*Article* **A Planar Parallel Device for Neurorehabilitation**

### **Jawad Yamine 1, Alessio Prini 2, Matteo Lavit Nicora 2, Tito Dinon 2, Hermes Giberti <sup>3</sup> and Matteo Malosio 2,\***


Received: 15 October 2020; Accepted: 27 November 2020; Published: 3 December 2020

**Abstract:** The patient population needing physical rehabilitation in the upper extremity is constantly increasing. Robotic devices have the potential to address this problem, however most of the rehabilitation robots are technically advanced and mainly designed for clinical use. This paper presents the development of an affordable device for upper-limb neurorehabilitation designed for home use. The device is based on a 2-DOF five-bar parallel kinematic mechanism. The prototype has been designed so that it can be bound on one side of a table with a clamp. A kinematic optimization was performed on the length of the links of the manipulator in order to provide the optimum kinematic behaviour within the desired workspace. The mechanical structure was developed, and a 3D-printed prototype was assembled. The prototype embeds two single-point load cells to measure the force exchanged with the patient. Rehabilitation-specific control algorithms are described and tested. Finally, an experimental procedure is performed in order to validate the accuracy of the position measurements. The assessment confirms an acceptable level of performance with respect to the requirements of the application under analysis.

**Keywords:** parallel kinematic architecture; kinematic optimization; rehabilitation robotics; assist-as-needed control algorithms

### **1. Introduction**

Stroke is one of the main causes of long-term disability worldwide and the most common in Western countries [1]. The number of patients having difficulties in performing daily-living activities due to physical disabilities is constantly increasing, making the availability of therapists and caregivers more and more inadequate and, therefore, creating an unmet market need.

Robotic devices for neurorehabilitation have been widely investigated, developed and introduced in the market to offer a valid alternative to conventional therapy and fill the constantly growing gap between supply and demand [2,3]. Since the invention of the MIT-Manus [4], robot-assistance, force-feedback and force-based control are sought after features of neurorehabilitation devices [5], enabling them to sense the patient's interaction with the robot, react accordingly and adapt the level of physical assistance provided. Most of the proposed robots are technically advanced, but are relatively expensive and designed for clinical settings, which makes it hard for patients to afford such treatment. There are also examples of commercial general-purpose industrial manipulators, properly equipped with force-based control algorithms, exploited in rehabilitation scenarios [6–8]. They can be very flexible and useful for testing purposes but, on the other side, they are inherently relatively expensive with respect to rehabilitation budget requirements.

Focusing on upper-limb rehabilitation, a number of low-cost rehabilitation devices are currently available, typically passive or passively gravity-balanced [9]. Nevertheless, the lack of actuation, of an assist-as-needed support and of haptic capabilities, preclude them to be effectively used by patients with low/medium motion capabilities. The development of low-cost rehabilitation devices also meets the need of low-income countries where the healthcare system is lacking and the medical personnel is insufficient. In these countries, where even hospitals cannot afford expensive mechanical devices, the challenge is to conceive and develop low-cost and easily-replicable systems for rehabilitation, as far as possible.

Some tabletop actuated devices have been specifically developed with the aim of satisfying economic and installation requirements in out-of-clinic environments. These solutions often rely on reduced complexity and optimized costs by limiting the number of degrees of freedom with respect to complex rehabilitation devices, such as exoskeletons [10–12], in order to partially meet the affordability requirements. However, strictly reducing the number of degrees of freedom of exoskeletons can sometimes lead to drawbacks. The authors of [13] developed an interesting elbow rehabilitation device; but, since the architecture is not supported or constrained to a fixed structure, the device weighs on the shoulder of the patient with a consequent lack of rehabilitation for that specific body part.

The large majority of tabletop devices are constituted by rigid links and joints. Nevertheless, it is worth to mention the existence of alternative solutions. CUBE is a tabletop cable-driven device enabling 3D-movements of the upper limb [14]. Despite its peculiar and interesting kinematic architecture, it does not provide a steady support for the hand in spatial movements since its end-effector is constrained only by two groups of three wires. MOTORE is an interesting mobile robot for upper-limb rehabilitation, but the need of resting completely the forearm on the device can constrain the upper arm excessively and lead to a high elevation angle of the elbow [15].

Alternative solutions can mobilize the upper arm for specific movements, but do not allow a wide movement of the upper limb, both in terms of shoulder and elbow. For instance, Nam et al. developed a portable device, capable of mobilizing the pronosupination of the forearm, unusual capability for tabletop devices [16]. However, its kinematic structure does not allow the rehabilitation of the upper limb in an extensive range of motion, since it does not provide any mobilization of the shoulder.

Moreover, planar movements are largely used for upper-limb neurorehabilitation and they represent the basis for interesting works such as the one proposed by Zadravec et al., in which a solution to model the planar movement trajectory formation [17] is suggested. By referring to articulated kinematic structures, it is possible to highlight a characteristic shared among different devices. The human body is inherently symmetric with respect to the sagittal plane. Nevertheless, several devices are characterized by a non-symmetric structure that could cause kinetostatic performance and manipulability ellipses to also be asymmetric with respect the sagittal plane. Asymmetrical kinematic structures produce asymmetrical shapes of the manipulability ellipsis, leading to an asymmetric kinetostatic behaviour for right-handed and left-handed patients. This is true for the kinematic structure of several rehabilitation devices, such as MIT-Manus [4], Braccio di Ferro [18] and NURSE [19].

Focusing on symmetrical kinetostatic behaviour with respect to the sagittal plane, some paradigmatic devices can be found. Some of them exploit Cartesian kinematic architectures, both serial and parallel. Wu et al. developed an admittance-controlled Cartesian serial kinematic architecture [20], while Zollo et al. proposed a planar orthogonal parallel rehabilitation device [21]. Both these devices are characterized by an inherent isotropic kinetostatic behaviour. However, in the opinion of the authors, such architectures are relatively cumbersome and complex and would not allow an effective commercial exploitation, especially in low-budget rehabilitation scenarios. An additional solution is provided by the H-MAN [22], a differential-based isotropic planar device for upper-limb rehabilitation. Although the authors consider its design outstanding, the goal of this work was to develop a device able to exploit extensively the range of motion of the upper limb, without leading to a relatively bulky structure. In these terms, the notable architecture of H-MAN would have resulted in

a big and not straightforwardly portable device if properly scaled to allow large movements of the upper limb, mainly because of its Cartesian structure.

The aim of the present work was to present a rehabilitation device, namely PLANarm2, developed specifically to achieve an acceptable compromise in terms of (1) workspace symmetry with respect to the sagittal plane, (2) relatively large workspace, (3) portability and (4) affordability (Figure 1). The well-known 5R planar kinematic chain was considered a promising solution [23]. It is a matter of fact that this architecture has already been adopted to realize the haptic device developed by Klein et al. [24]. Starting from the parametric model of the 5R kinematics, link lengths of PLANarm2 have been optimized to have good kinematic performances in the large majority of its workspace, properly dimensioned to overlap the range of motion of the upper limb. Its symmetric kinematic structure is inherently characterized by a symmetrically distributed kinetostatic behaviour with respect to the sagittal plane. Moreover, in order to reduce the total cost of the device, it has been designed to be clamped quite easily on a standard table and to facilitate both portability and fast installation inside already furnished environments. As opposed to the device described in [24], which is characterized by a self-supported manipulandum, the PLANarm2 manipulandum slides on a table or an a desk, whose surface supports the gravitational load. The links of the parallel structure only transmit horizontal forces, limiting bending loads. This allowed the device to be realized by additive manufacturing techniques with plastic material, in line with the affordability requirement.

The paper is organized as follows: the kinematic architecture is presented in Section 2; the mechanical design and its optimization are described in Section 3; the main components of the prototype and a brief cost analysis are reported in Section 4; the control framework is presented in Section 5; results of an experimental assessment are outlined in Section 6; conclusions are drawn in Section 7.

**Figure 1.** 3D model of the prototype.

### **2. Kinematics**

The forward and inverse kinematics presented in this section were developed to provide a less general but more efficient formulation than the one in [25]. The model proposed in the mentioned work provides the solution to the inverse kinematics problem for each of the four a possible configurations depicted in Figure 2. In addition, the forward kinematic problem leads to two solutions, one for the up-configuration and one for the down-configuration. In order to reduce the computational burden, the model presented in the following pages has been developed specifically for the configuration of interest, which is configuration (a) in the up-configuration.

**Figure 2.** Four configurations of the planar 5R parallel architecture. (**a**): "+ −", (**b**): "− −", (**c**): "− +", (**d**): "+ −", by denoting the convex (+) or the concave (−) configuration of the left and the right elbow joints, respectively.

### *2.1. Architecture*

The device described in this paper is a 2-DOF parallel kinematic manipulator. It is characterized by a structure made up of four links and a fixed frame connected by five revolute joints. The main reason for the choice of this kind of closed-loop architecture was the possibility of placing both motors on a fixed base. Thanks to this solution, the robot is characterized by a relatively high stiffness and lower moving masses if compared to serial manipulators, therefore providing higher dynamic performances, a lighter structure and, potentially, better positioning accuracy.

