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Dynamics and control of an orbiting space platform based mobile flexible manipulator Chan, Julius Koi Wah 1990

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D Y N A M I C S  A N D C O N T R O L  S P A C E M O B I L E  P L A T F O R M  F L E X I B L E  O F A N O R B I T I N G B A S E D  M A N I P U L A T O R  Julius Koi Wah Chan B.A.Sc.,  University of British Columbia, 1987  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in  The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF  B R I T I S H  COLUMBIA  April 1990 ©Julius Koi Wah Chan, 1990  In presenting this thesis in partial fulfilment of the  requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and  study. I further agree that permission for extensive  copying of this thesis for scholarly purposes may department or  by  his or  her  representatives.  be  granted by the head of  It is understood  that  copying  publication of this thesis for financial gain shall not be allowed without my permission.  Department of  Mechanical Engineering  The University of British Columbia Vancouver, Canada Date A p r i l 30 1990  DE-6  (2/88)  my or  written  ABSTRACT This paper presents a Lagrangian formulation for studying the dynamics and control of the proposed Space Station based Mobile Servicing System (MSS) for a particular case of in plane libration and maneuvers. The simplified case is purposely considered to help focus on the effects of structural and joint flexibility parameters of the MSS on the complex interactions between the station and manipulator dynamics during slewing and translational maneuvers. The response results suggest that under critical combinations of parameters, the system can become unstable. During maneuvers, the deflection of the MSS can become excessive, leading to positioning error of the payload. At the same time the libration error can also be significant. A linear quadratic regulator is designed to control the deflection of the manipulator and maintain the station at its operating configuration.  ii  T A B L E OF C O N T E N T S Abstract List of Tables List of Figures Acknowledgement List of Symbols  ii v vi x xi  1.  INTRODUCTION 1.1 Preliminary Remarks 1.2 A Brief Review of the Literature 1.3 Scope of the Investigation  2.  FORMULATION OF THE PROBLEM 2.1 System Model 2.2 Maneuvers and Generalized Coordinates 2.3 Modal Functions 2.4 Kinematics 2.5 Kinetic Energy 2.6 Potential Energy 2.7 Equations of Motion  9 9 11 14 20 26 29 30  3.  DYNAMICAL RESPONSE 3.1 Numerical Approach 3.2 Closed-Form Solution 3.2.1 A Simplified Model for Analytic Solution 3.2.2 Variation of Parameters Method 3.2.3 Accuracy of Solution 3.3 Numerical Simulation Results 3.3.1 Numerical Data 3.3.2 Rigid One-Link System 3.3.3 One-Link Flexible System 3.3.4 Two-Link Flexible System 3.3.5 Effect of Maneuver Profile 3.3.6 Effect of Modal Functions  32 32 34 34 37 43 46 48 49 60 69 79 81  4.  LINEAR ANALYSIS . 4.1 Linearization 4.2 Eigensolutions 4.3 Linear Response  86 86 87 90  5.  CONTROLLED RESPONSE 5.1 Control Inputs 5.2 State Space Representation 5.3 A Review of the L Q R 5.4 Controllability and Observability 5.5 Controlled System Response 5.5.1 Minimization of Payload Error  98 98 100 102 104 104 107  6.  CLOSING COMMENTS ' . . 6.1 Summary of Conclusions  117 117 iii  1 1 3 7  6.1  Recommendation for Future Work  118  Bibliography  120  Appendix I - Transformation matrices  127  Appendix II - Librational Energy  128  Appendix III - Kinetic Energy w.r.t. System Frame  129  Appendix IV - Angular Momentum Vector  131  Appendix V - Mode Shape Integrals  133  Appendix VI - Equations of Motion  136  iv  LIST O F T A B L E S 4-1  Effect of the payload mass on the system eigenvalues  88  4-2  Effect of payload mass on the system eigenvalues in the presence of joint flexibility.  89  Effect of an increase in the link and joint stiffness on the system eigenvalues  89  4-3  v  LIST O F FIGURES 1- 1 2- 1 2-2  Revised baseline configuration of the Space Station as of August 1987 A schematic diagram of the Space Station based Mobile Servicing System (MSS)  2 10  A schematic diagram showing deflections of the link and joint relative to the specified slew frame. .  13  2-3  Thefirstfour modes of the free cantilever beam  16  2-4  fa) Effect of tip load on thefirstmode of a cantilever beam (b) A comparison of thefirstmode of the free and tip loaded cantilever beams with tip defection normalized to 1 Effect of the tip mass on the higher cantilever modes: (a) second mode; (b) third mode  18 18  2- 6  Reference frames and coordinates for the mathematical model  21  3- 1  Flowchart for the MSS dynamics simulation program  33  3-2  A comparison of the responses obtained using 'exact' and simplified equations for a relatively large disturbance (7 = 10°, 6 = 0.1): fa) time history of the joint angle 7; (b) time history of the generalized coordinate 6 associated with theflexiblelink  38  2-5  3-3  3-4  3-5  3-6  3-7  19 19  38  A comparison between the closed form and numerical undamped solution for an initial link tip deflection equal to 10% of its length (6(0) = 0.1).  45  Correlation between damped response results as predicted by the closed form solution and numerical integration of the governing equations. The damping coefficient Cj corresponds to 1% of critical damping  47  Normalized time histories of the sine-ramp and cubic maneuvering profiles in normalized time showing displacement, velocity, and acceleration  50  Response of the rigid one-arm manipulator as affected by magnitude of the pitch disturbance. Note the period of oscillation lengthens with an increase in the initial condition due to the nonlinear characteristics of the system  52  Librational response of a rigid one-link manipulator to a translational maneuver of (h : 0 —• 20m). Note the equilibrium point after the translation is at ib = —0.11°  53  0  vi  3-8  Librational response of arigidone-link manipulator to a translational maneuver of (h : 0 —• —20m). Now the equilibrium shifts in the opposite direction to rj} = +0.11° b  3-9  3-10  Effect of the translational maneuvering speed on the librational response. The rigid one-link MSS travels a distance of 10, 20, 30, 40, and 50 m in 0.01 orbit (1 minute)  56  Librational response of the rigid one-link manipulator during a slewing maneuver through 180° (h = 0; &j : - 9 0 ° 90°) completed at several speeds  57  b  3-11  54  x  Effect of periodic maneuver on librational stability showing beat type and resonance responses: —#  —*  translational maneuver between /ij, = 15 m and h = — 1 5 m; . . . 58 slewing maneuver between f}j = 0° and f3j = 180°. The pitch period is 0.58 orbit compared to the maneuver periods of T = 0.01, 0.5, 0.6 orbit 58 Effect of the payload mass on the librational response of a one-link rigid manipulator executing a simultaneous translational and slewing maneuver (h : 0 30 m; /?J : 90° -*• 0°) 59 0  x  3-12  b  3-13  1  x  Effect of the armflexibilityon pitch response as the single link manipulator executes the combined translational and slewing maneuvers (h : 0 -»• 3Qm; Bj : 90° 0°). The payload mass of 32,000kg is 10 times the manipulator mass (m = 10)  61  Effect of the armflexibilityon the pointing accuracy of the one-link manipulator showing generalized coordinate corresponding to link deformation 6\, in dimensionless units and the tip deflection e in meters  63  Free oscillation of theflexibleone-arm system with an elastic joint. The initial vibrational disturbance at the joint of ("u^ = 10°) significantly excites the first mode of the link deflection 6i  64  Free oscillation of theflexibleone-arm system with the joint degree of freedom. The initial condition is a pitch displacement of 10°. . . .  66  Effect of jointflexibilityon the system response of one-link manipulator with an arm stiffness corresponding to 2 rad/s. Maneuver and payload are the same as before: librational response; payload deflection error  67 67  Comparison of payload deflection £ of the one-link manipulator with a 'stiff' joint (wj = 10rad/s) and arigidjoint (u>j = oo). Arm stiffness is 2 rad/s  68  b  t  p  3-14  3-15  3-16 3-17  3-18  vii  3-19  A comparison between the system response obtained using one and five modes to model the link deflection. The manipulator executes the same maneuver as before with a 32,000 kg payload (m = 10). p  3-20 3-21 3-22  3-23 3-24  3-25 3-26 3-27  Relative significance of the higher modes on the system response for the same maneuver and payload mass System response of the two-link flexible manipulator for a simultaneous translation of the mobile base and slew at joint 1.  4-2 4-3 4-4  4- 5  . . .  73  Response of the two-link flexible manipulator executing a 180° slew at the second joint  76  Effect of considering the higher modes of the first link on the response of the two-armflexiblemanipulator. The general maneuver involves a translation of the mobile base and slews at the two joints. . .  78  Comparison of the two-link manipulator system responses using cubic and sinusoidal profiles for the maneuver coordinates  80  Significance of the higher modes of the first arm on the system response. The cubic maneuver profile is used  82  Comparison of the damped responses using 1 and 3 free cantilever modes to discretize the deflections of the two links. The joint damping coefficients are: (Cj = 0.1; C } = 0.1)  84  Comparison of the damped responses using 1 and 3 modes of the loaded cantilever modes to discretize the deflections of the two links. The joint damping coefficients are: (Cj = 0.1; C } = 0.1)  85  Nonlinear free oscillation response for an initial condition of 6n = 0.05 corresponding to a 10% link deflection at the tip  91  Linear free oscillation response for an initial condition of 6n = 0.05 corresponding to a 10% link deflection at the tip  92  Comparison of linear and nonlinear undamped response for the MSS executing a 180° slew at joint 2  94  Comparison of linear and nonlinear undamped response for a general MSS maneuver consisting of simultaneous translation and slews at both joints. A cubic time history is used for all maneuver motions  95  Comparison of linear and nonlinear damped response to the same maneuver as in Figure 4-4 with 10% critical damping at each joint ( C j = 0.1; C} = 0.1)  97  x  5- 1  71  75  2  x  4- 1  70  Effect of the shift in center of mass as the flexible two-link manipulator performs a simultaneous translation and slewing maneuver with a 32,00kg payload  X  3- 28  . .  2  2  Block diagram of the centralized controller showing nominal reference trajectory and optimal feedback components viii  103  5-2  Optimal control of the two-link manipulator subjected to an initial condition of A/?jj = 5°. The state penalty matrix Q is diag { 10 . . . 10 ; 10 . . . 10 } and control weight matrix R is diag { 0.001, 0.1, 0.1 } 10  5-3  5-4  10  3  3  Nominal control of the two-link manipulator executing a general maneuver involving simultaneous translation of the mobile base and slew at the two joints. The cubic maneuvering profile is used.  . . 110  Optimal control of system in Figure 5-3. The state penalty coefficients are: Mi = 10 , M 2 = 100, /x - 10 , fi = 10 , fi = 1, Me = 1. R is diag {0.001,0.1,0.1}. a) Time histories of the generalized coordinates b) Optimal feedback control torques c) Total control torques (nominal -f feedback)  112 113 113  Effect of increased damping in LQR. The state penalty coefficients are: Mi = 10 , M 2 = 1000, / i = 10 , fi = 10 , Ms = 10, Me = 10. R is diag {0.001,0.1,0.1}. a) Time histories of the generalized coordinates b) Optimal feedback control torques c) Total control torques (nominal + feedback)  115 116 116  8  6  6  3  5-5  106  4  5  i  8  i  6  6  3  4  ix  ACKNOWLEDGEMENT The author wishes to express his sincere thanks to Prof. V.J. Modi for his guidance throughout the preparation of this thesis. The investigation reported here was supported by the Natural Sciences and Engineering Research Council of Canada, grant no. 5-80029.  x  LIST O F S Y M B O L S A  state space matrix  B  input matrix  G  optimal control gain matrix  H  linear gyroscopic matrix  K  linear stiffness matrix  M  system mass matrix ( nonlinear and linear )  Q  state penalty matrix  R  control penalty matrix  Cji  damping coefficient of joint t  ch  nondimensionalized damping coefficient of joint i  Ci  transformation matrix due to rotation of mobile base  c c c  2  transformation matrix for specific slew at joint 1  3  transformation matrix for elastic rotation at joint 1  4  transformation matrix for link 1 deflection  C  5  transformation matrix for specific slew at joint 2  C  6  transformation matrix for elastic rotation at joint 2  C  7  transformation matrix for link 2 deflection  {EI)  flexural rigidity of link t  Li  F  force vector in the equations of motion  HSF  total angular momentum vector w.r.t. the system frame  [I]  system inertia matrix  jX  jY  jZ  inertia of station w.r.t. local body frame inertia of mobile w.r.t. local body frame  jX  jY  TZ  inertia of joint t w.r.t. local body frame  jX  TY  TZ  inertia of link t w.r.t. local body frame  Kji  spring stiffness of joint t xi  Q  vector of generalized forces  Qi  component of the generalized force vector Q  Ri  position vector of mass element dm,- w.r.t. the inertial frame  To  orbital kinetic energy  TL  librational kinetic energy  TsF  total kinetic energy w.r.t. the system frame  UG  gravitational potential energy  Us  strain potential energy  Uj  joint elastic potential energy  a, a, a c  x  shift in the system center of mass w.r.t. the system frame  y  aji  vector defining the joint 1 frame in the mobile base frame  Sj2  vector defining the joint 2 frame in the link 1 frame  a  vector defining the position of the gripper/payload in link2 frame  p  hb, hb , hb x  y  specified translation of the mobile base  l^i  length of link »  m  mass of the space station platform  g  m  0  mass of MSS mobile base  mji  mass of joint t  mii  mass of link t  trip  mass of payload  n,-  number of modes used in the discretization of link i deflection  q  vector of generalized coordinates  u  controller input vector  x  state vector  f  position vector to the center of mass of the system in inertial frame  c  r  orbital velocity of the system center of mass  r,  position vector of mass element drrii w.r.t. system frame  c  xii  fl  librational velocity vector  ojji  nondimensionalized stiffness of joint i nondimensionalized stiffness of link i  CJI  angular velocity vector of mobile base rotation  £>2  angular velocity of specific slew at joint 1  u>3  angular velocity of elastic rotation at joint 1  CJ  angular velocity of link 1 deflection  £5  angular velocity of specific slew at joint 2  UQ  angular velocity of elastic rotation at joint 2  (U7  angular velocity of link 2 deflection  ctLi  rotation due to deflection of link t  (3b  specified rotation of the mobile base  (3j  specified rotation of joint i  4  i  7J.  elastic deflection of joint t  6ij  generalized coordinate associated with <f>ij  6x,.  vector of link t deflection generalized coordinates  Ei  Pointing error of arm »  s*r,.  position vector to a mass element in link t after deformation = pi + ^ .  p,  gravitational constant  p-i  state penalty coefficients in LQR controller  Pi  position vector to a mass element in body fixed frame before deforma-  t  tion 6  true anomaly  —#  £L  V  transverse deflection of link *  4>ij  3  ip  system libration, pitch  th  admissible function for deflection of link ».  xiii  1.  1.1  INTRODUCTION  Preliminary Remarks The United States has committed itself to the development of a manned space  station by the late nineties. In addition to being a prestigious symbol of man's permanent presence in space, the Space Station Freedom will serve as an operational base for scientific exploration, satellite launch and maintenance, manufacture and processing in the favorable microgravity environment, and enhancement of earth oriented technologies such as communication, meteorology, navigation, pollution monitoring, etc. The Shuttle based Space Transportation System will play a key role in the construction of the station. Around twenty-eight Shuttle flights will be required to ferry the material and construction crew with the first flight scheduled in 1996. Spanning over 150 meters in its Permanently Manned Capability (PMC) configuration (Figure 1-1), the Space Station is a highly flexible structure with the fundamental frequency in bending of approximately 0.1 Hz and closely spaced higher harmonics. A remote manipulator system such as that on the Space Shuttle, but now bigger, more versatile and mobile, is invaluable for handling cargo as well as releasing and retrieving satellites. Forming an integral part of the Space Station, the Mobile Servicing System (MSS) consists of highly flexible robotic manipulator arms supported by an essentially rigid mobile base which traverses the station through translational and rotational maneuvers. In addition to the above stated tasks, it will assist in the construction, operation, maintenance and future evolution of the Space Station itself. The Space Station based MSS is inherently a highly complex variable inertia system. It is further complicated by structural and joint flexibilities. Maneuvers of the MSS may involve complex slewing and translational time histories thus affecting 1  Figure 1-1 Revised baseline configuration of the Space Station as of August 1987.  the attitude stability of the station and hence other on-board missions. Stringent performance criteria on the pointing accuracy of the Space Station present challenging problems in dynamics, stability, and control of large flexible space structures. In its utmost generality, the problem is indeed quite formidable. It would involve an orbitingflexiblemanipulator, with elastic dissipative joints, operating on a highly flexible Space Station and carrying an elastic payload. It represents a challenge of a higher order of magnitude than ever encountered before. This class of problems has never been faced by the conventional (i.e., ground-based) robots which, until recently, tended to be rigid. The highly nonlinear, nonautonomous and coupled character of the system, together with a large number of variables, demand the problem to be approached in an increasing order of difficulty to gain some appreciation of the associated dynamics. With this as background, the thesis focuses on the dynamics and control of a two linkflexiblemanipulator, with elastic joints, free to traverse on an arbitrary rigid platform, negotiating a general trajectory and undergoing planar libration. 1.2  A Brief Review of the Literature In the early stages of space exploration, satellites tended to be relatively small  in size, simple in design, and essentially rigid withflexibilityarising only from appendages such as solar panels and antennae. Attitude dynamics and control of such rigid andflexiblespacecraft have received considerable attention as reflected in the enormous amount of literature on the subject. Comprehensive reviews of this vast body of literature have been presented by Likins, Modi, Williams, and others [1-6]. Environmental effects such as those due to gravity-gradient, solar radiation pressure, aerodynamic forces, magnetic field, etc., can cause significant perturbations of the orbital and attitude motions of satellites. On the other hand, these forces can also be utilized for libration stabilization and control. Since this type of control is passive 3  or semipassive, it played an important role in the development of satellite attitude control strategies. Literature related to the attitude dynamics and control in the presence of environment torques have been reviewed quite thoroughly by Shrivastava and Modi [7]. With the advent of large space structures, flexibility has become increasingly important, and accurate modelling of elastic behavior is fundamental to dynamics and control studies. The problem is further accentuated by the demanding performance requirements associated with these systems [8]. In treating flexibility, the continuous system is described in terms of discrete and distributed coordinates, thus rendering the system hybrid. The resultant governing equations of motion are normally transformed into a set of ordinary differential equations using finite element, lumped parameter, or assumed mode methods with time dependent generalized coordinates. The method of assumed modes, or more precisely constrained modes [9,10], is used in this thesis. Dynamic simulation codes such as DISCOS [11] and TREETOPS [12] were developed for multi-body systems consisting of rigid/flexible components. However, these programs have certain limitations. Kane et al. [13] investigated the dynamics of a moving elastic cantilever beam and observed conventional methods predicting erroneous divergence under extreme conditions. Of particular interest is the class of spacecraft with deployment of flexible appendages from the central body. This system involves transient inertia dynamics similar to that experienced by the Space Station during MSS maneuvers. Deploying systems have been analyzed, with simplifying assumptions, by Lang and Honeycutt [14], Cloutier [15], Hughes [16], Sellapan and Bainum [17], and others with the flexible members treated as point masses orrigidbodies. A general formulation for studying librational dynamics of a rigid spacecraft deploying an arbitrary number of flexible appendages is presented by Modi and Ibrahim [18]. The formulation accounts for the 4  shifting center of mass and allows extensible beam and plate type members. Slewing spacecrafts also impose transient inertias on the system dynamics and hence are relevant to the current investigation. Hablani [19] derived equations of motion for a spacecraft with a chain of hinge-connected bodies in a gravitational field. Conway and Widhalm [20] included dissipative elastic joints in their system subjected to arbitrary external forces. Both studies considered only rigid bodies and are based on the Newton-Euler method. The Canadarm on the space shuttle represents aflexiblemanipulator system capable of rapid large angle slewing maneuvers. The associated literature [21-27] would be of some value in understanding the dynamics of a large scale manipulator slewing in space although it is smaller than the MSS and lacks the mobile degree of freedom. Meirovitch and Quinn [28] derived the equations of motion for maneuvering flexible spacecraft using a perturbation approach. Longman et al. [29] addressed the problem of slew induced reaction moment on libration and modified the joint angle commands through the kinematic equations to position the end-effector of a rigid remote manipulator at its desired target. With the high precision positioning critical to the success of certain missions, the problem of maneuvering aflexiblespacecraft while suppressing the induced vibrations and attitude disturbances is becoming increasingly important. Hale et al. [30] have discussed simultaneous optimization of structural and control parameters in maneuveringflexiblespacecraft. Nonlinear feedback control was explored by Carrington and Junkins [31] while Yuan and Stieber [32] studied the robust beam-pointing and attitude control of aflexiblespacecraft. The research in controllingflexiblespacecraft intensified with two experiments proposed by NASA: Spacecraft Control Laboratory Experiment (SCOLE) [33] and Control of Flexible Spacecraft (COFS) [34], The associated literature [35-41] is useful 5  in understanding slewing dynamics and control of a longflexiblebeam with rigid tip reflector antenna in space. Despite substantial theoretical research in the control of large space structures, relatively few experiments have been conducted to verify the predicted dynamics and control algorithms [42]. Experimental results on control offlexiblebeam have been reported by several authors [43-45]. Being one of the first laboratory experiments, Schaechter and Eldred [43] have discussed at length, the effectiveness of several control strategies. Schafer and Holzach [44] employed direct velocity feedback to increase damping of the structure while Meirovitch et al. [45] applied independent modalspace control to minimize the amount of computational effort thus promising ease of implementation on a large structure. As pointed out before, unlike the largeflexibleMSS, factory robots used in automation are generally small and rigid. Hence, in general, literature pertaining to ground-based robot is of limited relevance here. However, recently in order to meet the demand of higher speed and efficiency, the new generation of fast robots are made of lighter materials and consequently areflexible.Lightweight systems can offer higher speed of operation, smaller actuator sizes, reduced energy consumption, higher productivity, and improved cost effectiveness. Generally,flexibilityarises from two sources: elasticity of the structural links and compliance of the motor and transmission units at the joints. The structural and jointflexibilitiesof the manipulator can cause end-effector oscillations thus limiting its ability to perform high precision manipulations required to follow a given trajectory in approaching a target point. The dynamic analysis of aflexiblerobot is complicated by the coupling between the nonlinear rigid body motion and the essentially linear elastic displacements of the deformed links [46]. Derivation of the coupled nonlinear equations of motion is 6  enormously lengthy and efforts have been made to simplify the process. Book [47] developed a recursive Lagrangian approach to generate full nonlinear dynamics of multilink flexible manipulators using homogeneous transformation matrices. Cetinkunt et al. [48] proposed the use of symbolic manipulation programs to overcome the algebraic complexities and thus facilitate the formulation. Control of flexible robots has been investigated by many researchers [49-64]. Control performance can be improved by using end-effector sensing feedback as suggested by Sweet and Good [49]. Cannon and Schmitz [50] demonstrated the effectiveness of end-point sensing feedback in their experiment with a flexible one-link robot. In studying control of flexible robots, many researchers have considered elasticity only at the joints [51-54] and regarded the links to be rigid. Link flexibility in robots is often investigated for the one arm case only [55-58]. Literature incorporating structural and joint flexibility [59-62] has stressed the importance of including both parameters in the analysis. Of particular interest is the application of optimal control procedure to a flexible robot manipulator [63,64], although only the one arm case was considered in these studies. 