IROS2019国际学术会议论文集2429

Stiffness Bounds for Resilient and Stable Physical Interaction of Articulated Soft Robots Riccardo Mengacci1, Franco Angelini1,2, Manuel G. Catalano2, Giorgio Grioli2, Antonio Bicchi1,2and Manolo Garabini1 AbstractIt is widely recognized that impedance modulation is a key aspect in applications in which robots signifi cantly inter- act with the environment or humans. Either active impedance controllers or actuators with passive variable impedance can be exploited to modulate the impedance. However, methods capable of determining the right (constant or time-varying) impedance profi le in order to guarantee task performance as well as resilience and stability are required. In this work, we discuss how task-related aspects such as uncertainties, contact surface shapes, and interaction forces, set bounds on the admissible Cartesian stiffness. We recall that an upper bound on the stiffness is required to prevent high forces exchanged during the interaction to guarantee adaptability and safety. Despite this, however, there is also a lower bound to be considered in order to preserve stability during the interaction. To this purpose, we study the interaction of a robot, with controllable Cartesian impedance, with a curved surface. Thus, we provide an analytic lower bound for the Cartesian stiffness that guarantees the stability of such interaction task, and we prove that this bound directly depends on task parameters, namely contact force and surface curvature. Theoretical results are experimentally validated on robots powered by variable stiffness actuators and compliance controlled industrial robots. I. INTRODUCTION For decades, robots were employed in industries for the execution of positioning operations of perfectly known parts, in completely structured environments, usually surrounded by cages to minimize the risk of human injuries and robot damages. One of the primary objectives was to reduce task completion time. This favored bulky and jerky machines, designed and controlled according to the-stiffer-the-better paradigm. Today, robots are asked to move out of their cages to achieve new and more ambitious goals, including: to safely share their workspace with humans, to operate in unstruc- tured or unknown environments 1, to grasp objects with different shape/size/mechanical properties 2. These fi elds This work has received funding from the European Unions Horizon 2020 research and innovation program under agreement no. 732737 (ILIAD) and agreement no. 780883 (THING). The content of this publication is the sole responsibility of the authors. The European Commission or its services cannot be held responsible for any use that may be made of the information it contains. This work has supplementary downloadable material available at , provided by the authors. The Sup- plementary Materials contain a single movie fi le in MPEG-4 format. This material is 7 MB in size. 1 CentrodiRicercaE.PiaggioeDipartimentodiIngegneria dellInformazione, Universit a di Pisa, Pisa, Italia 2 Istituto Italiano di Tecnologia, via Morego, 30, 16163 Genova, Italia riccardo.mengacci (a) Button approaching (b) Low Ky: failure(c) High Ky: success Fig. 1. A soft robotic arm equipped with variable stiffness actuators (VSAs) executing a button pushing task (approaching phase) is shown in (a). Kyis the Cartesian stiffness in the y direction. The task fails in the case of low stiffness (b), while it is accomplished in the case of high stiffness (c). of use, share a common challenge: to manage potential interactions in a stable, resilient and safe way. Indeed, this challenge adds, to the classical objective of minimizing the task completion time, upper bounds on the forces that the robot can exchange during an interaction 3. This change of problem led to the development of new designs and control approaches. In this regard, Hogan in 4 proposed to change the-stiffer- the-better paradigm of designing and controlling robots, by suggesting to develop systems capable of modulating their impedance, in a way similar to what humans do by varying their joint stiffness when performing movements 5. This simple but effective principle sprouted out in two main different technological approaches: Impedance Control fi rst and Soft Robotics later. According to the fi rst approach impedance modulation can be obtained by the use of suitable controllers, often combined with force/torque (F/T) sensors, e.g. in the case of Cartesian impedance control proposed in 6. Conversely, in the Soft Robotics approach, elastic elements with high-compliance are included into the robot structure, either by inserting lumped elasticity in the joints (articulated soft robots 7) or by realizing structures with continuum soft materials 8. Both approaches require the planning of a desired IEEE Robotics and Automation Letters (RAL) paper presented at the 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 Copyright 2019 IEEE (a) Button approaching (b) Low Ky,z: failure(c) High Ky,z: success Fig. 2.The Panda arm executing a button pushing task (approaching phase) is shown in (a). Ky,zare the Cartesian stiffness components in the y,z plane. The task fails in the low stiffness case (b), while is accomplished in case of high stiffness (c). impedance profi le (constant or time-varying), which has to consider several task-related aspects. Despite low impedance values would be preferrable to reduce the forces exchanged with the environment and to better adapt to the environment profi le 9, we show that there are cases in which a lower bound on the stiffness exists, under which the stability of the interaction may not be guaranteed. The interactions that may occur during robot operations can be unexpected or intentional. Among the examples of unexpected