This paper describes a dimensional synthesis method used in the design of a passive finger exoskeleton that takes into account the user limb anthropometric dimensions and contact requirements for grasping objects. The paper is the first step in our current efforts on the design of wearable devices that use a common slider at the hand to passively drive each exofinger. The finger exoskeleton is comprised of a 3R serial limb and is constrained to multiloop eight-bar slider mechanism using two RR constraints. To design the exolimb, the pose of the limb was captured using an optical motion capture and its dimensions were determined using a constrained least square optimization, which takes into account human skin movement. To illustrate the generality of our approach, an example of the design of an index and middle finger exolimb is described.

## Introduction

Human hands are highly dexterous and are capable of grasping a wide variety of objects of different shapes, textures, and orientations based on heuristics [1]. As shown in Fig. 1, the hand consists of five articulated serial chains that comprised of 19 links and 19 joints with 24 degrees-of-freedom (DoF). As viewed from the distal end for each finger except for the thumb, the kinematic chain comprises of a 1DoF joint at the distal interphalangeal (DIP), 1DoF joint at the proximal interphalangeal (PIP), and a 2DoF joint at the metacarpophalangeal (MCP), forming a TRR serial chain with its base at the respective MCP joint. If we neglect the abduction/adduction movement of each finger at the MCP joints, the kinematic model simplifies to that of a planar 3R chain. The thumb also has a similar kinematic structure from the distal end, which comprises a 1DoF joint at the interphalangeal (IP), 1DoF joint at the MCP, and a 2DoF joint at the carpometacarpal (CMC). Similarly, if we neglect abduction/adduction movement at the CMC joint, the thumb can also be simplified as a planar 3R chain. This interesting kinematic structure can be used to design exoskeleton devices that parallel the limb. Together with the advent of additive manufacturing, this biological structure provides opportunities for the design of a new breed of bioinspired wearable underactuated devices for a variety of purposes such as assistive, rehabilitative, or augmentative grasping.

Fig. 1
Fig. 1
Close modal

Over the past several decades, wearable exoskeletons had become a common part of our daily lives [26]. Many improvements have been made over the last decade, resulting in enhanced novel materials, as well as methods for alignment and fitting to the human wearer, as well as having biosensing and biofeedback capability. The most important design requirement for wearable exoskeleton devices is safety and is usually achieved through some form of mechanical range stopper or through the design itself [7]. The key approach is to design the wearable device with its rotation axis to coincide with that of the human joint so as to mimic its workspace. This way, even though there is a failure on the device controller, the exoskeleton will not force the user to move in an unnatural manner, resulting in damage to his limbs. There are a few ways in which one could achieve this. The most common manner is to match the joint centers directly [8]. However, this approach requires structural space on the side of the limbs. Alternatively, a remote center of rotation can be considered [9,10]. Alignment and adjustment for human limb size plays a significant challenge for the design of these devices as well [11].

Besides safety, the overall weight of the wearable device also plays an important role in the design. Hence, there has been a desire to minimize the number of actuators, due to the constraints on size and mass as well as the greater reliability and lower cost of simplified mechanical components. This has been accomplished by coupling the motion of multiple joints, with many designs having fewer actuators than degrees-of-freedom. Such devices termed “underactuated” have shown significant benefits in grasping applications, due to the passive adaptability between the degrees-of-freedom. Under certain conditions, the unconstrained freedoms allow these devices the ability to conform to the environment shape, without any need for sensing. Two general types of under-actuated wearable exoskeleton have been proposed in previous works. The first group is based on tendon-driven systems [12,13], while the second group is based on mechanical linkages [14]. Tendon-driven exoskeletons can be usually designed with compact size and dexterous operation. However, these systems lead to friction and elasticity issues during operation [1517] and are generally limited to small grasping forces. Linkage mechanisms on the other hand are preferable for applications in which high stability and large grasping forces are required [1820].

## The Geometric Design Process

Our motion generation for wearable hand device can be described by the flow chart shown in Fig. 2. It consists of anthropometric backbone chain specification, physiological task specification, dimensional synthesis, linkage topology selection, and design candidate evaluation. This process is applied to each of the 3R limb (in black) as shown in Fig. 1 to yield a wearable hand device.

Fig. 2
Fig. 2
Close modal

To obtain the limb anthropomorphic dimensions, instead of taking the measurements directly, we formulate a synthesis procedure that determines a planar 3R chain which best approximates its trajectory. This is achieved by using the data, captured from an optical tracker with markers attached to the segments of the various moving limbs as shown in Fig. 3. The synthesis of planar 3R chains, also known as triads, for motion generation has been first studied by Chase et al. [25] and Subbian and Flugrad [26]. They used triad synthesis to design planar 3R chains with up to seven task positions. Perez and McCarthy [27] used Clifford algebra for the synthesis of planar 3R chains, and their approach is generalizable for the design of spatial serial chains. In contrast to the previous work, which deals with exact synthesis, we built upon the work by Chang and Pollard [28,29] and consider the over-constrained synthesis problem of planar 3R chains. This approach, unlike the previously mentioned triad synthesis approaches, has the benefit of not requiring the limb to maintain rigid body motion relative to each other. This is important as the human skin tissue flexes during motion, which affects the accuracy of the result.

