# A Low Cost Bin Picking Solution for E-Commerce Warehouse Fulfillment Centers

Avnish Gupta, Akash Jadhav and Pradyot VN Korupolu
Addverb Technologies Pvt. Ltd., India
{avnish.gupta, akash.jhadav<EMAIL_ADDRESS>
###### Abstract

In recent years, the throughput requirements of e-commerce fulfillment
warehouses have seen a steep increase. This has resulted in various automation
solutions being developed for item picking and movement. In this paper, we
address the problem of manipulators picking heterogeneous items placed
randomly in a bin. Traditional solutions require that the items be picked to
be placed in an orderly manner in the bin and that the exact dimensions of the
items be known beforehand. Such solutions do not perform well in the real
world since the items in a bin are seldom placed in an orderly manner and new
products are added almost every day by e-commerce suppliers. We propose a
cost-effective solution that handles both the aforementioned challenges. Our
solution comprises of a dual sensor system comprising of a regular RGB camera
and a 3D ToF depth sensor. We propose a novel algorithm that fuses data from
both these sensors to improve object segmentation while maintaining the
accuracy of pose estimation, especially in occluded environments and tightly
packed bins. We experimentally verify the performance of our system by picking
boxes using an ABB IRB 1200 robot. We also show that our system maintains a
high level of accuracy in pose estimation that is independent of the
dimensions of the box, texture, occlusion or orientation. We further show that
our system is computationally less expensive and maintains a consistent
detection time of 1 second. We also discuss how this approach can be easily
extended to objects of all shapes.

##  1 Introduction

Industrial robots used for tasks like assembly operations or pick and place
operations can work only on objects whose location and orientation are
predefined. These robots without vision system hence are used only for
repetitive tasks. This design of a blind robot system increases the cost of
automation considering the requirement of expensive object feeding conveyor
systems.

An efficient solution to this situation can be provided by considering a
manual assembly environment that has several objects placed randomly in bins
around the work area from which required objects can be picked for assembly.
Automation of this process requires developing an intelligent robot capable of
recognizing and handling objects in bins. This problem, called bin picking,
thus consists of picking an object from a group of similar (homogeneous) or
different (heterogeneous) objects. The objects in the bin are generally in
random orientation and occlude each other; this poses a great challenge to the
vision sensors in segmenting the objects to be picked.

A great deal of previous research on vision-based bin picking has been limited
to recognition of objects and STL matching where the object is predefined due
to the complexity involved in segmenting the bin images. In recent years, much
work has been proposed on the recognition of objects for the bin picking
problem. Some techniques are limited to detection of the objects while others
concentrate on picking an object and analyzing the object for pose
determination and recognition using the object database [?], [?], [?].

(a) RGB Image

(b) Point Cloud

(c) Fused Data

Figure 1: Illustration of data fusion of RGB Image and Pointcloud

Image segmentation is another well-researched area that involves the division
of an image into meaningful sub-regions. If all the objects are segmented
individually, then identifying them is relatively easy; however, when the
objects are partially occluded more than one object will be considered as a
single object and thus making it difficult to identify each one of them. In
contrast to the approaches mentioned above, our algorithm attempts to provide
a solution for segmentation and localization of boxes of different sizes in a
partially occluded bin using a dual vision system based approach. In the dual
vision system, the RGB image is fused with a 3D point cloud as shown in Figure
1. Our algorithm proves to work efficiently on textured and non-textured boxes
of random sizes in cluttered environments without the requirement of an object
database. The accuracy of the algorithm is further evaluated on various test
cases which confirms the robustness of the system. The following section
compares some of the previous approaches to solving this problem to the
proposed algorithm.

##  2 Related Work

There has been an immense increase in the demand for a bin picking solution
for the past few years. Researchers have proposed various approaches to
tackling the bin packing problem. We have discussed some of the approaches
that built motivation for our work.

The most known solution to this problem is STL or CAD matching. Many
industrial robots are utilized to pick objects of the same type whose CAD file
is already known to the perception system. Using these CAD files, the
perception system accurately recognizes the shape and grasp point. [?]
demonstrated a flexible assembly of industrial parts where stereo matching and
structured light were used for the picking process while CAD models were used
to determine the orientation of the objects. Another similar research by [?]
uses the Interactive Closest Point algorithm(ICP) on 3D range data for finding
the best fitting cloud point with respect to the reference CAD file. These
types of instance recognition solutions are limited to organized environments
and are dependent on sensor accuracy. They also pose a major issue of
maintaining the object CAD database for all the objects which we need to pick.

Research has also focused on matching extracted 2D features with geometries of
known 3D objects using chamfer matching technique [?]. One approach discusses
Oriented Chamfer Matching (OCM) on an object database containing 360 images of
one object each one representing an orientation at equal intervals created
using CAD file of an object [?]. These solutions have a satisfactory
performance in the case of unoccluded and uncluttered objects in the bin and
are not quite robust [?].

Another approach to tackle the problem of cluttered object heap is the use of
Convolutional Neural Networks (CNNs) trained on a large dataset of images to
recognize the object [?]. With the development of 3D RGB-D sensors, there was
an increase in the point cloud data. This motivated researchers to further use
CNNs for the segmentation of object classes directly from the point cloud of
the scene. [?] and [?] trained CNNs on 3D dense point clouds from synthetic
datasets to predict antipodal grasps. These methods have a very good
performance if the data is collected from various viewpoints and is large
enough so that the algorithm can generalize well. The drawbacks of these
methods are the requirements of a powerful GPU system and several months taken
in the collection of datasets.

The major issue with all of the above-mentioned approaches is scalability for
a large number of classes of objects. [?] have discussed an approach using
Reinforcement Learning to solve the problem of scalability. Our approach
addresses the similar problem of scalability but is rather a computationally
less intensive and cost-effective solution.

The objective of this research is to develop a robust vision system for a
manipulator robot to perform automated bin-picking operations generalized for
boxes of any sizes and textured surfaces in cluttered environments. The task
of bin picking is split into two subtasks, namely box segmentation, and box
localization. The two subtasks and the experimental setup is detailed in
Section 3, which describes the proposed segmentation algorithm using robust
edge detection and elaborates on the box localization process to obtain 6DOF
of the box detected. Section 4 discusses the experimental results obtained
from the proposed algorithm and real-time bin picking experiments conducted on
an ABB IRB 1200 robot. Finally, section 5 and section 6 presents the
conclusion and future work. This paper presents a 3D vision based bin-picking
for cuboidal objects but this can be further extended for randomly shaped
objects.

##  3 Methodology

The proposed dual vision system consists of a monocular RGB camera with a 3D
depth sensor that provides an organized point cloud. A calibration technique
using perspective transform [16] is implemented to fuse the RGB image from the
monocular camera with the organized point cloud of the 3D depth sensor such
that for every point in the point cloud we have a specific RGB value for it.

Figure 2: Dual Vision System consisting of RGB Camera and 3D Depth Sensor

The dual sensor setup shown in Figure 2 consists of a monocular camera and the
3D depth sensor placed parallel to each other on a rigid mount. The
segmentation of the box is done using the RGB information from the monocular
camera. A binary mask is generated for the segmented boxes, where the detected
boxes have a value of one and rest as zero. Further for the pixels in the
mask, corresponding points in the point cloud are considered for fitting a
plane to obtain its centroid and orientation. Finally, for sending obtained
6DOF information of the box detected by the dual vision system to the robot, a
calibration procedure [?] is carried out between the ABB IRB 1200 robot and
the 3D depth sensor to transform the coordinates obtained with respect to the
3D depth sensor frame to the robot base frame.

Figure 3: Dual Vision System consisting of RGB Camera and 3D Depth Sensor

###  3.1 Box Segmentation

To segment out boxes, we have incorporated a robust edge detection algorithm,
which detects all possible edges in the image and filters out the unnecessary
edges, while preserving the structural properties of the detected boxes to be
used for further processing. The proposed algorithm also performs efficiently
for differentiating two objects kept very close to each other. The algorithm
further consists of a parent-child hierarchy, such that if a small box is
placed on top of a large box, the identity of the smaller box should be
preserved within the larger box. The smaller box is termed as a child and the
larger box is termed as a parent by the algorithm. The child box is given a
priority to be picked first by the algorithm.

