#  _Efficient Multi-Prize Lottery Tickets_ : Enhanced Accuracy, Training, and
Inference Speed

Hao Cheng  Northeastern University  Pu Zhao  Northeastern University  Yize Li
Northeastern University  Xue Lin  Northeastern University  James Diffenderfer
Lawrence Livermore National Laboratory  Ryan Goldhahn  Lawrence Livermore
National Laboratory  Bhavya Kailkhura  Lawrence Livermore National Laboratory

###### Abstract

Recently, Diffenderfer & Kailkhura [1] proposed a new paradigm for learning
compact yet highly accurate binary neural networks simply by pruning and
quantizing randomly weighted full precision neural networks. However, the
accuracy of these multi-prize tickets (MPTs) is highly sensitive to the
optimal prune ratio, which limits their applicability. Furthermore, the
original implementation did not attain any training or inference speed
benefits. In this report, we discuss several improvements to overcome these
limitations. We show the benefit of the proposed techniques by performing
experiments on CIFAR-10.

##  1 Introduction

It is found that randomly-initialized dense networks contain subnetworks that
can achieve test accuracy comparable to the trained dense network [5]. Based
on this, the authors in [1] propose (and prove) a stronger Multi-Prize Lottery
Ticket Hypothesis:

A sufficiently over-parameterized neural network with random weights contains
several subnetworks (winning tickets) that (a) have comparable accuracy to a
dense target network with learned weights (prize 1), (b) do not require any
further training to achieve prize 1 (prize 2), and (c) is robust to extreme
forms of quantization (i.e., binary weights and/or activation) (prize 3).

This provides a new paradigm for learning compact yet highly accurate binary
neural networks simply by pruning and quantizing randomly weighted full
precision neural networks. The proposed algorithm can find multi-prize tickets
(MPTs) with SOTA accuracy for binary neural networks without ever updating the
weight values.

The high accuracy of subnetworks can be achieved at a specific sparsity ratio.
Since the model weights are not trained, if the sparsity ratio is not
appropriately configured, the accuracy performance may suffer. Besides,
although the model weights are not updated, the MPTs algorithm needs to train
the scores (each score corresponding to one weight) with moderate training
efforts. Moreover, during inference, the sparsity of the subnetwork is not
fully exploited for efficient inference so the inference speed is still the
same as the full dense model without any acceleration effects.

To address the above issues, we further consider how to improve the MPTs
algorithm on these fronts. We propose four improvements to enhance the
accuracy, training, and inference speed. Our improvements include:

  * •

Power-propagation [6] parameterization of scores to improve the accuracy
performance on a wider range of sparsity ratios,

  * •

Fine-tuning the model to further improve the accuracy,

  * •

Thresholding instead of sorting-based pruning to accelerate the training, and

  * •

Structured pruning to remove kernels and improve the inference speed.

Next, we discuss these improvements in detail and provide empirical support to
justify their superiority over the original MPT method111Although [1] supports
both binary weight and binary activation MPTs, we only focus on binary weight
MPTs in this report for simplicity..

##  2 Incorporating Power-Propagation into MPTs

###  2.1 Incorporating Power-Propagation Parameterization

We incorporate Power-propagation [6] into MPTs [1]. Different from other model
training methods, MPT does not train the model weights. Instead, it keeps the
model weights untouched as their initial random values. Note that we only
focus on the binary weight instead of binary activations. For each weight, it
needs to determine whether to prune or binarize this parameter. MPT uses a
score for each model weight to indicate its importance. Thus, it trains and
sorts the scores to determine which weights should be pruned and which weights
should be preserved. We can see that score updating plays an important role in
MPTs. Thus, we adopt Power-propagation parameterization to further improve the
score updating performance. Specifically, the original score $\bm{S}$ is
replaced with a mapping function
$\Psi(\bm{s})=\bm{s}|\bm{s}|^{\alpha-1}=\bm{S}$. We only apply this
transformation to the scores in MPT, leaving other parameters untouched. With
the loss $\mathcal{L}(\Psi(\bm{s}))$, the gradients w.r.t. to $\bm{s}$ can be
obtained as follows,

$\frac{\partial\mathcal{L}(\Psi(\bm{s}))}{\partial\bm{s}}=\frac{\partial\mathcal{L}}{\partial\Psi(\bm{s})}\frac{\partial\Psi(\bm{s})}{\partial\bm{s}}={\frac{\partial\mathcal{L}}{\partial\Psi(\bm{s})}}{\mathop{\mathrm{diag}}(\alpha|\bm{s}|^{\circ\alpha-1})},$
(1)

where $\mathop{\mathrm{diag}}$ denotes a diagonal matrix, and
$|\bm{s}|^{\circ\alpha-1}$ indicates computing the power $\alpha-1$ of
$|\bm{s}|$ element-wise. $\frac{\partial\mathcal{L}}{\partial\Psi(\bm{s})}$ is
the derivative w.r.t. to the original score
$\bm{S}=\bm{s}|\bm{s}|^{\circ\alpha-1}$. In our update, it is additionally
multiplied (element-wise) by a factor ${\alpha|\bm{s}|^{\alpha-1}}$, which
scales the step taken proportionally to the magnitude of each entry. Note that
this update is different from simply scaling the gradients of $\bm{S}$ by the
magnitude of the parameter (raised at $\alpha-1$), since the update is applied
to $\bm{s}$ not $\bm{S}$, and is scaled by $\bm{s}$. The update rule (1) has
the following properties:

  * (i)

If $\alpha>1$, $\frac{\partial\mathcal{L}}{\partial\bm{s}}=0$ whenever
$\bm{s}=0$ due to the $\alpha|\bm{s}|^{\alpha-1}$ factor.

  * (ii)

In addition, 0 is surrounded by a plateau and hence scores are less likely to
change sign (gradients become vanishingly small in the neighborhood of 0 due
to the scaling). This should negatively affect initialization which allows for
both negative and positive values, but it might have bigger implications for
biases.

  * (iii)

This update is naturally obtained by the backpropagation algorithm since
backpropagation implies applying the chain-rule from the output towards the
variable of interest, and Eq. (1) simply adds another composition (step) in
the chain before the variable of interest.

###  2.2 Performance of Power-Propagation MPTs

We show the performance of MPTs with power-propagation in Fig. 1. We conduct
experiments on CIFAR-10 for CNNs with varying depths (including CONV-4 with 4
CONV layers, CONV-6 with 6 CONV layers and CONV-8 with 8 CONV layers). We show
the results of different hyperparameter $\alpha$ values
($\alpha=1,2,3,\text{or},4$). Note that when $\alpha=1$, it is just the case
of the original score $\bm{S}$ training without power-propagation. We adopt
the default hyperparameter setting from the original MPT paper to train the
model 222Note that the code support both global and layerwise pruning. Since
global pruning usually performs better, we adopt global pruning as our main
focus..

