Skip to main content
Loading Explore guides…
Mobile Robot Kinematics · Acedra
Home
Explore
Engineering
Mobile Robot Kinematics
Sign in to save
Chat
If YouTube asks you to sign in, open the video on YouTube.
Watch on YouTube
Reload player
Transcript
377 segments
Copy
Hide video
0:13
Welcome back to the lecture on, you can see, Introduction to Mobile Robot Kinematics. So,
0:18
the course on Wheel the Mobile Robots. As I already mentioned in the last lecture,
0:23
we were actually like talking about locomotion and types of locomotion. At the end of
0:27
the lecture, I told that we would be talking more about kinematics in the third
0:31
lecture. So, that is what we are going to focus here. So, in this particular
0:35
third lecture, we would be more focus on what is land based mobile robot. So,
0:41
what would be the kinematic relationship? So, how we can obtain the kinematic relationship? So,
0:45
based on the kinematic relationship, how we can actually go forward in the further robot
0:51
kinematic aspects. For example, forward and inverse differential kinematic. So, in For example, forward and
0:53
inverse differential kinematics. So, in that sense, as we did in the last two lectures
0:59
a similar way, so, this is the note. So, let us move to the particular
1:03
topic called lecture 3 in this way. So, this particular topic or lecture would be
1:07
focusing as I already mentioned, it would be focusing mainly on the mobile robot kinematics,
1:12
as I already told mobile robot means in general it is a land based. So,
1:17
land based means it is actually like having you call planar movement. to having you
1:19
call planar movement so we will not be seeing the off planar movement so that
1:24
is what the overall idea so let us start with the basic introduction about mobile
1:29
robot kinematics and then we move forward to what is degree of freedom and what
1:34
is differential kinematics so this is what the overall flow which we planned for this
1:38
lecture three so let us start with the kinematics so kinematics you already know it
1:43
is one of the branch of you call physics so where you talk about statics
1:47
and dynamics so inside dynamics you know one of the you can say subsection called
1:52
kinematics, but what this all about kinematics means study of motion without considering
1:58
the forces or effects that affect the motion. So, this is what we have seen.
2:02
So, now in that sense what we are actually trying to bring here is the
2:07
mathematical relation which is bringing the motion or you can say the geometrical relationship that
2:12
govern the at govern the motion of the system this is what we are trying
2:16
to correlate but robot motion of the system. kinematics means it is little more than
2:20
this so what that mean we are actually trying to map so we are trying
2:25
to map actually like two spaces or we are trying to map between the input
2:30
and output although here the input is not pose or you can say moment but
2:35
the input in the same the control parameter and the system parameter what you call
2:39
motion parameter we are trying to make a mapping so this mapping what we call
2:43
kinematics in robotics so that is what we are trying to cover as i already
2:48
mentioned you can see that kinematics what it it deals with the geometrical relationship that
2:53
govern the system so the other one is it deals with the relationship between control
2:57
parameters and the behavior of the system in state space this is what actually like
3:02
one important thing so let us actually like move forward in that case so why
3:07
it is required so the kinematic model or mathematical model why it is required there
3:12
are three purposes which we are putting forward so one is actually like to understand
3:16
the system so definitely so what kinematics means it is study of motion so in
3:20
the sense we are trying to understand the system motion so that is one thing
3:24
second thing is what happens so if you are study about the motion so what
3:28
one can see you can actually like see how to design that particular system for
3:33
example if i actually like make a two wheel mobile robot, so, how that two
3:37
wheel supposed to be located? Whether this is what the length or you can say
3:41
the distance between these two wheel, I need to put it like this. For example,
3:45
now you take it as a cycle, okay, bicycle. So the front wheel and the
3:49
back wheel, if I have probably the length or you can say distance between these
3:54
two wheel, so that parameter change the overall system study of motion, right? So, in
4:00
the sense you can actually try to do. to try to do in that sense
4:02
you can see the need of mathematical model come the broad In that sense, you
4:02
can see the need of mathematical model come the... way so which is nothing but
4:06
the to understand and design the system right to understand the behavior of the system
4:12
and design the mechanical system that is what i mean to say here the locomotion
4:16
system the second point is very straight forward since i already told the kinematic model
4:22
is deals about the you can see ah relation between the control parameter and this
4:26
the system parameter in the sense what you can see this mathematical model can be
4:31
used for be used you can say design the proper motion controller further what you
4:36
can see since it's a mobile robot even we can extend for navigational system design
4:41
and there you can say performance tuning so these are actually two broad category any
4:46
mathematical model for you can say mathematical model of a robot definitely these two are
4:51
the primos the third thing which is actually like very one one of the important
4:55
thing very you can say specific we can one of the important thing very you
4:56
can see a specific we can actually try to you can say predict or estimate
5:01
