mod01lec03 - Introduction to Mobile Robot Kinematics

Summary

This lecture delves into the fundamentals of mobile robot kinematics, focusing on the kinematic relationships of land-based robots. It explains the concept of kinematics as the study of motion without considering forces, and highlights the importance of understanding these relationships for system design and motion control. The lecture introduces the degree of freedom for mobile robots, discusses the mapping between control parameters and system behavior, and differentiates between forward and inverse differential kinematics. The course aims to provide a comprehensive understanding of wheeled mobile robot kinematics, setting the stage for further exploration into wheel configurations and kinematic simulations.

Takeaways

• 📘 Mobile Robot Kinematics is a branch of physics that studies motion without considering the forces affecting it, focusing on the geometrical and control parameter relationships governing the system's motion.
• 🔄 The course specifically discusses land-based mobile robots, which are assumed to move on a plane, and does not cover off-planar movements.
• 🗺 Kinematics in robotics involves mapping between two spaces: the input (control parameters) and the system's motion parameters, aiming to understand and design the system's motion.
• 🛠 The kinematic model serves three main purposes: understanding the system's motion, aiding in the design of the mechanical system and motion controller, and predicting or estimating system parameters for system identification.
• 🔢 The concept of 'degree of freedom' is introduced as the minimum number of variables required to uniquely describe the system's motion, which for a land-based mobile robot is typically three: two translations and one rotation.
• 📏 The importance of coordinate systems is highlighted for describing motion, with the Cartesian coordinate system being a simple and commonly used framework in the lecture.
• 📐 The body-fixed coordinate system is used to relate the robot's instantaneous velocities (u, v, r) to its position in the inertial frame, allowing for the description of motion with respect to a fixed point.
• 🔄 The kinematic relationship between the robot's control parameters (u, v, r) and its motion (x, y, psi) is established through the Jacobian matrix, which maps input velocities to the time derivatives of the generalized coordinates.
• 🔠 The lecture distinguishes between 'forward' and 'inverse' differential kinematics, where the former finds system motion given input velocities, and the latter finds the required input velocities for a desired motion.
• 🔧 Understanding and applying differential kinematics is crucial for controlling a mobile robot's motion at the velocity level, which can be open-loop or closed-loop with feedback.
• 🚀 The next steps in the course will cover the types of wheels used in mobile robots, the concept of maneuverability, and the kinematic simulation, building on the foundational knowledge provided in this lecture.

Q & A

What is the main focus of the third lecture on Mobile Robot Kinematics?

The main focus of the third lecture is on mobile robot kinematics, specifically the kinematic relationship of land-based mobile robots, and how to obtain that relationship to further explore aspects such as forward and inverse differential kinematics.

What is kinematics in the context of physics and robotics?

Kinematics is a branch of physics that studies motion without considering the forces causing the motion. In robotics, kinematics involves mapping the geometrical relationships that govern the motion of a system and the relationship between control parameters and the system's behavior in state space.

Why is a kinematic model necessary for mobile robots?

A kinematic model is necessary for understanding the system's motion, aiding in the design of the mechanical system, and for designing proper motion controllers. It can also be used for navigation system design, performance tuning, and system identification or parameter estimation.

What are the three purposes of a kinematic model in robotics as mentioned in the lecture?

The three purposes are to understand the system's motion, to design the mechanical system (locomotion system), and to design motion controllers and navigation systems for performance tuning.

What is meant by 'degree of freedom' in the context of mobile robots?

The 'degree of freedom' refers to the minimum number of independent variables required to describe the motion of a system in a unique way. For land-based mobile robots, the degree of freedom is typically 3, involving two translations and one orientation in a plane.

How many degrees of freedom are required to describe motion in a 3-dimensional space?

In a 3-dimensional space, six degrees of freedom are required, which include three translations and three orientations.

What is the difference between forward and inverse differential kinematics?

Forward differential kinematics is about finding the system's motion (time derivatives of generalized coordinates) given the input velocity commands. Inverse differential kinematics, on the other hand, is the process of finding the required input velocity commands for a desired motion trajectory (time derivatives of generalized coordinates).

What is the role of the Jacobian matrix in mobile robot kinematics?

