Compliant joint and joint with one or more degrees of freedom using two or more of said joints

Abstract

The invention relates to a computer-assisted method for the design and production of a flexible joint (10), said flexible joint being of the cross-axis double-beam type, having two beams (18, 19) of predetermined length measured along a longitudinal axis of the beam. Said beams are crossed and connected at a first end to a first rigid base (11) of the flexible joint and at a second end, opposite the first end, to a second rigid base (12) that moves relative to the first rigid base. Said beams are arranged so as to constrain the first base to the second base, maintaining a degree of freedom along a path oriented according to an inclination angle between the respective longitudinal axes of the first base and the second base. The invention also relates to a flexible joint of the cross-axis double-beam type, wherein the flexible joint comprises beams with variable sections, a joint with two degrees of freedom, also used as a prosthetic substitute for joints such as a prosthetic wrist or a prosthetic knee or as a mechanical transmission organ in a universal joint.

Description

COMPLIANT JOINT AND JOINT WITH ONE OR MORE DEGREES OF FREEDOM WITH TWO OR MORE OF SAID JOINTSDESCRIPTION

[0001] The present invention relates to a method for the design of flexible crossed-axis joints and the devices ob- tainable through this method, including certain specific configurations and devices that make use thereof, which are also the subject of the invention.Background of the Invention

[0002] The relentless progress in mechanical and robotic en- gineering has continuously pushed the boundaries of innova- tion in its essential components. Among these, two-degree- of-freedom joints, such as the Cardan joint, play a crucial role in enabling the smooth transfer of motion between in- terconnected parts within sophisticated devices.

[0003] Flexible joints (or flexure joints, in English) are used in mechanics to enable relative motion between two bod- ies composing a mechanism. They are employed as a replacement for traditional kinematic pairs, typically mechanical joints. Compliant mechanisms are those whose operation re- lies on the use of flexible joints or, more broadly, on parts subjected to bending, enabled by their particular sections and geometries.Description of the Invention

[0004] The present invention represents a significant leap forward in the field of two-degree-of-freedom joint design,introducing an innovative approach based on flexible or "com- pliant" elements. The use of these elements not only rede- fines the design paradigm for joints but also ensures a wide range of motion (Range of Motion - ROM), precision in move- ments, and a monolithic design that combines functionality and form.

[0005] Traditional joints present inherent challenges re- lated to the elimination of backlash and friction. These factors can undermine the precision and operational lifespan of complex devices such as industrial robots or prostheses.

[0006] The introduction of the concept of "compliance", i.e., the ability to elastically deform under load, opens new prospects in joint design, allowing smooth and precise adaptation to the complex movements required by these appli- cations. The approach adopted in this study relies on the use of compliant joints of the crossed-axis type or "cross- axis flexural pivot" (CAFP), a possible geometry of which is illustrated in Figure 1.

[0007] In general, a double crossed-beam joint comprises two beams of predetermined length, crossed and connected at one end to a first rigid base of the joint and at the opposite end of the beam to a second rigid base that moves relative to the first rigid base. The beams are arranged so as to constrain the first base to the second base while maintaining a degree of freedom along an arched path oriented at an angle of inclination (0) between the respective longitudinal axes of the first base and the second base.

[0008] These joints allow for wide rotations, ensuring ex- cellent precision without introducing undesired friction or backlash. The innovative use of CAFP joints provides a so- lution to the limitations of traditional rigid Cardan joints.

[0009] In a preferred embodiment, the present invention em- ploys an innovative optimization methodology developed by the inventors themselves, using the Bi-BCM analytical tech- nique described in "Bilancia, P., Baggetta, M., Hao, G. and Berselli, G., 2021. <>", International Journal of Mechanical Sci- ences, 205, p.106587.

[0010] In another embodiment, which proves particularly ad- vantageous when combined with Bi-BCM optimization, variable section beams are employed, suitably designed based on the resistance and motion parameters desired for the joint.

[0011] However, the use of the technique known as Bi-BCM is not indispensable; the optimization methodology can utilize other analytical techniques, such as BCM and CBCM. These techniques, described respectively in "Awtar, Shorya, and Shiladitya Sen., 2010 <<A generalized constraint model for two-dimensional beam flexures: non-linear load-displacement formulation .>>" and "Ma, Fulei, and Guimin Chen. <<Modeling large planar deflections of flexible beams in compliant mech- anisms using chained beam-constraint-model.>> Journal of Mechanisms and Robotics 8.2," would allow for optimization without the ability to employ CAFP with variable section beams, using only uniform section beams. Other alternatives include the use of finite element solvers which, although significantly increasing computational load during the op- timization phase, constitute a feasible option.

[0012] The Bi-BCM methodology redefines the process of de- signing and optimizing CAFP joints characterized by variable section beams (Figure 1), enabling rapid joint production based on the objective functions required by the specific application, such as the space constraints, the requiredROM, and the generated resisting torque, while ensuring the necessary structural resistance. The result is a rapid and efficient optimization of compliant joints, tailored to each specific case study.

[0013] The ability to perform computer-aided multi-ob ective optimization represents a significant tool for designers, enabling them to dimension the joint based on the results obtained from the combination of Pareto-optimal outcomes de- rived from the optimization of the objective functions se- lected during the design phase. For example, Figure 9 illus- trates the Pareto front of the joint obtained by simultane- ously minimizing the joint's height and the inverse of the generated resisting torque.

[0014] The invention is therefore the result of the inven- tors' commitment to improving CAFP joints in general and particularly for certain specific applications, from which more general concepts have been developed, capable of provid- ing advantages even beyond the specific applications / case studies presented here.Case Study 1: Cardan and Double Cardan Joints

[0015] The double Cardan joint represents a fundamental com- ponent in mechanical engineering and applied kinematics. This type of joint, also known as a universal joint, is widely used to enable the smooth transmission of motion be- tween two misaligned shafts, allowing them to rotate in dif- ferent directions. Over the years, this joint has been sub- ject to continuous research and development, leading to var- ious modifications and improvements. It consists of two sim- ple Cardan joints connected in series, with reciprocal an- gles between them (Figure 2). This arrangement compensates for angular deviations, allowing uniform movement even whenthe input and output shafts are out of alignment. Double Cardan joints are widely employed in vehicles, industrial machinery, complex mechanical devices, and robotic applica- tions.

