A study of a novel modular variable geometry frame arranged as a robotic surface

The novel concept of a variable geometry frame is introduced and explored through a three-dimensional robotic surface which is devised and implemented using triangular modules. The link design is optimized using surplus motor dimensions as firm constraints, and round numbers for further arbitrary constraints. Each module is connected by a passive six-bar mechanism that mimics the constraints of a spherical joint at each triangle intersection. A three dimensional inkjet printer is used to create a six-module prototype designed around surplus stepper motors powered by an old computer power supply as a proof-of-concept example. The finite element method is applied to the static and dynamic loading of this device using linear three dimensional (6 degrees of freedom per node) beam elements to calculate the cartesian displacement and force and the angular displacement and torque at each joint. In this way, the traditional methods of finding joint forces and torques are completely bypassed. An efficient algorithm is developed to linearly combine local stiffness matrices into a full structural stiffness matrix for the easy application of loads. This is then decomposed back into the local matrices to easily obtain joint variables used in the design and open-loop control of the surface. Arbitrary equation driven surfaces are approximated ensuring that they are within the joints limits. Moving shapes are then calculated by considering the initial position of the surface, the desired position of the surface, and intermediate shapes at discrete times along the desired path. There are no sensors on the prototype, but feedback models and state estimators are developed for future use. These models include shape sampling methods derived from existing meshing algorithms, trajectory planning using sinusoidal acceleration profiles, spline-based path approximation to allow lower curvature paths able to be traversed more quickly and/or able to be travelled with a constant velocity and optimized by iteratively calculating actuator saturation with no discontinuities, and the optimal tracking of a desired path (modeled with a time-varying ricatti equation)