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I will explain how you might visualize a Quaternion as well as explain the different operations that can be applied to quaternions. I will also compare applications of matrices, euler angles, and quaternions and try to explain when you would want to use quaternions instead of Euler angles or matrices and when you would not. In computer graphics, we use transformation matrices to express a position in space translation as well as its orientation in space rotation.

In this article, I will not discuss the details of transformation matrices. For a detailed description of transformation matrices, you can refer to my previous article titled Matrices. In this article, I want to discuss an alternative method of describing the orientation of an object rotation in space using quaternions. Hamilton was on his way to the Royal Irish Academy with his wife and as he was passing over the Royal Canal on the Brougham Bridge he made a dramatic realization that he immediately carved into the stone of the bridge.

Before we can fully understand quaterions, we must first understand where they came from. The root of quaternions is based on the concept of the complex number system. In addition to the well-known number sets Natural , Integer , Real , and Rational , the Complex Number system introduces a new set of numbers called imaginary numbers. Imaginary numbers were invented to solve certain equations that had no solutions such as:.

Complex numbers can be added and subtracted by adding or subtracting the real, and imaginary parts. A complex number is multiplied by a scalar by multiplying each term of the complex number by the scalar:. We can use the conjugate of a complex number to compute the absolute value or norm , or magnitude of a complex number. To compute the quotient of two complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator. We can also map complex numbers in a 2D grid called the Complex Plane by mapping the Real part on the horizontal axis and the Imaginary part on the vertical axis.

If we plot these complex numbers on the complex plane, we get the following result. We can also perform arbitrary rotations in the complex plane by defining a complex number of the form:. Which is the method to rotate an arbitrary point in the complex plane counter-clockwise about the origin.

Using this notation, we can more easily show the similarities between quaternions and complex numbers. Which results in another quaternion. If we multiply through with the quaternion unit and extract the common vector components, we can rewrite this equation in this way:. This equation gives us the sum of two ordered pairs. The first ordered pair is a Real quaternion and the second is a Pure quaternion. These two ordered pairs can be combined into a single ordered pair:.

And the product of two Real Quaternions is another Real Quaternion:. We can confirm this by using the product or Real Quaterions shown above to multiply a quaternion by the scalar as a Real Quaternion:. Similar to Real Quaterions, Hamilton also defined the Pure Quaternion as a quaternion that has a zero scalar term:. We can also express quaternions as an addition of the Real and Pure quaternion parts:. We can now combine the definitions of the unit quaternion and the additive form of a quaternion, we can create a representation of quaternions which is similar to the notation used to describe complex numbers:.

With the definition of a quaternion norm, we can use it to normalize a quaternion. Then, we must divide the quaternion by the norm of the quaternion to compute the normalized quaternion:. To compute the inverse of a quaternion, we take the conjugate of the quaternion and divide it by the square of the norm:. Similar to vector dot-products, we can also compute the dot product between two quaternions by multiplying the corresponding scalar parts and summing the results:. We can also use the quaternion dot-product to compute the angular difference between the quaternions:.

If you recall we defined a special form of the complex number called a Rotor that could be used to rotate a point through the 2D complex plane as:. Then by its similarities to complex numbers, it should be possible to express a quaternion that can be used to rotate a point in 3D-space as such:.

We can also confirm that the magnitude of the resulting vector is maintained:. However, all is not lost. Section 4. Independent of the selected method, the influence of the flexure hinge contour on the elasto-kinematic hinge properties can be generalized, especially for thin hinges.

In Figure 9 , the analytical results are exemplarily presented for a force load. The following order can be concluded from the lowest to the highest stiffness: the corner-filleted, power function, elliptical, and circular contour Figure 9a. Because the maximum strain value limits the deflection, the maximum rotation angle of a flexure hinge is always possible with a corner-filleted contour, while a circular contour leads to the lowest possible angles Figure 9c.

Furthermore, the asymmetric strain distribution due to the transverse force load is obvious, especially for a corner-filleted contour Figure 9d. Due to the notch effect, the strain is concentrated in the hinge center for a circular and elliptical contour, while the other contours lead to a more even strain distribution along the hinge length.

With regard to a high rotational precision or a small axis shift, the following order is existing for thin hinges: the circular contour, elliptical contour or power function contour to the same extend, and corner-filleted contour Figure 9b. Thus, the power function contour of the fourth order simultaneously provides a large angular deflection and a high rotational precision.

The influence of the basic hinge dimensions is further investigated in [ 33 ]. In this section, the synthesis method presented in Section 2. Therefore, a symmetric four-bar Roberts mechanism with four hinges, realizing the guidance of the coupler point P on an approximated rectilinear path cf. The rigid-body model and the compliant mechanism are shown in Figure 1 in the initial and deflected positions.

