## Abstract

This paper presents a new design concept for a morphing triangle-shaped compliant mechanism. The novel design is a bistable mechanism that has one changeable side. These morphing triangles may be arrayed to create shape-morphing structures. The mechanism design was based on a six-bar dwell mechanism that can fit in a triangle shape and has stable positions at the motion-limit (dead-center) positions. An example of the triangle-shaped compliant mechanism was designed and prototyped: an isosceles triangle with a vertex that changes from 120 deg to 90 deg and vice versa. Three of these in the 120-deg configuration lie flat and when actuated to the 90-deg configuration become a cube corner. This design may be of use for folding and packaging assistance. The mechanism was designed using geometric constraint programming. Force and potential energy analyses characterize the triangle mechanism’s stability. Because of its dead-center motion limits, the vertex angle of the triangle cannot be extended past the range of 90–120 deg, in spite of the mechanism’s compliant joints. Furthermore, because it is a dwell mechanism, the vertex angle is almost immobile near its stable configurations, although other links in the mechanism move. This makes the stable positions of the vertex angle robust against stress relaxation and manufacturing errors. We believe this is the first demonstration of this kind of robustness in bistable mechanisms.

## 1 Introduction

The objective of this research was to design a morphing triangle-shaped bistable compliant mechanism with impassible motion limits. A triangular morphing element is useful because any polygon shape can be built from a combination of triangles, and one polygon may be morphed into another by morphing its constituent triangles. One application of this is to produce polygon designs that are fabricated from a flat sheet of material and morph into their desired shape. Our morphing triangle mechanisms may have applications in folding and packaging processes [1,2] as they can be attached to a cardboard box while in its flat position and then deployed to fold it into a closed box shape. Another possible application may be portable boxes that can be deployed as cages for small animals and stored efficiently while flat. The need for morphing triangles may occur in other applications including aerospace devices [3], locking devices [4], and shape-changing structures [57].

For this paper, we developed a design concept for a bistable triangle compliant mechanism that, when three such triangles are arrayed in a circular pattern, can morph to create a cube-corner shape.

Shape-morphing structures have been investigated as morphing wings, automobile structures, and structural actuators [810]. One challenging part of shape-morphing structures is actuation. Researchers have studied many solutions for actuating shape-morphing structures, and most of them are complicated and are external to the structures [9]. Compliant mechanisms may be a viable solution because their members’ flexibility allows for appropriate mobility and actuation [1012]. Compliant mechanisms provide many advantages for shape-morphing structures as they help overcome actuation problems [5] and have simpler manufacturing processes [13]. In addition, compliant mechanisms can be manufactured as a single layer with compliant joints that function as pin joints. One way of simulating pin joints in compliant mechanisms is to use a small-length flexural pivot (living hinge), which are very short, very thin flexures that have a negligible resistance to bending so that they can be modeled as pin joint without needing a torsional spring to model their stiffness, as their stiffness is negligible [13].

Motion limits are useful in compliant mechanisms because they prevent overstress and premature fatigue failure. Dead-center motion limits are positions in which the mechanism input cannot continue moving in the same direction. In these positions, the kinematic coefficients (the instantaneous ratios of output velocity to input velocity) become infinity; the output is non-zero, and the input is zero. The dead-center positions of a crank-slider, with the slider as the input, are shown in Fig. 1. In both of the images, the slider cannot move any further in the direction of the arrow.

Fig. 1
Fig. 1
Close modal

This behavior is considered an obstacle in some industrial and machine designs. However, it is useful for other designs, such as in lock devices and motion-limit mechanisms [11,14,15].

