Experimental validation of a constant-force mechanism and analysis of its performance with a calibrated multibody model

Mechanism and Machine Theory 173 (2022) 104819 Available online 23 March 2022 0094-114X/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt Research paper Experimental validation of a constant-force mechanism and analysis of its performance with a calibrated multibody model Silvia Sánchez-Salinas a, Alfonso García-Agúndez b, Javier López-Martínez a,∗, Daniel García-Vallejo b aCIMEDES Research Center (CeiA3), Department of Engineering, University of Almería, Spain bDepartment of Mechanical Engineering and Manufacturing, Universidad de Sevilla, Spain ARTICLE INFO Keywords: Constant-force mechanisms Multibody model Experimental validation ABSTRACT The potential applications of constant-force mechanisms (CFMs) have increased in recent years with the development of new designs. This paper focuses on the experimental evaluation and the mathematical modelling of a CFM. The mechanism is based on the use of cams, rollers and springs, which allows for a large stroke and user-adjustable force value. The evaluation and validation of the mechanism is carried out by performing a series of tests examining the constancy of the input force, and where the influence of friction forces and the manufacturing process on the performance of the mechanism is revealed. A multibody model of the constantforce mechanism is developed and calibrated with the help of the experimental results. This model is based on the assumption of rolling without sliding between the cam and the roller, what makes the multibody model to include velocity constraints. The calibration of the model pursues the development of a mathematical tool to study the performance of the mechanism in conditions that are different from those of the experiments presented. The CFM model is used to perform a sensitivity analysis of some important design parameters of the mechanism. 1. Introduction Constant-force mechanisms (CFMs) are designed to achieve a constant force value on the input link for a certain prescribed range of displacement. These mechanisms are applied in several fields of engineering, where the design solution given in each case will depend on the intended use. Constant-force mechanisms can be classified into two main groups according to the type of components that provide mobility to the mechanism [1]. A first group includes those mechanisms based on flexible links, the so-called compliant mechanisms [2–4], where the mobility of these mechanisms is provided by flexible elements and can be designed in one single piece. The flexible parts are usually designed with relatively complex geometries to achieve the characteristic of constant-force in a certain range of movement of the mechanism. The geometry of the flexible element can be obtained from optimization techniques [5–9] and from the pseudo-rigid body method [5,10–13]. Compliant CFMs can be designed for a one directional constant force [14,15] or for a constant torque [7,16]. Some CFMs proposals are based on the use of a positive stiffness element together with a negative stiffness element, so that when acting together the resulting force–displacement curve presents some region with null or almost null slope, i.e. nearly null stiffness [13,17–20]. One way to achieve negative stiffness is by using beam elements subjected to buckling, where ∗Correspondence to: CIMEDES Research Center (CeiA3), Department of Engineering, University of Almería, 04120, Almería, Spain. E-mail addresses: [email protected] (S. Sánchez-Salinas), [email protected] (A. García-Agúndez), [email protected] (J. López-Martínez), [email protected] (D. García-Vallejo). https://doi.org/10.1016/j.mechmachtheory.2022.104819 Received 2 December 2021; Received in revised form 2 February 2022; Accepted 3 March 2022 Mechanism and Machine Theory 173 (2022) 104819 2 S. Sánchez-Salinas et al. the stiffness of the beam can be known from the Euler equation [21–23]. Compliant CFMs are of particular interest in applications that do not require large displacements, have the advantage of being simple and compact construction, and have no friction losses. The second group of CFMs includes those that use springs together with a non-linear transmission. Compared to compliant CFMs, they have the advantage of being designed for larger displacements [15,24], although in most cases the influence of the friction must be taken into account in the design [14,24,25]. Some proposals are based on the use of rigid links together with springs arranged in a particular configuration [26–28], as is the case of the well known desk lamp holder, which compensates the weight of the lamp for a wide range of positions [29]. In other cases springs are used together with cams with a particular geometry. In some designs a non-circular cam rotates winding a cable around its surface, which lengthens a linear spring. By combining the linear spring characteristic with the variable radius cam, a constant torque can be achieved on the cam’s axis of rotation [30,31]. Other designs use springs and rollers, where the springs are lengthened by rolling the rollers over a cam surface [14,15,24,32,33]. The particular geometry of the cam allows maintaining a constant force value on the input link of the mechanism. Some solutions allow the force value to be modified by presetting the initial spring tension [14] or by arranging several springs in parallel [24]. A generalization of the CFMs are mechanisms with user-defined stiffness. Instead of designing the mechanism for zero stiffness (constant force), the same design methods can be applied to obtain a non-linear spring with an specific stiffness curve [34,35]. Bidgoly et al. [36] derive the geometry of several cams to achieve constant torque, cubic, hyperbolic tangent and sinusoidal springs. Constant-force mechanisms are undergoing considerable development in the field of robotics, where they are used as gravitybalancing mechanism [28], as safety systems against overloads [17], or in the manipulation of objects to limit the gripping force [8,19,37,38], thus avoiding the use of complex force control systems. CFMs have found new applications in fields such as surgery, with the design of grasping forceps to control the clamping force [10,39] or in cardiac ablation catheter [40]; in the development of 2-DOF grippers [41]; in assistive exoskeletons [42]; in end-effectors for polishing/deburring operations [43]; in microgripers for biological cell micromanipulation [44]; in body building machines, where they seek to eliminate the inertia forces present in weight-stack machines [45,46] or for their use in low-gravity environments [47]; in snap-fit connectors [48]; in automotive clutches [49] or in electrical contacts [50]. The multibody modelling of cam mechanisms is a problem that is definitively influenced by the contact modelling between the cam and the follower. Using multibody system dynamics theory, Cardona et al. [51] presented a methodology for the optimal design of the cams of motor engine valve trains including nonlinear geometrical, kinematical and dynamical constraints. Ouyang et al. [52] also utilized a dynamic multibody model of a cam-follower mechanism in the optimal design problem of the cam profile. The contact problem between circular and non-circular bodies, as that of the cam and the follower in general cam mechanisms, has been treated in detail by Xu [53], who proposed a general methodology for planar multibody systems which is applicable in case single and multi-point contact. Since cam profiles are often numerically represented, it is common to use interpolation to reconstruct the cam profile between data points. In this sense, the influence of interpolation errors in the kinematics and dynamics of cam mechanisms is studied by Borboni et al. [54]. Analyses of computational multibody models’ results of cam mechanisms in comparison with experimental performance data can be found in the works of [55,56]. In the present work, a CFM based on a cam and springs is experimentally evaluated and mathematically modelled and studied. The CFM prototype evaluated here is based on one of the three conceptual designs of CFMs proposed in a previous work [24], and is intended for the future fabrication of a full-scale CFM for bodybuilding machines [45]. This paper is organized as follows: Section 2includes, first, a brief description of the conceptual design of the CFM developed, the force equation condition for constant force and the numerical derivation for the cam profile; second, the description of the prototype developed; and third, the evaluation of the manufacturing errors in the fabricated cam. Section 3contains a series of tests carried out to evaluate the performance of the mechanism and includes some simulations for comparison. Next, Section 4deals with the formulation of the multibody model of the CFM and Section 5discusses the influence of different parameters on the constant force characteristic. Lastly, Section 6draws the main conclusions. 