Statics FAQ

From WikIRI

This module corresponds to Chapter 2 of Duffy's book, on statics of planar robot manipulators. Here there are some comments on the material in this module, which you might find helpful. Anyone is invited to contribute to this page, by adding extra issues you would like to be discussed or rising particular questions (open an extra subsection for this) or providing answers to other posted questions (write the answer in its corresponding subsection).

Errata and summary of concepts

Errata detected

Please add any errata to this page.

Concepts related to this chapter

Line bound vector
Free vector
Force
Pure couple
A pure couple is a special case of a force
Pencil of lines
Euclidean group of motions
Change of coordinates of a line, under translations and rotations
Vector representation of a force, as a wrench $\hat{w}=\{\mathbf{f},\mathbf{c}_0\}$
Forward static analysis
Reverse static analysis
Singularity configuration

Frequently-asked questions

Can couples be seen as particular cases of forces?

Yes. A central idea of this chapter is that forces and couples are essentially the same thing. They are line bound vectors. We are used to think of a force as a vector sliding along a line of action, and a couple as a free vector (i.e., as two different things). However, a couple can also be viewed as a force. It is a vector of infinitesimal length, sliding along the line at infinity. One way to visualize this is explained below.

In general, we represent a force (including the special case of a couple) with the triple $(L, M, R)$ , which, as seen in the chapter, provides the force vector $(L, M)$ and determines the line of action $L y - M x + R z = 0$ . We deliberately say $L y - M x + R z = 0$ , instead of $L y - M x + R = 0$ (as done by Duffy in Eq. (2.6)), to also include the line at infinity. This is common in projective geometry: a line of the real projective plane is represented by a plane of $\mathbb{R}^3$ through the origin (see this short note on the real projective plane if you are unfamiliar with it). To obtain the equation of a usual line of the Euclidean plane, simply intersect $L y - M x + R z = 0$ with the plane $z = 1$ (you then obtain formula (2.6) in page 42). The form $L y - M x + R z = 0$ is preferable, though, because it also accommodates the case of a couple $(0,0, R)$ . In this case, the equation of the line is $0 y - 0 x + R z = 0$ , i.e., $z = 0$ , which is the plane representing the line at infinity.

What is the resultant of a system of forces?

With the unified view of forces and couples just given, we may say that a system of forces applied to an object is equivalent to a single force acting along a certain line of action, possibly at infinity. This is formula (2.20) on page 50.

Here's a nice picture that illustrates this, taken from the classic book by Beer and Johnston.

The forces exerted by the four tugboats on the submarine USS Pasadena could be replaced with a single equivalent force exerted by one tugboat!

Anticipating future readings, we note that this is only true for systems of planar forces, but not for systems of spatial forces. Several forces (including couples as special cases) acting on a spatial rigid body, do not reduce to a single force (including a couple) in general. They reduce to a wrench, which in turn can be decomposed as a force and a pure couple, both aligned along a same axis.

Are L, M, and R true coordinates for a line-bound vector?

Yes. Note that (2.4) in page 41 proves that the moment of the line-bound vector S is constant along any point on the line 1-2, whereas the initial derivation with determinants in page 42 proves that such moment can only be constant along such line. Thus (L,M,R) are true coordinates for the line bound vector S.

What is a:b:c?

At several places in this chapter, the book refers to the ratio $a : b : c$ , where at least one of the elements is non-null. The notation is used to refer to the set of values:

$\{(x,y,z) : (x,y,z) = \lambda (a,b,c), \lambda \in \mathbb{R} \}$

i.e., a line through the origin of $\mathbb{R}^3$ with direction vector $(a, b, c)$ .

Note that such set is in fact characterized by two values, since we can describe it using the vector $(a / c, b / c,1)$ instead (assuming c is non-null).

That's why in page 43, 2nd line from the bottom, Duffy says that the ratios $L : M : R$ "are equivalent to two quantities".

What does it mean to express the point of intersection of two lines as y:x:1?

Continuing with the previous comment, we note that in page 48, Eq. (2.18) the book expresses the point (x,y) of intersection of two lines in the form:

$y : x :1 = Δ 1 :Δ 2 :Δ 3$

where the $Δ i$ are $2\times2$ determinants. This in fact means that:

$y = \frac{\Delta_1}{\Delta_3}$

$x = \frac{\Delta_2}{\Delta_3}$

Why can a force be applied to any point on its line of action?

In page 50 (top), Duffy says that because the lamina is rigid, the point of application can be moved anywhere along the line. This fact is a consequence of the principle of transmissibility of forces.

This principle, which is based on experience, states that two forces $F 1$ and $F 2$ are mechanically equivalent (they have the same effect on a rigid body) if, and only if, they are equal (they have the same magnitude and direction) and have the same moment about a given point O.

How can I visualize a couple as a force on the line at infinity?

In page 50, Duffy says that a pure couple can be considered as a force of infinitesimal magnitude, acting on the line at infinity. This interpretation is nice, as it shows that couples are actually forces too, but it may be difficult to understand it at a first glance. One way to do so is to think of a system of antiparallel forces, and see what happens when the magnitude of one of the forces tends to coincide with the other, as shown next. (The following is a shorter version of the discussion in page 52 and 53.)

Consider the following two forces acting on a rigid lamina:

$\hat{w}_1 = (0,M_1,M_1 \cdot p_1)$

$\hat{w}_2 = (0,-m,-m \cdot p_2)$

Their resultant is:

$\hat{w}=\hat{w}_1+\hat{w}_2 = (0,M_1-m,M_1 p_1-m p_2)$ .

The line of action of the resultant is parallel to the y-axis. The x position of such line, as a function of $m$ is:

$p(m) = \frac{M_1p_1 - mp_2}{M_1-m}$

The resultant force has a magnitude $M$ , in terms of $m$ , of:

$M (m) = M 1 - m$

Finally, the moment of the resultant force, with respect to the origin is:

$R(m) = M(m) \cdot p(m) = (M_1-m) \cdot \frac{M_1p_1 -mp_2}{M_1-m}$

Now observe that for $m \rightarrow M_1$ we have:

$p(m)\rightarrow \infty$

$M(m)\rightarrow 0$

but their product tends to the finite value:

$M(m) p(m) \rightarrow M_1(p_1-p_2)$

What is the geometric condition for singularity on a parallel 3-RPR manipulator?

Note that all types of singular configurations in figures 2.26, 2.27, 2.28, and 2.30 in Duffy's book can be summarized by saying that the three leg lines must have (at least) one point in common. In all cases the condition to be checked is that the determinant of the leg lines vanishes.

What is the geometrical meaning of the inverse of j?

Section 2.8 provides an interpretation of the the matrix $\mathbf{j}^{-1}$ : its rows are, to a scalar multiple, the unitized coordinates of the vertical lines $$ 23,$ 31,$ 12$ (one can also see such lines as the corresponding "points" in this case). See top of page 75, and Figure 2.25. What happens with such lines in a singularity?

It is possible to give a kinematic interpretation of the lines (points) $$ 23,$ 31,$ 12$ . This will be done later in the course. But, ... can you anticipate this interpretation?

Application: A wrench-sensitive touchpad

As an application of the material learned in this Module, here's a paper on the design and implementation of a wrench sensor that can also act as a touchpad. (A wrench is a six-component vector encoding the resultant of a system of forces and torques acting on a rigid body.)

R. Frigola, L. Ros, F. Roure, and F. Thomas. A Wrench-Sensitive Touchpad Based on a Parallel Structure Proc. of the IEEE International Conference on Robotics and Automation Pages 3449-3454. May 19-23, 2008. Pasadena, USA. IEEE Computer Society Press.

Statics FAQ

From WikIRI

Contents