*Overall Framework*

The research activity was conducted at the Mechatronics and Dynamic Modeling Laboratory (MDM Lab) of the Department of Industrial Engineering of Florence (DIEF). The MDM Lab has been active in the field of wearable robotics since 2013. In that year, the very first prototype started to be developed. A patient affected by Spinal Muscolar Atrophy (SMA) was the first user of the device, which was specifically developed for his needs and basing on his requirements. This first version of the hand exoskeleton prototype represented a first embodiment of the novel 1-DOF kinematic mechanism architecture which has then been later developed during the following years. In 2016, a collaboration with the Don Carlo Gnocchi Foundation Rehabilitation Center of Florence allowed to enlarge the target of possible users of the device. This scenario demands for the adaptation of the designed robotic system to different patients' hands. Exploiting the Motion Capture (MoCap) system available at the Don Gnocchi Rehabilitation Center, several studies focusing on the hand kinematics were carried out and a new Acrylonitrile Butadiene Styrene (ABS) exoskeleton was developed in accordance with the necessity of tailoring different fingers gestures. Currently, the collaboration with the Don Gnocchi Foundation deals with the study of innovative control strategies for hand exoskeleton systems based on surface ElectroMyoGrapchic (sEMG) signals. Preliminary studies have been successfully concluded and some patients have already been enrolled for the testing campaign, which is about to start. At the time of writing, two projects are ongoing: HOLD, funded by the University of Florence and BMIFOCUS, funded by the Tuscany region.

## **2. Kinematic Architecture**

Assistive robotic devices are, in general, made of both mechanical parts and electronics (e.g., sensors, power supply circuits, micro-processors and motors). They need thus to be carefully controlled in order to provide an intuitive and safe utilization. Achieving a smooth, comfortable, and robust control is a requirement that has to be kept in mind since the very beginning of the whole design process. The accurate development of a novel mechanism, characterized by a single DOF per finger allowed to precisely and comfortably reproduce the complex hand kinematics. Exploiting a single-DOF mechanism per finger granted for the control of only one variable (per finger) and resulted in the exploitation of less sensors and in the reduction of the computational burden.

The overall architecture of the system is split into two parts: a fixed frame, integral with the back of the hand, which houses motors and electronics, and four mobile finger mechanisms which act on the four long fingers. Motion and forces are transferred from the motors to the fingers by means of a cable transmission. An in-depth analysis of the kinematics of the single-DOF finger mechanism is presented in this paper for the first time and detailed in the following. For the sake of brevity, what reported below is related to just one finger mechanism, but the same analysis can be applied to all long fingers mechanisms as well.

Figure 1 shows the single-DOF kinematic chain exploited to move each long finger: The center of the reference system *<sup>x</sup>*1*y*1 related to the body A is fixed to the hand, roughly right above the MetaCarpoPhalangeal (MCP) joint. The other reference systems *<sup>x</sup>*2*y*2, *<sup>x</sup>*3*y*3, *<sup>x</sup>*4*y*4, *<sup>x</sup>*5*y*5 and *<sup>x</sup>*6*y*6 are integral with the bodies C, B, E, D and F. To simplify the notation each reference frame will also be related to a specific joint following the numerical progression (e.g., joint 1 is related to frame *<sup>x</sup>*1*y*1, joint 2 to frame *<sup>x</sup>*2*y*2 and so on). Component F is a thimble which has been added in the first version of the presented hand exoskeleton whose presence does not modify the 1-DOF kinematic chain of the device. For this reason, even if the thimble introduces a second connection point with the hand, this will not be considered a proper end-effector and the attention will mainly focus on component E.

**Figure 1.** The figure shows the kinematic chain of the finger mechanism exploited in the presented work. While component A is integral with the back of the hand, the other parts are all mobile and their pose is uniquely identified once known the joint coordinate of joint 1.