With reference to the generic planar parallel five-bar mechanism depicted in Figure 3, the end-effector *P*(*x*, *y*) is connected to the base by two legs, each of which consists of three revolute joints and two links. Joints *A*<sup>1</sup> and *A*<sup>2</sup> are connected to the base where they are actuated. The joints at the other end of each actuated link are denoted as *B*<sup>1</sup> and *B*2. A fixed global reference system *O* − *xy* is located in the midpoint of the segment *A*1*A*<sup>2</sup> with the *y* axis normal to *A*1*A*<sup>2</sup> and the *x* axis directed along *A*1*A*2. The mechanism is characterized by a symmetric structure where *OA*<sup>1</sup> = *OA*<sup>2</sup> = *R*3(*r*3), *A*1*B*<sup>1</sup> = *A*2*B*<sup>2</sup> = *R*1(*r*1) and *B*1*P* = *B*2*P* = *R*2(*r*2). The notation *Ri*(*i* = 1, 2, 3) represents the link lengths with dimensions while *ri*(*i* = 1, 2, 3) represents dimensionless lengths of the links. Given:

$$D = \frac{R\_1 + R\_2 + R\_3}{3} \tag{1}$$

One can obtain the three non-dimensional parameters:

$$r\_1 = R\_1/D, \quad r\_2 = R\_2/D, \quad r\_3 = R\_3/D \tag{2}$$

**Figure 3.** The planar 5R parallel mechanism.

It is important to stress the fact that such an architecture is characterized by four possible configurations "*a*, *b*, *c* and *d*", as shown in Figure 2. However, only configuration "*a*" will be considered in the scope of this paper. Moreover, on the basis of the singularity analysis done in [25], the following constraints must be applied:


### *2.2. Inverse Kinematics*

The joint variables *θ* = [*θ*1, *θ*2] *<sup>T</sup>* are expressed as a function of the end-effector position *P* = [*x*, *y*] *T* using the following inverse kinematic equations:

$$
\begin{bmatrix} \theta\_1\\ \theta\_2 \end{bmatrix} = \begin{bmatrix} a\_1 + \arccos\left(\frac{r\_1^2 - r\_2^2 + (\sqrt{(x - r\_3)^2 + y^2})^2}{2r\_1\sqrt{(x - r\_3)^2 + y^2}}\right) \\ a\_2 - \arccos\left(\frac{r\_1^2 - r\_2^2 + (\sqrt{(x + r\_3)^2 + y^2})^2}{2r\_1\sqrt{(x + r\_3)^2 + y^2}}\right) \end{bmatrix} \tag{3}
$$

where

$$\mu\_1 = \begin{cases} \arctan\left(\frac{y}{x - r\_3}\right) & \text{if } \arctan\left(\frac{y}{x - r\_3}\right) \ge 0\\ \arctan\left(\frac{y}{x - r\_3}\right) + \pi\_\prime & \text{otherwise} \end{cases} \tag{4}$$

$$\mu\_2 = \begin{cases} \arctan\left(\frac{y}{x+r\_3}\right) & \text{if } \arctan\left(\frac{y}{x+r\_3}\right) \ge 0\\ \arctan\left(\frac{y}{x+r\_3}\right) + \pi\_\prime & \text{otherwise} \end{cases} \tag{5}$$

### *2.3. Forward Kinematics*

The forward kinematic relations are derived using the variables described in Figure 4. Regarding the notation, *m* is the midpoint of segment *B*1*B*2, *β* is the angle between segment *B*1*B*<sup>2</sup> and the x-axis, *d* is the distance between the end-effector *P* and segment *B*1*B*2, *a* represents the distance from *m* to *B*<sup>2</sup> and, finally, *γ* is the angle between segment *B*1*B*<sup>2</sup> and *B*2*P*.

**Figure 4.** Forward kinematics scheme for up-configuration.

The end-effector position can be calculated as:

$$\mathbf{P} = \begin{bmatrix} m\_x + d \cos \left( \frac{\pi}{2} + \beta \right) \\ m\_y + d \sin \left( \frac{\pi}{2} + \beta \right) \end{bmatrix} \tag{6}$$

where:

$$m\_x = \frac{r\_1(\cos(\theta\_1) + \cos(\theta\_2))}{2} \tag{7}$$

$$m\_{\mathcal{Y}} = \frac{r\_1 \left(\sin(\theta\_1) + \sin(\theta\_2)\right)}{2} \tag{8}$$

$$d = r\_2 \sin(\cos^{-1}(\frac{\sqrt{(r\_1(\cos(\theta\_1) - \cos(\theta\_2)) - 2r\_3)^2 + (r\_1(\sin(\theta\_1) - \sin(\theta\_2))^2}}{2r\_2})) \tag{9}$$

$$\beta = \arctan\left(\frac{r\_1(\sin(\theta\_1) - \sin(\theta\_2))}{(r\_1(\cos(\theta\_1) - \cos(\theta\_2)) - 2r\_3)}\right) \tag{10}$$

