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Errata file to Duffy's book

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This is a draft file gathering some errata found in the book

Joseph Duffy. "Statics and Kinematics with Applications to Robotics". Cambridge University Press.

Contents

1 Chapter 1. Displacement analysis of planar robot manipulators
2 Chapter 2. Statics of plane rigid systems
3 Chapter 3. First-order instantaneous planar kinematics
4 Chapter 4. Dualities of parallel and serial devices
5 Chapter 5. The stiffness mapping for a parallel manipulator

Chapter 1. Displacement analysis of planar robot manipulators

Lluís Ros and Oriol Bohigas: Page 10, Figure 1.6 is incorrect. The chain 12'3'4' (top configuration) is not congruent to the chain 1"2"3"4 (bottom configuration). Moreover, since the figure intends to exemplify two different inversions of the 3R manipulator, in the bottom configuration one of its links should be held fixed to the ground. But the figure wrongly fixes the last joint, not a link. The correct figure should have the link 34 fixed horizontally, and the rest of the configuration should exhibit the same relative angles as the top configuration, as shown next:

Oriol Bohigas: Page 12, line 6 should say "are reduced by $\Sigma_{i=1}^j(3-f_i)$ , ..." instead of "are reduced to ..."

Oriol Bohigas: Page 21, Figure 1.14 is confusing. On the previous page, equation 1.21 gives $S_1=\pm(x^2+y^2-a_{23}^2)^{1/2}$ . If $S_1^{(a)}$ is positive then $S_1^{(b)}$ is negative and $\theta_1^{(b)}$ should look like:

Lluís Ros: Page 42, sentence after Eq. (2.5), the $r$ should be $\mathbf{r}$ .

Chapter 2. Statics of plane rigid systems

Montse Manubens: Page 41, line 4 from the bottom. The third term $|\mathbf{S}| \mathbf{k}$ should be removed, since it is not equivalent to $|\mathbf{p}||\mathbf{S}| \mathbf{k}$ .

Lluís Ros: Adding to the previous comment, note that the fourth term $|\mathbf{p}||\mathbf{S}| \mathbf{k}$ should be $p|\mathbf{S}| \mathbf{k}$ in fact, where $p$ is the signed distance from line 1-2 to the origin $O$ , because the cross-product $\mathbf{p}\times\mathbf{S}$ is a vector that can have both positive and negative z coordinates. The same error occurs in the three previous formulas (lines 6, 7, and 8 from the bottom).

Lluís Ros: Page 42, sentence after Eq. (2.5). The last r in the "components of r" should be bold $\mathbf{r}$ .

Lluís Ros: Page 46, third line from the bottom. The p in $p = x Q sinθ - y Q cosθ$ should be in italic font.

Lluís Ros: Page 46, Fig. 2.5, subfigure b. The angle $θ$ should be $θ'$ . Instead of $\mathbf{S}=(c',s')$ it should say $\mathbf{S'}=(c',s')$ . Instead of $\mathbf{p}=-\mathbf{p}$ it should say $\mathbf{p}'=-\mathbf{p}$ .

Lluís Ros: Page 52, Fig. 2.10. The vector $M$ should point downwards, not upwards, because it is the resultant of the forces $M 1$ and $M 2$ , when $M 2$ is larger than $M 1$ . Moreover, the distance from the line of $M$ to the origin, indicated with an arrow, should be labeled with a $p$ . The figure would look better if the length of $M 2$ were increased.

Pablo Jimenez: Page 58, fifth line from the bottom. The second and third equations should be $\mathbf{r} \times \mathbf{j} = a\mathbf{k}$ and $\mathbf{r} = a\mathbf{i} + b\mathbf{j}$ .

Lluís Ros: Page 59, bottom footnote: the first basis vector should be $[1,0,0] T$ .

Lluís Ros: Page 62, Eq. (2.45): the first comma should be removed.

Lluís Ros: Page 66. Eq. (2.54) should be written as $\hat{w}=\{\mathbf{f};\mathbf{c}_o\}$ .

Montse Manubens: Page 66, line -2 (from bottom), after "Furthermore,". The $f$ should be f, since it denotes a vector.

Montse Manubens: Pages 67-68. Do not confuse the use of $\mathbf{c}_0, \mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3$ and $\hat{s}_1, \hat{s}_2, \hat{s}_3$ at the end of page 67 with $c 1, c 2, c 3, s 1, s 2, s 3$ in formula (2.66).

Oriol Bohigas: Pages 67-68. In formulas (2.60), (2.62) and (2.65) the subscript of the unit moment vectors should be with an o (letter) instead of 0 (number) because they refer to point O. For example, equation (2.62) should be $\mathbf{c}_o = f_1 \mathbf{S}_{o1}+f_2 \mathbf{S}_{o2}+f_3 \mathbf{S}_{o3}$ .