The forward kinematics equations of the mechanism can be obtained starting from the revolute constraints, identifying rotational joints, in *Ox*1*y*1, *Ox*2*y*2, *Ox*4*y*4:

$$\mathbf{0} = \mathbf{^1p}\_2 + \mathbf{R^1}\_2 \mathbf{^2p}\_1 \tag{1}$$

$$\mathbf{p}^1 \mathbf{p}\_2 = \mathbf{^1p\_3} + \mathbf{R\_3^1} \mathbf{^3p\_2} \tag{2}$$

$$\mathbf{p}^1 \mathbf{p}\_4 = \mathbf{^1p}\_3 + \mathbf{R}\_3^{1\ 3} \mathbf{p}\_4. \tag{3}$$

where, referring to Figure 1 and according to the mathematical notations reported in [6], the position of the origin of the *i*-frame with respect to *j*-frame has been denoted by the vector *<sup>j</sup>***p***i* = *j pxi j pyi T* ∈ R<sup>2</sup> (the component on **zi** axis has been omitted as the proposed mechanism acts on a plane) and **R***ji* represents the orientation of *i*-frame with respect to *j*-frame, which, in this case, results in a rotation about **zi**axis through an angle *αi*.

By analyzing the two cylindrical joints 3 and 5, the constraints equations are:

$$a\_1^{-1}p\_3^x + b\_1^{-1}p\_3^y + c\_1 = 0\tag{4}$$

$$a\_2 \,^4 p\_5^x + b\_2 \,^4 p\_5^y + c\_2 = 0 \tag{5}$$

where 
$$\mathbf{^1p\_5} = \mathbf{^1p\_2} + \mathbf{^2l\_2} \mathbf{^2p\_5} \tag{6}$$

and

$$\mathbf{p}^1 \mathbf{p}\_5 = \mathbf{^1p}\_4 + \mathbf{R\_4}^1 \mathbf{^4p\_5}.\tag{7}$$

In Equations (4) and (5), *a*1, *b*1, *c*1 and *a*2, *b*2, *c*2 represent the two linear constraints of the mechanism and Equations (6) and (7) have been obtained considering the rotational joint in 5. Finally,

even if it does not alter the kinematics of the device, an additional reference system (i.e., *<sup>x</sup>*6*y*6) has been considered in the kinematic synthesis.

$$\mathbf{^1p\_6} = \mathbf{^1p\_4} + \mathbf{R\_4^1} \mathbf{^4p\_6}.\tag{8}$$

Referring to Equations (1)–(8), the state of the system is represented by the vector

$$\mathbf{q} = \begin{bmatrix} ^1\mathbf{p}\_2^T \, ^1\mathbf{p}\_3^T \, ^1\mathbf{p}\_4^T \, ^1\mathbf{p}\_5^T \, ^1\mathbf{p}\_6^T \, ^1\mathbf{p}\_2 \, ^1\boldsymbol{a}\_2 \, \boldsymbol{a}\_3 \, \boldsymbol{a}\_4 \end{bmatrix}^T \in \mathbb{R}^{13} \tag{9}$$

and depends on the control variable *α*2. The unknowns representing the state of the system can be thus calculated as a function of only *α*2 by solving Equations (1)–(8). All the interesting points of the mechanism (included in the state vector **q**) are in fact completely described as functions of the angle *α*2 and of the geometrical parameters **S** ∈ R16:

$$\mathbf{S} = \begin{bmatrix} ^2 \mathbf{p}\_1^T \, ^3 \mathbf{p}\_2^T \, ^2 \mathbf{p}\_5^T \, ^3 \mathbf{p}\_4^T \, ^4 \mathbf{p}\_6^T \, ^4 \mathbf{p}\_1 \, b\_1, b\_1, c\_1, a\_2, b\_2, c\_2 \end{bmatrix}^T. \tag{10}$$

All these parameters are completely known because they represent geometric quantities, depending only on the design of the exoskeleton parts. Consequently, it is possible to solve the extended direct kinematic model **q˜** = **<sup>f</sup>**(*<sup>α</sup>*2, **S**) ∈ R<sup>12</sup> (see Equation (11)) of the mechanism writing a function of *α*2 and **S**, where **q˜** is the unknown part of the state vector **q**:

$$\mathbf{\tilde{q}} = [^1p\_{2'}^ {x} \; ^1p\_{2'}^ {y} \; ^1p\_{3'}^ {x} \; ^1p\_{3'}^ {y} \; \_ {a}a\_{3'}^ {1} \; ^1p\_{4'}^ {x} \; ^1p\_{4'}^ {y} \; ^1p\_{5'}^ {x} \; ^ 1p\_{5'}^ {y} \; ^ 1p\_{6'}^ {x} \; ^ 1p\_6^ {y}]^T = \mathbf{f}(a\_{2'}, \mathbf{S}).\tag{11}$$