### *2.4. Jacobian*

Differentiating Equation (6) with respect to time and rearranging the terms one can obtain:

$$\mathbf{J} = \begin{bmatrix} J\_{11} & J\_{12} \\ J\_{21} & J\_{22} \end{bmatrix} \tag{11}$$

where:

$$J\_{11} = -\frac{r\_1 s\_1}{2} - 2\cos(\mathcal{C}) \frac{Ar\_1 c\_1 - Br\_1 s\_1}{D\_{\\_}} - \frac{E}{B\_{\\_}^2} (Br\_1 c\_1 + Ar\_1 s\_1) \tag{12}$$

$$J\_{12} = -\frac{r\_1 s\_2}{2} + 2\cos(\mathcal{C}) \frac{Ar\_1 c\_2 + Br\_1 s\_2}{D} - \frac{E}{B^2} (-Br\_1 c\_2 + Ar\_1 s\_2) \tag{13}$$

$$J\_{21} = -\frac{r\_1 c\_1}{2} - 2\sin(\mathcal{C}) \frac{Ar\_1 c\_1 - Br\_1 s\_1}{D} + \frac{E'}{B^2} (Br\_1 c\_1 + Ar\_1 s\_1) \tag{14}$$

$$J\_{22} = -\frac{r\_1 c\_2}{2} + 2\sin(\mathcal{C})\frac{Ar\_1 c\_2 + Br\_1 s\_2}{D} + \frac{E'}{B^2}(-Br\_1 c\_2 + Ar\_1 s\_2) \tag{15}$$

in which:

$$s\_i = \sin(\theta\_i), \quad i = 1, 2 \tag{16}$$

$$c\_i = \cos(\theta\_i), \quad i = 1, 2 \tag{17}$$

$$A = r\_1 s\_1 - r\_1 s\_2 \tag{18}$$

$$B = 2r\_3 + r\_1c\_1 + r\_1c\_2\tag{19}$$

$$C = \frac{\pi}{2} + \underbrace{\tan^{-1}(\frac{A}{B})}\_{\overbrace{1}^{\circ} \qquad \tag{20}$$

$$D = 8r\_2 \sqrt{1 - \frac{B^2 + A^2}{4r\_2^2}}\tag{21}$$

$$E = \frac{r\_2 \sin(\mathcal{C})}{1 + \frac{A^2}{B^2}} \sqrt{1 - \frac{A^2 + B^2}{4r\_2^2}}\tag{22}$$

$$E' = \frac{r\_2 \cos(\mathcal{C})}{1 + \frac{A^2}{B^2}} \sqrt{1 - \frac{A^2 + B^2}{4r\_2^2}}\tag{23}$$

### **3. Mechanical Design**

### *3.1. Workspace*

The theoretical reachable workspace for upper arm neurorehabilitation in Cartesian coordinates was defined in [26] through a transformation from articular to Cartesian space, performed using the direct kinematics of the human arm. The inclusive theoretical platform was defined as the union between the workspace defined for minimum limb lengths and the workspace defined for the maximum limb lengths. The resulting workspace is identified by an ellipse with centre c = [0, 513.5] mm, minor axis = 222 mm and major axis = 502.75 mm.

Since the population under study in [26] was right-handed, the authors of that research centred the reachable workspace at x = 55.75 mm. Consequently, the y-axis of PLANarm2 has been translated in order to have it aligned with the centre of the reachable workspace, as shown in Figure 5a. Since the manipulator is designed to be home based, it will be installed on a regular home table or desk. An average sized table is assumed to have a length of, at least, 1500 mm and a width of about 800 mm. Furthermore, the patient must be located at a distance of 200 mm away from the table, as shown in Figure 5b.

**Figure 5.** (**a**) Sketch showing the total reachable workspace by combining the workspace of patients with minimum limb lengths and those with maximum limb lengths. (**b**) Sketch showing the placement of the rehabilitation device in accordance with the reachable workspace.

### *3.2. Kinematic Optimization*

The link lengths have been optimized in order to provide the best kinetostatic performance. The lower *r*<sup>3</sup> is, the larger the theoretical workspace is [25]. The maximum workspace is obtained when the joints connected to the ground are coaxial (*r*<sup>3</sup> = 0). However, due to mechanical constraints, the lowest possible value of *R*<sup>3</sup> was chosen to be equal to 45 mm. Based on the reachable workspace, it was sufficient to choose *R*<sup>1</sup> + *R*<sup>2</sup> = 800 mm. Finally, in order to determine the values of *R*<sup>1</sup> and *R*2, the minimum stiffness and isotropy were optimized over the radial direction. Both indexes are radially symmetric [23] and therefore they are plotted against the y-direction in Figures 6 and 7. The interval of interest is y = [300 mm, 700 mm], which includes the reachable workspace.
