# Design, analysis, and simulation of a new variable stiffness joint based on antagonist principle

María Guadalupe Contreras-Calderón, Med Amine Laribi, Marc Arsicault, Eduardo Castillo-Castañeda

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María Guadalupe Contreras-Calderón, Med Amine Laribi, Marc Arsicault, Eduardo Castillo-Castañeda. Design, analysis, and simulation of a new variable stiffness joint based on antagonist principle. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2022, 236 (18), pp.9991-10000. 10.1177/09544062221088714. hal-03830005

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# Design, analysis, and simulation of a new variable stiffness joint based on antagonist principle

María Guadalupe Contreras-Calderón<fnref n="1" /> ID, Med Amine Laribi<fnref n="2" /> ID, Marc Arsicault<fnref n="2" /> and Eduardo Castillo-Castañeda<fnref n="1" />

**Abstract**

In robotics, the variable stiffness joints have been developed as an alternative to include flexibility to a rigid mechanism. Variable Stiffness Joint (VSJ) allows for varying the stiffness in a mechanism during the performance of a trajectory. The development of this joint has increased, and various applications have emerged, most to improve human-robot interaction. This paper presents a VSJ based on the principle of antagonistic springs with a set of gears and racks that allow the extension or compression of the springs. The joint provides constant torque or variable torque to obtain a constant or variable force at the output of the link, so a relationship is between the torque provided by the spring's effort and the output force; however, it is possible to achieve a desired variable force with this joint design. Simulations are realized to obtain the behavior for both a constant and a variable force.

**Keywords**

Variable stiffness joint, antagonist springs, human-robot interaction, compliant joint

Date received: 16 November 2021; accepted: 1 March 2022

**Introduction**

Mechanisms that interact with humans must be safe against all possible accidents; for this reason, stiffness control is essential to ensure safe operation. In recent years, the development of variable stiffness joints has increased, and with the numerous applications, the most studied are those where robots share the workspace with a human. Traditional mechanisms are rigid, while devices with flexible elements enhance human-robot interaction and include high-precision, low-weight, and low-friction movements.<fnref n="1" /> It is possible to achieve mechanical compliance mainly by two mechanisms using flexible transmission elements or incorporating links of lightweight or elastic materials.<fnref n="2" /> A traditional mechanism can control its rigidity through the actuator's force control with force sensors as feedback; however, these mechanisms are still mechanically rigid compared to the human arm. The characteristic of each joint depends on the mechanism that is implemented to modify the stiffness. Various designs have been developed, and they are grouped into joints with a series configuration³⁻⁷ and an antagonistic configuration.⁴,⁸,⁹ Joints with series configuration incorporate a flexible element in the transmission mechanism and can be adjusted independently by a second actuator. Therefore, the stiffness depends on the location of the transmission element and the deflection of the lever.¹⁰ The design in Figure 1 uses a flexible bar with the cantilever beam principle, in which the stiffness changes according to the distance from movable support. Joints with an antagonistic configuration (Figure 1(b)) have elements of

flexible nature, or flexible transmission elements. They are coupled to the same joint but oppositely.<fnref n="2" /> Antagonistically joints employ actuators associated with compliant elements to provide the stiffness and torque behavior in the joint. This design is biologically inspired, a joint actuated by two muscles arranged in an antagonistic manner;⁴ for example, the biceps and the triceps during the elbow flexion, the biceps extends while the triceps contracts, and it is the opposite for elbow extension.

VSJ can be controlled by controlling each joint's stiffness, using active elements to change the property behavior of passive elements such as springs, flexible bars, and cables. The main characteristic property of VSJ is the output stiffness, which can be varied independently of the exit position. By replacing the conventional joints with the variable stiffness joints, the device can perform various tasks while varying its output stiffness or keeping constant. One of the difficulties of using these actuators is the increased complexity in trajectory planning and control, particularly at high

**Corresponding author:**

María Guadalupe Contreras-Calderón, Instituto Politécnico Nacional, CICATA-IPN, Cerro blanco #141, Colinas del Cimatario, Querétaro 76090, México.

