**4. Handwheelchair.q02**

The Handwheelchair.q02 is a wheelchair for everyday life. The main goals of Handwheelchair.q02 are:


The Hanwheelchair.q02 has two configurations. The first configuration, shown in Figure 6a, is employed for indoor spaces. In this configuration, the wheelchair is used as a common wheelchair, where the propulsion is obtained by the handrim. The foldable links, left and right, that support a pair of return pulleys, are folded in order to minimise the visual impact and to allow for sitting on the wheelchair. The second configuration, shown in Figure 6b, is employed outdoors to facilitate movement, to extend accessible places and to practice motor activities. The user rotates the foldable link, left and right, around the joint C in order to obtain the innovative configuration.

**Figure 6.** (**a**) Handwheelchair.q02 indoor configuration; (**b**) Handwheelchair.q02 outdoor configuration.

Figure 7 show the components that enable the use of the wheelchair in the innovative configuration. The red component is the hub of the wheel and the blue one is the pulley around which the cable is wrapped. The green one is the power spring that connects the pulley with the chassis of the wheelchair, in grey. In the first configuration, the mechanism of transmission of motion allows the user to use the wheelchair as a common wheelchair. In fact, the hub of the wheel rotates around the shaft as in the classic wheelchair, as sketched in Figure 8a.

**Figure 7.** (**a**) Components of the innovative system of propulsion; (**b**) Section of the mechanism.

When the user takes the handles from their site, a mechanism keeps in contact the two front ratchets, as visible in Figure 8b.

**Figure 8.** Particulars of the mechanism in the (**a**) Classic configuration; (**b**) Innovative configuration.

One ratchet is integrated with the pulley and the other one is mounted into the hub of the wheel. In this configuration, the propulsion is obtained by the rowing stroke previously described. During the traction phase, the user pulls the cables with a force Fu the cables are wrapped around a pair pulleys of radius rp. The user transmits a torque to the wheel through the ratchet system. During the traction phase, the user loads a power spring that connects the pulley with the chassis of the wheelchair. In Figure 9, the free-body diagrams during the traction phase of the pulley and of the hub are reported and in Tables 1 and 2 the description of the torques and the reference systems are described.


**Table 1.** Description of the torques in the free-body diagrams.



During the traction phase, the angular speed of the pulley is the same as the angular speed of the wheel and the equation can be written as follows:

$$
\omega\_P = \omega\_W > 0.\tag{1}
$$

The torque equilibrium of the pulley results:

$$F\mu \* rp = I\_P \* \dot{\omega}\_P + \text{C}\sigma + \text{K}\varepsilon \* \theta\_P + \text{C1}.\tag{2}$$

While the torque equilibrium of the hub is:

$$\mathbf{C1} = \mathbf{C}f\mathbf{1} + I\_W \* \dot{a}\_W.\tag{3}$$

**Figure 9.** Free-body diagram during the traction phase.

During the recovery phase the user stops pulling, and we assume that Fu ~ 0, Figure 10. The power spring, previously loaded, has to generate a torque Ceo + Ke\*θP to rotate the pulley in the opposite direction in order to rewind the cable around the pulley in a specific time TR. The recovery time TR is the parameter of the project that defines the torque of the power spring.

During the recovery phase, the angular velocities are:

$$
\omega\_P < 0; \; \omega\_W > 0. \tag{4}
$$

The torque equilibrium of the pulley is:

$$
\dot{P}\mathbf{p}\*\dot{\omega}\mathbf{p} + \mathbf{C}\mathbf{e}\mathbf{o} + \mathbf{K}\mathbf{e}\*\boldsymbol{\theta}\mathbf{p} = \mathbf{C}f\mathbf{2}.\tag{5}
$$

The torque equilibrium of the hub results:

$$
\mathbb{C}f\mathbf{1} + \mathbb{C}f\mathbf{2} + I\_W \ast \dot{\boldsymbol{\omega}}\_W = \mathbf{0}.\tag{6}
$$

**Figure 10.** Free-body diagram during the recovery phase.

#### **5. Dynamic Model of Handwheelchair.q**

In this section, according to the previous paragraph, a simplified dynamic model, Figures 11 and 12, and the description of the free-body diagram, Tables 3 and 4, of Handwheelchair.q are presented, with the following hypothesis and assumptions:



.