There are many ways to find dead-center positions of mechanism [17]. When the determinant of the mechanism’s Jacobian matrix goes to zero, the mechanism is at dead-center position [18]. In addition, the graphical instant centers (IC) method has been used to define the dead-center positions of planar linkages. It makes kinematic chains and their properties, such as displacement, straightforward to analyze [19]. The instant center, I(α, β), is a location at which there is no relative velocity between the two links α and β. The kinematic coefficients for rotating links can be computed based on the locations of their instant centers as
$dθodθi=hoi=|I(i,o)−I(g,i)||I(i,o)−I(g,o)|$
(1)
where i is the input link, o is the output link, and g is the ground link, and |I1–I2| is the distance between instant centers 1 and 2.

The procedure for finding ICs (based on the kinematic pairs in a mechanism and the Aronhold-Kennedy theorem) is given in several texts on mechanism design [20]. In Fig. 2, we illustrate an example of a dead-center position occurring in a crank-slider when the instant centers in the denominator of Eq. (1), e.g., I(2, 4) and I(1, 2), become coincident [18] resulting in a motion limit. In addition to the dead-center approach, a complimentary technique for motion limits is to use a dwell mechanism (which has a dead center).

Fig. 2
Fig. 2
Close modal

Dwell mechanisms have an interesting behavior in which the output of a mechanism to become momentarily stationary while the input still changes [20]. Figure 3 shows a coupler curve path of a four-bar mechanism. The coupler link is attached to link 6 by a pin and slider. When the input is applied, the coupler point goes through the collinear motion with link 6 for two different ranges of motion. In these regions, link 6 does not rotate; i.e., it dwells at a certain angle.

Fig. 3
Fig. 3
Close modal

In addition to dead-center motion limits and dwell mechansims, shape-morphing arrays are benefitted by bistability because the distinct array configurations may be stable, i.e., the distinct shapes may be held without actuation. A mechanism is called “bistable,” when the mechansim has two distinct configurations which are local minimums of potential energy. One of the best ways to describe bistability is using the “ball-on-a-hill” analogy [13], which shows the analogy between the potential energy (strain energy) of a compliant mechanism and the potential energy (gravitational) of a ball-on-a-hill. In Fig. 4, a ball on an uneven surface is shown. The ball is at equilibrium states at positions A, B, and C. At positions A and C, the ball is at a local minimum of potential energy and will oscillate about that minimum, if it is perturbed slightly from the positions shown. At position B, the ball is at a maximum of potential energy, and if disturbed, it will not return to its original position but will move to one of the stable positions.

Fig. 4
Fig. 4
Close modal

## 2 Polyhedral Surfaces With Triangle-Shaped Mechanisms

Any polyhedral surface can be defined geometrically as an assemblage of triangles. For example, when three triangles of a configuration of (90–45–45 deg angles) are joined to each other by the sides adjacent to the right angle, they form a corner of a cube. If the triangles have one changeable side, the assembly of the three triangles can be designed to lay flat (120–30–30 deg angles) using a shape-morphing triangle, as shown in Fig. 5.

Fig. 5
Fig. 5
Close modal

A triangle-shaped mechanism can be designed using a one-degree-of-freedom (DOF) mechanism (i.e., a single-loop four-bar, a two-loop six-bar, etc.). Four-bar mechanisms have limits in the number of dead-centers that can be used for stability (discussed in Sec. 3); moreover, the definition of dead-center in the four-bar mechanisms is when the transmission angle (the angle between the two-unactuated links) becomes either zero or 180 deg [21]. These limitations make four-bar designs behave somewhat like a hard stop. [22]. Thus, we chose to investigate six-bar one-DOF mechanisms. Six-bar mechanisms can have multiple dead-center positions [19], which allowed flexibility in satisfying our design objectives. We chose Stephenson’s chain for the six-bar triangle mechanism, which has two ternary links, two binary links, and one slider link as shown in Fig. 6(a). Figure 6(b) shows that using symmetry, the six-bar becomes a shape-morphing triangle, where link 2 and its reflection are the constant sides of the triangle and the changeable side is the virtual distance between the joint between link 2 and link 6, I(2,6), and its reflection.

