Track-stair and Vehicle-manipulator Interaction Analysis
for Tracked Mobile Manipulators Climbing Stairs
Yugang Liu and Guangjun Liu
∗
, Senior Member, IEEE
Abstract—This paper analyzes interactions between the
tracks and the stairs, as well as those between the tracked
mobile robot and the onboard manipulator for tracked mobile
manipulators (TMMs) climbing stairs. Combining a tracked
mobile robot, which has the ability to climb stairs, with an
onboard manipulator, a TMM extends the workspace and
scope of applications of the robot dramatically. However, this
combination gives rise to complex track-stair and vehicle-
manipulator interactions, because the configuration of the
onboard manipulator affects load distribution, which will
further influence the track-stair interactive forces. Unlike the
wheeled mobile robots, which are normally assumed to obey the
nonholonomic constraints, slippage is unavoidable for a tracked
mobile robot, especially when climbing stairs. The track-stair
interactive forces are complicated, which may take the forms
of grouser-tread hooking force, track-stair edge frictional force,
grouser-riser clutching force, and even their compositions. In
this paper, the track-stair and vehicle-manipulator interactions
are analyzed systematically, which are essential for tip-over
prediction and prevention, as well as for automatic control
of TMMs in autonomous and semi-autonomous stair-climbing.
Simulations for a TMM being developed in our laboratory have
demonstrated the usefulness of the presented analysis results.
I. INTRODUCTION
Tracked vehicles have attracted attentions from numerous
researchers in recent years since they provide better floatation
and traction than wheeled ones [1], and this characteristic
brings them substantial application potentials in explosive
ordnance disposing, searching, rescuing, mining, logging,
farming, earth moving and planetary exporting among others.
Traditionally, most tracked vehicles are designed to be bulky
and heavy to suit for field operations on natural terrain, and
they are normally insusceptible to tipping-over. Nowadays,
many lightweight tracked mobile robots are developed for
artificial environments, such as stairways [2]–[8].
In related research works, an autonomous stair-climbing
algorithm is developed for a linkage mechanism actuator
(LMA) tracked mobile robot, and the relevant technical
problems are identified and analyzed [2]. The stair-climbing
process is divided into three steps in terms of riser climbing,
riser crossing and nose-line climbing [3]. The tumbling
of a tracked mobile robot is attributed to the drift of
heading angles, and a feedback system is developed for
automatic heading control to stabilize stair climbing [5].
An autonomous stair climbing algorithm is presented by
introducing extended Kalman filters to estimate the rotational
This work is supported in part by the Canada Research Chair Program and
in part by Natural Sciences and Engineering Research Council of Canada.
The authors are with the Department of Aerospace Engineering, Ryerson
University, 350 Victoria St., Toronto, Ontario, Canada, M5B 2K3.
*The corresponding author (
[email protected])
velocity measurements provided by a 3-axial gyroscope and
the line parameters acquired from stair-edges’ projection on
a camera image [7].
A tracked mobile manipulator (TMM) integrates a tracked
mobile robot with an onboard manipulator. This combination
extends workspace and the scope of applications of the
robot dramatically. It is difficult to imagine how effective an
explosive ordnance disposal (EOD) robot or a rescue robot
would be without an onboard manipulator. Furthermore, the
terrain adaptability and the stair-climbing ability of tracked
mobile robots have expanded the workspace of the tradi-
tional wheeled mobile manipulators. However, the complex
track-stair and vehicle-manipulator interactions introduced
by this integration make autonomous or semi-autonomous
stair-climbing complicate and intractable tasks. The motion
of the onboard manipulator will lead to load transfer, which
makes the TMM vulnerable to tipping over, and the track-
stair slippage can give rise to failure of stair-climbing.
As well known, slippage is almost unavoidable for a
tracked mobile robot while climbing stairs. However, this
important issue has never been addressed in the literature.
While tracked vehicles and wheeled mobile manipulators [9]
have been extensively studied, few work has been reported
on TMMs [10]. In this paper, the track-stair and vehicle-
manipulator interactions for a TMM in stair-climbing are
analyzed systematically, which lays a solid foundation for
tip-over prediction and prevention, as well as automatic
control of TMMs in stair-climbing.
The rest of this paper is organized as follows: the stair-
climbing procedure is presented in the following section.
Slippage and track-stair interactive motions are analyzed
systematically in Section III. the track-stair interactive forces
are analyzed for a TMM climbing stairs with consideration
of vehicle-manipulator interactions in Section IV. To demon-
strate the applications of the proposed analysis methods,
simulations have been conducted on a TMM being developed
in our laboratory, and the simulation results are presented in
Section V. Finally, concluding remarks are given and future
researches are discussed in the last section.
