Final Report for Experiential Advanced Control Design

1. 1
NEW ERA FOR PUBLIC TRANSPORATION: MODEL PREDICTIVE CONTROL FOR URBAN
AIR MOBILITY IN THE BAY AREA
Sonny Li (3035300139), Muireann Spain (3033516594), Shawn Marshall-Spitzbart (3035289739), David Tondreau
(3035296317), Anthony Yan (3035300282)
ABSTRACT
Using Model Predictive Control (MPC), an Unmanned
Arial Vehicle (UAV) is simulated flying in ℝ3
between set
destinations. The Constrained Finite Time Optimal Control
problem (CFTOC) is weighted to prioritize optimization of input
power, velocity and deviation from the reference trajectory.
Furthermore, the UAV is constrained to fly within an admissible
tube, limiting the UAV’s maximum displacement from the
reference trajectory. A link to the source code is provided below
in Appendix 8.1. Our project video is linked here:
https://www.youtube.com/watch?v=tJv9LQBCpm0&feature=yo
utu.be&fbclid=IwAR1J4QYwMSt9eecqPHOYh_UXFUAJWNx_
mTUNrLnsxF8mUO9pUgAc6m-Dc6Y
1. INTRODUCTION
To solve current challenges with dense traffic in the Bay
Area, our team strives to redefine public transportation by
utilizing UAVs. For this project, the optimization problem seeks
to minimize input power for a multiple passenger-sized UAV
along a reference trajectory between Berkeley, San Francisco,
and Palo Alto. The system dynamics of the UAV are defined by
a discretized linear dynamic model and holonomic constraints,
which include the displacement of the UAV to be within an
admissible tube from the reference trajectory.
2. QUADCOPTER PARAMETERS
Figure 1: Body-fixed frame UAV
The parameters of the UAV are outlined with the assumption
that the UAV is symmetric about the body-fixed 1 and 2 axes as
shown in Fig.1. Drawing inspiration from Boeing-Aurora Flight
Science’s Passenger Aerial Vehicle [11], the mass of the fully-
loaded passenger UAV was defined as 800 kg. The UAV
traverses using four motors orthogonal to one another, three
meters radially from the COM. Next, the mass moment of
inertias (𝑱𝑱) are defined as follows:
𝑱𝑱 = �
𝐽𝐽𝑥𝑥𝑥𝑥 0 0
0 𝐽𝐽𝑥𝑥𝑥𝑥 0
0 0 𝐽𝐽𝑧𝑧𝑧𝑧
� (1)
where,
𝐽𝐽𝑥𝑥𝑥𝑥 = ∑ 𝑚𝑚 𝑀𝑀[ (𝑠𝑠𝑖𝑖
2)2
+ (𝑠𝑠𝑖𝑖
3
)2]4
𝑖𝑖=1 (2)
𝐽𝐽𝑧𝑧𝑧𝑧 = ∑ 𝑚𝑚 𝑀𝑀[ (𝑠𝑠𝑖𝑖
1)2
+ (𝑠𝑠𝑖𝑖
2)2]4
𝑖𝑖=1 (3)
note that 𝐽𝐽𝑥𝑥𝑥𝑥 is equal to 𝐽𝐽𝑦𝑦𝑦𝑦 due to the symmetry of the UAV.
3. QUADCOPTER DYNAMICS
The state vector (z) defines the UAV’s position (s), velocity
(v), roll (ϕ), pitch (θ), and yaw (φ) angles, roll rate (p), pitch rate
(q), and yaw rate (r) in ℝ3
Euclidean space:
𝑧𝑧 =
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
𝑠𝑠1
𝑠𝑠2
𝑠𝑠3
𝑣𝑣1
𝑣𝑣2
𝑣𝑣3
𝜙𝜙
𝜃𝜃
𝜑𝜑
𝑝𝑝
𝑞𝑞
𝑟𝑟 ⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
, 𝑧𝑧̇ =
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
𝑣𝑣1
𝑣𝑣2
𝑣𝑣3
𝜃𝜃 ∙ 𝑔𝑔
− 𝜙𝜙 ∙ 𝑔𝑔
𝐶𝐶Σ− 𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∙𝑔𝑔
𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑝𝑝
𝑞𝑞
𝑟𝑟
𝑛𝑛1
𝐽𝐽𝑥𝑥𝑥𝑥
�
𝑛𝑛2
𝐽𝐽𝑥𝑥𝑥𝑥
�
𝑛𝑛3
𝐽𝐽𝑧𝑧𝑧𝑧
� ⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
(4)
In Eqn.4, 𝑧𝑧̇ defines the linearized state derivatives, g is the
norm of the gravity tensor, C∑ is the sum of the four input forces,
and n are the input torques. To transform quadcopter propeller
forces into torques (n), a mixer matrix is introduced where l is
the perpendicular distance from each motor to the body-fixed 1
axis, and k is a proportionality constant (Eqn. 5).