2. The constant-force mechanism The CFM developed basically comprises a symmetric cam profile, two rollers, and a spring attached to both roller axes (Fig. 1). The pulling force is exerted on a cable (or a rod) aligned with the axis of symmetry of the mechanism, where this cable bifurcates into two braces that are connected to both roller axes respectively. As the cable moves vertically, the rollers move upward by rolling over the cam surface and the spring elongates by exerting a force opposing the movement. With this arrangement, a certain cam profile can be found that allows a constant force value to be maintained throughout the entire stroke of the mechanism. Once the conceptual design of the CFM has been introduced, this section begins with a summary of the input force equation, the constant force condition, and the numerical integration required to obtain the roller centre trajectory (a more detailed description of this procedure can be found in our former work [24]). Subsequently, the CFM prototype is described. Finally, the manufacturing errors of the cam profile are measured and discussed. 2.1. Force equation and cam profile Fig. 1 shows the forces acting on the right roller when the roller moves upward. Note that, due to the symmetry of the mechanism, both rollers move at the same time and the force diagrams are also the same for both rollers. From the equilibrium equations in Mechanism and Machine Theory 173 (2022) 104819 3 S. Sánchez-Salinas et al. Fig. 1. Conceptual design of the CFM and forces diagram for the roller. the horizontal and vertical directions, the moment equilibrium equation 𝐹𝑓𝑟=𝑁𝑑, and some basic relations, the input force can be written as [24] 𝐹= 4𝐾 𝛥𝑥 (1 − 𝜇𝑟tan 𝛽 tan 𝛽+𝜇𝑟 − tan 𝛼)−1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝐴 ,(1) where the input force 𝐹results as a function of the spring stiffness coefficient 𝐾and its elongation 𝛥𝑥, the rolling friction 𝜇𝑟, the slope of the curve 𝛽and the angle 𝛼of the brace. Eq. (1) shows that the force 𝐹will be constant if the trajectory of the roller centre verifies that the term 𝐴keeps constant. It must be noted that Eq. (1) has been obtained for the upward movement of the rollers, while for the downward displacement the sign preceding 𝜇𝑟must be changed. This means that, due to friction, the cam can only be designed for an exactly constant force curve in one of the two directions of displacement. Next, from Eq. (1), the following nonlinear first order differential equation can be derived [24] d𝑥 d𝑦=1 − 𝜇𝑟𝑓(𝑥) 𝜇𝑟+𝑓(𝑥),(2) where 𝑓(𝑥) = 1 𝐴(𝑥−𝑥0) + 𝑥 𝑙𝑡(1 − (𝑥 𝑙𝑡)2)−1 2 .(3) To obtain Eq. (2) from Eq. (1), the relations tan 𝛽=d𝑥 d𝑦and tan 𝛼=𝑥 𝑙𝑡(1 − (𝑥 𝑙𝑡)2)−1 2 have been used. Then, a first order implicit integration formula can be used to solve the differential Eq. (2) [24], resulting in the trajectory of the roller centre. Finally, the cam profile can be obtained with the CAD software AutoCAD from AUTODESK. First, the points of the roller centre path obtained from the numerical integration are joined by means of a spline. Secondly, the cam curve is obtained using the equidistance tool provided by the software, where the radius of the roller is used for the equidistance. 2.2. Prototype In order to experimentally validate the design proposed and to analyse its performance, a prototype of the CFM has been elaborated. Fig. 2 shows the CAD model and a picture of the prototype. The cam profile of the mechanism has been designed for a maximum stroke of 250 mm and to achieve a constant force value 𝐹of 50 N for a spring stiffness 𝐾of 1.38 N/mm. Table 1 summarizes the main characteristics of the CFM. Due to the initial uncertainty of the value of the rolling friction coefficient, the Mechanism and Machine Theory 173 (2022) 104819 4 S. Sánchez-Salinas et al. Fig. 2. (a) CAD design of the CFM and (b) an image of the prototype mounted in a H5KS Hounsfield testing machine. Labels in (a): 1 cam, 2 pulling link, 3 and 5 braces (length =200 mm), 4 and 6 rollers (radius =16 mm). Table 1 CFM design parameters. Constant force, 𝐹(see Eq. (1))𝐹= 4𝐾𝐴, with 𝐴= 9.058 Number of springs in parallel 2/4/6 Possible stiffness, 𝐾(N/mm) 1.38/2.76/4.14 Possible force