This is a general purpose dynamics library and tutorial.
Dynamics describe any process and how systems change over time. The purpose of this library is to give a in depth description and set of tools to deal with these dynamic systems and how to decompose and analyze them. The cheif practical usage of this library is to give a set of tools to analyze systems (whether it be through a model or a set of data) and produce a controller that would allow the system to reach any desired point in the state space.
The general formula for continuous time dynamics is below:
This function
Transformation. Transformations can take a number of different forms.
The following sections will describe the set of possibles models for your system and its change of
state
Generally a linear system can be described by the following equation:
The dynamics of the system are here modeled by the linear transformation A, where A is a matrix that
has the same rank as x(t).
This library provides functionality to establish any linear transformation you would like. The perferred interfaces are "Eigen" matrices.
Nonlinear dynamics are the core of this library and its tools. Non linear dyanmics are defined by systems in the form:
Where f() is an arbitrary function.
Control is broadly used to change the dynamics of a system and produce a desired state of the system. Depending on the system, you may have more or less control of the dynamics of the system. The distinction here is made for two such systems:
This form of system is broadly used in robotics and other communities. Fully actuated robotic systems are capable of controlling all aspects of the state space. For instance, with a 3D printer, the state space might look like this:
Where the state x is defined as just the cartesian coordinates of any point in space. A 3D printer is able to indendently actuate all of these states, that is, if torque is applied to the x-axis, the y-axis, or the z-axis, the system will reach its goal (any other point in the box of a 3D printer) pretty quickly and there is nothing else to worry about. This library has support for these systems, but likely the dynamics are highly linear and a simple PID controller would suffice for any of your control needs.
Underactuated systems by comparison have much more limitations to their dynamics. These systems are such that they are only able to be changed in only about one or more dimension at a time. These systems look closer to traditional biological systems, like walking robots (like Boston Dynamics walking dog, Spot) or bipedal robots (like Atlas).
source and more reading material
NOTE: The control algorithms presented here will likely work well for the fully actuated system as well as the under actuated system but the general rule of thumb is to stick with the simplest solution and work up from there. i.e. PID controllers, then PID controllers with Gain Scheduling, then LQR, then the more complicated algorithms.