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The following paper talks about the dynamic modeling and control of robot manipulators using Hamilton's equations in the screw theory framework. The difference between the proposed work with diverse methods in the literature is the ease of obtaining the laws of control directly with screws and co-screws, which is considered modern robotics by diverse authors. In addition, geometric algebra (GA) is introduced as a simple and iterative tool to obtain screws and co-screws. On the other hand, such as the controllers, the Hamiltonian equations of motion (in the phase space) are developed using co-screws and screws, which is a novel approach to compute the dynamic equations for robots. Regarding the controllers, two laws of control are designed to ensure the error's convergence to zero. The controllers are computed using the traditional feedback linearization and the sliding mode control theory. The first one is easy to program and the second theory provides robustness for matched disturbances. On the other hand, to prove the stability of the closed loop system, different Lyapunov functions are computed with co-screws and screws to guarantee its convergence to zero. Finally, diverse simulations are illustrated to show a comparison of the designed controllers with the most famous approaches. [ABSTRACT FROM AUTHOR]

The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6 × n or n × n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C , such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation. [ABSTRACT FROM AUTHOR]

• A new overconstrained 2R1T parallel manipulator with fixed actuators. • Only 13 single-DOF joints, simple inverse kinematics and few singularities. • Great performances in the levels of kinematics, stiffness and dynamics. • Suitable parallel module of five-axis hybrid machines through machining experiment. • Addition and multiplication in the geometric algebra avoid complex calculations. Five-axis hybrid machine tools are important equipment for highly efficient precision machining of complex workpieces, in which a key part is parallel manipulators (PMs) with two rotations and one translation (2R1T). In view of many single-degree-of-freedom (DOF) joints and virtual output rotational axes in most existing 2R1T PMs, a new 2R1T 2PRU-PSR PM with two certain rotational axes is proposed, which is composed of 13 single-DOF joints and only one S joint and is fully actuated with stationary actuators. Using geometric algebra as a mathematical tool, the mobility and kinematic performance evaluation considering the motion/force transmissibility are presented. With the motion/force transmissibility taken as the optimal index, the dimensional parameters of the 2PRU-PSR PM are optimized to obtain an improved transmission workspace without singular configurations. Based on the optimized parameters, stiffness and dynamic analyses of the 2PRU-PSR PM are presented in the framework of geometric algebra. The results reveal that the 2PRU-PSR PM exhibits excellent performance in the levels of kinematics, stiffness, and dynamics. Finally, a five-axis hybrid prototype is built based on the optimized 2PRU-PSR PM and two serial guides. Machining experiments show that the 2PRU-PSR PM is suitable as a parallel module for five-axis hybrid machines. [ABSTRACT FROM AUTHOR]

This work addresses the inverse kinematics of serial robots using conformal geometric algebra. Classical approaches include either the use of homogeneous matrices, which entails high computational cost and execution time, or the development of particular geometric strategies that cannot be generalized to arbitrary serial robots. In this work, we present a compact, elegant and intuitive formulation of robot kinematics based on conformal geometric algebra that provides a suitable framework for the closed-form resolution of the inverse kinematic problem for manipulators with a spherical wrist. For serial robots of this kind, the inverse kinematics problem can be split in two subproblems: the position and orientation problems. The latter is solved by appropriately splitting the rotor that defines the target orientation in three simpler rotors, while the former is solved by developing a geometric strategy for each combination of prismatic and revolute joints that forms the position part of the robot. Finally, the inverse kinematics of 7 DoF redundant manipulators with a spherical wrist is solved by extending the geometric solutions obtained in the non-redundant case. • The IK of serial robots with a spherical wrist is solved in closed-form. • The position problem is solved by the manipulation of different geometric entities. • The orientation problem is solved by the algebraic manipulation of a rotor. • The developed method is extended to 7 DoF serial robots with a spherical wrist. [ABSTRACT FROM AUTHOR]