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LYAPUNOVBASED CONTROL OF A ROBOT AND MASSSPRING SYSTEM UNDERGOING AN IMPACT COLLISION By CHIENHAO LIANG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007 O 2007 ChienHao Liang To my wife YenChen and son HsuChen who constantly filled me with love and joy. ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, mentor, and friend, Dr. Warren E. Dixon, for introducing me to the interesting field of Lyapunovbased control. As an advisor, he provided the necessary guidance and allowed me to try some stupid ideas during my research. As a mentor, he helped me understand the high pressure of working in a professional environment and was willing to give me time to learn and adjust. I feel fortunate in getting the opportunity to work with him. I also appreciate my committee members Dr. Carl D. Crane III and Dr. Gloria J. Wiens for the time and help they provided. I would like to thank all my friends for their support and encouragement. I especially thank my friend Guoqiang Hu for being my listener and mentor both in the research and daily life for these two years. I would also like to thank my colleagues K~eith Stegath and Gun Lee for helping me out on those difficult days when I was doing my experiments. Without them, I cannot accomplish these experiments. I would like to thank my colleague Darren Aiken for his caring during my first year in U.S. I express my gratitude to K~eith Dupree for helping me out on my research. I would like to thank Siddhartha Mehta for helping me in thesis editing. I also express my gratitude to Parag Patre and Will Mackunis who filled my daily life with joy. Finally I would like to thank my parents for their unconditional love and support, my wife YenChen for the support and joy she always brought to me, and my son HsuChen for his hug, kiss, and smile. Their love, understanding, patience and personal sacrifice made this thesis possible. TABLE OF CONTENTS page ACK(NOWLEDGMENTS ................... .......... iv LIST OF FIGURES ................... ............. vii ABSTRACT .................. .................. i CHAPTER 1 INTRODUCTION ................... ........... 1 1.1 Introduction ................... ............ 1 1.2 Dynamic Model ................... .......... 8 2 LYAPUNOVBASED CONTROL OF A ROBOT AND MASSSPRING SYSTEM UNDERGOING AN IMPACT COLLISION .. . 11 2.1 Error System and Control Development .. .. .. .. .. .. .. 11 2.1.1 ControlObjective .................. ..... 11 2.1.2 ClosedLoop Error System .. .. .. . .. 12 2.2 Stability Analysis.................. ......... 14 2.3 ExperimentalResults ................... ....... 20 2.4 Concluding Remarks ................... ....... 26 3 ADAPTIVE LYAPUNOVBASED CONTROL OF A ROBOT AND MASS SPRING SYSTEM UNDERGOING AN IMPACT COLLISION .. .. 27 3.1 Properties and Assumptions .. .. .. .. .. .. .. .. .. .. 28 3.2 Error System and Control Development .. .. .. .. .. .. .. 29 3.2.1 ControlObjective.................. ..... 29 3.2.2 ClosedLoop Error System .. .. .. . .. 30 3.3 Stability Analysis.................. ......... 34 3.4 ExperimentalResults ................... ....... 40 3.5 Concluding Remarks ................... ....... 47 4 AN IMPACT FORCE LIMITING ADAPTIVE CONTROLLER FOR A ROBOTIC SYSTEM UNDERGOING A NONCONTACTTOCONTACT TRANSITION ................... ............. 48 4.1 Properties ................... ............. 49 4.2 Error System and Control Development .. .. .. .. .. .. .. 49 4.2.1 ControlObjective.................. ..... 50 4.2.2 ClosedLoop Error System .. .. .. .. .. .. .. .. .. 51 4.3 Stability Analysis ................... ......... 52 4.4 ExperimentalResults ................... ....... 58 4.5 Concluding Remarks ................... ....... 64 5 CONCLUSION AND RECOlillr~DAIENDAIOS .. .. .. .. .. .. 65 APPENDIX A THE EXPRESSION OF x,d, IN SECTION 2.1 .. .. .. .. .. .. 67 B THE EXPRESSION OF x,dl IN SECTION 3.2 .. .. .. .. .. .. 69 REFERENCES ................... ................ 70 BIOGRAPHICAL SKETCH ................... ......... 74 LIST OF FIGURES Figure page 11 The MoI~Spring Robot (MSR) system is an academic example of an impact between two dynamic systems. .. .. .. .. .. .. .. 7 21 Top view of experimental testbed. .. .. .. .. .. .. .. .. .. .. 21 22 Side view of experimental testbed .. .. .. .. .. .. .. .. .. .. 21 23 Springmass and robot errors e(t). Plot (a) indicates the position error of the robot tip along the X1axis (i.e., ery(t)), (b) indicates the position error of the robot tip along the X28XiS (i.e., er2(t)), and (c) indicates the position error of the springmass (i.e., em(t)). .. .. .. .. 24 24 Computed control torques J (q)F(t) for the (a) base motor and (b) sec ond link motor.................... ............ 24 25 Applied control torques J (q)F(t) (solid line) versus computed control torques (dashed line) for the (a) base motor and (b) second link motor. .25 26 Computed desired robot trajectory, xay (t). .. .. .. .. .. .. .. .. 25 27 Contact (A = 1) and noncontact (A = 0) conditions for the robot and massspring system. ................... ........... 26 31 Massspring and robot errors e(t). Plot (a) indicates the position error of the robot tip along the X1axis (i.e., er,(t)), (b) indicates the position error of the robot tip along the X28XiS (i.e., er2(t)), and (c) indicates the position error of the massspring (i.e., em (t)). .. .. .. .. 42 32 Massspring and robot errors e(t) during the initial two seconds. .. .. 43 33 Computed control torques J (q)F(t) for the (a) base motor and (b) sec ond link motor.................... ............ 43 34 Applied control torques J (q)F(t) (solid line) versus computed control torques (dashed line) for the (a) base motor and (b) second link motor. .44 35 Computed desired robot trajectory, Xas(t). . . . . 44 36 Unitless parameter estimate Odk (t) introduced in (313). .. .. .. .. 45 37 Estimate for the unknown constant parameter vector Orttl (a) Brlo(t) K,, (b) Or4 1)= m, C) ,1(t) Km, and (d) 6,7(t)= m, where mi, m2 ERI denote the mass of the first and second link of the robot, m, ERI denotes the mass of the motor connected to the second link of the robot, and m ERI denotes the mass of the massspring system. .. 45 38 Estimate for the unknown constant parameter vector Or (f (a) Grs (t) = ",(b) r2~(t) s, and (c) 8,s(t) s. ............... 46 39 Estimate for the unknown constant parameter vector Or (1) (a) Gre (t) m,, (b) 0r3(t) = 1, and (c) 0r9@t) m2. .. .. .. .. .. .. .. .. 46 41 Massspring and robot errors e(t). Plot (a) indicates the position error of the robot tip along the X1axis (i.e., ery(t)), (b) indicates the position error of the robot tip along the X28XiS (i.e., er2(t)), and (c) indicates the position error of the massspring (i.e., em(t)). .. .. .. .. 60 42 Applied control torques J (q)F(t) for the (a) base motor and (b) second link motor. ................... ............. 60 43 Applied control torques J (q)F(t) for the (a) base motor and (b) second link motor during the first 0.8 second. .. .. .. . . 61 44 Computed desired robot trajectory, Xay (t). . . . . 61 45 Parameter estimate Odk (t) introduced in (313). .. .. .. .. .. .. 62 46 Estimate for the unknown constant parameter vector Orttl (a) Brlo(t) K,, (b) Or4 1)= m, C) ,1(t) Km, and (d) 6,7(t)= m, where mi, m2 ERI denote the mass of the first and second link of the robot, m, ERI denotes the mass of the motor connected to the second link of the robot, and m ERI denotes the mass of the massspring system. .. 62 47 Estimate for the unknown constant parameter vector Or (f (a) Grs (t) = ",(b) r2~(t) s, and (c) 8,s(t) s. ............... 63 48 Estimate for the unknown constant parameter vector Or (1) (a) Gre (t) ms, (b) 0r3t) = 1, and (c) 0r9@) m2. .................. 63 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science LYAPUNOVBASED CONTROL OF A ROBOT AND MASSSPRING SYSTEM UNDERGOING AN IMPACT COLLISION By ChienHao Liang May 2007 Chair: Warren E. Dixon Major: Mechanical Engineering The problem of controlling a robot during a noncontacttocontact transition has been a historically challenging problem that is practically motivated by applications that require a robotic system to interact with the environment. If the contact dynamics are not properly modeled and controlled, the contact forces could result in poor system performance and instabilities. One difficulty in controlling systems subj ect to noncontact tocontact transition is that the dynamics are different when the system status changes suddenly from the noncontact state to a contact state. Another difficulty is measuring the contact force, which can depend on the geometry of the robot, the geometry of the environment, and the type of contact. The appeal of systems with contact conditions is that shortduration effects such as high stresses, rapid dissipation of energy, and fast acceleration and deceleration may be achieved from lowenergy sources. Over the last 2 decades, many researchers have investigated the modeling and control of contact systems. Most controllers target contacts with a static environment for a fully actuated system. Many researchers also exploit switching or discontinuous controllers to accommodate contact conditions. Motivation exists to explore alternative control strategies because impacts between the robot and the static environment cannot represent all the impact system applications such as the capture of disabled satellites, spaceport docking, manipulation of nonrigid bodies, and so on, and discontinuous controllers require infinite control frequency (i.e., exhibit chattering) or yield degraded stability results (i.e., uniformly ultimately bounded). As stated previously, it is necessary to consider the impact control between two dynamic systems. Our study considered a class of fully actuated dynamic systems that undergo an impact collision with another dynamic system that is unactuated. Our study is specifically focused on a planar robot colliding with a massspring system as an academic example of a broader class of such systems. The control objective is defined as the desire to command a planar robot to collide with an unactuated system and regulate the resulting coupled massspring robot (MSR) system to a desired compressed state. The collision is modeled as a differentiable impact. Lyapunovbased methods are used to develop a continuous controller that yields asymptotic regulation of the mass and robot links. A desired timevarying robot link traj ectory is designed that accounts for the impact dynamics and the resulting coupled dynamics of the MSR system. A force controller is then designed to ensure that the robot link position tracking error is regulated. Unlike some other results in literature, the force controller is continuous and does not depend on measuring the impact force or the measurement of other acceleration terms. Experimental results are provided to illustrate the successful performance of the controller in each chapter. CHAPTER 1 INTRODUCTION 1.1 Introduction The problem of controlling a robot during a noncontacttocontact transition has been a historically challenging problem that is practically motivated by applications that require a robotic system to interact with the environment. If the contact dynamics are not properly modeled and controlled, the contact forces could result in poor system performance and instabilities. One difficulty in controlling systems subject to noncontacttocontact transition is that the dynamics are different when the system status changes suddenly from the noncontact state to a contact state. Another difficulty is measuring the contact force, which can depend on the geometry of the robot, the geometry of the environment, and the type of contact. As stated by Tornambe [1], the appeal of systems with contact conditions is that shortduration effects such as high stresses, rapid dissipation of energy, and fast acceleration and deceleration may be achieved from lowenergy sources. Over the last 2 decades, many researchers have investigated the modeling and control of contact systems including: [1][32]. 