INTRODUCTION

In many physical and technological problems, it is necessary to ensure that without additional adjustment, a complex mechanical system, after disassembly and reassembly, returns to the original state with a high degree of accuracy regarding its constructional properties. Such problems, in particular, continuously arise when designing multi-mirror reflectors for space telescopes. However, similar problems arise in optics as well upon assembly and adjustment of various systems on an optical bench, where they were considered by Maxwell and Kelvin. In these classical papers [1], fixtures, ensuring a high accuracy and a high repeatability of installation of optical elements, i.e., lenses, reflectors, prisms, and photographic plates, on an optical bench, were proposed. Later, this method of combining elements of construction became known as kinematic couplings in English-language literature (see, for example, in Wikipedia https://en.wikipedia.org/wiki/Kinematic_coupling). Fixtures were modernized, refined, and used to solve various scientific and applied problems. In recent decades, these devices have found application in nanotechnologies. Here, with high-quality manufacturing of fixture elements, it is possible to provide micron and submicron levels of accuracy and repeatability of assembly of construction elements [2–6].

This paper is aimed at showing that these constructions of laboratory optics (under a certain natural modification) will be useful for problems of constructing multi-mirror reflectors of large space telescopes. Such a non-obvious connection of classical optics and mechanics of space constructions seems worthy of discussion and examination.

We will call these combinations kinematic couplings and study the possibility of developing mechanical self-setting locks for space applications on their basis. The locks may find application, for example, for solving problems of assembly or automatic deployment of large multi-mirror reflector antennas [7–10] (see also electronic resources www.jwst.nasa.gov, http://safir.jpl.nasa.gov, http://www.asc.rssi.ru/millimetron).

The basis of a kinematic coupling in the Maxwell’s version is three V-shaped grooves (Fig. 1a), placed on one of the combined elements (the base), and three spherical supports, connected with a tripod, which is the second combined element (Fig. 1b). In its operating condition, the spherical supports are at the bottom of the grooves and retained at this position by gravity (Figs. 1c and 1d).

Fig. 1.
figure 1

Kinematic coupling: (а) V-shaped grooves on a bearing base, (b) three ball bearings, connected with a tripod, (c) assembled kinematic coupling, and (d) modern kinematic coupling of an optical quality.

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Three features of the fixture ensure important mechanical properties.

First, the operating position of the device is in an equilibrium state of construction. At small displacements of the tripod, there is a force, returning the device to an equilibrium operating state.

Second, the equilibrium state is unique, which provides a high repeatability of installation of the tripod on the base.

Finally, the construction turns out to be statically determinate at the equilibrium state, which ensures a non-stressed assembly of the fixture. Indeed, each spherical supports, retained at the bottom of a V-shaped groove, introduces two kinematic constraints into the construction. In a nondegenerate case, the total number of kinematic constraints in the construction is equal to six, which ensures its static determinacy. Nondegeneracy of the system is accomplished by the directions of V-shaped grooves.

From the point of view of the couplings’ mechanical properties, it is natural to distinguish the following cases. The coupling can be in an equilibrium state and a state displaced with respect to the state of equilibrium (a nonequilibrium coupling). If the equilibrium state is unique, then the construction is geometrically unchangeable. Changeable couplings can be instantaneously changed; i.e., they allow infinitesimal displacements relative to the equilibrium state, but can admit changes, which are not small, but finite. In the latter case, the coupling turns into a mechanism.

From an applied point of view, unchangeable couplings are of principal interest, since, in this case, the coupling possesses a self-setting property, i.e. spontaneously returns to the equilibrium position, which is desirable under its application. It is this property that was used by Kelvin and Maxwell in the construction and subsequent alignment of optical systems.

1. 2D KINEMATIC COUPLINGS FOR OPERATION IN ZERO GRAVITY

It is natural to begin the study of kinematic couplings with the case when all V-shaped grooves lie in the same plane. We call such couplings two-dimensional (2D couplings).

1.1. The Physical Model of a 2D Coupling

It is necessary to overcome two difficulties in order to use the idea of the classical kinematic coupling for assembling composite space constructions.

First, it is necessary to propose a way of retaining spherical supports at the bottom of V-shaped grooves under the conditions of zero gravity. The supports in the classical kinematic coupling are retained at the bottom of grooves by gravity. Under new conditions, it is necessary to preserve the kinematics of the couplings and direction of retaining forces in the equilibrium state. To this end, construction of a spring capture, presented in Fig. 2a, was proposed and its physical model was manufactured. Here, on the one hand, four degrees of freedom in mutual displacements of the combined elements are preserved and on the other hand, the retaining force is directed along the normal to the base of the groove. The assembled physical model of a 2D kinematic coupling with spring captures to operate under conditions of zero gravity is presented in Fig. 2b. We used the physical model of a spring capture, which was proposed and manufactured by A.Yu. Tondrik (Lebedev Physical Institute, Russian Academy of Sciences).

