtain (for notmeasurable states) a decoupled estimation from the unknown disturbance effects, and improve its noisereduction capabilities through the new optimal RUIO design. The robotic manipulator is controlled with astate-feedback control law that, making use of the LPV paradigm, has been designed to seamlessly avoidtorque saturation on manipulatorβs joints through a gain shifting strategy that modifies its compliant behaviour.Stability and performance requirements are imposed in both RUIO and state-feedback control synthesisproblems stated using the LMI framework, applying Polyaβs theorems on positive forms of the standard simplexto reduce its overall conservatism. Experiments, using a simulated head system of the TIAGo robot as a testbedin various realistic scenarios, show the benefits when compared to the existing joint-independent PD controlstrategy and state-of-art EKF disturbance estimation.
. Introduction
Leaning on the recent advances in Artificial Intelligence and Com-uter Vision, robotic platforms are being increasingly introduced intondustrial and domestic environments, where their workspace, andven the tasks to be performed, are shared with humans. For supportingβhuman-in-the-loopββ, a key factor is to ensure their dependability underny possible scenario that might arise (Alami et al., 2006), being onef them the existence of unknown effects (disturbances) that mightinder task fulfilment. Thus, in this work, we address the motionroblem of a robotic manipulator under the effect of exogenous a-priorinknown disturbances that should be rejected or compensated on-line.oreover, we are going to assume that some of the required states are
ot measurable, as e.g. due to faulty behaviour. We are focusing on-Degrees of Freedom (DoF) serial robotic manipulators for regulation
asks in their joint space, i.e. ββpoint-to-pointββ movements.Disturbances in robotic systems correspond to multiple phenomena
rom internal sources (friction, unmodelled dynamics) or interactionith the environment (collisions), presenting different behaviours and
haracteristics, usually non-predictable or difficult to model. Greatfforts have been made within the research community to tackle this
β Corresponding author.E-mail address: [email protected] (A. San-Miguel).
problem, some of them aiming at making the control strategy robustagainst disturbances (as e.g. with SMC). These strategies are definedunder the category of Disturbance Observers (DO), and have been usedin a wide range of platforms and for multiple purposes: from lowerlimb exoskeletons for estimating the exerted torque by the wearer(Mohammed, Huo, Huang, RifaΓ―, & Amirat, 2016) to humanoid robotsto compensate for unmodelled dynamics (Bae & Oh, 2017) or evento improve boundary control approach in flexible manipulators (Zhao,Ahn and Li, 2019; Zhao & Liu, 2020). In robotic manipulators, theyhave been used for both single (Agarwal & Parthasarathy, 2016) andmulti-agent systems (Ren, Du, Li, & Shi, 2018). Moreover, its integra-tion with the control strategy is currently being explored, for examplein Kim, Kim, Kim, Sim, and Park (2018), where it is presented as a newconcept for achieving a compliant behaviour under external contacts.On the other hand, another approach is to make a design robust againstdisturbances (He, Wang, He, Yang, & Kaynak, 2020), but this usuallyinvolves making certain assumptions on the disturbance characteristics.
Due to the highly non-linear phenomena in robotic manipulators,the use of linear analysis and design techniques for DO has beenlimited to certain standard strategies, such as linearisation around
ttps://doi.org/10.1016/j.conengprac.2021.104887eceived 9 November 2020; Received in revised form 2 July 2021; Accepted 6 Julvailable online 23 July 2021967-0661/Β© 2021 The Authors. Published by Elsevier Ltd. This is an open accesshttp://creativecommons.org/licenses/by-nc-nd/4.0/).
y 2021
article under the CC BY-NC-ND license
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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the operating point. This is the case of the Extended Kalman Filter(EKF), which can deliver optimal performance but only if the sys-tem behaves linearly around the operation point, as in Mohammadi,Tavakoli, Marquez, and Hashemzadeh (2013). During the last decades,several techniques have emerged to deal with the control of non-linearsystems, such as, e.g. Sliding Mode Control (SMC), Linear ParameterVarying (LPV) or TakagiβSugeno (TS) gain-scheduling control, amongother. In this work, we focus on the LPV paradigm (Shamma, 2012),which consist on characterising systems through a set of parametersdefined by exogenous (or endogenous) signals in a gain-schedulingfashion, allowing a straightforward extension of existing linear controltechniques to non-linear systems. Its success has been also motivatedby the increasing use of Linear Matrix Inequalities (LMI) together withthe application of Lyapunov theory to formulate multiple analysis anddesign problems in control (e.g. stabilisation and β) for LPV-suitableformulations. In these cases, as including all the reachable states leadsto an infinite number of constraints, LPV systems are usually describedby confining all the possible trajectories of the varying parametersdefined by a set of vertices, leading to the polytopic LPV formulationthat considers only a family of vertex systems. Polytopic LPV togetherwith LMI provide a systematic approach to guarantee off-line certainproperties, in contrast to ad hoc construction of Lyapunov candidatesnd trial and error tuning of the controller parameters required by otheron-linear control techniques (Zhao, He and Ahn, 2019).
DO exploit all the measurements that can describe the state ofhe system, but, as any physical entity, robots might be subjected toisturbances that affect the closed-loop behaviour. Control literatureas provided many model-based observers to estimate non-availablenformation, but in the case of system disturbances, the technique haso be robust enough against their effects. A well-known approach iso assume the dynamics of the disturbance (Ha & Back, 2019), but,n a discrete-time domain, its effectiveness is affected by the samplingeriod. Considering that this work aims at being applied to real roboticlatforms that operate in discrete time, we will study the applicationf Unknown Input Observers (UIO) structures, which can obtain a statestimation disregarding the effect of disturbances and without a-priorissuming its behaviour.
Therefore, in this work, we aim at providing a solution to the dis-urbance rejection problem considering a LPV formulation of a π-DoFerial manipulator. This allows extending linear theory for analysingnd designing structures for a non-linear system, which, up to ournowledge, has not been performed for disturbance rejection problemsn robotic platforms. Solutions provided in the literature rely on modelinearisation or neglecting non-linear dynamic terms, as e.g. in Kimt al. (2018). Considering that not all the system states are measurablen-line, we have explored the Robust-UIO formulation (RUIO for short)rom Chadli and Karimi (2012) to provide a decoupled state estimationrom the disturbance effects. Using this estimation, a model-basedO generates a compensating feed-forward action within the control
cheme. To enhance its performance, RUIO noise reduction propertiesave been improved through a novel formulation of its LMI designroblem based on Kalman Riccati equations. To completely address theotion regulation problem, we have described the synthesis of an op-
imal state-feedback controller through the Linear Quadratic RegulatorLQR) LMI problem. Additionally, we have introduced a gain shiftingtrategy (Rotondo, Nejjari and Puig, 2015) aimed at seamlessly modi-ying control compliance to avoid joint torque saturation effects (whichepresents a worst-case scenario for the feedforward compensation ofisturbances).
It should be pointed out that the LPV paradigm has not been widelypplied to robotics yet, mainly due to the large number of varyingarameters of the generated models. These issues are tackled withinhe LPV literature from two different perspectives: (1) by increasinghe flexibility of problem constraints and/or (2) by modifying theodel used. Latter ones consist on reducing over-boundness of the
PV model description by reshaping the set of varying parameters
A2
Fig. 1. TIAGo robotic platform developed by PAL Robotics performing a manipulationtask.
according to their behaviour (Bruzelius, Breitholtz, & Pettersson, 2002)or approximating the system to an LPV model with a reduced numberof parameters as e.g. in Hashemi, Abbas, and Werner (2012), which isalso one of the few application examples of LPV techniques in roboticmanipulators. On the other hand, reducing problem conservatism byrelaxing constraints lies in state-of-art methods from the field of math-ematics, usually trading-off an increase in the number of constraints. Inthis work, we propose to further exploit the use of these techniques inrobotic applications by making use of Polyaβs theorems (Sala & Arino,2007) in both the RUIO and state-feedback controller LMI synthesisproblems.
