Barrios, Asier

Dombóvári, Zoltán

Habib, Giuseppe

English

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ENOC 2020, July 5-10, 2020, Lyon, FranceBistability in nonlinear elastic robotic arms subject to delayed feedback controlGiuseppe Habib∗, Asier Barrios‡, Zoltan Dombovari∗∗MTA-BME Lendület Machine Tool Vibrations Research Group, Department of Applied Mechanics,Budapest University of Technology and Economics, Budapest, Hungary‡Dynamics and Control Department, Ideko, Elgoibar, SpainSummary. Stability and bifurcation analysis of a non-rigid robotic arm controlled in a time delayed feedback loop is addressed inthis work. The study aims at revealing the dynamical mechanisms leading to the appearance of limit cycle oscillations existing in thestable region of the trivial solution of the system, which are related to the combined dynamics of the robot control and its structuralnonlinearities. A numerical study of the bifurcations occurring at the loss of stability enables the development of strategies to eliminatethis undesired bistable phenomenon by the implementation of special additional nonlinearities in the control force.IntroductionRobots are increasingly adopted in modern manufacturing facilities, thanks to their versatility and relatively low cost [1].Milling operation is one of the operations robots are intended to be used for, where complicated tool trajectories canbe realized in a large workspace with a relatively low cost. The relative vibrations between workpiece and tool are atroublesome phenomenon in milling that is mainly caused by so-called regenerative vibration. The main solution to avoidthem is to increase stiffness and damping and try to disturb delays introduced by the regenerative effect [2]. Increasingstiffness is hardly achievable in robots, since robotic arms are naturally slender and not especially stiff [3]. This makesthem particularly prone to vibrations. The main method to mitigate these vibrations consists in implementing an activecontroller working in a feedback loop. In most of cases, this controller reads in input the acceleration of the end effector(EE, see Fig. 1a) and sends a proportional signal to the robot controller in order to counteract and suppress the vibrations.This signal is added to the position controller of the robot, required to make the robotic arm follow the prescribed pathduring machining.Although this procedure is rather straightforward to be implemented, there are several aspects, which might underminethe system stability if not properly accounted for. (i) Robotic arms are naturally slender and they cannot be assumedrigid, especially if they are subject to strong forces, as in the case of machining. (ii) Moreover, since actuators are placedat the joints of the arm, the system is underactuated. Depending on the position of the sensors, either near the motoror near the EE, the system can be considered as collocated or non-collocated, which have relevant consequences on thesystem stability [4, 5]. (iii) Robot configuration changes continuously during operation and the drive components of therobot generates non-negligible nonlinearities; as we will illustrate in this study, these nonlinearities might have importantconsequences on the system robustness. (iv) Robot’s controller is unavoidably subject to time delay in the feedback loop;although this is often negligible, if large control gains are required to counteract strong forces, time-delay can still generateinstabilities.This study is motivated by the appearance of unexpected vibrations in a real industrial robotic arm for milling opera-tions. This robot is equipped with a built-in controller (most probably a proportional-derivative controller) for its correctpositioning and with an additional controller proportional to the EE acceleration, to counteract machine tool vibrations(Fig. 1a). Although the control parameters of the system were set such that the system was stable, when subject to verysmall external forcing, in some occasions the robotic arm exhibited either large or small oscillations, which suggests thatit was in bistable conditions. The objective of this work is to define and study a general simplified model for this system inorder to understand the origin of the bistability and define methods to avoid it. From a broader prospective, this researchaim at providing a reliable modelling of robotic manufacturing.Mathematical modelThe mathematical model adopted is a two-degrees-of-freedom (DoF) mechanical system (Fig. 1a), consisting in twolumped masses, m1 and m2, connected by a linear damper c, and a nonlinear spring knl. The two DoF of the systemrepresent the two dominant DoF measured for the actual robotic arm at hand in a certain frequency bandwidth. Thenonlinearity models a stiffness nonlinearity observed during measurement, most probably originated in the joints. Thissimplified mechanical model captures the most important features of the real robotic arm considered in the study. Aprescribed reference trajectory xd is programmed, such that, in idea circumstances, an identical constrained motion xr isimposted to m1 via a spring of