Fig. 2.
figure 2

The physical model of a 2D kinematic coupling: (а) spring capture of a spherical support and (b) assembled kinematic coupling with spring captures.

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Second, it is necessary to find out at what mutual positions of grooves the design retains self-setting properties. In the classical kinematic coupling, using the Maxwell scheme, V-shaped grooves are placed on the plane, symmetrically with respect to the center of the coupling. Our goal is to combine solid reflectors using kinematic couplings, while preserving the property of repeatability of assembly. At an arbitrary shape of combined reflectors, the kinematic coupling can not provide for preservation of the classical position of V-shaped grooves, the grooves and spherical supports need to be placed on lateral sides of the combined elements, at significant distance from each other. The question arises, at what mutual position of the grooves are self-aligning properties of the structure preserved. We develop a mathematical model describing the operation of a 2D coupling at an arbitrary position of V-shaped grooves in order to answer this question.

1.2. Mathematical Model of the Equilibrium State of 2D Coupling

We consider three segments ω1, ω2, and ω3 on the OX1X2 plane, the mutual position of which coincides with the mutual position of the bases of V-shaped grooves of the 2D coupling (Fig. 3a). We call these segments, guides. The triangle a1a2a3 simulates the position of the coupling’s tripod; its vertices in the equilibrium state are held by constraints on the guides.

Fig. 3.
figure 3

Equilibrium (а, b) and nonequilibrium (c) positions of 2D kinematic couplings.

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Let αi be the angle between the guide ωi and axis OX1, n is the vector of the normal to ωi, f  i is the vector of the constraint force (directed along the normal to ωi), and fi is a numerical value of the constraint force. Then,

$${{f}^{i}} = ( - {{f}_{i}}\sin {{\alpha }_{i}},{{f}_{i}}\cos {{\alpha }_{i}}) = {{f}_{i}}{{n}^{i}},$$
$${{n}^{i}} = ( - \sin {{\alpha }_{i}},\cos {{\alpha }_{i}}),\quad i = 1,2,3.$$

The conditions of equilibrium, i.e., the sum of force projections on the OX1 and OX2 axes and the sum of moments with respect to the origin of coordinates, have the form

$$ - {{f}_{1}}\sin {{\alpha }_{1}} - {{f}_{2}}\sin {{\alpha }_{2}} - {{f}_{3}}\sin {{\alpha }_{3}} = {{F}_{1}},$$
$${{f}_{1}}\cos {{\alpha }_{1}} + {{f}_{2}}\cos {{\alpha }_{2}} + {{f}_{3}}\cos {{\alpha }_{3}} = {{F}_{2}},$$
((1))
$$\begin{gathered} \text{[}{{a}^{1}} \cdot {{f}^{1}}] + [{{a}^{2}} \cdot {{f}^{2}}] + [{{a}^{3}} \cdot {{f}^{3}}] \\ = {{f}_{1}}[{{a}^{1}} \cdot {{n}^{1}}] + {{f}_{2}}[{{a}^{2}} \cdot {{n}^{2}}] + {{f}_{3}}[{{a}^{3}} \cdot {{n}^{3}}] = M. \\ \end{gathered} $$

Here, vector products determine moments of constraint forces f 1, f 2, and f 3 with respect to the origin of coordinates, F1 and F2 are projections of the sum of external forces on the OX1 and OX2 axes, respectively, and M is the sum of moments of external forces with respect to the origin of coordinates.

We rewrite system (1) in matrix form

$$A \cdot f = F,$$
((2))

where

$$A = \left( {\begin{array}{*{20}{c}} { - \sin {{\alpha }_{1}}}&{ - \sin {{\alpha }_{2}}}&{ - \sin {{\alpha }_{3}}} \\ {\cos {{\alpha }_{1}}}&{\cos {{\alpha }_{2}}}&{\cos {{\alpha }_{3}}} \\ {[{{a}^{1}} \cdot {{n}^{1}}]}&{[{{a}^{2}} \cdot {{n}^{2}}]}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right),$$
$$f = \left( \begin{gathered} {{f}_{1}} \\ {{f}_{2}} \\ {{f}_{3}} \\ \end{gathered} \right),\quad F = \left( \begin{gathered} {{F}_{1}} \\ {{F}_{2}} \\ M \\ \end{gathered} \right).$$

Here, A is the equilibrium matrix. If the determinant of equilibrium matrix A is nonzero, system (2) is nondegenerate and has a unique solution, i.e., the structure is geometrically unchangeable. In the nondegenerate case, the coupling possesses self-setting properties.

