Formal Verification of Robot Rotary Kinematics

Abstract

With the widespread application of robots in aerospace, medicine, automation, and other fields, their motion safety is essential for the well-being of humans and the accomplishment of vital socially beneficial programs. Conventional robot hardware and software designs mainly rely on experiential knowledge and manual testing to ensure safety, but this fails to cover all possible testing paths and adds risks. Alternatively, formal, mathematically rigorous verifications can provide predictable and reliable guarantees of robot motion safety. To demonstrate the feasibility of this approach, we formalize the mathematical coordinate transformation of a robot’s rigid-body kinematics using the Coq Proof Assistant to verify the correctness of its theoretical design. First, based on record-type matrix formalization, we define and verify a robot’s spatial geometry by constructing formal expressions of the matrix’ Frobenius norm, trace, and inner product. Second, we divide rotary motion into revolution and rotation construct and provide their formal definitions. Next, we formally verify the rotational matrices of angle conventions (e.g., roll–pitch–yaw and Euler), and we complete the formal verification of the Rodriguez formula to formally verify the correctness of the motion theory in specific rotating kinematics problems. The formal work of this paper has a variety of essential applications and provides a generalizable kinematics analysis framework for robot control system verification. Moreover, it paves the way for automatic programming capabilities.