Series-parallel dualities on planar robots FAQ
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This module corresponds to Chapter 4 of Duffy's book on dualities of parallel and serial devices. Some material introducing the principle of virtual power will be provided in class, to better understand such chapter. Additional material on duality diagrams will also be provided in class.
There are some notions whose meaning is only implicit in the book. This section tries to clarify them.
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Frequently-asked questions
What are twists and wrenches?
At several places in this chapter, J. Duffy uses the words "twist" and "wrench" without introducing them. In the context of this book a twist is a rotor, and a wrench is a force. Perhaps unconsciously, Duffy uses the more general notions of twist and wrench that arise in the kinetostatic analysis of spatial linkages, when to be coherent with previous chapters it might have been better to use the terms rotor and force throughout.
What is reciprocity?
A force 
is said to be reciprocal to the rotor 
of a lamina, if such force generates no instant power under such rotor, i.e., if 
.
From a physical standpoint, this means that the force cannot change the amount of kinetic energy of the lamina, or, in other words, it cannot contribute to change its motion state.
What are rotors of freedom?
Sometimes Duffy says twist of freedom, but, as said, it should be rotor of freedom.
The set of rotors of freedom of a lamina is the set of all possible rotors under which the lamina can move. For example, on a serial robot with 3 joints, the end effector rotor is a linear combination of the intermediate relative rotors:
-
.
-
Thus, the set of rotors of freedom of the end effector is the set of all linear combinations
-
.
-
The set of rotors of freedom of a lamina is always a vector (sub)space of 
. We say that a lamina has n rotors of freedom if the dimension of such subspace is n.
The set of rotors of freedom of a lamina is also referred to as the rotor system of the lamina.
What are forces of constraint?
Given a lamina with a set 
of rotors of freedom, the set 
of forces of constraint is
-
,
-
i.e., the set of forces that are reciprocal to all rotors in .
From a physical standpoint the vectors in are the forces that can be applied on the lamina, without changing the motion state of the lamina, irrespectively of the particular motion that the lamina has in that instant of time. In particular, if the lamina was in equilibrium before the application of the force, it will remain in equilibrium after the application of the force.
The set of forces of constraint of a lamina is also referred to as the force system of the lamina.
What is an n-system rotor? and n-system force?
An n-system rotor is the set of rotors generated by the linear combination of n independent rotors. Analogously, an n-system force is the set of all forces generated by the linear combination of n independent forces.
In a singularity, can the trajectory of the end effector follow the direction of an impossible velocity?
This question was posed in class by Sergi Culubret. Consider the 2R manipulator of the figure, and assume that the actuators are rotating with constant angular velocities ω1 = 1 rad/s, and ω2 = − 2 rad/s.
As we proved in class, this is an internal motion of the manipulator because it produces a null velocity for point P in the singular configuration. We don't know what the trajectory of P will be a priori but, departing from the shown configuration, it will decrease the distance OP in the next instant of time. Isn't this inconsistent with the fact that at the singular configuration horizontal velocities are not possible?
I recommend you to think on this apparent paradox, and to try to explain it.
The example is also useful to illustrate the fact that, if we want point P to keep a constant horizontal velocity (say of 1 m/s), then we need larger and larger angular velocities ω1, and ω2 as we approach the singularity.
Do you see it?
