Design and Analysis of a Novel Compliant Actuator With Variable Stiffness by Spring Pretension Adjustment

Variable stiffness actuators (VSA) are an emerging advanced driving method for robotic joint in physical human-robot interaction scenarios. This paper proposes a VSA based on adjusting the pre-tightening force of the driving ropes by rotating the linear springs to change the angle of its axis relative to the input frame. By adjusting the spring angle within a small relative rotation angle stroke, the angle between the spring axis and the transmission rope can be changed, thereby achieving rapid stiffness adjustment in a large range. The characteristics analysis shows that the proposed VSA has good stiffness adjustment independence. The mechanical design scheme of the VSA is introduced in detail. Simulations based on a proportional derivative (PD) controller demonstrated the effectiveness of the design.


I. INTRODUCTION
Robots are increasingly integrated into the human environment to enable human-machine interaction.The motivation for the growing interest in these topics can be found in the possibilities of using robots for humans, such as exoskeletons, operating in hazardous environments, or helping the elderly or disabled [1], [2].In order to improve the safety and reliability of physical interaction between robots and humans, robotic systems with abilities of variable stiffness and intrinsic compliance as humans are investigated [3].
In robotic systems, there are generally two approaches to mimic the compliance of human limb movements: one is to use active compliance control at the control algorithm level, such as impedance control, and the other is to adjust the The associate editor coordinating the review of this manuscript and approving it for publication was Yangmin Li .mechanical stiffness of robot joints [4], [5].Active compliance control for stiff joint relies on proper sensors, parameter identification, accurate dynamic modeling, robust control, etc.Some successful implementations in robotic systems can be found in KUKA [6].The introduction of elastic elements into the drive chain of robot joint actuators to form a series elastic actuator (SEA) can easily make the robot system intrinsically flexible [7].Compared to conventional rigid actuators, the elastic element in a SEA significantly reduces the mechanical impedance of the joint, improves the ability to absorb shock loads and peak power [6], and the passive mechanical energy stored by the deformation of the elastic element can be reused, which helps to save energy during cyclic movements [8].However, the fly in the ointment is that the mechanical stiffness of a traditional SEA is often constant and un-adjustable, which greatly limits its control bandwidth and application.
To overcome the above limitations, joints with variable stiffness actuators (VSA) that are able to actively change position and stiffness of the output is one of the promising solutions [9].Classified in terms of the adjustment independence of the output stiffness and position, VSA includes antagonistic motor setup and independent motor setup [10].In an antagonistic system, two motors and a set of nonlinear spring mechanisms are used to work antagonistically to adjust stiffness and position, just like human skeletal muscles drive joints [11].Usually, two motor transmission chains are symmetrically arranged, and when they move in the same direction, the joint output position can be controlled.When they move in reverse or differential direction, the joint output stiffness can be adjusted [12].The antagonistic VSA belongs to the category of adjusting spring preload to change stiffness [13].Although it can closely mimic the driving mode of human skeletal muscles, its biggest drawback is severe energy consumption and a small range of stiffness adjustment.In addition, the coupling between the output stiffness and position adjustment of the joint is severe, and the control is complex [14].
There are three methods for achieving stiffness adjustment in the independent motor setup VSAs: (1) by adjusting the spring preload, (2) by changing the transmission ratio between the output and elastic element, (3) by changing the physical properties of an elastic elements.
For example, the DLR team used the differential motion of a pair of roller cam mechanisms and the screw to adjust the preload of the spring group to change its balance position, and developed a VS-joint for application on a humanoid robotic arm [15].In order to make the joint design more compact, the FSJ (Float Spring Joint) was also designed by DLR, a cam roller spring mechanism was adopted as well [16].Compared to the VS-joint that uses four compression springs, the FSJ only uses one linear floating spring, which effectively improves the compactness of the joint.In addition, the single-sided cam structure in VS-joint has been optimized into a double-sided oblique symmetric cam structure, which can make the stiffness adjustment process smoother.By changing pretension of a nonlinear spring, Mechanically Adjustable Compliance and Controllable Equilibrium Position Actuator (MACCEPA) Has been studied by many scholars [17], [18].By designing a rope driven winding mechanism, Z. Li et al. proposed a reconfigurable variable preload flexible robot joint named as JVSR [18].
As a typical method of adjusting stiffness based on variable transmission ratio, the lever principle is widely used in VSA.A lever has three principal points: the pivot [19], the spring attachment point, the force point [20].An actuator with adjustable stiffness (AwAS) was developed based on the method of moving the spring action point while the force action point and the pivot point are relatively fixed [21].In order to improve the stiffness adjustment range and the compactness of joints, [22], CompAct-VSA [23], [24], [25], [26], etc. fix the spring and the point of force relatively, only changing the position of the lever pivot, achieving a theoretical stiffness adjustment range from zero to infinity.The softer spring and shorter lever make this type of VSAs more compact with a fast stiffness adjustment speed.In addition, the movement direction of the stiffness adjustment mechanism of this type of VSAs is often approximately perpendicular to the direction of lever deflection, which is beneficial for reducing the stiffness adjustment resistance [27].Liu Z. et al. recently proposed a new VSA based on a second-order lever mechanism by changing the force point, which has wide stiffness regulation range.By employing a novel symmetric structure design and improving the load capacity of the stiffness regulation module, the proposed actuator also shows well performance in load capacity, stiffness regulation response, and elastic hysteresis, which was applied to a multi degree of freedom humanoid robotic arm [28].Although the principle of adjusting stiffness with variable transmission ratio is relatively simple and significantly improves the range and speed of stiffness adjustment, there is still significant room for improvement in the compactness, power density, and independence of VSA.
Recently, there has been increasing research on the use of changing the physical characteristics of flexible components to achieve stiffness adjustment.From the perspective of material mechanics theory, the bending stiffness EI/L of a beam with small deflection can be adjusted by changing its geometric or material parameters [29].Adjusting the effective length L of a leaf spring is the easiest method to achieve variable stiffness.Based on this, L. Liu et al. proposed a mechanical-rotary variable impedance actuator (MeRIA) [30].S. Yang et al. proposed a 2-DOF actuator with variable stiffness by varying the effective length of the springs, which is uniformly distributed in the variable stiffness unit [31].Y. Xu et al. proposed two compact VSAs based on S-shaped springs, which enables independent, continuous, fast, large range stiffness adjustment by rotating S-springs to change their effective curved beam length [32], [33].To achieve faster stiffness adjustment, Tyler Morrison et al. present a novel design concept for a robot arm link with variable stiffness based on a rotating beam link (RBL) mechanism to change the inertia I of some paired reverse RBL [34].In order to change the elastic modulus E of elastic component materials, functional materials are widely used, such as the shape memory alloy [35], the magnetorheological fluid [36], etc.As an example, a linear digital variable stiffness actuator (LDVSA) with variable stiffness mechanism (VSM) based on specially designed S-springs made from shape memory alloy (SMA) is developed by Y. Xu et al.By controlling the temperature of the SMA S-springs with heating wires for different states switching, the actuator stiffness could be discretely configured online [38].However, changes in material properties E often require a period of power input, so the slow corresponding speed of stiffness is the biggest flaw.
Despite these existing VSA concepts, there remains room for considerable improvement.In this study, a variable stiffness mechanism is designed and analyzed in Section II.Based on this, in Section III, a new VSA is designed.The performance analysis and simulations to evaluate the proposed VSA are presented in Sections IV.Finally, the conclusion is presented in Section V.