2 trends are apparent after a comprehensive survey of contact systems in control literature. Most controllers target contacts with a static environment for a fully actuated system. Many researchers also exploit switching or discontinuous controllers to accommodate for the contact conditions. The regulation of a onelink robot that undergoes smooth or nonsmooth impact dynamics was examined by Tornambe [1]. A class of switching controllers were examined by Brogliato et al. in [2] for mechanical systems with differentiable dynamics subject to an algebraic inequality condition and an impact rule relating the interaction impulse and the velocity. The analysis in [2] used a discrete Lyapunov function that required the use of the Dini derivative to examine the stability of the system. A simple mechanical system subject to nonsmooth impacts is considered by Menini and Tornambe [3], where the desired timevarying planar motion of a mass is controlled within a closed region defined by an infinitely massive and rigid circular barrier. In [4], Sekhavat et al. utilized LaSalle's Invariant Set Theorem to prove the stability of a discontinuous controller that is designed to regulate the impact of a hydraulic actuator with a static environment where no knowledge of the impact dynamics is required. Volpe and K~hosla developed a nonlinear impact control strategy for a robot manipulator experiencing an impact with a static environment [5]. The controller in [5] was based on the concept that negative proportional force gains, or impedance mass ratios less than unity, can provide impact response without bouncing. Tornambe [6] also proposed a switching controller to globally asymptotically regulated a two degreeoffreedom (DOF) planar manipulator to contact an infinitely rigid and massive surface. Pagilla and Yu [7] proposed a discontinuous stable transition controller to deal with the transition from a noncontact to a contact state where explicit knowledge of the impact model is not required. A discontinuous model based adaptive controller was proposed by Akella et al. [8] to asymptotically stabilize the contact transition between a robot and static environment. Tarn et al. [9] proposed a sensorreferenced control method using positive acceleration feedback with a switching control strategy for impact control for a robot and a constrained surface. A switching controller was also proposed by Wu et al. in [10] to eliminate the bouncing phenomena associated with a robot impacting a static surface. The structure of the switching controller in [10] was dependent on impact feedback from a force sensor. Lee et al. developed a hybrid bangbang impedance/timedelay controller that establishes a stable interaction between a robot with nonlinear joint friction and a stiff environment in [11] and [12]. Nelson et al. [13][15] proposed a nonlinear control strategy that considers force and vision feedback simultaneously and then switches to pure force control when it is unable to accurately resolve the location of the robot's endeffector relative to the surface to be contact. Motivation exists to explore alternative control strategy for the impact systems because impacts between the robot and the static environment cannot represent all the impact system applications such as the capture of disabled satellites, spaceport docking, manipulation of nonrigid bodies, and so on, and discontinuous controllers require infinite control frequency (i.e., exhibit chattering) or yield degraded stability results (i.e., uniformly ultimately bounded). As stated previously, it is necessary to consider the impact control between two dynamic systems. Several controllers have been developed for underactuated dynamic systems that have an impact collision. For example, a family of deadbeat feedback control laws were proposed by Brogliato and Rio [16] to control a class of juggling like systems. One of the contributions in [16] is a study of the intermediate controllability properties of the object's impact Poincari: mapping. A proportional derivative (PD) controller was developed by Indri and Tornambe [17] to address global asymptotic stabilization of underactuated mechanical systems subject to smooth impacts with a static object. In our previous work in [18], a nonlinear energybased controller is developed to globally asymptotically stabilize a dynamic system subject to impact with a deformable static mass. The contribution in [18] is that the underactuated states are coupled through the energy of the system as a means to mitigate the transient response of the unactuated states. Our study considered a class of fully actuated dynamic systems that undergo an impact collision with another dynamic system that is unactuated. Our study is specifically focused on a planar robot colliding with a massspring system as an academic example of a broader class of such systems. The control objective is defined as the desire to command a planar robot to collide with an unactuated system and regulate the resulting coupled massspring robot (MSR) system to a desired compressed state. The collision is modeled as a differentiable impact as in recent work in [1], [17], and our previous efforts in [18][20]. Lyapunov based methods are used to develop a continuous adaptive controller that yields asymptotic regulation of the mass and robot links. A desired timevarying robot link trajectory is designed that accounts for the impact dynamics and the resulting coupled dynamics of the MSR system. The desired link trajectory converges to a setpoint that equals the desired mass position plus an additional constant that is due to the deformation of the mass. A force controller is then designed to ensure that the robot link position tracking error is regulated. Unlike some other results in literature, the continuous force controller does not depend on measuring the impact force or the measurement of other acceleration terms: only the position and velocity terms of the springmass system and the joint angles and the angular velocities terms of the planar robotic arm are needed for the proposed controller. Chapter 2 and our preliminary efforts in [19] provide a first step at controlling the proposed impact system. The control development in Chapter 2 is based on the assumption of exact model knowledge of the system dynamics. The nonlinear con tinuous Lyapunovbased controller is proven to regulate the states of a planar robot colliding with the unactuated massspring system and yields global asymptotic result. In Chapter 3 and our preliminary results in [20], the dynamic model for the system is assumed to have uncertain parameters. The control objective is defined as the desire to command the planar robot to collide with the massspring system and regulate the resulting coupled massspring robot (MSR) system to a desired compressed state while compensating for the constant, unknown system parameters. Two linear parameterizations are designed to adapt for the unknown robot and massspring parameters. An adaptive nonlinear continuous Lyapunov based controller is proven to regulate the states of the systems and yields global asymptotic regulation result. When the controllers in Chapter 2 and Chapter 3 were implemented in the presence of large initial conditions, violent impacts between the robot and the massspring system resulted. In fact, the controller was artificially saturated (the saturation effects were not considered in the stability analysis) to reduce the impact forces so that the mass deflection would not destroy a capacitance probe. Various researchers have investigated methods that prescribe, reduce, or control the interaction forces during or after the robot impact with the environment such as [6][15] and [21][34] because large interaction forces can damage both the robot and/or the environment or lead to degraded performance or instabilities. Walker and Gertz et al. exploited kinematic redundancy of the manipulator to reduce the impact force in [21][23]. A two degreeoffreedom (DOF) planar manipulator was globally asymptotically regulated to contact an infinitely rigid and massive surface by Tornambe [6] where the impact force was estimated using a reducedorder observer. Pagilla and Yu [7] proposed a stable transition controller to deal with the transition from a noncontact to a contact state which can improve transition performance and force regulation. A discontinuous modelbased adaptive controller was proposed by Akella et al. [8] to asymptotically stabilize the contact transition between a robot and static environment. The controllers for each contact state were tuned independently to reduce contact force during the process of making contact with the environment. Tarn et al. [9], [10] proposed a sensorreferenced control method using positive acceleration feedback with a switching control strategy for impact control and force regulation for a robot and a constrained surface where the peak impulsive force and bouncing caused directly by overshooting and oscillation of the transient force response can be reduced. Lee et al. developed a hybrid bangbang impedance/timedelay controller that establishes a stable contact and achieves the desired dynamics for contact or noncontact conditions in [11] and [12], where the force overshoots can be minimized. Nelson et al. [13][15] proposed a switching nonlinear controller that combines force and vision control. When the robot's endeffector approaches the target, the controller switches to force control to minimize impact force and to regulate the contact force. By modeling the impact dynamics as a state dependent jump linear system, Chiu and Lee were able to apply a modified stochastic maximum principle for state dependent jump linear systems to optimize the approach velocity, the force transient during impact and the steady state force error after contact is established [24]. Hyde and Cutkosky [25] proposed an approach, based on input command shaping, to suppress vibration during the contact transition of switching controllers by modifying feedforward information. Various other applications also focused on the reduction of impact force between different systems during the control process such as impact force reduction of hopping robot considered by Shibata and Natori [26] and Ohnishi et al. [27], [28], bilateral telerobotic system considered by Dubey et al. [29], human robot symbiotic environment considered by Yamada et al. [30] and Li et al. [31], and space manipulator and freeflying target considered by Huang et al. [32]. Exploring alternative methods is motivated because kinematic redundancy is not always possible, and again the discontinuous controllers require infinite control frequency (i.e., exhibit chattering) or yield degraded stability results (i.e., uniformly ultimately bounded). These results provide the motivation for the control development in Chapter 4. Specifically, the feedback elements for the continuous controller in Chapter 3 are contained inside of hyperbolic tangent functions as a means to limit the impact forces resulting from large initial conditions as the robot transitions from noncontact to contact states. Although saturating the feedback error is an intuitive X I~ r91 ~I I Figure 11: The Me I~Spring Robot (MSR) system is an academic example of an impact between two dynamic systems. solution that has been proposed in previous literature for other types of robotic systems with limited actuation, several new technical challenges arise due to the impact condition. The main challenge is that the use of saturated feedback does not allow some coupling terms to be canceled in the stability analysis, resulting in the need to develop state dependent upper bounds that reduce the stability to a semiglobal result (as compared to the global results in Chapter 2 and Chapter 3). The semiglobal result is problematic in the current applicative context because certain control terms do not appear in the closedloop error system during the noncontact condition, resulting in a uniformly ultimately bounded result until the robot makes contact. Hence, the result hinges on new development within the semi global stability proof for an error system that is only uniformly ultimately bounded during the noncontact phase. This problem is exacerbated by the fact that the Lyapunov function contains radially unbounded hyperbolic functions of some states that only appear inside of saturated hyperbolic terms in the Lyapunov derivative. New control development, closedloop error systems, and Lyapunovbased stability analysis arguments are used to conclude the result. Experimental results are provided to illustrate the successful performance of the controller in each chapter. 