Robotics Primer: Kinematics, Dynamics - Lecture 02

[00:00:00][Music] [Music] good morning today we have the lecture entitled robotics primer where we focus on kinematics and dynamics of robotic systems the outline of this lecture will be first we'll discuss on robot kinematics in that we'll be seeing both for kinematics with the examples and inverse kinematics with examples then we see the Dynamics of a robotic system in that again we see the classification of Dynamics which is for Dynamics and iners Dynamics along with the model based robot Dynamic control control coming to the kinematics what is kinematics kinematics is a study of motion considering the relationship between the robot joints and the cartisian positions reached by the robotic system it is classified into for kinematics and inverse kinematics in F kinematics we have the inputs being the joint angles and the output being the end effector position so there is only one to one relationship you give the joint angle definitely it will go to to one position whereas in the case of inverse kinematics we have the input being the end effector position and the output of the inverse kinematics is the joint angles so having seen the description of forward kinematics and inverse kinematics we can say that for kinematics is simpler as compared to that of the inverse kinematics for the control of robotic system inverse kinematic can be used now let's go deep into for kinematics as seen from the SCH schematic you can see that Q General coordinate of the robotic system it has three link robotic system for that we have the joint variables Theta 1 theta2 theta 3 so here as I mentioned we provide some joint angle and it reaches here that means it is earlier lying here this robotic system L1 L2 L3 and one we give Theta 1 theta2 theta 3 this one comes here to position a now for for kinematics the important analysis that we have to do is through DH parameters denavit and H andberg parameters they are four in numbers one is AI di TI and Alpha as the name indicates AI di are the link length are precisely the L length m length parameters and Alpha thet are the angular parameters when we say AI let us Define that what is ai ai is the distance along XI from oi to the intersection of x i and z i minus one axis similarly di is the distance along Z ius1 from o ius1 to the intersection of x i and z i minus one axis di is the variable if the joint I is Prismatic whereas in the case of revolute joint di can be a constant similarly Alpha I is angle between Z IUS 1 and Z I measured around zi axis finally thet is the angle between x ius1 and x i axis measured around Z ius1 and T is the variable if the joint I is revolute joint we can see through the schematics the representation of Alpha and thet you can see that between z i minus one and Zi measured around XI we can get Alpha whereas in the case of th the thet is angle around Z IUS 1 between Z IUS 1 and XI so now Theta 4 is an angle around Z3 similarly Theta 9 is an angle around Z8 okay between axis X8 and X9 in for kinematics through the DH parameters we must need to have the homogeneous transformation matrix which is given by a combination of rotation transformation rotation translation translation and rotation which means it's a combination of rotation and translation more accurately it is first having the rotation around z- axis by an amount th then translation Along z-axis by an amount di then translation Along xaxis by an amount a and finally rotation around xaxis by an amount Alpha I with this combination we can get the homogeneous transformation 4 cross 4 Matrix okay so finally having substituted what is rotation around Z axis by Theta and similarly for translation and rotation along around xaxis we have this for these four matrices combinedly they form this one the final Matrix where the last row 0 0 0 1 is the scale scale value that will help us in having the analysis based on DH parameters in order to have for n degrees of freedom robotic system in order to have the position and orientation of the Endor with respect to the base or the position and orientation of the anth frame with respect to the zero or base frame we can have the simplification that is aided by this scalar term or the scalar row placed in the homogeneous transformation matrix fourth row now we are going to see the algorithm for deriving for kinematics first we need to establish the joint axis of the given manipulator physical manipulator what are the joint axis and term them as Z Z1 up to ZN minus one and form the base frame such that it obeys the right hand screw rule label the origin of the base frame as o0 that means origin o and0 frame locate the origin Y at each joint I that means joint one will have the origin joint o1 joint 3 will have origin O3 then establish XI along the direction normal to the plane formed by the intersection intersecting Z axis Z IUS one and z i or along the common normal through y if the z- axis are parall to each other that means if they are parallel you can have in this way s XI if they are like this Z IUS one and Z you can have the XI parallel to this plane found by z i minus one Z then form y as per the right hand screw rule at each joint now coming to the example of robot manipulators towards the for kinematics we first take a two revolute jointed nonplanar robotic system and from 0 to one frame we can have the first table uh first row of the DH table and from zero from one to two frames we can have the second row of the DH table finally we can say that first row contributes to this homogeneous transformation matrix represented by T 01 and the second homogeneous transformation matrix for the second row is T12 finally t02 that is the position of the tip frame with respect to the base frame zero is given by the product of these two individual homogeneous transformation matrices and finally we get this from which we can say that the three elements of the last column represents the position of the in Defector of the robot and the Three cross 3 rotational Matrix will give the orientation of the end effect where from the 3 cross 3 we can find alpha beta and gamma of the end effect and here we can see the simulation of this non-planar two degrees of freedom robotic system this basically we have simulated this in matlb programming platform also we have