This paper is organised as follows: Section 2 describes the LPV mod-elling of the manipulator and its polytopic representation. Section 3is devoted to the estimation of exogenous force effects using statevariables. The improved formulation of the RUIO and the derivation ofdesign conditions to minimise noise effects are presented in Section 4,together with the introduction of Polyaβs theorems for its synthesisproblem. In Section 5, the shifting control paradigm and its formulationunder Polyaβs method are described. Section 6 illustrates the applica-tion of the complete approach to the head of TIAGo robot, providingsimulation results under different settings. The final section concludesthe presented work and highlights future research developments.
2. System modelling
2.1. Description
The TIAGo humanoid robot (Fig. 1) developed by PAL Robotics1 hasbeen used in this paper to exemplify the described methodology andtechniques. This robotic platform is meant to operate in both domesticand industrial anthropic domains as a service robot, featuring a wideset of capabilities related with navigation, perception and humanβrobotinteraction, making it suitable for research purposes. It is also fully-integrated within the well-known Robotic Operating System (ROS)ecosystem,2 which offers realistic simulations through Gazebo3 using thesame communication interface for controlling the real robot.
Particularly, the focus has been put in the 2-DoF TIAGo headystem. This responds to an existing issue with its pose control strategy,
1 TIAGo robotic platform by PAL Robotics: http://tiago.pal-robotics.com/Accessed August 18, 2021).
2 ROS software development framework for robotics https://www.ros.org/Accessed August 18, 2021).
3 Gazebo http://gazebosim.org/ simulator for robotic applications (Accessedugust 18, 2021).
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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Table 1Description and value of inertial and distance parameters of the TIAGo head subsystem
Parameter Description Value
πΌ1 Inertial tensor of first link diag(1.192, 1.402, 0.889) β 10β3 [kg m2]πΌ2 Inertial tensor of second link diag(4.620, 4.861, 3.132) β 10β3 [kg m2]π1 Mass of first link 0.622 [kg]π2 Mass of second link 0.661 [kg]
πΏ Dist. in 1οΏ½οΏ½1 between {1} and {2} 0.098 [m]π·1,π§ Dist. of CoG1 in 1οΏ½οΏ½1 w.r.t. {1} 0.072 [m]π·2,π₯ Dist. of CoG2 in 2οΏ½οΏ½2 w.r.t. {2} 0.047 [m]π·2,π¦ Dist. of CoG2 in 2π2 w.r.t. {2} 0.055 [m]
based on joint-independent PD control, which was not able to reachdesired positions when a device (e.g. an additional RGB-D camera) wasattached, i.e. under the effect of a force associated to a mass. Headpositioning is crucial to ensure the required vision and 3D mappingfeatures of the TIAGo robot, becoming paramount to overcome thisissue. Either way, a valid description of the methods is given in thiswork for any π-DoF mechanism with rotational joints, such as a serialrobotic manipulator.
2.2. Analytical model
As any model-based approach, a set of analytical expressions thatdescribe the behaviour of the system has to be determined. For any π-DoF robotic manipulator with rotational joints, applying well-knownNewtonβEuler formulation (Craig, 2009), joint torque vector π can bedefined as a function of the joint acceleration π, velocity οΏ½οΏ½ and positionπ vectors:
π = π(π) π + π(π) [οΏ½οΏ½οΏ½οΏ½] + π(π) [οΏ½οΏ½2] +π(π) (1)
where π(π) β RπΓπ is the mass matrix, π(π) β Rπ the gravity effects,π(π) β RπΓπ the centrifugal coefficients, and π(π) β RπΓπ(πβ1)β2
the Coriolis terms. Joint friction phenomenon has been considered asa-priori modelled, such that their influence on the control scheme iseither fully cancelled or minimised (being a source of system noisein this latter case), and therefore it has been be omitted in Eq. (1).This assumption has been made considering that there exist severaltechniques that, from different approaches, successfully tackle thisproblem as e.g. in ColomΓ©, Planells, and Torras (2015).
Following the guidelines included in Appendix A for the TIAGo headsystem, depicted in Fig. 2 along with its abstracted link scheme, weobtain the terms of Eq. (1):
π(π) =
[
πΌπ§π§1 + π2(π 2π·2,π¦ β π2π·2,π₯)2 + πΌπ₯π₯2 π 22 + πΌπ¦π¦2 π
22 0
2 2
]
, (2a)
0 πΌπ§π§2 + π2(π·2,π₯ +π·2,π¦)3
π(π) =[
2π2(π 2π·2,π₯ + π2π·2,π¦)(π 2π·2,π¦ β π2π·2,π₯) + 2(πΌπ₯π₯2 β πΌπ¦π¦2 )π 2π20
]
, (2b)
(π) =
[
0 0π2(π2π·2,π₯ β π 2π·2,π¦)(π 2π·2,π₯ + π2π·2,π¦) 0
]
(2c)
(π) =
[
0π2(π2π·2,π₯ β π 2π·2,π¦)π
]
(2d)
eing ππ and π π abbreviations for πππ (ππ) and π ππ(ππ), respectively, andfor the Earth gravity constant. Regarding remaining parameters, πΌππ
epresent inertial tensor term on the π = {π₯, π¦, π§} axis for the πth link,π the mass of the πth link and π·2,π the distance along axis π from thease of frame {2} of to the CoG2, all in SI units. The value of thesearameters are given by the manufacturer and have been gathered inable 1.
.3. LPV model
As proposed in Kwiatkowski, Boll, and Werner (2006), non-inearities can be embedded within varying parameters using the LPVaradigm. This technique provides an exact representation of theriginal non-linear system (1), and its applicability is constrained tohe following set of conditions βπ = {1,β¦ , ππ·}:
1. Varying parameter ππ(π₯1, β¦ , π₯ππ₯ ) has to be bounded in [ππ, ππ]for the (bounded) variable set {π₯1, β¦ , π₯ππ₯} and continuouswithin this interval.
2. Controllability of the system has to be ensured for any value ofππ.
From Eq. (1), expressions for joint angular accelerations can be ob-ained as function of joint velocities, positions and torques. Without lossf generality, considering a state vector π₯(π‘) = [οΏ½οΏ½1, οΏ½οΏ½2,β¦ , οΏ½οΏ½π , π1, π2,β¦ ,ππ ]π , and the joint torque vector π’(π‘) β‘ π(π‘), the following stateβspacelike representation can be obtained
οΏ½οΏ½(π‘) = π΄π (π₯(π‘)) π₯(π‘) + π΅π (π₯(π‘)) π’(π‘) + gπ (π₯(π‘)), (3)
here
π (π₯(π‘)) π₯(π‘) = βπ(π)β1[
(π(π)[οΏ½οΏ½οΏ½οΏ½] + π(π)[οΏ½οΏ½2])02πΓπ
]
+[
0πΓπ 0πΓππΌπΓπ 0πΓπ
]
π₯(π‘)
π (π₯(π‘)) = [βπ(π)β1 0πΓπ ]π
π (π₯(π‘)) = [βπ(π)β1 π(π₯(π‘)) 0πΓ1]π .
ote that using this formulation, state matrix π΄π (π‘) has not a straight-orward definition decoupled from π₯(π‘), due to Coriolis and Centrifugalffects. Therefore, its definition is not unique and has to be made a-osteriori on Eq. (1) applied to each particular system. Subscript ββπββenotes matrices for continuous-time systems, meaning its absence thathey correspond to discrete-time forms.
Eq. (3) corresponds to a stateβspace alike formulation where statend input matrices π΄π and π΅π , respectively, depend on state vector π₯(π‘).PV framework formalises this type of systems by considering stateβpace matrices as function of a set of varying parameters π· β Rππ· thatre function of some scheduling variables that are on-line measured orstimated.
Regarding the TIAGo head system, from the joint configurationpace described in (2a)β(2d), its stateβspace alike representation arises:
π (π2, οΏ½οΏ½1) =
β‘
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1 0 0 00 1 0 0
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A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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Table 2State, input torque and scheduling variables limits for TIAGo head subsystem.