Fig. 3
Fig. 3
Close modal

To acquire the required task for synthesis, we identify the desired human motions, track them using optical tracker system, and then mathematically describe them as physiological task. The idea of physiological task specification is inspired by Howard and Kumar [30], who show that the stability of a grasp depends not only on the magnitude and direction of the contact forces but also on the local curvature of the contacting bodies. Another inspiration for defining the physiological contact task using higher order derivatives came from the works of Rimon and Burdick [31,32] who analyze the mobility of bodies in contact to show that first-order theories based on positions and velocities are not sufficient to define the local behavior of a linkage mechanism. Previous work of the authors [33,34] show that the acceleration properties of movement are related to the curvature of the contacting bodies and can be used to effectively constrain a rigid body for grasping applications. In order to define the movement direction and contact task constraints for each finger, we specify velocity and acceleration task requirements in the vicinity of two specified locations. The velocity vector specified in the start position gives the direction of the desired tangent motion, while the velocity and acceleration specified in the second position are related to the finger-object contact and curvature specifications. These are specified in addition to the pose requirements as a set of physiological task for synthesis.

The dimensional synthesis of an eight-bar slider linkage begins with the specification of physiological task and the 3R-PRR parallel chain that performs the desired movement. Two RR chains are then added to yield an eight-bar slider linkage. This is an exact synthesis procedure, where we seek to determine all eight-bar slider linkage solutions. This can yield as many as 69 design candidates. To select feasible linkage topologies that perform the desired physiological task, kinematic analysis is performed to study the overall movement of the linkage including potential branches or singularity configurations. If the linkage moves smoothly through the specified task, the candidate will be evaluated further using an assessment criterion taking into account its wearability.

## Determination of Joint Rotational Axes and Centers of Rotation

To reconstruct the planar 3R serial chain, termed the “backbone” chain, we use the marker locations, as measured by a motion capture system, to identify the joint axis of rotation at each of the limb joints. This is done by minimizing the planar measurement error, described later, and then locating a point on the joint axis using sphere fitting techniques.

### Determination of Joint Rotation Axes.

For an ideal hinge joint, each of the markers traces a circular path on a plane orthogonal to the axis of rotation n as shown by the PIP joint in Fig. 4. Hence, in a multiple markers setup, each of this markers will be constrained to circles of different radius that lie on parallel planes normal to the axis n. However, due to measurement errors, tissue movements, and the approximated anatomic model of the joints, slight deviation of the markers from their respective circles will occur. Let this deviation be denoted by δ. Now, the goal is to find a common rotation axis n that minimizes the δ error for all markers.

Fig. 4
Fig. 4
Close modal
Denote the pth marker location as measured by an optical motion tracking system during the k instance as $vkp$. The deviation $δkp$ of this marker from a circle that lies within a common normal plane n can be calculated as the difference between the projected component of $vkp$ onto n and the plane distance from the origin of the reference frame
$δkp=vkp·n−μp$
(1)
Hence for P markers, the total deviation fplane can be calculated as the squared planar error over all marker trajectories
$fplane=∑p=1P∑k=1N(δkp)2$
(2)
The optimal value of μp that minimizes fplane can be obtained by taking the derivative of Eq. (2) and substituting Eq. (1) to yield
$μp=(1N∑k=1Nvkp)·n=v¯p·n$
(3)
where $v¯p$ denotes the centroid displacement of the pth marker over all N measured displacement $vkp$. This means that the optimal value of μp for the pth marker is the projected component of its centroid vector with the rotation axis n. Now the cost function in Eq. (2) can be rewritten as
$fplane=∑p=1P∑k=1N(ukp·n)2$
(4)

where $ukp=vkp−v¯p$ and solved using the least square fitting of algebraic surfaces.

Let $[Up]$ denotes the acquired optical data matrix with its rows corresponding to the coordinates of the pth marker's location relative to its centroid displacement
$[Up]=[(u1p)T⋮(uNp)T]$
(5)
Given P markers, Eq. (4) can be written in terms of the acquired optical data as an optimization problem to estimate the common rotation axis n
$minimizenfplane=nT[D]T[D]n$
where $[D]=[[U1]⋮[UP]]$. The optimal solution can be determined easily by finding the singular value decomposition of $[D]T[D]$ such that
$[D]T[D]=[S][Σ][V]T$
(6)

The optimal n is the right singular vector that corresponds to the smallest singular value.

