In this study, we analyze the balance of quadruped walking robots. For this purpose, a simplified polygonal model of a quadruped walking configuration is considered. A boundaryrange based balance margin is used for determining the system stability of the polygonal walking configuration considered herein. The balance margin enables the estimation of the walking configuration’s balance for effective walking. The usefulness of the balance margin is demonstrated through exemplary simulations. Furthermore, balance compensation by means of foot stepping is addressed.
1. Introduction
Given their ability to walk, legbased mobile manipulation systems are very convenient to work with in normal as well as in irregular environments
[1

3]
. For achieving the desired mobility, it is important to design a multipleleg mechanism and manage system balance in a manner wherein there is adequate coordination among the legs. For instance, if any of the legs in a quadruped walking configuration is incompatible with the others, the robot may not remain in equilibrium. Therefore, proper coordination among the legs is required for achieving system balance and the desired walking performance in the stationary state as well as while maneuvering. A performance measure is helpful for identifying system balance during robot walking. From a geometric viewpoint, a few criteria have been put forth for ensuring stable quadruped walking
[4

7]
. In addition, an energybased stability margin has been considered for the potential stability of legged walking robots
[8
,
9]
. Recently, Bretl and Lall
[10]
attempted to evaluate the static equilibrium by considering the foot placement. However, the body’s falling direction may not have been considered in the aforementioned approaches. In contrast, when a robot is off balance, suitable foot stepping or walking steps can be executed for restoring the system balance. Additionally, for taking a step, the robot needs to know the location of the foot to be moved. In fact, the next foot location for effective balance compensation could be determined by testing an effective measure. However, comprehensive studies on identifying and compensating for the balance of quadruped walking systems are rare.
The objective of this study is to analyze the balance of quadruped walking configurations using a boundaryrangebased balance margin. In addition, suitable compensation for system balance through foot stepping is addressed.
2. QuadrupedWalking Configuration Model
Consider a representative quadruped robot system such as a dog, shown in
Figure 1
a. The AIBO robot is a wellknown example of fourlegged mobile manipulation systems. Thus, many researchers have been interested in the locomotion of the robot that can be used as a test bed for walking algorithms. The fourlegged robot shown in
Figure 1
b is well known and is actually used for delivery and exploration tasks in rough terrain. A stable walking strategy is usually required for quadruped tasks, and the walking mechanism’s stability is closely related to the shape of the polygon formed by its supporting feet. That is, system balance is influenced by the placement of the mechanism’s feet. If the walking configuration is well balanced, the given task is executed with ease.
Representative fourlegged robots.
Polygonal model of quadruped walking configuration.
For dealing with such a quadruped walking problem, we consider a simple model of a fourlegged robot, shown in
Figure 2
, where
X_{b}
,
Y_{b}
,
Z_{b}
denote the
x
,
y
, and
z
directional inertial frames, respectively, of the robot system. In practice, a manipulator can be attached onto the body platform for mobile manipulation tasks. However, here, we assume that the robot body includes such an interfacing manipulator. Furthermore, the freedom of the leg mechanism is adequate for the threedimensional motion considered in this study.
So long as the vertical projection of a multilegged robot’s center of mass (COM) lies inside the supporting foot polygon
[1]
, it can walk stably. During walking, the foot polygon varies based on the location (A, B, or C for f1) of the moving foot, as shown in
Figure 2
. If the location of the first foot (f1) is assigned as A, the system balance improves intuitively. Therefore, our goal is to devise a feasible measure for identifying the degree of balance of any quadruped walking configuration and demonstrate the applicability of said measure to the analysis of quadruped robotic walking.
3. BoundaryRangeBased Balance Margin
Consider an arbitrary motion of a quadruped robot in a landing situation, as shown in
Figure 3
. The landing situation implies that all feet are stably on ground, and the entire range of body motion within the operating range of the robot system is available. For effective description, we approximated body motion to the movement of the robot system’s COM.
Boundary range and balance margin of fourlegged robots. COM, center of mass; COFP, centroid of the foot polygon.
