Conversion of a Hydraulic System into an Autonomous Excavator

Transcription

1 Conversion of a Hydraulic System into an Autonomous Excavator H.L.J.Kerkhof DCT Master's thesis Coach(es): Supervisor: dr.ir. W. Post prof.dr.ir. M. Steinbuch Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology Group Eindhoven, August, 2007

2 i

3 Preface After graduating at the Hogeschool 's-hertogenbosch, I decided to continue my study at the Eindhoven University of Technology. I graduated from Hogeschool 's-hertogenbosch at the division Construction Engineering and continued my study with the master course of Mechanical Engineering at the division Powertrains. I have nished my master course with a graduation project at the section of Fluid Power Transmissions. This thesis is a conclusion of the work performed during my graduation project. I would like to thank the people who gave me support and assistance. Without their help, I would not have nished this project successfully. Firstly, sincere gratitude goes to my supervisor: dr.ir. Wil Post, who introduced me to this interesting topic. Considering my background, for me this topic was a whole new eld of Mechanical Engineering to explore. At rst it was hard to understand the world of uid power, but ir. Wil Post helped me out with critical notes and by giving me the opportunity to explore. This thesis made me an engineer with a broader scope. My graduation project was a great learning process, which pointed out to be an added value at my current job at DAF Trucks Eindhoven. Also, I want to thank my colleagues at the Oil Laboratory for their great cooperation, but most of all for the pleasant working atmosphere. I have experienced my time at the hydraulics lab as pleasant, for instance the short physical exercises during the breaks. Finally, I wish to express my thanks to my family and girlfriend who have supported me during my time at the university. Especially my parents for giving me the opportunity to continue my study at the university. ii

4 iii

5 Abstract Many studies have been performed about autonomous control of an excavator. However, there is still a lot of research to do to optimize it. Autonomous control has great benets, for instance at chemical or nuclear waste disposal. An advanced hydraulic motion system is build in the Fluid Power Laboratory at the Eindhoven University of Technology. The hydraulic motion system shows great similarities to the boom-bucket construction of an excavator. The aim of this study is to convert the hydraulic motion system into a model of the boombucket construction, so it can be used for further research about autonomous controlled excavation tasks. The hydraulic motion system is analyzed and modeled thoroughly. A trajectory generator toolbox is developed to attain autonomous control. The hydraulic valves and actuators will be discussed and modeled mathematically. After discussing the hydraulics, the mechanical system which connects the two actuators will be treated. The mechanical system is regarded as two rigid bodies connected with massless beams and is assumed to be innite sti. The center-point of gravity, inertia's, forces and geometric parameters will be determined as well as the kinematical models. The parameters and models will be veried with aid of the hydraulic motion system. To completely describe the system, the dynamics of the mechanical system will be described with the Lagrangian Equation's of Motion. After describing the hydraulic valves, actuators and mechanical system, a linear and nonlinear numeric model will be developed in Matlab Simulink. The linear model will be used by the identication of the hydraulic parameters. The nonlinear model will be used to accurately model the hydraulic motion system. The model enables to simulate excavation tasks on a remote computer. One of the goals of this thesis is to let the hydraulic motion system fulll an autonomous task. Therefore, a trajectory generator toolbox will be developed. The toolbox will generate a specic trajectory based on predetermined setpoints. The trajectory consists of a path, velocity, acceleration and jerk prole. The hydraulic motion system must execute the trajectory autonomously with aid of the trajectory generator toolbox and the control system. The numeric model will be veried with aid of the hydraulic motion system. The result of this study will be a numeric model of the hydraulic motion system, which represents an excavator that can be autonomous controlled. iv

6 v

7 Nomenclature Symbol Description Unit A area of piston m 2 B viscous damping coecient N s/m C spring stiness N/m C d discharge coecient [-] C ep external leakage coecient m 3 /s/p a C ip internal leakage coecient m 3 /s/p a C r radial clearance m C v viscous friction coecient Ns/m C 01 spring stiness chamber 1 N/m C 02 spring stiness chamber 2 N/m C 0min minimum spring stiness N/m e error signal V E oil bulk modulus N/m 2 F c coulomb friction force N F f friction force N F l load force N F m net force acting on piston of tilt actuator N F p net force acting on piston of horizontal actuator N F v viscous friction force N g gravitational constant m/s 2 i input current A J m1 mass moment of inertia kgm 2 k c valve ow pressure coecient m 3 /s/p a k q valve ow gain m 3 /s/m M 1 mass of the triangular shaped beam kg M 2 mass of the linear guide kg M total mass of the load kg M m dynamic mass of the titl actuator kg M p dynamic mass of the horizontal actuator kg vi

8 Symbol Description Unit P N normalized pressure P a P s supply pressure P a P l load pressure P a P r return pressure P a P 1 pressure in chamber 1 P a P 2 pressure in chamber 2 P a q generalized coordinates Q nc nonconservative generalized external forces q l load ow m 3 /s q N nominal ow m 3 /s q 1 ow to chamber 1 m 3 /s q 2 ow to chamber 2 m 3 /s r position vector m S stroke of the cilinder m T kinetic energie N m t time sec u plant input signal V v piston velocity m/s V potential energie N m V t total cylinder volume m 3 V 1 cylinder volume chamber 1 m 3 V 2 cylinder volume chamber 2 m 3 V 01 dead volume chamber 1 m 3 V 02 dead volume chamber 2 m 3 w noise signal V x m piston displacement of the tilt actuator m x p piston displacement of the horizontal actuator m x v valve displacement m Z x x-coordinate center-point of gravity m Z y y-coordinate center-point of gravity m β dimensionless damping coecient [-] ρ density of the oil kg/m 3 ω 0 natural frequency rad/s µ absolute viscosity kg/ms vii

9 Contents Preface Abstract Nomenclature ii iv vi 1 Introduction Setup of the Hydraulic Motion System Conguration Degrees of Freedom Problem Denition Report Outline Hydraulic System Introduction Electrohydraulic Servovalves Two-Stage Electrohydraulic Servovalve with Position Feedback Mathematical Model of the Servovalve Linear Hydraulic Actuator Hydraulic Double-Acting Double Ended Rod Cylinder Mathematical Model of a Valve controlled Hydraulic Actuator Summary Mechanical System Introduction Measurement Instrumentation Transducers Data Acquisition and Analysis Control System Center-Point of Gravity Mass Moment of Inertia Forces at Static Situation Kinematical Models Forward Kinematic Model Backward Kinematic Model Verication Kinematical Models Working Space viii

10 3.7 Mathematical Model of the Mechanical System Generalized Coordinates Global Coordinate System Kinetic Energie Potential Energie Nonconservative Generalized External Forces Lagrange's Equation of Motion Summary Modeling Introduction Linear Model Frequency Response Functions Measurements Parameter Identication Nonlinear Model Actuators Friction Lagrangian Model Summary Trajectory Planning Purpose Introduction on Trajectory Generation Spaces and Domains Trajectory Generator Input Path Description Time Mapping Derivatives Optimization Results Summary Final Results Simulations Hydraulic Motion System Numeric Model Numeric Model vs. Hydraulic Motion System Summary Conclusions and Recommendations Conclusions Recommendations A Parameters 68 ix

11 B Calculation of the Kinematical Models 71 B.0.1 Forward Kinematics B.0.2 Backward Kinematics C Models 73 C.1 Numeric Matlab Simulink Model C.1.1 Controller Module C.1.2 Valve Controlled Actuator Module C.1.3 Friction Module C.1.4 Lagrangian Module D Friction Measurements 77 D.1 Measurements D.2 Normality Test E Determining the Cubic Spline Coecients 83 F Calculation of the Time Derivatives of the Path 85 F.0.1 First Order Time Derivative F.0.2 Second Order Time Derivative F.0.3 Third Order Time Derivative G Files Trajectory Generator 87 G.1 Trajectory Generator G.1.1 Constraints G.1.2 Setpoints G.1.3 Path Description G.1.4 Time Mapping G.1.5 Time Parameters G.1.6 Derivatives G.1.7 Optimization x

12 xi

13 Chapter 1 Introduction Excavators are machines used in earth work activities as delving and mining. An example of an excavator is depicted in gure 1.1. When performing excavation tasks the bucket follows a certain trajectory or path. Excavators are equipped with a boom-bucket construction. The boom-bucket construction consists of a bucket and a number of beams. The beams and the bucket are connected with hinges. They are controlled by hydraulic actuators. The excavators are usually operated by a human machinist. The machinist uses a set of levers to operate valve controlled hydraulic actuators to position the bucket. Figure 1.1: Excavator The productivity and eciency of these excavators are of great importance as well as the safety of the operator. For these reasons automatic control would be a major step forwards. It would not only increase the productivity, but also allowing for remote control. In case of chemical or nuclear waste disposal, automatic control becomes an interesting subject. Other advantages are the reducing of time and costs for training the operator and reducing 1

14 dependence on operating skills. To accurately operate a manually controlled excavator requires a lot of experience and skills. The training of a human operator usually takes about two years. When an excavator is automatic controlled it can be controlled by a less trained or skilled operator. Automatic control is important in further developments of excavation. Before automatic control can be implemented on a real excavator, it must be studied with aid of a model and simulations. A hydraulic motion system, which shows a great similarity to the boom-bucket construction of an excavator, is present in the Fluid Power Laboratory. Figure 1.2 shows the setup of this system. The aim of this study is to convert the hydraulic motion system into the boom-bucket construction of an excavator. A numeric model of the hydraulic motion system can be used to investigate automatic control by means of simulations, so the hydraulic motion system can be used for research on excavation tasks. Figure 1.2: Setup of a Hydraulic Motion System 1.1 Setup of the Hydraulic Motion System The setup of the hydraulic motion system is similar to the boom-bucket system of an excavator. The bucket of an excavator, marked with (L) in gure 1.3a, is moveable in the x/y-plane. The movement of the bucket is controlled by the human operator using a set of levers. The levers indirectly position the bucket by actuating valve controlled hydraulic actuators. The hydraulic motion system, shown in gure 1.3b, has an end-eector marked with (L), mounted on a truss beam. The truss beam is driven by two hydraulic actuators. The positions of the pistons of the actuators are controlled in a closed loop system by means of digital control. 2

15 (a) Excavator (b) Hydraulic Motion System Figure 1.3: schematic side view The end-eector can be monitored with a computer while moving in x/y-plane. The two main similarities of the excavator and the hydraulic motion system are; 1. the correspondence between the boom-bucket construction and the truss beam-end effector construction. 2. the actuation by hydraulic actuators. Part of this study is to investigate the possibility to simulate the trajectory of the bucket by the end-eector, and to let the hydraulic motion system fulll an autonomous excavation task. The rotation of the bucket around point (D), gure 1.3a, and the rotation of the cabin around the y-axle are left out of account. In the next paragraphs the construction of the hydraulic motion system will be discussed Conguration The hydraulic motion system consists of two hydraulic actuators which are connected with a triangular latticed beam, called truss beam. The setup consist of a horizontal and tilt actuator. The tilt actuator is mounted on a swivel, point (A) of gure 1.3b, to enable rotational movements. The rod of the tilt actuator is connected to the truss beam. The horizontal actuator is xed and the rod of the horizontal actuator is connected to the truss beam by a pivot on a linear guide, point (B). This construction enables movements in x/y-plane with the end-eector (L), in a to be determined operative range Degrees of Freedom The position of a body can be determined by six independent coordinates, see gure 1.4. There are three translational coordinates, depicted with (x), (y) and (z), and three rotational coordinates, depicted with (α), (β) and (θ). If an independent coordinate of a body is not constrained it is called a degree of freedom (dof). The hydraulic motion system is designed so the truss beam has 3 degrees of freedom, without being statically over-determined. A body is statically over-determined when a independent coordinate is restricted in more than one mode. The three degrees of freedom of the truss beam are the translational movements in x- and 3

16 Figure 1.4: 6 Independent Coordinates y-direction and the rotational movement (α). The three independent constrained coordinates are the translational movement in z-direction and the rotational movements (β) and (θ). The constrained coordinates are restricted by the linear guide. It is imperative to choose the joints in points (A) and (C) of gure 1.3b well-considered, to prevent the system to be over-determined. Therefore, the tilt actuator is connected with universal joints at these points. The universal joints are capable to make three dimensional movements and thus do not over-determine the system. 1.2 Problem Denition Many studies have been performed on autonomous control of an excavator, but there are still a lot of possibilities to improve it. The aim of this study is to convert the hydraulic motion system into a boom-bucket construction of an excavator, to develop a numeric model of the hydraulic motion system and to enable autonomous control. The numeric model can be used in future studies at excavation tasks. 1.3 Report Outline Before the hydraulic motion system can be converted, several aspects must be considered. It is necessary to analyse the hydraulic parts of the motion system thoroughly. The methode and results of that analysis are described in chapter 2. The dierent components will be addressed and modeled in a linear and nonlinear fashion. Chapter 3 covers the measurement instrumentation of the motion system, the kinematic relationships of the complete motion system and the mechanical dynamics of the truss beam. The properties and relations should correspond closely with those of an excavator. Several problems might arise when performing an excavating task. For example the bucket could be overloaded, the desired excavating time could not be reached, obstacles could appear etc. To trace or investigate such problems a numeric model of an excavator should be developed. Chapter 4 covers the parameter identication and the modeling of the motion system. Additional aspects like friction and constraints will be discussed. 4

17 The bucket has to follow a prescribed trajectory when performing an autonomous controlled excavating task. To control the excavator the trajectory must be generated in advance. In Chapter 5 a trajectory generator is developed and discussed. This generator will determine a third order continuous trajectory by means of predened setpoints. The generated trajectory will be entered as reference trajectory at the hydraulic motion system and at the numeric model. The results of both simulations will be discussed and compared in chapter 6 Chapter 7 covers the nal conclusions of this thesis and the recommendations for any possible further studies. 5

18 Chapter 2 Hydraulic System 2.1 Introduction The hydraulic motion system is digitally controlled with aid of a computer. The actual position of the piston is compared with the demanded position. If the actual position is dierent from the demanded position the error signal sends an electric current to the electrohydraulic servovalves. The electric current results in a valve spool displacement which results in a volume ow to or from the actuator. The position of the piston is determined by the volume ow to or from the actuator. A schematic overview of the complete setup is given in gure 2.1. Figure 2.1: Schematic Overview 6

19 The actual position is measured by a transducer and compared with the demanded position. The hydraulic motion system is controlled with closed loop position feedback, the control system will be discussed in section 3.2. Before analyzing the complete motion system, it is important to understand the behavior of the hydraulic subsystems. The subsystems can be separated into electrohydraulic servovalves and hydraulic actuators. The type of servovalves used in the setup, the working principle and the mathematical model will be discussed in section 2.2. In section 2.3 dierent types of linear actuators are discussed and the mathematical model of a valve controlled actuator. Two separate hydraulic pumps are used for the hydraulic power supply. The type of pumps and their performances are not considerate because it is irrelevant for the system. The only aspect taken into consideration is the supply pressure which is set to 70 bar. The mechanical system which connects the two hydraulic actuators will be investigated in chapter Electrohydraulic Servovalves According to Merritt [7], electric devices are ideally suited for measuring signals, while hydraulic actuators are ideally suited for power output. An electrohydraulic servovalve can be considerate as the interface between those elds. The servovalve converts low power electric signals into motion of the spool in the valve, which controls the ow to and from a hydraulic actuator. The basic function of a servovalve is to control a high power output using a low power input. There are dierent types of servovalves. The servovalves, used in this setup, are two-stage electrohydraulic servo valves with internal position feedback Two-Stage Electrohydraulic Servovalve with Position Feedback An example of a two-stage electrohydraulic servovalve is shown in gure 2.2. Figure 2.2: Servovalve with Position Feedback 7

20 As the name indicates, the spool consists of two stages. The main part of the rst stage is the torque motor, consisting of a armature mounted on a pivot point (A) and the main part of the second stage is the spool. Spool position feedback is obtained by use of a feedback wire. Deecting the wire produces a counteracting torque, which result in position feedback. The wire connects the spool and torque motor to provide a torque balance. Working Principle As shown in gure 2.1, the input of the electrohydraulic servovale is an electric current. This electric signal is converted into a mechanical signal by a torque motor. The torque motor consist of an armature mounted on a pivot (A), which is suspended in the air gap of a magnetic eld produced by a pair of permanent magnets, see gure 2.3. When an electric current is owing through the two coils, the armature ends become polarized and are attracted to one of the magnet pole pieces and repelled by the other, generating a torque. Figure 2.3: Servovalve Torque Motor with Flapper and Feedback Wire The torque causes a rotation angle of the armature and apper around pivot point A. The rotation angle results in a transposition of the ball-end of the apper, changing the ow balance between two opposing nozzles, see gure 2.2. The excursion of the apper decreases the ow from one nozzle and increases the ow from the other nozzle. This results in respectively a lower pressure side and a higher pressure side between apper and nozzles. The apper and nozzles can be considered as a half wheatstone bridge. As shown in gure 2.2 the nozzles are connected to the chambers at the left and right side of the spool. The pressure dierence between the nozzles cause a pressure dierence over those chambers, resulting in a spool displacement. The spool displacement forces the ball-end of the feedback wire to displace. The feedback wire deects and generates a counteracting torque to the torque motor. The servovalve reaches an equilibrium state, when the feedback torque balances the torque produced by the magnetic forces. The armature is centered and the spool is stationary but deected to one side. The electric signal is now converted into a mechanical displacement. The displacement of the spool opens valve ports allowing oil to ow from and to the actuator. 8

21 2.2.2 Mathematical Model of the Servovalve A servovalve is a complex device which exhibits high order nonlinear response. Knowledge of a large number of internal valve parameters is required to formulate an accurate mathematical model. To be able to describe the dynamical behavior without knowledge of these parameters, amplitude-frequency responses can be used. It is assumed that the dynamics of the valve can be described by transfer functions, see Merritt [7] and Viersma [14]. The dynamic characteristics of the servovalves are usually provided by the supplier in the form of a transfer function. One of the risks of assigning a transfer function is a signicant deviation at higher frequencies because the valve shows a high order non linear response. According to Poley [8], the servovalve is usually not the primary dynamic element of the hydraulic motion system. This means that it is sucient to reproduce the valve response in a relative low frequency range compared to its bandwidth. This assumption depends on the dynamic characteristics of the mechanical system. The assumption is valid if the bandwidth of the valve and the bandwidth of the mechanical system agree to the following condition: ω 0(actuator) ω 0(valve) 1 (2.1) For this particular system, it will be sucient to reproduce the valve response in a low frequency range even without investigating the bandwidths. Since the hydraulic motion system will be converted into a model of an excavator and obviously excavation tasks are not performed in high frequency range. Figure 2.4: Transfer Functions Servovalves 9

22 An overview of the dynamic characteristics of the servovalves used in the setup are shown in gure 2.4. The servovalve used for controlling the horizontal actuator is the MTS Model and the servovalve used for controlling the tilt actuator is the MTS Model The characteristics of gure 2.4 can be modeled as a second order transfer function: x v i = q N c 1 s 2 + c 2 s + 1, (2.2) With c 1 = 1 and c ω0 2 2 = 2 β ω 0 where ω 0 is the natural frequency and β the damping coecient. q N indicates the nominal ow. The natural frequency and the nominal ow can be obtained from the characteristics shown in gure 2.4. The estimated values are shown in table 2.1. Table 2.1: Estimated Parameter Values Actuator Valve q N [lt/min] ω 0 [rad/s] Horizontal MTS Tilt MTS The natural frequency is the frequency where the slope of the characteristics at low frequency intersect with the slope of the characteristics at high frequencies. The natural frequency of the MTS Model is approximately 150 [Hz], and the natural frequency of the MTS Model is approximately 90 [Hz]. The nominal ow of servovalves depend on the size of the valve, and can be directly determined from the characteristics at gure 2.4. The nominal ow is indicated in gallons per minute, converting to SI-units results in a nominal ow of q N = [lt/min] for the MTS Model and q N = [lt/min] for the MTS Model The hydraulic motion system will be converted into a model of an excavator and therefore only used at low frequencies. Figure 2.4 shows that the dynamic response of the servovalves can be neglected when using at frequencies smaller than the natural frequency. It can be concluded that the servovalve response can be neglected for this particular situation. The only parameter of importance is the nominal ow q N. 2.3 Linear Hydraulic Actuator Servovalves are used for controlling ows and hydraulic actuators transform ows into mechanical velocity or rotational speed. Dierent types of hydraulic actuators exist. They can be classied as either linear or rotary actuators. The actuators used in this setup are linear. 10

23 There are several types of linear actuators, the most common types are depicted in gure 2.5. Figure 2.5: Basic Actuator Types Actuator (A) is a single acting single rod actuator. The actuator can move in one direction under the action of supplied uid. It requires an additional force to return the piston. These actuators are unsuitable for closed loop control. Actuator (B) and (C) return the piston under the action of a uid ow and are called double acting. Actuator (B) is called a double-acting single ended rod cylinder. It has a rod connected to one side of the piston. The rod has a signicant smaller diameter than the piston. Actuator (C) is called a double-acting double ended rod cylinder. It has rods on both sides of the piston. This cylinder is also referred as symmetric cylinder. The hydraulic actuators used in this setup are hydraulic double-acting double ended cylinders Hydraulic Double-Acting Double Ended Rod Cylinder There are two actuators used in the hydraulic motion system. Both actuators are symmetrical cylinders. The positions of the pistons of the actuators are measured with position transducers and the pressure dierences between the two chambers of the actuator are measured with pressure transducers. The tilt actuator is a cylinder of the MTS series This cylinder is specially designed for long life, low friction and exceptional performance in high frequency and low displacement applications. Since the cylinder is equipped with hydraulic cushions it has an eective stroke (284 mm) and a dynamic stroke (250 mm). The hydraulic cushions protect the actuator from adverse eect of accidental high velocity impacts between the piston and end caps. The servovalve and position transducer are integrated into the actuator. The cylinder is standard equipped with a PTFE seal which has low friction. A cross section of the cylinder is shown in gure 2.6. A small amount of hydraulic uid is allowed to ow past the high pressure seal (2) for continuous bearing lubrication. Drainback line (3) returns the hydraulic uid passed by the high pressure seal back to the system drain line. The low pressure seal (1) prevents leaking of the 11

24 cylinder. Figure 2.6: MTS actuator The horizontal actuator is an actuator manufactured by Parker Hannin and is of the 2M series. The position transducer is attached externally. The cylinder has a stroke of 400 [mm]. To achieve a relative good performance, even when operating on low pressure, the common industrial Parker actuator is equipped with special low friction seals, see Steinbuch [11] Mathematical Model of a Valve controlled Hydraulic Actuator Mathematical models of hydraulic actuators have been developed over the past decades, for example by Merritt [7] and Viersma [14]. A schematic sketch of a valve controlled actuator is depicted in gure 2.7. This section contains a linear and a nonlinear mathematical model of a valve controlled hydraulic actuator. The linear model will be used for investigating performance. Many systematic design and test procedures are designed for linear modeling. Performance criteria as bandwidth are fundamental to system understanding and analysis. The nonlinear model will be developed for describing the valve controlled hydraulic actuator more precisely and for all operating points. This model can be used for research on excavation. The mathematical description regards three dierent equations; the servovalve ow equation, the continuity equation and the second law of Newton. The Servovalve Flow Equation The ow through a valve can be described with the orice equation, see equation P q = C d wx v (2.3) ρ 12

25 Figure 2.7: Piston Controlled Actuator The discharge coecient C d implies that the jet formed from the ow through the ori- ce is smaller than the theoretical jet. This deviation is caused by turbulent ow. The discharge coecient is determined experimentally. The geometry of the port of the valve is indicated by the width w. It is assumed that the geometry of the orices are symmetrical and equal on both sides, so w is equal for each orice. The valve port opening is indicated by x v. The pressure dierence over the valve ports is indicated with P and the density of the oil is indicated with ρ. The pressure dierence P over the valve ports can be expressed by P 1, P 2, P s and P r, see gure 2.7. This results in the following equation: P port1 = P s P 1 (2.4) P port2 = P 2 P r The return pressure P r is the reference pressure and is assumed to be zero. The load pressure is equal to the pressure dierence between P 1 and P 2. The pressures P 1 and 13

26 P 2 can be calculated according equation 2.5. P 1 = 1 2 P s P l (2.5) P 2 = 1 2 P s 1 2 P l The load ow can be described by substituting equations 2.4 and 2.5 in the orice equation. The leak ows of the servovalves are negligible. To add directionality to the load ow equation, the factor x v x v is included. This result in the following equation: 1 q l = C d wx v ρ (P s x v P l ) (2.6) x v The servovalve is controlled with an electric current, as indicated in gure 2.1. The relation between the valve port opening and the electric current is assumed to be linear, and can be described as follows: x v x max = i i max (2.7) where x max is the maximum valve spool displacement and i max the maximum electric current. Manufacturers of valves usually indicate the nominal ow as a parameter. The nominal ow is the ow at maximum valve opening and maximum pressure. The nominal ow is indicated with equation q N = x max C d w ρ (P s) (2.8) The load pressure varies between -Ps and Ps. The normalized pressure can be expressed as: P N = 1 P s P l (2.9) Substituting equation 2.8 and 2.9 in equation 2.6 results in the nonlinear servovalve equation. x v q l = q N (1 x v P N ) (2.10) x max x v where x v is the valve port opening. The Continuity Equation The continuity equation is regarded with the equation of state. It is applied on both chambers of the symmetric actuator. Qin Q out = dv dt + V E dp (2.11) dt 14

27 The density and temperature are assumed to be constant. Line phenomena and cavitation are assumed to be absent. The continuity equations according gure 2.7 are: q 1 C ip (P 1 P 2 ) C ep P 1 = dv 1 dt + V 1 E dp 1 dt C ip (P 1 P 2 ) C ep P 2 q 2 = dv 2 dt + V 2 E dp 2 dt (2.12) (2.13) C ip indicates he internal piston leakage coecient and C ep the external piston leakage coecient. The external leakage is very small and will be neglected. The load ow can be described as mean of the ows on both sides, see equation q l = q 1 + q 2 2 (2.14) Substituting equations 2.12 and 2.13 in equation 2.14 results in the load ow described with the continuity equation: q l = C ip P l + dv 1 dt + V 1 1 E dt dv 2 dt V 2 2 E dt 2 (2.15) Assuming the supply pressure to be constant and considering equation 2.5, the following relation can be derived: dp 1 dt dp 2 dt = 1 dp l 2 dt = 1 dp l 2 dt (2.16) The volumes on both sides of the symmetric piston are divided into a dead volume and a dynamic volume. The volumes can be described with the following equations: V 1 = V 01 + x p A (2.17) V 2 = V 02 + (S x p ) A The position of the piston is indicated with x p, see gure 2.7, and the full stroke of the actuator is indicated with S. Substituting equations 2.16 and 2.17 in equation 2.15 results in the nonlinear load ow equation [ V01 + x p A q l = C ip P l + A ẋ p + 4E + V 02 + (S x p ) A ] P l (2.18) 4E Simplifying this equation and rewriting it to the normalized pressure results in: P N = 4E [ ] q l C ip P N Aẋ p (2.19) V t P s 15

28 The Second Law of Newton The second law of Newton can be described as the summation of all acting forces. It can be written in Laplace Domain as follows: A P l = M s 2 x p + B s x p + F l (2.20) where B indicates the viscous damping coecient, M the total mass and F l the load force. In this section the hydraulic system is discussed without interaction of the mechanical system and without an external damper. Therefore the magnitude of the load force is zero and B only indicates the internal viscous type of damping. It is generally assumed that hydraulic actuators have small viscous damping so B is assumed to be negligible, see Merritt [7] and Viersma [14]. Linear Mathematical Description A transfer function will be used as a linear mathematical model of the actuator. The system equations are linearized around operating point and transformed to a transfer function. The operating point will be based on a worst-case scenario. According to article Martin [6] worst-case scenario is at equal uid volume on both sides of the actuator. Therefore, the operating point of the cylinder will be taken at centered position. V 1 = V 2 = 1 2 V t (2.21) The nonlinear servovalve ow equation can be expressed as a Taylor's expansion series. Expressing equation 2.6 as a Taylor's expansion series and neglecting the higher order terms results in the linear description shown with equation q l = δq l xv + δq l pl δx v δp l q l = k q x v k c P l (2.22) where k q indicates the ow gain coecient and k c the ow pressure coecient. For a critical center valve yields: 1 k q = C d w ρ (P s P l ) (2.23) k c = C d wx v 1 ρ (P s P l ) 2(P s P l ) (2.24) The load ow, derived in equation 2.18, can be described in Laplace Domain with the following equation: q l = C ip P l + A s x p + V t 4E s P l (2.25) The transfer function of a valve controlled hydraulic actuator can be derived by substituting three dierent equations, the load ow equation 2.25, the servovalve ow

29 and the second law of Newton equation Because the servovalve ow equation is linearized, the transfer function is only valid for small excursions around operating point. The transfer function of the valve spool displacement and piston displacement can be described with equation x p = 1 x v s c 3 c 4 s 2 + c 5 s + 1 (2.26) Where c 3 = kq A, c 4 = 1 ω 2 0 and c 5 = 2β ω 0. The damping coecient β can be described as: β = 2(c ip + k c ) M (2.27) A V t The natural frequency of a hydraulic system can be described with the following equation: C ω 0 = (2.28) M where C is the spring stiness and M is the mass. The minimal spring stiness for a symmetric actuator in centered position can be described according equation C 0min = 4EA2 V t (2.29) Substituting equation 2.29 into 2.28 results in the minimal natural frequency which can be described as: 4EA ω 0min = 2 (2.30) MV t Beside the position of the piston x p the pressure in the actuator is also a quantity which can be measured. With this quantity the forces produced by the actuator can be controlled. The transfer function from the valve port opening to the pressure dierence can be derived using the second law of Newton. The second law of newton is described with equation 2.31 F l = M x p s 2 (2.31) The transfer function of the valve spool displacement and the load pressure can be expressed as follows: P l s = c 6 x v c 4 s 2 + c 5 s + 1 (2.32) with c 6 = k qm A 2 17

30 2.4 Summary This chapter contains a detailed descriptions of the hydraulic components used in the motion system. These descriptions can be used for modeling the motion system. The mathematical description of the servovalves is based on the information supplied by the manufacturer. The hydraulic actuators can be expressed by a linear and nonlinear mathematical model. The linear description results in a third order transfer function. The transfer function is only valid for small excursions around operation point. Dierent operating conditions results in variations in the hydraulic natural frequency and the damping ratio. The lowest natural frequency is at midpoint of the stroke. The mathematical description of the hydraulic actuators is based on three equations: 1. the orice ow equation 2. the continuity equation 3. the second law of newton/force balance 18

31 Chapter 3 Mechanical System 3.1 Introduction The hydraulic motion system consists of two actuators which are connected by a mechanical system. The servo controlled actuators have been discussed in chapter 2. This chapter will focus on the properties and the geometric relations of the mechanical system. To simplify the problem, the mechanical system is modeled as two rigid bodies connected with massless beams, see gure 3.1. Figure 3.1: Kinematical Model 19

32 The rigid bodies are indicated with m 1 and m 2. The massless beams are the beams which connects the points (B), (C) and (L). The mechanical system is assumed to be innite sti. The rst mass m 1 is the mass of the truss beam and is located at the center point of gravity. The second mass m 2 is the mass of the linear guide, connecting the truss beam and the horizontal actuator. The coordinates of the center point of gravity, the forces in the operating point and the inertia's of the system will be calculated in the following sections. As described in previous chapters, the point marked with (L) should represent the bucket of an excavator. Therefore it is of great importance that the position of point (L) is described unambiguously, this will be discussed in section 3.6. There are two vector spaces for describing the position of point (L), namely Cartesian Space and Joint Space. The position in Cartesian Space is expressed in x- and y- coordinates and the position in Joint Space is expressed by the positions of the pistons of the actuators. Table 3.1 illustrates the coordinates of the vector spaces. Table 3.1: Vector Spaces Joint Space x p x m Cartesian Space x-coordinate y-coordinate Where x p is the position of the piston of the horizontal actuator and x m the position of piston of the tilt actuator. The required set of equations to transform coordinates between the two vector spaces is called, the kinematical model. The kinematical model will be described in section 3.6. The geometric parameters and the kinematical model will be veried with aid of the hydraulic motion system. Finally the Equations of Motion of the system are deduced according to Lagrange's methode, this is discussed in section 3.7. This chapter comprehends results of operations executed with the hydraulic motion, therefore the measurement instrumentation will be discussed in section Measurement Instrumentation The measurement instrumentation is classied into two groups. One group contains transducers, to measure the analogue quantities, and one group contains both hardware and software. The hardware is used to gather and convert the analogue signals and the software is used to store and process the collected data Transducers There are two types of transducers used in the analysis of the hydraulic motion system. Each servo controlled hydraulic actuator is equipped with a position and pressure 20

33 transducer. Of course position transducers are used for measuring the positions of the pistons and pressure transducers are used for measuring the pressure dierence between the two chambers of the actuator Data Acquisition and Analysis The system used to gather and process measuring signals is a so called dspace system. The dspace system comprises hardware and software. The hardware consist of a DS1103 PPC Controller Board which is specially designed for real-time simulations. The DS1103 PPC Controller Board collects and processes all gathered signals and transfers them to a personal computer. The personal computer is installed with dspace software tools, for further processing and analyzing collected data. With aid of the software tools, data analysis, visualization as well as data import and export are possible Control System The hydraulic motion system is controlled with aid of transducers, a dspace system and a personal computer. The control system of the hydrauloc motion system will be explained in this section. The reference values, in this case the positions of the pistons, are entered at the dspace realtime user interface. The realtime user interface is a graphical software tool that forms the interactive visualization interface, and is called ControlDesk. The operator has complete control of the experiments running with aid of the ControlDesk environment. The model used by the dspace System converts the entered positions of the pistons from meters to voltage. The positions (in voltage) are compared with the measured positions of the pistons (in voltage). An electric current is send to the electrohydraulic servovalves if the entered and measured position signals dier. The servovalves converts the electric signal into a spool displacement, which results in a hydraulic volume ow, see section The hydraulic volume ow drives the pistons to the desired position. The positions of the pistons are measured by transducers. These transducers send a signal (in voltage) to the dspace System, where it is further processed. The measured and the entered positions are displayed in the ControlDesk environment. 21

34 3.3 Center-Point of Gravity The mechanical system is build up from thirteen steal mechanical elements and an additional mass load at the end-eector, see gure 3.2. Each element has a centerpoint of gravity in relation to the xy-coordinate system. The following formulas are used to calculate the coordinates of the center-point of gravity of the mechanical system; P m Z x = P i X i m tot with i ɛ [1, 2,... 14] (3.1) m Z y = i Y i m tot Where m i is the mass of the i-th element and X i or Y i are the x- or y-distance from center-point of the i-th component to the coordinate system. Figure 3.2: Centre-Point of Gravity The mass of the load varies when performing an excavating task, for example the load can represent the amount of earth during a digging trajectory. This can be simulated by adjusting the input m (14) in the mass array shown in Appendix A. Altering the magnitude of the mass load automatically will lead to change of the coordinates of the center point of gravity. The calculated coordinates of the center-point of gravity for the setup are; Z = (1.026; 0.004) [m]. 22

35 3.4 Mass Moment of Inertia At section a linear model of a single hydraulic actuator was derived. The model is formulated as a transfer function, see equation The quantity M, at that transfer function, indicates the massload of a single actuator. The hydraulic motion system consists of two hydraulic actuators connected by a mechanical system, as shown in gure 3.1. The load of the actuators of the motion system is the translating/rotating truss beam. The truss beam can not be regarded as a massload for the actuators, but must be dened with dynamic masses for the horizontal and tilt actuator. At this section the load for both actuators will be described with an equivalent dynamic mass to represent the truss beam. The dynamic masses can be calculated with the mass moment of inertia of the mechanical system. The moment of inertia of a point mass rotating about a known axis is dened by: N J = m i ri 2 (3.2) i=1 The horizontal actuator experiences an inertia rotating around point A and the tilt actuator experiences an inertia rotating around point B. The dynamic mass of the horizontal actuator can be calculated according equation 3.3. M p = J p e 2 (3.3) M p = m 1 X 2 AM + m 2 e 2 e 2 M p = m 1 X 2 AM e 2 + m 2 where X AM is the distance from point (A) to the center point of gravity m 1, see gure 3.1. Length X AM can be calculated from coordinates and actual length's of elements with aid of the cosine rule, see equation 3.4. X AM = e 2 + l 2 m1 2 e l m1 cos(π θ δ) (3.4) The mass moment of inertia of the horizontal actuator is position dependant, as the length X AM alters when the position of the piston of the tilt actuator changes. The tilt actuator experiences an inertia rotating around point B, and the dynamic mass can be calculated according equation 3.5. M m = J m l 2 3 M m = m 1 l 2 m1 + m 2 0 l 2 3 (3.5) M m = m 1 l 2 m1 l

36 The dynamic mass of the tilt actuator is independent of the position, because l 3 is constant. However, it is inuenced by the length l m1 which means that the equivalent dynamic mass depends on the center point of gravity of the mechanical system and thus on the magnitude of the load mass. Equations 3.3 and 3.5 can be used to describe the equivalent dynamic masses for every possible conguration of the setup. Nevertheless, the linear model will be used mainly for simulating the hydraulic motion system when the pistons are at center position. The equivalent dynamic masses at center position (x p = 0.2 [m] and x m = [m]) are: M p = [kg] and M m = [kg]. 3.5 Forces at Static Situation In this section the occurring forces for static operating points will be discussed. These forces can be used as verication for practical tests and as validation for later models. The static situation is sketched in gure 3.3. Figure 3.3: Forces at Static Situation From equilibrium of forces and torque follow three equations, point B is chosen as rotation point. The three equations are: Fy = 0; F x = 0; M B = 0 With these equations the two actuator forces F C and F B can be calculated for static 24

37 operating points. F C = m 1 g sin θ Z x l 3 sin φ cos(θ + α + γ) + l 3 cos φ sin(θ + α + γ) F B,x = sin φ F C (3.6) The force F B,y is a reaction force from the linear guide at point B. 3.6 Kinematical Models The bucket, indicated with point (L) in gure 3.3, should be positioned at a specic location and orientation for operating an excavator. The positioning of point (L) can be accomplished by adjusting the positions of the pistons of the hydraulic actuators. The mathematical expressions that relate the orientation and position of the mechanical system to the position of the pistons are called kinematics. Kinematical models consider purely geometric relationships, dynamic eects are not taken into account. If the positions of the pistons are known, the position and the orientation of the mechanical system can be determined by the forward kinematic model. If the position and orientation are specied, the positions of the pistons of the actuators can be determined by the backward kinematic model. The kinematical models of the hydraulic motion system are described in a Matlab script le, which is shown in Appendix B Forward Kinematic Model The input for the forward kinematic model are the positions of the pistons of the actuators, the output is the position of end-eector (L), see gure 3.1. The truss beam is assumed to be rigid and is represented as a massless beam with a point mass. The formula's to calculate the position of the end-eector are geometric equations, as well as the cosine rule to calculate the angle of a non equilateral triangle. The cosine rule is a generalization of Pythagorean Theorem for non equilateral triangles. The cosine rule relates the lengths e, x 2, l 3 and the angle β shown in gure 3.1. The cosine rule is shown in equation 3.7. ( ) e β = cos l3 2 x 2 2 (3.7) 2e l 3 The cosine rule shown in equation 3.7 and other relations used in the forward kinematic model are described in a script le. The script le is shown in Appendix B.0.1. It is important to derive equations which are unambiguous, otherwise singularities could exist. Singularities could result in wrong solutions or errors when calculating the equations. The forward kinematical model contains several equations with a denominator. These equations could result in a error if the denominator is zero. However, the denominator will be larger than zero for positive values of x 1 and x 2. Therefore now singularities exist. The script le will be used to simulate two distinctive trajectories. The two tra- 25

38 Figure 3.4: Results Forward Kinematic Model jectories will be used to verify the model with aid of the hydraulic motion system. The verication will be described in the next section, the two trajectories will be discussed in this paragraph. At both trajectories the position of the piston of one actuator changes from minimum into maximum value by discrete steps, and the position of the piston of the other actuator is constant. The results of the script le are shown in gure 3.4. The continuous graph in gure 3.4 shows the trajectory where the position of the piston of the tilt actuator has a constant value of x m = [m], and the position of the piston of the horizontal actuator changes from maximum into minimum value. The dashed graph shows the trajectory where the position of the piston of the horizontal actuator is set to a constant value of x p = 0.02 [m] and the position of the piston of the tilt actuator changes from minimum into maximum value Backward Kinematic Model The input for the backward kinematical model are the 'x' and 'y'- coordinates of the end-eector (L) shown in gure 3.1. The positions of the pistons x p and x m can be calculated by the backward kinematic model. The backward kinematic model will be applied for trajectory planning. Again, it is important to derive equations which are unambiguous to avoid singularities. The formula's used are geometric relations and the same cosine as applied at the forward kinematic model. Only now the length x 2 is calculated; x 2 = (e 2 + l3 2 (2e l 3 )) cos β (3.8) 26