1.2 Dynamic Model The subsequent development is motivated by the academic problem illustrated in Fig. 11. The dynamic model for the twolink revolute robot depicted in Fig. 11 can be expressed in the jointspace as M~(q)q + C(q, q)q + h(q) = 7, (11) where q(t), q(t), q(t) E RW2 represent the angular position, velocity, and acceleration of the robot links, respectively, M~(q) E RW2x2 represents the uncertain inertia matrix, C(q, q) E RW2x2 represents the uncertain centripetalCoriolis effects, h(q) n [hi (q), h2(~l E I2 repreSentS uHCertain COHSerVatiVe forCeS (e.g., gravity), and 7(t) E RW2 represents the torque control inputs. The Euclidean position of the end point of the second robot link is denoted by x,(t) n [x,l(t), x(tr2T )F 6 2, which can be related to the jointspace through the following kinematic relationship: Xr = J(q)q, (12) where J(q) E RW2x2 denotes the manipulator Jacobian. The unforced dynamics of the massspring system in Fig. 11 are mim + ks (Xm Xo) = 0, (13) where xm(t), xm(t), xm(t) ERI represent the displacement, velocity and acceleration of the mass m E R, so ERI represents the initial undisturbed position of the mass, and ks ERI represents the stiffness of the spring. Assumption 1.1: We assume that x,z (t) and xm(t) can be bounded as <~i I Xr(t) wZm7 < Xm(t)l < Om (14) where P0, ERI is a known constant that is determined by the minimum coordinate of the robot along the X1axis, and pm ERI is a known positive constant. The lower bound assumption for x,z (t) is based on the geometry of the robot, and the upper bound assumption for xm(t) is based on the physical fact that the mass is attached by the spring to some object, and the mass will not be able to move past that object. In the following, the contact model is considered as an elastic contact with finite stiffness. An impact between the second link of the robot and the springmass system occurs when xwe(t) > xm(t) (see Fig. 11). The impact will yield equal and opposite force reactions between the robot and massspring system. Specifically, the impact force acting on the mass, represented by Fm(x,, xm) E RW, is assumed to have the following form [1], [17] Fm = Kc,A(xrl Xm), (15) where K, ERI represents a positive stiffness constant, and A(x,, xm) ERI is defined A = 1 we > Xm(16 0=( X,1 < Xm1~ The impact force acting on the robot links produces a torque, denoted by 7d(X,, Xm, q) E RW2, aS follows. 7a = KIAn (sX, Xm) I I sin(ql) + 12 sinq 92 ql 17 where IIl, 12 ER denote the robot link lengths. Based on (11), (13), and (15)(1 7), the dynamic model for the MSR system can be expressed as M~(q)q + C(q, q)q + h(q) 7d = 7 (18) mxm + k~s(Xm Xo) = Fm After premultiplying the robot dynamics by the inverse of the Jacobian transpose and utilizing (12), the dynamics in (18) can be rewritten as [19], [20] M~ (x,) x, + C (x,, x,) x, + h(x,) + Fm = F (19) mxm + ks (Xm Xo) = Fm, (110) where F(t) n JT(q)7(t) E RW2 denotes the manipulator force. The dynamic model in (19) and (110) exhibits the following properties that will be utilized in the subsequent analysis. Property 1.1: The inertia matrix M~(x,) is symmetric, positive definite, and can be lower and upper bounded as alI1 2 I TM(E < a8I12, ~ E2 211 where al, a2 ERI are positive constants. Property 1.2: The following skewsymmetric relationship is satisfied T( 12I(XT) C(XT, XT))( = 0 V( e RW2. (112) CHAPTER 2 LYAPUNOVBASED CONTROL OF A ROBOT AND MASSSPRING SYSTEM UNDERGOING AN IMPACT COLLISION This chapter and our preliminary efforts in [19] provide a first step at con trolling the proposed impact system in Section 1.2. The academic example of a planar robot colliding with an unactuated massspring system is used to represent a broader class of such systems. The control development in this chapter is based on the assumption of exact model knowledge of the system dynamics. The control objective is to command a robot to collide with an unactuated massspring sys tem and regulate the springmass to a desired compressed state. Lyapunovbased methods are used to develop a continuous controller that yields global asymp totic regulation of the springmass and robot links. Unlike some other results in literature, the developed continuous force controller does not depend on sensing the impact, measuring the impact force, or the measurement of other acceleration terms. Experimental results are provided to validate our analysis. This chapter is organized as follows. Section 2.1 describes the error system and control development followed by the stability analysis in Section 2.2. Section 2.3 describes the experimental set up and results that indicate the successful perfor mance obtained by implementing the proposed controller followed by conclusion in Section 2.4. 2.1 Error System and Control Development 2.1.1 Control Ob jective One goal in this chapter is to regulate the states of a dynamic system (i.e., a twolink planar robot) that has an impact collision with another dynamic system (i.e., a massspring). A regulation error, denoted by e(t) E RW3, is defined to quantify the control objective as e= E e" em~ (21) where e,(t) n [ery,er2FT E I2 and em(t) ERI denote the regulation error for the robot and massspring system, respectively, and are defined as e, = Xwe X, em, = sma Xm. (22) In (22), sma ERI denotes the constant known desired position of the springmass. The desired position of the endpoint of the second robot link, denoted by xwa(t) a [X,dl, Xrd2] I2, is selected so that the robot will produce the desired spring mass position while accounting for the impact dynamics. Specifically, xay (t) (i.e., the desired horizontal Euclidean coordinate in Fig. 11) is a timevarying signal that is subsequently designed to account for the impact condition and the coupled dynamic response of the MSR system, and Zrd2 (i.e., the desired vertical Euclidean coordinate in Fig. 11) is selected as a constant. Filtered tracking errors, denoted by r,(t) E RW2 and rm(t) E RW, are defined as r, = e, + ae, rm = em + aem (23) to facilitate the subsequent control design and stability analysis where a ERI is a positive control parameter. 2.1.2 ClosedLoop Error System By taking the time derivative of mrm(t) and utilizing (22) and (23), the following openloop error system can be obtained: m~im = k(xm xo) KA(x,l xm) + amem (24) where the springmass dynamics in (110) were substituted for mxm(t). Motivated to design the desired robot link trajectory to position the springmass, (22) is used to rewrite the openloop system in (24) as m~im = k(xm xo) + amem + Knery + Knxm KAXar. (25) Based on (25), the desired robot link position is designed as 1 1 wa (amem ,+ k(xm ro)) + ( +k)m+ m(26) Xd K K Xrd2 = where seREI is an appropriate positive constant (i.e., e is selected so the robot will impact the massspring system), and ki and k2 ERI are positive constant control gains. After substituting (26) into (25), the closedloop error system for rmit) can be obtained as m~im = (1 A) (k(xm xo) + amem) + Kners A (ki + k2) rm. (27 As xm(t) z ma, (22) and (23) can be used to conclude that rm(t) 0, em(t) 0, and em(t) 0 Hence, (26) can be used to conclude that Xay (t) K ,(Xm Xo) + zma. (28) The physical meaning of (28) is that the desired robot position varies in time to account for the impact dynamics and the coupled dynamic system, and the desired steadystate value is a constant that equals the desired springmass position plus the mass deformation. After taking the time derivative of r,(t) and premultiplying by the robot inertia matrix, the following openloop error system is obtained: MFi, = M~xre F + Cx, + h + KA(x,l xm) + al~e,, (29) where (110) and (22) were utilized. Based on (29) and the subsequent stability analysis, the robot force control input is designed as F n M~Xre + CXre + h + + k37r x) S11 + a MeC, + Ce,.) + e, (2 10) where k3 ERI is a positive constant control gain. Based on the use of the backstep ping method, the robot force control input in (210) requires the first and second derivative of Xrd(t). As described in the Appendix A, the first and second derivative of X,d(t) exists (i.e., x,d(t) is continuously differentiable) and do not depend on acceleration terms. The closedloop error system for rr (t) can be obtained after substituting (210) into (29) as M~i, = k3Tr Crr Br (211) 2.2 Stability Analysis Ti;,~ ..;~, n: The controller given by (210) ensures global asymptotic stability of the robot and springmass regulation errors in the sense that I, (t) 0 e,(t)I 0 as t oo (212 ) provided a, ky, and k2 are selected sufficiently large and the following gain condi tion is satisfied: K2 a > (213) In the following proof, a Lyapunov function and its derivative are provided. The analysis is then separated into two cases: contact and noncontact. For the noncontact case, the stability analysis indicates the controller and error signals are bounded and converge to an arbitrarily small region. Additional analysis indicates that within this region, contact must occur. When contact occurs, a Lyapunov analysis is provided that illustrates the MSR system asymptotically converges to the desired setpoint. Proof: Let V(rm, Tr, e,) ERI denote the following continuously differentiable, nonnegative, radially unbounded function (i.e. a Lyapunov function candidate) 1 1 1 VA mr2 + r,"l~r, + e 'er. (214) 2"2 2 The Lyapunov function candidate in (214) can be lower and upper bounded as 11 1< V < yz : 215) where yl, y2 ERI are positive constants, and z(t) E R"5 is defined as The time derivative of (214) can be determined as where (112), (23), (27), and (211) were utilized. The remainder of the analysis is divided into two cases: Case 1the robot and massspring systems are not in contact (i.e., wel < xm and A = 0), and Case 2the robot and massspring systems are in contact (i.e., wel > xm and A = 1). Case la: Before the initial contact, the massspring system is at rest and the spring is not compressed; hence, k(xm xo) = 0 em = 0. (217) Based on (217), (216) can be expressed as V = k3r,~  0, Br. (218) The expressions in (214) and (218) can be used to prove that e, (t), r, (t), rm(t) E C, and that e,(t), r,(t) E 2. Based on the fact that e,(t), r,(t), rm(t) E C,, standard signal chasing arguments can be used to prove that e,(t), r,(t), Xwa(t), F(t) E Coo along with all of the other closedloop signals. Since e,(t), r, (t) E C, n 2 and are uniformly continuous, Barbalat's Lemma can be applied to conclude that  r, (t)I 0 e,(t)I 0 as t oo. (219) The result in (219) can be used along with (26) to conclude that X,dl(t) (ki1 + k2 tem 0 x leading to an impact with the massspring system. Case lb: After an impact, the robot may loose contact with the springmass. In this case k(xm xo) / 0 em, = j:m. (220) Based on (220), (216) can be expressed as For this case, the initial velocity of the springmass is denoted by ( ERI and the initial position is denoted by xm E RW. Given the aforementioned initial conditions, the solution for am (t) can be obtained from (110) as xm(t) = zo + (xm xo) cos(l t) + (iI sin(l t). (222) The time derivative of (222) can be expressed as j:m(t) = (xm xo) II sin(l t)+(o( t.