Assessment of a tracked vehicle’s ability to traverse stairs
Endo and Nagatani Robomech J
Assessment of a tracked vehicle's ability to traverse stairs
Daisuke Endo 0
0 Department of Aerospace Engineering, Graduate School of Engineering, Tohoku University , Aramaki-aza Aoba 468-1, Aoba-ku, Sendai 980-0845 , Japan
In some surveillance missions in the aftermath of disasters, the use of a teleoperated tracked vehicle contributes to the safety of rescue crews. However, because of its insufficient traversal capability, the vehicle can become trapped upon encountering rough terrain. This may lead to mission failure and, in the worst case, loss of the vehicle. To improve the success rate of such missions, it is very important to assess the traversability of a tracked vehicle on rough terrains based on objective indicators. From this viewpoint, we first derived physical conditions that must be satisfied in the case of traversal on stairs, based on a simple mechanical model of a tracked vehicle. We then proposed a traversability assessment method for tracked vehicles on stairs. In other words, we established a method to evaluate whether or not a tracked vehicle can traverse the target stairs. To validate the method, we conducted experiments with an actual tracked vehicle on our simulated stairs, and we observed some divergences between our calculation and the experimental result. Therefore, we analyzed possible factors causing these divergences, estimated the influence of the factors quantitatively by conducting additional experiments, and identified the reasons for the deviation. In this paper, we report the above-described assessment method, the experiments, and the analyses.
Tracked vehicle; Stairs; Stability; Traversing ability assessment
Teleoperated small-sized tracked vehicles have two
advantages compared to other vehicles: high
traversability on rough terrain and a simple mechanism. Therefore,
they are ideal for surveillance tasks to replace rescue
crews in exploring hazardous environments in search
and rescue missions. Well-known examples of tracked
vehicles for practical use missions include Quince [1, 2]
and Survey Runner . These robots explored the
buildings affected by the meltdown of the Fukushima Daiichi
nuclear power plant. These robots provided significant
information during surveillance missions, particularly
related to damage inspection of plants and acquisition of
dose distribution. However, in their last missions, both
tracked vehicles got stuck in rough terrain and could not
return. To prevent such situations, various approaches
have been proposed to improve the usability of tracked
vehicles; these include semi-autonomous control of
subtracks [4–6] and consideration of robot stability in path
planning . However, no fundamental study has been
conducted on assessing the ability of a tracked vehicle
to traverse rough terrains from the point of view of the
interaction between the tracks and the ground, directly.
To improve the success rate of such surveillance missions,
a prior assessment is crucial. For example, in the case of
surveillance missions in buildings, tracked vehicles are required
to traverse stairs for moving to another floor. However,
stair-climbing and stair-descending are obstacles that cause
various problems in operation, and an assessment of
traversability on stairs is important. Figure 1 shows the motion
flow for a tracked vehicle traversing stairs for the
stairclimbing case. The motion flow is divided into three steps:
Entering step the motion state from contacting
the first step of the stairs to finishing its traversal
Traversing step the intermediate state between (i)
and (iii) (Fig. 1(ii)), in which the pitch angle of the
robot matches the inclination angle of the stairs.
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Fig. 1 Motion flow of a tracked vehicle traversing stairs
(iii) Landing step the motion state from contacting
the final step of the stairs to finishing its traversal
In previous research, Guo et al. stated that, to ensure
traversability of a tracked vehicle on stairs, initially, the
first step should be clear . However, they did not
consider the situation after the second step of the
stairstraversal. Jingguo et al.  conducted and examined a
derivation of the physical condition to traverse stairs
stably. However, they did not sufficiently validate its
performance. Liu et al. proposed an online prediction system
for monitoring the physical stability of a tracked vehicle
, and Martens et al. proposed a practical control law
to improve stability . However, no previous study
has verified a physical model to predict the stability of a
tracked vehicle on stairs.
In this light, we develop a method for assessing whether
or not a tracked vehicle can traverse stairs. Among the
three steps [(i)–(iii)] described above, a physical condition
of the entering step has been proposed and evaluated by
Guo et al. . However, after the robot completes the
traversing step (ii), the landing step (iii) has not been
considered from the point of view of traversability. Therefore, in
this research, we focus on the traversing step (ii), and we
propose an assessment method to evaluate, in advance,
whether or not a tracked vehicle can traverse stairs.
Failure modes of tracked vehicles traversing stairs
First, to model a stair traversal of a tracked vehicle, we
classified failure modes of a tracked vehicle that may
occur during its traversal on stairs, and we derived
physical conditions that cause each failure mode.
Figure 2 shows a model of a tracked vehicle that has
mass M in a two-dimensional plane as it traverses a
flight of stairs. The angle of the stairs is θs, and the
distance between the leading edges of adjacent steps (pitch
Fig. 2 Physical model of a tracked vehicle traversing stairs
between the edges of the stairs) is p. The length of the
flat area of the track is L, and it has n + 1 (where n is
an integer greater than 2) contact points with the stairs.
The contact points are defined as t0, t1, . . . , tn from the
bottom to the top. At each contact point, there are a
tractive force fkL and a vertical force fkN. Here, k is an
arbitrary integer between 0 and n. In addition, the robot
is subject to an upward acceleration a along the stairs.
Figure 3 shows the state transition of the tracked
vehicle while it traverses target stairs. Transitions occur when
the number of contact points changes. In Fig. 3, we assume
np ≤ L < (n + 1)p. The tracked vehicle traverses the stairs
by transitioning through states A−→B−→C−→D−→A.
When the track is in contact with the edge of a step, it
is defined as state A. After that, it transitions to state B.
When the track detaches from the edge of the step, it is
defined as state C. After that, it transitions to state D.
To traverse the stairs successfully, a tracked vehicle
should avoid the following three failure modes:
1 Slipping the case in which the tractive force FkL is
insufficient, and the track slides down on the stairs,
as shown in Fig. 4.
2 Falling backward the case in which the robot’s body
tips over around the center t0, the direction of
rotation being counterclockwise, as shown in Fig. 5.
3 Falling forward the case in which the robot’s body
tips over around the center tn, the direction of
rotation being clockwise, as shown in Fig. 6.
The last mode (falling forward) occurs very rarely. It only
occurs when the robot’s centroid is located well forward
in its body, and the robot is acts on by a large downward
acceleration. In addition, this failure mode tends to cause
less damage to the robot and its surrounding
environment than the falling-backward mode.
This paper addresses and describes the physical
conditions required to prevent the occurrence of these failure
Fig. 3 State transitions of a tracked vehicle traversing stairs
Fig. 4 Slipping mode
When the traction force generated at the contact
points between the tracks and the stairs does not
exceed the static friction force, the tracked vehicle
traverses the stairs without slipping. The condition is
where fkL and fkN are the tractive force and normal
reaction force, respectively, at contact point tk (where
k is an arbitrary integer between 0 and n) and μS is
the coefficient of friction between the tracks and the
stairs. In this paper, it is assumed that μS has the same
value at each contact point.
The equilibria of the lateral and longitudinal forces that
describe the condition at which the robot is prevented
from slipping can be expressed as
Fig. 5 Falling backward mode
Fig. 6 Falling forward mode
where M is the mass of the tracked vehicle, a is the lateral
acceleration of the tracked vehicle, g is the gravitational
acceleration, and θs is the angle of inclination of the stairs.
Substituting Eqs. (2) and (3) into Eq. (1) and rearranging
to obtain acceleration a gives
Ma + Mg sin θs <μsMg cos θs.
∴ a <(μs cos θs − sin θs)g .
