On the Line-Symmetry of Self-motions of Linear Pentapods

Abstract

We show that all self-motions of pentapods with linear platform of Type 1 and Type 2 can be generated by line-symmetric motions . Thus this paper closes a gap between the more than 100 year old works of Duporcq and Borel and the extensive study of line-symmetric motions done by Krames in the 1930s. As a consequence we also get a new solution set for the Borel Bricard problem. Moreover we discuss the reality of self-motions and give a sufficient condition for the design of linear pentapods of Type 1 and Type 2, which have a self-motion free workspace.

1 Introduction

The geometry of a linear pentapod is given by the five base anchor points $\mathsf{M}_i$ in the fixed system $\varSigma _0$ and by the five collinear platform anchor points $\mathsf{m}_i$ in the moving system $\varSigma$ (for $i=1,\ldots ,5$). Each pair $(\mathsf{M}_i,\mathsf{m}_i)$ of corresponding anchor points is connected by a SPS-leg, where only the prismatic joint is active.

If the geometry of the linear pentapod is given as well as the lengths $R_i$ of the five pairwise distinct legs, it has generically mobility 1. This degree of freedom corresponds to the rotational motion about the carrier line $\mathsf{p}$ of the five platform anchor points. As this rotation is irrelevant for applications with axial symmetry (e.g. 5-axis milling, laser or water-jet engraving/cutting, spot-welding, spray-based painting, etc.), these manipulators are of great practical interest. Nevertheless configurations should be avoided where the linear pentapod gains an additional uncontrollable mobility, which is referred as self-motion.

1.1 Review on Self-motions of Linear Pentapods

The self-motions of linear pentapods represent interesting solutions to a problem posed 1904 by the French Academy of Science for the Prix Vaillant, which is also known as Borel-Bricard problem (cf. [2, 3]). This still unsolved kinematic challenge reads as follows: “Determine and study all displacements of a rigid body in which distinct points of the body move on spherical paths.”

For the special case of five collinear points the Borel-Bricard problem was studied by Darboux [5, p. 222], Mannheim [6, p. 180ff] and Duporcq [7] (see also Bricard [3, Chap. III]). A contemporary and accurate reexamination of these old results, which also takes the coincidence of platform anchor points into account, was done in [1] yielding a full classification of linear pentapods with self-motions.

Beside the architecturally singular linear pentapods [1, Corollary 1] and some trivial cases with pure rotational self-motions [1, Designs $\alpha$, $\beta$, $\gamma$] or pure translational ones [1, Theorem 1] there only remain the following three designs:

Under a self-motion each point of the line $\mathsf{p}$ has a spherical (or planar) trajectory. The locus of the corresponding sphere centers is a cubic space curve $\mathsf{P}$, where the mapping from $\mathsf{p}$ to $\mathsf{P}$ is named $\sigma$. $\mathsf{P}$ intersects the ideal plane in one real point $\mathsf{W}$ and two conjugate complex ideal points, where the latter ones are the cyclic points $\mathsf{I}$ and $\mathsf{J}$ of a plane orthogonal to the direction of $\mathsf{W}$. $\mathsf{P}$ is therefore a so-called straight cubic circle. The following subcases can be distinguished:

• $\mathsf{P}$ is irreducible:

  • $\sigma$ maps the ideal point $\mathsf{U}$ of $\mathsf{p}$ to $\mathsf{W}$ (Type 5 according to [1]).

  • $\sigma$ maps $\mathsf{U}$ to a finite point of $\mathsf{P}$ (Type 1 according to [1]).

• $\mathsf{P}$ splits up into a circle and a line, which is orthogonal to the carrier plane of the circle and intersects the circle in a point $\mathsf{Q}$. Moreover $\sigma$ maps $\mathsf{U}$ to a point on the circle different from $\mathsf{Q}$ (Type 2 according to [1]).