the system parameter for example you take a car the car i am assuming in
5:06
kinematic or even you take a general thing certain parameter you cannot actually like measure
5:11
or estimate you can see accurately so then what we can do we can actually
5:16
like use a mathematical model which is you predominantly based on the first principle and
5:21
you can actually like adjust the parameter based on the real, you can say output
5:26
and as well as your model performance. You can see, you can actually tune and
5:30
you can actually like identify. In the sense, what people call it is, can be
5:34
used for system identification or parameter estimation. For example, you take in the other way
5:39
around. So you take a open system and give input and take output. So what
5:45
one can see from the input and output relationship, you can understand the system, in
5:48
the system right so what would be the system so that is what we are
5:52
actually saying that right? So, what would be the system? So, that is what. to
5:53
predict or estimate in the sense the mathematical model definitely can be used for these
5:58
three purpose to design to understand this is the combined fact and to
6:03
design controller this is another fact and the third fact which we call to predict
6:08
and estimate so now we move little forward so what that mean so we will
6:13
talk the mobile robot kinematics so for that one of the important thing, so what
6:18
that, so for example, now you are talking one of the important thing so what
6:18
that so for a about the description, that is what nothing but right. So what
6:24
the description, so you are going to describe the motion. So now in order to
6:28
describe the motion, you have to see certain parameters right. For example, you want to
6:34
describe about me, imagine, so you know like one of the faculty in your institute,
6:40
he come from probably IIT Indore, imagine. so i was working in iit indore for
6:45
for example seven years i was there so now So, I was working in IIT
6:45
Indore, for example, seven years. you forget my name somehow but you are
6:51
actually like know some of the credentials of me so the person who is coming
6:55
from iit indore definitely put that credential you can understand now for putting that credentials
7:01
you have to put the minimum number of you can say credentials for example you
7:04
can say that the guy actually like worked in mechanical you want to describe me
7:09
ok this is one credential second thing is he graduated from iit madras then you
7:13
can see that even in iit indore currently there are four faculty working in mechanical
7:18
engineering who graduated from iit madras these are not sufficient right then you put a
7:23
another keyword probably is working in robotics then also you can see that in iit
7:27
indore there are two professors are working in robotics then that too like from iit
7:32
madras itself then you can actually put one more credential you can say that this
7:36
particular person who was actually like in germany for more than a year Then you
7:40
can see that the person who is actually like listening in the other side, he
7:44
can understand oh, this guy is talking about Shantakmar, okay, now I got it. So
7:48
now what you put, you put a, you can say list of description in such
7:52
a way that the opposite person can understand and identify the, you can say,
7:58
the current person. So now the same way, when you talk about, you can say
8:03
description, so when you talk about the study of motion the motion description so how
8:08
many variable about the study of motion, the motion description. required to describe in a
8:10
unique way so this is what we are calling as a minimum number of variable
8:15
to describe the system this minimum number of variable what we are going to call
8:19
as degree of freedom why it is called degree of freedom because we are talking
8:23
about the system which is in moving so what are the possible motions or what
8:28
are the possible movement happened on the system that is what we are trying to
8:32
give a idea so in the sense what in the other way around you can
8:36
say number of independent variable to describe the system in unique manner this is what
8:40
degree of freedom so now you got a very general definition right so now we
8:45
can actually see if you talk about a mobile robot so what are the possible
8:49
motion so mobile robot mean it is a land base so you can see that
8:53
the robot can move in longitudinal way lateral way and it can rotate about its
8:58
own vertical axis in the sense of what are the possible motions so there are
9:03
three motions so now in order to describe these three motions what are the number
9:07
of parameter required at least three parameter right so that is what we are going
9:11
to call as a degree of freedom in the sense for land based you can
9:15
say mobile robot in general wheel ok in specific wheel and whatever you call
9:21
water surface vehicle all are actually like coming under a planner case so we are
9:26
assuming that the off planner move coming under a planar case so we are assuming
9:27
that the off planar movement is not there so in that sense what are the
9:30
possible motions to translations and one you can say orientation in a plane so in
9:36
the sense you can see there are three motion primitives are required in the sense
9:39
the degree of freedom is three now you take a very general case you can
9:44
take a space so now you take for example this chalk this chalk actually like
9:48
I can put it on the space so now I want to describe this chalk
9:52
motion for example. how many variables required? This chalk can actually like move in three
9:58
-dimensional space. So in the sense three translation and three orientations required. In the sense
10:03
the degree of freedom for a general case in three -dimensional space, it required six.
