IROS2019国际学术会议论文集1203

Robust Impedance Shaping of Redundant Teleoperators with Time-Delay via Sliding Mode Control Davide Nicolis1, Fabio Allevi1, and Paolo Rocco1 AbstractThis paper presents a robust impedance shaping controller for teleoperation systems. An integral sliding mode control law (ISM) is employed together with standard robot inverse dynamics to reject disturbances and uncertainties acting on the robot model and obtain an ideal fully-decoupled system. Higher level optimization-based controllers are responsible for enforcing the desired end effector impedance on master and slave manipulators, as well as for solving possible kinematic re- dundancies and satisfying control constraints. A three-plus-one channel teleoperation architecture is proposed, with an in-depth analysis of its stability and transparency properties in presence of variable communication delays, based on Llewellyns ab- solute stability theorem. Impedance parameters tuning criteria are derived and the proposed scheme performance is compared in simulation with a time-domain passivity approach. The validation of the proposed controller is carried out on a ABB YuMi dual-arm redundant robot, with one arm employed as a master and the other one as a slave device. I. INTRODUCTION Impedance control plays an important role in robotics whenever contact with the environment is expected. While many of its applications involve hands-on human interaction, such as the handling of heavy materials, the use of this established control strategy is of primary importance in bilateral teleoperation systems. The slave-to-master force feedback generated by envi- ronment interaction creates closed-loop dynamics that may become unstable in presence of communication delay, thus requiring a thorough design of the underlying controller. This phenomenon is apparent in long distance teleoperation, such as in-orbit servicing, where latency may disrupt system stability. However, teleoperation and impedance control play a major role also in more common applications like robotic excavators 1. Different approaches have been proposed to overcome stability limitations while maintaining an acceptable degree of transparency. The most established techniques resort to damping injection or wave variables, while recent approaches favor time-domain passivity, in order to reduce performance only upon loss of passivity. In 2 passivity observers and controllers monitor and dissipate part of the system energy, when the communication channel displays an active behavior that might destabilize contact. A two-layer approach is employed in 3, where virtual reservoirs store surplus energy that would be otherwise dissipated and drain it when the channel becomes active, to preserve overall passivity. Absolute stability criteria are often employed to tune the control gains so that the system remains stable whatever 1The authors are with Politecnico di Milano, Dipartimento di Elettronica, Informazione e Bioingegneria, Piazza L. Da Vinci 32, 20133, Milano, Italy (e-mail:davide.nicolispolimi.it; fabio.allevimail.polimi.it; paolo.roccopolimi.it). the external environment and operator dynamics are, as long as they satisfy a reasonable assumption of passivity. In 4, the authors proposed an in-depth analysis of two and four-channel teleoperation control structures, based on Llewellyns criterion and Lawrence formalism. Although four-channel controllers provide substantially better trans- parency results, two-channel architectures were shown to be remarkably more stable and easier to tune, especially position-force schemes. The same authors analyzed the tun- ing of three-channel architectures 5, giving insights on the parameters choice depending on the application and the local feedback controller. Indeed, these provide improved trans- parency via operator force feed-forward and still manageable tuning. Nonetheless, a force/torque sensor also on the master device becomes necessary. Sliding Mode Control (SMC) techniques have seen some recognition in robotics for robust trajectory tracking in presence of disturbances and uncertainties. In practice, these algorithms have been considered in their second-order form to attenuate control input chattering, and therefore mechan- ical wearing and high frequency excitation. In the context of teleoperation, SMC has been fi rst em- ployed in 6. The authors discussed the defi nition of a classical sliding surface with a linear combination of position and velocity tracking errors, but delays were not considered. Cho et al. tackled the communication delay problem in 7, where they defi ned a sliding surface based on the integral of the desired impedance relation, in order to obtain both accurate tracking of the master and compliance during contact. A similar approach has been proposed by 8, but with a higher order sliding mode to avoid chattering. The method unfortunately requires the use of an observer to estimate the acceleration for the sliding surface computation. While previous attempts consider simple 1 d.o.f. devices, an operational space sliding mode was applied to a multi- dof robot in 9, but manipulator redundancies were not considered. In this paper we present a novel teleoperation controller that relies on sliding mode control theory to ensure accurate impedance tracking, and on Llewellyns absolute stabil- ity criterion to tune the impedance parameters in case of variable communication delays. An integral sliding mode controller allows the complete rejection of model uncertain- ties and disturbances due to imperfect inverse dynamics. Unlike previous approaches, chattering of control torques is alleviated thanks to a second order formulation without estimating robot accelerations, while the integral nature of the proposed sliding manifold guarantees robustness since the fi rst time instant, without a reaching