Stability-Guaranteed Impedance Control of Hydraulic Robotic Manipulators

Transcription

1 Tampere University o Technology Stability-Guarantee Impeance Control o Hyraulic Robotic Manipulators Citation Koivumäki, J., & Mattila, J. (217). Stability-Guarantee Impeance Control o Hyraulic Robotic Manipulators. IEEE - ASME Transactions on Mechatronics, 22(2), DOI: 1.119/TMECH Year 217 Version Peer reviewe version (post-print) Link to publication TUTCRIS Portal ( Publishe in IEEE - ASME Transactions on Mechatronics DOI 1.119/TMECH Copyright 217 IEEE. Personal use o this material is permitte. Permission rom IEEE must be obtaine or all other uses, in any current or uture meia, incluing reprinting/republishing this material or avertising or promotional purposes, creating new collective works, or resale or reistribution to servers or lists, or reuse o any copyrighte component o this work in other works. Take own policy I you believe that this ocument breaches copyright, please contact tutcris@tut.i, an we will remove access to the work immeiately an investigate your claim. Downloa ate:

2 This is the author's version o an article that has been publishe in this journal. Changes were mae to this version by the publisher prior to publication. The inal version o recor is available at IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 1 Stability-Guarantee Impeance Control o Hyraulic Robotic Manipulators Janne Koivumäki an Jouni Mattila, Member, IEEE Abstract In challenging robotic tasks, high-banwith closeloop control perormance o the system is require or successul task completion. One o the most critical actors inhibiting the wie-sprea use o close-loop contact control applications has been the control system stability problems. To prevent unstable system behavior, the nee or rigorously aresse manipulator ynamics is substantial. This is because the contact ynamics between a manipulator an its environment can be rastic. In this paper, a novel Cartesian space impeance control metho is propose or hyraulic robotic manipulators. To aress the highly nonlinear ynamic behaviour o the hyraulic manipulator, the system control is esigne accoring to the subsystem-ynamics-base virtual ecomposition control (VDC) approach. The unique eatures o VDC (virtual power low an virtual stability) are use to analyze the interaction ynamics between the manipulator an the environment. Base on the esire impeance parameters an stability analysis, an explicit metho to esign the control gains or the propose impeance control law is evelope. The L 2 an L stability is guarantee in both ree-space motions an constraine motions. Experimental results emonstrate that the hyraulic robotic manipulator is capable o ajusting its ynamic behaviour accurately in relation to the impose target impeance behaviour. This provies compliant system behaviour, which is neee in many ynamically challenging robotic tasks. Inex Terms hyraulic manipulators, impeance control, nonlinear moel-base control, stability analysis, virtual ecomposition control, virtual power low. I. INTRODUCTION ADVANCED robotic systems, such as humanoi robots, legge robots an exoskeletons, are currently receiving substantial attention in inustry an acaemia. From a mechanical esign perspective, hyraulic actuators provie an attractive solution or robotic systems because they can prouce signiicant orces/torques or their size, are robust an can provie accurate motions. Inee, hyraulic robotic systems, such as Boston Dynamics BigDog, Cheetah an Atlas, an SARCOS humanois an exoskeletons, have alreay avance the state-o-the-art in robotics. Acaemic inepth research is also ongoing (e.g., IIT s HyQ an Shanong University s SCal). For robotic systems, articulate limbs are crucially important subsystems because they can provie many versatile abilities, such as legge locomotion or manipulation o the environment. However, successul completion o these interactive tasks requires that the robotic system is capable Manuscript Submitte October 14, 215; Accepte October 1, 216. J. Koivumäki an J. Mattila are with the Department o Intelligent Hyraulics an Automation, Tampere University o Technology, FI-3311 Finlan (janne.koivumaki@tut.i, jouni.mattila@tut.i; tel: ). This work was supporte by the Acaemy o Finlan uner the project Cooperative heavy-uty hyraulic manipulators or sustainable subsea inrastructure installation an ismantling, grant no o accurately controlling its interaction with the surrouning environment, with humans or with other evices. However, orce control o a single hyraulic actuator is challenging ue to its highly nonlinear ynamic behaviour [1]. Moreover, control esign or articulate robotic systems is greatly complicate by the nonlinear nature o the associate multiboy ynamics. Impee by these nonlinearities, an accurate contact control or articulate hyraulic robots becomes an extremely challenging task. In robotic contact control, system stability issues have rawn consierable attention since the installation o the irst inustrial robots, an numerous reasons or the unstable responses have been ientiie [2] [6]. One reason is that contact ynamics between the robotic system an the environment can be rastic while robot nonlinear ynamics are not consiere rigorously [4], [7]. The esign challenges mentione above have le to the utilization o nonlinear moel-base control (NMBC) methos to achieve better ynamic perormance or hyraulic robots, which is neee in ynamically challenging contact tasks. In contrast to linear control methos, NMBC methos (where the speciic eeorwar control term can be use or system nonlinearities) can theoretically provie ininite control banwith, as long as proper eeorwar control is esigne [8]. As introuce in many books on the control o robots, such as [9] [11], typical NMBC esigns are base on the complete ynamic moels o robots using the Lagrangian ormulation. However, or complex robots (such as humanois), the implementation o complete-ynamicsbase control becomes substantially challenging, because with these methos, the complexity (computational buren) o robot ynamics is proportional to the ourth power o the number o egrees o reeom (DOF) in motion [8]. Virtual ecomposition control (VDC) [8], [12] is a unique subsystem-ynamics-base control metho using the Newton- Euler ormulation. A number o signiicant state-o-the-art control perormance improvements have been reporte with VDC with electrically-riven robots (see [13] [18]) an with hyraulically-riven robots (see [19] [23]). The subsystemynamics-base control o the VDC enables NMBC esign with many attractive special eatures or (complex) robotic systems, incluing the ollowing: 1) control computations are proportional to the number o subsystems an can be perorme even by locally embee harware/sotware, 2) subsystem ynamics remain relatively simple with ixe ynamic structures invariant to the target system, 3) changing the control (or ynamics) o one subsystem oes not aect the control equations within the rest o the system, 4) parameter uncertainties in the subsystem ynamics can be aresse with a parameter aaptation an 5) system stability analysis can be Copyright (c) 216 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine rom the IEEE by ing pubs-permissions@ieee.org.

