Bond graph Based Approach To Passive Teleoperation Of A Hydraulic Backhoe

Transcription

1 Bond graph Based Approach To Passive Teleoperation Of A Hydraulic Backhoe Kailash Krishnaswamy and Perry Y. Li Abstract Human operated, hydraulic actuated machines are widely used in many high-power applications. Improving productivity, safety and task quality (eg. haptic feedback in a teleoperated scenario) has been the focus of past research. For robotic systems that interact with the physical environments, passivity is a useful property for ensuring safety and interaction stability. While passivity is a well utilized concept in electromechanical robotic systems, investigation of electrohydraulic control systems that enforce this passivity property are rare. This paper proposes and experimentally demonstrates a teleoperation control algorithm that renders a hydraulic backhoe / force feedback joystick system as a two-port, coordinated, passive machine. By fully accounting for the fluid compressibility, inertia dynamics and nonlinearity, coordination performance is much improved over a previous scheme in which the coordination control approximates the hydraulic system by its kinematic behavior. This is accomplished by a novel bond graph based three step design methodology: ) energetically invariant transformation of the system into a pair of shape and locked subsystems; 2) inversion of the shape system bond graph to derive the coordination control law; ) use of the locked system bond graph to derive an appropriate control law to achieve a target locked system dynamics while ensuring the passivity property of the coordinated system. The proposed passive control law has been experimentally verified for its bilateral energy transfer ability and performance enhancements. I. INTRODUCTION A dynamic system with input u(t) and output y(t) is said to be passive with respect to s(u(t), y(t)), called the supply rate, if for all u( ) and time τ >, : τ s(u(t),y(t))dt c 2. When s(u(t),y(t)) is the (scaled) physical power input, a passive system is one from which the net (scaled) energy that can be extracted is finite, i.e. passive system only stores and dissipates energy but cannot generate energy of its own. Passive systems are easier and safer for humans to control and they interact with arbitrary passive systems, including many physical objects, stably []. The inherent safety that passive systems provide has been exploited by researchers in the development of human interacting machines like smart exercise machines [2], passive bilateral teleoperators [], [], COBOTS [] and PTER [6]. While passivity is widely used in electromechanical systems, it is relatively rare in electrohydraulic systems. Our research in the past few years has been directed toward developing passive hydraulic systems [7], [8], [9], [], []. To enable the passivity analysis of electrohydraulic systems under closed loop control, methods for enabling single-stage [7], [2] and multistage [9] hydraulic valves (through hardware redesign or feedback) to behave like passive two port systems (a command port and a hydraulic port) respect to the total scaled power input have been developed. Based on these passive valves, bilateral teleoperation control laws for force feedback joysticks and hydraulic actuators [8], [], and multi-dof hydraulic machines (such as a backhoe) [] have been proposed. These control laws render the closed loop controlled hydraulic machine passive, and ensure that the joystick and the hydraulic machine are coordinated. In our previous approaches to teleoperation of hydraulic machines [8], [], [], passivity of the closed loop control system was always maintained (to ensure safety). However, perfect coordination between the joystick and the machines was ensured only if a kinematic model (i.e. ignoring fluid compressibility and inertial dynamics) of the latter could be assumed. This requirement led to poor coordination performance under certain operating conditions. In the present paper, a bond graph based passive teleoperation algorithm design methodology that includes the previously neglected fluid compressibility and inertial effects is proposed. By utilizing bond graphs inherent passivity property, passive control can be designed for systems with complex dynamics. Although the approach is applied a hydraulic system here, it should be applicable to other mechatronic systems as well. The key aspects of the design methodology is presented in this paper. The readers are referred to [] for other aspects such as robustness and performance in the presence of uncertainties. Similar to [], [], [], the system dynamics are first decomposed using an energetically invariant transformation into a pair of shape and locked subsystems that correspond to the coordination and the overall system aspects. The subsystems are represented in bond graphs so that more complex dynamics can be handled. Stable coordination control is then derived Materials based on research supported by the National Science Foundation ENG/CMS To appear in the ASME Journal of Dynamic Systems, Measurement and Control: Special Issue on Novel Robotics and Control. First submitted in March 2, revised September 2. K. Krishnaswamy is with the Honeywell Labs, 66 Technology Dr, Minneapolis, MN 8, USA. kailash.krishnaswamy@honeywell.com P. Y. Li is with Department of Mechanical Engineering, University of Minnesota, Church St. SE, Minneapolis, MN, USA. pli@me.umn.edu. Please send all correspondence to him.