(a) ROI of bin

(b) Smoothening

(c) Gradients

(d) Refining Gradient

(e) Mask Generation

Figure 4: Results of box segmentation algorithm on individual object

The algorithm runs in 6 separate steps:

Set Bin ROI: In the image taken from the camera, the region of interest(ROI)
of the bin is extracted and used in the further processing of the algorithm.
Bin ROI is shown in Figure 4a.

Smoothing: Images taken from a camera might contain some amount of noise. To
prevent that noise from being mistaken for edges, a gaussian filter with
kernel 3x3 is applied to smoothen the input image as shown in Figure 4b.

Finding Gradients: Popular Canny edge algorithm [?] is known for finding edges
whose intensity gradients fall in a range of predefined threshold values. We
have incorporated an auto canny algorithm [?] that adaptively decides the
threshold based on the region of interest of the bin in the image. The edges
are determined by applying gradient at each pixel in the smoothened image
using Sobel operator as shown in Figure 4c. To approximate the gradient in the
x- and y-direction respectively the kernels shown in Equation 1 and 2 are
used.

$\centering KGX=\begin{bmatrix}\hskip 2.84544pt-1&\hskip 5.69046pt0&\hskip
5.69046pt1\hskip 4.26773pt\\\ \hskip 2.84544pt-2&\hskip 5.69046pt0&\hskip
5.69046pt2\hskip 4.26773pt\\\ \hskip 2.84544pt-1&\hskip 5.69046pt0&\hskip
5.69046pt1\hskip 4.26773pt\\\ \end{bmatrix}\@add@centering$ (1)

$KGY=\begin{bmatrix}\hskip 5.69046pt1&\hskip 5.69046pt2&\hskip
5.69046pt1\hskip 4.26773pt\\\ \hskip 5.69046pt0&\hskip 5.69046pt0&\hskip
5.69046pt0\hskip 4.26773pt\\\ -1&-2&-1\hskip 4.26773pt\\\ \end{bmatrix}$ (2)

Refining Gradients: Over the edges obtained in the image from the auto canny
edge detection, contours are detected. These are further refined over the
contour area of 2500 pixel-sq in a 2K pixel density image to remove
unnecessary edges. Contour detection aims at quantifying the presence of a
closed boundary at a given image location through local measurements. This
step results in the identification of surfaces of boxes as shown in Figure 4d.

Parent-Child hierarchy: A contour hierarchy is implemented to solve the
stacking of smaller boxes over larger boxes. Contour detection is used to
detect objects in the image, but in some cases, objects are present on top of
another object, just like nested figures. In this case, we call the outer one
as a parent and the inner one as a child. This way, contours in an image
constitute a relationship with each other. Representation of this relationship
is called the Hierarchy. An individual object is determined as a parent
contour as shown in Figure 4d.

Mask Generation: A binary mask is generated for the child and parent boxes
individually, where the full surface of the detected box has a value of one
and rest part of the bin as zero. While the child contour is given a priority,
the parent contour has a zero value. The mask is generated for parent contours
only after all the child contours have been picked by the ABB 1200 robot as
shown in Figure 4e. This binary mask is further used on the point cloud to
detect planes and generate the 6DOF of the plane.

###  3.2 Box Localization

This section discusses plane fitting and 6 DOF pose estimation of the object.
The masks calculated in the previous section are fused with the raw 3D point
cloud obtained from the depth sensor. The results will depend on the vision
system calibration step performed [?]. Finally, we have a cloud scene with
separated planes for all the objects as shown in Figure 6a. As this is raw
from the sensor, it tends to be a bit noisy. We will further process this data
in five steps to finally segment out planes and estimate centroid and Euler
angles for the robot to grasp. These steps are described in the following
subsections.

Input: Masked Point Cloud Clusters (clusters)

Output: Planes, Model Coefficients, Plane Normals

$uni\\_plane,uni\\_plane\\_coeff,normal\\_vecs=[]$

for  _$cluster\in clusters$_ do

$prev\\_coeff,planes,plane\\_coeff=[]$

while  _$cluster.size()\geq min\\_cluster\\_size$_ do

$points,coeff=ransac\\_seg\\_plane(cluster)$

if  _$prev\\_coeff==coeff$_ then

$break$

end if

if  _$points.size()\geq min\\_object\\_size$_ then

$plane.append(points)$

$plane\\_coeff.append(coeff)$

$cluster=rm\\_seg\\_plane(cluster,points)$

end if

$plane\\_num=planes.size()$

if  _$plane\\_num==0$_ then

continue

end if

else

$dp\\_mat=dot\\_product(normal\\_vecs)$

$group,visited=[]$

for  _$m\in range(plane\\_num)$_ do

if  _$m\not\in visited$_ then

$visited.append(m)$

$g=[]$

$g.append(m)$

for  _$n <=size(plane\\_num)$_ do

if  _$n\not\in visited\ \ &\ dp\\_mat[m][n]\sim 1.0$_ then

if  _$check\\_overlapping(planes[m],planes[n])$_ then

$g.append(n)$

end if

end if

end for

end if

end for

$group.append(g)$

end if

for  _$g\in group$_ do

$uni\\_plane.append(comp\\_grp\\_plane(g))$

$uni\\_plane\\_coeff.append(comp\\_grp\\_plane\\_coeff(g))$

$normal\\_vecs.append(comp\\_grp\\_plane\\_normal(g))$

end for

end while

end for

return _$uni\\_plane,uni\\_plane\\_coeff,normal\\_vecs$_

Algorithm 1 Plane Segmentation and Extraction

#### Filtering

Since the data consists of thousands of points, it will take a huge amount of
time to directly compute planes on this data. Moreover, the data is prone to
noise points because of sensors’ measurement capability and repeatability.
There are various methods available for filtering the point cloud [?], [?]. We
have selected two filters based on their efficiency and time complexity to
reduce the number of points and to minimize noise, after which an output shown
in Figure 6b is obtained.

Firstly, we apply the Voxel Grid filter to downsample the cloud. This method
assumes a 3D voxel grid over the cloud and then in for each voxel, a centroid
is assumed to approximate all the points in that voxel. This method robustly
reduces the number of points based on the leaf size which defines the size of
voxel and it represents the underlying surface more accurately. We can
optimally set this parameter so that geometric features don’t get lost and the
number of points also gets significantly reduced.

Secondly, we use Statistical Outlier Removal (SOR) filter [?]. Generally,
point clouds registered from depth sensors have some sparse outliers which can
corrupt the estimation of characteristics like surface normals. SOR computes a
distribution of the mean distance of a point to all its k nearest neighbors
specified by a parameter. The resulting distribution is assumed to be Gaussian
with a mean and a standard deviation. All the points whose mean distances are
above the threshold defined by the global distances mean and standard
deviation parameters are classified as outliers and removed from the cloud.

#### Resampling and Difference of Normals

Filtering, discussed in the above section, is incapable of removing some
erroneous points caused by measurement errors. Thus we perform resampling
using Moving Least Squares (MLS) surface reconstruction method [?]. This
resampling technique also helps in accounting irregularities in the cloud
created due to glossy surfaces and occlusions. The algorithm estimates missing
points in the surfaces by using higher-order polynomial interpolations between
the neighboring data points.

Following this, we perform a Difference of Normals based segmentation [?] to
remove the edges between the planes of the cuboidal objects. The normals
calculated on the plane point cloud are shown in Figure 6c. This method
enhances the performance of plane segmentation algorithm discussed later by
separating the faces of boxes. We first calculate two unit point normals
n̂(p,rl), n̂(p,rs) for every point p such that rl $>$ rs as shown in Figure 5.
Succeeding this we calculated a normalized Difference of Normals vector field
using equation 3, where rs, rl $\in\mathbb{R}$, rs $<$ rl.

