Aerospace and Electronic Systems Magazine September 2016 - 33

Liu and Foina

(5)

Use Pontryagin's maximum principle [34], we obtain the optimal
acceleration for t Î [t0, 0].
(6)

Secondly, to answer the question on when to apply the optimal
control, we need to compute the avoid set ∂Kh = {(x(t0), y(t0))}
by kinematics [38]. Define the relative velocity
. Given a
particular and
on ∂Sh, we have

(7)

Figure 1.

The safety set Sh is in gray and the avoid set Kh is in red.

complement of Kh is the maximal controlled invariant set [29], in
which there are no restrictions on control. If optimal control is
applied during [t0, 0], then the quadrotor would fly tangent to the
boundary of the safety set ∂Sh at the present time t = 0, and does
not enter Sh.
For convenience, the relative frame is rotated such that
and
on ∂Kh. In other words, points on ∂Kh only have velocity components in the -y axis. For legibility, we abuse the notation here by hiding this rotation. A backward rotation should be
made after the optimal control angle is computed [34].
Define state variables
, the relative kinematics are

From Figure 1, we can see that is only defined on certain parts of
for some
shown in (8). At
(or x(t0) = 0) and speed v, we have the worst case scenario,
giving us the worst case minimum maneuver distance (WCMMD)
to be
in (7).
(8)
Each point on ∂Kh maps uniquely to a on ∂Sh. Therefore, we first
generate ∂Kh using (7) for a dense set of
,
then compare the quadrotor's relative position x(t) to points on ∂Kh.
If x(t) is very close to a particular point on ∂Kh, then we can find the
corresponding and apply the optimal acceleration
.

HORIZONTAL CONTROL CONVERSION
(3)

Consider the dynamical system (3) over the time interval [t0, 0] and
define a continuously differentiable function
in (4). If
l(xh(0)) ≥ 0, then we are outside of Sh [29].

To convert the horizontal kinematic control to horizontal control
inputs U2 and U3, we assume that the quadrotor is in vertical equilibrium when performing the avoidance. Then the inputs that could
affect the horizontal motions are the desired roll (fd), pitch (qd),
and yaw (yd). It turns out that yaw is not effective, so we choose
roll and pitch. The conversion is straight forward. First, convert the
optimal acceleration [a*h , q*h ] from the relative frame to the quadrotor body frame , and then convert it to the desired pitch (qd) and
roll (fd). Equation (9) summarizes the process.

(4)
The objective is to maximize the value function J(xh(t), uh(t), t) =
l(xh(0)) subjected to dynamical and input constraints, defined in
(5). Note that the value function only has a terminal value at time
t = 0.
SEPTEMBER 2016

(9)
Lastly, a position controller with dynamic surface control
(DSC) filters [39] is implemented to track qd and fd defined in (9)

IEEE A&E SYSTEMS MAGAZINE

33



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