Optimizing dynamics and behavior of
multirobotsystems with hybrid automata and
MILPtechniques
Christian Reinl
Simulation and Systems Optimization Group
Technische Universität Darmstadt
Hohenwart,September/04/2006
Introduction
Modeling with automata
MILP formulation
Results,future work
Overview
Introduction
Investigated problems
Scope of activities
Modeling with automata
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Formulating a mixedinteger linear program (MILP)
Deﬁnition and characteristics of MILP
Techniques to formulate the linear program
Modeling the robotic soccer scenario
Results,future work
Results for the UAV benchmark problem
Future work
1/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Introduction
Investigated problems
Scope of activities
Modeling with automata
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Formulating a mixedinteger linear program (MILP)
Deﬁnition and characteristics of MILP
Techniques to formulate the linear program
Modeling the robotic soccer scenario
Results,future work
Results for the UAV benchmark problem
Future work
2/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Example problems
RoboCup scenario
Robots
together
try to
score a goal
Problem:optimal
trajectories and task
assignment
source:http://www.cometsuavs.org
Unmanned Air Vehicles (UAVs)
UAVs
cooperate
in monitoring
an area
Fire monitoring,trafﬁc
surveillance,...
COMETS project
(www.cometsuavs.org)
3/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Example problems
RoboCup scenario
Robots
together
try to
score a goal
Problem:optimal
trajectories and task
assignment
source:http://www.cometsuavs.org
Unmanned Air Vehicles (UAVs)
UAVs
cooperate
in monitoring
an area
Fire monitoring,trafﬁc
surveillance,...
COMETS project
(www.cometsuavs.org)
3/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Robotic games (Challenges)
attacker