### Determination of Limb Joint Centers of Rotation.

For an ideal spherical joint, each of the markers traces a path along a spherical surface of radius r with center m as shown at the DIP joint in Fig. 4. This means, that for multiple markers, each of them will be constrained to lie on spherical surface with common center m and radius rp. However, as mentioned earlier, due to measurement, skin movement, and modeling errors, slight radial variation $εkp$ will exist. This can be calculated by summing the difference of the radial squared lengths over all marker trajectories
$fsphere=∑p=1P∑k=1Nεkp=∑p=1P∑k=1N(||vkp−m||2−(rp)2)$
(7)
By letting $m=(xc,yc,zc)$ and $vkp=(xkp,ykp,zkp)$, Eq. (7) can be rewritten in terms of its variables $u=(a,b,c,d,e1,…,eP)$ as follows:
$fsphere=∑p=1P∑k=1Nawkp+bxkp+cykp+dzkp+ep$
(8)
where $wkp=(xkp)2+(ykp)2+(zkp)2$, a is a scaling parameter and the rest of the variables are
$b=−2xcc=−2ycd=−2zcep=xc2+yc2+zc2−rp2$
(9)
Given that P markers are used, Eq. (8) can be written in matrix form such that
$fsphere=[[D1]1⋮⋱[Dp]1⋮⋱[DP]1]u=[D]u$
(10)
where
$[Dp]=[w1px1py1pz1p⋮⋮⋮⋮wkpxkpykpzkp⋮⋮⋮⋮wNpxNpyNpzNp] and 1=[1⋮1⋮1]$
(11)
A suitable normalization constraint was introduced by Pratt [35] for the robust estimation of the center of rotation
$b2+c2+d2−4aep=1$
(12)
This can be extended for multiple markers, by requiring
$∑p=1P(b2+c2+d2−4aep)=1$
(13)
Now, the above can be formulated as a constrained optimization problem where the algebraic squared distances are minimized to estimate the sphere center
$minimizeuf=([D]u)T([D]u)=uT[D]T[D]usubject touT[C]u=1$
where
$[C]=[0−2…−2PPP−20⋮⋱−20]$
(14)
An equivalent unconstrained optimization problem can be formulated using the Lagrangian that minimizes
$L=uT([D]T[D])u−λ(uT[C]u−1)$
(15)
where λ is the Lagrangian multiplier. The solution to this problem can be solved by differentiating Eq. (15) with respect to u. This yields the generalized eigenvalue problem
$([D]T[D])u=λ[C]u$
(16)
The components of the rotation center m and the associated radius rp can be calculated from the eigenvectors of Eq. (16) using
$m=(xc,yc,zc)T=−12a(b,c,d)T$
(17)
and
$(rp)2=||m||2−epa$
(18)

### Reconstructing the Limb Dimensions.

To determine the limb dimensions of the planar 3R chain from the attached markers, we first find the relative displacements between consecutive limbs for each data set by using
$[Bi,i+1]k=[Bi]k−1[Bi+1]k, i=0,1,2$
(19)
We solve for the rotation center $mi$ using Eq. (17) and transform to a global reference frame F using
$Mki=[Bi]kmi, i=0,1,2$
(20)
Finally, to determine the limb dimensions, we solve for the shortest distance between two successive revolute joints and take its mean
$ai,i+1=1N∑k=1N|MkiMki+1×n||n|, i=1,2$
(21)

From the performed grasping task, recorded by motion capture system, two critical poses along the measured markers trajectory, where the local motion is very important for the finger performance, are selected. Typically, for exact synthesis, a velocity at the first pose was defined to keep the finger moving as tangent as possible to the acquired task trajectory. Acceleration at the second pose is defined at the point of contact between the fingertip and the object as shown in Fig. 5, where grasping occurs. This built upon the results of Robson and McCarthy [33], who showed that higher order derivatives, such as velocities and accelerations, defined in the synthesis task (M) relate to the finger-object contact and curvature specifications.

Fig. 5
Fig. 5
Close modal

Thus, the physiological task consists of two poses P, two velocities V, and one acceleration A specifications, compatible with the fingertip–object local contact and curvature constraints. These task specifications are parametrized in terms of the frame orientation θ, angular velocity $θ˙$, angular acceleration $θ¨$, as well as the frame origin dx and dy, velocity $d˙x$ and $d˙y$, and acceleration $d¨x$ and $d¨y$.

## Motion Generation of Eight Bar Sliders Based on the Physiological Task

The motion generation of eight bar slider mechanisms seeks to constrain a parallel 3R-PRR chain, in which the 3R chain laterally parallels the finger as shown in Fig. 6. This can be achieved by attaching two RR chains to yield an eight bar slider linkage. The process consists of three steps. First, we specify the various link dimensions for the parallel $C1−C5$ chain such that the resulting parallel robot moves through the specified physiological task. Note that the dimensions a12, a23, and l3 are computed based on the attached markers and they approximate the human limb anthropometric dimensions. Next, we solve for the robot joint parameters at each of the specified poses to determine the various links positions, velocity, and acceleration at each of the specified PVA poses. These matrices then form the task for the motion generation of two RR chains $G1W1$ and $G2W2$ to yield an eight-bar slider-based exoskeleton. Figure 7 shows the four ways of attaching two RR links if not constraining a link to the ground and to the newly formed RR link is preferred. Figure 8 shows the four ways of attaching two RR links if not constraining a link to the ground or to the new link formed by the first RR chain is preferred. For more details, refer to Soh and Ying [36].

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

### The Kinematics of a Planar 3R Chain Incorporating Physiological Motion Task.

The kinematic equations of a planar 3R chain equate the 3 × 3 homogeneous transformation $[T]$ between the end-effector and the base frame to the sequence of local coordinate transformations around the joint axes and along the links of the serial chain [37]
$[T]=[G][Z(θ1)][X(a12)][Z(θ2)][X(a23)][Z(θ3)][H]=[D]$
(22)

The parameters θi define the movement of each joint and $ai,j$ is the length of each link. The transformation $[G]$ defines the position of the base C1 of the chain relative to the fixed frame, and $[H]$ locates the task frame M relative to the end-effector frame (see Fig. 6).