In Fig. 1, we can observe that power-propagation can increase the accuracy
performance when the pruning ratio is small (smaller than 0.5) with less
pruned weights. When the pruning ratio is above 0.5, the accuracy performance
is similar to the original case without power-propagation. When $\alpha=1$,
the performance is very similar to the original case without power-
propagation, demonstrating the equivalence of $\alpha=1$ and no power-
propagation. When $\alpha$ increases, the accuracy performance under small
pruning ratios also improves.

Figure 1: Performance of Power-Propagation. The left figure shows the results
of CONV-6 with 6 CONV layers and the right figure shows the results of CONV-4
with 4 CONV layers.

As shown in Fig. 1, since power-propagation can help to improve the accuracy
when the pruning ratio is small, the performance of MPT becomes more robust
with high accuracy performance achieved on a wider range of sparsity ratios.

###  2.3 Score distribution with Power-Propagation

To explore why power-propagation can improve the MPT performance, we show the
distribution of scores of certain convolution layer in the model in Fig. 2. We
can see that after adopting power-propagation, the integral value range of the
score shrinks. It means that the scores are distributed in a smaller value
range. Therefore, updating the scores becomes more efficient and a small
change can effectively update the status of pruned or preserved.

Figure 2: Distribution of scores for Power-Propagation. The left figure shows
the distribution CONV-8 with 8 CONV layers. The right figure shows the
distribution of CONV-6 with 6 CONV layers.

##  3 Incorporating Weights Finetuning in MPTs

###  3.1  Finetuning Method

In the original MPTs, the weights are not trained and are kept as randomly
initialized values. MPT assigns a score for each weight and trains the scores
to determine whether to prune or binarize the weights. In this section, we
further consider whether finetuning the weights can further improve the
performance [3, 2]. Specifically, we first use MPTs to obtain a sparse
binarized model. This model can be represented by $\bm{W}_{b}\cdot\bm{M}$,
where $\bm{W}_{b}$ is the binarized weights and $\bm{M}$ is the mask with 0 or
1 values to determine whether to prune each weight. Based on MPTs, we have the
following,

$\displaystyle\alpha^{(j)}$
$\displaystyle=\|\bm{M}^{(j)}\odot\bm{W}^{(j)}\|_{1}/\|\bm{M}^{(j)}\|_{1}$ (2)
$\displaystyle\bm{W}_{b}^{(j)}$
$\displaystyle=\alpha^{(j)}\cdot\text{sign}(\bm{W}^{(j)})$ (3)

where $\bm{W}^{(j)}$ is the original weights in the $j$-th layer. After we run
MPTs, we can obtain $\bm{W}_{b}$ and $\bm{M}$. Next, when we start to finetune
the MPTs model, we keep the mask $\bm{M}$ fixed and only update the model
weights $\bm{W}$ with gradients. Note that different from MPTs where the
gradients are computed for scores to update mask, here the gradients are
computed for model weights. Besides, although model weights are updated,
during the inference, they are still transformed into binary values following
Eq. (2).

###  3.2 Performance of Finetuning

Figure 3: Performance of Finetuning. The three figures from left to right show
the results of finetuning the first layer, the last layer, and the whole
model.

We show the performance with weights finetuning in Fig. 3. We conduct
experiments on CIFAR-10 with Resnet-18. For the finetuning, we try three cases
including finetuning the first layer, the last layer, or the full model with
all layers.

In Fig. 3, we report the performance of the three cases discussed above. We
can see that finetuning can help to slightly improve the accuracy performance
in all three cases. Usually finetuning the last layer can obtain the largest
accuracy improvement. However, we should not finetune for too many epochs to
avoid the accuracy drop. Usually, the finetuning epochs can be smaller than
50.

For the funetuning, we perform a grid search over different hyperparameters
summarized in Table 1. The best accuracy performance is achieved with the SGD
optimizer, 0.001 learning rate with cosine scheduler, 256 batch size, and
finetuning the last layer, which is highlighted in Table 1.

Optimizer | SGD | Adam |  |
---|---|---|---|---
lr scheduler | multi-step | cosine | constant |
lr | 0.1 | 0.01 | 0.001 | 0.0001
Batch size | 64 | 128 | 256 | 512
Fine-tune | full model | last layer | first layer |
Table 1: Hyper-parameter settings for funetuning.

##  4 Replacing sorting with thresholding to accelerate the training

###  4.1  Thresholding Method

Original MPT implementation requires sorting of the scores to determine top
weights that are important and should be preserved. The sorting operation is
usually time-consuming, especially for the DNN layers with a large number of
parameters. Thus, to accelerate the training, we adopt a thresholding method
to replace the sorting operation. Specifically, once the scores are updated,
we do not sort the scores. Instead, we directly compare the scores with a
predefined threshold. If the score is larger than the threshold, the
corresponding weight is preserved. Otherwise, the corresponding weight is
pruned. Thus, we do not need to sort the scores in each iteration during
training and the training speed can be accelerated effectively.

###  4.2 Performance of Thresholding Based MPT

We conduct experiments on CIFAR-10 with Resnet-18 and show the results of
thresholding in Table 2 and 3. We adopt the default hyperparameter setting in
the original MPTs paper to train the model. We can observe that the training
time for each epoch can be reduced by 13%. Here we incorporate the power-
propagation into the MPTs for efficient score training. We also show the
accuracy performance with thresholding in Fig. 4. We can see that the accuracy
can be even higher than the original MPT implementation.

Figure 4: Performance of thresholding-based MPT. | $\alpha=1$ | $\alpha=2$ | $\alpha=3$
---|---|---|---
Sorting | 24.3 s/epoch | 25.4 s/epoch | 26.8 s/epoch
Thresholding | 21.5 s/epoch | 22.3 s/epoch | 24.5 s/epoch
Table 2: Training time comparison between sorting and thresholding. | $\alpha=1$ | $\alpha=2$ | $\alpha=3$
---|---|---|---
Sorting | 4.05 min | 4.23 min | 4.47 min
Thresholding | 3.58 min | 3.73 min | 4.07 min
Table 3: Training time comparison for 10 epochs between sorting and
thresholding.

##  5 Structured pruning to accelerate the inference

###  5.1  Posthoc Structured Pruning Method

Although MPTs use unstructured pruning for training, we find that, in the case
with a high pruning ratio, there is a certain sparsity structure that can be
utilized to accelerate the inference. For example, if most weights are pruned,
there are also many all-zero $3\times 3$ convolutional (CONV) kernels in each
CONV layer. To demonstrate this, we plot the number of zero kernels in each
layer for a MPT model of 95% pruning ratio in Fig. 5. We can see that there
are 67% kernels that are all zeros among all layers. This implies that we can
exploit the kernel sparsity to accelerate the training [7, 8, 4].

Figure 5: Distribution of sparse kernels of all layers.

Specifically, if certain kernels are all zeros, meaning the whole kernel can
be pruned, we can try to remove the whole kernel in the computation graph,
thus avoiding the computations related to this pruned kernel. The acceleration
rate can be computed as

$\displaystyle AR=\frac{\sum N\times M\times D\times D\times k\times k}{\sum
P\times D\times D\times k\times k}$ (4)

where $N$ and $M$ are the input and out channel numbers in one layer, $D$ is
the size of the feature map and $k$ is the kernel size of that layer. The sum
is taken over all layers. The acceleration rate is the rate between the
original computation counts and the sparse computation counts after removing
all-zero kernels, which shows the change of the computation number, and the
corresponding inference speed.

###  5.2 Performance of Structure Pruning

We show the performance with structure pruning in Table 4. We conduct
experiments on CIFAR-10 with Resnet-18. The default hyperparameter setting in
the original MPTs paper [1] is adopted to train the model. We can see that by
exploring the structure sparsity in the MPT model, we can remove the all-zero
kernels and accelerate the inference effectively.

Sparsity | 70% | 90% | 95%
---|---|---|---
Acceleration rate | 1.4$\times$ | 2.3$\times$ | 3.0$\times$
Table 4: Acceleration rate of various sparsities.

##  6 Conclusion

We proposed various enhancements for making MPTs more efficient. Our results
demonstrated that power-propagation and finetuning can help to improve the
accuracy performance. Further, the thresholding can accelerate the MPT
training and the structure pruning can accelerate the inference speed.

## Acknowledgement

This work was performed under the auspices of the U.S. Department of Energy by
the Lawrence Livermore National Laboratory under Contract No. DE-
AC52-07NA27344 and was supported by LLNL-LDRD Program under Project No.
20-SI-005.

## References

  * [1] Diffenderfer, J., Kailkhura, B.: Multi-prize lottery ticket hypothesis: Finding accurate binary neural networks by pruning a randomly weighted network. In: International Conference on Learning Representations (2021)
  * [2] Li, H., Chaudhari, P., Yang, H., Lam, M., Ravichandran, A., Bhotika, R., Soatto, S.: Rethinking the hyperparameters for fine-tuning. In: International Conference on Learning Representations (2020), https://openreview.net/forum?id=B1g8VkHFPH
  * [3] Li, S., Chen, D., Chen, Y., Yuan, L., Zhang, L., Chu, Q., Liu, B., Yu, N.: Unsupervised finetuning. CoRR abs/2110.09510 (2021), https://arxiv.org/abs/2110.09510
  * [4] Ma, X., Guo, F., Niu, W., Lin, X., Tang, J., Ma, K., Ren, B., Wang, Y.: PCONV: the missing but desirable sparsity in DNN weight pruning for real-time execution on mobile devices. AAAI (2020)
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# Anthropomorphic finger for grasping applications: 3D printed endoskeleton in
a soft skin ††thanks: Citation: Tavakoli, M., Sayuk, A., Lourenço, J. et al.
Anthropomorphic finger for grasping applications: 3D printed endoskeleton in a
soft skin. Int J Adv Manuf Technol 91, 2607–2620 (2017). DOI:
10.1007/s00170-016-9971-8

Mahmoud Tavakoli, Andriy Sayuk, João Lourenço, Pedro Neto
University of Coimbra
Coimbra

###### Abstract

Application of soft and compliant joints in grasping mechanisms received an
increasing attention during recent years. This article suggests the design and
development of a novel bio-inspired compliant finger which is composed of a 3D
printed rigid endoskeleton covered by a soft matter. The overall integrated
system resembles a biological structure in which a finger presents an
anthropomorphic look. The mechanical properties of such structure are enhanced
through optimization of the repetitive geometrical structures that constructs
a flexure bearing as a joint for the fingers. The endoskeleton is formed by
additive manufacturing of such geometries with rigid materials. The geometry
of the endoskeleton was studied by Finite Element Analysis (FEA) to obtain the
desired properties: high stiffness against lateral deflection and twisting,
and low stiffness in the desired bending axis of the fingers. Results are
validated by experimental analysis.

_K_ eywords Anthropomorphic fingers $\cdot$ Compliant joints $\cdot$ Grasping
mechanisms

##  1 Introduction

Integration of compliance in mechanical systems in general and in robotic
systems in particular received an increasing attention in the robotics
community during the recent years. For the design and manufacturing of robotic
hands, compliance can result in a passive adaptability in articulated fingers
of a grasping mechanism [1] [3]. In addition, integration of the compliance
enhances the safety in the human-robot interaction process. An important
subject of research in this domain is the design and development of joints
with a compliant behaviour, which can be formed by a proper controller. The
DLR HAND II [4] integrates the compliance into the control loop by using an
appropriate impedance controller. Yet, this solution demands for a
sophisticated controller with a fast response time. The development of
grippers made of soft materials has been in studied in the last few years,
including the design of anthropomorphic fingers [5] [6].

Compliance can also be integrated in the actuator. Meka-H2 compliant hand [7]
integrates a Series Elastic Actuator (SEA). It has a total of 12 Degrees Of
Freedom (DOF) controlled by 5 SEA actuators. Another approach is to directly
integrate compliance into the joints. Pisa-IIT Softhand [8], and the UB hand
[9] are examples of recent development of anthropomorphic hands that directly
integrate the compliance into the joints. In Pisa-IIT Softhand several
elements are interconnected with elastic elements, while in UB hand the joint
compliance is achieved by integration of springs into the joints. Flexirigid
[10], SDM hand [11], and ISR-Softhand [12] integrate elastomeric joints
between their rigid phalanges. The elastic joints of the SDM hand [11], are
formed by casting a urethane rubber into the moulds that are built in the 3D
printed finger. The shape deposition manufacturing originally reported in [16]
was later updated with the Hybrid Deposition Manufacturing (HDM) [14], in
which the multipart moulds can be reused instead of destroying the temporary
mould features, making this method even less labour intensive. Another
approach on development of the Softhand was presented in RBO hands [15], in
which authors developed a continuum finger which is pneumatically actuated.