(3 Sm m m23 Based on (222) and (223), rm(t) can be expressed as follows: rm = (xm xo) ii sin(l t) ( cos(l t) + max axm(t). (224) Vm m Vm The expressions in (222)(224) can be upper bounded as zm < zoI + Im Xo + (A I < (1 (225) sml < Im xo i + IEl P2 (226) Sm rm < I m Xo i + C + a~ (xmm + (1) < P3 (227) Sm where (1, (2, P3 ERI denote positive constants. After utilizing (225)(227), an upper bound for (221) can be developed as V I V + 6 228> where 6 ERI is a positive constant, and p ERI is defined as 8 min(k3, a, 1) Standard techniques can be used to solve (228) for V(t) as V~t < (0) e~xp(t)f + . (229) Based on (229), it is clear that during the transient case (Case lb) that e, (t), r,(t), rm(t) E C,. Based on the fact that e,(t), r,(t), rm(t) E C,, standard signal chasing arguments can be used to prove that e,(t), i,(t), x,d(t), F(t) E C, along with all of the other closedloop signals. As V(t) 0 eventually PV < 6. The previous development can be used to conclude that for the noncontact case V and hence,  r, (t)   e, (t)  rm(t)  as t oo. (230) Further analysis is required to prove that the manipulator makes contact with the massspring system. Contact between the manipulator and the massspring system occurs when xmi(t) > xm(t). Based on (230), a sufficient condition for contact can be developed as After using (26), the sufficient condition in (231) can be expressed as 1 1 6 (amem + k(xm ro)) + (k 2 m' .(232) K K P By using (23) and performing some algebraic manipulation, the inequality in (232) can be expressed as 6K (ki + k2) rm '> + m Tm, I 2m  + k~(Xma + I'. + Xo). (233) From Assumption 1.1, em (t) can be upper bounded as em < m (234) where, Em ER denotes a known positive constant. If em(t) < 0, then the sufficient condition in (233) may not be satisfied. The condition that em(t) < 0 will only occur if an impact collision occurs that causes the mass to overshoot the desired position. However, even if an impact occurs and the mass overshoots the desired position, the dynamics of the massspring system will force the mass position error to return to the initial condition in the noncontact case. That is, em(t) zma~ Xo > Em where mERI denotes a known positive constant. Based on (224), (227) and the fact that em,(t) will eventually be lower bounded by m in a noncontact condition, rm(t) can be upper and lower bounded as P3 > m m 0 X O I a, m r (235) Sm where 8, ERI denotes a known positive constant, provided the following gain condition is satisfied zm Xo i + I( a > m (236) m Based on (233), (234), (235), and the previous development, the control parameter ki and k2 can be selected according to the following sufficient condition to ensure the robot and massspring system make contact ki~" + k2 t + 2mem + kC(Xma + Em + Xo) (7 Case 2: When the robot and massspring systems are in contact, (216) can be upper bounded as V' < kI rm 2l k3 Ilr 2l Q lr 12 + [K rm e7 ki2 Il, 12]. (238) After completing the squares on the bracketed terms, the following expression is obtained: V <k rm2s i 3T 20 4k r 2 .i (239) Provided the gain condition given in (213) is satisfied, (215) can be used to upper bound (239) as Vi < YV (240) where y3 ERI is a positive constant. The expression in (240) indicates that while in contact, the robot and springmass position errors are exponentially regulated, and e,(t), r,(t), rm(t) E C,. Based on the fact that e,(t), r,(t), rm(t) E m,, standard signal chasing arguments can be used to prove that e,(t), i,(t), X,d(t), F(t) E C, along with all of the other closedloop signals. 2.3 Experimental Results The testbed depicted in Fig. 21 and Fig. 22 was developed for experimental demonstration of the proposed controller. The testbed is composed of a mass spring system and a twolink robot. The body of the massspring system includes a Ushaped aluminum plate (item (8) in Fig. 21) mounted on an undercarriage with porous carbon air bearings which enables the undercarriage to glide on an air cushion over a glass covered aluminum rail. A steel core spring (item (1) in Fig. 21) connects the undercarriage to an aluminum frame, and a linear variable displacement transducer (LVDT) (item (2) in Fig. 21) is used to measure the position of the undercarriage assembly. The impact surface consists of an aluminum plate connected to the undercarriage assembly through a stiff spring mechanism (item (7) in Fig. 21). A capacitance probe (item (3) in Fig. 21) is used to measure the deflection of the stiff spring mechanism. The twolink robot (items (46) in Fig. 21) is made of two aluminum links, mounted on 240.0 Nm (base link) and 20.0 Nm (second link) directdrive switched reluctance motors. The motors are controlled through power electronics operating in torque control mode. The motor resolvers provide rotor position measurements with a resolution of 614400 pulses/revolution, and a standard backwards difference algorithm is used to numerically determine velocity from the encoder readings. A Pentium 2.8 GHz PC operating under QNX hosts the control algorithm, which was implemented via Qmotor 3.0, a graphical userinterface, to facilitate realtime graphing, data logging, and the ability to adjust control gains without recompiling the program (for further information on Qmotor 3.0, the reader is referred to [40]). Data acquisition and control implementation were performed at a frequency of 2.0 kHz using the ServoToGo I/O board. Figure 21: Top view of experimental testbed. Figure 22: Side view of experimental testbed 1.* ` e*lr'. ~ :~q The parameters for the dynamic model in (19) and (110) have the following values : mi = 7.10 [kg] m2 = 1.11 [kg] m =13.6 [kg] m, = 10.0 [kg] lI = 0.37 [m] 12 = 0.17 [m] k = 2000 [N/m] K = 1.8 x 106 [N/m] , where mi, m2 ERI represent the mass of the first and second link, m ERI is the mass of the cart, and m, ERI is the mass of the motor that drives the second link. The control gains a and k~3, defined as scalars in (23) and (210), were implemented (with nonconsequential implications to the stability result) as diagonal gain matrices to provide more flexibility in the experiment. Specifically, the control gains were selected as ki= k = 10000 k3 = ding {250, 12} a = ding {95, 100, 90} . The initial conditions for the robot coordinates and the springmass position were (in [m]) [r,(0) i;r2(0) 21m(0) ]= 0016 0.487 0.203  The initial velocity of the robot and springmass were zero, and the desired spring mass position was tin [mml Xma = 233. That is, the tip of the second link of the robot was initially 217 mm from the desired setpoint and 187 mm from impact along the X1axis (see Fig. 21). Therefore, once the initial impact occurs, the robot is required to depress the spring (item (1) in Fig. 21) to move the mass 30 mm along the X1axis. The massspring and robot errors (i.e., e(t)) are shown in Fig. 23. The peak steadystate position error of the robot tip along the X1axis (i.e., erl) and along the X2aXiS (i.e., er2 ) are 9.6 pm and 92 pm, respectively. The peak steadystate position error of the springmass (i.e., '~. ) is 7.7 pm. The 92 pm is due to the lack of the ability of the model to capture friction and slipping effects on the contact surface. In this experiment, a significant friction effect is present along the X28XiS between the robot tip and the contact surface due to a large normal spring force that is applied along the X1axis. The input control torques (i.e., J (q)F(t)) are shown in Fig. 24 and Fig. 25. To constrain the impact force to a level that ensured the aluminum plate did not flex to the point of contact with the capacitance probe, the computed torques are artificially saturated. Fig. 24 depicts the computed torques, and Fig. 25 depicts the actual torques (solid line) along with the computed torques (dashed line). The resulting desired trajectory along the X1axis (i.e., xar (t)) is depicted in Fig. 26, and the desired trajectory along the X2axis was chosen aS Xrd2 = 368 mm. Fig. 27 depicts the value of A(x,, xm) that indicates contact (A = 1) and noncontact (A = 0) conditions for the robot and massspring system. A video of the experi ment is provided at http://www.mae.ufl.edu/~dixon/projects/robaimcthm 1000 2000 0 05 1 15 2 25 3 35 4 (b) 50 0 05 1 15 2 25 3 35 4 500 0 05 1 15 225 (b) 3 35 4 0 20 005s1 15 225 (c) 3 35 4 S20 200 05 1 15 2 25 3 35 4 [sec] Figure 23: Springmass and robot errors e(t). Plot (a) indicates the position error of the robot tip along the X1axis (i.e., en (t)), (b) indicates the position error of the robot tip along the X2aXiS (i.e., er2(t)), and (c) indicates the position error of the springmass (i.e., em(t)) (a) Figure 24: Computed control torques J (q)F(t) second link motor. for the (a) base motor and (b) 02 04 06 08 (b) 02 04 06 [sec] Figure 25: Applied control torques J (q)F(t) (solid line) versus computed control torques (dashed line) for the (a) base motor and (b) second link motor. 240 235 230 15 2 [sec] 3 35 4 Figure 26: Computed desired robot trajectory, xay (t). 08 06 04 02 0 00 05 1 15 2 25 3 35 4 [sec] Figure 27: Contact (A = 1) and noncontact (A = 0) conditions for the robot and massspring system. 2.4 Concluding Remarks A nonlinear Lyapunovbased controller is proven to regulate the states of a planar robot colliding with an unactuated massspring system. The continuous controller yields global asymptotic regulation of the springmass and robot links. Unlike some other results in literature, the developed continuous force controller does not depend on sensing the impact, measuring the impact force, or the mea surement of other acceleration terms. Innovative analysis methods are used to prove the stability of the system during contact and during different noncontact states. Experimental results are provided to illustrate the successful controller performance. CHAPTER 3 ADAPTIVE LYAPUNOVBASED CONTROL OF A ROBOT AND MASSSPRING SYSTEM UNDERGOING AN IMPACT COLLISION Similar to Chapter 2, this chapter and our preliminary results in [20] consider a fully actuated planar robot colliding with an unactuated massspring system, but unlike Chapter 2 the dynamic model for both the massspring and robot systems and the impact force are assumed to have uncertain parameters. The control objective is also defined as the desire to command the robot to collide with an unactuated system and regulate the resulting coupled MSR system to a desired compressed state while compensating for the constant, unknown system parameters. Specifically, in the dynamic model of Section 1.2, M\(q) E RW2x2 represents the uncertain inertia matrix, C(q, q) E RW2x2 represents the uncertain centripetalCoriolis effects, and h (q) n [hi (q), h 2 9)l E 6 2 repreSentS uHCertain conservative forces (e.g., gravity), to ERI represents the unknown mass and ks ERI represents the unknown stiffness of the spring, and K, ERI represents the unknown positive stiffness constant. To compensate for the uncertainty, adaptive Lyapunovbased methods are used to develop a continuous adaptive controller that yields global asymptotic regulation of the mass and robot links. Two linear parameterizations are designed to adapt for the unknown robot and mass parameters. A desired timevarying robot link trajectory is designed that accounts for the impact dynamics and the resulting coupled dynamics of the MSR system. The desired link trajectory converges to a setpoint that equals the desired mass position plus an additional constant that is due to the deformation of the mass. A force controller is then designed to ensure that the robot link position tracking error is regulated. This chapter is organized as follows. The associated properties and assump tions are provided in Section 3.1. Section 3.2 describes the error system and control development followed by the stability analysis in Section 3.3. Section 3.4 describes the experimental results that indicate the successful performance of the proposed controller followed by conclusion in Section 3.5. 