hey guys in this lecture we will expand our consideration of robot
manipulators beyond static positioning problems because the robot is always
busy and they're doing some interesting tasks so we have to discuss instantaneous
kinemics so we will examine the notions of linear and angular velocity of a
rigid body and then we will use these concepts to analyze the motion of a
manipulator through velocity propagation it turns out that the study of
velocities leads to a matrix called Jacobian matrix another very important
matrix comes after we have learned rotation matrix and transformation
matrix and the Jacobian also will be used in dynamics and the robot control
and two lectures the Jacobian matrix will show up in some important dynamics
equations and then I will use your file to show you singular configurations and
you will know what is singularity so this lecture is also a little bit
difficult because there are a lot of equations and math so this lecture
depends on the previous four lectures it depends on the transformation matrix and
the relationship between the Cartesian space and the joint space so I hope I
can successfully guide you to cross through this lecture and then after this
lecture you can survive a little bit okay so let's start in forward kinematics
we learn how to use the joint angles the joint configuration in joint space to
calculate the location of the energy factor in the Cartesian space right so
this is the relation from joint space to Cartesian space and then we learn
inverse kinematics we can given the configuration of the energy factor and
then we can get the joints in the joint space so this is the relationship from
Cartesian space to joint space and then if we make a very small displacement of
theta in joint space then we will get this sort of data theta that we are
introducing to each of those joint angles and then we are going to have a
small displacement of called data X so in Cartesian space another question is
what is the relationship between data theta and data X and you know it is not
only data X in position but also in orientation this is a location of the
energy factor we need to consider these two parts so when we talk about the
relationship between data theta and the data X if the displacement tends to be
zero it's very small very quick this is the instantaneous kinematics then we're
going to discuss the differential motion the velocity of the joints and the
velocity of the end effector so dot theta means the time derivative of the
joint theta and dot X means the time derivative of the location of the
end effector so now we want to find the relationship between data theta and
data X always say we want to find the relationship between the velocity of the
joint and also the end effector velocities and the answer is we will
use a linear relationship to connect these two and the the linear
relationship is Jacobian so before we talk about the Jacobian the Jacobian
matrix which can connect the joint velocity and any effector velocity
connects the joint space and the Cartesian space we'll start by looking
at the differential motion and we're going to discuss the velocities how to
calculate the velocity of rigid body calculate the velocity of the end effector
and the joint and then remember our representation the representation of X
involves the position and orientation so that we will discuss linear velocity and
angular velocity of the joint and the end effector okay so now let's start to
look at the velocity first the first one is a linear velocity you see here is
the trajectory and there is a point P in space and then a P is time varying so we
can use a time varying the position vector to represent the location of P
in space then one time is t0 then we can get the P of t0 and then after data t
time then we can get P of t0 plus or data t and the P is with respect to frame A
so if we want to calculate the linear velocity of a point in space then we
need to calculate the velocity of the position vector PA so do you get me so the
velocity of the linear velocity of a point is the velocity of a position
vector so if you want to calculate a velocity the linear velocity of a point
then you need to get the position vector of this point then you can get the time
derivative of the position vector and then you can get the linear velocity so
this one equals to the limit of delta P divided by delta t by delta t tends to 0
then we can get this stuff so this is the basic definition of the linear
velocity now we are presenting the velocity P in frame A if we present the
velocity another frame B then we can use the rotation matrix to accomplish
the change in reference frame and in this case we assume the frame A and B
the configuration between them are fixed so R B A remains invariant during the
motion of point P then now we calculate we P in B then we still can go back to
the time derivative of the position vector so we want to change P A to P B
then we need the rotation matrix in here but because the rotation matrix is time
independent so we can put the rotation matrix out of the derivative part then
we can get the result is the rotation matrix multiply the velocity but rather
than considering a general point velocity relative to an arbitrary frame A
or B we will often consider the velocity of the origin of a frame relative to some
understood universal reference frame some the map frame or the word frame for
example if P is the origin of a frame C the origin of a frame and P is moving
we typically we will use the small V that we see to denote that the linear
velocity of the origin of frame C with respect to reference from U so you it's
like fix the universal frame like word or map and if we use we see in A means
the velocity of origin of C with respect to U expressed in A maybe it's a
little bit confusing and we will see some example later okay so now let's see
the angular velocity when we use linear velocity we always use it to describe
the attribute of a point but we use angular velocity to describe an