3 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 2 aresse at a subsystem level using the unique eatures o VDC, virtual power low (VPF) an virtual stability. In this paper, a novel non-switching impeance control metho is propose. To aress the highly nonlinear ynamic behavior o the hyraulic manipulator an to obtain the rigorous ynamic perormance neee in contact control tasks, the manipulator s internal control is esigne base on VDC. Interaction ynamics between the manipulator an the environment are analyze using a VPF locate at the contact point. Base on the analysis, an interconnection between the esire impeance parameters (characterizing system target impeance behaviour) an the parameters o the propose novel impeance control metho can be oun such that impeance behavior can be esigne or the system. This paper provies the ollowing contributions. 1) A novel non-switching impeance control metho, incluing the VDC esign an the propose Cartesian space impeance control laws, is evelope or hyraulic manipulators. 2) The impeance control is esigne using the ramework o VDC. 3) A rigorous stability proo or a hyraulic manipulator is provie or the irst time to cover both ree-space motions an constraine motions. 4) The experiments emonstrate the eiciency o the propose metho an rigorously support the mathematical theorems on the stability-guarantee system behaviour an the target impeance behaviour. This paper is organize as ollows. Section II escribes contact control strategies propose or hyraulic robotic systems. Section III introuces the essential mathematical ounations neee in system control esign. Section IV escribes the propose impeance control esign an its VDC-base implementation. Section V concentrates on system stability issues. Section VI emonstrates contact control perormance o the propose controller. Conclusions are outline in Section VII. II. RELATED WORKS The basic approaches or robotic orce control are base on hybri position/orce control by Raibert [24] an impeance control by Hogan [25]. Historical overviews o robot orce control can be oun in [7], [26], [27]. With electrically-riven manipulators, the orce control, as presente, e.g., in [24], [25], [28] [32], has been extensively stuie. Typically, hyraulic manipulators are built to operate heavy objects (e.g., logs) or to exert large orces on the physical environment (e.g., in excavation). Thus, it is rather surprising that only a ew stuies exist regaring orce control in hyraulic robotic manipulators. Heinrich et al. [33] implemente the impeance control technique or hyraulically-actuate manipulators or the irst time. A nonlinear proportional-integral (NPI) controller was evelope or joint control. The stability proo o the propose controller esign was not given. Taazoli et al. [34] (see also relate stuies in [35], [36]) stuie the impeance control o a teleoperate miniexcavator, base on a simple proportional-erivative (PD) controller. Stability proo or a simple PD impeance controller was provie, but it was limite to a single-dof hyraulic cyliner acting on the environment. Zeng an Sepehri [37] propose a nonlinear tracking control or multiple hyraulic manipulators hanling a rigi object, where internal orces o couple manipulators were controlle. The control esign or the system was base on a backstepping methoology an the stability o the system was proven. However, the stability analysis was limite to situations where connection to the hel object was alreay establishe. The experiments were carrie out with two single-axis electro-hyraulic actuators, which were connecte rigily to the common object with spring mechanisms [38], preventing unilateral constraint. Semini et al. [39] reporte their recent results on the active impeance control o hyraulic quarupe robot HyQ. They use input-output eeback linearization to construct their moel-base control esign or the hyraulic leg. Rigorous stability proo or their control esign was not provie. A major step orwar rom the existing solutions was taken by Koivumäki an Mattila [23], who propose a stabilityguarantee contact orce/motion control or heavy-uty hyraulic manipulators. In this stuy, the highly nonlinear behaviour o the hyraulic manipulator was aresse with the VDC approach, an hybri motion/orce control was use to control en-eector motions an orces in their own subspaces. In the experiments, superior motion an orce tracking perormance were reporte. In the control esign, switching rom ree-space motion to constraine motion was utilize. In summary, NMBC or hyraulic manipulators with nonswitching contact control law an with rigorous stability proo in ree-space an constraine motions is still an open problem. This problem is aresse in the present paper. III. MATHEMATICAL FOUNDATION This section provies essential mathematical ounations neee in control system esign. A. Linear/Angular Velocity Vectors an Force/Moment Vectors Consier an orthogonal, three-imensional coorinate system {A} (calle rame {A} or simplicity) attache to the rigi boy. Let the linear/angular velocity vector o rame {A} be written as A V = [ A v A ω] T, where A v R 3 an A ω R 3 are the linear an angular velocity vectors o rame {A}. Similarly, let the orce/moment vector in rame {A} be written as A F = [ A A m] T, where A R 3 an A m R 3 are the orce an moment vectors applie to the origin o rame {A}, expresse in rame {A}. Then, consier two given rames, enote as {A} an {B}, ixe to a common rigi boy. The ollowing relations hol B V = A U T B A V (1) A F = A U B B F (2) where A U B R 6 6 enotes a orce/moment transormation matrix that transorms the orce/moment vector measure an expresse in rame {B} to the same orce/moment vector measure an expresse in rame {A}. B. Parameter Aaptation The ollowing projection unction rom [8] is use or parameter aaptation:

4 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 3 Deinition 1: A projection unction P(s(t), k, a(t), b(t), t) R is a ierentiable scalar unction eine or t such that its time erivative is governe by with κ = P = ks(t)κ (3), i P a(t) an s(t), i P b(t) an s(t) 1, otherwise where s(t) R is a scalar variable, k > is a constant an a(t) b(t) hols. The projection unction eine in (3) has the ollowing property: For any constant P c subject to a(t) P c b(t), it ollows that ( (P c P) s(t) 1 P k ). (4) C. Virtual Cutting Points an a Simple Oriente Graph In the VDC approach, the original system is virtually ecompose into the subsystems by placing conceptual virtual cutting points (VCPs). A cutting point orms a virtual cutting surace on which three-imensional orce vectors an threeimensional moment vectors can be exerte rom one part to another. The VCP is eine as shown in Deinition 2. Deinition 2 [8]: A cutting point is a irecte separation interace that conceptually cuts through a rigi boy. At the cutting point, two parts resulting rom the virtual cut maintain equal positions an orientations. The cutting point is interprete as a riving cutting point by one part an is simultaneously interprete as a riven cutting point by another part. A orce vector R 3 an a moment vector m R 3 are exerte rom one part to which the cutting point is interprete as a riving cutting point to