2 2 JOYSTICK HUMAN OPERATOR BAC KHOE PC Fig.. valves. Teleoperated backhoe consists of a motorized joystick, and a 2-DOF hydraulic backhoe actuated by a set of hydraulic cylinders and proportional by inverting the shape system bond graph. An appropriate control law can be derived by comparing the locked system bond graph with that of the target locked system. Locked system control algorithms that achieve second order and fourth order target dynamics will be presented. Experimental results demonstrate the bilateral energy transfer ability and performance enhancements. The rest of this paper is organized as follows. Section II presents the models of all the subsystems of the teleoperated backhoe and formulates the control design problem. In section III, the bond graph approach to the design of passive teleoperation controller is given. Experimental results of the implementation are presented in section IV. Concluding remarks are given in section V. Notations: Matrices / vectors are bolded, and scalar elements are un-bolded. Superscripts i =, 2 denotes the link number. II. SYSTEM MODELING AND CONTROL OBJECTIVE The teleoperated backhoe system is shown in Fig.. It consists of a 2-DOF master motorized joystick in a horizontal plane and a -DOF Backhoe of which -DOF (the boom) is constrained. Backhoe motions are actuated by single-rod hydraulic actuators which are in-turn driven by single-stage proportional valves. Hydraulic power is provided by a pressure compensated pump operating at constant supply pressure P s. The system models are summarized below with details contained in []. A. Single-stage passive valve and Hydraulic actuators We consider a single stage passive valve connected to a single-rod actuator (Fig. 2). The essential step in passifying a single stage proportional valve to obtain the passive valve is load pressure feedback. Details of this procedure can be found in [7], []. The spool dynamics of the two passive valves, which have been experimentally verified are given by: ẋ v = Ω x v + F x ΓF L () where x v = [x v,x 2 v] T denotes the valve spool displacements, F x = [F x,f 2 x] T denotes the passive valve control input, F L = [F L,F 2 L ]T ; F i L = Ai cp i c A i rp i r denotes the differential hydraulic load force, Γ = Diag[γ,γ 2 ] denotes the load force gain acting on the spool, Ω = Diag[ω,ω 2 ]. Consider now the pressure dynamics of the corresponding single rod hydraulic actuators: β V c Ṗ c = Q c A c ẋ, β V r Ṗ r = Q r + A r ẋ (2) where V c = Diag[V c,v 2 c ], V r = Diag[V r,v 2 r ] are the total volumes of the cap and the rod side chamber and hose volumes, β is the fluid compressibility, P c = [P c,p 2 c ], Q c = [Q c,q 2 c], P r = [P r,p 2 r ], Q r = [Q r,q 2 r] are the pressures and flows in

3 F x Actuator Supply P s P c P c Q c F L = A c P c A r P r A c x 2 Return P r Spool Q r x v P r A r F e Fig. 2. Single stage passive valve connected to a single-rod actuator the actuator chambers, A c = Diag[A c,a 2 c], A r = Diag[A r,a 2 r] are the cap and rod side piston cross-section areas and x is the piston position. Since Vr i and Vc i are dominated by the hose volumes, Vc/r i min(v c/r i ) max(v c/r i ) are assumed to be constants for both the cap side and rode sides. Using the relationships Q i c/q i r = A i c/a i r and the matched and symmetric properties of the four way directional valve [], and by decomposing the valve flows into the no-load flows and a shunt flow components [6]: where Q i c = A i c(k i Qx i v K i T( x i v,f i L) F i L) () Q i r = A i r(k i Qx i v K i T( x i v,f i L) F i L) () K i q KQ(sgn(x i i v)) = As P s () (A i c ) + (A i r) KT( x i i v,fl) i KQ i = xi v As P s As P s sign(x { i v)fl i A s (sgn(x i A i v)) = c, x i v A i r, x i v < (6) (7) Here KQ i (sgn(xi v)) and KT i ( xi v,fl i ) are proportional to the no-load flow gain, and the nonlinear shunt flow conductance. Notice that KT i ( xi v,fl i ) F L i2 and it can be considered to be the load induced energy dissipation within the valve. By differentiating the energy function, W av = 2 x2 v + 2β PT c V c P c + 2β PT r V r P r it can be shown that the combination of the passive valve and the hydraulic actuator is passive with respect to the supply rate s av ((F x,x v ),(F L,ẋ)) = x v T K Q Γ F x ẋ T F L where F L = A P c A 2 P r. This supply rate is the difference between the fictitious command power x v T K Q Γ F x and the output mechanical power (ẋ T F L ). B. Backhoe inertia dynamics and motorized joystick dynamics Both the backhoe and the motorized joysticks are modeled as planar, rigid 2-link robotic systems. The former lies in the vertical plane and the joystick lies in the horizontal plane. Their dynamics are given by, M x (x)ẍ + C x (x,ẋ)ẋ = F L F e, (8) M q (q) q + C q (q, q) q = F q + T q, (9) where M = M T > are the respective inertia matrices, and Ṁ 2C are skew-symmetric. For the backhoe, F L = A c P c A r P r is the differential hydraulic force acting on each backhoe link and F e is the net environment force (including

