Tutorial: Geometric design of a robot arm using Denavit-Hartenberg using SGLib

Tim Lueth, Professor at Technical University of Munich, Department of Mechanical Engineering, July 2018

This tutorial uses the symbollcs toolbox

1 Introduction into 6DoF serial Robot-Arms

At the end of the 1950s, the two kinematics experts Denavit and Hartenberg calculated inverse kinematics for serial robots, robot arms with 6 degrees of freedom (DoF), for the first time.

This possible calculation of inverse kinematics, i.e., the calculation of the required and desired joint positions from a specified position and tool orientation (6DoF) and an additional arm configuration, made it possible to program and use robots for industrial tasks and thus necessitated the production of robots from the end of the 1960s onwards.

A 6 Dof-robot consists of a frame and six subsequent links, which are connected at their joints via rotational degrees of freedom. This is followed by a robot hand or a normal tool at the "wrist" or "tool center point" (TCP).

loadweb('JACO_robot.mat');           % Load a robot arm example
loadweb: Access path to changed from "www.mimed.mw.tum.de" to "www.mw.tum.de/mimed/" in 2020 Aug.
loadweb: Access path to changed from "www.mw.tum.de/mimed/" to "www.mec.ed.tum.de/mimed/" in 2021 Nov.
Downloading "https://www.mec.ed.tum.de/fileadmin/w00cbp/mimed/Matlab_Toolboxes/JACO_robot.mat" into: /Volumes/LUETH-WIN/WIN AIM Matlab Libraries/SolidGeometry-Code!

The robot stored in a cell array. Each link of the robot is a Solid Geometry. JACO{1} ist the base, JACO{2} is the following link, and so on

whos JACO % The robot is a cell array of Soli

  Name      Size              Bytes  Class    Attributes

  JACO      1x8             6369520  cell               

SGfigure(-30,30); SGTplotalpha(JACO(1:7)); % Plot the links 1-7 of the arm ,the 7th link is the hand of JACO

If you remove your hand, you can see the six links and joints of the arm:

SGfigure(-30,30); SGTplotalpha(JACO(1:6)); % Plot the links of the arm

The fact that this clear illustration of the robot arm is already an assembled representation can be seen when the links of the robot arm are arranged in a box.

[~,SGall]=SGboxpacking(JACO);
binpacking3D: Packing 8 objects (h=398): 
VLFLtextattachVLUL: Text "06-Aug-2025 22-01-36" attached to union Nr: 2

SGfigure(-30,30); SGTplotalpha(SGall);

2 The usage of Coordinate Frames attached to the Robot arm links

This raises the question of how these individual links of the robot arm can be assembled into a movable robot kinematic or at least arranged spatially.

This is done using frames, local coordinate systems that are attached to the SG and move with it during translation or rotation.

In SGLib, solids with frames are abbreviated as SGT.

JACO{1}
ans = struct with fields:
       VL: [20231×3 double]
       FL: [40670×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[86×1 double]  [184×1 double]}
      col: 'k'
    alpha: 0.9000

You can see that the member JACO{1} has two frames named "B" and "F" in addition to VL and FL, as well as two corresponding homogeneous transformation matrices that describe these frames relative to the surface coordinates VL.

The designation "B" stands for the frame that points to the base or is the base of the solid. The designation "F" is the follower frame to which the next link can be coupled.

The z-axis of all frames always points out of a body. This is the default notation in SGLib.

If you want these frames to be displayed, simply write "SGTplot" or "SGTplotalpha" instead of "SGplot" or "SGplotalpha."

SGfigure(-30,30); SGplotalpha(JACO{2},'',0.5);

SGfigure(-30,30); SGTplotalpha(JACO{2},'',0.5);

When a body is moved or rotated, the frames rotate with it so that their position relative to the body remains unchanged.

SGfigure(-30,30); SGTplotalpha(SGtransR(JACO{2},[pi/3,pi/3,pi/3]),'',0.5);

3 Spatial alignment of solids to their frames

If you use the frames of the solids for alignment, for example to move the body to the coordinate origin, there are two useful options:

1) The completely congruent alignment of one frame with another coordinate system "SGTmatchT". This is often not the desired result for frames.

SGTmatchT(JACO{2},'B'); % Green is the color of the solid when 'B' matches eye(4)

2) The alignment of a frame with a different coordinate system that is identical in Y but opposite in Z and X is called "SGTalignT." This is the standard alignment for connecting frames that point outward.

SGTalignT(JACO{2},'B'); % Green is the color of the solid when 'B' matches eye(4)

The coordinates of the frames can be obtained with "SGTget."

SGTget(JACO{2},'B')

ans = 4×4

    1.0000   -0.0000   -0.0000    0.0000
   -0.0000   -1.0000   -0.0000    0.0000
         0    0.0000   -1.0000   94.5000
         0         0         0    1.0000

You can also align a body so that its frame is aligned relative to another coordinate system:

SGTalignT(JACO{2},'B', TofR([pi/3,pi/2,pi/6])); % Green is the color of the solid when 'B' matches eye(4)

4 Relative spatial orientation of two links of a robot

SGfigure(-30,30); h={};

JACO{2}=SGTalignT(JACO{2},'B', eye(4));

SGTplotalpha(JACO{2},'',0.5); % Align "B" of JACO 2 to origin

JACO{3}=SGTalignT(JACO{3},'B', SGTget(JACO{2},'F')); % Align "B" of JACO-3 to Follower frame "F" of JACO-2

h{2}=SGTplotalpha(JACO{3},'',0.5);

You will quickly want to be able to rotate your body by a defined angle while aligning it.

delete(h{2});

JACO{3}=SGTalignT(JACO{3},'B', SGTget(JACO{2},'F'),pi); % Align "B" of JACO-3 to Follower frame "F" of JACO-2 and turn bu 180 deg i.e. pi rad

h{3}=SGTplotalpha(JACO{3},'',0.5);

or turn link 3 a little further

delete(h{3});

JACO{3}=SGTalignT(JACO{3},'B', SGTget(JACO{2},'F'),pi+0.5); % Align "B" of JACO-3 to Follower frame "F" of JACO-2 and turn bu 180 deg i.e. pi rad

h{3}=SGTplotalpha(JACO{3},'',0.5);

The fourth link can then be added in the same way, and so on.

JACO{4}=SGTalignT(JACO{4},'B', SGTget(JACO{3},'F'),pi+0.5); % Align "B" of JACO-3 to Follower frame "F" of JACO-2 and turn bu 180 deg i.e. pi rad

h{4}=SGTplotalpha(JACO{4},'',0.5);

JACO{5}=SGTalignT(JACO{5},'B', SGTget(JACO{4},'F'),pi+0.5); % Align "B" of JACO-3 to Follower frame "F" of JACO-2 and turn bu 180 deg i.e. pi rad

h{5}=SGTplotalpha(JACO{5},'',0.5);

JACO{6}=SGTalignT(JACO{6},'B', SGTget(JACO{5},'F'),pi+0.5); % Align "B" of JACO-3 to Follower frame "F" of JACO-2 and turn bu 180 deg i.e. pi rad

h{6}=SGTplotalpha(JACO{6},'',0.5);

The concept of frames should now be understood. The frames can be assigned using the "SGTset" function. In the imported robot geometry, the frames were already part of the solids. There are also some functions such as "SGTui" for interactive creation of frames.

5 Creation of robot links according to Denativ and Hartenberg.

From a kinematic point of view, it is not the geometry of the robot arm that is important, but only the relative transformation matrix between the respective frames "B" and "F," or the alignment of the "B" frame of a link with the "F" frame of the previous link. The movement of the robot is just a shift of frames relative to the frame of a previous frame in the serial chain of links. The graphical representation of the robot, which is so interesting to us humans, or the motor movement of the real body, which is so important for the movement of a real robot, is mathematically just a "realization problem."

However, it is helpful for us engineers if we can not only "stick" frames onto existing body geometries, but conversely, automatically generate the corresponding geometry of a robot based on spatial specifications of frames. At least a first draft.

The "SGTofDenavitHartenberg" function is used for this purpose.

In Denavit-Hartenberg notation, there is a height D (1st parameter, here 100) from the plane and a distance a (2nd parameter, here zero) from the z-axis, and a rotation around the x-axis (3rd parameter, here 90 degrees) to define the position of the follower frame "F," i.e., the connection point for the next link. The rotation clearly shows that the connection for the next member is actually perpendicular above the base frame.

The base frame "B" lies at the origin.
D is the height of "F" above the base frame
A is the distance of "F" in X from the base frame
phi (1) is the fixed rotation around the X-axis
phi (2) is the fixed rotation around the Y-axis
The next parameter allows you to specify a cross-section
The last parameter allows rotation around the base, which is only useful for asymmetrical cross-sections and has no effect on the frames.

SGTofDenavitHartenberg(100,0,pi/2); figurerotate('',90)

To understand how this function works, I recommend simply trying out the parameters:

SGTofDenavitHartenberg(100,0,[0 0]); %

SGTofDenavitHartenberg(100,0,[pi/2 0]);

SGTofDenavitHartenberg(100,0,[0 pi/2]);

SGTofDenavitHartenberg(100,40,[0 0]);

SGTofDenavitHartenberg(100,40,[pi/2 0]);

SGTofDenavitHartenberg(100,40,[0 pi/2]);

And for those who enjoy complicated cross-sections, there is also:

CPLsample(38);

because the 4th parameter processes a centered cross section as a contour.

SGTofDenavitHartenberg(100,40,[0 pi/2],CPLsample(38)); view(42,11); figurerotate('',90);

SGTofDenavitHartenberg(100,40,[0 pi/2],CPLsample(30)); view(42,11); figurerotate('',90)

6 The homogeneous transformation matrix with DH parameter for symbolic calculation

The upper section describes the Denavit-Hartenberg notation for decomposing a kinematic system into individual links, which then results in the structure of the homogeneous transformation matrix (HT matrix) that describes each link.

If there is a sequential chaining of these links, the F-frame of the last link relative to the B-frame of the first link in this kinematic chain can be calculated by multiplying the HT matrices in the order of the links.

Each of these HT matrices contains a variable that describes the degree of freedom of the link. In this example, this is an angle that describes the position of a rotary joint or, in the case of a robot, the position that a servo motor sets to move the rotary joint.

The SGTofDenavitHartenberg function not only provides the geometry of the arm but also, with the help of the Symbolics Toolbox, the symbolic equations for describing the link.

[SG1,T1,TT1]=SGTofDenavitHartenberg(100,40,[0 pi/2]), SGfigure(SG1); view(42,11);

SG1 = struct with fields:
       VL: [1404×3 double]
       FL: [2808×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[]  []}

T1 =

TT1 =

[SG2,T2,TT2]=SGTofDenavitHartenberg(0,100,[pi 0]), SGfigure(SG2); view(42,11); VLFLplotlight(1,0.5);

SG2 = struct with fields:
       VL: [2496×3 double]
       FL: [4992×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[]  []}

T2 =

TT2 =

7 The homogeneous transformation matrix with DH parameter for serial-link robot-arm

It is easy to imagine that there should be a function for a serial chain of arm members that generates the entire chain and stores it in a cell list or cell array.

This function is called SGTofDHset. In this function, the DH parameters are passed in a [n x 4] array [d1 a1 wx1 wy2], and it is also possible to specify a cross-section profile.

SGTofDHset([50 0 0 0; 100,0,pi/2 0;0 100 pi*0.99 0;0 100 0 pi/2],[PLsquare(20,30);nan nan; PLcircle(7)]);

This illustration also shows why robot arms always have a circular profile at the joint in practice. This avoids unsightly open pivot surfaces, which occur at angles when the cross-sectional areas do not happen to fit together.

SGTofDHset([50 0 0 0; 100,0,pi/2 0;0 100 pi*0.99 0;0 100 0 pi/2],[PLcircle(10);nan nan; PLcircle(7)]);

Let's take a look at the output parameters

[SG,Ti,SGN,Tii,E]=SGTofDHset([50 0 0 0; 100,0,pi/2 0;0 100 pi*0.99 0;0 100 0 pi/2],[PLcircle(10);nan nan; PLcircle(7)]);

The first output parameter is the geometry of the links. It is a cell list containing all links for which the B-frame is located at the coordinate origin. The robot is therefore not assembled.

SGfigure(-30,30); SGplotcell(SG); SG{:}

ans = struct with fields:
       VL: [252×3 double]
       FL: [504×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[]  []}

ans = struct with fields:
       VL: [2100×3 double]
       FL: [4200×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[]  []}

ans = struct with fields:
       VL: [2688×3 double]
       FL: [5376×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[]  []}

ans = struct with fields:
       VL: [2100×3 double]
       FL: [4200×3 double]
    Tname: {'B'  'F'}
        T: {[4×4 double]  [4×4 double]}
     TFiL: {[]  []}

The second output parameter is a cell array containing the respective symbolic DH matrices of the individual elements, in which, however, the geometric constants d and a of the DH matrix are incorporated.

Ti{:}

ans =

ans =

The third output parameter is SG with the "assembled" kinematic chain and the transformation of the links to the correct position.

SGfigure(SGN); SGplotcell(SGN);

The fourth output parameter is a cell array containing the respective symbolic DH matrices of the individual elements, in which, however, the geometric constants d and a of the DH matrix are still present in the equations.

Tii{:}

ans =

ans =

The fifth output parameter contains the symbolic equations for the forward kinematics up to the last F frame. This equation can be used to calculate the backward kinematics symbolically or to reduce the complexity.

E =

R=E(1:3,1:3)

R =

t=E(1:3,4)

t =

In this equation, you can then reinsert the values for d1=50, d2=100, a3=100, a4=100 as required.

t=subs(t,'d1',50)

t =

t=subs(t,'d1',100)

t =

t=subs(t,'a3',100)

t =

t=subs(t,'a4',100)

t =

The inverse kinematics would be calculated with:

% syms x y z
% solve(t==[x y z]',w1,w2,w3,w4)   % Due to time constraints, this should not be done.

8 Direct calculation of numerical equations for forward kinematics from symbolic formulas

We can also obtain numerical equations for calculating the forward kinematics from the symbolic equations.

[e,f,g]=symbforwkine(SG,Ti);

e{:}

ans =

ans =

f = 
f(:,:,1) =
     0     0    50
     0     0   150
   100     0   150
   200     0   150
f(:,:,2) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,3) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,4) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,5) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,6) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,7) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,8) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,9) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,10) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,11) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,12) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,13) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,14) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  200.0000   -0.0000  150.0000
f(:,:,15) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,16) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,17) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,18) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,19) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,20) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  200.0000   -0.0000  150.0000
f(:,:,21) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,22) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,23) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,24) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,25) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,26) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  200.0000   -0.0000  150.0000
f(:,:,27) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,28) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,29) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,30) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,31) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,32) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  200.0000   -0.0000  150.0000
f(:,:,33) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,34) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,35) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,36) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,37) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,38) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  200.0000   -0.0000  150.0000
f(:,:,39) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,40) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,41) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,42) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,43) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
  124.6980 -156.3663  150.0000
f(:,:,44) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  200.0000   -0.0000  150.0000
f(:,:,45) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  124.6980  156.3663  150.0000
f(:,:,46) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -44.5042  194.9856  150.0000
f(:,:,47) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -180.1938   86.7767  150.0000
f(:,:,48) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -180.1938  -86.7767  150.0000
f(:,:,49) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -44.5042 -194.9856  150.0000
f(:,:,50) =
         0         0   50.0000
         0         0  150.0000
   62.3490         0  228.1831
  124.6980         0  306.3663
f(:,:,51) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,52) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,53) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,54) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,55) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,56) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,57) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,58) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,59) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,60) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,61) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,62) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,63) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  124.6980   -0.0000  306.3663
f(:,:,64) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,65) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,66) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,67) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,68) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,69) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  124.6980   -0.0000  306.3663
f(:,:,70) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,71) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,72) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,73) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,74) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,75) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  124.6980   -0.0000  306.3663
f(:,:,76) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,77) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,78) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,79) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,80) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,81) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  124.6980   -0.0000  306.3663
f(:,:,82) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,83) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,84) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,85) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,86) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,87) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  124.6980   -0.0000  306.3663
f(:,:,88) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,89) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,90) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,91) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,92) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   77.7479  -97.4928  306.3663
f(:,:,93) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  124.6980   -0.0000  306.3663
f(:,:,94) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
   77.7479   97.4928  306.3663
f(:,:,95) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -27.7479  121.5715  306.3663
f(:,:,96) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -112.3490   54.1044  306.3663
f(:,:,97) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -112.3490  -54.1044  306.3663
f(:,:,98) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -27.7479 -121.5715  306.3663
f(:,:,99) =
         0         0   50.0000
         0         0  150.0000
  -22.2521         0  247.4928
  -44.5042         0  344.9856
f(:,:,100) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,101) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,102) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,103) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,104) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,105) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,106) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,107) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,108) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,109) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,110) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,111) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,112) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000  247.4928
  -44.5042    0.0000  344.9856
f(:,:,113) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,114) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,115) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,116) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,117) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,118) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000  247.4928
  -44.5042    0.0000  344.9856
f(:,:,119) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,120) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,121) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,122) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,123) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,124) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000  247.4928
  -44.5042    0.0000  344.9856
f(:,:,125) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,126) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,127) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,128) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,129) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,130) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000  247.4928
  -44.5042    0.0000  344.9856
f(:,:,131) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,132) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,133) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,134) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,135) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,136) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000  247.4928
  -44.5042    0.0000  344.9856
f(:,:,137) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,138) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,139) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,140) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,141) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974  247.4928
  -27.7479   34.7948  344.9856
f(:,:,142) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000  247.4928
  -44.5042    0.0000  344.9856
f(:,:,143) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974  247.4928
  -27.7479  -34.7948  344.9856
f(:,:,144) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942  247.4928
    9.9031  -43.3884  344.9856
f(:,:,145) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548  247.4928
   40.0969  -19.3096  344.9856
f(:,:,146) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548  247.4928
   40.0969   19.3096  344.9856
f(:,:,147) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942  247.4928
    9.9031   43.3884  344.9856
f(:,:,148) =
         0         0   50.0000
         0         0  150.0000
  -90.0969         0  193.3884
 -180.1938         0  236.7767
