Bond Graph Based Approach to
Passive Teleoperation
of a Hydraulic Backhoe1
Kailash Krishnaswamy
Honeywell Labs,
3660 Technology Dr.,
Minneapolis, MN 55418
e-mail: kailash.kvishnaswamy@honeywell.com
Perry Y. Li
Department of Mechanical Engineering,
University of Minnesota,
111 Church St. SE,
Minneapolis, MN 55455
e-mail: pli@me.umn.edu
Human operated, hydraulic actuated machines are widely used in
many high-power applications. Improving productivity, safety and
task quality (e.g., 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: (1) 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; (3) 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.
关DOI: 10.1115/1.2168475兴
1
Introduction
A dynamic system with input u共t兲 and output y共t兲 is said to be
passive with respect to s共u共t兲 , y共t兲兲, called the supply rate, if for
all u共·兲 and time ␶ ⬎ 0,
−
冕
␶
s共u共t兲,y共t兲兲dt 艋 c2
0
When s关u共t兲 , y共t兲兴 is the 共scaled兲 physical power input, a passive
system is one from which the net 共scaled兲 energy that can be
1
Materials based on research supported by the National Science Foundation ENG/
CMS-00889640. First submitted to the ASME Journal of Dynamic Systems, Measurement and Control in March 2005, Revised September 2005.
Contributed by the Dynamic Systems, Measurement, and Control Division of
ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 15, 2005; final manuscript received November 19,
2005. Assoc. Editor: Sunil K. Agrawal. Paper presented at the 2004 ASME International Mechanical Engineering Congress 共IMECE2004兲, November 13–19, 2004,
Anaheim, California, USA.
176 / Vol. 128, MARCH 2006
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
关1兴. The inherent safety that passive systems provide has been
exploited by researchers in the development of human interacting
machines like smart exercise machines 关2兴, passive bilateral teleoperators 关3,4兴, Cobot 关5兴 and Passive Trajectory Enhancing Robot 关6兴. 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 关7–11兴. To enable the passivity analysis of electrohydraulic systems under closed loop control, methods for enabling single-stage 关7,12兴 and multi-stage 关9兴 hydraulic valves
共through hardware redesign or feedback兲 to behave like passive
two-port systems 共a command port and a hydraulic port兲 with
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 关8,10兴,
and multi-degree of freedom 共DOF兲 hydraulic machines 共such as a
backhoe兲 关11兴 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 关8,10,11兴, passivity of the closed loop control system was
always maintained 共to ensure safety兲. However, perfect coordination between the joystick and the machines was ensured only if a
kinematic model 共i.e., ignoring fluid compressibility and inertial
dynamics兲 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 to a hydraulic system here, it should be
applicable to other mechatronic systems as well. The key aspects
of the design methodology are presented in this paper. The readers
are referred to 关13兴 for other aspects such as robustness and performance in the presence of uncertainties.
Similar to 关3,4,14兴, 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 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 2 presents the models of all the subsystems of the teleoperated backhoe
and formulates the control design problem. In Sec. 3, the bond
graph approach to the design of passive teleoperation controller is
given. Experimental results of the implementation are presented in
Sec. 4. Concluding remarks are given in Sec. 5.
Notations: Matrices/vectors are bold, and scalar elements are
light face. Superscripts i = 1 , 2 denotes the link number.
2
System Modeling and Control Objective
The teleoperated backhoe system 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 共the boom兲 is constrained. Backhoe motions are actuated by single-rod hydraulic actuators which
are in turn driven by single-stage proportional valves. Hydraulic
Copyright © 2006 by ASME
Transactions of the ASME
⍀1 = Diag关␻11 , ␻21兴.
Consider now the pressure dynamics of the corresponding
single-rod hydraulic actuators
␤−1VcṖc = Qc − Acẋ
␤−1VrṖr = Qr + Arẋ
Vc = Diag关V1c , V2c 兴,
共2兲
Vr = Diag关Vr1 , Vr2兴
where
are the total volumes
of the cap and the rod side chamber and hose volumes, ␤ is the
fluid compressibility, Pc = 关P1c , P2c 兴, Qc = 关Q1c , Q2c 兴, Pr = 关Pr1 , Pr2兴,
Qr = 关Qr1 , Qr2兴 are the pressures and flows in the actuator chambers,
Ac = Diag关A1c , A2c 兴, Ar = Diag关Ar1 , Ar2兴 are the cap and rod side piston cross-section areas and x is the piston position. Since Vri and
i
i
Vic are dominated by the hose volumes, Vc/r
⬇ min共Vc/r
兲
i
⬇ max共Vc/r兲 are assumed to be constants for both the cap side and
rode sides.