4 friction and gravity). ẋ is a vector of the actuator piston velocities. For the motorized joystick, F q and T q are the motor actuated control torque and the human input torque, and q is a vector of the link angular velocities. By differentiating the kinetic energy functions W b = 2ẋT M x (x)ẋ, and W j = 2 qt M q (q) q, the mechanical backhoe and the joystick can be shown to be passive with respect to the supply rates C. Control objectives s b ((F L,ẋ),( F e,ẋ)) = ẋ T (F L F e ) s j ((F q, q),(t q, q)) = q T (F q + T q ). ) Passivity The closed loop teleoperated system should be passive with respect to the supply rate, s tele ((ρt q, q),(f e,ẋ)) = ρ q T T q ẋ T F e. () where q T T q and ẋ T F e are the human and work environment power inputs, and ρ is the desired power scaling factor so that the human power input is amplified (attenuated) when ρ > (ρ < ). 2) Coordination The backhoe and the joystick motion should mimic each other : i.e. E := αq x, () where α is a specified kinematic scaling. ) Target dynamics The desired target dynamics for the teleoperator system after coordination has been achieved can also be specified by the designer. III. PASSIVE TELEOPERATION CONTROLLER DESIGN The design procedure consists of the following steps: ) The system is decomposed using an energetically invariant transformation into a pair of shape and locked subsystems and represented in bond graphs; 2) inversion of the shape system bond graph to derive the coordination control law; ) use of the locked system bond graph to derive an appropriate control law to achieve a target locked system dynamics while ensuring the passivity property of the coordinated system. The dynamics of the joystick and hydraulic backhoe systems are given by M(q,x,K Q ) { }} { ρm q (q) q M x (x) β V c d ẋ β V r dt P c P r = K Q Γ x v ρc q (q, q) C x (x,ẋ) A c A r A c A 2 c K T A c A r K T A c K Q A r A c A r K T A 2 r K T A r K Q A c K Q A r K Q Ω q ẋ P c P r x v ρ(f q + T q ) + F e K Q Γ F x where Ω = K Q Γ Ω. In order to simplify the passive control synthesis and analysis, the dynamics of the master and slave systems (2) are decomposed using an energy invariant transformation: Ė αi I q q L F L F := I αψ Ψ ẋ A c A r P c () L A r (I A c Φ) Φ P r x v I x v where Ψ = α(ρm q (q)m x (x) + α 2 I) () Φ = A c + A r V c β A r A c V r β. () (2)

5 I : M L S e : ρt q F e q L I : K Q Γ C : I/α TF K Q Γ F x : S e x v K Q TF F L R : K T (αψ T I) TF GY : C EL Ψ T TF S e : ρf q R : Ω Ė SS I : M E S e : ρψ T T E Fig.. Bond graph of the teleoperator after energy invariant coordinate transformation. Using the above decomposition, the dynamics of the master and slave are transformed into M { }} { M E Ė C E C EL (αψ T I) Ė ρψ T F q ρψ T T E M L d q L dt F L = C LE C L αi q L (I αψ) αi K T K Q F L + ρf q + ρt q F e K Q Γ x v K Q Ω x v K Q Γ F x (6) d 2 dt F L =. (7) Eq.(6) and the new variables therein can be derived by substituting Eq.() into Eq. (2) (see [] for details). The transformation in Eq. () has the following properties: ) The coordination error Ė is explicitly one of the new variables. It represents the shape of the teleoperator. 2) The other coordinates [ q L,F L,x v ] and F L lie within the submanifold defined by Ė. Thus they describe the locked system dynamics when the teleoperator has been coordinated. In particular, q L = α q = ẋ when Ė ; ) the total energy of the system can be invariantly expressed in terms of M in (2) or ( M, 2 ) in (6): W total := ρ 2 qt M q (q) q + 2ẋT M x (x)ẋ + 2β P c T V c P c + 2β P r T V r P r + x v T K Q Γ x v = 2ĖT M E Ė + 2 qt LM L q L + 2 F L T F L + 2 F L T 2 F L + 2 x v T K Q Γ x v. (8) ) F L corresponds to the zero dynamics resulting from pressures in the actuator chambers that cancel out each other, and thus do not have any effects on any net mechanical motion. Eq. (7) shows that it is marginally stable. Eq.() is similar to the isometric decomposition proposed in [], [] except that Eq.() is proposed for N-DOF fourth order hydraulic systems whereas the decomposition proposed by [], [] is applicable to N-DOF second order simple (electro)mechanical systems. The bond graph describing the above dynamics is given in Fig.. Notice that the shape and the locked systems are still coupled. A. Coordination control design We now design a control law that ensures that E. To do this, consider the inverse dynamics of the system with output Ė. In Fig., the path between the joystick control input F q and Ė is the shortest causal path. Thus F q should be the coordination control input. The corresponding inverse dynamics (with input Ė and the output F q ) obtained using bicausal bonds [7], [8] suggests the coordination control law: F q = T E + Ψ T ρ ( ) C EL q L (αψ T I)F L K E E B E Ė, (9)

6 6 PSfrag replacements I : M L S e : ρt q F e q L K Q Γ F x : S e I : K Q Γ x v K Q C : I TF F L I/α TF S e : (αψ T I) Ė S e : E F (Ψ T I/α) R : Ω R : K T LOCKED SYSTEM DYNAMICS R : K E R : B E C : I E Ė I : M E Fig.. Ψ T C EL q L. COORDINATION ERROR DYNAMICS Bond graph of the master and slave systems with the coordination control law Eq. (9). Here E F = ρt E ρ K E E + B E Ė C LE Ė where B E = B E T > and K E = K E T >. This results in coordination error dynamics, M E (x,q)ë + C E ((x,q),(ẋ, q))ė = B E Ė K E E. (2) Since it can be shown that M E 2C E is skew symmetric, E exponentially. The effect of the coordination control law on the bond graph in Fig. is shown in Fig.. Notice that the shape and locked system dynamics appear in separate bond graphs. The shape system correspond to the coordination error dynamics, and the locked system that determines the haptic interaction between the teleoperator and the human and the work environment. The coordination control law (9) also affects the locked system bond graph through potentially active elements as indicated by the signal bond between F L to q L and the additional effort source S e. These active elements can equivalently be represented by Effort (AE) or Flow Amplifiers (AF) as defined in [9]. Signal (active) bonds are used here as they visually certifying the non-passive property of the bond graph. B. Locked System Design The locked system in Fig., has an additional control input F x which has not been utilized. This will be used to achieve two goals ) to render the locked system to be passive with respect to the supply rate in Eq. (); 2) to assign the locked system to have useful haptic properties. Both issues can be addressed simultaneously by ensuring that the locked system achieve some suitable target dynamics that respect the desired passvity property. While a variety of target dynamics can be specified, we consider two: th order dynamics as shown in Fig. : M L K Q Γ 2nd order dynamics as shown in Fig. 6: d q L z = C L Ψ T Ψ K T K Q dt z 2 K Q Ω q L ρt q α F e z + (2) z 2 M L (x,q) q L + C L ((x,q),(ẋ, q)) q L = ρt q α F e. (22) Remark Since q L = q = α ẋ (see Eq. ()) after coordination ((E,Ė) (,)) has been achieved, both Eq. (2) and Eq.(22) imply that the teleoperator system is passive w.r.t. the desired supply rate () when it is coordinated.