10:08
So now you take a aerial robot or underwater robot or space robot, these are
10:13
all required which will like six parameter described. This is the degree of freedom is
10:17
what you call six. what you call 6. So, 3 translation and 3 orientations are
10:22
what you call actually like a degree of So, three translation and three orientations are.
10:25
freedom for this particular system. So, this is what we are actually like mentioning. So,
10:29
now you know what is degree of freedom? It is nothing but minimum number of
10:32
variable to describe the system in unique sense. In that case, we are focusing on
10:37
the land base in that sense the degree of freedom is 3. That is what
10:41
we are focusing. So, in the sense what we are doing in this particular course?
10:45
This particular course is all about wheel This particular course is all about wheeled mobile
10:47
robot in the sense I am putting the same keyword again. So ability move around
10:53
with the help of wheels. So in the sense the degree of freedom in this
10:56
case is 3 right. So now two translation and one orientation we need to
11:02
describe in a plane. So now we are taking that into a general case. So
11:06
now imagine I have a magnetic duster and there is a steel board in the
11:11
behind. So now I put that magnetic duster on the steel board. What will happen?
11:16
to that? steel board will not allow the magnetic duster come away.
11:21
But if you actually like move around what happened the two translation and one orientation
11:26
is possible. So, that is what I am trying to put it in this particular
11:29
slide. You can see that this blue color box what I drawn imagine as a
11:34
magnetic duster. Now this magnetic duster can actually like move lateral in the sense in
11:39
this case it this is in this case it is longitudinal lateral and as well
11:42
as rotate about it right now i is longitudinal lateral and as well as rotate
11:42
about it. call that is as a mobile robot in a land ok land based
11:48
system this is a mobile robot now in order to describe the system what i
11:52
need so first of all in order to describe some standard thing is required for
11:57
example you are actually like describing about me for example brown color skin you
12:03
can say long hair short height so these are description our description but now the
12:08
same description if i go in a different country for example but now the same
12:08
description if i go in a different if i go a mongolian country i may
12:13
not be a short height right if i go probably in a korean country i
12:17
may not be having a longer if i go probably africa i may not be
12:20
a you can say a brown fair right so in the sense you can see
12:24
that there is some reference you have to maintain so what that so you need
12:28
to have some reference so in that case here we are going to describe about
12:32
the motion so what would be the motion motion can be described in two translation
12:37
and one in two orientation that is what here so in the sense what you
12:40
need to have you need to have a frame so in the sense you need
12:44
to have a coordinate system so there are several coordinate systems are possible but we
12:49
are taking one of the simplest one which is what we call cartesian coordinate system
12:53
there are three mutually perpendicular axis which is connected with a single point that is
12:58
what we call cartesian coordinate system since it is in a plane we can take
13:02
one of the Cartesian coordinate can take one of the cartesian coordinates system here and
13:05
now you take even Cartesian coordinate system, still I cannot define the system right, for
13:10
example if I take a duster on the board, so I need to define the
13:14
position of the duster with respect to something, so for that what I am bringing,
13:19
I am bringing one of the point which I call I, so why I call
13:23
I, this is inertial, so I am going to call I means inertial, in the
13:27
sense it is fixed on the wall or fixed on the board, it is fixed.
13:27
In that sense, it is fixed on the wall or fixed on the board. It
13:30
is fixed. So, now, I am actually putting the Cartesian coordinate system, where Z i,
13:36
in the sense Z axis, that is actually coming out of the screen. Okay. So,
13:41
that is what we are going to use. In that sense, we are going to
13:43
use the right hand coordinate or right hand rule. So, where thumb finger goes X,
13:48
you can say four finger goes Y and Z axis would be represented with the
13:52
middle finger. So, now, this is fine. so still you may have a doubt so
13:57
how can i represent the motion of the robot here because this is one of
14:01
the coordinate but the mobile robot is having infinite number of points right so how
14:06
i will define so for that what we are bringing one more coordinate which is
14:10
we are going to call body fixed coordinate so this is i am putting one
14:15
of the convenient point here which is nothing but b that b is having another
14:19
coordinate system call you can say x b y b and z b since it
14:23
is in a plane so i am not showing a you call z b and
14:27
z i so now you can see that i put one point b another point
14:31
is i so definitely in a cartesian way i can define the point b with
14:36
respect to i so what that would be so i can actually like a draw
14:40
imaginary line i can actually like write it x would be the x axis displacement
14:45
or x axis position with respect to i right so and y would be the
14:50
y axis position of b with respect to i and i am putting another of
14:51
b with respect to i right so and y is the x axis position with
14:52
respect to i right so variable called psi which is nothing but the rotation
14:57
about z axis now you can see that i have already done right what i
15:02
have done i have actually like represent the point b with respect to the fixed
15:06
frame i but what the issue comes the mobile robot is actually like moving system
15:11
and mostly what happened the mobile robot would be connected with a what would be
15:15
connected with a wheel or leg that would be energizing the mobile base for wheel
15:16
or leg that would be energized example even you take a duster
15:22
the duster will not be having any input with respect to i the input would
15:26
be with respect to the b frame in the sense what we know from the
15:30
body frame so there would be a longitudinal velocity which i am representing as u
15:35
and there would be a lateral velocity which i am representing as a v and
15:40
there is a angular velocity which i am representing as r so these are related
15:45
with the body ok so now if i take b is instantaneously
15:50
frozen so now what would be the instantaneous velocity of b along with xb
15:56
yb that would be u and v and what would be the instantaneous velocity of
16:01
b with respect to zb which is the angular velocity that is r in the
16:05
sense i have a instantaneous velocity u v r which i am assuming that to
16:09
u v r which i am assuming that this information is known with respect to
16:14
frame b but what i wanted i wanted x y psi so how i will
16:20
get this is what the motion variable right so how i will get so you
16:24
know the equation right a equal to b when this would be valid so there
16:29
are two variable i am writing a equal to b when this equation is valid
16:33
both a and b dimensions are right and units are right so in this case
16:38
you can see that u v r is one set of variable and units are
16:37
right so in this case you can see that u v r so x
16:43
y psi is another set of variable but you can see x y psi r
16:48
you can say positional variable there are linear position and angular position whereas u v
16:54
r is the velocity information in the sense linear and angular velocities both are
17:00
actually distinct so these are not straight away same so then what one can see
17:04
either you can actually like integrate that velocity bring it to you can say positional
17:09
domain or you can actually like elevate the position by differentiating and bring it to
17:14
the you can say time derivative in the sense what one can see. So, you
17:20
can actually like do either one. So, which is easy the instantaneous velocity is already
17:25
instantaneous I cannot get the instantaneous position right. So, obviously, one can actually like
17:31
see so, I can elevate the x y. one can actually see, so I can
17:32
elevate the x y psi into x dot y dot and psi dot, whatever I
17:37
put x top of dot that represent dx by dt, okay. So this is what
17:43
we are trying to see. So now you can see that the x dot y
17:47
dot and psi dot would be along with x axis that would be x dot,
17:51
along with y axis that would be y dot and about the z axis rotation
17:55
that is what psi dot. but what we have u v r so now the
18:01
u can be actually like represented on the frame of i so how we can
18:06
represent it so we can project so you know law of cosine so now i
18:11
am actually like taking the u and projecting on the frame you can say i
18:16
so now what would be there the angle already i know psi so if i
18:20
projected on the x axis that would be equivalent to u cast psi if i
18:24
projected on the y axis that would be equivalent to u a side if i
18:24
projected on the y axis that would be equivalent to u sine side so now
18:29
similarly i take v vector so if i actually projected on x axis and y
18:34
axis its v sine side and v cos side so now i projected so then
18:40
why you projected so this is one of the question will come now since the
18:45
system is you can say static at instantaneous point in the sense whatever the motion
18:51
happening on the B, if I actually see with respect to I, that supposed the
18:51
motion happening on the b if i actually see with respect to i to be
18:56
same right. So, in the sense you can see that whatever the longitudinal velocity which
19:01
is u cos psi and v sin psi, the vector addition supposed to be equivalent
19:07
to x dot that is what we are doing it. So, I am taking u
19:10
cos psi and v sin psi, I am doing the vector addition what would be
19:14
that is what x dot. Now, similarly you can see y axis vector addition vectors.
