http://www.diva-portal.org Postprint This is the accepted version of a paper presented at 25th IEEE\RSJ International Conference on Intelligent Robots and Systems (IROS), OCT 07-12, 2012, Algarve, Portugal. Citation for the original published paper : La Hera, P., Rehman, B., Morales, D. (2012) Electro-hydraulically actuated forestry manipulator: Modeling and Identification. In: 2012 IEEE/RSJ Iinternational Conference on Intelligent Robots and Systems (IROS) (pp. 3399-3404). New York: IEEE conference proceedings IEEE International Conference on Intelligent Robots and Systems http://dx.doi.org/10.1109/IROS.2012.6385656 N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-72717
Transcript
http://www.diva-portal.org
Postprint
This is the accepted version of a paper presented at 25th IEEE\RSJ International Conference onIntelligent Robots and Systems (IROS), OCT 07-12, 2012, Algarve, Portugal.
Citation for the original published paper:
La Hera, P., Rehman, B., Morales, D. (2012)
Electro-hydraulically actuated forestry manipulator: Modeling and Identification.
In: 2012 IEEE/RSJ Iinternational Conference on Intelligent Robots and Systems (IROS) (pp.
3399-3404). New York: IEEE conference proceedings
IEEE International Conference on Intelligent Robots and Systems
http://dx.doi.org/10.1109/IROS.2012.6385656
N.B. When citing this work, cite the original published paper.
Permanent link to this version:http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-72717
Electro-hydraulically actuated forestry manipulator:Modeling and Identification
Pedro La HeraSwedish University of Agricultural Sciences
Abstract—We present results of modeling dynamics of aforestry manipulator, in which we consider its mechanics, aswell as its hydraulic actuation system. The mathematical modelof its mechanics is formulated by Euler-Lagrange equations, forwhich the addition of friction forces is straightforward. Dynam-ics of the hydraulic system is modeled upon first principle laws,which concern flow through orifices and fluid compressibility.These models lead to a set of equations with various unknownparameters, which are related to the inertias, masses, locationof center of masses, friction forces, and valve coefficients. Thenumerical values of these parameters are estimated by the useof least-square methods, which is made feasible by transformingthe models into linear representations. The results of simulationtests show a significant correspondence between measured andestimated variables, validating our modeling and identificationapproach.
I. INTRODUCTION
Modern forestry operations use state-of-art systems andhigh-tech machinery to meet the mechanical and engineeringchallenges of harvesting and logging trees in a safe andenvironmentally responsible manner. The cranes of thesemachines are an especial type of mechanical manipulators,which employ hydraulic actuation to produce motions. Theyare engineered to be manually maneuvered by joysticks,having the human operator as the control unit. During dailywork, the driver is demanded to perform various tasks simul-taneously, which include visualization, recognition, selection,controlling the crane, and positioning the vehicle. This levelof work demand is stressful, since the operator has to dealwith an excess of information, and take various decisions athigh pace.
As an attempt to support human drivers, and in viewof increasing the machine productivity [1], forestry basedindustry and researchers have analyzed the possibility toautomate the routine motions performed in these tasks [2],[3]. The forwarder family of cranes used for logging hasbeen established as a benchmark platform to understand thefundamental challenges of designing this (semi) automaticsolution. From the industrial point of view, crane manufac-turers have entrepreneur the design of forwarder cranes thatcan support the implementation of feedback control systems[4]. Researchers, on the other hand, have provided differentsolutions regarding automatic control design, study of humancrane operation, and trajectory planning techniques. Many of
these concepts have been experimentally validated in modernlaboratories dedicated to the development of future forestrytechnologies [5], [6], [7], [8].