Variable Minimum Maximum
οΏ½οΏ½π [rad/s] β3 3π1 [β¦] β75 75π2 [β¦] β60 45
ππ [N m] β6 6
π1 0.0055 0.0091π2 β0.0110 0.0110
where π(π,π) refers the element of matrix π in the πth row and π-column.Notice that from the comparison between (2b) and (2c), a furthersimplification can be made by considering π(1,1)(π2) = β2π(2,1)(π2).
hus, the scheduling variables have been defined as follows:
β [π1, π2] = [π1,1(π2), π2,1(π2)οΏ½οΏ½1]. (4)
Bounds for ππ can be obtained from the state variables limits, also giveny the operation conditions of the TIAGo robot, both listed in Table 2.onditions for non-linear embedding method are held for this definitionf π·: (1) ππ are bounded considering the limits of state variables,2) controllability property depends only on the selection of the outputodel, which will be addressed in Section 4, as matrix π΄ does not lose
ank for any value of ππ.
emark 1. It is worth to remark that the selection of π· is generallyade (and so in this work for the TIAGo head example) such that theinimal set of scheduling variables is obtained, as the polytopic recon-
truction has a complexity that grows with (2ππ ). As aforementioned,urther methods and criteria can be applied to reduce the complexityf a LPV polytopic representation of a system through the reductionn the number of varying parameters, obtaining an approximatedepresentation of the system.
emark 2. In LPV literature, if any of the varying parameters isunction of endogenous variables, the model is formally denoted asuasi-LPV. Although this is the case for the considered π-DoF ma-ipulator whose π· set is defined according to joint position and ve-ocities, throughout this work we refer to LPV to avoid misleadingnterpretations for non-expert readers.
.4. Polytopic LPV model
In order to get the presented techniques closer to their implementa-ion on a real robotic manipulator, all of them have been designed forhe discrete-time domain. According to Toth, Heuberger, and Van denof (2010), the discrete-time form of a continuous-time LPV modelan be obtained if systemβs matrices have been evaluated accordingo a discretisation method for a sampling time ππ . More details on the
discrete-time implementation are given in Section 6.Considering the discrete-time form of (3), the LPV representation of
the system is the following one:
π₯(π + 1) = π΄(π·) π₯(π) + π΅(π·) π’(π). (5)
Regarding gravity effects gπ (π₯(π)) in (3), they appear as an indepen-dent term and will be omitted for the upcoming modelling process,considering that in our approach this term is counteracted through afeed-forward action, which will be addressed in Section 3.
In order to avoid considering all the reachable states for the LPVsystem, all the possible trajectories of π· can be confined in a polytopeπΊ defined by a set of vertices:
π·(π) β πΊ βΆ= πΆπ{π ,β¦ , π } (6)
1 π r4
being π = {1,β¦ , 2ππ·}. Thus, the stateβspace matrices are confined inthe polytope defined by their image at these vertices:[
π΄(π·) π΅(π·)]
β πΆπ{[
π΄π π΅π]
βΆ=[
π΄(ππ) π΅(ππ)]}
. (7)
Using this approach, system representation Eq. (5) can be stated as aweighted function of their images on polytope vertex as follows:
π₯(π + 1) =2ππ·β
π=1ππ(π·)[π΄π(π·)π₯(π) + π΅π(π·)π’(π)] (8)
where ππ(π·) are the polytopic coefficients, such that:
2ππ·β
π=1ππ(π·) = 1, ππ(π·) β₯ 0 (9)
From the set of applicability conditions, we know that ππ β [ππ, ππ], and
so the set of vertices will correspond to all the possible combination ofthese limits. Thus, ππ(π·) can be defined as
ππ(π·) =ππ·β
π=1πππ(ππ0 , π
π1 ) (10)
here,
ππ(π, π) =
{
π if πππ(π, 2π) β {1,β¦ , 2π91}π else
onsidering a linear interpolation between the limits.
. Model-based DO
This work aims at obtaining a control structure that is able to adaptts behaviour according to an exogenous disturbance such that it doesot interfere the task fulfilment and/or some undesired effects arevoided. Particularly, a robotic mechanism is meant to track certainoint trajectories and might be affected by an external force at anyime, which will (presumably) hinder this task. Therefore, the chosentrategy is to compensate its effects such that an admissible trackingerformance is obtained, which requires of their estimation through aO.
Recalling the joint configuration space in Eq. (1), will exert a setf joint torques π :
+ π = π(π) π + π(π, οΏ½οΏ½) +π(π) (11)
odifying Eq. (5) as follows:
(π + 1) = π΄(π·) π₯(π) + π΅(π·) (π’(π) + π (π)). (12)
ollowing a similar approach than the one proposed in Witczak (2014)or fault estimation, and considering the discrete-time difference be-ween current system state π₯(π) at time instant π and the expected valueiven by the available model of the system considering previous state(π 9 1) using Eq. (12), the exerted torque can be estimated as follows
(π 9 1) = π΅β (π 9 1)[π₯(π) β π΄(π·(π 9 1)) π₯(π 9 1)
+ π΅(π·(π 9 1)) π’(π 9 1)], (13)
here π΅β represents the pseudoinverse of matrix π΅. Note that usinghe available information at time step π it is only possible to obtain πt time instant π 9 1. This is often solved in the literature consideringhat π (π) = π π (π 9 1), where the term π corresponds to a gain designarameter. In this work, it has been considered equal to a identityatrix, i.e. a one time-step predictor.
It is worth to mention that through this approach π (π) is generatedsing joint position and velocities from the system, considering thevailable sensors on the TIAGo head system (and the vast majority of
obotic mechanisms)
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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4. State estimation
As the generation of π (π) depends on the system states, thesemeasurements have to be available online during the complete ex-ecution of a task. Although this premise is generally accepted, therobotics community points out that as robotic platforms have to operateuninterruptedly during long periods, and considering their inherentcomplexity, they are prone to different types of adverse events thatshould be overcome. Thus, in this work we have extended the problemto consider a limited number of measurements during the whole task.In the particular case of the TIAGo head, system output π¦ has beenconsidered as follows:
π¦ = πΆ π₯ = [οΏ½οΏ½2, π1, π2]π . (14)
where matrix πΆ β Rππ¦Γππ₯ . In this case, πΆ is constant disregardingthe operating point of the system, so no LPV formulation is required.Recalling the conditions described in Section 2 to define the schedul-ing vector π· through the non-linear embedding approach, the givendefinition of πΆ ensures the observability property for any value of ππ.
.1. Robust unknown input observer
The UIOs allow to obtain a decoupled estimation from the effects ofn unknown input, disregarding its behaviour. Particularly, the Robust-IO for TakagiβSugeno models (RUIO for short) presented in Chadlind Karimi (2012) has been adapted to this problem, making use ofxisting strong analogies between TakagiβSugeno and polytopic LPVepresentation (Rotondo, Puig, Nejjari and Witczak, 2015).
The RUIO assumes the following polytopic LPV discrete-time stateβpace representation
(π + 1) =2ππ·β
π=1ππ(πΉ )[π΄ππ₯(π) + π΅π π’(π) + π π π (π)], (15a)
π¦(π) = πΆ π₯(π), (15b)
where π (π) β Rππ stands for the unknown input vector, being ππ thenumber of unknown inputs considered. Matrix π π β Rππ₯Γππ representshe influence of π (π) in the system behaviour. These matrices areetermined according to the desired effect to represent, and provide thexistence of a solution as described in Chadli and Karimi (2012). Re-alling the stateβspace form including the exerted torque from Eq. (12)nd defined output model in Eq. (14):
π β π΅π.
he structure for the RUIO is:
(π + 1) =2ππ·β
π=1ππ(π·)[πππ§(π) + πΊππ’(π) + πΏππ¦(π)] (16a)
π§(π) = οΏ½οΏ½(π) + πΈπ¦(π) (16b)
where π§(π) corresponds to RUIOβs state vector, with the same dimen-sions of π₯(π), which embeds the given estimation οΏ½οΏ½(π). Matrices ππ,πΊπ, πΏπ and πΈ are the observer gains to be designed by ensuring anasymptotic stability of the observer dynamics, i.e. the estimation errorconverges to zero as time tends to infinite disregarding the unknowninputs magnitude. Fig. 3 graphically represents the RUIO structurewithin the complete architecture.
Remark 3. The original formulation from Chadli and Karimi (2012)does differentiate between ββunknown inputsββ and ββdisturbancesββ. Inthis paper, we refer to the first type as either ββdisturbancesββ or ββun-known inputsββ, and the second type corresponds to ββnoiseββ. We usethis nomenclature to help the reader link these terms to the realimplementation.
