Kinematic Analysis and Inverse Dynamics-based Control of Nondeterministic Multibody Systems

Transcription

1 Kinematic Analysis and Inverse Dynamics-based Control of Nondeterministic Multibody Systems Item Type text; Electronic Thesis Authors Sabet, Sahand Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 06/06/ :06:54 Link to Item

2 KINEMATIC ANALYSIS AND INVERSE DYNAMICS-BASED CONTROL OF NONDETERMINISTIC MULTIBODY SYSTEMS By: Sahand Sabet Copyright Sahand Sabet 2016 A Thesis Submitted to the Faculty of the DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE WITH A MAJOR IN MECHANICAL ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA 2016

3 Statement by Author The thesis titled Kinematic Analysis and Inverse Dynamics-based Control of Nondeterministic Multibody Systems prepared by Sahand Sabet has been submitted in partial fulfillment of requirements for a master s degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that an accurate acknowledgment of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Signed: Sahand Sabet APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below: Defense Date Mohammad Poursina Assistant Professor at the 05/06/2016 Department of Aerospace and Mechanical Engineering i

4 Acknowledgements I would like to sincerely thank Professor Mohammad Poursina for all of his mentorship, guidance, and thoughtfulness over the past two years. What I have achieved is greatly due to your continuous support. I am very grateful for what you have taught me in the field of uncertainty analysis, control, and multibody systems. I can t thank you enough for all you have done. Professor Nikravesh, it was a great pleasure to be one of your students. Your impressive teaching skills in the field of multibody dynamics has been a basis for my understanding in this area. It is astonishing how you simplified this complex field of science into a sensible subject. You are a great person and a true engineer. I would like to also thank Professor Gaylor for serving on my committee. I appreciate your time and your interest in the topic of this thesis. At the end, a special thanks to my family, friends, and everyone who has taught me. ii

5 Contents Statement by Author i Acknowledgements ii Contents List of Figures Abstract iii vi viii 1 Introduction SCOPE OF RESEARCH Polynomial Chaos Expansion For Uncertainty Analysis INTRODUCTION POLYNOMIAL CHAOS EXPANSION FOR UNCERTAINTY ANALYSIS 5 3 Forward Kinematic Analysis of Non-deterministic Articulated Multibody Systems in Polynomial Chaos Expansion Scheme INTRODUCTION DETERMINISTIC EQUATIONS FOR CONSTRAINED FORWARD KINEMATIC ANALYSIS Position Analysis of Deterministic Systems with Closed Chains Velocity Analysis of Deterministic Systems with Closed Chains Acceleration Analysis of Deterministic Systems with Closed Chains Position Analysis of Deterministic Systems with Open Chains Velocity Analysis of Deterministic Systems with Open Chains Acceleration Analysis of Deterministic Systems with Open Chains NONDETERMINISTIC EQUATIONS FOR CONSTRAINED FORWARD KINEMATIC ANALYSIS OF SYSTEMS WITH CLOSE CHAINS Position Analysis of Nondeterministic Systems with Closed Chains Velocity Analysis of Nondeterministic Systems with Closed Chains 21 iii

6 Contents iv Acceleration Analysis of Nondeterministic Systems with Closed Chains SIMULATION RESULTS FOR A SYSTEM WITH CLOSED CHAINS Accuracy Analysis of a Four-bar Mechanism with a Single Random Variable Efficiency Analysis of a Four-bar Mechanism with a Single Random Variable Accuracy Analysis of a Four-bar Mechanism with Multiple Random Variables Efficiency Analysis of a Four-bar Mechanism with Multiple Random Variables NONDETERMINISTIC EQUATIONS FOR CONSTRAINED FORWARD KINEMATIC ANALYSIS OF SYSTEMS WITH OPEN CHAINS Position Analysis of Nondeterministic Systems with Open Chains Velocity Analysis of Nondeterministic Systems with Open Chains Acceleration Analysis of Nondeterministic Systems with Open Chains SIMULATION RESULTS FOR A SYSTEM WITH OPEN CHAINS Accuracy Analysis of a SCARA Robot Efficiency Analysis of a SCARA Robot CONCLUSIONS Framework for the Computed Torque Control of Nondeterministic Multibody Systems INTRODUCTION COMPUTED TORQUE CONTROL LAW FOR DETERMINISTIC SYS- TEMS Controller Design COMPUTED TORQUE CONTROL LAW FOR NONDETERMINISTIC SYSTEMS Evaluating Nondeterministic Generalized Driving Forces SIMULATION RESULTS Evaluating the Joints Motion Evaluating the Nondeterministic Generalized Driving Forces Using PCE and Monte Carlo Method Conclusions 63 A MATLAB Files 64 A.1 Monte Carlo Files for One-dimensional Uncertainty A.2 PCE Files for One-dimensional Uncertainty A.3 Monte Carlo Files for the Control of the SCARA Robot A.4 PCE Files for the Control of the SCARA Robot

7 Contents v B Results for the Four-bar Mechanism with Single Uncertainty 90 B.1 APPENDIX Bibliography 92

8 List of Figures 3.1 A schematic of kinematically constrained multibody with a cut-joint Schematic of a scara robot [1] The schematics of a four-bar linkage Logarithm of the relative error in the expected value of the states from different numbers of one-dimensional Monte Carlo simulations Logarithm of the relative error in the standard deviation of the states from different numbers of one-dimensional Monte Carlo simulations Logarithm of the relative error in the expected value of the states of the system from different orders of one-dimensional PCE Logarithm of the relative error in the standard deviation of the states of the system from different orders of one-dimensional PCE Expected value of θ 3 from different orders of two-dimensional PCEs and Monte Carlo simulations Expected value of ω 3 from different orders of two-dimensional PCEs and Monte Carlo simulations Expected value of α 3 from different orders of two-dimensional PCEs and Monte Carlo simulations Standard deviation of θ 3 from different orders of two-dimensional PCEs and Monte Carlo simulations Standard deviation of ω 3 from different orders of two-dimensional pces and Monte Carlo simulations Standard deviation of α 3 from different order of two-dimensional PCEs and Monte Carlo simulations Logarithm of the relative error in the expected value from different Monte Carlo simulation Logarithm of the relative error in the standard deviation from different Monte Carlo simulation Logarithm of the relative error in the expected value from different orders of PCE Logarithm of the relative error in the standard deviation from different orders of PCE Expected value of x 4 (m) Expected value of y 4 (m) Standard deviation of x 4 (m) Standard deviation of y 4 (m) vi

9 List of Figures vii 3.22 Relative error in the expected value of x 4 (m) Relative error in the expected value of y 4 (m) Relative error in the standard deviation of x 4 (m) Relative error in the standard deviation of y 4 (m) Schematic of a scara robot [1] Nonlinear and linear components of CTCL [2] Joints trajectories Joints trajectories Mean value of the generalized forces Standard deviation of the generalized forces

10 THE UNIVERSITY OF ARIZONA Abstract DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING Master of Science KINEMATIC ANALYSIS AND INVERSE DYNAMICS-BASED CONTROL OF NONDETERMINISTIC MULTIBODY SYSTEMS by Sahand Sabet

11 Abstract ix Multibody dynamics plays the key role in the modeling, simulation, design, and control of many engineering problems. In practice, such problems may be encountered with the existence of uncertainty in the system s parameters and/or excitations. As the complexity of these problems in terms of the number of the bodies and kinematic loops (chains) increases, the effect of uncertainty in the system becomes even more significant due to the accumulation of inaccuracies. Therefore, considering uncertainty is inarguably a crucial aspect of performance analysis of a multibody problem. In fact, uncertainty needs to be propagated to the system kinematics and dynamics for the better understanding of the system behavior. This will significantly affect the design and control process of such systems. For this reason, this research presents a detailed investigation on the use of the Polynomial Chaos Expansion (PCE) method for both control and kinematic analysis of nondeterministic multibody systems. In the first part of this thesis, a detailed formulation for the kinematics analysis of constrained multibody systems at the position, velocity, and acceleration levels in the PCE scheme is presented. Herein, multibody problems with kinematically closed chains and open chains are considered for the uncertainty propagation. This analysis is performed by projecting the governing kinematic constraint equations of the system onto the space of appropriate polynomial base functions. Time efficiency and accuracy of the intrusive PCE approach are compared with those from the traditionally used Monte Carlo method for several mechanical systems. The results demonstrate a drastic increase in the computation time of Monte Carlo method when analyzing complex systems with a large number of uncertain parameters, while the intrusive PCE provides better accuracy with much less computational complexity. The second part of this thesis considers the control of non-deterministic multibody systems using Computed Torque Control Law (CTCL). In the field of robotics, a required task may be to force the robot to follow desired paths. Controlling such systems usually leads to difficulties since the dynamic equations of multibody problems are highly nonlinear. Computed Torque Control Law (CTCL) is able to overcome these difficulties by

12 Abstract x using feedback linearization to evaluate the required torque/force at any time to make the system follow a desired trajectory. In this research, a mathematical framework is introduced to apply the Computed Torque Control Law to a fully actuated multibody system with uncertainty. Surprisingly, it is shown that using this control scheme, uncertainty in geometry does not affect the closed-loop equations of the controlled system. Both the intrusive PCE method and the Monte Carlo approach are used to control a 4-DOF SCARA (Selective Compliance Assembly Robot Arm) robot where each arm is required to follow a specified trajectory. At the end, a comparison of the time efficiency and accuracy between the traditionally used Monte Carlo method and the intrusive PCE is presented. The results indicate that the intrusive PCE approach can provide better accuracy with much less computation time than the Monte Carlo method.

