Distributed Reactive Collision Avoidance - University of Washington

Distributed Reactive Collision Avoidance
Emmett Lalish
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
University of Washington
2009
Program Authorized to Offer Degree: Aeronautics and Astronautics

University of Washington
Graduate School
This is to certify that I have examined this copy of a doctoral dissertation by
Emmett Lalish
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the final
examining committee have been made.
Chair of the Supervisory Committee:
Kristi A. Morgansen
Reading Committee:
Kristi A. Morgansen
Mehran Mesbahi
Juris Vagners
Date:

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University of Washington
Abstract
Distributed Reactive Collision Avoidance
Emmett Lalish
Chair of the Supervisory Committee:
Professor Kristi A. Morgansen
Aeronautics and Astronautics
Collision avoidance is an important aspect of multivehicle coordination because it prevents
vehicles from disrupting or destroying each other. The work contained in this dissertation
concerns a novel approach to the n-vehicle collision avoidance problem. The vehicle model
used here allows for three-dimensional movement and represents a wide range of vehicles.
The algorithm works in conjunction with any desired controller to guarantee all vehicles
remain free of collisions while attempting to follow their desired control. This algorithm
is reactive and distributed, making it well-suited for real-time applications, and explicitly
accounts for actuation limits. A robustness analysis is presented which provides a means to
account for delays and unmodeled dynamics. Robustness to an adversarial vehicle is also
presented.

TABLE OF CONTENTS
Page
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Chapter 1: Introduction to Collision Avoidance . . . . . . . . . . . . . . . . . . 1
1.1 Definitions of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Introduction to DRCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2: Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 3: Conflicts and Collisions . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 4: DRCA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1 Deconfliction Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 Constant-Speed Maneuver . . . . . . . . . . . . . . . . . . . . . . 25
4.1.2 Variable-Speed Maneuver . . . . . . . . . . . . . . . . . . . . . . 29
4.1.3 Heuristic Performance . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Deconfliction Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.2 Guarantee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.3 Priority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.4 Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.5 Liveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 5: Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1 Robust Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Robust Avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
i

Chapter 6: Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.1 Variable Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2 Preferential Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3 Three-dimensional Avoidance . . . . . . . . . . . . . . . . . . . . . . . . 58
6.4 Heuristic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.5 Computational Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.6 Formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.7 Comparison to an SGT Algorithm . . . . . . . . . . . . . . . . . . . . . . 65
Chapter 7: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
ii

LIST OF FIGURES
Figure Number
Page
1.1 Air traffic control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 Geometric definition of the collision cone. . . . . . . . . . . . . . . . . . . 18
4.1 Flow chart of the DRCA algorithm. . . . . . . . . . . . . . . . . . . . . . 21
4.2 Example optimization of a constant-speed maneuver. . . . . . . . . . . . . 25
4.3 Example of an infeasible constant-speed maneuver problem. . . . . . . . . 30
4.4 Example optimization of a variable-speed maneuver. . . . . . . . . . . . . 31
4.5 Example of an infeasible variable-speed maneuver problem. . . . . . . . . 34
4.6 Block diagram of the deconfliction maintenance controller. . . . . . . . . . 36
4.7 Geometry of the conflict measures p t and p n . . . . . . . . . . . . . . . . . 37
4.8 Example of the control function, F . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Geometry of worst case adversary . . . . . . . . . . . . . . . . . . . . . . 53
6.1 Five vehicle simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Excess separation of vehicles in Fig. 6.1. . . . . . . . . . . . . . . . . . . . 55
6.3 Preferential routing comparison for three vehicles. . . . . . . . . . . . . . . 57
6.4 Three-dimensional collision avoidance. . . . . . . . . . . . . . . . . . . . 58
6.5 Excess separation of vehicles in Fig. 6.4. . . . . . . . . . . . . . . . . . . . 59
6.6 Comparison of the constant and variable-speed deconfliction maneuvers. . . 60
6.7 Merging two groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.8 Fifty-vehicle swarm avoiding a static obstacle. . . . . . . . . . . . . . . . . 64
6.9 Collision avoidance of 32 aircraft using the DRCA algorithm. . . . . . . . 66
6.10 Collision avoidance of two perpendicular flows. . . . . . . . . . . . . . . . 68
6.11 Excess separation for vehicles in 6.10. . . . . . . . . . . . . . . . . . . . . 68
iii

LIST OF TABLES
Table Number
Page
6.1 Algorithmic computation times. . . . . . . . . . . . . . . . . . . . . . . . 63
iv

ACKNOWLEDGMENTS
The author wishes to express sincere appreciation to Katie Marshall and the reading
committee for editing and improving this dissertation, to the many scholarships that made
his education possible, to Kristi Morgansen for employing him, and to the rest of the Department
of Aeronautics & Astronautics for the superb education received.
v

DEDICATION
To my parents, Greg and Paula Lalish.
vi

1
Chapter 1
INTRODUCTION TO COLLISION AVOIDANCE
The study of multi-vehicle systems has become a major field in control theory. The
fundamental reason that groups of vehicles are interesting is that often a group can perform
tasks that a single agent cannot (for instance an ant colony, or human organization). These
groups are also more immune to single point failures, since the loss of one agent does not
necessarily doom the group. Unfortunately, causing a group of individual agents to cooperate
in a useful manner is also more complex than controlling a single agent. Sharing
pertinent information between members of the group through some combination of sensing
and communication is nontrivial, not to mention finding the proper maneuvers to accomplish
the assigned task efficiently.
As multi-vehicle autonomous systems are studied and implemented, the issue of conflict
resolution becomes increasingly important. From mobile robots performing a cooperative
search to air traffic control for unmanned aerial vehicles, collision avoidance is of utmost
importance for safety. Much of the work so far on collision avoidance has been sponsored
by the FAA to support a potential move to free-flight air traffic control [1, 2], whereby
aircraft can avoid each other in a decentralized manner rather than relying on a land-based
controller. Similar concepts have been discussed regarding autonomous harbor control
for ships [3, 4]. These scenarios will become more important as unmanned vehicles are
introduced, because safety will need to be guaranteed for them to be accepted by their
manned counterparts.
As more vehicles are added to a system, collision avoidance gets significantly more
difficult. Air traffic control is already a difficult problem given the number of aircraft
currently in operation (Fig. 1.1), but as this number increases it may be necessary to develop

2
Figure 1.1: A snapshot of the US airspace showing the magnitude of the air traffic control
task (photo courtesy of NASA).
automated algorithms to decrease the workload of human operators.
Collision avoidance is a difficult and interesting problem in the study of control theory
because the goal is fundamentally different from that of a classical control system. The vast
majority of classical control problems (and their current nonlinear counterparts) demand
the system attain a particular point in the state space (be it stationary or moving). Most
of the methods for this type of task use asymptotic analysis, whereby the system can be
said to approach the goal regardless of the starting position or the route taken. Collision
avoidance is completely different in that there is no inherent end goal. The goal instead is
to not enter particular regions of the state space at any time. Collision avoidance therefore
has the effect of making the allowable set in the state space nonconvex, and in the case of
avoiding moving vehicles, the set is also time varying. Adding these restrictions to the state
space makes otherwise simple problems extremely difficult. One of the primary reasons is
that transient response is difficult to completely characterize using classical control, yet it
is exactly these transients that cause most collisions. Now, instead of only considering the

3
system state as time goes to infinity, one has to consider the system state at all times.
When one factors in the highly nonlinear nature of the nonholonomic dynamics that
govern most real vehicles, the problem truly balloons in complexity. Control authority limitations
become far more important than in most classical control problems because control
saturation primarily affects transient behavior. When two vehicles are on a collision course,
it is exactly that transient behavior that determines their ability to avoid each other. If saturation
is not accounted for explicitly, then collision avoidance becomes a trivial problem
where a vehicle simply applies infinite control in the direction away from a collision. Of
course this concept is unrealistic in practice, but nonetheless, many papers have been written
on collision avoidance which rely on exactly this principle for their proofs.
The typical goal for designing a nonlinear control algorithm is to prove global asymptotic
stability. Unfortunately, guaranteed global collision avoidance does not exist because
all real vehicles have inertia and control authority limits, which place fundamental limits on
their safe set of initial conditions. Two vehicles heading towards each other can always be
placed close enough that nothing can be done to avoid a collision. Therefore no approach
can possibly guarantee collision avoidance for arbitrary initial conditions. Determining the
exact set of initial conditions for which avoidance is impossible becomes extremely complex
as the number of vehicles increases, so generally the best that can be done is to define
a bound or test, which if passed, yields a conservative guarantee.
1.1 Definitions of Terms
A wide range of methods have attempted to solve the collision avoidance problem, and
the authors of a survey paper [5] split those methods into three categories: prescribed,
optimized, and force field. In prescribed maneuver approaches, all vehicles follow a set
protocol, not unlike the rules of the road. This approach tends to yield a discrete event
controller, which when combined with the vehicle’s continuous dynamics, forms a hybrid
system. Optimization methods attempt to find the best route for all the vehicles to take to
avoid each other while minimizing a cost function. Generally, these methods use a look-

4
ahead or time horizon so that the solution does not have to be recalculated often. Force field
approaches use continuous feedback mechanisms to compute the control. Commonly, the
force field between two vehicles is akin to the repulsion between two like-charged particles,
however there are many possible alternatives for feedback schemes. These approaches are
generally reactive, in that the control reacts to the current state of the system, rather than
planning a trajectory ahead of time.
The next differentiator between collision avoidance algorithms is their degree of centralization.
Centralized means that all of the information about the vehicles is sent to a
single server where the controls are calculated for each vehicle and sent back. The current
air traffic control system uses centralization (with human operators), as do most optimization
schemes. The other extreme is decentralization, in which no central server exists and
each vehicle computes its own control based only on other vehicles within a particular range
of itself. This situation is ideal from a computational perspective because there is a limit to
how many vehicles can be within range at once, and therefore there is a limit to how much
time the algorithm will take to run, regardless of the total number of vehicles involved. In
between is a category termed distributed, in which there is still no central server and each
vehicle computes its own control but may need more than local information to do so. In this
case, computations scale with the number of vehicles involved (n), but the scaling is at least
O(n) better than the centralized case, because now n separate processors are computing in
parallel, rather than one server doing all the work.
The vehicle model used in each formulation is also of utmost importance to the algorithm’s
applicability to real systems. First, for collision avoidance to be nontrivial, the
dynamics must be at least second order (acceleration-level control input), and the control
inputs must be bounded to model the difficulties of overcoming inertia. Many vehicles
(cars, ships, airplanes, etc.) have nonholonomic constraints, and as such are often modeled
at a high level by unicycle dynamics. The inputs to this model are forward acceleration
and heading rate (which is equivalent to acceleration normal to the velocity). All vehicles
have a maximum speed, but some vehicles (notably airplanes) also have a positive min-

5
imum speed. The constant-speed unicycle is one of the most broadly applicable vehicle
models because it is the most constrained (less constrained vehicles can still duplicate its
movements, but not vice-versa). Ideally an algorithm should use all of the capabilities a
vehicle has to offer, but still function under the constraints of a constant-speed unicycle if
necessary.
Another important aspect of a collision avoidance algorithm is whether it works for
a heterogeneous group of vehicles. Heterogeneity can be due to differences in vehicle
size, dynamics, speed range, control authority, etc. Many algorithms are developed for a
homogeneous group of vehicles for convenience and ease of notation, but could be easily
extended to a heterogeneous case. Others, especially constant-speed algorithms, can have
difficultly generalizing from a homogeneous scenario. Another important consideration is
that vehicle size, maximum speed, and control authority are often important parameters
for the other vehicles in the system to know in order to properly avoid each other. In a
homogeneous system these parameters are known implicitly because they are the same
for every vehicle, but in a heterogeneous system, these parameters must be exchanged.
Communication is the most obvious tool, though sensors could be employed to identify
the vehicle type and compare it to some known list, the same way a captain can identify a
sailboat or a ferry and infer their size and maneuverability. Until such technology is fully
mature, heterogeneity may require some degree of inter-vehicle communication.
A final aspect important to collision avoidance is liveness, which denotes the ability
of the vehicles to attain their goals. Liveness is important to consider because one simple
way to avoid collisions is to have everyone stop moving. While this method may avoid
collisions, it is not useful because the vehicles cannot arrive at their destinations. This type
of situation is called a deadlock. Another problem scenario is a livelock, where the vehicles
continue to move, but in such a way that they cannot reach their goals.