f(:,:,149) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,150) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,151) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,152) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,153) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,154) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,155) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,156) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,157) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,158) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,159) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,160) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,161) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  193.3884
 -180.1938    0.0000  236.7767
f(:,:,162) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,163) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,164) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,165) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,166) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,167) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  193.3884
 -180.1938    0.0000  236.7767
f(:,:,168) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,169) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,170) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,171) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,172) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,173) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  193.3884
 -180.1938    0.0000  236.7767
f(:,:,174) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,175) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,176) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,177) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,178) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,179) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  193.3884
 -180.1938    0.0000  236.7767
f(:,:,180) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,181) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,182) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,183) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,184) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,185) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  193.3884
 -180.1938    0.0000  236.7767
f(:,:,186) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,187) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,188) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,189) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,190) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  193.3884
 -112.3490  140.8812  236.7767
f(:,:,191) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  193.3884
 -180.1938    0.0000  236.7767
f(:,:,192) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  193.3884
 -112.3490 -140.8812  236.7767
f(:,:,193) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  193.3884
   40.0969 -175.6759  236.7767
f(:,:,194) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  193.3884
  162.3490  -78.1831  236.7767
f(:,:,195) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  193.3884
  162.3490   78.1831  236.7767
f(:,:,196) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  193.3884
   40.0969  175.6759  236.7767
f(:,:,197) =
         0         0   50.0000
         0         0  150.0000
  -90.0969         0  106.6116
 -180.1938         0   63.2233
f(:,:,198) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,199) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,200) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,201) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,202) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,203) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,204) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,205) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,206) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,207) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,208) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,209) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,210) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  106.6116
 -180.1938    0.0000   63.2233
f(:,:,211) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,212) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,213) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,214) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,215) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,216) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  106.6116
 -180.1938    0.0000   63.2233
f(:,:,217) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,218) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,219) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,220) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,221) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,222) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  106.6116
 -180.1938    0.0000   63.2233
f(:,:,223) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,224) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,225) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,226) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,227) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,228) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  106.6116
 -180.1938    0.0000   63.2233
f(:,:,229) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,230) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,231) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,232) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,233) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,234) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  106.6116
 -180.1938    0.0000   63.2233
f(:,:,235) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,236) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,237) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,238) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,239) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   70.4406  106.6116
 -112.3490  140.8812   63.2233
f(:,:,240) =
         0         0   50.0000
         0         0  150.0000
  -90.0969    0.0000  106.6116
 -180.1938    0.0000   63.2233
f(:,:,241) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -70.4406  106.6116
 -112.3490 -140.8812   63.2233
f(:,:,242) =
         0         0   50.0000
         0         0  150.0000
   20.0484  -87.8380  106.6116
   40.0969 -175.6759   63.2233
f(:,:,243) =
         0         0   50.0000
         0         0  150.0000
   81.1745  -39.0916  106.6116
  162.3490  -78.1831   63.2233
f(:,:,244) =
         0         0   50.0000
         0         0  150.0000
   81.1745   39.0916  106.6116
  162.3490   78.1831   63.2233
f(:,:,245) =
         0         0   50.0000
         0         0  150.0000
   20.0484   87.8380  106.6116
   40.0969  175.6759   63.2233
f(:,:,246) =
         0         0   50.0000
         0         0  150.0000
  -22.2521         0   52.5072
  -44.5042         0  -44.9856
f(:,:,247) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,248) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,249) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,250) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,251) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,252) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,253) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,254) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,255) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,256) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,257) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,258) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,259) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000   52.5072
  -44.5042    0.0000  -44.9856
f(:,:,260) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,261) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,262) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,263) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,264) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,265) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000   52.5072
  -44.5042    0.0000  -44.9856
f(:,:,266) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,267) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,268) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,269) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,270) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,271) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000   52.5072
  -44.5042    0.0000  -44.9856
f(:,:,272) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,273) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,274) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,275) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,276) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,277) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000   52.5072
  -44.5042    0.0000  -44.9856
f(:,:,278) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,279) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,280) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,281) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,282) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,283) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000   52.5072
  -44.5042    0.0000  -44.9856
f(:,:,284) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,285) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,286) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,287) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,288) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   17.3974   52.5072
  -27.7479   34.7948  -44.9856
f(:,:,289) =
         0         0   50.0000
         0         0  150.0000
  -22.2521    0.0000   52.5072
  -44.5042    0.0000  -44.9856
f(:,:,290) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -17.3974   52.5072
  -27.7479  -34.7948  -44.9856
f(:,:,291) =
         0         0   50.0000
         0         0  150.0000
    4.9516  -21.6942   52.5072
    9.9031  -43.3884  -44.9856
f(:,:,292) =
         0         0   50.0000
         0         0  150.0000
   20.0484   -9.6548   52.5072
   40.0969  -19.3096  -44.9856
f(:,:,293) =
         0         0   50.0000
         0         0  150.0000
   20.0484    9.6548   52.5072
   40.0969   19.3096  -44.9856
f(:,:,294) =
         0         0   50.0000
         0         0  150.0000
    4.9516   21.6942   52.5072
    9.9031   43.3884  -44.9856
f(:,:,295) =
         0         0   50.0000
         0         0  150.0000
   62.3490         0   71.8169
  124.6980         0   -6.3663