Fig. 6
Fig. 6
Close modal
The shape-morphing triangle shown in Fig. 5 may be adapted to a variety of shape-morphing tasks. Because the triangle’s angles can change by a specified amount, different curved surfaces may be approximated. A plane may be tessellated with triangles, and when the triangles morph, the change in angles can result in a curved surface. If a point in a tessellation is the meeting point of n triangles, a measure of curvature at that point is given by Refs. [2325]:
$Curvaturemeasure(C)=[360deg−∑i=1n(θi)]$
(2)
Where θi refers to the included angles, where the n triangles join. The result of the equation can be either C = 0 (planar), C > 0 (spherical), or C < 0 (hyperbolic). Our cube-corner mechanism has C = 90 deg and is part of the spherical category. Because the triangle mechanism uses symmetry, the first and second configurations of the mechanism are when θ2 = 60 deg and θ2 = 45 deg, respectively.

## 3 A Dead-Center’s Utility in Compliant Mechanisms

Because bistability requires two local minimums of potential energy, the well-known Kuhn–Tucker criterion [26] for a local minimum applies: local minimums occur where a derivative equals zero or at the boundary of the independent variable. The motion limits provided by dead-centers create the possibility that the motion limits could be minimums of potential energy. This is distinctly different from most compliant bistable mechanisms, which have potential energy that tends to a maximum as the motion goes toward the extremes (i.e., motion variables are theoretically unbounded, but stresses and actuation loads become large at either motion extremes.)

To analytically study this behavior, we apply the principle of virtual work [13,27] to the triangle-shaped six-bar mechanism. The force is applied vertically on the joint between link 3 and link 4 and a small-length flexural pivot (with stiffness k6) at the joint between links 2 and 6 as shown in Fig. 7. We assume the stiffness associated with all other joints (living hinges) to be negligible. The force equation is found to be (see Appendix  A for the derivation):
$F=−k6[(θ2−θ20)−(θ6−θ60)]h62r2cos(θ2)−r3cos(θ3)h32$
(3)
where h32 = 3/2 and h62 = 6/2 are the kinematic coefficients and k6 is the torsional spring constant. All the angles and link lengths are shown in Fig. 7. The angles θ20 and θ60 are the undeflected angles of the compliant joints, and at the fabricated position, θ2 = θ20 = 60 deg, and θ6 = θ60, and the numerator and hence the force vanishes. The typical approach to bistability involves designs in which multiple configurations occur at which the spring angles are at their undeflected angles (resulting in equilibrium). Our approach is different, we use dead-centers to make the kinematic coefficient h32 infinite at the boundaries of motion, which always produces an equilibrium configuration and, in this case, produces a local potential energy minimums, i.e., stable points.
Fig. 7
Fig. 7
Close modal

The required condition on the kinematic coefficient, h32, is produced by applying constraints on the relative positions of the instant centers as described in Sec. 4.

## 4 Designing Dead-Center Motion Limits Using the Instant Centers Method

In this section of the paper, the dead-center position for first and second required configurations is found by using the instant center method, and the dimension synthesis of the six-bar mechanism is described. For a six-bar mechanism, the number of ICs, N, is calculated based on the equation N = n(n − 1)/2, where n is the number of links, yielding 15 ICs, as shown in Fig. 8.

Fig. 8
Fig. 8
Close modal
Because the IC of any two bodies in a planar mechanism is the point where those two bodies have the same velocity at a given instant in time, we can obtain the kinematic coefficients based on the ICs, considering θ2 as the input, and using Eq. (1)
$dθ3dθ2=h32=|I(2,3)−I(1,2)||I(2,3)−I(1,3)|$
(4)
$dθ4dθ2=h42=|I(2,4)−I(1,2)||I(2,4)−I(1,4)|$
(5)
$dθ6dθ2=h62=|I(2,6)−I(1,2)||I(2,6)−I(1,6)|$
(6)
$dr1dθ2=h12=|I(2,5)−I(1,2)|$
(7)
when the denominator in h32, h42, and h62 goes to zero (the two ICs are coincident), the kinematic coefficients become infinity, which indicates a dead-center configuration.

To apply this process on the mechanism, we used parametric computer-aided design (cad) software that allowed us to draw the mechanism in the two required configurations and to construct the locations of their instant centers using geometric constraint programming [28,29]. Although the desired topology of the mechanism and the values of the input angle were known, the specific lengths and ratios of the lengths of the links were not known. To determine these ratios, the skeleton of the mechanism was drawn in its two required (dead-center) configurations. The drawings of the two required configurations are constrained to have equal link lengths (a necessary, if obvious step). The input angle, θ2, in the first configuration was set to 45 deg and 60 deg for the second configuration. Then, the pairs of ICs I(2, 3), I(1, 3) and I(2, 4), I(1, 4) are constrained to be, respectively, coincident for both configurations. The configurations after applying the {I(2, 3),I(1, 3)} coincidence constraints and the {I(1, 4),(2, 4)} coincidence constraints are shown in Fig. 9, which shows the six-bar mechanism at the different dead-center configurations 45 deg and 60 deg.

Fig. 9
Fig. 9
Close modal

The coincidence constraints guarantee that the two required configurations are dead-center positions for the mechanism. The topology, the input angle constraints, and the dead-center constraints are necessary but are not sufficient to specify the entire mechanism design. The overall size of the mechanism may be varied, and there is some leeway in the lengths of r3 and r4, but these are limited to a small range of lengths or the links may interfere; interference would make the mechanism not apt for fabrication as a single-layer-monolithic compliant mechanism. We chose to constrain the mechanism so that no interference occurs by setting r2= r5, which allows more room inside the vector loops for the links to move. All the chosen design parameters are shown in Table 1.

Table 1

The six-bar mechanism’s parameters, ξ and γ are constant angles, as shown in Fig. 7

 r2 6.4 mm r8 5.3 mm r3 4.5 mm γ 25.66 deg r4 8.6 mm ξ 31.14 deg r5 6.4 mm I 0.454 mm4 r6 6.6 mm E 999.7 MPa r7 4.4 mm l 1 mm
 r2 6.4 mm r8 5.3 mm r3 4.5 mm γ 25.66 deg r4 8.6 mm ξ 31.14 deg r5 6.4 mm I 0.454 mm4 r6 6.6 mm E 999.7 MPa r7 4.4 mm l 1 mm

## 5 Kinematic Analysis

The next two sections of the paper give a numerical analysis of the six-bar mechanism and provide a numerical perspective on the dead-center positions that are found from the IC method. Furthermore, because the geometric constraint method, used in Sec. 4, only considers the end points of the motion, this section describes the behavior of the mechanism’s kinematic coefficients between those endpoints. The behaviors of the kinematic coefficients are interesting because they are the main sources of nonlinearity (e.g., bistability) in compliant mechanisms. The parameters that are used for the position analysis and the kinematic coefficient calculations are given in Table 1. Figure 7 shows the parameters and angles required for the calculations. The length δ is a virtual parameter that we used as the input to simplify the position analysis, which ranges from 9 mm to 10.7 mm; its change between the two dead-centers. Appendix  B gives the loop-closure position equations. The kinematic coefficients are calculated based on the vector loops in Appendix  C as
$dθ4dθ2=h42=r5sin(θ2−θ3)−(r5+r2)(tanθ6cosθ3−sinθ3)sinθ2r7sin(θ7−θ3)+r8(tanθ6*cosθ3−sinθ3)cosθ8$
(8)
$dθ3dθ2=h32=−r2cosθ2+r4h42cosθ2r3cosθ6$
(9)
$dθ6dθ2=h62=(r2+r5)cosθ2−r8h42cosθ8r6cosθ6$
(10)
$dr1dθ2=h12=−r2sinθ2−r3h32sinθ3−r4h42sinθ4$
(11)

Because the kinematic coefficients are expected to have infinite values, and this would be inconvenient to plot, the reciprocal of the kinematic coefficients (except h62 and h12) are shown and have, as expected, values of zero at the dead-centers, as shown in Fig. 10.