5
4.2. Optimal RUIO
One of the main contributions of this work is the introduction ofconditions for optimal design based on Riccati equations of the Kalmanfilter for the RUIO. In the application to the real platform, processand sensor noise cannot be avoided, therefore their effects have tobe minimised such that do not hinder the estimation of the unknownstates. This formulation allows to introduce process and sensor noisecovariance through matrices ππ and π π, that define the quadraticriterion π½π according to the estimation error π(π) = π₯(π) 9 οΏ½οΏ½(π):
π =ββ
π=0(π(π)π ππ π(π) + ππ (π)πΎππΆ π π πΆππΎπ
π π(π)) < πΎπ (17)
roposition 1 (Optimal RUIO Design). Given ππ = πππ = π»ππ π»π > 0 and
π = π ππ > 0 matrices, and the optimal performance upper bound πΎπ > 0, thebserver from Eq. (16) converges asymptotically to the state of the polytopiciscrete-time LPV model from Eq. (15) if there exists matrices sets
π = {π1,β¦ , π2ππ· }; ππ β Rππ₯Γππ₯ β£ ππ = πππ > 0 β π = 1,β¦ , 2ππ· ,
(18a)
= {π1,β¦ ,π2ππ· }; ππ β Rππ₯Γππ¦ , (18b)
π = {π1,β¦ , π2ππ· }; ππ β Rππ₯Γππ¦ , (18c)
such that the following conditions hold βοΏ½οΏ½ β I(2,2ππ· ):
β‘
β’
β’
β’
β’
β£
βππ1 ππ1π΄π1 + ππ1πΆπ΄π1 βππ1πΆ ππ1π»ππ ππ1
(β) βππ2 0 0(β) 0 βπΌ 0(β) 0 0 βπ β1
π
β€
β₯
β₯
β₯
β₯
β¦
< 0, (19a)
(ππ1 + ππ1πΆ)π π1 = 0, (19b)[
πΎπ πΌ πΌπΌ ππ1
]
> 0 (19c)
where the symbol (β) denotes the transpose of the element in the symmetricposition, and
I(π,β) = {οΏ½οΏ½ = (π1,β¦ , ππ) β Nπ |1 β€ ππ β€ β βπ = 1,β¦ , π}.
The obtained solution for these matrices defines RUIOβs gains for π =1,β¦ , 2ππ· :
πΈ = πβ1π ππ, (20a)
πΊπ = (πΌ +πβ1π πππΆ)π΅π, (20b)
ππ = (πΌ +πβ1π πππΆ)π΄π βπβ1
π πππΆ, (20c)
πΏπ = πβ1π ππ βπππΈπ. (20d)
roof. The proof is given in Appendix B.
Note that I(π,β) is a multi-index notation to represent all the combina-ions of β elements that can take (natural) values from 1 to π. Therefore,I(2,2ππ· ) represents all the possible pairwise combinations between the2ππ· vertex imposed by RUIO condition in Eq. (19a). For the π-DoFmechanism case, the number of states ππ₯ = 2π , and remaining termsare defined by the number of varying parameters and the chosen outputmodel.
4.3. Polyaβs theorems formulation for the RUIO
In the proposed extension of the RUIO method including optimalityconditions, some feasibility issues could arise depending on the numberof polytope vertices and the similarity between them (i.e. the variabilityin terms of the scheduling parameters image). Also equality constraintsfrom Eq. (19b) have been found to highly increase complexity ofthe overall problem, as it imposes a strict relation between terms.
Although a set of positive definite variables π = {π1,β¦ , ππ} has been
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
Fig. 3. Schematic representation of the complete control architecture for the polytopic LPV formulation of the TiaGO head system, including the Robust Unknown Input Observer(RUIO), the State-feedback Shifting Control and Active Compensation strategies.
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π
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R
included in the formulation to reduce overall problem conservatism,we have additionally applied Polyaβs theorems on positive forms of thestandard simplex to obtain sufficient conditions, as proposed by Salaand Arino (2007). This solution introduces a new set of constraintsthrough dimensionality expansions of the LMI conditions. These newinequalities are obtained as sums of matrices evaluated in a certainindex combination according to a multi-index vector in I(π,2ππ· ), whereπ represents a complexity parameter of the method: larger values implygreater number of conditions, increasing the associated computationburden but reducing the overall conservatism of the problem. A briefdescription of the method has been included in Appendix C. Thus,combinatorial constraints from Eq. (19a) can be substituted by thefollowing one for a given π β₯ 2:
β
πβ(οΏ½οΏ½)
β‘
β’
β’
β’
β’
β£
βππ1 ππ1π΄π + ππ1πΆπ΄π1 βππ1πΆ ππ1π»ππ ππ1
(β) βππ2 0 0(β) 0 βπΌ 0(β) 0 0 βπ β1
π
β€
β₯
β₯
β₯
β₯
β¦
< 0
β οΏ½οΏ½ β I+(π,2ππ· ) (21)
5. Shifting control design
5.1. Feedback control law
The RUIO gives a decoupled estimation for system states from theeffects of an unknown exogenous force exerted on the system, allowingto use an state-feedback control strategy. For the polytopic discrete-time LPV system, which can be derived from Eq. (8), the correspondingstate-feedback control law is:
π’ππ(π) = βπΎππ(π·) οΏ½οΏ½(π) = β2ππ·β
π=1ππ(π·)πΎππ,π οΏ½οΏ½(π). (22)
Joint trajectory references are assumed to be given by an externalplanning module, according to a specific high-level objective, as e.g. forthe TIAGo head system, an object tracking algorithm based on live-image input. Associated control actions to the trajectory referenceshas been computed by means of the Feedforward Scaling matrix (π·),
a6
defined for the considered system:
(π·) =2ππ·β
π=1ππ(π·)[πΆπ (πΌ + π΅π πΎππ,π β π΄π)β1 π΅π]β1. (23)
Note that matrix πΆπ is determined considering the system states to beregulated through the control law, in this case, joint reference positionsgiven by vector ππ(π) = [π1(π),β¦ , ππ (π)]π . Thus, the input controlaction π’π for the system can be obtained as
π’π (π) = π’ππ(π) + (π·)ππ(π). (24)
Besides stability, control gains can be obtained according to certaincriteria regarding desired performance. Also from the Riccati equations,the well-known quadratic optimal control, referred in the literatureas Linear Quadratic Regulator (LQR) problem, aims at obtaining acontroller according to the quadratic index π½π similarly defined as inEq. (17)
π½π =ββ
π=0(π₯(π)π ππ π₯(π) + π₯π (π)πΎππ π π πΎπ
ππ π₯(π)) < πΎπ . (25)
There is not a straightforward definition to be considered for tuning ofππ and π π like in the observer synthesis problem. Generally, LQR de-sign approach is used to reduce the oscillatory behaviour of the systemresponse by penalising the magnitude increase of states and control ac-tions. In this work, it has been done through a weighted Brysonβs Rule,which gives their definition according to the squared maximum admis-sible values of variables ππ and user-defined
eights πΏπ:
π = diag{ πΏ1π1, β― , πΏ2π
π2π
}
, π π = diag{
πΏ2π+1π2π+1
,β¦ ,πΏ2π+ππ’π2π+ππ’
}
, (26)
where ππ’ corresponds to the number of control inputs for the system.The selection of ππ has been done for the TIAGo Head system recallingthe limits imposed by the real platform and summarised in Table 2 forstate variables and control input actions. The selection of πΏπ will beetailed in Section 6.1, as it corresponds to a tuning procedure on theystem.
emark 4. Brysonβs rule defines ππ and π π on desirable maximumdmissible values, as LQR conditions do not restrict state and action
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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values but allow design the controller ensuring stability accordingto π½ .