Using the relationships Qic / Qri = Aic / Ari and the matched and
symmetric properties of the four way directional valve 关15兴, and
by decomposing the valve flows into the no-load flows and a
shunt flow components 关16兴
Fig. 1 Teleoperated backhoe consists of a motorized joystick,
and a 2-DOF hydraulic backhoe actuated by a set of hydraulic
cylinders and proportional valves
i i
xv − KTi共兩xvi 兩,FLi兲 · FLi兲
Qic = Aic共KQ
共3兲
i i
xv − KTi共兩xvi 兩,FLi兲 · FLi兲
Qri = − Ari共KQ
共4兲
where
power is provided by a pressure compensated pump operating at
constant supply pressure Ps. The system models are summarized
below with details contained in 关13兴.
2.1 Single-Stage Passive Valve and Hydraulic Actuators.
We consider a single-stage “passive valve” connected to a singlerod actuator 共Fig. 2兲. The essential step in passifying a singlestage proportional valve to obtain the passive valve is load pressure feedback. Details of this procedure can be found in 关7,11兴.
The spool dynamics of the two passive valves, which have been
experimentally verified, are given by
ẋv = − ⍀1xv + Fx − ⌫FL
共1兲
where xv = 关x1v , x2v兴T denotes the valve spool displacements, Fx
= 关F1x , F2x 兴T denotes the passive valve control input, FL = 关FL1 , FL2 兴T;
FLi = Aic Pic − Ari Pri denotes the differential hydraulic load force, ⌫
= Diag关␥1 , ␥2兴 denotes the load force gain acting on the spool,
i
共sgn共xvi 兲兲 =
KQ
KTi共兩xvi 兩,FLi兲 =
Kiq
冑共Aic兲3 + 共Ari兲3
冑AsPs
i i
KQ
兩xv兩
冑AsPs冑AsPs − sign共xvi 兲FLi
As共sgn共xvi 兲兲 =
再
Aic , xvi 艌 0
Ari , xvi ⬍ 0
冎
共5兲
共6兲
共7兲
i
共sgn共xiv兲兲 and KTi共兩xiv兩 , FLi兲 are proportional to the no-load
Here KQ
flow gain, and the nonlinear shunt flow conductance. Notice that
KTi共兩xiv兩 , FLi兲 · FLi2 艌 0 and it can be considered to be the load induced energy dissipation within the valve.
By differentiating the energy function
1
1 T
1 T
Wav = xv2 +
P c V cP c +
P V rP r
2
2␤
2␤ r
it can be shown that the combination of the “passive valve” and
the hydraulic actuator is passive with respect to the supply rate
sav共共Fx , xv兲 , 共FL , ẋ兲兲 = xTv KQ⌫−1Fx − ẋTFL where FL = A1Pc − A2Pr.
This supply rate is the difference between the fictitious command
power xTv KQ⌫−1Fx and the output mechanical power 共ẋTFL兲.
2.2 Backhoe Inertia Dynamics and Motorized Joystick
Dynamics. Both the backhoe and the motorized joysticks are
modeled as planar, rigid two-link robotic systems. The former lies
in the vertical plane and the joystick lies in the horizontal plane.
Their dynamics are given by
Mx共x兲ẍ + Cx共x,ẋ兲ẋ = FL − Fe ,
共8兲
Mq共q兲q̈ + Cq共q,q̇兲q̇ = Fq + Tq ,
共9兲
M* = MT* ⬎ 0
Fig. 2 Single-stage “passive valve” connected to a single-rod
actuator
Journal of Dynamic Systems, Measurement, and Control
are the respective inertia matrices, and Ṁ*
where
− 2Cⴱ are skew symmetric. For the backhoe, FL = AcPc − ArPr is
the differential hydraulic force acting on each backhoe link and
−Fe is the net environment force 共including friction and gravity兲;
ẋ is a vector of the actuator piston velocities. For the motorized
joystick, Fq and Tq are the motor actuated control torque and the
human input torque, and q̇ is a vector of the link angular
MARCH 2006, Vol. 128 / 177
velocities.