For a tracked vehicle to successfully traverse stairs, the
summation of its angular moments must be zero. When a
tracked vehicle has n + 1 contact points without any
rotation, as shown in Fig. 2, the balance of its moment around
point t0 can be described as follows:
where α is the distance between the centroid C and the
contact point t0, and θα is the angle formed by tn and t0 and
C (∠tnt0C). When the robot rotates around the point t0,
there are no other contact points. Therefore, in this case,
fkN is equal to 0 (where k is an arbitrary integer between
1 and n) in Eq. (5), and the left-hand side of this formula
becomes greater than 0. Thus, we can derive
∴ a >
The complementary condition to that described by Eq.
(6) is required to prevent the robot from falling backward.
Therefore, we derive
where α is the distance between centroid C and contact
point t0; β is the distance between centroid C and contact
point tn; d is the robot’s progress, which is equal to 0 at
state A in Fig. 3; CL is the distance between the tip in the
flat area of the track and the centroid C in the front–back
direction; and CN is the distance between the bottom of
the track and the centroid C in the up–down direction.
As well as the falling-backward mode, when a tracked
vehicle has n + 1 contact points without any rotation, as
shown in Fig. 2, the balance of its moments around point
tn can be described as follows:
In the same way as the falling backward, the requirement
to prevent the robot from falling forward is derived from
a ≥ −
Traversing ability assessment method
Figure 7 shows a schematic diagram of the d–a plane.
The vertical axis indicates the acceleration a of the robot,
and the horizontal axis indicates the distance d that the
robot progresses. The diagram includes (a) the state
transition as a tracked vehicle traverses a flight of stairs
and (b) the requirements for acceleration to prevent the
robot from entering the failure modes described above.
Furthermore, aS, ar0 and arn are the marginal
accelerations required to prevent the vehicle from slipping,
falling backward, and falling forward, respectively. They are
arn = −
These equations imply that slipping occurs when the
acceleration of the robot is greater than aS, falling
Fig. 7 Stability profile of a tracked vehicle traversing stairs
backward occurs when the acceleration of the robot is
greater than ar0, and falling forward occurs when the
acceleration of the robot is less than arn. Here, amax is
defined as the lesser of the value of aS and the minimum
value attained by ar0, and amin is the maximum value
attained by arn, as expressed, respectively, by
amax = min(aS , min(ar0)),
amin = max(arn).
The stability can be assessed based on the schematic
diagram shown in Fig. 7. The d–a plane is divided into the
following five areas, and the robot’s stability can be
categorized into one of the following states:
A. Slipping area When the acceleration a does not
satisfy Eq. (4), the robot’s body slips along the flight of
B. Falling-backward area When the acceleration a
satisfies Eq. (4) but does not satisfy Eq. (7), the robot’s
body rotates around the contact point t0.
C. Falling-forward area When the acceleration a does
not satisfy Eq. (12), the robot’s body rotates around
the contact point tn.
D. Stable area When the acceleration of the robot
satisfies the condition amin < a < amax, it can traverse
stairs without any slipping or rotation.
E. Semi-unstable area This area cannot be categorized
as any of (A)–(D) above; however, it satisfies Eqs. (7)
and (12). The robot can traverse the stairs.
The robot needs to be controlled to maintain its
acceleration a in the semi-unstable area, according to the robot’s
position on the stairs. If the robot cannot control its
acceleration accurately according to its position on the
stairs, this area should be considered unstable. In this
case, the robot should be controlled in the stable area
(D). Additionally, note that the robot slips prior to
rotation if the acceleration a does not satisfy Eqs. (4) and (7).
In other words, the robot can traverse the stairs
without any slipping or rotation if its acceleration is
controlled in the stable area (D).
We conducted a case study with a tracked vehicle, called
“Kenaf ” [12, 13], that traverses stairs using the
traversability assessment method described in the previous
section. In this case study, we assumed that the robot’s
speed was constant (acceleration a = 0). There were two
reasons for this assumption. One is because it is
difficult for Kenaf to maintain its acceleration when moving
on stairs owing to the power restriction of its actuators.
