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Robotica User Manual

1. Out 7 I al S1 a2 512 a2 512 al CI a2 C12 a2 C12 0 0 Jacobian 0 0 0 1 1 Figure 3 5 The Jacobian for the two degree of freedom robot Another useful display function is EPrint it prints the elements of the matrix one per line EPrint takes either two parameters or three parameters the matrix to be displayed required a text label to print alongside the elements required and a filename in quotes in which to save the matrix elements optional Figure 3 6 shows how EPrint is used to display the matrix T 0 1 20 In 8 MPrint T 0 1 TOL Out 8 C1 S1 0 al Cl S1 Cl 0 al Sl TOl 0 0 1 0 In 9 TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL TOL 0 0 0 1 EPrint T 0 1 T01 1 1 C12 2 1 S12 3 1 0 4 1 0 1 2 S1 29 C1 3 2 0 4 2 0 1 3 0 2 3 0 33 1 4 3 0 1 4 al Cl 2 4 al S1 3 4 0 AAT 1 Figure 3 6 Output from EPrint In addition to EPrint and MPrint there are several other functions useful for displaying information about the forward kinematics TPrint prints all of the T matrices to a file if specified e g TPrint tmatrices or to the screen if no parameter is given APrint does the same thing for A matrices PrintInputData takes no parameters and displays all of the information which Robotica knows about the current data set previously read in with Da
2. The filename must be in quotes and specifies the file to take input from The input file should be in the following format which is supported directly by SIMNON tdi ql 0 1 0 02 1 1 0 04 The first line is a comment line containing the variables separated by spaces for which data are listed in this file There must be one space between the inital quotation mark and t Time must always be the first variable and there must be at least one other variable Following the varable list there may be any number of comment lines beginning with a quotation mark The data are listed after the last comment line with each line containing one set of input values The time parameter tis always assumed to be the first variable in the list Once the data are read in the graphics are generated and output The data and the variable list are stored for further analysis so that to replot with the same data the filename can be specified with The options for SimDrive are put in a list which may contain any of the following 84 trace arm frame animate print xprint mprint new image of the robot is drawn whenever the end effector moves a certain absolute distance which defaults to 0 2 A different distance can be specified by appending a number to the options list For example SimDrive simnon trace 0 11 would update the plot whenever the end effector move more that 0 1 absolute units The plot produc
3. base AffineShapelbase 5 5 5 point AffineShape point 5 5 5 point TranslateShape point 0 0 1 put the pyramid on top of the cube pris base point create a list with both shapes pris gt gt newpris save the shape in a file Figure 4 4 shows the new prismatic joint shape and Figure 4 5 shows how the new joints appear in a graphic Of course the commands PrisJoint and RevJoint must be used to load the new joint shapes before rendering File V View V Sound Figure 4 4 New prismatic joint 4 4 Planar Arm Interjoint Shapes Planar arms can be drawn with link shapes connecting the joint shapes The function LinkShape loads a shape definition from a file or if no filename is given uses a simple rectangular shape The function ClearLinkShapel causes Robotica to clear the link definition and use a line to represent links 28 File V View V J Sound V fat 0 0 2 30 Deggee OF tds 3 37 0 i 0 0 Figure 4 5 Stanford arm with new joint shapes Figure 4 6 shows a two link planar arm with the standard revolute joint shape and link shape Figure 4 7 shows the same arm with a diamond link shape this shape was stored in a file as Line 0 0 0 5 5 0 1 0 0 5 5 0 0 0 0 Much of the confusion in creating a link shape stems from the fact that the shape is scaled by either a in x or d in z before being placed
4. file The file will be called xrelp out This uses the internal Mathematica command Display to con vert the output graphic to a raw postscript file Some extra pro cessing may need to be done before the file is sent to a postscript output device See the Mathematica users manual for details The file will be called mrelp out 77 file Save the ellipsoid measures in a file called file if either mea sures or monly is specified in the option list H the file already exists the information is appended to the end of the file other wise a new file is created Use as the filename to indicate that the output should go to the current default file Options must be in a list and can be given in any order For example RElp an scale single causes REIp to draw a single scaled ellipsoid at the start values of the parame ters in the list an If no arm graphics have been loaded simple lines will be used to show the arm The functions RevJoint and PrisJoint can be used to load the standard joint shapes to provide more detail Example RElp ql 20Degree 90Degree 8 monly measures Response time init Response calculates the time response from the dynamics equations given an input torque vector loaded with GetInputTau Of course ELDynamics must have been run and all of the symbols in the equations must have been assigned numerical values The first parameter to Resp
5. msimplot out In addition appending a number to the end of the options list forces the plot to have an aspect ratio height divided by width equal to that number The plot generated by SimPlot is saved in the variable simplot Example SimPlot t ql qld print 1 TPrint filename This function prints all of the T matrices for the most recent run of FKin The file argument is optional and specifies a filename in which to store the matrices The filename must be a string e g Tmatrices which represents a valid filename on the system used If the file already exists the T matrix information is appended to the end of the file otherwise a new file is created Also once a file has been specified it becomes the default file for all the printing functions To specify that the output should go to the default file use as the filename The file created is a simple text file and can be sent to any printer or edited with a standard text editor If no filename is given the T matrices are displayed on the screen only 87 Example TPrint 88 APPENDIX A KINEMATICS AND DYNAMICS EQUATIONS Forward Kinematics Equations Figure A 1 this is Figure 3 4 in 3 shows the Denavit Hartenberg representation defined or successive links Joint i 1 Figure A 1 Denavit Hartenburg frame assignment 3 89 There are four parameters of interest a the distance along z from o to the intersect
6. 20 New prismatic joint 24 Stanford arm with new joint shapes 25 Arm with standard shapes a a aaa 26 Arm with new link shape 26 Successful read operation 29 Results of ELDynamics 30 The elements of the mass matrix 31 Results of running Simplify TrigNotation 32 Examining the Christoffel symbols 33 The Jvc matrices 33 G the gravity vector oaoa 34 Reading the input torque vector 36 Joint parameter ql tracking a Sine wave 39 Circular motion control definitions 41 Joint variables as a function of time 42 Arm location from circular controller 42 End effector location from circular controller 43 Planar manipulability example 48 Scaling the ellipsoids 49 Three link example 50 Three link nonplanar example 50 A change of ViewPoint 51 Loading a SIMNON file 55 Two lnk planar arm driven with dataset 56 Plot of the end effector location 57 Time t versus
7. and z coordinates so that the shape can be used for three dimensional plots as well as planar plots Of course if the robot is planar in x y for example then the z coordinate could always be zero but it still needs to be there For example Line 0 0 2 2 could not be used as part of the joint but the following could Line 0 0 0 2 2 0 For non planar arms Joints are connected together with a straight line representing the link between joints In the case of planar arms an interjoint shape can be specified with the LinkShape command see section 4 3 for more details For a revolute joint the following steps are executed before the joint shape is placed in the output graphic Stepl The joint shape is rotated by the value of the joint variable For example if the second joint in the Stanford arm is at 30Degree then the joint shape is rotated about its z axis 30Degree Step2 The joint shape is rotated to the orientation of its frame relative to the base frame Essentially this aligns the shape with its own coordinate axes Step3 The joint is translated out to the proper location determined by the T matrices 25 Step4 If the robot is planar then the appropriate coordinate is dropped and two dimensional plots result when drawing the robot For example an x y planar robot would have its z coordinate dropped The steps for a prismatic Joint are very similar Stepl The joint shape is rotat
8. revolute 07 al al 08 alphal 0 09 di 0 10 thetal ql 11 joint2 revolute 12 a2 a2 13 alpha2 0 14 d2 0 12 15 theta2 q2 16 17 The dynamics information 18 19 DYNAMICS 20 21 gravity vector 0 2 0 22 massl ml 23 center of mass 1 2 al 0 0 24 inertia matrix 0 0 0 0 0 11 25 mass2 m2 26 center of mass 1 2 a2 0 0 27 inertia matrix 0 0 0 0 0 12 In this example lines 01 through 03 are a general comment and are ignored In general there can be any number of such comment lines prefacing the data Line 04 tells Robotica that Denavit Hartenberg parameters are coming The keyword DOF is used by Robotica to decide when to start reading the parameters DOF should be followed by an equals sign then the number of links in the robot Line 05 is provided as a comment line it is skipped over when reading the data If no comment is wanted leave this line blank Lines 06 through 10 contain the Denavit Hartenberg parameters for the first link The line joint revolute tells Robotica that the first link is a revolute type Two types of links are supported by Robotica revolute and prismatic sliding If the link had been prismatic line 06 would have read Joint prismatic Line 07 specifies the a parameter for the link line 08 gives the alpha parameter line 09 supplies the d parameter and line 10 specifies the theta variable
9. 4 for more information on manipulability Manipulating Ellipsoid Figure 7 5 A change of ViewPoint 99 CHAPTER 8 EXTERNAL SIMULATION COMPATIBILITY Robotica has the ability to read data from external simulations such as SIMNON 1 and generate graphical output showing the result of applying the joint motion data in the file to the current robot In addition the data in the simulation file can be displayed plotted and printed in numerous ways There are two functions provided to handle simulation compatibility The first Sim Drivel begins by reading a simulation input file It then graphically shows the results of applying the joint parameter values specified by the simulation file to the current robot The second command SimPlot is used to plot the simulation input data in familiar x y format by specifying the various quantities to be graphed 8 1 Loading Simulation Data The Robotica function SimDrive is used to read the simulation output file it must adhere to the format shown in Section 8 2 If necessary a text editor can be used to modify the file so that it matches the required format SimDrivel requires that an input file of Denavit Hartenberg parameters has been created for the robot that the simulation output will drive This input file should be loaded with DataFile and the forward kinematics should then be generated with FKin The format for the SimDrivel command is SimDrivelfilename options
10. The filename must be in quotes and designates the file to acquire input from Once the data are read in the data in each set are used to update robot arm parameters and 56 the graphics are generated and output The data and variable list is stored for further analysis To indicate another plot with the same data use as the filename The options for SimDrivel are held in a list which may contain any of the following trace Show a plot of the final coordinate frame location arm Draw the arm graphics as the arm moves in response to the input data frame Put a frame around graphics animate Take all of the generated frames and show them sequentially in movie fashion The animation can be shown again with ShowAnim or saved with SaveAnim print If Mathematica has a LaserPrint function defined it will be used to print the graph xprint Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graph ics output window to save as a postscript compatible file The output file will be called xsim out mprint This uses the internal Mathematica command Display to convert the output graphic to a raw postscript file Some extra processing may need to be done before the file is sent to a postscript output device See the Mathematica users manual for details The output f
11. The parameters must be given in that order a alpha d and theta The format for each line is label parameter The label is ignored and the parameter is assigned to its respective Denavit Hartenberg parameter for that link There must be exactly one space or tab after the equals sign 13 Lines 11 through 15 give the corresponding data for link two The comments in the above paragraph apply There should be no space between consecutive data segments for neighboring links Lines 16 through 18 are a general comment and are ignored by Robotica In general there can be any number of such comment lines as a preface to the dynamics data Line 19 contains the required keyword DYNAMICS to indicate to Robotica that dynamics input data are to follow Line 20 is provided as a comment line it is skipped over when reading the data If no comment is wanted leave this line blank Line 21 gives the gravity vector in standard x y z form This vector is understood by Robotica to be referenced to the base frame Lines 22 through 24 give the dynamics data for the first link In line 22 mass1 ml assigns the variable ml to be the mass of the first link Line 23 supplies the location of the center of mass for link one in terms of the coordinate frame for link one The format is a list of x y and z offsets In Mathematica a list is surrounded by braces thus the input here is 1 2 al 0 0 which means that the center of mass for
12. a sphere with this radius has the same volume as the ellipsoid see Chap ter 4 of 4 Causes RElp to print only manipulability measures no graphics are generated Save the ellipsoid measures in a file called file if either measures or monly is specified in the option list The filename must be a string e g measures which represents a valid filename on the system being used If the file already exists the ellipsoid measure information is appended to the end of the file otherwise a new file is created Also once a file has been specified it becomes the default output file To specify that output should go to the default file use as the filename If no filename is given the ellipsoid measure information is displayed on the screen only Specify this option to scale the ellipsoids such that the largest axis is always two units long This helps to keep the plot from becoming too cluttered with overlapping ellipses If Mathematica has a LaserPrint function defined it will be used to print the graph 50 xprint Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graph ics output window to save as a postscript compatible file The output file will be called xrelp out mprint This uses the internal Mathematica command Display to convert the o
13. as well as convert a solid to a wireframe To create a custom joint with Shapes m and Polyhedra m the functions would be used to generate a series of shapes which would be assigned to some variable in Mathematica When the design is complete the variable should be written to a file and loaded with PrisJoint or RevJoint 4 3 An Example Creating a new joint shape is relatively easy solids from the package Shapes m are the most useful is defining new joints Assuming that the Shapes m package has been loaded the commands given below will create a new revolute shape that looks like a cylinder This shape differs from the default revolute shape in that it is rendered as a solid with hidden surface elimination which makes it more realistic looking A solid generated with Shapes m can be converted to wireframe with the WireFramel command see the Shapes m file for more information rev Cylinder rev AffineShapelrev 5 5 5 make it half size rev gt gt newrev save the shape in a file Other shapes can be incorporated into the Joint design as well For example adding a pyramid that points in the direction of motion for a prismatic joint is relatively straight forward The following commands create a new prismatic joint with a square base and a pyramid whose apex indicates direction of motion base Cylinder 1 1 4 4 sided cylinder is a cube point Cone 1 1 4 4 sided cone is a pyramid 27
14. in movie fashion The animation can be shown again with ShowAnim or saved with SaveAnim print If Mathematica has a LaserPrint function defined it will be used to print the graph xprint Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graphics output window to save as a postscript compatible file The output file will be called xrobotplot out mprint This uses the internal Mathematica command Display to con vert the output graphic to a raw postscript file Some extra pro cessing may have to be done before the file is sent to a postscript output device See the Mathematica users manual for details The output file will be called mrobotplot out In addition a variable called robotplot is created to hold the contents of the current output of ShowRobot It can displayed at any time once generated with Show robotplot Any of the various display options available from Show e g ViewPoint can be used to view the plot Example ShowRobot ql 20Degree 90Degree 8 83 SimDrive filename options SimDrive is provided to show graphically the results of output created by external simulation programs such as SIMNON The first step is to create an input file of DH parameters for the robot to display Then load the robot with DataFile and run FKin
15. pressed the command is sent to Mathematica and the output is posted in the output window The command input line allows the processing of any Mathematica command For example entering Plot Sin x x 0 Pi will produce a plot of Sin x from 0 to Pi In general any Mathematica command can be executed in this way 9 5 Robotica Function Selection Menu The right side of RFE consists of the Robotica Function Selection Menu In the menu are buttons corresponding to all of the available Robotica functions To execute a Robotica procedure simply click on the button that contains the function name For every function a screen will pop up showing the available options as well as a Send button and a Cancel button If the function takes no options such as FKin the pop up screen contains only the Send and Receive buttons Figure 9 3 shows an example of the option screen for the DataFile function Here the only parameter is a filename pressing Send transmits the function to Robotica while pressing Cancel prevents the function from being executed Each pop up screen remembers what was entered the last time the function was executed and sets the options to those values 67 Filename Figure 9 3 DataFile option screen Figure 9 4 shows the option screen for the SimDrive command The data source field is for the file name from which data are loaded and below that are listed the various options t
16. SimplifyTrigNotation takes no parameters Its purpose is to shorten the notation used for the Sin and Cosine functions to match that found in most robotics texts Enter SimplifyTrigNotation to obtain the complete list of reductions Example SimplifyTrigNotation SimPlot indep deplist options SimPlot plots data loaded with SimDrive The first parameter is the independent variable i e the x axis The second parameter is a list of dependent variables to be plotted against the independent variable For example SimPlot t q1 plots t vs ql and SimPlot t ql q2 plots t vs ql and q2 The following options are available for use with SimPlot they should be put in the options list 86 print If Mathematica has a LaserPrint function defined it will be used to print the graph frame Put a frame around the plot xprint Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graphics output window to save as a postscript compatible file The output file will be called xsimplot out mprint This uses the internal Mathematica command Display to con vert the output graphic to a raw postscript file Some extra pro cessing may have to be done before the file is sent to a postscript output device See the Mathematica users manual for details The output file will be called
17. called xrobotplot out mprint This uses the internal Mathematica command Display to con vert the output graphic to a raw postscript file Some extra pro cessing may have to be done before the file is sent to a postscript output device See the Mathematica users manual for details The output file will be called mrobotplot out In addition a variable called robotplot is created to hold the contents of the cur rent output of SeqShowRobot It can displayed at any time once generated with Show robotplot Any of the various display options available from Show e g View Point can be used to view the plot Example SeqShowRobot ql 20Degree 90Degree d2 3 6 d2 6 3 ql 90Degree 20Degree 81 81 SetRanges xrange yrange zrange This function is used to set the range of display for the animation frames To ensure that each frame s view is consistent with every other frame the view limits are set to xrange yrange zrange Each of the ranges should be a list containing a min max pair For example SetRanges 5 5 0 10 10 0 sets the animation views to include x values from 5 to 5 y values from 0 to 10 and z values from 10 to 0 Example SetRanges 5 5 2 4 0 9 Show Anim ShowAnim shows an animation sequence that was generated with RElp SimDrive etc or loaded with LoadAnim The package Animation m is required for this to function to execute Example Sh
18. computed Figure 3 2 shows a sample run of FKin on the data file of Figure 3 1 18 In 4 F Kin A Matrices Formed Jacobian Formed Jacobian 6x2 Figure 3 2 Results of FKin on input data 3 3 Printing Results After the forward kinematics equations have been generated showing the results is accomplished through a set of display functions provided within Robotica MPrint is such a function it takes up to three parameters the matrix to be displayed required a text label to print alongside the matrix required and a filename in quotes in which to save the matrix optional Figure 3 3 shows how MPrint is used to display the matrix A 2 In 5 MPrint A 2 A2 Out 5 A2 Cos q2 Sin q2 0 a2 Cos q2 Sin q2 Cos q2 0 a2 Sinld2 1 0 0 1 Figure 3 3 MPrint usage 19 It is sometimes convenient to change the trigonometric notation to be somewhat more succinct The function SimplifyTrigNotation takes no parameters and modifies the display of Sin and Cos to an abbreviated form Figure 3 4 shows the results of running Simplify TrigNotation In 5 Simplify TrigNotation In 6 MPrint A 2 A2 Out 7 C2 S2 0 a2 C2 S2 C2 0 a2 S2 A240 0 10 0 0 01 Figure 3 4 Results of Simplify TrigNotation Similarly the Jacobian can be displayed with a simple call to MPrint Figure 3 5 shows the results In 7 MPrint J Jacobian
19. dynamics equations can be calculated The inertia matrix Coriolis and centrifugal terms Christoffel symbols and gravity vectors are all available to the user once the dynamics routines have run Utilizing the forward kinematics results Robotica can calculate the manipulability ellipsoids when supplied with a range of joint variable values It is possible to generate and save a list of manipulability measures as well as display the ellipsoids with the robot on the screen 1 Robotica is a trademark of The Board of Trustees of the University of Illinois In addition Robotica has the capability of reading external simulation e g SIM NON 1 output files and displaying the motion of the robot when subjected to the sequence of Joint variable changes described in the file This requires that the robot has been input as a table of Denavit Hartenburg parameters and that the forward kinematics routines have been executed Robotica contains several functions that can be used to draw the robot in a spe cific configuration or show the robot moving through a range of joint parameter val ues Most of the graphics output can be animated if the Animation m package is loaded this includes the graphics produced with SimDrive REIp ShowRobot and SeqShowRobot The animations can be saved and later restored and viewed again See the functions SaveAnim LoadAnim ShowAnim and SetRanges in Appendix A for more details To simplify
20. freedom planar robot has been loaded with joint variables matching those specified in the SIMNON data ql and q2 here we can see the effects of the controller as specified by the data set For this simulation al and a2 were set at 5 For example entering SimDrive S arm results in the output shown in Figure 8 2 59 Figure 8 2 Two link planar arm driven with dataset Note the use of to tell SimDrive to utilize the data currently loaded and the drawing of a revolute arm shape loaded with RevLink Figure 8 3 shows a plot of the origin of the last coordinate frame which makes the circular motion much easier to see This plot was generated with SimDrive trace Now that the data are loaded we can plot the information using SimPlot Figure 8 4 shows an example of plotting time t against q and q2 The command used to generate this graph was SimPlot t ql q2 Finally Figure 8 5 shows a plot of q1 against q2 produced by SimPlot ql q2 60 Simnon results Figure 8 3 Plot of the end effector location 61 Figure 8 4 Time t versus ql and q2 62 Figure 8 5 Joint parameter ql versus q2 63 CHAPTER 9 ROBOTICA FRONT END The Robotica front end RFE is a graphical user interface for Robotica that allows the user to access the capabilities of the package without having to deal directly with Mathematica The front end insulates the user from the inconvenience and id
21. these varl start end var2 start end The sublists specify a DH parameter such as d1 q2 or al for the variable and the starting value and ending value for that variable For example d1 3 8 causes the DH parameter d1 to vary from 3 to 8 The range from start to end is by default divided into five pieces although this can be changed by appending the number of divisions to the variable list To use eight steps the variable list would look like this ql 3 4 dn 0 3 8 It should be noted that angular parameters are specified in radians Joint parameters that have a constant value should be assigned with a Mathematica command e g d3 5 entered at the command line The options are given in the form of a list as well The available options are 80 single Plot only one arm at the start values of the parameters frame Put a frame around each graphics frame animate Take all of the generated frames and show them sequentially in movie fashion The animation can be shown again with ShowAnim or saved with SaveAnim print If Mathematica has a LaserPrint function defined it will be used to print the graph xprint Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graphics output window to save as a postscript compatible file The output file will be
22. 2 M 2 1 aq1 M 2 2 aq2 CM 2 1 q1 t G 2 om 1 kp 1 400 kp2 400 kd1 40 kd2 40 Figure 6 3 Circular motion control definitions 45 Figure 6 4 Joint variables as a function of time y Sim results Figure 6 5 Arm location from circular controller 46 y Sim results 3 Ha 2 1 5 1 0 5 2 3 43 5 6 Figure 6 6 End effector location from circular controller 47 CHAPTER 7 MANIPULABILITY Another unique feature of Robotica is its ability to compute and display the ma nipulability ellipsoids for a robot described by Denavit Hartenburg input parameters Complete information on manipulability can be found in 4 The manipulability ellip soids are related to how quickly the end effector can move in a given direction For a planar arm the ellipsoids are ellipses in a plane and the distance from the center of the ellipse to the ellipse is one of the factors that determines how quickly the arm can move in that direction The case is similar in three dimensional space except that the ellipse is an actual three dimensional ellipsoid Of course the manipulability also depends on other factors such as the strength of the joint motors etc The manipulability measures that can be computed are valuable pieces of information to have when planning the workspace in which the robot will operate 7 1 Generating the Ellipsoids REIp is the function responsible for the calculation and
23. 6 1 Defining the Torque Vector The input torque vector represents 7 in M q C q d 9 q T wherer Ta a vector containing a number of elements equal to the number of degrees of freedom in the robot GetInput Taul is the function responsible for reading a definition of the input torque use a filename parameter to obtain an input vector from a file or use no parameter to set the input torque vector to all zeros For example GetInputTau taul reads the torque definition from the file taul and 39 Get Input Taul sets the torque vector to all zeros The input file format is rather flexible as it can contain variable name and function definitions that can be used in later assignments within the file Mathematica interprets all of the expressions found in the file and keeps track of them so that they may be modified later Note that definitions loaded from a file can interfere with user defined variables essentially the result of reading the input file is the same as if each line had been entered directly to Mathematica Consider the input file in Figure 6 1 qlr 3Sin t qldr 3Cos t qlddr 3Sin t tau 1 kl qlr q1 t k2 qldr ql t qlddr Figure 6 1 Reading the input torque vector The variables qlr qldr and qlddr are defined in the first three lines then later used in the definition of the first element of the tau vector tau 1 The only requirements on the format of the input
24. 9 LoadAnim eee 69 LinkShape eee 69 MPrint eee 70 Planar eee 70 PrintInputData eee 71 PrisJoint eee 71 REIp eee 71 Response 74 RevJoint eee 74 SDynamies aoaaa 5 Save nim aoaaa 75 SaveResponse 75 SeqShowRobot oaoa 6 SetRanges 8 ShowAnim 4e 8 ShowRobot oaao 8 SimDrive aaa aaa 80 SimplifyExpression eeen 82 SimplifyTrigNotation 82 SimPlot 4 4 40000 ee ei 82 TPrint eee 83 A KINEMATICS AND DYNAMICS EQUATIONS REFERENCES Table 2 1 Robotica notation LIST OF TABLES Figure 2 1 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 4 7 5 1 5 2 5 3 5 4 9 9 9 6 5 7 6 1 6 2 LIST OF FIGURES Page The c o m vectors for a two link arm 11 Successful read operation 14 Results of FKin on input data 15 MPrint usage 15 Results of Simplify TrigNotation 16 The Jacobian for the two degree of freedom robot 16 Output from EPrint 17 Standard revolute joint 19 Standard prismatic Joint 20 Stanford arm
25. SaveAnim will save an animation generated with RElp SeqShowRobot etc in the file specified Files are usually quite large due to the amount of rendering information stored Example SaveAnim anim1 SaveResponseffile time inc After a successful run of Response SaveResponse will store numerical values for the solution to the dynamics equations in the file specified as the first parameter The second parameter is a list containing a start time and a stop time over which to save the solutions The range specified must of course be included in the range solved for with Response The third parameter gives the time increment to be used when saving the data it defaults to 0 1 The first and second parameters are required and the third 79 is optional The output file is stored in a SIMNON compatible format as a result SimDrivel and SimPlot can be used to plot and analyze the data Example SaveResponse sinres 0 5 0 01 SeqShowRobot vars options This function displays the robot over a range of joint parameters it differs from ShowRobot in that each joint specification in vars is executed sequentially rather than simultaneously i e one joint at a time moves First though a robot must have been loaded with DataFile and FKin must have been run The first argument is a list containing a series of sublists The sublists are of the following form var start end and vars is a sequence of
26. TABLE OF CONTENTS CHAPTER PAGE 1 OVERVIEW en 5 1 1 Capabilities of Robotica 5 1 2 Requirements 6 2 BASIC CONCEPTS 7 2 1 Loading Robotica 7 2 2 Lists in Mathematica 7 2 3 Input File Format 8 2 4 Robotica Notation 0 ee ee 11 2 5 The Help Command 11 3 FORWARD KINEMATICS 13 3 1 Loading a Robot 13 3 2 Generating the Equations 14 3 3 Printing Results 15 3 4 Summary 18 4 ARM GRAPHICS 19 4 1 Arm Construction 21 4 2 Design Guidelines for Joints 22 4 3 An Example 23 4 4 Planar Arm Interjoint Shapes 24 4 5 Animation and other Options 27 5 DYNAMICS 28 5 1 Loading a Robot 28 5 2 Generating the Equations 29 5 3 Printing Results 30 5 4 Summary 31 6 DYNAMICS RESPONSE COMPUTATION 35 6 1 Defining the Torque Vector 35 6 2 Calling Resp
27. adians Not all parameters need to be specified as a range If there is a DH parameter that will have a constant value it can be assigned with a normal Mathematica command e g d3 5 The variable list and constant variables should leave no undefined variables in the Jacobian or T matrices The options are given in the form of a list as well The available options are single Plot only one ellipsoid at the start values of the parameters measure Print a list of manipulability measures for each ellipsoid The measures include volume eccentricity minimum radius and ge ometric mean 76 monly frame animate scale print xprint mprint Do not generate any graphics only print manipulability infor mation or save it to a file Put a frame around graphics Take all of the generated frames and show them sequentially in movie fashion The animation can be shown again with ShowAnim or saved with SaveAnim Specify this option to scale the ellipsoids such that the largest axis is always two units long This helps to keep the plot from becoming too cluttered If Mathematica has a LaserPrint function defined it will be used to print the graph Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graphics output window to save as a postscript compatible
28. cified by the ranges In addition boxes can be drawn around each frame if the frame option is specified Usually however adding the frame clutters the image too much to be useful 31 CHAPTER 5 DYNAMICS Another feature included in Robotica is the ability to compute the Euler Lagrange dynamics description of a robot Robotica computes the mass matrix Christoffel symbols C matrix and the gravity vector when an input file with dynamics data is loaded and the appropriate commands are executed 5 1 Loading a Robot The first step in the process of generating the equations is loading the data file representing the robot The Robotica command DataFile is used for this purpose it can be used in two ways DataFile filename or DataFile The first method specifies the filename from which to load the data it must be a quoted string For example DataFile newtwo loads the file newtwo if it exists in the file system Entering DataFile without the filename parameter will cause Robotica to prompt for the filename Simply type the filename without quotes in response to the query and Robotica will load the file In 2 DataFile newtwo or In 2 DataFile Enter Data File Name newtwo If the data file is in the correct format and DataFile successfully reads the parameters a table will be displayed showing what Robotica has stored internally from the file If the file does not exist or is inco
29. circl 0 10 Next SimDrive was used to load the data the following command was used In 30 SimDrive circl With the data in memory the following plots were created the values of the joint variables the location of the arm and the trace of the end effector location The com mands used were SimPlot t q1 q2 to generate Figure 6 4 SimDrive arm to generate Figure 6 5 and SimDrive trace to generate Figure 6 6 44 x 2 Cos om t y 2 Sin om t vx om Sin om t vy om Cos om t ax om om Cos om t ay om om Sin om t DD x x y y al al a2 a2 2 al a2 qld ArcTan x y ArcTan al a2 Cos q2d a2 Sin q2d q2d ArcCos DD jl1 al Sin qld a2 Cos qld q2d j12 a2 Sin qld q2d j21 a1 Cos q1d a2 Cos qld q2d j22 a2 Cos qld q2d detj al a2 Sin q2d vid 1 detj G22 vx J12 vy v2d 1 detj j21 vx jll vy jd11 al Cos q1d vld a2 Cos qld q2d v1d v2d jd12 a2 Cos qld q2d v1d v2d jd21 al Sin qld vld a2 Sin qld q2d v1d v2d jd22 a2 Sin qld q2d v1d v2d bbx ax jd11 vl1d jd12 v2d bby ay jd21 vl1d jd22 v2d ald 1 detj G22 bbx j12 bby a2d 1 detj j21 bbx j11 bby aql ald kp1 qld q1 t kd1 v1d q1 t aq2 a2d kp2 q2d q2 t kd2 v2d q2 t tau 1 M 1 1 aq1 M 1 2 aq2 CM 1 1 q1 t CM 1 2 q2 t G 1 tau
30. display of the ellipsoids The format of the RElp command is In 5 RElp vars options This function calculates and displays manipulability ellipsoids for a robot when a range of input parameters is specified First of course a robot must have been loaded with DataFile in addition FKin must have been run RElIp makes heavy use of the Jacobian and T matrices calculated with FKin The first argument called vars above is a list containing a series of sublists The sublists are of the following form variable start end and vars is a sequence of these 48 variablel start end variable2 start end The sublists specify three things first a Denavit Hartenberg parameter for the variable such as dl q2 and al these are the variable entries in the input parameter file second the starting value for that variable and finally the ending value for that variable For example d1 3 8 causes the Denavit Hartenberg parameter d1 to run from 3 to 8 The range from start to end is by default divided into five pieces although this can be modified by appending the number of divisions to the end of the variable list For example to use eight steps the variable list would look like this ql 3 4 dn 0 3 8 It should be noted that angular parameters are specified in radians however using the built in degrees to radians converter Degree a value can be given in degrees e g q2 30 Degree 180 Degre
31. e The converter can be placed anywhere after the value and must be spelled with a capital D Not all of the input parameters that were specified in the input file have to be assigned a range however each must have some definite numerical value If there is a Denavit Hartenberg parameter that will have a constant value it can be assigned with a normal Mathematica command for example In 6 d3 5 This definition can be cleared later with Clear d3 at which time the variable will be purely symbolic again Variables should be cleared before loading a new robot as well The variable list and constant parameter assignments should leave no variables in the Jacobian or T matrices with purely symbolic values 7 2 Options The options are also given in the form of a list The available options are single RElp plots only the ellipsoid at the start values of the parameters in the sublists frame Put a frame around graphics 49 animate measures monly file scale print Take all of the generated frames and show them sequentially in movie fashion The animation can be shown again with ShowAnim or saved with SaveAnim Print a set of manipulability measures for each ellipsoid i e at each step of parameter values The measures include the actual axes cal culated volume of the ellipsoid eccentricity closer to one means more spherical minimum radius and the geometric mean of the radii
32. ed Show a plot of the end effector location Draw the arm graphics as the arm moves in response to the input data Put a frame around each graphics frame Take all of the generated frames and show them sequentially in movie fashion The animation can be shown again with ShowAnim or saved with SaveAnim If Mathematica has a LaserPrint function defined it will be used to print the graph Using this option assumes that Mathematica is running under X windows including zprint in the options list will attempt to utilize the X windows utilities xpr and xwd to allow the user to pick a graphics output window to save as a postscript compatible file The output file will be called xsim out This uses the internal Mathematica command Display to con vert the output graphic to a raw postscript file Some extra pro cessing may have to be done before the file is sent to a postscript output device See the Mathematica users manual for details The output file will be called msim out by SimDrivel is saved in the variable driveplot 85 Example SimDrivel simtwo trace SimplifyExpression exp SimplifyExpression applies various trigonometric identities and other simplifying reductions to an expression passed in as the argument The length of time required to reduce the expression varies with the complexity of the expression Example SimplifyExpression c 1 1 1 Simplify Trig Notation
33. ed to the orientation of its frame relative to the base frame Essentially this aligns the shape with its own coordinate axes Step2 The joint is translated out to the proper location determined by the T matrices Step3 H the robot is planar then the appropriate coordinate is dropped and two dimensional plots result when drawing the robot For example an x y planar robot would have its z coordinate dropped 4 2 Design Guidelines for Joints To design a joint graphic which will work with Robotica the following points should be observed 1 The joint should be laid out in the coordinate system of the base frame This is why all of the rotations are done to align the shape with its local coordinate system 2 revolute joint probably should have its origin centered around 0 0 0 in the base frame so that the rotations applied will produce the expected results 3 Any joint shape even if planar should include value for all three coordinates x y z Thus the components of an x y planar prismatic joint could look like Line 0 1 0 1 1 0 26 and not like Line 0 1 1 1 The packages Shapes m and Polyhedra m can provide building blocks for custom joint shapes These two standard Mathematica packages contain functions for generating and manipulating many types of shapes such as cones cylinders and spheres In addition the packages include functions that can stretch rotate and translate shapes
34. f 3 the Euler Lagrange equations of motion for an n link rigid robot are given by M q q Clq a g q T where q R is a vector of joint variables M q is the nen inertia matrix C q q q contains the Coriolis and centrifugal terms g q contains the gravitational terms and 7 is the input torque The inertia matrix is calculated as M Y m JE Jis Ii RER Jun i 1 where m is the mass of link z J is the inertia tensor for link z R is the rotation matrix defined above and J and Jw are given by z 7 Ji J Ji 0 0 Pr J JE Ji 0 0 91 where J p Ji o ZX oi 0 1 J Ja Zj 1 if joint 7 is revolute oe is the vector to the center of mass of link 7 and j Uci Zil j Ji 0 if joint j is prismatic The C matrix is an nen matrix containing the Coriolis and centrifugal terms The k 5 element is calculated from cj D Gijkq 1 where the Christoffel symbols are given by _ 1 dk dpi Odi KT da dg ti where d is the 7 j element of M The j element of the gravity vector is simply g gt mig Ji i 1 The material in this appendix is largely taken from Chapter 3 of 3 92 REFERENCES 1 H Elmqvist SIMNON An Interactive Simulation Program for Nonlinear Systems User s Manual Department of Automatic Control Lund Institute of Technology April 1975 Report 7502 2 5 Wolfram Mathematica A System for Doing Mathematics by Compute
35. file are that each element of the tau vector is assigned with the following semantics tau n EXPRESSION and that there are n entries in tau one for each degree of freedom The example in Figure 5 1 defines the necessary element in the tau vector for a single link arm it applies feedback to try to force the arm to track a Sine wave of amplitude 3 Note that the variables kl and k2 are left undefined these must be assigned values before the response function can be called They are left as symbols in the input file to allow greater flexibility of parameter adjustment 6 2 Calling Response Assuming that the dynamics have been generated with ELDynamics the response can be calculated with a call to Response First however all symbolic entries must be assigned numerical values this includes setting the inertias link masses etc The following examples use the tau vector for the one degree of freedom revolute arm There 40 are two parameters to response each a list The first list can contain two or three numbers the first is the start time of the response the second is the end time of the response and the last is the maximum number of steps Response should take when calculating solutions The start and stop times are required the number of steps should be set to a large value if the total response time is long The default number of steps is 500 One way to handle choosing the number of steps to specify is to assign it some arbi
36. hat can be used with SimDrive Each option has a check box to its right to indicate whether or not that option is turned on A check mark in the box indicates that the option will be turned on when the function is sent while an empty box means the option is turned off In Figure 9 4 the options trace arm and animate have been turned on The Value field is a text entry field used to give a value for the step parameter described in the SimDrivel section of Appendix A Data Source frame O trace Mi AW sseveveres Mi animate prints iii L xprint csisesi O mprint D Value 4 Figure 9 4 SimDrive option screen 68 When entering parameters into the option screens the delimiters that usually sur round items are automatically added by RFE For example no quotation marks are needed around the filename parameter in the DataFile option screen As another ex ample when specifying lists of parameters to RElp or ShowRobot only the individ ual elements inside the variable list have to have braces around them The delimiters around the entire list are added automatically In other words entering q1 0 90Degree d3 2 4 on the variables line will be sent to Robotica as q1 0 90Degree d3 2 4 which has had the brace delimiters added The function selection menu also contains a vertical scrollbar At this point in time however all of the available functions fit on one screen therefore th
37. hown again with ShowAnim or saved with SaveAnim 42 File VJl View V J Sound V Figure 6 2 Joint parameter ql tracking a Sine wave 6 5 Two Link Example Figure 6 3 shows the tau file definition of a controller that tries to make a two link arm with revolute joints move in a circle centered about 2 2 The DH description is given in Figure 3 1 The input tau file defines the circle in Cartesian space in the first two lines then proceeds to define quantities which are used as references in the control equations The parameters om kpl kp2 kdl and kd2 are set at the end of the file although they could be assigned values at any time prior to the running of Responsel In addition the values al and a2 were set to 3 ml set to 10 m2 set to 5 g set to 9 8 I1 set to 0 833 and 2 set to 0 417 Note that the definitions of tau 1 and tau 2 include quantities that are generated by the dynamics commands namely elements of the simplified mass matrix M elements of the simplified C matrix CM and elements of the gravity vector G The commands given to calculate the response over the time period 0 10 and save the response in a file called circl are shown below In 25 Responsel 0 10 100000 q1 0 0 q2 0 0 q1 0 0 q2 0 0 43 Solving equations Out 25 al gt InterpolatingFunction 0 10 lt gt q2 gt InterpolatingFunction 0 10 lt gt In 26 SaveResponse
38. ile will be called msim out A new image of the robot is drawn whenever a specified sample time has elapsed see Section 8 2 the default being 2 units A different time can be specified by appending a number to the options list For example SimDrive simnon trace 1 would update the plot whenever more than 1 time units elapsed 57 If neither arm nor trace is a member of the options list or an option list is not given at all SimDrive will merely read the input data set and not produce any graphics of course the data set is now available to SimPlot for analysis 8 2 Simulation Input File Format The input file should be in the following format the line numbers are for reference only they should not actually exist in the input file 11 td ql 2 3 0 1 0 02 4 1 1 0 04 SIMNON directly supports this output format Line 1 is a comment line containing the names of the variables for which data is listed in this file In the above example data will be specified for t d1 and qi It is always assumed that t will be the first of the input parameters it is used in the determination of elapsed time mentioned above The third line is just a comment line and is ignored there can be any number of comment lines following the variable list The data are listed after the last comment line with each line containing one set of input values Thus if there are three input variables each line will con
39. ilename is not specified the routine will query for the name which should be typed in without quotes Example DataFile twodof ELDynamics ELDynamics computes the dynamics information for the current robot data set First however a data set must have been loaded with DataFile and the forward kine matics must have been computed with FKin See Table 2 1 for a complete listing of the quantities generated by Robotica Example FLDynamics EPrint matrix label filename This function prints one per line all elements of the matrix or vector specified as the first parameter which is required The label argument also required although one could make it for no label is a string that will be printed in front of individual matrix elements The file argument is optional and specifies a filename in which to store the matrices The filename must be a string e g elements which represents a valid filename on the system used If the file already exists the matrix information is appended to the end of the file otherwise a new file is created Also once a file has been specified it becomes the default file for all of the printing functions To specify that the output should go to the default file use as the filename If no filename is given the elements of the matrix are displayed on the screen only Example EPrint T 0 2 T02 tfile 72 FKin FKin computes the forward k
40. in the graphic where 7 is the joint number If the a parameter is non zero then the link shape is scaled by a in x This means that the link shape should begin at x 0 and end at x 1 so that after scaling the link shape starts at x 0 and ends at x a This was the case for the two link planar arm in Figure 4 6 and Figure 4 7 The situation is similar for a non zero d except that the arm is scaled in z These scalings must be done in order to insure that the links and joints appear connected on the graphic 29 y fa1 0 03 tq2 45 Degree oF Figure 4 6 Arm with standard shapes y fa1 0 03 q2 45 Degree 0 Figure 4 7 Arm with new link shape 30 4 5 Animation and other Options To produce animations with Robotica the package Amimation m must be loaded this package is provided by Wolfram Research as part of the standard Mathematica system Specifying the animate option to the various procedures will suppress the generation of the composite image and use each frame to generate a frame by frame animation Writing all of the frames to the animation program is not a quick process eventually however the animation output will stop flashing and the frames will flow smoothly together Each frame of the animation must have consistent plot ranges so that the animation flows smoothly The x y and z ranges are set with the SetRanges command which forces each part of the animation to include x y and z values as spe
41. inematics for the current robot data set It generates all of the A and T matrices as well as the Jacobian See Table 2 1 for a complete listing of quantities generated by Robotica Example FKin GetInputTauffilename GetInputTaul loads an input vector for the Response function The filename pa rameter is optional If specified a vector definition is loaded from that file otherwise the torque vector is set to all zeros Example GetInputTau taul LoadAnim filename LoadAnim will load an animation previously saved with SaveAnim Loading a large animation may take several minutes due to the amount of rendering information stored Example LoadAnim animl LinkShapelfilename This function loads an interjoint shape from a file to be used in planar arms The argument to LinkShapel is a string containing a filename from which to load the shape If no filename is given a standard rectangular column shape is loaded as the default link Currently shapes must be three dimensional and can include sets of Line Polygon and 73 Point primitives the shapes defined in the package shapes m satisfy this requirement See Chapter 4 for complete information about how arm graphics are handled Example LinkShapel link2 MPrint matrix label filename This function prints in matrix form the matrix or vector specified as the first param eter which is required The label argument also required alth
42. interaction with Robotica an X Windows based interface has been de signed The interface insulates the user from the inconvenient textual interface Mathe matica provides see Chapter 9 for details 1 2 Requirements Running Robotica directly through Mathematica requires Mathematica release 2 0 or better while running Robotica through the X Windows front end requires Mathematica 2 1 or better Much of Robotica deals with text only input and output however there are functions which produce graphical output which requires a machine capable of producing the displays 10 CHAPTER 2 BASIC CONCEPTS 2 1 Loading Robotica Once Mathematica has been started check that the version is 2 0 or greater the Robotica package can be loaded with the following command Inf1 lt lt robotica m At this point all of the functions provided by the package are available to the user although some may require that other actions have been performed first e g running FKin before ELDynamics Mathematica is case sensitive thus it is important to type the function names exactly as they appear in the text e g FKin is entered as capital F capital K lower case z lower case n open close J 2 2 Lists in Mathematica There are several functions in Robotica that take lists as input parameters There are few basic rules governing lists that should be understood when specifying them to functions First a list begins with an o
43. ion of the x and z 4 axes d the distance along z _1 from 0 to the intersection of the z and x _1 axes d is the joint variable if joint 2 is prismatic a the angle between z _1 and z along z 6 the angle between x _1 and x measured about z _1 9 is the joint variable if joint 2 is revolute The homogeneous transformation matrix A is then given by Coi SaCo Sai aio Sa Coil CoiSai aishi 0 Sai Cai d 0 0 0 1 where Cangle Cos angle and Sangle Sin angle The matrix A can be written as R di 0 1 Ai where Coi SeiCai sssi R Sa Cal CoiSai 0 Soi Cxi and aso di G Soi d With these quantities defined the T matrices are given by for k gt 7 T Aja Ar R d O 1 Tk I 90 Once the A and T matrices are generated the Jacobian J can be computed where the th column J is given by J Jv _ Zr X On O 1 Jw Zi if Joint z is revolute and Ju Zi J 1 Jw 0 if joint is prismatic Here z is the unit vector along the z axis but expressed in the orientation of the base frame Le zin RER where k is the unit vector for z in the local coordinate frame The vector o is the vector d from the origin O to the origin O for each 7 The vector z is merely the first three elements in the third column of T and o is given by the first three elements of the fourth column of T Dynamics Equations As detailed in Chapter 6 o
44. iosyncrasies associated with typing commands directly and provides an environment far easier to use than the standard Mathematica textual interface The interface was written in C using the Open Look widgets 5 6 9 1 The Structure of RFE The front end Figure 9 1 consists of four major elements a menu bar located across the top of the window it contains the file fonts and packages menus a Mathematica output window located in the left half of the window a function selection window located in the right half of the window and a command entry field located below the Mathematica output window 9 2 Menu Bar The menu bar contains pulldown menus called file fonts and packages The menus offer convenient access to features which are frequently needed 9 2 1 File Activating the file menu displays the following choices Clear Text Abort and Quit selecting Clear Text erases everything in the Mathematica output window and sets the 64 File N Packages V OPEN LOOK graphics initialized APrint In 1 Robotica version 3 50 Copyright 1993 Board of Trustees University of Illinois All rights reserved Email questions comments or concerns to robotica robot1 ge uiuc edu In 2 CPrint ClearLinkShape ClearPrisJoint ClearRevJoint DataFile gt DisplayTau ELDynamics EPrint Fkin Get InputTau LinkShape LoadAnim MPrint Planar Response RevJoint SDynanics Sa
45. is scrollbar always stays at the top When additional functions are added the scrollbar would be used to move up and down the list of functions which would be larger than the screen display 69 CHAPTER 10 COMMAND REFERENCE APrint file This function prints all the A matrices for the most recent run of FKin The file argument is optional and specifies a filename in which to store the matrices The filename must be a string e g matrices which represents a valid filename on the system being used If the file already exists the A matrix information is appended to the end of the file otherwise a new file is created Also once a file has been specified it becomes the default file for all of the printing functions To specify that the output should go to the default file use as the filename If no filename is given the A matrices are displayed on the screen only Example APrint amatrix ClearLinkShape ClearLinkShapel clears any definition for an interjoint shape The result is that the package will use a simple line to show links until a new interlink shape is defined with LinkShape ClearLinkShape takes no arguments Example ClearLinkShapel 70 ClearPrisJoint ClearPrisJoint clears any definition for a prismatic joint shape The result is that the package will use a simple line to show prismatic joint connections until a new prismatic link shape is defined with PrisJoint Clea
46. mber of links this process could take a very long time due to the symbolic nature of the quantities involved If there are only a few elements of interest in the mass matrix they can be simplified independently For example Simpli fyExpression MU 1 1 reduces only the first element The Christoffel symbols and C matrix will in general also need to be simplified Running SDynamics will simplify using SimplifyExpression all of the dynamics quantities generated and will also display which elements are currently going through the simplification process 5 3 Printing Results Once the dynamics equations have been generated showing the results is accom plished through a set of display functions provided within Robotica EPrint is one such function it takes up to three parameters the matrix to be displayed required a text label to print alongside the matrix required and a filename in quotes in which to save the matrix optional Figure 5 3 shows how EPrint is used to display the elements of the simplified mass matrix 34 In 6 EPrint M Mass matrix al ml a2 m2 Mass matrix 1 1 I1 12 Tr al m2 al a2 m2 Cos q2 2 a2m2 ala2 m2 Cos q2 2 Mass matrix 2 1 12 2 a2m2 ala2 m2 Cos q2 2 Mass matrix 1 2 12 a2 m2 Mass matrix 2 1 12 Figure 5 3 The elements of the mass matrix It is sometimes convenient to change the trigonometric notation to be somewhat more succi
47. mes Robotica uses are as close as possible to standard mathematical notation To access the variable in Robotica one merely types the correct Robotica notation at the prompt 2 5 The Help Command Mathematica provides a built in help command It may be used to request infor mation about any Mathematica command or function as well as all Robotica functions 15 available to the user For example to obtain information about the REIp function type RElp and Mathematica will respond with a brief help message about the function Table 2 1 Robotica notation Corresponding Corresponding Nota Nota ME EN Es CE e el Leal Jci mar CE KE Can Ee eea OE Sil STS esi PE ORS Sint PE EL ost KCE Sint Fin EL S 16 CHAPTER 3 FORWARD KINEMATICS One of the primary features of Robotica is its ability to generate the forward kinemat ics equations for any robot described by Denavit Hartenberg parameters The mathe matics behind this process is described in Appendix B Essentially Robotica will compute the A matrices T matrices and Jacobian see 3 for the arm 3 1 Loading a Robot The first step in the process of generating the equations is to load the data file The Robotica command DataFile is used for this purpose it can be used in two ways DataFile filename or DataFile The first method specifies the filename from which to load the data it must be a quo
48. nct The function SimplifyTrigNotation takes no parameters and modifies the display of Sin and Cos to an abbreviated form Figure 5 4 shows the results of running Simplify TrigNotation The Christoffel symbols are stored in a three dimensional array called c They can be examined by entering c with a subscript or by using the CPrint function Figure 5 5 shows how to examine individual elements Other quantities of interest that can be displayed include the Jacobians Jvc and Jwc For example Figure 5 6 shows the Jvc matrices for the two link robot In addition G the gravity vector can be displayed with a simple call to MPrint Figure 5 7 shows G for the two link robot 5 4 Summary In summary there are only three steps required to generate the Euler Lagrange dynamics information 1 Run DataFile to load the set of Denavit Hartenburg parameters 2 Run FKin to make Robotica do the forward kinematics 3 Run ELDynamics to generate the dynamics information 35 Once these steps are performed any of the display functions can be invoked to show the results SimplifyExpression can also be used at this point to simplify any required information In 7 Simplify TrigNotation In 8 EPrint CM C Matrix al a2 m2 Dq2 S2 C Matrix 1 1 2 al a2 m2 Dql 52 C Matrix 2 1 2 al a2 m2 Dal 82 al a2 m2 Dq2 2 C Matrix 1 2 2 2 C Matrix 2 2 0 Figure 5 4 Results of running Sim
49. nction loads a prismatic joint shape from a file The argument to PrisJoint is a string containing a filename from which to load the shape If no filename is given a standard rectangular column shape is loaded as the default prismatic joint Currently shapes must be three dimensional and can include sets of Line Polygon and Point primitives the shapes defined in the package shapes m satisfy this requirement See Chapter 4 for complete information about how arm graphics are handled Example PrisJoint RElp vars options This function calculates and displays manipulability ellipsoids for a given robot over a range of input parameters First though a robot must have been loaded with DataFile and FKin must have been run The first argument is a list containing a series of sublists The sublists are of the following form var start end and vars is a sequence of these varl start end var2 start end The sublists specify a DH parameter such as dl q2 or al for the variable and the starting value and ending value for that variable For example d1 3 8 causes the DH parameter dl to vary from 3 to 8 The range 75 from start to end is by default divided into five pieces although this can be changed by appending the number of divisions to the variable list To use eight steps the variable list would look like this ql 3 4 dn 0 3 8 It should be noted that angular parameters are specified in r
50. nd parameter is a list containing the time 4 range from the solution to save it is also required The time range given must lie within the range initially solved for with Response of course The third paramter is optional and specifies the size of time step to take when saving the data it defaults to 0 1 A value for the joint variables will be stored starting at the start time incremented by the step and ending at the stop time The following is a valid call to SaveResponsel SaveResponse sinres 0 5 0 05 6 4 Viewing Results Once the solutions have been saved the functions SimDrive and SimPlot can be used to visualize the results Assuming that we have generated the response and saved it to file sinres as described above SimDrivel can be called to load the data set into memory SimPlot can then be used to display the joint trajectory For example SimDrive sinres SimPlot t q1 results in the graph of q1 shown in Figure 6 2 All of the options available from SimDrivel can be specified of course For example SimDrive arm animate would show an animation of the arm respsonding to the input torque Some of the more useful options include trace Show a plot of the end effector location arm Draw the arm graphics as the arm moves in response to the input data animate Take all of the generated frames and show them sequentially in movie fashion The animation can be s
51. ng the option scale normalizes the size of each ellipsoid such that the largest axis is 2 units long this is useful for judging the relative amount of manipulability in each configuration RElp ql 0 90Degree d2 3 8 scale generated the results shown in Figure 7 2 92 Manipulating Ellisoid Figure 7 2 Scaling the ellipsoids The input data file for a more complicated robot is shown in Figure 7 3 The manip ulability for the arm was calculated with RElp ql 0 180Degree d3 2 8 3 The variable q2 was fixed at 50 Figure 7 4 shows the results 53 DOF joint al alphal dl thetal joint2 a2 alpha2 d2 theta2 joint3 a3 alpha3 d3 theta3 revolute 0 Pi 2 5 ql revolute 0 Pi 2 10 q2 prismatic 0 0 d3 Pi 2 Figure 7 3 Three link example MD View V Sound V Manipulating Ellipsoid Figure 7 4 Three link nonplanar example With three dimensional pictures it can be difficult to see just what is happening as the robot moves With the ViewPoint option it is possible to view the plot from any point allowing a more reasonable perspective to be presented Figure 7 5 shows the results of moving the viewpoint with Show manplot ViewPoint gt 1 0 5 0 Even with the ability to adjust the viewpoint it is usually more informative to generate the numerical measures of manipulability rather than extracting this information from the graphic See Chapter 4 of
