What does it mean to say that the coordinate axes are decoupled?

ENGR 2120: Dynamics Project #1b

Project Assignment The robot you are analyzing is a general purpose robot, and it is important to understand the position, velocity, an acceleration of different points on the machine. Please complete the following tasks for your project:

1. Create a new section in your report titled “General Curvilinear Motion.” In this section, explain what curvilinear motion is and how it is different from rectilinear motion.

2. Create a sub-section in the general curvilinear motion section entitled “Rectangular Coordinate Systems.” Within the rectangular coordinate systems section, please answer the following questions in paragraph form. Be sure to provide a clear explanation for each part.

a. What is a rectangular coordinate system and why do we use them in dynamics? b. Where do you place the origin of a rectangular coordinate system? c. What does it mean to say that the coordinate axes are decoupled?

3. Add a rectangular (Cartesian) coordinate system with the origin at the center of the revolute joint. The 𝑥𝑥-axis should be pointing to the right, and the 𝑦𝑦-axis should be pointing upward. This is a standard rectangular coordinate system. Please simply update the figure that is already posted on D2L. You do not need to re-create the figure from scratch. You will need to include this figure in your report. Imagine now that you are an engineer working for a company that builds robots to perform robotic welding tasks. The figure below shows the robotic welding trajectory as well as the welding point. The welding trajectory is typically specified in terms of Cartesian coordinates, but the machine is typically controlled by the variables 𝑟𝑟 and 𝜃𝜃. Thus, you will need to perform a transformation to express the position, velocity, and acceleration of the end-point in terms of 𝑟𝑟, 𝜃𝜃, and their derivatives. You can ignore any distance between the end-point of the manipulator and the actual welding tip.

4. Derive an expression for the position of the end-point of the machine as expressed in your

rectangular coordinate system. Keep your answer in variable form. It should be expressed in terms of 𝑟𝑟 and 𝜃𝜃.

𝑟𝑟 = _________𝚤𝚤̂ + ___________ 𝚥𝚥̂

5. Derive an expression for the velocity of the end-point of the machine as expressed in your rectangular coordinate system. Keep your answer in variable form. It should be expressed in terms of 𝑟𝑟, 𝜃𝜃, �̇�𝑟, and �̇�𝜃.

�⃗�𝑣 = _________𝚤𝚤̂ + ___________ 𝚥𝚥 ̂

6. Derive an expression for the acceleration of the end-point of the machine as expressed in your rectangular coordinate system. Keep your answer in variable form. Is should be expressed in terms of 𝑟𝑟, 𝜃𝜃, �̇�𝑟, �̇�𝜃, �̈�𝑟, and �̈�𝜃.

�⃗�𝑎 = _________𝚤𝚤̂ + ___________ 𝚥𝚥̂

7. Document your results from steps 3-6 in your report.

8. Assuming that 𝜃𝜃 = 55∘ and 𝑟𝑟 = 4 m, determine the magnitude of the position vector at this instant. Please provide the numeric value in your report (with units).

9. Assuming that 𝜃𝜃 = 55∘, 𝑟𝑟 = 4 m, �̇�𝜃 = 5 rad/s and �̇�𝑟 = 10 m s

, determine the speed of the end-point at this instant. Please provide the numeric value in your report (with units).

10. Assuming that 𝜃𝜃 = 55∘, 𝑟𝑟 = 4 m, �̇�𝜃 = 5 rad/s, �̇�𝑟 = 10 m s

, �̈�𝜃 = 5 rad/s2, �̈�𝑟 = 10 m s2

, determine the magnitude of the acceleration of the end-point at this instant. Please provide the numeric value in your report (with units). Be sure to document steps 8-10 in your report.

What to Turn In Please turn in your assignment packet that contains the following stapled from top to bottom:

1. Assignment cover sheet for Part 1 of the project 2. A copy of your project report that includes parts 1-6 from project part 1a and parts 1-10 from project

part 1b.