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Robust energy aware output controller for uncertain plant ⁎
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Available online at www.sciencedirect.com Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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Robust Robust Robust Robust
energy energy aware aware output output controller controller for for energy aware output controller controller for uncertain plant energy aware output for uncertain plant uncertain plant uncertain plant Dmitrii Dobriborsci, ∗∗ Alexey Margun, ∗∗ Sergey Kolyubin ∗,∗∗ ∗,∗∗
Dmitrii Dobriborsci, ∗∗ Alexey Margun, ∗∗ Sergey Kolyubin ∗,∗∗ ∗,∗∗ Dmitrii Dobriborsci, ∗ Alexey Margun, ∗ Sergey Kolyubin ∗,∗∗ Dmitrii Dobriborsci, Alexey Margun, Sergey Kolyubin ∗ ∗ Control systems and Robotics faculty, ITMO University, St. ∗ Control systems and Robotics faculty, ITMO University, St. ∗ Control systems and Robotics faculty, ITMO University, St. Petersburg, Russia, ([email protected]) ∗ Petersburg, Russia, ([email protected]) ∗∗ Control systems and in Robotics faculty,Mechatronics ITMO University, St. ∗∗ Center for Technologies Robotics Petersburg, Russia, ([email protected]) for Technologies in Robotics and and Mechatronics Components, Components, ∗∗ Center Petersburg, Russia, ([email protected]) ∗∗ Innopolis University, Innopolis, Russia Center for Technologies in Robotics and Mechatronics Components, ∗∗ Innopolis University, Innopolis, Russia Center for Technologies in RoboticsInnopolis, and Mechatronics Innopolis University, Russia Components, Innopolis University, Innopolis, Russia Abstract: Abstract: Nowadays, Nowadays, safety safety guarantee guarantee during during physical physical Human-Robot Human-Robot Interaction Interaction (pHRI) (pHRI) is is important problem. Control over energy becomes very useful approach and provides possibility Abstract: Nowadays, safety guarantee during physical Human-Robot Interaction (pHRI) is important problem. Control over energy becomes very useful approach and provides possibility Abstract: Nowadays, safetyover guarantee during measurement physical Human-Robot Interaction (pHRI) is important problem. Control energy becomes very useful approach provides of safety implementation but of variables. In this of safety conditions conditions implementation but requires requires measurement of energy energyand variables. In possibility this paper paper important problem. Control over for energy becomes veryoutput useful control approach and provides possibility we extend our previous results discrete robust algorithm by insertion of of safety conditions implementation but requires measurement of energy variables. In this paper we extend our previous results for discrete robust output control algorithm by insertion of of safety conditions implementation but requires measurement ofisenergy variables. Ininsertion this paper energy aware mechanism in the structure of control system. It proven earlier that this way we extend our previous results for discrete robust output control algorithm by of this way energy aware mechanism in the structure of control system. It is proven earlier that we extend ourmechanism previous results forenergy discrete robust output control algorithm by insertion of make the controller aware of the that they inject into the plant. By combining this energy aware in the structure of control system. It is proven earlier that this way make the controller awareinofthe the energy that they system. inject into the plant.earlier By combining this energy aware mechanism structure of control It is proven that this way make the controller aware of the energy that they inject into the plant. By combining this approaches we here a robust discrete-time output controller for plant approaches we introduce introduce here robust discrete-time output controller for uncertain uncertain plant that that make theexponential controller aware of a energy thaterror they to inject into the plant. By passivity combining this provides convergence of tracking the bounded area the approaches we introduce here athe robust discrete-time output controller for and uncertain plantof that provides exponential convergence of tracking error to the bounded area and passivity of the approaches we introduce here a robust discrete-time output controller for uncertain plant that closed loop system. The proposed approach is validated through simulation and experiments provides exponential convergence of tracking error to the bounded area and passivity of the closed loop system. The proposed approach is validated through simulation and experiments provides exponential convergence trackingiserror to thethrough bounded area andand passivity of the closed loop system. The proposed ofapproach validated simulation experiments using Ball-and-Plate setup. using setup. closedBall-and-Plate loop system. The proposed approach is validated through simulation and experiments using Ball-and-Plate setup. © 2019, IFAC (International using Ball-and-Plate setup.Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: robust robust control, control, energy energy aware, aware, energy energy budget, budget, ball ball and and plate, plate, passive passive systems systems Keywords: robust control, energy aware, energy budget, ball and plate, passive systems Keywords: robust control, energy aware, energy budget, ball and plate, passive systems 1. INTRODUCTION INTRODUCTION and 1. and safe safe contact contact resemblance resemblance simultaneously simultaneously by by introducintroduc1. INTRODUCTION ing a controller shaping function that robustly handles and safe contact resemblance simultaneously by introducing a controller shaping function that robustly handles 1. INTRODUCTION and contact resemblance simultaneously by introducing asafe controller shaping function that robustly handles unexpected contact loss and avoids chattering behavior Nowadays, robots take part not only in industry, but step contact loss and avoids chattering behavior Nowadays, robots take part not only in industry, but step unexpected ing a controller shaping function that robustly unexpected contact loss and avoids chattering behavior that generated by switching based approaches. Nowadays, robots not life. only Therefore in industry, butsafety step that generated by switching based approaches. handles by step step enter enter intotake ourpart casual life. Therefore the safety by into our casual the contact loss andbased avoids chattering behavior Nowadays, robots take part not only in industry, but step unexpected that generated by switching approaches. by step enter into our casual life. Therefore the safety problem becomes becomes major. major. Especially Especially when when there there is is aa physphysproblem In Lee and Huang (2010) a passive set-position that generated by switching based approaches. by step enter into our casual life. Therefore the safety Lee and Huang (2010) a passive set-position modulation modulation problem becomes major. Especially when there is”safety” a phys- In ical human-robot human-robot interaction (pHRI). Definition ”safety” In Lee and Huang (2010) a passive It set-position modulation ical interaction (pHRI). Definition framework (PSPM) is enables to problem becomes major. Especially when there is a physframework (PSPM) is presented. presented. It enables users users to conconical human-robot interaction (pHRI). Definition ”safety” in pHRI was discovered in Alami et al. (2006). We note In Lee and Huang (2010) a passive set-position modulation in pHRI was discovered in Alami et al. (2006). We note nect a continuous-time robot’s position to a sequence of framework (PSPM) is presented. It enables users to conical human-robot interaction (pHRI). Definition ”safety” nect a continuous-time robot’s position to a sequence of in pHRI was discovered Alamiindex, et al. (2006). Wemean note framework (PSPM) is presented. It enables users to consome of them: them: maximuminpower power index, maximum mean some of maximum maximum slowly updating/sparse set-position signal (discrete-time) nect a continuous-time robot’s position to a sequence of in pHRI was discovered in Alami et al. (2006). We note slowly updating/sparse set-position signal (discrete-time) some of them: maximum power index, maximum mean strain injury injury criterion, criterion, head head injury injury criterion. criterion. Some Some of of them them slowly nect athe continuous-time robot’s position to(discrete-time) a sequence of updating/sparse set-position signal strain using simple spring coupling with damping injection, some of them: maximum power index, maximum mean using the simple spring set-position coupling with damping injection, strain injuryto injuryand criterion. Some ofbased them slowly are related related tocriterion, energy: head velocities and accelerations based updating/sparse signal (discrete-time) are energy: velocities accelerations while enforcing passivity of the closed-loop robotic sysusing the simple spring coupling with damping injection, strain injurydepend injuryand criterion. Some ofstratethem while enforcing of the closed-loop robotic sysare related tocriterion, energy: velocities accelerations based approaches on head the energy transfer. Control using the simplepassivity spring coupling with damping injection, approaches depend on the energy transfer. Control stratetem. The PSPM modulates set-position signal while enforcing passivity ofthe theoriginal closed-loop robotic sysare related to energy: velocities and accelerations based tem. The PSPM modulates the original set-position signal approaches depend on the energy transfer. Control strategies that that limit limit this this energy energy (mostly (mostly kinetic) kinetic) are are successful successful while enforcing passivity ofthe theoriginal closed-loop robotic sysgies in such a way that the modulated signal is as close to tem. The PSPM modulates set-position signal approaches depend on the energy transfer. Control stratesuch aPSPM way that the modulated signal is as close to gies that limit energy (mostly kinetic) arevelocity successful in robotics: the this position setpoint or maximum velocity are in tem. The modulates the original set-position signal in such a way that the modulated signal is as close to in robotics: the position setpoint or maximum are the original signal as possible (i.e. maximum information gies that limit this energy (mostly kinetic) are successful the original signal as possible (i.e. maximum information in robotics: the position setpoint or maximum velocity are adapted to stay within safe energy levels by estimating in such a way that the modulated signal is as close to adapted to stay within safe energy levels by estimating recovery for better performance), yet only to the extent the original signal as possible (i.e. maximum information in robotics: the position setpoint or joint maximum velocity are recovery for signal betterasperformance), yet only toinformation the extent adapted to stay within safe energy levels by estimating the energy of the robot based on velocities Tadele the original possible (i.e. maximum the energy thewithin robot safe based on joint velocities Tadele recovery permissible the available the system (i.e., for by better yetin only the extent adapted to of stay energy levels by estimating by the performance), available energy energy the to (i.e., the energy of the robot based on joint velocities Tadele et al. (2014), Laffranchi et al. al. (2009). Moreover, safety permissible recovery for better performance), yetin only tosystem the extent et al. (2014), Laffranchi et (2009). Moreover, safety passivity constraints). permissible by the available energy in the system (i.e., theal. energy of the robot based on joint of velocities Tadele passivity constraints). et (2014), Laffranchi et al. (2009). Moreover, safety may be achieved by ensuring passivity the robot. For by the available energy in the system (i.e., passivity constraints). may be(2014), achieved by ensuring passivity of the robot. For permissible et al. Laffranchi et al. (2009). Moreover, safety may be achieved by ensuring the if For passivity In this we instance, in Stramigioli Stramigioli (2015)passivity is proven provenofthat that ifrobot. system constraints). instance, in (2015) is aa system this research research we modify modify discrete discrete robust robust output output conconmay be isachieved by ensuring passivity ofthat the ifrobot. For In instance, in Stramigioli (2015) is proven a system troller presented earlier in Dobriborsci et al. (2018b), DoIn this research we modify discrete robust output con(robot) not strictly passive, it is always possible to find (robot) is not strictly passive, it is always possible to find troller presented earlier in Dobriborsci et al. (2018b), DoIn thispresented research we modify discrete of robust output coninstance, in Stramigioli (2015) isisproven that if a system troller earlier in Dobriborsci etenergy al. (2018b), Do(robot) is not strictly passive, it always possible to find briborsci et al. (2018a) by including energy tank into such passive environment that destabilizes the system and briborsci et al. (2018a) by including of tank into such passive environment that destabilizes the system and troller presented earlier in Dobriborsci et al. (2018b), Do(robot) isinfinite notenvironment strictly passive, it is always possible to find such passive that destabilizes the system and controller structure. briborsci et al. (2018a) by including of energy tank into extracts energy from it As result system becomes extracts infinite energy from itdestabilizes As result system becomes controller structure. briborsci et al. (2018a) by including of energy tank into such passive environment that the system and extracts energy fromoperating it As result system becomes controller structure. unstable.infinite Therefore, robots operating in an an unpredictable unstable. Therefore, robots in unpredictable Proposed controller controller structure. was extracts infinite energy fromoperating it Astoresult system becomes controller was tested tested with with computer computer simulation simulation unstable. Therefore, robots in an unpredictable environment can be guaranteed remain stable if and and Proposed environment can be guaranteed to remain stable if and experimental setup. Experimental researches were Proposed controller was tested with computer simulation unstable. Therefore, robots operating in an unpredictable and experimental setup. Experimental researches were environment can be guaranteed to remain stable if and only if if they they are are passive passive Folkertsma Folkertsma et et al. al. (2018). (2018). There There Proposed controller was tested with computer simulation only conducted for ”Ball and plate system” (plate with two and experimental setup. Experimental researches were environment can be guaranteed to remain stable if and conducted for ”Ball and plate system” (plate with two only if they are passive Folkertsma et al. (2018). There were published a number of papers in this field of study. In and experimental setup. Experimental researches were were published a number of papers in et thisal.field of study. In conducted for ”Ball and plate system” (plate with two rotational degrees of freedom and ball located on the plate) only if they are passive Folkertsma (2018). There rotational degrees of freedom and ball located on the plate) were published a number of papers in this field of study. In Schindlbeck and Haddadin (2015) a novel hybrid Cartesian conducted for tasks: ”Ball and plate system” (plate with two Schindlbeck and Haddadin (2015) a novel hybrid Cartesian rotational degrees of freedom and ball located on the plate) for following object stabilization in the specific were published number of(2015) papers thishybrid fieldwith ofCartesian study. In for following tasks: object stabilization in the specific Schindlbeck anda Haddadin novel force/impedance controller that is isain equipped energy rotational degrees of freedom and ball located on thespecific plate) force/impedance controller that equipped with energy point on the plate, optimal point-to-point motions and for following tasks: object stabilization in the Schindlbeck and Haddadin hybrid Cartesian on the plate, optimal point-to-point and force/impedance controller that isa novel equipped with energy tanks to preserve preserve passivity(2015) is proposed. proposed. It provides provides ac- point for following tasks:tracking. object stabilization in motions the specific tanks to passivity is It acdesired Such kind of systems are point ontrajectory the plate, optimal point-to-point motions and force/impedance controller that is equipped with energy desired trajectory tracking. Such kind of systems are tanks to preserve passivity is proposed. It provides accurate force tracking, full compliant impedance behavior, point on theinplate, optimal point-to-point and curate force tracking, full compliant impedance behavior, desired trajectory tracking. Suchautomobile kind of motions systems are widely used flight simulators, simulators, tanks to preserve passivity is proposed. It provides acused in flight simulators, automobile simulators, curate force tracking, full compliant impedance behavior, widely desired trajectory tracking. Such kind of systems are widely used in flight simulators, automobile simulators, industrial automation such as fast sorting Dobriborsci and curate forcewas tracking, fullsupported compliant This work financially by impedance Government behavior, of Russian industrial automation as fast sorting Dobriborsci and widely used in flight such simulators, automobile simulators, This work was financially supported by Government of Russian Kolyubin (2017). industrial automation such as fast sorting Dobriborsci and Federation (Grant 08-08). The reported by study was partially funded This work was financially supported Government of Russian Kolyubin (2017). Federation (Grant 08-08). The reported study was partially funded industrial automation such as fast sorting Dobriborsci and This work was financially supported by Government of Russian Kolyubin (2017). by RFBR according to the research The Federation (Grant 08-08). The reported project study was18-38-20037. partially funded by RFBR according to the research project 18-38-20037. The The paper is Kolyubin (2017). Federation (Grant 08-08). The reported study was partially funded The paper is organized organized as as follows. follows. Section Section 22 is is devoted devoted to to development of the modified version ofproject the energy-aware output by RFBR according to the research 18-38-20037. The development of the modified version of the energy-aware output the mathematical problem statement. Section 3 contains The paper is organized as follows. Section 2 is devoted to by RFBR was according to the research project 18-38-20037. The controller supported by the Ministry of energy-aware Science and output Higher the mathematical problem statement. Section 3 contains development of the modified version of the The paperdescription is organized as follows. Section 2 is devoted to controller was supported by the Ministry of Science and Higher controller and stability analysis. The Embedthe mathematical problem statement. Section 3 contains development of the modified version of the energy-aware output Education of Federation controller description and stability analysis. The Embedcontroller supported by the(8.8885.2017/8.9) Ministry of Science and Higher Education was of Russian Russian Federation (8.8885.2017/8.9) the mathematical problem statement. Section 3 contains controller description and stability analysis. The Embedcontroller was supported by the(8.8885.2017/8.9) Ministry of Science and Higher Education of Russian Federation controller description and stability analysis. The EmbedEducation of Russian Federation (8.8885.2017/8.9)
2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 264 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 264 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 264 10.1016/j.ifacol.2019.11.178 Copyright © 2019 IFAC 264
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ded Energy Aware algorithm in Section 3 as well. Section 4 presents the experimental setup and its mathematical model. Results of simulation and experiments are presented in Section 5. We conclude the paper with discussion on achieved results and its application. 2. PROBLEM STATEMENT
σ > α+β is a chosen by developer coefficient that regulates estimation convergence speed. Choose observation error as follows: η(t) = LT e(t) − ξ(t).
(6)
Taking into account (5), calculate derivative of (6):
Consider linear plant
η(t) ˙ = σΓη(t) + LT e(t). ˙
(7)
Q(p)y(t) = R(p)(u(t) + f (t)), (1) where Q(p) and R(p) are linear differential operators of degrees n and m respectively with unknown coefficients, y(t) is a plant output, u(t) is a control signal, f (t) is a bounded piece wise smooth external disturbance, ρ = n − m is a plant relative degree.
Closed loop system consists of plant (1), control law (4) and observer (5) takes the form: ε(t) ˙ = Aε(t)+B(−βe+ (α+ β)(e(t) − eˆ(t)) + B1 ϕ(t), (8) η(t) ˙ = σΓη(t) + LT e(t). ˙
Reference model is described by linear differential equation
According to Bobtsov (2002) closed loop system (8) exponentially converges to the zero stable state in the case of disturbance absence. Otherwise, (8) exponentially convergence to the bounded area.