[0016] Recently, research has focused on devising solutions to make double Cardan joints more efficient, precise, and robust. Advanced technologies such as 3D printing and the use of composite materials have opened new horizons in the design and fabrication of such joints. Additionally, the adoption of compliance concepts in double Cardan joints is progressively gaining traction, generating increasing inter- est. A common issue with joints in the literature concerns their considerable size and limited ROM. This invention pro- poses an innovative solution capable of overcoming these limitations, thus contributing to advancements in this field.Case study 2: Prosthetic wrist

[0017] Upper limb prostheses have long been a field of study for the scientific community and numerous industrial sec- tors, aiming to overcome the limitations that upper limb amputees face in daily activities. A crucial element in such devices is the wrist, a complex structure that provides the upper limb with the essential degrees of freedom (DOFs) for manipulation .

[0018] The human wrist has three DOFs: flexion / extension (FE), ulnar / radial deviation (UR), and pronation / supination (PS). Among these, the third DOF is usually actuated at the forearm level and is often not considered in the design of wrist prostheses, while the first two are fundamental for performing various tasks.

[0019] In this framework, the present invention focuses onthe design and realization of prostheses with 2 DOFs to provide an artificial wrist capable of oscillating according to the two degrees of freedom, namely flexion / extension (FE) and ulnar / radial deviation (UR). In this context, one of the objectives of the invention is a design method that enables the creation of prosthetic wrists and, more generally, sim- ple, durable, economical, compact, lightweight joint pros- theses with an adequate degree of freedom, aimed at improving the performance of known prostheses.

[0020] Robotic wrists can be classified as passive or active, depending on whether they are powered by motors. Active ro- botic wrists, powered by electromechanical actuators, offer several advantages over passive or body-powered prostheses.

[0021] However, their development has been limited by vari- ous factors, including size, weight, the number of DOFs, and the range of motion (ROM). To achieve optimal functionality, the component must be small enough to allow the passage of power and sensor cables, as well as actuation tendons, from the arm to the hand, where the battery, motors, and control- ler are usually located.

[0022] Despite numerous examples of robotic wrists reported in the literature, only a few have achieved the desired combination of functionality and design, and it has not yet been possible to develop a robotic wrist capable of repli- cating the maximum ROM of a human wrist, which is approxi- mately [-76°, +73°] for FE motion and [-25°, +45°] for UR motion, while maintaining the same dimensions (approximately 43mm x 63mm for men and 37mm x 56mm for women). Moreover, many existing robotic wrists consist of numerous components, making assembly, maintenance, and replacement difficult.

[0023] In summary, the development of a robotic wrist thatcombines desired functionality with a compact design for prosthetic applications remains a significant challenge. This invention proposes an innovative solution that ad- dresses some of the limitations of existing designs, thereby contributing to advancements in this field.SUMMARY OF THE INVENTION

[0024] The subject of the present invention is a method for the realization of flexible joints of the crossed-axis type and the related flexible joint that this method enables to be realized. This type of joint leverages the use of compli- ant elements of the cross-axis flexural pivot (CAFP) type. As mentioned, the proposed design methodology utilizes a pioneering analytical optimization procedure, developed by the inventors themselves, which ensures a rapid and effec- tive multi-objective optimization of CAFP joints.

[0025] The cardan joint thus realized is suitable for mul- tiple applications, including mechanical transmission cardan joints, double cardan joints, and the two-degree-of-freedom prosthetic robotic wrist, which are also the subject of the invention and are described below in terms of the results achieved .

[0026] In a particularly advantageous embodiment, the ob- tained component features a monolithic design which, com- bined with the absence of friction and backlash, makes it ideal for high-precision robotic applications. Furthermore, the lack of lubrication requirements contributes to its sim- plicity and long-term reliability, as well as its potential use in challenging environments where maintenance is partic- ularly burdensome.

[0027] According to a further variant, the application of passive contacts, as better described below, allows for theelimination of one of the main intrinsic problems of the CAFP joint, namely sensitivity to compression loads and, if required by the application, to torsion (Figure 9).

[0028] Another noteworthy aspect concerns the production method, which is carried out through additive manufacturing, i.e., 3D printing, preferably using the technology known as SLS (Selective Laser Sintering).

[0029] Preferably, the material used for production in pros- thetic applications is powder-printed nylon; this material proves perfectly suited for the realization of compliant joints, offering a combination of lightness, robustness, and flexibility necessary to ensure the proper functioning of the devices. For industrial applications involving high loads, titanium would be the best solution in terms of flex- ibility / mechanical strength ratio.

[0030] In summary, the combination of compliant CAFP joints and the proposed analytical optimization methodology repre- sents a promising approach for the creation of highly effi- cient, compact, and reliable two-degree-of-freedom cardan joints, with potential applications in fields such as indus- trial robotics, medicine, and prosthetics.

[0031] The proposed invention thus addresses optimized CAFP flexible joints through the aforementioned analytical tech- niques such as BCM, CBCM, and Bi-BCM. The Bi-BCM technique, in particular, effectively solves CAFP joints with variable- section beams, as shown in Figure 1. This optimization tech- nique can also be implemented with other analytical solution methodologies, such as BCM and CBCM, provided it is limited to considering only beams with uniform sections.

[0032] Details of the invention can be better understood with reference to the figures, in which:- Fig. 1 shows a 3D example of a flexible joint with a variable section;- Fig. 2 shows a beam of a flexible joint with a variable section;- Fig. 3 shows the construction steps of two possible two-degree-of-freedom joints including flexible joints with variable sections and reinforcement passive con- tacts;- Fig. 4 shows two possible configurations of joints with tie rods for their actuation;- Fig. 5 shows a schematic side view of a joint in the resting position and in a flexed condition with related descriptive parameters;- Fig. 6 presents a beam modeled as an assembly of two symmetrical half-beams;- Fig. 7 shows real images of a prosthetic joint realized according to the present invention and involved in move- ment and load tests;- Fig. 8 shows a one-degree-of-freedom joint with passive contacts supporting torsion and compression;- Fig. 9 shows three different joint configurations and the Pareto front of various joints obtained by the sim- ultaneous minimization of the joint height and the re- ciprocal of the generated resisting torque;- Fig. 10 shows two cardan or universal joints realized with one or two flexible two-degree-of-freedom joints, respectively;- Fig. 11 depicts a block diagram of the method steps according to one embodiment of the invention with CAD+ FEM analytical solution of the joint equations;- Fig. 12 depicts an alternative variant of the analyti- cal solution step from the diagram in Figure 11, where the solution of the equations does not resort to finite element modeling but instead uses BCM, CBCM, or Bi-BCM techniques ;- Fig. 13 shows a subvariant of Figure 12 where the so- lution of the joint equations employs the Bi-BCM tech- nique with variable-section beams, each composed of two symmetrical sections.

[0033] In the following the invention will also be described below with reference to the embodiments illustrated in the figures, which are not, however, limiting of the scope of protection .

[0034] The invention achieves the intended objectives through a computer-aided method as claimed in claim 1. The invention further includes flexible joints with one or more degrees of freedom obtained using the method of claim 1, as well as mechanical transmission elements such as prostheses 100' and 100' ' (Figure 3) and universal or cardan joints 300, 400 (Figure 10) comprising one or a combination of flexible joints 10 made using the method of claim 1.

[0035] In general, a flexible joint 10 thus comprises two bases (first base, 11, and second base, 12) , connected and partially constrained by a pair of crossed beams, 18 and 19. The two beams are made of flexible material, allowing move- ment between the bases along a path oriented at an angle of inclination 0 between the respective longitudinal axes of the first and second bases.

[0036] The geometric characteristics of the joint are de- scribed, among other parameters, by β, which measures thesemi-angle between the beams 18 and 19, and by the relative position where the beams intersect, determined by X, which describes the position in relation to the beam length. These values vary with the inclination between the bases when the joint 10 is subjected to torsional forces acting on the bases. In the rest condition, the joint appears as shown in Figure 1, while Figure 5 shows the joint in the resting condition (left) and in the flexed condition (right).

[0037] The above-mentioned optimization was performed on a computer using Matlab® software; however, other present or future software tools capable of performing such optimiza- tion are not excluded.

[0038] Other geometric parameters include the width of the first base 12 (lower), the width of the second base 12(upper), , the length of the beam measured along the lon-gitudinal axis X, L, and the overall height of the CAFF, h, measured between the beams in the resting condition. One of these parameters (e.g.,) can be considered as an independ- ent variable, along withand, to determine the other parameters as follows:

[0039] In particular, the value of h thus determined is the initial value of the distance between the bases in the rest- ing condition of the joint (shown on the left in Figure 5) and is subject to optimization as described below to achieve the final value of h.

[0040] A multi-objective optimization was then implemented to minimize the dimensions of the CAFP without necessarilysacrificing its rigidity. In summary, the joint design op- timization problem can be formulated as follows:

[0041] As can be seen from the formula, stress values are measured in the specific condition of inclination as-suming the joint is in the condition of maximum relative inclination between the bases (maximum stress condition).

[0042] In a first embodiment, the method works with beams 18 and 19 having variable sections, with a profile symmetric about a symmetry plane orthogonal to the longitudinal axis X (Figure 2) of the single beam and passing through the midpoint C of the beam measured along said longitudinal axis X, such that each beam can be modeled as an assembly con- sisting of a pair of half-beams 191, 192 (Figure 6) with mutually symmetrical and opposite profiles. In this embodi- ment, it is advantageous to use the previously mentioned Bi- BCM analytical technique, which allows reformulation of the equations describing the movement of the joint components and the forces acting on them, particularly the torque be- tween the bases and the material stress, as the inclination between the two bases of the joint varies.

[0043] Specifically, the single beam can have a variable section profile with a linear, parabolic, or polynomial pat- tern along the longitudinal direction X, thereby varying the width w and height t of the beam as a function of the beam's length coordinate.

[0044] The joint design optimization problem can again be formulated as follows:

[0045] Where, unlike the previous case, the geometric param- eters are reflecting the concept of a beam with avariable section.

[0046] In the case of parabolic profiles (second-order pol- ynomial), the beam shape is adjusted by varying both width and thickness through the application of second-order (par- abolic) laws. This choice is based on their demonstrated greater efficiency in limiting stress compared to first- order (linear) laws, as shown in "Bilancia, P., Baggetta, M., Hao, G. and Berselli, G., 2021. <>, " International Journal of Mechanical Sciences, 205, p.106587.

[0047] The equations for the variable section beam can be formulated as follows:Where e are the shape coefficients of the beamsection, and are the initial width and thickness ofthe beam, respectively,and are char-acteristic functions of the variable section beam.

[0048] The optimization procedure begins by calculating theBCM coefficients, denoted as following the same nomen-clature used in Awtar et al., (S. Awtar and S. Sen, "A generalized constraint model for two-dimensional beam flex- ures: nonlinear load-displacement formulation," 2010), which presented the analytical solution for uniform section beams.

[0049] It is essential to note that the BCM coefficients depend strictly on the beam's shape (and not its size). In other words, these coefficients must be modified whenever changes are made to the characteristic functions,. ofthe variable section beam.

[0050] Considering each CAFP beam as two BCM elements (Figure6) and defining the width coefficient and the thick-ness coefficient for the first beam element it willbe, which leads to:While, for the second element, which instead leadsto:

[0051] When considering a CAFP subject to a set of loads consisting of a moment M and forces FxandFy acting on the movable base (following the left-hand scheme in Figure 6 and assumingwhere the following normalizedgoverning BCM equations can be applied:

[0052] Here, the problem is dimensionless with respect to the beam length L, is the final planar displacement ofthe beam along y, is the displacement function of thebeam along y, are the dimensionless loads appliedat the beam ends.

[0053] Equation 4 can be solved using numerical techniques within the Matlab software via the ODE45 solver. The algo- rithm takes as input the values and andreturns as output and its derivatives. However, sinceis initially unknown, an iterative approach is required to obtain accurate results. Specifically, the numerical value of the vertical displacement at maximum deflection, denoted as at the i-th iteration step, is compared withthe corresponding value obtained from the solution of Equa- tion 4 at x = 1 to determine the updated input value for theiteration step, givenis a user-defined parameter. The algorithm converges when the calculated error between these two values is less than a specified tolerance, denoted as a. For precise and effi- cient results, it is recommended to set and

[0054] To calculate the BCM coefficients for a specific beam shape, as defined in Equation 4 using the function itis necessary to compute the stiffness matrix of the beam. This matrix is determined at discrete points within the range of values, from —0.5 to 0.5:However, since this matrix is not directly available, it can be obtained by solving and inverting the damping matrix C:

[0055] Specifically, when = 0.5 and mz1= 0, the final dis- placement and the rotation provide the terms c11and for a specific value of fx1. Similarly, whenand = 0.5, the terms c12and c22are obtained. The matrix K is then determined by inverting matrix C. At the end of this iterative process for various values of fx1, the collected data can be interpolated to establish the following rela- tionships:From these, the BCM stiffness coefficients can be easilyextracted.

[0056] Once the variable section beam coefficients are cal- culated, the Bi-BCM equations can be used to solve the CAFP and obtain the cost function values. Referring to Figure 6, to ensure the correct load distribution between the two beam elements, it is essential to maintain local static equilib- rium. Examining the free-body diagram, the following equa- tions can be written for the first beam section:For the second beam section, the equations become:dove

[0057] Subsequently, for both beams, the following equationsprovide the final displacements of points A and B relative to the coordinate systems CS1 and CS2, as shown in Figure 5:

[0058] Equation (14) results from the connection between the beams and the movable base of the CAFP. When vector rotations of and are applied, corresponding to theorientation angles of CS1 and CS3 (Figure 5), to the quanti- ties expressed in Equations (12) and (13), we can write:The displacements of the movable base, and , are thusgiven by:where and represent the planar final dis-placements of points A and B, respectively.

[0059] To complete the Bi-BCM model, the global equilibrium equations must be defined, taking into account all forces and moments acting on the movable base, as illustrated in Figure 5:

[0060] Finally, the Bi-BCM equations provide the joint stiffness functionand the normal stress of the beam where is the bending stress alongthe x-axis of the element, andis the tensile stress due to axial force. The bending stress is given by:

[0061] where is the local bending moment function.Thetensile stress can be calculated using the following equa- tion:

[0062] Once the optimizer converges or the maximum number of iterations is reached, the optimization process terminates. Each simulation takes approximately 0.6 seconds on a work- station with an Intel(R) Core(TM) processor @2.5 GHz and 16 GB of RAM, compared to the 480 seconds required to solve the same model using finite element analysis with at least three element layers across the beam thickness.

[0063] Thanks to the reformulation of the equations, pref- erably via Bi-BCM, it is possible to perform an interactive multi-objective optimization procedure, wherein the dimen- sions in terms of width and height of the beam sections (w , t) are described as a function of the half-beams 191 and 192, and particularly in terms of the width and height of thesections andat the two longitudinal ends of thesame.

[0064] In an embodiment, an additional step is included to assign one or more parameters according to the following numerical values:- Coordinate of the crossing point of the beams relative to their length = 0.5- Amplitude of the semi-angle / between the beams at the crossing point and in the resting condition: = 70 de-grees- Maximum permissible inclinationbetween the first and second bases:=40 degrees, measured by the angle (0) between the respective longitudinal axes of the first and second bases

[0065] This configuration is optimal for creating prosthetic wrists, as described in Case Study 2. As will also be seen later, this combination of parameters allows for a joint suitable for inclusion in a prosthesis capable of replicat- ing the range of motion of a human wrist.

[0066] In an embodiment of the method, the optimization pro- cedure includes an additional step of repeatedly calculating the joint's target parameters while varying the conditions of reciprocal inclination between the bases, specifically by varying the angle (0) between the respective longitudinal axes (A1,A2') of the first base (11) and the second base (12) for a predetermined number of-angle values, i.e., for a predetermined number of reciprocal positions between the ba- ses. This calculation, as the angle varies, is performedduring each cycle, i.e., within each iteration cycle aimed, as said, at minimizing the cost functions.

[0067] At each iteration, the optimization step selects new values for and t (in the case of variable section beams,the values are . The selection of these values isdetermined by the chosen optimization algorithm (in the case of a genetic algorithm, the process begins by exploring ran- dom values and then retaining the best combinations until the optimal ones are selected).

[0068] When the values of the independent variables are changed, h is already available as it is a dependent varia- ble, while Mrmust be calculated once the CAFP is solved. The value of h, initially determined through andis re-evaluated at each iteration during the optimization process to exclude designs that exhibit excessive stress and to combine the minimization of h with the maximization of Mr.

[0069] The repetition of optimization calculations while varying the inclination angle occurs during the determina- tion of the joint's geometric parameters within a single iteration. Compared to embodiments that operate with a sin- gle inclination value, at the maximum inclination, the multi- step solution allows the evaluation of these quantities throughout the motion, thereby also obtaining the trajectory of the instantaneous center of rotation.

[0070] A particular embodiment provides for a total of 10 repetitions in increments of 1 / 10 of the maximum angleThis choice is advantageous as it allows achieving optimal results comparable to those obtained via FEM but with sig- nificantly lower computational impact.

[0071] In one embodiment, the material used for constructing the joint 10 or an assembly of multiple joints, such as the composite joints 100' and 100'', is PA11 polyamide powder,with the maximum allowable stress set at 35 MPa. This choice is particularly advantageous for applications in the field of wrist prosthetics.

[0072] In a further embodiment, the minimization of the ob- jective function is achieved using a multi-objective genetic algorithm, such as the one made available by the ga function implemented in Matlab®. In general, a genetic algorithm is a stochastic algorithm based on a population of values that performs random searches through mutation and crossover among the population members, iteratively seeking the com- binations of values that best solve the optimization prob- lem.

[0073] Compared to other possible minimization algorithms, such as the known Matlab®'s fmincon function, genetic algo- rithms can avoid the local minima of the objective function.

[0074] According to an embodiment, the method includes the additional steps of:- Providing the flexible joint with a pair of lateral walls that extend in an axial direction of the joint, along an axis orthogonal to the two said bases, wherein the lateral walls, respectively in combination with said bases, form a substantially channel-shaped seat that houses at least part of said beams;- Assigning each of said walls a sectional edge profile of an arcuate type, with a radius such that the opposing sectional edges of the two lateral walls are in mutual contact, while the curvature is such that, during the relative motion between the bases according to the oscil- latory movement allowed by the beams connecting said ba- ses, the opposing sectional edges are in mutual contact at an edge portion that follows the path of the crossingpoint between the beams.

[0075] According to a further embodiment, as already men- tioned, an additional step is provided to manufacture, via 3D printing, a joint with two degrees of freedom 100' or 100'' by printing two or more single-degree-of-freedom joints 10, preferably using a continuous printing process to produce the complete element. The result is advantageous for applications designed to withstand relatively low loads, such as prosthetic wrists. Alternatively, combined solutions with metal alloys such as titanium, or solutions entirely made of metal alloys, may be of interest for applications involving higher working stress.

[0076] The invention also concerns a flexible joint 10 with a single degree of freedom, characterized by being designed using one or more of the method embodiments claimed herein. In its simplest form, such a flexible joint is shown in Figure 1, the left part of Figure 3, and Figure 9, in three different configurations depending on the optimization re- sult according to the method.

[0077] Furthermore, Figure 8 illustrates an application with a single-degree-of-freedom joint 200 obtained by combining a plurality 201 of flexible joints 10 arranged such that the oscillation planes of each joint coincide, effectively re- sulting in a composite joint capable of oscillating with one degree of freedom within the same oscillation plane.

[0078] In addition, this embodiment also includes a pair of opposing lateral walls 202 and 203 that extend in an axial direction of the joint along an axis orthogonal to the two bases which lateral walls, respectively in combination with the said bases, form a substantially channel-shaped seat that houses at least part of the beams. These walls 202 and203 have an arcuate sectional edge profile with a radius such that the opposing sectional edges 211 and 212 of the two lateral walls 202 and 203 are in mutual contact for any possible inclination position of the joint, whereas the cur- vature is such that, during the relative movement between the bases according to the oscillatory motion permitted by the beams connecting the bases, the opposing sectional edges 211 and 212 come into mutual contact at an edge portion following the path of the crossing point between the beams of the single joint 10 or the centroid of the body 201 formed by the set of joints 10.

[0079] Additionally, the embodiment of Figure 8 also in- cludes a configuration of the bases to provide reinforcement elements for the composite joint 200, designed to increase resistance to lateral torsions, defined as torsional forces with at least one component not parallel to the oscillation plane of the joint. Specifically, projections 220 are pro- vided on one of the bases, with a known thickness, which extend substantially in a plane parallel to the oscillation plane of the joint 10 or joints 201 and are housed in cor- responding recesses 204 in the opposing base. The recesses204 have a width close to and slightly greater than the thickness of the projections 220, opposing torsional forces except for those oriented within the oscillation plane of the joint. The shape of the projections has a curved profile so as not to impede the oscillation of the joint within its oscillation plane.

[0080] Figure 11 provides a block diagram of the method steps according to an embodiment of the invention, which imposes certain geometric and physical parameters used to determine the joint characteristics. In Step 1, the values of the crossing point coordinate A of the beams 18 and 19 relativeto their length, in the resting condition of the joint 10, and the semi-angle amplitude between the beams at theircrossing point and in the resting condition are set. Addi- tionally, the Young's Modulus of the material is set to 1400 MPa, representing the material's mechanical rigidity prop- erty.

[0081] The independent variables, to be determined as output from the optimization process in subsequent steps, and the dependent variables, calculated based on the independent variables, are also defined. The independent variables are:- Length of the first base,- Width and height of the beam sections,The dependent variables are:- Length of the second base, r2, calculated as:- Distance between the bases, h, calculated as:- Beam length, calculated as:- Lengths of the half-beams and calculated respec-tively as:Initial transverse moment of inertia of the first beam element :Initial transverse moment of inertia of the second beam element :- Width coefficient of the half-beam:- Thickness coefficient of the half-beam:

[0082] In Step 2, the optimization problem, the cost func- tion, and the constraints to be respected are defined. Spe- cifically, the maximum number of iterations is set (104in this specific case, but this choice can be freely varied heuristically depending on computational resources and pro- cessing times), along with the maximum inclination angleand the optimization algorithm, in this specific case, a genetic algorithm ("ga" function in Matlab®).

[0083] Also, in Step 2, the cost functions to be optimized are defined using multi-objective optimization. As stated, the cost functions are the distance between the bases h and the inverse of the torque Mrapplied to the beams when the joint is in an inclined condition.

[0084] The optimization process is subject to constraints on valid results, particularly concerning the yield stress of the material composing the beams, when the beams are in the maximum inclination condition, the stress must be less than the maximum allowable stressof the material.

[0085] The routine then iterates within a loop (Blocks 3 and 4), exiting when either the result converges or the maximumnumber of iterations set in Block 2 is exceeded. Convergence is achieved when the final dispersion between the optimal solutions of the two cost functions differs from the disper- sion obtained in the previous iteration by a value smaller than the chosen tolerance, typically on the order of 10-4.

[0086] In Block 5, the CAFP is solved using either CAD-CAE solution engines or analytical techniques such as BCM, CBCM, or Bi-BCM. In this block, all geometric values of the beam (updated at each iteration) are provided as input, and the following are returned as output: stiffness values, resist- ing torque, coordinates of points A and B of the movable base, and the position of the instantaneous center of rota- tion of the CAFP. If CAD-CAE solvers are selected, Matlab would handle the optimization and the selection of geometric values at each iteration. Using a script that varies depend- ing on the chosen software, the CAD-CAE solver would be opened in batch mode to update the model geometry, solve it via finite element simulation, and import the results into Matlab.

[0087] In Block 6, the results obtained in Block 5 are saved, and if analytical techniques have been used, the maximum stress acting on the CAFP is calculated according to Equa- tions (21) and (22). Specifically, the maximum value between the two is considered as •

[0088] The results processed in the previous blocks, partic- ularly the stress values to which the beams are subjected, are evaluated in Block 7, where it is decided whether to retain or discard the processing results from the previous steps.

