## 2 Materials and Methods
The speed value was determined with a basis of 30 ms using an independent counter. Figure 1: Dimensions of the track used in the research. In the research, two modes were tested: hill-climbing assistance (t3) and amplification of the driving torque (t4) generated by the upper limb. In the first mode, the system controller used an accelerometer and a gyroscope. The acceleration values measured by the accelerometer and angular velocity were converted to angular position with the use of a complementary filter. In addition, the acceleration values were corrected using a low-pass filter. The measured values were then sent to the controller, where the amplification coefficient \(w\) was added. Due to vibrations generated during movement, especially when moving on uneven surfaces (e.g., cobblestone) and a sharp tilt of the wheelchair due to inertia forces, the system was equipped with two independent software angle measurement tools. The first checked if the value of the angular position was greater than \(1^{\circ}\) for a time period longer than 300 ms. Only then was movement assistance initiated. The second one verified if the angular position was below \(1^{\circ}\) for a time period longer than 300 ms. If it was, assistance was switched off. The value of the voltage controlling the motor for the hill-climbing assistance mode was determined using the following Formula (1):  
\[u\ =\ \alpha\cdot c+r_{3}\cdot w+\delta \tag{1}\]  
where: \(u\)--signal given to the BLDC motor controller, \(\alpha\)--wheel chair angle [\({}^{\circ}\)], \(c\)--angle coefficient (0.0075) \(r_{3}\)--gain coefficient in mode 3 (0.0016), \(w\)--assistance amplification coefficient [\(\%\)], \(\delta\)--voltage offset (1.102) [V]. The amplification of the driving torque mode (t4) was used to assist propulsion with the use of hand rims. When a change in the rotational speed of the drive wheel was detected, the controller set the current value proportional to the value of the assistance amplification coefficient \(w\). The control  
Figure 3: Block diagram of the control system. Figure 2: Tested AR-405 wheelchair equipped with a hybrid propulsion module. (**a**) BrushLess Direct-Current motor; (**b**) incremental encoder; (**c**) electronic controller; (**d**) touch screen. was carried out with a time base of 100 ms. The voltage controlling the motor for this mode was determined using the following Formula (2):  
\[u\;=\;r_{4}\cdot w+\delta \tag{2}\]  
where: \(u\)--signal given to the BLDC motor controller, \(r_{4}\)--gain coefficient in mode 4 (0.008), \(w\)--assistance gain coefficient [%], \(\delta\)--voltage offset (1.102) [V]. Four values of the assistance amplification coefficient were tested. The list of controlling voltages for these values is shown in Table 2.  
The test procedure consisted of each subject moving four times along the designated track with five assistance amplification coefficient settings (0%, 25%, 50%, 75%, 100%) for each of the two assisted modes tested. The user had a 10-minute break to rest between each test. The subjects were asked to accelerate the wheelchair on the first horizontal section, and then to maintain a constant rhythm of pushing the hand rims on the slope. Then, on the last horizontal section of the track, they had to brake the wheelchair to a standstill. The data measured with the use of incremental encoders was used to analyze the kinematics of the wheelchair. The analyzed track included only the section where the wheelchair was moving uphill. During the climbing up the slope, the patients repelled the pushrims with an interval equal to 1 s. The interval value was chosen experimentally. For the examined patients, the assumed length of the interval prevents the wheelchair from rolling down. Based on the data collected for each measurement test, graphs of the speed and acceleration of the wheelchair were prepared (Figure 4). Analyzing the measured waveforms of speed and acceleration, the end points of the propulsive cycle (PC) were detected.