done the for kinematics of the hand exos developed in our laboratory which is meant for grasping task and here we can see that in the previous case we have seen kinematics of a serial link here it is a kinematic of kinematic model of a closed loop link or a closed loop you can say which is a combination of both physical link or metallic link and the human being link because the human link is a part of the hand exoskeleton and hence we have two closed Loops coming into picture to form the kinematic model of the hand ex skeleton and we have tested the kinematic validation we have first simulated it with the kinematic model that we have made and then we have validated it we have moved the real system placed on the human hand and taken their trajectory that we could see that the two trajectories almost matches perfectly and hence our kinematic model is validated now coming to the inverse matics here we have the input is the ector pose which means the position and orientation of the ector and we need to find the joint variable Q Theta and D as the output that means given a position B in the cartisian space we need to move this end Vector from position A to B so at present we have Theta 1 Theta 2 theta 3 is a joint angular set we need to find a new joint set in order to reach B so that is the task of inos kinematics there are three ways we can do that one is through direct approach another one is to geometric approach and the third one is close loop inverse kinematics based which involves the jacoban Matrix computation the direct or algebraic approach has equating the final value of the ector post and orientation posst and IND posst Matrix homogeneous Matrix with the expression and by equating it with the nonlinear functions of the joint variables based homogeneous transformation matrices expressions and the values we can get the values of theta the another one is using the geometrical relationship between the joints and the links that is called geometric approach okay now coming to the Clos Loop inverse kinematics approach we can see that the block diagram of the close loop iners kinematics approach for iners kinematics is given here where given the desired angle or decide position of the cartisian position of the end Defector we need to have the joint angle formed by this close invers kinematics so that the robot tip can reach that XD so as for the block diagram we have the first block is the control gain and then we have the control law and that law will give Q dot which is the control input to the robotic system that is fed back to the robotic system in terms of simulation it is the for kinematic model in terms of the real system it will be a robotic system so the control gain is multiplied by the error and then added to the Joint inverse Okay so Jacobin inverse so we can say that Q do equal to J inverse into XD do+ KP into error that's a control law and this control law will lead to first order error Dynamics e do plus KP equal to Z okay this differentially equation will have the solution e of T which is going to zero As Time tends to Infinity okay so this approach of inverse kattic ensures that this is point a point B you need to reach from point A to point B through various way points so this is the trajectory of the actual robotic system so this one this C guarantees that the trajectory between the desired and the actual is zero almost because this law control law takes into or the invers kinematic C law takes into account the error between the cartisian actual position and the cartisian Desir position of the end of this is how we get the first order aerodynamic equation okay now now coming to the jacoban Matrix the relationship between the end effector velocities and the joint angular velocities is given by a jacoban matrix where the end effector velocity is represented by linear velocity and angular velocity combination which is equal to J into angular velocity or joint angle velocity The jacoban Matrix has two components in that it has two components one is Jacobin Matrix responsible for the position Jaan Matrix responsible for the orientation jacoban Matrix is of size in general 6 CR n where R stands for the row which is 6 degrees of freedom in the cartisian space and n stands for the colum which is the degrees of freedom of the robotic the N degrees of freedom robotic system will have 6 cross n so for four degrees of freedom Scara robot it will be 6 cross 4 will be the jaob Matrix the individual component each column of a robotic system or jacoban Matrix is given by JP ji J JPI and ji being Z and Z Vector for a prismatic joint because the orientation part will be zero here whereas for a re joint it is z i-1 cross P minus p i -1 where z i -1 z i - one are the third column of a rotational Matrix for IUS one frame and P is a end effector position and p i - 1 is the fourth column of the homogeneous transformation matrix for the frame IUS one so this Jacobian matrix it's called geometric ja Jacobian matrix where we represented the end Vector velocity as X do equal to 6 cross one vector which is a linear combination of the linear velocities X do y do Z Dot and instead of angle derivative we give the angular velocity as the velocity part then we can say that J Matrix is a geometri ja Matrix if we represent the end effect velocity in terms of alpha do beta dot gamma dot which means the orientation derivative then we can say that jacoban involved is analytical jacoban Matrix now the example for in kinematics is given for a Scara manipulator which has 4 deges of Freedom so when projecting the manipulator's posture onto the XY plane of this Scara manipulator we get from the two rute jointed robotic system easily we get theta2 joint angle which is given by this expression and from this Theta 2 we can find Theta 1 and then we can go for computing Theta 4 finally we can find the Prismatic joint variable D3 from the above computed joint angle Theta 2 Theta 1 and Theta 4 okay now coming to the Statics before we enter into Dynamics we just see what is Statics Statics means the study of force considering no motion of the robotic system whereas we consider only the forces an object mobile manipulator is in contact with the wall so there is no movement whereas there is force interaction so there we will be having Statics involved the goal of Statics