Details

Drones (quadcopters or unmanned aerial vehicles) are popular because of their agility and ease of control. The ability to derive the optimal control sequence to reach a given position (or alternately, to escape a current position) is useful for collision avoidance, path planning and evasion. This Demonstration presents analytical equations to calculate the time-limited reachable set, which is the set of all position states reachable by the quadcopter at time . To make the problem tractable, the simplified two-dimensional model of quadcopter dynamics proposed in [1, 2] is used. A three-dimensional model can be approximated by revolving the two-dimensional plot about the axis.

This model is integrated using the set of candidate optimal controllers. The drone state is described by the quadcopter vertical position , horizontal position and pitch angle . Specifically, this model assumes that the pitch angular velocity can be controlled directly, justifying this assumption because modern quadcopters can reach high angular accelerations, but angular velocity is generally limited by the on-board gyroscopic sensors. This simplification reduces the model order.

The dynamic model equation assumes gravity , drone mass , and two inputs, the thrust and rotation rate :

,

,

.

The system is nondimensionalized by scaling by the maximum rotation rate ω_:

,

,

.

The nondimensionalized dynamic model equations are then:

,

,

.

This reduces the model to have just two parameters. The thrust control input is the ratio of force over drone mass

,

and is bounded:

.

The nondimensionalized rotation control input is bounded to .

However, the candidate optimal control inputs are always either turning at the maximum rate or flying straight: UR={1,-1,0}, and the optimal thrust inputs are full or minimum . To compute the final position of the drone, the pitch is integrated and the accelerations are double-integrated. The optimal control inputs that generate the boundary of the reachable set are drawn in bold colors on both the plot and the control inputs.

References

[1] R. Ritz, M. Hehn, S. Lupashin and R. D’Andrea, “Quadrocopter Performance Benchmarking Using Optimal Control,” in IEEE International Conference on Intelligent Robots and Systems (IROS), San Francisco, CA, Piscataway, NJ: IEEE, 2011 pp. 5179–5186. doi:10.1109/IROS.2011.6094775.

[2] M. Hehn, R. Ritz and R. D'Andrea, "Performance Benchmarking of Quadrotor Systems Using Time-Optimal Control," Autonomous Robots, 33(1–2), 2012 pp. 69–88. doi:10.1007/s10514-012-9282-3.