The closed form resolution of the aforementioned forward kinematic is given hereinafter. Each component of vector **q˜** is highlighted in blue when it is solved in terms of only *α*2 and elements of **S**. Starting from Equation (1), it is possible to obtain Equations (12) and (13):

$$\mathbf{p}\_2^x p\_2^x(\mathbf{a}\_2) = -\left(c\mathbf{a}\_2 \cdot ^2 p\_1^x - s\mathbf{a}\_2 \cdot ^2 p\_1^y\right) \tag{12}$$

$$\mathbf{p}\_2^{\mathbf{y}} p\_2^{\mathbf{y}}(\mathfrak{a}\_2) = -\mathfrak{ca}\_2 \cdot \mathbf{^2} p\_1^{\mathbf{y}} - \mathfrak{sa}\_2 \cdot \mathbf{^2} p\_1^{\mathbf{x}}.\tag{13}$$

From Equations (2) and (4), the following equations can be written:

$$\begin{cases} \;^1p\_2^x - ^1p\_3^x = c a\_3 \cdot ^3p\_2^x - s a\_3 \cdot ^3p\_2^y\\ \;^1p\_2^y - ^1p\_3^y = c a\_3 \cdot ^3p\_2^y + s a\_3 \cdot ^3p\_2^x\\ a\_1 \cdot ^1p\_3^x + b\_1 \cdot ^1p\_3^y + c\_1 = 0. \end{cases} \tag{14}$$

and solving the system:

$$\mathbf{p}^1 p\_3^y(a\_2) = \frac{-\begin{pmatrix} \frac{1}{2} p\_2^x \frac{b\_1}{a\_1} + \frac{b\_1 \cdot c\_1}{a\_1} - 1 \cdot p\_2^y\\ \left(\frac{b\_1}{a\_1}\right)^2 + 1 \end{pmatrix}}{\left(\frac{b\_1}{a\_1}\right)^2 + 1} + \frac{\sqrt{\begin{pmatrix} \frac{1}{2} p\_2^x \frac{b\_1}{a\_1} + \frac{b\_1 \cdot c\_1}{a\_1} - 1 \cdot p\_2^y\\ \left(\frac{b\_1}{a\_1}\right)^2 + 1 \end{pmatrix} \cdot H}}{\left(\frac{b\_1}{a\_1}\right)^2 + 1} \tag{15}$$

$$\mathbf{p}\_3^T p\_3^x(\mathbf{a}\_2) = -\frac{1}{a\_1} \cdot \left( b\_1 \cdot {}^1 p\_3^y + c\_1 \right) \tag{16}$$

where

$$H = \left(^{1}p\_{2}^{x}\right)^{2} + \left(^{1}p\_{2}^{y}\right)^{2} - \left(^{3}p\_{2}^{x}\right)^{2} - \left(^{3}p\_{2}^{y}\right)^{2} + 2 \cdot ^{1}p\_{2}^{x} \cdot \frac{b\_{1}}{a\_{1}} + \left(\frac{c\_{1}}{a\_{1}}\right)^{2} \cdot \tag{17}$$

and depends only on known values. Now *α*3 can be computed as:

$$a\_3(a\_2) = a \tan 2 \left( \frac{\left(-\:^1p\_2^x + ^1p\_3^x + \:^3\_{\overline{p}\_2^y} \colon ^1p\_2^y - \:^3\_{\overline{p}\_2^y} \colon ^1p\_3^y\right)}{\:^3p\_2^y + \:^3\_{\overline{p}\_2^y} \colon ^3p\_2^y} , \frac{^1p\_2^y - \:^3\_{\overline{p}\_2^y} \colon ^3p\_2^y - \:^1p\_3^y}{\:^3p\_2^y} \right) . \tag{18}$$

At this point, Equations (19) and (20) can be written from Equation (3):

$$\mathbf{p}^1 p\_4^x(\alpha\_2) = \,^1 p\_3^x + c\mathbf{a}\_3 \cdot \,^3 p\_4^x - s\mathbf{a}\_3 \cdot \,^3 p\_4^y \tag{19}$$

$$\mathbf{p}^1 p^y\_4(\mathbf{a}\_2) = \,^1 p^y\_3 + c\mathbf{a}\_3 \cdot \,^3 p^y\_4 + s\mathbf{a}\_3 \cdot \,^3 p^x\_4. \tag{20}$$