By differentiating the kinetic energy functions Wb
1
1
= 2 ẋTMx共x兲ẋ, and W j = 2 q̇TMq共q兲q̇, the mechanical backhoe and
the joystick can be shown to be passive with respect to the supply
rates
冤冥冢
q̇L
s j共共Fq,q̇兲,共− Tq,q̇兲兲 = q̇T共Fq − Tq兲.
FL
FL⬜
Control Objectives
1. Passivity. The closed loop teleoperated system should be
passive with respect to the supply rate
stele共共␳Tq,q̇兲,共Fe,ẋ兲兲 = ␳q̇ Tq − ẋ Fe
T
T
where ⍀̄1 = KQ⌫−1⍀1. In order to simplify the passive control synthesis and analysis, the dynamics of the master and slave systems
共12兲 are decomposed using an energy invariant transformation
Ė
sb共共FL,ẋ兲,共− Fe,ẋ兲兲 = ẋT共FL − Fe兲
2.3
共12兲
T
共10兲
T
where q̇ Tq and −ẋ Fe are the human and work environment power inputs, and ␳ is the desired power scaling
factor so that the human power input is amplified 共attenuated兲 when ␳ ⬎ 1 共␳ ⬍ 1兲.
2. Coordination. The backhoe and the joystick motion
should mimic each other, i.e.,
E ª ␣q − x → 0
xv
ª
0
−I
0
␣I
0
I − ␣⌿ ⌿
0
0
0
− Ar
Ac
0
0 A−1
⌽
r 共I − Ac⌽兲
0
0
0
0
0
0
0
0
I
冣冤 冥
q̇
ẋ
Pc
Pr
xv
共13兲
where
⌿ = ␣共␳Mq共q兲Mx共x兲−1 + ␣2I兲−1
共14兲
−1
⌽ = Ac + ArVc␤−1ArA−1
c V r␤ .
共15兲
Using the above decomposition, the dynamics of the master and
slave are transformed into
共11兲
where ␣ is a specified kinematic scaling.
3. Target dynamics. The desired target dynamics for the
teleoperator system after coordination has been achieved
can also be specified by the designer.
3
Passive Teleoperation Controller Design
The design procedure consists of the following steps:
1. 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; and
3. 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
共16兲
⌬2
d ⬜
F = 0.
dt L
共17兲
Equation 共16兲 and the new variables therein can be derived by
substituting Eq. 共13兲 into Eq. 共12兲 共see 关11兴 for details兲.
The transformation in Eq. 共13兲 has the following properties:
1. the coordination error Ė is explicitly one of the new variables. It represents the shape of the teleoperator;
2. the other coordinates 关q̇L , FL , xv兴 and F⬜
L lie within the submanifold defined by Ė ⬅ 0. Thus they describe the locked
system dynamics when the teleoperator has been coordinated. In particular, q̇L = ␣q̇ = ẋ when Ė ⬅ 0;
3. the total energy of the system can be invariantly expressed in
terms of M in Eq. 共12兲 or 共M̄ , ⌬2兲 in Eq. 共16兲
1
1 T
␳
P V cP c
Wtotal ª q̇TMq共q兲q̇ + ẋTMx共x兲ẋ +
2
2
2␤ c
+
1 T
P VrPr + xTv KQ⌫−1xv
2␤ r
1
1
1
1
= ĖTMEĖ + q̇LT MLq̇L + FLT ⌬1FL + FL⬜T⌬2FL⬜
2
2
2
2
1
+ xTv KQ⌫−1xv ,
2
178 / Vol. 128, MARCH 2006
共18兲
Transactions of the ASME
Fig. 3 Bond graph of the teleoperator after energy invariant coordinate transformation
4. 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, Equation 共17兲 shows that it is marginally stable.
Equation 共13兲 is similar to the isometric decomposition proposed in 关3,14兴 except that Eq. 共13兲 is proposed for N-DOF fourth
order hydraulic systems, whereas, the decomposition proposed by
关3,14兴 is applicable to N-Degrees of Freedom second order simple
共electro兲mechanical systems.
The bond graph describing the above dynamics is given in Fig.
