Aerospace and Electronic Systems Magazine September 2016 - 32
An Autonomous Quadrotor Avoiding a Helicopter in Low-Altitude Flights
integer quadratic programming (MIQP) becomes computationally
expensive when the number of binary variables increases.
In [34], the collision avoidance problem between multiple
quadrotors is solved by a decentralized optimal control approach.
The optimal control is defined as the maximum acceleration with
a constant optimal heading. Consequently, a real-time avoid set
can be computed. The avoidance is activated only when collision
becomes inevitable. However, the safety controller is derived from
a simplified 2D model. Once the controller is applied to a full 3D
dynamic model, the optimality is no longer guaranteed due to delays from rotational inertia (RI) and aerodynamic drag. Although
the model difference is almost negligible in the low-acceleration
(~ 1.5 m/s2) low-speed (~ 2 m/s) drone-drone avoidance scenario
[34], it is no longer valid in the high-acceleration (~ 10 m/s2) highspeed (~ 100 m/s) drone-helicopter scenario.
In this article, we modified the safety controller in [34] to fit
our drone-helicopter scenario, and demonstrated that the avoidance could be handled much more efficiently. First, we only use the
safety set from [34] as an initial guess, and enlarge the safety set
via an iterative algorithm to account for uncertainties already mentioned. Second, the optimal control is only applied to the quadrotor, and the helicopter flies freely as if the quadrotor does not exist.
These are the control inputs for altitude and Euler angles. The constraints in (1) represent the physical capacity of the motors.
Other forces are gravity mg, translation drag ctV, rotational
drag crw2, and Coriolis forces from quadrotor body rotation and
motor rotations. Define
to be the rotation matrix from to
, and Rv to be a linear transformation from to w. The statespace equations could be written compactly as (2). We only give
a brief summary of the model to introduce enough notations for
controller conversions. For details, refer to [37].
(1)
(2)
DYNAMIC MODEL
First, we need a quadrotor dynamic model, which is used later. We
adopt the same model introduced in [37], with a different notation.
Define a fixed north-east-down (NED) inertial world frame and
a noninertial body frame attached to the center of gravity of the
quadrotor. The following is a list of variables used to describe the
dynamics of a quadrotor.
X = [X Y Z]T: quadrotor position in
HORIZONTAL SAFETY CONTROLLER FROM KINEMATIC
MODEL
;
V = [VX VY VZ]T: quadrotor velocity in
;
Q = [f q y] : Euler angles roll, pitch, and yaw in , respectively;
T
w = [wx wy wz]T: quadrotor angular velocity in ;
m: quadrotor mass;
I = diag(Ix, Iy, Iz): mass moment of inertia in ;
: motor speeds, i = 1, 2, 3, 4;
: sum of motor speeds;
kf, km: motor thrust and torque coefficients, respectively;
ct, cr: translational and rotational friction coefficients, respectively;
l: moment arm from the origin of
to each motor.
g: gravitational acceleration, [0 0 g]T in NED frame with g =
9.81 m/s2.
Assume that the motor forces are proportional to , and the
control inputs U satisfy (1) with input constraints. Physically, the
control inputs U1, U2l, U3l, U4 represent the total thrust and the total motor torques along the roll, pitch, and yaw axes, respectively.
32
SAFETY CONTROLLER
To keep the system implementable in real time, the safety controller is derived from a simplified horizontal kinematic model [34].
The safety controller includes two components. The first component is an optimal horizontal avoidance maneuver,
,
which represents the quadrotor's acceleration
with magnitude
and direction
, where is measured from
the positive x-direction in the relative frame centered at the helicopter (Figure 1). The second component is a switching surface, or
the avoid set boundary ∂Kh, in which the optimal avoidance maneuver must be initiated to avoid intrusion into a safety set Sh.
We elaborate the two components below in details.
Figure 1 highlights some position trajectories in solid red lines
under optimal control . The helicopter is at the origin, and the
quadrotor has a relative position x(t) = (x(t), y(t)) and is heading
toward the helicopter with relative velocity
. The quadrotor
should not enter the safety set Sh in the present time t = 0, defined
by the solid gray circle with radius rmin around the helicopter. The
set of all x(0) lies on the boundary of ∂Sh.
To avoid entering Sh, the quadrotor starts the avoidance maneuver at some initial time t = t0 in the past when touching the avoid
set Kh. The set of all possible x(t0) defines the boundary of the
avoid set, ∂Kh, indicated by the dashed red curve in Figure 1. The
IEEE A&E SYSTEMS MAGAZINE
SEPTEMBER 2016
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