dynkin

A toolkit for 3D dynamics and kinematics of rigid bodies using the YPR euler angle convention.

General

dynkin is a set of tools for handling the dynamics and kinematics of rigid bodies in 3 dimensions (6DOFs). Some features:

Homogenous transformation matrices
Chained reference frames
Idealized rigid body implementation

The fundamentals of reference frames and the kinematic relations of these are based on Theory of Applied Robotics (Reza N. Jazar) , the idealized rigid body implementation follows the outline suggested in the lectures by Fossen.

Theory intro

Some basic notions:

A reference Frame is defined in dynkin as an object which positions, vectors, velocities, accelerations etc can be decomposed in. dynkin represents reference Frames by (4x4) homogenous transformation matrices. A Frame is aligned (positioned, rotated) and moves (linear and angular velocity) in relation to another Frame or the inertial frame (represented by None).
The pose of a Frame is its generalized position and the twist of a Frame is its generalized velocity, both in relation to the inertial frame.
All rotations in dynkin are internally represented by rotation matrices but the external API, so far, deals only with Euler angles of the YPR (Yaw-Pitch-Roll) convention.
A kinematic chain is a single-linked list of Frames, where each Frame holds a reference to its closest parent. Any number of Frames may be attached into such a chain and the chain may also have any number of branches, it is however the user´s responsibility to ensure no kinematic loops occur.
A transform is an object relating two Frames enabling transformation of positions, vectors, velocities etc from one Frame to the other. The Frames do not need to be part of the same kinematic chain.
A RigidBody is a 3D body with arbitrary extent that may be described by a generalized inertia matrix (6x6). It accelerates when subject to generalized external forces (wrenches) and rotational velocities give rise to inertial forces (coriolis and centripetal contributions).

Installation

dynkin can be included as per below in your cmake workflow if using the CPM.cmake package manager:

CPMAddPackage(
  NAME dynkin
  GITHUB_REPOSITORY freol35241/dynkin
  VERSION 0.1.0
)

Or, since dynkin is a single-file and header-only file library, just copy dynkin.hpp into your project.

Examples

Single frame

{c++}
#include <Eigen/Dense>
#include <dynkin/dynkin.hpp>

#define DEG2RAD M_PI/180.0f

using namespace dynkin;

Frame frame1 = create_frame();
frame1->position() << 1, 2, 3;
frame1->set_attitude({0,0,90*DEG2RAD});

// Find transformation from the inertial frame to frame1
Transform ti1 = transform(nullptr, frame1);

// Transformation of vector
Eigen::Vector3d v1_decomposed_in_inertial_frame, v1_decomposed_in_frame1;
v1_decomposed_in_frame1 = ti1.apply_vector(v1_decomposed_in_inertial_frame);

// Transformation of position
Eigen::Vector3d p1_decomposed_in_inertial_frame, p1_decomposed_in_frame1;
p1_decomposed_in_frame1 = ti1.apply_position(p1_decomposed_in_inertial_frame);

// Transformation of wrench
Eigen::Vector6d w1_decomposed_in_inertial_frame, w1_decomposed_in_frame1;
w1_decomposed_in_frame1 = ti1.apply_wrench(w1_decomposed_in_inertial_frame);

// Find the inverse transformation
Transform t1i = ti1.inverse();

// Pose of this frame, decomposed in inertial frame
Eigen::Vector6d pose = frame1.get_pose();

// Twist of this frame, decomposed in inertial frame
Eigen::Vector6d = frame1.get_twist();

Two frames

{c++}
#include <Eigen/Dense>
#include <dynkin/dynkin.hpp>

#define DEG2RAD M_PI/180.0f

using namespace dynkin;

Frame frame1 = create_frame();
frame1->position() << 1, 2, 3;
frame1->set_attitude({0,0,90*DEG2RAD});

Frame frame2 = create_frame();
frame2->position() << 3, 2, 1;
frame2->set_attitude({0,0,-90*DEG2RAD});

// Find transformation from frame1 to frame2
Transform t12 = transform(frame1, frame2);

// Transformation of vector
Eigen::Vector3d v1_decomposed_in_frame1, v1_decomposed_in_frame2;
v1_decomposed_in_frame2 = t12.apply_vector(v1_decomposed_in_frame1);

// Transformation of position
Eigen::Vector3d p1_decomposed_in_frame1, p1_decomposed_in_frame2;
p1_decomposed_in_frame2 = t12.apply_position(p1_decomposed_in_frame1);

// Transformation of wrench
Eigen::Vector6d w1_decomposed_in_frame1, w1_decomposed_in_frame2;
w1_decomposed_in_frame2 = t12.apply_wrench(w1_decomposed_in_frame1);

// Find the inverse transformation
Transform t21 = t12.inverse();

Kinematic chains

{c++}
#include <Eigen/Dense>
#include <dynkin/dynkin.hpp>

#define DEG2RAD M_PI/180.0f

using namespace dynkin;

Frame frame1 = create_frame();
frame1->position() << 1, 2, 3;
frame1->set_attitude({0,0,90*DEG2RAD});

Frame frame2 = frame1->create_child();
frame1->position() << 3, 2, 1;
frame1->set_attitude({0,0,-90*DEG2RAD});

Frame frame3 = frame2->create_child();
frame3->position() << 1, 1, 1;


// Find transformation from inertial frame to frame3
Transform ti3 = transform(None, frame3);

// Transformation from frame3 and frame1
Transform t31 = transform(frame3, frame1);

...

Rigid body

{c++}
TODO

License

Distributed under the terms of the MIT license, dynkin is open source software.