Formally, forwarder cranes can be regarded as electro-hydraulically actuated mechanical systems. Theoretically,control design for such systems has proven to be challenging,due to the complex nonlinear nature of the process dynamics.In literature, we recognize the establishment of two welldefined brands of development for these systems. On onehand, there exist a vast number of publications with quitemature content regarding modeling and controller design forhydraulic servo systems. Such terminology is attributed tothe interconnection between valve and cylinder (or rotator),and the main objective is to control the actuator dynamics,e.g. [9], [10], [11], [12], [13], [14]. On the other hand,we have hydraulic manipulators, for which mainly model-free controllers, resembling conventional decentralized jointcontrol, are found [15], [16], [17], [6], [8], [18], [7]. Thisclass of controllers are usually desirable due to their simplic-ity of implementation. However, experimental studies revealthat their efficiency is limited to joint trajectories with slowvelocity profiles [17]. As the speed of motion reaches high(human) profiles, or when the payload changes dramatically,they lack of damping capabilities and stability.
To attain improving performance, it is of interest to applymodel-based motion control strategies, employing models ofrobot kinematics and dynamics [19]. Within this context,modeling and parameters identification are key elementsto realize such a design. Therefore, the initial interest isto verify that, despite the complex mechanical structureand hydraulic hardware, standard engineering procedures forrobot modeling and parameter estimation are applicable fordescribing and simulating dynamics of these machines. Thefollowing presentation can be regarded as a technical reportof results along these lines, which presents a detailed reviewof our experience modeling dynamics of the mechanics of aforwarder crane and its experimental validation.
II. MODELING
The manipulator used for our study is a downsized versionof a typical forwarder crane1, but similar in configuration and
1All dimensional parameters required in all the coming set of equationsare reported in [15].
dynamics (see Fig. 1). This machine has been fully equippedto realize various experimental studies, as reported in [20].
q1
q2
q3
q4
Fig. 1. Laboratory crane installed at the Department of Applied Physicsand Electronics, Umea University.
A. Dynamics modelingWe apply the Euler-Lagrange equation as a formalism used
to systematically describe the robot dynamics [19], [21],[22]. Based on such a procedure, the computation of theEuler Lagrange equations of motion leads to a second orderset of differential equations of the form
M(q)q + C(q, q)q +G(q) = B(q)τ, (1)
where M(q) denotes a symmetric and positive-definite ma-trix of inertias, G(q) the gravity vector, C(q, q) the matrixof Coriolis forces, and B(q) is a matrix that allocates theexternal forces to joint torques.
0r1r
2d
3d
1d
4d
2r
3r
4r
5r
5d
6d
7d
6r
7r
d
2c
1c
mFLgmTLg
q2
-q3
mSLg
mGrg
rcmSL
rcmTL+q4
rcmFL
Fig. 2. Crane dimensions and masses description in the sagittal plane.
Considering motions in the plane, as shown in theschematics of Fig. 2, the model (1) takes the form
M(qi, ξ)
q2q3q4
+C(qi, qi, ξ)
q2q3q4
+G(qi, ξ) =
τ2τ3F4
,
(2)
where q = [q2, q3, q4]T , i.e. the slewing angle q1 is not
considered, and ξ represents the crane inertial parameters.In addition, and assuming no frictional forces, the set ofequations in (2) can be transformed and represented linearlyin the elements of the base parameters ξ that constitutes theminimum set [19], i.e.
ϕ(q(t), q(t), q(t)) · θ = τ, (3)
where ϕ(·) denotes the regressor written in terms of themeasurable variables [q, q, q], θ = [θ1, ..., θ7]
T the minimumset of inertial parameters, which components are
and τ the vector of generalized input torques and forces,which is equal to the right hand side of (2).
B. Modeling friction forcesA model often used to represent friction forces considers
the static Coulomb - viscous friction, and it is formulated as
τFi = fc,i · (qi) + fv,i · qi, (11)
where i = {2, 3, 4}, and corresponds to the ith link ofthe robot, fv denotes the viscous friction, and fc theCoulomb friction [23]. Hydraulic systems exhibit appreciableasymmetric friction forces, i.e. the constant parameters usedto represent friction change according to the direction ofmotion, i.e.
where the terms with a ¯bar denote the mean values, and theirvariational values ∆ depend on the direction of motion givenby the (q). Introducing the above equations into the generalform (11) yields a more complete model for friction forces:
which is able to capture the most relevant effects of thenonlinear friction phenomena.
