In The Name Of God
Robot Motion Planning
Under Unertainty
Presenter: Fatemeh Zamani
Introduction
One of the ultimate goals of robotics
research is to create easily autonomous
robots.
 Such robots will accept high-level
descriptions of tasks specifying what the
user wants done, rather than how to do it.

Introduction
One of the key topics in robotics is motion
planning.
 A major limitation of these planners,
however, is that they assume complete
and accurate geometric models of the
robot and its workspace, and perfect
control of the robot and robot can sense
perfectly.

Introduction
There exists a variety of operations which
cannot be achieved reliably by simply
executing preplanned paths.
 These operations require uncertainty to be
taken into account at the planning stage in
order to generate motion strategies that
combine motion and sensing commands.
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Introduction
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Motion planning with uncertainty is a critical
problem in robotics.
Even the most complex models of the physical
world cannot be perfectly accurate.
Increasing model complexity often adversely
afects the ability of a robot to plan its actions
efficiently.
The use of simplied models seems to be the
only practical way.
All details omitted from these models unite to
form uncertainty.
Introduction

Two main application domains have been
considered in Motion planning with
uncertainty:

part mating for mechanical assembly
 mobile robot navigation

Various approaches have been proposed
which are applicable to one domain or the
other or both.
Taking uncertainty into account at planning
time is essential when potential errors are
comparable to or larger than the
tolerances.
 mechanical: in partmating tasks where
both errors and tolerances are usually
small.
 mobile robot: in navigation tasks where
both errors and tolerances may be large.
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For such tasks classical path planning methods
which use simple geometric models while
assuming null uncertainty are clearly insufficient.
At best they produce paths that require frequent
replanning to deal with discrepancies detected by
sensors during execution.
Due to errors in sensing they may also lead the
robot to incorrectly believe that it has achieved
some expected state or on the contrary that it has
failed to achieve this state.

To generate reliable plans the planner
must choose actions whose execution is
guaranteed to make enough knowledge
available to allow the robot to correctly
identify the states it traverses despite
execution time errors in control and
sensing.
Reasoning in the presence of uncertainty:
 This attitude has often led to engineering
brittle complicated systems.
 Experiments may show that these
systems work beautifully on some tasks
but fail on simpler ones.
 Prediction is impossible.

