## Abstract

This paper proposes a compact and reconfigurable variable stiffness actuator (VSA) using disc spring which is named as SDS-VSA (symmetrical disc spring variable stiffness actuator). To enhance the actuator’s torque density, symmetric compression springs are integrated into the cam-roller-spring mechanism, and a disc spring combination design is employed instead of conventional springs. The disc spring configuration is tailored to achieve a broader stiffness range within a limited size, aligned with stiffness and dimensional requirements. Subsequently, the dynamics of the cam-roller-spring mechanism are derived. To tackle the challenge of strong coupling dynamics, a decoupled modeling method by introducing mismatched and matched disturbances is proposed. A back-stepping tracking controller and proportional-derivative (PD) controller with feedforward are proposed to track the link-side and stiffness motor-side trajectories, respectively. Tracking experiments under significant stiffness alteration are conduced to verify the performance of the prototype SDS-VSA.

## 1 Introduction

Inspired by the human’s musculoskeletal structure [1], variable stiffness actuators (VSAs) have been proposed and applied in humanoid robotic arms, exoskeletons, and legged robot systems. The fundamental idea behind VSAs is to introduce passive compliant elements into rigid actuators and regulate their stiffness through redundant actuation mechanisms. The incorporation of variable compliance enhances robustness, safety [2], and dynamic performance [3] in human–robot interactions. Therefore, variable stiffness actuators hold significant potential for resolving the trade-off between safety and energy efficiency in human–robot physical interactions.

The initial designs of VSAs were inspired by the human musculoskeletal structure, simulating the arrangement of the biceps and triceps muscles at the human elbow joint. These types of actuators employ pairs of elastic elements and motors, arranged in an antagonistic manner on both sides of the output link. Antagonistic actuators control stiffness when the motors move in opposite directions and control torque when they move in the same direction [4]. However, this also leads to a strong coupling between position and stiffness, making them difficult to control independently, reducing controllability, and limiting their adaptability to application scenarios.

Therefore, the utilization of independent motor-based antagonistic mechanism became prevalent [5–9], where position and stiffness are controlled by separate motors. This decouples the control of position and stiffness, simplifying the design and control complexity. Building upon this, various designs emerged based on different principles to achieve variable stiffness, the mechanisms that change spring preload, such as the cam-roller-spring structure found in VSJ [10], FSJ [11], and LVSA [12], the mechanisms that alter the output transmission ratio, like lever-based systems such as AWAS [13], AWAS-II [14], CompAct-VSA [15], and RJVS [16], and the mechanisms that modify the physical characteristics of flexible elements [17].

The mechanical structure of variable stiffness actuators varies widely, and their variable stiffness performance needs to be tailored to specific application demands. The range of stiffness adjustment is a critical factor affecting the performance of periodic motions, where frequency and amplitude are limited by the system’s modal frequency. Impact-type movements, on the other hand, require the ability to generate instantaneous bursts of force, making peak torque and stiffness adjustment speed crucial [18]. Key performance indicators for variable stiffness actuators include adjustable stiffness range, stiffness adjustment speed, rated torque, peak torque, and energy density. These performance metrics are interconnected and sometimes conflicting due to the constraints of the stiffness adjustment structure. For example, the use of springs in stiffness adjustment mechanisms often leads to the output torque being proportional to the spring’s elastic potential energy. Larger elastic potential energy can increase the size of the actuator, reducing its energy density.

To address these challenges, a novel mechanism based on the lever mechanism principle and a change in the method of wire winding is proposed, which exhibits multiple modes and stiffness-torque characteristics. This actuator is capable of adapting to the diverse interaction requirements of exoskeleton wearers [19,20]. Others have suggested lever-based and reconfigurable plate spring mechanisms to achieve different stiffness variation ranges [21–24]. The research of reconfigurable mechanisms and the synthesis of multi-objective performance mechanisms are promising in enhancing the requirement of variable stiffness actuators in different scenarios. However, the previously proposed reconfigurable structures still have limitations in achieving a balance between output torque and mechanism size. Larger output torque often results in increased dimensions of flexible components, which in turn enlarges the overall mechanism size, thereby limiting the generality of their applications.

