# On the Self-Mobility of Point-Symmetric Hexapods

## Abstract

**:**

## 1. Introduction

_{i}with coordinates M

_{i}:= (A

_{i}, B

_{i}, C

_{i})

^{T}with respect to the fixed frame and by the six platform anchor points m

_{i}with coordinates m

_{i}:= (a

_{i}, b

_{i}, c

_{i})

^{T}with respect to the moving frame (for i = 1, …, 6). Each pair (M

_{i}, m

_{i}) of corresponding anchor points of the fixed body Σ

_{0}(base) and the moving body Σ (platform) is connected by an SPS-leg, where only the prismatic joint (P) is active and the spherical joints (S) are passive. Note that for a hexapod, (M

_{i}, m

_{i}) ≠ (M

_{j}, m

_{j}) holds for pairwise distinct i, j ∈ {1, …, 6}. Moreover, a hexapod is called planar, if M

_{1}, …, M

_{6}, as well as m

_{1}, …, m

_{6}are coplanar; otherwise, it is called non-planar.

_{i}≥ 0 of the six legs, the hexapod is generically rigid. However, under particular conditions, this body-bar framework can perform an n-dimensional motion (n > 0), which is called self-motion (=continuous flexion).

**Definition 1.**A hexapod is called point-symmetric if it has the following properties (after a possible necessary relabeling of anchor points):

- M
_{i}and M_{i}_{+3}are symmetric with respect to a point O of the fixed frame for i = 1, 2, 3; - m
_{i}and m_{i}_{+3}are symmetric with respect to a point o of the moving frame for i = 1, 2, 3; - None of the following three distance conditions are satisfied for all distinct i, j ∈ {1, 2, 3} : (a)${\overline{\mathrm{OM}}}_{i}$; (b)${\overline{\mathrm{om}}}_{i}$; (c)${\overline{\mathrm{OM}}}_{i}$.

**Corollary 2.**A PSH is architecturally singular if and only if one of the following cases hold (after a possible necessary renumbering of anchor points and the exchange of the platform and the base):

- m
_{1}, …, m_{6}are collinear; - m
_{1}, m_{2}, m_{4}, m_{5}are collinear, M_{1}, M_{2}, M_{4}, M_{5}are collinear and CR(m_{1}, m_{2}, m_{4}, m_{5}) = CR(M_{1}, M_{2}, M_{4}, M_{5}) holds, where CR denotes the cross-ratio; - The platform and base are planar, and there exists a regular affinity with M
_{i}↦ m_{i}for i = 1, …, 6.

**Proof.**The corollary can directly be concluded from the literature on architecturally singularity (see, e.g., Theorem 1 of [6] and Theorem 3 of [10]). □

**Definition 3.**A PSH is called orthogonal if it fulfills the following conditions:

- OM
_{1}, OM_{2}, OM_{3}are pairwise orthogonal; - om
_{1}, om_{2}, om_{3}are pairwise orthogonal.

**Remark 1.**For better imagining the OPSH’s geometry (with${\overline{\mathrm{OM}}}_{i}$ for i = 1, 2, 3), the anchor points can be interpreted as the vertices of an ellipsoid in the fixed/moving space.

#### 1.1. Results on OPSHs

_{1}(resp. m

_{1}) is located on the x-axis, M

_{2}(resp. m

_{2}) on the y-axis and M

_{3}(resp. m

_{3}) on the z-axis of the fixed frame (resp. moving frame). Therefore, we get:

_{1}, R

_{2}, R

_{3}are involved. If these two conditions are fulfilled, the OPSH has a one-dimensional flexion, which we call the Dietmaier self-motion.

_{1}, R

_{2}, R

_{3}if and only if A = a, B = b and C = c hold. As a consequence, there exists an orientation preserving congruence transformation τ with:

**Remark 2.**If τ of Equation (8) is an orientation reversing congruence transformation, we get a so-called reflection-congruent hexapod. In the more general case of τ being an equiform transformation, we name the hexapod an equiform one.

