Operational Space Control for Planar PAN–1 Underactuated Manipulators Using Orthogonal Projection and Quadratic Programming

Abstract

In this paper, we propose an operational space control formulation for a planar N-link underactuated manipulator (PA <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N–1</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> with a passive first joint subject to actuator constraints (N ⩾ 3), covering both stabilization and tracking tasks. Such underactuated manipulators have an inherent first-order nonholonomic constraint, allowing us to project their dynamics to a space consistent with the nonholonomic constraint. Based on the constrained dynamics, we can design operational space controllers with respect to tasks assuming that all joints of the manipulator are active. Due to underactuation, we design a Quadratic Programming (QP) based controller to minimize the error between the desired torque commands and available motor torques in the null space of the constraint, as well as involve the constraint of motor outputs. The proposed control framework was demonstrated by stabilization and tracking tasks in simulations with both planar PA <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and PA <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> manipulators. Furthermore, we verified the controller experimentally using a planar PA <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> robot.