1.3  Scope of the Investigation In its utmost generality the Space Station based Mobile Servicing System presents  a rather formidable problem in dynamics and control of maneuvering large space structures. The purpose of this thesis is to consider, purposely, a simplified model to obtain insight into the intricate dynamics involved, and to assess the performance of a central controller in simultaneously suppressing vibration and attitude disturbances. To this end a mathematical model for studying the dynamics and control of the Space Station based MSS is developed. The model accounts for both the structural and joint flexibility of the manipulator as well as its translational and slewing maneu7  ver capabilities. Each flexible manipulator link is treated as an Euler-Bernoulli beam with flexural deformation discretized using admissible functions in conjunction with generalized time dependent coordinates. Jointflexibilityis modelled by a torsional spring and a damper. Despite the Space Station being highlyflexibleas stated, it is purposely assumed rigid in this investigation to focus on the coupling between libration and the MSS' maneuvers and flexibility. The mobile base, joints, gripper, and payload are also treated as rigid bodies. Shift in the system center of mass due to MSS maneuvers is included in the formulation. In order to focus on the complex interactions among link and jointflexibilities,translational and slewing maneuvers, and librational motion, the system dynamics are confined to the plane of a circular orbit. The governing equations of motion, derived using the Lagrangian procedure, are highly nonlinear, nonautonomous and coupled. As can be expected, the system of equations, in general, are not amenable to any closed-form solution - hence one is forced to resort to a numerical approach. However, search for an analytical solution is also attempted using the variation of parameters method for a particular case. The simulation results for the dynamical response show that the manipulator maneuvers can significantly excite both librational and vibrational motions. Hence, an effective controller is required for the successful operation of the MSS while performing high accuracy tasks. A centralized optimal controller is designed to simultaneously suppress the attitude disturbance and manipulator deflections. The thesis ends with some concluding remarks and recommendations for future extension of the present study.  8  2. F O R M U L A T I O N O F T H E P R O B L E M In general, the interactive dynamics involving the Space Station and the Mobile Servicing System (MSS) is indeed challenging as it involves relative slewing and translational motion offlexibleappendages on a highlyflexibleplatform. For the manipulator, both the structural elasticity and joint compliance may adversely affect its performance in following a desired trajectory as well as pointing and positioning of payloads. A formulation for studying the coupled dynamics of the Space Station and Mobile Servicing System, with emphasis on theflexiblecharacter of the manipulator, is presented here. The development of the formulation starts with the selection of a mathematical model for the Space Station based Mobile Servicing System, a model amenable to realistic measure of time and effort, yet sufficiently comprehensive to include the essential features of the system. Reference frames and coordinates, specified and generalized, are then assigned to indicate the state of the system. The deformations of the continuous flexible members are described by appropriate modal functions, thus truncating the infinite degrees of freedom to afinitenumber and transforming the partial differential equations into a set of ordinary differential equations. The classical Lagrangian procedure, which involves evaluation of the system kinetic and potential energies, is employed in deriving the governing equations of motion. 2.1  System Model Consider the Space Station based Mobile Servicing System depicted in Figure 2-  1. The simplified, yet effective model retains the essential features of the MSS by including both structural and jointflexibilityin addition to translational and slewing maneuver capabilities. Despite the Space Station beingflexibleas stated earlier, it is purposely assumed to berigidin this study to focus attention on the influence of the MSS' structural and joint flexibilities. 9  Rigid Payload  Mobile Base, Joint 1  System Center Of Mass Rigid Space Station Figure 2-1 A schematic diagram of the Space Station based Mobile Servicing System (MSS).  In the model, the Space Station is treated as an arbitrarily shaped rigid body in the gravity gradient orientation while the MSS is modelled as a two link robot mounted on a mobile base. The system consists of seven structural members, referred to as bodies 1 to 7: space platform, mobile base, joint 1, link 1, joint 2, link 2, and the payload. Directly attached to the mobile base is joint 1 to which link 1 is connected through an elastic and dissipative transmission unit. Joint 2 and link 2 are similarly connected at the tip of link 1. The payload is considered as an extension of the gripper and both are treated together as one rigid body. The system is free to undergo planar librational or pitch motion about the system center of mass which describes a circular orbit about the Earth. In case of a large motion maneuver with a massive payload, the system center of mass can shift significantly. The formulation accounts for a shift in the center of mass due to slewing and translational motions; however, the contribution offlexibilityon the shift is assumed to be small and hence neglected. All maneuvers and elastic deformations are also confined to the orbital plane. 2.2  Maneuvers and Generalized Coordinates The mobile base, treated as rigid, can traverse the Space Station platform execut-  ing translational as well as rotational maneuvers. The joints 1 and 2 can perform slew maneuvers to impart any desired orientation to the MSS thus positioning the payload as required in the orbital plane. All maneuvers — translation  and rotation 0b{t)  of the mobile base as well as slew of the two links, 0j (t) and (3J (t), respectively — x  2  are prescribed trajectories consistent with the system constraints. They are treated as specified functions of time in the dynamical analysis. In performing an actual task, the maneuvering time histories of the MSS may be quite involved. Note that even during the planar motion, translation of the mobile base along a curved platform will involve rotation of the mobile base. 11  The elasticity of the drive mechanism at the joints will affect the manipulator performance and accuracy by allowing oscillations about the nominal specified slew values. To account for this effect in the model, the joints are consideredflexibleand dissipative. Theflexibilityis modelled by torsional springs while viscous dampers provide for the dissipation of energy. The elastic rotation, denoted by the generalized coordinate  is superimposed on the corresponding specified slew, /?j ., to give the t  total rotation of link » at joint i (Figure 2-2): rotation of link 1 at joint 1 = Sj + 7jj; l  1  rotation of link 2 at joint 2 = /?j + ij . 2  2  The twoflexiblelinks, treated as Euler-Bernoulli beams, are free to deform transversely in the orbital plane. The discretization of theflexuraldeformations is accomplished using assumed modes in conjunction with generalized time dependent coordinates. Thus transverse deflections £z, (£, t), where x is the coordinate along the t  longitudinal axis of the link i, is represented by a series of the modes <f>ij(x) and the generalized coordinates 6ij(t): i  n  deflection of link 1: CLi{x,i) =  <pij[x)6ij{t)', n  deflection of link 2: £ L  ( >*) X  2  2  f 2j{x)62j(t);  < >  =  where rii is the number of admissible functions used to represent the deflection of —*  —*  link i. Expressing the 6^- and 62j collectively as vectors 6r and 6T, of dimensions ll  n-i and n , respectively, the deflections can be written as: 2  tL {x,t)=i {t) $ {x)T  1  Ll  l  & (M) =  1  h (t) '&(*); 7  a  2  where ^i(x) and $2{ ) are vectors containing the mode shapes. x  12  2  Figure 2-2 A schematic diagram showing deflections of the link and joint relative to the specified slew frame.  The librational degree of freedom, i.e., the pitch, given by the generalized coordinate tb, describes orientation of the Space Platform in orbit with respect to the local vertical. 2.3  Modal Functions Theflexiblelinks of the MSS are continuous members possessing infinite degrees  of freedom. Rigorously, the problem will be governed by partial differential equations with both spatial and time independent variables. The method of assumed modes, which transforms the problem into ordinary differential equations, is used here to facilitate the analysis. The selection of the appropriate modal functions is by no means an easy task, especially for a time varying system such as the MSS manipulator. The method of unconstrained modes [10], incorporating the time varying boundary conditions, have been applied to one-link systems by many authors [50]. Its application to the MSS is not as straightforward since quite complex boundary conditions are encountered. In this analysis, mode shapes of a cantilevered Euler-Bernoulli beam are used as the admissible functions. Thus, the rotary inertia and shear deformation effects, both pronounced for large deflections and higher frequencies [65], are neglected here. Determination of the mode shapes for an Euler-Bernoulli beam starts with the beam equation which can be obtained by applying Newton's law to a differential beam element  where EI is theflexuralrigidity of the beam and p, the mass per unit length. Assuming EI to be constant and rearranging the terms, the beam equation takes the form: dt  2 +  dx*  = 0; 14  with the appropriate boundary conditions (displacement, slope, moment, and shear) at the two ends. For a free cantilever beam, the boundary conditions are zero displacement and slope at the fixed end, and zero moment and shear at the free end:  i=0  = 0;  = 0; dx z=0 d £ dx x=l 3  EI  dx  2  x=£  EI  0;  3  0.  The problem can easily be solved by the method of separation of variables, £ ( M ) = *(s)A(t), where <j>(x), the mode shape, depends only on the spatial coordinate x and X(t) is a function of time. This leads to: d <j>  .  -*-*V =0 4  where k = w / a = pu /EI. 4  2  2  2  l  4  n  , dX and ^ 2  ^ + "A = 0;  Assuming the solutions as:  <f>(x) = Ci sin kx + C cos kx + C 3 sinh kx + C cosh A;x; 2  A(i) =  C 5 sin  + CQ C O S  4  CJ<;  and introducing the boundary conditions yield a transcendental equation for the spatial frequency parameter A; and the coefficients of the mode shape as cos kt cosh kl + 1 = 0, <t>(x) = (sinkx — sinhfcx) — ff f f, (cos kx — coshfcx). ' c o s k £ + coshfc£ v  ;  v  v  y  The modes obtained are orthogonal and self-adjoint. The first four modes are shown in Figure 2-3. Addition of a point mass payload at the tip of the cantilever will force the shear boundary condition at x = I to be dependent on the acceleration of the mass. The 15  moment will remain zero since the inertia of the point mass is zero. Thus the boundary conditions are modified to:  ''  0  =  1=0  EI  dx  2  JT  ox  EI  x=£  = 0;  i=0  ae 3  dx  3  x=l  =  tn.  dt  2  where m is the mass of the payload. Solving the equations as before yields the p  characteristic equation, cos kt cosh kl + 1 = —-k ^sin kl cosh kl — cos kt sinh fc^J, which reverts to the free cantilever frequency equation if the mass m is set to zero. p  The modal function is identical to that for the free cantilever case, sin kt + sinh kt -(cos kx — cosh kx). <f>(x) = (sinfcx — sinhfci) — cos kt + cosh kt However, since <f> is a function of A;, the resultant mode shapes are different. The first six roots of the characteristic equations k for different payload mass ratios are given below: m /pt  0  1  2  5  kit  1.8751  1.2479  1.0762  0.8700  kt  4.6941  4.0311  3.9826  3.9500  kt 3  7.8548  7.1341  7.1027  7.0825  kt 4  10.996  10.257  10.234  10.220  kt 5  14.137  13.388  13.370  13.359  kt  17.279  16.523  16.508  16.499  p  2  6  Figure 2-4a compares the first mode of the free cantilever with that loaded with a tip mass. After scaling (Figure 2-4b), the modes are almost coincident. The second and third modes are affected by the payload mass as depicted in Figure 2-5. For the range of payload mass considered (m /pt p  functions appear very similar. 17  = 1 — > 5), the corresponding modal  (a)  x/l  Figure 2-4a Effect of tip load on thefirstmode of a cantilever beam.  z/l  Figure 2-4b A comparison of the first mode of the free and tip loaded cantilever beams with tip defection normalized to 1. 18  (a)  x/l  ure 2-5 Effect of the tip mass on the higher cantilever modes: (a) second mode; (b) third mode. 19  Since the MSS links can undergo slewing motion, it may be more appropriate to model the fixed end as a pin connection instead treating it asfixedas a cantilever. Experiments withflexibleground based manipulators have revealed that for high gear ratios at the joints, the deflection can be better approximated by the cantilever modes [59]. Actually the torsional spring accounting for the joint compliance imposes a moment boundary condition at x = 0. The moment is dependant on the deflection of the spring, which in turn, is governed by the system dynamics. Furthermore, the finite inertia of the payload will introduce a moment boundary condition at x = £. Taking these effects into consideration will make the boundary conditions enormously complex and determination of the mode shapes extremely difficult. Therefore, for simplicity, the cantilever modes are used as the admissible functions for discretizing the transverse deflection of the manipulator links. It should be emphasized that this does not affect the system dynamics if appropriate number of modes are used to discretize the system. The choice of modes only influences the number of them required to adequately model the response. 2.4 Kinematics Application of the Lagrangian procedure to formulate the governing equations of motion requires evaluation of the system kinetic and potential energies. The fundamental kinematic expressions for displacement and velocity are developed in this section. The reference frames and coordinates selected for the kinematic analysis are shown in Figure 2-6. The center of mass of the system, consisting of seven structural members, undergoes a prescribed orbit about the Earth's center where the inertial frame is located. The position vector f and true anomaly 6 define the location of the system c  center of mass (cm.) with respect to the inertial frame. There are two frames with origin at the system center of mass, the orbital frame FQ and the system frame Fs20  Figure 2-6 Reference frames and coordinates for the mathematical model.  The orbital frame consists of the local vertical parallel to f , the local horizontal perc  pendicular to r in the orbital plane, and the orbit normal. The system frame is a c  body fixed reference frame attached to the platform-manipulator complex at its c m . Thus the librational motion u) is represented by the rotation of the system frame with respect to the orbital frame. Local body fixed reference frames are defined on each of the seven bodies constituting the system. Frame F is fixed to the station, F& to the mobile base, Fj\ to p  joint 1, Fx,i to link 1, Fj  2  to joint 2, FL2 to link 2, and F to the assembly of gripper g  and payload. The position vector  locates the mass element dm-i w.r.t. the local  body frame in a rigid body. In the case of a flexible body, the position vector that locates a mass element w.r.t. the local frame is £ f =  P+£  —#  which is a superposition of p, defining the undeformed position, and £, the transverse deflection. For the rigid bodies, it is convenient to fix the origin of the local frame at the center of mass of the body such that J  M  pi drrii = 0.  For the two flexible links,  however, this is not practical since the center of mass is continuously shifting even w.r.t. the local frame. Thus frames F L \ and Fr, for links 1 and 2, respectively, have 2  origins at the centers of rotation. This choice also facilitates the use of cantilever modal functions. As a mass element dm in the link undergoes a transverse deflection £(x, t), it will also experience a rotation a, corresponding to the slope along the beam: n  t  22  Hence joint 2 located at the tip of link 1 is rotated by an amount ai({.L ,t) w.r.t. 1  frame FLI- Similarly, the payload will be rotated by 02(^2>0 - -t- frame FLIw  r  Displacement of a mass element dm with respect to the inertial frame, Ri, is first expressed in the bodyfixedlocal coordinates (pi for arigidbody and £ if the body is flexible) and then projected onto the system frame through a series of transformations. The position vector of a mass element dm,- on body i with respect to the inertial frame —#  is given by Ri, Ri = f + i\,  i = 1,2,..., 7,  c  where r defines the position of the center of mass in the inertial frame and fi locates c  the mass element dm,- in the system frame. Expanding fi in terms of the local position vectors: f1 = a + ps\ c  r*2 = o, + hb + Cipb] c  r*3 = o,c + hb + Cxdjx + C1C2PJ1; f = a + hb + CISJI + CxC^aLi 4  +  c  f& = o-c + hb + C\aj\  C\C2CZ$L \ X  + C\Cio,ij\ + C1C2C3CLJ2  + C1C2C3C4C5PJ2; fe = a + hb + C\cijx + C\C2<i-L\ +  C\C Cza.j  c  2  2  fi = o-c + hb + CIUJI + CXCZ&LX + C\C"ICZ<LJI + C\C CzC C§aj <i 2  4  +  J  C\C2CzC C^C%a 4  + C\C2CZC CZCGC7P ', 4  p  (2.1)  P  where SJI is the vector that identifies the origin of frame FJI in the mobile base frame Fb', aj2 locates joint 2 in frame FLI', and a positions the gripper and payload p  23  in frame FLI- The C s represent rotation matrices [Appendix I]: C\ = Ci(hb), rotation of the mobile base w.r.t. frame F ; p  C = C2{/3J ), 2  specified slew at joint 1;  1  C 3 = C (7J ), elastic rotation of joint 1; ,  3  1  rotation at tip of link 1 due to bending;  C4 = C^CLL^),  C = Cs(/3j ), specified slew at joint 2; 6  2  CQ = CQ(IJ ),  elastic rotation of joint 2;  C =  rotation at tip of link 2 due to bending.  2  C7(0:2,2)5  7  Several simplifying assumptions are made at this stage to make the system more manageable. These assumptions reduce the kinematic expressions considerably without affecting the physics of the problem: (t) Joint 1 is at the center of mass of the mobile base, thus the two body frames (Fb, FJI) have a common origin, i.e., aji = 0. (»t) Joints 1 and 2 have their centers of rotation at their respective centers of mass giving SLI = 0.1,2 = 0. {iii) Payload center of mass at origin of gripper frame, i.e., J Now the position vectors in Eq.(2.l) simplify to: = a + Ps, c  f = a + hb + C\Pb', c  2  —* = a + hb + C1C2PJ1', c  U = a + hb +  C\C CZ^L \  c  r  5  2  X  —* — a + hb + C1C2C3SJ2 + C 1 C 2 C 3 C 4 C 5 P J 2 ; c  re = a + hb + C\C2Cz(Lj2 + C\C CzC c  rV =  2  a + hb + C\C Czo.j2 c  +  2  +  C\C CZC CSCQC7P . 2  4  P  \CsC^h  C\C"iCzC4C§C§a  p  2  p dm — 0. p  p  Velocity of the mass element dm, with respect to the inertial frame is given by  — * Ri, dfRi='r  + {nxr^-r-^,  c  t = l,2,...,7,  where: f = orbital velocity of the system; c  —*  fl = librational velocity; df'  = time derivative of  w.r.t. the system frame.  dt  In the planar case, the librational velocity n reduces to 6 + rjJ. The term dfi/dt  represents velocity of the mass element drrii w.r.t. the system  frame. Differentiating ri w.r.t. time in the system frame, the dfi/dt  (i = 1,...,7)  can be written as:  dfi _ da c ; ~dt ~ "dT —*  dfi  da  dhb  c  „  -dT = -dT -dT^ +  c+  —*  dfz da dhb -> _ _ —f- = — + —- + (Ciwi + CiC wa) x CiC i; at at at c  2  2PJ  —t  itt  =  lit i r +  + ( C W l + C l  x CiC Cz^Li  ° * 2  + CiC Cz  2  2  2 + c  » » »a») c  c  ^ ; 1  —*  dt  dt  dt daj  2  x CiC Czdj 2  +  (CiWj  + CiC Cz  2  +  2  CIC UJ 2  + CIC CZC4<JJ4 +  ^  =^  2  CiCzCzC^sWs)  2  x  CiC Cz&z  +  2  ^  CiC CzC C pj ', 2  +^  5  2  + (duJx + d C a ^ +  x CiC Czaj 2  4  2  + CiC Cz 2  25  ^  2  CiC Cz*z) 2  xpb)  >  + (CitDi + C1C2W2  +  C\C Cz^oz 2  + C1C2C3C4W4 + C1C2C3C4C5W5  + C1C2C3C4C5C6W6)  CICICZCICZCQCI.^  x  + C1C2C3C4C5C6 ^ ; —*  at  at  at  x C\CiCzaj<2,  + C1C2C3  + ( C 1 W 1 + C1C2W2 + C1C2C3CD3 + C1C2C3C4W4 + C1C2C3C4C5CD5 + C\C CZC C^CQQQ) 2  +  x  4  C\CICZC C^CQO,P 4  del  + C\Ci&i  C1C2C3CUCSCQ —-^- + (CiU\  at  + C1C2C3W3 + C'1C2C3C' cD + 4  + CIC^CZC^C^CQCJQ  +  4  C1C2CZC4C5UJS  CXC^CZC^C^C^C-JCJT)  x C\C2CZC C^CQCTPP\  (2-3)  4  where the terms involving w- arise from differentiation of the C,'s w.r.t. time [Apt  pendix I]. 2.5 Kinetic Energy With the position and velocity of each elemental mass dm,- established, the kinetic energy T and potential energy U of the system can be evaluated by integrating over the mass M of the entire system:  = - I Ri-Ri 2  dM  JM  4/[(r;.r,)  +  (nxr-;).(nxr-;) (§.§) +  + 2(f -(nxr ) + f . | + | . ( n x r ) ) ] dM c  <  c  i  26  ^•t/  m  = T +T +T 0  .(^l)^  (  L  SF  + U- HsF'  (2.4)  where: To = orbital K.E.; T = librational K.E.; L  TSF — ^^TsF;, total K.E. w.r.t. the system frame; HSF = YL^ i' SF  angular momentum w.r.t.  the system frame; [I] = instantaneous inertia matrix w.r.t. the system frame.  The kinetic energy To is dependent only on the orbital motion of the center of mass which is governed by the classical Keplerian equations. The energy TL is the rotational energy of the Space Station librating about the center of mass [Appendix II]. Energy due to vibrations and maneuvers is given by TSF- Finally, ft • HSF represents the contribution due to coupling of the librational motion with the manipulator degrees of freedom. The TsFi and HsFi can be obtained readily by substituting the kinematic relations from eqs. (2.2) and (2.3) in (2.4). For example, contribution to TSF from the first four bodies (platform, mobile base, joint 1, and link 1), after substantial algebraic 27  manipulations, can be written as: ' da da. (ddc oa \  1 i  r  c  2 \-b7'-bTP m  -=?[(tr-(f)^(t 4)] TsF  =  + j M A +^  2  + 7, ) +[- f+-^] 2  1  1  5  (cDi + cD + w ) x C C C 2  3  X  2  /  Z  J m  -f C1C2C3 / £  i l  dm  4  4  dm + (cDi + tD + cD ) 2  4  3  Jm  4  • C CC X  2  * «'m  x  Jm  f  1  where  I  3  t )dm Ll  4  4  4  is the inertia of body » in its local body frame. The remaining Tsjr. can be t  obtained in a similar manner [Appendix III]. Substituting for fi and dfi/dt in the angular momentum vector HSF and simplifying, the first four terms are obtained as: —  /  HF S  da \ c  = {a + h ) x m  2  c  #SF  b  ,_  3  -» .  = (a + Afc) x m c  + ^) —#  2  t dd  3  ^—  c  +  + /m ^i; 2  dhb\  _  ,_  _.  J + i m ( w i + wa); 3  — *  -± < r (da dhb\ ,_ _ . ffsf = (o + h ) x |m ^ — + — J + + w + w) c  4  b  c  x C CC X  2  3  I  Jm  4  tf dm + C C C Ll  4  X  2  3  j  Jm  4  28  2  Z  Ll  4  3  rfm ] 4  c /  + C l C 2  3  + CCC X  +  2  Z  J m ( w i 4  I  m / l l  $  Ll  + u5  2  ^,(§ §) +  x£  L l  dm  4  + W ). 3  Expressions for the HSF pertaining to joint 2, link 2, and payload can be obtained {  similarly [Appendix IV]. 2.6  Potential Energy The potential energy of the system arises from three sources: gravitational field;  elastic spring energy of the joints; and strain energy of the links; U = U + Uj + U . G  s  The librational response is dominated by the gravitational energy UQ while the vibrations are primarily dependent on Uj and Us, the elastic and strain energies stored at the joints and links, respectively. The gravitational potential energy is given by  to a fourth order approximation, where : H = gravitational constant; l  rc  = direction cosines of f w.r.t. the system frame. c  Since the station is assumed to be in the gravity gradient orientation in this planar study, the gravitationalfieldis critical for its stabilization in the uncontrolled case. The joint elastic potential energy is given by  t=i  t = i  29  where kj = spring stiffness of joint i. i  The strain energy in bending of the links can be written as  where:  2  i=i  i=i  0  —fiexuralrigidity of link t; V dx  2.7  2  = curvature of link i.  