Robotics is a field that relies heavily on a variety of mathematical concepts and techniques. In this blog post, we will give you an introduction to some of the essential maths that are used in robotics, including

- linear algebra
- calculus
- geometry
- probability and statistics
- optimization
- control theory
- graph theory

We will also provide examples of how these concepts are applied in different robotics applications and how they can be implemented in code. Don’t worry if you are new to these topics – we will be breaking down each concept in an easy-to-understand way. And for those who are looking for more in-depth coverage, stay tuned for future posts where we will delve into specific subjects such as probability and statistics, control theory, machine learning, and kinematics.

All the code can be found on our GitHub!

#### The Eigen library

In the following sections, we will provide examples of how to implement these mathematical concepts in code. For that, you will need to set up your C++ environment (in my case VSCode with MinGW, we will do a separate post on setting up your coding environment) and download the Eigen library. Eigen is a C++ library for linear algebra, including matrix and vector operations, decompositions, and numerical solvers. To use the library follow these steps:

- Download Eigen and extract it
- Move the extracted folder into your C++ project (in the case of this article the include folder)
- Add the Eigen folder to the include path of your workspace

Then you should be able to follow along with the examples provided.

## Linear Algebra

Linear algebra is a fundamental mathematical discipline that is essential for understanding and solving many problems in robotics. It is used to describe and manipulate linear relationships between variables and to represent and manipulate data in a linear way.

One of the key concepts in linear algebra is the vector, which is a mathematical representation of a quantity that has both magnitude and direction. It can be thought of as an arrow in space, with the length of the arrow representing the magnitude of the quantity and the direction of the arrow representing the direction of the quantity. Vectors can be used to represent quantities such as position, velocity, and acceleration, and they can be added, subtracted, and multiplied using special rules.

Another important concept in linear algebra is the matrix, which is a rectangular array of numbers arranged in rows and columns. Matrices can be used to represent and manipulate data in a variety of ways, including transformations, systems of equations, and linear transformations.

One common application of linear algebra in mobile robotics is the use of matrix transformations to represent and manipulate the positions and orientations of objects in 3D space. Matrix transformations can be used to translate, rotate, and scale objects in a way that is consistent with the laws of physics and geometry. This is useful for tasks such as obstacle avoidance, path planning, and localization in mobile robotics.

For example, consider a mobile robot that is navigating through a cluttered environment and needs to avoid obstacles while following a predetermined path. To do this, the robot needs to be able to accurately represent and manipulate the positions and orientations of both itself and the obstacles in 3D space. This can be achieved using matrix transformations, which allow the robot to calculate the necessary movements and orientations to avoid collisions and stay on course.

In summary, linear algebra is a vital tool for mobile robotics, as it allows robots to represent and manipulate data in a linear way, which is essential for tasks such as obstacle avoidance, path planning, and localization.

#### Application of Linear Algebra in Robotics

One of the many applications of linear algebra in robotics is a state vector. It is a mathematical representation of the state of a system at a given time. In robotics, a state vector is often used to describe the position, orientation, velocity, and other properties of an object or robot at a specific point in time. In the case of SLAM for example, it also describes the state of all the objects in the environment.

For example, consider a robot that is navigating through a 2D space. The robot’s state at any given time can be represented by a state vector \(\mathbf{x} = [x, y, \theta]\), where \(x\) and \(y\) represent the robot’s position in the x and y dimensions, and \($\theta$\) represents the robot’s orientation in the plane.

State vectors are often used in robotics to track and predict the movement and behavior of objects and robots. They can be used in conjunction with mathematical models and algorithms to estimate the future state of a system, which is useful for tasks such as path planning and localization.

In summary, a state vector is a mathematical representation of the state of a system at a given time, and it is often used in robotics to track and predict the movement and behavior of objects and robots.

```
#include <Eigen/Dense>
int main()
{
// Create a 2D state vector with position (x, y) and orientation (theta)
Eigen::Vector3f state;
state << 1.0, 2.0, 3.14; // x = 1.0, y = 2.0, theta = 3.14
// Print the state vector
std::cout << "state = " << state << std::endl;
return 0;
}
```

Compile it using g++

`g++ -I .\include\eigen-3.4.0\eigen-3.4.0\ .\linearalgebra\linearalgebra.cpp -o .\linearalgebra\linearalgebra `

And execute:

`.\linearalgebra\linearalgebra `

It should result in this output:

`state = 1 2 3.14 `

## Calculus

Calculus is a branch of mathematics that deals with rates of change and the study of continuously changing processes. It is used to model and analyze physical phenomena, such as motion, force, and energy, which are essential concepts in the field of robotics.

