IROS2019国际学术会议论文集2607

IEEE TRANSACTIONS ON ROBOTICS, VOL. 35, NO. 2, APRIL 2019317 Control of Nonprehensile Planar Rolling Manipulation: A Passivity-Based Approach Diana Serra, Fabio Ruggiero, Member, IEEE, Alejandro Donaire, Member, IEEE, Luca Rosario Buonocore, Vincenzo Lippiello, Senior Member, IEEE, and Bruno Siciliano, Fellow, IEEE AbstractThis paper presents a new procedure to design a control law using the classical interconnection and damping as- signment technique within the passivity-based port-Hamiltonian framework. The sought goal is to reduce the complexity of solv- ing the so-called matching equations. The proposed approach is applied to two case studies of planar rolling nonprehensile ma- nipulation, namely, the ball-and-beam and the eccentric disk-on- disk. The performance of the resulting controllers is illustrated through both simulations and experimental results, showing the applicability of the design in a real setup. Index TermsDynamic manipulation, nonprehensile rolling manipulation, passivity-based control, underactuated systems. I. INTRODUCTION T HE port-Hamiltonian (pH) formalism has gained the at- tention of the control and robotics research communities in the last decade as a methodology for modeling and control Manuscript received May 2, 2018; accepted November 27, 2018. Date of publication January 16, 2019; date of current version April 2, 2019. This paper was recommended for publication by Associate Editor R. Carloni and Editor T. Murpheyuponevaluationofthereviewerscomments.Thisworkwassupported by the RoDyMan project, funded by the European Research Council FP7 Ideas underAdvancedGrant320992.Theauthorsaresolelyresponsibleforthecontent of this paper. (Corresponding author: Fabio Ruggiero.) D. Serra was with the CREATE Consortium and the Department of Electrical Engineering and Information Technology, University of Naples Federico II, 80125, Naples, Italy (e-mail:,diana.serraunina.it). F. Ruggiero,V. Lippiello, and B.Siciliano arewith the CREATEConsortium and the Department of Electrical Engineering and Information Technology, University of Naples Federico II, 80125 Naples, Italy (e-mail:,fabio.ruggiero unina.it; vincenzo.lippiellounina.it; bruno.sicilianounina.it). A. Donaire is with the University of Newcastle, Callaghan, NSW 2308, Australia (e-mail:,.au). L. R. Buonocore is with the European Organization for Nuclear Research, 1211 Gen eve, Switzerland (e-mail:,luca.rosario.buonocorecern.ch). Thispaperhassupplementarydownloadablematerialavailableat , provided by the author. The material consists of a video, viewable with all players supporting .mp4 format. The video shows ex- perimental results in stabilizing the upright nonactuated disk of an eccentric disk-on-disk setup. The center of mass of the actuated disk does not coincide with the center of rotation. The underactuated setup is in full gravity, and the resulting dynamic model is with a nonseparable Hamiltonian. The proposed solution aims at reducing the complexity of solving the so-called matching equations in the classical interconnection and damping assignment technique within the passivity-based port-Hamiltonian framework. In this approach, the target potential matching equation depends on a parameterization of the de- sired closed-loop mass matrix. The desired energy function for the closed-loop system is selected as well. The designed solution can be applied to any underac- tuated mechanical system with both separable and nonseparable Hamiltonians. Contact fabio.ruggierounina.it for further questions about this work. Color versions of one or more of the fi gures in this paper are available online at . Digital Object Identifi er 10.1109/TRO.2018.2887356 design of a complex system 15. Rooted in the classical mechanics, the pH formalism is a representation of the system dynamics that explicitly reveals energy and physical proper- ties related to the energy exchange, power fl ow, and intercon- nection structure. Such physical information is exploited for the design of control algorithms within nonnegligible dynamics tasks. In particular, the method of interconnection and damping assignment passivity-based control (IDA-PBC) is here consid- ered 2, 5. The IDA-PBC aims at fi nding a control law such that the closed loop preserves the Hamiltonian structure, with a minimum of the potential energy at the desired equilibrium, and a further