Email: ma.gpe.contreras.c@gmail.com

<footnote n="1">¹Instituto Politécnico Nacional, CICATA, Querétaro, QRO, MX</footnote>

<footnote n="2">²PPrime, Departement Génie Mécanique Systémes Complexes, Futuroscope Chasseneuil Cedex, France</footnote>
speeds;¹¹ however, it simplifies the control problem since interaction control is conducted using position control rather than torque control. VSJs are used in tasks to damp motions that cannot avoid some amount of friction in the system,¹² these devices are used in tasks that require a specific force at the output of the device; therefore, they can restore forces without the use of motors.

High torque motors are required for a device to offer variable resistance to motion. However, VSJ improves the safety in the robot-human interaction providing intrinsic compliance, allowing robots to be rigid and accurate adding additional redundancy in control, improving the energy efficiency as the included motor provides the necessary torque to modify the passive elements properties, the motor does not provide the necessary torque in joints and dynamic range, also offering the possibility to achieve a high dynamic range with smaller motors.¹¹ Some VSJ designs, such as AwAS,¹³ in a traditional two-spring design placed in an antagonistic configuration, change the stiffness by varying the lever length. The extension of both springs makes the joint stiffer; relaxing both springs makes the joint more flexible; this joint has been tested in the ASIBOT¹² service robot. Some joints in refs. 8, 14–16 are based on antagonist configuration. The case of a VSJ used for human-robot interaction has a transmission belt that is tensioned by springs and non-linearly connects to the main shaft to the pair of opposing pulleys that are connected to motors DC with reverse-actuated position control. Concordant angular variations generate displacements and variations in stiffness.¹⁷ Other joints in refs. 18–22 are based on the cantilever beam principle where the support point of the beam is movable. This distance is related to the variation of the stiffness. Work²³ presents a joint using a lever, in which the deflection depends on the position of the pivot. The design of a VSJ proposed in ref. 14 provide

stiffness to the joint regardless of the rotation angle; it consists of an output shaft connected to a link, and the stiffness variation is a displacement function of two blocks with springs that are located below a bar attached to the shaft. In addition, the springs push the bar in both directions. The proposed VSJ is based on a single actuator. As a consequence, the motion profile (position or trajectory) and the force profile (based on desired stiffness) are enforced separately since changes in the joint position affect the stiffness and vice versa. This represents the behavior of antagonistic VSJ with an interesting contrast and requests a joint hybrid control. The present work presents the design and analysis of a new variable stiffness joint based on the antagonistic principle springs coupled to a set of gears and racks. Also, it presents the joint simulation in a rotation case to analyze its performance.

## Variable stiffness joint working principle

The design of the variable stiffness joint (VSJ) can provide variable torque and maintain a constant torque independently of the rotation angle. With variable torque, a variable force at the end of the link is obtained and with constant torque is obtained constant force. The proposed VSJ incorporates two antagonist springs $R_1$ and $R_2$, with lengths $l_1$ and $l_2$, respectively, and are attached to a link (Figure 2). It considers three positions of the VSJ, equilibrium, clockwise, and counterclockwise. In the equilibrium position (Figure 2(b)), the springs are at their initial length, and from this position, the joint can rotate counterclockwise (Figure 2(c)), and the spring $R_1$ is extended, and $R_2$ is compressed. When the joint rotates clockwise (Figure 2(a)), the spring $R_1$ is compressed while $R_2$ is extended.

The spring’s length changes according to the direction of rotation. The new VSJ considers the extension and compression lengths of the springs to vary the stiffness. For linear springs behavior, the compression length is $l_c = l_i - x_m$ and $l_e = l_i + x_m$ is the extension length (Figure 2(b) and (c)), where the length $x_m$ depends on the rotation angle amplitude and the link radius. It follows then that the greater the angle of $\theta$ is, the torque $\tau$ increase. A

<figure><img src="image_2.png" /><figcaption>Figure 1. (a) Serial configuration; (b) Antagonistic configuration.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. VSJ diagram: (a) Clockwise rotation; (b) Equilibrium position; (c) counterclockwise rotation.</figcaption></figure>
relationship is established between the torque and the output force at the end of the link. When the torque is variable, the output force is also variable. If the torque is constant, then the output force is constant. If the extension/compression of the spring is controlled, then the constant force at the end of the link is possible. The length variation is obtained from the springs' ground displacement. The extension or compression length is $x_{li}; i \in \{1,2\}$ as shown in Figure 3. So the total length ($X_{Tl}; i \in \{1,2\}$) of the spring is the sum of $x_m$ and $x_{li}$, as shown in Figure 3. Adjusting the distance from the ground allows modifying the spring length and can be achieved using linear or rotational actuators; for this VSJ, a rotational motor is chosen to transmit the movement through a set of gears (as reported in Figure 4).

