Position of robot after given movements

The position of a robot after a series of movements can be calculated mathematically by tracking its displacements along the coordinate axes. Assuming the robot starts at the origin $(x_{0}, y_{0})$, its position can be updated based on a sequence of movements in a 2D plane.

Math Formula for Robot Position

Let:

Ø $(x_{0}, y_{0})$ be the initial position of the robot.

Ø $( d_{x}, d_{y})$ represent the displacements in the x and y directions for a single movement.

Ø (x, y) be the final position of the robot after n movements.

The final position can be expressed as:

$$ x=x_{0}+ \sum_{i=1}^{n}d_{x,i} $$

$$ y=y_{0}+ \sum_{i=1}^{n}d_{y,i} $$

Special Cases: Movement Directions

For typical directions (e.g., up, down, left, right), the displacement can be defined as:

Ø Up: $d_{x}=0,d_{y}=+step size$

Ø Down: $d_{x}=0,d_{y}=-step size$

Ø Left: $d_{x}=−step size,d_{y}=0$

Ø Right: $d_{x}=+step size,d_{y}=0$

Example

Suppose the robot starts at (0,0) and follows these movements:

Move Up by 2 units
Move Right by 3 units
Move Down by 1 unit
Move Left by 4 units

Calculation:

Up: x = 0, y = 0 + 2 = 2
Right: x = 0 + 3 = 3, y = 2
Down: x = 3, y = 2 - 1 = 1
Left: x = 3 - 4 = -1, y = 1

Final Position: (−1,1)

Generalization in Vector Form

If the robot's movement is represented as a vector $M_{i}=(d_{x,i},d_{y,i}),$ the position after n steps is:

$(x,y)=(x_{0},y_{0})+ \sum_{i=1}^{n}M_{i}$

This formula can be extended to 3D or higher dimensions by including additional axes (e.g., z).

To track the position of a robot after a given sequence of movements in Java, you can define a simple program that handles movement commands (e.g., 'U' for up, 'D' for down, 'L' for left, 'R' for right). Here's an example implementation:

java

public class RobotMovement {

public static void main(String[] args) {

// Define the initial position of the robot

int x = 0;

int y = 0;

// Define the sequence of movements

String movements = "UUDDLRLR";

// Process each movement command

for (char move : movements.toCharArray()) {

switch (move) {

case 'U':

y += 1; // Move up

break;

case 'D':

y -= 1; // Move down

break;

case 'L':

x -= 1; // Move left

break;

case 'R':

x += 1; // Move right

break;

default:

System.out.println("Invalid movement command: " + move);

}

// Print the final position of the robot

System.out.println("Final position: (" + x + ", " + y + ")");

}

Explanation:

Initial Position: The robot starts at the origin (0, 0).
Movement Sequence: The movements are provided as a string (e.g., "UUDDLRLR").
Processing Movements: Each character in the movement string is processed in a loop:

'U' increments the y-coordinate (moving up).
'D' decrements the y-coordinate (moving down).
'L' decrements the x-coordinate (moving left).
'R' increments the x-coordinate (moving right).

Final Position: After processing all the movements, the final position of the robot is printed.

Running the Program:

You can compile and run this program using a Java compiler. Save the code in a file named RobotMovement.java, and then use the following commands to compile and run it:

javac RobotMovement.java

java RobotMovement

This will output the final position of the robot after executing the given sequence of movements. For the input "UUDDLRLR", the output will be:

Robo final position

Final position: (0, 0)

This is because the movements cancel each other out, bringing the robot back to the origin.

Problem Statement 2

Examples:

Statement 1

int[][] mat = {

{ 1, 2, 3, 4,17 },

{ 5, 6, 7, 8,18 },

{ 9, 10, 11, 12,19 },

{ 13, 14, 15, 16,20 }

};

String []dir={"LEFT","RIGHT","DOWN","LEFT","LEFT","LEFT","LEFT","RIGHT","RIGHT","DOWN",

"DOWN","DOWN","RIGHT","RIGHT","UP","UP","UP","UP","LEFT","LEFT","LEFT","LEFT","LEFT"};

move(mat, dir);

out put (0,0) 1

Statement 2

int[][] mat = {

{ 1, 2, 3, 4,17 },

{ 5, 6, 7, 8,18 },

{ 9, 10, 11, 12,19 },

{ 13, 14, 15, 16,20 }

};

String []dir={"DOWN","DOWN","DOWN","DOWN","DOWN"};

move(mat, dir);

out put (3,0) 13

Input : move = "UDDLRL"

Output : (0, 0)

Move U : (0, 0)-->(0, 1)

Move D : (0, 1)-->(0, 0)

Move D : (0, 0)-->(0, -1)

Move L : (0, -1)-->(-1, -1)

Move R : (-1, -1)-->(0, -1)

Move L : (0, -1)--.(-1, -1)

Therefore final position after the complete

movement is: (0, 0)

java

package com.kartik.org;

import java.util.Scanner;

/**

* @author kmandal

public class MatrixMove {

public static void main(String ...args){

int[][] mat = {

{ 1, 2, 3, 4,17 },

{ 5, 6, 7, 8,18 },

{ 9, 10, 11, 12,19 },

{ 13, 14, 15, 16,20 }

};

String []dir={"LEFT","RIGHT","DOWN","LEFT","LEFT","LEFT","LEFT","RIGHT","RIGHT","DOWN","DOWN","DOWN","RIGHT",

"RIGHT","UP","UP","UP","UP","LEFT","LEFT","LEFT","LEFT","LEFT"};

move(mat, dir);