The Jacobian matrix is used to map the body-fixed instantaneous velocities (u, v, r) to the time derivatives of the generalized coordinates (x dot, y dot, psi dot). It represents the kinematic transformation between the input velocity commands and the motion of the robot.

What types of motion are considered when discussing the kinematics of a land-based mobile robot?

The kinematics of a land-based mobile robot considers two types of translations (longitudinal and lateral) and one type of orientation (rotation about its own vertical axis) in a plane.

What is the significance of the coordinate systems in describing the motion of a mobile robot?

Coordinate systems are essential for defining the position and orientation of a mobile robot. They provide a reference frame for describing motion relative to a fixed point (inertial frame) and for relating the robot's body-fixed velocities to its motion in the plane.

What will be the topics covered in the subsequent lectures following Lecture 3?

The subsequent lectures will cover topics such as types of wheels used in mobile robots, degree of maneuverability, kinematic simulation, and the relationship between the angular velocity of the wheels and the input command velocities.

Outlines

00:00

🤖 Introduction to Mobile Robot Kinematics

This paragraph introduces the topic of mobile robot kinematics within the context of wheeled mobile robots. It highlights the focus on understanding the kinematic relationships and how they can be used to advance in robot kinematics, including forward and inverse differential kinematics. The lecturer emphasizes the importance of kinematics in robotics, which involves mapping control parameters to motion parameters, and outlines the lecture's flow, which includes basic kinematics, degrees of freedom, and differential kinematics.

05:02

🔍 Purpose and Requirements of Kinematic Modeling

The second paragraph delves into the reasons for using kinematic models in robotics, which include understanding system motion, aiding in system design, and developing motion controllers. It also touches on the use of mathematical models for system identification and parameter estimation. The lecturer uses an analogy of describing a person to explain the concept of degrees of freedom, which are the minimum number of variables required to uniquely describe the motion of a system.

10:07

📏 Understanding Degrees of Freedom in Mobile Robots

This paragraph explains the concept of degrees of freedom in the context of mobile robots. It discusses how many parameters are needed to describe the possible motions of a mobile robot, such as translations and rotations, and how these relate to the robot's maneuverability. The lecturer provides examples to illustrate the point, including the comparison of different types of robots and their respective degrees of freedom in various environments.

15:10

📐 Establishing Coordinate Systems for Robot Motion

The fourth paragraph focuses on the establishment of coordinate systems necessary to describe robot motion. It introduces the inertial frame and the body-fixed coordinate system, explaining their roles in defining the position and orientation of a mobile robot. The lecturer discusses the importance of having a reference for motion description and how these coordinate systems help in mapping the robot's motion in a standardized way.

20:11

🔄 The Relationship Between Input Velocities and Robot Motion

This paragraph explores the relationship between the input velocities (longitudinal, lateral, and angular) and the motion of a mobile robot. It describes how these velocities can be represented in terms of the robot's body-fixed frame and then mapped to the fixed frame of reference. The lecturer explains the process of projecting velocities onto the fixed frame and how this leads to the formulation of the robot's kinematic equations.

25:15

🔄 Kinematic Mapping and Differential Kinematics

The sixth paragraph introduces the concept of kinematic mapping, which is the process of relating the input velocity commands to the time derivatives of the generalized coordinates of a robot. It discusses the forward and inverse differential kinematics, explaining how the former predicts the motion given the input velocities, while the latter determines the required input velocities for a desired motion. The paragraph also introduces the Jacobian matrix as a key component in this mapping process.

Keywords

💡Mobile Robot Kinematics

Mobile Robot Kinematics is the study of the motion of mobile robots without considering the forces that cause the motion. It is a fundamental aspect of robotics that deals with the geometrical relationships governing the motion of a robot. In the video, this concept is central to understanding how a mobile robot moves and how its motion can be mathematically modeled and controlled.

💡Locomotion

Locomotion refers to the ability to move from one place to another. In the context of the video, it is used to describe the different ways a mobile robot can move, such as translational and rotational movements. The script discusses types of locomotion and how they relate to the design and kinematics of mobile robots.