3.1 Properties and Assumptions Property 3.1: The robot dynamics given in (19) can be linearly parameter ized as Y(XT, in, XT)0 = M~ (XT) XT + C (XT, XT) XT + h(XT) + Fm where 8 E R~p contains the constant unknown system parameters, and Y(X,, in, XT) E RW2xp denotes the known regression matrix. Assumption 3.1: We assume that the mass of the massspring system can be upper and lower bounded as mi < m where mi, m, ERI denote known positive bounding constants. The unknown stiffness constants K, and ks are also assumed to be bounded as , where ( i:,1 (K~ Cks ERI denote known positive bounding constants. Assumption 3.2: During the subsequent control development, we assume that the minimum singular value of J(q) is greater than a known, small positive constant 6 > 0, such that max {Jl(q)} is known a priori, and hence, all kinematic singularities are always avoided. 3.2 Error System and Control Development The subsequent control design is based on integrator backstepping methods. A desired trajectory is designed for the robot (i.e., a virtual control input) to ensure the robot converges to and impacts with the mass, and to ensure that the robot regulates the mass to the desired position. Since we can not directly control the robot trajectory, a force controller is developed to ensure that the robot tracks the desired trajectory despite the transition from free motion to an impact collision and despite uncertainties throughout the MSR system. As is typical of the backstepping design method, the derivative of the desired robot trajectory is required to develop the force controller. Taking the derivative of the desired trajectory could lead to unmeasurable higher order terms (i.e., acceleration). The subsequent development exploits the hyperbolic filter structure developed in [38] to overcome the problem of injecting higher order terms in the controller and to facilitate the development of sufficient gain conditions used in the subsequent stability analysis. 3.2.1 Control Ob jective The control objective is to regulate the states of an uncertain dynamic system (i.e., a twolink planar robot) that has an impact collision with another uncertain dynamic system (i.e., a massspring). A regulation error, denoted by e(t) E RW3, iS defined to quantify this objective as e= I em e" where e,(t) n [ery,er2FT E I2 and em(t) ERI denote the regulation error for the endpoint of the second link of the robot and massspring system (see Fig. 11), respectively, and are defined as e, = Xwe X, em = sma xm. (32) In (32), sma ERI denotes the constant known desired position of the mass, and X~d(t) n [X,ax(t), Xrd2] I2 denotes the desired position of the endpoint of the second link of the robot. The subsequent development is based on the assumption that q(t), q(t), xm(t), and j:m(t) are measurable, and that x,(t) and j:,(t) can be obtained from q(t) and q(t). To facilitate the subsequent control design and stability analysis, filtered tracking errors, denoted by rm (t) ERI and r, (t) E RW2, are defined as [38] rlm = em + al tanh(em) + a2 tanh(ef) (33) Tr e, + ae,, where a, al, a2 ERI are positive, constant gains, and ef(t) ERI is an auxiliary filter variable designed as [38] ef = a3 tanh(ef) + a2 tanh(em) ki cosh2 6f )rm,(34 where ki ERI is a positive constant control gain, and a3 ERI is a positive constant filter gain. 3.2.2 ClosedLoop Error System By taking the time derivative of mrlm(t) and utilizing (15), (110), (32), and (33), the following openloop error system can be obtained: milm = Yabla KAi (sX, 2m) + r2m cosh2 (ep)if f olmcosh2(em)em,. (35) In (35), Yd(Xm) n (Xm Xo) and Odn A k. To facilitate the subsequent analysis, the following notation is introduced [38]: Ydla = YaKK,'94 = Ydkcedk (36) = K,,, (m o After using (33) and (34), the expression in (35) can be rewritten as milm = Ydea + K, (Xway Az,2) + Kc,Amm KIsX~d + X a2mk10lm, (37) where X(em, ef, rim, t) ERI is an auxiliary term defined as S=aim cosh2( 6m) trm 01tanh(em,)) tit~m cosh2(em) tanh(ef) (38) + Ll2mn cosh2 6/~i) (3 tanh(e7~i)) + az2m cosh2 6/.) (L2~ tanh(em))  The auxiliary expression X(em, ef, rim, t) defined in (38) can be upper bounded as IX  (1  ,39) where (1 ERI is a positive bounding constant, and z(t) E RW3 is defined as z = Im tanh(em,) tanh(e7) (310) Based on (37) and the subsequent stability analysis, the desired robot link position is designed as Ild ~ ddk + 1Cm + k2z tanh(em) kik2L~ cosh2(eyi) tanh(ef) (311) Xrd2 = ~ In (311), seREI is an appropriate positive constant (i.e., e is selected so the robot will impact the massspring system in the vertical direction), k2 ERI is a positive constant control gain, and the control gain ki ERI is defined as S1 kiC ^ (3 + k,l1(f) (3 12) where kni ERI is a positive constant nonlinear damping gain. The parameter estimate Odk (t) ERI in (311) is generated by the adaptive update law Ojdk npTOjTFY / ). (3li In (313), r ERI is a positive constant, and proj() denotes a sufficiently smooth projection algorithm [39] utilized to guarantee that Odk (t) can be bounded as , dk e Od 8dkc (314) where Odkc 8dkc ERI denote known, constant lower and upper bounds for Odkct) respectively. After substituting (311) into (37), the closedloop error system for rlm(t) can be obtained as mrilm = KI (Zpar Azl,l) + KI (Amm, Ism) + Klklik2 cosh2 (e ) tanh(e y) (315) + Ydkcedk KI~k2 tanh(em) + X a2mklrlm In (315), the parameter estimation error Odk (t) ERI is defined as edkc edk idk The openloop robot error system can be obtained by taking the time deriva tive of r, (t) and premultiplying by the robot inertia matrix as M~i, = Y8, Cr, F, (316) where (19), (32), and (33) were utilized, and Y,, = M~xre + al~e, + h + C~rd + aCzC~r + KI~ m aCx,, (317) where Y(x,, x,, xm, im, ef, rlm, t) E RW2xP denotes a known regression matrix, and 8, E RW~denotes an unknown constant parameter vector. See Appendix A for a linearly parameterizable expression for M~ (x,) Xrd(t) that does not depend on acceleration terms. Based on (316) and the subsequent stability analysis, the robot force control input is designed as FA n ,, + e, + k37r, 38 where k3 ERI is a positive constant control gain, and 8,(t)l E RWP is an estimate for 8, generated by the following adaptive update law 8, T rjFYr) (319) In (319), r, E RWPxP is positive definite, constant, diagonal, adaptation gain matrix, and proj() denotes a projection algorithm utilized to guarantee that the i th element of Or (t) can be bounded as where 8,i, 8,i ERI denote known, constant lower and upper bounds for each element of 8,(t)l, respectively. The closedloop error system for rr (t) can be obtained after substituting (318) into (316) as M~i, = Y,0, k~37r Cr, er. (320) In (320), the parameter estimation error Or t)l E RWP is defined as 8, 8, 8,. (321) Remark 3.1: Based on (318), the control torque input can be expressed as 7 = JT (Y, + e, + k37r (32) where J(q) denotes the manipulator Jacobian introduced in (12). 3.3 Stability Analysis Ti;,~ ..;~, u: The controller given by (311), (313), (318), and (319) ensures global asymptotic regulation of the MSR system in the sense that I, (t) 0 Ie,(t) 0I~ as t 00o provided ky, k2, and km1 are selected sufficiently large and the following sufficient gain condition is satisfied: a2 > maX1 , PK (323) where pGm, Pir, PK, and a are defined in (14) and (31), respectively. In the following proof, a Lyapunov function and its derivative are provided. The analysis is then separated into two cases as in Chapter 2. For the noncontact case, the stability analysis indicates the controller and error signals are bounded and converge to an arbitrarily small region. Additional analysis indicates that within this region, contact must occur. When contact occurs, a Lyapunov analysis is provided that illustrates the MSR system asymptotically converges to the desired setpoint . Proof: Let V(r,, e,, em, ef, rlm, 8r, Odk, t) ERI denote the following nonnegative, radially unbounded function (i.e. a Lyapunov function candidate): V n r,"1R, + Of r',le + 1 k,,Fc'edk (324) 2 T1l y12 2 1 1 + k2K, [ln (cosh(em)) + In (cosh (ef))] + 2ee 2m . The time derivative of (324) can be determined as Vj = r,TMi, + ~rr"Mr, + 6T~, C'0 + OfkK;Flledke (325) + kK, tan~em4m+ tanh(ef)ef] + e, i: + mml~mj, After using (112), (33), (34), (312), (313), (315), and (319)(321), the expression in (325) can be rewritten as  k,~I1(f 1'/. 2 ae, er + rim [K, (X1 '7' 1 /,,~ k2Kt3 tanh2(ei) dl Az,,) + K, (Amm Xm) + x]  (326) 0) The expression in (326) will now be examined under two different scenarios. Case 1Noncontact: For this case, the systems are not in contact (A = and (326) can be rewritten as T~< k3TrTTr 0lk2K, tanh2(em) k2Kyt3~ tanh2(ef) 3 ~11/ Rewriting Xgaz(t) and substituting for X(em, ef, rim, t) yields <~j kTrTr 01k2Kl tanh2(em) k2Kyt3a tanh2(ef) 2 1 't 3 [0 e,2" CK Ijm, Br] [knla5t2 l ~ 1 I Om17n1 (327) After completing the square on the bracketed terms, (327) can be expressed as T~< k3TrTTr 0lk2Ky; tanh2(em) k2Kyt3a tanh2(eyf) n y;1:: (328) CK 2 I PK(Z, m rl (2 02 2 4a m 4a2kni 4a2 Provided kni is selected according to the sufficient condition 4a2 min (a1k2( ,' k2 ,3,0 the expression in (328) can be further reduced as As  m ~ k3C~ Fr 2 2 K 02 (329) where Az ERI is defined as X1 = min arl k2 K, k2i K3, na2 Based on (14) in Assumption 1.1, for the noncontact case i'< 1 <~l sm <, m.330 where A ERI is defined as A = min A z, kF3, P2_K and y(t) E R"5 and e, ERI are defined as [ T 2z r" ex = K s~2m (332) r 0 where e, can be made arbitrarily small by making a2 large. Based on (324) and (3 31), either A y2 E Or X 1 2 1 Er If 212 > c,, then Barbalat's Lemma can be used to conclude that Vilt) 0 since V(t) is lower bounded, V/(t) is negative semidefinite, and V/(t) can be shown to be uniformly continuous. As V'(t) 0 eventually A y2 E. Provided the sufficient gain condition in (323) is satisfied (i.e., e, < 1), then (310), (332), and the facts that 8,(t) and Odkct 00Co from the use of a projection algorithm, can be used to conclude that V() E foo; hence, y(t),  :(t), rr(t), e,(t), r1m(t), ef(t), em(t) E foo. Signal chasing arguments can be used to prove the remaining closedloop signals are also bounded during the noncontact case. The previous development can be used to conclude that for the noncontact case y(t)' and hence, r,(t) ~A as t oo. (333) Based on (333), linear analysis methods (see Lemma A.19 of [37]) can be applied to (33) to prove that Ilr (t)l 6 Ie(0)  exp(at) + I (1 exp(at)) (334) as t 00o for the noncontact case. Further analysis is required to prove that the manipulator makes contact with the massspring system. Contact between the manipulator and the massspring system occurs when xmi(t) > xm(t). Based on (334), a sufficient condition for contact can be developed as Xwdl > Xm + r1. (335) After using (311), the sufficient condition in (335) can be expressed as Ydeldk + kF2 tanh(em,) kik2 cosh2(ef) tanh(ef) '> (336) By using (32) and (36) and performing some algebraic manipulation, the inequality in (336) can be expressed as ki2 tanh(em) kik2h~ cosh2(r)> mCd dk + em 0o edk (3 37 where ~dk(t) and Odkc(t) are defined in (314). From Assumption 1.1, em(t) can be upper bounded as em < m (338) where, Em ER denotes a known positive constant. If em(t) < 0, then the sufficient condition in (337) may not be satisfied. The condition that em(t) < 0 will only occur if an impact collision occurs that causes the mass to overshoot the desired position. However, even if an impact occurs and the mass overshoots the desired position, the dynamics will force the mass position error to return to the initial condition. That is, em(t) ama~ xo > Em where ,Em ER denotes a known positive constant. Based on (338) and the fact that em(t) will eventually be lower bounded by m, in a noncontact condition, the inequality in (337) can be simplified as k2~ (tanh(Em) ki cosh2~f) _r md xdkd m 0e ) ~dk. (339) To further simplify the inequality in (339), an upper bound can be determined for ef(t). The inequality in (333) along with (310) and (332) can be used to conclude that as the manipulator approaches the mass, ef(t) will eventually be upper bounded as ef < tanh ( e I < ef, (340) where af yER is a known positive constant. Based on (332) and (340), the control parameter k2 can be selected according to the following sufficient condition to ensure the robot and massspring system make contact k2> > V (341) tanh(em) kcoh2lOh(f where ki is chosen as tanh(Em) cosh2(f Case 2Contact: Provided the sufficient condition in (341) is satisfied, the robot will eventually make contact with the mass. For the case when the dynamic systems collide (A = 1), and the two dynamic systems become coupled then 1 The dynamic systems can separate after impact, however this case can still be analyzed under the Noncontact section of the stability analysis. (326) can be rewritten as V/ < k~3,T~r 01k2KI tanh2(em) 3a029@ k2Kyt3 tanh2 (f [(y e,2 K ljOm Br~] [k,1s(Y2 29 1 O where (39) was substituted for X(em, ef, rim, t). Completing the square on the two bracketed terms yields V' < k3,7,~ 011k2eK tanh2 6m) 03k2~ Ktanh2(eyS) (342) s 4a 4a2kni A final bound can be placed on (342) as 1'<min alk2 KZ,; 3k2 K, ~ l n 2a  2 2 _K 7 3T Because (324) is nonnegative and its derivative is negative semidefinite, Tr(t), 8r(1), 8dk 1t), er(t), em(t), ef(t), and rlm(t) E C,. Due to the fact that em(t), ef(t), and rlm(t) E Coo, the expression in (33) can be used to conclude that em(t) E C, (and hence, em(t) is uniformly continuous). Due to the fact that em(t) E C,, (32) can be used to conclude that xm(t) E C,. Previous facts can be used to prove that X,d(t) E Coo, and since e,(t) E C,, then x,(t) E C,. Due to the fact that ef(t), em(t), rlm(t) E C,, (34) can be used to conclude that ef(t) E Co. The expression in (35) can be used to conclude that ilm(t) E C, (and hence, rlm(t) is uniformly continuous). Given that r,(t), em(t), ef(t), and rlm(t) E Coo, Yr() E C,. Since e,(t) E C,, (321) can be used to prove that 8,(t) E C. The expression in (318) can then be used to prove that F(t) E C, The expression in (320) can be used to conclude that ri, (t) E C, (and hence, r,(t) is uniformly continuous). Due to the fact that em(t), Tr(t), rlm(t) E C2 and uniformly continuous, Barbalat's Lemma can be used to conclude that t)  r, (t)   Om (t)l  0 as t oo. Based on the fact that  r, (t)  0 as t oo, standard linear analysis methods (see Lemma A.15 of [37]) can then be used to prove that  ez,(t)l  0 as t oo. As is typical in literature, the controller developed in (311), (313), (318), and (319) is based on the underlying assumption that an arbitrarily large (but finite) control authority is available. A potential disadvantage of the controller is that the gain conditions developed above and in (323) indicate that kn1, k2, and a2, respectively, should be selected sufficiently large. As demonstrated by the subsequent experimental results, the gains may be selected much lower in practice (i.e., the gains conditions are the result of a conservative Lyapunov analysis). However, the subsequent experimental section also illustrates that even when the gain conditions are violated, large initial conditions and a high stiffness coefficient result in a high gain controller that initially saturates the actuators. The control torque in the experiment was artificially saturated to reduce the magnitude of the impact to protect a capacitance probe from contact by excessive bending of the aluminum rod/spring assembly. Research that can limit the required control torque for systems that undergo an impact collision will be discussed in Chapter 4. 3.4 Experimental Results With the testbed in Section 2.3, the control gains a and k~3, defined as scalars in (33) and (318), were implemented (with nonconsequential implications to the stability result) as diagonal gain matrices to provide more flexibility in the experiment. Specifically, the control gains were selected as ki = 0.18 k2 = 0.9 k3 = diag {185, 170} (343) or = 45 02 8 a3 = 0.01 0 = diag {60, 90} . The control gains in (343) were obtained from a trial and error process and may not satisfy the sufficient gain conditions developed in the theorem proof. The sufficient gain conditions are the result of a conservative stability analysis and were used as a guide in the tuning process. The subsequent results indicate that the developed controller can be applied despite the fact that some gain conditions are not satisfied. The adaptation gains were selected as r = 90 r, = diag {4.01 x 1012, 1.2 x 10 0.2, 3.3 x 1012, (344) 6 x 106, 0.1, 2.4 x 101, 7 x 105, 0.1, 2.35 x 1011) The adaptation gains r, in (344) are used to enable the adaptive estimate to sufficiently change relative to the large values of the uncertain parameters in Or  Smaller adaptation gains could be used to obtain different results. The initial conditions for the robot coordinates and the massspring position were (in [m]) [r,(0) iir2(0) 21m(0) ]= 0008 0.41 0.202  The initial velocity of the robot and massspring were zero, and the desired mass spring position was tin [mml Xma = 232. That is, the tip of the second link of the robot was initially 224 [mm] from the desired setpoint, and 194 [mm] from xo along the X1axis (see Fig. 21). Once the initial impact occurs, the robot is required to depress the spring (item (1) in Fig. 21) to move the mass 30 [mm] along the X1axis. The massspring and robot errors (i.e., e(t)) are shown in Fig. 41 and Fig. 32. The peak steadystate position error of the endpoint of the second link of the robot along the X1axis (i.e., erl(t)) and along the X2aXiS (i.e., ler2(t)) are 0.212 [mm] and 5.77 [pm], respectively. The peak steadystate position error of the mass (ie,.(t)) is 2.72 [pm]. The 0.212 [mm] maximum steady state error in erl(t) 42 soo 500 5 10 15 200 200 5 10 15 50 5 10 15 [sec] Figure 31: Massspring and robot errors e(t). Plot (a) indicates the position error of the robot tip along the X1axis (i.e., en (t)), (b) indicates the position error of the robot tip along the X2aXiS (i.e., er2(t)), and (c) indicates the position error of the massspring (i.e., em(t)) is due to the Ydeldk(t) term of XAdl(t) in (311) where Yd(Xm) is approximately 0.03 [m], and Odkc(t) has a maximum steady state value of 0.007 [ '], yielding 0.21 [mm] error. All of the other terms in erl(t) are negligible at steady state. The input control torques in (322) are shown in Fig. 33 and Fig. 34. To constrain the impact force to a level that ensured the aluminum plate did not flex to the point of contact with the capacitance probe, the computed torques are artificially saturated. Fig. 33 depicts the computed torques, and Fig. 34 depicts the actual torques (solid line) along with the computed torques (dashed line). The resulting desired trajectory along the X1axis (i.e., Xay (t)) is depicted in Fig. 35, and the desired trajectory along the X2aXiS was chosen aS Xrd2 = 370 [mm]. Fig. 36 depicts the value of Odkct W and Fig. 37 Fig. 39 depict the values of 8,(t)l E RIo1. The order of the curves in the plots is based on the relative scale of the parameter estimates rather than numerical order in 8,(t). A video of the experiment is provided at http: //www.mae.ufl.edu/~dixon/proj ects /robman/adaptiveimpact .htm. 0 5 10 1 (b) 200 200 400 200 \ E100 o 0 50 100 is 40 E 20 EO Figure 32: 1\assspring and robot errors e(t) during the initial two seconds. S500H J (q)F(t) for the (a) base motor and (b) Figure 33: Computed control torques second link motor. (a) 50 100~ i, 0 005 01 015 02 025 03 035 04 (b) 005 01 015 02 [sec] 025 03 035 04 Figure 34: Applied control torques J (q)F(t) (solid line) versus computed control torques (dashed line) for the (a) base motor and (b) second link motor. 2335 232 2325 231 5 E 231 Figure 35: Computed desired robot trajectory, xay (t). 0 06 0 05 0 04 0 03 0 02 0 01 Figure 36: Unitless parameter estimate Odk (t) introduced in (313). 1 Figure 37: Estimate for the unknown constant parameter vector Or (1) (a) erio(t) K,, (b) Or4 1) Km, (C) ,1(t) m, and (d) ,7(t) lm, where mi, m2 E RW denote the mass of the first and second link of the robot, m, E RW denotes the mass of the motor connected to the second link of the robot, and m ERI denotes the mass of the massspring system. Figure 38: Estimate for the unknown constant parameter vector Or (f ) (a) Brs (t) ", (b) Or2(t) ~1, and (c) B,s(t)  parameter vector Or (1) (a) Gre (t) Figure 39: Estimate for the unknown constant m,, (b) Or3 1) = 1, and (c) Or9 1) = 2 3.5 Concluding Remarks An adaptive nonlinear Lyapunovbased controller is proven to regulate the states of a planar robot colliding with an unactuated massspring system. The continuous controller yields global asymptotic regulation of the springmass and robot links. New control design, error system development and stability analysis techniques were required to compensate for the fact that the dynamics changed from an uncoupled state to a coupled state. Experimental results are provided to illustrate the successful performance of the controller. Sufficient conditions developed in the stability analysis indicate that the control gains should be selected large enough to minimize the closedloop steady state error, but high gains could result in large torques for large initial errors. The high gain problem is exacerbated in the developed result because of the presence of the estimated impact stiffness coefficient. The experimental results were obtained by artificially saturating the torque to prevent damage to the capacitance probe. These issues point to a need to develop controllers that account for limited actuation which we will discuss in Chapter 4. CHAPTER 4 AN IMPACT FORCE LIMITING ADAPTIVE CONTROLLER FOR A ROBOTIC SYS TEM UNDERGOING A NONCONTACT TO CONTACT T TRANSITION Similar to Chapter 2 and 3, the objective of this chapter is also to control a robot from a noncontact initial condition to a desired (contact) position so that the massspring system is regulated to a desired compressed state. When the controllers in Chapter 2 and Chapter 3 were implemented in the presence of large initial conditions, violent impacts between the robot and the massspring system resulted. In fact, the controller was artificially saturated (the saturation effects were not considered in the stability analysis) to reduce the impact forces so that the mass deflection would not destroy a capacitance probe. These results provide the motivation for the control development in this chapter. Specifically, the feedback elements for the continuous controller in Chapter 3 are contained inside of hyperbolic tangent functions as a means to limit the impact forces resulting from large initial conditions as the robot transitions from noncontact to contact states. The main challangfe of this work is that the use of saturated feedback does not allow some coupling terms to be canceled in the stability analysis, resulting in the need to develop state dependent upper bounds that reduce the stability to a semiglobal result. New control development, closedloop error systems, and Lyapunovbased stability analysis arguments are used to conclude the result. Experimental results are provided that successfully demonstrate the control objective. This chapter is organized as follows. The associated properties are provided in Section 4.1. Section 4.2 describes the error system and control development followed by the stability analysis in Section 4.3. Section 4.4 describes the experi mental results that indicate the successful performance of the proposed controller followed by conclusion in Section 4.5. 