Application Of Matrix Multiplication In Robotics

There are a lot of applications of matrices in engineering. One such field is robotics In the field of robotics the application of matrix algebra is used for different calculations. The orientation of robotic arms with respect to a fixed frame can be represented using a matrix. Also the rotation of the robotics arm about an axis can be represented by an elementary rotational matrix.

What Is The Importance Of Learning Matrix Multiplication

Transformation Matrix

If you want to transform one vector into another then a transformation matrix can be used. The transformation matrix changes the coordinate system by mapping the coordinate of the vectors to the new coordinate system via matrix multiplication. That is the transformation of space can be done through a transformation matrix.

Example- A vector x_i+y_j can be transformed to another vector x’_i+y’_j using a transformation matrix

$\begin{pmatrix} a &b \\ c & d \end{pmatrix}$

That is

$\begin{pmatrix} a &b \\ c & d \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}=\begin{pmatrix} x'\\ y' \end{pmatrix}$

Idea of this transformation is used in robotics to change the coordinates. For example the velocity of the end effector of a robotic arm can be controlled by using a transformation matrix known as a Jacobian matrix. The Jacobian matrix converts angular velocities of the joints to the velocity of the end effector of a robotic arm. After transformation the matrix gives both linear and angular components of velocities. This is one example of usage of matrix in robotics. Matrices of higher order(6 x 6) will be dealt with in robotics. The calculations can easily be done using MATLAB, a software where matrix calculations and robotic programmes can be easily done. There are other coding platforms like Python and more are available for robotics.

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Application Of Matrices In Machine Learning

Machine learning prediction algorithms require a huge amount of data for accurate output. Dealing with a huge amount of data is easy if it is represented as a matrix. Let us take an example of linear regression. Predictions are done through linear regression. In linear regression a variable Y is predicted based on input variable X.

For example if we want to predict the salary of employees based on experience, linear regression can be used.

Y=a+bX

Where Y is the target variable salary, X is the year of experience, a and b are linear coefficients.

If we have n number of data

Then

y₁=a₁+b₁x₁

y₂=a₂+b₂x₂

y_n=a_n+b_nx_n

This can be represented as matrix

$\begin{pmatrix} y_1\\ y_2 \\ . \\ . \\ y_n \end{pmatrix}=\begin{pmatrix} a_1\\ a_2\\ .\\ .\\ a_n\end{pmatrix}+\begin{pmatrix} b_1 &b-2 &. &. &b-n \end{pmatrix}\begin{pmatrix} x_1\\ x_2\\ .\\ .\\ x_n\end{pmatrix}\\i.e\\Y=a+b^T X$

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In linear regression the value of a and b is selected such that the error between actual value and predicted value should be minimum. The error is minimized using a cost function. The main aim of introducing a bit idea of linear regression here is to show the use of matrix multiplication and addition in representation of data. The method of calculating a and b will be studied in the higher classes if you are taking statistics of studying AI or Machine Learning.

What we have discussed above are applications of matrix multiplication in two important technologies. There are many other examples for matrix multiplication that are used in statistics and related calculations. Here the discussion is done to show examples of how matrix multiplication is used in real life, that is robotics and prediction.