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Research Papers

# A New Parallel Actuated Architecture for Exoskeleton Applications Involving Multiple Degree-of-Freedom Biological JointsPUBLIC ACCESS

[+] Author and Article Information
Justin Hunt

School for Engineering of Matter,
Transport, and Energy,
Arizona State University,
Tempe, AZ 85281
e-mail: jphunt3@asu.edu

Hyunglae Lee

School for Engineering of Matter,
Transport, and Energy,
Arizona State University,
Tempe, AZ 85281
e-mail: Hyunglae.Lee@asu.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received January 17, 2018; final manuscript received June 25, 2018; published online August 6, 2018. Assoc. Editor: K. H. Low.

J. Mechanisms Robotics 10(5), 051017 (Aug 06, 2018) (10 pages) Paper No: JMR-18-1018; doi: 10.1115/1.4040701 History: Received January 17, 2018; Revised June 25, 2018

## Abstract

The purpose of this work is to introduce a new parallel actuated exoskeleton architecture that can be used for multiple degree-of-freedom (DoF) biological joints. This is done in an effort to provide a better alternative for the augmentation of these joints than serial actuation. The new design can be described as a type of spherical parallel manipulator (SPM) that utilizes three 4 bar substructures to decouple and control three rotational DoFs. Four variations of the 4 bar spherical parallel manipulator (4B-SPM) are presented in this work. These include a shoulder, hip, wrist, and ankle exoskeleton. Also discussed are three different methods of actuation for the 4B-SPM, which can be implemented depending on dynamic performance requirements. This work could assist in the advancement of a future generation of parallel actuated exoskeletons that are more effective than their contemporary serial actuated counterparts.

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## Introduction

The majority of exoskeleton robotic devices utilize a serial chain architecture of actuated joints and linkages to accomplish a desired task [1]. Such designs may in part be bio-inspired by our own anatomy of bones and ligaments, and therefore might seem a logical choice for mirroring or augmenting our own movements. While this rationale holds for simple one degree-of-freedom (1DoF) joints like the elbow and knee, it does not for multiple DoF ball and socket joints like the shoulder, hip, wrist and ankle. Unlike 1DoF biological joints that can share an axis of rotation with a corresponding artificial joint, ball and socket joints operate about a center of rotation that cannot be shared in an anatomically similar manner. Therefore, in order for a serial actuated exoskeleton to interface with a ball and socket joint, its design must diverge from our own anatomy.

The most common solution to augment the motion of a ball and socket joint with serial actuation is to use a multiple linkage robotic armature. The armature works to bridge the ball and socket joint by connecting the structures on either side. While this approach works from a kinematic standpoint, it can be problematic from a dynamic one. In order to bridge a three degree-of-freedom (3DoF) ball and socket joint using a serial armature, the armature must contain at least three joints and two linkages. The moment arm associated with these linkages has the inherent disadvantage of amplifying torques applied to the joint by external forces. Furthermore, each motorized joint in the armature must be capable of not only actuating the corresponding human limb, but also able to lift and manipulate the joints that follow in series. This requires larger motors that consume more energy.

Despite its problems, serial actuation is still arguably the most popular solution for emulating ball and socket joint motion. It has been utilized extensively in exoskeleton robotics with several notable examples including the XOS by Sarcos/Ratheon2, HAL by CyberDyne3 [2], Body Extender by Perceptual Robotics Laboratory [3], HULC by Ekso Bionics4 [4], and H-Wex by Hyundai5 [5]. It could be contended that the popularity of using serial actuation for the purpose augmenting ball and socket joint motion is not the result of ideal performance, but rather a lack of viable alternatives.

When considering alternatives to the conventional use of serial actuation to augment ball and socket joint motion, parallel actuation may come to mind. A parallel architecture can offer many benefits over a serial one. Below is a list of the most prominent advantages of parallel actuation that are relevant to exoskeleton robotics.

1. (1)Short moment arm: The end effector of a parallel actuated system has a considerably shorter moment arm than that of a serial system with comparable DoF [6,7]. This can help reduce the resultant torque applied at a joint by external forces.
2. (2)Low-end effector inertia: Parallel actuated systems have a low center of mass, which results in low-end effector inertia [6,7]. This can reduce energy cost during operation and is an important consideration from a mobility perspective.
3. (3)High-end effector acceleration: As a result of low end effector inertia, parallel actuated systems are typically capable of high-end effector acceleration [6,7]. This can be advantageous for high performance augmentative applications.
4. (4)Minimal positioning error: Unlike serial actuated systems, which have position errors that accumulate and amplify from base to end effector, parallel systems work to attenuate errors found in one or more linkages through cooperative end effector control [68].
5. (5)High potential stiffness: The multiple mounting point locations on both the mobile platform and stationary base of a parallel system can lead to high mechanical stiffness [6,7,9,10]. However, this result is highly dependent on both the parallel architecture type and degree of actuation chosen.
6. (6)May not occupy center of rotation: Parallel actuated systems can be designed to rotate about a point without occupying it [7,11]. This has important application when interfaced with ball and socket joints like the shoulder, hip, wrist, and ankle.

Several exoskeleton robotic devices have incorporated parallel actuation in prior work. A few notable examples include the RiceWrist [12] and the MR-SoftWrist [13], both of which utilize a three revolute-prismatic-spherical (3RPS) architecture [14] and the Anklebot [15], which uses a two spherical-prismatic-spherical (2SPS) architecture [16]. These robot architectures have been used quite successfully to study certain properties of the human neuromuscular system [17,18]. However, these robots require the human joint to be used in the closed-form kinematic solution. This is not ideal for certain augmentation applications where dangerously high forces could be transmitted through the system and thus the user.

Despite a few examples of its use, parallel actuation has not yet been widely adopted in the field of exoskeleton robotics. The primary reason for this is that parallel actuated architectures that do not include the human joint as part of the closed-form kinematic solution are disadvantaged by a relatively small workspace when compared to serial actuation. This can make it difficult to meet the design requirements of many applications. This limitation is largely a result of mechanical interference between the parallel actuated linkages [7]. A good example of this interference can be observed in the 6DoF Stewart-Gough platform with its six parallel actuators [19].

One method of achieving a larger workspace with parallel actuation is to utilize an architecture that contains fewer parallel linkages and thus less mechanical interference. Such architectures are typically designed to operate with either purely translational or rotational motion and are commonly referred to as translational parallel manipulators and spherical parallel manipulators (SPM), respectively [11,20]. In the case of a ball and socket joint that only requires spherical motion, the use of an SPM is possible. Several SPM architectures have been proposed over the years, with the most common being the Three Universal Parallel Universal (3 UPU) [21] and the three rotational rotational rotational (3RRR) [22]. Both of these designs have three decoupled rotational DoF and do not occupy the center of rotation. Therefore, either design could theoretically be used for an exoskeleton robotic application. However, both have been found to have limitations. The 3 UPU, though sound from a kinematic standpoint, has been shown to have poor dynamic performance as a result of a high number of passive DoF [23]. The 3RRR, while also not ideal from a dynamic standpoint [22,24], has the additional problem of its curved linkages passing through the majority of the spherical workspace. This makes mechanical interference with the limb a concern.

