Deriving Analytical Inverse Kinematics of 6R Robots with Three Parallel Joints with AI Agents

This paper presents analytical closed-form inverse kinematics (IK) solvers for 6R robots with parallel joints via prompting large language models (LLMs). Current IK solvers face a fundamental trade-off: general-purpose algebraic methods become intractable for robots with special architectures, while specialized solvers impose restrictive assumptions such as zero link offsets or fixed twist angles, limiting applicability to real-world robots with arbitrary Denavit-Hartenberg parameters. We develop production-quality analytical solvers tailored to the parallel joint architecture while remaining general across all other kinemaatic parameters using a two-stage LLM-assisted development workflow. In the first stage, we leverage LLM prompting to develop robust helper functions for solving four canonical trigonometric equation systems with degeneracy handling. In the second stage, we inject expert guidance through structured prompts to direct the LLM in deriving and implementing the full IK solver, which strategically decouples the 6R problem into two sub problems: (1) bilinear trigonometric system solving for joints 1 and 6, and (2) planar 3R kinematics for the parallel joints. Validation across 119 real industrial collaborative robots and 100 randomly generated configurations demonstrates 100% success rates with low millisecond solve times and numerical errors below 10^-9, confirming suitability for real-time robotic control.