Solution Manual for Introduction to Robotics Mechanics and Control 4th Edition Craig 0133489795 9780133489798
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Chapter 4 Solutions for Introduction to Robotics
1. • a accuracy
• α accuracy
• d accuracy
• θ both accuracy and repeatability
2. Use 3P =3 Pheel = [0 50 0]T . Since
Now, starting with (4.6), one follows a method of§4.4 to get
(The other solution would not be feasible for the human knee joint.)
3. Start with (4.17):
and regroup to get
from which two values of
1 can be found by
If a similar solution using (4.18) is performed then there will be four values for θ1, two of which are distinct and one being repeated. The latter is the correct angle.
4. The subspace has only five degrees freedom, which can result in either one of two scenarios for a general point. The first option is to meet the desired orientation and two of the three position cordinates. The second option yields the desired point coordinates but limits the orientations to those attainable through Z-X-Z Euler angle rotations where α is not a free choice.
5. Generate the x and y point coordinates by using random joint angles in the equations x = l1c1 + l2c12 + l3c123 y = l1
123.
Once a reachable point is found, whether or not it is in the dexterous workspace can be checked by using x3ORG and y3ORG in (4.14), where all values of φ in the range of -π to π (to some resolution) are used to find
x3ORG= x l3 cosφ
y3ORG = y l3 sinφ.
6. We have x
where
and y can take any values. A Z-X-Z Euler rotation defines, 0R, as
Substituting these values into 0R yields the subspace as
must be rotated into alignment with
[Henceforth
the solution is unchanged.]
9. (4.16) Solution is unchanged through "(other terms are zero)". After that it proceeds as follows.
1.866 = √ 2 2 s 2 3 = s3 + 1
2 2 2 s3 = 1.866 1 = 0.866
Since 180 < θ3 < 180, there are 2 sols:
10. (4.17) Since a1 = 0, we use Pieper’s method with r = k3 Substituting a, α, and d values yields
5.6390 = 4.4142+ 1.4142s3
s3 = 0.866
Since 180 < θ3 < 180, there are 2 sols: