Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity of Planar Linkages

Baokun Li and Guangbo Hao

## Abstract

The theory of nonlinear stiffness characteristic by employing the kinematic limb-singularity of planar mechanisms with attached springs is proposed. After constructing the position formula with closed-loop form of the mechanism, the kinematic limb-singularity can be identified. The kinetostatic model can be obtained based on the principle of virtual work. The influences of spring stiffness on the force-displacement or torque-angle curve are analysed. Different spring stiffness results in one of four types of stiffness characteristic, which can be used to design an expected stiffness characteristic. After replacing corresponding joints with flexures, the pseudo-rigid-body model of the linkage with springs is obtained. The compliant mechanisms with nonlinear stiffness characteristic can further be synthesised based on the pseudo-rigid-body model.

Keywords: kinematic singularity, nonlinear stiffness, kinetostatic model, planar linkage with springs, compliant mechanism

### 1. Introduction

A planar linkage always arrives at some several special positions, which may decrease the stability, disable the motion ability or change the degree of freedom of the linkage. These special positions are called kinematic singular configuration or kinematic singularity. It is one of intrinsic properties of the linkage [1].

Kinematic singularity attracted many scholars' attention since it affects the performance of the linkage. Kinematic singularity classification, singularity identification and singularity property, with a particular emphasis on eliminating or avoiding the singularities, are discussed [2–8].

However, everything has two sides. Kinematic singularity of linkages can also be applied to create new useful devices, such as fixture based on the dead-point singularity. In recent years, some compliant mechanisms with new performance are constructed using the kinematic singularity [9–12].

For a generic planar linkage, kinematic limb-singularity and kinematic actuation-singularity may often be exhibited. In this chapter, we mainly introduce how to use the kinematic limb-singularity of the linkage with placing springs at corresponding joints to generate the kinetostatic nonlinear stiffness, which can also be used to synthesise the nonlinear stiffness compliant mechanisms using the pseudo-rigid-body model (PRBM) [13, 14].

### 2. Position analysis and kinematic singularity identification

Position analysis is the base of the kinematic singularity determination and singularity classification. Many approaches can be used to carry out the position analysis. Graphical methods often provide a fast and efficient means of analysing mechanisms. Analytical methods are currently more common because of the ease in which they are programmed. Here we mainly use the closed-form solution to present the position analysis of the planar linkage followed by the kinematic singularity identification.

By a generic planar double-slider linkage as an example, the position analysis using the closed-form equation and the kinematic singularity identification are introduced.

Consider a planar double-slider linkage with given structure parameters as shown in Figure 1.

The right-hand rule fixed frame O-XYZ is attached on the base, where the intersection point of two paths of points A and B is set as the origin O. We suppose that the moving direction of the input slider is the negative of X-axis. The rotation angle from the negative moving direction of point A to the initial moving direction of point B is defined by α. Line AC is the vertical line from point A to line OB, where point C is the foot. The position vectors of points A and B with respect to the fixed frame O-XYZ are defined by r<sup>A</sup> and rB, respectively. The position vector from point A to point B is defined by rAB. Thus the closed-loop vector equation of the linkage as shown in Figure 1 can be obtained as

$$
\mathbf{r\_A} + \mathbf{r\_{AB}} = \mathbf{r\_B} \tag{1}
$$

Considering the symmetry, the case of α > 180° can be treated as the case of

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>A</sup> sin <sup>2</sup>α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where Eq. (5) exists when the output slider is located at the right side of point C and moves right. With considering the symmetry, here we only discuss the case that the output slider is located at the left side of point C and moves left, which is

The initial distance between origin O and point B corresponding to Eq. (4) is

A0 sin <sup>2</sup>α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼ � <sup>r</sup><sup>A</sup> sin <sup>2</sup><sup>α</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 AB � r<sup>2</sup>

> dr<sup>B</sup> dr<sup>A</sup>

dr<sup>B</sup> dr<sup>A</sup>

It shows that the instant velocity ratio between the output and the input is equal to infinity, which is the kinematic actuation-singularity. Here we are limited to use the kinematic limb-singularity to design a mechanism with nonlinear stiffness by placing appropriate springs at corresponding joints. Therefore, the coordinate of the

In order to pass the kinematic limb-singularity position, the initial coordinate/

It indicates that if AB⊥OA, then the linkage is in kinematic limb-singularity which occurs when the instant velocity ratio between the output and the input is

<sup>A</sup> sin <sup>2</sup>α

0<α< 180<sup>∘</sup> (3)

þ r<sup>A</sup> cos α (4)

þ r<sup>A</sup> cos α (5)

þ rA0 cos α (6)

<sup>A</sup> sin <sup>2</sup><sup>α</sup> <sup>p</sup> <sup>þ</sup> cos <sup>α</sup> (7)

r<sup>A</sup> ¼ rAB= tan α (8)

¼ 0 (9)

¼ ∞ (10)

�rAB= sin α <r<sup>A</sup> <rAB= sin α (11)

rA0 >rAB= tan α (12)

α < 180°. Therefore, α is set to satisfy the following condition:

The solution of Eq. (2) for r<sup>B</sup> with eliminating θ<sup>A</sup> yields

q

r2 AB � <sup>r</sup><sup>2</sup>

> r2 <sup>A</sup> � <sup>r</sup><sup>2</sup>

q

r2 AB � r<sup>2</sup>

q

r<sup>B</sup> ¼

DOI: http://dx.doi.org/10.5772/intechopen.85009

or r<sup>B</sup> ¼ �

rB0 ¼

dr<sup>B</sup> dr<sup>A</sup>

If r<sup>A</sup> = �rAB/sinα, which occurs when AB⊥OB, then

input slider should satisfy the kinematic constraint as follows:

position of the input slider, rA0, should satisfy

According to Eq. (4), we can further obtain

described by Eq. (4).

Eq. (7) shows that if

then

equal to zero.

67

The X-axis coordinate of point A is defined by rA, the rotation angle from vector r<sup>A</sup> to vector r<sup>B</sup> is defined by θA, the distance between point B and origin O is defined by scalar rB, and the length of the coupler AB is defined by scalar rAB. Thus, based on Eq. (1), two algebraic equations can be transformed as follows:

$$\begin{cases} r\_{\rm A} + r\_{\rm AB} \cos \theta\_{\rm A} = r\_{\rm B} \cos a \\ r\_{\rm AB} \sin \theta\_{\rm A} = r\_{\rm B} \sin a \end{cases} \tag{2}$$

Figure 1. Double-slider linkage.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

Considering the symmetry, the case of α > 180° can be treated as the case of α < 180°. Therefore, α is set to satisfy the following condition:

$$0 < a < 180^{\circ} \tag{3}$$

The solution of Eq. (2) for r<sup>B</sup> with eliminating θ<sup>A</sup> yields

$$r\_{\rm B} = \sqrt{r\_{\rm AB}^2 - r\_{\rm A}^2 \sin^2 a} + r\_{\rm A} \cos a \tag{4}$$

$$\text{or } r\_{\text{B}} = -\sqrt{r\_{\text{A}}^2 - r\_{\text{A}}^2 \sin^2 a} + r\_{\text{A}} \cos a \tag{5}$$

where Eq. (5) exists when the output slider is located at the right side of point C and moves right. With considering the symmetry, here we only discuss the case that the output slider is located at the left side of point C and moves left, which is described by Eq. (4).

