Bondgraph Based Approach to Passive Teleoperation of a Hydraulic ...

DSC TOC Proceedings of IMECE04 2004 ASME International Mechanical Engineering Congress and Exposition November 13-20, 2004, Anaheim, California USA Proceedings of ASME-IMECE 2004 ASME International Mechanical Engineering Congress and RD&D Expo November 13-19, 2004, Anaheim, CA USA

IMECE2004-61244

IMECE2004-61244

BONDGRAPH BASED APPROACH TO PASSIVE TELEOPERATION OF A HYDRAULIC BACKHOE∗

Perry Y. Li† Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis, MN 55455 Email: [email protected]

Kailash Krishnaswamy Honeywell Labs 3660 Technology Dr Minneapolis, MN 55418 Email: [email protected]

1

Introduction Passive systems possess the beneficial properties of ease of human control, and stable interaction with arbitrary passive systems. Motivated by these properties, our research has been directed towards developing passive hydraulic systems [1–5]. A new hydraulic element called the ‘passive valve’ was developed for both single-stage, [1] and multi-stage [3] valve configurations. The passive valves are mechatronic two-port devices with a command port and a hydraulic port such that they are passive with respect to the total scaled power input to the device. The single stage and multi-stage passive valves were used in passive bilateral teleoperation of a hydraulic actuator, [2] and [4]. The single stage passive valve was experimentally used in the teleoperation of a 2-DOF kinematic modeled hydraulic backhoe, [5]. Passive teleoperation ensured rigid, mechanical, two-port passive behavior of the motorized joystick teleoperated hydraulic backhoe. The benefit of ensuring passivity of the teleoperated system is that the system can interact stably with other passive elements like humans and work environments. A dynamic system with input u(t) and output y(t) is passive with respect to a supply rate R s(u(t), y(t)) ∈ L1e if ∃c such that, for all u(·) and time τ > 0, 0τ s(u(t), y(t))dt ≤ −c2 , [6].The inherent safety that passive systems provide has been exploited by researchers in the development of human interacting machines like smart exercise machines [7], passive bilateral teleoperators [8] and in other passive systems such as COBOTS [9] and PTER [10]. In a previous paper [5], we developed a control design method for 2-DOF passive teleoperation of a hydraulic backhoe using a motorized joystick. Although passivity of the teleoperated system was ensured, the algorithm ensured coordination of

ABSTRACT Human operated, hydraulic actuated machines are widely used in many high-power applications. Improving productivity, safety and task quality (eg. force feedback to the operator in a teleoperated scenario) has been the focus of past research. In addressing these issues, our research proposes and experimentally demonstrates a control technique that renders a hydraulic machine (teleoperated backhoe in this case) as a two-port, coordinated, passive machine. The passive teleoperated backhoe is driven by a human operator at one-port and interacts with the environment at the other. It guarantees interaction stability and safety with the human / work environment as the latter are usually passive. In previous work, a passive teleoperation algorithm was proposed for multi degree of freedom teleoperation of a hydraulic backhoe approximated by its kinematic behavior. The approximation led to severe performance deterioration under certain operating conditions. In this paper, a bondgraph based passive teleoperation architecture is proposed for the nonlinear dynamic modeled backhoe. Passive control is designed in two stages. In the first stage, bondgraph based system inversion ideas are used to determine a coordination control law. In the second stage, a desired locked system (desired dynamics of the coordinated teleoperator) is defined and an appropriate control law is determined to ensure the passivity property of the locked teleoperator. The proposed passive control law is experimentally verified for its bilateral energy transfer ability and performance enhancements.

† Associate

Professor and Corresponding Author

1

c 2004 by ASME Copyright °

JOYSTICK

can follow the development of the models as outlined in [11]. Throughout this paper, matrices / vectors are represented as bold face symbols and the elements are represented as the same symbol in normal font with a superscript i denoting the link number. Single-stage passive valve: The development of a single stage passive valve, we use in our design and experimental work was presented in [5]. Due to space constraints we include only the dynamics of the passive valve as given in Eq. (1).

HUMAN OPERATOR

BAC KHOE

x˙ v = −Ω1 xv + Fx − ΓFL

(1)

PC

Figure 1.

where xv = [xv1 , . . . , xvi ]T denotes the valve spool position, Fx = [Fx1 , . . . , Fxi ]T denotes the passive valve control input, FL = [FL1 , . . . , FLi ]T ; FLi = Aic Pci −Air Pri denotes the differential hydraulic load force, Γ = Diag[γ1 , . . . , γi ] denotes the load force gain acting on the spool, Ω1 = Diag[ω11 , . . . , ωi1 ]; ωi > 0. The valve is passive w.r.t. the supply rate: sv ((Fx , xv ), (P1 , Q1 ), (P2 , Q2 )) = xv T KQ Γ−1 Fx − Q1 T P1 − Q2 T P2 , where P1 , Q1 , P2 , Q2 are the pressures and flows in the cap and rod side actuator chambers. The first term in the supply rate is the power input by the exogenous control Fx , the sum of the second and the third terms is the net hydraulic power withdrawn by the hydraulic load (single-rod actuator). Hydraulic actuator: The pressure dynamics of a hydraulic actuator associated with link i are given by:

Schematic of the Teleoperated Backhoe

the joystick and the backhoe assuming a kinematic model of the latter. That is, dynamic effects due to a) hydraulic fluid compressibility and b) inertia of the hydraulic backhoe were ignored in order to determine a coordination control law. These assumptions led to poor performance during teleoperation under certain operating conditions. In the present paper, we propose a bond graph based passive teleoperation algorithm for the 2-DOF hydraulic backhoe by including the previously neglected fluid compressibility and inertial effects. The objective is to design a teleoperation controller, interacting with a 2-DOF master joystick at one port and a 2-DOF hydraulic backhoe at the other port. The controller ensures passive two-port behavior of the teleoperated backhoe w.r.t. a scaled power input as its supply rate. The twoport machine interacts with the human operator at one port and the environment at the other. The teleoperation controller also ensures the coordination of the master joystick and the backhoe at all times. The rest of this paper is organized as follows. Section 2 presents the models of all the subsystems of the teleoperated backhoe and formulates the control design problem. In section 3 we design the passive teleoperation controller. Experimental results of implementation are presented in section 4. Conclusions and future work are given in section 5.

V1 β−1 P˙1 = Q1 − A1 x˙ ,

V2 β−1 P˙2 = Q2 + A2 x˙

(2)

where V1 , V2 are the total volumes of the capside and the rodside chambers including the hose volume, respectively, β is the fluid compressibility, P1 = Diag[Pc1 , ..., Pci ], Q1 = Diag[Q1c , ..., Qic ], P2 = Diag[Pr1 , ..., Pri ], Q2 = Diag[Q1r , ..., Qir ] are the pressures and flows in the actuator chambers, A1 , A2 are the cap and rod side piston cross-section areas and x is the piston position. Since the actuator chamber volumes are primarily composed of hose volumes, although the actuator volumes V1 ,V2 are dependant on the piston position x, it is assumed that Vi β−1 ≈ min(Vi )β−1 ≈ max(Vi )β−1 , Qic = Aic (KQi xvi − KTi (|xvi |, FLi ) · FLi )

= −Air (KQi xvi − KTi (|xvi |, FLi ) · FLi ), p Kqi KQi = p As Ps i3 i3 Ac + A r K i |xvi | q Q KTi (|xvi |, FLi ) = √ As Ps As Ps − sign(xvi )FLi Qir

2

System modeling and control objective A schematic of the teleoperated backhoe is shown in Fig. 1. It consists of a 2-DOF master motorized joystick in a horizontal plane and a 3-DOF Backhoe of which 1-DOF (the boom) will be constrained. Backhoe motion is actuated by single-rod hydraulic actuators which are in-turn driven by single-stage proportional valves. A submerged constant displacement vane pump can provide upto 19 LPM (5 GPM) flow at 10.3 MPa (1.5 KPSI). In this section, we present models of the teleoperated backhoe subsystems and formulate the control objective. Interested readers

( Aic , As = Air ,

2

xvi ≥ 0 xvi < 0

(3) (4) (5) (6)

(7)


SLAVE (BACKHOE) DYNAMICS

Notice that KTi (|xvi |, FLi ) · FLi2 ≥ 0 for all xvi , FLi . The hydraulic actuator is a passive 3-port subsystem with respect to the supply rate: sa ((P1 , Q1 ), (P2 , Q2 ), (FL , x˙ )) = Q1 T P1 + Q2 T P2 − x˙ T FL where FL = A1 P1 − A2 P2 . This supply rate is the difference of cumulative input hydraulic power at the cap, (Q1 T P1 ) and rod, (Q2 T P2 ) side ports and the output mechanical power, (x˙ T FL ). Backhoe dynamics: The backhoe is modeled as a planar, rigid 2-link robotic system. Its dynamics are given by, Mx (x)¨x + Cx (x, x˙ )˙x = FL − Fe ,

C : V 2β−1

I : KQ KQ

: Se

xv

1

(KQ A2 )−1

z}|{ TF

P1

0

GY

TF |{z}

(KQ A1 )−1

P2

0

A−1 2

z}|{ TF R : KT A22 R : KT A21 TF |{z}

C : V 1β−1

I : Mx (x) x˙

1

Se : −Fe

A−1 1

ρFq : Se

I : ρMq (q) 1

(8)

q˙ Se : ρTq

MASTER (JOYSTICK) DYNAMICS

Figure 2.

Bondgraphs of Master and Slave systems.

guarantee stable interaction with the passive power scaled human and work environment. In addition, our teleoperation control should also ensure that the scaled coordination error E := αq − x → 0,

(11)

asymptotically, where α is a kinematic scaling. This ensures that the backhoe motion mimics the scaled joystick motion.

(9)

3

Passive teleoperation controller design The control design procedure consists of the following steps in sequence. 1. The dynamics of the teleoperated hydraulic backhoe will be represented using bondgraphs. 2. Bondgraph based system inversion ideas will be used to determine a coordination control law which will ensure that the master and slave systems of the backhoe remain coordinated at all times. 3. The coordination control law will be represented on the bondgraph of the hydraulic teleoperated backhoe and 1) a suitable coordinate transformation and 2) a passive control law will be determined so that the coordinated teleoperator (locked system) is passive w.r.t. the desired supply rate. The bondgraph representation of the passive motorized joystick with power scaling (9) and the passive hydraulic backhoe (1), (2) and (8) is shown in Fig. 2. The dynamics of the master and slave systems represented in the above bond graph, Fig. 2 are given by (12).

where Mq = Mq T > 0 and M˙ q − 2Cq is skew-symmetric, Fq and Tq are the motor actuated control torque and the human input torque on the joystick. q˙ is a vector of the link angular velocities. Notice that gravity does not affect the dynamics as the joystick motion is constrained to the horizontal plane. The joystick is a passive 2-port subsystem with respect to the supply rate, ˙ (Tq , q)) ˙ = q˙ T (Fq + Tq ). s j ((Fq , q), In order to analyze the passivity property of the interconnection of the subsystems, we use the interconnection lemma, [2] which states that the cascade interconnection of two passive systems results in a passive system with a modified supply rate. By virtue of this property the passive valve, hydraulic actuators and the backhoe form a cascade interconnected passive system. In order to ensure the passivity property of the teleoperated machine, we need to determine a teleoperation controller which ensures that he teleoperated system is passive with respect to the supply rate, ˙ (Fe , x˙ )) = ρq˙ T Tq − x˙ T Fe . stele ((ρTq , q),

x

¯1 R:Ω

where Mx = Mx T > 0 and M˙ x − 2Cx is skew-symmetric, FL = A1 P1 − A2 P2 is the differential hydraulic force acting on each backhoe link and Fe is the net environment force acting on the backhoe. x˙ is a vector of the actuator piston velocities. The environment force includes effects due to friction (links and actuators) and gravity. The mechanical backhoe is a passive 2-port subsystem with respect to the supply rate, sb ((FL , x˙ ), (−Fe , x˙ )) = x˙ T (FL − Fe ). Motorized joystick: The joystick is modeled as a planar, rigid 2-link robotic system. Its dynamics are: ˙ q˙ = Fq + Tq , Mq (q)q¨ + Cq (q, q)

Γ−1 F

Γ−1

   ρMq (q) 0 0 0 0 q˙ 0 Mx (x) 0 0 0  x˙   d    0 0 V1 β−1 0 0  P 1  =  dt P  −1  0 0 0 V2 β 0 2 xv 0 0 0 0 KQ Γ−1      ˙ −ρCq (q, q) 0 0 0 0 ρ(Fq + Tq ) q˙   0 −Cx (x, x˙) A1 −A2 0   x˙  −Fe        0 0 −A1 −A21 KT A1 A2 KT A1 KQ  P1  +      P   2   0 0 A2 A1 A2 KT −A2 KT −A2 KQ 2 ¯1 xv KQ Γ−1 Fx 0 0 −A1 KQ A2 KQ −Ω (12)

     

(10)

where ρ is the desired power scaling factor. In Eq. (10), the first term is the scaled human power and the second term is the environment power input to the teleoperated backhoe. Human power input to the teleoperator is scaled using ρ. Satisfying the above passivity condition will ensure that the joystick teleoperated backhoe will behave like a passive 2-port device and thus

¯1 = KQ Γ−1 Ω1 . In order to simplify the passive conwhere Ω trol synthesis and analysis, the dynamics of the master and slave 3


systems (12) are transformed using an orthogonal decomposition (w.r.t. the metric M) similar to the one proposed by [12], conceptually. The difference is that the following decomposition is proposed for N-DOF fourth order hydraulic system whereas the PSfragsecond replacements decomposition proposed by [12] is applicable to N-DOF order systems. The metric, M of the orthogonal decomposition KQ Γ−1 Fx : Se is given by  ρMq (q) 0 0 0 0 Mx (x) 0 0 0   0  0 V1 β−1 0 0  M :=  0   0 −1 0 0 V2 β 0  0 0 0 0 KQ Γ−1 

I : ML q˙L 1 I/α

I : KQ Γ−1 KQ −1

xv

1

z}|{ TF

C : ∆1 z}|{ TF

0 FL

GY−1: CELΨ−T (αΨT −I)

R : KT

¯1 R:Ω

SS

−I 0 0 Ψ 0 0 0 A1 −A2 0 A−1 2 (I − A1 Φ) Φ 0 0 0

I : ME

I : ML

  0 q˙ 0  x˙    0  P1   0 P2  xv I

q˙L 1 KQ −1

KQ Γ−1 Fx : Se

where

Se : ρΨT TE

xv

1

¯1 R:Ω

Se : ρTq − Fe

I/α

I : KQ Γ−1

(13)

Se : ρFq

z}|{ TF

Bondgraph of coordinate transformed teleoperator

and the decomposition is given in Eq. (13).  ˙   αI E  q˙ L  I − αΨ    0  FL  :=   ⊥  0 FL 0 xv

z}|{ TF

E˙ 1

Figure 3.

Se : ρTq − Fe

z}|{ TF

C : ∆1 z}|{ TF

0 FL

GY : CELΨ−T z}|{ z}|{ TF TF

(αΨT −I)−1

R : KT

Se : ρFq

E˙ 1

Ψ = α(ρMq (q)Mx (x)−1 + α2 I)−1 Φ = A1 + A2V1 β

−1

−1 A2 A−1 1 V2 β .