The transmission ratio of the gears allows that during the counterclockwise rotation (as shown in Figure 4), the spring $R_1$ extends while the $R_2$ that should remain in compression is also extended, the spring does not exceed the compression limit. The joint design reported in Figure 4 considers the variation of the length of the spring; the link has a

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Adjusting the distance from the ground.</figcaption></figure>

gear ($P_e$) in the rotational joint, two springs ($R_1$ and $R_2$) are attached to one of its ends, while the other end is attached to the mechanical ground. The gear ($P_e$) engages with two racks ($C_3$ and $C_4$), from it is held the springs ($R_1$ and $R_2$), and they are positioned parallel to the link, which allows the linear transmission of the joint rotation to the springs. When the link rotates counterclockwise, rack $C_4$ extends the spring $R_1$, while rack $C_3$ compresses the spring $R_2$ because of the antagonist principle. When the rotation is clockwise, then spring $R_2$ is extended while $R_1$ is compressed. The other springs end, $R_1$ and $R_2$, are attached to two racks $C_1$ and $C_2$, respectively; are engaged these racks with a set of gears ($P_1$, $P_2$, and $P_3$). Gears $P_1$ and $P_2$ engage to racks $C_1$ and $C_2$, respectively. The gear $P_3$ has a coupled motor, which allows its rotation; in this way, the pinion $P_3$ transmits the movement to the gears $P_1$ and $P_2$, which allows racks $C_1$ and $C_2$ to extend the springs, modifying their length. The racks $C_1$, $C_2$, $C_3$, and $C_4$ supports are fixed to the ground. The motor rotation moves the gears set; in this way, the spring lengths vary with the control of the position and allow a constant force at the link output. In the next section, a relationship is established between the length of the spring and the force at the end of the link.

## Variable stiffness joint mathematical model

The spring that extends is considered as the one that provides the force during the rotation. In the case of counterclockwise rotation, as reported in Figure 5, the spring $R_1$ is the one that provides the force to the joint. The length by gears set $x_{li}$ must vary through the gear $P_3$ rotation to obtain a constant force at the end of the link. Hooke's law for

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Adjusting the spring length with a motor.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5.** VSJ diagram.</figcaption></figure>
spring analysis states that the force F is proportional to the extension

$$F = kx$$

where k is the spring constant and x corresponds to the spring length that is extended or compressed. Total extension/compression of the spring can be written as follow

$$X_{Ti} = x_{li} \pm x_{m} \quad i = 1, 2 \quad (1)$$

where $x_{li}$ = length by gears set and $x_{m}$ = rθ.

Therefore, in a counterclockwise rotation as in Figure 5, one obtains:

Extension: $X_{T1} = x_{l1} + x_{m}$

Compression: $X_{T2} = x_{l2} - x_{m}$

So the force is

$$F_{i} = k(x_{li} \pm x_{m}) \quad (2)$$

According to the rotation (Figure 5), the forces for each spring are

$$F_{1} = k(x_{l1} + x_{m}) \quad (3)$$

$$F_{2} = k(x_{l2} - x_{m})$$

<figure><img src="image_7.png" /><figcaption>Figure 6. Numerical simulation of VSJ: (a) Angular displacement of θ (b) $x_{li}$ length of springs by a set of gears.</figcaption></figure>
Using the desired force at the end of the link the distance $x_{li}$ can be calculated as

$$x_{li} = \left[ \frac{F_i}{k} \right] \pm x_m, i = 1, 2 \quad (4)$$

The applied torque due to the springs is

$$\tau = r(F_1 - F_2)$$

$$\tau = rk(x_{l1} - x_{l2}) + 2kr^2\theta \quad (5)$$

The gear rotation controls that length; for this reason, it is important to determine the relationship between the length