The initial distance between origin O and point B corresponding to Eq. (4) is

$$r\_{\rm B0} = \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2 \sin^2 a} + r\_{\rm A0} \cos a \tag{6}$$

According to Eq. (4), we can further obtain

$$\frac{dr\_\text{B}}{dr\_\text{A}} = -\frac{r\_\text{A}\sin^2a}{\sqrt{r\_\text{AB}^2 - r\_\text{A}^2\sin^2a}} + \cos a \tag{7}$$

Eq. (7) shows that if

$$r\_{\rm A} = r\_{\rm AB} / \tan a \tag{8}$$

then

be used to synthesise the nonlinear stiffness compliant mechanisms using the

Position analysis is the base of the kinematic singularity determination and singularity classification. Many approaches can be used to carry out the position analysis. Graphical methods often provide a fast and efficient means of analysing mechanisms. Analytical methods are currently more common because of the ease in which they are programmed. Here we mainly use the closed-form solution to present the position analysis of the planar linkage followed by the kinematic singu-

By a generic planar double-slider linkage as an example, the position analysis using the closed-form equation and the kinematic singularity identification are

Consider a planar double-slider linkage with given structure parameters as

The right-hand rule fixed frame O-XYZ is attached on the base, where the intersection point of two paths of points A and B is set as the origin O. We suppose that the moving direction of the input slider is the negative of X-axis. The rotation angle from the negative moving direction of point A to the initial moving direction of point B is defined by α. Line AC is the vertical line from point A to line OB, where point C is the foot. The position vectors of points A and B with respect to the fixed frame O-XYZ are defined by r<sup>A</sup> and rB, respectively. The position vector from point A to point B is defined by rAB. Thus the closed-loop vector equation of the linkage as

The X-axis coordinate of point A is defined by rA, the rotation angle from vector r<sup>A</sup> to vector r<sup>B</sup> is defined by θA, the distance between point B and origin O is defined by scalar rB, and the length of the coupler AB is defined by scalar rAB. Thus, based

> r<sup>A</sup> þ rAB cos θ<sup>A</sup> ¼ r<sup>B</sup> cos α rAB sin θ<sup>A</sup> ¼ r<sup>B</sup> sin α

on Eq. (1), two algebraic equations can be transformed as follows:

r<sup>A</sup> þ rAB ¼ r<sup>B</sup> (1)

(2)

2. Position analysis and kinematic singularity identification

pseudo-rigid-body model (PRBM) [13, 14].

Kinematics - Analysis and Applications

larity identification.

shown in Figure 1.

shown in Figure 1 can be obtained as

introduced.

Figure 1.

66

Double-slider linkage.

$$\frac{dr\_\mathrm{B}}{dr\_\mathrm{A}} = \mathbf{0} \tag{9}$$

It indicates that if AB⊥OA, then the linkage is in kinematic limb-singularity which occurs when the instant velocity ratio between the output and the input is equal to zero.

If r<sup>A</sup> = �rAB/sinα, which occurs when AB⊥OB, then

$$\frac{dr\_\mathrm{B}}{dr\_\mathrm{A}} = \infty \tag{10}$$

It shows that the instant velocity ratio between the output and the input is equal to infinity, which is the kinematic actuation-singularity. Here we are limited to use the kinematic limb-singularity to design a mechanism with nonlinear stiffness by placing appropriate springs at corresponding joints. Therefore, the coordinate of the input slider should satisfy the kinematic constraint as follows:

$$-r\_{\rm AB} / \sin a < r\_{\rm A} < r\_{\rm AB} / \sin a \tag{11}$$

In order to pass the kinematic limb-singularity position, the initial coordinate/ position of the input slider, rA0, should satisfy

$$r\_{\rm A0} > r\_{\rm AB} / \tan a \tag{12}$$

#### 3. Force equilibrium equation

We suppose that each of the two prismatic joints is attached a translational spring and each of the two rotational joints is attached a torsional spring. The planar double-slider linkage with springs is obtained as shown in Figure 2. The stiffnesses of the translational springs placed at prismatic joints A and B are defined by kPA and kPB, respectively, and the stiffnesses of the translational springs placed at prismatic joints A and B are defined by kPA and kPB, respectively.

Substituting Eq. (16) into Eqs. (14) and (15), the potential energy, U, and the

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

rAB

rAB

� � q

From Eq. (18), we can predict that the variation of driving force exerted on the input slider versus input displacement, i.e., Fd-S curve, would present nonlinear

It is evident that the springs placed at joints are the cause of nonlinear stiffness characteristic generation, so it is necessary to discuss the influence of spring stiffness on the Fd-S curve characteristic. When the influence of one specific spring

Substituting kPA = 0 and kRA = kRB = 0 into Eqs. (17) and (18), the following can

� � q <sup>2</sup>

� � q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 CA:

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup>

� rA0 cos α �

� rA0 cos α �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 CA:

4. Cause of nonlinear stiffness characteristic generation

stiffness is analysed, every other spring stiffness is set to zero.

r2

r2

q

q

4.1 Influences of translational spring stiffness placed at output slider

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup>

<sup>r</sup>AB � �<sup>2</sup>

� �<sup>2</sup>

þ ð Þ rA0 � S cos α �

� arccosrA0 sin <sup>α</sup>

q

� arccosrA0 sin <sup>α</sup> <sup>r</sup>AB � � sin <sup>α</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� rA0 cos α �

r2 AB � <sup>r</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>A</sup><sup>0</sup> sin <sup>2</sup>α

r2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A0 sin <sup>2</sup>α

q

r2 AB � r<sup>2</sup> � rA0 cos α

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup>

:

(17)

(18)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A0 sin <sup>2</sup>α

(19)

A0 sin <sup>2</sup>α

(20)

r2 AB � r<sup>2</sup>

> r2 AB � r<sup>2</sup>

driving force, Fd, with respect to S can be obtained as

DOI: http://dx.doi.org/10.5772/intechopen.85009

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup>

dS <sup>¼</sup> <sup>k</sup>PA<sup>S</sup> <sup>þ</sup> ð Þ <sup>k</sup>RA <sup>þ</sup> <sup>k</sup>RB arccosð Þ <sup>r</sup>A0 � <sup>S</sup> sin <sup>α</sup>

r2

q

ð Þ <sup>k</sup>RA <sup>þ</sup> <sup>k</sup>RB arccosð Þ <sup>r</sup>A0 � <sup>S</sup> sin <sup>α</sup>

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2

> þ 1 2 kPB

<sup>F</sup><sup>d</sup> <sup>¼</sup> dU

<sup>k</sup>PAS<sup>2</sup> <sup>þ</sup>

1 2

r2

þ kPB ð Þ rA0 � S cos α þ

r2

stiffness characteristic.

be obtained, respectively:

kPB ð Þ rA0 � S cos α þ

dS <sup>¼</sup> <sup>k</sup>PB ð Þ <sup>r</sup>A0 � <sup>S</sup> cos <sup>α</sup> <sup>þ</sup>

� ð Þ <sup>r</sup>A0 � <sup>S</sup> sin <sup>2</sup><sup>α</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>L</sup><sup>2</sup> � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup> <sup>q</sup> � cos <sup>α</sup>

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>F</sup><sup>d</sup> <sup>¼</sup> dU

69

0 B@

0 B@

� ð Þ <sup>r</sup>A0 � <sup>S</sup> sin <sup>2</sup><sup>α</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup> <sup>q</sup> � cos <sup>α</sup>

q

The potential energy of the whole mechanism as shown in Figure 2 can be derived as

$$U = \frac{1}{2}k\_{\rm PA}(r\_{\rm A} - r\_{\rm A0})^2 + \frac{1}{2}(k\_{\rm RA} + k\_{\rm RB})(\theta\_{\rm A} - \theta\_{\rm A0})^2 + \frac{1}{2}k\_{\rm PB}(r\_{\rm B} - r\_{\rm B0})^2 \tag{13}$$

where θA0 is the initial angle between positive direction of X-axis and coupler AB. After differentiating θA, θA0, r<sup>B</sup> and rB0, which can be derived from the geometry of the linkage as shown in Figure 1, with respect to rA, and substituting them into Eq. (13), the following can be further derived:

$$\begin{split} U &= \frac{1}{2}k\_{\rm PA}(r\_{\rm A} - r\_{\rm A0})^2 + \frac{1}{2}(k\_{\rm RA} + k\_{\rm RB})\left(\arccos\frac{r\_{\rm A}\sin a}{r\_{\rm AB}} - \arccos\frac{r\_{\rm A0}\sin a}{r\_{\rm AB}}\right)^2 \\ &+ \frac{1}{2}k\_{\rm PB}\left(\sqrt{r\_{\rm AB}^2 - r\_{\rm A}^2\sin^2 a} + r\_{\rm A}\cos a - \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2\sin^2 a} - r\_{\rm A0}\cos a\right)^2. \end{split} \tag{14}$$

