Modern Robotics Errata

The errata below are for the updated first edition of Modern Robotics (as well as the practice exercises and linear algebra refresher appendix). The updated first edition (also called "version 2") was originally published by Cambridge University Press in late 2019 (marked "3rd printing 2019" or later) and the corresponding online preprint is dated December 2019. The updated first edition includes several corrections and minor additions to the original first edition, which was originally published by Cambridge in May 2017, with a corresponding online preprint dated May 2017. (The errata for the original first edition can be found here.)

Please click here to report any corrections for the updated first edition of the book, the practice exercises, or the linear algebra refresher appendix.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\text{x}}_{\text{b}} = (1/2, \sqrt{3}/2)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\text{y}}_{\text{b}} = (-\sqrt{3}/2, 1/2)} .
• Under "Representing a configuration" in Chapter 3.3.1.2, the text says "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, $T_{ce}$, of Example 3.19. The term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 130/\sqrt{2}} should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 160/\sqrt{2}} . (Also, there is no need to use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{ad}} in this example.)
• In the displayed equation just after Equation (3.76), the left-hand side should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{q,\hat{s},h\}} representation" but it should say "a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{q,\hat{s},h\}} representation" since $q$ 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_i} should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, -L_2, 0)} (the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -} sign is missing).

Chapter 5

• Equation (5.7): The first two terms on the right-hand side of the equation should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $T_{sd}^{-1}{\dot {T}}_{sd}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{K}(\theta)} should be replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{K}(\theta,\dot{\theta})} .

• Figure 8.5 says the volume of the rectangular parallelepiped is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle abc} but it should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle hlw} .

• End of first sentence after Eq (8.26) in Section 8.2.1 should read "columns of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{bc}} correspond to the eigenvectors of ${\mathcal {I}}_{b}$." (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{ad}_{J_i}(J_j)} has the indices switched; the correct expression is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{ad}_{J_j}(J_i)} .

• Exercise 8.7: The expression should be written:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $so(3)$ form. Also, the term written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{\text{T}}(d)} should be written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{\text{T}}(t)} .

• In Equation (11.33), the right-hand side should be zero (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3} are erroneously written in the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_x,f_y,m_z)} ; they should be written in the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (m_z,f_x,f_y)} , i.e., $(-2,0,1),(1,-1,0),(1,1,0)$, respectively.

Appendix C

• Section C.3, Equation (C.5): All instances of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_{i-1}} should be replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i} .

• In Equation (C.12), two instances of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{i-1}} should be replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_0} should be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta ^{0}} (five times) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_1} should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta^1} (one time).
• Chapter 6.2.2, Example 6.1: just before the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_i \geq 0} ..." should read simply "... if there exist Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_i \geq 0} ..."

Chapter 13

• Equation (13.29): Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{p}} should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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