💡Degree of Freedom

The degree of freedom is a concept that describes the number of independent parameters needed to determine the position and orientation of an object in space. For a land-based mobile robot, the script explains that there are typically three degrees of freedom: two for translational movement (forward and sideways) and one for rotational movement around the vertical axis.

💡Differential Kinematics

Differential Kinematics is a subset of kinematics that deals with the instantaneous rates of change of position, rather than the positions themselves. The video discusses differential kinematics in the context of mobile robots, explaining how to calculate the relationship between input velocities (like wheel speeds) and the resulting motion of the robot.

💡Forward and Inverse Kinematics

Forward and Inverse Kinematics are two types of kinematic analysis. Forward kinematics involves calculating the motion of a robot given certain input parameters, while inverse kinematics is the process of determining the input parameters needed to achieve a desired motion. The script explains these concepts in the context of mobile robot motion control.

💡Kinematic Model

A kinematic model is a mathematical representation of the motion of a mechanical system without considering the forces that cause the motion. In the video, the kinematic model is essential for understanding and predicting how a mobile robot will move in response to different inputs.

💡Control Parameters

Control parameters are the inputs to a system that determine its behavior. In the context of mobile robots, control parameters might include the speed and direction of the wheels. The script discusses how these parameters are used in the kinematic model to predict and control the robot's motion.

💡System Identification

System identification is the process of determining the parameters of a system model based on input-output data. The script mentions system identification in the context of using a mathematical model to estimate or predict system parameters that may be difficult to measure directly.

💡Jacobian Matrix

The Jacobian Matrix is a matrix of all first-order partial derivatives of a vector-valued function. In robotics, it is used to map between the velocities of the robot's joints (or wheels) and the resulting velocities of the robot's movement in space. The script explains the role of the Jacobian Matrix in the kinematics of mobile robots.

💡Magnetic Duster

The magnetic duster serves as an example in the script to illustrate the concept of motion and degrees of freedom. It is described as being able to move laterally and rotate on a steel board, demonstrating the translational and rotational movements that are key to understanding mobile robot kinematics.

💡Coordinate Systems

Coordinate systems are used to describe positions and movements in a defined space. The video script discusses two types of coordinate systems: the inertial (or fixed world) coordinate system and the body-fixed coordinate system. These systems are essential for mapping the motion and control inputs of a mobile robot.

Highlights

Introduction to Mobile Robot Kinematics, focusing on the kinematic relationship for land-based mobile robots.

Kinematics is the study of motion without considering the forces affecting it, applied to robotics to understand the geometrical relationship governing motion.

Robot kinematics involves mapping between input control parameters and motion parameters, essential for system understanding and design.

The importance of kinematic models for designing motion controllers and navigation systems in mobile robots.

Kinematic models can predict and estimate system parameters, aiding in system identification.

The concept of 'degree of freedom' in mobile robots, defining the minimum number of variables needed to describe the system's motion uniquely.

For land-based mobile robots, the degree of freedom is generally three, including two translations and one orientation in a plane.

Description of motion using a Cartesian coordinate system and the establishment of an inertial frame I for reference.

Introduction of body-fixed coordinate system B for defining the robot's motion with respect to its own frame.

The relationship between the robot's instantaneous velocities (u, v, r) and its position in the inertial frame (x, y, psi).

The kinematic transformation matrix, or Jacobian matrix, which maps the robot's velocity inputs to its motion derivatives.

Forward differential kinematics involves finding the system motion given velocity input commands.

Inverse differential kinematics is the process of determining the necessary velocity input commands for a desired motion.

The practical applications of differential kinematics in controlling mobile robots at the velocity level.

Upcoming lectures will cover types of wheels used in mobile robots and their classification.

Future lectures will also address kinematic simulation and the relationship between wheel angular velocities and input command velocities.

The conclusion of Lecture 3 with a preview of Lecture 4 focusing on wheel types and maneuverability.

Transcripts

Welcome back to the lecture on you can see Introduction to Mobile Robot Kinematics. So,

the course on Wheeled Mobile Robots. As I already mentioned in the last lecture we were

actually like talking about locomotion and types of locomotion, at end of the lecture

I told that we would be talking more about kinematics in the third lecture. So, that

is what we are going to focus here. So, in this particular third lecture, we would

be more focused on what is land based mobile robot, so what would be the kinematic relationship?

