This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this paper, a novel variable stiffness mechanism is presented, which is capable of achieving an output stiffness with infinite range and an unlimited output motion,

Traditional robotic systems have stiff structures and are actuated using stiff joints,

Many different implementations of VSA mechanisms have been presented in the literature, which can be roughly classified into three major groups [

In this paper, the novel design of a variable stiffness mechanism (VSM) is presented, which fits in the group in which the transmission between load and spring is changed. The novelty lies in the property that when the mechanism is in its zero stiffness configuration, the output is completely decoupled from the input (the rotor) and an unlimited output motion is then possible (unconstrained, infinite rotation). Therefore, the output behavior will only be a result of the dynamics of the output load. Moreover, the actuator is capable of achieving an infinite stiffness range (from truly zero stiffness to infinite stiffness,

This paper is structured as follows: first, the conceptual design is presented in

This section elaborates on the conceptual design, showing the working principle of the VSM and the key design features that enable an infinite stiffness range and completely decoupled and infinite output motion from the input.

The novel VSM, shown in

The novel variable stiffness mechanism creates compliance between the rotor and output by means of leaf springs, and it is made variable by means of support pins that can be moved in between the leaf springs by using a hypocycloid gearing mechanism.

The working principle of the concept (of only one leaf spring) is shown in

Note that the device presented here is not yet an actuator in the sense that it has motors and can deliver power to a load. Currently, it is a conceptual mechanism to which actuators can be connected.

The working principle of the variable stiffness mechanism. The deflection shape of the leaf spring is varied by changing the location of the supports, and therefore, the perceived stiffness at the location of the force application points is changed.

The support pins, at location

The supports pins should be moved in a straight line along the leaf spring, which is achieved using a hypocycloid gearing mechanism (similar to a planetary gearing system, previously proposed in [

The pin used as one of the two supports for the leaf spring.

Schematic representation of the motion of the supports in between the leaf springs. Contact with both leaf springs is always ensured, because of the pin shape (shown in red).

This section elaborates on the VSM modeling. A leaf spring was modeled as a beam to analyze its deflection for specific support locations and applied forces. Although the mechanism consists of two leaf springs, only one leaf spring is modeled, since an otherwise highly complex model would likely result from the modeling of the coupling between the leaf springs by the clamps. A finite element model (FEM) of the support pin was used to analyze the stress on the pins as a result of the actuator torque.

The hypocycloid gearing mechanism used to move the supports in a straight line along the leaf spring.

The leaf spring is modeled as a doubly-supported, doubly-overhanging rectangular Euler–Bernoulli beam with a force couple on the ends due to the mechanism output torque (refer to

The geometry of the beam and the coordinate frame that is used, which is placed in the center of the beam, is shown in

Geometry and coordinate usage of a generic beam.

The general linear Euler–Bernoulli beam equation is given by:

Expanding the terms, this equation can be given by:

This is a fourth order, first degree, ordinary differential equation with variable coefficients.

For a uniform beam,

This is a fourth order, first degree, ordinary differential equation with constant coefficients.

The beam deflection can be solved by separately taking into account the individual loading segments and choosing proper boundary conditions to form a continuous solution. However, the loading profile can also be specified in a way such that it is valid over the complete beam length and solving it for the complete beam at once. The loading on the leaf spring consists of four forces, as shown in

Force profile on the leaf spring, consisting of four forces: two applied forces,

The loading

The fourth order differential equation of (

Young’s modulus

The leaf spring (thick solid line) is fixed to the output of the variable stiffness actuator (VSA) mechanism (grey ring), which causes the leaf spring slope at its ends (dashed line) to be coincident with the mechanism center and perpendicular to the tangent of the output ring.

When solving the equation in (

The beam shape for several settings of the support location,

When the leaf spring is not uniform, but either variable over length

In

Note that the model presented here is based on linear beam theory, and therefore, the model is less accurate for larger deflections.

The support pins should have a special shape to allow two of those pins to join together, while still ensuring that the complete space between two leaf springs in the mechanism is filled, such that there is no play between the output and the rotor. The support pins should also be able to withstand the forces as a result of force

The force on a support pin is equal to

Support force

The maximum force,

The stress analysis of the support pin fixed to a gear of the hypocycloid gearing mechanism. The applied loading is given by the maximum force on the pin under a certain angle.

