Dynamics of Rigid and Flexible Manipulators Using Screw Theory
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This thesis presents contributions within dynamic modeling of rigid and flexible open-chain holonomic multibody systems, as well as contributions in force analysis of such systems. Although application of the modeling and force analysis procedures to offshore cranes is discussed, the presented procedures are given in a general and systematic manner, such that they can be applied for various robotic manipulators. A dynamic modeling procedure for rigid multibody systems is based on Kane’s equation of motion where velocities and angular velocities are represented by twists, while partial velocities and partial angular velocities are represented as lines or screws. Kinematics are conveniently derived using screw transformations, which leads to a systematic procedure with a clear geometric interpretation. Screws and lines associated with partial velocities and partial angular velocities are arranged as columns in projection matrices, which serve both as velocity Jacobians and force/torque projection operators on the directions of the velocities associated with the generalized speeds. A method for the determination of constraint forces (i.e., reaction forces) in rigid multibody systems is given as an extension of the dynamic modeling procedure. Constraint forces are represented by constraint wrenches, which are given as a product of an orthogonal screw basis and a vector of unknown magnitudes of constraint forces and torques. Orthogonal screw bases can be transformed using screw transformations, and are arranged as columns of auxiliary projection matrices, which serve as force/torque projection operators on the directions of the unknown constraint forces. A method for the dynamic modeling of flexible manipulators driven by linear actuators is an extension of the rigid body formulation, where, in addition to lines and screws describing rigid body velocities, screw matrices for modeling of elastic motion are introduced. This way both rigid body velocities and velocities associated with elastic deformation are conveniently unified in a framework of screw theory. This allows for using the same mathematical procedures developed for rigid body dynamics. A special case of flexible manipulators driven by linear actuators is considered, where only the flexibility of links in the actuator planes is taken into account. A small deformation assumption is used. A method for the determination of constraint forces in flexible manipulators driven by linear actuators is again given as an extensions of the dynamic modeling procedure and is directly derived from the method for determination of constraint forces in rigid body systems. The final development of this work is a method for derivation of linearized equations of motion for flexible multibody systems based on screw theory and dual algebra. Mathematical framework of the method is based on the proposed novel Lie groups of dual rotation matrices and dual homogeneous transformation matrices. The method is based on Kane’s linearized equation of motion for flexible systems, where velocities and angular velocities are represented by dual twists, which are a vector form of Lie algebras associated with the group of dual homogeneous transformations. Then partial velocities and partial angular velocities are given by dual lines or dual screws, which are arranged as columns in dual projection matrices. Dual projection matrices are given of two types: the Jacobian form and the projecting form, which are velocity Jacobians and force/torque projection operators respectively. Dual screw transformations are introduced for performing the derivations. Detailed examples of implementation of the proposed methods for a single-boom crane are given, as well as more complex examples are given in the articles included in this thesis.