**Figure 11.** Handwheelchair.q: Free-body diagram.

**Figure 12.** Free-body diagram of wheel 1, wheel 2, the pulley and the return pulley.


**Table 3.** Description of the torques and force of the dynamic model.

**Table 4.** Reference system's dynamic model.


The user's force is modelled as follows:


$$
\mathbf{x}(t') - \mathbf{x}(t0) = lh\\_m \mathbf{x} \ast \mathbf{r} \tag{7}
$$

$$t'-t0=Tp\tag{8}$$

where, lh is the stroke length of the gesture and τ = 3 is the transmission ratio defined by:

$$\mathbf{x} = l\mathbf{h} \ast \frac{rw\mathbf{1}}{rp} = l\mathbf{h} \ast \mathbf{r}.\tag{9}$$

The Figure 14, shows the traction phase during the indoor test of the first prototype Handwheelchair.q01.

*Machines* **2019**, *7*, 31


**Figure 13.** Time of the traction phase.

**Figure 14.** Innovative gesture during the indoor test of the first prototype Handwheelchair.q01. (**a**) t0, lh = 0; (**b**) t0 < t < t1, 0 < lh < lh\_max; (**c**) t1, lh = lh\_max.

**Figure 15.** User's force model.

According to the free-body diagram previously reported, and with the abovementioned hypothesis and assumptions:

The equilibrium of the wheelchair along the axis *x* results

$$T1 = m\ddot{\mathbf{x}} + F\mathbf{a} = m\ddot{\mathbf{x}} + k\_d \mathbf{x}^2. \tag{10}$$

.

The equilibrium of the wheelchair along the axis *z* is

$$N1 = Fp = mg.\tag{11}$$

The rotation equilibrium of wheel 1 around the joint Cw1 results

$$T1\*r\_{\mathbb{N}1} + \mathbb{N}1\*u\_1 = \mathbb{C}1.\tag{12}$$

The rotation equilibrium of the pulley around the joint CP is

$$T \ast r\_P = \mathbb{C}1 + \mathbb{C}\varepsilon.\tag{13}$$

The rotation equilibrium of the return pulley around the joint CRP is

$$T \ast r\_{RP} = F\mathfrak{u} \ast r\_{RP}.\tag{14}$$

By replacing Equations (10), (11), (13) and (14) in Equation (12), it results:

$$F u \* r \mathbb{R} P = m \mathbb{g} \* u\_1 + \left( m \ddot{\mathbf{x}} + k\_4 \dot{\mathbf{x}}^2 \right) \* r\_{W1} + \text{Ce}.\tag{15}$$

By rewriting Equation (15), the acceleration can be evaluated as:

$$\ddot{\mathbf{x}} = \frac{Fu}{\pi m} - \frac{k\_{\rm a}}{mr\_{\rm W1}} \dot{\mathbf{x}}^2 - \frac{mg \ast u\_1}{mr\_{\rm W1}} - \frac{Ce}{mr\_{\rm W1}}.\tag{16}$$

Equation (16) and the model of the user's power have been implemented in Matlab/Simulink.

During the steady-state phase, the average wheelchair speed is constant. Figure 16 show respectively the wheelchair speed and the user's power during a complete cycle: traction phase and recovery phase.

**Figure 16.** *Cont*.

**Figure 16.** Wheelchair speed (**a**) and user's power (**b**) in a complete cycle during the steady-state phase.

The average power of 23 W and the average speed of 1.57 m/s = 5.65 km/h obtained with the simulation is in accordance with the tests carried out in [12,13,15].

By employing the user's force model, we obtain a Fu\_max = 120 N as shown in Figure 17, while the average force Fu\_avg = 43 N. The force Fu is the sum of the right and left arm contribution: Fur + Ful. Then, the average force for each arm is approximately

$$
\overline{Fu\_l} = \overline{Fu\_l} = 22 \text{ N.}\tag{17}
$$

**Figure 17.** User's force during the steady-state phase.
