# Coupling between the Output Force and Stiffness in Different Variable Stiffness Actuators

## Abstract

**:**

## 1. Introduction

## 2. SAM Classifications

#### 2.1. Antagonistic Mechanisms

#### 2.1.1. Simple Antagonistic Mechanism (C1)

#### 2.1.2. Cross-coupled Antagonistic Mechanism (C2)

#### 2.1.3. Bidirectional Antagonistic Mechanism (C3)

#### 2.2. Series Mechanisms

#### 2.2.1. Changing pretension of the nonlinear spring (C4)

#### 2.2.2. Changing the ratio of a compliant lever (C5)

**Figure 6.**Series class of stiffness adjustment mechanism based on changing the spring’s position (lever ratio).

## 3. Performance Analysis

_{Min}, the maximum output stiffness K

_{Max}and the maximum output deflection q

_{Max}. This parameters are, in fact, the most essential ones to define the energy characteristics of a SAM. Therefore, in this analysis, K

_{Min}, K

_{Max}and q

_{Max}are set to be 100 (Nm/rad), 1000 (Nm/rad) and 0.2 (rad), respectively. Furthermore, in some cases, the mentioned parameters in Table 1 are not adequate to define the unique performance of a VSA, as other additional parameters are also required. In this case, the performance of the SAM is optimized with respect to those additional parameters.

Parameter | Description |
---|---|

A | Radius of the link’s pulley |

θ_{1,2} | Motor position |

K | Output link stiffness |

p | Spring’s pretension |

x | Spring’s deflection |

K_{s} | Spring’s constant |

q | deflection of the output link |

f(x) | Spring’s force |

_{ext}also needs to be set equally for all different classes of SAM. Here, it is assumed that an external load of 20 (Nm) that can deflect the output link to its maximum deflection q

_{Max}= 0.2 (rad) at its minimum stiffness K

_{Min}= 100 (Nm/rad). Therefore:

_{ext}= K

_{Min}q

_{Max}

- (1)
- Only the principle of the mechanisms are considered, regardless of the properties of the actuation unites, e.g., motors and how the principle is realized. Therefore, some actuation-related parameters, such as output power and torque, are not considered in this study.
- (2)
- The work presented here draws upon our previous work on the modeling and stiffness characterization of different SAMs [31].
- (3)

#### 3.1. Simple Unidirectional

_{1}= θ

_{2}= θ. Furthermore, each spring has an initial pretension p when both motors are at rest position θ

_{min}= 0. In this case, the link stiffness is denoted as K

_{p}. The springs are considered to be extension nonlinear springs; therefore, the deflection of each spring, x

_{1}and x

_{2}, can be expressed as:

_{1}= p + θ − Aq

x

_{2}= p + θ + Aq

_{min}and the link deflection is maximum (q = q

_{max}), the deflection of Spring 1 becomes zero (x

_{1min}= 0). Therefore, based on Equation (5), the pretension p is:

_{max}

_{1}) − f(x

_{2}]

_{min}should be considered. The minimum stiffness occurs when both motors are at the zero position (θ

_{1}and θ

_{2}= 0); therefore:

_{min}= K

_{p}

_{Min}= 2g(p)A

^{2}

#### 3.2. Cross-coupling

_{3}is formulated as:

_{3}= p

_{3}− 2θ

_{3}is the pretension when both motors are at rest positions. To simplify the analysis, it is assumed that the constant of Spring 3 is the same as the other two. With the motors at the rest position, the net force applied to each motor is zero, and since all of the springs have the same constant, thus their deflection also has to be the same (p

_{3}= p). The stiffness of the link when both motors are at the zero position can be tuned by applying different pretension to Spring 3. Furthermore, in this case, when the link stiffness is minimum (K = K

_{Min}) and the link deflection is maximum (q = q

_{Max}), the deflection of Spring 1 becomes zero (x

_{1min}= 0). Therefore, based on Equation (5), the deflection of the third spring x

_{3}is:

_{Max}− θ

_{Min}

_{p}= 2g(p)A

^{2}

_{Min}= 2g(Aq

_{Max})A

^{2}

_{p}can be written as:

#### 3.3. Bidirectional

_{1}= p + θ − Aq

x

_{2}= p + θ − Aq

x

_{3}= p + θ + Aq

x

_{4}= p + θ + Aq

_{Min}) and the link deflection is maximum (q = q

_{Max}), the deflection of Springs 1 and 2 becomes zero (x

_{1min}= x

_{2min}= 0). Therefore, based on Equation (18), the pretension p is:

_{max}− θ

_{min}

_{1}) + f(x

_{2}) − f(x

_{3}) − f(x

_{4})]

^{2}

_{min}, pretension p and motor position θ can be formulated as functions of link stiffness and deflection [31].

#### 3.4. Changing pretension of the nonlinear spring

_{2}. At a deflection of the link equal to q, the torque acting on the output link is:

_{2}+ q)

_{2})

#### 3.5. Changing position of linear spring (lever ratio)

_{2}. Springs are inserted with a pretension p. The linear deflection of each spring while the link is deflected as q can be approximated as:

_{s}(p + ∆x) − K

_{s}(p − ∆x)] r cos q = 2K

_{s}r

^{2}sin q cos q

_{s}r

^{2}

## 4. Results and Discussion

_{Max}= 1000 (Nm/rad), as the output link is exposed to the external load of 20 (Nm). Cubic springs, however, show noticeable coupling between the load and stiffness in antagonistic classes. In the class C4, where stiffness is tuned by changing the spring’s pretension, quadratic and cubic springs present almost the same coupling, but again, using exponential springs can lead to a high coupling. Finally, the class C5 with a linear spring presents a weak coupling; however, the stiffness is not completely decoupled from the external force.

## 5. Conclusions

**Table 2.**Coupling between external force and the stiffness in different stiffness adjustment mechanism (SAM) classes using different types of springs.

SAM Class | Spring Type | Coupling Measure α |
---|---|---|

C1 | quadratic | 0 |

Exponential | 6.3 | |

cubic | 0.93 | |

C2 | quadratic | 0 |

Exponential | 6.3 | |

cubic | 0.93 | |

C3 | quadratic | 0 |

Exponential | 9 | |

cubic | 2.32 | |

C4 | quadratic | 0.76 |

Exponential | 6.6 | |

cubic | 0.89 | |

C5 | linear | 0.21 |

## Acknowledgment

## Conflicts of interest

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**MDPI and ACS Style**

Jafari, A.
Coupling between the Output Force and Stiffness in Different Variable Stiffness Actuators. *Actuators* **2014**, *3*, 270-284.
https://doi.org/10.3390/act3030270

**AMA Style**

Jafari A.
Coupling between the Output Force and Stiffness in Different Variable Stiffness Actuators. *Actuators*. 2014; 3(3):270-284.
https://doi.org/10.3390/act3030270

**Chicago/Turabian Style**

Jafari, Amir.
2014. "Coupling between the Output Force and Stiffness in Different Variable Stiffness Actuators" *Actuators* 3, no. 3: 270-284.
https://doi.org/10.3390/act3030270