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- 1. MAE 493/593 Mathematical Methods in Robotics Final Project Bryan Jones Jared Kuhl Wenxuan Zhai
- 2. Serial Chain: Our serial chain manipulator was an PRPR. Using D-H parameterization we decided on having technically having 3 links. D1, D3, and R4. R2 was combined into D3 as the minimum length for D3. The figure below displays our intial configuration. Phi R4 D3 Theta 4 Theta 2 D1 Y X Theta 2 and Theta 4 are also represented in the D-H parameterization form, and then there is the end effector angle phi. Forward Kinematics: Analytically, they were extremely easy to solve, as long as the joint variables were known. Inverse Kinematics: In order to solve for the inverse kinematics we used the matlab function, FMINCON along with ODE45 to solve for the independent joint variables. In order to solve these equations completely we had to hold the joints steady. The inverse kinematics were needed to solve for the tracjectory path planning part of our program. As a result, when testing the program to get an accurate tracjectory, we would switch the values of what joints we were holding steady so that the manipulator would already be in the range of the tracjectory. Otherwise, which you can see in figures below, the final path would not follow the initial path closely at all.
- 3. Figure 1: An example of a correct path planning. The star region represents the outermost region of the workspace. The circles represent the path of the end effector. As shown here, the manipulator was already in the range of the path that was planned. Our final path matches the the initial path that was placed on the screen. This also agrees with the results of FCONMIN, which when close to zero, means that the planned path matches the initial path nicely.
- 4. Figure 2: An example of bad path planning. The trail of circles represents the path of the end effector, where as the smoother line represents the initial target. In this case, our path lied in a portion of the workspace we could not reach due to our current configuration. By changing which joint variables and where they are fixed we can change the area where the path planning comes out correctly. Our inverse kinematics involved using FMINCON to figure out the initial joint variables, and then taking the data from a spline that was fitted to the path to integrate a velocity function using ODE 45. This can be seen in the Kinematics.m code file.
- 5. Figure 3: We are capable of tracing a circle or any other type of line given to us. Redundancy: Using ODE45 allowed us to solve for all variables, thus eliminating the problem of redundancy. The derivative data from the spline Parallel Chain: For the parallel manipulator there were 3 chains consisting of PRR. We used the technique of naming the links by 1-1,1-2,1-3,2-1,2-2,2-3.. etc. We labeled the angles in the same format. The lengths of the
- 6. primastic joints were kept at default as 2 and the link lengths were kept at 1. The equations wer e developed from the diagram below. 3-1 3-2 1-1 3-3 1-2 2-2 2-3 1-3 2-1 The forward kinematic equations were done analytically below. Xb2 and Xb3 represent the coordinates of the fixed positions. Inverse Kinematics. The inverse kinematics were done in the same way as the serial chain manipulator. However, the results wer e not as good. The path planning was not nearly as accurate and was confined to much more limited range. Our visual results show a disjointed path that is very hard to follow, however our indicator from FMINCON shows a result approximately at zero.
- 7. The redundancy was taken care of the same way as the serial manipulator. Specifically we had to change from absolute to relative coordinates. GUI: The gui has the capability to switch between the serial chain and parallel chain manipulator. Also a capability with the GUI is the ability to change link lengths. The other option are to show the work space of the respective option and to generate a trajectory by clicking on the screen to create a spline that will allow the program to run the inverse kinematics.

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