defender
models
speciﬁc
dynamical
abilities need adjusted tactic planning
discrete
roles like gotoball,dribble,kick,
...
reaction
on
perturbations
(e.g.caused by the opponent’s actions)
4/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Multiple UAVs (Challenges)
conﬁguration
of the UAVs
(heterogeneous/homogeneous,central controlled/
selfgoverned)
task assignment and optimal trajectory planning
collision
avoidance
(particularly
obstacle
avoidance)
reaction
on perturbations
(e.g.after the breakdown of an UAV)
balance between
optimality
and
time complexity
5/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Benchmarkproblems
circular tours for unmanned aerial
vehicles
two vehicles
collision avoidance
intention:minimize time or
energy
cooperative task allocation and
trajectory planning in the RoboCup
I
II
passing the ball
playerI
playerII
passing the ball
defender
two players
one (simple) defender
intention:improve the
attackers chances
6/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Approaches fromEarl and D’Andrea for the RoboFlag
scenario
considered problem:
attacker
trying to enter the
defense zone,
defenders
try to
intercept them
each vehicle has a threemotor
omnidirectorial drive
move along any direction
irrespective
of its orientation
7/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
RoboFlag  vehicle dynamics
dynamic:
¨
x
¨
y
¨
θ
+
˙
x
˙
y
2mL
2
I
˙
θ
=u(θ(t),t)
with
u(θ(t),t) =P(θ(t))U(t)
equations are
coupled
and
nonlinear
!
with some restrictions the
problem can be formulated
with linear dynamics
discrete time governing
equations
can be written as
x
u
[k +1] =A[k]x
u
[k] +B[k]u[k]
cost function:
J =
N
u
−1
∑
k=0
(z
x
[k] +z
y
[k])
(−z
♦
[k] ≤u
♦
[k] ≤z
♦
[k])
together with the polygon
constraints:complete
MILP
formulation for the dynamics
8/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
RoboFlag  vehicle dynamics
dynamic:
¨
x
¨
y
¨
θ
+
˙
x
˙
y
2mL
2
I
˙
θ
=u(θ(t),t)
with
u(θ(t),t) =P(θ(t))U(t)
equations are
coupled
and
nonlinear
!
with some restrictions the
problem can be formulated
with linear dynamics
discrete time governing
equations
can be written as
x
u
[k +1] =A[k]x
u
[k] +B[k]u[k]
cost function:
J =
N
u
−1
∑
k=0
(z
x
[k] +z
y
[k])
(−z
♦
[k] ≤u
♦
[k] ≤z
♦
[k])
together with the polygon
constraints:complete
MILP
formulation for the dynamics
8/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
RoboFlag  results
defender
is governed by the
introduced discrete time
dynamical system
attacker
has two states
9/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Comparison between MILP and NLP techniques
(Francesco Borelli et al)
min
n
∑
i =1
J
i
(v
(t
0
,t
f
)
x,i
,v
(t
0
,t
f
)
y,i
,v
(t
0
,t
f
)
z,i
)
subject to
˙
x(t)
=
v
x
(t),
x
i
(t
0
) = x
i 0
,x
i
(t
f
) = x
if
,
˙
y(t)
=
v
y
(t),
y
i
(t
0
) = y
i 0
,y
i
(t
f
) = y
if
,
˙
z(t)
=
v
z
(t),
z
i
(t
0
) = z
i 0
,z
i
(t
f
) = z
if
g(x
i
,y
i
,z
i
,x
j
,y
j
,z
j
) ≤0 ∀1 ≤i <j ≤n
g
obst
(x
i
,y
i
,z
i
) ≤0
v
x,i
≤ v
x,i
(t) ≤
v
x,i
i ∈ {1,2,...n}
y
x,i
≤ v
y,i
(t) ≤
y
x,i
z
x,i
≤ v
z,i
(t) ≤
z
x,i
10/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
i) Solution via nonlinear programming
objective function:
Z
t
f
t
0
[v
x
(t)
1
+v
y
(t)
1
+v
z
(t)
1
] dt
collision (obstacle) avoidance constraint g(x
i
,y
i
,z
i
,x
j
,y
j
,z
j
):
x
i
(t)−x
j
(t) ≥R
∨
y
i
(t)−y
j
(t) ≥R
∨
z
i
(t)−z
j
(t) ≥H (i =j )
transformation of the disjunctive equation according to:
g
1
≥0 ∨ g
2
≥0 ∨ g
3
≥0
⇔
λ
1
g
1
+λ
2
g
2
+λ
3
g
3
≥ 0
λ
1
+λ
2
+λ
3
= 1
λ
1
,λ
2
,λ
3
≥ 0
time
discretization
and
collocation
approach lead to a
ﬁnite dimensional NLP
11/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
ii) Solution via mixed integer linear programming
discretization of cost function and dynamics with a
sampling time T
s
:
x
i
(k +1) = x
i
(k) +T
s
v
x,i
y
i
(k +1) = y
i
(k) +T
s
v
y,i
z
i
(k +1) = z
i
(k) +T
s
v
z,i
reformulation of the disjunctions describing the protection
zones with binary variables and
’bigM’ technique
leads
to a
MILP
min
ε
b
,ε
c
f
c
ε
c
+f
b
ε
b
subject to G
c
ε
c
+G
b
ε
b
≤b
ε
b
∈ {0,1}
n
b
ε
c
∈R
n
c
solved with
’Branch &
Bound’
provided by
CPLEX
12/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
iii) Comparison between NLP and MILP approach
MILP is
faster
for all
solved instances,particularly for big
problem size.
(average solution times:4.32 s ↔46.53 s)
J(z
∗
MILP
) −J(z
∗
NLP
)
J(z
∗
NLP
)
≈10
−1
no other correlation between problem instance and related
time complexity
(for ﬁxed number of vehicles and obstacles)
13/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
iii) Comparison between NLP and MILP approach
MILP is
faster
for all
solved instances,particularly for big
problem size.
(average solution times:4.32 s ↔46.53 s)
J(z
∗
MILP
) −J(z
∗
NLP
)
J(z
∗
NLP
)
≈10
−1
no other correlation between problem instance and related
time complexity
(for ﬁxed number of vehicles and obstacles)
13/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
iii) Comparison between NLP and MILP approach
MILP is
faster
for all
solved instances,particularly for big
problem size.
(average solution times:4.32 s ↔46.53 s)
J(z
∗
MILP
) −J(z
∗
NLP
)
J(z
∗
NLP
)
≈10
−1
no other correlation between problem instance and related
time complexity
(for ﬁxed number of vehicles and obstacles)
13/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Investigated problems
Scope of activities
Hybrid optimal control (Markus Glocker)
modelled with
hybrid
automata
transformation into a ﬁnite
dimensional problem with
direct collocation
solved with SNOPT
Branchandbound