6
1.2 Literature Review
Much work has been done on the collision avoidance problem to date, and a wide variety
of solutions have been proposed. While each of these answers solves a piece of the puzzle,
they also have limitations that keep them from solving the wide range of applications that
the work in this dissertation seeks to address. However, they also form the foundation of
tools and ideas on which this work has been built.
An early theoretical work on collision avoidance [6] showed a method of designing
avoidance controllers for a general class of vehicle models and constraints. The mathematical
foundation is solid in that it presents a proof to avoid not just another vehicle, but
an adversary. The method is limited because it requires finding an appropriate Lyapunov
function, which is often a nontrivial task, and it only works for two-vehicle systems.
The collision cone concept, introduced in [7] and subsequently used in the deconfliction
literature [8, 9, 10, 11, 12], is a first order look ahead for detecting conflicts. The collision
cone is a set of velocities for one vehicle that will cause it to collide with another, assuming
each of their velocities are constant. While most algorithms using the collision cone define
collision by the distance between two points (as though each vehicle is a disk or sphere),
[7] shows how the method also works for irregularly shaped objects. An extension of the
collision cone concept to accelerating vehicles was shown in [9], but the resulting conflict
region is nonlinear and difficult to describe analytically, which limits its use for collision
avoidance.
A three-dimensional version of the original collision cone was used in [10] to avoid
conflicts pairwise. Separate solutions were found using speed changes, heading changes,
and flight path angle changes, but not in combination. An extension of that work to handle
simultaneous speed, heading, and flight path angle changes was shown in [11]. Extensive
real-world simulations were shown for two aircraft, but the algorithm is not designed for
larger numbers of interacting vehicles.
An application of the collision cone to three-dimensional air traffic control is presented

7
in [12], where the rates of change of the collision cones are used to prioritize the evasion of
potentially adversarial vehicles. However, the protocols for handling multiple vehicles are
not completely described, and no guarantees are given for safety, even in the two vehicle
case.
A combination of repulsive and vortex functions are used as force fields in [13], causing
vehicles to form roundabout avoidance maneuvers, as tend to crop up in many of the collision
avoidance methods. The algorithm provides no guarantees regarding safe separation,
especially in the presence of control authority limits. Often parameters of the functions can
be adjusted to give adequate results for a given situation, but it is unclear how to choose
these parameters in general.
A function involving more look-ahead is presented in [14], which shows good qualitative
performance, but again lacks guarantees. Their simulations consistently show a significant
percentage of the vehicles violating their separation bounds. A somewhat similar
approach was taken by this author in [15], which avoided three dimensional conflicts by
maneuvering out of the plane of the conflict. Again, simulations showed promise, but no
guarantees could be found for safety.
Gyroscopic and braking forces are used for collision avoidance in [16], resulting in a
decentralized control law. However, safety can only be guaranteed so long as each vehicle
only sees one other vehicle at a time within the detection range. Therefore no guarantees
exist for systems with more than two vehicles, though simulations are shown for manyvehicle
cases.
A simple potential function method is employed for collision avoidance in [17], while
simultaneously attaining a desired formation of the agents involved. The guarantee of collision
avoidance given proves that the vehicles never attain the exact same position at the
same time by using unbounded control authority. Problematically, no minimum separation
distance or any concept of vehicle size is used in this method.
Another force field approach that guarantees collision avoidance is presented in [18].
Unfortunately, the guarantee is based on a potential function that goes to infinity at the mini-

8
mum vehicle separation. Effectively, for the guarantee to apply, the vehicle must be capable
of infinite control authority, which is an unreasonable assumption for any real vehicle.
A realistic situation in which force-field collision avoidance is necessary is given in
[19]. The idea is to use electrostatic charges on spacecraft to create enough repulsion force
to avoid collisions in high orbit and deep space. This situation results in a different kind
of underactuated system, but one for which collision avoidance is guaranteed, even in the
case of control saturation, given restrictions on the initial conditions. These results use a
Lyapunov function and only apply to the two-vehicle case.
The work in [20] uses a navigation function to combine the tasks of collision avoidance
and waypoint navigation into a single gradient-following routine. The navigation function
is useful in its ability to guarantee liveness in a static environment. Unfortunately, adequate
separation is not proven because the dynamics of the vehicle make it unable to follow the
gradient in all cases. A vortex is added to the function to better the heuristic performance,
but its effect on safety is unclear.
A collision avoidance approach for a three-dimensional unicycle model that uses dipole
navigation functions to avoid static obstacles while maneuvering to a goal is presented in
[21]. While the dipole navigation functions provide smooth controls for the vehicle, still no
way is given to choose the parameters such that particular rate limits are observed. These
control limits can therefore cause collisions in certain situations.
A high-performance force-field method is presented in [22], which uses a concept similar
to an abbreviated collision cone to take into account how much space an avoidance
maneuver will need based on the relative velocity between the vehicles. Simulations show
promising results for large numbers of vehicles, and the computation time is kept low by
the decentralized nature of the algorithm. While this algorithm does guarantee safety for a
homogeneous system of two or three vehicles with acceleration constraints, its performance
in larger systems is heuristic, and it does not allow unicycle-type dynamics in its vehicle
model.
The optimization scheme presented in [23] is performed over the reachable set of a

9
hybrid system that describes each aircraft involved. Safety is guaranteed, but only in the
two-vehicle case. An n-vehicle optimization approach is presented in [8] that uses a relaxation
approach to turn the collision avoidance problem into a semidefinite program, solved
with convex optimization. This achievement is impressive given the highly nonconvex
nature of collision avoidance. Unfortunately, the solution must be centralized, and the algorithm
is randomized, which means that to get an optimal solution, the optimization must
be run several times. While convex optimization is computationally efficient compared to
general nonlinear optimization, it is still less scalable than prescribed maneuver or force
field approaches.
A method showing better computation time is presented in [24], which uses mixed
integer programming. This method allows for either speed changes or heading changes,
but not both concurrently. It is still a centralized solution, and like many of the other
optimization approaches, no mention is made of restrictions on the initial conditions to
ensure that a feasible solution exists.
Another convex relaxation approach to optimizing collision avoidance is shown in [25]
for an n-spacecraft system using 3D double integrator dynamics. The algorithm works by
computing an optimal set of new velocities each time any vehicles get too close together.
These new velocities cause the vehicles to bounce away from each other, and in this way
collision avoidance is guaranteed for the system as is liveness of the solution. However,
in addition to the optimization being centralized, the control is designed to attain the new
velocity in a single time step, meaning the magnitude of the control force is unbounded
as the time step goes to zero. Alternatively, one could choose the time step to enforce a
control limit, but given the discrete time nature of the constraints, motion during the time
step would become significant and could cause collisions.
An approach to distributing an optimization scheme across the vehicles involved is
presented in [26]. This approach is much broader than collision avoidance, as the optimization
occurs across a whole host of constraints, from visiting targets to avoiding danger
zones. The authors use market-based task trading and evolutionary optimization to find

10
paths dynamically for the vehicles. While most of the computations are distributed, a central
server is still required to make the trades. Additionally, no guarantee has been made
that a collision-free set of paths will be found.
The authors of [27] also use a negotiation process to optimize collision avoidance maneuvers,
but another algorithm must be present to give collision-free paths for the vehicles
to start from for the guarantees to work. The idea is that this other algorithm need not be
optimal, but merely feasible. Then the second stage enables the vehicles to negotiate a more
optimal solution derived from the first. The algorithm for finding a set of feasible paths is
not given, assuming instead that one of the other approaches discussed here will suffice.
A major improvement in the computation time of optimization methods is shown in
[28], which uses satisficing game theory to make a truly decentralized optimization scheme.
This algorithm is efficient enough to make real-time computation reasonably scalable to
large numbers of vehicles. The price paid for efficient computation is a loss of the safety
guarantees, as simulation results show that collisions are rare, but that they do occur.
In [29], a distributed collision avoidance algorithm with guaranteed safety is presented
for a homogeneous group of n aircraft. This prescribed maneuver approach involves two
discrete heading and speed changes for each aircraft, where each aircraft turns the same
way. The method is based upon perturbations from an exact conflict, where all vehicles
are headed toward a single collision point. This algorithm was extended in [30] to work in
3D and to account for bounded acceleration. Since the algorithm is computed only once,
it is unclear how it could account for a dynamically changing environment where vehicles
are not necessarily capable of performing the exact maneuver prescribed. Additionally, the
algorithm may run into trouble with extremely inexact conflicts, since it is based upon a
transformation to an exact conflict.
Probably the safest collision avoidance algorithm to date is that of [31], which considers
a homogeneous group of constant-speed unicycles. This method uses a completely decentralized,
prescribed maneuver algorithm for n vehicles which has an absolute guarantee of
safety in the presence of constrained control authority. While this solution might appear

11
at first to be the holy grail of collision avoidance, it has some undesirable performance
characteristics. To achieve this guarantee, the vehicles must keep a region the size of their
turning diameter free of other vehicles at all times. For mobile robots this requirement may
be reasonable, but for aircraft and ships that tend to have a turning radius much larger than
their physical size, this requirement becomes overly conservative. Additionally, the system
can suffer from deadlocks and livelocks.
1.3 Introduction to DRCA
The Distributed Reactive Collision Avoidance (DRCA) algorithm presented here fills an
important gap in the current collision avoidance literature. Namely, this algorithm presents
a distributed, computationally efficient method to guarantee collision avoidance between n
vehicles in the presence of arbitrary control authority limitations. Plus, liveness of the solutions
can be guaranteed, as well as robustness to modeling errors and adversarial behavior.
The vehicle model used for the formulation of the problem is the 3D double-integrator.
Specific vehicles are then modeled by restricting the control authority in particular directions
by time- or state-dependent functions, rather than by adjusting the dynamics themselves.
In this way, this single model can represent two-dimensional systems, or even
nonholonomic vehicles of the constant-speed unicycle variety. Likewise, static obstacles
can be avoided since they can be represented as a vehicle with zero speed and zero control
authority. Higher order dynamics (like an airplane banking to turn) can be accounted for
with the robustness analysis.
The DRCA algorithm is a two-step process, consisting of an optimization-based deconfliction
maneuver, followed by the longer-term deconfliction maintenance phase, which is a
reactive, force-field type approach. Both of the steps are based upon the collision cone concept.
The optimization schemes are uniquely distributed and scale well computationally.
Though they do not find a global optimum, they do guarantee finding a feasible solution
under given restrictions on the initial conditions. The deconfliction maintenance controller
ensures that the vehicles follow their arbitrary outer-loop controllers as much as possible

12
without sacrificing safety. This framework also allows vehicles to be given different priorities,
such that lower priority vehicles will make way for higher priority ones.
This algorithm requires little (though nonzero) communication between vehicles, and
can safely add new vehicles to the system as they reach sensor range. While the algorithm
is distributed, it is not quite decentralized, in that all-to-all information is needed within a
connected group. The information exchange can be easily implemented through a broadcast,
and if there is enough bandwidth for vehicles to rebroadcast the information they
receive, then a quasi-all-to-all topology can be achieved. The robustness analysis accounts
for the corresponding time delays. Additionally, the algorithm can be run in a completely
decentralized fashion, and while some of the guarantees no longer apply, simulations are
given that demonstrate significant capability nonetheless.
No homogeneity is required among the vehicles that make up the system. The DRCA
algorithm allows the vehicles to perform completely different tasks and to have different
size, speed, actuation limitations, and control gains.
The remainder of this dissertation is organized as follows. The problem statement is
given in Chapter 2, including the vehicle model. Specific definitions of collision and the
collision cone are given in Chapter 3. In Chapter 4 the DRCA algorithm is described, which
forms the bulk of the dissertation. A robustness analysis is given in Chapter 5. Finally,
simulation results are presented in Chapter 6 and concluding remarks in Chapter 7.

13
Chapter 2
PROBLEM STATEMENT
The work here presents a method for deconflicting n vehicles.
Each vehicle has a
nominal desired control input, u d (t), which comes from an arbitrary outer-loop controller.
This controller is designed for the vehicle to perform a desired task, which could range
from target tracking to waypoint navigation, to area searching, etc. The goal of the DRCA
algorithm is to adjust the control input on each vehicle to guarantee collision avoidance
while simultaneously staying close to the desired control input (keeping in mind that this
desired control can change with time).
For this approach to collision avoidance, the only vehicle states that matter are position
and velocity. Orientations affect performance, as they often have bearing on the magnitude
of acceleration available in a particular direction, but they do not affect the underlying
concepts of conflict and collision. In this way, many different vehicle models work equivalently
with this approach. To simplify the math, a simple vehicle model will be used for
most of the following analysis: a 3D double integrator, which for the i th vehicle is given by
⎡ ⎤ ⎡ ⎤
r
d
i v i
⎢ v
dt ⎣ i ⎥
⎦ = ⎢ u
⎣ i ⎥
⎦ , (2.1)
Θ i Ω i Θ i
where r i , v i , u i ∈ R 3 are the position, velocity, and control input, respectively. The matrix
Θ i
= [ˆt i , ˆn i , ˆb i ] defines the orientation and Ω i is the cross product matrix of the body
rotation vector ω = [ω t , ω n , ω b ] T . The notation throughout this paper will use bold face
for vectors, hats over unit-vectors, script capital letters for sets, standard capital letters for
matrices and functions, and everything else is assumed scalar. Quantities subscripted with
t, n, or b refer to the tangent, normal, or binormal direction, respectively.

14
Note that the orientation (defined by the ˆt, ˆn, and ˆb vectors) is only used as a local
coordinate frame for the DRCA algorithm. The orientation does not directly affect the
dynamics (r and v), and as such can be arbitrary. However, many vehicle’s input constraints
are related to their orientation, and so it can be useful to tie this local coordinate frame to
the actual body coordinates of the vehicle.
We constrain the input by use of an arbitrarily varying constraint set, u i ∈ C i . The only
requirement is that C i always contain the origin. A simple example of an input constaint set
that limits maximum acceleration and velocity is
C i =
{
u i ∈ R 3∣ }
∣ ‖ui ‖ ≤ u max , ‖v i ‖ ≥ v max =⇒ u T i v i ≤ 0 . (2.2)
For the DRCA algorithm, one must choose a set of rectangular constraints R i (which
can also vary with time, state, etc.)
for each vehicle that encloses its C i , as well as a
corresponding saturation function, S i : R i → C i . The function S i must be continuous, must
become the identity map for any u ∈ C i , and must preserve the sign of each component of
u when projected onto the ˆt, ˆn, and ˆb directions. In this example, one can choose
R i = { u i ∈ R 3∣ ∣ − umaxi ≤ u ti ≤ u maxi , . . . } , (2.3)
and
⎧
u
u max i
⎪⎨
, ‖u ‖u i ‖ i‖ > u max
S i = u i − v iu T i v i
v max
, ‖v i ‖ ≥ v max , u T i v i ≥ 0
⎪⎩ u i , otherwise.
(2.4)
An example of how more complex vehicle dynamics can be represented by this simple
model with an appropriate choice of input constraint set follows. Let us model a vehicle
which can go forward with variable speed, can turn in two axes (a 3D unicycle model), and
has limits on its turn rate, forward acceleration, and maximum speed. One way to describe
the model mathematically is by

15
⎡ ⎤ ⎡ ⎤
r sˆt
d
⎢ s ⎥
dt ⎣ ⎦ = ⎢ u
⎣ a ⎥
⎦ , (2.5)
Θ ΩΘ
where |u a | ≤ u amax , |ω n | ≤ ω nmax , |ω b | ≤ ω bmax , and |s| ≥ s max =⇒ u a s ≤ 0.
Alternatively, an equivalent representation of the system is (2.1) with u = u aˆt+‖v‖ ω bˆn−
‖v‖ ω nˆb. The tangent vector must be initialized to the same direction as the velocity vector,
but the dynamics will keep them aligned from then on. In this case, R can be defined by
u tmax = −u tmin = u amax
u nmax = −u nmin = ‖v‖ ω bmax
u bmax = −u bmin = ‖v‖ ω nmax , (2.6)
and the accompanying saturation function is
⎧
⎪⎨ u − vuT v
s
S =
max
, ‖v‖ ≥ s max , u T v ≥ 0
⎪⎩ u, otherwise.
(2.7)
Normally one would not equate a holonomic model to a nonholonomic one, largely
because of differences in controllability. However, controllability is not essential to the
DRCA algorithm since orientations are arbitrary, and only position and velocity matter.
The DRCA algorithm is designed to use any control authority available, but it does not
require the state space to be locally accessible.
The relative position vector from vehicle i to vehicle j is denoted ˜r ij ≡ r j − r i , while
the relative velocity vector is defined in the opposite sense: ṽ ij ≡ v i − v j . Note that these
definitions imply that ˙˜r ij = −ṽ ij , and ˙ṽ ij = u i − u j .
It is useful to define the dimensionless Deconfliction Difficulty Factor, η, to compare
different systems in which collision avoidance is to be implemented. This factor is defined
as
η =
v2 max
u max d sep
. (2.8)

16
Conceptually, this factor is the ratio of the worst case turning radius to the required separation
distance. It can also be thought of in terms of stopping distance.
The DRCA algorithm was designed primarily for systems with large η (greater than
unity) such as aircraft and ships, where the collision avoidance task is difficult because
of the dominance of vehicle inertia. Vehicles with small η (significantly less than unity)
such as mobile robots can often be modeled as having a direct velocity command, since
the inertia becomes insignificant. In such cases, potential function methods for collision
avoidance may be preferable to the DRCA algorithm because of their simplicity and ability
to closely pack vehicles. The downside of potential function methods is their general lack
of guarantees, especially when inertia is considered.

17
Chapter 3
CONFLICTS AND COLLISIONS
The vehicles considered here are modeled as point masses, however physical vehicles
have finite size. Therefore, to account for physical constraints in the theoretical model, the
condition for collision is not to occupy the same position in space at the same time, but
rather to come within a minimum allowed distance at some point in time. This minimum
distance could be, for example, the five nautical mile separation between aircraft required
by the FAA or the sum of the radii of two vehicles.
Definition 1 (Collision). A collision occurs between vehicles i and j when
‖˜r ij ‖ < d sep,ij , (3.1)
where d sep,ij is the minimum allowed separation distance between the vehicles’ geometric
centers.
For two vehicles not actively in a collision, the next question is whether they will collide
if they remain on their present headings. This situation will be called a conflict.
Definition 2 (Conflict). A conflict occurs between vehicles i and j if they are not currently
in a collision, but with zero control input (i.e. constant velocity), at some future point in
time they will enter a collision:
d min,ij min
t>0 ‖˜r ij‖ < d sep,ij . (3.2)
The following lemma provides a useful way to check for conflicts. To simplify the
notation in the rest of this paper, the ij subscripts will generally be suppressed (for example,
˜r ij will be written as ˜r).

18
ṽ ij
d sep,ij
d sep,ij
β
Vehicle j
α
˜r ij
α
Vehicle i
Figure 3.1: A 2D section of the collision cone along the ˜r ij -ṽ ij plane. The area between
the two dotted lines is the collision cone; a conflict occurs when the relative velocity vector,
ṽ ij , lies within this area.
Lemma 1. Let β = ∠ṽ − ∠˜r 0 , α = arcsin
( )
dsep
‖˜r 0
and ˜r
‖ 0 be the relative position vector
at the time conflict is being checked. A necessary and sufficient condition for no conflict to
occur is
|β| ≥ α. (3.3)
The angle α represents the half-width of the collision cone ([7, 8, 10]), which is depicted
in Fig. 3.1.
Proof. First, define ˜r min as the position vector corresponding to the closest approach of
one vehicle to another in (3.2). By definition, at ˜r min the time derivative of ‖˜r‖ 2 = 0.
Therefore:
Next, note that for constant velocity, ṽ:
d (˜rT˜r ) = 0
dt
ṽ T˜r min = 0. (3.4)
˜r min = ˜r 0 − ṽt, (3.5)

19
where t is the time to closest approach. To find t, multiply by ṽ T on both sides of (3.5),
apply (3.4) and solve:
ṽ T˜r min = ṽ T˜r 0 − ṽ T ṽt
0 = ‖ṽ‖ ‖˜r 0 ‖ cos β − ‖ṽ‖ 2 t
t = ‖˜r 0‖
‖ṽ‖
cos β. (3.6)
Now, using (3.4), (3.5) and (3.6), a concise expression for the closest approach distance
is given by
d min =
√
˜r T min˜r min
= √ (˜r 0 − ṽt) T˜r min
√
= ˜r T 0 (˜r 0 − ṽt)
√
=
=
‖˜r 0 ‖ 2 − ‖ṽ‖ ‖˜r 0 ‖ cos β
√
‖˜r 0 ‖ 2 (1 − cos 2 β)
( ‖˜r0 ‖
‖ṽ‖ cos β )
= ‖˜r 0 ‖ |sin β| . (3.7)
For no conflict to occur, the converse of (3.2) must be true:
d sep ≤ d min
= ‖˜r 0 ‖ |sin β| . (3.8)
Therefore, to remain free of conflict it is necessary that:
( )
dsep
|β| ≥ arcsin = α. (3.9)
‖˜r 0 ‖
exists.
For sufficiency, if |β| ≥ α, then (3.9) and (3.8) still hold, thus implying that no conflict
For notational simplicity below, ˜r 0 will be denoted simply by ˜r, as only its immediate
value is used for the control (because the controller is reactive).

20
Chapter 4
DRCA ALGORITHM
The DRCA algorithm uses a two-step process: a deconfliction maneuver and a deconfliction
maintenance phase. The guarantees of n-vehicle collision avoidance rest upon three
results. The first is that a deconfliction maneuver will keep the vehicles from colliding so
long as they are each separated by at least a given bound. The second is that this deconfliction
maneuver will result in a conflict-free state for the vehicles. The final result is that
once a group of vehicles is conflict-free, the deconfliction maintenance controller will keep
them that way.
These steps yield the flow chart for the DRCA algorithm that is shown in Fig. 4.1.
When the vehicles are sufficiently separated, no avoidance is necessary, so the vehicles
can simply use their desired control. The deconfliction maintenance phase also takes the
desired control into account, though there is no guarantee that it is followed at all times.
In general, vehicles will not all see each other at the same time and perform one synchronous
maneuver. Rather, they must avoid each other as they come into sensing/communication
range (this will be referred to as communication from here on since some communication
is required even if sensing is used). This maneuver does not require all-to-all communication,
but does require some assumptions about connectivity and separation in order for the
full collision avoidance guarantees to hold. However, if these assumptions are relaxed, the
DRCA algorithm will still function as a high-performance heuristic, albeit without the full
safety guarantee.
The first step of the DRCA algorithm is to decide when a vehicle needs to perform its
deconfliction maneuver. Let D be the set of vehicles who are currently deconflicting (either
performing the deconfliction maneuver or deconfliction maintenance). Define a binary

21
Enter
Desired
Control
Deconfliction
Maintenance
Yes
Yes
Separated
No
Conflict−Free
No
Deconfliction
Maneuver
Figure 4.1: Flow chart of the DRCA algorithm.
condition, χ i : D → {true,false}, which designates that the separation between vehicle
i and the vehicles in D is enough to guarantee that vehicle i can avoid those vehicles. This
condition will be derived later. Define a second binary condition, ξ i (D), which designates
when vehicle i executes avoidance. The condition ξ i can be chosen for a longer or shorter
look-ahead, but at the crossover point, χ i must be true.
The DRCA algorithm causes vehicle i to follow its desired control until ξ i (D) becomes
false. At this point vehicle i becomes a member of D and performs its deconfliction maneuver,
which amounts to choosing a conflict-free velocity vector, broadcasting it to the
group, and then accelerating to attain it as quickly as possible. Once this vehicle attains
a conflict-free state with the other vehicles in D, it switches to deconfliction maintenance.
The other vehicles in D that are already performing deconfliction maintenance avoid this
new vehicle using its broadcast velocity instead of its actual velocity until its maneuver is
complete. Conceptually, this method is akin to a turn signal, where the new vehicle tells
the group how it will be accelerating, and the group can then work around that decision. As
such, each vehicle will see no conflicts with a newly added member of the group. To implement
this algorithm, each vehicle in D must be able to broadcast to all the other vehicles

22
in D.
If D is empty, then each vehicle uses ξ i (j) to determine when to maneuver, where j is
the nearest vehicle. If vehicle i is less maneuverable than vehicle j, then ξ i (j) will turn
false before ξ j (i) does. In this case, vehicle i becomes an element of D, making D = {i},
but vehicle i does not perform a deconfliction maneuver, since vehicle j will be able to
safely deconflict even if the vehicles move closer together. Now that D is nonempty, the
system then follows the previous directions. Once ξ i (D) becomes true again, vehicle i is
removed from D and solely follows its desired control.
The deconfliction maneuvers used here are in fact simple optimization schemes with
the goal of finding the smallest velocity change necessary to attain a conflict-free state.
In this sense, the maneuvers are most similar to [8, 24] (in fact, because these authors’
optimization schemes use the same definition of conflict, they could also be used as deconfliction
maneuvers here). However, these two optimization schemes are centralized and
computationally expensive. More importantly, they do not give a bound on how far apart
the vehicles must be for a feasible solution to exist. This bound is of the utmost importance
for designing a safe system.
This algorithm is greedy in the sense that each vehicle minimizes its own cost function
(‖∆v i ‖), meaning that the solution will not be a global optimum for some overarching cost
function of the group. However, given the nonconvexity of the problem, finding any global
optimum is nontrivial. For instance [8] used a semidefinite relaxation, but still required a
random initial guess, and hence could not guarantee a global optimum either. Additionally,
the actual cost function for the system is unknown in general, since any desired controller
can have its own unique cost function associated with it, depending on the task. Therefore,
instead of attempting to minimize an arbitrary cost function, a suboptimal solution will be
allowed, with the focus instead on feasibility and low computation load.
The only information the DRCA algorithm needs on a continuous basis is the position
and velocity of each vehicle in D, which can either come from broadcast communication
(e.g. a transponder) or sensing (e.g. radar). If the system is heterogeneous, then the vehicles

23
must know a bound on the maximum speed and the minimum separation distance of the
other vehicles in D, which can be accomplished by a broadcast communication or vehicle
identification. Finally, the vehicles must know who is in D, and must receive the intended
velocity of any vehicle performing a deconfliction maneuver. This information cannot be
attained through sensing, so some form of broadcast communication is necessary to disseminate
it. Any vehicle not yet in D only needs the information at a range just outside of
where ξ i (D) becomes false. In the case of D being empty, this information is required only
of the nearest vehicle.
There are two caveats with respect to the assumptions in this algorithm. First, two
separate deconfliction groups cannot safely merge. The reason is that vehicles performing
deconfliction maintenance are in general separated by as little as d sep , for which χ i (D) is
false, but vehicles joining D must have χ i (D) true.
In many applications, vehicles are added to a group singly. For example in an airport,
if one considers the congested airspace of the airport itself as a group, wherein all vehicles
are in the deconfliction maintenance phase with each other, then new aircraft arrive from
all around where the airspace is not as congested. These aircraft can safely integrate one at
a time as long as they stay separated from each other by enough space such that when one
joins D it does not cause χ i (D) of the others to become false.
The second caveat concerns communication range. While the communication range
must be chosen such that vehicle i can talk to vehicle j by the time ξ i (j) becomes false,
this range only ensures that D is connected, not necessarily all-to-all. This problem can be
overcome by having vehicles rebroadcast information they have received. This multi-hop
method will tend to accrue more time delay, however the robustness analysis in Chapter 5
shows how the algorithm can be made robust to these delays.
4.1 Deconfliction Maneuvers
The purpose of the deconfliction maneuver is to bring a vehicle to a conflict-free state as
quickly as possible with a guarantee that no collisions will occur during this maneuver,

24
given certain bounds on the separation. The separation bounds are necessary because a
wide class of initial conditions exist for which collision avoidance is impossible, usually
due to a lack of sufficient control authority.
While the deconfliction maintenance phase can operate with arbitrarily small control
authority, the deconfliction maneuver requires control authority to get out of conflict before
a collision occurs. Therefore it is important when designing a maneuver that the vehicle
will have sufficient control authority to complete it. The problem with using speed changes
in the maneuver is that all vehicles have a maximum speed, so their control authority to
accelerate forward goes to zero as this maximum speed is approached.
Therefore, turning as a means of deconfliction is applicable to a wider range of vehicles,
since there is almost never a maximum heading angle. In addition, most vehicles have
more acceleration available in turning than in speeding up or slowing down, since the latter
requires work and former does not. Vertical acceleration can be as easy as turning for a
buoyant vehicle, or require a significant change in potential energy for a vehicle such as
an airplane. Therefore it may be prudent to restrict some vehicles to a 2D deconfliction
maneuver in order to ensure they are capable of completing it.
Two deconfliction maneuvers will be presented here, first a constant-speed maneuver
that nearly any vehicle is capable of, and second a variable speed maneuver that has better
performance at the cost of computational complexity. Both of these maneuvers are 2D, but
deconfliction maintenance works for 3D as well. In order to use these maneuvers in a 3D
application one would have to project the vehicle’s positions and velocities onto the plane,
perform the maneuver, then add back in the original vertical component to the resulting
velocity. The guarantees of attaining conflict-freedom will still apply (though the solutions
will be more conservative) and once deconfliction maintenance takes over, the 3D nature of
the system will be accounted for in a less conservative way.

25
Figure 4.2: Example optimization of a constant-speed maneuver. The black vector is v i ,
the red vector is v ′ i, the circle is the allowable space of solutions and the blue regions are
the collision cones.
4.1.1 Constant-Speed Maneuver
The optimization vehicle i performs is to find a new velocity vector, v i, ′ which is free of
conflict with the other vehicles in D and minimizes ‖∆v i ‖, where ∆v i = v i ′ − v i . While
this minimization is nonconvex in general (see Fig. 4.2), it can be solved relatively quickly
because only a small number of possible points must be checked. The algorithm for solving
this works as follows.
The allowable space for v i ′ is a circle, centered at the origin, of radius ‖v i ‖. The collision
cones split this space into several disjoint regions, each of which is an arc of the circle.
The optimal solution to each region must lie on one of its two ends, unless there are no
conflicts, in which case v i ′ = v i . The only possible end points are points where a collision
cone intersects the allowable circle. Therefore the first step of the optimization is to calculate
each of the intersections for each collision cone. This is done by first computing the
unit-vector ĉ for each side, representing the direction of the edge of the collision cone (see

26
Figure 3.1):
where R is the 2 × 2 rotation matrix. Now v ′ i = v j − aĉ where
a = ĉ T v j ±
ĉ = R(±α) ˜r
‖˜r‖ , (4.1)
√
(ĉ T v j ) 2 − v T j v j + v T i v i, (4.2)
and only solutions with real and positive values of a are valid. The list of v is ′ are ordered
by increasing ∆v i and checked consecutively for conflicts with the other vehicles. Because
of the ordering, as soon as a point is found which is conflict free for all j, it is the optimal
solution and the algorithm terminates.
There are a finite number of points to check as possible optima, which bounds the
maximum possible time the deconfliction maneuver will take to compute. For a vehicle
deconflicting with n other members of D, there are a maximum of 4n points to check. Each
of these much be checked against a maximum of n − 1 other collision cones. Therefore
the maximum computation time is upper-bounded by cn 2 , where c is related to the time
each type of computation requires. In general, the computation times are less than this
bound because as soon as a feasible point is found, the algorithm terminates, so most points
are never computed or checked. This bound is better than a general “polynomial time”
guarantee (as with convex optimization, for instance), since in this case the polynomial is
known to be quadratic.
This analysis would guarantee a conflict-free solution if the vehicle could attain its
desired velocity vector instantaneously. However, the limited control authority available
makes this impossible. Instead, it takes a finite amount of time for the vehicle to attain
its desired velocity, and during that time it and the other vehicles move, which causes the
collision cones to move. In order to ensure that the system is still conflict-free after this
motion, the initial collision cones must be enlarged to the point of enclosing all possible
movements.
To bound the collision cone, one must simply bound ‖∆˜r‖ ≤ δ, or how much the vehicles
can change position before the maneuver is complete. Then the width of the collision

27
cone is enlarged from (3.9) to
( )
dsep + δ
α e = arcsin
. (4.3)
‖˜r‖
Note that this expression implies an initial separation of ‖˜r‖ ≥ d sep + δ, or else α e will be
undefined.
Lemma 2. Let there be two vehicles (i and j), each modeled by a planar version of (2.1).
Vehicle j is subject to the maximum speed constraint ‖v j ‖ ≤ v j,max . Vehicle i is subject
to ‖u i ‖ ≤ u i,max and ‖v i ‖ constant and is turning as quickly as possible from its initial
velocity, v i , to its desired velocity, v ′ i. The relative motion between the vehicles in the time
it takes vehicle i to attain its desired velocity is bounded by ‖∆˜r‖ ≤ δ, where
δ = ‖v i‖
u i,max
(2 ‖v i ‖ + πv j,max ) . (4.4)
Proof. The longest time, t, the maneuver could take is if v ′ i = −v i , so t ≤ π ‖v i ‖ /u i,max .
Likewise, since vehicle i is tracing an arc, its motion is bounded by ‖∆r i ‖ ≤ 2 ‖v i ‖ 2 /u i,max .
Finally, vehicle j is bounded by ‖∆r j ‖ ≤ v j,max t. Therefore,
‖∆˜r‖ ≤ ‖∆r i ‖ + ‖∆r j ‖ ≤ 2 ‖v i‖ 2
u i,max
+ π ‖v i‖ v j,max
u i,max
= ‖v i‖
u i,max
(2 ‖v i ‖ + πv j,max ) . (4.5)
This bound can in turn be used in (4.3) to size the enlarged collision cone. Note that for
a homogeneous group of vehicles this bound can be written in terms of the deconfliction
difficulty factor as δ = (2 + π)ηd sep . The following theorem states how this bound can be
used to keep vehicles from colliding during a deconfliction maneuver.

28
Theorem 1. Let there be a set of vehicles, D, which are not in conflict with each other.
When another vehicle, i, is in conflict with some or all members of D and performs the
constant-speed maneuver, the system will be conflict-free in time t, where
t ≤ π ‖v i‖
u i,max
, (4.6)
and no vehicles will collide during the maneuver. The vehicles are all modeled by a planar
version of (2.1), have speed constraints ‖v j ‖ ≤ v j,max and vehicle i has the input constraint
‖u i ‖ ≤ u i,max . It is assumed that a feasible solution to the optimization problem, v ′ i, exists
and that the vehicles in D maintain a conflict-free state with v ′ i, using a cone with width
defined by (4.3) and (4.4).
Proof. If a feasible point exists for the optimization problem, then the optimal solution is
guaranteed to be found and this point will satisfy
( )
|∠ṽ ′ dsep + δ
− ∠˜r| ≥ arcsin
. (4.7)
‖˜r‖
The maximum amount of time required for vehicle i to get from its initial v i to v ′ i is t =
π ‖v i ‖ /u i,max , and during this time ∆˜r ≤ δ from Lemma 2. Therefore, once the desired
velocities have been attained, one still has
|∠ṽ ′ − ∠(˜r + ∆˜r)| ≥ arcsin
( )
dsep
, (4.8)
‖˜r‖
meaning the vehicles are not in conflict. The vehicles cannot collide during this time because
as stated earlier, the pairs must be initially separated by at least ‖˜r‖ ≥ d sep + δ,
which means after the maneuver, they still must be outside of collision because ‖˜r + ∆˜r‖ ≥
d sep .
Of course, all of this is for naught if a feasible solution does not exist for the optimization.
The following theorem gives a conditional bound, χ i (D), on initial separation that is
sufficient to guarantee the existence of a solution.

29
Theorem 2. For vehicle i of an n-vehicle system in the plane, let vehicle i’s speed be
the constant ‖v i ‖, while each other vehicle’s speed is constrained by the uniform bound
‖v j ‖ ≤ v max . There exists an admissible velocity vector, v ′ i, which is conflict-free with the
other n − 1 vehicles, given that vehicle i is separated from the other vehicles such that
⎧ (
∑
√ )
‖v
⎪⎨ arccos cos(2α e ) − sin(2α e ) i ‖ 2
− 1 , ‖v
vmax
2 i ‖ < v max
j∈D
π ≥
(4.9)
∑
⎪⎩ 2α e , ‖v i ‖ ≥ v max .
j∈D
Proof. In order for a conflict-free point to exist, the collision cones from the other vehicles
cannot completely cover the admissible circle of possible headings of vehicle i. When
another vehicle’s maximum speed is higher, the largest range of headings, θ, it can cover
occurs when one edge of its collision cone is tangent to the admissible circle (see Fig. 4.3).
Some geometry can show that the relationship between the width of the expanded collision
cone, α e , and θ is given by
⎛
θ ≤ 2 arccos ⎝cos(2α e ) − sin(2α e )
√
‖v i ‖ 2
v 2 max
⎞
− 1⎠ . (4.10)
As long as the sum of those θs is less than 2π, then not all of the possible headings can be
in conflict. When the other vehicle’s maximum speeds are less, then θ takes on its limiting
value of 4α e . Putting this all together yields the full bound (4.9).
4.1.2 Variable-Speed Maneuver
If a vehicle has no minimum speed (i.e. it can stop), then it can use a variable-speed deconfliction
maneuver, which has better performance than the constant-speed maneuver. The
better performance comes from the fact that any constant-speed maneuver is admissible for
a variable-speed vehicle, so this optimization is less constrained than the previous, meaning
the solution must be equal to or better than the constant-speed solution. The cost of this
optimality is higher computational load.

30
Figure 4.3: Example of a borderline case for five vehicles where no conflict-free constantspeed
maneuver exists. The circle is the allowable set (radius ‖v i ‖) and the blue regions
are the collision cones (shown with all equal velocities, of magnitude greater than ‖v i ‖).
The allowable space for v ′ i is a disk, centered at the origin, of radius v i,max . The optimal
solution to each region must lie on its boundary because the zero-cost point is not
feasible (otherwise there would be no conflicts). This optimal solution must either be a
vertex between two collision cones, the nearest point on a single collision cone, or the vertex
between one collision cone and the edge of the allowable space. An example of this
optimization is shown in Fig. 4.4.
The first step of the algorithm is to compute each of the nearest points on each collision
cone (there is a right and a left solution for each cone), which is v ′ i = ĉĉ T ṽ + v j . This
list of points is checked to make sure each ‖v ′ i‖ ≤ v i,max . If this is not the case, then v ′ i is
replaced with
√
v i ′ = v j − ĉĉ T v j ± ĉ (ĉ T v j ) 2 − vj Tv j + vi,max 2 , (4.11)
where only the solution with the smaller ∆v i is kept. These points account for the only
possible optima on the intersection of a cone and the edge of the space. Next, all of the twocone
intersection points are found using linear equations (keeping only those with ‖v ′ i‖ ≤
v i,max ). All of these points are ordered by increasing ∆v i and checked consecutively for

31
Figure 4.4: Example optimization of a variable-speed maneuver. The black vector is v i , the
red vector is v ′ i, the circle is the edge of the allowable disk of solutions and the blue regions
are the collision cones.
conflicts with the other vehicles. Because of the ordering, as soon as a point is found which
is conflict free for all j, it is the optimal solution and the algorithm terminates.
As with the constant-speed maneuver, this optimization has a bound on its computation
time since there are still a finite number of possible optima to check. However, since there
are now points that are a combination of two cones, there are O(n 2 ) points to check. Since
these still need to be checked for conflict with the other n−2 collision cones, the maximum
computation time is now upper-bounded by cn 3 , rather than cn 2 , where c is related to the
time each type of computation requires.
Just as with the previous optimization, this algorithm usually terminates far before its
maximum computation time. However, if a particular application has very little computational
resource available, the constant-speed maneuver can still be used on a variable-speed
vehicle. The solution will not in general be as optimal as the variable-speed maneuver, but
it will be feasible and does scale better in computation time.

32
Once again the time required to attain this new velocity vector must be accounted for
in order to ensure the system actually comes to a conflict-free state and that there are no
collisions. The following lemma parallels Lemma 2, but obtains a smaller bound by use of
the now 2D acceleration vector.
Lemma 3. Let there be two vehicles (i and j), each modeled by a planar version of (2.1).
Vehicle j is subject to the maximum speed constraint ‖v j ‖ ≤ v j,max . Vehicle i is subject to
‖u i ‖ ≤ u i,max and ‖v i ‖ ≤ v i,max and is accelerating as quickly as possible from its initial
velocity, v i , to its desired velocity, v ′ i. The relative motion between the vehicles in the time
it takes vehicle i to attain its desired velocity is bounded by ‖∆˜r‖ ≤ δ, where
δ = v i,max
u i,max
(v i,max + 2v j,max ) . (4.12)
Proof. Let the angle between v i and v ′ i be 2γ and let t be the time required for the maneuver.
Then ‖∆v i ‖ = u i,max t, ‖∆r i ‖ = t ‖v i + v ′ i‖ /2, and ‖∆r j ‖ ≤ v j,max t. Also,
‖v i + v ′ i‖ ≤ 2v i,max cos γ and ‖∆v i ‖ ≤ 2v i,max |sin γ|. Therefore,
‖∆˜r‖ ≤ ‖∆r i ‖ + ‖∆r j ‖ ≤ 2v2 i,max |cos γ sin γ|
u i,max
+ 2v i,maxv j,max |sin γ|
u i,max
≤ v i,max
u i,max
(v i,max + 2v j,max ) . (4.13)
This bound can in turn be used in (4.3) to size the enlarged collision cone. Note that for
a homogeneous group of vehicles this bound can be written in terms of the deconfliction
difficulty factor as δ = 3ηd sep . Also note that if one decides to use the constant-speed
maneuver on a vehicle that fits the assumptions of Lemma 3, then one can still use the δ
bound of (4.12), since the proof is only based on the vehicle’s capabilities.
Theorem 1 still applies to the variable-speed maneuver, with the slight adjustment of
(4.6) to t = 2v i,max /u i,max (see above). However, Theorem 2 requires some adjustments to
the bound χ i (D):

33
Theorem 3. For vehicle i of an n-vehicle system in the plane, let vehicle i’s speed be constrained
by ‖v i ‖ ≤ v i,max , while each other vehicle’s speed is constrained by the uniform
bound ‖v j ‖ ≤ v max . There exists an admissible velocity vector, v ′ i, which is conflict-free
with the other n − 1 vehicles, given that vehicle i is separated from the other vehicles such
that
⎧ ( )
∑
⎪⎨
vi,max
arcsin
v
α e ≤
max
, v i,max < v max
j∈D
⎪⎩ π
, v 2 i,max ≥ v max .
(4.14)
Proof. In order for a conflict-free point to exist, the collision cones from the other vehicles
cannot completely cover the admissible disk of possible velocities of vehicle i. The worst
case for this coverage when the other vehicles are capable of higher speed than vehicle i is
when all the vehicles have the same velocity, with their maximum speed, and their positions
splay out their collision cones so as to form one large, continuous cone (see Fig. 4.5). The
interior angle of this large cone must be less than 2 arcsin(v i,max /v max ) to ensure that it
cannot cover the entire admissible circle. When vehicle i’s speed is higher than the other
vehicles, the arcsine takes on its limiting value of π/2 and becomes conservative because
it only allows the combined collision cone to cover a half-space within the disk. Together
these constraints form (4.14).
In the special case where all of the vehicles are exactly the same distance away (as in
Fig. 4.5), the bound (4.14) can be simplified to a bound on distance. First note that the sum
of the collision cones is just (n − 1)α e and rearranging (4.3) gives ‖˜r‖ ≥ (d sep + δ)/ sin α e .
Then the distance bound is
⎧
⎪⎨
‖˜r‖ ≥
⎪⎩
sin
d sep+δ
arcsin ( v i,max
vmax )
n−1
d sep+δ
π
sin(
!, v i,max < v max
2(n−1)) , v i,max ≥ v max .
(4.15)