So, how we can obtain that kinematic relationship? So, based on the kinematic relationship how

we can actually like go forward in the further robot kinematic aspects; for example, forward

and inverse differential kinematics. So, in that sense as we did in the last two lectures

a similar way. So, this is the note. So, let us move to the particular topic called

lecture 3 in this way. So, this particular topic or lecture would be focusing as I already

mentioned, it would be focusing mainly on the mobile robot kinematics. As I already

told mobile robot means in general it is a land based. So, land based means it is actually

like having you call planner movement. So, we will not be seeing the off planner

movement. So, that is what the overall idea. So, let us start with the basic introduction

about a mobile robot kinematics and then we move forward to what is degree of freedom

and what is differential kinematics. So, this is what the overall flow which we planned

for this lecture 3. So, let us start with the kinematics. So,

kinematics you already know it is one of the branch of you call physics, so where you talk

about statics and dynamics. So, inside dynamics you know one of the you can say subsection

called kinematics. But what this is all about; kinematics means study of motion without considering

the forces or effects that affect the motion. So, this is what we have seen.

So, now, in that sense what we are actually like trying to bring here is the mathematical

relation which is bringing the motion or you can say the geometrical relationship that

govern the motion of the system. This is what we are trying to correlate. But robot kinematics

means it is little more than this. So, what that mean? We are actually trying to map.

So, we are trying to map actually like two spaces or we are trying to map between the

input and output. Although here the input is not force or you can say moment, but the

input in the sense the control parameter and the system parameter what you call motion

parameter we are trying to make a mapping. So, this mapping what we call kinematics in

robotics. So, that is what we are trying to cover.

As I already mentioned you can see that kinematics what it it deals with the geometrical relationship

that govern the system. So, the other one is it deals with the relationship between

control parameters and the behavior of the system in state space. This is what actually

like one important thing. So, let us actually like move forward in that

case. So, why it is required? So, the kinematical model or mathematical model, why it is required?

There are 3 purposes which we are putting forward. So, one is actually like to understand

the system. So, definitely so what kinematics means, it is study of motion. So, in the sense

we are trying to understand the system motion. So, that is one thing.

Second thing is what happens; so, if you are study about the motion, so what one can see

you can actually like see how to design that particular system. For example, if I actually

like make a two wheel mobile robot, so how that two wheels supposed to be located, whether

this is what the length or you can say the distance between these two wheel I need to

put it like this. For example, now you take it as a cycle, ok

bicycle. So, the front wheel and the back wheel if I have properly the length or you

can say distance between these two wheel, so that parameter change the overall system

study of motion, right. So, in the sense you can actually try to do. In that sense you

can see the need of mathematical model come the broad way, so which is nothing, but the

to understand and design the system, right. To understand the behavior of the system and

design the mechanical system that is what I mean to say here the locomotion system.

The second point is very straightforward. Since, I already told the kinematic model

is deals about the, you can say relation between the control parameter and this the system

parameter. In the sense, what you can see, his mathematical model can be used for you

can say design the proper motion controller. Further what you can see since it is a mobile

robot, even we can extend for navigational system design and there you can say performance

tuning. So, these are actually like two broad category. Any mathematical model for, you

can say mathematical model of robot, definitely these two are the prime most.

The third thing which is actually like very one of the important thing very, you can say

specific we can actually try it to you can say predict or estimate the system parameter.

For example, you take a car. The car I am assuming in kinematic or even you take a general

thing, certain parameter you cannot actually like measure or estimate you can see accurately.

So, then what we can do? We can actually like use a mathematical model which is you predominantly

based on the first principle and you can actually like adjust the parameters based on the real

you can say output and as well as your model performance. You can see you can actually

tune and you can actually like identify. In the sense what people call it is can be used

for system identification or parameter estimation. For example, you take in the other way around.

So, you take a open system and give input and take output. So, what one can see from

the input and output relationship? You can understand the system, right, so what would

be the system. So, that is what we are actually saying that to predict or estimate.

In the sense, the mathematical model definitely can be used for these 3 purpose, to design,

to understand, this is the combined fact and to design controller this is another fact,

and the third fact which we call to predict and estimate. So, now we move a little forward.