Fig. 10
Fig. 10
Close modal

The kinematic coefficients h24 and h23 are equal to zero when θ2 is at 45 deg and 60 deg; in other words, the mechanical advantage (the ability of link 4 and link 3 to affect link 2) of the mechanism when θ2 is 45 deg or 60 deg is equal to zero.

## 6 Force and Potential Energy Analysis

Using the principle of virtual work, as described in Eq. (3), the force analysis was performed on the six-bar compliant mechanism with the parameters given in Table 1. All the mechanism’s joints (except joint A in Fig. 7) are designed to be thin enough (living hinges) that they do not have a significant impact on the mechanism’s stiffness. The characteristic stiffness of joint A is referred to as k6 and is given by
$k6=EIl$
(12)
where E is the modulus of elasticity, I is the cross section’s second moment of area, and l is the joint’s length. The force-input angle diagram is shown in Fig. 11. It can be seen that the force goes to zero at the first position and at the second position which indicates the required equilibrium states at these positions. The force also becomes zero at 48 deg, which an unstable equilibrium position.
Fig. 11
Fig. 11
Close modal

The potential energy associated with the mechanism is calculated and plotted in Fig. 11. The potential energy curve has two distinct local minimums (stable configurations) at the 60 deg and 45 deg configurations.

At 48 deg, the mechanism is at a maximum of potential energy and, if disturbed, will move to one of the local minimum configurations. This maximum is the dividing line between the “basins of attraction” of the two stable equilibrium configurations; i.e., it is the boundary between the mechanism returning to its first stable equilibrium configuration and returning to its second. The fact that the 45 deg equilibrium point is nearer to the unstable equilibrium point in position and energy, as shown in Fig. 13, shows that the cube-corner configuration is less stable than the planar position.

## 7 Design Robustness at Bistable Positions

The ability of other bistable mechanisms to move past their equilibrium point can be a problem in shape-morphing structures when rigidity is desired at that configuration. The bistable triangle-shaped compliant mechanism is designed, using dead-center motion limits, so that it does not allow significant motion beyond its designed stable configurations. The 45 deg dead-center motion limit occurs when link 4 becomes parallel with link 1 (the mirror symmetry line), and the 60 deg motion limit occurs when links 4 and 6 contact/become parallel.

Furthermore, the bistable triangle-shaped compliant mechanism has dwell behavior in the vertex angle θ2 near the bistable configurations. Near the bistable configurations, the vertex angle θ2 does not change even when the rest of the mechanism moves due to the dwell-based design. This allows compliant mechanism prototypes to be made from viscoelastic materials such as polypropylene that exhibit creep and plastic deformation in their links and joints (which may result from large stresses in compliant mechanism) and, in spite of those deformations, maintain the designed stable configurations. To better show how the dwell aspect of the mechanism works, we show (in Fig. 12) the inversion in which r5 is the ground link and r1 is a moving link. In this inversion, the pin on the slider link 5 moves as the coupler point in a four-bar mechanism (links 3, 4, 5, and 6). Near the stable configurations, the slider pin moves radially (changing the length of r1) but not tangentially (no change in θ2). The fact that θ2 (labeled 60 deg in the first configuration and 45 deg in the second configuration in Fig. 12) does not change for small motions near the equilibrium points is what gives the mechanism its robustness.

Fig. 12
Fig. 12
Close modal

This lack of motion produces dwell behavior in θ2 (the vertex of the triangle) and means that even if the stable configurations change a little due to creep or plastic deformation, it is unlikely to significantly alter to values of θ2 at the stable configurations.