(2) Qm (p)ym (t) = Rm (p)r(t), where Qm (p) and Rm (p) are linear differential operators with known coefficients, ym (t) is a reference plant output, r(t) is a piece-wise smooth input of reference plant. The control goal is to design controller that provides tracking of the plant (1) output for the output of reference model (2) with pre-specified accuracy for finite time: (3) |y(t) − ym (t)| ≤ δ, ∀t > T, where δ is a tracking accuracy, T is a transient time. We don’t try to provide necessary transient time but it is necessary to provide its existence (condition (3)). Let us introduce following assumptions • Unknown coefficients of plant (1) belong to the known compact set Ξ. • R(λ) is Hurwitz polynomial, where λ is an imaginary unit. • Plant (1) and model (2) are passive. 3. CONTROLLER SYNTHESIS 3.1 Continuous consecutive compensator In Bobtsov (2002) continuous control law is proposed u = −(α + β)D(p)ˆ e(t), (4) where α,β > 0 are coefficients chosen by developer (greater value of α and β correspond to higher stability of closed loop system), D(λ) is Hurwitz polynomial of degree ρ − 1, eˆ(t) is an estimate of tracking error e(t) = y(t) − ym (t).
It is necessary to know ρ − 1 unmeasured derivatives of tracking error e(t) for implementation of control law (4). Thus, introduce observer for its estimation: ˙ = σΓξ(t) + σGe(t), ξ(t) (5) eˆ(t) = Lξ(t),
where ξ(t) ∈ Rρ−1 is an observer state vector, Γ = 0 Iρ−2 is Hurwitz matrix, G = [0 0 c1 ]T , −c1 ... −cρ−1 Iρ−2 is a unit matrix of ρ − 2 order, L = [1 0 ... 0],
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3.2 Discrete consecutive compensator Lets obtain discrete realization of controller for its implementation on digital computing devices. Rewrite tracking error as e(t) = e(tk ) + hψ(ξ(tk ), tk ), where e(tk ) is a discrete value of tracking error e(t) on k-th step, h = tk − tk−1 is a sampling time, ψ(ξ(tk ), tk ) is a bounded function. Moreover ψ(ξ(tk ), tk ) is a Lipschitzian function because of plant is linear and disturbance is a piece-wise smooth. Discrete version of observer with use of right derivatives and replacement of tracking error by its discrete measurement takes the form ξ1 (tk+1 ) = ξ1 (tk ) + hσξ2 (tk ), ξ2 (tk+1 ) = ξ2 (tk ) + hσξ3 (tk ), ............................ (9) ξρ−1 (tk+1 ) = σ(−c1 ξ1 (tk ) − c2 ξ2 (tk ) − ... + cρ−1 e(tk )) eˆ(tk ) = ξ1 (tk ). Rewrite discrete derivative (9) observer in matrix representation ξ(tk+1 ) = (I + hσΓ)ξ(tk ) + σhGe(tk ), (10) eˆ(tk ) = Lξ(tk ).
¯>0 From lemma I.B. Furtat (2015) it follows existence of h ¯ limt→∞ ||ξ(t) − ξ(tk )|| < C, where such, that ∀h < h, C > 0. That’s mean that we always can find sampling time h satisfying required accuracy. Dynamics of discrete observer error is described by equations η(t) = LT e(t) − ξ(k) + hψ(t),
(11) η(t) ˙ = LT e(t) ˙ + σΓη(t) + σΓhψ(t). Remark 1.Last terms of equations (11) can be considered as additional bounded disturbance in continuous time linear system. So, here to reduce disturbances we have to choose h such small as possible (see statement below) Control law (4) takes the form :
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u(tk ) = −(α + β)(d1 ξ1 (tk ) + d2 σξ2 (tk ) + +d3 σ 2 ξ3 (tk ) + ...). (12)
where υ is a small positive number.
Closed loop system takes the form
(17) where R1 , R2 are positive defined matrices due to the choose of α, β and σ; θ is a bounded function.
˙ = Aε(t) + B(βe(t) + (α + β)Lη(t)) + Bhψ(t)+ ε(t) B1 ϕ(t), η(t) ˙ = LT e(t) ˙ + σΓη(t) + σhΓψ(t) (13)
Replacement of continuous value e(t) by sum of its discrete measurement e(tk ) and bounded function allows us to consider continuous plant with discrete controller as a continuous system with additional disturbances in the case of linear plant (1) and controller (12), (9). Proposed replacement sufficiently simplifies stability analysis of closed loop system. 3.3 Stability analysis Statement ¯ > 0 such that for any h < h ¯ discrete There exist α, β, σ, h observer (10) and control law (12) provide exponential convergence of tracking error e(t) to the bounded area.
R1 = Q1 + 2βLT B T P1 − υP1 B1 B1T P1 −
hυP1 BB T P1 − (α + β)υP1 BB T P1 + (β −1 + βυ)I, R2 = σQ2 − 2(α + β)P2 BL − υ −1 I − 2σ
hυP2 ΓΓT P2 − βP2 AAT P2 − βυP2 BLLT B T P2 −
(18)
υhP2 BB T P2 − υP2 B1 B1T P2 , 2h T 2 + 2σh T ϕ ϕ+ ψ ψ). θ = sup( υ υ
We always can replace bound on derivative of Lyapunov function (17) by inequality λmin R1 , V˙ ≤ −ζV + θ, ζ = λmax (P1 ) (19) θ θ e−ζt − . V ≤ V (0) − ζ ζ Taking into account λmin (P1 )e2 ≤ λmin (P1 )ε2 ε ≤ V we obtain
Prove of Statement Consider Lyapunov function T
Taking into account (16) rewrite (15) in the form V˙ ≤ −εT R1 ε − η T R2 η + θ,
T
V (t) = ε (t)P1 ε + η (t)P2 η(t), (14) where P1 , P2 are solutions of Lyapunov equations AT P1 + P A = −Q1 , ΓT P2 + P Γ = −Q2 , Q1 and Q2 are positive defined symmetric matrices. Obtain derivative of Lyapunov function (14) along trajectories (13) V˙ = −εT (Q1 + 2βLT B T P1 )ε + 2εT P1 B1 ϕ + 2hεT P1 Bψ+
|e| ≤
1 λmin (P1 )
θ V (0) − ζ
e−ζt
θ + ζ
(20)
Thereby, tracking error exponentially converges to the area bounded by inequality
|e| = |y − ym | ≤
1 θ λmin (P1 ) ζ
(21)
2(α + β)εT P1 BLη − η T (σQ2 − 2(α + β)P2 BL)+
3.4 Energy Aware algorithm
Bound terms of the right part of (15) by inequalities
To ensure effective regulation of the plant, it is proposed to use the energy budget approach introduced in Folkertsma et al. (2018).
2σhη T P2 Γψ + 2η T P2 Aε − 2βη T P2 BLε + 2hη T P2 Bψ+ 2ηP2 B1 ϕ. (15)
1 2εT P1 B1 ϕ ≤ υεT P1 B1 B1T P1 ε + ϕT ϕ υ 1 2εT P1 Bψ ≤ υεT P1 BB T P1 ε + ψ T ψ, υ 1 2εT P1 BLη ≤ υεT P1 BB T P1 ε + η T η, υ 1 T T T T 2η P2 Γψ ≤ υη P2 ΓΓ P2 η + ψ ψ, υ 1 T T T 2η P2 Aε ≤ βη P2 AA P2 η + εT ε, β 1 2η T P2 BLε ≤ υη T P2 BLLT B T P2 η + εT ε, υ 1 T T T T 2η P2 Bψ ≤ υη P2 BB P2 η + ψ ψ, υ 1 T T T 2η P2 B1 ϕ ≤ υη P2 B1 B1 P2 η + ϕT ϕ. υ
Let us assume that at moment t = 0 controller has energy budget Em and cannot inject to plant more energy than Em . Consider energy supplied to the system by controller for the time (t0 , t1 ): ∆E = (16)
t1
u(τ )v(τ )dτ,
(22)
t0
where v is a co-vector of u, i.e. product uv is an energy. For example, in mechanical systems if u is a force than v is a velocity. If uv > 0 than energy is transferred from controller to plant Otherwise, plant returns energy to controller. For discrete controller input signal u(t0 ) is a constant for a time (t0 , t1 ). Therefore
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∆E = u(t0 )
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t1 t0
v(τ )dτ = u(t0 )(q(t1 ) − q(t0 )),
(23)
where q is a coordinate. Energy Aware algorithm that don’t allows controller (12) to give plant more energy than energy budget: • Calculating of control value u(tk ) by discrete consecutive compensator algorithm. • Calculating of ∆E. • If ∆E ≥ 0 and current energy of controller Em (tk ) > ∆E than controller injects its energy into plant: u = u(tk ), Em (tk+1 ) = Em (tk ) − ∆E. • If ∆E ≥ 0 and current energy of controller Em (tk ) < ∆E than controller don’t have enough energy: u(tk ) = 0, Em (tk+1 ) = Em (tk ). • If ∆E < 0 than plant returns (virtually) its energy to controller: u = u(tk ), Em (tk+1 ) = Em (tk ) − ∆E. • go to beginning.