13 Chapter 1 Introduction Multibody systems are composed of rigid and/or flexible components constrained by different kinematic pairs. These systems have a wide applicability in real world engineering problems including: robotics, airplanes, molecular systems, and the automotive industry. Therefore, a deep understanding of the behavior of such problems is crucial for the better design and control of these systems. In real applications, articulated multibody systems are subjected to uncertainties. This may originate from the lack of knowledge about the physical and geometrical properties of the system such as mass, location of the mass center, damping coefficient, and spring stiffness. Uncertainty may also exist in mathematical modeling or in the measurement and control inputs. In articulated biomolecular systems, for instance, uncertainty may exist in the bond length and the coefficients used to model the force field. In practice, excitations applied to the system may be the source of uncertainty. This type of uncertainty is found in many electromechanical systems, such as robots and vehicles following rough trajectories or terrain. Uncertainty may also originate from the difference between the physical system and corresponding computational model. For instance, model reduction through the elimination of high frequency modes of motion of a system is a source of uncertainty. In complex systems with a larger number of stochastic elements, uncertainty can significantly affect the system s functionality. This problem highlights the inevitable role 1

14 Chapter 1. Introduction 2 of reliability assessment in the design and analysis of such systems. Great researches have been conducted to analyze the dynamics of multibody systems without uncertainty. However, including uncertainty in the design, control, modeling and simulation of multibody systems is relatively new. Polynomial chaos expansion (PCE) provides a powerful tool to appropriately propagate uncertainty through the system s kinematics, dynamics, and control leading to a more reliable design. In 2006, for the first time, Sandu and Ahmadian [3, 4] applied the PCE theory to study the uncertainty in multibody systems. In the work published in [5], the mass, center of mass, and tensor of inertia of the uncertain rigid body are replaced with random variables. Then, by applying the maximum entropy principle under the constraints expressed by the available information, prior probability distributions of the stochastic model are constructed. Bayesian networks have been used for the state estimation of mechanisms [6]. The method of divide-and-conquer algorithm [7] has been generalized to accommodate PCE to study the uncertainty propagation in multibody systems in a time effective manner [8]. In this thesis, the framework to use PCE in the kinematics and inverse dynamics-based control of multibody systems is provided in two separate chapters. In the first part of this research, the PCE method is extended and used for the detailed kinematic analysis of nondeterministic articulated multibody systems. This contains the mathematical formulations to form and solve the equations governing the kinematics of nondeterministic constrained multibody problems at the position, velocity, and acceleration levels. Computational costs, accuracy, and convergence of the PCE method for the forward kinematic analysis of a four-bar mechanism and a SCARA robot with uncertainty in the systems parameters will be compared with those from the Monte Carlo technique. The second part of this thesis considers the control of nondeterministic systems using Computed Torque Control Law (CTCL). PCE was first applied to control of nondeterministic systems in 2003 [9]. In 2009, Voglewede et al, applied Polynomial Chaos Theory to the control of SCARA robot manipulator with uncertainties in the mass of

15 Chapter 1. Introduction 3 the links of the system [1]. In that work, the authors treated the nonlinearities of the system by using Taylor s series approximation. Then they used a proportional-derivative (PD) controller to control the joints motion. In this research, the method of PCE is integrated with CTCL to control nondeterministic fully-actuated multibody systems. A mathematical framework is introduced to solve the nondeterministic equations of motion, while nonlinear terms are not linearized. At the end, the CTCL method is applied to a fully actuated SCARA robot in both PCE and Monte Carlo schemes. To conclude the analysis, a comparison between the time efficiency and accuracy of the Monte Carlo and PCE methods is provided. In chapters two and three, the joint coordinate method is used to develop the formulations in the PCE scheme; however, the presented approach can be applied to different types of coordinates such as body coordinates. 1.1 SCOPE OF RESEARCH This research focuses on the extension and use of Polynomial Chaos Expansion (PCE) method in the kinematics and inverse dynamics-based control of nondeterministic multibody systems. The PCE method is compared to traditionally used Monte Carlo approach in order to evaluate the accuracy and efficiency of both methods. It is shown that PCE is able to provide a better accuracy and efficiency compared to Monte Carlo specifically when the complexity of the system in terms of the number of uncertain parameters increases. The detailed comparison of Monte Carlo and PCE method is accomplished by providing a framework to introduce uncertainty to the kinematics and inverse dynamics-based control of multibody systems using PCE approach, then comparing it with the Monte Carlo method, demonstrating how these methods are different algorithmically,

16 Chapter 1. Introduction 4 analyzing how PCE is advantageous over Monte Carlo in predicting the system s behavior demonstrating how the tendency of PCE being more efficient than Monte Carlo will scale, investigating the source of efficiency of PCE method over Monte Carlo approach, from a theoretical perspective. In Chapter 2, Polynomial Chaos Expansion theory is introduced and a mathematic framework is provided to demonstrate how uncertainty can be introduced to different systems using PCE method. In Chapter 3, the PCE method is extended and used for the detailed kinematic analysis of nondeterministic articulated multibody systems. Systems with kinematically closedchains and open-chains are considered in this thesis. In Chapter 4, the method of PCE is integrated with Computed Control Torque Law (CTCL) to control nondeterministic fully-actuated multibody systems. A mathematical framework is introduced to solve the nondeterministic equations of motion. Finally, in Chapter 5, conclusions of this research are provided.

17 Chapter 2 Polynomial Chaos Expansion For Uncertainty Analysis 2.1 INTRODUCTION In this chapter, the method of PCE is introduced and a mathematical framework to use this approach for uncertainty analysis is provided. 2.2 POLYNOMIAL CHAOS EXPANSION FOR UNCER- TAINTY ANALYSIS In general, the methods to analyze the uncertainty in dynamic systems are categorized in two types: Intrusive and non-intrusive. In non-intrusive methods, uncertainty assessment is performed by studying multiple solutions of the original deterministic model. For instance, in Monte Carlo approach statistical properties of the response of the system are calculated using large number of independent simulation runs [10, 11]. This method provides a very good understanding of the dynamics of the system under uncertainty. However, it is computationally expensive since the estimation of the variance converges 5

18 Chapter 2. PCE FOR UNCERTAINTY ANALYSIS 6 with the inverse square root of the number of runs [3]. For multibody problems, in particular, as the complexity of the system increases, undesirable cost may be imposed on each simulation if the system s equations of motion are not formed and solved wisely. In intrusive methods, the distribution of the response due to the uncertainty propagation is provided through the formulation and analysis of a stochastic version of the original model. One of the most important intrusive approaches of uncertainty analysis is Polynomial Chaos Expansion (PCE) which will be described in the next section as it is the core of this thesis. Note that PCE can also be done in a non-intrusive manner through the so-called collocation methods [12, 13]. In these approaches, a set of points are defined to serve as a design of experiments which is used to find the values of the expansion coefficients. In its basic approach, non-intrusive PCE might require a large number of function evaluations [14]. For this reason, a sparse version of PCE was developed. PCE is a convenient and powerful tool for presenting uncertainty in nondeterministic systems. PCE was initially proposed by Wiener [15]. Hermite polynomials were applied in this method for the modeling of stochastic processes with Gaussian random variables. The generalized chaos has been studied and developed by Xiu [16] and applied to various continuous and discrete distributions using orthogonal polynomials from the so-called Askey-scheme. In addition, convergence in the corresponding Hilbert functional space has been established. This method, in which uncertainties in systems with large magnitudes can be adequately captured, has been successfully applied to different fields including stochastic finite elements [17], flow-structure interactions[18], thermo-fluid applications [19], nonlinear estimation [20], probabilistic robust control [21], systems biology [22], control [8], and solid mechanics [23]. The fact that the PCE method is successfully applied to the various fields [17]-[23] is due to the following advantages:

19 Chapter 2. PCE FOR UNCERTAINTY ANALYSIS 7 the polynomial chaos expansion methods, as opposed to non-intrusive samplingbased methods, desirably provide the distribution of the response due to the propagation of the uncertainty, using the polynomial chaos expansion approach, one can find coefficients for known orthogonal polynomial base functions, and easily compute the mean and variance of the output random variables of the system, as expressed in [4], the method allows the quantification of uncertainty distribution in both time and frequency domains. PCE provides a compact representation of random variables. Using this method, each stochastic response output and random input is projected onto the space of appropriate independent orthogonal polynomial base functions as [3, 24] R(t, ζ) = r i (t)λ i (ζ). (2.1) i=0 In this definition, r i consist of time-dependent modal coefficients, while Λ i (ζ 1, ζ 2,..., ζ n ) (usually denoted as λ i (ζ) in case of one-dimensional random variable) are time-invariant generalized polynomial chaos of order of n, in terms of multidimensional random variables ζ(ζ 1, ζ 2,..., ζ n ). These polynomials form a complete orthogonal basis for the Hilbert space of square integrable random variables [16]. The series shown in Eqn. (2.1) converges to any random process in L 2 sense [25]. In the application process, one truncates the infinite expansion at a finite number of terms as R(t, ζ) N t = r i (t)λ i (ζ). (2.2) i=0 In general, the polynomial chaos expansion includes a complete basis of polynomials up to a fixed total order specification. In this case, for a system with N u finite number of uncertain parameters ζ = (ζ 1, ζ 2,..., ζ Nu ), using an expansion of total maximum order