39 The backward kinematical model contains a square root function, see equation 3.8. The square root function could result in singularities if the square root is applied on a negative term. However, the term will never be negative for positive values of x 1 and x 2. Therefore now singularities exist at the backward kinematical model. The script le of the backward kinematical model is shown in Appendix B Verication Kinematical Models The verication of the kinematic models is separated into two parts. 1. The positions of the pistons of the actuators and the main geometric parameters will be veried. 2. The trajectory, described in gure 3.4, will be simulated on the hydraulic motion system and compared with results of the kinematical model. Position of Piston and Constant Parameters In this section the positions of the pistons will be veried with aid of the motion system. The positions of the pistons will be measured with the dspace system and will be checked at the setup. The measured values will be compared with the values entered at the model. Each actuator will be checked at 4 dierent static situations, see table 3.2. The fully retracted situation is used as a reference for measuring the position of the piston at the setup. The measurements will be repeated several times and the means are shown in table 3.2. Table 3.2: Piston Position (mm) Horizontal Actuator Stroke Entered x p Measured x p dspace Measure x p at setup fully retracted percent percent percent fully extended Tilt Actuator Stroke Entered x m Measured x m dspace Measured x m at setup fully retracted percent percent percent fully extended

40 The dierence between the entered value and the value of the dspace system is caused by friction in the hydraulic motion system. It can simply be solved by using a better controller. The used P-controller is to weak to compensate the error. The dierence between the dspace system and the measured value at the setup is caused by the calibration and accuracy of the signals and measuring instruments. The dierence at the horizontal actuator is signicant larger than at the tilt actuator. This can be explained by the dierent calibration methods. The position transducer of the tilt actuator sends a maximum allowed voltage signal of +10 [V] to the dspace system if the actuator is fully extended and a signal of -10 [V] if the actuator is fully retracted. The position transducer of the horizontal actuator sends a signal of +10 [V] if the actuator is fully extended and a signal of 0 [V] if the actuator is fully retracted. The signal of the position transducer of the tilt actuator can directly be used as input for the dspace System. While the signal of the position transducer of the horizontal actuator has to be calibrated between -10 [V] and +10[V] on an analoge board. It can be concluded that the dierence between the horizontal actuator and tilt actuator is mainly caused by a calibration error. It is dicult to accurately calibrate the signal on the analoge board. The hydraulic motion system will be converted into an excavator. When operating an excavator it is allowed to have a deviation of several centimeters with regard to the digging trajectory. The maximum error, derived from the values of table 3.2, is smaller than 5 percent. The error is therefore accepted. Before the kinematical models can be veried, the quantities [b, d, l 1, l 2, l 3, x max ] shown in gure 3.1 will be discussed. The lengths (b) and (d) have been calculated at Rodermond [10] by the so called "estimation method". The values of these quantities are shown in table 3.3. Table 3.3: Estimation Method of Rodermond [10] Parameter [m] b d The quantities [l 1, l 2, l 3 ] are measured and checked with the technical drawings. The measurements correspond with the drawings, the values can be found in Appendix A. The quantity x max can be measured with a spirit level and with aid of the ControlDesk interface. Using a spirit level the actuators are positioned, so the truss beam is leveled horizontally and the tilt actuator is set perpendicular to the truss beam. The following equation applies at that particular conguration: x max = l 1 + x p (3.9) 28

41 The length l 1 can be obtained from the technical drawings and x p can be read from the ControlDesk interface. x max results in: x max = = A rough verication can be made with a tape line. Measuring the magnitude of x max veries that x max = The six quantities [b, d, l 1, l 2, l 3, x max ] are determined and veried. They can be used in the kinematic models. Trajectory To verify the kinematical model with aid of the hydraulic motion system, the trajectory described in gure 3.4, will be simulated with the hydraulic motion system. A marker will be attached to the end-eector so it will draw the trajectory on a whiteboard. The results are shown in gure 3.5a. (a) Photograph of the Trajectory (b) Comparison Trajectory Figure 3.5: Trajectory Hydraulic Motion System Specic points will be measured and compared with the corresponding theoretical points. The measurements will be performed out of a explicit xy-coordinate system, which is drawn on the board with a spirit level. The measurements are compared with the theoretical values by projecting them on the trajectory. Point (0.75, 0) of gure 3.5b is used as reference point for projecting the measurements. The positions of the pistons of the actuators at reference point are x p = 0.02 and x m = From table 3.2 can be concluded that the relative error of the position of the piston is smallest at low values. The measured values match the theoretical trajectory quite well. The only large deviation arises around maximum value of the x-coordinate of the end-eector. The 29

42 maximum measured value is [m] whereas the maximum value of the theoretical kinematical model is [m]. The dierence can be explained with the calibration error. The measurements of table 3.2 illustrate, that when the dspace system indicates a fully extended horizontal actuator, x p = [m], the actual position of the piston is not fully extended but x p = [m]. It can be concluded that the error shown in gure 3.5b is not caused by errors in the kinematical model but due to a calibration error Working Space The script le of the forward kinematic model can be used to determine the working space of the hydraulic motion system. The boundaries can be simulated by adjusting the position of the piston of one actuator to maximum or minimum value and varying the position of the other piston. The working space has four boundaries, see table 3.4 and gure 3.6. The boundaries Figure 3.6: Working Space can be determined with aid of the script le, when the values suggested in table 3.4 are used as input. The working space must be compared with the working space of an excavator for converting the hydraulic motion system. The motion of the bucket during a actual digging trajectory of a small excavator, is measured by Quang [9]. A Komatsu mini excavator was tted with additional sensors for the project described in that thesis. The digging of a trench is the most common operation, this operation will be compared with the working space. The digging tra- 30

43 Table 3.4: Boundaries Position of the Piston Boundary Horizontal Actuator Tilt Actuator I maximum value minimum to maximum value II maximum to minimum value maximum value III minimum value maximum to minimum value IV minimum to maximum value minimum value jectory is performed according to an operator's experience. The operator performs the following steps: 1. operating the motion system to move the bucket to the ground. 2. operating the system so the boom-bucket construction performs a digging operation. 3. operating the system to lift the bucket up. 4. nally operating the system back to starting position. Figure 3.7: Digging Trajectory This typical digging trajectory is shown in gure 3.7. Obviously the working space of the hydraulic motion system is too small to simulate an actual excavator. Comparing 31

44 the working space of the hydraulic motion system with the digging trajectory, it can be concluded that the hydraulic motion system can simulate an excavator down scaled by a factor of approximately Mathematical Model of the Mechanical System Kinematical models comprehend geometric relationships. They do not include the dynamical behavior of the mechanical system. Therefore the mechanical system will be represented with the Lagrangian equations of motion. The methode to derive the Lagrangian equations of motion is described in Kraker/van Kampen [5]. The methode will be applied to the motion system and the results will be discussed in this section. The hydraulic motion system requires two independent coordinates to describe the motion, it is called a two-degree-of-freedom motion system. The mathematical formulation results in two equations of motion. These equations are derived using the Lagrange's Methode Generalized Coordinates At the rst step of Lagrange's Methode, the generalized coordinates are determined. For each degree of freedom one coordinates must be described, so in this case 2. The dened generalized coordinates are x 1 and θ: [ ] x1 q = (3.10) θ Global Coordinate System The position of the two masses must be determined with respect to a global coordinate system. The origin of the dened global coordinate system is shown in gure 3.8. The position vector must be derived with respect to time ( r m ) also with respect to the generalized coordinates ( r m,q ). [ x1 + l r m1 = m1 sin θ l m1 cos θ [ 1 lm1 cos(θ) r m1,q = 0 l m1 sin(θ) ] [ lm1 θ cos(θ) ẋ1, rm1 = l m1 θ sin(θ) ] ], (3.11) r m2 = [ x1 0 ] [ ẋ1, rm2 = 0 ] [ 1 0, r m2,q = 0 0 ] (3.12) 32

45 3.7.3 Kinetic Energie Figure 3.8: Free Body Diagram The kinetic energie can be calculated with the vectors derived in section The kinetic energie is dened as: T = 2 2 1/2m 1 r m1 + 1/2m 2 r m2 + 1/2J z θ2 (3.13) T = 1/2(m 1 + m 2 ) x 2 1 m 1 l m1 x 1 θ cos θ + 1/2(m1 lm1 2 + J m1 ) θ 2 To derive Lagrange's equations of motion, two derivatives of the kinematic energie have to be calculated: T, q and d dt (T, q ). T, q = d dt (T, q ) = Potential Energie [ [ 0 m 1 l m1 ẋ 1 θ sin(θ) ] (m 1 + m 2 )ẍ 1 m 1 l m1 θ cos θ + m1 l m1 θ2 sin θ m 1 l m1 ẍ 1 cos θ + m 1 l m1 ẋ 1 θ sin θ + (m1 l 2 m1 + J m1 ) θ ] (3.14) (3.15) The potential energie must be calculated for the mass m 1. The total potential energie of mass m 2 does not change with variation of the generalized coordinates and can be 33

46 neglected. The potential energie function can be expressed as follows: V (q) = U(q) in + V(q) ex (3.16) [ ] 0 V (q) = m 1 gl m1 cos(θ) The potential energie should also be derived with respect to the generalized coordinates: [ ] 0 V, q = (3.17) m 1 gl m1 sin(θ) Nonconservative Generalized External Forces The nal step in Lagrange's Methode is the calculation of the nonconservative generalized external forces: Q nc = ( r m2, q) T F1 + ( r c, q) T F2 (3.18) For calculating the nonconservative generalized external forces the position vector at point (C), see gure 3.8, is determined according equation The actuator forces are written with respect to the global coordinate system. r c = [ x1 + l 3 sin(θ + γ + α) l 3 cos(θ + γ + α) ] [ 1 l3 cos(θ + γ + α), r c,q = 0 l 3 sin(θ + γ + α) ] (3.19) F 1 = [ Fp 0 ] [, F2 Fm sin(ϕ) = F m cos(ϕ) ] (3.20) With these vectors the nonconservative generalized external forces can be written as: [ ] Q nc F = p F m sin ϕ (3.21) F m l 3 sin(ϕ (θ + γ + α)) Lagrange's Equation of Motion The expression for Lagrange's equations of motion can be derived. The number of equations of motion is equal to the number of independent generalized coordinates. The two Lagrangian Equation of Motions can be described with the following equations: d dt (T, q ) T, q +V, q = (Q nc ) T (3.22) (m 1 + m 2 )ẍ 1 m 1 l m1 θ cos θ + m1 l θ2 m1 sin θ = F p F m sin ϕ (m 1 l 2 m1 + J m1 ) θ m 1 l m1 ẍ 1 cos θ m 1 gl m1 sin θ = F m l 3 sin(ϕ (θ + γ + α)) The mathematical expressions of equation 3.22 can be used to model the dynamical behavior of the mechanical system. 34

47 3.8 Summary The properties of the mechanical system are described, as well as the kinematic relations between the positions of the pistons of the actuators and the position of the end-eector. The kinematical models are veried with aid of the hydraulic motion system. The working area of the hydraulic motion system can be determined with the forward kinematical model, see gure 3.6. The working area is too small compared to an actual excavator and therefore represents a scaled model. The load variations of an excavation task can be simulated by adjusting the load mass m (14) of the mass array. Adjusting the Load mass aects the position coordinates of the center-point of gravity. The length l m1 is related to the coordinates of the center-point of gravity and is processed in the Lagrangian Equations of Motions. The Lagrangian Equation of Motion model the dynamical behavior of the mechanical system. Thus the load variations can be simulated by adjusting the load mass array. 35

48 Chapter 4 Modeling 4.1 Introduction This chapter discusses the numeric modeling of the hydraulic motion system. The hydraulic motion system must be converted into a model of a small excavator. The main reason for developing a numeric model, is the possibility to simulate a digging action. It is dicult to simulate variations of an excavation task, like soil interaction or load displacements, in a real time environment. A numeric model of the hydraulic motion system could bring a solution. Predictions of the behavior can be made with aid of this model. Other advantages are the ability to predict the outcome of a task before it is executed on the motion system and the ability to use the model for research studies, for example MIMO control. The hydraulic valves, actuators and mechanical system where described in chapter two and three. By means of those descriptions, a numeric model of the hydraulic motion system can be developed. This chapter discusses a linear and nonlinear model. The linear model will be used to obtain more insight in system properties and behavior, and the nonlinear model will be used to describe the hydraulic motion system accurately. The nonlinear model will be used for converting the system into an excavator. 4.2 Linear Model A servovalve controlled hydraulic actuator can be modeled with a transfer function as discussed in section The transfer function is described with equation To determine the coecients c 3 to c 5, the frequency response magnitude and phase of the valve controlled hydraulic actuators must be obtained. Frequency Response Functions (FRF) measurements will be used to obtain the dynamic characteristics of the valve controlled actuators. Both actuators are closed loop controlled with position feedback. The block diagram of a feedback controlled system is shown in gure 4.1, where (C) is the controller and 36

49 (H) is the valve controlled actuator. Figure 4.1: Control System FRF-measurements excite the system in all frequencies by injecting white noise (w) between controller (C) and plant (H). By measuring the Sensitivity (S) and the Process Sensitivity (PS), the Frequency Response Functions of the plant (H) can be extracted, see equation 4.1. S = u w = 1 CH+1 P S = e w = H CH+1 H = P S S (4.1) The Sensitivity of a system can be obtained by measuring the signals (u) and (w), the Process Sensitivity can be obtained from the signals (e) and (w). A transfer function can be tted on the Frequency Respons Functions measurements. In section 4.2.2, the natural frequency, damping coecient and bulk modulus will be identied with aid of the tted transfer functions Frequency Response Functions Measurements The transfer functions are obtained from equations which are linearized around the center position, therefore the FRF-measurements are performed around the center position. The FRF-measurements and the tted transfer functions are shown in gure 4.2a and gure 4.2b. The sensitivity of the plant and the coherence of the sensitivity are illustrated to verify the measurements reliability. The measurements are reliable when the coherence has a value larger than 0.9. The coherence of the sensitivity of the horizontal actuator indicates reliability at frequencies above 8 [rad/s] and the coherence of the tilt actuator at frequencies above 6 [rad/s]. The gures show a signicant phase lag at high frequencies. This deviation is caused by time delay or due to the phase change caused by the position transducers or valves. 1. Time delay can arise from delays in the process itself see Sugiyama [12], or from delays in the processing of the sensed signals. 2. The phase response of the position transducer or valve can have a signicant contribution to the phase response obtained from the FRF-measurements. The measured phase respons will show a deviation with respect to the phase respons of the tted transfer function, if the bandwidth of the position transducer or valve is near the bandwidth of the actuator. 37