the other part to which the cutting point is interprete as a riven cutting point. Ater the original system is virtually ecompose into subsystems by placing VCPs, the system can be represente by a simple oriente graph. A simple oriente graph is eine, as shown in Deinition 3. Deinition 3 [8]: A graph consists o noes an eges. A irecte graph is a graph in which all eges have irections. An oriente graph is a irecte graph in which each ege has a unique irection. A simple oriente graph is an oriente graph in which no loop is orme. D. L 2 an L Stability Deinition 4 [8]: Lebesgue space, enote as L p with p being a positive integer, contains all Lebesgue measurable an integrable unctions (t) subject to p = lim T T (t) p τ 1 p < +. (5) Two particular cases are consiere: (a) A Lebesgue measurable unction (t) belongs to L 2 i an only i lim T T (t) 2 τ < +. (b) A Lebesgue measurable unction (t) belongs to L i an only i max t [, ) (t) < +. The ollowing Lemma 1 (Lemma 2.3 in [8]) provies that a system is stable with its ailiate vector x(t) being a unction in L an its ailiate vector y(t) being a unction in L 2. Lemma 1 [8]: Consier a non-negative ierentiable unction ξ (t) eine as ξ (t) 1 2 x(t)t Px(t) (6) with x(t) R n, n 1 an P R n n being a symmetric positive-einite matrix. I the time erivative o ξ (t) is Lebesgue integrable an governe by ξ (t) y(t) T Qy(t) s(t) (7) where y(t) R m, m 1 an Q R m m being a symmetric positive-einite matrix an s(t) is subject to s(t)t γ (8) with γ <, then it ollows that ξ (t) L, x(t) L an y(t) L 2 hol. The ollowing Lemma 2 provies that L 2 an L signal retains its properties ater passing through a irst-orer multipleinput-multiple-output (MIMO) ilter. Lemma 2 [8]: Consier a irst-orer MIMO system escribe by ẋ(t) + Kx(t) = u(t) (9) with x(t) R n, u(t) R n, an K R n n being symmetrical an positive-einite. I u(t) L 2 L hols, then x(t) L 2 L an ẋ(t) L 2 L hol. The ollowing Lemma 3 provies an asymptotic convergence or an error signal e(t). Lemma 3 [4]: I e(t) L 2 an ė(t) L, then lim e(t) =. t E. Virtual Stability The unique eature o the VDC approach is the introuction o a scalar term, namely the virtual power low (VPF); see Deinition 5. VPFs uniquely eine the ynamic interactions among the subsystems an play an important role in the einition o virtual stability, which is eine in Deinition 6. Deinition 5 [8]: The VPF with respect to rame {A} can be eine as the inner prouct o the linear/angular velocity vector error an the orce/moment vector error as p A = ( A V r A V ) T ( A F r A F) (1) where A V r R 6 an A F r R 6 represent the require vectors o A V R 6 an A F R 6, respectively. Deinition 6 [8]: A subsystem with a riven VCP to which rame {A} is attache an a riving VCP to which rame {C} is attache is sai to be virtually stable with its ailiate vector x(t) being a virtual unction in L an its ailiate vector y(t) being a virtual unction in L 2, i an only i there exists a non-negative accompanying unction such that ν(t) 1 2 x(t)t Px(t) (11) ν(t) y(t) T Qy(t) + p A p C s(t) (12)

5 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 4 (a) (b) {B} Open chain 2 (c) Open chain 1 Object 1 Object {B} Close chain 2 Close chain 1 Remaining subsystem Object Open chain 2 Open chain 1 VCP = Virtual Decomposition Simple oriente graph Object 1 {T O2 } Object = Object 2 Open chain 4 {O 2 } Open chain 3 = VCP Open chain = Fig. 1. The stuie system. Subigure (a) shows the original two-dof hyraulic manipulator. Subigure (b) shows a virtual ecomposition o the system. Note the parallelism (//) in the VCPs. Subigure (c) shows a simple oriente graph o the virtually ecompose system. hols, subject to M {G} {G} Object 2 s(t)t γ s (13) with γ s <, where P an Q are two block-iagonal positive-einite matrices an p A an p C enote the VPFs (by Deinition 5) at rames {A} an {C}, respectively. IV. CONTROL OF THE MANIPULATOR This section aresses a esign o novel impeance control metho or a two-dof hyraulic manipulator, which is shown in Fig. 1(a). Even though a two-dof system is stuie in this paper, the evelope approach is extenable to systems with any number o actuators. First, to aress the highly nonlinear behaviour o the hyraulic manipulator, Section IV-A shows the manipulator s internal control esign base on the VDC approach. Then, Section IV-B introuces the novel impeance control metho, esigne using the ramework o VDC. A. Virtual Decomposition Control The irst step in the VDC approach is to virtually ecompose an original system into subsystems (i.e., objects an open chains) by placing conceptual VCPs (see Deinition 2). Then, the ecompose subsystems are represente by a simple oriente graph imposing ynamic interactions among subsystems. The system s virtual ecomposition an the simple oriente graph presentation are aresse in Section IV-A1. Then, the subsystem-ynamic-base control is esigne to make each subsystem qualiie to be virtually stable. The virtue o the VDC is that when all subsystems are virtually stable, the stability o the entire system can be guarantee. This is aresse in Section V. T O2 F VCP {T O2 } Fig. 2. Object 2 in the constraine motion. G F {O 2 } {G} 1) Virtual Decomposition an Simple Oriente Graph: The virtually ecompose manipulator is shown in Fig. 1(b). As iscusse, changing the control (or ynamics) o one subsystem oes not aect the control equations within the rest o the system. In this stuy, only the control equations subject to Object 2 have been change in relation to [23]. For this reason, only the control esign or Object 2 (subsystem interacting with the environment) has been stuie in etail in this paper. Control esigns or the remaining subsystems, shown at the ashe line in Fig. 1(b), can be oun in [23]. The simple oriente graph or the manipulator is shown in Fig. 1(c). In this paper, the subsystems insie the ashe line in Fig. 1(c) are consiere as one subsystem; see the corresponing lines in Fig. 1(b). Each subsystem correspons to a noe, an each VCP correspons to a irecte ege whose irection eines the orce reerence irection. Thus, a VCP is simultaneously interprete as a riving VCP by one subsystem (rom which the orce/moment vector is exerte or the irecte ege is pointing away) an as a riven VCP by another subsystem (towar which the orce/moment vector is exerte or the irecte ege points) [8]. Next, in Sections IV-A2 through IV-A4, the kinematics, ynamics an control o Object 2 are given. 2) Object 2 Kinematics: Fig. 2 shows the Object 2, to which rame {O 2 } is ixe to escribe the orce an motion speciications. Frame {T O2 } exists at the riven VCP o Object 2, an rame {G} is the en-eector target rame where the contact occurs an in which the Cartesian motion an orce control is speciie. The linear/angular velocity vector G V R 6 in rame {G} can be written as G V = N c χ (14) where χ R 2 is the Cartesian velocity vector an the mapping matrix N c can be written as [ ] T 1 N c =. 1 The ollowing relations hol or Object 2: O 2 V = G U T O 2 G V = T O2 U T O 2 T O2 V. (15) 3) Object 2 Dynamics: The en-eector orce/moment vector in rame {G} can be written as G F = N c G (16) where G R 2 is a Cartesian contact orce vector (exerte by the manipulator on the environment).

