Exercise 1

Explain what a Rotation Matrix is, how it is derived and what are its uses in robotics. Explain and give example on the di↵erence between performing the compounding of Rotation Matrices with respect to the current frame or respect to the fixed frame. Finally, explain how we can describe the rate of change in time of a Rotation Matrix and how this is connected to the definition of the Angle and Axis representation. Use equations, schematics and examples. (maximum 500 words)

Exercise 2

Consider the RRR planar manipulator in Fig. 1, where L1  = 0.3, L2  = 0.2 and L3  = 0.1 and where angles ✓ 1 , ✓2 , ✓3  respectively belong to joint-1, joint-2 and joint-3. Noticing carefully the orientation of the end-e↵ector:

1. derive and explain each passage to compute the forward kinematic of this kind of manipulator using pure geometrical reasoning (i.e only using Homogenous Transformation Matrices),

2. derive and explain each passage to compute the forward kinematic of this kind of manipulator using Screw Theory,

3. following the solution with Screw Theory, then write a Matlab code that solves the Forward Kinematic and draws the workspace when ✓ 1  = 40◦ , ✓2  = 20◦  and −90◦  < ✓3  < 90◦ ,

4. finally solve the Forward Kinematic to draw the full workspace when the joint angles are allowed to span every possible angle, i.e. 0◦  < ✓ 1 , ✓2 , ✓3  < 360◦

L3 

L2

L1

joint-2

Figure 1: (From exercise 2) Schematic of 3R planar manipulator.

Exercise 3

Explain the meaning and usage of the screw  axis, helping yourself with drawings and schematics to support your explanation. Then solve the following:

1. Given a vector q~s  = (1.5, 0, 0) representing the distance between a frame {s} and a screw axis Ss  whose axis of rotation is s  = (0, 0, 1) and whose pitch is h = 2 mm/rad, draw this screw axis and the motion induced by it on {s} by ✓ = 0.3rad using Matlab. Support the code with suitable explanation of each step of the calculation.

2. Given a reference frame {b} represented by,

20  −1   0    2 3

Tsb  =6(6) 7(7)

40   0    0    1 5

compute the screw axis Ss expressed with respect to {b}, i.e. Sb using the adjoint operator AdTbs . Show this calculation by hand in your report and write a Matlab script to check your results.

3. Given a twist V = (0, 2, 2, 4, 0, 0), derive the associated screw axis and draw it using Matlab. Support the code with suitable explanation of each step of the calculation.

Exercise 4

Given an initial frame {s} and a Screw Axis S1  (expressed with respect to {s}) defined by a rotation unit axis 1  = (0.7, 0.0, 0.7), a vector q~1  = (1.0, 1.0, 0.5) and a screw pitch h = 1, write and upload a Matlab code that can perform the following operations:

1. Predict the new pose {b} of the original frame {s} after a rotation ✓ = 250◦  around screw axis S1  and plot the new frame {b}. Provide detailed written explanation of the various passages required to compute S1  and Tsb .

Figure 2: (From exercise 5) Schematic of an RPRP spatial manipulator.