Oriol Bohigas: Page 68. The denominator at the right side of equations (2.64) and (2.56) should be $f$ or $|\mathbf{f}|$ , but not $|\boldsymbol{f}|$ .

Montse Manubens: Page 73, end of line before formula (2.73) should be "can be expressed...".

Lluís Ros: Page 76, 6th line: Instead of "(see Section 2.4)" it should say "(see Section 2.3)".

Lluís Ros: Page 77, 5th line: Again, instead of "(see Section 2.4)" it should say "(see Section 2.3)".

Oriol Bohigas: Page 77, first line after (2.92) and last line: instead of det $j$ it should say only $j$ , i.e. "The first two rows of $j$ are identical..." and "The first two columns of $j$ are...", respectively.

Chapter 3. First-order instantaneous planar kinematics

Pablo Jimenez: Page 82, third line of Section 3.1, the $ω$ at the right side of the equation is a scalar, thus it should not be bold.

Lluís Ros: Page 83, last line, it says "where $|\mathbf{S}|=N$ ", but it should say something like "where $N$ is the signed length of the line segment".

Pablo Jimenez: Page 84, Equation 3.7, right side, should be $P\mathbf{i}+Q\mathbf{j}$ .

Montse Manubens: Page 84, line -4 (from bottom). The sentence in this line should include "then" right after $N = 1$ .

Lluís Ros: Page 87, line 12. It says "As $\delta t \rightarrow O$ " but it should say "As $\delta t \rightarrow 0$ ".

Lluís Ros: Page 87, line 4 from the bottom. It is perhaps more clear to say "The infinitesimal displacement of the rigid lamina will be defined as", than saying "The infinitesimal displacement will be denoted by", because this is the first place where such a displacement is introduced.

Lluís Ros: Page 87, line 2 from the bottom. It should be "rotor" instead of "twist", to be consistent with the previous text.

Lluís Ros: Page 88, lines 4 and 5 from the bottom. The " $O$ " should be a " $0$ ", i.e., a non-bold zero.

Lluís Ros: Page 89, line 2 from the bottom. It should be $\hat{t}$ instead of $t$ .

Montse Manubens: Page 91, line 1, should say "the Plücker axis coordinates". Line 2 should say "plane through point U".

Montse Manubens: Page 91, line 2. The coordinates ${0, y, - x;0,0,1}$ are wrong and should be ${y, - x,0;0,0,1}$ .

Lluís Ros: Page 95, lines 5 and 6 from the bottom. Note that $ω$ is a signed scalar in this context, and hence cannot be defined as the norm of the vector in (3.41). Thus, these two lines should say "From (3.41), the signed magnitude $ω$ of vector $\mathbf{\omega}$ is $ω = ω 1 + ω 2 + ω 3$ ".

Pablo Jimenez: Page 100, ``(d) Show that $\hat{s}^T_{23} \hat{S}_2 = \hat{s}^T_{23} \hat{S}_3=0$ ´´.

Lluis Ros: Page 100. Part 4 of Exercise 3.2 is exactly the same as Part 4 of Exercise 4.2, on page 134. This part, together with Fig. 3.2(4) in the following page, should be removed from here.

Lluis Ros: Page 103. This is not an error. The first paragraph in this page would perhaps read better as follows. ``It is not possible for the end effector to move on a rotor $T$ with coordinates $\mathbf{\omega}\{y,-x,1\}$ whose axis passes through a point $U$ outside the X axis, no matter what the intensities $ω 1,ω 2,ω 3$ of the three rotors $T 1$ , $T 2$ , and $T 3$ are. Clearly, from (3.63) $ω y = 0$ , and since $\omega \neq 0$ , then it must be $y = 0$ .´´

Montse Manubens: Page 110, formula (3.72) should be an equation ending by $... = [0,0] T$ .

Chapter 4. Dualities of parallel and serial devices

Montse Manubens: Page 118, formula (4.1) should be denoted as $\hat{s_i}$ and not $\hat{s}$ (for coherence with Fig. 4.1 and Eqs. (4.3) and (4.4)). Moreover, the third coordinate for $\hat{s}$ in this formula would look better if denoted as $p i + p i'$ instead of as $r i + r i'$ , in order to be consistent with (2.13) and with the rest of the book, where unitized line coordinates are denoted $[c i, s i, p i] T$ .

Lluís Ros: This is a consideration, rather than an erratum. 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 keep the terms rotor and force throughout.

Lluís Ros: Page 122, Fig. 4.4: $(1,0)$ should be (a_{12},0).

Lluís Ros: Page 122. Second paragraph. Substitute "The lamina has two degrees of freedom...", by "The end-effector lamina has two degrees of freedom...".

Lluís Ros: Page 125, first sentence. Replace "There must be a single force", by "There must be a single force of constraint".

Lluís Ros: Page 128, Exercise Figure 4.1(2). Replace $(2,0)$ by $(0, - 2)$ , and $(1.2,0)$ by $(2 / tan60,0)$ . Strictly, the latter coordinates are unnecessary to solve Exercise 4.1-2(a).

Lluís Ros: Page 129, first sentence. Replace "and the corresponding reciprocal space" by "and the corresponding reciprocal space of the end effector".

Lluís Ros: Page 131, Fig. 4.9. The arrows of $τ 1,τ 2,τ 3$ should be counterclockwise, in order to indicate the common direction of positive torque.

Lluís Ros: Page 131, second paragraph. The last sentence in this paragraph asks to premultiply both sides by $(γ T) - 1$ . Since vectors are not invertible, the intended meaning is to premultiply both sides by $\frac{\gamma}{||\gamma||}$ .

Lluís Ros: Page 134, second line. Replace "vanish. The line..." by "vanish, because the line...".

Lluís Ros: Page 134, Exercise 4.2, 1(b). Replace "lbf = ft" by "lbf·ft".

Lluís Ros: Page 134, line 8 from the bottom. Replace "We show" by "We have shown". Also, on the last line in this page, replace "pair configurations" by "pair of configurations".

Lluís Ros: Page 134. Part 4 of Exercise 4.2 is exactly the same as part 4 of Exercise 3.2, page 100. This is quite probably an error, and it is recommended to remove this part from Exercise 3.2.

Lluís Ros: Page 135, Exercise Figure 4.2(2). The point of coordinates (2,1), should be (1,2) instead, as this figure must match Figure 3.2(1b) on page 100.

Lluís Ros: Page 136, first equation. Replace $s \; \hat{} \; _1$ by $\hat{s}_1$ .

Lluís Ros: Page 139, third line from the bottom. The (a) should be (b).

Lluís Ros: Page 140, Exercise Figure 4.3(2). There are three missing measures in this figure. The distances $(1) 1$ to $(1) 2$ , and $(1) 2$ to $(1) 3$ , and the angle of the second leg relative to the base. One possibility is to fix them as 2", 0.893", and 90º, respectively, in order to have collinear platform points.

Lluís Ros: Page 142, Figure 4.13. Replace $ω l i$ by $ω 1 i$ .

Lluís Ros: Page 142, Equation (4.32). Replace $c_1^2$ by $c_i^2$ .

Lluís Ros: Page 143, first line after Eq. (4.36). Replace "Equation (4.36) can also be derived by..." by "Equation (4.35) can also be derived by...".

Lluís Ros: Page 143, end of first paragraph after Eq. (4.36). Replace "premultiplying both sides by $(λ T) - 1$ ..." by "premultiplying both sides by $\frac{\mathbf{\lambda}}{||\mathbf{\lambda}||}$ ...". Note that $\mathbf{\lambda}$ is a vector, and, hence, it is non-invertible.

Lluís Ros: Page 143, (eq. 2.82). To be in accordance with Eq. (2.82) on page 74, this equation should be written as $j^{-1}=[s_{3-2}\hat{S}_{23}, s_{1-3}\hat{S}_{31}, s_{2-1}\hat{S}_{12}]^T / det(j)$ , i.e., the inner transposes should be removed.

Lluís Ros: Page 144, Sentence "It is also helpful to compute the component...". Add "(see Fig. 4.13)" to the end of this sentence.

Lluís Ros: Page 144, Sentence "This line intersects...". This sentence would be more clarifying if it said "This line intersects the axis of the revolute joint at $C i$ , hence $\hat{s}_{ic}^T \cdot \hat{T}_{3i}=0$ , and ...".

Lluís Ros: Page 144, last equation. Replace $\hat{T}_{li}$ by $\hat{T}_{1i}$ .

Lluís Ros: Page 149, Figure 4.16. The displacements $δ$ should be $δ l i$ to agree with the text. Also, the displacements indicated as $δθ i$ should be $l i δθ i$ .

Lluís Ros: Page 152, equations (4.55) and (4.56). Replace $O B$ by $O B i$ .

Chapter 5. The stiffness mapping for a parallel manipulator

Oriol Bohigas: Page 164, at the end of the second paragraph in section 5.4 should say "given in subsection 1.8.3" instead of 1.6.3, which do not exists.

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