WEBVTT - autoGenerated
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hey guys in this lecture we will expand our consideration of robot
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manipulators beyond static positioning problems because the robot is always
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busy and they're doing some interesting tasks so we have to discuss instantaneous
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kinemics so we will examine the notions of linear and angular velocity of a
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rigid body and then we will use these concepts to analyze the motion of a
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manipulator through velocity propagation it turns out that the study of
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velocities leads to a matrix called Jacobian matrix another very important
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matrix comes after we have learned rotation matrix and transformation
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matrix and the Jacobian also will be used in dynamics and the robot control
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and two lectures the Jacobian matrix will show up in some important dynamics
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equations and then I will use your file to show you singular configurations and
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you will know what is singularity so this lecture is also a little bit
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difficult because there are a lot of equations and math so this lecture
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depends on the previous four lectures it depends on the transformation matrix and
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the relationship between the Cartesian space and the joint space so I hope I
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can successfully guide you to cross through this lecture and then after this
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lecture you can survive a little bit okay so let's start in forward kinematics
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we learn how to use the joint angles the joint configuration in joint space to
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calculate the location of the energy factor in the Cartesian space right so
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this is the relation from joint space to Cartesian space and then we learn
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inverse kinematics we can given the configuration of the energy factor and
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then we can get the joints in the joint space so this is the relationship from
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Cartesian space to joint space and then if we make a very small displacement of
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theta in joint space then we will get this sort of data theta that we are
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introducing to each of those joint angles and then we are going to have a
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small displacement of called data X so in Cartesian space another question is
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what is the relationship between data theta and data X and you know it is not
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only data X in position but also in orientation this is a location of the
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energy factor we need to consider these two parts so when we talk about the
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relationship between data theta and the data X if the displacement tends to be
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zero it's very small very quick this is the instantaneous kinematics then we're
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going to discuss the differential motion the velocity of the joints and the
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velocity of the end effector so dot theta means the time derivative of the
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joint theta and dot X means the time derivative of the location of the
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end effector so now we want to find the relationship between data theta and
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data X always say we want to find the relationship between the velocity of the
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joint and also the end effector velocities and the answer is we will
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use a linear relationship to connect these two and the the linear
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relationship is Jacobian so before we talk about the Jacobian the Jacobian
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matrix which can connect the joint velocity and any effector velocity
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connects the joint space and the Cartesian space we'll start by looking
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at the differential motion and we're going to discuss the velocities how to
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calculate the velocity of rigid body calculate the velocity of the end effector
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and the joint and then remember our representation the representation of X
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involves the position and orientation so that we will discuss linear velocity and
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angular velocity of the joint and the end effector okay so now let's start to
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look at the velocity first the first one is a linear velocity you see here is
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the trajectory and there is a point P in space and then a P is time varying so we
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can use a time varying the position vector to represent the location of P
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in space then one time is t0 then we can get the P of t0 and then after data t
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time then we can get P of t0 plus or data t and the P is with respect to frame A
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so if we want to calculate the linear velocity of a point in space then we
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need to calculate the velocity of the position vector PA so do you get me so the
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velocity of the linear velocity of a point is the velocity of a position
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vector so if you want to calculate a velocity the linear velocity of a point
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then you need to get the position vector of this point then you can get the time
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derivative of the position vector and then you can get the linear velocity so
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this one equals to the limit of delta P divided by delta t by delta t tends to 0
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then we can get this stuff so this is the basic definition of the linear
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velocity now we are presenting the velocity P in frame A if we present the
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velocity another frame B then we can use the rotation matrix to accomplish
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the change in reference frame and in this case we assume the frame A and B
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the configuration between them are fixed so R B A remains invariant during the
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motion of point P then now we calculate we P in B then we still can go back to
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the time derivative of the position vector so we want to change P A to P B
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then we need the rotation matrix in here but because the rotation matrix is time
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independent so we can put the rotation matrix out of the derivative part then
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we can get the result is the rotation matrix multiply the velocity but rather
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than considering a general point velocity relative to an arbitrary frame A
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or B we will often consider the velocity of the origin of a frame relative to some
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understood universal reference frame some the map frame or the word frame for
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example if P is the origin of a frame C the origin of a frame and P is moving
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we typically we will use the small V that we see to denote that the linear
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velocity of the origin of frame C with respect to reference from U so you it's
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like fix the universal frame like word or map and if we use we see in A means
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the velocity of origin of C with respect to U expressed in A maybe it's a
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little bit confusing and we will see some example later okay so now let's see
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the angular velocity when we use linear velocity we always use it to describe