The main contribution of this work is the proposition of a compact and reconfigurable variable stiffness mechanism utilizing disc springs (symmetrical disc spring variable stiffness actuator (SDS-VSA)). In contrast to prior research that utilized a single extension spring [11], a novel cam-roller-spring mechanism with two opposing and symmetric compression springs is proposed to enhance the output torque. Furthermore, the disc springs are combined to achieve a compact structure and a broader stiffness range according to stiffness and dimensional requirements. The major difference between the proposed SDS-VSA and the FSJ in Ref. [11] lies in the reconfigurability of the VSA achieved through the combination of disc springs. Different stiffness coefficient springs can be configured without altering the internal space, sparing VSAs from repetitive and tedious design processes based on various application requirements and indirectly enhancing energy density and output torque. To the best of authors’ knowledge, it is the first time disc springs are employed in VSA-related research. What is more, due to the strongly coupled dynamics of VSAs, a decoupled modeling approach is introduced to decouple the dynamics into two subsystems: the link-side subsystem and the stiffness motor-side subsystem. Subsequently, a robust back-stepping tracking controller is proposed to track the link trajectories under highly dynamic stiffness alterations.

The remaining sections of the paper are organized as follows. Section 2 introduces the cam-roller-spring mechanism, the combination design method of disc springs, along with mechanical design and prototype construction. The force analysis of the cam-roller-spring mechanism is presented in Sec. 3. Subsequently, the stiffness model and decoupled dynamics model are derived. In Sec. 4, the back-stepping tracking controller for the link-side subsystem and a PD controller for the stiffness motor-side are introduced. Section 5 presents the experiments and their results. In Sec. 6, the advantages and limitations of this work are discussed. Finally, the conclusion is presented in Sec. 7.

## 2 Mechanical Design

### 2.1 Principle of SDS-VSA.

SDS-VSA is designed and implemented based on the principle of cam-roller-spring mechanism. Its main components of the stiffness adjustment mechanism consist of two symmetrically arranged compression springs, cam-guided rollers with their roller retainers, and two cylindrical cams with linear guided cam sliders as depicted in Fig. 1(a). The primary feature lies in its differentially arranged cam disc layout, achieving a compact overall structure. The working principle involves fixing the cam-guided rollers onto the roller retainers and mounting the cylindrical cams on cam sliders with linear guides on both sides. The presence of cam sliders ensures axial movement of the cams. Simultaneously, the two cam discs are tightly compressed together using compression springs. The force exerted by the springs maintains close contact between the cam discs and rollers. When the rollers press the cam discs, generating axial movement, the cam discs move axially along the cam sliders, compressing the springs and generating spring force. The stiffness adjustment of SDS-VSA is achieved by altering the relative angles of the two cylindrical cams, thus adjusting the preloading force of the springs and achieving different joint output stiffness characteristics.

As shown in Fig. 1(a), the variable stiffness mechanism in this work employs symmetrically arranged compression springs on both sides. When the cams on both sides undergo axial movement due to roller compression, they generate spring forces from the compressed springs, thus producing joint output torques that counteract the loading torque. The rollers in the variable stiffness joint are fixed to roller retainers, which are also connected to the output end of the joint position motor’s reducer. Due to the initial spring force of the compression spring, the rollers are tightly enveloped by the cams on both sides. The cam discs on both sides are connected to the joint link as a whole. When the link experiences external forces or loads, the force acts directly on the cam discs on both sides. As a result, the rollers, serving as the power input, drive the link to change position and complete the corresponding motion tasks. On the other hand, the joint stiffness output is achieved by changing the relative angle of the cam discs on both sides. One cam disc is connected to the output end of the stiffness motor’s reducer, while the other cam disc is fixed to the link. When the stiffness motor drives the relative position of the cam discs on both sides, the initial value of the joint stiffness output is changed.

In the initial position of the variable stiffness joint, as shown in Fig. 1(a), the rollers are in balance with the cam discs on both sides, resulting in zero output torque. At this point, the joint’s stiffness output is provided by the initial preload force of the spring, and a larger preload force leads to a higher initial stiffness. As shown in Fig. 1(b), under static conditions, the position of the cam discs deviates from the original equilibrium position due to external torque. Since the cam discs on both sides tightly compress the rollers through spring force, the axial movement of the cam discs on both sides generates spring forces due to the compression of the springs caused by roller compression. The axial displacement of the cam discs is determined by the cam curve, and thus the output characteristics of the joint can be customized by designing the cam curve.

The operation principle of the symmetric compression spring stiffness adjustment structure is shown in Fig. 1(c). At the beginning of the adjustment process, the rollers remain at the equilibrium position, and the cam discs on both sides experience relative rotation due to the driving force. The relative position of the cam discs on both sides is determined by the output angle of the stiffness motor. During the rotation process, the rollers still remain at the equilibrium position due to the symmetric nature of the cylindrical cam surface. However, the cam discs in space experience axial displacement due to the pressure exerted by the rollers, resulting in axial movement. Assuming consistent cam surface shapes, the symmetric compression springs provide symmetric spring forces on both sides. Because the cam curve is incrementally increasing, when the cam discs on both sides rotate more toward the roller direction, the axial displacement of the cam discs increases, leading to an increase in spring force. This results in a steeper curve for the joint stiffness output. Conversely, when the cam discs on both sides rotate away from the roller direction, the preload force of the spring decreases, leading to a flatter stiffness output curve.

### 2.2 Disc Spring of SDS-VSA.

Disc springs offer a compact design and high stiffness, allowing substantial force to be generated with minimal deflection within confined spaces. This characteristic makes disc springs highly suitable for utilization as elastic elements in variable stiffness joint structures. Moreover, combining disc springs enables the creation of equivalent compression springs with diverse stiffness coefficients, introducing modularity and reconfiguration to the elastic components.

As shown in Fig. 2, stacking and facing are the fundamental arrangements for disc springs. In Fig. 2(a), a facing disc spring can be regarded as two regular springs connected in serial, with an equivalent stiffness coefficient of *K*_{1}*K*_{2}/(*K*_{1} + *K*_{2}). Additionally, as illustrated in Fig. 2(b), a stacked disc spring can be seen as two regular springs connected in parallel, yielding an equivalent stiffness coefficient of *K*_{1} + *K*_{2}. The stiffness coefficient of disc springs exhibits nonlinear characteristics. For computational convenience, this study assumes a fixed value for the disc spring’s stiffness coefficient. Thus, *K*_{1} and *K*_{2} represent the stiffness coefficients of the disc springs. Through various combinations of stacking and facing arrangements, the disc springs can be flexibly combined to create equivalent compression springs with different stiffness coefficients. In the case of the combined spring shown in Fig. 2(c), the stiffness coefficient is (*K*_{1} + *K*_{2})/2. In stiffness-adjustable mechanisms based on altering the preloading force of springs, selecting and calculating the spring parameters becomes crucial to balance overall joint dimensions and torque output. However, due to the wide range of spring options, choosing suitable spring parameters can be challenging. While variable stiffness joints find widespread applications, each application requires customized selection and calculation of the variable stiffness joint, adding significant repetition to the design process. The following section introduces a design method for equivalent springs based on a combination of disc springs.