34
Figure 4.5: Example of a borderline case for six vehicles where no conflict-free variablespeed
maneuver exists. The circle is the edge of the allowable disk (radius v i,max ) and the
blue regions are the collision cones, shown with all equal velocities, of magnitude greater
than v i,max .
Note that for large n, the small angle approximation can be used to simplify (4.15) to
⎧
⎪⎨
‖˜r‖ ≥
⎪⎩
(n−1)(d
„ sep+δ)
vi,max
«,
arcsin v j,max
v i,max < v j,max
2(n−1)
π
(d sep + δ), v i,max ≥ v j,max ,
(4.16)
however the result is not actually conservative, so (4.15) should still be used for safety
purposes. It is interesting to note though, that as n gets large, the bound approaches a linear
relationship with n − 1. This makes intuitive sense in that as more vehicles are in a space,
they must be spaced out more to ensure a safe trajectory exists between them.
4.1.3 Heuristic Performance
The guarantees provided by the previous maneuvers are ideal if one can ensure the vehicles
always have enough warning of each other’s presence. However, some systems may not
always be that capable. We know from the previous analysis that these requirements are
conservative, so it may still be possible to deconflict the system, though not in a guaranteed

35
fashion. The goal then is to have a heuristic backup algorithm that will do a qualitatively
good job of attempting to deconflict vehicles which do not meet the requirements of the
guarantees. Handily, this heuristic backup algorithm can be seamlessly joined with the
normal deconfliction maneuvers. Basically it operates the exact same way, but guarantees
that a solution is returned regardless of feasibility.
The first thing that could cause a solution to not exist is if ‖˜r‖ < d sep + δ, causing
α e to be undefined. In this situation, let the heuristic be to define α e = π/2 (its limiting
value). The only other thing that could cause a solution to not exist is if the collision cones
cover the entire admissible space, making the optimization find no feasible solution. In this
case, let the heuristic return the solution v i ′ = −v i . This solution was chosen because it is
a velocity the vehicle can attain and it will take maximum acceleration to get there. The
intuition behind the choice is that when vehicles are in conflict, it is better to do something
than nothing, and by continuing to use maximum acceleration, there is more chance that
the system will stumble upon a conflict-free state.
The vehicle in this case does not join D until it attains a conflict-free state, since the
deconfliction maintenance controller that the other vehicles in D are running cannot cope
with vehicles in conflict. As a worst case, it is better for the new vehicle to collide with one
of the members of D than to cause them all to collide with each other.
Now the DRCA algorithm can be compared to other collision avoidance algorithms
that are completely decentralized or have other assumptions that violate the requirements
for guaranteed avoidance. Most of these algorithms are also heuristic, in that they do not
guarantee collision avoidance in the presence of limited available acceleration and a minimum
separation distance, d sep > 0. Therefore the algorithms can be compared qualitatively
based on how often collisions occur and how well the vehicles attain their goals. This performance
comparison is shown in Chapter 6.

36
Desired
Control
u d
Deconfliction
Maintenance
u
Vehicle
z i
z
Figure 4.6: Block diagram of the system when using the deconfliction maintenance controller.
The vector z = [z 1 , z 2 , . . . , z n ] T and z i = [r i , v i ] T . Note that the deconfliction
maintenance block acts as a type of saturation on the desired control, u d .
4.2 Deconfliction Maintenance
Once the deconfliction maneuver has been performed and the system is in a conflict-free
state, then the deconfliction maintenance controller can be used to keep the system conflictfree.
This controller allows each vehicle to use its desired control input unless that input
would cause the vehicle to come into conflict with another vehicle. A basic block diagram
of this setup is shown in Fig. 4.6.
4.2.1 Control Law
In order to smoothly transition from the desired control to the avoidance control, each
vehicle needs a way to measure how close its velocity vector is to causing a conflict. The
first step is to construct a unit-vector, ĉ, representing the side of the collision cone nearest
ṽ (like (4.1), but now in 3D). The vector ĉ is found by rotating ˜r by α around a vector
q = ˜r × ṽ and normalizing:
ĉ =
˜r ( ) q × ˜r
‖r‖ cos α + sin α. (4.17)
‖q‖ ‖r‖
Next, construct a normal vector, e, from the collision cone to the relative velocity vector,
ṽ (see Fig. 4.7). If ĉ T ṽ > 0, then e = (I − ĉĉ T )ṽ, but if ĉ T ṽ ≤ 0 (the vehicles are headed
away from each other), then no normal exists, and the nearest point on the collision cone is

37
v ĩ
v
e
p t,ijˆt i
v j
p n,ijˆn i
d sep
ĉ
β
α
˜r
α
d sep
Figure 4.7: Geometry of the e and ĉ vectors, as seen in an ˜r-ṽ section through the 3D
collision cone (dotted lines). The conflict measures p t and p n are shown, but ˆb points into
the page, so p b is infinite (ĉ T ṽ > 0 in this example).
the tip, so e = ṽ. Therefore:
⎧
⎪⎨ ṽ, ĉ T ṽ ≤ 0
e =
⎪⎩ (I − ĉĉ T )ṽ, ĉ T ṽ > 0.
(4.18)
In order to combine the effects of multiple collision cones, it is helpful to decompose
the system into three component directions and analyze those directions separately. Let the
coordinate system be defined by the orthonormal vectors ˆt, ˆn, and ˆb. The orientation of this
coordinate system is arbitrary, but the convention of using tangent, normal, and binormal
notation is chosen since fixing the coordinates to the body of the vehicle often simplifies
analysis.
The next step is to determine how much control (change in velocity) can be applied in
each of these directions before a conflict forms. For simplicity, a conservative approach
is taken whereby the signed distance is found from ṽ to the tangent plane enclosing the
collision cone (defined by the normal vector e) in each of the ˆt, ˆn, and ˆb directions. These
signed distances are
p t,ij = ‖e ij‖ 2
, p
e T n,ij = ‖e ij‖ 2
ijˆt i e T ijˆn , and p b,ij = ‖e ij‖ 2
, (4.19)
i
e T ijˆb i

38
which are described graphically in Fig. 4.7.
Define ɛ t , ɛ n , ɛ b > 0 as thresholds such that when |p t | > ɛ t , the conflict is far enough
away that it can be ignored (and likewise for p n and p b ). The n-vehicle deconfliction maintenance
controller running on vehicle i computes p t , p n , and p b to each of the other vehicles
and then finds the closest conflict in each direction, i.e.
p + t i
p − t i
= min
j∈D {p t,ij > 0, ɛ ti }
= − max
j∈D {p t,ij < 0, −ɛ ti } ,
(4.20)
and likewise for p n and p b . Note that by definition 0 < p ± ≤ ɛ. To simplify notation, in any
case where a relation holds in the tangent, normal, and binormal directions, the subscript
will be suppressed.
The input is constructed using a function, F , such that in each direction u = F (p + , p − ).
The control function chosen for this implementation of the DRCA algorithm is
F (p + , p − ) = u min
ɛ
p + + u max
p − + u d − u max − u min
p + p − , (4.21)
ɛ
ɛ 2
because it is a bilinear interpolation of the following ordered triples of the form (p + , p − , u):
P 1 = (0, 0, 0) P 2 = (ɛ, 0, u min )
P 3 = (0, ɛ, u max ) P 4 = (ɛ, ɛ, u d ).
(4.22)
An example of this control function is shown in Fig. 4.8. Because F depends on the desired
control, u d must be saturated such that
u min ≤ u d ≤ u max . (4.23)
This choice of control function means that once the u vector is constructed from its
three components, then u ∈ R. Then the saturation function S will give the final resultant
control vector, which will be in C. The octant-preserving nature of S will ensure this final
saturation step does not violate the requirements for the proof of collision avoidance.
A more intuitive way to choose values for ɛ is to relate it to a gain-like parameter,
k = u max − u min
. (4.24)
ɛ

39
Figure 4.8: Example of the control function, F . Note that P 4 moves up and down with
changing u d .
Note that k has units of inverse seconds. Also, one can see that the magnitude of the
gradient of the control function will always be less than or equal to k, regardless of the
desired control.
Assuming there are n vehicles in the deconfliction group, D, this algorithm’s computation
time on each vehicle scales as O(n), since it only requires the computation of each
other vehicle’s collision cone and then substituting these results into the control function.
Technically there are O(n 2 ) computations happening in the entire group, but since these
computations are independent of each other (only linked by the sensed or communicated
states), they can happen in parallel in a distributed fashion, so only the per-vehicle scaling
matters.
4.2.2 Guarantee
The following theorem and proof guarantee that the system can maintain its conflict-free
state despite arbitrary control authority restrictions. Each time a new vehicle is added to the

40
deconfliction group, D, it performs its deconfliction maneuver, broadcasting a conflict-free
velocity to the group. Since it is this broadcasted velocity that is used by the deconfliction
maintenance algorithm until the new vehicle really is conflict-free, the deconfliction group
never sees any new conflicts.
Theorem 4. The deconfliction maintenance controller described above, when implemented
on n vehicles with dynamics (2.1) and inputs constrained by C i , will keep the system collision
free for all time if the system starts conflict-free.
Proof. To measure the distance to a collision, define m as a signed version of ‖e‖ (in terms
of ṽ from the geometry in Fig. 4.7):
⎧
⎪⎨ ‖ṽ‖ , ĉ T ṽ ≤ 0
m =
⎪⎩ ‖ṽ‖ sin(|β| − α), ĉ T ṽ > 0.
Note that m is negative during conflict and positive during no conflict.
(4.25)
To ensure that a conflicted state is never reached (i.e. m is always greater than zero), it
is sufficient to show that for every pair of vehicles there is a neighborhood on the positive
side of m = 0 where ṁ ≥ 0. This condition implies that as the boundary of a collision cone
approaches, it will either stop approaching or recede before a conflict is formed. The fact
that this is a one-sided neighborhood is important because ṁ does not exist at m = 0 for
the same reason that d ‖ṽ‖ does not exist at ṽ = 0. However, if this condition is satisfied,
dt
then m > 0 for all time, ensuring that m ≠ 0.
For ĉ T ṽ ≤ 0 (using the ij notation again briefly for clarity),
ṁ = eT ˙ṽ
m = êT ˙ṽ, (4.26)
where ê = e
‖e‖ . Expanding u i into its components yields u i = [u ti t i + u ni n i + u bi b i ]. Next
substitute for ṽ ij in (4.26) to get
ê T ij ˙ṽ ij = u ti ê T ijˆt i + u ni ê T ijˆn i + u bi ê T ijˆb i − u tj ê T ijˆt j − u nj ê T ijˆn j − u bj ê T ijˆb j . (4.27)

41
Because of the symmetry of the problem e ji = −e ij , so
ê T ˙ṽ ij ij =u ti ê T ijˆt i + u ni ê T ijˆn i + u bi ê T ijˆb i + u tj ê T jiˆt j + u nj ê T jiˆn j + u bj ê T jiˆb j
(
uti
=m ij + u n i
+ u b i
+ u t j
+ u n j
+ u )
b j
. (4.28)
p t,ij p n,ij p b,ij p t,ji p n,ji p b,ji
For ĉ T ṽ > 0, the derivative of (4.25) becomes
ṁ = sin(|β| − α) d ‖ṽ‖
dt
+ ‖ṽ‖ cos(|β| − α) d (|β| − α). (4.29)
dt
The derivative exists because m > 0 implies ‖ṽ‖ ≠ 0, |β| ≥ α > 0, and ‖˜r‖ ≥ d sep > 0.
From the geometry,
d |β|
dt
( d∠ṽ
= sgn(β) − d∠˜r )
dt dt
= sgn(β) d∠ṽ
dt
The derivative of α is somewhat less straight-forward:
dα
dt = d ( ( ))
dsep
arcsin
dt ‖˜r‖
= d dt
( ) (
dsep
1 −
‖˜r‖
⎛
= d sep ‖ṽ‖ cos β
‖˜r‖ 2
Combining the above two terms gives
+ ‖ṽ‖ |sin β| . (4.30)
‖˜r‖
( ) ) 2 −1/2
dsep
‖˜r‖
⎞
⎝
‖˜r‖
√ ⎠
‖˜r‖ 2 − d 2 sep
= ‖ṽ‖ cos β tan α. (4.31)
‖˜r‖
d
‖ṽ‖
(|β| − α) = (|sin β| − cos β tan α) + sgn(β)d∠ṽ
dt ‖˜r‖ dt , (4.32)
which can be substituted into (4.29) to get
ṁ = sin(|β| − α) d ‖ṽ‖
dt
+ cos(|β| − α) sgn(β) ‖ṽ‖ d∠ṽ
dt
+ cos(|β| − α) ‖ṽ‖2 (|sin β| − cos β tan α). (4.33)
‖˜r‖

