IN order to design, improve performance, simulate the behavior

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1 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 Bond Graph Modeling o Mechanical Dynamics o an Excavator or Hydraulic System Analysis and Design Mutuku Muvengei and John Kihiu, Abstract This paper ocuses on the development o bond graph dynamic model o the mechanical dynamics o an excavating mechanism previously designed to be used with small tractors, which are abricated in the Engineering Workshops o Jomo Kenyatta University o Agriculture and Technology. To develop a mechanical dynamics model o the manipulator, orward recursive equations similar to those applied in iterative Newton-Euler method were used to obtain kinematic relationships between the time rates o joint variables and the generalized cartesian velocities or the centroids o the links. Representing the obtained kinematic relationships in bondgraphic orm, while considering the link weights and momenta as the elements led to a detailed bond graph model o the manipulator. The bond graph method was ound to reduce signiicantly the number o recursive computations perormed on a DOF manipulator or a mechanical dynamic model to result, hence indicating that bond graph method is more computationally eicient than the Newton-Euler method in developing dynamic models o DOF planar manipulators. The model was veriied by comparing the joint torque expressions o a two link planar manipulator to those obtained using Newton- Euler and Lagrangian methods as analyzed in robotic textbooks. The expressions were ound to agree indicating that the model captures the aspects o rigid body dynamics o the manipulator. Based on the model developed, actuator sizing and valve sizing methodologies were developed and used to obtain the optimal sizes o the pistons and spool valve ports respectively. It was ound that using the pump with the sized low rate capacity, the engine o the tractor is able to power the excavating mechanism in digging a sandy-loom soil. Keywords Actuators, bond graphs, inverse dynamics, recursive equations, quintic polynomial trajectory. I. INTRODUCTION IN order to design, improve perormance, simulate the behavior and inally control a system or plant, it is necessary to obtain it s dynamics. To develop the dynamics o a manipulator, a kinematic model o the manipulator is required irst. The kinematic modeling is done irst by attaching coordinate rames to every link. The usual convention applied to attach rames in the links o a manipulator is the Denavit-Hartenberg procedure []. The dynamics o a manipulator can be modeled using various methods namely; Newton-Euler ormulation, Lagrangian ormulation, Kane s method, and others [], []. The three methods; Newton-Euler, Lagrangian and Kane s methods tend to hide the physical interactions between the M. Muvengei is with the Department o Mechanical Engineering, Jomo Kenyatta University o Agriculture and Technology, P.O BOX - Kenya (Tel: +7, mmuvengei@eng.jkuat.ac.ke.) J. Kihiu is with the Department o Mechanical Engineering, Jomo Kenyatta University o Agriculture and Technology, P.O BOX - Kenya ( kihiusan@yahoo.com) elements involved []. The relatively new bond graph modeling technique, has been proposed to successully model the dynamics o manipulators and mechanisms. Since bond graph method is based on the interaction o power between elements, it can be used to model multi-energy domains also, or example the actuator system o the manipulator which may be electrical, pneumatic, hydraulic or mechanical. Once the bond graph model is ready, the system equations can be derived rom it algorithmically in a systematic manner. This process is usually automated using appropriate sotwares such as ENPORT, CAMP-G, TUTSIM -SIM, SYMBOLS, etc which support bond graphs. For mechanical manipulators and mechanisms, the bond graph model can be developed based on kinematic relationships between the time rates o joint variables and the generalized cartesian velocities (translational and angular velocities). It is not necessary to have higher order time rates o variables involved, that is translational and angular accelerations. The concept o bond graphs was originated by Paynter []. The idea was urther developed by Karnopp and Rosenberg in their textbooks [] [8], such that it could be used in practice. By means o the ormulation by Breedveld [9] o a ramework based on thermodynamics, bond graph model description evolved to a systems theory. More inormation about bond graphs can be ound in [] []. The Bond graph method can be used to obtain more intricate inormation such as the power required to drive each joint actuator, or the power interaction at the interace with the environment. Such inormation can also be used to study the stability o the manipulator system during contact interaction with the environment. Modiications and additions to the system can be easily incorporated by connecting suitable bondgraphic sub-systems to its existing bond graph. In this paper, the mechanical dynamics o the excavating manipulator designed in [] to be used with the small tractors (abricated in the Engineering Workshops o Jomo Kenyatta University o Agriculture and Technology) in digging mediumheight trenches or small scale armers, is modeled using the bond graph method. The excavating manipulator is shown in Fig.. Inverse dynamics is perormed on the developed dynamic model or purposes o analyzing the hydraulic system design. In this work, orward recursive equations or motion o manipulators similar to those used in Newton-Euler method are used to derive the kinematic relationships between the time rates o joint variables, and the generalized cartesian velocities 8

2 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 Fig.. Schematic drawing o a part o assembled excavator (translational and angular velocities) o mass centers o the links. These kinematic relations are urther used or graphical representation o the system dynamics using Bond graphs. II. MODELING THE MANIPULATOR DYNAMICS Assuming that; the inertial eects o cylinders and their pistons are negligibly small compared to those o manipulator links, the hydraulic cylinders transmit axial orces only, the revolute joints have no riction, and all the links and supports are rigid, a bond graph model representing the mechanical dynamics o the excavating manipulator was developed. A. Kinematic Analysis and Forward Recursion Kinematic analysis was perormed on the excavating manipulator to relate the translational velocities o the center o masses o the links (v Gi ) to the time rates o the joint variables ( θ i ), or i =,,. The choice o center o mass velocities or rigid bodies leads to a highly systematic approach or constructing bond graphs and is recommended [7]. The homogeneous transormation matrices () to () were obtained by irst attaching world coordinate rames to the three links as shown Fig. by using Denavit-Hartenberg procedure as described in []. Fig.. O O, A y y E I x B G y F Coordinate System assignment or excavator. A () = x O,C G x J L O, D O, N cos θ sin θ a cos θ sin θ cos θ a sin θ x K G y G y x () A () = A () = A () = cos θ sin θ a cos θ sin θ cos θ a sin θ cos θ sin θ a cos θ sin θ cos θ a sin θ cos θ sin θ a cos θ sin θ cos θ a sin θ () () () Kinematic relationships between the translational velocities o the center o masses o the links (v Gi ) to the time rates o the joint variables ( θ i ) (where i =,, ) can be obtained by using orward recursive equations which were proposed by Luh et al. []. The translational velocity vgi i o the center o mass o the i th link as speciied in the i th coordinate rame is given recursively by; v (i) Gi = v(i) i + ω(i) i (P (i) Gi P (i) i ) () Where P (i) Gi is the vector rom the origin o the base coordinate system to the center o mass o the i th rame as expressed in the i th coordinate system. is the vector rom the origin o the base coordinate system to the origin o the i th coordinate system as expressed in the i th coordinate system. is the rotational velocity o link i as speciied in the i th coordinate rame, and is given recursively as; P (i) i ω (i) i v (i) i ω (i) i = R (i ) i ω (i ) i + `Z q i () where R (i ) i is the rotational matrix relating two adjacent rames and is obtained rom the respective homogeneous transormation matrix. `Z = q i = θ i or revolute joints. is the translational velocity o the origin o the i th link coordinate rame as expressed in the i th coordinate rame and is given recursively as, v (i) i where P (i) i = R (i ) i v (i ) i + ω(i) i (P (i) i P (i) i ) (7) is the vector rom the origin o the base coordinate system to the origin o the (i ) th coordinate system as expressed in the i th coordinate system. is the vector rom the origin o the base coordinate system to the origin o the i th coordinate system as expressed in the i th coordinate system. P (i) i 9

3 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 B. Modeling the Bucket Digging Force A model that accounts or the material being retained in the bucket, which was developed by Cannon [] using orce equilibrium and undamental earthmoving equation in soil mechanics was applied in this study to determine the orce F exerted by the excavator bucket to the soil. From Fig., the orce F is given in (8). z F α x δ ρ Q d W W β φ cl R remain constant throughout the digging process. The critical value o the cutting angle is given by [7], α c = ( ) π σ sin (sin σ sin ρ) () The soil-tool orce F is assumed to be applied at the cutting edge o the bucket. From the Newton s third law o motion, the soil applies an opposite and equal reaction orce at the bucket, which can be resolved to a normal and tangential orce components as shown in Fig. G D,O c a L t y x Fig.. bucket. Wedge model that accounts or the material being retained in the b F N N,O F = d wγgn w + cwdn c + V s γgn q (8) V s is the swept volume, Q is the surcharge, W is the weight o the material above the bucket, W is the weight o the rest o material in the wedge, L t is the length o the tool, L is the length o the ailure surace, R is the orce o the soil resisting the moving o the wedge, F is the orce exerted by the tool on the wedge, c a is the adhesion between the soil and the blade, c is the cohesiveness o the soil media, β is the ailure surace angle (slip angle), α is the surace terrain slope (cutting angle), φ is the soil-soil riction angle, ρ is the rake angle o the tool relative to the soil surace, σ is the soil-tool riction angle, d is the depth o the bucket tool perpendicular to the soil surace, w is the width o the bucket, γ is the bulk density o the soil media and g is the gravitational acceleration. N w, N c N q are N-actors which depend on: the soil s rictional strength, the bucket tool geometry and soil-to-tool strength properties, and are given by the ollowing equations. ( )( ) cot β tan α cos α +sinαcot(β + φ) N w = ( ) (9) cos(ρ + σ)+sin(ρ + σ) cot(β + φ) +cotβcot(β + φ) N c = () cos(ρ + σ)+sin(ρ + σ) cot(β + φ) cos α +sinαcot(β + φ) N q = () cos(ρ + σ)+sin(ρ + σ) cot(β + φ) Equations (8) to () show that the magnitude o the digging orce depends on many actors such as the cutting angle, speciic resistance to cutting, volume o the bucket, amount o the the material ripped into the bucket and the volume o the material surcharged. These actors are always varying during the bucket digging operation and indicate complicated interactions o the bucket and the soil, hence making modeling o the bucket digging orce throughout the digging process a complex and bulk process. In this study, a simpliied model is presented by considering the situation o critical orce, and then assuming the orce to Fig.. Bucket digging orce at the tip. x The horizontal and vertical components o the bucket reaction orce are given as; F x = (F T cos θ b F N sin θ b ) cos(θ + θ + θ ) +(F T sin θ b + F N cos θ b )sin(θ + θ + θ ) () F y = (F T cos θ b F N sin θ b )sin(θ + θ + θ ) (F T sin θ b + F N cos θ b ) cos(θ + θ + θ ) () These orces are included at the translational velocity o the origin o the th link coordinate rame but reerenced to the base coordinate rame. C. Overall Bond Graph Model o the Manipulator The kinematic relationship obtained rom () or each link was represented in bond-graphic orm while considering the weights and momenta o the links as the bond graph elements. All the bond graph sub-models were then assembled to an overall non causal bond graph o the manipulator as shown in Fig.. Where r = L GO sin(θ σ ) L OO sin θ r = L GO cos(θ σ )+L OO cos θ r = L OO sin θ + L GO sin(θ + θ σ ) L OO sin(θ + θ ) r = L GO sin(θ + θ σ )+L OO sin(θ + θ ) r = L OO cos θ L GO cos(θ + θ σ )+ L OO cos(θ + θ ) r = L GO cos(θ + θ σ )+L OO cos(θ + θ ) r 7 = L OO sin θ L OO sin(θ + θ ) L OO sin(θ + θ + θ )+L GO sin(θ + θ + θ σ ) b FT y

4 G y International Journal o Mechanical, Industrial and Aerospace Engineering : 9 I : M I : M : : vg x vg y g I : J MTF : r MTF : r I : M MTF MTF : r : : r : vg y g MSE : F x MTF : r MTF : r 7 I : v O x : MTF I vg x : J MTF : r I MTF : r 8 : r : M MTF MTF : r : r : MTF : r MTF : r 9 I : M : vg x MTF : r I : J MTF : r 7 MTF : r : MTF : r I : M : v g MTF : r 8 : v MSE : Fy O y Fig.. A non causal bond graph model representing the mechanical dynamics o the manipulator. r 8 = L OO sin(θ + θ ) L OO sin(θ + θ + θ )+ L GO sin(θ + θ + θ σ ) r 9 = L OO sin(θ + θ + θ )+L GO sin(θ + θ + θ σ ) r = L OO cos θ + L OO cos(θ + θ ) + L OO cos(θ + θ + θ ) L GO cos(θ + θ + θ σ ) r = L OO cos(θ + θ )+L OO cos(θ + θ + θ ) L GO cos(θ + θ + θ σ ) r = L OO cos(θ + θ + θ ) L GO cos(θ + θ + θ σ ) r = L OO sin θ L OO sin(θ + θ ) L OO sin(θ + θ + θ ) r = L OO sin(θ + θ ) L OO sin(θ + θ + θ ) r = L OO sin(θ + θ + θ ) r = L OO cos θ + L OO cos(θ + θ ) + L OO cos(θ + θ + θ ) r 7 = L OO cos(θ + θ )+L OO cos(θ + θ + θ ) r 8 = L OO cos(θ + θ + θ ) D. Checking the Model One way to check the bond graph model developed in Fig. is to compare results with those available in the literature. A two link manipulator shown in Fig., moving in a ree space, and whose links are uniorm and o equal lengths was