Control of Quadrotors Using the Hopf Fibration on SO(3)

Image-based estimation, planning, and control for high-speed flying through multiple openings

“…For example, Mellinger and Kumar (2011) found that the 3D position and yaw angle represent flat outputs in Cartesian space. The snap and second derivative of yaw, which relates the input moment, were minimized (Bry et al, 2015; de Almeida and Akella, 2017; Hehn and D’Andrea, 2011; Liu et al, 2018; Loianno et al, 2017; Mellinger and Kumar, 2011; Mellinger et al, 2012; Thomas et al, 2016b; Vitus et al, 2012; Watterson and Kumar, 2018). In addition, a process for rapid trajectory generation and verification of feasibility of motion primitives for quadcopters was developed by Mueller et al (2015).…”

“…Linear control schemes such as proportional–integral–derivative (PID) (Bouabdallah et al, 2005), linear–quadratic regulator (LQR) (Foehn and Scaramuzza, 2018; Hoffmann et al, 2007), and H ∞ (Raffo et al, 2010) have been developed to stabilize the vehicle during hover, and these techniques assume a linearized robot model. Nonlinear flight control has been developed using feedback linearization (Lee et al, 2009; Mokhtari et al, 2006), NMPC (Neunert et al, 2016), backstepping (Bouabdallah and Siegwart, 2005; Guo et al, 2018; Hamel and Mahony, 2002; Klausen et al, 2017; Mebarki et al, 2015; Serra et al, 2016), and geometric control to deal with the nonlinear dynamics (Bry et al, 2015; Liu et al, 2018; Loianno et al, 2017; Mellinger and Kumar, 2011; Mellinger et al, 2012; Thomas et al, 2016b; Watterson and Kumar, 2018). In backstepping control, the translational and rotational dynamics are stabilized automatically instead of canceling the nonlinear terms as in feedback linearization (Härkegård and Glad, 2001), but the convergence of the position errors depends heavily on, and are limited by, the speed of convergence of the attitude error (Thomas et al, 2016b).…”

“…Finally, the proposed control approach guarantees precise tracking of the invariant feature trajectories and achieves agile maneuvers in three dimensions by actively observing the target. In contrast, state-of-the-art nonlinear control techniques for aggressive flight have been designed in the Cartesian coordinate (Bry et al, 2015; Liu et al, 2018; Loianno et al, 2017; Mellinger and Kumar, 2011; Mellinger et al, 2012; Thomas et al, 2016b; Watterson and Kumar, 2018). Furthermore, existing image-based flight controllers are either limited to low-speed maneuvers (Hamel and Mahony, 2002; Mebarki et al, 2015; Serra et al, 2016) or are constrained by planar motion (Thomas et al, 2014).…”

PANTHER: Perception-Aware Trajectory Planner in Dynamic Environments

“…This closeness between the hovering condition and the singularity can limit the set of possible accelerations in aggressive flights, since an optimal solution that passes through or close to this singularity can provoke numerical instabilities and/or lead to artificial large changes in orientation. Recently, the Hopf map was leveraged in [36] to place the singularity in the inverted ("upside-down") configuration, which is independent of ψ and has the farthest possible angle away from the hovering condition. Although flying highly aggressive trajectories is not the main goal of this work, we decide to use the Hopf map (as opposed to the commonly-used maps presented in [33], [37]) since it automatically maximizes the distance to the singularity by simply changing the definition of the map.…”

“…Although flying highly aggressive trajectories is not the main goal of this work, we decide to use the Hopf map (as opposed to the commonly-used maps presented in [33], [37]) since it automatically maximizes the distance to the singularity by simply changing the definition of the map. In [36], however, the Hopf fibration was only used in the controller to track predefined trajectories. It was also leveraged in [38] to find the set of charts for a previously-optimized position trajectory, which are then used for the controller and to obtain the ψ trajectory.…”