There are two main branches of calculus: differential calculus and integral calculus. Differential calculus is concerned with the study of rates of change and slopes of curves, while integral calculus is concerned with the study of the accumulation of quantities and the area under and between curves.

One example of how calculus is used in robotics is in the analysis and control of robot motion. For instance, the position of a robot can be represented by a function of time, \(x(t)\), which describes the robot’s location in space at any given time. The derivative of this function, denoted as \(x'(t)\) or \(\frac{dx}{dt}\), gives the rate of change of the robot’s position with respect to time, which is its velocity. The second derivative denoted as \(x”(t)\) or \(\frac{d^2x}{dt^2}\), gives the rate of change of the velocity with respect to time, which is the robot’s acceleration. These quantities can be used to design controllers that accurately track a desired trajectory or manipulate objects with precise force and torque.

#### Application of Calculus in Robotics

Here is an example of how you could use a two-dimensional vector and the Eigen library to compute the acceleration of a robot using the finite difference approximation:

```
#include <iostream>
#include <Eigen/Dense>
// Time step size
const double dt = 0.1;
int main()
{
// Previous position (two-dimensional vector)
Eigen::Vector2d x_prev;
x_prev << 0.0, 0.0;
// Current position (two-dimensional vector)
Eigen::Vector2d x;
x << 0.1, 0.2;
// Compute the derivative of the position function using the finite difference approximation
Eigen::Vector2d dx_dt = (x - x_prev) / dt;
std::cout << "Derivative of position function: " << dx_dt.transpose() << std::endl;
return 0;
}
```

Compile it using g++

`g++ -I .\include\eigen-3.4.0\eigen-3.4.0\ .\calculus\calculus.cpp -o .\calculus\calculus `

And execute:

`.\calculus\calculus `

It should result in this output:

`Derivative of position function: 1 2 `

## Geometry

Geometry is a branch of mathematics that deals with the study of shapes, sizes, relative positions of figures, and the properties of space. It includes the study of points, lines, angles, surfaces, and solids.

One example application of geometry in mobile robotics is the use of geometric algorithms to navigate and move around in a given environment. For example, a mobile robot may need to determine the shortest path between two points in order to navigate efficiently through a cluttered space. Geometry can be used to calculate the distance between the robot and various objects in its environment, and to determine the orientation of the robot relative to these objects. This information is essential for the robot to be able to avoid collisions and to accurately reach its desired destination.

#### Useful Geometry Functions for Robotics

Here are some examples of geometric equations that may be useful:

- Distance between two points:
- The distance between two points \(P1\) and \(P2\) with Cartesian coordinates \([x_1, y_1]\) and \([x_2, y_2]\), respectively, can be calculated using the Pythagorean theorem:
- \(d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}\)

- Angle between two lines:
- The angle theta between two lines \(L_1\) and \(L_2\) can be calculated using the dot product of the normal vectors \(n_1\) and \(n_2\) of the lines:
- \(\theta = \arccos{\frac{n_1 \cdot n_2} {||n_1|| \cdot ||n_2||}}\)

- Intersection of two lines:
- The intersection \(I\) of two lines \(L_1\) and \(L_2\) can be calculated using the equations of the lines and the system of linear equations:
- \(L_1: y = m_1 \cdot x + b_1\)
- \(L_2: y = m_2 \cdot x + b_2\)
- \(I: (x, y) = \frac{ b_2 – b_1} { m_1 – m_2}, \frac{m_1 \cdot b_2 – m_2 \cdot b_1} {m_1 – m_2}\)

## Probability and Statistics

**Probability** is the study of random events and the likelihood of their occurrence. It is a mathematical framework for representing uncertain events or outcomes and quantifying their likelihood of occurring.

**Statistics** is the study of collecting, analyzing and interpreting data. It involves using statistical methods to make inferences about a population based on a sample of data from that population.

Probability and statistics are important in robotics because they provide tools for understanding and modeling uncertainty and variability in the data that robots collect and process.

In mobile robotics, probability and statistics can be applied in a variety of ways. For example:

Probability can be used to

**model the uncertainty in the location and orientation of a robot**as it navigates through an unknown environment. This can be important for tasks such as localization (determining the robot’s position and orientation) and mapping (creating a representation of the environment).Statistics can be used to

**analyze data collected by sensors on the robot**, such as laser scanners or cameras. For example, a robot might use statistical techniques to identify patterns in the data that can be used to classify objects or identify features in the environment.Probability and statistics can also be used to

**model and predict the behavior of other agents in the environment**, such as pedestrians or other robots. This can be important for tasks such as motion planning (determining a safe and efficient path for the robot to follow) and collision avoidance (preventing the robot from colliding with other objects).

Overall, probability and statistics are essential tools for helping robots make sense of the complex and uncertain world around them, and for making informed decisions in real-time.

#### Application of Probability and Statistics in Robotics

One example of a probabilistic algorithm that is commonly used in mobile robotics is the Kalman filter. The Kalman filter is an algorithm that is used to estimate the state of a system based on noisy, incomplete, or uncertain observations.

The Kalman filter is often used in robotics for tasks such as sensor fusion (combining data from multiple sensors to estimate the state of the robot) and motion estimation (estimating the motion of the robot based on its sensors and actuators).

To use the Kalman filter, the robot must be able to model the uncertainty in its state (such as its position and velocity) and the noise in its observations (such as sensor noise or measurement errors). The Kalman filter then uses this information to iteratively update its estimate of the state based on new observations.

One example of how the Kalman filter might be used in mobile robotics is in a self-driving car. The car might use sensors such as cameras, lidar, and radar to gather data about its surroundings. The Kalman filter could then be used to estimate the position and orientation of the car based on this data, taking into account factors such as sensor noise and the uncertain dynamics of the car’s motion.

Another example of a statistical algorithm that is commonly used in mobile robotics is the Monte Carlo localization (MCL) algorithm. MCL is an algorithm that is used to estimate the pose (position and orientation) of a robot in an unknown environment using probabilistic techniques.

To use MCL, the robot must be able to generate a set of possible poses (called “particles”) that represent the uncertainty in its pose. The robot then uses its sensors to update the probability of each particle based on how well it matches the observations. The final estimate of the robot’s pose is then obtained by taking the weighted average of the particles based on their probabilities.

MCL is often used in robotics for tasks such as localization, mapping, and simultaneous localization and mapping (SLAM), where it is important to estimate the pose of the robot accurately and efficiently in real-time. We will do a separate post only dedicated to SLAM and its implementation, as this is both a really important and a quite complicated topic.

The following code will demonstrate how a Kalman filter implementation in C++ might look like. Note that here, only the positon and velocity are within the state vector, so it only serves as a “slim downed” version, compared to an acctual implementation. On the post abour SLAM, we will go into more detail on what the state and measurement matrices represent, and how to model them.

```
#include <Eigen/Dense>
// State vector is (x, y, vx, vy)
// Measurement vector is (x, y)
class KalmanFilter
{
public:
KalmanFilter()
{
// Initialize state and measurement matrices
x << 0, 0, 0, 0;
P << 1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1;
F << 1, 0, 1, 0,
0, 1, 0, 1,
0, 0, 1, 0,
0, 0, 0, 1;
H << 1, 0, 0, 0,
0, 1, 0, 0;
R << 0.1, 0,
0, 0.1;
Q << 0.1, 0, 0, 0,
0, 0.1, 0, 0,
0, 0, 0.1, 0,
0, 0, 0, 0.1;
}
void predict()
{
// Predict state and covariance
x = F * x;
P = F * P * F.transpose() + Q;
}
void update(const Eigen::Vector2d &z)
{
// Compute Kalman gain
Eigen::Matrix2d S = H * P * H.transpose() + R;
Eigen::Matrix<double, 4, 2> K = P * H.transpose() * S.inverse();
// Update state and covariance
x += K * (z - H * x);
P -= K * H * P;
}
Eigen::Vector4d x; // State vector
Eigen::Matrix4d P; // Covariance matrix
private:
Eigen::Matrix4d F; // State transition matrix
Eigen::Matrix2d H; // Measurement matrix
Eigen::Matrix2d R; // Measurement noise covariance matrix
Eigen::Matrix4d Q; // Process noise covariance matrix
};
```

To use this Kalman filter, you would first create an instance of the `KalmanFilter`

class and initialize the state and covariance matrices to appropriate values. Then, you can call the `predict`

and `update`

methods in a loop to perform the prediction and update steps of the Kalman filter algorithm.

For example, to implement a simple tracking application, you might do something like this:

```
KalmanFilter kf;
while (true) {
// Predict state
kf.predict();
// Get measurement from sensors
Eigen::Vector2d z = getMeasurement();
// Update state with measurement
kf.update(z);
// Use updated state to control motors or perform other actions
controlMotors(kf.x);
}
```

This is just a simple example of how you might use a Kalman filter in a mobile robotics application. In practice, you would need to customize the Kalman filter to suit the specific needs of your application. This might involve adjusting the dimensions of the state and measurement vectors, as well as the matrices that represent the system dynamics, measurement noise, and process noise.