20 Chapter 2. PCE FOR UNCERTAINTY ANALYSIS 8 Table 2.1: Relation between base functions, supports, and distribution functions Distribution Density function Base function Support range Weight function 1 Uniform 2 Legendre [ 1, 1] 1 Beta Jacobi [ 1, 1] (1 x) α (1 + x) β (1 x) α (1+x) β 2 ( α+β+1)b(α+1,β+1) Normal 1 2π e x2 2 Hermite (, ) e x2 2 Exponential e x Laguerre [0, ) e x Gamma x α e x Γ(α+1) Generalized Laguerre [0, ) x α e x P yields the total number of terms in the summation [3, 26], i.e. N t + 1, as N t + 1 = (N u + P )!. (2.3) N u!p! Table 2.1 presents the orthogonal polynomial base functions used for different types of random variables and support ranges for a system containing a single random variable [16, 24]. For a nondeterministic system with multiple random variables (N u ), the polynomial base function Λ i (ζ 1, ζ 2,..., ζ Nu ) is a tensor product of one-dimensional base functions as [27] N u Λ i (ζ 1, ζ 2,..., ζ Nu ) = λ βij (ζ j ) i = 1,..., N t + 1, (2.4) j=1 where β is an (N t + 1) N u matrix including multi-indices satisfying the following [27] N u j=1 β ij κ i i = 1,..., N t + 1, (2.5) in which κ is an (N t + 1) 1 column matrix, the elements of which are natural numbers starting at 0 and ending at P. The elements of this column matrix will appear consecutively for a definite number of times. Defining σ h as the number of repetitions of the

21 Chapter 2. PCE FOR UNCERTAINTY ANALYSIS 9 element with value h in the matrix κ, one can evaluate this repetition number as σ h σ h = (N u + h)! N u!h! (N u + h 1)!, N u!h 1! h = 0, 1,..., P, where σ 0 = 1. (2.6) To indicate how to construct base functions for systems with multiple random variables, this chapter demonstrates the process of finding the generalized polynomials up to second order (P = 2) for a system with two random variables (N u = 2), ζ 1 and ζ 2, using twodimensional Legendre base functions. For this particular case, using Eqn. (2.3) yields to N t + 1 = 6. The elements of matrix κ will be 0, 1, and 2, and the repetition of each of these values, as obtained from Eqn.(2.6), is respectively σ 0 = 1, σ 1 = 2, and σ 2 = 3, as can be seen in Eqn. (2.7) κ = [ 0, 1, 1, 2, 2, 2 ] T. (2.7) Based on Eqn. (2.5), the (6 2) matrix β is calculated as β =. (2.8) It can be seen that Eqn. (2.5) is satisfied for each row of matrix β. In other words, adding all elements within each row of matrix β will yield to the (N t + 1) 1 column matrix κ. The terms in the first and second column of matrix β designate the order of the Legendre base functions corresponding to ζ 1 and ζ 2, respectively. We define a new

22 Chapter 2. PCE FOR UNCERTAINTY ANALYSIS 10 Table 2.2: Procedure of calculating two-dimensional legendre polynomial base functions i β i1 β i2 κ i λ β1 λ β2 Λ i ζ 1 1 ζ ζ 2 ζ ( 3ζ ) 1 ( 3ζ ) ζ 1 ζ 2 ζ 1 ζ ( 3ζ ) ( 3ζ ) matrix λ β mapping each term of β to its corresponding one-dimensional Legendre base function, as can be seen in the Eqn. (2.9) λ β = 1 1 ζ ζ 2. (2.9) 2 ) 1 ζ 1 ζ 2 1 ( 3ζ ) ( 3ζ2 1 1 According to Eqn. (2.4), the six corresponding polynomial base functions Λ i (i = 1,..., 6) can be calculated in a matrix form by multiplying all terms appearing in each row of matrix λ β, as Λ = [ 1, ζ 1, ζ 2, ( 3ζ ), ζ 1 ζ 2, ( 3ζ2 2 1 ) ] T. (2.10) 2 The entire procedure for finding the generalized base functions is presented in Tab The weighted inner product using the joint probability density function w(ζ) associated with each specific distribution for a one-dimensional random variable is determined by

23 Chapter 2. PCE FOR UNCERTAINTY ANALYSIS 11 Eqn.(2.11) < λ i (ζ), λ j (ζ) > = λ i (ζ)λ j (ζ)w(ζ)dζ = h 2 i δ ij, (2.11) where h 2 i is a positive number while δ ij represents the Kronecker delta with the following definition 1 for i = j δ ij = 0 for i j. (2.12) In case of n-dimensional random variables, Eqn. (2.11) is expressed as < Λ i (ζ), Λ j (ζ) > =... Λ i (ζ)λ j (ζ)w(ζ 1 )...w(ζ n )dζ 1...dζ n = c 2 i δ ij, (2.13) where c 2 i is a positive number. Using the PCE method, stochastic parameters of the system such as the expected value and variance of R(t, ζ) are easily calculated as E[R] = r 0, (2.14) V ar(r) = E[R 2 ] (E[R]) 2 = ri 2 < Λ 2 i >. (2.15) i=1

24 Chapter 3 Forward Kinematic Analysis of Non-deterministic Articulated Multibody Systems in Polynomial Chaos Expansion Scheme 3.1 INTRODUCTION In real applications, articulated multibody systems are subjected to uncertainties. This may originate from the inherent uncertainty in mathematical modeling, system s complexity, lack of information in physical and geometrical properties, and uncertainty in the measurement and control inputs [3]. In complex systems with a larger number of kinematic chains, uncertainty in the geometry of the system elements such as the length of links due to manufacturing processes, can significantly affect the system s kinematics. This problem highlights the inevitable role of reliability assessment in the design and analysis of such systems. In this research, the PCE method is extended and used for the 12

25 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 13 detailed kinematic analysis of nondeterministic articulated multibody systems containing open chains and closed chains. This includes the necessary formulations to form and solve the equations governing the kinematics of nondeterministic constrained multibody systems at the position, velocity, and acceleration levels. Computational costs, accuracy, and convergence of the PCE method for the forward kinematic analysis of a four-bar mechanism and a SCARA robot with uncertainty in the parameters of the system are compared with those from the Monte Carlo technique. 3.2 DETERMINISTIC EQUATIONS FOR CONSTRAINED FORWARD KINEMATIC ANALY- SIS In this section, an overview of the kinematic analysis of deterministic multibody systems is provided. This analysis will be helpful when nondeterministic systems are considered Position Analysis of Deterministic Systems with Closed Chains The kinematic constraint equation for a deterministic articulated multibody system containing kinematically closed chains (Fig. 3.1) is generally expressed by equating the position vector at both sides of the constraints Φ(q; s) = 0. (3.1) In the above expression, Φ designates the n c linearly independent constraint equations of the system in terms of q and s, where q represents n generalized coordinates, and s is the set of all system s design parameters. When performing forward kinematic analysis of the system, Φ(q, s) is iteratively solved for the generalized coordinates for iteration i + 1 as

26 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 14 Φ, q(q i, s)(q i+1 q i ) = Φ(q i, s), (3.2) where Φ, q denotes the constraint Jacobian, and q i and q i+1 represent the column matrices of the generalized coordinates in iterations i and i + 1, respectively. Figure 3.1: A schematic of kinematically constrained multibody with a cut-joint Velocity Analysis of Deterministic Systems with Closed Chains The generalized coordinate column matrix q introduced in Eqn. (3.1) can be expressed in a partitioned form [28] as q = [u T, v T ] T, (3.3) where u represents the k dependent coordinate column matrix with unknown values, while v provides the column matrix of r known independent coordinates of the closedchain multibody problem. At the velocity level, the kinematic constraint equation for a deterministic articulated multibody system is obtained by differentiating Eqn. (3.1) as Φ, q q = 0, (3.4)

27 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 15 where q represents the first time derivative of the generalized coordinates and the Jacobian matrix Φ, q is the partial derivative of the constraint equations with respect to the generalized coordinates. By defining v and u [28] as u = [ u 1,..., u k ] T, (3.5) v = [ v 1,..., v r ] T, (3.6) and setting q = [ u T, v T ] T, (3.7) one can express Eqn. (3.4) in the partitioned form [28] as Φ, u u = Φ, v v, (3.8) where Φ, v and Φ, u represent two submatrices of Φ, q associated with v and u, respectively. In Eqn. (3.8), all parameters of the system have known values except for the velocity column matrix u. Since the kinematic constraints mentioned in Eqn. (3.1) are independent, the matrix Φ, u is nonsingular. As such, Eqn. (3.8) is solved for the unknown generalized speeds of the system Acceleration Analysis of Deterministic Systems with Closed Chains At the acceleration level, the kinematic constraint equation for a deterministic articulated multibody system containing closed chains can be obtained by differentiating Eqn. (3.4) as