In general, a quadruped robot system can attain a balanced posture so long as the vertical projection of its COM lies within the boundary of its foot polygon. The balance worsens if the vertical projection of the robot system’s COM lies outside the foot polygon. Based on this concept, we propose a boundaryrange based balance margin that can be used for checking the degree of balance of a quadruped walking configuration.
Figure 3
shows a schematic diagram of the process of defining the boundary range based on a quadruped robot’s motion. Here, each foot lands on the
xy
plane, and the robot’s COM projected on that plane can be located as being inside, on the edge, or outside of the foot polygon according to the motion trajectory in the stationary and mobile situations. As shown in
Figure 3
, the boundary range is defined as the distance between the centroid of the foot polygon (COFP) and
p
^{*}
.
p
^{*}
lies at the point of intersection of the line between the COFP and the COM projected on the supporting plane, andary of the polygon. The supporting plane refers to the plane that is formed by the two neighboring feet, f1 and f4, and the centroid in
Figure 3
. If the position of the COM is the same as that of the COFP, the boundary range is assigned as the minimum distance between the COPF and the edge of the foot polygon.
In order to present the balance margin during walking, we first check the point of intersection of
p
^{*}
(
x
^{*}
,
y
^{*}
) for the boundary range based on the previous research
[7]
. We used the following procedure to effectively determine the point of intersection. First, if the coordinates of the COM are different from that of the COFP, the
x
 and
y
directional positions of the point of intersection can be determined as follows:
where
a_{ij}
(
i
= 1, 2, 3, 4,
j
= 2, 3, 4, 1) and
b_{ij}
(
i
= 1, 2, 3, 4,
j
= 2, 3, 4, 1) denote the slope of the line connecting the robotic feet f
i
and f
j
, and the
y
intercept, respectively. The parameter
a_{cc}
denotes the slope of the line projecting from the COFP to the COM, and
b_{cc}
denotes the
y
intercept. The angle
α
indicates the angle between the horizontal axis and the projection line, and it can physically be considered as the body’s potential direction of fall during walking.
Second, if the
x
directional positions of the COM and the COFP are identical but their
y
directional positions are different, the position of the point of intersection can be determined as follows:
where
x
_{COM}
is the
x
coordinate of the COM.
Third, if the
x
directional positions of the COM and the COFP are different but their
y
directional positions are identical, the position of the point of intersection can be determined as follows:
where
y
_{COM}
is the
y
coordinate of the COM.
Finally, if all positions of the COM and the COFP are identical, the intersection point is determined to be on the boundary of the foot polygon such that the distance between the COFP and the edge of the polygon is at its minimum.
Following the abovementioned procedure, the
x
 and
y
 directional boundary ranges,
B_{x}
and
B_{y}
, can be expressed as follows:
Given that the boundary ranges include the directional information of walking motion, they are useful for analyzing the directional balance of the quadruped robot in the stationary and walking situations.
We finally define the
x
 and
y
directional balance margins,
M_{x}
and
M_{y}
, as a useful performance measure for identifying quadruped balance as follows:
where the values of
M_{x}
and
M_{y}
denote the difference between each directional boundary range and the current COM position. Therefore, regardless of the sign of the boundary range (positive or negative), we can conclude that the system balance of the robot is within the stable range so long as the sign of the margin is the same as that of the boundary range.
4. Simulation for Balance Analysis
This section shows exemplary simulation results to analyze the balance of the quadruped robot system doing some stationary motions or walking.
 4.1 Balance Trend as Stationary Motions
The effort to identify the system’s balance is basically important to achieve the performance of a walking robot system performing a stationary motion or a mobile manipulation. So, the goal of the first simulation is to analyze the stationary balance of the robot system by estimating the boundary range and the balance margin described in Section 3. through some exemplary motions.
The three trajectories shown in
Figure 4
have been considered for the first simulation. These trajectories imply the body motions for some manipulation tasks of the quadruped robot and they specified as the trajectories of the COM projected on the planar ground space. Those COM trajectories are represented by
where
c_{x}
and
c_{y}
represent the
x
 and
y
directional center positions of the trajectory, and they have been assigned as 0.65 and 0.70, respectively. The parameters of
r_{x}
and
r_{y}
represent the
x
 and
y
directional radius of the trajectory, respectively, and they have been specified in
Table 1
for the case studies. Also, the circular time for the trajectories
t_{f}
has been set as 2.0 s.
