Quaternion-based state-dependent differential Riccati equation for quadrotor drones: Regulation control problem in aerobatic flight

Abstract

The quaternion is a powerful and common tool to avoid singularity in rotational dynamics in three-dimensional (3D) space. Here it has been particularly used as an alternative to Euler angles and rotation matrix. The application of the quaternion is exercised in quadrotor modeling and control. It changes the dynamics and represents a singularity-free attitude model. Here for the first time (for the best knowledge of authors), the state-dependent differential Riccati equation (SDDRE) control has been implemented on the quaternion-based model of a quadcopter. The proposed control structure is capable of aerobatic flight, and the Pugachev’s Cobra maneuver is chosen to assess the capability of the quaternion-based SDDRE approach. The introduced control simulator is validated by comparison with conventional dynamics based on Euler angles, controlled using a proportional-derivative (PD) controller on a normal regulation flight. The simulator successfully performed the Cobra maneuver and also validated the proposed structure. The more precision in regulation along with lower energy consumption demonstrated the superiority of the introduced approach.

Design and Development of an Air–Land Amphibious Inspection Drone for Fusion Reactor

“…The above equation is a Riccati matrix differential equation which is nonlinear, while P(t) is symmetric [42]. Therefore, only n(n+1) 2 sets of first-order differential equations need to be solved to obtain the value of P(t).…”

Approximate optimal control design for quadrotors: A computationally fast solution

“…where R (∶,3) (q) is the third column of R(q), T B (N) is thrust aligned with z axis of the body frame, 𝔤(m∕s 2 ) is gravitational constant, m(kg) is the quadrotor mass, e 3 ≜ [0, 0, 1] T , and D(kg∕s) ∈ ℜ 3×3 is a diagonal matrix collecting the drag and aerodynamic parameters. 38 The attitude dynamics can be described as…”

“…The translational dynamic equation can be expressed as $$$$ m\ddot{\boldsymbol{\xi}}={\mathbf{R}}_{\left(:,3\right)}\left(\mathbf{q}\right){T}_B-m\mathfrak{g}{\mathbf{e}}_3-\mathbf{D}\dot{\boldsymbol{\xi}}, $$$$ where $$$ {\mathbf{R}}_{\left(:,3\right)}\left(\mathbf{q}\right) $$$ is the third column of $$$ \mathbf{R}\left(\mathbf{q}\right) $$$, $$$ {T}_B\left(\mathrm{N}\right) $$$ is thrust aligned with $$$ z $$$ axis of the body frame, $$$ \mathfrak{g}\left(\mathrm{m}/{\mathrm{s}}^2\right) $$$ is gravitational constant, $$$ m\left(\mathrm{kg}\right) $$$ is the quadrotor mass, $$$ {\mathbf{e}}_3\triangleq {\left[0,0,1\right]}^T $$$, and $$$ \mathbf{D}\left(\mathrm{kg}/\mathrm{s}\right)\in {\mathfrak{\Re}}^{3\times 3} $$$ is a diagonal matrix collecting the drag and aerodynamic parameters 38 …”

Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control

“…The SDDRE is a nonlinear closed-loop suboptimal controller with the power of finite-time control by penalizing a terminal boundary condition [13,14]. Tripathy used the state-dependent differential Riccati equation for manipulator control [15].…”

“…The SDDRE is a nonlinear closed-loop suboptimal controller with the power of finite-time control by penalizing a terminal boundary condition. 13,14 Tripathy et al 15 used the SDDRE for manipulator control. There are several solution methods to the differential Riccati equation, such as backward integration, 16 Lyapunov-based method, 17,18 and state transition matrix approach.…”