Kinematic Models of Mobile Robots

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13 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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EE 5325/4315 – Kinematics of Mobile Robots, Summer 2004
Jose Mireles Jr.,
Selected notes from 'Robótica: Manipuladores y Robots móviles' Aníbal Ollero.

Kinematic Models of Mobile Robots

Assumptions:
a) The robot moves in a planar surface.
b) The guidance axis are perpendicular to floor.
c) Wheels rotate without any slippery problems.
d) The robot does not have flexible parts.
e) During small amounts of time, which direction is maintained constant, the
vehicle will move from one point to other following a circumference arc.
f) The robot is considered as a solid rigid body, and any movable parts are the
direction wheels, which are moved following a commanded control position.

Kinematic restrictions.
Consider an inverted pendulum as shown in figure 1. Its movement is restricted
by the following equation

x
2
+ y
2
– l
2
= 0 (1)











Figure 1. Simple pendulum

Similar restrictive equations are found in kinematic equations of mobile robots.










Figure 2. Speed of wheels.

Wheel movement (speed) in direction x is calculated by radius r and angle speed
rotation θ by.
x’ = r θ’ (2)
x
y l
l
θ'
r
x’











Figure 3. kinematic restrictions of wheels in a 2D plane.

This is, the speed in the x direction is directly proportional by the angular velocity of the
wheel. However, other restrictions appear in wheels when the movement is restricted to
a 2D plane (x,y). Assume the angular orientation of a wheel is defined by angle φ. Then,
while the wheel is following a path and having no slippery conditions, the velocity of
the wheel at a given time, which is set by rθ’, has the following restrictive velocity
components (x’,y’) with respect to coordinate axes X and Y.













Figure 4. Remark restrictions for 2D plane movement.

rθ’ = -x’ sin φ + y’ cos φ (3)
0 = x’ cos φ + y’ sin φ (4)

θ
x
y
z
φ
φ
rθ’
X
Y
φ
x’
y’













Figure 5. Circumference movement of the vehicle.


Consider now that the mobile robot (or vehicle) follows a circular trajectory as shown in
figure 5. Notice that the lineal and angular velocities of the vehicle are given by

v =
t
s


(5)
and


t∆

=
φ
ϖ
(6)

where
s∆
and
φ

are the arc distance traveled by the wheel, and its respective
orientation with respect to the global coordinates.
The arc distance
s∆
traveled in
t∆
time is obtained by:


φ
∆=∆ Rs
(7)
where
R
is the circumference radius of the wheel.

The
curvature
is defined as the inverse of the radius
R
as:


sR ∆

==
φ
γ
1
(8)

The movement equations in the initial position are given by the following two
expressions:


)1)cos(()( −∆=∆
φ
Rx (9)
)sin()(
φ
∆=∆ Ry (10)
An extension of the later equations is provided in the next expressions, considering an
specific initial orientation of angle
φ
⸠⁔桩猠楳⁡捣潭p汩獨敤⁢礠牯瑡瑩l朠瑨攠敡牬楥爠
楮楴楡氠捯潲ii湡瑥猠⠹⤠慮搠⠱〩⸠
=
=
φ
φ
φ
φ
獩s)獩渨捯c]1)捯猨[




=∆ RRx (11)
X
Y
y

x

s

R
φ


φ
φ
φ
φ
捯c)獩渨獩s]1)捯猨[

+


=∆ RRy (12)
Assuming now that the control interval is sufficiently small, then we can assume that
the orientation change would be small enough, and
1)cos( ≅∆
φ
(13)

φ
φ
∆≅∆ )sin( (14)

Substituting (13) and (14) into (11) and (12), we got

φ
φ
獩s∆−=∆ Rx (15)

φ
φ
捯c∆=∆ Ry (16)

Now, considering (7), we got

φ
獩ssx ∆−=∆ (17)

φ
捯csy ∆=∆ (18)

Dividing both sides of equations (17) and (18) by
t

, and considering also (5), if
t


tends to zero, we finally got

φ
獩s∆ vx −=
(17)

φ
捯c∆ vy =
(18)

Also, using (6) we can obtain the complimentary equation

ϖ
φ
='
(19)


Jacobian Model.
Assume that p represents a point in the space having n generalized coordinates,
and q a vector of m actuation variables (for n>m), and assume p’ and q’ are the
respective derivatives of such vectors, then the direct model is obtained by the Jacobian
matrix, J(p) by

')('qpJp =
(20)

This jacobian expression can be written in the form (Zhao and Bennet):

i
m
i
i
qpgpfp')()('
1

=
+=
(21)


ϖφ
φ










+











=
1
0
0
0
cos
sin
'vp
(22)
where
v
is the linear velocity of the vehicle, and
ϖ
=楳⁩瑳⁡湧畬慲⁶敬潣楴礮⁔桥獥i
敱畡瑩潮猠捡渠慬獯⁢攠捨慮来搠瑯 ⁴桥⁦潲m映敱畡瑩潮
㈰⤠慳㨠
=

















