# 3D-OGSE: Online Safe and Smooth Trajectory Generation using Generalized
Shape Expansion in Unknown 3-D Environments

Vrushabh Zinage1, Senthil Hariharan Arul2, Dinesh Manocha2, Satadal Ghosh1
1Vrushabh Zinage and Satadal Ghosh are with the Department of Aerospace
Engineering, Indian Institute of Technology Madras, Chennai, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>Hariharan Arul and
Dinesh Manocha are at University of Maryland, College Park, USA<EMAIL_ADDRESS>
###### Abstract

In this paper, we present an online motion planning algorithm (3D-OGSE) for
generating smooth, collision-free trajectories over multiple planning
iterations for 3-D agents operating in an unknown obstacle-cluttered 3-D
environment. Our approach constructs a safe-region, termed ’generalized
shape’, at each planning iteration, which represents the obstacle-free region
based on locally-sensed environment information. A collision-free path is
computed by sampling points in the generalized shape and is used to generate a
smooth, time-parametrized trajectory by minimizing snap. The generated
trajectories are constrained to lie within the generalized shape, which
ensures the agent maneuvers in the locally obstacle-free space. As the agent
reaches boundary of ’sensing shape’ in a planning iteration, a re-plan is
triggered by receding horizon planning mechanism that also enables
initialization of the next planning iteration. Theoretical guarantee of
probabilistic completeness over the entire environment and of completely
collision-free trajectory generation is provided. We evaluate the proposed
method in simulation on complex 3-D environments with varied obstacle-
densities. We observe that each re-planing computation takes $\sim$1.4
milliseconds on a single thread of an Intel Core i5-8500 3.0 GHz CPU. In
addition, our method is found to perform 4-10 times faster than several
existing algorithms. In simulation over complex scenarios such as narrow
passages also we observe less conservative behavior.

##  I INTRODUCTION

Due to their small size and superior agility, UAVs (unmanned aerial vehicles)
are increasingly being used for various applications including search and
rescue, surveillance, and exploration. Generating smooth feasible collision-
free trajectories for such 3-D agents, especially in obstacle-dense or
cluttered 3-D environments, forms a key problem. This becomes more challenging
when the operating environment is not known prior to the flight [1, 2] as it
is not possible to pre-compute a collision-free trajectory in such scenarios.
As a consequence, we need efficient online planning algorithms that can
rapidly generate a smooth and safe trajectory for 3-D agents using only local
sensor data.

There has been extensive research over last few decades on trajectory
generation [2, 3, 4]. Prior trajectory generation methods can broadly be
grouped into three categories: (1) optimization-based algorithms [4, 5]; (2)
search-based/motion primitives [6, 7, 8]; and (3) sampling-based methods [9,
10, 11]. Except few of these algorithms like [7, 8], most of them work based
on a-priori known complete information of environment, thus posing a
limitation for real-time implementation in navigation of 3-D agent through
unknown environments.

Figure 1: A 3-D agent maneuvering in a random forest environment using
3D-OGSE. The agent has a sensing region of edge 10m, and the multicolored
obstacles represent the obstacles inside the agent’s sensing region. The brown
obstacles are not visible to the agent at the current instance. The red curve
represents an illustration of a planned trajectory generated by our approach,
which can handle obstacle-dense 3-D environments.

To this end, for online smooth and safe trajectory generation, optimization-
based [2, 12] and search-based methods [13, 14, 15] have been explored in
recent literature. In the latter approach, the free-space is approximated as a
sequence of convex shapes to reduce the complexity and enable fast
computation, and a notion of safe corridor has been adopted for safe online
trajectory generation. It is also important to design online trajectory
generation methods that can provide guarantees of algorithm completeness.

Main Contributions: We present a 3D online GSE (named as ’3D-OGSE’) algorithm
for generating collision-free smooth dynamically feasible trajectories for 3-D
agents operating in obstacle-cluttered a-priori unknown 3-D environments. Our
approach is based on recent offline algorithms that use the notion of the
generalized shape [11, 16]. We exploit the property that sampling-based
algorithms are computationally more efficient than search-based methods for
higher dimensions. Based on the locally available information on environment
at every planning iteration, a receding horizon planning (RHP) mechanism is
framed for trajectory re-planning. While horizon of the RHP mechanism is
restricted by the sensing region of on-board sensors of 3-D agent, the re-
planning trigger is planned based on a novel notion of ’sensing shape’ defined
using sensing region and the notion of the generalized shape [16]. Once a
feasible path is generated at every planning iteration, snap minimization
quadratic problem is formulated for generating smooth optimal trajectories
that are dynamically feasible [17]. We show that the use of generalized shape
in sampling points during path planning and in constraining trajectory points
in trajectory optimization step at every planning iteration renders the
generated trajectory completely collision-free. The novel components of our
online approach include:

  1. 1.

Fast and safe path planning leveraging a novel notion of ‘sensing shape’
derived from on-board sensing region and 3D generalized shape at every
planning iteration.

  2. 2.

A practically realizable sensing-restricted receding horizon planning
mechanism and fast trajectory optimization for generating dynamically feasible
safe trajectory generation.

  3. 3.

Theoretical guarantee of probabilistic completeness of the presented 3D-OGSE
for generation of collision-free feasible path from start point to goal point
over multiple planning iterations. Besides, we provide guarantee on the safety
of generated overall trajectory.

  4. 4.

Extensive evaluation of the efficacy of the presented 3D-OGSE algorithm under
FOV-restricted, noisy sensing scenario in simulated complex environments.
Extensive comparison study of the same with several other well-studied
algorithms in literature reveals that the trajectory cost of the 3D-OGSE is
comparable to them, while in terms of computation time, the 3D-OGSE is about
$4-10\times$ faster than them.

The outline of the paper is as follows. We discuss relevant literature in
Section II, and introduce the preliminaries and problem formulation in Section
III. Subsequently, 3D-OGSE algorithm is presented in Section IV, followed by
detailed proofs of its probabilistic completeness and guarantee of collision-
free trajectory in Section V. proves the probabilistic completeness guarantee
that the agent converges to the goal position. Finally, extensive numerical
simulations and comparison with state-of-the-art methods are given in Section
VI, followed by concluding remarks in Section VII.

##  II Related Work

In this section, we summarize the related work in the areas of path planning
and online trajectory generation.

Optimization-based methods: Sequential convex programming (SCP) has been used
in [2] and [3] for generating smooth trajectories, while mixed-integer
optimization methods have been explored in [4, 5] for generating reliable,
collision-free, dynamically feasible trajectories. Several of these methods
suffered from higher computational burden.A minimum snap formulation was
presented in [17] that used differential flatness and pose the trajectory
generation problems as Quadratic program (QP) problem. Online 3-D trajectories
can be generated using QP methods [14, 15, 18]. These techniques use convex
safe corridors and provide a considerable performance improvement over mixed-
integer methods. The trajectory generation problem was formulated as a
nonlinear optimization over penalty of smoothness and safety to obtain a
locally optimal solution in [19]. B-spline optimization was also utilized for
generation of dynamically feasible smooth trajectories in [12, 20].

Search-based methods: Search-based planning methods [6, 7, 8] were used for
fast generation of trajectories for UAVs. They build a graph by discretizing
the state-space of the UAVs, where states form the nodes, and motion
primitives form the edges in the graph. These methods rely on searching a
large pre-computed lookup table that contains the motion primitives to
identify the suitable maneuver. This problem was reduced greatly in [21] by
exploiting differential flatness property of the quadrotor. B-spline-based
kinodynamic (EBK) search algorithm was presented in [22] to find a feasible
trajectory with minimum control effort. Sequences of axes-aligned cubes in
free-space were used in [13] to generate safe corridor, while [14] leveraged
collection of connected convex polyhedrons in the free-space and [15] used
connected spheres of a radius equal to minimum distance to any local obstacle
for this purpose. While selection of cubes made the approach in [13] efficient
mainly for specific shape of obstacles, selection of restricted radius spheres
made the approach in [15] quite conservative in nature.

Sampling-based methods: Many sampling methods like PRM [23] and RRT [24] have
been used for collision-free path computation. Sampling-based methods such as
[9, 10, 11] avoid the explicit construction of configuration space and have
been successfully used for collision-free path computation in high dimensional
configuration spaces. These methods compute sampling points in the free space
and tend to grow the graph towards the goal configuration. [9] propose RRT*
and PRM*, which provide asymptotic optimality. Generalized Shape Expansion
(GSE)-based algorithm was recently proposed in [11, 16] for 2-D and 3-D
environments, respectively, which used a novel notion of generalized shape for
fast and efficient free-space exploration to generate a collision-free path.
It was found to show superior performance in computational time than several
well-established algorithms. An even faster planner was presented in [25] for
2-D environments and in [26] for 3-D environments by embedding directional
sampling scheme to the GSE for 2-D and 3-D environments, respectively. Path
planning in our 3D-OGSE algorithm relies on the generalized shape-driven [16]
fast sampling-based method, while we use snap minimization [17] for fast
trajectory generation.

##  III Problem Description and Preliminaries

(a) ${\mathcal{R}_{1,\boldsymbol{X}}}$ and ${\mathcal{R}_{2,\boldsymbol{X}}}$

(b) $\widehat{\mathcal{R}_{1,\boldsymbol{X}}}$,
$\widehat{\mathcal{R}_{2,\boldsymbol{X}}}$ and $\mathcal{I}$

(c) $\widetilde{\mathcal{R}_{12,\boldsymbol{X}}}$

(d) $\mathcal{S}_{\boldsymbol{X}}$

Figure 2: An illustration of 2-D projection of 3-D generalized shape
$\mathcal{S}_{X}$ about $\boldsymbol{X}$ in an environment having two
obstacles.

Notations |  Description
---|---
$\mathbb{X},\boldsymbol{X}$ |  3-D workspace and a point in 3-D workspace, respectively
$\mathbb{V},\mathbb{E}$ |  Vertex and edge set, respectively
$m_{\text{loc}}$ |  Number of obstacles after segmentation of local map in $\mathbb{X}$
$\boldsymbol{p}$ |  Current position of agent
$r_{\text{i,X}}$ |  Minimum distance of $\boldsymbol{X}$ from obstacle $i$
$p_{\text{i,X}}$ |  Point on obstacle $i$ at a distance of $r_{\text{i,X}}$ from $\boldsymbol{X}$
$n_{\text{i,X}}$ |  Vector from $\boldsymbol{X}$ to $p_{\text{i,X}}$
$\mathbb{D}_{\text{i,X}}$ |  set of points on obstacle $i$
$\mathbb{Q}_{\text{i,X}}$ |  points of intersection of plane with normal $\boldsymbol{n}_{\text{i,X}}$, passing through $\boldsymbol{p}_{\text{i,X}}$ and the lines joining $\mathbb{D}_{\text{i,X}}$ and $\mathbf{X}$
$l_{\text{i,X}}$ |  Maximum distance of point $\boldsymbol{p_{\text{i,X}}}$ from the points in $\mathbb{Q}_{\text{i,X}}$
$\mathcal{S}_{\boldsymbol{X}}$ |  3-D generalized shape about $\boldsymbol{X}$
$\mathcal{R}_{i,X}$ |  Spherical sector with vertex at the sampled point $\boldsymbol{X}$ radius equal to $r_{i,X}$, cone angle equal to $\text{tan}^{-1}({l_{i,X}/r_{i,X}})$ and axis along $\boldsymbol{n_{\text{i,X}}}$
$\widehat{\mathcal{R}_{i,X}}$ |  Intersection of $\mathbb{X}$ and the spherical sector with vertex at sampled point $\boldsymbol{X}$, cone angle $\texttt{tan}^{-1}(l_{i,X}/r_{i,X})$, axis along $\boldsymbol{n_{\text{i,X}}}$ and radius being infinite
$\widetilde{\mathcal{R}_{ij,X}}$ |  $\widehat{\mathcal{R}_{i,X}}\cap\widehat{\mathcal{R}_{j,X}}$ for $i<j$
$C_{\boldsymbol{p}}$ |  Sensing region about $\boldsymbol{p}$ (assumed to be a cube with center at $\boldsymbol{p}$ and dimensions $10m\times 10m\times 10m$)

TABLE I: Notations used in the paper.