#### 1.2. Outline of the Article

## 2. Bond Theory

_{0}: e

_{1}: e

_{2}: e

_{3}: f

_{0}: f

_{1}: f

_{2}: f

_{3}) for solving the forward kinematics of hexapods. Note that the first four homogeneous coordinates (e

_{0}: e

_{1}: e

_{2}: e

_{3}) are the so-called Euler parameters. Now, all real points of the Study parameter space P

^{7}(seven-dimensional projective space), which are located on the so-called Study quadric $\mathrm{\Psi}$, correspond to an Euclidean displacement, with the exception of the three-dimensional subspace E of Ψ given by e

_{0}= e

_{1}= e

_{2}= e

_{3}= 0, as its points cannot fulfill the condition N ≠ 0 with $N$. The translation vector

**t**:= 2(t

_{1}, t

_{2}, t

_{3})

^{T}and the rotation matrix

**R**of the corresponding Euclidean displacement x ↦

**R**x +

**t**are given by:

^{7}, which cannot fulfill this normalizing condition, are located on the so-called exceptional cone N = 0 with vertex E.

_{i}is located on a sphere centered in M

_{i}with radius R

_{i}is a quadratic homogeneous equation according to Husty [17]. This so-called sphere condition Λ

_{i}has the following form:

_{1}, …, Λ

_{6}, N = 1. In general, V consists of a discrete set of points, which correspond with the (at most) 40 solutions of the forward kinematic problem.

_{1}, …, Λ

_{6}. If the manipulator has an n-dimensional self-motion, then the algebraic motion also has to be of this dimension. Now, the points of the algebraic motion with N ≠ 0 equal the kinematic image of V. However, we can also consider the points of the algebraic motion, which belong to the exceptional cone N = 0. An exact mathematical definition of these so-called bonds can be given as follows (cf. Remark 5 of [15]):

**Definition 4.**For a hexapod, the set$\mathcal{B}$ of bonds is defined as:

^{3}by the elimination of f

_{0},…,f

_{3}. This projection is denoted by π

_{f}. In the second step, we determine those points of the projected point set π

_{f}(V), which are located on the quadric N = 0; i.e.,

**Remark 3.**A more sophisticated bond theory for pentapods and hexapods is based on a special compactification of SE(3), where the sphere condition Λ

_{i}is only linear in 17 motion parameters. This approach, which was presented in [18], has many theoretical advantages compared to the method described above, but it is not suited for direct computations due to the large number of motion parameters. In contrast, the approach of the paper at hand was already successfully used for direct computations in [15,19–21].

_{f}equals the group of translational motions. As a consequence, a component of V, which corresponds to a pure translational motion, is projected to a single point O (with N ≠ 0) of the Euler parameter space P

^{3}by the elimination of f

_{0}, …, f

_{3}. Therefore, the intersection of O and N = 0 equals Ø, which warrants the exclusion of pure translational motions within this approach. However, this does not cause any trouble, as all hexapods with pure translational self-motion were already characterized by the author in [15] as follows:

_{1}= M

_{1}into a pose, where the vectors $\overrightarrow{{\mathrm{M}}_{i}{\mathrm{m}}_{i}}$ for i = 2, …, 6 fulfill the condition $r$. Moreover, all one-dimensional self-motions are circular translations, which can easily be seen by considering a normal projection of the manipulator in the direction of the parallel vectors $\overrightarrow{{\mathrm{M}}_{i}{\mathrm{m}}_{i}}$ for i = 2, …, 6. If all of these six vectors are zero-vectors, which corresponds with the case that the platform and the base are congruent, we get the already mentioned two-dimensional translational self-motion of a hexapod (cf. [19]).

**Corollary 5.**A PSH possesses a pure translational self-motion, if and only if the platform can be rotated about the center o = O into a pose, where the vectors${\overrightarrow{{\mathrm{M}}_{i}\mathrm{m}}}_{i}$ for i = 1, 2, 3 fulfill the condition$r$.

**Proof.**The proof follows directly from the fact that the midpoint property is invariant under parallel projections. □

**Corollary 6.**A non-architecturally singular PSH can only have an n-dimensional self-motion with n > 1 if it is congruent. The self-motion is the two-dimensional translation.

