High-Performance Six-DOF Flight Control of the Bee$^{++}$: An Inclined-Stroke-Plane Approach

Abstract

We present a new method for synthesizing and implementing high-performance <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">six-degree-of-freedom</i> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{6}$</tex-math></inline-formula> -DOF) flight controllers for the Bee <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{++}$</tex-math></inline-formula> , an insect-scale flying robot driven by four independently-actuated flapping wings. Each wing of the Bee <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{++}$</tex-math></inline-formula> is installed with a preset orientation such that the stroke plane generated during flight is inclined, thus enabling reliable roll, pitch, and yaw torque generation. Leveraging this capability, we propose a Lyapunov-based nonlinear control architecture that enables closed-loop position and attitude regulation and tracking. The control algorithms presented in this article simultaneously stabilize position and attitude by independently varying the wingstroke amplitudes of the four flapping wings of the Bee <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{++}$</tex-math></inline-formula> . We use this particular control architecture to exemplify the process of controller synthesis and real-time implementation; however, the aerodynamic design of the Bee <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{++}$</tex-math></inline-formula> is compatible with a great variety of control structures and performance objectives. As a main result, we present the first set of experimental data demonstrating sustained and robust high-performance tracking of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{6}$</tex-math></inline-formula> -DOF reference signal during flight at the insect scale, which has been a long-standing control problem in the field of flapping-wing microrobotics. Furthermore, using data obtained through a series of systematic flight tests, we show that the Bee <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{++}$</tex-math></inline-formula> can achieve the highest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{6}$</tex-math></inline-formula> -DOF performance ever recorded for an insect-scale flapping-wing flying robot during sustained flight.