C. Combining the models of dynamics and friction forcesDue to the linearity of the friction parameters in (14), i.e.
τFi = [(qi) 1 qi |qi|]
︸ ︷︷ ︸
ϕfi
·
fc,i
∆fc,i
fv,i
∆fv,i
︸ ︷︷ ︸
θfi
, (15)
the introduction of frictional forces into the linear model (3)is straightforward. To show this, we form a complete setof friction forces for the three links considered, i.e. q =[q2, q3, q4]
T , as follows
τF =
ϕf2 0 0
0 ϕf3 0
0 0 ϕf4
︸ ︷︷ ︸
ϕf
·
θf2
θf3
θf4
︸ ︷︷ ︸
θf
, (16)
where the zero vectors have dimension 1-by-4. Thus, con-sidering that frictions forces act opposed to the torques inthe right hand side of (3), a more complete linear model isgiven by
[ϕ(q(t), q(t), q(t)) ϕf (q(t))] ·
[θθf
]
= τ, (17)
which results by the combination of (3) and (16).
D. Mapping cylinder forces to joint torquesUsing the geometry of the machine, the mapping between
the motion of the hydraulic cylinders given angles of thejoints can be explicitly found. This mapping xi = f(qi), canbe used for a) deriving the joint torques, and b) computingthe velocity of the pistons given angular velocity. To thisend, we consider the virtual work principle [19], to definethat
τi · dqi = Fi · dxi, (18)
which yields an equality for the joint torque as
τi = Fi ·dxi
dqi, (19)
for the links i = 2, 3, as required in (2)2. The actuators forces(see Fig. 3) can be calculated as [13]
Fi = AA,ipA,i −AB,ipB,i, i = {2, 3, 4}, (20)
where AA,i and AB,i denote the areas of chambers Aand B respectively, and p(·) the measurements of theircorresponding pressures, which in the machine are availablethrough pressure transducers.
Fig. 3. Hydraulic single-rod cylinder [13].
2Recall that the telescopic displacement q4 is linear, and therefore onlyits cylinder force F4 is needed in the dynamics (2)
E. Modeling the hydraulic actuation systemIn this machine, the hydraulic servo system consists of a
directional proportional control valve, and differential singlerod cylinders used as actuators to produce forces. Eachpiston’s motion can be controlled by regulating the oil flowrates QA,i and QB,i in each chamber, which mathematicalapproximation is given by [13]:
QA,i = c1 ui
√
Ps − PA,i
QB,i = −c2 ui
√
PB,i − Pr
}
ui ≥ 0 (21)
QA,i = c3 ui
√
PA,i − Pr
QB,i = −c4 ui
√
Ps − PB,i
}
ui < 0 (22)
where i = {2, 3, 4}, and corresponds to either of the links,and ui is the electrical control input. Ps and Pr are thesupplied and return pressures respectively, while PA,i andPB,i are the chamber pressures in the hydraulic cylinders, asshown in Fig. 3. The valve coefficients are denoted by c1, c2,c3 and c4, and they are related to the physical properties ofthe valve, which numerical values are usually not provided bythe manufactures. The hydraulic pressures at both chamberscan be defined by the differential equations:
PA,i =β
(VhA,i +AA,ixp,i)(QA,i − xp,iAA,i), (23)
PB,i =β
(VhB ,i +AB,i(si − xp,i))(QB,i + xp,iAB,i),
with VhA,i and VhB ,i being the volumes of oil on eachchamber when the cylinder is fully closed, β the fluid Bulkmodulus, si the cylinder’s maximum length, and xp,i thepiston position, which is calculated according to:
xp,i = (xi − xm,i), (24)
where xi is the current length of the hydraulic cylinder, andxm,i its minimum length. The linear displacements of thecylinders are derived in [20], from which the piston’s linearvelocity can be calculated as3:
xp,i = xi =∂xi
∂qiqi. (25)
The force produced by the hydraulic actuator can be calcu-lated as (20), which time derivative is:
Fi = PA,iAA,i − PB,iAB,i. (26)
Substituting (23) into (26) allows to define that:
Fi = βxp,i
(
A2A,i
VA,i
+A2
B,i
VB,i
)
+β
(AA,i
VA,i
QA,i +AB,i
VB,i
QB,i
)
,
(27)where
VA,i = (VhA,i +AA,ixp,i),
VB,i = (VhB ,i +AB,i(si − xp,i)).
