"jansen-linkage" is a simulator of a linkage mechanism with gnuplot.

Let

$$ \begin{array}{lll} a=38.0 & b=41.5 & c=39.3 & d=40.1 & e=55.8\\ f=39.4 & g=36.7 & h=65.7 & i=49.0 & j=50.0\\ k=61.9 & l=7.8 & m=15.0 & & \\ \end{array} $$

These numbers are called "Holy Numbers". Then

$$ \begin{eqnarray*} \overrightarrow{\mathrm{OA}} &=& \begin{bmatrix}m \cos\theta\\ m\sin\theta \end{bmatrix}\\ \overrightarrow{\mathrm{OB}} &=& \begin{bmatrix} -a\\ -l \end{bmatrix}\\ \overrightarrow{\mathrm{OC}} &=& \overrightarrow{\mathrm{OB}} +\overrightarrow{\mathrm{BC}} &=& \left[ \begin{array}{c} -a + b\cos(\alpha+\beta)\\ -l + b\sin(\alpha+\beta) \end{array} \right] \end{eqnarray*} $$

where

$$ \begin{eqnarray*} \alpha &=& \arctan2\left(\mathrm{A}_y-\mathrm{B}_y, \mathrm{A}_x-\mathrm{B}_x\right)\\ \beta &=& \arccos\left(\frac{\mathrm{AB}^{2}+b^{2}-j^{2}}{2\cdot \mathrm{AB}\cdot b}\right) \end{eqnarray*} $$

$$ \begin{eqnarray*} \overrightarrow{\mathrm{OD}} &=& \overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{BD}}&=&\left[ \begin{array}{c} -a + b\cos(\alpha+\beta+\gamma)\\ -l + b\sin(\alpha+\beta+\gamma) \end{array} \right]\\ \overrightarrow{\mathrm{OE}} &=& \overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{BE}}&=&\left[ \begin{array}{c} -a + c\cos(\alpha-\delta)\\ -l + c\sin(\alpha-\delta) \end{array} \right] \end{eqnarray*} $$ where $$ \begin{eqnarray*} \gamma &=& \arccos\left(\frac{b^{2}+d^{2}-e^{2}}{2bd}\right)\\ \delta &=& \arccos\left(\frac{\mathrm{AB}^{2}+c^{2}-k^{2}}{2 \cdot \mathrm{AB} \cdot c}\right) \end{eqnarray*} $$

$$ \begin{eqnarray*} \overrightarrow{\mathrm{OF}} &=& \overrightarrow{\mathrm{OE}}+\overrightarrow{\mathrm{EF}}\nonumber\\ &=&\left[\begin{array}{c} -a + c\cos(\alpha-\delta)+g\cos(\varepsilon+\zeta)\\ -l + c\sin(\alpha-\delta)+g\sin(\varepsilon+\zeta) \end{array} \right]\\ \overrightarrow{\mathrm{OG}} &=&\overrightarrow{\mathrm{OE}}+\overrightarrow{\mathrm{EG}}\nonumber\\ &=&\left[ \begin{array}{c} -a + c\cos(\alpha-\delta)+i\cos(\varepsilon+\zeta+\eta)\\ -l + c\sin(\alpha-\delta)+i\sin(\varepsilon+\zeta+\eta) \end{array} \right] \end{eqnarray*} $$ where $$ \begin{eqnarray*} \varepsilon &=& \arctan2\left(\mathrm{D}_y-\mathrm{E}_y, \mathrm{D}_x-\mathrm{E}_x\right)\\ \zeta &=& \arccos\left(\frac{\mathrm{DE}^{2}+g^{2}-f^{2}}{2\cdot \mathrm{DE} \cdot g}\right)\\ \eta &=& \arccos\left(\frac{g^{2}+i^{2}-h^{2}}{2gi}\right) \end{eqnarray*} $$

Changing the number of legs, you can increase the number of robot legs.

LEG_SET_NUM = 6

Altering the setting of plot area and disappearing coordinate plane, 6-legged robot appears to walk.

unset xlabel # or set xlabel "..." tc(=textcolor) rub 'white' 
unset ylabel # or set ylabel "..." tc rub 'white'
unset tics # or set tics tc rub 'white'
unset grid # or set grid lc rub 'white'
unset border # or set border lc rub 'white'

If you make gradual color change, you use the following code.

WHITEN_VAL = int(180./LEG_SET_NUM) # If numerator is not 180 but 255, the last pair of legs turns too white.
color = 0x000000 + (WHITEN_VAL*k << 16)+(WHITEN_VAL*k << 8)+(WHITEN_VAL*k << 0) # If base color is black
color = 0xff0000 + (WHITEN_VAL*k << 8)+(WHITEN_VAL*k << 0) # If base color is red

On the other hand, if you specify the color used for each leg, you use the following code. (k=0 : Red, k=1 : Green, k=2 : Blue)

color = (k==0 ? 0xff0000 : (k==1 ? 0x00ff00 : 0x0000ff))

When transparency is used in png terminal, the "truecolor" option must be selected [3]. Then you use transparent color if you represent color by 32 bits AARRGGBB [4]. So, I modified the code as follows.

set term png truecolor enhanced size 960, 720 # truecolor is important!
CIRC_R = 4 # Largen radius
color = (j==0 ? 0x80ff3434 : (j==1 ? 0x80759dff : 0x8056bf56)) # 0x80RRGGBB = 50% transparency, rgb RRGGBB
plot_command = plot_command.sprintf(", outputfile u %d:%d every ::%d::%d w l lw LW*0.7 lc rgb 'gray30'", 16+2*k, 17+2*k, start, end) # Change color

Thanks to truecolor, all of the circles representing joint $\mathrm{A}$ are transparent as shown in the below figure. However, arrows drawn without a head are not transparent. So, if you want to draw transparent not only circles but also lines, I suggest you to set the qt terminal. Please note that you can't save image by using command 'set output' in qt terminal.

Terminal type : 'png'	Terminal type : 'qt' (in MacOS)

• Hiro Shigeyoshi
• Twitter: https://twitter.com/hiroloquy

http://www.youtube.com/watch?v=rSbgDpycorc

"jansen-linkage" is under MIT license.