robot.fodt

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   <text:h text:style-name="Heading_20_1" text:outline-level="1">How to calibrate a 6-axes articulated robot.</text:h>
   <text:p text:style-name="Standard">CADSM robot convention</text:p>
   <text:p text:style-name="Standard">X points to forward direction</text:p>
   <text:p text:style-name="Standard">Y points to the left</text:p>
   <text:p text:style-name="Standard">Z points up</text:p>
   <text:p text:style-name="Standard">The reference configuration is the robot standing at its tallest with its belly pointing forward.</text:p>
   <text:p text:style-name="Standard">Assembly and all part frames coincide at the reference configuration.</text:p>
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   <text:p text:style-name="Standard">/Assembly1/Marker0/</text:p>
   <text:p text:style-name="Standard">X is unknown</text:p>
   <text:p text:style-name="Standard">Y is unknown</text:p>
   <text:p text:style-name="Standard">Z is zero (axis 1 pierces assembly XY plane)</text:p>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Part 1</text:h>
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   <text:p text:style-name="Standard">/Assembly1/Part1/Marker1/ coincides with /Assembly1/Marker0/</text:p>
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   <text:p text:style-name="Standard">X is unknown</text:p>
   <text:p text:style-name="Standard">Y is zero (axis 2 pierces part XZ plane)</text:p>
   <text:p text:style-name="Standard">Z is unknown</text:p>
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=
left lbrace 
matrix{ 
x ## 
0 ## 
z} 
right rbrace_{p1,p1m2,p1}</annotation>
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    </draw:frame></text:p>
   <text:p text:style-name="Standard">Bryant1 is unknown (near minus half pi)</text:p>
   <text:p text:style-name="Standard">Bryant2 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant3 is is zero when motor encoder is zero.</text:p>
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   <text:p text:style-name="Standard">Relative configuration between Marker1 and Marker2.</text:p>
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=
left [{bold nitalic A} right]_{p1,p1m1}^T
left({bold nitalic r}_{p1,p1m2,p1} - {bold nitalic r}_{p1,p1m1,p1}
right)</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Part 2</text:h>
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   <text:p text:style-name="Standard">Axis 2 rotation is about the common z axes which are almost parallel to world Y.</text:p>
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=
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   <text:p text:style-name="Standard">/Assembly1/Part2/Marker2/</text:p>
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=
left lbrace 
matrix{ 
x ## 
0 ## 
z} 
right rbrace_{p2,p2m2,p2}</annotation>
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   <text:p text:style-name="Standard">Bryant1 is unknown (near minus half pi)</text:p>
   <text:p text:style-name="Standard">Bryant2 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant3 is is zero when motor encoder is zero.</text:p>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Part 3</text:h>
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   <text:p text:style-name="Standard">/Assembly1/Part3/Marker1/ coincides with /Assembly1/Part2/Marker2/</text:p>
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=
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   <text:p text:style-name="Standard">Axis 3 rotation is about the common z axes which are almost parallel to world Y.</text:p>
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             <mtd>
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             <mtd>
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            </mtr>
            <mtr>
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             <mtd>
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             <mtd>
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            </mtr>
            <mtr>
             <mtd>
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             </mtd>
             <mtd>
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             <mtd>
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              </mrow>
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            </mtr>
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        <annotation encoding="StarMath 5.0">left [{bold nitalic A}(%theta_3) right]
=
left [
matrix
{
cos %theta_3 # 0 # sin %theta_3 ##
0 # 1 # 0 ##
-sin %theta_3 # 0 # cos %theta_3
}
right]
=
left [{bold nitalic A} right]_{p2m2,p3m1}
</annotation>
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    </draw:frame></text:p>
   <text:p text:style-name="Standard">/Assembly1/Part3/Marker2/</text:p>
   <text:p text:style-name="Standard">X is unknown</text:p>
   <text:p text:style-name="Standard">Y is unknown</text:p>
   <text:p text:style-name="Standard">Z is zero (axis 4 pierces assembly XY plane)</text:p>
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       <semantics>
        <mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">r</mi>
          </mstyle>
          <mrow>
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           <mn>3</mn>
           <mi>,</mi>
           <mi>p</mi>
           <mn>3</mn>
           <mi>m</mi>
           <mn>2</mn>
           <mi>,</mi>
           <mi>p</mi>
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          </mrow>
         </msub>
         <mo stretchy="false">=</mo>
         <msub>
          <mrow>