II. GENERAL PROCEDURE FOR STAIR-CLIMBING
The proposed analysis methods are presented on the basis
of a TMM being developed in our laboratory, which is
composed of a tracked mobile robot and a n-rotary-joint
onboard manipulator, as shown in Fig. 1, and the results
can be easily extended to suit for other TMMs. The two
driving wheels can be controlled independently to realize
steering; the two planetary wheels are attached at the tip of
4th IEEE Conference on Automation Science and Engineering
Key Bridge Marriott, Washington DC, USA
August 23-26, 2008
978-1-4244-2023-0/08/$25.00 ©2008 IEEE. 157
(a) Flippers are put forward (b) Flippers are raised to climb onto the stairs
Fig. 1. The prototype for a tracked mobile manipulator
d
N
d
F d
R
s
N
m
m g
0
m g
1
m g
2
m g
3
m g
4
m g
5
m g
1
q
2
q
3
q
4
q
5
q
f
m g
(
)
p
p
L
θ
2 L 2 L r
m
h
0
h
b
h
s
θ
nose
grouser
0
l
1
l
2
l
3 l
4
l
5
l
nose line
s
F s
R
B
X
B
O
B Z
p m g
g
h
z
ext
F x
ext
F
g
l
(a) Interacting with the lower level
d
N d
F d
R
s θ
1
N
1
F
θ
p θ
m
O
S
P
D
p L
(b) Climbing onto the stair
d
N d
F
d
R
1
N
1
F
p
F
p
N
p
R
m X
m O
m Z
B
Z
B
X
B
O
θ p θ
D
S
P
(c) Setting back the flippers
p
N p
F p
R
s θ
1
N
1
F
S
P
D
m O
p
θ
2
N
2
F
(d) Climbing onto the stair again
p
N
p
F
p
R
1
N 1
F
2
F
2
N
D
S
P
(e) Starting to move along the nose line
1
N
1
F
2
F
2
N
p θ
p L
θ
m X
m O
m Z
P
S
D
(f) Landing on the upper floor
Fig. 2. The general procedure for a TMM climbing stairs
the flippers via spring loaded prismatic joints to retain tension
in each track; and the two flippers can be driven by the same
pitch motor to ensure synchronization of the two tracks. The
tracks are equipped with grousers, which are designed to
improve the stair-climbing ability of the TMM, as shown in
Fig. 1(b). For the convenience of presentation, the general
stair-climbing procedure for the TMM is detailed first.
At the very beginning of stair climbing, the pitch joint
raises the flippers and planetary wheels, and the TMM is
driven by tractive forces between the lower part of the tracks
and the terrain at the lower level to get close to the first
stair, as shown in Fig. 1(b), and also in Fig. 2(a). This step
belongs to planar motion and is not studied in this paper.
The stair-climbing procedure can be broken down into four
steps, namely “climbing onto the stairs”, “setting back the
flippers”, “going on the nose line” and “landing on the upper
floor”, as shown in Fig. 2.
Step 1: climbing onto the stairs. When the upper part of the
tracks begins to interact with the stairs, the supporting wheels
leave the ground, and the chassis, together with the onboard
manipulator, turns anti-clockwise, as shown in Figs. 2(a)-(b).
In this step, the TMM is driven by both the tractive force
and the track-stair interactive forces.
Step 2: setting back the flippers. To climb the stairs
effectively, the flippers and the planetary wheels are set back
to put the robot’s center of gravity (COG) forward, as shown
in Fig. 2(c); then the TMM climbs onto the stairs with the
flippers in the back, as shown in Figs. 2(c)-(e). There are both
tractive force and track-stair interactive forces in Step 2.
Step 3: going on the nose line. For nose-line climbing, the
pitch joint is locked at the position θ
p
= π, and the planetary
wheels work as supporting wheels, as shown in Fig. 2(e). The
TMM is driven by the track-stair interactive forces.
Step 4: landing on the upper floor. After the center of
the chassis passing over the edge of the last stair, the pitch
joint rotates anti-clockwise to a definite position, as shown
in Fig. 2(f). Then the driving wheels continue to drive the
TMM, until it plunges onto the upper level.
The motion after belongs to planar motion again, and the
robot is driven by the tractive forces between the tracks and
the terrain of the upper level, which is not studied here.
III. SLIPPAGE AND INTERACTIVE MOTION ANALYSIS
The interactive forces between the tracks and the stairs are
complicated, which can take the forms of hooking between
the grousers and the tread, as shown in Fig. 3(a), friction
between the track and the edge of the stairs, as shown
in Fig. 3(b)–(c), clutching between the grousers and the
riser, as shown in Fig. 3(d), and even their compositions. In
Figs. 3(b)–(c), the track works in the same way as pure track
without grousers, and the sufficient and necessary condition
for non-slipping can be given by −f
s
≤
F1
N1
≤ f
s
, where f
s
is the coefficient of static friction; F
1
and N
1
are track-stair
interactive forces, which will be detailed in Section IV.
From Fig. 3(a), the sufficient and necessary firm-hooking
condition, which ensures non-slipping between the grouser
and the tread, can be given by
−fs ≤
F11
N11
=
F1 cos θs −N1 sin θs
F1 sin θs +N1 cos θs
≤ fs (1)
where N
1
> 0 and F
1
> −N
1
· cot θ
s
.
Similarly, the sufficient and necessary condition for firm
clutching as shown in Fig. 3(d) can be given by
−fs ≤ −
F11
N11
=
F1 sin θs +N1 cos θs
N1 sin θs −F1 cos θs
≤ fs (2)
where N
1
> 0 and F
1
< N
1
· tan θ
s
.
If f
s
< cot θ
s
, solving (1) yields
sin θs −fs cos θs
cos θs +fs sin θs
≤
F1
N1
≤
sin θs +fs cos θs
cos θs −fs sin θs
(3)
158
1
N
11
F
11
N
s
θ
Track Grouser
Tread
Riser
T
r
e
n
d
1
F
(a) Hooking the tread: F1 > −N1 cot θs
1
F
1
N
11
F
11
N
s
θ
Track Grouser
Tread
Riser
T
r
e
n
d
(b) Leaving from the tread: F1 ≤ 0
1
F 1
N
11
F
11
N
s
θ
Track
Grouser
Tread
Riser
T
r
e
n
d
(c) Leaving from the riser: F1 ≥ 0
1
F
1
N
11
F
11
N
s
θ
Track
Grouser
Tread
Riser
T
re
n
d
(d) Clutching the riser: F1 < N1 tan θs
Fig. 3. Track-stair interactive force classification. F
1
and N
1
represent the
equivalent tractive and supporting forces; F
11
and N
11
denote actual forces
generated at the grouser-tread (riser) or track-stair edge contact points; the
hollow arrows show the moving trend of the track.
On the other hand, if f
s
≥ cot θ
s
, we can only obtain the
left half part of (3).
If f
s
< tan θ
s
, solving (2) yields
−
cos θs +fs sin θs
sin θs −fs cos θs
≤
F1
N1
≤
fs sin θs −cos θs
sin θs +fs cos θs
(4)
Furthermore, the left half part of (4) can be set free on
the condition that f
s
≥ tanθ
s
.
Remark 1: From the analysis above, we can see that the
grouser can always hook the tread of the stair firmly to
avoid sliding down on the condition that f
s
≥ cot θ
s
;
even if f
s
< cot θ
s
, the track-stair engagement has been
improved dramatically than the track without grousers since
sin θs+fs cos θs
cos θs−fs sin θs
>> f
s
. Similarly, the grouser can always
clutch the riser firmly to avoid sliding up on the condition
that f
s
≥ tan θ
s
; even if f
s
< tan θ
s
, the track-stair
engagement can also be improved over the track without
grousers due to −
cos θs+fs sin θs
sin θs−fs cos θs
<< −f
s
. This remark
explains how grousers can help improve stair-climbing.
In the first two steps, the track-stair interactive forces
can take any form as shown in Fig. 4, and the slippage
is negligible as long as the grouser can hook the tread
of the first stair firmly. However, the slippage is difficult
to determine, and the performance of the TMM cannot be
guaranteed when the track slides up from the riser or slides
down from the tread, which should be avoided or the stair
climbing may have to be terminated. On the basis of the
firm-hooking and firm-clutching condition analysis, we can
derive the slippage and track-stair interactive motion for the
first two steps, as illustrated in Fig. 4, in which d
g
represents
the distance between two adjacent grousers.
