hi everyone so in these few videos we

will be looking at the Jacobian matrix for a robotic system and a few methods

that we can use to calculate it so the Jacobian matrix or a Jacobian for short

is used in a variety of fields in mathematics and physics but here in

robotics we will use it mainly to find singularities the forces and torques that

can act on a joint and the joint velocities now this topic has a

reasonable amount of calculations and some parts can be confusing or difficult

to understand but I will do my best to simplify it down for you without waiting

any longer let’s get into it so if you guys like this video don’t forget to

give me a thumbs up by clicking the like button below and if you haven’t done so

already go ahead and click that subscribe button right now I’m Chams and

welcome to that’s engineering all right so the Jacobian matrix the

first thing you should know is that it is a multi

dimensional matrix meaning it can have multiple rows

multiple columns and it would be made up of the derivatives so practically the

Jacobian is all about the velocity the Jacobian matrix would be made up of

the equations or the velocity vectors but when we do a question or a

calculation the easiest thing to find would be the position of the joints and

the links but by taking the differentiation or the derivative of

this position we can find the velocity and thus the Jacobian matrix so what you

have to know is that the minute someone talks about the Jacobian they’re

referring to the velocity of the joints in the system let me give you an example

this can be generalized for almost any robot or any system involved in the

Jacobian you can have a general set of equations that describe the position of

the system such as y1 equals f1 some function with variables x1 x2 and so on

y2 would be f2 and you can have multiple equations depending on the size of the

system which represent the position of the joints or the links in this robot

when you take the differentiation or the derivative and here you will be doing

partial differentiation because you have multiple variables in the function you

would get something that looks like daba y equals daba f over dhaba x

x double x and this would actually be a matrix containing y 1 dot y 2 dot and so

on followed by the matrix holding the derivatives of the function times the

matrix holding X 1 dot X 2 dot and so on so this derivative of the function would

correspond to this matrix and that matrix would be the Jacobian

now I’m not going to go into detail on this because we will have a look at it

again when we are doing partial differentiation in my next video so

let’s keep this aside for now right and let’s have a look at the definition of

the Jacobian so the Jacobian can be used to relate the joint velocities to the

cartesian velocities in a robot manipulator so let me show you this in

an equation format so it would be easy for you to understand a joint or a link

usually a joint for a robot can have a velocity and this velocity can be

expressed in terms of the position vector for the three coordinates XYZ the

derivative of those position vectors and this would be the velocity in the

Cartesian space this would also be equal to the velocity

in the joint space which would comprise of two values

and this would be joint space where V zero would be the

linear velocity and Omega would be the angular velocity or the rotational

velocity and this would actually be equal to the Jacobian times whatever

variables are there in the system I’m taking theta one and theta two for

example the derivatives of that so the Jacobian can be used to really at the

joint velocities to the Cartesian velocities right so before we move on to the next

topic there’s one more thing I would like to discuss and that is the frame

the jacobian is in now just imagine a simple robot sort of something like this with the zero to joint or the base frame

being zero this being the first frame first joint second joint and the third

frame for the tool to perform most of the calculations involving torques forces

and singularities you would need the Jacobian in the base frame or j0 but

when performing the calculations to find the Jacobian you may end up with the

Jacobian in the final frame j-3 that being said you can also Express the

Jacobian in any of the other frames as well and from these other frames we

would need to use that value to find the Jacobian in the base frame so how do you

do that well it’s actually quite simple the Jacobian in the base frame is equal

to the rotation matrix from the base to any frame I’m using three here as an

example times the Jacobian in that particular frame right so you have to remember this

equation practically what you have to knwois that if you have a Jacobian

in the higher frame simply multiply it by its rotation matrix which comes from

the transformation matrix so you have your position rotation so

you multiplied by the rotation matrix to get the Jacobian in the base frame all right so now let’s have a look at

singularities I’m not going to go into depth on this topic but I will give you

a brief introduction by definition a singularity is a point where the robot

loses one or more degrees of freedom and at this point it would be impossible to

move the tool of the robot in a particular direction regardless of the

joint rates now there are two main types of singularities workspace boundary and

workspace interior if you’re not quite familiar with the term workspace

practically it means the area that the robot can operate in so if this is the tool of the robot it

can operate within this entire area and reach any object within this area so

that would be the workspace of the robot so for work space boundary singularities

there are two cases the first one being that the robot can either be fully

stretched out or folded back on itself and for both of these conditions the

pool of the robot would be at the boundary

of the workspace in workspace interior singularities the tool is no longer at

the boundary but two or more joint axes would line up with each other you for this particular type of singularity

singularity to take place so alright so to help you visualize singularities

let’s just take this example of a very simple robot wait three joints and this

joint can be rotated by an angle theta now let’s say we rotated this robot or

rotated that joint such that the robot was in a straight line here theta would

be equal to 180 degrees and at this point the robot has lost some degrees of

freedom and therefore it is at a singularity so the value of theta equals 180 degrees

would be the point at which the robot reaches its singularity and it

would be of interest to calculate singularities because we would know at

what points the robot loses its degrees of freedom so we can program it to not

reach those particular angles or to not enough the joints to rotate those

particular angles to stop it from getting stuck for example right so to calculate or to find a

singularity you will look at the determinant of the Jacobian so if the

determinant of the Jacobian is equal to zero then a singularity or singularities

would exist for the robot if however the determinant of the

Jacobian is not equal to zero there would be no singularities for that robot

system right so these are two things that you

have to keep in mind now when performing calculations if the

question asks you to find the singularity of a robot system then you

would find the determinant of the Jacobian and equate it to zero and you

would get an equation with multiple different variables such as theta 1

theta 2 and so on and by equating this whole equation to zero you would be able

to calculate some values for theta 1 theta 2 or whatever the variables are

there and those values so just for example as a theta 1 is 30 degrees theta

2 is 60 degrees these values would be the singularities

so you would not want the robot or the joints to reach these particular angles

all right so now let’s have a look at the two methods that we will use to find

the Jacobian I’m not going to do an example to explain this method in detail

rather just an introduction to each method and I will go through each method

separately in the following videos so before we do that just keep in mind that

the Jacobian involves finding the velocity of the joints and the links of

the robot right so the first method is the partial differentiation method so

let’s start from the very beginning you have a robot you assign your frames you

analyze the system and you’re able to form a DH table

and from this d-h table you would be able to get your transformation matrices so you would have something at t0 1 T 1

2 and so on and using these transformation matrices you can

calculate the transformation matrix from 0 to the tool or the end effector by

multiplying all the previous transformation matrices together

1 2 2 and so on until you have n 2 e and this final transformation matrix

would be in the form of a 4×4 matrix with a position vector and a rotation

matrix so this position vector here would comprise of three values you would

have a position in X Y & Z so this would be the position of your end effector

relative to your base frame and if you were to differentiate each of these

three and combine it to a matrix you would have px dot py dot and PZ dot and the equations that you get can be

factorized into the form of the Jacobian relative to the base frame and whatever

the variables were there in the equations for example theta 1 theta 2

and theta 3 so in this case you can directly calculate the Jacobian in the

base frame now let’s have a look at the velocity propagation method so in the

velocity propagation method you would first need to find the linear velocity denoted by B then you will need to

calculate the angular velocity denoted by Omega keeping in mind that some

people or some textbooks will use this to denote the angular velocity this

symbol and there are already equations that you can use straight away to

calculate the linear and angular velocity you do not really have to take

any derivatives or do any sort of differentiation as the name suggests

velocity propagation method you start from the first frame or the zeroth frame

and work your way up frame by frame until you reach the final frame or the

tool frame so here you would finally have the velocity in the tool frame and

that velocity would be equal to the Jacobian in the tool frame times

whatever variables you had in your system so from this we can calculate the

Jacobian in the base frame using the rotation matrix from the base frame to

the tool frame times the Jacobian in the tool frame alright guys so the final thing we are

going to look at in this video would be torques and forces so we have two main

types of joints that we looked at in this course we had revolute joints and prismatic giants so a revolute joint would actually

produce a rotation and so there would be a torque while a prismatic joint would

produce some sort of force so for example we had a system like this with two revolute joints and one

prismatic joint you would have 2 torques in this system tau 1 and tau 2 better

use a different color tau 1 tau 2 and a force from the

prismatic joint F 3 so you can relate the forces and the talks to each other

by using a Jacobian and that is done using this general

equation is equal to the Jacobian from the base frame the transpose of that

Jacobian times so what you would actually get from this is related to

this particular robot system you would have Tao one torque 2 and f3 equal to the

to the Jacobian of the system in the base frame the transpose of that times

the three unit vectors FX FY and FZ alright guys so I will have three more

videos on the Jacobian matrix the first video will talk about partial

differentiation method the second the velocity propagation method and the

third video about stalks and forces so this video is just an introduction to

the Jacobian matrix but we will get into the examples and the calculations in the

coming videos right so that’s all for this video if

you guys like this video don’t forget to give me a thumbs up by clicking the like

button below if you have any questions or clarifications on the work we just

did involving an introduction to the Jacobian matrix please feel free to

leave a comment and I will do my best to get back to you I hope this video helped

you and if you haven’t done so already please hit that subscribe button right

now thank you for watching and I’ll see you all in the next video

Cheers