Necessary and Sufficient Polynomial Constraints on Compatible Triplets of Essential Matrices

Abstract

The essential matrix incorporates relative rotation and translation parameters of two calibrated cameras. The well-known algebraic characterization of essential matrices, i.e. necessary and sufficient conditions under which an arbitrary matrix (of rank two) becomes essential, consists of a single matrix equation of degree three. Based on this equation, a number of efficient algorithmic solutions to different relative pose estimation problems have been proposed in the last two decades. In three views, a possible way to describe the geometry of three calibrated cameras comes from considering compatible triplets of essential matrices. The compatibility is meant the correspondence of a triplet to a certain configuration of calibrated cameras. The main goal of this paper is to give an algebraic characterization of compatible triplets of essential matrices. Specifically, we propose necessary and sufficient polynomial constraints on a triplet of real rank-two essential matrices that ensure its compatibility. The constraints are given in the form of six cubic matrix equations, one quartic and one sextic scalar equations. An important advantage of the proposed constraints is their sufficiency even in the case of cameras with collinear centers. The applications of the constraints may include relative camera pose estimation in three and more views, averaging of essential matrices for incremental structure from motion, multiview camera auto-calibration, etc.

Appendix

We collect here several technical lemmas that we used in the proof of Theorem 5.

Recall that the radical of an ideal J, denoted $\sqrt{J}$, is given by the set of polynomials which have a power belonging to  J: OR to the ideal J:

$$\begin{aligned} \sqrt{J} = \{p \mid p^k \in J \hbox { for some integer}\ k \ge 1\}. \end{aligned}$$

It is known that $\sqrt{J}$ is an ideal and the affine varieties of J and $\sqrt{J}$ coincide. The following lemma gives a convenient tool to check whether a given polynomial is in the radical or not.

Lemma 1

(Cox et al. 2007) Let $J = \langle p_1, \ldots , p_s \rangle \subset {\mathbb {C}}[\xi _1, \ldots , \xi _n]$ be an ideal. Then a polynomial $p \in \sqrt{J}$ if and only if $1 \in {{\widetilde{J}}} = \langle p_1, \ldots , p_s, 1 - \tau p \rangle \subset {\mathbb {C}}[\xi _1, \ldots , \xi _n, \tau ]$.

By Lemma 1, a polynomial $p \in \sqrt{J}$ if and only if the (reduced) Gröbner basis of ${{\widetilde{J}}}$ is $\{1\}$. In the proof of Theorem 5, we used the computer algebra system Macaulay2 (Grayson and Stillman 2002) to compute the Gröbner bases. The computation time did not exceed 3 seconds per basis.

Lemma 2

Let essential matrices $E_{12}$, $E_{23}$, $E_{31}$ be represented in form $E_{ij} = [b_{ij}]_\times R_{ij}$. If the matrices $R_{ij}$ and vectors $b_{ij}$ are constrained by

$$\begin{aligned} R_{12}R_{23}R_{31}&= I, \end{aligned}$$

(64)

$$\begin{aligned} R_{12}^\top b_{12} + R_{23}b_{31} + b_{23}&= 0, \end{aligned}$$

(65)

then the triplet $\{E_{12}, E_{23}, E_{31}\}$ is compatible.

Proof

Let $R_{ij}$ and $b_{ij}$ satisfy Eqs. (64)–(65). Then Eq. (13) has the following possible solution for $R_i$ and $b_i$:

$$\begin{aligned} \begin{aligned} R_1&= I,&\quad b_1&= 0,\\ R_2&= R_{12}^\top ,&\quad b_2&= -b_{12},\\ R_3&= R_{31},&\quad b_3&= R_{31}^\top b_{31}. \end{aligned} \end{aligned}$$

(66)

It follows that the triplet $\{E_{12}, E_{23}, E_{31}\}$ is compatible. Lemma 2 is proved. $\square$

Lemma 3

The following triplet of essential matrices with collinear epipoles is compatible:

$$\begin{aligned} E_{12} = \gamma _1 [s]_\times ,\quad E_{23} = \gamma _2 [s]_\times ,\quad E_{31} = (\epsilon _1\gamma _1 + \epsilon _2\gamma _2) [s]_\times ,\nonumber \\ \end{aligned}$$

(67)

where $\epsilon _i = {\pm } 1$, $s = \begin{bmatrix} 0&\delta&1 \end{bmatrix}^\top$, and $\delta$ is an arbitrary parameter.