[0089] If the stress values (maximum tension) are lower thanthe set threshold, in Step 8, the cost functions are evalu- ated. Specifically, the values of h and (cost functions)from the current iteration are stored and added to the set of possible optima. These values are considered candidates for inclusion in the Pareto front. The set of optimal can- didates will be updated during the optimization process by selecting the best candidates at each step.

[0090] Upon exiting the cycle, whether due to convergence (Block 3) or reaching the iteration limit (Block 4), the Pareto front is reported, representing the set of solutions that form the Pareto optimum based on the optimization pro- cess results. Since there is no dominant solution to the multi-objective problem, the method offers a set of solu- tions, leaving the selection of the specific configuration to the user based on their analytical expertise.

[0091] In Figure 12, an embodiment of the method is illus- trated in which, at Block 53, the use of the analytical techniques BCM, CBCM, or Bi-BCM is provided. At Block 51, the beam equations are solved as described earlier (see

[0053] ), and at Block 52, the determination of the coeffi- cient matrix k is performed (see

[0054] ). Specifically, if the BCM or CBCM techniques are chosen, the optimization pro- cedure would include geometric constraints such thatand

[0092] In Figure 13, an embodiment of the method is shown in which the Bi-BCM analytical technique, as described earlier, is specifically used. In particular, at Block 531, the Bi- BCM equations are defined; at Blocks 532 and 533, the free- body diagram for the two symmetric sections 18 and 19 of the beam is analyzed (as described in

[0056] ); at Block 534, therelative displacements between the joint bases are deter- mined (

[0057] ); at Block 535, the overall geometric equation is defined (

[0058] ); and at Step 536, the global equilibrium equations are established (

[0059] ).

[0093] The invention also concerns a composite joint 199 with two degrees of freedom, comprising two or more joints 10 designed in accordance with the method claimed herein. Preferably, such a joint 100' or 100'' is produced in a single casting or additive manufacturing operation, result- ing in a one-piece element that is easy to produce and highly durable compared to other known mechanical solutions that require a high number of components, making automation in assembly and maintenance difficult and costly.

[0094] The combination of two or more single joints 10 to form composite joints of type 100' or 100'', as shown in Figure 3, can be achieved by combining single joints in sequence such that a second base 11b of a first joint 10b is in contact with or coincides with the first base 12a of a subsequent joint 10a in the sequence. As shown in Figure 3, a sub-embodiment of this configuration provides for the use of three single joints, preferably produced as a single piece, combined such that the second base of the first joint is in contact with the first base of the second joint, and the second base of the second joint is in contact with the first base of the third joint and the three joints are ar- ranged according to one of the following two alternatives:- The oscillation plane of the second joint is orthogonal to the oscillation plane of the first joint, while the oscillation plane of the third joint is orthogonal to the oscillation plane of the second joint and parallel to the oscillation plane of the first joint; or- The oscillation plane of the second joint is orthogonal to the oscillation plane of the first joint, while the oscillation plane of the third joint is orthogonal to the oscillation plane of the second joint and parallel to the oscillation plane of the first joint.

[0095] In an embodiment of the invention specifically de- signed for prosthetic wrists, the bases have an ellipsoidal section in plan view, where the ellipse of each base has axes measuring 40 and 30 millimeters, respectively, and the distance between the two bases is 40 millimeters. Thanks to the method of the present invention, it is possible to obtain joints of such compact dimensions that are fully compatible with the anatomical characteristics of the users.

[0096] Preferably, at least one of the bases of the joints composing the prosthesis is connected to tie rods that act as traction and motion elements of the prosthesis, at least partially substituting for the tendons of the human organ being replaced. The said tie rods slide and / or are connected through seats arranged at the opposite ends of the major and minor axes of the ellipse representing the plan view of the joint bases.

[0097] Alternatively, the said tie rods slide and / or are connected through seats arranged at the vertices of a rec- tangle inscribed within the perimeter of the ellipse repre- senting the plan view of the joint bases, with said rectangle preferably oriented with its sides parallel to the axes of the ellipse.

[0098] In Figure 10, two mechanical transmission organs with universal joints 300 and 400 are shown, respectively with one and two joints, for the mechanical transmission of motionbetween two rotating components, particularly between an in- put shaft 32 or 42 and an output shaft 31 or 41. In the first configuration, identified as 300, the transmission organ comprises two shafts 31 and 32, each rotatable around a rotation axis, while in the second configuration 400, an intermediate shaft 43 is present. This latter configuration is commonly used as a constant velocity joint between input and output shafts. The use of flexible joints 35, 46, and 47 designed according to the present method advantageously al- lows for reducing the overall dimensions of the components and improving the maximum range of motion permitted between the shafts.

[0099] Thanks to this optimization methodology, it was pos- sible to minimize the dimensions and the stiffness required for component deflection while ensuring that the stresses remained sufficiently below the material's breaking point (for joints made of nylon 11 or polyamide 11, 35 MPa compared to the material's maximum stress of 55 MPa).

[0100] However, this analytical solution allows for calcu- lating component deflections up to 50°. Therefore, for the realization of an embodiment of the invention related to a prosthetic wrist, modules with a maximum deflection of 40° were designed, using two for the first degree of freedom (flexion / extension) and one for the second degree of freedom (ulnar / radial deviation), to achieve the required ROM for the robotic wrist case study (±80° for the first degree of freedom and ±40° for the second).

[0101] Once the individual modules are optimized, they can be combined to compose the final device.

[0102] In a preferred embodiment, appropriate passive con- tacts are finally added to protect the flexible joint fromexternal compressive forces. For the first case study, to achieve the same ROM for both degrees of freedom, the modules were optimized such that the double joint exhibited half the ROM of a single joint.

[0103] Figure 3 illustrates the design process of this com- ponent. It should be noted that the modules of the component can be assembled in various ways, depending on the specific requirements. FEA analyses on the individual modules vali- dated the analytical procedure used, as demonstrated by the excellent overlap of the results obtained.

