**5. Conclusions**

This paper has presented novel variations of RPMs with similar structure and a different number of kinematic chains. The mobility analysis of the mechanisms carried out via screw theory has provided a calculation of controllable and uncontrollable DOFs. It was found that 4-DOF tripod has three uncontrollable DOFs, 3-DOF quadropod has two uncontrollable DOFs, 2-DOF pentapod has one uncontrollable DOF and 1-DOF hexapod is fully controllable.

The reconfigurability of the mechanisms allows for changing the output link trajectories without using additional kinematic chains or drives. The proposed mechanisms have such design advantages as a drive fixed on the base, absence of the possibility of collisions between the adjacent carriages due to correctly chosen cranks lengths and diverse reconfigurable capabilities due to having a flexible coupling in each kinematic chain.

Based on the obtained mechanisms structures, the hexapod one (the mechanism with six kinematic chains) has been developed as an assembling CAD model, which allowed us to perform a numerical kinetostatic analysis and then fabricate an actuated physical prototype. The developed CAD model was adapted to using 3D printing technologies in prototype production.

The conducted research confirms that virtual and physical prototyping processes relate to each other, and their cooperative execution is very effective in product development. In addition, rapid prototyping technologies, which are one of the most optimal ways to develop physical prototypes, can be often realized only with virtual prototyping.

Having the physical prototype built, we can aim our further research at its experimental study, including accuracy and repeatability tests, rigidity and load capacity analyses, tests for maximum operating speed and others. Additional research can be associated with an analysis of the presented hexapod for specific practical applications, such as rehabilitation procedures, where the hexapod platform could provide various types of cyclic motions. In this regard, the analysis would aim to calculate the initial cranks configuration and the motion laws of the driving link based on the predetermined platform displacements.

**Supplementary Materials:** The following are available online at: https://www.mdpi.com/2076-341 7/11/15/7158/s1, Video S1: Movie of the CAD model operation, Video S2: Movie of the physical prototype operation.

**Author Contributions:** Conceptualization, A.F. and A.A.; methodology, A.F. and A.A.; software, A.F. and D.P.; validation, A.F. and D.P.; formal analysis, A.F. and A.A.; investigation, A.F., D.P. and A.A.; resources, A.F., D.P., A.A. and V.G.; writing—original draft preparation, A.F., A.A. and M.C.; writing—review and editing, A.F. and A.A.; visualization, A.F. and D.P.; supervision, A.F., A.A. and V.G.; project administration, A.F.; funding acquisition, M.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by Russian Science Foundation (RSF) under gran<sup>t</sup> № 21-79- 10409, https://rscf.ru/project/21-79-10409/.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare that they have no conflict of interest.

#### **Appendix A. Basics of Instantaneous Screw Theory**

The major idea of screw theory relies on Chasles' theorem [32] (p. 26), by which any finite motion of a rigid body can be represented as a combination of rotation about some axis and translation parallel to it. This axis is called a screw axis. If the motion is infinitesimal, we can consider it instantaneous, and the screw axis becomes an instantaneous screw axis. This instantaneous screw motion can be characterized by three basic elements: the axis itself, the angular speed about this axis and the linear speed along it. We can combine these elements into a 6-D vector called (instantaneous) *screw* ξ [32] (p. 40):

$$\mathfrak{E} = \mathfrak{w} \begin{bmatrix} \mathfrak{s} \\ \mathfrak{r} \times \mathfrak{s} \end{bmatrix} + \mathfrak{v} \begin{bmatrix} \mathfrak{0} \\ \mathfrak{s} \end{bmatrix} \tag{A1}$$

where **sˆ** is a unit vector directed along the screw axis; **r** is a vector pointed from the origin of the coordinate system to any point on the screw axis; ω is an angular speed relative to the screw axis; and υ is a linear speed about the screw axis.

