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Inverse Kinematic Modeling of Bending Response of Ionic Polymer Metal Composite Actuators
Published in Srijan Bhattacharya, Ionic Polymer–Metal Composites, 2022
Siladitya Khan, Gautam Gare, Ritwik Chattaraj, Srijan Bhattacharya, Bikash Bepari, Subhasis Bhaumik
The various steps involved in the computation of the IK solution with the help of Tractrix-based IK algorithm in the global reference frame are pictorially represented in Figure 5.3. The described IK solver had originated in order to resolve redundancy in serial multi-body mobile platforms. However, in the context of a manipulator, the base being fixed mandates the displacement of the tail location of the link l1 be subtracted from the position of all the links, which results in the joint motion to be constrained to execute solely rotary movements. This action results in a positional error between the end effector of the manipulator and its designated target location, which is minimized by executing the said algorithm iteratively. In this context, it is worth mentionable that the notion of ‘Tractrix’ emerged from the perspective of an object beginning its motion with a vertical offset and being dragged along a horizontal line along the X-axis. The described Tractrix algorithm extended the resulting motion in three-dimensional Cartesian space, for which an arbitrary position of the end effector was seen to drive the motion of all other links using the Tractrix solution. Once the end effector positions are known, the angular alignments of the manipulator are computed by resorting to simple vector algebra, thereby creating an effective solution to the incumbent redundancy. The algorithm was validated on an eight-link hyper-redundant mobile prototype, in addition to performing simulations of a moving snake and tying knots with a rope as examples of its potential applications [67]. The animated visuals were found to be more realistic since the displacement of the links diminished gradually from head to tail.
A 3D-printable machine for conics and oblique trajectories
Published in International Journal of Mathematical Education in Science and Technology, 2022
Pietro Milici, Frédérique Plantevin, Massimo Salvi
Descartes' foundation limited geometry to algebraic curves. However, already in the second half of the 17th century, this boundary was no longer widely accepted by mathematicians, who looked for an appropriate geometrical legitimation to introduce non-algebraic curves. Among other less powerful geometrical methods, a general problem that originated a broad class of transcendental curves was the inverse tangent problem (that, in a modern setting, involves the geometrical solution of differential equations). Although to find an object tangent to a given curve while satisfying specific properties (the direct tangent problem) is present at least since classical Greek geometry, it was only after Descartes that mathematicians tried to consider new curves given the properties that their tangents have to satisfy. The first documented appearance of an inverse tangent problem is the ‘constant-subtangent problem’ studied by Florimond Debeaune in 1638 Scriba (1961). In 1672 the architect Claude Perrault proposed a first operative construction of a curve given its tangent: a new curve, the tractrix, is introduced as the trace of a pocket watch on a plane while moving the endpoint of its chain along a straight line. The role of traction made such constructions termed ‘tractional’. Until the first half of the 18th century, tractional constructions interested many influential mathematicians, from Leibniz to Euler (see Bos, 1988; Tournès, 2009 or also, for a brief introduction, Crippa & Milici, 2019). Quite curiously, it seems that the geometrical principle behind tractional constructions hides something interesting yet (Dawson et al., 2020).