# Dynamics Notes

This will be the localized place for all future dynamics notes, however, much of my dynamics notes are currently located on the notes page for my analytical dynamics class.

## Table of Contents

## Equations of Motion

Cannonical Form Given in Featherstone's Book

- Where:

- H is the generalized inertia matrix
- C is the generalized bias force
- q is the generalized coordinate and its derivatives

Unlabeled Form in Kanes Book

- where

- are undefined functions of time
- Sentence in book "set of v first-order differential equations of the form". Doesn't give name to the form.

Forms Labeled in Robinovitch's Slides

- inertia (or mass) matrix
- net muscle moment vector
- vector of torques due to centripetal and coriolis forces
- vector of joint torques due to gravity
- Equations appear to be more centered on rotational systems

Forward Dynamics

Inverse Dynamics

## Mass Matrix

The formal definition for a mass matrix M is

where

- is the kinetic energy of the system
- is first time derivative of the generalized coordinate vector q

## Nonholonomic system

A nonholonomic system is a system whose state depends on the path taken in order to achieve it. (Path dependent system)

A constraint which does not reduce the degrees of freedom (Knowing one generalized coordinate and the constraint allows solving for the other coordinate is a holonomic system, ie two points connected by a massless rod)

## Sources

- Dynamics: Theory and Applications (Thomas Kane and David Levinson)
- Dynamics pt. 2: Equations of motion and system identification (Stephen Robinovitch)
- Mass Matrix, Wikipedia
- Nonholonomic systems, Wikipedia
- Nonholonomic Constraints
- Rigid Body Dynamics Algorithms (Roy Featherstone)