52. ocated for it and therefore scrollbars are provided to allow access to nondisplayed portions of the window Figure 9 2 shows the output window after the DataFile and FKin commands have been executed When the display fills the vertical scrollbar is automatically moved down to keep the most recent information visible Of course the scrollbars can be moved to another position to display previous output This window consists of read only text to send a command to Mathematica use the command input line File V Fonts V Packages V APrint ClearPrisJoint ClearRevJoint CPrint Matrices Formed T Matrices Formed T 0 0 T 0 1 T 0 2 T 1 2 Jacobian Formed Jacobian J 6x2 Mathematica Command gt DataF ile EL Dynamics EPrint FKin Loadfn im MPrint Planar PrisJoint Print InputData RElp RevJoint SaveAnim SeqShouRobot SetRanges SDynam ics ShowAn im ShowRobot SimDrive SimplifyExpression SimplifyTrigNotat ion SimPlot TPrint Figure 9 2 Output in the Mathematica output window 66 9 4 Command Input Line While the function selection menu provides access to all of the Robotica routines any function or Mathematica command can be entered on the command input line at the bottom of the screen When a text string is entered on the line and the jreturn key
53. onse 36 6 3 Saving Results 37 6 4 Viewing Results 38 6 5 Two Link Example 39 7 MANIPULABILITY 44 7 1 Generating the Ellipsoids 44 7 2 Options 45 7 3 Examples 47 8 EXTERNAL SIMULATION COMPATIBILITY 52 8 1 Loading Simulation Data aa a 52 8 2 Simulation Input File Format 54 8 3 Plotting Simulation Data 54 8 4 Examples 55 9 ROBOTICA FRONT END 60 9 1 The Structure of RFE 60 9 2 Menu Bar 60 9 2 1 File 60 9 2 2 Packages 61 9 3 Mathematica Output Window 62 9 4 Command Input Line 63 9 5 Robotica Function Selection Menu 63 10 COMMAND REFERENCE 66 APrint 4 44 40e eee eee eee 66 ClearLinkShape ee 66 ClearPrisJoint r ee ee s se se s s osos ie oas 2 67 ClearRevJoint o ooo aa 67 CPrint eeen 67 DataFile eee 68 ELDynamics 68 EPrint eee 68 FKin eee 69 GetInput Tau a aaa 6
54. onse is a list containing a start and end time over which to solve Additionally a third number may be added to the end of the list this number sets the maximum number of steps that the algorithms can take to solve the equations The second parameter is a list containing a set of initial conditions that determines a unique solution to the equations When successful the output can be saved with SaveResponsel See Chapter 6 for complete details Example Responsel 0 5 q1 0 0 ql 0 0 RevJoint filename This function loads a revolute joint shape from a file The argument to RevJoint is a string containing a filename from which to load the shape If no filename is given a 78 standard circular column shape is loaded as the default revolute joint Currently shapes must be three dimensional and can include sets of Line Polygon and Point primitives the shapes defined in shapes m satisfy this requirement See Chapter 4 for complete information on how link graphics are handled Example RevJoint revolute SD ynamics Since dynamics quantities are not simplified when generated this function is provided to instruct Robotica to attempt to reduce the entire set of dynamics information De pending on the number of links in the robot this process could take a large amount of time Specific pieces of information can be simplified with SimplifyExpression instead Example SDynamics SaveAnim filename
55. ough one could make it for no label is a string that will be printed in front of the matrix The file argument is optional and specifies a filename in which to store the matrices The filename must be a string e g matrix which represents a valid filename on the system used If the file already exists the matrix information is appended to the end of the file otherwise a new file is created Also once a file has been specified it becomes the default file for all the printing functions To specify that the output should go to the default file use as the filename If no filename is given the matrix is displayed on the screen only Example MPrint A 2 A2 Planar Planar uses the Jacobian constructed with FKin and returns an integer which de scribes the space in which the robot moves The return codes are The robot can not move in x y or z The robot moves only in x The robot moves only in y The robot moves in x and y 1 e an x y planar robot arm The robot moves only in z The robot moves in x and z 1 e an x z planar robot arm D Oo FP WW N The robot moves in y and z 1 e a y z planar robot arm 74 7 The robot moves in x y and z 1 e a general motion in 3 space Example Planar PrintInputData This function prints the input data set to the screen It does not take any parameters Example PrintInputDatal PrisJoint filename This fu
56. owAnim ShowRobot vars options This function displays a robot over a range of joint parameters First though a robot must have been loaded with DataFile and FKin must have been run The first argument is a list containing a series of sublists The sublists are of the following form var start end and vars is a sequence of these varl start end var2 start end The sublists specify a DH parameter such as d1 q2 or al for the variable and the starting value and ending value for that variable For example d1 3 8 causes the DH parameter dl to vary from 3 to 8 The range from start to end is by default divided into five pieces although this can be changed by appending the number of divisions to the variable list To use eight steps the variable list would look like this ql 3 4 dn 0 3 8 It should be noted that angular parameters are specified in radians Not all parameters need to be specified as a range If there is a DH parameter that will have a constant value it can be assigned with a normal Mathematica command e g 82 d3 5 The variable list and constant variables should leave no undefined variables in the T matrices The options are given in the form of a list as well The available options are single Plot only one arm at the start values of the parameters frame Put a frame around each graphics frame animate Take all of the generated frames and show them sequentially
57. pening ends with a closing and contains elements separated by commas if there is more than one For example trace is a list containing one element called trace and trace print is a list containing the two elements trace and print In addition Lists can include just about anything as elements symbols numbers even other lists For example q1 0 3 1415 6 is a list containing another list 11 q1 0 3 1415 and the number 6 This inner list contains three elements q1 0 and 3 1415 If there isa list that is used frequently it is sometimes more convenient to assign some variable name to it than to retype it every time it is used For example In 5 simlist d1 q2 In 6 SimPlot t simlist Whenever Mathematica sees simlist it will substitute the list d1 q2 which can save much unnecessary user input See 2 for more information about lists and Mathematica in general 2 3 Input File Format convenient way to specify input data to Robotica is through a data file Denavit Hartenberg parameters and dynamics data for the robot can be given in such a file A representative input data file is shown below there are line numbers at the beginning of each line which are for reference only they should not actually be present in a datafile 01 A Robotica input data file for 02 a two degree of freedom planar robot 03 04 DOF 2 05 The Denavit Hartenberg parameters 06 jointl
58. plify TrigNotation 36 In 10 c 1 1 1 Out 10 0 In 11 c 1 2 1 al a2 m2 82 Out 11 2 In 12 c 1 1 2 al a2 m2 S2 Out 12 TD Figure 5 5 Examining the Christoffel symbols In 14 EPrint Jvc 1 Jvel Jvaj LLS 2 al C1 Jvel 2 1 2 Jvc113 1 0 Jvc1 1 2 0 Jvc1 2 2 0 Jvc1 3 2 0 In 16 MPrint Jve 2 Jve2 a2S12 a2 12 Out 16 al S1 wij ats 5 Jvc2 a2 C12 a2 C12 al Cl 2 0 0 Figure 5 6 The Jvc matrices 37 In 25 MPrint G Gravity Vector g Out 25 2 2 Gravity Vector g a2 g m2 C12 2 Figure 5 7 G the gravity vector 38 CHAPTER 6 DYNAMICS RESPONSE COMPUTATION Complementing the ability to compute the dynamics equations governing a robotic arm is the ability to solve the equations numerically for the time response of the joint variables given an input torque The torques are supplied to Robotica through an input file which defines the elements of the torque vector as well as other variables which may be used in the definition of the torque vector The time response once computed can then be saved in SIMNON compatible format for later analysis with Simplot SimDrive or SIMNON itself See SimDrivel in the command reference for a description of the file format Before any of these calculations can proceed however both ELDynamics and SDynamics must have been run
59. ql and q2 58 Joint parameter ql versus q2 59 Robotica Front End 61 Output in the Mathematica output window 62 DataFile option screen 64 SimDrive option screen 64 Denavit Hartenburg frame assignment 3 85 CHAPTER 1 OVERVIEW 1 1 Capabilities of Robotica Robotica is a collection of useful robotics problem solving functions encapsulated in a Mathematica package Utilizing Mathematica s computational features allows results to be generated in a purely symbolic form The benefit of the symbolic representation is that the problem is solved for all possible input parameter values at once and the user can then substitute and experiment with actual numbers to discover the effects of various parameters Robotica requires input in the form of a table of Denavit Hartenberg parameters describing the robot to be analyzed Once the table has been entered Robotica can generate the forward kinematics for the robot The A and T matrices as well as the velocity Jacobian J are generated Of course it is possible to display and save to an external file all of the data generated If the dynamics equations of the robot are also to be generated the input must include the dynamics description data as detailed in Chapter 2 Once the forward kinematics are produced Euler Lagrange
60. r Addison Wesley Publishing Company Inc 1991 3 M W Spong and Vidyasagar Robot Dynamics and Control New York NY John Wiley amp Sons 1989 4 T Yoshikawa Foundations of Robotics The MIT Press 1990 5 D A Young The X Window System Programming amp Applications with Xt Prentice Hall 1992 6 Hewlett Packard Programming with the Xt Intrinsics Hewlett Packard Company 1989 93
61. rPrisJoint takes no arguments Example ClearPrisJoint ClearRevJoint ClearRevJoint is similar to ClearPris except that it works on the revolute joint shape definition The revolute joint shape is cleared and the system will use a line to show revolute joint connections A new revolute shape can be defined with RevJoint ClearRevJoint function takes no arguments Example ClearRev CPrint label filename CPrint prints one per line all of the elements of the C matrix called CM in Robot ica The label parameter is required and is specified as a text string that will be printed in front of the individual elements The filename is optional and specifies a filename in which to store the elements The filename must be a string e g elements which represents a valid filename on the system used If the file already exists the matrix in formation is appended to the end of the file otherwise a new file is created In addition once a file has been specified it becomes the default file for all of the printing functions To specify that the output should go to the default file use as the filename If no filename is given the elements of the matrix are displayed on the screen only Example CPrint C symbols 71 DataFile filename DataFile loads the DH parameters from a file The filename argument is optional but if included must be a string containing the name of the source file If the f
