# Torque Measurement and Control for Electric-Assisted Bike Considering Different External Load Conditions

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## Abstract

**:**

## 1. Introduction

## 2. E-Bike Pedaling Dynamic

#### 2.1. Cyclist Pedaling Behavior

#### 2.2. Pedal Crank Angle Effort

#### 2.3. Parameters of Analyzed E-Bike

## 3. E-Bike Dynamics

_{w}and acceleration α

_{w}of the E-bike considering different torque control methods with external loads. An analytical E-bike model in Figure 2 is developed to investigate the ω

_{w}and α

_{w}performance of the E-bike under these external loads. These external loads include the wheel friction torque T

_{roll}, the windage torque T

_{wind}, and the climbing-reflected torque T

_{slope}. It can be shown that:

_{dis}is the summation of all external loads. In addition, the road slope angle θ

_{slope}, the angular speed and acceleration ω

_{w}and α

_{w}, the tire pressure P

_{T}, the bike wheel radius R, the wind speed V

_{wind}, the E-bike mass M

_{e}, the cyclist mass M

_{c}, the gravitational constant g, the density of air $\mathsf{\rho}$, the aerodynamic drag coefficient ${\mathrm{C}}_{\mathrm{d}}$, and the frontal area A are all parameters used for the calculation of external loads. The maximum climbing angle of the E-bike can also be obtained under a specific value for T

_{pdl}and T

_{M}.

#### 3.1. Synergetic Torque

_{total}combines the cyclist pedaling torque T

_{pdl}and the motor-assisted torque T

_{M}. It can be calculated by:

_{total}is assumed to drive the rear wheel. It is noted that the pedaling torque generated by the cyclist is assumed to be a rectified sinusoidal torque, illustrated in Figure 1b. Under this effect, the synergetic T

_{total}can be either the constant torque or the sinusoidal torque, depending on the manipulation of the motor torque T

_{M}.

#### 3.2. Wheel Friction Torque

_{roll}on the overall cycling torque output. This is given by:

_{slope}is the road slope angle in Figure 3. In addition, R is the E-bike rolling radius in Table 2. K

_{roll}is the resistance coefficient affected by the road’s surface shape, the tire’s structure, material, and pressure, as well as the wheel speed.

_{roll}is strongly influenced by tire pressure. The tire deformation is visible with considerable rolling resistance when the tire pressure is low [41,42]. In general, the resistance coefficient is calculated by:

_{w}is the wheel angular speed, and P

_{T}is the tire pressure.

_{roll}versus the wheel speed. In this simulation, a constant acceleration of 1.334 rad/s

^{2}is assumed, in which the wheel speed is increased from 0 to 20 rad/s within 15 s. Two cyclists, weighing 70 kg and 50 kg, are compared. Although the friction torque T

_{roll}is slightly increased as the wheel speed increases, the influence of the cyclist’s weight is more visible than the wheel speed. Based on Figure 4, it can be concluded that T

_{roll}is mainly dominated by the cyclist’s weight. Thus, the motor-assisted torque T

_{M}can be manipulated depending on the current cyclist’s weight.

#### 3.3. Windage Torque

_{wind}, which is shown to be:

_{bike}is calculated by the wheel angular speed, and V

_{wind}is the corresponding wind speed depending on the airflow condition. In addition, ρ is the air density, ${\mathrm{C}}_{\mathrm{d}}$ is the aerodynamic drag coefficient, and A is the frontal area of airflow. For the analyzed E-bike in this paper, these three parameters are listed in Table 2.

^{2}acceleration is assumed. Within 15 s, the wheel angular speed is increased from 0 to 20 rad/s. In the case of no wind, the windage torque is equivalent to a quadratic function proportional to ${\mathrm{v}}_{\mathrm{b}\mathrm{i}\mathrm{k}\mathrm{e}}^{2}$. Even at zero wheel speed V

_{bike}= 0, there is a windage T

_{wind}for the headwind with V

_{wind}= 10 km/h. However, T

_{wind}is only 2 Nm based on the calculated parameters in Table 2. The influence of the windage T

_{wind}is relatively less than the friction torque analyzed in Figure 4.

_{wind}= −10 km/h, the airflow can be used to generate an assisted torque. However, as shown in (5), once V

_{bike}exceeds V

_{wind}, the assisted torque is converted to resistive torque. Nevertheless, the T

_{wind}is sufficiently low during tailwind conditions.

#### 3.4. Climbing-Reflected Torque

_{slope}once the slope angle θ

_{slope}≠ 0. Depending on the slope angle, the T

_{slope}can be shown to be:

_{slope}with respect to the slope angle. Different from T

_{roll}in (4) and T

_{wind}in (6), the climbing torque T

_{slope}is only dependent on the slope angle and cyclist weight. Comparing two different cyclists of 70 kg and 50 kg on the same bike, the heavier cyclist results in a higher T

_{slope}. However, compared to the T

_{roll}simulation in Figure 4, T

_{slope}is mainly affected by the slope angle θ

_{slope}instead of the cyclist’s weight. As a result, the motor-assisted torque should be manipulated with respect to the slope angle θ

_{slope}for different E-bike trekking conditions.

#### 3.5. E-Bike Dynamic Model

_{drv}, the wheel angular acceleration α

_{w}, and the speed ω

_{w}of the E-bike can be respectively modeled by (8) and (9):

_{w}is the corresponding wheel inertia. Considering the E-bike with different cyclist weights, J

_{w}can be modeled by:

_{e}and M

_{c}are, respectively, the weight of the E-bike and the cyclist.

_{dis}. It is noted that the torque control for this E-bike system is equivalent to an open-loop control system in this paper. As seen in Figure 7, the total torque input T

_{total}consists of the cyclist pedaling torque T

_{pdl}and the motor torque T

_{M}. In this paper, the motor torque magnitude is manually adjusted. When the external load is increased, the cyclist is expected to generate more pedaling torque as well. In this case, the overall control stability of the assisted E-bike system is only dependent on the motor torque regulation.

_{d}, i

_{q}are the stator current of the d- and q-axis. V

_{d}, V

_{q}are the stator voltage of the d- and q-axis, K

_{pd}and K

_{pq}are the corresponding proportional gains, and K

_{id}and K

_{iq}are the corresponding integral gains.

_{d}is controlled to be zero, and the torque T

_{M}is directly proportional to the q-axis current i

_{q}. Regarding the torque controller design, pole/zero cancellation technology is used. PI controller gains are designed to be equal to:

## 4. Proposed E-Bike Torque Control

_{total}, wheel acceleration α

_{w,}and speed ω

_{w}.

_{pdl}in Figure 1b is used. Figure 7 illustrates the control process of the E-bike model. Four motor torque-assisted methods are implemented. These four assisted methods are individually added to the original pedaling torque under the E-bike model in (3). After obtaining the total synergetic torque T

_{total}, the actual torque can be obtained under the influence of three external load torques. The actual wheel driving torque T

_{drv}, angular acceleration α

_{w}, and speed ω

_{w}are obtained from Equations (8)–(10). It is noted that the E-bike cycling performance can be evaluated based on the E-bike wheel speed ω

_{w}and acceleration α

_{w}conditions.

#### 4.1. No Motor-Assisted Torque (NMT)

_{w}simulation in Figure 9a, the α

_{w}wheel acceleration waveform is the same as the pedaling torque T

_{pdl}, since α

_{w}is directly proportional to T

_{pdl}. Considering the wheel inertia J

_{w}= 5.80 kg/${\mathrm{m}}^{2}$ with M

_{c}= 70 kg, the average α

_{w}is 0.48 rad/s

^{2}with a peak-to-peak acceleration ripple of 2.38 rad/s

^{2}. By contrast, a wheel speed ω

_{w}simulation based on (8) is also analyzed in Figure 9b. The average speed is 7.69 rad/s, with a 0.28 rad/s peak-to-peak speed ripple. The corresponding α

_{w}and ω

_{w}waveforms in Figure 9 can be used as a benchmark to compare the different torque control methods listed below.

#### 4.2. Constant Motor-Assisted Torque (CT)

_{M}, T

_{pdl}, and T

_{total}under the same 30 cpm cadence. The motor-rated torque is 45 Nm. Considering the average pedaling torque after the transmission, the ratio between T

_{M}and T

_{pdl}is T

_{M}= 1.83 T

_{pdl.}To easily compare different torque waveforms, a zoom-in figure is also added in Figure 10 in this simulation.

_{w}for E-bike torque control with the CT method. Due to the additional constant T

_{M}, the average α

_{w}is increased from 0.48 to 1.90 rad/s

^{2}. For the speed simulation in Figure 11b, the average ω

_{w}is increased due to the additional T

_{total}. It is noteworthy that the ω

_{w}ripple is increased to 0.60 rad/s compared to NMT due to the higher average α

_{w}based on (8). By applying the CT method, it is concluded that both the average α

_{w}and ω

_{w}can be increased for a better E-bike trekking performance. However, the visible ripple in ω

_{w}might degrade the cyclist’s riding experience.

#### 4.3. Same Phase as Pedaling Torque (SPPT)

_{M_SPPT}is manipulated by:

_{pdl}and T

_{pdl_peak}are, respectively, the instantaneous and peak value of the pedaling torque, depending on the pedaling torque sensor performance. Further, T

_{M_rated}is the rated motor torque.

