Erratum

The online version of the original Book can be found at http://dx.doi.org/10.1007/978-3-319-01851-5

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The publisher regrets the error in the print and online versions of this book. Listed below are the corrections.

Introduction: In order to ease the finding of items in this document, we have kept the page format and the original fonts of the book; we have also typeset with typewriter font--the one used in this Introduction--text that does not belong to the book.

p. 93: The correct matrix [R ₂]c is

$\displaystyle\begin{array}{rcl}[\mathbf{R}_{2}\,]_{\mathcal{C}}& =& \left [\begin{array}{ccc} \mathtt{0.373} &\mathtt{-0.926}& \mathtt{0.043}\\ \mathtt{0.902} &\mathtt{0.352 } &\mathtt{-0.249} \\ \mathtt{0.215}&\mathtt{0.132}& \mathtt{0.967}\end{array} \right] \end{array}$

p. 129: Line below Eq. (3.133a): in light of Eq. (2.39), should read: in light of Eq. (2.40)

p. 137: In Exercise 3.20, the expression for MA is faulty. The correct expression is

M_A = M _C + mPP^T

p. 144: The last line of text, “One thus has, using subscripted brackets as introduced in Sect. 2.2,”, should read:

“One thus has, using subscripted brackets as introduced in Sect. 2.3,”

p. 171: The third line of text below eq. (4.33), “From Definition 2.2.1, then [u]1 = [e7]1 = [e6]1”, should read:

“From Definition 2.2.1, then [u]₁ = [e ₇]₁ = [e6]₁”

p. 178: The correct expression for Q₁₂₃ is

$\displaystyle{\mathbf{Q}_{123} = \mathbf{Q}_{1}\mathbf{Q}_{2}\mathbf{Q}_{3} = \left [\begin{array}{ccc} 0& 1& 0\\ \\ \\ -1 & 0 & 0\\ \\ \\ 0 & 0 &1 \end{array} \right ]}$

[e₆]₄ should read:

$\displaystyle{[\,\mathbf{e}_{6}\,]_{4} = \mathbf{Q}_{1}\mathbf{Q}_{2}\mathbf{Q}_{3}^{T}[\,\mathbf{e}_{ 6}\,]_{1} = \left [\begin{array}{ccc} 0 & -1 & 0\\ \\ \\ - 1 & 0 & 0\\ \\ \\ 0 &0&1 \end{array} \right ]\left [\begin{array}{c} 2/3 \\ - 2/3 \\ - 1/3 \end{array} \right ] = \left [\begin{array}{c} 2/3 \\ 2/3 \\ - 1/3 \end{array} \right ]}$

θ_4,2 should be

$\theta_{4,2} = -80,26438967^\circ$

p. 182: The caption of Fig. 4.26 is faulty. The correct caption is Motoman-EA1400N welding robot: (a) top view; (b) side view; (c) orthographic projection; (d) view A, as per side view; (e) view B, as per side view. All dimensions in mm

p. 200 Where it reads: (b) the moments of the three lines about any point on the intersecting line are all zero, the correct wording should read:

(b) the moments of the three lines with respect to the intersecting line are all zero.

p. 202: The expression for α is faulty. The correct expression is

$\displaystyle{\alpha = \frac{\sqrt{a_{3 }^{2 } + b_{4 }^{2}}} {{\sqrt{a_{2}^{2 } + d^{2}}} + \sqrt{a_{3 }^{2 } + b_{4 }^{2}}}}$

Please refer to Appendix A for details.

p. 211: Where it reads: with τ_a and τ_w defined as the wrist and the arm torques, respectively, the correct wording should read:

with τ_a and τ_w defined as the arm and the wrist torques, respectively.

p. 219: Equation (5.67c) should read:

$\ddot{\theta}_1 = \ddot{\phi}-(\ddot{\theta}_2+\ddot{\theta_3})$

p. 291: The second line of the expression for $\ddot{\rm c}_3$ should read:

$\displaystyle\begin{array}{rcl} -\frac{1}{2}m_{3}a_{3}(a_{1}s_{23} + 2a_{2}s_3){\dot{\theta}_1}{\dot{\theta}_3} - m_{3}a_{2}a_{3}s_{3} {\dot{\theta}_2}{\dot{\theta}_3}-\frac{1}{2} m_{3}a_{2}a_{3}s_{3} \dot{{\theta }}_{3}^2\end{array}$

p. 321: Caption of Fig. 7.7 should read:

Mass-center location of the robot of Fig. 4.17

p. 324: The second line of the expression for $\dot{\rm t}_{11}$ should read:

$= \left[\begin{array}{lcl}&\dot{e}_1 \\ &\dot{e}_1\times\rho_1+e_1\times\dot{\rho}_1 \end{array}\right]$

The second line of the expression for $\dot{\rm t}_{21}$ should read:

$= \left[\begin{array}{lcl} &0 \\ &{e}_1\times{\omega_1+a_1+\omega_2\times\rho_2} \end{array}\right] = p \left[\begin{array}{lcl}&0\\ &({a}/{2})({i-3j})\end{array}\right]$

The fourth line of the expression for $\dot{\rm t}_{31}$ should read:

$\begin{array}{lcl} = p [&0\\ &({a}/{2})({i-3j}) ] \end{array}$

The second line of the expression for $\dot{\rm t}_{32}$ should read:

$= \left[\begin{array}{lcl} & pe_1 \times e_2 \\& (pe_1 \times e_2) \times (a_2 + \rho_3)+ e_2 \times [p(e_1 + e_2) \times a_2 + p(e_1 + e_2 + e_3) \times \rho_3] \end{array}\right]$

p. 325: Entry (3,1) of matrix T ^T M $\dot{\rm T}$ is flawed. The correct expression for this matrix is:

$\displaystyle{{\mathbf{T}}^{T}\mathbf{M}\mathbf{\dot{T}} = p\left [\begin{array}{ccc} - (1/4){a}^{2}m& (7/4){a}^{2}m & - (1/2){a}^{2}m - I \\ - (1/4){a}^{2}m& 0 & (1/4){a}^{2}m + I \\ (3/4){a}^{2}m &(1/4){a}^{2}m - I & 0\\ \end{array} \right ] \equiv \overline{\mathbf{P}}}$

p. 326: Entries (1,3), (2,3) and (3,1) of matrix $\dot{\rm I}$ are faulty. The correct expression of the matrix is:

$\displaystyle{\mathbf{\dot{I}} = p\left [\begin{array}{ccc} - (1/2){a}^{2}m &(5/4){a}^{2}m& - I+(1/4)a^2 m \\ (5/4){a}^{2}m & 0 &{(1/2)a^2} m\\ - I+(1/4)a^2 m &(1/2){a}^{2}m& 0\\ \end{array} \right ]}$

p. 327: The second-to-last line of text, “Now we have”, should read:

“Now, thematrix C of Coriolis and centrifugal forces is obtained as shown below:”

The last equation displayed should read:

$\rm{C = T^T M\dot{T} + T^T WMT = pA}$

p. 328 Entry (1,1) of matrix A is flawed. The correct expression is

$\displaystyle{\mathbf{A} \equiv \left [\begin{array}{ccc} - (1/4){a}^{2}m & (7/4){a}^{2}m + I & - (1/2){a}^{2}m - 2I \\ - (1/2){a}^{2}m - I & 0 & (1/4){a}^{2}m + 2I \\ (3/4){a}^{2}m + I &(1/4){a}^{2}m - 2I & 0\\ \end{array} \right ]}$

The first entry of the vector array in the second equation display has a “(1/2)” too much. The correct display is

$\displaystyle{({\mathbf{T}}^{T}\mathbf{M}\mathbf{\dot{T}}+{\mathbf{T}}^{T}\mathbf{W}\mathbf{M}\mathbf{T})\boldsymbol{\dot{\theta }} = {p}^{2}\left [\begin{array}{c} {a}^{2}m - I \\ - (1/4){a}^{2}m + I \\ {a}^{2}m - I\\ \end{array} \right ]}$

p. 329: The second line of the expression for $\ddot{\rm c}_3$ has an “D” too much. It should read:

$\displaystyle\begin{array}{rcl} & & +\,\boldsymbol{\omega }_{2} \times (\boldsymbol{\omega }_{2} \times \boldsymbol{\rho }_{2}) = \frac{1} {2}a{p}^{2}(-4{\rm j + k}) + - \frac{1} {2}a{p}^{2}(\mathbf{\rm j} + \frac{1} {2}a{p}^{2}(2{\rm i - j + k}) \end{array}$

The expressions for f^P ₂ , n^P ₂ , and f^P ₁ are faulty. They should read:

p. 330: The second equation display, that of τ ₁, is faulty. The correct expression reads:

$\displaystyle\begin{array}{rcl} \tau _{1}& =& \mathbf{n}_{1}^{P} \cdot \mathbf{e}_{ 1} = -I{p}^{2} + {a}^{2}m{p}^{2} {}\\ \end{array}$

The first component of vector C.(θ, $\ddot{\theta}$)$\ddot{\theta}$ is faulty. The correct expression is

$\displaystyle{\mathbf{C}(\boldsymbol{\theta },\dot{\boldsymbol{\theta }})\dot{\boldsymbol{\theta }} = \left [\begin{array}{c} - I{p}^{2} + {a}^{2}m{p}^{2} \\ I{p}^{2} - (1/4){a}^{2}m{p}^{2} \\ - I{p}^{2} + {a}^{2}m{p}^{2}\\ \end{array} \right ]}$

Appendix A

FIGURE 1.

Elbow singularity of the Puma robot

With reference to the figure above, the relations below can be derived: