Coordinate Frames

A real robot doesn’t have just one coordinate system — it has many. The world has a fixed frame, the robot base has its own frame, each joint defines a frame, the camera sees in its own frame, and the gripper tip has yet another. Every piece of data — positions, velocities, forces — is expressed relative to some frame, and confusing which frame you’re in is one of the most common (and dangerous) bugs in robotics software.

This lesson is about managing that complexity rigorously.

What Is a Coordinate Frame?

A coordinate frame (or reference frame) is an origin point plus a set of orthogonal basis vectors that define directions. In 3D, a frame $\{A\}$ consists of:

• An origin $\mathbf{o}_A$ — the “zero point”
• Three unit vectors $\hat{x}_A, \hat{y}_A, \hat{z}_A$ — the axis directions

A point $\mathbf{p}$ can be described in frame $\{A\}$ or frame $\{B\}$, and the coordinates will generally be different numbers representing the same physical point.

We write ${}^A\mathbf{p}$ to mean “the coordinates of point $\mathbf{p}$ expressed in frame $\{A\}$.”

Transforms Between Frames

The homogeneous transformation $T_B^A$ (or equivalently ${}^A T_B$) converts a point from frame $\{B\}$ coordinates to frame $\{A\}$ coordinates:

This $4 \times 4$ matrix encodes both the rotation and translation of frame $\{B\}$ relative to frame $\{A\}$:

where:

• $R_B^A$ is the $3 \times 3$ rotation: the columns are the unit vectors of $\{B\}$‘s axes expressed in $\{A\}$
• $\mathbf{d}_B^A$ is the position of $\{B\}$‘s origin expressed in $\{A\}$

Chaining Transforms

To go from frame $\{C\}$ to frame $\{A\}$ via frame $\{B\}$, chain the transforms:

Read right-to-left: first convert from $\{C\}$ to $\{B\}$, then from $\{B\}$ to $\{A\}$.

The subscript/superscript indices “cancel” like fractions: $T_{\cancel{B}}^A \cdot T_C^{\cancel{B}} = T_C^A$. This mnemonic helps catch mistakes — if adjacent indices don’t match, the chain is wrong.

For a full kinematic chain with $n$ frames:

Common Frames in Robotics

A typical robotic system involves several standard frames:

World Frame $\{W\}$

The global or map frame. Fixed to the environment. All global planning and mapping happens here. Gravity points along $-z$ (or $-y$, depending on convention).

Base Frame $\{B\}$

Fixed to the robot’s base. For a mobile robot, it moves with the robot. For a fixed manipulator, it’s often coincident with the world frame.

Joint Frames $\{J_i\}$

Each joint in a serial manipulator defines a frame according to the DH convention (from Module 3). The chain $T_{J_1}^B \cdot T_{J_2}^{J_1} \cdots T_{J_n}^{J_{n-1}}$ gives forward kinematics.

Tool/End-Effector Frame $\{T\}$

Attached to the gripper or tool tip. This is what you want to control — positioning $\{T\}$ at the desired pose in $\{W\}$ is the goal of motion planning.

Sensor Frames

Each sensor has its own frame:

• Camera frame $\{C\}$: origin at the optical center, z-axis along the viewing direction
• LiDAR frame $\{L\}$: origin at the scanner, measurements in its local coordinates
• IMU frame $\{I\}$: measures accelerations and angular velocities in its own axes

Frame Trees

In a complex system, frames form a tree structure rooted at the world frame. Every frame has exactly one parent, and the path from any frame to any other frame is unique.

{World}
├── {Map}
│   └── {Odom}
│       └── {Base}
│           ├── {Joint1}
│           │   └── {Joint2}
│           │       └── {EE}
│           │           └── {Tool}
│           ├── {Camera}
│           ├── {LiDAR}
│           └── {IMU}
└── {Target}

To transform between any two frames, walk up the tree to their common ancestor, then back down:

Or equivalently: $T_{\text{Target}}^{\text{Camera}} = T_{\text{World}}^{\text{Camera}} \cdot T_{\text{Target}}^{\text{World}}$.

Extrinsic Calibration

Extrinsic calibration is the process of determining the fixed transform between two frames — most commonly between a sensor and the robot body.

