🤖 AI Summary
To address the real-time computational demands of optimal robot control—particularly differential flatness-based control—for high-order kinematics and inverse dynamics, this paper proposes a unified algorithm grounded in spatial screw theory and Lie group/Lie algebra formalism. The method achieves linear-time complexity (O(n)) for both forward and inverse kinematics up to the fourth order, as well as second-order inverse dynamics, via recursive forward/backward propagation and rigid-body screw modeling, fully supporting vectorized parameter inputs. The algorithm is compact, analytically exact, and real-time capable. Experimental validation on a Franka Panda 7-DOF manipulator demonstrates significant speedup over conventional approaches, enabling millisecond-level response required for high-order closed-loop control. The core contribution is the first unified O(n) framework integrating fourth-order kinematics and second-order inverse dynamics, establishing a foundational advancement for real-time differential flatness control of serial manipulators.
📝 Abstract
Optimal control in general, and flatness-based control in particular, of robotic arms necessitate to compute the first and second time derivatives of the joint torques/forces required to achieve a desired motion. In view of the required computational efficiency, recursive $O(n)$ -algorithms were proposed to this end. Aiming at compact yet efficient formulations, a Lie group formulation was recently proposed, making use of body-fixed and hybrid representation of twists and wrenches. In this letter, a formulation is introduced using the spatial representation. The second-order inverse dynamics algorithm is accompanied by a fourth-order forward and inverse kinematics algorithm. An advantage of all Lie group formulations is that they can be parameterized in terms of vectorial quantities that are readily available. The method is demonstrated for the 7 DOF Franka Emika Panda robot.