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Coverage, Connectivity and Mobility in Wireless Mobile Sensor Robots. Youn-Hee Han [email protected] Korea University of Technology and Education Laboratory of Intelligent Networks Introduction. Review: Sensor Node Architecture.

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Coverage connectivity and mobility in wireless mobile sensor robots

Coverage, Connectivity and Mobility in Wireless Mobile Sensor Robots

Youn-Hee Han

[email protected]

Korea University of Technology and EducationLaboratory of Intelligent Networks

Review sensor node architecture
Review: Sensor Node Architecture

  • System architecture of a typical wireless sensor node

    • i) a computing subsystem consisting of a microprocessor or microcontroller

    • ii) a communication subsystem consisting of a short range radio for wireless communication

    • iii) a sensing subsystem that links the node to the physical world and consists of a group of sensors and actuators

    • iv) a power supply subsystem, which houses the battery and the dc-dc converter, and powers the rest of the node.

Mobile sensors
Mobile Sensors

  • Mobile Sensor Capabilities [1,2]

    • Sensing

    • Communication

    • Computation

    • Locomotion

      • Self-deploy function

  • Mobile Robots with Sensors

Static Sensor’ Capabilities

[Similar to a Tank]

[Eight Legged Robot of LEGO mindstorm]


Mobile sensor robots
Mobile Sensor Robots

Single Sensor vs. Distributed Multiple Sensors

Single Robot vs. Distributed Multi-Robots

Mobile Sensor Robots

: Distributed Multi-Robots with Sensing Capability

What issues in mobile robots
What Issues in Mobile Robots?

  • Issues in Distributed Multi-Robots [3]

    • Biological Inspirations

      • Use of the local control rules of biological societies, such as ants, bees, and birds to the development of similar behaviors in multi-robot systems.

      • behavior-based robotics

        • robot architectures are built on activity-generating building blocks rather than on centralized representations and deductive logic.

    • Communication

      • Network robotics and Inter-robot interaction

      • How to handle non-deterministic time delays in communications and achieve robust performance in faulty communication environment

        • E.g., the remote tele-operation of space exploration robots

      • Connectivity Issues

What issues in mobile robots1
What Issues in Mobile Robots?

  • Issues in Distributed Multi-Robots [3]

    • Localization, Mapping, and Exploration

      • Enables robot team members to track positions of autonomously moving objects

      • Navigate between places of interest in an initially unknown environment

    • Motion Coordination

      • Multi-robot path planning, formation generation

    • Reconfigurable Robotics

    • Architecture, Task Allocation, and Control

    • Object Transport and Manipulation

What issues in mobile sensor robots
What Issues in Mobile Sensor Robots?

  • Then, what issues in Mobile Sensor Robots ?

    • Environmental Robotics

      • the deployment of distributed sensors and supported mobile sensor robots to observe, monitor, and assess the state of complex environmental processes.

      • It involves many different types of distributed sensing in land, sea, and air, and the coordination of mobile sensors through adaptive redeployment and adaptive sampling of environmental phenomena.

      • Coverage Issues


Mobile sensor robots1
Mobile Sensor Robots

[조선일보 2008-09-22]

떼지어 군사작전 '로봇' 나왔다

정찰·독성물질 탐지 수행… 英 내년 상용화

벌이나 개미처럼 무수한 소형로봇들이 하나의 군사작전을 수행하는 '로봇떼(swarm of robots)'가 곧 현실화한다. 영국 국방부가 16~18일 영국 솔즈베리에서 개최한, 새 군사 기술의 경연대회인 '그랜드 챌린지'에서 특히 '소형로봇떼' 개념이 떠오르는 신기술로 주목을 받았다고 BBC 방송이 보도했다. 전체 11개 팀 중에서 3개 팀이 '로봇떼'를 선보였다. 작은 곤충로봇들이 땅에서 움직이는 '마인드시트(Mindsheet)', 날아다니는 비행로봇들의 집단인 '로커스트(Locust)', 그리고 미니 헬리콥터 8개가 나는 '아울스(Owls)'등이다. 영국 국방부는 '아울스'의 기술을 참가 팀들 중 '가장 혁신적인 아이디어'로 선정했다.

