Motion planning

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Submitted on 15 May 2019

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Motion planning

Florent Lamiraux

To cite this version:

Florent Lamiraux. Motion planning. Doctoral. GdR Robotics Winter School: Robotica Principia,

Centre de recherche Inria Sophia Antipolis – Méditérranée, France. 2019. �cel-02130128�

Introduction Definitions Random Sampling Collision testing Software

Motion planning

Florent Lamiraux

CNRS-LAAS, Toulouse, France

Introduction Definitions Random Sampling Collision testing Software

Motion planning

Introduction

Definitions

Random Sampling

Collision testing

Software

Introduction Definitions Random Sampling Collision testing Software

Context

industrial robot

aerial vehicle

autonomous

vehicle

Autonomous mobile systems

I

moving in an environment cluttered by obstacles

I

possibly subject to kinematic or dynamic constraints

Motion planning : automatically compute a feasible collision-free

path between two given configurations.

Introduction Definitions Random Sampling Collision testing Software

Context

industrial robot

aerial vehicle

autonomous

vehicle

Autonomous mobile systems

I

moving in an environment cluttered by obstacles

I

possibly subject to kinematic or dynamic constraints

Motion planning : automatically compute a feasible collision-free

path between two given configurations.

Introduction Definitions Random Sampling Collision testing Software

Context

industrial robot

aerial vehicle

autonomous

vehicle

Autonomous mobile systems

I

moving in an environment cluttered by obstacles

I

possibly subject to kinematic or dynamic constraints

Motion planning : automatically compute a feasible collision-free

path between two given configurations.

Introduction Definitions Random Sampling Collision testing Software

Robot

Set of rigib bodies B

0

, · · · B

m

, linked to one

another by joints.

Joint : mobility of a body in the reference

frame of its parent, parameterized by one or

several numbers.

Introduction Definitions Random Sampling Collision testing Software

Robot configuration

The configuration q of a robot is represented

by the concatenation of the parameters of

each joint.

Introduction Definitions Random Sampling Collision testing Software

Forward kinematics

Computation of the position of each joint in

world frame.

M

i

(q) = M

parent(i )

(q) M

i /parent

T

i

(q)

Introduction Definitions Random Sampling Collision testing Software

Definitions

I

_{Workspace : W = R}

2

_{or R}

3

: space in which the robot moves

I

Workspace obstacle : compact subset of W, denoted by O.

I

Configuration space : C.

I

Position in configuration q of a point M ∈ B

i

: x

i

(M, q).

I

Configuration space obstacle :

C

_obst

= {q ∈ C,

∃i ∈ {1, · · · , m}, ∃M ∈ B

_i

, x

i

(M, q) ∈ O ou

∃i , j ∈ {1, · · · , m}, ∃M

_i

∈ B

_i

, ∃M

j

∈ B

j

,

x

i

(M

i

, q) = x

j

(M

j

, q)}

I

Free configuration space : C

free

= C \ C

obst

.

Introduction Definitions Random Sampling Collision testing Software

Definitions

I

_{Workspace : W = R}

2

_{or R}

3

: space in which the robot moves

I

Workspace obstacle : compact subset of W, denoted by O.

I

Configuration space : C.

I

Position in configuration q of a point M ∈ B

i

: x

i

(M, q).

I

Configuration space obstacle :

C

_obst

= {q ∈ C,

∃i ∈ {1, · · · , m}, ∃M ∈ B

_i

, x

i

(M, q) ∈ O ou

∃i , j ∈ {1, · · · , m}, ∃M

_i

∈ B

_i

, ∃M

j

∈ B

j

,

x

i

(M

i

, q) = x

j

(M

j

, q)}

I

Free configuration space : C

free

= C \ C

obst

.

Introduction Definitions Random Sampling Collision testing Software

Definitions

I

_{Workspace : W = R}

2

_{or R}

3

: space in which the robot moves

I

Workspace obstacle : compact subset of W, denoted by O.

I

Configuration space : C.

I

Position in configuration q of a point M ∈ B

i

: x

i

(M, q).