is to determine the relationship between the general IED forces applied to the end of vector and the generalized forces appli to the joints forces for a prismatic jointed robotic system and talks for the reute joints with the manipulator at an equilibrium configuration and hence the relationship t equal to J transpose gamma where gamma is the tip Force to is the joint TOS of the robot now coming to the Dynamics what is Dynamics Dynamics is the study of motion considering the force associated with the robotic system here in this lecture we have seen three definitions including Dynamics first one is kinematics what is kinematics it is a study of motion considering no forces associated with the robotic system then we have the definition of Statics which means the study of forces when there is no movement of the robotic system involved then we see Dynamics Dynamics means the study of motion considering the forces associated with the robotic system the general highly coupled nonlinear dynamic equation of n degrees of freedom robotic system is given by t equal to M of theta into Theta dot plus v of thetaa theta dot into Theta dot plus G of theta where m is the inertia Matrix which is of size n cross n and V is called Coralis and centrifugal Matrix which is also n cross n size and G of theta is gravity Vector which is of size n cross one and obviously Theta double dot is a joint acceleration vector of size n cross one and Theta dot is a joint angular velocity of size n cross one so with this we have the left hand side joint TP which is also of size n cross one now the Dynamics has been classified further into forward Dynamics and inverse Dynamics forward Dynamics means the input is torque that's the input whereas the output is joint trajectory Theta double dot Theta Dot and Theta whereas in inverse kinematics inverse Dynamics we have the input being the joint trajectories and the output will be joint T forward Dynamics is used for simulation case simulation whereas inverse Dynamics you meant for controlling robotic system now having seen that inverse Dynamic model is going to be helpful for controlling robotic system let's see model based manipulator control system which means we have a dynamic model given by this expression if you assume friction also it becomes one more term which is friction then the model based manipulator control system we'll have the procedure as such the first one is we can have the partition control law for this Dynamic based robot control given by to equal to Alpha to dash plus beta let that be the partition control law for this case again it is given by tal Alpha t- plus beta where T is the n cross1 Vector of joint TOs and we choose Alpha being the mass Matrix or the initia Matrix M of theta and beta being the rest of the dynamic model which is given by V of theta comma Theta dot comma multipied by Theta dot plus G of theta plus F of thetaa theta Dot and we consider the servo law which is given by T Dash equal to thet D do plus KV into e do plus KP into e where KP KV are control gain matrices with e being the difference between decide joint angle and actual joint angle and E dot being the difference between Eed Theta D dot minus Theta dot so now we know the value of alpha we know what is T Dash and what is beta so substituting back into this Dynamic model that means the control law is now substituted back to the dynamic model equation two then we can have this second order error Dynamic equation which is given by E dot plus KV into e dot plus KP into e equal to Z which means that the error t to Zer As Time tends to infinity and the matrices k v and KP are diagonal matrices so if we go by joint by joint basis we have the error vectors coming out to be EI dble dot plus KV e do plus KP i e equal to Z when we have simulated for a given Theta D and Theta D dot and Theta D dot we can obtain these values for giving here as input from interpolation that means take Theta KN equal to say 20° and Theta final being say 45° so this is your desired angle and you have one initial angle for the robotic system say 20° between this and this you can have an interpolated trajectory from which we can get Theta of T Theta dot of T and Theta double do of T which can be given as input to this case thus in this lecture we have seen robot kinematics Statics and Dynamics with the concluding part as the model based robot Dynamic control as a short we can say that kinematics involving both forward and inverse kinematics has been seen have been seen with examples for both kinematics and forward kinematics and inverse kinematics as I want to mention that compared to inverse kinematics forward kinematics is a simpler one the problem or the challenges associated with inverse kinematics or in some situation we may get for one position of the end Defector we get several or multiple joint angles and we need to optimize which one is the best joint angle by considering the objective functions to minimize energy or to maximize manipulability measure then coming to the Statics we we have seen the expression of that and it is advised by me to read The Duality property of robot Statics to have the relationship between the end effector forces and the joint top Vector then we have seen robot Dynamics with forward Dynamics and inverse Dynamics involved in the model based control in the model based control given the input as joint trajectories Theta D Theta D dot Theta d double dot we are finding the joint torque Vector to by partition control law given by t equal to Alpha T Das plus beta where T Das is a Servo law thus we can involve both forward Dynamics and inverse Dynamics where is forward Dynamics involved forward Dynamics is involved in getting the feedback angles that means given the desired joint angle Theta D Theta D dot and Theta dble dot we can get the actual systems joint variables through forward Dynamics because the forward Dynamics receives the joint torque and we get the joint angle by inversing from the dynamic model thus we have seen three topics here in the robotics primer involving robot Dynamics and robot kinematics we have seen kinematics Statics and Dynamics along with control in this lecture the readers are requested to go through several books and example problems in order to understand these topics thank you very much [Music] w [Music]