Using a combination of stacking or facing with *N*_{1} disc springs, an equivalent compression spring can be obtained. Each pair of disc springs can be connected in only one way, resulting in *M*_{1} possible stacking arrangements within *N*_{1} disc springs. When the spatial dimensions and stiffness coefficients of the flexible components in the variable stiffness joint are specified, a comparison table of combined disc spring stiffness coefficients can be established. Then, based on Fig. 3, the required disc spring combination and model can be determined. The free combination of disc springs essentially boils down to an integer partition problem. When dealing with a large number of disc springs, dynamic programming or generating functions can be employed to compute the partition combinations.

Taking the example of six disc springs, there are 11 distinct partitioning ways: (1,1,1,1,1,1), (2-2-2), (2-2-1-1), (2-1-1-1-1), (3-3), (3-2-1), (3-1-1-1), (4-2), (4-1-1), (5-1), and (6). Here, 2, 3, 4, 5, and 6 indicate the number of stackings. By increasing the number of disc springs in a combination, a wider range of equivalent stiffness coefficients can be achieved. In the case of variable stiffness joints, where space is limited, achieving the desired equivalent stiffness coefficient within confined dimensions is crucial for engineering implementation. Apart from increasing the number of spring elements, choosing disc springs with different outer dimensions can also expand the range of stiffness coefficient options. Disc springs with the same outer and inner diameters but varying thicknesses exhibit different stiffness coefficients, as shown in Table 1. Thus, by maintaining the outer dimensions and selecting disc springs with varying stiffness coefficients, it is possible to achieve the desired equivalent stiffness coefficient for the compression spring. For a configuration with *N*_{1} disc springs and *M*_{2} different stiffness coefficients, the calculation expression for the number of free combinations is $CM21+CM22CN1\u221211+\u2026+CM2M2CN1\u22121N1\u2212M2+1$. $Cji$ denotes the combination where *j* represents the total number of items and *i* represents the number of items being chosen at a time. For example, with six disc springs having five different stiffness coefficients, there are 210 distinct combinations. Therefore, the equivalent stiffness coefficient of the combined disc springs, as shown in Fig. 3, falls within ranges of [114.5397, 64,432] N/mm, [68.7179, 26,347] N/mm, and [37.9722, 10,946] N/mm for three typical outer dimensions of disc springs. Thus, the utilization of combined disc springs enables the design of compact structures with a wide stiffness range, facilitating a simplified design process for variable stiffness joint structures and enhancing their modularity and scalability.

Outer diameter, D (mm) | Inner diameter, d (mm) | Thickness (mm) | Stiffness (N/mm) | Maximum force (N) |
---|---|---|---|---|

31.5 | 16.3 | 0.8 | 687.2 | 721.6 |

31.5 | 16.3 | 1.25 | 2621.1 | 2359 |

31.5 | 16.3 | 1.5 | 4530 | 4077 |

31.5 | 16.3 | 1.75 | 7194.3 | 5036 |

31.5 | 16.3 | 2.0 | 10,739 | 8054 |

Outer diameter, D (mm) | Inner diameter, d (mm) | Thickness (mm) | Stiffness (N/mm) | Maximum force (N) |
---|---|---|---|---|

31.5 | 16.3 | 0.8 | 687.2 | 721.6 |

31.5 | 16.3 | 1.25 | 2621.1 | 2359 |

31.5 | 16.3 | 1.5 | 4530 | 4077 |

31.5 | 16.3 | 1.75 | 7194.3 | 5036 |

31.5 | 16.3 | 2.0 | 10,739 | 8054 |

### 2.3 Mechanical Design of SDS-VSA.

As depicted in Fig. 4, the SDS-VSA can be divided into three main modules: the position motor module, the stiffness adjustment module, and the stiffness motor module. The position motor module includes the motor (1) mounted on the base (4), the harmonic reducer (2), and the position output shaft (3). The position output shaft connects to the roller retainer (6) within the stiffness adjustment mechanism. The section outlined with a dashed box in Fig. 4 represents the stiffness adjustment module. It comprises the roller retainer (6), the two side cam discs (5), and the dual sets of disc spring combinations (7). The stiffness adjustment module is secured to the base using crossed roller bearings, ensuring both axial load capacity and compactness within the joint. Simultaneously, the stiffness motor module consists of the motor (9) and the harmonic reducer (8). Compared to other variable stiffness joints based on cam mechanisms, the approach using symmetric compression disc springs not only achieves higher output torque but also offers modularity to accommodate diverse application requirements.

