This paper proposes a novel method for analyzing linear series elastic actuators (SEAs) in a parallel-actuated Stewart platform, which has full six degrees-of-freedom (DOF) in position and orientation. SEAs can potentially provide a better human–machine interface for the user. However, in the study of parallel-actuated systems with full 6DOF, the effect of compliance in series with actuators has not been adequately studied from the perspective of wrench capabilities. We found that some parameters of the springs and the stroke lengths of the linear actuators play a major role in the actuation limits of the system. This is an important consideration when adding SEAs into a Stewart platform or other parallel-actuated robots to improve their human usage.

## Introduction

Parallel mechanisms have been extensively used in different robotic applications due to their closed-loop architecture. This feature contributes to structural rigidity, high load capacity for a smaller mass, and higher position accuracy [1]. Therefore, intrinsic compliance within a parallel robotic structure is undesirable in most circumstances.

Nevertheless, from the perspective of human–robot interaction, it can be useful to decouple the stiff interface of the actuation system from the end effector by intentionally adding compliant elements in series between the load and the gear train. This arrangement of actuator and compliance is also called a series elastic actuator (SEA) [2].

A highly compliant element between the load and the gear train allows the actuation system to control the output force based on the force–displacement characteristics of the springs. In addition, the spring acts as a mechanical low pass filter to help reduce the effects of the backlash nonlinearity of the gearbox and reduce impact forces from the environment [3]. Due to the lower stiffness of the system, the controller gain can be set higher, which can improve the force tracking performance [4]. These features potentially benefit most parallel manipulators designed to interact with humans, such as our Robotic Spine Exoskeleton, which has Stewart platforms as the underlying architecture [5].

However, only a few studies have explored parallel robots using SEAs [69]. Furthermore, these mechanisms have at most three degrees-of-freedom (DOF), while the range of motion in a Stewart platform spans a six-dimensional space of position and orientation.

Some previous works were reported where a 6DOF passive parallel platform was designed to be mounted on the end effector of a series industrial robot for passive force control [10,11]. These studies demonstrate the design of Stewart platform legs with in-line linear elastic elements. However, the workspace is limited to a small area around an equilibrium point since the unloaded limb lengths cannot be actively changed. In contrast, the Stewart platform with SEAs inherits more independent control of its configuration and output wrench.

The increase in kinematic complexity of the architecture raises a fundamental but unaddressed question: Since the compliant actuation limit relies on the deflection of the springs, how can closure kinematic constraints impact the actuation capabilities of the robot in a six-dimensional workspace? To the best of the authors' knowledge, this essential design problem has not been explored sufficiently in 6DOF parallel manipulators.

This paper proposes basic mathematical models of a general 6-6 Stewart platform to which series elastic actuators have been added. We focus on the configuration where a linear spring has been added in series with a linear actuator. The effects on the workspace and wrench capabilities of the system were analyzed based on a kinematic model of our Robotic Spine Exoskeleton.

## Mathematical Formulation

The kinematic diagram in Fig. 1 represents the fixed base and the end effector of a Stewart platform. These two bodies lie in different planes. Each limb consists of one universal joint at point $Ai$, a linear series elastic actuator represented by vector $qi,tot$ and a spherical joint at point $Bi$, which makes a universal–prismatic–revolute kinematic chain.

### Kinematic Constraints.

A closed kinematic loop equation for each limb in Fig. 1 can be expressed in the vector form as
$qi,tot=pB−ai+RBbi, i=1,…,6$
(1)
where $pB$ denotes a position vector from the origin of the base frame to the origin of the end-effector frame, while $ai$ is a vector from the origin of the base frame to the point of attachment of the universal joint $Ai$. The vector $bi$, expressed in the platform frame, runs from the origin of the platform to the spherical joint $Bi$, and $RB$ is a rotational matrix computed from the pitch, roll, and yaw angles $(ψx,θy,ϕz)$ rotating about the fixed axes of the base platform. The singularities in this angle representation are kept outside the robot workspace. The vector $qi,tot$ is in the direction from the universal joint to the spherical joint, relative to the base frame, and has a magnitude of $ℓi,tot$. The total length of each leg can be expressed as
$ℓi,tot=ℓi,a+ℓi,s$
(2)
where $ℓi,a$ and $ℓi,s$ denote the actuator length and the spring length, respectively. In addition, Eq. (2) can be expressed in the Euclidean norm, namely,
$ℓtot,i2=qi,totTqi,tot=(pB−ai+RBbi)T(pB−ai+RBbi), i=1,…,6$
(3)

### Force and Moment Equilibrium.

The output wrench of the mechanism can be computed by using the principle of virtual work in a configuration. In this case, we assume that the linear actuators do not exert any force, and the system is in quasi-static condition. Therefore, the resulting virtual work done by the springs and the external wrench is
$δW=−∑i=16∂V∂qiδqi−FTδX=0$
(4)
where $F=[fxfyfzmxmymz]T$ denotes a wrench applied to the environment at the center of the moving platform. Here, the virtual displacements $δqi$ are for the generalized coordinates, i.e., the joint displacement for each leg of the robot, and $δX$ denotes the virtual displacement in the Cartesian coordinate frame. Since we assume that the spring stiffness is constant and dominates the stiffness of each limb, V denotes the total potential energy of the springs as a result of the external force. The total potential energy is a function of the joint displacements in the generalized coordinates, namely,
$V=V(q1,q2,q3,…,q6)=∑i=1612ks,i(qi−qi,0)2$
(5)
where $qi=ℓi,tot$ from Eq. (2) and $qi,0$ is the rest length of each spring added to the current actuator length. Note that the negative sign in the first term of Eq. (4) is due to the decrease in potential energy of the system when the springs exert force on the environment. From Eqs. (4) and (5), we can express
$δW=τTδq−FTδX=0$
(6)
where
$τ=[−ks,1(q1−q1,0)−ks,2(q2−q2,0)⋮−ks,6(q6−q6,0)], δq=[δq1δq2⋮δq6]$
(7)
where $τ=[τ1τ2⋯τ6]T$ is the static force from the deflection of the springs along the limb axes, and $δq$ is a vector of virtual displacements in the generalized coordinates. $ks,i$ denotes the spring stiffness for each limb, which is assumed to be equal in every leg ($ks,i=ks$). Positive and negative values of $τi$ indicate compression and tension in the springs, respectively. Next, the Jacobian matrix of the parallel manipulator provides a relationship between the two virtual displacements as
$δq=JδX$
(8)
Substituting Eq. (8) into Eq. (6), we get
$(τTJ−FT)δX=0$
(9)
This leads to the static wrench solution
$F=JTτ$
(10)
The Jacobian transpose matrix JT can be written in terms of the limb axis [12], as
$JT=[q̂1q̂2⋯q̂6b1×q̂1b2×q̂2⋯b6×q̂6]6×6$
(11)
where $q̂i$ is a unit vector of limb i from the universal joint on the fixed based to the spherical joint on the moving platform, and the vector $bi$ is now expressed in the base frame. Note that the solution could also be carried out by using free body diagrams of the moving platform, springs, and limbs of the Stewart platform. In addition, the Hooke's law for each limb from Eq. (7) can also be expressed in terms of spring length
$τi=−ks(ℓi,s−ℓi,s0), i=1, …,6$
(12)

where $ℓi,s$ represents the actual spring length (solid-line spring in Fig. 1) and $ℓi,s0$ is the resting spring length in the module (dotted-line spring), i.e., the length of the linear spring when there is no force applied on the module.

## Kinematic and Static Wrench Solutions

Due to the force–displacement relationship of the spring components, the kinematic and force solutions need to be solved simultaneously. Given all geometric parameters of the mechanism, the system contains 36 variables. The variables are the six total lengths $(ℓi,tot)$, six actuator lengths $(ℓi,a)$, six spring lengths $(ℓi,s)$, six limb forces $(τi)$, three positions in Cartesian coordinate $(xB,yB,zB)$, three orientation angles $(ψx,θy,ϕz)$, and six components of the output wrench $(fx,fy,fz,mx,my,mz)$. Also, there are 24 scalar equations, 12 of which are kinematic constraint equations, Eqs. (2) and (3), and the remaining twelve are force/moment equilibrium equations, Eqs. (10) and (12). Therefore, one has to define twelve variables in order to solve for every unknown in the system.