###  III-A 3-D Environment and 3-D Agent

Consider a 3-D obstacle-cluttered environment denoted by
$\mathbb{X}\subset\mathbb{R}^{3}$. The environment is assumed to be path-
connected, that is, there always exists a collision-free path from any point
in the free-space $(\mathbb{X_{\text{free}}})$ to the goal point
$\boldsymbol{X}_{\text{goal}}$ with sufficient clearance from obstacle-space
$(\mathbb{X_{\text{obs}}})$. Thus,
$\mathbb{X_{\text{free}}}=\mathbb{X}\setminus\mathbb{X_{\text{obs}}}$.
However, the environment is not known a-priori. Instead it is assumed that
point cloud information about the local environment within the sensing region
of on-board sensor of 3-D agent is available with sufficient accuracy for
planning of safe trajectory that is dynamically feasible for the 3-D agent.
Additionally, we assume that the trajectory planning module of the 3-D agent
has access to point cloud data segmented into distinct obstacles in local
environment. Several computer vision and geometry processing techniques [27,
28] perform similar real-time segmentation. The sensing region is assumed to
be a cube of dimension $10m\times 10m\times 10m$ as shown in Fig. 1. We also
assume that the 3-D agent can localize itself in the locally available
environment information.

In subsequent discussion we assume the 3-D agent as a point mass. Note that
the approach developed in this paper could be easily extended to deal with
agent geometry by considering $\mathbb{X_{\text{safe}}}$ as the available
space for collision-free paths instead of $\mathbb{X_{\text{free}}}$, where
$\mathbb{X_{\text{safe}}}=\mathbb{X_{\text{free}}}\setminus\delta\mathbb{X_{\text{obs}}}$,
where $\delta\mathbb{X_{\text{obs}}}$ encapsulates the 3-D agent’s geometric
shape.

###  III-B Problem Formulation

Given initial and goal states
$\boldsymbol{X}_{\text{init}},\boldsymbol{X}_{\text{goal}}\in\mathbb{X}_{\text{free}}$,
respectively, the task is to plan a time-parametrized trajectory
$\sigma(t):[t_{0},t_{f}]\longrightarrow\mathbb{R}^{3}$ based on the locally
sensed environment information, until the agent reaches
$\boldsymbol{X}_{\text{goal}}$ starting form $\boldsymbol{X}_{\text{init}}$,
where $t_{0}$ is the initial time and $t_{f}$ is the time at which the agent
reaches $\boldsymbol{X}_{\text{goal}}$. Generated trajectory also needs to
ensure that the total snap incurred $c(\sigma(t))$ by the 3-D agent is
minimized subject to constraints on maximum magnitudes of velocity and
acceleration of the agent. Snap minimization takes care of the smoothening of
trajectory to make it dynamically feasible to follow for any realistic vehicle
[14]. The problem can be expressed as:

$\displaystyle\underset{{{\sigma(t)}}}{\operatorname{minimize}}\;\;c({\sigma(t)})$
$\displaystyle\text{subject
to,}\quad{\sigma(t_{0})}\;=\;\boldsymbol{X_{\text{init}}};\quad{\sigma(t_{f})}\;=\;\boldsymbol{X_{\text{goal}}},$
$\displaystyle\|\dot{\sigma}(t)\|\leq{v}_{\text{M}};\|\ddot{\sigma}(t)\|\leq{a}_{\text{M}},{\sigma(t)}\;\in\;\mathbb{X}_{\text{free}};\forall
t\in[t_{0},t_{f})$ (1)

Due to the availability of local environment information only, a systematic
re-planning mechanism also needs to be devised for the above trajectory
planning.

###  III-C Generalized Shape: 2-D GSE and 3-D GSE

The overall idea of the GSE algorithm in 2-D and 3-D environments is described
in [11], [16]. They present an offline method to generate collision-free
feasible paths for an agent in a-priori known 2-D and 3-D environments,
respectively. Expansion of generated graph over generalized shape, an
efficient approximator of safe region, was instrumental for computational
advantage of the GSE and 3D-GSE algorithms. For details on the 3D-GSE
algorithm, refer to [16]. In the 3D online GSE (3D-OGSE) algorithm presented
in this paper, we use the same notion of generalized shape, but present a
scheme to extend the 3D-GSE to perform computations in an online manner for
unknown environments that is sensed locally only. Consider an agent at a point
$\boldsymbol{X}$ in the 3-D environment. Let there be $m_{\text{loc}}$
obstacles in the workspace. For each obstacle
$i\in\\{1,2,\dots,m_{\text{loc}}\\}$ in the local map, we now have the
parameters $r_{\text{i,X}}$, $l_{\text{i,X}}$, and
$\boldsymbol{n_{\text{i,X}}}$ (refer to Table I) computed about
$\boldsymbol{X}$. Without loss of generality we assume that $r_{\text{1,X}}$
$\leq$ $r_{\text{2,X}}\dots\leq r_{\text{m}_{\text{loc}},\text{X}}$. The 3-D
generalized shape about $\boldsymbol{X}$ is represented as,

$\mathcal{S}_{\boldsymbol{X}}=\mathcal{R}_{1,X}\cup(\mathcal{R}_{2,X}-\widetilde{\mathcal{R}_{12,X}})\cup(\mathcal{R}_{3,X}-(\widetilde{\mathcal{R}_{23,X}}+\widetilde{\mathcal{R}_{13,X}}))\\\
\cup\dots\cup(\mathcal{R}_{m_{\text{loc}},X}-\cup_{j=1}^{j=i-1}\widetilde{\mathcal{R}_{jm_{\text{loc}},X}})\cup\mathcal{I},$
(2)

where, the set $\mathcal{I}$ is given as,
$\mathcal{I}\subset\mathbb{X}|\mathcal{I}\cap\\{\cup_{i=1}^{i=m_{\text{loc}}}\widehat{\mathcal{R}_{i,X}}\\}=\emptyset$.
Fig. 2 shows an illustration of 2-D projection of
$\mathcal{S}_{\boldsymbol{X}}$.

##  IV 3-D Online Generalized Shape Expansion (3D-OGSE) Algorithm

Now, we present 3D-OGSE for online trajectory planning having the following
steps at $k^{\text{th}}$ planning iteration.

###  IV-A Generating a Collision-free Path

The $k^{\text{th}}$ planning iteration is initiated at point
$\boldsymbol{p}_{k,0}$, which is $\boldsymbol{X}_{\text{init}}$ for $k=1$ and
$\boldsymbol{p}_{k-1,f}$, otherwise. Information about $m_{\text{loc}}$ number
of obstacles in the sensing region ($C_{\boldsymbol{p}_{k,0}}$) about
$\boldsymbol{p}_{k,0}$ is first obtained from local point cloud data. Because
of no information about obstacles outside $C_{\boldsymbol{p}_{k,0}}$, we treat
unexplored $\mathbb{X}\setminus C_{\boldsymbol{p}_{k,0}}$ as obstacle-free.
With this set-up, a collision-free path from $\boldsymbol{p}_{k,0}$ to
$\boldsymbol{X}_{\text{goal}}$ is then generated using the notion of 3-D
generalized shape given in [16]. Once feasible paths are obtained, shortest
path among them is found using Dijkstra’s algorithm. This is given in Lines 5
to 7 of Algorithm 1. The pseudo code of
$\texttt{3D-GSE}(\boldsymbol{p}_{k,0},\boldsymbol{X}_{\text{goal}},m_{\text{loc}})$
(Line 5 of Algorithm 1) for the generation of feasible path is given in
Algorithm 2. Note that the use of generalized shape facilitates fast path
planning. Let the finally obtained path from $\boldsymbol{p}_{k,0}$ to
$\boldsymbol{X}_{\text{goal}}$ generated consist of directed edges through set
of vertices
$\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2}\dots\boldsymbol{X}_{h_{k}}\\}$,
where $h_{k}\in\mathbb{N}$. An overall graph $G$ is updated with the graph
generated at $k^{\text{th}}$ planning iteration in Line 6 of Algorithm 1.

Algorithm 1 3D Online GSE Algorithm (3D-OGSE)

1:$k\leftarrow
1\;\;,\mathbb{E}\leftarrow\emptyset\;\;,G_{k}\leftarrow\emptyset,\;\;\boldsymbol{p}\leftarrow\boldsymbol{p}_{1,0}\leftarrow\boldsymbol{X}_{\text{init}}$

2:while $\|\boldsymbol{p}-\boldsymbol{X}_{\text{goal}}\|$$\geq\epsilon$ do

3:
$\boldsymbol{p}_{k,0}\leftarrow\boldsymbol{p},\;\;\mathbb{V}\leftarrow\\{\boldsymbol{p}_{k,0},\boldsymbol{X_{\text{goal}}}\\}\;j\leftarrow
1$

4: $m_{\text{loc}}\leftarrow\texttt{SegmentLocalMap}(\textbf{p})$ // local
obstacles

5:
$(\mathbb{V},\mathbb{E})\leftarrow\texttt{3D-GSE}(\boldsymbol{p}_{k,0},\boldsymbol{X}_{\text{goal}},m_{\text{loc}})$

6: $G\leftarrow(\mathbb{V},\mathbb{E})\cup G_{k}$

7:
$\texttt{Path}\leftarrow\texttt{djikstra}(\boldsymbol{p}_{k,0},\boldsymbol{X_{\text{goal}}},G)$

8: $\sigma^{k}\leftarrow\texttt{TrajOpt}(\texttt{Path},\texttt{Shapes})$

9: while
$\texttt{Trigger}(\boldsymbol{p}_{k,0},C_{\boldsymbol{p}_{k,0}},\mathcal{S}_{\boldsymbol{p}_{k,0}},j)=0$
do

10: $\boldsymbol{p}\leftarrow\sigma^{k}(j\Delta t)$ //update agent position

11: $j\leftarrow j+1$

12: end while

13: $\boldsymbol{p}_{k,f}\leftarrow\boldsymbol{p}$

14:
$GP_{k}\leftarrow\texttt{Prune}(G,C_{\boldsymbol{p}_{k,0}},\mathcal{S}_{\boldsymbol{p}_{k,0}})$

15: $\mathbb{V}\leftarrow\emptyset,\;\mathbb{E}\leftarrow\emptyset$ // clear
vertex and edge Set

16: $G_{k}\leftarrow G_{k}\cup GP_{k}$

17: $k\leftarrow k+1$

18:end while

Algorithm 2
$\texttt{3D-GSE}(\boldsymbol{p}_{k,0},\boldsymbol{X}_{\text{goal}},m_{\text{loc}})$

1:$\mathbb{V}\leftarrow\\{\boldsymbol{p}_{k,0},\boldsymbol{X_{\text{goal}}}\\}$

2:while No connected graph is generated from $\boldsymbol{p}_{k,0}$ to
$\boldsymbol{X_{\text{goal}}}$ do

3: $\boldsymbol{X_{\text{rand}}}\leftarrow\texttt{SamplePoint}$

4:
$\boldsymbol{X_{\text{nearest}}}\leftarrow\texttt{Nearest}(\mathbb{V},\boldsymbol{X_{\text{rand}}})$

5:
$\boldsymbol{X}_{\text{new}}\leftarrow\texttt{Steer}(\boldsymbol{X_{\text{rand}}},\boldsymbol{X_{\text{nearest}}})$