6 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 5 The net orce/moment vector O 2F R 6 o Object 2, expresse in rame {O 2 }, can be written in view o [8] as M O2 t (O 2 V ) + C O2 ( O 2 ω) O 2 V + G O2 = O 2 F (17) where M O2 R 6 6 enotes the mass matrix, C O2 ( O 2ω) R 6 6 enotes the matrix o Coriolis an centriugal terms an G O2 R 6 enotes the gravity terms. On the other han, the net orce/moment vector (i.e., orce resultant equation) or Object 2 can be written as O 2 F = O 2 U To2 T o2 F O 2 U G G F. (18) 4) Object 2 Control: This section aresses the VDCbase rigi boy control o Object 2. Similar to (14), the require linear/angular velocity vector in the en-eector target rame {G} can be written as G V r = N c χ r (19) where χ r R 2 is the require Cartesian space velocity (esign) vector speciie later in (31). In view o (15), the require velocity transormations in Object 2 can be written as O 2 V r = G U T O 2 G V r = T O2 U T O 2 T O2 V r. (2) Similar to (16), the require orce/moment vector in the en-eector target rame {G} can be obtaine as G F r = N c G (21) where G R 2 is a esire Cartesian contact orce vector. Then, in view o [8], the require net orce/moment vector or Object 2 can be written as with O 2 F r = Y O2 θ O2 + K O2 ( O 2 V ) (22) Y O2 θ O2 = M O2 t (O 2 V r ) + C O2 ( O 2 ω) O 2 V r + G O2 (23) where regressor matrix Y O2 R 6 13 an parameter vector θ O2 R 13 can be solve as shown in [8, in Appenix A]. Moreover, in (22), θ O2 enotes the estimate o θ O2 an K O2 is a symmetric positive-einite matrix characterizing the velocity eeback control. The estimate parameter vector θ O2 in (22) nees to be upate. Deine s O2 = Y T O 2 ( O 2 V ). (24) Then, (3) can be use to upate the ith element o θ O2 as θ O2 i = P(s O2 i,ρ O2 i,θ O2 i,θ O 2 i,t), i {1,2,...,13} (25) where θ O2 i enotes the ith element o θ O2, s O2 i enotes the ith element o s O2, ρ O2 i > is the upate gain, an θ O2 i an θ O2 i enote the lower boun an the upper boun o θ O2 i. In relation to (18), the require orce resultant equation can be written as O 2 F r = O 2 U TO2 T O2 F r O 2 U G G F r. (26) Finally, the ollowing Lemma 4 is use to prove the virtual stability o Object 2. Lemma 4: Consier Object 2, escribe by (15), (17) an (18), combine with its control equations (2), (22) an (26) an with the parameter aaptation (24) an (25). Let the nonnegative accompanying unction ν O2 be ν O2 = 1 2 (O 2 V ) T M O2 ( O 2 V ) i=1 (θ O2 i θ O2 i) 2 ρ O2 i Then, the time erivative o (27) can be expresse by (27) ν O2 ( O 2 V ) T K O2 ( O 2 V ) + p TO2 p G (28) where p TO2 is the VPF by Deinition 5 at the riven VCP o Object 2, an p G characterizes the VPF between the eneector an the environment. Proo: The proo is similar to Appenix B in [23]. Remark 1: Note that Object 2 has only one VCP (see Fig. 2) but two VPFs exists in (28). The VPF p TO2 locates at the VCP in Object 2. Thus, or the virtual stability o Object 2, a solution (which satisies Deinition 6) must be oun or the VPF p G in (28). This will be aresse later in Section V-A. B. The Design o the Propose Impeance Control In this section, the impeance control law by Hogan [25] is introuce irst in Section IV-B1. Then, the propose Cartesian space impeance control laws are esigne in Section IV-B2. 1) Impeance Control Law: In view o Hogan [25], the target impeance or the manipulator can be escribe as G G = M ( χ χ) D ( χ χ) K (χ χ) (29) where M R 2 2, D R 2 2 an K R 2 2 are iagonal positive-einite matrices an characterize the esire inertia, amping an stiness, respectively. Neglecting the inertia term in (29), the target impeance can be written as G G = D ( χ χ) K (χ χ). (3) Then, the ollowing Assumption 1 is mae or the esire impeance parameters D an K. Assumption 1: The esire amping D an stiness K are selecte such that: 1) their ratio an magnitues are not subject to unstable behavior in the overall system; an 2) the target impeance in (3) is attainable or the manipulator. Assumption 1 imposes the conition that the impeance parameters D an K must be selecte within the ynamic range o achievable impeance, so-calle Z-with [41], which eines the combination o stiness an amping that can be passively achieve by a certain mechanism. One metho to eine Z-with or hyraulic articulate systems is given in [42]. 2) Propose Impeance Control Laws: In the ramework o VDC, the require velocity 1 serves as a reerence trajectory or a system an the control objective is to make the controlle actual velocities track the require velocities. In this section, the 1 The general ormat o a require velocity inclues a esire velocity (which usually serves as a reerence trajectory or a system) an one or more terms that are relate to control errors [8].

7 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 6 G. χ χ The propose impeance control; see eqs. (31) (33).. χ r q r Jacobian Matrix Object 2 is aresse in Section IV.A.4. Remaining subsystem is aresse in [23] Physical system, aresse in etail in [23] u c p VDC Hyraulic ynamics Manipulator Dynamics G χ. q, q Fig. 3. Diagram o the esigne novel impeance control or the hyraulic manipulator. The require Cartesian velocity χ r is compute rom (31). Then, the require joint velocity vector q r is solve using the Jacobian matrix. The etaile structure or the remaining subsystem (see Fig. 1) insie the VDC block can be oun in [23]. The output u c o the VDC block is the control signal or the hyraulic valves, which control the hyraulic cyliners. Finally, the motion ynamics o the manipulator is prouce by the output orce p o the hyraulic cyliners. Please see [23] or more etails. objective is to esign such a control or the require Cartesian velocity vector χ r R 2 in (19), which 1) realizes the Cartesian impeance behaviour or the manipulator an 2) qualiies Object 2 as virtually stable in the sense o Deinition 6. Let χ r or the manipulator be esigne as χ r = χ + Λ χ (χ χ) + Λ ( G G ) (31) where Λ χ R 2 2 an Λ R 2 2 are two iagonal positiveeinite matrices characterizing Cartesian position an orce control, an let them be eine accoring to Conition 1. Conition 1: The iagonal positive-einite matrices Λ an Λ χ are eine as Λ = D 1 (32) Λ χ = K D 1. (33) Then, the ollowing Theorem 1 provies that the target impeance behaviour (3) can be achieve or the system. Theorem 1: Consier the propose control law (31), which eines the require velocity behaviour or the system. I an only i the iagonal positive-einite matrices Λ an Λ χ in (31) are eine accoring to (32) an (33) in Conition 1, then the control law (31) equals the target impeance law (3). Proo: See Appenix A. A iagram or the propose control esign is shown in Fig. 3. Remark 2: Compare to the typical Cartesian ree-space control law, term Λ ( G G ) is inclue in (31). In this stuy, G = [ ] T is use. This enables a high control banwith in ree space (see ree-space results in [22]), provie that zero contact orce is measure in ree-space motions, i.e., ( G G ) = hols. Alternatively, time-variant solutions or G are not exclue. Remark 3: Note that the actual target impeance behavior in (3) is not irectly involve in the system control; rather, the propose impeance control law (31) with Conition 1 provies an input or the control system (see Fig. 3). In principle, the propose control law (31) iers rom the target impeance law (3). However, as Theorem 1 shows, the target impeance behaviour (3) can be esigne or (31) with D an K by using Conition 1. Theorem 1 satisies the irst objective above: realization o the target impeance behaviour or the require Cartesian velocity vector χ r in (19). The realization o the secon objective (Object 2 qualiies as virtually stable in the sense o Deinition 6) using the esigne control laws (31) (33) is analyze next in Section V-A. V. STABILITY ANALYSIS The conceptual VPFs (see Deinition 5) are a unique eature o VDC an they are use to aress the ynamic interaction between subsystems. The virtual stability (see Deinition 6) o every subsystem ensures that, at every place VCP, a negative VPF (at a riving VCP o a subsystem) is connecte to its corresponing positive VPF (at a riven VCP o the ajacent subsystem). Thus, VPFs act as stability connectors between subsystems; eventually, they cancel each other out at every VCP [8]. Finally, as aresse in Theorem 2.1 in [8], the virtual stability o every subsystem ensures the L 2 an L stability (see Lemma 1) o the entire system. As can be seen rom Fig. 2, only one VCP ( having a VPF p TO2 ; see (28) ) is speciie or Object 2. However, in (28), there also exists another VPF p G (characterizing the ynamic interaction between the manipulator an the environment), which has no stabilizing counterpart. Consequently, it is control esigner s obligation to ensure that a control is esigne or the system such that Object 2 qualiies to be virtually stable. Next, in Section V-A, the VPF p G between the manipulator an the environment is analyze using Deinitions 5 6, eventually leaing to the stabilizing solution or p G an virtual stability o Object 2. Finally, the stability o the entire system is proven in Section V-B. A. Virtual Stability o Object 2 Using the esigne control law (31), the ollowing Lemma 5 can be erive. Lemma 5: Deinition 5, (14), (16), (19), (21), (3) an (31) yiels p G = ( χ χ) T (D Λ D D )( χ χ) +(χ χ) T (K Λ K Λ χ K )(χ χ) +(χ χ) T (2D Λ K Λ χ D K )( χ χ). (34) Proo: See Appenix B. Then, the ollowing Lemma 6 is eine or p G to provie suicient conitions or the virtual stability o Object 2. Lemma 6: Let the ollowing constraints hol or the iagonal positive-einite matrices Λ an Λ χ : Λ D 1 (35) Λ χ K Λ. (36) Then, it ollows rom Lemma 5 that t p G (τ)τ γ G (37) hols with γ G <. Proo: See Appenix C. Finally, the ollowing Theorem 2 ensures that Object 2 qualiies as virtually stable in the sense o Deinition 6.

8 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 7 Theorem 2: Let Conition 1 hol. Then, using Lemma 4, Lemma 5 an Lemma 6, Object 2 qualiies as virtually stable in the sense o Deinition 6. Proo: Let Conition 1 hol. Substituting Λ an Λ χ (in Conition 1) into Lemma 5 yiels p G = t p G(τ)τ =, which satisies (37) in Lemma 6 (note that Conition 1 satisies the constraints in (35) an (36) in Lemma 6). Consier the act that Object 2 has one riven VCP associate with rame {T O2 }. Then, using Lemma 4 an (37), Object 2 qualiies as virtually stable in the sense o Deinition 6. Remark 4: The esign process or the propose impeance control laws (31) (33) is iterative. The virtual stability, i.e, Deinition 6, o Object 2 etermines the general constraints or this process. In aition, the esigne control input shoul be written in the orm o require velocity (see the ootnote in Section IV-B2), constraining the esign o (31). Then, given the target impeance law (3), Deinition 5 is use to esign the control laws (31) (33), which realizes the Cartesian impeance behavior or the system (see Theorem 1) an provies virtual stability or Object 2 (see Theorem 2). Remark 5: Assumption 1, Conition 1 an Theorem 1 eine that Λ an Λ χ in (31) are not subject to unstable system behaviour. As the constraints (35) an (36) in Lemma 6 suggest, in aition to Conition 1, other stable solutions or Λ an Λ χ can also exist. This is true; a Z-with [41] can be etermine or the impeance-controlle systems [42]. Note that selecting Λ an Λ χ with (35) an (36) (excluing solution in Conition 1) will result to ierent impeance behaviour in relation to the speciie D an K. VI. EXPERIMENTS This section emonstrates the contact control perormance o the propose controller. System set-up an implementation issues are outline in Section VI-A. Section VI-B emonstrates the controller s ability to prevent excessive contact orces when the manipulator collies with an object. The main results o this stuy are presente in Section VI-C, where the propose control metho is veriie an the accuracy o Theorem 1 is shown in practice. A. Experimental Set-up an Implementation Issues The experimental set-up or the contact experiments is shown in Fig. 4. For the environmental contact, a set o wooen pallets was place on a rubber mat. The set-up consiste o the ollowing harware components: Space DS113 system, with 3 ms sample time 475 kg payloa, enote as M in Fig. 1(a) Bosch 4WRPEH1 proportional valve (1 m 3 p = 3.5 MPa per notch) or cyliners Heienhain ROD 456 incremental encoer (5 inc/rev) with IVB interpolation units or joints 1 an 2, proviing a theoretical piston position resolution < mm Druck PTX14 pressure transmitters (range o 25 MPa) B. Stability o the Entire System In relation to [23], only the control laws or Object 2 have been change in this paper. Thus, in view o [23], the nonnegative accompanying unction an its time-erivative or the remaining subsystem, shown with ashe lines in Fig. 1(b) (c), can be written as ν R (38) ν R p TO2 (39) where p TO2 is the VPF at the riving VCP o this subsystem. The ollowing Theorem 3 guarantees the stability o the entire system, in view o Lemma 1. Theorem 3: Consier Object 2, shown in Fig. 2 an escribe by Lemma 5. Furthermore, let the remaining subsystem be aresse with (38) an (39), an let Assumption 1 an Conition 1 hol. Then, using (27), (28), (38), (39) an Lemma 6, it can be shown that O 2V r O 2V L 2 L hols, in view o Lemma 1. Consequently, this yiels that Λ 1 ( χ χ)+λ 1 Λ χ (χ χ)+( G G ) L 2 L hols. In the special case ( G G ) =, it ollows rom Lemma 2 that χ χ L 2 L an χ χ L 2 L hol, with an asymptotic convergence or χ χ (in the sense o Lemma 3). Proo: See Appenix D. Remark 6: In Theorem 3, ( G G ) enotes constraine motion an ( G G ) =, with G = [ ] T, enotes ree-space motions. Y {B} X 1st pallet 2n pallet 3r pallet Y {G} X Fig. 4. The experimental set-up. Manipulator s base rame {B} an eneector target rame {G} are shown in re. The manipulator s position in this igure shows the starting point o the riven motion trajectories. In the experiments, the manipulator shoul be sti along the rame {G} X-axis (the irection o unconstraine motion) an compliant along the rame {G} Y-axis (the irection o constraine motion). For this reason, in D an K, high gains were impose along the X-axis an minor gains were impose along the Y-axis. Table I shows Λ an Λ χ base on the etermine D an K. TABLE I GAINS FOR THE CARTESIAN CONTROL; SEE (31) Speciie D an K or the manipulator D = iag(1 1 5, ) K = iag(1 1 6, ) Conition 1 Λ = iag(1. 1 5, ) Λ χ = iag(1, 5)

9 [m] [m] Force [kn] Cart. Y [m] IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 8 As Fig. 4 shows, a orce sensor that measures the exact siximensional contact orces/moments at the en-eector was not use. The contact orces (applie by the manipulator on the environment) were estimate rom the cyliner chamber pressures by removing the gravitational orces. This methoology to estimate the contact orces is iscusse in more etail in [23]. Even though this metho has some limitations, e.g., in orce estimation accuracy, it may provie a practical solution or many heavy-uty operations where a ragile six-dof orce/moment sensor cannot be place at the en-eector. B. Collision Experiment To test the propose controller interaction ynamics behaviour, in the irst set o experiments, a collision between the manipulator en-eector an an obstacle was arrange with three ierent Cartesian velocities. The manipulator position in Fig. 4 shows the starting point or the test trajectories. Then, a set o wooen pallets was place in the way o the riven trajectories, so that the en-eector collie with the surace o the secon pallet (when pallet 1 was remove); see Fig. 4. In the irst trajectory, the en-eector was instructe to travel rom.5 m to 1. m along the Y-axis o the system base rame {B} in one secon. Then, the same path was riven over three secons an, inally, over ive secons. The plots in the irst row o Fig. 5 show the manipulator position paths in blue, re an green, using a one-secon trajectory, three-secon trajectory an ive-secon trajectory, respectively. In these plots, the esire path is shown in black an the surace o the secon pallet is epicte with a ashe line. As these plots show, the propose controller limits the en-eector path when contact with the environment is establishe. The collision velocities between the en-eector an the environment were.52 m/s,.26 m/s an.18 m/s or the one-secon trajectory, three-secon trajectory an ive-secon trajectory, respectively. Note that the X-axes o the plots are scale signiicantly smaller compare to the Y-axes. The plot in the secon row o Fig. 5 shows the measure en-eector contact orces in blue, re an green, using the one-secon trajectory, three-secon trajectory an ive-secon trajectory, respectively. The contact points are shown by blue circles. As this plot shows, the propose controller eiciently limits the contact orce (applie by the manipulator on the environment along the rame {G} Y-axis) to approximately 28 N when contact with the environment is establishe. Because contact orces were estimate rom the cyliner chamber pressures, inaccuracies exist in the estimates; thereore, the measure en-eector orce beore contact is not zero. This is ue to the act that system inertia an piston riction were not consiere in the contact orce estimation. Thus, the smaller velocity yiels a better contact orce estimation in ree space. One practical metho to estimate link accelerations (an to aress the system inertia in the contact orce estimation propose in [23]) can be oun in [43]. C. Target Impeance Behaviour In the secon set o experiments, three test cases were chosen to evaluate the ability o the propose controller to Cartesian path (1-sec. traj.) Pallet 2 surace Measure path Desire path Cart. X [m] Cart. X [m] Cart. X [m] Cartesian orces along Y-axis o rame {G} Contact point Cartesian path (3-sec. traj.) Measure path Pallet 2 surace Desire path Measure path Pallet 2 surace Fig. 5. The collision experiments. The plots in the irst row show the measure en-eector paths in color, the esire paths in black an the surace o the environment with the ashe line. The test cases in the irst row are riven using 1-secon, 3-secon an 5-secon trajectories. The plot in the secon row shows the measure contact orces in the test trajectories Cartesian path (5-sec. traj.) Desire path 1-secon trajectory 3-secon trajectory 5-secon trajectory perorm the target impeance behaviour. In test case 1, contact occurre on the surace o pallet 1. In test case 2, contact occurre on the surace o pallet 2 (ater removing pallet 1). In test case 3, contact occurre on the surace o pallet 3 (ater removing pallets 1 an 2). The manipulator position in Fig. 4 shows the starting point or the test trajectories. Fig. 6 shows the test trajectory in the Cartesian coorinates (along the Y- an X-axes o the system base rame {B}). The suraces o wooen pallets 1 3 are marke in the irst plot (see relation to Fig. 4). As Fig. 6 shows, the en-eector was irst instructe to travel a istance o.5 m along the Y-axis in three secons (see time interval 3 s rom the irst plot). Then, the en-eector was instructe to travel a istance o.5 m along the X-axis in three secons (see time interval 3 6 s rom the secon plot). Finally, the en-eector was instructe to travel a istance o.5 m along the Y-axis in three secons (see time interval 6 9 s rom the irst plot). The main results o this paper are shown in Figs The irst plots in Figs. 7 9 show the esire Cartesian paths in black an the measure Cartesian paths in re. The contact point is shown by a blue circle in these plots. As all these plots show, the propose controller elimits the en-eector position when contact with the environment occurs, thus preventing excessive contact orces rom being applie to the environment Desire Cartesian position trajectory along Y-axis Pallet 1 surace Pallet 2 surace Pallet 3 surace Desire Cartesian position trajectory along X-axis Fig. 6. Desire Cartesian position trajectories. The irst plot shows the esire Cartesian position trajectory along the Y-axis, an the secon plot shows the the esire Cartesian position trajectory along the X-axis. The suraces o wooen pallets 1 3 are marke in the irst plot (see the pallets in Fig. 4).

10 Force [kn] Position error [mm] Y [m] Force [kn] Position error [mm] Y [m] Force [kn] Position error [mm] Y [m] IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 9 The secon plots in Figs. 7 9 show the position tracking error along the X-axis o rame {G}, in which irection the manipulator was mae sti. The contact point is shown by a blue circle in these plots. As these plots show, the maximum position tracking errors are all below 4 mm or the test trajectory. This can be consiere a signiicant result in light o the manipulator scale, which has a reach o about 3.2 meters. The last plots in Figs. 7 9 show the measure contact orce exerte by the manipulator on the environment (shown in black) along the Y-axis o rame {G}, in irection which the manipulator was mae compliant. The contact point is shown by a blue circle in these plots. As aresse in Theorem 1, the target impeance behavior (3) shoul be achieve or the manipulator when the iagonal positive-einite matrices Λ an Λ χ in (31) are eine accoring to Conition 1. Consequently, the measure contact orce shoul correspon to the contact orce G y solve rom (3), i.e, 2 G y = y [D ( χ χ) + K (χ χ) + G ] (4) where y = [ 1], G = [ ] T, an ( χ χ) an (χ χ) are variables rom the riven test cases. As the last plots in Figs. 7 9 show, the measure contact orce (shown in black) correspons accurately to the contact orce suggeste by (4) (shown in green). The nearly perect matching o the two lines in these plots emonstrates the accuracy o Theorem 1. The maximum eviations between the measure contact orce an (4) are 136 N, 17 N an 146 N in Figs. 7 9, respectively. VII. CONCLUSIONS In this paper, a novel impeance control metho in the Cartesian space was evelope or hyraulic manipulators. To acquire high-banwith close-loop perormance or the system, the internal control or the manipulator was esigne accoring to the subsystem-ynamics-base virtual ecomposition control (VDC) approach. Interaction ynamics between the hyraulic manipulator en-eector an the environment were analyze in a novel manner, using the virtual power low, which is a unique eature o VDC. From the analysis, an explicit metho (see Conition 1) was evelope to eine parameters or the propose impeance control law (external control or the manipulator), so that target impeance behavior can be esigne or the hyraulic manipulator. The L 2 an L stability was guarantee or both ree-space an constraine motion control. The experimental results support the mathematical theorems on the stability-guarantee (see Theorem 3) target impeance behaviour (see Theorem 1). Even though a two-dof system was stuie in this paper, the evelope approach is extenable to systems with any number o actuators. The results o this stuy can be use to realize compliant behaviour or complex an nonlinear systems, not limite only to hyraulic manipulators. The control metho, which rigorously aresses the nonlinear ynamic behaviour o the system, can be applie to many ynamically challenging (robotic) tasks, such as legge locomotion. 2 Note that (4) is solve rom the target impeance law (3) by Hogan [25], which is not irectly involve in the control laws; rather it is use to eine Λ an Λ χ in control law (31); see Conition 1 an Remark D = 22 K = 11 Cartesian path X [m] Cartesian position tracking along X-axis o rame {G} Contact point Cartesian orce along Y-axis o rame {G} Measure rom (4) Fig. 7. Test Case 1: Environmental contact on the surace o the irst pallet (see Figs. 4 an 6). The collision velocity with pallet 1 was approx..22 m/s. The irst plot shows the esire Cartesian position path in black an the measure Cartesian path in re. The secon plot shows the Cartesian position tracking along the rame {G} X-axis. The last plot shows the measure Cartesian contact orce (along the rame {G} Y-axis) in black. The contact orce suggeste by the target impeance law (4) is shown in green D = 22 K = 11 Cartesian path X [m] Cartesian position tracking along X-axis o rame {G} Contact point Cartesian orce along Y-axis o rame {G} Measure rom (4) Fig. 8. Test Case 2: Environmental contact on the surace o the secon pallet (see Figs. 4 an 6). The collision velocity with pallet 2 was approx..27 m/s. The irst plot shows the esire Cartesian position path in black an the measure Cartesian path in re. The secon plot shows the Cartesian position tracking along the rame {G} X-axis. The last plot shows the measure Cartesian contact orce (along the rame {G} Y-axis) in black. The contact orce suggeste by the target impeance law (4) is shown in green D = 22 K = 11 Cartesian path X [m] Cartesian position tracking along X-axis o rame {G} Contact point Cartesian orce along Y-axis o rame {G} Measure rom (4) Fig. 9. Test Case 3: Environmental contact on the surace o the thir pallet (see Figs. 4 an 6). The collision velocity with pallet 3 was approx..28 m/s. The irst plot shows the esire Cartesian position path in black an the measure Cartesian path in re. The secon plot shows the Cartesian position tracking along the rame {G} X-axis. The last plot shows the measure Cartesian contact orce (along the rame {G} Y-axis) in black. The contact orce suggeste by the target impeance law (4) is shown in green.

