Kinematic Accuracy Method of Mechanisms Based on Tolerance Theories

Traditional tolerance analysis is mostly restricted to static analysis. However, tolerances of different components also affect the movement accuracy in a mechanism. In this paper, the idea of kinematic tolerance analysis is advanced. In the interest of achieving movement precision considering tolerance, a kinematic Jacobianmodel is established on the basis of a traditional dimensional chain and an original Jacobianmodel..e tolerances of functional element (FE) pairs are expressed as small-displacement screws. In addition, joint clearances resulting from tolerance design also influence the kinematic accuracy, and they are modeled by FE pairs. Two examples are presented to illustrate the rationality and the validity of the kinematic tolerance model. .e results of the two examples are shown, and the discussion is presented. A physicalmodel of the 2D example is also built up in 3DCS software. Based on the discussion, a comparison between the statistical and physical models is carried out, and the merits and demerits of both are listed.


Introduction
For an assembly, component tolerance reflects the actual relationships between mating parts. Conventionally, tolerance analysis is used to estimate the accumulation of the assembly dimensions. As a consequence, tolerance analysis is mostly restricted to static tolerance analysis in an assembly. However, the tolerances of different parts also affect the accuracy of movement in an assembly. With this need for tolerance analysis, the effects of component tolerances on the kinematic accuracy of mechanisms require study.
In terms of tolerance propagation and analysis, the traditional approach is a dimensional chain. e main methods that have been developed in recent years are the matrix model [14,15], vector loop model [16][17][18][19], SOV model [20,21], and Jacobian model [22,23]. Concerning the traditional method, a tolerance stack-up function is established on the basis of the relevant dimensional chain. e first-order differential value of the corresponding dimension is taken as the deviation. is method is mainly used in static tolerance analysis and has been applied in many cases [24,25]. Utilizing homogeneous matrix transformation, a matrix model transfers the displacements from local reference frames to a global reference frame. e matrix propagation and representation models are used together to perform the tolerance analysis. A vector loop model uses vectors to represent the dimensions in an assembly [17,19]. e vectors are arranged in chains or loops representing those dimensions that stack together to determine the resultant assembly dimensions [14]. In an SOV model, the dimensional deviations that represent the influences of tooling and part errors (including part-to-fixture, part-topart, and interstation interactions) are considered as product and process factors; the whole assembly process is modeled as a state-space model. e accumulations of dimensional deviations are represented by state transfer functions. e SOV model is suitable for complex assemblies. In recent decades, increasingly more attention has been directed to the development of three-dimensional (3D) tolerance analysis, and a Jacobian model is advanced with the tide. Laperrière and Lafond adopted virtual joints in robotics to explore a tolerance transformation process, and the Jacobian model was obtained by first associating a coordinate frame to every virtual joint [23,26]. In this model, functional element (FE) pairs and functional requirement (FR) are used to represent the relationship between parts. Based on this innovation, many other studies ensued [27][28][29][30][31][32][33]. In addition to the mainstream tolerance analysis methods, new ideas are also burgeoning [18,34,35]. Wang et al. [36] invented an assembly dimensional model based on the shortest path. e tolerance propagation function can be acquired through the shortest path, as described in their theory. Based on equivalence of the deviation source, Zhao et al. [18] proposed a combined deviation accumulation method. Although the angles of view are new, the core propagation approaches are mentioned above. Among the propagation models introduced in recent years, the Jacobian model is the most suitable one for the kinematic tolerance analysis.
Regarding tolerance representation, different models exist in the relevant literature. Requicha introduced the mathematical definition of tolerance semantics and proposed a solid offset approach initially [37]. Salomons et al. specified tolerances based on technologically and topologically related surfaces (TTRS) [38,40]. Desrochers and Riviere presented a homogeneous matrix approach coupled with the notion of constraints for the representation of tolerance zones [15]. T-Maps were set up as well as a hypothetical volume of points that correspond to all possible locations and variations of a segment of a plane that can arise from tolerances on size, form, and orientation [41]. e skin model is a basic concept within GeoSpelling and the ISO standards. Skin model shapes [42][43][44][45][46][47] are skin model representatives that comprise various kinds of geometric deviations. With this method, geometric deviations are represented by discretizing features into points and measuring the distances between the points of the nominal model and the skin model shape. e main advantage of this method is to model geometric tolerance more practically. To apply geometric tolerances in the Jacobian model, Polini and Corrado [48][49][50] presented an approach and, whereafter, also integrated manufacturing signature and operation conditions into the Jacobian model based on the skin model shapes. In recent years, there has been an increasing amount of literature on small-displacement screw models [51][52][53][54][55]. A small-displacement screw model was first discussed by Bourdet and Clement [56]. As variations of a surface and its features from the nominal position can be represented by a screw, a small-displacement model was gradually developed for tolerance analysis in the following years.
Most of the tolerance studies are limited to static tolerance analysis. However, for a mechanism, the tolerance of every part in an assembly also affects the kinematic performance, e.g., the accuracy of movement. Walter et al. [57] used "the integrated tolerance analysis in motion" approach to obtain the effects of manufacturing-caused and operationdepending deviations on a system's FKCs. e propagation function is derived from a dimensional chain, while in the three-dimensional space, the ideal propagation method for kinematic tolerance analysis is the Jacobian matrix; meanwhile, the study concerns more about the operationdepending effect on the tolerance, not the tolerance influence on the kinematic accuracy. Utilizing the skin model shapes, Schleich et al. [58,59] constituted a framework for the deviation analysis of contact and mobility in an assembly. e skin model shapes are a tolerance representation method, and their study mainly focuses on the geometric modeling of the contact in motion. e mobility is involved; however, the propagation method is also a dimensional chain, and the application is limited to some extent. Zhou et al. [60] proposed a kinematic accuracy method based on DP-SDT theory. Although the influences of motional displacements, force direction, and vibration can be calculated using the method, the propagation function is complicated. e purpose of this paper is to propose a novel kinematic accuracy method for mechanisms based on tolerance analysis. First, joint clearances resulting from the fit tolerances are represented as tolerances by small-displacement screws. en, the kinematic Jacobian model is established by combining an original tolerance Jacobian model and a traditional dimensional chain. Finally, the kinematic movement error can be obtained through the analysis of the kinematic tolerance Jacobian model. ese ideas will generate important insights into tolerance design and kinematic accuracy; a small constituent part of the establishment method can also be used independently and integrated with other theories (e.g., robust theory) so that several kinds of practical problems in engineering can be solved [61]. e outline of this work is as follows: Section 2 reviews related theory. Section 3 discusses the details of the proposed method. Section 4 presents a case study to verify the feasibility and efficiency of the new method, and a physical model is also given to prove the legitimacy of the results in this section. Meanwhile, a practical mechanism in engineering is also chosen as an example in Section 5, and the discussion of the results is given. Section 6 summarizes the work and points out the future study.

Jacobian Model.
A Jacobian model adopting kinematic theory has been advanced in robotics. Several basic concepts of an original Jacobian model are listed in Table 1 [61,62].
In an assembly, it is essential to establish the tolerance chain first and then define FE pairs in the established tolerance chain.

Deviation of a Functional Element Pair from Its Nominal
Value (Based on Dimensional Chain eory). A tolerance chain is a representation of how a functional requirement x depends on a known set of functional dimensions d i [61,63]. Each dimension has a deviation from its nominal value: Δd i � d i − d i , and, as a consequence, a deviation will occur on the functional requirement: In the general case, x � f(d 1 , d 2 , . . . , d n ) is nonlinear and possibly unknown, and the above assumption Δx is difficult to calculate. However, the equation can be linearized by a first-order Taylor approximation (as the deviations on the dimensions are small compared to nominal values, higher-order terms of a Taylor series can be omitted): en, FR and FE pairs are both established between two FEs. Even the length of an FE pair can be seen as a FR of its elements, letting x � f(d 1 , d 2 , . . . , d n ) be the length of an FE pair, then. en, where ∆x is the small deviation of the length of the functional element pair and ∆d i is its deviation from the nominal value of d i (in tolerancing, it can be seen as a designed tolerance of dimension d).
For an FE pair in Cartesian coordinates, the projections of lengths along three axes are and the respective angles about three axes are respectively, where Δx Δy, Δz, Δθ x , Δθ y , and Δθ z are, respectively, small deviations along and about the three Cartesian axes.

Jacobian Model for Kinematic Tolerance Analysis (Kinematic Jacobian Model)
In this article, the generic forms of small-displacement screws are used to model the tolerances associated with features and gaps in an assembly.

Tolerance Modeling Using the Small-Displacement Screw.
] is a small translational displacement and [θ 1 θ 2 θ 3 ] is a small rotational displacement. Concerning the existence of tolerances, when mechanisms move, FE pairs move, the lengths of FE pairs change, and the small-displacement screw models change as well. We let η be the input variable of a mechanism; then, the projections of lengths and angles, respectively, along and about three axes are Table 1: Basic concepts of a Jacobian model.

Parameter Definition
Functional element (FE) Points, curves, or surfaces that belong to parts in the assembly. An FE can be real, e.g., the plane surface of a block, or constructed, e.g., the axis of a cylinder. Functional requirement (FR) An important condition to be satisfied between two FEs on different parts, e.g., a fitting condition.
Kinematic pair If two FEs are on different parts and there is physical contact between them, then they constitute a kinematic pair.
Internal pair If two FEs are on the same part and both of them participate in a contact relation with some other parts, then they form an internal pair. Functional element pair (FE pair) Two FEs on different parts or on the same part, including kinematic pairs and internal pairs.

Virtual joints
For the purpose of tolerancing, some coordinate frames are associated with the toleranced FEs in an FE pair, assuming a set of virtual joints exist in each FE pair and can make the toleranced FEs "move" relative to the other FEs, in order to simulate manufacturing inaccuracies. Global reference frame (GRF) e frame in which the FR and global dimensional chain are established.
Local reference frame (LRF) e frames in which FE pairs are established. , and then, respectively, where Δx Δy, Δz, Δθ x , Δθ y , and Δθ z are, respectively, small deviations that are functions of the input variables η and Δd i . As a consequence, the small-displacement screw can be expressed as In the LRF, For the ith FE pair of a mechanism which contains n FE pairs, the small-displacement screw model is Here, the clearance in an assembly is seen as an FE pair of the tolerance chain.

Jacobian
Matrix. For the Jacobian model, FE pairs are used to represent the dimensions and variations in an assembly. e representations of virtual joints and coordinate frames in an FE pair are shown in Figure 1.
e transformation matrix can be deduced to be Figure 1: Virtual joints and coordinate frames to FE pairs.
e Jacobian matrix introduced previously is written as [J 1 J 2 . . .J 6 ] is the 6 × 6 Jacobian matrix associated with the FE of the ith FE pair (internal or kinematic) to which the tolerances are applied, with i � 1∼n.
For small rotational virtual joints, the ith column of the Jacobian matrix J i is computed as where Z →i−1 0 is the third column of T i−1 0 and d →i−1 0 is the last column of T i−1 0 . In a mechanism, a kinematic chain moves consistently. If a kinematic chain is constrained to a plane, changes with the movement of the kinematic chain; therefore, J i changes with the movement of the kinematic chain. Here, ( d →n For small rotational virtual joints, the ith column of the Jacobian matrix J i can be changed to while regarding small translational virtual joints, they do not contribute to the small rotational displacements of the final FR, and the ith column of the Jacobian matrix J i is written simply as For a mechanism containing n FE pairs, the Jacobian model for kinematic tolerance analysis in the GRF can be expressed as is model is called the kinematic Jacobian model, where is the small displacement of the FR in the GRF. e Jacobian matrix (J) and small-displacement screws of FE pairs (∆FEi) change with input variable η in the mechanism. e clearance between parts is seen as an FE pair and modeled as a small-displacement screw.

