automation in construction -...
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Automation in Construction xxx (2013) xxx–xxx
AUTCON-01549; No of Pages 11
Contents lists available at SciVerse ScienceDirect
Automation in Construction
j ourna l homepage: www.e lsev ie r .com/ locate /autcon
Dynamically optimal trajectories for earthmoving excavators
Young Bum Kim, Junhyoung Ha ⁎, Hyuk Kang, Pan Young Kim, Jinsoo Park, F.C. Park
⁎ Corresponding author.E-mail address: [email protected] (J. Ha).
0926-5805/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.autcon.2013.01.007
Please cite this article as: Y.B. Kim, et al., Dynadx.doi.org/10.1016/j.autcon.2013.01.007
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Article history:Accepted 10 December 2012Available online xxxx
Keywords:ExcavatorEarthmovingMotion optimizationMinimum torque
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OWe consider the problem of generating time-efficient, minimum torque motions for earthmoving excavators.Limits on actuator forces, as well as on joint velocities and accelerations, are assumed given, and the objectiveis to produce the fastest possible minimum torque motions while avoiding torque saturation limits. Relyingon a set of recursive geometric algorithms for calculating the excavator dynamics that also calculates analyticgradients and Hessians of the dynamics as a by-product, we develop robust, reliable algorithms that generatesuch optimal trajectories. The resulting trajectories are compared with actual digging motions of experiencedoperators.
© 2013 Elsevier B.V. All rights reserved.
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1. Introduction
Interest in unmanned excavators is driven by many consider-ations, from a desire to reduce the numerous fatal accidents thatoccur regularly to human operators—particularly in dangerous envi-ronments involving highly uneven terrain and unstable ground—toimproving task efficiency by automating excavation in an optimalmanner. To realize such potential benefits, an unmanned excavatormust have the capability to produce optimal motions in a reliableand efficient manner. While there are many possible notions of opti-mality, our focus in this paper is on excavator motions that are fast,power-efficient, and smooth. These criteria naturally require carefulconsideration of the excavator dynamics.
In some relevant previous work, [1] reports on a fully automatedexcavator that senses the dig face, recognizes and localizes the truck, de-tects any obstacles, andfinally produces a feasible digging path via threeplanning algorithms: a coarse-to-fine dig point planning, template-based dump planning, and script-based planning. It is important tonote that none of these planning algorithms take into account the dy-namics of the excavator; only the geometry of the terrain and kinematicconstraints are considered. More recently in [2], a high-level hierarchi-cal framework for autonomous construction machinery is proposed, inwhich the planning, measurement, and control functions are clearlyidentified and defined. While the general issues involved in planningare addressed, no specific planners are proposed.
In this paper we report on a class of algorithms for generatingdynamically optimal excavator trajectories, and more broadly, on anoffline simulation andmotion planning system for unmanned excava-tors. The benefits of a computer simulation and planning tool for suchpurposes are obvious: a means of evaluating and optimizing systemperformance, from design to motion planning and control, as well
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mically optimal trajectories fo
EDas reduced development time and costs with fewer iterations of pro-
totype construction and testing.Our main technical contribution is a class of motion optimization
algorithms for generating the fastest possible minimum torque mo-tions. As has been reported in some of our earlier work, [3], thatmotionoptimization taking into account the dynamics is, although seeminglystraightforward in principle, computationally and numerically highlychallenging in practice. For one thing, the availability of analytic gradi-ents of the objective function and constraints impacts numerical con-vergence in a significant way; finite difference approximations of thegradient often result in numerical instabilities and poor convergencebehavior. Moreover, the presence of closed loop constraints—the kine-matics of a typical excavator involves several closed loops and passivejoints that are not actuated—complicates the calculation of the dynam-ics and the gradients.
Drawing upon some of our previous work [3–5], we develop anoptimization algorithm that robustly and efficiently obtains the fastestpossible minimum torquemotion—that is, without violating any actua-tor saturation limits, or other constraints on cylinder velocities or accel-erations; we call such motions minimum torque/time motions—takinginto account the closed-loop structure of excavators, and the soil dy-namics during digging and dumping, i.e., in a way that reflects the influ-ence of soil–tool interaction forces during digging, and the effects of soilloads during lifting and dumping. Such motions avoid the undesirablebang-bang characteristics of typical minimum-time trajectories, whilebeing fast, smooth, and energy-efficient. The key to our algorithm liesin the ability to efficiently calculate analytic gradients (and if necessaryanalytic Hessians) of the objective function, all as a simple by-product ofthe recursive dynamics calculation—this is achieved by drawing uponthe geometric algorithms first derived in [5].
More broadly, our motion optimization algorithm is part of anoffline planning and simulation system that integrates disparate com-ponents, ranging from a dynamics simulation engine to methods forsoil model parameter identification, into a coherent and easy-to-useframework. We describe the basic elements of our overall motion
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LiftingDigging
DumpingReturning
Soil Parameter Estimation
Dynamics
Planning andOptimization
Boundary conditions
Optimal Trajectories
Measurementdata
Comparison resultsbetween simulationand measurement
Fig. 1. Architecture for our simulation and motion planning system.
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planning and simulation architecture. In particular, using soil modelsbased on the fundamental earth moving equations (FEE) [6] and itsvariants [7], we describe a soil model parameter identification proce-dure that relies on measurements of digging forces in actual excava-tion tasks.
The optimal motions generated by our method are compared withthe actual excavation trajectories of experienced human operators.We find that the optimal digging motions, while for the most part sim-ilar to those performed by an experienced human operator, do showsome important qualitative differences, particularly in the dumpingand returning motions. These and other observations are discussed indetail.
The remainder of the paper is organized as follows. Section 2 pro-vides a high-level description of our simulation and motion planningarchitecture, together with a more detailed description of the maincomponents. Section 3 derives the excavator dynamics for the fourmain tasks involved in earthmoving, and also describes both the soilmodel used for the study and a procedure for estimating its parameters.Section 4 presents details of the algorithm for generating minimumtorque and minimum time motions. Section 5 presents a comparisonof the optimal motions generated by our planner with the actual exca-vation trajectories performed by experienced human operators. We
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Fig. 2. Excavator in
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
ED Pconclude in Section 6 with a discussion of possible future extensions
to our simulator and planner.
2. Simulation and planning architecture
Fig. 1 depicts the overall architecture for our simulation and motionplanning system. The three basic functions are dynamics simulation,motion planning and optimization, and soil parameter estimation. Thedynamic simulation engine formulates the equations of motion for thefour basic excavator tasks (digging, lifting, dumping, and returning).Motions obtained from dynamic simulations can be compared withmeasurement data taken from field experiments; a direct comparisonbetween, e.g., the force profiles is onemeans of evaluating the accuracyand reliability of the dynamics simulation.
Digging simulation requires a computational model for the soildynamics, which in turn requires a method for estimating the soilmodel parameters. The soil parameter estimation module takes asinput measurement data (obtained from field experiments), and esti-mates the soil model parameters via an optimization procedure.Among the external forces exerted on the excavator, digging forcesare typically the largest, so that careful estimation of the soil modelparameters is essential for realistic dynamics simulation.
reduced form.
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Bucket angle
Vs
α i α f
Fig. 5. Dumping work.
λi
-λi
Fig. 3. Forces at i-th cut points.
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The planner invokes the dynamics simulation engine when gener-ating motions, and takes as input the boundary conditions, i.e., theinitial and final bucket pose and velocity, and produces optimal cylin-der trajectories for the minimum torque/time criterion. Embeddedwithin the planner are the requisite optimization algorithms, whichrequire analytic gradients; these gradients can be obtained as a by-product of the dynamics simulation as noted earlier.
3. Excavator dynamic modeling
For the purposes of this study we model only the rigid-bodydynamics of the excavator, without taking into account the hydraulicactuator dynamics. The methodology presented in this paper can bestraightforwardly extended to this case, although the governingequations of motion become more involved.
The dynamic modeling of the excavator is complicated by thepresence of four closed loops, and several passive joints. There aretwo main approaches to the dynamic modeling of such constrainedmultibody systems: (i) modeling the system by a set of differentialequations subject to algebraic constraints (the differential-algebraicapproach), and (ii) modeling the system exclusively by a set of differ-ential equations formulated in terms of an independent set of gener-alized coordinates. For purely dynamic simulation purposes there islittle difference between the two approaches. However, for motionoptimization purposes, where analytic gradients of the objectivefunction are extremely useful, the latter approach is computationallyadvantageous and more straightforward.