7 7 th ORDER LOCKED SYSTEM I : M L q L S e : ρt q F e I : K Q Γ C : I Ψ T TF z 2 K Q TF z R : Ω R : K T R : K E R : B E C : I E Ė I : M E DISSIPATIVE Fig.. The fourth order desired locked system in Eq.(2) The design process involves comparing the dynamics of the actual and desired locked systems and choosing the appropriate coordinate transformation which makes choice of the passive control input F x almost trivial. The coordinate transformation and control input ensure that the actual dynamics mimic the desired dynamics. We illustrate this procedure for the th order target dynamics Eq. (2). The dynamics of the actual locked system shown in Fig. are: M L K Q Γ d q L F L = C L Ψ T q L ρt q F e αi K T K Q F L + dt x v K Q Ω x v ( ) ρt E ρ K E E + B E Ė C LE Ė + Ψ T C EL q L + (I αψ)ė (2) K Q Γ F x Note that z, z 2 in Eq.(2) are different from F L, x v in Eq.(2) even though they have the same dynamics. We will determine the relationship between them later. In general, the desired locked dynamics (2) need not have the same parameters as the actual locked system (2). However, the proposed locked system minimizes additional control effort F x which would be necessary to modify the natural behavior of the coordinated hydraulic teleoperator and hence the actual locked system. The required control F x is obtained by comparing the actual dynamics of q L in (2) and the desired dynamics as given by (2), using a procedure similar to the bond graph based passification procedure proposed in [2]. The actual locked system dynamics (2) and desired dynamics (2) will be identical if and only if z := F L + Ψ T D(t). (2) ( ) where D(t) = ρt E ρ K E E + B E Ė C LE Ė+Ψ T C EL q L. By substituting this transformation for z into the desired dynamics (2) and comparing the actual and desired dynamics of F L it can be noted that the dynamics will be identical if

8 8 and only if [ z 2 := K Q (Ψ T αi) q L + x v + K Q K T Ψ T D(t) + d ] dt [ Ψ T D(t)]. (2) Proceeding in this manner, the above analysis leads to the following coordinate transformation [ q L F L x v ] T [ q L z z 2 ] T : q L I q L z = I F L + Ψ T D(t) z 2 K Q (Ψ αi) I x v K ( Q K T Ψ T D(t) + d dt [ Ψ T D(t)] ), (26) ( ) where D(t) = ρt E ρ K E E + B E Ė C LE Ė + Ψ T C EL q L. Applying Eq. (26) to the actual locked system dynamics (2) results in the following transformed dynamics: M L d q L C L Ψ T q L ρt q F e z = Ψ T K T K Q z +, (27) K Q Γ dt z 2 K Q Ω z 2 K Q Γ F x + D 2 (t) where D 2 (t) = K Q [z F L ] + K Q Γ (Ω[z 2 x v ] + d ) dt [z 2 x v ], (28) z F L := Ψ T D(t), z 2 x v := K Q (Ψ αi) q L + K Q ( K T Ψ T D(t) + d ) dt [ Ψ T D(t)]. Notice that the signals z F L and z x v depend on D(t) and its successive derivatives which are functions of the external variables T E, q L,F L,E and Ė. If the locked system control F x in (27) is chosen as follows: then the target locked system dynamics Eq. (2) will be achieved. In summary, K Q Γ F x = D 2 (t), (29) the coordination control law given in Eq. (9) ensures that the teleoperator will be coordinated so that E = αq x exponentially (see (2)) the locked system control law given by F x in Eq. (29) and the preceding transformations ensure that the desired target system dynamics (2) are achieved. Since q L = q = α ẋ and Eq. (2) is passive with respect to the supply rate, s tele ((T q, F e ),( q L, q L )) = q T L (ρt q F e ) which is equivalent to s tele ((T q, F e ),( q L, q L )) = q T L(ρT q F e ) = ρ qt q ẋf e A similar locked system design procedure can be applied to achieve the 2nd order target dynamics Eq.(22). Compared to the th order target dynamics, the high frequency content of human and environment forces are not filtered as much, resulting in a more transparent and responsive haptic feel. Since the original dynamics (2) is th order, the locked system control involves lowering the relative degree through feedforward actions. Due to space limitations, the algorithm is not presented here. Readers are referred to [] for details. For both the th and 2nd order target dynamics cases, the valve and the joystick control forces F x and F q assume accurate knowledge of Ψ, T E which depend on the backhoe / joystick inertia and external forces respectively. In reality, it may not be possible to know these parameters exactly. It is however possible to modify the control algorithms (such as some form of discontinuous control) so as to ensure the coordination and passivity properties []. IV. EXPERIMENTAL RESULTS In the experimental setup in Fig., the joystick is powered by a set of DC motors and instrumented with angular position encoders, and a JR force sensor for measuring the operator input force. The hydraulic system consists of a pressure compensated 9 LPM ( GPM) flow pump operating at 6.9 MPa ( PSI); a set of Vickers KBFDGV- series proportional valves that have been actively passified [7] and have bandwidths of about Hz; single rod hydraulic actuators instrumented with displacement sensors and chamber pressures sensors. A PC running MATLAB (MA) xpc Target provides real time control at a sample rate of KHz. The conducted experiments are typical of a digging task. The backhoe is teleoperated to dig into a sand box. A wooden box is buried in the sandbox to mimic an underground obstacle. A kinematic scaling of α = in./rad and a power scaling ρ = 2 are used. Both the th and 2nd order locked system teleoperation algorithm are tested. For comparison, the algorithm in [] that assumes a kinematic model of the hydraulic backhoe and neglects the compressibility and inertial dynamics was implemented as well.