So, what that mean? So, we will talk the mobile robot kinematics. So, for that one of the

important thing. So, what that? So, for example, now you are talking about the description

ok that is what nothing but, right. So, what the description? So, you are going to describe

the motion. So, now, in order to describe the motion you

have to see certain parameters, right. For example, you you want to describe about me.

Imagine, so you know like one of the faculty in your institute. He comes from probably

IIT, Indore imagine. So, I was working in IIT, Indore, for example, 7 years, I was there.

So, now, you forget my name somehow, but you are actually like know some of the credentials

of me. So, the person who is coming from IIT, Indore definitely put that credential you

can understand. Now, for putting that credentials you have to put the minimum number of you

can say credentials. For example, you can say that the guy actually

like worked in mechanical, you want to describe me, ok. This is one credential. Second thing

is he graduated from IIT, Madras. Then you can see that even in IIT, Indore currently

there are 4 faculty working in mechanical engineering who graduated from IIT, Madras.

These are not sufficient, right. Then you put another key word probably he is working

in robotics. Then, also you can see that in IIT, Indore

there are two professors are working in robotics then that to like from IIT, Madras itself.

Then you can actually like put one more credential. You can say that this particular person who

was actually like in Germany for more than a year, then you can see that the person who

is actually listening in the other side he can understand oh this guy is talking about

Shanta Kumar. Ok, now I got it. So, now what you put, you put you can say

list of description in such a way that the opposite person can understand and identify

the, you can say the correct person. So, now, the same way when you talk about you can say

description, so when you talk about the study of motion the motion description. So, how

many variable required to describe in a unique way. So, this is what we are calling as a

minimum, number of variable to describe the system.

This minimum number of variable what we are going to call as degree of freedom. Why it

is called degree of freedom? Because we are talking about the system which is in moving.

So, what are the possible motions or what are the possible movement happened on the

system that is what we are trying to give a idea. So, in the sense what in the other

way around you can say, number of independent variable to describe the system in unique

manner. This is what degree of freedom. So, now you got a very general definition,

right. So, now, we can actually like see if you talk about a mobile robot, so what are

the possible motion? So, mobile robot mean it is a land base. So, you can see that the

robot can move in longitudinal way, lateral way, and it can rotate about its own vertical

axis. In the sense of what are the possible motions?

So, there are 3 motions. So, now, in order to describe these 3 motions, what are the

number of parameter required? At least 3 parameter, right. So, that is what we are going to call

as a degree of freedom. In the sense for land based you can say mobile robot in general

wheeled, ok in specific wheel and whatever you call water surface vehicle all are actually

like coming under a planner case. So, we are assuming that the off planner movement is

not there. So, in that sense what are the possible motions?

Two translations and one you can say orientation in a plane. So, in the sense you can see there

are 3 motion primitives are required. In the sense the degree of freedom is 3. Now, you

take a very general case you can take a space. So, now you take for example, this chalk.

This chalk actually like I can put it on the space. So, now I want to describe this chalk

motion for example. So, how many variables required? This chalk and actually like move

in 3 dimensional space. So, in the sense 3 translation and 3 orientations

required. In the sense the degree of freedom for a general case in 3 dimensional space,

it required 6. So, now you take a aerial robot or underwater robot or space robot these are

all required actually like 6 parameter described this is the degree of freedom is what you

call 6. So, 3 translation and 3 orientations are what you call actually like a degree of

freedom for this particular system. So, this is what we are actually like mentioning.

So, now, you know what is degree of freedom. It is nothing, but minimum number of variable

to describe the system in unique sense. In that case, we are focusing on the land base;

in that sense the degree of freedom is 3, that is what we are focusing.

So, in the sense what we are doing in this particular course? This particular course

is all about wheeled mobile robot. In the sense, I am putting the same keyword again.

So, ability move around with the help of wheels. So, in the sense the degree of freedom in

this case is 3, right. So, now, two translation and one orientation we need to describe in

a plane. So, now we are taking that into a general

case. So, now, imagine I have a magnetic duster and there is a steel board in the behind.

So, now, I put that magnetic duster on the steel board what will happen to that? That

steel board will not allow the magnetic duster come away. But if you actually like move around

what happened? The two translation and one orientation is possible.

So, that is what I am trying to put it in this particular slide. You can see that this

blue color box what I drawn imagine as a magnetic duster. Now, this magnetic duster can actually

like move lateral, in the sense in this case it is a longitudinal lateral and as well as

rotate about it, right. Now, I call that is as a mobile robot in a

land, ok land based systems this is a mobile robot. Now, in order to describe this system

what I need? So, first of all in order to describe some standard thing is required.

For example, you are actually like describing about me for example,brown color skin, you

can say long hair, short height. So, these are description.

But now the same description if I go in a different country, for example, if I go a

Mongolian country I may not be a short height, right. If I go probably in a Korean country

I may not be having a longer. If I go probably Africa, I may not be a you can say brown fair,

right. So, in the sense you can see that there is some reference you have to maintain. So,

what that? So, you need to have some reference. So, in that case here we are going to describe

about the motion. So, what would be the motion? Motion can be described in two translation

and one orientation that is what here. So, in the sense what you need to have? You need

to have a frame. So, in the sense you need to have a coordinate system.

So, there are several coordinate systems are possible, but we are taking one of the simplest

one which is what we call Cartesian coordinate system. There are 3 mutually perpendicular

axes which is connected with a single point that is what we call Cartesian coordinate

system. Since, it is in a plane we can take one of the Cartesian coordinate system here.

And now you take even Cartesian coordinate system.

Still I cannot define the system, right. For example, if I take a duster on the board.

So, I need to define the position of the duster with respect to something. So, for that what

I am bringing? I am bringing one of the point which I call I. So, why I call I? This is

inertial. So, I am going to call I means inertial, in that sense it is fixed on the wall or fixed

on the board, it is fixed. So, now, I am actually like putting the Cartesian

coordinate system where Zi in the sense Z axis that is actually coming out of the screen,

ok. So, that is what we are going to use. In that sense, we are going to use the, right

hand coordinate or right hand rule, so where thumb finger goes x, you can say forefinger

goes y and Z axis would be represented with the middle finger. So, now, this is fine.

So, still you may have a doubt. Sir, how can I represent the motion of the robot here?

Because this is one of the coordinate, but the mobile robot is having infinite number

of points, right. So, how I will define? So, for that what we are bringing? One more

coordinate which is we are going to call body fixed coordinate. So, this is I am putting

one of the convenient point here which is nothing, but B. That B is having another coordinate

system call you can say xb, yb and a Zb. Since it is in a plane, so I am not showing a you

call Zb and Zi. So, now, you can see that I put one point B, another point is I. So,

definitely in a Cartesian way I can define the point B with respect to I.

So, what that would be? So, I can actually like a draw imaginary line. I can actually

like write it x would be the x axis displacement or x axis position with respect to I, right.

So, and y would be the y axis position of b with respect to I. And I am putting another

variable called psi which is nothing, but the rotation about Z axis.

Now, you can see that I have already done, right. What I have done? I have actually like

represent the point b with respect to the fixed frame I. But what the issue comes the

mobile robot is actually like moving system, and mostly what happened the mobile robot

would be connected with a wheel or leg that would be energizing the mobile base.

For example, even you take a duster the duster will not be having any input with respect

to I, the input would be with respect to the B frame in the sense what we know from the

body frame, so there would be a longitudinal velocity which I am representing as u and

there would be a lateral velocity which I am representing as a v, and there is a angular

velocity which I am representing as r. So, these are related with a body, ok.

So, now, if I take B is instantaneously frozen, so now, what would be the instantaneous velocity

of B along with xb yb? That would be u and v and what would be the instantaneous velocity

of B with respect to is Z b which is the angular velocity that is r. In the sense, I have a

instantaneous velocity u, v, r, which I am assuming that this information is known with

respect to frame B. But what I wanted? I wanted x y psi.

So, how I will get? This is what the motion variable, right. So, how I will get? So, you

know the equation, right a equal to b when this would be valid. So, there are two variable

I am writing a equal to b. When this equation is valid? Both a and b dimensions are, right

and units are, right. So, in this case you can see that u, v, r

is one set of variable. So, X, Y, psi is another set of variable, but you can see X, Y, psi

or you can say positional variable, there are linear position and angular position;

whereas, u, v, r is the velocity information in the sense linear and angular velocities.

Both are actually like distinct. So, these are not straightaway same.

So, then what one can see? Either you can actually like integrate that velocity bring

it to you can say positional domain or you can actually like elevate the position by

differentiating and bring it to the you can say time derivative. In the sense, what one

can see, so you can actually like do either one, so which is easy the instantaneous velocity

is already instantaneous, I cannot get the instantaneous position, right.

So, obviously, one can actually like see, so I can elevate the X, Y, psi into x dot,

y dot and psi dot. Whatever I put x top of dot that represent dx by dt, ok. So, this

is what we are trying to see. So, now, you can see that the x dot, y dot, and psi dot

would be along with the x axis that would be x dot, along with y axis that would be

y dot, and above the Z axis rotation that is what psi dot, but what we have u, v, r.

So, now, the u can be actually like represented on the frame of I. So, how we can represented?

So, we can project, so you know law of cosine. So, now, I am actually like taking the u and

projecting on the frame you can say I. So, now what would be there? The angle already

I know psi, so if I projected on the x axis that would be equivalent to u cos psi, if

I projected on the y axis that would be equivalent to u sin psi. So, now, similarly I take v

vector, so if I actually like projected on x axis and y axis, it is v sin psi and v cos

psi. So, now, I projected. So, then why you projected? So, this is one of the question

will come. Now, since the system is you can say static

at instantaneous point, in the sense whatever the motion happening on the B, if I actually

like see with respect to I that supposed to be same, right. So, in the sense you can see

that whatever the longitudinal velocity which is u cos psi and v sin psi, the vector addition

supposed to be equivalent to x dot that is what we are doing it. So, I am taking u cos

psi and v sin psi, I am doing in the vector addition what would be that is what x dot.

Now, similarly you can see y axis vector addition vectors, so u sin psi and v cos psi. So, if

I take vector addition that would be equivalent to y dot. And what one can see easily the

psi dot and r, are actually like in the same you can see direction and as well as it is

parallel. So, in the sense r is straightaway equivalent to psi dot.

So, in that sense what one can see, so x dot I can write as u cos I minus v sin psi and

y dot I can write as u sin psi plus v cos psi and psi dot I can write as r, that I am

writing in a vector form. You can see this particular slide, so x dot y dot psi dot I

am writing as one of the vector that later on I am going to call as eta dot, so where

eta is vector of x, y and psi. So, eta dot is actually like x dot y dot and

psi dot that is equivalent to u cos psi minus v sin psi u sin psi plus v cos psi and r.

So, now, what one can see I can actually like group it this u, v, r into one side and x

dot y dot psi dot is another side. So, whatever the in between that is what you call mapping.

So, now, what you can see u, v, r is the body fixed instantaneous velocity and x dot, y

dot, and psi dot is the initial fixed time derivatives of the generalized coordinate.

Now, I can map. So, this is what the next slide is telling. You can see this is the

relation I got it and I am rewriting into a matrix and vector form. So, what one can

see, I can map x dot y dot psi dot with respect to u, v, r. So, this is what we call kinematic

relation. So, this is what the kinematic model. So, now, this particular matrix I am going

to call as one of the generalized matrix called mapping matrix, some people call kinematic

transformation matrix, but this kinematic transformation as actually like given in a

time domain in the sense time derivative domain given by Jacobian, so this particular matrix

called Jacobian matrix. So, now J of psi what you call Jacobian matrix.

What this particular relationship is giving a idea? Where you consider u, v, r as one

of the vector called zeta that is what nothing but velocity input. So, now, you assume that

there is a command. So, now, velocity input command is mapped to the you call the time

derivatives of the generalized coordinate eta dot. So, now you can see that this is

mapping between these two, right. So, that is what we call mobile robot kinematics.

Now, what you obtain this equation nothing, but the robot kinematic equation. So, as I

already told robot kinematics means not only study of motion it is mapping further, right.

So, now, you can see that the mapping between happening either actually like velocity input

command to the time derivative of generalized coordinate. So, now, based on that it can

be classified into two, ok. So, what that mean?

