Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Contents
1. Control
2. Perception
3. Planning
4. Knowledge
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
51
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Outline
Control
Perception
Planning
Knowledge
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
52
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Exam Prep. Session: Control Frameworks
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
53
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Control Frameworks, Example 1
Assume, that we have already designed a controller for positioning our robot
arm. But, after that, we bought a hand and equipped the robot with the new
hand. How would you account for the disturbances from the added hand in your
controller?
1. Which one would you use: feedback or feed-forward control? Give a short
explanation for your answer!
2. Assume the task of positioning the arm and hand at a static position: Are
these new disturbances mainly configuration-dependent? Choose yes or no,
and explain why!
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
54
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Control Frameworks, Example 1
1. Which one would you use: feedback or feed-forward control? Give a short
explanation for your answer!
2. Assume the task of positioning the arm and hand at a static position: Are
these new disturbances mainly configuration-dependent? Choose yes or no,
and explain why!
Answers:
1. The hand should be incorporated as a feed-forward term, because it is a
disturbance that can be modeled fairly accurately.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
55
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Control Frameworks, Example 1
1. Which one would you use: feedback or feed-forward control? Give a short
explanation for your answer!
2. Assume the task of positioning the arm and hand at a static position: Are
these new disturbances mainly configuration-dependent? Choose yes or no,
and explain why!
Answers:
1. The hand should be incorporated as a feed-forward term, because it is a
disturbance that can be modeled fairly accurately.
2. Yes, the added disturbances are mainly configuration-dependent: In a static
setup the main source of disturbance will be the additional effect of gravity
caused by the added weight of the hand. Since gravity compensation is
usually modelled as just configuration-dependent, the hand weight should
be compensated in the same fashion
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
56
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Control Frameworks, Example 2
Calculate the Jacobian for this given two-link planar robot!
The forward kinematics of this robot is given as follows:
x
L1 cos q1 + L2 cos(q1 + q2 )
=
y
L1 sin q1 + L2 sin(q1 + q2 )
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
57
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Control Frameworks, Example 2
Calculate the Jacobian for this given two-link planar robot!
The forward kinematics of this robot is given as follows:
x
L1 cos q1 + L2 cos(q1 + q2 )
=
y
L1 sin q1 + L2 sin(q1 + q2 )
Solution:
J(q) =
Control
Michael Beetz
Summer Term 2012
∂fx (q)
∂q1
∂fy (q)
∂q1
∂fx (q)
∂q2
∂fy (q)
∂q2
!
=
−L1 sin(q1 ) − L2 sin(q1 + q2 ) −L2 sin(q1 + q2 )
L1 cos(q1 ) + L2 cos(q1 + q2 ) L2 cos(q1 + q2 )
Perception
Planning
Knowledge
Technical Cognitive Systems
58
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Outline
Control
Perception
Planning
Knowledge
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
59
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Exam Prep. Session: Perception
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
60
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 1
Describe the algorithmic steps of the ICP algorithm!
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
61
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 1
Describe the algorithmic steps of the ICP algorithm!
1. [optional] select subsets of points / keypoints.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
62
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 1
Describe the algorithmic steps of the ICP algorithm!
1. [optional] select subsets of points / keypoints.
2. give the initial guess (transformation), and apply.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
63
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 1
Describe the algorithmic steps of the ICP algorithm!
1. [optional] select subsets of points / keypoints.
2. give the initial guess (transformation), and apply.
3. for each point from 1st dataset, find best matching (e.g. closest) point
from 2nd dataset
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
64
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 1
Describe the algorithmic steps of the ICP algorithm!
1. [optional] select subsets of points / keypoints.
2. give the initial guess (transformation), and apply.
3. for each point from 1st dataset, find best matching (e.g. closest) point
from 2nd dataset
4. compute a new transformation that when applied to 1st dataset minimizes
the error (e.g. least squares error (SVD) or non-linear optimizers
(Levenberg-Marquardt).
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
65
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 1
Describe the algorithmic steps of the ICP algorithm!