6: $\mathbb{V}\leftarrow\mathbb{V}\cup\\{\boldsymbol{X}_{\text{new}}\\}$

7:
$\mathbb{E}\leftarrow\mathbb{E}\cup\\{(\boldsymbol{X_{\text{nearest}}},\boldsymbol{X}_{\text{new}})\\}$

8:
$\mathbb{X_{\text{near}}}\leftarrow\texttt{NearIntersectedShapes}(G,\boldsymbol{X_{\text{new}}})$

9: for $\boldsymbol{X_{\text{n}}}\in\mathbb{X_{\text{near}}}$ do

10:
$\mathbb{E}\leftarrow\mathbb{E}\cup\\{(\boldsymbol{X_{\text{n}}},\boldsymbol{X_{\text{new}}})\\}$

11: end for

12:end while

###  IV-B Trajectory Optimization

The collision-free path from $\boldsymbol{p}_{k,0}$ to
$\boldsymbol{X}_{\text{goal}}$ obtained in Section IV-A is piece-wise linear
between the set of vertices
$\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2}\dots\boldsymbol{X}_{h_{k}}\\}$. In
order to obtain a dynamically feasible smooth trajectory along this path that
could be followed by a realistic vehicle, inspired by [17], a QP for snap
minimization is framed using time-parametrized polynomial in (3a) subject to
the constraints (3b)-(3f). Thus, a smooth and locally collision-free
trajectory $\sigma^{k}$ is obtained (see the function TrajOpt in Line 8 of
Algorithm 1). The snap minimization problem then becomes:

$\displaystyle\underset{\sigma^{k}(t)}{\operatorname{argmin}}\quad
J=\sum_{i=1}^{l_{k}}\int_{t_{i}}^{t_{i+1}}\|\frac{d^{4}}{dt^{4}}\sigma^{k}_{i}(t)\|^{2},\;\;\text{subject
to,}$ (3a) $\displaystyle{\text{Boundary
conditions:}}\;\;\boldsymbol{p}_{1,0}=\boldsymbol{X}_{\text{init}},\;\boldsymbol{p}_{k,0}=\boldsymbol{p}_{k-1,f}$
(3b)
$\displaystyle\sigma^{k}(t_{l_{k}})=\boldsymbol{X}_{\text{goal}},\;\sigma^{k}(t_{1})=\boldsymbol{p}_{k,0}$
(3c)
$\displaystyle\frac{d}{dt}\sigma^{k}(t)\big{|}_{\sigma^{k}(t_{1})=\boldsymbol{p}_{k,0}}=\frac{d}{dt}\sigma^{k-1}(t)|_{\sigma^{k-1}(t)=\boldsymbol{p}_{k-1,f}}\quad$
(3d) $\displaystyle\text{Other constraints for all }i=1,2,\dots,l_{k}:$
$\displaystyle\frac{d^{z}}{dt^{z}}\sigma^{k}_{i}(t_{i})=\frac{d^{z}}{dt^{z}}\sigma^{k}_{i+1}(t_{i})\quad
z=0,\dots,4$ (3e)
$\displaystyle\|\frac{d}{dt}\sigma^{k}_{i}(t)\|\leq{v}_{\text{M}},\;\|\frac{d^{2}}{dt^{2}}\sigma^{k}_{i}(t)\|\leq{a}_{\text{M}},\;\;\sigma^{k}_{i}(t)\in\mathcal{S}_{\boldsymbol{X}_{i}}$
(3f)

Thus, the generated trajectory is a collection of segments as
$\sigma^{k}(t)=\\{\sigma^{k}_{i}(t)=\sum\limits_{\ell=1}^{n}\sigma^{k}_{i\ell}(t)^{\ell}\;\text{for}\;{t_{i}\leq
t<t_{(i+1)}}\\}_{i=1}^{l_{k}}$. For each $i=\\{1,\dots,l_{k}\\}$,
$\sigma^{k}_{i}$ connects $\boldsymbol{X}_{i}$ to $\boldsymbol{X}_{i+1}$ and
belongs to $\mathcal{S}_{\boldsymbol{X}_{i}}$ (see (3f)) ensuring that
$\sigma^{k}_{i}$ lies in $\mathbb{X}_{\text{free}}$. As a consequence, inside
$k^{\text{th}}$ planning horizon, each points on $\sigma^{k}$ also lies in
$\mathbb{X}_{\text{free}}$. It also facilitates in fast QP solving by
restricting the search-space to $\mathcal{S}_{\boldsymbol{X}_{i}}$. Eq. (3e)
and velocity and acceleration constraints in (3f) provide the dynamical
feasibility to $\sigma^{k}$. Moreover, Eqs. (3b) - (3d) renders smoothness to
trajectory even at the transition point from $(k-1)^{\text{th}}$ to
$k^{\text{th}}$ planning iteration.

Algorithm 3
$\texttt{Trigger}(\boldsymbol{p}_{k,0},C_{\boldsymbol{p}_{k,0}},\mathcal{S}_{\boldsymbol{p}_{k,0}},j)$

1:$\boldsymbol{p}^{\prime}\leftarrow\sigma^{k}(j\Delta t)$

2:if $\boldsymbol{p}^{\prime}\in
C_{\boldsymbol{p}_{k,0}}\quad\text{and}\quad\boldsymbol{p}^{\prime}\in\mathcal{S}_{\boldsymbol{p}_{k,0}}$
then

3:
$\texttt{Trigger}(\boldsymbol{p}_{k,0},C_{\boldsymbol{p}_{k,0}},\mathcal{S}_{\boldsymbol{p}_{k,0}},j)=0$

4:else

5:
$\texttt{Trigger}(\boldsymbol{p}_{k,0},C_{\boldsymbol{p}_{k,0}},\mathcal{S}_{\boldsymbol{p}_{k,0}},j)=1$

6:end if

7:$\text{return}\quad\texttt{Trigger}(\boldsymbol{p}_{k,0},C_{\boldsymbol{p}_{k,0}},\mathcal{S}_{\boldsymbol{p}_{k,0}},j)$

Figure 3: Illustration of first planning iteration (2-D projection). The area
inside the dotted square shows projection of the sensing region
$C_{\boldsymbol{X}_{\text{init}}}$ about ${\boldsymbol{X}_{\text{init}}}$, and
the blue region shows the generalized shape
$\mathcal{S}_{\boldsymbol{X}_{\text{init}}}$ about
${\boldsymbol{X}_{\text{init}}}$. The agent moves along the smooth trajectory
$\sigma^{1}$ until it reaches the boundary of sensing shape about
$\boldsymbol{X}_{\text{init}}$
(=$C_{\boldsymbol{X}_{\text{init}}}\cap\mathcal{S}_{\boldsymbol{X}_{\text{init}}}$).

###  IV-C Replanning Trigger for RHP Mechanism

Sensing shape about a point $\boldsymbol{X}$ is defined as the region
$C_{\boldsymbol{X}}\cap\mathcal{S}_{\boldsymbol{X}}$, where
$C_{\boldsymbol{X}}$ and $\mathcal{S}_{\boldsymbol{X}}$ are sensing region and
generalized shape, respectively, about $\boldsymbol{X}$. Starting from
$\boldsymbol{p}_{k,0}$, as the 3-D agent moves along $\sigma^{k}(t)$, at every
time-instant it checks whether the next point on $\sigma^{k}$ lies inside the
sensing shape about $\boldsymbol{p}_{k,0}$. This check helps in reducing
computational overhead since $C_{\boldsymbol{p}_{k,0}}$ and
$\mathcal{S}_{\boldsymbol{p}_{k,0}}$ are already computed at the beginning of
the $k^{th}$ planning iteration. The agent keeps on following $\sigma^{k}$ if
the next point lies inside the sensing shape about $\boldsymbol{p}_{k,0}$. A
re-planning is triggered only when the agent reaches some point
$\boldsymbol{p}_{k,f}$ on the boundary of
$C_{\boldsymbol{p}_{k,0}}\cap\mathcal{{S}}_{\boldsymbol{p}_{k,0}}$. Once a re-
planning is triggered the RHP mechanism shifts the planning horizon to
$(k+1)^{\text{th}}$ planning iteration (restricted by
$C_{\boldsymbol{p}_{k,0}}$) such that
$\boldsymbol{p}_{k+1,0}=\boldsymbol{p}_{k,f}$ is satisfied.

The fore-mentioned RHP scheme helps in avoiding re-planning trajectory in
every time-point, thus saving much computational time. Clearly, this RHP
scheme is easily real-time applicable for stationary obstacles scenario.
However, for dynamic obstacles scenario the re-planning rate might increase
based on moving obstacles’ states in neighborhood of the sensing shape of
$\boldsymbol{p}_{k,0}$. Even in that scenario also because of significantly
low computational time requirement (see Table II) of the presented 3D-OGSE
algorithm, it would not add significantly high computational overhead and thus
would be real-time applicable as well.

##  V Analysis of the 3D-OGSE Algorithm

###  V-A Probabilistic Completeness of the 3D-OGSE Algorithm

The probabilistic completeness proof for the offline GSE algorithm is given in
[29]. However in the following proof, the local sensing region is accounted
for.

Let the region of intersection of the sensing region $C_{\boldsymbol{X}}$ and
generalized shape $\mathcal{S}_{\boldsymbol{X}}$ i.e.
$C_{\boldsymbol{X}}\cap\mathcal{S}_{\boldsymbol{X}}$ be termed as the Sensing
Shape about $\boldsymbol{X}$. Consider a sensing restricted visibility
function $g:2^{\mathbb{X}_{\text{free}}}\rightarrow
2^{\mathbb{X}_{\text{free}}}$, defined as follows:

$\displaystyle g(x)=\\{\boldsymbol{X}\in\mathbb{X}_{\text{free }}\mid
C_{\boldsymbol{X}}\cap\mathcal{S}_{\boldsymbol{X}}\cap x\neq\emptyset\\}$ (4)

where, $2^{\mathbb{X}_{\text{free}}}$ is the power set of the free-space.
Thus, this function maps from the power set of the free-space to itself and
provides the set of points that contain any point of the argument in the
sensing shape.

(a) Planning iteration 1 ($k=1$)

(b) Planning iteration 2 ($k=2$)

(c) Planning iteration 3 ($k=3$)

(d) Planning iteration 4 ($k=4$)

(e) Planning iteration 5 ($k=5$)

Figure 4: Illustration of planning iterations of the 3D-OGSE algorithm for
dynamically feasible safe trajectory generation (2-D projection). Here,
$g^{i}\triangleq g^{(i)}(\boldsymbol{X}_{\text{goal}})\forall i\in\\{1,2\dots
5\\}$ denotes $i^{\text{th}}$ iterate of sensing-restricted visibility
function. (a) In planning iteration 1, the agent starts from
$\boldsymbol{p}_{1,0}=\boldsymbol{X}_{\text{init}}\in g^{4}$ and generates a
locally safe and smooth trajectory $\sigma^{1}$ and moves along it until it
reaches a point $\boldsymbol{p}_{1,f}$ on the boundary of sensing shape about
$\boldsymbol{p}_{1,0}$
(=$C_{\boldsymbol{p}_{1,0}}\cap\mathcal{S}_{\boldsymbol{p}_{1,0}}$). (b)-(e)
In planning iteration ’k’ for $k=2,3,4,5$, the agent starts from
$\boldsymbol{p}_{k,0}=\boldsymbol{p}_{k-1,f}$ and generates a locally safe and
smooth trajectory $\sigma^{1}$ and moves along it until it reaches a point
$\boldsymbol{p}_{k,f}$ on the boundary of sensing shape about
$\boldsymbol{p}_{k,0}$
(=$C_{\boldsymbol{p}_{k,0}}\cap\mathcal{S}_{\boldsymbol{p}_{k,0}}$). During
this time, in planning iterations 2, 4 and 5 the agent moves from higher
iterate of $g$ to lower iterate of $g$ (from $g^{4}$ to $g^{3}$, from $g^{3}$
to $g^{2}$, and from $g^{2}$ to $g^{1}$, respectively). Finally, at the end of
planning iteration 5 the agent reaches $\boldsymbol{p}_{5,f}\in
g(\boldsymbol{X}_{\text{goal}})$, that is $\boldsymbol{X}_{\text{goal}}\in
C_{\boldsymbol{p}_{1,0}}\cap\mathcal{S}_{\boldsymbol{p}_{1,0}}$ ensuring
existence of a straight-line edge connectivity between $\boldsymbol{p}_{5,f}$
and $\boldsymbol{X}_{\text{goal}}$.

The main intuition for proposing the following two lemmas is as follows. As
the agent moves in the free space to reach the goal position, the union of the
sensing shapes of the agent grows in volume and at some iteration this union
covers the entire free space and it also contains the goal point
$\boldsymbol{X}_{\text{goal}}$. At this stage one can claim that there is a
collision-free path from the agent’s current position to the goal position.

Let the function $g^{(2)}(\boldsymbol{X}_{\text{goal}})\triangleq
g(g(\boldsymbol{X}_{\text{goal}}))$. In the following two lemmas, we prove
that the function $\mu({{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})})$ increases
by a positive constant with every $i$ and equals
$\mu(\mathbb{X}_{\text{free}})$ for some value of $i\geq q$. Consequently,
this would imply that the entire free-space $\mathbb{X}_{\text{free}}$ is also
contained in ${{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})}$ for the
corresponding finite value of $i$.

###### Lemma V.1

For $g$ defined in (4), for all $i$,
${g}^{(i)}({\boldsymbol{X}_{\text{goal}}})\subseteq{g}^{(i+1)}({\boldsymbol{X}_{\text{goal}}})$

###### Proof:

By definition of $g$, clearly, ${g}^{(i+1)}({\boldsymbol{X}_{\text{goal}}})$
includes all points $\boldsymbol{X}\in\mathbb{X}_{\text{free}}$ such that
$C_{\boldsymbol{X}}\cap\mathcal{S}_{\boldsymbol{X}}\cap{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})\neq\emptyset$.
Now, consider any point
$\boldsymbol{X}\in{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})$. Recall from the
definition of the sensing shape that
$\mathbf{X}\in{C_{\boldsymbol{X}}}\cap\mathcal{S}_{\boldsymbol{X}}$. This
implies that
$C_{\boldsymbol{X}}\cap\mathcal{S}_{\boldsymbol{X}}\cap{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})\supseteq\\{\mathbf{X}\\}$.
Hence, the lemma follows. ∎

###### Lemma V.2

For $g$ defined in Eqn. (4),
$\mu({g}^{(i+1)}({\boldsymbol{X}_{\text{goal}}})\setminus{g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))>c$
for a fixed point positive constant $c$, if
${g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))\subset\mathbb{X}_{\text{free}}$.

###### Proof:

Consider any $i$ such that
${g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))\subset\mathbb{X}_{\text{free }}$.
Let the union of $\epsilon$-neighbourhoods of all points within the set
${g}^{(i)}({\boldsymbol{X}_{\text{goal}}})$ be denoted by $E_{i}$. From Lemma
V.1, it follows that there exists a fixed positive $\epsilon$ such that
$((E_{i}\setminus{g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))\setminus\mathbb{X}_{\text{obs}})\subseteq{g}^{(i+1)}({\boldsymbol{X}_{\text{goal}}})$.
We can then conclude that
$\mu({g}^{(i+1)}({\boldsymbol{X}_{\text{goal}}})\setminus{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})>\mu((\partial{g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))*\epsilon$
where $\partial\mathbb{S}$ denotes the boundary of a set $\mathbb{S}$. Since
the free-space is assumed to be path-connected, there exist no isolated points
surrounded by obstacles in $\mathbb{X}_{\text{free}}$. This guarantees a lower
bound on the rate of growth of the volume of
${g}^{(i)}({\boldsymbol{X}_{\text{goal}}})$, and proves the lemma. ∎

From Lemma V.2, measure of $g$ grows by at least $c$ at every iteration $i$ of
the sensing restricted visibility function $g$. Considering that the free-
space $\mathbb{X}_{\text{free}}$ is bounded, thus, Lemmas V.1 and V.2 imply
that for sufficiently large value of $i$,
$\mu({g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))=\mu(\mathbb{X}_{\text{free}})$.
Hence, we define a sensing-restricted visibility index as follows,

$q=\underset{i\in\mathbb{N}}{\operatorname{argmin}}(\mu({g}^{(i)}({\boldsymbol{X}_{\text{goal}}}))=\mu(\mathbb{X}_{\text{free}}))$
(5)

Due to construction of the sensing shape above, from Eqn. (4) it can be shown
easily that $g^{(q)}(\boldsymbol{X}_{\text{goal}})=\mathbb{X}_{\text{free}}$.

###### Theorem V.3

The probability that the 3D-OGSE algorithm fails to generate a feasible path
from the current position $\boldsymbol{X}_{\text{init}}$ to
$\boldsymbol{X}_{\text{goal}}$ after $N$ planning iterations is at most
$ae^{-bN},$ for some positive real numbers $a$ and $b$. This ensures a
probabilistic guarantee that a collision-free path is generated from
$\boldsymbol{X}_{\text{init}}$ to $\boldsymbol{X}_{\text{goal}}$ over multiple
planning iterations using the 3D-OGSE algorithm.

###### Proof:

Let $\widehat{\mathbb{V}}_{k}$
($G_{k}=(\widehat{\mathbb{V}}_{k},\mathbb{E}_{k})$ in Line 16 of Algorithm 1
denote the set of vertices excluding the goal point
$\boldsymbol{X}_{\text{goal}}$ generated by the 3D-OGSE algorithm after $k$
planning iterations. Note that the 3D-OGSE algorithm succeeds generating a
collision-free path after $N$ planning iterations if
$\widehat{\mathbb{V}}_{N}\cap g(\boldsymbol{X}_{\text{goal}})\neq\emptyset$.

From construction of $G_{k}$ (Line 14 and 16 of Algorithm 1) all the vertices
in $\widehat{\mathbb{V}}_{k}$ are guaranteed to lie in the free-space. Now,
define $i_{k}$ to be the minimum value of $i$ such that
$\widehat{\mathbb{V}_{k}}\cap{g}^{(i)}({\boldsymbol{X}_{\text{goal}}})\neq\emptyset$.
In addition, define a random variable $A_{k}$, which is equal to one if
${i}_{k+1}<i_{k}$ or zero otherwise. $A_{k}=1$ indicates that some of the
additional vertices of the extended graph generated at the $(k+1)^{\text{th}}$
planning iteration lies in the set $g^{(i)}(\boldsymbol{X}_{\text{goal}})$
with a lower $i$ than before. Now, let us consider a random variable $A$ which
is equal to $A_{1}+A_{2}\dots+A_{N}$. From the definition of $q$ in Eqn. (5),
clearly, the maximum value of $A$ is $q$.

Consider a random variable $\mathcal{P}_{N}$ such that $\mathcal{P}_{N}=1$
denotes the event of success, that is, a collision-free path is generated from
initial to goal point after $N^{\text{th}}$ planning iteration, and
$\mathcal{P}_{N}=0$ denotes the event of failure, that is, even after
$N^{\text{th}}$ planning iteration a collision-free path is not generated from
initial to goal point.

If $A_{k}=1$ at least $q$ times for various $k\leq N$, then the graph
generated till the $N^{\text{th}}$ planning iteration of the 3D-OGSE algorithm
has progressed from ${g}^{(q)}({\boldsymbol{X}_{\text{goal}}})$ to
$g({\boldsymbol{X}_{\text{goal}}})$. Hence, we have
$\widehat{\mathbb{V}}_{N}\cap{g}({\boldsymbol{X}_{\text{goal}}})\neq\emptyset$,
which indicates that $\mathcal{P}_{N}=1$ implying success in $N^{\text{th}}$
planning iteration. However, it could be noted that success could also be
achieved even for $A<q$. Therefore, it can be concluded that
$\\{\mathcal{P}_{N}=1\\}\supseteq\\{A=q\\}$. Taking complement on both sides
gives $\\{\mathcal{P}_{N}=0\\}\subset\\{A<q\\}$.

###### Proposition 1

For $k^{\text{th}}$ planning iteration of 3D-OGSE, $\mathbb{P}(A_{k}=1)$ is
bounded below by a positive fraction $p<1$ for all $k$ such that
$\widehat{\mathbb{V}}_{k}\cap{g}({\boldsymbol{X}_{\text{goal}}})=\emptyset$.

###### Proof:

At the $k^{\text{th}}$ iteration, let there be $w$ vertices in
$\widehat{\mathbb{V}}_{k}$, which belong to
${g}^{(i_{k})}({\boldsymbol{X}_{\text{goal}}})$ also. It is needed to be shown
here that there exists a minimum probability of extending the graph in the
next planning iteration such that it would contain vertices belonging to
${g}^{(i_{k}-1)}({\boldsymbol{X}_{\text{goal}}})$. Clearly, if $w$ is more,
then the probability of getting vertices in the $(k+1)^{\text{th}}$ planning
iteration that belongs to the region
${g}^{(i_{k}-1)}({\boldsymbol{X}_{\text{goal}}})$ also increases. In other
words, as $w$ increases, the probability that $A_{k}=1$ also increases. Hence,
considering $w=1$ suffices for this proof.

Now, for $w=1$, let the corresponding vertex be denoted by $\boldsymbol{v}$.
Let $\text{Vor}(\boldsymbol{v})$ denote the Voronoi cell of the vertex
$\boldsymbol{v}$ in the Voronoi diagram consisting of $|{\mathbb{V}}_{k}|$
vertices of the graph $G_{k}$ generated by the 3D-OGSE so far. Because of the
finite number of vertices in $G_{k}$, for the vertex
$\boldsymbol{v}\in{g}^{(i_{k})}({\boldsymbol{X}_{\text{goal}}})$, it follows
that $\mu(\text{Vor}(\boldsymbol{v}))$ is bounded below by some positive
number, say $s_{i_{k}}$. Now, note that for any arbitrary location of
$\boldsymbol{v}$ in the set ${g}^{(i_{k})}({\boldsymbol{X}_{\text{goal}}})$,
there is a subset of $\text{Vor}(\boldsymbol{v})$, in which, if the new vertex
is sampled, leads to extension of the 3D-OGSE-generated graph to at least one
point in ${g}^{(i_{k}-1)}({\boldsymbol{X}_{\text{goal}}})$. Let the minimum
size of such a subset over the set
${g}^{(i_{k})}({\boldsymbol{X}_{\text{goal}}})$ be denoted as $\zeta_{i_{k}}$.
We can then conclude that for $p=\underset{i_{k}\leq
m}{\text{min}}{\frac{{s_{i_{k}}\zeta_{i_{k}}}}{{\mu(\mathbb{X}_{\text{free}})}}}$,
the proposition holds true. ∎

Now, consider a Bernoulli distributed random variable $B_{k}$ with parameter
$p$ as obtained in Proposition 1. Note that $B_{k}$ thus defined could be
thought as a Bernoulli random variable used to model the worst-case
probability of the 3D-OGSE-generated graph extending from a set
${g}^{(i_{k})}({\boldsymbol{X}_{\text{goal}}})$ to a set
${g}^{(i_{k}-j)}({\boldsymbol{X}_{\text{goal}}})$ in the $k^{\text{th}}$
iteration for $j\leq i_{k}-1$. Next, define random variable
$B=B_{1}+B_{2}+\dots+B_{N}$. Following the arguments used in the proof for RRT
[24], $\mathbb{P}(A<q)$ is obtained as upper-bounded as given in the following
proposition.