II. VARIABLE STIFFNESS MECHANISM DESIGN AND ANALYSIS A. ABBREVIATIONS AND ACRONYMS
Fig. 1 shows the stiffness adjustment principle of the proposed design.For the convenience of observation, the output frame in Fig. 1 (b)-(d) has been temporarily hidden.As shown in Fig. 1 (a) and (b), a stiffness adjustment frame is coaxially installed in the middle of the input frame and output frame.On this stiffness adjustment frame, a pair of guide rail sliders are symmetrically arranged in space.The sliders are installed on the stiffness adjustment frame, and the guide rail can slide on the sliders within a certain range.One end of the guide rail is equipped with a compression spring, and the other end is connected to a rope.The rope is fixedly connected to the edge fixing holes of the input or output frame through the edge fixing hole of the input frame or output frame.
If the output frame does not rotate relative to the input frame, the spring and rope states are shown in Fig. 1 (a), where the angle between the rope and the extension line of the guide rail is defined as α.As shown in Fig. 1 (b), rotate the stiffness adjustment frame by a certain angle q s relative to the input frame, because the rope length is constant, then one end of the guide rail compresses the spring and extends a certain length along the slider, while α also increases.As shown in Fig. 1 (d), as q s increases, α also continues to increase to 90 • , at which point the rope and guide rail are perpendicular.Overall, during the process of α changing from 0 to 90 • , the output stiffness of the variable stiffness mechanism (VSM) also varies from the minimum value to approximately infinite.Therefore, by controlling the angle q s of the stiffness adjustment frame relative to the input frame, the output stiffness of the VSM can be continuously adjusted over a large range.

B. STIFFNESS MODELING OF THE VSM
Due to the symmetry of the structure, for the convenience of analysis, as shown in Fig. 2 (a), only the force situation of the unilateral spring during the stiffness adjustment process is analyzed, where there is no relative deflection between the input and output frames.Assuming that the elastic expansion and contraction rate of the rope is negligible.The spring is rotated by a certain angle q s relative to the input frame, and the guide rail extends a certain length δ s relative to the slider.The spring will be compressed during this process, thus generating resistance torque on the stiffness adjustment frame.Therefore, the required stiffness adjustment torque τ s can be calculated by where n is the number of springs in the VSM, F rope is the tension of the rope, α 0 is the angle between the guide rail and the rope without deflection.
According to Hooke's law, the force F k exerted by the spring on the guide rail is where k and δ 0 are the stiffness constant and the pre-compression length of the spring, respectively.
According to the force balance relationship, there are Thus, the required stiffness adjustment torque τ s can be rewritten as According to the sine theorem of a triangle, the relationship between α 0 and q s can be derived as where R and r 0 are the distance of OA and the length of the rope, respectively.Based on the above analysis, the adjustment range of q s to change the output stiffness of the joint can be calculated by Further based on the triangle sine theorem, it can be obtained that Obviously, α 0 = 0 o when q s = 0 o , and if there is no deflection (γ = 0 o ), δ s = 0 which means no load is applied on the joint actuator.
However, as shown in Fig. 2 (b), α 0 will increase to α 1 with the actuator deflection.Then, the compression of the spring will become where δ e is the additional spring compression caused by elastic deflection (external load force).
The chord length AB of the elastic deflection section between the input and output frames can be calculated by where γ = ̸ AOB = q 2 − q 1 is the deflection angle of the actuator, q 1 and q 2 are the angles of the input frame and the output frame, respectively.Because the total length of the rope r 0 is a constant, after ignoring the stretching of the rope, it can be obtained that Combining the sine theorem of a triangle again, the relationship between α 1 and q s can be defined as Referring to Eq. ( 2), the force exerted by the spring on the guide rail can be recalculated by Referring to Eq. ( 3), and ignoring the frictional force when the rope passes through hole A, the elastic driving torque τ e of the tension on the rope on the output frame can be calculated by And δ e is caused by γ , which can be defined as Referring to Eqs. ( 9) and ( 13), and the geometric relationship, it can be obtained that In fact, the length of the AB mapped to the axial elongation of the guide rail can be defined as Usually, the output stiffness of the actuator can be calculated by

C. ANALYSIS OF THE PROPOSED VSM
In order to determine the key design parameters of the VSA, it is necessary to analyze its characteristics to determine design constraints.Referring to Eq. ( 7), when adjusting the spring angle q s independently, the spring compression can vary nonlinearly, as shown in Fig. 3(a).Similarly, as shown in Fig. 3(b), the resistance torque of the VSM transmission chain increases sharply after the spring angle exceeds 30 • , which means that the stiffness adjustment motor drive requires a larger peak torque or reduction ratio.Referring to Eq. ( 17), the output stiffness change trend of the VSA without deflection shown in Fig. 4 is similar to that of τ s .Although a high transmission ratio of the stiffness adjustment drive chain may lead to a decrease in the response speed of the stiffness adjustment, the adjustment limit range  of q s is approximately within 35 • (refer to Fig. 3), and the stiffness change is sharp when q s is greater than 30 • , making rapid stiffness adjustment possible.In addition, as an adjustable passive compliant robot joint, from an application perspective, the resolution of adjustment in high stiffness areas (q s ≥ 35 0 ) has sharply decreased, and its regulation has almost no practical significance.
Referring to Eq. ( 17), when the output frame of the VSA is subjected to a load, it will deflect, and the output stiffness of the actuator will be additionally affected by the coupling effect of the deflection, as shown in Fig. 5. From a macro perspective, in most of the coupling regulation regions of q s and γ , the linearity of stiffness changes is smooth, which is conducive to achieving accurate and rapid stiffness adjustment response.Therefore, the design and selection of the adjustable stiffness drive chain do not need to overly emphasize the global control response speed.
Referring to Eq. ( 13), when the actuator deflects, the ropes are pulled by the input and output frames, and the elastic  torque generated can be used to drive the load.The dynamic characteristics of τ e are shown in Fig. 6, where q s has a minimal impact on τ e .This means that the online stiffness adjustment has a slight impact on stable elastic torque output, which is beneficial for improving the independence of the stiffness adjustment and the output position q 2 .In addition, the transmission design of the main drive motor can refer to the peak torque and change rate along γ in Fig. 6.
Under the coupling effect of the elastic deflection and spring angle, the resistance torque suffered by the VSM defined in Eq. ( 4) can be rewritten as Under this working condition, the change in resistance torque reflected on VSM is shown in Fig. 7. Obviously, the impact of changes in q s on the required stiffness adjustment torque τ s shows an approximate linear trend, but the impact of γ on it shows a significant nonlinearity.This indicates that changes in load will have a certain impact on the VSM.When designing the stiffness adjustment drive chain, the issue of stiffness maintenance should be considered (for example, worm gear and worm transmission mechanisms with self-locking effects can be adopted).
Referring to Eqs. ( 3) and ( 12), in addition, when designing the VSA, the ropes used need to meet a certain level of strength and toughness, so the tension change of the ropes under the coupling effect of load and stiffness adjustment should be considered.Fig. 8 shows that the tension change of the rope is relatively mild in most of the stiffness adjustment areas.Referring to the tension change of the rope in extreme cases (q s > 0.3rad, γ > 0.1rad), it can provide a reliable basis for the selection of the ropes.
The elastic mechanism of a VSA can store energy and reuse this part of energy during periodic motion, achieving energy-saving motion control and shock absorption, achieving a safe physical interaction effect of buffering and vibration absorption [9].In this design, the elastic energy storage E s of the proposed VSA can be calculated by Fig. 9 shows that the change in E s is mainly influenced by the deflection γ , but when there is no deflection, a small amount of elastic energy is still stored in the spring compression due to the change in preload (caused by q s adjustment), which is a characteristic of VSA based on the principle of preload adjustment.
In order to ensure the reasonable design of the springs in the VSM, the selection of the spring should not only comprehensively consider its total length, stiffness constant, material, etc., but also consider the deformation situation after assembly under the coupling effect of the pre-compression, load, and the stiffness adjustment.Fig. 10 shows the change in spring compression of the VSM during the dynamic process of the VSA.