phase. On top of this fi rst control level, an optimization-based controller 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE2740 Fig. 1: The overall control architecture. The same block diagram can be sketched also for the master. enforces the desired end effector impedance dynamics of master and slave manipulators, while taking into account control and kinematic constraints. Differently from 7, 9 the sliding mode control is not in charge of enforcing the desired impedance, but to compensate the unmodeled robot dynamics and uncertainties, this allows to split the control effort in a component responsible for robustness, and another for the nominal dynamic behavior through a cascade of optimizations that solve the manipulators redundancies, extending the approach to platforms with an arbitrary number of d.o.f. 10. In the resulting ”three-plus-one” channel teleoperation architecture, only the slave force is fed back to the master station, while velocity, user interaction force and also a delayed version of slave contact forces are fed forward to the slave device. We provide an in-depth discussion of the ar- chitecture stability and transparency properties, highlighting some aspects that have been neglected in previous works, that are nonetheless critical to ensure the system overall stability. The performance of the proposed algorithm is also compared with a classical time-domain passivity (TDP) approach. Fig. 1 provides an overall diagram of the control structure. The paper is organized as follows. Section II presents the second order integral sliding mode controller for robust inverse dynamics and chattering alleviation. In Section III the hierarchical optimization-based approach is proposed to achieve the desired impedance and solve redundancies. Stability and transparency properties of the teleoperation controller are discussed in Section IV, and compared with a passivity-based algorithm in simulation. Results of the experimental validation are given in Section V, while Section VI concludes the paper. II. ROBUST INVERSE DYNAMICS In the following we present the sliding mode control strategy employed to obtain robust feedback linearization. For both master and slave devices we consider generic n d.o.f. robots, possibly redundant, characterized by the well- known rigid model B(q) q + n(q, q) = JT(q)F(1) where q, q, q Rnare the joint position, velocity and accelerations, respectively, B(q) Rnnis the robot inertia matrix, n(q, q) Rnthe term accounting for Coriolis, gravitational and friction effects, Rnis the robot joint actuation torque. J Rmn, m n is the robot Jacobian, and F Rmthe vector of external forces. For simplicity we consider end effector forces and operational space impedance, however the remainder of the paper is valid also for joint space impedance controllers, as long as torque measurements are available for each joint. A. Inverse dynamics control The objective of inverse dynamics control is to completely compensate the robot model with a torque feedback that linearizes and effectively decouples the joint dynamics. Such torque is given by = Bv + n + JT(q)F(2) where the hat indicates estimated values, and v is an auxiliary control. Note that to fully decouple the system and completely assign the robot impedance, external force measurements are necessary. Substituting (2) in (1) we obtain the partially feedback- linearized dynamics q = B1Bv + B1 n(3) where n = n n is the estimation error of the Coriolis, gravitational and friction terms. Clearly, the system is fully decoupled only with perfect knowledge of the dynamics. Therefore, in a teleoperation system, the application of a standard impedance controller does not guarantee neither a zero tracking error of the slave device with respect to the master, nor the enforcement of the desired dynamics, invali- dating absolute stability properties, and possibly resulting in instability due to communication delay. B. Integral sliding mode control In order to make the inverse dynamics control robust against uncertainties, we propose to use Sliding Mode Control. Its main drawback is that in the time interval necessary for the system to reach the specifi ed manifold (reaching phase) 11, the system evolution is uncontrolled and we cannot ensure full inverse dynamics compensation, therefore losing any guarantee that a desired end effector impedance will be correctly enforced. Moreover, fi rst order sliding modes result in unwanted chattering that can limit the lifetime of mechanical components. Compared to previous results that did not consider the reaching phase 7 or required acceleration information to minimize chattering 8, here we employ an Integral Super-Twisting algorithm that removes these drawbacks. Let us consider the following auxiliary control to be applied to system (3) v = v0+ vsmc()(4) where v0is a nominal control that will be detailed in the next Section, and is responsible for enforcing the desired impedance, while vsmc() is the sliding mode control that compensates for the inverse dynamics uncertainties and keeps the system on the sliding manifold = 0. The sliding vector and control vsmc() have to be designed so that applying (4) guarantees the appearance of an integral sliding mode, i.e. that the system evolves with the following completely feedback linearized and decoupled dynamics, obtained from (3) z = f(z,v0) = ?z 2 v0 ? (5) where z = z1z2 = q q. To do so, we propose to use the following sliding vector: 2741 Proposition 1 Consider the partially feedback linearized system (3) and the control (4). Let (t) = q(t) q0(t)(6) be the selected sliding vector, with q0(t) = t t0v0d + q(t0). On the sliding manifold = 0, the system evolves with the ideal dynamics (5). Moreover, this holds from the initial time instant t0, effectively removing the reaching phase. Proof To prove the proposition we