11 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 1 APPENDIX A PROOF FOR THEOREM 1 Deine Λ = D 1 an Λ χ = K Λ, as propose in Conition 1. Then, it ollows rom the iagonal positive-einite property o D, K an Λ χ that D 1 Λ χ D = Λ χ (41) D K D 1 = K (42) hol. Substituting (3) into (31) an using Conition 1 an (41) yiels χ r = χ + Λ χ (χ χ) Λ [D ( χ χ) + K (χ χ)] = χ + Λ χ (χ χ) D 1 D ( χ χ) D 1 Λ χ D (χ χ) = χ + Λ χ (χ χ) χ + χ Λ χ (χ χ) = χ. (43) Then, using Conition 1, (42) an (43) yiels χ r = χ + Λ χ (χ χ) + Λ ( G G ) G G = Λ 1 ( χ χ r ) Λ 1 Λ χ (χ χ) G G = D ( χ χ) D K D 1 (χ χ) G G = D ( χ χ) K (χ χ). (44) Note that the irst row in (44) is equal to (31), whereas the last row is equal to (3). This completes the proo or Theorem 1. APPENDIX B PROOF FOR LEMMA 5 It ollows rom Deinition 5, (14), (16), (19), (21), (3), (31), N T c N c = I 2 2 an the iagonal positive-einite property 3 o matrices Λ χ, Λ, D an K that p G = ( G V r G V ) T ( G F r G F) = [( χ χ) + Λ χ (χ χ) + Λ ( G G )] T N T c N c ( G G ) = ( χ χ) T ( G G ) + (χ χ) T Λ T χ ( G G ) + ( G G ) T Λ T ( G G ) = ( χ χ) T [D ( χ χ) + K (χ χ)] (χ χ) T Λ T χ [D ( χ χ) + K (χ χ)] + [D ( χ χ) + K (χ χ)] T Λ T [D ( χ χ) + K (χ χ)] = ( χ χ) T (D Λ D D )( χ χ) + (χ χ) T (K Λ K Λ χ K )(χ χ) + (χ χ) T (2D Λ K Λ χ D K )( χ χ) (45) hols, which valiates Lemma 5. 3 The iagonal property o matrices Λ χ, Λ, D an K yiels Λ T χ = Λ χ, Λ T = Λ, D T = D an K T = K. APPENDIX C PROOF FOR LEMMA 6 Let Λ D 1 an Λ χ K Λ hol, as eine in (35) an (36). Then, it ollows rom (45) that p G (χ χ) T A 1 ( χ χ) (46) hols, where A 1 = (2D Λ K Λ χ D K ) hols, in view o (35) an (36). Integrating (46) over time yiels that t p G (τ)τ 1 2 (χ (t) χ(t)) T A 1 (χ (t) χ(t)) 1 2 (χ () χ()) T A 1 (χ () χ()) 1 2 (χ () χ()) T A 1 (χ () χ()) = γ G (47) hols with γ G <, which valiates Lemma 6. APPENDIX D PROOF FOR THEOREM 3 Let the remaining subsystem, shown with ashe lines in Fig. 1(b) (c), be aresse with (38) an (39), an let Assumption 1 an Conition 1 hol. Then, in view o (27) an (38), the non-negative accompanying unction ν tot or the entire manipulator can be chosen as ν tot = ν R + ν O2 1 2 (O 2 V ) T M O2 ( O 2 V ) i=1 (θ O2 i θ O2 i) 2. (48) ρ O2 i Then, in view o (28) an (39), the time erivative o (48) can be written as ν tot ( O 2 V ) T K O2 ( O 2 V ) p TO2 + p TO2 p G = ( O 2 V ) T K O2 ( O 2 V ) p G (49) where, in view o Lemma 6, t p G (τ)τ γ G (5) hols with γ G <. It ollows irectly rom Lemma 1 an (48) (5) that O 2 V L 2 L (51) hols. Using (14), (15), (19), (2) an (51) yiels χ r χ L 2 L. (52) Then, subtracting χ rom both sies o (31) an using (52) yiels Λ 1 ( χ χ) + Λ 1 Λ χ (χ χ) + ( G G ) L 2 L. (53) Let ( G G ) = hol. Subtracting χ rom both sies o (31) yiels χ r χ = ( χ χ) + Λ χ (χ χ). (54)

12 IEEE/ASME TRANSACTIONS ON MECHATRONICS: FOCUSED SECTION ON DESIGN AND CONTROL OF HYDRAULIC ROBOTS 11 Then, it ollows irectly rom Lemma 2, (52) an (54) that χ χ L 2 L (55) χ χ L 2 L (56) hol. Then, Lemma 3 yiels lim t [ χ (t) χ(t) ] =. REFERENCES [1] A. Alleyne an R. Liu, On the limitations o orce tracking control or hyraulic servosystems, J. Dyn. Syst. T. ASME, vol. 121, no. 2, pp , [2] C. An an J. Hollerbach, Dynamic stability issues in orce control o manipulators, in Proc. American Control Con., 1987, pp [3], Kinematic stability issues in orce control o manipulators, in Proc. IEEE Int. Con. Robotics an Autom., vol. 4, 1987, pp [4], The role o ynamic moels in cartesian orce control o manipulators, Int. J. Robot. Res., vol. 8, no. 4, pp , [5] S. Eppinger an W. Seering, Unerstaning banwith limitations in robot orce control, in Proc. IEEE Int. Con. Robotics an Autom., vol. 4, 1987, pp [6], Three ynamic problems in robot orce control, in Proc. IEEE Int. Con. Robotics an Autom., 1989, pp [7] T. Yoshikawa, Force control o robot manipulators, in Proc. IEEE Int. Con. Robotics an Autom., vol. 1, Apr. 2, pp [8] W.-H. Zhu, Virtual Decomposition Control - Towar Hyper Degrees o Freeom Robots. Springer-Verlag, 21. [9] C. An, C. Atkeson, an J. Hollerbach, Moel-base control o a robot manipulator. MIT press Cambrige, MA, 1988, vol [1] L. Sciavicco an B. Siciliano, Moeling an Control o Robot Manipulators. Lonon: Springer, 2. [11] R. N. Jazar, Theory o applie robotics: kinematics, ynamics, an control. Springer Science & Business Meia, 21. [12] W.-H. Zhu et al., Virtual ecomposition base control or generalize high imensional robotic systems with complicate structure, IEEE Trans. Robot. Autom., vol. 13, no. 3, pp , [13] W.-H. Zhu, Z. Bien, an J. De Schutter, Aaptive motion/orce control o multiple manipulators with joint lexibility base on virtual ecomposition, IEEE Trans. Autom. Control, vol. 43, no. 1, pp. 46 6, [14] W.-H. Zhu an J. De Schutter, Aaptive control o mixe rigi/lexible joint robot manipulators base on virtual ecomposition, IEEE Trans. Robot. Autom., vol. 15, no. 2, pp , [15], Control o two inustrial manipulators rigily holing an egg, IEEE Control Systems, vol. 19, no. 2, pp. 24 3, [16], Experimental veriications o virtual-ecomposition-base motion/orce control, IEEE Trans. Robot. Autom., vol. 18, no. 3, pp , 22. [17] W.-H. Zhu an G. Vukovich, Virtual ecomposition control or moular robot manipulators, in Proc. IFAC Worl Congress, Sep. 211, pp [18] W.