## Delta robot kinematic

Hi

// robot geometry
// (look at pics above for explanation)
const float e = 115.0; // end effector
const float f = 457.3; // base
const float re = 232.0;
const float rf = 112.0;

// trigonometric constants
const float sqrt3 = sqrt(3.0);
const float pi = 3.141592653; // PI
const float sin120 = sqrt3/2.0;
const float cos120 = -0.5;
const float tan60 = sqrt3;
const float sin30 = 0.5;
const float tan30 = 1/sqrt3;

// forward kinematics: (theta1, theta2, theta3) -> (x0, y0, z0)
// returned status: 0=OK, -1=non-existing position
int delta_calcForward(float theta1, float theta2, float theta3, float &x0, float &y0, float &z0) {
float t = (f-e)*tan30/2;
float dtr = pi/(float)180.0;

theta1 *= dtr;
theta2 *= dtr;
theta3 *= dtr;

float y1 = -(t + rf*cos(theta1));
float z1 = -rf*sin(theta1);

float y2 = (t + rf*cos(theta2))*sin30;
float x2 = y2*tan60;
float z2 = -rf*sin(theta2);

float y3 = (t + rf*cos(theta3))*sin30;
float x3 = -y3*tan60;
float z3 = -rf*sin(theta3);

float dnm = (y2-y1)*x3-(y3-y1)*x2;

float w1 = y1*y1 + z1*z1;
float w2 = x2*x2 + y2*y2 + z2*z2;
float w3 = x3*x3 + y3*y3 + z3*z3;

// x = (a1*z + b1)/dnm
float a1 = (z2-z1)*(y3-y1)-(z3-z1)*(y2-y1);
float b1 = -((w2-w1)*(y3-y1)-(w3-w1)*(y2-y1))/2.0;

// y = (a2*z + b2)/dnm;
float a2 = -(z2-z1)*x3+(z3-z1)*x2;
float b2 = ((w2-w1)*x3 - (w3-w1)*x2)/2.0;

// a*z^2 + b*z + c = 0
float a = a1*a1 + a2*a2 + dnm*dnm;
float b = 2*(a1*b1 + a2*(b2-y1*dnm) - z1*dnm*dnm);
float c = (b2-y1*dnm)*(b2-y1*dnm) + b1*b1 + dnm*dnm*(z1*z1 - re*re);

// discriminant
float d = b*b - (float)4.0*a*c;
if (d < 0) return -1; // non-existing point

z0 = -(float)0.5*(b+sqrt(d))/a;
x0 = (a1*z0 + b1)/dnm;
y0 = (a2*z0 + b2)/dnm;
return 0;
}

// inverse kinematics
// helper functions, calculates angle theta1 (for YZ-pane)
int delta_calcAngleYZ(float x0, float y0, float z0, float &theta) {
float y1 = -0.5 * 0.57735 * f; // f/2 * tg 30
y0 -= 0.5 * 0.57735 * e; // shift center to edge
// z = a + b*y
float a = (x0*x0 + y0*y0 + z0*z0 +rf*rf - re*re - y1*y1)/(2*z0);
float b = (y1-y0)/z0;
// discriminant
float d = -(a+b*y1)*(a+b*y1)+rf*(b*b*rf+rf);
if (d < 0) return -1; // non-existing point
float yj = (y1 - a*b - sqrt(d))/(b*b + 1); // choosing outer point
float zj = a + b*yj;
theta = 180.0*atan(-zj/(y1 - yj))/pi + ((yj>y1)?180.0:0.0);
return 0;
}

// inverse kinematics: (x0, y0, z0) -> (theta1, theta2, theta3)
// returned status: 0=OK, -1=non-existing position
int delta_calcInverse(float x0, float y0, float z0, float &theta1, float &theta2, float &theta3) {
theta1 = theta2 = theta3 = 0;
int status = delta_calcAngleYZ(x0, y0, z0, theta1);
if (status == 0) status = delta_calcAngleYZ(x0*cos120 + y0*sin120, y0*cos120-x0*sin120, z0, theta2); // rotate coords to +120 deg
if (status == 0) status = delta_calcAngleYZ(x0*cos120 - y0*sin120, y0*cos120+x0*sin120, z0, theta3); // rotate coords to -120 deg
return status;
}