3The telescopic link q4 is directly proportional to its piston’s displace-ment.
Furthermore, replacing (21)-(22) into (27) defines the forcesaccording to the direction of motion:
F+i = −β xp,i
(AB,i
2
VB,i
+AA,i
2
VA,i
)
+ βψ1 (ui, PA,i, PB,i) ,
F−i = −β xp,i
(AB,i
2
VB,i
+AA,i
2
VA,i
)
+ βψ2 (ui, PA,i, PA,i) ,
where (·)± represents positive or negative input signal, and
ψ1 =
(
c1 ui
√Ps − PA,iAA,i
VA,i
+c2 ui
√PB,i − PrAB,i
VB,i
)
ψ2 =
(
c3 ui
√PA,i − PrAA,i
VA,i
+c4 ui
√Ps − PB,iAB,i
VB,i
)
such that, the force can be denoted in a compact form as:
Fi = ˙F+i
(1 + sign(ui)
2
)
︸ ︷︷ ︸
up
+ ˙F−i
(1− sign(ui)
2
)
︸ ︷︷ ︸
un
. (28)
To define (28) in the linear form:
A ·X = Υ, (29)
we can represent the parameter vector:
X =[
1/β c1 c2 c3 c4
]T, (30)
and consider that,
z1,i(t) = −AA,i ui(t)
√
Ps − PA,i(t)
(VhA,i + AA,ixp,i(t)),
z2,i(t) = −AB,i ui(t)
√
PB,i(t)− Pr
(VhB ,i + AB,i(si − xp,i(t))),
z3,i(t) = −AA,i ui(t)
√
PA,i(t)− Pr
(VhA,i + AA,ixp,i(t)),
z4,i(t) = −AB,i ui(t)
√
Ps − PB,i(t)
(VhB ,i + AB,i(si − xp,i(t))),
to define the regression matrix:
A =[
Fi(t) z1,i(t)up z2,i(t)up z3,i(t)un z4,i(t)un
],
(31)and the observation vector as:
Υ = −xp,i
(AB,i
2
VB,i(xp(t))+
AA,i2
VA,i(xp,i(t))
)
. (32)
III. ESTIMATION OF MODEL PARAMETERS
Recalling that dynamics of the robot, as well as thehydraulic system, can be linearly written in the form
Φ ·Θ = Σ, (33)
and that measurements are recorded at each ti, with i ={1, ..., T}, an overdetermined matrix of the form
Φ(t1)Φ(t2)
...Φ(T )
·Θ =
Σ(t1)Σ(t2)
...Σ(T )
, (34)
can be formed for finding an estimate Θ of Θ that fitsthe model (34). There are various mathematical forms todefine this concept, such as the least-square estimate, whichis conceptually the value of Θ that minimizes the residualof the vector |Σ− ΦΘ|. This can be formulated as
min ||Σ− ΦΘ||2. (35)
The classical least-square estimate has the unique solutiongiven by
Θ = (ΦT Φ)−1
ΦT Σ = Φ†Σ, (36)
where (ΦT Φ)−1
Φ = Φ† is known as the pseudo-inverse ofΦ.