Coordinate

Experiments have been performed to measure the stiffness characteristic of the proposed variable stiffness mechanism. As stated before, the stiffness characteristic that is a function of the support location,

Three different leaf spring shapes, shown in the x-y-plane. Leaf Spring 1 is a uniform beam, and Leaf Springs 2 and 3 are beams with a negative and positive parabola cutout, respectively.

The experimental setup used to measure the stiffness characteristic of the variable stiffness mechanism. A magnetic sensor is placed over the device to measure the output rotation, while a force/torque sensor is placed under the device to measure the applied torque. The support pins can be placed at predefined locations.

Since the model assumes only one leaf spring, also one leaf spring was first measured in the setup. Since the mechanism is designed for using a double-leaf spring, a single-leaf spring was fixed to the output and placed in between the two supports with a slight pretension, such that it is possible to measure the torque/deflection characteristic in one direction. See

Deflection/torque measurement results using a single-leaf spring (for clarity, only half of the measurements are shown). The dashed line represents the simulated deflection/torque data, and the dotted line is the linear least squares approximation to the deflection/torque data. (

Stiffness/support location plots for a single-leaf spring based on a linear least squares approximation to the deflection/torque data. (

Using the same measurement procedure, the situation for the regular configuration of two leaf springs was measured, and these measurements were again used to obtain a support location/stiffness plot. Modeling the effect of a double-leaf spring constrained by clamps would likely result in a highly complex model, so this effect was empirically tested by normalizing the double-leaf spring stiffness at each support location to the stiffness obtained when using only one leaf spring, for every leaf spring shape. This results in

Normalized stiffness derived from the double-leaf spring measurement, normalized to the single-leaf spring measurement, which shows the effect of the double-leaf spring and clamps. This effect is approximated by a linear regression for increasing support location

Deflection/torque measurement results using a double-leaf spring (for clarity, only half of the measurements are shown). The dashed line represents the simulated deflection/torque data, and the dotted line is the linear least squares approximation to the deflection/torque data. (

Stiffness/support location plots for a double-leaf spring based on a linear least squares approximation to the deflection/torque data. The simulated line is adapted using the measured effect of the double-leaf spring and the clamps with respect to a single-leaf spring. (

Stiffness plots for the three different leaf spring shapes, making the possibility of shaping the stiffness characteristic of the variable stiffness mechanism (VSM) explicit.

In

In this paper, a novel variable stiffness mechanism has been presented, which is capable of an infinite stiffness range not before encountered in this class of variable stiffness realization and completely decoupled unlimited output motion with respect to the rotor for safe passive behavior. An important feature of this mechanism is the ability to easily change the stiffness characteristic by shaping the leaf springs. A Euler–Bernoulli beam model is proposed to model one of the two leaf springs that are present in the mechanism. Experiments using PETG copolyester show the validity of this model for single-leaf springs of various shapes. The stiffness measurement using the double-leaf spring is normalized to the single-leaf spring stiffness measurement, to measure the effect of coupling a leaf spring by clamps to another leaf spring. This effect is approximated and used to adapt the model data to correspond to the double-leaf spring measurements. In this way, a close approximation of the measurements to the model data is achieved. The result of shaping the leaf springs is shown and agrees with the expectation.

A different realization of the same concept might be explored in future research, possibly one that uses only one leaf spring and one circular support with a groove, in which the leaf spring is placed, that is mounted with a bearing on the planet gear. Furthermore, a fixed support with two quarter cylinder cut-outs may be possible. Note that these different realizations do not change the properties of the mechanism, and it should be investigated whether these realizations have advantages over the presented mechanism. See

Alternative realizations of the VSM concept. One uses a pin with a groove; the other uses a fixed cylindrical pin with two quarter cylinders cut away (white parts).

Gerben ter Riet o/g Scholten is acknowledged for his contributions in the prototype design.

The research and development of the prototype was done by Stefan Groothuis, based on the preliminary conceptual foundation by Stefano Stramigioli. Raffaella Carloni co-advised Stefan Groothuis and she was involved in the previously published conceptual designs that have led to the work presented in this manuscript.

The authors declare no conflict of interest.