damping injection to ensure asymptotic stabil- ity. The IDA-PBC method differs from other nonlinear control methodologies, typically applied in robotics, such as feedback linearization,wherealineardynamicsisimposedattheexpense of exact cancelation of the nonlinear system dynamics, which maycauserobustnessproblems.Thecontrollawisthenobtained by matching the open-loop and desired closed-loop dynamics. Such a match is guaranteed by solving a set of partial differen- tial equations (PDEs), the so-called matching equations, which is also a bottleneck of the IDA-PBC approach despite the exis- tence of constructive and explicit solutions for many structured problems (see, e.g., 68). Inthispaper,anewproceduretosolvethematchingequations for a class of mechanical systems is proposed. The carried-out approach reduces the complexity of the IDA-PBC design, while preserving its effectiveness. Under certain conditions, the pro- posed method consists of giving the explicit solution of a subset of PDEs resulting from the matching equations, while trans- forming the remaining PDEs in a set of algebraic equations. This novel procedure for the IDA-PBC design can be applied to underactuated planar mechanical systems with separable and nonseparable Hamiltonians, i.e., with constant and nonconstant mass matrices, respectively. Such a class of systems includes nonprehensile planar rolling manipulation tasks, which are here proposed as robotic case studies to illustrate the design procedure outlined in this paper. Nonprehensile planar rolling manipulation systems address those tasks that involve an actuated manipulator referred to as hand and an object that is manipulated without form or force closure grasps 9. The disk-on-disk 1, 10, 11, the ball- and-beam 1215, and the butterfl y robot 1618 are some robotic benchmarks used to simulate different nonprehensile planar rolling manipulation tasks. In detail, the disk-on-disk is composed of an upper disk (object) free to roll without slipping 1552-3098 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See /publications standards/publications/rights/index.html for more information. IEEE Transactions on Robotics (T-RO) paper, presented at IROS 2019. It should be cited as a T-RO paper. 318IEEE TRANSACTIONS ON ROBOTICS, VOL. 35, NO. 2, APRIL 2019 on the rim of a lower actuated disk (hand). The ball-and-beam consists of a beam (hand) actuated by a torque around its center of mass (CoM) together with a ball (object) rolling on it. The butterfl y robot is composed of an actuated butterfl y-shaped link (hand) on whose rim a ball (object) can freely roll. In all these cases, the control objective is to balance the object and drive the hand toward the desired confi guration. In this paper, two nonprehensile planar rolling robotic systems with nonseparable Hamiltonian are considered as case studies: the ball-and-beam and the eccentric disk-on-disk. This last example is a variant of the disk-on-disk system, where the center of rotation of the hand and its geometric center are not coincident. Simulation tests on the ball-and-beam and experiments on the real physical prototype of the eccentric disk-on-disk system are presented to confi rm the performance of the proposed control methodology. Theoutlineofthispaperisasfollows.SectionIIhighlightsthe novelties proposed within this paper. Existing control designs for the selected case studies are described in Section III. A summary of the IDA-PBC is presented in Section IV. The main result of this paper is shown in Section V. The general dynamic model for nonprehensile planar rolling manipulation systems is derived in Section VI. The ball-and-beam and the eccentric disk-on-disk case studies are deeply analyzed in Sections VII and VIII, respectively. Section IX concludes this paper. II. NOVELTIES As will be detailed in Section IV, the matching equations are split into two subsets of PDEs, namely the kinetic and the po- tential energy matching equations. In this paper, under certain conditions and through a suitable parameterization of the de- sired closed-loop mass matrix, an explicit solution is provided for the potential energy matching equation. This new procedure reduces the complexity of the IDA-PBC design by simulta- neously fi nding the desired potential energy function for the closed-loop system and simplifying the choice of the desired mass matrix. Once