have to show that (6) produces an integral sliding mode. Let us defi ne an auxiliary generic sliding vector (t) = ( q(t) qr(t) + (q(t) qr(t)(7) where is a positive gain and the subscript r indicates a reference joint trajectory. By applying the defi nition of integral sliding mode to 11, we may write (t) = (t) Z t t0 ? z f(z,v0) + zr zr ? d (t0) (8) Substituting (5), (7) in the integral and simplifying, we obtain (t) = q(t) Z t t0 v0d q(t0)(9) which is exactly (6). Hence, the proposed sliding vector generates an integral sliding mode and it is independent of the reference trajectory. Indeed, we have that (t0) = 0, therefore the system will remain on the manifold since the initial time instant if vsmcis chosen appropriately, the reaching phase will be completely removed, guaranteeing disturbance compensation and perfect inverse dynamics since the beginning. To show that (5) describes the system dynamics when = 0, let us compute = 0, which is a necessary condition to remain on the manifold. Substituting (3) in = 0, and making vsmcexplicit using (4), we have vsmc= B 1( Bv0+ n)(10) with B = B B. Substituting back in (4) and then in (3), it is clear that the system will evolve with the nominal dynamics (5).? Unfortunately, the equivalent control (10) cannot be com- puted since it depends on unknown quantities. A common approach in fi rst order SMC is to select the discontinuous function vsmc() = ksgn() with k large enough to com- pensate the uncertainties effects, and immediately drive the system back on the manifold whenever a small detachment happens. Therefore, model uncertainties and nominal control must be bounded to ensure the sliding mode (see (10). In the following we prefer to consider the second order Super-Twisting algorithm 12, in order to reduce chattering. The second order formulation is given by the following choice of the sliding control vsmc() = k1 p |sgn() k2 Z t t0 sgn()d(11) where we considered entry-wise square root and vector prod- uct, with k1, k2design gains. By doing so, we can guarantee = = 0, even without knowledge of the equivalent control (10), and the use of a discontinuous control, thanks to the sign integral and the sliding vector square root in the fi rst term. However, gain tuning becomes more complex, so that in practice they should be progressively increased until an acceptable performance level is reached. III. IMPEDANCE CONTROL The application of control (4) with the sliding mode component (11) grants us the ability to simply consider the nominal system (5) for both master and slave robot, without worrying about unmodeled effects or reaching phase that are completely compensated, and precisely assign the desired dynamics since the system is switched on. The impedance controllers are detailed here focusing on the translation components, but the extension to full 6D pose is straightforward. A. Master device For the user operated master device we consider an impedance model where the slave interaction force is used for haptic feedback. This allows to avoid the refl ection of slave dynamics onto the master, as well as high robustness in free motion. Then: Mm xm+ Dm xm+ Kmxm= Fh kfF d e (12) where Mm, Dmand Kmare the desired inertia, damping and stiffness respectively, with subscript m indicating master quantities. xmis the end effector position, while Fhis the force applied by the user on the master end effector. F d e(t) = Fe(t d2) is the force exerted on the slave, with d2(t) the slave-to-master variable delay, and kfa scaling factor. Considering the relation between master end effector ac- celeration and joint velocity and acceleration via the robot Jacobian xm= Jm qm+ Jm qm, using (5), and then substi- tuting in (12), we obtain the impedance equation Im= 0 Im= MmJmv0,m bm= 0(13) bm= Mm Jm qm Dm xm Kmxm+ Fh kfF d e where the dependency on the auxiliary nominal control v0,m (i.e. the robot joint accelerations) has been highlighted. B. Slave device In the design of the slave impedance we must ensure the correct tracking of the reference master trajectory and compliance in case of contact forces. Hence, we select Ms x + Ds x + Ks x = Fe(14) where subscript s indicates slave quantities, and x= xs kpxd m, with kp a scaling factor accounting for the possibly different robot workspaces. xd m(t) = xm(t d1) is the delayed master position with d1(t) the master-to-slave variable delay. Similar to the master case, we have Is= MsJsv0,s bs= 0(15) bs= Ms Js qs+ Mskp xd m Ds x Ks x Fe While selection of the slave impedance as in (12) resembles a two-channel position-force architecture, the presence of the 2742 master acceleration in the previous equation is critical. To solve this problem we can simply compute the acceleration from (12) xd m= M 1 m ? Dm xd m+ Kmx d m F d h+ kfF dd e ? (16) where F dd e (t) = Fe(t d1 d2) is the twice delayed slave contact force forwarded by the master and necessary for a correct computation of the acceleration. Substituting the pre- vious equation in (15), the nominal auxiliary control depends only on the master position/velocity and operator force, plus the additional feed-forward of the delayed slave external force, obtaining a ”three-plus-one” channel controller. Note that the computation of the master acceleration in (16) is valid only because the sliding mode controller cancels out the disturbing terms remaining from the inverse dynam- ics, and (5) is enforced. If a robust feedback linearization is not employed at master side, the accuracy of (16) depends on the magnitude of the uncertainties. It should also be remarked that the overall teleoperation control approach can be applied to manipulators with differ- ent kinematics and number of d.o.f. Indeed, the only infor- mation exchanged between the two devices are Cartesian end effector quantities, such as position, velocities, and forces, therefore each robot