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SPACAR User Manual

1. keywords INPUTS 1 input number 1 2 element number 3 deformation parameter number 1 2 3 4 5 or 6 INPUTF 1 input number x1 2 node number 3 coordinate number 1 2 3 or 4 OUTE 1 output number 1 OUTEP 2 element number OUTEDP 3 deformation parameter number 1 2 3 4 5 or 6 OUTX 1 output number 1 OUTXP 2 node number OUTXDP 3 coordinate number 1 2 3 or 4 NOTES x1 The input numbers and output numbers are the positions of the specified input or output in the input and output vectors respectively They need not be identical to the nominal input vector and reference output vector specified during the generation of setpoints see Sect 2 4 2 and or Sect 2 5 but for a quite straightforward comparison it is convenient to use at least partially the same numbering scheme 44 Chapter 2 Keywords 2 7 Visualization and animation To adjust the default settings of SPAVISUAL the user can type VISUALIZATION after the last two END commands in the dat file All commands after the command VISUALIZATION are read by SPAVISUAL as an adjustment on the default settings BEAMVIS HINGEVIS TRUSSVIS TRANSPARENCY VIBRATIONMODE BUCKLINGMODE ENLARGEFACTOR RECORDMOVIE MOVIENAME UNDEFORMED VIBREND STEPLINE STEPVIBRATION LIGHT JOINTS TRAJECTVIS TRAJECTNODE
2. Figure 1 2 Typical overview with MATLAB analyses and a SIMULINK simulation Read Uo Dee vo Unom 4 SPASIM fi e fi Read Yo ta control ew u actuator o mechanism sensor y Yref system model model L gt model read setpoints amp ee coeff matrices _ i control Ih fo Read gt parameters MO e g M function Figure 1 3 Block diagram of a typical closed loop simulation in SIMULINK The left blocks read setpoints and coefficient matrices stored in data files during previous SPACAR analyses Fig 1 1 1 SPASIM the non linear open loop model of the manipulator with its actuators and sen sors It operates in a way comparable to the forward dynamics mode in SPACAR as dis cussed for the MATLAB interface in Sect The mechanism is defined in an input data file of file type dat The filename of the input file must be specified An output log file is written Note that in a SIMULINK simulation the integration is determined by the SIMULINK environment e g the kind of solver the step size and tolerances The degrees of freedom of the mechanism and their first time derivatives are the states of the SPACAR S function The dimensions of the input and output vectors are determined from the input file and should match the requirements of the other SIMULINK blocks they are connec
3. Adjusts the height and the width of a beam element Adjusts the radius and the length of the hinge element Sets the visualization of the truss element on or off Adjusts the transparency of the elements Selects the vibration modes Selects the buckling modes Sets the amplitude of the vibration or buckling modes Sets recordmovie on or off the movie is saved as an avi file in the workspace Sets the name of the recorded movie Sets the visualization of the undeformed mechanism on or off Sets the period of the sine function for the vibration mode Sets the size of the line elements that are used to draw the elements Sets the number of steps in the vibration visualization Sets the light on or off Sets the joints on or off Sets the trajectory on or off Selects the node for the trajectory Section 2 7 Visualization and animation 45 The parameters for these keywords are listed below KEYWORD DESCRIPTION DEFAULT SETTINGS BEAMVIS x1 1 size of all beam elements in the local y direction 0 006 2 size of all beam elements in the local z direction 0 006 BEAMVIS x1 1 element number 2 size of the element for the local y direction 0 006 3 size of the element for the local z direction 0 006 HINGEVIS 1 element number
4. END END 58 Chapter 3 Examples Note that in the initial configuration the input shaft is rotated by a right angle with respect to the configuration in Figure 3 21 However the visualization of this simulation is quite poor This can be improved by adding some beams to the input and output rotational nodes numbers 2 and 4 respectively The com plete input file cardan dat becomes INGE INGE INGE INGE rg ne BEAM BEAM BEAM BEAM tj tj x x KM MX INPUTE Wr O Dd ona 0 Bw WN EF O1 F BW NY EF Oe W NH HN 9 N oM who I 707 707 0 707 0 707 RLSE RLSE RLSE END HALT INPUTE w TIMESTEP 100 END END oo a 0 l 0 707 0 707 707 rae One 0 0 14 0 0 707 0 707 The initial configuration of this mechanism is shown in Fig Figures and 4 e respectively The MATLAB commands to plot these data are gt gt plot e le 1 1 e le 1 1 e le 1 1 e show the zeroth first and second order geometric transfer functions from input ef to output le 4 1 Section 3 3 Cardan joint mechanism 59 p 7 0 Figure 3 22 Initial configuration of the cardan joint 3 el rad Figure 3 24 First order geometric transfer func tion for the cardan joint gt gt grid
5. XF 5 1 0 0 0 END END In a MATLAB session we obtain gt gt spacar 8 column2 gt gt edit column2 log Load multipliers and normalized buckling modes Load multiplier no 1 iO 3 9 94384680E 00 4 00000000E 01 1 28722820E 02 Buckling mode nro 1 to 3 0 2596610869 1 0000000000 0 0519056301 0 8141747968 0 0000000000 0 9932416764 0 5193221738 0 0000000000 0 1038112603 Hence we find a load multiplier A Fy Fo 9 944 Since Fo 1 we have Fu Fin 9 944 7 1 0075 68 Chapter 3 Examples 3 7 Cantilever beam subject to concentrated end force of N 2 gt D EN o o Figure 3 29 Cantilever beam loaded by a concentrated force at the free end Consider a slender cantilever beam with a circular cross section of diameter d 1cm and length l 10m The material properties for this example are EI 102Nm The beam is subdivided into 5 planar finite elements as shown in Fig A point force F of 14N is applied along the vertical axis at the free end of the beam It generates an elastic deformation as shown in the figure The deformation is reached in ten steps of loading For each step the residual vector converges in 4 Newton Raphson iterations with an accuracy equal to 0 5K 6 A SPACAR input file 0 lbeam5 dat for this case is PLBEAM 112 3 4 PLBEAM2 34 5 6 PLBEAM 356 7 8 PLBEAM 4 7 8 9 10 PLBEAM 5 9 10 11 12 x 1 0 0 Xx 3 1 66
6. In mode 4 the LINEAR module is an extension of the forward dynamic analysis node 1 where coefficient matrices of the linearized equations are calculated as functions of the set of degrees of freedom q If there are only holonomic deformations in a system the linearized equations are generated in the form M dq Co Do 64 Ko No Go dq DFO Sf DFO 50 1 1 where My is the reduced mass matrix Cy the velocity sensitivity matrix Dp the damping ma trix Ko the structural stiffness matrix and No and Go are the dynamic and geometric stiffness matrices respectively External and internal driving forces are represented by the vectors f and do respectively In addition if input and output vectors u and dy are defined also the linearized state equations and output equations are computed see mode 9 In mode 3 locally linearized models are generated about a predefined nominal trajectory where the output data setpoints from the inverse dynamics module i e a previous mode 2 run are used In addition to the coefficient matrices a complete state space system is generated and written to a so called 1tv file see Sect 1 5 In the case of a flexible mechanism additional degrees of freedom describing the elastic behaviour of the mechanism have to be included in the dynamic models both mode 2 and 3 At this stage in the so called rigidified model these flexibilities are prescribed zero i e e In mode 7 eigenva
7. For the keywords in the third block it is important to remark that there are no keywords to fix a deformation or to release a coordinate These are the default settings So a deformation is fixed unless a RLSE INPUTE or DYNE keyword specifies otherwise Similarly a coordinate is calculable unless a F X INPUTX or DYNX keyword specifies otherwise For systems with non holonomic deformations dependent coordinates or deformations can be specified as generalized configuration coordinates by the keywords KINX and KINE these are called the kinematic generalized coordinates and the corresponding velocities are not dynamic degrees of freedom With the keywords of the fourth optional block the calculation of some non linear terms in the expressions for the deformations of planar or spatial beams can be suppressed and geometric properties for PINBO beam can be specified The keyword in the fifth section is not really a kinematic keyword as it sets the level of output from the program DY elements and their cognates rigid beam planar pinbody planar rigid 18 Chapter 2 Keywords KEYWORDS KINEMATICS PLBEAM Planar beam element PLTRUSS Planar truss element PLTOR Planar hinge element PLBEAR Planar bearing element not supported PLPINBOD Planar pinbody element PLRBEAM Planar rigid beam element
8. ndof ndof 6 Only available for mode 4 and mode 9 In mode 2 3 4 and 9 a so called 1tv file is created The contents of this file varies and is not automatically imported to the MATLAB workspace From a mode 2 run the following data are available the names identitify the data used in the file data marked with are available at each time step NNOM number of actuator inputs NY number of outputs T time U0 nominal input for the desired motion YO reference output of the desired motion In the addition the linearization runs yield additional setpoints state space matrices and other data in the 1t v file not all data are always present NNOM number of actuator inputs NX number of states 2xndof NU number of inputs length of U0 NY number of outputs length of YO NRBM number of rigid body DOFs NYS number of outputs with 2 order expression 8 Chapter 1 The SPACAR program KOB combined stiffness matrix Ko No Go SIGO generalized stress resultants The getss tool can be used to read the state space matrices from the 1tv file see page 9 Other utilities are available to use parts of these data in a SIMULINK environment e g to read setpoints or to simulate a linear time varying LTV system see Sect 1 4 NYSI index array for outputs with 2 order expression DFT direct feedthrough flag D 0 X0 initial state vector T time 5 A state space system matrix i B state s
9. Figure 3 33 Second buckling mode for a can tilever beam with 5 elements Fer 22 715N theoretically 22 651 N Figure 3 35 Third buckling mode for a cantilever beam with 5 elements Fer3 64 798 N theoreti cally 62 919 N Section 3 8 Short beam 71 3 8 Short beam Figure 3 36 Short Timoshenko beam loaded in shear In this example the influence of shear deformation on the behaviour of short beams is studied A square plate is loaded in shear in its plane as shown in Figure 36 The beam has unit height h length l and Young s modulus F and a small unit width t With Poisson s ratio v 0 27 the shear correction value is k 10 1 v 12 11v 0 8484 The deflection if shear deflection is taken into account is F 1 v Fl l Y IEI kEth ens with J th 12 So the compliance is l 2 1 l 2 1 Z a En Mo 3 14 F12EI kEth The moment of inertia per unit of length is J pth 12 An input file in which the beam is modelled by two planar beams of equal length shear2 dat in this case is PLBEAM 1 PLBEAM 2 3 23 4 45 6 X 1 x 3 x Oo 00 ouo oOoO6o0 ooo Hj j RY Ry be bd bd bd O1 oy NO RLSE 1 RLSE 2 DYNX 3 72 Chapter 3 Examples END HALT EM 1 DYNX DYNX EM 2 ESTI FF 0 0833333333 ESTEI EE ITERSTEP 10 1 END END 0 1 0 0 0833333333 1 2 1 0 0 0833333333 0 2495 1 0 0 083
10. On Ww N o w N A x OX X OW e O N H amp tj INPUTX 2 1 DYNE L DYNE 31 H RLSE 2 1 END HALT XM 2 1 XM 3 0 8 XM 5 0 5 ESTIFF 1 130 ESTIFF 2 105 INPUTX 2 1 0 0 10 0 END END In a MATLAB session we find for the stationary configuration 791 and ro2 in agreement with Eqs and 3 10 gt gt spacar 7 massspring gt gt x lnp 3 1 lt ans 0 1184 gt gt x lnp 5 1 lt lt ans 0 2891 The linearized equations of motion in terms of the dynamic degrees of freedom are m 0 Ory ky ko mi Lh sr 0 allerlei om Section 3 5 Rotating mass spring system 65 The associated frequency equation is given by 0 k k m Q ko det w 7 en 0 3 12 i 0 ma k ka ko mad en where the quantities w are the natural frequencies of the system In a MATLAB session we obtain gt gt spacar 7 massspring gt gt m0 m0 0 8000 0 0 0 5000 gt gt k0 k0 2000 700 700 700 gt gt nod n0 80 0000 0 0 0000 50 0000 The complex eigenvalues and associated eigenvectors can be found in the log file Complex eigenvalues and normalised eigenvectors of the stat system matrix space Notation real imaginary Eigenvalue numbers 1 to 4 0 00000E 00 5 55511E 01 0 00000E 00 2 47806E 01 Eigenvector numbers 1 to 4 0 0141650 0 0000000
11. UTEP x7 3 deformation parameter number UTEDP 1 2 3 4 5 or 6 UTF x8 1 output number 1 UTX 2 node number UTXP 2 x9 3 coordinate number 1 2 3 or 4 UTXDP OOO O O CO CO Section 2 5 Linearization 41 NOTES 1 2 3 x4 x5 6 7 8 9 The input numbers and output numbers are the positions of the specified inputs or outputs in the input and output vectors respectively Associated with dynamic DOFs e or dependent coordinates e Associated with prescribed deformations e For INE only holonomic deformations are allowed Associated with calculable coordinates x or dynamic DOFs x Associated with prescribed nodal coordinates 2 Can be associated with prescribed deformations e or e but can also be associated with the free types in which case the output stress is calculated from the constitutive equations and possibly from the input stress Associated with calculable deformations e or dynamic DOFs e or prescibed defor mations e Associated with prescribed nodal coordinates x or a Associated with calculable coordinates 2 or dynamic DOFs a or prescribed coordi nates x 42 Chapter 2 Keywords 2 6 Non linear simulation of manipulator control To simulate the behaviour of a manipulator with a control system the SPACAR program is also accessible as an S function block SPASIM from SIMULINK SIMULINK
12. 1 total trajectory time total trajectory time 2 number of time steps number of time steps for an extended analysis 3 number of time steps within the previous step For three arguments the total number of time steps is a multiplication of the last two arguments In intermediate points a standard analysis is done 4 The keywords TRVMAX and TREPMAX have an optional third argument to express the ex treme velocity creation of a zero acceleration period If no extreme is given it can be calculated from the total time and path length The second argument contains the rise time The period of deceleration is calculated from the a total time b rise time c total path length d extreme velocity In this way the velocity profile is fully determined For asymmetrical velocity profiles the rise time can be calculated too To indicate the symmetry of the profile the second argument is given a dummy argument a non positive value The default velocity profile is symmetrical without constant velocity period The keyword TRFRONT has a second argument for the type of acceleration and deceleration function of time There are three types of fronts 0 constant acceleration 1 sine function half period 2 quadratic sine function half period The default velocity front has a constant acceleration type 0 6 The keyword TRM has only for 3 D orientation nodes a real list of parameters For 2 D orientation and po
13. gt no gt gt gO spacar 4 fourbar mO 1 0000 15 1423 Section 3 5 Rotating mass spring system 63 3 5 Rotating mass spring system Figure 3 27 Rotating mass spring system Consider the system shown in Fig A smooth horizontal tube containing masses m and mz connected with springs k FA l and kg EAs lz is mounted on a rotating shaft The shaft rotates at constant angular speed The unstretched lengths of the springs are denoted by l and l2 The equations of motion in terms of the generalized coordinates r and T2 are fe 0 4 _ lima k lr l1 ka r2 r u 3 7 0 m To MP ra ka r2 r l The stationary solution ro1 792 is obtained by substituting a 7 2 0 li k2 m ko Re lige g ky ka mad T02 Kaly 3 8 from which the stationary configuration 191 792 is obtained analytically as mak l Makala J o k kal MimQ koma kgm k ma k ka 3 9 ro m d kala k k l l2 3 10 02 m mad kama kom k ma kiko This result can also be obtained numerically from a SPACAR analysis E g with the following numerical values L 0 10m ky 1 3kN m ly 0 15m ko 0 7kN m m 0 80kg 10rad s m 0 50kg A SPACAR input file massspring dat describing this case is 64 Chapter 3 Examples PLBEAM 1 PLBEAM 23 PLTRUSS 3 1
14. gt gt xlabel e_1 1 rad gt gt ylabel e_4A 1 rad gt gt gt gt plot e gt gt grid gt gt xlabel e_1 7 1 rad gt gt ylabel DF e_l 1 gt gt gt gt plot e le 1 1 edd 1 gt gt grid gt gt xlabel e_1 1 rad gt gt ylabel D 2F e_l 1 i i i 3 4 5 6 ef rad Figure 3 23 Zeroth order geometric transfer function for the cardan joint 3 4 5 6 ef rad Figure 3 25 Second order geometric transfer function for the cardan joint 60 Chapter 3 Examples 3 4 Planar four bar mechanism In examples 5 7 1 and 12 4 1 of the lecture notes 1 the planar four bar mechanism of Fig is analysed analytically The mechanism has one degree of freedom The mechanism is mod elled by four rigid truss elements denoted by 1 2 4 and 5 which are joined together at their nodal points to form a rhombus As Fig 3 26limplies these four bars are set at right angles Figure 3 26 Four bar mechanism to one another The diagonal element 3 represents a spring with stiffness k EA Io A con centrated mass m is attached to node 4 The deformation parameter e3 has been chosen as the generalized coordinate The equation of motion is m s V2m 3 kez mg 3 1 Using the coefficient matrices from the lecture notes the linearized equation of motion is mo 3 2Y 2megz ez k V2mg 2V2