19:20
So, u sin psi and v cos psi. So, if I take vector addition that
19:25
would be equivalent to y dot and what one can see easily the psi dot
19:30
and r are actually like in the same you can see direction and as well
19:35
as it is parallel. So, in the sense r is straight away equivalent to psi
19:40
dot. So, in the sense of what one can see. So, x dot i can
19:45
write as u cos psi minus v sine psi and y dot i can write
19:49
as u sine psi plus dot I can write as u cos psi minus v
19:47
sin psi and v cos psi and psi dot
19:53
i can write as r that i am writing in a vector form you can
19:56
see this particular slide so x dot y dot psi dot i am writing as
20:00
a one of the vector that later on i am going to call as eta
20:04
dot so where eta is vector of x y and psi so eta dot is
20:09
actually like x dot y dot and psi dot that is equivalent to u cos
20:13
psi minus v sine psi u sine psi plus v cos psi and r so
20:18
now u cos psi minus v sine psi u sine psi what one can see
20:19
i can actually like group it this u v r into one psi and x
20:24
dot y dot and psi dot is another psi so whatever the in between that
20:27
is what you call mapping so now what you can see u v r is
20:31
the body fixed instantaneous velocity and x dot y dot and psi dot is the
20:36
initial fixed time derivatives of the generalized coordinate now i can map so this is
20:41
what the next slide is telling you can see this is the relation i got
20:45
it and i am rewriting into a matrix and vector form so what one can
20:49
see i can map x dot y dot psi dot with respect to u v
20:54
r so this is what we call kinematic relation so this is what the kinematic
20:59
model so now this particular matrix i am going to call as one of the
21:03
generalized matrix called mapping matrix some people call kinematic transformation matrix
21:08
but this kinematic transformation matrix but this is actually given in time domain in the
21:13
sense time derivative domain given by jacoby so this particular matrix called jacobian matrix
21:18
so now j of psi what you call jacobian matrix what this particular
21:24
relationship is giving a idea where you consider u v r as one of the
21:29
vector called zeta that is what nothing but velocity input so now you assume that
21:35
this is the command so now velocity input command is mapped to the you call
21:40
the time derivatives of the generalized coordinate eta dot so now you can see that
21:45
this is mapping between these two right so that is what we call mobile robot
21:50
kinematics now what you obtain this equation nothing but the robot kinematic equation so as
21:56
i already told robot kinematics means not only study of motion it is mapping further
22:02
right so now you can see that the mapping between happening right so now you
22:03
can see that either actually like velocity input command to the time derivative of
22:09
generalized coordinate so now based on that it can be classified into two okay so
22:14
what that mean so one is actually you give input command velocities or input velocity
22:20
command you give it in the sense you take the mobile robot connect the wheel
22:24
and connect the motor and run the motor with certain speed so what you will
22:28
see you will see how the robot moves right so in the speed so what
22:28
you will see you will see how the robot moves sense what you are trying
22:32
to understand you are giving the input command and seeing the you can say the
22:37
time derivative and if you integrate that what you will get the generalized derivative you
22:42
can say generalized coordinate so this is what we are trying to see for a
22:46
given velocity input command finding the derivatives of generalized coordinate nothing but finding the system
22:51
motion this is what you call forward differential kinematic so why it is so called
22:56
forward because your input is straight given and you are seeing the motion variable how
23:01
it moves so this is nothing but one scenario if you are doing mathematically it
23:06
is try to simulate you take a real robot and do it nothing but analyzing
23:10
right so that is what we call so simulating or analyzing the system in velocity
23:14
level that is what forward differential kinematics so now what would be the next case
23:19
you just imagine you just imagine so you are having some desired
23:25
derivative of generalized coordinate you want to see what would be the input so in
23:29
the sense you are trying to do the reverse way so that's why it is
23:33
called inverse differential kinematics where you can see that for a given you can say
23:38
decided derivative of generalized coordinate you are trying to find out you can see that
23:43
corresponding velocity input command in the sense what you are trying to see for a
23:47
given position trajectory you are trying to see what would be the you can say
23:51
body a given position trajectory you are trying to see what would be the you
23:51
can say fixed velocities so this is what we call you can say inverse differential
23:56
kinematics you know already the equation you see eta dot is straight away jacobian
24:02
of you can say j of psi into zeta now we are trying to find
24:06
out the zeta so obviously what one usually you see this is known and this
24:11
is actually like known and this is what unknown so you the inverse of this
24:15
so that is what we are doing it so you can even imagine that this
24:16
see this is what we are doing it so you can even you is the
24:19
way you can actually take remember that inverse means j inverse will exist right this
24:24
is a blind it is not right but i can actually give a idea so
24:28
zeta is j inverse of psi into eta dot so now what we are trying
24:32
to see the given i already mentioned give and decide so in the sense what
24:37
we are trying to see i want the robo supposed to move in this manner
24:41
i want certain motion this way can i achieve So, I want certain motion this
24:44
way can I actually like make it in the sense I am trying to navigate
24:47
in particular way in the sense what you are trying to do you are trying
24:51
to do a controlling action right. So, that is what you call controlling the system
24:55
in velocity level. So, this is what the inverse differential kinematics further on this inverse
25:01
differential kinematics what we call control the control can be open loop in the sense
25:06
you just take eta dot and J inverse of psi this is actually like very
25:12
open, but you can actually even make a close loop you take a feedback and
25:17
make it then that is actually like feedback control. But these all we would be
25:21
seeing in the end of this course, but right now what you understood is actually
25:25
like what is mobile robot kinematics? It is nothing but mapping between two you can
25:30
say velocity spaces. So one is actually like input ah can say velocity input commands.
25:35
So, the other one is derivative that to time derivative of generalized coordinate you are
25:35
So, the other one is ah derivative that to time derivative. trying to map
25:41
why it is mapping because in mobile robot it cannot be directly find it. So,
25:47
you have done with the mapping. So, this is what the mapping further divide into
25:52
two. So, one is forward differential kinematics another one is inverse differential kinematics in the
25:57
next lecture we will see how the zeta is obtained because the zeta is actually
26:02
like a slightly different right because the zeta is actually like what you a slightly
26:03
different right? call velocity input commands the velocity input
26:09
commands is actually like coming from the wheeled configuration so then we have to bring
26:14
one more mapping where the zeta would be coming with you can say angular velocity
26:19
of the wheel so that is what we are trying to address in the next
26:22
lecture so before that probably we will see the one of the important aspects you
26:27
should know like what are the types of wheels which are used in the mobile
26:31
robot how that can be classified and how we can how that can actually make
26:35
it in the sense the next slide or next lecture would be talking about more
26:39
about a real mobile robot so right now we talk about degree of freedom
26:45
but since it is a mobile robot the maneuverability will come into a picture so
26:50
the next lecture would be talking about degree of maneuverability and we will see how
26:54
to obtain that so based on that the further lecture will come about the wheeled
26:58
you can say locomotion. in the sense you will be bringing the kinematic relationship between
27:04
the angular velocity of the wheel to the input command velocities and then you can
27:10
make it to the time derivative of generalized coordinate. So with that you can see
27:14
that this particular lecture 3 is over. So the lecture 4 would be talking about
27:18
types of wheel and then lecture 5 would be talking about the kinematic simulation. So
27:23
with that we would be finishing this course here, so we will see. So, with
27:23
that, we would be finishing this course here. So, we will see in the lecture
27:27
4 later. Bye. Thank you.
Here to help you learn
Sign in to keep a private conversation with your course tutor.
Ask me anything about the material…
Sign in to chat