50 (a) Horizontal Actuator (b) Vertical Actuator Figure 4.2: Frequency Response Measurements 38

51 The hydraulic motion system will be used at low frequency range because it will simulate an excavator. The phase lag is only disturbing at high frequency range and therefore not relevant in this thesis. The model is presumed to be accurate when simulating below the bandwidths of the actuators. The bandwidth of the actuators are discussed in the next section. The coecients c 3 to c 5 of equation 2.26 are determened from the tted transfer functions. The estimated values of the coecients are shown in table 4.1. Table 4.1: Transfer Function Coecients Actuator C3 C4 C5 Horizontal actuator e-5 1.5e-3 Tilt actuator 1 1.0e-5 4.0e Parameter Identication Some basic parameters can be derived from the coecients shown in table 4.1. Theoretically the coecients c 3 to c 5 can be expressed as; c 3 = q N, c A 4 = 1 and c ω0 2 5 = 2β ω 0, see section The damping coecient β and the natural frequency ω 0 can be calculated with c 4 and c 5. The normalized ow gain q N on the contrary, can not be obtained from the measured coecients. The measurements are performed at various valve port openings. The normalized ow gain can only be measured at maximum valve port opening. The estimate normalized ow gain, the natural frequency, the damping coecient and the bulk modulus are shown in table 4.2. They are discussed in the following paragraphs. Table 4.2: Estimated Parameter Values Actuator q N [m3/s] β [-] ω 0 [rad/s] E [N/m2] Horizontal 6.69e e9 Tilt 2.77e e9 Natural Frequency The natural frequency is the frequency where the system oscillates without being subjected to a continuous or repeated external force. When the system is forced to oscillate at that frequency the amplitude of the oscillation will increase dramatically, when not damped. The natural frequency of the hydraulic actuators can be derived from the measured coecient c 4. As shown in table 4.2 the natural frequencies of the actuators are smaller 39

52 than the natural frequencies of the valves, see table 2.1. The condition of equation 2.1 is validated, so the assumption that the valves are not the dominant dynamic elements is conrmed. Dimensionless Damping Coecient The damping coecient β depends on internal leakage and friction, such as friction of seals or viscous friction. It is wel known that hydraulic actuators are usually poorly damped see Viersma [14]. The damping can be improved with a bypass channel for example a hole through the piston, which increases the internal leak. The damping coecients obtained from the FRF measurements are shown in table 4.2. A system is critically damped at a damping ratio of β = 0.707, see Franklin [4]. Both actuators are under damped as expected. Weaker damping results in oscillatory time response. The damping will be illustrated with the eect of an impuls response on the actuators. Compressibility of uid results in an integrating eect. The integrator, indicated with 1 in equation 2.26, will be neglected to illustrate the time response in a understandable gure. Neglecting the integrating eect is acceptable because the integrator only s inuence the static error and does not inuence damping. The time responses of the resulting second order systems are plotted in gure 4.3. From gure 4.3 can be con- Figure 4.3: Decay of Pulse Shaped Signal cluded that the tilt actuator is better damped than the horizontal actuator, as expected. 40

53 Normalized Flow Gain The normalized ow gains will be derived from measurements where the actuators are accelerated with a constant value. The signals measured are the positions of the pistons of the actuators. The time derivative of those signals are calculated according to the backward dierence methode, and result in the velocities of the rod. The maximum velocity can be converted into the normalized ow gain by means of piston area; q N = v max A (4.2) The maximum velocity is reached when the measured position signal diverges from the reference sinal, see gure 4.4. The graph of the measured position versus time shows a linear relation at the interval 7 to 10 seconds. It is assumed that the velocity of the piston has reached its maximum value at that interval. The maximum velocity is calculated via the backward dierence methode. Figure 4.4: Maximum Velocity The measured ow gains are shown in table 4.2. They approximate the values of the manufacturer. The relative error for the tilt actuator is 12.4% and for the horizontal actuator 5.7%. The error is probably caused by leakage in the valve. Leakage occurs because the valvespool is not perfectly centric with the valveports. Bulk-Modulus No liquid is fully incompressible. The ratio of a uid's decrease in volume as a result of increase in pressure is given by the isothermal bulk modulus of elasticity, dened as; E dv dp = (4.3) V Here dp is the pressure dierence, V the initial volume, dv the dierence of the volume and E the bulk modulus. The bulk modulus is a quantity which is dicult to calculate. It depends, for example, on the percentage of gas in the uid, the pressure and the temperature. The bulk modulus of the hydraulic system will be estimated, based on the FRF measurements. First the mathematical description will be derived. There is a relation between the natural frequency, the mass and the oilstiness. The natural frequency can be expressed according equation 4.4. C ω 0 = (4.4) M 41

54 Where C is the oilstiness and M the dynamic mass. The dynamic masses of the hydraulic motion system can be calculated with equation 3.3 and 3.5. According to Viersma [14], the oilstiness of the symmetric actuators can be described with the following equation. [ 1 C = C 01 + C 02 = AE + 1 ] (4.5) x i S x i Where A the piston surface, x i the position of the piston and S the stroke of the actuator. Substituting equation 4.5 into equation 4.4 results in the mathematical description of the bulk modulus, see equation 4.6. E = ω2 0 M [ x S x ] i A S (4.6) The bulk modulus can be calculated with the estimate natural frequencies derived from gures 4.2a and 4.2b. The estimated bulk modulus are shown in table 4.2. The hydraulic motion system consists of two valve controlled actuators each connected to an oil pump. The two separate oil pumps use one oil reservoir. Therefore the initial conditions, like percentage gas dissolved and temperature, are approximately equal at both actuators. The dierence between the values of the estimated bulk modulus is likely caused by inaccuracies of the t. The estimated bulk modulus of the hydraulic system should be equal. With aid of the FRF measurements two values are estimated. The mean of these two values will be used as nal estimated bulk modulus; E = 0.95e9 [N/m2]. 4.3 Nonlinear Model The nonlinear model is based on equations 2.10, 2.19 and the Lagrangian Equations of Motion This model is divided in three parts. The rst part contains the valves and actuators, the second part the friction and the third part the Lagrangian Model Actuators The nonlinear Simulink model of a servocontrolled actuator is shown in Appendix C.1.2. As discussed earlier the model contains the orice equation and the continuity equation applied on the actuator. The actuators also experience friction. This will be covered in the next section Friction The friction force of the actuators depends on piston velocity, the net force acting on the piston and oil temperature. One standard experimental method of modeling friction is modeling as function of velocity. The total friction force is divided in three parts, static friction, coulomb friction and viscous friction. The static friction is a transient 42

55 term, present when the piston begins to move. In this thesis the assumption is made that viscous and coulomb friction dominate. The coulomb friction is assumed to be a constant force that depends on the direction of movement. The viscous friction is assumed to be a linear term which is proportional to the velocity and contributes to stability and damping. According to Poley [8] the total friction can be modeled as: F f = F v + F c F f = C v ẋ + sign(ẋ) F c (4.7) Where F v is the viscous friction, F c the coulomb friction, C v the viscous friction coecients and ẋ is the piston velocity. Frictional eects are very dicult to measure and accurate values are unlikely to be found, but simple measurements give at least an order of magnitude estimates. The set of data of the friction measurements is shown in Appendix D. Coulomb Friction The coulomb friction can be calculated by measuring the pressure dierence at a low velocity of the piston. Due to the low velocity, the viscous friction can be neglected see equation 4.7. The input signal of the measurements, results in building up a pressure dierence P c. When P c overcomes the coulomb friction the piston breaks away and starts to move. The coulomb friction can be calculated with the following equation. F c = P c A (4.8) The mean of the set of measurements is normative for the friction, when the set of measurements is a sample of a normal distributed population. The assumption of normality is tested with the Anderson-Darlin Normality test and is shown in Appendix D.2. Both sets of measurements are from a normal distributed population. This implies that the mean can be used as a measure of friction. The average measured coulomb friction of the horizontal actuator is: F cp = P cp A p = 46.0 [N] (4.9) The average measured coulomb friction of the tilt actuator is: F cm = P cm A m = 16.2 [N] (4.10) As expected the coulomb friction of the horizontal actuator is higher then the coulomb friction of the tilt actuator, see section Viscous Friction Viscosity is an important property of a uid. A denition of viscosity, referring to a piston and cylinder, was given by Sir Isaac Newton. He observed that a force is 43

56 necessary to cause a relative motion. This force is a measure of the internal friction, and is called viscous friction. The denition states; F v = µa C r ẋ (4.11) Where C r is the radial clearance between piston and cylinder and µ is the absolute viscosity, which is a property of the uid. In this section the viscous friction coecient C v will be estimated. Theoretically the viscous friction coecient can be written as follows: C v = µa (4.12) C r To measure the viscous friction of the actuator the piston is accelerated to a relatively large velocity. The measured pressure dierence at a large velocity, is caused by coulomb and viscous friction. Rewriting equation 4.7 to the viscous friction coecient results in equation 4.13 ; F f = P 0 P 1 A C v = P 0 P 1 A F c (4.13) ẋ The viscous measurements and normality test are shown in Appendix D. Again both sets of measurements are from a normal distributed population. The mean can be calculated and used as an estimate for the viscous friction coecient: C vp = 749[N/m] C vm = 945[N/m] The estimate viscous friction can be modeled as: The model of the friction is shown in Appendix C.1.3. (4.14) F vp = 749 ẋ p F vm = 945 ẋ m (4.15) Lagrangian Model The dynamics of the truss beam are described with the Lagrangian Equations of Motion, as discussed in section 3.7. These equations are modeled in Matlab Simulink, see Appendix C.1.4. This Simulink model will be validated with aid of the forces in steady state condition derived in section 3.5. The actuator forces will be calculated for the steady state condition where the truss beam is horizontally leveled and angle φ, shown in gure 3.1, is zero. The static forces calculated according to equation 3.6 are; F B = 198 [N] F C = 1135 [N] The same conditions are applied to the Matlab Simulink model, the actuator forces are plotted in gure

57 (a) Horizontal Actuator (b) Tilt Actuator Figure 4.5: Forces in Steady State Condition The horizontal actuator force F B is 194[N] and the tilt actuator force F C is 1106[N]. The calculated and modeled forces are compared at several steady state conditions, it can be concluded that the Matlab Simulink model corresponds well with the theoretical static forces. To simulate the hydraulic motion system, boundaries and the initial conditions must be inserted into the model. The boundaries represent the maximum displacement and orientation of the truss beam. The initial conditions represent the conditions when starting a simulation. Boundaries and Initial Conditions The maximum and minimum positions of the pistons of the actuators limits the generalized coordinates, and thus the working range. The generalized coordinate x 1 is limited by the horizontal actuator. When the horizontal actuator is fully retracted x 1 reaches its maximum value. When the horizontal actuator is fully extended x 1 reaches its minimum value. The boundaries of the generalized coordinate x 1 are; xmax xp x 1 xmax 0 (4.16) The generalized coordinate θ reaches minimum value when the angle ϕ is zero and the tilt actuator is fully extended. The maximum value is reached when the horizontal actuator is fully extended and the tilt actuator fully retracted. The maximum and minimum value of the angle θ can be derived with the forward kinematical model, the values are shown in Appendix A. Besides the boundary conditions the initial conditions must be described as well. The system starts from stagnation, which means that the initial conditions for the velocity ẋ 1 and the angular velocity θ are zero. When starting a simulation, the hydraulic motion system is set in the position where the truss beam is leveled horizontally and 45

58 the tilt actuator aligned perpendicular to the truss beam. This starting position results in the following initial conditions; θ = 0.5π γ x 1 = xmax 0.02 (4.17) Summary This chapter covers the parameter identication and numeric modeling of the hydraulic motion system. The hydraulic parameters are identied with aid of FRF-measurements and the friction coecients are measured with pressure measurements. The numeric model is developed, so it will be possible to simulate and investigate tasks without performing them in realtime environment. A linear model is developed to obtain more insight in system properties and behavior, and a nonlinear model is developed to simulate the desired tasks. The nonlinear model will be used at converting the system into an autonomous excavator. 46

59 Chapter 5 Trajectory Planning 5.1 Purpose The goal of this thesis is to convert the hydraulic motion system into an autonomous excavator. Meaning, the hydraulic motion system must be able to fulll an excavation task without human intervention. For realizing an autonomous task, the operator should prescribe the trajectory of the hydraulic motion system in advance. A digital control system must ensure that the motion system performs the trajectory within a certain degree of accuracy. Prescribing the trajectory comprehends two steps: 1. The user denes specic points of the excavating task. 2. A numeric Trajectory Generator determines the complete trajectory The trajectory will be used as input for the model at the ControlDesk environment so the autonomous task can be realized. The Trajectory Generator can be used for converting the hydraulic motion system into an autonomous excavator. From a practical point of view it is not desired to design a specic trajectory planner for each task. Therefore a Trajectory Generator must be designed which can deal with several types of trajectories Introduction on Trajectory Generation This chapter discusses a methode which computes a trajectory in 2 dimensional space for any desired possible motion of the end-eector. To make the discussion in this chapter transparent, an excavation task will be used as an example. The example will simulate the excavation taks described in section which is shown in gure 5.1. In many cases it is enough to specify the initial and the nal condition of the actuators. But in this case it is necessary to specify the trajectory in more detail. The Trajectory Generator will be used to generate a trajectory involving setpoints. The setpoints consist of a beginpoint, intermediate points and an endpoint. The setpoints of the example are illustrated with black dots in gure 5.1. It is desired that the excavator can pass the intermediate setpoints without stopping. 47