In Fig. 5, three pairs of cam rollers (10) are evenly mounted on the roller retainer (9), with a 15-deg spacing between each pair. Due to the force of the springs, the rollers snugly envelop and maintain contact with the cam surfaces on both sides of the joint. As the cam rollers compress the cam discs (4) during motion, the cam discs move axially. To facilitate this axial motion, slidable keyways are present on the outer perimeter of the cam discs, acting as guides. The relative position of the two cam discs is adjusted by a stiffness motor, with the stiffness motor secured to the keyway shaft sleeve (2) via a harmonic reducer mounting flange (8). The motor’s output shaft is connected to one cam disc through a harmonic reducer primary reduction, supported by crossed roller bearings (6).

Moreover, to ensure the reliability of joint motion, the cam disc design incorporates stiffness-adjustable limit protection. The limit protection is realized in three ways. First, the presence of stoppers on the cam disc acts as a hard limit for reverse motion during adjustment of the cam disc’s relative angle by the stiffness motor, as shown in Fig. 5. These stoppers prevent the failure of soft limits during control. Second, the cam profile on the cam disc employs an exponential curve for limit protection. The high slope of the exponential curve at points far from the origin obstructs roller movement beyond the travel limit, serving as an indirect limit. Third, the limit protection is further ensured through the deformation limit of the elastic element. At maximum deformation, the compression springs on both sides cannot be further compressed, indirectly establishing a limit.

These limit protection measures enhance the reliability and robustness of the variable stiffness joint, preventing damage from human–machine interactions or impact-induced movements.

### 2.4 Prototype Construction.

An SDS-VSA prototype based on a symmetrical compression spring stiffness adjustment mechanism is developed in this paper and is shown in Fig. 6. The stiffness motor and position motor both employ Panasonic A6 series servo motors, specifically the model MSMF012L1U2M, featuring a rated power of 100 W, a rated torque of 0.32 N m, and a peak torque of 0.95 N m. The motor output shaft transmits power through a harmonic reducer, with the chosen reducer being the LHD-17-100-C-I-S132 from LeadDrive, providing a reduction ratio of 100. The stiffness adjustment mechanism incorporates disc springs with dimensions of *D*16.3 × *d*31.5, where a single disc spring of this size yields a minimum elastic force of 687 N and a maximum of 8054 N.

By employing different combinations, a set of six assembled disc springs can achieve a wide range of stiffness coefficients, spanning from a minimum of 114 N/mm to a maximum of 64,300 N/mm. In comparison, a similarly sized heavy-duty die spring has a stiffness coefficient of 182 N/mm, and a tension spring has a stiffness coefficient of 26.48 N/mm. This demonstrates that the stiffness adjustment mechanism proposed in this paper can achieve higher output torque and a broader range of stiffness variations while maintaining the same external dimensions. In addition to utilizing combined disc springs to enhance the stiffness coefficient of the elastic element, the choice of cam profile can meet different requirements for output torque and stiffness. A comparison of the proposed SDS-VSA variable stiffness joint with typical variable stiffness joints is presented in Table 2. When comparing with the FSJ variable stiffness joint, while maintaining similar external dimensions, the SDS-VSA achieves a 4.3 times increase in stiffness variation range and a 40$%$ increase in peak torque, all while reducing input power by 45$%$. From these comparative results, it is evident that the proposed SDS-VSA variable stiffness joint features a compact structure, a broad range of output stiffness, and low input power requirements.

## 3 Modeling of SDS-VSA

### 3.1 Stiffness Model.

As shown in Fig. 7, the roller pair, serving as the active components, comes into contact with the cam surface during the joint motion driven by the rollers. When the spatial cam surface is compressed due to roller pressure, it compresses the compression spring along the axial direction of the joint, generating a torque that resists the load torque until equilibrium is reached with the load torque.

*σ*

_{0}+

*σ*, where

*σ*

_{0}represents the initial angle between the two cam discs and

*σ*represents the angle by which the two cam discs change due to input from the stiffness motor. The initial angle of the cam discs is determined by the roller diameter and the angle between the roller pair. The roller pair is held between the two cam discs by spring force, and we define the contact points between the roller pair in its initial equilibrium position as points A and C, with the rotation centers of each roller defined as points 1 and 2. The angle between the two rollers is defined as 2

*φ*

_{0}, and due to the symmetric arrangement of the rollers, the angle on one side starting from the centerline is referred to as the initial deformation angle

*φ*

_{0}, and the angle formed between contact point A and the centerline of the roller pair is defined as

*φ*

_{A}. We define the rollers as cylindrical with a radius of

*r*. Additionally, since cylindrical spatial cams are used, the radius of the cam surface in contact with the rollers is defined as

*R*. Therefore, to determine the position of contact point A on the cam surface, the angle at which the contact point of the roller pair lies on the circumference needs to be calculated as follows:

*φ*, also referred to as the deformation of the variable stiffness joint. Additionally, due to the change in the relative position of the roller pair to the cam discs, new contact points between the rollers and the two cam discs are defined as points B and D in order to analyze the force situation. Thus, the angle of the contact point on the arc cam profile can be calculated using the following expression:

*F*

_{N}at the contact point, which can be decomposed into a tangential force

*F*

_{a}and an axial force $F\tau $. At the new equilibrium position, the axial force is counteracted by the spring forces applied to the two cam discs, and the output torque formed by the tangential forces along the circumference balances the load torque. Therefore, the relationship between the output torque and the spring force can be expressed as follows:

*α*

_{B}represents the angle between the tangent line and the horizontal line at the contact point of the lower cam disc.

*α*

_{D}represents the angle between the tangent line and the horizontal line at the contact point of the upper cam disc.

*K*

_{1}and

*K*

_{2}represent the stiffness coefficients of the springs used on the two sides of the cam discs. In general,

*K*

_{1}=

*K*

_{2}.

*Y*

_{1′}and

*Y*

_{1}represent the positions of the center 1′ and center 1 of the roller, respectively, along the

*Y*-axis.

*Y*

_{2′}and

*Y*

_{2}represent the positions of the center 2′ and center 2 of the roller, respectively, along the

*Y*-axis.

*ρ*

_{char}represents the shape parameter of the cam curve, which is used to adjust the curvature of the curve.

*φ*′ represents the angle at the contact point of the roller when the joint deformation is

*φ*.

*φ*

_{A},

*φ*

_{B},

*φ*

_{C},

*φ*

_{D}, satisfy Eqs. (2) and (6). Therefore, for a single contact point, the angular values can be obtained by solving the above equations simultaneously. Taking contact point B as an example, the system of equations is represented as follows:

In Eq. (8), when given *φ*_{0} and *φ*, the system of equations above represents a transcendental system in terms of *φ*_{B} and the roller angle *α*_{B}. This system of equations does not have an analytical expression for its solution. Therefore, we use numerical methods to obtain solutions for this nonlinear system of equations. Let us proceed with a specific example.

First, we determine the cam profile shape with shape parameters defined as *a* = 1.7 and *ρ*_{char} = 8. The stiffness coefficient of the spring is defined as *K* = 1000 N/mm. In this variable stiffness mechanism design, the spring stiffness coefficient can be changed as needed. The initial angles of the two cam discs are *σ*_{0} = 0 and the initial deformation angle is *φ*_{0} = *π*/12. The contact radius of the cam disc is *R* = 22 mm and the roller radius is *r* = 4 mm.