28 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 16 Φ,q q = Φ, q q. (3.9) Assuming that q = [ü T, v T ] T, (3.10) one can rewrite Eqn. (3.9) as Φ, u ü = Φ, q q Φ, v v, (3.11) where all quantities appearing in Eqn. (3.11) are known except the acceleration column matrix ü. Since the kinematic constraints mentioned in Eqn. (3.1) are independent, the matrix Φ, u is nonsingular. Therefore, this equation can be solved for the first time derivative of the generalized speeds of the system ü Position Analysis of Deterministic Systems with Open Chains In the position analysis of multibody systems with closed chains, the kinematic constraint equation Eqn. (3.1), is iteratively solved for the unknown generalized coordinates. While for open-chain systems such as the one shown in Fig. 3.2, the position of desirable points can be written in terms of the system s parameters, states, and generalized coordinates using the corresponding constraints of the system as Z = Ψ(q; s). (3.12)

29 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 17 In the above expression, Z is the position vector of a set of the desirable points, Ψ designates the n c linearly independent constraint equations of the system in terms of q and s. When performing forward kinematic analysis of the system, Eqn. (3.12) is solved for the desirable position vector Z. Figure 3.2: Schematic of a scara robot [1] Velocity Analysis of Deterministic Systems with Open Chains The velocity of any point in the open-chain system can be found by taking the first time derivative of the position vector Eqn. (3.12) as Ż = Ψ, q q, (3.13)

30 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 18 where q represents the first time derivative of the generalized coordinates and the Jacobian matrix Ψ, q is the partial derivative of the constraint equations with respect to the generalized coordinates Acceleration Analysis of Deterministic Systems with Open Chains The acceleration of any point in the open-chain system can be found by taking the first time derivative of the velocity vector Eqn. (3.13) as Z = Ψ,q q + Ψ, q q, (3.14) where q represents the second time derivative of the generalized coordinates and Ψ, q is the first time derivative of the Jacobian matrix. 3.3 NONDETERMINISTIC EQUATIONS FOR CONSTRAINED FORWARD KINEMATIC ANALY- SIS OF SYSTEMS WITH CLOSE CHAINS In this section, kinematic analysis of closed-chain systems is considered and the effect of uncertainty on the equations governing the kinematics of the system is fully explained Position Analysis of Nondeterministic Systems with Closed Chains In the PCE-based kinematic analysis of a nondeterministic multibody problem at the position level, uncertainty may exist in the system s parameters and/or inputs. Consider a nondeterministic multibody problem including N u uncertain parameters as s = {s w } w=nu w=1. The existence of uncertainty in the system s parameter in Eqn. (3.1) affects

31 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 19 the system s states as well as generalized coordinates, speeds, and first time derivative of generalized speeds. Hence, generalized coordinates of the system and their first and second time derivatives will be projected onto the space of appropriate orthogonal base functions. Due to the existence of uncertainty, Eqn. (3.1) can be reformulated as Φ( q; s) = 0. (3.15) Where the tilde sign indicates that the corresponding terms contain nondeterministic values. Φ represents nc linearly independent constraint equations in terms of nondeterministic generalized coordinates ( q) and parameters of the system ( s). As a result, Eqn. (3.2) is reformulated as Φ, q( q i, s)( q i+1 q i ) = Φ( q i, s). (3.16) The above equation can then be solved for the non-deterministic generalized coordinates at iteration i + 1 as a function of random variables of the system. The nondeterministic generalized coordinates of the system q can be expressed as q i (ζ) = N t j=0 q ij Λ j (ζ), i = 1,..., n. (3.17) Therefore, the generalized column matrix q can be expressed as q = [ q 1,..., q n ] T = N t m=0 Q m Λ m (ζ), (3.18) where

32 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 20 Q m = [ q 1m,..., q nm ] T, m = 0,..., N t. (3.19) Finally, the Galerkin projection is used to compute the modal values in Eqn. (3.19) associated with each generalized coordinate. For instance, assume that the j th generalized coordinate at iteration i + 1 ( q j i+1 ) is computed as a function of ζ as f i+1 (ζ). Therefore, this generalized coordinate is expressed in terms of its modal values as j q i+1 j = N t m=0 q i+1 jm Λ m(ζ) = f i+1 j (ζ). (3.20) The subscripts j and m in Eqn. (3.20) denote the components along the deterministic and the stochastic dimension, respectively. By projecting the above relation with the designated base function Λ l, this equation is expressed as < q i+1 j, Λ l (ζ) > = < f i+1 j, Λ l (ζ) >. (3.21) On the other hand, < q i+1 j, Λ l (ζ) > = = c 2 l N t m=0 N t m=0 < q i+1 jm Λ m(ζ), Λ l (ζ) > q i+1 jm δ ml = c 2 l qi+1 jl l = 0,..., N t, and j = 1,..., n, (3.22) where c 2 l is defined as c 2 l = < Λ l (ζ), Λ l (ζ) >. (3.23)

33 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 21 Therefore, the modal values of q i+1 j can be calculated as q i+1 jl = 1 c 2 l < f i+1 j, Λ l (ζ) > l = 0,..., N t, and j = 1,..., n. (3.24) Velocity Analysis of Nondeterministic Systems with Closed Chains In the study of nondeterministic articulated multibody systems at the velocity level, uncertainty will propagate to the system due to nondeterministic information at the position and velocity levels. Considering the uncertainty, Eqn. (3.7) is reformulated as q = [ ũ T, ṽ T ] T, (3.25) where ũ = [ ũ 1,..., ũ k ] T (3.26) ṽ = [ ṽ 1,..., ṽ r ] T. (3.27) Vector ũ is the unknown nondeterministic dependent velocity of the system and vector ṽ is the known deterministic or nondeterministic independent velocity of the system. Finally, one can express the Eqn. (3.8) as Φ,ũ ũ = Φ,ṽ ṽ. (3.28) The right-hand side of Eqn. (3.28) includes the known values of the system (in terms of deterministic or nondeterministic system s states and parameters). The left-hand side

34 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 22 of Eqn. (3.28) shows the unknown non-deterministic velocity column matrix ( ũ) and the known Jacobian matrix Φ,ũ, in terms of deterministic or nondeterministic system s states and parameters. Since the kinematic constraints mentioned in Eqn. (3.1) are independent, the matrix Φ,ũ is nonsingular and by premultiplying the inverse of Φ,ũ to both sides of Eqn. (3.28), the independent generalized speed ( ũ) can be simply evaluated in terms of ζ. The vector ũ will be projected onto the space of appropriate polynomial base functions as ũ = [ ũ 1,..., ũ k ] T = N t h=0 Ũ h Λ h (ζ), (3.29) where Ũ h = [ ũ 1h,..., ũ kh ] T, h = 0,..., N t, (3.30) To obtain the modal values in Eqn. (3.30) associated with non-deterministic generalized speeds, the Galerkin projection is applied to the results of Eqn. (3.28). For instance, assume that the j th generalized speed of the system ũ j is found as a function of ζ as g j (ζ). Following the strategy provided in Eqns. ( ), one can evaluate the corresponding modal coefficients as ũ jl = 1 c 2 l < g j, Λ l >, l = 0,..., N t, and j = 1,..., n. (3.31)

35 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems Acceleration Analysis of Nondeterministic Systems with Closed Chains In the study of nondeterministic articulated multibody systems, uncertainty in the system parameters and states will affect the kinematic analysis at the acceleration level. Considering the uncertainty, Eqn. (3.10) is reformulated as q = [ ũ T, ṽ T ] T (3.32) where ũ = [ ũ 1,..., ũ k ] T (3.33) ṽ = [ ṽ 1,..., ṽ r ] T. (3.34) Vector ũ is the unknown nondeterministic dependent acceleration of the system and vector ṽ is the known deterministic or nondeterministic independent acceleration of the system. Finally, one can express the Eqn. (3.11) as Φ,ũ ũ = Φ, q q Φ,ṽ ṽ. (3.35) In the study of nondeterministic systems, the term Φ,ṽ ṽ on the right-hand side of Eqn. (3.35) consists of known parameters of the system (in terms of deterministic and non-deterministic of the system s states and parameters), the Jacobian matrix Φ,ṽ associated with the generalized coordinates ṽ and the acceleration column matrix ṽ. Meanwhile, the expression Φ, q q presents known deterministic or nondeterministic system s states and parameters, the first time derivative of the Jacobian matrix Φ, q and the first time derivative of the generalized coordinates q, respectively. Since all the system