Since the specified physiological task consists of higher order motion constraints, velocity and acceleration kinematic equations are obtained. The kinematics equation for the velocity task $[Ω]$ equates the joint angular and linear velocities of the 3R chain to the velocity of the end-effector in the vicinity of the task pose
$[Ω]=[D˙][D]−1=[ωv00]$
(23)

where $ω=[0−(θ˙1+θ˙2+θ˙3)θ˙1+θ˙2+θ˙30]$ is the angular velocity matrix and $v=−ωd+d˙$ is the linear velocity vector of the end-effector.

Similarly, the kinematics equation for the acceleration task $[Λ]$ equates the joint angular and linear acceleration of the 3R chain to the acceleration of the end-effector in the vicinity of the task pose
$[Λ]=[D¨][D]−1=[αa00]$
(24)

where $α=[−(θ˙1+θ˙2+θ˙3)2−(θ¨1+θ¨2+θ¨3)θ¨1+θ¨2+θ¨3−(θ˙1+θ˙2+θ˙3)2]$ is the angular acceleration matrix and $a=−ω2d−ω˙d+d¨$ is the linear acceleration vector.

### The Dimensional Synthesis Equations for the Two RR Chains.

The synthesis of an RR link to reach a start location, defined by pose and velocity specification $P1V1$ and an end location $P2V2A2$, defined by pose, velocity, and acceleration specifications, can be found by solving the following set of design equations [33]
$(W2−G2)·(W2−G2)−(W1−G1)·(W1−G1)=0([Ωk,1]W1−[Ωl,1]G1)·(W1−G1)=0([Ωk,2]W2−[Ωl,2]G2)·(W2−G2)=0([Λk,2]W2−[Λl,2]G2)·(W2−G2)+([Ωk,2]W2−[Ωl,2]G2)([Ωk,2]W2−[Ωl,2]G2)=0$
(25)
where the fixed $G2$ and moving $W2$ pivots of the kth and lth moving link at the second position are related to the start location by
$G2=[Bk,2][Bk,1]−1G1 and W2=[Bl,2][Bl,1]−1W1$
(26)
The matrices $[Ωk,j]$ denote the velocity of the kth moving link, and $[Ωl,j]$ and $[Λl,j]$ denote the velocity and acceleration of the lth moving link at the jth task location, where
$[B1j]=[G][Z(θ1)][X(a12)][B2j]=[B1j][Z(θ2)][X(a23)][B3j]=[B2j][Z(θ3)][H][Ω1j]=[B1j.][B1j]−1[Ω2j]=[B2j.][B2j]−1[Ω3j]=[B3j.][B3j]−1[Λ1j]=[B1j..][B1j]−1[Λ2j]=[B2j..][B2j]−1[Λ3j]=[B3j..][B3j]−1, j=1,2$
(27)

## Incorporating Linkage Assessment Criteria for Hand Exolimb at the Conceptual Design Level

Based on the previously mentioned approach, the motion generation of eight bar sliders can yield a variety of design candidates that fulfill the grasping task requirements. For practical reasons, it is important to choose linkage structures that would interface well with the human hand and not impede with the intended task. To achieve this, we identify $1−3$ regions around the finger to sort the two synthesized RR links $(G1W1$ and $G2W2)$ according to their undesirable or desirable traits as shown in Fig. 9.

Fig. 9
Fig. 9
Close modal
Region 1 denotes area beside or underneath the finger which are undesirable locations for the fixed and moving pivots $GiWi$. Having pivots beside the finger causes a wider linkage envelope and increases the chance of collision between adjacent fingers. Similarly, if the pivots are located underneath the finger, they are likely to collide with the finger itself or the object during flexion. Region 2 denotes designs that have the pivots extending beyond the tip of the finger. This is undesirable as the linkage would most likely contact with the object first instead of the fingertip during the grasping of larger objects. After eliminating designs with pivots in regions 1 and 2, we rank the remaining linkage candidates, with pivots located in region 3, according to their proximity to the finger. This can be formulated as the sum of the perpendicular distance of $Gi$ and $Wi$ to the nearest limb segment $C1C2$ or $C2C3$. To determine the perpendicular distance di of any pivot $Pi$, we use the following condition:
$di={|PiC2×C3C2||C3C2| if0≤∠PiC2C3≤12∠PIP|PiC1×C2C1||C2C1| if12∠PIP≤∠PiC2C3≤∠PIP$

## Design of a One Degree-of-Freedom Passive Index and Middle Finger Exoskeleton

To illustrate the benefit of this design process, we obtain experimental data from an optical motion capture of a subject's left hand performing a full range of precise thumb-two finger grasping [38] of a pen task. Specifically, the subject was sitting at a table with their left arm resting on the table in a relaxed position with a pen placed within their reach, and when prompted grasps it precisely using the thumb, index, and middle fingers. The collected motion capture data allows to further obtain the various finger anthropomorphic dimensions and determine the required physiological task for motion generation. A small object with high curvature was specifically chosen for the task to later test the extent of the proposed synthesis method to generalize to grasping of objects with different size and geometry.

The obtained left index finger full range of motion trajectory can be seen in Fig. 10. Based on the trajectory, following the steps described in Sec. 5, a planar 3R chain was synthesized, which best fits the pose trajectory of each consecutive limb section. The various section dimensions are found to be $a12=42.07$ mm and $a23=18.02$ mm, and the joint rotation axis $[B0]$ was found to be $n=(0.0424942,−0.950104,0.309026)T$. For the PRR chain, the fixed and moving pivots were chosen to lie within region 3 of the finger with $a45=77.42$ mm and slide angle $β=23deg$. For its kinematics details, see Ref. [36].