So, one is actually like you give input command velocities are input velocity command you

give it, in the sense you would take the mobile robot, connect the wheel, and connect the

motor and run the motor with certain speed. So, what you will see? You will see how the

robot moves, right. So, in the sense what you are trying to understand?

You are giving the input command and seeing the you can say the time derivative. And if

you integrate that what you will get? That generalized you can say generalized coordinate.

So, this is what we are trying to see. For a given velocity input command finding

the derivatives of generalized coordinate nothing, but finding the system motion this

is what you call forward differential kinematics. So, why it is so called forward? Because your

input is straightforward given and you are seeing the motion variable how it moves.

So, this is nothing, but one scenario. If you are doing mathematically it is tried to

simulate, you take a real robot and do it nothing, but analyzing, right. So, that is

what we call, so simulating or analyzing the system in velocity level that is what forward

differential kinematics. So, now, what would be the next case? You

just imagine, you just imagine, so you are having some desired derivative of generalized

coordinate you want to see what would be the input. So, in the sense you are trying to

do the reverse way. So, that is why it is called inverse differential kinematics, where

you can see that for a given you can say it desired a derivative of generalized coordinate

you are trying to find out, you can sees that corresponding velocity input command.

In the sense, what you are trying to see? For a given position trajectory you are trying

to see what would be the you can say body fixed velocities. So, this is what we call

you can say inverse differential kinematics. You know already the equation. You have see

eta dot is straight away Jacobian ofyou can say J of psi into zeta.

Now, we are trying to find out the zeta, so obviously, what one usually you see this is

known, and this is actually like known and this is what unknown, so you take the inverse

of this. So, that is what we are doing it. So, you can even imagine that this is the

way you can actually take, remember that inverse means J inverse will exist, right this is

a blind it is not, right, but I can actually like give a idea.

So, zeta is J inverse of psi into eta dot. So, now, what we are trying to see? The given

I already mentioned give and decide. So, in the sense what we are trying to see? I want

the robo supposed to move in this manner, I want certain motion this way, can I actually

like make it. In the sense, I am trying to navigate in particular

way. So, in the sense what you are trying to do? You are trying to do a controlling

action, right. So, that is what you call controlling the system in velocity level. So, this is

what the inverse differential kinematics. Further on, this inverse differential kinematics

what we call control, the control can be open loop, in the sense you just take eta dot and

J inverse of psi. This is actually like very open, but you can actually like even make

a closed loop, you take a feedback and make it then that is actually like feedback control,

but these all we would be seeing in the end of this course.

But, right now what you understood is actually like what is mobile robot kinematics. It is

nothing, but mapping between two you can say velocity spaces. So, one is actually like

input you can say velocity input commands, so the other one isderivative, that to time

derivative of generalized coordinate. You are trying to map why it is mapping because

in mobile robot it cannot be directly find it. So, you have done with the mapping. So,

this is what. The mapping further divided into two. So,

one is forward differential kinematics another one is inverse differential kinematics. In

the next lecture, we will see how the zeta is obtained because the zeta is actually like

a slightly different, right, because the zeta is actually like what you call velocity input

commands. The velocity input commands is actually like

coming from the wheeled configuration. So, then we have to bring one more mapping where

the zeta would be coming with you can say angular velocity of the wheel. So, that is

what we are trying to address in the next lecture.

So, before that probably will see the one of the important aspect you should know like

what are the types of wheels which are used in the mobile robot, how that can be classified,

and how we can actually like make it. In the sense, the next slide or next lecture would

be talking about more about a real mobile robot. So, right now we talk about degree

of freedom, but since it is a mobile robot the maneuverability will come into a picture.

So, the next lecture would be talking about degree of maneuverability and we will see

how to obtain that. So, based on that the further lecture will come about the wheeledyou

can say locomotion in the sense, you will be bringing the kinematic relationship between

the angular velocity of the wheel to the input command velocities and then you can make it

to the time derivative of generalized coordinate. So, with that you can see that this particularlecture

3 is over. So, then lecture 4 would be talking about types of wheel, and then lecture 5 would

be talking about the kinematic simulation. So, with that we would be finishing this course

here. So, we will see in the lecture 4 later. Bye.

Thank you.