# Expansion of MK Circle Theory for Dyads and Triads

^{*}

## Abstract

**:**

## 1. Introduction

^{1}[1,2,3] circles are a useful tool for both finding solutions to mechanism design problems and for visualizing properties of the possible pivot location solution space. The name MK circle stems from the German words, “Mittelpunkt” and “Kreispunkt”, which refer to the center point (ground pivot) and circle point (moving pivot) of a dyad. This information is compiled and updated in a new way, and then expanded to show its applications to triads in several prescribed positions. The property of kinematic synthesis that dyad pivot location solutions settle into circles was first observed by Loerch, who demonstrated the existence of these circles, identified some of their properties, and found solutions for up to five prescribed positions of path generation [1]. Mlinar expanded on these findings to include triad synthesis, but primarily focused on the existence of forbidden regions in triad synthesis and did not lean into the potential implications for identifying solution triads [3,4].

_{1}joints.

## 2. Body

**δ**, α

_{2}_{2},

**δ**, and α

_{3}_{3}are prescribed and both the angles β

_{2}and β

_{3}are free choices. If a designer holds the value of β

_{2}constant in Equation (1), then iterates through many values of β

_{3}, plotting the XY coordinates of either the ground pivot (vector origin for W) or the moving pivot (vector origin for Z) will cause a circle to emerge. Each point on this circle represents a unique pivot location which is a solution to Equation (1). The standard form equations for a dyad in four positions are included in Appendix A.

_{3}. If no satisfactory solutions exist on the circle (for example, an acceptable ground or moving pivot location), they may choose a new value of β

_{2}, which then generates a new circle of β

_{3}values. One such pair of MK circles is seen in Figure 1. Consistently throughout this paper, figures use red points to indicate ground pivot locations, and blue points to indicate moving pivot locations.

## 3. Poles

_{23′}is found by reflecting the point P

_{23}across the line $\overline{{\mathrm{P}}_{12}{\mathrm{P}}_{13}}$.

_{3}, the solution density of ground and moving pivots is highest around certain poles, and the density is the lowest at the opposite end of the circle. Notice that for the ground pivot circle of a three-position problem, this high-density region forms around pole P

_{13}. While the circles are continuous if an infinite number of points are plotted, this indicates that a larger range of β

_{3}values have their solution adjacent to the given pole.

_{23}to find P

_{23′}), and then using Equation (4).

_{12}and P

_{23}, m is the slope of that same line, and d is the expression shown in Equation (5). Real() and imag() denote the real and imaginary components of the pole [14].

_{2}has been held constant, while β

_{3}was varied. If β

_{2}is also varied, an interesting pattern emerges. In Figure 2, six values of β

_{2}are plotted with the same several hundred values of β

_{3}as before. Each unique value of β

_{2}generates a new circle. What makes this outcome so interesting, though, is that each of these circles passes through the same two points, which happen to be the poles.

_{13}is a consistent ground pivot solution because of the case when β

_{3}= α

_{3}. In this instance, the dyad will behave similarly to a single link as it rotates from position one to three, and therefore the only place a ground pivot could exist is P

_{13}. Similarly, P

_{13}is a consistent moving pivot location because of the special case where β

_{3}= 0, in which case all motion between positions one and three would have to stem from the rotation of the moving plane about the pivot point K

_{1}. The poles are also useful as they mark points that the Burmester curves pass through for the four specified positions, allowing for a rough sketch of the shape of the curves with limited calculation [2,15].

_{23′}using Equations (2)–(4). P

_{13}and P

_{23}are associated with the M circle, and P

_{13}and P

_{23′}are associated with the K circle. To find the circle centers, draw a perpendicular bisector through each of these two pairs of poles. These two lines are the centerlines on which the center points of each circle will lie. For each unique value of β

_{2}, use the angular relationships established in Equations (6) and (7) to find the center points (C

_{M}and C

_{K}) of the M and K circles, respectively.

