Stability of an Immersed Object

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Stability of an Immersed Object

By: Song Yoong Siang (PhD Student)

 

When an object is completely immersed in a liquid, its stability depends on the relative positions of the centre of gravity and the centre of buoyancy. There are three conditions of equilibrium of immersed object, which are stable equilibrium, unstable equilibrium, and neutral equilibrium.

When the centre of buoyancy, B is above the centre of gravity, G, the body is in stable equilibrium state. Figure 1 (a) shows an example of object in stable equilibrium state. If this object is given s small angular displacement and then released, it will be able to returns to its original position by retaining the originally vertical axis as vertical.

When the centre of buoyancy, B is belowthe centre of gravity, G, the body is in unstable equilibrium state. Figure 1 (b) shows an example of object in unstable equilibrium state. If this object is given s small angular displacement and then released, it will not return to its original position but moves further from it.

When the centre of buoyancy, B is at same point with the centre of gravity, G, the body is in neutral equilibrium state. Figure 1 (c) shows an example of object in neutral equilibrium state. If this object is given s small angular displacement and then released, it will simply maintain at its new position.

 

2a may15

Figure 1: (a) stable equilibrium (b) unstable equilibrium (c) neutral equilibrium

 

Reference(s):

  1. http://nptel.ac.in/courses/112104118/lecture-5/5-6_stability_submerged_fluid.htm
  2. http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops /statics/node24.html

Restoring Forces and Moments of an Underwater Vessel

 

button2Restoring Forces and Moments of an Underwater Vessel

By: Song Yoong Siang (PhD Student)

 

 

All submerged vessel will affected by weight, W and buoyancy force, B. These two forces are called restoring forces. Weight is calculated using Eq. (1) whereas buoyancy force is calculated using Eq. (2), where m is the mass of vessel, g is gravitational acceleration, 2a june15 is the density of sea water, and V is the total volume of vessel.

  2b june15

The weight and buoyancy force can be transformed to the body frame by using Eq. (3) and Eq. (4), where 2c june15 is the Euler angle coordinate transformation matrix.

2d june15

The restoring moment, M is calculated using Eq. (5), where r is the displacement vector from body frame to the restoring force.

2e june15


Finally, the vector of gravitational/buoyancy forces and moments 2f june15in the body fixed frame can be represented by Eq. (6), where 2g june15are the same three rotational angles, roll, pitch, and yaw of the body fixed frame, 2h june15is the distance of center of buoyancy from body fixed frame, and 2i june15is the cistance of center of gravity form body fixed frame.

 

2j june15

Reference(s):

  1. T. I. Frossen, "Marine Control System: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles," Marina Cybernetics AS. 2002


 

 

One Obstacle Avoidance Method-Artificial Potential Field (Part II)

button6One Obstacle Avoidance Method-Artificial Potential Field
(Part II)

By: Mei Jian Hong (PhD Student)

 

(From Part I)

From Eqn. (1) we can see that the repulsive force becomes greater when the robot move towards obstacle. Generally a minimum safety margin d0 is set to prevent collision. When  6a aug15  Thus the repulsive potential function is 

6b aug15     (2)

 

Attractive potential field

The robot is compelled away from obstacles by repulsive potential, while it is able to track target by attractive potential. The attractive potential function is also related to the distance between target and robot. It could be expressed as,

6c aug15      (3)

Where Ka is attractive potential constant, XR is location of robot and XG is location of goal.

The robot is affected by both attractive and repulsive force for autonomous navigation. Where the attractive force is

6d aug15     (4)

The process of robot navigated by artificial potential field is illustrated in Fig. 2. The final heading and speed of robot is determined by the resultant force of attractive force and repulsive force.

6e aug15

Fig. 2. Artificial Potential Field illustration.

 

The simulation result of obstacles avoidance by artificial potential field is shown in Figure 3. From the result we can see that the APF method is effective for reactive obstacles avoidance, however, it is not an optimal path.

6f aug15

 

Reference(s):

  1. O. Khatib, Real-time Obstacle Avoidance for Manipulators and Mobile Robots. International Journal of Robotics Research, 1986, 5(1): 90-98.
  2. http://www.cs.mcgill.ca/~hsafad/robotics/

 

Solving First Order Linear Differential Equation

button2Solving First Order Linear Differential Equation

By: Song Yoong Siang (PhD Student)

 

Given a first order linear differential equation:

2a aug15

It can be solved by finding an integrating factor, 2b aug15 such that;

2c aug15

Below are the step-by-step solution to solve a first order linear differential equation:

2d aug15

Example: Solving the equation:2e aug15

2f aug15

               

Reference(s):

  1. http://www.sosmath.com/diffeq/first/lineareq/lineareq.html

 

One Obstacle Avoidance Method-Artificial Potential Field (Part 1)

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One Obstacle Avoidance Method-Artificial Potential Field (Part 1)

By: Mei Jian Hong (PhD Student)

 

1.Introduction

 

Artificial Potential Field (APF) is a reactive method for robot path planning, which was proposed by Khatib in 1986. It uses a position related potential function to achieve obstacles avoidance, and it makes the robot flexibly cope with the changing environment.

Assuming that the there are obstacles standing between start and target in robot motion space, the robot is required to plan a free collision path from start to target. The idea of APF method mainly consists of virtual force vectors, where obstacles repulsive forces and target attractive forces work on the robot simultaneously. Thus the motion of robot is determined by the resultant of forces, as shown in Figure 1.

 

6a june15

Figure 1. Illustration of virtual repulsive and attractive forces

 

The repulsive forces induced by obstacles are represented by repulsive potential function which is related to the distances from robot to obstacles. The potential function can be expressed as,

6b june15

6c june15

Reference(s):

  1. O. Khatib, Real-time Obstacle Avoidance for Manipulators and Mobile Robots. International Journal of Robotics Research, 1986, 5(1): 90-98.
  2. http://www.cs.mcgill.ca/~hsafad/robotics/

           

 

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