User:Ckalafatis24/sandbox

You are working for Amazon and need to go on a break. The company has developed some robots that can do basic tasks, so while you are gone, it can do some of the heavy lifting for you. The robot is a mobile robot with a 3-revolute joint arm which is similar to the sketch of one in the figure.

We want to find the end-effector velocity from the joint positions and rates slightly before the robot picks up a certain package. In order from link 1 to 3, the lengths are 4 $$m$$, 2.5 $$m$$, and 3 $$m$$, the angles ($$\theta$$) with respect to the x-axis or the floor are 30 degrees, 45 degrees, and -50 degrees, and the joint velocities are 2 degrees/sec, -2 degrees/sec, and -6 degrees/sec, respectively.

First, we needed to form the transformation matrices between each joint to the next and post multiply them, starting with the last transformation matrix and the end coordinates. This gave us the change in position coordinates.

X: 4.0*cos(theta1) + 2.5*cos(theta1 + theta2) + 3.0*cos(theta1 + theta2 + theta3)], [4.0*sin(theta1) + 2.5*sin(theta1 + theta2) + 3.0*sin(theta1 + theta2 + theta3)], [0], [1 Then we had to differentiate each coordinate with respect to each joint angle, symbolically, and have their numerical joint angle values substituted, which needed to be done before inserting them into the Jacobian Matrix.

J: -5.68266935094477 -3.68266935094477 -1.26785478522210] [6.83007258900401 3.36597097386625 2.71892336110995 The Jacobian Matrix was then dotted with the joint angular velocities which gave us our final result for the End-Effector velocity components in $$\frac{m}{s}$$. As you can see, the x-component is positive (0.0630 $$\frac{m}{s}$$ ) and the y-component is negative (-0.1638 $$\frac{m}{s}$$ ). This is expected, since we want the robot to inch forward while descending its arm down to the package to then clamp onto it.

f0_v_4: 0.0629562725559737] [-0.163805030948942