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           <mrow>
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              <mtd>
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             </mtr>
             <mtr>
              <mtd>
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              </mtd>
             </mtr>
             <mtr>
              <mtd>
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              </mtd>
             </mtr>
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           </mrow>
           <mo fence="true" form="postfix" stretchy="true">}</mo>
          </mrow>
          <mrow>
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           <mi>,</mi>
           <mi>p</mi>
           <mn>3</mn>
           <mi>m</mi>
           <mn>2</mn>
           <mi>,</mi>
           <mi>p</mi>
           <mn>3</mn>
          </mrow>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_{p3,p3m2,p3}
=
left lbrace 
matrix{ 
x ## 
y ## 
0} 
right rbrace_{p3,p3m2,p3}</annotation>
       </semantics>
      </math>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Bryant1 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant2 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant3 is is zero when motor encoder is zero.</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object18" text:anchor-type="as-char" svg:width="1.322in" svg:height="0.2425in" draw:z-index="17"><draw:object>
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       <semantics>
        <msub>
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          <mrow>
           <mrow>
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            <mrow>
             <mo fence="true" form="prefix" stretchy="false">(</mo>
             <mrow>
              <mrow>
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                <mi>θ</mi>
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               </msub>
               <mi>,</mi>
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           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">]</mo>
         </mrow>
         <mrow>
          <mi>p</mi>
          <mn>3</mn>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Part 4</text:h>
   <text:p text:style-name="Standard">Part4 frame coincides with the Assembly frame at the reference configuration.</text:p>
   <text:p text:style-name="Standard">/Assembly1/Part4/Marker1/ coincides with /Assembly1/Part3/Marker2/</text:p>
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=
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">/Assembly1/Part4/Marker2/</text:p>
   <text:p text:style-name="Standard">X is unknown</text:p>
   <text:p text:style-name="Standard">Y is zero (axis 3 pierces part XZ plane)</text:p>
   <text:p text:style-name="Standard">Z is unknown</text:p>
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       <semantics>
        <mrow>
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          </mstyle>
          <mrow>
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           <mi>,</mi>
           <mi>p</mi>
           <mn>4</mn>
           <mi>m</mi>
           <mn>2</mn>
           <mi>,</mi>
           <mi>p</mi>
           <mn>4</mn>
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         </msub>
         <mo stretchy="false">=</mo>
         <msub>
          <mrow>
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             </mtr>
             <mtr>
              <mtd>
               <mn>0</mn>
              </mtd>
             </mtr>
             <mtr>
              <mtd>
               <mi>z</mi>
              </mtd>
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           </mrow>
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          </mrow>
          <mrow>
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           <mn>4</mn>
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           <mi>p</mi>
           <mn>4</mn>
           <mi>m</mi>
           <mn>2</mn>
           <mi>,</mi>
           <mi>p</mi>
           <mn>4</mn>
          </mrow>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_{p4,p4m2,p4}
=
left lbrace 
matrix{ 
x ## 
0 ## 
z} 
right rbrace_{p4,p4m2,p4}</annotation>
       </semantics>
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       BQAAAP////8AhgABAAQAAAAAAAAAcgACACAAAACcDAAATgYAAAQAcDRtMjMCAAAAAP//BABw
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Bryant1 is unknown (near minus half pi)</text:p>
   <text:p text:style-name="Standard">Bryant2 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant3 is is zero when motor encoder is zero.</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object28" text:anchor-type="as-char" svg:width="1.3256in" svg:height="0.2425in" draw:z-index="28"><draw:object>
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       <semantics>
        <msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">[</mo>
          <mrow>
           <mrow>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">A</mi>
            </mstyle>
            <mrow>
             <mo fence="true" form="prefix" stretchy="false">(</mo>
             <mrow>
              <mrow>
               <msub>
                <mi>θ</mi>
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                 <mi>x</mi>
                </mrow>
               </msub>
               <mi>,</mi>
               <msub>
                <mi>θ</mi>
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             </mrow>
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           </mrow>
          </mrow>