III. VARIABLE STIFFNESS ACTUATOR DESIGN
Based on the VSM described above, a VSA was designed, as shown in Figs.6-9.The actuator consists of the following three parts, which will be explained from the outside to the inside of the VSA, and its key design parameters are shown in Table 1.(1) Protective cases: As shown in Fig. 11, parts 1-4 form the protective bases of the whole VSA, which take into account the protection of internal moving components and the role of force transmission in robot applications.Among them, the end face of the part 2 can be used to connect the remaining links of a robot.Part 4 is used to protect the encoder (part 25).Part 2 is used to withstand all loads and weights.
(2) Main drive module: As shown in Fig. 11 and Fig. 12, parts 8-24 can be considered as a whole rotating relative to part 2 based on a pair of large bearings (part 5).The position motor (part 6) is mounted on the protective case (part 2).The main drive power from the position motor (part 6) is transmitted to the large bevel gear (part 8) installed at one end of the input frame (part 9) through a small bevel gear (part 7), and then to the output frame (part 10) through a pair of ropes (parts [21][22].The load can be connected to one end of the output frame (part 10).In addition, the inner ring of the  encoder (part 25) is fixed on the output frame and the reading head is installed at the protective case (part 2) end, which can be used to obtain the position of the output frame.The position of the input frame can be calculated by the encoder and reduction ratio of the position motor.
(3) Stiffness adjustment mechanism: As shown in Figs.11-14, the stiffness adjustment motor (part 11) with self-locking characteristics is installed on the large bevel gear (part 8).Then, a pair of small gears (part 12 shown in Figure 13 and part 13) are installed on the stiffness adjustment motor and transmission beam (part 14) which is fixed on a stiffness adjustment frame (part 15), respectively.As shown in Fig. 11 and Fig. 13, a pair of the stiffness adjustment frames (part 15) symmetrically installed at respective end faces with respect to the origin point in space.In order to provide adjustable stiffness characteristics for the ropes (parts 21-22) located between the input frame and output frame, a linear telescopic elastic module is installed on a stiffness adjustment frame (part 15) shown in Figs.[13][14], which consists of a slider (part 16), a guide rail (part 17), a guide rail limit bolt (part 18), a spring (part 19) and a spring limit bolt (part 20).The spring is installed coaxially with the guide rail, and its two ends are respectively limited by the slider and the spring limit bolt.The slider is mounted on the one end of the stiffness adjustment frame and rotates together with it.The guide rail can move back and forth on the slider and is limited by the guide rail limit bolt and the spring.
As shown in Fig. 12 and Fig. 13, one end of a rope (part 21) is fixed at the position of the guide rail limit bolt (part 18), and the other end of this rope is fixed at the corresponding hole on the output frame (part 10) through the hole on the edge of the input frame (part 9).On the contrary, one end of the another rope (part 22) is fixed at the position of the guide rail limit bolt (part 18), and the other end of this rope is fixed at the corresponding hole on the input frame (part 9) through the hole on the edge of the output frame (part 10).Obviously, the drive power between the input and output frames is transmitted in series through the ropes (parts [21][22].When there is a deflection between the input frame and the output frame, the ropes are pulled, which in turn pull the guide rails to move along the sliders and ultimately compress the springs.Therefore, by adjusting the angle of the stiffness adjustment frames (part 15) relative to the input and output frames (parts 9 and 10), the angle between the spring (part 19) axis (or the guide rail (part 17)) and the rope (part 21 or 22) can be changed, and the pre tightening force of the rope (part 21 or 22) can be adjusted, and the output stiffness of the actuator can be adjusted ultimately.The spring angle relative to the input frame can be calculated by combining the encoder of the stiffness adjustment motor with the reduction ratio.
As shown in Figs.12-14, a pair of elastic modules (parts 15-20) are supported and installed in the input and output frames respectively through a pair of small bearings (part 23), which can be can be driven by the stiffness adjustment motor (part 11) and rotated relative to the input frame or the output frame.To avoid motion interference, a ring (part 24) is installed between the two elastic modules.Referring to Eqs. ( 6), ( 16), and Fig. 1, the VSA can achieve a larger stiffness adjustment bandwidth within a small rotation range, which is beneficial for improving the stiffness adjustment response speed.In addition, in order to avoid jamming between the input frame and the output frame when the ropes are completely relaxed, the length of the rope components should be controlled during assembly, so that the spring produces a certain amount of pre compression.

IV. PERFORMANCE SIMULATION OF THE PROPOSED VSA A. DYNAMIC MODEL OF THE VSA
As mentioned above, referring to Fig. 14, the proposed VSA is essentially a series elastic driving configuration.The VSM is supported on the input and output frames, but its relative position is mainly constrained by the input frame.Therefore, the dynamic model of the entire VSA is shown in Fig. 15.As shown in Fig. 15, the actuator dynamics can be modeled by [24] J 2 q2 + D 2 q2 + G(q 2 ) + τ d = τ e J 1 q1 + D 1 q1 + τ e = τ m1 J s qs + D s qs + τ s = τ m2 (20) where q 1 , q 2 and q s are the angular displacements of the input frame, the output frame, and the VSM, respectively.J 1 , J 2 and J s are the reflected inertias of the input frame, the output frame, and the VSM, respectively, D 1 , D 2 and D s are the equivalent damping coefficients for the input frame, the output frame, and the VSM, respectively, G(q 2 ) is the gravitational torque acting on the output frame, τ d is a lumped disturbance torque that includes the load torque and model uncertainties such as hysteresis, friction, and internal damping, τ s is the resistant torque on the VSM, which represents the influence of the deflection γ , τ m1 and τ m2 are the equivalent driving torque acting on each motor, and τ e is the spring elastic torque given by In order to conduct dynamic performance simulation analysis on the VSA, its main dynamic parameters are shown in Table 2.

B. CONTROLLER DESIGN
To verify the basic motion performance of the proposed VSA, PD controllers were designed for the main drive chain (with 108218 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.gravity compensation) and the VSM drive chain, respectively, which is shown in Fig. 16.The output frame of the VSA is connected with a load and set as a whole to the horizontal state of the joint axis.The position information on the input and output frames is used to calculate the deflection angle.Then, based on the stiffness model defined by Eq. ( 17), the required spring angle can be calculated.