Assuming that the excavator mechanism consists of a total of mone degree-of-freedom joints, the first step in the dynamic modelingis to distinguish the n actuated joints, denoted qa∈Rn; from the m-npassive joints, denoted qp∈Rm−n (for typical excavators, n=4 andm=16). Define q ¼ qa; qp
� �∈Rm. The kinematic loop closure con-
straints can then be expressed in the following differential form(see, e.g., [8]):
_q ¼ Φ _qa ð1Þ
Φ ¼ I−J†pJa
� �ð2Þ
€q ¼ _Φ _qa þΦ€qa: ð3Þ
The matrices Ja∈R m−nð Þ�n and Jp∈R m−nð Þ� m−nð Þ above are obtainedby time-differentiating the m-n kinematic loop closure equationsg(qa,qp)=0. _q can be further differentiated to obtain €q; [8] describes
Ubucket-tip path
Ground
Vs
Fig. 4. Bucket-tip path.
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an efficient screw-theoretic method for this differentiation. Belowwe shall have occasion to use the constraint Jacobian Jc, defined asJc ¼ Ja Jp
� �∈R m−nð Þ�m:
Having modeled the excavator kinematics in this form, we nowconvert the excavator into a topological tree structure (which wecall a reduced system) by appropriately cutting all closed loops inthe original system as shown in Fig. 2. For each cut point, we imaginea generalized force λi exerted at the passive joint that causes the re-duced system to undergo exactly the same movement as the originalsystem (see Fig. 3). Physically this generalized force can be identifiedwith the internal force and bending moment at the correspondingpoint of the original closed-loop system.
Denoting by the vector λ the collection of all the generalizedforces λi, the dynamics of the excavator can be expressed in the form
M qð Þ€q þ C q _; qð Þ _q þ V qð Þ ¼ τ þ JTcλ ð4Þ
whereM qð Þ∈Rm�m is themassmatrix,C q _; qð Þ∈Rm�m andV qð Þ∈Rm arethe terms associated with the Coriolis and gravitational forces of thereduced system, τ∈Rm is the vector of the torques, and λ∈Rm−n isthe vector of constraint forces. Explicit formulations of these matrices,as well as efficient recursive algorithms for the evaluation of thedynamic equations, are provided in [5]. Following [5], τ can then be cal-culated for the reduced system provided q _; q €; qð Þ are assumed given:
τ≜M qð Þ€q þ C q _; qð Þ _q þ V qð Þ ð5Þ
Measurementdata
Soil ParameterEstimation
Soil-ToolInteraction Model
Digging motiondata
Diggingforce
Fig. 6. Block diagram illustrating calculation of digging forces.
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fs
Rd
terrain surface
bucket blade
failure planecaLt
cLf
Q
Fig. 7. Modified FEE model: pseudo-static model of earthmoving at critical point justbefore failure. α is the slop of the terrain, Lt is length of the tool, Lf is the length ofthe surface along which the wedge slides, Q is the surcharge, or the displaced soilthat rests on the wedge, ca is the adhesion between the soil and blade, R is the forceof the soil resisting the movement of the wedge, and fs is the force exerted by thetool to cause failure. The material in the shaded region constitutes all of the materialsthat have passed over the top of the bucket tip during the bucket's motion, and is referredto as the swept volume, Vs [9].
Table 1 t1:1
t1:2Optimization constraint for each working tasks.
t1:3Working Equation number Pmax Qmax_Wmax Vsmin N.P.
t1:4Digging Eq. (12) ○ ○ ○ ○ ×t1:5Lifting Eq. (15) ○ ○ ○ × ○t1:6Dumping Eq. (16) ○ ○ ○ × ○t1:7Returning Eq. (4) ○ ○ ○ × ○
Table 2 t2:1
t2:2Minimum torque optimization algorithm.
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τ ¼ τ−JTcλ: ð6Þ
τ here is the sum of the original torques τ with the constraint forces λ.Eq. (6) can be divided into two parts corresponding to the active
and passive joint torques:
τa ¼ τa−JTaλ ð7Þ
0 ¼ τp−JTpλ: ð8Þ
Note that the left-hand side of Eq. (8) is zero because the torques atpassive joints are zero. From Eqs. (7) and (8), λ and τa can be obtainedas follows:
λ ¼ J†Tp τp ð9Þ
τa ¼ τa−JaJ†Tp τp: ð10Þ
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ΔF(t)
Simulation
Measurement
tfti
Digging force
Digging time
Fig. 8. Estimation of soil parameters.
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
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The final remaining step is to compute τa (or λ) from the givenq; _qa; €qa. Here _q; €q can be obtained from Eqs. (1) and (3), from whichτa and λ are then obtained from (5), (9), and (10). Note that Eq. (4)does not account for the soil forces. Taking these into account, the exca-vator dynamics equations assume the form
M qð Þ€q þ C q _; qð Þ _q þ V qð Þ ¼ τ þ JTcλþ JTs f s þ JTk f k; ð11Þ
where Js is the soil–tool Jacobian, fs is the soil force exerted on the bucketduring the digging, Jk is the soil–bucket Jacobian and fk is the forceexerted on the bucket arising from soil weight during the lifting (detailsare given below).
Thus farwe have derived the rigid-body dynamics of the excavator inan independent context;we nowexamine the task-specific dynamics fortruck loading, which consists of digging, lifting, dumping, and returningto the original digging position. The next subsections describe, for eachsubtask, how to formulate the inverse dynamics, which is needed inthe motion optimization procedure.
3.1. Digging dynamics
During digging, the soil lifting force fk in Eq. (11) is not exerted(that is, fk should be zero). In this case the equations of motion canbe rewritten as follows:
M qð Þ€q þ C q _; qð Þ _q þ V qð Þ ¼ τ þ JTcλþ JTs f s: ð12Þ
The digging force fs can be calculated via the soil model, which wedescribe later in Section 3.5.
3.2. Lifting dynamics
During lifting, the bucket is assumed to be filled with the soilobtained during the digging phase. The volume of soil, Vs, is calculatedfrom the closed volume between the ground and bucket-tip path as
t2:3Minimun torque optimization
t2:41 Input: Measurement data M, Motion script Φ=[Φdig,Φlift,Φdump,Φreturn], tf=[tf,dig,tf,lift,tf,dump,tf,return]
2 Soil parameter Ψ ← Soil parameter estimation(M) described in Section 3.5.3 (Pdig∗ ,soil volume Vs)← Digging motion minimum torque optimization (Φdig,Ψ,tf,dig)
3.1 Objective function is given in Eq. (27) with the dynamics specified inSection 3.1.
3.2 Constraints are given in Table 1.3.3 The soil volume Vs is the closed volume as shown in Fig. 4.
4 Plift∗ ← Lifting motion minimum torque optimization (Φlift,Vs,tf,lift)4.1 Objective function is given in Eq. (27) with the dynamics specified in
Section 3.2.4.2 Constraints are given in Table 1.
5 Pdump∗ ← Dumping motion minimum torque optimization (Φdump,tf,dump)5.1 Objective function is given in Eq. (27) with the dynamics specified in
Section 3.3.5.2 Constraints are given in Table 1.
6 Preturn∗ ← Returning motion minimum torque optimization (Φreturn,tf,return)6.1 Objective function is given in Eq. (27) with the dynamics specified in
Section 3.4.6.2 Constraints are given in Table 1.
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Table 3 t3:1
t3:2Minimum time optimization algorithm.
t3:3Minimun time optimization
t3:41 Input: Measurement data M, Motion script Φ=[Φdig,Φlift,Φdump,Φreturn]2 Soil parameter Ψ← Soil parameter estimation(M) described in Section 3.5.3 (Pdig∗ ,tf,dig,soil volume Vs)← Digging motion minimum time optimization
(Φdig,Ψ)3.1 Objective function is given in Eqs. (28) and (29) with the dynamics spec-
ified in Section 3.1.3.2 Constraints are given in Table 1.3.3 The soil volume Vs is the closed volume as shown in Fig. 4.
4 (Plift,∗ tf,lift)← Lifting motion minimum time optimization (Φlift,Vs)4.1 Objective function is given in Eqs. (28) and (29) with the dynamics spec-
ified in Section 3.2.4.2 Constraints are given in Table 1.
5 (Pdump,∗ tf,dump)← Dumping motion minimum time optimization (Φdump)5.1 Objective function is given in Eqs. (28) and (29) with the dynamics spec-
ified in Section 3.3.5.2 Constraints are given in Table 1.
6 (Preturn,∗ tf,return)← Returning motion minimum time optimization (Φreturn)6.1 Objective function is given in Eqs. (28) and (29) with the dynamics spec-
ified in Section 3.4.6.2 Constraints are given in Table 1.
Table 4 t4:1
t4:2Specifications for the hydraulic excavator used in experiments.
t4:3Item Value
t4:4Operating weight 29,300 kgt4:5Boom length 6.25 mt4:6Arm length 3.05 mt4:7Bucket capacity 1.27 m3
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shown in Fig. 4. The soil carried in the bucket exerts a force fk on thebucket, whereas digging force fs should be zero during lifting phase. Ingeneral it is difficult to analytically calculate fk, since it depends on theinterior shape of bucket and the motion of the bucket. However, sincethe soil mass can be readily computed, one can augment the bucketmass with the soil mass for a reasonable approximation of fk. To de-rive the equations of motion when the soil mass is augmented tothe bucket mass in this form, the corresponding Euler–Lagrangeequations can be written as follows:
ddt
∂Lk∂ _q −∂Lk
∂q ¼ Mk qð Þ€q þ Ck q _; qð Þ _q þ Vk qð Þ ð13Þ
¼ JTk f k; ð14Þ
where Lk is the Lagrangian for the augmented soil. Using both Eqs.(11) and (14), the equations of motion for lifting can be written asfollows:
M qð Þ þMk qð Þð Þ€q þ C q; _qð Þ þ Ck q; _qð Þð Þ _q þ V qð Þ þ Vk qð Þ¼ M qð Þ€q þ C q; _qð Þ _q þ V qð Þ ¼ τ þ JTcλ: ð15Þ
Note the similarity between Eqs. (15) and (11).
3.3. Dumping dynamics
During dumping, the soil volume in the bucket Vs decreases as afunction of bucket angle α(q). As shown in Fig. 5, we assume thatthe soil volume Vs linearly decreases as the bucket angle α increases,in which case the equations of motion (16) can now be written
M q;αð Þ€q þ C q _; q;αð Þ _q þ V q;αð Þ ¼ τ þ JTcλ; ð16Þ
and the boundary conditions for each of terms in Eq. (16) are given asfollows:
M q;αð Þ ¼ M qð Þ þMk qð Þ at α qð Þ ¼ αiM qð Þ at α qð Þ ¼ αf
�ð17Þ
C q _; q;αð Þ ¼ C q _; qð Þ þ Ck q _; qð Þ at α qð Þ ¼ αiC q _; qð Þ at α qð Þ ¼ αf
�ð18Þ
V q;αð Þ ¼ V qð Þ þ Vk qð Þ at α qð Þ ¼ αiV qð Þ at α qð Þ ¼ αf
;
�ð19Þ
where αi,αf respectively denote the start and final angles for dump-ing. Note that each of the terms M , C and V in Eq. (16) depends onthe bucket angle α as well as the joint angle vector q.
3.4. Returning dynamics
If during the returning task no external forces other than gravityare exerted on the excavator, then the dynamic equations in thiscase are given by Eq. (4).
3.5. Soil modeling and identification
In general, calculating soil digging forces is complicated by the dif-ficulty in constructing accurate models that capture the static and dy-namic properties of the soil. The choice of soil–tool interaction modelcrucially affects the accuracy and performance of digging simulation.The soil–tool interaction process is summarized in Fig. 6.