the solution of the potential energy match- ing equation is found, the procedure to solve the kinetic energy matching equation takes inspiration from 19, without solving any PDEs. Theapproachhereproposeddiffersfrom3,wherethePDEs derived from the kinetic energy matching condition are trans- formed into a set of ordinary differential equations. Moreover, in 3 and 14, a necessary condition for the validity of the methods is that the mass matrix of the system depends only on the unactuated variable. Differently, in the proposed control ap- proach, the open-loop mass matrix and the desired closed-loop mass matrix can be dependent on both actuated and unactuated variables. It is worth underlining that the proposed control approach can be applied to many two-dimensional (2-D) underactuated mechanical systems having the structure outlined in Section V. Nonprehensile planar rolling manipulation systems fi t into such a class. Therefore, the generalized dynamic model of non- prehensile rolling between arbitrary shapes in 2-D, presented in20,isextendedbyformulatingthedynamicsinthepHform. Besides, the assumption that the center of actuation of the hand and its geometric center are coincident is dropped. This small technical contribution, in addition to the above-outlined control approach, overcomes the limitation in 20. In that work, non- prehensile planar rolling manipulation systems are shown to be differentiallyfl atonlyiftheyhaveaconstantmassmatrix(i.e.,a separable Hamiltonian). The method proposed in this paper, in- stead,canbeappliedtosystemshavingaseparableHamiltonian or a nonseparable Hamiltonian indifferently. The proposed con- trolcanbethuselectedasaunifyingapproachtosolvingthesta- bilization problem of nonprehensile planar rolling manipulation systems. As sketched in 21, fi nding general strategies to settle aclassofproblemsisyetanopenissuewithinthenonprehensile manipulation domain. III. EXISTINGCONTROLDESIGNS FOR THE SELECTEDCASESTUDIES In the following, a brief state-of-the art about the modeling and the control of the ball-and-beam and the eccentric disk-on- disk is provided. These case studies are considered to bolster the proposed control approach. A more comprehensive analysis about nonprehensile manipulation is tackled in 21. On the one hand, the ball-and-beam system has been exten- sively studied in the past years due to its peculiar feature: it fails to have a well-defi ned relative degree. Hence, feedback linearization cannot be applied. The authors of 13 propose an approximate inputoutput linearization, whereas an output feedback controller is introduced in 22. The authors of 12 show a technique for obtaining stable and robust oscillations for suchasystemconsistingoftwosteps:theformeraimsatfi nding a control law such that the closed loop of a reduced model of the dynamicsisasecond-orderHamiltoniansystem,whichpresents stable oscillations; in the latter step, the controller is extended to the full system using backstepping. A control method for a redundant manipulator to balance the ball-and-beam system is shown in 15. A force/torque sensor attached to the end effec- tor of the manipulator is used for estimating the ball position. Since it involves signifi cant noise, a state-feedback controller is employed along with an observer. On the other hand, the eccentric disk-on-disk has some char- acteristics that make it attractive as a benchmark. In 1, an IDA-PBCcontrollerisdesignedadhocviaacoordinatetransfor- mation for the traditional disk-on-disk (separable Hamiltonian), but it cannot be directly extended for the eccentric disk-on- disk (nonseparable Hamiltonian). It is worth noticing that the dynamic behavior and the stability properties of the eccentric disk-on-disk are similar to the circular ball-and-beam investi- gated in 2325. In 24, the Jordan form of the model of the circular ball-and-beam is linearized near the unstable equilib- rium to design a linear controller. A linear control approach is alsousedin23,wherethelimitsofthebeamactuatoraretaken into account. A geometric passivity-based control approach for this system is presented in 25. Also, in that work, the authors proposeatechniquetoavoidthesolutionofthematchingcondi- tions. Nevertheless, the gyroscopic term is not addressed for the control design within 25, since the energy shaping is applied IEEE Transactions on Robotics (T-RO) paper, presented at