###### Proposition 2

$\mathbb{P}(A<q)<\mathbb{P}(B<q)$

###### Proof:

From Proposition 1, we know that
$\mathbb{P}(A_{i}=1)\geq\mathbb{P}(B_{i}=1)=p$ for $i$ less than or equal to
$N$. Considering the nature of variation of the probability distribution of
Binomial distribution with variation in distribution parameter, it could be
inferred that $\mathbb{P}(B<q)$ provides a worst-case (lower) bound on
$\mathbb{P}(A>q)$, that is, $\mathbb{P}(A>q)\geq\mathbb{P}(B>q)$. Taking
complement on both sides proves the proposition. ∎

Using a theorem for Chernoff bounds for sum of Bernoulli random variables
$B_{1},B_{2}\dots B_{N}$ [30],

$\displaystyle\mathbb{P}(A<q)\leq\mathbb{P}(B<q)\leq
e^{(-((N/p)+(({(qp)}^{2})/N)-2q))}$ (6)

The upper bound of $\mathbb{P}(A<q)$ obtained above could be given in a more
compact form as $\mathbb{P}(A<q)\leq ae^{-bN}$, where
$a=e^{-((({(qp)}^{2})/N)-2q)}$ and $b=\frac{1}{p}$.

Recall that the event $\\{\mathcal{P}_{N}=0\\}$ is a subset of the event
$\\{A<q\\}$. In addition, from Eqn. (6) since $\mathbb{P}(A<q)\leq ae^{-bN}$,
therefore it can be concluded that $\mathbb{P}(\mathcal{P}_{N}=0)\leq
ae^{-bN}$. Considering limit supremum operation on both sides gives the
following,

$\limsup_{N\to\infty}\;{\mathbb{P}(\mathcal{P}_{N}=0)}\leq\limsup_{N\to\infty}\;{\mathbb{P}(A<q)}=0$
(7)

Thus, the 3D-OGSE algorithm satisfies the probabilistic completeness
criterion, i.e., there is a probabilistic guarantee that the 3D-OGSE generates
a collision-free feasible path from $\boldsymbol{X}_{\text{init}}$ to
$\boldsymbol{X}_{\text{goal}}$ over multiple planning iterations. ∎

The following corollary presents an upper bound of the number of such planning
iterations required to generate such a feasible path in expectation.

###### Corollary V.3.1

Since the expectation of the random variable $B$ i.e. $\mathbb{E}(B)=Np$
(since it is a Binomial distribution), we can conclude that the expected
number of iterations it takes for the 3D-OGSE algorithm to succeed is bounded
above by $\frac{q}{p}$, for a space with restricted visibility index $q$.

###  V-B Guarantee of Collision-free Trajectory by the 3D-OGSE Algorithm

Theorem V.3 above gives the probabilistic guarantee of obtaining a collision-
free feasible path generated over multiple planning iterations, in which all
the vertices belong to $\mathbb{X}_{\text{free}}$. Now, let’s consider the
overall trajectory obtained over multiple planning iterations. The following
proposition gives a guarantee that this trajectory is also collision-free.

###### Proposition 3

(Collision-Free Trajectories): The 3D-OGSE guarantees collision-free
trajectories.

###### Proof:

Once a feasible path is generated at each planning phase, a smooth trajectory
is computed by solving the snap minimization problem given in (3a). Note that
the last constraint in the optimization problem given in (3f) ensures that any
point on the generated trajectory at that planning phase lies in free-space.

Now, let’s consider the transition from one planning phase to the subsequent
one. This re-planning is set by the Trigger function module given in Line 9 of
Algorithm 1 and in Algorithm 3. From these, it could be easily observed that
the re-planning trigger decision is taken in one-step ahead manner, that is,
if the next point (Line 1 of Algorithm 3) is beyond the sensing shape about
the present point, then a re-planning is triggered even before the agent
reaches the surface of the sensing shape, and subsequent planning phase
begins. This ensures that the agent does not traverse to any obstacle-space
even at the transition from one planning phase to another. This completes the
proof. ∎

Depending on the computational time required to generate a trajectory and also
ensuring that the agent lies in the collision-free zone at every point, a
multiple-step look ahead can be incorporated in Line 1 of the Trigger
algorithm (Algorithm 3). In other words instead of using the one-step look
ahead $\sigma^{k}(j\Delta t)$, a multiple-step look ahead can be incorporated
$\sigma^{k}((j+w)\Delta t)$ (in Line 1 of Algorithm 3), where $w$ steps are
looked ahead.

Besides this, also note that in several existing literature safe corridors are
formed in a half-space generated by a hyperplane touching the closest point on
an obstacle. In non-convex obstacle shape scenario, such half-spaces and safe
corridors contained in it still might have intersection with obstacle-space
making their approaches not suitable for non-convex obstacle shapes. To the
contrary, the 3D-OGSE presented in this paper considers the notion of
generalized shape for the formation of safe corridor, which is devoid of such
problem of intersection with obstacle-space and hence, can be equally
effectively leveraged for non-convex shape of obstacles also.

##  VI Results

###  VI-A Implementation

Our method is implemented on an Intel Core i5-8500 CPU at 3.0 GHz with 32GB
memory. We use ROS Melodic and Rviz for our simulation experiments. The
trajectory optimization is performed using Mosek. We evaluate our method in
simulation with prior state-of-the-art online trajectory generation methods
presented by Liu et al. [14], Gao et al. [15] and Usenko et. al [12]. We
present the benefits of the 3D-OGSE in terms of its low computation time,
trajectory cost, and better performance in narrow passages and in high
obstacle-dense environments.

####  VI-A1 Computation Time and Trajectory Cost

Method |  | Computation Time (ms) |  Trajectory Cost (snap (m/s4))
---|---|---|---
|  Total Time |  Path Computation |  Trajectory Generation
3D-OGSE (FOV:$360\degree$) |  Mean |  1.345 |  0.536 |  0.809 |  51.562
Max |  4.745 |  0.934 |  3.811 |  96.562
Std dev |  1.452 |  0.441 |  1.011 |  17.357
3D-OGSE (FOV:$120\degree$) |  Mean |  1.726 |  0.678 |  1.048 |  53.367
Max |  4.923 |  1.127 |  3.796 |  102.378
Std dev |  3.167 |  0.978 |  2.189 |  20.126
Liu et al. [14] |  Mean |  9.324 |  3.371 |  5.952 |  60.577
Max |  15.729 |  6.437 |  9.292 |  101.578
Std dev |  2.245 |  0.378 |  1.873 |  20.178
Gao and Shen [15] |  Mean |  13.547 |  5.509 |  8.038 |  62.189
Max |  25.265 |  10.638 |  14.627 |  105.272
Std dev |  4.352 |  1.283 |  3.069 |  12.467
Usenko et. al [12] |  Mean |  5.512 |  - |  5.512 |  58.986
Max |  7.653 |  - |  7.653 |  96.463
Std dev |  2.568 |  - |  2.568 |  16.578

TABLE II:  We tabulate the computation time and trajectory cost (snap)
averaged over 100 generated trajectories for an agent operating in the random
forest environment (Fig. 1). We highlight the breakdown of the computation
time between path generation and trajectory optimization for our algorithm
3D-OGSE and compare it with Liu et al. [14], Gao and Shen [15], and Usenko et.
al [12]. Overall, we observe a $4-10$ times improvement in the performance
over the prior methods, while our trajectory cost are comparable.  Since [12]
is a trajectory optimization based method, there is no path generation, thus
the table entries for this case is empty.

Table II summarizes the computation time and trajectory cost for 3D-OGSE and
other algorithms under comparison.We evaluate two implementations of 3D-OGSE,
one with a sensor field-of-view (FOV) of $360\degree$, while the other one is
for a restricted sensor FOV of $120\degree$. The values tabulated are averaged
over 100 generated trajectories, and the methods were compared in the random
forest environment (Fig. 1) with the same start and goal positions. The
maximum velocity and acceleration limits were taken as $3m/s$ and $2m/s^{2}$,
respectively. It can be observed that the trajectory cost of 3D-OGSE is
comparable to prior methods under comparison, while the computation time for
our method is significantly lower than them. The main advantage derived from
the consideration of sensing shape based on the notion of generalized shape is
that it leads to having fewer edges in the overall connected path.
Subsequently, the number of connected segments in the QP formulation are few,
thereby significantly reducing the computation time for optimizing trajectory
about the generated path as is reflected in Table II.

####  VI-A2 Restricted FOV with Noisy Sensing

To simulate realistic scenarios, we consider restricted FOV with noisy sensing
by on-board sensor. While the FOV is limited to $120\degree$, the noisy
sensing is characterized by the dimension of the sensing region
$\boldsymbol{x}$ being chosen from a truncated Gaussian distribution with mean
as its nominal sensing region dimension $(\mu_{s})$ and standard deviation
$(\sigma_{s})$ as below.

$\displaystyle
f_{s}(\boldsymbol{x},\mu_{s},\sigma_{s},a)=\frac{1}{\sigma_{s}}\frac{\Phi\left(\frac{\boldsymbol{x}-\mu_{s}}{\sigma_{s}}\right)}{\Phi\left(\frac{a}{\sigma_{s}}\right)-\Phi\left(\frac{-a}{\sigma_{s}}\right)}$
(8)

where, $\Phi$ is the cumulative Gaussian distribution. For simulations, we
choose $\mu_{s}=5$m (half of side-length 10m of nominal cubic sensing region)
and $\sigma_{s}=1m$. And, $a$ is selected as 0.1m, 0.3m and 0.5m indicating
varied level of sensing uncertainty. From Table III it can be observed that
with increase in $a$ or sensing uncertainty, for low obstacle-density the
number of collided trajectories increase linearly. However, for higher sensing
uncertainty and higher obstacle densities the number of collided trajectories
increase superlinearly. In this context, it is important to note that for
ideal sensing with full and restricted FOV, our 3D-OGSE leads to no collided
trajectory (result of which is given in Table II), which can also be verified
from the theoretical guarantees given in Theorem V.3 and Proposition 3.

3D-OGSE | Low (1 obs./$m^{2}$) | Med.(2 obs./$m^{2}$) | High(3 obs./$m^{2}$)
---|---|---|---
a=0.1m | 91 | 101 | 139
a=0.3m | 98 | 107 | 185
a=0.5m | 106 | 135 | 289

TABLE III: Table indicates the number of trajectories that collided with
obstacles in a scenario with sensor noise (8) and sensor FOV of $120\degree$.
The number reported are out of a 1000 generated trajectories, in different
environments with obstacle density varying from $1$ to $3obs/m^{2}$, and $a$
between 2% and 10% of $\mu_{s}$.

####  VI-A3 Obstacle Density

Fig. 5 shows the variation in computation time with an increase in the
obstacle density in the random forest environment. For this evaluation, we
varied the obstacle density from $1$ to $3$ obstacles$/\text{m}^{2}$. We
observe that even with the increase in obstacle density, the corresponding
increase in computation time is significantly lower in our method than the
methods under comparison. The computation times reported are averaged over 100
generated trajectories.

Figure 5: We observe the trajectory computation time to increase with an
increase in the obstacle density (obstacles per unit area) of the random
forest environment. But, the observed increase in computation time for the
3D-OGSE is considerably smaller compared to prior methods by Liu et al. [14],
Gao and Shen [15], and Usenko et al. [12]. The figure summarizes the
computation times for the four methods for 1, 2, and 3 obstacles per Sq.m. The
values stated are averaged over 100 trajectories generated for randomly chosen
start and goal positions.

####  VI-A4 Narrow Passages

(a)

(b)

Figure 6: (a): We observe that our method samples lower number of points for
the narrow passage than Liu et al. [14] and Gao et al. [15]. We evaluate on
scenarios with passage width ranging from $1$ to $6m$.(b): The average number
of replanning triggers averaged over 100 trajectories in random forest
environment using our method, Liu et al. [14], Gao et al. [15] and Usenko et
al. [12].

