China
Africa
Explore chapters and articles related to this topic
Manipulator Kinematics
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
The question of how to solve the inverse kinematics problem in general, (that is, in the absence of any intersections of the axes of the manipulator) for both planar and spatial mechanisms is an extremely active area of current research. In particular, there are many interesting questions about the number of inverse kinematics solutions and how the computations can be mechanized in real-time. In this subsection, we give a brief summary of some of the newest and most general approaches in this regard, drawn from [58], [65] and [96]. Our development closely parallels that of Manocha and Canny [65]. The approaches are primarily based on classic elimination theory from algebraic geometry; this is a systematic procedure for simultaneously eliminating n − 1 of the variables in a system of n polynomials in n variables to obtain a single polynomial in one variable. This procedure is a general procedure but also a “brute-force” procedure, in that it only takes very simple properties of the manipulator kinematics into account (unlike the solutions based on the subproblems listed above). We illustrate this procedure, sometimes referred to as dialytical elimination, in the following example of three nonhomogeneous polynomials in three variables.
Performance and robustness trade-offs in PIR control of uncertain second-order systems with input disturbances
Published in International Journal of Control, 2023
While recent studies have demonstrated the benefits of delays in feedback control, these studies have also recognised that designing delay-based controllers is not trivial. One of the main difficulties arising is that delay brings about infinitely many system poles, and traditional pole placement techniques are no longer applicable. To overcome this difficulty, authors rely on mathematical tools; e.g., elimination theory (Ramírez & Sipahi, 2018a), algebraic geometry (Ramírez et al., 2016), Lambert W functions (Ramírez & Sipahi, 2018b), dimensionless quantities (Zítek et al., 2013), multiplicity induced dominancy properties (Castillo-Zamora et al., 2022), convex directions for quasipolynomials (Gomez & Ramírez, 2022) and Nyquist plots (Mammadov et al., 2022), under which the problem of assigning a small group of closed-loop rightmost poles becomes solvable by radicals. With closed-form formulas at hand, delay-based controllers can be precisely calculated and furthermore, these controllers have been shown to outperform their counterparts, PID, PI, PD and P controllers, in benchmark tests, in terms of noise mitigation capabilities and transient response specifications (Mert Özer & İftar, 2022; Özer & İftar, 2021).
General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings
Published in International Journal of Mathematical Education in Science and Technology, 2020
A. Cuida, F. Laudano, E. Martinez-Moro
Note that the topics stated above can be also stated in terms of resultants (see Lang, 2002, Chapter IV, § 8 for their precise definition), both the bezoutian and the resultant matrix came in the eighteenth century from Eulers work in elimination theory (Wimmer, 1990). It is well known that the resultant of two polynomials can be expresed in terms of the diferences of their roots and it is 0 if and have a common root. It is easy to check that if is a monic polynomial, moreover, if both and are monic Resultants can be computed in any unique factorization domain and they behave well under ring specialization. As mentioned before they are commonly defined as the determinant of a Sylvester matrix whose entries are related to the coefficients of the polynomials. When one wants to compute the resultant in an efficient way one has to take into accout the so called ‘invariance under change of polynomials ’ property that involves the Euclidean algorithm for polynomials. That is, the resultant of two successive remainders differs from the resultant of the initial polynomials by a controlable factor involving the leading coefficients and this leads again to our approach. Following the example above it is clear that Thus indeed a deep understanding of the remainder-division algorithm is a key point in an abstract algebra course involving elimination (resultants) theory.
Coercive polynomials: stability, order of growth, and Newton polytopes
Published in Optimization, 2019
Various algebraic and analytic properties of polynomials are encoded in the properties of their Newton polytopes. To name some of them, for example the number of isolated roots of n polynomial equations in n unknowns can be bounded by the (mixed) volumes of their Newton polytopes (cf. e.g. [4–6]), absolute irreducibility of a polynomial is implied by the indecomposability of its Newton polytope in the sense of Minkowski sums of polytopes [7], and there are also some results dealing with Newton polytopes in elimination theory [8].