A compliant mechanism with individually shaped power function flexure hinges is synthesized according to the synthesis method based on the relative rotation angles in the rigid-body model cf. Section 2. The resulting compliant mechanism is shown in Figure 10d. Furthermore, the mechanism properties are compared with three compliant mechanisms using identical hinges designed with circular, corner-filleted, or power function contours of the fourth order see Figure 10a—c.

Following the rigid-body replacement approach, the flexure hinge centers are designed identical to the revolute joints. Next, suitable flexure hinge orientations are chosen with respect to the link orientations of the crank and the coupler cf. The exponents are exemplarily determined as even numbers, while rational exponents are also possible for a more specific design.

For the quasi-static structural and geometrically nonlinear FEM simulation of the compliant Roberts mechanisms, the same settings as for a separate hinge are used cf. The results for the motion path of the coupler point P are shown in Figure 11 for a given x -displacement in dependence of the used flexure hinge contours and additionally for the rigid-body model.

Regarding a consistent modeling, the coordinate system is defined at the fixed support in the following. Furthermore, analyzing the straight-line deviation, it becomes obvious that a more precise rectilinear motion can be realized using the compliant mechanism with individually shaped flexure hinges.

The analytical modeling of the compliant mechanisms is also based on the nonlinear theory for large deflections of rodlike structures described in Section 3. To consider the coupler point P , a branched mechanism has to be modeled, and the rod is split into three sections in K , each with its own rod axis s 1 — s 3 see Figure The straight-line deviation, the maximum strain, and the necessary deflection force are determined, too.

From investigations on separate hinges [ 49 ] and flexure hinge-based compliant mechanisms [ 50 , 51 ], it is known that the flexure hinge orientation strongly influences the elasto-kinematic properties of compliant mechanisms.

Therefore, a study of the Roberts mechanism is done, while the hinges A 0 and B 0 and A and B are modeled equally mirrored see Figure The FEM results and analytical results for the four investigated compliant Roberts mechanisms are in a very good correlation see Table 7.

Generally, all four compliant mechanisms exhibit a very small straight-line deviation in the low micrometer range. With respect to the path deviation compared to rigid-body model, the values differ from the straight-line deviations. However, as for the separate hinge cf. Figure 9b , the mechanism with circular contours provides the smallest path deviation.

With regard to the maximum admissible strain, the desired stroke cannot be realized when using identical circular or power function hinges of the fourth order cf. Figure 11b. In contrast, the full stroke is possible when using the corner-filleted hinges and, as expected, also with the synthesized mechanism with individually shaped hinges.

Furthermore, the input force varies considerably, and, thus, a required stiffness can be achieved, too. Hence, the result method independently confirms the practicability and impact of the angle-based synthesis method for different hinges in one mechanism. Moreover, the presented nonlinear analytical approach is suitable to accurately model the elasto-kinematic properties of planar flexure hinge-based compliant mechanisms under consideration of the specific hinge contour without simulations.

The influence of the scale on the deformation and motion behavior is a further relevant aspect regarding the similitude of mechanisms [ 52 ]. Based on investigations of a separate flexure hinge and a compliant parallel linkage [ 53 ], the uniform geometric scaling may also be a suitable synthesis approach for compliant mechanisms if the change ratios of the elasto-kinematic properties are known. The geometric scaling approach is exemplified for the Roberts mechanism with different power function hinges based on analytical calculations.

Therefore, different scaling factors are regarded see Figure The results are mentioned in Table 8 , while the input stroke u xP is scaled as well. Based on the results, geometric scaling is an appropriate approach for the accelerated synthesis through the adjustment of an initially designed or used compliant mechanism with known elasto-kinematic properties to each required scale of the new application through the use of the property change ratios concluded in Table 9.

Therefore, performing a new calculation is not necessary anymore. The results are independent of the hinge contour, while a nonuniform scaling is possible, too [ 53 ]. Furthermore, this approach can be used to significantly increase the stroke by increasing the mechanism size or to reduce the straight-line deviation by miniaturization because strain values and angles are independent of the scale.

Influence of uniform geometric scaling with the factor g on the change ratios of the elasto-kinematic properties of planar flexure hinge-based compliant mechanisms. Flexure hinge-based compliant mechanisms offer a high-precise and large-stroke guidance motion with straight-line or path deviations in the single-micrometer range if they are purposefully designed.

It is shown that the synthesis of a compliant mechanism with individually shaped flexure hinges based on the relative rotation angles in the rigid-body model is a suitable and general synthesis method which is easy to use without the need of numerical calculations, FEM simulations, or a multi-criterial optimization cf. Therefore, this chapter provides a survey of several approaches, guidelines, and aids for the accurate and comprehensive design of notch flexure hinges using various hinge contours, while power function contours are particularly suitable.