Limitations with current SPM architectures available led the authors to develop the first SPM specifically intended for exoskeleton robotics in prior work [25]. The principal concept of this early design was to minimize the number of parallel substructures while maximizing the number of active DoF in order to maintain good dynamic performance. This was done through a new technique the authors developed call Modular Motion Coupling. This technique involved the use of a specially designed gearbox with one input and multiple outputs connected to each parallel substructure. The multiple outputs would be used to actively control multiple DoF of each substructure dependently. With more active DoF, dynamic performance overall would be improved while still maintaining a large workspace. A prototype of the design was developed to validate its kinematics and investigate its dynamic behavior.

Building and testing this new parallel architecture helped identify two issues with it. First, the need for a complex gearbox with two different specifically geared outputs was found to be significantly more expensive than a traditional gearbox. This limits the design's adoptability from an economic standpoint. Second, while the design was able to provide good dynamic performance along the DoF that were actively controlled, the passive roll DoF of each substructure was not directly actuated, but instead determined by the synergistic motion of all three substructures and proved susceptible to positioning error under load. While likely not an issue for low force tasks, this result could be problematic for high force augmentative applications.

In an effort to resolve the issues with the author's prior work, a novel SPM was developed and is presented in this work. The two driving design principles of this new architecture are affordability and fully active DoF control of all substructures so as to provide good dynamic performance. The design varies from any other SPM, including the author's prior work, through its utilization of multiple four bar linkages as part of a greater mechanism to generate spherical motion. Therefore, the design will be referred to in this work by the acronym 4 bar spherical parallel manipulator (4B-SPM).

The rest of this paper focuses on the 4B-SPM architecture and its potential application for exoskeleton robotics. Section 2 begins with an overview of the 4B-SPM design, followed by an analysis of its forward and inverse kinematics. Techniques to actuate all DoF of each substructure in order to improve dynamic performance are also introduced. Last, a cooperative control strategy is discussed that allows the 4B-SPM to operate with highly flexible joints, like the shoulder and hip, without experiencing mechanical interference between operator and device. Section 3 demonstrates four different 4B-SPM exoskeleton simulations that establish both the range of application and validate the kinematic methods describe in Sec. 2. These simulations include an ankle, hip, wrist, and shoulder exoskeleton robot. Also included in Sec. 3 is a case study, in which the development of a shoulder exoskeleton is considered in order to discuss the real world cost and feasibility of the 4B-SPM. Finally, Sec. 4 concludes the paper with a discussion of the results and a summary of contribution.

## Methods

###### Overview of the 4 Bar Spherical Parallel Manipulator Architecture.

The architecture of the 4B-SPM is shown in Fig. 1. The design consists of three actuated substructures that couple a stationary base to a mobile platform. The mobile platform has three independent rotational DoF and operates tangential to the spherical workspace shown in Fig. 1. Each substructure is comprised of a 4 bar mechanism connected to a grounded revolute joint at its base. The axes of the grounded joints are designed to intersect at a desired point in space. This point represents the center of rotation of the mobile platform. The top linkage in each 4 bar mechanism is extended to reach the mobile platform. Each top linkage is coupled to the mobile platform using a 3DoF spherical joint. Depending on the dynamic performance requirements, three versions of the 4B-SPM that use different actuation methods are presented. These include the three revolute actuator version presented in Fig. 1, along with a six revolute actuator version and a three revolute two prismatic actuator version discussed in Sec. 2.4. While all three architecture share similar kinematic solutions, each has advantages and disadvantages. The three revolute actuator version presented first is the simplest and thus most economic, but is susceptible to low torque output in certain configurations. Therefore, this architecture is not ideal for high load applications. The two alternate designs discussed are more complicated, but offer benefits that include more homogenous torque output and/or higher stiffness. More details regarding the dynamic advantages of these two architectures are provided in Sec. 2.4.

###### Forward Kinematics.

In order to implement practical control of the 4B-SPM, its forward kinematics must be solved so that any differences between commanded and actual position can be mitigated using a closed-loop controller. Solving the forward kinematics require a unique solution to exist for a given set of commands. A singular kinematic solution was shown to exist for the author's prior SPM design that functioned using the same number of substructures with the same number of active and passive DoF [25]. Therefore, the 4B-SPM can be classified under the same category of SPM and is subject to the same mobility.

As with many parallel architectures, the forward kinematics of the 4B-SPM are not observable using only actuator feedback [7,26]. Instead, it is necessary to integrate additional sensors into each 4 bar substructure so that the roll, in addition to the pitch, can be directly measured. By knowing the roll and pitch of each substructure, the corresponding end points (i.e., platform mounts) shown in Fig. 2 can be determined. This is most easily done by first realizing that the platform mount vector $P¯i$ that originates at the global origin frame at the center of the spherical workspace is equal to the 4-bar linkage vector $C¯i$. This is because $P¯i$ is not a physical linkage and consequently its magnitude and direction are dictated by the parallel vector $C¯i$. Therefore, if the pitch $βi$ and roll $γi$ sensors of a given substructure are zeroed with respect to the global frame, then the resulting spherical coordinates $(C¯i,βi,γi)$ of $C¯i$ can be directly converted into the Cartesian coordinates $(xi,yi,zi)$ of platform mount $P¯i$.

Once the three platform mounts $P¯1,2,3$ are identified, they can be used to generate a transformation matrix $T$ that describes the position $P¯c$ and orientation $R$ of the platform at its centroid. Referring again to Fig. 2, the transformation matrix $T$ can be obtained by first considering the platform orientation component $R$. The 3 × 3 matrix $R$ can be determined by first generating an intermediate rotation matrix $R′$ that is dependent only on the position of the platform mounts $P¯1,2,3$ and does not consider the desired $XY$ planar orientation needed to align the local platform frame with the direction of a user limb. Assume that matrix $R′$ is orientated as shown in Fig. 2, then $R′$ can be calculated as follows: Display Formula

(1)$R′=[R¯x′R¯y′R¯z′]$

where Display Formula

(2)$R¯z′=(P¯2–P¯1)×(P¯3–P¯1)(P¯2–P¯1)×(P¯3–P¯1)$
Display Formula
(3)$R¯y′=R¯z′×P¯1R¯z′×P¯1$
Display Formula
(4)$R¯x′=R¯y′×R¯z′$

If $R′$ is assumed to not have the same desired $XY$ planar alignment as $R$, then it must be multiplied by the rotation matrix $Rz$ about the $z$-axis such that Display Formula

(5)$R=RzR′$

where Display Formula

(6)$Rz=cos(α)−sin(α)0sin(α)cos(α)0001$

The angle $α$ is taken to be measured from the axis $R¯′x$ as shown in Fig. 2. The direction of the new axis $R¯x$ is intended to be aligned with the user limb. Therefore, $α$ is dependent on the particular geometric design of the mobile platform.