Such switching algorithm allows to bound total enegy of closed loop system.Therefore system is passive. Schematically control algorithm with Energy Aware Mechanism is illustrated on Fig. 1.
the screen, pressing the two layers together. This actions create an electrical connection between layers. However, both X and Y axes cannot be read simultaneously, because the concept works by applying a voltage across one layer, and looking for the voltage that appears on the another. Therefore, the system operates by alternately applying a voltage to one layer and reading off of the another. The voltages are then read in by an analog-to-digital (A/D) converter to be used as a coordinate value. There are several types of the resistive touch-screens, e.g. 4wire, 5-wire and 8-wire. These labels refer to number of wires between the screen and the controller. In the current study we chosen a 4-wire resistive touch AST150C. It features the typical advantages of the resistive type such as operational stability, detection accuracy, ease of introducing, competitive cost and so on. It have great durability with up to as much as 10 million touches, due to its structure. 4.1 Mathematical model We consider the following equations of motion for experimental setup Ball and Plate to design the robust output controller. In this part we introduce the following assumptions: • There is no slipping for ball. • The ball is completely symmetric and homogeneous. • Friction forces are neglected.
Fig. 1. Control scheme with Energy Aware mechanism 4. EXPERIMENTAL SETUP We built a tiltable platform with two degrees of freedom consisting of the following components: resistive touchscreen, servo-drives, square plate and connecting links. The control system was implemented on the Arduino single-board computer. Resistive touch-screen determines the coordinates of the object. The kinematic scheme of the developed system is presented in the Fig.2 Each tilting axis is operated on by a servomotor HITEC HS-5685mh with JR connector.
Fig. 2. Kinematic scheme of the mechanism
The platform was specially built to conduct the following experiments: object stabilization in the specific point on the plate, its optimal point-to-point motions and desired motion trajectory tracking as well as dynamics object-toplate contact model identification. In contrast to previous papers here we detect the ball position using resistive touch-screen system. An analog resistive touch-screen consist of a glass or acrylic panel that is coated with electrically conductive and resistive layers made with indium tin oxide (ITO). These layers are separated by invisible spacers/ To generate a position, the user or an object must exceed the activation force of 267
Fig. 3. Parallel kinematics robot Ball and Plate
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• The ball and plate are in contact all time.
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Simulation results are on Fig.
The angles of servo arms θx , θy are assumed to be the inputs, while the ball position on x, y axis are assumed to be the output. Here we derive dynamical equations of ballon-plate system, by the help of Lagrangian. The following mathematical equations are based on . So the nonlinear differential equations for the ball and plate system are presented below. I 2 ˙ m+ 2 x ¨ − m xα˙ + y α˙ β + mgsinα = 0 (24a) r I 2 ˙ ˙ m + 2 y¨ − m y β + xα˙ β + mgsinβ = 0 (24b) r where m, r, I are mass of the ball, radius of the ball, mass moment of inertia x, y are position of axis, α, β, α, ˙ β˙ are inclination angles of the plate, angular velocity of the plate, g, L, d are gravitation, plate side length, length between the joint and the center of the gear, respectively. The relations between inclination angles of the plate α, β and θx , θy are the following sinθx d (25a) sinα = L sinθy d sinβ = (25b) L In the case of a slow rate of change for the plate angles equations (24a), (24b) can be linearized 2mgd I θx = 0 (26a) ¨− m+ 2 x r L 2mgd I θy = 0 (26b) m + 2 y¨ − r L The equations (26a), (26b) are equivalent because of the symmetry of the plate. With the Laplace transformation, the following transfer functions can be obtained x 2mgdr2 = (27a) Px (s) = θx L(mr2 + I)s2 x 2mgdr2 Py (s) = = (27b) θy L(mr2 + I)s2
It should be noted, observer needs some time for convergence of eˆ˙ to Ball velocity v. However integral of error is bounded. Therefore, closed loop system remains passive. 5. EXPERIMENTAL VALIDATION The experiment validation was performed on the developed 2 DOF Ball and Plate laboratory setup. The control goal is to stabilize a ball on the plate. Sample time h = 0.02s. The adjusted robust controller for a ball stabilization is as follows: θx (tk ) = −0.17(d1 ξ1 (tk ) + 0.95ξ2 (tk ) (31) u(k) − u(k))). +0.952 ξ3 (tk ) + 0.045(ˆ Energy Aware mechanism for discrete system is chosen as ∆E(tk ) = σmg(sin α(tk ) − sin α(tk−1 )).
(32)
Results can be seen in Fig. 4-7
Fig. 4. Output signal
where s is the Laplace operator. The control algorithm was first verified using the continuous time simulation model in Matlab/Simulink for ball stabilizing in desired point mode. Model parameters are assumed unknown. For system input signal is a rotation angle of the platform. For Ball input signal is a projection of gravity force. Controller calculates force that should acting on Ball. Therefore, we can calculate required angles from condition u = mg sin α.
(28)
In this case co-vector v for force is unmeasured velocity. We can find it from observer: v = eˆ˙ = ξ˙1 = σξ2 .
(29)
Energy aware measurement takes the form ∆E = σmg
The input signals of the servos are constrained with the ˆ ≤ 180 due to its characteristics. Thats limits 0 ≤ u(t) corresponds to platform rotation limits −15 ≤ α ≤ 15
The maximum values of the tracking errors are ex (t) = 10 mm, ey (t) = 10 mm, average tracking error for both axes does not exceed 15 mm, which corresponds to necessary quality. 6. CONCLUSION
t1
sin α(τ )ξ2 (τ )dτ.
Fig. 5. Control signal
(30)
t0
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Output energy aware robust discrete controller for uncertain plants is presented. Proposed algorithm provides sta-
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output robust controller for uncertain plant. 2018 European Control Conference (ECC), 533–538. Folkertsma, G.A., Groothuis, S.S., and Stramigioli, S. (2018). Safety and guaranteed stability through embedded energy-aware actuators. 2018 IEEE International Conference on Robotics and Automation (ICRA), 2902– 2908. I.B. Furtat, A.L. Fradkov, D.L. (2015). Compensation of disturbances for mimo systems with quantized output. Automatica, 239–244. Laffranchi, M., Tsagarakis, N.G., and Caldwell, D.G. (2009). Safe human robot interaction via energy regulation control. 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, 35–41. Lee, D. and Huang, K. (2010). Passive-set-positionmodulation framework for interactive robotic systems. IEEE Transactions on Robotics, 26(2), 354–369. Schindlbeck, C. and Haddadin, S. (2015). Unified passivity-based cartesian force/impedance control for rigid and flexible joint robots via task-energy tanks. 2015 IEEE International Conference on Robotics and Automation (ICRA), 440–447. Stramigioli, S. (2015). Energy-aware robotics. Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, ser. Lecture Notes in Control and Information Sceinces, M.K. Camlibel, A.A. Julius, R. Pasumarthy, and J.M. Scherpen, Eds. Springer International Publishing, 37–50. Tadele, T.S., de Vries, T., and Stramigioli, S. (2014). The safety of domestic robotics: A survey of various safety-related publications. IEEE Robotics Automation Magazine, 21(3), 134–142.
Fig. 6. Tracking error
Fig. 7. Energy injected by the controller to the ball bility and passivity of closed loop system. Controller synthesis is based on consecutive compensator method. Energy aware mechanism is based on energy budget method. Computer simulation and experimental results were conducted for Ball and Plate mechatronic setup. Obtained results confirm performance of proposed approach. REFERENCES Alami, R., Albu-Schaeffer, A., Bicchi, A., Bischoff, R., Chatila, R., Luca, A.D., Santis, A.D., Giralt, G., Guiochet, J., Hirzinger, G., Ingrand, F., Lippiello, V., Mattone, R., Powell, D., Sen, S., Siciliano, B., Tonietti, G., and Villani, L. (2006). Safe and dependable physical human-robot interaction in anthropic domains: State of the art and challenges. 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, 1–16. Bobtsov, A. (2002). Robust output-control for a linear system with uncertain coefficients, automation and remote control. Automation and Remote Control, 1794–1802. Dobriborsci, D. and Kolyubin, S. (2017). Design and control of parallel kinematics platform for nonprehensile manipulation. 2017 IEEE International Workshop of Electronics, Control, Measurement, Signals and their Application to Mechatronics (ECMSM), 1–6. Dobriborsci, D., Margun, A., and Kolyubin, S. (2018a). Discrete robust controller for ball and plate system. 2018 26th Mediterranean Conference on Control and Automation (MED), 1–9. Dobriborsci, D., Margun, A., and Kolyubin, S. (2018b). Theoretical and experimental research of the discrete 269
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Robust Robust Robust Robust
energy energy aware aware output output controller controller for for energy aware output controller controller for uncertain plant energy aware output for uncertain plant uncertain plant uncertain plant Dmitrii Dobriborsci, ∗∗ Alexey Margun, ∗∗ Sergey Kolyubin ∗,∗∗ ∗,∗∗
Dmitrii Dobriborsci, ∗∗ Alexey Margun, ∗∗ Sergey Kolyubin ∗,∗∗ ∗,∗∗ Dmitrii Dobriborsci, ∗ Alexey Margun, ∗ Sergey Kolyubin ∗,∗∗ Dmitrii Dobriborsci, Alexey Margun, Sergey Kolyubin ∗ ∗ Control systems and Robotics faculty, ITMO University, St. ∗ Control systems and Robotics faculty, ITMO University, St. ∗ Control systems and Robotics faculty, ITMO University, St. Petersburg, Russia, ([email protected]) ∗ Petersburg, Russia, ([email protected]) ∗∗ Control systems and in Robotics faculty,Mechatronics ITMO University, St. ∗∗ Center for Technologies Robotics Petersburg, Russia, ([email protected]) for Technologies in Robotics and and Mechatronics Components, Components, ∗∗ Center Petersburg, Russia, ([email protected]) ∗∗ Innopolis University, Innopolis, Russia Center for Technologies in Robotics and Mechatronics Components, ∗∗ Innopolis University, Innopolis, Russia Center for Technologies in RoboticsInnopolis, and Mechatronics Innopolis University, Russia Components, Innopolis University, Innopolis, Russia Abstract: Abstract: Nowadays, Nowadays, safety safety guarantee guarantee during during physical physical Human-Robot Human-Robot Interaction Interaction (pHRI) (pHRI) is is important problem. Control over energy becomes very useful approach and provides possibility Abstract: Nowadays, safety guarantee during physical Human-Robot Interaction (pHRI) is important problem. Control over energy becomes very useful approach and provides possibility Abstract: Nowadays, safetyover guarantee during measurement physical Human-Robot Interaction (pHRI) is important problem. Control energy becomes very useful approach provides of safety implementation but of variables. In this of safety conditions conditions implementation but requires requires measurement of energy energyand variables. In possibility this paper paper important problem. Control over for energy becomes veryoutput useful control approach and provides possibility we extend our previous results discrete robust algorithm by insertion of of safety conditions implementation but requires measurement of energy variables. In this paper we extend our previous results for discrete robust output control algorithm by insertion of of safety conditions implementation but requires measurement ofisenergy variables. Ininsertion this paper energy aware mechanism in the structure of control system. It proven earlier that this way we extend our previous results for discrete robust output control algorithm by of this way energy aware mechanism in the structure of control system. It is proven earlier that we extend ourmechanism previous results forenergy discrete robust output control algorithm by insertion of make the controller aware of the that they inject into the plant. By combining this energy aware in the structure of control system. It is proven earlier that this way make the controller awareinofthe the energy that they system. inject into the plant.earlier By combining this energy aware mechanism structure of control It is proven that this way make the controller aware of the energy that they inject into the plant. By combining this approaches we here a robust discrete-time output controller for plant approaches we introduce introduce here robust discrete-time output controller for uncertain uncertain plant that that make theexponential controller aware of a energy thaterror they to inject into the plant. By passivity combining this provides convergence of tracking the bounded area the approaches we introduce here athe robust discrete-time output controller for and uncertain plantof that provides exponential convergence of tracking error to the bounded area and passivity of the approaches we introduce here a robust discrete-time output controller for uncertain plant that closed loop system. The proposed approach is validated through simulation and experiments provides exponential convergence of tracking error to the bounded area and passivity of the closed loop system. The proposed approach is validated through simulation and experiments provides exponential convergence trackingiserror to thethrough bounded area andand passivity of the closed loop system. The proposed ofapproach validated simulation experiments using Ball-and-Plate setup. using setup. closedBall-and-Plate loop system. The proposed approach is validated through simulation and experiments using Ball-and-Plate setup. © 2019, IFAC (International using Ball-and-Plate setup.Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: robust robust control, control, energy energy aware, aware, energy energy budget, budget, ball ball and and plate, plate, passive passive systems systems Keywords: robust control, energy aware, energy budget, ball and plate, passive systems Keywords: robust control, energy aware, energy budget, ball and plate, passive systems 1. INTRODUCTION INTRODUCTION and 1. and safe safe contact contact resemblance resemblance simultaneously simultaneously by by introducintroduc1. INTRODUCTION ing a controller shaping function that robustly handles and safe contact resemblance simultaneously by introducing a controller shaping function that robustly handles 1. INTRODUCTION and contact resemblance simultaneously by introducing asafe controller shaping function that robustly handles unexpected contact loss and avoids chattering behavior Nowadays, robots take part not only in industry, but step contact loss and avoids chattering behavior Nowadays, robots take part not only in industry, but step unexpected ing a controller shaping function that robustly unexpected contact loss and avoids chattering behavior that generated by switching based approaches. Nowadays, robots not life. only Therefore in industry, butsafety step that generated by switching based approaches. handles by step step enter enter intotake ourpart casual life. Therefore the safety by into our casual the contact loss andbased avoids chattering behavior Nowadays, robots take part not only in industry, but step unexpected that generated by switching approaches. by step enter into our casual life. Therefore the safety problem becomes becomes major. major. Especially Especially when when there there is is aa physphysproblem In Lee and Huang (2010) a passive set-position that generated by switching based approaches. by step enter into our casual life. Therefore the safety Lee and Huang (2010) a passive set-position modulation modulation problem becomes major. Especially when there is”safety” a phys- In ical human-robot human-robot interaction (pHRI). Definition ”safety” In Lee and Huang (2010) a passive It set-position modulation ical interaction (pHRI). Definition framework (PSPM) is enables to problem becomes major. Especially when there is a physframework (PSPM) is presented. presented. It enables users users to conconical human-robot interaction (pHRI). Definition ”safety” in pHRI was discovered in Alami et al. (2006). We note In Lee and Huang (2010) a passive set-position modulation in pHRI was discovered in Alami et al. (2006). We note nect a continuous-time robot’s position to a sequence of framework (PSPM) is presented. It enables users to conical human-robot interaction (pHRI). Definition ”safety” nect a continuous-time robot’s position to a sequence of in pHRI was discovered Alamiindex, et al. (2006). Wemean note framework (PSPM) is presented. It enables users to consome of them: them: maximuminpower power index, maximum mean some of maximum maximum slowly updating/sparse set-position signal (discrete-time) nect a continuous-time robot’s position to a sequence of in pHRI was discovered in Alami et al. (2006). We note slowly updating/sparse set-position signal (discrete-time) some of them: maximum power index, maximum mean strain injury injury criterion, criterion, head head injury injury criterion. criterion. Some Some of of them them slowly nect athe continuous-time robot’s position to(discrete-time) a sequence of updating/sparse set-position signal strain using simple spring coupling with damping injection, some of them: maximum power index, maximum mean using the simple spring set-position coupling with damping injection, strain injuryto injuryand criterion. Some ofbased them slowly are related related tocriterion, energy: head velocities and accelerations based updating/sparse signal (discrete-time) are energy: velocities accelerations while enforcing passivity of the closed-loop robotic sysusing the simple spring coupling with damping injection, strain injurydepend injuryand criterion. Some ofstratethem while enforcing of the closed-loop robotic sysare related tocriterion, energy: velocities accelerations based approaches on head the energy transfer. Control using the simplepassivity spring coupling with damping injection, approaches depend on the energy transfer. Control stratetem. The PSPM modulates set-position signal while enforcing passivity ofthe theoriginal closed-loop robotic sysare related to energy: velocities and accelerations based tem. The PSPM modulates the original set-position signal approaches depend on the energy transfer. Control strategies that that limit limit this this energy energy (mostly (mostly kinetic) kinetic) are are successful successful while enforcing passivity ofthe theoriginal closed-loop robotic sysgies in such a way that the modulated signal is as close to tem. The PSPM modulates set-position signal approaches depend on the energy transfer. Control stratesuch aPSPM way that the modulated signal is as close to gies that limit energy (mostly