The principle of virtual work, which does not need to determine the inner forces between two connected links, is a simplified useful method to construct the force equilibrium equation. According to the principle of virtual work, the required driving force, Fd, applied on the input slider can be obtained as

$$F\_{\rm d} = \frac{dU}{dr\_{\rm A}} = -k\_{\rm PA}(r\_{\rm A} - r\_{\rm A0}) + (k\_{\rm RA} + k\_{\rm RB}) \left( \arccos \frac{r\_{\rm A} \sin a}{r\_{\rm AB}} - \arccos \frac{r\_{\rm A0} \sin a}{r\_{\rm AB}} \right) \frac{\sin a}{\sqrt{r\_{\rm AB}^2 - r\_{\rm A}^2 \sin^2 a}}$$

$$+ k\_{\rm PB} \left( \sqrt{r\_{\rm AB}^2 - r\_{\rm A}^2 \sin^2 a} + r\_{\rm A} \cos a - \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2 \sin^2 a} - r\_{\rm A0} \cos a \right) \left( \frac{r\_{\rm A} \sin^2 a}{\sqrt{r\_{\rm AB}^2 - r\_{\rm A}^2 \sin^2 a}} - \cos a \right). \tag{15}$$

Here the input slider displacement is denoted by S, which satisfies

$$S = -r\_{\mathcal{A}} + r\_{\mathcal{A}0} \ge 0 \tag{16}$$

where S ≥ 0 means the input slider moves along the negative direction of X-axis.

Figure 2. Planar double-slider linkage with springs.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

Substituting Eq. (16) into Eqs. (14) and (15), the potential energy, U, and the driving force, Fd, with respect to S can be obtained as

$$\begin{aligned} U &= \frac{1}{2}k\_{\rm PA}S^2 + \frac{1}{2}(k\_{\rm RA} + k\_{\rm RB})\left(\arccos\frac{(r\_{\rm A0} - S)\sin\alpha}{r\_{\rm AB}} - \arccos\frac{r\_{\rm A0}\sin\alpha}{r\_{\rm AB}}\right)^2 \\ &+ \frac{1}{2}k\_{\rm PB}\left(\sqrt{r\_{\rm AB}^2 - (r\_{\rm A0} - S)^2\sin^2\alpha} + (r\_{\rm A0} - S)\cos\alpha - \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2\sin^2\alpha} - r\_{\rm A0}\cos\alpha\right)^2. \end{aligned} \tag{17}$$

$$\begin{aligned} F\_{\rm d} &= \frac{dU}{dS} = k\_{\rm PA}S + \left(k\_{\rm RA} + k\_{\rm RB}\right) \left(\arccos\frac{\left(r\_{\rm A0} - S\right)\sin a}{r\_{\rm AB}} - \arccos\frac{r\_{\rm A0}\sin a}{r\_{\rm AB}}\right) \frac{\sin a}{\sqrt{r\_{\rm AB}^2 - \left(r\_{\rm A0} - S\right)^2 \sin^2 a}} \\\\ &+ k\_{\rm PB} \left(\left(r\_{\rm A0} - S\right)\cos a + \sqrt{r\_{\rm AB}^2 - \left(r\_{\rm A0} - S\right)^2 \sin^2 a} - r\_{\rm A0}\cos a - \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2 \sin^2 a}\right) \\\\ &\times \left(\frac{\left(r\_{\rm A0} - S\right)\sin^2 a}{\sqrt{r\_{\rm AB}^2 - \left(r\_{\rm A0} - S\right)^2 \sin^2 a}} - \cos a\right). \end{aligned} \tag{18}$$

From Eq. (18), we can predict that the variation of driving force exerted on the input slider versus input displacement, i.e., Fd-S curve, would present nonlinear stiffness characteristic.

#### 4. Cause of nonlinear stiffness characteristic generation

It is evident that the springs placed at joints are the cause of nonlinear stiffness characteristic generation, so it is necessary to discuss the influence of spring stiffness on the Fd-S curve characteristic. When the influence of one specific spring stiffness is analysed, every other spring stiffness is set to zero.

#### 4.1 Influences of translational spring stiffness placed at output slider

Substituting kPA = 0 and kRA = kRB = 0 into Eqs. (17) and (18), the following can be obtained, respectively:

$$U = \frac{1}{2}k\_{\rm PB} \left( (r\_{\rm A0} - S)\cos a + \sqrt{r\_{\rm AB}^2 - (r\_{\rm A0} - S)^2 \sin^2 a} - r\_{\rm A0}\cos a - \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2 \sin^2 a} \right)^2 \tag{19}$$

$$F\_{\rm d} = \frac{dU}{dS} = k\_{\rm PB} \left( (r\_{\rm A0} - S) \cos \alpha + \sqrt{r\_{\rm AB}^2 - (r\_{\rm A0} - S)^2 \sin^2 \alpha} - r\_{\rm A0} \cos \alpha - \sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2 \sin^2 \alpha} \right). \tag{20}$$

$$\times \left( \frac{(r\_{\rm A0} - S) \sin^2 \alpha}{\sqrt{L^2 - (r\_{\rm A0} - S)^2 \sin^2 \alpha}} - \cos \alpha \right). \tag{20}$$

3. Force equilibrium equation

Kinematics - Analysis and Applications

kPAð Þ r<sup>A</sup> � rA0

kPAð Þ r<sup>A</sup> � rA0

them into Eq. (13), the following can be further derived:

<sup>A</sup> sin <sup>2</sup>α

¼ �kPAð Þþ <sup>r</sup><sup>A</sup> � <sup>r</sup>A0 ð Þ <sup>k</sup>RA <sup>þ</sup> <sup>k</sup>RB arccosr<sup>A</sup> sin <sup>α</sup>

driving force, Fd, applied on the input slider can be obtained as

r2 AB � r<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>A</sup> sin <sup>2</sup><sup>α</sup> <sup>p</sup> <sup>þ</sup> <sup>r</sup><sup>A</sup> cos <sup>α</sup> �

r2 AB � r<sup>2</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

derived as

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>F</sup><sup>d</sup> <sup>¼</sup> dU drA

Figure 2.

68

Planar double-slider linkage with springs.

þ kPB

þ 1 2 kPB

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2

We suppose that each of the two prismatic joints is attached a translational spring and each of the two rotational joints is attached a torsional spring. The planar double-slider linkage with springs is obtained as shown in Figure 2. The stiffnesses of the translational springs placed at prismatic joints A and B are defined by kPA and kPB, respectively, and the stiffnesses of the translational springs placed at prismatic joints A and B are defined by kPA and kPB, respectively.

The potential energy of the whole mechanism as shown in Figure 2 can be

ð Þ kRA þ kRB ð Þ θ<sup>A</sup> � θA0

where θA0 is the initial angle between positive direction of X-axis and coupler

AB. After differentiating θA, θA0, r<sup>B</sup> and rB0, which can be derived from the geometry of the linkage as shown in Figure 1, with respect to rA, and substituting

ð Þ kRA þ kRB arccos

þ r<sup>A</sup> cos α �

r2 AB � r<sup>2</sup>

q

Here the input slider displacement is denoted by S, which satisfies

� arccos

A0 sin <sup>2</sup>α

� �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� arccosrA0 sin <sup>α</sup> rAB

� rA0 cos α

S ¼ �r<sup>A</sup> þ rA0 ≥ 0 (16)

� � sin α

r<sup>A</sup> sin α rAB

> r2 AB � r<sup>2</sup>

q

� �<sup>2</sup>

The principle of virtual work, which does not need to determine the inner forces between two connected links, is a simplified useful method to construct the force equilibrium equation. According to the principle of virtual work, the required

rAB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where S ≥ 0 means the input slider moves along the negative direction of X-axis.

� � r<sup>A</sup> sin <sup>2</sup>α

A0 sin <sup>2</sup>α

kPBð Þ r<sup>B</sup> � rB0

rA0 sin α rAB

� rA0 cos α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 AB � <sup>r</sup><sup>2</sup> <sup>A</sup> sin <sup>2</sup><sup>α</sup> <sup>p</sup>

r2 AB � r<sup>2</sup> <sup>A</sup> sin <sup>2</sup><sup>α</sup> <sup>p</sup> � cos <sup>α</sup> !

<sup>2</sup> (13)

:

(14)

:

(15)

When Eq. (20) is zero, solving this equation with respect to S obtains

$$\mathbf{S\_1 = 0, \quad S\_2 = -r\_{\rm AB}/\tan a + r\_{\rm A0}, \quad S\_3 = -2r\_{\rm A0}\cos^2 a - 2\sqrt{r\_{\rm AB}^2 - r\_{\rm A0}^2\sin^2 a}\cos a + 2r\_{\rm A0} \tag{21}$$

where S<sup>2</sup> is the kinematic limb-singularity position (based on Eqs. (8) and (16)). From Eq. (19), we know that

$$\left.U\right|\_{\mathbb{S}=\mathbb{S}\_1} = \left.U\right|\_{\mathbb{S}=\mathbb{S}\_\flat} = \mathbf{0} \tag{22}$$

Eq. (23) shows that when S = S2, the potential energy, U, reaches the local maximum. Therefore, S = S2, the kinematic limb-singularity position, is also the

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

Therefore, according to [15], when kPA = 0, kRA = kRB = 0 and kPB 6¼ 0, the kinematic limb-singularity position, S2, is the mechanism's unstable equilibrium

slider position, rA0, is 10 mm; and the unit of kPB is N/mm (here the unit of translational spring stiffness is N/mm, and the unit of torsional spring stiffness is N�mm/rad); the stiffness characteristic produced by the mechanism is shown in

exclusively zero, the mechanism produces the bistable characteristic.

4.2 Influences of translational spring stiffness placed at input slider

Figure 3 confirms that when the spring stiffness placed at output slider is

Substitution of kPB = 0 and kRA = kRB = 0 into Eqs. (17) and (18) obtains the

kPAS<sup>2</sup> (24)

F<sup>d</sup> ¼ kPA S (25)

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2

Stiffness characteristic with different kPA when kRA = kRB = kPB = 0. (a) Driving force versus input

displacement. (b) Potential energy versus input displacement.

If the coupler length, rAB, is 100 mm; intersection angle, α, is 100°; initial input

unstable equilibrium position [15].

DOI: http://dx.doi.org/10.5772/intechopen.85009

Figure 3.

Figure 4.

71

expressions as follows:

position and S<sup>1</sup> and S<sup>3</sup> are the stable equilibrium points.

Substitution of Eq. (16) into the differentiation of Eq. (20) with respect to S leads to

$$\left. \frac{dF\_{\rm d}}{d\mathcal{S}} \right|\_{\mathcal{S}=\mathcal{S}\_{\rm 2}} = \frac{d^2U}{d\mathcal{S}^2} \Big|\_{\mathcal{S}=\mathcal{S}\_{\rm 2}} = -k\_{\rm PB} \left( \frac{r\_{\rm AB}}{\sin a} - r\_{\rm B0} \right) \cdot \frac{1}{r\_{\rm AB} \sin a} < 0. \tag{23}$$

#### Figure 3.

The bistable characteristic with different kPB when kPA = kRA = 0 and kRB = 0. (a) Driving force versus input displacement. (b) Potential energy versus input displacement.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

Eq. (23) shows that when S = S2, the potential energy, U, reaches the local maximum. Therefore, S = S2, the kinematic limb-singularity position, is also the unstable equilibrium position [15].

Therefore, according to [15], when kPA = 0, kRA = kRB = 0 and kPB 6¼ 0, the kinematic limb-singularity position, S2, is the mechanism's unstable equilibrium position and S<sup>1</sup> and S<sup>3</sup> are the stable equilibrium points.

If the coupler length, rAB, is 100 mm; intersection angle, α, is 100°; initial input slider position, rA0, is 10 mm; and the unit of kPB is N/mm (here the unit of translational spring stiffness is N/mm, and the unit of torsional spring stiffness is N�mm/rad); the stiffness characteristic produced by the mechanism is shown in Figure 3.

Figure 3 confirms that when the spring stiffness placed at output slider is exclusively zero, the mechanism produces the bistable characteristic.

#### 4.2 Influences of translational spring stiffness placed at input slider

Substitution of kPB = 0 and kRA = kRB = 0 into Eqs. (17) and (18) obtains the expressions as follows:

$$U = \frac{1}{2}k\_{\rm PA} \mathcal{S}^2 \tag{24}$$

$$F\_{\rm d} = k\_{\rm PA} \,\, \mathcal{S} \tag{25}$$

#### Figure 4.

Stiffness characteristic with different kPA when kRA = kRB = kPB = 0. (a) Driving force versus input displacement. (b) Potential energy versus input displacement.

When Eq. (20) is zero, solving this equation with respect to S obtains

<sup>S</sup>¼S<sup>1</sup> ¼ Uj

¼ �kPB

The bistable characteristic with different kPB when kPA = kRA = 0 and kRB = 0. (a) Driving force versus input

displacement. (b) Potential energy versus input displacement.

Uj

where S<sup>2</sup> is the kinematic limb-singularity position (based on Eqs. (8) and (16)).

Substitution of Eq. (16) into the differentiation of Eq. (20) with respect to S leads to

rAB sin <sup>α</sup> � <sup>r</sup>B0 � �

α � 2

r2 AB � r<sup>2</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>S</sup>¼S<sup>3</sup> ¼ 0 (22)

� <sup>1</sup>

A0 sin <sup>2</sup>α

<sup>r</sup>AB sin <sup>α</sup> <sup>&</sup>lt;0: (23)

cos α þ 2rA0 (21)

<sup>S</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>, S<sup>2</sup> ¼ �rAB<sup>=</sup> tan <sup>α</sup> <sup>þ</sup> <sup>r</sup>A0, S<sup>3</sup> ¼ �2rA0 cos <sup>2</sup>

¼ d2 U dS<sup>2</sup> � � � � S¼S<sup>2</sup>

From Eq. (19), we know that

Kinematics - Analysis and Applications

dF<sup>d</sup> dS � � � � S¼S<sup>2</sup>

Figure 3.

70

Eqs. (24) and (25) show that the mechanism only generates positive-stiffness characteristic when the mechanism has only one minimal potential energy point, which can be confirmed by Figure 4, where specific parameters are given as rAB = 100 mm, α = 100° and rA0 = 10 mm.

Figure 5 demonstrates that when the torsional springs are placed at the pin joints, the mechanism only produces positive-stiffness characteristic but does not

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

Section 4 showed that spring placed at the output slider causes the bistable characteristic with the negative domain, while springs placed at other joints produce the corresponding positive-stiffness characteristic. It can be predicted that if more than one spring are placed at joints, when the mechanism moves from nonsingular position (Figure 6(a)) to another non-singular position (Figure 6(c)) while passing through the limb-singularity position (Figure 6(b)), the stiffness characteristic of the mechanism is the superposition of the corresponding stiffness characteristic caused by the joints. For instance, if kPB = 1 N/mm, α = 100°,

Positions of the mechanism. (a) Initial kinematic non-singular position. (b) Kinematic limb-singularity

produce other stiffness characteristic.

DOI: http://dx.doi.org/10.5772/intechopen.85009

Figure 6.

73

position. (c) End kinematic non-singular position.

5. Nonlinear stiffness characteristic construction

#### 4.3 Influences of torsional spring stiffness placed at pin joints

When kPA = kPB = 0 and kRA = kRB 6¼ 0, from Eq. (18), we can obtain

$$F\_{\rm d} = (k\_{\rm RA} + k\_{\rm RB}) \left( \arccos \frac{(r\_{\rm A0} - \rm S) \sin a}{r\_{\rm AB}} - \arccos \frac{r\_{\rm A0} \sin a}{r\_{\rm AB}} \right) \frac{\sin a}{\sqrt{r\_{\rm AB}^2 - (r\_{\rm A0} - \rm S)^2 \sin^2 a}} \tag{26}$$

where only and only if S = S<sup>1</sup> = 0, i.e., r<sup>A</sup> = rA0, then F<sup>d</sup> = 0.

