M7-T7 — Rotation Composition and Inverse

Learning objectives

After this lesson, you should be able to:

• Compose a chain of transforms in the correct order using the inner-subscripts rule.
• Invert a chain by reversing it and inverting every factor, without a general matrix inverse.
• Compute the relative rotation and relative pose between two poses.
• Explain the difference between composing about fixed axes and about moving axes, and spot the usual bugs.

Introduction and motivation

A LiDAR scan appears 5 m to the left of the wall in the map. Nothing in the sensor is broken: someone wrote T_BL * T_WB instead of T_WB * T_BL. Because both are valid 4 × 4 matrices the code runs, and the result is plausible-looking but wrong. A written rule that every engineer on the team applies to every product removes this whole class of bug.

The intuition

Treat the subscripts as a chain of labels: T_WB converts from B to W, T_BL from L to B. To convert from L to W, you want B to appear once on the right of the first matrix and once on the left of the second, so they meet and disappear: T_WB T_BL = T_WL. The matrices are applied to the point from the right: the point is first mapped from L to B, then from B to W. If the labels do not meet, the product has no meaning.

Geometric and physical interpretation

Geometric view. T_WB T_BL first re-expresses the point relative to the base axes, then places the base in the world. Reading left to right instead describes the same motion as a sequence of moves about the moving frame: start at W, move to B, then from there move to L. Reading right to left describes it as moves about the fixed world axes, applied in reverse. Both views give the same matrix. Physical view. “The robot is here, and the sensor is mounted this far from the robot” is one product; reversing it would mount the sensor relative to the world origin.

The mathematics

T_AB T_BC

composition; the inner B cancels, giving T_AC.

T⁻¹

inverse, [[Rᵀ, −Rᵀt], [0, 1]].

R_AB

rotation of frame B relative to frame A.

R_rel

relative rotation between two poses: R_WAᵀ R_WB.

Composition of a chain

TWL = TWB TBL,      ʷp = TWB TBL ᴸp

What it means. Apply the right-most matrix to the point first. The product is valid only when adjacent labels match; if they do not, a transform or an inverse is missing.

Inverse of a chain

(T₁T₂ ⋯ Tₙ)⁻¹ = Tₙ⁻¹ ⋯ T₂⁻¹ T₁⁻¹        TBW = TWB⁻¹

What it means. To undo a chain of operations, undo the last first. Each factor is inverted with the closed form from the SE(3) lesson. A transform with swapped labels is the inverse of the original.

Relative rotation

RAB = RWAᵀ RWB        (relative orientation of B with respect to A)

What it means. Rotations do not subtract. The difference between two orientations is the rotation that takes one to the other, obtained by composing one with the inverse (transpose) of the other. Its angle is the true angular distance between them.

Match inner labels, apply right to left, and invert a chain by reversing it. The difference of orientations is RₐᵀR_b, not Rₐ − R_b.

Step-by-step derivation: why the order is what it is

1. The point passes through the frames. A point ᴸp is converted to base coordinates by ᴮp = T_BL ᴸp, then to the world by ʷp = T_WB ᴮp.
2. Substitute. ʷp = T_WB (T_BL ᴸp) = (T_WB T_BL) ᴸp, using associativity. So the matrix acting on ᴸp is T_WB T_BL, and the inner B has cancelled.
3. Inverse of a product. We want (AB)⁻¹. Check (AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = AA⁻¹ = I, so (AB)⁻¹ = B⁻¹A⁻¹: the order reverses. For a chain, the same argument applied repeatedly gives the reversed list of inverses.
4. Relative rotation. Orientations of A and B in the world satisfy R_WB = R_WA R_AB (B is A followed by the relative turn). Left-multiply by R_WAᵀ = R_WA⁻¹ to get R_AB = R_WAᵀ R_WB.
5. Relative angle. The angle of R_AB follows from its trace, cos θ = (tr R_AB − 1)/2. This is the true angular distance between two attitudes, the quantity used to measure an orientation error.
6. Fixed versus moving axes. Starting from an orientation R, a turn Q expressed in the world axes is applied as Q R (pre-multiply); a turn expressed in the body axes is applied as R Q (post-multiply). The two coincide only when Q commutes with R, for instance both are rotations about the same axis.

Sensors on a moving robot. A measurement stamped at time t must use the pose T_WB(t) at that time, not the latest pose. For a spinning LiDAR, each point has its own timestamp and its own pose (motion distortion), so the single product above is applied with a different T_WB per point.

The loop-closure error. Going around a loop of frames and returning to the start should give the identity: T₁T₂T₃T₄ = I. The deviation from identity is the residual that pose-graph optimisation drives to zero.