I

Configuration space obstacle :

C

_obst

= {q ∈ C,

∃i ∈ {1, · · · , m}, ∃M ∈ B

_i

, x

i

(M, q) ∈ O ou

∃i , j ∈ {1, · · · , m}, ∃M

_i

∈ B

_i

, ∃M

j

∈ B

j

,

x

i

(M

i

, q) = x

j

(M

j

, q)}

I

Free configuration space : C

free

= C \ C

obst

.

Introduction Definitions Random Sampling Collision testing Software

Definitions

I

_{Workspace : W = R}

2

_{or R}

3

: space in which the robot moves

I

Workspace obstacle : compact subset of W, denoted by O.

I

Configuration space : C.

I

Position in configuration q of a point M ∈ B

i

: x

i

(M, q).

I

Configuration space obstacle :

C

_obst

= {q ∈ C,

∃i ∈ {1, · · · , m}, ∃M ∈ B

_i

, x

i

(M, q) ∈ O ou

∃i , j ∈ {1, · · · , m}, ∃M

_i

∈ B

_i

, ∃M

j

∈ B

j

,

x

i

(M

i

, q) = x

j

(M

j

, q)}

I

Free configuration space : C

free

= C \ C

obst

.

Introduction Definitions Random Sampling Collision testing Software

Definitions

I

_{Workspace : W = R}

2

_{or R}

3

: space in which the robot moves

I

Workspace obstacle : compact subset of W, denoted by O.

I

Configuration space : C.

I

Position in configuration q of a point M ∈ B

i

: x

i

(M, q).

I

Configuration space obstacle :

C

_obst

= {q ∈ C,

∃i ∈ {1, · · · , m}, ∃M ∈ B

_i

, x

i

(M, q) ∈ O ou

∃i , j ∈ {1, · · · , m}, ∃M

_i

∈ B

_i

, ∃M

j

∈ B

j

,

x

i

(M

i

, q) = x

j

(M

j

, q)}

I

Free configuration space : C

free

= C \ C

obst

.

Introduction Definitions Random Sampling Collision testing Software

Definitions

I

_{Workspace : W = R}

2

_{or R}

3

: space in which the robot moves

I

Workspace obstacle : compact subset of W, denoted by O.

I

Configuration space : C.

I

Position in configuration q of a point M ∈ B

i

: x

i

(M, q).

I

Configuration space obstacle :

C

_obst

= {q ∈ C,

∃i ∈ {1, · · · , m}, ∃M ∈ B

_i

, x

i

(M, q) ∈ O ou

∃i , j ∈ {1, · · · , m}, ∃M

_i

∈ B

_i

, ∃M

j

∈ B

j

,

x

i

(M

i

, q) = x

j

(M

j

, q)}

I

Free configuration space : C

free

= C \ C

obst

.

Introduction Definitions Random Sampling Collision testing Software

Motion

I

Motion :

I

continuous mapping from [0, 1] into C.

I

Collision free motion :

I

continuous mapping from [0, 1] into C

free

.

Introduction Definitions Random Sampling Collision testing Software

Motion

I

Motion :

I

continuous mapping from [0, 1] into C.

I

Collision free motion :

I

continuous mapping from [0, 1] into C

free

.

Introduction

Definitions

Random Sampling Collision testing Software

Motion planning problem

initial configuration

goal configuration

C = [−2π, 2π]

6

Introduction Definitions Random Sampling Collision testing Software

History

I

before the 1990’s : mainly a mathematical problem

I

Real algbraic geometry

I

Decidability : Schwartz and Sharir 1982

I

Tarski theorem, Collins decomposition

I

Approximate cell decomposition

I

from the 1990’s : an algorithmic problem

I

random sampling (1993)

I

asymptotically optimal random sampling (2011)

Introduction Definitions Random Sampling Collision testing Software

History

I

before the 1990’s : mainly a mathematical problem

I

Real algbraic geometry

I

Decidability : Schwartz and Sharir 1982

I

Tarski theorem, Collins decomposition

I

Approximate cell decomposition

I

from the 1990’s : an algorithmic problem

I

random sampling (1993)

I

asymptotically optimal random sampling (2011)

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Random sampling motion planning methods appeared in the

early 1990’s

I

Principle

I

sample random configurations

I

test whether they are collision-free

I

build a roadmap the nodes of which are the free configurations,

I

and the edges of which are collision-free linear interpolations.