4.1 Properties Remark 4.1: To aid the subsequent control design and analysis, we define the vector Tanh() E RW" as follows Tanh(G) n [tanh(bl), ..., tanh(6,)]" (41) where 6 n [61, ..., 6,]T E R". Property 4.1: The following inequalities are valid for all ( = [(1, ..., s]T E RW" [36] ( > Tanh(()  (  +1 >(42) Stanh( ) (2 > In(osh((4)) > In(cosh(()) (43) i=1 ( Tanh(() > Tank (()Tanh(() = Tanh(()2 (44) > tanh2(11) 4.2 Error System and Control Development The subsequent control design is based on integrator backstepping methods. A desired trajectory is designed for the robot (i.e., a virtual control input) to ensure the robot converges to and impacts with the mass, and to ensure that the robot regulates the mass to the desired position. Since the robot trajectory can not be controlled directly, a force controller is developed to ensure that the robot tracks the desired trajectory despite the transition from free motion to an impact collision and despite uncertainties throughout the MSR system. 4.2.1 Control Ob jective As in Chapter 2 and 3, the control objective is to regulate the states of the masssping system via a twolink planar robot that transitions from noncontact to contact with the massspring through an impact collision. An additional objective in this chapter is to limit the impact force to prevent damage to the robot or the environment (i.e., the massspring system). A regulation error, denoted by e(t) E RW3, is defined to quantify this objective as e 4 em e" ,' where e,(t) n [erz(t),er2(~l I2 and em(t) ERI denote the regulation error for the endpoint of the second link of the robot and massspring system (see Fig. 11), respectively, and are defined as e, A go X, em = sma xm. (45) In (45), sma ERI denotes the constant known desired position of the mass, and X~d(t) n [X,ax(t), Xrd2] I2 denotes the desired position of the endpoint of the second link of the robot. The subsequent development is based on the assumption that q(t), q(t), xm(t), and xm(t) are measurable, and that x,(t) and x,(t) can be obtained from q(t) and q(t). To facilitate the subsequent control design and stability analysis, filtered tracking errors, denoted by rm (t) ERI and r, (t) E RW2, are defined as [38] rlm = em ,+ az tanh(em,) + a2 tanh(ef)(46 Tr e, + ae,, where a, al, a2 ERI are positive, constant gains, and ef(t) ERI is an auxiliary filter variable designed as [38] ey a a3 tanh(ep) + a2 tanh(em) kl cosh26f )llm, (4~7 where ki ERI is a positive constant control gain, and a3 ERI is a positive constant filter gain. 4.2.2 ClosedLoop Error System In this section, the closedloop error system for rlm is exactly the same as in (35)(315) in Chapter 3. The openloop robot error system can be obtained by taking the time derivative of r, (t) and premultiplying by the robot inertia matrix M~i, = Y,, Cr, F, (48) where (19), (45), and (46) were utilized, and Y,, n M~xre + ale, + h + C~rd + aCzrd + I aCX,, where Y(x,, x,, xm, im, ef, rlm, t) E RW2xP denotes a known regression matrix, and 8, ER Idenotes an unknown constant parameter vector. By making substitutions from the dynamic model and the previous error systems, xra(t) can be expressed without a dependance on acceleration terms. Based on (48) and the subsequent stability analysis, the robot force control input is designed as FA n r, + Tanh(e,) + k3T Thr,), (49) where k3 ERI is a positive constant control gain, and 8,(t)l E RWP is an estimate for 8, generated by the following adaptive update law 8r A proj(r, ,r,). (410) In (410), r, E RWPxP is positive definite, constant, diagonal, adaptation gain matrix, and proj() denotes a projection algorithm utilized to guarantee that the i th element of Or (t) can be bounded as Ori < ari < Or, (4111) where 8,i, 8,i ERI denote known, constant lower and upper bounds for each element of 8,(t)l, respectively. The closedloop error system for rr (t) can be obtained after substituting (49) into (48) as M~i, = Y,, Cr, Tanh(e,) k3T Thr,). (412) In (412), the parameter estimation error Or t)l E RWP is defined as 8, 8, 8,. (413) 4.3 Stability Analysis Ti;,~ ..;~, u: The controller given by (311), (313), (49), and (410) ensures semiglobal asymptotic regulation of the MSR system in the sense that S.(t)I 0 e,(t)I 0 as t 00o provided the following sufficient gain conditions are satisfied: k,2 > 1(414) 402 mint a1k2 CG3k29K ,02 02 ma (415) k3a > 12V 2~ (416) where V, ERI is defined as V, max Asl (0)2 +(o, 5Az(tanh'( ))2V A ERI is a known positive constant, and al, a, al, a2, a3, k,1, k2, k~3, X1, P$cm) PCc, K. and (Ki are defined in (14), (111), (31), (311), (312), (46), (47), and (49). As in the previous chapters, the subsequent analysis is separated into two cases: contact and noncontact. For the noncontact case, the stability analysis indicates the controller and error signals are bounded and converge to an arbitrarily small region where contact must occur. When contact occurs, a Lyapunov analysis is provided that illustrates the MSR system asymptotically converges to the desired setpoint . Proof: Let V(t) ERI denote the following nonnegative, radially unbounded function (i.e. a Lyapunov function candidate): 1 1" 1" 1 V n r,"Air, + ,"0,1, + ~ "k,d 6B m (417) 2, 2iref~~~ 2 2l, + k2KI [ln (cosh(em)) + In (cosh (ef))] + In(cosh(ery)) + In(cosh(er2) where (111) and (43) can be utilized to bound V(t) as 4811,1 2 < V < Az  rl + Co (418) where Az,P Co eR and E IR are defined as a2 m, Az Amax{ 3, ", kF2 K 2 2 CO KC5d a p O (4~19) z[TrT ep lm em, ey]T where Amx,( { E denotes the maximum eigenvalue of a matrix, and Peodkl Pa r are the knownI uIpTper bolunds of dk (t) anrd H,(t) resp~ectively. After usiing (1 12), (312), (313), (315), (44), (46), (47), (410), and (412), the time derivative of (417) can be determined as V~ < k3 tanh2 lr, I) 0lk2K, tanh2 6m) 03sk2K, tanh2(ef) + 2e r, 2ae, e, 3 1 I,I k,1(fI 11/y, atanh2( l erI + rlm [K, (Xway Az,,) + K(n, (A Xm) + x]  (420) The expression in (420) will now be examined under two different scenarios. Case 1Noncontact: For this case, the systems are not in contact (A = 0) and (420) can be rewritten as Vj < kJ3 tanh2( Tr, ) 01k2K~1tanh2 (em)  + 2e r, 2ae e, 3a292, k,lp(f t24, + rlm [Kp~rdl Kymm + X]  (421) Based on (31), (39), and (45), the expression in (421) can be rewritten as V' < k3 tanh2( Tr ) 01k2CKze tanh2 6m,) 03k2CK tanh2(ey) (422a) 2 oa /,I a tanh2( li, r [0 e,2" Kl 77m1 Br.] [k,~1(f;a21 t29 I 1 111m1 After completing the square in the bracketed terms, (422) can be expressed as 4( 23) l' < 8  I  a tanh2(le Ir 02 _~ K ,, 4a 2 2C2~ 40X, X12 a tanh2( I ,IIITr 1 lh i where ,8 ERI is defined as :3 = min asrk2zli K~Eari 3k2 ) K 2 > 03k2K~ltanh2(ef) a tanh2( IIerI provided kni is selected according to (414). Provided a2 is selected according to the sufficient condition in (415), then (14), (42), (418), and the fact that <~i I Xr < Xm < pOm (424) for the noncontact case, can be used to rewrite (423) as V < ,8   a tanh2 Br l)~ + (4~25) 3_ 2~3 tanh2( Fr,  In (425), ,Ex ER is a known positive arbitrarily small constant that is defined as Provided the following sufficient condition is satisfied k~t > 1 2 (426) the expression in (425) can be expressed as V' < A 12 + E 427) where A is a known constant and ye R is defined as = z' tanh(e, tanh(r, .j (428) In (427), e, can be made arbitrarily small by making a2 large. Based on (4 17) and (4227), if A y(t)2" > c,, then Bar2bala~t's T;Lemma can be uLsed to conclude~C that Vj(t) 0 since V(t) is lower bounded, Vj(t) is negative semidefinite, and Vj(t) can be shown to be uniformly continuous. As 11'(t) 0, eventually A y(t)2 < x Wi~hile X y(t)2 > Ez, (417), (418), and (427) canl be used to conclude that V(t) E Em, and the sufficient condition given in (426) can be expressed as k~t > 2 (AI (0)2 + 1 (429) Provided a2 >~ C thlen eventually: A y(t)2 < E, < A. Based on (310) anld (428), the fact, that Ayl(t)2 < e, < A can be: used to conclude: that em,(t), eyS(t), e,(t), r,(t), flm(t) E C,. Since e,(t) and Odkct E a frOm the use of a projection algorithm, the previous facts can be used to conclude that V(t) E Em. Signal chasing arguments can be used to prove the remaining closedloop signals are also bounded during the noncontact case provided (426) is satisfied. If the initial conditions for V(0) are large enough that A y(t)2" > ,, then the condition in (429) is sufficient. However, if the initial conditions for V(0) are inside the region defined by e,, then V(t) can growv larger unttil A yl(t)2 ex. Therefore, further development is required to determine how large V(t) can grow so that the sufficient condition in (426) can be satisfied. When V(0) is inside the region defined by e,, then y(t) < .J (430) The expression in (430) can be used along with (310), (419), and (428) to conclude that 7(t) The inequality in (431) can be used along with (418) and (419) to rewrite the sufficient condition in (426) as ~a > 2 (5Az(tanh ())2 +~)(2 Hence, the final sufficient condition for (426) is given by (416). That is, provided k,1, a, a2, and kF3 are selected larger than known constants (that depends on the initial conditions of the states) according to (414)(416) then the all the states converge to an arbitrarily small neighborhood. Development has been provided (see Section 3.3) to prove that contact with the massspring system is ensured within this neighborhood. Case 2Contact: For the case when the dynamic systems collide (A = 1) and the two dynamic systems become coupled then (420) can be rewritten as Vi < k~3 tanh2 Flrl> 01k2 K tanh2 6m) 03k2CK tanh2(f where (31) was substituted for KI and (39) was substituted for X(em, ef, rim, t) Completing the square on the three bracketed terms yields 1' < I  a tanh2( Ilr P 2_K 93 Because (417) is nonnegative, as long as (414)(416) are satisfied, (433) is negative semidefinite, and r,(t), 8,ttel(t), Od 1, Bt em t ef(t), and rlm(t) E Co Due to the fact that em(t), ef(t), and rlm(t) E C,, the expression in (46) can be used to conclude that em(t) E C, (and hence, em(t) is uniformly continuous (UC)). Due to the fact that em(t) E C,, zim(t) E C,. Based on (14), xm(t) E C, Previous facts can be used to prove that x,d(t) E C,, and since e,(t) E C,, then x,(t) E C,. Due to the fact that ef(t), em(t), rlm(t) E C,, (47) can be used to conclude that e7 (t) E Co. The expression in (35) can then be used to conclude 1 The dynamic systems can separate after impact, however this case can still be analyzed under the NonImpact section of the stability analysis. that ilm(t) E C, (and hence, rlm(t) is UC). Based on (46) and the fact that r,(t) and e,(t) E C,, e,(t) E C,. Also, based on (314) and the fact that xm(t), em(t), ef (t), rlm(t), and xm(t) E C,, the expression in (311) can be used to prove that X,d(t) E Co. Based on the fact that e,(t) and x,d(t) E C,, the expression in (45) can be used to prove that x,(t) E C,. Given that x,(t), x,(t), xm(t), xm(t), ef(t), and rlm(t) E Coo, Yr() E C,. The expression in (49), (411) can then be used to prove that F(t) E C,. The expression in (412) can be used to conclude that i,(t) E C, (and hence, r,(t) is UC). Since em(t) and r,(t) are UC which implies tanh(em(t)) and tanh(r,(t))> are also UC, and the fact that tanh(em(t)), tanh(r, (t)), and rlm(t) are square integrable, Barbalat's Lemma can be used to conclude that tanh(em(t)), tanh(r,(t)), 9m,(t) 0 as t oo, which also implies S. (t), r,(t) 0 as t oo. Based on the fact that r,(t) 0 as t oo, standard linear analysis methods (see Lemma A.15 of [37]) can then be used to prove that  e, (t)l  0 as t oo. 4.4 Experimental Results With the testbed in Section 2.3, the control gains a and k~3, defined as scalars in (46) and (49), were implemented (with nonconsequential implications to the stability result) as diagonal gain matrices to provide more flexibility in the experiment. Specifically, the control gains were selected as ki = 0.28 k2 = 0.855 k~3 = ding {110, 10} al = 60 a2 = 2.8 a3 = 0.06 a = diag {40, 8} . The adaptation gains were selected as r = 0.06 r, = diag {1.8 x 1011, 1.5 x 105, 0.2, 7 x 1010, 7 x 104, 0.1, 1 x1010, 1 x104, 0.01, 2.8 x 109) The initial conditions for the robot coordinates and the massspring position were (in [m]) [r,(0) i;r2(0) 21m(0) ]= 0070 0.285 0206  The initial velocity of the robot and massspring were zero, and the desired mass spring position was tin [mml Xma = 236. That is, the tip of the second link of the robot was initially 166 mm from the desired setpoint and 136 mm from the initial impact point along the X1axis (see Fig. 21). Therefore, once the initial impact occurs, the robot is required to depress the spring (item (1) in Fig. 21) to move the mass 30 mm along the X1axis . The massspring and robot errors (i.e., e(t)) are shown in Fig. 41. The peak steadystate position error of the robot tip along the X1axis (i.e., er, (t)i) and along the X2aXiS (i.e., lr2 (t) ) are 0.855 mm and 0.179 mm, respectively. The peak steadystate position error of the massspring (i.e., I~. (t) ) is 0.184 mm. The input control torques (i.e., J (q)F(t)) are shown in Fig. 42 and Fig. 43. The resulting desired trajectory along the X1axis (i.e., Xar (t)) is depicted in Fig. 44, and the desired trajectory along the X2axis was chosen aS xrd2 = 357.5 mm. Fig. 45 depicts the value of Odk (t) ERI and Fig. 46 Fig. 48 depict the values of 8,(t)l E RIo1. The order of the curves in the plots comes from their scales rather than their numerical order in 8,(t). (a) 100 100 200 0 24 6 8 10 0 24 6 8 10 100 0i 100I 0 24 6 8 10 Figure 41: Massspring and robot errors e(t). Plot (a) indicates the position error of the robot tip along the X1axis (i.e., en (t)), (b) indicates the position error of the robot tip along the X2aXiS (i.e., er2(t)), and (c) indicates the position error of the massspring (i.e., em(t)) 10 Figure 42: Applied control torques ond link motor. [sec] J (q)F(t) for the (a) base motor and (b) sec o 200 0 57 01 02 03 04 05 06 07 (b) 01 02 03 04 [sec] 05 06 07 08 Figure 43: Applied control torques J"(q)F(t) for ond link motor during the first 0.8 second. the (a) base motor and (b) sec 8 10 420 400 380 360 340 320 300 280 260 240 220 0 Figure 44: Computed desired robot trajectory, xay (t). x 10 8 6 05 4 3 [sec] Figure 45: Parameter estimate Odk (t) introduced in (313). [sec] Figure 46: Estimate for the unknown constant parameter vector Or (1) (a) orio(t) Kr,,(b) Or4 1) K", C) ,1(t) "', and (d) 6,7(t) I", where mi, m2 E RW denote the mass of the first and second link of the robot, m, E RW denotes the mass of the motor connected to the second link of the robot, and m ERI denotes the mass of the massspring system. S800 600 400 Figure 47: Estimate for the unknown constant parameter vector Or (1) (a) As () "", (b) Or2(t) m, and (c) 8,s(t)= s Figure 48: Estimate for the unknown constant parameter vector Or (1) (a) Gre t) m,, (b) Or3(t = m1, and (c) Or9 1) = 2 4.5 Concluding Remarks In this chapter, we consider a two link planar robotic arm that transitions from free motion to contact with an unactuated massspring system. An adaptive nonlinear Lyapunovbased controller with bounded torque input amplitudes is proven to regulate the states of the system. The feedback elements for the controller in this chapter are contained inside of hyperbolic tangent functions as a means to limit the impact forces resulting from large initial conditions as the robot transitions from noncontact to contact. The continuous controller yields semiglobal asymptotic regulation of the springmass and robot links. Experimental results are provided to illustrate the successful performance of the controller. CHAPTER 5 CONCLUSION AND RECOli lli lENDATIONS Motivated by the fact that impacts between the robot and the static environ ment cannot represent all the impact system applications, and discontinuous con trollers require infinite control frequency (i.e., exhibit chattering) or yield degraded stability results (i.e., uniformly ultimately bounded), continuous Lyapunovbased controllers for a fully actuated dynamic systems undergoing an impact collision with another unactuated dynamic system are developed. Lyapunovbased methods are used to prove the asymptotic regulations of the mass and robot links. Unlike some other results in literature, the continuous force controller does not depend on measuring the impact force or the measurement of other acceleration terms: only the position and velocity terms of the springmass system and the joint angles and the angular velocities terms of the planar robotic arm are needed for the proposed controller. Chapter 2 provides a first step at controlling the proposed impact system. The control development in Chapter 2 is based on the assumption of exact model knowledge of the system dynamics. The controller is proven to regulate the states of the system and yields global asymptotic result. In Chapter 3, the dynamic model for the system is assumed to have uncertain parameters. The control objective is defined as the desire to both regulate the system to a desired compressed state and compensate for the constant, unknown system parameters. Two linear parameteri zations are designed to adapt for the unknown robot and massspring parameters. The controller is proven to yield global asymptotic regulation result. An extension of the developed regulation controller of Chapter 3 is presented in Chapter 4 where the feedback elements for the controller in this chpater are contained inside of hy perbolic tanget functions as a means to limit the impact forces resulting from large initial conditions as the robot transitions from noncontact to contact states. The controller yields semiglobal asymptotic regulation of the system. Experimental results are provided to illustrate the successful performance of the controller in each chapter. Future efforts can focus on utilizing nonmodel based control methods such as neural networks, fuzzy logic, or robust control methods to compensate the more complex uncertainties of the system. The developed model can also be modified to control the impact between an actuated dynamic system and an unactuated dynamic system with nonlinear flexibilities. Future work could also focus on the specific application of the developed methods for applications such as docking space vehicles, walking robots, etc. Efforts could also focus on obtaining a global result of the feedback saturated case in Chapter 4. APPENDIX A THE EXPRESSION OF x,d, IN SECTION 2.1 Due to the fact that taking the time derivative of any expression with A except for KA(x,l xm) is undefined, some care has to be taken in taking the time derivative of Xar (t). This is done by first taking the time derivative of (26) and utilizing (23) and (27), to obtain the following expression: Xsdl = K[(1 A) (k(xm xo) + amem)] + KA [Kers (ki + k2) rm] (A1) K KmK 1 1 + (ki + k2) (1 A) amrem + (i+k)A[ey(i+k)E mK mK In order to derive X,4, (t), the second time derivative of (26) is taken rather than the first time derivative of (A1) to obtain Xwdl = ,(am'e m + kim + (ki + k2)rm) + xm. (A2) The expression for em(t) can be obtained by rewriting (23) as em = rm II. (A3) Differentiating (A3) yields the following expressions for em(t) and 'e mi): em = im ... (A4) em = m .. .(A5) By using (27), the expression for im(t) can be written as 1 1 rm [(1 A) (k(xm xo) + amem)] + [Knery A(ki + k2)rm] (A6) m m The time derivative of (24) is given by m KA(xry xm) + amem). (A7) The dynamics for the massspring system xm(t) can be written as xm (KA(x,l xm) k(xm ro)). m By using (A2) and (A5), the following expression can be obtained: am k(i+k Xwdl = (Tm aem) + ( + 1)xm + Em. K K Substituting (A4) and (A7)(A9) yields (A8) (A9) Q20 in1 xm) + amim ) im, + SK KA (xin xm) + amem]  Q3m K Xwdl = (km,  KA (j k1 + ( + 1) (KA ( K m (ki + k2) + [kim  Km After using (A4) and (A6), a simplified expression for Argy (t) can be obtained as follows: Xay = (kim K KA (xin xm)) k1 K m (ki + k2) + [kim K Km (ki + k2) a + (1 A) Km (ki + k2) 0 + Aery  m 1  Xm) k(Xm Xo)) A (X,1 Xm)] (k(xm Xo)) (A10) (A11) APPENDIX B THE EXPRESSION OF x,d, IN SECTION 3.2 Since Xrd2 is a constant, the subsequent development is only focused on determining xrdl(t). After using (32), (34), (311), and (313), the first time derivative of Xar (t) can be determined as nral = Y (prloj (rY, t; )) + Hdk+1 k2 coshl(2 6m im (B1) kik~2 (sinh2(ep) + cosh2 By)) (a3 tanh(ei) + cy2 tanh(e~m) ki cosh2 6f )rlm Based on the fact that the projection algorithm for Odkc(t) is designed to be suffi ciently smooth [39], the expressions in (313) and (B1) can be used to determine the second time derivative of Xar (t) as 8(poj(Y(t.) 8(proj(rY, ,.)) .r,r: )X B2 2k2 cosh3(r m) sinhjem)i r + Hdk+ 1 0Sk2( cos2 m 4kik~2 (sinh(ef) cosh(eyf)) e)~ lkzk (sinh2(ef) + cosh2 By)) (a3 cosh2(ef) 2kil cosh(ej) sinh(ef)7lm) eyS + lkzk (sinh2(ef) + cosh2 By)) (n2 cosh2(em)em, + ki cosh2 6f il) m After substituting (34) and (35) into (B2) for ef(t) and rjm(t), respectively, and substituting (15) and (18) into (B2) for xm(t), the expression for M~ (x,) X44(t) in the linear parameterization in (317) can be determined without requiring acceleration measurements. REFERENCES [1] A. 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Nagarkatti, Nonlinear Control of Engineering Systems: A L~lrainli nor'Based Approach, Birkhituser Boston, 2003. [38] W. E. Dixon, E. Zergeroglu, D. M. Dawson, and M. W. Hannan, "Global Adaptive Partial State Feedback Tracking Control of RigidLink FlexibleJoint Robots," Robotica, Vol. 18. No 3. pp. 325336, 2000. [39] Z. Cai, M.S. de Queiroz, and D.M. Dawson, "A Sufficiently Smooth Projection Operator," IEEE Transactions on Automatic Control, Vol. 51, No. 1, pp. 135 139, 2006. [40] M. Loffler, N. Costescu, and D. Dawson, "Qmotor 3.0 and the Qmotor Robotic Toolkit An Advanced PCBased RealTime Control Platform," IEEE Control Systems M~agazine, Vol. 22, No. 3, pp. 1226, 2002. BIOGRAPHICAL SKETCH ChienHao Liang was born in Taipei, Taiwan. He completed his Bachelor in Ocean Engineering in the year 2001 from National Taiwan University, Taipei. After finishing his military service, he joined the nonlinear controls and robotics research group in the University of Florida for Master of Science in mechanical and aerospace engineering in the year 2005. 