Displacement analysis FAQ
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The contents of this module corresponds to Chapter 1 of Duffy's book (displacement analysis of planar manipulators). This page contains frequently asked questions on this module. Feel free to add your question here, or provide answers to questions posted here by others.
Errata detected
Please add any errata you may find in Duffy's book to this page.
What is a link, a joint, a linkage, and a mechanism?
Just to have some vocabulary in mind:
- A link is a rigid body.
- A joint is a connection between two bodies that restricts the relative motion of these bodies.
- A linkage is a set of bodies assembled by lower pair joints (like revolute or prismatic joints). An interesting linkage is the double butterfly. See here how it moves. Other interesting linkages have been analyzed with the software package CUIK, developed by the GMR group at IRI.
- Sometimes a linkage is also called a mechanism. However, the term mechanism has a broader sense, since it refers to assemblies that can contain higher pairs too (like gears or cams).
What does it mean for two figures to be congruent?
Two sets of points are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. See wikipedia.
What is the definition of mobility?
The text doesn't give it precisely. Can you define what is the mobility of a mechanism? This is also referred to as the number of internal degrees of freedom of the mechanism.
The mobility of a linkage is usually understood as the lowest number of kinematic pairs that must be actuated to fix the linkage configuration.
More precisely the mobility of a linkage is the dimension of its configuration space at a regular configuration (one where the differential of the linkage equations has full rank, as explained in this note). If at a regular configuration 
the mobility is m, this means that the configuration space can be parametrized by m parameters in a neighborhood of x (or, in other words, that it is diffeomorphic to an open set of 
.
Why does the Grübler-Kutzbach formula fail to predict the mobility sometimes?
The Grübler-Kutzbach formula to which we refer here is Equation (1.1) in page 12 of Duffy's book.It doesn't predict the mobility of a linkage correctly in some situations. It is worth understanding which are such situations.
Think on the following examples, and determine the mobility in an intuitive manner, by counting how many joints should be locked in order to make the linkage rigid. Then do the same using Formula (1.1). Verify whether the two methods coincide. Note that each example on the left bears some similitude with its neighbor on the right. Can you tell why Duffy's formula correctly predicts the mobility of all examples on the left, but fails to do so on all examples on the right? (Each shaded polygon represents a single rigid body.)
A detailed answer can be found in this short note on mobility formulas.
Inverse kinematics problems on parallel manipulators
Francisco Javier Fernández: For a parallel manipulator, how do we compute the joint-space trajectory that corresponds to a given end-effector trajectory?
Lluís Ros: Suppose that the manipulator is the 3RPR manipulator shown in the class. Let the pose of the platform by given by a tuple (x,y,θ), where x and y are the coordinates of a point on the platform, and θ is the orientation angle of the platform, given relative to some line. Suppose we have a desired trajectory for the platform, expressed as a curve of poses parameterized in time, of the form γ(t) = (x(t),y(t),θ(t)). Then, for each point of γ(t) we only need to solve the inverse kinematic problem of this platform. The inverse kinematic problem of parallel manipulators of this kind is explained in Section 1.8.2 of Duffy's book.
If you have a spatial fully-parallel manipulator, the procedure is analogous to the planar case.
Note that the inverse kinematics problem is easy to solve in general parallel manipulators, as it admits closed-form solutions (it is possible to derive explicit formulas giving the leg lengths in terms of the platform pose). However, the problem is difficult in serial manipulators, as it involves computing the solutions of a system of polynomial equations. A good book where efficient solutions to the problem are described is
- L. W. Tsai. "Robot Analysis. The Mechanics of Serial and Parallel Manipulators". John Wiley and Sons. 1999.
The inverse and direct kinematic problems can also be solved for general mechanisms too, as explained next.
Is it possible to solve the forward/inverse kinematic problem on general mechanisms?
Yes. The following link contains the details of a method which is able to solve general position analysis problems (including forward/reverse analysis of parallel/serial manipulators):
- Box Approximations of Planar Linkage Configuration Spaces. J. M. Porta, L. Ros, T. Creemers, and F. Thomas. ASME Journal of Mechanical Design. Vol. 129, n. 4, pp. 397-405. April 2007.
This method has recently been extended to also deal with general spatial mechanisms:
- A Linear Relaxation Technique for the Position Analysis of Multi-loop Linkages. J. M. Porta, L. Ros, F. Thomas. IEEE Transactions on Robotics. Vol. 25, n. 2, pp. 225-239. April 2009.
See also some example problems solved by this method
and the web page of the related project:
