Python Robotics Jacobian - read description

Robot Control Inverse Kinematics - Velocity control From the previous task you can acquire an estimation of the joint states and the location of the correct target. With this information we can construct a Jacobian on how the robot's state will change given changes to the joints. Therefore, through combining these two pieces of information, we can use a technique called Inverse Kinematics (IK) to tell the robot how to reach the desired state. Simply put, given the joint angles of the links, where is the end eec- tor's position, and how does the change in these angles aect the end eector's position minimise the error between actual and desired locations. This part will involve implementing IK of the four link robot. As stated in section A, two of the links rotate around the y-axis and the other two around the z -axis. In the labs you constructed Jacobians that used the rotation matrix for the z -axis, and here you will also need to use the rotation matrix corresponding to the y-axis. You should focus rst on implementing this in the velocity control mode. This can be done by setting the mode via: self.env.controlMode="VEL" The whole aim will to be to reach the right target and remain stable until the target (the star target) moves to a new location. Inverse Kinematics - Gravity compensated torque con- trol Once you have implemented IK, you will add gravity. This will mean that you will need to calculate the torques required to overcome gravity. Each link has a weight of 1kg. Uncomment line 34 to include gravity: [login to view URL]((0,0,-9.81)) This will set gravity to be in the z -axis 8 You should reimplement the IK as a task space proportional-derivative (PD) controller with gravity compensation to reach the same target using the torque mode: self.env.controlMode="TORQUE" To do this you will have to enable gravity by adding this before the loop in the go method: [login to view URL] gravity(True) It is best to rst nd the torques needed to move the robot to the correct position without gravity initially. Once the robot is performing IK successfully then enable gravity as above and add the additional torque required. You can then explore this system by adjusting the parameters of the PD controller to observe if you can speed up the system without reducing accuracy. Diculty with vision If you had diculty with the previous section, you may use the ground truths instead of your estimated values to test your controller. These values being:  [login to view URL] truth joint angles  [login to view URL] truth joint velocities  [login to view URL] valid target You will need to report if you are using these values instead. Expected outcome By the end your robot should be moving to the correct target. Not only this you should be able to report quantitatively:  Accuracy of the arm movement 9  Time it takes to reach the target  Dierences with and without gravity  The aects of the PD parameters For some of the above points it is perhaps best to use graphs to illustrate potential trends within the data. You can see how many times your arm reaches the valid target with: [login to view URL] And the number of times your robot goes for the invalid target with: [login to view URL] To get a measurement of time you should multiply the step size with the number of timesteps. In your code it should look like this: number of timesteps*[login to view URL] The output will be in seconds.