15. type e 1 1 1 type ez 1 1 4 2 o 0 2 o 0 type e3 1 1 4 mass per length 0 2225 0 2225 0 2225 EA 5 65 106 5 65 10 5 65 10 EI 13 4 13 4 13 4 damping 0 0 0 T translational R rotational Section 3 3 Cardan joint mechanism 57 3 3 Cardan joint mechanism In section 11 1 of the lecture notes a cardan joint is described Cardan joints also known as Hooke s joints have been used as a shaft coupling in a wide range of machinery which includes locomotive as well as automotive drive lines A drive line connected by a Cardan joint may exhibit torsional oscillations due to fluctuating angular velocity ratios inherent in such systems e9 NY 7 u a b Figure 3 21 Schematic of Cardan joint system Figure shows a one degree of freedom shaft system incorporating a Cardan joint The Cardan joint is modelled by four spatial hinge elements as shown in Figure B 21b The ro tating shaft axes having an steer misalignment of 3 45 is driven at a constant angular speed Q The quantities e and e 4 represent the input and output angles of the hinge elements D and 4 respectively The essential behaviour of the joint can be simulated with the following input file cardansimp dat 1 0 0 0 a 0 0 0 ze 0 707 0 707 0 INGE INGE INGE INGE Oe W NY D DG E DUNE DUNE FIX 1 FIX INPUTE RLSE RLSE RLSE BW DN END HALT INPUTE TIMESTEP 1 0 100
16. 0 XF 9 1 5 TIMESTEP 0 2 100 END END VISUALIZATION VIBRATIONMODE 1 ENLARGEFACTOR 0 2 By running SPACAR with mode 7 it can be checked that the initial position is indeed an equilibrium position but it is unstable One mode has an eigenvalue equal to zero and in the corresponding mode shape the distal phalanx remains horizontal The other mode has an exponentially increasing or decaying motion and is shown in Figure 58 Note that the current version of SPAVISUAL does not draw the pulleys A simulation with an initial perturbation specified by STARTDX STARTDE 9 1 21 0 001 0 04 0 004 0 16 shows this unstable behaviour Note that the initial perturbation is approximately in the direction of the eigenvector corresponding to the unstable eigenvalue 92 Chapter 3 Examples 0 04 0 02 0p 0 02 0 04 0 06 L L 1 1 1 L 0 05 0 0 05 0 1 0 15 0 2 Eigen Frequency 0 6 3191iHz Figure 3 58 Unstable mode Section 3 14 Tricycle 93 3 14 Tricycle Figure 3 59 Tricycle A tricycle has two rear wheels on acommon axle and a front wheel in a fork that can rotate about a steering axis with respect to the rear frame The steering axis can be vertical A 0 or be inclined see Fig The input file for the planar version trike2 dat is PLRBEAM 1 1 2 3 PLWHEEL 2 3 2 4 0 3 0 0 1 0 PLRBEAM 3 1 2 5 PLWHEEL 4 5 2 6 0
17. 0 03 LIGHT 1 TRAJECT 1 TRANSPARENCY 0 6 TRANS 8 0 536 0 0 TRAJECT 1 TRTIME 0 2 20 TRAJECTNODE 8 TRAJECT 2 TRANS 8 0 13 Che TRVMAX 8 0 2 1 76 TRFRONT 8 0 TRTIME 1 0 100 The inverse dynamics analysis yields the stresses that have to be applied at the hinges and the deformations of the hinges Fig shows the stresses Figures and show the deformations which are the relative rotations of the hinges and the first time derivatives respectively Clearly to accomplish the quite simple trajectory of the end effector of this non Section 3 11 Rigid spatial manipulator mechanism 81 Figure 3 41 SPAVISUAL output for the spatial manipulator mechanism linear mechanism rather complicated functions for the rotation of the hinges are needed Note that the input file defines the inputs and outputs that will be used in a SIMULINK simu lation The nominal inputs are computed to accomplish the deformations of the hinges The outputs include the six sensor signals with the rotations and the speeds of rotation of the hinges Nine more outputs are defined to obtain extra information on the performance of the manipula tor the acceleration of the rotation of the hinges and position and velocity of the end effector At the end of the file visualization settings for SPAVISUAL are defined In figure 3 4 1 the output of SPAVISUAL is presented 82 Chapter 3 Examples 600 T T T 1 4 T 500 12h J 400 4 1P 4 300
18. 0 1 0 0 0 WHEEL 11 11 12 13 0 0 1 0 0 0 HINGE 12 14 2 0 0 0 0 1 0 X 1 0 3 0 0 0 9 X 3 0 0 0 35 0 3 X 5 0 0 0 35 0 0 X 6 0 0 0 35 0 3 X 8 0 0 0 35 0 0 X 9 1 03 0 0 O53 X 11 1 00 0 0 0 25 X 13 00 0 0 0 0 FIX 14 RLSE 12 2 3 RLSE 6 6 DYNE 2 1 DYNE 8 1 KINE 5 KINE 10 1 KINE 12 1 KINX 1 KINX 1 2 END HALT GRAVITY 0 0 0 0 9 81 XM 1 80 0 XM 2 2 0 0 0 0 0 3 0 0 0 5 0 XM 3 2 0 XM 4 0 0 0 0 0 0 0 1 0 0 0 0 XM 6 2 0 XM 7 0 0 0 0 0 0 0 1 0 0 0 0 XM 11 1 5 XM 12 0 025 0 0 0 0 0 05 0 0 0 025 STARTDE 2 1 0 0 10 0 STARTDE 8 1 0 5 0 0 TIMESTEP 1 0 100 END END Note that hinges elements 2 5 and 10 are used to connect the wheels to the rigid beams and an additional hinge element 12 is introduced in order to make the yaw angle available With mode 7 the same eigenvalues are found as for the planar model If variables are saved from the run with the planar model it will be seen that the results of a simulation are very nearly the same The three dimensional model has the advantage that the normal forces at the wheels are calculated which are the first components of the stress of the wheel elements Fig B 60 gt gt plot time sig le 3 1 k time sig le 6 1 k time sig le 11 1 k grid on 96 Chapter 3 Examples 100 4 d 200 Er 4 Pa er RER lesen adem wes normal force N 400 500 _600 l l N N l
19. 3 0 0 1 0 PLRBEAM 5 1 2 7 PLTOR 6 2 8 PLRBEAM 7 7 8 9 PLWHEEL 8 9 8 10 0 25 0 0 1 0 x 1 0 3 0 0 x 3 0 0 0 35 X 5 0 0 0 35 X 7 1 05 0 0 X 1 00 0 0 RLSE 4 2 DYNE 6 1 KINX 1 1 2 KINX 2 1 DYNX 4 KINX 6 1 KINX 10 1 END o A Chapter 3 Examples HALT XM 1 80 0 XM 2 5 0 XM 3 2 0 XM 4 0 1 XM 5 2 0 XM 6 O 1 XM 8 0 025 XM 9 1 5 XM 10 0 05 STARTDX 4 1 0 0 10 0 STARTDE 6 1 0 5 0 0 TIMESTEP 1 0 100 END END The origin of the coordinate system is initially located at the centre of the rear axle with the x axis pointing in the forward direction and the y axis pointing to the left The centre of mass of the frame is at a distance of 0 3 m in front of the rear axle The rear wheels elements 2 and 4 have a radius of 0 3 m and are connected to the centre of mass of the frame by two rigid beams elements 1 and 3 Another rigid beam element 5 connects the centre of mass of the frame to the steering head where the hinge element 6 makes the connection to the front fork The rigid beam 7 represents the rigid connection between the steering head and the the front wheel element 8 with radius 0 25 m which is conncted to the front fork All wheels can rotate freely about their spin axis The frame and the wheels have mass but the front fork is assumed to be massless The system has two degrees of freedom the rotation angle of the left rear wheel and the steering angle are
20. 30 END EM 4 4 EM 5 2 XF 1 0s 0 14 XE 5 0 u 121 XF 8 Ou 0 307 END HALT The other blocks in the block diagram are standard SIMULINK blocks and are used to export data to workspace and to display results on the screen The Selector blocks select only specified components from an input vector They are e g used to select only the first three components of the output vector deformations of the hinges as displaying all components makes the graphs unreadable SIMULINK s ode45 solver is used with a relative tolerance of 1075 an absolute tolerance of 1078 and a maximum time step of 0 01s With these parameters the simulation of the motion from t 0 0s tot 1 5s is completed after 172 time steps The size of many time steps is 86 Chapter 3 Examples robotinvlin a 125 15 T gt Selector 3 7 botsi Sel E Setpoint UO SPASM elector Scope E 3 unom 15 y Unom To Workspace r Y To Workspace 15 15 1 Selector 3 gt Selector Ytip Scope dYtip ER ag saeco Reference YO Selector Eref Scope Eref Display Time t Yref To Workspace Clock Time To Workspace Figure 3 48 Block diagram for an open loop simulation of the motion of the manipulator mechanism using SIMULINK 2 14 0 2 f 1 fi fi L fi 0 i time s i Figure 3 49 Deformation of the hinges of spatial manipul
21. 4 SPASIM and SIMULINK The behaviour of a manipulator mechanism with e g closed loop control can be simulated using SIMULINK The closed loop simulation is defined as the problem of computing the actual trajectory of e g the manipulator tip with controlled actuation of the motion Tracking errors with respect to a nominal prescribed trajectory can be calculated Figure 1 2 shows an overview of a typical simulation scheme The simulation is characterized by the inverse dynamics stage based on a rigid link model and a forward dynamic stage At the forward dynamics stage the tracking behaviour of the manipulator system is studied In the case of flexible manipulators additional generalized coordinates describing the elastic behaviour of the manipulator links can be used in the dynamic system The block diagram in Fig 1 3 shows a typical closed loop simulation in more detail Blocks are used from the SPACAR SIMULINK library spacar_1ib that is part of the SPACAR package These blocks are front ends to so called S functions in SIMULINK 3 The following blocks are provided Section 1 4 SPASIM and SIMULINK 11 Rigid or rigidified link model Flexible link model prescribed trajectory actual trajectory Uo Pe read y eee Bre setpoints yY C simulation LINEAR es _ read coeff Mo Co Ko control matrices parameters MATLAB analyses SIMULINK simulation S SS E INVDYN
22. 8 30 END EM 4 4 END EM 5 2 XF 1 0 O 14 XF 5 0 Qa 124 XF 8 0 O 30H END HALT Note that the setpoints are read from the sbd data file of which the name is the longest substring of the name of the input file name robotinvlin The file from the previous inverse dynamics run robot inv is a likely candidate Section 3 11 Rigid spatial manipulator mechanism 85 Open loop simulation The behaviour of the manipulator mechanism without feed back control is simulated using SIMULINK for the open loop configuration of Fig B 48 Two blocks from the SPACAR library spacar_lib are used to read the Setpoint UO and Reference YO data respectively from the inverse dynamics run file name robotinv In this open loop configuration the nominal input is fed directly into the SPASIM block also available in the library In the in put file robot sim for this block the actual inputs and outputs are identical to the previously defined inputs and outputs HINGE 1 1 2 O 0 1 INPUTS 1 1 1 HINGE 2 2 3 0 0 INPUTS 2 2 BEAM 44356 0 1 0 INPUTS 3 3 HINGE 3 6 7 0 1 0O OUTE 1 1 1 BEAM 55789 0 10 OUTE 2 2 d OUTE 3 od X 4 0 0 0 OUTEP 4 1 1 X 5 0 268 0 0 6467 OUTEP 52 X 8 0 536 0 0 OUTEP 63 OUTEDP 7 1 1 FIX l OUTEDP 8 2 FIX 4 OUTEDP 9 3 DYNE 1 OUTX 10 8 DYNE 2 1 OUTX 11 8 2 DYNE 31 OUTX 12 8 3 OUTXP 13 8 1 END OUTXP 14 8 2 HALT OUTXP 15 8 3 XM 5 10 END XM 8
23. 88 Chapter 3 Examples 2 2 ef l gt Eee e1 1 e1 ref 1 e1 2 e1 ref 2 e1 3 e1 ref 3 rad x xref y yref z zref 8 m Pa 1 1 ah ey 1 ref T 8 3 a ei ret 02 ari 38 is i 12 1 4 time s Figure 3 52 hinges of spatial manipulator mechanism in a closed loop simulation Error in the deformation of the 100 200 0 6 time s Figure 3 54 feedforward part uo Input applied to the manipulator al 8 8 4l yl Yref al gt i i i i fi 0 0 2 0 4 0 6 0 8 1 1 2 1 4 time s Figure 3 53 Position error of the end effector of spatial manipulator mechanism in a closed loop simulation 40 1 1 eal 30 i 20 Z 104 0 zol f 3 3 uau 236 0 0 2 0 4 0 6 0 8 time s Figure 3 55 Input applied to the manipulator feedback part u uo Section 3 12 Flexible spatial manipulator mechanism 89 3 12 Flexible spatial manipulator mechanism To be added 90 Chapter 3 Examples 3 13 Chord driven underactuated robotic finger proximal phalanx second pulley first pulley distal phalanx Figure 3 56 Robotic finger In order to illustrate the use of the planar belt gear element a model for a chord driven underac tuated robotic finger with two phalanges as in a thumb is considered see Fig The distal phalanx can rotate with
24. EDAMP element number EA longitudinal damping for beam truss and belt ele ments Sai torsional damping for hinge elements Sai first damping coefficient for pinbody and cognates Gak torsional damping for beam elements Eal bending damping for planar beams Sa2 second damping coefficient for pinbody and cognates x5 EaI bending damping in y direction for spatial beams Saz third damping coefficient for pinbody and cognates x5 Eal bending damping in z direction for spatial beam 5 Section 2 3 Dynamics 27 TIMESTEP length of time period number of time steps INPUTX node number position or orientation node 6 coordinate number 1 2 3 or 4 start value 7 start rate acceleration constant INPUTE verurwveve element number x8 deformation mode coordinate number 1 2 3 4 5 or 6 9 start value 10 start rate acceleration constant STARTDX node number coordinate number 1 2 3 or 4 start value 7 start rate STARTDE element number deformation mode coordinate number 1 2 3 4 5 or 6 start value 10 start rate AU Ne WN RN B W USERINP Name of the MATLAB M file with user defined input func tions 11 GRAVITY x component of the acceleration of gravity y component of the acceleration of gravity z component of the acceleration of gravity INTEGRAT Sp
25. PLWHEEL Planar wheel element PLBELT Planar belt gear element BEAM Beam element TRUSS Truss element HINGE Hinge element PINBODY Spatial pinbody element RBEAM Spatial rigid beam element WHEEL Spatial disk wheel element TWHEEL Spatial torus wheel element X Specification of the initial Cartesian nodal positions FIX Support coordinates x RLSE Calculable deformations e INPUTX Prescribed DOF x INPUTE Prescribed DOF e DYNX Dynamic DOF x DYNE Dynamic DOF e KINX Configuration coordinate x KINE Configuration coordinate e LDEFORM Suppresses the calculation of non linear elastic strains of a beam element due to possibly large curvatures and twists of the elastic line ORP INBOD Defines the orientations of the generalized deforma tions for the PINBODY elements and cognates DRP INBOD Defines the undeformed reference distances for the PINBODY elements and cognates OUTLEVEL Sets the level of output generated in the log file and in the SPACAR binary data sbd file The parameters for these keywords are listed below xi refers to note i listed at the end of the keywords Section 2 2 Kinematics 19 PLBEAM element number first position node first orientation node second position node second orientation node PLTRUSS element number first position node second position node PLTOR element number first orientation node second orientation node PLPINBOD
26. REFXDP NOTES x1 The nominal input numbers and output numbers are the positions of the specified input or output in the input and output vectors respectively 40 Chapter 2 Keywords KEYWORDS INPUT VECTOR u mode 4 9 INPUTS Specification of input stresses INPUTF Specification of input forces INE Specification of input deformation parameters INEP The same first time derivative INEDP The same second time derivative INX Specification of input nodal coordinates INXP The same first time derivative INXDP The same second time derivative KEYWORDS OUTPUT VECTOR y mode 4 9 OUTS Specification of output stresses OUTF Specification of output forces OUTE Specification of output deformation parameters OUTEP The same first time derivative OUTEDP The same second time derivative OUTX Specification of output nodal coordinates OUTXP The same first time derivative OUTXDP The same second time derivative see note The parameters for these keywords are listed below xi refers to note 7 listed at the end of the keywords INPUTS x2 1 input number 1 INE 2 element number INEP 4x3 3 deformation parameter number INEDP 1 2 3 4 5 or 6 INPUTF x4 1 input number 1 INX 2 node number INXP 4x5 3 coordinate number 1 2 3 or 4 INXDP OUTS x6 1 output number 1 UTE 2 element number
27. al a 20 8 3 E a 200 Ten 8 gt gt ee 06H ia 5 10l oe 0 r 0 2 100 i I pr i i i i i 0 0 2 0 4 0 6 0 8 1 1 2 1 4 0 0 2 0 4 0 6 0 8 1 12 1 4 time s time s Figure 3 42 Stresses to be applied at the hinges uo 05r d dt x y 2 8 m s A fi fi L fi fi fi 0 0 2 0 4 0 6 time s Figure 3 44 effector 0 8 1 1 2 1 4 Velocity components of the end 2 a e1 1 1 2 e1 3 rad o 0 5 time s Figure 3 46 Deformations relative rotations of hinges 1 2 and 3 Figure 3 43 effector Position coordinates of the end 10 i i i 0 0 2 0 4 0 6 time s 0 8 Figure 3 45 Acceleration components of the end effector d dt e1 1 e1 2 e1 3 rad s time s Figure 3 47 Velocities of deformation of hinges 1 2 and 3 Section 3 11 Rigid spatial manipulator mechanism Rotational nodes for the spatial manipulator node 1 node2 node3 node6 node7 node9 type 2 2 2 2 2 2 Xo 2 2 2 2 2 2 A 1 2 2 2 2 2 A2 1 2 2 2 2 2 Ag 1 2 2 2 2 2 forces 0 0 0 0 0 0 Tri 0 0 0 0 0 0 Translational nodes for the spatial manipulator node4 node5 node 8 type 1 1 1 x coordinate 0 0 268 0 536 y coordinate 0 0 0 z coordinate 0 0 647 0 BC type x 1 2 3 Zo 0 536 o 0 BC type y 1 2 3 Yo 0 Yo 0 BC
28. chosen as generalized coordinates The lateral slip of the right rear wheel is released because otherwise the system whould be overconstrained The other five slips at the wheels are prescribed as zero to impose the non holonomic constraints of pure rolling Five kinematic coordinates are defined as the two position coordinates and the yaw angle for the rear frame and the two rotation angles at the other wheels The moments of inertia at the nodes 4 6 and 10 are the moments of inertia about the spin axes of the wheels The stationary motion and the linearized equations can be found by running SPACAR with mode 7 It appears that there are seven eigenvalues equal to zero with eigenvectors which correspond to the three rotations of the wheels and the two positions and yaw angle of the rear frame and a change in the forward velocity The other two eigenvalues are real and negative corresponding to exponentially decaying motion A simulation can be made with mode 1 A three dimensional model of the same tricycle is file trike3v dat RBEAM 1 1 2 3 0 0 1 0 0 0 HINGE 2 2 4 D0 1 40 0 80 WHEEL 3 3 4 5 0 0 1 0 0 0 RBEAM 4 1 2 6 0 0 1 0 0 0 HINGE 5 2 7 0 0 1 0 0 0 WHEEL 6 6 7 8 0 0 1 0 0 0 RBEAM 7 1 2 9 0 0 1 0 0 0 HINGE 8 2 10 00 1050 1150 Section 3 14 Tricycle 95 RBEAM 9 9 10 11 0 0 1 0 0 0 HINGE 10 10 12 0
29. element number first position node first orientation node second position node PLRBEAM element number first position node first orientation node second position node PLWHEEL SI element number position node first orientation node yaw angle second orientation node spin angle wheel radius initial direction of the spin axis i e the y axis PLBELT element number first position node first orientation node second position node second orientation node first pulley base circle radius second pulley base circle radius BEAM A PO DHNSIO NHB WNRKINDN HP WN RFP WNRI PWN RI WN RI WN KH NB WN oo 6 9 element number first position node first orientation node second position node second orientation node initial direction of the principal y axis of the beam cross section 1 torsion elongation coupling parameter f 1 TRUSS element number first position node second position node HINGE BWNRI WN eR element number first orientation node second orientation node initial direction of the x axis of rotation 2 20 Chapter 2 Keywords PINBODY run SI element number first position node first orientation node second position node initial direction of the principal y axis of the beam cross section 3 RBEAM MA BWN Re SI element number first position node first orientation nod
30. elements 2 and 3 gt gt plot x 1inp 2 1 e 1 gt gt grid gt gt xlabel phi 2 rad gt gt ylabel v eps 2 3 eps 3 2 2 e 2 3 te 3000 T T T 2000 1000 F 1000 2000 3000 x 8 m s didt 4000 5000 U 6000 0 001 002 0 03 004 0 05 0 06 007 008 0 09 01 Figure 3 18 Case 3 Horizontal acceleration of the sliding block x0 6 m amp Figure 3 20 Case 3 Difference in the horizontal position of the sliding block compared to case 1 le 3 2 2 m Figure shows the small vibration of the sliding block due to the bending by comparing its position with the rigid simulation of case 1 Fig B 6 56 Chapter 3 Examples Nodal points for the planar slider crank mechanism nodel node2 node3 node 4 5 7 node 6 node 8 node 9 node type T R T R T T R x coordinate 0 0 15 0 45 0 30 y coordinate 0 0 0 0 BC type x 1 2 2 2 BC type y 1 2 1 2 BC type 3 2 2 Qo 0 wo 150 w 0 forces moment 0 0 0 0 0 0 0 mass inertia 0 0 0 0 0 033 0 0 T translational R rotational BC boundary condition The numbers of the BC type refers to the numbers of the groups mentioned on page Elements for the planar slider crank mechanism element 1 element 2 element 2a 3 element type beam beam beam T nodes 1 3 3 6 3 8 8 6 R nodes 2 4 5 7 5 9 9 7
31. linear time varying LTV system A SPACAR mode 3 run generates time varying state space matrices that are well suited for this purpose Then a typical SPACAR analysis and linearized simulation procedure is as follows e Use e g an inverse dynamics run mode 2 to define the nominal motion for the rigidified manipulator Inputs and outputs of the system may be specified e Next the system is linearized with a mode 3 call The system is analysed along the nominal path computed previously The elastic deformations are defined with INPUTE commands Inputs and outputs must be specified e Finally the linearized simulation can be run with a SIMULINK model of which a typical example is shown in Fig In comparison with the non linear simulation of Fig 1 3lthe spasim block is replaced by an LTV block that uses the linearized equations of motion Note that now only the differences compared to the nominal motion are computed Only the difference u of the manipulator s input compared to the nominal input is needed In addition the generalized stress resultants 0 are part of the input of the LTV block 14 Chapter 1 The SPACAR program Figure 1 4 Block diagram of a typical closed loop simulation in SIMULINK based on the perturbation method In addition to the above outlined standard implementation some further extensions are provided It is possible to include the effect of proportional controller gain i e a proportional control
32. matrix K into the stiffness matrix Ko Of course in that case this part of the control action should no longer be included in the controller in the block scheme This approach offers advantages when subsequently a modal analysis is applied to the linear time varying state space system Such an analysis discriminates quasi static behaviour of the system low frequency vibrational modes and high frequency vibrational modes Mostly the latter do not significantly affect the output of the system while they can have a detrimental effect on the computational efficiency even for a linearized system With a modal analysis it is possible to eliminate these high frequency modes A more profound description of the latter two techniques is currently outside the scope of this manual Keywords 2 1 Introduction In this chapter the user is informed about the creation of correct input data for the software package SPACAR The input must have a specific form Behind a number of permitted keywords the user supplies a list of arguments The arguments behind a keyword are well defined Each module of SPACAR except mode 4 of LINEAR has its own list of available keywords They form blocks that are separated by the following pair of keywords END HALT The final closure of the input is effected by END END The first block contains the kinematic data The input of the mechanism model by means of keywords is treated in the Kinematics section
33. number 2 10 direction vectors x7 DRP INBOD 1 PINBODY RBEAM PLPINBOD or PLRBEAM element number 2 undeformed projection of x x on the first direction vec tor 3 undeformed projection of x on the second direction vector 4 undeformed projection of 27 x on the third direction vector for spatial elements OUTLEVEL 1 level of output in log file x8 2 level of output in the SPACAR binary data sbd file x8 NOTES 22 Chapter 2 Keywords 1 The direction vector lies in the local x y plane of the beam element If no direction is specified the local direction vector is chosen as the standard basis vector that makes the largest angle with axis of the beam in case of a draw the vector with the highest index is chosen The torsion elongation coupling parameter takes into account the shortening of the beam due to torsion such that for a twisted axially unloaded beam the axial strain is fia where a is the specific twist of the beam For thin walled open cross sections f Iy I A but it may have a different value or even be negative for solid cross sections x2 The local y and z unit vectors are chosen as follows First the standard basis vector with the largest angle with the hinge axis is chosen in case of a draw the vector with the highest index is chosen Then the local y is chosen in the direction of the cross product of the local x direction with this basis vector The
34. or the coordinate number x 3 The current value of the deformation e or coordinate x 4 The current rate of the deformation or velocity x 30 Chapter 2 Keywords 5 The current acceleration of the deformation or coordinate 7 The user has to assure the correctness of the derivatives SPACAR does not carry out any checks but the results depend heavily on these derivatives 12 Available integrator types are 0 130 135 140 155 220 225 310 320 330 410 420 430 Default Shampine Gordon Explicit third order Runge Kutta fixed step size Explicit third order Runge Kutta variable step size Explicit fourth order Runge Kutta fixed step size Explicit fifth order Runge Kutta variable step size Explicit Runge Kutta for second order systems second order accurate fixed step size Explicit Runge Kutta for second order systems second order accurate variable step size Semi implicit Runge Kutta Rosenberg first order accurate fixed step size Semi implicit Runge Kutta Rosenberg second order accurate fixed step size Semi implicit Runge Kutta Rosenberg third order accurate fixed step size Singly diagonally implicit Runge Kutta implicit Euler first order accurate fixed step size Singly diagonally implicit Runge Kutta second order accurate fixed step size Singly diagonally implicit Runge Kutta third order accurate fixed step size Change this only if you know what
35. the differences in joint positions and velocities which are computed by comparing the actual motion and the nominal output The output of the subsystem is K e K e with well chosen matrices K and K see e g the lecture notes 1 This output is multiplied with the time dependent 3 x 3 reduced mass matrix M gt using a block from the spacar_lib library Finally the nominal input vector uo is added as a feedforward signal NIE E robotinvlin Un 3 unom Setpoint UO Unom To Workspace 6 Omega 28 3 6 beta 0 85 robotinvlin T Pj Selector 5 P Selector 3 gt Scope E robotsim Kp Kv control times MO SPASIM Selector E Ed Selector E rs yref Yref To Workspace gt 1 5 Display Time 3 robotinvlin HP 15 Pj Selector 8 5 P Selector Reference YO Selector Eref Edref Selector Eref Scope Eref 3 LD t T Sd rs Selector H gt Selector Ytip Scope dYtip Clock Time To Workspace Figure 3 51 Block diagram for a closed loop simulation of the motion of the manipulator mechanism using SIMULINK Most signals are vectors and the numbers indicate the size of the vectors The motion is simulated with the same parameters as in the open loop simulation see page 85 In this case the actual size of the varia
36. you are doing 13 The error tolerances are used for integration methods with a variable step size in the in tegrators of type 0 135 155 and 255 For the integrators of type 410 420 and 430 the absolute tolerance is used as the tolerance for the modified Newton Raphson iteration Section 2 4 Inverse dynamics setpoint generation 31 2 4 Inverse dynamics setpoint generation For clarity the keywords for the inverse dynamics including the generation of setpoints are dis cussed in two subsections In the input file keywords from both subsections must be combined into one part so there should be no END HALT pair in between 2 4 1 Trajectory generation There are three essential keyword blocks KEYWORDS TRAJECTORY GENERATION 1 TRAJECT Trajectory header the given trajectory number is valid for all keywords before the next TRAJECT 2 TROT Definition of the actual trajectory TRANS the number and type of DOFs determine which key words and TRCIRL how many of them have to be specified TRE TROT TRANS and TRCIRL for nodes and TRE for elements maximum of 6 USERTRAJ Trajectory defined by a user function 3 TRTIME Definition of trajectory time and number of time steps and there are two blocks of optional keywords i TREPMAX Specification of velocity profile rise time TRVMAX and maximum velocity TRFRONT Specification of a
37. 0 0177403 0 0000000 0 0111041 0 0000000 0 0362089 0 0000000 0 0000000 0 7868804 0 0000000 0 4396164 0 0000000 0 6168430 0 0000000 0 8972801 From the eigenvalues numbers in this table we find w 24 78 rad s and wa 55 55 rad s 66 Chapter 3 Examples 3 6 Cantilever beam in Euler buckling Figure 3 28 Cantilever beam loaded axially by a force F at the free end Consider a slender cantilever beam or column with suppressed rotation of the free end loaded axially by a force F The smallest load that produces buckling is called the critical or Euler load Fa For a load equal to or greater than the critical load the beam is unstable The bent shape shown represents the buckling mode Euler s theoretical buckling load for the above beam end conditions is Fn MEI l where EI is the flexural rigidity and l the length of the beam This result can also be obtained numerically from a SPACAR analysis e g with the following numerical values 1 EJ 1 Fy 1 The beam is modelled by two equal planar beam elements as shown in Figure B 28 A SPACAR input file column2 dat for this case is PLBEAM 112 3 4 PLBEAM 2 3 4 5 6 xX 1 X X5 l w oO O ouno oO ooo FIX 1 FIX 2 FIX 6 DYNX DYNX DYNX RLSE RLSE NHEOUOUORMAW NM MNF NW w w END HALT EM 1 1 EM 2 1 Section 3 6 Cantilever beam in Euler buckling 67 ESTIFF 1 0 1 ESTIFF 2 0 1
38. 0 7m The distributed mass per length is p4 4kg m for element 4 and p 2 kg m for element 5 The concentrated masses in nodes B and C are 10 kg and 30kg respectively The effect of gravity is accounted for by applying external forces m g in negative z direction where g 10 m s Inverse dynamics problem First the inverse dynamics problem is analysed Figure B 44 shows the velocity components of the end effector that are computed for the trajectory defined in the input file The position and acceleration components of the end effector are shown Fig 3 43 and Fig respectively The following input file robotinv dat is used SPACAR mode 2 80 Chapter 3 Examples HINGE 112 0071 TRAJECT 3 HINGE 2 23 0 1 0 TRANS 8 0 Tes 0 BEAM 44356 0 1 0 TRTIME 0 2 20 HINGE 3 6 7 0 1 0 BEAM 557829 0 1 Q NOMS 111 NOMS 221 x 4 0 0 0 NOMS 33 l X 5 0 268 0 0 6467 X 8 0 536 0 O REFE 111 REFE 22 1 FIX 1 REFE 337 FIX 4 REFEP 41 1 INPUTX 8 1 REFEP 5 2 1 INPUTX 8 2 REFEP 6 3 1 INPUTX 8 3 REFEDP 7 1 1 RLSE 1 1 REFEDP 8 2 1 RLSE 2 REFEDP 9 3 RLSE 31 REFX 10 8 1 REFX 11 8 2 END REFX 12 8 3 HALT REFXP 13 8 1 REFXP 14 8 2 XM 5 10 REFXP 15 8 3 XM 8 30 EM 4 4 END EM 5 2 END XF 1 0 0 14 XF 5 0 One 121 VISUALIZATION XF 8 0 0 307 BEAMVIS 0 01 0 01 HINGEVIS 1 0 01 0 03 END HINGEVIS 2 0 01 0 03 HALT HINGEVIS 3 0 01
39. 1 0 000 0 000 X 3 0 150 0 000 X 6 0 300 0 000 X 8 0 450 0 000 FIX 1 FIX 8 2 INPUTX 2 1 DYNE 2 2 3 DYNE 3 2 3 END HALT XM 8 0 033375 EM 2 0 2225 EM 3 0 2225 ESTIEF 2 0 000000 13 359623 ESTIEF 3 0 000000 13 359623 INPUTX 2 1 0 000000 150 000000 0 000000 TIMESTEP 0 100000 100 STARTDE 2 2 0 000000 0 000000 STARTDE 2 3 0 000000 0 000000 STARTDE 3 2 0 000000 0 000000 STARTDE 3 3 0 000000 0 000000 END END The second order contributions of the bending deformations on the elongation Eq 6 4 22 in the lecture notes are taken into account The initial configuration of case 3 is depicted in Fig The horizontal acceleration of the sliding block as function of time is given in Fig The bending of the slider given by v Lie e as function of the crank angle is presented in Fig The MATLAB commands used to plot these results are gt gt plot time xdd 1lnp 8 1 gt gt grid gt gt xlabel time s gt gt ylabel d dt 2 x 8 m s 2 gt gt Section 3 2 Planar slider crank mechanism 55 0 2 0 15 0 1 0 05 0 1 3 2 6 3 Ze 0 05 0 1 0 15 0 2 1 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 0 45 Figure 3 17 Case 3 Initial configuration of the slider crank mechanism y i I Bin A eps 2 3 gt it i i i i i i 0 2 4 6 8 10 12 14 16 phi 2 rad Figure 3 19 Case 3 Bending of the flexible con necting rod
40. 2 2 A second block of input is reserved for the dynamics module The keywords for this block are presented in the Dynamics section The solution of inverse dynamics problems demands additional input for the trajectory descrip tion and for the definition of the input and output vectors uo and y Trajectory keywords and system keywords are treated in the Inverse dynamics section The keywords for the lin earization of mode 3 mode 4 and mode 9 are given in the Linearization section 2 5 At the end of the file custom settings for SPAVISUAL can be added The keywords for SPAVISUAL are presented in section 2 7 The simulation of mechanisms using SIMULINK is controlled by the keywords described in the Simulation section Some general remarks e Keywords and arguments can be separated by one or more spaces tabs or line breaks e Lines must not contain more than 160 characters 15 16 Chapter 2 Keywords e Any text in a line following a or is treated as a comment e All input is case insensitive e Data read from the input file are echoed in the log file after the comments have been removed and all text is transformed into upper case capitals e Angles are always specified in radians e For some commands such as XF and STARTDE not all arguments have to be specified Default values are zero unless otherwise specified 2 2 Kinematics A kinematic mechanism model can be built up with finite eleme
41. 2 length of the hinge 0 003 3 radius of the hinge 0 009 TRUSSVIS 1 sets the visualization of the truss elements on 1 or off 0 on TRANSPARENCY 1 adjusts the transparency between 0 and 1 1 VIBRATIONMODE 1 switch between the 10 lowest vibration modes 2 1 BUCKLINGMODE 1 switch between the 10 buckling modes x2 1 ENLARGEFACTOR 1 amplitude of the vibration or buckling modes 1 RECORDMOVIE 1 sets record movie on 1 or off 0 off MOVIENAME 1 filename for the recorded movie filename UNDEFORMED 1 sets the visualization of the initial on un deformed mechanism configuration on 1 or off 0 VIBREND 1 sets the period of the sin function for the 2r x3 vibration mode STEPLINE 1 sets the size of the line elements that are used 0 2 to draw the elements STEPVIBRATION 1 sets the number of intermediate steps in the vibration ip 1 3 visualization LIGHT 1 sets the light source on 1 or off 0 off JOINTS 1 sets the joints on 1 or off 0 on TRAJECTVIS 1 sets the trajectory on 1 or off 0 off TRAJETCNODE 1 selects the node for the trajectory 1 Notes x1 The BEAMVI S command has two variations The one with only two parameters adjusts all beam elements The variant with three parameters can be used to adjust only a single beam element 2 Only the lowest vibration and buckling modes are available with a maximum of 10 modes 3 Only numerical values are allowed no symbols or functions 46 Chapter 2 Keywords
42. 3333333 0 2495 0 000000000001 In a MATLAB session the compliances and eigenfrequencies can be found as follows gt gt spacar 8 shear2 gt gt xcompl 1Inp 5 2 ans 3 9940 gt gt spacar 7 shear2 gt gt type shear2 log 0 Eigenvalue numbers 5 to 8 00000E 00 2 54107E 00 Eigenvector numbers 5 to 8 0 0000000 0 0000000 0 1544078 0 0000000 0 2825588 0 0000000 0 1744141 0 0000000 0 0000000 0 0000000 0 0000000 0 3923612 0 0000000 0 7180016 0 0000000 0 4431985 a ee NN RS Se oh Le ee ee be eee oe lt Q OQ DILDO 00000E 00 0000000 4065978 2566711 6173727 0000000 0000000 0000000 0000000 7 95645E 01 oOo OO OO OO DW DD 000 000 000 000 000 235 042 4912 000 000 000 000 000 074 190 The compliance based on thin plate theory is 3 8822 m N so the approximation with a short beam overrates the compliance by about 3 If the shear flexibility were nor included the compliance would be 1 0 m N The lowest numerical eigenfrequency w 0 795645 rad s compares well with a value from plate theory w1 p 0 7987 rad s If shear flexibility nor rotational inertia is included the first numerical eigenfrequency is 1 6168 rad s Section 3 9 Lateral buckling of cantilever beam 73 3 9 Lateral buckling of cantilever beam In this example lateral buckling is considered of a cantile
