Combine individually working cartesian coordinates

Question

I am trying to control a Dobot arm. The arm moves with angles whereas I need to work with cartesian coordinates. From inverse kinematics equations and polar coordinates I have implemented x,y and z coordinates working very well on their own. But I can not combine the coordinates in order to work all together. When I add them up it is not going to the desired place. How can I combine these coordinates? I got some help from (https://github.com/maxosprojects/open-dobot) but could not manage to successfully move dobot.

Edit: I've written the codes in Qt and also I've added the triangles used for angle calculations.

//foreArmLength=160mm rearArmLEngth=135mm

float DobotInverseKinematics::gotoX(float x) //func for x-axis
float h=qSqrt(qPow(lengthRearArm,2)-qPow(x,2)); //height from ground
QList<float> zEffect=gotoZ(h); //trying to find the effect of x movement on z-axis
float cosQ=h/lengthRearArm; //desired joint angle
float joint2=qRadiansToDegrees(qAcos(cosQ));    
//move in range control
if(joint2 != joint2)
{joint2=0;
    qDebug()<< "joint2NAN";}
return joint2;

QList<float> DobotInverseKinematics::gotoY(float y) //func for y-axis
QList<float> result ;
float actualDist=lengthForeArm+distToTool; //distance to the end effector
float x=(qSqrt(qPow(actualDist,2)+qPow(y,2)))-actualDist; //calculating x movement caused by y movement
float joint1=qRadiansToDegrees(qAcos(actualDist/(actualDist+x))); //desired joint angle
float joint2=gotoX(x);  //the angle calculation of y movement on x axis
if(joint1 != joint1)
{joint1=0;
    qDebug()<< "joint1NAN";}
result.append(joint1);
result.append(joint2);
return result;

QList<float> DobotInverseKinematics::gotoZ(float z) //func. for z-axis
QList<float> result ;
float joint3=qSqrt(qPow(160,2.0)-qPow(z,2.0))/ 160; //desired joint angle
float temp=160-qSqrt(qPow(160,2.0)-qPow(z,2.0));
float joint2=qSqrt(qPow(lengthRearArm,2)-qPow(temp,2.0))/lengthRearArm; //desired joint angle
if(joint3 != joint3)
{joint3=0;
    qDebug()<< "joint3NAN";}
joint2=qAcos(joint2)*(180/M_PI);
joint3=qAcos(joint3)*(180/M_PI);
result.append(joint2);
result.append(joint3);
return result;

Answer 1

$a_1, a_2$ : Link lengths

$p_x,p_y$ : Target $x$ and $y$ coorditate

$\theta_1=atan2(p_y,p_x)$

$p_r=\sqrt{p_x^2+p_y^2}$ : Target radial distance

$p_z$ : Target $z$ coorditate

$\kappa=\frac{p_r^2+p_z^2-a_1^2-a_2^2}{2a_1a_2}$

If $\kappa>1$ the position cannot be reached.

$\theta_3=cos^{-1}(\kappa)$

There are 2 solutions, elbow up or elbow down. Below use $\theta_3^*=\theta_3$ or $\theta_3^*=-\theta_3$

$\Delta=a_1^2+a_2^2+2a_1a_2cos(\theta_3^*)$

$c\theta_2=\frac{p_r(a_1+a_2cos(\theta_3^*))+p_za_2sin(\theta_3)}{\Delta}$

$s\theta_2=\frac{p_z(a_1+a_2cos(\theta_3^*))-p_ra_2sin(\theta_3)}{\Delta}$

$\theta_2=atan2(s\theta_2,c\theta_2)$

These are joint angles. To get motor angles, you will have to adjust the angles based on how the motors are connected to the robot. $\theta$ is 0 when horizontal, $\theta_3$ is 0 when straight (in-line) with Link 2.

A detailed explanation can be found in Robot Analysis by Tsai. ISBN 9780471325932 pp 71