$\Delta\hat{n}(p,r_{s},r_{l})=\frac{\hat{n}(p,r_{s})-\hat{n}(p,r_{l})}{2}$ (3)

Taking the norm of this vector field yields values in the range (0,1) for all
the points. We select the points for which the norm has value less than a
user-defined threshold parameter.

Figure 5: DoN : Illustration of effect of support radius on surface normal
estimates in 1D

#### Clustering

Clustering is an important step as it helps ensure that a common plane is not
fitted for two parallel faced cuboidal objects. Our goal here is to accurately
generate separate clusters for all the objects. An algorithm is required which
is robust to varying density in the cloud and selects clusters based on
hierarchy. Thus we use HDBSCAN [?] algorithm which predicts stable clusters in
noisy environments and is not prone to outliers. The algorithm works by
transforming the cloud-based on sparsity and then constructs a minimum
spanning the tree of the distance weighted graph. Following this, cluster
hierarchy is generated for connected components in the graph. Finally, stable
clusters are selected by condensing the cluster hierarchy based on the minimum
cluster size. This minimum cluster size is a user-defined parameter and should
be set equal to the number of points on the surface of the smallest object.
For our application, we have set a minimum cluster size equal to 30 points and
obtained the output for a plane as shown in Figure 6d.

#### Plane Segmentation

Plane Segmentation is the most critical part of the algorithm. The main
challenge here is to accurately segment unique planes such that no two planes
belong to the same surface. The clusters generated in the previous section may
contain more than one plane. Thus plane fitting is done iteratively to segment
out all the planes in that cluster [?]. We use the Random Sample Consensus
(RANSAC) algorithm [?] with a plane model over the cloud clusters.

RANSAC algorithm starts by selecting a random subset of points from the
cluster, called hypothetical inliers. Model is fitted for these hypothetical
inliers and all the remaining points are checked against this model to be
classified as inliers or outliers. Then this model is again re-estimated by
using all classified inliers. This process is repeated until we get a
segmented plane with its coefficients from the cluster. The coefficients of
this segmented plane are saved and all the points belonging to this plane are
removed from the cluster. The updated cluster is again processed for plane
segmentation. This process keeps on repeating until the number of points in
the cluster becomes less than the user-defined parameter for minimum points in
the cluster. A similar kind of processing is done for all the clusters. The
parameters of the RANSAC algorithm should be chosen wisely such that the
points which are not on the surface to be grasped don’t become part of the
segmented plane.

Following the segmentation of all the planes from all the clusters, we need to
extract unique planes. Generally, due to measurement noise in sensors, the
same surface can be detected as 2 different planes. In order to address this
problem, we group all segmented planes which are parallel to each other. The
two parallel planes found from different clusters (Figure 6d) after RANSAC but
are part of the same surface are combined into a single plane as shown in
Figure 6e. This is done by combining all those planes which are overlapping
using the following checks:

  * •

Distance between the centroid is less than a certain threshold. This threshold
is defined based on the minimum object size to be picked.

  * •

Perpendicular distance between planes is less than the measurement noise of
the depth sensor. Correct selection of this parameter ensures that planes
which are part of same surface gets combined.

The pseudo code shown in 1 contains the functions _check_overlapping_ to check
for overlapping planes and _comp_grp_plane_ to group overlapping planes.

#### 6DOF Pose Estimation

After finding all the unique planes from the cloud, we need to generate a 6DOF
pose over each plane for the robot to grasp the object as shown in Figure 6f.
Since in our implementation we used a vacuum gripper, we need to calculate the
centroid of the plane and Euler angles to represent the orientation of the
plane.

The centroid of the plane is calculated using Mean Shift clustering [?]
method. The advantage of using this algorithm over others is that it is a
centroid based algorithm. This algorithm almost accurately calculates the
center of the region even in the case of planes having sparse points. The
centroid is calculated with reference to the depth sensor frame.

We estimated Euler angles by attaching a reference frame at the previously
computed centroid for each plane. For each plane, we have a normal vector from
estimated plane coefficients from which we approximate an axis along the plane
by finding a unit vector along the plane. Finally, we compute the cross
product between the unit normal vector and unit vector along the plane to find
the third axis of the reference frame. Following this, a rotation matrix is
computed for defining the rotation of this object frame with respect to the
depth sensor. Further, this rotation matrix is converted into Z-Y-X Euler
angles with respect to the depth sensor frame.

$Z_{1}Y_{2}X_{3}=\begin{bmatrix}c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}&s_{1}s_{3}+c_{1}c_{3}s_{2}\\\
c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}\\\
-s_{2}&c_{2}s_{3}&c_{2}c_{3}\\\ \end{bmatrix}$ (4)

,where c represents cos(), s represents sin(), 1,2 and 3 are Euler angles
about X, Y and Z axes respectively.

(a)Mask Fused Cloud

(b)Filtering

(c)Re-sampling and DON

(d)Clustering

(e)Plane Segmentation

(f)Pose Estimation

Figure 6: Results of Box Localisation algorithm for individual object

##  4 Experiment Test-bench

We performed our experiments with ABB IRB 1200 robot and dual vision system.
The dual vision system consisted of a Monocular RGB camera with 2K resolution
and a Time of Flight based 3D depth sensor which gives an organized point
cloud, of resolution 224x172. This visual system was mounted at a height of
1.2m above the bin having dimensions 400mm x 600mm. All the experiments were
performed on a laptop having Ubuntu 16.04 with a 2.30GHz Intel Core i5-8300H
CPU and 4GB DDR3 RAM.

##  5 Results

This section is divided into three subsections:

###  5.1 Verification

To evaluate the accuracy of the 3D pose estimated by our system, a plate with
an Aruco marker [?] is attached to the last link of the robot with a known
static transform as shown in Figure 7.

Figure 7: Experimental setup for performing verification procedure

The robot is moved by the teach pendant such that the plate gets all possible
orientations of a box in the bin. The Aruco maker is detected in the RGB image
of the camera and a mask is generated for it [?]. Further, a plane is fitted
in the point cloud for this mask and its 6DOF pose is estimated by our system
which is compared with the ground truth values of the teach pendant. In our
experiment, the performance was tested on a total of 9 orientation cases of
the plate, For each test case, we have recorded 20 samples and taken the
average of the error in pose calculated by the algorithm. All the Euler angles
are measured with respect to the robot base frame and the z-axis of the robot
is pointing upward with respect to the ground. The results obtained for the 9
test cases of the plate are shown in Table 1. Additionally, the system
demonstrated stable performance with an overall mean translation error of less
than 3.3 mm in all directions and a mean rotation error of less than 3.26
$\deg$ on all axes.

Description Angle Angle Trans Err Trans Err Trans Err Rot Err Rot Err X Y X(in
mm) Y(in mm) Z(in mm) X(deg) Y(deg) +ve X 135 0 2.7 2.3 3.4 2.8 2.9 -ve X -135
0 2.4 2.5 3.1 2.9 2.6 +ve Y 180 45 3.1 2.7 2.9 2.7 3.1 -ve Y 180 -45 2.9 2.9
2.8 2.8 3.2 +ve X and +ve Y 135 45 4.7 4.8 3.7 3.5 4.1 -ve X and +ve Y -135 45
2.7 3.1 3.4 3.1 3.8 +ve X and -ve Y 135 -45 2.6 3.2 3.7 3.3 3.4 -ve X and +ve
Y -135 -45 4.6 4.7 3.5 3.4 3.9 Parallel to bin surface 180 0 1.6 2.1 3.2 2.1
2.4 Average Error 3.03 3.27 3.3 2.95 3.26

Table 1: Computation of system error for 6DoF pose.