5.2. Active compensation and shifting paradigm
As aforementioned, the set of torques exerted by the exogenousforce π is meant to be actively compensated without jeopardising taskfulfilment. Recalling the stateβspace alike representation from Eq. (3),gravity effects denoted by g where omitted to construct the polytopicLPV model, on the assumption that this term will be also counteracted.Thus, using the decoupled state estimation computed by the RUIO οΏ½οΏ½(π),the counter values of both π and g can be obtained and activelynjected in π’π such that joint torque vector π is generated:
(π) β‘ π’(π) = π’π (π) β [π (π) + g(π)]. (27)
Active compensation relieves the state-feedback control strategy ofenerating the required effort to simultaneously track desired jointrajectories and compensate for additional effects. This allows to ensureome stability and desirable performance properties on the controllerynthesis problem, considering the nominal operation instead of allhe possible ones that might arise from the effects of unknown inputs,hich in practice will not be tractable. However, this paradigm under-
ies the assumption that control strategy and compensation mechanisman operate individually disregarding any further constraint implyingoth of them. In this work, we have considered actuator saturation,hich imposes that the control torque π(π) obtained as the sum ofoth the active compensation and state-feedback actions has to beounded in order to avoid ill-posing control performance and stability,r damaging the actuator itself. Therefore, we have introduced a designtrategy where the controller acknowledges this saturation effects bydapting control compliance according to the magnitude of the effectso be compensated, following the shifting paradigm as applied by Ruiz,otondo, and Morcego (2019).
This paradigm for LPV formulations considers an augmentation ofhe varying parameter set π· with a new one πΉ β RππΉ , hencefortheferred as shifting parameter set, used to alter the state-feedbackontrol law behaviour in Eq. (22) as follows
ππ = βπΎππ(π·) οΏ½οΏ½(π) = β2ππ·β
π=1ππ(π·)
2ππΉβ
π=1ππ(πΉ )πΎππ,ππ οΏ½οΏ½(π) (28)
where ππ(πΉ ) are the associated polytopic weights defined followingonditions from Eq. (9). This structure has been also graphically de-icted within the complete control architecture in Fig. 3. Shiftingariables have to be defined considering the phenomena on whichhe control behaviour has to be modified, in this particular case, theagnitude of the compensated effects with respect to the torque limits.efining the input saturated action for the πth actuator
π =
{
sign(π’π) π’π if abs(π’π) > π’ππ’π if abs(π’π) β€ π’π,
(29)
here π’π is the saturation limit (assuming a symmetry on the effect),hifting variable ππ is defined according to the difference between thexerted torque and gravity effects and input saturation limits, scaleduch that ππ β [0, 1]:
π =π’2π β [π (π) + g(π)]2
π’2π. (30)
Defining the behaviour associated with the shifting strategy relieson the synthesis problem of state-feedback gains πΎππ,ππ. Following thepolytopic approach, different performances have to be associated withthe limits of ππ such that any value within these bounds has the cor-esponding ββinterpolatedββ one. To avoid reaching actuator saturationimits, it is desired that as the corresponding torque of the activelyompensated effects increases, the state-feedback control compliance isncreased, i.e. control effort is decreased through a slower response of
7
the closed-loop system. Control theory has proven for the LPV systemsthe influence of placing the systemβs poles in some particular regionson the performance characteristics (Ruiz et al., 2019). Particularly,system response can be defined by its time constant ππ , for a givensampling time ππ , which can be determined by the magnitude of itsoles. Thus, in this work, pole placement conditions that constraintoles to certain regions have been introduced into the control synthesisroblem to define the system response, depending on the values ofhifting variables.
In order to ease a straightforward applicability of this method, polelacement has been defined in circular regions according to the settlingime of the system ππ ππ‘π‘,π for a certain band with a width of π % referencealue. Thus, given desired maximum and minimum values, respectivelyπ π ππ‘π‘,π % and π π ππ‘π‘,π %, their radii ππ and centre ππ in the complex planecan be defined as follows βπ = 1,β¦ , 2πΉ :
π =πβπΌπππ β πβπ½πππ
2, (31a)
ππ = ππ + πβπ½πππ , (31b)
where
πΌπ =π|π| β ππ (1 β |π|)
π, π½π =
π|π| β ππ (1 β |π|)π
,
being
ππ = βlog(π %)ππ ππ‘π‘,π %
, π =ππ β ππ
2Β± π.
The term ππ = {π β {1,β¦ , 2ππΉ } βΆ πππ(π, 2ππΉ ) β {1,β¦ , 2ππΉβ1}},eing |ππ| its cardinality, and the user-defined variable π determine theverlapping between fastest and slowest LMI regions. Note that all theseegions are defined such that closed-loop system poles lie in the Right-alf Plane (RHP) of the discrete complex plane to avoid undesiredscillatory behaviours (Isermann, 2013).
The LMI framework gives a general description of regions in theomplex plane under the definition of D-stabilisation (Chilali, Gahinet,Apkarian, 1999). The Dπ,π circular regions considered in the shifting
trategy are characterised by matrices
π,π =[
βππ ππππ βππ
]
, ππ,π =ππ =[
0 10 0
]
.
Thus, under the shifting control strategy, LQR design problem (Os-tertag, 2011) defined βοΏ½οΏ½ β I(2,2ππ· ) has to be stated βπ = 1,β¦ , 2ππΉ asfollows:β‘
β’
β’
β’
β’
β£
βπ π π΄ππ1 βπππ1 ,ππ΅ππ2 ππ»π
π π ππ1 ,π
(β) βπ 0 0(β) 0 βπΌ 0(β) 0 0 βπ β1
π
β€
β₯
β₯
β₯
β₯
β¦
< 0, (32a)
[
πΎπ πΌ πΌπΌ π
]
> 0 (32b)
and the corresponding LMI pole placement conditions have to beincluded intro the control synthesis problem:
πΏπ,π β π +ππ β (π΄π1π βππ1 ,ππ΅π2 )
+πππ β (π΄π1π βππ1 ,ππ΅π2 )
π < 0, (33)
being πΎππ,ππ = ππ,ππ β1.
Remark 5. The presented shifting approach assumes that the sumof gravity and exogenous force effects to be compensated will notbe greater than the actuator saturation limits, which will limit theiroperation to the linear region. Although it might seem like a strongassumption to be made, robotic manipulators are generally designedtowards increasing their payload-weight ratio, i.e. reducing its inertiawhile increasing the maximum load it can handle. Also, the design pro-cess of robotic manipulators generally considers the maximum forces to
be exerted by the robot according to their context of operation.
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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Remark 6. Under the Shifting Control paradigm, the definition givenin Eq. (23) for the Feedforward Scaling matrix (π·) has to be slightlymodified to include terms πΎππ,ππ, akin to reformulation of the state-feedback control law in Eq. (28) from Eq. (22).
Remark 7. Design problems of the observer (RUIO) and controlstrategy (State-Feedback Shifting Controller) have been considered sep-arately, which is customary in control implementation problems, ase.g. in Rotondo, Nejjari, and Puig (2013). However, the stability guar-antees would be only guaranteed either by proving the separationprinciple or considering a joint design, being any of these two optionsinvestigated in a future research.
5.3. Polyaβs theorems formulation for the polytopic state-feedback shiftingcontroller
The LPV description given in Eq. (5) sets an input matrix π΅ func-tion of π·, which under the polytopic approach depicted in Eq. (8)corresponds to a set of π΅π matrices. Thus, conditions from Eq. (32a)including the closed-loop form of the system, involve the combinationof every π΅π with all ππ or ππ,π, meaning that each gain has to be robustwith respect to every possible input matrix, increasing the overall con-servatism of the design problem. Moreover, control shifting approachincludes additional pole placement conditions from Eq. (33), whichinvolve same combinations. Similarly to Section 4.3, we can relaxthe LMI conditions on the design process applying Polyaβs theorems.Therefore, for π β₯ 2 for conditions from Eqs. (32a) and (33) we obtainthe following sufficient conditions β οΏ½οΏ½ β I+(π,2ππ· ) and βπ = 1,β¦ , 2ππΉ :
β
πβ(οΏ½οΏ½)
β‘
β’
β’
β’
β’
β£
βπ π π΄ππ1 βπππ1 ,π
π΅ππ2 ππ»ππ π π
π1 ,π(β) βπ 0 0(β) 0 βπΌ 0(β) 0 0 π β1
π
β€
β₯
β₯
β₯
β₯
β¦
< 0 (34a)
β
πβ(οΏ½οΏ½)
πΏπ,π β π +ππ β (π΄π1π βππ1 ,ππ΅π2 )
+πππ β (π΄π1π βππ1 ,ππ΅π2 )
π < 0. (34b)
Remark 8. In the controller synthesis problem, a common variableπ is used for the candidate Lyapunov function, instead of a polytopicset as with the RUIO. Although it increases the degree of conservatism,it will also add an additional combinatorial index to the existing onedue to variant π΅. Polyaβs theorems allow a definition for any π-degreecombination, but we have preferred to provide a unique applicationcase to avoid misunderstandings.