The design of a reliable system dealing
with uncertainty must be based on:
 bounded and well-defined expectations
of what may actually happen in the real
world.
A Sound, Complete and Polynomial
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Sound : The planner only generates correct
plans which are guaranteed to succeed if some
predefined assumptions bounding uncertainty
are satisfied.
complete : The planner returns a correct plan
whenever one exists otherwise it declares
failure.
Time complexity a function of the size of the
input problem (typically the complexity of the
robots environment).
.
Several motion planners with uncertainty
have been proposed.
 soundness and completeness are
provable for only a few of them.
 Most known sound and complete planning
algorithms take exponential time.
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A powerfull approach
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The most powerful known approach to this kind
of planning problem is the preimage
backchaining.
Preimage backchaining consists of iteratively
computing preimages of the goal , preimages of
computed preimages taken as intermediate
goals, for various selected motion commands,
until a preimage contains the initial subset.
Assumptions
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We assume that the robot is the only agent in a
static workspace,
Error Sources (Uncertainty):
•
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Control )the robot does not perfectly execute the
motion commands
Sensing )the sensors do not return accurate data(
Task Modeling
Configuration space
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We are interested in planning the motion of an
object A - the robot - in a workspace W
populated by obstacles Bi, i  [1, q].
A configuration of A is a specification of the
position of every point in A with respect to a
coordinate system embedded in W.
The configuration space of A, denoted by C, is
the set of all the possible configurations of A.
Reference point
We assume that A is a two-dimensional
object that can only translate in the plane
R2.
 A configuration is represented as q = (x, y)
 R2, where x and y are the coordinates of
a specific point of A, known as the
reference point, with respect to the
coordinate system embedded in W.
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C-obstacle
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Each obstacle Bi maps in C to the subset
CBi of configurations where A intersects Bi
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The region CBi is called a C-obstacle.
C-obstacle
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Each obstacle Bi maps in C to the subset
CBi of configurations where A intersects Bi,
i.e.:
where A(q) denotes the subset of W
occupied by A at configuration q.
 The region CBi is called a C-obstacle.
peg-into-hole task
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The goal of the task is to insert A in B's
depression.
Free space, contact space, valid space
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The union of the C-obstacles, is called the C-obstacle
region and is denoted by CB.
The complement of the C-obstacle region in C is called
the free space and is denoted by Cfree.
The subset of configurations q where A(q) intersects with
the obstacle region without overlapping its interior is
called the contact space and is denoted by Ccontact .
The union of the free space and the contact space is
called thevalid space and is denoted by Cvalid.
Motion command
A motion command M is a pair (CS, TC).
 CS is called the control statement.
 TC is called the termination condition. The
controller stops the motion when TC
evaluates to true. TC's arguments may be
sensory inputs during the execution of the
motion and the elapsed time since the
beginning of the motion.
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generalized damper compliance model
The generalized damper control model
has been used in most work related to
preimage backchaining.
 It is less sensitive to errors than pure
position control.
 CS is parameterized by a vector v1 called
the commanded velocity.
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Moving in this model
As long as the configuration of A is in the
free space, it moves at constant velocity
v1.
 When it is in a contact edge, the
configuration may move away from the
edge, slide along it, or stick to it.
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Moving away
If v1 projects positively along the outgoing
normal of the edge, it moves away.
 Otherwise, the Reaction is dependent on
the friction of the surface.
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Friction
Let  be the friction coefficient.
 A generalized damper motion along v1
sticks to a contact edge if the magnitude of
the angle between - v1 and the outgoing
normal of the edge is less than or equal to
tan- 1.
 Otherwise it slides.
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Uncertainty in control
Let v1 be the commanded velocity.
 The effective commanded velocity is a
vector v2 .
 The magnitude of the angle between v1
and v2 is less than a fixed angle  < ½ 
(control uncertainty cone).
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For simplification, we therefore assume
that there is no uncertainty in the velocity
modulus, i.e., || v1 ||=|| v2 ||.
 This assumption has no significant impact
on the methods we will describe and
could be easily retracted.
 For further simplification we assume that
it is a unit vector,
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Motion in free space
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During motion, v2 may vary arbitrarily, but
continuously, between the two extreme
orientations determined by v1 and  if A is
in the free space, it moves along a
trajectory that is contained in the control
uncertainty cone.
Motion in contact space
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if all vectors v1 project positively on the outgoing
normal of the edge, then it moves away from the
edge.
Instead, if the inverted control uncertainty cone
is entirely contained in the friction cone, A sticks
to the edge.
