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A simplified dynamic model for front-end loader design M D Worley‡ and V La Saponara∗ Department of Mechanical and Aeronautical Engineering, University of California, Davis, California, USA The manuscript was received on 01 April 2007 and was accepted after revision for publication on 09 May 2008. DOI: 10.1243/09544062JMES688
Abstract: The front-end loader is an indispensable machine for the off-road construction equipment industry. It is a classic example of a working machine with complex interactions between its subsystems (hydraulic, mechanical, and electrical). Dynamic models of the full-scale vehicle coupled with event-based operator models are currently used to help quantify the overall system performance, efficiency, and operability. However, these models are complex and not always necessary to characterize the response of individual subsystems. There is great value added to the design process – especially in prototyping of new vehicle platforms – in development of simpler models that can quickly and accurately define first-order measures of system loads and performance. This paper presents a subscale dynamic model, which isolates the boom and bucket manipulator systems of the front-end loader for the purpose of design load characterization. The model includes state equations governing the hydraulic dynamics across the control valves and in the cylinders, as well as soil–tool interaction loads (passive earth loads) at the bucket cutting edge. The governing equations of motion for the multi-rigid body model of the bucket linkage are developed using Kane’s method. The proposed model is intended to accelerate the structural design and analysis of the boom and bucket linkage subsystems and may yield useful information for optimization purposes. The output from the dynamic simulation is compared with the field test data of the machine. Keywords: front-end loader, dynamic model, Kane’s method
1
INTRODUCTION
Davis, California 95616, USA. email:
[email protected] ‡ Now at: DJH Engineering Centre, Inc., Holladay, Utah, USA.
complex and may not always be necessary for characterizing the response of individual subsystems for the purposes of design. The development of accurate, simplified models that accelerate the design process – especially in prototyping of new vehicle platforms – are of great value to the analyst, and provide the motivation for this study. This paper proposes a new subscale dynamic model for the boom and the bucket manipulator of the frontend loader, with the goal of adequately characterizing the subsystem response for design and optimization purposes. The interaction of the subsystem with the rest of the machine via the hydraulic control system is modelled using state equations governing flow across the control valves and in the hydraulic cylinders. The interaction of the subsystem with the work environment via passive earth loads on the digging tool (i.e. bucket) is modelled using soil–tool reaction loads that make heavy use of earth moving equations developed by Terzaghi [2] and McKeys [3]. The equations of motion for the multi-rigid body model
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The front-end loader is a classic example of a working machine that has complex, often non-linear, interactions among its subsystems (hydraulic, mechanical, and electrical) [1]. Full-scale vehicle dynamic models and event-based operator models represent the current state-of-the-art dynamic simulations, carried out with the purpose of quantifying the overall performance, efficiency, and operability of the system. However, the correct simulation of these factors is complicated by the variability of the machine’s work environment and task profile. While the full-scale simulation may offer the best insight in the machine’s overall performance and efficiency, these models are ∗ Corresponding
author: Department of Mechanical and Aeronau-
tical Engineering, University of California, One Shields Avenue,
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of the manipulator system are obtained using Kane’s method. Some of the primary assumptions of the model should be highlighted. First, it should be noted that the proposed model is limited in application to the digging phase of the front-end loader’s work cycle. Though this is certainly not a comprehensive crosssection of the structural loading experienced by the entire machine, it is the primary loading cycle for the boom and bucket manipulator subsystem and is therefore of primary interest. Next, it should be noted that the proposed model ignores the pitch/bounce dynamics of the tires, owing to the fact that during the digging operation the vehicle speeds are generally low. In more general models that wish to capture vehicle dynamics during transport or articulation, the tire–terrain interaction dynamics cannot be ignored. Lastly, as indicated above, the proposed model ignores the elastic behaviour of the boom, bucket, and links, opting to model the system using rigid bodies. The goal of the proposed model is not to mirror the measured load–time history from an actual machine, which inexorably leads one to build more and more complicated machine and operator simulation models. Rather, the goal of this model is to aptly predict peak loads during the primary work cycle (dig cycle). The elastic response of the bucket, boom, and link structures, due to either symmetric or asymmetric application of these peak loads, is expected to be captured in subsequent structure and finite element analyses (FEA).
Fig. 1
The remainder of this paper is organized as follows. Section 2 provides a brief, general introduction to the composition, operation, and application of the front-end loader. Section 2.1 provides a description of the particular boom and bucket mechanism modelled. Section 2.2 outlines the simplified hydraulic system operation and equations governing the hydro-mechanical dynamics of the actuator cylinders. Section 2.3 describes the interaction forces between the bucket and the dirt pile. A review of the literature revealed extensive discussion regarding soil–tool interaction models but little information regarding a detailed procedure for applying them to wheel loaders and sloped pile digging. As such, this section concerning bucket–soil interaction is exhaustive in detail, hoping to document the procedure used here. Section 3 provides information concerning assumptions about operator input (cylinder actuation). In section 4, the simulation results are correlated and compared with the field data gathered on an actual machine. Discussion about results and areas of future improvement is also included. Section 5 provides a brief summary of the paper, drawing conclusions about the model and its applicability.
2 THE FRONT-END LOADER MODEL Figure 1 shows a generic machine level description of a four-wheel articulated-steer front-end-loader. The articulation axis is the pivot axis between the forward
General global (planar) schematic used for four-wheel articulated front-end loader model
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and rear frames. With the exception of the front axle and its connecting driveshaft, the powertrain system components are located in the rear of the machine. The operator cabin is typically mounted on the rear frame, but in a location (forward and above the articulation axis) that allows maximum visibility of the working tool (e.g. a loading bucket). The forward frame provides a platform for the boom and bucket manipulator linkage, which are collectively labelled in this study as the bucket mechanism. Several bucket manipulator linkage types exist for controlling bucket orientation. Also, it should be noted that the tool at the end of the boom need not necessarily be a loading bucket – the versatile front-end loader is increasingly being used and designed to handle a variety of applications. However, the loading bucket remains the traditional and primary work tool for the wheel loader. There are typically three sets of hydraulic cylinders and associated systems for kinematic control: (a) steering cylinders (not shown in Fig. 1) for vehicle articulation; (b) lift cylinder(s) for controlling the boom position, and (c) tilt cylinder(s) for controlling bucket orientation. Fluid pressure and flow in these systems are delivered by hydraulic pumps, which are in turn powered by the vehicle’s engine. In fact, as all onboard power generation ultimately originates in the combustion chamber of the engine, this available power must be distributed to, and balanced among the machine’s electrical, hydraulic, and powertrain demands. Detailed discussion regarding the power balance between the hydraulic and powertrain systems can be found in references [1] and [4] to [6]. Furthermore, all system control ultimately originates with the operator located in the cabin, who controls the subsystems using various levers, pedals, switches, etc. Dynamic operator models have been developed to capture this ‘human element’ in the overall system performance of the machine [1, 4, 5]. It is recognized that current state of the art in dynamic simulation of these machines employ full-scale vehicle models and eventbased operator models to quantify the overall system performance, efficiency, and operability. However, for the purposes of this study, simplified operator logic and hydraulic system models suffice to simulate the machine operations considered. This will be discussed in more detail later in this paper. As suggested by Fig. 1, a comprehensive planar description of the machine dynamics would include translation and pitch motion of the vehicle, also taking into account the compliance of the tires. Moreover, general loader operations include turning and transporting the payload, often over uneven terrain. Properly constructing a vehicle model to handle a wide variety of loading scenarios obviously requires increasing generality and complexity. Examples of various approaches and levels of detail to the modelling of loader dynamics can be found in earlier work
by Hemami and Daneshmend [7], Sarata et al. [8], and Scheding et al. [9]. In this paper, the machine is assumed fixed in the inertial frame and the bucket mechanism analysed as a planar mechanism grounded in the forward frame. In reality, forces on the bucket during loading/digging are created, in part, by global motion (penetration) of the vehicle into the terrain. A method for estimating these loads and applying them to the bucket as time-varying forces is given later. This paper is concerned primarily with the digging phase of the loader’s work-cycle and developing a simplified model.
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2.1
Bucket mechanism
The bucket mechanism considered for this study is shown in Fig. 2. It has a relatively new bucket manipulator linkage type, introduced by John Deere in 2004 on its Powerllel™ machines, and was selected because of interest in its unorthodox design: the centre pivot joint of the bellcrank is not affixed to the boom structure as in other linkage designs. The bellcrank pivots on a separate link, labelled here as the ‘Y-link’ because the shape of the part’s actual design resembles a ‘y’. The Y-link pivots in the forward frame. For clarity, the underlying kinematic geometry that defines the boom structure and lift cylinder(s) is shown using dotted lines. Similarly, dashed lines indicate the geometries of the bucket manipulator linkage and tilt cylinder. Figures 3 and 4 show the boom and bucket linkage systems separated, and define the geometrical (kinematic) values required for the overall mechanism model. The mechanism has ten rigid bodies and two degrees of freedom. The equations of motion for the system may be derived with the aid of any commercially available symbolic manipulator. In this case, the program Autolev [10] was used to generate and solve the equations. This program is especially suited for employing Kane’s method [11]. Due to their length, the equations are not included here. However, a brief description of their set-up is appropriate. The trivial set of generalized speeds is chosen as shown below U1 = U2 = U3 = U4 = U5 =
N
d (LLCYL ) = L˙ LCYL = vLCYL_ROD dt N d (LTCYL ) = L˙ TCYL = vTCYL_ROD dt N d (φLCYL ) = ωLCYL dt N d (φTCYL ) = ωTCYL dt N d (φBOOM ) = ωBOOM dt
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(1) (2) (3) (4) (5)
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Fig. 2
Schematic of bucket mechanism (and attached bucket) used for this study
N N
v J11_GDLK = N v J11_BOOM
(13)
= v
(14)
v
J12_BKT
N
J12_BOOM
(12)
Equations (11) to (14) represent eight planar motion constraints, dictating continuity of velocities for joint J5 (belonging to lift cylinder and boom), joint J7 (belonging to bellcrank and Y-link), joint J11 (belonging to guide link and boom), and joint J12 (belonging to bucket and boom). Substituting the generalized speeds U1 –U10 in equations (11) to (14) and solving for U1 and U2 as the independent generalized speeds (i.e. the system degrees of freedom), one obtains the necessary dependencies for the generalized speeds U3 –U10 . The contributing active and inertial forces in the system include: (a) the weights and inertial loads of the linkage bodies (mass) due to gravity and motional acceleration; (b) the lift cylinder force acting on the boom at joint J5; (c) the tilt cylinder force acting on the bellcrank at joint J8; and (d) the reactive forces on the bucket tool due to digging operations. Therefore, the planar velocities and accelerations of the system bodies and these idealized points of active loading are constructed and used to determine the constrained partial velocities and partial angular velocities. Dot products of these partial velocities with the corresponding active and inertial forces produce expressions for the generalized active and inertial forces, which comprise the equations of motion [11]. The following sections provide the details for the forces generated by the hydraulics as well as the passive earth pressure loads generated on the bucket during digging.
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Fig. 3
Kinematic variables of the boom system
N
d (φBLCRK ) = ωBLCRK dt N d (φBKTLK ) = ωBKTLK U7 = dt N d (φBKT ) = ωBKT U8 = dt N d (φGDLK ) = ωGDLK U9 = dt N d (φYLINK ) = ωYLINK U10 = dt
U6 =
(6) (7) (8) (9) (10)
The following set of motion constraints apply to the system N N
v J5_LCYL = N v J5_BOOM v
J7_YLINK
= v N
J7_BLCRK
(11)
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Dynamic model for front-end loader design
Fig. 4
2.2
Kinematic variables of the bucket linkage system
Hydraulic dynamics
Figure 5 shows a schematic of the simplified hydraulic system model used to actuate the tilt and lift cylinders. The hydraulic pump is considered as a positive displacement device that maintains a supply pressure PS a fixed P above the cylinder load pressure, unless the system relief pressure is reached. Fluid flow to the cylinders is regulated by turbulent flow across the control valve orifice. The flow control valves and cylinders are modelled as in references [12] and [13], with cylinder extension designated as the positive convention. Therefore, flow into the piston-side volume of the cylinder and flow out of the rod-side volume are respectively given by QPIS = AVALVE C1
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2|PS − PPIS | sign(PS − PPIS ) ρhyd
(15)
The equations governing pressure inside of the cylinders are given by P˙ ROD =
βF (QROD + AROD vROD ) VolROD
(17)
and P˙ PIS =
βF (QPIS − APIS vROD ) VolPIS
(18)
where VolROD and VolPIS are the instantaneous volumes of the rod-side and piston-side chambers, respectively; βF is the bulk modulus of the hydraulic fluid; AROD and APIS are the fluid areas of the rod-side and piston-side cross-sections of the cylinder, respectively; vROD is the velocity of the rod (positive for extension). Equations (15) to (18) are solved at every timestep in the dynamic simulation of the bucket mechanism.
