Abstract
Current serial robots, encountered not only in research laboratories but also in production or construction environments, include features that deserve a chapter apart. We will call here general serial robots all non-redundant serial robots that do not fall in the category of those studied in Chap. 4. Thus, the chapter is devoted to manipulators of the serial type that do not allow a decoupling of the positioning and the orientation problems. The focus of the chapter is, thus, the inverse displacement problem (IDP) of general six-revolute robots. While redundant manipulators of the serial type fall within this category as well, we will leave these aside, for their redundancy resolution calls for a more specialized background than what we have either assumed or given here.
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- 1.
N.B. Lee and Li of the references in this chapter are one and the same person, namely, Dr.-Ing. Hongyou Lee (a.k.a. Dr.-Ing. Hongyou Li).
- 2.
By power product we mean terms with their coefficients deleted; for example, the power products of the polynomial \(5{x}^{2}y + 3xz + 9{y}^{2} + 4z = 0\) are the terms x 2 y, xz, y 2 and z.
- 3.
Neither Li nor Raghavan and Roth disclosed the geometric interpretation of this fourth equation, first proposed by Lee and Liang (1988).
- 4.
while the last row of Z 3 is free of θ 3, the last row of \(\mathbf{X}_{2}\mathbf{Z}_{3}\) is \([\mu _{2}s_{3},\ -\mu _{2}c_{3},\ \lambda _{2}]\).
- 5.
Formulation singularities occur when, in the absence of a kinematic singularity—characterized by the vanishing of det(J), for J defined as in Eq. (5.10b)—two or three contours \(\mathcal{C}_{i}\) are tangent at an intersection. When this is the case, and a pair of functions (9.62) is chosen to find their roots, whose contours are tangent, the numerical computation of the coordinates of the intersection point becomes impossible.
- 6.
- 7.
Graphical methods of mechanism analysis rely on this form of linear-equation solving.
- 8.
The Italian mathematicians Niccolò Tartaglia—meaning the “stammerer,” his real name believed to have been Fontana—(1535) and Girolamo Cardano (1545), independently, or so each claimed, found the formula for the three roots of the cubic equation, now known as Cardan’s formula. Ferrari’s formula—so named after the Italian mathematician Ludovico Ferrari, a disciple of Cardano’s—provides the four roots of a quartic polynomial.
- 9.
The ODA library is available on www.mcgill.ca/~/rmsl/Angeles_html/courses/MECH577/.
- 10.
The accompanying CD includes a GUI allowing the user to automate the computation of accurate values of the joint variables by clicking at the visual estimates of the intersections of all four contours.
- 11.
The self-motion is not readily detected by contour-intersection using this procedure.
- 12.
This idea was proposed by Dr. Kourosh Etemadi Zanganeh, CANMET (Nepean, Ontario, Canada).
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Angeles, J. (2014). Geometry of General Serial Robots. In: Fundamentals of Robotic Mechanical Systems. Mechanical Engineering Series, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-01851-5_9
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