1. [optional] select subsets of points / keypoints.
2. give the initial guess (transformation), and apply.
3. for each point from 1st dataset, find best matching (e.g. closest) point
from 2nd dataset
4. compute a new transformation that when applied to 1st dataset minimizes
the error (e.g. least squares error (SVD) or non-linear optimizers
(Levenberg-Marquardt).
5. apply transformation and repeat (goto step 3) until convergence!
(convergence criteria: for example a small transformation distance, etc..)
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
66
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 2
Name common uses for the Iterative Closest Point (ICP) algorithm?
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
67
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 2
Name common uses for the Iterative Closest Point (ICP) algorithm?
1. Dataset Registration
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
68
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 2
Name common uses for the Iterative Closest Point (ICP) algorithm?
1. Dataset Registration
2. Pointcloud registration
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
69
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 2
Name common uses for the Iterative Closest Point (ICP) algorithm?
1. Dataset Registration
2. Pointcloud registration
3. Panorama stitching
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
70
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Perception, Example 2
Name common uses for the Iterative Closest Point (ICP) algorithm?
1. Dataset Registration
2. Pointcloud registration
3. Panorama stitching
4. Scan matching in SLAM.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
71
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Outline
Control
Perception
Planning
Knowledge
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
72
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Exam Prep. Session: Planning/Searcing
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
73
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning, Example
Robby can move across the rooms only the according to the above graph. She
can pick the mail only from the mailbox, and drop it only at johns office. Given
the world state definitions, restrictions, and the initial state below: Draw a task
tree, and solve it by depth first search. Show nodes expanded.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
74
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning, Example
State of the world:
1. Robot-Location (RG—E—JO—JM)
2. John-has-received-mail (True—False)
3. Mailbox-has-mail (True—False)
4. Robot-has-mail (True—False)
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
75
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning, Example
Actions and restrictions:
ACTIONS
Move-to-E
Move-to-JO
Move-to-JM
Move-to-RG
Put-down-mail
Pick-up-mail
Control
Michael Beetz
Summer Term 2012
PRECONDITIONS
None
Robot-Loc=E
Robot-Loc=E
Robot-Loc=E AND Robot-has-mail=False
Robot-Loc=JO AND Robot-has-mail=TRUE
Robot-Loc=JM AND Robot-has-mail=FALSE
Perception
Planning
Knowledge
Technical Cognitive Systems
76
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning, Example
Initial State:
1. Robot-Location (RG)
2. John-has-received-mail (False)
3. Mailbox-has-mail (True)
4. Robot-has-mail (False)
Goal State:
1. Robot-Location (JO)
2. John-has-received-mail (True)
3. Mailbox-has-mail (False)
4. Robot-has-mail (False)
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
77
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning, Example
Nodes:
Name
A
B
C
.
.
GS
Location
E
JO
RG
.
.
JO
Control
Michael Beetz
Summer Term 2012
John-has-mail
False
False
False
.
.
True
Perception
Robot-has-mail
False
False
False
.
.
False
Planning
Mailbox-has-mail
True
True
True
.
.
False
Knowledge
Technical Cognitive Systems
78
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning, Example
Graph Solution:
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
79
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Final Solution
DFS Expands:
IS
A
B|D
D
F
H
I
GS
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
80
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Planning: Resources
Online Resources:
1. AI appplets
2. airspace.org
3. artint.info
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
81
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Outline
Control
Perception
Planning
Knowledge
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
82
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
A Knowledge Base for Spatial Reasoning
You are supposed to implement a logical knowledge base (KB) for a cognitive
robot. The robot has a perception system that is able to recognize objects in
the world by reference to their position in (x, y , z) ∈ R3 and their dimensions
(∆x, ∆y , ∆z) ∈ R3 in cartesian coordinates, where the position coordinates
define the centroid of the bounding box of the respective object.