The most straightforward condition is case A, shown in Table 1, where the desired variables are position, orientation angles, and the output wrench. Once we know the position and the orientation matrix, we can solve for the force in the limbs and the total limb lengths by using Eqs. (3) and (10), respectively. We then use the Hooke's law from Eq. (12) to find the actual spring lengths, and finally all total lengths from Eq. (2). Therefore, this condition allows us to use the inverse kinematics approach and provides a closed-form solution, which is useful for specifying setpoints for the controller.

For case B, given the lengths of both the actuators and springs through a pair of sensors on each limb, one can specify both position/orientation of the platform by solving forward kinematics. Despite the absence of analytical solutions for Stewart platforms with general geometries [13], we can find the numerical solution using the Newton–Raphson method [14]. Once we know the configuration of the robot, we can employ Eq. (10) to find the output wrench.

One of the most challenging situations is when the position and the orientation of the end effector are not given, and other given variables do not let us explicitly specify all total lengths of limbs. For example, in case C, where the given variables are only an output wrench and all six actuator limb lengths (e.g., the spring deflections are not measured, and we know only the resting spring lengths), we cannot calculate either the Jacobian matrix or use the total length of the limbs to find position and orientation via the forward kinematic approach. Since this case will give us more than 20 constraint equations, traditional iterative solvers may not be able to handle these efficiently. One way to find all possible solutions is to use more advanced numerical solvers like polynomial homotopy continuation [15]. This scenario can be avoided in real applications by using additional sensors on the springs.

Practically, cases A and B can be employed in a position controller to regulate the position/orientation or the output wrench of the mechanism, and we notice only a slight increase in the computational complexity when using within control loops.

## Reachable Workspace

The reachable workspace of the architecture was found by checking the range of motion of the actuators at each discrete point. We assume no mechanical interference as well as no universal and spherical joint limits. Owing to additional compliant components, the workspace can be separated into two main types. The first type is the “active workspace” where the mechanism can reach while still having all springs in the resting positions, and the second type is the “passive workspace” where the mechanism can only reach by having at least one spring preloaded with its linear actuator fully extended or retracted. Figure 2 shows the translational workspace at zero orientation angles. The volume inside the inner surface represents the active workspace, while the extra volume between the outer and the inner surfaces shows the passive workspace. The passive workspace is the area with nonzero minimum potential energy of the springs, as illustrated in Fig. 3. The volume enclosed by the inner surface can also be seen as the workspace for rigid actuators with the same total length as the SEAs with resting spring lengths.

## Output Wrench Capability

Unlike rigid actuators, which have a constant force saturation limit regardless of the current position inside the workspace, SEAs control the deformation of springs to generate the force. This means that the actuation limit of each limb will vary within the workspace based on a relationship between the calculated total length of each limb and dimensions of the spring and the linear actuator. In the case of the 6DOF Stewart platform, the effect on each linear SEA cannot be explicitly determined without solving the kinematic and force equations. Before finding the wrench capability in the workspace, we need to first specify the actuation limit of each actuator throughout the workspace.

### Feasible Actuation Set.

From Sec. 4, the active workspace is the area which the moving platform can reach without any spring deflection, whereas in the passive workspace, at least one actuator is fully extended or retracted with its spring deflected. Hence, we categorize actuation limits into seven cases. Consider the scenarios in Fig. 4. Given total lengths from desired position/orientation and an output wrench (case A from Table 1), the bidirectional actuation limits {I, II, III} can occur in every area of the workspace, while the unidirectional ones {IV, V, VI, VII} only occur in the passive workspace. Considering case {I}, the position of the actuator still has enough clearance to reach the maximum spring deflection on both directions, while cases {II} and {III} cannot achieve the maximum compression and extension forces, respectively, due to the smaller gaps. In cases {IV, V}, the actuator has reached its position limit, and the spring has to be preloaded to satisfy a fixed total length ($ℓi,tot$). Hence, the actuator can move only in the direction that increases the magnitude of the force. Finally, in the extreme cases, both the actuator and the spring are either fully retracted, applying the maximum compression force (case {VI}), or are fully extended (case {VII}), producing the maximum tension force. Regarding the hardware design, $ℓi,fix$, has already included the fully compressed length of the spring.