The other is that the increase and decrease in
acceleration are equivalent to that of the inclination of stairs
based on the physical model shown in Fig. 2. In addition,
Kenaf originally has two main tracks and four sub-tracks.
However, to improve the accuracy of the verification tests
described in the next section, the robot’s mechanical
system should be simple. Therefore, we used Kenaf without
all of its sub-tracks; only the two main tracks were used
in this case study (Fig. 8). The physical parameters of the
robot, as used for the calculation, are listed in Table 1.
Kenaf has multiple convex-shaped grousers made of
chloroprene rubber on the surface of the tracks (Fig. 9).
Therefore, the friction between the tracks and the ground
is sufficient to prevent slipping, and falling backward
occurs prior to slipping on stairs with large inclinations
in the preliminary experiments. In other words, in case
of aS > min(ar0), slipping does not occur. Furthermore,
falling forward did not occur at constant speed.
Consequently, we only consider the falling-backward
phenomenon for assessing problems related to the shape of the
stairs for Kenaf.
The shape of the stairs can be described by two
parameters: the angle of inclination of the stairs, θs, and the
pitch between the edges of the stairs, p. The condition
for whether falling backward occurs (described as
“margin” in this paper) in the θs–p plane can be derived by
substituting a = 0 in Eqs. (7)–(10). This margin can be
described by the following set of equations:
Table 1 Robot specifications
dC = L − np.
dC + (n − 1)p − CL
These new symbols in Eqs. (19)–(23) can also be
denoted like as Fig. 10. Figure 11 shows the predicted
margin obtained by substituting Kenaf ’s parameters
(Table 1) into Eqs. (19)–(23). This means that,
theoretically, the robot falls backward in the area on the right, but
not in the area on the left in the plane bounded by the
To verify the traversability assessment method, we
conducted verification tests to compare with the predicted
margin obtained in the previous section.
We fabricated the simulated stairs shown in Fig. 12. The
setup allowed us to change the inclination θs of the stairs
to any value between 0° and 70°, and the pitch between
Fig. 8 The target tracked vehicle used in our verification test
Fig. 9 Shape of the main track
Fig. 10 Definition of symbols
the edges of the stairs, p, could be changed to any value
up to 2400 mm. The inclination and the pitch between
the edges of the stairs were adjusted by changing the
positions of fixing nuts.
We conducted traversal tests under different (θs, p)
conditions. For each trial, the tracked vehicle was placed on
the simulated stairs and operated to climb up vertically
to the end of the stairs at a constant speed, 100 mm/s
(acceleration a = 0). This value of speed is the upper limit
for the Kenaf to keep on steep stairs. We then observed
its behavior and judged whether or not falling backward
occurred. At the tip of the robot, a safety tether was
attached to prevent the robot from falling and crashing.
The pitch between the edges of the stairs, p, was changed
to four different values—150, 180, 200 and 220 mm—and
the inclination of the stairs θs was changed discretely. At
Fig. 11 Predicted margin of Kenaf
each pitch between the edges of the stairs, we evaluated
the marginal inclination θssup above which falling
backward occurred. We performed five trials under the same
Figure 13 shows the results of the above tests. In the
graph, the symbols have the following meaning:
: The robot did not fall at any time.
×: The robot fell down all five times.
: The robot fell backward sometimes. The index
represents the frequency of falling backward.
Fig. 12 Changeable stairs simulation
The results of the tests described in the previous
section are in good agreement with the predicted values in
those areas where the inclination θs is relatively small.
These results indicate that the falling-backward
phenomenon predominantly depends on the relation between the
robot’s centroid and the shape of the stairs, as described
in Eqs. (19)–(23). This fact is the good evidence of
necessity to consider the centroid position exactly, in case not
only for stability assessment but also stage of design for
the tracked vehicles. However, in those areas where θs
is large, the predicted values diverged from the
measured values. The largest difference was observed when
p = 150 mm: the robot fell down at 55.0°–57.0° in the
experiment; however, the predicted margin was 60.2°.