In Steps 3–4, the interactive forces can only be in the
form of Figs. 3(a), (c) or their compositions because
¸
n
i=1
F
i
keeps positive; and slippage is still almost unavoidable even
Y
N
N
Y
N
N
Y
N
Y
1
1
sin cos
?
cos sin
s s s
s s s
f F
N f
θ θ
θ θ
+
>
−
1
1
sin cos
?
cos sin
s s s
s s s
f F
N f
θ θ
θ θ
−
≥
+
1
1
?
s
F
f
N
≥ −
tan ?
s s
f θ ≥
1
1
cos sin
?
sin cos
s s s
s s s
f F
N f
θ θ
θ θ
+
≥ −
−
Start
The grouser cannot hook
the tread, and the track will
slide down from the tread
Y
N
The grousers can clutch the
riser firmly, and the slippage is
12 g
s d − ≃
The grouser cannot clutch the
riser, and the track will slide up
from the riser
Report error, send the
stop request, and return 1
Exit
Store the slippage
and return 0
N
1
1
sin cos
?
sin cos
s s s
s s s
f F
N f
θ θ
θ θ
−
≤
+
Y
Y
cot ?
s s
f θ ≥
The grouser can hook the
tread firmly, and there is no
slippage, i.e., 0 12
s =
The grouser can keep contacting with the tread with the
aid of the grouser-tread hooking force and the track-stair
frictional force; and the slippage is negligible, i.e., 0 12
s ≃
Fig. 4. Slippage and interactive motion analysis for the first two steps.
if the grouser can hook the tread firmly due to the alternation
of hooking points. If
¸¸
n
i=1
F
i
¸¸¸
n
i=1
N
i
¸
≤ f
s
, the
TMM can climb on the stairs without slippage, even without
grousers. If
¸¸
n
i=1
F
i
¸¸¸
n
i=1
N
i
¸
> f
s
and if the firm
hooking condition given by (3) can be ensured, slippage
may occur only at the beginning of Step 3. Otherwise, the
track will slide down from the treads, and the stair-climbing
process has to be terminated. In the following, we will
analyze the slippage of the track in Step 3 on the condition
that the grousers can hook the treads firmly. Let ⌈⋆⌉ denotes
the ceil function, which gives the smallest integer larger than
or equal to “⋆”; define k
rem
= ⌈
√
b
2
+h
2
dg
⌉ −
√
b
2
+h
2
dg
, the
slippage at the beginning of Step 3 can be determined by:
s3,0 =
0, s12 = 0
dg krem, s12 = 0 & krem = 0
dg, s12 = 0 & krem = 0
(5)
In view of the grousers interacting with the stairs in the
same way as that between gear teeth and racks, the slippage
of the track when a grouser releases from the k
th
stair s
3,k
can be derived as follows:
(a) If {k
rem
= 0}∪
¸
k
rem
=
1
2
¸
, there is no slippage when
a grouse releases from the tread of the k
th
stair, i.e., s
3,k
= 0;
(b) If 0 < k
rem
<
1
2
, when one grouser releases from the
k
th
stair, another grouser will hook the (k + 1)
th
one; and
the slippage in such a small duration is s
3,k,1
= d
g
k
rem
;
(c) If
1
2
< k
rem
< 1, when one grouser releases from
the k
th
stair, another grouser will hook the (k + 2)
th
one;
and the slippage in this small duration can be calculated as
s
3,k,2
=
2
√
b
2
+h
2
dg
d
g
− 2
√
b
2
+h
2
.
From the analysis above, we can obtain a uniform expres-
sion for the slippage of the TMM in Step 3 on the condition
that {s
12
= 0} ∪ {k
rem
= 0} as follows:
s3 (φ) =
0, krem = 0,
1
2
r (φ−φ
30
)−
√
b
2
+h
2
√
b
2
+h
2
dg
dg
¸
s
3,k,1
, 0 < krem <
1
2
r (φ−φ
30
)−
√
b
2
+h
2
2
√
b
2
+h
2
dg
dg
¸
s
3,k,2
,
1
2
< krem < 1
(6)
159
where φ
30
=
Lp(0)
r
−
L
2r
−
tan(
θs
2
)
cos θs
represents the rotating
angle of the driving wheels at the beginning of Step 3.
In the same way, the uniform slippage expression in Step 3
on the condition that {s
12
= 0} can be derived as follows:
s3 (φ) =
0, krem = 0,
1
2
r (φ−φ
30
)−s
12
−s
30
−2
√
b
2
+h
2
√
b
2
+h
2
dg
dg
¸
s
3,k,1
, 0 < krem <
1
2
r (φ−φ
30
)−s
12
−s
30
−2
√
b
2
+h
2
2
√
b
2
+h
2
dg
dg
¸
s
3,k,2
,
1
2
< krem < 1
(7)
Remark 2: In track-stair interaction analysis presented in
this section, the height of the grousers is assumed to be
negligible comparing to the grouser distance, the riser height
and the tread width; furthermore, the slippage caused by
hooking-point difference is not considered.