2. 2
�
𝐶𝐶Σ
𝑛𝑛1
𝑛𝑛2
𝑛𝑛3
� = �
1 1 1 1
𝑙𝑙 −𝑙𝑙 −𝑙𝑙 𝑙𝑙
−𝑙𝑙 −𝑙𝑙 𝑙𝑙 𝑙𝑙
𝜅𝜅 −𝜅𝜅 𝜅𝜅 −𝜅𝜅
� ∙ �
𝐶𝐶P1
𝐶𝐶P2
𝐶𝐶P3
𝐶𝐶P4
� (5)
Thus, using the mixer matrix, the four motor forces are utilized
as inputs to the system. Finally, the system propagation was
performed using the Forward Euler method:
𝑧𝑧𝑘𝑘+1 = 𝑧𝑧𝑘𝑘 + 𝑇𝑇𝑠𝑠 ∙ 𝑧𝑧𝑘𝑘̇ (6)
where Ts is the simulation time step and 𝑧𝑧̇ is the state derivative
vector.
To calculate the state derivative vector, 𝑧𝑧̇, the dynamics of
the UAV were linearized about a hover state following the
approach in [9]; in doing so, small angles, velocities, and angular
accelerations were assumed. To analyze this assumption, a
CFTOC was formulated using the linearized dynamics, and the
resultant optimal inputs were simulated with both the linear and
nonlinear dynamics as similarly completed in [1]. For roll and
pitch angles less than 30 degrees, the error between the linear
and nonlinear dynamics was found to be small (Fig.2). However,
when small angles were added as a constraint to the main
CFTOC described in Section 5, the solution was found to be
infeasible. Thus, creating a dynamically feasible trajectory,
given small angles, velocities, and angular accelerations, has
been left for future work.
Figure 2: Analysis of linearized dynamics vs nonlinear dynamics
where the y axes of the x and y position graphs is in meters, the y axes
of the roll and pitch graphs is in radians, and the x axes are in seconds
4. QUADCOPTER TRAJECTORY
The reference trajectory shown in Fig.3 was defined in ℝ3
.
The “drop-off locations” illustrated were derived by importing
geographical data of the Bay Area into MATLAB and manually
selecting the X and Y coordinates of our three destinations.
Terminal waypoints were discretized from the reference
trajectory to be implemented into our MPC algorithm. These
waypoints were propagated forward along the trajectory at
constant intervals of 5 m by utilizing a reference velocity of 50
m/s along with a time step of 0.1 s. This reference velocity
ensures that the UAV continues moving forward along the
trajectory at a constant rate. However, the implementation of
such a reference velocity disables the maximization of UAV
velocity.
Figure 3: Map of the Bay Area with the tube constraint shown in red
and the drop off locations labeled with triangles
In anticipation of a multitude of UAVs flying various
trajectories across the Bay area, a holonomic constraint was
designed to ensure that the radial displacement of the UAV from
the reference trajectory is within a virtual tube. To define this
constraint, a cubic interpolation is performed along the reference
trajectory for each iteration of the CFTOC function, with the
number of interpolated values equal to the prediction horizon, N.
Next, the position of the UAV is constrained to be within the
radius of the tube minus half of the width of the UAV at each of
the aforementioned interpolated steps along the trajectory. The
tube constraints are therefore manifested as cubes constraining
the X, Y, and Z positions of the UAV in the inertial frame to the
waypoints calculated along the reference trajectory at each step
from zero to the prediction horizon, N.
5. MODEL PREDICTIVE CONTROL FORMULATION
(7)

3. 3
The MPC function utilizes four input propeller forces and
the 12 UAV states as decision variables. First, a CFTOC problem
is solved using the discretized linear dynamics described in
Section 3, the tube constraints described in Section 4, and a
quadratic objective function. Abstractly, the objective function
was weighted to minimize fuel consumption by primarily
minimizing the UAV’s propeller forces, while still penalizing
lateral and vertical deviation from the reference trajectory at each
discrete time step. The weights for P, Q, W, and R are given in
Appendix 8.3.
After solving each CFTOC iteration, the optimal inputs for
the initial time step are applied to the system. The system is then
simulated using the discretized linear dynamics for one time
step. The state of the system at the end of one sample time is
retrieved, then applied as the initial state for the next CFTOC
formulation. This procedure is repeated in a loop for the duration
of the MPC simulation horizon, M. In our case, the MPC
algorithm terminates when the UAV reaches within a defined
tolerance of the final waypoint.
