Delta robot kinematics, help with understanding the equations

[urban sandal]

I have had a problem with some texts I’ve been reading, I need some explanation on the equations and logic behind, if someone could help:
(Part of an AI resume)
Let's work with arm 1 first (the one aligned with x) to make the math clearer. The joint position on the base for arm 1 is at B₁ = (Rb, 0, 0). The corresponding position on the moving platform is at P₁ = (x + Rp, y, z). Now comes the trick: you'll be working in a 2D coordinate system in the vertical plane of arm 1. In this plane, the horizontal distance from the base joint point to the projection point on the platform is E₁ = √[(x + Rp - Rb)² + y²], and the vertical distance is simply z.

[urban sandal]

It refers to the one aligned with X cause there’s 3 arms but one goes through the tip of the triangle,which is aligned with the triangles tip and the x coordinate

[urban sandal]

Before diving into the equations, you need to define your coordinate system. We typically place the origin of the Cartesian coordinate system (x, y, z) at the center of the upper triangular base, with the z-axis pointing downwards. The triangular base has a radius Rb (from the center to each vertex), and the moving platform has a radius Rp, which is usually smaller. Each of the three arms is positioned symmetrically at 120 degrees to each other around the z-axis. If you number the arms 1, 2, and 3, arm 1 might be aligned with the x-axis, arm 2 at 120°, and arm 3 at 240°.

X
Y
Z
Rb
Arm 1 (0°)
Arm 2 (120°)
Arm 3 (240°)
120°
Figure 2: Coordinate system with origin at the center of the base

Base Parameters

Rb: Radius of the triangular base (center to vertex)

L: Length of the upper arm

Platform Parameters

Rp: Radius of the mobile platform

l: Length of the lower arm

Each arm has two segments: the upper arm of length L (connected to the motor on the base) and the lower arm of length l (connected to the end effector). The upper arm rotates in only one direction, creating that arc-shaped movement you see when the robot is working. The junction between the two segments is the robot's "elbow," and this part is super important because it's where things get mathematically interesting. The lower arms are usually composed of parallelograms or double rods to keep the platform always parallel to the base, but for basic kinematic analysis, we can simplify by thinking of them as rigid segments of length l.

[weak flicker]

What exactly do you need clarification with? Just wondering about the general area of confusion!

I think the only nontrivial equation is the horizontal distance - clearly, that is using the Pythagorean theorem for distance between B1 and P1.

[urban sandal]

Can you explain the rb and rp calculus

[weak flicker]

Sorry, do you mean what they are? They're the radii of the base and of the arm.

Since the arm is aligned with the x axis, and the base is at the origin, there's only the radius "between" the joint position and the origin. So you get 0,0,0, but x += Rb effectively.

Similarly, the moving platform is at x,y,z, and the arm is aligned with x axis, so you just add the radius of the shoulder along x axis (Rp).

Is that what you were asking? Being clearer would really help.

[urban sandal]

I wanna know how the radii intersect, if you could make like a simple drawing with lines i would understand,i cant quite make out the intersections in my head, and what is Rp exactly, is it how much it's tilted in Z axis compared to the x axis?

[weak flicker]

@urban sandal sorry, to be more precise I guess I'll need to have some more context about what the 'moving platform' is. I don't have a background in robotics, so I don't know offhand!

[urban sandal]

my bad

I can send more stuff

[weak flicker]

No worries!