We propose an effective method for solving the inverse kinematic problem of a specific model of 6-degree-of-freedom (6-DOF) robot manipulator using computer algebra. It is known that when the rotation axes of three consecutive rotational joints of a manipulator intersect at a single point, the inverse kinematics problem can be divided into determining position and orientation. We extend this method to more general manipulators in which the rotational axes of two consecutive joints intersect. This extension broadens the class of 6-DOF manipulators for which the inverse kinematics problem can be solved, and is expected to enable more efficient solutions. The inverse kinematic problem is solved using the Comprehensive Gr obner System (CGS) with joint parameters of the robot appearing as parameters in the coefficients to prevent repetitive calculations of the Gr obner bases. The effectiveness of the proposed method is shown by experiments.

Camera-to-robot (also known as eye-to-hand) calibration is a critical component of vision-based robot manipulation. Traditional marker-based methods often require human intervention for system setup. Furthermore, existing autonomous markerless calibration methods typically rely on pre-trained robot tracking models that impede their application on edge devices and require fine-tuning for novel robot embodiments. To address these limitations, this paper proposes a model-based markerless camera-to-robot calibration framework, ARC-Calib, that is fully autonomous and generalizable across diverse robots and scenarios without requiring extensive data collection or learning. First, exploratory robot motions are introduced to generate easily trackable trajectory-based visual patterns in the camera's image frames. Then, a geometric optimization framework is proposed to exploit the coplanarity and collinearity constraints from the observed motions to iteratively refine the estimated calibration result. Our approach eliminates the need for extra effort in either environmental marker setup or data collection and model training, rendering it highly adaptable across a wide range of real-world autonomous systems. Extensive experiments are conducted in both simulation and the real world to validate its robustness and generalizability.