*Published in the IEEE Robotics and Automation Magazine*

Authors: Jesse Haviland Peter Corke

# Abstract

Kinematics, derived from the Greek word for motion, is the branch of mechanics that studies the motion of a body, or a system of bodies, without considering mass or force. This two-part tutorial is about the kinematics of robot manipulators, and in that context, it is concerned with the relationship between the position of the robot's joints and the pose of its end effector as well as the relationships between various derivatives of those quantities. Kinematics is a fundamental concept in the study or application of robot manipulators, and our audience for Part I is students, practitioners, or researchers encountering this topic for the first time or looking for a concise refresher.

# Synopsis

Manipulator kinematics is concerned with the motion of each link within a manipulator without considering mass or force. A serial-link manipulator, which we refer to as a manipulator, is the formal name for a robot that comprises a chain of rigid links and joints, it may contain branches, but it can not have closed loops. Each joint provides one degree of freedom, which may be a prismatic joint providing translational freedom or a revolute joint providing rotational freedom. The base frame of a manipulator represents the reference frame of the first link in the chain, while the last link is known as the end-effector.

In Part I of the two-part Tutorial, we provide an introduction to modelling manipulator kinematics using the elementary transform sequence (ETS). Then we formulate the first-order differential kinematics, which leads to the manipulator Jacobian, which is the basis for velocity control and inverse kinematics. We describe essential classical techniques which rely on the manipulator Jacobian before exhibiting some contemporary applications.

In Part II of the Tutorial, we formulate the second-order differential kinematics, leading to a definition of manipulator Hessian. We then describe the differential kinematics' analytical forms, which are essential to dynamics applications. Subsequently, we provide a general formula for higher-order derivatives. The first application we consider is advanced velocity control. In this Section, we extend resolved-rate motion control to perform sub-tasks while still achieving the goal before redefining the algorithm as a quadratic program to enable greater flexibility and additional constraints. We then take another look at numerical inverse kinematics with an emphasis on adding constraints. Finally, we analyse how the manipulator Hessian can help to escape singularities.

# Citation

```
@article{haviland2023dkt1,
author={Haviland, Jesse and Corke, Peter},
title={Manipulator Differential Kinematics: Part I: Kinematics, Velocity, and Applications},
journal={IEEE Robotics \& Automation Magazine},
year={2023},
pages={2-12},
doi={10.1109/MRA.2023.3270228}
}
```