To clarify conditions of degeneration of the construction, we introduce the system of coordinates, at which the OX1 axis is parallel to ω1, and the origin of coordinates coincides with a point of intersection of normals to ω1 and ω2 at points a1 and a2 (Fig. 3a). In a new system of coordinates, matrix A takes the form

$$A = \left( {\begin{array}{*{20}{c}} 0&{ - \sin {{\alpha }_{2}}}&{ - \sin {{\alpha }_{3}}} \\ 1&{\cos {{\alpha }_{2}}}&{\cos {{\alpha }_{3}}} \\ 0&0&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right).$$

Hence, we find

$$\det A = \sin {{\alpha }_{2}}[{{a}^{3}} \cdot {{n}^{3}}].$$

Therefore, if

$$[{{a}^{3}} \cdot {{n}^{3}}] \ne 0$$

and

$$\sin {{\alpha }_{2}} \ne 0,$$

then, the construction is geometrically unchangeable, the state of equilibrium is unique, and kinematic couplings have self-setting properties.

1.3. Degenerate Constructions

We now turn to degenerate constructions, which are, at least, instantly changeable. We consider the possible cases of degeneration. In practice, it is convenient to check the nondegeneracy by making sure that the system under study does not belong to any of the types listed below.

1. [a3 ⋅ n3] = 0, normals n1, n2, and n3 intersect at one point, the construction is instantly changeable and admits an infinitesimal rotation (Fig. 4a) around the point of intersection of the normals.

Fig. 4.
figure 4

Degenerate states of the coupling: (а) degenerate state 1, (b) degenerate state 2.1, (c) degenerate state 2.2, and (d) degenerate state 3 (the construction turns into a mechanism).

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2. sin α2 = 0, the guides are parallel. Two cases are possible:

2.1. Normals n1 and n2 lie on the same straight line (Fig. 4b). The construction is degenerate, instantly changeable; an infinitesimal rotation around the point of intersection of the normals is possible.

2.2. Normals n1 and n2 do not lie on the same straight line. We introduce the system of coordinates, in which the OX1 axis is parallel to ω1 and ω2, and the origin of coordinates coincides with the point of intersection of normals ω1 and ω3 (Fig. 4c). Then, matrix A takes the form:

$$\begin{gathered} A = \left( {\begin{array}{*{20}{c}} { - \sin {{\alpha }_{1}}}&{ - \sin {{\alpha }_{2}}}&{ - \sin {{\alpha }_{3}}} \\ {\cos {{\alpha }_{1}}}&{\cos {{\alpha }_{2}}}&{\cos {{\alpha }_{3}}} \\ {[{{a}^{1}} \cdot {{n}^{1}}]}&{[{{a}^{2}} \cdot {{n}^{2}}]}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right) \\ = \left( {\begin{array}{*{20}{c}} 0&0&{ - \sin {{\alpha }_{3}}} \\ 1&1&{\cos {{\alpha }_{3}}} \\ 0&{[{{a}^{2}} \cdot {{n}^{2}}]}&0 \end{array}} \right). \\ \end{gathered} $$

Hence, we find

$$\det A = - \sin {{\alpha }_{3}}[{{a}^{2}} \cdot {{n}^{2}}] \ne 0.$$

The construction is geometrically unchangeable.

3. Lines ω1, ω2, and ω3 are parallel (Fig. 4d). Matrix A of system (1) takes the form

$$\begin{gathered} A = \left( {\begin{array}{*{20}{c}} { - \sin {{\alpha }_{1}}}&{ - \sin {{\alpha }_{2}}}&{ - \sin {{\alpha }_{3}}} \\ {\cos {{\alpha }_{1}}}&{\cos {{\alpha }_{2}}}&{\cos {{\alpha }_{3}}} \\ {[{{a}^{1}} \cdot {{n}^{1}}]}&{[{{a}^{2}} \cdot {{n}^{2}}]}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right) \\ = \left( {\begin{array}{*{20}{c}} 0&0&0 \\ 1&1&1 \\ {[{{a}^{1}} \cdot {{n}^{1}}]}&{[{{a}^{2}} \cdot {{n}^{2}}]}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right). \\ \end{gathered} $$

Hence,

$$\det A = 0.$$

The construction is degenerate, i.e., movement of the triangle along the guides is possible. We illustrate the use of the considered concepts on a specific example. In the classical papers of Maxwell, V-shaped grooves are on the same plane, symmetrically with respect to the center of the coupling. In many cases, it is necessary to abandon this symmetric arrangement of elements of construction without losing the self-setting properties. The physical model of two solid surfaces, joined by a kinematic coupling with separated V-shaped grooves, is presented in Fig. 5. The construction is equipped with spring captures of spherical supports. Directly calculating the determinant of the equilibrium matrix, we are convinced that it is nonzero at the chosen position of V-shaped grooves. This is found in the physical experiment as well, i.e., the construction is fixed in configuration. Thus, the proposed construction is geometrically unchangeable and possesses a high repeatability of assembly.

