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 Abstract This 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 Llewellyn s 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 nicolis polimi it fabio allevi mail polimi it paolo rocco polimi 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 Llewellyn s 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 Llewellyn s 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 Rn nis the robot inertia matrix n q q Rnthe term accounting for Coriolis gravitational and friction effects Rnis the robot joint actuation torque J Rm n 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 B 1 Bv B 1 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 does not require knowledge of each other specifi c kinematics which are solved locally as we will show next C Hierarchical optimization controller To implement the impedance dynamics and take into account redundancies and control constraints we employ a hierarchical QP hQP optimization approach After the application of the sliding mode control the dynamics es sentially become a chain of integrators and the optimization has the effect of computing the device inverse kinematics by providing the joint accelerations that are then used in the auxiliary control 4 Sincetheobjectiveistoensuretheendeffector impedances 12 14 in a fi rst optimization stage we minimize the impedance errors 13 15 subject to the desired constraints v0 0 argmin v0 kI k2 Q 17a s t v0 vsmc 17b Cv0 d 17c The star stands for master or slave quantities v0 0 is the solution to the fi rst op