values, 𝐹(N) 50/100/150 Stroke (mm) 240 Brace length, 𝑙𝑡(mm) 200 Spring rest length, 2𝑥𝑜(mm) 136 Roller radius, 𝑟(mm) 16 cam profile has been derived for 𝜇𝑟= 0. To achieve a low rolling friction coefficient, the cam is made of stainless steel ASTM A 240 (AISI 304) and two different materials have been tested for the rollers (steel and nylon). The shaft of the rollers has been prolonged to be able to use up to six springs arranged in parallel. In this way, it is easy to modify the constant force value of the mechanism. According to Eq. (1), the force 𝐹is proportional to the spring stiffness. Then, the force value can be changed by adding several springs in parallel or replacing the springs by other with different stiffness constant. In Table 1, three different values of force 𝐹are listed (50 N, 100 N, and 150 N), which are achievable by arranging 2, 4, or 6 springs in parallel, with a stiffness of 0.69 N/mm stiffness in each spring. Moreover, others values of force can be obtained using springs with different stiffness. Note that all the springs must have the same rest length, since the distance between the right side and the left side of the cam is conditioned by the spring rest length (defined as 2𝑥𝑜in Fig. 1). To compensate for possible differences in spring length or manufacturing errors, individual spring tensors (M4 ×0.7 mm screws) have been used for each spring to adjust the spring rest length. 2.3. Evaluation of the manufactured cam profile A precise fabrication of the cam profile is essential to get the constant-force characteristic of the mechanism. The prototype cam has been obtained with a CNC laser cutting machine from the CAD drawing. This manufacturing method has been preferred instead of milling, due to the much lower price of the former. The cam has been obtained from a 10 mm thick steel plate. To evaluate the manufactured cam profile, it has been measured with an optical measuring machine with vision system TESA-VISIO 300 (1 μm resolution) and then compared with the designed profile. Due to the dimensions of the cam and the limited working area of the measuring machine, only the lower half of the cam profile has been measured (first 150 mm of the cam profile). Cam profile measurements were taken at 0.5 mm intervals. The best fit between the set of measured points and the set of model points has been obtained with an iterative closest point (ICP) algorithm, where the registration of the sets of points has been done by iterative reweighted least squares [57]. After the transformation (translation plus rotation) of the measured points sets, a very good fit between both curve points has been obtained, with a root-mean-square error (RMSE) of 0.023 mm and standard deviation (SD) of 0.017 mm. Fig. 3 shows the error in 𝑥coordinate between the measured points and the model, for the right and left sides of the cam. Negative values indicate that the measured points of the fabricated cam profile are inside the design profile (lower cam width). These results, which show small manufacturing errors, together with the experimental results described in Section 3, confirm the feasibility of the CNC laser cutting. Mechanism and Machine Theory 173 (2022) 104819 5 S. Sánchez-Salinas et al. Fig. 3. Error in 𝑥coordinate between the measured and the theoretical cam profile. Table 2 Measured stiffness constant of the springs. Spring #1 #2 #3 #4 #5 #6 Stiffness (N/mm) 0.683 0.682 0.681 0.686 0.686 0.679 3. Experimental results A series of tests have been carried out to analyse the performance of the mechanism. The constant-force mechanism has been tested for three different spring stiffness values and two rollers materials. Previously, some initials tests have been done to obtain the real stiffness constant of the springs and to adjust the spring rest length in the mechanism. All force tests have been done in a H5KS Hounsfield testing machine (see Fig. 2b). This model H5KS includes a 5 kN load cell with a 0.5% load measurement accuracy and 0.001 mm position accuracy. The machine has a maximum stroke of 750 mm and a test speed range from 0.001 mm/min to 1500 mm/min. Tests have been conducted starting with the rollers located at the bottom of the cam and moving upwards. Once the maximum stroke of the mechanism has been reached, with the rollers at the top of the cam, the rollers begin to move downward until they return to the initial position. Most of the tests