Fig. 5.
figure 5

Two plates joined by a kinematic coupling with spring captures.

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1.4. Mathematical Model of a Nonequilibrium State of the Coupling and Optimization of Stiffness

It follows from the developed mathematical model that, for arbitrary placement of V-shaped grooves, the design, as a rule, has self-setting properties. Therefore, in the class of self-setting couplings, one can formulate the problem of optimizing certain characteristics of a joint. We consider the optimization problem of one element of the stiffness matrix of the multi-mirror reflector, i.e., the torsional stiffness of a coupling.

For this purpose, we construct a stiffness matrix and find the expression for torsional stiffness of a 2D coupling as a function of angles α1, α2, and α3, i.e., directions of V-shaped grooves.

The elements of construction (tripod, base, and V-shaped grooves) will be considered as solid bodies. With “deformation” of the construction, that is displacing the tripod from the equilibrium state, there are restoring forces, the values of which are determined by the spring captures of spherical supports. The restoring forces are applied to the vertices of the triangle and directed along the normals to guides ω1, ω2, and ω3. The magnitudes of forces are proportional to distances from the vertex to the corresponding line ω1, ω2, or ω3.

Figures 3b and 3c show schemes of the equilibrium and nonequilibrium position of the coupling. We construct the stiffness matrix G of the system, which relates generalized vector F of external forces (forces and moments) to generalized vector d of displacements of the triangle (displacements and rotations):

$$F = G \cdot d,$$
((3))
$$\begin{gathered} F = \left( \begin{gathered} {{F}_{1}} \\ {{F}_{2}} \\ M \\ \end{gathered} \right), \\ G = \left( {\begin{array}{*{20}{c}} {{{g}_{{1,1}}}}&{{{g}_{{1,2}}}}&{{{g}_{{1,3}}}} \\ {{{g}_{{2,1}}}}&{{{g}_{{2,2}}}}&{{{g}_{{2,3}}}} \\ {{{g}_{{3,1}}}}&{{{g}_{{3,2}}}}&{{{g}_{{3,3}}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{{\partial {{F}_{1}}}}{{\partial {{x}_{1}}}}}&{\frac{{\partial {{F}_{1}}}}{{\partial {{x}_{2}}}}}&{\frac{{\partial {{F}_{1}}}}{{\partial \varphi }}} \\ {\frac{{\partial {{F}_{2}}}}{{\partial {{x}_{1}}}}}&{\frac{{\partial {{F}_{2}}}}{{\partial {{x}_{2}}}}}&{\frac{{\partial {{F}_{2}}}}{{\partial \varphi }}} \\ {\frac{{\partial M}}{{\partial {{x}_{1}}}}}&{\frac{{\partial M}}{{\partial {{x}_{2}}}}}&{\frac{{\partial M}}{{\partial \varphi }}} \end{array}} \right), \\ \end{gathered} $$
$$d = \left( \begin{gathered} d{{x}_{1}} \\ d{{x}_{2}} \\ d\varphi \\ \end{gathered} \right).$$

Along with equations of equilibrium (1), we consider the system of compatibility of deformations of a 2D coupling:

$$B \cdot d = s,$$
((4))

where

$$B = \left( {\begin{array}{*{20}{c}} { - \sin {{\alpha }_{1}}}&{\cos {{\alpha }_{1}}}&{[{{a}^{1}} \cdot {{n}^{1}}]} \\ { - \sin {{\alpha }_{2}}}&{\cos {{\alpha }_{2}}}&{[{{a}^{2}} \cdot {{n}^{2}}]} \\ { - \sin {{\alpha }_{3}}}&{\cos {{\alpha }_{3}}}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right),$$
$$d = \left( \begin{gathered} d{{x}_{1}} \\ d{{x}_{2}} \\ d\varphi \\ \end{gathered} \right),\quad s = \left( \begin{gathered} d{{l}_{1}} \\ d{{l}_{2}} \\ d{{l}_{3}} \\ \end{gathered} \right).$$

Vector d determines small displacements of the triangle (displacements along the coordinate axes and rotation) and vector s determines the distance from the displaced vertices of the triangle to the guides. It is important to note that

$$B = A{\text{*}}.$$
((5))

Assuming that all spring captures are identical, we write Hooke’s law

$$f = k \cdot s,$$
((6))

where k is a coefficient of elasticity. Substituting (6) into (1), we have

$$k \cdot A \cdot s = F.$$
((7))

Then, substituting (4) into (7), we obtain

$$k \cdot A \cdot B \cdot d = F.$$
((8))