techniques
circular tours:
RoboCup scenario:
14/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Introduction
Investigated problems
Scope of activities
Modeling with automata
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Formulating a mixedinteger linear program (MILP)
Deﬁnition and characteristics of MILP
Techniques to formulate the linear program
Modeling the robotic soccer scenario
Results,future work
Results for the UAV benchmark problem
Future work
15/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Describing the systemwith a hybrid automaton
s1
ballingoal
s4
kick
s2
s3
playerIdribbling
ballfree
playerIIdribbling
kick
kick
catch
catch
initialconditions
jumpconditions
eventconditions (e.g.”kick” )
ﬁxed number of possible
transitions at unknown points
in time t
i
dynamics in the states is
described by differential
equations of motion
˙
x =f (x,u,t)
Complete physical description of
the multivehicle system from a
global view!
16/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Clocked hierarchical automaton
transitions only occur on a
time discretization
hierarchy allows description of
behavior on different levels
heuristic rules are implemented
to control the behavior,
to guarantee coordinated
actions and
to control the motions
The sumof these automata des
cribes the system from an inter
nal view!
17/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Structure of discretetime hybrid systems
HYSDEL  Hybrid System
Description Language
(Bemporad,Morari et al.
ETHZ)
equivalent classes for
modeling:
L
inear
C
omplementary,
P
iece
W
ise
A
fﬁne and
M
ixed
L
ogical
D
ynamical
systems
18/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Introduction
Investigated problems
Scope of activities
Modeling with automata
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Formulating a mixedinteger linear program (MILP)
Deﬁnition and characteristics of MILP
Techniques to formulate the linear program
Modeling the robotic soccer scenario
Results,future work
Results for the UAV benchmark problem
Future work
19/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Formal deﬁnition of MILP
mixed integer linear program (MILP)
min
x,z
f
T
1
x+f
T
2
z
subject to G
1
x+G
2
z ≤b
x ∈B
1
⊆R
n
c
z ∈B
2
⊆
N
n
d
simpliest form of a mixedinteger optimization problem with
constraints
several approaches to
transforma complex problem
in
this formalism need to be investigated
these techniques are decisive for the
quality of the entire
procedure
20/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Mixedinteger linear programs
solution of linear problems can be
computed directly
without iterating the objective function and derivations
many (tricky) methods to combine this with
integeroptimization
even for simple problems a huge number of variables is
needed
structure
is more important than the number of variables
best available solver
CPLEX
21/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Mixedinteger linear programs
solution of linear problems can be
computed directly
without iterating the objective function and derivations
many (tricky) methods to combine this with
integeroptimization
even for simple problems a huge number of variables is
needed
structure
is more important than the number of variables
best available solver
CPLEX
21/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Mixedinteger linear programs
solution of linear problems can be
computed directly
without iterating the objective function and derivations
many (tricky) methods to combine this with
integeroptimization
even for simple problems a huge number of variables is
needed
structure
is more important than the number of variables
best available solver
CPLEX
21/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Mixedinteger linear programs
solution of linear problems can be
computed directly
without iterating the objective function and derivations
many (tricky) methods to combine this with
integeroptimization
even for simple problems a huge number of variables is
needed
structure
is more important than the number of variables
best available solver
CPLEX
21/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Mixedinteger linear programs
solution of linear problems can be
computed directly
without iterating the objective function and derivations
many (tricky) methods to combine this with
integeroptimization
even for simple problems a huge number of variables is
needed
structure
is more important than the number of variables
best available solver
CPLEX
21/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Handling nonconvex environments
considered problems are (mostly)
nonconvex
use
linear approximations
to subdivide them in convex
subproblems
logical constraints
describe the (approximated) region
22/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Handling nonconvex environments
considered problems are (mostly)
nonconvex
use
linear approximations
to subdivide them in convex
subproblems
logical constraints
describe the (approximated) region
22/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Handling nonconvex environments
considered problems are (mostly)
nonconvex
use
linear approximations
to subdivide them in convex
subproblems
logical constraints
describe the (approximated) region
22/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Collision Avoidance
static obstacles:
(x −x
Obst
)
2
+(y −y
Obst
)
2
>r
2
Obst
⇓
6
_
i =1
k
i,1
(x −x
Obst
) +k
i,2
(y −y
Obst
) >r
k
i,1
=sin
i
3
π k
i,2
=cos
i
3
π
avoiding collisions with moving objects:
4
_
i =1
k
i,1
(x
1
−x
2
) +k
i,2
(y
1
−y
2
) >d
k
i,1
=sin
i
2
π k
i,2
=cos
i
2
π
d
d
d
(x
2
,y
2
)
(x
1
,y
1
)
d
23/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Collision Avoidance
static obstacles:
(x −x
Obst
)
2
+(y −y
Obst
)
2
>r
2
Obst
⇓
6
_
i =1
k
i,1
(x −x
Obst
) +k
i,2
(y −y
Obst
) >r
k
i,1
=sin
i
3
π k
i,2
=cos
i
3
π
avoiding collisions with moving objects:
4
_
i =1
k
i,1
(x
1
−x
2
) +k
i,2
(y
1
−y
2
) >d
k
i,1
=sin
i
2
π k
i,2
=cos
i
2
π
d
d
d
(x
2
,y
2
)
(x
1
,y
1
)
d
23/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Modeling logical constraints
Logical operators like ∨,∧,¬,⇒and ifthenexpressions
cam be translated into linear constraints.
popular method:
’bigM’
example:
either (g
1
≥b
1
) or (g
2
≥b
2
)
⇔
g
1
≤ b
1
+Mδ
1
g
2
≤ b
2
+Mδ
2
δ
1
+δ
2
≤ 1
δ
i
∈ {0,1}
,M >0,M >max{g
1
−b
1
},M >max{g
2
−b
2
}
not unique
L
1
∨(L
2
∧L
3
) ⇔δ
1
+δ
2
≥1,δ
1
+δ
3
≥1 ⇔2δ
1
+δ
2
+δ
3
≥2
not
equivalent for δ
i
∈R!
24/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Modeling logical constraints
Logical operators like ∨,∧,¬,⇒and ifthenexpressions
cam be translated into linear constraints.
popular method:
’bigM’
example:
either (g
1
≥b
1
) or (g
2
≥b
2
)
⇔
g
1
≤ b
1
+Mδ
1
g
2
≤ b
2
+Mδ
2
δ
1
+δ
2
≤ 1
δ
i
∈ {0,1}
,M >0,M >max{g
1
−b
1
},M >max{g
2
−b
2
}
not unique
L
1
∨(L
2
∧L
3
) ⇔δ
1
+δ
2
≥1,δ
1
+δ
3
≥1 ⇔2δ
1
+δ
2
+δ
3
≥2
not
equivalent for δ
i
∈R!
24/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Modeling the vehicles’ dynamic
introducing a
sampling time
t
s
˙
x(t) =f (x(t),u(t)) x
k+1
=x
k
+t
s
(A
k
x
k
+B
k
u
k
)
decouplingand linearization must be done carefully
additonal constraints may be needed
physical characteristics
must be considered
25/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Modeling the robotic soccer benchmark problem
After a timehorizon [t
o
,t
f
]
the position of the player on
the ball (K
1
),
his distance to the defender
(K
2
) and
the position of the supporter
(K
3
)
should be optimized.
A simple objective function to
guarantee this:
J =c
1
K
1
+c
2
K
2
+c
3
K
3
K
1
=y
B,f
 +
1
3
(x
goal
−x
B,f
)
K
2
=x
B,f
−x
D
 +x
B,f
−y
D