A major problem associated with compliant joints which are fabricated merely
with an elastomer is that they suffer from lateral deflections and twisting.
The stiffness of the joint on all axes is dependant to the low Young’s modulus
of the elastomer. In both of our previous attempts in the development of
grasping mechanisms with soft joints, i.e., in Flexirigid [10] and ISR-
Softhand [12], we faced this problem, Figure 1. This problem was reported also
in the development of a soft miniature grasper for surgical applications. The
integration of a second material into the joint might be a solution to this
problem. This was suggested for instance in [17], but in a different context,
i.e. for an interventional MRI device. To develop the multi-material joint,
authors used a polyjet additive manufacturing unit and combined two materials
with a relatively large difference in their Young’s modulus. Using the polyjet
additive manufacturing it is possible to combine different materials in the
same part, which gives a great freedom on design of dual material flexure
bearing. This is still an expensive technology and the range of materials
available is yet limited.

Figure 1: The ISR-Softhand grasping a spherical object. The exerted forces
cause undesired deflections on the compliant joint of the thumb, affecting the
grasp stability. Grasping heavier objects is impossible, even if the thumb can
exert enough normal force

In the context of prosthetic hands and grasping mechanisms, a multi-material
approach was also proposed by Matasuoka’s group [18, 19], where a design for
an artificial finger joint for anthropomorphic robotic hands was suggested and
tested. In this highly bio-inspired design, each finger is composed of several
ABS parts that closely resemble the skeleton of a human finger. It includes
ball joints and an elastic capsule around the ball joints. This is a close
replica of a human finger which can be used for further studies of such.
However, its fabrication and implementation procedure is more labour intensive
compared to the other methods such as SDM and HDM.

The goal of our research is to maintain the fabrication simplicity, reduced
cost, and at the same time contribute toward two main objectives. The first
objective is related with the development of fingers with a better
anthropomorphic look and feel. To do so, we proposed a silicone skin to cover
the whole finger structure. The current practice of using cosmetics gloves on
the prosthetic hands is expensive and it is reported that the cosmetic glove,
which covers a hand prosthesis, negatively affects the mechanical efficiency
of a prosthesis [20]. The second objective is to enable the design of the
finger joints in a way that its stiffness in different axes could be defined
within the design. It is desired to establish a trade-off between the desired
low stiffness in the flexion axis and the higher stiffness on all other axes
of the joints. The former is a very important factor that directly affects the
size, weight and power consumption of the actuators. Reducing the required
force for flexion of the fingers is one of the most important steps in
reducing the overall size and weight of a prosthetic hand.

Our proposed approach and the main novelty of this work for achieving both
goals is to decouple the two challenges, i.e., the cosmetic appearance and the
stiffness. Another particular novelty of this work is that the desired
compliance of the soft fingers is achieved with a rigid matter endoskeleton.
This has 3 main advantages. First, to design a finger, one has a broader range
of material choices and is not limited only to elastic and soft materials.
This is important because 3D printing of arbitrary geometries of rigid
materials is currently more affordable and accessible than 3D printing of soft
materials. Second, this method allows researchers to fine-tune the properties
of the finger joints based on their grasping application, mainly by optimizing
the geometry of the endoskeleton and not solely by selection of the joint
material. Third, the material for soft skin is decoupled from the material of
the joints, while a continuous soft skin is formed. This provides a good
freedom for the researchers in choosing the materials and geometry of the soft
skin. The soft skin has an important effect on cosmetics and functionality
since it provides a larger contact area and a higher friction.

##  2 Compliant joints

Compliant joints and flexural joints have been widely considered in the
precision control elements and also in the design of Microelectromechanical
systems (MEMS) [21, 22]. When exposed to external forces, the resulting pose
of the flexural joints can be precisely modelled and controlled. However, they
have not received enough attention in the robotics community, probably because
of their non-conventional nature, i.e., due to the difficulties in their
design, optimization and fabrication. Among these, the fabrication problem is
currently being addressed by additive manufacturing methods.

The topology optimization of the flexural joints concentrated mostly on the
objective of distributing a limited amount of material in the design domain
such that the output displacement is optimized. This conducts to the
maximization of the joint’s sensitivity but did not consider the other effects
such as the stiffness on other axes, which is an irrelevant subject in that
domain due to the generally low external forces. An important factor of
flexural joint’s is the ratio of $\sigma y/E$ (yield strength/Young’s modulus)
[23]. Smaller tensile modulus is beneficial since it reduces the required
force for bending of the joints in the desired direction. However, for the
material to stay in its elastic zone and not entering the permanent plastic
deformation, a high $\sigma y$ is advantageous.

Lotti et. al. compared the $\sigma y/E$ of some materials, and based on the
results, they developed a single piece finger with Teflon (PTFE) material
[23]. The single piece finger was developed by removal manufacturing from a
single block. In other research work the same team utilized coil springs as
joint of the fingers of the previously mentioned UB-Hand [24]. Despite the
fact that in the latter case the hand’s digits are not anymore composed of a
single piece endoskeleton, the concept of utilization of coil springs was
interesting due to the low stiffness against the flexion. However, in both
cases, the undesired deflection and undesired torsion were not discussed. This
aspect was later discussed, in which researchers developed a small compliant
grasping mechanism with the shape deposition moulding of compliant materials
as joints. In order to reduce the undesired deflections in the fingers they
integrated a thin steel reinforcement piece into the joints. In summary, an
ideal compliant joint to be used in the digits of a bionic hand or an
industrial gripper should be flexible about the flexure axis and have the
following characteristics:

  * •

Low stiffness in the direction of the desired bending;

  * •

High stiffness against undesired deflection and torsions;

  * •

Higher ratio of $\sigma y/E$ and thus a high elastic property range.

In this section, we will discuss several possible versions of the compliant
joints and their different properties, as well as their fabrication. This
includes joints formed by highly elastic compliant materials, with and without
the reinforcements, and joints formed by rigid materials.

###  2.1 Compliant joints

As discussed, the lateral deflection is the undesired feature of the elastic
joints, especially for the thumb, as can be seen in Figure 1. We can use the
beam deflection formula in order to model the joint deflection on both axes,
i.e. the desired flexion and the undesired deflection. Figure 2 demonstrates
the applied forces to the finger for a tip pinch grasp, where $F_{n}$
indicates the normal force and $F_{f}$ indicates the resulting tangential
force. Figure 3 demonstrates the normal and lateral forces applied to one
finger. The deflection of the joint [25] can be estimated by:

$\delta=\frac{Fl^{3}}{3EI}$ (1)

where $\delta$ determines the amount of the deflection of the beam at the
length of $l$, which is caused by a force of $F$, $E$ is the material Young’s
Modulus and $I$ is moment of inertia of the beam. For a rectangular beam $I$
is calculated as:

$I=\frac{bh^{3}}{12}$ (2)

where $h$ is the dimension in the plane of bending, i.e. in the axis in which
the bending moment is applied. Here we call the inertia of the beam in the
desired flexion bending as $I_{f}$ and the inertia of the beam on the
undesired lateral bending as $I_{l}$. Also $\delta_{f}$ and $\delta_{l}$
demonstrate the deflection on the flexion axis (desired) and lateral
deflection (undesired). From figure 3, we have:

$\displaystyle I_{f}=\frac{wt^{3}}{12};I_{l}=\frac{tw^{3}}{12}$ (3)
$\displaystyle\delta_{f}=\frac{F_{f}L^{3}}{36wt^{3}};\delta_{l}=\frac{F_{n}L^{3}}{36tw^{3}}$
(4)

Thus:

$C=\frac{\delta_{l}}{\delta_{f}}=\frac{F_{f}wt^{3}}{F_{n}tw^{3}}=\mu(\frac{t}{w})^{2}$
(5)

where $C$ shows the ratio between the undesired lateral deflection and desired
deflection, which should be minimized. At the same time, to have a stable
grasp the actual amount of the undesired deflection ($\delta_{l}$) should be
limited to a predetermined value (e.g. less than $1mm$). By simply increasing
the thickness of the joint $t$, we can address both conditions. However,
increasing the $t$ results in the cubic growth of the required force for
flexion of the joint in the desired direction. Therefore, it is highly
desirable to keep the $t$ as low as possible.

Figure 2: Forces applied to the fingers in a tip pinch Figure 3: Model of the
joint bending

###  2.2 Reinforced joints

One method to increase the joint stiffness against the undesired lateral
deflection is to integrate into the joint a thin layer of a second material
with a higher Young’s modulus than the elastic material, and with a high
$\frac{w}{t}$ width to thickness value. If this new layer is inserted in the
centre of the elastic beam, and considering that its thickness is
significantly smaller than the elastic material, the moment of inertia can be
calculated as the sum of the moments of the inertia of both beams (for more
information please refer to [25]). That is:

$\displaystyle I_{fn}=I_{f}+\alpha\frac{w_{r}t_{r}^{3}}{12}$ (6)
$\displaystyle I_{ln}=I_{l}+\alpha\frac{t_{r}w_{r}^{3}}{12}$ (7)

In which $I_{fn}$ and $I_{ln}$ are the new moment of inertia of the joint
against the desired flexion and the undesired lateral bending, $w_{r}$ and
$t-r$ are the width and the thickness of the enforcement layer, and
$\alpha=frac(Er/Ec)$ is the ratio between the Young’s modulus of the
integrated reinforcement and the original highly elastic material with low
Young’s modulus. It should be noted that this model works only if the
reinforcement is located in middle of the joint, since the centre of mass
stays unchanged in both cases (with or without reinforcement). Otherwise, it
becomes necessary to find out the new centre of mass and recalculate the
moment of inertia around the new shifted axis.

Figure 4 shows a model of a thumb which integrates a thin layer of
reinforcement. This was applied and tested in the second version of the ISR-
Softhand. Despite effectively reducing the deflections, this model increased
substantially the required force for closing the thumb (this will be further
analysed in the next section.) For the rest of the fingers, the problem of
deflection is less significant than for the thumb, since in grasping of
heavier objects, usually all 4 fingers distribute loads among all of them.
Therefore, as can be seen in Figure 5, the reinforcement for the fingers were
redesigned and optimized for having less stiffness in the axis of flexion,
thus reducing the total power required to close the fingers.

Figure 4: A thin rigid element is integrated to the thumb’s flexible joint in
order to increase the thumb’s lateral stiffness Figure 5: A 3D printed
endoskeleton integrating rigid joint reinforcements (a); The DIP joint is
filled with some sponge and sealed with an elastic tube (b); This set is then
placed in a mould (Figure 6) which is formed from 3D scanning of a human
finger and a resin is cast around the endoskeleton (c) (d). Figure 6: The
internal surfaces of the mould are created from the 3D scanning of a human
finger

##  3 Flexible Joints through geometrical features on rigid materials

While the solution of integration of a reinforcement could reduce the
deflections, it increased also the joint’s stiffness in the flexion direction
and the required force for closing of the finger. It is therefore interesting
to study alternatives for the rigid reinforcement such as non-rectangular
shapes

###  3.1 Rigid endoskeleton and the soft skin

It was performed a study which intends to analyse other possible rigid
reinforcements. However, we decided to take a different approach. In the
previous approach, the elastomer plays the role of the joint. In the new
joint, the elastomer would only play the role of the soft skin, which can be
designed for better cosmetics with minimal effect on the joint’s stiffness. We
selected an Ecoflex series silicon (Smooth-on), which is highly stretchable,
has a low chemical reactivity, and exhibits very similar mechanical properties
to the human skin. Similarities between the mechanical properties of the
silicone with human tissues and also the safety of silicone in contact with
human body are the reason for being used not only in medical implants, but
also in beauty and hair products. For the same reasons, along PDMS, different
types of silicone are the most used materials in researchers on soft
electronics for wearable technologies that aim to build products in touch with
the human skin and human organs. In the specific case of Ecoflex this material
is certified for skin touching products.

The rigid reinforcement in the previous approach is the actual joint in the
new approach. One advantage of the previous elastomer joint was that during
the bending, it enables a continuous bending profile which helps in encircling
the object. This is an important feature that should be maintained in the new
approach. Based on that, the novel joints of the endoskeleton were designed
based on repetitive patterns that enable a continuous bending profile. As can
be seen in figure 7, this includes triangular, rectangular and circular
patterns in which the final flexion of the joint results from superposing
small flexion of each of the patterns. The considered patterns were:

  * •

A flat reinforcement, as was already used in the ISR-Softhand (Sample A);

  * •

A triangular geometry that is repeated for four times with the width of 13mm
and thickness of 1mm (Sample B);

  * •

A rectangular geometry that is repeated for four times with fillets of 1mm
radius applied at some edges with the width of 13mm and thickness of 1mm
(Sample C);

  * •

A circular geometry that is repeated for four times with the width of 10mm and
thickness of 1mm (Sample D);

  * •

Same geometry of Sample D with the width of 13mm and 15mm (Sample H (only for
simulation) and Sample E);

  * •

An hybrid design with circular geometry as well as flat reinforcements (Sample
F);

  * •

A circular pattern with a different geometry from Sample D used to analyse the
effect of the geometry (Sample G, Figure 8).

Figure 7: Different joint models used in tests Figure 8: Geometry used in
circular pattern for joint samples D, E and H (top), and for sample G (bottom)

These joints were studied in terms of the desired flexion, undesired lateral
deflection and twisting, as well as the stress concentration. In this way the
continuous curve of the finger while bending would be similar to the
elastomeric joints. It is difficult to use analytical models to calculate
joint’s deflection, except for the flat geometry. Therefore a Finite Element
Analysis (FEA) was applied to compare several characteristics of the joints. A
non-linear solver for large deformations was used in the SolidWorks simulation
toolbox. For each joint model we performed the simulations according to the
following loads:

  1. 1.