f(:,:,296) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,297) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,298) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,299) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,300) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,301) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,302) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,303) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,304) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,305) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,306) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,307) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,308) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000   71.8169
  124.6980   -0.0000   -6.3663
f(:,:,309) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,310) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,311) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,312) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,313) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,314) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000   71.8169
  124.6980   -0.0000   -6.3663
f(:,:,315) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,316) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,317) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,318) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,319) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,320) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000   71.8169
  124.6980   -0.0000   -6.3663
f(:,:,321) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,322) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,323) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,324) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,325) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,326) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000   71.8169
  124.6980   -0.0000   -6.3663
f(:,:,327) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,328) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,329) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,330) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,331) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,332) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000   71.8169
  124.6980   -0.0000   -6.3663
f(:,:,333) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,334) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,335) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,336) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,337) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464   71.8169
   77.7479  -97.4928   -6.3663
f(:,:,338) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000   71.8169
  124.6980   -0.0000   -6.3663
f(:,:,339) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464   71.8169
   77.7479   97.4928   -6.3663
f(:,:,340) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858   71.8169
  -27.7479  121.5715   -6.3663
f(:,:,341) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522   71.8169
 -112.3490   54.1044   -6.3663
f(:,:,342) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522   71.8169
 -112.3490  -54.1044   -6.3663
f(:,:,343) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858   71.8169
  -27.7479 -121.5715   -6.3663
f(:,:,344) =
         0         0   50.0000
         0         0  150.0000
  100.0000         0  150.0000
  162.3490   -2.4558   71.8554
f(:,:,345) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,346) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,347) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,348) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,349) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,350) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,351) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,352) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,353) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,354) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,355) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,356) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,357) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  162.3490   -2.4558   71.8554
f(:,:,358) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,359) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,360) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,361) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,362) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,363) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  162.3490   -2.4558   71.8554
f(:,:,364) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,365) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,366) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,367) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,368) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,369) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  162.3490   -2.4558   71.8554
f(:,:,370) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,371) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,372) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,373) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,374) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,375) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  162.3490   -2.4558   71.8554
f(:,:,376) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,377) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,378) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,379) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,380) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,381) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  162.3490   -2.4558   71.8554
f(:,:,382) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,383) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,384) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,385) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,386) =
         0         0   50.0000
         0         0  150.0000
   62.3490  -78.1831  150.0000
   99.3029 -128.4607   71.8554
f(:,:,387) =
         0         0   50.0000
         0         0  150.0000
  100.0000   -0.0000  150.0000
  162.3490   -2.4558   71.8554
f(:,:,388) =
         0         0   50.0000
         0         0  150.0000
   62.3490   78.1831  150.0000
  103.1429  125.3984   71.8554
f(:,:,389) =
         0         0   50.0000
         0         0  150.0000
  -22.2521   97.4928  150.0000
  -33.7318  158.8250   71.8554
f(:,:,390) =
         0         0   50.0000
         0         0  150.0000
  -90.0969   43.3884  150.0000
 -145.2058   72.6532   71.8554
f(:,:,391) =
         0         0   50.0000
         0         0  150.0000
  -90.0969  -43.3884  150.0000
 -147.3369  -68.2280   71.8554
f(:,:,392) =
         0         0   50.0000
         0         0  150.0000
  -22.2521  -97.4928  150.0000
  -38.5203 -157.7321   71.8554
f(:,:,393) =
         0         0   50.0000
         0         0  150.0000
   62.3490         0  228.1831
  162.3188   -2.4558  228.2072
f(:,:,394) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,395) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,396) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,397) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -147.3097  -68.2149  228.2072
f(:,:,398) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -38.5136 -157.7027  228.2072
f(:,:,399) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,400) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,401) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,402) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,403) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -147.3097  -68.2149  228.2072
f(:,:,404) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -38.5136 -157.7027  228.2072
f(:,:,405) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,406) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  162.3188   -2.4558  228.2072
f(:,:,407) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,408) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,409) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -147.3097  -68.2149  228.2072
f(:,:,410) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -38.5136 -157.7027  228.2072
f(:,:,411) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,412) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  162.3188   -2.4558  228.2072
f(:,:,413) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,414) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,415) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -147.3097  -68.2149  228.2072
f(:,:,416) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -38.5136 -157.7027  228.2072
f(:,:,417) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,418) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  162.3188   -2.4558  228.2072
f(:,:,419) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,420) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,421) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -147.3097  -68.2149  228.2072
f(:,:,422) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -38.5136 -157.7027  228.2072
f(:,:,423) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,424) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  162.3188   -2.4558  228.2072
f(:,:,425) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,426) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,427) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,428) =
         0         0   50.0000
         0         0  150.0000
  -13.8740  -60.7858  228.1831
  -38.5136 -157.7027  228.2072
f(:,:,429) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,430) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  162.3188   -2.4558  228.2072
f(:,:,431) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,432) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,433) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,434) =
         0         0   50.0000
         0         0  150.0000
  -56.1745  -27.0522  228.1831
 -147.3097  -68.2149  228.2072
f(:,:,435) =
         0         0   50.0000
         0         0  150.0000
   38.8740  -48.7464  228.1831
   99.2841 -128.4371  228.2072
f(:,:,436) =
         0         0   50.0000
         0         0  150.0000
   62.3490   -0.0000  228.1831
  162.3188   -2.4558  228.2072
f(:,:,437) =
         0         0   50.0000
         0         0  150.0000
   38.8740   48.7464  228.1831
  103.1241  125.3748  228.2072
f(:,:,438) =
         0         0   50.0000
         0         0  150.0000
  -13.8740   60.7858  228.1831
  -33.7251  158.7956  228.2072
f(:,:,439) =
         0         0   50.0000
         0         0  150.0000
  -56.1745   27.0522  228.1831
 -145.1787   72.6401  228.2072
f(:,:,440) =

The third output parameter is a cell array with numerical functions for directly calculating the HT matrix of the respective F-frame directly from the given angular positions of the (here 4) rotary joints.

g{:}
ans = function_handle with value:
    @(w1)reshape([cos(w1),sin(w1),0.0,0.0,-sin(w1),cos(w1),0.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,5.0e+1,1.0],[4,4])
ans = function_handle with value:
    @(w1,w2)reshape([cos(w1).*cos(w2)-sin(w1).*sin(w2),cos(w1).*sin(w2)+cos(w2).*sin(w1),0.0,0.0,0.0,0.0,1.0,0.0,cos(w1).*sin(w2)+cos(w2).*sin(w1),-cos(w1).*cos(w2)+sin(w1).*sin(w2),0.0,0.0,0.0,0.0,1.5e+2,1.0],[4,4])
ans = function_handle with value:
    @(w1,w2,w3)reshape([cos(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),cos(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),sin(w3),0.0,sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),-cos(pi./1.0e+2).*cos(w3),0.0,-cos(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+sin(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),cos(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))+sin(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),-sin(pi./1.0e+2).*cos(w3),0.0,cos(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)).*1.0e+2,cos(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)).*1.0e+2,sin(w3).*1.0e+2+1.5e+2,1.0],[4,4])
ans = function_handle with value:
    @(w1,w2,w3,w4)reshape([cos(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))-sin(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-cos(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-sin(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),sin(pi./1.0e+2).*cos(w3),0.0,cos(w4).*(sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)))-cos(w3).*sin(w4).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-cos(w4).*(sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)))-cos(w3).*sin(w4).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),-sin(w3).*sin(w4)-cos(pi./1.0e+2).*cos(w3).*cos(w4),0.0,sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)))+cos(w3).*cos(w4).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)))+cos(w3).*cos(w4).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),cos(w4).*sin(w3)-cos(pi./1.0e+2).*cos(w3).*sin(w4),0.0,sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))).*1.0e+2+cos(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)).*1.0e+2+cos(w3).*cos(w4).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)).*1.0e+2,sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))).*-1.0e+2+cos(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)).*1.0e+2+cos(w3).*cos(w4).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)).*1.0e+2,sin(w3).*1.0e+2+cos(w4).*sin(w3).*1.0e+2-cos(pi./1.0e+2).*cos(w3).*sin(w4).*1.0e+2+1.5e+2,1.0],[4,4])

The function for determining the F-frame of the fourth limb at angles (0,pi,0,pi) is

g{4}
ans = function_handle with value:
    @(w1,w2,w3,w4)reshape([cos(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))-sin(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-cos(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-sin(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),sin(pi./1.0e+2).*cos(w3),0.0,cos(w4).*(sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)))-cos(w3).*sin(w4).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-cos(w4).*(sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)))-cos(w3).*sin(w4).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),-sin(w3).*sin(w4)-cos(pi./1.0e+2).*cos(w3).*cos(w4),0.0,sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)))+cos(w3).*cos(w4).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)),-sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)))+cos(w3).*cos(w4).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)),cos(w4).*sin(w3)-cos(pi./1.0e+2).*cos(w3).*sin(w4),0.0,sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))+cos(pi./1.0e+2).*sin(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))).*1.0e+2+cos(w3).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)).*1.0e+2+cos(w3).*cos(w4).*(cos(w1).*cos(w2)-sin(w1).*sin(w2)).*1.0e+2,sin(w4).*(sin(pi./1.0e+2).*(cos(w1).*cos(w2)-sin(w1).*sin(w2))-cos(pi./1.0e+2).*sin(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1))).*-1.0e+2+cos(w3).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)).*1.0e+2+cos(w3).*cos(w4).*(cos(w1).*sin(w2)+cos(w2).*sin(w1)).*1.0e+2,sin(w3).*1.0e+2+cos(w4).*sin(w3).*1.0e+2-cos(pi./1.0e+2).*cos(w3).*sin(w4).*1.0e+2+1.5e+2,1.0],[4,4])
HT4=g{4}(0,pi,pi,0)
HT4 = 4×4
    0.0000   -0.0000    1.0000  200.0000
    0.9995    0.0314   -0.0000   -0.0000
   -0.0314    0.9995    0.0000  150.0000
         0         0         0    1.0000

Unfortunately, it is also quite clear that the numerical accuracy of these calculations with HT matrices leads to angular errors in the degree range even with just four concatenations. The correct Euler angles should be 90 0 90.

This is why numerical calculations usually switch to quaternions and use these instead of homogeneous transformation matrices in the calculations.

rotm2eul(HT4(1:3,1:3))/pi*180
ans = 1×3
   90.0000    1.8000   90.0000

9 Graphical representation of robot poses, based on defined joint angle values

The graphical representation of the pose assumed by a robot can be displayed using SGTchain. SGTchain moves all links of a kinematic chain according to the joint angles and stores the position of the links in the SG cell array at the correct position.

To align the zero position of the joint angles with the zero position of a real robot, there is the function "SGTcalibchain."

SGTchain(JACO(1:6),[0 0 +60 +90 +90] /180*pi); SGN=ans; view(-80,20); %

SGTchain(SG(1:4),[0 0 +60 +90 +90] /180*pi); view(-30,30); %

SGN=SGTchain(SG(1:4),[0 0 +60 +90 +90] /180*pi); SGplotcell(SGN);

10 Animating the movement of a serial kinematics (robot arms)

If you want to represent a movement of a robot's rotary joint, the "handles" of the plotted solid, which are the output parameters of a plot function, must be saved.

The handles are an array, where the first handle is for the first link/base, the second for the second link, and so on. For a movement, part of the robot's representation must then be deleted, the geometry rotated, and redrawn.

This is done by the "Videoquickanimatego" function, which is passed the handles to be moved, a coordinate system for rotation, the step size in degrees of rotation, and the movement principle ("rotation around the z-axis").

SGfigure(-30,30); axis4animation([0 1 0 1 0 1]); h=SGplot(JACO(2:6)); shg; view(-30,30); Videoquickanimatego('',{h([1:end]),SGTget(SGN{1},'B'),5,'rotz'})

ans = 10.0009

Hier wird der Roboter an der Basis gedreht. Um es besser sichtbar zu machen, ergänzen wir zur Geometrie einen kleinen Sockel an der sonst zylindrischen Basis.

SGN{1}=SGconcat(SGN{1},SGbox([40 20 10]));

SGfigure(-30,30); axis4animation([0 1 0 1 0 1]); h=SGplot(SGN); shg; view(-30,30); Videoquickanimatego('',{h([1:end]),SGTget(SGN{1},'B'),5,'rotz'})

ans = 10.0010

Hier wird der Roboter am zweiten Gelenk gedreht.

clf; h=SGplot(SGN); shg; view(-30,30); Videoquickanimatego('',{h([2:end]),SGTget(SGN{2},'B'),5,'rotz'})

ans = 10.0011

Hier wird der Roboter am dritten Gelenk gedreht.

clf; h=SGplot(SGN); shg; view(-30,30); Videoquickanimatego('',{h([3:end]),SGTget(SGN{3},'B'),5,'rotz'})

ans = 10.0012

Hier wird der Roboter am vierten Gelenk gedreht.

clf; h=SGplot(SGN); shg; view(-30,30); Videoquickanimatego('',{h([4:end]),SGTget(SGN{4},'B'),5,'rotz'})

ans = 10.0013

Show the execution time of this script

stamp
stamp: Mac OSX 15.5 | R2024b Update 6 | SG-Lib 5.4 | Java 11.0.24+8-LTS | M3-PRO 12-Core CPU, 18-Core GPU 36GByte RAM | 06-Aug-2025 22:02:55