ALGORITHM FOR ROBOT NAVIGATION AT ENVIRONMENT WITHOUT COLLISION
 
ALGORITHM REPRESENTATION FOR NAVIGATION OF MOBILE ROBOT WITHOUT OBSTACLE COLLISON
Mobile robot It is a kind of robot that has the ability to travel Relative to the environment (i.e. locomotion), and one of the actuators of the robot is the locomotive system
This chapter of my bachelor thesis is to develop algorithms that will help the autonomous mobile robot in visual navigation. g the robot. Then, the robot tries to understand their environment to extract data from a sequence of image data, in this case, optical, and then uses this information as a guide for the movement. The strategy adopted to avoid collisions with obstacles during movement – a balance between the right and left optical flow vectors.

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An integral part of any navigation scheme is the desire to reach a destination and do not get lost or bump into any of the objects. There may be other restrictions on a given route, such as speed limits or zones of uncertainty, where in theory, of course, can pave the path, but not desirable. Often, the way is to move the robot autonomously planned, ie on the basis of previous input and without interference in real time. It can work effectively, but only on condition that the environment is perfectly known and does not change and the robot can travel on the route perfectly. However, in the real world everything is much more complicated.
Note that navigation will include:

Location of robot
Environment perception and his model
Methods of traffic planning
Robot motion control algorithms

The problem of mobile robot navigation is a very complex issue characteristic at both ends. The implementation of tasks by moving a mobile robot requires obtaining information about the surrounding-limiting environment – hence the importance of having AD sensory system that allows the observation of the environment and its perception, For this purpose, both simple rangefinder systems and contact sensors, which correspond with collision detection.
Using a constant speed of 4m/s for the algorithm and a step size of 0.125m which was obtained by the multiplication of the speed by interval in which information is received. = 0.125m. The algorithm is given below.

Set iteration values K equals K =1. Tolerance factor δ
Using the following sequence inside the loop for K

If ||ΔT||δ, if not set K=1 And repeat step 2.otherwise terminate

But considering the above algorithm it’s still going to encounter some problems. For example saw tooth pattern that occurs at the along the path, shown below:
Saw-tooth pattern
Saw-tooth happens due to fixed step size at some point in the navigation of the mobile robot reduction in step size is necessary which also means reduction in the speed of the robot . The reason for this effect is because the present point of the robot is not always the best point possible. Meaning that point after that will guide the path back, resulting in a saw=tooth pattern zig -zagging along the path. The reason this problem occurs is because the robot has a constant speed.
ADDING CONSTAINTS:
To determine the new point of the robot the speed and acceleration needs to be known if we have a speed of and an acceleration of
The constraints are |speed|
Now starting speed will be set has speed(K=0)=0m/s, which means is assumed that robot is in a static state
Determining position of robot
All points in the line represent the Newton’s Direction. Robot needs to move to one of its point so we can determine the speed and acceleration of robot
OVERSHOOT SCENARIO:
This is a scenario when the acceleration that is generated is not large enough to get to the point on the newton direction, solution to this can’t be found, the only way out is that the point closer to the line will move .I.e. line perpendicular to the newton’s direction must be found and the lie should intercept in the center.

Now considering the new algorithm

Setting values at start point, target point and obstacle location

MATHEMATICAL BACKGROUND OF ALGORITHM
FUNCTION OF TARGET:
Every robot has its starting point and it has its destination that to say its target point and to accomplish this task it needs a target function: Target function is

Where the position of the mobile robot is at present is and the destination of mobile robot is . A mobile robot has reached its minimum function when current position of the robot is equal to the target position.
Fig 1: Position of Target
BOUNDARY FUNCTION:
Every Mobile robot has its environment and areas that are out of mobile robots environment is therefore represented with a boundary. What the boundary represents is the size, shape and location of an object. Boundary function and function of target will both give an optimization problem when finding the minimum.
BARRIER FUNCTION:
The most difficult part of mobile robot navigation is generating its path without going out of its environment that is where the barrier function comes in
The barrier function and the target function are added up, and this leads to the following function:

PENALTY FUNCTION:
What the penalty function does is that it controls the importance of obstacles on the path of a mobile robot. It show if an obstacle is of high priority or isn’t. This is where distance comes to play how close the obstacle to the robot is to the obstacle. When calculating the penalty function of a mobile robot the most important obstacles are the obstacles closer to the robot. The penalty function is obtained by the calculation of the distance between the obstacle and the mobile robot. The result of the calculation shows the increases or decreases considering the movement of the robot away or towards the obstacle
This represents the variation is the distance between the obstacle and mobile robot.
NEWTON DIRECTION:
Mobile robot optimization is very important in robot navigation. Choosing the most efficient path to follow to from robot’s current position to the target point around its environment, this is called Newton method. Newton direction is calculated by the optimal direction in which a step should be taken, ithis is given in the equation below:

Where is the gradient of target function and the inverse of hessian matrix is: which is used to describe the second order derivative of the function of target, that is evaluated at point (delta t) is used in describing the change in the first order derivative of function of target.
THEORITICAL EXPERIMENT
After considering the algorithm it will be right to do some experiments based on the algorithm to investigate and test whether it does what we want it to. I will be using static obstacles to test.
ONE STATIONARY OBSTACLE:
Stationary point;