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the attribute of a point but we use angular velocity to describe an
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attribute of a rigid body we always attach some frame a frame on a rigid
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body then we can analyze the configuration of this rigid body so
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basically angular velocity describes rotation motion of a frame of a body and
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we use Omega to represent angular velocity Omega B in A means the angular
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velocity of frame B respect to frame A and also there have a shorthand notion
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called the small Omega the small Omega C it means the angular velocity of C with
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respect to some reference frame U and in at any instant the direction of Omega B
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in A indicates the instantaneous axis of rotation the rotation axis is the
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direction of Omega and the magnitude you see the vector the magnitude of Omega B
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in A indicates the speed of rotation so angular velocity is also a vector is a
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three times one vector and the position vector is also three times one vector
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and they have the direction and also magnitude okay so now we introduce the
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notions notations for time varying position and orientation these two
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velocities and then we'll investigate the description of motion of rigid body
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so we always touch a coordinate system to any rigid body that we wish to
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describe and then the motion of rigid bodies can be equivalent started as the
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motion of frames relative to one another so now we will analyze the velocity of
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rigid body so actually we are analyzing the velocity between frames so for
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example now we start from linear velocity now if there is a fixed frame A
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and there is a frame B and frame B attached to a sound rigid body in this
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example from B is the frame of this train and assume this train is driving
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on a street railway the driver now needs to use the steering wheel it's
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a straight railway and there is a passenger Q is working on the train in
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a fixed orientation so what's the velocity of the passenger respect to the
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frame A then we want to know the linear velocity of the passenger with respect
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to the fixed the frame and requiring the velocity we can start from the
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position that we can get the position vector and then we can get the time
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derivative of the position then we can get a linear velocity right so now this
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is from A and from B okay now this is point P not point Q you can think this
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is the passenger in here and if we want to know the velocity of P in A then now
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we first we want to get the position vector of P in A and this depends on two
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parts the first part is we need to know the position vector from the origin of A
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to the origin of B so here is a vector P B in A now we know the position vector
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of these two origins and then we need to know the from P we know P B we know the
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vector position vector in frame B so then we from P B in A we plus P B P in B
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but we want to represent in frame A so we need to use the rotation matrix being
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A multiply P B right and in this case we only assume there only have a linear
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motion of a B respect to A so there is no rotation relative rotation between A
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and B and also P not P is moving also P is only have linear motion then now we
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want to get the linear velocity so we will use a differentiation to get the
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velocity now this is a time derivative okay and then so the okay to click the P
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the velocity of P in A is a velocity of B in A and then we want to get the time
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derivative of this part we have seen this part before right so the rotation
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matrix is time independent so we can get the rotation matrix out of the derivative
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part then we can get the velocity so the linear velocity of P with respect to
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frame A is the sum of the velocity with respect to frame B and the velocity of
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origin of frame B with respect to frame A and here we only discuss the
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translational motion of the frame and but if they also have a rotation of this
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frame then since will become more complicated then the angular velocity
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comes in order to make sense simpler now we assume three conditions the first one
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is there is no linear velocity of frame B with respect to frame A and so like in
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here I this figure so these two frames they have coincident origins and their
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origins will remain coincident for all the time and there is no linear relative
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velocity between these two frames and there is a rotational velocity of frame
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B with respect to frame A so frame B is always is rotating and so the rotation
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matrix of frame B with respect to frame A is time where is always changing
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and there is a point P is fixed in frame B so frame B actually it is attached on
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a rigid body and the P is one of the point on the rigid body so P non P is
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fixed and you see this is similar system is a similar to the fan we
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attach from B on one of the fan blade so they have four blades so I attach the
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frame B on the fan blade and then A is a fixed frame then we when the fan is the
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blades is rotating so they have in this case they have a fixed rotational
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velocity or maybe they also can change and P is like one of the point on the
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blade so now if we if the fan is rotating different points imagine they
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have so many points on the blades and then different points on the blades will
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move at different linear velocities depends on the distance between the
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point and the center of the rotation axis
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you can understand this sentence look for we when we use linear velocity it
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always used to describe a point but angular velocity used to describe a
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rigid body the attribute of a rigid body so for this fan for the for the fan
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blades they always have for the whole blades for the for the whole body the
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angular velocity is the same for the linear velocity of the points on the
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blades and they are different right and if some points is far away from the
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center then the the points have bigger linear velocity similar to the car in a
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larger radiance tire a larger tire rotating at the same angular velocity it
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will generate greater generated bigger linear speed of the car so one of the
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reason and why the big trunk have so big tire this is one of the reason because
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then they can't make the trunk more faster but also to have big friction now
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I would like to ask you do you know where is the point on this system or on
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this system so which points have zero linear velocity you know the angular
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velocity of the the whole rigid body they are same but the linear velocity
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are different and there have some points have linear zero linear velocity do you
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know which points
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the points that lines on the axis of rotation right also in this figure the
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points on the rotation axis they have zero linear velocity because there's no
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distance from these points to the rotation axis and so now we know the
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basic ideas let's see this one this figure is a little bit abstract and we
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will understand it okay so let's pick a point P of a rigid body and this
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rigid body is attached by frame B and the angular velocity measured above is
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Omega B with respect to frame A and Omega itself it is a 3 by 1 vector
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representing the direction and the magnitude of the angular velocity and
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you see the rotation axis in here is perpendicular to the rotation plane so
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it is perpendicular to the this line the line which connects the the axis this
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center to point P and also you see this is 90 degree and here is also a 90
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degree and then this is the linear velocity and now we P with respect to
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frame A then we want to calculate the linear velocity based on the angular
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velocity the given and because we also know now we gave the location of point P
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so now we need to locate P to some frame and now we look at P at frame A and then
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you see here is a theta in here and this this vector can be the position vector of
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point P right it's also this also a 3 times 1 vector and this is 90 degrees this is
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90 degrees so if we already know this position vector then we can get the
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magnitude or the vector of this one the magnitude of this one right what's this
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so this is sine theta of P with respect to A right and if we connect this to the
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points to the origin of frame A then we also can get the VA the linear velocity
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of P is also perpendicular to this position vector then we know some features
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about this linear velocity it is the magnitude is proportional to the omega
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the angular velocity so if the angular velocity the omega increase and the
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linear velocity in here will also increase and also it is proportional to
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the distance in here if the P is closer to this point and then P is in here and
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then the linear velocity will smaller and so it also it is depends on the
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distance this this distance is P with respect to A and sine theta and also we
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have two perpendicular relation in here so now we want to get the result of
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linear velocity P with respect to frame A now I give you these conditions then
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can you write the equation of linear velocity P this vector is perpendicular
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to this one and the perpendicular to this one and it is proportional to the
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magnitude of the angular velocity and the magnitude of the distance so in mass
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which operator can represent the relationship like this way
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cross product all right so the linear velocity of P with respect to A is the
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cross product of omega B the relative to A and the position vector with respect
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to A so now we know we have this relation the linear velocity is the angular
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velocity across the distance it is represented as a vector model in here so
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the angular velocity is a three times one vector and also the position vector
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is also three times one also the linear velocity is also three times one vector
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but we want something we want some matrices then we can use linear algebra
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to to calculate then instead of you writing a vector representation there
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also have a matrix form so in this case we can write if C is the product of A
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and B then C can write like A is known as a skew symmetric matrix then multiply B
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so now then we can use this skew symmetric matrix to change from vector
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to matrices and then we can write like this way so this matrix the three
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diagonals are zero but there this matrix it is not really symmetric here has
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negative the negative sign here but then in this form we can convert vectors to
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matrices and then similar to our velocity case and then we can write the
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omega B relative to A to skew the symmetric matrix format and then we can
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get this now this is the angular velocity part you can see this is three
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zero and then this is omega Z omega Y omega X then we can get linear velocity
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okay so now we know how to calculate the linear velocity based on the given
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omega velocity and the location of point P but all these results now they are
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built on these three assumptions so now I delete the assumption three because at
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the beginning we assume P is fixed in frame B but maybe like the fan there
00:26:04.000 --> 00:26:12.000
have a small insect on the fan blades and now the insect is working on the
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fan working on the blade now can you figure out how to update this
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equation now the point P you hear the point P is fixed but now the point P is
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also have the velocity right so we need to add this velocity together with this
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part with the part from the angular velocity right so this is the velocity
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of V P in B right and then we can get you see the part is in here this is V P
00:26:56.000 --> 00:27:03.000
in B but then because the all the system all the frames are with respect to A so
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we need a rotation matrix then can convert the velocity of P to a new
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reference frame A then this is now the point P is moving but sometimes there
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may be the B and A there have linear velocity of between B and A so then we
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also need to add this linear velocity together I think we have learned how to
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write this right how to get the linear velocity of a point
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one part is the velocity of the point in the self and the other part is the
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origin the velocity of the origin of frame B with respect to frame A right so
00:27:57.000 --> 00:28:03.000
this is V B with respect to A then we will get the whole the general part of
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this then this we call this is the simultaneous linear and angular motion
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and in this slide this is the summary of the previous slides okay these are very
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interesting hope you can quickly digest this information and in this video we
00:28:24.000 --> 00:28:28.000
will learn the motion of the links of a manipulator