Equations of Motion Using the Lagrangian procedure, the governing equations of motion can be ob-  tained from £(21.) dt\dqjJ  - — + — - Q. dqj dqj '  i-i  J  n  >•••>>  where T and U are the kinetic and potential energies of the system with q - and Qj as 3  the generalized coordinates and the associated generalized forces, respectively. The generalized coordinates are {VS7J 5^15 7J '^ ^ ,  /  2  1  V> 7jj 6u  W N E R E :  2  system pitch; elastic rotation of joint 1; generalized coordinate associated with the I mode of link 1, t = th  l,...,ni; 7J  2  ^2t  elastic rotation of joint 2; generalized coordinate associated with the i  th  l,...,n ; 2  while the specified time dependent maneuver coordinates are: hb{t)  translation of the mobile base;  0b(t)  rotation of the mobile base;  PJ (t)  slew of joint 1;  PJ (t)  slew of joint 2.  1  2  30  mode of link 2, t =  The differentiation process and grouping of terms can be facilitated by recognizing the following facts: T = T {rp, h ,/3 , /3J , pj , u , 7J , 6 , 6 ); L  L  b  b  2  1  l  2  —•  X  2  —* —#  TSF = TsF[h , Pb, 0J , b  Pj >lJiilJ2>tii,62,  x  •  •  2  •  —»  —*  h ,P ,Pj Pj ,ij , ij ,Si,6 )\ t  b  HF S  b  v  = HsF\h ,  2  l  2  2  0 , /3j ,  b  b  PJ ,1J ,1J ,SI,6 ,  1  2  1  2  2  h , 0 , 0J > PJ ) ij\ > IJ >*i > ^ 2 ) ; b  b  =  X  2  2  U {xl),h ,p ,Pj ^j ,^j ^j JiJ )\ G  b  b  l  2  1  2  2  Uj = Uj(<u ,u ); 1  Us =  2  Us(6 S ). u  2  The resultant equations of motion have the form M(q,t)q+F(q,q,t) = Q, where q and Q are vectors of the generalized coordinates and generalized forces, respectively. The mass matrix M is symmetric and positive definite. It is a function of both q and t due to the nonlinear and time varying nature of the system inertia. The force vector F includes all other forces not associated with accelerations of the generalized coordinates, such as coriolis and centrifugal forces. Details of the equations of motion are presented in Appendix V and VI.  31  3.  DYNAMICAL RESPONSE  The governing equations of motion formulated in the previous chapter are nonlinear, nonautonomous, and coupled. In scalar form, these equations are rather lengthy and analytic closed-form solutions are not possible unless substantial simplifications are introduced. Hence the equations are coded in a computer program and simulated to yield the system response to given disturbances and maneuvers. To complement the numerical analysis, an approximate closed-form solution is obtained for a simplified system consisting of a flexible link with aflexiblejoint. 3.1  Numerical Approach As mentioned before, the governing equations of motion, not being amenable to  any closed-form solution, are solved numerically. The FORTRAN computer program is written in a modular fashion to help isolate the effect of system parameters such as inertia properties, maneuvers,flexibility,number of admissible functions, shift in the center of mass, etc. Numerical solution is obtained using the IMSL differential equation subroutine DGEAR [66]. This routine is capable of integrating stiff systems of equations, i.e., systems with vastly different time constants. For the Space Station based MSS system, the orbital period is about 100 minutes while the structural natural frequency can be around several Hertz. Figure 3-1 is aflowchartof the Space Station based MSS dynamics simulation program. After obtaining the initial conditions, maneuver specifications, inertia properties, and stiffness as well as damping parameters in subroutine SYSPAR, the main program calls the numerical integrator DGEAR. The only routine DGEAR calls is FCN, which provides it with the dynamics of the system. The governing equations of motion are contained in the subroutine EQN. Time histories of the specified transla32  DGEAR  SYSPAR  FCN  TRAJ  EON  Figure 3-1 Flowchart for the MSS dynamics simulation program  33  tion and slew of the manipulator are handled by TRAJ. Closed Form Solution  3.2  A closed-form solution of the governing equations, if available, can provide more insight into the system dynamics. It can be a valuable tool in the parametric study of the problem predicting system response to changes in important variables. Furthermore, an analytic solution can save the computational time and effort, especially during long-term predictions. 3.2.1  A Simplified Model for Analytic Solution  To reduce the system to a manageable form for analytical approach while retaining the desirable degrees of freedom to preserve the more significant characteristics of system, the following simplifications are introduced: • System with one arm; m^2 = J2 — 0. M  • Controlled libration ;  = rj} = rb = 0.  • Straight platform; /i& = h , h = 0, Pb = Pb = Pb = 0. x  y  • Point mass payload; I* = Ip = I* = 0. The period of the circular orbit was taken to be 100 minutes. Considering only the first mode of deflection, the one-link system has two degrees of freedom: elastic joint rotation 7, and link deflection coordinate corresponding to thefirstmode 6. With these simplifications, the equations of motion reduce to: 7 equation [II + m (l/4 + < M ) £ + m (l + * 6 )e ](0j 2  2  2  L  p  2  e  2  + 7)  + [m $ + m $ ] £ 6 + C J 7 + i C j 7 L  p  c  + (m /2 + m )l[-h p  L  2  e  sin(/?j + 7) +  x  + ( m * + m $ )£6[-h L  m  p  e  x  cos(£/ + 7) + 0 h sin(/?j + 7)] 2  x  cos(/?j + 7) - 26h sin(/?j + 7) x  + 6 h cos(/?j + l)] + 2(0 + PJ +  +  2  x  34  m $ )£ 66 2  p  2  e  + 0 {z{ll  - 1$) C O B ( / ? J  2  7) sin(/?j + 7)  +  + 3[(m /4 + m ) - (m $ + m $ ) 6 ] £ cos(/3j + 7) sin(/?j + 7) 2  L  p  L  a  2  2  p  + 3(m $ + m $ )£ £(cos(/?j + 7 ) - sin(/?j + 7 ) ) 2  L  c  p  2  sin(/?j + 7)  + 2(m /2 + m )th L  2  e  p  x  + 2(m $ + m $ ) £ £ / i cos(/?j + 7 ) } = Q \ L  m  p  e  z  n  6 equation + 7) +  [m $ + m $ }l 0j 2  L  c  p  + (mi$  e  + m $ )£ £ + K 6 2  2  p  L  + m $ ) £ [ - / i sin(/?j + 7) + 2«A cos(/?j + 7)  m  p  e  x  X  + 9 h sin(/?j + 7)] - (m *. + m $ )£ ^(^ + 0j + 7 ) 2  2  x  L  2  2  p  + 0 {(m $ + m $ ) £ £ ( l - 3sin(/?j + 7 ) ) 2  2  L  a  2  2  p  + 3(m $ + m $ e K cos(/?j + 7) sin(/?j + 7) 2  L  c  p  + 2(m $ + m 9 )th sin(/?j + 7)} = L  m  p  e  (3.1)  t  Here, the $'s are integrals of the mode shape assumed for the link deflection [Appendix V]. The subscript '1' is dropped from the equations for conciseness since the system under consideration has only one link and one joint. The equations are still coupled, nonlinear and nonautonomous. The time varying nature of the coefficients makes the search for a closed form solution virtually impossible. To make some progress, only the autonomous case is considered here. The system can be made autonomous byfixingthe translation and slew orientations, i.e., setting h = h = J3j = 0j = 0. This means that the manipulator is stationary and x  x  the arms arefixedin any arbitrary orientation. The resultant autonomous, however, highly nonlinear and coupled, equations can be written as: 7 equation [m /3 + m + (m $ + m $ )<5 )]£ 7 + [m $ + m $ ] £ 6 L  p  L  p  a  2  2  L  + C J 7 + KJI + 3(m /2 + m )M h 2  L  2  2  p  x  35  c  sin(/?j + 7)  p  e  cos(/?j + 7)  + 3(m $ + m $ )l60 h 2  L  m  p  e  x  + 2(0 + i){m $ L  +  e  m $ )l 66 2  p  2  e  + 30 [(m /3 + m ) - ( m $ „ + m $ )6 \l 2  2  L  p  L  2  cos(fij + 7) sin(/?j + 7)  2  p  + 30 (m $ + m $ )£ £(cos(/?j + 7) - sin(&, + 7) ) = 2  2  L  c  p  2  2  e  6 equation [m $ + m $ ] £ 7 + (m $ + m $ ) £ £ + K 6 2  L  c  p  + (mi$  2  e  L  + m $ )t6 h p  e  x  - ( m $ + m $ ) £ 6 ( 0 + 7) 2  L  a  + 0 {m $  L  2  2  p  + m $ ) £ £ ( l - 3sm{fij  2  L  2  p  sm(fij + 7)  2  m  a  2  s  2  p  + 7) ) 2  + 0 3(m $ + m $ ) £ cos(/?j + 7) sin(/?j + 7) = Qs2  (3.2)  2  L  c  p  e  It is thought appropriate at this stage to assess the relative significance of the nonlinear terms prior to searching for a solution. It is found that, in general, terms involving second power of the generalized coordinates (6 ,#7,7 ) have negligible con2  2  tribution to the system response. The exception is the coefficient of 7, which contains a term proportional to 6 , (mz,$« + m $ ) £ . Despite its being second order, the sys2  2  2  p  tem response appears to be quite sensitive to its presence, especially for large initial conditions. Deleting all the insignificant (^ ,^7,7 ) nonlinear terms in the original system, 2  2  a simpler set of equations which retains the essential characteristics of the original system is obtained: 7 equation [m /3 + m + (m $ + m $ ) 6 ) ] £ 7 + [m $ + 2  L  p  L  8  2  2  p  L  c  m $ ]i 6 2  p  + C J 7 + i 0 7 + 3(m /2 + m )£0 /i (sin fi + 7 cos fi) 2  L  + 2(0 + 7) (m $ + L  p  m $ )L 66 2  8  + 3(m $ + m $ )l60 h 2  L  m  p  e  2  p  x  cos fi 36  x  e  + 30* (m /3 + m ) £ (sin /? cos /? + 7 cos 2/?) 2  p  L  + 30 (m $ + m $ ) £ 5 cos 2/3 = Q ; 2  2  L  c  p  e  7  6 equation [m $ + m $ ] £ 7 + (m $ + m $ ) £ 5 + iiC 6 2  L  c  p  2  e  L  8  2  m  p  e  L  (sin/9 + 7 cos/?)  + 3(m $ + m $ )££ h L  2  p  x  - (m $ + m $ ) £ * ( 2 7 + 7* + 3 sin (3) 2  L  a  2  2  p  + 0 3(m $ + m $ )£ (sin/?cos/? + 7 cos 2)9 = Q . 2  2  L  c  p  e  5  (3.3)  These simplified equations are still nonlinear because of the terms involving products of velocities in addition to the 8 coefficient in 7. Figure 3-2 compares the 2  numerical responses using the approximate equations (3.3) and the original 'exact' nonlinear equations (3.1). Note, the system is subjected to a rather large initial conditions of: 7(0) = 10°, 6(0) = 0.2; corresponding to a 20° rotation at the joint and a transverse tip link deflection of 20%. It should be pointed out that since only one mode is used here in the analytical analysis, it is convenient to normalize the mode shape to yield unit displacement at the end. This simplifies the equations further as $ = 1 and gives the coordinate 6 a physical meaning: ratio of tip deflection to e  the arm length. From Figure 3-2, it is apparent that the character of the system is captured rather well by the approximate equations. Even with such large initial conditions, the agreement is excellent in phase, frequency, and amplitude. This suggests that a large number of nonlinear terms taken together contribute little in the present case thus permitting, through a careful study, simplification of the system equations. 3.2.3  Variation of Parameters Method  Several methods have been reported in literature to solve, approximately, a set of nonlinear equation. Since there is damping in the system, methods that give multiplicative corrections, such as the variation of parameters approach, may be more 37  0.000  0.002  0.004  0.006  0.008  0.010  ORBITS  Figure 3-2 A comparison of the responses obtained using 'exact' and simplified equations for a relatively large disturbance (7 = 1 0 ° , 6 = 0.1): (a) time history of the joint angle 7; (b) time history of the generalized coordinate 6 associated with the flexible link. 38  suitable than those with additive corrections, as in series perturbation techniques. The variation of parameters method used here is a modification of the original approach proposed by Butenin [67] which can tackle nonlinearities involving dependent variables and their first derivatives. The present set of equations involve nonlinearities in the coefficient of second derivative of 7. After normalizing w.r.t. m£,£ 0 , the system is cast into a form similar to that 2  2  discussed by Butenin: the linear term are collected on the left hand side (LHS) while the nonlinear terms are grouped on the right hand side (RHS). The equations now take the form: Tf + Aid + B16 + w <7 = /x/(7,7,6,6) + fip(i, 7,6,6)r, 2  6 + A i+B i 2  where:  + ulS = pg{ ,^6,6y,  2  (3.3)  1  A i = -.— : (1/3 + m)' ;  r  3[(S + m) cos2/?j + ($ + m$ )h cos 0j]  „  C  m  2 W  l  e  (1/3 +  1  ro)  ;  _ Kj + 3[(l/2 + m)h cos PJ + (1/3 + m) cos2/?j] ~  (1/3+ m)  5  fif = -{3(1/3 + m) sin PJ COS PJ + 3(1/2 + m)hsmPj + Cj^  + 2($ + m)«i + 2($ + m)w}/(l/3 + ro); 8  _ ^ ~  8  ($ + ro)6 (1/3 + ro) - (^c + m) + m)' R = l ( ^ c + m) cos2/?j + ( $ + ro)/tcos/?j] (*. + m) ul = K /($ + ro) - 3sin 0j; 2  a  ;  4  2  3  m  2  5  2  L  a  ^ = - J3($ + m) sinPj cos PJ +.3($ + m)hsmP c  m  - (*. + m)(27<5 + 7 <5)}/($ + ro). 2  8  39  Here m is the normalized mass of the payload, m /mi,; and h the normalized distance, p  h /L  This system differs from the classical Butenin's format by virtue of the nonlinear  x  term involving the second derivative 7. The terms on the RHS are assumed to be of order fi, a small parameter. Hence, by continuity, solution of the nonlinear system may be expected to have, approximately, the same form as that of the linear homogeneous system obtained by setting // = 0. This linear solution can serve as a generating function to which a particular integral can be added to take into account contribution of the nonlinear terms /if, fip^f, and LKQ. It is apparent that the solution to the linear homogeneous system can be written as: 7 = asin(A;it + 0\) + 6sin(A;2< + #2); 6 = aiasin(M + 0i) + o. bs\n(k t + #2); 2  (3.4)  2  where k\ and k are the eigenvalues satisfying the characteristic equation 2  (1 - A A )k x  while [l  ai]  T  + (UJ\ + u\ - B A  A  2  2  and [l  ct ] 2  2  2  2  X  2  = 0,  form the corresponding eigenvectors:  T  _ A k\ - B 2  U  - A Bi)A; + ufu} - B B  X  (JJ  L  2  2  — k\  _ A k\ - B u)\ — k\ 2  '  a  2  2  The terms a, b, &\, and 6 are constants of integration found from the initial condi2  tions. With the generating solution in hand, the coefficients a, b, 6\, and 6 are now 2  considered to be slowly varying functions of time to account for the effect of the nonlinear terms on the RHS. Because there are four unknown variables in the system (a, b, 0i, and 02) and only two equations (3.3), two constraints conditions are imposed on the system to obtain a unique solution. The constraint conditions are such that the first derivatives are the same as if the parameters were constant: <xsin(M + 0i) + bsm(k t + 0 ) + a0 cos(M + 0i) + 60 cos(fc t + 02) = 0; 2  aiasin(kit  2  x  2  2  + 0i) + a bsin(k t + 0 ) + ctiaOi cos(kit + 0i) + 0:2602 cos^J + 02) = 0. 2  2  2  40  These constraints relations, together with the two original equations, form a linear set in [a,  6,  a0\,  b0 ] , T  2  sin ©2  sin©i aisin©i  a  ki(l + ctiAi — up) cos@i  k (l +  2  k (a  2  2  s m  2  2  b  2  2  2  where ©i = k\t + 0\, 0 not dependent on [a,  2  a  2  2  x  2  COS ©2  a cos ©2 —k (l + ct A\ — HP) sin©2 X —k (a + A ) sin ©2 2  COS © ! Ct\ COS © i  ©2 — HP)  cos ©2 —ki(l + a Ai — p,p) sin©! —&i(ai + A )sin©i + A ) cos©  OL A\  2  fci(ai + A ) cos©!  2  bd  —  0 0 HJ + HP*  2  J = k t + 8 , and HP* is the terms in the expansion of HP1  2  2  2  b, a6\,  b$ ] , ie., T  2  HP* = HP{—o.K\  sin©i — 6fc|sin© ) 2  The system of equations can be solved for [d, b\ ad\,  6 } using Cramer's T  2  Rule: da dt db dt adO\ dt bd0 dt  ki(a  -M k (a 2  + P*)  if HP  + P*)  if  + P*)  if  + P*)  - «i)  2  - «i)  2  ki(a  - «i) Vk (a -  1 + aiAi - HP  2  HP  2  where a = 1 — A\A .  a - HP 1 + a Ai - HP a - HP 1 + a Ai — HP 2  2  2  1 + a Ai - HP a - HP  cos 01;  2  if  2  o -— HP uv a  7]  cos © ; 2  sinQj;  -g \ sin© ; J 2  Assuming slowly varying parameters, one can use average values  2  of the RHS taken over the periods corresponding to ©i and ©2: da dt ~~ ki(a db _ dl ~~ k (a d9i _ dt ~~ aki(a d$ ~dt ~~bk (a 2  2  - ai)l  2  -  2  -  2  -  [  \  2  2  41  r2ir  j  where:  r2n  4^ Jo Jo j r2w r2w F  =  F  =  2  3  4^Jo j  r2ir  2  2  r2*r r2iT — - — c o s © i d 0 i d 0 ; a- HP  j  /•2»r />27r  2  2  a- HP  j  ^2»r  i^/o  =  4  f + P* — cos © rf©irf0 ;  J  ^  G  tiG\ =  ^2JT  y  —-—sin0id0id©2; a- HP  0  4^y  0  y  0  4^/0  y  0  — - — s i n ©2^01^025 a- HP  -t2L- cos © i0id0 ;  /-2JT  4^/0  4^ Jo j  LtG* = 4  4TT y  0  /•2»r  J  g  cos 02d©i<f©2;  —ffEi— i © dQ!rf©2; s  o-- /xp  Jo  r2jr  P  a- HP  0  r2lt  2  2  / - 2 7 T  y  r2ir  tiG% =  lt  a- HP M  2  2  /-2ir />2TT  j  HG* =  x  — s i n © rf©1rf©2. 2  0--  0  n  and  terms in it, the averaging 2o-(l/3 + m)' Cj6A; , 2a(l/3 + m)' $ + m)a r  1  2  2  a  2 2  « . + m)6  =  4  2  —-—cos0 d0id0 ;  =  3  1  a- up  =  2  G  r2ir  7O  4?r 7  0  G  i - t ^ _ cos 0 d 0 d 0 ; a - /xp  /-2ir •^-^-8^0^01^02; /o a- up  2  Gi =  - /ip  / " 2 7 T  i^/o  =  4  Or  2  Jo  j  F  J-—^— cos ©id©id©2;  r  2  8(1/3+ m)<7L  G  x  =G  G  3  =—  2  2  2  2 2 2  2 2  2  2  ^  2  = 0;  a^A; + 2ct ab k 2  2  1  80" L  42  ]  i 2  J  ;  a  f c  1  2  2l  J '  uG\ = uG* = uG* = uG* = 0. 3  4  Substituting these into the expressions for a, b, 6\, and 6 , the amplitude and 2  phase angles can be evaluated subjected to initial conditions. The amplitude parameters a and b were found to exhibit exponential decay dependent on the damping at the joint Cj. The phase parameters ©j and © 2 were linear functions of time: Q = 0\t + 0i ; r  G = 0t + 0 ;  o  2  2  2o  leading to frequency perturbations of the linear frequency parameters k\ andfc .The 2  final form of the analytic solution is: 7 = a(t) single! + 6\)t + 6 } + b(t) sin{(fc + 0 )t + 0 }\ lo  2  2  2o  6 = «ia(*) sinl^x + 0\)t + 0 } + a b(t) s'm{(k + 6 )t + 6 }. lo  2  2  2  2o  It is evident that the nonlinear analytic solution has the same form as the linear one except for the frequency perturbations (?i and 0 . Furthermore, <?i and 0 are 2  2  dependent on the amplitude coefficients a and 6 which are determined from the initial conditions. The nonlinear system will oscillate faster or slower than the linear homogeneous system depending on the contribution of the nonlinear terms to the frequency. 3.2.4  Accuracy of Solution  Accuracy of the approximate closed form solution was assessed by comparing it with that obtained through numerical integration of the original nonlinear equations (3.1) corresponding to the one arm case. The nondimensionalized stiffness values KT, and Kj for the arm and joint are chosen to be 2.2 x 10 and 5.5 x 10 , respectively. 4  The mass payload parameter m = ro /mx, is taken to be unity. p  43  4  Figure 3-3 compares the analytical and numerical undamped responses with the system subjected to the initial 10% tip deflection of the arm length, (7 = 0, £ = 0.10). The frequency and amplitude parameters predicted by the approximate closed-form solution are: ki  0\ 0io  = 112 = -5 =90°  = 2159 0 = -181 02o = - 9 0 °  ai 0  k  2  2  = 2.42 a = 0.029 b  = -1.04 = 0.029  2  The frequencies are in dimensionless units of cycles per orbit (cpo). The predicted frequency of vibration is the sum of the linear frequency A: and the perturbation frequency 0: Lower vibration frequency = k + 0\ = 107cpo; x  Higher vibration frequency = k + 0 = 1987cpo. 2  2  It is evident that the linear homogeneous system over-estimates both the lower and higher frequencies. The nonlinear contribution provides a 5% correction to ki and 8% to k . The phase predicted by the analytic solution is slightly off. However they 2  seem to be in step after the completion of one low frequency cycle in 0.01 orbit. This is indeed encouraging as the error in phase does not accumulate for more than one low frequency cycle. Amplitudes and frequencies match rather well. When damping is present in the system, the amplitude coefficients a and 6 are simple exponential functions, (-{ct2 + A )  Cj  2  «(«) = «oexp{  ( t t a  6(0 = 6 e x p ( o  _  ( Q 2  a i )  _  \  «);  ( g ( 1 / 3 +  Q i ) ( g ( i / 3 +  ro))  m ) )  t).  As pointed out before, the frequency perturbations are functions of the amplitude coefficients: 0i = 0 i ( a , 6 ) ; 2  0 = 0 (a , 6 );  2  2  2  2  2  hence the phase perturbations are not constant as in the undamped case. In fact they vanish as the amplitudes decay to zero. This is logical since for very small amplitude 44  Figure 3-3  A comparison between the closed form and numerical undamped solution for an initial link tip deflection equal to 10% of its length (5(0) = 0.1). 45  motion, the system is adequately governed by the linear parameters. Figure 3-4 compares the damped responses for a Cj corresponding to 1% of the critical damping and an initial condition of 6(0) = 0.10. As in the undamped case, the low frequency is predicted rather well. Note that after 0.04 orbit, the higher frequency component amplitude is virtually zero and the response is dominated by the lower frequency mode. Considering the highly nonlinear character of the system, the closed form solution is remarkably good. It should be pointed out that the high frequency component of the numerical solution does not have a constant period. This is apparent in Figure 3-2 which shows the system response for a large initial condition. The frequency modulation is due to the change in the system inertia caused by the deflection of the arm, 6. Since 6 itself is oscillatory, the frequency has a periodic variation. In the analytical solution, the phase perturbations are given by a pair of first order uncoupled linear equations from which it is impossible to obtain any periodic variations. The analytical solution can achieve, at best, an average of the maximum and minimum values in the periodic variation so that the phase/frequency error cancels out in one cycle instead of accumulating. This explains the inability of the analytic approximation to follow the phase/frequency character of the response closely despite good correlation in amplitude. In any case, accuracy of the closed form solution is indeed adequate for a preliminary design. 3.3  Numerical Simulation Results and Discussion For better appreciation of the dynamical response, the system is analyzed in an  increasing order of complexity: starting with a one link rigid system, followed by a one linkflexiblesystem andfinallythe two linkflexiblesystem. The amount of information obtained through a planned variation of system parameters and initial conditions is rather extensive. For conciseness, only the typical results useful in establishing trends are presented here.  46  orbits  Figure 3-4 Correlation between damped response results as predicted by the closed form solution and numerical integration of the governing equations The damping coefficient Cj corresponds to 1% of critical damping  47  3.3.1  Numerical Data  In the following simulation results, the period of the circular orbit is assumed to be 100 minutes. The platform mass is taken as 214,000kg while the manipulator system has a mass of 3200kg. The length of the platform is 115m, compared to the full extension length of 15m for the manipulator. Cylindrical geometry, with an axial to transverse inertia ratio of 0.003, is used for calculating the inertia of the platform and manipulator links. When not specified explicitly, the mass and length of the manipulator are assumed to be evenly distributed between the two links, with the mobile base and joints massless. Furthermore, the payload is considered as a point mass unless stated otherwise. The mass of the payload is often expressed as a ratio of the manipulator mass (3200kg). The elastic character of the manipulator links is determined by their flexural rigidity (EI) which is assumed uniform. In expressingflexuralrigidity, it is more convenient to use the nondimensionalized form of natural frequency LJT,, EI(kl)  4  m£  '  3  where kl is equal to 1.875 for the first free cantilever mode. Therefore, given the natural frequency of the first free cantilever mode, the corresponding EI for the beam can be obtained from EI = w?  7.  1.875  L  4  For example, using the aforementioned inertia values for the manipulator arm, a natural frequency of UT, = 1 rad/s will correspond to an EI of 5.5 x 10 Nm . When 4  2  not specified, only the first free cantilever mode is used to discretize the transverse link deflection. To be consistent, the spring stiffness of the joints (KJI, Kj ) and the damping 2  coefficients (CJI, Cj ) are also defined in terms of natural frequencies. The model 2  48  here is that of a beam attached to a torsional spring: 1 + Cji + KJI = 0.  where m£ /3 is the inertia of the beam about the joint. Comparing this with the 2  standard second order equation 7 + 2C}wj<7 + w}7 = 0;  the natural frequency of oscillation u>j and nondimensionalized critical damping factor Cj are given by,  The advantage of expressing these parameters in terms of natural frequency is that it provides immediate insight as to the type of response expected. Two time histories are considered for maneuvers: cubic and sinusoidal on ramp. Figure 3-5 shows the displacement, velocity and acceleration associated with these profiles. Despite similarities between the displacement and velocity time histories, the acceleration characteristics are drastically different. The sinusoidal acceleration is 'smooth' with zero initial andfinalvalues while the cubic function reaches maximum acceleration at these points. In order to minimize excitation of theflexiblesystem, the sinusoidal profile is used for the maneuvers unless stated otherwise. 3.3.2  Rigid One-Link System  In a one-link system, the mass and length of the single arm are m  L  = 3200 kg,  l  L  = 15 m.  For a rigid manipulator system, the only degree of freedom is the planar librational pitch response. With the inertia values selected, the system can undergo sustained periodic oscillations in the gravity gradient orientation. Consider the system with 49  Legend Sine-ramp Cubic  Time  Figure 3-5 Normalized time histories of the sine-ramp and cubic maneuvering profiles in normalized time showing displacement, velocity, and acceleration.  