Figure 14 shows the behavior of the robot when the
robot traverses a flight of stairs for which θs = 44.0° and
p = 200 mm. The falling-backward phenomenon did not
occur in this case; however, the robot’s body started to
exhibit a swinging motion, as shown in Fig. 14(2). This
situation occurred when the edge of the track detached
from the contact point. Moreover, the robot satisfies Eq.
(7) at this instant, because the projecting point of the
robot’s centroid is located within the polygon formed
from the contact points without a detaching point .
Therefore, the other phenomenon must have occurred
exactly when the edge of the track detached. In the next
section, we discuss the reasons for this divergence.
Deformation of the track
Generally, a tracked vehicle, including Kenaf, has multiple
grousers across the surface of its tracks to increase the
friction between the track’s surface and the ground, thereby
preventing the track from slipping. Typically, for small
tracked vehicles, the tracks are made of a nonrigid material.
Fig. 13 Experimental results
Therefore, bending deformation of the track occurs at the
point where the grouser comes into contact with the edge
of the stairs, particularly at the lowermost contact point.
As a result, the deformation increases the pitch angle of the
robot’s body. When the load-sharing ratio at the bottom of
the contact point is maximized, the pitch angle of the body
is also maximized, as shown in Fig. 15. In this paper, we
describe the amount of increase in the pitch angle of the
robot’s body caused by the above reason as θM. To
eliminate the influence of this factor, we measured the
maximum increment of the pitch angle θd at p = 150, 180 and
200 mm. In each condition, the angle of inclination of the
stairs was approximately marginal to falling backward.
Figure 15 shows a side view of the robot’s state on stairs
with (p, θs) = (150 mm, 56.8°). In this case, the result was
2.6° greater than the inclination angle of the stairs.
Therefore, we concluded that the deformation of the track
grousers is one of the reasons for the divergence.
In this paper, we evaluated the influence of the
deformation factor for Kenaf only (Fig. 16). However, the amount
of deformation is determined by the rigidity of the tracks,
the relative positions of the centroid of the robot, and the
edge of the stairs. This deformation problem is common
and can emerge for various tracked vehicles. Therefore,
our proposal and verification tests are applicable not only
to this particular robot but to tracked vehicles in general.
Lowest contact point angular moment generation
When the lowermost contact point of a track detached
from the edge of a step, it takes some time because
of the action of the grouser. During this period, the
grouser causes the circular part of the track to move,
and it generates an angular moment that pushes the
lower part of the robot’s body down, as shown in
Fig. 17(3). The moment at the lowest contact point
angular moment is abbreviated as LCM in this paper.
This period is very short, but it increases the pitch
angle of the robot’s body. When the robot is just about
Fig. 16 Measurement results for θd, between the pitch angle of the
robot and the pitch p of stairs
to tip over, the LCM may provide the impetus for the
robot to fall backward.
However, with regard to the descent of a tracked
vehicle, an LCM is not generated owing to the relative
positions of the grouser and the edge of the stairs (Fig. 18).
Therefore, the influence of the LCM on the divergence
can be evaluated by comparing the marginal angles for
the ascent and descent of the same stair configuration.
Figure 19 shows the results of these additional tests. In
this figure, the vertical axis θM is the difference between
the marginal angle for descending and the marginal angle
for ascending. The error range in Fig. 19 describes the
variance in the experiments, which were conducted five
times for each condition.
The tests were conducted at three conditions: p = 150,
180 and 200 mm. In each condition, the marginal angle
for descending was larger than that for ascending.
Focusing on the result of p = 150 mm, we see that θM was
between 1.2° and 2.6°. We can recognize this value as the
influence of the LCM on the divergence between the
theoretical and the measured marginal angles.
