Dual Quaternion Numpy and CasADi

This is a brief explanation of how to install and run this package on another machine.

Create a new conda environment using Python 3.8 (for example, named "dual_quat") and activate it with the following commands:

conda create -n dual_quat python=3.8
conda activate dual_quat

This project includes an installation file that contains all the necessary dependencies for Python and ROS. Inside the conda environment, execute the following commands:

chmod +x install_python.sh
./install_python.sh

ROS Workspace

Create a ros workspace using the following commands:

mkdir -p ~/catkin_ws/src
cd ~/catkin_ws/

Inside the ROS workspace /src must be included: dual_quaternion.

Since the creation of the conda virtual environment, it should be activated before compiling the packages; therefore, execute the following command:

catkin_make -DPYTHON_EXECUTABLE=~/miniconda3/envs/dual_quat/bin/python
source devel/setup.bash

Tutorial: DualQuaternion Operations

Conjugate

Norm

Addition

Retrieving Elements of the DualQuaternion (Real, Dual, Quaternion, and Translation)

Tutorial Rigid Body

Defining Different Rigid Body Frames

This section explains how to use the library to define the pose of a rigid body as a DualQuaternion object and how to display its information within rviz

The proper way to define the pose of a rigid body is described below:

#Defining the Orientation Using Quaternions
theta1 = ca.SX([np.pi/2])
n1 = ca.SX([0.0, 0.0, 1.0])
q1 = ca.vertcat(ca.cos(theta1/2), ca.sin(theta1/2)@n1)

# Defining the translation
t1 = ca.SX([0.0, 1.0, 1.0, 0.0])

#Defining the Orientation Using Quaternions
theta2 = ca.SX([ca.pi/4])
n2 = ca.SX([1.0, 0.0, 0.0])
q2 = ca.vertcat(ca.cos(theta2/2), ca.sin(theta2/2)@n2)

# Defining the translation
t2 = ca.SX([0.0, 0.0, 2.0, 0.0])

# Init Dualquaternion
Q1 = DualQuaternion.from_pose(quat = q1, trans = t1)
Q2 = DualQuaternion.from_pose(quat = q2, trans = t2)

The proper representations of these body frames are presented below:

It is also possible to verify the norm of the DualQuaternion elements; the user can obtain the norm using the following commands:

#Obtaining the norm of the DualQuaternion elements
Q1_norm = Q1.norm
Q2_norm = Q2.norm

The norm property of the elements returns the is a dual number whose real part and dual part are both positive

Sequential Pose Transformations

In order to perform sequential pose transformations, it is only necessary to multiply the DualQuaternions. For instance, the user can define the following transformation:"

# Computing a sequential transformation
Q3 = Q1 * Q2

The result of this operation can be visualized in the following image:

It is posible to replicate theese results by executing the following commad:

python Tutorial_1.py

Differential Kinematics Quaternions Control

It is possible to execute quaternion-based kinematic control, where the purpose is to converge to a desired orientation.

To run the controller taking the shortest path based on numpy, execute the following command:

roslaunch dual_quaternion quat_numpy.launch

To run the controller taking the shortest path based on Casadi, execute the following command:

roslaunch dual_quaternion quat_casadi.launch

The results of this formulation are presented in the following images.

Differential Kinematics DualQuaternions Control