considered. Such a problem is studied using Newton-Euler, Lagrangian, and d Alembert s methods in the standard robotic textbooks [], []. All the rotation axes at the joints are along the z-axis normal to the paper surace. Let; L OO = L OO = l L GO = L GO = l σ = σ = The bond graph model o the two link planar manipulator can be represented as shown in Fig. 7; I : M I : M : vg x I : M : vg y g MTF : r MTF : r g SE : : I : M : vg x : vg y MTF : r MTF : r I : J MSE : F x MTF : r 7 MTF : r MTF : r 9 I : J : : vo y vo x MTF : r 8 MTF : r MSE : F y SE : : MTF : r Fig. 7. A non causal bond graph model representing the mechanical dynamics o a -link manipulator. Where, r = l sin θ r = l cos θ r = l sin θ l sin(θ + θ ) r = l sin(θ + θ ) r = l cos θ + l cos(θ + θ ) r = l cos(θ + θ ) r 7 = l sin θ l sin(θ + θ ) r 8 = l sin(θ + θ ) r 9 = l cos θ + l cos(θ + θ ) r = l cos(θ + θ ) The joint torques applied to each o the joint by the respective actuator can be obtained systematically rom the bond graph using the constitutive relations, and noting that F x = F y =since the manipulator is moving in ree space. The external torque applied to move link can be obtained rom the bond graph as; τ = m l θ + m l θ + m l θ cos θ + m l θ sin θ + m gl cos(θ + θ ) () The external torque applied to move link can be obtained rom the bond graph as; τ = m l θ + m l θ + m l θ + m l θ cos θ + m l θ cos θ m l θ θ sin θ m l θ sin θ + m gl cos θ + m gl cos θ + m g cos(θ + θ )() Fig.. A two link planar manipulator. The equations o external torque given in () and () correspond to those obtained using Newton-Euler and Lagrangian

5 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 methods or the same planar manipulator, as illustrated in [], []. This indicates that, the model developed captures the essential aspects o rigid body dynamics o the manipulator. III. HYDRAULIC SYSTEM ANALYSIS AND DESIGN USING INVERSE DYNAMICS In inverse dynamics, the generalized joint torques are computed given the desired joint trajectories. The joint trajectories are obtained through the trajectory planning schemes which generally interpolate or approximate the desired manipulator path by a class o polynomial unctions and generates a sequence o time-based set-points or the manipulator rom the initial position and orientation to its destination []. Quintic trajectory is used to size the spool valves and the pistons o the hydraulic cylinders, and also to check the total power required when the bucket o the manipulator is digging a sandy-loom soil. As described in [], all the cylinders will be considered to be extending simultaneously during the digging operation. Model parameters are needed to run simulations. Parameters like lengths, masses and angles were ound rom design drawings and trigonometric calculations. But parameters like the location o a center o mass, link moments o inertia, products o inertia o a link could not be estimated rom blueprints. Auto CAD with Advanced Modeling Extension Package was used to estimate the mass properties o all the links and the locations o center o masses. A. Trajectory Planning In a typical trajectory, all joints move simultaneously. For the typical trajectory selected here, the boom, arm, and bucket links move rom their minimum to maximum positions and all joints start and inish moving at the same time, although dierent time limits can be programmed. Three common trajectories namely, trapezoidal trajectory, cubic polynomial trajectory, and quintic polynomial trajectory have been previously applied in trajectory planning or hydraulic manipulators. Sarkar [8] used the three trajectories to size the valves and power requirement or an articulated orestry machine. Among the three methods, the quintic polynomial trajectory has advantage in that; the velocity trajectory is smooth unlike in trapezoidal trajectory whose velocity proile has discontinuities where the link motion starts to settle at a constant velocity, and where the link starts to decelerate. the acceleration proile has values equal to zero at starting and inishing times o the trajectory, unlike in the other trajectories where the acceleration values at the start and inal times have non zero values. Thereore the trajectory to be adopted in this work is the quintic polynomial trajectory which is given by; x(t) =a + a t + a t + a t + a t + a t (7) The desired boundary conditions are; x t= = x min, x t=t = x max, ẋ t= =, ẋ t=t =, ẍ t= = and ẍ t=t =. By taking the irst and second derivatives o (7) and satisying these boundary conditions, the coeicients o the polynomial can be obtained and substituted in the polynomial equation (7) to get the required displacement, velocity and acceleration