아울스는 8개의 소형 헬리콥터 로봇이 한 팀이 돼 움직인다. 로봇 1개당 프로펠러 4개가 달려 있고 무게는 1㎏ 미만. 이 로봇떼는 다양한 각도에서 고해상도의 영상을 찍어 적의 위협을 감지한다. 대기에 뿌려진 독성물질을 탐지할 수도 있다. 8개 중 일부가 파괴되거나 고장 나도, 나머지 로봇들이 없어진 로봇들의 임무를 대신하도록 프로그램 돼 있다. 내년에 '새떼'의 움직임을 모방한 알고리즘까지 아울스에 내장된다. 영국 일간지 가디언은 "아울스는 내년쯤 상용화해 영국군에 배치될 전망"이라고 보도했다.

이 밖에, 현재 미군이 개발 중인 '마이크로 자동 시스템기술(MAST)' 프로그램은 병사 1명에게 하나의 로봇떼를 제공하는 것이 목표다. MAST의 로봇떼는 시가전(市街戰)상황에서 건물이나 모퉁이 너머로 몰래 다가가 적의 동태를 살피는 '정찰병' 역할을 수행한다.

Change of research issues in sensor networks
Change of Research Issues in Sensor Networks

  • Hardware (2000)

    • CPU, memory, sensors, etc.

  • Protocols (2002)

    • MAC layers

    • Routing and transport protocols

  • Applications (2004)

    • Localization and positioning applications

  • Management (2005)

    • Coverage and connectivity problems

    • Power management

    • Etc.

FromDr. Yu-Chee Tseng(Associate Dean),College of Computer Science, National Chiao-Tung University

Study of coverage problem
Study of Coverage Problem

  • Coverage Problem

    • In general, determine how well the sensing field is monitored or tracked by sensors.

  • Objectives of the problem

    • Determine the coverage hole (or targets)

    • Minimize the number of sensors deployed

    • Make the whole area covered by three or more sensors

      • Location determination by “Triangulation”

    • Maximize the network lifetime

      • [Def.] Sensor Network Lifetime

        • The time interval that all points (or targets) in the given area is covered by at least one sensor node.

    • Etc.

Review art gallery problem
Review: Art Gallery Problem

  • Victor Klee (1973)

    • Place the minimum number of cameras such that every point in the art gallery is monitored by at least one camera

  • Chvátal's art gallery theorem (1975)

    • guards (cameras) are always sufficient and sometimes necessary to guard a simple polygon with vertices

42 vertices  upper bound:

Review power saving



* 2Mb/s IEEE 802.11 Wireless LAN

Energy Consumption




Review: Power Saving

  • Make the sensor node sleep!!! [13]

  • Rockwell’s WINS Nodes

  • Medusa II Nodes

It is highly recommended to “schedule” the wireless sensor nodes to alternate between active (Tx, Rx, Idle) and sleep mode

Review power saving1
Review: Power Saving

  • Make the sensor node intelligent!!! [13]

    • The ratio of the energy spent in sending one bit of information to the energy spent in executing one instruction.

      • 1500~2700 for Rockwell’s WIN nodes

      • 220~2900 for the MEDUSA II nodes

      • 1400 for the WINS NG 2.0

    • So, local data processing, data fusion and data compression are highly desirable.

Problem design methodology
Problem Design Methodology

  • Algorithm Characteristics

    • 1) Centralized

    • 2) Distributed

    • 3) Self-*

      • Self-determination

        • free choice of one’s own acts without external compulsion

      • Self-organization (Self-configuration)

        • a process of evolution where the effect of the environment is minimal, i.e. where the development of new, complex structures takes place primarily in and through the system itself

      • Self-healing

        • For example, a mobile sensor can move to an area with a coverage hole or routing void and significantly improve network performance.

Problem design criteria 1 2
Problem Design Criteria (1/2)

  • Sensor Deploy Method

    • Deterministic (planned) vs. Random

  • Coverage Types

    • Area coverage vs. Target (Point) coverage

















Problem design criteria 2 2
Problem Design Criteria (2/2)

  • Coverage Modeling

    • Binary Model vs. Probability Model

  • Communication Range ( ) & Sensing Range ( )

    • vs. vs.

    • Homogeneous vs. heterogeneous?

Probabilistic sensing model

Binary, unit disc sensing model

Coverage modeling
Coverage Modeling

  • Binary Model [1]

    • Each sensor’s coverage area is modeled by a disk

    • Any location within the disk is perfectly monitored by the sensor located at the center of the disk; otherwise, it is not monitored by the sensor.

