Matrix, Quaternion, and Biquaternion Methods of Sequential Composition of Kinematic Transformations in Robotics

This paper presents and compares three approaches to solving forward kinematics in serial-chain robotic manipulators: homogeneous transformation matrices, quaternions, and biquaternions. The study addresses stringent demands for speed and accuracy in modern robotic systems — particularly in safety-critical domains like medicine and nuclear industry, where delays or computational errors may lead to critical consequences. Each method’s mathematical framework is described. A detailed, operation-level analysis of computational complexity is conducted, accounting for arithmetic and trigonometric operations. Emphasis is placed on practical performance when implemented on the ARM Cortex-M4F microcontroller — widely used in robotic control. Experimental comparisons are performed across compilation modes and single/double-precision data. Results show that, despite higher theoretical operation counts, quaternion and biquaternion methods outperform the classical matrix approach in execution speed. This stems from fewer memory accesses, simpler structure enabling efficient pipelining, and faster trigonometric evaluations via half-angle formulations. Moreover, these methods offer compact representation and key advantages: elimination of gimbal lock, smooth orientation interpolation, robustness against numerical error accumulation, and efficient Jacobian computation — making them especially valuable for real-time robotic systems with constrained computational resources.