Figure 5 shows an example of the actuation limits by using the same set of spring parameters on each module. Each point on an elevation in the translational workspace (bounded by the outer line) projects to a pair of upper and lower actuation limits. Hence, the upper and lower surfaces enclosing the feasible actuation set are the limits of each actuator throughout this elevation of the workspace. The surface shapes vary from one actuator to another. The slopes of upper and lower surfaces (cases {II, IV} and {III, V}, respectively) are on the opposite lateral sides, due to the fact that each actuator will either have to extend or retract more when the robot reaches a corner of the workspace. Intuitively, the regions where the six pairs of the slopes will appear on the workspace correspond to different actuator mounting points, as illustrated by the representative three actuators in Fig. 5. Around the boundary of the active workspace (inner line), we can see those slopes crossing the zero to the unidirectional limits in the passive zone (case {II} becomes cases {IV, VII} or case {III} becomes cases {V, VI}).

In order to translate these characteristics to the output wrench, we can apply a selected method on discretized points in a workspace, which will be discussed in Sec. 5.2.

### Explicit Analysis.

The explicit method was first proposed by Zibil et al. [16] to find the wrench capability of a redundantly actuated planar mechanism. Firmani et al. showed the geometric interpretation of the method in terms of projection of facets from a hypercube in joint space to facets of a polytope in the task space [17]. Using this method, one can prescribe a moment vector as a constraint and then find the maximum achievable force or vice versa.

We evaluated the wrench capabilities in two cases: the maximum force with a prescribed moment and the maximum moment with a prescribed force. For example, in the former case, given the direction of the force by three directional cosine angles $(αf,βf,γf)$ at a platform position/orientation and a prescribed moment vector $m$, there are twelve cases in total for each configuration, which result from selecting one motor to have its maximum output force as $±τi,m$ (in addition to the five prescribed variables). Then, we can rearrange Eq. (10), which is a linear system of six equations, to solve for the magnitude of the remaining actuators and the magnitude of the force, namely,
$[−q̂1−q̂2⋯−q̂5f̂−b1×q̂1−b2×q̂2⋯−b5×q̂503×1]︸A[τ1τ2⋮τ5f]︸x=[q̂5b5×q̂5][±τ6,m]−[03×1m]︸b$
(13)

where $f̂=[ cos αf cos βf cos γf]T$ and $m=[mxmymz]T$.

After solving x from the linear equation (13), all solved actuator forces τi are checked to see if they are still within limits, and f has to be positive as well. Then, the maximum magnitude of f will be selected from the feasible set of solutions, and the algorithm will move on to another adjacent position or force direction.

Note that the upper and lower actuation limits can have the same sign in the passive workspace (denoted by $τi,m,upper$ and $τi,m,lower$ instead of $±τi,m$). The lower magnitude (closer to zero) among these two values will contribute to the nonzero lower limit of the force/moment capability. If there are more than one actuator that has unidirectional actuation limit in the same end-effector configuration, the algorithm will select the minimum value of f among the feasible solutions.

For simplicity, we assume that all six actuators are identical, including the spring parameters, the actuator stroke lengths (50 mm), and the actuator fixed lengths.

To illustrate the results using this approach, Fig. 6 shows upper and lower magnitudes of force in the positive x direction, (a) and (b), and moment in the positive z direction, (d) and (e), using a particular set of spring parameters. Also, these results are compared with the case where the linear spring is replaced by a rigid element (Figs. 6(c) and 6(f)), using a fixed bidirectional actuation limit of the motor, bounded by the same amount of force from the maximum spring deflection.

The lower magnitude limits in Figs. 6(b) and 6(e) are always zero inside the active workspace (dotted line) since all springs can stay unloaded, while the results in the passive workspace (between dotted and solid lines) may not be zero since there exists at least one motor with a unidirectional actuation limit.

Comparing the results between the SEA case and the rigid actuator case, the decrease in wrench capabilities that occurs is related to the desired direction of the force/moment vector. In Fig. 6(a), the area with higher force magnitudes shifts to the opposite direction (−x) of the desired force (+x). The results imply that bias forces from the preloaded springs in the passive workspace make the mechanism more capable of producing forces in the direction that restores the spring equilibrium. It is also reasonable that the desired moment direction (+z), normal to the sampled workspace, causes the reduction of the upper limit in every direction further from the center of the workspace as shown in Fig. 6(d).