42
To simplify the above, note that in the case of ĉ T ṽ > 0, the geometry of the vectors (Fig.
4.7) gives
Therefore (4.33) reduces to
ê T ˙ṽ = sin(|β| − α) d ‖ṽ‖
dt
+ cos(|β| − α) sgn(β) ‖ṽ‖ d∠ṽ
dt . (4.34)
ṁ = ê T ˙ṽ + cos(|β| − α) ‖ṽ‖2 (|sin β| − cos β tan α). (4.35)
‖˜r‖
This expression can be further simplified by recognizing that
m
‖ṽ‖ cos α
=
sin |β| cos α − sin α cos |β|
cos α
= |sin β| − cos β tan α. (4.36)
Therefore
ṁ = ê T ˙ṽ + m
‖ṽ‖ cos(|β| − α)
. (4.37)
‖˜r‖ cos α
The second term is always positive because ĉ T ṽ > 0 implies that cos(|β| − α) > 0, and
α ≤ π/2 by definition. Combining this result with (4.26) implies that
for any value of ĉ T ṽ. Recalling (4.28),
ṁ ≥ ê T ˙ṽ (4.38)
(
uti
ṁ ij ≥ m ij + u n i
+ u b i
+ u t j
+ u n j
+ u )
b j
. (4.39)
p t,ij p n,ij p b,ij p t,ji p n,ji p b,ji
As long as the controller ensures that u ti has the same sign as p t,ij , etc. then ṁ ≥ 0 for that
pair of vehicles. Note that each vehicle need only calculate its control from its own point of
view, and this rule will automatically cause the vehicles to cooperate in avoiding conflicts.
Combining this result with the definitions (4.20), any continuous control function that
satisfies
lim
p + t i
→0 + u ti ≥ 0,
lim u ni ≥ 0,
p + n i →0 +
lim u bi ≥ 0,
p + b →0 + i
lim
p − t i
→0 + u ti ≤ 0,
lim u ni ≤ 0,
p − n i →0 +
lim u bi ≤ 0,
p − b →0 + i
(4.40)

43
also ensures that there is a neighborhood on the positive side of m = 0 for which ṁ ≥ 0,
guaranteeing the system cannot enter a conflicted state.
The control function used in this implementation (4.21) satisfies (4.40), so the deconfliction
maintenance controller will cause the n-vehicle system to remain conflict-free for
all time, assuming it started that way.
Note that this result holds for arbitrary (even time varying) u d , u min and u max , so long
as they satisfy (4.23) and R contains the origin at every instant. In addition, u can be
further saturated using S in order conform to the non-rectangular constraint set C without
affecting the guarantee. The key for this saturation to work is that S preserve the octant of
u, such that (4.40) is still satisfied.
4.2.3 Priority
Vehicle priority can be assigned through the choice of the vehicles gains, k. If a vehicle has
a higher gain, it has a higher priority, meaning it gets routing preference over vehicles with
lower gains. When the gains are equal, the system will tend to come to quasi-equilibrium
where each vehicle’s control goes to zero and m stays roughly constant. This equilibrium
will tend to occur near m = ɛ/2 (though many factors will alter it slightly). By raising the
gain, one shrinks the corresponding ɛ, meaning the higher gain vehicle will approach an
equilibrium m value that is smaller than the lower gain vehicle. This situation is characterized
by the higher gain vehicle always getting slightly closer to conflict with the lower gain
vehicle, while the lower gain vehicle moves away. In the overall solution, the lower gain
vehicle will give way so that the higher gain vehicle can follow its desired control more
closely, but without causing conflict among any of the vehicles.
Likewise, each vehicle can also have different gains in different directions, i.e. k t ≠ k n ,
etc. For instance, if k t > k n , the vehicle will tend to turn more (accelerate in the normal
direction) rather than speed up or slow down in trading off with the other vehicles. This
choice gives the control designer another knob to tune how the system behaves.

44
4.2.4 Behavior
One thing to notice about the DRCA algorithm is that the deconfliction maneuvers are set
up in a conservative way (by enlarging the collision cones). Since it is unlikely that the
vehicles will actually move in a worst-case way, once the maneuver is complete and the
new vehicle switches to the deconfliction maintenance phase, it will likely not be near the
edges of the real collision cones. However, at this point the deconfliction maintenance
algorithm acts like a second stage optimization, whereby the desired controller can bring
the vehicles together until they are near, but not quite in, conflict.
In a qualitative sense, the DRCA algorithm works in the following way. First, the
deconfliction maneuver solves the nonconvex optimization problem to find a feasible optimum.
Next, the deconfliction maintenance algorithm continuously adjusts that solution
to keep it locally optimal, accounting for changing desires, vehicle priority, etc. The same
idea was used in [27], though implemented completely differently. This method is much
more computationally efficient than the method in [8, 24], wherein the entire optimization
routine was rerun periodically to adjust the trajectories, because the deconfliction maintenance
algorithm requires only O(n) computation time instead of O(n 2 ) or O(n 3 ) for the
deconfliction maneuvers (or unknown scaling for [8, 24]).
Additionally, while the deconfliction maneuver is fundamentally two-dimensional, the
deconfliction maintenance algorithm is more general. If one considers a three-dimensional
group of vehicles in conflict, they would have to be projected onto the plane in order to compute
a deconfliction maneuver. One can then add back the z-components of their velocities
to the solution obtained without affecting the guarantees, and while this three-dimensional
solution is grossly conservative, the deconfliction maintenance algorithm will effectively
reoptimize it while taking into account the three-dimensional nature of the system.
Likewise, vehicles with a minimum speed must use the constant-speed deconfliction
maneuver even if they have the ability to vary their speed. The original optimization is
therefore quite conservative, but the deconfliction maintenance phase will again allow them

45
to adjust their trajectories to a more optimal solution once a conflict-free state has been
reached.
4.2.5 Liveness
Liveness of a deconfliction algorithm refers to the ability of the vehicles to attain their
goals in addition to not colliding. Liveness is difficult to prove in general because of its
dependence on the desired controller’s interaction with the DRCA algorithm. For instance,
if the desired control is for two vehicles to attain the same waypoint the DRCA algorithm
will always choose safety over performance, so the goal will not be met in favor of keeping
the vehicles from colliding.
Likewise, if the desired controller is a cooperative search
algorithm, it becomes unclear how one would define the liveness of a solution. In light
of these facts, liveness will be shown here for a very strict set of assumptions, though the
arguments should hold qualitatively for a wider range of situations.
Theorem 5. Let the deconfliction group, D, contain n constant-speed vehicles that are not
in conflict. Let their desired controllers be heading regulators where each vehicle has some
constant desired heading. If each vehicle has an avoidance gain, k n , such that its speed is
separated from the other’s speeds by at least the avoidance threshold, ɛ n , then all of the
vehicles will attain their desired headings while remaining conflict-free.
Proof. Theorem 4 already guarantees that the system will remain conflict-free for all time.
Any vehicle for which p n ≥ ɛ n will follow its desired control exactly, and hence will attain
its heading as long as the collision cones remain at least ɛ away.
Any vehicle for which p n < ɛ n must have ĉ T ṽ > 0, because if ĉ T ṽ ≤ 0 then from
(4.18), e = ṽ and ‖ṽ‖ ≥ ɛ n and since p n ≥ ‖e‖, then p n ≥ ɛ n , which is a contradiction.
Since ĉ T ṽ > 0, (4.37) applies, so
ṁ = ê T ˙ṽ + m
‖ṽ‖ cos(|β| − α)
. (4.41)
‖˜r‖ cos α
The only reason a conflict is a problem is if the desired controllers want to accelerate
the vehicles such that the e T ˙ṽ term is negative, since the DRCA algorithm adjusts those

46
inputs such that ṁ ≥ 0. However, we already know that the second term is strictly positive
since the vehicles are not in conflict and ĉ T ṽ > 0 (see proof of Theorem 4). Therefore
e T ˙ṽ can be negative while allowing the DRCA algorithm to keep ṁ ≥ 0. Recall from
(4.28) that e T ˙ṽ ij ij = u ni e T ijˆn i + u nj e T ijˆn j . Even though each vehicle wants its own term to
be negative, and even though the sum is allowed to be negative, the DRCA algorithm may
cause one vehicle’s term to be positive (go away from its goal) in order to allow the other
to be more negative (approach its goal faster), which is the idea behind preferential routing
and vehicle priority. However, once the vehicle with priority attains its goal heading, then
its u n goes to zero, which allows the lower priority vehicle to start moving toward its goal.
Even though these conflicts can be daisychained such that each vehicle is in conflict
with the next vehicle over, this analysis still guarantees that at least one vehicle in the
group is moving toward its goal at all times, and so some vehicle must eventually attain its
goal and allow another to progress. Therefore all the vehicles in the group must eventually
attain their desired headings.
This theorem may seem limited, but attaining a fixed heading is effectively equivalent
to moving towards a distant waypoint, which is a common goal for a wide range of applications.
Additionally, even for changing desires this analysis still shows some net progress
will be made, but it is unclear how that progress will be distributed between the vehicles.
The main limiting assumption in this theorem is the speed separation. The reason for
this assumption is to guarantee that ĉ T ṽ > 0, because otherwise ṁ = ê T ˙ṽ and there
may be no net progress. This situation can only occur if the relative velocity is small:
‖ṽ‖ < ɛ. Most desired controllers tested so far on equal-speed vehicles or variable-speed
vehicles have shown this situation to be unstable, such that the vehicles will move to a
different configuration where progress toward goals can be made. However, it is possible
to stabilize this situation and form a livelock, which must be avoided in the design of a
desired controller or else goals may never be attained.
One whole class of desired controllers is eliminated by this logic: flocking or schooling

47
controllers. The goal of these controllers is to make ṽ = 0, which amounts to m = 0,
which the DRCA algorithm does not allow. The vehicles cannot progress toward their goal
of ṽ = 0 because the DRCA algorithm sees this situation as being on the edge of not safe.
On the plus side, for vehicles which can stop, a deadlock is impossible because only one
vehicle is allowed to stop at a time (since two stopped vehicles would also have m = 0).
However, that is not sufficient to prove liveness in general.

48
Chapter 5
ROBUSTNESS ANALYSIS
Robustness for this system can be split into two areas: robust stability and robust avoidance.
Robust stability encompasses the traditional ideas of linear stability analysis, focused
on how much gain can be applied to the system before time delays or other unmodeled
dynamics drive the system to oscillate explosively. Robust avoidance is a higher-level idea
relating to how relaxation of the assumptions (in this case cooperation) affects the validity
of the collision avoidance guarantees.
5.1 Robust Stability
As complex as this control algorithm may appear, it can still be analyzed for robustness
using linear tools. The deconfliction maneuver is a feed-forward control, so robust stability
does not apply, since the input is static. The desired controller is assumed to exhibit an adequate
degree of robust stability, since this controller is designed for the system in question.
Therefore, the deconfliction maintenance controller is the part of this algorithm that needs
to be analyzed for robust stability. Depending on the system in question, uncertainties can
take many different forms. The following analysis is based upon a complex multiplicative
uncertainty bounded by the transfer function w(jω).
Stability in this case is analyzed about an equilibrium in the conflict space, i.e. where
ṁ = 0 for the pair of vehicles in question. This equilibrium refers to the period from
when the vehicles end their deconfliction maneuver to when they begin to pass each other,
characterized by nearly constant-velocity trajectories.
Theorem 6. Let k ij = k ti + k ni + k bi + k tj + k nj + k bj . The system is stable in the presence

49
of a complex multiplicative uncertainty bounded by w(jω) if k ij satisfies
k ij
√k 2 ij + ω2 <
1
|w(jω)|
∀ω. (5.1)
Proof. Recalling (4.26), (4.37), and (4.28), one has that
ṁ ij = m ij
(
uti
p t,ij
+ u n i
p n,ij
+ u b i
p b,ij
+ u t j
p t,ji
+ u n j
p n,ji
+ u b j
p b,ji
+ h ij
)
, (5.2)
where h ij is a positive quantity given by
⎧
⎪⎨ ‖ṽ‖ cos(|β|−α)
, |β| − α ≤ π ‖˜r‖ cos α 2
h ij =
(5.3)
⎪⎩ 0, |β| − α > π. 2
The worst case for robust stability is when the feedback gain is the largest (most negative),
and one can see by looking at Fig. 4.8 that the largest gain in terms of p + will occur when
p − = ɛ and u d = u min . One can assume without loss of generality that the nearest conflict
is in the positive direction, i.e p > 0. In this case,
Applying to (5.2) yields
m u t
= m F (p t, ɛ)
p t
= m
p
( t
ut,max
p t
+ u )
t,min − u t,max
ɛ t
= −k t m + ê Tˆtu t,max . (5.4)
ṁ ij = − (k ij − h ij ) m ij + ê T Θ i u i,max + ê T Θ j u j,max , (5.5)
where u i,max = [u ti,max , u ni,max , u bi,max ] T . The second two terms are always positive (p t >
0 =⇒ ê Tˆt > 0, etc.) and only serve to push the equilibrium to an m that is greater than
zero. Only the first term has bearing on the robust stability of the system. One can open
the loop on this negative feedback system and examine the loop gain as a transfer function,
L(s):
L(s) = k ij − h ij
. (5.6)
s

50
Note that h ij
analysis does not apply.
< k ij or else this system does not have an equilibrium and the stability
The complementary sensitivity function, T (s), is given by
T (s) =
L
1 + L = k ij − h ij
s + k ij − h ij
. (5.7)
For a system perturbed by a complex multiplicative uncertainty, the perturbed loop gain,
L p (s), is given by
L p (s) = L(1 + w∆), |∆(jω)| ≤ 1, ∀ω. (5.8)
In this case, robust stability to an uncertainty bounded by w is guaranteed if
|T (jω)| <
1
|w(jω)|
∀ω, (5.9)
as shown in [32]. From (5.7),
|T (jω)| ≤
k ij − h ij
√
(kij − h ij ) 2 + ω 2 . (5.10)
Since 0 ≤ h ij ≤ k ij , then
k ij − h ij
√
(kij − h ij ) 2 + ω 2 ≤
k ij
√k 2 ij + ω2 , (5.11)
which implies that (5.1) is a sufficient bound for stability.
Depending on the nature of the system on which this algorithm is implemented, the
uncertainty, w, could take many forms. As an example, one common type of unmodeled
dynamics is time delay. Delays can be caused by sensors, communication, or even the
discretization of this continuous-time system.
Lemma 4. The system is stable in the presence of a pure time delay no greater than τ, if
k < 1.47
τ . (5.12)

51
Proof. A multiplicative uncertainty, w, that bounds this time delay is given in [32]:
⎧
⎪⎨ |1 − e −jωτ | , ω < π/τ
w(ω) =
⎪⎩ 2, ω ≥ π/τ.
(5.13)
If the limiting case is when ω ≥ π/τ, then (5.1) becomes
k
√
k2 + ω 2 < 1 2
If the other case is limiting, then
k < ω √
3
<
π
τ √ 3 . (5.14)
k
√
k2 + ω < 1
2 |1 − e −jωτ |
k
√
k2 + ω < 1
√ 2 2 − 2 cos(ωτ)
k 2 (1 − 2 cos(ωτ)) < ω 2
kτ <
ωτ
√
1 − 2 cos(ωτ)
. (5.15)
The right hand side is now in terms of a single variable (ωτ), and the minimum can be
calculated numerically:
min
ωτ
ωτ
√
1 − 2 cos(ωτ)
≈ 1.4775. (5.16)
This case is indeed limiting because 1.4775 < π/ √ 3. Therefore (5.12) is sufficient to
guarantee stability in the presence of the time delay τ.
This type of analysis can be used to find similar bounds on the gain of the system to
guarantee stability to other types of unmodeled dynamics and uncertainty, for instance noise
or lags caused by filtering.
5.2 Robust Avoidance
Another possible problem for this system is a vehicle which does not cooperate, i.e. does
not satisfy (4.40). This vehicle is regarded as antagonistic, regardless of its actual goals.

52
Note that a vehicle which does not run the DRCA algorithm, but rather holds a constant
velocity always satisfies (4.40) and is therefore not considered antagonistic.
One can see that in a multivehicle scenario, several antagonistic vehicles could surround
another vehicle’s velocity vector with their collision cones, bringing them together
until no conflict-free region was left, leaving the hapless vehicle with no guarantee of collision
avoidance. Indeed, in some situations collision avoidance may be impossible against
adversaries. Likewise even a single adversary can pin another vehicle between itself and
one or more constant-velocity vehicles (or vehicles with very small control authority). This
problem is fundamental to pursuit-evasion within groups of vehicles, and this algorithm
does not provide an easy solution. However, for a two-vehicle system with a single adversary,
the DRCA algorithm still provides a guarantee.
Theorem 7. For a two-vehicle system in which the vehicle’s dynamics are restricted by
(2.3), and vehicle one runs the deconfliction maintenance controller, and vehicle two is an
adversary using an unknown controller, the vehicles will remain conflict-free for all time
provided they started that way and that
which implies also that v 1,max > v 2,max .
u 1,max ≥ u √
2,maxv 1,max 3
√
, (5.17)
v1,max 2 − v2,max
2
Proof. The choice of the control function (4.21) ensures that at the edge of a conflict,
not only does vehicle one not approach the conflict any further, but it actually applies its
maximum control authority in the opposite direction. Rederiving (5.5) for only vehicle one
using deconfliction maintenance yields
ṁ = − (k t + k n + k b − h) m + ê T Θ i u 1,max + ê T Θ j u 2 , (5.18)
where one can see that the first term goes to zero as m → 0. The second term is always
greater than or equal to u 1,max because ê Tˆt 1 ≥ 0, etc. However, the last term is upperbounded
by u 2,max
√
3, so u1,max ≥ u 2,max
√
3 is sufficient to guarantee limm→0 + ṁ ≥ 0
and thus that conflict is avoided.

53
u 1
u 2
θ
γ
v 2,max
v 1,max
Figure 5.1: Geometry of the worst situation for avoiding an adversary. The dotted lines
represent the collision cone and the circles represent the set of allowable velocity vectors
for the two vehicles.
A problem with this analysis is that the control authority limits of the vehicles are not
actually fixed, but get restricted when the maximum speed, v max , is reached. The worst
case occurs when vehicle one is at its maximum speed and is on the edge of a conflict with
vehicle two, which is nearly at its own maximum speed. The angle that the collision cone
makes with the boundary of vehicle one’s allowable velocity set defines the ratio of control
authority needed to avoid vehicle two’s worst action. As can be seen in Fig. 5.1, the worst
angle for the collision cone to have is when γ is maximized. The law of sines gives
sin γ
= sin θ , (5.19)
v 2,max v 1,max
so γ is maximized when sin θ = 1, and the maximum is γ = arcsin(v 1,max /v 2,max ). The
required control authority to both stay out of conflict and stay within v 1,max is then
u 1,max ≥ u 2,max sec γ √ 3. (5.20)
Substituting the maximum γ from above and simplifying yields the bound (5.17).

54
Chapter 6
PERFORMANCE
This chapter will demonstrate the performance of the DRCA algorithm in simulation.
The following sections are intended to illustrate a variety of possible applications as well
as showcase the nominal and off-nominal behavior of the algorithm. The first section
shows the standard way the algorithm functions, comforming to all of the requirements
for the guarantees of safety in a 2D, variable-speed, heterogeneous system. The next section
demonstrates how vehicle priority can be changed and how it affects the trajectories.
Most of the simulations shown in this chapter are two-dimensional in order to make them
clear when printed, however section 6.3 shows a three-dimensional case for completeness.
Section 6.4 shows a number of simulations where various requirements of the guarantees
are broken and demonstrates the limits of the heuristic performance of the algorithm. This
section also serves to compare the constant and variable-speed deconfliction maneuvers.
Section 6.5 discusses computational scaling and compares the various parts of the DRCA
algorithm to another collision avoidance algorithm in terms of computation time. The next
section discusses how the DRCA algorithm can be adjusted to work with swarms of vehicles
and shows an example. The final section compares the DRCA algorithm to a Satisficing
Game Theory (SGT) algorithm in two classically difficult situations.
6.1 Variable Speed
This simulation (Fig. 6.1) is of a heterogeneous group of five vehicles performing target
tracking. Each vehicle has a target that moves at constant velocity (0.4 m/s), whose position
the vehicle is attempting to attain using a standard target following controller. Circles
around each vehicle in Fig. 6.1a denote their size, in that overlapping circles imply colli-

55
speed (m/s)
(a)
0.5
0.4
0.3
0.2
0.1
0
0 10 20 30 40 50 60
Time (s)
‖u‖ (m/s 2 )
0.25
0.2
0.15
0.1
0.05
(b)
0
0 10 20 30 40 50 60
Time (s)
(c)
(d)
Figure 6.1: Five vehicle simulation: (a) shows the vehicle trajectories. Target paths are
shown as dotted lines. Vehicle images are plotted at 30 seconds, while their initial positions
are denoted by small circles. (b) shows a snapshot of the collision cones from the
perspective of the red vehicle at 30 seconds. The red arrow denotes the red vehicle’s velocity.
This illustration effectively overlaps several diagrams like Fig. 4.7. Subplots (c) and
(d) show the corresponding speeds and control magnitudes of the vehicles, respectively.
‖˜r‖ − dsep (m)
10
5
0
0 10 20 30 40 50 60
Time (s)
Figure 6.2: Excess separation of vehicles in Fig. 6.1.

56
sion. In this example k = 5 s -1 , each vehicle’s maximum speed is 0.5 m/s, and its maximum
control authority is 0.2 m/s 2 . Figure 6.2 shows the excess separation distances for each pair
of vehicles, demonstrating that no collisions occur.
The initial conditions in this example are contrived such that the vehicles enter the
deconfliction group one at a time, counter-clockwise starting from the red vehicle. Each
vehicle starts its variable-speed deconfliction maneuver just before the bound (4.14), to
ensure all the safety guarantees are met. The deconfliction maneuvers can be seen as short
plateaus in the control magnitude plot (Fig. 6.1d).
The snapshot of the simulation in Figs. 6.1a and b shows just how small a conflict-free
region can be. While Theorem 4 guarantees that this region will never disappear, one can
see how if any of the green, yellow or purple vehicles became antagonistic, they could easily
cause a collision despite the best efforts of the other vehicles to avoid. This type of situation
is why the analysis of antagonistic vehicles was kept to only two-vehicle scenarios.
6.2 Preferential Routing
A feature of the DRCA algorithm is that vehicles can be given different priority by adjusting
the gain, k, of one vehicle relative to the other vehicles in the group. Vehicles with lower
gain values will effectively get out of the way of higher gain vehicles, allowing those with
priority to follow their desired controllers more closely.
In order to isolate the effects of preferential routing, the pair of simulations in Fig.
6.3 use identical vehicles and initial conditions. This case is geared more toward aircraft,
as the three vehicles are restricted to a constant speed of 1 m/s. The vehicles start in an
exact conflict (i.e. without control, all would reach the same point at the same time), and
immediately begin their constant-speed deconfliction maneuver. The red vehicle is the
first vehicle, and so maintains its heading throughout the maneuver. After one second, a
conflict-free state is attained, after which the vehicles attempt to get back to their original
straight paths.
Even though this simulation is symmetric and homogeneous, the red vehicle stays closer

57
un (m/s 2 )
0.2
0
(a)
-0.2
0 5 10 15 20 25
(c)
Time (s)
un (m/s 2 )
0.2
0
(b)
-0.2
0 5 10 15 20 25
(d)
Time (s)
Figure 6.3: Preferential routing comparison for three vehicles. Desired paths are depicted
by dotted lines. The symmetric case is shown in (a) where each vehicle has k n = 5 s -1 ,
while (b) shows the difference when the blue vehicle is given routing preference (k n = 10
s -1 ). (c) and (d) show the corresponding control inputs (vehicles are unit speed).
to its path than the others because it had order preference during the deconfliction maneuver.
In Fig. 6.3b, the blue vehicle was given priority by doubling its gain, k n , relative to the
others. As can be seen, the blue vehicle still performs the same deconfliction maneuver,
but once a conflict-free state is achieved, it begins to turn back toward its path. The other
vehicles are forced to make way for it, while never allowing a conflict to form.

58
5
0
-5
5
0
-5
0
5
-5
Figure 6.4: Four vehicles performing collision avoidance in 3D (vehicle paths are denoted
by thick lines). The initial conditions (black circles) form a tetrahedron and the targets the
vehicles follow (thin lines) intersect exactly at the origin. The vertical position along the
paths are denoted by their fading colors, corresponding to the colorbar on the left.
6.3 Three-dimensional Avoidance
The simulation in Fig. 6.4 demonstrates the DRCA algorithm’s abilities in three-dimensions.
The vehicles immediately perform a deconfliction maneuver by projecting their positions
and velocities onto the horizontal plane, solving a variable-speed maneuver, then adding
back the original vertical component of their velocities. The initial conditions form a tetrahedron,
oriented such that its projection onto the horizontal plane leaves the points adequately
separated for the maneuver.
In this example k = 2 s -1 , the vehicle’s maximum speed is 1 m/s, and their maximum
control authority is 0.2 m/s 2 . Their targets move at 0.7 m/s and the minimum separation
distance is d sep = 2 m. Fig. 6.5 shows the excess separation distances for each pair of
vehicles, reinforcing that the vehicles do not collide.

59
10
8
‖˜r‖ − dsep (m)
6
4
2
0
0 5 10 15 20 25
Time (s)
Figure 6.5: Excess separation of vehicles in Fig. 6.4.
6.4 Heuristic Performance
Unlike the previous sections, these simulations violate some of the guarantees of safety,
forcing the algorithm to use its heuristic backup to try and avoid collisions. The first simulation
(Fig. 6.6) has ten vehicles arranged in a circle of radius 14 m. Their maximum speed
is 1 m/s, their target speed is 0.6 m/s, and their maximum control authority is 0.5 m/s 2 ,
giving these variable-speed vehicles δ = 6 m (4.12). The minimum separation is d sep = 2
m, so the vehicles are initially spaced such that the maximum α e is nearly π/2. Therefore,
(4.14) is not satisfied, so Theorem 3 cannot guarantee that a deconfliction maneuver
solution can be found.
Nonetheless, both the constant-speed maneuver (a) and the variable-speed maneuver
(b), do find solutions for each vehicle that quickly lead to a conflict-free state without any
collisions (see (c) and (d)). However, using cost functions, one can get an idea of the relative
efficiency of the two maneuvers. The first cost function considered is an H 2 -type (c 1 =
∑ n
∫
i=1 ‖ri − r i,target ‖ dt), while the second is an H ∞ -type (c 2 = max i,t ‖r i − r i,target ‖).

60
‖˜r‖ − dsep (m)
10
5
(a)
0
0 5 10 15 20 25 30 35 40
(c)
Time (s)
‖˜r‖ − dsep (m)
10
5
(b)
0
0 5 10 15 20 25 30 35 40
(d)
Time (s)
Figure 6.6: Ten vehicles performing collision avoidance in the plane. The constant-speed
maneuver is shown in (a), with corresponding excess separation in (c). The variable-speed
maneuver and corresponding excess separation are shown in (b) and (d), respectively. The
randomized initial conditions are identical for both simulations and the vehicles are shown
at 25 seconds.