Comparing (3) and (8) and considering (4), we find an expression for the stiffness matrix of the coupling

$$\begin{gathered} G = k \cdot A \cdot B = k \cdot A \cdot {{A}^{{\text{T}}}} = \left( {\begin{array}{*{20}{c}} {{{g}_{{1,1}}}}&{{{g}_{{1,2}}}}&{{{g}_{{1,3}}}} \\ {{{g}_{{2,1}}}}&{{{g}_{{2,2}}}}&{{{g}_{{2,3}}}} \\ {{{g}_{{3,1}}}}&{{{g}_{{3,2}}}}&{{{g}_{{3,3}}}} \end{array}} \right) \\ = k\left( {\begin{array}{*{20}{c}} { - \sin {{\alpha }_{1}}}&{ - \sin {{\alpha }_{2}}}&{ - \sin {{\alpha }_{3}}} \\ {\cos {{\alpha }_{1}}}&{\cos {{\alpha }_{2}}}&{\cos {{\alpha }_{3}}} \\ {[{{a}^{1}} \cdot {{n}^{1}}]}&{[{{a}^{2}} \cdot {{n}^{2}}]}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right) \\ \times \left( {\begin{array}{*{20}{c}} { - \sin {{\alpha }_{1}}}&{\cos {{\alpha }_{1}}}&{[{{a}^{1}} \cdot {{n}^{1}}]} \\ { - \sin {{\alpha }_{2}}}&{\cos {{\alpha }_{2}}}&{[{{a}^{2}} \cdot {{n}^{2}}]} \\ { - \sin {{\alpha }_{3}}}&{\cos {{\alpha }_{3}}}&{[{{a}^{3}} \cdot {{n}^{3}}]} \end{array}} \right). \\ \end{gathered} $$
((9))

As an example, we consider the problem of optimization of torsional stiffness for a 2D coupling.

Let the centers of V-shaped grooves be determined for reasons of construction, i.e., a1, a2, and a3 are given. It is required to choose directions of grooves (angles α1, α2, and α3), at which torsional stiffness of the coupling takes the largest value.

The torsional stiffness is given by element g3,3 of stiffness matrix (3). From (9), we have for g3,3:

$${{g}_{{3,3}}} = \frac{{\partial M}}{{\partial \varphi }} = k({{[{{a}^{1}} \cdot {{n}^{1}}]}^{2}} + {{[{{a}^{2}} \cdot {{n}^{2}}]}^{2}} + {{[{{a}^{3}} \cdot {{n}^{3}}]}^{2}}).$$

We introduce notations:

$$\begin{gathered} U({{\alpha }_{i}}) = {{[{{a}^{i}} \cdot {{n}^{i}}]}^{2}} = {{(a_{1}^{i} \cdot \cos {{\alpha }_{1}} + a_{2}^{i} \cdot \sin {{\alpha }_{1}})}^{2}}, \\ i = 1,2,3. \\ \end{gathered} $$

Then, the following conditions are fulfilled at extreme points:

$$\begin{gathered} \frac{{\partial U}}{{\partial {{\alpha }_{1}}}} = 2(a_{1}^{1} \cdot \cos {{\alpha }_{1}} + a_{2}^{1} \cdot \sin {{\alpha }_{1}}) \\ \, \times ( - a_{1}^{1} \cdot \sin {{\alpha }_{1}} + a_{2}^{1} \cdot \cos {{\alpha }_{1}}) = 0. \\ \end{gathered} $$

Hence, two possibilities appear. The first of them

$$(a_{1}^{1} \cdot \cos {{\alpha }_{1}} + a_{2}^{1} \cdot \sin {{\alpha }_{1}}) = 0$$
((10))

corresponds to minimal torsional stiffness of the coupling. In this position, the guide is orthogonal to the line a1o (Fig. 6a), the construction is instantly changeable.

Fig. 6.
figure 6

Extremal states of a 2D kinematic coupling and a spatial kinematic coupling, (а, b) states of minimal and maximal torsional stiffness of the 2D coupling, (c) spatial kinematic coupling.

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The second possibility is

$$( - a_{1}^{1} \cdot \sin {{\alpha }_{1}} + a_{2}^{1} \cdot \cos {{\alpha }_{1}}) = 0.$$
((11))

We calculate the second derivative of function U(α1):

$$\begin{gathered} \frac{{{{\partial }^{2}}U}}{{\partial \alpha _{1}^{2}}} = - 2[{{(a_{1}^{1} \cdot \cos {{\alpha }_{1}} + a_{2}^{1} \cdot \sin {{\alpha }_{1}})}^{2}} \\ \, + {{(a_{1}^{1} \cdot \sin {{\alpha }_{1}} - a_{2}^{1} \cdot \cos {{\alpha }_{1}})}^{2}}]. \\ \end{gathered} $$

If condition (10) is fulfilled and condition (9) is not fulfilled, then

$$\frac{{{{\partial }^{2}}U}}{{\partial \alpha _{1}^{2}}} = - 2[{{(a_{1}^{1} \cdot \cos {{\alpha }_{1}} + a_{2}^{1} \cdot \sin {{\alpha }_{1}})}^{2}}] < 0.$$

Namely, the position corresponds to a maximum of torsional stiffness of the coupling (Fig. 6b), directions of a1o and ω1 coincide.