60 Figure 5.1: Digging Trajectory The trajectory must contain a path and time information of that path. The discrete path is dened as the set of coordinates of the generated motion. The path will be discussed in section The time information in combination with the set of coordinates will be used to derive the velocity, acceleration and jerk proles of the pistons. These quantities will be used to optimize the trajectory. The time information will be discussed in section The time derivatives of the path are discussed in section 5.3. To make an autonomous excavator protable it should work as fast as possible, therefore the trajectory must be optimized. The optimization will be discussed in section 5.4. So, the generator must be able to specify a trajectory which consists of an exact path, time information, velocity, acceleration and jerk of the piston of the actuators. Research about trajectory generation at Brink [13] resulted in a generic Path Generator Toolbox. That Path Generator Toolbox will be adapted to meet the requirements needed for this study. Before the Trajectory Generator will be treated, it is important to understand the dierent spaces and domains used in the toolbox. The spaces and domains will be explained in the next section Spaces and Domains Spaces There are two coordinate systems to describe the position and orientation of the end eector, Cartesian Space (CS) and Joint Space(JS). Cartesian Space describes the coordinates of the end-eector in x/y-plane. The Joint Space coordinate system describes the position and orientation of the end eector by indicating the positions of the pistons of the actuators. 48

61 In most trajectory generators the desired trajectory is calculated in Cartesian Space and transformed into Joint Space, this methode is for instance described by Craig [1]. The advantage of calculating the trajectory in such manner is the ability to regard system constraints in Cartesian as well as Joint Space. As discussed in section 3.6, the kinematic model of the hydraulic motion system is nonlinear. To avoid dicult nonlinear kinematical transformations, the path and time mapping will solely be calculated in Joint Space. Since the properties of the hydraulic motion system include no constraints in Cartesian Space, this methode is considerate adequate. To check whether the trajectory is satisfying, with respect to the entered setpoints, the calculated discrete path in Joint Space will be transformed to end-eector coordinates in Cartesian Space. Domains There are two domains used at the trajectory generator toolbox, spline domain and time domain. Spline domain is used to describe the path from setpoint to setpoint, when the trajectory does not contain any time information. Time domain is used when time information is included at the trajectory. Table 5.1 represents the used domains and spaces and their notation, see table 5.1. Table 5.1: Domains and Spaces CS time domain JS spline domain JS time domain notation t CS k JS t JS position x(t) Φ(k) Φ(t) 5.2 Trajectory Generator An algorithm is developed to describe the methode used for generating the trajectory. The algorithm describes the methode step by step, which lead to a satisfying trajectory. A schematic overview of the algorithm is shown in gure 5.2. The algorithm includes the following steps; 1. The user enters the input in the trajectory generator toolbox. The input consist of setpoints and constraints. 2. The setpoints will be transformed from Cartesian Space to Joint Space with aid of the backward kinematical model. 3. A path will be calculated between the setpoints in Joint Space spline domain. 4. Time will be mapped on that path by using a fth order polynomial and by using an estimated end time. At the rst iteration loop the user dened end time will used. At the following iteration loops the end time will be calculated, based on a correction factor. 49

62 5. The time derivatives of the path will be calculated. The rst, second and third order time derivative determine respectively the velocity, acceleration and jerk proles. 6. The velocity, acceleration and jerk proles will be compared with the constraints of the hydraulic motion system. A correction factor will be calculated based on the dierence between the proles and the constraints. 7. If the correction factor is smaller than a predetermined tolerance the algorithm will end. If the correction factor is larger than a predetermined tolerance, the endtime will be optimized which invokes a iteration loop, see gure 5.2. The last two steps are implemented to optimize the trajectory. The iteration loop will continue till at least one constraint becomes active. Each step of the algorithm will be explained in the following sections. The algorithm is processed in a Matlab toolbox, the main le of the Trajectory Generator is shown in Appendix G.1. Figure 5.2: Trajectory Generation Algorithm 50

63 5.2.1 Input The input of the generator can be classied into two groups, constraints and setpoints, see gure 5.3. Input Constraints Setpoints Geometric Dynamic Static Dynamic Figure 5.3: Input Constraints The constraints dene the limitations of the excavator. There are two types of constraints, geometric and dynamic constraints. The geometric constraints are the minimum and maximum position of the piston of the actuators. The geometric constraints restrict the working space with so called boundaries, see section The path of the trajectory is restricted by the geometric constraints. The dynamic constraints are the maximum piston velocity, acceleration and jerk. The dynamic constraints have eect on the time needed to complete the trajectory. Because all constraints are properties of the actuators, they are described in Joint Space. The values of the constraints are shown in the Matlab scriptle at Appendix G.1.1. The constraints are entered as input for the generator. The Trajectory Generator ensures that the trajectory does not contain any velocity, acceleration or jerk which exceeds the related constraint. Setpoints As discussed in the introduction, the setpoints are specic points of the trajectory which are dened by the user. The setpoints consists of a beginpoint, intermediate points and an endpoint. It is assumed that the begin and endpoint have zero velocity, therefore they are called static setpoints. The intermediate points are the points the end-eector has to pass through without stopping. The end-eector has a non-zero velocity when passing the intermediate point, therefore they are called dynamic setpoints. The setpoints are entered at the trajectory generator toolbox in Cartesian Space. The coordinates of the setpoints are transformed into Joint Space with aid of the backward kinematical model. The setpoints in Joint Space are used as input to generate the path. The values of the setpoints are shown in the Matlab scriptle at Appendix G

64 5.2.2 Path Description The motion of the end-eector from an initial position to an end-position must be stored in a path. The set of coordinates in Joint Space which correspond to the setpoints in Cartesian Space, are calculated by the backward kinematics. To describe the complete motion, a continuous path in Joint Space spline domain ( k JS) between these coordinates is desired. The trajectory generator constructs the intermediate path with splines. A continuous spline is created between two setpoints. When the user enters (n) setpoints the generator constructs (n-1) spline segments. Together these splines form the continuous path. The path in spline representation is mathematically dened as: P i (k) = (Φ p,i (k), Φ m,i (k)), with i = 1, 2,..., n 1 (5.1) Φ p,i (k) is the path in spline domain for the piston of the horizontal actuator and Φ m,i (k) is the path for the piston of the tilt actuator. To obtain a continuous path, the splines can be represented in several ways. In this thesis an algebraic equation of cubic splines is chosen. Cubic splines are chosen because they are the lowest degree of polynomials that result in a smooth curve, see Croft[3]. Cubic Spline Polynomial The path is a piecewise continuous third-order function. Piecewise indicates that for every section between two setpoints, an other cubic spline polynomial is valid. The cubic spline polynomials for the horizontal actuator are indicated with Φ p,i (k) and the cubic spline polynomials for the tilt actuator with Φ m,i (k). The equation of the splines are shown in equation 5.2. Φ i (k) = a i + b i k + c i k 2 + d i k 3, with i = 1, 2,..., n 1 (5.2) Where a i, b i, c i and d i are the constant coecients of the cube spline polynomials and k the spline parameter k ɛ [0, 1]. The methode for deriving the coecients is discussed in Appendix E. The coecients are stored in a matrix indicated with P i. The path can be expressed in matrix form, see equation 5.3. Φ i (k) = [ a xp i a xm i b xp i b xm i c xp i c xm i d xp i d xm i ] = P i k (5.3) The disadvantages of cubic spline polynomials, in relation to a linear point to point methode, are; 1. The path can show oscillations when setpoints are located close to each other 1 k k 2 k 3 52

65 2. The trajectory may move out of area spanned by the constraints 3. Splines are curved, so straight lines are dicult to compute with polynomials Dening a desired motion for the end-eector, these disadvantages have to be regarded. The discrete path is calculated with aid of the cubic spline polynomials. The path is calculated in the Matlab script of Appendix G Time Mapping So far the path is described. The discrete path contains the set of coordinates of the pistons of the actuators. The coordinates are calculated with aid of the derived cubic splines. As mentioned in section 5.1.1, the coordinates of the path do not hold any time information. Time information must be added to determine the velocity, acceleration and jerk proles of the pistons. There are several options for time mapping, in this theses a fth-order polynomial is used. The reason for using a fth order polynomial is simple, the number of unknown parameters will equal the number of equations. The fth order polynomial can be described with equation 5.4. k t = p 0 + p 1 t + p 2 t 2 + p 3 t 3 + p 4 t 4 + p 5 t 5 (5.4) Where p 0, p 1, p 2, p 3, p 4 and p 5 are constant coecients and t the time parameter. As discussed in section the path is divided into segments, it is obvious that the time information can not be divided. The polynomial k t is continuously dened for k ɛ [0, n] and must be strictly increasing. The rst and second order time derivatives of the polynomial are described with equation 5.5. dk dt = p 1 + 2p 2 t + 3p 3 t 2 + 4p 4 t 3 + 5p 5 t 4 d 2 k dt 2 = 2p 2 + 6p 3 t + 12p 4 t p 5 t 3 (5.5) Using equation 5.4 and 5.5 for the start and endtime, results in 6 equations. There are 8 unknown parameters, p 0 to p 5, t 0 and t n. These parameters can be calculated with the boundary conditions of the mapping, the boundary conditions are; for t 0 k = 0 dk = ( dk dt d 2 k = 0 dt 2 dt ) 0 for t n k = n dk = ( dk dt d 2 k = 0 dt 2 dt ) n To maintain the demand of being second-order continuous, the values ( d2 k) dt 2 0 and ( d2 k) dt 2 n are set at zero, see Brink [13]. The values of ( dk) dt 0 and ( dk) dt n depend on the type of setpoints. Because the rst and last setpoint are static, as stated in section 5.2.1, both values can be set at zero. 53

66 To determine the unknown parameters, the start time t 0 is zero and the endtime t n is estimated. This results in 6 unknown parameters and 6 independent equations. The 6 parameters can easily be determined using the 6 independent equations. The time mapping is calculated in the Matlab script shown in Appendix G Derivatives The time derivatives of the path are required to determine the feasibility of the generated trajectory regarding the constraints of the hydraulic motion system. The rst, second and third order time derivative of the path determine respectively the velocity, acceleration and jerk proles. The magnitudes of the generated velocity, acceleration and jerk may not exceed the magnitude of the constraints, see section The path of the trajectory is calculated in section and is described with equation 5.3. The path derivatives with respect to k can easily be calculated and will be indicated with Φ i(k). The path derivative with respect to t will be indicated with Φ i (t). The rst, second and third order derivatives with respect to k are described with equations 5.6, 5.7 and 5.8. Φ i(k) = P i k (5.6) Φ i (k) = P i k (5.7) Φ i (k) = P i k (5.8) The time derivatives can be calculated by applying the chain rule. The time derivatives are derived in Appendix F. The derivation leads to the following equations: 1. Velocity: 2. Acceleration: 3. Jerk: Φ i (t) = P i k dk dt Φ i (t) = P i (k... Φ i (t) = P i (k 3 dk dt 2 dk dt (5.9) + k d2 k dt 2 ) (5.10) + 3k dk d 2 k dt dt + d3 k 2 k dt ) (5.11) 3 The time derivatives are calculated with the Matlab script shown at Appendix G.1.6. The calculated time derivatives of the path are compared with the entered dynamic constraints. The comparison reveals if it is possible to execute the trajectory, with the hydraulic motion system, within the estimated endtime. If not, a correction factor will be calculated and used for optimizing the estimate endtime to a denite endtime. The optimization will be discussed in the next section. 54

67 5.4 Optimization The optimization methode is based on the comparison between the time derivative pro- les and the corresponding dynamic constraints. The rst order time derivative of the path corresponds to the velocity constraint. The second order time derivative corresponds to the acceleration constraint. And the third order time derivative corresponds to the jerk constraint. The endtime will be optimized so at least one dynamic constraint becomes active. With the phrase "active constraint" is meant that the velocity, acceleration or jerk prole reaches the same value as the corresponding dynamic constraint, within a certain tolerance margin. That dynamic constrained is called active. To accomplish an active constrain the following actions are performed; 1. The maximum absolute values of the time derivative proles are calculated. The derivatives are calculated for a discrete number of positions. The position where the derivative is maximal is regarded as the maximum absolute value. The accuracy of this method is limited by the discretization step, since the magnitude of the derivative prole can exceed the corresponding dynamic constraints between two discrete positions. This will not be visible with this methode. However it is assumed that most constrains are determined within a certain safety margin, therefore the methode is assumed to be adequate. 2. The relative error between the maximum absolute value and the dynamic constraints are determined. The most critical error of each order derivative is determined with respect to both actuators, see equation 5.12 ɛ 1 = (ˆv ẋ)/ˆv (5.12) ɛ 2 = (â ẍ)/â ɛ 3 = (ĵ... x )/ĵ where ˆv, â and ĵ are the the velocity, acceleration and the jerk constraints. The most critical constraint is selected with; ɛ i = min(ɛ i ) with i = [1, 2, 3] (5.13) 3. The most critical error is transformed into an endtime correction factor. According to Brink [13], the transformation of the error to the correction factor is determined by; The endtime correction factor is selected with; ν 1 = 1 ɛ 1 (5.14) ν 2 = 1 ɛ 2 ν 3 = 3 1 ɛ 3 ν = max(ν 1, ν 2, ν 3 ) (5.15) 55

