Transforms & Poses
Fundamental Concepts
Before we can create autonomy applications for our robot, we need to understand where our robot and other points of reference are. The following concepts are critical to understand if you wish to develop autonomous robotics applications with your Amiga.
Frames of Reference
In robotics, a frame of reference (often called a "frame", coordinate frame, or "reference frame") is a description of a coordinate system of 3 orthogonal axes (x, y, & z) defined by the position and orientation of the object.
The two primary frames you need to be aware of are:
World Frame (world):
Conventionally, this is a fixed frame representing the environment in which the robot operates.
Think of it as an anchor point that doesn't move.
If using RTK GPS, a typical world frame coordinate system is defined at the location
of your RTK base station.
Robot Frame (robot): This frame is attached to the robot.
As the robot moves, this frame moves with it.
Considering the robot is not a single point, it is important to define where on the robot
is considered the center of the robot frame axes.
At farm-ng we choose the center of the robot (in length & width) at ground level.
Additional relevant frames for your Amiga-based robotics applications
may include the camera frame, the imu frame, the gps_antenna frame, and so on.
Each reference frame is represented below as a set of red-green-blue axes. The frames are connected by 6 degree-of-freedom transforms, represented below by yellow arrows.
Image Credit: https://foxglove.dev/blog/understanding-ros-transforms
Transformations
A transformation, or transform, describes how to move from one reference frame to another.
Typically in robotics we represent these transforms as an isometry transformation in 3D space. This is a distance-preserving 6 degree-of-freedom (DOF) transformation that includes a translation (3 DOF) and a rotation (the other 3 DOF).
For instance, we can represent the transformation from the world reference frame
to the robot reference frame.
Our naming convention at farm-ng is to call this the world_from_robot transformation,
following a <parent>_from_<child> or <frame_a>_from_<frame_b> naming convention.
This transform contains the translation along the world x, y, z axes,
as well as the rotation required to align the axes.
The translation, a 3-dimensional linear offset,
is represented as a vector of [x, y, z] coordinates in the parent reference frame.
The rotation, a 3-dimensional rotation, can be represented in a number of ways, but typically is represented as a quaternion.
Quaternions
Quaternions are a type of mathematical object used to represent rotations in 3D space.
Quaternions consist of four numbers (x, y, z, w) (or sometimes in order (w, x, y, z)).
w represents the scalar (or real) part of the rotation
and x, y, and z are the vector (or imaginary) parts.
Quaternions are an alternative to other methods like Euler angles or rotation matrices. Quaternions are particularly useful because they are compact, avoid certain problems like gimbal lock, and can be more computationally efficient.
Transform math
Transforms can be mathematically manipulated to understand where coordinate frames are in relation to one another. Most commonly, you will invert transforms and you will multiply transforms.
If you know the transform from the world coordinate frame to the robot coordinate frame (world_from_robot),
you can invert that transform to get the transform from the robot coordinate frame
to the world coordinate frame (robot_from_world).
world_from_robot = robot_from_imu^-1
If you know two transforms with a common frame, you can multiply them.
Say you know world_from_robot and the robot_from_imu transform from your
robot frame to your imu frame (where the IMU is on your robot).
You can calculate the transform from the world frame to the imu frame (world_from_imu)
with transform multiplication.
world_from_imu = world_from_robot * robot_from_imu
Poses
We tend to think of where our robot is as a pose, a combination of position and orientation. The position being where the robot is, and the orientation being which way the robot is facing.
A pose is, however, undetermined as there needs to be a frame of reference the position and orientation are in. Queue transforms!
We can define the pose of our robot as the 6-DOF transformation from the world frame
to our robot frame (world_from_robot).
We are not limited to representing the world_from_robot transformation as a pose.
Any transform can be represented as a pose by correctly specifying the frame_a (parent frame)
and frame_b (child frame) in our
Pose protobuf definition.
The farm-ng Pose proto
For the latest definition of the Pose structure, please refer to our
Pose protobuf definition.
Each pose has an Isometry3F64, which is a representation of rigid body transformations in 3D space.
In simple terms, it describes how an object moves and rotates in three-dimensional space
without changing its shape.
The term "isometry" implies that distances between points remain unchanged during the transformation.
In the context of robotics, Isometry3F64 is used to describe the movement
and rotation of a robot in 3D space.
Properties of Isometry3F64
Translation: This represents the linear movement of the robot. This is represented as a 3D vector, where each component (x, y, z) describes movement along that axis.
Rotation: This represents the angular movement of the robot.
This rotation is represented using Rotation3F64, which, in this context, uses the Rz method
to denote a rotation about the z-axis.
Rz method: The Rz method, when applied to Rotation3F64, denotes a rotation about the z-axis. In 3D space, the z-axis typically points upwards, perpendicular to the ground plane (assuming the x-y plane represents the ground). When you rotate an object about the z-axis, it turns around this vertical line, much like how a spinning top rotates around its central axis.
Resources
The use and multiplication of coordinate frame transforms is a fundamental concept in robotics! As such, there is an abundance of quality, free resources on the topic.
For a slightly-less-brief introduction you can refer to Understanding ROS Transforms.
If you wish to dive deeper on this topic, one option is MIT OpenCourseWare - Introduction To Robotics.