In other words, if and only if S = S<sup>1</sup> = 0, the mechanism is in equilibrium position without external force. When the mechanism is located in any other positions, it is unstable except when applied by external force. Meanwhile, the potential energy, U, has no local maximum but has only one minimum which is located at S = S<sup>1</sup> = 0.

For kPA = kPB = 0, kRA = kRB ¼6 0, when rAB = 100 mm, α = 100° and rA0 = 10 mm, the force-displacement characteristic and the potential energy curve are shown in Figure 5.

#### Figure 5.

Behaviours with different kRA = kRB = k<sup>R</sup> when kPB = 0 and kPA = 0. (a) Driving force versus input displacement. (b) Potential energy versus input displacement.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

Figure 5 demonstrates that when the torsional springs are placed at the pin joints, the mechanism only produces positive-stiffness characteristic but does not produce other stiffness characteristic.

### 5. Nonlinear stiffness characteristic construction

Section 4 showed that spring placed at the output slider causes the bistable characteristic with the negative domain, while springs placed at other joints produce the corresponding positive-stiffness characteristic. It can be predicted that if more than one spring are placed at joints, when the mechanism moves from nonsingular position (Figure 6(a)) to another non-singular position (Figure 6(c)) while passing through the limb-singularity position (Figure 6(b)), the stiffness characteristic of the mechanism is the superposition of the corresponding stiffness characteristic caused by the joints. For instance, if kPB = 1 N/mm, α = 100°,

#### Figure 6.

Positions of the mechanism. (a) Initial kinematic non-singular position. (b) Kinematic limb-singularity position. (c) End kinematic non-singular position.

Eqs. (24) and (25) show that the mechanism only generates positive-stiffness characteristic when the mechanism has only one minimal potential energy point, which can be confirmed by Figure 4, where specific parameters are given as

� arccos

In other words, if and only if S = S<sup>1</sup> = 0, the mechanism is in equilibrium position without external force. When the mechanism is located in any other positions, it is unstable except when applied by external force. Meanwhile, the potential energy, U, has no local maximum but has only one minimum which is located at S = S<sup>1</sup> = 0. For kPA = kPB = 0, kRA = kRB ¼6 0, when rAB = 100 mm, α = 100° and rA0 = 10 mm, the force-displacement characteristic and the potential energy curve are shown in Figure 5.

� � sin α

rA0 sin α rAB

r2

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

AB � ð Þ <sup>r</sup>A0 � <sup>S</sup> <sup>2</sup> sin <sup>2</sup><sup>α</sup>

(26)

4.3 Influences of torsional spring stiffness placed at pin joints

rAB

where only and only if S = S<sup>1</sup> = 0, i.e., r<sup>A</sup> = rA0, then F<sup>d</sup> = 0.

When kPA = kPB = 0 and kRA = kRB 6¼ 0, from Eq. (18), we can obtain

Behaviours with different kRA = kRB = k<sup>R</sup> when kPB = 0 and kPA = 0. (a) Driving force versus input

displacement. (b) Potential energy versus input displacement.

rAB = 100 mm, α = 100° and rA0 = 10 mm.

Kinematics - Analysis and Applications

<sup>F</sup><sup>d</sup> <sup>¼</sup> ð Þ <sup>k</sup>RA <sup>þ</sup> <sup>k</sup>RB arccosð Þ <sup>r</sup>A0 � <sup>S</sup> sin <sup>α</sup>

Figure 5.

72

rA0 = 10 mm and rAB = 100 mm, several nonlinear stiffness characteristics with different spring stiffness are shown in Figure 7.

Figure 7 shows that after assigning appropriate spring stiffness placed at the corresponding joints, the mechanism can generate one of four types of nonlinear

stiffness characteristics including bistable characteristic, partial negative-stiffness characteristic, partial zero-stiffness characteristic and positive-stiffness characteristic which are shown in Figure 8, as it works around the kinematic limb-singularity

Four types of nonlinear stiffness characteristic of the mechanism. (a) Bistable characteristic, (b) partial negative-stiffness characteristic, (c) partial zero-stiffness characteristic, and (d) positive-stiffness characteristic.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

DOI: http://dx.doi.org/10.5772/intechopen.85009

The above-mentioned analysis illustrates the case that the mechanism moves from a non-singular position. When the mechanism moves from the kinematic limb-singularity position (Figure 6(b)) to a non-singular position (Figure 6(c)), every spring transforms with zero potential energy to a position with a certain amount of potential energy. The total potential energy of the mechanism starts from zero to nonzero without local minimal energy point except the initial position. Every spring force/torque increases in the process of the mechanism's motion. From Eqs. (18) or (20), the driving force is to overcome the resistance caused by every spring, so the driving force increases when the mechanism moves from nonsingular position. In other words, when mechanism moves from non-singular position with no deflected springs, the mechanism only generates the positive-stiffness

characteristic as shown in Figure 8(d). If kPA = kPB = 0, kRA = kRB 6¼ 0 and rAB = 100 mm, α = 100° and rA0 = 10 mm, the stiffness characteristic is shown in

it only generates positive-stiffness characteristic.

Figure 9 demonstrates that when the mechanism starts from the kinematic limb-singularity position with no deflected springs towards a non-singular position,

As the final stiffness characteristic is determined by the superposition of stiffness characteristic caused by each spring, an expected stiffness characteristic can be constructed by assigning appropriate values to kPB, kRA, kRB and kPA on the condition of kRB 6¼ 0 when the mechanism moves from one non-singular position to another non-singular position with passing through the kinematic limb-singularity position. The method for designing an expected nonlinear stiffness design is

position.

Figure 8.

Figure 9.

proposed in [16].

75

Figure 7.

Nonlinear stiffness characteristic when kPB = 1 N/mm. (a) Driving force versus input displacement for minor change of kRA = kRB = k<sup>R</sup> when kPA = 0. (b) Driving force versus input displacement for large change of kRA = kRB = k<sup>R</sup> when kPA = 0. (c) Driving force versus input displacement for different spring stiffness when kRA = kRB = 0.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

Figure 8.

rA0 = 10 mm and rAB = 100 mm, several nonlinear stiffness characteristics with

Figure 7 shows that after assigning appropriate spring stiffness placed at the corresponding joints, the mechanism can generate one of four types of nonlinear

Nonlinear stiffness characteristic when kPB = 1 N/mm. (a) Driving force versus input displacement for minor change of kRA = kRB = k<sup>R</sup> when kPA = 0. (b) Driving force versus input displacement for large change of kRA = kRB = k<sup>R</sup> when kPA = 0. (c) Driving force versus input displacement for different spring stiffness when

different spring stiffness are shown in Figure 7.

Kinematics - Analysis and Applications

Figure 7.

74

kRA = kRB = 0.

Four types of nonlinear stiffness characteristic of the mechanism. (a) Bistable characteristic, (b) partial negative-stiffness characteristic, (c) partial zero-stiffness characteristic, and (d) positive-stiffness characteristic.

stiffness characteristics including bistable characteristic, partial negative-stiffness characteristic, partial zero-stiffness characteristic and positive-stiffness characteristic which are shown in Figure 8, as it works around the kinematic limb-singularity position.

The above-mentioned analysis illustrates the case that the mechanism moves from a non-singular position. When the mechanism moves from the kinematic limb-singularity position (Figure 6(b)) to a non-singular position (Figure 6(c)), every spring transforms with zero potential energy to a position with a certain amount of potential energy. The total potential energy of the mechanism starts from zero to nonzero without local minimal energy point except the initial position. Every spring force/torque increases in the process of the mechanism's motion. From Eqs. (18) or (20), the driving force is to overcome the resistance caused by every spring, so the driving force increases when the mechanism moves from nonsingular position. In other words, when mechanism moves from non-singular position with no deflected springs, the mechanism only generates the positive-stiffness characteristic as shown in Figure 8(d). If kPA = kPB = 0, kRA = kRB 6¼ 0 and rAB = 100 mm, α = 100° and rA0 = 10 mm, the stiffness characteristic is shown in Figure 9.