IV. TRACK-STAIR INTERACTIVE FORCE ANALYSIS
In this section, we analyze the track-stair interactive forces
for TMMs climbing stairs with consideration of vehicle-
manipulator interactions. To simplify the calculations, the
tracked mobile robot is assumed to be symmetrical, and the
wheel-track slippage is assumed to be negligible. Further-
more, the grousers are assumed to be rigid.
The coordinate system is defined as follows: an inertial
base frame O
B
-X
B
Y
B
Z
B
is fixed on the lower level, and a
frame O
m
-X
m
Y
m
Z
m
is attached to the tracked mobile robot,
as shown in Fig. 2(a),(c). In frame O
m
-X
m
Y
m
Z
m
, O
m
is
selected as the COG of the tracked mobile robot, and O
m
Z
m
is selected to be vertical with the tracked mobile robot. Then,
during the course of stair climbing, the motion of the tracked
mobile robot can be determined by the position O
m
(x
m
, z
m
),
and the turn angle θ. Furthermore, the axis O
B
Z
B
is selected
to be consistent with the initial position of O
m
Z
m
.
To derive the forces N
d
, N
1
and F
1
for Step 1, we need to
analyze the kinematics first. The rotating angles for the two
driving wheels should be the same in stair climbing to avoid
drift of heading angle. Let the rotating angle of the driving
wheels be φ, which is set as zero at the beginning of stair
climbing. With the assumption that the stretch of the tracks
is negligible, from Figs. 2(a)-(b), we can obtain
xm = r · φ −s (φ) −
L
2
· (1 −cos θ)
zm = r +hg +
L
2
· sin θ
θ = arcsin
¸
[r·φ−s(φ)] sin θs
L
¸
(8)
where r is the radius of the wheels; L is the distance between
the supporting wheels and the driving wheels, as shown in
Fig. 2(a); s (φ) is the slippage of the TMM, which will be
detailed in the next section; and 0 ≤ φ ≤
L+s12
r
for Step 1.
The trajectory of the planetary-wheel center P is an
ellipse, and the length of the flippers can be determined by:
Lp (θp) =
Lp (0) · Lp
π
2
L
2
p
π
2
· cos
2
θp +L
2
p
(0) · sin
2
θp
(9)
where θ
p
is the pitch joint angle; L
p
(0) and L
p
π
2
are the
lengths of the flippers at horizontal and vertical positions,
respectively, which are constants and can be determined by
the track length L
t
and the wheel radius r by L
p
(0) =
Lt−L−2π r
2
, L
p
π
2
=
√
(Lt−2π r−2 L)(Lt−2π r)
2
.
The tractive force and resistant force can be calculated by
F
d
= µx · N
d
· [1 −exp(−Ks)] , R
d
= fr · N
d (10)
where µ
x
and f
r
are the coefficient of adhesion and the
coefficient of external motion resistance, respectively; and
K
s
can be determined by a pull slip test [11].
According to the D’Alembert’s principle of inertial forces,
the TMM will become an equivalent static system if inertial
forces and torques are added to the corresponding COG.
Furthermore, a force and moment equilibrium analysis yields
N
d
=
M
I
+F
IX
·(h−zm)−F
IZ
·(xs+b−xm)
{µx[1−exp(−K)]−fr}·h−[xs+
L cos θ
2
+b−xm]
F1 = FIX · cos θs +FIZ · sin θs −N
d
· sin θs
−N
d
· {µx [1 −exp(−K)] −fr} · cos θs
N1 = FIZ · cos θs −FIX · sin θs −N
d
· cos θs
+N
d
· {µx [1 −exp(−K)] −fr} · sin θs
(11)
where x
s
=
L
2
+r tan
θs
2
; F
IX
, F
IY
, M
I
are detailed by
FIX =
mm¨ xm+
5
¸
i=0
{m
i
¨ x
ci
}
+2mp¨ xcp+2m
f
¨ x
cf
−F
x
ext
2
FIZ =
mm(¨ zm+g)+
5
¸
i=0
{m
i
(¨ z
ci
+g)}
−F
z
ext
2
+mp (¨ zcp +g) +m
f
(¨ z
cf
+g)
MI =
5
¸
i=0
I
i
+m
i [(x
ci
−xm)
2
+(z
ci
−zm)
2
]
2
¨
θ +
i
¸
k=0
¨ q
k
¸
−mpLp [¨ xcp sin (θ +θp) −(¨ zcp +g) cos (θ +θp)]
−m
f
L
f
[¨ x
cf
sin (θ +θp) −(¨ z
cf
+g) cos (θ +θp)]
+
Im
¨
θ+2(Ip+I
f
)(
¨
θ+
¨
θp)+F
x
ext
·(ze−zm)+F
z
ext
·(xm−xe)
2
−
5
¸
i=0
m
i
[¨ x
ci
(z
ci
−zm)−(¨ z
ci
+g)(x
ci
−xm)]
2
¸
(12)
where m
m
denotes mass of the mobile platform; m
i
denotes
mass of the i
th
link; m
f
and m
p
denote masses of the flippers
and the planetary wheels; I
i
I
m
, I
f
and I
p
are corresponding
moments of inertia; L
f
is the distance between O
m
and the
COG of the flippers; F
x
ext
and F
z
ext
are external forces added
to the end-effector along O
B
X
B
and O
B
Z
B
; the items with
double dots represent the corresponding accelerations.