Fig. 10
Fig. 10
Close modal

To define the physiological task for the motion generation, described in Sec. 4, we perform a coordinate transformation of the captured pose, such that the resulting base frame is located at the MCP joint with the z-axis parallel to n. Next, we chose two critical poses on that trajectory, that is a start location, with the finger fully extended and a fingertip–object contact end location, at the instance of grasp, where the local behavior is important. A velocity constraint in the starting pose was defined to control the tangent motion of the finger upon flexion. The second pose, located at the instance of the contact between the fingertip and the pen, required the specification of both velocity and acceleration compatible with finger-object contact and curvature constraints. The velocity and acceleration constraints were computed from the optical motion capture data, which was sampled at 100 Hz. The computed task for the index finger at each of the two selected poses is shown in Table 1. Each of the two PVA task poses were assembled into a 3 × 3 homogenous transformation matrix using

Table 1

The PVA task data based on the movement of the subject's left index finger. The task consists of two positions, at the fully extended and grasp state, with velocities specified in each pose and acceleration in the second pose. The unit for d is mm, $d˙$ is mm/s, and $d¨$ is mm/s2.

Pose ($θdeg,dx,dy$)$(−145.60,48.75,46.57)$$(111.86,54.72,−24.50)$
Velocity ($θ˙deg/s,d˙x,d˙y$)$(−583.14,258.96,−392.63)$$(−370.87,−121.42,−199.12)$
Acceleration ($θ¨deg/s2,d¨x,d¨y$)$(11956.9,3515.97,7444.27)$
Pose ($θdeg,dx,dy$)$(−145.60,48.75,46.57)$$(111.86,54.72,−24.50)$
Velocity ($θ˙deg/s,d˙x,d˙y$)$(−583.14,258.96,−392.63)$$(−370.87,−121.42,−199.12)$
Acceleration ($θ¨deg/s2,d¨x,d¨y$)$(11956.9,3515.97,7444.27)$
$[T]=[ cos θ−sin θdx sin θ cos θdy001]$
(28)
$[Ω]=[0−θ˙d˙x+θ˙dyθ˙0d˙y−θ˙dx000]$
(29)
$[Λ]=[−θ˙2−θ¨d¨x+θ¨dy+θ˙2dxθ¨−θ˙2d¨y−θ¨dx+θ˙2dy000]$
(30)

As a next step, following Sec. 5, for each of the topologies shown in Figs. 7 and 8, we compute an RR chain $G1W1$ and $G2W2$ using Eq. (25). This yields 18 out of a 69 possible candidates shown in Table 2. As a last step, the proposed linkage assessment criteria, described in Sec. 6, was used to sort the design solutions. The synthesis procedure was repeated for the subject's left middle finger. Note that for the middle finger the final design was slightly different, due to ergonomic reasons related to integrating both fingers within a hand exoskeleton. More specifically, it was not desirable to have a link that directly parallels the MCP joint within the hand exoskeleton design. For this case, a scissors joint was used to create a remote revolute joint. Figure 11 shows the image sequence for the most preferred index and middle finger exolimb designs. Figure 12 shows the computer-aided design (CAD) drawing of the resulting one degree-of-freedom eight bar slider index and middle finger exoskeleton. Figure 13 shows the three-dimensional-printed exohand prototype and the image sequence of the two fingers performing the desired grasping physiological task.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Table 2