6. Case study
In this section, the implementation process of the control architec-ture presented in this paper is described for the TIAGo head example,providing general guidelines to be applied for any π-DoF mechanismwith rotational joints. First, the design problem of both the shiftingcontroller and RUIO will be stated, endorsing made choices. Then,simulation results are provided for a real scenario involving TIAGo ina environment simulated in ROS.
6.1. Design problem
The polytopic set of stateβspace matrices for the π-DoF manip-ulator system has been discretised using the Zero-order Hold (ZoH)method, considering the discretisation behaviour of AnalogβDigitalconversion in both sensing and acting devices (Toth et al., 2010).Sampling time ππ has been chosen according to the required execution
and communication times of the platform for a worst-case scenario. p8
Table 3LMI circular regions radii π and centres π for observer and control synthesis problemsof the TIAGo head system.
Problem π π
Shifting control [0.0243,0.0246,0.0246,0.0249] [0.9490,0.9610,0.9610,0.9732]RUIO 0.2890 0.2990
TIAGo head uses Dynamixel servomotors from ROBOTIS,4 specificallyMX-64 model, which features current-based torque control. Executingthe code in MATLAB (2016b version), using the available DynamixelSDK library and considering TTL serial communication, the executiontime per cycle has been determined to be below 0.5 ms under theseconditions. Therefore, ππ = 10 [ms] has been used, where idle timecontemplates the simultaneous execution of other processes that wouldincrease computation time, as e.g. an object tracking algorithm.
Considering the overall control architecture, RUIO synthesis prob-lem has to be extended with a D-stabilisation constraint such that theredoes not exist any dynamical coupling between closed-loop system andobserver. Therefore, dominant poles of the RUIO have been set to beat least 10 times faster than the fastest pole of the shifting controller.Following D-stabilisation definition, LMI constraint is stated for ππmatrix, as it determines the error dynamics of the RUIO:
πΏπ βππ +ππ βππππ +πππ βπππ
ππ < 0, (35)
here
π =[
βππ ππππ βππ
]
, ππ =[
0 10 0
]
.
efine a circular region, being ππ and ππ determined according toaforementioned criterion after the controller synthesis problem, andrecalling that they should lie on the RHP. For the TIAGo head system,these values have been included in Table 3.
A quadratic criterion has been introduced in both controller andobserver design, being upper bounded by parameter πΎ for any reach-able state, according to matrices π and π . Optimality of the solu-tion is obtained through the minimisation of πΎ such that the set ofLMI constraints holds. Hence, synthesis problems have been stated asfollows:
Optimal RUIO Design
Given ππ, π π, πΏπ, ππ
minimise πΎπsubject to (21), (19b), (19c), (35)
Shifting Controller Design
Given ππ , π π , ππ and πΏπ,π βπ = 1,β¦ , 2ππΉ
minimise πΎπsubject to (34a), (34b), (32b)
To achieve noise reduction effects for the RUIO, ππ and π π have beendefined according to sensor and process noise. For the TIAGo headsystem sensors, noise follows a zero-mean Gaussian distribution, withstandard deviations ππ = 1 β 10β4 [rad] and ππ£ = 3 β 10β4 [rad/s] for jointposition and velocity, respectively. In the shifting controller design ππand π π are defined according to a weighted Brysonβs rule given a setof maximum admissible values for actions and states. Weights πΏπ havebeen obtained through a tuning process aimed at reducing oscillatorybehaviours in the system response to step references. These matrixdefinitions for the TIAGo head example have been gathered in Table 4.
Regarding the shifting strategy, closed-loop settling time for π = 2%has been chosen to lie in [0.5,20] [s], defining through Eqs. (31) sets ππ
4 Dynamixel Actuator Series from ROBOTIS: http://en.robotis.com/model/age.php?co_id=prd_dynamixel (Accessed August 18, 2021).
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
Fig. 4. Placement of poles and defined LMI regions for closed-loop system with shifting state-feedback control (red) and RUIO (blue) solutions in the complex plane (a), and adetail of the closed-loop system placement of poles and LMI regions corresponding to the shifting strategy (b). (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)
g
o
Table 4Definition of tuning matrices π and π for RUIO and shifting control synthesis problemsof the TIAGo head system.
Matrix Definition
π π diag(32, 12, 12)β 10β8
πππ₯ππ
=diag(0.99, 0.99, 0.01, 0.01)
π·ππ= diag(1β32 , 1β32 , 1β1.312 1β1.052
π ππ₯π π = diag(0.9, 0.9)π·π π = diag(1β62 , 1β62)
Table 5Number of LMI constraints for the control synthesis problem of the TIAGo headsystem, regarding different formulations and performance constraints. In bold, thecorresponding one to the complete scheme.
Formulation Def. Pole plac. Main Total
LMI Equal.
Standard Stability 3 16 16 β 35
Polyaβs (π = 3) LQR & No shifting 3 20 20 β 43LQR & Shifting 3 68 68 β 139
Table 6Number of LMI constraints for the RUIO synthesis problem of the TIAGo headsystem, regarding different formulations and performance constraints. In bold, thecorresponding one to the complete scheme.
Formulation Def. Pole plac. Main Total
LMI Equal.
Standard Stability 4 4 16 4 28
Polyaβs (π = 5) LQR 5 4 57 4 70
and ππ of the four LMI regions for the TIAGo head case, also includedin Table 3. Lower values of π determine greater overlapping betweenregions in the discrete complex plane, and for the considered system amaximum of π = β1.3 has been found.
A
9
LMI conditions have been stated using YALMIP toolbox (LΓΆfberg,2004) for MATLAB, and solved by the Semi-Definite Programming(SDP) algorithms provided by MOSEK.5 Feasible solutions have beenfound in both cases, and the number of LMI constraints has beendetailed for the TIAGo head case in Tables 5 and 6 for the ShiftingControl and RUIO Designs, respectively. It should be pointed out thatin these Tables, column ββDef.ββ considers all the constraints regardingthe definition of the problem and/or the bound of πΎ, ββPole plac.ββ forthose defining LMI regions, and ββMainββ for the ones that determinethe overall objective of the design, e.g. stability, optimal conditionsand/or specific conditions linked to the structure definition, distin-guishing between LMI and equality (ββEqual.ββ) constraints. These tablesadditionally include the same information for each problem with andwithout the application of Polyaβs theorems and regarding differentperformance constraints.
Obtained solution for the TIAGo head has been graphically re-produced through the representation of chosen LMI regions and thecorresponding poles. Fig. 4(a) distinguishes the LMI regions imposedfor the observer (blue) and closed-loop system (red) within the unitcircle. Shifting strategy is detailed in Fig. 4(b), where the distributionof obtained 4 LMI regions (2ππ’ ) is depicted.
6.2. Simulation results
In this part, we focus on key aspects of the presented control archi-tecture through different scenarios to assess its overall performance inthe joint tracking problem. All of them correspond to different settingsin the GAZEBO environment, where the TIAGo tracks an ArUco cubemarker6 by moving its head (which embeds a RGB-D camera) in thepresence of exogenous forces (Fig. 5). Results provided in this part con-sist on combinations of angular trajectories (Step References, Curve Ref-erences) and exogenous force behaviours (Mass Force, Variant Force,
5 MOSEK optimisation software: https://www.mosek.com/ (Accessed Au-ust 18, 2021).
6 ArUco: a minimal library for Augmented Reality applications basedn OpenCV: https://www.uco.es/investiga/grupos/ava/node/26 (Accessedugust 18, 2021).
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
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Fig. 5. Gazebo environment for the cube following task, together with the RGB-Damera output and the ArUco marker detection.
alf-Sinus Force), which are thoughtfully described in Appendix D.dditionally we have included a video that shows the performancef our approach (ββPolytopic State-Feedback Shifting Control with Ac-ive Compensationββ) against the default joint-independent PD control,hich can be also found in the related website.7
Remark 9. It should be pointed out that under the same force be-haviour, different control strategies and trajectories influence its effect,as exerted joint torques by the exogenous force depend on the relativedistances to joint axes, that change according to angular positions.