Next, we define the range of variation for the deformation angle as *φ* ∈ [0, 5*π*/36]. By substituting the above coefficients into Eqs. (7) and (8), the joint output torque characteristic and output stiffness characteristic based on the variable stiffness principle in this paper are shown in Figs. 8(a) and 8(b), respectively.

The output characteristics of the stiffness adjustment mechanism using independent extension springs are calculated with Eq. (9). The maximum output torque for this configuration is 43.5 N m. However, for the same dimensions and spring stiffness coefficient, and using the symmetric compression springs as proposed, the maximum output torque of the joint increases to 63.2 N m. This represents an increase of over 30$%$ in the maximum torque compared to the independent extension spring solution. This makes the proposed mechanism promising for designing high-power-density variable stiffness joint actuators with small size and high energy output, showing strong potential for practical applications.

### 3.2 Decoupled Dynamic Model of SDS-VSA.

## 4 Control of SDS-VSA

In this work, the link-side and stiffness motor-side dynamics are decoupled directly by introducing disturbances into the system dynamics. Therefore, the variable stiffness actuated robot system is treated as two individual and decoupled subsystems and the tracking controllers of them are also designed separately. Therefore, the back-stepping control method [26] is employed to design the trajectory tracking controller of link-side subsystem and traditional PD controller is employed for the stiffness motor-side subsystem. The tracking control diagram of SDS-VSA is depicted in Fig. 9.

*c*_{1},

*c*_{2},

*c*_{3},$c4\u2208R$ are user-defined control gains. $\kappa stiff\u2208R$ is user-defined and denotes the stiffness coefficients of PD controller. $\xi stiff\u2208R$ is user-defined and denotes the damping coefficients of PD controller.

## 5 Experiment and Results

*ω*

_{p}= 0.5 and

*C*

_{v}=

*ω*

_{p}

*π*. The link command trajectory is depicted in Fig. 11(a). The motion command trajectory for the stiffness motor is defined as follows:

The hardware of control platform for SDS-VSA is depicted in Fig. 10. The control platform is based on matlab/simulink and consists of two PCs: a regular PC referred to as the Host-PC and an industrial PC known as the Target-PC. The Host-PC runs software of matlab/simulink, while the Target-PC is chosen based on the X86 architecture with PCI bus peripheral expansion capabilities. The operation based on the EtherCAT protocol involves running the protocol master station on the Target-PC. Other hardware components in the system are chosen to support the EtherCAT protocol as slave devices, including input and output (IO) signal boards, motor drives, and external sensors. The IO signal board, produced by Qianjiang Robotics, is a QJ-EE6120A model, responsible for connecting digital switch signals and supporting RS232 or RS485 peripherals such as emergency stop switches, limit proximity switches, and absolute encoders. An external encoder from YuHeng Optics, a 17-bit absolute optical encoder, is employed for feedback linkage position. The motor drives in the system utilize RC series EtherCAT slave servo drives from Tsino-Dynatron, which come with built-in vibration suppression algorithms and feedforward capabilities, significantly enhancing robot positioning accuracy and dynamic characteristics. As shown in Figs. 11(c) and 11(d), the maximum tracking error of the stiffness motor is $0.3287deg$, with an average error of $0.0006deg$ and a root mean square error of $0.0804deg$. However, the accuracy of the link-side trajectory tracking is slightly lower than, as depicted in Fig. 11(b), a maximum tracking error of $8.8456deg$, an average error of $0.2406deg$, and a root mean square error of $3.4217deg$. The variable stiffness joint exhibits low damping and strong flexibility, which affects the accuracy of the link’s trajectory tracking due to the joint’s flexibility during the tracking control process. Additionally, the link-side trajectory tracking performance is not only influenced significantly by the dynamics of the stiffness adjustment side but also by unmodeled factors in the system, resulting in poorer trajectory tracking accuracy for the link-side. Additionally, as depicted in Figs. 11(e) and 11(f), the control torque of position motor is saturated at 0.3 N m which is constrained by the physical limitation of motor’s output torque. Furthermore, it is evident that there is significant noise in the control torque of the position and stiffness motors. This can be attributed to vibrations occurring in the gear transmission of the external encoder of the link. However, despite this issue, the control laws proposed in this paper are capable of effectively tracking the commanded trajectory with stability and converging to a certain level of precision.