1.2 Basics on Line-Symmetric Motions

Krames (e.g. [4, 10]) studied special one-parametric motions (Symmetrische Schrotung in German), which are obtained by reflecting the moving system $\varSigma$ in the generators of a ruled surface of the fixed system $\varSigma _0$, which is the so called basic surface. These so-called line-symmetric motions were also studied by Bottema and Roth [8, Sect. 7 of Chap. 9], who gave an intuitive algebraic characterization in terms of Study parameters $(e_0:e_1:e_2:e_3:f_0:f_1:f_2:f_3)$, which are shortly repeated next.

All real points of the Study parameter space $P^7$ (7-dimensional projective space), which are located on the so-called Study quadric $\Psi :\,\sum _{i=0}^3e_if_i=0$, correspond to an Euclidean displacement with exception of the 3-dimensional subspace $e_0=e_1=e_2=e_3=0$, as its points cannot fulfill the condition $N\ne 0$ with $N:=e_0^2+e_1^2+e_2^2+e_3^2$. The translation vector ${\mathbf s}:=(s_1,s_2,s_3)^T$ and the rotation matrix ${\mathbf R}$ of the corresponding Euclidean displacement ${\mathbf m}_i\mapsto {\mathbf R}{\mathbf m}_i + {\mathbf s}$ are given for $N=1$ by:

$$\begin{aligned} {\begin{matrix} s_1&{}=-2(e_0f_1-e_1f_0+e_2f_3-e_3f_2), \quad s_2=-2(e_0f_2-e_2f_0+e_3f_1-e_1f_3), \\ s_3&{}=-2(e_0f_3-e_3f_0+e_1f_2-e_2f_1), \\ {\mathbf R} &{}= \begin{pmatrix} r_{11} &{} r_{12} &{} r_{13} \\ r_{21} &{} r_{22} &{} r_{23} \\ r_{31} &{} r_{32} &{} r_{33} \end{pmatrix}= \begin{pmatrix} e_0^2+e_1^2-e_2^2-e_3^2 &{} 2(e_1e_2-e_0e_3) &{} 2(e_1e_3+e_0e_2) \\ 2(e_1e_2+e_0e_3) &{} e_0^2-e_1^2+e_2^2-e_3^2 &{} 2(e_2e_3-e_0e_1) \\ 2(e_1e_3-e_0e_2) &{} 2(e_2e_3+e_0e_1) &{} e_0^2-e_1^2-e_2^2+e_3^2 \end{pmatrix}. \end{matrix}} \end{aligned}$$

There always exists a moving frame (in dependence of a given fixed frame) in a way that $e_0=f_0=0$ holds for a line-symmetric motion. Then $(e_1:e_2:e_3:f_1:f_2:f_3)$ are the Plücker coordinates (according to the convention used in [8]) of the generators of the basic surface with respect to the fixed frame.

1.3 Line-Symmetric Self-motions of Linear Pentapods

It is well known (cf. [7, Sect. 15], [3, Sect. 12]) that the self-motions of Type 5 are obtained by restricting the Borel-Bricard motions¹ (also known as BB-I motions) to a line. Note that Krames gave a detailed discussion of this special case in [4, Sect. 5], where he also pointed out the line-symmetry of BB-I motions.

Beside these BB-I motions, there also exist line-symmetric motions (so-called BB-II motions), where every point of a hyperboloid carrying two reguli of lines has a spherical path. It is known (cf. [9, p. 24] and [10, p. 188]) that the corresponding sphere centers of lines, belonging to one regulus,² constitute irreducible straight cubic circles, which imply examples of Type 1 self-motions. It should be noted that there also exist degenerated cases where the hyperboloid splits up into the union two orthogonal planes, which contain examples of Type 2 self-motions.

A simple count of free parameters shows that not all self-motions of Type 1 (5-parametric set³ of motions where all points of a line have spherical paths) can be generated by BB-II motions (which produce only a 4-parametric set⁴). The same argumentation holds for Type 2 self-motions and the mentioned degenerated case.

As a consequence the question arise whether all self-motions of linear pentapods of Type 1 and Type 2 can be generated by line-symmetric motions . If this is the case we can apply a construction proposed by Krames [4, p. 416], which is discussed in Sect. 4, yielding new solutions to the Borel-Bricard problem.

Finally it should be noted that a detailed review on line-symmetric motions with spherical trajectories is given in [11, Sect. 1].

2 On the Line-Symmetry of Type 1 and Type 2 Self-motions

For our calculations we do not select arbitrary pairs $(\mathsf{m}_i,\mathsf{M}_i)$ of $\mathsf{p}$ and $\mathsf{P}$, which are in correspondence with respect to $\sigma$ ($\Leftrightarrow$ $\sigma (\mathsf{m}_i)=\mathsf{M}_i$), but choose the following special ones:

$\mathsf{M}_4$ equals $\mathsf{W}$, $\mathsf{M}_2$ coincides with $\mathsf{I}$ and $\mathsf{M}_3$ with $\mathsf{J}$. The corresponding platform anchor points are denoted by $\mathsf{m}_4$, $\mathsf{m}_2$ and $\mathsf{m}_3$, respectively. As $\mathsf{M}_i$ are ideal points the corresponding points $\mathsf{m}_i$ are not running on spheres but in planes orthogonal to the direction of $\mathsf{M}_i$. Therefore these three point pairs imply three so-called Darboux conditions $\varOmega _i$ for $i=2,3,4$. Moreover we denote $\mathsf{U}$ as $\mathsf{m}_5$ and its corresponding finite point under $\sigma$ by $\mathsf{M}_5$. This point pair describes a so-called Mannheim condition $\varPi _5$ (which is the inverse of a Darboux condition). The pentapod is completed by a sphere condition $\varLambda _1$ of any pair of corresponding finite points $\mathsf{m}_1$ and $\mathsf{M}_1$.

In [1] we have chosen the fixed frame $\mathscr {F}_0$ in a way that $\mathsf{M}_1$ equals its origin and $\mathsf{M}_4$ coincides with the ideal point of the z-axis. Moreover we located the moving frame $\mathscr {F}$ in a way that $\mathsf{p}$ coincides with the x-axis, where $\mathsf{m}_1$ equals its origin.

For the study at hand it is advantageous to select a different set of fixed and moving frames $\mathscr {F}_0^{\prime }$ and $\mathscr {F}^{\prime }$, respectively:

• As $\mathsf{M}_2$ and $\mathsf{M}_3$ coincides with the cyclic points, we can assume without loss of generality (w.l.o.g.) that $\mathsf{M}_5$ is located in the xz-plane (as a rotation about the z-axis does not change the coordinates of $\mathsf{M}_1,\ldots ,\mathsf{M}_4$). Moreover we want to apply a translation in a way that $\mathsf{M}_5$ is in the origin of the new fixed frame $\mathscr {F}_0^{\prime }$. Summed up the coordinates with respect to $\mathscr {F}_0^{\prime }$ read as:

  $$\begin{aligned} {\mathbf M}_5=(0,0,0),\quad {\mathbf M}_1=(A,0,C) \quad \text {with} \quad A\ne 0 \end{aligned}$$

  (1)

  as $A=0$ implies a contradiction to the properties of $\mathsf{P}$ for Type 1 and Type 2 pentapods given in Sect. 1.1. Moreover, $\mathsf{M}_2$, $\mathsf{M}_3$ and $\mathsf{M}_4$ are the ideal points in direction $(1,i,0)^T$, $(1,-i,0)^T$ and $(0,0,1)^T$, respectively.

• With respect to $\mathscr {F}_0^{\prime }$ the location of $\mathsf{p}$ is undefined, but the coordinates ${\mathbf m}_i$ of $\mathsf{m}_i$ can be parametrized as follows for $i=1,\ldots , 4$:

  $$\begin{aligned} {\mathbf m}_i= {\mathbf n}+ (a_i-a_r){\mathbf d} \quad \text {with}\quad a_1=0,\,\, a_2=a_r+ia_c,\,\, a_3=a_r-ia_c \end{aligned}$$

  (2)

  where $a_r,a_c\in {\mathbb R}$ and $a_c\ne 0$ holds. $\mathsf{m}_5$ is the ideal point in direction of the unit-vector ${\mathbf d}=(d_1,d_2,d_3)^T$, which obtains the rational homogeneous parametrization of the unit-sphere, i.e.

  $$\begin{aligned} d_1=\tfrac{2h_0h_1}{h_0^2+h_1^2+h_2^2}, \quad d_2=\tfrac{2h_0h_2}{h_0^2+h_1^2+h_2^2}, \quad d_3=\tfrac{h_1^2+h_2^2-h_0^2}{h_0^2+h_1^2+h_2^2}. \end{aligned}$$

  (3)

Now we are looking for the point ${\mathbf n}=(n_1,n_2,n_3)^T$ and the direction $(h_0:h_1:h_2)$ in a way that for the self-motion of the pentapod $e_0=f_0=0$ holds. We can discuss Type 1 and Type 2 at the same time, just having in mind that $a_4\ne 0\ne C$ has to hold for Type 1 and $a_4=0=C$ for Type 2 (according to [1]).

By setting ${\mathbf r}_i:=(r_{i1},r_{i2},r_{i3})^T$ for $i=1,2,3$ the Darboux and Mannheim constraints with respect to $\mathscr {F}_0^{\prime }$ and $\mathscr {F}^{\prime }$ can be written as:

$$\begin{aligned} \varOmega _2:\,&(s_1+{\mathbf r}_1{\mathbf m}_2)-i(s_2+{\mathbf r}_2{\mathbf m}_2)-p_2N=0,&\quad \varOmega _4:\,&(s_3+{\mathbf r}_3{\mathbf m}_4)-p_4N=0, \end{aligned}$$

(4)

$$\begin{aligned} \varOmega _3:\,&(s_1+{\mathbf r}_1{\mathbf m}_3)+i(s_2+{\mathbf r}_2{\mathbf m}_3)-p_3N=0,&\quad \varPi _5:\,&({\mathbf R}{\mathbf d})({\mathbf s}+{\mathbf R}{\mathbf p}_5)N^{-1}=0, \end{aligned}$$

(5)

with ${\mathbf p}_5={\mathbf n}+(p_5-a_r){\mathbf d}$, which is the coordinate vector of the intersection point of the Mannheim plane and $\mathsf{p}$ with respect to $\mathscr {F}^{\prime }$. Moreover $(p_j,0,0)^T$ for $j=2,3$ (resp. $(0,0,p_4)^T$) are the coordinates of the intersection point of the Darboux plane and the x-axis (resp. z-axis) of $\mathscr {F}_0^{\prime }$.

Remark 1

As from the Mannheim constraint $\varPi _5$ of Eq. (5) the factor N cancels out, all four constraints $\varOmega _2,\varOmega _3,\varOmega _4,\varPi _5$ are homogeneous quadratic in the Study parameters and especially linear in $f_0,\ldots ,f_3$. $\diamond$

According to [1, Theorems 13 and 14] the leg-parameters $p_2,\ldots ,p_5,R_1$ have to fulfill the following necessary and sufficient conditions for the self-mobility (over ${\mathbb C}$) of a linear pentapod of Type 1 and Type 2, respectively:

$$\begin{aligned}&p_2=\tfrac{Aa_3v}{(a_3-a_4)^2},\qquad p_3=\tfrac{Aa_2v}{(a_2-a_4)^2},\qquad p_4=-\tfrac{Ca_4v}{(a_2-a_4)(a_3-a_4)}, \end{aligned}$$

(6)

$$\begin{aligned}&\quad (a_2-a_4)^2(a_3-a_4)^2\left[ 2wp_5-vR_1^2-(2w-va_4)a_4 \right] + vw^2(A^2+C^2)=0, \end{aligned}$$

(7)

with $v:=a_2+a_3-2a_4$ and $w:=a_2a_3-a_4^2$. Therefore if we set $p_2,p_3,p_4$ as given in Eq. (6) then only one condition in $p_5$ and $R_1$ remains in Eq. (7). Therefore these pentapods have a 1-dimensional set of self-motions.