and QROD = AVALVE C2
2|PROD − PE | sign(PROD − PE ) (16) ρhyd
where AVALVE is the exposed control valve orifice area, which is a function of the throw distance of the control lever by the operator and, moreover, the following holds: (a) For cylinder extension: C1 = C1in ; C2 = C2out ; PS = pump (supply) pressure; PE = tank (exhaust) pressure; (b) For cylinder retraction: C1 = C1out ; C2 = C2in ; PS = tank (exhaust) pressure; PE = pump (supply) pressure. JMES688 © IMechE 2008
Fig. 5
Schematic of simplified hydraulics system
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The operator input and hydraulic pump assumptions determine the flowrates for the cylinders while the system dynamics determine the velocity of the cylinder rods. Time integration of equations (17) and (18) produce the pressures in the cylinders. Finally, the net cylinder forces applied to the system are calculated as Fcyl = APIS PPIS − AROD PROD 2.3
(19)
Digging forces on the bucket
One of the challenging aspects in modelling the behaviour of a wheel loader – or any earthmoving equipment – is capturing the forces produced during digging operations. These forces are primarily dependent on the properties of the soil or rock that is being excavated, which can vary drastically depending on size, composition, temperature, moisture content, compaction, etc. This variability, however, should and does pose a greater challenge to efforts for autonomous excavation and productivity studies, since soil conditions can greatly alter the method of scooping required to achieve optimal performance [14]. For the purposes of structural design and baseline comparison studies, the order of magnitude predictions for standard design cases should suffice. Based on previous work by Hemami [15], Singh [16], and Ericsson and Slattengren [17], it appears possible to achieve reasonable estimates of the interaction forces between the cutting tool and soil. Furthermore, a two-dimensional model is adequate for the geometries of most bucket tools [16]. Their methods utilize the empirical and analytical studies in soil mechanics done by Terzaghi [2], Bekker [18], and McKeys [3], to name a few. Singh [16] provides a good synopsis of basic soil mechanics and associated literature. Figure 6 shows a diagram of the bucket tool digging/scooping in a dirt pile. The path of the bucket cutting edge is determined by the global vehicle motion
Fig. 6
Sketch of bucket tool digging in work pile
(penetration) into the pile as well as the bucket lift and rotation due to actuation of the lift and tilt cylinders, respectively. The path is not necessarily known a priori and will depend ultimately on operator commands. As pointed out in references [16] and [19], the ‘best’ or most productive path may vary from cycle to cycle due to the variability of forces in the digging process created by the soil and/or rock properties. Hemami [15] describes the bucket–soil interaction as consisting of five force components: (a) f1 , the weight of the loaded material; (b) f2 , the resistance force created if the bucket motion causes it to compress the dirt pile; (c) f3 , the net friction force between tool and bucket; (d) f4 , the digging resistance acting at the bucket cutting edge; (e) f5 , the inertial force required to accelerate the accumulated mass. In attempting to describe a path of minimal energy input, he suggests that such a path should cause force f2 to be zero. This point might be debatable, however, as it is at least conceivable that this force could be used beneficially in the loading process: if the line of action of force f2 is such that it creates a positive moment (with respect to the machine frame M1 , M2 , M3 ) about the bucket hinge pin (joint J12 in Fig. 3), then this force is aiding the tilt cylinder in curling the bucket and payload. This point is made only to ask: ‘Does such a beneficial scenario exist?’ That is, since such a force f2 would require a proper combination of tractive effort and bucket orientation during the scooping process, it becomes a matter of weighing the additional energy consumed in the powertrain to the energy saved in the hydraulic cylinder(s). The authors believe that all the five force components described are included in this proposed model. However, they exist in a form more consistent with the soil wedge model (originally proposed by Coulomb in 1772 and revisited by Terzaghi [2]) detailed by McKeys [3]. As previously pointed out, the nature and activation of these loads may change during the digging process due to the variability in soil properties, changes in the contact interface between the bucket and terrain, and the variability in the operator reaction/preference. To simulate loads imparted to the bucket tool throughout the digging process requires making assumptions for all the three. The inaccuracy due to nominal soil properties is considered here as unavoidable. Real time, in situ measurements of the soil properties appear to be the only way to accurately develop these parameters [16]. The items concerning changing tool–soil contact and operator reaction are considered interconnected and are thus treated together in the following discussion.