C. SIMULATION RESULTS
The step response of the output frame position of the main drive module under a moderate stiffness level is shown in Fig. 17.In the small travel step response, the tracking response is relatively fast at no-load, and the steady-state time is significantly longer with the increase of load.In the step response of larger travel, due to the acceleration, the steady-state time under load condition decreases, which also shows the effectiveness of gravity compensation mention above.
For a VSA, its stiffness response speed is a very important indicator.Therefore, for the proposed VSA, the spring angle step response of small stroke and large stroke is carried out, as shown in Fig. 18 (a), and the corresponding actuator output stiffness based on Eq. ( 17) is shown in Fig. 18 (b).It can be seen that there is not much difference in the response speed of the VSM under no-load and load conditions, which once  again verifies the results shown in Fig. 5 (the deflection angle γ has a slight impact on K a ).Fig. 19 shows the position and stiffness synchronization tracking control effect of the VSA.The main drive output frame tracks a sinusoidal trajectory signal with a frequency of 1 Hz and an amplitude of 50 degrees under no-load and load conditions.The output position tracking performance and tracking error are shown in Fig. 19  Obviously, the load has a certain deteriorating effect on the tracking performance.In addition, to verify the performance of synchronous variable stiffness, under load conditions, the VSM is also tracking a sinusoidal signal with a frequency of 1 Hz and an amplitude of 35 Nm/rad, with a tracking error of approximately 4.7 Nm/rad, as shown in Fig. 19(c).Because the synchronous tracking effect of the stiffness under no-load conditions is relatively ideal and has little reference significance, it has not been shown yet.

V. CONCLUSION
A new type of VSA that can simultaneously adjust the position and output stiffness independently is developed in this paper.The VSM based on the spring angle adjustment and rope pre-tension adjustment makes smooth and fast stiffness adjustment possible.The relative angle between the rope and the spring axis allows the actuator to have a large stiffness adjustment range.The entire VSA has modular design features.The model accuracy and the design performance are demonstrated by regulation and tracking simulations.Our next research direction is to manufacture a prototype of the proposed VSA for performance testing and apply it to a multi degree of freedom flexible robot.More advanced or personalized control algorithms will be designed for the VSA control.

FIGURE 1 .
FIGURE 1. Schematic diagram of the proposed variable stiffness mechanism: (a) Description of the positional relationship between the output frame and input frame; (b) Minimum stiffness state; (c) Operating state of the variable stiffness mechanism; (d) Maximum stiffness state.

FIGURE 2 .
FIGURE 2. Schematic diagram of force analysis: (a) Force analysis of the stiffness adjustment process when there is no relative deflection between the input and output frames; (b) Force analysis when there is relative deflection γ between input and output frames.

FIGURE 3 .
FIGURE 3. Theoretical analysis of independent motion of the VSM: (a) Change in spring compression; (b) The required stiffness adjustment torque.

FIGURE 4 .
FIGURE 4. Stiffness of the SVSA in different spring angle q s in zero deflection angle.

FIGURE 5 .
FIGURE 5. Theoretical stiffness under the coupling effect of the spring angle and deflection angle.

FIGURE 6 .
FIGURE 6.The elastic driving torque of the actuator under the coupling effect of the spring angle and deflection angle.

FIGURE 7 .
FIGURE 7. Resistance torque of VSM under the coupling effect of the spring and deflection angles.

FIGURE 8 .
FIGURE 8.The tension of the rope under the coupling effect of the spring angle and deflection angle.

FIGURE 9 .
FIGURE 9.The elastic energy storage of the VSA under the coupling effect of the spring angle and deflection angle.

FIGURE 10 .
FIGURE 10.The change in spring compression of the VSM during the dynamic process of the VSA.

FIGURE 15 .
FIGURE 15.Schematic diagram of the kinetic model of the proposed VSA.

FIGURE 16 .
FIGURE 16.Control block diagram and joint simulation settings.

FIGURE 17 .
FIGURE 17.Step response for the main position control.

FIGURE 18 .
FIGURE 18. Step response for the main position control: (a) Spring angle; (b) Corresponding stiffness.

FIGURE 19 .
FIGURE 19.Trajectory tracking performance: (a) Output position; (b) Position tracking error; (c) Stiffness tracking results under the load condition.

TABLE 1 .
Key design parameters of the proposed VSA.

TABLE 2 .
Key dynamic parameters of the proposed VSA.