IROS 2019. It should be cited as a T-RO paper. SERRA et al.: CONTROL OF NONPREHENSILE PLANAR ROLLING MANIPULATION: A PASSIVITY-BASED APPROACH319 to a modifi ed dynamics resulting from a geometric feedback transformation. IV. IDA-PBCIN ANUTSHELL The pH framework allows modeling of mechanical systems including the information about the energy transfer explicitly. The canonical Hamiltonian equations of motion are ? q p ? = ? OnIn InOn ? H(q,p) + ? Onm G(q) ? u(1) where q Rn is the confi guration vector, p Rnis the mo- mentavector,u Rmisthecontrolinput,G(q) Rnmisthe input mapping vector, In,On Rnnare the identity and the zero matrices, respectively, and Onm Rnmis an n m matrix with all-zero entries. The function H : R2n R is the Hamiltonian, which represents the total energy (kinetic plus potential) stored in the system, having the form H(q,p) = 1 2p T M1(q)p + V (q) where V (q) R is the potential energy function and M(q) = MT(q) Rnn is the positive-defi nite mass matrix. Stabilization of (1) to the desired equilibrium (q,p) = (q?,0n), where 0n Rnis the zero vector, is achieved us- ing the IDA-PBC by assigning the target dynamics to the closed loop 14 ? q p ? = ? OnM1(q)Md(q) Md(q)M1(q)J2(q,p) ? Hd(q,p) (2) where J2(q,p) Rnnis the desired interconnection matrix, and Md(q) Rnnis the desired mass matrix. The desired total energy function is given by Hd(q,p) = 1 2p T M1 d (q)p + Vd(q) with Vd(q) R being the desired potential energy function. Then, (q?,0n ) will be a stable equilibrium confi guration of the closed loop (2) if: C.1 Md (q) is symmetric and positive defi nite; C.2 q?= argminVd(q); C.3 J2(q,p) is skew-symmetric. The stabilization of the desired equilibrium is achieved by identifyingtheclassofHamiltoniansystemsthatcanbeobtained via feedback. The conditions under which this feedback law exists are the matching conditions, i.e., matching the original dynamic system (1) and the target dynamic system (2) ? OnIn InOn ? H + ? Onm G ? u = ? OnM1Md MdM1J2 ? Hd(3) where the dependence of the functions on their argument has been dropped to simplify the notation. The fi rst line in (3) is straightforwardly satisfi ed, while the second line in (3) corresponds to the following set of PDEs: G ? qH(q,p) Md(q)M 1(q)qHd(q,p) +J2(q,p)M1 d (q)p?= 0(4) where Gis the full-rank left annihilator of G. The PDEs (4) can be separated into the two subsets of PDEs as G ? q(pTM 1(q)p) M d(q)M 1(q)q(pT M1 d (q)p) + 2J2(q,p)M1 d (q)p?= 0(5) G ? qV (q) Md(q)M 1(q)qVd(q)? = 0(6) where(5)and(6)arethekineticandthepotentialenergymatch- ing equations, respectively. By solving (5) and (6) for Md(q), Vd(q), and J2(q,p), subject to C.1C.3, the energy shaping control is given by ues= (GTG)1GT(qH(q,p) Md(q)M1(q)qHd(q,p) + J2(q,p)M1 d (q)p). (7) It is worth remarking that not every desired Md(q), Vd(q), and J2(q,p) can be chosen, but only those solving (5) and (6) subject to the conditions C.1C.3. By applying (7) to the Hamiltonian dynamics (1), the closed-loop target dynamics (2) is obtained. Damping aimed at achieving asymptotic stability is then injected through udi= KvGTpHd(q,p)(8) whereKv Rmm isasymmetricandpositive-defi nitematrix. The damping injection (8) and the energy shaping control (7) are then assembled to generate the IDA-PBC u = ues+ udi.(9) Therefore,throughthisadjustment,theclosed-loop dynamics (2) is modifi ed as follows: ? q p ? = ? OnM1(q)Md(q) Md(q)M1(q)J2(q,p) Rd ? Hd(q,p) (10) in which dependencies have been dropped, and Rd= GKvGT Rnn is the positive-(semi)defi nite dissipation matrix 4, 14. V. MAINRESULT Consider the class of underactuated Hamiltonian systems (1) with n = 2, m = 1, G = e1= ? 1 0 ?T , and, consequently, G= eT 2 = ? 0 1 ? . The fi rst step toward the proposed resolu- tion to solve the matching conditions is related to the poten- tial energy PDEs and the conditions of symmetry and positive defi niteness of the desired closed-loop mass matrix. Let q = ? q1q2 ?T be the confi guration vector, and let M(q) = ? b11(q)b12(q) b12(q)b22(q) ? (11) betheexpressionofthemassmatrixin(1).Tolookforasolution of the potential energy matching equation, the desired inertia IEEE Transactions on Robotics (T-RO) paper, presented at IROS 2019. It should be cited as a T-RO paper. 320IEEE TRANSACTIONS ON ROBOTICS, VOL. 35, NO. 2, APRIL 2019 matrix is parameterized as follows: Md(q,c1) = ? a11(q,c1)a12(q,c1) a12(q,c1)a22(q,c1) ? (12) where = b11(q