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Fig. 9
Fig. 7 Three phases of the digging operation
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Force diagram of bucket tool
orientation α is altered. Inertial effects on the soil mass have been included. At each timestep, a mass m (shown crosshatched in Fig. 8) is being added to the bucket and must be accelerated to match the speed of the soil wedge. Due to the relative motion between the wedge and the bucket tool and referring to Fig. 8, the speed V of the soil mass m is related to the average speed V of the bucket tool cutting plane as follows [3] V =
motion of soil mass, m = time
χ cos β
1 t
where Fig. 8
Force diagram of soil wedge
χ = χ + ε = χ (1 + tan β cot ρ) The sketch in Fig. 7 describes the three phases of bucket loading that are assumed in this study. Figures 8 and 9 describe the interaction loads assumed between the bucket, the soil wedge, and the bulk dirt pile. Not all these forces are necessarily active at the same time during any given phase. Detailed descriptions of active loads and assumptions for each phase are given in the following sections. At this point, a discussion of the soil wedge model and associated forces is required. Referring to Figure 8, force F is the combined net normal and frictional force acting between the soil wedge and tool along boundary AB, acting at the angle δ of soil-to-metal friction. Force Ca LT is the force due to cohesion between the soil and metal along boundary AB. Force R is the combined net normal and frictional force acting between the soil wedge and ‘undisturbed’ soil along boundary BC, acting at the angle of internal shearing resistance for the soil, φ. Force CLF is the force due to the (apparent) cohesion of the soil along the failure boundary BC. Load q is the surcharge pressure. The angle ρ = θ − α is the rake angle of the cutting edge (commonly referred to as the ‘bolt-on-cutting edge’, hence the use of the abbreviation BOC in Figs 9 and 11) relative to the dirt pile surface. This angle changes as the cutting edge JMES688 © IMechE 2008
(20)
Therefore
V =
V χ = t cos β(1 + tan β cot ρ) cos β(1 + tan β cot ρ) (21)
where motion of bucket tool = V = time
χ cos β
1 t
Therefore, the inertial force FA , acting in opposite direction to V , can be written as γ · χ · dw V = V = V γ · V · dw t t γ · V 2 · dw = cos β(1 + tan β cot ρ)
FA = m
(22)
where γ is the total soil density. Continuing with the method presented by McKeys [3], the unknown R can Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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be eliminated and solve for the force F as F = γ · g · d 2 · Nγ + C · d · NC + Ca · d · NCa + q · d · N q + γ · V 2 · d · Na · w
(23)
where w is the width of the bucket tool cutting edge and N γ , NC , NCa , Nq , and Na are given by Nγ =
(cot ρ + cot β) · [sin θ cot(β + φ) + cos θ] (24) 2 · cos(ρ + δ) + sin(ρ + δ) cot(β + φ)
NC =
1 + cot β cot(β + φ) cos(ρ + δ) + sin(ρ + δ)cot(β + φ)
NCa =
1 − cot ρ cot(β + φ) cos(ρ + δ) + sin(ρ + δ) cot(β + φ)
Nq = 2Nγ Na =
(25) (26) (27)
tan β + cot(β + φ) [cos(ρ + δ) + sin(ρ + δ) cot(β + φ)] · (1 + tan β cot ρ)
(28)
Noting that LT = d/sin ρ and LF = d/sin β, it is seen that the loads F and Ca LT are functions of the following: (a) the soil and soil–tool properties, i.e. δ, φ, γ , C, and Ca ; (b) the kinematics of the bucket, i.e. ρ, d, and V ; and (c) the unknown angle β, which is found by minimization of N γ with respect to β [3]. One disadvantage of the wedge model is that it is meant to model steady-state loading of soil under imminent failure. In reality, the compressive load F builds over a period of time, drops as the soil fails along boundary BC, and then builds again until the next failure. This produces a saw-like profile for the load F , whereas F is constant in the wedge model. Nevertheless, the model provides reasonable results with considerable simplicity. The bucket is assumed to have three points where active loading may occur: the mass centre of the bucket, the mass centre of the developing payload, and the cutting edge of the bucket. Referring to Fig. 9, loads F and Ca LT are applied to the bucket tool by the soil wedge. Moreover, a load P and possibly loads Fsupport and Ca LBOC are applied to the bucket by the dirt pile. The load P is the penetration load that exists whenever the cutting edge is forced further into the dirt pile. The magnitude of P is determined as the product of the face area of the bucket cutting edge and the pressure acting on this area. This pressure is approximated using the empirical load-sinkage formula proposed by Bekker [18] kC p= + kφ z n (29) b where z is the sinkage depth, n is the exponent of deformation, b is the width of the penetrating surface, and
kC and kφ are the cohesive and frictional moduli of deformation, respectively. It should be observed that the value for b has considerable influence on the magnitude of force P and the resulting depth of penetration prior to initiating wheel slip. This, in turn, influences the initial payload at the start of the scooping phase (digging ‘Phase 2’, described in section 2.3.2). Figure 10 shows a close-up of a typical cutting edge for a loader bucket. The chamfer profile makes straightforward use of the cutting edge thickness for the dimension b. It is reasoned that using the full thickness is incorrect, as the soil tends to slide up and off the chamfer slope. On the other hand, it is also probably not correct to use the smaller dimension b1 since the weight of the soil above provides constraint to this process. Several iterative studies for the penetration phase were run to help identify a reasonable value for b. The value that best simulated the penetration depth and pressure transition from digging ‘Phase 1’ to digging ‘Phase 2’ (discussed in next section) was determined to be approximately one-half of the cutting edge thickness. Further discussion regarding the setting of b may be found in the next sections on the three digging phases. To determine when and how the aforementioned forces are assumed to be active, a more detailed discussion regarding the three phases of digging and their respective assumptions is appropriate. These three phases are Phase 1 ‘crowding’, Phase 2 ‘bucket filling’, and Phase 3 ‘rollback’. 2.3.1
Phase 1: crowding
The bucket tool penetrates into the pile at an initial rake angle ρ0 . The cutting edge motion is due only to tractive effort creating forward translation of the vehicle. In reality, some degree of soil cutting occurs owing to the bucket’s sloping profile, and the bucket is being loaded by the developing soil wedge (creating forces F and Ca LT , Fig. 8). However, since the cutting edge is supported by the soil underneath it, the components
Fig. 10
Close-up of cutting edge
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of F and Fsupport (Fig. 9) that are normal to the cutting edge cancel. Equation (29) is used to calculate the penetration force P, where the value z is estimated using the depth d. For all phases, the terrain loads acting on the bucket are transformed into the loads FV_P3 , FH_P3 , and MP3 , which are, respectively, the vertical force component, horizontal force component, and moment, and act at the cutting edge (i.e. designated as point P3) with respect to the cutting edge frame (Fig. 11). Based on the assumptions aforesaid, the expressions for each of these forces during Phase 1 are FV_P3 = F cos δ − Fsupport cos δ = 0
(30)
FH_P3 = P + 2F sin δ + Ca (LBOC + LT ) · w
(31)
MP3 = F cos δ
LT 2
(32)
Consideration was given as to how this phase should end. Most operators do not simply crowd the pile until they lose traction as this represents wasted effort – it generates in excess the force that Hemami [15] labelled as f2 . However, sufficient penetration is required to ensure a full bucket. Therefore, the following simplifying assumptions have been made: 1. It is assumed that the operator desires to keep the machine moving forward at a relatively constant speed throughout the majority of the loading process. That is, the operator will crowd the pile until he/she senses the machine has slowed to some nominal speed, at which he/she begins using the lifting and tilting functions to fill the bucket in a
Fig. 11 JMES688 © IMechE 2008
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manner that allows him/her to maintain this speed (approximately). This assumption is made based on observations of the loading process and with the intent to provide some autonomy to the operator commands in Phase 2. The critical transition speed assumed in this study is 0.5 m/s, after which a maintenance speed range of 0.25–0.5 m/s is set as the target parameter. 2. If the soil is very loose and non-cohesive, the operator may penetrate deeper than desired or necessary following rule 1. Instead, it is assumed he/she would begin Phase 2 (discussed in the next section) after reaching some nominal or critical depth. Though somewhat arbitrary, this critical depth is determined in the following way. A critical or maximal length LTcrit of tool–soil engagement is assumed to define the point where the bucket profile disrupts the flow of soil in the wedge. The maximum length assumed here is the curve length shown in Fig. 11. As part of the simplifications of this study, the state equations for the powertrain response have been omitted. Instead, the forward velocity of the vehicle is governed by the following linearized load–velocity relationship VFWD = Vgear1 −
Vgear1 H Rimpullmax
(33)
where Vgear1 is the maximum travel speed in first gear, Rimpullmax is the maximum rimpull force at full slip, and is approximated as Rimpullmax = g µd Mvehicle
(34)
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Moreover, H is the total horizontal force imparted to the bucket by the terrain. In general, H is given by H = FV_P3 sin α + FH_P3 cos α
(35)
It is recognized that the maximum available rimpull may vary with time – increasing due to vertical loading on the bucket and/or an increasing mass due to payload. However, staying with equation (34), this could in fact easily lead to an estimate for rimpull that exceeds the power capability of the vehicle’s engine. Since the powertrain response is omitted and a simpler
Fig. 12
approximation desired, this detail is not considered in this study. As mentioned previously, several preliminary studies were undertaken to determine a reasonable value for the parameter b in equation (29). After some iteration, it was decided that setting b equal to one-half of the cutting edge thickness best matched the available field data. Though not measured, the soil properties intrinsic to the collected pressure data are most similar to those of sandy soils. The soil properties given in reference [20] were used for comparison, with the dry sand and sandy loam soils most heavily influencing the choice of b. Figure 12 shows the calculated
Plots of calculated crowd loads (rimpull) and lift cylinder pressures for various soils using b equal to one-half cutting edge thickness
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rimpull and lift cylinder pressures for crowding into various soils. Figure 12(a) shows the two critical transition parameters (the critical forward speed and critical penetration depth) that trigger the onset of Phase 2. Figure 12(b) shows the estimated jump in pressure as Phase 2 begins, i.e. the supporting force from the pile vanishes. Note that the ‘clayey soil’ and the ‘sandy loam4’ transitions are triggered by the critical penetration distance rather than the critical speed.
area of the soil wedge – defined by A1 B1 C1 – represents the mass accumulated thus far. At some time later, the bucket has moved through an arbitrary path, and the new soil wedge has area defined by A2 B2 C2 . The mass accumulated from the first state to the second – not counting the current soil wedge – is represented by the area A1 A2 B2 B1 . This area is calculated as
2.3.2
where the area E1 E2 B2 B1 is calculated using the trapezoid rule as
Phase 2: bucket filling
The motion of the cutting edge is due to a combination of forward motion of vehicle, lift cylinder actuation, and tilt cylinder actuation. The transition from Phase 1 to Phase 2 involves, by definition, the actuation of either the tilt cylinder or the lift cylinder. Either case results in loss of contact between the terrain and the underside of the cutting edge, eliminating the forces Fsupport and Ca LBOC . The forces P, Ca LT , and F remain active throughout Phase 2. Expressions for the loads at point P3 take the form FV_P3 = F cos δ
(36)
FH_P3 = P + F sin δ + Ca LT w
(37)
MP3 = F cos δ
LT 2
A1 A2 B2 B1 = E1 E2 B2 B1 + A1 E1 B1 − A2 E2 B2
E1 E2 B2 B1 =
1 χi di + di−1 2
(39)
(40)
This reasoning holds for any two wedges from one timestep to the next. Therefore the mass accumulated in sweeping the bucket from a configuration ψ1 = f (ρ1 , d1 , L1T ) to a configuration ψ2 = f (ρ2 , d2 , L2T ) is approximated as Maccum = γ · w ·
N
Ai
(41)
i=1
(38)
In addition, the accumulating mass of the payload must be tracked and added to the payload mass centre. For simplicity, this payload mass centre is assumed to be constant throughout the digging process and is defined per SAE J742. This is a reasonable assumpcrit tion for 0.65L crit T LT L T , as the mass of soil being pushed around the bucket curvature will tend to accumulate above the soil wedge near to this point. Referring to Fig. 13, accumulation of mass is calculated in the following way. At the end of Phase 1, the
Fig. 13
Geometry for cutting edge motion
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Fig. 14
Operator command template
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where N is the number of timesteps taken between configurations, and where Ai , χi , and di are given below i−1 2 2 d cotρ i−1 − d i cot ρ i 1 Ai = χi di − di−1 + 2 2 (42) N P3 χi = v i−1 · DP 1 + VFWDi−1 cos θ t (43) N P3 di = di−1 + VFWDi−1 sin θ − v i−1 · DP 2 t (44) Note that the velocity component in brackets in equation (43) is not, in general, equal to the cutting plane speed V used in equations (21) to (23). For general planar motion of the bucket, the determination
Fig. 15
of V requires taking an average of the velocities along boundary AB at the soil wedge. The mass accumulated at each timestep is transferred to the payload mass centre. At the end of Phase 2, all of the swept mass has accumulated at the payload centre, with the exception of the most recent soil wedge. Phase 2 ends when one of the following events occurs: (a) the rake angle ρ drops below a nominal limit; (b) the accumulated mass plus the current soil wedge mass exceeds a nominal limit; (c) the volume of the accumulated mass plus the volume of the current soil wedge exceeds a nominal limit.
(a) Lift cylinder pressures during control boom cycle (‘plot 1’); (b) tilt cylinder pressures during control boom cycle (‘plot 2’). Field data is represented by solid lines
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Fig. 16
2.3.3
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(a) Lift cylinder pressures during control boom cycle (‘plot 3’); (b) tilt cylinder pressures during control bucket cycle (‘plot 4’). Field data is represented by solid lines
3
Phase 3: rollback
The tilt cylinder is actuated until the bucket is fully rolled, which, depending on boom height, will occur when either the tilt cylinder reaches full extension or the mechanical roll stop is contacted. The mass of the remaining soil wedge from Phase 2 is transferred to the payload mass centre. Inherent in this approach is the assumption that there is negligible soil cutting force at this point in the digging process.
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OPERATOR INPUT
As mentioned previously, the operator action to end crowding and begin filling the bucket is automated by achieving either the critical penetration depth or the critical forward speed of the vehicle. During bucket filling, the operator response is somewhat autonomous as the only constraint is that his/her actions keep the forward speed of the vehicle within a set bandwidth around the nominal critical forward speed. The model
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is provided by a standard operator template that is followed if the velocity requirement is met. Otherwise, deviations are made from the template to try and correct the forward velocity. Figure 14 shows the template operator input for the digging portion of the short work cycle. 4
COMPARISON WITH FIELD DATA AND DISCUSSION
Two control simulations are performed and compared with field test data measured on an actual machine, to verify that the hydraulic cylinder and valve models are working together with the bucket mechanism dynamics. These control cases do not include terrain loads from the previous section in order to isolate the hydraulics and bucket mechanism. The first control case is referred to here as the boom cycle case. In this scenario, the lift cylinders are activated to raise the boom to its full height, hold it at full height for a certain period of time, then lower it back to its original starting position. The bucket tool has no payload during the operation and pressures in the tilt cylinder are purely reactionary. The second control case is referred to here as the bucket cycle case. In this scenario, the tilt cylinder is activated to roll the bucket tool from a full dump position to a fully rolled position, hold it at fully rolled for a certain period of time, then dump it back to its original starting position. Again, the mechanism has no payload and, in this case, the pressures in the lift cylinders are purely reactionary. In either case, cylinder actuation is controlled through changes in the control valve orifice area. Data for this orifice area as a function of the control lever throw distance was used to generate equations that govern the exposed area as function of lever throw. Comparison of simulation output and measured values is shown in Figs 15 and 16. Table 1
Parameters used in simulation
Parameter
Value
Maximum supply pressure, PS (N/m2 ) Tank (exhaust) pressure, PE (N/m2 ) Fluid bulk modulus, βF (N/m2 ) Fluid density, ρhyd (kg/m3 ) Lift cylinder bore (mm) Tilt cylinder bore (mm) Dirt pile angle of repose, θ (◦ ) Soil internal friction angle, φ (◦ ) Soil–tool friction angle, δ (◦ ) Bekker parameters n KC (kN/m(n+1) ) Kφ (kN/m(n+1) ) Soil density, γ (T/m3 ) Vehicle speed in low gear, Vgear1 (m/s)
24.8 E6 0 E6 2.0 E9 900 125 160 37.5 29 17.5 0.7 5.27 1515.04 1.4 2.0
Based on the results, the simulation shows to be in reasonable agreement with test results. There are primarily two areas where the simulation and field data differ substantially. However, these can be explained as follows: 1. Plot 1 (the top plot in Fig. 15) and plots 3 and 4 (the plots in Fig. 16) show different responses over the dwell regions, where the initial cylinder actuation has reached its extent and the operator reverses the control lever to return to original position (e.g. at 5.75–6.50 s for the boom cycle case and 3.0-3.75 s for the bucket cycle case). It should be noted that in reality this will entail striking a mechanical stop – either inside the cylinder itself or between designated stop pads on the mechanism links. This event was not modelled by the simulation. Therefore, it is expected that the measured data and simulation output will differ in these regions. 2. Referring to Plot 2 (the bottom plot in Fig. 15), the lift cylinder pressures oscillate wildly about the measured data. This makes sense considering that the hydraulic cylinder model provides little damping when fluid is not flowing into or out of it (i.e. across the flow control valves). This is the case here, where only the tilt cylinder is being ‘activated’ and the lift cylinders are reacting more or less like springs. In reality, viscous friction in the cylinder and mechanism joints, as well as the compliance of the tires, provide damping to the system and help mitigate these oscillations. The model appears under-damped without including some of these phenomena. Next, simulations including the terrain forces on the bucket were conducted. Table 1 provides an abridged summary of the simulation parameters used in the correlation and dig studies. (Intellectual property prevents detailed publication of mass properties data and actual geometries.) Figure 17 shows a comparison of the measured lift cylinder and tilt cylinder pressures to the simulation output. The results show promise, but are certainly not as accurate as one would like, even for simplified design schemes. The different results for both cylinders between the interval 3 and 4 s are believed to be due to a lack of modelling the compaction force called ‘f2 ’ by Hemami. Furthermore, it is believed that the difference in results for the tilt cylinder over the interval 4–7 s is due to a lack of the extra soil weight existing on the inclined slope that is not captured by the wedge model, but was hoped to be ‘averaged’ out by keeping the Bekker sinkage force P active throughout the loading process. The rise in tilt cylinder pressure over the interval 7–9 s is thought to be due to the immediate transfer of the soil wedge mass to the bucket
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Dynamic model for front-end loader design
Fig. 17
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Comparisons of the lift cylinder and tilt cylinder pressures (piston side) during digging cycle. Field values are represented by solid lines
mass centre during the final rollback. In reality, the mass transfer is more gradual (though, sometimes only slightly), and the pile provides some support to the bucket while excess soil spills out of the bucket.