But if it is completely outside the friction cone
with all the vectors in the control uncertainty
cone, A is guaranteed to slide.
Reaction force
In the generalized damper compliance
control model, the robot exerts a force
proportional to the effective commanded
velocity v1 , say Bv.
 When the robot is in free space, no
reaction force is applied to the robot and
the force exerted by the robot is entirely
used to create motion.
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Reaction force
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When the robot is in contact space and
pushes on an obstacle, the obstacle
applies a reaction force.
Uncertainty in sensing
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Generally the robot A is equipped with two
sensors - a position sensor and a force
sensor.
The position sensor measures the current
configuration of A.
The sensed configuration is denoted by q*, while
the current actual configuration is denoted by q.
The uncertainty in the measurement is modeled
as an open disc of fixed radius pq centered at
the sensed configuration q*.
The force sensor measures the reaction
force exerted on A.
 f lies in an open cone.
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Preimage backchaining
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Let S be a subset of Cvalid will be the initial region of the
motion plan.
Let G be another subset of Cvalid as the goal region.
Let L be a subset of Cvalid and and M = (v1 , TC) be a
motion command.
A preimage of L for M is defined as any subset P of Cvalid
such that: if A's configuration is in P when the execution
of M starts, then it is guaranteed that A will reach L (goal
teachability) and that it will be in L when TC terminates
the motion (goal recognizability).
Example1(peg-into-hole problem)
preimage backchaining
Now, suppose that an algorithm is
available for computing preimages.
 Given the initial and goal, preimage
backchaining consists of constructing a
sequence of preimages p1,p2,…,pp such
that:
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preimage backchaining
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If the backchaining process terminates
successfully, the inverse sequence of the motion
commands which have been selected to
produce the preimages, [M1, M2,…, Mp] , is the
generated motion strategy.
This strategy is guaranteed to achieve the goal
successfully, whenever the errors in control and
sensing remain within the ranges determined by
the uncertainty intervals.
Example 2
generating the sequence of preimages
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The problem of generating the sequence of
preimages can be transformed into that of
searching a graph by selecting motion
commands from a discretized set.
The root of this graph is the goal region g, and
each other node is a preimage region; each arc
is a motion command, connecting a region to a
preimage for this command.
maximal preimage
For a given commanded velocity the ideal
preImage computation method would
compute the maximal preimage.
 maximal preimage : It is not contained in
any other prelmage of the state for the
defined command.
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maximal preimage(Advantages)
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A large preimage has more chance to include
the initial region than a small one.
if it is considered recursively as an intermediate
goal, a large preimage has more chance to
admit large preimages than a small one. Thus,
considering larger preimages may reduce the
size of the search graph. it may also produce
"simpler" strategies,
Computation of preimages
Two methods for computing backprojection:
 backprojection from sticking edges.
 backprojection from goal kernel.
 None of these methods compute maximal
preimages but both of them are easily
implementable.
Backprojection from sticking edges
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It consists of:
Backprojection from sticking edges
The region L (Js) is called the
backprojection of Jsfor v1.
 Erdmann gave a simple algorithm to
compute the maximal backprojection of a
single contact edge.
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Step 1
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Consider every vertex in Ccontact Mark
every non-goal vertex at which sticking is
possible. Mark every goal vertex if sliding
away from the vertex in the non-goal edge
is possible.
Step 2
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At every marked vertex erect two rays
parallel to the sides of the inverted control
uncertainty cone. Compute the
intersection of these rays among
themselves and with Ccontact . Ignore each
ray beyond the first intersection. The
remaining segment in each ray is called a
free edge.
Step 3
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Trace out the backprojection region. To
that purpose, trace the edge L in the
direction that leaves the C-obstacle region
on the right-hand side, until an endpoint is
attained; then, let all the contact and free
edges incident at this point be sorted
clockwise in a circular list; trace the first
edge following L in this list.
An Example
PreImage backchaining is a very general
approach, however, raises difficult
computational issues which still prevent its
widespread application.
 On solution is using Landmarks in the
workspae.
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Introduction of Landmark
A typical notion for planning with
uncertainty is that of a landmark, an
element of the workspace that can be
detected reliably.
 Sometimes, a direct path to the goal may
seem attractive, but if it does not allow
enough landmarks to be detected along
the way, the robot may fail to attain the
goal due to the errors in its planning
models.
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Introduction of Landmark
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The overall task of the planner is:
 select
an adequate set of states.
 associate appropriate motion commands with
these states.
 synthesize state recognition functions.
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All three subtasks are interdependent.
The Concept of a landmark
Landmark : an “island of perfection” in the