It is also important to note that this example implementation is very basic and does not include many of the advanced features that are often used in real-world Kalman filter applications. For example, it does not handle nonlinear systems, time-varying dynamics, or correlated noise.

## Optimization

Optimization is the process of finding the optimal solution to a problem by considering multiple variables or constraints. In mobile robotics, optimization is often used to find the best path or trajectory for a robot to follow, given certain constraints such as avoiding obstacles, minimizing energy consumption, or maximizing performance.

There are several types of optimization algorithms that can be used in mobile robotics, including linear programming, gradient descent, and genetic algorithms. These algorithms can be used to optimize various aspects of robot motion, such as path planning, trajectory generation, and control.

For example, a robot may need to navigate through a cluttered environment to reach a specific location. In this case, an optimization algorithm can be used to find the shortest or most efficient path for the robot to follow, taking into account the positions and shapes of the obstacles in the environment. The algorithm might consider factors such as the robot’s maximum speed, acceleration, and turning radius, as well as the costs associated with different types of motion (e.g., straight versus curved paths).

Optimization can also be used to generate smooth and energy-efficient trajectories for robots. For example, a robot moving on a flat surface might be able to move more efficiently by following a smooth, sinusoidal path rather than a jerky, zig-zag path. Optimization algorithms can be used to find the optimal shape and amplitude of the sinusoidal path, given the robot’s maximum speed, acceleration, and other constraints.

Overall, optimization is a powerful tool for mobile robotics, allowing robots to make intelligent and efficient decisions about how to move and interact with their environment. By using optimization algorithms, robots can navigate complex environments, follow smooth and efficient trajectories, and make other types of decisions that maximize performance and minimize energy consumption.

#### Application of Optimization in Robotics

One common optimization algorithm used in mobile robotics is gradient descent. Gradient descent is an iterative optimization algorithm that is used to find the minimum of a differentiable function by moving in the direction of the steepest descent.

The basic idea of gradient descent is to start at a random point on the function and then iteratively move in the opposite direction of the gradient at that point until the minimum of the function is found. The gradient at a point gives the slope of the function at that point, and moving in the opposite direction of the gradient will take the algorithm down the steepest slope.

Mathematically, the update rule for gradient descent can be written as:

\(\boldsymbol{x}_{t+1} = \boldsymbol{x}_t – \alpha \nabla f(\boldsymbol{x}_t)\)Where \(\boldsymbol{x}_t\) is the current estimate of the minimum, \(\alpha\) is the learning rate, and \(\nabla f(\boldsymbol{x}_t)\) is the gradient of the function \(f\) at point \(\boldsymbol{x}_t\). The learning rate determines the size of the step taken at each iteration, and it is typically set to a small value (e.g., 0.01) to ensure that the algorithm converges to the minimum in a stable manner.

Gradient descent is widely used in mobile robotics, particularly for tasks such as path planning and trajectory optimization. For example, it can be used to find the optimal path for a robot to follow through a cluttered environment, by minimizing a cost function that takes into account the positions and shapes of the obstacles, as well as the robot’s maximum speed and acceleration.

Here is an example of how gradient descent could be implemented in C++ for a simple one-dimensional optimization problem:

```
#include <iostream>
#include <cmath>
// Function to optimize (parabola with minimum at x = 2)
double f(double x) {
return (x - 2) * (x - 2);
}
// Derivative of f(x)
double df(double x) {
return 2 * (x - 2);
}
int main() {
// Initialize variables
double x = 0; // Starting point
double alpha = 0.1; // Learning rate
double tolerance = 1e-6; // Stopping criteria
int max_iterations = 1000; // Maximum number of iterations
// Run gradient descent
for (int i = 0; i < max_iterations; i++) {
// Compute gradient at current point
double gradient = df(x);
// Update x using gradient descent rule
x -= alpha * gradient;
// Check stopping criteria
if (std::abs(gradient) < tolerance) {
std::cout << "Minimum found at x = " << x << std::endl;
break;
}
}
return 0;
}
```

Compile it using g++

`g++ .\optimization\optimization.cpp -o .\optimization\optimization `

And execute:

`.\optimization\optimization `

It should result in this output:

`Minimum found at x = 2 `

This code defines a simple parabola with a minimum at x = 2, and then uses gradient descent to find the minimum of this function. The function `f(x)`

is the function to be minimized, and the function `df(x)`

is the derivative of `f(x)`

, which is used to compute the gradient at each iteration. The code initializes the starting point, learning rate, tolerance, and maximum number of iterations, and then iteratively updates the value of `x`

using the gradient descent rule until the minimum is found or the maximum number of iterations is reached.