interactions, there are the cases of impacts against objects, or human beings, where the robot has to guarantee safety and to minimize exchanged forces. The problem of safety is largely discussed in 10, while 11 reviews solutions to identify, detect and react to collisions. In the case of intentional interactions the problem is different. Indeed, in such cases, the robot must exert a certain amount of force on the environment/object to successfully execute the task, e.g. in physical human robot co-operation 12, in locomotion when the feet have to be placed on given footholds 13 or in manipulation when the fi ngers come in contact with the object surface, just to name a few examples. Along with this requirement, the interaction task must cope with uncertainties, with complex surfaces and still must preserve stability 14,15. The effect of controller impedance, environment impedance and time delay in the control loop on the interaction stability were investigated in 16 and 17. Instead, the problem of being resilient in case of poor knowledge of the environment is addressed in 18, where the authors show that, in case of uncertainties, the impedance value should be low, if low interaction forces must be guaranteed. However, there exist interaction tasks that may fail if performed with too low impedance profi les, e.g. a legged robot walking on an uneven terrain. Indeed, in such case, at the feet it is desired a compliant behavior to adapt to the unknown ground, but simultaneously the feet must carry the weight of the robot and exert a suitable force on the environment such that the robot is allowed to go forward while keeping the stability of the foothold. An additional example is the case of a button pushing task, executed with a compliant robot. Indeed, it is possible to observe that, due to the geometry of the button, the task outcome may result in a failure if performed with a low Cartesian stiffness (Fig. 1(b), Fig. 2(b), while in case of a high value the task can be successfully achieved (Fig. 1(c), Fig. 2(c). In this last case, however, a stiffness too high would result in excessive interaction forces. Motivated by these experimental observations and by a lack of literature in this regard (to the best of authors knowl- edge), we characterize the feasible set of desired Cartesian stiffness for a robot that: i) interacts with a curved surface in presence of uncertainties, ii) must exert a force constrained by lower (given by the task specifi cations) and upper (given by resilience issues) bounds on the environment, and iii) must guarantee a stable interaction. The outcomes of this analysis and major contributions of the paper are that: i) the stiffness along the axis aligned with the one of the interaction force has an upper and lower bound, ii) the stiffness along axes normal to the one of the interaction force has a lower bound coming from the stability condition. The analysis has been carried out for the planar and the three dimensional cases. The second contribution of the paper is a thorough experimental validation of the theoretical results. The tests have been carried out on a robotic arm powered by variable stiffness actuators and on an actively compliance controlled arm. The experimental validation showed that there are practical upper bounds on the stiffness that can be achieved by the robot due to stability issues of the control loop in case of actively compliance controlled arms, and motor torque limitations in case of robot powered by antagonistic variable stiffness actuators (VSA). II. PROBLEMFORMULATION Consider the planar (2D) case shown in Fig. 3(a), where the end-effector of the robot is represented as a spherical tip of radius r and mass ms(brown circles), while the environment has an uneven shape (grey area). The contact surface profi le can be locally approximated by circles with constant radius Ri(dashed blue curves). We assume to be able to control the end-effector reference position xp, yp R (represented by the yellow carts) and to modulate its Cartesian impedance along the x and y axes. Assuming that a nominal surface curvature and position are given, the problem we address in this paper is to under- stand which values of the Cartesian impedance guarantee: i) interaction forces lower than upper bounds given by resilience (or safety) constraints; ii) interaction forces higher than lower bounds given by task requirements; iii) interaction stability. (a)(b) Fig. 3.Sketches of the analyzed 2D system. In (a) are highlighted: the radius of curvature of the surface R, the impedance parameters of the robot Kx,y,dx,y, the contact point P and its angular position , the parameters of the end-effector tip ms,r and the exerted force F. In (b) is shown the contact scenario at the analyzed equilibiurm in case of position uncertainties, where Fmaxis the maximum force exchanged during the interaction. A. Non-linear system model Consider the system in Fig. 3(a), in which the Cartesian position of the mass msis represented by the vector qs. Using the polar representation, we obtain qs= ? x y ? = ? (R + r)cos() (R + r)sin() ? ,(1) where is the angular coordinate. It is worth remarking that the radius R is supposed to be constant, meaning that the mass is constrained to move along the contact surface. We choose R as Lagrangian coordinate to derive the system dynamics (see Appendix I for the details) that results = (, ) + ()xpypT,(2) with (,) = ? (R + r)(dyRcos()2+ dxRsin()2) + d ms(R + r)2 ? + sin(2)(Kx Ky) 2ms , () = ? Kxsin() ms(R + r) Kycos() ms(R + r) ? , (3) where and are the fi rst and second time-derivative of , respectively; while Kx,Ky R+and dx,dy R+ are Cartesian stiffness and damping terms, respectively, as shown in Fig. 3(a). The effect of the friction at the contact surface is modeled by the damping coeffi cient d R+. Considering the state z , , T, z1,z2Tand the input u , xp,ypT, u1,u2T, it is possible to rewrite (2) in state form as ( z1= z2 z2= (z1,z2) + (z1)u ,(4) with (,) and () defi ned in (3). B. Linearized model To study the system stability it is useful to derive the linearized model of the system (4) at the desired equilibrium point z = 0,0T, u = xp,0T: = ? 