43. 6 0 XS 84333 0 Section 3 7 Cantilever beam subject to concentrated end force 69 J K Z a OB WN FP MB MN N NH WWW WW Ww m Z VBWMNH n N 102 102 102 102 102 ESTIFF ESTIFF ESTIFF ESTIFF ESTIFF oe WN EF O O00 0 oOo O00 0 oOo OO 0 0 XF 11 0 0 14 END END In a MATLAB session we get gt gt spacar 8 plbeam d inp 11 1 w IL AR 6808 theoretically 3 8109 8 4897 theoretically 8 4044 gt gt xcompl 1np 11 1 ans 70 Chapter 3 Examples 0 4859 gt gt xcompl 1Inp 11 2 ans 0 0633 undeformed configuration undeformed configuration 0 3 268 To show the usefulness of SPAVISUAL the first three free vibration modes no external loads and buckling modes axially loaded by an end force are displayed for the cantilever beam of this example in figures to S665 Figure 3 30 First vibration mode for a cantilever beam with 5 elements w 0 355131 rad s the oretically 0 355100 rad s Figure 3 32 Second vibration mode for a can tilever beam with 5 elements wa 2 2266 rad s theoretically 2 22537 rad s eso RRON g l Figure 3 34 Third vibration mode for a cantilever beam with 5 elements w3 6 25198 rad s theo retically 6 23111 rad s Figure 3 31 First buckling mode for a cantilever beam with 5 elements Fer 2 516776 N theo retically 2 516749 N
44. EX modules These files are checksbf m combsbd m getfrsbf m getss m loadsbd m loadsbm m ltv m mrltv m repinsbf m spacar m spacntrl m spasim m e Files with extension mdl are SIMULINK models There is only one file which is actually a library from which the SPACAR modules for use in SIMULINK can be copied spacar_lib mdl The optional data files from spadata zip can be extracted in a separate working directory The files in the SPACAR subdirectory should be in the MATLAB path when MATLAB is running There are two ways to accomplish this 1 Make sure that the SPACAR subdirectory is the local directory You can verify this by typing pwd If necessary change your local directory by typing cd Matlab Toolbox Spacar or whatever directory you chose to store your files 2 Another possibility is to change the settings of the MATLAB environment by adding the SPACAR subdirectory to the MATLAB path This modification is either temporary or per manent The path can be modified from the pulldown menu with File Set Path or by using the MATLAB commands path or addpath Now you are ready to run SPACAR in MATLAB and SIMULINK SPACAR error messages An analysis with SPACAR in MATLAB or a simulation with SPASIM in SIMULINK can suffer from errors These errors can be divided into fatal errors that cause an immediate terminations and less severe errors which may report unexpected conditions in the 1og file while the calculation continues Mos
45. Examples The data files used to run the examples in this chapter can be downloaded from the SPACAR web site see Appendix A 3 1 Planar sliding bar In example 4 3 1 of the lecture notes 1 the sliding bar of Fig 3 1lis described A rigid bar pq of length 2 m is suspended from two sliders The bar is driven by the condition x vt 0 where v v is the constant horizontal velocity component of point p Thus lt v and r 0 We want to compute 7 and ij for 0 lt t lt 2v3 s and v 1 m s The position y can be computed easily from the symbolic expression y y 4 V3 zP 2 so B 30 Figure 3 1 Sliding bar 47 48 Chapter 3 Examples Differentiating once and twice with respect to the time t yields a Y3 t i 4 q O yra 1 23 y1 2V3t t The mechanism has one degree of freedom and there is only one element This is the planar truss element denoted by 1 that connects nodal points 1 and 2 in the following SPACAR input file slider dat PLTRUSS 1 1 2 X 1 0 0 X 2 1 7321 1 FIX 1 2 FIX 2 1 INPUTX 1 1 END HALT INPUTX 1 1 0 T 0 TIMESTEP 3 4641 100 END END Both symbolic and numeric results are shown in Figs B 2 and B 3 with the Matlab commands gt gt t time dt y 2 mvs 5 a 1 5 time s time s Figure 3 2 Vertical velocity 1 of the sliding bar Figure 3 3 Acce
46. N N N 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 time s Figure 3 60 Normal forces at road contact points The fully drawn line is for the left rear wheel the dotted line for the right rear wheel and the dashed line for the front wheel gt gt mlabel time s gt gt ylabel normal force N It is seen that all forces are negative which means that the normal force is compressive as it should be The normal force in the right rear wheel is initially much higher than the corre sponding force at the left rear wheel Because the wheel planes remain perpendicular to the road surface the third components of the stresses are also equal to the normal force at the road Lateral forces are in the second components as well as in the sixth in a scaled version The fourth components of the stresses are zero as they should be The fifth components represent scaled longitudinal tyre forces at the contact points A model with an inclined steering axis as shown in Fig 3 59 is in the input filetrike3i dat which differs from t rike3v dat in the definition of the hinge at the steering head element 8 and the position of node 9 HINGE 8 2 10 0 30901699437495 0 0 0 95105651629515 X 9 0 96 0 0 0 3 Section 3 14 Tricycle 97 100 4 ee 200 We d 300 l 4 normal force N 400 500 ll l ll l li 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 time s 600 ji l Figure 3 61
47. Normal forces at road contact points The fully drawn line is for the left rear wheel the dotted line for the right rear wheel and the dashed line for the front wheel Note that for this case the normal force in the right rear wheel is initially just compressive Figure 3 61 so for a slightly higher speed the wheel would lose contact with the ground This loss of contact cannot directly be included in the model The front wheel no longer stays perpendicular to the road surface so the first and third stress components are no longer equal Morreover the rotation angle of the hinge with element number 12 is not exactly equal to the yaw angle in this case 98 Chapter 3 Examples SPACAR installation Prerequisites Before installing SPACAR on a computer system it is advisable to check that the system is suitable of running the software and to have MATLAB installed This SPACAR version has been developed and tested with MATLAB 7 0 4 and SIMULINK 6 2 Release 14SP2 It is expected to work with any modern version of MATLAB SIMULINK since R12 but in case of problems we can offer only limited support The system requirements depend heavily on the version of MATLAB you are using Consult the accompanying Installation Guide or check The Mathworks You may expect that SPACAR will run on any Microsoft 32 bit Windows PC on which MATLAB SIMULINK are running Only the base systems of MATLAB and SIMULINK are required to run SPACAR but additional
48. Phase deg 225 7 180 gi pt ge ooi 10 10 10 10 10 Frequency rad sec Figure 3 39 Bode diagram for the lever system Section 3 11 Rigid spatial manipulator mechanism 79 3 11 Rigid spatial manipulator mechanism Figure gives an example of a simplified manipulator The prescribed motion of the end effector C is represented by the coordinates x y and z as functions of time Z U 1 76 m s 1 2 y ea 0 0 02 0 4 10 12 14 C t I Figure 3 40 Spatial manipulator mechanism trajectory and velocity profile of the end effector The end effector must follow the straight trajectory from point I to point II Three trajectories are distinguished Initially the manipulator is at rest for 0 2 s Next during 1 0 s the mo tion is carried out according to the velocity profile in Fig 3 40 with constant acceleration and deceleration during the first and final 0 2 s Finally the manipulator is at rest again The motion of manipulator is determined by the rotation of three hinges Hinge 1 enables rotations about the z axis while hinge 2 enables motions perpendicular to the xy plane Hinge 3 takes care of motions in the same plane wherein hinge 2 is active The hinges are driven by internal actuators For control purposes we assume that sensors are available that measure the rotations and the speed of rotation of the hinges The manipulator consists of two beams elements 4 and 5 which are equal in length l4 l
49. SIG Specification of preloaded state EDAMP Specification of viscous damping coefficients 4 TIMESTEP Duration and number of time steps INPUTX Specification of simple time functions for the INPUTE prescribed degrees of freedom STARTDX Specification of initial values for the dynamic degrees STARTDE of freedom USERINP Specification of MATLAB M file for user functions with input for the degrees of freedom 5 GRAVITY Specification of the gravitational acceleration vector INTEGRAT Selection of integrator ERROR Specification of error tolerances for the integrator ITERSTEP Specification of number of iterations and steps and error tolerance for static calculations in modes 7 8 and 9 Section 2 3 Dynamics 25 The parameters required with these keywords are listed below xi refers to note i listed at the end of the keywords XM 1 node number 2 concentrated mass for position nodes rotational inertia for planar orientation nodes for spatial orientation nodes the inertia components 3 Jay 1 4 Fen ely 5 Jyy 1 6 Jyz 1 EM 1 element number 2 mass per unit of length 3 rotational inertia Jy per unit of length for spatial beam 2 rotational inertia J per unit of length for planar beam x2 angle over which the belt is initially wound over the first pulley for a planar belt 4 rotational inertia Jy per unit of length for spatial beam 2 angle over which the belt is initially wound
50. SPACAR User Manual dr ir R G K M Aarts dr ir J P Meijaard and prof dr ir J B Jonker 2009 Edition March 1 2009 Report No WA 1180 of contents Preface iii 1 The SPACAR progra 1 1 1 Introduction 2 2 oz moon 1 1 2 SPACAR amp MATLAB oo oo on 1 13 S5SPAVBUA L 3 22232 22 224 aaa ie nd 10 1 4 __SPASIM and SIMULIN YN Corres y PETTEE ITET 12 15 2 1 Introduction 2 22mm men 15 ee ee Se a es we Mi Gr ec an Minen He Arne Ge Be ied 16 1S u u a So ni ee nr Sele ip Gees cb Se oh Seo hb Sn tw Bes bee 24 a he eho he a 31 2 4 1 Trajectory generation 0 0 0 a ee ee 31 2 4 2 Nominal inputs and reference output 2 02 36 bb ate eek ee es Be SE ea Gee pe en 38 2 6 Non linear simulation of manipulator control 2 4 42 2 7__ Visualization and animation 2 0400 a 44 47 3 1 Planar sliding bay 2222 CC mo nen 47 ne 57 Laie ee ete ee een baren 60 EEE EEE ee chee es 63 ee 66 3 9 Lateral buckling of cantilever bea 3 10 State variable and output equations ii Table of contents 3 11 Rigid spatial manipulator mechanis 3 13 Chord driven underactuated robotic finge A SPACAR installatio B_ SPACAR error message C _ MATLAB tutoria C l Basic MATLAB graphics commands C 2 Quitting and saving the workspace Reference 3 12 Flexible spatial manipu
51. am Limitations The SPACAR package has some built in limitations on the size of the manipulators that can be analysed Table 1 1 shows the limits for the so called Student version that can be downloaded as describes in Appendix In case your requirements are larger you need to contact the authors The licence for the freely downloadable software is time limited Maximum number of coordinates deformations 175 Maximum number of DOFs 20 Maximum number of elements nodal points 50 Maximum number of inputs 12 Maximum number of outputs 25 Table 1 1 Built in limitations of the Student version of the SPACAR package 1 3 SPAVISUAL SPAVISUAL is the visualization tool for SPACAR It can visualize deformation vibration and buckling modes SPAVISUAL shows beams trusses and hinges in 2 D as well as in 3 D It works with default settings which can be adjusted by the user The only input of SPAVISUAL is a filename This file has to be a dat file which has been analysed with SPACAR This is necessary because SPAVISUAL needs the sbd files for the deformation modes and also the sbm files for the vibration and the buckling modes There are some keywords that can adjust the default settings These keywords are listed in section SPAVISUAL is a stand alone function in MATLAB To run SPAVISUAL the user has to type the next command gt gt spavisual filename Here filename refers to the dat file that is executed by SPACAR 1
52. antities as are represented by the corresponding variables in the SPACAR binary data see the overview on page 5 The script should return again time t user defined stresses sig and user defined nodal forces fx Either sig or fx or both may be empty in the case no stresses and or forces are prescribed Otherwise each row in sig and or fx should define one stress value or force component at the specified time t Three columns should be provided with 1 The element number e or the node number x 2 The deformation mode number e or the coordinate number x 3 The current value of the stress or force component Two more columns can be provided which specify the diagonal elements of the stiffness and damping matrices respectively coresponding to the stress or force component x5 Unspecified values for the stiffness and damping are assumed to be zero by default The meaning of the variables is elasticity modulus Young s modulus G E 2 2v shear modulus v Poisson s ratio Ea damping modulus in Kelvin Voigt model Ga shear damping modulus in Kelvin Voigt model A cross sectional area I Iy I second area moment about y axis and z axis T Saint Venant s torsion constant k ky and kx shear correction factor in y direction and z direction The shear correction factors are about 0 85 a table of values for various cross sections can be found in 5 The generalized stresses are calculated according to
53. ation used in a previous mode 2 run and the current mode 3 run ERROR in subroutine ORDEO IFLAG 2 and ERROR in subroutine ORDEO No convergence indicate problems with the zeroth order iteration In SPASIM this may be avoided by setting or decreasing the maximum time step of SIMULINK s solver ERROR in subroutine SOLDYN is usually caused by a singular mass matrix PRPARE NUMBER OF NXC NOT EQUAL TO NEO NEM is caused by an ill defined mechanism ERROR in subroutine PRPARE Too many means that the mechanism that is defined is too large to be handled by the SPACAR version you are using see Ta ble 1 1 on page 10 Simplify the mechanism or contact the authors The messages written to the log file may be self explanatory but also a somewhat cryptic messages ERROR OR POSSIBLE ERROR CODED lt code gt ITEM lt number gt can occur The lt code gt is related to a procedure in the software Typical examples are INVOERi input for the kinematics Sect 2 2 SINVOERi input for the inverse dynamics setpoint generation Sect 2 4 LIMVOEi input for the linearization Sect 2 5 SIMVOEi input for the non linear simulation of manipulator control Sect 2 6 PREPTR trajectory data processing Note that errors in the input file are often reported one line later than the actual error position MATLAB tutorial C 1 Basic MATLAB graphics commands MATLAB provides a variet
54. ator mechanism in an open loop simula tion time s i Figure 3 50 Position of the end effector of spa tial manipulator mechanism in an open loop sim ulation dictated by the specified maximum value The results from the simulation are plotted using the MATLAB commands gt gt plot t yref 1 r oes A bt yret 2 3 D t7y 2 1 EI pass tay eee 9 ae a pole be gt gt plot t yref 10 r t yref 11 Ey VEER ipl 2 7 bes tery Ce lO pee pe es t y ad Gs pty y irda bs Figures and 3 50 show the deformation of the hinges and the position coordinates of the end effector from this simulation The solid lines are the reference data yref and the dotted lines are from the actual simulation y Clearly small errors during the integration lead to relatively large position errors at the end of the motion The error can be decreased by increasing the integration accuracy e g by enlarging the number of computed setpoints More reliable results can be obtained by applying feedback control as will be discussed next Section 3 11 Rigid spatial manipulator mechanism 87 Closed loop simulation The block diagram of Fig B 48 is extended with a feedback controller as shown in Fig A feedback signal is computed by a controller that is implemented as a subsystem block and a multiplication with the reduced mass matrix Mo The subsystem assumes that the input is a vector with both de and d These are
55. been generated in an inverse dynamics run mode 2 The set of DOFs used in the inverse dynamics computation represent the actuator joint coordinates e In case of a flexible manipulator mechanism additional DOFs ei 0 describing the elastic behaviour of the mechanism links should be included in the dynamic model both in mode 2 and mode 3 Clearly the mechanisms used in both runs have to be closely related If the manipulation task is prescribed in terms of relative DOFs TRE the list of keywords is identical with those used in the inverse dynamics run mode 2 If the manipulation task is prescribed as a motion of some nodal points triads TROT TRANS TRCIRL then the corresponding RLSE command of the actuators should be replaced by INPUTE commands in the kinematic block In the software some checks are carried out to verify that data from the inverse dynamics run can be reasonably used during the linearization The nominal input vector u and the reference output vector yy are again defined in the fol lowing blocks These blocks are optional but as before omitting one or both blocks means that no input and or output vectors are defined and hence no state space matrices can be gen erated and written to the 1tv file The keywords are similar to the input and output keywords in Sect In the output 1tv file of a mode 3 run the setpoints of the input and output vector are stored in the same way as for a mode 2 run In addition the state space matr