$x_{li}$ and racks displacement and obtain angle rotation of the gear $P_{1,2}$. Where $r_{Pi}$ is the radius of the gear

$$\theta_{P_i} = \frac{x_{li}}{r_{Pi}} \quad (6)$$

The rotation angle $\theta_{P3}$ of gear $P_3$ is determined by

$$\theta_{P3} = \theta_{P1} \frac{Z_{P1,2}}{Z_{P3}} \quad (7)$$

where $Z_{P1,2}$ is the number of gears teeth $P_1$ and $P_2$. $Z_{P3}$ is the number of gear teeth $P_3$.

<figure><img src="image_8.png" /><figcaption>Figure 7. Numerical simulation of VSJ: (a) $P_3$ angular position; (b) Torque $\tau$ behavior.</figcaption></figure>
Simulation

The mathematical model of VSJ is verified through a
simulation under MATLAB of an oscillatory rotation, from
60° to -60° as reported in Figure 6(a)) to evaluate the joint
stiffness behavior. The simulation considers the antagonist
springs identical with a length of 0.0385 m and a constant
k = 340 N/m. The input value of force at the end of the link is
considered as F = 30 N constant. The behavior of x li is in
Figure 6(b), with the maximum length from -0.03 m to
0.03 m.

The design of the set of gears P₁ and P₂ is considered
identical. These gears have a module of M<sub>P1,2</sub> = 1.25, with a
number of teeth Z<sub>P1,2</sub> = 20. Gear P<sub>3</sub> has a number of teeth Z<sub>P3</sub> =
10. The relationship between the length x<sub>li</sub> and the rotation

angle gear P<sub>3</sub> is calculated. Figure 7(a) shows an oscillatory
behavior of θ<sub>P3</sub> with values between -272° and 273.3°.
Figure 7(b) describes the expected torque τ behavior at the
given trajectory of θ. One observes the constant torque τ value.

The joint also allows a variable torque and adjusts the
spring length to increase or decrease. This latter is obtained
by keeping the ground fixed at a specific value. A simu-
lation is realized with the same values as the previous
simulation. The difference is that the ground is left fixed at
three different values, as reported in Figure 8(a) x<sub>li</sub> = 0.01 m
(blue line), 0.02 m (red line), and 0.03 m (yellow line).
Figure 8(b) shows the corresponding position of gear P<sub>3</sub> to
reach the length of x<sub>li</sub>, which corresponds to 91.67°, 183.3°,
and 275°, respectively.

<figure><img src="image_9.png" /><figcaption>**Figure 8.** (a) Fixed values for x<sub>li</sub>; (b) Gear position P<sub>3</sub> to keep x<sub>li</sub> fixed.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 9.** Springs length at different values, $x_{li} = 0.01, 0.02$ and $0.03$ m.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10.** Torque $\tau$ behavior at any value of $\theta$, with different lengths of $x_{li}$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 11.** Stiffness behavior versus $\theta$, with different lengths of $x_{li}$.</figcaption></figure>

Figure 9 shows the VSJ configuration for each $x_{li}$ value, where the gears set extends both springs, modifying their length.

Figure 10 shows the torque $\tau$ behavior for each value of $\theta$. It is observable that the torque $\tau$ increases when $\theta$ increases. Conversely, torque $\tau$ decreases when $\theta$ decreases and the behavior is reversed from zero.

The torque $\tau$ behavior for value $x_{li} = 0.02$ m (blue line) is from -4.09 to -4.7Nm. When the $x_{li} = 0.03$ m (red line), the torque is from -3.9 to -4.6Nm. At value $x_{li} = 0.04$ m (yellow line) the torque is -3.8 to -4.47Nm.

Figure 11 shows the joint stiffness behavior at different lengths of $x_{li}$. The blue line represents the stiffness with a value of $x_{li} = 0.01$ m, the red line corresponds to $x_{li} =$
<figure><img src="image_13.png" /><figcaption>**Figure 12.** VSJ CAD Design: (a) Variable stiffness joint CAD proposal; (b) Exploded view.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 13.** VSJ prototype.</figcaption></figure>

0.02 *m*, and the yellow line to $x_i = 0.03 m$. It can be seen that the stiffness increases as $x_i$ increases.