3. Notice that the shape and the locked systems are still coupled.
3.1 Coordination Control Design. We now design a control
law that ensures that E → 0. To do this, consider the inverse dy-
namics of the system with output Ė. In Fig. 3, the path between
the joystick control input Fq and Ė is the shortest causal path.
Thus Fq should be the coordination control input. The corresponding inverse dynamics 共with input Ė and the output Fq兲 obtained
using bicausal bonds 关17,18兴 suggests the coordination control
law
Fq = − TE +
⌿−1
共CELq̇L − 共␣⌿T − I兲FL − KEE − BEĖ兲,
␳
共19兲
BE = BTE ⬎ 0
where
error dynamics
and
KE = KTE ⬎ 0.
This results in coordination
Fig. 4 Bond graph of the master and slave systems with the coordination control law
Eq. „19…. Here EF1 = −␳TE − ␳„KEE + BEĖ… − CLEĖ − ⌿−TCELq̇L.
Journal of Dynamic Systems, Measurement, and Control
MARCH 2006, Vol. 128 / 179
Fig. 6 Second order desired locked system
Fig. 5 The fourth order desired locked system in Eq. „21…
ME共x,q兲Ë + CE共共x,q兲,共ẋ,q̇兲兲Ė = − BEĖ − KEE.
共20兲
Since it can be shown that ME − 2CE is skew symmetric, E → 0
exponentially.
The effect of the coordination control law on the bond graph in
Fig. 3 is shown in Fig. 4. 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 共19兲 also affects the locked system bond graph through potentially active elements as indicated by the signal bond between
FL to q̇L and the additional effort source Se. These active elements
can equivalently be represented by effort 共AE兲 or flow amplifiers
共AF兲 as defined in 关19兴. Signal 共active兲 bonds are used here as
they visually certify the nonpassive property of the bond graph.
3.2 Locked System Design. The locked system in Fig. 4, has
an additional control input Fx which has not been utilized. This
will be used to achieve two goals: 共1兲 to render the locked system
to be passive with respect to the supply rate in Eq. 共10兲; 共2兲 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:
fourth order dynamics as shown in Fig. 5
冢
0
ML
0
0
⌬1
0
0
0
KQ⌫−1
冣冤冥冢
q̇L
d
z1 =
dt
z2
+
− CL
⌿
−1
− ⌿ − KT
0
冤
0
KQ
− KQ − ⍀̄1
␳Tq − ␣ Fe
0
0
−1
冥
冣冤 冥
q̇L
180 / Vol. 128, MARCH 2006
0
0
0
⌬1
0
0
0
KQ⌫−1
冢
− CL
冣冤冥
q̇L
d
FL
dt
xv
⌿−T
0
= − ␣I − KT KQ
0
− KQ − ⍀̄1
+
+
冤 冥
冣冤 冥
q̇L
FL
xv
␳Tq − Fe
0
0
冤
− ␳TE − ␳共KEE + BEĖ兲 − CLEĖ + ⌿−TCELq̇L
共I − ␣⌿兲Ė
KQ⌫−1Fx
冥
共23兲
z2
Note that z1 , z2 in Eq. 共21兲 are different from FL , xv in Eq. 共23兲
even though they have the same dynamics. We will determine the
relationship between them later. In general, the desired locked
dynamics 共21兲 need not have the same parameters as the actual
locked system 共23兲. 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 required control Fx is obtained by comparing the actual
dynamics of q̇L in Eq. 共23兲 and the desired dynamics as given by
Eq. 共21兲, using a procedure similar to the bond graph based pas-
共21兲
共22兲
Remark 1. Since q̇L = q̇ = ␣ ẋ 共see Eq. 共13兲兲 after coordination
共共E , Ė兲 ⬅ 共0 , 0兲兲 has been achieved, both Eq. 共21兲 and Eq. 共22兲
−1
冢
ML
z1
second order dynamics as shown in Fig. 6
ML共x,q兲q̈L + CL共共x,q兲,共ẋ,q̇兲兲q̇L = ␳Tq − ␣−1Fe .
imply that the teleoperator system is passive w.r.t. the desired
supply rate 共10兲 when it is coordinated.
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. We illustrate this procedure for the fourth order target
dynamics Eq. 共21兲.
The dynamics of the actual locked system shown in Fig. 4 are
Transactions of the ASME
sification procedure proposed in 关12兴. The actual locked system
dynamics 共23兲 and desired dynamics 共21兲 will be identical if and
only if
z1 ª FL + ⌿ D共t兲
where D共t兲 = −␳TE − ␳共KEE + BEĖ兲 − CLEĖ + ⌿−TCELq̇L. By substituting this transformation for z1 into the desired dynamics 共21兲
and comparing the actual and desired dynamics of FL, it can be
noted that the dynamics will be identical if and only if
冋
册
d
KT⌿TD共t兲 + 关⌬1⌿TD共t兲兴 .
dt
共25兲
Proceeding in this manner, the above analysis leads to the following coordinate transformation 关q̇L FL xv兴T 哫 关q̇L z1 z2兴T
冤冥冢
I
q̇L
z1
0
=
z2
+
−1
KQ
共⌿−1
冤
0 0
I 0
− ␣I兲 0 I
冉
0
0
⌬1
0
0
0
K Q⌫
−1
冣冤冥冢
− CL ⌿−T
0
q̇L
d
T
z1 = − ⌿ − KT KQ
dt
0
− KT − ⍀̄1
z2
冣冤 冥
FL
xv
⌿ D共t兲
T
d
KT⌿TD共t兲 + 关⌬1⌿TD共t兲兴
dt
+
where
冤
␳Tq − Fe
0
−1
KQ⌫ Fx + D2共t兲
冉
D2共t兲 = KQ关z1 − FL兴 + KQ⌫−1 ⍀关z2 − xv兴 +
z1 − FL ª ⌿TD共t兲,
冉
−1
−1
z2 − xv ª KQ
共⌿−1 − ␣I兲q̇L + KQ
KT⌿TD共t兲 +
冣冤 冥
q̇L
z1
z2
冥
共27兲
冊
d
关z2 − xv兴 ,
dt
冊
d
关⌬1⌿TD共t兲兴 .
dt
共28兲
Notice that the signals z1 − FL and z1 − xv depend on D共t兲 and its
successive derivatives which are functions of the external variables TE, q̇L, FL, E and Ė. If the locked system control Fx in Eq.
共27兲 is chosen as follows
q̇L
0
−1
KQ
0
共24兲
T
−1
−1
共⌿T − ␣I兲q̇L + xv + KQ
z2 ª KQ
冢
ML
冊
冥
KQ⌫−1Fx = − D2共t兲
共26兲
where D共t兲 = −␳TE − ␳共KEE + BEĖ兲 − CLEĖ + ⌿−TCELq̇L. Applying Eq. 共26兲 to the actual locked system dynamics 共23兲 results in
the following transformed dynamics
共29兲
then the target locked system dynamics Eq. 共21兲 will be achieved.
In summary, the coordination control law given in Eq. 共19兲
ensures that the teleoperator will be coordinated so that E = ␣q
− x → 0 exponentially 共see Eq. 共20兲兲 the locked system control law
given by Fx in Eq. 共29兲 and the preceding transformations ensure
that the desired target system dynamics 共21兲 are achieved. Since
q̇L = q̇ = ␣−1ẋ and Eq. 共21兲 is passive with respect to the supply
rate, stele共共Tq , −Fe兲 , 共q̇L , q̇L兲兲 = q̇TL共␳Tq − Fe兲 which is equivalent to
Fig. 7 Displacement trajectories „scaled joystick-solid, backhoe-dashed… during a digging
task
Journal of Dynamic Systems, Measurement, and Control
MARCH 2006, Vol. 128 / 181
stele共共Tq,− Fe兲,共q̇L,q̇L兲兲 = q̇LT 共␳Tq − Fe兲 = ␳q̇Tq − ẋFe
A similar locked system design procedure can be applied to
achieve the second order target dynamics Eq. 共22兲. Compared to
the fourth order target dynamics, the high frequency content of
human and environment forces is not filtered as much, resulting in
a more transparent and responsive haptic feel. Since the original
dynamics 共23兲 is fourth 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 关13兴 for details.
For both the fourth and second order target dynamics cases, the
valve and the joystick control forces Fx and Fq assume accurate
knowledge of ⌿, TE 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 共such as some form of discontinuous control兲 so as to ensure the coordination and passivity properties 关13兴.
4
Experimental Results
In the experimental setup in Fig. 1, the joystick is powered by a
set of dc motors and instrumented with angular position encoders,
and a JR3 force sensor for measuring the operator input force. The
hydraulic system consists of a pressure compensated 19 Liters per
Minute 共5 Gallons per minute兲 flow pump operating at 6.9 MPa
共1000 PSI兲; a set of Vickers KBFDG4V-5 series proportional
valves that have been actively passified 关7兴 and have bandwidths
of about 50 Hz; single rod hydraulic actuators instrumented with
displacement sensors and chamber pressures sensors. A PC running MATLAB 共MA兲 xPC Target provides real time control at a
sample rate of 1 KHz.
The conducted experiments are typical of a digging task. The
backhoe is teleoperated to dig into a sandbox. A wooden box is
buried in the sandbox to mimic an underground obstacle. A kinematic scaling of ␣ = 5 in./ rad and a power scaling ␳ = 12 are used.
Both the fourth and second order locked system teleoperation algorithm are tested. For comparison, the algorithm in 关11兴 that
assumes a kinematic model of the hydraulic backhoe and neglects
the compressibility and inertial dynamics was implemented as
well.
4.1 Previous Algorithm [11]. The teleoperated trajectories of
the bucket and the stick 共and their corresponding joy stick links兲
are shown in Fig. 7. The corresponding haptic force 共Fq兲 experienced by the operator is shown in Fig. 8.
The maximum observed coordination error as shown in Fig. 7
is within 0.5 in.. When the backhoe hits the wooden box 共t = 218
and t = 228兲, severe oscillations in the haptic force are experienced
by the operator as shown in Fig. 8. This is a result of neglecting
compressibility and inertia dynamics.
4.2 Fourth Order Locked System. The results for the control law that mimics the fourth order target locked system 共21兲 are
shown in Figs. 9 and 10. The maximum observed coordination
error in Fig. 9 is within 0.1 in. which is significantly better than
that in Fig. 7 for the previous controller. Also, as the backhoe hits
the underground wooden box 共about t = 18 s兲, there are no oscillations 共Fig. 10兲. Notice that the operator force is balanced by the
work environment force when interacting with the wooden box as
indicated by the net force being nearly zero 共Fig. 10兲.
4.3 Second Order Locked System. The results for the control law that mimics the second order target locked system 共22兲
are shown in Figs. 11 and 12. The maximum observed coordination error in Fig. 11 is within 0.01 in. which is even better than
that in Fig. 9 for the fourth order locked system controller. There
are again no oscillations when the backhoe hits the underground
wooden box 共about t = 16 and t = 19 s兲 共Fig. 10兲.
It is interesting to note that for similar operating speeds, the
force range for the second order locked system 共Fig. 12兲 is only
20% of that for the fourth order locked system 共Fig. 10兲. This is an
indication that the second order locked system is easier 共in terms
of necessary power兲 to teleoperate than the fourth order locked
system. The operator also reported that the teleoperator is more
responsive and more able to feel the environment force using the
second order target system controller than the fourth order target
system controller.
Fig. 8 Haptic torque „Fq… trajectories „stick-solid, bucket-dashed… during a digging
task
182 / Vol. 128, MARCH 2006
Transactions of the ASME
Fig. 9 Link 1 and link 2 displacement trajectories for fourth order locked system „joysticksolid, backhoe-dashed… during a digging task
Fig. 10 Locked system Force „␳Tq − Fe… trajectories during a digging task for fourth order
locked system „stick-solid, backhoe-dashed…
Journal of Dynamic Systems, Measurement, and Control
MARCH 2006, Vol. 128 / 183
Fig. 11 Link 1 and link 2 displacement trajectories for the second order locked system
„joystick-solid, backhoe-dashed… during a digging task
Fig. 12 Locked system force „␳Tq − Fe… trajectories during a digging task for second order
locked system „stick-solid, backhoe-dashed…
184 / Vol. 128, MARCH 2006
Transactions of the ASME
5
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 verified in experiments. Moreover, the
present approach allows the user to specify a desired target locked
system dynamics which directly affects transparency and haptic
sensation of the system. In particular, the second order target system is more responsive and more transparent than the fourth 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 共IPC兲, unlike the
algorithm in 关8兴. 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.
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