Inverse Kinematics

7. Recall that the “reachable” workspace is defined as the set of locations that the endpoint can reach without regard to the orientation of the end-effector. ( this is often called the “fingertip” workspace). The “dextrous” workspace is the set of locations the the endpoint can reach and for which any arbitrary end-effector orientation can be achieved. However, note that this definition of the dextrous workspace is often qualified such that it only really considers the appropriate orientations, that is, the orientations in the ‘subspace” of possible motion. So, for example. with a planar robot, we would generally not consider the fact that out-of-plane rotations are not possible; so, for this case, we would say that there 15 a dextrous workspace if there is a set of endpoint locations that that exist such that any end-effector z-axis rotation is possible. Note that determining the workspace of a robot can be oddly difficult and non-intuitive, especially when arbitrary joint limits are considered.

X et-
Consider the 2-link planar manipulator shown abovle. a. For the robot geometry L1=50 cm, L2=25 cm, draw the ‘reachable’ workspaces for each of the following cases:
I Case I Thetal Range I Theta2 Range I1 I 0 degrees — 360 degrees I 0 degrees — 360 degrees 2 I -90 degrees –+90 degrees I -90 degrees — 90 degrees I 3 I 0 degrees — 180 degrees II 0 degrees — 150 degrees Act”a+
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8. For the same arm as in #7:
a) Find the forward kinematics (e.g., the symbolic T matrix from the base to the end effector, given the numeric values for Ll and L2).
b) Find the inverse kinematic functions.
c) Compute the joint angle solutions for the values of X and Y in each of the cases listed below. Plot both solutions for case a. Do this using toolbox functions. You’ll need to understand the use of ikine! In particular, for this problem, you’ll need a mask vector. And – you’ll need to use appropriate solution guesses to generate both solutions.
Right-Handed Left-Handed Case X Y Theta 1 (deg) Theta 2 (deg) Theta 1 (deg) Theta 2 (deg) a 50 25 b 43.3 50 c 0 775 d -46.7 55.8
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Sample Solution