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)
Austrian Academy of Sciences (ÖAW)

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

      Linkage drawing an ellipse


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):

Linkage drawing an ellipse

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.

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):

3D view of the ellipse linkage

On the other hand, we describe a procedure how to realize such linkages in a completely collision-free way, by allowing U-shaped and Z-shaped links, in addition to the flat links (F-links) which are just straight rods located in a single layer. The linkage shown here has three F-links (green, red, magenta), three U-links (white, yellow, orange), and two Z-links (cyan, blue). See also the animation.

Link graph

Example 2: John Hancock's J

Next we present an example in connection with a popular formulation of Kempe's Theorem, stating that There is a linkage that signs your name. Henry King attributes this formulation to William Thurston. On the other hand, as remarked by Joseph O'Rourke, it is very implausible that a concrete signing linkage has ever been constructed due to the complexity, in terms of links and joints, of the linkages produced by Kempe's procedure. As an example to support his claim, O'Rourke points out that already constructing a linkage drawing the J of John Hancock's famous signature would be very difficult. We approximate this letter by a rational curve that is given by a parametrization of the form (f/h, g/h) where f,g,h are polynomials of degree 5,5,6, respectively. Our implementation of the algorithm ConstructStrongLinkage, which we develop in the paper, produces a linkage with 26 links and 37 joints drawing this curve. The motion it realizes is a translation, which we visualize by using a quill pen whose shape is a line segment in direction (5,6). The spatial arrangement of the links is indicated by hue: darker links lie below brighter ones (see the animation).

Linkage drawing the letter 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.


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


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.
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