## 6 Discussions

The innovation of SDS-VSA is the utilization of disc spring combinations instead of general spring. Based on the principle of the cam-roller-spring mechanism with symmetrical compression springs, a design approach for compressed springs with equivalent specified stiffness coefficients of combined disc springs was proposed. This approach enabled the realization of a compact, reconfigurable, and high-torque stiffness adjustment mechanism, leading to the development of the variable stiffness joint prototype SDS-VSA, characterized by scalability and high output torque. Compared with the previous research about reconfigurable VSAs, a wider stiffness range is obtained with the proposed design method. For example, with six disc springs having five different stiffness coefficients, there are 210 distinct combinations. Therefore, the equivalent stiffness coefficient of the combined disc springs, as shown in Fig. 3, falls within ranges of [114.5397, 64432] N/mm, [68.7179, 26347] N/mm, and [37.9722, 10946] N/mm for three typical outer dimensions of disc springs.

Compared to the cam-roller mechanism with a single extension spring [11], the proposed mechanism with symmetric compression springs can provide higher output torques for the same joint dimensions. Furthermore, it is important to note that for springs with the same stiffness coefficient, extension springs have hooks on both ends that can limit their use in confined spaces. Additionally, their length tends to be longer than regular compression springs. Therefore, in the stiffness adjustment mechanism using the cam-roller-spring configuration, the symmetric disc springs not only offer expandability but also effectively enhance the maximum output torque of the joint under size constraints.

The force distribution in the cam-roller-spring mechanism was analyzed, leading to the derivation of stiffness and dynamic models for the variable stiffness joint. By introducing disturbance terms, the coupled dynamic model was decoupled into subsystems for the link and stiffness motor, respectively. A decoupling control strategy based on back-stepping was introduced, allowing stable trajectory tracking even under varying stiffness dynamics and demonstrating robust performance.

Although the proposed algorithms in this study can resist the impact of disturbances and saturation on tracking errors compared with PD controller, they are model-based methods and are susceptible to model uncertainty and unmodeled factors. Therefore, the next step will involve identifying the variable stiffness joint model and designing adaptive and robust control laws to enhance the system’s robustness and adaptability to model errors.

## 7 Conclusion

In this work, a compact and reconfigurable variable stiffness actuator using disc spring was proposed. The output torque of the proposed mechanism using symmetric compression springs was enhanced by approximately 30$%$ compared to the cam-roller-spring mechanism that employs a single spring. The disc spring configuration was tailored to achieve a broader stiffness range within a limited size. With a disc spring combination with six pieces, the maximum stiffness range reaches [114.5397, 64432] N/mm which is far more wider than the regular spring. Subsequently, the dynamics of the cam-roller-spring mechanism were derived. To tackle the challenge of strong coupling dynamics, a decoupled modeling method by introducing mismatched and matched disturbances was proposed. A back-stepping tracking controller and PD controller with feedforward were proposed to track the link-side and stiffness motor-side trajectories, respectively. The performance of the prototype SDS-VSA was verified with tracking experiments under highly dynamic stiffness alteration. The results show that the proposed tracking controller is robust to the external disturbances.

## Acknowledgment

This work is supported by Jiangsu Robotics and Intelligent Manufacturing Equipment Engineering Laboratory (Soochow University).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

No data, models, or code were generated or used for this paper.