9 9 I : M L DISSIPATIVE q L S e : ρt q F e I : K Q Γ C : I z K Q TF z S e : Ψ T z 2nd ORDER LOCKED SYSTEM R : Ω R : K T R : K E R : B E C : I E Ė I : M E DISSIPATIVE Fig. 6. Second order desired locked system A. Previous algorithm [] The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 7. The corresponding haptic force (F q ) experienced by the operator is shown in Fig. 8. The maximum observed coordination error as shown in Fig. 7 is within.in. When the backhoe hits the wooden box (t = 28 and t = 228), severe oscillations in the haptic force is experienced by the operator as shown in Fig. 8. This is a result of neglecting compressibility and inertia dynamics. B. th order locked system The results for the control law that mimics the th order target locked system (2) are shown in Fig. 9-Fig.. The maximum observed coordination error in Fig. 9 is within.in which is significantly better than that in Fig. 7 for the previous controller. Also, as the backhoe hits the underground wooden box (about t = 8s), there are no oscillations (Fig. ). Notice that the operator force is balanced by the work environment force when the interacting with the wooden box as indicated by the net force being nearly zero (Fig. ). C. 2th order locked system The results for the control law that mimics the 2th order target locked system (22) are shown in Fig. -Fig. 2. The maximum observed coordination error in Fig. is within.in which is even better than that in Fig. 9 for the th order locked system controller. There are again no oscillations when the backhoe hits the underground wooden box (about t = 6 and t = 9s) (Fig. ). It is interesting to note that for similar operating speeds, the force range for the 2nd order locked system (Fig. 2) is only 2 % of that for the th order locked system (Fig. ). This is an indication that the 2nd order locked system is easier (in terms of necessary power) to teleoperate than the th order locked system. The operator also reported that the teleoperator is more responsive is more able to feel the environment force using the 2nd order target system controller than the th order target system controller.

10 Bucket Displacement (inches) Stick Displacement (inches) Time (seconds) Fig. 7. Displacement trajectories (Scaled joystick - solid, Backhoe - dashed) during a digging task.. Feedback Torque (N m) time (sec) Fig. 8. Haptic torque (F q) trajectories (Stick - solid, Bucket - dashed) during a digging task. Bucket Displacement (inches) Stick Displacement (inches) Time (seconds) Fig. 9. Link and link 2 displacement trajectories for th Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task.

11 Haptic Force, ρ = 2 (N) 2 2 Haptic Force, ρ = 8 (N) time (sec) Fig.. Locked system Force (ρt q F e) trajectories during a digging task for th Order locked system (Stick - solid, Backhoe - dashed). Bucket Displacement (inches) Stick Displacement (inches) Time (seconds) Fig.. Link and link 2 displacement trajectories for the 2nd Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Haptic Force, ρ = 2 (N) Haptic Force, ρ = 8 (N) time (sec) Fig. 2. Locked system force (ρt q F e) trajectories during a digging task for 2nd Order locked system (Stick - solid, Backhoe - dashed).