Recent contributions to soil–tool interaction modeling include thefundamental earth-moving equation (FEE) [6], learning based models[10], spring-damper models [11], and energy-dissipation models [12].For our purposes we choose a modified version of the FEE model
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
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Rproposed in [7], illustrated in Fig. 7. This model specifies the soil–tool dynamic interaction in terms of the five soil parameters δ, ϕ, β,γ, and c:
• δ: the angle between bucket blade and soil force fs exerted on buck-et blade as shown in Fig. 7
• ϕ: the angle between the failure plane and resistance force R asshown in Fig. 7
• β: the angle between the terrain surface and the failure plane asshown in Fig. 7
• γ: the mass density of the soil• c: soil cohesion parameter with the same unit as pressure.
The soil force fs can be calculated as a function of five parameters,
f s ¼ d2wγgNw þ cwdNc þ VsγgNq ð20Þ
Nw ¼ cotβ−tanαð Þ cosα þ sinαcot β þ ϕð Þð Þ2 cos ρþ δð Þ þ sin ρþ δð Þcot β þ ϕð Þ½ � ð21Þ
Nc ¼1þ cotβcot β þ ϕð Þ
cos ρþ δð Þ þ sin ρþ δð Þcot β þ ϕð Þ ð22Þ
Nq ¼cosα þ sinαcot β þ ϕð Þ
cos ρþ δð Þ þ sin ρþ δð Þcot β þ ϕð Þ ð23Þ
where Γ=(d,w,α,ρ,Vs) is a set of additional geometric parameterswhich correspond to the depth of the bucket tip, the width of thebucket, the terrain slope, rake angle, and swept volume. Γ can be cal-culated once bucket pose, bucket shape and terrain shape are known.The derivation of fs is given in [9].
We note that this model describes the static force-displacement rela-tion between the soil and tool; while for our purposes this model is suffi-cient for the most part, for fast, highly dynamic movements of the bucketthe accuracy of the dynamic simulation can be expected to degrade.
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Fig. 9. Displacement and pressure sensors installed on the hydraulic excavator.
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Wenow address the estimation of the soil model parameters. In thisstudy we formulate this as an optimization problem, in which the fol-lowing objective function is minimized:
soil parameters ¼ arg minδ;Φ;β;γ;c
∫tftiΔF tð Þj j dt; ð24Þ
where ti is the time at which digging starts, tf is the time at which dig-ging ends, and ΔF(t) is the difference between the measured force andthe force calculated from the soil model. The optimization variablesare the soil model parameters, and the objective function ∫tf
tiΔF tð Þj j dt
is taken to be the integral of the absolute value of the force difference(which, as shown in Fig. 8, corresponds to the area between the twoindicated curves).
4. Minimum torque/time optimization
4.1. Formulation
The generation of minimum torque/time trajectories for excava-tors can be approached in a number of ways. One possible optimalcontrol formulation involves augmenting the minimum torque func-tional with a penalty term for the final time tf, i.e.,
minτ tð Þ
ϕ tf� �
þ 12∫tf0 jjτ tð Þjj2dt ð25Þ
subject to boundary conditions and various physical constraints assummarized in Table 1.
In [13] a local solution to the optimal control problem is found byassuming that the joint trajectories q(t) are parametrized in terms ofB-splines:
q t; Pð Þ ¼Xmi¼1
Bi tð Þpi; ð26Þ
where Bi(t) is a B-spline basis function and P=[p1,p2,⋯,pm] is the set ofcontrol points. The cost functional then reduces to a parameter optimi-zation problem of the form
minP
ϕ tf� �
þ 12∫tf0 jjτ P; tð Þjj2dt ð27Þ
• Equation number: governing dynamics equation number• Pmax: maximum pressure
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
ED P
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• Qmax: maximum flow rate• _Wmax: maximum power• Vsmin: minimum amount of soil volume• N.P. (No Penetration): whether penetration between ground andbucket is allowed (×) or not (○)
subject to the same constraints as described in Table 1. The basis functionsmust be chosen so as to satisfy all the terminal boundary conditions.
For the minimum torque problem in which the final time tf isgiven, it is intuitively clear that for larger values of tf, the minimumtorque profiles become increasingly flatter. If tf is allowed to vary,the first term ϕ(tf) acts as a penalty function to prevent the finaltime from becoming arbitrarily large in order to decrease torque con-sumption. The functional ϕ(tf) should be a monotonically increasingfunction of tf, and should be appropriately weighted for balancing be-tween final time and torque consumption.