01 12 ? + ? 00 0Ky/ms(R + r) ? v,(5) where R2and v R2represent state and force perturbations, respectively, around the equilibrium point, and 1= Ky(R + r) + Kx( xp R r) ms(R + r) ,2= d+ dyR(R + r) ms(R + r)2 . (6) It is worth nothing that the system is at the equilibrium xp 0; H.3 friction, we assume a friction-less case, i.e. d= 0; H.4 interaction force, we assume that the desired interac- tion force is normal to the tangent to the surface at the equilibrium point; H.5 impedance control, we assume that the input v of the linearized system is null since we prefer to rely on the autonomous behavior conferred to the robot by the impedance control (or the variable stiffness actuators) rather then to adopt an active force control that would require a feedback force loop. The reason of our choice is in the well known possible issues of the force control (e.g. typically the force measurements are affected by noise) 19. Moreover, we will assume that the stiffness matrix is diagonal. By considering these hypotheses, (5) simplifi es in the fol- lowing autonomous linear system = 01 (KyR + Kx( xp R) msR |z 1 dy ms |z 2 . (7) A. Stability analysis According to the Lyapunovs indirect method, the lin- earized system (7) has to be asymptotically stable, in order to conclude on the local stability of the non linear system (2). Furthermore, it is important to note that the state matrix in (7) has a particular form for which Cartesios criterion can be applied. Then, according to this criterion, to avoid eigenvalues with positive real part, the terms 1and 2must have equal signs. More in details, to be asymptotically stable, the system in (7) must have (Ky+ Kx( xp R)R1)/ms 0,dy/ms 0.(8) Since ms,R 0, and given a desired contact force Fx, we have that the stability of the linearized system induces a lower bound on Ky. If Kyis too low, then the linearized system is unstable and so is the equilibrium of the original Fig. 4. Simulated interaction forces Fxand Fyand the equilibrium angle for a numerical example. For the simulation, the following parameters have been used: Kx= 500 N/m, xp= 0.01 m, R = 0.0725 m and a uncertainty of yp 0,0.02m. Dashed and dotted red lines in the top plot refer to the maximum and minimum forces for the x component, respectively (Fmax= 40 N and Fmin= 22 N), while the vertical black dashed line is the stability lower bound (9) for the lateral stiffness component Ky= 552 N/m, in the case of maximum force. Note that, in case of a perception tolerance of 0.01m (violet line), comparable to state of the art vision sensors, the choice of the parameters verifi es both the stability and resilience conditions. non linear system. More in details, it is possible to show that, to perform a (locally) stable interaction task, the value of the Cartesian stiffness along the y axis must be Ky c|Fx|,dy 0,(9) with c = R1curvature of the contact surface. Finally, dy can be suitably modulated to guarantee pole allocation for the system. For instance, if the following condition is satisfi ed dy q 4ms(Ky+ FxR1),(10) then all eigenvalues are real and the system is not under- damped. From (9), we can conclude that, the less is the radius of curvature of the surface, the larger the Cartesian stiffness has to be to preserve stability, if we want to exert a specifi c force on the surface. B. Resilience analysis We consider the case of an interaction task affected by uncertainties on the positioning of the end-effector tip or on the estimate of the contact surface location at the equilibrium, Fig. 3(b). Thus, the uncertainties are modeled by assuming that the second component of the inputs is not at the equilibrium, namely u0= xp, ypTwith yp y, y where 0 y R to guarantee the contact. Notice that this corresponds to admit a positioning error in the y direction. Moreover, we assume that the static interaction force magnitudes are subject to following constraints Fmin |Fx| Fmax,|Fy| Fmax,(11) where Fmaxand Fminare two positive constants represent- ing the minimum among the maximum forces that the robot and the end-effector can support, and the maximum force that guarantees the task accomplishment, respectively, and Fyrepresents the interaction forces along the y axis. Eq. (11) in terms of stiffness and defl ection for the system (2) is Fmin Kx|( xp Rcos()| Fmax, xp Rcos(), Ky|( yp Rsin()| Fmax, yp R,R, (12) where is given by the following equation Ky|( yp Rsin()| Kx|( xp Rcos()| = tan(),(13) which is obtained by imposing the equilibrium of the system (2) for given stiffness Kx, Kyand reference positions xp, yp at the generic . The values of Kx, Kyand xpthat simultaneously satisfy (9), (12), and (13) yp y, y guarantee that: i) the interaction is stable, ii) both Fxand Fydo not exceed the upper bound Fmax, and iii) Fxis such that the interaction task is successfully accomplished. It is interesting to discuss which are the conditions that set the limits of these values. The minimum value of Kyis given directly by (9) with Fx= Fmax. The minimum value of the bound on xpis obtained by guaranteeing that the contact condition (domain defi nition for the fi rst of the (12) holds in the worst possible case, that is when = /2 and results xp 0. Interestingly enough, it is straightforward to verify that for the minimum stiffness for which the stability condition (9) holds (for all Fxin particular for Fx= Fmax) the second o