# Why using screws instead of homogeneous transforms in kinematics and dynamics?

Can anyone give a justification for using screws (twists, wrenches) instead of the traditional approach (rotation matrices, homogeneous transforms)? Even if screws are more compact, the situation gets complicated whenever we want to consider accelerations, so using screws in dynamics seems cumbersome for me.

According to Kevin Lynch in his video of Twists, "just like the time-derivative of a rotation matrix is not equivalent to the angular velocity, the time-derivative of a transformation matrix is not equivalent to the rigid-body velocity" (linear and angular).

Also he mentions that, instead, "any rigid-body velocity is equivalent to the instantaneous velocity around some screw axis".

Checking his book: Modern Robotics: Mechanics, Planning and Control can explain this in more detail.

About differences between Product of Exponentials (PoE) and Denavit-Hartenberg (DH) (from comments) :

Appendix C of the same book explains the difference between product of exponentials (PoE) and Denavit-Hartenberg (DH). For a more complete description go to the book.

Essentially:

• Once a home position for the robot is defined and the base and end effector frames set (which can be chosen arbitrarily), then the PoE formula is defined.
• No link frames are needed in PoE.
• PoE is supposed to be more natural and intuitive.
• DH needs to have link frames set in a way that DH parameters exist, they cannot be chosen arbitrarily.
• DH parameters can become ill-conditioned in some situations.
• PoE works with a joint value $theta$, while DH differentiates between prismatic and revolute joints, assigning different variables to it ($d$ and $theta$).

That said, one can choose to compute forward kinematics in both ways. It can also be proven that the PoE formula can be derived from the DH parameter representation.

Hope this helps.

• An equivalence to a velocity and clear physical interpretation maybe are not the most important for a computer program calculating kinematics. I have checked the book you recommended, and albeit the authors use an approach based on screws in kinematics, they admitted that Denavit-Hartenberg notation has the minimum number of parameters required to define the robot's kinematics. Still, I see no argument that completely convinces that the screws approach is better than the traditional one. – Kepler Aug 17 '18 at 3:48
• @Kepler I have expanded the answer with some bullet points extracted from Kevin Lynch's book. These are differences between DH and PoE. You can find them in appendix C of the book. – daniglezad Aug 17 '18 at 7:46
• All right, this is convincing. – Kepler Aug 20 '18 at 3:00