📦 tf2

Intro

tf2 (TransForm version 2) is a ROS library that provides an easy way to calculate transformations between different coordinate frames (e.g., the robot’s base frame, camera frame, map frame).

📦tf2

tf2 is the second generation of the transform library, which lets the user keep track of multiple coordinate frames over time. tf2 maintains the relationship between coordinate frames in a tree structure buffered in time, and lets the user transform points, vectors, etc between any two coordinate frames at any desired point in time.

geometryCoordinateFrameConventions

A robotic system typically has many 3D coordinate frames that change over time, such as a world frame, base frame, gripper frame, head frame, etc. tf2 keeps track of all these frames over time, and allows you to ask questions like:

• Where was the head frame relative to the world frame, 5 seconds ago?
• What is the pose of the object in my gripper relative to my base?
• What is the current pose of the base frame in the map frame?

Understanding ROS Transforms

https://foxglove.dev/blog/understanding-ros-transforms

Understanding ROS Transforms

In robotics, a frame refers to the coordinate system describing an object’s position and orientation, typically along x, y, and z axes. ROS requires each frame to have its own unique frame_id, and a typical scene consists of multiple frames for each robot component (i.e. limb, sensor), detected object, and player in the robot’s world.

A scene always has one base frame – usually named world or map – that is an unmoving constant. All other frames in the scene – including the robot’s own frame, which ROS typically calls base_link – are children to this base frame, and are positioned relative to this parent in some way.

Calculating an object’s position using transforms

tf2 under the hood example

Each child frame has a transform that represents its position vector and rotation in relation to its parent frame:

Parent: base_link

• sensor_link – Position: $(2,1)$, Rotation: $-45^{\circ }$ ($-\frac{\pi }{4}$ radians)
• arm_base_link – Position: $(1,-1)$, Rotation: $30^{\circ }$ ($\frac{\pi }{6}$ radians)

Parent: arm_base_link

• arm_end_link – Position: $(1,1)$, Rotation: $0^{\circ }$ (no rotation)

Calculation

To deduce the position of arm_end_link in the base_link frame, we need to follow the chain of transformations from the arm_end_link to the base_link. Here’s how to approach it step by step using the given transforms:

1. Transformation from base_link to sensor_link:

• Position: $(2,1)$ in the base_link frame.
• Rotation: $-45^{\circ }$ (or $-\pi /4$ radians) in the base_link frame.

This means that sensor_link is positioned at $(2,1)$ in base_link, and its orientation is rotated $-45^{\circ }$ from base_link.

2. Transformation from base_link to arm_base_link:

• Position: $(1,-1)$ in the base_link frame.
• Rotation: $30^{\circ }$ (or $\pi /6$ radians) in the base_link frame.

This means that arm_base_link is positioned at $(1,-1)$ in base_link, and it has a $30^{\circ }$ ($\pi /6$ radians) rotation with respect to base_link.

3. Transformation from arm_base_link to arm_end_link:

• Position: $(1,1)$ in the arm_base_link frame.
• Rotation: $0^{\circ }$ (no rotation).

This means that arm_end_link is positioned at $(1,1)$ in the arm_base_link frame and has no rotation.

Now, we need to compute the overall position of arm_end_link in the base_link frame by following these transformations step by step:

Step 1: Find the position of arm_end_link in the arm_base_link frame:

The position of arm_end_link in the arm_base_link frame is:

• Position: $(1,1)$ in arm_base_link.

Step 2: Apply the transform from base_link to arm_base_link:

The position of arm_base_link in base_link is $(1,-1)$, and it is rotated by $30^{\circ }$ ($\pi /6$ radians) with respect to the base_link frame. To apply this transform, we first need to rotate and then translate the position of arm_end_link$ (1, 1)$ in the arm_base_link$ frame.

1. Rotation: Rotate $(1,1)$ by $30^{\circ }$ ($\pi /6$ radians). Use a rotation matrix to rotate the point $(1,1)$:

   $$R_{30^{\circ }}=\left[\begin{array}{cc}\cos \left(\frac{\pi }{6}\right)& -\sin \left(\frac{\pi }{6}\right)\\ \sin \left(\frac{\pi }{6}\right)& \cos \left(\frac{\pi }{6}\right)\end{array}\right]=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]$$

   Applying the rotation to the position $(1,1)$:

   $$\text{Rotated position}=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}\frac{\sqrt{3}}{2}-\frac{1}{2}\\ \frac{1}{2}+\frac{\sqrt{3}}{2}\end{array}\right]=\left[\begin{array}{c}\frac{\sqrt{3}-1}{2}\\ \frac{1+\sqrt{3}}{2}\end{array}\right]$$

2. Translation: After rotation, translate by the position of arm_base_link in `base_link$,whichis$(1, -1)$:

This gives us the final position of arm_end_link in the base_link frame.