I hope this example helps to illustrate how gradient descent can be implemented in C++ for a simple optimization problem. Of course, in a real mobile robotics application, the function to be minimized would be more complex and would likely involve multiple variables and constraints.

## Control Theory

Control theory is a branch of engineering that deals with the design, analysis, and implementation of systems that control the behavior of dynamic systems. It is a fundamental concept in robotics and is used to design algorithms and controllers that allow robots to perform a variety of tasks, including navigation, localization, manipulation, and more.

In mobile robotics, control theory is used to design controllers that can stabilize and control the motion of the robot. This involves designing feedback controllers that can adjust the robot’s motion based on sensory information and desired goals. For example, a mobile robot may use control theory to maintain a stable and smooth trajectory while navigating through a cluttered environment or to follow a desired path.

Control theory is also used in mobile robotics to design controllers for tasks such as manipulation, where the robot must be able to accurately control the motion of its end effectors in order to pick up and manipulate objects. This can involve designing controllers that can accurately track and follow desired trajectories, as well as controllers that can adapt to changes in the environment or the robot’s internal state.

One of the key concepts in control theory is feedback control, which involves using a feedback loop to adjust the control signal based on the error between the desired output and the actual output of the system. This allows the controller to correct for any deviations from the desired behavior and ensure that the system remains stable and performs as intended.

Overall, control theory is a crucial aspect of mobile robotics and is used to design algorithms and controllers that allow robots to accurately and reliably perform a wide range of tasks.

#### PID Controllers

One of the most well known types of feedback controllers are PID controlers. A PID (Proportional-Integral-Derivative) controller is a type of feedback control system that is widely used in robotics and other fields to control the behavior of dynamic systems. It consists of three components: the proportional, integral, and derivative terms, which work together to adjust the control signal based on the error between the desired output and the actual output of the system.

The proportional term adjusts the control signal based on the current error, the integral term adjusts the control signal based on the accumulated error over time, and the derivative term adjusts the control signal based on the rate of change of the error. These three terms can be combined to form the following control signal equation:

\(u(t) = K_p \cdot e(t) + K_i \cdot \int_{0}^{t} e(\tau) ,d\tau + K_d \cdot \frac{de(t)}{dt}\)Where:

- \(u(t)\) is the control signal at time \(t\).
- \(K_p\), \(K_i\), and \(K_d\) are the proportional, integral, and derivative gains, respectively, which are constants that are chosen based on the desired characteristics of the control system.
- \(e(t)\) is the error between the desired output and the actual output at time \(t\).
- \(\int_{0}^{t} e(\tau) ,d\tau\) is the integral of the error over time, which represents the accumulated error.
- \(\frac{de(t)}{dt}\) is the derivative of the error with respect to time, which represents the rate of change of the error.

The values of the gains \(K_p\), \(K_i\), and \(K_d\) can be tuned to achieve the desired behavior of the system. For example, increasing the value of \(K_p\) will increase the proportionality of the control signal to the error, which can improve the system’s response to changes in the error but may also increase oscillations. Similarly, increasing the value of \(K_d\) will increase the damping effect of the derivative term, which can reduce oscillations but may also decrease the system’s ability to respond quickly to changes in the error. The value of \(K_i\) determines the contribution of the integral term to the control signal and can be used to eliminate steady-state errors in the system. In the latest Roboost project, a PID controller was used to control the speed of the motors.

Here is an example of a PID controller implemented in C++:

```
#include <cmath>
class PIDController
{
public:
// Constructor
PIDController(double kp, double ki, double kd)
: Kp(kp), Ki(ki), Kd(kd),
prevError(0), integral(0)
{
}
// Compute the control signal using the PID algorithm
double computeControlSignal(double error, double dt)
{
// Proportional term
double p = Kp * error;
// Integral term
integral += error * dt;
double i = Ki * integral;
// Derivative term
double d = Kd * (error - prevError) / dt;
// Save the current error for the next iteration
prevError = error;
// Return the sum of the three terms
return p + i + d;
}
private:
double Kp, Ki, Kd;
double prevError;
double integral;
};
```

This implementation includes a constructor that takes the proportional, integral, and derivative gains as arguments, and a `computeControlSignal`

function that computes the control signal using the PID equation described above. The function takes the current error and the elapsed time since the last iteration as arguments and returns the control signal.