Box Surface Type Mean Trans Error from box centre (in mm) No texture 2.1
Minimal Texture 2.8 Highly textured 3.6 Glossy Surface 5.7

Table 2: Average error for different box surface textures measured over 25
samples

(a) No Texture

(b) Min Texture

(c) High Texture

(d) Glossy Surface

Figure 8: Segmentation results on boxes with different surface types as
mentioned in test cases

###  5.2 Box Test Case

The surface details of the boxes used in experimentation are presented in
Table 2. We have used boxes with a minimum dimension of 50mmX75mmX20mm and
various boxes having a surface with no texture, minimal texture, maximum
texture, and gloss. Further, some special test cases are taken into
consideration where two objects of the same dimensions are kept adjacent to
each other and a smaller object is kept on top of a larger object. For planar
surfaces, the deviation along the x and y axes determines the accuracy of the
system. This accuracy further determines if the system is good enough to
perform random boxes picking tasks, whereas the deviation and orientation on
the z-axis with respect to the gripping tool are less important as the single
suction cup type gripper can easily search for the box within a certain range
in the z-direction. After the box being picked by the robot, it is put in the
barcode scanning zone where it is identified by the system. The maximum
average percentage error in the result of the translation estimate is found to
be 3.12% along the x and y axes of the box. This error is acceptable as the
box is not to be picked from its barcode which is generally present at the
edges of the box. The error computed ensured that the proposed method worked
well on the boxes in the cluttered environment for random bin-picking tasks.

###  5.3 Complete System Performance

The performance of our system for the box specifications mentioned in Table 2
is shown in Figure 8. The boxes are placed individually in the bin and the
image from the RGB camera is first processed using the box segmentation module
and then fused with the point cloud data of the depth sensor. This is given as
input to the box localization module to extract the 6DOF pose data of the box.

Figure 8a shows the performance for the box with no texture, Figure 8b shows
performance for the box with minimal texture, Figure 8c shows performance for
the box with high texture and Figure 8d shows performance for the box with a
glossy surface. The pose estimation error is slightly higher in the case of
glossy surfaces because of sensor limitations caused due to reflectivity.

(a)Adjacent Boxes

(b)Overlapping Boxes

Figure 9: Results of mask generation for Parent Child Hierarchy for Adjacent
and Overlapped test case

The performance of our system in the case of two adjacent boxes of the same
dimensions is shown in Figure 9a and the case of smaller box placed on top of
a larger box is shown in Figure 9b. Where the 1st row shows the input RGB
image ROI of the bin, 2nd row shows the smoothened image, 3rd row shows the
Edges found, 4th row shows the refined edges found, 5th row shows the mask of
child contour and the 6th row shows the mask of parent contour, after the
child was picked by the robot.

(a)Adjacent boxes

(b)Overlapped boxes

Figure 10: Parent Child Hierarchy

Figure 10 explains the parent-child hierarchy in both cases. When two adjacent
boxes are taken into consideration as shown in Figure 10a, they are not
identified as two individual boxes. Instead, their contours are detected as
one contour (child) is inside another contour(parent) and the child is given a
priority to be picked up. Once the child box is picked up, in the next
detection the parent box is picked up. When the case of a smaller box is kept
on top of a larger box is taken into consideration as shown in Figure 10b, the
smaller box is detected as child contour and the larger box is detected as
parent contour. The vision system first segments the child contour boxes and
computes their midpoint [x, y, and z coordinates] and orientation [Euler
angles] and further goes for the parent contour boxes when it has picked all
the child contour boxes. Figure 11 shows the performance of our system in the
cluttered bin. Our proposed algorithm was able to detect smaller boxes on the
top of larger boxes due to the parent-child hierarchy used in generating
masks. These masks were further used to extract an accurate plane from the
point cloud and estimate the 6DoF pose.

(a) Bin ROI

(b) Child Mask

(c) Parent Mask

Figure 11: Results of algorithm for Cluttered environment case

Process Computation time (in seconds) Mask Generation 0.015 Resampling and DON
0.153 Clustering 0.19 Plane Segmentation 0.0831 Filtering 0.301 6DOF Pose
Estimation 0.159 Total Time 0.901

Table 3: Results are averaged over 25 samples.

Table 3 shows the total processing time for both box segmentation and box
localization modules. The total processing time comes out to be 0.9 seconds.
These results are comparable to the results demonstrated in [?]. As opposed to
[?], which requires very high computation resources to generate pick points,
our approach works very well on a regular laptop. This makes our approach
cost-effective. In addition, our approach is easily scalable to multiple boxes
since it does not require any prior knowledge of the boxes or any form of
training.

##  6 Conclusion and Future Work

We presented a dual sensor system for box segmentation and box location
computation in bin picking applications. Two algorithms for box mask
segmentation and box localization are proposed. The box mask segmentation
algorithm is found to be robust in case of no texture, minimal and high
texture boxes. However, the performance for the glossy surface is limited by
3D depth sensor measurement. The unified vision algorithm is capable of
computing accurate poses in all three axes with an average error of +/- 3.3mm;
however, the picking operations were unaffected due to the error in z-axis
because of the tool with single suction type vacuum gripper which has a
tolerance of up to +/- 5 mm. The real-time experimental demonstrates a
consistent detection time of approximately 1 second on the specified hardware
and test cases. We have empirically proved the feasibility of the proposed
algorithms for bin-picking application and it can be adopted for many real-
world box picking operations in warehouses and e-commerce industry. This
solution has the potential to be extended for other objects such as cylinders
using shape decomposition algorithms.

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# Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent
Bundles

S. L. Druţă-Romaniuc

_The paper will appear in Czechoslovak Mathematical Journal_

Abstract: We obtain the natural diagonal almost product and locally product
structures on the total space of the cotangent bundle of a Riemannian
manifold. We find the Riemannian almost product (locally product) and the
(almost) para-Hermitian cotangent bundles of natural diagonal lift type. We
prove the characterization theorem for the natural diagonal (almost) para-
Kählerian structures on the total spaces of the cotangent bundle.

Key words: natural lift, cotangent bundle, almost product structure, para-
Hermitian structure, para-Kähler structure.

Mathematics Subject Classification 2000: primary 53C05, 53C15, 53C55.

##  1\. Introduction

Some new interesting geometric structures on the total space $T^{*}M$ of the
cotangent bundle of a Riemannian manifold $(M,g)$ were obtained for example in
[7], [19], [21]–[23] by considering the natural lifts of the metric from the
base manifold to $T^{*}M$. Extensive literature, concerning the cotangent
bundles of natural bundles, may be found in [12].

The fundamental differences between the geometry of the cotangent bundle and
that of the tangent bundle, its dual, are due to the construction of the lifts
to $T^{*}M$, which is not similar to the definition of the lifts to $TM$ (see
[27]).

In a few papers such as [2]-[6], [9]–[11], [17], [24], and [26], some almost
product structures and almost para-Hermitian structures (called also almost
hyperbolic Hermitian structures) were constructed on the total spaces of the
tangent and cotangent bundles.

In 1965, K. Yano initiated in [26] the study of the Riemannian almost product
manifolds. A. M. Naveira gave in 1983 a classification of these manifolds with
respect to the covariant derivative of the almost product structure (see
[20]). In the paper [25] in 1992, M. Staikova and K. Gribachev obtained a
classification of the Riemannian almost product manifolds, for which the trace
of the almost product structure vanishes, the basic class being that of the
almost product manifolds with nonintegrable structure (see [16]).

A classification of the almost para-Hermitian manifolds was made in 1988 by C.
Bejan, who obtained in [3] 36 classes, up to duality, and the
characterizations of some of them. P. M. Gadea and J. Muñoz Masqué gave in
1991 a classification à la Gray-Hervella, obtaining 136 classes, up to duality
(see [8]). Maybe the best known class of (almost) para-Hermitian manifolds are
the (almost) para-Kähler manifolds, characterized by the closure of the
associated 2-form, and studied for example in [1] and [15].

In the present paper we consider an $(1,1)$-tensor field $P$ obtained as a
natural diagonal lift of the metric $g$ from the base manifold $M$ to the
total space $T^{*}M$ of the cotangent bundle. This tensor field depends on
four coefficients which are smooth functions of the energy density $t$. We
first determine the conditions under which the tensor field constructed in
this way is an almost product structure on $T^{*}M$. We obtain some simple
relations between the coefficients of $P$. From the study of the integrability
conditions of the determined almost product structure, it follows that the
base manifold must be a space form, and two coefficients may be expressed as
simple rational functions of the other two coefficients, their first order
derivatives, the energy density, and the constant sectional curvature of the
base manifold. Then we prove the characterization theorems for the cotangent
bundles which are Riemannian almost product (locally product) manifolds, or
(almost) para-Hermitian manifolds, with respect to the obtained almost product
structure, and a natural diagonal lifted metric $G$. Finally we obtain the
(almost) para-Kähler cotangent bundles of natural diagonal lift type.

Throughout this paper, the manifolds, tensor fields and other geometric
objects are assumed to be differentiable of class $C^{\infty}$ (i.e. smooth).
The Einstein summation convention is used , the range of the indices
$h,i,j,k,l,m,r,$ being always $\\{1,\dots,n\\}$.

##  2\. Preliminary results

The cotangent bundle of a smooth $n$-dimensional Riemannian manifold may be
endowed with a structure of $2n$-dimensional smooth manifold, induced by the
structure on the base manifold. If $(M,g)$ is a smooth Riemannian manifold of
dimension $n$, we denote its cotangent bundle by $\pi:T^{*}M\rightarrow M$.
Every local chart on $M$, $(U,\varphi)=(U,x^{1},\dots,x^{n})$ induces a local
chart $(\pi^{-1}(U),\Phi)=(\pi^{-1}(U),q^{1},\dots,q^{n},$
$p_{1},\dots,p_{n})$, on $T^{*}M$, as follows. For a cotangent vector
$p\in\pi^{-1}(U)\subset T^{*}M$, the first $n$ local coordinates
$q^{1},\dots,q^{n}$ are the local coordinates of its base point $x=\pi(p)$ in
the local chart $(U,\varphi)$ (in fact we have
$q^{i}=\pi^{*}x^{i}=x^{i}\circ\pi,\ i=1,\dots n)$. The last $n$ local
coordinates $p_{1},\dots,p_{n}$ of $p\in\pi^{-1}(U)$ are the vector space
coordinates of $p$ with respect to the basis
$(dx^{1}_{\pi(p)},\dots,dx^{n}_{\pi(p)})$, defined by the local chart
$(U,\varphi)$, i.e. $p=p_{i}dx^{i}_{\pi(p)}$.

The concept of $M$-tensor field on the cotangent bundle of a Riemannian
manifold was defined by the present author in [7], in the same manner as the
$M$-tensor fields were introduced on the tangent bundle (see [18]).

We recall the splitting of the tangent bundle to $T^{*}M$ into the vertical
distribution $VT^{*}M={\rm Ker}\ \pi_{*}$ and the horizontal one determined by
the Levi Civita connection $\dot{\nabla}$ of the metric $g$:

(2.1)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}TT^{*}M=VT^{*}M\oplus
HT^{*}M.$

If $(\pi^{-1}(U),\Phi)=(\pi^{-1}(U),q^{1},\dots,q^{n},p_{1},\dots,p_{n})$ is a
local chart on $T^{*}M$, induced from the local chart
$(U,\varphi)=(U,x^{1},\dots,x^{n})$, the local vector fields
$\frac{\partial}{\partial p_{1}},\dots,\frac{\partial}{\partial p_{n}}$ on
$\pi^{-1}(U)$ define a local frame for $VT^{*}M$ over $\pi^{-1}(U)$ and the
local vector fields $\frac{\delta}{\delta q^{1}},\dots,\frac{\delta}{\delta
q^{n}}$ define a local frame for $HT^{*}M$ over $\pi^{-1}(U)$, where
$\frac{\delta}{\delta q^{i}}=\frac{\partial}{\partial
q^{i}}+\Gamma^{0}_{ih}\frac{\partial}{\partial p_{h}},\
\Gamma^{0}_{ih}=p_{k}\Gamma^{k}_{ih},$ and $\Gamma^{k}_{ih}(\pi(p))$ are the
Christoffel symbols of $g$.

The set of vector fields $\\{\frac{\partial}{\partial
p_{i}},\frac{\delta}{\delta q^{j}}\\}_{i,j=\overline{1,n}}$, denoted by
$\\{\partial^{i},\delta_{j}\\}_{i,j=\overline{1,n}}$, defines a local frame on
$T^{*}M$, adapted to the direct sum decomposition (2.1).

We consider the energy density defined by $g$ in the cotangent vector $p$:

$\displaystyle\begin{array}[]{c}t=\frac{1}{2}\|p\|^{2}=\frac{1}{2}g^{-1}_{\pi(p)}(p,p)=\frac{1}{2}g^{ik}(x)p_{i}p_{k},\
\ \ p\in\pi^{-1}(U).\end{array}$

We have $t\in[0,\infty)$ for all $p\in T^{*}M$.

In the sequel we shall use the following lemma, which may be proved easily.

######  Lemma 2.1.

If $n>1$ and $u,v$ are smooth functions on $T^{*}M$ such that

$\displaystyle ug_{ij}+vp_{i}p_{j}=0,\quad ug^{ij}+vg^{0i}g^{0j}=0,\quad or\
u\delta^{i}_{j}+vg^{0i}p_{j}=0,$

on the domain of any induced local chart on $T^{*}M$, then $u=0,\ v=0$. We
used the notation $g^{0i}=p_{h}g^{hi}$.

##  3\. Almost product structures of natural diagonal lift type on the
cotangent bundle

In this section we shall find the almost product structures on the (total
space of the) cotangent bundle, which are natural diagonal lifts of the metric
from the base manifold $M$ to $T^{*}M$. Then we shall study the integrability
conditions for the determined structures, obtaining the natural diagonal
locally product structures on $T^{*}M$.

An _almost product structure_ $J$ on a differentiable manifold $M$ is an
$(1,1)$\- tensor field on $M$ such that $J^{2}=I$. The pair $(M,J)$ is called
an _almost product manifold_. When the almost product structure $J$ is
integrable, it is called a _locally product structure_ , and the manifold
$(M,J)$ is a _locally product manifold_.

An _almost paracomplex manifold_ is an almost product manifold $(M,J)$, such
that the two eigenbundles associated to the two eigenvalues $+1$ and $-1$ of
$J$, respectively, have the same rank. Equivalently, a splitting of the
tangent bundle $TM$ into the Whitney sum of two subbundles $T^{\pm}M$ of the
same fiber dimension is called an _almost paracomplex structure_ on $M$.

V. Cruceanu presented in [6] two simple almost product structures on the total
space $T^{*}M$ of the cotangent bundle, obtained by considering on the base
manifold $M$ a linear connection $\nabla$ and a non-degenerate $(0,2)$-tensor
field $g$. If $\alpha$ is a differentiable 1-form and $X$ is a vector field on
$M$, $\alpha^{V}$ denotes the vertical lift of $\alpha$ and $X^{H}$ the
horizontal lift of $X$ to $T^{*}M$, one can consider

(3.1)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P(X^{H})=-X^{H},\
P(\alpha^{V})=\alpha^{V},$ (3.2)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}Q(X^{H})=(X^{\flat})^{V},\
Q(\alpha^{V})=(\alpha^{\sharp})^{H},$

where $X^{\flat}=g_{X}$ is the 1-form on $M$ defined by
$X^{\flat}(Y)=g_{X}(Y)=g(X,Y),\forall Y\in\mathcal{T}^{1}_{0}(M)$,
$\alpha^{\sharp}=g^{-1}_{\alpha}$ is a vector field on $M$ defined by
$g(\alpha^{\sharp},Y)=\alpha(Y),~{}\forall Y\in\mathcal{T}^{1}_{0}(M)$. $P$ is
a paracomplex structure if and only if $\nabla$ has vanishing curvature, while
$Q$ is paracomplex if and only if the curvature of $\nabla$ and the exterior
covariant differential $Dg$ of $g$, given by

$(Dg)(X,Y)=\nabla_{X}(Y^{\flat})-\nabla_{Y}(X^{\flat})-[X,Y]^{\flat},$

vanishes.

The results from [13] and [14] concerning the natural lifts, allow us to
introduce an $(1,1)$ tensor field $P$ on $T^{*}M$, which is a natural diagonal
lift of the metric $g$ from the base manifold to the total space $T^{*}M$ of
cotangent bundle. Using the adapted frame
$\\{\partial^{i},\delta_{j},\\}_{i,j=\overline{1,n}}$ to $T^{*}M$, we define
$P$ by the relations:

(3.3) $\displaystyle P\delta_{i}=P^{(1)}_{ij}\partial^{j},\quad
P\partial^{i}=P_{(2)}^{ij}\delta_{j},$

where the $M-$tensor fields involved as coefficients have the forms

(3.4)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}P^{(1)}_{ij}=a_{1}(t)g_{ij}+b_{1}(t)p_{i}p_{j},\quad
P_{(2)}^{ij}=a_{2}(t)g^{ij}+b_{2}(t)g^{0i}g^{0j}.$

$a_{1},\ b_{1},\ a_{2},$ and $b_{2}$ being smooth functions of the energy
density $t$.

The invariant expression of the defined structure is

(3.8)
$\displaystyle\begin{array}[]{c}PX^{H}_{p}=a_{1}(t)(X^{\flat})^{V}_{p}+b_{1}(t)p(X)p_{p}^{V},\\\
\\\
P\alpha^{V}_{p}=a_{2}(t)(\alpha^{\sharp})_{p}^{H}+b_{2}(t)g^{-1}_{\pi(p)}(p,\alpha)(p^{\sharp})_{p}^{H},\end{array}$

in every point $p$ of the induced local card $(\pi^{-1}(U),\Phi)$ on $T^{*}M$,
$\forall~{}X\in\mathcal{T}^{1}_{0}(M),\forall~{}\alpha\in\mathcal{T}^{0}_{1}(M)$.
The vector $p^{\sharp}$ is tangent to $M$ in $\pi(p)$,
$p^{V}=p_{i}\partial^{i}$ is the Liouville vector field on $T^{*}M$ , and
$(p^{\sharp})^{H}=g^{0i}\delta_{i}$ is the geodesic spray on $T^{*}M$.

######  Example 3.1.

When $a_{1}=a_{2}=1,$ $b_{1}$ and $b_{2}$ vanish, we have the structure given
by $(\ref{Q}).$

The following theorems present the conditions under which the above tensor
field $P$ defines an almost product (locally product) structure on the total
space of the cotangent bundle.

######  Theorem 3.2.

The tensor field $P$, given by (3.3) or (3.8), defines an almost product
structure of natural diagonal lift type on $T^{*}M$, if and only if its
coefficients satisfy the relations

(3.9)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{1}=\frac{1}{a_{2}}\
,\ \ \ a_{1}+2tb_{1}=\frac{1}{a_{2}+2tb_{2}}.$

###### Proof.

The condition $P^{2}=I$ in the definition of the almost product structure, may
be written in the following form, by using (3.3):

$P^{(1)}_{ij}P_{(2)}^{il}=\delta^{l}_{j},~{}~{}~{}~{}~{}~{}P_{(2)}^{ij}P^{(1)}_{il}=\delta^{j}_{l},$

and replacing (3.4) it becomes

$(a_{1}a_{2}-1)\delta^{l}_{j}+[b_{1}(a_{2}+2tb_{2})+a_{1}b_{2}]g^{0l}p_{j}=0.$

Using Lemma 2.1, we have that the above expression vanish if and only if

$a_{1}=\frac{1}{a_{2}},\ b_{1}=-\frac{a_{1}b_{2}}{a_{2}+2tb_{2}},$

which imply also the second relation in (3.9). ∎

######  Remark 3.3.

If $a_{1}=\frac{1}{\beta},\ a_{2}=\beta$, $b_{1}=\frac{u}{\alpha\beta}$ and
$b_{2}=-\frac{u\beta}{\alpha+2tu}$, where $\alpha$ and $\beta$ are real
constants and $u$ is a smooth function of $t$, the statements of Theorem 3.2
are satisfied, so the structure considered in [24] is an almost product
structure on the total space $T^{*}M$ of the cotangent bundle.

######  Theorem 3.4.

The natural diagonal almost product structure $P$ on the total space of the
cotangent bundle of an $n$-dimensional connected Riemannian manifold $(M,g)$,
with $n>2$, is a locally product structure on $T^{*}M$ (i.e. $P$ is
integrable) if and only if the base manifold is of constant sectional
curvature $c$, and the coefficients $b_{1},\ b_{2}$ are given by:

(3.10)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b_{1}=\frac{a_{1}a_{1}^{\prime}+c}{a_{1}-2ta_{1}^{\prime}},\
b_{2}=\frac{a_{1}a_{2}^{\prime}-a_{2}^{2}c}{a_{1}+2cta_{2}}.$

###### Proof.

The almost product structure $P$ on $T^{*}M$ is integrable if and only if the
vanishing condition for the Nijenhuis tensor field $N_{P}$,

$N_{P}(X,Y)=[PX,PY]-P[PX,Y]-P[X,PY]+P^{2}[X,Y],\forall
X,Y\in\mathcal{T}^{1}_{0}(T^{*}M),$

is satisfied.

Studying the components of $N_{P}$ with respect to the adapted frame on
$T^{*}M$, $\\{\partial^{i},\delta_{j}\\}_{i,j=\overline{1,n}}$, we first
obtain:

$N_{P}(\partial^{i},\partial^{j})=[P^{(1)}_{km}(\partial^{j}P_{(2)}^{mi}-\partial^{i}P_{(2)}^{mj})+\operatorname{Rim}^{0}_{kml}P_{(2)}^{mi}P_{(2)}^{lj}]\partial^{k}.$

Replacing the values (3.4), after some tensorial computations the above
expression becomes

(3.11)
$a_{1}(a_{2}^{\prime}-b_{2})(\delta^{h}_{i}p_{j}-\delta^{h}_{j}p_{i})-a_{2}^{2}\operatorname{Rim}^{h}_{kij}+a_{2}b_{2}(\operatorname{Rim}^{h}_{kjl}p_{i}-\operatorname{Rim}^{h}_{kil}p_{j})g^{0k}g^{0l}=0.$

We differentiate (3.11) with respect to $p_{h}$. Since the curvature of the
base manifold does not depend on $p$, we take the value of this derivative at
$p=0$, and we obtain

(3.12)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R^{h}_{kij}=c(\delta^{h}_{i}g_{kj}-\delta^{h}_{j}g_{ki}),$

where

$c=\frac{a_{1}(0)}{a_{2}^{2}(0)}(a^{\prime}_{2}(0)-b_{2}(0))$

is a function depending on $q^{1},...,q^{n}$ only. Schur’s theorem implies
that $c$ must be a constant when $M$ is connected, of dimension $n>2$.

Moreover, by using (3.12), the relation (3.11) becomes

(3.13)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[a_{1}a_{2}^{\prime}-a_{2}^{2}c-b_{2}(a_{1}+2ta_{2}c)](\delta^{h}_{i}g_{kj}-\delta^{h}_{j}g_{ki})=0.$

Solving (3.13) with respect to $b_{2}$, we obtain the second relation in
(3.10).

The Nijenhuis tensor field computed for both horizontal arguments is

$N_{P}(\delta_{i},\delta_{j})=(P^{(1)}_{li}\partial^{l}P^{(1)}_{hj}-P^{(1)}_{lj}\partial^{l}P^{(1)}_{hi}+\operatorname{Rim}^{0}_{hij})\partial^{h},$

which vanishes if and only if

$[b_{1}(2ta_{1}^{\prime}-a_{1})+a_{1}^{\prime}a_{1}+c](g_{hj}p_{i}-g_{hi}p_{j})=0,$

namely when $b_{1}$ has the expression in (3.10).