36 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 24 quantities at the position and velocity levels have already been calculated, and the acceleration column matrix ṽ is known, the right-hand side of the above equation is a known quantity. The left-hand side includes the unknown nondeterministic acceleration column matrix ũ and the known Jacobian matrix Φ,ũ in terms of deterministic or nondeterministic system s states and parameters. Finally, one can solve Eqn. (3.35) for the unknown non-deterministic accelerations ũ as a function of random variables ζ. The vector ũ will be projected in the space of appropriate polynomial base functions as ũ = [ ũ 1,..., ũ k ] T = N t h=0 Ũ h Λ h (ζ), (3.36) where Ũ h = [ ũ 1h,..., ũ kh ] T, h = 0,..., N t. (3.37) To obtain the modal values in Eqn. (3.37),Galerkin projection is applied to the results of Eqn. (3.35). For instance, assume that the j th second time derivative of a generalized coordinate ũ j is found as a function of ζ as h j (ζ). Following the Galerkin projection provided in Eqns. ( ), one can evaluate the modal accelerations as ũ jl = 1 c 2 l < h j, Λ l >, l = 0,..., N t and j = 1,..., n. (3.38)

37 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems SIMULATION RESULTS FOR A SYSTEM WITH CLOSED CHAINS Consider the four-bar mechanism shown in Fig This section studies the uncertainty analysis of this system for two different cases: single-degree uncertainty in the length of one link, and multi-degree uncertainty in the length of two links. Herein, the accuracy of the results and the time efficiency of the intrusive PCE and Monte Carlo methods are compared to each other. The method of Monte Carlo with various numbers of simulation runs as well as PCE with different orders are used for the kinematic analysis at the position, velocity, and acceleration levels of this closed-chain mechanism. In order to solve the kinematic constraints of the system with PCE and Monte Carlo, the iterative Newton-Raphson method is used until a relative error smaller than 10 9 is achieved [28]. L 2 ϴ 2 L 3 L 1 ϴ 1 ϴ 3 L 0 Figure 3.3: The schematics of a four-bar linkage Accuracy Analysis of a Four-bar Mechanism with a Single Random Variable In the study of the four-bar mechanism with uncertainty in a single parameter of the system, it is assumed that uncertainty with uniform distribution exists in the length of L 1 [1.8, 2.2] (Units) with the distribution function L 1 = ζ (Units) where ζ [ 1, 1]. Legendre polynomials are used due to the uniform distribution of uncertainty in L 1. For this analysis, PCE with Legendre polynomials of orders 1, 2, 3, and 4 has been used to find the values of the remaining generalized coordinates, generalized speeds,

38 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 26 and the time derivative of the generalized speeds for the given configuration with the following deterministic values; θ 1 = 60 o, ω 1 = 3 rad/s, α 1 = -2 rad/s 2, L 0 = 5 (Units), L 2 = 6 (Units), and L 3 = 4 (Units). Table 3.1: Expected value of θ 2, θ3, ω 2, ω 3, α 2, α 3 from different orders of onedimensional PCE Order of the PCE θ2 θ3 ω 2 ω 3 α 2 α 3 P = P = P = P = Table 3.2: Standard deviation of θ 2, θ 3, ω 2, ω 3, α 2, α 3 from different orders of onedimensional PCE Order of the PCE θ2 θ3 ω 2 ω 3 α 2 α 3 P = P = P = P = Table 3.3: Expected value of θ 2, θ 3, ω 2, ω 3, α 2, α 3 from different one-dimensional Monte Carlo simulations Number of Monte Carlo Simulations θ2 θ3 ω 2 ω 3 α 2 α Table 3.4: Standard deviation of θ 2, θ 3, ω 2, ω 3, α 2, α 3 from different one-dimensional Monte Carlo simulations Number of Monte Carlo Simulations θ2 θ3 ω 2 ω 3 α 2 α Tables 3.1 and 3.2, respectively, show the expected value and standard deviation obtained from different orders of PCE with seven decimal places. It can be seen that the expected value and standard deviation obtained from higher order PCE (order 3 and 4) converge to the same number. Since the obtained stochastic results from the PCE of

39 Logarithm of Expected Value Error Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems Mean Mean Mean! Mean! Mean, Mean, Numbers of Monte Carlo Simulations #10 5 Figure 3.4: Logarithm of the relative error in the expected value of the states from different numbers of one-dimensional Monte Carlo simulations order three and four present identical converged values, a PCE with order higher than four is not used in this research. In order to analyze the Monte Carlo results, different numbers of Monte Carlo simulations are conducted to examine if their converged values match the ones obtained from the PCE. Monte Carlo simulations with 10 4, 10 5, 10 6, and 10 7 runs are performed. Tables 3.3 and 3.4, respectively, present the expected value and standard deviation of different numbers of Monte Carlo simulations. It can be seen that the expected value and standard deviation shown by using the 3 rd and 4 th order PCE and 10 7 Monte Carlo simulations are identical (Tables 3.1, 3.2, 3.3, and 3.4 ). In other words the PCE of order 3 is equivalent to 10 7 Monte Carlo simulations. Since the expected value and standard deviation of the states of the system converge to a specific number, one can define the relative error by comparing the stochastic results from Monte Carlo or PCE simulations x 1 with the established 10 7 Monte Carlo simulations y 1 as the reference model. (largest number of Monte Carlo simulations for one dimensional uncertainty analysis) as Relative Error = x 1 y 1 y 1. (3.39) Figures 3.4 and 3.5 present the logarithm of the relative error in the expected value and standard deviation of the states of the system at the position, velocity, and acceleration

40 Logarithm of Standard Deviation Error Logarithm of Expected Value error Logarithm of Satndard Deviation Error Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems Standard Deviation Standard Deviation Standard Deviation! Standard Deviation! Standard Deviation, Standard Deviation, Numbers of Monte Carlo Simulations #10 5 Figure 3.5: Logarithm of the relative error in the standard deviation of the states from different numbers of one-dimensional Monte Carlo simulations Mean Mean Mean! Mean! Mean, Mean, Order of PCE Figure 3.6: Logarithm of the relative error in the expected value of the states of the system from different orders of one-dimensional PCE -1.2 Standard Deviation Standard Deviation Standard Deviation! Standard Deviation! Standard Deviation, Standard Deviation, Order of PCE Figure 3.7: Logarithm of the relative error in the standard deviation of the states of the system from different orders of one-dimensional PCE levels for 10 4, 10 5, and 10 6 simulations (Since the results of Fig. 3.5 overlap, the relative errors are shown in a table in the appendix of this thesis). These results indicate that in order to reach a desired level of accuracy, the number of simulations must be modified accordingly. An example of the effect of varying the numbers of simulations can be

41 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 29 seen in Fig. 3.5, where using 10 4 Monte Carlo simulations (relatively small compared to other stated number of simulations) yields a relative error of up to in standard deviation at the acceleration level, while using 10 6 Monte Carlo simulations yields a clearly smaller error of up to Table 3.5: Computation time for different numbers of one-dimensional Monte Carlo simulations Number of Monte Carlo simulations Computational Time (seconds) Table 3.6: Computation time for different orders of one-dimensional PCE PCE Order Computational Time (seconds) Figures 3.6 and 3.7 present the logarithm of the relative error in the expected value and standard deviation of the system s states at the position, velocity, and acceleration levels for PCE with different order Legendre polynomials compared against those from 10 7 Monte Carlo simulations. It is observed that the first order PCE can predict the expected value with an appropriately small error (up to ); however, the relative error in the standard deviation becomes more significant at the acceleration level (up to ). The PCE of order two can predict the expected value and standard deviation with an acceptable relative error (up to ), but the PCE of order three can provide the most accurate results, even more precise than conducting 10 4, 10 5, and 10 6 Monte Carlo simulations as can be seen in Figures 3.4, 3.5, 3.6, and 3.7. It is also observed that the error in the expected value and standard deviation becomes almost insensitive when higher order PCEs (more than 3) are used for this analysis. As such, the intrusive method has demonstrated more accurate results than the specified Monte Carlo simulations shown in Figures 3.4 and Efficiency Analysis of a Four-bar Mechanism with a Single Random Variable It is important to evaluate the computational efficiency of these two methods for forward kinematic analysis. Tables 3.5 and 3.6, respectively indicate the computation time to

42 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 30 perform Monte Carlo and PCE simulations. This computation time includes the entire computational effort to perform the kinematic analysis at the position, velocity, and acceleration levels, as well as the time required to find expected values and standard deviations of the system s states. As mentioned previously, the relative error becomes almost insensitive for PCEs with polynomials of orders higher than three. As such, in this analysis, a PCE of order three is chosen as the method which can provide accurate results. According to Tables 3.5 and 3.6, although the computation time for the PCE of order three (6.5 sec) is less than the one for the 10 6 Monte Carlo simulations (73 sec), the third order PCE provides more accurate results (Figures 3.4, 3.5, 3.6, and 3.7). In addition, a PCE of order three can provide the same expected value and standard deviation as those obtained from the 10 7 Monte Carlo simulations (Tables 3.1, and 3.2, 3.3, and 3.4) with a significantly shorter runtime, 6.50 vs seconds, as can be seen in Tables 3.5 and 3.6. It can be concluded that the PCE method with low order polynomials can be used when higher accuracy of the results is required at a faster computation time Accuracy Analysis of a Four-bar Mechanism with Multiple Random Variables In the kinematic analysis of the established system with two degrees of uncertainty, it is assumed that uncertainty with uniform distribution exists in the lengths of L 1 [1.8, 2.2] and L 2 [5.75, 6.25] (Units) with the distribution L 1 = ζ 1 (Units) and L 2 = ζ 2 (Units) where ζ 1 [ 1, 1] and ζ 2 [ 1, 1]. Therefore, in this analysis, two-dimensional Legendre polynomials up to the first, second, third, and fourth order are constructed and used for the intrusive forward kinematic analysis to find the values of the remaining generalized coordinates, velocities, and accelerations of the system for the given configuration with the following deterministic values; θ 1 = 60 o, ω 1 = 3 rad/s, and α 1 = -2 rad/s 2, L 0 = 5 (Units), and L 3 = 4 (Units).