(14)

SS

(15) Figure 4.

ME  0   0 0

0 ML 0 0

 0 0 0 0   d ∆1 0  dt 0 KQ Γ−1

      ML 0 0 −CL αI 0 ˙ q˙ L q L d  0 ∆1 0  FL  =  −αI −KT KQ  FL  ¯1 xv 0 −KQ −Ω 0 0 KQ Γ−1 dt xv     ρFq − CLE E˙ ρTq − Fe  +  (I − αΨ)E ˙  0 + −1 0 KQ Γ Fx −T ¢ £ ¤ ¡ d Ψ Fq = ME E˙ − TE + CEL q˙ L − (αΨT − I)FL . dt ρ (17) 

    −CE −CEL (αΨT − I) 0 E˙ E˙  q˙L  q˙L   −CLE C αI 0 L    = FL  (I − αΨ) −αI −KT KQ  FL  ¯ 0 0 −KQ −Ω1 xv xv     ρΨT Fq ρΨT TE  ρFq  ρTq − Fe   + +     0 0 −1 0 KQ Γ F x d ⊥ F = 0. dt L

Inverse dynamics bondgraph of the teleoperator, input Fq and

input E˙ and the output Fq is represented as shown in Fig. 4. The inverse dynamics from the bondgraph can be written as follows:



∆2

Se : ρΨT TE

˙ output E

The above decomposition transforms the master-slave dynamics ˙ 2) into dynamics involving the states; 1) coordination error, E, ˙ average motion of the master and slave systems, q˙ L = x˙ = αq, if E˙ = 0, 3) load force acting on the backhoe, FL and 4) its perpendicular component, F⊥ L . Using the above decomposition, the dynamics of the master and slave are transformed into 

I : ME

(16)

An interested reader is referred to [5] for details on the individual parameters. The bond graph describing the above dynamics is given in Fig. 3.

The following choice of joystick control Fq

3.1

Coordination control design The shortest causal path between the output E˙ and a control input is of length 1 to the joystick control input Fq . Using bicausal bonds [13], the inverse dynamics bondgraph between the

Fq = −TE +

4

¢ Ψ−T ¡ CEL q˙ L − (αΨT − I)FL − KE E − BE E˙ , ρ (18) c 2004 by ASME Copyright °

PSfrag replacements

C : ∆1 C : ∆1 I : ML

Se : ρTq − Fe

q˙L I : KQ KQ Γ−1 Fx : Se

xv

1

¯1 R:Ω

Γ−1

1

I/α KQ −1

z}|{ TF

C:I 0

z}|{ TF

FL R : KT

Se

I : KQ

Se : EF1 (Ψ−T − I/α)

z2

: (αΨT

− I)−1 E˙

1

¯1 R:Ω

Γ−1

KQ −1

z}|{ TF

C:I

z1

C:I

C:I

1

I : ME

R : BE E˙

E 0

1

I : ME

COORDINATION ERROR DYNAMICS

DISSIPATIVE

Figure 6.

Figure 5. Bondgraph of the master and slave systems with coordination control law

Fourth order desired locked system

section) proposed in this paper is a design variable. This approach has the advantage of letting the control designer choose the haptic behavior of the teleoperator while ensuring the control objectives listed above. Notice that the locked system shown in Fig. 5 is not passive w.r.t. the desired supply rate (10) due to the presence of 1) signal bonds (uni-directional arrows) and 2) additional effort sources Se in the bond graph Fig. 5. The task now is to determine the remaining control input Fx so that the bondgraph shown in Fig. 5 is passive w.r.t. the supply rate (10). As stated above, the desired locked system structure is up to the control system engineer. Towards that end, two locked system designs are proposed in this paper. The first design shown in Fig. 6 proposes a fourth order locked system. The second design shown in Fig. 7 proposes a second order locked system.

where BE = BE T > 0 and KE = KE T > 0 ensures asymptotic coordination error dynamics, ˙ E˙ = −BE E˙ − KE E. ME (x, q)E¨ + CE ((x, q), (˙x, q))

Se : ρTq − Fe

1

R : KE

E˙

E 0

0

q˙L

Ψ−T

z}|{ TF

R : KT

R : KE R : BE

LOCKED SYSTEM DYNAMICS

I : ML

4th ORDER LOCKED SYSTEM

(19)

In the above particular case, the shortest causal path led to the joystick control input, Fq . In a general case, the shortest causal path would be associated with the system (master or slave) with lower relative degree. We now determine the energy necessary to implement the coordination control law (18) and a passive control so that the coordinated teleoperator is passive w.r.t. the supply rate (10). This is achieved by first graphically representing the coordination control in the bondgraph Fig. 3 and analyzing the power flow.