56. ble time step is somewhat smaller and after 183 time steps the simulation is completed The differences between the prescribed and actual trajectory is much smaller in this case as is illustrated in Figs and 53 The maximum error of the tip position is less than 1 mm which is better than 0 1 Figs and 3 55 show the feedforward part uo and feedback part u uo of the input applied to the manipulator respectively Clearly the larger contribution is from the feedforward part The size of the feedback part is smaller and relatively large correction are only applied during limited periods of time However as is clear from this example this feedback is essential to keep the manipulator on track The simulation for 1 5 s now requires 182 time steps which is only slightly more than in the open loop simulation However the simulation takes much more time which is caused by the occurrence of a so called algebraic loop in the block diagram The reason for this algebraic loop is the presence of the joint accelerations in the output vector of the spasim block as accelerations depend algebraically on the input torques These accelerations are only exported to the workspace and are not used in the feedback loop so there is no real algebraic loop Unfor tunately SIMULINK has no means to detect this If you are not interested in the accelerations they can easily be removed from the output vector and the simulation speed will increase sig nificantly
57. cceleration front for each velocity profile 2 TRM Specification of extra masses and TRF forces on the end effector 32 Chapter 2 Keywords The trajectories can be constructed in two ways with a user function or with built in profiles The latter are defined below and are of course limited to combinations of the built in profiles On the other hand practically any input can be generated with user functions This feature is activated by defining exactly one TRAJECT with the USERTRAJ keyword The required parameter is the name of an MATLAB M script that is to be called With TRTI ME the total trajectory time and the number of time steps must be specified The calling syntax of the M script is exactly equal to that of the M script for the USER NP keyword see page Alternatively one can use the built in trajectory profiles The next scheme shows in more detail the combination possibilities of the setpoint generation keywords Essential keywords are accompanied by a number of optional keywords placed between brackets Other optional keywords than those mentioned are not allowed for that specific essential keyword TRAJECT TRE TROT TREPMAX TRFRONT TRVMAX TRM TRFRONT TRANS TRCIRL TRVMAX TRM TRF TRFRONT TRVMAX TRM TRF TRFRONT TRTIME The way to follow through the sche
58. d and sig matrices is like x t k where t is the time step and k ranges from 1 to nx for x xd xdd and fx fxtot and from 1 to ne for e ed edd and sig respectively Section 1 2 SPACAR amp MATLAB 7 2 The variables In it and rxyz are mainly intended for internal use in the drawing tool spadraw More user friendly information is available in the 1og file page 8 3 The large variables dec dxc de dx d2e and d2x are only created if the parameters of the LEVELLOG are set accordingly Sect 2 2 4 After a linearization run node 8 directional nodal compliances inverse stiffnesses are computed Using the location matrix xcompl Inp i j gives this quantity for the jt coordinate j 1 4 of node i After a linearization run mode 3 4 7 8 or 9 the coefficient matrices are stored in a SPACAR binary matrix file with extension sbm Besides nnom see infra and t ime the accompanying MATLAB matrices are moO reduced mass matrix Mo 5 bO input matrix Bo 5 6 co velocity sensitivity matrix Co gt do damping matrix Do gt k0 structural stiffness matrix Ko 5 no geometric stiffness matrix No 5 g0 geometric stiffness matrix Go ak0 kinematic matrix Ayo 5 bk0 kinematic matrix By 5 Notes 5 Storage of the time varying matrices is in a row for each time step so in mO t k index t is the time step and k ranges from 1 to ndof xndof To restore the matrix structure at some time step type e g reshape m0 t
59. deformations in each class as explained in the lecture notes 1 e The numbers NXO NXC NXMM and NXM indicating the numbers of position coordinates in each class as explained in the lecture notes 1 e The stiffness damping and mass of the elements Section 1 2 SPACAR amp MATLAB 9 e The nodal point forces mass and gyroscopic terms e The total mass of the system The zeroth first second and third order transfer functions are shown next each for the position parameters and deformation parameters respectively The amount of output can be controlled by the keyword OUTLEVEL in the input file Next for a forward analysis mode 1 and mode 4 the name of the integrator and accuracy settings are shown Finally a list with all time steps and the number of internal iterations are given For an inverse dynamics analysis the trajectories and input output definitions see also Sect are read and analysed In case of mode 3 the name of the data file of the previous mode 2 is shown In case of mode 7 the eigenvalues frequencies and normalized eigenvec tors of the state system matrix are shown In case of mode 8 load multipliers and normalized buckling modes are presented In addition the vector of directional nodal compliances is shown SPACAR binary data files Some utilities are available to show check load or replace the data in SPACAR binary data files SBF These are files with extensions sbd sbm and ltv checksbf checks and sh
60. dimensional spatial elements can be specified In the second block the initial configuration of the mecha nism is specified In the third block the coordinates and generalized deformations are divided into four groups depending on the boundary conditions fixed prescribed coordinates supports dependent calculable deformations prescribed time dependent coordinates dynamic degrees of freedom gt Dr Section 2 2 Kinematics 17 keyword type end node p end node q generalized a K a deformations PLBEAM planar beam xP oP x gp E1 E2 E3 PLTRUSS planar truss a x Ey PLTOR planar hinge oP ol amp y PLBEAR planar bearing xP oP xI gp E1 E2 E3 PLPINBOD planar pinbody xP oP xI _ E1 2 PLRBEAM planar rigid beam x ax E1 E2 PLWHEEL planar wheel rP o or 4 9 PLBELT planar belt gear oP x or e BEAM spatial beam ar AP at XN E E2 3 E4 E5 E6 TRUSS spatial truss x _ x E1 HINGE spatial hinge AP A amp 1 E2 3 PINBODY spatial pinbody AP a E1 E2 E3 RBEAM spatial rigid beam x AP af E1 E2 E3 WHEEL spatial disk wheel x AP ax E1 E2 E3 Ed Eb E6 TWHEEL spatial torus wheel A a E1 E2 E3 E4 E5 E6 Table 2 1 Nodes nodal coordinates and deformation parameters for the planar and spatial truss beam bearing hinge pinbody belt gear and wheel elements
61. e second position node initial direction of the principal y axis of the beam cross section 1 WHEEL SI element number first position node first orientation node second position node initial direction of the spin axis i e the z axis TWHEEL O element number first position node first orientation node second position node wheel radius in equatorial plane transverse wheel radius initial direction of the spin axis i e the z axis BWNRINDWDNSPWNRH AF WN e position node number x coordinate X 9 coordinate x3 coordinate 4 Section 2 2 Kinematics 21 FIX 1 node number 2 coordinate number 1 2 3 or 4 5 RLSE 1 element number 2 deformation mode coordinate number 1 2 3 4 5 or 6 x6 INPUTX 1 node number 2 coordinate number 1 2 3 or 4 x5 INPUTE 1 element number 2 deformation mode coordinate number 1 2 3 4 5 or 6 x6 DYNX 1 node number 2 coordinate number 1 2 3 or 4 5 DYNE 1 element number 2 deformation mode coordinate number 1 2 3 4 5 or 6 6 KINX 1 node number 2 coordinate number 1 2 3 or 4 5 KINE 1 element number 2 deformation mode coordinate number 1 2 3 4 5 or 6 6 LDEFORM 1 BEAM element number ORP INBOD 1 PINBODY RBEAM PLPINBOD or PLRBEAM element
62. ecify integrator type x12 Step size or initial step size ERROR Absolute error for the integrator Relative error for the integrator x13 i N N e e WN ITERSTEP N maximal number of iterations in calculating a stationary so lution default value 10 number of load steps default value 4 error tolerance default value 5 0E 7 NOTES x1 The inertia components are related to the global coordinate system x y z in the initial configuration The tensor components are needed so J etc represent the negative of the products of inertia 2 The distributed moments of inertia are lumped to the orientation nodes of the beam elements They represent the mass moments of inertia of the cross section of the beam so J and Jptz1 ave zero 28 Chapter 2 Keywords 3 The keyword MEE is used to add a fixed mass coupled to deformation mode coordinates If all five numbers are specified the mass is placed as a coupling between the two de formation mode coordinates if three numbers are specified the mass is placed on the diagonal 4 The required parameter of the USERSIG keyword is the name of a MATLAB M file without the extension m and with a maximum filename length of 8 characters The calling syntax of the M script is function time sig f pushsig t ne le e ep nx lnp xX xp The input parameters are the time t and a list of variables that store the instantaneous values of the same qu
63. efore utilities are provided to read and modify data in these files see page 9 Depending on the mode up to three binary output files may be created For all modes a SPACAR binary data file with filename identical to the input file and extension sbd is written The contents of this file are also stored in MATLAB arrays that are of course immediately available in the MATLAB workspace e g to be visualized with the standard MAT LAB graphics commands such as plot see e g Chapter B and Appendix C The following variables are created or overwritten mode SPACAR mode number ndof number of DOFs including rheonomic ones nddof number of dynamic DOFs nkdof number of configuration coordinates including rheonomic ones nkddof number of configuration coordinates nx number of coordinates ne number of deformation parameters nxp number of fixed calculable input dynamic and kinematic coordinates nep number of fixed calculable input dynamic and kinematic deformation parameters Inp location matrix for the nodes x1 The two location matrices provide information to find the location of a specific quantity in the data matrices inp The locations of undefined or unused coordinates and deformations equal zero For example the x and y coordinates of node 7 can be shown as function of time in a 6 Chapter 1 The SPACAR program le location matrix for the elements I In connection matrix for the nodes in the elements 2 it list
64. ference output vector Y The difference of these vectors is the input of the control sys tem The state matrices can be used to develop and tune a controller of any type e g lin ear non linear discrete continuous by means of the available software tools in MATLAB and SIMULINK The output of the controller u is added to the nominal input vector uo to actuate the mechanism An example is discussed in Sect When using blocks from the SPACAR SIMULINK library spacar_1ib note the following e Using any of the LTV Setpoint U0 Setpoint Sigma0 Reference YO and Times MO blocks at times beyond the last time step found in the data file may lead to unexpected results e In the current version of the software all spasim blocks in a block diagram should refer to the same input filename Analogously all LTV Setpoint U0 Setpoint Sigma0 Reference YO and Times MO must use the same 1tv file 1 5 Perturbation method and modal techniques For systems with a larger number of degrees of freedom the required computer time for a SPASIM simulation may be unacceptable in particular when high eigenfrequencies play a role Then the perturbation method may provide a numerically efficient solution strategy Consider e g the motion of the flexible manipulator depicted in Fig In the case the flex ibility is taken into account the generalized coordinates or degrees of freedom can be written as q ea 1 11 Section 1 5 Perturbation method and modal tec
65. formed into the linearized state space form dz Adz Bou 18 dy C z Dou where A is the state matrix B the input matrix C the output matrix and D the feed through matrix The state vector 6z is defined by dz dq 5q where dq is the vector of dynamic degrees of freedom The matrices B C and D depend on the chosen input vector du and the output vector dy Details of the linearization are discussed in Chapter 12 of the lecture notes Systems with non holonomic deformations For systems with non holonomic deformations arising from wheel elements the above descrip tion has to be modified in several respects Only mode 0 mode 1 mode 4 mode 7 and mode 9 are supported The state vector consists of the coordinates describing the configura tion q and the velocity coordinates q The configuration coordinates are split in coordinates whose derivatives are velocity coordinates and coordinates that have no corresponding velocity coordinates the latter are called kinematic coordinates The dynamic equations consist of two parts the kinematic differential equations defining the derivatives of the configuration coordi nates and the equations of motion defining the time derivatives of the velocity coordinates The linearized equations have the form I O s n Bio AN g 7 0 O Mo gq Ko No Go Co Do oq DF Sf DFO 5a 1 9 where Axo and Byo are kinematic matrices For ordinary systems Byo is a zero matri
66. gram SPACAR is based on the non linear finite element theory for multi degree of freedom mechanisms as described in Jonker s lecture notes on the Dynamics of Machines and Mechanisms 1 The program is capable of analysing the dynamics of planar and spatial mechanisms and manipulators with flexible links and treats the general case of coupled large displacement motion and small elastic deformation The motion can be simulated by solving the complete set of non linear equations of motion or by using the so called perturbation method The computational efficiency of the latter method can be improved further by applying modal techniques In this chapter an outline of the SPACAR package for use with MATLAB and SIMULINK is given in the next sections For instance for the design mechanical systems involving automatic con trols such as robotic manipulators interfaces with MATLAB are provided for open loop system analyses Section 1 2 Open loop and closed loop simulations can be carried out with blocks from a SIMULINK library Section 1 4 A special visualization tool SPAVISUAL is de scribed in Section Additional tools are available for using the perturbation method and the modal techniques in SIMULINK Section 1 5 Installation notes for SPACAR are given in Appendix A A graphical user interface GUI for generating input files for spatial systems is under devel opment and will be available in near future People interested in rigid planar mechani
67. gure from the figure s menu File Print or by pressing Ctr1 P Output to several graphics formats can be carried out as well File Export Alternatively MATLAB s print command can be used at the MATLAB command prompt E g you can generate PostScript output of the contents the current MATLAB figure window The print command sends the output directly to your default printer or writes it to the specified file if you supply a filename You can also specify the type of PostScript file Supported types include e PostScript dps e Color PostScript dpsc e Encapsulated PostScript deps e Encapsulated color PostScript depsc For example the statement print dataplot deps saves the contents of the current figure window as Encapsulated PostScript in the file called dataplot eps Depending on your MATLAB installation other graphics formats are sup ported try help print C 2 Quitting and saving the workspace To quit MATLAB type quit or exit Terminating a MATLAB session deletes the variables in the workspace Before quitting you can save the workspace for later use by typing save This command saves all variables in a file on disk named mat lab mat The next time MAT LAB is invoked you can execute load to restore the workspace from mat lab mat You can use save and load with other filenames or to save only selected variables The com mand save temp stores the current variables in the file named temp mat The command save
68. hniques 13 where e represent the large relative displacements and rotations and e are the flexible de formation parameters Due to the flexibility the actual trajectory motion will deviate from the prescribed motion If the deviations are small compared with the large scale motion then the small vibrational motion of the manipulator can be modelled as a first order perturbation q of the nominal rigid link motion qo by writing for the degrees of freedom q qo q 1 12 The perturbation method involves two steps 1 Compute nominal rigid link motion qo from the non linear equations of motion with all flexible deformation parameters e 0 This analysis will also provide the nominal input wo of the manipulator necessary to carry out the nominal motion and the general ized stress resultants Langrange multipliers 05 of the rigidified deformations i e the flexible deformations that are prescribed as zero 2 Compute the vibrational motion q from linearized equations of motion where Mo is the reduced mass matrix Co includes the velocity sensitivity and damping matrices and all stiffness matrices are combined into Ko The right hand side equals 1 14 where du u Uo is the actual control action u minus the nominal input uo The pre viously computed generalized stress resultants o are now applied as internal excitation forces To solve the linearized equations of motion 1 13 these are expressed as a