Steps to Establish a Kinematic Jacobian Model in Kinematic Tolerance Analysis.
e basic steps of the kinematic Jacobian model for tolerance analysis are as follows: (1) Identify FE pairs and define the LRF for each FE and the virtual joints (2) Create the tolerance chain in the mechanism, and obtain small-displacement screws for FE pairs; then, establish the Jacobian model for kinematic tolerance analysis (3) Compute the Jacobian model using a statistical method, worst-case method, or Monte Carlo method (4) Analyze the results

Case Study 1 for the Kinematic Tolerance Analysis
In Section 4, a crank-slider mechanism is taken as an example to analyze the tolerance effect. In addition to the results obtained using the kinematic Jacobian model introduced above, a software simulation is also performed and is discussed in Section 4. Figure 2 shows the vector representation of a classic crank-slider mechanism with a clearance r e in the revolving joint resulting from the tolerance design. B and C are centers of the hole and shaft, Mathematical Problems in Engineering 5 respectively.
e distance between A and D is seen as a functional requirement. e frame established at point A is seen as the GRF, while the other frames are taken as LRFs.
e effect of small angle change is omitted in this case.
As r e is the clearance resulting from the fit tolerances predesigned, considering the contact theories stated by Johnson, the following two assumptions are provided before the analysis: (1) ere is no deformation caused by the contacts in joints (2) Contacts between the holes and shafts are supposed to be line contacts Based on the above two basic assumptions, the angle between r e and the horizontal line can be concluded to be equal to the angle between r 2 and the horizontal line. e parameters of the crank-slider are listed in Table 2.
In this case, the Monte Carlo method is used in the calculation; therefore, the tolerance of the connecting rod obeys norm distribution, as shown in Table 2. e functional pairs of the crank-slider are listed in Table 3. It is worth noting that the z directions of small-displacement screws are different from the original virtual joints; therefore, for functional pairs, small displacements in the z directions are negative.
In Table 4
Letting θ to change from 0 to 2π, calculations are taken every 15°. For the tolerances of the crank-slider, the Monte Carlo method is used here, and the test number is chosen as 100,000. e histograms for each angle are shown in Figures 3-8. PD represents the probability density of deviation with every δ, while the small possible deviations of the functional requirements are denoted as offset values in Figures 3-8. Meanwhile, mean offset values are plotted in Figure 9.

Kinematic Tolerance Analysis Using the Physical Model.
As tolerance analysis can be processed with 3DCS software, here a physical model has been established, and numerical simulation has also been completed ( Figure 10). In 3DCS, the fits are endowed with tolerances, and the analysis is carried out. Basic parameters are the same as analysis using the kinematic Jacobian model. e mean values are listed in Figure 11.

Discussion and Comparison.
When the kinematic Jacobian model is used as the stack-up function, Figures 3-8 show the probability densities of deviations for functional requirements at every angle. Meanwhile, the mean values of all offset values are drawn in Figure 9, from which it can be seen that the offset values of the functional requirements are strongly dependent on the input variable of the crank-slider; that is, they change as the input variable changes. e figure reaches the peak when θ is near π•i/2(i � 2k + 1, k � 1, 2,. . ., n). Figure 10 illustrates the results obtained when the crankslider is analyzed using 3DCS software. In general, the results are influenced by the input variable of the crank-slider as well, and the figure also reaches the peak when θ is near π·(i/ 2 ± 1/12) (i � 2k + 1, k � 1, 2,. . ., n).
In Figure 12, the results obtained from the kinematic Jacobian model and the results obtained by 3DCS software are both plotted. From the comparison in Figure 12, it can be concluded that the trends of two curves are almost the same. However, for the curve obtained by 3DCS, the peak of the curve is brought forward or postponed compared with the curve of the kinematic Jacobian model results. e reason is that the tiny unalterable tolerance setting in software will inevitably affect the tolerance results.
It can be seen from the above results that the kinematic Jacobian model obtains comparatively stable results and clearly shows the trend of error change. e results obtained with the kinematic Jacobian model are conservative, however, and the basic error-change tendency is the same as that found via the software analysis.  x 1 � r 1 · cos θ y 1 � 0 z 1 � −r 1 · sin θ δ 11 � −Δr 1 · sin θδ 12 � 0 δ 11 � Δr 1 · cos θ FE pair 2 (B and C) x 2 � r e · cos c y 2 � 0 z 2 � −r e · sin c δ 21 � −Δr e · sin c δ 22 � 0 δ 23 � Δr e · cos c FE pair 3 (C and D) x 3 � r 2 · cos c y 2 � 0 z 1 � −r 2 · sin c δ 33 � −Δr 2 · sin c δ 32 � 0 δ 33 � Δr 2 · cos c FR (A and D) Unknown and must be calculated       the kinematic Jacobian model can be used as the tolerance stack-up function in kinematic tolerance analysis, and its validity for this purpose is proved.

Case Study 2 for the Kinematic Tolerance Analysis
In this section, a three-dimensional landing gear retraction mechanism is taken as an example to illustrate the theory introduced. e overall structure and representation of the landing gear retraction system are shown in Figures 13 and  14. e mechanism is composed of five components: AB is the upper connecting joint, BC is the upper brace, CD is the lower brace, DE is the lower connecting joint, and GF is the main strut. A and G are localized. e main strut rotates around GH. When the landing gear mechanism retracts or extends, the movement of the wheel is not stable. It closely relates to the tolerances of components, and the kinematic tolerance analysis is performed next. e landing gear retraction mechanism holds the same assumptions as Section 4. e parameters of the landing gear retraction mechanism are listed in Table 5. On the ground of the experience and the mechanical engineering handbook, the tolerances of the components and fits are chosen as medium level. e Monte Carlo method is used in the calculation, and the tolerances of the components obey norm distribution, as shown in Table 5.
e FE pairs of the landing gear retraction mechanism are listed in Table 6. e ( d →n 0 − d →i−1 0 )s of FE pairs are listed in Table 7.
In Figure 15, mean offset values for δ 1 increase from first to last as the input variable θ y5 increases. However, the absolute values for δ 1 decrease as θ y5 increases, they drop from 0.04 to 0, and then increase to 0.01. In the beginning of the curve, the fluctuations are comparatively larger.
As shown in Figure 16, when the input variable θ y5 is within the range of −90°∼ −70°, mean offset values have no significant change for δ 2 . When θ y5 is −70°, they begin to increase gradually from 0.035 to 0.07. From the beginning to the end, the fluctuations have no apparent change.
Looking at Figure 17, at the very beginning of the curve, no significant change can be observed. en, the mean values for δ 3 increase gradually and continually. Meanwhile, the absolute values for δ 3 decrease first and then increase. It can be seen from the curve that the fluctuations are comparatively smaller when the input variable θ y5 is within the range of −90°∼ −50°.
Similarly, major trends are also easily observed using the kinematic Jacobian model in this three-dimensional case, implying that this model can show the trends of the mechanism more distinctly.

Conclusion
In this article, the idea of kinematic tolerance analysis was advanced, and a kinematic Jacobian model was built up. Two study cases are given as examples to expand traditional static tolerance analysis to kinematic tolerance analysis. Major contributions and comments are summarized as follows: (1) By combining an original tolerance Jacobian model and a traditional dimensional chain, a kinematic Jacobian model is established. e kinematic movement error resulting from the tolerance design can be obtained through the analysis of the kinematic Jacobian model.
(2) An elementary crank-slider was used to demonstrate the proposed methodology. Meanwhile, a physical model was also built up, and simulations were performed. Using the final results, a comparison between the statistical model and physical model was made. e comparison indicates the rationality, intrinsic logic, and its validity.
(3) A three-dimensional landing gear retraction mechanism is also taken as an example to illustrate the theory introduced. Two examples simultaneously show that the Jacobian model can exhibit the inner regularity between the input variable and the movement accuracy more easily. (4) In general, the proposed kinematic accuracy method is plain and distinct from the computational point of view. Utilizing this method, joint clearances resulting from the tolerance design and tolerance of components are modeled by small-displacement screws; their effects on the final movement error can be estimated. is method can strongly enhance the efficiency of kinematic tolerance analysis. (5) is work presents the key idea and the first step of expanding traditional static tolerance analysis to kinematic tolerance analysis. Future work aims to further modify the existing Jacobian model and make it more suitable to consider variables (such as geometrical tolerance and angle tolerance) in the kinematic analysis.
Data Availability e data supporting the conclusions of this study are included within the article, and these are also available from the corresponding author upon request.

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Mathematical Problems in Engineering