KINEMATICS MODELS OF KINEMATICS MODELS OF MOBILE ROBOTS MOBILE ROBOTS

2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
KINEMATICS MODELS OF
KINEMATICS MODELS OF
MOBILE ROBOTS
MOBILE ROBOTS
Maria Isabel Ribeiro
Pedro Lima
mir@isr.ist.utl.pt
pal@isr.ist.utl.pt
Instituto Superior Técnico (IST)
Instituto de Sistemas e Robótica (ISR)
Av.Rovisco Pais, 1
1049-001 Lisboa
PORTUGAL
April.2002
All the rights reserved
MOBILE ROBOTICS course
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
References
• Gregory Dudek, Michael Jenkin, “Computational Principles of
Mobile Robotics”, Cambridge University Press, 2000 (Chapter 1).
• Carlos Canudas de Wit, Bruno Siciliano, Georges Bastin (eds),
“Theory of Robot Control”, Springer 1996.
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Kinematics for Mobile Robots
• What is a
kinematic
kinematic
model ?
• What is a
dynamic
dynamic
model ?
• Which is the difference between kinematics and
dynamics?
•
Locomotion
is the process of causing an autonomous
robot to move.
– In order to produce motion, forces must be applied to the
vehicle
•
Dynamics
– the study of motion in which these forces are
modeled
– Includes the energies and speeds associated with these
motions
•
Kinematics
– study of the mathematics of motion withouth
considering the forces that affect the motion.
– Deals with the geometric relationships that govern the system
– Deals with the relationship between control parameters and
the beahvior of a system in state space.
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Notation
• {X
m
,Y
m
} – moving frame
• {X
b
, Y
b
} – base frame
X
m
Y
m
P
X
b
Y
b
x
y
θ










θ
=
y
x
q
robot posture in base frame










θθ−
θθ
=θ
100
0cossin
0sincos
)(R
Rotation matrix expressing
the orientation of the base
frame with respect to the
moving frame
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Wheeled Mobile Robots
• Idealized rolling wheel
y axis
y axis
x axis
z motion
• If the wheel is free to rotate about its axis (x axis), the
robot exhibits preferencial rollong motion in one direction
(y axis) and a certain amount of lateral slip.
• For low velocities, rolling is a reasonable wheel model.
– This is the model that will be considered in the kinematics
models of WMR
Wheel parameters:
• r = wheel radius
• v = wheel linear velocity
• w = wheel angular velocity
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Differential Drive
• v
r
(t) – linear velocity of right wheel
• v
l
(t) – linear velocity of left wheel
• r – nominal radius of each wheel
• R – instantaneous curvature radius of the robot trajectory, relative
to the mid-point axis
L
R
ICC
x
y
θ
2
L
R
−
2
L
R
+
Curvature radius of trajectory
described by LEFT WHEEL
Curvature radius of trajectory
described by RIGHT WHEEL
2
L
R
)t(v
)t(w
r
+
=
2
L
R
)t(v
)t(w
l
−
=
L
)t(v)t(v
)t(w
lr
−
=
))t(v)t(v(
))t(v)t(v(
2
L
R
rl
rl
−
+
=
))t(v)t(v(
2
1
R)t(w)t(v
lr
+==
• 2 drive rolling wheels
)Rcos y,sinRx(ICC
θ+θ−=
control variables
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Differential Drive
•
Kinematic model in the robot frame


















−
=










θ
)t(w
)t(w
LrLr
00
2r2r
)t(
)t(v
)t(v
r
l
y
x

• w
r
(t) – angular velocity of right wheel
• w
l
(t) – angular velocity of left wheel
Useful for velocity control
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Differential Drive


















θ
θ
=










θ
)t(w
)t(v
10
0)t(sin
0)t(cos
)t(
)t(y
)t(x



)t()q(S)t(q
ξ
=

control
variables
))t(v)t(v(
2
1
R)t(w)t(v
lr
+==
( )
( )
σσ=θ
σσθσ=
σσθσ=
∫
∫
∫
d)(w)t(
d)(sin)(v)t(y
d)(cos)(v)t(x
t
0
t
0
t
0
L
)t(v)t(v
)t(w
lr
−
=
)t(w)t(
)t(sin)t(v)t(y
)t(cos)t(v)t(x
=θ
θ=
θ=



Kinematic model in the world frame
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Differential Drive
• Particular cases:
– v
l
(t)=v
r
(t)
•
Straight line trajectory
– v
l
(t)=-v
r
(t)
•
Circular path with ICC (instantaneous center of curvature) on
the mid-point between drive wheels
.cte)t( 0)t( 0)t(w
)t(v)t(v)t(v
lr
=θ⇒=θ⇒=
==

)t(v
L
2
)t(w
0)t(v
R
=
=
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Synchronous drive
• In a synchronous drive robot (synchro drive) each wheel is
capable of being driven and steered.
• Typical configurations
– Three steered wheels arranged as vertices of an equilateral
triangle often surmounted by a cylindrical platform
– All the wheels turn and drive in unison
• This leads to a holonomic behavior
•
Steered wheel
– The orientation of the rotation axis can be controlled
y axis
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Synchronous drive
• All the wheels turn in unison
• All of the three wheels point in the same direction and turn
at the same rate
– This is typically achieved through the use of a complex
collection of belts that physically link the wheels together
• The vehicle controls the direction in which the wheels point
and the rate at which they roll
• Because all the wheels remain parallel the synchro drive
always rotate about the center of the robot
• The synchro drive robot has the ability to control the
orientation θ of their pose diretly.
• Control variables (independent)
– v(t), w(t)
• The ICC is always at infinity
• Changing the orientation of the wheels
manipulates the direction of ICC
( )
( )
σσ=θ
σσθσ=
σσθσ=
∫
∫
∫
d)(w)t(
d)(sin)(v)t(y
d)(cos)(v)t(x
t
0
t
0
t
0
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Synchronous Drive
• Particular cases:
– v(t)=0, w(t)=w=cte. during a time interval
•
The robot rotates in place by an amount
– v(t)=v, w(t)=0 during a time interval
•
The robot moves in the direction its pointing a distance
t
∆
t w
∆
t
∆
t v
∆
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Tricycle
• Three wheels and odometers on the two rear wheels
• Steering and power are provided through the front wheel
• control variables:
– steering direction
α
(t)
– angular velocity of steering wheel w
s
(t)
ICC
ICC
The ICC must lie on
the line that passes
through, and is
perpendicular to, the
fixed rear wheels
R
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Tricycle
If the steering wheel is set to
an angle
α
(t) from the
straight-line direction, the
tricycle will rotate with
angular velocity w(t) about a
point lying a distance R along
the line perpendicular to and
passing through the rear
wheels.
d
x
y
θ
α
R
(
)
)t(
2
tg d)t(R
α−
π
=
22
s
)t(Rd
r )t(w
)t(w
+
=
X
b
Y
b
r = steering wheel radius
angular velocity of the moving frame
relative to the base frame
r )t(w)t(v
ss
=
linear velocity of steering wheel
)t(sin
d
)t(v
)t(w
s
α=
2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Tricycle
Kinematic model in the robot frame
Kinematic model in the world frame
)t(sin
d
)t(v
)t(
)t(sin)t(cos)t(v)t(y
)t(cos)t(cos)t(v)t(x
s
s
s
α=θ
θα=
θα=



)t(sin
d
)t(v
)t(
0)t(v
)t(cos )t(v)t(v
s
y
sx
α=θ
=
α=

with no splippage
)t(sin
d
)t(v
)t(w
)t(cos)t(v)t(v
s
s
α=
α=


















θ
θ
=










θ
)t(w
)t(v
10
0)t(sin
0)t(cos
)t(
)t(y
)t(x



2002 - © Pedro Lima, M. Isabel Ribeiro
Robótica Móvel
Kinematics Models
Omnidireccional
X
m
Y
f
θ
30º
1
2
3
L
Y
m
X
f


























−
−
=










θ
3
2
1
y
x
w
w
w
L3
r
L3
r
L3
r
r
3
1
r
3
1
r
3
2
r
3
1
r
3
1
0
V
V

Kinematic model in the robot frame
w
1
, w
2
, w
3
–
angular
velocities of the three
swedish wheels
Swedish wheel