The use of design graphs, design equations, a computational design tool, or a geometric scaling approach is briefly presented. The results are verified by analytical calculations and FEM simulations, and also, not mentioned, by experimental investigations e. Moreover, especially the used nonlinear analytical approach has a great potential for the future work, for example, the implementation of a GUI for the compliant mechanism synthesis.

We acknowledge support for the research by the DFG Grant no. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Edited by Joseph Mizrahi. Published: September 4th, Impact of this chapter. Abstract A compliant mechanism gains its mobility fully or partially from the compliance of its elastically deformable parts rather than from conventional joints. Keywords compliant mechanism flexure hinge deformation behavior motion behavior modeling design.

Introduction A mechanism is generally understood as a constrained system of bodies designed to convert forces or motions. Table 1. Table 2. Table 3. Contour-independent closed-form design equations based on analytical modeling.

Table 4. Table 5. Table 6. Table 7. Scaling factor Stroke u xP [mm] Straight-line deviat. Table 8. Table 9. References 1. Handbook of Compliant Mechanisms. Chichester: Wiley; Zentner L. Nachgiebige Mechanismen. Large stroke ultra-precision planar stage based on compliant mechanisms with polynomial flexure hinge design. In: Proceedings of the 17th Euspen; Hannover, Germany. A large deflection and high payload flexure-based parallel manipulator for UV nanoimprint lithography: Part I.

Modeling and analyses. Precision Engineering. DOI: Compliant microgripper with parallel straight-line jaw trajectory for nanostructure manipulation. Static behavior of weighing cells. Journal of Sensors and Sensor Systems. Lobontiu N. Compliant Mechanisms: Design of Flexure Hinges. Compliant mechanism design for realizing of axial link translation. Mechanism and Machine Theory. Ltd; Flexure pivot for aerospace mechanisms. Multiobjective design optimization of flexure hinges for enhancing the performance of micro-compliant mechanisms.

Journal of the Chinese Institute of Engineers. On polynomial flexure hinges for increased deflection and an approach for simplified manufacturing. Mechanical Sciences. Howell LL, Midha A. A method for the design of compliant mechanisms with small-length flexural pivots. Journal of Mechanical Design. Topological synthesis of compliant mechanisms using multi-criteria optimization. Optimal selection of the compliant mechanism synthesis method. Motion characteristics of the compliant four-bar linkages for rectilinear guiding.

Journal of Mechanical Engineering Design. Hricko J. Straight-line mechanisms as one building element of small precise robotic devices. Applied Mechanics and Materials. Wan S, Xu Q. Design and analysis of a new compliant XY micropositioning stage based on Roberts mechanism.

Li J, Chen G. A general approach for generating kinetostatic models for planar flexure-based compliant mechanisms using matrix representation. Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy. Ilmenau: TU Ilmenau; URN: urn:nbn:de:gbv:ilm Synthesis of compliant mechanisms based on goal-oriented design guidelines for prismatic flexure hinges with polynomial contours.

Meng Q. A design method for flexure-based compliant mechanisms on the basis of stiffness and stress characteristics [doctoral thesis]. Design and simulation of a binary actuated parallel micro-manipulator. Wuest W. The modeling of cross-axis flexural pivots. Dimensionless design graphs for three types of annulus-shaped flexure hinges. Paros JM, Weisbord L. How to design flexure hinges. Machine Design. General design equations for the rotational stiffness, maximal angular deflection and rotational precision of various notch flexure hinges.

Design of single-axis flexure hinges using continuum topology optimization method. Optimization of compliant mechanisms by use of different polynomial flexure hinge contours. The influence of asymmetric flexure hinges on the axis of rotation. URN: urn:nbn:de:gbv:ilmiwk On systematic errors of two-dimensional finite element modeling of right circular planar flexure hinges.

Review of circular flexure hinge design equations and derivation of empirical formulations. Elasto-kinematic comparison of flexure hinges undergoing large displacement. A novel technique for position analysis of planar compliant mechanisms. Raatz A. Braunschweig: TU Braunschweig; Optimized flexural hinge shapes for microsystems and high-precision applications.

On Modeling the bending stiffness of thin semi-circular flexure hinges for precision applications. Tseytlin YM. Notch flexure hinges: An effective theory. The Review of Scientific Instruments. Dirksen F, Lammering R.

On mechanical properties of planar flexure hinges of compliant mechanisms. Campanile LF, Hasse A. A simple and effective solution of the elastica problem. Error analysis of a flexure hinge mechanism induced by machining imperfection. Contour-independent design equations for the calculation of the rotational properties of commonly used and polynomial flexure hinges.

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