Constant force of 0,8N acting on a Z direction applied at the distance of
0,055m from the fixed base to measure the maximum flexion of all joints for a
constant force (Figure 9 top);

  2. 2.

Constant force of 4N acting on a Y direction applied at the distance of 0,04m
from the fixed base to simulate the lateral deflection (Figure 9 middle);

  3. 3.

A constant torque of 0,12 Nm to simulate the twisting of the joint due to the
tangential force on the fingertip (Figure 9 bottom).

Figure 9: Three types of loads analysed during FEA, maximum bending (top),
lateral deflection (middle) and displacement due to torsion (bottom)

It should be also mentioned that the absolute value of the forces and torques
applied do not affect the conclusions obtained from this research. This is
because this is a comparative study and the parameters that will be compared
are unitless. For the best functionality of the joints, it is desired to
minimize the lateral deflection and also the torsion of the joint due to
lateral forces. It is also desirable to have the minimum bending force for
flexion of the joint to reduce the size of the required actuators. We
therefore created and compared two new criteria:

  1. 1.

LD/FL denotes the amount of the lateral deflection divided by the displacement
due to the flexion;

  2. 2.

Tor/FL denotes the amount of the displacement due to the torsion divided by
the displacement due to the flexion.

Both rates should have a minimum value and the amount of absolute flexion
(displacement due to flexion forces) should be also as high as possible.
Furthermore, the FEA analysis allows us to inspect and compare the joints for
stress values due to the forces applied on different axes. In all simulations,
we considered the properties of the PA2200 material from EOS 3D printers which
is used by most of SLS printing service providers. This will be discussed
further in the section of materials.

###  3.2 Materials and printing method

Before the discussion of the results of the simulation and experiments on
different joints, it is important to discuss the role of the materials for the
3D printed joints. The choice of material is important since the achievement
of high strains without overcoming the yield strength requires a material with
a high $\sigma y/E$ ratio. In [23], the authors calculated this ratio for
several plastics and metals, and found out that PTFE has a very good $\sigma
y/E$ ratio of 66.7 compared to 7.1 in Aluminium 7075. Here we made a
comparison between the widely used 3D printed methods and materials to find
out the best trade-off. 3D printed materials are vast, materials are evolving,
and the selection of the best material is not the main focus of this article.
For instance, photo-cured polymers can be produced with very different
properties. In this study we compare some of the most used materials.

Table 1 compares the properties of commonly used 3D printed materials selected
from a large 3D printing service provider. As can be seen in Table 1, a
polyamide produced with selective laser sintering (SLS) of a powder has the
best $\sigma y/E$ ratio. In the 3D printing process, not only the material is
important, but also the production method makes a difference in the properties
of the final prototype, since these properties depend on the density of the
object and bonding between the layers. A full comparison of the 3D printed
materials is out of scope of this paper. However, considering that the 3D
printed parts are generally inferior in properties compared to a machined
plastic block, the $\sigma y/E$ value of the polyamide 3D printed parts seems
to be good enough when compared to the list of materials provided in Table 1.
Therefore, we used this material for prototyping of the fingers. We made some
tests to reproduce the joints with ABS with a professional grade fused
deposition modelling (FDM) 3D printer, as well as an StereoLithography (SLA)
printer, Figure 10. However, none of these parts were usable in our work. The
SLA part presents a relatively low maximum yield stress of the material. On
the other hand, the FDM part is low-cost and more available than SLS. However,
our tests to reproduce the joints with the FDM printer were unsuccessful. One
main problem was that the parts made by FDM are usually with many defects, and
these defects are where the joint starts to enter in plastic deformation (some
of these effects are visible in Figure 10). The other problem of the FDM
method is that the produced part is often non isotropic, and the tensile
strength of the prototype differs in different directions. In both cases, the
materials and methods are evolving. However, currently we found out that the
SLS method suits better for this application.

Table 1: Comparison between $\sigma y/E$ ratio of 3D printed materials [26, 27]. Material and processing method |  Young’s modulus $E$ (MPa) |  Yield Strength $\sigma y$ (Mpa) |  $\sigma y/E$ x1000
---|---|---|---
Polyamide PA 2200/SLS |  48 |  1700 |  28
Polyamide PA3200/SLS |  51 |  3200 |  16
Resin example/SLA |  26 |  1100 |  23
Alumide/SLS |  48 |  3800 |  12
ABS/FDM |  36 |  2265 |  16
Figure 10: Finger printed with FDM method (top) and SLA method (bottom). In
the FDM part the visible defects on the joints reduce the actual yield
strength of the joint. The SLA part do not presents the appropriate elastic
properties

###  3.3 Experimental setup

In order to validate the results of the simulations, we 3D printed some
fingers as a test benchmark in order to compare them with the simulation
results. As can be seen in Figure 11, the test model includes a flexible joint
(1), a rigid part (2), two perpendicular bars (3) designed to add a mass to
simulate the deflection and torsion of the joint in different directions, a
sharp tip (4) used for taking the measurements, and a base (5) for fixing the
model. In the experimental setup, we could not simulate/visualize the pure
torsion as it was simulated in FEA because the added mass would cause a
torsion accompanied by a lateral deflection. In any case, this is actually
what happens in the actual grasping, i.e. a tangential force is applied to the
fingers that cause the deflection and the torsion at the same time. Figure 12
shows the experimental setup for all finger joint models considered in this
study.

Figure 11: Model of a finger prepared for experimental evaluation. m1 and m2
show the direction of the applied forces for deflection and torsion
experiments, respectively Figure 12: Experimental setup for measuring the
undesired deflection

##  4 Results and discussion

###  4.1 FEA simulations

A curvature based mesh was used for the FEA simulation. The element size is
determined mathematically by the minimum number of elements that fit in a
hypothetical circle, while taking into account the user specified minimum and
maximum element size. For all models the same values for minimum and maximum
element sizes of 0,6mm and 3,0mm were applied. Due to the differences in joint
geometries, the number of mesh elements is different for each model falling
always in the range between 35000 and 45000 elements.

Table 2 shows the absolute values for the lateral deflections and the
undesired displacement due to twisting for each of the joints. Both values are
obtained by FEA and measured on the tip of each finger, Figure 13. Table 3
shows the absolute value for flexion of each joint based on the FEA. It
presents the ratio between the undesired lateral deflection and desired
displacement (LD/FL), as well as the ratio between the undesired displacement
as a result of twisting (Tor) and the desired flexion (FL). In this study, a
minimum value for the ratio LD/FL and Tor/FL is desirable. The stress
concentration during the flexion of the joints is depicted in Figure 14. The
maximum stress value that occurs in every joint is shown in Table 4.