Locked System Design : 4th Order, 2-DOF Haptic Behavior We need to determine the passive valve exogenous control input Fx in Fig. 5 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired locked system dynamics shown in Fig. 6. 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 Fx almost trivial. The coordinate transformation and control input ensure that the actual dynamics mimic the desired dynamics. The dynamics of the actual locked system shown in Fig. 5 are:

3.2

Closed loop passivity property The coordination control (18) is represented in the bondgraph shown in Fig. 5. Notice the use of a signal bond to indicate the flow of effort from FL to q˙ L . Other researchers have used regular power bonds with one of the port variables set to 0 (Effort (AE) or Flow Amplifiers (AF) in [14]). Signal (active) bonds are used here as they visually certifying the non-passive property of the bond graph. In Fig. 5, EF1 is given by: ¡ ¢ EF1 = −ρTE − ρ KE E + BE E˙ − CLE E˙ − Ψ−T CEL q˙ L .



        ML 0 0 ρTq − Fe q˙L q˙L −CL Ψ−T 0 d  0 ∆1  0  FL  =  −αI −KT KQ  FL  +  0 dt x ¯1 0 xv 0 −KQ −Ω 0 0 KQ Γ−1 v ¡ ¢   −ρTE − ρ KE E + BE E˙ − CLE E˙+ Ψ−T CEL q˙L  ˙ + (I − αΨ)E KQ Γ−1 Fx (20)

The bondgraph shown in Fig. 5 represents the ‘Locked System’ behavior; i.e., the behavior of the coordinated teleoperator and ‘Shape System’ behavior; i.e., the behavior of the coordination dynamics of the teleoperator. The bottom disconnected bond graph in the figure represents the dynamics of the coordination error. The top disconnected bond graph in the figure represents the dynamics of the locked system and determines the haptics experienced by the human operator. Unlike the locked system in [12] which is the result of an isometric energetic decomposition, the desired locked systems (introduced in the following

The dynamics of the desired locked system shown in Fig. 6 are: 

        ML 0 0 q˙L q˙L ρTq − Fe −CL Ψ−T 0 d  0 ∆1  0   z1  =  −Ψ −KT KQ   z1  +  0 dt z ¯1 z2 0 0 −KQ −Ω 0 0 KQ Γ−1 2 (21)

5


Note that z1 , z2 are different from FL , xv even though they have the same dynamics. We will determine the relationship between them later. In general, the desired locked dynamics (21) need not have the same parameters as the actual locked system (20). However, the proposed locked system minimizes additional control effort Fx which would be necessary to modify the natural behavior of the coordinated hydraulic teleoperator and hence the actual locked system. The passive control objective is to design the passive valve control Fx in (20) (also in bondgraph in Fig. 5) so that the actual locked system behaves like the desired locked system in Eq. (21) (also in bondgraph in Fig. 6). By doing so, we can ensure that the locked system is passive with respect to the supply rate (10). In order to achieve this, compare the actual dynamics of q˙ L in (20) and the desired dynamics as given by (21), using a procedure similar to the procedure proposed by [15] where a bondgraph approach to passification of a single-stage valve is presented. Notice that the actual locked system dynamics (20) and desired dynamics (21) will be identical if and only if z1 := FL + ΨT D(t).

where ¶ µ d D2 (t) = KQ [z1 FL ] + KQ Γ−1 Ω[z2 xv ] + [z2 xv ] , dt

z1 FL := ΨT D(t),

¶ µ d z2 xv := KQ −1 (Ψ−1 − αI)q˙L + KQ −1 KT ΨT D(t) + [∆1 ΨT D(t)] . dt

Notice that the signals z1 FL and z1 xv are dependant on D(t) and its successive derivatives which are functions of external ˙ If the locked system control Fx variables TE , q˙ L , FL , E and E. in (25) is chosen as follows: KQ Γ−1 Fx = −D2 (t),



        ML 0 0 −CL Ψ−T 0 q˙L q˙L ρTq − Fe d  0 ∆1 . 0   z1  = −ΨT −KT KQ   z1  +  0 dt z ¯1 z2 0 0 0 KQ Γ−1 0 −KQ −Ω 2 (28)

The above analysis provides a sketch of the proof of the following theorem which summarizes the passive teleoperation design procedure. Interested readers are refered to [11] for a proof of the theorem.

(22)

Theorem 1 4th Order, 2-DOF Locked System, perfect model: The teleoperated hydraulic backhoe composed of a) the passive valve (1), b) the dynamically modeled hydraulic actuators (2), c) the mechanical backhoe (8) and d) the motorized joystick (9) which is concisely represented in Eq. (16) transforms to an interconnected locked system (25) under the choice of joystick control input Fq as given in (30) and the coordinate transformation (24). The interconnected locked system (25) is passive with respect to the supply rate:

· ¸ d z2 := KQ −1 (ΨT − αI)q˙L + xv + KQ −1 KT ΨT D(t) + [∆1 ΨT D(t)] . (23) dt

Proceeding in this manner, the above analysis leads to the following coordinate transformation [q˙ L FL xv ]T 7→ [q˙ L z1 z2 ]T : 





stele ((Tq , −Fe ), (q˙L , q˙L )) = q˙TL (ρTq − Fe ),



I 00 q˙ L q˙ L  z1  =  0 I 0 FL  z2 xv KQ −1 (Ψ−1 − αI) 0 I   0  ΨT D(t) + ¢ , ¡ d −1 T T KQ KT Ψ D(t) + dt [∆1 Ψ D(t)]

(27)

then the locked system is passive w.r.t. the supply rate (10) and its dynamics are given by:

¡ ¢ where D(t) = −ρTE − ρ KE E + BE E˙ − CLE E˙ + Ψ−T CEL q˙ L . By substituting this transformation for z1 into the desired dynamics (21) and comparing the actual and desired dynamics of FL it can be noted that the dynamics will be identical if and only if



(26)

(29)

if the passive valve control input, Fx is chosen as follows: Fq = −TE + (24)

¤ Ψ−T £ CEL q˙ L − (αΨT − I)FL − KE E − BE E˙ , ρ

Fx = −ΓKQ −1 D2 (t),

(30)

where D2 (t) is given by (26). The choice of joystick control input Fq in (30) ensures that the coordination error E → 0 asymptotically.

¡ ¢ where D(t) = −ρTE − ρ KE E + BE E˙ − CLE E˙ + Ψ−T CEL q˙ L . This transformation (Eq. (24)) when applied to the actual locked system dynamics (20) results in the following transformed dynamics:

˙ → (0, 0) Remark 1 Since the coordination error states (E, E) (as stated in theorem 1), the energy associated with the error dynamics V → 0. Owing to this property, ensuring that the locked system given by Eq. (28) is passive w.r.t. the supply rate (29) implies that the teleoperated backhoe is also passive w.r.t. the same supply rate, (29). Moreover, ensuring that the teleoperated backhoe is passive w.r.t. the 1-port supply rate

       ML 0 0 −CL Ψ−T 0 q˙ L q˙ L d  0 ∆1 0   z1  = −ΨT −KT KQ   z1  ¯1 z2 0 0 KQ Γ−1 dt z2 0 −KQ −Ω   ρTq − Fe  , (25) 0 + −1 KQ Γ Fx + D2 (t)

stele1 ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ), 6

(31)


PSfrag replacements

C : ∆1 I : ML

also ensures passivity of the teleoperated backhoe w.r.t. the 2port supply rate

q˙L

I : KQ Γ−1 Workenv.input power

˙ x˙ )) = stele2 ((Tq , −Fe ), (q,

ρq˙ T Tq | {z }

Operatorinput power

−

z }| { x˙ T Fe

z3

.

1

¯1 R:Ω

KQ −1

z}|{ TF

C:I

Se

: Ψ−T z

1

2nd ORDER LOCKED SYSTEM

z1

0

R : KT R : KE R : BE

(32) C:I

This is significant as, the environment consisting of the human and the work environment interact with the teleoperated backoe via a supply rate given by stele2 and not stele1 . Hence, ensuring passivity of the teleoperated backhoe (16) w.r.t. the supply rate (32) not only has better physical relevance but is also necessary to ensure 2-port passivity property of the teleoperated backhoe. Ensuring that the teleoperated backhoe behaves like a 2-port passive device is in turn beneficial as it is then possible to analyze the stability of interconnection of the passive backhoe with other passive systems by invoking properties of interconnection of passive systems stated in section 1.

E˙

E 0

1

I : ME DISSIPATIVE

Figure 7.

Second order desired locked system

depend on the backhoe / joystick inertia and external forces respectively. In reality, it may not be possible to know these parameters exactly. As a result it is necessary to analyze the passivity and coordination property of the proposed algorithm in the presence of such uncertainties. Due to space constraints, the formal results are not presented in this paper. We just state the result that both the algorithms undergo minor implementation modifications in order to ensure the coordination and passivity properties of the teleoperation control algorithm. The design procedure is exactly the same as outlined in this paper. Theorem 1 demonstrates the ability of the control design method in achieving different order locked systems and hence designing the haptic behavior of the passive teleoperator. This is beneficial as the locked system behavior is exactly the haptic dynamics of the teleoperator. Being able to flexibly design the order of the teleoperator will improve the efficiency of the teleoperator depending on its purpose.

Theorems 1 presented a passive teleoperation control design which ensures that the master-slave teleoperator mimics a passive 4th order 2-DOF system operated upon by the human operator and the environment. Since the locked system dynamics are of fourth order, the teleoperator behaves like a fourth system when operated upon by the operator, interacting with the environment. While the system order has certain advantages in terms of filtering high frequency content of environment forces, it may be disadvantageous in other circumstances when the operator requires an agile teleoperator. Hence, it would be benefecial to control (using passive valve control Fx ) the locked system to behave like a lower order dynamic system. Such a method is proposed in the following section where, the teleoperator behaves like a second order locked system. Locked System Design : 2nd Order, 2-DOF Haptic Behavior The locked system control objective remains the same which is to determine Fx in Fig. 5 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired 2nd order locked system dynamics as shown in Fig. 7. The design process is similar to that outlined in the previous section and involves comparing the dynamics of the actual and desired locked systems and choosing the appropriate control input Fx so that the actual dynamics mimic the desired dynamics. Due to space limitations, the algorithm is not presented here. The design results in the following closed loop dynamics: ˙ q˙ L = ρTq − Fe . ML (x, q)q¨ L + CL ((x, q), (˙x, q))