## Variable stiffness joint design and prototype

**CAD design.** Following the proposed principle, the conceptual design consists of two springs attached to a set of gears and racks. The stiffness is controlled by the motor placed in gear $P_3$. The rotation of $P_3$ allows the racks to slide and stretch the springs. The gear ($P_e$) in the link transmits the rotation to the racks to generate a linear spring movement.

Figure 12(a) shows a design top view of the variable stiffness joint. Figure 12(b) shows a design exploded view, where all the elements that compose the joint.

**Variable stiffness joint prototype.** Figure 13 shows the VSJ prototype. It is made of aluminum alloy; the pinions and

<figure><img src="image_15.png" /><figcaption>**Figure 14.** VSJ experimental test.</figcaption></figure>

racks are made of acetal plastic (Delrin – POM). The motor that generates the rotation in the gear $P_3$ is a Dynamixel with a torque of 8Nm. The link length is 0.150 *m*. The gears corresponding to $P_1$, 2 have $Z_{p1,2} = 20$ teeth, and the gear coupled to the motor $Z_{p3} = 10$ teeth, with a transmission ratio of $i = 2$. The prototype uses extension springs,²⁴ which absorb and store elastic energy and create resistance to a pulling force. Both springs are considered identical, and
<figure><img src="image_16.png" /><figcaption>**Figure 15.** Force behavior at the end of the link.</figcaption></figure>

their characteristics are constant $k = 340 \ N/m$ and a free length of $0.0385 \ m$.

*Experimentation.* Experimental tests were carried out on the variable stiffness joint. In this case, the springs ground is fixed to obtain a variable force at the link output. Gear $P_3$ is turned with the motor to move the ground of the springs a length $x_{li}$. $P_3$ is positioned in $184^\circ$ to obtain a fixed $x_{li}$ distance ranging of $0.02 \ m$. A sensor is fixed at the end of the link (as shown in Figure 14) to measure the force during the joint rotation of $\theta = -20^\circ$ to $20^\circ$. The force behavior in the function of $\theta$ is shown in 15 (blue line). Also, was carried out a joint simulation with the same conditions as the experimental test; the theoretical force behavior is the orange line in 15. With this measurement, it is expected that the force increase when the angle $\theta$ increase.

It is noticeable, in Figure 15, the comparison of the experimental and the theoretical curves are similar. The behavior is symmetrical for negative and positive values of $\theta$.

## Conclusions

In this paper, a joint of variable stiffness has been presented. It was designed with the principle of antagonistic springs attached to a joint that, when rotating, one spring is compressed while the other spring is extended. The length of the springs is adjusted depending on desired output force at any angle through gears and racks set. If the length is fixed, it is possible to obtain a variable output force, and it can be increased by increasing the prestressing of the springs if they are fixed. The joint mathematical model was established based on Hooke's law, where the spring stiffness is related to the force obtained at the end of the link, considered as the joint output. A trajectory was simulated to obtain a constant output, which corresponds to a behavior of gear $P_3$ leading to a constant force. A variable force was also simulated by setting the length of the springs to a specific value, where the $P_3$ gear behavior is constant. The prototype was designed and fabricated. The obtained results through simulation as well as experimentation both show the feasibility of the proposed

joint. The output angle was measured at different weights with a fixed-length $x_{li}$. The obtained results validate the proposed concept and disseminate that the angle range decrease as the length $x_{li}$ increases. In the same way, if the length $x_{li}$ increases, the force increases. Future work will be focused on the motor control to maintain a constant joint torque, also on choosing the optimal elements for customizing the VSJ, such as the size of the gears and racks, the optimal spring characteristics, and the motor that should have the ability to stretch the springs. One of the applications for this type of joint is to design mechanisms for arm rehabilitation in which the force of the end effector must be controlled.

## Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

## Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the program EcosNord with number M19M03 and entitled: "Optimal Design and advanced control of rehabilitation devices for upper aRm (Odor)".

## ORCID iDs

María Guadalupe Contreras-Calderón https://orcid.org/0000-0003-3916-9893

Med Amine Laribi https://orcid.org/0000-0003-0797-7669

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