Figure 9 demonstrates that when the mechanism starts from the kinematic limb-singularity position with no deflected springs towards a non-singular position, it only generates positive-stiffness characteristic.

As the final stiffness characteristic is determined by the superposition of stiffness characteristic caused by each spring, an expected stiffness characteristic can be constructed by assigning appropriate values to kPB, kRA, kRB and kPA on the condition of kRB 6¼ 0 when the mechanism moves from one non-singular position to another non-singular position with passing through the kinematic limb-singularity position. The method for designing an expected nonlinear stiffness design is proposed in [16].

6. Further discussion

here).

represents a crank-slider linkage with springs.

DOI: http://dx.doi.org/10.5772/intechopen.85009

the output slider and its stiffness is denoted by KPC.

the principle of virtual work can be constructed as

� �ð Þ 1 � r<sup>1</sup> cos θA=a

a ¼

curve can be described.

Figure 10.

77

Crank-slider with springs.

<sup>þ</sup> <sup>K</sup>RC arcsin <sup>r</sup><sup>1</sup> sin <sup>θ</sup><sup>A</sup> � <sup>e</sup>

<sup>T</sup><sup>d</sup> <sup>¼</sup> <sup>K</sup>RAð Þþ <sup>θ</sup><sup>A</sup> � <sup>θ</sup>A0 <sup>K</sup>RB �θ<sup>A</sup> � arcsin <sup>r</sup><sup>1</sup> sin <sup>θ</sup><sup>A</sup> � <sup>e</sup>

r2

coordinate of output slider and e is the offset. Moreover, a and a<sup>0</sup> are defined by

r2

q

� �

þ KPCðr<sup>1</sup> cos θ<sup>A</sup> þ a � r<sup>1</sup> cos θA0 � a0Þ� �ð Þ r<sup>1</sup> sin θ<sup>A</sup> � b=a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � ð Þ r<sup>1</sup> sin θ<sup>A</sup> � e

From Sections 2–5, it can be shown that if a planar linkage exhibits a kinematic limb-singularity, it generates different nonlinear spring stiffness characteristics if the linkage is added with springs at corresponding joints with the condition that the spring stiffness corresponding to output slider is nonzero. Nonlinear stiffness characteristic generation using the kinematic limb-singularity of a planar linkage can be demonstrated by another planar linkage with springs as shown in Figure 10, which

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

The Cartesian coordinates system, O-xyz, is constructed as shown in Figure 10, where crank AB rotates about joint A in an anticlockwise direction, the slider moves along the x-axis, and coupler BC connects link AB and slider by two rotation joints B and C. The three rotation joints are added with torsional springs with spring stiffnesses denoted by KRA, KRB and KRC, respectively. An extension spring is placed at

The position formula with closed-loop form, whose derivation process is similar to one of the double-slider four-bar linkages, can be established easily (not shown

Based on the position analysis, the kinetostatic model of the mechanism by using

� arcsin <sup>r</sup><sup>1</sup> sin <sup>θ</sup>A0 � <sup>e</sup>

where r<sup>1</sup> and r<sup>2</sup> are crank length and coupler length, respectively, r<sup>3</sup> is the X-axis

, a<sup>0</sup> ¼

According to Eq. (27), the Td-θ<sup>A</sup> (driving torque versus input position angle)

r2

q

2

r2

r2

� �

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � ð Þ r<sup>1</sup> sin θA0 � e

r<sup>1</sup> cos θ<sup>A</sup> a

<sup>þ</sup> <sup>θ</sup>A0 <sup>þ</sup> arcsin <sup>r</sup><sup>1</sup> sin <sup>θ</sup>A0 � <sup>e</sup>

2

r2

(27)

#### Figure 9.

Stiffness characteristic with initial non-singular position. (a) Stiffness characteristic for different kPB when kRA = kRB = 0 and kPA = 0. (b) Stiffness characteristic for different kRA = kRB = k<sup>R</sup> when kPB = 1 N/mm and kPA = 0. (c) Stiffness characteristic for kPA when kPB = 1 N/mm and kRA = kRB = 0.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

### 6. Further discussion

From Sections 2–5, it can be shown that if a planar linkage exhibits a kinematic limb-singularity, it generates different nonlinear spring stiffness characteristics if the linkage is added with springs at corresponding joints with the condition that the spring stiffness corresponding to output slider is nonzero. Nonlinear stiffness characteristic generation using the kinematic limb-singularity of a planar linkage can be demonstrated by another planar linkage with springs as shown in Figure 10, which represents a crank-slider linkage with springs.

The Cartesian coordinates system, O-xyz, is constructed as shown in Figure 10, where crank AB rotates about joint A in an anticlockwise direction, the slider moves along the x-axis, and coupler BC connects link AB and slider by two rotation joints B and C. The three rotation joints are added with torsional springs with spring stiffnesses denoted by KRA, KRB and KRC, respectively. An extension spring is placed at the output slider and its stiffness is denoted by KPC.

The position formula with closed-loop form, whose derivation process is similar to one of the double-slider four-bar linkages, can be established easily (not shown here).

Based on the position analysis, the kinetostatic model of the mechanism by using the principle of virtual work can be constructed as

$$\begin{aligned} T\_{\rm d} &= K\_{\rm RA}(\theta\_{\rm A} - \theta\_{\rm A0}) + K\_{\rm RB} \left( -\theta\_{\rm A} - \arcsin \frac{r\_1 \sin \theta\_{\rm A} - \varepsilon}{r\_2} + \theta\_{\rm A0} + \arcsin \frac{r\_1 \sin \theta\_{\rm A0} - \varepsilon}{r\_2} \right) \\ &\times (-1 - r\_1 \cos \theta\_{\rm A}/a) \\ &+ K\_{\rm RC} \left( \arcsin \frac{r\_1 \sin \theta\_{\rm A} - \varepsilon}{r\_2} - \arcsin \frac{r\_1 \sin \theta\_{\rm A0} - \varepsilon}{r\_2} \right) \times \frac{r\_1 \cos \theta\_{\rm A}}{a} \\ &+ K\_{\rm PC} (r\_1 \cos \theta\_{\rm A} + a - r\_1 \cos \theta\_{\rm A0} - a\_0) \times (-r\_1 \sin \theta\_{\rm A} - b/a) \end{aligned} \tag{27}$$

where r<sup>1</sup> and r<sup>2</sup> are crank length and coupler length, respectively, r<sup>3</sup> is the X-axis coordinate of output slider and e is the offset.

Moreover, a and a<sup>0</sup> are defined by

$$\mathfrak{a} = \sqrt{r\_2^2 - (r\_1 \sin \theta\_\mathcal{A} - \varepsilon)^2}, \\ \mathfrak{a}\_0 = \sqrt{r\_2^2 - (r\_1 \sin \theta\_\mathcal{A} - \varepsilon)^2}$$

According to Eq. (27), the Td-θ<sup>A</sup> (driving torque versus input position angle) curve can be described.

Figure 10. Crank-slider with springs.

Figure 9.

Kinematics - Analysis and Applications

76

Stiffness characteristic with initial non-singular position. (a) Stiffness characteristic for different kPB when kRA = kRB = 0 and kPA = 0. (b) Stiffness characteristic for different kRA = kRB = k<sup>R</sup> when kPB = 1 N/mm and

kPA = 0. (c) Stiffness characteristic for kPA when kPB = 1 N/mm and kRA = kRB = 0.

#### Figure 11.

Different positions of the crank-slider mechanism with springs. (a) Non-singular initial position. (b) Kinematic limb-singularity position. (c) Non-singular end position.

#### Figure 12.

Case that the mechanism moves from non-singular position. (a) Input torque variation when KRA = KRB = K<sup>R</sup> is small. (b) Input torque variation when KRA = KRB = K<sup>R</sup> is large.

Figure 13.

79

Case that the mechanism moves from kinematic limb-singularity position. (a) Case of different KPC when KRA = KRB = KRC = 0. (b) Case of different KRA when KRB = KRC = 0 and KPC = 0. (c) Case of different KRB

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

DOI: http://dx.doi.org/10.5772/intechopen.85009

when KRA = KRC = 0 and KPC = 0. (d) Case of different KRC when KRA = KRB = 0 and KPC = 0.