The dwell mechanism enhances the stability of the design because the stable configurations do not have to be at a particular mechanism configuration, and they can fall within a range of configurations that have the same vertex angle.

To further illustrate this behavior, we plot the position variables of the mechanism with θ2 is the output and input r1 as shown in Fig. 13. These plots show that near the equilibrium configurations of 45 deg and 60 deg, motion occurs, yet the vertex angle θ2 remains very close to the stable positions of the vertex angle.

Fig. 13
Fig. 13
Close modal

Figure 16 shows that the vertex angle θ2 is limited between 45 deg and 60 deg with the various inputs meaning that the mechanism’s vertex angle θ2 is locked between these limits. Figures 14(a) and 14(b) (near θ2 = 45 deg) and then Figs. 14(f) and 14(g) (near θ2 = 60 deg) show how portions of the mechanism can move while the vertex angle θ2 is nearly constant.

Fig. 14
Fig. 14
Close modal

## 8 Design Prototypes

In this section of the paper, two examples are prototyped to illustrate the use of the bistable triangle-shaped compliant mechanism in shape-morphing mechanisms. The first design is a cube corner that may be used for packaging assistance. As mentioned in Sec. 2, the cube corner consists of three triangular units of configuration (120–30–30 deg) that transform to cube-corner configuration (90–45–45 deg). Our design was chosen to have r2= r5 = 76 mm. All parameters of the design were derived from Sec. 2 and parametric CAD and are given in Table 2. Figure 15 shows the prototype of the bistable triangle-shaped compliant mechanism in both configurations. The prototype was laser-cut from a 1/8-in.-thick polypropylene co-polymer material, as shown in Fig. 15. The pin joints in the rigid-body mechanism (e.g., Fig. 5) are replaced with living hinges, except for joint A in Fig. 7, which is considerably thicker. The specific widths of these flexures can vary somewhat without impacting the kinematic behavior of the mechanism, provided that the portions representing rigid links are sufficiently rigid and those representing joints are sufficiently flexible.

Fig. 15
Fig. 15
Close modal
Table 2

The prototype’s parameters

 r2 38 mm r8 33 mm r3 28.5 mm γ 25.66 deg r4 50 mm ξ 31.14 deg r5 38 mm I 1.14 μm4 r6 40.5 mm E 999.7 Mpa r7 28 mm l 3 mm
 r2 38 mm r8 33 mm r3 28.5 mm γ 25.66 deg r4 50 mm ξ 31.14 deg r5 38 mm I 1.14 μm4 r6 40.5 mm E 999.7 Mpa r7 28 mm l 3 mm

Because the prototype was designed to be fabricated from a flat sheet, the initial configuration was 120–30–30 deg. The three units were connected using groove joints [30] that produce a constrained tight fold as shown in Fig. 16. Figure 17 shows the cube corner glued to wood sheets in (a) a flat position and (b) folded to the package shape.

Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal

The second example is a portable box that can be flattened for easy storage and deployed as a closed box as needed. The box consisted of six bistable triangle-shaped compliant mechanisms as shown in Fig. 20. Figure 21 shows the deployment process of the portable box. All parameters of the design were derived from Sec. 2 and parametric CAD in the following Table 2.

The prototype was designed to be fabricated from a flat sheet, and it was laser-cut from a 1/8-in.-thick polypropylene co-polymer material as shown in Fig. 18. The box was designed of two corner units that were connected with two links and fold opposite to each other as shown in Fig. 19. Each corner unit is actuated independently, and a latch mechanism is used to lock the two stable corners into the box shape. Groove joints were used to help guide the folding pattern. Figure 18 shows the portable box in a flat position and folded to the closed box as shown in Fig. 20.

Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal
Fig. 20
Fig. 20
Close modal

## 9 Conclusion

This paper introduces a new type of bistable mechanism that utilizes dead-center motion limits and dwell-mechanism stability enhancement. The mechanism is designed in a triangle shape with one side changeable. It can be tessellated for shape-morphing compliant mechanisms. The mechanism has a dwell behavior at the bistable configurations that enhances stability. Both the force analysis and potential energy were calculated to elucidate the stability at the designed configurations. The bistable triangle-shaped compliant mechanism was prototyped for two tasks: a cube corner and a portable box, which demonstrate the concept.

## Acknowledgment

The authors of this work gratefully acknowledge the support of the National Science Foundation (Grant No. CMMI-1053956).

### Appendix A

The generalized coordinate (input) is chosen to be θ2. The applied force is vertical on the joint between link 3 and link 4
$F=FinJ^$
(A1)
The vector from the origin to placement to the applied force is
$Z=r2sin(θ2)−r3sin(θ3)j^$
(A2)
The virtual displacement is found by differentiating the position vector as
$δZ=dZdθ2δθ2=(r2cos(θ2)−r3cos(θ3)*h32)j^$
(A3)
The virtual work due to the force is given by
$δW=F⋅δZ$
$δW=−Fin(r2cos(θ2)−r3cos(θ3)h32)δθ2$
(A4)
The virtual work due to the potential energy of the torsional spring in link 6 is given by
$δW=−k6[(θ2−θ20)−(θ6−θ60)h62]δθ2$
(A5)
By summing the virtual work terms and applying the principle of virtual work (δW = 0)
$F=−k6[(θ2−θ20)−(θ6−θ60)]h62r2cos(θ2)−r3cos(θ3)h32$
(A6)

### Appendix B

The position analysis for the triangle-shaped mechanism was carried out using closed-form equations. The input was the virtual distance δ. The law of cosines was used to calculate the internal angles from Fig. 21
$φ=cos−1(r32+r22−δ22r2r3)$
(B1)
$ω=π−φ$
(B2)
$θ2=μ+∅$
(B3)
$δ1=(r52+r32−2r3r5cosω)1/2$
(B4)
$β=cos−1(δ12+r32−r522δ1r3)$
(B5)
$α=cos−1(r72+δ12−r622r7δ1)$
(B6)
$η=cos−1(r32+δ2−r222r3δ2)$
(B7)
$ϑ=2π−ξ−η−β−α$
(B8)
$μ=cos−1(r22+δ2−r322r2δ)$
(B9)
$r1=(δ2+r42−2δr4cosϑ)1/2$
(B10)
$∅=cos−1(r12+δ2−r422r1δ)$
(B11)
$Ψ=cos−1(r62+δ12−r722r6δ1)$
(B12)
$θ2=μ+∅$
(B13)
$θ3=2π+θ2−ω$
(B14)
$θ4=θ2+φ+ϑ+η$
(B15)
$θ6=2π+θ2−ω−β+Ψ$
(B16)
Fig. 21
Fig. 21
Close modal

### Appendix C

The kinematic coefficients are calculated from the vector loops shown in Fig. 7. The complex number method may be used for the kinematic analysis as
$R2+R3+R4=R1$
$r2eiθ2+r3eiθ3+r4eiθ4=r1eiθ1$
(C1)
$R4+R5+R6+R8=R1$
$r4eiθ4+r5eiθ2+r6eiθ6+r8eiθ8=r1eiθ1$
(C2)
By taking the derivative of each equation with respect to θ2, and writing derivatives as kinematic coefficients, o/i = hoi. The kinematic coefficients can be found as
$h42=r5sin(θ2−θ3)−(r5+r2)(tanθ6*cosθ3−sinθ3)sinθ2r7sin(θ7−θ3)+r8(tanθ6cosθ3−sinθ3)cosθ8$
(C3)
$h32=−r2cosθ2+r4h42cosθ2r3cos(θ2)$
(C4)
$h62=(r2+r5)cos(θ2)−r8h42cos(θ8)r6cos(θ6)$
(C5)
$h12=−r2sin(θ2)−r3h32sin(θ3)−r4h42*sin(θ4)$
(C6)

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