Your KB has a Tell-Ask interface, by means of which the perception system
is able to assert new facts about objects it has recognized in the environment,
using the predicates
position(ObjectID, X, Y, Z)
dimensions(ObjectID, X, Y, Z)
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
83
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
We want to implement a Prolog KB that is able to perform spatial reasoning on
the facts from the perception system. In particular, the robot should be able to
infer
• if an object o1 is contained in some other object o2 (e.g. milk in the
refrigerator)
• or if o1 is located on o2 (e.g. a cup on the table).
Implement the respective Prolog predicates
in(Object, Containter) and
on(Object, SuppObject),
using only the predicates position and dimensions (you are allowed to use
arithmetic operators, though).
Assume that an object is in/on sth. if its bounding box is completely inside/on
the other object.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
84
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
We define a helper predicate for checking if some interval I1 given by its
centroid and dimension is contained in some other interval I2 :
contains(CI1,DI1,CI2,DI2) :-
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
85
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
We define a helper predicate for checking if some interval I1 given by its
centroid and dimension is contained in some other interval I2 :
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
86
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
We define a helper predicate for checking if some interval I1 given by its
centroid and dimension is contained in some other interval I2 :
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
87
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):-
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
88
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
89
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
dimensions(O, XDIMO, YDIMO, ZDIMO),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
90
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(C,XDIMC, YDIMC, ZDIMC),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
91
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(C,XDIMC, YDIMC, ZDIMC),
contains(XO, XDIMO, XC, XDIMC),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
92
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(C,XDIMC, YDIMC, ZDIMC),
contains(XO, XDIMO, XC, XDIMC),
contains(YO, YDIMO, YC, YDIMC),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
93
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(C,XDIMC, YDIMC, ZDIMC),
contains(XO, XDIMO, XC, XDIMC),
contains(YO, YDIMO, YC, YDIMC),
contains(ZO, ZDIMO, ZC, ZDIMC),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
94
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
in(O, C):position(O, XO, YO, ZO), position(C, XC, YC, ZC),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(C,XDIMC, YDIMC, ZDIMC),
contains(XO, XDIMO, XC, XDIMC),
contains(YO, YDIMO, YC, YDIMC),
contains(ZO, ZDIMO, ZC, ZDIMC),
C \= O.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
95
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):-
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
96
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
97
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
98
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
99
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
(contains(XO, XDIMO, XO, XDIMO),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
100
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
(contains(XO, XDIMO, XO, XDIMO),
contains(YO, YDIMO, YS, YDIMS);
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
101
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
(contains(XO, XDIMO, XO, XDIMO),
contains(YO, YDIMO, YS, YDIMS);
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
102
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
(contains(XO, XDIMO, XO, XDIMO),
contains(YO, YDIMO, YS, YDIMS);
contains(XS, XDIMS, XO, XDIMO),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
103
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
(contains(XO, XDIMO, XO, XDIMO),
contains(YO, YDIMO, YS, YDIMS);
contains(XS, XDIMS, XO, XDIMO),
contains(YS, YDIMS, YO, YDIMO)),
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
104
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
contains(CI1,DI1,CI2,DI2) :CI1 + DI1 / 2 =< CI2 + DI2 / 2,
CI1 - DI1 / 2 >= CI2 - DI2 / 2.
on(O, S):position(O, XO, YO, ZO), position(S, XS, YS, ZS),
dimensions(O, XDIMO, YDIMO, ZDIMO),
dimensions(S,XDIMS, YDIMS, ZDIMS),
(contains(XO, XDIMO, XO, XDIMO),
contains(YO, YDIMO, YS, YDIMS);
contains(XS, XDIMS, XO, XDIMO),
contains(YS, YDIMS, YO, YDIMO)),
ZS + ZDIMS / 2 =:= ZO - ZDIMO / 2.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
105
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
A Knowledge Base for Spatial Reasoning (ctd)
The relations in and on both are transitive, so our knowledge base should take
this into account. So we add
(1) in(X,Z) :- in(X,Y), in(Y,Z).
to our knowledge base. Would this work?