Stationary motion trajectories of the quadruped robot: (i) Case 1, (ii) Case 2, and (iii) Case 3.
Parameters of rxand ryfor the case studies
Parameters of r_{x} and r_{y} for the case studies
Positions of four feet standing on planar ground
Positions of four feet standing on planar ground
In particular, the first case in
Figure 4
implies that the stationary motion of the robot’s body is performed in the inside of the foot polygon and its pattern is an elliptical shape. The second case shows that the motion approaches to the boundary of the foot polygon. The third case shows that some parts of the motion are performed in the outside of the foot polygon. The current positions of the four feet have been specified in
Table 2
. In this situation, the
x
 and
y
directional centroid of the foot polygon,
x
_{COFP}
and
y
_{COFP}
, can be calculated as 0.65 m and 0.7 m, respectively
[11]
.
Boundary ranges and balance margins for the stationary balance: (i) Case 1, (ii) Case 2, and (iii) Case 3.
Figure 5
shows the
x
 and
y
directional boundary ranges and margins checked by (7)(10). If the sign of the margin is reverse compared with that of the corresponding boundary range, the motion can be unstable and then the balance of the walking robot is eventually collapsed in the ranges of AE pointed in the third case of
Figure 5
. It is natural because those parts of the motion in the third case are performed in the outside of the foot polygon. In practice, a foot stepping is considerable in order to recover the system balance. In the next section, we discuss on the system balance compensated by the fundamental foot stepping.
 4.2 Balance Compensation by Foot Stepping
In order to consider the issue of balance compensation by foot stepping, we especially used the two trajectories shown in
Figure 6
. Actually, the second trajectory is to consider a moving situation for a walk so that some parts of the body’s motion are out of the stable range of the foot polygon and the walking system is to be unbalanced.
Figure 7
shows the boundary ranges and margins for the given motions in
Figure 6
. It is found that the quadruped robot has some margin for the stability in the motion of the first case. However, the balance margins at the regions of AD in
Figure 7
are inadequate to achieve the second motion stably without walking.
Test motions to identify the system balance in a moving situation for a walk: (i) Case 1 and (ii) Case 2. COM, center of mass; COFP, centroid of the foot polygon.
Boundary ranges and balance margins for the given motions: (i) Case 1 and (ii) Case 2.
Now, we tried to move a foot in order to compensate the unbalance.
Figure 8
shows that the first foot has been moved for the balance compensation during the second motion. The objective of this simulation is to show the improvement of the system balance when the first foot f1 is moving to a suitable position of
xy
(0:43, 1.05). Intuitively, such an effort of foot stepping can contribute to improve the balance margin compared to the second case of
Figure 7
. That is, the unbalanced regions of A and C in
Figure 7
can be recovered stably. But it is still unsatisfactory for the robot to follow the trajectory between f1 and f2.
In order to compensate such an unbalance, the robot can try an additional foot stepping by the second foot f2 as shown in
Figure 9
.
Figure 10
shows the resultant boundary ranges and balance margins by the second foot stepping. Finally, we can confirm that the system balance has been made stably. Therefore, our analysis of balance compensation through appropriate foot stepping can be applied for effective gait generation of quadruped robots.
Movement of the first foot for the second motion. COFP, centroid of the foot polygon.
Movement of the second foot for the balance compensation. COFP, centroid of the foot polygon.
5. Concluding Remarks
The main conclusion of this study is that the balance of quadruped walking robots can be identified using the proposed boundaryrange based balance margin. Through exemplary simulations using certain standing and footstepping motions, we demonstrated the effectiveness of the proposed measure. Additionally, the effect of balance compensation through appropriate foot stepping was analyzed. We finally concluded that the boundaryrange based balance margin can be applied to determine the foot to be moved and its optimum landing location for effective balance restoration, and that the measure can contribute to improving the system balance over the course of a reasonably long walk. In addition, the proposed measure can be applied to the development of a footstepplanning strategy for ensuring effective quadruped walking.
Boundary ranges and balance margins by second foot stepping.
Acknowledgements
This research was supported by Kyungsung University ResearchGrants in 2013.
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