  • Probability Model [2]

    • An event happening in the coverage of a sensor is either detected or not detected by the sensor depending on a probability distribution

    • Hence even if an event is very close toa sensor, it may still by missed by the sensor.

Binary model k coverage in 2 d
BinaryModel: K-coverage in 2-D

  • K-coverage (only within Binary Model)

    • [Definition] covered

      • A location in an area is said to be covered by if it is within 's sensing range.

    • [Definition] k-covered (location or area)

      • A location in an area is said to be k-covered if it is within at least K sensors' sensing ranges.

      • “k” is called coverage level

  • WhyK>1?

    • Fault-tolerance in case of the dismissal of some sensors

    • Power saving and enlarge network lifetime

    • Triangulation: getting location of a targeted object

    • Uplift the confidence level on gathering information

Binary model k coverage in 2 d1
BinaryModel: K-coverage in 2-D

  • Problems about K-coverage [1]

    • [Definition] k-NC problem

      • Given a natural number k, the k-Non-unit-disk Coverage (k-NC) problem is a decision problem whose goal is to determine whether all points in an area are k-covered or not.

    • [Definition] k-UC problem

      • Given a natural number k, the k-Unit-disk Coverage (k-UC) Problem is a decision problem whose goal is to determine whether all points in an area are k-covered or not, subject to the constraint that r1 = r2 = · · · = rn.

k-UC (k=1)

k-NC (k=1)

BinaryModel: K-coverage in 2-D

Is this area 1-covered?

So this area is not 1-covered!

This region is not covered by any sensor!

This area is not only 1-covered, but also 2-covered!

What is the coverage level of this area?

1-covered means that every point in this area is covered by at least 1 sensor

2-covered means that every point in this area is covered by at least 2 sensors

Coverage level = k means that this area is k-covered

Binary model k coverage in 2 d2
BinaryModel: K-coverage in 2-D

  • Algorithm to determine coverage level, k, in a given sensor network? [1]

    • [Definition] k-perimeter-covered

      • Consider any two sensors si and sj. A point on the perimeter of si is perimeter-covered by sj if this point is within the sensing range of sj

    • [Theorem]

      • An area A is k-covered iff each sensor in A is k-perimeter-covered.

    • 2차원 문제를 1차원 문제로 바꾸어 해결

    • Partially self-determination, but a central node determines the coverage level (k) finally.

Binary model coverage configuration in 2 d
BinaryModel: Coverage Configuration in 2-D

  • Coverage Configuration Protocol (CCP) [3]

    • 1) a coverage level (k) is allocated to all sensors

    • 2) all sensors are deployed randomly at the target area

    • 3) Each sensor makes itself sleep or active to achieve the coverage level

    • [Theorem]

      • A given area is “k-covered” if the following conditions are satisfied

        1) All intersection points between each pair of sensors are "k-covered"

        2) All intersection points between each sensor and boundary of the area are "k-covered”

Active nodes

Intersection points

Binary model coverage configuration in 2 d1
BinaryModel: Coverage Configuration in 2-D

  • Coverage Configuration Protocol (CCP) [3]

    • A node becomes “sleep” if all intersection points inside its coverage is already K-covered by other active nodes in its neighborhood.

    • A node becomes “active” if there exists an intersection point inside its sensing circle that is not K-covered by other active nodes.


Active nodes

Sleeping nodes

Intersection points

Binary model k coverage in 3 d
BinaryModel: K-coverage in 3-D

  • K-coverage in 3-D [4]

    • [Definition] k-BC Problem

      • Given a natural number k, the k-Ball-Coverage (k-BC) Problem is a decision problem whose goal is to determine whether all points in a 3-D cuboid sensing area are k-covered or not.

    • How to determine k?

      • (3D2D) Determine whether the sphere of a sensor is sufficiently covered

      • (2D1D) Determine whether the circle of each spherical cap of a sensor intersected by its neighboring sensors is covered

Probability model
Probability Model

  • Why Probability Coverage Model? [2]

    • Quality of sensor surveillance may be much affected by sensing distances, signal propagation characteristics, obstacles, and environmental factors.

    • Probability coverage model may be more realistic!