43 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 31 Table 3.7: Expected value of θ 2, θ3, ω 2, ω 3, α 2, α 3 from different orders of twodimensional PCE Order of the PCE θ2 θ3 ω 2 ω 3 α 2 α 3 P = P = P = P = Table 3.8: Standard deviation of θ 2, θ 3, ω 2, ω 3, α 2, α 3 from different orders of twodimensional PCE Order of the PCE θ2 θ3 ω 2 ω 3 α 2 α 3 P = P = P = P = Tables 3.7 and 3.8, respectively, show the expected value and standard deviation obtained from different orders of PCE. It can be seen that the evaluated data associated with higher orders of PCE (order 3 and 4) are almost the same. Since the obtained stochastic results from the PCE of order three and four present almost identical converged values, a PCE with order higher than four is not used in this research. In order to analyze the Monte Carlo results, the values obtained from different numbers of Monte Carlo simulations are compared with the converged PCE results. The number of Monte Carlo simulations is increased until an relative error less than is achieved between the results of the PCE of order four and the Monte Carlo results is achieved (it will be shown that reaching this relative error requires a significantly large computation time for the Monte Carlo simulations). Several Monte Carlo simulations, including (500 samples of ζ 1 and 500 samples of ζ 2 ), ( ), ( ), ( ), ( ), and ( ) have been evaluated. Tables 3.9 and 3.10, respectively, present the expected value and standard deviation of the states of the system for different numbers of Monte Carlo simulations. This research also studies the hypothesis that as the order of PCE and the number of Monte Carlo simulation increases, the results of the Monte Carlo tend to converge to the results of the PCE or vice verse. Figures respectively show

44 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 32 the expected values of θ 3, ω 3, and α 3. Figures respectively show the standard deviations of θ 3, ω 3, and α 3. In these figures, the results of different numbers of Monte Carlo simulations are displayed with red color and PCEs are represented in blue. It is observed that in all of these figures, the Monte Carlo results get closer to the converged PCE results as the number of Monte Carlo simulations increases. In addition, it can be seen that only a significantly large number of Monte Carlo simulations ( ) is able to provide almost identical results as the PCE of order four. However, this number of Monte Carlo simulation provides the result with a notably higher computation time (eleven hours and fifty eight minutes) compared to the PCEs as will be discussed in the next section. Since the same trend is observed in the expected value and standard deviation of θ 4, ω 4, and α 4, the corresponding figures are not shown to avoid redundancy. Table 3.9: Expected value of θ 2, θ 3, ω 2, ω 3, α 2, α 3 From different two-dimensional Monte Carlo simulations Number of Monte Carlo Simulations θ2 θ3 ω 2 ω 3 α 2 α Table 3.10: Standard deviation of θ 2, θ 3, ω 2, ω 3, α 2, α 3 from different two-dimensional Monte Carlo simulations Number of Monte Carlo Simulations θ2 θ3 ω 2 ω 3 α 2 α In order to evaluate the accuracy of the results, relative error is calculated against Monte carlo simulations as our reference model. Figures 3.14 and 3.15 respectively show the logarithm of the relative error in the expected value and standard deviation of the states of the system for different Monte Carlo simulations. Since the results of Fig overlap, the relative errors are shown in a table in the appendix of this research. The logarithm of the relative error in the expected value and standard deviation of the

45 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems PCE of Order 3 PCE of Order 2 PCE of Order 3 PCE of Order 4 PCE of Order 4 PCE of Order 4 Expected Value of x10 5 Monte Carlo Simulations 5x10 5 Monte Carlo Simulations 1x10 6 Monte Carlo Simulations 4x10 6 Monte Carlo Simulations 2.5x10 7 Monte Carlo Simulations 6.25x10 8 Monte Carlo Simulations PCE Results Monte Carlo Results Figure 3.8: Expected value of θ 3 from different orders of two-dimensional PCEs and Monte Carlo simulations Expected Value of! x10 5 Monte Carlo Simulations 5x10 5 Monte Carlo Simulations 1x10 6 Monte Carlo Simulations 4x10 6 Monte Carlo Simulations Monte Carlo Results PCE Results 2.5x10 7 Monte CarloSimulations 6.25x10 8 Monte Carlo Simulations PCE of Order 1 PCE of Order 2 PCE of Order 3 PCE of Order 4 PCE of Order 4 PCE of Order Figure 3.9: Expected value of ω 3 from different orders of two-dimensional PCEs and Monte Carlo simulations PCE of Order 1 PCE of Order 2 PCE of Order 3 PCE of Order 4 PCE of Order 4 PCE of Order 4 Expected Value of, x10 8 Monte 2.5x10 7 Monte Carlo Simulations Carlo Simulations 4x10 6 Monte Carlo Simulations 1x10 6 Monte Carlo Simulations 5x10 5 Monte Carlo Simulations x10 5 Monte Carlo Simulations Monte Carlo Results PCE Results Figure 3.10: Expected value of α 3 from different orders of two-dimensional PCEs and Monte Carlo simulations system s states from the PCE compared to those from Monte Carlo simulations are respectively shown in Figures 3.16 and Although the first order PCE has the advantage of predicting the expected value of the states of the system with an order of up to , the error in standard deviation becomes more significant (up to ) at the acceleration level. The PCE of order two can evaluate the expected value of the system s states with a higher accuracy compared to all the stated Monte Carlo methods presented in Fig The PCE of order three can predict both the

46 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 34 expected values and standard deviations with better accuracy (up to at acceleration level) than all the stated Monte Carlo methods presented in Figures 3.14 and Standard Deviation of x10 5 Monte Carlo Simulations 5x10 5 Monte Carlo Simulations 1x10 6 Monte Carlo Simulations 4x10 6 Monte Carlo Simulations Monte Carlo Results PCE Results 2.5x10 7 Monte Carlo Simulations 6.25x10 8 Monte Carlo Simulations PCE of Order 2 PCE of Order 3 PCE of Order 4 PCE of Order 1 PCE of Order 4 PCE of Order 4 Figure 3.11: Standard deviation of θ 3 from different orders of two-dimensional PCEs and Monte Carlo simulations Standard Deviation of! 3 2.5x10 5 Monte Carlo Simulations 5x10 5 Monte Carlo Simulations PCE of Order 3 1x10 6 Monte Carlo Simulations 4x10 6 Monte Carlo Simulations 2.5x10 7 Monte Carlo Simulations 6.25x10 8 Monte Carlo Simulations PCE of Order 2 PCE of Order 3 PCE of Order 4 PCE of Order 4 PCE of Order 4 Monte Carlo Results PCE Results Figure 3.12: Standard deviation of ω 3 from different orders of two-dimensional pces and Monte Carlo simulations Standard Deviation of, x10 5 Monte Carlo Simulations PCE of Order 1 5x10 5 Monte Carlo Simulations PCE of Order 2 1x10 6 Monte Carlo Simulations 4x10 6 Monte Carlo Simulations 2.5x107 Monte Carlo Simulations 6.25x10 8 Monte Carlo Simulations PCE of Order 3 PCE of Order 4 PCE of Order 4 PCE of Order 4 Monte Carlo Results PCE Results Figure 3.13: Standard deviation of α 3 from different order of two-dimensional PCEs and Monte Carlo simulations A key result observed by comparing Figures 3.5 and 3.15 is that in a single degree of uncertainty model, the Monte Carlo simulation with 10 4 runs yields an error of up to at the acceleration level in the standard deviation of the results (Fig. 3.5), whereas in the case of two degrees of uncertainty, a relatively larger number of Monte Carlo simulations ( ) yields to a larger standard deviation error of up to

47 Logarithm of Expected Value Error Logaritm of Standard Deviation Logarithm of Expected Value Error Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems Mean Mean Mean! Mean! Mean, Mean, Number of Monte Carlo Simulation #10 7 Figure 3.14: Logarithm of the relative error in the expected value from different Monte Carlo simulation Standard Deviation Standard Deviation Standard Deviation! Standard Deviation! Standard Deviation, Standard deviation, Numbers of Monte Carlo Simulations #10 7 Figure 3.15: Logarithm of the relative error in the standard deviation from different Monte Carlo simulation -4.6 Mean Mean Mean! Mean! Mean, Mean, Order of PCE Figure 3.16: Logarithm of the relative error in the expected value from different orders of PCE at the acceleration level (Fig. 3.15). This comparison shows that by increasing the degree of uncertainty in a system, a higher number of Monte Carlo simulations (higher computation time) is required to obtain a desirable accuracy.