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Random sampling motion planning methods appeared in the

early 1990’s

I

Principle

I

sample random configurations

I

test whether they are collision-free

I

build a roadmap the nodes of which are the free configurations,

I

and the edges of which are collision-free linear interpolations.

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Random sampling motion planning methods appeared in the

early 1990’s

I

Principle

I

sample random configurations

I

test whether they are collision-free

I

build a roadmap the nodes of which are the free configurations,

I

and the edges of which are collision-free linear interpolations.

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Random sampling motion planning methods appeared in the

early 1990’s

I

Principle

I

sample random configurations

I

test whether they are collision-free

I

build a roadmap the nodes of which are the free configurations,

I

and the edges of which are collision-free linear interpolations.

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Random sampling motion planning methods appeared in the

early 1990’s

I

Principle

I

sample random configurations

I

test whether they are collision-free

I

build a roadmap the nodes of which are the free configurations,

I

and the edges of which are collision-free linear interpolations.

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_init

q

_goal

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_init

q

_goal

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM) 1994

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Probabilistic roadmap (PRM)

I

Numerous useless nodes are created

I

this makes the connection of new nodes more time consuming

I

Variant : visibility-based PRM

I

only interesting nodes are kept.

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_init

q

_goal

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Visibility-based probabilistic roadmap (Visi-PRM) 1999

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_init

q

_goal

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly exploring Random Tree (RRT) 2000

q

_goal

q

_init

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Pros :

I

no explicit computation of the configuration space

I

easy to implement,

I

robust.

I

Cons :

I

no completeness, only probrabilistic completeness

I

difficult to find narrow passages.

I

required operators :

I

_{collision checking}

I

for configurations (static)

for linear interpolation (dynamic)

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Pros :

I

no explicit computation of the configuration space

I

easy to implement,

I

robust.

I

Cons :

I

no completeness, only probrabilistic completeness

I

difficult to find narrow passages.

I

required operators :

I

_{collision checking}

I

for configurations (static)

for linear interpolation (dynamic)

Introduction Definitions Random Sampling Collision testing Software

Random sampling

I

Pros :

I

no explicit computation of the configuration space

I

easy to implement,

I

robust.

I

Cons :

I

no completeness, only probrabilistic completeness

I

difficult to find narrow passages.

I

required operators :

I

_{collision checking}

I

for configurations (static)

I

for linear interpolation (dynamic)

Introduction Definitions

Random Sampling

Collision testing Software

Asymptotically optimal random sampling

Variants of PRM and RRT exist, and are asymptotically optimal :

I

when the number of nodes tends to infinity,

I

the solution computed by the algorithm tends to the optimal

collision-free path.

Introduction Definitions Random Sampling Collision testing Software

PRM*

PRM

V ← ∅, E ← ∅

for i ∈ {0, · · · , n} do

x

rand

← SampleFree

i

U ← G .Near(x

rand

, r )

for all u ∈ U in order of

increa-sing ku − x

rand

k do

if x

rand

and u in different

connected

components

then

TryConnect (x

rand

, u)

end if

end for

end for

PRM*

V ← SampleFree

_{i =1,···n}

, E ← ∅

for v ∈ V do

U ← G .Near(v ,

r

∗

) \ v

for all u ∈ U do

TryConnect (v , u)

end for

end for

r

∗

= γ

PRM

(log(n)/n)

Introduction Definitions Random Sampling Collision testing Software

PRM*

PRM

V ← ∅, E ← ∅

for i ∈ {0, · · · , n} do

x

rand

← SampleFree

i

U ← G .Near(x

rand

, r )

for all u ∈ U in order of

increa-sing ku − x

rand

k do

if x

rand

and u in different

connected

components

then

TryConnect (x

rand

, u)

end if

end for

end for

PRM*

V ← SampleFree

_{i =1,···n}

, E ← ∅

for v ∈ V do

U ← G .Near(v ,

r

∗

) \ v

for all u ∈ U do

TryConnect (v , u)

end for

end for

r

∗

= γ

PRM

(log(n)/n)

Introduction Definitions Random Sampling Collision testing Software

kPRM*

kPRM

V ← ∅, E ← ∅

for i ∈ {0, · · · , n} do

x

rand

← SampleFree

i

U ← G .Nearest(x

rand

, k)

for all u ∈ U in order of

increa-sing ku − x

rand

k do

if x

rand

and u in different

connected

components

then

TryConnect (x

rand

, u)

end if

end for

end for

kPRM*

V ← SampleFree

_{i =1,···n}

, E ← ∅

for v ∈ V do

U ← G .Nearest(v ,

k

∗

) \ v

for all u ∈ U do

TryConnect (v , u)

end for

end for

k

∗

= k

PRM

log(n), k

PRM

> e(1 +

_d

1

)

Introduction Definitions Random Sampling Collision testing Software

kPRM*

kPRM

V ← ∅, E ← ∅

for i ∈ {0, · · · , n} do

x

rand

← SampleFree

i

U ← G .Nearest(x

rand

, k)

for all u ∈ U in order of

increa-sing ku − x

rand

k do

if x

rand

and u in different

connected

components

then

TryConnect (x

rand

, u)

end if

end for

end for

kPRM*

V ← SampleFree

_{i =1,···n}

, E ← ∅

for v ∈ V do

U ← G .Nearest(v ,

k

∗

) \ v

for all u ∈ U do

TryConnect (v , u)

end for

end for

k

∗

= k

PRM

log(n), k

PRM

> e(1 +

_d

1

)

Introduction Definitions Random Sampling Collision testing Software

PRM*, kPRM*

Note that :

I

PRM*, kPRM* are not iterative anymore,

I

making them iterative is not trivial.

Introduction Definitions

Random Sampling

Collision testing Software

Rapidly Exploring Random trees

There exists also asymptotically optimal variants of RRT

I

RRG, RRT*

but they are specific to a given problem (q

_init

, q

_goal

).

Introduction Definitions Random Sampling Collision testing Software

Collision tests

I

static : for configurations

I

problem : given

I

_{two rigid objects made of triangles}

I

the relative position of one with respect to the other one

determine whether they are colliding.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Bounding volume hierarchies

I

binary tree of bounding volumes such that

I

each node has two children,

I

leaves are triangles.

Introduction Definitions Random Sampling

Collision testing

Software

Collision testing for configurations

I

Algorithm

I

test root nodes of each tree,

I

if two bounding volumes collide, test one with the children of

the other one.

Introduction Definitions Random Sampling

Collision testing

Software

Collision testing for configurations

I

Algorithm

I

test root nodes of each tree,

I

if two bounding volumes collide, test one with the children of

the other one.

Introduction Definitions Random Sampling

Collision testing

Software

Collision testing for configurations

I

Algorithm

I

test root nodes of each tree,

I

if two bounding volumes collide, test one with the children of

the other one.

Introduction Definitions Random Sampling

Collision testing

Software

Collision testing for configurations

I

Algorithm

I

test root nodes of each tree,

I

if two bounding volumes collide, test one with the children of

the other one.

Introduction Definitions Random Sampling Collision testing

Software

Open source software platform

Several open-source platforms for motion planning are available

I

OMPL (Rice University)

I

no kinematic chain,

I

no collision checking.

I

Openrave (CMU)

I

MoveIt (ROS)

I

Integration in ROS of

I

fcl (collision checking), KDL (kinematic chain)

I

Humanoid Path Planner

I

numerical constraints (quasi-static equilibrium)

I

advanced manipulation planning

Introduction Definitions Random Sampling Collision testing

Software

Humanoid Path Planner

https ://humanoid-path-planner.github.io/hpp-doc