String []dir1={"UP"};

move(mat, dir1);

String []dir2={"DOWN","DOWN","DOWN","DOWN","DOWN"};

move(mat, dir2);

/*Scanner in = new Scanner(System.in);

try {

int T = in.nextInt();

if (T < 1 && T >= 20) {

return;

} else {

int[][] mat = new int[T][T];

for (int j = 0; j < T; j++) {

for (int k = 0; k < T; k++) {

int N = in.nextInt();

if (N < Integer.MIN_VALUE && N > Integer.MAX_VALUE) {

return;

} else {

mat[j][k] = N;

}

int D = in.nextInt();

if (D < 1 && D >= 10000) {

return;

}

String [] dir=new String[D];

for (int i = 0; i < D; i++) {

String direction=in.next();

if (direction.equalsIgnoreCase(DIRECTION.LEFT.name())

|| direction.equalsIgnoreCase(DIRECTION.RIGHT.name())

|| direction.equalsIgnoreCase(DIRECTION.UP.name())

|| direction.equalsIgnoreCase(DIRECTION.DOWN.name())) {

dir[i] = direction.toUpperCase();

}else{

return;

}

move(mat, dir);

}

} catch (Exception e) {

}*/

}

private static enum DIRECTION

{

LEFT, RIGHT, UP,DOWN;

}

/**

* @param mat

* @param dir

private static void move(int [][]mat,String []dir){

int pontX=0;

int pontY=0;

int []pos=new int[2];

for (String string : dir) {

pos=move(mat,string,pontX,pontY);

pontX=pos[0];

pontY=pos[1];

}

//System.out.println(pontX+" "+pontY);

System.out.println(pontX+" "+pontY+"===="+mat[pontX][pontY]);

}

/**

* @param x

* @param y

* @param row

* @param col

* @return

private static boolean isSafeToGo(int x, int y, int row, int col) {

// check if x and y are in limits and cell is not blocked

if (x >= 0 && y >= 0 && x <= row-1 && y <= col-1) {

return true;

}

return false;

}

/**

* @param mat

* @param dir

* @param x

* @param y

* @return

private static int[] move(int[][] mat, String dir, int x, int y) {

int[] pont = new int[2];

int row = mat.length;

int col = mat[0].length;

if (isSafeToGo(x, y, row, col)) {

if (dir.equalsIgnoreCase(DIRECTION.LEFT.name())) {

pont[0] = x;

pont[1] = safePoint(y - 1, col);

} else if (dir.equalsIgnoreCase(DIRECTION.RIGHT.name())) {

pont[0] = x;

pont[1] = safePoint(y + 1, col);

} else if (dir.equalsIgnoreCase(DIRECTION.UP.name())) {

pont[0] = safePoint(x - 1, row);

pont[1] = y;

} else if (dir.equalsIgnoreCase(DIRECTION.DOWN.name())) {

pont[0] = safePoint(x + 1, row);

pont[1] = y;

}

} else {

pont[0] = x < 0 ? 0 : x;

pont[1] = y < 0 ? 0 : y;

}

return pont;

}

/**

* @param cur

* @param max

* @return

private static int safePoint(int cur, int max) {

int ret = 0;

if (cur < 0) {

ret = 0;

} else if (cur > max - 1) {

ret = max - 1;

} else {

ret = cur;

}

return ret;

}

This Java program moves a pointer in a 2D matrix according to a given list of directions (LEFT, RIGHT, UP, DOWN). The program also ensures that the pointer stays within the matrix boundaries, which are checked in the isSafeToGo and safePoint methods.

Explanation of Key Components

Enum DIRECTION: Defines valid directions as constants (LEFT, RIGHT, UP, DOWN).
Method move: This is the main method that takes a matrix and an array of directions, then calculates the final position of the pointer and prints the value at that position.
Method isSafeToGo: Checks if a given position (x, y) is within the matrix boundaries.
Method safePoint: Adjusts a position to ensure it doesn’t exceed matrix boundaries.

Sample Output

This program will print the final position and matrix value at that position for each set of directions provided in main.

Explanation of the Code

Matrix and Starting Position: Initialize the matrix and set the starting coordinates at (0,0).
Direction Processing: For each command in the direction array:

UP: Moves up one row if the row is within bounds.
DOWN: Moves down one row if it’s within bounds.
LEFT: Moves left by one column if it’s within bounds.
RIGHT: Moves right by one column if it’s within bounds.

Output: After processing all movements, print the final coordinates and the matrix value at that position.

Example Output

For the first test case:

plaintext

Final Position: (0, 0) Value: 1

For the second test case:

plaintext

Final Position: (3, 0) Value: 13

Explanation:

Initial Position: The robot starts at the origin (0, 0).
Movement Sequence: The movements are provided as a string (e.g., "UUDDLRLR").
Processing Movements: Each character in the movement string is processed in a loop:

'U' increments the y-coordinate (moving up).
'D' decrements the y-coordinate (moving down).
'L' decrements the x-coordinate (moving left).
'R' increments the x-coordinate (moving right).

Final Position: After processing all the movements, the final position of the robot is printed.

Running the Program:

You can compile and run this program using a Java compiler. Save the code in a file named RobotMovement.java, and then use the following commands to compile and run it:

javac MatrixMove.java

java MatrixMove

This will output the final position of the robot after executing the given sequence of movements. For the input "UUDDLRLR", the output will be:

Robo final position

Final position: (0, 0)

This is because the movements cancel each other out, bringing the robot back to the origin.