kinetic) arevelocity successful in robotics: the this position setpoint or maximum velocity are in tem. The modulates the original set-position signal in such a way that the modulated signal is as close to in robotics: the position setpoint or maximum are the original signal as possible (i.e. maximum information gies that limit this energy (mostly kinetic) are successful the original signal as possible (i.e. maximum information in robotics: the position setpoint or maximum velocity are adapted to stay within safe energy levels by estimating in such a way that the modulated signal is as close to adapted to stay within safe energy levels by estimating recovery for better performance), yet only to the extent the original signal as possible (i.e. maximum information in robotics: the position setpoint or joint maximum velocity are recovery for signal betterasperformance), yet only toinformation the extent adapted to stay within safe energy levels by estimating the energy of the robot based on velocities Tadele the original possible (i.e. maximum the energy thewithin robot safe based on joint velocities Tadele recovery permissible the available the system (i.e., for by better yetin only the extent adapted to of stay energy levels by estimating by the performance), available energy energy the to (i.e., the energy of the robot based on joint velocities Tadele et al. (2014), Laffranchi et al. al. (2009). Moreover, safety permissible recovery for better performance), yetin only tosystem the extent et al. (2014), Laffranchi et (2009). Moreover, safety passivity constraints). permissible by the available energy in the system (i.e., theal. energy of the robot based on joint of velocities Tadele passivity constraints). et (2014), Laffranchi et al. (2009). Moreover, safety may be achieved by ensuring passivity the robot. For by the available energy in the system (i.e., passivity constraints). may be(2014), achieved by ensuring passivity of the robot. For permissible et al. Laffranchi et al. (2009). Moreover, safety may be achieved by ensuring the if For passivity In this we instance, in Stramigioli Stramigioli (2015)passivity is proven provenofthat that ifrobot. system constraints). instance, in (2015) is aa system this research research we modify modify discrete discrete robust robust output output conconmay be isachieved by ensuring passivity ofthat the ifrobot. For In instance, in Stramigioli (2015) is proven a system troller presented earlier in Dobriborsci et al. (2018b), DoIn this research we modify discrete robust output con(robot) not strictly passive, it is always possible to find (robot) is not strictly passive, it is always possible to find troller presented earlier in Dobriborsci et al. (2018b), DoIn thispresented research we modify discrete of robust output coninstance, in Stramigioli (2015) isisproven that if a system troller earlier in Dobriborsci etenergy al. (2018b), Do(robot) is not strictly passive, it always possible to find briborsci et al. (2018a) by including energy tank into such passive environment that destabilizes the system and briborsci et al. (2018a) by including of tank into such passive environment that destabilizes the system and troller presented earlier in Dobriborsci et al. (2018b), Do(robot) isinfinite notenvironment strictly passive, it is always possible to find such passive that destabilizes the system and controller structure. briborsci et al. (2018a) by including of energy tank into extracts energy from it As result system becomes extracts infinite energy from itdestabilizes As result system becomes controller structure. briborsci et al. (2018a) by including of energy tank into such passive environment that the system and extracts energy fromoperating it As result system becomes controller structure. unstable.infinite Therefore, robots operating in an an unpredictable unstable. Therefore, robots in unpredictable Proposed controller controller structure. was extracts infinite energy fromoperating it Astoresult system becomes controller was tested tested with with computer computer simulation simulation unstable. Therefore, robots in an unpredictable environment can be guaranteed remain stable if and and Proposed environment can be guaranteed to remain stable if and experimental setup. Experimental researches were Proposed controller was tested with computer simulation unstable. Therefore, robots operating in an unpredictable and experimental setup. Experimental researches were environment can be guaranteed to remain stable if and only if if they they are are passive passive Folkertsma Folkertsma et et al. al. (2018). (2018). There There Proposed controller was tested with computer simulation only conducted for ”Ball and plate system” (plate with two and experimental setup. Experimental researches were environment can be guaranteed to remain stable if and conducted for ”Ball and plate system” (plate with two only if they are passive Folkertsma et al. (2018). There were published a number of papers in this field of study. In and experimental setup. Experimental researches were were published a number of papers in et thisal.field of study. In conducted for ”Ball and plate system” (plate with two rotational degrees of freedom and ball located on the plate) only if they are passive Folkertsma (2018). There rotational degrees of freedom and ball located on the plate) were published a number of papers in this field of study. In Schindlbeck and Haddadin (2015) a novel hybrid Cartesian conducted for tasks: ”Ball and plate system” (plate with two Schindlbeck and Haddadin (2015) a novel hybrid Cartesian rotational degrees of freedom and ball located on the plate) for following object stabilization in the specific were published number of(2015) papers thishybrid fieldwith ofCartesian study. In for following tasks: object stabilization in the specific Schindlbeck anda Haddadin novel force/impedance controller that is isain equipped energy rotational degrees of freedom and ball located on thespecific plate) force/impedance controller that equipped with energy point on the plate, optimal point-to-point motions and for following tasks: object stabilization in the Schindlbeck and Haddadin hybrid Cartesian on the plate, optimal point-to-point and force/impedance controller that isa novel equipped with energy tanks to preserve preserve passivity(2015) is proposed. proposed. It provides provides ac- point for following tasks:tracking. object stabilization in motions the specific tanks to passivity is It acdesired Such kind of systems are point ontrajectory the plate, optimal point-to-point motions and force/impedance controller that is equipped with energy desired trajectory tracking. Such kind of systems are tanks to preserve passivity is proposed. It provides accurate force tracking, full compliant impedance behavior, point on theinplate, optimal point-to-point and curate force tracking, full compliant impedance behavior, desired trajectory tracking. Suchautomobile kind of motions systems are widely used flight simulators, simulators, tanks to preserve passivity is proposed. It provides acused in flight simulators, automobile simulators, curate force tracking, full compliant impedance behavior, widely desired trajectory tracking. Such kind of systems are widely used in flight simulators, automobile simulators, industrial automation such as fast sorting Dobriborsci and curate forcewas tracking, fullsupported compliant This work financially by impedance Government behavior, of Russian industrial automation as fast sorting Dobriborsci and widely used in flight such simulators, automobile simulators, This work was financially supported by Government of Russian Kolyubin (2017). industrial automation such as fast sorting Dobriborsci and Federation (Grant 08-08). The reported by study was partially funded This work was financially supported Government of Russian Kolyubin (2017). Federation (Grant 08-08). The reported study was partially funded industrial automation such as fast sorting Dobriborsci and This work was financially supported by Government of Russian Kolyubin (2017). by RFBR according to the research The Federation (Grant 08-08). The reported project study was18-38-20037. partially funded by RFBR according to the research project 18-38-20037. The The paper is Kolyubin (2017). Federation (Grant 08-08). The reported study was partially funded The paper is organized organized as as follows. follows. Section Section 22 is is devoted devoted to to development of the modified version ofproject the energy-aware output by RFBR according to the research 18-38-20037. The development of the modified version of the energy-aware output the mathematical problem statement. Section 3 contains The paper is organized as follows. Section 2 is devoted to by RFBR was according to the research project 18-38-20037. The controller supported by the Ministry of energy-aware Science and output Higher the mathematical problem statement. Section 3 contains development of the modified version of the The paperdescription is organized as follows. Section 2 is devoted to controller was supported by the Ministry of Science and Higher controller and stability analysis. The Embedthe mathematical problem statement. Section 3 contains development of the modified version of the energy-aware output Education of Federation controller description and stability analysis. The Embedcontroller supported by the(8.8885.2017/8.9) Ministry of Science and Higher Education was of Russian Russian Federation (8.8885.2017/8.9) the mathematical problem statement. Section 3 contains controller description and stability analysis. The Embedcontroller was supported by the(8.8885.2017/8.9) Ministry of Science and Higher Education of Russian Federation controller description and stability analysis. The EmbedEducation of Russian Federation (8.8885.2017/8.9)
2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 264 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 264 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 264 10.1016/j.ifacol.2019.11.178 Copyright © 2019 IFAC 264
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ded Energy Aware algorithm in Section 3 as well. Section 4 presents the experimental setup and its mathematical model. Results of simulation and experiments are presented in Section 5. We conclude the paper with discussion on achieved results and its application. 2. PROBLEM STATEMENT
σ > α+β is a chosen by developer coefficient that regulates estimation convergence speed. Choose observation error as follows: η(t) = LT e(t) − ξ(t).
(6)
Taking into account (5), calculate derivative of (6):
Consider linear plant
η(t) ˙ = σΓη(t) + LT e(t). ˙
(7)
Q(p)y(t) = R(p)(u(t) + f (t)), (1) where Q(p) and R(p) are linear differential operators of degrees n and m respectively with unknown coefficients, y(t) is a plant output, u(t) is a control signal, f (t) is a bounded piece wise smooth external disturbance, ρ = n − m is a plant relative degree.
Closed loop system consists of plant (1), control law (4) and observer (5) takes the form: ε(t) ˙ = Aε(t)+B(−βe+ (α+ β)(e(t) − eˆ(t)) + B1 ϕ(t), (8) η(t) ˙ = σΓη(t) + LT e(t). ˙
Reference model is described by linear differential equation
According to Bobtsov (2002) closed loop system (8) exponentially converges to the zero stable state in the case of disturbance absence. Otherwise, (8) exponentially convergence to the bounded area.