69. ices for the linearized equations of motion Sect are generated Obviously the input matrix B and output matrix C depend on the chosen input and output vectors In a usual state space system the output vector is computed from a linear expression In the case a larger accuracy is required SPACAR can be instructed to use a second order expression This feature is available for all deformation parameters and coordinates not for the time derivatives with the keywords REFES and REFXS The use of these keywords will generate elements in the output reference vector that are the same as the elements from REFE and REFX respectively Also the associated row in the output matrix C is the same but in addition a tensor denoted G in the 1tv file is computed with the second order geometric transfer function Linearization in mode 7 8 and 9 is around a pre computed static equilibrium configuration or a state of steady motion In addition in mode 9 the state space matrix A the input matrices Bo and B the output matrix C and the feedthrough matrix D are calculated Obviously the matrices Bo B C and D depend on the chosen input and output vectors du and dy respec tively These vectors are again defined in the blocks on page 40 These blocks are optional but as before omitting one or both blocks means that no input and or output vectors are defined and hence no state space matrices can be generated and written to the Itv file Section 2 5 Lineariza
70. ion of the sliding block 100 i i i i i i i i i 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 1 time s Figure 3 10 Case 1 Supporting forces on the sliding block 32 Chapter 3 Examples gt gt gt gt plot time xdd 1np 6 1 gt gt grid gt gt zlabel time s gt gt ylabel d dt 2 x 6 m s 2 gt gt gt gt plot time fxtot 1Inp 2 1 gt gt grid gt gt xlabel time s gt gt ylabel M 2 Nm gt gt gt gt plot time fxtot 1np 6 1 2 gt gt grid gt gt xlabel time s gt gt ylabel Fx 6 Fy 6 N Case 2 The input file of case 1 page 50 is modified to account for the shortened connecting rod Only the initial position of node 6 in the second block of the kinematic definition has to be changed X 6 0 35 0 The initial configuration of case 2 is depicted in Fig The horizontal position velocity and acceleration of the sliding block as a function of time are given in Figs The driving moment in node 2 versus time is shown in Fig and the supporting forces acting on the sliding block are presented in Fig The MATLAB commands used to plot these results are the same as in case 1 page 50 Case 3 To take the flexibility of the connecting rod into account with a reasonable accuracy the planar beam element used for this rod see Fig is split into two parts One translati
71. ion of the mechanism connectivity the configuration and the degrees of free dom DOFs q 2 e 2 System preparation 3 Calculation of the geometric transfer functions DYN is the dynamics module that generates the equations of motion and performs numerical integration in the forward dynamic analysis in the so called mode 1 of SPACAR Fur thermore it generates and solves the equations for the kinetostatic analysis INVDYN is the inverse manipulator dynamics module that performs the inverse kinematics and dynamics mode 2 and generates the setpoints for the simulation of manipulator motion with closed loop control in SIMULINK see Sect 1 4 The system inputs represented by the nominal input vector ug are to be varied by the control system actuators The system outputs represented by the reference output vector Yo consist of the coordinates to be monitored by control sensors Coordinates that are not measured may be added to check the performance of the manipulator in the simulation STATIO computes stationary solutions of autonomous systems Stationary solutions are solu tions in which the vector of dynamic degrees of freedom qf has a constant value This can represent a static equilibrium configuration or a state of steady motion LINEAR is a forward dynamics stage for the generation of linearized equations and state space matrices It can be used in different modes as described below Section 1 2 SPACAR amp MATLAB 3
72. lator mechanism Preface This is the 2009 edition of the manual that describes the use of the SPACAR package in a MAT LAB SIMULINK environment This software is being developed at the Laboratory of Mechanical Automation of the Department of Engineering Technology University of Twente and is partly based on work carried out at the Department of Engineering Mechanics Delft University of Technology This manual accompanies the 2009 UT release of SPACAR With respect to the previous editions of this manual new keywords have been included reflecting changes in the software In partic ular systems with non holonomic constraints resulting from the rolling contact of wheels can now be modelled As before the SPAVISUAL manual is integrated The examples are updated to show the use of SPAVISUAL The references to sections and examples in the lecture notes 1 are updated for the 2005 edition of these lecture notes They may be only approximate for other editions The visualisation tool SPAVISUAL has been implemented by Jan Bennik who also provided the description of the keywords and the examples for the part of the manual related to this tool Corrections of errors suggestions for improvements and other comments are welcome March 1 2009 dr ir R G K M Aarts Email R G K M Aarts utwente nl dr ir J P Meijaard and prof dr ir J B Jonker ill iv Preface The SPACAR program 1 1 Introduction The computer pro
73. leration 7 of the sliding bar Section 3 2 Planar slider crank mechanism 49 gt gt plot it was inp 2 37 1 2 t 1 2 3 1 2 t t 72 7 1 2 gt gt grid gt gt xlabel time s gt gt ylabel d dt y 2 m s gt gt figure gt gt plot t xdd 1Inp 2 2 4 1 2 37 1 2 t t 2 1 2 3 1 2 t t 72 2 7 1 2 gt gt grid gt gt xlabel time s gt gt ylabel d dt 2 y 2 m s 2 Obviously in both graphs the symbolic and numeric results are practically identical which illustrates the good agreement between both solutions Note that in this example no masses are defined There are no dynamic degrees of freedom either so effectively only a kinematic problem is solved 3 2 Planar slider crank mechanism The slider crank mechanism is frequently applied as a subsystem in the design of a mechanism It finds its applications in combustion engines compressors and regulators Figure 3 4 presents a slider crank mechanism for which three dynamics computations have to be carried out In the first problem case 1 see also example 5 7 2 in the lecture notes 1 the crank and the connecting rod are assumed to be rigid In the second computation case 2 the connecting rod is shorter but still somewhat longer than the crank In case 3 the flexibility of the connecting rod with the dimensions of case 1 is taken into account
74. local z direction is chosen so as to complete an orthogonal right handed coordinate system 3 If no direction is specified directions initially aligned with the global coordinate axes are chosen otherwise the line connecting the translational nodes is chosen as the local x direction and the specified vector is in the local x y plane The directions used are made orthonormal The directions can also be specified with the keyword ORPINBOD 4 The specification of the initial positions with the keyword X is only required for non zero position coordinates The initial orientations cannot be chosen freely x5 If the keywords INPUTX DYNX FIX and KINX are used without an explicit specification of the coordinate all independent coordinates will be marked as degrees of freedom or supports This means that x x2 and x3 are marked for position nodes and 8 or A Aa and 3 for orientation nodes If more than one coordinate is specified each of the speci fied coordinates is chosen as a degree of freedom or a support 6 Ifthe keywords INPUTE DYNE RLSE and KINE are used without an explicit specification of the deformation mode coordinate all deformation mode coordinates will be marked as degrees of freedom or released If more than one deformation mode coordinate is speci fied each of the specified coordinates is chosen as a degree of freedom or as released 7 There are four distinct cases two for the plana
75. lues frequencies and corresponding eigenvectors of the state space ma trix A are computed for a static equilibrium configuration or a state of steady motion The associated frequency equation of the undamped system is given by det w Mg KE NG GG 0 1 2 where the quantities w are the natural frequencies of the system In mode 8 a linear buckling analysis is carried out for a static equilibrium configuration or a state of steady motion Critical load parameters A are determined by solving the eigenvalue problem det KE 4 44 0 1 3 where the load multipliers satisfy fi Afo 1 4 Here K is the structural stiffness matrix and Gi is the geometric stiffness matrix due to the reference load fo giving rise to the reference stresses oo f represents the bucking load that corresponds with A In addition directional nodal compliances are computed In mode 9 linearized equations for control system analysis are computed for a static equilib rium configuration or a state of steady motion and are generated in the form My sg CE DE q Ki NG GE q Bodu 1 5 where By Du FS Da FP Mg CF DV K NG GY 1 6 is the input matrix and u so dm T T T 1 7 4 Chapter 1 The SPACAR program is the input vector The vectors 6g dq dq represent the prescribed input accelerations velocities and displacements respectively The linearized equations can be trans
76. m 3 5M E3 de3 0 3 2 These results can also be obtained numerically from a SPACAR analysis E g with numerical values for m 1 g 10 and k 1 and initial conditions e3 0 and 3 1 the acceleration is according to Eq B I 10 v2 8 59 A SPACAR input file fourbar dat for this case is PLTRUSS 1 1 2 PLTRUSS 2 1 3 PLTRUSS 3 2 3 PLTRUSS 4 2 4 PLTRUSS 5 3 4 X 1 0 0 Section 3 4 Planar four bar mechanism 61 X 2 0 7071 0 7071 X 3 0 7071 0 7071 X 4 0 1 4142 FIX iL FIX 4 DYNE 3 d END HALT XM 4 1 XF 4 0 10 ESTIEF 3 1 4142 STARTDE 3 1 0 1 END END In a MATLAB session we get the literal text of the session is modified somewhat to get a more compact presentation gt gt spacar 1 fourbar gt gt e le 3 1 ans 0 lt gt gt ed le 3 1 ans 1 gt gt edd le 3 1 ans 8 5858 Substituting the numerical values of the parameters into the linearized equation of motion Eq gives 5 3 2V 26 3 1 10V2 2V2 10 V2 5 de3 0 or d 3 2 83 d 3 16 14 de3 0 The stiffness term is a combination of Ko k 1 Go V2kes 0 No V29 z k m eg 15 14 where the solution of Eq 3 g V2 3 k m es has been used In a MATLAB session we get 3 3 3 4 3 5 3 6 62 Chapter 3 Examples gt gt gt gt mO gt gt co gt gt kO gt
77. me is almost fully dictated by the number and type of degrees of freedom Each trajectory is defined for the same DOF and therefore runs through the same branch of the scheme Only TRANS and TRC trajectory RL may be changed into one another after each Section 2 4 Inverse dynamics setpoint generation 33 At this stage it is useful to mention the way in which degrees of freedom are declared Position and orientation coordinates are declared as DOF by input command INPUTX node number component number Deformation mode coordinates are declared as DOF by input command INPUTE element number component number INPUTX and INPUTE are kinematic keywords Sect 2 2 So degrees of freedom are declared separately For generation of setpoints in relative coor dinates such as joint angles each INPUTE in the kinematics input prepares one TRE in the setpoint generation input only the first relative coordinate per element is allowed as input for the setpoint generation For the positions and orientations the situation is more complex be cause a trajectory in two or three dimensions is defined on node level not on coordinate level The keywords TROT TRANS and TRC RL prescribe the motion of one node keyword description node type and type number DOF cae rotation about a 2 D orientation 1 fixed axis in space 3 D orientation 4 o hy ha hg translati iti TENS
78. n Figs The driving moment in node 2 versus time is shown in Fig and the supporting forces acting on the sliding block are presented in Fig The MATLAB commands used to plot these results are gt gt plot time x 1np 6 1 gt gt grid gt gt xlabel time s gt gt ylabel x 6 m gt gt gt gt plot time xd 1lnp 6 1 gt gt grid gt gt xlabel time s gt gt ylabel d dt x 6 m s Section 3 2 Planar slider crank mechanism 51l 0 157 0 1 Ll L 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 0 45 Figure 3 5 Case 1 Initial configuration of the slider crank mechanism 30 d dt x 6 m s f 3 o re 20b 30 n L L f L L L f n 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 1 time s Figure 3 7 Case 1 Horizontal velocity of the sliding block 40 ET EEE M 2 Nm o i i i i i i i i i 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 1 time s Figure 3 9 Case 1 Driving moment in rotational node 2 i i i i i i i i i 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 1 time s Figure 3 6 Case 1 Horizontal position of the sliding block 3000 EN Bi Ly 1000 a n 3 TEE 8000 i i i i i i i i i 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 1 time s diat x 6 m s4 Figure 3 8 Case 1 Horizontal accelerat
79. nts by letting them have nodal points in common The nodal coordinates of the finite elements are described by position and orientation coordinates Therefore two types of nodes are distinguished position or trans lational nodes denoted by p for node p and orientation or rotational nodes denoted by p The nodes nodal coordinates and deformation parameters for the truss beam planar bearing hinge pinbody rigid beam planar belt gear element and wheel elements are summarized in Table 2 1 Usually the convention is made that node p of an element is assigned to the lower number of the element nodes and that node q is assigned to the higher node number The interconnections between the elements are accomplished by indicating common nodes between the elements For instance with a pin joint connection only the translational nodes are shared In case of a hinge joint connection only the rotational nodes are shared whereas translational coordinates can either be shared or unshared When elements are rigidly connected to each other both the translational and rotational nodes are shared see Fig It can be observed from Table that a truss element and a hinge element do not have common nodal types and therefore cannot be connected to each other gt N pin joint hinge joint rigid joint Figure 2 1 Joint connections between finite elements In the first block of the kinematics module either two dimensional planar or three
80. number REFEDP 3 deformation parameter number 1 2 3 4 5 or 6 REFX 1 reference output number 1 2 REFXP 2 node number REFXDP 3 coordinate number 1 2 3 or 4 NOTES x1 The nominal input numbers and reference output numbers are the positions of the specified input or output in the input and output vectors respectively Section 2 4 Inverse dynamics setpoint generation 37 2 The keywords REFES and REFXS that are defined for the linearization module Sect 2 5 are accepted as well and do not give errors Their meaning and usage is identical to the normal keywords REFE and REFX respectively 38 Chapter 2 Keywords 2 5 Linearization As mentioned in Sect 1 2 the module LINEAR is a forward dynamics stage for the generation of linearized equations of motion and state space matrices that can be used in two different modes mode 4 is basically an extension of the forward dynamic analysis of mode 1 No further keywords are required to obtain the coefficient matrices of the linearized equations as functions of the set of dynamic degrees of freedom q These matrices are stored in a SPACAR binary matrix data file with extension sbm This file can be loaded with the utility Loadsbm If input and output vectors du and dy are defined also the linearized state equations and output equations are computed see mode 9 Linearization in mode 3 is around a predefined nominal trajectory and takes place after that trajectory has
81. o that end the specified filename is truncated with at least one character at the right until a valid output data file is found So e g spacar 3 testlin can use data from an earlier spacar 2 test computation If no data file can be found in this way the linearization is aborted During the computation a plot of the mechanism is shown in a separate window While the simulation is running an Abort button is activated in the plot area Pressing this button will terminate the simulation possibly after some delay To speed up the computation the plot can be disabled by specifying the mode with a minus sign e g mode 2 for an inverse dynamics computation without a continually updated plot The plotting utility spadraw can also be used after the simulation to visualize the results see page D During the computations the results are stored in one or more data files and in MATLAB arrays A log file is always created when SPACAR Starts processing the input dat file This Log file contains an analysis of the input and possible errors and warnings It is described in more detail on page 8 Some errors in the input file do not lead to an early termination of the SPACAR computation but nevertheless give unusable results Therefore it is advisable to check the log file for unexpected messages All other data files are so called SPACAR binary data files SBF which implies that these are in a binary format and cannot be easily read by a user Ther
82. of e0 t 0 kols la 1 ls la 827 B 16 e D which have the desired form These results can also be obtained numerically from a SPACAR analysis E g with numerical values for ma 1 b EaA l 5 ka ks 1000 and l4 l3 2 A SPACAR input file Lever dat for this case is PLTRUSS 1 1 2 PLTRUSS 2 2 3 PLBEAM 3 3 4 5 6 PLBEAM 45 6 7 8 PLTRUSS 5 7 9 X 10 0 0 0 Section 3 10 State variable and output equations 77 x K K KX oO IO WD WNHNNE S0000 WWN OOo SOs S S Hj RY RY Ry be id WoO ond FP DYNX 2 1 INPUTX 7 1 RESE 1 1 RLSE 2 1 RLSE 5 1 END HALT XM 2 1 0 ESTIFF 2 1000 ESTIFF 1000 EDAMP 15 Oo END HALT INX 17 1 OUTF 1 5 1 END END In a MATLAB session we get gt gt spacar 9 lever gt gt A getfrsbf lever ltv A 1 A 0 1 1000 5 gt gt B getfrsbf lever ltv B 1 78 Chapter 3 Examples 0 2000 gt gt C getfrsbf lever ltv C 1 C 3000 0 gt gt D getfrsbf lever ltv D 1 6000 The state space matrices can also be obtained with the command get ss lever A Bode diagram Figure 3 39 can be made by the command gt gt bode getss lever Bode Diagram 100 rrr Ia ry 80 7 70r 60 50 40 7 Magnitude dB 30 7 20 7 360 F SSS FH u 315 270 F
