Geometric Robot Kinematics

Beyond D-H Parameters with Screw Theory

L01_ADV_KIN

ROBOTICS_RL_CORE

The Hook

Why is the mathematics of a screw essential for describing the motion of a 7-degree-of-freedom robotic arm?

D-H Limitations

Coordinate frames can be ambiguous and singular.

Geometric Intuition

Every rigid motion is a helical displacement.

AI Readiness

Global coordinate-free state representations.

Chasles' Theorem

"Every rigid body displacement can be represented by a screw motion: a rotation about an axis followed by a translation along that same axis."

1

Screw Axis \(\mathcal{S} = (\omega, v)\)

2

Screw Pitch \(h\)

3

Displacement Magnitude \(\theta\)

AXIS OF MOTION

Mathematical Abstraction

Rigid Body Transformation SE(3)

The Exponential Map

Transformation Matrix

\[ T = e^{[\mathcal{S}]\theta} \]

Where \([\mathcal{S}]\) is the matrix representation of the screw in \(se(3)\).

For Rotation:

Rodrigues' Formula maps \(\mathbb{R}^3 \to SO(3)\)

\[ R = e^{[\hat{\omega}]\theta} = I + \sin\theta [\hat{\omega}] + (1-\cos\theta)[\hat{\omega}]^2 \]

For General Motion:

Maps \(se(3) \to SE(3)\)

Includes rotation and the complex coupling of translation.

Product of Exponentials (PoE)

Forget the recursive frame assignments of D-H. With PoE, we define everything relative to the fixed space frame or the end-effector body frame.

Space Frame FK

\[ T(\theta) = e^{[\mathcal{S}_1]\theta_1} \dots e^{[\mathcal{S}_n]\theta_n} M \]

Only 2 Frames Required

Fixed base frame and the end-effector frame at home.

Global Geometric Clarity

Screw axes \(\mathcal{S}_i\) are defined in the home configuration.

Analytic Jacobians

Easily derived by differentiating the PoE formula.

Redundancy & AI States

AI State Definitions

RL agents need robust state spaces. Screw coordinates (\(S, \theta\)) provide a cleaner, more continuous mapping than Euler-based D-H parameters.

Avoiding Gimbal Lock
Unitary representations of poses
Easier gradient flow for policies

The Redundancy Paradox

For a 7-DOF arm, IK is under-constrained. Classical robotics uses null-space projection.

\(\dot{\theta} = J^\dagger \mathcal{V} + (I - J^\dagger J) \phi\)

RL "learns" \(\phi\) implicitly through rewards.

Workshop Time

Now, let's derive the screw axes for a 3-DOF planar system and generalize to the Panda 7-DOF arm.

Derive

Calculate

Document

Robotic Dynamics and RL Control

Advanced Kinematics and Screw Theory

Dynamics Formulation for Control

Introduction to RL in Robotics

Training Agents in Simulators

Adaptive Control and Path Planning

Screw Theory Workshop Document

Screw Theory Kinematics Slides

Dynamics Derivation Worksheet

Robotic RL Fundamentals Slides

MDP Mapping Activity Worksheet

Simulator Training Slides

Adaptive Control Comparison Slides

Control Benchmark Guide

Robotic RL Grading Rubric

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