_{2}is negative, an observer viewing pole P

_{13}from the center of the circle would rotate clockwise by the angle $\left|{\mathsf{\beta}}_{2}\right|$ to arrive at pole P

_{23}(as in Figure 3). If the value of β

_{2}is positive, the observer rotates counterclockwise from P

_{13}to P

_{23}.

_{2}is found without using the standard form equation.

_{2}, iterating through values of β

_{3}will generate the 123 circle, and iterating through values of β

_{4}will generate the 124 circle. The pairs of circles will intersect zero, one (if the circles are tangent to one another), or two times, indicating the number of solutions available for the given value of β

_{2}. These intersecting circle pairs for both M and K circles can be seen in Figure 4.

_{2}. The procedure is as follows.

_{2}using Equations (6)–(9). If the inter-center distance between any two circles is greater than r

_{M1}+ r

_{M2}, the circles do not intersect, and no solutions exist for that given value of β

_{2}. Second, the distance from each circle center to the radical line must be found. For any two non-congruent circles, the radical line is the line passing through their intersection points [16,17]. These distances are given in Equation (10), where d is the distance between the circle’s center-points.

_{1}determined, all the required information is known to identify the intersection points. The last step is to adjust the coordinate system to align the local system with the global coordinate system. This is achieved by multiplying by unit vectors set in the proper directions. The two vectors are given in Equations (12) and (13) [18].

_{2}and the pole positions, which are set by the problem definition. The simplest way to implement this expression for the five-position case is to identify the intersection points between the first two circles, and then find the intersections between the third circle and either of the first two. A viable value of β

_{2}is determined in any case where a matching intersection point is found.

## 4. Triad MKT Circles

**Z**or the tail of

**V**which extends out to the precision point. Figure A2b shows a triad in two positions with the appropriate labeling.

## 5. Triad Circles

_{2}, β

_{3}, and β

_{4}, as opposed to β

_{2}and β

_{3}for the dyad). An example of varying multiple free choice variables is shown in Figure 6. In this Figure, α

_{2}and α

_{3}are free choices that locate the center point location of the circles. For each new value of α

_{2}or α

_{3}that is plotted, an additional circle emerges. The points on these pivot circles are found by varying the value of α

_{4}from 0 to 360°. Figure 6 shows six circles of each type (M ground, K

_{1}moving pivot, T

_{1}moving pivot).

_{2}or α

_{2}. The solution is found at the three-way intersection of circles formed from positions 1,2,3,4, 1,2,3,5 and 1,2,3,6. A designer may make a free choice for the value of β

_{2}, then solve for the value of β

_{3}which produces a triple intersection point (if indeed any exist, not all values of β

_{2}are guaranteed to produce a valid solution) [5] (p. 43). Figure A5 depicts a triple intersection point for a triad solved for six prescribed positions. In Figure A6, a solution triad is shown for a problem defined by seven precision positions. In the seven-position case, four circles must intersect at a single point, and the designer has no free choices. This makes the seven-position case quite difficult to implement practically, as solutions are extremely limited.

## 6. Example

**Solution**: first, a basic linkage topology must be selected. Based on observation and experience it is doubtful whether a basic four-bar linkage would be able to achieve the desired translation and rotation while still fitting inside a reasonably small space. The Watt 1 six-bar mechanism is known for producing complex motions, so that is the linkage topology selected. This mechanism is shown with its dyad and triad loops highlighted in Figure 8.

#### 6.1. Loop One

_{j}and α

_{j}angles are initially free selections for this dyad, but their values have great implications for the other two chains. Thus, even for four prescribed positions, there are many more possible solutions for this dyad than can be represented by a single selection of input parameters. In Figure 10, the MK circles for the selected four-precision position dyad problem are shown.

#### 6.2. Loop Two

_{1–4}are given in the problem definition, and β

_{1–4}must match the β values of the dyad calculated in loop one or else they cannot combine into the single ternary link 2 in Figure 8a. This means that only the α angles remain unprescribed. Any of these three α

_{j}angles may be chosen as a free choice, but the other two must be solved for. One way to do this with the MKT circles is to iterate through many values of α

_{3}and α

_{4}for a given value of α

_{2}, then pick out the set of pivots based on whichever set produces a matching ground pivot to loop one. While inexact, designers should be able to quickly find matching solutions within four or five decimal point accuracy by increasing the number of selections considered for α

_{3}and α

_{4}. This procedure is applied in Figure 11. Only the ground pivot circles are shown, with 50 circles plotted comprised of 200 points each.

_{2}will cause these regions to shift, but if a particular value of α

_{2}is desired for the triad, it may be necessary to reassign the free choice values of the loop one dyad to make finding a viable solution possible.

_{3}, and each pivot point on the circles is plotted via a unique value of α

_{4}(or vice versa). So, once the matching ground pivot point is identified, the values of α

_{3}and α

_{4}are inherently known—they are whichever two values were used to create that point. In this example, the solution is found when α

_{3}= 90.062 and α

_{4}= 132.141 degrees, based on a free-choice value of α

_{2}= 24.919 degrees. This value of α

_{2}was selected after some trial and error with the final mechanism construction, as the value appeared to produce favorable motion results.

#### 6.3. Loop Three

_{j}angles for the loop three dyad are prescribed, as they must match with the α

_{j}values of the triad to form link five, shown in Figure 8. It is also important to note that the precision positions are no longer those specified in the first two loops, but rather the distal precision position minus the

**V**vector of the triad at this position. This new distal displacement is symbolized by ${\mathsf{\delta}}_{\mathbf{j}}^{\mathbf{\prime}}$, determined as shown in Equation (16).

_{j}angles to define the motion, the α

_{j}angles of the triad define the motion.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Position # | Position Coordinates | β | α |
---|---|---|---|

1 | 0 + 0i ^{1} | 0 | 0 |

2 | 0.130 + 0.274i | −44.775° ^{2} | 24.919° |

3 | −0.246 + 0.396i | Unknown | 90.062° |

4 | −0.586 + 0.167i | Unknown | 132.141° |

^{1}The position coordinates are uniformly adjusted to have PP1 as (0, 0).

^{2}The value of the free choice ${\mathsf{\beta}}_{2}$ used in the MK solution for loop three was −45.508. The value was modified by 0.733° for this example to account for rounding errors within the solution software and to produce an identical dyad solution.

_{2}is a free choice.

**W**and

**Z**. The compatibility linkage method ensures that the values of β

_{3}and β

_{4}will be compatible with all four positions, not just the first three. The resultant dyad is shown in its four positions in Figure A1 and is identical to the loop three dyad shown in Figure 12 and Figure 13.

## Appendix B

**Figure A2.**(

**a**). A dyad shown in two prescribed positions, with key vectors and angular displacements labeled. (

**b**). A triad shown in two prescribed positions, with key vectors and angular displacements labeled. Note that this triad notation assigns the vector link “V” to the third link of the triad chain, as opposed to the intermediate link where it has classically been assigned. We feel this updated nomenclature is more intuitive for designers who are well acquainted with dyads and hope to incorporate triads into their designs [5,26,27].

**Five Prescribed Position Synthesis for Dyads:**

_{2}, the quest is for a value of β

_{2}that yields a motion generation dyad for all five positions. This requires a nonlinear solution or an optional search [2,5]. Figure A3 shows one triple intersection and the resulting dyad.

**Figure A3.**A depiction of a triple intersection point for the MK circles of a dyad in five positions. PP1 = 0 + 0.0i, PP2 = 1 + 2i, PP3 = 2.5 + 4i, PP4 = 4 + 3.5i, and PP5 = 4.6865 + 3.0306i. Note that when a solution point exists on the K circle, a matching solution will exist on the M circle as well. Here,

**W**= 2.67 − 2.59i, and

**Z**= −0.64 − 2.19i, α

_{2}= 0, α

_{3}= 45, α

_{4}= 70, and α

_{5}= 85. The solution is found when β

_{2}= 35.002. It is possible but not guaranteed that other values of β

_{2}may also produce solutions.

**Figure A4.**This Figure depicts the same circles as Figure 5, but here the vectors spanning between the circles are shown, demonstrating how each point on the circles represents a unique triad chain. PP1 = 0 + 4i, PP2 = 2.553 + 3.329i, PP3 = 5.023 + 1.957i, PP4 = 8.564 + 6.014i. β

_{2}= 10, β

_{3}= 70, β

_{4}= 140, α

_{2}= 60, γ

_{2}= 30, γ

_{3}= 90, γ

_{4}= 145.

**Six Position Case:**

**Figure A5.**The triad MKT circles are shown in six prescribed positions, with one possible solution triad highlighted. The solution requires three circles intersecting at a single point to find solutions. PP1 = 0 + 0.0i, PP2 = 1.553 + 1.329i, PP3 = 3.023 + 2.957i, PP4 = 4.564 + 2.014i, PP5 = 7.432 + 2.016i, PP6 = 8.670 + 0.737, β

_{2}= 30, β

_{3}= 70, β

_{4}= 140, β

_{5}= 150, β

_{6}= 130, γ

_{2}= −30, γ

_{3}= −60, γ

_{4}= −65, γ

_{5}= −90, γ

_{6}= −110. A solution vector is found for α

_{2}= −15.

**W**= 1.651 + 1.567i,

**Z**= −2.134 + 0.998i,

**V**= −6.326 + 1.137i.

**Seven Position Case:**

**Figure A6.**The triad MKT circles are shown in seven prescribed positions, with one possible solution triad highlighted. PP1 = 0 + 0.0i, PP2 = 1.553 + 1.329i, PP3 = 3.023 + 2.957i, PP4 = 4.564 + 2.014i, PP5 = 7.432 + 2.016i, PP6 = 8.670 + 0.737, PP7 = 11.174 − 0.659i, β

_{2}= 30, β

_{3}= 70, β

_{4}= 140, β

_{5}= 150, β

_{6}= 130, β

_{7}= 160, γ

_{2}= −30, γ

_{3}= −60, γ

_{4}= −65, γ

_{5}= −90, γ

_{6}= −110, γ

_{7}= −130. A solution vector is found for α

_{2}= −15.

**W**= 1.651 + 1.567i,

**Z**= −2.134 + 0.998i,

**V**= −6.326 + 1.137i.

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**Figure 1.**A depiction of the MK circles for a dyad in three positions. PP1 = (0, 0).

**δ**= 1.75 + 1i.

_{2}**δ**= 3.5 + 3i. α

_{3}_{2}= −50, α

_{3}= −90, and β

_{2}= −110. The red circle represents potential ground pivot locations, while the blue circle represents potential moving pivot locations.

**Figure 2.**Six values of β

_{2}are plotted with several hundred values of β

_{3}each. β

_{2}has an initial value of −110 degrees, as in Figure 1, and the increment between each value of β

_{2}is 60 degrees.

**Figure 3.**A depiction of a single ground pivot circle with the key geometric parameters labeled for the identification of the center point coordinates and radius of the circle in terms of β

_{2}.

**Figure 4.**Sample view of the MK circles for a dyad in four precision positions, with the poles,

**W**, and

**Z**vectors shown. Here, PP1 = 0 + 0i, PP2 = 1.75 + 1i, PP3 = 3.5 + 2i, PP4 = 5.5 + 1i. α

_{2}= −50, α

_{3}= −90, α

_{4}= −110. Pictured circles are for β

_{2}= −110.

**W**= 0.514 + 0.382i,

**Z**= −3.141 + 1.245i.

**W***= 1.232 + 0.894i,

**Z***= −4.895 + 1.036i.

**Figure 5.**A depiction of the MKT circles of a triad in four precision positions. In this example, the γ

_{j}and β

_{j}angles are prescribed in the problem, indicating the base triad case is a motion generation problem with prescribed timing, and making the α

_{j}angles the free choices. However, any of the three sets of angles can be taken as a free choice by defining the problem differently. Here, PP1 = 0 + 4.0i, PP2 = 2.552 + 3.329i, PP3 = 5.023 + 1.957i, PP4 = 8.564 + 6.014i, α

_{2}= 60. γ

_{2}= 30, γ

_{3}= 90, γ

_{4}= 145, β

_{2}= 10, β

_{3}= 70, β

_{4}= 140.

**Figure 6.**A triad in four positions. Testing six values of α

_{3}(resulting in six circles in each M, K and T set) and numerous values of α

_{4}(creating each unique point on the circles). PP1 = 0 + 4i, PP2 = 2.553 + 3.329i, PP3 = 5.023 + 1.957i, PP4 = 8.564 + 6.014i. β

_{2}= 60, β

_{3}= 70, β

_{4}= 140, α

_{2}= 40, γ

_{2}= 30, γ

_{3}= 90, γ

_{4}= 145.

**Figure 7.**The triad MKT circles are shown in five prescribed positions, with one possible solution triad highlighted. PP1 = 0 + 0.0i, PP2 = 1.552 + 1.329i, PP3 = 3.023 + 2.957i, PP4 = 4.564 + 2.014i, PP5 = 6.564 + 1.014i. β

_{2}= 30, β

_{3}= 70, β

_{4}= 140, β

_{5}= 150, α

_{2}= 60, γ

_{2}= −30, γ

_{3}= −60, γ

_{4}= −65, γ

_{5}= −90. The highlighted solution is

**W**= −1.634 + 0.695i,

**Z**= 1.694 + 0.901i,

**V**= −5.351 + 2.733i.

**Figure 8.**(

**a**) The Watt 1 topology is shown, with the individual loops highlighted. (

**b**) The loops are separated to show the individual synthesis chains more clearly. Loop one and loop three are dyads shown in black and blue, respectively, while loop two is a triad shown in red.

**Figure 10.**The MK circles for the loop one dyad of the final mechanism, with a dyad solution shown. W = 0.786 − 0.208i, and Z = −0.230 + 0.644i. The β

_{j}angles were set as β

_{2}= −18.582, β

_{3}= −21.767, β

_{4}= −12.517. The free choice was chosen as α

_{2}= 5.862.

**Figure 11.**The ground pivot circles of a triad, plotting numerous values of α

_{3}and α

_{4}. The K

_{1}and T

_{1}circles are hidden to simplify the figure. The matching ground pivot location is highlighted.

**Figure 12.**A depiction of the MK circles for the loop three dyad. PP1 = 0 + 0i, PP2 = 0.130 + 0.274i PP3 = −0.246 + 0.396i, PP4 = −0.586 + 0.167i. α

_{2}= 24.919, α

_{3}= 90.062, α

_{4}= 132.141. Solution found for β

_{2}= −45.508, β

_{3}= −90.331, β

_{4}= −95.758.

**Figure 13.**An assembled prototype visualized at a few points in its motion. The loops are labeled in the first position. Clockwise from top left, positions one to four.

**Table 1.**A classification of the prescribed data and maximum number of potential solutions for each number of prescribed positions of a dyad.

