# Library Work Cantos ```Christian Lawrence C. Cantos
METE260
Library Work (Robot Motion Analysis)
1. Discuss the forward and backward kinematics of a 2D Manipulator.
- Forward kinematics involves calculating the position of the end-effector given the joint
parameters namely the length of the links and the joint angles. In a 2D manipulator, two
methods can be used to obtain the position and orientation of the end-effector. For a simple
2-DOF manipulator with 2 revolute joints, a geometric solution approach can be utilized. For
much more complex manipulators involving a lot of joints, Denavit-Hartenberg method can
be utilized.
Figure 1
The position of the end-effector of the given figure is:
Px = l1CosꝊ1 + l2 cos (Ꝋ1+ Ꝋ2)
Py = l1sin Ꝋ1 + l2 sin (Ꝋ1+ Ꝋ2)
On the other hand, backward kinematics is the inverse operation of forward kinematics.
That is, given the desired position of the end-effector, the appropriate joint configuration
can be obtained. There are two methods to obtain the inverse kinematic equation:
analytical and numerical solution. The analytical method is used for manipulators with a low
number of DOF whereas the numerical method is used for complex manipulator
configurations.
Based on figure 1, the inverse kinematic equation used is:
2. Discuss the forward and backward kinematics of a 3D Manipulator.
- The forward and backward kinematics of a 3D manipulator used the same approach as a 2D
manipulator, but the z-axis must be taken into consideration when doing calculations.
For a 3D manipulator with two revolute joints, the forward kinematics equation is:
Px = l2cosꝊ1 cosꝊ2+ l3 cosꝊ1cos (Ꝋ2+ Ꝋ3)
Py = l2sinꝊ1 cosꝊ2+ l3 sinꝊ1cos (Ꝋ2+ Ꝋ3)
Pz = l1+ l2cosꝊ2 + l3 cosꝊ1cos (Ꝋ2+ Ꝋ3)
3. Differentiate the kinematics motion analysis of a 2D and 3D Manipulator.
- The forward and backward kinematic equation of a 2D compared to the 3D Manipulator is
much simpler. For a much more complex 3D manipulator, the equation in item no. 2 can’t
be used. Instead, Transformation matrices are applied. A numerical method is utilized when
dealing with 3D manipulators since there is no single approach that would describe the
configuration of the manipulator. That is, the equation involves a lot of solutions. Hence
complex manipulator configurations require a computer to obtain the equation both for the
forward and backward kinematic equation.
4. Explain Denavit-Hatenberg Convention in Kinematics.
- DH Convention in kinematics is a convention where each joint of the manipulator is
breakdown into four parameters, which each are taken with reference to the previous joint.
The 4 parameters are namely:
1. Distance between the previous and the current x-axis, along the previous z-axis.
2. The angle around the z-axis between the previous and current x-axis.
3. The distance between the previous and current z-axis, which is also called the
common normal.
4. The angle between the previous and current z-axis and the common normal.
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