To use the PID controller, you would create an instance of the `PIDController`

class and call the `computeControlSignal`

function at regular intervals, passing in the current error and elapsed time as arguments. The returned control signal can then be used to adjust the behavior of the system being controlled.

Note that this is just one possible implementation of a PID controller and there are many variations and refinements that can be made to the basic algorithm. For example, you may want to implement additional features such as windup prevention, saturation limits, or gain scheduling. You may also want to experiment with different types of controllers, such as an LQR (Linear-Quadratic Regulator) or a sliding mode controller, depending on the specific requirements of your application.

## Graph Theory

Graph theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures used to model pairwise relationships between objects. A graph consists of a set of vertices, or nodes, and a set of edges that connect these vertices. The edges may be directed, meaning that they have a specific direction, or they may be undirected, meaning that they have no specific direction.

Graph theory has a wide range of applications in various fields, including computer science, engineering, and biology. In mobile robotics, graph theory is used to represent and analyze the relationships between different locations or landmarks in an environment, as well as the movements of a robot within that environment.

For example, a mobile robot may be required to navigate from one location to another in an unknown environment. In order to do this, the robot may use sensors to gather information about its surroundings and construct a map of the environment in the form of a graph. The vertices of the graph represent the different locations or landmarks in the environment, and the edges represent the paths or movements that the robot can take to travel between these locations.

Once the graph has been constructed, the robot can then use graph theory algorithms to plan its path from the starting location to the destination. These algorithms may include shortest path algorithms, which find the shortest route between two locations, or motion planning algorithms, which determine the optimal movements of the robot in order to avoid obstacles and reach its destination.

Graph theory is also used in mobile robotics to analyze and optimize the efficiency of the robot’s movements. For example, a graph may be used to represent the energy consumption of the robot at different locations in the environment, and algorithms may be used to find the most efficient path that minimizes the total energy consumption of the robot.

#### Implementation of the A* Algorithm

The **A* algorithm** is a popular graph search algorithm that is used to find the shortest path between two vertices in a graph. It combines the benefits of both the breadth-first search (BFS) and the best-first search algorithms, allowing it to explore the graph efficiently while prioritizing the paths that are most likely to lead to the goal.

At each step of the algorithm, the A* algorithm expands the vertex with the lowest cost, which is determined by a cost function known as the “f-value”. The f-value is calculated as the sum of two other values: the “g-value”, which represents the cost of reaching the current vertex from the start, and the “h-value”, which represents an estimate of the cost of reaching the goal from the current vertex. The g-value is calculated using the actual cost of the edges between the vertices, while the h-value is calculated using a heuristic function that estimates the cost based on some characteristics of the vertices and the graph.

The A* algorithm can be formally defined as follows:

- Initialize the open list, which contains the starting vertex and its f-value.
- Initialize the closed list, which is initially empty.
- While the open list is not empty:
- Select the vertex with the lowest f-value from the open list and add it to the closed list.
- If the selected vertex is the goal, construct the path from the start to the goal using the information in the closed list and return it.
- Expand the selected vertex by adding its neighbors to the open list, if they are not already in the open list or the closed list. Update the f-values of the neighbors using the g-values and h-values of the selected vertex.

The f-value of a vertex v can be calculated using the following equation:

\(f(v) = g(v) + h(v)\)where \(g(v)\) is the cost of reaching the vertex v from the start, and \(h(v)\) is the estimated cost of reaching the goal from the vertex v.

The g-value of a vertex v can be calculated using the following equation:

\(g(v) = g(u) + cost(u, v)\)where \(g(u)\) is the g-value of the parent vertex u of v, and \(cost(u, v)\) is the cost of the edge between u and v.

The h-value of a vertex v can be calculated using a heuristic function \(h(v)\), which estimates the cost of reaching the goal from the vertex v. The heuristic function can be any function that satisfies the following conditions:

- It must be admissible, meaning that it never overestimates the true cost of reaching the goal.
- It must be monotonic, meaning that the cost of any path using only vertical or horizontal movements is not greater than the cost of any other path using the same movements.

Examples of common heuristic functions include the Euclidean distance, the Manhattan distance, and the diagonal distance.

An **occupancy grid** is a data structure used in robotics and computer vision to represent the occupied and unoccupied areas in an environment. It is typically implemented as a two-dimensional grid of cells, where each cell represents a small region in the environment. Each cell can be marked as occupied, meaning that it is not passable by the robot, or unoccupied, meaning that the robot can pass through it.

Occupancy grids are used in mobile robotics to map the environment and plan the movements of the robot. They are particularly useful for environments that are partially unknown or that change over time, as they allow the robot to reason about the occupied and unoccupied areas in a flexible and efficient way.

To construct an occupancy grid, the robot typically uses sensors, such as lidars or cameras, to gather information about its surroundings and update the grid accordingly. The robot may also use additional algorithms, such as probabilistic techniques or machine learning methods, to improve the accuracy and robustness of the grid.

Here is an example of how the A* algorithm for a occupancy grid could be implemented in C++.

First, we would need to include the necessary headers:

```
#include <Eigen/Sparse>
#include <unordered_map>
#include <queue>
#include <algorithm>
#include <iostream>
```

Next, we would define the occupancy grid map class using a sparse Eigen matrix:

```
template <typename T>
class OccupancyGrid {
public:
// Constructor
OccupancyGrid(int width, int height) : _width(width), _height(height) {
// Initialize the sparse matrix with the given width and height
_grid = Eigen::SparseMatrix<T>(height, width);
}
// Set the occupancy value at a given cell
void SetOccupancy(int x, int y, T occupancy) {
_grid.coeffRef(y, x) = occupancy;
}
// Get the occupancy value at a given cell
T GetOccupancy(int x, int y) const {
return _grid.coeff(y, x);
}
// Get the width of the grid
int GetWidth() const {
return _width;
}
// Get the height of the grid
int GetHeight() const {
return _height;
}
// overload the '<<' operator
friend std::ostream& operator<<(std::ostream& os, const OccupancyGrid<T>& grid) {
// Iterate through all cells of the grid
for (int y = 0; y < grid.GetHeight(); y++) {
for (int x = 0; x < grid.GetWidth(); x++) {
// Print the occupancy value at the current cell
os << grid.GetOccupancy(x, y) << " ";
}
os << std::endl;
}
return os;
}
private:
// The width of the grid
int _width;
// The height of the grid
int _height;
// The grid as a sparse matrix
Eigen::SparseMatrix<T> _grid;
};
```

Now we can then define the A* class:

```
template <typename T>
class AStar {
public:
// Constructor
AStar(const OccupancyGrid<T>& grid) : _grid(grid) {}
// Perform the A* search
std::vector<std::pair<int, int>> Search(const std::pair<int, int>& start,
const std::pair<int, int>& goal) {
// Set up the open list (using the f value as the key) and closed list
std::unordered_map<int, Node*> openList;
std::unordered_map<int, Node*> closedList;
// Create the start node and add it to the open list
Node* startNode = new Node {
start.first, start.second, // Coordinates
0, // g value
heuristic(start, goal), // h value
0 + heuristic(start, goal), // f value
nullptr // Parent node
};
openList[startNode->f] = startNode;
// Continue the search until the open list is empty or the goal is found
while (!openList.empty()) {
// Get the node in the open list with the lowest f value
int lowestF = openList.begin()->first; // TODO: Is it really the lowest that way?
Node* current = openList[lowestF];
// If the current node is the goal node, we have found the path
if (current->x == goal.first && current->y == goal.second) {
// Reconstruct the path by following the chain of parent pointers back to the start node
std::vector<std::pair<int, int>> path;
while (current != startNode) {
path.emplace_back(current->x, current->y);
current = current->parent;
}
path.emplace_back(start.first, start.second); // Add the start node to the path
// Reverse the path so that it goes from start to goal
std::reverse(path.begin(), path.end());
// Delete the nodes to free up memory
deleteNodes(openList);
deleteNodes(closedList);
return path; // Return the path
}
// Remove the current node from the open list and add it to the closed list
openList.erase(lowestF);
closedList[coordToInt(current->x, current->y)] = current;
// Check the neighbors of the current node
for (int i = -1; i <= 1; i++) {
for (int j = -1; j <= 1; j++) {
// Ignore the current cell and cells outside the map
if (i == 0 && j == 0 || current->x + i < 0 || current->x + i >= _grid.GetWidth() || current->y + j < 0 || current->y + j >= _grid.GetHeight())
continue;
// Ignore cells that are not traversable or are already in the closed list
if (_grid.GetOccupancy(current->x + i, current->y + j) || closedList.count(coordToInt(current->x + i, current->y + j)))
continue;
// Calculate the g, h, and f values for the neighbor
int g_new = current->g + 1;
int h_new = heuristic({current->x + i, current->y + j}, goal);
int f_new = g_new + h_new;
// If the neighbor is not in the open list, add it
if (!openList.count(coordToInt(current->x + i, current->y + j))) {
Node* neighbor = new Node {
current->x + i, current->y + j, // Coordinates
g_new, // g value
h_new, // h value
f_new, // f value
current // Parent node
};
openList[f_new] = neighbor;
}
// If the neighbor is already in the open list, update its g value if the new g value is lower
else if (g_new < openList[coordToInt(current->x + i, current->y + j)]->g) {
openList[coordToInt(current->x + i, current->y + j)]->g = g_new;
openList[coordToInt(current->x + i, current->y + j)]->f = g_new + openList[coordToInt(current->x + i, current->y + j)]->h;
openList[coordToInt(current->x + i, current->y + j)]->parent = current;
}
}
}
// Add the current cell to the closed list
closedList[coordToInt(current->x, current->y)] = current;
}
std::cout << "no valid path found" << std::endl;
// If the search fails, return an empty path
return {};
}
private:
// A node in the graph
struct Node {
int x, y; // Coordinates of the node
int f, g, h; // f = g + h, where g is the cost of the path from the start node to this node, and h is the estimated cost of the path from this node to the goal node
Node* parent; // Pointer to the parent node, used for reconstructing the path
};
// Free the associated Nodes
void deleteNodes(std::unordered_map<int, Node*>& list) {
for (auto& [key, value] : list) {
delete value;
}
list.clear();
}
// The grid to be searched
const OccupancyGrid<T>& _grid;
// Convert a pair of coordinates to a single integer
int coordToInt(int x, int y) const {
return x + y * _grid.GetWidth();
}
// Convert a single integer to a pair of coordinates
std::pair<int, int> intToCoord(int n) const {
return {n % _grid.GetWidth(), n / _grid.GetWidth()};
}
// Calculate the heuristic value between two cells
int heuristic(const std::pair<int, int>& a, const std::pair<int, int>& b) const {
// Use the Manhattan distance as the heuristic
return std::abs(a.first - b.first) + std::abs(a.second - b.second);
}
};
```