With the rotation matrix $R$ of the platform known, the only component of transformation $T$ left to determine is platform translation. If the translation $P¯c$ of the platform is taken to be its centroid, then calculating $P¯c$ is a simple matter of finding the mean vector described by $P¯1,2,3$, which is Display Formula

(7)

With both the rotation and translation of the platform known, the complete transformation matrix $T$ that describes the platform position and orientation as a function of the three platform mounts $P¯1,2,3$, with respect to the global frame, can be written as Display Formula

(8)$T=RP¯c0¯1x31$

To clarify the components of the transformation matrix $T$, $0¯n×m$ represents a zero vector comprised of $n$ rows and $m$ columns, $R$ represents the desired 3 × 3 platform rotation matrix, and $P¯c$ represents the calculated 3 × 1 platform centroid vector.

###### Inverse Kinematics.

The second kinematic operation that must be considered in order to implement effective control over the 4B-SPM is its inverse solution. For this operation, the rotation matrix $R$ of the platform is given and the substructure pitch and roll angles are to be calculated. As with the forward kinematics, identifying the mounting point positions should be considered a first step toward obtaining the substructure pitch angles. To find $P¯1,2,3$ that correspond to the commanded $R,$ a series of transformations are used. These transformations in order are: (1) rotational transformation $T1$ from the global frame orientation to the mobile platform orientation $R,$ (2) translational transformation $T2$ from the center of rotation to the platform centroid at radius $r,$ (3) translational transformation $T3,i$ from the mobile platform centroid to the platform mounting point position. Therefore, the transformation matrix that describes the position of each platform mount $Tp,i$ as a function of the platform orientation $R,$ platform centroid radius of operation $r$, and the distance from the platform centroid to the platform mounting points $p¯i$ can be expressed as Display Formula

(9)$Tp,i=T1T2T3,i$

where Display Formula

(10)$T1=R0¯3x10¯1x31$
Display Formula
(11)$T2=I00r0¯1x31$
Display Formula
(12)$T3,i=Ip¯i0¯1x31$

Here, $I$ represents a 3 × 3 identity matrix and $0¯n×m$ represents a zero vector comprised of $n$ rows and $m$ columns. The platform mounts $P¯1,2,3$ with respect to the origin are given as the translational component of the transformation $Tp,i.$ The platform centroid radius of operation is given as $r.$ The pitch $βi$ and roll $γi$ angles of a given substructure $i$ can be identified using $P¯i$, $B¯i$ and the global reference frame vector $z¯$ such that Display Formula

(13)$βi=acosP¯i·B¯iP¯i·B¯i$
Display Formula
(14)$γi=acosB¯i×z¯·B¯i×P¯iB¯i×z¯·B¯i×P¯i$

It should be noted that since Eqs. (13) and (14) utilize a dot product operation, the sign of each angle is undetermined. Therefore, it is important to observe the location of $P¯i$ so that the correct sign convention can be applied.

###### Roll Actuation of the 4 Bar Spherical Parallel Manipulator Substructures to Improve Dynamic Performance.

Many SPM designs have been shown to have sound kinematics, but do not function well under real world conditions. The reason for this is that certain DoF of a given SPM may not be directly actuated, but instead are determined by the synergistic motion of its substructures [22,24,27]. This is problematic, because certain substructure orientations may produce low torques or forces along the DoF that are not directly actuated. In the case of the 4B-SPM, this is potentially the roll DoF of each substructure. Thus, for certain applications where high loading is a possibility, it may be necessary to actuate the roll DoF of each substructure in a more direct way in order to produce a more homogeneous output torque. Here, two methods to do so are discussed.

###### Roll Actuation Using Three Revolute Motors.

The simplest and most direct method of actuating the 4B-SPM substructure roll is to actuate each ground revolute joint as shown in Fig. 3. Thus, for this version of the 4B-SPM, both pitch angle $βi$ and the roll angle $γi$ found in Sec. 2.3 would be directly controlled. This method of roll actuation would, in effect, produce three simple serial architectures. Each of the serial architectures has its own independent singular kinematic solution, but share the same coupled solution described in Secs. 2.1 and 2.2.

While the addition of three revolute motors works to actuate roll, it also brings the total motor count to six. This may raise questions as to whether a more traditional six actuator parallel architecture, such as the Stewart Platform (i.e., hexapod) could be used instead of the 4B-SPM. The answer to this goes back to the workspace limitations. Because the 4B-SPM has only three substructures instead of six, it encounters less mechanical interference between substructures and therefore affords a larger spherical workspace.

###### Roll Actuation Using Two Interconnecting Motorized Linkages.

A second approach to roll actuation is to include linkages between substructures 1–2 and 2–3 as shown in Fig. 4. By connecting one end of a linkage to a motorized slider mounted on one of the two corresponding substructures, it is possible to adjust the linkage position such that it does not violate the kinematic solution found in Sec. 2. This approach has two advantages. First, it only requires two additional motors instead of three. Second, because the linkage is mounted away from the roll axis of each substructure, it helps to further reduce deflection. On the other hand, the use of motorized sliders makes this approach more complex from a design standpoint than the three and six motor configurations described in Secs. 2.1 and 2.4.1, respectively.

To find the slider positions that comply with the kinematic solution found in Sec. 2, a system of equations must be solved in a simultaneous manner. Referring to Fig. 5, consider the system Display Formula

(15)$B¯1+C¯1+k1D¯1+L¯1=B¯2+C¯2+k2D¯2$

Here, $k1$ and $k2$ are scalars of the corresponding D vectors such that $kiD¯i. Assume the slider is placed along $D¯1$, then $k1D¯1$ describes the position of the slider and $k2D¯2$ is a fixed position along $D¯2$. The position of $k2D¯2$ is a design choice and therefore known. The vectors $B¯1,C¯1,$ and $D¯1$ are also known from Sec. 2. The unknown terms are $k1$ and the interconnecting linkage vector $L¯1$. Thus, from Eq. (15), we have three equations and four unknowns. The design choice of magnitude of $L¯1$ must be made before the unknowns can be determines, such that Display Formula

(16)$L¯12=L1,x2+L1,y2+L1,z2$

By solving Eqs. (15) and (16) simultaneously, it is possible to identify the slider position $k1$ that complies with the kinematic solution given in Sec. 2. In a similar manner, $k3$ can be determined by choosing $k2$ to remain fixed.

###### Cooperative Control Techniques to Improve Effective Workspace.

As mentioned in the Introduction, one of the disadvantages of parallel manipulation is limited workspace. Although the 4B-SPM architecture presented here provides one of the largest known spherical workspaces, it can still have difficulty matching the range of motion of a highly flexible joint, such as the shoulder or hip. Depending on the task, the mechanical limits of the 4B-SPM may be reached before that of the human joint. In order to improve the practical workspace of the 4B-SPM, a new method of cooperative control is presented. It should be noted that this new method shares some techniques with a similar method that the authors developed in a prior work [25].

The cooperative control technique discussed here involves allocating one of the three DoF of the 4B-SPM to the robot controller. This is done so that the robot controller can determine the solution to this DoF so as to keep the 4B-SPM within its workspace while still following the two degree-of-freedom (2DoF) commands sent by the operator. To better understand this idea, the shoulder will be used as an example moving forward. The shoulder arguably has the greatest range of motion of any joint in the body and therefore is an ideal candidate for such a technique.