stiffness k with a certain time delay τr. This enables the robot to follow the prescribedpath. The equation of motion has the following form:m1x¨1 + c (x˙1 − x˙2) + knl(∆x) (x1 − x2) + kx1 = kxr,m2x¨2 + c (x˙2 − x˙1) + knl(∆x) (x2 − x1) = 0,(1)where knl(∆x) = k2 + κ∆x2 (∆x := x2 − x1) and c is the damping coefficient. Apart from the position controllerintegrated in the robot, an additional signal xf, proportional to the acceleration of the EE, is added to xr. This generates afinal constrained motion xr(t) = xd(t− τr) +Kx¨2(t− τf), where τf is the delay of the acceleration feedback.ENOC 2020, July 5-10, 2020, Lyon, FranceKFigure 1: a) shows the sketch of the proposed additional control ; b) stability chart in the (τ,K) space for various for ω1 = ω2 =2pi rad/s, χ =: (50, 10, 5, 1)%, Kcr = 0.0253 s2, c) time evolutions for different initial conditions (x2,0 = 0, x˙2,0 6= 0) whenµ = −1000000 (s m)−2, K = 0.01 s2, τ = 0.6594 s.In order to focus on the instabilities generated by the acceleration feedback control, in this work we assume constantdesired position, that is, xd(t) := xd, which results in the equilibrium (x1, x2) = (xd, xd). By introducing the per-turbation x1 := x1 + u1 and x2 := x2 + u2 the stability of the local equilibrium can be studied. Via a standardnon-dimensionalization procedure, equations of motion around the equilibrium are reduced tou¨1 + 2χrω2 (u˙1 − u˙2) + ω22r (u1 − u2) + µr(u1 − u2)3 + ω21u1 = ω21Ku¨2τ ,u¨2 + 2χω2 (u˙2 − u˙1) + ω22 (u2 − u1) + µ(u2 − u1)3 = 0,(2)where r := m2/m1, ω21 := k1/m1, ω22 := k2/m2, χ := c/(2m2ω2), µ := κ/m2, u¨2τ := u¨2(t− τ) and τ := τr + τf.Stability and bistable behaviorBy linearising (2) and setting χ := 0 the linear stability of the corresponding neutral equation can be investigated (seeFig. 1b). A necessary condition for stability is that the neutral coefficient (here ω21K) has magnitude less than one, thatis |Kcr| ≤ ω−21 . It can be seen that stability boundaries always lie between the critical values Kcr and −Kcr, with apattern that depends on the complete time delay of the feedback control. In the case of small damping, this repeating lobestructure significantly erodes the stable region, which is limited to a narrow region around K = 0.Apart from the local stability of the trivial solution, time simulations show that the nonlinearity of the system has signif-icant impact on its global stability. In particular, if the system is subject to a softening nonlinearity (µ < 0) simulationsfor parameter values within the stable region tend to diverge if initial conditions are sufficiently large. This phenomenonis caused by the subcritical characteristic of the bifurcations occurring at the stability limit and it is probably directlyrelated to the bistable behaviour observed in real robotic arms. The detailed analysis of the bifurcation behaviour of thesystem enables us to design additional nonlinearities to be included in the control force algorithm in order to control thecharacteristic of the bifurcations and enforce supercritically, therefore eliminating the bistable behavior.ConclusionsThe stability and bifurcation analysis of a simplified model of robotic arm subject to acceleration feedback was performed.Results illustrated that the stability chart is characterized by a critical value of the control gain, which is a necessarycondition for guaranteeing stability, and by a repeating stability limit pattern, which strongly depends on the time delayand on system damping. The mechanism connecting bistable behavior and softening nonlinearity was also identified. Thefull understanding of this mechanism enables the development of a control algorithm, based on nonlinear functions, whichforces the bifurcation to be supercritical, suppressing bistable behaviour. The next step of the study consists in includingin the model regenerative forces generated by machining processes and verifying if the results so far obtained persist.References[1] Hagele M. (2016) Robots Conquer the World [Turning Point]. IEEE Robot Autom Mag 23:120-118.[2] Munoa J., Beudaert X., Dombovari Z., Altintas Y., Budak E., Brecher C., Stepan G. (2016) Chatter suppression techniques in metal cutting. CIRPAnnals 65(2):785-808.[3] Mousavi S., Gagnol V., Bouzgarrou B.C., Ray P. (2018) Stability Optimization in Robotic Milling through the Control of Functional Redundancies.Robot Com-Int Manuf 50:181-192.[4] De Luca A., Schroder D., Thummel M. (2007) An Acceleration-Based State Observer for Robot Manipulators with Elastic Joints. Proc IEEE IntConf Robot Autom:3817-3823.[5] Habib G., Rega G., Stepan G. (2015) Stability Analysis of a Two-degree-of-freedom Mechanical System Subject to Proportional-Derivative DigitalPosition Control. J Vib Control 21:1539-1555.[6] Stépán, G. (1989) Retarded Dynamical Systems: Stability and Characteristic Functions. Longman Scientific & Technical, London.

Bistability in nonlinear elastic robotic arms subject to delayed feedback control

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Bistability in nonlinear elastic robotic arms subject to delayed feedback control

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