vRotR

vRotR stands for vector Rotation of Rotations. It is a computational technique used in computer graphics, robotics, and simulation to apply incremental rotations to vectors and coordinate frames. The approach emphasizes real-time performance and numerical stability, making it suitable for systems that update orientations every frame or control cycle.

Conceptually, vRotR represents the current orientation as a unit quaternion (or an equivalent rotation representation) and applies small, composite rotations using simple operations. A rotation command is provided as a small rotation vector or angular velocity. The method computes a delta rotation quaternion dq from the magnitude and direction of the input, typically using dq = [cos(theta/2), sin(theta/2) * n], where theta is the rotation angle and n is the unit axis. The updated orientation is obtained by multiplying the delta quaternion with the current orientation (dq * q or q * dq, depending on convention). The orientation is then re-normalized to counter numerical error.

Advantages include composability of rotations, avoidance of gimbal lock, and efficient updates suitable for real-time pipelines. Limitations involve careful handling of large rotations, order of multiplication, and occasional drift if normalization is neglected. In practice, vRotR is often used in conjunction with explicit normalization and, when necessary, re-orthogonalization of rotation representations.

Applications span real-time rendering, camera control, character animation, robotics actuation, and VR/AR head tracking. Related concepts include quaternions, rotation matrices, axis-angle representations, and Rodrigues' rotation formula.

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