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         </mrow>
         <mrow>
          <mi>p</mi>
          <mn>4</mn>
          <mi>,</mi>
          <mi>p</mi>
          <mn>4</mn>
          <mi>m</mi>
          <mn>2</mn>
         </mrow>
        </msub>
        <annotation encoding="StarMath 5.0">left [{bold nitalic A}(%theta_1x, %theta_2y) right]_{p4,p4m2}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Part 5</text:h>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">/Assembly1/Part5/Marker2/</text:p>
   <text:p text:style-name="Standard">X is unknown</text:p>
   <text:p text:style-name="Standard">Y is unknown</text:p>
   <text:p text:style-name="Standard">Z is zero (axis 4 pierces assembly XY plane)</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object34" text:anchor-type="as-char" svg:width="1.7063in" svg:height="0.678in" draw:z-index="34"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mstyle mathvariant="bold">
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          </mstyle>
          <mrow>
           <mi>p</mi>
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           <mi>,</mi>
           <mi>p</mi>
           <mn>5</mn>
           <mi>m</mi>
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           <mi>,</mi>
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         <msub>
          <mrow>
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              <mtd>
               <mi>x</mi>
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             </mtr>
             <mtr>
              <mtd>
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             </mtr>
             <mtr>
              <mtd>
               <mn>0</mn>
              </mtd>
             </mtr>
            </mtable>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">}</mo>
          </mrow>
          <mrow>
           <mi>p</mi>
           <mn>5</mn>
           <mi>,</mi>
           <mi>p</mi>
           <mn>5</mn>
           <mi>m</mi>
           <mn>2</mn>
           <mi>,</mi>
           <mi>p</mi>
           <mn>5</mn>
          </mrow>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_{p5,p5m2,p5}
=
left lbrace 
matrix{ 
x ## 
y ## 
0} 
right rbrace_{p5,p5m2,p5}</annotation>
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      </math>
     </draw:object><draw:image>
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       hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAATQkAAAgEAAABAHm6AAAAAAD//wEA
       eQCMAAEAAAAAAIsAAQACAAAAHwCKAAEAOQAAAAIAMwAAAAkAVGhvcm5kYWxlAAAAAAAApwEA
       AAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACABcAAAAzCQAA/wUAAAEAMNQAAAAAAP//AQAwAIwAAQAAAAAAiwAB
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       AAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAPUK
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       AAAJAFRob3JuZGFsZQAAAAAAAP0AAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAACI
       AAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAMgwAAE4GAAABACxA
       AAAAAAD//wEALACMAAEAAAAAAIsAAQACAAAAHwCKAAEAOQAAAAIAMwAAAAkAVGhvcm5kYWxl
       AAAAAAAA/QAAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAIgAAQACAAAAAQCHAAEA
       BQAAAP////8AhgABAAQAAAAAAAAAcgACACAAAAC2DAAATgYAAAQAcDVtMjMCAAAAAP//BABw
       ADUAbQAyAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA5AAAAAgAzAAAACQBUaG9ybmRhbGUAAAAA
       AAD9AAAAAAADAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Bryant1 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant2 is unknown (near zero)</text:p>
   <text:p text:style-name="Standard">Bryant3 is is zero when motor encoder is zero.</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object35" text:anchor-type="as-char" svg:width="1.3252in" svg:height="0.2425in" draw:z-index="35"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">[</mo>
          <mrow>
           <mrow>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">A</mi>
            </mstyle>
            <mrow>
             <mo fence="true" form="prefix" stretchy="false">(</mo>
             <mrow>
              <mrow>
               <msub>
                <mi>θ</mi>
                <mrow>
                 <mn>1</mn>
                 <mi>x</mi>
                </mrow>
               </msub>
               <mi>,</mi>
               <msub>
                <mi>θ</mi>
                <mrow>
                 <mn>2</mn>
                 <mi>y</mi>
                </mrow>
               </msub>
              </mrow>
             </mrow>
             <mo fence="true" form="postfix" stretchy="false">)</mo>
            </mrow>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">]</mo>
         </mrow>
         <mrow>
          <mi>p</mi>
          <mn>5</mn>
          <mi>,</mi>
          <mi>p</mi>
          <mn>5</mn>
          <mi>m</mi>
          <mn>2</mn>
         </mrow>
        </msub>
        <annotation encoding="StarMath 5.0">left [{bold nitalic A}(%theta_1x, %theta_2y) right]_{p5,p5m2}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAA0wMAABECAAACADF4/gAAAAAA//8C
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       AAAAcgACABoAAADKBQAApwEAAAEAAAC+4OUAAAAAAP//AQC+4IwAAQAAAAAAiwABAAIAAAAf
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       AAAAAAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAA
       AJ0GAAARAgAAAgAyefYAAAAAAP//AgAyAHkAjAABAAAAAACLAAEAAgAAAB8AigABADoAAAAC
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       AAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAF8IAADcAQAAAQAAAF0AZgAA
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Relative configuration between Marker1 and Marker2.</text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">r</mi>