Hand-Eye Calibration

A camera is mounted on a robot’s end-effector. You know:

• $T_{EE}^B$ from forward kinematics (changes with joint angles)
• $T_{\text{target}}^C$ from the camera (sees a calibration target)

You need to find the unknown fixed transform $T_C^{EE}$ (camera relative to end-effector). The key observation: for any pose $i$, the chain from calibration target to base through the camera and end-effector must equal the fixed (unknown) target-to-base transform:

By taking two poses $i, j$ and eliminating the unknown constant, you get the classic $AX = XB$ problem, where $A$ is the relative end-effector motion and $B$ is the relative camera observation. It requires at least two poses with non-parallel rotation axes, and in practice you use 10+ poses with a least-squares solver.

LiDAR-to-IMU Calibration

For autonomous vehicles, the LiDAR and IMU frames must be precisely aligned. Errors here cause the point cloud to “smear” when the vehicle turns, corrupting the map. The fixed transform $T_L^I$ is typically found by:

1. Collecting data while performing varied motions (turns, accelerations)
2. Optimizing $T_L^I$ to minimize inconsistency between the LiDAR-based motion estimate and the IMU-based motion estimate

Velocity and Force Transformations

Transforms don’t just apply to positions. Velocities and forces also need frame conversions, but the rules differ.

Velocity Transform (Adjoint)

A twist (linear + angular velocity) in frame $\{B\}$ transforms to frame $\{A\}$ via the adjoint of $T_B^A$:

The adjoint is a $6 \times 6$ matrix — not the same as simply applying $T$ to a position vector. This is because angular velocity doesn’t transform like a point.

Force/Wrench Transform

A wrench (force + torque) transforms with the inverse transpose of the adjoint. Forces and velocities are dual — they transform differently to preserve the power relationship $P = \mathcal{F}^T \mathcal{V}$.

Common Pitfalls

1. Frame Mismatch

Combining data from different frames without transforming first. Every vector and matrix has an implicit frame — treat frame labels as rigorously as you treat units (meters vs. millimeters).

2. Transform Direction

Applying $T_B^A$ when you needed $T_A^B$ (or vice versa). If the result looks mirrored or the robot moves in the opposite direction, check the transform direction.

3. Pre-Multiply vs. Post-Multiply

For body-fixed (intrinsic) operations, post-multiply: $T_{\text{new}} = T_{\text{old}} \cdot T_{\text{delta}}$. For world-fixed (extrinsic) operations, pre-multiply: $T_{\text{new}} = T_{\text{delta}} \cdot T_{\text{old}}$.

4. Stale Transforms

On a moving robot, transforms change over time. Using a camera image from time $t_1$ with a robot pose from time $t_2$ introduces error proportional to the robot’s velocity and the time gap.

Practice Problems

1. Frames $\{A\}$, $\{B\}$, $\{C\}$ are arranged such that $T_B^A = \begin{bmatrix} R_z(90°) & (1, 0, 0)^T \\ 0 & 1\end{bmatrix}$ and $T_C^B = \begin{bmatrix} I & (0, 2, 0)^T \\ 0 & 1\end{bmatrix}$. What is $T_C^A$?

2. A camera sees an object at ${}^C\mathbf{p} = (0, 0, 1.5)$ m. The camera-to-base transform is $T_C^B = \begin{bmatrix} I & (0.5, 0, 0.8)^T \\ 0 & 1\end{bmatrix}$. What is ${}^B\mathbf{p}$?

3. You have transforms $T_B^A$, $T_C^B$, and $T_D^C$. Write the expression for $T_A^D$ (from $\{A\}$ to $\{D\}$).

4. A LiDAR measures a wall point at ${}^L\mathbf{p} = (3, 0, 0)$. The LiDAR is mounted 0.2 m above and 0.1 m forward of the robot base, with no rotation offset. Write $T_L^B$ and compute ${}^B\mathbf{p}$.

5. A mobile robot has base-to-world transform $T_B^W$ and a camera with $T_C^B$. The camera sees an AprilTag at ${}^C\mathbf{p}_{\text{tag}}$. Write the full expression for the tag position in the world frame.

Next Steps

You can now manage multiple coordinate frames confidently. The final lesson in this module tackles rotation representations — axis-angle, rotation vectors, and quaternions — representations that overcome Euler angle limitations and enable smooth interpolation for motion planning.