Table 2: FEA simulation results for the lateral deflection measured after applying a constant force of 4N (F2) and the total displacement on the tip of fingers after applying a torsional torque (M) of 0,12N.m Joint model |  Lateral Deflection (mm) |  Displacement due to torsion (mm)
---|---|---
A |  2,3 |  46,0
B |  3,8 |  26,5
C |  5,0 |  13,4
D |  21,7 |  38,6
E |  7,6 |  19,4
F |  1,1 |  25,7
G |  9,1 |  17,4
H |  11,2 |  22,4
Table 3: FEA simulation results for the ratio LD/FL and the ratio Tor/FL. displacement. Joint model |  Flexion (mm) |  LD/FL |  Tor/FL
---|---|---|---
A |  22,5 |  0,10 |  1,07
B |  23,4 |  0,17 |  1,19
C |  21,2 |  0,23 |  0,63
D |  37,3 |  0,58 |  1,04
E |  27,6 |  0,28 |  0,70
F |  18,2 |  0,06 |  1,07
G |  21,2 |  0,43 |  0,82
H |  30,9 |  0,36 |  0,82
Table 4: FEA simulation results for the maximum stress value on each of the joints as a result of an equal load on each joint for flexion and lateral deflection Joint model |  Flexion (MPa) |  Lateral Deflection (MPa)
---|---|---
A |  24,8 |  18,9
B |  17,6 |  13,8
C |  18,4 |  54,5
D |  29,2 |  95,7
E |  21,7 |  52,5
F |  25,7 |  25,1
G |  15,8 |  42,0
H |  23,6 |  68,1
Figure 13: FEA results for lateral displacement (top) and torsion (bottom). A
refers to the initial position and B to the final position Figure 14: Stress
concentrations from FEA

###  4.2 Experimental tests

Table 5 shows the results of the undesired deflections for two different
masses of $150g$ and $300g$ applied in m1 and m2, Figure 11. To test the
flexion of each joint under the effect of a constant force we measured the
desired displacement with the masses of $40g$ and $80g$. Table 6 shows the
amount of the flexion on each of the joints when a mass of $40g$ and $80g$ is
applied in the direction of m3, Figure 11. Here, higher displacement is
desired since this means that the joint offers less resistance against the
desired flexion. It is also desired to minimize LD/FL and Tor/FL, and to
maximize the displacement due to flexion for a constant force. The values of
LD/FL and Tor/FL for the experimental tests are also presented in Table 6.

Table 5: Experimental results for undesirable deflection Joint model |  Parameter |  m1.1= 150g |  m2.1= 150g |  m1.2=300g |  m2.2=300g
---|---|---|---|---|---
|  |  Lateral deflection |  Displacement due to torsion |  Lateral deflection |  Displacement due to torsion
A |  y, mm |  0,0 |  4,8 |  0,0 |  18,3
|  z, mm |  1,2 |  2,4 |  3,8 |  13,0
|  delta, mm |  1,2 |  5,4 |  3,8 |  22,5
B |  y, mm |  0,0 |  2,9 |  0,0 |  7,3
|  z, mm |  5,2 |  6,4 |  11,3 |  15,0
|  delta, mm |  5,2 |  7,0 |  11,3 |  16,6
C |  y, mm |  0,0 |  -2,2 |  0,0 |  2,2
|  z, mm |  5,9 |  5,8 |  12,3 |  12,5
|  delta, mm |  5,9 |  6,2 |  12,3 |  12,6
D |  y, mm |  0,0 |  -6,7 |  0,0 |  8,9
|  z, mm |  20,0 |  19,7 |  32,0 |  32,9
|  delta, mm |  20,0 |  20,8 |  32,0 |  33,5
E |  y, mm |  0,0 |  1,5 |  0,0 |  14,9
|  z, mm |  9,6 |  9,0 |  15,3 |  2,1
|  delta, mm |  9,6 |  9,1 |  15,3 |  14,1
F |  y, mm |  0,0 |  3,0 |  0,0 |  22,0
|  z, mm |  1,8 |  2,7 |  2,4 |  9,4
|  delta, mm |  1,8 |  4,0 |  2,4 |  24,0
G |  y, mm |  0,0 |  2,7 |  0,0 |  5,4
|  z, mm |  10,4 |  9,4 |  18,4 |  18,2
|  delta, mm |  10,4 |  9,8 |  18,4 |  18,9
Table 6: Joints flection, LD/FL and Tor/FL Joint model | Parameter | Flexion | LD/FL | Tor/FL
---|---|---|---|---
|  | m3.1= 40g | m3.2=80g |  |
A | x, mm | 8 | 4,8 |  |
| z, mm | 10,8 | 18,2 |  |
| delta, mm | 13,4 | 23,6 | 0,16 | 0,85
B | y, mm | 10,4 | 20,2 |  |
| z, mm | 12,1 | 18,3 |  |
| delta, mm | 16,0 | 27,3 | 0,41 | 0,56
C | y, mm | 8,7 | 17,3 |  |
| z, mm | 14,6 | 23,5 |  |
| delta, mm | 17,0 | 29,2 | 0,42 | 0,40
D | y, mm | 17,2 | 27,5 |  |
| z, mm | 15,3 | 19,1 |  |
| delta, mm | 23,0 | 33,5 | 0,96 | 0,95
E | y, mm | 10,4 | 20,8 |  |
| z, mm | 14,9 | 22,0 |  |
| delta, mm | 18,2 | 30,3 | 0,50 | 0,44
F | y, mm | 7,6 | 12,6 |  |
| z, mm | 11,2 | 18,0 |  |
| delta, mm | 13,5 | 22,0 | 0,11 | 0,97
G | y, mm | 7,6 | 14,8 |  |
| z, mm | 13,4 | 20,1 |  |
| delta, mm | 15,4 | 25,0 | 0,74 | 0,71

###  4.3 Discussion

Analysing Table 3 with results from simulations and Table 5 with results from
experimental tests, it can be concluded that joint F expresses the overall
best LD/FL ratio, Figure 15. However, the amount of flexion that it can
achieve is lower than all of the other joints. This is due to the two lateral
supports parallel to the circular joint, which limits the lateral deflections,
but also increases the stiffness against the desired flexion. Therefore, it
does not reflect a good trade-off. On the other hand, joint model D offers the
highest amount of flexion among all joints, both in simulation and
experimental results. However, results of the stress analysis reflected on
table 4 show that it also should bear the highest stress concentration
compared to other models. As can be seen in Table 6, where experimental
results are reflected, joint model C presents the best Tor/FL ratio. This is
closely followed by Joint B and E. However, joint B does not reflect the
relatively good characteristics regarding the torsional stiffness. Looking at
the simulation results of the stress analysis, Table 4 and Figure 14, we can
see that joints C and E are also very similar in terms of the stress
concentration. Both of these joints (C and E) express good trade-offs. In this
case joint C is slightly a better trade-off for the first objective (minimize
undesired deflections - e.g. for the thumb) and joint E is a slightly better
trade-off toward the second objective (minimize joints stiffness against the
desired flexion - e.g. for other digits).