An alternative common notation for a screw implies using scalar parameters: magnitude *s* and pitch *h* = <sup>υ</sup>/<sup>ω</sup>. In this case, a screw can be defined as follows [22] (p. 19):

$$\begin{aligned} \mathfrak{E} &= s \left[ \begin{array}{c} \hat{\mathbf{s}} \\ \mathbf{r} \times \hat{\mathbf{s}} + h\mathfrak{E} \end{array} \right], \text{ if } h \neq \infty, \\\mathfrak{E} &= s \left[ \begin{array}{c} \mathbf{0} \\ \mathbf{\dot{s}} \end{array} \right], \text{ if } h = \infty. \end{aligned} \tag{A2}$$

Similarly, all the forces and moments acting on a rigid body can be reduced to a force directed along some axis and a moment about the axis. We can define screw ζ representing this resulting force and moment using relations similar to (A1) and (A2). To differ between these screws, terms *twist* and *wrench* are usually applied for ξ and ζ, respectively. Twists

and wrenches of zero pitch (*h* = 0) correspond to pure rotations and forces. Twists and wrenches of infinite pitch (*h* = ∞) correspond to pure translations and moments.

Screws can be summed with each other and multiplied by a scalar—these properties allow screws to form vector spaces. Thus, the space of twists can define the possible velocities of the rigid body, while the space of wrenches represents all the forces and moments acting on it. If each twist ξ from the twist space and each wrench ζ from the wrench space satisfy the following condition [22] (p. 24):

$$
\boldsymbol{\mathfrak{E}} \circ \boldsymbol{\mathfrak{L}} = \left(\mathbf{P}\boldsymbol{\mathfrak{E}}\right)^{\mathrm{T}} \boldsymbol{\mathfrak{L}} = \boldsymbol{0},\tag{A3}
$$

where **P** is a permutation matrix of the form:

$$\mathbf{P} = \begin{bmatrix} \mathbf{0}\_{3 \times 3} & \mathbf{I}\_{3 \times 3} \\ \mathbf{I}\_{3 \times 3} & \mathbf{0}\_{3 \times 3} \end{bmatrix} \tag{A4}$$

then the wrench space represents all forces and torques that constrain the body motion. Operation "◦" is known as a *reciprocal product*, and screws ξ and ζ that satisfy condition (A3) are *reciprocal*.

If two bodies are connected by a series of joints, we can associate unit twist ξ*j* (*s* = 1 in expression (A2)) *j* = 1 ... *m* with each joint, where *m* is a number of joints. Then, twist system T comprising twists ξ1, ξ2, ... , ξ*m* will define the space of possible velocities of one body relative to the other. The reciprocal wrench system W (each wrench of which is reciprocal to each twist of system T) corresponds to constraints between the bodies.

For a parallel mechanism with *n* kinematic chains, each having *mi* joints, *i* = 1 ... *n*, we can define twists systems T*i* of unit twists ξ*ij*, *j* = 1 ... *mi*, associated with each chain. Wrench system W*i* reciprocal to T*i* corresponds to constraints that the *i*-th chain imposes on the motion of the output link. Considering all the kinematic chains, the output link motion will be constrained such that its resulting wrench system W is a union of all wrench systems W*i*. Finally, twist system T reciprocal to W will define all the possible velocities the output link can attain. One should never forget that all the screws are instantaneous, and the results of the analysis performed above depend on a particular mechanism configuration. In some configurations, twist or wrench systems can include linearly dependent screws. Such situations are known as *singularities*, and the mechanism can lose or gain degrees of freedom in singular configurations, or even change its motion type when passing through them. Authors of [34] present a thorough analysis of this topic.

#### **Appendix B. Hexapod Elements Geometry**

Figure A1 shows hexapod links with identified mass centers and axes used to calculate the inertia parameters. The links shown in Figure A1a–d,g rotate around the fixed axes, and their inertia moments are calculated relative to these axes. The inertia tensors for the links having spatial motion shown in Figure A1e,f are calculated with respect to the depicted coordinate systems placed at the mass centers. The links in Figure A1a,b have displaced mass centers because of the shaft keys.

**Figure A1.** Definition of the inertia terms based on the hexapod elements geometry.