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

#### Figure 13.

Case that the mechanism moves from kinematic limb-singularity position. (a) Case of different KPC when KRA = KRB = KRC = 0. (b) Case of different KRA when KRB = KRC = 0 and KPC = 0. (c) Case of different KRB when KRA = KRC = 0 and KPC = 0. (d) Case of different KRC when KRA = KRB = 0 and KPC = 0.

Figure 11.

Figure 12.

78

limb-singularity position. (c) Non-singular end position.

Kinematics - Analysis and Applications

Different positions of the crank-slider mechanism with springs. (a) Non-singular initial position. (b) Kinematic

Case that the mechanism moves from non-singular position. (a) Input torque variation when KRA = KRB = K<sup>R</sup> is

small. (b) Input torque variation when KRA = KRB = K<sup>R</sup> is large.

kinematic limb-singularity position, spring stiffness determines one of four types of stiffness characteristics. On the other hand, when the mechanism moves from the kinematic limb-singularity position, it only produces the positive-stiffness

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

The authors gratefully acknowledge the support of the National Natural Science Foundation of China under Grant No. 51605006, the Research Foundation of Key Laboratory of Manufacturing Systems and Advanced Technology of Guangxi

characteristic.

Acknowledgements

Author details

Huainan, China

81

Baokun Li<sup>1</sup> and Guangbo Hao<sup>2</sup>

\*

2 School of Engineering, University College Cork, Cork, Ireland

\*Address all correspondence to: g.hao@ucc.ie

provided the original work is properly cited.

1 School of Mechanical Engineering, Anhui University of Science and Technology,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Province, China, under Grant No. 17-259-05-013K.

DOI: http://dx.doi.org/10.5772/intechopen.85009

Figure 14.

Nonlinear stiffness characteristic compliant mechanisms based on the PRBM. (a) Compliant double-slider mechanism and (b) compliant crank-slider mechanism.

When the mechanism with springs moves from one non-singular position to the kinematic limb-singularity position and then stops at another non-singular position as shown in Figure 11, it may produce one of four types of nonlinear stiffness characteristics including the bistable characteristic, partial negative-stiffness characteristic, partial zero-stiffness characteristic and positive-stiffness characteristic. For illustration, in Figure 12, crank length, r1, is 10 cm; coupler length, r2, is 50 cm; offset e is 3 cm; input initial position angle, θA0, is 5°; and KPC = 1 N/cm.

It can also be shown that, similarly to the double-slider four-bar mechanism with springs, when the crank-slider mechanism with springs moves from the kinematic limb-singularity position, it only generates the positive-stiffness characteristic, which is shown in Figure 13, where the geometry parameters are given as the same as shown in Figure 12.

Therefore, we can conclude that after placing springs at different pair combinations, a planar linkage which has the kinematic limb-singularity can generate corresponding nonlinear stiffness characteristic in condition that the mechanism moves from initial non-singular position with no deflected springs to the kinematic limb-singularity position and then stops at another non-singular position. If the mechanism moves from the kinematic limb-singularity position with no deflected springs, it only generates the positive-stiffness characteristic.

It is worth to point out that the nonlinear stiffness characteristic generation method can also be applied to design the nonlinear stiffness characteristic compliant mechanism by using the PRBM as shown in Figure 14.

In Figure 14, the equivalent stiffness of compliant rotational joint and compliant translational joint can be calculated by referring to previous work [17, 18].

#### 7. Conclusions

The kinematic limb-singularity positions of planar linkages with attached springs can be used to generate nonlinear characteristics. After assigning different spring stiffness, the mechanism may exhibit one of four types of nonlinear stiffness characteristics. These are the bistable characteristic, partial negative-stiffness characteristic, partial zero-stiffness characteristic and positive-stiffness characteristic. The type of stiffness characteristic is determined by the motion model and the value of spring stiffness. On the condition that the mechanism moves from the initial non-singular position to another non-singular position with passing though the

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

kinematic limb-singularity position, spring stiffness determines one of four types of stiffness characteristics. On the other hand, when the mechanism moves from the kinematic limb-singularity position, it only produces the positive-stiffness characteristic.

## Acknowledgements

When the mechanism with springs moves from one non-singular position to the kinematic limb-singularity position and then stops at another non-singular position as shown in Figure 11, it may produce one of four types of nonlinear stiffness characteristics including the bistable characteristic, partial negative-stiffness characteristic, partial zero-stiffness characteristic and positive-stiffness characteristic. For illustration, in Figure 12, crank length, r1, is 10 cm; coupler length, r2, is 50 cm;

Nonlinear stiffness characteristic compliant mechanisms based on the PRBM. (a) Compliant double-slider

It can also be shown that, similarly to the double-slider four-bar mechanism with springs, when the crank-slider mechanism with springs moves from the kinematic limb-singularity position, it only generates the positive-stiffness characteristic, which is shown in Figure 13, where the geometry parameters are given as the same

Therefore, we can conclude that after placing springs at different pair combina-

offset e is 3 cm; input initial position angle, θA0, is 5°; and KPC = 1 N/cm.

tions, a planar linkage which has the kinematic limb-singularity can generate corresponding nonlinear stiffness characteristic in condition that the mechanism moves from initial non-singular position with no deflected springs to the kinematic limb-singularity position and then stops at another non-singular position. If the mechanism moves from the kinematic limb-singularity position with no deflected

It is worth to point out that the nonlinear stiffness characteristic generation method can also be applied to design the nonlinear stiffness characteristic compliant

translational joint can be calculated by referring to previous work [17, 18].

The kinematic limb-singularity positions of planar linkages with attached springs can be used to generate nonlinear characteristics. After assigning different spring stiffness, the mechanism may exhibit one of four types of nonlinear stiffness characteristics. These are the bistable characteristic, partial negative-stiffness characteristic, partial zero-stiffness characteristic and positive-stiffness characteristic. The type of stiffness characteristic is determined by the motion model and the value of spring stiffness. On the condition that the mechanism moves from the initial non-singular position to another non-singular position with passing though the

In Figure 14, the equivalent stiffness of compliant rotational joint and compliant

springs, it only generates the positive-stiffness characteristic.

mechanism by using the PRBM as shown in Figure 14.

as shown in Figure 12.

mechanism and (b) compliant crank-slider mechanism.

Kinematics - Analysis and Applications

Figure 14.

7. Conclusions

80

The authors gratefully acknowledge the support of the National Natural Science Foundation of China under Grant No. 51605006, the Research Foundation of Key Laboratory of Manufacturing Systems and Advanced Technology of Guangxi Province, China, under Grant No. 17-259-05-013K.

### Author details

Baokun Li<sup>1</sup> and Guangbo Hao<sup>2</sup> \*

1 School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, China

2 School of Engineering, University College Cork, Cork, Ireland

\*Address all correspondence to: g.hao@ucc.ie

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

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[2] Gosselin CM, Angeles J. Singularity analysis of closed loop kinematic chains. IEEE Transactions on Robotics and Automation. 1990;6(3):281-290. DOI: 10.1109/70.56660

[3] Amine S, Mokhiamar O, Caro S. Classification of 3T1R parallel manipulators based on their wrench graph. ASME Journal of Mechanisms and Robotics. 2017;9(1):011003. DOI: 10.1115/1.4035188

[4] Huang Z, Cao Y. Property identification of the singularity loci of a class of Gough-Stewart manipulators. The International Journal of Advanced Robotics Research. 2005;24(8):375-685. DOI: 10.1177/0278364905054655

[5] Boudreau R, Nokleby S. Force optimization of kinematicallyredundant planar parallel manipulators following a desired trajectory. Mechanism and Machine Theory. 2012; 56(10):138-155. DOI: 10.1016/j. mechmachtheory.2012.06.001

[6] Saglia J, Dai JS, Caldwell DG. Geometry and kinematic analysis of a redundantly actuated parallel mechanism that eliminates singularity and improves dexterity. ASME Journal of Mechanical Design. 2008;130(12): 124501. DOI: 10.1115/1.2988472