Number of Positions | Prescribed Values | Unknowns | Free Choices | Number of Solutions |
---|---|---|---|---|

2 | ${\mathit{\delta}}_{\mathbf{2}}^{}$, α_{2} | 5 *—W, Z, β_{2} | 3—β_{2}, W or Z | ∞ ^{3} |

3 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{3}}^{}$, α_{2–3} | 6—W, Z, β_{2–3} | 2—β_{2–3} | ∞ ^{2} |

4 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{4}}^{}$, α_{2–4} | 7—W, Z, β_{2–4} | 1—β_{2} | ∞ ^{1} |

5 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{5}}^{}$, α_{2–5} | 8—W, Z, β_{2–5} | 0 | Finite |

**W**and

**Z**are both vector quantities, they account for two unknowns each. Table shows the number of free choices available for a dyad based on the number of prescribed positions. Notice that for three positions (Equation (1)) there are two free choices, which are represented by the MK circles as β

_{2}and β

_{3}are varied. For the cases of four and five positions, this paper shows how multiple sets of MK circles are combined to yield dyad solutions.

**Table 2.**A classification of the prescribed data and maximum number of solutions for each number of prescribed positions of a triad.

Number of Positions | Prescribed Values | Unknowns | Free Choices | Number of Solutions |
---|---|---|---|---|

2 | ${\mathit{\delta}}_{\mathbf{2}}^{}$, α_{2}, γ_{2} | 7 *—W, Z, V, β_{2} | 5—β_{2}, W and Z | ∞ ^{5} |

3 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{3}}^{}$, α_{2–3,} γ_{2–3} | 8—W, Z, V, β_{2–3} | 4—β_{2–3}, W | ∞ ^{4} |

4 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{4}}^{}$, α_{2–4,} γ_{2–4} | 9—W, Z, V, β_{2–4} | 3—β_{2–4} | ∞ ^{3} |

5 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{5}}^{}$, α_{2–5,} γ_{2–5} | 10—W, Z, V, β_{2–5} | 2—β_{2–3} | ∞ ^{2} |

6 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{6}}^{}$, α_{2–6,} γ_{2–6} | 11—W, Z, V, β_{2–6} | 1—β_{2} | ∞ ^{1} |

7 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{7}}^{}$, α_{2–7,} γ_{2–7} | 12—W, Z, V, β_{2–7} | 0 | Finite |

**W**and

**Z**are both vector quantities, they account for two unknowns each.

Loop # | Prescribed Values | Unknowns | Connection |
---|---|---|---|

Loop 1 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{4}}^{}$ | Oa, β_{2–4}, α_{2–4} | - |

Loop 2 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{4}}^{}$, β_{2–4}, Oa | α_{2–4} | β_{2–4}, Oa from Loop 1 |

Loop 3 | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{4}}^{\mathit{\prime}}$, α_{2–4} | β_{2–4} | ${\mathit{\delta}}_{\mathbf{2}\mathbf{-}\mathbf{4}}^{\mathit{\prime}}$, α_{2–4} from Loop 2 * |

_{j}minus V

_{j}, as shown in Equation (9).

Loop | W ^{1} | Z | V |
---|---|---|---|

1 | −0.725 + 0.369i | 0.617 − 0.296i | - |

2 | −0.504 + 0.603i | 0.369 + 0.129i | 0.017 − 0.651i |

3 | −0.164 + 0.239i | 0.562 + 0.090i | - |

^{1}All values are for the mechanism in its first position and rounded to the third decimal point.

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## Share and Cite

**MDPI and ACS Style**

Mather, S.; Erdman, A.
Expansion of MK Circle Theory for Dyads and Triads. *Machines* **2023**, *11*, 841.
https://doi.org/10.3390/machines11080841

**AMA Style**

Mather S, Erdman A.
Expansion of MK Circle Theory for Dyads and Triads. *Machines*. 2023; 11(8):841.
https://doi.org/10.3390/machines11080841

**Chicago/Turabian Style**

Mather, Sean, and Arthur Erdman.
2023. "Expansion of MK Circle Theory for Dyads and Triads" *Machines* 11, no. 8: 841.
https://doi.org/10.3390/machines11080841