attribute of a rigid body we always attach some frame a frame on a rigid
body then we can analyze the configuration of this rigid body so
basically angular velocity describes rotation motion of a frame of a body and
we use Omega to represent angular velocity Omega B in A means the angular
velocity of frame B respect to frame A and also there have a shorthand notion
called the small Omega the small Omega C it means the angular velocity of C with
respect to some reference frame U and in at any instant the direction of Omega B
in A indicates the instantaneous axis of rotation the rotation axis is the
direction of Omega and the magnitude you see the vector the magnitude of Omega B
in A indicates the speed of rotation so angular velocity is also a vector is a
three times one vector and the position vector is also three times one vector
and they have the direction and also magnitude okay so now we introduce the
notions notations for time varying position and orientation these two
velocities and then we'll investigate the description of motion of rigid body
so we always touch a coordinate system to any rigid body that we wish to
describe and then the motion of rigid bodies can be equivalent started as the
motion of frames relative to one another so now we will analyze the velocity of
rigid body so actually we are analyzing the velocity between frames so for
example now we start from linear velocity now if there is a fixed frame A
and there is a frame B and frame B attached to a sound rigid body in this
example from B is the frame of this train and assume this train is driving
on a street railway the driver now needs to use the steering wheel it's
a straight railway and there is a passenger Q is working on the train in
a fixed orientation so what's the velocity of the passenger respect to the
frame A then we want to know the linear velocity of the passenger with respect
to the fixed the frame and requiring the velocity we can start from the
position that we can get the position vector and then we can get the time
derivative of the position then we can get a linear velocity right so now this
is from A and from B okay now this is point P not point Q you can think this
is the passenger in here and if we want to know the velocity of P in A then now
we first we want to get the position vector of P in A and this depends on two
parts the first part is we need to know the position vector from the origin of A
to the origin of B so here is a vector P B in A now we know the position vector
of these two origins and then we need to know the from P we know P B we know the
vector position vector in frame B so then we from P B in A we plus P B P in B
but we want to represent in frame A so we need to use the rotation matrix being
A multiply P B right and in this case we only assume there only have a linear
motion of a B respect to A so there is no rotation relative rotation between A
and B and also P not P is moving also P is only have linear motion then now we
want to get the linear velocity so we will use a differentiation to get the
velocity now this is a time derivative okay and then so the okay to click the P
the velocity of P in A is a velocity of B in A and then we want to get the time
derivative of this part we have seen this part before right so the rotation
matrix is time independent so we can get the rotation matrix out of the derivative
part then we can get the velocity so the linear velocity of P with respect to
frame A is the sum of the velocity with respect to frame B and the velocity of
origin of frame B with respect to frame A and here we only discuss the
translational motion of the frame and but if they also have a rotation of this
frame then since will become more complicated then the angular velocity
comes in order to make sense simpler now we assume three conditions the first one
is there is no linear velocity of frame B with respect to frame A and so like in
here I this figure so these two frames they have coincident origins and their
origins will remain coincident for all the time and there is no linear relative
velocity between these two frames and there is a rotational velocity of frame
B with respect to frame A so frame B is always is rotating and so the rotation
matrix of frame B with respect to frame A is time where is always changing
and there is a point P is fixed in frame B so frame B actually it is attached on
a rigid body and the P is one of the point on the rigid body so P non P is
fixed and you see this is similar system is a similar to the fan we
attach from B on one of the fan blade so they have four blades so I attach the
frame B on the fan blade and then A is a fixed frame then we when the fan is the
blades is rotating so they have in this case they have a fixed rotational
velocity or maybe they also can change and P is like one of the point on the
blade so now if we if the fan is rotating different points imagine they
have so many points on the blades and then different points on the blades will
move at different linear velocities depends on the distance between the
point and the center of the rotation axis
you can understand this sentence look for we when we use linear velocity it
always used to describe a point but angular velocity used to describe a
rigid body the attribute of a rigid body so for this fan for the for the fan
blades they always have for the whole blades for the for the whole body the
angular velocity is the same for the linear velocity of the points on the
blades and they are different right and if some points is far away from the
center then the the points have bigger linear velocity similar to the car in a
larger radiance tire a larger tire rotating at the same angular velocity it
will generate greater generated bigger linear speed of the car so one of the
reason and why the big trunk have so big tire this is one of the reason because
then they can't make the trunk more faster but also to have big friction now