The mixed components of the Nijenhuis tensor field, have the forms

$N_{P}(\delta_{i},\partial^{j})=-N_{P}(\partial^{j},\delta_{i})=(P^{(1)}_{mi}\partial^{m}P_{(2)}^{hj}+P_{(2)}^{hl}\partial^{j}P^{(1)}_{li}-P_{(2)}^{lh}P_{(2)}^{jm}\operatorname{Rim}^{0}_{lim})\delta_{h},$

which after replacing (3.4) and (3.12) become

$(a_{1}a_{2}^{\prime}+a_{2}b_{1}-a_{2}^{2}c+2ta_{2}^{\prime}b_{1})g^{hj}p_{i}$
$+(a_{2}b_{1}+a_{1}b_{2}+2tb_{1}b_{2})\delta^{j}_{i}g^{0h}+(a_{1}^{\prime}a_{2}+a_{1}b_{2}+a_{2}^{2}c+2cta_{2}b_{2})\delta^{h}_{i}g^{0j}$
$+(a_{2}b_{1}^{\prime}+a_{1}^{\prime}b_{2}+3b_{1}b_{2}+a_{1}b_{2}^{\prime}-a_{2}b_{2}c+2tb_{1}^{\prime}b_{2}+2tb_{1}b_{2}^{\prime})p_{i}g^{0h}g^{0j}.$

Taking (3.9) into account, the above expression takes the form

$\frac{(a_{1}-2a_{1}^{\prime}t)b_{1}-a_{1}a_{1}^{\prime}-c}{a_{1}^{2}}g^{hj}p_{i}+\frac{a_{1}a_{1}^{\prime}+c-(a_{1}-2a_{1}^{\prime}t)b_{1}}{a_{1}(a_{1}+2tb_{1})}\delta^{h}_{i}g^{0j}$
$+\frac{(a_{1}a_{1}^{\prime}+c)b_{1}-(a_{1}-2ta_{1}^{\prime})b_{1}^{2}}{a_{1}^{2}(a_{1}+2tb_{1})}p_{i}g^{0h}g^{0j}$

and it vanishes if and only if $b_{1}$ is expressed by the first relation in
(3.10).

One can verify that all the components of the Nijenhuis tensor field vanish
under the same conditions, so the almost product structure $P$ on $T^{*}M$ is
integrable, i.e. $P$ is a locally product structure on $T^{*}M$. ∎

######  Remark 3.5.

If the coefficients involved in the definition of $P$ have the values
presented in Remark 3.3, the relations (3.10) take the form
$u=c\alpha\beta^{2}$, so Theorem 3.4 implies the results stated in [24,
Theorem 4.2].

##  4\. Natural diagonal Riemannian almost product and almost para-Hermitian
structures on $T^{*}M$

Authors like M. Anastasiei, C. Bejan, V. Cruceanu, H. Farran, A. Heydari, S.
Ishihara, I. Mihai, G. Mitric, C. Nicolau, V. Oproiu, L. Ornea, N. Papaghiuc,
E. Peyghan, K. Yano, and M. S. Zanoun considered almost product structures and
almost para-Hermitian structures (called also almost hyperbolic Hermitian
structures) on the total spaces of the tangent and cotangent bundles.

A Riemannian manifold $(M,g)$, endowed with an almost product structure $J$,
satisfying the relation

(4.1)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}g(JX,JY)=\varepsilon
g(X,Y),\ \forall X,Y\in\mathcal{T}^{1}_{0}(M),$

is called a _Riemannian almost product manifold_ if $\varepsilon=1$, or an
_almost para-Hermitian manifold_ (called also an _almost hyperbolic Hermitian
manifold_) if $\varepsilon=-1$.

In the following we shall find the Riemannian almost product (locally product)
and the (almost) para-Hermitian cotangent bundles of natural diagonal lift
type.

To this aim, we consider a natural diagonal lifted metric on the total space
$T^{*}M$ of the cotangent bundle, defined by:

(4.2)
$\left\\{\begin{array}[]{l}G_{p}(X^{H},Y^{H})=c_{1}(t)g_{\pi(p)}(X,Y)+d_{1}(t)p(X)p(Y),\\\
\\\
G_{p}(\alpha^{V},\omega^{V})=c_{2}(t)g^{-1}_{\pi(p)}(\alpha,\omega)+d_{2}(t)g^{-1}_{\pi(p)}(p,\alpha)g^{-1}_{\pi(p)}(p,\omega),\\\
\\\ G_{p}(X^{H},\alpha^{V})=G_{p}(\alpha^{V},X^{H})=0,\end{array}\right.$

$\forall~{}X,Y\in\mathcal{T}^{1}_{0}(M),$
$\forall~{}\alpha,\omega\in\mathcal{T}^{0}_{1}(M),\forall~{}p\in T^{*}M$,
where the coefficients $c_{1},\ c_{2},\ d_{1},\ d_{2}$ are smooth functions of
the energy density.

The conditions for $G$ to be nondegenerate are assured if

$c_{1}c_{2}\neq 0,\ (c_{1}+2td_{1})(c_{2}+2td_{2})\neq 0.$

The metric $G$ is positive definite if

$c_{1}+2td_{1}>0,\quad c_{2}+2td_{2}>0.$

Using the adapted frame $\\{\partial^{i},\delta_{j}\\}_{i,j=\overline{1,n}}$
on $T^{*}M$, (4.2) becomes

(4.3)
$\displaystyle\begin{cases}G(\delta_{i},\delta_{j})=G^{(1)}_{ij}=c_{1}(t)g_{ij}+d_{1}(t)p_{i}p_{j},\\\
G(\partial^{i},\partial^{j})=G_{(2)}^{ij}=c_{2}(t)g^{ij}+d_{2}(t)g^{0i}g^{0j},\\\
G(\partial^{i},\delta_{j})=G(\delta_{i},\partial^{j})=0,\end{cases}$

where $c_{1},\ c_{2},\ d_{1},\ d_{2}$ are smooth functions of the density
energy on $T^{*}M$.

Next we shall prove the following characterization theorem:

######  Theorem 4.1.

Let $(M,g)$ be an $n$-dimensional connected Riemannian manifold, with $n>2$,
and $T^{*}M$ the total space of its cotangent bundle. Let $G$ be a natural
diagonal lifted metric on $T^{*}M$, defined by (4.2), and $P$ an almost
product structure on $T^{*}M$, characterized by Theorem 3.2. Then
$(T^{*}M,G,P)$ is a Riemannian almost product manifold, or an almost para-
Hermitian manifold if and only if the following proportionality relations
between the coefficients hold:

(4.4)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{c_{1}}{a_{1}}=\varepsilon\frac{c_{2}}{a_{2}}=\lambda,\
\frac{c_{1}+2td_{1}}{a_{1}+2tb_{1}}=\varepsilon\frac{c_{2}+2td_{2}}{a_{2}+2tb_{2}}=\lambda+2t\mu,$

where $\varepsilon$ takes the corresponding values from the definition
$(\ref{defprodpara})$, and the proportionality coefficients $\lambda>0$ and
$\lambda+2t\mu>0$ are some functions depending on the energy density $t$.

If moreover, the relations stated in Theorem 3.4 are fulfilled, then
$(T^{*}M,G,$ $P)$ is a Riemannian locally product manifold for
$\varepsilon=1$, or a para-Hermitian manifold for $\varepsilon=-1$.

###### Proof.

With respect to the adapted frame
$\\{\partial^{i},\delta_{j}\\}_{i,j=\overline{1,n}}$, the relation (4.1)
becomes:

(4.5) $~{}~{}~{}~{}G(P\delta_{i},P\delta_{j})=\varepsilon
G(\delta_{i},\delta_{j}),\ G(P\partial^{i},P\partial^{j})=\varepsilon
G(\partial^{i},\partial^{j}),\ G(P\partial^{i},P\delta_{j})=0,$

and using (3.3) and (4.3) we have

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{array}[]{c}(-\varepsilon
c_{1}+a_{1}^{2}c_{2})g_{ij}+[-\varepsilon
d_{1}+a_{1}^{2}d_{2}+2b_{1}c_{2}(a_{1}+tb_{1})+4tb_{1}d_{2}(a_{1}+tb_{1})]p_{i}p_{j}=0,\\\
\\\ (a_{2}^{2}c_{1}-\varepsilon c_{2})g^{ij}+[-\varepsilon
d_{2}+a_{2}^{2}d_{1}+2b_{2}c_{1}(a_{2}+tb_{2})+4tb_{2}d_{1}(a_{2}+tb_{2})]g^{0i}g^{0j}=0.\end{array}$

Taking into account Lemma 2.1, the coefficients which appear in the above
expressions vanish. Due to the first relation in (3.9), we get by equalizing
to zero the coefficients of $g_{ij}$ and $g^{ij}$, the first relation in
(4.4).