  • Methodology

    • Simple Model [5]

    • Signal-strength-based Model [2]

임의의 센서와 가까운 지역이 특수한 요인 (장애물)에 의하여 센싱이 되지 않을 수 있거나 그 센서와 먼 지역이 특수한 요인 (다수의 센서의 감지)에 의하여 센싱이 될 수도 있다.

Probability model1
Probability Model

  • Simple Model [5]

    • : the probability that a sensor can sense a event happened at a location

    • : the detection probability contributed by the sensors


  • Basic Policy

    • Sensor should be active or sleep?

    • Scheduling (related to the coverage issue)

      • An interval: is active

      • Another interval: is active

      • So, the battery power can be saved


  • Scheduling Type

    • Centralized

      • All sensors send “their location information” to the centralized sink node.

      • The sink node performs “its scheduling algorithm” for the sensors

      • The sink node broadcasts “the scheduling information” to all sensor nodes

      • Each sensor becomes active or sleep according to the information

    • Distributed

      • Each sensor self-determies its scheduling time

      • # of messages reduced

Centralized scheduling
Centralized Scheduling

  • MDSC (Maximum Disjoint Set Covers) [9]

[Definition] Maximum Disjoint Set Covers Problem

Centralized scheduling1
Centralized Scheduling

  • MDSC (Maximum Disjoint Set Covers) [9]

    • For example,

      • C={S1, S2, S3, S4}, TARGETS={t1, t2, t3}

      • A sensor’s battery lifetime: 1

      • Network Lifetime without any scheduling: 1

      • By MDSCScheduling

        • Two Set Covers, C1 and C2

          • C1={S1, S2} with active time=1

          • C1={S3, S4} with active time=1

        • So that, network lifetime: 2















Centralized scheduling2
Centralized Scheduling

  • MSC (Maximum Set Covers) [10]

[Definition] Maximum Set Covers Problem




MDSC problem is a special case of MSC problem.!

Centralized scheduling3
Centralized Scheduling

  • MSC (Maximum Set Covers) [10]

    • For Example,

      • By MSCScheduling

        • Network Lifetime: 2.5








active time=0.5

active time=0.5

active time=1

active time=0.5

Centralized scheduling4
Centralized Scheduling

  • MSC (Maximum Set Covers) [10, 11]

    • Existing Algorithms

      • Linear Programming [10]

      • Greedy [10] (Complexity: )

      • Branch-and-Bound [11]

i: # of setcovers, m: # of targets, n: # of sensors

Centralized scheduling5
Centralized Scheduling

  • MSC (Maximum Set Covers) [10, 11]

    • Existing Algorithms

      • Linear Programming [10]

      • Greedy [10] (Complexity: )

      • Branch-and-Bound [11]

i: # of setcovers, m: # of targets, n: # of sensors

Distributed scheduling
Distributed Scheduling

  • 1-Coverage Preserving Scheduling (1-CP) [12]

    • For Example

Init Phase: 1) Each sensor exchange its location and Ref. value

2) Each sensor get its schedule (active) time

The set of intersection points within ‘s area


The set of sensorscovering the target p

Ref1=2, Ref2=9, Ref3=11

Distributed scheduling1
Distributed Scheduling

  • 1-Coverage Preserving Scheduling (1-CP) [12]







  • Why Connectivity?

    • Any sensing data should be sent to gateway (sink, base station) node

    • Multi-hop routing

Base Station


K connectivity

  • Connected Graph of Sensor Networks

    • Vertex: each sensor nodes

    • Edge: direct communication path for pairs of sensors

      • there exists an edge between two vertices iff the distance between them is less or equal to the transmission range r.

K connectivity1

  • [Definition]k-connectivity

    • The network will remain connected after removing any arbitrary k-1 sensors from network.

    • It is also called “vertex k-connectivity” (not “edge k-connectivity”)

  • k-connected:  any pair of nodes are connected by k indep. paths

    • Independent paths:

K connectivity2

  • Examples



K edge connectivity

  • [Definition]k-edge-connectivity

    • The network will remain connected after removing any arbitrary k-1 edges from network.