48 Logarithm of Standard Deviation Error Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems Standard Deviation Standard Deviation Standard Deviation! Standard Deviation! Standard Deviation, Standard Deviation, Order of PCE Figure 3.17: Logarithm of the relative error in the standard deviation from different orders of PCE Efficiency Analysis of a Four-bar Mechanism with Multiple Random Variables Computation time for performing the kinematic analysis at the three kinematic levels, including the required time for evaluating the expected value and the standard deviation of the states of the system for different numbers of Monte Carlo and PCE simulations, are provided in Tables 3.11 and Efficiency analysis of the intrusive PCE method is evaluated by comparing both computation time and accuracy of various orders of PCE with different Monte Carlo simulations. For instance, a comparison between the PCE of order two with Monte Carlo simulations reveals that the computation time for Monte Carlo simulations is seconds (Tab. 3.11). This Monte Carlo computation time is approximately 17 times slower than the one of the PCE of order two, which has the computation time of 8.40 seconds (Tab. 3.12). Not only is the second order PCE computationally more efficient, but this technique also provides more accurate results in the expected value and standard deviation at the position, velocity, and acceleration levels as provided in Figures 3.14, 3.15, 3.16, and For example, to achieve more accurate results, the same comparison between the PCE of order three and the largest number of Monte Carlo simulations ( ) demonstrates that the PCE of order three can provide more accurate results in a significantly smaller period of time (21.02 sec vs sec). As it was mentioned before the Monte Carlo with runs and PCE of order three, and four almost provide a same result.

49 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 37 However, it is worth mentioning that stated PCEs can provide the results in and second while Monte Carlo have a second simulation time. Table 3.11: Computation time for different numbers two-dimensional of monte carlo simulation Number of Monte Carlo simulations Computational Time (seconds) Table 3.12: Computation time for different orders of two-dimensional pce PCE Order Computational Time (seconds) NONDETERMINISTIC EQUATIONS FOR CONSTRAINED FORWARD KINEMATIC ANALY- SIS OF SYSTEMS WITH OPEN CHAINS In this section, kinematic analysis of open-chain systems is considered and the effect of uncertainty on the equations governing the kinematics of the system is fully explained Position Analysis of Nondeterministic Systems with Open Chains Consider a nondeterministic multibody problem including N u uncertain parameters. The position vector of the desirable points can be written in terms of nondeterministic parameters of system as Z = Ψ( q; s). (3.40) In these equations, the right-hand of the relation is a known function in terms of polynomials. In other words, the nondeterministic value of the position of the desired points of the system is a known function of ζ. In order to find the modal values associated

50 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 38 with the position of this desired points ( Z), the Galerkin projection is conducted using Eqns. ( ) Velocity Analysis of Nondeterministic Systems with Open Chains The velocity of any point in the open-chain system can be found by taking the first time derivative of the position vector Eqn. (3.40) as Z = Ψ, q q. (3.41) Then, the Galerkin projection is used to compute the modal values associated with velocity of desirable points Z using Eqns. ( ) Acceleration Analysis of Nondeterministic Systems with Open Chains The velocity of any point in the open-chain system can be found by taking the first time derivative of the position vector Eqn. (3.41) as Z = Ψ, q q + Ψ, q q. (3.42) Then, Galerkin projection is used to compute the modal values associated with the acceleration of desirable points Z using Eqns. ( ). 3.6 SIMULATION RESULTS FOR A SYSTEM WITH OPEN CHAINS Kinematic analysis of the SCARA robot shown in Fig. 3.2 is conducted in this section. The position of the end effector (x 4,y 4 ), center of link 4, in x and y direction is set to

51 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems 39 be found. Both PCE and Monte Carlo methods are conducted to find the stochastic results of the system for the desirable points Accuracy Analysis of a SCARA Robot Consider the SCARA robot with 4 degrees of freedom shown in Fig Links 1 and 2 rotate in x y plane. Link 3 slides in z direction and link 4 rotates in x y plane. The geometrical properties of the system are shown in the Tab Where L i is the length, and θ i is angular position of the i th link. Uncertainty in the lengths of the links might be due to the design procedure while uncertainty in the joints angular positions might originate from inaccuracies in the actuators. These values are found from commercial SCARA robot brochures. Four-dimensional Legendre polynomials and several Monte Carlo simulations are separately used for the kinematic analysis of the system. Monte Carlo simulations are chosen as: (41 samples of ζ 1, 41 samples of ζ 2, 41 samples of ζ 3 and 41 samples of ζ 4 ), ( ), ( ), ( ), ( ), and ( ). Table 3.13: Geometrical properties of the SCARA mechanism Parameter Value Distribution of Uncertainty L ζ 1 (m) ζ 1 [ 1, 1] L ζ 2 (m) ζ 2 [ 1, 1] L 3 0.5(m) None π θ ζ 3(m) ζ 3 [ 1, 1] θ ζ ζ 2 (m) ζ 4 [ 1, 1] Figure 3.18 shows the expected value of x 4 obtained from different orders of PCE and various numbers of Monte Carlo simulations. Figure 3.19 presents the expected value of y 4 obtained from both PEC and Monte Carlo. The standard deviation of x 4 from both PCE and Monte Carlo method is shown in Fig and standard deviation of y 4 is presented in Fig It can be seen that the expected value and the standard

52 Standard Deviation of X 4 (m) Expected Value of Y 4 (m) Expected Value of X 4 (m) Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems e-01 PCE of Order 1 PCE of Order 2 PCE of Order 2 PCE of Order 2 PCE of Order e e e e e e-01 PCE of Order x10 9 Monte 2.38x10 9 Monte Carlo Simulations Carlo Simulations 2.52x10 8 Monte 6.72x10 8 Monte 1.04x10 8 Carlo Simulations Carlo Simulations Monte Carlo Simulations e e e x10 6 Monte Carlo Simulations Figure 3.18: Expected value of x 4 (m) e-01 PCE of Order 1 PCE of Order 2 PCE of Order 2 PCE of Order 2 PCE of Order e-01 PCE of Order e e x x10 9 Monte 2.38x10 9 Monte Monte Carlo Simulations Carlo Simulations Carlo Simulations 1.04x10 8 Monte 2.52x10 8 Monte Carlo Simulations Carlo Simulations e e x10 6 Monte Carlo Simulations e-01 Figure 3.19: Expected value of y 4 (m) 7.250e e x10 6 Monte Carlo Simulations 7.150e x10 8 Monte Carlo Simulations 2.52x10 8 Monte Carlo Simulations 6.72x108 Monte Carlo Simulations 1.63x109 Monte Carlo Simulations 2.38x109 Monte Carlo Simulations 7.100e-03 PCE of Order e-03 PCE of Order 2 PCE of Order 2 PCE of Order 2 PCE of Order 2 PCE of Order 2 Figure 3.20: Standard deviation of x 4 (m) deviation obtained from PCE of order 2 and 3 converge to the same number. Also, It can be observed that while the PCE results have converged, the Monte Carlo ones have not reached a converged value yet. As it was shown before, higher number of Monte Carlo

53 Relative Error of the Expected Value of X 4 (m) Standard Deviation of Y 4 (m) Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems e e e e e x10 6 Monte Carlo Simulations 4.140e-03 PCE of Order x10 8 Monte Carlo Simulations 2.52x10 8 Monte Carlo Simulations 6.72x108 Monte Carlo Simulations 1.63x109 Monte 2.38x10 9 Monte Carlo Simulations Carlo Simulations 4.120e e-03 PCE of Order 2 PCE of Order 2 PCE of Order 2 PCE of Order 2 PCE of Order 2 Figure 3.21: Standard deviation of y 4 (m) 3.50e e e x10 6 Monte Carlo Simulations 2.00e e e x10 8 Monte Carlo Simulations 2.52x10 8 Monte Carlo Simulations 6.72x10 8 Monte Carlo Simulations 1.63x10 9 Monte 2.38x10 9 Monte Carlo Simulations Carlo Simulations 5.00e-07 Figure 3.22: Relative error in the expected value of x 4 (m) simulations is required when a system with large number of uncertain parameters is considered. Since the expected value and standard deviation of the states of the system converged to a specific number (PCE of order 2 results), one can define the relative error in the stochastic results obtained from different Monte Carlo simulations as Relative Error (in percent) = γ 1 γ 2 γ 2 100, (3.43) where γ 1 is the quantity obtained from a specific number of Monte Carlo simulation and γ 2 is the converged value obtained from the PCE of order 2. The relative error in the expected value of x 4 and y 4 using different Monte Carlo simulations are respectively shown in Figures 3.22 and It can be seen that all of