-H. Zhu et al., Precision control o moular robot manipulators: The VDC approach with embee FPGA, IEEE Trans. Robot., vol. 29, no. 5, pp , 213. [19] W.-H. Zhu an J. Pieboeu, Aaptive output orce tracking control o hyraulic cyliners with applications to robot manipulators, J. Dynamic Syst., Meas. Control, vol. 127, no. 2, pp , Jun. 25. [2] J. Koivumäki an J. Mattila, The automation o multi egree o reeom hyraulic crane by using virtual ecomposition control, in Proc. IEEE/ASME Int. Con. Av. Intell. Mechatr., 213, pp [21], An energy-eicient high perormance motion control o a hyraulic crane applying virtual ecomposition control, in Proc. IEEE/RJS Int. Con. Intell. Robots Syst., 213, pp [22], High perormance non-linear motion/orce controller esign or reunant hyraulic construction crane automation, Automation in Construction, vol. 51, pp , 215. [23], Stability-guarantee orce-sensorless contact orce/motion control o heavy-uty hyraulic manipulators, IEEE Trans. Robot., vol. 31, no. 4, 215. [24] M. Raibert an J. Craig, Hybri position/orce control o manipulators, J. Dyn. Syst. T. ASME, vol. 13, no. 2, pp , [25] N. Hogan, Impeance control: An approach to manipulation: Parts I- III, J. Dyn. Syst. T. ASME, vol. 17, no. 1, pp. 1 24, [26] D. Whitney, Historical perspective an state o the art in robot orce control, in Proc. IEEE Int. Con. Robotics an Autom., vol. 2, 1985, pp [27] M. Vukobratović an A. Tuneski, Contact control concepts in manipulation robotics an overview, IEEE Trans. In. Electron., vol. 41, no. 1, pp , Feb [28] O. Khatib, A uniie approach or motion an orce control o robot manipulators: The operational space ormulation, IEEE Trans. Robot. Autom., vol. 3, no. 1, pp , [29] J. De Schutter an H. Van Brussel, Compliant robot motion: Parts I-II, Int. J. Robot. Res., vol. 7, no. 4, pp. 3 17, [3] N. McClamroch an D. Wang, Feeback stabilization an tracking o constraine robots, IEEE Trans. Autom. Control, vol. 33, no. 5, pp , [31] S. Chiaverini an L. Sciavicco, The parallel approach to orce/position control o robotic manipulators, IEEE Trans. Robot. Autom., vol. 9, no. 4, pp , [32] J. Pratt et al., Virtual moel control: An intuitive approach or bipeal locomotion, Int. J. Robot. Res., vol. 2, no. 2, pp , 21. [33] B. Heinrichs, N. Sepehri, an A. Thornton-Trump, Position-base impeance control o an inustrial hyraulic manipulator, IEEE Control Systems, vol. 17, no. 1, pp , [34] S. Taazoli et al., Impeance control o a teleoperate excavator, IEEE Trans. Control Syst. Technol., vol. 1, no. 3, pp , 22. [35] S. Salcuean, S. Taazoli, P. Lawrence, an I. Chau, Impeance control o a teleoperate mini excavator, in Proc. 8th Int. IEEE Con. on Avance Robotics, 1997, pp [36] S. Salcuean et al., Bilateral matche-impeance teleoperation with application to excavator control, IEEE Control Systems, vol. 19, no. 6, pp , [37] H. Zeng an N. Sepehri, On tracking control o cooperative hyraulic manipulators, Int. J. Control, vol. 8, no. 3, pp , 27. [38] H. Zeng, Nonlinear control o co-operating hyraulic manipulators, Ph.D. issertation, University o Manitoba, 27. [39] C. Semini et al., Towars versatile legge robots through active impeance control, Int. J. Robot. Res., vol. 34, no. 7, pp , 215. [4] G. Tao, A simple alternative to the barbalat lemma, IEEE Trans. Autom. Control, vol. 42, no. 5, p. 698, may [41] J. Colgate an J. Brown, Factors aecting the z-with o a haptic isplay, in Proc. IEEE Int. Con. Robotics an Automation, 1994, pp [42] T. Boaventura et al., Stability an perormance o the compliance controller o the quarupe robot HyQ, in Proc. o IEEE/RSJ Int. Con. on Intel. Robots an Syst., 213, pp [43] J. Vihonen, J. Honkakorpi, J. Tuominen, J. Mattila, an A. Visa, Linear accelerometers an rate gyros or rotary joint angle estimation o heavy-uty mobile manipulators using orwar kinematic moeling, IEEE/ASME Trans. Mechatronics, vol. 21, no. 3, pp , 216. Janne Koivumäki receive his M.Sc. egree in Automation Engineering rom the Tampere University o Technology (TUT), Finlan, in 212. He has worke at IHA, TUT, since 211 an is currently pursuing a octoral egree in Machine Automation. From 1/215 his work has been supporte by the Acaemy o Finlan uner the project Cooperative heavy-uty hyraulic manipulators or sustainable subsea inrastructure installation an ismantling. His research interests inclue a control o electrohyraulic systems an hyraulic robotic manipulators, nonlinear moel-base control, contact orce/motion control an energyeiciency o lui power systems. Jouni Mattila receive his M.Sc. (Eng.) in 1995 an Dr. Tech. in 2, both rom the Tampere University o Technology (TUT), Finlan. He has been involve in numerous inustrial research projects, incluing the Sanvik AutoMine with Hermia Group Lt. He is currently a Proessor in Machine Automation in the Dept. o Intelligent Hyraulics an Automation (IHA), TUT. For the past ten years, he has been a program manager on ITER usion reactor maintenance projects involving research on heavyuty hyraulic robotic manipulators. His research interests inclue machine automation, nonlinear moel-base control o hyraulic robotic manipulators an energy-eiciency o lui power systems. View publication stats