trajectories as; x(t)=x min + (x max x min ) t t + (x min x max ) t t + (x max x min ) t t (8) ẋ(t)= (x max x min ) t t + (x min x max ) t t + (x max x min ) t t (9) ẍ(t)= (x max x min ) t t + 8(x min x max ) t t + (x max x min ) t t () B. Simulink Model or the Inverse Dynamics o the do System In Fig. 8, the complete Simulink model o the inverse dynamics o do excavating manipulator is shown. This model is the test-bed or all inverse dynamics simulations done on the system. The whole system is run in the same time rame, and as a result the outputs must match the generated trajectories. joint_traje joint trajectory generator Fig. 8. em thetas To Workspace joint traj torques To Workspace emu joint torque inverse dynamics piston traj theta traj J joint to piston level emu cylinder_orces To Workspace cyl orce piston traj The overall Simulink block or the inverse dynamics emu u u u pressure drops Hydraulics power To Workspace low rates pressure_drops To Workspace low_rates To Workspace7 The joint trajectory generator block produces the desired boom, arm and bucket quintic polynomial trajectories, that is, the angular displacements, angular velocities and angular accelerations o the links according to (8)-(). The three signals rom the joint trajectory generator block each containing the corresponding signals or the boom, arm and bucket link are ed into the inverse dynamics block to compute the joint torques or the bucket, arm and boom links respectively. The joint torque proiles when the bucket is digging a sandy-loom soil is shown in Fig. 9. C. Power Proiles The dynamic model obtained also permits either sizing o the system power supply or checking whether the desired

6 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 Boom joint Torque (Nm) 8 (a) 8 Arm Joint Torque (Nm) - (b) - 8 Bucket Joint Torque (Nm) - - (c) - 8 Fig. 9. The torque proiles at manipulator joints, when the bucket is digging a sandy-loom soil; (a) Boom joint torque (b) Arm joint torque (c) Bucket joint torque manipulator trajectory can be ollowed without exceeding the power capacity o the supply. The power required or the boom, arm and bucket motion is respectively given as; P bo = τ bo θ () P a = τ a θ () P bu = τ bu θ () Where τ is the torque required to move a link at an angular velocity o θ. The total power required or a given trajectory can be obtained by; P total = P bo + P a + P bu () assuming there are no power losses. The power requirement proiles or the manipulator are given in Fig. when the bucket is digging a sandy-loom soil. Boom link Power (kw) Bucket link Power (kw)... (a) (c) -. 8 Arm link Power (kw) (b) -. 8 Total Power (kw) (d) 8 Fig.. (a) Boom joint power (b) Arm joint power (c) Bucket joint power (d) Total joints power ;when the bucket is digging a sandy-loom soil and at the initial cycle times The peak value or the total power consumption when the bucket is digging a sandy-loom soil is shown to be approximately kw (.7hp). This value is greater than the power rating o the engine (.hp) which is to drive the hydraulic pump, implying that, the digging operation cannot be achieved under the given link trajectories without exceeding the power capacity o the prime mover. The power consumption as seen in ()-() depends on the joint torque requirements and the angular velocities o the links. Two options are available to reduce the total power requirement o the manipulator. These are; Reduce the joint torque requirements or the manipulator, by reducing the mass properties o the links and/or reducing the orce exerted to the ground by the bucket. Increase the cycle times o the link trajectories. This means that the pump low rate capacity will be reduced. In this work, the second option o increasing the cycle times o the link trajectories was used. Generally, the cycle time has a direct impact on the low rate requirements relative to the link/actuator motion requirements, and on the power requirement o the manipulator. As seen in (9) increasing the cycle time t will decrease the link velocities, and this will subsequently reduce the power requirement and also the low rates requirement. The optimum cycle time necessary to ensure that the power requirement or the manipulator motion when the bucket is digging a sandy-loom soil is reduced was ound to be s. Fig. shows the new power requirement proiles or the manipulator. Boom Link Power (kw) Bucket Link Power (kw)... (a) (c) -. 8 Arm Link Power (kw) (b) - 8 Total Power (kw) (d) 8 Fig.. (a) Boom joint power (b) Arm joint power (c) Bucket joint power (d) Total joints power ;when the manipulator is digging a sandy-loom soil and at the optimal cycle times D. Sizing the Hydraulic Actuators and Valves based on Inverse Dynamics An important application o the inverse dynamic modeling o hydro-mechanical systems is the sizing o hydraulic components. In this section, the optimal sizes o the actuator pistons as well as the optimal sizes o the spool valve oriice ports o the boom, arm, and bucket cylinders are determined. For this purpose, the pressure drop proiles across the cylinders and valves need to be plotted rom the inverse dynamics. ) Pressure Drop Proiles: The hydraulic orces required to produce the torques necessary or manipulator motion results in corresponding pressure drops across the hydraulic cylinders. The expressions or the pressure drop in the boom cylinder,

7 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 Δp bo, arm cylinder, Δp a, and bucket cylinder, Δp bu are approximated using equations below; Δp bo Δp a Δp bu = F cy bo A pbo () = F cy a A pa () = F cy bu A pbu (7) The cylinder orces are related to the joint torques by the manipulator jacobians which or the manipulator in consideration are derived in [9]. The pressure drop proiles across the cylinders when the bucket is digging a sandy-loom soil are shown in Fig.. x (a) x (b) x 7 (c) - 8 Fig.. The Pressure drop proiles across the cylinders, when the bucket is digging a sandy-loom soil and at original piston diameters; (a) Boom cylinder (b) Arm cylinder (c) Bucket cylinder Neglecting line pressure drops, the pressure drops at the corresponding valves are approximated as; Δp bov = p s Δp bo (8) Δp av = p s Δp a (9) Δp buv = p s Δp bu () Where p s is the operating pressure. The pressure drop proiles at the valves or the case when the bucket is digging a sandyloom soil is shown in Fig.. I Δp max <p s, then this implies that the supply pressure is not used maximally and hence there is no need o such a high pump pressure since the cylinder is oversized. I Δp max >p s, then the pressure drop across the valve becomes negative and this will result to a negative low rate through the valve. This is not possible practically and implies that the cylinder is undersized. As seen in Fig., the maximum pressure drops across the cylinders (points with asterisks) are less than the supply pressure o p s =.MPa. This is also shown in Fig. where at the point o maximum pressure drop or all the cylinders (points with asterisks), the pressure drops across the corresponding valves do not equal to zero. Thereore it can be concluded that the boom, arm and bucket cylinders are oversized. The optimal cylinder piston sizes were determined by tuning the piston sizes until that instant when the maximum pressure drops equaled the supply pressure or all the cylinders. These values were ound to be precisely.mm,.mm and.9mm or the boom, arm and bucket cylinders respectively, as shown in Fig.. The optimal piston diameters values o the cylinders were rounded to the next imperial values which are available in the market as 7.mm ( in),.8mm (in) and.mm ( in) or the boom, arm and bucket cylinders respectively. Pressure drop (MPa). (a) p MAXbo p Vbo -. 7 Piston diameter (mm) (Boom cylinder) Pressure drop (MPa). (b) p MAXa p Va -. 7 Piston diameter (mm) (Arm cylinder) Pressure drop (MPa). (c) p MAXbu p Vbu -.9 Piston diameter (mm) (Bucket cylinder) Fig.. The pressure drops versus piston diameters; (a) Boom cylinder (b)arm cylinder (c) Bucket cylinder x (a) 8 8. x 7 (b) x 7 (c) Fig.. The Pressure drop proiles across the cylinder valves, when the bucket is digging a sandy-loom soil and at original piston diameters; (a) Boom cylinder valve (b) Arm cylinder valve (c) Bucket cylinder valve ) Actuator Sizing: As shown in ()- (7) the cross sectional area o the piston o a cylinder determines the pressure drop across the cylinder during the working stroke, which is considered to be the extension stroke. And rom (8)-() the maximum possible pressure drop (Δp max ) across a given cylinder should be equal to the supply pressure (p s ), that is, Δp max = p s () ) Valve Sizing: A valve is properly sized when it can supply the demanded low at the required pressure drop across it. Thereore to size a valve, low and pressure requirements must be obtained as a unction o time or a given task. Obviously, the task becomes more demanding when the manipulator s bucket is digging a trench. The low through the valves or the three actuators is obtained rom the ollowing equations, Q bo = A pbo ẋ pbo () Q a = A pa ẋ pa () Q bu = A pbu ẋ pbu () where Q is the low to the cylinder, A p is the average area o the cylinder piston, and ẋ p is the velocity o the cylinder piston. The pressure drops across the valves is obtained rom (8)- (). These equations can be used to plot valve low versus valve pressure drop or the desired end-point trajectories.

8 International Journal o Mechanical, Industrial and Aerospace Engineering : 9 The resulting Q Δp curve should lie below the valve pressure-low characteristics at ull valve opening, Q v Δp v, typically a curve described by a relationship o the orm; Q v = C d A O ρ Δp v () Flow rate(m /s) 8 x (a) - Q - p v v Flow rate (m /s) 8 x (b) - Q v - p v Flow rate (m /s) 8 x (c) - Q v - p v Where Q v is the low rate through a valve, Δp v is the pressure drop across the valve, C d is the discharge coeicient, ρ is the luid density and A O is the area o the oriice opening. I Q Δp curve does not lie below the valve pressure-low characteristic curve at ull valve opening, then the pressure drop across the valve is less since the pressure drop across the actuator is large. Thereore, the valve low rate is not able to provide the motion to the manipulator at the speciied speed at a particular operating pressure. In this case a valve o larger capacity must be speciied, or the value o the operating pressure increased. Fig., shows the typical plots o such curves when the bucket is digging a sandy-loom soil. Flow rate(m /s) x (a) - Q - p v v Pressure drop x Flow rate (m /s) x (b) - Q - p v v Pressure drop x Flow rate (m /s) x (c) - Q - p v v Pressure drop x Fig.. Pressure drops versus low curves, when the bucket is digging a sandy-loom soil and at the initial sizes o the valve s oriices; (a) Boom cylinder valve (b) Arm cylinder valve (c) Bucket cylinder valve As seen in Fig., the Q Δp curves or all the cylinder valves are ar below the valve characteristic curves, hence it can be concluded that the manipulator will be able to operate with the selected valves, although the valve oriice ports are seen to be oversized. An optimal oriice port size should ensure that the peak value o the Q Δp curve is well near the valve characteristic curve. The optimal oriice port sizes were determined by tuning the radii o the ports until that instant when the peak values o the Q Δp curves or all the valves are well below the characteristic curves. These values were ound to be precisely equal to.mm or all the cylinder valves. The optimal port diameter values o mm were rounded to the next imperial values which are available in the market as.7mm ( 8 in). The resulting pressure drop versus low curves are shown in Fig.. Pressure drop x Pressure drop x Pressure drop x Fig.. Pressure drops versus low curves, when the bucket is digging a sandy-loom soil and at the optimal sizes o the valve s oriices; (a) Boom cylinder valve (b)arm cylinder valve (c) Bucket cylinder valve IV. CONCLUSION Bond graph modeling tool has been applied to model the mechanical dynamics o an excavating manipulator which was modeled as a degree o reedom planar manipulator. This was done by applying orward recursive equations similar to those applied in iterative Newton-Euler method only to determine the centroid velocities o the links, unlike in Newton- Euler method which requires extra recursive computations to determine the centroid accelerations o the links. A dynamic model resulted ater representing the horizontal and vertical velocities o the links in bond-graphic orm, while considering the momenta and weights o the links as the bond graph elements. On the other hand, Newton-Euler method requires backward recursion to be perormed in order to obtain a dynamic model. This showed that the bond graph method reduces signiicantly the number o recursive computations required to be perormed to a manipulator or a dynamic model to result, and thereore it can be concluded that bond graph method is more computationally eicient than the Newton- Euler method in developing dynamic models o manipulators. 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