A subscale dynamic model of a unique front-end loader bucket mechanism type was developed, that included the hydraulic and mechanism dynamics, as well as simplified estimations for the terrain
loads imparted to the bucket tool during the scooping/loading portion of the loader’s short work-cycle. The equations of motion were developed using Kane’s method and programmed with the aid of a commercial symbolic manipulator. Test simulations isolating the hydraulic and mechanism models were run and compared with field data for proper correlation of control valve dynamics, a step deemed necessary because of the extreme simplification of the hydraulic system. Furthermore, several digging simulations were run to verify the capability of the model to predict the loads generated by the terrain on the bucket. The current model assumptions appear
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CONCLUSIONS
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to lack in the ability to capture these loads adequately over the time history of the digging operation, most notably during the bucket-filling phase. The model does show promise in being able to predict peak operation loads, seen most notably for the lift cylinder reaction loads. Future work and improvements should focus on improving the simulation of the bucket-filling phase. Also, similar simulations that vary the mechanism type, operator command, and soil properties should be done to verify the robustness of the model. The field data required to do this is unfortunately not available at this time. While this model’s accuracy cannot eclipse the accuracy provided by more detailed, full-scale models (e.g. those that utilize sophisticated operator logic to guide the vehicle), it is however promising for use as a preliminary performance analysis and for rapid characterization of design loads.
15
ACKNOWLEDGEMENTS
16
The authors thank Professor Don Margolis, Department of Mechanical and Aeronautical Engineering, University of California, Davis, for his feedback and review of this paper.
REFERENCES
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1 Filla, R. Operator and machine models for dynamic simulation of construction machinery. Licentiate Thesis, Department of Mechanical Engineering, Linkopings Universitet, Linkoping, Sweden, 16 September 2005. 2 Terzaghi, K. Theoretical soil mechanics, 1947 (Wiley, New York). 3 McKyes, E. Soil cutting and tillage, 1985 (Elsevier Press, Amsterdam). 4 Filla, R., Ericsson, A., and Palmberg, J.-O. Dynamic simulation of construction machinery: towards an operator model. In the IFPE 2005 Technical Conference, Las Vegas, NV, USA, 16–18 March 2005, pp. 429–438. 5 Filla, R. An event-driven operator model for dynamic simulation of construction machinery. In the Ninth Scandinavian International Conference on Fluid Power, Linkoping, Sweden, 1–3 June 2005, available from http://arxiv.org/ftp/cs/papers/0506/0506033.pdf. 6 Filla, R. and Palmberg, J.-O. Using dynamic simulation in the development of construction machinery. In the Eighth Scandinavian International Conference on Fluid Power, Tampere, Finland, 7–9 May 2003, vol. 1, pp. 651–667. 7 Hemami, A. and Daneshmend, L. Force analysis for automation of the loading operation in an LHD loader. In Proceedings of the 1992 IEEE International Conference on Robotics and Automation, Nice, France, 1992, pp. 645–650. 8 Sarata, S., Sato, K., and Yuta, S. Motion control system for autonomous wheel loader operation. In Proceedings
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of the International Symposium on Mine Mechanization and Automation, Golden, CO, USA, June 1995, 12 pages. Scheding, S., Dissanayake, G., Nebot, E., and DurrantWhyte, H. Slip modeling and aided inertial navigation of an LHD. In Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, New Mexico, USA, 1997, pp. 1904–1909, vol. 3. Available from http://www.autolev.com Kane, T. R. and Levinson, D. A. Dynamics, theory and applications, 1985 (McGraw-Hill, New York) Margolis, D. and Shim, T. Instability due to interacting hydraulic and mechanical dynamics in backhoes. ASME J. Dyn. Syst. Meas. Control, 2003, 125, 497–504. Zhu, W. and Piedboeuf, J. Adaptive output force tracking control of hydraulic cylinders with applications to robot manipulators. ASME J. Syst. Meas. Control, 2005, 127, 206–217. Singh, S. State of the art in automation of earthmoving. ASCE J. Aerosp. Eng., 1997, 10(4), 179–188. Hemami, A. Motion trajectory study in the scooping operation of an LHD-loader. IEEE Trans. Ind. Appl., 1994, 30(5), 1333–1338. Singh, S. Synthesis of tactical plans for robotic excavation. PhD Thesis, The Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, 1995. Ericsson, A. and Slattengren, J. Predicting digging and cutting forces in granulated material. In the 15th European ADAMS Users’ Conference, Rome, Italy, 15–17 November 2000, available from http://www. mscsoftware.com/support/library/conf/adams/euro/ 2000/Volvo_Predicting_Digging.pdf. Bekker, M. G. Introduction to terrain-vehicle systems, 1969 (University of Michigan Press, Michigan). Marshall, J. A. Towards autonomous excavation of fragmented rock: experiments, modelling, identification, and control. MS Thesis, Queen’s University, Kingston, Ontario, Canada, 2001. Wong, J. Y. Theory of ground vehicles, 1979 (John Wiley & Sons, New York).
APPENDIX Notation abucket_cg apayload_cg APIS AROD AVALVE b (BKT1 , BKT2 ) (BKTCG1,
acceleration of the bucket centre of gravity acceleration of the centre of gravity of accumulated soil area of piston-side cross-section of hydraulic cylinder area of rod-side cross-section of hydraulic cylinder area of hydraulic control valve orifice effective width of penetrating surface reference frame local to bucket location of centre of gravity
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BKTCG2) (BKTLK1 , BKTLK2 ) (BKTLKCG1, BKTLKCG2) (BLCRK1 , BLCRK2 ) (BLCRKCG1, BLCRKG2) (BOC1 , BOC2 ) (BOCHORIZ , BOCVERT )
(BOOM1 , BOOM2 ) (BOOMCG1, BOOMCG2) C C1 , C2 Ca d
N
d()/dt
(DP1 , DP2 ) F
FA Fcyl FH_P3
Fsupport
FV _P3
g (GDLK1 , GDLK2 ) (GDLKCG1,
of bucket in its local reference frame reference frame local to bucket link location of centre of gravity of bucket link in its local reference frame reference frame local to bellcrank location of centre of gravity of bellcrank in its local reference frame reference frame local to bucket bolt-on cutting edge location of load application point (P3) on the bucket in the local bolt-on cutting edge reference frame reference frame local to boom location of centre of gravity of boom in its local reference frame cohesion factor for the soil correlation constants for flow across control valve orifice cohesion factor between soil and cutting tool projection normal to dirt pile free boundary of bucket cutting edge penetration depth derivative with respect to time in reference system N reference frame local to the dirt pile free boundary net normal and friction force along boundary between cutting tool and soil wedge inertial force generated in accelerating soil mass net force generated by hydraulic pressure in cylinder component of total digging load acting on the bucket cutting edge, resolved parallel to the cutting edge motion net supporting force of dirt pile on underside of cutting tool (i.e. bucket) component of total digging load acting on the bucket cutting edge, resolved normal to the cutting edge motion gravitational constant reference frame local to guide link location of centre of