For the example of a shoulder exoskeleton, the operator would control the inclination and azimuth of the arm, while the robot would control the roll of the shoulder plate (i.e., mobile platform) in order to keep the 4B-SPM from hitting a mechanical limit. This allows for greater range of motion with the 4B-SPM, but the division of control has a drawback. Allowing the robot control of the shoulder plate roll limits operator ability to reach a desired position with a particular roll orientation of the arm. Therefore, for the device to remain effective, a 1DoF revolute joint interface between the human operator and the 4B-SPM must be added. This joint could be passive or active depending on the application.

One novel method of determining the roll solution of the shoulder plate is presented here. For this method, the roll of the shoulder plate is considered as a function of the inclination and azimuth of the arm. From Sec. 2, the rotation matrix $R$ that defines the orientation of the platform can be described by a set of three Euler angles. These three angles will be defined as the azimuth $φ$, inclination $θ$, and roll $ψ$ of the arm. Therefore, using the global coordinate frame shown in Fig. 2, the matrix $R$ can be defined as Display Formula

(17)$R=RzRyRx$

where Display Formula

(18)$Rz=cos(φ)−sin(φ)0sin(φ)cos(φ)0001$
Display Formula
(19)$Ry=cos(θ)0sin(θ)010−sin(θ)0cos(θ)$
Display Formula
(20)$Rx=1000cos(ψ)−sin(ψ)0sin(ψ)cos(ψ)$

The matrices $Rz$ and $Ry$ are functions of $φ$ and $θ$, respectively, and therefore are determined by the operator. To determine the shoulder plate roll matrix $Rx$, it is necessary to consider how one would desire the shoulder plate to move. For example, Fig. 6 shows a set of shoulder plate orientations $R$ selected for a particular range of motion defined by arm at rest, 90 deg flexion, and 90 deg abduction. Each of these arm directions has a corresponding roll matrix $Rx$ associated with it. Since $Ry$, $Rz$, and $R$ are known, the roll matrix $Rx$ can be determined by Display Formula

(21)$Rx=Ry−1Rz−1R$

The angle $ψ$ that defines $Rx$ can be determined by comparing similar terms between Eqs. (20) and (21). Now, with all the Euler angles known, it is possible to choose a function for $ψ$ that describes its solution for any location within the desired range of motion. The chosen solution must not only satisfy the three boundary solutions already known, but also take into consideration whether the desired roll motion between boundary solutions should be linear, sinusoidal, etc. For example, assume linear roll motion between points is desired; then the function $ψ$ that also satisfies the boundary solutions can be expressed as Display Formula

(22)$ψ=2θφπ−θ+π2rad/s$

Therefore, the Euler angle $ψ$ that describes the shoulder plate roll is a linear function of the desired azimuth and inclination Euler angles $φ$ and $θ$ as determined by the operator. This relation satisfies the three rotation matrices shown in Fig. 6. As a result, it should be noted that since Eq. (22) was developed using the desired workspace boundary orientations in Fig. 6, it is only applicable for that set orientations. For example, consider the alternative case where the desired shoulder plate orientation for 90 deg abduction is above rather than behind the shoulder, then a new function for $ψ$ would need to be developed using this new rotation matrix, along with the other two shown in Fig. 6.

## Results

###### Kinematics.

To demonstrate the versatility and kinematics of the 4B-SPM, a series of exoskeleton simulations created using Matlab (MathWorks, Natick, MA) are presented. These include an ankle, a wrist, a hip, and a shoulder exoskeleton. For each simulation, both methods of roll actuation described in Sec. 2.4 are demonstrated. Additionally included for the shoulder exoskeleton is the cooperative control technique discussed in Sec. 2.5. Each simulation shows the 4B-SPM linkages as red lines and the mobile platform as a red plane defined by the three mounting points. It should be noted that for a real device, the mobile platform would likely represent a shoulder plate for the shoulder, a rigid cuff for the hip, a shoe for the ankle, and a glove for the wrist. One final aspect of these simulations to be aware of is that since placement of both the base mounts and platform mounts is flexible, each simulation only demonstrates one possible configuration of the 4B-SPM. The configurations shown were chosen because each interfaces well with the anatomy of the corresponding joint while still reaching the desired workspace.

###### Ankle Exoskeleton.

A 4B-SPM ankle robot is shown in Fig. 7. A simulation of its kinematics for both roll actuation methods described in Sec. 2.4 is shown in Table 1, Extension 1. In the video, the ankle range of motion is shown with bounds of 45 deg planter flexion, 15 deg dorsiflexion, 20 deg inversion, 20 deg eversion, 20 deg pronation, and 20 deg supination. These bounds were chosen to approximately match accumulative average range of motion data [28], but could be changed if needed. In this configuration, the base mounts are positioned about the tibia bone and the mobile platform mounts are positioned atop the front and back of the foot. Depending on the application, the mobile platform may be taken as a shoe and the stationary base as either part of a greater exoskeleton or possibly a knee orthosis brace.

###### Wrist Exoskeleton.

The 4B-SPM wrist robot is shown in Fig. 8. A simulation of its kinematics for both roll actuation methods described in Sec. 2.4 is shown in Table 1, Extension 2. In the video, the wrist range of motion is shown with bounds of 65 deg flexion, 65 deg extension, 75 deg pronation, 75 deg supination, 20 deg ulnar deviation, and 20 deg radial deviation. These bounds were chosen to approximately match accumulative average range of motion data [28], but could be changed if needed. In this configuration, the base mounts are positioned about the forearm and the mobile platform mounts are positioned on the back of the hand opposite the palm. Depending on the application, the mobile platform could be taken as a partially rigid glove and the stationary base as either part of a greater exoskeleton or possibly an elbow orthosis brace.

###### Hip Exoskeleton.

The 4B-SPM hip robot is shown in Fig. 9. A simulation of its kinematics for both roll actuation methods described in Sec. 2.4 is shown in Table 1, Extension 3. In the video, the hip range of motion is bounded by the three operator leg configurations of 90 deg flexion, 30 deg abduction, and at rest. These bounds were chosen to approximately match accumulative average range of motion data [28], but could be changed if needed. In this configuration, the base mounts are positioned about the waist and the mobile platform mounts are positioned about the femur. Depending on the application, the mobile platform could be taken as a cuff about the upper leg and the stationary base as either part of a greater exoskeleton or possibly a partially rigid vest.

###### Shoulder Exoskeleton.

The 4B-SPM shoulder robot is shown in Fig. 10. A simulation of its kinematics for both roll actuation methods described in Sec. 2.4 using the cooperative control technique for improving effective workspace discussed in Sec. 2.5 is shown in Table 1, Extension 4. To help demonstrate the importance of cooperative control for large range of motion joints, Extension 5 in Table 1 shows what happens when it is not implemented for the shoulder exoskeleton. In these videos, the shoulder range of motion is bounded by the three operator arm configurations of 90 deg flexion, 90 deg abduction, and at rest. While the range of human shoulder motion on average exceeds these bounds [28], this workspace is sufficient to demonstrate both the architecture's potential application for a shoulder exoskeleton device and the cooperative control techniques descripted in Sec. 2.5. The range of motion could be increased by utilizing bounded nonlinear multi-objective optimization to configure the parallel substructures so as to maximize the workspace [29]. In this configuration, the base mounts are positioned behind the operator and the mobile platform mounts are positioned on a shoulder plate. Depending on the application, the stationary base could be taken as either part of a greater exoskeleton or possibly an electric wheel chair.

###### Case Study: 4 Bar Spherical Parallel Manipulator Shoulder Exoskeleton Design and Development Considerations.

In order to have the opportunity to discuss 4B-SPM exoskeleton design aspects such as cost feasibility, material selection, and control methods, a case study is performed here. The objective of this case study is to demonstrate viability of the 4B-SPM architecture through a dialogue regarding aspects critical to its development. A 4B-SPM shoulder exoskeleton was chosen as the subject of this case study. The shoulder was chosen because its complexity relative to the hip, wrist, and ankle would suggest that the design methods involved could likely be extrapolated and applied to these simpler joints as well. For this case study, the device is assumed to not be part of a greater exoskeleton system, but instead mounted to a support structure, such as a stationary frame or electric wheel chair.

For this case study, the roll actuation method described in Sec. 2.4.1 was chosen in order to provide good dynamic performance. This method was selected because the top linkage connecting each four bar mechanism to the shoulder plate is relatively short for the chosen actuator configuration and therefore does not lend itself well to the roll actuation method described in Sec. 2.4.2.

The 4B-SPM shoulder exoskeleton for this case study is shown in Figs. 11 and 12. The design consists of three substructures, each equipped with two Dynamixel MX-106R smart servo motors (Robotis, Seoul, South Korea) with the actuated roll configuration shown in Fig. 3. Each four-bar mechanism is built using a combination of rod end bolts, threaded rod, ball bearings, and 6061 aluminum tubing. Each four-bar mechanism is coupled to a shoulder plate constructed from 6061 aluminum using a ball joint rod end. The position and orientation of each substructure are held fixed with respect to one another through the use of static Stewart-Gough platforms (SSGP) built using threaded rod and ball joint rod ends. The use of SSGPs is an affordable positioning solution that also offers high stiffness and the ability to alter the substructure configuration if desired. A third SSGP (not shown in Figs. 11 and 12) is used to rigidly connect the shoulder exoskeleton to a stationary support frame. It should be noted that the use of SSGP is only one possible solution for the positioning of each actuated substructure. Alternatively, a single piece of custom fabricated material with mounts to connect the three substructures could be used.

It is important to mention not only where the actuated substructures are positioned, but how they are positioned. Proper alignment of the actuated substructures is important. If all substructures do not rotate about the same center of rotation, then the workspace would be ellipsoidal instead of spherical. Additionally, mechanical stress due to misalignment will occur. Two methods to ensure proper alignment of the substructures have been considered: (1) With a CAD rendition of the desired 4B-SPM, the designer could extend rods along the roll axis of each substructure and intersect them at the center of rotation. These rods could be encapsulated about the center of rotation by a cubic structure. By subtracting the rod geometry from the cubic geometry, the designer would be left with a part that could be used to orient all three substructures. This cubic part could be 3D printed or fabricated out of more robust materials. The designer would only need to fix temporary rods along the axis of rotation of each substructure to intersect the cubic part during installation of the SSGPs. (2) As previously mentioned, one alternative to SSGPs would be to use a single piece of custom fabricated material with mounts to connect the three substructures. If this option were used, then the designer would only need a CAD rendition of the desired 4B-SPM with known substructure mounting locations to design and commission a single piece component that would ensure that the roll axis of each substructure intersects at the same point.

To couple the shoulder plate to the operator, a cuff positioned at the upper arm is used. As briefly mentioned in Sec. 2.5, in order for cooperative control techniques that improve the effective workspace of the shoulder exoskeleton to be applied, an additional DoF must be added at the cuff. This is accomplished by constructing the cuff from two concentric tubes. The outer tube is connected to the shoulder exoskeleton and the inner tube is positioned around the upper arm with a padded interface between operator and cuff, as shown in Figs. 11 and 12. Not only does this design permit the rotational 1DoF required for cooperative control, but it also allows 1DoF of minor translational slip motion between the cuffs. This slip is critical to preventing possible joint misalignment between the operator and shoulder exoskeleton from applying dangerously high force to the human shoulder joint [3032]. More details regarding this slip mechanism can be found in the authors' prior work [25] where it was first introduced.

While the cost of the 4B-SPM will vary depending on design choices such as the servos or the controller used, this case study demonstrates the majority of the parallel substructure components can be purchased off-the-shelf and therefore do not invoke high fabrication costs. In addition to this, as discussed in the Introduction, parallel actuation has the inherent ability to rectify positioning errors. This means that cheaper, lower tolerance sensors and components can be used without a significant impact on position accuracy. So, even if the six motor 4B-SPM configuration is chosen for high dynamic performance, the architecture may still prove to be more affordable than a serial actuated shoulder exoskeleton with a comparable 3DoF.

To operate the shoulder exoskeleton, an admittance control scheme [33,34] that utilizes force feedback from the cuff could be used to determine user commands in terms of position. These commands could be sent to the motors using a Dynamixel CM-700 (Robotis, Seoul, South Korea) motion controller. Force feedback could be measured using an array of piezoresistive force sensors placed between the inner and outer cuff to capture both the magnitude and direction of the contact force vector. This vector could be converted to angles $φ$ and $θ$ in order to calculate the desired operator command rotations $Rz$ and $Ry$ described in Sec. 2.5.

Finally, a comment regarding the dynamic performance of this 4B-SPM shoulder exoskeleton. Even without a complete dynamic model for this particular actuator configuration, the output torque can still be estimated within a given range. It should be noted that a torque estimate is only possible for the six motor configuration shown in Fig. 3 and used for this case study. This is because each substructure of the parallel system is fully actuated and thus can operate independently as a 2DoF serial manipulator. The torque output of a 2DoF serial manipulator is well understood and is easily calculate based on the motor outputs. Therefore, estimates of static output torque for three 2DoF serial manipulators operating synchronously can be given. Each MX-106R servo has a rated stall torque of 10 Nm. Therefore, if each servo were fully engaged, ∼60 Nm of torque output would be expected for the static case. However, most shoulder plate orientations will not utilize all six servos fully at once. As shown Sec. 2.3, operation is possible with only three servos. Therefore, depending on the commanded shoulder plate motion, operating torques can be approximated to be between ∼30 and ∼60 Nm of output at the shoulder for the static case. If conversely, three motors were used, then the approximated torque output would be between ∼0 and ∼30 Nm of static torque output. The occurrence of ∼0 Nm torque output for certain shoulder configurations can be mitigated by carefully considering the placement of each actuator or by constraining the workspace.

## Discussion

The development of the 4B-SPM was motivated by the need for an exoskeleton robotic architecture that can operate effectively with complex ball and socket joints like the shoulder, hip, wrist, and ankle. Conventionally, these spherical joints have been augmented using serial actuation as it can offer a large workspace. However, this approach requires the use of a long serial moment arm that in turn results poor dynamic performance. Not only does the unique architecture of the 4B-SPM provide a short moment arm while still maintaining a large workspace, but its use of multiple mounting points and fully active DoF offers some of the dynamic advantages inherent to parallel actuation.

In this work, an analysis of the 4B-SPM forward and inverse kinematics was performed, along with multiple methods of actuation that can be implemented depending on dynamic performance requirements. These actuation types include a three revolute actuator, a six revolute actuator, and a three revolute two prismatic actuator version of the 4B-SPM. Also, discussed is a technique of cooperative control that allows the 4B-SPM a greater effective workspace by preventing mechanical interference between the device and operator from occurring.

The results of this paper detail four example exoskeleton types that could be built using the 4B-SPM. These include a shoulder, hip, wrist, and ankle device. Each was designed to match the approximate range of motion of the respective joints. Video simulations of each exoskeleton were included in this work. Also, included in all four simulations were the different methods of roll actuation to improve dynamic performance. Furthermore, for the shoulder simulation, the cooperative control technique discussed in Sec. 2.5 was demonstrated.

The four embodiments of the 4B-SPM presented in this work help to establish its versatility as a result of flexible placement of the actuator mounting points. This versatility means the 4B-SPM substructures could be positioned to accommodate not only the human anatomy but also aspects of a greater exoskeleton system. Therefore, certain design decisions such as placement of a power source, life support, and onboard sensors could be made before 4B-SPM substructure placement is considered.

In conclusion of this work, the 4B-SPM can be realized as a viable parallel actuated design that allows for the augmentation complex biologicals joints like the shoulder, hip, wrist, and ankle. With advantages such as a short moment arm, low inertia, and rigidity associated with parallel structures, the 4B-SPM can be considered a viable alternative to the traditional use of serial actuation to the same end. Along with other architectures such as the Stewart-Gough platform, which has been suggested to be an effective solution for augmenting torso movement [35], the 4B-SPM could assist in the development of a future series of parallel actuated exoskeletons that are more effective than their contemporary serial counterparts.

## References

Craig, J. J. , 2005, Introduction to Robotics: Mechanics and Control, Pearson/Prentice Hall, Upper Saddle River, NJ. [PubMed] [PubMed]
Bogue, R. , 2009, “Exoskeletons and Robotic Prosthetics: A Review of Recent Developments,” Ind. Rob. Int. J, 36(5), pp. 421–427.
Marcheschi, S. , Salsedo, F. , Fontana, M. , and Bergamasco, M. , 2011, “Body Extender: Whole Body Exoskeleton for Human Power Augmentation,” IEEE International Conference on Robotics and Automation, Shanghai, China, May 9–13, pp. 611–616.
Young, A. J. , and Ferris, D. P. , 2017, “State of the Art and Future Directions for Lower Limb Robotic Exoskeletons,” IEEE Trans. Neural Syst. Rehabil. Eng., 25(2), pp. 171–182. [PubMed]
Toxiri, S. , Ortiz, J. , and Caldwell, D. G. , 2018, “Assistive Strategies for a Back Support Exoskeleton: Experimental Evaluation,” Mechanisms and Machine Science, Springer, Cham, IL, pp. 805–812.
Hunt, K. H. , 1983, “Structural Kinematics of in-Parallel-Actuated Robot-Arms,” ASME J. Mech. Des, 105(4), pp. 705–712.
Stechert, C. , Franke, H. J. , and Wrege, C. , 2006, “Task-Based Modular Configurations for Hybrid and Redundant Parallel Robots,” IFAC Proceedings Volumes, 39(15), pp. 218–223.
Merlet, J. P. , 2006, “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots,” ASME J. Mech. Des, 128(1), pp. 199–206.
Gosselin, C. , 1990, “Stiffness Mapping for Parallel Manipulators,” IEEE Trans. Rob. Autom., 6(3), pp. 377–382.
Pashkevich, A. , Chablat, D. , and Wenger, P. , 2009, “Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44(5), pp. 966–982.
Gosselin, C. M. , and Lavoie, E. , 1993, “On the Kinematic Design of Spherical Three-Degree-of-Freedom Parallel Manipulators,” Int. J. Rob. Res, 12(4), pp. 394–402.
Gupta, A. , O'Malley, M. K. , Patoglu, V. , and Burgar, C. , 2008, “Design, Control and Performance of RiceWrist: A Force Feedback Wrist Exoskeleton for Rehabilitation and Training,” International Journal of Robotics Research, Sage Publications, London, pp. 233–251.
Erwin, A. , O'malley, M. , K., Ress , D. , and Sergi, F. , 2015, “Development, Control, and MRI-Compatibility of the MR-SoftWrist,” IEEE International Conference on Rehabilitation Robotics (ICORR), Singapore, Aug. 11–14, pp. 187–192.
Kim, H. S. , and Tsai, L.-W. , 2002, “Kinematic Synthesis of Spatial 3-RPS Parallel Manipulators,” ASME Paper No. DETC2002/MECH-34302.
Roy, A. , Krebs, H. I. , Williams, D. J. , Bever, C. T. , and W, L. , 2009, “Robot-Aided Neurorehabilitation: A Novel Robot for Ankle Rehabilitation,” IEEE Trans. Rob., 25(3), pp. 569–582.
Alici, G. , and Shirinzadeh, B. , 2004, “Topology Optimisation and Singularity Analysis of a 3-SPS Parallel Manipulator With a Passive Constraining Spherical Joint,” Mech. Mach. Theory, 39(2), pp. 215–235.
Pehlivan, A. U. , Sergi, F. , Erwin, A. , Yozbatiran, N. , Francisco, G. E. , and O'Malley, M. K. , 2014, “Design and Validation of the RiceWrist-S Exoskeleton for Robotic Rehabilitation After Incomplete Spinal Cord Injury,” Robotica, 32(8), pp. 1415–1431.
Lee, H. , Ho, P. , Rastgaar, M. A. , Krebs, H. I. , and Hogan, N. , 2011, “Multivariable Static Ankle Mechanical Impedance With Relaxed Muscles,” J. Biomech., 44(10), pp. 1901–1908. [PubMed]
Husty, M. L. , 1996, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380.
Kong, X. , and Gosselin, C. M. , 2004, “Type Synthesis of 3-DOF Translational Parallel Manipulators Based on Screw Theory,” ASME J. Mech. Des., 126(1), pp. 83–92.
Tsai, L.-W. , and Joshi, S. , 2000, “Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator,” ASME J. Mech. Des., 122(4), pp. 439–446.
Kong, X. , and Gosselin, C. M. , 2004, “Type Synthesis of 3-DOF Spherical Parallel Manipulators Based on Screw Theory,” ASME J. Mech. Des., 126(1), pp. 101–108.
Walter, D. R. , Husty, M. L. , and Pfurner, M. , 2009, “A Complete Kinematic Analysis of the SNU 3-UPU Parallel Robot,” Contemp. Math., 496, p. 331.
Wu, J. , Wang, J. , and You, Z. , 2011, “A Comparison Study on the Dynamics of Planar 3-DOF 4-RRR, 3-RRR and 2-RRR Parallel Manipulators,” Rob. Comput. Integr. Manuf., 27(1), pp. 150–156.
Hunt, J. , Lee, H. , and Artemiadis, P. , 2016, “A Novel Shoulder Exoskeleton Robot Using Parallel Actuation and a Passive Slip Interface,” ASME J. Mech. Rob., 9(1), p. 011002.
lee , 1988, “Dynamic Analysis of a Three-Degrees-of-Freedom in Parallel Actuated Manipulator,” IEEE J. Rob. Autom., 4(3), pp. 354–360.
Liu, X. J. , Jin, Z. L. , and Gao, F. , 2000, “Optimum Design of 3-DOF Spherical Parallel Manipulators With respect to the Conditioning and Stiffness Indices,” Mech. Mach. Theory, 35(9), pp. 1257–1267.
Hoffman, J. , 2006, Norms for Fitness, Performance, and Health, Human Kinetics, Champaign, IL.
Deb, K. , 2001, Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, New York, NY.
Schiele, A. , and Van Der Helm, F. C. T. , 2006, “Kinematic Design to Improve Ergonomics in Human Machine Interaction,” IEEE Trans. Neural Syst. Rehabil. Eng., 14(4), pp. 456–469. [PubMed]
Cempini, M. , De Rossi, S. M. M. , Lenzi, T. , Vitiello, N. , and Carrozza, M. C. , 2013, “Self-Alignment Mechanisms for Assistive Wearable Robots: A Kinetostatic Compatibility Method,” IEEE Trans. Rob., 29(1), pp. 236–250.
Stienen, A. H. A. , Hekman, E. E. G. , van der Helm, F. C. T. , and van der Kooij, H. , 2009, “Self-Aligning Exoskeleton Axes Through Decoupling of Joint Rotations and Translations,” IEEE Trans. Rob., 25(3), pp. 628–633.
Glosser, G. D. , and Newman, W. S. , 1994, “The Implementation of a Natural Admittance Controller on an Industrial Manipulator,” IEEE International Conference Robot Automation, San Diego, CA, May 8–13, pp. 1209–1215.
Hogan, N. , 1988, “On the Stability of Manipulators Performing Contact Tasks,” IEEE J. Rob. Autom., 4(6), pp. 677–686.
Park, J.-H. , Stegall, P. , and Agrawal, S. K. , 2015, “Dynamic Brace for Correction of Abnormal Postures of the Human Spine,” IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, May 26–30, pp. 5922–5927.
View article in PDF format.

## References

Craig, J. J. , 2005, Introduction to Robotics: Mechanics and Control, Pearson/Prentice Hall, Upper Saddle River, NJ. [PubMed] [PubMed]
Bogue, R. , 2009, “Exoskeletons and Robotic Prosthetics: A Review of Recent Developments,” Ind. Rob. Int. J, 36(5), pp. 421–427.
Marcheschi, S. , Salsedo, F. , Fontana, M. , and Bergamasco, M. , 2011, “Body Extender: Whole Body Exoskeleton for Human Power Augmentation,” IEEE International Conference on Robotics and Automation, Shanghai, China, May 9–13, pp. 611–616.
Young, A. J. , and Ferris, D. P. , 2017, “State of the Art and Future Directions for Lower Limb Robotic Exoskeletons,” IEEE Trans. Neural Syst. Rehabil. Eng., 25(2), pp. 171–182. [PubMed]
Toxiri, S. , Ortiz, J. , and Caldwell, D. G. , 2018, “Assistive Strategies for a Back Support Exoskeleton: Experimental Evaluation,” Mechanisms and Machine Science, Springer, Cham, IL, pp. 805–812.
Hunt, K. H. , 1983, “Structural Kinematics of in-Parallel-Actuated Robot-Arms,” ASME J. Mech. Des, 105(4), pp. 705–712.
Stechert, C. , Franke, H. J. , and Wrege, C. , 2006, “Task-Based Modular Configurations for Hybrid and Redundant Parallel Robots,” IFAC Proceedings Volumes, 39(15), pp. 218–223.
Merlet, J. P. , 2006, “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots,” ASME J. Mech. Des, 128(1), pp. 199–206.
Gosselin, C. , 1990, “Stiffness Mapping for Parallel Manipulators,” IEEE Trans. Rob. Autom., 6(3), pp. 377–382.
Pashkevich, A. , Chablat, D. , and Wenger, P. , 2009, “Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44(5), pp. 966–982.
Gosselin, C. M. , and Lavoie, E. , 1993, “On the Kinematic Design of Spherical Three-Degree-of-Freedom Parallel Manipulators,” Int. J. Rob. Res, 12(4), pp. 394–402.
Gupta, A. , O'Malley, M. K. , Patoglu, V. , and Burgar, C. , 2008, “Design, Control and Performance of RiceWrist: A Force Feedback Wrist Exoskeleton for Rehabilitation and Training,” International Journal of Robotics Research, Sage Publications, London, pp. 233–251.
Erwin, A. , O'malley, M. , K., Ress , D. , and Sergi, F. , 2015, “Development, Control, and MRI-Compatibility of the MR-SoftWrist,” IEEE International Conference on Rehabilitation Robotics (ICORR), Singapore, Aug. 11–14, pp. 187–192.
Kim, H. S. , and Tsai, L.-W. , 2002, “Kinematic Synthesis of Spatial 3-RPS Parallel Manipulators,” ASME Paper No. DETC2002/MECH-34302.
Roy, A. , Krebs, H. I. , Williams, D. J. , Bever, C. T. , and W, L. , 2009, “Robot-Aided Neurorehabilitation: A Novel Robot for Ankle Rehabilitation,” IEEE Trans. Rob., 25(3), pp. 569–582.
Alici, G. , and Shirinzadeh, B. , 2004, “Topology Optimisation and Singularity Analysis of a 3-SPS Parallel Manipulator With a Passive Constraining Spherical Joint,” Mech. Mach. Theory, 39(2), pp. 215–235.
Pehlivan, A. U. , Sergi, F. , Erwin, A. , Yozbatiran, N. , Francisco, G. E. , and O'Malley, M. K. , 2014, “Design and Validation of the RiceWrist-S Exoskeleton for Robotic Rehabilitation After Incomplete Spinal Cord Injury,” Robotica, 32(8), pp. 1415–1431.
Lee, H. , Ho, P. , Rastgaar, M. A. , Krebs, H. I. , and Hogan, N. , 2011, “Multivariable Static Ankle Mechanical Impedance With Relaxed Muscles,” J. Biomech., 44(10), pp. 1901–1908. [PubMed]
Husty, M. L. , 1996, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380.
Kong, X. , and Gosselin, C. M. , 2004, “Type Synthesis of 3-DOF Translational Parallel Manipulators Based on Screw Theory,” ASME J. Mech. Des., 126(1), pp. 83–92.
Tsai, L.-W. , and Joshi, S. , 2000, “Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator,” ASME J. Mech. Des., 122(4), pp. 439–446.
Kong, X. , and Gosselin, C. M. , 2004, “Type Synthesis of 3-DOF Spherical Parallel Manipulators Based on Screw Theory,” ASME J. Mech. Des., 126(1), pp. 101–108.
Walter, D. R. , Husty, M. L. , and Pfurner, M. , 2009, “A Complete Kinematic Analysis of the SNU 3-UPU Parallel Robot,” Contemp. Math., 496, p. 331.
Wu, J. , Wang, J. , and You, Z. , 2011, “A Comparison Study on the Dynamics of Planar 3-DOF 4-RRR, 3-RRR and 2-RRR Parallel Manipulators,” Rob. Comput. Integr. Manuf., 27(1), pp. 150–156.
Hunt, J. , Lee, H. , and Artemiadis, P. , 2016, “A Novel Shoulder Exoskeleton Robot Using Parallel Actuation and a Passive Slip Interface,” ASME J. Mech. Rob., 9(1), p. 011002.
lee , 1988, “Dynamic Analysis of a Three-Degrees-of-Freedom in Parallel Actuated Manipulator,” IEEE J. Rob. Autom., 4(3), pp. 354–360.
Liu, X. J. , Jin, Z. L. , and Gao, F. , 2000, “Optimum Design of 3-DOF Spherical Parallel Manipulators With respect to the Conditioning and Stiffness Indices,” Mech. Mach. Theory, 35(9), pp. 1257–1267.
Hoffman, J. , 2006, Norms for Fitness, Performance, and Health, Human Kinetics, Champaign, IL.
Deb, K. , 2001, Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, New York, NY.
Schiele, A. , and Van Der Helm, F. C. T. , 2006, “Kinematic Design to Improve Ergonomics in Human Machine Interaction,” IEEE Trans. Neural Syst. Rehabil. Eng., 14(4), pp. 456–469. [PubMed]
Cempini, M. , De Rossi, S. M. M. , Lenzi, T. , Vitiello, N. , and Carrozza, M. C. , 2013, “Self-Alignment Mechanisms for Assistive Wearable Robots: A Kinetostatic Compatibility Method,” IEEE Trans. Rob., 29(1), pp. 236–250.
Stienen, A. H. A. , Hekman, E. E. G. , van der Helm, F. C. T. , and van der Kooij, H. , 2009, “Self-Aligning Exoskeleton Axes Through Decoupling of Joint Rotations and Translations,” IEEE Trans. Rob., 25(3), pp. 628–633.
Glosser, G. D. , and Newman, W. S. , 1994, “The Implementation of a Natural Admittance Controller on an Industrial Manipulator,” IEEE International Conference Robot Automation, San Diego, CA, May 8–13, pp. 1209–1215.
Hogan, N. , 1988, “On the Stability of Manipulators Performing Contact Tasks,” IEEE J. Rob. Autom., 4(6), pp. 677–686.
Park, J.-H. , Stegall, P. , and Agrawal, S. K. , 2015, “Dynamic Brace for Correction of Abnormal Postures of the Human Spine,” IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, May 26–30, pp. 5922–5927.