Penaltymethods are highly sensitive toweighting parameters, and inthis paper we adopt an approach initially developed for open chain armsinitially described in [14]. Specifically, we minimize the peak torque
maxtf
τ P�; t
�� ���� ��∞ ð28Þ
P� ¼ argminP
∫tf0 jjτ P; tð Þjj2 dt ð29Þ
where ||⋅ ||∞ denotes the L∞ norm, and subject to the same constraintsgiven in Table 1. This formulation is based on the observation that thepeak torque attained by the minimum torque trajectory decreasesmonotonically as tf is increased. A specific algorithm for solving thisproblem is described in Section 4.2.
4.2. Algorithms
Numerical procedures for solving the minimum torque problemand minimum time problem are described in Tables 2 and 3. HereM denotes the measurement data containing the cylinder trajectoriesand corresponding cylinder forces for several digging motion trials, Φis the motion script which contains the initial and final configurationsof digging, lifting, dumping and returning motions specified by theuser, and tf is the duration time for each motion.
For the minimum time problem described in Table 3, the motionoptimization procedure is as follows
1. Optimize theminimum torque for a fixed tf and check all constraints.2. Using an appropriate line search algorithm, decrease the final time
tf if all constraints are satisfied, otherwise increase tf.
r earthmoving excavators, Automation in Construction (2013), http://
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7Y.B. Kim et al. / Automation in Construction xxx (2013) xxx–xxx
3. Repeat the routine from step 1 to step 2 until some terminationcondition is satisfied.
The final time tf can be determined via a standard line search pro-cedure, e.g., golden section search, with an upper bound on admissi-ble values of tf.
To handle the constraints specified in Table 1, it is desirable toenforce hard constraints as much as possible, and to convert someconstraints into soft constraints as required.
5. Experimental results
In this section we present experimental results verifying the fidelityof our dynamics simulations, and a comparison of the trajectories
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Fig. 10. A comparison of simulated and measured
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
produced by our planner with those produced by an experiencedhuman operator.
Physical specifications for the excavator used in the experimentsare listed in Table 4.
Displacement sensors and pressure sensors attached each of thecylinders at the boom, arm, and bucket measure the correspondingcylinder displacement and force (Fig. 9).
5.1. Verification of excavator dynamics simulation
We first compare the cylinder force trajectories produced by ourdynamic simulation with actual measurements. The cylinder force tra-jectories are measured for the two cases: (1) without soil–tool interac-tion, and (2) with soil–tool interaction. Fig. 10 compares the simulated
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cylinder forces without soil–tool interaction.
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measuresimulation
Fig. 11. A comparison of simulated and measured cylinder forces with soil–tool interaction.
8 Y.B. Kim et al. / Automation in Construction xxx (2013) xxx–xxx
Uand measured forces for repeated trials of a circular reference bucketmotion in three dimensions, performed by a human operator. The sim-ulated forces show good agreementwith themeasurements; the offsetsin arm and bucket forces can be traced to small discrepancies in themodel parameters (i.e., masses and inertias of the links), as well asunmodelled dynamic effects such as friction, bending, and hysteresis.
For the case with soil–tool interactions, we first estimate the five pa-rameters in the modified FEE soil model using the procedure describedearlier. Measurement data has been obtained for eight excavation cycles,in which each cycle consists of digging–lifting–dumping–returning. Weuse the first digging cycle to estimate the soil parameters, and then com-pare the predicted and measured cylinder forces over the remainingseven cycles; the results are shown in Fig. 11.
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
5.2. Optimized motion versus operator motion
We now generate optimal motions using the models and algo-rithms described earlier, and compare them to the motions generatedby an experienced human operator. Minimum torque/time motionsare obtained for initial and final times that correspond to those ofthe human operator motion (Fig. 12). Table 5 compares the task exe-cution times and average torque sums of the human operator motion,the minimum-time motion (using a minimum-time motion planner),and the minimum torque/time motions generated by our planner (Figs.15 and 16). The results suggest that the operator uses excessive torquescompared to our planner's motions. Note that the minimum-timemotion only results in a small improvement in task execution time
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Fig. 12. Cylinder forces over a single cycle. Dotted lines separate digging, lifting, dumping, and returning periods from left to right.
t5:1
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9Y.B. Kim et al. / Automation in Construction xxx (2013) xxx–xxx
Ccompared to our planner's motion, while requiring larger torques. Theresulting force profiles are shown in Figs. 13 and 14.
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RR6. Conclusions
As a prelude to the realization of fully unmanned excavators, thisstudy has attempted to determine, based on dynamic models of theexcavator as well as soil–tool interaction models, excavator motionsthat are fast, power-efficient, and smooth. Themain technical contribu-tion of our paper is a set of analytic gradient-basedmotion optimizationalgorithms, specialized to earthmoving excavators, for generating thefastest possible minimum torque motions —that is, without violatingany actuator saturation limits, or other constraints on cylinder velocitiesor accelerations; we call suchmotionsminimum torque/timemotions—taking into account the closed-loop structure of excavators, the soildynamics during digging and dumping, and any collision avoidance con-straints that arise during typical excavation tasks. Such motions avoidthe undesirable bang-bang characteristics of typical minimum-time tra-jectories, while being fast, smooth, and energy-efficient. The optimalmotions generated by the planner are also compared with the actualexcavation trajectories of experienced human operators.
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Table 5Comparison of motions.
Criteria Measurement Min time Min torque
tf 14.9 12.6 14.9Average ||τ(t)||2 1.0145 0.61483 0.61128
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories fodx.doi.org/10.1016/j.autcon.2013.01.007
One of the interesting findings of our experiments is that there isnotable difference between the generated optimal motion and actualexcavation trajectories performed by a human operator, particularlyduring the arm return motion. We suspect that these differencescan be mitigated somewhat by imposing more physically realisticconstraints on the hydraulics, and taking into account collision avoid-ance between the excavator and truck.
At a higher level, this paper has also presented an offline simula-tion and planning system for unmanned excavators, that integratesan excavator dynamic simulation engine with our optimal motionplanner. One of the key elements in both the planning and simulationphases is the use of a soil model. In this paper we have described anonlinear least-squares soil model parameter identification proce-dure that relies on measurements of digging forces in actual excava-tion tasks.
The presented work can be extended in a number of ways. Takinginto account the dynamics of the hydraulic actuators is clearly useful,in particular considerations on, e.g., hydraulic actuator limits and heatloss considerations. Although [4] presents one realization of such amodel, accurate hydraulic actuator dynamic modeling is a highlycomplicated task, and it may be profitable to examine hierarchicalmodels that capture features of the dynamics at different resolutionscales. Improved soil models, as well as excavation tasks in moreextreme unstructured environments, are also being examined in thecurrent context, as well as taking into account ground collision avoid-ance constraints that arise during typical excavation tasks.
There are also several means of bootstrapping the above procedureinto a class ofmethods for generating real-time suboptimal trajectories;one particular method is described in [14] that involves principal com-ponent analysis of a dataset of optimally generated motions.
r earthmoving excavators, Automation in Construction (2013), http://
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Fig. 14. Cylinder forces.
10 Y.B. Kim et al. / Automation in Construction xxx (2013) xxx–xxx
Please cite this article as: Y.B. Kim, et al., Dynamically optimal trajectories for earthmoving excavators, Automation in Construction (2013), http://dx.doi.org/10.1016/j.autcon.2013.01.007
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Fig. 15. Minimum time solution. Red curve shows the bucket-tip path of the optimal motion; the blue curve shows the human operator's motion. (For interpretation of the refer-ences to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 16. Minimum torque solution. Red curve shows the bucket-tip path of the optimal motion; the blue curve shows the human operator's motion. (For interpretation of the ref-erences to color in this figure legend, the reader is referred to the web version of this article.)
11Y.B. Kim et al. / Automation in Construction xxx (2013) xxx–xxx
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Acknowledgment
This researchwas supported by a research grant fromHyundai HeavyIndustries, with further partial support from SNU-IAMD, ROSAEC-ERC,AIM, and SNU-BK21 Program in Mechanical Engineering.
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