Figure 15: Comparison of LD/FL obtained from experimental tests and FEA
simulation

###  4.4 Hand’s digits

As it was discussed earlier, Ecoflex-30 (smooth-on) was used as the soft skin
that covers the endoskeleton. The low Young’s modulus of Ecoflex has a minimal
effect on the overall stiffness of the joints. In this way, one can optimize
further the geometrical features for the best properties, without concerning
much about the effect of the enveloping polymer. The enveloping polymer keeps
its other important roles on formation of larger contact areas, a high
friction coefficient, and the desired cosmetics.

Figure 16 shows the fabrication method of the fingers with the novel approach.
In contrary to the previous method, Figure 5, here we do not fill the joint
with the sponge prior to casting. Since the enveloping polymer benefits from a
low Young’s modulus, it can fill the joint without having a significant effect
on the joint behaviour. A recent study on the cosmetics gloves for prosthetics
hands considered that silicone gloves are more suitable for application on
articulating hand prosthesis, as they had a lower joint stiffness and required
a lower maximum joint torque [20].

Compared to the previous approaches, the new method is significantly simpler
and less labour intensive, resulting in a more uniform skin that fully
envelops the endoskeleton. In this way a more anthropomorphic look and feel is
achieved. Figure 17 shows a sample prototype of the fingers. As can be seen in
Figure 16 and Figure 17, the mould shape is designed to minimize the amount of
the silicone around the endoskeleton. The reason for this specific shape is to
avoid the buckling effect of the silicone, which does not allow a full flexion
of the joint.

Based on our previous tests with the ISR-Softhand [12], we found out that the
thumb should have the highest stiffness against lateral deflection and
undesired twisting, since it is suffering higher tangential and normal forces
than the other fingers. That is, to grasp a heavy object the thumb should
oppose the force that on the other side is divided by the other 4 fingers. For
the other fingers such stiffness can be lower, in favour of lower required
force for full flexion of the fingers. This is important to reduce the
required force for closing the fingers, thus reducing the size and weight of
the actuators on a prosthetic terminal. Therefore, joint C is the preferred
configuration for the thumb and joint E is the preferred configuration for the
other fingers.

Figure 16: Fabrication process of the finger. The prepared 3D printed
endoskeleton is placed into the mould and filled with silicone resin Figure
17: A sample prototype of the finger

Figure 18 shows an example of application of these joints on a body-actuated
prosthetic hand which was developed for an 8 year old amputee based on the
E-nable community concept. The thumb was selected from the joint type C
(rectangular) and the other 4 fingers were selected from the joint type D
(circular). The dimensions of the joints were then adjusted based on the size
of the hand.

Figure 18: A prosthetic hand developed for an 8 years old amputee using the
fingers with the developed compliant joint. It is based on the E-nable
community designs (actuated by the wrist/elbow movement)

###  4.5 Integrated finger comparison

A sample thumb digit was developed by this method with the same size of the
previous thumb based on elastomeric joint in the ISR-Softhand. On the ISR-
Softhand the joint is made with an elastomer (PMC-780-smooth-on). In the new
version, an endoskeleton with a rectangular geometry was embedded in a
silicone skin. Both thumbs were then compared in terms of required force for
flexion and torsion. Figure 19 shows the setup used for measuring the flexion
and torsion. For flexion, the tendon is pulled by a cable and the bending
angle at the tip of the finger is measured. For torsion, a tangential force is
applied in the fingertip and the bending angle is measured. In both cases the
force is measured by a digital force gauge.

Figure 20 shows the comparison between the two fingers on flexion. The
endoskeleton design in this finger is rectangular which is specially selected
for the thumb. It is not the best option in terms of bending, but it is better
than the circular geometry in terms of lateral deflection. The endoskeleton
finger covered by the silicone requires a higher force in the beginning of the
bending range and a lower force at the end of the bending range. Indeed, the
elastomer finger has a non-linear behaviour, and reaches to a threshold around
60 degrees, after which additional bending of the joint becomes very
insensitive to the increase of the force. The reason for this is that the
elastomer’s bending has a tension component on the upper layer of the
elastomer, and a compression component in the lower layer of the joint, and
after this threshold more compression cannot be achieved. This is also
referred as the “buckling effect”, as this happens in the lower half of the
elastomeric joint

Figure 21 shows the undesired deflection of each of the fingers, as a result
of a tangential force applied at the finger tip. Such force resembles the
undesired deflection and torsion at the same time, and also resembles the
force which is being applied to the finger during the grasping. As can be
seen, the endoskeleton joint behaves considerably better than the PMC
elastomeric joint.

Figure 19: The setup for measuring the flexion force (left) and the undesired
lateral torsion (right) of the two versions of the thumb Figure 20: Angular
displacement of the two types of fingers against the pulling force applied to
the tendon Figure 21: Undesired angular displacement due to torsion for the
two types of fingers. The load is applied tangential to the finger, on its tip

##  5 Conclusion

In this article we introduced a novel method for fabrication of soft fingers
for grasping applications. Compared to the previous methods of fabrication of
soft fingers, this method presents several advantages. This includes easier
fabrication (e.g. one step casting), more anthropomorphic look and a more
uniform soft skin cover for the endoskeleton. In addition, this approach
addresses an important problem in the fabrication of soft fingers. That is, by
optimization of the endoskeleton’s geometry, the undesired lateral deflections
and the undesired twisting are minimized, and furthermore the joint’s
stiffness against the desired flexion can be reduced. The latter is especially
important, since it results in reducing the size of the actuators which
contributes to reduction of the size and weight of the prosthetic terminals.
We also compared several geometries of the flexural joints both by simulation
and experimental results, suggesting that a rectangular geometry for the thumb
and a circular geometry for the other fingers are preferred.

## Acknowledgments

This research work was partially supported by the Portuguese Foundation of
Science and Technology, the Carnegie Mellon-Portugal under contracts CMUP-
EPB/ TIC/ 0036/ 2013 and CMUP-ERI/TIC/0021/2014, and by Portugal 2020 project
POCI-01-0145-FEDER-016418 by UE/FEDER through the program COMPETE2020.

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