have been done with a travel speed of 500 mm/min and a sampling rate of 35 Hz. As described above, the prototype can incorporate up to six springs arranged in parallel (2, 4, or 6 springs are possible). The stiffness of each spring has been obtained through tensile tests. Table 2 lists the stiffness constant values for each spring. The spring rest length will have influence in the force curve of the mechanism, specially in the first section of the cam [24]. For each pair of springs, it has been adjusted the spring rest length from tensile–compression tests results, where the spring rest length can be modified through a screw. Fig. 4 shows the force curves obtained for five different adjustments of the spring rest length of the pair of springs #1 and #2. The step between two consecutive adjustments corresponds to two complete turns of the screw. In Fig. 4, it is represented the upwards and downwards displacements of the mechanism in the same curve, where the resultant hysteresis loop for each test is appreciated. The curve with the best approximation to the constant-force characteristic has been defined as ‘adjust =0 mm’ in Fig. 4. Variations in the spring rest length result in deviations in the force curve, being more pronounced in the first section of the cam. These results are in accordance with the theoretical results obtained in Ref. [24]. The force value can be modified by adding springs arranged in parallel. Fig. 5 shows the force hysteresis loop measured for 2, 4, and 6 springs. Force curves show some fluctuations due to the errors of the manufactured cam profile and also some noise mainly due to the surface roughness of the cam. In the three force curves, the same fluctuation pattern can be observed. The force curves show larger hysteresis loop for higher spring stiffness, where the area within the hysteresis loop mainly depends on the rolling friction coefficient and the force value. It is interesting to note that the area within the hysteresis loop is not linear with the force (Table 3). For each section of the hysteresis loop (upward and downward displacement), Table 3 summarizes the mean force value, the standard deviation and the maximum and minimum values. Additionally, it is included the theoretical value of the forces, calculated from Eq. (1) with the measured spring stiffness constant and for null friction coefficient. These force values have been represented with dashed lines in Fig. 5. All force curves obtained in the different tests showed high repeatability. As example, Fig. 6 shows five repetitions of the force test for four springs in parallel. The tests described so far have been done with rollers of steel. With the aim of studying the effect of other materials in the rolling friction, rollers of nylon have been tested. The curves for the three force levels obtained for steel rollers have been compared with the curves obtained for nylon rollers, showing small differences between the curves. As expected, since the rolling friction for nylon material is slightly higher, the hysteresis loop is somewhat larger when nylon rollers are used. This difference is more easily Mechanism and Machine Theory 173 (2022) 104819 6 S. Sánchez-Salinas et al. Fig. 4. Force curves for variations in the spring rest length. Fig. 5. Force curves for 2, 4, and 6 springs in parallel. Table 3 Parameters of the force curves in Fig. 5. Springs arranged in parallel 2 (#1 and #2) 4 (#1 to #4) 6 (#1 to #6) Spring equivalent stiffness (N/mm) 1.365 2.732 4.097 Theoretical force (N) 49.46 98.99 148.4 Displacement up/down up/down up/down Mean force (N) 49.67/48.27 98.94/96.62 148.27/144.17 Standard deviation (N) 0.427/0.425 0.949/1.02 1.38/2.12 Max. force (N) 51.00/49.33 101.0/99.00 151.33/148.00 Min. force (N) 48.00/46.33 95.33/92.17 143.17/136.00 Hysteresis loop area (N m) 0.406 0.555 1.069 observed in the curves of higher value of force. Fig. 7 shows the curve force for the two type of rollers and for 6 springs, with hysteresis loop areas of 1.069 Nm and 1.220 Nm for steel and nylon rollers, respectively. Test displacement speed may influence the force curve results. Previous results were conducted at 500 mm/min. Next, Fig. 8 shows the force curves for five different velocities, from 100 mm/min to 1500 mm/min (maximum allowed by the testing machine). Since the displacement velocity is low in all tests, no appreciable influence of inertia forces is expected. All