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GDLKCG2) H J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 kC
kφ
L1A, L1B, L1C L2 L3 L4 L5 L6 L7 L8 L9 LBOC (LCYL1 , LCYL2 ) LCYLPISCG1
LCYLRODCG1
LF
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gravity of guide link in its local reference frame total horizontal load on bucket cutting edge fixed pivot joint of body boom fixed pivot joint of tilt cylinder fixed pivot joint of lift cylinder fixed pivot joint of ‘Y-link’ pivot joint between lift cylinder and boom pivot joint between bellcrank and bucket link pivot joint between bellcrank and ‘Y-link’ pivot joint between bellcrank and tilt cylinder pivot joint between guide link and bucket link pivot joint between bucket link and bucket pivot joint between guide link and boom pivot joint between boom and bucket cohesive modulus of deformation – empirical factor used in Bekker’s load-sinkage formula frictional modulus of deformation – empirical factor used in Bekker’s load-sinkage formula lengths along boom length between joints J7 and J8 length between joints J6 and J7 length between joint J5 and point P1 length between joint J11 and point P2 length between joints J4 and J7 (i.e. length of ‘Y-link’) length between joints J6 and J9 length between joints J9 and J10 length between joints J9 and J11 (i.e. length of guide link) length of boundary between soil wedge and bolt-on cutting edge reference frame local to lift cylinder distance to centre of gravity of barrel section of lift cylinder in its local reference frame distance to centre of gravity of rod section of lift cylinder in its local reference frame length of boundary between soil wedge and dirt pile
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LLCYL LTCYL LT mbucket mpayload MP3 Mvehicle n
(N1 , N2 ) Na NC NCa Nq Nγ p P P3 PE PPIS PROD ·
P PIS ·
P ROD PS q QPIS QROD R
Rimpullmax
t (TCYL1 , TCYL2 )
M D Worley and V La Saponara
overall length of the lift cylinder (joint-to-joint) overall length of the tilt cylinder (joint-to-joint) length of boundary between cutting tool and soil wedge mass of the bucket mass of the accumulated soil net moment acting on the bucket cutting edge total mass of the vehicle exponent of deformation – empirical factor used in Bekker’s load-sinkage formula inertial reference frame Terzaghi factor for soil cutting force (acceleration term) Terzaghi factor for soil cutting force (soil cohesion term) Terzaghi factor for soil cutting force (soil–tool cohesion term) Terzaghi factor for soil cutting force (surcharge term) Terzaghi factor for soil cutting force (soil shear term) penetration pressure on bucket cutting edge penetration force on bucket cutting edge load application point on the bucket hydraulic system exhaust (tank) pressure hydraulic pressure on piston side of cylinder hydraulic pressure on rod side of cylinder change in piston-side hydraulic pressure with respect to time change in rod-side hydraulic pressure with respect to time hydraulic system supply (pump) pressure surcharge pressure flowrate on piston side of cylinder flowrate on rod side of cylinder net normal and friction force along boundary between soil wedge and dirt pile maximum push force that vehicle can develop (either powertrain or traction limited) time reference frame local to tilt cylinder
TCYLPISCG1
TCYLRODCG1
U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 vLCYL_ROD vROD vTCYL_ROD V V VFWD Vgear1 VolPIS VolROD w Wbucket Wpayload Wwedge (X 1, Y 1) (X 2, Y 2) (X 3, Y 3) (X 4, Y 4) (YLINK1 , YLINK2 )
distance to centre of gravity of barrel section of tilt cylinder in its local reference frame distance to centre of gravity of rod section of tilt cylinder in its local reference frame generalized speed set as the velocity of lift cylinder rod generalized speed set as the velocity of tilt cylinder rod generalized speed set as the angular velocity of lift cylinder generalized speed set as the angular velocity of tilt cylinder generalized speed set as the angular velocity of boom generalized speed set as the angular velocity of bellcrank generalized speed set as the angular velocity of bucket link generalized speed set as the angular velocity of bucket generalized speed set as the angular velocity of guide link generalized speed set as the angular velocity of ‘Y-link’ linear velocity of lift cylinder rod linear velocity of cylinder rod (general case) linear velocity of tilt cylinder rod average speed of cutting tool relative speed of cut soil mass forward speed of vehicle during penetration maximum forward speed of vehicle in first gear hydraulic fluid volume in piston-side chamber of cylinder hydraulic fluid volume in rod-side chamber of cylinder width of the bucket cutting edge weight of the bucket weight of the accumulated mass (soil) in the bucket weight of the soil wedge location of joint J1 in inertial reference frame location of joint J2 in inertial reference frame location of joint J3 in inertial reference frame location of joint J4 in inertial reference frame reference frame local to ‘Y-link’
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(YLINKCG1, YLINKCG2) z •
α
βF β γ
location of centre of gravity of ‘Y-link’in its local reference frame penetration depth of cutting tool centre of gravity geometrical point of interest in figures defining kinematics, also indicates connection point in hydraulic schematic figures a planar pivot joint in figures defining kinematics, also indicates a ball check valve in hydraulic schematic figures inclination angle of bucket cutting edge relative to ground (global horizontal). This is equivalent to orientation angle of the bolt-on cutting edge coordinate system reference, φBOC bulk modulus of hydraulic fluid break angle of soil wedge soil density
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γBKTLK γBLCRK δ θ µd ρ ρhyd φ χ χ
BKTLK
BKT
BLCRK
BOOM
GDLK
LCYL
TCYL
YLINK
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fixed angle of bucket link geometry fixed angle of bellcrank geometry soil–tool friction angle angle of repose of soil dynamic coefficient of friction between vehicle tires and ground rake angle of bucket during soil cutting density of hydraulic system fluid soil internal friction angle distance along dirt pile free boundary travelled by cutting tool over a given instant of time distance along dirt pile free boundary travelled by cut mass of soil over instant of time angular velocity of bucket link angular velocity of bucket angular velocity of bellcrank angular velocity of boom angular velocity of guide link angular velocity of lift cylinder angular velocity of tilt cylinder angular velocity of ‘Y-link’
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