A. Recording data and averaging
In the machine, different trajectories can be realized bydecentralized PD feedback gains [17], i.e.
ui = −Kp(qi − qrefi )−Kd(qi − q
refi ) + Ff (qi), (37)
where Kp denotes the values of proportional gains, Kd thevalues of derivative gains, and Ff is a feedforward term usedfor avoiding dead-zones present in the hydraulic system. Thecontrol signal ui is the electrical input level to the electro-hydraulic servo valve [17], and the estimation of velocitiesand accelerations is done by the use of Kalman filtering, assuggested in [24]. The reference trajectories are designed tobe periodic, and the data is recorded for various periods, sothat it can be averaged to improve the signal-to-noise ratio.
B. Estimation and validation of parameters from (17)
Various trajectories were recorded to evaluate differentranges of motion. Considering (36), a set of parameters[θ, θf ]T was found. Some additional data sets were recordedfor validation tests, two of which are presented in Fig. 4 - 5.The simulation results are shown in Fig. 6 - 7. The validationshows a comparison of the averaged measured torque τm,versus the torque computed by the model (17), once thevalues of the unknown parameters have been estimated.We can see that major dynamics are successfully recoveredby the model, with minor uncertainty not captured due tounmodeled dynamics.
C. Estimation and validation of parameters (30)
For simplicity, we present results for the first link, i.e.q2. The data used for estimation is presented in Fig. 8. Inthis figure, the first column shows three different referencetrajectories qref
2 (t), at different frequencies. The secondcolumn shows the control input (37), while the estimationof (26) is shown in the third column. Applying the set ofestimated parameters (30), a comparison of the left handside of (29), with its right hand side is shown in Fig. 9.These results show agreement between the recorded and theestimated responses of the hydraulic model.
0 10 20 30 40 50 600.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
First Link Angle
0 10 20 30 40 50 60
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
Second Link Angle
0 10 20 30 40 50 60
0.5
0.6
0.7
0.8
0.9
1
1.1
Telescopic Link Opening
0 10 20 30 40 50 60
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
First Link Velocity
0 10 20 30 40 50 60
−0.1
−0.05
0
0.05
0.1
Second Link Velocity
0 10 20 30 40 50 60
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Telescopic Link Velocity
0 10 20 30 40 50 60−0.2
−0.1
0
0.1
0.2First Link Acceleration
0 10 20 30 40 50 60−0.2
−0.1
0
0.1
0.2Second Link Acceleration
0 10 20 30 40 50 60−0.2
−0.1
0
0.1
0.2Telescopic Link Acceleration
Fig. 4. First example of data used for validation. First row: Averagedtrajectories in [rad]. Second row: Averaged velocity in [rad/sec]. Third row:Estimated acceleration in [rad/secˆ2]. The x-axis is time in seconds.
0 10 20 30 40 50
0.4
0.5
0.6
0.7
First Link Angle
0 10 20 30 40 50
−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
Second Link Angle
0 10 20 30 40 50
0.5
0.6
0.7
0.8
0.9
1
Telescopic Link Opening
0 10 20 30 40 50
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
First Link Velocity
0 10 20 30 40 50
−0.2
−0.1
0
0.1
0.2
Second Link Velocity
0 10 20 30 40 50
−0.1
−0.05
0
0.05
0.1
0.15
Telescopic Link Velocity
0 10 20 30 40 50−0.2
−0.1
0
0.1
0.2First Link Acceleration
0 10 20 30 40 50−0.2
−0.1
0
0.1
0.2Second Link Acceleration
0 10 20 30 40 50−0.2
−0.1
0
0.1
0.2Telescopic Link Acceleration
Fig. 5. Second example of data used for validation. First row: Averagedtrajectories in [rad]. Second row: Averaged velocity in [rad/sec]. Third row:Estimated acceleration in [rad/secˆ2]. The x-axis is time in seconds.
IV. CONCLUSIONS
We have presented results on modeling and parametersestimation, which were successfully carried on a manipulatorused in forestry machines. It is shown that despite thecomplex interaction between the hydraulic and mechanicalprocesses of the robot, the system’s dynamical responsecan be described by first principle laws. To model frictionforces we have considered a map consisting of Coulomband viscous friction, which are relevant to describe nonlineardynamics near zero velocities. The geometry of the machineis used to explicitly map the forces of the cylinders to torqueson the joints, and to compute their linear velocities givenmeasurements of angular velocities.To calibrate the model, and to find reliable approximations
of unknown parameters, we apply the conventional least-square method. To this end, we proposed a transformation
0 10 20 30 40 50 60 708000
8500
9000
9500
10000
10500
11000Simulation with found parameters First Link
Tau2
LHS Regressor First Link
0 10 20 30 40 50 60 703000
4000
5000
6000
7000Simulation with found parameters Second Link
Tau3
LHS Regressor Second Link
0 10 20 30 40 50 60 70−4000
−3000
−2000
−1000
0
1000
2000
3000Simulation with found parameters Telescopic Link
Tau4
LHS Regressor Telescopic Link
Fig. 6. Validation with the first data set from Fig. 4. The plot shows themeasured averaged torque (black line) vs. estimated torque (red line), wherethe two graphs on top are in [Nm], and the bottom graph is in [N]. Thex-axis is time in seconds.
0 10 20 30 40 50 600.7
0.8
0.9
1
1.1
1.2
1.3
1.4x 10
4 Simulation with found parameters First Link
Tau2
LHS Regressor First Link
0 10 20 30 40 50 602000
3000
4000
5000
6000
7000
8000Simulation with found parameters Second Link
Tau3
LHS Regressor Second Link
0 10 20 30 40 50 60−4000
−3000
−2000
−1000
0
1000
2000
3000Simulation with found parameters Telescopic Link
Tau4
LHS Regressor Telescopic Link
Fig. 7. Validation with the second data set from Fig. 5. The plot showsthe measured averaged torque (black line) vs. estimated torque (red line),where the two graphs on top are in [Nm], and the bottom graph is in [N].The x-axis is time in seconds.
of the system dynamics into a linear form, possible due tothe linearity of the model parameters. To capture the data,we apply decentralized PD control, which has been designedonly for the purpose of system identification.The results allow to assess the correctness of the estimated
parameters, and let us conclude that despite minor differ-ences, the model found is able to capture the most relevantdynamics involved in a motion. This statement is valid forthe mechanics, as well as for the hydraulic actuation system.
REFERENCES
[1] B. Lofgren, “Automation of forestry machines - an important piecein precision forestry,” in Proceedings of the International PrecisionForestry Symposium, (Stellenbosch University, South Africa), 2006.
0 10 20 30 40 50 60−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Time[s]
q[r
ad
]
0 50 100 150
−0.2
0
0.2
0.4
0.6
Time[s]
q[r
ad
]
0 20 40 60 80 100
−0.1
0
0.1
0.2
0.3
0.4
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Time[s]
q[r
ad
]
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−0.1
−0.05
0
0.05
0.1
0.15
Time[s]
u(t
)
0 50 100 150−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Time[s]
u(t
)
0 20 40 60 80 100
−0.1
−0.05
0
0.05
0.1
Time[s]
u(t
)
0 10 20 30 40 50 60
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−1000
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1500
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0 50 100 150
−2000
−1500
−1000
−500
0
500
1000
1500
Time[s]
N/s
0 20 40 60 80 100
−500
0
500
Time[s]
N/s
Fig. 8. The data used for estimation. First column: Reference trajectoriesq
ref2
(t)[rad]. Second column: Control signal u2(t). Third column: Esti-mated derivatives of force F2. The x-axis is time in seconds.
0 10 20 30 40 50 60
−4
−3
−2
−1
0
1
2
3
x 10−3
Time[s]
Measured
Simulated
0 20 40 60 80 100 120 140 160
−4
−2
0
2
4
x 10−3
Time[s]
Measured
Simulated
0 10 20 30 40 50 60 70 80 90 100
−2
−1
0
1
2
x 10−3
Time[s]
Measured
Simulated
Fig. 9. Validation results: Plot shows the recorded responses (red line) vs.the estimated responses (black line). The x-axis is time in seconds.
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