Homework 1 Introduction to Robotics Solution

What are the mechanical degrees of freedom of a standard, four-wheel, hand-pushed lawnmower like the one pictured? What are its environmental degrees of freedom? Why are you still able to mow an entire lawn?



What are the maximum non-redundant degrees of freedom for orientable robots on the X-Y plane?
(a) Calculate the angle between vectors ( 45, − 45, 0) and ( 45, 45,0) .



Provide a third vector that forms a coordinate system with the other two.



4. (a) Write out the entries of a rotation matrix

assuming basis vectors , , , and , , .
(b) Express = [0, 1, 0]




in frame {A}.



(c) Write out the entries of rotation matrix




.






Consider a tricycle with two independent standard wheels in the rear and a steerable, actuated front-wheel. Assume to be the radius of the front wheel and to be the distance between the front and rear axle. Chose a suitable coordinate system, use as the steering wheel angle, and ̇as angular velocity (only the front-wheel is driven). Provide the forward kinematics of the mechanism.



A robot using a local coordinate frame B detects an object Q at position (8, -4). In



coordinate frame A, the robot shows odometry readings of (6, 10, 135). Using a homogenous transform, find the position of Q in coordinate frame A.