Optimizing dynamics and behavior of multi-robot-systems with hybrid automata and MILP-techniques

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13 Νοε 2013 (πριν από 4 χρόνια και 1 μήνα)

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Optimizing dynamics and behavior of
multi-robot-systems with hybrid automata and
MILP-techniques
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 mixed-integer 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 multi-robot-systems
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 mixed-integer 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 multi-robot-systems
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.comets-uavs.org
￿
Unmanned Air Vehicles (UAVs)
￿
UAVs
cooperate
in monitoring
an area
￿
Fire monitoring,traffic
surveillance,...
￿
COMETS project
(www.comets-uavs.org)
3/33
Christian Reinl
Optimizing multi-robot-systems
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.comets-uavs.org
￿
Unmanned Air Vehicles (UAVs)
￿
UAVs
cooperate
in monitoring
an area
￿
Fire monitoring,traffic
surveillance,...
￿
COMETS project
(www.comets-uavs.org)
3/33
Christian Reinl
Optimizing multi-robot-systems
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 go-to-ball,dribble,kick,
...
￿
reaction
on
perturbations
(e.g.caused by the opponent’s actions)
4/33
Christian Reinl
Optimizing multi-robot-systems
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/
self-governed)
￿
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 multi-robot-systems
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 multi-robot-systems
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 three-motor
omni-directorial drive
￿
move along any direction
irrespective
of its orientation
7/33
Christian Reinl
Optimizing multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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

2
g
2

3
g
3
≥ 0
λ
1

2

3
= 1
λ
1

2

3
≥ 0


￿
time
discretization
and
collocation
approach lead to a
finite dimensional NLP
11/33
Christian Reinl
Optimizing multi-robot-systems
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
’big-M’ technique
leads
to a
MILP
min
ε
b

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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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
￿
Branch-and-bound
-
techniques
circular tours:
RoboCup scenario:
14/33
Christian Reinl
Optimizing multi-robot-systems
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 mixed-integer 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 multi-robot-systems
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
ball-in-goal
s4
kick
s2
s3
playerI-dribbling
ball-free
playerII-dribbling
kick
kick
catch
catch
￿
initial-conditions
￿
jump-conditions
￿
event-conditions (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 multi-vehicle system from a
global view!
16/33
Christian Reinl
Optimizing multi-robot-systems
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Hybrid and hierarchical automata
Systematics introduced by Bemporad et al
Structure of discrete-time 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 multi-robot-systems
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 mixed-integer 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 multi-robot-systems
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 mixed-integer 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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Mixed-integer linear programs
￿
solution of linear problems can be
computed directly
without iterating the objective function and derivations
￿
many (tricky) methods to combine this with
integer-optimization
￿
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Mixed-integer linear programs
￿
solution of linear problems can be
computed directly
without iterating the objective function and derivations
￿
many (tricky) methods to combine this with
integer-optimization
￿
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Mixed-integer linear programs
￿
solution of linear problems can be
computed directly
without iterating the objective function and derivations
￿
many (tricky) methods to combine this with
integer-optimization
￿
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Mixed-integer linear programs
￿
solution of linear problems can be
computed directly
without iterating the objective function and derivations
￿
many (tricky) methods to combine this with
integer-optimization
￿
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Mixed-integer linear programs
￿
solution of linear problems can be
computed directly
without iterating the objective function and derivations
￿
many (tricky) methods to combine this with
integer-optimization
￿
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Handling non-convex environments
￿
considered problems are (mostly)
non-convex
￿
use
linear approximations
to subdivide them in convex
subproblems
￿
logical constraints
describe the (approximated) region
22/33
Christian Reinl
Optimizing multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Handling non-convex environments
￿
considered problems are (mostly)
non-convex
￿
use
linear approximations
to subdivide them in convex
subproblems
￿
logical constraints
describe the (approximated) region
22/33
Christian Reinl
Optimizing multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Handling non-convex environments
￿
considered problems are (mostly)
non-convex
￿
use
linear approximations
to subdivide them in convex
subproblems
￿
logical constraints
describe the (approximated) region
22/33
Christian Reinl
Optimizing multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Modeling logical constraints
Logical operators like ∨,∧,¬,⇒and if-then-expressions
cam be translated into linear constraints.
￿
popular method:
’big-M’
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

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

2
≥1,δ
1

3
≥1 ⇔2δ
1

2

3
≥2
not
equivalent for δ
i
∈R!
24/33
Christian Reinl
Optimizing multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Modeling logical constraints
Logical operators like ∨,∧,¬,⇒and if-then-expressions
cam be translated into linear constraints.
￿
popular method:
’big-M’
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

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

2
≥1,δ
1

3
≥1 ⇔2δ
1

2

3
≥2
not
equivalent for δ
i
∈R!
24/33
Christian Reinl
Optimizing multi-robot-systems
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 multi-robot-systems
Introduction
Modeling with automata
MILP formulation
Results,future work
Definition
Techniques
Modeling the robotic soccer scenario
Modeling the robotic soccer benchmark problem
After a time-horizon [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 multi-robot-systems
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 multi-robot-systems
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 mixed-integer 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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
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 multi-robot-systems
Thank you for your attention!