6.2.1. Noise effect reductionThe first scenario is built for Curve References and Variant Force to
evaluate the noise reduction effect of the Optimal RUIO formulation.For the TIAGo case, noise is propagated from the estimation of π1o the state-feedback control and active compensation, i.e. the inputction to the system, amplifying its effect on measured variables, usedn the RUIO estimation at next time step. Fig. 6 shows a side-by-sideomparison between solutions under the same conditions, showing themprovement in estimating both π1 and torque exerted in Pan Jointπ ,1, where the noise effect is most significant. Nevertheless, RUIOestimation at time instant π is based on current system output, unlikeclassical Kalman Filters where it is done for output at πβ1. Thus, noisereduction cannot be as effective for RUIO as it is for Kalman Filter,as measurement noise is directly injected into state estimation withoutbeing filtered out by observer dynamics.
6.2.2. Overall performancePerformance of the complete control scheme is evaluated on Curve
References for Mass and Variant Force scenarios. To validate our ap-proach, we have included in these experiments a comparison againsttwo variations:
β’ EKF variation: The complete control scheme, except for theRUIO, replaced by an Extended Kalman Filter (EKF). As afore-mentioned, it is considered in the unknown disturbance rejectionliterature as the state-of-art technique. Using this approach, stateβspace is augmented to include the disturbance effects into theobserver model, which requires to a-priori assume its dynamicbehaviour. In discrete-time scenarios, the rate of change of the
7 http://www.iri.upc.edu/people/asanmiguel/Projects/DO_LPVShift/.
10
Table 7Actuator PD gains for TIAGo head system.
Joint P Gain D Gain
π1 1 32π2 5 34
Table 8Root Mean Squared Error (RMSE) (in [β¦]) for compared control strategies underdifferent force scenarios.
Scenario Joint Complete scheme EKF PD & Act. Comp.
No force π1 1.5413 1.3146 0.1192π2 1.3196 1.2035 0.1545
Mass applied π1 1.3863 1.3166 0.1078π2 1.3085 2.2115 1.5209
Variant force π1 1.9658 3.8871 1.9888π2 2.8069 6.0832 4.9082
disturbance in EKF is assumed to be constant throughout theoperation, i.e. οΏ½οΏ½ (π‘) = 0 (Mohammadi et al., 2013), which hasbeen also considered for these experiments.
β’ PD and Active Compensation variation: using the estimationof the RUIO and the disturbance compensation mechanism, State-feedback Shifting controller is substituted by an independent-jointPD controller. This constitutes the off-the-shelf solution for regula-tion tasks, and is the one implemented as the pose control strategyfor the TIAGo head system. Current PD gains embedded in jointactuators are presented in Table 7.
Figs. 7 and 8 include the results for considered force scenarios,summarised in Table 8 using the Root Mean-Squared Error (RMSE)over the whole simulation together with its value when there is not anexogenous force acting on the system. Under the Mass Force, appliedin π‘ β [25β35] [s] as it can be seen in Fig. 7(c), only π2 is affected,and except both the Complete Design and the EKF variation are able tocompensate for the exogenous force effects that are rejected during thistime. It should be pointed out that the force effect on the EKF variationproduces a higher deviation from the reference trajectory when theforce is firstly applied at 25 [s] and released at 35 [s], approximately.For the Variant Force case depicted in Fig. 8, differences between themethods become more significant, and the Complete Scheme presentedin this paper outperforms the variations of the method. Overall, PDcontrol with Active Compensation presents less deviation than the EKFvariation, which even reaches the upper π1 limit in the interval π‘ =[20, 25] [s]. Greatest fluctuations of the EKF variation correspond toπ‘ β [18β28] [s], when the exogenous force has a magnitude peak, as itcan be seen in Fig. 8(c). This arises from the EKF design, which requiresassuming the dynamics of the unknown disturbance, constant in theseexperiments. When a significant mismatch with its real behaviouroccurs, the compensation of the estimated effects hinders the rejectioncapabilities of the control scheme.
To further analyse the differences between the EKF and RUIO, Ta-ble 9 includes the RMSE for οΏ½οΏ½1, π ,1 and π ,2 for the Variant Force case.RUIO has lower RMSE in all estimations, being one order of magnitudelower than EKF for the exogenous force torques. Together with theprevious results, this highlights the importance of avoiding any a-prioriassumption on the disturbance behaviour, moreover if it is meant tobe rejected by a feedforward compensation strategy. Thus, using RUIOhas been proved to be a more suitable option for disturbance rejectionproblems than state-of-art EKF technique.
6.2.3. Actuator saturationFinally, Shifting control is evaluated for Step References and Half-
Sinus Forces in order to emulate an scenario where the required controltorque is close to actuator saturation limits. The complete scheme fea-turing the shifting strategy is compared against two static state-feedback
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
s
Fig. 6. Evolution of οΏ½οΏ½1 and π ,1 and their estimation for RUIO standard formulation (a,c) and optimal one for noise reduction (b,d) under the Variant Force. (For interpretationof the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 9Root Mean Squared Error (RMSE) of estimated values given RUIO and EKF (for thesame control strategy) under the Variant Force scenario, excluding measurement noiseeffects.
Variable RUIO EKFπ1 [β¦/s] 0.7528 0.8731π ,1 [N m] 0.0043 0.0663π ,2 [N m] 0.0041 0.0665
Table 10Root Mean Squared Error (RMSE) (in [β¦]) for compared shifting and pole placementtrategies under different force scenarios.Scenario Joint Complete scheme Fast cont. Slow cont.
No force π1 17.7408 15.6746 32.1372π2 11.1072 9.9150 19.5654
Half-Sinus force π1 19.8564 15.6929 33.2427π2 12.4489 9.9723 21.2979
controllers, designed also according to their settling time for a singleLMI circular Region: a slower controller with ππ ππ‘π‘,2% = [0.5, 20] [s] anda faster one with ππ ππ‘π‘,2% = [0.5, 1] [s]. Fig. 9 shows the Pan trajectorytogether with a flag signal to indicate whether joint actuator saturatesor not, for the complete scheme, and the slower and faster controllers.Table 10 presents the RMSE over the whole simulation for all these
11
control approaches under aforementioned scenario and without anexogenous force. Faster controller reaches saturation limits in Pan Jointfor 0.07 [s] (Fig. 9(c)) when the Half-Sinus Force is applied, whilethe complete shifting controller does for 0.01 [s] (Fig. 9(a)). Lookingat the RMSE values of each strategy it can be seen that while Fastcontroller presents similar values in both scenarios, complete schemeones slightly increase under the Half-Sinus Force, i.e. there exist arelaxation in the tracking task to deal with the active compensationwhile avoiding actuator saturation. Note that for the slow controlleractuator saturation is reached for 0.27 [s] (Fig. 9(b)), although itpresents the slower response time among the compared strategies. As ithas been aforementioned, exerted torques depend on the manipulatorpose, which determines the relative distances between the point ofapplication of the force and joint axes. Thus, when the applied forcegets closer to its maximum value, slow controller drives the system toan angular position where force is applied further from the joint axes,increasing its exerted torque. Additionally Fig. 10 has been included tovisualise the Shifting strategy in this scenario, including the evolutionof shifting variables πΉπ and weights ππ.
7. Conclusions and future work
Existing joint-independent PD control in the TIAGo head systemcannot deal with exogenous forces in a regulation task. The approachproposed in this paper successfully tackles this problem, as provided
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
ssto
raarsmLwcosu
ytartectplf
Fig. 7. Desired trajectories and evolution of π1 (a) and π2 (b) under the Mass Forcecenario, depicted in terms of magnitude and orientation (c), for compared controltrategies. Grey areas in (a,b) represent the physical joint limits. (For interpretation ofhe references to colour in this figure legend, the reader is referred to the web versionf this article.)
esults in a realistic simulation case study. Moreover, guidelines to bepplied on any π-DoF robotic manipulator are given for the designnd implementation phases. Embedded optimal conditions for the RUIOepresent an improvement in the noise reduction properties of the ob-erver, enhancing the overall performance. Shifting approach allows toodify system response through the placement of closed-loop poles in
MI regions, which has been shown to help avoiding actuator saturationhen actively compensated effects are close to input limits. The appli-
ation of Polyaβs theorems has been used to reduce the conservatismf both control and RUIO synthesis problems through the definition ofufficient constraints, such that feasible solutions can be found evennder highly conservative constraint and definitions.