(2) Qm (p)ym (t) = Rm (p)r(t), where Qm (p) and Rm (p) are linear differential operators with known coefficients, ym (t) is a reference plant output, r(t) is a piece-wise smooth input of reference plant. The control goal is to design controller that provides tracking of the plant (1) output for the output of reference model (2) with pre-specified accuracy for finite time: (3) |y(t) − ym (t)| ≤ δ, ∀t > T, where δ is a tracking accuracy, T is a transient time. We don’t try to provide necessary transient time but it is necessary to provide its existence (condition (3)). Let us introduce following assumptions • Unknown coefficients of plant (1) belong to the known compact set Ξ. • R(λ) is Hurwitz polynomial, where λ is an imaginary unit. • Plant (1) and model (2) are passive. 3. CONTROLLER SYNTHESIS 3.1 Continuous consecutive compensator In Bobtsov (2002) continuous control law is proposed u = −(α + β)D(p)ˆ e(t), (4) where α,β > 0 are coefficients chosen by developer (greater value of α and β correspond to higher stability of closed loop system), D(λ) is Hurwitz polynomial of degree ρ − 1, eˆ(t) is an estimate of tracking error e(t) = y(t) − ym (t).
It is necessary to know ρ − 1 unmeasured derivatives of tracking error e(t) for implementation of control law (4). Thus, introduce observer for its estimation: ˙ = σΓξ(t) + σGe(t), ξ(t) (5) eˆ(t) = Lξ(t),
where ξ(t) ∈ Rρ−1 is an observer state vector, Γ = 0 Iρ−2 is Hurwitz matrix, G = [0 0 c1 ]T , −c1 ... −cρ−1 Iρ−2 is a unit matrix of ρ − 2 order, L = [1 0 ... 0],
265
3.2 Discrete consecutive compensator Lets obtain discrete realization of controller for its implementation on digital computing devices. Rewrite tracking error as e(t) = e(tk ) + hψ(ξ(tk ), tk ), where e(tk ) is a discrete value of tracking error e(t) on k-th step, h = tk − tk−1 is a sampling time, ψ(ξ(tk ), tk ) is a bounded function. Moreover ψ(ξ(tk ), tk ) is a Lipschitzian function because of plant is linear and disturbance is a piece-wise smooth. Discrete version of observer with use of right derivatives and replacement of tracking error by its discrete measurement takes the form ξ1 (tk+1 ) = ξ1 (tk ) + hσξ2 (tk ), ξ2 (tk+1 ) = ξ2 (tk ) + hσξ3 (tk ), ............................ (9) ξρ−1 (tk+1 ) = σ(−c1 ξ1 (tk ) − c2 ξ2 (tk ) − ... + cρ−1 e(tk )) eˆ(tk ) = ξ1 (tk ). Rewrite discrete derivative (9) observer in matrix representation ξ(tk+1 ) = (I + hσΓ)ξ(tk ) + σhGe(tk ), (10) eˆ(tk ) = Lξ(tk ).
¯>0 From lemma I.B. Furtat (2015) it follows existence of h ¯ limt→∞ ||ξ(t) − ξ(tk )|| < C, where such, that ∀h < h, C > 0. That’s mean that we always can find sampling time h satisfying required accuracy. Dynamics of discrete observer error is described by equations η(t) = LT e(t) − ξ(k) + hψ(t),
(11) η(t) ˙ = LT e(t) ˙ + σΓη(t) + σΓhψ(t). Remark 1.Last terms of equations (11) can be considered as additional bounded disturbance in continuous time linear system. So, here to reduce disturbances we have to choose h such small as possible (see statement below) Control law (4) takes the form :
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261
u(tk ) = −(α + β)(d1 ξ1 (tk ) + d2 σξ2 (tk ) + +d3 σ 2 ξ3 (tk ) + ...). (12)
where υ is a small positive number.
Closed loop system takes the form
(17) where R1 , R2 are positive defined matrices due to the choose of α, β and σ; θ is a bounded function.
˙ = Aε(t) + B(βe(t) + (α + β)Lη(t)) + Bhψ(t)+ ε(t) B1 ϕ(t), η(t) ˙ = LT e(t) ˙ + σΓη(t) + σhΓψ(t) (13)
Replacement of continuous value e(t) by sum of its discrete measurement e(tk ) and bounded function allows us to consider continuous plant with discrete controller as a continuous system with additional disturbances in the case of linear plant (1) and controller (12), (9). Proposed replacement sufficiently simplifies stability analysis of closed loop system. 3.3 Stability analysis Statement ¯ > 0 such that for any h < h ¯ discrete There exist α, β, σ, h observer (10) and control law (12) provide exponential convergence of tracking error e(t) to the bounded area.
R1 = Q1 + 2βLT B T P1 − υP1 B1 B1T P1 −
hυP1 BB T P1 − (α + β)υP1 BB T P1 + (β −1 + βυ)I, R2 = σQ2 − 2(α + β)P2 BL − υ −1 I − 2σ
hυP2 ΓΓT P2 − βP2 AAT P2 − βυP2 BLLT B T P2 −
(18)
υhP2 BB T P2 − υP2 B1 B1T P2 , 2h T 2 + 2σh T ϕ ϕ+ ψ ψ). θ = sup( υ υ
We always can replace bound on derivative of Lyapunov function (17) by inequality λmin R1 , V˙ ≤ −ζV + θ, ζ = λmax (P1 ) (19) θ θ e−ζt − . V ≤ V (0) − ζ ζ Taking into account λmin (P1 )e2 ≤ λmin (P1 )ε2 ε ≤ V we obtain
Prove of Statement Consider Lyapunov function T
Taking into account (16) rewrite (15) in the form V˙ ≤ −εT R1 ε − η T R2 η + θ,
T
V (t) = ε (t)P1 ε + η (t)P2 η(t), (14) where P1 , P2 are solutions of Lyapunov equations AT P1 + P A = −Q1 , ΓT P2 + P Γ = −Q2 , Q1 and Q2 are positive defined symmetric matrices. Obtain derivative of Lyapunov function (14) along trajectories (13) V˙ = −εT (Q1 + 2βLT B T P1 )ε + 2εT P1 B1 ϕ + 2hεT P1 Bψ+
|e| ≤
1 λmin (P1 )
θ V (0) − ζ
e−ζt
θ + ζ
(20)
Thereby, tracking error exponentially converges to the area bounded by inequality
|e| = |y − ym | ≤
1 θ λmin (P1 ) ζ
(21)
2(α + β)εT P1 BLη − η T (σQ2 − 2(α + β)P2 BL)+
3.4 Energy Aware algorithm
Bound terms of the right part of (15) by inequalities
To ensure effective regulation of the plant, it is proposed to use the energy budget approach introduced in Folkertsma et al. (2018).
2σhη T P2 Γψ + 2η T P2 Aε − 2βη T P2 BLε + 2hη T P2 Bψ+ 2ηP2 B1 ϕ. (15)
1 2εT P1 B1 ϕ ≤ υεT P1 B1 B1T P1 ε + ϕT ϕ υ 1 2εT P1 Bψ ≤ υεT P1 BB T P1 ε + ψ T ψ, υ 1 2εT P1 BLη ≤ υεT P1 BB T P1 ε + η T η, υ 1 T T T T 2η P2 Γψ ≤ υη P2 ΓΓ P2 η + ψ ψ, υ 1 T T T 2η P2 Aε ≤ βη P2 AA P2 η + εT ε, β 1 2η T P2 BLε ≤ υη T P2 BLLT B T P2 η + εT ε, υ 1 T T T T 2η P2 Bψ ≤ υη P2 BB P2 η + ψ ψ, υ 1 T T T 2η P2 B1 ϕ ≤ υη P2 B1 B1 P2 η + ϕT ϕ. υ
Let us assume that at moment t = 0 controller has energy budget Em and cannot inject to plant more energy than Em . Consider energy supplied to the system by controller for the time (t0 , t1 ): ∆E = (16)
t1
u(τ )v(τ )dτ,
(22)
t0
where v is a co-vector of u, i.e. product uv is an energy. For example, in mechanical systems if u is a force than v is a velocity. If uv > 0 than energy is transferred from controller to plant Otherwise, plant returns energy to controller. For discrete controller input signal u(t0 ) is a constant for a time (t0 , t1 ). Therefore
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∆E = u(t0 )
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t1 t0
v(τ )dτ = u(t0 )(q(t1 ) − q(t0 )),
(23)
where q is a coordinate. Energy Aware algorithm that don’t allows controller (12) to give plant more energy than energy budget: • Calculating of control value u(tk ) by discrete consecutive compensator algorithm. • Calculating of ∆E. • If ∆E ≥ 0 and current energy of controller Em (tk ) > ∆E than controller injects its energy into plant: u = u(tk ), Em (tk+1 ) = Em (tk ) − ∆E. • If ∆E ≥ 0 and current energy of controller Em (tk ) < ∆E than controller don’t have enough energy: u(tk ) = 0, Em (tk+1 ) = Em (tk ). • If ∆E < 0 than plant returns (virtually) its energy to controller: u = u(tk ), Em (tk+1 ) = Em (tk ) − ∆E. • go to beginning.