From Figs. 6(b) and 6(e), the lower magnitude limits start from zero at the boundary of the active workspace (dotted line) and then approach closer to the upper magnitude values at further distances from the boundary. This characteristic results from a shorter range of unidirectional actuation limits, of which the upper bound of the magnitude of force is always the maximum value. However, both upper and lower force/moment limits do not expand all the way to the outer edge of the passive workspace. This outcome is due to the fact that when the end effector reaches beyond a certain distance, it will not be possible to create a force/moment in an exact desired direction while still satisfying all actuation limits.

In addition, the inner dashed line in Figs. 6(a), 6(b), 6(d), and 6(e) shows the full actuation zone, that is, the area where all six actuators have full actuation limits, i.e., every actuator belongs to condition {I}. It is reasonable that the decrease in wrench capabilities happens outside this dashed line.

### Isotropic Force/Moment.

Next, we will refer to the isotropic force/moment to get a more comprehensive analysis of the wrench capability at a given position/orientation. We define an isotropic force as the smallest force magnitude among the maximum allowable force in every direction. In other words, at a given position/orientation, the isotropic force is the maximum force that the end effector can apply in every direction in the Cartesian coordinate of the fixed frame. The same definition applies to the isotropic moment.

A comparison of the isotropic force/moment between two sets of spring parameters in a translational workspace and a pitch-roll-yaw workspace is shown in Fig. 7. The first row in the figure shows the isotropic force with the zero prescribed moment while the second row shows the isotropic moment with the zero prescribed force. Each surface layer represents the outer most boundary where the robot can apply the isotropic force/moment up to the specified value.

Despite having the same maximum actuating force in both sets of spring parameters ($ksℓs,0=±$ 50 N), we can see noticeable differences in the isotropic force/moment between these two sets of spring parameters. The first set of spring parameters in Figs. 7(a)7(d), with a higher compliance and a longer maximum spring deflection, provides a smaller volume of isotropic force/moment as compared to the other spring set. For example, the 75 N isotropic force surfaces in Figs. 7(e) and 7(f) and the 14 N·m isotropic moment surface in Figs. 7(g) and 7(h) expand to around half of the size of the active workspace when using a spring that is twice as stiff as the first one.

Clearly, the force–moment capabilities of the end effector will increase with the spring constant, based on the fact that each actuator needs a shorter traveling length to generate a certain amount of force. In addition, we can see that the passive workspace is no longer useful because the brace cannot reach these regions without crossing through an area of zero isotropic force at the boundary of the active workspace.

## Discussion

The selection of spring parameters has a significant effect on the wrench capability of a Stewart platform with SEAs. This analysis can be a useful guideline for selecting an appropriate set of parameters to get a desirable isotropic force/moment profile. Although the increase in the spring stiffness can improve the wrench capability, we have to consider the position accuracy of spring deflection sensors and also the resulting stiffness of the mechanism. Furthermore, the maximum force from the spring deflection should be lower than the back-drive force of the linear actuator to minimize the power consumption as well as extend the lifetime of the actuators.

Even though the entire workspace of the robot is expanded by the passive workspace, the robot cannot generate any useful output wrenches inside that area according to the isotropic force/moment analysis. Nevertheless, this kind of workspace still increases the mobility of the mechanism and reduces impact forces on actuators around the workspace boundary.

## Conclusions

Our formulated mathematical models of a 6-6 Stewart platform with SEAs solve the position and the static wrench of the robot simultaneously. We also find that the computation of our solutions for the SEA system with extra position sensors on the springs is not significantly more complicated than a system with rigid actuation. This characteristic allows easy implementation within the controller.

Our observation that the compliant actuation limit in the joint space is strongly influenced by the actuator stroke length, the spring stiffness, and the maximum traveling length of the spring paves the way for a more systematic approach of designing parallel mechanisms with SEAs. In addition, the implementation of the explicit analysis to convert this actuation limit into the wrench capability of the robot in the form of isotropic force/moment expands the guideline for choosing appropriate sets of the spring parameters corresponding to required applications. The novel methodology presented here is applicable to other parallel architectures with linear SEAs, including haptic devices and rehabilitation robots.

## Funding Data

• National Science Foundation (Grant No. IIS-1527087).

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