50  the station aligned with the local vertical and the manipulator located at the station center with the arm extended in the local horizontal direction. In this configuration,  tb = 0 is the equilibrium point for the pitch oscillation. The system is subjected to several different initial disturbances in pitch angle to observe the steady state response. Figure 3-6 shows, for the undamped system, periodic oscillations with amplitude set by the initial conditions. The frequency of oscillation, however, depends slightly on the initial condition. Note that the period of the librational oscillation increases with amplitude, from 0.58 orbit for xp = 5° to 0.62 orbit for xb = 30°. This dependence of the response on initial conditions is attributed to the nonlinear nature of the system. To assess the effect of maneuvers on the station libration, the dynamic response for various combinations of translation-slew maneuvers and payload are simulated. All maneuvers start with the configuration described above unless stated otherwise. Using the sinusoidal on ramp profile for the specified coordinate, the maneuvers are specified by their magnitude and speed or duration (time to finish). Figure 3-7 attempts to assess the effect of duration of a translational maneuver on the station libration. A translation of 20 meters along the station in completed in time r, ranging from 0.01 to 0.8 orbit (1 to 80 minutes). Note that a short duration requires more energy to execute the maneuver, hence the disturbance on libration is generally greater for smaller r's. The resultant platform librations are not symmetric with respect to tb = 0 since it no longer represents the equilibrium state after the maneuver due to the change in system inertia. In fact the oscillations now take place about xb = —0.11°. Applying a 20 meter translation in the other direction (Figure 3-8) yields a similar response plot, but now inverted with the new equilibrium at xb = +0.11°. The effect of translation magnitude on the station response is displayed in Fig51  CD  to  1^  -40  Orbit Figure 3-6 Response of the rigid one-arm manipulator as affected by magnitude of the pitch disturbance. Note the period of oscillation lengthens with an increase in the initial condition due to the nonlinear characteristics of the system.  0  0.5  1  1.5  T =  0.01  T =  0.10  T =  0_.5 0_  T =  0.80  2  Orbit Figure 3-7 Librational response of a rigid one-link manipulator to a translational maneuver of (hj, : 0 —• 20m). Note the equilibrium point after the translation is at ^ = —0.11°.  Cn  T  =  0.01  T  =  0.10  T  =  0.50_  T  =  0.80  Orbit Figure 3-8 Librational response of a rigid one-link manipulator to a translational maneuver of (hb: 0 — > —20m). Now the equilibrium shifts in the opposite direction to ip = +0.11°.  ure 3-9. A duration of r = 0.01 orbit (1 minute) is used and the translation distance is varied from 10m to 50m. As can be expected, a longer displacement in a predefined time, corresponding to a higher translation speed, causes a greater disturbance. From Figure 3-9, the maximum pitch amplitude is 0.18° for the 10 m translation, increasing to 1.5° for 50m. Next, the attention is focused on the influence of slew maneuvers. With the mobile basefixedat the center of the station, the manipulator arm is slewed through 180° (PJ^.  —90° —• +90°) with the maneuver beginning and ending perpendicular  to the local vertical. Several slewing durations (r = 0.01, 0.1, 0.5, 0.8 orbit) are used to assess the effect of slewing speed on the response. The results in Figure 3-10 show essentially the same trend established earlier by the translational maneuvers (Figure 3-7) - a faster maneuver induces a greater libration amplitude. The pitch response is now symmetrical w.r.t. xb = 0 because, unlike the translation disturbance, this slew maneuver does not alter the equilibrium state. It is of interest to recognize that as the arm swings in a counterclockwise sense, the station rotates clockwise (corresponding to negative xb) to conserve the angular momentum. It is likely that during construction and operation phases of the Space Station, the MSS may be called upon to load or unload Shuttle cargo, or transfer a payload from one location to another on a periodic basis. Depending upon the frequency of the periodic maneuver, this may lead to a beat response or even resonance. This is clearly indicated in Figure 3-11 for both translational and slewing periodic maneuvers as the manipulator performs the task in periods of 0.01, 0.5, and 0.6 orbit. As the maneuver period approaches the natural period of the pitch, which for small pitch amplitudes is approximately 0.58 orbit from Figure 3-6, a beat type response is observed. Operating at a frequency quite close to the natural frequency leads to instability of the platform. Figure 3-12 shows the effect of payload mass as the manipulator executes a general 55  Orbit Figure 3-9 Effect of the translational maneuvering speed on the librational response. The rigid one-link MSS travels a distance of 10, 20, 30, 40, and 50m in 0.01 orbit (1 minute).  oo  5==o Si]  CD  Legend T  =  0.01  r = 0.10 T  =  0.50_  T  =  0.80  Orbit Figure 3-10 Librational response of the rigid one-link manipulator during a slewing maneuver through 180° (h = 0; fij : —90° — > 90°) completed at several speeds. b  x  Figure 3-11 Effect of periodic maneuver on librational stability showing beat type and resonance responses: (a) translational maneuver between h = 15m and h = -15 m; (b) slewing maneuver between = 0 ° and pj = 180°. The pitch period is 0.58 orbit compared to the maneuver periods of r = 0.01, 0.5, 0.6 orbit. b  b  x  10  CD  Legend  CO  m —p = 0  Q2, = 2_  -10  -f  m = 5 —p m = 10 —p  Figure 3-12 Effect of the payload mass on the librational response of a one-link rigid manipulator executing a simultaneous translational and slewing maneuver (hb: 0 -»• 30 m; : 90° -»• 0°).  translational and slewing maneuver. Here the one-link mobile manipulator translates a distance of 30m and the arm simultaneously slews through 90° in 0.01 orbit. The payload mass is expressed as a ratio to the mass of the manipulator (3200 kg). As can be expected, a larger payload produces a greater pitch amplitude. A payload of 32,000kg, 10 times the manipulator mass, causes a maximum platform pitch response of 8.4° while the maximum amplitude attained without a payload is merely 0.6°. 3.3.3  One-Link Flexible System  The next logical step would be to assess the influence of the armflexibilityduring translational and slewing maneuvers. The maneuver considered here involves a 30 m displacement of the mobile base and an angular rotation of the arm through 90° (hb'. 0 ->• 30m; ySjj: 90° -> 0°) in 0.01 orbit. A 32,000kg payload is purposely considered to simulate extreme effects. Figure 3-13 shows the librational response for several values of the arm stiffness given in frequency (as a cantilever infirstmode): 1.0, 1.2, 1.5, 2.0, and 5.0 rad/s. The rigid arm response case (UT, = oo) is also included to facilitate comparison. Note, the flexible pitch response lags the rigid case initially until approximately the mid-point of the maneuver when it starts to overshoot. After the maneuver is completed in 0.01 orbit, the pitch exhibits a high frequency oscillation about the rigid response. Therefore, introduction offlexibilityin the manipulator link only modulates the rigid arm pitch response with a superposition of a low amplitude, high frequency component.  With an arm stiffness of 5 rad/s, the two responses  are found to be almost identical. For this maneuver and payload combination, an arm stiffness corresponding to a frequency of 1.0 rad/s is rather 'soft' and leads to a significant level of librational motion which would be unacceptable for most applications. The high frequency modulation of the librational response is the result of the arm deflecting from its nominal position. As the link becomes moreflexible,the increased 60  v  i  I  l  0.000  0.005  0.010  :  I  I  0.015  0.020  :  I  f  0.025  0.030  Orbit Figure 3-13 Effect of the armflexibilityon pitch response as the single link manipulator executes the combined translational and slewing maneuvers (hb : 0 30 m; : 90° -> 0°). The payload mass of 32,000kg is 10 times the manipulator mass (m = 10). p  deformation leads to greater payload positioning error. For a one-link manipulator, the payload error is the total tip deviation of link 1 w.r.t. the reference position. Only the transverse component of this error, denoted by ei, is significant. It is the sum of the joint and link deformations,  1  Figure 3-14 assesses the effect of link flexibility on the payload error for the above one-link rigid joint system executing the same maneuver. The time histories of the generalized link deformation coordinate 61 and tip deflection £1 as well as the libration tb have significance dependence on the arm stiffness. For the rigid joint manipulator considered here, £\ has contribution from only the link deflection, hence the two have identical forms. Although the error is found to be large during the maneuver, it reduces in the post-maneuver phase when the system oscillates at its natural frequencies subjected to the initial conditions at the end of the maneuver. For a 1.5 rad/s link, the maximum payload error is 2.4 m during maneuver and 1.0 m in the free oscillation phase. With an arm length of 15 m, these correspond to errors of 16% and 7%, respectively. Now the joint compliance is introduced into the manipulator. In analyzing the coupling between the link-joint vibration and libration, the system is first excited by initial conditions in the nominal gravity gradient orientation. Figure 3-15 shows the time histories of the libration tb, joint deflection 7 ^ , and link deformation generalized coordinate 8\ when subjected to an initial condition of 10° joint deflection ( 7 ^ = 10°). Both the joint and link stiffnesses are selected to be 0.2 rad/s. It is apparent that the responses are made up of two frequency components present: a low frequency of 128 cycles per orbit (cpo) and a high frequency of 1400 cpo, corresponding to 0.134 rad/s and 1.47 rad/s, respectively. Thus one frequency is lower than the component natural frequencies of 0.2 rad/s while the other is substantially higher. The two vibration related coordinates exhibit similar behavior. The joint 7.7. and link 61 deflections 62  0.0  1  /A  J T  V  -0.2 5  1  \ ^ /  0  (m)  \  XJ  Legend  7 \ Mi  = 1,2r/s  cj] = 1,5r/s ej,. = 2J)r/s  -5 0.000  Mi = 5.0r/s  0.005  0.010  0.015  0.020  0.025  0.030  orbits Figure 3-14 Effect of the arm flexibility on the pointing accuracy of the one-link manipulator showing generalized coordinate corresponding to link deformation 6\, in dimensionless units and the tip deflection £ in meters. 63  Parameters 0.2 rad/s 0.2 rad/s  Jl  Ll W  J2  oo  =  oo  L2  = 0 ™L2 =  0  = 0  m, = 0 L,"o = 0 L2  U  P  =°  0 -> 0 m 90° -> 90° 0° -» 0°  hi T  L  on  Maneuver History  B =  T  J1 = J2= T  0  0  1  Initial Conditions - F i r s t Free Mode ,(0)=10° 7  0.03  0.000  0.002  0.004  0.006  0.008  0.010  orbits Figure 3-15 Free oscillation of the flexible one-arm system with an elastic joint. The initial vibrational disturbance at the joint of (IJ = 10°) significantly excites thefirstmode of the the link deflection 6 . X  X  64  are in phase at the low frequency but 180° out of phase at the high frequency. The system pitch response is confined to the low frequency and is out of phase with the vibrations. The librational natural frequency of 1.72 cpo, which is much lower than the two observed frequencies, is not visible. With a pitch initial condition of ip = 10° (Figure 3-16), the librational response has a period of 0.58 orbit as in the case of the rigid system. The amplitudes of 7 ^ and 6\ are 0.005° and 5 x 10 , respectively. The two frequency components present now -5  are the librational frequency at 1.7 cpo and the lower vibration frequency of 130 cpo observed before. All the degrees of freedom are in phase at the librational frequency. As before, 7J and 61 are also in phase at the 130 cpo frequency. The high vibration x  frequency of 1400 cpo, prominent in the case with vibratory initial condition, is not excited here by the librational initial condition. Next, the jointflexibilityis varied to investigate its effect on the libration and positioning error during the same simultaneous translational and slew maneuver, h : b  0j : t  0 —• 30m,  r = 0.01 orbit;  90° -»• 0°,  T = 0.01 orbit;  with a payload of 32,000kg (Figure 3-17). The link stiffness isfixedat 2 rad/s while the jointflexibilityis varied. Five different values of joint stiffness wj are considered: 1.5, 2, 4, 8, and 00 rad/s. Note that even with a massive payload, the effect of jointflexibilityon the pitch response is virtually negligible if it is higher than the arm frequency (Figure 3-l7a). On the other hand, it affects the payload position error significantly (Figure 3-17b). For a joint stiffness around 10 times greater than the link stiffness, the response was virtually the same as that of a rigid joint system (Figure 3-18). The same was also found to be true for the arm stiffness approximately 10 times higher than the joint stiffness. Obviously, the information can be used to advantage during the MSS design. 65  n  0.00  i  0.05  i  0.10  orbits  i  -0.15  r  0.20  Figure 3-16 Free oscillation of the flexible one-arm system with the joint degree of freedom. The initial condition is a pitch displacement of 10°. 66  * -I  0.000  1  1  1  1  I  I  0.005  0.010  0.015  0.020  0.025  0.030  Orbit Figure 3-17 Effect of joint flexibility on the system response of one-link manipulator with an arm stiffness corresponding to 2 rad/s. Maneuver and payload are the same as before: (a) librational response, (b) payload deflection error. 67  1.5  QX  A  — * h  0  —  £  0.5  A f\ / A  JV ^ A  0  V  h v /  /A> / A ii  ¥ vV  \ \ // * \  -0.5  M  /  1 Legend  ' 1  •  <y, - 10 r/s  ,  0.000  0.005  0.010  0.015  0.020  0.025  0.030 ,  Orbit re 3-18 Comparison of payload deflection £ of the one-link manipulator with a 'stiff' joint (uj = 10 rad/s) and a rigid joint (uj = oo). Arm stiffness is 2 rad/s.  Until now, only the first mode is used to model the link deformation. To investigate the significance of the higher order modes, the response of the system with the same maneuver and payload (32,000kg) is now studied using the first five modes. Both the joint and link stiffness values are set at 2.0 rad/s. Figure 3-19 suggests that there is virtually no difference in the pitch degree of freedom. However, the higher modes affect the payload position slightly, increasing the amplitude of deflection by a maximum of about 5% during the maneuver. A small deviation in the 6\ time history indicates that only a small fraction of the energy of the maneuver is transformed into oscillation at the higher modes. To assess the relative significance of the higher modes in modelling the system, simulation results employing one to five modes are presented together in Figure 320. Note, contribution from the third and higher modes is negligible. Therefore, the first two modes are sufficient to describe the link flexibility for this set of maneuver and payload conditions. The relative insignificance of the higher order modes can be attributed to the 'smoothness' (zero initial andfinalaccelerations) of the sine-ramp profile used for the maneuver. As a matter of interest, the computation of the two mode case took 42 seconds of CPU time while the five mode simulation demanded 297 seconds on an IBM 3070. 3.3.4  Two-Link Flexible System  Finally, the manipulator is treated as a flexible two-link system. The Space Station based MSS system now has a total of (3 + n\ + n ) degrees of freedom corresponding 2  to the libration rp, elastic rotations at the joints  7J ), and the number of modes 2  used to model the transverse link deflections (6u, 6 ). Two coordinates are required 2t  to describe the two-link manipulator payload position error which is dependent on the configuration of the two arms as well as the deflections of the deformable bodies. Instead of expressing the payload error in terms of system frame (x, y) coordinates, 69  Parameters W  J1  "Ll W  =  =  J2 =  "L2 =  Q on  Maneuver History  2.0 rad/s 2.0 rad/s  fill  CO  $12  oo  0 90° 0° JI  30 m 0° 0° T = 0.01 J 2  /lb  0 m, = 0 0 L2 = 0 "12 = = 10 ™3  =  L  "V  o.ooo  Initial Conditions - 1 to 5 Modes A l l 1C = 0  0.005  0.010  Q= ud  0.020  0.015  orbits Figure 3-19 A comparison between the system response obtained using one and five modes to model the link deflection. The manipulator executes the same maneuver as before with a 32,000 kg payload (m = 10). p  70  W  J1  =  W  L1  =  "J2 = L2  mg  2.0 rad/s 2.0 rad/s oo  fill T  "12 =  0  m, L  L2  = 0 =  0 90° 0°  0J2  =  =0  Q on  Maneuver History  Parameters  0  B  =  T  Jl  30 m 0° 0°  = J2= T  0  0  Initial Conditions - 1 to 5 Modes All IC = 0  1  a  Legend  -0.05 0.000  1 mode 2 modes 3 modes 4 modes 5 modes  0.005  0.010  0.015  0.020  orbits Figure 3-20 Relative significance of the higher modes on the system response for the same maneuver and payload mass. 71  it is more informative to consider the total tip deviations of the two arms  (E\,E ). 2  EI = li sin'yjrj + ^ t W i » ( i ! i ) 1  n  2  £2 = £2 sin(a i + 7J ) ]jP 6 <f> (£ ) L  2  2i  2i  2  1  As before in the one-arm case, the total tip deflection or pointing error of arm 1, £1, is dependent on the deformations of joint 1 and link 1. The definition of E , 2  however, includes the rotation at joint 2 due to deflection of link 1 (ar,i) as well as the deflection of joint 2 and link 2. The actual payload position error is a function of these two deviations and the manipulator geometry. Figure 3-21 shows the response time histories of the two-link flexible system executing the same translational and slew maneuvers performed in the one-link case: £ :0->30ro, b  pj  x  : 90°  0°,  /?j = 0° 2  in 0.01 orbit with a 3200kg payload. Natural frequencies of the joints and links are set to 2 rad/s: IJJJI =  2rad/s,  ULI =  2rad/s,  u>j = 2rad/s,  WL2 =  2  2rad/s.  The platform pitch (xb), tip deflections of the two arms (E , E ) as well as the time X  variations of the flexibility generalized coordinates (7J  1?  2  611, 7J , £21) are presented 2  thus providing a comprehensive picture of the system behavior. At the outset it is apparent that the system undergoes significant librational motion which may affect operation of the on board antennae, telescopes and other equipment demanding a high degree of pointing accuracy. As observed in the one-link flexible manipulator case (Figure 3-19), the arm deflections reach peak values during the maneuver and settle down to smaller amplitude free oscillations in the post-maneuver phase. The generalized coordinates also show the same trend. This is attributed to the peak maneuvering accelerations of the sinusoidal profile which occur at times corresponding 72  Parameters  Maneuver  uj | = 2 . 0 r a d / s t>L,  = 2 . 0 rad/s  u  = 2 . 0 rad/s  J Z  CJ  L Z  =  2.0  = 0 n\2  =  1  0 -> 3 0 m  fill fiiZ  rad/s  m. U  L2  on  History  T  = 0  B =  T  Initial -  J  9 0 ° -»  0°  0° -  0°  =  L  T  j  = 0.01  2  Conditions  First mode  -  A.  Cn  A l l IC = 0  -01 CO  o.ooo  0.005  0.010  0.015  0.020  0.000  0.005  0.010  0.015  0.020  orbits orbits Figure 3-21 System response of the two-linkflexiblemanipulator for a simultaneous translation of the mobile base and slew at joint 1.  to 0.025 and 0.075 of an orbit (the maneuver is completed in 0.01 of an orbit). The maximum tip deviations of arms 1 and 2 (ei,e ) are approximately 2 m compared to 2  the 7.5 m length of each of the links. Maximum elastic joint deflections are 5° and 2.2° for  and qj respectively while the generalized coordinates for the first mode 2  deflection of the links (611,621) reach peak values of 0.067 and 0.025 respectively in dimensionless units. It is of interest to recognize that the significantly larger deformations of arm 1 (8\ 1, IJ ) as compared to those of arm 2 are attributed to the larger X  effective load associated with the former (outer arm + payload). For a large translational maneuver of the manipulator mobile base, the shift in the system center of mass can be significant and it may influence the dynamical response, especially if a heavy payload is involved. Figure 3-22 assesses the effect of a center of mass shift on the system response for the same maneuver and payload configuration as in Figure 3-21. Note the effect appears to be confined to the station libration if) only, and even here the deviation is rather small. Ignoring the cm. shift increases the pitch amplitude by approximately 3.3% in this case. Accounting for the shift in cm. leads to an increase in the effective system inertia which results in smaller librational motion. The vibrational response remains essentially unaffected by the center of mass displacement. This is important as the shifting center of mass, particularly due to flexibility, significantly complicates the governing equations of motion thus increasing the computational effort. The system response to a slew maneuver applied at the second joint is presented in Figure 3-23 with the same stiffness parameters as before but no payload. The second arm slews through a total angle of 180°: h = 0m, b  0^=90°,  pj : +90° 2  -90°;  in 0.01 orbit. Without the payload, the libration if) reaches a maximum of 0.04° as the maneuver is completed and effects offlexibilitymodulation are not noticeable. 74  Maneuver  Pa r a m e t e r s = 2.0  Uj,  rad/s  u  = 2.0 rad/s  u  = 2.0 rad/s  Ll  JZ  u  L 2  =2.0 = 0  n\2  =  1  0.000  rad/s mj  Li  =  o =  History  90° 0 ° -»  T  B =  Tj,  Initial -  A/,  0 -» 3 0 m  B  =  0°  pn  /  0° 0.01  TJ 2  Conditions  First mode  Q  -  o  U  p  A l l IC = 0  0.005  0 010  0.015  0.020  0.000  0.005  0.010  0.015  0.020  orbits orbits Figure 3-22 Effect of the shift in center of mass as theflexibletwo-link manipulator performs a simultaneous translation and slewing maneuver with a 3200kg payload.  -0.20.000  1  0.005  1  0.010  —i— 0.015  0.020  -1.00.000  0.005  I 0.010  orbits orbits Figure 3-23 Response of the two-link flexible manipulator executing a 180° slew at the second joint.  — i —  0.015  0.020  The amplitudes of the vibrational degrees of freedom are rather small with maximum values of 7J  X  = 0.29°,  $11 = 3.4 x 10~ ,  -yj = 0.08°,  3  6  2  = 7.8 x 10~ ; 4  2X  resulting in arm tip errors of only 0.1m. Despite the deflections 7J and 621 being 2  much smaller than their arm 1 counterparts, the error e is comparable to e i because 2  the definition of e includes the rotation of link 1. Of particular interest is the presence 2  of high frequency modulations in the responses of the second link and joint degrees of freedom. This high frequency component is excited during the maneuver and persists in the post-maneuver free oscillations. When the manipulator executes a general maneuver involving a slewing motion of the second link, the induced moment may attempt to bend the first link in such a way contrary to that imposed by the first mode deflection assumed so far. Hence higher order modes are necessary to accurately describe the structural deformation of the link. Figure 3-24 assesses the effect of the higher order modes on the dynamical response for a general maneuver involving translation and slewing at the two joints: £ :-15^15ro, 6  /3J : 135° -> 45°,  0j : 45° - • -45°;  X  2  in 0.01 orbit. As before, the links and joints have natural frequencies of 2 rad/s and payload mass is zero. The system responses employing up to 3 modes for discretizing the link 1 deflection are presented. Similar to the results obtained in Figure 3-20 which investigates the significance of higher order modes on a one-link manipulator, the responses obtained in Figure 3-24 using 2 and 3 modes are very close, sometimes indistinguishable. Reg-ardJ^ss "the number of modes used for discretization of link 1 deflection, the maximum responses are: tb = 0.31°, 7jj = 0.85°,  E = 0.28 m, X  *n = 0.0106,  e = 0.28 m; 2  7J = 0.21°, 2  77  6  = 2.06 x 10 ; -3  21  oc ; 0.000  i —  0.005  0.010  0.015  0.020  —  -1  -z.o i 0  0  0  i — 0  0  0  0  5  i  i 0 0  1  0  0  0  1  5  °  (  orbits orbits Figure 3-24 Effect of considering the higher modes of the first link on the response of the two-arm flexible manipulator. The general maneuver involves a translation of the mobile base ,and slews at the two joints.  