61
‖˜r‖ − dsep (m)
4
2
0
(a)
0 5 10 15 20 25 30
(c)
Time (s)
‖˜r‖ − dsep (m)
(b)
4
2
0
0 5 10 15 20 25 30
Time (s)
(d)
Figure 6.7: Merging two groups of vehicles. The constant-speed maneuver (a) creates a
collision between the green and yellow vehicles (shown at 11 seconds). The excess separation
going negative in (c) shows the collision to last three seconds. The variable-speed
maneuver (b) creates no such collision, as shown in the excess separation plot (d). Vehicles
shown at 19 seconds.
Indeed, using c 1 , the constant-speed maneuver has a cost of 936 while the variable-speed
maneuver attains 711. Likewise, for c 2 , constant-speed yields 9.01 to the variable-speed
7.10. So indeed, the extra computational expense of the variable-speed maneuver does buy
better performance by optimizing over a larger set.
The next simulation (Fig. 6.7) shows an even more difficult situation. The vehicle
properties remain the same as in Fig. 6.6, but now the vehicles are started much closer
together and in two opposite groups. The first problem is that the nearest vehicles have
‖˜r‖ < d sep + δ, so α e is set to π/2, rather than forcing it to be undefined. The second
problem is that the vehicle’s communication range is only 10% more than d sep + δ (or 8.8
m). These vehicles are set up to rebroadcast information about their neighbors, such that
any connected group will have full information (pseudo all-to-all), which means that the

62
two groups know about their own members, but not about each other until they get close
enough to communicate. This leads to two deconfliction maneuvers: one at the beginning
to avoid the other vehicles in the group, and later a second to avoid everyone.
The constant-speed maneuver is used in Fig. 6.7a, and while the first maneuver works
fine, by the time the groups are close enough to perform the second maneuver the vehicles
are so close that no solution can be found for the two green vehicles, so the heuristic commands
them to turn around. Unfortunately, with no safety guarantees, this maneuver causes
two vehicles to collide before a conflict-free state is reached (see Fig. 6.7c). However, by
using the variable-speed maneuver with its larger optimization set, solutions are found for
each vehicle (Fig. 6.7b) which keep the system collision-free, as shown in Fig. 6.7d.
6.5 Computational Scaling
Though bounds on the computational scaling of the various pieces of this algorithm have
already been shown, the actual time taken for some realistic scenarios is also of interest.
This actual time taken is especially important because the worst-case scaling is conservative.
The simulations run were based on those from [22], to facilitate comparison. Each
scenario consisted of n vehicles in a circle with randomized velocities headed generally
toward the center (similar to Fig. 6.6). Table 6.1 gives results on the time taken for each
part of the algorithm for various numbers of vehicles.
Interesting to note is the wide variety of computation time for the maneuvers; often
there is as much as a factor of six between the average time and the maximum time for a
particular vehicle. Part of the reason is that the first vehicle needs to avoid no one, while
the last vehicle needs to avoid everyone. Also note the difference in scaling between the
constant-speed and variable-speed maneuvers (recalling that the worst case is O(n 2 ) and
O(n 3 ), respectively): for eight vehicles the variable-speed maneuver costs only about three
times the computations, but for 200 vehicles it costs 16 times more than constant-speed.
These results also compare favorably to the computation times listed in [24] (which is a
centralized optimization scheme), though the setups are not exactly the same.

63
Table 6.1: Comparison of computation times between collision avoidance algorithms.
Times reflect computations on a per vehicle basis.
n CS Maneuver VS Maneuver Maint- [22]
Mean Max Mean Max enance
8 2.8 ms 18 ms 20 ms 56 ms 0.96 ms 0.41 ms
32 4.4 ms 25 ms 82 ms 507 ms 3.5 ms 1.6 ms
64 8.3 ms 32 ms 489 ms 4.0 s 7.2 ms 3.6 ms
100 17 ms 67 ms 2.5 s 19 s 11 ms 4.6 ms
200 44 ms 246 ms 44 s 244 s 23 ms 9.2 ms
Computation times for deconfliction maintenance are comparable to those listed in [22],
since both of these algorithms must be computed at each time step (as opposed to the
deconfliction maneuvers, which are only computed once). While the DRCA algorithm is
the slower of the two, these data show that they both scale linearly with n. Also, these times
are off by some factor, since the DRCA algorithm was run on one core of an Intel Core2
Quad 2.4 GHz computer while [22] used a Core Duo 2.16 GHz computer. Additionally, it
is unlikely that either implementation was fully optimized for speed.
6.6 Formations
One limitation of the DRCA algorithm is that vehicles moving with identical velocities are
right on the edge of conflict (m = 0). Because the deconfliction maintenance controller is
continuous, it will tend to push the vehicles to an equilibrium where m > 0, which means
the vehicles move slightly away from one another. This behavior precludes the use of the
DRCA algorithm for collision avoidance within a formation of vehicles because the desired
state is for all of the vehicles to move with the same velocity.
One way around this problem is to split up the duties of the desired controller and the
DRCA algorithm. In the case of formation flight, the relative motion of the vehicles can

64
(a)
(b)
(c)
Figure 6.8: Fifty-vehicle swarm avoiding a static obstacle, shown at 20 second intervals.

65
be stabilized with any of a variety of controllers from the literature [33, 34, 35]. By using
a desired controller that guarantees intervehicle spacing, the DRCA algorithm can simply
be turned off between the vehicles in the formation. However, for the more difficult task of
avoiding objects moving at high relative speed, the DRCA algorithm can take over.
A simulation of this situation for a fifty-vehicle swarm avoiding a static obstacle is
shown in Fig. 6.8. In this case the communication topology is broken into two pieces: the
DRCA algorithm uses a star graph, from the static obstacle to each other vehicle, while the
desired controller uses everything else (the complete graph minus the star graph). Therefore
the desired controller has no knowledge of the static obstacle; it simply keeps the vehicles
spaced out using potential functions. In this way the vehicles avoid the static obstacle
in a guaranteed fashion using the DRCA algorithm while the desired controller maintains
intervehicle spacing.
The maximum vehicle speed is 1 m/s while the swarm moves at 0.5 m/s and the vehicle’s
maximum control authority is 0.2 m/s 2 . The vehicle diameters are 1 m and the obstacle’s
diameter is 6 m. The vehicles begin their deconfliction maneuver when they have 6.5 m of
excess separation from the obstacle (13 m center-to-center). The swarm is unzipped around
the obstacle, but the vehicles avoid it by no more than they have to. One can also choose
the distance at which vehicles deconflict such that α e is no more than some threshold,
thus keeping the ‖∆v‖ of the maneuver small enough to ensure the desired controller can
adequately attain intervehicle spacing.
6.7 Comparison to an SGT Algorithm
To demonstrate the performance strength of the DRCA algorithm, two systems were simulated
to compare with the results of a Satisficing Game Theory (SGT) algorithm presented
in [28]. The first scenario is based originally on a model used in [24]). The aircraft have
a constant speed of 500 mph, start evenly spaced on a circle of radius 50 miles, and each
is attempting to reach a waypoint on the exact opposite side of the circle (the starting point
of the opposite aircraft). This scenario is referred to as a choke point because without

66
Figure 6.9: Collision avoidance of 32 aircraft using the DRCA algorithm. Vehicles are
shown 30 seconds before attaining their waypoints. Note that no near misses occur even
though the vehicles start without sufficient separation to guarantee deconfliction (4.14).
deconfliction, all the vehicles would meet at the center (Figure 6.9).
A near miss is defined as two aircraft coming within five miles of each other, so d sep was
set to this value. The efficiency of the maneuver is defined as the average of the percentage
of time delay in arrival at the waypoints:
Efficiency = 1 n
n∑
i=1
T r
T i
, (6.1)
where T r = 12 minutes is the reference time (flying straight), and T i is the actual time taken
for the ith vehicle. In [28], the maximum turn allowed was 5 degrees per second, which
corresponds to u max = 64 ft/s 2 . Additionally, the gain chosen was k n = 0.4 s -1 .
In order to fairly compare these two algorithms, the DRCA algorithm has been set up
for this simulation such that each vehicle has information only from those vehicles within a
50 mile radius of itself, just as in [28]. While this breaks some of the guarantees of safety,

67
the heuristic performance will be shown to be more than adequate. Additionally, since the
vehicles start off such that ‖r‖ < d sep + δ, this simulation used half of the standard δ value,
because the safety guarantees are already gone, so it makes sense to choose δ such that
‖r‖ > d sep + δ.
Figure 6.9 shows a simulation of the DRCA algorithm deconflicting 32 vehicles, the
densest choke pattern demonstrated in [28]. The DRCA algorithm attained an efficiency
of 87.6%, compared to 85.7% in [28]. The real achievement was that there were zero near
misses with the DRCA algorithm, as opposed to 19 in [28].
The second scenario is that of two perpendicular flows of the same aircraft as before.
Like the worst case shown in [28], the aircraft streams are constant, with a starting separation
of seven miles between the aircraft. Because the DRCA algorithm does not allow
vehicles to maintain the same velocity as each other, the algorithm was tweaked such that
vehicles only avoid each other within a circle of radius 50 miles from the initial crossing
point. That is, for two vehicles to detect each other, they must both be in the circle and
be within 50 miles of each other. Unlike the standard implementation of the DRCA algorithm,
these vehicles must frequently perform deconfliction maneuvers as new vehicles are
spotted. The δ used is 1 mile, but everything else is the same as the previous simulation.
It was unclear exactly what the goal of each vehicle was in the simulation in [28], so
for the simulation in Fig. 6.10 the desired controller points each vehicle toward a distant
waypoint along their original trajectory, which is effectively equivalent to attaining their
original heading. Likewise, the definition of efficiency was unclear in this example, and so
is omitted. Fig. 6.11 shows that there were no near misses while [28] recorded 29. The
algorithms were qualitatively similar, forming alternating waves of vehicles that flowed
past each other.

68
Figure 6.10: Collision avoidance of two perpendicular flows. Vehicles only perform the
DRCA algorithm when they are inside of the black dashed circle (radius 50 miles).
‖˜r‖ − dsep (m)
20
10
0
0 200 400 600 800 1000
Time (s)
Figure 6.11: Excess separation for vehicles in 6.10. Note that no near misses occurred.

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Chapter 7
CONCLUSION
The primary contribution of this dissertation is a widely applicable, provably safe collision
avoidance algorithm. The DRCA algorithm is designed for the exceedingly simple
3D double-integrator, which can model a vast array of vehicles through the application of
appropriate control saturation functions. Instead of using more complex dynamics to model
nonholonomic constraints, state-dependent control saturation can be used to accomplish the
same goal. As shown in the robustness analysis, higher order dynamics can be accounted
for through the proper choice of gains.
The most important aspects of the DRCA algorithm are the complete guarantees of
safety at every phase. Almost no other collision avoidance algorithms in the literature
give complete safety guarantees for n vehicles in the presence of acceleration or force
limitations. Even the other optimization schemes frequently give no mention of when a
feasible solution exists.
The information required by this algorithm in order to maintain safety guarantees is
higher than for some others, and may be higher than is practical for certain systems. However,
it has been demonstrated that even when these requirements are relaxed, for instance
by making the system truly decentralized with only local information, the heuristic performance
is still better than other algorithms. Part of the reason is that though the deconfliction
maneuver may not be proven safe, if the system becomes conflict-free, the deconfliction
maintenance controller will guarantee it stays that way.
As the name Distributed Reactive Collision Avoidance implies, this algorithm ensures
real-time operability by distributing the computations among the agents, bounding the computational
scaling, and requiring no central server. The computation time has been shown

70
to not only scale well, but to compare favorably with the computation time of other fast,
distributed algorithms.
7.1 Future Work
There are many avenues for potential future research building on what has been presented
here. Some are direct extensions to the presented work while others include humanmachine
interaction and even human decision modeling.
The section on robust avoidance of a single adversary came about from looking at the
DRCA algorithm from a game theory point of view. An interesting extension would be to
further this analysis to more than two vehicles, and perhaps more than one adversary and
see what can be proven about avoidance. The inverse problem could also be interesting:
what adversarial behaviors can guarantee a collision Perhaps this would give some insight
into how the DRCA algorithm can be improved to better handle adversaries.
One major restriction on this implementation is that the algorithm does not allow vehicles
to move in formation, because zero relative velocity is on the tip of the collision cone.
It might be possible to cut the tip off of the collision cone, perhaps where the amount cut off
is proportional to the distance between the vehicles, such that vehicles moving slowly toward
each other at enough distance would no longer be considered in conflict. This method
would allow formation flight within the DRCA architecture, but the guarantees would have
to be reproven.
All of the bounds given in this dissertation are conservative in one way or another.
Fundamentally, this is why the algorithm still works so well heuristically when the bounds
are violated. New methods of proof that allow less conservative bounds will bring the most
significant improvement to the guaranteed performance and will correspondingly help any
implementation.
The three-dimensional case needs a better deconfliction maneuver the most, and perhaps
this can be accomplished through some other optimization. Granted, there are fewer
systems that require true three-dimensional deconfliction, but for those that do, the current

71
system of projecting everything to a plane is insufficient. Some type of relaxation should be
developed to make optimizing deconfliction in three-space tractable. Keeping in mind that
feasibility is far more important than optimality will probably make solving the problem
easier.
The whole point of developing this algorithm is to implement it on real systems to
improve safety. Perhaps one day it can join the ranks of TCAS and ADS-B as air traffic
becomes more automated and the skies become denser-trafficked with UAVs and such. It
could even serve in a centralized role as a load relief for air traffic controllers by creating
default flight paths and giving warnings associated with conflicts. Along this line, an
interesting research topic would be how to make this algorithm effectively interact with a
human, to reduce the operator’s work load while utilizing his skill at nonlinear optimization
to improve performance.
The original inspiration for this algorithm came from observing crowds of people walking
across a large square and avoiding collisions with each other. Therefore it may be that
people and other animals use a thought process akin to this algorithm to control their own
motion. Using the DRCA algorithm as a benchmark to study human behavior may give
insight into how our brains process control and planning.

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VITA
Emmett Lalish was born and raised on the tiny island of Marrowstone in the Puget
Sound. He dropped out of high school and moved away from home at the age of 16 to
attend Olympic College in Bremerton, where he earned an Associate of Science in Engineering.
After transferring to the University of Washington, he earned his B.S. in the
Aeronautics & Astronautics department in 2005. He stayed for graduate school, earning
his Master’s in 2007 and Ph.D. in 2009, all the while employed at Kristi Morgansen’s Nonlinear
Dynamics & Control Lab. From 2006 to 2009 he also lead the UW’s Design, Build,
Fly team, culminating in a 10 th place finish in Tucson, AZ against 54 other teams. At the
age of 24, he is among the youngest to receive a doctorate from the UWAA department.