6. RESULTS AND DISCUSSION
The UAV successfully lifted off from Berkeley and
followed a user-defined flight path, landing on various
destinations without violating the radial tube constraint of 50
m. Fig.4 illustrates the UAV, depicted as a ‘*’, with the
reference trajectory depicted in red and the MPC path traveled
in black. It was necessary to set the prediction horizon value, N,
to the sufficiently high value of 35 to guarantee feasibility of
the MPC function with constraints. The simulation horizon was
defined to terminate the MPC loop once the UAV reached
sufficiently close to the final waypoint at 6100 steps. Fig.5
illustrates an example of the UAV following the reference
trajectory while obeying the virtual tube constraints.
The tube radius was initially defined at 10 m, but the MPC
formulation was infeasible as the rather sharp ascension/
descension portions of the ℝ3
curve were unsolvable to solve
given the UAV system dynamics and the tight constraints. A
tube radius of 50 meters enabled the desired path to be feasible
and therefore solvable.
Figure 4: The UAV makes three stops – Berkeley, San Francisco, and
Palo Alto – flying at height of 100m in-between stops
Figure 5: A sample trajectory of the UAV following part of the desired
trajectory within the tube constraint with axes in meters.
7. FUTURE WORK AND IMPROVEMENTS
The initial approach taken to formulate this problem was to
employ the Serret-Frenet frame attached to the reference
trajectory. Specifically, this frame would allow the UAV state
to be described by a curvilinear abscissa and displacement from
the abscissa to the UAV as done in [6] [7] [10] [12] [13] [14].
This frame greatly simplifies the formulation of vertical and
lateral position constraints along the reference trajectory by
creating an admissible tube where the UAV is permitted to
traverse. This approach was taken by [12] with great success
where a constant radius tube formed the width of the race track.
By using the Serret-Frenet frame, the constrained finite-time
optimal control problem effectively transforms a reference-
tracking problem to a reference-free trajectory-optimization
problem. The authors believe future work entails utilizing a
curvilinear abscissa in ℝ3
to formulate the MPC and solve for
the optimal trajectory between Berkeley, San Francisco, and
Palo Alto in real-time. Next, the authors believe that leveraging
Learning MPC to train for time-optimality [13], given
passenger-comfort constraints and an admissible tube radius,
could greatly reduce commute times between the three cities.

4. 4
8. APPENDIX
8.1 Github: https://github.com/smarshall-
spitzbart/Advanced-Control-Design-Final-Project
8.2 Table 1: Numerical Values of UAV Parameters
Variable Numerical Value
R 3 m
α 45°
l 3/√2 m
k 0.01
mB 600 kg
mM 50 kg
𝐽𝐽𝑥𝑥𝑥𝑥 1.44E+04 kg-m2
𝐽𝐽𝑧𝑧𝑧𝑧 2.88E+04 kg-m2
8.3 Table 2: Numerical Values of Cost Function Weights
Variable Numerical Value
P I3
Q I3
W I3
R 10*I4
ACKNOWLEDGEMENTS
Special thanks to Francesco Borrelli and Mark Mueller for their
consultation during this project and their instruction over the
course of the semester.
REFERENCES
[1] Anand, Raghav. “MPC Control of Multiple Quadcopters
Cooperatively Lifting an Object,” 2018
[2] Brown, Matthew, “Safe Driving Envelopes for Path
Tracking in Autonomous Vehicles,” 2016
[3] Gray, Andrew, “Predictive Control for Agile Semi-
Autonomous Ground Vehicles using Motion Primitives,”
2012
[4] Hehn, Markus, “Quadcopter Trajectory Generation and
Control,” 2011
[5] Islam, Maidul, “Dynamics and Control of Quadcopter
Using Linear Model Predictive Control Approach,” 2017
[6] Lorenzetti, Joseph, “A Simple and Efficient Tube-based
Robust Output Feedback Model Predictive Control
Scheme,” 2019
[7] Lot, Robert, “A Curvilinear Abscissa Approach for the Lap
Time Optimization of Racing Vehicles,” 2014
[8] Micaelli, Alain, “Trajectory tracking for two-steering-
wheels mobile robots,” 1994
[9] Mueller, Mark. “Quadcopter Dynamics,” 2019
[10]Oulmas, Ali. “Closed-loop 3D path following of scaled-up
helical microswimmers,” 2016
[11]“PAV – Passenger Air Vehicle,” accessed 10th
December
2019, https://www.aurora.aero/pav-evtol-passenger-air-
vehicle/
[12]Rosollia, Ugo, “Autonomous Vehicle Control: A Nonconvex
Approach for Obstacle Avoidance,” 2016
[13]Samson, Claude, “Path Following And Time-Varying
Feedback Stabilization of a Wheeled Mobile Robot,” 1992
[14]Wang, Ye. “Nonlinear Model Predictive Control with
Constraint Satisfaction for a Quadcopter,” 2016
[15]Zhang, Xiaojing, “Optimization-Based Collision
Avoidance,” 2018