Assuming that the change of turn angle for the chassis is
negligible and the stretch of the tracks can be neglected, from
Figs. 2(c)–(e) we can obtain the kinematics of the tracked
vehicle in Steps 2–3 as follows:
xm =
L(1+cos θ)
2
+ [r · φ −L −s (φ)] cos θ
zm = r +
Lsin θ
2
+ [r · φ −L −s (φ)] sin θ +hg
θ = θs
(13)
According to the number of track-stair interactive points,
Step 2 is divided into one-point and two-point sub-processes.
For the one-point sub-process, N
p
, N
1
and F
1
can be
calculated in the same way as that in Step 1. Assuming
that the differences of supporting forces at all the points are
consistent, from force and moment equilibrium analysis, N
p
,
N
1
, N
2
and F
12
= F
1
+F
2
for the two-point sub-process of
160
Step 2 can be calculated, as follows:
N1 =
[MI −r (FIX cos θs +FIZ sin θs)] · (A −2)
+(FIZ cos θs −FIX sin θs) [d2 −(A −1) B]
¸
(A·B−d
1
)·(A−2)+(A+1)·[d
2
−(A−1)·B]
N2 =
¸
(A+ 1) N1 −FIZ cos θs +FIX sin θs
¸
(A−2)
Np =
¸
A· N1 −(A −1) · N2
¸
{cos θs −kµ · sin θs}
F12 = FIX cos θs +FIZ sin θs −Np (sin θs +kµ cos θs)
(14)
where k
µ
= µ
x
1 −e
−K
−f
r
; d
1
,d
2
can be calculated by
substituting k = 1 into (20); and d
p
,A,B can be detailed by
dp =
Lcos θs−
Lcos θp+
√
4L
2
p
−L
2
sin
2
θp
cos(θp+θs)
2
A =
1
√
b
2
+h
2
¸
L
2
−rφ
cos θs +dp
¸
cos θs + 2
B =
[µx·(1−e
−K
)−fr]·(zm−r·cos θs)−dp−r·sin θs
cos θs−[µx(1−e
−K
)−fr] sin θs
(15)
From Fig. 2(e), we can obtain the geometry constraints for
successful alternation of track-stair contact points as follows
2b
cos θs
−r · tanθs ≥ Lp (0) +
L
2
<
3b
cos θs
−r · tanθs (16)
In Fig. 2(e), letting the summation of forces along O
m
X
m
and O
m
Z
m
be zeros yields
n
¸
i=1
Fi =
1
2
· {FIX · cos θs +FIZ · sin θs}
n
¸
i=1
Ni =
1
2
· {FIZ · cos θs −FIX · sin θs}
(17)
where n = 2 or n = 3 are corresponding to the cases with
two or three interactive points, respectively.