## 3. OPSH With Self-Motions

#### 3.1. Necessary Condition for Translational Self-Motions

**Theorem 7.**A pure translational self-motion of an OPSH can only exist in one of the following five cases:

**Proof.**Due to Corollary 5, we compute the vectors $\overrightarrow{{\mathrm{M}}_{i}{\mathrm{m}}_{i}}$ by

**Rm**

_{i}− N

**M**

_{i}and express the condition for their linear dependency by

**K**

_{k}=

**o**, where

**o**denotes the zero-vector and

**K**

_{k}is given by:

**K**

_{k}=

**o**by ${K}_{k}^{1}$ and ${K}_{k}^{3}$, respectively. Therefore, we have to show that this resulting system of nine equations ${K}_{1}^{1}$ can only be fulfilled in one of the five cases α, …, ε. This can be done as follows:

_{1}and w

_{4}can only be fulfilled in the following two cases:

- e
_{1}= e_{2}= 0: Then, w_{6}is either fulfilled for c = C (⇒ case γ) or for:- e
_{0}= 0: We get ${G}_{3}^{3}$, which cannot vanish without contradiction (w.c.). - e
_{3}= 0: In this case, we get:$${G}_{1}^{1}$$

- e
_{3}= 0: For the discussion of this case, we can assume that e_{1}= e_{2}= 0 does not hold. We distinguish the following cases:- e
_{1}e_{2}≠ 0: Under this assumption, w_{5}yields c = −C. Then, we get:$${G}_{1}^{1}$$_{0}= 0 (⇒ case δ) or the reflection-congruent OPSH (⇒ case ε). - e
_{1}= 0, e_{2}≠ 0: Now, w_{3}can only vanish either for b = B (⇒ case β) or e_{0}= 0. In the latter case, we get ${G}_{3}^{3}$, which also implies b = B (⇒ case β). - e
_{2}= 0, e_{1}≠ 0: This case can be discussed analogously to the last one, with the sole exception that we always end up in the case α.

_{i}, m

_{i}, m

_{i}

_{+3}, M

_{i}

_{+3}forms a parallelogram during the translatoric self-motion of case α (i = 1), β (i = 2), γ (i = 3) and δ (i = 3).

_{3}= 0, a real translatoric self-motion. For more details on this reflection-congruent case, we refer to [20].

#### 3.2. Necessary Condition for Non-Translational Self-Motions

**Theorem 8.**An OPSH can have a non-translational self-motion only if either Equation (6) is fulfilled or abcABC = 0 holds.

**Proof.**The proof of this theorem is done by the bond theory presented in Section 2; i.e., we determine the conditions for the existence of a projected bond. For the computational proof, we need the two linear combinations:

_{1}and G

_{2}do not depend on f

_{0}, f

_{1}, f

_{2}, f

_{3}. Besides these two expressions, also the following three equations:

_{0}, f

_{1}, f

_{2}, f

_{3}. Now, we split the proof up into a general case and a special one.

**General case:**For this case, we assume that ν ≠ 0 holds, where ν is given by:

_{1,4}, Δ

_{2,5}, Δ

_{3,6}, which is linear in f

_{0}, f

_{1}, f

_{2}, f

_{3}for these variables. Plugging the obtained expressions in Λ

_{1}, we obtain in the numerator the expression NG

_{0}[5064], where G

_{0}is homogeneous of degree six in the Euler parameters. Note that the number in the brackets gives the number of terms.

_{0}= 0 and the two quadrics G

_{1}= G

_{2}= 0 have a curve in common. If this is the case, there also has to exist at least one intersection point (projected bond) of this curve with the exceptional quadric N = 0.

_{0}by computing the resultant H

_{i}of G

_{i}and N for i = 0, 1, 2. We get:

_{k}of H

_{i}and H

_{j}for pairwise distinct i, j, k ∈ {0, 1, 2}. Then, we eliminate e

_{1}by computing the resultant Q

_{k}of L

_{i}and L

_{j}for pairwise distinct i, j, k ∈ {0, 1, 2}. Now, the necessary condition for the existence of a projected bond is that the greatest common divisor of Q

_{0}, Q

_{1}, Q

_{2}vanishes, which reads as follows (up to powers of the given factors):

_{2}= 0 implies e

_{1}= e

_{3}= e

_{0}= 0 (a contradiction), the theorem is proven for the general case.

**Special case:**Now, we consider the special case ν = 0. Therefore, we have to determine the condition for the existence of a common point of the four quadrics G

_{1}= G

_{2}= ν = N = 0.

_{0}by ν, the same elimination procedure as in the general case yields the following greatest common divisor (up to powers of the given factors) of the resulting final expressions:

_{2}= 0 implies again e

_{1}= e

_{3}= e

_{0}= 0 (a contradiction), the theorem is proven. □

**Remark 4.**A consequence of the Theorems 7 and 8 is that an OPSH with$0$ for i = 1, 2, 3, which does not fulfill Equation (6), is free of self-motions. This is of importance for practical applications, as self-motions are dangerous, because they are uncontrollable and, thus, a hazard to man and machine. Therefore, being able to avoid OPSH designs that engender self-motion is of interest to engineers. Note that also the later given Theorem 4.1 should be read in this context.

_{1}= m

_{4}= o:

- Butterfly self-motions: If the platform is located in a way that the y-axis (z-axis) of the moving frame coincides with the z-axis (y-axis) of the fixed frame, then the platform can rotate freely around this line.
- Spherical four-bar self-motion: If the platform is in a configuration where the centers of the moving frame and fixed frame coincide (⇔ o = O), then the manipulator can perform a spherical four-bar motion with center o = O.

#### 3.2.1. Dietmaier Self-Motions

_{1,4}, Δ

_{2,5}, Δ

_{3,6}are homogeneous linear equations in f

_{0}, …, f

_{3}. Therefore, we distinguish the following two cases:

_{1,4}, Δ

_{2,5}, Δ

_{3,6}for f

_{0}, …, f

_{3}, which yields f

_{0}= f

_{1}= f

_{2}= f

_{3}= 0. As a consequence, we can only have a pure spherical self-motion, as o = O holds.

_{1}= G

_{2}= 0 are three homogeneous quadratic equations in the Euler parameters without mixed terms e

_{i}e

_{j}for i 6= j ∈ {0, …, 3}. Under the Dietmaier conditions given in Equations (6) and (7), the three quadrics in the Euler parameter space are linearly dependent, which is already sufficient for the existence of a self-motion over ℂ. Based on this observation, we will determine a new set of PSH with non-translatoric self-motions in Section 4.2.

**Example 1.**For a detailed example of a Dietmaier self-motion, please see [13]. We only want to note in this context an example with rational coordinates fulfilling Equation (6):

**Remark 5.**It is still an open problem to determine all non-translational self-motions of an OPSH besides the known Dietmaier self-motion, spherical four-bar self-motion and the butterfly self-motion. It is only known (cf. last paragraph of Section 1.1) that congruent OPSHs also have Schönflies self-motions.

## 4. PSHs with Self-Motions

_{0}, Q

_{1}, Q

_{2}running Maple 17 on a computer with 12 GB RAM (not even for planar PSHs). Therefore, we restrict ourselves to equiform/congruent PSHs, which are studied next.

#### 4.1. Equiform/Congruent PSHs

**Theorem 9.**If an equiform PSH has a non-translational self-motion, then it is either congruent or architecturally singular.

**Proof.**W.l.o.g., we can choose coordinate systems in the platform and the base in a way that we have:

_{1}= c

_{1}= c

_{2}= B

_{1}= C

_{1}= C

_{2}= 0 and the remaining coordinates are coupled by:

_{1}= 0 contradicts the definition of a PSH, and B

_{2}= 0 results in an architecturally singular design (cf. Item 2 of Corollary 2). Moreover, ${B}_{3}^{2}$ implies B

_{3}= C

_{3}= 0, and we get again Item 2 of Corollary 2. The last factor can only be fulfilled for C

_{3}= 0 and A

_{2}B

_{3}− A

_{3}B

_{2}= 0, which also implies Item 2 of Corollary 2.

_{1}= 0 contradicts the definition of a PSH, μ = 1 yields a congruent PSH and B

_{2}= 0 results in Item 2 of Corollary 2. Therefore, we can assume B

_{2}≠ 0, which implies that F

_{i}can only be fulfilled over ℝ for C

_{3}= 0 for i = 1, 2, 3, 4. Therefore, C

_{3}= 0 has to hold, which implies a planar equiform hexapod. These manipulators were already studied in detail in [23,24] with the result that they cannot have self-motions if they are not architecturally singular. This closes the proof of the theorem. □

**Definition 10.**A self-motion of a PSH is called generalized Dietmaier self-motion, if Equation (5) holds and the equations Ψ, Δ

_{1,4}, Δ

_{2,5}, Δ

_{3,6}are linearly dependent with respect to f

_{0}, …, f

_{3}.

**Theorem 11.**A congruent PSH, which is not architecturally singular, can have the following two types of non-translation self-motions:

- Line-symmetric self-motions (including Schönflies self-motions as a special case),
- Generalized Dietmaier self-motions.