2. 3D KINEMATIC COUPLINGS

We now abandon the condition of the two-dimensional coupling. This is necessary since when joining surfaces of a complex shape it is unrealistic to require all grooves to be arranged in the same plane. The classical kinematic coupling, using V-shaped grooves, assumes placement of the grooves on a flat base. It is necessary to arrange the grooves in space and ensure self-setting properties of the construction in order to join curved reflectors. Unfortunately, in the general three-dimensional case, it is not yet possible to obtain analogs of the conditions, obtained above for two-dimensional couplings, which are appropriate for practical use. However, this can be done for several practically important cases. We will look at one of them in detail.

We consider a spatial kinematic coupling with its grooves arranged according to Fig. 6c. This coupling is apparently not two-dimensional, however, as we now show, its investigation can be performed using the methods of study of two-dimensional couplings considered above. We show that upon this spatial arrangement of V-shaped grooves, the construction is geometrically unchangeable and the 3D coupling possesses self-setting properties as well. Positions of points a1, a2, and a3 and directions of V-shaped grooves are determined by vectors

$${{a}^{1}} = (a_{1}^{1},0,0),\quad {{a}^{2}} = (0,a_{2}^{2},0),\quad {{a}^{3}} = (0,0,a_{3}^{3}),$$
$${{\omega }^{1}} = (0,1,0),\quad {{\omega }^{2}} = (0,0,1),\quad {{\omega }^{3}} = (1,0,0).$$

Equations of equilibrium of forces and moments have the form

$${{f}^{1}} + {{f}^{2}} + {{f}^{3}} = F,$$
$$[{{a}^{1}} \cdot {{f}^{1}}] + [{{a}^{2}} \cdot {{f}^{2}}] + [{{a}^{3}} \cdot {{f}^{3}}] = M,$$

where f  1, f  2, and f  3 are constraint forces at a1, a2, and a3, F* = (F1, F2, F3) is a vector of the sum of external forces, M* = (M1, M2, M3) is a vector of the sum of external moments.

At chosen a1, a2, and a3 and ω1, ω2, and ω3, we have for the constraint force

$${{f}^{1}} = (f_{1}^{1},0,f_{3}^{1}),\quad {{f}^{2}} = (f_{1}^{2},f_{2}^{2},0),\quad {{f}^{3}} = (0,f_{2}^{3},f_{3}^{3}).$$

Here, there are six unknown quantities that need to be found from the equilibrium conditions. We find three equations from the equilibrium condition of forces

$$f_{1}^{1} + f_{1}^{2} = {{F}_{1}},$$
((12))
$$f_{2}^{2} + f_{2}^{3} = {{F}_{2}},$$
((13))
$$f_{3}^{1} + f_{3}^{3} = {{F}_{3}}.$$
((14))

To calculate moments, we use expression

$$\begin{gathered} \text{[}a \cdot f] = \left| {\begin{array}{*{20}{c}} i&j&k \\ {{{a}_{1}}}&{{{a}_{2}}}&{{{a}_{3}}} \\ {{{f}_{1}}}&{{{f}_{2}}}&{{{f}_{3}}} \end{array}} \right| \\ = \left| {\begin{array}{*{20}{c}} {{{a}_{2}}}&{{{a}_{3}}} \\ {{{f}_{2}}}&{{{f}_{3}}} \end{array}} \right| \cdot i - \left| {\begin{array}{*{20}{c}} {{{a}_{1}}}&{{{a}_{3}}} \\ {{{f}_{1}}}&{{{f}_{3}}} \end{array}} \right| \cdot j + \left| {\begin{array}{*{20}{c}} {{{a}_{1}}}&{{{a}_{2}}} \\ {{{f}_{1}}}&{{{f}_{2}}} \end{array}} \right|. \\ \end{gathered} $$