54 Relative Error of the Expected Value of Y 4 (m) Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems e e x10 6 Monte Carlo Simulations 2.50e e e e x10 8 Monte Carlo Simulations 2.52x10 8 Monte Carlo Simulations 6.72x10 8 Monte Carlo Simulations 1.63x10 9 Monte Carlo Simulations 2.38x109 Monte Carlo Simulations 5.00e-07 Figure 3.23: Relative error in the expected value of y 4 (m) the Monte Carlo simulations are able to provide accurate results in the expected values (with order of 10 6 ). Figures 3.24 and 3.25 respectively present the relative error in the standard deviation of x 4 and y 4. It can be observed that the relative error in the standard deviation becomes more significant. For instance, the Monte Carlo results with the highest number of simulations ( ) have approximately the relative error of 0.5 percent (Fig. 3.24). As it will be shown later the computation time for this Monte Carlo simulation is approximately 400 times more than run time for the PCE of order Efficiency Analysis of a SCARA Robot So far it has been shown that a PCE of order two can provide better accuracy than all Monte Carlo simulations stated in this research. Also it was observed that Monte Carlo results get closer to the PCE ones as the number of Monte Carlo simulation increases. In order to evaluate the efficiency of both methods, the computation times are also needed to be compared. The PCE of order 2 has the computation time of 9.96 (sec). The computation time for Monte Carlo simulations are shown in Tab It can be seen the even though a Monte Carlo with simulation has the computation time of (sec) (higher than PCE of order 2), it has approximately an error of 1 percent (Fig. 3.24). When more accurate results are required, a Monte Carlo with higher number of

55 Relative Error of the Standard Deviation of Y 4 (m) Relative Error of the Standard Deviation of X 4 (m) Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems x10 6 Monte Carlo Simulations x10 8 Monte Carlo Simulations 2.52x10 8 Monte Carlo Simulations 6.72x10 8 Monte Carlo Simulations 1.63x10 9 Monte 2.38x10 9 Monte Carlo Simulations Carlo Simulations 0 Figure 3.24: Relative error in the standard deviation of x 4 (m) x10 6 Monte Carlo Simulations x10 8 Monte Carlo Simulations 2.52x10 8 Monte Carlo Simulations 6.72x10 8 Monte Carlo Simulations1.63x10 9 Monte 2.38x10 9 Monte Carlo Simulations Carlo Simulations 0 Figure 3.25: Relative error in the standard deviation of y 4 (m) Table 3.14: Computation time for different numbers of four-dimensional monte carlo simulation Number of Monte Carlo simulations Computational Time (seconds) simulations can be used. For instance, using Monte Carlo simulations yield to a maximum relative error of 0.5 percent with computation time of (sec). This computation time is approximately 400 times more than run time for the PCE of order 2. Therefore, it can be concluded that a PCE with a small order (in this problem 2) can provide acceptable accuracy in a timely effective manner.

56 Chapter 3. Kinematic Analysis of Nondeterministic Multibody Systems CONCLUSIONS In this research, the polynomial chaos expansion has been extended to kinematically analyze multibody systems. Kinematic analysis of both open-chain and closed-chain systems are presented. The mathematical framework of the method has been explained for the detailed kinematic analysis at the position, velocity, and acceleration levels. The intrusive PCE with various polynomial orders has been used for the kinematic analysis of a non-deterministic four-bar mechanism when uncertainty exists in the length of the links of the system. The same analysis has been conducted on an open-chain system (a SCARA robot with 4 degrees of freedom). The simulation efficiency and the accuracy of the stochastic properties of the states of the system from the PCE simulations has been compared to those from the Monte Carlo simulations. The obtained results demonstrate that the computation time for the traditional Monte Carlo approach increases drastically when implemented on systems with a large number of uncertain parameters. This occurs since in the Monte Carlo method, the kinematic constraints need to be solve for each value of the uncertain parameter, while the PCE solves the nondeterministic kinematic constraints for the unknown parameters/states of the system only once. Consequently, for complex systems, it is expected that the PCE method can provide more accurate results with much less computation time than the Monte Carlo technique.

57 Chapter 4 Framework for the Computed Torque Control of Nondeterministic Multibody Systems 4.1 INTRODUCTION In the field of robotics, a required task may be to force the robot to follow desired paths. Therefore, a controller is usually required to force the robot s arms to follow the desirable trajectories. Dynamics equations of multibody systems are usually highly nonlinear. Controlling nonlinear systems often leads to difficulties. One solution to this problem is to linearize the system and then find the required forces/torques to control the system. The CTCL is a method which use feedback linearization by using inverse dynamics of a system to calculate external forces/torques necessary to drive the system along a desired trajectory [29]. 45

58 Chapter 4. Computed Torque Control of Nondeterministic Multibody Systems 46 Considering uncertainties in the equations of motion of multibody systems is relatively new. PCE was first applied to control nondeterministic systems in 2003 [9]. Templeton applied the method of Polynomial Chaos Expansion (PCE) to an H 2 optimal control problem [30]. In 2009, Voglewede et al, applied Polynomial Chaos Theory to the control of SCARA robot manipulator with uncertainties in the linkages mass [1]. In that work, the authors treated the nonlinearities of the system by using Taylor s series approximation. Then they used a proportional-derivative (PD) controller to control the joints motion. PCE has also been applied to the stability analysis of stochastic systems [31], where the Galerkin projection is used to find the corresponding coefficients of dynamical system equations. Then, the stochastic stability is defined based on the convergence of the coefficients. In this thesis, the method of PCE is integrated with Computed Control Torque Law (CTCL) to control nondeterministic fully-actuated multibody systems. A mathematical framework is introduced to solve the nondeterministic equations of motion. Also, a general scheme for the control of nondeterministic multibody systems is demonstrated. At the end, the CTCL method is applied to a fully actuated SCARA robot with 4 degrees of freedom Fig. 4.1 in both PCE and Monte Carlo schemes. To conclude the analysis, a comparison between the time efficiency and accuracy of the Monte Carlo and PCE method is provided. 4.2 COMPUTED TORQUE CONTROL LAW FOR DE- TERMINISTIC SYSTEMS In this section, the Computed Torque Control Law (CTCL) is introduced for deterministic multibody systems [32]. The CTCL is a method of feedback linearization utilizing the inverse dynamics of a system to calculate external forces necessary to drive the system along a desired trajectory [29]. The dynamic equation of a fully actuated open chain

59 Chapter 4. Computed Torque Control of Nondeterministic Multibody Systems 47 Figure 4.1: Schematic of a scara robot [1] multibody system in the joint coordinate frame can be written as M(q) q + N(q, q) + f d = f, (4.1) where M(q) is the inertia matrix of the entire system, q is the joint variable vector, f d is the disturbance vector, and f is the generalized force vector at each joint. The term N(q, q) includes the Coriolis, centrifugal, and gravity terms [2]. Defining the prescribed joint trajectory as q d (t), one can express the tracking error as e(t) = q d (t) q(t). (4.2) The first and second time derivative of the tracking error can be defined as ė(t) = q d (t) q(t), (4.3) ë(t) = q d (t) q(t). (4.4)

60 Chapter 4. Computed Torque Control of Nondeterministic Multibody Systems 48 Equation (4.1) can be solved for q and the result can be substituted into Eqn. (4.4). Finally, one can represent the second time derivative of the Eqn. (4.4) as ë(t) = q d (t) + M 1 (q)(n(q, q) + f d f). (4.5) The control input function can be defined as w = q d (t) + M 1 (q)(n(q, q) f), (4.6) while the disturbance is represented as s = M 1 (q)f d. (4.7) One can define the state x as x = e(t). (4.8) ė(t) By using Eqns. ( ), the time derivative of x is expressed as a linear combination of the tracking error at the position and velocity levels, input function, and disturbance function in Brunovsky canonical form [2] as ė(t) = 0 I e(t) + 0 w + 0 s. (4.9) ë(t) 0 0 ė(t) I I Choosing an appropriate input function w, the above equation can be stabilized and the tracking error e is driven to zero. Equation (4.6) can then be solved to find the generalized driving forces f as f = M(q)( q d w) + N(q, q). (4.10)

61 Chapter 4. Computed Torque Control of Nondeterministic Multibody Systems 49 Figure 4.2: Nonlinear and linear components of CTCL [2]. This is called Computed Torque Control Law. In this approach, there is no state-space transformation in going from Eqn. (4.1) to Eqn. (4.10). Hence, by using a control input in Eqn. (4.9) that drives the error e to zero, the nonlinear control input given by f drives the open chain multibody system to follow the prescribed trajectory [2]. One may substitute Eqn. (4.10) in Eqn. (4.1) to represent the dynamic equation of system as M(q) q + N(q, q) + f d = M(q)( q d w) + N(q, q), (4.11) which after some simplifications can also be stated as ë = w + M(q) 1 f d. (4.12) This closed-loop control scheme shown in Fig. (4.2) is comprised of two components. The first component consists of a nonlinear inner loop where the generalized applied forces (actuator torques) are calculated and the equations of motion are solved by an integrator. This part of the controller design is essentially the inverse dynamics of the system. The second component is an outer loop which stabilizes the system by using PI, PID, or any other type of controller that makes the system follow the specified trajectory [29]. The outer-loop signal w can be chosen using many approaches, including robust and adaptive control techniques [2].