Kinematics and rigid transformations

Abstract¶

Robotics is about producing motion. In this lecture, we dive into the mathematical representation of robots as articulated systems of rigid bodies, and define formally what motion means. We start by defining the concepts of translation, rotation, rigid transformation and velocities in 2D, where these operations naturally define the groups SO(2) and SE(2). From there, we move to 3D with the groups SO(3) and SE(3), equipping ourselves with rotation matrices, quaternions, exponential and logarithm maps. These groups give us a principled language to define forward kinematics, how a robot moves in the world given its joint angular motions, and inverse kinematics, the inverse problem of figuring out a robot motion that achieves some targets in the world. As a bonus, for those interested in digging deeper, we show how these Lie-group tools extend beyond single-frame geometry to higher-dimensional groups that couple frames, body velocities, and sensor biases, revealing hidden linear structure in major problems that appear nonlinear at first sight.

Content¶

Slides

References¶

	A Mathematical Introduction to Robotic Manipulation. Richard M. Murray, Zexiang Li, S. Shankar Sastry. CRC Press, 1994.
	Representing attitude: Euler angles, unit quaternions, and rotation vectors. James Diebel. Stanford University, 2006.

Lectures¶

This lecture was given in the following courses:

• Fall 2025 class at Master MVA, Paris.