From Equation (6), one can ge<sup>t</sup>

$$\mathbf{p}\_5^x p\_5^x(\mathbf{a}\_2) = \,^1 p\_2^x + \mathbf{c} \mathbf{a}\_2 \cdot \,^2 p\_5^x - \mathbf{s} \mathbf{a}\_2 \cdot \,^2 p\_5^y \tag{21}$$

$$\mathbf{p}^1 p^y\_5(\mathfrak{a}\_2) = \mathbf{^1p^y\_2} + \mathfrak{a}\mathfrak{a}\_2 \cdot \mathbf{^2p^y\_5} + \mathfrak{a}\mathfrak{a}\_2 \cdot \mathbf{^2p^x\_5} \tag{22}$$

and from Equations (5) and (7):

$$\begin{cases} \;^1p\_5^x - \;^1p\_4^x = c\boldsymbol{\alpha}\_4 \cdot \!^4p\_5^x - s\boldsymbol{\alpha}\_4 \cdot \!^4p\_5^y\\ \;^1p\_5^y - \;^1p\_4^y = c\boldsymbol{\alpha}\_4 \cdot \!^4p\_5^y + s\boldsymbol{\alpha}\_4 \cdot \!^4p\_5^x\\ \;^2\boldsymbol{\alpha}\_2 \cdot \!^4p\_5^x + \boldsymbol{b}\_2 \cdot \!^4p\_5^y + c\_2 = 0. \end{cases} \tag{23}$$

As in the previous cases, it can be obtained

$$p\_5^{\text{v}}(a\_2) = \frac{-\left(b\_2 \cdot c\_2\right) + \sqrt{\left(b\_2 \cdot c\_2\right)^2 - \left(b\_2^{\text{v}} + a\_2^{\text{v}}\right) \cdot T}}{b\_2^{\text{v}} + a\_2^{\text{v}}} \tag{24}$$

$$\mathbf{p}^4 p\_5^x(a\_2) = -\frac{1}{a\_2} \cdot \left( b\_2 \cdot {}^4 p\_5^y + c\_2 \right) \tag{25}$$

where:

$$T = -a\_2^{-2} \cdot \left[ \left(^1 p\_5^x - ^1 p\_4^x\right)^2 + \left(^1 p\_5^y - ^1 p\_4^y\right)^2 \right] + c\_2^{-2}.\tag{26}$$

*α*4 is now calculated as:

$$\log\_4(a\_2) = \operatorname{atan2}\left(\frac{\left(-{}^{1}p\_4^y + {}^{1}p\_5^y - \frac{4}{4}\frac{p\_5^y}{p\_5^y}}{{}^{1}p\_5^y}\cdot {}^{1}p\_5^y + \frac{4}{4}\frac{p\_5^y}{p\_5^y}\cdot {}^{1}p\_4^x}{{}^{1}p\_5^x}\right)}{-{}^{4}p\_5^x + \frac{{}^{4}p\_5^y}{{}^{4}p\_5^y}}\right) \;. \tag{27}$$

Finally, <sup>1</sup>**p**6 results:

$$\int \prescript{1}{}{p\_6^x}(a\_2) = \prescript{1}{}{p\_4^x} + c\kappa\_4 \cdot \prescript{4}{}{p\_6^x} - s\kappa\_4 \cdot \prescript{4}{}{p\_6^y} \tag{28}$$

$$\mathbf{p}^1 p\_6^y(a\_2) = \,^1 p\_4^y + \alpha \mathbf{a}\_4 \cdot \,^4 p\_6^y + s \mathbf{a}\_4 \cdot \,^4 p\_6^x \tag{29}$$

At this point, the motion is completely described and the positions of every joint relative to the 1-frame is given. Figure 1 also shows the trajectories of the finger mechanism joints, providing a qualitative overview of the resulted kinematics of the mechanism when fingers are actuated.

#### **3. First Prototype: Kinematic Validation**

A first version of the hand exoskeleton (shown Figure 2 and detailed in [34,35]) has been designed and manufactured to test the embodiment of the kinematic model proposed in Section 2. A patient affected by SMA was the first user of the device, specifically developed for him.

**Figure 2.** The figure shows, on the left, the first version of the hand exoskeleton prototype worn by the patient and, on the right, the corresponding kinematic chain and Computer Aided Manufacturing (CAD) model. Colors and names (capital letters) of the components, and joints enumeration are reported as introduced in Section 2.