From the force and moment equilibrium analysis with two
track-stair interactive points, N
1
, N
2
can be calculated as
N1 =
1
d
2
−d
1
·
d2 ·
2
¸
i=1
Ni
−r ·
2
¸
i=1
Fi
+MI
N2 =
1
d
1
−d
2
·
d1 ·
2
¸
i=1
Ni
−r ·
2
¸
i=1
Fi
+MI
(18)
For the case with three track-stair interactive points, with
the assumption that the difference of supporting forces at the
three interactive points are consistent, N
1
, N
2
and N
3
can be
calculated from force and moments equilibrium as follows:
N1 =
2
3
3
¸
i=1
Ni
−N3, N2 =
1
3
3
¸
i=1
Ni
N3 =
1
d
3
−d
1
r
3
¸
i=1
Fi
−MI −
d
2
+2d
1
3
3
¸
i=1
Ni
(19)
where
d1 =
{xm −(xs +k · b)}
2
+ (zm −k · h)
2
−r
2
d2 =
{xm −[xs + (k + 1) b]}
2
+ [zm −(k + 1) h]
2
−r
2
·sgn{xm −[xs + (k + 1) · b] −r · sin θs}
d3 =
{xm −[xs + (k + 2) b]}
2
+ [zm −(k + 2) h]
2
−r
2
(20)
The landing process consists of two sub-processes: as the
first subprocess, the pitch joint rotates anti-clockwise for
π
9
to raise the driving wheels, as shown in Fig. 2(f); in the
second subprocess, the driving wheels continue to drive the
TMM until it plunges onto the upper floor.
From Fig. 2(f), the kinematics of the TMM in the first
TABLE I
DESIGN PARAMETERS FOR THE TMM
Parameter Value Masses Value Length Value
L (m) 0.51 m
m
(kg) 42 l
0
(m) 0.10
L
t
(m) 2.05 m
0
(kg) 4 l
1
(m) 0.20
h
0
(m) 0.17 m
1
(kg) 4 l
2
(m) 0.46
h
g
(m) 0.01 m
2
(kg) 4 l
3
(m) 0.46
d
g
(m) 0.07 m
3
(kg) 4 l
4
(m) 0.30
m
p
(kg) 1.30 m
4
(kg) 3 l
5
(m) 0.10
m
f
(kg) 1.10 m
5
(kg) 3 l
g
(m) 0.20
subprocess can be derived as follows:
xm = xs +ns b −Lp (0) cos θs −Lp (θp) cos (θp +θ)
zm = ns h +
r
cos θs
−Lp (0) sin θs −Lp (θp) sin (θp +θ)
θ = θs −arctan 2
¸
Lp (θp) sin θp, Lp (θp) cos θp −
L
2
¸
(21)
The kinematics of the TMM in the second subprocess can
be described as follows
xm = x41 +r · (φ −φ41) · cos θs
zm = z41 +r · (φ −φ41) · sin θs
θ = θ41
(22)
where x
41
,z
41
,θ
41
can be calculated by substituting θ
p
=
10π
9
into (21); φ
41
=
ns b
r cos θs
+
L
2r
+
tan
θs
2
cos θs
+
Σs3
r
.
The supporting forces N
1
and N
2
can be calculated as
that in Step 3, but replacing r with r +
L
2
sin (θ
s
−θ), and
k with n
s
− 1.
V. SIMULATION RESULTS
In this section, we present the simulation results using a
TMM being developed in our laboratory, which is composed
of a reconfigurable tracked mobile robot and a 5-DOF on-
board manipulator, as shown in Fig. 1. The design parameters
for the TMM under consideration are listed in Table I.
To demonstrate the application of the proposed analysis
methods, simulations are conducted for three different cases:
in Case 1, the stair parameters are selected as h = 0.23 m,
θ
s
= 50
◦
, f
r
= 0.06, µ
x
= 0.8, K
s
= 13, f
s
= 0.9,
n
s
= 12 and the onboard manipulator is locked at q =
[0 −
π
4
0 0 0]
T
; the stair parameters and configuration of the
onboard manipulator in Case 2 are given by h = 0.16 m,
θ
s
= 30
◦
, f
r
= 0.03, µ
x
= 0.3, K
s
= 11, f
s
= 0.6, n
s
=
12, q = [0 −
π
6
0 0 0]
T
; the same stair parameters as that in
Case 2 are adopted in Case 3, but replacing the configuration
of the onboard manipulator with q = [0 −
π
3
0 0 0]
T
.
The simulation results are presented in Fig. 5. According
to the relationship between the height and the incline angle of
the stairs, regular stairs are distinguished from the irregular
ones, as shown in Fig. 5(a). Fig. 5(b) gives the slippage
occurred in the first two cases, from which we can see
that the slippage may take quite different forms. The total
slippage is 0.127 m for Case 1 and 0.33 m for Case 2,
which cannot be neglected for a TMM in stair-climbing.
Figs. 5(c)–(d) present the load distribution for Case 1. The
load distribution for Case 2 can be found in Figs. 5(e)–(f).
