UBC BMEG $400$F $2021$W$2$ Assignment $2$ Instructions Due Sunday, $06-$Feb$-22$ at $11\mathrm{.}59$ PM$.$ This is a graded homework assignment, worth $15\%$ of the class grade. Do not discuss the problems with anyone else other than the instructor or the TA$.$ You can discuss clarifications with other students or work with them, but do not share or discuss solutions. Submit the assignment as a single pdf file. You can use software such as Word or LaTeX, scan written work, or a combination of the two. Total: $15$ points $1.$ Introducing the Selective Compliance Articulated Robot Arm $($SCARA$).$ This $4$ degreeof freedom pick$-$and$-$place robot RRPR design is a design template used by many manufacturers $($Fig$.1$A$).$ Figure $1$: $($A$)$ Epson SCARA $($B$)$ Kinematic Diagram of SCARA The kinematic diagram with assigned D$-$H axes is shown in Fig. $1$B$.$ Frame $0$ is the base frame, and frame $4$ is the end effector frame. For simplicity, we are going to set many of the offsets that exist in a real$-$world robot to be zero. For instance, the offset from the base to the elbow, d$1=0,$ and the offset from the last revolute joint to the end effector, d$4=0$ The D$-$H parameters are: Link ri ai di $?$i $1$ L$100?12$ L$2$ p $0?2300$ d$304000?4$ The corresponding forward kinematics matrix is given here $($although you will have to redo some of this work for part $($e$).1$ of $2$ UBC BMEG $400$F $2021$W$2$ Assignment $2????$ c$12$c$4+$ s$12$s$4-$c$12$s$4+$ s$12$c$40$ L$1$c$1+$ L$2$c$12$ s$12$c$4-$ c$12$s$4-$s$12$s$4-$ c$12$c$40$ L$1$s$1+$ L$2$s$1200-1-$d$30001????($a$)(1$ pt$)$ Look at the translation component of the forward kinematics. Why does this look familiar? Can you figure out these terms just by looking at the structure of the manipulator? $($b$)(1$ pt$)$ Based on this intuition, what is the shape of the reachable workspace of the end effector? $($assuming L$1>$ L$2)($c$)(1$ pt$)$ Based on this intuition, write down the inverse kinematics equations of the robot. i$.$e$.$ given $($x$,$ y$,$ z$)$ return possible solutions for the three joint variables. $(?4$ does not matter since it does not affect end effector position$).($d$)(2$ pt$)$ Find the linear velocity Jacobian $(34)$ of the robot. $($e$)(3$ pts$)$ Find the angular velocity Jacobian $(34)$ of the robot. $($f$)(1$ pt$)$ Consider only the first three joint variables $($including $?4$ will make the Jacobian non$-$square and it does not affect position anyway$),$ what are the singularities of the robot? There are two singularities that are not obvious from $|$J$|=0.$ What are they? $2.$ Write a program that makes the end effector of this robot move in a cubic polynomial trajectory from initial position p$0$ to final position pf over time tf $,$ using the algorithm discussed in class. The program should take p$0,$ pf and tf as its inputs, and return a set of joint variables qi$($t$),$ desired end effector positions p$($t$),$ and actual end effector positions x$($t$)$ as outputs, where t are equally spaced time intervals $?$t from $0$ to tf $,$ e$.$g$.?$t $=0\mathrm{.}01$ s$.$ Remember that you have to first drive the robot differentially from its zero configuration to the starting point p$0.$ Alternatively, you can use inverse kinematics to start the robot at p$0.$ MATLAB or Python preferred. Please clear it with the instructor if you plan to use another language. $($a$)(3$ pts$)$ Include the source code for the file. You can add them as pages in the solution pdf$.($b$)(1$ pt$)$ Plot two examples of the robot moving from an initial position to a final position. Specifically, plot the desired polynomial trajectory as a dashed line and the actual trajectory as a solid line. You dont have to animate the joints and links of the robot $($although that would be fabulous!$).($c$)(1$ pt$)$ Would this program work if you try to input any two points within the workspace of the robot? In what conditions would it fail? $($d$)(1$ pt$)$ Increase the sampling frequency $($decrease $?$t$).$ What happens to the desired trajectory vs$.$ actual trajectory? $2$ of $2$