We evaluate the performance of the algorithms in multiple narrow passages of
fixed length and different widths. Fig. 6(a) shows the variation in the number
of sampled points (in the narrow passage) when computing a feasible path and
the passage width. We observe in our method the number of sampled points
remain almost constant even with decrease in passage width, while the number
of sampled points increases in [14, 15]. Further from Fig. 7, we observe our
algorithm performs well even in complex scenarios such as the ‘horseshoe’
environment.

####  VI-A5 Replanning Triggers

We compare the number of replanning triggers required to reach the goal
position for our proposed method, [14], [15], and [12]. We considered the
random forest environment for this scenario and generated 100 trajectories
from a start to a goal position. The average number of replanning triggers
required to reach the goal is then calculated for these 100 trajectories and
is shown in Fig. 6(b). We observe that our methods required a lower number of
replanning triggers on average than other methods under comparison.

(a)

(b)

Figure 7: A horseshoe environment and a thin corridor is considered. We
observe that our proposed 3D-OGSE algorithm is able to generate a feasible
trajectory from the start to goal location in these scenarios. The width of
the passage in horseshoe environment and thin corridor is 1m.The green lines
denote the shortest path from $\boldsymbol{X}_{\text{init}}$ to
$\boldsymbol{X}_{\text{goal}}$ and black lines denote the smooth trajectory
between these points

##  VII Conclusion

In this paper, we propose a novel online safe and smooth trajectory generation
algorithm, 3D-OGSE, for autonomous navigation of a 3-D agent in an unknown 3-D
obstacle-cluttered environment. Our algorithm uses local information about
environment to continuously re-plan trajectories to direct the agent towards a
goal position over multiple planning horizons / iterations that are mechanised
by re-planning trigger of a receding horizon planner. Our algorithm is fast
and requires $1.4$ ms on average to re-plan trajectories, which is
approximately 4-10 times faster than [14], [15], and [12]. Also, in complex
environments like narrow passages also the efficiency of the 3D-OGSE is found
evident. We plan on extending our algorithm to noisy sensor settings, where
trajectories are to be generated by taking sensor noise also into account.
Further, we plan on extending this method for scenarios with dynamic obstacles
and evaluating the performance on test-bed with real-world 3-D agents .

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# Conditions of coincidence of central extensions of von Neumann algebras and
algebras of measurable operators

S. Albeverio1, K. K. Kudaybergenov2 and R. T. Djumamuratov3 1 Institut für
Angewandte Mathematik and HCM, Rheinische Friedrich-Wilhelms-Universität Bonn,
Endenicher Allee 60, D-53115 Bonn, Germany<EMAIL_ADDRESS>2 Department
of Mathematics, Karakalpak state university
Ch. Abdirov 1, 230113, Nukus, Uzbekistan
<EMAIL_ADDRESS>3 Department of Mathematics, Karakalpak state university
Ch. Abdirov 1, 230113, Nukus, Uzbekistan
<EMAIL_ADDRESS>
###### Abstract.

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of
$M.$ We describe class of von Neumann algebras $M$ for which the algebra
$E(M)$ coincides with the algebra $S(M)$ – the algebra of all measurable
operators with respect to $M,$ and with $S(M,\tau)$ – the algebra of all
$\tau$-measurable operators with respect to $M.$

###### Key words and phrases:

von Neumann algebras, measurable operators, central extensions

###### 2000 Mathematics Subject Classification:

46L51, 46L10

##  1\. Introduction

In the series of paper [1, 3, 2, 4] we have considered derivations on the
algebra $LS(M)$ of locally measurable operators affiliated with a von Neumann
algebra $M,$ and on various subalgebras of $LS(M).$ A complete description of
derivations has been obtained in the case of von Neumann algebras of type I
and III. A comprehensive survey of recent results concerning derivations on
various algebras of unbounded operators affiliated with von Neumann algebras
is presented in [4]. The general form of automorphisms on the algebra $LS(M)$
in the case of von Neumann algebras of type I has been obtained in [2]. In the
proof of the main results of the above papers the crucial role is played by
the central extensions of von Neumann algebras and also by various topologies
considered in [3].

Let $M$ be an arbitrary von Neumann algebra with the center $Z(M)$ and let
$LS(M)$ denote the algebra of all locally measurable operators with respect
$M.$ We consider the set $E(M)$ of all elements $x$ from $LS(M)$ for which
there exists a sequence of mutually orthogonal central projections
$\\{z_{i}\\}_{i\in I}$ in $M$ with $\bigvee\limits_{i\in I}z_{i}=\textbf{1},$
such that $z_{i}x\in M$ for all $i\in I.$ It is known [3] that $E(M)$ is a
*-subalgebra in $LS(M)$ with the center $S(Z(M)),$ where $S(Z(M))$ is the
algebra of all measurable operators with respect to $Z(M),$ moreover,
$LS(M)=E(M)$ if and only if $M$ does not have direct summands of type II. A
similar notion (i.e. the algebra $E(\mathcal{A})$) for arbitrary *-subalgebras
$\mathcal{A}\subset LS(M)$ was independently introduced by M.A. Muratov and
V.I. Chilin [8]. The algebra $E(M)$ is called the central extension of $M.$

In section 2 we recall the notions of the algebras $S(M)$ of measurable
operators and $LS(M)$ of locally measurable operators affiliated with a von
Neumann algebra $M.$ We also consider the central extension $E(M)$ of the von
Neumann algebra $M.$

In section 3 we describe the classes of von Neumann algebras $M$ for which the
algebra $E(M)$ coincides with the algebras $S(M),S(M,\tau)$ and $M.$

##  2\. Central extensions of von Neumann algebras

In this section we recall the notions of the algebras $S(M)$ of measurable
operators and respectively $LS(M)$ of locally measurable operators affiliated
with a von Neumann algebra $M.$ We also consider the central extension $E(M)$
of the von Neumann algebra $M.$

Let $H$ be a complex Hilbert space and let $B(H)$ be the algebra of all
bounded linear operators on $H.$ Consider a von Neumann algebra $M$ in $B(H)$
with the operator norm $\|\cdot\|_{M}.$ Denote by $P(M)$ the lattice of
projections in $M.$

A linear subspace $\mathcal{D}$ in $H$ is said to be _affiliated_ with $M$
(denoted as $\mathcal{D}\eta M$), if $u(\mathcal{D})\subset\mathcal{D}$ for
every unitary $u$ from the commutant

$M^{\prime}=\\{y\in B(H):xy=yx,\,\forall x\in M\\}$

of the von Neumann algebra $M.$

A linear operator $x$ on $H$ with the domain $\mathcal{D}(x)$ is said to be
_affiliated_ with $M$ (denoted as $x\eta M$) if $\mathcal{D}(x)\eta M$ and
$u(x(\xi))=x(u(\xi))$ for all $\xi\in\mathcal{D}(x)$ and for every unitary
$u\in M^{\prime}.$

A linear subspace $\mathcal{D}$ in $H$ is said to be _strongly dense_ in $H$
with respect to the von Neumann algebra $M,$ if

1) $\mathcal{D}\eta M;$

2) there exists a sequence of projections $\\{p_{n}\\}_{n=1}^{\infty}$ in
$P(M)$ such that $p_{n}\uparrow\textbf{1},$ $p_{n}(H)\subset\mathcal{D}$ and
$p^{\perp}_{n}=\textbf{1}-p_{n}$ is finite in $M$ for all $n\in\mathbb{N},$
where 1 is the identity in $M.$

A closed linear operator $x$ acting in the Hilbert space $H$ is said to be
_measurable_ with respect to the von Neumann algebra $M,$ if $x\eta M$ and
$\mathcal{D}(x)$ is strongly dense in $H.$

Denote by $S(M)$ the set of all linear operators on $H,$ which are measurable
with respect to the von Neumann algebra $M.$ If $x\in S(M),$
$\lambda\in\mathbb{C},$ where $\mathbb{C}$ is the field of complex numbers,
then $\lambda x\in S(M)$ and the operator $x^{\ast},$ adjoint to $x,$ is also
measurable with respect to $M$ (see [10]). Moreover, if $x,y\in S(M),$ then
the operators $x+y$ and $xy$ are defined on dense subspaces and admit closures
that are called, correspondingly, the strong sum and the strong product of the
operators $x$ and $y,$ and are denoted by $x\stackrel{{\scriptstyle.}}{{+}}y$
and $x\ast y,$ respectively. It was shown in [10] that
$x\stackrel{{\scriptstyle.}}{{+}}y$ and $x\ast y$ belong to $S(M)$ and these
algebraic operations make $S(M)$ a $\ast$-algebra with the identity 1 over the
field $\mathbb{C}.$ Here, $M$ is a $\ast$-subalgebra of $S(M).$ In what
follows, the strong sum and the strong product of operators $x$ and $y$ will
be denoted in the same way as the usual operations, by $x+y$ and $xy,$
respectively.

A closed linear operator $x$ in $H$ is said to be _locally measurable_ with
respect to the von Neumann algebra $M,$ if $x\eta M$ and there exists a
sequence $\\{z_{n}\\}_{n=1}^{\infty}$ of central projections in $M$ such that
$z_{n}\uparrow\textbf{1}$ and $z_{n}x\in S(M)$ for all $n\in\mathbb{N}$ (see
[11]).

Denote by $LS(M)$ the set of all linear operators that are locally measurable
with respect to $M.$ It was proved in [11] that $LS(M)$ is a $\ast$-algebra
over the field $\mathbb{C}$ with identity $\textbf{1},$ the operations of
strong addition, strong multiplication, and passing to the adjoint. In such a
case, $S(M)$ is a $\ast$-subalgebra in $LS(M).$ In the case where $M$ is a
finite von Neumann algebra or a factor, the algebras $S(M)$ and $LS(M)$
coincide. This is not true in the general case. In [7] the class of von
Neumann algebras $M$ has been described for which the algebras $LS(M)$ and
$S(M)$ coincide.

Let $\tau$ be a faithful normal semi-finite trace on $M.$ We recall that a
closed linear operator $x$ is said to be $\tau$-measurable with respect to the
von Neumann algebra $M,$ if $x\eta M$ and $\mathcal{D}(x)$ is $\tau$-dense in
$H,$ i.e. $\mathcal{D}(x)\eta M$ and given $\varepsilon>0$ there exists a
projection $p\in M$ such that $p(H)\subset\mathcal{D}(x)$ and
$\tau(p^{\perp})<\varepsilon.$ Denote by $S(M,\tau)$ the set of all
$\tau$-measurable operators with respect to $M$ (see [9]).

It is well-known that $S(M,\tau)$ is a $\ast$-subalgebras in $LS(M)$ (see
[9]).

Consider the topology $t_{\tau}$ of convergence in measure or measure topology
on $S(M,\tau),$ which is defined by the following neighborhoods of zero:

$V(\varepsilon,\delta)=\\{x\in S(M,\tau):\exists\,e\in
P(M),\tau(e^{\perp})\leq\delta,xe\in M,\|xe\|_{M}\leq\varepsilon\\},$

where $\varepsilon,\delta$ are positive numbers, and $\|\cdot\|_{M}$ denotes
the operator norm on $M$.

It is well-known [9]) that $S(M,\tau)$ equipped with the measure topology is a
complete metrizable topological $\ast$-algebra.