A worked example: base pose, lidar mount and a point

• Setup (2D). The robot base is at (2, 1) with heading 90°: T_WB = [[0, −1, 2], [1, 0, 1], [0, 0, 1]]. The LiDAR is 0.3 m forward on the base, unrotated: T_BL = [[1, 0, 0.3], [0, 1, 0], [0, 0, 1]].
• The return. A return 4 m away at 30°: ᴸp = (3.464, 2, 1).
• Correct order. T_BL ᴸp = (3.764, 2, 1), then T_WB (3.764, 2, 1) = (−2 + 2, 3.764 + 1, 1) = (0, 4.764, 1). The world point is (0, 4.764).
• Wrong order. T_BL T_WB ᴸp: T_WB ᴸp = (0, 4.464, 1), then T_BL adds 0.3 to x: (0.3, 4.464, 1). It is off by (0.3, −0.3), about 0.42 m, a believable-looking error.
• Inverse of the chain. T_LW = (T_WB T_BL)⁻¹ = T_BL⁻¹ T_WB⁻¹. Applying it to the world point (0, 4.764) returns (3.464, 2), the original lidar point ✓.
• Relative rotation. A drone yaws from 30° to 80°: R_AB = Rz(30°)ᵀRz(80°) = Rz(50°), relative angle 50°, with trace 1 + 2 cos 50° = 2.2856.

Robotics application

SLAM front ends. The scan-to-map pipeline multiplies sensor mount, odometry pose and map correction; each scan point goes lidar → base → odom → map; see TF frames and consistency and scan registration.

Pose graphs. Edge residuals are products of inverses of measured transforms and relative poses; loop closure is the identity-deviation of a chain; see pose graphs.

UAV state estimation. Camera, IMU and LiDAR calibrations chain to the body pose; attitude error is the relative rotation R_estᵀR_true, and its angle is the headline accuracy metric; see extrinsic calibration and the error-state Kalman filter.

Manipulators. Forward kinematics is a product of link transforms in order from base to tool; inverting it gives the tool-to-base chain (Module 8).

Engineering implications and limitations

Practical rules

• Write the labels, always. Name every transform T_target_source and check that the inner names match in each product. If they do not, find the missing inverse.
• Compose in one place. Build the chain with one helper function and unit tests with a known point; do not scatter manual multiplications across the code base.
• Test with a known point. Choose a configuration with an obvious answer (robot turned 90°, sensor 1 m forward) and assert the world coordinates.
• Look up the transform at the right time. Use the pose interpolated at the measurement timestamp, not the latest. Moving fast makes a 50 ms mismatch visible.
• Closed-form inverse only. Use [[Rᵀ, −Rᵀt], [0, 1]]; a general inv loses orthogonality over time.
• Measure attitude error with the relative rotation. The angle of R_estᵀR_true, not the difference of Euler angles, which can jump by 360° or be wrong near gimbal lock.

Limits

Composition assumes exact, time-aligned transforms. Calibration error, timestamp error and drift in the chain are not captured by the algebra; they must be modelled as uncertainty (Modules 9 and 10).

Common misconceptions

Engineering takeaways

1. Compose transforms so that inner frame labels cancel: T_WB T_BL = T_WL; apply the right-most factor to the point first.
2. Invert a chain by reversing it and inverting each factor: (T₁T₂)⁻¹ = T₂⁻¹T₁⁻¹.
3. The relative orientation is R_AB = R_WAᵀR_WB; its angle, from the trace, is the true angular error.
4. Pre-multiply for fixed-axis turns, post-multiply for body-axis turns.
5. Test chains with a known point, and use the pose at the measurement timestamp.

References and further reading

Primary papers and authoritative documentation.

• Modern Robotics (Lynch and Park), free preprint: chapter 3.3.1, uses of transformation matrices: changing frames and composing
• Barfoot, State Estimation for Robotics (author-hosted first edition, free to read): chapter 6, composing and inverting poses
• Solà, Deray and Atchuthan, A micro Lie theory for state estimation in robotics: A micro Lie theory: composition and the inverse of the group
• UCSD ECE276A: Sensing and Estimation in Robotics: transform chains and the pose graph

Knowledge check

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What’s next

Rotation matrices are exact but use nine numbers. The next lesson compares the compact alternatives (Euler angles, axis-angle and quaternions), and explains the singularity that makes three-angle descriptions risky.

Related content

Nothing below is required to finish this lesson.

• Before
  M7-T6: SE(3)

  Describe the group of rigid-body motions and use the structure of a 4×4 transform.

• Lab
  M7-L1: Robot Coordinate Frame Explorer

  Chain frames from the world to a sensor on an arm, find the wrong-order and inverse bugs, and bring a measured point back to the world.