introduction to robotics - universidad veracruzana · introduction to robotics ... • the value of...

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Introduction to Robotics

Ph.D. Antonio Marin-Hernandez

Artificial Intelligence Department Universidad Veracruzana Sebastian Camacho # 5

Xalapa, Veracruz Robotics Action and Perception

LAAS-CNRS 7, av du colonel Roche

Toulouse, France

Topics

•  Introduction: Types of robots •  Locomotion •  Kinematics of Mobile Robots •  Perception •  Navigation •  Localization •  Path Planning •  Task Planning

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Mobile Robots: Kinematics

• Kinematics is the basic study of a dynamic system behavior

•  It is necessary to understand the mechanical behavior in order to: – Design tasks – Create control software

•  The problems with kinematics of mobile robots are very similar to those on robotics manipulation:

•  Workspace. It defines the range of possible solutions – Manipulators. All poses of end-effectors relative

to their fixture to the environment – Mobile Robots. All poses of the robot in

reference to their environment


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• The main difference between mobile robots and manipulators is the way as position is estimated.

•  Manipulators: – Arm’s pose is a simple matter of understanding

the kinematics and measuring the position for their joints.

– Those measures can be obtained directly from their sensors


• For a Mobile robot, there is not a direct way to compute their instantaneous position.

•  It must be integrate by motion over time – Slippage problems

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• To understand the whole motion it’s n e c e s s a r y t o u n d e r s t a n d t h e contribution of each wheel. – Each wheel has motion constrains. – All those constrains and the geometry of the

robot are combined to restrict the complete motion of the mobile robot


• Robot pose representation • Considerations

– A rigid body with wheels – Works on an horizontal plane

• Three dimensions (x, y, θ) • There are more degrees of freedom

– Wheel axes, turn axes (castor wheels), etc.

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x

y

xr

y r

θ OL

PG

OG

OG= Global reference frame OL = Local reference frame PG = Position on OG

Mobile Robots: Kinematics •  Axis x & y define the Global Reference

Frame OG •  Axis xr & yr define the local reference frame

fixed to the robot •  Pose of the robot in OG is given by :

–  P = [ x y θ ]Τ

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•  To describe the motion in terms of their individual component it is necessary to map the motion in the local reference frame to the global reference frame.

•  This mapping is a function of the current state of the robot.

•  We must use the orthogonal rotation matrix



R(θ ) =cosθ sinθ 0−sinθ cosθ 00 0 1

"

#

$$$

%

&

'''

Orthogonal rotation Matrix

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•  This matrix is used to map between two reference frames: global OG and local OL •  And is given by:

!pL =R(θ ) !pG


!pL =R(π2) !pG =

0 1 0−1 0 00 0 1

"

#

$$$

%

&

'''

!x!y!θ

"

#

$$$

%

&

'''=

!y− !x!θ

"

#

$$$

%

&

'''

Basic example:

Be

θ =π /2 & !p = !x !y !θ!"#

$%&

T

then:

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x

y xr

yr

θ

OL

PG

OG

• Forward Kinematic Models • Mapping is described by the previous

equation •  Lets consider the following example :


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•  A diferential model •  Each wheel has a radio r and these are at

distance l •  Given r, l, θ and the angular speeds of each

wheel, we have :

!p =!x!y!θ

!

"

###

$

%

&&&= f l, r,θ, !ϕ1, !ϕ2( )


•  Then, its possible to compute the motion in the Global reference frame from local reference frame by :

!pG =R(θ )−1 !pL


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x

y

xr

y r

θ OL

ω(t)

OG

v(t)

•  Lets begin with some examples. •  Suppose the local reference frame is

initially aligned with the global reference frame.

•  Then, if only one wheel moves, the geometric center of the robot who is in the center of the axis between wheels, will move at half the speed of the moving wheel.


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x

y

yr

OL

OG

V1

Vr

V2= 0

•  For each wheel we have: & Then the tangential speed of the robot is

given by:


!xr1 =12r !ϕ1 !xr2 =

12r !ϕ2

!xr = !xr1 + !xr2

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•  If

•  then:


!xr =12r !ϕ1

!ϕ2 = 0

•  If

•  then:

•  So the robot doesn’t move but it spins


!xr = 0

!ϕ1 = − !ϕ2

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•  The value of is simpler to compute, as neither wheel contribute to the motion over this axe.

•  Now, we must compute the rotational speed

•  In a similar way, we can compute the contribution of each wheel and then added.


!θr

!y1

•  The rotational motion of each wheel is given by:

•  &

•  As when one wheel spin clockwise the other is opposite


ω1 =r !ϕ12l

ω2 = −r !ϕ22l

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•  In a similar way, we can compute the independent contribution of each wheel and then added.


!θ = r !ϕ12l

−r !ϕ22l

And then the complete motion will be given by:


!pG =R(θ )−1

r !ϕ12+r !ϕ22

0r !ϕ12l

−r !ϕ22l

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

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And R(θ)-1 is given by:


R(θ )−1 =cosθ −sinθ 0sinθ cosθ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Be the a robot at θ = π/2 with r =1 & l =1, if their wheel speeds are ϕ1= 4 y ϕ2= 2, then:


!pG =!x!y!θ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

0 −1 01 0 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

301

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

031

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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•  Wheels kinematic constraints •  As we have seen, first of all the kinematic

constraints of each wheel are modeled and then is possible to combine them to get the kinematic model of the whole robot.

•  Basically there are 4 types of wheels. •  Then we are going to analyze their

constraints.


• Several simplifications will be taking into account – The plane of the wheel is always vertical – There is only one single contact point for

each wheel – There is no slippage at the contact point.


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•  Fixed standard wheel •  This wheel has not vertical axis of rotation •  It’s angle to the robot is fixed •  Motion is then limited along the wheel plane

and rotation along it’s contact point


•  Another approach for kinematics computation

•  Consider a Differential Drive Robot – Independently of which wheel turns, the robot

rotates about a point that lies on the common axis between two wheels.

– Varying the relative speed of the two wheels, the rotation point will change.


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– At each instant of time, the point at which the robot rotates must have the property that, left and right wheels follow a path that moves around the ICC at the same angular speed ω ant then:

– Where l is the de distance between the center of two wheels and R is the signed distance from the ICC to the midpoint between wheels


ω R+ l2

⎛

⎝⎜

⎞

⎠⎟= vr ω R− l

2⎛

⎝⎜

⎞

⎠⎟= vl

•  Note that vr, vl, ω and R are functions of time.

•  Solving for ω and R we get:


R = l2vl + vr( )vr − vl( )

ω =vr − vll

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•  If vr = vl the radius R is infinite, the the robot moves in a straight line.

•  If vr = -vl the radius R becomes zero and then the robot turns around the middle point between the wheels


R = l2vl + vr( )vr − vl( )

ω =vr − vll

•  For other values of vr, vl , the robot moves around a curved trajectory about a distance R away from the center of the robot.

•  The kinematics structure of the vehicle prohibits certain motions. For example there are not combination of vr and vl to move the robot in the wheel common axis.


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•  A differential drive robot is very sensitive to the relative speeds of the wheels.

•  Small errors in the velocity provided to each wheel result in different trajectories.

•  Differential drive robots are sensitive to slight variations in the ground plane.


•  Forward Kinematics •  Suppose the robot at the given position

(x, y, θ) •  Throughout manipulation of control

parameters vr and vl, the robot can be on different poses.

•  Determining the pose that is reachable given the control parameters is known as the forward kinematics problem


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•  If a given pose (x, y, θ), speeds vr and vl are applied over a period of time t + δt, then the ICC is given by:


ICC = x − Rsin θ( ), y+ Rcos θ( )( )

•  The pose of the robot is given by

•  This equation describes the motion of the a

rotating robot a distance R about its ICC with an angular velocity ω.


ʹxʹyʹθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

cos ωδt( ) −sin ωδt( ) 0

sin ωδt( ) cos ωδt( ) 0

0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

x − ICCx

y− ICCy

θ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+

ICCx

ICCy

ωδt

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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•  Integrating from some initial condition (x0, y0, θ0), it is possible to compute the position at any time t based on control parameters vr and vl.


•  In general, for a robot capable of moving in a particular direction θ(t) at a given speed v(t)


x t( ) = v t( )cos θ t( )( )0

t∫ dt

y t( ) = v t( )sin θ t( )( )0

t∫ dt

θ t( ) = ω t( )0

t∫ dt

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•  And for the special case of a differential robot we have:


x t( ) = 12

vr t( )+ vl t( )( )cos θ t( )( )0

t∫ dt

y t( ) = 12

vr t( )+ vl t( )( )sin θ t( )( )0

t∫ dt

θ t( ) = 1l

vr t( )− vl t( )( )0

t∫ dt

•  However, what is more interesting is: •  How to select the parameters to make the

robot to arrive at a desired position or to follow a given path.

•  That is know as inverse kinematics •  This is also related with path planning


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•  Inverse Kinematics •  The previous equations describe the

constraints on the velocity of the robot that cannot be integrated into a position constraint

•  This is know as non-holonomic constraint and is very difficult to solve.

•  There are solutions for limited classes of the control functions.


•  For example, if it is assumed vl(t) = vl, vr(t) =vr and vr≠ vl, we get:

where (x0, y0, θ0)= (0,0,0)


x t( ) = l2vr + vlvr − vl

sin tlvr − vl( )

⎛

⎝⎜

⎞

⎠⎟

y t( ) = − l2vr + vlvr − vl

cos tlvr − vl( )

⎛

⎝⎜

⎞

⎠⎟

θ t( ) = tlvr − vl( )

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•  Given a goal time t and a position (x, y), previous equations can be solved for vr and vl but does not provide for independent control of θ.

•  Rather to invert previous equations to solve parameters that lead to a specific robot pose, it is possible to consider two special cases.

where (x0, y0, θ0)= (0,0,0)


•  If vr = vl = v, then the robot’s motion simplifies to:

•  The robot moves in a straight line


ʹxʹyʹθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

x + vcos θ( )δty+ vsin θ( )δt

θ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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•  If we choose -vr = vl = v, then we get:

•  The robot rotates in place


ʹxʹyʹθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

xy

θ + 2vδt / l

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  In this way, it is possible to drive the robot to any pose (x, y, θ): – We can turn the robot in site until it drives

thought the pose (x, y) – The we drive in straight line the robot – Then we turn the robot to the desired position θ

•  These are not the only solution for inverse kinematics, other solutions are possible.

•  The robot rotates in place


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Bibliography

•  Siegwart R. and I. Nourbakhsh, “Introduction to Autonomous Mobile Robots”,MIT Press, 2004.

•  Dudek G. and M. Jenkin, “Computational Principles of Mobile Robotics”, Cambridge University Press, 2000.

Bibliography •  Ulrich Nehmzow, “Scientific Methods in Mobile

Robotics”, Springer, 2006. •  Sebastian Thrun, Wolfram Burgard, and Dieter

Fox, “Probabilistic Robotics”, MIT Press, 2005. •  Howie Choset, Kevin M. Lynch, Seth Hutchinson,

George Kantor, Wolfram Burgard, Lydia E. Kavraki, Sebastian Thrun,“Principles of Robot Motion: Theory, Algorithms, and Implementations”, MIT Press, 2005

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