K
3
=x
supp,f
−x
goal
 −x
supp,f
−x
D

−y
supp,f
−y
D
 −
1
3
x
supp,f
−x
B,f

−
1
3
y
supp,f
−y
B,f

26/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Deﬁnition
Techniques
Modeling the robotic soccer scenario
Dynamics
robots:
x
k+1
=x
k
+t
s
v
k
v
k+1
=v
k
+t
s
u
k
ball (free):
x
k+1
=x
k
+t
s
v
k
v
k+1
=v
k
−t
s
Dv
k
additional constraints,e.g.
size of the
ﬁeld
collision avoidance
conditions for
taking the ball
just one
player can be on the
ball
a ball must be dribbled before
it can be kicked
...
27/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
Introduction
Investigated problems
Scope of activities
Modeling with automata
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Formulating a mixedinteger linear program (MILP)
Deﬁnition and characteristics of MILP
Techniques to formulate the linear program
Modeling the robotic soccer scenario
Results,future work
Results for the UAV benchmark problem
Future work
28/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
The fastest tour
objective function:
min J =t
f
+
∑
u
i,x
 +u
i,y

number of time steps:25
MILP size:746 x 1692
computational time:47min 35s
29/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
A tour with minimized accelerations
objective function:
min J =
∑
u
i,x
 +u
i,y

number of time steps:25
MILP size:746 x 1692
computational time:30min 41s
30/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
A tour with an obstacle
objective function:
min J =
∑
u
i,x
 +u
i,y

number of time steps:17
MILP size:536 x 1188
computational time:2min 18s
31/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
Introduction
Modeling with automata
MILP formulation
Results,future work
Results
Future work
introducing a
mesh adaption
improving the internal
data management
testing more detailed
dynamics
comparison with
MIQP
modeling
completing and augmenting the
robotic soccer
benchmark problem
automatic translation
of
XABSL
code into a set of linear
constraints
modeling of
hierarchies
and testing their effects
implementations and tests with
HYSDEL
improving the direct collocation method with
convex
underestimators
an initial guesses
32/33
Christian Reinl
Optimizing multirobotsystems
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