curves show the same main fluctuation due to the small deviation in the manufacturing process, but the noise due to surface roughness increases with decreasing speed. As representative values, the standard deviations for the upward displacement section of the curves are 0.693 N, 0.551 N, 0.427 N, 0.359 N and 0.318 N for the velocities between 100 mm/min and 1500 mm/min shown in Fig. 8. Finally, this section presents a comparison between experimental results and simulations using the measured cam profile described in Section 2.3. The simulations were performed with SOLIDWORKS motion commercial software, which uses the ADAMS/Solver simulation engine to solve the equations of motion and calculate the forces on each component of the assembly. The real model has been simplified for simulations. Braces, springs and shafts have been included as virtual links without mass. In addition, no mass has been considered for the rollers. Therefore, no inertial effects are considered, which have no significant Mechanism and Machine Theory 173 (2022) 104819 7 S. Sánchez-Salinas et al. Fig. 6. Force curves for five repetitions of the force test (4 springs). Fig. 7. Force curves for steel rollers and for nylon rollers (6 springs). Fig. 8. Force curves for different displacement velocities (2 springs). Mechanism and Machine Theory 173 (2022) 104819 8 S. Sánchez-Salinas et al. Fig. 9. Force curves from experimental result and from simulations with the measured cam profile and different values of the rolling friction coefficient. (a) raw data and (b) filtered curves. influence due to the low accelerations. As in the tests, the simulations were performed with a low displacement velocity of 500 mm/min. Lastly, the contacts between the rollers and the measured cam profile have been imposed as non-penetrating and with allowable separation, and the rolling friction coefficient 𝜇𝑟has been included. Fig. 9(a) shows the measured force as well as the simulated results for rolling friction coefficient values of 𝜇𝑟= 0.002 and 𝜇𝑟= 0.004, and for no friction. For an easier comparison of the results, the data reported in Fig. 9(a) have been back and forth filtered with a sixth order Butterworth filter with cut-off frequency of 2 Hz, leading to the smoother curves drawn in Fig. 9 (b). From the similarity in the fluctuations shown in the experimentally measured and simulated curves, it is confirmed that the force fluctuations are due to manufacturing errors. The error induced by the machining process defined as the difference between the force measured in the test conducted on the manufactured mechanism and the force obtained by simulation using the profile sent to the manufacturer is studied later on in Section 4, where a dedicated multibody model of the CFM mechanism is developed. Besides the similarity of the curves, a drop in the experimental curve with respect to the simulated curves can be appreciated in the initial section of the displacement. This may be indicative that the spring rest length introduced in the simulation (the theoretical one) does not fit the real one (see Fig. 4). In Fig. 10, the simulation has been repeated for 𝜇𝑟= 0.002, but a spring rest length of 136.5 mm (0.5 mm longer than the theoretical) has been used instead. In this way, the simulated curves of Fig. 10 follow a more similar pattern in the initial section. In addition, the results of the simulations allow obtaining an indicative value of the rolling friction coefficient. In Fig. 9(a), the areas of the hysteresis loops in the section between 40 mm and 210 mm of the input displacement result in 0.233 Nm and 0.410 Nm for 𝜇𝑟= 0.002 and 𝜇𝑟= 0.004, respectively, and 0.259 Nm for the experimental measurement. In Fig. 10(a), for 𝜇𝑟= 0.002, the area is 0.232 Nm (almost the same as for Fig. 9(a)). These values indicate that, for this case, the real rolling friction coefficient will be slightly higher than 0.002. Once the value of the rolling friction coefficient has been identified, the cam profile could be designed Mechanism and Machine Theory 173 (2022) 104819 9 S. Sánchez-Salinas et al. Fig. 10. Force curves from experimental result and from simulation with the measured cam profile, a spring rest length of 136.5 mm and a rolling friction coefficient of 0.002. (a) raw data and (b) filtered curves. by including that value in Eq. (1). This allows maintaining a more accurate constant force curve in one of the directions of the displacement. 