83. of element types 2 rxyz initial orientations of elements gt gt time time column vector x coordinates nodal displacements xd nodal velocities xdd nodal accelerations fx prescribed nodal forces moments fxgrav gravity nodal forces moments fxtot reaction forces moments e generalized deformations ed velocities of generalized deformations edd accelerations of generalized deformations sig generalized stress resultants dec first order geometric transfer function for the deformations DF ate 3 dxc first order geometric transfer function for the coordinates DF 3 de first order geometric transfer function for the deformations DF e 3 dx first order geometric transfer function for the coordinates DF 93 d2e second order geometric transfer function for the deformations DPF 3 d2x second order geometric transfer function for the coordinates DPF 3 xcompl location vector for directional nodal compliances 4 Notes location matrix for the nodes The matrix element Inp i j denotes th the location of the j coordinate j 1 4 of node i location matrix for the elements The matrix element le i j denotes th the location of the 3 generalized deformation j 1 6 of element i graph by typing gt gt plot time x lnp 7 1 2 and the first generalized stresses in elements 1 2 and 3 can be plotted by typing gt gt plot time sig le 1 3 1 Obviously storage in the x xd xdd fx e ed ed
84. on function for mode 4 7 8 and 9 For the second parameter SPACAR binary data sbd file are defined 0 1 2 Default for all modes except mode 7 8 and 9 All normal output Default for mode 7 8 and 9 Additional output of the first order geometric transfer functions in de and dx Additional output of the second order geometric transfer functions in d2e and d2x Additional output of the first and second order geometric transfer functions a combination of 1 and 2 24 Chapter 2 Keywords 2 3 Dynamics With the keywords of the dynamics module the following blocks of information can be supplied Blocks 1 and 2 are optional If deformable elements have been defined in the kinematics block 3 has to be filled lest the stiffness and damping are zero If the motion is not prescribed by trajectories block 4 has to be used to define the input motion Finally with the keywords from the 5 block miscellaneous settings can be adjusted KEYWORDS DYNAMICS XM Inertia specification of lumped masses EM Inertia specification of distributed element masses XGYRO Inertia specification of gyrostat MEE User defined mass put into M 2 XF External force specification of the mechanism in nodes USERSIG Specification of MATLAB M file for user functions with input for forces and stresses 3 ESTIFF Specification of elastic constants E
85. onal node and one rotational node are inserted and the numbers of the nodes in the sliding block C are changed The bending stiffness of the connecting rod is computed using the moment of inertia I rd 64 The input file crank f1 dat is now Section 3 2 Planar slider crank mechanism 53 0 15 0 1 0 05 T A 77 0 05 0 1 0 15 i 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 Figure 3 11 Case 2 Initial configuration of the slider crank mechanism 1 1 1 1 1 1 1 1 1 20 N a 10 J F E ot 3 il 10H 4 20 j J A 30 0 0 01 0 02 003 0 04 005 006 0 07 008 0 09 01 time s Figure 3 13 Case 2 Horizontal velocity of the sliding block M 2 Nm o Ben Figure 3 15 Case 2 Driving moment in rota tional node 2 0 4 0 35 4 0 25 e 4 zl T a oo 0 15 0 1F et 0 05f 4 001 0 02 0 03 0 08 0 06 007 0 08 0 09 0 1 0 05 time s Figure 3 12 Case 2 Horizontal position of the sliding block 6000 4000F g 2000 N 6 rv s ne x OP 3 z 2000 4000 8000 0 0 01 0 02 0 03 0 04 0 05 006 007 0 08 009 O14 Figure 3 14 Case 2 Horizontal acceleration of the sliding block Figure 3 16 Case 2 Supporting forces on the sliding block 54 Chapter 3 Examples PLBEAM L 1 2 3 4 PLBEAM 2 3 5 6 7 PLBEAM 3 6 7 8 9 X
86. over the second pulley for a planar belt 5 rotational inertia Jy per unit of length for spatial beam 2 6 rotational inertia Jy per unit of length for spatial beam 2 XGYRO 1 node number 2 Q components of absolute angular rotor velocity free 3 Qz gt rotor motion or components of constant angular rotor 4 Qs velocity relative to the carrier body prescribed rotor motion 5 rotor inertia J 6 type of rotor motion 0 free 1 prescribed MEE 1 first element number 2 deformation coordinate of first element 3 second element number 4 deformation coordinate of second element 3 5 entry in the mass matrix M x3 XF 1 node number 2 forces dual with the 1 nodal coordinate 3 5 forces dual with the 2 4 3 and 4 nodal coordinate USERSIG 1 Name of the MATLAB M file with user functions with forces and stresses x4 26 Chapter 2 Keywords ESTIFF element number EA for beam truss and belt elements S S for hinge elements S first stiffness coefficient for pinbody and cognates GI for spatial beam EI for planar beam Sy second stiffness coefficient for pinbody and cognates 5 EI for spatial beam EI GAk for planar beam S3 third stiffness coefficient for pinbody and cognates x5 EI for spatial beam 5 EI GAk for spatial beam 5 EI GAk for spatial beam 5 ESIG ISO W element number preloaded generalized stresses x5
87. ows the contents of a SPACAR binary data file The output for each variable is the name Id the type 1 for integer 2 for real 3 for text and the size number of rows and columns First the header variables are shown with their values Long vectors may be truncated Between TDEF and TDAT the time varying data are given The number of time steps is equal to the number of rows specified for TDEF getfrsbf extracts a variable from a SPACAR binary data file The Id must be specified and for time varying data the time step as well repinsbf replaces the value of a variable in a SPACAR binary data file The Id must be specified and for time varying data the time step as well loadsbd loads all data from a SPACAR binary data sbd file into MATLAB s workspace loadsbm loads all data from a SPACAR binary matrix data sbm file into MATLAB s workspace getss loads the state space matrices at one time instant from a SPACAR 1t v file into a state space system in MATLAB s workspace combsbd combines data from two or more SPACAR binary data sbd files into a single output file The specified output file is overwritten without a warning spadraw is the plotting utility used internally by SPACAR It can also be used to visualize results after a simulation has been completed For all utilities additional online help is available by typing help command at the MATLAB prompt 10 Chapter 1 The SPACAR progr
88. pace input matrix i C state space output matrix D state space direct feedthrough matrix G second order output tensor E MO mass matrix Mo COB combined damping matrix Co Do z The 1og file The log file contains an analysis of the input and possible errors and warnings that are encoun tered The error and warning messages are explained in more detail in Appendix B The other output can be separated into a number of blocks The first lines indicate the version and release date of the software and a copyright note Next the lines from the input file read by the KIN module are shown not showing comments present in the input file see also Sect From the analysis is written e The elements used in this model The deformations of all elements are shown with the internal numbers according to the le array and the classification of each deformation O fixed C calculable and M DOF e The nodal point information with the internal numbers of the coordinates according to the Inp array and the classification as above e A list showing the degrees of freedom in which dynamic degrees of freedom are indi cated e The condition number of the part of the difference matrix that has to be inverted which shows how well the degrees of freedom have been chosen The DYN module reads the next data block and processed input lines are shown From the analysis we get e The numbers NEO NEMM NEM and NEC indicating the numbers of
89. r elements and two for the spatial elements For the planar elements if two numbers are specified this is the direction of the local x axis and an orthogonal y direction is found by rotating by a right angle in the positive direction and the directions are normalized if four numbers are specified these are taken as the direction vectors in the local x and y directions as they are For the spatial elements if six numbers are specified these are taken as the direction of the x axis and a direction in the local x y plane which are made orthonormal and completed by a local z axis if nine numbers are specified these are taken as the three direction vectors as they are Section 2 2 Kinematics 23 8 Both parameters for the output level are integers of which the values are the sum of the desired outputs A value of 0 implies the least output an output level of 1 means maxi mum output to obtain multiple outputs the specified values for the parameters should be added For the first parameter for the log file are defined 0 1 2 4 Default All normal output Additional output of the first order geometric transfer functions in de and dx Additional output of the second order geometric transfer functions in d2e and d2x formode 4 7 8 and 9 Additional output of the third order geometric transfer functions in d3e and d3x for mode 4 7 8 and 9 Additional output of the derivative of the global deformati
90. rdinates of circle end point x1 2 4 3D c ca and c3 coordinates of circle centre point 1 5 7 3D b b2 and bz coordinates of circle end point x1 TRE 1 element number 2 total displacement relative angle or elongation USERTRAJ 1 name of M script 2 TRTIME 1 total time for the trajectory 2 number of time steps 3 number of intermediate time steps 3 TREPMAX 1 element number 2 rise time period of acceleration 3 extreme velocity x4 TRVMAX 1 node number position or orientation node 2 rise time period of acceleration 3 extreme value of the velocity x4 TRFRONT 1 node or element number 2 acceleration front type x5 TRM 1 node number position or orientation node 2 extra mass m I or Jz 3 Jo 4 Jez 5 Jyy 6 Jyz 7 Jey 6 TRF 1 node number position node 2 f coordinate of external force 3 f2 coordinate 4 f3 coordinate Section 2 4 Inverse dynamics setpoint generation 35 NOTES x1 The positions of the parameters of keyword TRCIRL are different in 2 D and in 3 D cases Places 2 5 are used for 2 D places 2 7 for 3 D Note that the endpoint of the circle cannot be taken literally as it is over determined The second point defines a line through the centre on which the circle ends 2 See the note for the USERINP keyword on page 3 The keyword TRTIME has an optional third argument that influences the meaning of the second argument 2 arguments 3 arguments
91. respect to the proximal phalanx and the proximal phalanx can rotate with respect to the palm which is assumed to be immobile A pulley is rigidly connected to the distal phalanx centred at its rotation point which carries a chord that is slung over a second pulley centred at the rotation point of the proximal phalanx but free to rotate The chord is finally attached to a third pulley which is driven by a motor The finger is underactuated for it has two degrees of freedom but a single motor The degrees of freedom are chosen as the relative rotation angle between the two phalanges and the rotation angle of the motor pulley A force F 10N acts near the tip of the finger which is balanced by a moment M 1 5 Nm delivered by the motor Owing to the choice of the dimensions of the radii of the pulleys and the lengths of the phalanges this is an equilibrium position Figure 3 57 Finite element model for the robotic finger The finite element model is shown in Fig The input file finger dat is PLBEAM 112 3 4 PLTOR 24 5 PLBEAM 3 3 5 6 7 PLBELT 4 8 9 1 10 0 015 0 015 Section 3 13 Chord driven underactuated robotic finger 91 PLBELT 5 1 10 3 5 0 015 0 009 PLRBEAM 6 6 7 11 FIX 112 FIX 8 12 DYNX 9 1 DYNE 2 1 x 10 0 X 3 0 06 0 0 X 6 0 15 0 0 X 8 0 0 06 x 11 0 15 0 015 END HALT EM 3 1 0 EM 5 1 0 STARTDX 9 1 0 0 0 0 STARTDE 2 1 0 0 0 0 XF 11 0 0 10
92. see also example 8 3 1 in the lecture notes 1 Case 1 First of all the nodal coordinates must be specified In the initial configuration the crank and the connecting rod are horizontal The crank length is 0 15 m the length of the connecting rod Figure 3 4 Planar slider crank mechanism 50 Chapter 3 Examples is 0 30 m For the dynamic analysis the following parameters are needed The connecting rod has a circular cross section with diameter d 6mm The mass density is p 7 87 10 kg m and the Young s modulus is E 2 1 10 N m Consequently the mass per unit length is 0 2225 kg m and its total mass m 0 06675kg The mass of the sliding block or plunger C is given by mc Im 0 033375kg The crank is driven at a constant angular velocity wo 150rad s The total simulation should comprise two crank rotations Node B must be defined as a single translational node and a double rotational node since the rotations of the slider and the crank are not the same The mass of the crank is taken as zero An input file crank dat describing this case is PLBEAM 1 1 2 3 4 PLBEAM 2 3 5 6 7 X 1 0 00 0 X 3 015 0 X 6 0 45 0 FIX 1 FIX 6 2 END HALT XM 6 0 033375 EM 2 0 2225 INPUTX 2 1 0 150 0 TIMESTEP 0 1 100 END END The initial configuration of case 1 is depicted in Fig The horizontal position velocity and acceleration of the sliding block as function of time are given i
93. sition nodes one mass parameter is sufficient In the 3 D case six values determine the symmetric rotational inertia matrix A des 123 es 45 Tes 6 36 Chapter 2 Keywords 2 4 2 Nominal inputs uo and reference outputs yo The nominal input vector uo and the reference output vector Yo are defined in the following blocks These blocks are optional but omitting one or both blocks means that no input and or output vectors are defined and hence no setpoints for that input and or output vector are gener ated and written to the Lt v file KEYWORDS NOMINAL INPUT VECTOR u mode 2 NOMS Specification of actuator elements NOMF Specification of actuated nodes KEYWORDS REFERENCE OUTPUT VECTOR y mode 2 REFE Specification of the deformation parameters to be sensed REFEP The same first time derivative REFEDP The same second time derivative REFX Specification of the nodal coordinates to be sensed REFXP The same first time derivative REFXDP The same second time derivative The parameters for these keywords are listed below xi refers to note 7 listed at the end of the keywords NOMS 1 nominal input number x1 2 element number 3 deformation parameter number 1 2 3 4 5 or 6 NOME 1 nominal input number x1 2 node number 3 coordinate number 1 2 3 or 4 REFE 1 reference output number 1 2 REFEP 2 element
94. slation alonga 2 D position 2 1 T2 straight line 3 D position 3 1 T2 T3 translati iti ROTAL i on alonga 2 D position 2 1 T2 circle segment 3 D position 3 1 2 T3 For the administration of trajectories two numbers are of main importance the trajectory number and the node or element number The trajectory number has to be given once after TRAJECT node numbers or element numbers follow immediately after all other keywords In this way information about the path the velocity profile and additional loads can be grouped and worked up by node element number Taking as starting point the type of DOF the picture becomes DOF PATH VELOCITY PROFILE LOADS ELEMENT amp TRE TREPMAX TRFRONT TROT TRVMAX TRFRONT TRM NODE Ti TRANS TRVMAX TRFRONT TRM TRF TRCIRL TRVMAX TRFRONT TRM TRF 34 Chapter 2 Keywords The parameters required with these keywords are listed below i refers to note i listed at the end of the keywords TRAJECT 1 trajectory number TROT 1 node number orientation node 2 total angle in rad 3 h coordinate of fixed rotation axis 4 h gt coordinate 5 h3 coordinate TRANS 1 node number position node 2 x coordinate of end position 3 Y2 coordinate 4 x3 coordinate TRCIRL 1 node number position node 2 3 2D c and ca coordinates of circle centre point 1 4 5 2D b and bz coo
95. sms may consider the use of the commercially available package SAM by ARTAS 4 It has a nice graph ical interface for the definition of mechanisms and it provides more elements than SPACAR 1 2 SPACAR amp MATLAB The SPACAR program system for use in the MATLAB environment contains five modules which obtain their input from format free user supplied data In the following a short description of every module will be given The functional connections between the modules are illustrated in Fig 2 Chapter 1 The SPACAR program mechanism connectivity dynamic configuration properties trajectory path DOFs q forces velocity profile x e f oe ab 25 mode 1 f KIN gt DYN i gt DOFs q 4q mode Joint variables eg a INVDYN gt lt nominal inputs uo reference outputs Yo mode 3 gt bh State space matrices mode 4 i LINEAR gt Linearized equation Penn mode 7 I Eigen frequencies gt STATIO gt LINEAR mode 8 Buckling loads mode 9 State space matrices Figure 1 1 Functional relations between modules in SPACAR The indicated modes are available in the MATLAB environment KIN is the kinematics module that analyses the configuration of the mechanism The kinematic properties of the motion are specified by the geometric transfer functions The following steps are provided by the KIN module 1 Definit
96. sponds with the specified nodal position 7 The default position start value for INPUTX and STARTDX is the value specified by the kinematic keyword X which has a default zero 8 Stiffness and damping properties of the corresponding element are not used for the dynamic computations In amode 7 8 or 9 run a deformed mechanism configuration is computed which corresponds with the specified element deformation x9 Rotational deformations are defined in radians 10 Note that the keyword X defines an initial configuration in which the deformations are zero An exception is an element for which the keyword DRPINBOD has been used A start value defined with INPUTE or STARTDE defines a deformation with respect to the initial configuration x11 The required parameter of the USERINP keyword is the name of a MATLAB M file with out the extension m and with a maximum filename length of 8 characters The calling syntax of the M script is function t e x mymotion t is The input parameters are the time t and time step number is The script should return again time t prescribed deformations e and prescribed coordinates x Either e or x may be empty in the case no deformations or coordinates are prescribed Otherwise each row in e and or x should define one deformation or coordinate at the specified time t Five columns should be provided with 1 The element number e or the node number x 2 The deformation mode number e
97. t fatal error have a clear error message e SPACAR requires 2 or 3 input arguments SPACAR requires no output argument CCONST must be 1 x Nor N x 1 vector CCONST contains too many parameters MODE has an invalid value and FILENAME contains illegal characters indicate an incorrect call of SPACAR from MATLAB The last error can also occur in SPASIM SIMULINK e Wrong number of input arguments Flag must be a scalar variable Too many output arguments Time must be a scalar variable State vector of wrong size Input vector of wrong sizeand Not a valid flag number indicate an incorrect call of SPASIM from SIMULINK and should not occur during normal operation e ERROR opening file means that SPACAR can not open the specified file for output e ERROR opening existing file means thata file from a previous run is not found e ERROR in subroutine DINVOE is caused by an error in the dynamics input see Sect 2 3 101 102 Appendix B SPACAR error messages PREPTR Illegal velocity profile is reported when no valid velocity profile can be determined Can not determine valid and existing input file names from means that no mode 2 output data file with extension sbd matching the current mode 3 data file can be found Mechanisms are different Configuration mismatch LE and Configuration mismatch LNP arise from an error during the comparison between a the configur
98. ted to 2 LTV simulation of a linear time varying system as defined in an Lt v file see Sect 12 Chapter 1 The SPACAR program 3 Setpoint UO reads the nominal input from an 1tv file with setpoints generated e g with mode 2 or mode 3 The filename must be specified The setpoints are inter polated between the specified time steps The interpolation method can be chosen from Stepwise Linear default and Spline The block has no input and the dimension of the output vector equals the number of nominal inputs found in the file 4 Setpoint Sigma0 reads oo from an 1tv file generated with e g mode 3 see Sect L5 5 Reference YO reads the reference output from an 1tv data file with setpoints The filename must be specified Interpolation is as above This block has no input and the dimension of the output vector equals the number of reference outputs found in the file 6 Times MO reads the square reduced mass matrix Mo from an 1tv file generated with e g mode 3 The output of the block equals the input of the block is multiplied with the mass matrix The filename must be specified In the case not the full dimension of Mo in the 1tv is used the reduced dimension has to be specified All elements of Mo are interpolated linearly default or stepwise The dimension of the output vector equals the dimension of the input vector In the block diagram in Fig 1 3 the output vector y of the SPASIM block is compared with the re
99. temp X saves only variable X while save temp X Y Z saves X Y and Z load temp retrieves all the variables from the file named temp mat 106 Appendix C MATLAB tutorial 1 2 3 4 5 References Jonker J B Dynamics of Machines and Mechanisms A Finite Element Approach Lecture notes Department of Mechanical Engineering University of Twente vakcode 113130 October 2001 The Math Works Inc Getting Started with MATLAB version 7 Revised for MATLAB 7 1 Release 14SP3 September 2005 The Math Works Inc SIMULINK Getting Started version 6 New for SIMULINK 6 3 Release 14SP3 September 2005 SAM Version 4 2 5 0 or 5 1 ARTAS Engineering Software The Netherlands http www artas nl 2001 2005 Cowper G R The shear coefficient in Timoshenko s beam theory ASME Journal of Applied Mechanics 33 1966 pp 335 340 107
100. the Kelvin Voigt model as follows All first stresses are calculated as o Sye Sar co where S EA lo and Sa EaA lo for the truss and beam elements where l is the undeformed length of the element and the first stiffness and damping coefficients as defined in the input for the other types of elements co is the preload defined by the keyword ESIG For hinge and pinbody elements the other stresses are calculated in an analogous way For a planar beam element the bending stresses are calculated as o So 4 2 0 Jeo Sa2 4 6 246 03 146 2 9 4 E3 1 2 0 4 9 E3 i where S gt EI I 12EI GAkI and Sag Eal l For a spatial beam element the torsional stress is calculated as 02 Saca Saa amp a where Sy GI i and Sao Section 2 3 Dynamics 29 Gal l3 For bending along the local y and z axes the stresses are analogous to the planar case o3 93 44 2 P 3 y Sa3 4 2 3 oca 1 98 2 444 e 1 d 2 9 4 9 amp and 05 S4 4 2 P E5 i Sa4 4 P 2 P E5 os 1 0 2 44 4 les 14 2 4 4 3 amp where 53 EIp l 12E1 GAkyl Sa3 Ealy l S4 EL R P 12E1 GAk l5 and Saa Ealz l To all stress components a preload can be added by the key word ESIG x6 In amode 7 8 or 9 run a deformed mechanism configuration is computed which corre
101. the theoretical value is 5 6752 N If the warping is constrained at the clamped end the first element is effectively shorter for torsion by a distance b 1 v 24 where b is the height of the beam here b 0 2m and v is Poisson s ratio here v 0 The torsional stiffness of the first beam element now increases with a factor l b 2 3 1 19517 The input line for the stiffness of the first beam element now becomes ESTIFF 1 0 0 2 39034 0 0 1 0 The critical load is now increased to 6 1694 N Section 3 9 Lateral buckling of cantilever beam 73 0 15 0 1 0 05 0 05 0 1 0 15 Figure 3 37 Cantilever beam lateral buckling buckling mode 1 76 Chapter 3 Examples 3 10 State variable and output equations Figure 3 38 Lever system Find the state space variable and output equations for the system shown in Fig The input is the displacement x of the left end of spring ka EA lz it affects the mass ma through spring ks EA ls and the lever which is modelled by the planar beam elements 3 and 4 The lever has a fixed pivot at node 5 and is assumed to be massless yet rigid Its angular orientation is small so that only horizontal motion need be considered We will select dx and t as state variables with dx being the input and reaction force f as output With these definitions the state variable and output equations are then bx 0 1 dr 0 5 t m e i t eama a 3 15 TI
102. tion 39 KEYWORDS NOMINAL INPUT VECTOR u mode 3 i NOMS Specification of actuator elements NOMF Specification of actuated nodes KEYWORDS REFERENCE OUTPUT VECTOR y mode 3 REFE Specification of the deformation parameters to be sensed REFES The same with second order expression REFEP The same first time derivative REFEDP The same second time derivative see note REFX Specification of the nodal coordinates to be sensed REFXS The same with second order expression REFXP The same first time derivative REFXDP The same second time derivative see note Note Specifying second derivatives in the output vector implies an algebraic coupling between input and output i e a non zero state space matrix D This is currently not implemented and the keywords REFEDP and REFXDP are ignored for the linearization The parameters for these keywords are listed below xi refers to note i listed at the end of the keywords NOMS 1 nominal input number x1 2 element number 3 deformation parameter number 1 2 3 4 5 or 6 NOMF 1 nominal input number x1 2 node number 3 coordinate number 1 2 3 or 4 REFE 1 reference output number 1 REFES 2 element number REFEP 3 deformation parameter number 1 2 3 4 5 or 6 REFEDP REFX 1 reference output number 1 REFXS 2 node number REFXP 3 coordinate number 1 2 3 or 4
103. toolboxes like the Control System Toolbox may be helpful to develop and analyse control systems The installation of SPACAR uses less than 4 MB extra disk space The SPACAR files are stored in ZIP archives or in Microsoft Windows XP a compressed folder In Windows XP you can easily open such archives but of course you may chose to use your favourite unzipper The ZIP archives can be downloaded from http www wa ctw utwente n1l Software SPACAR In addition to the software there is a ZIP archive with the data files that are used for the examples in Chapter B Installation First of all you should create a subdirectory e g Matlab Toolbox Spacar Next you extract the files from the SPACAR software ZIP archive spacar2007_bin zip into this subdirectory There are three types of files e Files with the extension d11 are the actual executables of the SPACAR package The original SPACAR code not provided is written in C and FORTRAN77 compiled and linked into so called MEX modules that are executables for use within the MATLAB environment The following files must exist 29 100 Appendix A SPACAR installation checksbf dll combsbd dl1 getfrsbf dll loadsbd dl1ll loadsbm dl Iitv dliil mritv dll repinsbf dll spacar dll spacntrl dll spasim dll e Files with extension m are the MATLAB files necessary to use the SPACAR program The following file must exist spadraw m Other m files provide help text for the M
104. treats this block like a non linear state space system which has a state vector Z an input vector u and an output vector y Each of these vectors has a well defined meaning in the SPACAR block the states correspond to the degrees of freedom and their first time derivatives The input and output are coupled to actuators and coordinates as specified by keywords in the SPACAR input data file see below In the SIMULINK graphical user interface the input and output vectors must be coupled to other blocks e g the control system The states are used internally in SIMULINK and are usually not available to the user That implies that any coordinate or deformation parameter that is used for control purposes or is monitored in a graph must be included in the output vector y block 2 KEYWORDS INPUT VECTOR u SPASIM i INPUTS Specification of actuator elements INPUTF Specification of actuated nodes KEYWORDS OUTPUT VECTOR y SPASIM i OUTE Specification of the deformation parameters to be sensed OUTEP The same first time derivative OUTEDP The same second time derivative OUTX Specification of the nodal coordinates to be sensed OUTXP The same first time derivative OUTXDP The same second time derivative Section 2 6 Non linear simulation of manipulator control 43 The parameters for these keywords are listed below xi refers to note 7 listed at the end of the
105. type z 1 2 3 Zo 0 o 0 force x 0 0 0 force y 0 0 0 force z 14 121 307 mass 0 10 30 including the element masses Elements for the spatial manipulator element 1 element2 element3 element4 element 5 element type hinge hinge hinge beam beam T nodes 4 5 5 8 R nodes 1 2 2 3 6 7 3 6 7 9 x local y axis 0 0 0 0 0 y local y axis 0 1 1 1 1 z local z axis 1 0 0 0 0 type e 2 2 2 1 1 type ez 1 1 1 1 1 type e3 1 1 1 1 1 type e4 1 1 type 5 1 1 type ee 1 1 T translational R rotational 84 Chapter 3 Examples Linearization In one of the next sections the design of a closed loop controller for this manipulator will be discussed This controller depends on parameters derived from the linearized equations of mo tion Therefore a linearization is needed in terms of the DOFs corresponding to the actuator joints An input file robotinvlin dat for this analysis SPACAR mode 3 is HINGE 1 1 2 Ov uO AL NOMS Te ab HINGE 2 2 3 0 1 0 NOMS OF DM BEAM 44356 0 1 0 NOMS 3 831 HINGE 3 6 7 0 1 0 BEAM HS 8 9 Od 0 REFE de REFE 2 R2 x 4 0 Ou 10 REFE 331 X 5 0 268 0 0 6467 REFEP 41 x 8 0 536 105 O REFEP 5 2 REFEP 6 3 FIX L REFEDP 7 1 FIX 4 REFEDP 8 2 INPUTE 1 REFEDP 9 3 INPUTE 2 REFX 10 8 1 INPUTE 3 REFX 11 8 2 REFX 12 8 3 END REFXP 13 8 1 HALT REFXP 14 8 2 REFXP 15 8 3 XM 5 10 XM
106. ver beam with a narrow rectangular cross section which is loaded by a transverse force Fkipp at its free end in the direction of the larger flexural rigidity The theoretical buckling load is Fj 4 013 ETS 1 where EI is the smaller flexural rigidity S the torsional rigidity and the length of the beam For numerical analysis the beam is divided into four equal spatial beam elements in which the second order terms in the bending deformations are included in the analysis In a MATLAB session we get gt gt spacar 8 lateral4 gt gt spavisual lateral4 An input file lateral4 dat describing this case is BEAM 1 1 2 3 4 Oanig t0 BEAM 2 3 4 5 6 Oi L 0 BEAM 3 5 6 7 8 Os 0 BEAM 4 7 8 9 10 0 1 0 X 10 00 0 00 0 00 X 30 25 0 00 0 00 X 5 0 50 0 00 0 00 X 70 75 0 00 0 00 Xx 9 1 00 0 00 0 00 DYNE 12 5 6 DYNE 2 2 5 6 DENE 23 2 2556 DYNE 4 2 5 6 FIX 1 FIX 2 OUTLEVEL O 1 END HALT EM 1 1 0 0 0033 EM 2 1 0 0 0033 EM 3 1 0 0 0033 EM 4 1 0 0 0033 ESTIFF 1 0 0 2 0 0 0 1 0 ESTIEF 2 0 0 2 0 0 0 1 0 ESTIFF 3 0 0 2 0 0 0 1 0 ESTIFF 4 0 0 2 0 0 0 1 0 74 Chapter 3 Examples XF 9 0 0 0 0 1 0 END END VISUALIZATION BUCKLINGMODE 1 TRANSPERANCY 0 9 BEAMVIS 0 01 0 1 LICHT 1 STEPLINE 0 01 ENLARGEFACTOR 0 04 The 3D visualization of this file is presented in figure 3 37 The buckling load found is 5 7619 N whereas
107. versus x Line styles markers and color You can pass a character string as an argument to the plot function in order to specify various line styles plot symbols and colors In the statement plot x y S s is a l 2 or 3 character string delineated by single quotes constructed from the characters in the following table Symbol Color Symbol Linestyle y yellow point m magenta o circle cyan x x mark ie red plus g green x star b blue solid W white dotted k black ae dashdot dashed For example plot x y c plots a cyan plus symbol at each data point If you do not specify a color the plot function automatically uses the colors in the above table For one line the default is yellow because this is the most visible color on a black background For multiple lines the plot function cycles through the first six colors in the table Adding lines to an existing graph You can add lines to an existing graph using the hold command When you set hold to on MATLAB does not remove the existing lines instead it adds the new lines to the current axes It may however rescale the axes if the new data fall outside the range of the previous data For example plot f1 hold on BLOE LEZ TR plot 3 7 hold off These statements produce a graph displaying three plots Appendix C MATLAB tutorial 105 Creating hardcopy of MATLAB figures You can make a hardcopy of a fi
108. x and Axo is an identity matrix For mode 7 a stationary solution is first obtained with the module STATIO and the eigenvalues are obtained by solving the characteristic equation 1 10 det ag ate ee Ko No Go MoA Co Do For mode 9 the linearized state equations are obtained as in equation 1 8 with the difference that the variations in the states are now dz dg 6q Definition of a mechanism model A model of a mechanism must be defined in an input file of file type or file name extension dat This input file consists of a number of keywords with essential and optional parameters The input file can be generated with any text editor In Chapter P the meaning of the keywords and their parameters is discussed in detail In the examples in Chapter 3lcomplete input files are presented Section 1 2 SPACAR amp MATLAB 5 Running SPACAR in the MATLAB environment Once the mechanism is defined and this information is saved to a dat input file SPACAR can be activated with the MATLAB command gt gt spacar mode filename Here mode indicates the type of computation as shown in Fig filename is the name of the input file without the extension dat The filename is limited to 20 characters from the set 0 9 a z A Z and _ so it can not include drive or path specifications The linearization with mode 3 needs data from a previous inverse dynamics computation T
109. y of functions for displaying data This section describes some of these functions For a complete survey of graphics functions available in MATLAB we refer to the official MATLAB documentation 2 or to the online help utility Elementary plotting functions The following list summarizes the functions that produce basic line plots of data These func tions differ only in the way they scale the plot axes Each accepts input in the form of vectors or matrices and automatically scales the axes to accommodate the input data plot creates a plot of vectors or columns of matrices loglog creates a plot using logarithmic scales for both axes semilogx creates a plot using a logarithmic scale for the x axis and a linear scale for the y axis semilogy creates a plot using a linear scale for the x axis and a logarithmic scale for the y axis You can add titles axis labels grid lines and text to your graph using e title adds a title to the graph xlabel adds a label to the x axis y label adds a label to the y axis text displays a text string at a specified location gtext places text on the graph using the mouse grid turns on off grid lines 103 104 Appendix C MATLAB tutorial Creating a plot If y is a vector plot y produces a linear graph of the elements of y versus the index of the elements of y If you specify two vectors as arguments plot x y produces a graph of y

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