[7] Li B, Cao Y, Zhang Qiuju, et al. Position-singularity analysis of a special class of the Stewart parallel mechanisms with two dissimilar semi-symmetrical hexagons. Robotica. 2013;31(1):123-136. DOI: 10.1017/S0263574712000148

[8] Karimia A, Masouleha MT, Cardoub P. Avoiding the singularities of 3-RPR parallel mechanisms via dimensional

synthesis and self-reconfigurability. Mechanism and Machine Theory. 2016; 99:189-206. DOI: 10.1016/j. mechmachtheory.2016.01.006

30–September 2, 2009; San Diego, California, USA; DETC2009–86845; 2009. DOI: 10.1115/DETC2009-86845

28473; 2010. DOI: 10.1115/

11(5):566-573. DOI: 10.1109/

JMEMS.2002.803284

[15] Baker MS, Howell LL. On-chip actuation of an in-plane compliant bistable micromechanism. Journal of Microelectromechanical Systems. 2002;

[16] Li B, Hao G. Nonlinear behaviour design using the kinematic singularity of a general type of double-slider four-bar linkage. Mechanism and Machine Theory. 2018;129:106-130. DOI:

10.1016/j.mechmachtheory.2018.07.016

[17] Zhao H, Bi S, Yu J. Nonlinear deformation behavior of a beam-based

[18] Pei X, Yu J, Zong G, et al. The modeling of cartwheel flexural hinges. Mechanism and Machine Theory. 2009; 44(10):1900-1909. DOI: 10.1016/j. mechmachtheory.2009.04.006

flexural pivot with monolithic arrangement. Precision Engineering. 2011;35(2):369-382. DOI: 10.1016/j.

precisioneng.2010.12.002

83

DETC2010-28473

[14] Olsen BM, Issac Y, Howell LL, et al. Utilizing a classification scheme to facilitate rigid-body replacement for compliant mechanism design. In: ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; August 15–18, 2010; Montreal, Quebec, Canada; DETC2010–

DOI: http://dx.doi.org/10.5772/intechopen.85009

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity…

[9] Abhilash N, Li H, Hao G, et al. A reconfigurable compliant four-bar mechanism with multiple operation modes. In: ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; August 6–9, 2017; Cleveland, Ohio, USA; DETC2017–67441; 2017. DOI: 10.1115/ DETC2017-67441

[10] Rubbert L, Caro S, Gangloff J, et al. Using singularities of parallel manipulators for enhancing the rigidbody replacement design method of compliant mechanisms. ASME Journal of Mechanical Design. 2014;136(5): 051010. DOI: 10.1115/1.4026949

[11] Rubbert L, Renaud P, Caro S, et al. Design of a compensation mechanism for an active cardiac stabilizer based on an assembly of planar compliant mechanisms. Mechanics and Industry. 2014;15(2):147-151. DOI: 10.1051/meca/ 2014013

[12] Quentin B, Marc V, Salih A. Parallel singularities for the design of softening springs using compliant mechanisms. In: ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; August 2–5, 2015; Boston, Massachusetts, USA; DETC2015-47240; 2015. DOI: 10.1115/DETC2015-47240

[13] Gallego JA, Herder J. Synthesis methods in compliant mechanisms: An overview. In: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference & Computers and Information in Engineering Conference; August

Kinetostatic Nonlinear Stiffness Characteristic Generation Using the Kinematic Singularity… DOI: http://dx.doi.org/10.5772/intechopen.85009

30–September 2, 2009; San Diego, California, USA; DETC2009–86845; 2009. DOI: 10.1115/DETC2009-86845

References

(in Chinese)

10.1109/70.56660

10.1115/1.4035188

[4] Huang Z, Cao Y. Property

[5] Boudreau R, Nokleby S. Force optimization of kinematically-

following a desired trajectory.

56(10):138-155. DOI: 10.1016/j. mechmachtheory.2012.06.001

[6] Saglia J, Dai JS, Caldwell DG. Geometry and kinematic analysis of a

redundantly actuated parallel

redundant planar parallel manipulators

Mechanism and Machine Theory. 2012;

mechanism that eliminates singularity and improves dexterity. ASME Journal of Mechanical Design. 2008;130(12): 124501. DOI: 10.1115/1.2988472

[7] Li B, Cao Y, Zhang Qiuju, et al. Position-singularity analysis of a special class of the Stewart parallel mechanisms with two dissimilar semi-symmetrical hexagons. Robotica. 2013;31(1):123-136. DOI: 10.1017/S0263574712000148

[8] Karimia A, Masouleha MT, Cardoub P. Avoiding the singularities of 3-RPR parallel mechanisms via dimensional

82

identification of the singularity loci of a class of Gough-Stewart manipulators. The International Journal of Advanced Robotics Research. 2005;24(8):375-685. DOI: 10.1177/0278364905054655

[1] Huang Z, Zhao YS, Zhao TS. Advanced Spatial Mechanism. Beijing, China: Higher Education Press; 2005.

Kinematics - Analysis and Applications

[2] Gosselin CM, Angeles J. Singularity analysis of closed loop kinematic chains. IEEE Transactions on Robotics and Automation. 1990;6(3):281-290. DOI:

synthesis and self-reconfigurability. Mechanism and Machine Theory. 2016;

[9] Abhilash N, Li H, Hao G, et al. A reconfigurable compliant four-bar mechanism with multiple operation modes. In: ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; August 6–9,

99:189-206. DOI: 10.1016/j. mechmachtheory.2016.01.006

2017; Cleveland, Ohio, USA;

Using singularities of parallel

DETC2017-67441

2014013

DETC2017–67441; 2017. DOI: 10.1115/

[10] Rubbert L, Caro S, Gangloff J, et al.

manipulators for enhancing the rigidbody replacement design method of compliant mechanisms. ASME Journal of Mechanical Design. 2014;136(5): 051010. DOI: 10.1115/1.4026949

[11] Rubbert L, Renaud P, Caro S, et al. Design of a compensation mechanism for an active cardiac stabilizer based on an assembly of planar compliant mechanisms. Mechanics and Industry. 2014;15(2):147-151. DOI: 10.1051/meca/

[12] Quentin B, Marc V, Salih A. Parallel singularities for the design of softening springs using compliant mechanisms. In: ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; August 2–5, 2015; Boston, Massachusetts, USA; DETC2015-47240; 2015. DOI: 10.1115/DETC2015-47240

[13] Gallego JA, Herder J. Synthesis methods in compliant mechanisms: An overview. In: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference & Computers and Information in Engineering Conference; August

[3] Amine S, Mokhiamar O, Caro S. Classification of 3T1R parallel manipulators based on their wrench graph. ASME Journal of Mechanisms and Robotics. 2017;9(1):011003. DOI: [14] Olsen BM, Issac Y, Howell LL, et al. Utilizing a classification scheme to facilitate rigid-body replacement for compliant mechanism design. In: ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; August 15–18, 2010; Montreal, Quebec, Canada; DETC2010– 28473; 2010. DOI: 10.1115/ DETC2010-28473

[15] Baker MS, Howell LL. On-chip actuation of an in-plane compliant bistable micromechanism. Journal of Microelectromechanical Systems. 2002; 11(5):566-573. DOI: 10.1109/ JMEMS.2002.803284

[16] Li B, Hao G. Nonlinear behaviour design using the kinematic singularity of a general type of double-slider four-bar linkage. Mechanism and Machine Theory. 2018;129:106-130. DOI: 10.1016/j.mechmachtheory.2018.07.016

[17] Zhao H, Bi S, Yu J. Nonlinear deformation behavior of a beam-based flexural pivot with monolithic arrangement. Precision Engineering. 2011;35(2):369-382. DOI: 10.1016/j. precisioneng.2010.12.002

[18] Pei X, Yu J, Zong G, et al. The modeling of cartwheel flexural hinges. Mechanism and Machine Theory. 2009; 44(10):1900-1909. DOI: 10.1016/j. mechmachtheory.2009.04.006

Section 3

Kinematics for Spacecrafts

and Satellites

85

Section 3
