This project is being done by me individually under the Guidance of Dr. Arshad Javed from the department of Mechanical Engineering in BITS-Pilani, Hyderabad Campus.

Materials I've used:

• Neema 17 stepper motors (3)
• 3D printed Lower arms (3)
• Carbon fibre upper arms (6)
• 3D printed end-effector
• L298N motor driver shields (3)
• Ball joints (12)
• Arduino Uno
• Jumper wires
• Bread-board

Delta robot is a type of parallel robot. Parallel robots have certain advantages over serial ones (like a robotic arm):

1. higher stiffness
2. strong bearing capacities
3. high precesion
4. fast pick-ups

Delta robots are usually ==3-DOF parallel robots==. They have the following parts:

1. 3 equal arms
2. 1 base (fixed)
3. 1 end-effector (moving) The arms are usually attatched at 120 degrees of each other. The end-effector is attatched to the arms witha revolute joint. Due to it's parallelogram like nature, the delta robot is unable to rotate, only translate.

Delta robots are used for:

1. pick-and-place operations.

We need to find the 3D coordinate: $P[x y z]$ by knowing the $\theta_{i}$'s of the arms (typical FK) The 3D positional coordinates of B:

$$ B_i=\begin{bmatrix}R+L_1cos\theta_{i} \ 0 \ L_1sin \ theta_{i} \end{bmatrix}----(1) $$

the rotation matrix (transformation matrix)-- about Z

$$ R=\begin{bmatrix}cos\alpha & -sin\alpha & 0\ sin\alpha & cos\alpha & 0\ 0&0&1\end{bmatrix} $$

The installation angles of the 3 arms are:

$$ \alpha_{1}=0, \alpha_{2}=\frac{2\pi}{3}, \alpha_{2}=\frac{4\pi}{3} $$

using pythagoras theorem we can write the following:

$$ \begin{gather*} (x-cos\alpha_1[(R-r)+L_{1}cos\theta_1])^2+(y-sin\alpha_1[(R-r)+L_{1}sin\theta_1])^2+(z-L_1sin\theta_1)^2=L_2^2----(2.1)\\ and\\ (x-cos\alpha_2[(R-r)+L_{1}cos\theta_2])^2+(y-sin\alpha_2[(R-r)+L_{1}sin\theta_2])^2+(z-L_1sin\theta_2)^2=L_2^2----(2.2)\\ and\\ (x-cos\alpha_3[(R-r)+L_{1}cos\theta_3])^2+(y-sin\alpha_3[(R-r)+L_{1}sin\theta_3])^2+(z-L_1sin\theta_3)^2=L_2^2----(2.3)\\ \end{gather*} $$

in a simpler manner we can write something of the sorts of:

$$ (x-k_{ii})+(y-k_{ij})+(z-k_{ik})=L_2^2----(3) $$

where $k$ will be a matrix containing the simplified terms of the equations (2.1-2.3)

$$ k=\begin{bmatrix} (R-r)+L_1cos\theta_{1}&0&-L_1sin\theta_1\\ 0.5[(R-r)+L_1cos\theta_{2}]&{\frac{-\sqrt{3}}{2}[(R-r)+L_1cos\theta_{2}]}&-L_1sin\theta_2\\ 0.5[(R-r)+L_1cos\theta_{3}]&{\frac{\sqrt{3}}{2}[(R-r)+L_1cos\theta_{3}]}&-L_1sin\theta_3 \end{bmatrix} $$

Eqn (1) is substracted from eqns (2) and (3) (==why??==) and then merged and [[MATRIX DECOMPOSITION]].

$$ \begin{gather*} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2----(4) \end{gather*} $$

This in turn can also be written as:

$$ \begin{bmatrix} a_{1}&b_{1} \ a_{2}&b_2 \end{bmatrix} \begin{bmatrix} x\ y \end{bmatrix} = \begin{bmatrix}d_1-c_1z \ d_2-c_2z \end{bmatrix}----(5) $$

Solve this linear equation for x and y.

$$ \begin{gather*} x=f_1+f_xz \\ y=f2+f_yz----(6) \end{gather*} $$

For a given delta robot, the vector loop closure equations are:

$$ {B^B_i}+L^B_i+I^B_i=P^B_P+[R^B_P]P^P_i=P^B_P+P^B_i $$

where:

$R = I_3$,
$B_i =$ the revolute joints of the fixed base,
$P_i=$ connection joints (vertices of the equilateral triangle of the base and end-effector

The basic IPK solution for the 3 legs are modelled as:

$E_i { cos \theta_i }+ F_i sin \theta_i + G_i=0$

where:

$$ \begin{gather*} E_1=2L(y+a) \\ F_1=2zL \\ G_1=x^2+y^2+z^2+a^2+L^2+2ya-l^2 \\ E_2=-L(\sqrt3(x+b)+y+c) \\ F_2=2zL \\ G_2=x^2+y^2+z^2+b^2+c^2+L^2+2(xb+yc)-l^2 \\ E_3=L(\sqrt3(x+b)-y-c) \\ F_3=2zL \\ G_3=x^2+y^2+z^2+b^2+c^2+L^2+2(-xb+yc)-l^2 \\ \end{gather*} $$

then we put these individually in the equation and then finally solve using tangent half angle substitution.

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