68 4. A new endtime is calculated by multiplying the old endtime with the correction factor. With the new endtime a new time mapping will be made and new derivatives will be calculated. The endtime is optimized if the value of one derivative is equal to the value of the corresponding constraint, within a certain tolerance. If not, the iteration loop continues till that condition is reached. The avoidance of geometric constraints can be done by inserting proper setpoints, afterwards it can be checked if the trajectory satises. 5.5 Results The Trajectory Generator, derived from the generic Generator Toolbox treated in Brink [13], is adjusted to the hydraulic motion system. In this section the generator will be tested and the results will be discussed. The hydraulic motion system will be converted into a model of an excavator. Therefore a typical excavation task will be simulated. The trajectory generated with the toolbox, simulates the trajectory of the excavation task shown in gure 5.1. The simulation is shown in gure 5.4. Figure 5.4: End-Eector Path x in t CS domain 56

69 The trajectory includes the following steps of the excavation tasks: I The movement of the bucket to the ground. II The digging operation and the lifting of the bucket. III The movement of the bucket back to starting position. The movement of the bucket to the ground is simulated with the path between the setpoints depicted with (1) and (3). The digging operation and lifting of the bucket is simulated with the path between the setpoints (3) and (7). The movement of the bucket to starting point is simulated with the path between the setpoints (7) and (1). Setpoints (1), (3) and (7) are called static setpoints, the other setpoints are dynamic setpoints. It can be seen from the gure that the path of the end-eector include the setpoints as desired. The trajectory consists of three sections. The section between the setpoints (3) and (7) will be further claried. As shown in gure 5.4 the section consists of four segments viz.; segment (3)-(4), (4)-(5), (5)-(6) and (6)-(7). For each segment a spline path is dened in Joint Space spline domain ( k JS), as discussed in section The spline paths are dened with respect to a parameter k, over an interval [0,1]. Merging the four spline paths by adding up the intervals, results in the path Φ(k) in k JS domain for that section of the trajectory. The path Φ(k) is shown in gure 5.5. The path of the horizontal actuator is indicated with Φ p (k) and Figure 5.5: Path Φ(k) in k JS domain the path of the tilt actuator with Φ m (k). When time information is added to the paths, 57

70 they can be expressed as Φ i (t) in Joint Space time domain ( t JS), see gure 5.6. The gure also illustrates the rst, second and third order time derivative proles. At the Figure 5.6: Path Prole Φ(t) in t JS domain upper left corner of gure 5.6 the paths of the pistons are plotted against the time. The black vertical continuous lines distinguish the four segments. The red vertical interrupted line indicates the time where the rst derivative of one of the paths equals the velocity constraint of the corresponding piston. As mentioned earlier the prase "active constrained" is valid at that instant. At the upper right corner of gure 5.6 the rst order time derivative proles of the paths are depicted. The velocity constraints are illustrated with the blue interrupted vertical lines. It can be seen from the gure that the rst order time derivative prole of the horizontal actuator indeed equals the velocity constrained. The lower left gure illustrates the second order time derivative proles. The acceleration constraints are not illustrated while the constraints have larger values than shown on the current scale of the y-axle. The lower right gure illustrates the third order time derivative proles. The jerk constraints are also not illustrated. 58

71 5.6 Summary A trajectory generator toolbox has been developed for the hydraulic motion system. The trajectory generator toolbox can generate a reference trajectory involving a path, velocity, acceleration and jerk prole. The reference trajectory is optimized till at least one constrained is active (the phrase "active constrained" is explained at section 5.4). The toolbox enables to investigate the feasibility of a trajectory before operated on the hydraulic motion system. It can be regarded as an important tool for studying the possibilities to convert the hydraulic motion system into an autonomous excavator. An additional Graphical User Interface could be programmed to make the toolbox more user friendly. 59

72 Chapter 6 Final Results To be able to study the possibilities of autonomous excavation, the hydraulic motion system is converted into a model of an excavator. This makes it possible to simulate an excavation task with aid of the hydraulic motion system. In chapter 1 to 4, the hydraulic motion system is investigated and a numeric model of the system is created in Matlab Simulink. In chapter 5 a trajectory generator toolbox is developed, the toolbox can generate a trajectory based on user dened setpoints. With aid of the trajectory generator toolbox and the control system, discussed in section 3.2, it is possible to execute an autonomous task with the hydraulic motion system. Hence, the end-eector of the hydraulic motion system can simulate the bucket of an autonomous controlled excavator. 6.1 Simulations This section discusses the simulation of an excavation. Section discusses the results of the simulation executed with the hydraulic motion system and section discusses the results of the simulation executed with the Matlab Simulink model. The results of both simulations will be compared in section Hydraulic Motion System This section discusses the response of the hydraulic motion system when simulating the excavation task shown in gure 5.4. Figure 6.1 illustrates the generated trajectory and the trajectory of the end-eector of the hydraulic motion system. The trajectories are shown in Cartesian Space. Figure 6.2 illustrates the motion of the pistons of the actuators and the position error of the trajectory with respect to the reference trajectory. The error of the horizontal actuator is indicated as e p and the error of the tilt actuator is indicated as e m. To evaluate the results, the prole of the error is studied. The maximum absolute error and the relative error are shown in table 6.1. It can be concluded from the error prole that it is related to the velocity. Namely, high piston velocity result in a large error, independent of the direction. High accuracy is usually not demanded when operating an excavator. The relative 60

73 Figure 6.1: Generated Trajectory vs Hydraulic Motion System Figure 6.2: Response of the Hydraulic Motion System error is smaller than 10 percent and therefore assumed to be negligible. However when higher accuracy is desired a better controller can bring a solution. At Rodermond [10] controllers are developed which are applicable for the hydraulic motion system. It is 61

74 Table 6.1: Trajectory Error Actuator Max. Absolute Error Relative Error Horizontal 18 [mm] 6% Tilt 14 [mm] 9% recommended to investigate the controllers, since it could result in better performance Numeric Model This section illustrates the response of the numeric Matlab Simulink model, when simulating the similar excavation task. In the following section, the response will be compared with the response of the hydraulic motion system to verify the numeric model. Figure 6.3 illustrates the generated trajectory and the trajectory simulated by the model. Figure 6.3: Generated Trajectory vs Numeric Model Figure 6.4 shows the simulation in Joint Space time domain. The gure illustrates the positions of the pistons and the position error with respect to the generated trajectory. The largest absolute error and the relative error are shown in table

75 Figure 6.4: Response of the Numeric Model Table 6.2: Trajectory Error Actuator Absolute Error Relative Error Horizontal 36 [mm] 17% Tilt 10 [mm] 6% Numeric Model vs. Hydraulic Motion System The response of the numeric model, gure 6.4, agrees with the response of the hydraulic motion system, gure 6.2. The error is of the same order of magnitude and shows the same prole. However, the error of the model is rather larger and shifted in time compared to the error of the hydraulic motion system. The dierences between the two responses is presumed to be caused by inaccuracies at the parameter identication (section 4.2.2) or friction model. Comparing gure 6.2 with gure 6.4 shows that the prominent dierence between the response of the model and the motion system appears around t=15 and t=30 seconds at the horizontal actuator. The reason for these deviations seems to be inaccuracies at the friction model. As mentioned at section 4.3.2, frictional eects are very dicult to measure so the modeled friction can deviate from the real friction. Since the velocity of the horizontal actuator has maximum value around these points, the viscous friction dominates. Therefore the largest contribution to the error, between the numeric model 63

76 and the hydraulic motion system, is assumed to be caused by an inaccurate viscous friction coecient. Nevertheless, the model corresponds well with the motion system and is hereby veried. The numeric model can be used for further research about excavation tasks. 6.2 Summary As a nal result of this thesis, a nonlinear numeric model of the hydraulic motion system and a trajectory generator have been developed. The numeric model of the hydraulic motion system is developed to save time and costs on realtime simulations. To investigate a trajectory with the model is less expensive, faster and takes no time to prepare. The model is veried with aid of the hydraulic motion system. After numerous simulations, the hydraulic servovalve controlling the horizontal actuator, seemed to be the limiting component. As mentioned in chapter 5, the performance of the hydraulic motion system is limited by the active constrained. The simulations indicated that in most circumstances, the velocity prole of the horizontal actuator contained the active constrained. The velocity of the piston can be directly coupled to the maximum volume ow q N, see equation 4.7. The maximum volume ow q N is related to the dimensions of the valve e.g. valveport. Replacing the valve controlling the horizontal actuator with a bigger type of valve could result in better performance of the hydraulic motion system. 64

77 Chapter 7 Conclusions and Recommendations 7.1 Conclusions As nal result of this thesis, a numeric nonlinear model of the hydraulic motion system and a trajectory generator have been developed. The hydraulic motion system represents an excavator and can be autonomous controlled with aid of the trajectory generator. A rst step of converting the hydraulic motion system into the boom-bucket construction of an excavator is investigated. The hydraulic motion system can represent a down scaled boom-bucket construction. The hydraulic system is studied thoroughly and described with mathematical equations. Aspects like conguration and working space are compared with a small excavator. Unfortunately an extensive research at excavation tasks was nor realizable in the time span of one graduation project. However, this thesis contains a well based fundament for further investigations at autonomous controlled excavation. A numeric model of the motion system is developed to investigate tasks without performing them in realtime environment. The model saves time and costs compared to realtime simulations. To investigate an excavation task with the help of a model is faster, less expensive and takes shorter time to prepare. The numeric model is veried with aid of the hydraulic motion system. After numerous simulations, the hydraulic servovalve controlling the horizontal actuator, seemed to be the limiting component. Replacing it with a bigger type of valve, results in better performance when simulating an excavation task. 7.2 Recommendations At this state, the numeric model can simulate the path of an excavation task. Interaction between the excavating tool and soil are not discussed. During a digging trajectory or by repetitive digging the forces acting on the bucket can vary. The variation occurs 65

78 due to dierent layers of soil, contaminations in the soil, excavation dept etcetera. The numeric model can simulate these variations if the load mass array is adjusted. It is recommended to add a subsystem to the Matlab Simulink model, which permits a variable load mass array as input. The nonlinear model contains a friction model. It is dicult to accurately determine friction of a hydraulic system. The friction model is a rst order estimation. Using the hydraulic motion system for studies at excavation tasks, the friction model appears to be accurate enough. However, further research about the friction model could be interesting. When further investigating excavation tasks, simple trajectory control is insucient for automatically controlling the digging process. Feedforward control can be used to reduce the error, when the variations of the digging process are known. At Feedforward control, the dynamic mass and acceleration of the pistons must be known in advance. The dynamic mass can be calculated with the mass moment of inertia's. The acceleration of the pistons are calculated with aid of the trajectory generator toolbox. It is recommended to implement Feedforward design. 66

79 Bibliography [1] J.J. Craig. Introduction to Robotics. Addison-Wesly Publishing Company, Inc., [2] C. Croarkin. Handbook of Statistical Methods. [3] E.A. Croft. Smooth and Time-Optimal Trajectory Planning for Industrial Manipulators along Specied Paths. University of Britisch Columbia, Columbia, [4] G.F. Franklin. Feedback Control of Dynamic Systems. Prentice-Hall Inc., [5] A. Kraker. Mechanical Vibrations. Shaker Publishing BV, [6] K.F. Martin. Stability and step response of a hydraulic servo. Journal of Mechanical Engineering Science, 12, [7] H.E. Merritt. Hydraulic Control Systems. John Wiley and Sons, Inc., [8] R. Poley. Dsp control of electro-hydraulic servo actuators [9] Nguyen Hong Quang. Robust low level control of robotic excavation. Master's thesis, University of Sydney, [10] S. Rodermond. Modelling, identication and control of a multivariable hydraulic servo system. Master's thesis, Eindhoven university of technology, The Netherlands, [11] M. Steinbuch. Onderzoek hydraulische servosystemen , [12] T. Sugiyama. Gain scheduling control for electro-hydraulic servo system considering time-delay modeling error [13] S.N. van den Brink. Trajectory generation for a four axis roboth with linear kinematics. Master's thesis, Eindhoven university of technology, The Netherlands, [14] T.J. Viersma. Analysis, Synthesis and Design of Hydraulic Servosystems and Pipelines. Delft University of Technology,

80 Appendix A Parameters The model of the hydraulic motion system consists of several Simulink models and Matlab script les. The models and les are used to analyse and describe the hydraulic motion system This Appendix shows the script le which contains the parameters. 1 % %%%%%%%%%%%%%%%%%%%%%%% % %% mimoparameters.m %%% % %%%%%%%%%%%%%%%%%%%%%%% % %% Zwaartepunt MIMO arm, inclusief Load m = %%% 6 M =[ * * ]; Mt = sum (M); X =[700; 375; 1075;359;1108;700;700;1450; 50; 18;700;1450]; Y =[25;25;25;274;274;245;477.5;25;25;25; -20; -50]; 11 Zx =(( M*X)/ Mt +40) /1000; Zy =(( M*Y)/Mt -25) /1000; % %% Kinematical Parameters %%% 16 m1 = sum ( M); % mass of the triangle [ kg ] m2 = 26; % mass of the sledge [ kg ] mload = ; % mass of the load [ kg ] alpha = atan (45/740) ; % angle between triangle and point B [ rad ] 21 gamma = atan ( Zy / Zx ); % angle between centerline beam and m1 [ rad ] theta_min = ; % calculated with kinematics. m [ rad ] theta_max = ; l = 1.490; % lenght of the triangle [ m] 26 l1 = 0.740; % right half of triangle from point B [ m] l2 = l - l1 ; % left side of triangel [ m] l3 = l1 / cos ( alpha ); % length BC [m] lm1 = ( Zx ^2+ Zy ^2) ^0.5; % length M1B [ m] b = ; % length of vertical c [ m] 31 d = ; % height between lower joint and joint [ m] xmax = ; % max. distance between point A and B [ m] g = 9.81; % gravitational acceleration [ ms ^{ -2}] Jm1 = m1 * lm1 ^2; % inertia m1 from point B [ kgm ^2] 36 % %%%%%%%%%%%%%%%% % %% ACTUATORS %%% % %%%%%%%%%%%%%%%% % General hydraulic parameters 41 Ps = 70 e5 ; % system presure [ Pa ] E = 1 e9 ; % oil stiffness [ Pa ] 68

81 % Parker Hannifin parameters Mp = 185.4; % virtuele massa Parker centerposition [ kg ] 46 Ap = 10.7e -4; % piston surface [ m ^2] Sp = 0.4; % piston stroke [ m] Cp = 1e -11; % leakage coefficient Vtp = Ap * Sp ; % total volume [ m ^3] V0lp = 2.4e -4; % dead oil volume of left compartiment [ m ^3] 51 V0rp = 3e -4; % dead oil volume of right compartiment [ m ^3] % MTS 244 parameters Mm = Jm1 / l3 ^2; % virtuele massa MTS [ kg ] Am = 13.5e -4; % piston surface [ m ^2] 56 Sm = 0.284; % piston stroke [ m] Cm = 0; % leakage coefficient Vtm = Am * Sm ; % total volume [ m ^3] V0lm = 2.052e -5; % dead oil volume of left compartiment [ m ^3] V0rm = 2.052e -5; % dead oil volume of right compartiment [ m ^3] 61 % %%%%%%%%%%%%% % %% VALVES %%% % %%%%%%%%%%%%% xmax_valve = 1; % maximum valve spool position [ -1,1] 66 i_v = 1/0.045; % nomalization inputcurrent valves [1/ A] % MTS valve properties, valve mounted on Parker i_ap = 0.045/5; % 30e -3 voltage to amperage [ A/ V] i_sp = 5/0.4; % conversion from position to voltage [ V/ m] 71 zeta_p = 1; % valve dimensionless damping coefficient [ -] wn_p = 942; % valve undamped eigenfrequency [ rad / s] kq_p = 3.8/60/1 e3 ; % valve gain [ m3 / s] Hc_p = i_sp * i_ap ; % gain 1 c1p = 1/ wn_p ^2; % transfer function constant 76 c2p = 2* zeta_p / wn_p ; % transfer function constant % MTS valve properties, valve mounted in MTS i_am = 0.045/5; % 28.33e -3 from voltage to amperage [ A/ V] i_sm = 5/0.25; % conversion from position to voltage [ V/ m] 81 zeta_m = 1; % valve dimensionless damping coefficient [ -] wn_m = 565; % valve undamped eigenfrequency [ rad / s] kq_m = 19/60/1 e3 ; % valve gain = qn / xmax_valve [ m3 / s] Hc_m = i_am * i_sm ; % gain 2 c1m = 1/ wn_m ^2; % transfer function constant 86 c2m = 2* zeta_p / wn_m ; % transfer function constant % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% Transfer Functions Actuators %%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 91 % Parker C3p = 0.85; C4p = 1.6e -5; C5p = 1.2e -3; 96 C6p = ( kq_p * Mp )/ Ap ^2; % MTS C3m = 1; C4m = 1.0e -5; 101 C5m = 4.0e -3; C6m = ( kq_m * Mm )/ Am ^2; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% Limits For Model ; Calculation max / min speed %%% 106 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vmax_p = kq_p / Ap ; % max. speed Parker ( extension ) [ m/ s] 69

82 vmin_p = - kq_p / Ap ; % min. speed Parker ( retraction ) [ m/ s] vmax_m = kq_m / Am ; % max. speed MTS ( extension ) [ m/ s] 111 vmin_m = - kq_m / Am ; % min. speed MTS ( retraction ) [ m/ s] omega_max = vmax_m / l1 ; % angular speed point B, approximation [ rad / s] omega_min = vmin_m / l1 ; % angular speed point B, approximation [ rad / s] 70

83 Appendix B Calculation of the Kinematical Models This Appendix shows the Matlab script les which describe the forward and backward kinematical models. B.0.1 Forward Kinematics % %%%%%%%%%%%%%%%%%%%%%%%%%%% 2 % %% forward_kinematics.m %%% % %%%%%%%%%%%%%%%%%%%%%%%%%%% function [LX,LY, THETA, PHI ]= forward_kinematics (X1, X2 ); 7 run g :\ Parameters. m % Geometric Calculations LX = []; LY = []; THETA = []; PHI = []; for i = 1: min ( max ( size ( X1 )), max ( size ( X2 ))); 12 x1 = abs ( X1 (i)); x2 = X2 (i); alpha = atan (0.045/0.740) ; gamma = atan ( Zy / Zx ); delta = atan ( x1 /d); 17 e = x1 / sin ( delta ) ; beta = acos (( e ^2+ l3 ^2 - x2 ^2) /(2* e* l3 )); theta = pi - delta - beta - alpha - gamma ; THETA = [ THETA, theta ]; lamda = acos (( e ^2+ x2 ^2 - l3 ^2) /(2* e* x2 )); 22 phi = lamda - delta ; PHI = [ PHI, phi ]; Lx = -x1 +l* sin ( theta + gamma ); Ly = l* cos ( theta + gamma ); 27 LX = [LX, Lx ]; LY = [LY, Ly ]; end LX = LX '; LY = LY '; THETA = THETA '; PHI = PHI '; B.0.2 Backward Kinematics 1 % %%%%%%%%%%%%%%%%%%%%%%%%%%% % %% Backward_kinematics.m %%% % %%%%%%%%%%%%%%%%%%%%%%%%%%% 6 function [X1,X2, THETA, PHI ]= backward_kinematics (LX, LY ); run g :\ Parameters. m; 71

84 % Geometric Calculations X1 = []; X2 = []; THETA = []; PHI = []; 11 for i = 1: min ( max ( size ( LX )), max ( size ( LY ))); Lx = LX (i,1) ; Ly = LY (i,1) ; alpha = atan (0.045/0.740) ; gamma = atan ( Zy / Zx ); 16 theta = acos ( Ly /l) - gamma ; x1 = Lx -l* sin ( theta + gamma ); delta = atan ( abs ( x1 )/d); e = abs ( x1 )/ sin ( delta ); beta = pi - theta - delta - alpha - gamma ; 21 x2 = sqrt (e ^2+ l3 ^2 -(2* e* l3 )* cos ( beta )); lamda = acos (( e ^2+ x2 ^2 - l3 ^2) /(2* e* x2 )); phi = lamda - delta ; X1 = [X1, x1 ]; X2 = [X2, x2 ]; THETA = [ THETA, theta ]; PHI = [ PHI, phi ]; 26 end X1 = X1 '; X2 = X2 '; THETA = THETA '; PHI = PHI '; 72

85 Appendix C Models This Appendix discusses the Matlab Simulink of the hydraulic motion system. C.1 Numeric Matlab Simulink Model As shown in gure C.1, the numeric model can be separated into four dierent modules. Figure C.1: Numeric Matlab Simulink Model 1. The module containing the proportional controllers. The controller x p is used for controlling the horizontal actuator and the controller x m is used for controlling the tilt actuator. 2. The module containing the nonlinear equations which describe the valve controlled actuators. The module for the horizontal actuator is indicated with (P) and for the tilt actuator with (M). 73

86 3. The module containing the friction model, indicated with (f) and (f1). 4. The module containing the nonlinear Lagrangian Equations of Motion. The kinematical module at the right hand side of the Lagrangian module is added for the visual interpretation when using the numeric model. C.1.1 Controller Module The input of the controller module is the error in millimeters, indicated with (e). The output is the valve current in ampere, indicated with (i). The conversion from spool position into input current of the valve is integrated into the module. C.1.2 Valve Controlled Actuator Module Figure C.2: Detail: Nonlinear Model of a Valve controlled Actuator The nonlinear model of a valve controlled actuator is derived in chapter The Matlab Simulink model is shown in gure C.2, the sign-function of equation 2.10 is replaced by a subsystem, indicated with (sign). The subsystem contains a function to approximate the sign-function, resulting in a non innitive derivative. The function is shown in equation C.1. 2 π arctan(e dx m) e >> 1 (C.1) The oilstiness described by equation 4.5 is modeled in the subsystem indicated with (oilstiness). C.1.3 Friction Module Figure C.3 shows the model used to simulate the friction of the hydraulic actuators. The magnitude of the friction coecients C c and C v are derived in section The model is based on equation 4.7, the sign-function is again replaced by a subsystem containing the softer equation C.1. 74

87 C.1.4 Lagrangian Module Figure C.3: Friction Model As shown in gure C.4, the input for the Lagrangian module are the actuator forces. The output are the position and velocity of the pistons. The two nonlinear LEM's described in section are used to convert the actuator forces into the length x 1 and angle θ, shown in gure 3.8. Figure C.4: Lagrangian Model The positions and velocities of the pistons can be calculated with aid of the subsystems (s1) and (s2). The rst subsystems converts x 1 and θ into ϕ and x m. The following 75

88 equation is used: ϕ = l 3 sin(θ + α + γ) x 1 d + l 3 cos(θ + α + γ) (C.2) The second subsystem is used to calculate the velocity of the tilt actuator. The input of the subsystem are dx1, θ, α, γ, ϕ and dθ the output is the velocity of the tilt actuator dx m. The velocity dx m results from two vectors, the calculation is divided in two parts. dx m = dx m,a + dx m,b (C.3) The velocity dx m,a is caused by the velocity of the piston of the horizontal actuator, the velocity dx m,b is caused by the velocity of the piston of the tilt actuator. The velocity dx m,b can be calculated with the angular speed dθ, see gure C.5. Figure C.5: Velocity Tilt Actuator The nal equation to calculate the velocity of the vertical actuator can be described with equation C.4. dx m = dx m,a + dx m,b dx m = dx1 sin(ϕ) + ω l 3 cos(i) dx m = dx1 sin(ϕ) + dθ l 3 cos(0.5π (θ + γ) + ϕ) (C.4) 76

89 Appendix D Friction Measurements This Appendix discusses the friction measurements at the hydraulic motion system. D.1 Measurements Coulomb Friction Table D.1 shows the coulomb friction measurements by means of pressure dierences over the chambers of the actuators. The pressure dierence is measured at two conditions. Firstly when the actuator is in static situation, indicated with P 0, and secondly when the piston breaks away and starts to move, indicated with P 1. An example of a measurement of the coulomb friction is shown in gure D.1. The gure Figure D.1: Estimate Coulomb Friction indicates a pressure dierence P 0 and a pressure dierence P 1. The pressure dierence P 0 can be explained with the fact that the actuators are subjected to gravitational forces, see section

90 Table D.1: Coulomb Friction Pressure Measurements [bar] Horizontal Tilt measurement P 0 P 1 P cp P 0 P 1 P cm mean Theoretically the pressure dierence P 0 should have a constant value depending on the position of the piston. After several measurements, the occurring P 0 seemed to be sensitive for inuences like; oil temperature and the direction of approaching the operating point. The piston breaks away and starts to move at a pressure dierence P 1. Theoretically the force generated by P (P 1 P 0 ) is equal to three forces; P A = F c + F v + M i a (D.1) As mentioned in section 4.3.2, the coulomb friction is measured at a low and constant velocity of the piston. Due to the low velocity the viscous friction can be neglected, so the second term on the right hand side of equation D.1 is assumed to be zero. Because the velocity is constant the third term on the right hand side is also zero. Viscous Friction Table D.2 shows the coulomb friction measurements. The force generated by P is calculated according equation D.1. The third term on the right hand side is again zero, but obviously the second term on the right hand side is not. An example of a measurement of the viscous friction is shown in gure D.2. 78

91 Table D.2: Viscous Friction Pressure Measurements [bar] Horizontal Tilt measurement xp[m/s] P 0 P 1 P vp xm[m/s] P 0 P 1 P vm mean The gure shows a peak value of the pressure dierence when the piston starts to move. The peak value can be explained with the pressure dierence which must be generated to accelerate the piston, see the third term on the right hand side of equation D.1. The pressure dierence decreases after the piston reaches a constant velocity. To derive the viscous friction the pressure dierence P between the value just before and after the peak value is calculated. Verication A simple verication of these measurements can be performed by calculating the forces of the actuators and comparing them with the theoretical values. The pressure difference at stagnation is assumed to be caused by gravity. With equations (3.6) the theoretical forces for the steady state condition can be calculated. The average pressure dierence at stagnation is P 0 = 8, 07 [bar]. The actuator force derived from the measurements are: F A = P 0 A m = 1089[N] The theoretical force for the steady state condition is F A = 1102 [N]. The relative error between the measurements and the theoretical value is smaller than 2%, so it can be 79

92 Figure D.2: Estimate Viscous Friction stated that the measurements are satisfying. D.2 Normality Test The assumption of normality is tested with the Anderson-Darlin Normality test. The test checks whether the null hypothesis is valid or not. The null hypothesis "H0" and the alternative hypothesis "Ha" are stated as follows; 1. H0: The data of the measurements follow a normal distribution. 2. Ha: The data of the measurements do not follow the normal distribution. The test uses a "P-value" to measure whether the null hypothesis is valid. If the P-value is less than the generally-accepted standard of 0.05, the null hypothesis is assumed to be false and dierences between the samples are likely to exist. For a complete description of the Anderson-Darlin Normality test see [2]. The results of the normality tests of the coulomb measurements are shown in gure D.2. The P-values are indicated in table D.3. The value's are above the standard of 0.05, thus the null hypothesis can not be rejected. Both coulomb friction measurements are part of a normal distributed population. The results of the normality tests of the viscous measurements are shown in gure D.2. The P-values of the viscous friction measurements are indicated in table D.4. The values are again above the standard of So the viscous friction measurements are also part of a normal distributed population. 80

93 (a) Horizontal Actuator (b) Vertical Actuator Figure D.3: Normality Test Coulomb Friction Measurements Table D.3: P-values Normality Test I Actuator P-value [-] Horizontal actuator Vertical actuator

94 (a) Horizontal Actuator (b) Vertical Actuator Figure D.4: Normality Test Viscous Friction Measurements Table D.4: P-values Normality Test II Actuator P-value [-] Horizontal actuator Tilt actuator