=










ϖ
φ
φ
φ
v
y
x
10
0cos
0sin
'
'
'
(23)

for q’ =[v
ϖ
]
T

Notice that combining the first two equations from (23), and eliminating v, we got back
the restricted relationship (4) by
x’ cos
φ
+ y’ sin
φ
= 0 (24)

This is due that the vehicle can only move along its longitudinal axis by

tg
φ
=
'
'
y
x

(25)
In other words, the vehicle position (x,y) and its orientation
φ
⁡牥潴⁩湤数敮摥湴⸠
=
周攠楮癥牳攠T潤敬映瑨攠獹獴敭⁩湶潬癥猠 瑨攠楮癥牳攠潦⁴桥⁊慣潢楡渮†th敮⁴桥e
䩡捯扩慮⁩γ潴⁡⁳煵慲攠= a瑲楸Ⱐ楴⁩猠湥捥獳a特⁴漠捡汣r污瑥⁩瑳⁰獥l摯楮癥牳攬⁢礠
mu汴楰汹楮朠扯瑨⁳楤敳⁢y= J
T
, and solving for q’, to obtain:

')(})()({'
1
ppJpJpJq
TT −
=
(26)

Then, for model (23) and using (20), we obtain



















=






'
'
'
100
0cossin
φ
φφ
ϖ
y
x
v
(27)
From the first relationship, we obtain the earlier restricted condition (3) by

v = -x’ sin
φ
+ y’ cos
φ

㈸⤠
=
=
Configurations of Mobile Robots.












Figure 6. Typical mobile robot configurations.


The two sketches shown in figure 6 show the differential and the classical three-
wheeled vehicles. The differential configuration use independent velocities in both
wheels left and right (v
L
, and v
R
, respectively) to move in the 2D plane to a specific
point (x,y) and specific orientation
φ
⸠周攠瑨牥攠.h敥汥搠癥ei捬攠畳敳⁡⁳楮g汥l
捯湴牯汬敤⁡湧汥⁡湤⁳灥敤⁷桥敬⁴漠浯癥c 瑯⁡⁤敳楲敤⁰潳楴楯渠慮搠潲楥湴慴楯渮†
=
t
ϖ
l
(x,y)
v
L

Classic Three-wheeled vehicle
α
=
φ
v
R

b
Differential
φ
⡸ⱹ⤠
v
d

b
α
R
Differential Configuration.
Assume for differential configuration model, that
L
ϖ
and
R
ϖ
are the corresponding
angular velocities of the left and right wheels. Given the radius of the wheels as r, the
corresponding linear and angular velocities of the vehicle are given by


r
vv
v
LRLR
22
ϖ
ϖ
+
=
+
=
(29)

r
bb
vv
LRLR
ϖ
ϖ
ϖ

=

=
(30)

where b is the bias of the vehicle (separation of the two central wheels). Also, if the
linear and angular velocities are provided, then the angular velocities of the wheels can
be obtained by


r
bv
L
ϖ
ϖ
)2/(

=
(31)

r
bv
R
ϖ
ϖ
)2/(
+
=
(32)

Substituting equations (29) and (30) into the model of mobile robots (22), we found


RL
br
r
r
br
r
r
y
x
ϖφ
φ
ϖφ
φ
φ











+












=










/
2/)cos(
2/)sin(
/
2/)cos(
2/)sin(
'
'
'
(33)



















−−
=










R
L
brbr
rr
rr
y
x
ϖ
ϖ
φφ
φφ
φ
//
2/)cos(2/)cos(
2/)sin(2/)sin(
'
'
'
(34)

Three-wheeled Configuration.
This configuration is the Romeo 3R configuration. For this vehicle the control angle for
direction is defined by angle
α
=⡯爠批⁩瑳⁡湧(污爠癥汯捩瑹=
α
ϖ
), and the angular
velocity of the wheel itself
t
ϖ
(or by its total velocity
tt
rv ϖ=
. Assume that the
guidance point of the vehicle is in the back part of the control wheel (central back axis).
For this configuration, the corresponding model is obtained by

v
=
v
t
cos
α
= r
t
ϖ
cos
α

(35)
and

'
α
=

α
ϖ
(36)

Also, the angular velocity orientation is given by

αα
ϖ
φ
sinsin'
l
v
l
r
tt
==
(37)

Substituting these equations into the model (22), we found the model by


α
ϖ
α
αφ
αφ
α
φ












+













=












1
0
0
0
0
/)(sin
coscos
cossin
'
'
'
'
t
v
l
y
x
=



















α
ϖ
α
αφ
αφ
t
v
l
10
0/)(sin
0coscos
0cossin
(38)

Notice that once known the desired lineal
v
and angular velocities
ϖ
潦⁴桥⁶敨楣汥
慳o
獨潷渠楮
㈷⤬⁴桥⁣潮瑲潬⁶慲楡扬敳s
α
慮搠
t
ϖ
can be obtained by


)arctan()arctan(
v
l
R
l
ϖ
α ==
(39)

r
lv
r
v
t
t
222
ϖ
ϖ
+
==
(40)