# Modern Robotics Errata, First Edition Version 1

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Errata below are for the original first edition hardcover book published by Cambridge University Press and the corresponding preprint version of the book posted on May 3, 2017. Errata for the updated first edition of the book, late 2019, can be found here.

## Book: Significant corrections

### Chapter 2

• Proposition 2.2 says that Grubler's formula "provides a lower bound on the number of degrees of freedom" of a mechanism to account for the case where the joint constraints are not independent. This statement is misleading, as it assumes a linear analysis of mobility. In fact, some mechanisms may have fewer dof than implied by Grubler's formula by a higher-order analysis. A more correct statement is that Grubler's formula yields the correct number of dof in the "generic" case (where the joint constraints are independent and a linear analysis of mobility suffices), but a mechanism may have more dof (e.g., if the joint constraints are not independent everywhere) or fewer dof (e.g., the zero-dof four-bar linkage where the sum of the lengths of three links equals the length of the fourth link). This comment also applies to the sentence immediately after Example 2.6, the fifth bullet of the Summary in Chapter 2.6, and the second-to-last sentence of the third paragraph of Chapter 1's summary of Chapter 2.

### Chapter 3

• In the caption of Figure 3.3, it should read ${\displaystyle {\hat {\text{x}}}_{\text{b}}=(1/2,{\sqrt {3}}/2)}$ and ${\displaystyle {\hat {\text{y}}}_{\text{b}}=(-{\sqrt {3}}/2,1/2)}$.
• Under "Representing a configuration" in Chapter 3.3.1.2, the text says "${\displaystyle T_{bc}=(R_{bc},p_{bc})}$ represents {b} relative to {c}" but it should say "represents {c} relative to {b}".
• There is a typo in the final matrix, ${\displaystyle T_{ce}}$, of Example 3.19. The term ${\displaystyle 130/{\sqrt {2}}}$ should be ${\displaystyle 160/{\sqrt {2}}}$. (Also, there is no need to use ${\displaystyle T_{ad}}$ in this example.)
• In the displayed equation just after Equation (3.76), the left-hand side should be ${\displaystyle [{\mathcal {V}}_{s}]}$ (the brackets are missing).
• At the end of the introduction Exercise 3.16, it says "origin of {b} is at (0,2,0) is {s}" but "is {s}" should be "in {s}".
• Exercise 3.16(i) asks for "the ${\displaystyle \{q,{\hat {s}},h\}}$ representation" but it should say "a ${\displaystyle \{q,{\hat {s}},h\}}$ representation" since ${\displaystyle q}$ is not unique.
• Exercise 3.20, Figure 3.26: In the figure, the y and z axes for the {a}, {b}, and {c} frames are switched (y should point forward and z should point up). Also, the space frame is located at the bottom of the small wheel, directly below the {a} frame.
• Exercise 3.25(a): the element in the third row and third column of the matrix ${\displaystyle A}$ should be 0 (it is incorrectly written as 1).

### Chapter 4

• Example 4.1, last row of the table (screw axis for joint 3): The linear component ${\displaystyle v_{i}}$ should be ${\displaystyle (0,-L_{2},0)}$ (the ${\displaystyle -}$ sign is missing).
• Exercise 4.21: The question should begin "For each ${\displaystyle T}$ below..." (instead of "For each ${\displaystyle T\in SE(3)}$ below...") since the first part of the problem is determining whether ${\displaystyle T}$ is indeed an element of ${\displaystyle SE(3)}$.

### Chapter 5

• Equation (5.7): The first two terms on the right-hand side of the equation should be ${\displaystyle J_{s1}{\dot {\theta }}_{1}+J_{s2}(\theta ){\dot {\theta }}_{2}}$.
• Exercise 5.2(b): The wrench applied by the last link, expressed in the {s} frame, is 5 N in the ${\displaystyle {\hat {\text{y}}}_{\text{s}}}$ direction, with zeros in all other components. (The wording seems to imply that the force is through the tip of the last link, which is incorrect.)
• Exercise 5.16 and Figure 5.26: The robot is referred to as PRRRRR, but it is PRPRRR.

### Chapter 6

• Chapter 6.3, first line after Eq (6.7): the matrices ${\displaystyle T_{sd}^{-1}{\dot {T}}_{sd}}$ and ${\displaystyle {\dot {T}}_{sd}T_{sd}^{-1}}$ are referred to as twists, but these are the se(3) matrix representations of the twists.

### Chapter 8

• In Equation (8.14), ${\displaystyle {\mathcal {K}}(\theta )}$ should be replaced by ${\displaystyle {\mathcal {K}}(\theta ,{\dot {\theta }})}$.
• Figure 8.5 says the volume of the rectangular parallelepiped is ${\displaystyle abc}$ but it should be ${\displaystyle hlw}$.
• End of first sentence after Eq (8.26) in Section 8.2.1 should read "columns of ${\displaystyle R_{bc}}$ correspond to the eigenvectors of ${\displaystyle {\mathcal {I}}_{b}}$." (The word "eigenvalues" should be replaced by "eigenvectors.")
• Equation (8.74): the first two plus signs should be minus signs.
• In the parenthetical text a few lines after Equation (8.94): "...there is an equality constraint..." should instead be "...there is also an inequality constraint..."
• Caption of Figure 8.10: The operating region is light gray and the continuous operating region is dark gray.
• Exercise 8.6(a): The expression ${\displaystyle {\text{ad}}_{J_{i}}(J_{j})}$ has the indices switched; the correct expression is ${\displaystyle {\text{ad}}_{J_{j}}(J_{i})}$.
• Exercise 8.7: The expression should be written:

${\displaystyle {\dot {M}}=-{\mathcal {A}}^{\rm {T}}{\mathcal {L}}^{\rm {T}}{\mathcal {W}}^{\rm {T}}[{\mbox{ad}}_{{\mathcal {A}}{\dot {\theta }}}]^{\rm {T}}{\mathcal {L}}^{\rm {T}}{\mathcal {GLA}}-{\mathcal {A}}^{\rm {T}}{\mathcal {L}}^{\rm {T}}{\mathcal {GL}}[{\mbox{ad}}_{{\mathcal {A}}{\dot {\theta }}}]{\mathcal {WLA}}}$

### Chapter 10

• Equation (10.4) should read ${\displaystyle u=F(q)-B{\dot {q}}}$ (the plus sign in the book should be a minus sign).
• First line of chapter 10.6.4: The equation should be ${\displaystyle u=F(q)-B{\dot {q}}}$ (plus sign should be a minus sign). Similarly, in the last sentence of chapter 10.6.4, the plus sign should be a minus sign.

### Chapter 11

• In the displayed equation after Equation (11.18), the vector ${\displaystyle X_{e}(t)}$ is a six-vector. The bottom three elements are written correctly, but the top three elements, an angular velocity, are written instead in their 3x3 ${\displaystyle so(3)}$ form. Also, the term written ${\displaystyle R^{\text{T}}(d)}$ should be written ${\displaystyle R^{\text{T}}(t)}$.
• In Equation (11.33), the right-hand side should be zero (${\displaystyle c=0}$).
• Figure 11.24, the Robonaut 2 series elastic actuator, 4th sentence of the caption: The words "outer" and "inner" should be switched, so the new sentence reads "The inner ring of hole mounts connects to the harmonic gearhead output, and the outer ring of hole mounts is the output of the SEA, connecting to the next link."

### Chapter 12

• In Example 12.4, the elements of the wrenches ${\displaystyle {\mathcal {F}}_{1},{\mathcal {F}}_{2},{\mathcal {F}}_{3}}$ are erroneously written in the order ${\displaystyle (f_{x},f_{y},m_{z})}$; they should be written in the order ${\displaystyle (m_{z},f_{x},f_{y})}$, i.e., ${\displaystyle (-2,0,1),(1,-1,0),(1,1,0)}$, respectively.

### Appendix C

• Section C.3, Equation (C.5): All instances of ${\displaystyle \phi _{i-1}}$ should be replaced by ${\displaystyle \phi _{i}}$.
• In Equation (C.12), two instances of ${\displaystyle M_{i-1}}$ should be replaced by ${\displaystyle M_{i}}$.

## Book: Minor typos, etc., no danger of misunderstanding

### Chapter 1

• (online version only) Description of Chapter 6: "jont positions" should be "joint positions."

### Chapter 2

• Chapter 2.2.2, Example 2.3: "Substituting" is misspelled.

### Chapter 3

• Near the end of Chapter 3.6 Software, the function "AxisAng" should be written "AxisAng6."

### Chapter 5

• Figure 5.15(b): The circular arrow indicating the rotation of joint 3 is slightly misplaced.
• Exercise 5.25: This problem instructs you to use the space Jacobian, so you should use it for the manipulability ellipsoids in this problem (even though it is generally preferred to use the body Jacobian for manipulability ellipsoids).

### Chapter 6

• (online version only) Figure 6.7: ${\displaystyle \theta _{0}}$ should be ${\displaystyle \theta ^{0}}$ (five times) and ${\displaystyle \theta _{1}}$ should be ${\displaystyle \theta ^{1}}$ (one time).
• Chapter 6.2.2, Example 6.1: just before the matrix ${\displaystyle T_{sd}}$, "corresponds to to" should be "corresponds to."
• Chapter 6.3, first sentence after Equation (6.7): "however small" should be written "however, small" to avoid ambiguity.

### Chapter 7

• (online version only) Last line of first paragraph of Chapter 7.1: "valuees" should be "values."
• Just below caption for Figure 7.8: There is an extraneous dot.

### Chapter 8

• (online version only) Chapter 8.1.1, last paragraph: ${\displaystyle {\dot {\theta }}_{1}}$ is improperly typeset.

### Chapter 12

• Caption of Figure 12.14, second-to-last sentence: "rotation if possible" should be "rotation is possible."
• (online version only) Example 12.11: The sentence "... can remain standing if there is a consistent solution if there exist ${\displaystyle k_{i}\geq 0}$ ..." should read simply "... if there exist ${\displaystyle k_{i}\geq 0}$ ..."

### Chapter 13

• Equation (13.29): ${\displaystyle y_{p}}$ should be ${\displaystyle y_{P}}$.

## A partial list of errata contributors

Thanks to the following people who provided corrections, starting from the preliminary version of the book posted in October, 2016:

H. Andy Nam, Eric Lee, Yuchen Rao, Chainatee Tanakulrongson, Mengjiao Hong, Kevin Cheng, Jens Lundell, Elton Cheng, Michael Young, Jarvis Schultz, Logan Springgate, Sofya Akhmametyeva, Aykut Onol, Josh Holcomb, Yue Chen, Mark Shi, AJ Ibraheem, Yalun Wen, Seongjae Jeong, Josh Mehling, Felix Wang, Drew Warren, Chris Miller, Clemens Eppner, Zack Woodruff, Jian Shi, Jixiang Zhang, Shachar Liberman, Will Wu, Dirk Boysen, Awe Wang, Ville Kyrki, John Troll, Andrew Taylor, Nikhil Bakshi, Yunzhe Pan, Barrett Ames, Marcel Bonnici, Mahdiar Edraki, Jose Capco