during the maneuver. The post maneuver vibrational amplitudes reduce by approximately 50% when more than one mode is considered. The effect of higher order modes is most prominent in the pointing error of arm 2 e though little change appears in 2  e\. With 2 modes, e reaches 0.30m, an increase of 8% compared to the 0.277 m 2  obtained using one mode. Obviously this is attributed to the excitation of the second mode of link 1, 6 , which has a large contribution on the slope of the link 1, an, 12  thus influencing the pointing accuracy of the second arm. 3.3.5  Effect of Maneuver Profile  The sinusoidal-ramp profile has been used exclusively for the maneuver time histories of the previous dynamics simulations. From Figure 3-5, the cubic profile is more abrupt with nonzero initial andfinalaccelerations thus imparting more excitation to theflexiblesystem. The responses of theflexibledegrees of freedom (7J 6LI, 15  7J > 2  f>L2)  obtained before approximately follow the acceleration time histories of the  maneuver coordinates (a sine curve) during the maneuver. The amplitudes decrease significantly when the manipulator becomes stationary in the free oscillation phase where the amplitude is determined by the initial conditions present at the completion of the maneuver. Therefore, a cubic profile for the maneuvering coordinates, with its nonzero initial andfinalaccelerations, can influence both phases of the responses of the degrees of freedom pertaining to flexibility. The MSS system in Figure 3-24 is simulated again using a cubic time history in place of the sinusoidal for the simultaneous translational and slewing maneuvers. Fig-:xre 3-25 compares the responses given by the sinusoidal and cubic maneuvering functions. At the outset it is apparent that the system responses are significantly different. During the maneuver, the vibrational coordinates (7J  1?  6LI, 7 J , 6L ) and 2  2  arm pointing errors (t?i, e ) follow the acceleration of the maneuver, which in the cubic 2  case is a straight line. The peak amplitudes of t?i and e during maneuver increase 2  79  Parameters  Maneuver  History  Cubic =2.0  rad/s  =2.0  rad/s  =2.0  rad/s  =2.0  rad/s  = 0 "V2  = 1 = 0  k =1 L„  2  = 0  B =  15m  135° - 45° 45°  T  = 0  rrij  -15 -  B  Initial  —45° r  ' J I  0.01  J2  Conditions  — First mode  -  A l l IC = 0  00  o  0.000  0.005  0.010  0.015  0.020  0  0  0  0  0 0. 00 0 55  0  0  0  1  0  0.015  0.020  orbits orbits Figure 3-25 Comparison of the two-link manipulator system responses using cubic and sinusoidal profiles for the maneuver coordinates.  from 0.28roand 0.28 m to 0.35 m and 0.39 m, respectively. Unlike the sinusoidal case, the post maneuver free oscillations of the cubic response are of approximately the same magnitude as those during maneuver: 0.30 m and 0.34 m for E\ and e respectively. Contrasting these with the corresponding sinusoidal response pointing error amplitudes of 0.05 m and 0.05 m, the cubic maneuver indeed provides more excitation to the flexible MSS. The platform libration xb is also affected by the cubic maneuver because of its coupling with the vibrational motion. Since the cubic maneuver intensifies the disturbance on the flexible MSS, it is possible that the higher order modes are significantly excited. Figure 3-26 assesses the influence of the second and third mode of link 1 on the dynamic response of the above system subjected to cubic maneuvers. It is evident that the third mode has negligible contribution on the pointing errors (E\, E2), a conclusion reached earlier in Figure 3-24 which performs the same analysis with the sinusoidal profile. Contrary to the sinusoidal results, the higher order modes increase the post maneuver free oscillation amplitudes when a cubic time history is used. With the inclusion of the second mode, the steady state amplitude of ei increases by 17% to 0.35 m while E  2  reaches 0.43 m, an increase of almost 26%. The time histories of the vibrational coordinates, 7 ^ , 6n, 7J , and 621, exhibit the same trend when the second mode is 2  considered.  3.3.6  Effect of Modal Functions  In chapter 2.3 the modal functions for a free and tip loaded cantilever beam have been derived. Only the free mode shapes have been used thus far for simulating the MSS dynamics. As mentioned in chapter 2.3, the selection of mode shapes influences the number of them required to adequately model the response. In this section, the rate of convergence of the solutions obtained using the free and tip loaded cantilever modes are investigated. 81  2  1  |  0.000  0.005  .  !  |  0.010  0.015  fI  —  0.020  —  • I  -D.U' 0  0  0  I 0  0  °0  I 5  I 0  0  1  0  1 0  0  1  5  0  0  2  0  orbits orbits Figure 3-26 Significance of the higher modes of the first arm on the system response. The cubic maneuver profile is used.  Consider the flexible two-link manipulator described in the previous section executing the same general maneuver involving translation of the mobile base and slew at the two joints repeated here for clarity: h: b  -15 -> 15m,  0j  x  : 135° -> 45°,  /3j : 45° - • -45°. 2  The cubic profile is used to impart more excitation to the flexible system. The natural frequencies of the two links and joints remain at 2 rad/s and the payload is zero. Damping is also considered to illustrate its effect on the system response. wji = 2rad/s,  U>LI  = 2rad/s,  u)j = 2rad/s,  U>L,2  2  Ch = 0.1,  = 2rad/s;  C* = 0.1; J2  Figure 3-27 shows the system responses using 1 (ni = n = 1) and 3 (rii — n = 3) 2  2  free cantilever modes to discretize the transverse deflection of each link. For the (3, 3) mode case, there are 9 generalized coordinates in the system corresponding to 7Ji»  6 u , *>12, 6 , 13  7J , 2  6, 2i  6 2, 2  623.  As observed before, the magnitudes of the pointing errors of arms 1 and 2 (ei, e ) 2  increase when more discretization modes are used. The corrections can be as large as 25% for £"i and 37% for e . Because of the presence of damping, the high frequency 2  component prominent in the previous responses of 7J and 6 2  2i  in Figures 3-25 and 3-  26 are merely observable here. Since the loaded cantilever mode shapes appear to be the same regardless of the tip load mass (Figures 2-4, 2-5), the mode shapes corresponding to a unity payload/arm mass ratio are used to illustrate the convergence of the solution. The simulations in Figure 3-27 are repeated in Figure 3-28 using 1 and 3 loaded cantilever modes (ni = n = 1 and n.\ = n — 3) to describe the deformations of the two links. It 2  2  is apparent that the corrections when the higher modes are accounted for are much smaller than those encountered in the free cantilever mode case. The increase in amplitude of tri is 7% while e changed by 18%. 2  83  Parameters  Maneuver  History  Cubic  "Jl w U  J Z  L2  = 2.0  rad/s  =2.0  rad/s  = 2.0  rad/s  =2.0  rad/s  -15  B  I ft 2 B = 7  -  l&n  135° - 4 5 °  ft  45° Tjl  ma  = 0  Initial  "\? mp  = 1 = 0  -  =  —45° T  , = o.Ol  '| J 2  Conditions  1 and 3 Modes  -  A l l IC = 0  0.005  0.010  0.015  0.020  0  0  0  0  0 0. 00 0 55  0  0  0  1  0  0.015  0.020  orbits orbits Figure 3-27 Comparison of the damped responses using 1 and 3 free cantilever modes to discretize the deflections of the = 0.1). two links. The joint damping coefficients are: (C = 0.1; J 2 JX  U  Parameters  Maneuver  History  Cubic  = 2 . 0 rad/s = 2 . 0 rad/s = 2 . 0 rad/s = 2 . 0 rad/s  I* fill  =  Initial  n\z = =  0 1  0  m,  T  = 0  kz = 1  -  B =  -15 135° 45° r  Jl  =  15m  -> 4 5 ° —45° T  J Z =  0  0  1  Conditions  I and 3 Modes  -  A l l IC = 0  00  0.005  0.010  0.015  0.020  -5.00.000  0.010  0.005  0.015  0.020  orbits orbits Figure 3-28 Comparison of the damped responses using 1 and 3 modes of the loaded cantilever modes to discretize the deflections of the two links. The joint damping coefficients are: (C* = 0.1; Cj = 0.1). JX  2  4.  LINEAR  ANALYSIS  A linear dynamic model of the Space Station based MSS system is developed in this chapter. With the assumption that the characteristics of the original nonlinear system are retained in the linearization process, the eigenvalues and eigenvectors of the linear system can give significant insight into the system response. Furthermore, the linear model is essential to the design of the optimal controller for suppressing the libration and vibration disturbances. 4.1  Linearization As stated before, the equations of motion derived in Chapter 2 for the Space  Station based MSS system are highly nonlinear. However, the flexibility coordinates (lJi> ^ L I ,  7J , 2  define only small deviations about the reference trajectory of the  manipulator. Hence higher order terms of the flexibility coordinates may be assumed small and omitted from the equations. To incorporate the nonlinear effects associated with large motion maneuvers, the governing equations of motion are linearized about the planned MSS trajectory consisting of the desired translation hb(t), as well as the slew at the joints, f3j and f3j . x  2  Thus the nonlinear equation of motion in Chapter 2, M(q,t)q+F(q,q,t) = Q  can be written, after linearization, as: M ( i ) q + H(r)q + K(*)q + F ( i ) = Q  with the large nonlinear motions involving the maneuver coordinates (hb,  and  (3j ) retained in the form of nonautonomous terms. 2  The linear equation consists of the familiar mass, gyroscopic, and stiffness matrices, denoted by M , H , and K , respectively. As in the nonlinear case, the mass matrix  86  M is symmetric and positive definite. Coriolis and other velocity related forces due to orbital and maneuver motions are included in H . Flexibility effects, orbital and maneuver accelerations, as well as gravity potential terms are considered in matrix K . In the absence of maneuvers, H and K are skew symmetric and symmetric, respectively. The vector F consists of the forces attributed to the orbital motion and maneuvers independent of the generalized coordinates. Finally, Q represents the generalized force vector as before.  4.2  Eigensolutions The solution of a linear system can be obtained analytically by determining its  eigenvalues and eigenvectors which together completely characterize the response of the system. Of particular interest are the eigenvalues since their imaginary components correspond to the natural oscillation frequencies and the real components indicate the stability of the system. In this section, the eigenvalues for several system configurations are obtained to yield "the corresponding natural frequencies of oscillation. Effect of the payload mass on the eigenvalues is also investigated. In the absence of damping, all eigenvalues are purely imaginary, indicating sustained oscillatory responses without any amplitude variations. For a flexible two-link rigid joint system, there are (l + n i + n.2) degrees of freedom (Vs &Li, ^Li).  Considering only the first mode of deflection for each link, the linear  system has six eigenvalues. The eigenvalues are imaginary conjugate pairs, hence only the positive imaginary components are shown. Setting the stiffness of the links at 1 rad/s,  WJI = wj2 = 00;  un = UL2 = lrad/s 87  —* 955cpo;  Eigenvalues of the rigid joint system in the orientation corresponding to:  hb = 0;  fij  x  = 90°;  0j  2  = 0;  with a payload mass (m ) of 0 to 5 times the manipulator mass (3200kg) are shown p  in the table below.  Table 4-1  Effect of the payload mass on the system eigenvalues  0  1  2  5  Al,2  1.728  1.723  1.718  1.704  ^3,4  247.0  As.6  2000.  111.0 1340.  83.30 1300.  54.70 1270.  m  p  The eigenvalues have the dimensionless units of cycles per orbit. The lowest pair of eigenvalues (Ai 2 « 1.7cpo), essentially the librational frequency, are least influenced (  by the variation in payload mass. The higher pairs (A 4, Xs,6) decrease with the 3)  addition of the payload as the tip mass effectively makes the system more flexible thus reducing its natural frequencies. The middle frequency (\3,4) is always below the component natural frequency of the links (955 cpo) while the higher frequency (As.e) exceeds it. Allowing the joints to be compliant, the two-link flexible system has (3 + tii + n ) 2  degrees of freedom. The eigenvalues of this system with the stiffness properties of  (JJJI = Uj2  = lrad/s;  WLI  = ur,2 = lrad/s  —> 955cpo;  are, as before, pairs of imaginary conjugates. Table 4-2 presents the eigenvalues for this case in the standard gravity gradient configuration. 88  Table 4-2 Effect of payload mass on the system eigenvalues in the presence of joint flexibility 0  1  2  5  Al,2  1.728  1.723  1.718  1.704  ^3,4  196.0  90.40  67.50  44.40  As,6  1430.  1010.  982.0  963.0  ^7,8  5370.  4920.  4890.  4870.  ^9,10  12300  11000  11000  11000  Again, the lowest frequency (Ai^) is not affected by the payload mass. In fact, it is the same as the previous case with rigid joints. The remaining eigenvalues indicate that the system vibration can occur at frequencies much higher than the component frequency of 955 cpo chosen for the links. Using the same system configuration as before, the link and joint stiffness values are increased to 3.14 rad/s from 1 rad/s. Table 4-3  Effect of an increase in the link and joint stiffness on the system eigenvalues  0  1  2  5  Al,2  1.728  1.723  1.718  1.704  •^3,4  617.0  284.0  212.0  140.0  ^5,6  4500.  3170.  3090.  3030.  ^7,8  16900  15400  15400  15300  Ag.lO  38700  34700  34500  34400  The eigenvalues for this stiffer system, except for the lowest pair  (Ai^),  are greater  than those obtained before by approximately the same factor as the increase in component stiffness frequency values. For example, A3 4(m = 0) of 617 cpo is 3.148 times )  p  the previous value (Table 4-2), and Aa io(»Tip = 0) of 38700 cpo represents an increase (  by a factor of 3.146. 89  4.3  Linear Response Accuracy of the linear model is assessed by comparison with the nonlinear re-  sponses. As seen in the closed form analysis of the flexible arm-joint system in Chapter 3.2, it has been found that the linear solution always predicts a slightly higher frequency of oscillation than that obtained through numerical integration of the nonlinear equations. Although the nonlinear analytic analysis provides a frequency perturbation due to the nonlinear terms, there still exists phase differences in the higher frequency component. Hence it can be expected that the same problem of frequency and phase discrepancies, particularly at the higher frequencies, will persist here in the strictly linear solution. Consider the free oscillation of aflexibletwo-link system in the reference gravity gradient orientation subjected to an initial condition of $u(0) = 0.05 with the following stiffness parameters, WJI = lrad/s  Uj = Iradjsu^i  = lrad/s  2  U!r, = 2  lrad/s,  and no payload. From the eigenvalues in Table 4-2, the natural frequencies of this autonomous system are  1.728, 196, 1430, 5370,  and  12300  cpo. Since the initial  condition is arbitrary, the corresponding modes should be excited. The nonlinear system response in Figure 4-1 shows that the 196 cpo frequency component dominates the time histories of all the degrees of freedom. In addition, the responses of E\, E , 2  7Jj)  5  a  nd  $21  contain a  5000  cpo component while the oscillation at joint  2,  7 J , oscillates at a higher frequency of 9800 cpo. As concluded in Chapter 3.2, the 2  nonlinear vibration frequencies are lower than the corresponding results predicted from the linear analysis. The system response using the linear model is shown in Figure 4-2. As in the nonlinear case, the primary frequency of oscillation is 196 cpo for the pitch xb as well as the vibration coordinates: E\, E , 7 J , $n, ~u , and $21- Superimposed on the 196 cpo 2  x  2  90  Parameters  Maneuver  History  Cubic uj  = 1.0  rad/s  uL | = 1.0  rad/s  «  = 1.0  rad/s  =  rad/s  }  J 2  " L Z mg ,  N  = 0 =1 = 0  L Z  rn,,  nrij  90°  T  -  0 1  B =  T j ]  Initial -  -. 9 0 "  0°  0JZ  L2 = Lp = 0  L  0 -> 0 m  B  ->  =  T  Conditions  First mode  6,,(0)  0°  „ = 0.01 ' J2  -  = 0.05  0.015  0.010-  r  0.005  0.000  0.002  0.004  0.006  0.008  0.010  0.000  0.002  0.004  orbits  orbits  0.008  0.006  _  0.010  Figure 4-1 Nonlinear free oscillation response for an initial condition of 6 = 0.05 corresponding to a 10% link deflection at the tip. U  U I 0.000  uui |  |  0.002  0.004  0.006  0.008  0.010  0  0  I 0  0  0  0  0  I 2  0  0  0  I 4  0  0  0  I 6  0  0  0  I 8  0  0  1  0  orbits orbits Figure 4-2 Linear free oscillation response for an initial condition of 6u = 0.05 corresponding to a 10% link deflection at the tip.  oscillations of Ei, £2, 7 J $11, and $21 is a 5370 cpo component. The response of 7J 15  2  contains a frequency of 12,300 cpo which is also present in $21- These are exactly the frequencies predicted by the eigenvalues. As in the case of analytical solution obtained in Section 3.2, the amplitudes of the linear and nonlinear responses match extremely well while substantial discrepancies exist in the higher frequencies values of the response. It is of interest to note that the 1400 cpo frequency mode predicted by the eigenvalues of the linear model is not detectable in either the nonlinear or linear response. Figure 4-3 compares the linear and nonlinear responses of the above system undergoing a 180° slew maneuver at joint 2:  hb = 0;  PJ = 90°;  0j  X  : +90°  2  -90°;  using the sinusoidal maneuver time history. During the maneuver, some high frequency components are excited, especially in ij  2  and 6 i. 2  However, in the post  maneuver free oscillation, only a low frequency of approximately 260 cpo is present. Since the high frequencies are only mildly excited in this case, the linear response matches the nonlinear one almost perfectly. In order to excite the higher frequencies by the maneuver, a cubic time history is used in Figure 4-4. The maneuver applied here is a general maneuver involving translation of the mobile base and slew at both joints, h :-15-> b  15m,  fa  x  : 135° -> 45°,  0j  2  : +45° -> - 4 5 ° ;  with no payload. Discrepancies at the higher frequencies are now clearly visible in the linear solution, for the forced response during maneuver as well as the free oscillations after the maneuver is completed. In the linear responses of 7 J and 6 i, the high 2  2  frequency component excited during the maneuver persists in the post maneuver vibrations with relatively no reduction in amplitude. However, these high frequency 93  Figure 4-3 Comparison of linear and nonlinear undamped response for the MSS executing a  orbits Figure 4-4 Comparison of linear and nonlinear undamped response for a general MSS maneuver consisting of simultaneous translation and slews at both joints. A cubic time history is used for all maneuver motions. o r b l t s  amplitudes are subdued in the nonlinear results after 0.01 orbit when the maneuver is completed. If damping is present in the system, the higher frequencies will decay faster and the response will be dominated by the lower frequency modes. Figure 4-5 compares the nonlinear and linear responses to the same maneuver with 10% critical damping at each of the two joints: C  dl  = 0.10;  C* = 0.10. J2  The correlation between the nonlinear and linear responses are improved now that the high frequency components are suppressed. However, some errors associated with the higher frequencies are still present initially until they are damped out after 0.015 orbit.  96  0.000  1  i  1  0.005  0.010  0 015  1'  0.020  orbits •  '  -u.ui-i 0  0  0  1 0  0  0  0  5  1 0 0  1  0  orbits  1 0 0  1  5  r 0  0  2  0  Figure 4-5 Comparison of linear and nonlinear damped response to the same maneuver as in Figure 4-4 with 10% critical damping at each joint (C = 0.1; C = 0.1). JX  J2  CONTROL  5.  The dynamic simulation results clearly suggest that the manipulator can deform significantly under fast maneuvers especially when a large payload is involved. The MSS maneuvers can also affect libration adversely. To improve the performance of the manipulator and maintain the station at the desired orientation, a centralized optimal controller is designed to simultaneously suppress the structural deformations and pitch disturbances caused by the maneuvers. 5.1  Control Inputs The Space Station libration can be controlled directly using thrusters and/or con-  trol momentum gyros to generate a restoring torque when the operating configuration is disturbed. To reduce the manipulator tip deviation from the planned trajectory due toflexibility,the joint motors can oscillate slightly about their nominal positions to compensate for any elastic deformations at the joints and links. In this way the link deflections are not controlled directly but indirectly through the coupling with the joint rotations. This is possible due to the strong coupling between the joint and link degrees of freedom as revealed in the dynamic analysis. Therefore control of the system can be accomplished using only three control inputs: station pitch, rb\ motor angle at joint 1, fij^, and motor angle at joint 2, (5j . 2  With the joint motor coordinates as controlled variables, their equations of motion are added to the dynamic model. Ignoring the stiffness and damping of the motors themselves, the equations can be written as:  IJ2~PJ  2  Here IJI and Ij  2  = T  J 2  + C ij J2  2  (5.1)  + K j. J2l 2  are the inertias of the joint motors, TJI and Tj  2  are the applied  motor torques at joints 1 and 2, respectively, and the remaining terms correspond to 98  forces imposed by the manipulator through the joint elasticity coordinate. The joint elastic degrees of freedom, 7 ^ and 7J , have equations which can be put into the 2  simplified form:  + C J 2 7 J + Kj2lJ  { i)lJ  I  <  2  2  2  + f(q,q,q)  =  It is evident that the system dynamics directly affect 7 ^ and 7J . The manipulator 2  dynamics influences the joint motor coordinates  and /3j only through coupling 2  with the elastic coordinates 7 ^ and 7J , respectively. In the case of rigid joints 2  (Kji = 00), 7 ^ is constrained to be zero and the equation of motion for /?j . takes t  the form of 7.7. which involves the Space Station based MSS dynamics explicitly. Including the /Jj.'s as degrees of freedom of the system, the vector of the generalized coordinates, q, increases in dimension by two and becomes q=[^»  Pjt* 7JJ>  *Li>  Pj , t  6i  TJ > 2  2  ] , T  having a total dimension of 5 + n\ + n . The nonlinear equations of motion have the 2  same form as before, M(q,t)q+F(q,q,t) = Q.  (5.2)  The mass matrix M and force vector F are still time varying because the translation of the mobile base remains a specified coordinate. As afirststep in assessing the performance of controlling the link deflections with the joint motors, the joint elasticities are omitted. The generalized coordinate vector of dimension (3 + nj + /i, ) can be written as: 2  q=[0.  0J  PJ ,  P  2  $L  )T  2  Note that the form of the nonlinear governing equation of motion remains unchanged. 99  5.2  State Space Representation The optimal controller, based on linear state feedback design, requires a linear  model of the system to be controlled. The equations of motion derived in Chapter 2 are highly nonlinear, particularly due to the large maneuver motion of the manipulator which dictates the configuration of the Space Station based MSS system. Because the nonlinear effects associated with large motion maneuvers are important to the system dynamics, the governing equations of motion are linearized about the planned MSS trajectory consisting of the desired translation h (t), as well as the slew at the joints, 0  /Jjj and (3j . The generalized coordinate vector q is now divided into two parts, q 2  0  and Aq, such that q = q + Aq, 0  0  0  hi 0  — * PJ  u S  0 fij 0  2  U  2a  2  ~* L  +  2  J  2  6L  —»  2  0  Here q represents the large planned motion of the system and Aq describes the small 0  deviations from it. Substituting q + Aq for q into the nonlinear equation (5.2) yields 0  M(q + Aq, t) (q, + Aq) + F(q + Aq, q + Aq, q + Aq, t) = Q. 0  c  c  0  Regrouping the terms and linearizing with respect to the deviations Aq, which are assumed small, the linear equations take the form M(q , r) Aq + H(q ,6^, t) Aq + K(q , q,, q,, i) Aq 0  0  0  + f(q ,q ,qo^)-Q 0  0  0  (5.3)  + AQ  with the large nonlinear reference motions of /?J and PJ 1o  2O  retained in the form of  terms involving q c^, and q,,. Given a specific maneuver to execute, these quantities OJ  become functions of time. 100  T h e mass matrix M remains symmetric and positive definite. A l l deviation velocity related forces are represented by matrix H while K includes all forces dependent on the deviation.  T h e excitation force caused by the maneuver independent of the  deviations is given by f.  Vector Q  is the nominal torque required for the planned  0  trajectory and is equal to f for a rigid system. T h e control torque calculated by the feedback controller, A Q , is the supplementary control effort necessary to counteract the induced vibration and libration. In designing the control law, the force vector f, which accounts for the disturbance induced by the nominal maneuver, is not considered. Since it is a function of time, dependent only on the reference maneuvers, it can be computed and compensated by Q . A c t u a l l y this compensation is only partial because there is no direct control 0  input for the 6T, degrees of freedom. In other words, Q {  0  is the nominal control force  required to compensate for the maneuver of a rigid system. Deleting f and Q  0  from  E q n (5.3) yields  M(q , 0  t) A q + H ( q , q^, t) A q + K ( q , c^, b^, t) A q = A Q D  0  (5.4)  which is the linear model for the controller design. T h e linear set of equations is rearranged into state space form for the controller design. Defining the state vector x to consist of the deviation vector a n d its velocity,  x=[  Aq ,Aq ] , r  T  r  the state space equation can be written as:  x = Ax + Bu  "Aq" A*.  "Aq"  0 - M  -  1  K  + Bu.  Here A is the state space matrix; B , the input matrix; and u , the feedback control force vector. F o r the two link manipulator with joint control variables, B and u can  101  be written as: 0 B -  M  - 1  •1 0 0 0 .0  0 1 0 0 0  0] 0 0 1 0 J.  Defining the dimension of the mass matrix (3 + n\ + n ) as N, the dimensions of A 2  and B are 2N x 2N and 2N x 3, respectively. Figure 5-1 is a block diagram of the centralized optimal controller showing the nominal reference trajectory and optimal feedback components. 5.3  A Review of the LQR The linear quadratic regulator is chosen to control this multi-input multi-output  system since it yields a unique set of state feedback gains for a given performance criterion [68]. Applying negative state feedback to a state space system, x = Ax + Bu, the control force u can be written as u = —Gx, where G is the optimal gain matrix. The optimal gains minimize a quadratic cost function J which considers tracking error and energy expenditure  where Q is the symmetric state penalty matrix and R is the symmetric control penalty matrix. The matrix R is required to be positive definite while Q can be positive semidefinite. The optimal control force u is given by u = - R B K x = -Gx, 1  T  R  102  Nominal Control  Disturbance  q  4  o  CO  r  Nonlinear Station-MSS i Dynamics  Figure 5-1 Block diagram of the centralized controller showing nominal reference trajectory and optimal feedback components.  where ~KR is the solution to the Ricatti matrix equation, which for the infinite time case becomes, - K  R  A  - A  T  K  K BR- B K 1  +  R  - Q = 0.  T  f l  B  Since the system is time varying, the gains are updated continuously based on the current configuration. All states of the system are assumed available to the controller by direct measurement or an observer. 5.4  Controllability and Observability For a nonautonomous system controllability and observability are not as straight-  forward as the time invariant case. They are verified by checking the ranks of the controllability and observability matrices, respectively, at different intervals. This will provide information concerning the controllable and observable character of the system at those distinct points, and by the continuity assumption, at all points along the reference trajectory. 5.5  Controlled System Response The performance of the optimal controller is determined by the selected state  penalty matrix Q and control penalty matrix R. For the rigid joint MSS manipulator considered here, the Q and R matrices have dimensions of 2N x 2N and 3 x 3 , respectively, where N = (3 + n\ + n ) is the dimension of the mass matrix M . Thus 2  in the simplest one mode case (ni = n = 1), there are 100 elements in Q, of which 2  only 50 need to be specified because of the symmetry. The matrices can be treated as diagonal to facilitate the process. Defining Q and R initially as identity matrices and adjusting the diagonal weights, satisfactory performance can be obtained with the following set of weights: Q = diag {10 ,10 ,10 ,10 ,10 , 10  10  10  10  10  10 ,10 ,10 ,10 ,10 }; * 3  3  104  3  3  3  R = diag {0.01,1,1}. The control penalty on libration is relaxed to increase the speed of the pitch response to be comparable with the vibration. The closed loop eigenvalues with this set of weights in the gravity gradient orientation are: Imaginary  Real -6.851 x 10  ±4.307 x 10  -1.248 x 10  ±4.416 x 10  -6.755 x 10  ±9.585 x 10  -9.078 x 10  ±1.467 x 10  -1.371 x 10  ±1.383 x 10  3  3  3  3  2  2  2  1  1  1  The open loop poles of the system has unstable eigenvalues associated with the motor joint degree of freedom because motor stiffness is ignored (Eqn 5.1): Real  Imaginary  0  ±7.550 x 10  0  ±4.389 x 10  0  ±1.729 x 10°  ±1.732 x 10°  0  ±1.732 x 10°  0  3  3  Figure 5-2 presents the controller performance in correcting an initial joint 1 error A/3J of 5° in the gravity gradient orientation: 1  h = 0, b  /*j  lo  = 90°,  fr  2o  = 0°;  with the link stiffness values: W £ i = Irad/s,  u)L2 = 1 rad/s.  Updating of the optimal feedback gains are not required for this autonomous case. The inertia of the motors are neglected compared to the inertia of the manipulator arm. As the linear regulator returns Af3j to zero from its initial value of 5°, 6u x  105  Parameters u  ll  Wj  'rad/s  =  2  Maneuver  =  00  "L2 = 1 - 0 rad/s  0-0m 90° - 90° P) 2 0° - 0° T pB = T j , = TJ|2p = 0 .01  *B ft)  2-  r  Initial Conditions - First mode ft,(0)=5°  mg = 0 n\2 =  on  History  1  mp = 0 0.010  r  0.005  0.5£  1  (m)  60^05 H 0.000  0.000  0.002  0.004  0.006  0.008  -0.005 0.000  0.010  I  0.002  0.004  11  ! 0.006  — r — 0.008  0.010  orbits orbits Figure 5-2 Optimal control of the two-link manipulator subjected to an initial condition of A/?^ = 5°. The state penalty matrix Q is diag { 10 . . . 10 ; 10 . . . 10 } and control weight matrix R is diag { 0.001, 0.1, 0.1 }. 10  10  3  3  is excited to a maximum amplitude of 0.02 while Afij  and $21 reach magnitudes  2  of 1° and 0.006, respectively. The pointing error of arm 1, £\, initially at 0.65m due to the 5° Afij  initial condition, decreases to zero after 0.006 orbit. The arm 2  x  pointing error, £2, attains a maximum value of 0.4 m as the link vibrations and joint 2 motor coordinates are excited. All the vibration coordinates diminish to zero after 0.006 orbit while substantial error exists in tb. This is due to the slow decaying poles associated with the tb degree of freedom. If the speed of response is critical, the gain on tb can be increased to achieve a faster response at the expense of energy expenditure. 5.5.1  Minimization of Payload Error  The approach in the last section minimizes the cost function in the following form with Q and R as constant diagonal matrices, roo  J=  (x Qx + u Ru) dt. T  Jo  T  In this section a new cost function J is derived which involves the payload position error A r and velocity A f directly. p  p  J=  f°° M A f ; ) Jo  2  + u {Ar ) 2  2  p  + ^(A/ffjJ +  u (Apj ) 4  2  + MsfA/JjJ + Me(Aj9j ) + u Rudi  (5.5)  r  2  The /x's are constant coefficients or weights for corresponding elements. Also included in the cost function is the range of motion of the joints, A/3j and A/?j , as well as 1  2  their velocities, Aj3j and A/?j . The last term considers the penalty on control effort 2  1  as before. Displacement vector of the payload with respect to the system frame is: f = a + hb + C\C2Czdj2 p  e  (a + hbx) (a + h )  + C1C2C3  x  y  +  +  by  4  4  tL2 E,-Jl(*St-)jM$2i*L2  107  g  E?==i(*S -)Ll*li^H t  C\C2C C C^CQ 3  C\C2CzC CsCQa  The reference position of the payload only depends on the reference coordinates and is defined by (a  + hbx)  x  {a  +  y  +  h )  CiC  2  +  0  by  C\C C§ 2  lL2 0  Hence the payload error Afp is given by (f — (r ) ). Retaining only the quadratic p  p  0  terms, the square of this error can be written as: (AF ) = (Ar;, + Ar^) 2  p  = (A/JjJ [t  2  2  L1  + t\ + 2l t 2  Lx  cos(^j )  L2  2o  t=i y=i + 2{*E )Ll{*X )LltLltL2 i  n  + (A/3j ) ii 2  2  2  + 2A^  +  n  +x; x; 2  + 2A/3J APJ {1  s ^ ) ^ .) ti 6 2i  X  Ex  L2  E  +  +^I£L COS(^ 2  ^ 1 ^ 2 C O s ( ^  + ILXILI  2  2  n  )]  2  ^[(*^)LI(4X  1  2 o  L2  2  2j  t=iy=i  (*^)Ll(£L  + 2A$J  C08(/?J  i  L2  j  2  o  2  O  ))  ) ) ]  COS(/3J )) 2O  2  £ M * ^ ) L 2 ( * L 2 + lLltL2  COs(/3j )) 2o  1=1  »*2  +  2£L2  2  £E n  * 2 . - ( ^ - ) L 2 ( ( * B y ) L l / j l l COs(/3j ) 2o  T  +  ($X )Ll£L2)6iy t  i=l J= l  n  2  t=i  A similar quadratic expression can be obtained for the square of the payload error velocity (Ar ) . Since these expressions are quadratic, they can be implemented as 2  p  a product of the state vector x and a symmetric Q matrix such that /zx(Ar ) + (AFp) = x Qx 2  p  2  T  M2  The state penalty matrix Q is now time varying and block diagonal consisting of two 5 x 5 full matrices. The additional penalties on the controlled coordinates in J (Eqn. 108  5.5), /x A/3jr , u Afij 3  x  4  U$APJ ,  2  1  and U&PJ^  contribute diagonal elements to Q thus  retaining its symmetry. It should be emphasized that this cost function minimizes the payload error in the system frame which is parallel to the body frame of the Space Station platform thus making it useful for position payloads with respect to the station itself. For cases that absolute position and orientation with respect to the earth is important, the payload error in the orbital frame (defined by the local vertical and local horizontal) should be considered. Consider the MSS manipulator undergoing a general maneuver involving translation of the mobile base and slew at the two joints and link stiffness values corresponding to 1 rad/s: h : -15m  +15m,  b  (3j  lo  a>x,i = lrad/s,  : 135° -» 45°,  pj  : +90° - • -90°;  2o  wr, = lrad/s,  m = 0.  2  p  The cubic maneuver profile is intentionally used to agitate theflexiblemanipulator. The system response using only nominal control (Figure 5-1) without feedback is shown in Figure 5-3. As can be expected, the reference maneuver excites the vibrational degrees of freedom leading to high frequency oscillations in the pointing errors £i and  The vibrations excited by the maneuver persist after 0.01 orbit when the  maneuver is completed because of the lack of damping. The maximum pitch disturbance is 0.4°. The maximum torque required for the nominal control of P J 1000Nm while 120Nm is needed for PJ  2O  1 O  is  The libration xb is not considered in the  .  nominal controller. The optimal feedback controller is implemented to reduce the high frequency oscillations in the system response. Minimizing the cost function in Eqn (5.5) with the following coefficients: m = 10 , 8  u = 100, 2  u = 10 , 6  3  109  u = 10 , 6  4  / i = 1, 5  u = 1; 6  Parameters  Maneuver -  = =  rad/s  oo  "LE = 1 . 0  rad/s  ™B = 0 "\2  = = 0 1  0.000  History -  - 1 5 -> 15m  oo  "LI = 1 . 0 "J2  Cubic  = 0 L  L2 = 1  135° - 45°  fin  90°  fiiz T  B =  T  Initial -  J  =  ]  -*-90° ' J 2„ = 0 . 0 1  T  Conditions  First mode  -  A l l IC = 0  0.005  0.010  0.015  -0.02 0.000  0.005  0.010  0.015  orbits orbits Figure 5-3 Nominal control of the two-link manipulator executing a general maneuver involving simultaneous translation of the mobile base and slew at the two joints. The cubic maneuvering profile is used.  R = diag {0.001,0.1,0.1}; results in a satisfactory response. The closed-loop poles, which are time varying since Q is dependent on current system configuration, are evaluated at the standard gravity gradient orientation: Real  Imaginary  -7.632 x 10  ±7.515 x 10  -2.291 x 10  ±4.386 x 10  -3.077 x 10  ±2.794 X 10  -6.465 x 10  ±6.514 x 10  -1.084 x 10  ±1.098 x 10  2  3  2  3  2  2  1  1  1  1  Initially, the LQR controlled responses shown in Figure 5-4 resemble the nominal controlled results, with similar vibration amplitudes and frequencies. It takes the LQR approximately 0.005 orbit to suppress the vibrations excited at the start of the maneuver. As the maneuverfinishesat 0.01 orbit, there is an abrupt change in  — * the acceleration of hb, /?j ,  a n  lo  d  /?J  lading to large vibration amplitudes in the  2 o  free oscillation phase as observed in the nominal control case. These post maneuver oscillations are effectively suppressed by the optimal controller in 0.003 orbit. The maximum control torques required by the optimal feedback controller are approximately 700 Nm for ip, 50 Nm for A/3j , and 150 Nm for A/?j . The libration reaches x  2  a maximum amplitude of 0.165°. In order to further reduce the high frequency oscillations encountered in the responses of Figure 5-3, the penalty on the velocity of the payload error (/Lt ) is increased 2  to simulate more damping. Increasing /z by a factor of 10 to 1000, the penalty coef2  ficients are now: Hi = 10 , 8  H2 =  1000,  n = 10 , 6  3  HA =  10 , 6  Hs = 10,  He = 10;  R = diag {0.001,0.1,0.1}; with the following closed-loop poles at the standard gravity gradient orientation: 111  Maneuver  Pa r a m e t e r s  — Cubic  0>j, = oo CJ | = 1 . 0 L  CJj =  -15 rad/s  Pi l  0 0  =  1•0  mg  = 0  "\.2  =  "V =  1  rad/s  f  B  =  -> 4 5 °  90°  —90°  -  4,2  A/,  0.01  T J, I, = T  •12  Initial  m.  -» l & n  135°  Pi 2  2  "L2  History -  A/,  Conditions  First mode  -  A l l IC = 0  0  0.2  A/?S 0 2  (a) 0.000  0005  0.010  orbits  0.015  0.000  0.005  0.010  0.015  orbits  Figure 5-4 Optimal control of system in Figure 5-3. The state penalty coeffiecients aire: u\ = 10 , H2 = 100, A*3 = 1 0 , u = 10 , u = 1, /z = 1. R is diag {0.001,0.1,0.1}. (a) Time histories of the generalized coordinates. 8  6  4  5  6  6  500  500  CO  -100-J 200  T  A  ^ 2  (Nm)  0  -200 0.000  1  0.005  orbits  (»)  0.010  0.015  oooo  0.005  orbits (<=)  Figure 5-4 (b) Optimal feedback control torques, (c) Total control torques (nominal + feedback).  0.010  0.015  Real  Imaginary  -2.384 x 10  ±7.178 x 10  -7.055 x 10  ±4.359 x 10  -3.980 x 10  ±1.089 x 10  -6.590 x 10  ±6.393 X 10  -1.085 x 10  ±1.097 x 10  3  2  2  1  1  3  3  2  1  1  It is apparent that the closed-loop system is now more damped than before. In the previous case, the poles associated with the highest frequency have a real/imaginary ratio of 763.2/7515 = 0.10 while the corresponding ratio is now 2384/7178 = 0.33. Figure 5-5 presents the controlled response with the enhanced weight on the payload error velocity. Clearly, the high frequency oscillations in e , e%, A/3j $n, A/3j , x  15  and $21 are significantly reduced. Initial vibrations caused by the maneuver, which have taken 0.005 orbit to be suppressed in the previous case, vanish in under 0.002 orbit. The post maneuver vibrations diminish to zero in approximately 0.001 orbit. The librational response remain essentially the same as before. Maximum torques required by the optimal controller are 180 Nm for A/?jj and 90 Nm for A/?j . 2  114  2  >g,005  0.000  (a) 0.000  0.005  0.010  0.015  -0.005 °°0 0  °°0  5  0  0  1  0  0.015  orbits orbits Figure 5-5 Effect of increased damping in LQR. The state penalty coeffiecients are: n\ = 10 , Hi = 1000, Hz = 10 , fi = 10 , HB = 10, He = 10. R is diag {0.001,0.1,0.1}. (a) time histories of the generalized coordinates. 8  6  4  6  Figure 5-5 (b) Optimal feedback control torques, (c) Total control torques (nominal + feedback).  6.  6.1  CONCLUDING REMARKS  Summary of Conclusions A formulation for investigating the in-plane dynamics and control of a Space  Station based two-link flexible manipulator is developed. The formulation is also applicable to any space system withflexibleslewing appendages that can be modelled as beams. The simulation results indicate that the system may undergo significant librational motion and under critical conditions, the Space Station can become unstable. The manipulatorflexibilityinfluences the libration by superimposing a high frequency component on the rigid response. The positioning error of the manipulator due to its structural and jointflexibilitiescan be excessive, especially during the maneuver. The time history of the maneuver ( cubic or sinusoidal ) is found to have significant influence on the response of theflexiblemanipulator. The smoother sinusoidal maneuver profile leads to much less vibration of theflexiblesystem especially in the post maneuver phase when the amplitudes are essentially zero. Shift in the instantaneous system center of mass attributed to the maneuver has noticeable effect on the libration while the vibration degrees of freedom are virtually unaffected. For the maneuver cases considered, two cantilever modes seem sufficient for discretization of the deformations of each link. The cantilever modes obtained with a tip load mass appear to converge faster than the free modes. The correlation between the linearized equations and the nonlinear governing equations is extremely good at low frequencies. However, substantial discrepancies exist at the high frequency modes. From the analytical closed form solution analysis on a simplified nonlinear system, it is found that the hnear frequencies predicted by the eigenvalues always exceed the corresponding vibration frequencies of the non117  linear system. Using the variation of parameters method, a frequency perturbation correction based on the average contribution of the nonlinear terms in one cycle can be obtained to improve the accuracy of the linear solution. An optimal controller is designed to maintain the station at its operating orientation and minimize the position deviation of the payload from the nominal path. Using a set of time-varying penalty weights based on the minimization of the payload error, the feedback controller is found to be very effective in suppressing the vibrations of the flexible manipulator as well as maintaining the operating configuration of the space platform. 6.2  Recommendations for Future Work This study is a preliminary analysis that provides some insight into the dynamics  and control of the highly complex Space Station based Mobile Servicing System. In general, this system can involve a flexible mobile manipulator executing complex maneuvers with a flexible payload on a highly flexible platform. To investigate this challenging system in more detail, a few suggestions are listed below: • It is anticipated that the highly flexible nature of the Space Station platform can have significant influence on the operation and accuracy of the MSS, hence platform flexibility should be modelled. • When undergoing three dimensional motion, the torsional degree of freedom, as well as transverse deflections, of the beam can be important. Consideration of axial foreshortening may be necessary when deflections are large. • Control of link vibrations can be achieved using thrusters mounted along the link to directly suppress oscillation. This direct control may be more effective for longer beams. • Environmental forces (such as thermal radiation and free molecularflow)may have significant effects on the system dynamics. 118  • The proposed Space Station is not in the gravity gradient mode. Results should be obtained for the actual operational orientation. 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[68] Kuo, B.C., Automatic Control Systems, Third Edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975.  126  A P P E N D I X I - Transformation Matrices  The transformation matrices in Chapter 2 in the developement of the governing equations of motion are:  - sin  COS/?&  Ci =  sin/?b  3  C = 5  cos Tfjj sin 7.^  - sin 7 j cos 7 jj 0  COS (3j2  -sin/?j cos PJ  PJ  2  C = 7  x  C = 4  smPj  x  x  0  cos ax,i sina£i  2  C = 6  2  2  0  — sin ax, i 0 cosax,! 0  -sin7j cos 7 J 0  0  cos a 1,2 — sin ax,2 sin ctL2  COS CCL2  0  0  The angular velocites are: 0 0  "0 " 0  w = 3  A .  A W5 =  0 0  0 0  w = 4  0 0  >1UQ —  A .  0 " 0 UJ7 =  0 0 6tL2.  127  1  0  cos 7J sin7j  0  X  cos fij 0  0  2  0  - sin PJ  X  0  0  sin  Co =  0  0  C =  COS PJ  Pb cosfib  1  2  2  A P P E N D I X H - Librational Kinetic Energy The librational kinetic energy discussed in Chapter 2 has the following form: T  L  = {if  h  z  +  + ma  +1J,  1E1 +  +  &  + if* + if  + (mi + mji + rriLi + mj + rati + m )(a i i m i(4i/4 S(*Si )Li Wy^i) a  2  +£  L  y=i n  J  2  + m  + m )(l p  + 2 | m / 2 + (m i l  L1  +m  J2  bx  2  )LiSuhi  ^(^JLI^LI)  + m)  L2  p  ui)  6  (a  +  +  y  fc )  sin(/?  6y  i  n  +  i  n  Ei  \(a + h ) cos(/3 + Pj! + x  i  +  2  L2  2  c  y  »=i  + ( m  p  n  n  +  + hb)  2  n  m x i ^ ( ^ M ^ L i ^ i i ^ L i  +  (m  + m  J2  +  L2  +  6  pji +  IJI)]  i  m  p  »=i  i=i  [-(o, + h ) s\n(P + PJI + Ul) + {fly + V ) c<*(& + A " + T J i ) ] 2 2 bx  b  n  + m  L  2  n  (£/ 4 + £  ]T( s.y)"^2;£L) $  2  .=1  y=i  n  "•2 +  + £(*jO«*2t*L2  ™p(*L2  £ ( * ^ ) L 2 « 2 t * L 2 ) t=l  1=1  + 2(m /2  +  L 2  [(a  +  x  /i  f c l  m  ) £  p  I  (  2  cos(p + PJI + Ul  )  2  +  b  « L i +  #72  7J2)  +  + (a + h ) sin(p + PJI + 7JI + «ii + PJ2 + U2) y  +  by  li  cos(a ,  L  n  b  £  +  1  P  7J )  +  J2  2  i  + ^(QEJLISHZLI  s i n ( a  p  i +  L  +  J2  7 )] J 2  t=i  "2 + 2[m  n  ^($A/ )L2*2.^2 +  L 2  i  WlpXl( ^i)i2*2t42] $  t=l \-(a  +  x  +  (a  -  ti  y  t=l  6  x  +  ) sin(/?  L  l  +  p  J2  PJI  +  6  /ifcy) c o s ( / ?  s i n ( a  L  n  +  /i  2  6  +  + 7.71  PJI  +  +  7JI  +  +  « M  +  +  p  +  J2  PJ2  U2)  2  1 YL(*Ei)Li6idLi  +  U2)  7J )  i  i=l  a ^ i  cos(a  L  1  128  +  pj2 + U2)} ^  W  A P P E N D I X III - Kinetic Energy w.r.t. System Frame  The kinetic energy w.r.t. the system frame, TSF, includes the all the contributions from vibration and maneuvers of the MSS. For a rigid stationary manipulator, this component of the kinetic energy will be zero. In vector form, TSF  c  a  n  be- expressed  as: TSF = ^  TSFJ  ^[2) \dt  at)  m  1 • — -m (a )  1 + -\m  2  a  c  +  +m  b  + ( J2 + m  +m  JX  d  m  p  1 2~(  +  3  + [rn  mj2  X  +m  J2  + (a + h ) • C C C c  b  X  2  2  J2  +  m  L  2  [J t (  3  Ll  3  + p{&p) ](vi m  + \J  2  m  2  5 J  2  c  dm + (m  x f  Ll  b  +m  J2  + m )aj ]  L2  p  dm + (mj + m 2  3  p  2  2  3  \(m ){k ) p  4  p  4  5  6  2  6  X  2  3  4  5  ?r, dm  6  2  b  + [a + f\ ) • C C C C CSCQ c  b  X  2  3  4  [^  dm + ro a ] p  p  + (u + w + cD)(wi + w + u + cD + w + w )aj x  2  3  • C CsCe^J 4  2  2  7  p  5  c  J2  5  + m a j x (a + h ) p  2  + m )a  L 2  + (a?! + w + w + w + u> + w ) • C C C C C C 2  2  3  p)( 2)  Lx  4  +  2  2  2  b  + w + w + u3 + w + w + w )  2  ZL • 5i  c  + w + w)  x  +  p  x  + ^^(^l + w + w + w + w ) +  L2  [J ?L dm  3  p  3  2  J2  ~* + h)  + m }(a  + ro )aj J x (a + h )  L2  + (cDi + w + w ) • 2  +m  2  L2  + (wi + u> + w ) • C C C 2  LX  + m )(a ) ](u  M  1 f '— + 2/ '^i  +m  3  4  dm + m a p  129  p  5  6  2  x aj J 2  + (Qi + w + w ) • a 2  3  xC C C [ 4  J2  5  6  J  £L, dm + m p a p j 2  + (cDi + cD + u>3 + w + w + ufe) • C C C ^ fi, rfm+ m a, x a 2  +a  J 2  4  5  4  5  6  2  p  J2  • C C C [y tf dm + m a ] 4  5  6  L2  p  p  + (wi + w + w + u + u5 + w ) • j^y fr, x f 2  3  4  5  6  where { " } denotes local time derivatives.  130  2  Lz  dm + m a x a j p  p  p  A P P E N D I X I V - Angular Momentum Vector  — * The angular momentum w.r.t. the system frame, HSF, together with the librational vector fi gives the kinetic energy coupling between the libration with the  —• remaining degrees of freedom in the system. In vector form, HSF can be expressed as: HSF  =  X/  HSF  {  - t f = m (a B  c  x a) c  + (mi, + mji + m x  ™L2 + m )  i l + "V2 +  2  +  w) 2  3  + (mj2  +  c  m  2  $L dm  3  x  L 2 + »7»p)aj2]  + (a + hb) x  CiC C ^J  + C1C2C3  $ dm + (mj2 + m 2  2  c  £  3  dm + (mj +  Ll  dm + (mj2 + m  Ll  m )aj J  +  L  p  2  + m )a  L2  p  + m )aj  "^L2  2  Ll  + J fLj x £  p  x  X a  J2  J2  + m )(aj ) ](wi + u + w ) 2  + ( ^ 2 + rn 2 m  p  L  2  hb)  c  + (uJj + w + w )(a + hb) • CiC C  + Ij (&l  c  (a +  + i f Wl + IJI($I  +  j^(a +  p  2  2  3  +W +W +W + W) 2  + (uJj + w + 2  £3  3  4  5  + w + w + w ) (a + hb) 4  5  • CiC CsC4C5Ce^J c  2  + dCiCaC&Ce  3  A  b  c  Ct, rfm+ m a J  2  + (o +fcb)x CiC C C C C  6  2  p  [y tf dm + m a  Q  L2  [y & dm + m a 2  p  p  + [2(a5i + u> + w ) + u> + w + w ]aj 2  3  p  4  5  131  6  p  p  x (o + £ ) c  2  6  2  (a + h ) c  b  • CCC 4  +a  $ dm + m o ]  6  L2  p  p  x C C C [J C dm + m a ]  J2  4  + CCC 4  B  5  6  L2  p  p  $L dm + m a ] x a  6  + J ?L * 2  6  2  £L  p  + p(«P  rfm  m  2  p  X  J2  5  P)  + [ L 2 + m p ^ ) ] ^ + w + w + w + u> + w ) J  2  2  3  4  5  + i"5(wi + tD + ui + w + cD + <3 + u>) 2  3  4  5  132  6  7  6  A P P E N D I X V - Mode Shape Integrals The mode shape function for the free cantilever beam and one with a tip mass payload have the identical <p(x) = (sinfcx — sinhfcx) —  ^ ^ ^^1 ( os kx — cosh/cz) COS fct* *T~ COSiX fcc S U 1  s  n  C  where kl is the root of the characteristic equation cos kl cosh kl+ 1 = ^rkl sin kl cosh kl — cos pi  This transcendenal equation has an infinite number of roots thus leading to an inifinte number of mode shapes. The first five characteristic spatial frequencies (kl)i for the cantilever beam with various tip load mass ratios are shown in chapter 2.3. In formulating the governing equations of motion, it is necessary to evaluate the displacement and slope given by the modal functions at the end points as well as various forms of their integrals over the length of the beam. To make the results general, the length of the beam is normalized to 1. $E  =  {  M*)\x=l  *  I *  dx  lx=i  $ M = / 4>i{x)dx t  Jo  $c. = / x<f>i(x) dx Jo * tj 5  >  Vi  Here  and $x t  a r e  =  MrfMJ)  /  J  0  dx  2  dx  dx  2  the tip displacement and slope contribution, respectively, of  mode t while $A/ . represents the mean deflection of mode t . t  The numerical values of these functions are shown below for the free cantilever beam and the case with a unit mass payload. The integrals are calculated numerically. 133  = 0  trip — 1  m  = 0  trip = 1  2.00000  2.04831  2.75301  3.02473  $E  -2.00000  0.31444  *x  -9.56156  6.67489  $E  2.00000  0.18747  $  15.6973  10.8556  -2.00000  0.13218  $x  -21.9917  15.2627  2.00000  0.10214  *x  28.2754  24.0997  trip = 0  m  trip = 0  trip = 1  0.78299  0.77447  $C  0.56883  0.56691  0.43394  -0.83174  $c  2  0.09077  -0.44220  0.25443  0.09650  $c  3  0.03242  0.14766  0.18190  -0.32845  $c  4  0.01654  -0.09096  0.14148  0.04781  $c  5  0.01001  0.09062  trip = 0  trip = 1  1  **21 ^ 22  m  2  3  $E  4  *M  $  M  t  2  $M  3  p  p  2  x  3  4  5  =1 X  p  = 0  trip = 1  1  12.3624  12.6002  0  -0.64408  0  0  1  1  f 2  485.519  290.175  0  -0.38400  ^31  0  0  * 32  0  -0.05895  *^32  0  0  ^ 33  1  1  ^33  3806.55  2681.43  $s  0  -0.27075  0  0  0  -0.04156  0  0  0  -0.02478  *u  0  0  1  1  *V  14617.3  11260.0  0  -0.20927  0  0  0  -0.03212  ^52  0  0  0  -0.01915  **5S  0  0  0  -0.01350  0  0  1  1  39944.0  32459.2  *s  n  S  s  5  $  ^  *5  *5  $  $  5  4 1  4 3  4 4  5 2  5  4  *55  m  $  2  *V  41  43  44  $  ^55  p  The functions pertaining to thefirstmode are similar in the two cases as expected 134  since the two mode shapes are almost identical as shown in Figure 2-4. For higher modes, however, the mode shapes are quite different as indicated by the modal values and integrals. It is interesting to note that with a tip mass at the end, the mode shapes are no longer orthogonal. In the free cantilever case, there is no coupling between the modes: *5  i  y  =  0;  *tty = 0  for v£ j.  However, they will be coupled once the MSS system dynamics are accounted for.  135  A P P E N D I X V I - Equations of Motion  ip Equation  {  If + h + Ifi + III + I J. + Z  Z  IE2 + I  Z P  + m a\ + (m + mji + m i + m j + m i + m )(a + h )  2  a  b  L  n  i  n  2  2  + {m  + mpX^ +  L2  BI  t=i  LI  p  2  ^(^JIIMII)  )LISIULI  »=i  + 2|m /2 + (mj + mz,2 + >™ ) il  b  LI*I*ML)  t=i y = i  +m  J2  c  i  + m (£l j/4 + £ L1  p  [(o + h ) cos(/? + An + Ui) z  bx  6  + (a + h ) sin(/3 + /?JI + 7JI)] v  + 2\m  6  hy  X ^ ( * M ) L i 6 i . ^ i i + (m  L1  t  + m  J2  i 2  + m) p  t'=i  [-(a + /ibx) sin(/? + An + 7.71) 2  53($£ -)Ll*li*Ll]  t=i  6  + (a + h ) cos(/? + pji + 7JI)] y  by  6  "2  «2  n  n  2  2  £(*S -)M(*Sy)L2«2iML2)  + ™p(*L2 + £  t  1=1 y = i  + 2(m /2 + m ) £ [(c^ + h ) cos(/? + /3JX + 7 J + a p  i2  bx  L2  6  x  L 1  +p  J2  + (a + h ) sm(p + PJI + 7 j ! + a i + p y  by  b  + £1,1 cos(a i + P L  n  J2  L  J2  + U2) + 7J2)  + U2)  i  + Y2(® J t'=l E  n  + 2[m  2  L l 6 u f , L 1  s i n  ( i i + PJ* + a  n  2  X)( ^t)^262«^L2 + ™p ^ ( S E J M W M ] $  L2  Ui)]  t=l  »'=1  136  t  [ - ( a + h ) sm{p + PJI + Ui + « L I + PJ2 + U2) bx  z  b  + (a + h ) cos(p + PJI + ui + <*L\ + PJ2 + U2) y  - £  by  L 1  +  b  sin(a  + p  L 1  + mLx&xl*  ^  n  U2)  1  J^&EiUiSulLi  t=i  + UL  +  J2  cos(a  +D  L 1  Ue + tJ,)  + 7 )]  + pJ2  J2  D(Hy)^i)*«)M^i)  a  »=i y=i  + tfa + ' f a  t=iy=i i 2  2  t=iy=i + [»"LI/2 + ( m  J 2  + m  L2  + m)  l\  p  L  Y * + bx) cos(p + pji + Tfjj) a  h  b  sin(/? + / 5 J I + ui)\  + (a + h ) by  y  6  i  n  ^ ( $ A f ) L i ^ i i £ L i + ("»J2 + mL  + \m i L  i  n  t  +  2  m  p  t=l  i=l  j-(a„ +  sin(/? + PJI +  Ui)  6  + (a + h ) co&(p + PJI + y  + [m  by  b  +m  J2  + m ){£  L2  p  + (m /2 + m ) £ L2  p  2 L1  + ] T £(*J5jLi(*s )Li*itMLi) i=i y=i y  L 2  [ ( a + h ) cos(p + pji + 7 J I + a i + PJ + U2) bx  x  b  L  2  + (a + h ) sm(p + PJI + Ui + « L I + /?J2 + U2) y  by  b  + 2(£ii c o s ( a  L l  + p  J2  + 5^(*s -)n*i^Li t  + 7  J 2  )  s i n ( a i + pJ2  + 7J2))]  L  t=i n  + \rn  L2  n  2  2  J^($M )L25 ,£L2 + m t  i=l  2  j2($^ .) 2^2t£l,2] t=l t  p  137  L  [-(a  + (a  + h )  sin(/? + PJI + Ul  + C*LI + PJ2 +  U2)  + h )  cos(/? + PJI + U \ + <*LI + PJ2 +  U2)  bx  x  y  6  by  fc  + 2 [ - £ L I sin(a  + p  L1  + ^{QEJLISUILI + I  Pb + I^(Pb  Z  B  n  i  + 7.72)  J2  cos(a  + PJ ) +  I^PJ  X  + p  + 7 J ) J j >{Pb + fa  J2  L 1  2  x  + IJJ  2  f  + H* * i=l  1 ( Li{$Mi)Li ^  1  + (m  m  [(a* + h )  L  +  2  m )(^ )Li)hi p  Ei  + 7JI)  cos(/?6 +  bx  + m  J2  + (a + h ) sin(/? + PJI + 7./1)] y  by  6  + {m {$ )Li L1  + (mj  Ci  n  t=l sin(a + p L 1  L2  2t  £L2  cos(a i + p  L2  2  + P X^ m  $ B  ») * '^ J i 2  2 ,  L 2  + 7J )]  J2  2  + Y^Yl{9 ..)L2S S ll )  [+^J2 + / f 2 + mL2(£ L2/4 n  2  **  2  ii  s  n  £ 2  +  + 7J2)  J2  L  I  2  Ei  t=l  2  p  p  n $  + m {ll  m ){$ )Li)Zli  2  X^( A*,) * '  L 2  +  2  p  L2  X  [m  L  [ ( m / 2 + rn )l  + {$E )L\IL\  -  + m  2  2  X > £ , ) " ( * ^ ) ^  » M L 2 )  2  t=iy=i  + (™W + m )£  + Ip  2  p  L 2  (a* + h ) cos(p + PJI + 7 J I + "1,1 + PJ2 + U2) bx  b  + (a + h ) sin(p + PJI + 7 J I + «Li + PJ2 + U2) y  + £ n  L 1  by  cos(a  i  b  L X  + p  J2  +t'=lX^( ») $B  n  +  U) 2  ,  il<5l,£il s i n  ( ^ i + PJ* + ^ ) J a  2  n  2  2  t'=l  138  ij  2  -{a  + h ) sin(/?6 + PJI + Ui  x  + ct  bx  LX  + PJ + U2) 2  + (a + h ) cos{p + PJI + Ui + a x i + PJ2 + U2) y  -  t  by  b  + U2)  sin(ax,i + p  L1  J2  i  ll^i  n  + ^ ^ E ^ L i h i h i  p + 7J  cos(a i +  J2  L  2  )J J >  t=l  '  f  n  + m (ll /4 L2  n  n  2  2 s  L  6  i6  l  2  Y,{*E )L2(*E )L262iO-2 t 2)  + m {t 2 + L 2  p  n  Y,Y,(* ij) * * V LJ t=i j = i  +  2  2  + Ip  2  L  i  j  j  L  t = l j=l  + (m /2 [{a  m )l  +  L2  p  L2  p  + h ) cos(/? +  x  hx  + Ui  JX  6  + ct i  + PJ + U2)  L  2  + (a + h ) sm{pb + PJI + Ui + « L I + PJ2 + U2) by  y  + hi cos(a;  +  »=i  Yl(® i) E  n  + ]rn  L1  + P  J  2  + U2) s  Ll6uiL1  i  n  (  a  i i  n  2  t  i  2  2  i=l  t'=l +  x  2  5^($M -)L2<52i£i2 + m p ^ ( $ £ ; . ) L 2 < 5 , ' £ l , j  L2  [-{a  + PJI + f - » ) J  sin(/3 + 6  p  JX  + Ui  + <*LI + PJ2 + U2)  + (a + h ) cos{Pb + PJI + Ui + « L I + PJ2 + U2) y  by  - l \ sm(a + p + U2) L  +  L1  J2  cos(ai,i + /?j + U2)  ^{QBJLISUILI  2  HPJ  2  +  U) 2  1=1 n  2  + 2 ^ 2 t { [ " l L 2 ( * C , ) L 2 + " t p ( * K ) L 2 ]^L2 ^ t=l t  +  [m (  i 2  ($M -)l,2 +  (a  t  x  m  p( B )i2]£i2 $  t  + h ) c o s ( / ? + £71 + Ui + « L I + PJ2 + 7J2) bx  b  + ( a + h ) sin(p + PJI + ui + « L I + PJ2 + U2) y  by  b  + l \ c o s ( a i , i + PJ + U2) L  2  139  sin(«Li +  + ^[QEJLISUILI 1=1  PJ2 + U2)  )  +  +m  e  y  x  + (m + m j i + m b  - a a + 2(6 + rp)(a a + a d )  [a dy LX  + m  J  + m  2  x  L  2  x  x  y  y  + m ) p  [(a + h )(a + hb ) — (a + hb )(a + 2(0 + tp) [(a, + h )(a + h ) + (a„ + x  bx  y  y  y  bx  + 2(0 + if +  + PJI +  x  y  bx  iji)  t=i y=i  +  [ m i i / 2 + (m  + m  J2  L2  11  "1  t=i  t=i  + m) t p  L 1  sm(Pb + PJI {a + h ) + 2(0 + V-) [(a, + h ) cos(P + PJI + Ui)  [-(a + h ) z  + 7JI) +  bx  hx  b  b  by  +Pb + PJI (-(a + h ) sin(/? + PJI  + (2(0 + i)p  b  by  ) sm(p + PJI + 7 J i ) ]  + (d + h y  y  +  PJI + iji  x  bx  + [m i Y^^M^Li^idLi  6  + {mj2 + m  L  iji)  +  L2  +  7JI) +  ( %  + m)X ^ i ^ U i ^ L i J p  t=i  +  '  t=i  [-(a + hx) cos(p + PJI + 7 J i ) x  b  - (a  y  ) sin(/?6 + PJI +  +h  by  ) sm(p + PJI + 7.71)  + 2(6 + ip) [-(d + h x  + (d + h y  by  UI)  bx  b  ) cos(/? + PJI + 7 J I ) ] 6  - (2(0 + rP)p + PJI + 7 J i +Pb + PJI + 7 J I ) b  [(a + h x  bx  ) cos(p + PJI + 7 J I ) b  140  hb ) y  (a + hby) y  +  2  L  m\  5 3 ( * M ) L i * i t ^ i  L  t  {mj2  +  + m  i  2  + m)  LI6UILI\  p  t=l  {6 +  t=l  i> + 0 + Pji + iji) h  [-(a  + h ) sin{P + P  s  + (a  bx  B  JX  + Ui)  + h ) cos{p + PJI + 7JI)]  y  by  b  + 2(6 + ip + P + PJI  7JI + «ii + PJ2 + IJ2)  +  B  mL2 n  n  2  2  + P 52(^JS )L2*2.^L2 5 3 ( W ) i 2 ^ 2 t ^ L 2 j M  $  t  t  t'=l  i = l  + {m 2/2 +  m )l 2  L  -(a  L  p  + hb ) sm(Pb + PJI + Ui + a  x  x  Ji ) cos(/?  + (a +  6y  y  + PJI + 7JI + a  6  L  1  +p  +p  L 1  + 7J2)  J2  J2  + U2)  + 2(0 + xjj) [ ( a + hbx) cos(p + PJI + 7JI + « L i + PJ2 + U2) z  b  + (a + h ) sm(p y  by  + PJI  b  + 2(6 + xb' + P + PJI b  n  +  7JI +  +  ct  L X  + PJ  2  7J )]  +  2  7Ji)  i  Y^EJLISUZLI  sin(a i  p  +  L  + 7J2)  J2  1=1  + ((6 + xj> + p + PJX + 7Ji + «LI + &/2 + I J 2 )  2  b  [-(a  + /i  x  ) sin(/3 + /3j + 7 J + a b  6 l  X  x  L  -(0 + xb) ) 2  i + /3 + 7J2) J2  + ( a + h y) cos(p + PJI + Ui + a i + Pj2 + U2) y  b  b  L  + («L1 + PJ2 + 1J2)(2(6 + XJJ+ P + Pjl + iJl) + b  ( - £  L  X  sm(a  L  X  + Pj2  + ^ ( ^ E ^ L l h i t L l  +  COs(a  U2)  PJ  +  LX  2  t=l n  n  $  i=l  1J2))} ^  2  + [m 2 X^( A*,)i2^2.-42 + m L  +  p  2 J3($B .)L2*2t42j  i = l  141  t  "Ll  +  Pj2 + 1J2)  •{{a + h ) cos(/? + PJI + 7 J I + <*Li + PJI + 7 J ) x  bx  6  2  + (a„ + h ) sin(/? + PJI + Ux + a i + P J + U2)) hy  6  tjj) [ - ( a  + 2(0 +  L  h ) sin(/? +  +  x  + <*LI + PJ2  PJI + IJI  6  bx  2  + U2)  + (a + h ) cos{P + PJI + Ui + a + P + 2(0 + 4 +P + y  by  PJI  b  i  n  Ll  b  +  +  J2  IJI)  cos(a i + PJ2 + U2)  ^{QEJLISUILI  L  1=1  - [(0 + j, + P + Pj! + b  7  J  1  + C\ + P  (a + h ) cos(p + PJI + 7 J I + a x  hx  b  LX  + 7 J ) -{0 + 0 ) 2  J2  2  2  + PJ + 7 J )  L i  2  2  + (a + h ) sm(p + PJI + 7Ji + <*Li + PJ2 + 7J2;] y  -  hy  b  PJ2 + 7J2, 2(0 + 0 + p + PJI  («L1 +  cos(a  +p  L1  7Jl) + *L1 + PJ2 + 7J2]  +  b  + 7J )  J2  2  + Yl^E^Lihif-Li  + 7J2)  sin(a + p L1  J2  »'=i  n  + 2[m  n  2  Y2(*Mi) *kilL2  L 2  p  i=l (0 +  2  + m ]P($ .) <5 £ j  L  Mt  L2  2l  L2  i=l  j> + p + PJI b  + 7J1  + «Z,1  Pj2 + 7 J )  +  2  - [a + h ) sin(p + PJI + 7 J I + « L I + PJ2 + U2) x  bx  b  + 7JI + a  + ( y + ^6y) cos(/? + a  6  - £  L 1  sin(a + p L1  + X^ i) * * $s  L1  1  + /3 + 7 J ) J2  2  + 7J )  J2  2  £il c o s  3^{(J* - J*) + (I  X  + (tfi - tfi)  LX  ( £ i + A " + 7J2) a  - I?) [cos(/? ) - sin(/? ) ] 2  2  6  b  [™{Ph + PJI) - MPb + 2  + {ILI ~ III) [cos(/3 + PJI + 7 J i ) - sin(/3 + An + 2  6  + (^J2 " Ij2)  - sin(& +  6  ^  \t°s{Pb + A/1 + 7J1 + + 7jx + a i L  +  /3 ) ) J2  142  2  «L1+  /?J ) 2  2  PJI) } 2  + {fa ~ iL)  + PJI + 7 J I + a  [cos{p  b  L  + p  1  J  + -yj )  2  2  2  - sm{/3 + PJI + Ui + ctLi + PJI + 7J2, ] 2  b  + (J* - lJ)[cOB(0 +  + PJI  + 7J1 + « L 1  + A/2  Sm(/? + PJI + 7J1 + OJLl + /?J2 +  Tfja) ]  b  )  2  + (ra + mji  + m  6  + (cos(/? + pj! +  i / 4  L  n  \ + m  L  + u x )  6  - ( m  2  2  6  m«(<iy - a  + 1J2 + « L 2 )  M  L  + m )((a  2  p  sin(& + P  -  2  + ™L2  i J2 i  + m  J 2  + m ))l  JX  + h  y  +  b y  )  2  -  (a  x  + h  ) ) 2  b x  7JI) ) 2  2  p  L1  i  n  + mil t=i y=i  + (m  J  + m  2  i  + m ) ^  2  ^($£ -)Li(*£; )Li6i,-6iy£  p  t  y  L 1  )]  t=i y=i  + 4 [ m i E($C,)LI6I.£LI + ( m  J  L  2  +m  L2  +m ) J^BJU^LIJ^LI p  t=i  *'=i  cos(/? + An + Ui) sin(p fc  + PJI +  b  + 2[m i/2 +  (m  L  J  + m  2  L  2  p  + (a + fc ) sin(A + PJI y  ^ ( ^ M j L i ^ i t ^ i  L  x  L  + ( m  J  2  m  +  L 2  6  + h  b y  ) cos(p  + pji  b  +  + [cos(A, + PJI + Tfji + CCLI + PJ2 + U2)  2  + PJI + 1JI + a  - sin{P  B  .  [-(m  L 2  / 4 + m )£i p  n  + (m  2  n  L 1  + p  J2  2  + U2) ]  2  2  2 53( Siy)i262.-6 y42 $  L  fc  )  p  sin(/3 + An +  x  y  cos(/? + An +  Ei  t = l  [(o + (a  )  +m)  i=l  +  b x  + -7.71)]  by  + 2[m i  [-(a + h  + m )^£ i  2  143  2  Li^idLi\  «2 r*2  "22 n  + m $ 3 X;(*JB -)L2(*B )L2«2.-Mi2)] p  t  y  t = l J3== 1l n*2 4  +  L  ™L2 2 ( * C - ) j i 2 * 2 t * L 2  p$3(*fl,-U2*2t*L2]*L2 i=l  m  +  t  »=1  cos(/? +  + Ui + a  6  Sin(y9 + PJI + Ul  +  L2  [-{a  x  + (a -  y  m )l p  + h  b x  ) cos(/?  + h  b y  ) sm(P  + pji  6  J 2  1J2)  + a  + ui  + / S J X + 7.71 + a  B  L  2  L2  L  L  i + p i + p  J  J  +  U2)  +  7J2)  2  2  U2)  £ i c o s ( a i + Pj2 + n  + /3j + 7 )  L 1  + <*L1 + Pj2 +  6  + 2{m /2  2 2  n n  2  L  i  +  sin(ai,i + p  E ) LI^H^LI {  J  U2)  +  2  t=i  + 2sin(#, n  +  + 7J1)  + An + 7./1 + «ii + PJ2 +  sin(A  U2)  i  + $ ^ ( * s . ) L i 0 i t * L i cos(/3 t  6  + 7JI + a  +  L  i + /?j  2  +  Ut)]]  *=i "2  + 2^m  n  ^(QM^LihilM  L2  +  t=i [(a  z  + /i  6 a :  2  m Y^($ ) £,262^1,2 P  Ei  t'=i  ) sin(/3  + 7 J I + ct  +  6  + ( a + /ib ) cos(/? + y  y  + 7JI + a  6  n  + £ i isin(aLi + p  J  + /3  Ll  2  L  1  J2  +  7J2)  + /3 + 7 J ) J2  2  i  + 7J ) + ^ { ^ E ^ L i h i h i 2  cos(a  L1  + p  J  2  +  U2)  i=l + 2 sin(/? + 6  n  -  pj!  +  ui) [ h i  cos{p  b  + PJI  + Ui + "  L i + p  J  2  +  U2)  i  ^{^E^LISUILI  sin(/? + b  pji  +  ui + « L I +  PJ2 +  U2)]]  t=i  }  cos(t/>)  s'm(ip)  + 3^-|(J -/ )cos(^)sin(/? ) y  6  x  6  + {IJI ~ IJI)  6  c o s  ( & + PJI)  s i n  ( & + PJI)  + {III ~ !LI) ^os(P + PJI + ui) MPb + PJI + Ui) b  144  + {Ij2 ~ ft)  cos  sin(p + PJI b  +  (#> + +  ui  ~ 1*2)  Ui  PJI +  + <*LI +  + OCLI  PJ2)  + PJI  + Ui  sm{p + PJI + Ui + ct  +p  b  L1  + PJ2)  + OCLI + PJ2 +  U2)  + U2)  J2  + {Ij - If) COs{p + PJI + Ul + Ct l + Pj2 + lJ2 + Ct ) b  L  L2  sm{P + PJI + Ui + " L i + PJ2 + U2) H  +  maa 8  x  y  + (mb + mji + m i + m L  +m  J2  L 2  + m )(a p  x  + h ){ay + hb ) bx  y  + cos(/? + PJI + Ui) sin(/? + PJI + Ui) 6  fc  [(m i/4 + ( m L  n  n  il  n  n  J 2  +m  + m ))t\ 111  L2  p  il  62  i + m ) ^ ^(*£jii($EyUi£it*iy4i]] i=ij=i "i  + (m  J2  +m  L2  p  + {rriLi ^^{^C^LI^UILI i=l  +  2  J 2  {mj2 + mL2 + ™p )J2(* )Li6iilLi)t Ei  »=1  [cos(ft + PJI + Ui) + [n»Li/2 + ( m  n  ~ sin{pb + PJI + Ui) } 2  + ™L2 + rn ) £ i p  L  [(a* + h ) sin(/?& + PJI + Ui) bx  + (a + h ) cos(p + PJI + Ui)\ y  by  b  i  i  n  n  )*£(* )Li6uhi Ei  i=l  t=l  [(a + h ) cos(p + PJI + UI) bx  x  b  - (a + h ) sin(/? + PJI + ui) y  hy  6  + cos(p + PJI + Ui + «Li + PJI + U2) b  sin(/? + pji + u i + ctLi + PJ2 + U2) 6  145  Ll  [(m  /4 +  L 2  m )l  n  L2  n  2  n  2  p  2  n  2  2  + pJ2 '52{&E )L2{$E )L262i62jf L ) t = l j=l n n + {m ^T{$ )L262iZL2 + m ^{^ )L2hdL2)£L2 t=l t=l [cOs(/? + 0J1 + 7 J i + CX i + 0J + 7 J 2 ) m  2  i  j  2  2  2  Ci  L2  p  Ei  2  6  L  - sin(/? + 6  + (m /2  + 7 J I + ct i + PJ2 + 7 J 2 ) ]  PJI  +  L2  2  2  L  m )e p  L2  [(a + hbx) sin(/3 + p 6  z  JX  + 7JI + a  + 7.72)  +p  LX  J2  + (a + h ) cos(p + pji + 7 J X + a +p + 7J ) l $ + l sin(az,i + PJ2 + U2) + ]T]( E{)Lihit-Li cos(a y  by  b  LX  J2  2  n  LX  +p  LX  J2  +7  J 2  )  t=i + 2sin(/? + PJX + 771) [ £ i COS(/3 + p 6  6  L  i  - Y^&EJLISUILI n  JX  +  PJI  + 7.71 + a  LX  + <*LI  sin{Pb + Ui + 1=1 n n + [ m E($A/ )L2<52i^L2 + m E($S .)L2<52t^L2j t'=l »=1 2  L  +7 J )  +p  PJ2  2  J2  +  U2)]]  2  t  2  [ hxcos{P  + PJI + U\ + a  b  t  p  LX  + PJ + 7 J ) 2  2  - {dy + h ) sin(/? + pji + 7 J I + a n by  6  LX  + PJ2 + 7J2)  x  + £1,1 cos(a i + PJ2 + U2) ~ $ ^ ( $ J 5 - ) L \ h i t h \ sin(a£i + p + 7 J ) 1=1 - 2sin(/? + A n + 7 J i ) [in sin(/? + p + 7 J I + 0^1 + p + u) i Y^{*Ei)LlSiilLl ™ (P + 7J1 + CXLI + Pj2 + Ui)]] X i=l t  L  n  6  6  +  }cos(V>) 2  S  sin ^) 2  b  PJ  = Q^,  146  +  2  J2  JX  J2  2  Equation  III + m (t /4  +  2  Ll  L1  X;)($5 )Li)6i,)6iy)£Li) + l f + / £ 2  iy  2  i=i j = i n n 2  2  £, 6 ,-6 j£i ) 2  i  2  2  2  i=i y=i **  2  2  + » M « L + 2 2 ( * « , . ) L a ( # * ) M « M « 2 i ^ ) + If 2  y  t=iy=i + [mn/2 + ( m j + m 2  [(a  L 2  +  m ) £LI p  + / i ) cos(/? + PJI + 7 J I )  z  6 x  6  + (a + / i ) sin(/3 + PJI + 7 J I ) y  6  6y  M )LISHZLI  + [m i  + (m  {  L  J 2  + m  L 2  + m ) p  i=l [-(a  y  E i  )  t=l  sin(/? +  + /i )  b  6!/  7JI)  /?JI +  + (a + h ) cos(P + PJI + 7 J I ) ] y  by  b  n  + (m  J 2  + m  L  + m )(£  2  +  2  p  L1  i i Y,^Z(^E )Li{^E )LiSi S ( n  i  j  i  i=lj=l  + (m /2 + m ) £ L2  [(a  x  p  + h)  L 2  cos(/?6 + PJI + Ui + <*L\ + PJI +  bx  + (d + h ) s'm(Pb + PJI + ui + an y  + 2(£ i c o s ( a  + L  t'=i n  + [m  M  Yl(® i) E  + P Ll6uiL1  J2  +  s i n  x  + 7J2)  ( £ i + PJ2 +  U*))\  a  2  E( ^,)i2^L2 + m i=l  [-(d  J2  U2)  $  i 2  + p  by  U2)  p  E($ J S  L 2  $ £ 2 t  L 2  J  t'=l  + /i6i) sin(/3 + PJ + 7 J I + a i + p 6  X  L  + (dy + h y) cos(p + PJI + 7 J I + a b  b  147  L1  + U2)  J2  + pj + U2) 2  l j  2 L 1  )  + 2 [ - £ i sin(ai,i + 0 L  + U2)  J2  + j^pEihitiilLi t=l  cos(a + 0J + 1J2)] L1  +  2  '  |m (4j4 + ^)X;)(*^)Li)M^)^ L1  ^  t  =i y=i »»2  2  n  + m (££ /4 + £ L2  X;($s )L2*2i^4 ) t'=l y=l  2  t J  n  n  2  2  2  Y,l>Z&E )L2{*E )L262i82 ll2) t=i y=i j  i  + (m + m J2  + X)  + mp)(«i  £(*Si)Li(*s )LiMiy*£i) t=i y=i  x  L2  + 2(m /2 + m p ) £ L2  + Ip  j  y  jL2  ^x,! cos(a£,i + /?j + U2) + ^(^E^LiSuhi i=i  sin(a + 0  2  n  + 2[m  L 2  n  2  2($M )i2*2t£L2 + m i  L1  J2  +  2  j2($ ; .) L $ £i ]  p  £  i  J  2  2 t  2  t=l  »=1  J-£x,i sin(a i + /3j + 7J ) 2  L  +  £(*15 -)Ll*l.-*Ll f  2  CO (a S  + 0 J2 +  L 1  7J )J 2  t=i  + $ N [ m i ( $ ) i i + (mj2 + m t=i ^ L  +  C i  [(m 2/2 +  ($E,)I,I£LI  L  + m )($^)x,i]£  i 2  p  cos(ai,i +  m )££,2 p  n  n  2  -  rn  J3( Af )L2<52t^L2 + m t  i=l  0  J 2  L 1  + U2)  2  J3(*S )L2<52t£i2j  $  L 2  >(fc + /Sjj + '  p  t  »=1  sin(ai,i + /?jr + 7 J ) ] 2  2  n  2  [/J + J£ + m ( £ £ / 4 + £ 2  2  i 2  2  t=i i  + m (4 p  2  n  2  n  2  X)(*5i )L2*2.-«2y« ) y=i y  2  2  + X ; £(**,0L2(*S,.)l,a$Kfei*L) + #  148  7JJ  + (mW  +  2  m  p)h2  h i cos(ai.i  + p  J  2  + U2)  i  n  ,  + ^{QBJLISUILI  s i n ( a i + PJI + U2)\ L  i'=i n  n  2  + [m E ( L 2  [-In  «)  $ M  t=l  s i n ( a  + E(  $  «)  s  L  i  + p  1  *  l  *'  i 2 5 2  1  ,  £  J  £ L 2  +  M  2  P ]0  $  B  i )  L  2  <  5  2  t  '  £  i  2  t'=l + 7J ) 2  2  i  l  n  2 2  c  o  (  s  + & T 2 + 7J2)  £ i  a  i=i  /-  + 1 I&2 + m (£ /4 L2  n  +£  2  L2  ^  £(*s< )Mfc<fcy*L) y  t=iy=i  n  w  2  + mp(*L + £  2  S(«E -)La(«* )L2^,4 ) + t  a  y  t=i i = i  + (m /2 + m )t p  L2  [ L / lcos(a  L  i  L2  + / ?  + ^ { ^ E ^ L i h i h i  J  2  7J )  +  2  sin(a  L  i  + p  J  +  2  i )\ J2  i=l n  n  2  2  + [m 2 £ ( * M - ) M * 2 I * L 2 +  P E(  M  L  i=i [-£LI  +  $ £ ;  t)"*  sin(ax,i  + p  J  £ ( $ £ , ) Li£it£z,i  c  o  s  (  a  i i  + A*  2  + f-> ) 2  f  E{^  i=l  + (m  (  m L 2  L 2  $ C 7  .)  L 2  +  m  p(®E )L2)£ L2 2  i  ($M )L2+  rn ($ )L2)tL2  I  p  Ei  l  n  [E( i=l +  +  £  $ S  L  i  i)  c o s ( a  i  *'  i l < 5 l  ($X ) 2/ t  ]  2  2  +  £ L 2  + 7J )  2  l  2  '  1=1  + If2^ J2PJ n  2 l  P  L  ?  1  £ i l  + / ?  s i n  J  2  ( i l + PJ2 + 1J2) a  + 7  J 2  ) ]  }^  149  f ( ^  2  + ^  2  )  + [mn/2 + (m  + rn  J2  +m )  L2  ti  p  L  ^ - (a + h ) sin(/? + PJI + 7 J I ) x  bx  6  + (a + h ) cos(/?6 + PJI + 7 J I ) y  by  + 2(8 + i) [(a + h ) cos(p x  bx  + PJI +  b  Ui)  + (a + h y) sin(pb + PJI + 7 J I ) ] y  b  + (0 + if  (a + h ) sin(/? + PJI + x  bx  fc  - (a + /i ) cos(/? + PJ\ + 7 J I ) y  n  by  b  i  n  t=i  i  «'=i  j^-(a + h ) cos(p + PJI z  b  bx  +  Ui)  - (a + h ) sin(/?6 + PJI + 7.71) y  by  + 2(6 + i) [~(a + h x) sin(p + PJI + 7 J I ) x  b  b  + (d + h ) cos(/? + ySji + 7./!)] y  6  by  + (0 + 0 ) [(a + hbx) cos(p + Pj! + 7 J I ) 2  z  b  + (a + hby) sin(/? + An + 7 J i ) y  b  + 2(0* + ^ + p + PJI i i h  n  + 7Ji)  n  »=i y=i + (mj + m 2  + 2(0 + t£ +  L 2  + m) £(*tf )Li^iihi p  t=i  t  ft + / j , ! + 7 J I + « L I + PJI + 7 J ) 2  m 2 £ X^($5tj)i2*2»^'£i2 L  i=iy=i n  »»2  2  + P Yl(® J E  m  L262i  t=l  + (m /2 + L2  £ ( * L I ^ L I J  »=i  m )l p  ^  L2  X^( $M ») i2 * 2 * £L2  t=l  L2  150  - ( a + h x) sin(/? + PJI + U i + « L i + PJ2 + U2) x  + (a  b  6  cos(/5 +  + h y)  y  + 7JI  fc  h  <*LI + PJ2 +  +  U2)  + 2(0 + tp) [(o + h ) cos(/3 + fln + 7J1 + " L I + /?J2 + U2) x  hx  6  + (aj, + hby) sin(/? +  + ui + a ^ i + / ?  6  + 7 J I + « L I + PJ2 + U2)  + (0 + t/>) [(o + M sin(/? + 2  x  + U2)]  J 2  6  - {a + h ) cos(/9 + PJI + ui + « L i + PJ2 + U2) y  by  fc  i  n  + 2{6 + tj> + P + PJI + Ui)  ^{^EihiSulLi  b  t=i  sin(a£i + Pj2 + U2) -  CLLl +  PJ2 + 7J (2(0 + 2  if +  fa + PJI + Ul)  +  O L 1 +  PJ2 + 1J2)  J£ i sin(a ,i + PJ + U2) L  z  ~  2  cos(a i + PJI + 7j )j  i)LIMLI  B  L  2  1=1  2  2  n  n  + [mL2 X]( ^i)"^2t£L2 $  mp^(*S,)l.2*2t£L2j  +  t=l  i=l  |-(a  x  cos(pb + PJI + Ui  + h ) bx  + O-LX + PJ2 +  U2)  - (% + hby) sin(/? + PJI + U l + « L 1 + PJ2 + U2) b  + 2(0 + rj>) [ - ( a  x  s\n{Pb + PJI + Ui  + h ) bx  + "Li + PJ2 + U2)  + (a + h ) cos(/? + pji + 7 J I + a i + PJ + 7J2)] y  by  + {6 + rjj) [(o 2  + (a  y  6  x  + h )  + h ) bx  sm(p  by  b  + 2{6 + jt + 0 + PJI B  n  L  cos{Pb + PJI + Ui + PJI + Ui  +  2  + <* + P L1  J2  +  U2)  + « Z , i + PJ2 + U2)]  Ui)  i  ^{QEJLI&UILI  cos(a i + PJ2 + U2) L  1=1  -  ai  + p  £i  cos(a i + p  L  L  J2  + U2W L  J2  + if + P + PJI + Ui) b  +7J ) 2  151  + OCLI + Pj2 + U2)  +  B'm{a i + PJ2 +  ^{^B^LISUILI  n  2  _  t'=i  (0 + V> + ft + PJI L 1  2  n  #  t=i  [-£  U2)]}  L  Ui + a  +  i + PJ2 + U2)  L  s i n ( a i , i + £ / + T72) 2  + YLi^E^Li^uhi i  n  + 4{3 +  (/£ " Vj2  cos  ~ J2)  cos(a  + p  L 1  (#> + PJI + Ui) Bin{Pb + PJI +  C O s ( / ? + PJI  J  + U2)  J2  + Ul  6  sin(/3 + Pj! + Ui + a 6  +  L1  Ui)  PJ2)  + OLLl +  p ) J2  + {IL2 ~ II2) COs(/?6 + Pji + Ul + CtLl + PJ2 + U2) sin(/?  + PJI  6  + (J* - / J ) sin(/? +  cos(/?  sin(/?  cos(/?  + PJI  b  + 7j!  JX  L  + <*LI + PJ2 +  +  1  p  + U2 + oc )  J2  L2  U2)  +  [ - ( ^ i i / 4 + (mj2 + m i  n  + (m  + a  U2)  + 7JI)  +  6  p  +  +PJ2  + <*LI  +  6  + Ul  + PJI  6  UI  +  J  2  i  2  +  m ))i p  * i  + m  L  2  + m )^ p  5^(*£?i)iii(*£?y)Li*i»^Li]]  i=ij=i i  i  n  +  n  U Z L I ^^{^c^LibidLi t=l [sin(/?  b  + /5j! + 7J!)  + {mj  + rn  + m  L2  p  t=l 2  -  + c o s ( / ? + pjx + 7 J I + a 6  2  cos(/?  L  6  +  +  i + Pj2 + U2)  sin(/?6 + PJI + 7 J I + a L i + PJ2 + U2)  152  7  J  1  )  2  ]  [-(m  / 4 + m ) £^L2 2  L 2  p  n  +  n  2 2  2 2  \m  L2  t=lJ = l  n +  m  n  2  2  p  1  t=l; = i  + ( m  L  «  2  ^{QCJMML*  2  + m E( ^ )L2*2t£L )£L2 $  p  t  t'=l [sin(/?  -  -  COs(/?  + h)  + h)  y  r -  7JJ +  + P  6  6  J X  £  +  2  7J2)  2  L l  + 7JI)  +  +  7  J X  )]  n i  n i  t  +  z  -  2{a  fc )  cos(/3  6l  + h)  y  p  Z  + h)  y  +  7ji)  + An  6  + p  6  &  +  2  + m  ^ ( ^ S J L I ^ I . ^ I J 1=1  + m)  L2  p  + 7JX)]  + 7.71 + a i  JX  + #/  L  cos(/?  by  + 3 cos(/?  ( m j  L  + Ai6 ) s i n ( / ?  + 2(a  +  1  m )l 2  2  x  +  6  sin(/?  hy  - (mi /2 + [(a  L  p  [ m n £ ( * M ) z , i £ i » £ i , i 1=1 [(a  a i + PJ  + m)  L  cos(/?  hy  2  2  + m2  sin(/?  bx  + 2(a  \2  7 J i + « L I + /?J2 + 7 j )  + ^JX +  6  [ W L I / 2 + (mj2 [(a*  2  t=l  + /?JI +  6  2  6  + pjx  + 7^i)  [*  + 7.71 + a sin(/?  L 1  6  L i  + An  +  7  + /?  +  2  +  7  J2  J  1  J 2  ) 7  + a  J 2  L 1  ) + 0  J2  +  7 J 2  )  i  n  + £ ( * B - ) L I * H ^ L I COS(/3 + PJI + ~Ul + « L 1 + Pj2 + 1  6  U2)  1=1 + 3 sin(/?  6  + (3JX +  - Y^{*Ei)Li6iilLi  7Ji) [*LI c o s ( / ?  6  + pjx  s i n ( f t + PJI  +  + 7J1 + <*L1 + # 7 2 +  7 J I + « L I + Pj2 + 7J2)]]  1=1 n  -  [ m  L  2  2  n  £(*M )L2*2t£i2 + m t  p  1=1  2  £($E )l2<$2i4r,2j t  1=1  153  7J2)  [(as  -  + h )  cos(/?  bx  2{a  + hhy) s i n ( / ?  y  + Ui  + PJI  6  + ct  + PJX + ui  6  + PJI  L1  + *LI  +  U2)  + Pj2  +  U2)  + 3 cos{p + PJX + Ui) [hi cos(p + pjx + Ui + ct b  b  + pj + U2) 2  sin{P + pjx + Ui + a i + Pj2 + U2)\  - ^{^E^Lihihi i=l -  L1  b  3 s i n ( f t + PJX + Ui)  hi  L  sin(ft  + Ui  + pjx  + « L I + PJ2 +  U2)  COs[p + PJX + Ul + <*L1 + Pj2 + U2)\\ >  + ^{^E^LlSlihl i=l  b  + £ J6[(JLI - / £ ) cos{p + PJX + ui) MPb + PJI + Ui) b  + VJ2 ~ !j2)  cos  + PJI + Ui + < * M + PJ2)  iPb  sm(p + PJX + Ui + a-Li + PJ2) b  +  VI2 ~ IL2) C°S(/?6 + PJI + Ui  sm(p + pjx b  + Ui +  + ( / J - I*) cos{p + sin(/?  +  + PJX + Ui  b  Ul)  s i n ( / ? 6 + PJX +  Ui)  n  i  n  J2  +  Ul + OLLl  +m  +  L2  U2)  U2)  + « L I + PJ2 +  c o s ( # , + PJX +  [(mxi/4 + ( m  <*LI + PJ2  PJX +  b  + « L I + PJ2 +  + PJ2 +  U2 + Ct ) L2  U2)  m ))£  2  p  L1  i  t=i3=1 n  + (m  J2  i  i  ,, m^^^i^E^Lii^E^LiSiiSxjtlxll  n  +m2 + L  t=i y=i nx + {m x L  nx  ^ ( $ c , ) L i O u h i  + (m  J2  +m  L2  +m) p  t=i [cos(/?  +  + PJX + Ui)  2  b  C O s ( / ? + PJX + 6  ~ sin(/?  6  + PJX +  Ul + CtLl + PJ2 + U2)  154  i=i  Ui) } 2  ^{^E^Li^uhijhi  Ui  sm(p + PJI + b  [(m  L 2  + OCLI +  + m )££  /4  p  n  2  n  PJ2 + U2)  2  2  - {™L2 X ] X ^ * t i ) " * * * i * « »=iy=i n « 2 n2 5  71  2  +  m  2  2  2  £ X;(^,.)L (*^.)L2*2tML2)] t = i y=i n n p  2  2  2  + {m 2 Y^iPc^LifclL*  + ™  L  t=l X  fc  2t^L2)£L2  5  + Pj2 + U2)'  6  / 2 + ( J 2 + L2 m  L 1  <  \2  L l  - sin(/? + PJI + ui + a [m  2  i = l  [cOs(/? + PJ + 7 J ! + a  +3  X^(*^)i  P  m  L  i  + /? + J2  7  j )  2  2  +Wp)  [(o« + h ) sin(/?6 + & r i + Ui) bx  + {a + h ) cos(p + PJI + J I ) ] y  by  n  b  7  i  i  n  t=i  t=i  [(a. + /i ) cos(/3 + pji + 7^1) 6  fra;  - ( a + h ) sin(/? + by  y  + (m {a  L 2  x  /2 +  6  +  7 > 7 1  )]  m )£ 2 p  L  + h ) sm(Pb + PJI + Ui + « L I + PJ2 + U2) bx  + {a + h ) cos(p + PJI + Ui + «Li + PJ2 + U2) y  by  b  + 2 [cos(/? + p 6  JX  + 7JX) ( £  i + YJpEJLihitLX  L 1  sin(ft +  +  7  >  n  + a  L  1  + p  J2  n  i=l  cos{p + PJI + Ui + ct i + PJ2 + b  + sin(/? + Pj! + Ul) [hi 6  n  L  7J2  )  . U2))  cos(P + PJI + Ul + ct \ + Pj2 + U2) b  L  i  ~ ^(QEJLISUZLI  +  sin(ft + p  155  3 l  +  Ui  + a  L 1  + p  J2  +  U2))]]  i=l  i = l  [(a* + h ) cos(/? + p hx  -  fc  + Ui + cc  JX  (a + h ) sin(/3 + p y  by  6  + ui  JX  L1  +p  b  L  $ B  t)  i l 5 l , £ i l  sin  + U2)  + <*LI + PJI  + 2 [cos(ft + pjx + ui) (t i cos{p + - 53(  + U2)  J2  + Ui + cc  PJI  L1  +  PJT. +  U2)  (A> + &Ji + ^ 1 + U + ^ 2 + 7J2,) a  1=1  - sin(/? + A n + 7Ji) ( ^ L I sin(ft + PJI + Ui + a 6  + t=i YI&EJLISUILI  cos{p + b  ui + a  PJI +  L 1  L 1  + pj  2  + PJ2 +  + U2)  7J2,)]]j  s(V>) 2  + /3JI + ui) ~ MPb + PJI + Ui) )  + 34{( Ii ~ J  2  + {Ij2 ~ 1*2) [cos(/3 + p 6  - sm{p + P b  + Ui  JX  + ( L2 - L2) [™s{Pb J  2  + Ui + <* +  JX  p )  L1  2  J2  /3 ) ] 2  + OCLI +  J2  + 0 J 1 + Ul + OCLI + PJ2 +  J  U2?  - sin(/? + PJX + ui + « L I + PJ2 + U2) ] 2  6  + [I* - if) \cos{P + PJX + Ui + «ii + Pj2 + U2 + a b  L 2  )  2  - sm{pb + PJX + Ui + « L I + PJ2 + U2) ] 2  + [cos(ft + pjx + ui) - *HPb + 2  [(m i/4 + (m L  n  i  J2  +m  L 2  +  PJI +  Ui) ] 2  m ))t\ £ 1 p  «i  t=i y=i i  i  n  n  + ( m + m + m )X]X]( ^) ^j) * * J ^JJ $  J2  L 2  il  L1  1  6l  £  p  ,=iy=i nx ~ 4{m  L1  n  x  53(*C -)Ll*lt<Ll t  +  ( J 2 + ™ L 2 + rn ) m  p  1=1  cos(/? + 6  1=1  +  sin(/3 + PJX + 7Ji) 6  156  ^{^E^Llhif-Ll)^!  + [m i/2 L  [(o -  + h ) cos(p bx  x  + (3ji + ui)  b  (dy + kby) sin(/?6  + -7.il)]  +p  Jx  »i  -  rn  » i  £ ( $ M ) L i 5 i t ^ L i  Li  t  +  (m  J2  + m  + m)  L 2  p  t=i  {a + h ) sm(P + PJI + x  + (a  bx  b  + h ) cos(p  y  5^(  $ £  \)  i l 5 l  *'  £ i l  J  i = i  by  Ui)  + PJI +  b  Ui)\  + [cos(/3 + PJI + ui + « L i + PJ2 + U2)  2  b  - sm(p  + PJI + Ui + « L I + PJ2 + U2)  2  b  [{m /4  +  m )l  n  n  L2  n  2  2  t=ij=\ n 2  + P £  ^{$E )L2{$E )L262i62je L2)\ 2  i  i=l  - 4^m  L2  2  m  n  2  p  j  j'=l  n  2  ( $ C , ) i 2 ^2^1,2 +  L 2  t=l  + (m  L 2  +  p  i  + 7J1 + « i l + PJ2 + U2)  b  b  rn Y^{$E )L2&2i?-L2)LL2 i = l  COs(/? + sm[p  2  PJI  + Ui + <*Li + PJ2 + U2) m )£  /2 +  p  L2  [(d + h ) cos{P + PJI + Ul + <*Li + PJ2 + U2) x  bx  b  - (d + h ) sin(/? + pji + ui + «ii + PJ2 + U2) y  by  6  + 2[cos(ft + P  JX  +  Ui){hi  i - ^{$E )Lib~iit-Li  cos{p  b  +  PJI  + Ui +  oc  L1  + PJ2 + U2)  n  i  1=1  s i n ( A + PJI + Ul + &L1 + Pj2 + U?.))  + Ui + i + X)(*£,)LI5I,£LI cos(ft + PJX + Ul + CL  - sm{Pb + n  PJX + Ui){hi  sm{p + b  PJI  LX  1=1  157  "LI+ +  PJ2 + U2)  Pj2 +  U2))]]  »»2  N  t'=l  2  t=l  [(a + /i ) sin(/3 + /3j + 7 J I + a z  bl  6  X  + (a + Ai ) cos(/? + y  6y  6  + 2[cos(/? +  +  6  n  + Ifji +  +p  L1  Ct i L  +  J2  + U2) + 1J2)  sin(/? + An + IJI + a 6  +p  L1  J2  + 7 ) J2  i  + ^{QEJLISUZLI  COS(/3 + pji 6  + 7JI + a  Ll  + /?  J 2  +  U2))  t=i  + sin(/3 + pji + i 6  cos(/? + p b  JX  + ui + <*L\ + PJ2 + U2)  n  - 2 ( * J 5 - ) L i S u h i sin{p + PJI + 7 J I + a b  t  »=i  }  008(0)  sin(0) = Q  n j  158  Ll  + p  J2  +  U2)))]]  6u Equation |(m  L  1  ($A/.)  L  + ( m  1  J  + m  2  [(a* + h ) cos(/3 + p bx  6  rn )($ .) i)£  +  L 2  p  E  L  LX  +  JX  + (a + h ) sm(p + PJI + Ui) by  y  b  + {mLi{$Ci)Li  + {m  + (*£,)LI£LI [(m n -  \rn  +m  J2  p  2  p  n  2  +  J2  L1  j  7  )  2  2  m Y(^ ) 6 i£  +  2  Ei LX  +p  LX  L2  P  t'=l  Ei  L2  2  L2  t=l  sin(a i + P  + j )]  J2  L  +  m )(^ ) )£  cos(a  / 2 + m )e  L 2  y^{^Mi)L2S j£L2  L2  +  L2  7  2  (*X,)LI  [if2 + II2 + m {t\ j\ L2  n i 2  + ™ (£ P  $  L 2  Y  2  2  2  2  2  + X ; £ ( ^ ) " ( » = i j=i  2  E($5, )L 5 ^ y££ )  + f;  2  ^ ) i 2 ^  $  2  y £ i  2  ) +  + ("l£2/2 + m )(.L2 p  [(a +  + 7JI + a  cos(ft +  z  + pj + j )  LX  2  7  2  + {a + h ) sm(p + PJI + Ui + « L I + PJ2 + U2) y  + £  L 1  by  b  cos(a i + Pj2 + U2) L  + Yi^E^Lihihi  sin(a  L1  +p  J2  +  7 j 2  )j  t'=i 2  2  n  + [m 2 L  n  53( Af,)l,2* $  2 t  £l,  2  [-(a  rn **T($ ) 6 £ ]  +  t=i  p  + h ) sin(p + PJX + Ui + ct  x  bx  b  + (a + hby) cos(p + PJI + ui + a y  - £  L X  Ei  L2  2i  L2  t'=i  b  sin(ai,i + PJ + jr ) 2  7  2  159  L1  L 1  + PJ + U2) 2  + pj2 + j 7  2  )  + ^Z^E^LISHILI  cos(a  p  i +  L  J2  + U2)]]}  [0 + tj>)  1=1  +  j(mi,i ($0^)1,1  (rn  +  +m  J2  L  + m )(*s )ii)^ii  2  P  [(m /2 + m ) £  + [QEJLXILI  L2  p  L 2  2  n  + U2)  cos(a + p 2 L1  J2  n  • - [m 253( M,)L2*2t^L2 $  m '*r{$E )L262ih2\ t=l  +  L  +  t  p  sin(a i  i  L  +  (3 + J2  )  7j2  ($X,)L1  i  /J  2  ( £ / 4 + 53 E ( * s , ) L 2 *  +& + m  2  i  2  L2  2  y  -ML)  2 l  t=iy=i  n  n  2  2  I,2^2t*2j££ )  +  2  z  t=i y=i  + (mi, /2 + m )£z, [£LI 2  p  2  + 1^{^E )LihdLi =1  + /?j- + 7 J 2  )  2  sm{a x + /3 + U2)  x  +  COS^LI  L  J2  t'=i  1=1  [-£ sin(ax,i + p L1  J2  + 7J ) 2  111  n i  + 53(* i)LIMLI E  cos(a \ L  + P  + 7  J2  J 2  Uj3 + pj +  )  b  x  f  n i  + J3j  (  m  £ l (  $  S - j ) £ l  ("V2 +  +  t  mp){^E )Ll{^Ej)Ll)i Ll 2  "»I2 +  i  3=1 ^ +  (*B )Li($x )n^Li t  y  n  -  +  2  [("* /2 + m )t cos(aii + p + 7 ) 2 t=i ' + p]C( t) »' J ( il + ^ p  i2  L2  ["»L2 53(* t"=l  M  t)  L 2 5 2 l  m  £ L 2  $ £  i 2 5 2  £ i 2  s i n  (*X,)LI [ ( * X y ) L l [ / J  2  1I2  +/ i  n  n  2  2  + m ( £ £ / 4 + 53 5 3 ( s , y ) " * ' M L ) $  L2  2  2  y=i  t=l  n i 2  + m (££ + p  J 2  J2  n  2  53  2  53($E )L2(* t  S y  )L2^^£  i=l3=1 160  2  2  ) +J*  a  2 +  7J )J 2  +($£y)z,i -  p  2  n  cos(a  m )£ 2  [(mr, /2 +  n  2  [mL £($M )L25 ,£L 2  l  2  +  L  1  [/f  2  J2  + 1J ) 2  2  +m Y^{^E )L2hi£L2\  2  p  »'=1  s i n ( a  0  +  L1  L  i  t'=l  + 0  + 7J2)]]  J2  j^iy  m (£| /4 + £  +  L2  X > * y )  2  t=iy=i  i  i  2  2  + ™r( L + ]T J ^ f t ^ C ^ U a f e M M ) + 1=1i=i £  [(m  L  2  /2 + m  ) £  p  L  cos(a i  2  L  2  [m  L2  2  r [  n  5^(*M -)L2*2i^2 + m  5^($S )r2*2^L2j  i=l  t=l  L  i  <  p  1  s i n ( a n  + U2)  J2  2  n  -  fi  +  + / ? j + 7J2)]  }{PJ  2  +  2  .  lJ ) 2  ,  + Y,{[ L2{^C )L2+rn {^E )L2\£l (^X )Ll m  i  p  i  2  j  y=i +  ( m  L  2  ( $  +  M  ) L 2  i  ™p{$E )L2)£l  +  i  {^X )L2l {^X )Ll}s i  p  j  n  + 7| 9 2 /  j 2  2  c o s ( a n  + / ?  +  p  J  2  7J2){$E )LI  +  J  2j  l  +( E/) X;(*r/ )M i l  J  t i  3=1 + (m i($AfJi,i + ( m L  -{a  +  x  (o  + h)  sin(/3  + h)  cos(/?  bx  y  by  J 2  + m  L2  m )($ ) x)t x  6  +  +  ui)  6  +  +  7JI)  Ei  L  + 2(0 + ^) [(o + h ) cos{p + PJX + 7J1) x  +  (a  + h)  y  bx  sin(/?  by  b  6  +  +  7JI)]  + (0 + 0 ) [ ( a + h ) sm{/3 + 2  z  bx  b  - ( a + h ) cos((3 + pjx + Ui) y  -  (mn  £ ( $ 5  by  i  y  b  ) L i ^ i y £ L i  y=i  161  + 7JI)  L  l  n  + {m  +m  J2  + m )($ )Li  L2  p  E)  Ei  Lihihi)  t  t=i (0 + rp + 0  -{m /2  7JI) £LI  + PJX +  H  2  + 0 + Pb + PJI + 7 J I + " l - i + PJ2 + IJ2)  + m )t 2{0  L2  p  2  L  s i n ( a n + PJ + U2) 2  n  n  2  2  - ]rn 2 ^{^M^LiMLt +m ^ ( ^ i ) ^ ^ t=l *=1 (0 + 4 + Pb + PJ1 + IJI + 0CL1 + Pj2 + ij2) 2  L  2  p  2  + PJ2 + U2)  cos(a  L1  - 2 ' m 2 Y^{$Mi)L262itL2 i=l  +™ p  L  {0 + 1p + Pb + Pjl sin(a  ]  L1  +  IJI  J^2x^2^  M  i=l <*L1 + PJ2 + 7J2)  +  + Pj2 + U2) + 4 + p + PJI + iji + *Li + PJ2 + IJ2)  + (^E )L2^{0 J  b  n 2 {m 2 E i=lj=l n n  2  ^2(^S )L2^2jh2  L  IJ  »2  2  + P M  53( ,) » $£!  i252  i=l  £L2  i=l  Yl^ i) ^ ) M  L2  L2  m )l 2  + (m /2 +  p  L2  L  + ( - ( S i + hx) S\l\{p + PJX + Ul + <*L1 + Pj2 + 7J2; h  + (a + h y) cos[p + PJX + 7J1 + « L i + P.J2 + U2)) y  b  b  .+ 2(0 + j>){{a + h ) cos(p + PJX + x  bx  b  7JI +  <*LI +  Pj2 + U2)  + (a + h y) s\n(pb + PJX + 7 J I + «L1 + PJ2 + U2)) y  h  + {0 + 0) ((a + h ) sin(/? + PJX + Ui 2  x  bx  6  - [a + h y) cos(p + PJX + Ui + ct y  h  b  + {0 + rj> + Pb + Pji +  lJl)  2  162  L1  +  <*LI +  PJ2 + U2)  + PJ + U2)) 2  (£  sm(a i + PJI + UT)  L1  L  »I -  C O s ( d ! L l . + Pj2 + 7J2y)  ^{^E^LlhdLl t=l  l  n  + 2{6 + ip + p + PJX + ) 53( ^)LI^I^LI $  7jl  b  t'=l sin(aii  + p  +  J2  1J ) 2  2  n  n 2  t=l  i=l  {a + h ) cos(p  + PJI +  (a + h ) sin(p  + PJI +  x  -  bx  y  b  by  b  J I+ a  + p  + U2)  J I+ a  + p  + U2)  7  L1  7  J2  L1  + 2(0 + 4>){-{a + h ) sm{p + pjx + x  + (a  + h )  y  bx  cos(/?  by  + (0 + 0 ) ( ( O 2  X  + PJX +  6  7  J I+ a  7  b  L1  +  b  2  6  b  + PJ + j 2 ) )  L1  + PJX + 7 J I )  2  2  c o s ( a i + / ? j + 7.72) L  n  +  2  i  .  sin(a + p  ^2(^E )Li6xihi  L1  {  +  J2  7 j 2  )J  t=i n  i  + 2{0 + tjj + p + PJX + 7 J i ) X ) ( * * ? i ) " * H * " b  t=i c o s ( a  L  + /3  1  n  c  1  J 2  + 7J2)  i  }  y = i  L  n  + (m  J2  +m  L2  i  + mp)($ }Li  ^{QEJLISUZLI)  Ei  1=1  (1 - 3cos(/? + &n + 6  -  ) ) 2  7 J 1  3(m i($c )Li + ( m j + L  t  2  m  L2  +  m ){$ )Li)hi p  cos(p + pjx + Ux) sm{p + pjx + J I ) b  2  +fc&x)COs(/3 + A/1 + 7J1 + CLLX + Pj  6 y  (e + xi> + p  + PJ + j  b  7  2  J I + CLLX + /?J2 + 7J2))  + (a + / i ) s'm(p + PJX + 7 J i + ct y  J2  7  163  Ei  7  + J ) 7  2  )  -  ( m £ , i ( $ A f , ) n + (mj2  + m  +  L2  m )($ )Li) p  Ei  {{a + h ) sin(/? + PJX + Ux) x  bx  6  + 2{a + h ) cos(/? + pj + Ux)) y  [-(m  L 2  hy  6  + m )l  /2  p  X  [2 s i n ( a L i + PJ2 + U2)  L2  + 3 COs(/? + PJX + UX + <*L1 + PJ2 + U2)  SVX\{P + PJX +  6  n  -  2  n  H  2  53(*Af )L2*2t£i2 + m 5 3 ( ^ ) L 2 ^ 2 t £ t 2 J [2COS(O:LI + p  \m 2  $  L  p  t  t=l  V=l  - 3 sin(/? +  + 7 J i + « M + £/2 + U2) sin(ft + A n +  6  + 3($X,)LI(/^2 - #2)  cos(/? +  b  + 3{*Xi)Li  -  {I 2 X  +  6  sin{p + pji + Ui + ct  7.,! +  a  L 1  +  PJ ) 2  + pji)  L1  cos(ft + PJX + Ux + « L I + PJ2 + U2)  ID  s\n(p + PJI + 7 J I + *LX + PJ2 + U2) b  + {if - Ij) COs{P + PJX + 7J1 + « L 1 + /?J2 + 7J2 + * B  sin(/?  Ux + ct  + PJI +  b  + U2)  + Pj2  Ll  + cos(p + PJX + Ux + ct x + PJ2 + 7.72) b  L  sin(/? +  +  PJX + IJX  6  n  2  t=l n  n  m  -  (m n  »=1 /4  L 2  2  n  53(* .) ^ » )  i252l£L2  i 2 5 2 l  '  £ i 2  +  m  L  +  2  p53^ t)i2^L )£L2 t=l S  + /?J2 + 7 J 2 ;  b  L2  £i2  2  - cos(p + PJX + Ux + ct x i  2  2  n  53( ^t) t=l  (.$x )Lx{m /2  L2  t=l  p  (sin(ft + PJX + Ux +  -  S  2  + m )£| ]  2  $  i  2  L  j=l $E  2  ct x + PJ2 + 7 J )  2  + P 53( ») + (m  Ux)]  + PJ2  m )l p  L2  164  2  + U2) ) 2  L 2  )  ux)  J 2  +  U2)  (a + h ) sin(/? + PJX + ui + " L i + PJ2 + U2) x  bx  + 2{a  6  + h )  y  c o s ( f t + PJ\ + ui + « L i + PJ2 +  by  sin(aLi + PJI + U2) i + 2^2{9 ) i6 l i cos(a + p  U2)  + l  L1  n  Ei  L  u  L  L1  J2  + U2)  t=i  + 3sin(/? + PJI  Ui){hi  +  6  - ^[QBJLISUILI  ui + ctLi  cos[p + pji + b  sin[Pb + PJI + Ui + ct  L1  +P  J2  + PJ2 + U2) + U2))  t=i n  2  2  n  + ( $ X - ) i l [ ™ L 2 ^J&M^LthitL*  i=l  {a  + h )  x  cos(p  bx  y  i  p  L2  2  t=l + PJI + Ui  b  - 2{a + hby) sin(/? + y  + £  m £^(^E ) 6 ilL2  +  t  ui + <*LI +  PJI +  6  + OLLI + PJ2 +  U2)  PJ2 +  U2)  cos(a i + Pj2 + U2) i -2^2($ ) i6ul isaju i=l - 3 sin(/? + Pj! + Ui){hi sin(ft + PJI + Ul + <* + PJ + U2) i , <) + J2^ i^l uhl COs{P + PJI + Ul + *L1 + PJ2 + U2))\ > L l  L  n  Ei  L  L  6  L1  2  n  E  S  b  »'=1  +  '  MLI\(m  Ll  j^PsijUiSijlLi n  +  {m  + m  J2  L2  L1  cos(p  b  t  p  E  LISUILI)  L  - sm{p + PJI + Ui) ]  2  2  b  + [rn  +m  J2  L2  +  m ){$ ) )L x p  Mi  + (mj + m  L  2  L2  L1  + m )($£.)/, ) P  (a + h ) s'm(pb + PJI + Ui) x  bx  + (a„ + h y) cos(p + PJI + Ui)] b  +  Ei  + PJX + Ui) sin(/?6 + PJI + Ui)  + (m i(9 ) i L  .  + m )($ .) i  [cos(& + PJI + Ui)  + 2(m ($c )Li  i  b  {*Ei)Ll  165  1  L  [ ( m / 2 + m )l 2  [sin(ax,i + PJ + U 2 )  + 2 sin(#, +  ui) cos(/? + p  p  i 2  L  2  PJI +  6  + ui  JX  + « L i  + A/2 + 7.72)]  2 "2 53($Af,)l,2*2.£L2 + m 53($E .)L252»£L2J [ c O s ( a n + p n  +  m  L  2  t  p  1=1  J2  i = l  - 2 sin(ft + PJX + Ui) sin(A + PJI + ui + « L I + PJ2 + 7 J ) ] 2  +  6 ( $ x  t  ) i i ( / J  2  - ij )  (ft +  c o s  2  sin(P + PJX + 7 J I + a B  LX  +  PJI  +  +  PJ*)  + PJI)  + 6 ( * ) n (l£ - I&) c o s ( / 3 + A n + Ui + « L I + A/2 + 7 J ) x<  6  2  sin(A + A / i + 7 J I + a  2  L 1  +U2)  +P  J2  + (/Jf - / * ) COs(A> + A/1 + Ul + Ct + Pj2 + U2 + Ct ) Ll  L2  sin(p + PJI + ui + °-LI + PJ2 + U 2 ) b  + cos(p + PJI + Ui + <*Li + PJ2 + U 2 ) b  s i n ( A , + PJI + Ui + ct [{m /4  + m )t\ n n  L2  v  2  L1  + PJI + 7^2)  2  2  - \ L2 m  n  +  t = l j=l 2 2 . pY^Y.^E )L2{^E )L2hi62jtl )\ n  m  j  i  n  t = l j = l  n  2  2  + (m 53($c )i2*2t42 + i2  2  t  t=l  m Y^{^E )L2hih2)h2 P  i  1=1  (cC-s(A, + Pji + Ul + CtLl + Pj2 + U 2 )  2  - sin(/? + PJI + ui + «ii + A/2 + 7 J 2 / ) 2  6  + H$x )Li(m /2 t  +  L2  m )L p  L2  + (a + h ) sin(p + PJI + Ui + ct x  bx  b  Ll  + PJ2 + 7.72)  + (a + h ) cos(p + PJI + Ui + ct i + pj2 + U 2 ) y  + l  L1  by  b  L  s\i\{a + Pj2 + U 2 ) Ll  166  +U2)  E ) L l S u l L l CC-s(an + 0J2 + U2)  +  T  »'=1 + 2sin(/? + /?ji + -YJi) 6  [i x cos(ft + PJI + 7 J i + « L I + PJ2 + U2) L  sin(/?b + PJI + Ui + a  - ^{QEJLISHILI «=i n  n  2  + 3 ( $ ) L l [m 2 ^{^Mi)L2^2itL2 X i  L l  +p  + U2)) j  2  + ™p  L  J2  E^Lihifa^  [(flj + Ai ) cos(/? + /?jri + 7 J I + a n + /3j + U2) 2  6  bl  + 7 J I + O L I + A/2 + 7 J )  - {a + /i ) sin(/? + y  6y  2  6  + £ n cos(an + pj2 + 7J2) - 53(*£ -)M*ii*Li sin(an + / f o + 7 J ) t=i 2  t  -2Bin(/9 + /?ji + 7 J i ) 6  ^ £ n sin(#, + PJI + ui + L\ + PJ2 + U2) A  + J2[*Ei)Li6utn cos{p + pjx + 7J1 + «z-i + PJ2 + Tfjj))]J cos(V>) 2  b  r  +^|3£n + (m  J 2  'T1  n i  -2(m  L 1  53(»  5 < i  )LiMiii  + m i + m ) ( $ B . ) n 53(*B )ii*i.£ii) i'=i 2  p  t  t  sin(#, + PJX + Ui) cos(p + PJI + Ui) b  + ( m n ( $ c j L i + {rnj2 + m 2 L  +  LI)£Z,I  (cos(/3 + iSji + 7 J i ) ~ sin(ft + / 3 J I 2  6  +  K I ( $ M , ) I I +  (mj + m 2  L 2  +  7JI) ) 2  + m )($E .)n) p  t  [{a + hbx) cos(Pb + ?7i -f 7 J i ) x  - (ay + h ) sin(/?6 + A / i + 7 J i ) ] by  +  {QBJLI ( m / 2 + m ) £ i 2 [cos(an + p i 2  p  J2  + 7J ) 2  167  - 2 sin(/?6  + PJI  [  £ 2  m  i252  £i2  + p  2  L  -  L  [sin(a i  sin(/?  + PJI + Ui + a + ct  + ui  J*)(cos(ft  sin(/?  +  U2)]  +  p  J2  U2)  +  +  jSjj + Ul  +  L i  p)  +P  J2  +U2)  2  J2  Pj2?)  PJX +  + OCLX + Pj  + PJX + Ui  6  +  L1  2  JX  +  L1  " /L )(COS(A  + p  6  L  B  + Ui  JX  + 3(S*.) i + ( / I  + (I* -  oc i + PJI  + Ui) cos{P + PJX + Ui + a  + {*Xi)Li{Ij2 ~ &)  -  +  i  t  b  Ui  i = i  + 2sin(/J +  sin(/?  +  rn  t=i  -  PJI  B  53( ,) *' + pY{^E )L2hdL2\ $Af  ~  ui) sm{P +  +  Ul + « L 1  2  U2) )  +  2  Pj2 + Ui)  +  2  + OCLI  + PJ2 +  + OCL2)  2  1J2  U2) )  + « M + PJ2 +  2  + (cos{p + PJX + Ui + « M + PJ2 + U 2 )  2  b  -  sin(/?  6  + (m  L 2  + OCLI + PJ2 + U2) )  + PJX + Ui  m )l  /4 + n  2  p  n  2  L2  n n  2  2  $  2  L  t  i  t=iy=i  n 53( C,)L2*2.^i2  n  2  4 ( m  $  L  2  +  6  sii\{P + L 2  m )t  /2 +  p  [(a* + h )  cos(/?  bx  -  (a  y  + lx L  + h )  Ui  $  L  1  + OCLI + + OCLI +  Pj2 + U2)  Pj2 + U2)  L2  + PJX + T / i + « L I + A/2  6  sin(/?  by  cos(a  Ul  +  PJX +  b  + (m  2  p  t=l  + A/1  COS(/?  »=iy=i  m 53( ^)L2*2t42j  t=l £L2  2  [m 2 53 5 3 ( 5 y ) L 2 « 2 ^ 2 i £ L 2 + ™ P 53 Y,(*E )L2{*E )L262i62jt L2)  -  -  [  2  6  + PJX + Ui  + P  + « L I + PJ2 +  U2)  +  J2  +  nx - 53( i) $s  -2sin{p  b  (L I L  ii<  * * i  £Li  s i n  ( ii+fa*+ui) a  + PJI + UI)  sin{p + PJX b  +  Ui  + OCLI +  PJ2  168  + U2)  Ui)  U2)  j  j  l  n  + Y2{^E )L\ML\  COS(/?  X  6  PJX +  +  Ul  +  a  + (3j2  L X  + U2))]  t'=l n  [(a  z  2  + /i  2  N  6 a :  ) sm(pb + PJX +  Ui + « t i  + ( a + h ) cos(p + PJX + Ul y  by  b  + t sin(a + P  + U2)  + ^2{$E )Li6iitLi  cos(a +  L1  Ll  J2  L1  t  1  + PJ2  + U2)  + « L i + PJ2 + U2)  PJ2 + U2)  1=1  + 2sin(/? + pjx + Ui) 6  ( / i i cos(/3 + /?Ji + 7 J I + a i + / ? J 2 + U2) L  6  - 53($J5 -)Ll£li^Ll sin(/?6 + A/1 + Ul t  t=l  }cos(^) sin(0) = Qs  u  169  + CXLl + PJ2 +  U2))  7J  Equation  2  n  r  n  2  2  Il + m ( ^ , / 4 + £  53(**/)«*WMM) t=i y=i  M  2  >•  ri2  "2  + ^ ( £ £ + 5 3 ( * * , - ) " * « « 53(*s<)«*tt*«)+/p *=i t=i £  P  2  + (m 2/2 + m )£i2 L  p  (a* + /ifca;) cos(/3  + A/2 + 7J2)  + {a +  sin(/? + PJI + ui + L \ + PJ2 + U2)  + t  M  A  y  b  cos(a + P  LX  n  + 7J2)  J2  i  + 53(  $ B  n +  + 7J1 + CCLl  B  t)  ^ *'  L 1  1  £ i  '  1  s  i  (  n  a  L l + 0J2 + 7J2) n  2  [ ™ L 25 3 (  $  A  f  t)  L  2  5  2  »'  £  i  2  +  2  p53(  m  »=1  $  S  t)  L  2  6  2  l  '  £  L  2  j  «'=1  + 7.71 + a n + p  [-(a, + Ai ) sin(#> +  J2  6x  + U2)  + {a + hby) cos(p + PJI + ui + £ i + PJ2 + U2) a  y  b  - t \sin(a i L  + PJ + IJ2) 2  L  11 + ^{^BihlSlitLl  COs(a  L1  + PJ + 1J2)\ H6 + 4>) 2  '  1=1  .  ri2  n  2  L2^2t^2j'^I,2) ^  t=l j'=l n  +  m  p  + (m  ( £ £  L 2  "2  2  + 5 3 ( t=l  2  $  ^ ) ^ 2 ^ L 2  53(*^)«*2»«L2) t'=l  +  /2 + m )£ 2 p  L  J^£LI cos(a i + PJ2 + U2) L  n  i  + 53(  »=i  , $ £  ,-) n  +  i l 5 l  *  £ i l  s i n  ( i i + ^ 2 + 7J2,J a  n 2  [mi,2 53( t=l  $ M  t) '2^2t^L2 + I  2  m 53( Ei)i2*2t^L2j $  p  »=1  170  f  ?  sin(ai,i + p  [-£  L1  J 2  + ^2{*Ei)LiSulLi  +  1J ) 2  cos(a  + p  L 1  + Tfja)] }(#> + h  J2  i=l  x  + 1J ) 2  ' n  nj  n  2  2  1=1y=i  t=i n  n  2  t=l  t'=l  + (*s -)zii^Li [(m /2 + m ) £ f  -  p  i2  n  [m  L 2  2  cos(ax,! + p n  + 7J2)  J2  2  X)( ^)"*2t^L t=i  + rn t=i  $  L 2  2  2  E )LihdL^ {  p  s i n ( a £ , i + (3j + 7J2)] j ^ i t 2  .  n  + {l£  + m (£ /4  2  L2  n  2  2  + £  2  L2  ^  Y,^ )L2S i6 il ) Sij  2  2j  2  i=iy=i i  i  2  %  2  + m ( £ i + X > * \ ) ^ 2 , L 2 X ^ . O " * " * " ) + I f \{Pj £  p  +  2  2  t=l  t=l  i3{(^L2(*c )i t  + m ($ .) p  2  B  L 2  + 1J ) + / j A j  J  )£i  2  + (* .)i, lf j* , x  2  t=l  2  '  + C ij J2  +  2  K IJ J2  2  + 2(0 + tP + /3 + PJI + 7 J I + a b  n  n  2  L1  + pj  2  + 7J2)  2  ("112 X ) y3(^S. )L2^2i^2j£r,2 J  »=i y=i n  n 2  »=1  + (m /2 -(a  t=l  +  L2  2  m )L p  L2  + h ) sm(p + PJI + Ui + « L I + Pj2 + U2)  x  bx  b  + ( a + h ) cos{p + PJI + u i + " L i + Pj2 + U2) y  by  b  + 2(0 + rjj)((a + h ) cos(/? + p x  bx  6  JX  + 7J1 + « L I + Pj2 + U2)  + (d + h ) sin(/? + pjx + 7 J I + a by  y  fc  + {B + 4) {{a 2  x  L  i + PJ2 + U2))  + h ) sin(/? + pj! + 7 J i + ct bx  L1  6  171  + p  J2  + 7J2)  2  2  2  - (a + h y  b y  cos(ft + pji  )  + (0 + ^ + ft + PJI ( £ i sin(ax,i + p L  n  -  +  J 2  Ui + OCLI + 0J2 +  +  U2))  UI)  2  + U2)  i  ^ { ^ E ^ L l h i t L l  COs{a  + p  LX  J  2  + 1J2))  t=l + 2(0 + ^ + ft + PJI + Ul)  Yli^E^Llkihl t=l  sin(a£,i + /3j + U 2 ) 2  2  n  n  «=i  t=i {a  + /ibi) cos(ft + pji  x  2  + Ui  + a-Li + PJ2 +  U2)  - (a + h ) sin(ft + PJI + ui + L i + PJ2 + U 2 ) a  y  by  + 2(0 + tp){-(a  + h  x  + (a  + h  y  b y  + 0 + ip ((a  x  + (a  + h  y  + (0 + xb +  b y  )  sin(ft + PJI  ) cos(ft + PJI + Ui + h  2  b x  )  b x  Ui +  +  + a  i + Pj2 +  L  ) c o s ( f t + PJI + ui  sin(ft + pjx  Pb + PJI +  + Ui  Ui)  OCLI + PJ2 + U2))  + OCLI + PJ2 +  + oc i L  + Pj2  +  U2)  U2)  U2))  2  cos(aix + pj2 + U 2 )  (i i L  n  +  l Yl^E^Llhit-Ll  Sm{oc l  + Pj2 +  L  1J2))  1=1 n  + 2(0 + xb + ft + p  J  X  + ui)  l ^ { ^ E ^ L i h i h i  1=1  C0S((XL1  + 4{  + Pj2 + 1J2)  3  { L2 J  IDcos(ft  + PJI  + Ui  + OCLI + PJ2 +  U2)  sin(ft + PJI + Ui + « L I + PJ2 + U2) +  {if  -  sin(ft  1%) COs(ft + p + PJI + Ui  J  X  + Ul  + OCLI + Pj2 + 1J2 + OC 2)  + OCLI + Pj2 +  L  U2)  172  + cos(ft + PJI + 7 J I + a  + PJ + U2)  L1  2  sin(/? + PJI + 7 J I + * i + pj + U2) 6  L  n  2  "2  2  i = i y = i  n +  n  2  P53(  m  $ B  t)  L 2 5 2  »'  2  53( »'=1  £ i 2  t'=l  $ £  t)  L 2  * «' 2  £ L 2  )  - {m /4 + m )t ] 2  L2  p  L2  n  n  2  + (m  y3($c.)Llfaih2 + m 53( S )l,2*2t^Z,2j£L2 t=l 1=1 $  p  L2  t  + 7 J I + a n + pj + U 2 )  (sin(/? + 6  b  - {m /2 +  m )t  L2  p  7JI  +  + a  L  1  P  +  2  L2  7  hx  + 2{a +  + 7J2) )  J2  {a + h ) sin(ft + PJI + J I + a x  2  2  - cos(p + PJI  +P  L 1  + U2)  J2  cos{p + 0 J I + 7 J i + « L i + A/2 + 7J2)  y  b  sin(ax,i + PJ + U2)  + l  2  L1  n  2  i  + 2 5 3 ( * S , ) L I * I I £ L I c o s ( a i + PJ + 1=1 L  + 3 s\n(p + PJX + 7 J I ) (i b  $s  li5i  £li s  i n  ^  7  j ) 2  cos(0 + PJI + Ui +  L1  - 53( ») *'  2  + A/2 + U2)  B  6 +  +  7 J 1+  a  i  l +  / 9 j 2+  7 J 2  VJ  »'=i  n  n  2  i=l  i = l  cos(/3 + A / i + 7 J I + ct  [(a, +  6  y  L  hy  cos(a  L1  + A/2 + U2)  sin(/?6 + PJI + Ui + « L I + PJ2 + 7J2)  - 2{a + h ) + li  2  M  + pj  2  + U2)  t'=l  - 3 sin(/3 + PJX + Ui) [t \ sin(A> + PJX + Ui + Li + PJ2 + Ui) a  6  L  173  + ^{QEJLISUILI  + PJI  cos(ft  +  Ui + ct i L  p  +  + U2))  J2  \  i=l  +4(  6  ( "  1*2) cos(/?6 +  "  J  PJX +  Ui + ct i + p L  J2  + U2)  sm(p + PJI + Ui + « L I + PJ2 + U2) b  + ( / J - I*) COs{P +  Ul + CtLl + Pj2 + U2  PJX +  b  Ui + ct i + PJI  s'm(Pb + PJX +  +  L  + <*L2,  U2)  + cos[p + PJX + Ul + CtLl + PJ2 + U2) b  sin[p + PJI + Ui + ct i + PJ + U2) b  L  2  [(m /4 + m )££. L2  p  n  2  n  2  n  2  - {m 2 £ Yl^S^Llhi^jt^  + P 53{^E )L2hdL2  L  t  t=l j = l  n  +  (m  (cos(/?  -  n  53( C») i=l  L 2  t=l  2  $  i25  2»  £i2  +  m  sin(/?  6  2  J3( E )L2Mn) $  {  »=1  2  P 5Z( sjL2^2iiL2^(-L2 $  t=l  + & 7 1 + 7.71 + « Z , 1 + PJ2 + 7 J 2 )  6  n  2  m  + PJI +  Ui + ct i + L  PJ  2  2  U2) ) 2  +  m )£ 2  + 3(m /2 +  p  L2  L  [(fli + h ) sm(p + PJX + Ui + « L I + PJ2 + U2) bx  b  + (a + h ) cos(p + PJI + Ui + ctLi + PJ2 + U2) y  +  by  b  ILI s i n ( a £ . i + PJ  2  +  U2)  i + Y^(®E )Ll6ldLl COs(ct x + Pj2 + U2) n  t  L  t= l  + 2 s i n ( f t + pjx +  Ui)  {t i cos(p + PJX + Ux + « L i + Pj2 + U2) L  b  - ^[^E^Lihihi  sin(/?  6  + PJX +  Ui + ct i + PJ2 + L  t=i  n  + 3 [m  L2  n  2  J3 t=i  2  + m J^i^ p  E) {  t'=l  174  L2^L2  7.72))]  [(a -  + h  x  {a  + h  y  cos(/? +  )  b x  b y  sin(p +  )  Ui  PJI +  b  PJI +  b  + ct i  + PJ  L  2  Ui +  ct i  +p  +  +  L  +  U2)  Pj2 + U2)  + L i cos(an + PJ + U2) L  -  2  sin(a  ^2(^Ei)Lihihi  L1  J2  U2)  1=1  -2sin{P  + PJI + 1JI)  B  (t  sin(/? + fc  L l  +  Ui +  PJI +  ]C(*J5 -)Li«i.-*Li t  + 3 ^ j + (l£ - / £ ) (cos{P 2  - sin(/? + 6  ui  PJX +  6  +  (cos(0  +u i  + ct i  Pj2 + U2))]\  + PJ2 + U2  + oc )  7J2) ) 2  U2?  - sin(/? + PJX + Ul + «L1 + PJ2 + 7 J 2 ) ) 2  6  z  p  2  n2 n  n 2  n  2  2  2  P ^Psi^LihiSijliL2 2  £  - (rn  L2  i=l 3 = 1  n "2  n "2  2  53( t) * «' ' )j $E  i  £x  ' 2  2  £  m  2  s  i=i  sin(p + b  + (m /2 + L2  + h  b x  )  I  2<52  *' ) £i2  t=i  cos{p + b  2  n  2  L  [(a*  2  53(*c-,)^*«'^ + P 53(* t) '  -4\rn 2  i2  i2  «=1  n  £  \  2  + ™p^2($E )L262itL2 t=l 2  PJI + Ul  PJI + Ui  + CtLl +  + ct i L  Pj2 + U2)  + PJ2 + U2)  m )t p  L2  cos(p + PJI b  + Ui  + ctLi  +  PJ2  +  U2)  - (o + h ) sm(p + PJI + Ui + « L I + PJ2 + U2) y  by  2  2  + <*Li + PJ2 +  [(m , /4 + m ) £ |  U2)  U2) )  +  2  +  + <*LI + PJ2 +  + ctLi  + PJ  L  + PJI + ui  B  Ul + « L 1  PJI +  + <*LI + PJ2 +  Ui  + PJI +  PJ2 + IJ2)  + PJI + Ui  B  b  b  L  cos(/? +  + {if - If) (cos{p + pjy - sm(p  +  ct i  b  175  L2  2  cos ^) 2  + i  cos(a x  L1  L  + U2)  + p  J2  i  n  sin(a i + PJ2 + 7 J Z )  ~ ^{QEJLISUILI  L  -2sin(/3 + / ? + ui) 6  (l  L1  J1  sin(/3 + PJI + ui + a i b  + ^Zi^E^LlSldLl B  - [™L2 Y^{^Mi)L2kif-L2 »'=1 {a  + h )  x  bx  sm(p  b  + rn  LX  , + "Ll + Pj2 + TfJ ))J 2  Y^pE^LtMLi]  t=l  + PJX + IJX + a  L 1  X  n  2  2  p  + (o„ + fcfcy) COs(ft + /?j + 7J! + a  + l  + 1J )  b  N 2  J2  COs{P + PJX + Ul  i=l  n  +p  L1  + p  L l  +  J2  + p  J2  7 ) J2  + U2)  sin(a i + PJ + 7 J ) L  2  2  i  + ^i^E^LiSni-Li  cos(a i + p L  J2  +  7 ) J2  t'=i  + 2sm(p  b  + Pjx + lJx)  (i x cos{p + PJX + Ui + « L I + PJ2 + U2) L  b  - ^(^E^LISIULI t=i  sin(/?& + PJX + Ui + ctLx + PJ2 + U2))  2  176  > c o s ( ^ ) sin(t/>)  6u Equation  {[m  i  ($c,)l2 + m  2  ($£;.) L 2 ^ L 2  p  [mx, (*M )i2 +  +  (  m p ( $  i  2  {a  + h  x  b  . ) i  S  ) cos(/3 + p 6  x  ] ^ L 2  2  J  + 1JI + « L I + A/2 + 7J2)  X  + {a + h ) sin(/? + PJX + Ui + « L I + Pj2 + U2) y  by  6  + t x cos(a L  L1  + Pj2 + U2)  i=l + + { ( m  (*X )L2l }(0  + J>)  Z  i  L  2  +  p  ( $  , ) L 2  c  +  m  {m ($M )L2 L2  p  ( ^  . )  E  + m  i  L  { ^  p  ) l  2  L  . )  E  sin(a  E^Lihif-Li  2 2  L  ) l  2  L  2  + Pj2 + U2)  L1  t'=i  + l  cos(a i + PJ + j )  L l  L  2  + ($x )L2l }{Pb  +  Z  i  n  P  7  2  Pj +lJx) 1  i  + Y^{[ L2{$Ci)L2  +  m  m  p  { ^  E  . )  L  2  ] t  { ^  2 L  2  X  j  ) L l  3= 1 +  {m {^M )L2 L2  + + { ( m n  2  + m {<Sf ) )l  i  p  Ei  {*X )L2l {$X )Ll}6l  cos(az,i + PJ2 +  2  L2  L2  1J2){^E )LX J  Z  i  i  2  ( $  C  t  ) L 2  + m  p  p  { $  j  E  i  )  L  2  j  ) t  2 L  2  + {$  X i  )L2l }{Pj2 p  +  1J ) 2  r  -I _  + ^Z[(mL2{^S )L2  + m  ij  p  ( §  E  . )  L  2  { $  E  j  )  L  2  ) l  2 L  2  +  3=1 B  ( L2  J= l  -{0 + xb + Pb + PJI + 7 J I + cc + P32 + ij2? L1  177  {^X )L2{^X )L2l \62j i  j  p  "2 +  53(*F,)L2*2,-42  m ( $ 2 5 . ) i ( / ) p  2  J^L2  I'=1 +  [ m  L  2  ( $ M j L 2 ( / )  -(a  +  z  + ™  p  Ji ) sin(/3 6z  ( $  . )  B  L  ( / ) ] £ L 2  2  + PJI + 7 J I + a  6  + {a + h ) cos(p + pjx + 7 J I + a y  +  by  b  + rj>)[(a + k ) cos(P + PJI  2(9  x  B  + Pj2 + U2)  L 1  +  + Ui)  J2  Ui  +  <*LI +  PJI  U2)  +  sin{Pb + PJI + 7 J I + « L I + PJ2 + U2)}  + (a + h ) y  bx  +p  L l  by  + (9 + rp) [{a + h ) sm(p + PJI + Ui + ct i + PJ2 + U2) 2  x  hx  h  L  - (a + h ) cos(p + PJI + Ui + a y  +  by  b  + rjj + p + PJI i  (9  J2  sin(a i + P  1JI) [ZLI 2  +  b  + p  LX  L  J2  + U2)] + 7J ) 2  n  - ^{^E^LlSlif-Ll t=l  C O s ( a l + Pj2 + 1J2)] L  + 2(9 + 0 + P + PJX + 7 J i ) X ^ ^ z - i ^ l - i  sin  H  + 7J2,J  ( £i + a  t=i  t + 4i  r £  n  2  n  L2 + { L2 J 3 ( 5 ) L 2 * 2 i ^ L 2 + m {9 m  $  i y  p  ;=i  f c  ( 1 - 3 COs(/? + 0J1 + 7J1 + <*L1 + PJ2 + 6  - 3(m ($c )L2 i  L 2  sin(Pb + PJI  - (rn ($M )L2 L2  +  +  m (^E )L2)f-L2  7JI  p  x  +  +  t  OLLI  2  ) L2 £ ( $ E . ) L * , £ I ) t=i  Ei  2  2  cos(p + PJI b  +  Ul + CtLl + Pj2 + U2)  + PJ2 + 7J2) Ei  L2  (a + h ) sm(p + PJI + 7 J I + «Li + PJ2 + U2) x  bx  b  + 2(a + h ) cos(p + PJX + 7 J I + « i i + PJ2 + U2) y  + t  L1  by  sm(a nx  L1  b  + PJI + U2)  + 2^2(^ )LiSxdLi Ei  cos(a  M  + p  J2  1=1  + 3sin(/? + / 3 + 7 6  J1  J 1  ) 178  2  1J2) )  rn ($ ) ) p  2  +7 J ) 2  ( £  + PJX + 7 J I + a  cos(0  L 1  B  + pj  Ll  + U2)  2  »i  -  Y^E^LI^UILI  sm{p  + pji + 7 J X + a  b  L1  +p  J  2  + U2) )  i=i  + H*X )L2(I i  sin(/?  f  b  + PJX + 7 J I + ct  6  r  + 4] " 3 £  (cos(/3 -  ~ Ij) COs{P + Pj! + 7.71 + « L 1 + PJ2 + 1J2 + «  X  p  n  2  L2  n  p  L  + A n + 771 + a  6  Li  p  t  + PJI + 7 J I + a  cos(p  b  6  [(a + h ) sia(P bx  s  2  +  £  X  by  1  s i n ( a  P  Ei  M  J2  +7J2)  + PJ + 7 J ) 2  2  t  + PJX + Ui + a  b  + p  1  2  + 7J2) )  2  L1  + PJ + U2) 2  + &/2 + 7.72)  b  £  2  2  J  + (a + h ) cos{p + PJX + Ux + y  {  m (*£ )i2)  +  Mi  L2  + 7J )  J2  +p  i+p  L  s i n ( / 3 + PJX + 7.71 + a + (m {$ )L2  E  2  m ($ )L2)£L2  + 2(m ($c -)j.2 + L2  )  ]T($ E) L2*2^L )  + m {^ .)  ij  + A/i + T J I + ot x + p  sin(/?  2  + U2) j  J2  {rn Y^i^S )L262jt-L2  fc  L2  + p  L1  L  + 7J ) 2  J2  i  n  + YL(® J E  LL6LI  ^  L1  c o s  ( L i + PJ2 + U2) a  t=i  + 2sin(ft + /3ji + 7 J i ) (£  cos(/?6 + PJI + 7 J I + a  L 1  L1  - Yl^E^LihdLi n  + PJ2 + U2)  i  sin(/?  6  +  PJI + 7 J I + O H + PJ2 + U2))  t=i  + H$X )L2{lJ i  sin(/?  +4 r  c  i  £  L  6  + PJI + 7 J I + CLLX + pj2 + 7 J 2 ) | c o s ( ^ )  r ^2 -2(m L  n  L2  2  n  Y,&S )L2&2jlL2 ij  + m {^ .) 2 p  y = i  6  E  L  2  53( £ )L2*2t£L2) $  t  « = i  c o s ( f t + PJI + 7 J I + a L i + PJ2 + U2) sin(/?  L2  2  6  s 3  - if) C O s ( / ? + PJX + iJX + Ct x + pJ2 + lJ2 + 0C )  + PJX + 7 J i + « L I + PJ2 + U2)  179  + ( m L a ^ c J w + m ($£ .)x, )£L2 p  t  2  (cos(/? + 0ji + 7 J I + ct 6  + /?j + - y ) 2  Ll  J 2  - sin(/? + fln + TfJi + «xx + / 3 6  +  (m  L  2  ($M )t2 t  +  2  + 7J2) ) 2  J 2  rn ($ .) ) p  E  L2  (a + / l ) COs(ft + 0jy + J ! + Ct + 0J2 + 7 j ) 6x  x  -  (a  + h )  y  + t  cos(a£i + P  2  L1  sm{Pb + PJI + Ui  by  L1  n  7  + <*LI + PJI +  + U2)  J2  i  - Y^(^ i) i=l E  -2sm{p  L l S l i i L 1  s  i  n  (  a  M  +  PJ2 + U2)  + PJI + UI)  b  sin(ft + PJI + 7 J I + a +p + U2) i + ^2(9 .) iS tLi cos(ft + pj! + ni + a  (t  U2)  L1  L l  J2  n  E  L  u  i=i  L 1  + Pj2 + 7J ))J 2  J  - I*)(cos(ft + PJX + 7J1 + CtLl + PJ2 + 1J2 + c t )  +  L 2  - sm(p + PJI + Ui + an + PJ2 + U2) ) 2  b  180  \ sin(V>) cos(V>) = Qs  2i  2  

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