Modelling and Control of a Flexible Structure for Robotic Surgery

Kurzfassung

Introduction: Tongues and other mammalian organs are composed almost entirely out of muscles that realize motion such as bending, extension and torsion in a highly complex way. Within the robotic context, such structures are called continuous robots. They consist of a continuum that provides structural support and flexibility necessary for motion at the same time. The necessary actuators are placed in a certain pattern around them, depending on their type and properties. We assumed that planar motion of such a continuum can be modelled as the transverse motion of an elastic beam by the use of the classical beam theories. In the present work, we aimed to investigate those beam theories, built up a testbed with a compliant mechanism and, furthermore, implement a position controller to command the transverse motion. Methods: The equations of motions for the Euler-Bernoulli and Timoshenko beam theory were established using the Lagrangian formalism and solved analytically. As a proof, a modal analysis using the finite element method was carried out. The kinematics to describe the mechanism’s tip position for a concentrated tip moment were described using differential geometry. To verify the results, a testbed was built up consisting of a compliant mechanism and actuators which allowed the introduction of a moment or a transverse shear force at the mechanism’s tip. The compliant mechanism itself was designed in a way, that we were able to observe a significant difference between the two beam theories. We used silicone as the material of the mechanism’s continuum. To investigate the elastic properties of this elastomer, we conducted standardised and modified material tests, such as the compression or 3-point-bending test. Results: Within the material tests, the Young’s modulus showed a non-linear behaviour, while the Poisson’s ratio was found to be a constant. The accuracy of the tip angle using the Timoshenko theory and those material parameters never exceeded an error of 2% for the actuation via tip moment. Using the constant curvature approximation, we achieved a tip position accuracy with a maximum error of 4% for the lateral deflection and 12% for the axial compression of the mechanism. With higher pretension in the tendons, the results lost accuracy. By the use of the proposed position controller, we achieved an accuracy of 22% for a commanded tip angle. Discussion: We aimed to built up a testbed for a continuous robot and proofed accuracy for the tip angle and lateral position using the Timosehnko theory. In x-direction, the accuracy was lower. The reasons for those inaccuracy were found to be geometrical non-linearities resulting from the loading condition in the testbed, tendon elasticity and friction. Conclusion: Further steps are investigations concerning tendon elasticity and friction to improve accuracy of the position controller. As well, one should consider the use of the exact or Cosserat beam theory to include the undefined load case which was found to have influence for higher deflections.