Such switching algorithm allows to bound total enegy of closed loop system.Therefore system is passive. Schematically control algorithm with Energy Aware Mechanism is illustrated on Fig. 1.
the screen, pressing the two layers together. This actions create an electrical connection between layers. However, both X and Y axes cannot be read simultaneously, because the concept works by applying a voltage across one layer, and looking for the voltage that appears on the another. Therefore, the system operates by alternately applying a voltage to one layer and reading off of the another. The voltages are then read in by an analog-to-digital (A/D) converter to be used as a coordinate value. There are several types of the resistive touch-screens, e.g. 4wire, 5-wire and 8-wire. These labels refer to number of wires between the screen and the controller. In the current study we chosen a 4-wire resistive touch AST150C. It features the typical advantages of the resistive type such as operational stability, detection accuracy, ease of introducing, competitive cost and so on. It have great durability with up to as much as 10 million touches, due to its structure. 4.1 Mathematical model We consider the following equations of motion for experimental setup Ball and Plate to design the robust output controller. In this part we introduce the following assumptions: • There is no slipping for ball. • The ball is completely symmetric and homogeneous. • Friction forces are neglected.
Fig. 1. Control scheme with Energy Aware mechanism 4. EXPERIMENTAL SETUP We built a tiltable platform with two degrees of freedom consisting of the following components: resistive touchscreen, servo-drives, square plate and connecting links. The control system was implemented on the Arduino single-board computer. Resistive touch-screen determines the coordinates of the object. The kinematic scheme of the developed system is presented in the Fig.2 Each tilting axis is operated on by a servomotor HITEC HS-5685mh with JR connector.
Fig. 2. Kinematic scheme of the mechanism
The platform was specially built to conduct the following experiments: object stabilization in the specific point on the plate, its optimal point-to-point motions and desired motion trajectory tracking as well as dynamics object-toplate contact model identification. In contrast to previous papers here we detect the ball position using resistive touch-screen system. An analog resistive touch-screen consist of a glass or acrylic panel that is coated with electrically conductive and resistive layers made with indium tin oxide (ITO). These layers are separated by invisible spacers/ To generate a position, the user or an object must exceed the activation force of 267
Fig. 3. Parallel kinematics robot Ball and Plate
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• The ball and plate are in contact all time.
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Simulation results are on Fig.
The angles of servo arms θx , θy are assumed to be the inputs, while the ball position on x, y axis are assumed to be the output. Here we derive dynamical equations of ballon-plate system, by the help of Lagrangian. The following mathematical equations are based on . So the nonlinear differential equations for the ball and plate system are presented below. I 2 ˙ m+ 2 x ¨ − m xα˙ + y α˙ β + mgsinα = 0 (24a) r I 2 ˙ ˙ m + 2 y¨ − m y β + xα˙ β + mgsinβ = 0 (24b) r where m, r, I are mass of the ball, radius of the ball, mass moment of inertia x, y are position of axis, α, β, α, ˙ β˙ are inclination angles of the plate, angular velocity of the plate, g, L, d are gravitation, plate side length, length between the joint and the center of the gear, respectively. The relations between inclination angles of the plate α, β and θx , θy are the following sinθx d (25a) sinα = L sinθy d sinβ = (25b) L In the case of a slow rate of change for the plate angles equations (24a), (24b) can be linearized 2mgd I θx = 0 (26a) ¨− m+ 2 x r L 2mgd I θy = 0 (26b) m + 2 y¨ − r L The equations (26a), (26b) are equivalent because of the symmetry of the plate. With the Laplace transformation, the following transfer functions can be obtained x 2mgdr2 = (27a) Px (s) = θx L(mr2 + I)s2 x 2mgdr2 Py (s) = = (27b) θy L(mr2 + I)s2
It should be noted, observer needs some time for convergence of eˆ˙ to Ball velocity v. However integral of error is bounded. Therefore, closed loop system remains passive. 5. EXPERIMENTAL VALIDATION The experiment validation was performed on the developed 2 DOF Ball and Plate laboratory setup. The control goal is to stabilize a ball on the plate. Sample time h = 0.02s. The adjusted robust controller for a ball stabilization is as follows: θx (tk ) = −0.17(d1 ξ1 (tk ) + 0.95ξ2 (tk ) (31) u(k) − u(k))). +0.952 ξ3 (tk ) + 0.045(ˆ Energy Aware mechanism for discrete system is chosen as ∆E(tk ) = σmg(sin α(tk ) − sin α(tk−1 )).
(32)
Results can be seen in Fig. 4-7
Fig. 4. Output signal
where s is the Laplace operator. The control algorithm was first verified using the continuous time simulation model in Matlab/Simulink for ball stabilizing in desired point mode. Model parameters are assumed unknown. For system input signal is a rotation angle of the platform. For Ball input signal is a projection of gravity force. Controller calculates force that should acting on Ball. Therefore, we can calculate required angles from condition u = mg sin α.
(28)
In this case co-vector v for force is unmeasured velocity. We can find it from observer: v = eˆ˙ = ξ˙1 = σξ2 .
(29)
Energy aware measurement takes the form ∆E = σmg
The input signals of the servos are constrained with the ˆ ≤ 180 due to its characteristics. Thats limits 0 ≤ u(t) corresponds to platform rotation limits −15 ≤ α ≤ 15
The maximum values of the tracking errors are ex (t) = 10 mm, ey (t) = 10 mm, average tracking error for both axes does not exceed 15 mm, which corresponds to necessary quality. 6. CONCLUSION
t1
sin α(τ )ξ2 (τ )dτ.
Fig. 5. Control signal
(30)
t0
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Output energy aware robust discrete controller for uncertain plants is presented. Proposed algorithm provides sta-
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output robust controller for uncertain plant. 2018 European Control Conference (ECC), 533–538. Folkertsma, G.A., Groothuis, S.S., and Stramigioli, S. (2018). Safety and guaranteed stability through embedded energy-aware actuators. 2018 IEEE International Conference on Robotics and Automation (ICRA), 2902– 2908. I.B. Furtat, A.L. Fradkov, D.L. (2015). Compensation of disturbances for mimo systems with quantized output. Automatica, 239–244. Laffranchi, M., Tsagarakis, N.G., and Caldwell, D.G. (2009). Safe human robot interaction via energy regulation control. 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, 35–41. Lee, D. and Huang, K. (2010). Passive-set-positionmodulation framework for interactive robotic systems. IEEE Transactions on Robotics, 26(2), 354–369. Schindlbeck, C. and Haddadin, S. (2015). Unified passivity-based cartesian force/impedance control for rigid and flexible joint robots via task-energy tanks. 2015 IEEE International Conference on Robotics and Automation (ICRA), 440–447. Stramigioli, S. (2015). Energy-aware robotics. Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, ser. Lecture Notes in Control and Information Sceinces, M.K. Camlibel, A.A. Julius, R. Pasumarthy, and J.M. Scherpen, Eds. Springer International Publishing, 37–50. Tadele, T.S., de Vries, T., and Stramigioli, S. (2014). The safety of domestic robotics: A survey of various safety-related publications. IEEE Robotics Automation Magazine, 21(3), 134–142.
Fig. 6. Tracking error
Fig. 7. Energy injected by the controller to the ball bility and passivity of closed loop system. Controller synthesis is based on consecutive compensator method. Energy aware mechanism is based on energy budget method. Computer simulation and experimental results were conducted for Ball and Plate mechatronic setup. Obtained results confirm performance of proposed approach. REFERENCES Alami, R., Albu-Schaeffer, A., Bicchi, A., Bischoff, R., Chatila, R., Luca, A.D., Santis, A.D., Giralt, G., Guiochet, J., Hirzinger, G., Ingrand, F., Lippiello, V., Mattone, R., Powell, D., Sen, S., Siciliano, B., Tonietti, G., and Villani, L. (2006). Safe and dependable physical human-robot interaction in anthropic domains: State of the art and challenges. 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, 1–16. Bobtsov, A. (2002). Robust output-control for a linear system with uncertain coefficients, automation and remote control. Automation and Remote Control, 1794–1802. Dobriborsci, D. and Kolyubin, S. (2017). Design and control of parallel kinematics platform for nonprehensile manipulation. 2017 IEEE International Workshop of Electronics, Control, Measurement, Signals and their Application to Mechatronics (ECMSM), 1–6. Dobriborsci, D., Margun, A., and Kolyubin, S. (2018a). Discrete robust controller for ball and plate system. 2018 26th Mediterranean Conference on Control and Automation (MED), 1–9. Dobriborsci, D., Margun, A., and Kolyubin, S. (2018b). Theoretical and experimental research of the discrete 269