## Planar Linkages Following a Prescribed Motion

Matteo Gallet, Christoph Koutschan, Zijia Li, Georg Regensburger, Josef Schicho, Nelly Villamizar

Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Research Institute for Symbolic Computation (RISC)
Johannes Kepler University Linz (JKU)

### Abstract

Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.

The paper is published in Mathematics of Computation (DOI: 10.1090/mcom/3120). Here is the preprint version.

### Example 1: Ellipse

We first consider the ellipse defined by the equation (x + 1)2 + 4y2 = 1, i.e., its center is (-1, 0) and its radii are 1 and 1/2. Our algorithm ConstructStrongLinkage applied to the parametrization (-2, t) / (t2 + 1), yields a linkage with 10 links and 13 joints, which realizes a translational motion along the ellipse. If we skip the requirement that the motion is a translational one, i.e., if we just want to draw the ellipse, we can obtain an even simpler linkage, namely one with only 8 links and 10 joints (see the animation):

The link graph is given by the following figure (note that the vertices of this graph correspond to the links of the linkage, and the edges to the joints). The coloring is the same as in the figure above. The numbers correspond to the labeling chosen in the paper. Note that our algorithm always yields linkages with such a ladder-shaped link graph.

When we realize the links as straight rods, then we can find a spatial arrangement of them which produces only two self-collisions. Both happen at the same time, namely when the linkage is in its initial position, corresponding to the right-most point (0, 0) of the ellipse. This situation is visualized in the following 3D graphics, at an instant shortly before the collisions happen: the green link will collide with the joint connecting yellow and cyan, and the blue link with the joint connecting orange and red (see the animation):

### Example 2: John Hancock's J

While this linkage draws the depicted J, seven self-collisions occur. However, they are hardly visible, even in a 3D animation. The picture below shows the linkage at position t = –0.319038; we observe three collisions, namely the yellow link collides with the joint connecting blue and red, the red link collides with the joint connecting yellow and green, and the orange link with the joint connecting the two cyan links.

### Example 3: Astroid

The astroid is a star-like curve with four cusps that is obtained by rolling a circle of radius 1 inside a circle of radius 4. After the small circle is back to its original position (after travelling clockwise, say), it has performed three counter-clockwise revolutions. Hence we can obtain its motion by composing a clockwise revolution around the origin with four counter-clockwise revolutions (one to compensate the clockwise rotation!) around the point (3,0). This yields a motion polynomial of degree 5. In the animation, this decomposition of the motion can be we well observed: the black link is fix, i.e., it moves according to the identity (corresponding to zero linear factors). The brown link performs the clockwise revolution. The red link performs a circular translation, corresponding to the composition of one clockwise and one counter-clockwise revolution. The orange, yellow, and white link make one, two, resp. three counter-clockwise revolutions at the same time, corresponding to the product of three, four, resp. five linear factors.

### Animations

Here we list again all videos that were mentioned above, showing the linkage drawing an ellipse and the linkage signing a J.

### Software

All algorithms described in our paper have been implemented in the computer algebra system Mathematica, e.g., the factorization algorithm for motion polynomials and the construction of linkages with mobility one. In the notebook below we have collected a few instructive examples and the commands that produce some of the above pictures and animations. At ISSAC 2016, our software was awarded with the Distinguished Software Demonstration Award.