Proof

We denote $R_0 = \begin{bmatrix} -1 &{} 0 &{} 0 \\ 0 &{} -\cos \psi _0 &{} \sin \psi _0 \\ 0 &{} \sin \psi _0 &{} \cos \psi _0 \end{bmatrix}$, where $\cos \psi _0 = \frac{1 - \delta ^2}{1 + \delta ^2}$ and $\sin \psi _0 = \frac{2\delta }{1 + \delta ^2}$. The essential matrices from triplet (67) can be written in form $E_{ij} = [b_{ij}]_\times R_{ij}$, where matrices $R_{ij}$ and vectors $b_{ij}$ are defined as follows

$$\begin{aligned} \begin{aligned} b_{12}&= -\epsilon _2\gamma _1 s,&\quad R_{12}&= {\left\{ \begin{array}{ll}R_0, &{} \epsilon _2 = 1\\ I, &{} \text {otherwise}\end{array}\right. },\\ b_{23}&= -\epsilon _1\gamma _2 s,&\quad R_{23}&= {\left\{ \begin{array}{ll}R_0, &{} \epsilon _1 = 1\\ I, &{} \text {otherwise}\end{array}\right. },\\ b_{31}&= (\epsilon _2\gamma _1 + \epsilon _1\gamma _2) s,&\quad R_{31}&= {\left\{ \begin{array}{ll}R_0, &{} \epsilon _1\epsilon _2 = -1\\ I, &{} \text {otherwise}\end{array}\right. }. \end{aligned} \end{aligned}$$

(68)

It is straightforward to verify that constraints (64)–(65) hold. By Lemma 2, triplet (67) is compatible. Lemma 3 is proved. $\square$

Lemma 4

The following triplets of essential matrices are compatible:

• 1.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} -\gamma _1 &{} \beta _1 \\ \gamma _1 &{} 0 &{} -\alpha _1 \\ -\beta _1 &{} \alpha _1 &{} 0 \end{bmatrix},\quad E_{23} = \begin{bmatrix} 0 &{} -\gamma _2 &{} \beta _2 \\ \gamma _2 &{} 0 &{} \alpha _1 \\ -\beta _2 &{} -\alpha _1 &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \begin{bmatrix}0 &{} \gamma _1 + \gamma _2 &{} -\beta _1 - \beta _2 \\ -\gamma _1 - \gamma _2 &{} 0 &{} 0 \\ \beta _1 + \beta _2 &{} 0 &{} 0 \end{bmatrix}; \end{aligned}$$

  (69)
• 2.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\alpha _1 \\ 0 &{} \alpha _1 &{} 0 \end{bmatrix},\,\nonumber \\ E_{23}= & {} \begin{bmatrix} 0 &{} 0 &{} \beta _2 \\ 0 &{} 0 &{} -\alpha _1 \\ -\beta _2 &{} \alpha _1 &{} 0 \end{bmatrix},\, E_{31} = \begin{bmatrix} 0 &{} 0 &{} \beta _2 \\ 0 &{} 0 &{} 0 \\ \beta _2 &{} 0 &{} 0 \end{bmatrix}; \end{aligned}$$

  (70)
• 3.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} 0 &{} \beta _1 \\ 0 &{} 0 &{} -\alpha _1 \\ -\beta _1 &{} \alpha _1 &{} 0 \end{bmatrix},\, E_{23} = \begin{bmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\alpha _1 \\ 0 &{} \alpha _1 &{} 0 \end{bmatrix},\, \nonumber \\ E_{31}= & {} \begin{bmatrix} 0 &{} 0 &{} -\beta _1 \\ 0 &{} 0 &{} 0 \\ -\beta _1 &{} 0 &{} 0 \end{bmatrix}; \end{aligned}$$

  (71)
• 4.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} 0 &{} \beta _1 \\ 0 &{} 0 &{} -\alpha _1 \\ -\beta _1 &{} \alpha _1 &{} 0 \end{bmatrix},\quad E_{23} = \begin{bmatrix} 0 &{} 0 &{} -\frac{\alpha _1^2}{\beta _1} \\ 0 &{} 0 &{} -\alpha _1 \\ \frac{\alpha _1^2}{\beta _1} &{} \alpha _1 &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \begin{bmatrix} 0 &{} 0 &{} -\frac{\alpha _1^2 + \beta _1^2}{\beta _1} \\ 0 &{} 0 &{} 0 \\ -\frac{\alpha _1^2 + \beta _1^2}{\beta _1} &{} 0 &{} 0 \end{bmatrix}; \end{aligned}$$

  (72)
• 5.
  $$\begin{aligned} E_{12}&{=}&\begin{bmatrix} 0 &{} -\gamma _2 - \gamma _3 &{} 0 \\ \gamma _2 + \gamma _3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{bmatrix},\quad E_{23}{=} \begin{bmatrix} 0 &{} -\gamma _2 &{} \beta _2 \\ \gamma _2 &{} 0 &{} 0 \\ -\beta _2 &{} 0 &{} 0 \end{bmatrix},\nonumber \\ E_{31}&{=}&\begin{bmatrix} 0 &{} \gamma _3 &{} -\beta _2 \\ -\gamma _3 &{} 0 &{} 0 \\ {-}\beta _2 &{} 0 &{} 0 \end{bmatrix}; \end{aligned}$$

  (73)
• 6.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} -\gamma _1 &{} \beta _1 \\ \gamma _1 &{} 0 &{} 0 \\ -\beta _1 &{} 0 &{} 0 \end{bmatrix},\quad E_{23} = \begin{bmatrix} 0 &{} -\gamma _1 - \gamma _3 &{} 0 \\ \gamma _1 + \gamma _3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \begin{bmatrix} 0 &{} \gamma _3 &{} \beta _1 \\ -\gamma _3 &{} 0 &{} 0 \\ \beta _1 &{} 0 &{} 0 \end{bmatrix}; \end{aligned}$$

  (74)
• 7.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} -\gamma _1 &{} \beta _1 \\ \gamma _1 &{} 0 &{} 0 \\ -\beta _1 &{} 0 &{} 0 \end{bmatrix},\quad E_{23} = \begin{bmatrix} 0 &{} -\gamma _2 &{} \beta _2 \\ \gamma _2 &{} 0 &{} 0 \\ -\beta _2 &{} 0 &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \begin{bmatrix} 0 &{} \gamma _3 &{} \beta _3 \\ -\gamma _3 &{} 0 &{} 0 \\ \beta _3 &{} 0 &{} 0 \end{bmatrix}, \end{aligned}$$

  (75)

  where

  $$\begin{aligned} \begin{aligned} \gamma _1&= \delta \beta _1(\beta _1 + \beta _2 - \beta _3),\\ \gamma _2&= \delta \beta _2(\beta _1 + \beta _2 + \beta _3),\\ \gamma _3&= \delta \beta _3(-\beta _1 + \beta _2 + \beta _3), \end{aligned} \end{aligned}$$

  (76)

  and parameter $\delta$ is subject to

  $$\begin{aligned} \delta (\gamma _1 + \gamma _2 + \gamma _3) = -1. \end{aligned}$$

  (77)

Proof

Triplets (69)–(71), (73), and (74) are compatible by definition as the corresponding essential matrices can be represented in form (13). Namely,

• 1. for triplet (69):

  $$\begin{aligned} E_{12}= & {} I [b_1 - 0]_\times I,\quad E_{23} = I [0 - b_3]_\times I,\nonumber \\ E_{31}= & {} I[b_3 - b_1]_\times I, \end{aligned}$$

  (78)

  where $b_1 = \begin{bmatrix} \alpha _1&\beta _1&\gamma _1 \end{bmatrix}^\top$, $b_3 = \begin{bmatrix} \alpha _1&-\beta _2&-\gamma _2 \end{bmatrix}^\top$;

• 2. for triplet (70):

  $$\begin{aligned} E_{12}= & {} D_1 [0 - b_2]_\times I,\quad E_{23} = I [b_2 - b_3]_\times I,\nonumber \\ E_{31}= & {} I [b_3 - 0]_\times D_1, \end{aligned}$$

  (79)

  where $D_1 = {{\,\mathrm{diag}\,}}(1, -1, -1)$, $b_2 = \begin{bmatrix} \alpha _1&0&0 \end{bmatrix}^\top$, $b_3 = \begin{bmatrix} 0&-\beta _2&0 \end{bmatrix}^\top$;

• 3. for triplet (71):

  $$\begin{aligned} E_{12}= & {} I [b_1 - b_2]_\times I,\quad E_{23} = I [b_2 - 0]_\times D_1,\nonumber \\ E_{31}= & {} D_1 [0 - b_1]_\times I, \end{aligned}$$

  (80)

  where $b_1 = \begin{bmatrix} 0&\beta _1&0 \end{bmatrix}^\top$, $b_2 = \begin{bmatrix} -\alpha _1&0&0 \end{bmatrix}^\top$;

• 4. for triplet (73):

  $$\begin{aligned}&E_{12} = D_3 [b_1 - b_2]_\times I,\quad E_{23} = I [b_2 - 0]_\times I,\quad \nonumber \\&E_{31} = I [0 - b_1]_\times D_3, \end{aligned}$$

  (81)

  where $D_3 = {{\,\mathrm{diag}\,}}(-1, -1, 1)$, $b_1 = \begin{bmatrix} 0&\beta _2&-\gamma _3 \end{bmatrix}^\top$, $b_2 = \begin{bmatrix} 0&\beta _2&\gamma _2 \end{bmatrix}^\top$;

• 5. for triplet (74):

  $$\begin{aligned} E_{12}= & {} I [0 - b_2]_\times I,\quad E_{23} = I [b_2 - b_3]_\times D_3,\nonumber \\ E_{31}= & {} D_3 [b_3 - 0]_\times I, \end{aligned}$$

  (82)

  where $b_2 = \begin{bmatrix} 0&-\beta _1&-\gamma _1 \end{bmatrix}^\top$, $b_3 = \begin{bmatrix} 0&-\beta _1&\gamma _3 \end{bmatrix}^\top$.

Further, let

$$\begin{aligned} \begin{aligned} \cos \varphi _i&= \frac{\alpha _i^2 - \beta _i^2}{\alpha _i^2 + \beta _i^2},&\quad \sin \varphi _i&= \frac{2\alpha _i\beta _i}{\alpha _i^2 + \beta _i^2},\\ \cos \psi _i&= \frac{\beta _i^2 - \gamma _i^2}{\beta _i^2 + \gamma _i^2},&\quad \sin \psi _i&= \frac{2\beta _i\gamma _i}{\beta _i^2 + \gamma _i^2}. \end{aligned} \end{aligned}$$

(83)

The essential matrices from triplets (72) and (75) admit the representation $E_{ij} = [b_{ij}]_\times R_{ij}$, where the matrices $R_{ij}$ and vectors $b_{ij}$ are defined below in (84) and (85) respectively:

• 1.
  $$\begin{aligned} b_{12}&= \begin{bmatrix} -\alpha _1 \\ -\beta _1 \\ 0 \end{bmatrix},&\quad R_{12}&= \begin{bmatrix} \cos \varphi _1 &{} \sin \varphi _1 &{} 0 \\ \sin \varphi _1 &{} -\cos \varphi _1 &{} 0 \\ 0 &{} 0 &{} -1 \end{bmatrix},\nonumber \\ b_{23}&= \begin{bmatrix} -\alpha _1 \\ \frac{\alpha _1^2}{\beta _1}\nonumber \\ 0 \end{bmatrix},&\quad R_{23}&= \begin{bmatrix} -\cos \varphi _1 &{} -\sin \varphi _1 &{} 0 \\ -\sin \varphi _1 &{} \cos \varphi _1 &{} 0 \\ 0 &{} 0 &{} -1 \end{bmatrix},\\ b_{31}&= \begin{bmatrix} 0 \\ -\frac{\alpha _1^2 + \beta _1^2}{\beta _1} \\ 0 \end{bmatrix},&\quad R_{31}&= \begin{bmatrix} -1 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 1 \end{bmatrix}. \end{aligned}$$

  (84)
• 2.
  $$\begin{aligned} b_{12}&= -\begin{bmatrix} 0 \\ \beta _1 \\ \gamma _1 \end{bmatrix},&\quad R_{12}&= \begin{bmatrix} -1 &{} 0 &{} 0 \\ 0 &{} \cos \psi _1 &{} \sin \psi _1 \\ 0 &{} \sin \psi _1 &{} -\cos \psi _1 \end{bmatrix},\nonumber \\ b_{23}&= -\begin{bmatrix} 0 \\ \beta _2 \\ \gamma _2 \end{bmatrix},&\quad R_{23}&= \begin{bmatrix} -1 &{} 0 &{} 0 \\ 0 &{} \cos \psi _2 &{} \sin \psi _2 \\ 0 &{} \sin \psi _2 &{} -\cos \psi _2 \end{bmatrix},\nonumber \\ b_{31}&= -\begin{bmatrix} 0 \\ \beta _3 \\ \gamma _3 \end{bmatrix},&\quad R_{31}&= \begin{bmatrix} 1 &{} 0 &{} 0 \\ 0 &{} -\cos \psi _3 &{} \sin \psi _3 \\ 0 &{} -\sin \psi _3 &{} -\cos \psi _3 \end{bmatrix}. \end{aligned}$$

  (85)

It is straightforward to verify that constraints (64)–(65) hold for both cases. By Lemma 2, triplets (72) and (75) are compatible. Lemma 4 is proved. $\square$

Lemma 5

The following triplets of essential matrices are compatible provided that $\lambda ^2 + \mu ^2 = 1$:

• 1.
  $$\begin{aligned} E_{12}= & {} \beta _1\begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} -\frac{\lambda - 1}{\mu } \\ -1 &{} \frac{\lambda - 1}{\mu } &{} 0 \end{bmatrix},\nonumber \\&E_{23} = \begin{bmatrix} 0 &{} 0 &{} \beta _1 + \beta _3 \\ 0 &{} 0 &{} -\beta _1\frac{\lambda - 1}{\mu } \\ -\beta _1 - \beta _3 &{} \beta _1\frac{\lambda - 1}{\mu } &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \beta _3\begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ -\lambda &{} -\mu &{} 0 \end{bmatrix}; \end{aligned}$$

  (86)
• 2.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} 0 &{} \beta _2 + \beta _3\lambda \\ 0 &{} 0 &{} -\alpha _1 \\ -\beta _2 - \beta _3\lambda &{} \alpha _1 &{} 0 \end{bmatrix},\nonumber \\&E_{23} = \beta _2\begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} -\frac{\lambda - 1}{\mu } \\ -1 &{} \frac{\lambda - 1}{\mu } &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \beta _3\begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ -\lambda &{} -\mu &{} 0 \end{bmatrix}, \end{aligned}$$

  (87)

  where $\alpha _1 = (\beta _2 + \beta _3(\lambda + 1))\frac{\lambda - 1}{\mu }$;

• 3.
  $$\begin{aligned} E_{12}= & {} \begin{bmatrix} 0 &{} 0 &{} \beta _1 \\ 0 &{} 0 &{} -\alpha _1 \\ -\beta _1 &{} \alpha _1 &{} 0 \end{bmatrix},\nonumber \\&E_{23} = \begin{bmatrix} 0 &{} 0 &{} \beta _2 \\ 0 &{} 0 &{} \alpha _1\lambda + \beta _1\mu \\ -\beta _2 &{} -\alpha _1\lambda - \beta _1\mu &{} 0 \end{bmatrix},\nonumber \\ E_{31}= & {} \beta _3\begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ -\lambda &{} -\mu &{} 0 \end{bmatrix}, \end{aligned}$$

  (88)

  where

  $$\begin{aligned} \begin{aligned} \beta _2&= -\frac{(\alpha _1(\lambda - 1) + \beta _1\mu )(\alpha _1\lambda + \beta _1\mu )}{\alpha _1\mu - \beta _1(\lambda - 1)},\\ \beta _3&= \frac{(\alpha _1^2 + \beta _1^2)(\lambda - 1)}{\alpha _1\mu - \beta _1(\lambda - 1)}. \end{aligned} \end{aligned}$$

  (89)

Proof

Throughout the proof, $\cos \varphi _i = \frac{\alpha _i^2 - \beta _i^2}{\alpha _i^2 + \beta _i^2}$ and $\sin \varphi _i = \frac{2\alpha _i\beta _i}{\alpha _i^2 + \beta _i^2}$. The essential matrices from triplets (86), (87), and (88) can be written in form $E_{ij} = [b_{ij}]_\times R_{ij}$, where the matrices $R_{ij}$ and vectors $b_{ij}$ are defined below in (90), (91), and (92) respectively:

• 1.
  $$\begin{aligned} b_{12}&= -\begin{bmatrix} \alpha _1 \\ \beta _1 \\ 0 \end{bmatrix},&\quad R_{12}&= \begin{bmatrix} \cos \varphi _1 &{} \sin \varphi _1 &{} 0 \\ \sin \varphi _1 &{} -\cos \varphi _1 &{} 0 \\ 0 &{} 0 &{} -1 \end{bmatrix},\nonumber \\ b_{23}&= \begin{bmatrix} \alpha _1 \\ \beta _1 + \beta _3 \\ 0 \end{bmatrix},&\quad R_{23}&= I,\nonumber \\ b_{31}&= -\begin{bmatrix} 0 \\ \beta _3 \\ 0 \end{bmatrix},&\quad R_{31}&= R_{12}, \end{aligned}$$

  (90)

  where $\alpha _1 = \beta _1\frac{\lambda - 1}{\mu }$;

• 2.
  $$\begin{aligned} b_{12}&= \begin{bmatrix} \alpha _1 \\ \beta _2 + \beta _3\lambda \\ 0 \end{bmatrix},&\quad R_{12}&= I,\nonumber \\ b_{23}&= -\begin{bmatrix} \alpha _2 \\ \beta _2 \\ 0 \end{bmatrix},&\quad R_{23}&= \begin{bmatrix} \cos \varphi _2 &{} \sin \varphi _2 &{} 0 \\ \sin \varphi _2 &{} -\cos \varphi _2 &{} 0 \\ 0 &{} 0 &{} -1 \end{bmatrix},\nonumber \\ b_{31}&= -\begin{bmatrix} 0 \\ \beta _3 \\ 0 \end{bmatrix},&\quad R_{31}&= R_{23}, \end{aligned}$$

  (91)

  where $\alpha _1 = (\beta _2 + \beta _3(\lambda + 1))\frac{\lambda - 1}{\mu }$, $\alpha _2 = \beta _2\frac{\lambda - 1}{\mu }$;

• 3.
  $$\begin{aligned} b_{12}&= -\begin{bmatrix} \alpha _1 \\ \beta _1 \\ 0 \end{bmatrix},&\quad R_{12}&= \begin{bmatrix} \cos \varphi _1 &{} \sin \varphi _1 &{} 0 \\ \sin \varphi _1 &{} -\cos \varphi _1 &{} 0 \\ 0 &{} 0 &{} -1 \end{bmatrix},\nonumber \\ b_{23}&= -\begin{bmatrix} \alpha _2 \\ \beta _2 \\ 0 \end{bmatrix},&\quad R_{23}&= \begin{bmatrix} \cos \varphi _2 &{} \sin \varphi _2 &{} 0 \\ \sin \varphi _2 &{} -\cos \varphi _2 &{} 0 \\ 0 &{} 0 &{} -1 \end{bmatrix},\nonumber \\ b_{31}&= \begin{bmatrix} 0 \\ \beta _3 \\ 0 \end{bmatrix},&\quad R_{31}&= (R_{12}R_{23})^\top = \begin{bmatrix} \lambda &{} \mu &{} 0 \\ -\mu &{} \lambda &{} 0 \\ 0 &{} 0 &{} 1 \end{bmatrix}, \end{aligned}$$

  (92)

  where $\beta _2$ and $\beta _3$ are defined in (89).

By direct computation, constraints (64)–(65) hold for all three cases. By Lemma 2, triplets (86)–(88) are compatible. Lemma 5 is proved. $\square$