Solution for $G1W1$ and $G2W2$ for the various design candidates

Design (type)$G1$$W1$$G2$$W2$
1 (B15B25)(52.73, 72.15)(57.66, 80.47)(6.89, 31.14)(12.96, 33.55)
2 (B15B25)(52.73, 72.15)(57.66, 80.47)(11.51, 43.9)(–7.21, 46.30)
3 (B15B25)(52.73, 72.15)(57.66, 80.47)(19.07, 49.15)(15.45, 51.31)
4 (B15B25)(−2.59, 43.30)(−0.15, 57.37)(6.89, 31.14)(12.96, 33.55)
5 (B15B25)(−2.59, 43.30)(−0.15, 57.37)(11.51, 43.9)(−7.21, 46.30)
6 (B15B25)(−2.59, 43.30)(−0.15, 57.37)(19.07, 49.15)(15.45, 51.31)
7 (B15B37)(−30.87, 239.67)(−26.98, 150.44)(−16.83, 59.44)(−17.31, 65.58)
8 (B15B37)(−30.87, 239.67)(−26.98, 150.44)(−10.03, 66.15)(−9.22, 59.46)
9 (B15B37)(52.73, 72.15)(57.66, 80.47)(37.38, 117.85)(35.77, 122.65)
10 (B15B37)(52.73, 72.15)(57.66, 80.47)(36.01, 118.15)(36.27, 117.44)
11 (B15B37)(52.73, 72.15)(57.66, 80.47)(37.38, 117.85)(38.3, 115.1)
12 (B15B27)(−30.87, 239.67)(−26.98, 150.44)(−19.91, 72.83)(−18.67, 76.15)
13 (B15B27)(−30.87, 239.67)(−26.98, 150.44)(−13.66, 74.27)(−14.83, 70.24)
14 (B15B27)(52.73, 72.15)(57.66, 80.47)(43.96, 99.63)(247.08, −207.24)
15 (B15B27)(52.73, 72.15)(57.66, 80.47)(60.29, 90.97)(66.38, 104.4)
16 (B15B27)(52.73, 72.15)(57.66, 80.47)(44.77, 107.73)(44.75, 107.76)
17 (B15B27)(52.73, 72.15)(57.66, 80.47)(44.71, 107.74)(44.72, 107.71)
18 (B13B57)(29.80, 38.23)(32.33, 40.45)(25.69, 28.51)(22.78, 23.23)
Design (type)$G1$$W1$$G2$$W2$
1 (B15B25)(52.73, 72.15)(57.66, 80.47)(6.89, 31.14)(12.96, 33.55)
2 (B15B25)(52.73, 72.15)(57.66, 80.47)(11.51, 43.9)(–7.21, 46.30)
3 (B15B25)(52.73, 72.15)(57.66, 80.47)(19.07, 49.15)(15.45, 51.31)
4 (B15B25)(−2.59, 43.30)(−0.15, 57.37)(6.89, 31.14)(12.96, 33.55)
5 (B15B25)(−2.59, 43.30)(−0.15, 57.37)(11.51, 43.9)(−7.21, 46.30)
6 (B15B25)(−2.59, 43.30)(−0.15, 57.37)(19.07, 49.15)(15.45, 51.31)
7 (B15B37)(−30.87, 239.67)(−26.98, 150.44)(−16.83, 59.44)(−17.31, 65.58)
8 (B15B37)(−30.87, 239.67)(−26.98, 150.44)(−10.03, 66.15)(−9.22, 59.46)
9 (B15B37)(52.73, 72.15)(57.66, 80.47)(37.38, 117.85)(35.77, 122.65)
10 (B15B37)(52.73, 72.15)(57.66, 80.47)(36.01, 118.15)(36.27, 117.44)
11 (B15B37)(52.73, 72.15)(57.66, 80.47)(37.38, 117.85)(38.3, 115.1)
12 (B15B27)(−30.87, 239.67)(−26.98, 150.44)(−19.91, 72.83)(−18.67, 76.15)
13 (B15B27)(−30.87, 239.67)(−26.98, 150.44)(−13.66, 74.27)(−14.83, 70.24)
14 (B15B27)(52.73, 72.15)(57.66, 80.47)(43.96, 99.63)(247.08, −207.24)
15 (B15B27)(52.73, 72.15)(57.66, 80.47)(60.29, 90.97)(66.38, 104.4)
16 (B15B27)(52.73, 72.15)(57.66, 80.47)(44.77, 107.73)(44.75, 107.76)
17 (B15B27)(52.73, 72.15)(57.66, 80.47)(44.71, 107.74)(44.72, 107.71)
18 (B13B57)(29.80, 38.23)(32.33, 40.45)(25.69, 28.51)(22.78, 23.23)

## Conclusions

The paper describes a dimensional synthesis technique for the design of finger exoskeletons that are passively driven by a slider that is incorporated within the synthesis procedure. Each finger exoskeleton is comprised of a 3R serial chain, sized according to the user's anthropometric dimensions. This chain is later constrained to an eight-bar slider mechanism with common slider attachment at the hand that can be used by the wearer to passively drive the impaired fingers using their healthy arm.

The paper discusses how to capture the pose of an anthropomorphic limb using an optical motion capture system and determine its dimensions by a constrained least square optimization technique. Next, physiological task requirements were derived, compatible with finger-object contact and curvature constraints, to formulate the synthesis equations. The geometric synthesis of a left index exofinger slider mechanism is illustrated to show the applicability of the proposed design approach. A prototype is built to be used in the further assessment of the wearable device performance with respect to ergonomics, generalization to grasping of objects with different size and geometry, as well as ability to conform to different object shapes without the use of any sensing.

## Acknowledgment

The authors gratefully acknowledge the support of the National Science Foundation (NSF), Award No. 1404011 and Qatar National Research Fund, National Priorities Research Program Sub-award No. NPRP 7-1685-2-626, as well as the Agency of Science, Technology and Research (A*STAR) Singapore, Grant No. SERC 12251 00005 and ZJU-SUTD Research Collaboration Grant (Grant No. SUTD-ZJU/RES/03/2013).

## References

1.
Bekey
,
G. A.
,
Liu
,
H.
,
Tomovic
,
R.
, and
Karpuls
,
W. J.
,
1993
, “
Knowledge-Based Control of Grasping in Robot Hands Using Heuristics From Human Motor Skills
,”
IEEE Trans. Rob. Autom.
,
9
(
6
), pp.
709
722
.
2.
Dollar
,
A. M.
, and
Herr
,
H.
,
2008
, “
Lower Extremity Exoskeletons and Active Orthoses: Challenges and State-of-the-Art
,”
IEEE Trans. Rob.
,
24
(
1
), pp.
144
158
.
3.
Yang
,
C.
,
Zhang
,
J.
,
Chen
,
Y.
,
Dong
,
Y.
, and
Zhang
,
Y.
,
2008
, “
A Review of Exoskeleton-Type Systems and Their Key Technologies
,”
Proc. Inst. Mech. Eng., Part C
,
222
(
8
), pp.
1599
1612
.
4.
Bogue
,
R.
,
2009
, “
Exoskeletons and Robotic Prosthetics: A Review of Recent Developments
,”
Ind. Rob.: Int. J.
,
36
(
5
), pp.
421
427
.
5.
Gopura
,
R.
, and
Kiguchi
,
K.
,
2009
, “
Mechanical Designs of Active Upper-Limb Exoskeleton Robots: State-of-the-Art and Design Difficulties
,” IEEE International Conference on Rehabilitation Robotics (
ICORR
), Kyoto, Japan, June 23–26, pp.
178
187
.
6.
Gopura
,
R.
,
Kiguchi
,
K.
, and
Bandara
,
D.
,
2011
, “
A Brief Review on Upper Extremity Robotic Exoskeleton Systems
,” 6th IEEE International Conference on Industrial and Information Systems (
ICIIS
), Kandy, Sri Lanka, Aug. 16–19, pp.
346
351
.
7.
Heo
,
P.
,
Gu
,
G. M.
,
Lee
,
S.
,
Rhee
,
K.
, and
Kim
,
J.
,
2007
, “
Current Hand Exoskeleton Technologies for Rehabilitation and Assistive Engineering
,”
Int. J. Precis. Eng. Manuf.
,
13
(
5
), pp.
807
824
.
8.
Worsnopp
,
T. T.
,
Peshkin
,
M. A.
,
Colgate
,
J. E.
, and
Kamper
,
D. G.
,
2007
, “
An Actuated Finger Exoskeleton for Hand Rehabilitation Following Stroke
,” IEEE 10th International Conference on Rehabilitation Robotics (
ICORR
), Noordwijk, The Netherlands, June 13–15, pp.
896
901
.
9.
Nakagawara
,
S.
,
Kajimoto
,
H.
,
Kawakami
,
N.
,
Tachi
,
S.
, and
Kawabuchi
,
I.
,
2006
, “
An Encounter-Type Multi-Fingered Master Hand Using Circuitous Joints
,” IEEE International Conference on Robotics and Automation (
ICRA
), Barcelona, Spain, Apr. 18–22, pp.
2667
2672
.
10.
Fontana
,
M.
,
Dettori
,
A.
,
Salsedo
,
F.
, and
Bergamasco
,
M.
,
2009
, “
Mechanical Design of a Novel Hand Exoskeleton for Accurate Force Displaying
,” IEEE International Conference on Robotics and Automation (
ICRA
), Kobe, Japan, May 12–17, pp.
1704
1709
.
11.
Chiri
,
A.
,
Vitiello
,
N.
,
Giovacchini
,
F.
,
Roccella
,
S.
,
Vecchi
,
F.
, and
Carrozza
,
M. C.
,
2012
, “
Mechatronic Design and Characterization of the Index Finger Module of a Hand Exoskeleton for Post-Stroke Rehabilitation
,”
ASME/IEEE Trans. Mechatronics
,
17
(
5
), pp.
884
894
.
12.
Yeow
,
C. H.
,
Baisch
,
A. T.
,
Talbot
,
S. G.
, and
Walsh
,
C. J.
,
2014
, “
Cable-Driven Finger Exercise Device With Extension Return Springs for Recreating Standard Therapy Exercises
,”
ASME J. Med. Devices
,
8
(
1
), p.
014502
.
13.
Ma
,
Z.
, and
Pinhas
,
B. T.
,
2015
, “
Design and Optimization of a Five-Finger Haptic Glove Mechanism
,”
ASME J. Mech. Rob.
,
7
(
4
), p.
041008
.
14.
Birglen
,
L.
, and
Gosselin
,
C. M.
,
2005
, “
Geometric Design of Three-Phalanx Underactuated Fingers
,”
ASME J. Mech. Des.
,
128
(
2
), pp.
356
364
.
15.
Wege
,
A.
, and
Hommel
,
G.
,
2005
, “
Development and Control of a Hand Exoskeleton for Rehabilitation of Hand Injuries
,” IEEE/RSJ International Conference on Intelligent Robots and Systems (
IROS
), Edmonton, AB, Canada, Aug. 2–6, pp.
3046
3051
.
16.
Wang
,
J.
,
Li
,
J.
,
Zhang
,
Y.
, and
Wang
,
S.
,
2009
, “
Design of an Exoskeleton for Index Finger Rehabilitation
,” Annual International Conference of the IEEE Engineering in Medicine and Biology Society (
EMBC
), Minneapolis, MN, Sept. 3–6, pp.
2667
2672
.
17.
Fontana
,
M.
,
Fabio
,
S.
,
Marcheschi
,
S.
, and
Bergamasco
,
M.
,
2013
, “
Haptic Hand Exoskeleton for Precision Grasp Simulation
,”
ASME J. Mech. Rob.
,
5
(
4
), p.
041014
.
18.
Gosselin
,
C. M.
, and
Laliberte
,
T.
,
1998
, “
Underactuated Mechanical Finger With Return Actuation
,” Université Laval, Quebec City, QC, Canada, U.S. Patent No.
5762390
.
19.
Baek
,
S.
,
Lee
,
S. H.
, and
Chang
,
J.
,
1999
, “
Design and Control of Robotic Finger for Prosthetic Hands
,” IEEE/RSJ International Conference on Intelligent Robots and Systems (
IROS
), Kyongju, Korea, Oct. 17–21, pp.
113
117
.
20.
Fukaya
,
N.
,
Toyama
,
S.
,
Asfour
,
T.
, and
Dillmann
,
R.
,
2000
, “
Design of the TUAT/Karlsruhe Humanoid Hand
,” IEEE/RSJ International Conference on Intelligent Robots and Systems (
IROS
), Takamatsu, Japan, Oct. 31–Nov. 5, pp.
1754
1759
.
21.
Robson
,
N. P.
, and
Soh
,
G. S.
,
2016
, “
Geometric Design of Eight-Bar Wearable Devices Based on Limb Physiological Contact Task
,”
Mech. Mach. Theory
,
100
, pp.
358
367
.
22.
Sergi
,
F.
,
Accoto
,
D.
,
Tagliamonte
,
N. L.
,
Carpino
,
G.
, and
Guglielmelli
,
E.
,
2011
, “
A Systematic Graph-Based Method for the Kinematic Synthesis of Non-Anthropomorphic Wearable Robots for the Lower Limbs
,”
Front. Mech. Eng.
,
6
(
1
), pp.
61
70
.
23.
Accoto
,
D.
,
Sergi
,
F.
,
Tagliamonte
,
N. L.
,
Carpino
,
G.
,
Sudano
,
A.
, and
Guglielmelli
,
E.
,
2014
, “
Robomorphism: A Nonanthropomorphic Wearable Robot
,”
IEEE Rob. Autom. Mag.
,
21
(
4
), pp.
45
55
.
24.
Agarwal
,
P.
,
Hechanova
,
A.
, and
Deshpande
,
A. D.
,
2013
, “
Kinematics and Dynamics of a Biologically Inspired Index Finger Exoskeleton
,”
ASME
Paper No. DSCC2013-3893.
25.
Chase
,
T. R.
,
Erdman
,
A. G.
, and
Riley
,
D. R.
,
1978
, “
Triad Synthesis for up to Five Design Positions With Application to the Design of Arbitrary Planar Mechanisms
,”
ASME J. Mech. Trans. Autom.
,
109
(
4
), pp.
426
434
.
26.
Subbian
,
T.
, and
,
D. R.
,
1984
, “
Six and Seven Position Triad Synthesis Using Continuation Methods
,”
ASME J. Mech. Des.
,
116
(
2
), pp.
660
665
.
27.
Perez
,
A.
, and
McCarthy
,
J. M.
,
2005
, “
Clifford Algebra Exponentials and Planar Linkage Synthesis Equations
,”
ASME J. Mech. Des.
,
127
(
5
), pp.
931
940
.
28.
Chang
,
L. Y.
, and
Pollard
,
N. S.
,
2006
, “
Constrained Least-Squares Optimization for Robust Estimation of Center of Rotation
,”
J. Biomech.
,
40
(
6
), pp.
1392
1400
.
29.
Chang
,
L. Y.
, and
Pollard
,
N. S.
,
2007
, “
Robust Estimation of Dominant Axis of Rotation
,”
J. Biomech.
,
40
(
12
), pp.
2707
2715
.
30.
Howard
,
W. S.
, and
Kumar
,
V.
,
1996
, “
On the Stability of Grasped Objects
,”
IEEE Trans. Rob. Autom.
,
12
(
6
), pp.
904
917
.
31.
Rimon
,
E.
, and
Burdick
,
J.
,
1998
, “
Mobility of Bodies in Contact—Part I: A 2nd-Order Mobility Index for Multiple-Finger Grasps
,”
IEEE Trans. Rob. Autom.
,
14
(
5
), pp.
696
708
.
32.
Rimon
,
E.
, and
Burdick
,
J.
,
1998
, “
Mobility of Bodies in Contact—Part II: How Forces are Generated by Curvature Effects
,”
IEEE Trans. Rob. Autom.
,
14
(
5
), pp.
709
717
.
33.
Robson
,
N. P.
, and
McCarthy
,
J. M.
,
2007
, “
Kinematic Synthesis With Contact Direction and Curvature Constraints on the Workpiece
,”
ASME
Paper No. DETC2007-35595.
34.
Robson
,
N. P.
,
Allington
,
J.
, and
Soh
,
G. S.
,
2014
, “
Development of Underactuated Mechanical Fingers Based on Anthropometric Data and Anthropomorphic Tasks
,”
ASME
Paper No. DETC2014-34878.
35.
Pratt
,
V.
,
1987
, “
Direct Least-Squares Fitting of Algebraic Surfaces
,”
Comput. Graphics
,
21
(
4
), pp.
145
152
.
36.
Soh
,
G. S.
, and
Ying
,
F. T.
,
2015
, “
Motion Generation of Planar Six- and Eight-Bar Slider Mechanisms as Constrained Robotic Systems
,”
ASME J. Mech. Rob.
,
7
(
3
), p.
031018
.
37.
McCarthy
,
J. M.
, and
Soh
,
G. S.
,
2010
,
, 2nd ed.,
Springer Verlag
,
New York
.
38.
Feix
,
T.
,
Romero
,
J.
,
Schmiedmayer
,
H. B.
,
Dollar
,
A. M.
, and
Kragic
,
D.
,
2016
, “
The GRASP Taxonomy of Human Grasp Types
,”
IEEE Trans. Hum. Mach. Syst.
,
46
(
1
), pp.
66
77
.