[0104] Further experimental tests (Figure 9) validated the optimization and FEA tests. It is important to note that, although this invention is applied here as a prosthetic ro- botic wrist, it can be used and implemented in other mechan- ical solutions, such as a single- or double-joint universal compliant joint of the type shown in Figure 10.

[0105] Throughout the description, similar reference numbers may be used to identify similar elements.

[0106] While various aspects of the embodiments are pre- sented in the drawings, the drawings are not necessarily to scale unless specifically indicated.

[0107] It will be readily understood that the components of the embodiments as generally described here and at least partially illustrated in the accompanying figures could be arranged and designed in a wide variety of different config- urations.

[0108] Therefore, the following description, as also repre- sented in the figures, is not intended to limit the scope of the present invention but is merely representative of some of the possible embodiments.

[0109] The present invention can be implemented in other specific forms without departing from its spirit or essen- tial characteristics.

[0110] The embodiments described should be considered in all respects merely as illustrative and not restrictive.

[0111] All changes that fall within the meaning and range of equivalency of the claims should be considered as included within their scope.

[0112] References throughout this description to features, advantages, or similar language do not imply that all such features and advantages must be or are present in any single embodiment of the invention.

[0113] Rather, language referring to features, advantages, or specific characteristics described in relation to an em- bodiment is intended to mean that at least one embodiment of the present invention includes the described feature, ad- vantage, or characteristic.

[0114] Accordingly, discussions of features, advantages, and similar language throughout this description may, but do not necessarily, refer to the same embodiment.

[0115] Furthermore, the features, advantages, and character- istics described in relation to the invention can be combined in any suitable manner in one or more embodiments. Here is the literal and precise translation:

[0116] >

[0117] The person skilled in the art will recognize, in light of this description, that the invention can be implemented without one or more of the specific features or advantages of a particular embodiment.

[0118] In other cases, additional features and advantagesmay be found in certain embodiments that may not be present in all embodiments of the invention.

[0119] References throughout this description to "an embod- iment, " "a variant, " "an implementation, " or similar lan- guage mean that a particular function, structure, or char- acteristic described in connection with the mentioned embod- iment is included in at least one embodiment of the present invention.

[0120] Accordingly, references to "an embodiment, " "a vari- ant, " or similar language throughout this description may, but do not necessarily, refer to the same embodiment.

[0121] The components of the embodiments as generally de- scribed herein and illustrated in the accompanying figures could be used and designed in a wide variety of different configurations .

[0122] All changes that fall within the meaning and scope of equivalency of the claims should be considered as included within their scope.

[0123] Although the operations of the methods herein are shown and described in a particular order, the order of operations for each method may be altered such that some operations can be performed in a different order and / or such that some operations can be performed, at least in part, simultaneously with other operations.

[0124] In the foregoing description, specific details of various embodiments are provided.

[0125] However, some embodiments can be implemented with fewer than all these specific details.

[0126] In other cases, some methods, procedures, components,structures, and / or functions are described not comprehen- sively but only to the minimum detail necessary to understand the various embodiments of the invention, for the sake of brevity and clarity.

[0127] It is nevertheless evident that the invention should not be considered limited to the particular arrangements illustrated above, which constitute only exemplary implemen- tations of it, but that various modifications are possible, all within the reach of a person skilled in the art, without departing from the scope of protection of the invention, which is defined by the annexed claims.

Claims

CLAIMS1. Computer aided method for the design and production of a flexible joint (10), said flexible joint being of the double crossed beam type having two beams (18, 19) of predetermined length measured along a longitudinal axis (X) of the beam, said beams (18,19) being crossed and connected for a first end to a first rigid base (11) of the said flex- ible joint and for a second end, opposite to the said first end, to a second rigid base (12) which moves relative to the first rigid base, said beams (18, 19) being arranged so as to constrain said first base (11) with said second base (12), main- taining a degree of freedom along a path oriented ac- cording to an angle of inclination (0) between the re- spective longitudinal axes of first base and second base, said method comprising the steps of: a. Set the dimensions, in terms of length (rl), width and height, of said first rigid base (11); b. Assign to each individual beam a known section variation profile along a longitudinal axis of the beam; c. Set the shape characteristic of the flexible joint(10), in terms of:- coordinate of the crossing point (X) of the beams (18,19) in relation to their length and in rest conditions of the joint (10);- width of the semi-angle (p) between the beams at their crossing point and in the rest conditionof the joint. d. Determine the dimensions, in terms of length (r2), width and height, of said second rigid base ac- cording to the parameters set in the previous steps; e. Determine the length (L) of the single beam, for each of the two beams, based on the parameters set in the previous steps; f. Determine the initial dimension, in terms of dis- tance (h) between the first base (11) and the second base (12) in the joint's rest condition; g. Select a material of which the flexible joint will be made, determining the Young's Modulus (E) and the yield stress (os) of the material itself; h. Set the maximum inclination (QMAX) according to the design parameters between the first base (11) and the second base (12) in joint inclination conditions, measured in terms of the angle (0) between the respective longitudinal axes (Al, A2) of the first base (11) and second base (12); i. Set a range of allowable values (A) for the width and height of the beam section (w; t) along the longitudinal direction of the beam (19; 20); j. Set the maximum value (o) of admissible tension to which the chosen material can be subjected in the various positions that the flexible joint can assume while the inclination (0) between the ba- ses (11, 12) varies; k. Define a pair of cost functions to be minimizedby means of a multi-objective optimization pro- cedure reiterated in a predetermined number of steps;1. Minimize the said cost function by means of a multi-objective optimization procedure reiter- ated in a predetermined number of steps, with the target of optimizing: the final value of the distance (h) between the first base (11) and the second base (12) in the joint's rest condition;- the inverse of the torque (1 / Mr) between the first base (11) and the second base (12); in conditions of maximum relative inclination be- tween the bases (0 = 0MAX), thus determining a set of possible solutions in terms of width and height of the section of the beams (w; t) and length of the base (rl) and within the limits of the maximum value (o) of admissible tension to which the chosen material can be subjected and within the limits of admis- sible values (A) for the width and height of the beams section (w; t) for the base length (rl).

