Delta Parallel Robot — Building a Kinematics Library from Scratch | Notes / Kinematics

Recently I finished writing the kinematics library for a Delta parallel robot. Let me record the process and key data.

Mechanical Parameters

Parameter	Value	Description
Rb	115 mm	Static platform radius (center to motor shaft)
Re	75 mm	Moving platform radius (center to ball joint, 150/2)
L1	125 mm	Driving arm (big arm) length
L2	338 mm	Passive arm (small arm/rod) length

Core Kinematics

Inverse Kinematics (IK)

Given end-effector position (x, y, z), find three motor angles θ₁, θ₂, θ₃.

For each leg, transform the problem to a local coordinate system, simplifying to planar 2R linkage inverse kinematics:

x_local·sin(θ) - z_local·cos(θ) = (d² + L1² - L2²) / (2·L1)

Solution:

θ = φ + π/2 + α

where φ = atan2(z_local, x_local), α = acos((L1² + d² - L2²) / (2·L1·d)).

Key detail: The projection of the passive arm outside the rotation plane needs to be processed as effective length L2_eff = sqrt(L2² - y_local²).

Forward Kinematics (FK)

Given θ₁, θ₂, θ₃, find end-effector position. Use Newton iteration to solve for the three-sphere intersection point, converging to error < 1e-6 mm.

MATLAB Verification Results

Verified IK→FK consistency in MATLAB, error is 0.0000 mm at all test points.

Workspace

Scanned with 20mm grid, total 5928 reachable points:

Z (mm)	Max Radius (mm)
-460	14
-440	103
-400	192
-360	247
-300 ~ -200	247
-100	247

The workspace is dome-shaped, Z range approximately -460 ~ -100 mm, maximum horizontal radius approximately 250mm.

Code Organization

deltarobot/
├── inc/
│   ├── delta_common.h        → vectors, angles, utilities
│   ├── delta_param.h         → mechanical parameter struct
│   ├── delta_kinematics.h    → IK/FK/Jacobian
│   ├── delta_trajectory.h    → trajectory planning (interpolation + S-curve)
│   └── delta_control.h       → three-axis coordinated control interface
├── src/
│   ├── delta_kinematics.cpp
│   ├── delta_trajectory.cpp
│   └── delta_control.cpp
├── scripts/
│   ├── verify_delta.m         → kinematics verification + workspace calculation
│   └── workspace.m            → workspace visualization
└── README.md

Next Steps

Run IK calculation on STM32F407
Cooperate with stepper motors DRV8825/TMC2209 for open-loop position control
Later add encoder for closed-loop