161
20 25 30 35 40 45 50
0.05
0.1
0.15
0.2
0.25
0.3
The incline angle θs(◦)
H
e
i
g
h
t
o
f
t
h
e
s
t
a
i
r
s
(
m
)
Irregular stairs: too large
Regular stairs
Irregular stairs: too small
(a) Regular & irregular stairs
0 10 20 30 40 50
−0.1
0
0.1
0.2
0.3
0.4
Time, s
S
l
i
p
p
a
g
e
f
o
r
C
a
s
e
s
1
–
2
,
m
Case 1
Case 2
(b) Slippage for the first two cases
0 2 4 6 8 10 12
0
50
100
150
200
250
300
350
1
Time, s
S
u
p
p
o
r
t
i
n
g
f
o
r
c
e
s
f
o
r
S
t
e
p
s
1
–
2
,
N
2 3 4
N1
Nd
N1
Nd
Np
N1
N1
N2
Np
(c) Forces for steps 1–2: Case 1
20 30 40 50
−50
0
50
100
150
200
Time, s
S
u
p
p
o
r
t
i
n
g
f
o
r
c
e
s
f
o
r
S
t
e
p
s
3
–
4
,
N
N1
N2
N3
(d) Forces for steps 3–4: Case 1
0 2 4 6 8 10
0
50
100
150
200
250
300
Time, s
S
u
p
p
o
r
t
i
n
g
f
o
r
c
e
s
f
o
r
S
t
e
p
s
1
–
2
,
N
1
N1
2 3 4
N1
N1
N2
Nd
Nd
Np
Np
N1
(e) Forces for steps 1–2: Case 2
20 30 40 50
0
50
100
150
200
250
300
Time, s
S
u
p
p
o
r
t
i
n
g
f
o
r
c
e
s
f
o
r
S
t
e
p
s
3
–
4
,
N
N1
N2
N3
(f) Forces for steps 3–4: Case 2
0 2 4 6 8 10
−100
0
100
200
300
400
Time, s
S
u
p
p
o
r
t
i
n
g
f
o
r
c
e
s
f
o
r
S
t
e
p
s
1
–
2
,
N
1
4 3 2
Nd
N2
Nd
Np
Np
N1
N1
N1 N1
(g) Forces for steps 1–2: Case 3
20 30 40 50
−50
0
50
100
150
200
250
300
Time, s
S
u
p
p
o
r
t
i
n
g
f
o
r
c
e
s
f
o
r
S
t
e
p
s
3
–
4
,
N
N1
N2
N3
(h) Forces for steps 3–4: Case 3
Fig. 5. Simulation results
Time-varying supporting forces for Case 3 are illustrated by
Figs. 5(g)–(h). In Fig. 5(d), the minimum supporting force
N
3
has become negative, and the TMM may have already
tipped over from the stairs. Similarly, the minimum support-
ing force in Figs. 5(g) and 5(h) are negative, which means
that the TMM will fall up into the tread of the stairs, and the
stair-climbing process has to be terminated. Comparing the
load distributions in Case 2 with those in Case 3, we can see
that the configuration of the onboard manipulator is vital to
the load distribution, which will further affect the track-stair
interactive forces. With an optimal configuration, the TMM
can climb onto the stairs successfully; on the other hand, tip-
over may occur if the onboard manipulator is not configured
properly. From these figures, we can see that the track-stair
and vehicle-manipulator interactions can not be neglected for
autonomous and semi-autonomous stair-climbing.
VI. CONCLUSIONS AND FUTURE WORKS
In this paper, track-stair and vehicle-manipulator inter-
actions are analyzed systematically. Firm-hooking, firm-
clutching and non-slipping conditions are presented for a
TMM climbing stairs. The stair-climbing process is divided
into four steps. The slippage and track-stair interactive mo-
tions are analyzed for each step. Furthermore, track-stair
interactive forces are derived with consideration of vehicle-
manipulator interactions. To demonstrate the applications
of the proposed analysis methods, simulations have been
conducted on a TMM being developed in our laboratory.
The proposed analysis methods are presented on the basis
of a specific TMM, but the results are general and can be
easily extended to suit for some other TMMs.
As the first attempt to analyze track-stair and vehicle-
manipulator interactions for TMMs in stair-climbing, this
work lays a solid foundation for further investigations on
this important topic. Our next step is to develop an efficient
and reliable tip-over prediction algorithm on the basis of the
proposed track-stair interactive force analysis method, so as
to prevent the TMM from tipping-over in stair-climbing. Fur-
thermore, the determination of optimal onboard-manipulator
configuration and autonomous tip-over stability recovery for
TMMs in stair-climbing form another two promising topics
for future research and development.
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