Assignment #3

Due Wednesday, February 4

1) Here is a schematic of the arm used on space shuttles:

shuttle_arm.png

Write the forward kinematics that position the camera with respect to the shuttle frame, using homogeneous transform matrices. I see the following revolute joints: shoulder pitch joint, shoulder yaw joint, elbow pitch joint, wrist pitch, wrist yaw, wrist roll. Let the configuration of these be \theta_1 \ldots \theta_6, and let the lengths of the links be \l_1 \ldots \l_6. You will write out six homogeneous transform matrices, in terms of angles and lengths, but do not have to multiply out the matrices. Label each matrix clearly; for example, ^0_1T, etc.

You do not have to multiply out the matrices, but do describe (or give the equation) that determines the pose of the camera given the joint angles and measurements.

2) Here's a planar 4R arm:

arm_4R.png

a) Estimate the Jacobian at the current configuration. Don't do any computations; just eyeball the configuration and write down your estimate. (You can use a ruler if you like, but an estimate is fine.)

b) Now assume (\dot \theta_1, \dot \theta_2, \dot \theta_3, \dot \theta_4) = (1, 2, 1, 1). What is your estimate for (\dot x, \dot y)?

3) Spiderman's coordinates in the picture below are (x, y). (He's a point spider.) His configuration is (l_1, l_2), the lengths of the two strands of spider silk. You may assume that the two points where the strands connect to the buildings are the same height, and choose coordinates so that this height is zero. Choose units so that the distance between the two buildings is 1/2.

spider.png

a) Write the forward kinematics of spiderman, that, given silk strand lengths, compute his location. Big hint from MathWorld: the intersection of two circles, one of which has a center at the origin, and the other of which has a center at the coordinates (d, 0), with radii R and r respectively has x coordinate

(1)
x= \frac{d^2 - r^2 + R^2}{2d}

and y-coordinate

(2)
y = \sqrt{\frac{4 d^2R^2 - (d^2 - r^2 + R^2)^2}{4d^2}}.

b) Write the general differential kinematics of spiderman, that gives the mapping from (\dot l_1, \dot l_2) to (\dot x, \dot y) in terms of l_1, l_2. Full credit will only be given if you write it in matrix form.

c) Write the inverse kinematics of spiderman, so that given his desired location, he can compute the desired lengths of spider silk. Hint: inverse kinematics aren't difficult for parallel systems like this. Look at the geometry rather than trying to invert the forward equations.

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