12 2 V. CONCLUSIONS A passive teleoperation control algorithm for backhoe operation is proposed. The passivity property of the teleoperation scheme ensures stability of interaction of the teleoperated backhoe and a wide range of human / work environment. Using a bond graph based design procedure, neglected compressibility and inertial dynamics can be accounted for, thus rectifying a previously developed algorithm which assumed a kinematic modeled backhoe. This results in significant improvement in coordination performance that has been verifies in experiments. Moreover, the present approach allows the user to specify a desired target locked system dynamics which directly affect transparency and haptic sensation of the system. In particular, the 2nd order target system is more responsive and more transparent than the th order target system. Although the control design ensures that the closed loop system becomes passive after coordination has been achieved, the control law is not an Intrinsically Passive Controller (IPC), unlike the algorithm in [8]. An IPC has the advantage of guaranteeing passivity of the teleoperated backhoe even under significant system variation and uncertainties. Development of such a controller will greatly enhance the robustness properties of the teleoperated backhoe. REFERENCES [] Vidyasagar, M., 99. Nonlinear systems analysis. Prentice Hall []. [2] Li, P. Y., and Horowitz, R., 997. Control of smart machines, part : problem formulation and non-adaptive control. IEEE/ASME Transactions on Mechatronics, 2 () [December], pp [] Lee, D., and Li, P., 2. Passive coordination control of nonlinear mechanical teleoperator. IEEE Transactions on Robotics, 2 () [October], pp. xxx xxx. In press. [] Lee, D., and Li, P., 2. Passive bilateral feedforward control of linear dynamically similar teleoperated manipulators. IEEE Transactions on Robotics and Automation, 9 () [June], pp. 6. [] Colgate, J. E., Wannasuphoprasit, W., and Peshkin, M. A., 996. Cobots: Robots for collaboration with human operators. Proceedings of the ASME Dynamic Systems and Control Division, 8 [], pp. 9. [6] Gomes, M. W., and Book, W., 997. Control approaches for a dissipative passive trajectory enhancing robot. IEEE/ASME Conference on Advanced Intelligent Mechatronics []. [7] Li, P. Y., 2. Towards safe and human friendly hydraulics: The passive valve. ASME Journal of Dynamic Systems, Measurement and Control, 22 () [Sep.], pp [8] Li, P. Y., and Krishnaswamy, K., 2. Passive bilateral teleoperation of a hydraulic actuator using an electrohydraulic passive valve. International Journal of Fluid Power [August], pp. 6. [9] Krishnaswamy, K., and Li, P. Y., 22. Passification of a two-stage pressure control servo-valve. Proceedings of the American Control Conference, Anchorage, AK, 6 [May], pp [] Krishnaswamy, K., and Li, P. Y., 22. Single degree of freedom passive bilateral teleoperation of an electrohydraulic actuator using a passive multi-stage valve. Proceedings of the IFAC Mechatronics Conference, Berkeley, CA [Dec]. [] Krishnaswamy, K., and Li, P. Y., 2. Passive teleoperation of a multiple degree of freedom hydraulic backhoe using a dynamic passive valve. Proceedings of the ASME-IMECE []. [2] Li, P. Y., and Ngwompo, R. F., 2. Power scaling bondgraph approach to the passification of mechatronic systems - with application to electrohydraulic valves. ASME Journal of Dynamic Systems, Measurement and Control, 27 () [Dec.], pp. xxx xxx. In press. Also appeared in Proceedings of th IFAC World Congress, 22. [] Krishnaswamy, K., 2. Passive Teleoperation of Hydraulic Systems. PhD thesis, Department of Mechanical Engineering, University of Minnesota, May. PhD Thesis. Available at kk/thesis. [] Lee, D. J., 2. Passive Decomposition and Control of Interactive Mechanical Systems under Motion Coordination Requirements. PhD thesis, Department of Mechanical Engineering, University of Minnesota, May. [] Merritt, H. E., 967. Hydraulic Control Systems. John Wiley & Sons. [6] Li, P. Y., 998. Passive control of bilateral teleoperated manipulators. Proceedings of the 998 American Control Conference [], pp [7] Gawthrop, P. J., 99. Bicausal bond graphs. Proceedings of the International Conference on Bond Graph Modeling and simulation (ICBGM) [], pp [8] Ngwompo, R. F., Scavarda, S., and Thomasset, D., 2. Physical model-based inversion in control systems design using bond graph representation part : theory. Proceedings of the Institute of Mechanical Engineers, 2 [], pp. 9. [9] Ngwompo, R. F., and Gawthrop, P. J., 999. Bond graph based simulation of nonlinear inverse systems using physical performance specifications. Journal of the Franklin Institute, 6 [], pp