## Figures

Fig. 1

Architecture of the 4B-SPM. The design consists of three substructures, each comprised of a 4 bar linkage connected to a grounded revolute joint. The top linkage in each 4 bar mechanism is extended to reach a mobile platform. Each top linkage is coupled to the mobile platform using a spherical joint. The mobile platform is capable of spherical motion about the point in which the axes of all three grounded revolute joints intersect.

Fig. 2

Vector representation of the 4B-SPM. All vectors originate at the origin, but are drawn in a configuration similar to Fig. 1 to help the reader identify what each vector represents. The global origin frame is denoted by the RGB (xyz) frame shown at the center of the spherical workspace.

Fig. 3

Method of substructure roll actuation by the addition of three motors. Each of the three motor is fixed to the ground and coupled to the roll axis of each respective 4 bar mechanism.

Fig. 4

Method of substructure roll actuation by the addition of two linear actuated sliders. The sliders are positioned on the outermost substructure with each coupled to the central substructure using interconnecting linkages mounted with spherical joints on either end.

Fig. 5

Vector representation of the 4B-SPM with the inclusion of two linear actuated sliders and two interconnecting linkages

Fig. 6

Desired shoulder plate orientations corresponding to the three boundary arm configurations that define the workspace. These include: (a) arm at rest (φ=0, θ=π/2, and ψ=0 or φ=π/2, θ=π/2, and ψ=π/2), (b) 90 deg flexion (φ=π/2, θ=0, and ψ=π/2), and (c) 90 deg abduction (φ=0, θ=0, and ψ=π/2). For each arm configuration, the corresponding rotation matrix of the shoulder plate with respect to the global frame is shown.

Fig. 7

Four-bar spherical parallel manipulator ankle exoskeleton: (a) shows the 4B-SPM architecture used for three or six revolute motors as described in Secs. 2.1 and 2.4.1, respectively and (b) shows the 4B-SPM architecture used for three revolute motors and two linear actuators as described in Sec. 2.4.2. The interconnecting linkages that slide up and down the outermost substructures via linear actuation are shown in blue to help distinguish them from the rest of the 4B-SPM.

Fig. 8

Four-bar spherical parallel manipulator wrist exoskeleton: (a) shows the 4B-SPM architecture used for three or six revolute motors as described in Secs. 2.1 and 2.4.1, respectively and (b) shows the 4B-SPM architecture used for three revolute motors and two linear actuators as described in Sec.2.4.2. The interconnecting linkages that slide up and down the outermost substructures via linear actuation are shown in blue to help distinguish them from the rest of the 4B-SPM.

Fig. 9

Four-bar spherical parallel manipulator hip exoskeleton: (a) shows the 4B-SPM architecture used for three or six revolute motors as described in Secs. 2.1 and 2.4.1, respectively and (b) shows the 4B-SPM architecture used for three revolute motors and two linear actuators as described in Sec. 2.4.2. The interconnecting linkages that slide up and down the outermost substructures via linear actuation are shown in blue to help distinguish them from the rest of the 4B-SPM.

Fig. 10

Four-bar spherical parallel manipulator shoulder exoskeleton: (a) shows the 4B-SPM architecture used for three or six revolute motors as described in Secs. 2.1 and 2.4.1, respectively and (b) shows the 4B-SPM architecture used for three revolute motors and two linear actuators as described in Sec. 2.4.2. The interconnecting linkages that slide up and down the outermost substructures via linear actuation are shown in blue to help distinguish them from the rest of the 4B-SPM.

Fig. 11

Four-bar spherical parallel manipulator shoulder exoskeleton with a six motor configuration. Shown are back, side, and front views of the device.

Fig. 12

Four-bar spherical parallel manipulator shoulder exoskeleton with a six motor configuration as described in Sec. 2.4.1. With this configuration, each of the three substructures consists of two motors: one for the pitch of the four bar mechanism and one for the roll. Noteworthy components are as follows: (a) revolute motor, (b) four bar mechanism, (c) shoulder plate, (d) arm cuff with one translational and one rotational passive DoF, and (f) Static Stewart-Gough platform for rigid positioning of adjacent actuated substructures.

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