Se : ρTq − Fe

1

DISSIPATIVE

4

Experimental results Experimental results of teleoperation of the hydraulic backhoe using the algorithms presented in the preceding section are presented. A schematic of the experimental setup is shown in Fig. 1. The motors in the joystick are of the low speed hi torque type. These provide the necessary haptic force to the operator (Fq in section 3). The motors are also instrumented with co-axial encoders which provides angular position (q in section 3) reading of the joystick links and a JR3 force sensor which measures the operator input force (Tq in section 3) on the teleoperator. Since the master has only 2-DOF, the slave hydraulic backhoe is also restricted to operate as a 2-DOF robot. Of the three links of the backhoe, the boom, the stick and the bucket, the boom is constrained and the bucket and stick are teleoperated using the master joystick. The hydraulic actuation system of the backhoe involves 1) A pressure compensated constant flow pump regulated at a fixed 1000 psi of pressure (Ps ), 2) Vickers KBFDG4V5 series proportional valve which is passified and the exogenous valve command (Fx in section 3) to spool position (xv in section

(33)

Notice that the locked system given by (33) is 2nd order whereas the locked system given by (28) is 4th order. The valve and joystick control forces Fx and Fq that ensure both the 4th and 2nd order locked system assume accurate knowledge of Ψ, TE which 7


1.5

4 1

Feedback Torque (N−m)

Bucket Displacement (inches)

5

3

2

0.5

0

1 215

220

225

230

235

240 −0.5

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7 215

220

225

230

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240

time (sec)

6

Figure 9. Haptic torque (Fq ) trajectories (Stick - solid, Bucket - dashed) during a digging task.

5

4

3 215

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the controller, there is much scope for improvement in both the haptics and coordination performance. Dynamic Teleoperation We test the bondgraph based dynamic teleoperation control algorithms which were designed for a 2-DOF teleoperated system with included nonlinear models of the joystick and backhoe and compressible effects of the hydraulic actuator. 4th order locked system: The control law which ensured that the nonlinear teleoperated system (16) behaves like a 4th order locked system is tested. The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 10. The corresponding locked system force (ρTq − Fe ) experienced by the 4th order locked system while performing the task is shown in Fig. 11. A kinematic scaling, α = 5in./rad and two different power scaling ρ = 12Watt/Watt and ρ = 18Watt/Watt were used for the experiments. Notice from Fig. 10 that the maximum observed coordination error is within 0.1in for the backhoe stick. When the locked system hits the underground wooden box (about t = 18 seconds), the locked system force (Fig. 11) does not experience any oscillation as did the Haptic torque in Fig. 9 due to the fact that compressible dynamics of hydraulic fluid are modeled in the dynamic teleoperation scenario. When the backhoe hits the box, the net force on the locked system shown in Fig. 11 is about zero even though the operator is pushing on the joystick (ρTq > 0). This is because, the environment force (Fe ) provides an equal and opposite reaction force on the joystick thus indicating a bilateral transfer of power. The high order (4th) of the teleoperator is felt by the operator during hi frequency motion (oscillatory for example) of the teleoperator. The teleoperator (via the master joystick) feels resistant to instantaneous change in direction of motion. 2nd order locked system: The control law which ensures that the nonlinear teleoperated system (16) behaves like a 2nd order locked system is tested. The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 12 and the corresponding locked system haptic force (ρTq − Fe ) experienced by the 2nd order locked system is shown in Fig. 13. A kinematic scaling, α = 5in./rad and two different power scaling ρ = 12Watt/Watt and ρ = 18Watt/Watt were used for the experiments. The maximum observed coordination error is within

240

Time (seconds)

Figure 8. Displacement trajectories (Scaled joystick - solid, Backhoe dashed) during a digging task.

3) bandwidth of about 50 Hz, 3) Vickers customized hydraulic actuators instrumented with Hydroline sensors to measure the actuator position x and 4) Pressure sensors to measure the individual actuator chamber pressures (P1i , P2i in section 3). The RealTime Operating System used for the experiments is MATLAB xPC Target. The interface used are 1) AD card with 16 channels, 2) DA card with 8 channels, 3) decoder card with 4 channels and 4) Force sensor signal conditioning card. A sample rate of 1 KHz was used for all the command and sensor signals. The teleoperator control algorithms are experimentally verified in this section. The conducted experiments are typical of a digging task. The backhoe is teleoperated to dig into a sand box. A wooden box buried in the sandbox is supposed to mimic underground obstacles. The first and second experiments test the 4th and 2nd order locked system teleoperation algorithm presented in section 3.2. Both the algorithms are compared to the kinematic teleoperation experimental results presented in [5]. Kinematic Teleoperation In this section, experimental results obtained by implementing the intrinsically passive control law developed in [5] are presented. The controller was designed assuming a kinematic model of the the hydraulic actuators on the backhoe (fluid compressibility neglected). Backhoe inertia was also neglected. The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 8. The corresponding haptic force (Fq ) experienced by the operator is shown in Fig. 9. A kinematic scaling, α = 5in./rad and a power scaling ρ = 12Watt/Watt. Notice from Fig. 8 that the maximum observed coordination error is within 0.5in. Notice also that when the backhoe hits the wooden box (t = 218 repetitively and 228 seconds repeatedly), the operator experiences the backhoe contact force (Fe ) embedded in the bucket and stick haptic forces (Fq ). There are severe oscillations in the haptic force experienced by the operator primarily because compressible dynamics of the hydraulic backhoe have been neglected in the control design. Although the experiments indicate bilateral power transferability of 8




5 4 3 2 10

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Figure 12. Displacement trajectories for 2nd Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Sequentially from top : 1)-2) ρ = 12, 3)-4) ρ = 18.