I would like to ask you do you know where is the point on this system or on
this system so which points have zero linear velocity you know the angular
velocity of the the whole rigid body they are same but the linear velocity
are different and there have some points have linear zero linear velocity do you
know which points
the points that lines on the axis of rotation right also in this figure the
points on the rotation axis they have zero linear velocity because there's no
distance from these points to the rotation axis and so now we know the
basic ideas let's see this one this figure is a little bit abstract and we
will understand it okay so let's pick a point P of a rigid body and this
rigid body is attached by frame B and the angular velocity measured above is
Omega B with respect to frame A and Omega itself it is a 3 by 1 vector
representing the direction and the magnitude of the angular velocity and
you see the rotation axis in here is perpendicular to the rotation plane so
it is perpendicular to the this line the line which connects the the axis this
center to point P and also you see this is 90 degree and here is also a 90
degree and then this is the linear velocity and now we P with respect to
frame A then we want to calculate the linear velocity based on the angular
velocity the given and because we also know now we gave the location of point P
so now we need to locate P to some frame and now we look at P at frame A and then
you see here is a theta in here and this this vector can be the position vector of
point P right it's also this also a 3 times 1 vector and this is 90 degrees this is
90 degrees so if we already know this position vector then we can get the
magnitude or the vector of this one the magnitude of this one right what's this
so this is sine theta of P with respect to A right and if we connect this to the
points to the origin of frame A then we also can get the VA the linear velocity
of P is also perpendicular to this position vector then we know some features
about this linear velocity it is the magnitude is proportional to the omega
the angular velocity so if the angular velocity the omega increase and the
linear velocity in here will also increase and also it is proportional to
the distance in here if the P is closer to this point and then P is in here and
then the linear velocity will smaller and so it also it is depends on the
distance this this distance is P with respect to A and sine theta and also we
have two perpendicular relation in here so now we want to get the result of
linear velocity P with respect to frame A now I give you these conditions then
can you write the equation of linear velocity P this vector is perpendicular
to this one and the perpendicular to this one and it is proportional to the
magnitude of the angular velocity and the magnitude of the distance so in mass
which operator can represent the relationship like this way
cross product all right so the linear velocity of P with respect to A is the
cross product of omega B the relative to A and the position vector with respect
to A so now we know we have this relation the linear velocity is the angular
velocity across the distance it is represented as a vector model in here so
the angular velocity is a three times one vector and also the position vector
is also three times one also the linear velocity is also three times one vector
but we want something we want some matrices then we can use linear algebra
to to calculate then instead of you writing a vector representation there
also have a matrix form so in this case we can write if C is the product of A
and B then C can write like A is known as a skew symmetric matrix then multiply B
so now then we can use this skew symmetric matrix to change from vector
to matrices and then we can write like this way so this matrix the three
diagonals are zero but there this matrix it is not really symmetric here has
negative the negative sign here but then in this form we can convert vectors to
matrices and then similar to our velocity case and then we can write the
omega B relative to A to skew the symmetric matrix format and then we can
get this now this is the angular velocity part you can see this is three
zero and then this is omega Z omega Y omega X then we can get linear velocity
okay so now we know how to calculate the linear velocity based on the given
omega velocity and the location of point P but all these results now they are
built on these three assumptions so now I delete the assumption three because at
the beginning we assume P is fixed in frame B but maybe like the fan there
have a small insect on the fan blades and now the insect is working on the
fan working on the blade now can you figure out how to update this
equation now the point P you hear the point P is fixed but now the point P is
also have the velocity right so we need to add this velocity together with this
part with the part from the angular velocity right so this is the velocity
of V P in B right and then we can get you see the part is in here this is V P
in B but then because the all the system all the frames are with respect to A so
we need a rotation matrix then can convert the velocity of P to a new
reference frame A then this is now the point P is moving but sometimes there
may be the B and A there have linear velocity of between B and A so then we
also need to add this linear velocity together I think we have learned how to
write this right how to get the linear velocity of a point
one part is the velocity of the point in the self and the other part is the
origin the velocity of the origin of frame B with respect to frame A right so
this is V B with respect to A then we will get the whole the general part of
this then this we call this is the simultaneous linear and angular motion
and in this slide this is the summary of the previous slides okay these are very
interesting hope you can quickly digest this information and in this video we
will learn the motion of the links of a manipulator