Actuation Control for Nanostructured Origami™
Authors: Stellman Paul , Barbastathis George 
Published: April 2008

The fabrication of arbitrary nanostructured devices in 3D space is relevant to many areas of academic and industrial research. From hybrid systems with various physical features to complex 3D optical interconnects, the added functionality gained by 3D nanomanufacturing is promising for the development of novel applications. Nevertheless, the 2D nature of conventional nanomanufacturing processes (i.e. lithography) underutilizes the 3rd dimension since there is currently no infrastructure for 3D. Nanostructured Origami has been proposed [1-3] as one solution to the 3D nanomanufacturing problem. The two-step process consists of first patterning devices and creases (axes of rotation) in 2D followed by a subsequent folding step which actuates the origamis to its final 3D shape. Several actuation mechanisms have been investigated for the folding step, and the folding of simple origamis with an open kinematic chain has been successfully demonstrated experimentally [1-3]. Since the origami segments must be accurately aligned in the 3D folded state, the actuation mechanisms for Nanostructured Origami must be both controllable and repeatable. By developing analytical models of the origamis, control schemes can be simulated to aid in the manufacturing of devices in the laboratory. As an example, a PD control scheme is introduced to achieve set-point position control of an example origami, the corner cube. In the laboratory, a PD control system would be built using a magnetic feedback mechanism. A strip of gold is patterned as a hinge material, and electrical current passes through the wire. In the presence of a magnetic field, the Lorentz force acts upon the origami segments and the resulting torque is given by τ = Cicos α,[Equation] where C is a positive constant, i is the current, and α is the angle between the magnetic field and the current. The PD control law for Nanostructured Origami is equivalent to PD control of an articulated robotic manipulator, with the exception that gravity can be ignored due to the low masses of the membranes. Instead, the stiffnesses of the hinges must be balanced, resulting in a control torque of [Equation] where τ is the vector of joint torques, G is the constraint Jacobian, Kp is the proportional control constant, Kd is the derivative control constant, K is the hinge stiffness matrix, q is the vector of joint angles, and qd is the desired steady-state values of the joint angles. This input torque is applied to the origami device, and the response is calculated by integrating the system's equations of motion [3]. The angular response of the PD controller for the corner cube origami is plotted in Fig. 1, and Fig. 2 shows a schematic of the folding of the corner cube from flat to folded state. Note the well-behaved response for a Kp value of 1500, which demonstrates zero overshoot and a rise time of approximately 15 milliseconds. A plot of the joint torques as a function of time is shown in Fig. 3. This abstract has briefly introduced the use of a PD controller for the actuation of origami devices. For a Lorentz force actuation scheme, we have demonstrated through simulations that the PD control law is stable and robust. If complicated 3D origamis with multiple closed kinematic chains are to be built, detailed control laws must be implemented. Advanced control techniques, such as optimal control, will be investigated to explore improved actuation strategies.