Moreover, multiplying by $2t$ the coefficients of $p_{i}p_{j}$ and
$g^{0i}g^{0j}$ and adding them to the coefficients of $g_{ij}$ and $g^{ij}$,
respectively, we obtain:

(4.10)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{array}[]{l}-\varepsilon(c_{1}+2td_{1})+(a_{1}+2tb_{1})^{2}(c_{2}+2td_{2})=0,\\\
\\\
(a_{2}+2tb_{2})^{2}(c_{1}+2tb_{1})-\varepsilon(c_{2}+2td_{2})=0.\end{array}$

Using the second relation in (3.9), (4.10) leads to the second relation in
(4.4). ∎

######  Remark 4.2.

When the coefficients of the almost product structure $P$ have the expressions
in Remark 3.3, and the coefficients of the metric $G$ on $T^{*}M$ are
$c_{1}=a_{1},$ $d_{1}=b_{1},$ $c_{2}=-a_{2}$ and $d_{2}=-b_{2},$ Theorem 4.1
implies that $T^{*}M$ endowed with the almost product structure and with the
metric considered in [24] is an almost para-Hermitian manifold.

##  5\. Natural diagonal para-Kähler structures on $T^{*}M$

In the sequel we shall study the cotangent bundles endowed with para-Kähler
structures of natural diagonal lift type. This class of almost para-Hermitian
structures, studied for example in [1] and [15], is characterized by the
closure of the associated 2-form $\Omega$.

The 2-form $\Omega$ associated to the almost para-Hermitian structure $(G,P)$
of natural diagonal lift type on the total space of the cotangent bundle is
given by the relation

$\Omega(X,Y)=G(X,PY),\forall X,Y\in\mathcal{T}^{1}_{0}(T^{*}M).$

Studying the closure of $\Omega$, we may prove the following theorem:

######  Theorem 5.1.

The almost para-Hermitian structure $(G,P)$ of natural diagonal lift type on
the total space $T^{*}M$ of the cotangent bundle of a Riemannian manifold
$(M,g)$ is almost para-Kählerian if and only if

$\mu=\lambda^{\prime}.$

###### Proof.

The 2-form $\Omega$ on $T^{*}M$ has the following expression with respect to
the local adapted frame $\\{\partial^{i},\delta_{j}\\}_{i,j=1,\dots,n}$:

$\Omega(\partial^{i},\partial^{j})=\Omega(\delta_{i},\delta_{j})=0,\quad\Omega(\partial^{j},\delta_{i})=G_{(2)}^{jh}P^{(1)}_{hi},\quad\Omega(\delta_{i},\partial^{j})=G^{(1)}_{ih}P_{(2)}^{hj},$

By substituting (3.4) and (4.3) in the above expressions, and taking into
account the conditions for $(T^{*}M,G,P)$ to be an almost para-Hermitian
manifold (see Theorem 4.1), we have

(5.1)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega(\delta_{i},\partial^{j})=-\Omega(\partial^{j},\delta_{i})=\lambda\delta^{j}_{i}+\mu
p_{i}g^{0j},$

which has the invariant expression

$\Omega\left(X_{p}^{H},\alpha_{p}^{V}\right)=\lambda\alpha(X)+\mu
p(X)g^{-1}_{\pi(p)}(p,\alpha),$

for every $X\in\mathcal{T}^{1}_{0}(M),\ \alpha\in\mathcal{T}^{0}_{1}(M),\ p\in
T^{*}M,$

Taking into account the relation (5.1) we have that the $2$-form $\Omega$
associated to the natural diagonal para-Hermitian structure has the form

(5.2)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega=(\lambda\delta^{j}_{i}+\mu
p_{i}g^{0j})dq^{i}\wedge Dp_{j},$

where $Dp_{j}=dp_{j}-\Gamma^{0}_{jh}dq^{h}$ is the absolute differential of
$p_{i}$.

Moreover, the differential of $\Omega$ will be

$d\Omega=(d\lambda\delta^{j}_{i}+d\mu g^{0j}p_{i}+\mu dg^{0j}p_{i}+\mu
g^{0j}dp_{i})\wedge dq^{i}\wedge Dp_{j}-(\lambda\delta^{j}_{i}+\mu
p_{i}g^{0j})dq^{i}\wedge dDp_{j}.$

Let us compute the expressions of $d\lambda,\ d\mu,\ dg^{0i}$ and $dDp_{i}$:

$d\lambda=\lambda^{\prime}g^{0h}Dp_{h},\quad
d\mu=\mu^{\prime}g^{0h}Dp_{h},\quad
dg^{0i}=g^{hi}Dp_{h}-\Gamma^{i}_{j0}dq^{j},$
$dDp_{i}=\frac{1}{2}R^{0}_{ijh}dq^{h}\wedge dq^{j}-\Gamma^{h}_{ij}Dp_{h}\wedge
dq^{j}.$

Then, by substituting these relations into the expression of $d\Omega$, taking
into account the properties of the external product, the symmetry of $g^{ij}$
and $\Gamma^{h}_{ij}$ and the Bianchi identities, we obtain

$d\Omega=(\mu-\lambda^{\prime})p_{k}g^{kh}\delta^{i}_{j}Dp_{h}\wedge
Dp_{i}\wedge dq^{j},$

which, due to the antisymmetry of $\delta^{i}_{j}Dp_{i}\wedge dq^{j}$, may be
written as

$d\Omega=\frac{1}{2}(\mu-\lambda^{\prime})p_{k}(g^{kh}\delta^{j}_{i}-g^{kj}\delta^{h}_{i})Dp_{h}\wedge
Dp_{j}\wedge dq^{i},$

and it vanishes if and only if $\mu=\lambda^{\prime}$. ∎

Using the theorems 3.2, 3.4 and 5.1, we immediately prove:

######  Theorem 5.2.

An almost para-Hermitian structure $(G,P)$ of natural diagonal lift type on
the total space $T^{*}M$ of the cotangent bundle of a Riemannian manifold
$(M,g)$ is para-Kählerian if and only if $P$ is a locally product structure
(see Theorem 3.4) and $\mu=\lambda^{\prime}$.

######  Remark 5.3.

The almost para-Kählerian structures of natural diagonal lift type on $T^{*}M$
depend on three essential coefficients $a_{1},\ b_{1}$ and $\lambda$, while
the natural diagonal para-Kählerian structures on $T^{*}M$ depend on two
essential coefficients $a_{1}$ and $\lambda$, which in both cases must satisfy
the supplementary conditions $a_{1}>0,\ a_{1}+2tb_{1}>0,\ \lambda>0,\
\lambda+2t\lambda^{\prime}>0$, where $b_{1}$ is given by $(\ref{b}).$

######  Remark 5.4.

Taking into account Remark 4.2, we have that the constant $\lambda$ is equal
to $1$ and $\mu$ vanishes, so Theorem 5.1 leads to the statements in [24,
Theorem 3.1], namely the structure constructed in [24] is almost para-
Kählerian on $T^{*}M$. Moreover, taking into account Remark 3.5, it follows
that the relations (14) in [24] are fulfilled in the case when the constructed
structure is para-Kähler.

Acknowledgements. The author would like to thank Professor V. Oproiu for the
techniques learned and Professor M. I. Munteanu for the critical remarks
during the preparation of this work.

The paper was supported by the Program POSDRU/89/1.5/S/49944, ”Al. I. Cuza”
University of Iaşi, Romania.

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Author’s address: Simona-Luiza Druţă-Romaniuc, Department of Sciences, 54
Lascar Catargi Street RO-700107, ”Al. I. Cuza” University of Iaşi, Romania,
e-mail<EMAIL_ADDRESS>