How to find Theta 1 and theta 2 inverse of the two line connecting

Question

I'm doing the two axis arm robotics on the canvas which shown in the picture as red to blue circle is theta 1 and than blue to yellow is theta 2 here i need to find joint pair of them which is inverse as green and cyan circle is movable so i set according to inverse theta but seems not able to find inverse theta of them correctly here angle are theta1 = 80 theta2 = 50

and my math is bad so can't figure out how can i find their inverse i found the angle using the axis of lines but it seems to be wrong i guess these angle calculated using this function

function angle(cx, cy, ex, ey) {
    var dy = ey - cy;
    var dx = ex - cx;
    var theta = Math.atan2(dy, dx); // range (-PI, PI]
    theta *= 180 / Math.PI; // rads to degs, range (-180, 180]
    //if (theta < 0) theta = 360 + theta; // range [0, 360)
    return -theta;
  }

A1 : 80 & A2 : 50
A1' : 47.45904399010462 & A2' : 82.29636680333454

IMG

i.e. I made cyan circle move a bit on purpose to make yellow visible

red is base
blue,green is shoulder
yellow,cyan is tooltip

Answer 1

Let's consider the following simple robot, with all four arms the same length $L$, angle $\theta$ between $x$ axis and blue arm $O A$, angle $\phi$ between the blue arm $O A$ and the red arm $O B$, with $O$ at origin, and the robot tip at $C = (x, y)$:

Note the right triangles end-to-end. The angle at points $O$ and $C$ is $\phi/2$, and both hypotenuses are $L$. Therefore, $$\left\lVert O C \right\rVert = \sqrt{x^2 + y^2} = 2 L \cos\left(\frac{\phi}{2}\right)$$ Solving for $\phi$ yields $$\phi = 2 \operatorname{acos}\left(\frac{\sqrt{x^2 + y^2}}{2 L}\right) \tag{1}\label{1}$$
We can also see that the angle between the $x$ axis and $O C$ is $\theta + \phi/2$: $$\operatorname{atan2}(y, x) = \theta + \frac{\phi}{2}$$ solving for $\theta$ yields $$\theta = \operatorname{atan2}(y, x) - \frac{\phi}{2} \tag{2}\label{2}$$

In Python, you could use

from math import acos, atan2, sqrt, pi

two_pi = pi+pi

def robot_angles(x, y, L):
    half_phi = acos(sqrt(x*x + y*y) / (L + L))  # 0 <= half_phi <= 0.5pi
    theta = atan2(y, x) - half_phi              # -1.5pi < theta <= pi
    if theta <= -pi:
        theta += two_pi
    return (theta, half_phi+half_phi)           # -pi < theta <= pi

which returns the tuple $(\theta, \phi)$ in radians.

The inverse is e.g.

from math import sin, cos

def robot_tip(theta, phi, L):
    return ( L*cos(theta) + L*cos(theta + phi),
             L*sin(theta) + L*sin(theta + phi) )

which returns the tuple $(x, y)$.