           </mstyle>
           <mrow>
            <mi>p</mi>
            <mn>5</mn>
            <mi>m</mi>
            <mn>1</mn>
            <mi>,</mi>
            <mi>p</mi>
            <mn>5</mn>
            <mi>m</mi>
            <mn>2</mn>
            <mi>,</mi>
            <mi>p</mi>
            <mn>5</mn>
            <mi>m</mi>
            <mn>1</mn>
           </mrow>
          </msub>
          <mo stretchy="false">=</mo>
          <msubsup>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">[</mo>
            <mrow>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">A</mi>
             </mstyle>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">]</mo>
           </mrow>
           <mrow>
            <mi>p</mi>
            <mn>5</mn>
            <mi>,</mi>
            <mi>p</mi>
            <mn>5</mn>
            <mi>m</mi>
            <mn>1</mn>
           </mrow>
           <mi>T</mi>
          </msubsup>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">r</mi>
             </mstyle>
             <mrow>
              <mi>p</mi>
              <mn>5</mn>
              <mi>,</mi>
              <mi>p</mi>
              <mn>5</mn>
              <mi>m</mi>
              <mn>2</mn>
              <mi>,</mi>
              <mi>p</mi>
              <mn>5</mn>
             </mrow>
            </msub>
            <mo stretchy="false">−</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">r</mi>
             </mstyle>
             <mrow>
              <mi>p</mi>
              <mn>5</mn>
              <mi>,</mi>
              <mi>p</mi>
              <mn>5</mn>
              <mi>m</mi>
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              <mi>,</mi>
              <mi>p</mi>
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            </msub>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_{p5m1,p5m2,p5m1}
=
left [{bold nitalic A} right]_{p5,p5m1}^T
left({bold nitalic r}_{p5,p5m2,p5} - {bold nitalic r}_{p5,p5m1,p5}
right)</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Part 6</text:h>
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=
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=
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   <text:p text:style-name="Standard">/Assembly1/Part6/Marker2/ correspond to end-effect frame.</text:p>
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   <text:p text:style-name="Standard">Z is unknown</text:p>
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             </mtr>
             <mtr>
              <mtd>
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             <mtr>
              <mtd>
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          <mrow>
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           <mi>,</mi>
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           <mi>m</mi>
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=
left lbrace 
matrix{ 
x ## 
y ## 
z} 
right rbrace_{p6,p6m2,p6}</annotation>
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    </draw:frame></text:p>
   <text:p text:style-name="Standard">Bryant1 is zero or must be specified.</text:p>
   <text:p text:style-name="Standard">Bryant2 is zero or must be specified.</text:p>
   <text:p text:style-name="Standard">Bryant3 is zero or must be specified.</text:p>
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        </msub>
        <annotation encoding="StarMath 5.0">left [{bold nitalic A}(%theta_1x, %theta_2y) right]_{p6,p6m2}</annotation>
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   <text:p text:style-name="Standard">Relative configuration between Marker1 and Marker2.</text:p>
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=
left [{bold nitalic A} right]_{p6,p6m1}^T
left({bold nitalic r}_{p6,p6m2,p6} - {bold nitalic r}_{p6,p6m1,p6}
right)</annotation>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
{bold nitalic r}_{a,p6m2,a}
=
{bold nitalic r}_{a,m0,a}
+
left [{bold nitalic A} right]_{a,m0}
left [{bold nitalic A} (%theta_1) right]
\(
newline
&quot;                       &quot;
{bold nitalic r}_{p1m1,p1m2,p1m1}
+
left [{bold nitalic A} right]_{p1m1,p1m2}
left [{bold nitalic A} (%theta_2) right]
\(
newline
&quot;                              &quot;
{bold nitalic r}_{p2m1,p2m2,p2m1}
+
left [{bold nitalic A} right]_{p2m1,p2m2}
left [{bold nitalic A} (%theta_3) right]
\(
newline
&quot;                                      &quot;
{bold nitalic r}_{p3m1,p3m2,p3m1}
+
left [{bold nitalic A} right]_{p3m1,p3m2}
left [{bold nitalic A} (%theta_4) right]
\(
newline
&quot;                                              &quot;
{bold nitalic r}_{p4m1,p4m2,p4m1}
+
left [{bold nitalic A} right]_{p4m1,p4m2}
left [{bold nitalic A} (%theta_5) right]
\(
newline
&quot;                                                      &quot;
{bold nitalic r}_{p5m1,p5m2,p5m1}
+
left [{bold nitalic A} right]_{p5m1,p5m2}
left [{bold nitalic A} (%theta_6) right]
\(
newline
&quot;                                                              &quot;
{bold nitalic r}_{p6m1,p6m2,p6m1}
\)
\)
\)
\)
\)
\)</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Finding Robot Parameters</text:h>
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   <text:p text:style-name="Standard">Minimize</text:p>
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   <text:p text:style-name="Standard">Taking partial derivatives and setting them to zeros:</text:p>
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   <text:p text:style-name="Standard">Apply Newton Raphson</text:p>
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