4. Multibody model of the CFM mechanism The objective of this section is to develop a multibody model of the CFM that reproduces the experimental results presented before. Once the model is validated by comparison with the experimental results, it will be used to investigate the influence of inertial effects and manufacturing tolerances on the CFM performance. In order to be able to study a wide range of manufacturing errors, the mathematical model of the CFM does not make use of symmetry. Instead, the two cam profiles of the CMF are modelled separately. The model comprises a total of five moving bodies which are described as follows (see Fig. 11). Body 2 is a slider that translates vertically when pulled up by the user. It is the input body of the mechanism. Both bodies 3 and 5 are connecting rods articulated to the slider and the two rollers, bodies 4 and 6, which are assumed to roll without slipping over the left and right cam surfaces. The assumption of no slipping is reasonable as long as the rollers have low inertia torques and low friction torques. They just follow the cam profile as the slider is pulled upwards. Since the cam profile has a curved shape, the contact between the roller and the cam is implemented by using two non-generalized coordinates to locate the contact point of each roller. The vector of coordinates is as follows: 𝒙=(𝑠2𝜃3𝜃4𝜉4𝑦4𝜃5𝜃6𝜉6𝑦6)T,(4) where 𝜉4and 𝑦4are the non-generalized coordinates used to define the contact of roller 4 and the right cam profile, and 𝜉6and 𝑦6are the non-generalized coordinates used to define the contact of roller 6 and the left cam profile. This set of coordinates is Mechanism and Machine Theory 173 (2022) 104819 16 S. Sánchez-Salinas et al. [48] Y.-H. Chen, C.-C. Lan, Design of a constant-force snap-fit mechanism for minimal mating uncertainty, Mech. Mach. Theory 55 (2012) 34–50. [49] Z. Li-Jun, L. Tao, S. Bao-Yu, Optimum design of automobile diaphragm spring clutch, in: 2008 IEEE Vehicle Power and Propulsion Conference, VPPC 2008, 2008, pp. 1–4. [50] B. Weight, C. Mattson, S. Magleby, L. Howell, Configuration selection, modeling, and preliminary testing in support of constant force electrical connectors, Trans. ASME, J. Electron. Packag. 129 (3) (2007) 236–246. [51] A. Cardona, E. Lens, N. Nigro, Optimal design of cams, Multibody Syst. Dyn. 7 (3) (2002) 285–305, http://dx.doi.org/10.1023/A:1015278213069. [52] T. Ouyang, P. Wang, H. Huang, N. Zhang, N. Chen, Mathematical modeling and optimization of cam mechanism in delivery system of an offset press, Mech. Mach. Theory 110 (2017) 100–114, http://dx.doi.org/10.1016/j.mechmachtheory.2017.01.004. [53] L. Xu, A method for modelling contact between circular and non-circular shapes with variable radii of curvature and its application in planar mechanical systems, Multibody Syst. Dyn. 39 (3) (2017) 153–174, http://dx.doi.org/10.1007/s11044-016-9549-0. [54] A. Borboni, F. Aggogeri, I. Elamvazuthi, G. Incerti, P. Magnani, Effects of profile interpolation in cam mechanisms, Mech. Mach. Theory 144 (2020) http://dx.doi.org/10.1016/j.mechmachtheory.2019.103652. [55] P. Danakas, M. Bilodeau, N. Quaegebeur, Cam-follower mechanism for airborne ultrasound generation, Appl. Acoust. 182 (2021) http://dx.doi.org/10. 1016/j.apacoust.2021.108225. [56] R. Rayner, M. Sahinkaya, B. Hicks, Improving the design of high speed mechanisms through multi-level kinematic synthesis, dynamic optimization and velocity profiling, Mech. Mach. Theory 118 (2017) 100–114, http://dx.doi.org/10.1016/j.mechmachtheory.2017.07.022. [57] P. Bergström, O. Edlund, Robust registration of point sets using iteratively reweighted least squares, Comput. Optim. Appl. 58 (3) (2014) 543–561, http://dx.doi.org/10.1007/s10589-014-9643-2. [58] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comput. Methods Appl. Mech. Engrg. 1 (1) (1972) 1–16, http://dx.doi.org/10.1016/0045-7825(72)90018-7. [59] D.M. Frost, J.B. Cronin, R.U. Newton, A comparison of the kinematics, kinetics and muscle activity between pneumatic and free weight resistance, Eur. J. Appl. Physiol. 104 (6) (2008) 937–956. [60] S. Orange, J. Metcalfe, A. Liefeith, P. Marshall, L. Madden, C. Fewster, R. Vince, Validity and reliability of a wearable inertial sensor to measure velocity and power in the back squat and bench press, J. Strength Cond. Res. 33 (9) (2019) 2398–2408, http://dx.doi.org/10.1519/JSC.0000000000002574.