robot conguration space where position
sensing and motion control are accurate.
 However neither control nor sensing need
be perfect in landmark regions.
 It mainly reduces planning to selecting
motion commands to navigate from
landmark to landmark.
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Similar notions for Landmark
Similar notions have been previously
introduced as landmarks in the literature
with dierent names :
 atomic regions
 Signature neighborhoods
 perceptual equivalent classes
 sensory uncertainty field
 visual constraints
Creating landmarks
Creating landmarks requires some prior
engineering of the robot and its
environment.
 It is important that its cost be reduced
as much as possible.
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Assumptions
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The robot is a point moving in a plane.
Obstacles are considered as circular regions.
landmark disks are circular regions too.
Both the obstacles and the landmark disks are
stationary.
The number of landmark disks is finite(l)
The number of obstacle disks is also finite(O(l)).
We can consider l as a criteria for measuring the
size of the input problem.
Assumptions
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The robot has perfect position sensing in the
landmark disks and no sensing outside the
landmark disks.
The direction of motion at any instant differs
from the commanded direction of motion by an
angle bounded by  .
In the imperfect control mode the actual value
of teta can be controllable by the robot in a
connected interval
A motion command in the imperfect
control mode is specied as a tripletd (d, 
,L)
 L is a set of landmark disks defning the
termination condition.
 The direction of motion at any instant
differs from the commanded direction of
motion by an angle bounded by teta.
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Example 1
Assume a constant directional uncertainty
is given.
 The workspace contains 23 landmark
disks shown white or grey forming 19
landmark regions and 25 obstacle disks
 The white disks are those with which the
planner has associated imperfect control
motion commands to attain another
landmark disk.
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The algorithm
The algorithm assumes given constant
directional uncertainty(0.09 radian).
 It produces plans minimizing the number
of motion commands to be executed.
 The arrow attached to the initial region or
a whitedisk is the commanded direction of
a command
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Example 2
Assuming controllable directional
uncertainty.
 The directional uncertainty is controllable
in the interval [0.1 0.5].
 The planner produces a one-step plan, a
plan containing a single imperfect control
motion command (d ,  , L)
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extension of the goal
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If a connected set of landmark disks intersects
the goal region the robot can move into the goal
from any point in this set.
The union of the landmark regions intersecting
the goal is called the extension of the goal the
other landmark disks are called the intermediate.
If the goal does not intersect any landmark disk
then it is considered unachievabl.
zero step plan
Given a goal G we first compute its
extension E(G).
 If the initial region I lies entirely in E(G) no
further planning effort is needed.
 Indeed in a landmark region a plan is
simply a geometric path whose
computation is straightforward Such a plan
is called a zero step plan.
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one-step plan
If there does not exist a zerostep plan to
achieve G the planner may try to find a
pair (d,  ) such that the initial region I is
contained in the backprojection of E(G) for
(d, ).
 We call this plan a one-step plan.
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A one step plan may not exist or may not
be desirable if its cost is too high
 Then the planner can attempt to create a
multi-step plan iteratively.
 The backprojection of the new goal
extensions are computed.
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Building the Discrete Search Space
Each planning iteration requires selecting
a pair (d ) such that the backprojection of
the current goal extension either contains
the initial region I or intersects
intermediategoal disks.
 The d -space can be partitioned into an
arrangement of curves defining cells.
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The concept of cells
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Each cell is regular in the following sense:

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The backprojection of the goal extension for
any pair (d, ) in a cell C contains the same
initial position disks and intersects the same
intermediategoal disks as the backprojection for
any other pair (d’, ’) in C.
The number of cells is polynomial in the
number of landmark and obstacle disks
Creation the arrangement of cells
Let us assume that no landmark disk
intersects an obstacle disk.
 The arrangement of cells is created by a
network of curves corresponding to the
some critical events.
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I-cover event
An example of a critical point:
 I-cover event: A left ray of the
backprojection is tangent to an initial
region disk with this disk on its righthand
 sideLet s be the slope of the left ray
tangent to the initial position disk.
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The equation of the
curve denoted by this
event is :
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There are eleven critical point and eleven
corresponding equation.
Allowing landmark disks to intersect with
obstacle disks would simply require con- sidering
additional critical events It would not change the
asymptotic complexity of the cell arrangement.
D plane
References
Jean-Claude Latombe, Anthony Lazanas
and Shashank Shekhar,“ Robot motion
planning with uncertainty in sensing
and control.
 Anthony Lazanas, and Jean-Claude
Latombe, “Motion Planning with
Uncertainty:A Landmark Approach”
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Thanks