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
106
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
A Knowledge Base for Spatial Reasoning (ctd)
The relations in and on both are transitive, so our knowledge base should take
this into account. So we add
(1) in(X,Z) :- in(X,Y), in(Y,Z).
to our knowledge base. Would this work? No, since we end up in infinite
recursion!
(2) isIn(X,Z) :- in(X,Z).
isIn(X,Z) :- in(X,Y), isIn(Y,Z).
fixes it.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
107
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Infinite Recursion
We consider the seed-apple-box problem from the exercise.
We have the following Prolog program:
in(seed,apple).
in(apple,box).
in(X,Z) :- in(X,Y), in(Y,Z).
We want to compute the transitive closure of seed. Tracing the query yields:
?- trace,in(seed,Z).
Call: (7) in(seed, G259) ? creep
Exit: (7) in(seed, apple) ? creep
Z = apple ;
Redo: (7) in(seed, G259) ? creep
Call: (8) in(seed, G376) ? creep
Exit: (8) in(seed, apple) ? creep
Call: (8) in(apple, G259) ? creep
Exit: (8) in(apple, box) ? creep
Exit: (7) in(seed, box) ? creep
Z = box ;
Redo: (8) in(apple, G259) ? creep
Call: (9) in(apple, G376) ? creep
Exit: (9) in(apple, box) ? creep
Call: (9) in(box, G259) ? creep
Call: (10) in(box, G376) ? creep
Call: (11) in(box, G376) ? creep
Call: (12) in(box, G376) ? creep
Call: (13) in(box, G376) ? creep
Call: (14) in(box, G376) ? creep
Call: (15) in(box, G376) ? creep
Call: (16) in(box, G376) ? creep
Call: (17) in(box, G376) ? creep
Control
Call: (18) in(box,Perception
G376) ?
Michael Beetz
Summer Term 2012
Planning
Knowledge
Technical Cognitive Systems
108
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
DFS-Tree
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
109
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Recursion
As long as we find valid unifications, everything is fine. If we try to prove
something that cannot be proven, we run into infinite recursion with formula
(1), since the recursive formula is the only one, which potentially could prove it.
Formula (2) fixes this problem, since the ground/abort condition occurs before
the recursion takes place.
But we need to be careful with the ordering here, as well. Why?
in(seed,apple).
in(apple,box).
isIn(X,Z) :- in(X,Z).
isIn(X,Z) :- in(X,Y), isIn(Y,Z).
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
110
Department of Informatics
Intelligent Autonomous Systems
Technische Universität München
Trace yields:
?- trace,isIn(seed,Z).
Call: (7) isIn(seed, G259) ? creep
Call: (8) in(seed, G259) ? creep
Exit: (8) in(seed, apple) ? creep
Exit: (7) isIn(seed, apple) ? creep
Z = apple ;
Redo: (7) isIn(seed, G259) ? creep
Call: (8) in(seed, G376) ? creep
Exit: (8) in(seed, apple) ? creep
Call: (8) isIn(apple, G259) ? creep
Call: (9) in(apple, G259) ? creep
Exit: (9) in(apple, box) ? creep
Exit: (8) isIn(apple, box) ? creep
Exit: (7) isIn(seed, box) ? creep
Z = box ;
Redo: (8) isIn(apple, G259) ? creep
Call: (9) in(apple, G376) ? creep
Exit: (9) in(apple, box) ? creep
Call: (9) isIn(box, G259) ? creep
Call: (10) in(box, G259) ? creep
Fail: (10) in(box, G259) ? creep
Redo: (9) isIn(box, G259) ? creep
Call: (10) in(box, G376) ? creep
Fail: (10) in(box, G376) ? creep
Fail: (9) isIn(box, G259) ? creep
Fail: (8) isIn(apple, G259) ? creep
Fail: (7) isIn(seed, G259) ? creep
false.
Control
Michael Beetz
Summer Term 2012
Perception
Planning
Knowledge
Technical Cognitive Systems
111