Summary of discussion
Figure 20 summarizes the above discussion. The process
whereby the robot falls backward is explained as follows:
1. When the lowermost contact point is located under a flat part of the track, the tracked vehicle moves parallel to the stairs, keeping the summation of its angular moments at zero.
2. After the lowermost contact point reaches the end of
the flat area, the tractive force resulting from the
contact of the grouser affects the robot body tangentially
to the circular part of the track. According to the
tractive force, the LCM continues to affect the robot
until the grouser completely separates from the edge
of the stair.
3. When the load-sharing ratio at the bottom of the
contact point is maximized, the elevation angle is
also maximized. As a result, the location of the
centroid moves backward.
4. When the lowermost contact point separates completely from the edge of the stairs, the projecting point of the centroid is located more toward the rear of the robot than at the rotation center.
Given the above results, we conclude that falling
backward can actually happen relatively easily in comparison
with the results of our calculations, particularly when
the stairs are steep. Figure 21 shows the result obtained
from Fig. 13, reflecting the results shown in Figs. 16 and
19. We confirmed that the prediction accuracy for falling
Fig. 17 Falling-backward mode affected by track deformation and rib
Fig. 18 State transitions of a tracked vehicle going downstairs
backward improves upon including the effects of θd and
θM in Eq. (19).
The above results for assessing a robot’s ability to
traverse the stairs are going to be improved. In other words,
modification from Eqs. (19) to (24) is effective using the
parameters described in Eqs. (20)–(23):
As a supplement, the data denoted by red points with
error range in Fig. 21 are the estimation results based on
Conclusion and future work
Based on a mechanical model, we have derived the
physical conditions under which a tracked vehicle can traverse
stairs without falling, and we have performed
verification tests using a tracked vehicle to clarify the
backwardfalling failure mode. As a result, for stairs with relatively
small inclination, it is possible to assess whether they can
be traversed by the robot based on the robot’s centroid,
the length of track, inclination, and the pitch between
the edges of the stairs. However, for stairs with very large
inclination, we confirmed that the robot fell backward
more easily than indicated by our calculations. Therefore,
Fig. 20 Summary of cause investigation
we conducted additional experiments and identified two
primary factors for the robot to tip: the deformation of
the tracks caused by the increase of load sharing at the
lowermost contact point and the influence of the LCM
generated by the grouser that maintains contact when it
moves along the circular part of the track. The influence
of each primary factor was evaluated quantitatively.
Conclusively, we found a more accurate method to assess the
robot’s traversal ability described in Eq. (24).
The method requires not only the parameters of the
configuration of the target tracked vehicle and the stairs
but others as well. We can expect that the effect of θd can
be estimated by mechanical analysis, including some
factors such as the shape of stairs, the centroid of the robot,
and the shape and mechanical conductance of the tracks.
θM is determined by the interactive force between the
tracks and the stairs, and it may also be affected by the
grouser shape. However, quantitative methods to
estimate the effects of the grouser shape have not been
established yet. Therefore, more fundamental experiments and
analyses are required to account for these effects.
This research covers only stairs with a constant
inclination and a pitch between the contact points. In future
work, we will extend the derived physical model to
general terrains whose inclination and pitch between contact
points are not constant. In addition, we will establish an
assessment method of slipping phenomena to determine,
in advance, whether the robot can traverse the terrain.
DE contributed to the conception, experimental design, acquisition and
interpretation of data, and writing of the manuscript. KN took part in the
conception and in revising the manuscript. Both authors read and approved
the final manuscript.
This work was supported in part by a Grant-in-Aid for JSPS Fellows (No.
16J2554). Also, we gratefully acknowledge the work of past and present
members of our laboratory, especially, Mr. Kai Kudo, contributing to some
interpretation of data. Also, we would like to thank Editage
(http://www.editage.jp) for English language editing.
The authors declare that they have no competing interests.
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