And implement a main routine to test the functionality:

```
int main() {
// Create an OccupancyGrid object with a width of 10 and a height of 10
OccupancyGrid<float> grid(10, 10);
// Set some of the cells in the grid to be occupied
grid.SetOccupancy(2, 2, true);
grid.SetOccupancy(3, 3, true);
grid.SetOccupancy(4, 4, true);
grid.SetOccupancy(4, 5, true);
grid.SetOccupancy(5, 5, true);
grid.SetOccupancy(6, 5, true);
grid.SetOccupancy(7, 5, true);
grid.SetOccupancy(8, 5, true);
grid.SetOccupancy(8, 6, true);
grid.SetOccupancy(8, 7, true);
grid.SetOccupancy(8, 8, true);
grid.SetOccupancy(8, 9, true);
std::cout << "Current Occupancy Grid: " << std::endl
<< grid << std::endl;
// Create an AStar object using the OccupancyGrid object
AStar<float> astar(grid);
// Perform the A* search
std::vector<std::pair<int, int>> path = astar.Search({6, 3}, {2, 7});
// Print the resulting path
std::cout << "Path: " << std::endl;
for (const auto& coord : path) {
std::cout << "(" << coord.first << ", " << coord.second << ") ";
}
std::cout << std::endl;
return 0;
}
```

Now compile:

`g++ -I .\include\eigen-3.4.0\eigen-3.4.0\ '.\graph theory\AStar.cpp' -o '.\graph theory\AStar' `

And execute:

`& '.\graph theory\AStar' `

This should result in an output similar to this:

```
Current Occupancy Grid:
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 1 1 1 1 1 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1 0
Path:
(6, 3) (5, 4) (6, 4) (5, 3) (4, 3) (4, 2) (3, 2) (2, 3) (2, 4) (2, 5) (3, 6) (4, 7) (3, 8) (2, 9) (1, 8) (0, 7) (1, 6) (1, 5) (2, 6) (3, 5) (4, 6) (5, 7) (4, 8) (3, 9) (2, 8) (1, 9) (0, 8) (1, 7) (2, 7)
```

## Conclusion and Further Resources

In conclusion, math is an integral part of the field of robotics. From linear algebra and calculus to control theory and graph theory, a wide range of mathematical concepts and techniques are used to design, analyze, and control robotic systems. We hope this guide has given you a better understanding of the various types of math used in robotics and provided you with some hands-on examples of how to implement these concepts in C++.

If you’re interested in learning more about the role of math in robotics, there are plenty of resources available online. Some additional resources to check out include:

- Our introduction to AI post
- Our Robotics 101 guide
- “Introduction to Robotics” by MIT
- “Probabilistic Robotics” by Sebastian Thrun

Thanks for sticking around! 🙂