## *3.1. Mechanical Design*

All the mechanical parts have been 3D-printed using a Dimension Elite by Stratasys in ABS thermoplastic polymer. One of the main advantages of 3D-printing technology is that it allows to manufacture components without considering technological constraints due to the particular production method as well as it may happen using subtracting processes. Then, among different 3D-printable materials, ABS has been chosen because it represents a satisfying trade-off between good mechanical characteristics, lightness, and cheapness.

The design process was composed of three sequential phases. Firstly, 2D trajectories of the index finger of the specific user were acquired exploiting open source software Kinovea, https://www.kinovea.org/ (accessed on 26 March 2019). Secondly, an optimization MATLAB-based algorithm [36] minimized the constrained nonlinear multi-variable function describing the kinematics of the mechanism, modifying its geometrical parameters and leading the mechanism kinematics to fit the acquired trajectories. Thirdly, a scaling phase allowed to adapt the kinematic model to each patient's finger by resizing the geometry of the index finger mechanism accordingly to the dimensions of the other fingers. Then, once the mechanism features was defined, virtual tests exploiting SolidWorks Motion Simulation tools have then been carried out to assess, before the manufacturing of the device, the hand-exoskeleton kinematic coupling and interaction in simple opening and closing gestures. Finally, right before the manufacturing, the mechanical parts have undergone a further reshaping process which has changed their structure, but which has not changed their overall kinematics (as shown in Figure 2, the shapes of the real parts differ from the straight lines of the kinematic model, but the position of the joints has remained unchanged). This later step was required to avoid possible interpenetrations with the hand and, as described in the following lines, for reasons of mounting and loading conditions.

In particular, component C was split in two parts guaranteeing a symmetric load configuration during the use of the exoskeleton and obtaining a more stable solution. Components D was made in two parts as well, allowing them to be assembled together. Component E, which represents the hand-exoskeleton interface, has been designed to wrap only the back side of the finger phalanx not to reduce the sense of touch, while a Velcro held the finger tight achieving a solid connection. To reduce the lateral encumbrances of each mechanism, pins and shafts were directly integrated in the ABS components as lateral rods. All the aforementioned modifications are graphically reported in Figure 3.

**Figure 3.** The figure shows the adaptation of components C, F, and E.

#### *3.2. Actuation System and Control Strategy*

As visible in Figure 2, both the transmission and the actuation system are placed on the hand backside, as well as the mechanisms are positioned on the fingers, and they do not impede objects handling. The reduction of the total mass, which was one of the main requirements of the device, has led to the choice of high power density actuators to be directly mounted on the back of the hand.

Four Savox SH-0254 servomotors, http://www.savoxusa.com (accessed on 26 March 2019), one per long finger, have been selected for their characteristics: maximum torque of 0.38 Nm and maximum angular speed of 7.69 rad/s at 6.0 V, with a size of 22.8 × 12 × 29.4 mm and weight of 16 g. These motors have been modified to allow for the continuous rotation of the shaft despite the resulting loss of position feedbacks. The four servomotors are in charge of opening the fingers at the same time by pulling cables which have two connection points on each finger mechanism. Closing gesture is, instead, passively allowed releasing the same cables.

The control unit was based on a 6-channels MicroMaestro control board, https://www.pololu.com (accessed 26 March 2019) which has been chosen for its cheapness, its lightness (only 5 g), its small dimensions (21 × 30 mm ) and, above all, because its six channel matched the number of external devices that had to be connected to the board: the aforementioned four servomotors and two buttons, one for opening and one for closure triggering action. The control unit and the actuators were powered by a compact 4-cell Lithium battery (at 6.0 V), which was placed in an elastic band on the arm of the user, provided with a safety switch close to the buttons case, mounted instead on the forearm.

Regarding the control strategy, the system was controlled by a simple script, stored and running directly on the MicroMaestro chip-set. The code had to continuously check for one of the two buttons to be pushed and held down and then react by sending the corresponding command to the actuators. This version did not include sensors for fingers position feedback and the bounds of the exoskeleton range of motion were manually managed by the user (keeping pushed or releasing the buttons) in order not to overcome his anatomical limits. Even thought there was one motor per finger, the possibility to move each of them independently from the others has not been considered for simplicity and all the long fingers were moved together.