**Proof.**We use the same coordinatization as in the proof of Theorem 9 under consideration of μ = 1. Moreover, we can assume that the congruent PSH is non-planar, because it is well-known (cf. [24]) that planar ones can only have translational self-motions, if they are not architecturally singular.

- e
_{3}≠ 0: Under this assumption, we can solve Ψ, Δ_{1,4}, Δ_{2,5}for f_{1}, f_{2}, f_{3}. Plugging the obtained expressions into Δ_{3,6}yields in the numerator $N$ with:$$E$$_{2}of Equation (20), which now read as follows:$${G}_{1}$$$${G}_{2}$$_{1}after plugging the expressions for f_{1}, f_{2}, f_{3}into it. However, this condition is not of interest, as one can always solve it for f_{0}in case of a non-translational self-motion. Therefore, the two quadrics G_{1}= G_{2}= 0 and the plane E = 0 in the Euler parameter space have to have a curve s in common. We distinguish the following cases:- E = 0 is not fulfilled identically: In this case, we consider the expression of E after replacing e
_{i}by f_{i}for i = 1, 2, 3. This yields:$$F$$_{1}, f_{2}, f_{3}into F, the numerator factors into two factors, where one of it is E. Therefore, E = 0 implies F = 0; thus, we have two linear relations between the Study parameters. According to [25], this is one possible characterization of line-symmetric motions. In general, the curve s is a regular conic section, but it can also happen that it consists of one or two straight lines, implying Schönflies self-motions.Remark 6. Note that the geometric characterization of parallel manipulators with a Schönflies self-motion was already done in [26], and therefore, this case can only be a special case of it. - E = 0 is fulfilled identically: As A
_{1}B_{2}C_{3}≠ 0 holds, this can only be the case if Equation (5) holds. Therefore, we get a generalized Dietmaier self-motion, which corresponds in the generic case with a curve s of degree four in the Euler parameter space.

- e
_{3}= 0: Now, Δ_{1,4}and Δ_{2,5}do not depend on f_{0}, f_{1}, f_{2}. Elimination of f_{3}implies that E = 0 has to hold with:$$E$$For R_{1}= R_{4}and R_{2}= R_{5}, the equation E = 0 is fulfilled identically. Now, Δ_{1,4}and Δ_{2,5}can only vanish without contradiction for f_{3}= 0. Due to e_{3}= f_{3}= 0, we can only end up with a line-symmetric self-motion.

**Example 2.**We give an example of a line-symmetric self-motion, which is not a Schönflies motion, as this is the essential new possibility compared to the classification of congruent OPSHs. The geometry is given by:

_{1}, f

_{2}, f

_{3}from Ψ, Δ

_{1,4}, Δ

_{2,5}. Moreover, for the choice:

_{2}is a real-valued multiple of G

_{1}, which determines the conic s (and therefore, the self-motion) by the equation:

_{0}and plugged into Λ

_{1}. As this final equation is quadratic in f

_{0}, the self-motion is in general not rational.

_{3}= 1, the self-motion is real if e

_{2}is within the interval [−t, t] with:

#### 4.2. Generalized Dietmaier Self-Motions

_{1,4}, Δ

_{2,5}, Δ

_{3,6}are homogeneous linear in f

_{0}, …, f

_{3}. Therefore, a necessary and sufficient condition for a generalized Dietmaier self-motion is that the three quadrics G

_{1}= G

_{2}= ν = 0 in the Euler parameter space have a curve in common, where G

_{1}and G

_{2}are defined as in Equation (20) and ν is the remaining factor of the determinant of the coefficient matrix of Ψ, Δ

_{1,4}, Δ

_{2,5}, Δ

_{3,6}with respect to f

_{0}, …, f

_{3}after splitting away N.

_{1}and M

_{1}are on their x-axes and m

_{2}and M

_{2}are located in their xy-planes, which implies:

_{1}[48], G

_{2}[68], ν[64]. The resultant of each of these two equations with respect to any Euler parameter is a quartic in the remaining ones with more than 20,000 terms. Therefore, the author was not able to compute the general conditions for the existence of a common curve s of G

_{1}= G

_{2}= ν = 0 (as the calculation of the resultant of two such quartics failed by Maple 17 on a computer with 12 GB RAM).

_{1}= G

_{2}= ν = 0 are linearly dependent as for the original Dietmaier self-motion (cf. Section 3.2.1); i.e., there exists a linear combination:

**Theorem 12.**The following set D of non-architecturally singular PSHs with non-planar platform and non-planar base has a generalized Dietmaier self-motion:

**Proof.**Due to the non-planarity assumption of the platform and the base, we can assume a

_{1}A

_{1}b

_{2}B

_{2}c

_{3}C

_{3}6= 0. Moreover, we can even assume w.l.o.g. that a

_{1}, A

_{1}, b

_{2}, B

_{2}> 0 holds.

^{ijkl}. Then, we consider the following set of 10 necessary and sufficient equations:

_{1}depends on the leg lengths R

_{1}, R

_{2}, R

_{3}. W.l.o.g., we can solve w

_{2}, w

_{3}for λ

_{1}and λ

_{2}. As λ = 0 implies λ

_{1}= λ

_{2}= 0 a contradiction, we can set λ = 1. Then, w

_{5}implies b

_{3}of Equation (52), and w

_{8}can only vanish w.c. for:

_{9}implies a

_{2}of Equation (52). Substituting this expression into Equation (56) yields the condition for a

_{3}given in Equation (52). Now, one condition on the geometry of the PSH remains, namely w

_{3}, which is the equation given in Equation (53).

**Remark 7**. Note that w

_{3}implies the condition of Equation (6) under the additional assumption A

_{2}= A

_{3}= B

_{3}= 0 of an OPSH. Note that therefore the original Dietmaier self-motion is included as a special case.

_{1}is not fulfilled, which has the following structure: ${q}_{0}$. Therefore, each design of D has at least a two-parametric set of generalized Dietmaier self-motions. We only get a three-parametric set if this linear relation in ${R}_{1}^{2}$, ${R}_{2}^{2}$ and ${R}_{3}^{2}$ is fulfilled identically, which is studied next.

_{3}= 0 implies ${c}_{3}$. Then, q

_{2}can only vanish w.c. for a

_{1}= A

_{1}. Now, q

_{1}= 0 is already fulfilled identically, and we remain with the conditions w

_{3}and q

_{0}= 0. It can easily be seen that w

_{3}implies b

_{2}= B

_{2}; i.e., the congruent PSH. Now, q

_{0}= 0 is also fulfilled identically. □

**Remark 8**. Finally, it should be noted that a geometric interpretation of the algebraic conditions determining the set D is missing. This also includes the geometric meaning of Equation (6) (cf. Remark 7).

_{1}, …, w

_{10}used in the proof of Theorem 12, it can be shown by a straightforward discussion of cases that a linear combination of Equation (51) does not exist for non-architecturally singular PSHs, if the platform or the base or both are planar. Therefore, the only missing cases yielding a generalized Dietmaier self-motion are those where s is a cubic curve, a conic section or a straight line in the Euler parameter space. The latter case is of less interest, as it can only be a special case of the manipulators given in [26] (cf. Remark 6).

**Example 3.**In the following, we give an example for a generalized Dietmaier self-motion. The PSH of the set D is given by:

_{1}e

_{2}e

_{3}-plane by the elimination of e

_{0}yields the planar quartic curve s

^{′}given by:

## 5. Conclusions and Future Work

#### 5.1. Hexapodal Self-Motions Viewed under the Aspect of Symmetry

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The trajectories of the platform anchor points under (one branch of) the line-symmetric self-motion of Example 2 are displayed. An animation of this motion is provided as supplementary data (see Supplementary File S1).

**Figure 3.**We identify e

_{3}= 0 with the line at infinity and illustrate the affine part of the planar quartic s′; i.e., we set e

_{3}= 1 and plot e

_{2}horizontally and e

_{1}vertically.

**Figure 4.**The trajectories of the platform anchor points under a part of the self-motion of Example 3 are displayed. An animation of this motion is provided as supplementary data (see Supplementary File S2).

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**MDPI and ACS Style**

Nawratil, G.
On the Self-Mobility of Point-Symmetric Hexapods. *Symmetry* **2014**, *6*, 954-974.
https://doi.org/10.3390/sym6040954

**AMA Style**

Nawratil G.
On the Self-Mobility of Point-Symmetric Hexapods. *Symmetry*. 2014; 6(4):954-974.
https://doi.org/10.3390/sym6040954

**Chicago/Turabian Style**

Nawratil, Georg.
2014. "On the Self-Mobility of Point-Symmetric Hexapods" *Symmetry* 6, no. 4: 954-974.
https://doi.org/10.3390/sym6040954