2. Method according to claim 1 which provides the further steps of: a. Use variable section beams (18; 19) with a profile symmetric according to a symmetry plane arranged or- thogonal to the longitudinal axis (X) of the single beam and passing through the midpoint (C) of the beam measured along said longitudinal axis (X) so that each beam is modeled as a pair of half-beams(191, 192) with mutually symmetric and opposite pro- file; b. Set the shape profile of the single half-beam of each beam according to a profile of the linear, par- abolic or polynomial type along the longitudinal di- rection (X) and along the transversal direction (Y); c. Perform the said interactive multi-objective opti- mization procedure in which the beams are described as a function of the said half-beams (191, 192) and in particular in terms of width and height of the sections (wO, tO; wl, tl) of the half-beam at the two longitudinal ends of the same.

3. Method according to claim 1 or 2 wherein it is provided the further step of assigning one or more of the param- eters according to the following numerical values: a. coordinate of the crossing point of the beams in relation to their lengthequal to 0.5; b. width of the semi-anglebetween the beams at their crossing point and in the rest condition of the joint equal to 70 sexadecimal degrees; c. maximum inclinationpermissible between the first base (11) and the second base (12), as measured by the angle (0) between the respective longitudinal axes (Al, A2) of the bases, in an interval included between 5 and 50 sexadecimal degrees and preferably equal to 40 sexadecimal degrees.

4. Method according to one or more of the preceding claims, wherein the optimization procedure further comprises to perform object function minimization operations at dif- ferent positions of relative inclination between the bases of the flexible joint (10), i.e. by varying theanglebetween the respective longitudinal axes (Al, A2) of the first base (11) and the second base (12) for a predetermined number of positions and preferably for a number equal to 10 positions i.e. values of the said angle .

5. Method according to one or more of the previous claims, wherein the material used is PA11 polyamide powder and the maximum flow stress value (os) is set to 35 MPa.

6. Method according to one or more of the previous claims, wherein the minimization of the object function is ob- tained by means of a multi-objective genetic algorithm.

7. Method according to one or more of the previous claims, comprising the further steps of: a. Provide the flexible joint with a pair of lateral walls which grow in an axial direction of the joint, according to an orthogonal axis of the two said ba- ses, which lateral walls, respectively in combina- tion with the said bases, form a substantially chan- nel-shaped seat which houses at least part of the said beams; b. Assign for each of the said walls a sectional edge profile of the curved type with a radius such that the opposing sectional edges of the two side walls are in mutual contact, while the curve is such that, during the relative movement between the bases ac- cording to the oscillation movement permitted by the beams that connect the bases, the opposing sectional edges are in mutual contact at a portion of the edge that follows the path of the crossing point between the beams.

8. Method according to one or more of the previous claims,comprising the further step of producing, via 3D print- ing, a joint with two or more degrees of freedom by printing two or more joints with one degree of freedom, preferably according to a continuous printing process to print the overall item.

9. Flexible joint (10) of the double crossed beam type having two beams (18, 19) of predetermined length meas- ured along a longitudinal axis (X) of the beam, said beams (18,19) being crossed and connected for a first end to a first rigid base (11) of the said flex- ible joint and for a second end, opposite to the said first end, to a second rigid base (12) that can move with respect to the first rigid base, said beams (18, 19) being arranged so as to constrain said first base (11) with said second base (12), main- taining a degree of freedom along a path oriented ac- cording to an angle of inclination (0) between the re- spective longitudinal axes of first base and second base.

10. Flexible joint (10) according to the previous claim characterized in that it comprises beams with variable section.

11. Flexible joint (10) according to claim 9 or 10 char- acterized in that it is designed in accordance with the method according to one or more of the previous method claims.

12. Joint with two degrees of freedom, characterized in that it comprises two or more joints designed according to the method of one or more of the previous claims from 1 to 8 and preferably made in one piece.

13. Joint according to claim 12 wherein the individualjoints are mutually combined in sequence such that a second base of a first joint contacts or coincides with the first base of a subsequent joint in the sequence.

14. Joint according to claim 13 comprising three single joints, preferably made in one piece, mutually combined in such a way that the second base of the first joint (10c) is in contact with the first base of the second joint (10b) and the second base of the second joint is in contact with the first base of the third joint (10a) and the three said joints are arranged according to one of the two following alternatives:- The oscillation plane of the second joint is orthog- onal to the oscillation plane of the first joint whereas the oscillation plane of the third joint is orthogonal to the oscillation plane of the second joint and parallel to the oscillation plane of the first joint; or- The oscillation plane of the second joint is orthog- onal to the oscillation plane of the first joint whereas the oscillation plane of the third joint is orthogonal to the oscillation plane of the second joint and parallel to the oscillation plane of the first joint.

15. Articulation replacement prosthesis such as for exam- ple a prosthetic wrist or a prosthetic knee, charac- terized in that it comprises one or more flexible joints according to claims 12 to 14.

16. Prosthesis according to claim 13, characterized in that the bases have an ellipsoidal section in plan wherein the ellipse of each base has axes of size 40and 30 millimeters respectively and the distance be- tween the two bases is 40 millimetres.

17. Prosthesis according to claim 13 or 14 wherein at least one of the bases of the joints constituting the pros- thesis is connected to tie rods that act as traction and movement elements of the prosthesis at least par- tially replacing the tendons of the human organ that is replaced.

18. Prosthesis according to claims 14 and 15 wherein the said tie rods slide and / or are connected through seats arranged at the opposite ends of the major axis and the minor axis of the ellipse which represents the plan shape of the bases of the joints.

19. Prosthesis according to claims 14 and 15 wherein the said tie rods slide and / or are connected through seats arranged at the vertices of a rectangle inscribed in the perimeter of the ellipse which represents the plan shape of the bases of the joints, being said rectangle preferably arranged with the sides parallel to the axes of said ellipse.

20. Universal joint mechanical transmission member, for the mechanical transmission of motion between two ro- tating members, said transmission member comprising two or more shafts, each rotatable around an axis of rotation, characterized in that it comprises one or more universal joints i.e. of the cardanic type ac- cording to the previous claims, said joints being ar- ranged so as to transfer the rotary motion between the said rotating members.

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