  • k-edge-connected:  any pair of nodes are connected by k disjoint paths

    • disjoint paths:

Min power connectivity problem
Min-Power Connectivity Problem

  • Connectivity & Transmission Power

    • Nodes in the network correspond to transmitters

    • More power  larger transmission range  More Edges  More Connectivity

      • transmitting to distance r requires rpower

    • Battery operated  power conservation critical

  • [Definition]Min-Power Connectivity Problems

    • Find min-power range assignment so that the resulting communication network satisfies prescribed properties (k-connectivity)

Min power connectivity problem1















Min-Power Connectivity Problem

Range assignment

Communication network

K connectivity k coverage
K-Connectivity & K-Coverage

  • Relationbetween K-Coverage and K-Connectivity [3]

    • Communication Range:

    • Sensing Range:


      • If the given region is continuous and , “The region is k-covered” means “The region is k-connected”

    • For example, k=1

      • Assume that the requested coverage level, k, is one and

      • If The sensors covers the whole region completely, then

      • Any sensing data produced by a sensor can be delivered to the sink node.

Sensing and communication ranges
Sensing and Communication Ranges

  • Real Products’ Ranges [7]

Potential field based strategy
Potential Field-based Strategy

  • Self-deploy using Potential Field [4]

    • Problem Definition

      • How to maximize the sensor coverage in amodel-free environment

    • Assumption

      • each node is equipped with a sensor that allows it to determine the range and bearing of both nearby nodes and obstacles

        • sensors can be constructed using “scanning laser range-finder”, “supersonic” or “omni-camera”.

    • Procedure Summary

Deploy the sensor nodes randomly

Determine “the virtual forces” from nodes and obstacles

convert “the virtual forces” into a control vector to be sent to its motors.

Potential field
Potential Field

  • Potential Fields and Forces [4]

  • Potential Fields generated by Obstacles and Boundary [5]

The force vectors in the potential field generated by “AvoidObstacle” behavior

Force vectors from potential field
Force Vectors from Potential Field

  • Force Vectors

    • Force Vector due to obstacles

      • : coordinate of the current sensor node

      • : coordinate of obstacle

      • : distance from obstacle and the node

      • : constant describing the strength of the field

    • Force Vector due to other sensors

      • : coordinate of other sensor

      • : distance from sensor and the node

      • : constant describing the strength of the field

    • The compound force vector by the two components

How to determine the next position
How to determine the next position?

  • From Force Vectors to next location

    • Next Acceleration

      • : mass of the node

      • : friction force (마찰력)

        • : viscosity coefficient

        • : current velocity of node

  • Next Velocity

    • : unit time

  • Next Location

    • : current location of the node


  • Example



Performance evaluation
Performance Evaluation

  • Proto-typical deployment experiment for a 100-node network.

  • Initial network configuration.

  • Final configuration after 300 seconds.

  • Occupancy grid generated for the final configuration; visible space is marked in black (occupied) or white (free); unseen space is marked in gray.

Coverage hole based strategy
Coverage hole-based Strategy

  • Self-deploy using Coverage Hole [7]

    • Problem Definition

      • How to maximize the sensor coverage with minimal time and minimal movement distance in an obstacle-less model-free and finite environment

    • Procedure Summary

Deploy the sensor nodes randomly

Discover the coverage hole (the area not covered by any sensor)

Calculate the target positions of the moving sensors

Voronoi diagram
Voronoi Diagram

All positions inside are closer to the node O than to any other nodes

  • Voronoi polygon

    • : Voronoi polygon of sensor node O

      • isthe set of Voronoi vertices of O

      • is the set of Voronoi edges of O

    • : the set of Voronoi neighbors of O

  • example

Voronoi diagram1
Voronoi Diagram

  • Why Voronoi diagram?

    • All positions inside a Voronoi partition are closer to the generating node than to any other nodes

    • So, each sensor is responsible for the sensing task only within its Voronoi partition

      • One partition is small area to be monitored by one sensor

      • Each sensor just examine the coverage hole locally

Coverage hole
Coverage hole

  • How to find the coverage hole?

    • After constructing the Voronoi polygons, each sensor intersects it with the sensing circle of the containing sensor.

  • If it is found, next?

    • If any coverage hole exists in its Voronoi partition, the generating sensor decide where to move to eliminate it or reduce its size.

Movement protocols
Movement protocols

  • Three movement protocols

    • VEC (VECtor-based)

      • pushes sensors away from a densely covered area

    • VOR (VORonoi-based)

      • pulls sensors to the sparsely covered area

    • Minimax

      • moves sensors to their local center area

  • Features

    • Distributed Self-deployment protocols

Vec vector based
VEC (VECtor-based)


Final goal

  • Strategy

    • To find the overall virtual force as the vector summation of virtual forces from the boundary and all Voronoi neighbors.