Let $(\Omega,\Sigma,\mu)$ be a measure space and suppose that the measure
$\mu$ has the direct sum property, i.e. there is a family
$\\{\Omega_{i}\\}_{i\in J}\subset\Sigma,$ $0<\mu(\Omega_{i})<\infty,\,i\in J,$
such that for any $A\in\Sigma,$ $\mu(A)<\infty,$ there exist a countable
subset $J_{0}\subset J$ and a set $B$ with zero measure such that
$A=\bigcup\limits_{i\in J_{0}}(A\cap\Omega_{i})\cup B.$

It is well-known (see e.g. [10]) that every commutative von Neumann algebra
$M$ is $\ast$-isomorphic to the algebra $L^{\infty}(\Omega,\Sigma,\mu)$ of all
(equivalence classes of) complex essentially bounded measurable functions on
$(\Omega,\Sigma,\mu)$ and in this case $LS(M)=S(M)\cong
L^{0}(\Omega,\Sigma,\mu),$ where $L^{0}(\Omega,\Sigma,\mu)$ the algebra of all
(equivalence classes of) complex measurable functions on
$(\Omega,\Sigma,\mu).$

Further we consider the algebra $S(Z(M))$ of operators which are measurable
with respect to the center $Z(M)$ of the von Neumann algebra $M.$ Since $Z(M)$
is an abelian von Neumann algebra it is $\ast$-isomorphic to
$L^{\infty}(\Omega,\Sigma,\mu)$ for an appropriate measure space
$(\Omega,\Sigma,\mu)$. Therefore the algebra $S(Z(M))$ coincides with
$Z(LS(M))$ and can be identified with the algebra $L^{0}(\Omega,\Sigma,\mu).$

The basis of neighborhoods of zero in the topology of convergence locally in
measure on $L^{0}(\Omega,\Sigma,\mu)$ consists of the sets

$W(A,\varepsilon,\delta)=\\{f\in L^{0}(\Omega,\Sigma,\mu):\exists
B\in\Sigma,\,B\subseteq A,\,\mu(A\setminus B)\leq\delta,$ $f\cdot\chi_{B}\in
L^{\infty}(\Omega,\Sigma,\mu),\,||f\cdot\chi_{B}||_{L^{\infty}(\Omega,\Sigma,\mu)}\leq\varepsilon\\},$

where $\varepsilon,\delta>0,\,A\in\Sigma,\,\mu(A)<+\infty,$ and $\chi_{B}$ is
the characteric function of the set $B\in\Sigma.$

Let us recall the definition of the dimension functions $d$ on the lattice
$P(M)$ of projection from $M$ (see [6], [10]).

Let $L_{+}$ denote the set of all measurable functions
$f:(\Omega,\Sigma,\mu)\rightarrow[0,{\infty}]$ (modulo functions equal to zero
$\mu$-almost everywhere).

Let $M$ be an arbitrary von Neumann algebra with the center $Z(M)\equiv
L^{\infty}(\Omega,\Sigma,\mu).$ Then there exists a map $d:P(M)\rightarrow
L_{+}$ with the following properties:

(i) $d(e)$ is a finite function if only if the projection $e$ is finite;

(ii) $d(e+q)=d(e)+d(q)$ for $p,q\in P(M),$ $eq=0;$

(iii) $d(uu^{*})=d(u^{*}u)$ for every partial isometry $u\in M;$

(iv) $d(ze)=zd(e)$ for all $z\in P(Z(M)),\,\,e\in P(M);$

(v) if $\\{e_{\alpha}\\}_{\alpha\in J},\,\,\,e\in P(M)$ and
$e_{\alpha}\uparrow e,$ then $d(e)=\sup\limits_{\alpha\in J}d(e_{\alpha}).$

This map $d:P(M)\rightarrow L_{+},$ is a called the _dimension functions_ on
$P(M).$

The basis of neighborhoods of zero in the topology $t(M)$ of _convergence
locally in measure_ on $LS(M)$ consists (in the above notations) of the
following sets

$\displaystyle V(A,\varepsilon,\delta)=\\{x\in LS(M):\exists p\in
P(M),\,\exists z\in P(Z(M)),\,xp\in M,$
$\displaystyle||xp||_{M}\leq\varepsilon,\,\,z^{\bot}\in
W(A,\varepsilon,\delta),\,\,d(zp^{\bot})\leq\varepsilon z\\},$

where $\varepsilon,\delta>0,\,A\in\Sigma,\,\mu(A)<+\infty$ (see [11]).

The topology $t(M)$ is metrizable if and only if the center $Z(M)$ is
$\sigma$-finite (see [6]).

Given an arbitrary family $\\{z_{i}\\}_{i\in I}$ of mutually orthogonal
central projections in $M$ with $\bigvee\limits_{i\in I}z_{i}=\textbf{1}$ and
a family of elements $\\{x_{i}\\}_{i\in I}$ in $LS(M)$ there exists a unique
element $x\in LS(M)$ such that $z_{i}x=z_{i}x_{i}$ for all $i\in I.$ This
element is denoted by $x=\sum\limits_{i\in I}z_{i}x_{i}.$

We denote by $E(M)$ the set of all elements $x$ from $LS(M)$ for which there
exists a sequence of mutually orthogonal central projections
$\\{z_{i}\\}_{i\in I}$ in $M$ with $\bigvee\limits_{i\in I}z_{i}=\textbf{1},$
such that $z_{i}x\in M$ for all $i\in I,$ i.e.

$E(M)=\\{x\in LS(M):\exists z_{i}\in P(Z(M)),z_{i}z_{j}=0,i\neq
j,\bigvee\limits_{i\in I}z_{i}=\textbf{1},z_{i}x\in M,i\in I\\},$

where $Z(M)$ is the center of $M.$

It is known [3] that $E(M)$ is *-subalgebras in $LS(M)$ with the center
$S(Z(M)),$ where $S(Z(M))$ is the algebra of all measurable operators with
respect to $Z(M),$ moreover, $LS(M)=E(M)$ if and only if $M$ does not have
direct summands of type II.

A similar notion (i.e. the algebra $E(\mathcal{A})$) for arbitrary
*-subalgebras $\mathcal{A}\subset LS(M)$ was independently introduced recently
by M.A. Muratov and V.I. Chilin [8]. The algebra $E(M)$ is called the central
extension of $M.$

It is known [3], [8] that an element $x\in LS(M)$ belongs to $E(M)$ if and
only if there exists $f\in S(Z(M))$ such that $|x|\leq f.$ Therefore for each
$x\in E(M)$ one can define the following vector-valued norm

$||x||=\inf\\{f\in S(Z(M)):|x|\leq f\\}$

and this norm satisfies the following conditions:

$1)\|x\|\geq 0;\|x\|=0\Longleftrightarrow x=0;$

$2)\|fx\|=|f|\|x\|;$

$3)\|x+y\|\leq\|x\|+\|y\|;$

$4)||xy||\leq||x||||y||;$

$5)||xx^{\ast}||=||x||^{2}$
for all $x,y\in E(M),f\in S(Z(M)).$

Let $M$ be an arbitrary von Neumann algebra with the center $Z(M)\equiv
L^{\infty}(\Omega,\Sigma,\mu).$ On the space $E(M)$ we consider the following
sets:

$O(A,\varepsilon,\delta)=\left\\{x\in E(M):||x||\in
W(A,\varepsilon,\delta)\right\\},$

where $\varepsilon,\delta>0,\,\,\,A\in\sum,\,\,\,\mu\left(A\right)<+\infty$.

The system of sets

$\\{x+O(A,\varepsilon,\delta)\\},$ (2.1)

where $x\in E(M),\varepsilon>0,\delta>0,A\in\Sigma,\mu(A)<\infty$, defines on
$E(M)$ a Hausdorff vector topology $t_{c}(M),$ for which the sets (2.1) form
the base of neighborhoods of the element $x\in E(M).$ Moreover in this
topology the involution is continuous and the multiplication is jointly
continuous, i.e. $(E(M),t_{c}(M))$ is a topological $\ast$-algebra. It is
known [5, Proposition 3.2] that $(E(M),t_{c}(M))$ is a complete topological
$\ast$-algebra and $M$ is a $t_{c}(M)$-dense in $E(M).$

##  3\. Conditions of coincidence of central extensions of von Neumann
algebras and algebras of measurable operators

In this section we describe class of von Neumann algebras $M$ for which the
algebra $E(M)$ coincide with algebra $S(M),S(M,\tau)$ and $M.$

It should be noted that [3, Proposition 1.1] and [5, Theorem 3.1] imply the
following result.

######  Theorem 3.1.

The following conditions on a given von Neumann algebra $M$ are equivalent:

(1) $E(M)=LS(M);$

(2) $M$ does not have direct summands of type II.

In this case the topologies $t_{c}(M)$ and $t(M)$ coincide.

Now we describe a class of von Neumann algebras $M$ for which the algebras
$E(M)$ and $S(M)$ coincide.

######  Theorem 3.2.

The following conditions on a given von Neumann algebra $M$ are equivalent:

(1) $E(M)=S(M);$

(2) $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ be a von Neumann
algebra of type $I_{fin},$ $M_{k}$ be a factors of type $I_{\infty}$ or III,
$k=\overline{1,n}.$

In this case the topologies $t_{c}(M)$ and $t(M)$ coincide.

For the proof of Theorem 3.2 we need following result.

######  Lemma 3.3.

If $M$ be a von Neumann algebra of type II with a faithful normal semifinite
trace $\tau,$ then $E(M)\neq S(M,\tau)$ and $E(M)\neq S(M).$

###### Proof.

Suppose that $M$ is a type II von Neumann algebra. First assume that $M$ is of
type II1 and admits a faithful normal tracial state $\tau$ on $M.$ Without
loss generality we assume that $\tau(\textbf{1})=1.$ Let $\Phi$ be the
canonical center-valued trace on $M.$ Since $M$ is of type II, there exists a
projection $p_{1}\in M$ such that

$p_{1}\sim\textbf{1}-p_{1}.$

Then $\Phi(p_{1})=\Phi(p_{1}^{\perp}).$ From
$\Phi(p_{1})+\Phi(p_{1}^{\perp})=\Phi(\textbf{1})=\textbf{1}$ it follows that

$\Phi(p_{1})=\Phi(p_{1}^{\perp})=\frac{\textstyle 1}{\textstyle 2}\textbf{1}.$

Suppose that there exist mutually orthogonal projections
$p_{1},\,p_{2},\cdots,p_{n}$ in $M$ such that

$\Phi(p_{k})=\frac{\textstyle 1}{\textstyle
2^{k}}\textbf{1},\,k=\overline{1,n}.$

Set $e_{n}=\sum\limits_{k=1}^{n}p_{k}.$ Then
$\Phi(e_{n}^{\perp})=\frac{\textstyle 1}{\textstyle 2^{n}}\textbf{1}.$ Take a
projection $p_{n+1}<e_{n}^{\perp}$ such that

$p_{n+1}\sim e_{n}^{\perp}-p_{n+1}.$

Then

$\Phi(p_{n+1})=\frac{\textstyle 1}{\textstyle 2^{n+1}}.$

Hence there exists a sequence a mutually orthogonal projections
$\\{p_{n}\\}_{n\in\mathbb{N}}$ in $M$ such that

$\Phi(p_{n})=\frac{\textstyle 1}{\textstyle
2^{n}}\textbf{1},\,n\in\mathbb{N}.$

Note that $\tau(p_{n})=\frac{\textstyle 1}{\textstyle 2^{n}}.$ Indeed

$\tau(p_{n})=\tau(\Phi(p_{n}))=\tau\left(\frac{\textstyle 1}{\textstyle
2^{n}}\textbf{1}\right)=\frac{\textstyle 1}{\textstyle 2^{n}}.$

Since

$\sum\limits_{n=1}^{\infty}n\tau(p_{n})=\sum\limits_{n=1}^{\infty}\frac{n}{2^{n}}<+\infty$

it follows that the series

$\sum\limits_{n=1}^{\infty}np_{n}$

converges in measure in $S(M,\tau).$ Therefore there exists
$x=\sum\limits_{n=1}^{\infty}np_{n}\in S(M,\tau).$

Let us show that $x\in S(M,\tau)\setminus E(M).$ Suppose that $zx\in M,$ where
$z$ is a nonzero central projection. Since any $p_{n}$ is a faithful
projection we have that $zp_{n}\neq 0$ for all $n.$ Thus

$||zx||_{M}=1||zx||_{M}1=||p_{n}||_{M}\cdot||zx||_{M}\cdot||p_{n}||_{M}\geq||zp_{n}xp_{n}||_{M}=||zp_{n}n||_{M}=n,$

i.e.