Figure 10. Displacement trajectories for 4th Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Sequentially from top : 1)-2) ρ = 12, 3)-4) ρ = 18.

15

1.5 Haptic Force, ρ = 12 (N)

Haptic Force, ρ = 12 (N)

20 Time (seconds)

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15 10 5 0 −5 −10 −15

2 1 0 −1

−20 10

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Figure 13. Locked system force (ρTq −Fe ) trajectories during a digging task for 2nd Order locked system (Stick - solid, Backhoe - dashed).

Figure 11. Locked system Force (ρTq − Fe ) trajectories during a digging task for 4th Order locked system (Stick - solid, Backhoe - dashed).

9


0.01in and significantly better than the 4th order locked system. The behavior of the locked system when it hits an underground wooden box (t = 16,19 sec) is similar to the 4th order locked system behavior. A key factor to notice is that the range of locked system force acting on the 2nd order locked system (Fig. 13) is only about 20 % of that acting on the 4th order locked system (Fig. 11) at similar speeds. This is an indication that the 2nd order locked system is easier (in terms of necessary power) to teleoperate than the 4th order locked system. This 2nd order locked system has the haptic benefits similar to the kinematic teleoperation but with improved performance owing to the richer model (compressible effects + inertial effects) used to design the passive coordination control. Hence, it is very beneficial to be able to not only design passive teleoperators but also design their dynamic behavior as was shown in this paper and proven using experiments.

[4]

[5]

[6] [7]

[8]

5

Conclusions and Future Work A passive teleoperation control algorithm for backhoe operation is proposed. The controller provides amplification / attenuation of joystick motions and also scaled feedback force to the operator. The passivity property of the teleoperation scheme ensures stability of interaction of the teleoperated backhoe and arbitrary human / work environment (usually strictly passive). The current teleoperation approach rectifies a previously developed algorithm which assumed a kinematic modeled backhoe. Severe performance deterioration was observed in certain operating conditions with the previously developed algorithm. The proposed algorithm is shown to perform significantly better. The main advantage of the proposed algorithm is that the haptic dynamics of the teleoperated backhoe feature as a design variable. The procedure to ensure that the coordinated teleoperated backhoe behaves a 4th order locked system and a 2nd order locked system is shown in this paper. The proposed algorithm does not guarantee an Intrinsically Passive Controller (IPC). An IPC has the advantage of guaranteeing passivity of the teleoperated backhoe even under parametric uncertainties. This is because, cascade interconnection of passive systems (passive joystick + passive controller + passive backhoe) guarantees a passive system. Development of such a controller will greatly enhance the robustness properties of the teleoperated backhoe.

[9]

[10]

[11]

[12]

[13]

[14]

[15]

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two-stage pressure control servo-valve”. Proceedings of the American Control Conference, Anchorage, AK, 6 [May], pp. 4831–4836. Krishnaswamy, K., and Li, P. Y., 2002. “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., 2003. “Passive teleoperation of a multiple degree of freedom hydraulic backhoe using a dynamic passive valve”. Proceedings of the ASMEIMECE. Vidyasagar, M., 1993. “Nonlinear systems analysis”. Prentice Hall. Li, P. Y., and Horowitz, R., 1997. “Control of smart exercise machines, part 1: problem formulation and non-adaptive control”. IEEE/ASME Transactions on Mechatronics, 2, pp. 237–247. Lee, D. J., and Li, P. Y., 2001. “Passive control of bilateral teleoperated manipulators: Theory”. Proceedings of the American Control Conference, 6 [Jul.], pp. 4612–4618, Arlington, VA. Colgate, J. E., Wannasuphoprasit, W., and Peshkin, M. A., 1996. “Cobots: Robots for collaboration with human operators”. Proceedings of the ASME Dynamic Systems and Control Division, 58, pp. 433–439. Gomes, M. W., and Book, W., 1997. “Control approaches for a dissipative passive trajectory enhancing robot”. IEEE/ASME Conference on Advanced Intelligent Mechatronics. Krishnaswamy, K., 2004. “Passive teleoperation of hydraulic systems”. PhD Thesis. Available at http://www.me.umn.edu/˜ kk/thesis [May]. Lee, D., and Li, P., 2002. “Passive coordination control of nonlinear mechanical teleoperator”. Submitted to IEEE Transactions on Robotics and Automation. Gawthrop, P. J., 1995. “Bicausal bond graphs”. Proceedings of the International Conference on Bond Graph Modeling and simulation (ICBGM), pp. 83–88. Ngwompo, R. F., and Gawthrop, P. J., 1999. “Bond graph based simulation of nonlinear inverse systems using physical performance specifications”. Journal of the Franklin Institute, 336, pp. 1225–1247. Li, P. Y., and Ngwompo, R. F., 2002. “Passification of electrohydraulic valves using bond graphs”. Proceedings of 15th IFAC World Congress [July].
