Modeling and LQR Control of Insect Sized Flapping Wing Robot

Abstract

Flying insects can perform rapid, sophisticated maneuvers like backflips, sharp banked turns, and in-flight collision recovery. To emulate these in aerial robots weighing less than a gram, known as flying insect robots (FIRs), a fast and responsive control system is essential. To date, these have largely been, at their core, elaborations of proportional-integral-derivative (PID)-type feedback control. Without exception, their gains have been painstakingly tuned by hand. Aggressive maneuvers have further required task-specific tuning. Optimal control has the potential to mitigate these issues, but has to date only been demonstrated using approxiate models and receding horizon controllers (RHC) that are too computationally demanding to be carried out onboard the robot. Here we used a more accurate stroke-averaged model of forces and torques to implement the first demonstration of optimal control on an FIR that is computationally efficient enough to be performed by a microprocessor carried onboard. We took force and torque measurements from a 150 mg FIR, the UW Robofly, using a custom-built sensitive force-torque sensor, and validated them using motion capture data in free flight. We demonstrated stable hovering (RMS error of about 4 cm) and trajectory tracking maneuvers at translational velocities up to 25 cm/s using an optimal linear quadratic regulator (LQR). These results were enabled by a more accurate model and lay the foundation for future work that uses our improved model and optimal controller in conjunction with recent advances in low-power receding horizon control to perform accurate aggressive maneuvers without iterative, task-specific tuning.

Figures (6)

• Figure 1: RoboFly, an insect-sized flapping robot weighing 146 milligrams, hovers next to a flower using feedback from motion capture cameras. The robot performs this hovering maneuver using the LQR controller reported in this work.
• Figure 2: Torque and Thrust Generation Mechanism in FIRs: (Top) Inspired by the work in ddhingraTrimming, this figure shows that changing the signal parameters $\delta A$ and $V_o$ introduces roll and pitch torques, respectively. Here, $V_{bias}$ is the bias voltage. (Bottom) Mapping of (a) thrust, (b) roll torque, and (c) pitch torque of the RoboFly used in this work. The thrust mapping is obtained using a high-precision scale, while the torque mappings are obtained using the torque measurement device introduced in ddhingraTrimming. Pink dots represent the collected data points, and the green line represents the linear fit of the data. The corresponding equations for these linear fits are provided in Table \ref{['MAtable']}.
• Figure 3: Visualization of the collected data: The graph shows the robot achieving high attitude angles greater than 30$^\circ$ and corresponding lateral/longitudinal speeds exceeding 0.4 m/s in the collected data. This highlights significant perturbations, which will be used to validate the stroke-averaged dynamics developed in this work. The color intensity on the graph represents the density of data points.
• Figure 4: Model validation plots: Measured accelerations (green) from the RoboFly trajectories plotted with the predicted accelerations (pink) calculated using the theoretical model.
• Figure 5: LQR control loop and hovering trajectories: (a) The LQR control loop used to perform hovering and trajectory tracking maneuvers. Here, R represents the 3-2-1 rotation matrix. (b) The plot displays five different hovering trajectories, showing the robot maintaining a stable attitude and remaining close to the starting position. The mean RMS error and standard deviation for the trajectories are 4.17$\pm$ 0.37 cm.