Then,

$$\begin{gathered} \text{[}{{a}^{1}} \cdot {{f}^{1}}] = \left| {\begin{array}{*{20}{c}} i&j&k \\ {a_{1}^{1}}&0&0 \\ {f_{1}^{1}}&0&{f_{3}^{1}} \end{array}} \right| \\ = 0 \cdot i - \left| {\begin{array}{*{20}{c}} {a_{1}^{1}}&0 \\ {f_{1}^{1}}&{f_{3}^{1}} \end{array}} \right| \cdot j + \left| {\begin{array}{*{20}{c}} {{{a}_{1}}}&0 \\ {{{f}_{1}}}&0 \end{array}} \right| \cdot k = - a_{1}^{1} \cdot f_{3}^{1} \cdot j, \\ \end{gathered} $$
$$\begin{gathered} \text{[}{{a}^{2}} \cdot {{f}^{2}}] = \left| {\begin{array}{*{20}{c}} i&j&k \\ 0&{a_{2}^{2}}&0 \\ {f_{1}^{2}}&{f_{2}^{2}}&0 \end{array}} \right| = 0 \cdot i - 0 \cdot j \\ + \left| {\begin{array}{*{20}{c}} 0&{a_{2}^{2}} \\ {f_{1}^{2}}&{f_{2}^{2}} \end{array}} \right| \cdot k = - a_{2}^{2} \cdot f_{1}^{2} \cdot k, \\ \end{gathered} $$
$$\begin{gathered} \text{[}{{a}^{3}} \cdot {{f}^{3}}] = \left| {\begin{array}{*{20}{c}} i&j&k \\ 0&0&{a_{3}^{3}} \\ 0&{f_{2}^{3}}&{f_{3}^{3}} \end{array}} \right| \\ = \left| {\begin{array}{*{20}{c}} 0&{a_{3}^{3}} \\ {f_{2}^{3}}&{f_{3}^{3}} \end{array}} \right| \cdot i - 0 \cdot j + 0 \cdot k = - a_{3}^{3} \cdot f_{2}^{3} \cdot i. \\ \end{gathered} $$

The equilibrium conditions of moments take the form

$$\begin{gathered} \text{[}{{a}^{1}} \cdot {{f}^{1}}] + [{{a}^{2}} \cdot {{f}^{2}}] + [{{a}^{3}} \cdot {{f}^{3}}] \\ = - a_{3}^{3} \cdot f_{2}^{3} \cdot i - a_{1}^{1} \cdot f_{3}^{1} \cdot j - a_{2}^{2} \cdot f_{1}^{2} \cdot k = M \\ \end{gathered} $$

or in the coordinate form

$$ - a_{3}^{3} \cdot f_{2}^{3} = {{M}_{1}},$$
((15))
$$ - a_{1}^{1} \cdot f_{3}^{1} = {{M}_{2}},$$
((16))
$$ - a_{2}^{2} \cdot f_{1}^{2} = {{M}_{3}}.$$
((17))

We introduce into consideration a vector of desired unknown quantities \(\bar{f}^{*} = (f_{1}^{1}\), \(f_{3}^{1}\), \(f_{1}^{2}\), \(f_{2}^{2}\), \(f_{2}^{3}\), \(f_{3}^{3}\)) (the asterisk stands for transposition), a vector of the external actions Φ* = (F1, F2, F3, M1, M2, M3), and rewrite system (2)−(17) in matrix form:

$$B \cdot \bar {f} = \Phi ,$$
$$B = \left( {\begin{array}{*{20}{c}} 1&0&1&0&0&0 \\ 0&0&0&1&1&0 \\ 0&1&0&0&0&1 \\ 0&0&0&0&{ - a_{3}^{3}}&0 \\ 0&{ - a_{1}^{1}}&0&0&0&0 \\ 0&0&{ - a_{2}^{2}}&0&0&0 \end{array}} \right) \times \left( \begin{gathered} f_{1}^{1} \\ f_{3}^{1} \\ f_{1}^{2} \\ f_{2}^{2} \\ f_{2}^{3} \\ f_{3}^{3} \\ \end{gathered} \right) = \left( \begin{gathered} {{F}_{1}} \\ {{F}_{2}} \\ {{F}_{3}} \\ {{M}_{1}} \\ {{M}_{2}} \\ {{M}_{3}} \\ \end{gathered} \right).$$

If detB ≠ 0, then the system is nondegenerate, construction is geometrically unchangeable, the equilibrium state is unique, and the system possesses self-setting properties. Expanding the determinant, we find

$$\begin{gathered} \det B = \det \left( {\begin{array}{*{20}{c}} 1&0&1&0&0&0 \\ 0&0&0&1&1&0 \\ 0&1&0&0&0&1 \\ 0&0&0&0&{ - a_{3}^{3}}&0 \\ 0&{ - a_{1}^{1}}&0&0&0&0 \\ 0&0&{ - a_{2}^{2}}&0&0&0 \end{array}} \right) \\ = - a_{1}^{1} \cdot a_{2}^{2} \cdot a_{3}^{3} \cdot \left| {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right| = - a_{1}^{1} \cdot a_{2}^{2} \cdot a_{3}^{3}. \\ \end{gathered} $$

Therefore, if numbers \(a_{1}^{1}\), \(a_{2}^{2}\), and \(a_{3}^{3}\) are nonzero, then the coupling with its V-shaped grooves arranged according to Fig. 6c possesses self-setting properties.

The considered arrangement of 3D coupling grooves was used in the physical model presented in Fig. 7. Here, two parabolic petals, connected by self-setting locks on an assembly stock (Fig. 7a), two assembled petals (Fig. 7b), and spring captures of spherical supports (Figs. 7c and 7d) are shown.

Fig. 7.
figure 7

3D kinematic coupling of two parabolic petals, (а) assembly of two petals on a template, back view, (b) two petals, joined by the kinematic coupling, front view, (c) vertical capture of a spherical support, and (d) horizontal capture of a spherical support.

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The simulation confirmed geometric unchangeability and simplicity of assembly of a multi-mirror reflector.

3. KINEMATIC COUPLING IN THE PROBLEM OF DEPLOYMENT OF A MULTI-MIRROR REFLECTOR

It is remarkable that the methods of kinematic couplings for accurate assembly of multi-mirror reflectors can be slightly modified and used for another related problem, which is the problem of deployment of multi-mirror reflectors. This problem is especially important when constructing multi-mirror reflectors for orbital telescopes.

We studied the classical construction of a transformable petal reflector, produced according to a scheme from the Dornier Corporation [10].

The construction includes a central reflector and a set of petals, each of which is connected with the central reflector via a cylindrical hinge. The deployment of the reflector is carried out by synchronous rotation of petals around the axes of hinges (Fig. 8). It has been shown, and this is the key finding of the Dornier developers, that there exists such direction of the axes of cylindrical hinges in which the transition of the petals from the folded state to the opened one takes place without the petals being hooked.

Fig. 8.
figure 8

The classical scheme of deployment of a petal reflector. Successive phases of deployment of the reflector are shown at the top of the figure. The corresponding changes in position of an individual petal are shown at the bottom of the figure, and the deployed construction as a whole is depicted on the right image of the bottom row.

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This construction was used when developing an antenna for the space radio telescope of the Radioastron project [11]. This antenna efficiently operates in the centimeter spectral range. However, the achieved accuracy of the reflecting surface is not sufficient for operation of the reflector in millimeter and submillimeter spectral ranges.

We studied the possibility of improving the accuracy and repeatability of deployment of a single petal with the use of kinematic couplings [12–14]. To this end, a physical model of the system of deployment of a petal was constructed and its investigations were performed. In the model, three spherical supports were placed on the back side of the petal near its base (Fig. 9). Three V-shaped grooves were mounted on a bed, simulating the base of the central reflector. The conversion of the petal from the folded (transport) position into the deployed (operating) state was carried out by rotation of the petal around the axis, connecting the petal with the base of the central reflector. At the final stage of deployment, the ball bearings were fixed at the bottom of V-shaped grooves.

Fig. 9.
figure 9

Kinematic coupling in the system of deployment of a petal: (а) a scheme of a high-precision deployment. The stages of the deployment: I transport position of a petal, II stage of a low-precision deployment, III−IV stages of a high-precision fixation of a final state, (1) axis of rotation of a petal at the stage of deployment, (2) axis of rotation of a petal in fixing a final state, (3) V-shaped supporting elements, (4) spherical supports. (b) The physical model of the kinematic coupling of the petal deployment system: (5) spherical supports, (6) V-shaped grooves, (7) a simulator of a petal, (8) axis of rotation of the petal simulator.

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In the simulation, an important and interesting feature of using kinematic couplings in the problem of deployment was clarified. On the one hand, a kinematic coupling must be statically determinate in order to accurately fix the final state of the deployed construction. On the other hand, the deployment mechanisms introduce additional kinematic constraints into the system, and at the moment when the spherical supports reach the bottom of V-shaped grooves, the construction becomes indeterminate and the coupling loses self-setting properties.

To resolve this contradiction, a system with a variable structure was introduced into the deployment mechanism. The deployment was carried out in two stages. At the initial stage of low-precision deployment, the petal was converted from the transport state to a state close to the operating one by rotation of the petal around the axis of a cylindrical hinge according to the Dornier scheme. At the final stage of precise fixation of the operating state of the petal, the connection with the central reflector was deactivated. The spherical supports were held at the bottom of the grooves by a spring, which created a force corresponding to requirements of the kinematic coupling. In laboratory experiments, 10 μm accuracy of repeatability of petal model (simulator) deployment (along the normal to the reflecting surface) was achieved; the shift in a tangential plane did not exceed 30 μm. The stages of the physical simulation are presented in Fig. 10.

Fig. 10.
figure 10

The physical simulation of the deployment system of a petal: (а) the simulator of a petal at deployment, (b) the simulator of a petal in the deployed state.

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When performing a physical simulation, we replaced (without changing the essence of the construction) the petal by a simpler element, which is referred to below as a simulator of the petal.

CONCLUSIONS

Classical kinematic couplings and their modifications open new interesting opportunities when self-setting locks are developed to resolve problems of assembly and deployment of multi-mirror solid reflectors. In this paper, new constructions of self-setting locks of a statically determinate type are proposed for conditions of zero-gravity, and the results of their physical simulation are presented.