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Homework 1 Introduction to Robotics

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Homework 1
Introduction to Robotics
1. What are the degrees of freedom of a standard, four-wheel, handpushed lawnmower? Why are you still able to mow your entire lawn?
2. What are the maximum degrees of freedom for objects driving on the
plane?
3. (a) Calculate the angle between vectors (cos45o
, −sin45o
, 0)T and
(sin45o
, cos45o
, 0)T
.
(b) Provide a third vector that forms a coordinate system with the
other two.

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Homework 1
Introduction to Robotics
1. What are the degrees of freedom of a standard, four-wheel, handpushed lawnmower? Why are you still able to mow your entire lawn?
2. What are the maximum degrees of freedom for objects driving on the
plane?
3. (a) Calculate the angle between vectors (cos45o
, −sin45o
, 0)T and
(sin45o
, cos45o
, 0)T
.
(b) Provide a third vector that forms a coordinate system with the
other two.
4. (a) Write out the entries of a rotation matrix A
BR assuming basis
vectors XA, YA, ZA, and XB, YB, ZB.
(b) Express XˆB = [0, 1, 0]T
in frame {A}.
(c) Write out the entries of rotation matrix B
AR.
5. Consider a tri-cycle with two independent standard wheels in the rear
and the stearable, actuated front-wheel. Assume r to be the radius
of the front wheel and l be the distance between the front and rear
axle. Chose a suitable coordinate system and use φ as the steering
wheel angle and wheel-speed ω˙ (only the steared front-wheel is driven).
Provide the forward kinematics of the mechanism.
1

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