Theorem 1

Each self-motion of a linear pentapod of Type 1 and Type 2 can be generated by a 1-dimensional set of line-symmetric motions. For the special case $p_5=a_4=a_r$ this set is even 2-dimensional.

Proof

W.l.o.g. we can set $e_0=0$ as any two directions ${\mathbf d}$ of $\mathsf{p}$ can be transformed into each other by a half-turn about their enclosed bisecting line. Note that this line is not uniquely determined if and only if the two directions are antipodal.

W.l.o.g. we can solve $\Psi ,\varOmega _2,\varOmega _3,\varOmega _4$ for $f_0,f_1,f_2,f_3$ and plug the obtained expressions into $\varPi _5$, which yields in the numerator a homogeneous quartic polynomial G[1563] in $e_1,e_2,e_3$, where the number in the brackets gives the number of terms. Moreover the numerator of the obtained expression for $f_0$ is denoted by F[600], which is a homogeneous cubic polynomial in

$e_1,e_2,e_3$.

General Case $(v\ne 0)$ : The condition $G=0$ already expresses the self-motion as G equals $\varLambda _1$ if we solve Eq. (7) for $R_1$. Moreover $F=0$ has to hold if the self-motion of the line $\mathsf{p}$ can be generated by a line-symmetric motion. As for any solution $(e_1:e_2:e_3)$ of $F=0$ also $G=0$ has to hold, G has to split into F and a homogeneous linear factor L in $e_1,e_2,e_3$.

Now $L=0$ cannot correspond to a self-motion of the linear pentapod, but has to arise from the ambiguity in representing a direction of $\mathsf{p}$ mentioned at the beginning of the proof. This can be argued indirectly as follows:

Assumed $L=0$ implies a self-motion, then it has to be a Schönflies motion (with a certain direction $\mathsf{v}$ of the rotation axis) due to $e_0=0$. As under such a motion the angle enclosed by $\mathsf{v}$ and $\mathsf{p}$ remains constant⁵ the ideal point $\mathsf{U}$ of $\mathsf{p}$ has to be mapped by $\sigma$ to the ideal point $\mathsf{V}$ of $\mathsf{v}$. This implies that $\mathsf{V}$ has to coincide with $\mathsf{W}$, which can only be the case for pentapods of Type 5; a contradiction.

Therefore there has to exist a pose of $\mathsf{p}$ during the self-motion, where it is oppositely oriented with respect to the fixed frame and moving frame, respectively. As a consequence we can set $L=d_1e_1+d_2e_2+d_3e_3$ which yields the ansatz $\varDelta :$ $\lambda LF-G=0$. The resulting set of four equations arising from the coefficients of $e_1^3e_2$, $e_1^3e_3$, $e_1e_3^3$ and $e_2e_3^3$ of $\varDelta$ has the unique solution:

$$\begin{aligned} n_1=a_cd_2,\quad n_2=-a_cd_1, \quad n_3=(a_r-a_4)d_3, \quad \lambda =2(h_0^2+h_1^2+h_2^2). \end{aligned}$$

(8)

Now $\varDelta$ splits up into $(e_1^2+e_2^2+e_3^2)^2(h_0^2+h_1^2+h_2^2)H[177]$, where H is homogeneous of degree 4 in $h_0,h_1,h_2$. For more details on $H=0$ please see Remark 3, which is given right after this proof.

Remark 2

Note that all self-motions of the general case can be parametrized as the resultant of G and the normalizing condition $N-1$ with respect to $e_i$ yields a polynomial, which is only quadratic in $e_j$ for pairwise distinct $i, j\in \left\{ 1,2\right\}$.$\diamond$

Special Case $(v= 0)$ : If $v=0$ holds, we cannot solve Eq. (7) for $R_1$. The conditions $v=0$ and Eq. (7) imply $p_5=a_4=a_r$. Now G is fulfilled identically and the self-motion is given by $\varLambda _1=0$, which is of degree 4 in $e_1,e_2,e_3$. Moreover for this special case $F=0$ already holds for ${\mathbf n}$ given in Eq. (8). Therefore any direction $(h_0:h_1:h_2)$ for $\mathsf{p}$ can be chosen in order to fix the line-symmetric motion.    $\square$

Remark 3

$H=0$ represents a planar quartic curve, which can be verified to be entirely circular. Moreover $H=0$ can be solved linearly for $p_5$. The corresponding graph is illustrated in Fig. 1.

If we reparametrize the $h_0h_1h_2$-plane in terms of homogenized polar coordinates by:

$$\begin{aligned} h_0=(\tau _1^2+\tau _0^2)\rho _0,\qquad h_1=(\tau _1^2-\tau _0^2)\rho _1,\qquad h_2=2\tau _0\tau _1\rho _1, \end{aligned}$$

(9)

where $(\tau _0,\tau _1)\ne (0,0)\ne (\rho _0,\rho _1)$ and $\tau _0,\tau _1,\rho _0,\rho _1\in {\mathbb R}$ hold, then H factors into $(\tau _0^2+\tau _1^2)^3(H_2\tau _1^2 + H_1\tau _0\tau _1 +H_0\tau _0^2)$ with

(10)

Therefore this equation can be solved quadratically for the homogeneous parameter $\tau _0:\tau _1$. Note that the value $p_5$ is fixed during a self-motion.                                            $\diamond$

Fig. 1

For a type 1 pentapod with self-motion given by the parameters $a_4=2$, $A=-1$, $C=-5$, $a_r=7$ and $a_c=4$, the graph of $p_5$ in dependency of $h_1$ and $h_2$ with $h_0=1$ is displayed in the axonometric view on the left and in the front resp. top view on the right side. The highlighted point at height 6 corresponds to the values $h_1=-\tfrac{489262}{226525}+\tfrac{488}{226525}\sqrt{675091}$ and $h_2=\tfrac{535336}{226525}+\tfrac{446}{226525}\sqrt{675091}$

3 On the Reality of Type 1 and Type 2 Self-motions

A similar computation to [1, Example 1] shows that for any real point $\mathsf{p}_t\in \mathsf{p}$ with $t\in {\mathbb R}$ and coordinate vector ${\mathbf p}_t={\mathbf n}+ (t-a_r){\mathbf d}$ with respect to $\mathscr {F}^{\prime }$ the corresponding real point $\mathsf{P}_t\in \mathsf{P}$ has the following coordinate vector ${\mathbf P}_t$ with respect to $\mathscr {F}_0^{\prime }$:

$$\begin{aligned} {\mathbf P}_t=\left( \tfrac{A(a_r^2+a_c^2-ta_r)}{(t-a_r)^2+a_c^2}, -\tfrac{Aa_ct}{(t-a_r)^2+a_c^2}, \tfrac{Ca_4}{a_4-t}\right) ^T. \end{aligned}$$

(11)

As $L=0$ corresponds with one configuration of the self-motion we can compute the locus $\mathscr {E}_t$ of $\mathsf{p}_t$ with respect to $\mathscr {F}_0^{\prime }$ under the 1-parametric set of self-motions by the variation of $(h_0:h_1:h_2)$ within $L=0$. Moreover due to the mentioned ambiguity we can select an arbitrary solution $(e_0:e_1:e_2)$ for $L=0$ fulfilling the normalization condition $N=1$; e.g.:

$$\begin{aligned} e_1=\tfrac{h_2}{\sqrt{h_1^2+h_2^2}}, \quad e_2=-\tfrac{h_1}{\sqrt{h_1^2+h_2^2}} \quad \text {and} \quad e_3=0. \end{aligned}$$

(12)

Now the computation of ${\mathbf R}{\mathbf p}_t+{\mathbf s}$ yields a rational quadratic parametrization of $\mathscr {E}_t$ in dependency of $(h_0:h_1:h_2)$.

Note that this approach also includes the special case $(v= 0)$ as there always exists a value for $R_1^2$ (in dependency of $(h_0:h_1:h_2)$) in a way that $\varLambda _1=0$ holds.

For $t\ne a_4$ all $\mathscr {E}_t$ are ellipsoids of rotation (see Fig. 2a), which have the same center point $\mathsf{C}$ and axis of rotation $\mathsf{c}$. In detail, $\mathsf{C}$ is the point of the straight cubic circle (11) for the value $t=c$ with $c:=\tfrac{a_4^2-a_c^2-a_r^2}{2(a_4-a_r)}$ (for $a_4=a_r$ we get $c=\infty$ thus $\mathsf{p}_{\infty }=\mathsf{U}=\mathsf{m}_5$ holds, which implies $\mathsf{C}=\mathsf{M}_5$) and $\mathsf{c}$ is parallel to the z-axis of $\mathscr {F}_0^{\prime }$. Moreover the vertices on $\mathsf{c}$ have distance $|a_4-t|$ from $\mathsf{C}$ and the squared radius of the equator circle equals $(a_r-t)^2+a_c^2$. Note that for $a_4\ne a_r$ the only sphere within the described set of ellipsoids is $\mathscr {E}_c$. For $a_4= a_r$ no such sphere exists.

$\mathscr {E}_{a_4}$ is a circular disc in the Darboux plane $z=p_4$ (w.r.t. $\mathscr {F}_0^{\prime }$) centered in $\mathsf{C}$.

Remark 4

The existence of these ellipsoids was already known to Duporcq [7, Sect. 9], who used them to show that the spherical trajectories are algebraic curves of degree 4 (intersection curve of $\mathscr {E}_t$ and the sphere $\varPhi _t$ centered in $\mathsf{P}_t$ illustrated in Fig. 2b). $\diamond$

Fig. 2

Type 1 pentapod with self-motion given by $a_4=2$, $A=-1$, $C=-5$, $a_r=7$ and $a_c=4$. a The loci $\mathscr {E}_{a_4}$, $\mathscr {E}_c$ and $\mathscr {E}_{t}$ with $t=\tfrac{69}{20}$ are sliced (along the not drawn axis of rotation $\mathsf{c}$) in order to visualize their positioning with respect to the cubic $\mathsf{P}$ on which the points $\mathsf{P}_{\infty }=\sigma (\mathsf{U})$, $\mathsf{P}_c=\mathsf{C}$ and $\mathsf{P}_t$ are highlighted. Note that $\mathsf{P}_{a_4}=\mathsf{W}$ is the real ideal point of $\mathsf{P}$. b By setting $p_5=6$ a one-parametric self-motion $\mu$ is fixed. The trajectory of $\mathsf{p}_t$ under $\mu$ is illustrated as the intersection curve of $\mathscr {E}_t$ and the sphere $\varPhi _t$ centered in $\mathsf{P}_t$. c A strip of the basic surface of $\mu$ is illustrated for the value highlighted in Fig. 1. In addition $\mathsf{P}$ and $\overline{\mathsf{p}}$ are visualized, where the latter denotes the pose of $\mathsf{p}$ such that its half-turns about the generators of the basic surface yield the self-motion $\mu$. d Krames’s construction is illustrated with respect to the generator $\mathsf{g}$ of the basic surface: As $\mathsf{P}_{a_4}$ (resp. $\overline{\mathsf{p}}_{\infty }$) is the real ideal point of $\mathsf{P}$ (resp. $\overline{\mathsf{p}}$), the trajectory of $\mathsf{p}_{a_4}$ (resp. $\overline{\mathsf{P}}_{\infty }$) under $\mu$ is planar. The (Mannheim) plane $\in \varSigma$, which contains the point $\mathsf{P}_{\infty }$ (resp. $\overline{\mathsf{p}}_{a_4}$) and is orthogonal to the direction of the real ideal point $\mathsf{p}_{\infty }$ (resp. $\overline{\mathsf{P}}_{a_4}$) of $\mathsf{p}$ (resp. $\overline{\mathsf{P}}$) in the displayed pose, slides through the point $\mathsf{P}_{\infty }$ (resp. $\overline{\mathsf{p}}_{a_4}$) during the complete motion $\mu$

Based on this geometric property, recovered by line-symmetric motions, we can formulate the condition for the self-motion to be real as follows:

• $w\ne 0$: We can reduce the problem to a planar one by intersecting the plane spanned by $\mathsf{P}_0=\mathsf{M}_1$ and $\mathsf{c}$ with $\mathscr {E}_0$ and the sphere with radius $R_1$ centered in $\mathsf{P}_0$. Now there exists an interval $I_0=]I_-,I_+[$ such that for $R_1\in I_0$ the two resulting conics have at least two distinct real intersection points. It is well known (e.g. [14]) that the computation of the limits $I_-$ and $I_+$ of the reality interval $I_0$ leads across an algebraic problem of degree 4 (explicitly solvable). Thus for a real self-motion we have to choose $R_1\in I_0$ and solve Eq. (7) for $p_5$.

• $w=0$: Now $\mathsf{P}_0$ coincides with $\mathsf{C}$ and the interval collapses to the single value $R_1=|a_4|$, which can be seen from Eq. (7). Moreover $p_5$ can be chosen arbitrarily.

These considerations also show that any pentapod of Type 1 and 2 has real self-motions if the leg-parameters are chosen properly. Note that this is e.g. not the case for some designs of Type 5 pentapods described in [1, Sect. 6], where it was also proven that pentapods with self-motions have a quartically solvable direct kinematics. It is possible to use this advantage (closed form solution) of pentapods with self-motions without any risk,⁶ by designing linear pentapods of Type 1 and Type 2, which are guaranteed free of self-motions within their workspace.

A sufficient condition for that is that (at least) for one of the five legs $\mathsf{p}_t\mathsf{P}_t$ of the pentapod the corresponding reality interval $I_t$ is disjoint with the interval of the maximal and minimal leg length implied by the mechanical realization. This condition for a self-motion free workspace gets especially simple if $\mathsf{p}_c\mathsf{P}_c$ is this leg.

Remark 5

Due to limitation of pages, we refer for detailed examples to the paper’s corresponding arXiv version [13], which also show that for the general case $(v\ne 0)$ the basic surface is of degree 5 (see Fig. 2c) and that a general point has a trajectory of degree 6 under the corresponding line-symmetric motion.⁷ Note that the latter also holds for a general point of the cubic $\overline{\mathsf{P}}$ explained in the next section.                                      $\diamond$

4 Conclusion and Open Problem

Krames [4, p. 416] outlined the following construction (see Fig. 2d): Assume that $\mathsf{p}$ is in an arbitrary pose of the self-motion $\mu$ with respect to $\mathsf{P}$, where $\mathsf{g}$ denotes the generator of the basic surface, which corresponds to this pose. Moreover $\overline{\mathsf{p}}$ and $\overline{\mathsf{P}}$ are obtained by the reflexion of $\mathsf{p}$ and $\mathsf{P}$, respectively, with respect to $\mathsf{g}$, where $\overline{\mathsf{p}}$ belongs to the fixed system $\varSigma _0$ and $\overline{\mathsf{P}}$ to the moving system $\varSigma$. Then under the self-motion $\mu$ also the points of $\overline{\mathsf{P}}$ are located on spheres with centers on the line $\overline{\mathsf{p}}$.

We can apply this construction for each line-symmetric motion of Theorem 1, which yields new solutions for the Borel Bricard problem, with the exception of one special case where $\mathsf{W}\in \overline{\mathsf{p}}$ holds (i.e. $h_1=h_2=0$ or $h_0=0$), which was already given by Borel in [2, Case Fa4]. Moreover for this case Borel noted that beside $\mathsf{p}$ and $\overline{\mathsf{P}}$ only two imaginary planar cubic curves ($\in$ isotropic planes through $\mathsf{p}$) run on spheres. The example of [13] shows that this also holds true for the general case.

Thus the problem remains to determine all line-symmetric motions of Theorem 1 where additional real points (beside those of $\mathsf{p}$ and $\overline{\mathsf{P}}$) run on spheres. Until now the only known examples with this property are the BB-II motions (cf. Sect. 1.3).