62. rrectly formatted a message will be displayed to that effect and Robotica will not store anything 32 Figure 5 1 shows what a successful read operation looks like In 2 Datafile Enter data file name newtwo State Reset Kinematics Input Data Joint Type a alpha d theta 1 revolute al 0 0 ql 2 revolute a2 0 0 q2 Dynamics Input Data Gravity vector 0 g 0 Link mass com vector 1 ml al 2 0 0 2 m2 a2 2 0 0 0 Inertia 1 0 0 0 Il Inertia 2 0 0 2 Figure 5 1 Successful read operation 5 2 Generating the Equations Once the data file has been loaded a two step process is required to generate Euler Lagrange dynamics first FKin must be successfully run to compute the forward kine matics See Chapter 3 for complete information on this process The second step is to run ELDynamics the command which will produce the dynamics information ELDy 33 namics does not take any parameters Figure 5 2 shows the generation of dynamics equations with the input data from Figure 5 1 In 5 ELDynamics Mass Matrix MU 2 x 2 Formed No Trigonometric Simplification Christoffel Symbols Formed C Matrix CM 2 x 2 Formed Gravity Vector G 2 x 1 Formed Figure 5 2 Results of ELDynamics Due to its immense complexity the mass matrix produced MU will not be simpli fied unless it is explicitly requested with SimplifyExpression MU or SDynamics For arms with even a small nu
63. taFilel 21 3 4 Summary In summary there are only two steps required to generate the forward kinematics equations 1 Run DataFile to load the set of Denavit Hartenberg parameters 2 Run FKin to make Robotica do the actual computation Once these steps are performed any of the display functions can be invoked to show the results Consult Table 2 1 for the notation Robotica uses to represent the various quantities 22 CHAPTER 4 ARM GRAPHICS The graphics shapes used to draw robot arms are really quite simple but still very flexible Revolute joints and prismatic joints have separate models to represent them shapes for revolute and prismatic joints can be loaded with the RevJoint and PrisJoint commands If no filename is specified in these commands then a standard shape for the joint is loaded For revolute joints this shape is a cylinder Figure 4 1 for prismatic joints the shape is a rectangular column Figure 4 2 The Stanford arm is shown in Figure 4 3 at the ql 0 q2 90Degree d3 3 configuration with the standard joint shapes File V View V Sound V Figure 4 1 Standard revolute joint 23 Figure 4 2 Standard prismatic Joint tal 0 03 tne Kes as 03 td3 3 33 1 Figure 4 3 Stanford arm 24 4 1 Arm Construction Any three dimensional shape can be loaded from a file and used as a joint shape The first thing to be aware of is that each shape must specify x y
64. tain three values For example Line 3 would assign 0 to t 1 to dl and 0 02 to q1 The time parameter t is always assumed to be the first variable in the list 8 3 Plotting Simulation Data The other Robotica function dealing with external simulation data is called SimPlot Its format is shown below SimPlotfindep deplist options SimPlot plots data loaded through the function SimDrive The first parameter is the independent variable 1 e the x axis The second parameter is a list of dependent 58 variables to be plotted against the independent variable For example SimPlot t q1 plots t vs ql and SimPlot t q1 q2 plots t vs ql and q2 The parameters indep and deplist are required and options is optional The options must be in a list and can include print xprint and mprint These functions as described in Section 7 1 the files produced however are called xsimplot out for zprint and msimplot out for mprint 8 4 Examples Assume SIMNON has been used to simulate the effect of a controller attempting to move a two degree of freedom planar arm in a circle The joint variable data have been stored in the file simnon Figure 8 1 shows how the file would be loaded In 4 SimDrive Simnon Got header Attempting to read values for these parameters t ql q2 qld q2d Working Data file loaded Figure 8 1 Loading a SIMNON file Now assuming that a two degree of
65. ted string For example DataFile twodof loads the file twodof if it exists in the file system Entering DataFile without the filename parameter will cause Robotica to prompt for the filename Simply type the filename without quotes in response to the query and Robotica will load the file The following lines show the two possibilities In 2 DataFile twodof or In 2 DataFile Enter Data File Name twodof If the data file is in the correct format and DataFile successfully reads the parameters a table will be displayed showing what Robotica has stored internally from the file If the file does not exist or is incorrectly formatted a message will be displayed to that effect 17 and Robotica will not store anything Figure 3 1 shows what a successful read operation looks like In 2 Datafile Enter data file name newtwo State Reset Kinematics Input Data Joint Type a alpha d theta 1 revolute al 0 0 ql 2 revolute a2 0 0 q2 Dynamics Input Data Gravity vector 0 g 0 Link mass com vector 1 ml al 2 0 0 2 m2 a2 2 0 0 0 0 Inertia 1 0 0 0 0 Il 0 0 Inertia 2 0 0 2 Figure 3 1 Successful read operation 3 2 Generating the Equations Once the data file has been loaded it is very simple to generate the forward kinematics equations the command FKin is used for this purpose FKin takes no parameters and displays a report of the various quantities
66. this link is back 1 2 of the length of this link in the x direction 0 in the y direction and 0 in the z direction in terms of the coordinate frame for link one Figure 2 1 shows the c o m setup for a two link arm Line 24 gives the six unique components of the symmetric inertia tensor in list format These are assigned as follows el e2 e3 el e2 e3 e4 e5 e6 gt e2 e4 e5 e3 e5 eb The inertia tensor for link is a 3 x 3 matrix computed in the coordinate frame attached to link amp See Equation 6 2 19 in 3 for complete information The format for each line is label parameter The label is ignored and the parameter is assigned to its respective Denavit Hartenberg parameter for that link Lines 25 through 27 give the dynamics information for the next link The comments of the above paragraphs apply to these lines 14 link com 1 2 a2 0 0 Figure 2 1 The c o m vectors for a two link arm Of course if the generation of forward kinematics equations are all that Robotica will be used for the dynamics portion of the file need not exist Note also that for each degree of freedom specified by DOF n there must be a set of Denavit Hartenberg parameters and if dynamics data are given there must also be a set of data for each degree of freedom 2 4 Robotica Notation The notation that Robotica uses to store the various quantities and parameters is summarized in Table 2 1 In general the na
67. trary large value e g 1000000 although the number of steps can be determined by running Response then adjusting the steps to a greater value if the following message appears NDSolve mxst Maximum number of steps reached at the point xxx The second list contains the initial conditions used when solving the differential equa tions In general there should be 2n initial conditions where n is the number of degrees of freedom since the equations involve up to second derivatives for each joint parameter The following is a valid call to Responsel Responsel 0 5 q1 0 0 q1 0 0 This calculates the time response of the one link example over the time range 0 to 5 using the default number of steps with the input torque as defined in Figure 6 1 Notice the format of the initial condition assignment It is important that double equal signs be used when setting these parameters if a single equal sign is used Mathematica assumes that an assignnment is being made directly to the variables rather than a conditional test 6 3 Saving Results After Response has been run and solutions have been generated they may be saved to a file in SIMNON compatible format for analysis Putting the results in a file is accomplished with the SaveResponsel function Either two or three parameters are given to SaveResponse the first parameter is a string containing a file name in which to save the data this parameter is required The seco
68. utput graphic to a raw postscript file Some extra processing may have to be done before the file is sent to a postscript output device See 2 for details The output file will be called mrelp out Options must be in a list and can be given in any order For example RElp an scale single causes RElp to draw a single scaled ellipsoid at the start values of the param eters in the list an If no arm graphics have been loaded simple lines will be used to show the arm See Chapter 4 for information on the use of more complicated arm graphic representations Also a variable called manplot is created to hold the current manipulability plot It can be viewed at any time after a plot has been generated with Show manplot Any of the various display options available from the Mathematica function Show e g ViewPoint can be used when viewing the plot 7 3 Examples Figure 7 1 shows a sample of the output produced by RElp for a simple two link revolute joint prismatic joint arm First of course the data file was loaded with DataFile Next a prismatic joint shape was loaded with PrisJoint and a revolute joint shape with RevJoint The parameter al was set to 5 and the command RElp q1 0 90Degree d2 3 8 single was used to generate the graph 51 Manipulating Ellipsoid Figure 7 1 Planar manipulability example When many ellipsoids are drawn they tend to obscure one another on the graph Specifyi
69. vefinim SaveResponse SeqShowRobot SetRanges ShowAnim ShowRobot SimDrive SimPlot SimplifyExpression SimplifyTrigNotation TPrint PrintInputData PrisJoint REI lathematica Command gt Figure 9 1 Robotica Front End scrollbar slider back to the top ready for new input Interrupting an operation can be done with the Abort selection The currently executing Mathematica operation will be stopped and Mathematica will be ready for new input after the abort sequence The last operation available under File is Quit Not surprisingly this ends the session with RFE and returns the user to the environment that existed before RFE was run 9 2 2 Packages This menu contains a short list of Mathematica standard packages that contain func tions which add flexibility to Robotica The package Animation m when loaded allows the use of the movie like animation sequences that RElp ShowRobot etc can gen erate The other two packages Shapes m and Polyhedra m are primarily useful for designing new joint shapes See Chapter 4 for further information on designing joints 65 9 3 Mathematica Output Window When Mathematica has an output ready for display it sends the result to RFE and RFE posts the output in the Mathematica output window The window extends beyond the limits of the screen area all

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