$||zx||_{M}\geq n$

for all $n\in\mathbb{N}.$ From this contradiction it follows that $x\in
S(M,\tau)\setminus E(M).$

For a general type II von Neumann algebra $M$ take a non zero finite
projection $e\in M$ and consider the finite type II von Neumann algebra $eMe$
which admits a separating family of normal tracial states. Now if $f\in eMe$
is the support projection of some tracial state $\tau$ on $eMe$ then $fMf$ is
a type II1 von Neumann algebra with a faithful normal tracial state. Hence as
above one can construct an element $x\in S(M,\tau)\setminus E(M),$ moreover
$x\in S(M)\setminus E(M).$ The proof is complete. ∎

The proof of the theorem 3.2. (1) $\Rightarrow$ (2). If the algebra $M$ has a
direct summand of type II, then by Lemma 3.3 we have that $E(M)\neq S(M).$
Hence if $E(M)=S(M),$ then there exist mutually orthogonal central projections
$z_{0},z_{1},z_{2}$ with $z_{0}+z_{1}+z_{2}=\textbf{1}$ such that $z_{0}M$ is
a type I${}_{fin},$ $z_{1}M$ is a type I∞ and $z_{2}M$ is a type III.

Suppose that $zZ(M)$ is infinite dimensional, where $Z(M)$ is the center $M$
and $z=z_{1}+z_{2}.$ Then there exists a sequence of nonzero mutually
orthogonal projections $\\{p_{n}\\}_{n=1}^{\infty}$ in $zZ(M).$ Put

$x=\sum\limits_{n=1}^{\infty}np_{n}.$ (3.1)

Then $0\leq x\in E(M)$ and $e_{n}(x)=\sum\limits_{k=1}^{n}p_{k},$ where
$e_{n}(x)$ is a spectral projection $x$ corresponding to the interval $[0,n].$
Since $zM$ is a properly infinite algebra, then
$e_{n}(x)^{\perp}=\sum\limits_{k=n+1}^{\infty}p_{k}$ is an infinite projection
for all $n\in\mathbb{N}.$ This means that $x\notin S(M),$ i.e. $E(M)\neq
S(M).$ This is a contradiction with $E(M)=S(M).$ Therefore $zZ(M)$ is a finite
dimensional algebra. Thus

$zM=\bigoplus\limits_{k=1}^{n}M_{k},$

where $M_{k}$ is a factor of type I∞ or III, $k=\overline{1,n}.$ Put
$M_{0}=z_{0}M.$ Then

$M=z_{0}M\oplus zM=\bigoplus\limits_{k=0}^{n}M_{k},$

where $M_{0}$ is a type I${}_{fin},$ $M_{k}$ is a factor of type I∞ or III,
$k=\overline{1,n}.$

(2) $\Rightarrow$ (1) Let

$M=\bigoplus\limits_{k=0}^{n}M_{k},$

where $M_{0}$ be a type I${}_{fin},$ $M_{k}$ be a factor of type I∞ or III,
$k=\overline{1,n}.$ Since $M_{0}$ is a type I${}_{fin},$ then from [3,
Proposition 1.1] it follows that

$E(M_{0})=LS(M_{0})=S(M_{0}).$

Since $M_{k}$ are factors of type I∞ or III, $k=\overline{1,n},$ then by [7,
Theorem 1] we obtain that

$S(M_{k})=M_{k}=E(M_{k}),$

for all $k=\overline{1,n}.$ Hence

$E(M)=\bigoplus\limits_{k=0}^{n}E(M_{k})=\bigoplus\limits_{k=0}^{n}S(M_{k})=S(M),$

i.e. $E(M)=S(M).$

Now let $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ is a von Neumann
algebra of type I${}_{fin},$ $M_{k}$ are factors of type I∞ or III,
$k=\overline{1,n}.$ Then $LS(M)=S(M)=E(M)$ and $M$ does not have direct
summands of type II. Therefore from Theorem 3.1 we obtain that the topologies
$t(M)$ and $t_{c}(M)$ coincide. The proof is complete. $\Box$

We now describe a class of von Neumann algebras $M$ for which the algebras
$E(M)$ and $S(M,\tau)$ coincide.

######  Theorem 3.4.

Let $M$ be a von Neumann algebra with a faithful normal semifinite trace
$\tau.$ The following conditions are equivalent:

(1) $E(M)=S(M,\tau);$

(2) $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ be a type $I_{fin}$
algebra, $M_{k}$ be a factors of type $I_{\infty},$ $k=\overline{1,n},$ and
restriction of trace $\tau$ on $M_{0}$ is finite.

In this case the topologies $t_{c}(M)$ and $t_{\tau}$ coincide.

###### Proof.

(1) $\Rightarrow$ (2). If $M$ has a direct summand of type II, then by Lemma
3.3 we obtain that $E(M)\neq S(M,\tau).$ Hence if $E(M)=S(M,\tau),$ then there
exist orthogonal central projections $z_{0},z_{1}$ with
$z_{0}+z_{1}=\textbf{1}$ such that $z_{0}M$ is a type I${}_{fin},$ $z_{1}M$ is
a type I${}_{\infty}.$

If we assume that $z_{1}Z(M)$ is infinite dimensional then the element $x\in
E(M)$ defined similarly as in (3.1) we have that $x\notin S(M,\tau),$ i.e.
$E(M)\neq S(M,\tau).$ Hence, $z_{1}Z(M)$ is finite dimensional. Thus

$zM=\bigoplus\limits_{k=1}^{n}M_{k},$

where $M_{k}$ is of type I${}_{\infty},$ $k=\overline{1,n}.$ Put
$M_{0}=z_{0}M.$ Then

$M=z_{0}M\oplus z_{1}M=\bigoplus\limits_{k=0}^{n}M_{k},$

where $M_{0}$ is of type I${}_{fin},$ $M_{k}$ are factors of type
I${}_{\infty},$ $k=\overline{1,n}.$ Now by Theorem 3.2 we have that
$E(M_{0})=S(M_{0}).$ At the same time by conditions of theorem it follows that
$E(M_{0})=S(M_{0},\tau_{0}),$ where $\tau_{0}$ is the restriction of $\tau$ on
$M_{0}.$ Thus $S(M_{0})=S(M_{0},\tau_{0}).$ Therefore by [7, Proposition 9]
the restriction of trace $\tau$ on $M_{0}$ is finite.

(2) $\Rightarrow$ (1). Let

$M=\bigoplus\limits_{k=0}^{n}M_{k},$

where $M_{0}$ is an algebra of type I${}_{fin},$ $M_{k}$ are factors of type
I${}_{\infty},$ $k=\overline{1,n}$ and the restriction $\tau$ on $M_{0}$ be a
finite. Let $\tau_{0}$ be the restriction of $\tau$ on $M_{0}.$ Since $M_{0}$
is of type Ifin then by [3, Proposition 1.1] it follows that

$E(M_{0})=LS(M_{0})=S(M_{0}).$

Now since the trace $\tau_{0}$ is a finite then $S(M_{0})=S(M,\tau_{0}).$ Thus
$E(M_{0})=S(M_{0},\tau_{0}).$ Since $M_{k}$ is a factor of type
I${}_{\infty},$ $k=\overline{1,n},$ then from [6, Theorem 2.2.9] we obtain
that $S(M_{k},\tau_{k})=M_{k}=E(M_{k}),$ where $\tau_{k}$ is the restriction
of $\tau$ on $M_{k},$ $k=\overline{1,n}.$ Therefore

$E(M)=\bigoplus\limits_{k=0}^{n}E(M_{k})=\bigoplus\limits_{k=0}^{n}S(M_{k},\tau_{k})=S(M,\tau),$

i.e. $E(M)=S(M,\tau).$

Now let $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ is an algebra of
type I${}_{fin},$ $M_{k}$ are factors of type I${}_{\infty},$
$k=\overline{1,n},$ and the restriction $\tau_{0}$ of the trace $\tau$ on
$M_{0}$ is finite. Since the restriction of the trace $\tau$ on $M_{0}$ is
finite then $E(M_{0})=S(M_{0})=S(M_{0},\tau_{0})$ and the restrictions of the
topologies $t_{c}(M)$ and $t_{\tau}$ on $E(M_{0})$ coincide. Further since
$M_{k}$ are factors of type I${}_{\infty},$ $k=\overline{1,n},$ then

$E\left(\bigoplus\limits_{k=1}^{n}M_{k}\right)=S\left(\bigoplus\limits_{k=1}^{n}M_{k},\tau|_{\bigoplus\limits_{k=1}^{n}M_{k}}\right)=\bigoplus\limits_{k=1}^{n}M_{k}$

and the restrictions of the topologies $t_{c}(M)$ and $t_{\tau}$ on
$\bigoplus\limits_{k=1}^{n}M_{k}$ coincide with the uniform topology on
$\bigoplus\limits_{k=1}^{n}M_{k}.$ Therefore the topologies $t_{c}(M)$ and
$t_{\tau}$ coincide. The proof is complete. ∎

Finally we describe a class of von Neumann algebras $M$ for which the algebras
$E(M)$ and $M$ coincide.

######  Theorem 3.5.

Let $M$ be a von Neumann algebra. The following conditions are equivalent:

(1) $E(M)=M;$

(2) $M=\bigoplus\limits_{k=1}^{n}M_{k},$ where $M_{k}$ are von Neumann factors
for all $k=\overline{1,n}.$

In this case the topology $t_{c}(M)$ coincides with uniform topology.

###### Proof.

Let $E(M)=M.$ Suppose that $Z(M)$ is infinite dimensional. Then there exists a
sequence of nonzero central orthogonal projections
$\\{p_{n}\\}_{n=1}^{\infty}$ in $M.$ Put

$x=\sum\limits_{n=1}^{\infty}np_{n}.$

Then $x\in E(M)\setminus M,$ this is contradiction. This means that $Z(M)$ is
finite dimensional. Thus

$M=\bigoplus\limits_{k=1}^{n}M_{k},$

where $M_{k}$ are von Neumann factors for all $k=\overline{1,n}.$

(2) $\Rightarrow$ (1). Let

$M=\bigoplus\limits_{k=1}^{n}M_{k},$

where $M_{k}$ are von Neumann factors for all $k=\overline{1,n}.$ Then by the
definition of central extensions it follows that $E(M_{k})=M_{k}.$ Therefore

$E(M)=\bigoplus\limits_{k=1}^{n}E(M_{k})=\bigoplus\limits_{k=1}^{n}M_{k}=M,$

i.e. $E(M)=M.$

Now let $M=\bigoplus\limits_{k=1}^{n}M_{k},$ where $M_{k}$ are von Neumann
factors for all $k=\overline{1,n}.$ Since $M_{k}$ are von Neumann factors for
all $k=\overline{1,n}$ then the restriction of the topology $t_{c}(M)$ on
$M_{k}$ coincides with the uniform topology. Therefore the topology $t_{c}(M)$
coincides with the uniform topology. The proof is complete. ∎

## Acknowledgments

The second named author would like to acknowledge the hospitality of the
”Institut fur Angewandte Mathematik”, Universitat Bonn (Germany). This work is
supported in part by the DFG AL 214/36-1 project (Germany). This work is
supported in part by the German Academic Exchange Service – DAAD.

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