    • The virtual force will push sensors from the densely covered area to the sparsely covered area.

  • Terms

    • : the distance between two sensors ( , )

    • : the distance between a sensor and boundary

    • : the average distance betweentwo sensors when the sensors are evenly distributed in the target area

      • It should be calculated beforehand

Vec vector based1
VEC (VECtor-based)


센서 s1 과 s2 모두 자신의 Voronoi Partition을 Cover 하고 있지 못하므로 둘 다 전체 평균 거리에 그들 사이의 거리를 뺀 것에 대해 절반의 거리씩 이동

Boundary 로 부터 센서 s1까지의 거리는 전체 센서들의 평균 거리의 절반으로 유지해야 한다.

센서 s3는 자신의 Voronoi Partition을 Cover 하고있으므로 센서 s3는 이동하지 않고 센서 s1만 전체 평균 거리에서 그들 사이의 거리를 뺀 거리를 이동

  • E.g.) Vector Summation of the sensor s1

Vec vector based2
VEC (VECtor-based)

  • The execution of VEC

    • 35 sensors / 50m x 50m / random deployment

    • Coverage : 75.7% -> 92.2% -> 94.7%

Vor voronoi based
VOR (VORonoi-based)

  • Strategy

    • Pull sensors to their local maximum coverage holes

      • Sensors move toward its farthest Voronoi vertex ( )

      • In the above figure, Sensor si’s target location is B

        • is equal to the sensing range

    • It is a greedy algorithm

Vor voronoi based1
VOR (VORonoi-based)

  • The execution of VOR

    • Coverage : 75.7% -> 89.2% -> 95.6%


Circumcircle of 3 Voronoi vertices



  • Strategy

    • Choose the target location as the point inside the Voronoi polygon whose distance to the farthest Voronoi vertex ( ) is minimized

      • The target location is called “Minimax point ( )”

    • It reduces the variance of the distances to the Voronoi vertices, resulting in a more regular shaped Voronoi polygon

    • It considers distances to all the Voronoi vertices, rather than only to the farthest vertex.


Minimax point.

So, how to find it?

최소 크기를 가지는 외접원의 중심

  • VOR vs. Minimax


  • Terms

    • : Minimax point (target point)

    • : Minimax circle centered at the minimax point , with radius

    • : Circumcircle of three points

    • : Circumcircle of three points

  • Algorithm

    • 1) Find all the circumcircles of any 2 and any 3 Voronoi vertices.

    • 2) Among these circles, select the one having the minimum radius and covers all the vertices as the Minimax circle for that polygon.

    • 3) The center of the selected circle is the Minimax point


  • The execution of Minimax

    • Coverage : 75.7% -> 92.7% -> 96.5%

Performance evaluation2
Performance Evaluation

  • Coverage

    • Minimax performs best, VEC the worst.

      • Minimax fully utilizes the Voronoi polygon

      • VEC does not consider holes nor Voronoi polygon structure when choosing target location

    • Minimax better than VOR

      • since it considers more information.

Performance evaluation3
Performance Evaluation

  • Coverage vs. Communication Range

    • Performance is reduced when communication range is reduced.

      • This is because most sensors do not know all the neighbors, thus construct inaccurate Voronoi polygons.

      • Consequently get incorrect coverage holes and target locations.

    • VEC is least affected, since it does not use the Voronoi polygon to determine target location.

Performance evaluation4
Performance Evaluation

  • Moving Distance

    • Minimax moves longer distance than VOR, since not only fixes holes but tries to reach more regular shaped polygons.

    • For VEC, moving distance is similar under different sensor densities.

Innercenter vs circumcenter vs centroid
Innercenter vs. Circumcenter vs. Centroid

[Centroid vs. Center of Gravity]

- 도심(Centroid)와 무게중심(Center of Gravity)은 일반적으로 동의어로 쓰인다.

- 하지만, 도심의 계산은 기하학적인 모양에만 관련이 된다.

- 만약 물체가 균질하다면(homogeneous) 즉, 일정한 밀도를 가졌다면 무게중심과 도심은 일치한다.


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