Spatial descriptions: rotation matrices and homogeneous transforms

Why it matters

Every robot calculation — where the tool is, where the camera sees a part, how to reach a fixture — needs a precise way to say where one frame is relative to another. Rotation matrices and 4 × 4 homogeneous transforms are that language; forward kinematics, inverse kinematics, the Jacobian and vision calibration are all built on them.

Key ideas

Frames. A frame {B} is an origin plus three mutually perpendicular unit axes x̂_B, ŷ_B, ẑ_B (right-handed). A position vector is always written in some frame: ᴬp means "p expressed in {A}".

Rotation matrix. ᴬR_B is the 3 × 3 matrix whose columns are the unit axes of {B} written in {A}: ᴬR_B = [ᴬx̂_B ᴬŷ_B ᴬẑ_B]. It does two jobs:

• describes the orientation of {B} relative to {A};
• maps a vector: ᴬv = ᴬR_B · ᴮv.

Properties: columns (and rows) are orthonormal, so Rᵀ·R = I, R⁻¹ = Rᵀ, det R = +1. It preserves lengths and angles. Only 3 of its 9 numbers are independent. A matrix with det = −1 is a reflection, not a rotation.

Elementary rotations about x, y, z by angle θ are given below (counter-clockwise positive when looking down the axis towards the origin — right-hand rule).

Composing rotations. Order matters: rotations do not commute.

• Rotations about the current (moving) axes → post-multiply: R = R₁·R₂.
• Rotations about the fixed (original) axes → pre-multiply: R = R₂·R₁. Example: Z-Y-X Euler angles (yaw-pitch-roll about moving axes) give R = R_z(α)·R_y(β)·R_x(γ), which equals X-Y-Z about fixed axes (roll, pitch, yaw).

Other orientation descriptions. Euler angles (3 numbers, but singular when the middle angle makes two axes line up — "gimbal lock"), axis-angle, and unit quaternions (4 numbers, no singularity, used in ROS and controllers). The rotation matrix itself has no singularity.

Homogeneous transform. Position and orientation together:

ᴬT_B = [ ᴬR_B ᴬp_B ; 0 0 0 1 ], where ᴬp_B is the origin of {B} in {A}.

A point transforms as ᴬp = ᴬR_B·ᴮp + ᴬp_B, written compactly with 4-vectors [x y z 1]ᵀ as ᴬp = ᴬT_B·ᴮp. Transforms chain: ᴬT_C = ᴬT_B·ᴮT_C — the "cancelling" sub/superscripts are a quick check. The inverse is cheap (no general 4 × 4 inversion needed).

Free vectors (velocities, forces, directions) have a 0 as the fourth element, so translation does not affect them.

Formulas

R_x(θ) = [[1, 0, 0], [0, cos θ, −sin θ], [0, sin θ, cos θ]]

R_y(θ) = [[cos θ, 0, sin θ], [0, 1, 0], [−sin θ, 0, cos θ]]

R_z(θ) = [[cos θ, −sin θ, 0], [sin θ, cos θ, 0], [0, 0, 1]]

• θ — rotation angle (rad or degrees), positive by the right-hand rule about the named axis.

R⁻¹ = Rᵀ, det R = +1

ᴬp = ᴬR_B · ᴮp + ᴬp_B

• ᴬp, ᴮp — the same point in frames {A} and {B} (m); ᴬp_B — origin of {B} in {A} (m).

ᴬT_B = [[ᴬR_B, ᴬp_B], [0 0 0, 1]] (4 × 4)

ᴬT_C = ᴬT_B · ᴮT_C

ᴮT_A = (ᴬT_B)⁻¹ = [[ᴬR_Bᵀ, −ᴬR_Bᵀ · ᴬp_B], [0 0 0, 1]]

Rotation angle from R: cos φ = (r₁₁ + r₂₂ + r₃₃ − 1) / 2 (equivalent axis-angle)

Worked examples

Example 1 (standard). Frame {B} is rotated 90° about z_A and its origin is at (5, 0, 0) m in {A}. A point has ᴮp = (2, 1, 0) m. Find ᴬp.

1. ᴬR_B = R_z(90°) = [[0, −1, 0], [1, 0, 0], [0, 0, 1]].
2. ᴬR_B·ᴮp = (0·2 − 1·1 + 0, 1·2 + 0 + 0, 0) = (−1, 2, 0) m.
3. ᴬp = (−1, 2, 0) + (5, 0, 0) = (4, 2, 0) m.

Answer: ᴬp = (4, 2, 0) m.

Example 2 (GATE level). With ᴬT_B as in Example 1, frame {C} is rotated 90° about x_B with its origin at (0, 2, 0) m in {B}. Find (a) ᴬT_C, (b) the origin of {C} in {A}, (c) the origin of {A} expressed in {B}.

1. ᴮT_C: R = R_x(90°) = [[1, 0, 0], [0, 0, −1], [0, 1, 0]], p = (0, 2, 0).
2. ᴬR_C = R_z(90°)·R_x(90°) = [[0, 0, 1], [1, 0, 0], [0, 1, 0]] (check: det = +1).
3. ᴬp_C = ᴬR_B·(0, 2, 0) + (5, 0, 0) = (−2, 0, 0) + (5, 0, 0) = (3, 0, 0) m.
4. So ᴬT_C = [[0, 0, 1, 3], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]].
5. (c) ᴮp_A = −ᴬR_Bᵀ·ᴬp_B. ᴬR_Bᵀ = [[0, 1, 0], [−1, 0, 0], [0, 0, 1]]; ᴬR_Bᵀ·(5, 0, 0) = (0, −5, 0); negate → (0, 5, 0) m.

Answer: (a) as above, (b) (3, 0, 0) m, (c) (0, 5, 0) m.

Example 3 (order matters). The unit vector x̂ = (1, 0, 0) is rotated 90° about the fixed y-axis and then 90° about the fixed z-axis. Pre-multiply: R_z(90°)·R_y(90°)·x̂. R_y(90°)·x̂ = (0, 0, −1); R_z leaves a z-vector unchanged → (0, 0, −1), i.e. along −z. Doing the same rotations about the moving axes (post-multiply) gives R_y(90°)·R_z(90°)·x̂ = (0, 1, 0) — a different answer.

Common mistakes

• Putting the axes of {B} in the rows instead of the columns of ᴬR_B.
• Sign of sin θ in R_y: the minus sign is in the bottom-left, unlike R_x and R_z.
• Inverting T by inverting R and negating p separately: the translation of the inverse is −Rᵀ·p, not −p.
• Mixing fixed-axis and moving-axis rules (pre- vs post-multiplication).
• Applying translation to direction vectors — use a 0 in the fourth slot.
• Using degrees in one place and radians in another inside a calculator.

For GATE ME

Expect short numericals: transform a point between frames, compose two transforms, find the inverse, or check whether a given matrix is a valid rotation (orthonormal, det = +1). Conceptual MCQs test properties of R and the effect of rotation order. Practise multiplying 3 × 3 matrices quickly and checking results with det R = 1 and unit-length columns.

Quick check

1. What is the inverse of a rotation matrix?
2. Rotate the point (1, 0, 0) by 90° about z. Result?
3. In ᴬT_B, what does the fourth column (top three entries) represent?
4. Is [[1, 0, 0], [0, 1, 0], [0, 0, −1]] a rotation matrix?
5. For rotations about moving axes, do you pre- or post-multiply?

Answers: 1. Its transpose; 2. (0, 1, 0); 3. The origin of {B} expressed in {A}; 4. No — det = −1, it is a reflection; 5. Post-multiply.