Geometric modeling and inverse dynamics of flexible manipulators

Efficiency is a central question in robotic developments nowadays. Robotic manipulators are designed to be more energy efficient and safer for their surroundings. This leads to the integration of lightweight and flexible components in the manipulators. The resulting elastic behavior affects the overall precision of the system and needs to be considered in the design of control laws. This work aims at developing tools able to model the complex behavior of flexible spatial manipulators and generate appropriate controls to achieve a given task.
Flexible robotic manipulators are modeled using a finite element approach allowing the description of spatial systems in a general and systematic way. Differential geometry tools are used to describe the kinematics and dynamics of the flexible system: thanks to a formulation on the SE(3) matrix group, also known as the group of rigid body motions, the equations describing the complex spatial motion of the system are expressed in the local frame of the system and show reduced non-linearities compared to traditional methods.
For trajectory tracking of robotic manipulators, the de nition of appropriate control inputs to drive the system is often based on the inverse dynamics of the system. However, the inverse dynamics of flexible systems can exhibit unstable internal dynamic behavior that needs to be taken care of. Here, based on a model of the system, the inverse dynamics problem is solved by formulating it as an optimization problem: an objective function is minimized under constraints imposed by the tracking task. The proposed method is general enough to deal with systems having serial or parallel kinematic topologies, flexible joints and/or links and even systems with unstable internal dynamics.
The generated feedforward inputs are first applied on simulation examples. The resulting behaviors of the flexible systems are discussed and compared to examples available in the literature. Two real robotic system are then considered. It is shown that realistic inputs can be generated with the method and that a reduction of unwanted vibrations inside the arm is obtained. Also, the real output error from the prescribed trajectory is reduced.