In future works we will aim at (1) extending the system anal-sis, mainly regarding stability of the RUIO and Shifting controllerogether and (2) adapting this control architecture for new scenariosnd robotic manipulators with more than 2-DoF operating in a task-elevant space. One interesting research line that could be followed iso consider a disturbance corresponding to a force exerted by a human,ither accidentally or not. This consideration will require far moreomplex reaction and adaptation mechanisms from the control archi-ecture, that might be integrated through the presented gain shiftingaradigm. Furthermore, an in-depth comparison against other non-inear control strategies is needed, specially regarding their suitabilityor implementation in robotic platforms.
12
Fig. 8. Desired trajectories and evolution of π1 and π2 (a,b) under the Variant Forcescenario, depicted in terms of magnitude and orientation (c), for compared controlstrategies. Grey areas in (a,b) represent the physical joint limits. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web versionof this article.)
Declaration of competing interest
The authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
Acknowledgement
This work is supported by the Spanish State Research Agencythrough the MarΓa de Maeztu Seal of Excellence to IRI MDM-2016-0656.
Appendix A. Modelling assumptions for the TIAGo head system
To obtain the formulation of Eq. (1) for the TIAGo head system thefollowing customary guidelines have been considered to formulate itsmodel as :
β’ Base frame (Link 0) is placed at the top part of TIAGo torso, whichfeatures a vertical movement. Considering that is not movingduring the trajectory following task, head base frame has beenset as fixed.
β’ Each link is considered as a rigid body, whose mass distribution ischaracterised by the position of its Centre of Gravity (CoG), beingits inertia tensor and mass referred to it. We use the parameters
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
Fig. 9. Evolution of π1 under the Half-Sin Force scenario, together with the saturation limit flag (blue regions), using the designed shifting (a), the slow (b) and the fast (c)controller designs. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
as defined by PAL Robotics in the ROS simulation environment,listed in Table 1 and included in Fig. 2. On the inertia tensorsfor both links, off-diagonal terms are negligible with respect todiagonal ones (by a factor of 10β4), and therefore have beenconsidered as null.
β’ The pose depicted in Fig. 2 is the zero-angle configuration of thehead system, being aligned axis 2οΏ½οΏ½2 and 1οΏ½οΏ½1 with 0οΏ½οΏ½0; and 2π2and 1οΏ½οΏ½1 with 0οΏ½οΏ½0.
β’ External forces and torques acting on the head system have notbeen considered for its dynamic characterisation.
13
Appendix B. Proof of Proposition 1
Considering the set
π = {π1,β¦ , π2ππ· }; ππ β Rππ₯Γππ₯ β£ ππ = π ππ > 0 βπ = 1,β¦ , 2ππ·
for the candidate polytopic Lyapunov function
π(π(π)) =2ππ·β
π=1ππ(π·) π(π)π ππ π(π),
applying the Kalman filter Riccati equation to the RUIO descriptiongiven in Eq. (15), the following inequality can be obtained βοΏ½οΏ½ β I(2,2ππ· )
ππ π π β π +π +πΎ π πΎπ < 0.
π1 π2 π1 π1 π π1 π π1
A. San-Miguel, V. Puig and G. AlenyΓ Control Engineering Practice 115 (2021) 104887
A
β‘
β’
β’
β’
β’
β£
fl
AL
p
π―
Fig. 10. Evolution of estimated torques exerted by the exogenous force π ,π and gravityeffects οΏ½οΏ½ (a), shifting variables πΉπ (b) and the associated weights ππ (c) under the Half-Sin Force scenario, for the complete control design. (For interpretation of the referencesto colour in this figure legend, the reader is referred to the web version of this article.)
Multiplying both sides by ππ1 = πβ1π1
and introducing the variablechange ππ1 = ππ1πΎπ1 leads to
βππ1 β [(πππ1ππ1 )
π (π»πππ1 )π ππ1 ]
β‘
β’
β’
β£
βπβ1π2
0 00 βπΌ 00 0 βπ π
β€
β₯
β₯
β¦
β‘
β’
β’
β’
β£
πππ1ππ1
π»πππ1π ππ1
β€
β₯
β₯
β₯
β¦
< 0,
pplying Schur complement lemma, we obtain
βππ1 ππ1 ππ1 ππ1 π»ππ ππ1
(β) βππ2 0 0(β) 0 βπΌ 0(β) 0 0 βπ β1
π
β€
β₯
β₯
β₯
β₯
β¦
< 0.
and considering ππ = πππΈ results in Eq. (19b), following the standarddesign of the RUIO.
Lyapunov matrix set π is bounded for all the reachable state domainaccording to the optimality criterion π½π such that
2ππΉβ
π=1ππ(πΉ )ππ < πΎππΌ.
As aforementioned, multiplying at both sides by the change of variableππ = πβ1
π leads to
πΎ πΌ β πΌ πβ1 πΌ > 0,
π π14
rom which Eq. (19c) is obtained after applying the Schur complementemma.
ppendix C. Application of Polyaβs theorem on combinatorialMIs
Assessing positive-definiteness of the symmetric matrix π1 π2 for aolytopic LPV system of ππ· scheduling variables implies:
=2ππ·β
π1=1
2ππ·β
π2=1ππ1 (π·) ππ1 (π·) π₯
π π1 π2 π₯ > 0, β π β I+(2,2ππ· ) (C.1)
According to Sala and Arino (2007), application of Polyaβs theoremssets that for a fixed π β₯ 2, on the positive-definiteness conditions fromEq. (C.1), constraints defined as
οΏ½οΏ½ =β
πβ(οΏ½οΏ½)
π1 π2 > 0, β οΏ½οΏ½ β I+(π,2ππ· ) (C.2)
where (οΏ½οΏ½) denotes the permutation set of multi-index οΏ½οΏ½,
I+(π,β) = {οΏ½οΏ½ β I(π,β) β£ ππ β€ ππ+1, π = 1,β¦ , π 9 1}
are sufficient for positivity of π―. Note that this formulation is validfor both continuous and discrete-time system forms as it does notimpose any form of π1 π2 . Also, although it has been defined forpositive-definitive assessment, it can be equivalently used for negativecase.
Appendix D. Simulation scenarios
β’ Joint Angle TrajectoriesT.1 Step References given between two symmetric angular
positions, such that there exist an phase offset betweenjoints to stress the dynamical coupling effects. For theTIAGo head case, Pan angle trajectory is defined forβ 0.5 π
1and Tilt one for Β± 0.5 π2, with a period of 10 [s]
and existing a 5 [s] offset (πβ2) between them.T.2 Curve References to mimic trajectories given by an upper
layer to the control architecture, as e.g. path generator viaa set of points. In this case
π1,πππ (π‘) = 0.7 π1 sin(0.04π‘),π2,πππ (π‘) = 0.7 π2 sin(0.08π‘ β 26 β 10β4 π‘2),
evaluated each time step π as π‘ = π ππ .
β’ Exogenous force behaviours. Forces have been considered to beexerted on the CoG of the second link, defined as function of amagnitude | | and orientation π w.r.t. the vertical axis of themanipulator base.
F.1 Mass Force, i.e. the force corresponding to a mass attachedto the robot, addressing the issue that motivated the appli-cation of presented approach to the TIAGo head system.Force orientation is constant and equal to zero and themagnitude corresponds to a trapezoidal rounded-edge signalwith a 1 [s] rise time to emulate the mass placement effect.For these simulations, placed mass is of 1 [kg].
F.2 Variant Force in both magnitude and orientation, accord-ing to the following equations for the TIAGo case:
| |(π‘) = 15 β 10β2(π‘ β 20) + 3 sin(π‘ β 20)β(π‘ β 20) [N]
π(π‘) = 45 sin(0.17π‘) [rad]
F.3 Half-Sinus Force in terms of magnitude, to assess controlbehaviour from zero to a maximum value. For the TIAGohead case, considering a constant orientation of β45 [β¦],the maximum force value has been set to 8.5 [N].
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C
H
H
H
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L
M
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O
R
R
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R
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