_{M_SPPT}, T

_{pdl}, and T

_{total}under the same 30 cpm cadence. Comparing T

_{M_SPPT}with the CT in Figure 10, it is seen that the average T

_{M_SPPT}can be smaller, leading to better battery usage. However, Figure 13a demonstrates the corresponding α

_{w}resulting from the SPPT method. Compared to α

_{w}based on the CT method in Figure 11a, the average α

_{w}is reduced from 1.90 to 1.48 rad/s

^{2}, but with the ripple increased from 2.41 to 5.09 rad/s

^{2}. For the ω

_{w}speed waveform in Figure 13b, a similar decline in performance is also observed. A detailed performance comparison between the CT and SPPT methods will be explained in Section 4.6.

#### 4.4. Delay Phase as Pedaling Torque (DPPT)

_{M_DPPT}is formulated by:

_{pdl_d}is a 90° delay torque with respect to the measured instantaneous T

_{pdl}. For real-time implementation, T

_{pdl_d}can be obtained by:

_{pdl_d}can only be obtained after a 90° delay of θ

_{crank.}Due to this limitation, the E-bike might not be able to provide the motor-assisted torque during the initial startup. Nevertheless, the motor torque control can be operated after one-fourth of the pedaling cycle.

_{M_DPPT}, T

_{pdl}, and T

_{total}under the same cadence and slope situation. Since the motor torque magnitude is the same as the SPPT method, the average total torque should be the same. More importantly, because of the lower torque ripple for T

_{total}in Figure 14, peak-to-peak ripples are decreased for α

_{w}in Figure 15a and ω

_{w}in Figure 15b. It is expected that a relatively comfortable cyclist performance is achieved. However, in Figure 15, a certain amount of T

_{total}ripple is still observed, because T

_{pdl}cannot be equal to the motor T

_{M_DPPT}. The T

_{total}ripple should be increased due to the increase in T

_{pdl}under the same rated motor torque T

_{M_rated}. A detailed comparison of the performance with the SPPT method will also be explained in Section 4.6.

#### 4.5. Compensation for the Gap in the Pedaling Torque (CGPT)

_{M_CGPT}is derived from:

_{ref}is a synergy torque reference. It can be determined by the previously mentioned external load conditions. Based on the definition in (13), the manipulated motor torque T

_{M_CGPT}is disabled when T

_{pdl}is higher than T

_{ref.}By contrast, T

_{total}can be the same as T

_{ref}once T

_{pdl}< T

_{ref}. Figure 16 shows the torque waveform using this CGPT method. Compared to the prior torque control methods, the primary advantage is the lowest torque ripple.

_{w}ripple is only 0.84 rad/s

^{2}, which is also smaller than 1.15 rad/s

^{2}, resulting from the prior DPPT control method. A smaller ω

_{w}ripple performance can be observed in Figure 17b. However, since T

_{M_CGPT}is generated only at a low T

_{pdl}, a drawback is the reduced average speed in Figure 17b. Comparing CT control with the highest average ω

_{w}for trekking, CGPT control is well suited for commuter applications to maximize the E-bike’s battery usage.

#### 4.6. Performance Comparison

_{w}and ω

_{w}response. By contrast, Table 4 compares these torque control methods to reach the same final speed. In Table 4, the cycling time can be different depending on different torque methods. The key findings can be summarized as follows:

- (1)
- CT: The CT control method results in the highest α
_{w}and ω_{w}due to the highest motor torque output. However, the ripples in α_{w}are also the highest. This method is well suited for trekking applications under visible external loads. - (2)
- SPPT and DPPT: The highest α
_{w}ripple is the result of the SPPT method. When the α_{w}ripple is much higher than in the NMT case, the cyclist may have an uncomfortable experience. By contrast, for the DPPT method, a smaller α_{w}ripple is achieved under the same motor torque. Compared to SPPT control, the DPPT method can provide a comparable cycling experience as the original NMT. The DPPT method is well suited for standard E-bike torque management for different load conditions. - (3)
- CGPT: Because the CGPT method generates the lowest motor torque, the resulting α
_{w}ripple can be smaller than the original NMT condition. However, the lowest motor output might degrade the E-bike’s acceleration performance. As seen in Table 4, CGPT requires 18.72 s to reach a 15 rad/s final speed. By contrast, for CT control, only 4.72 s is spent. It is concluded that the CGPT is well suited for commuting cyclists. This control results in the best battery usage at the smallest α_{w}ripple. It is especially well suited for cyclists under a heavy daily urban traffic burden.

## 5. Experiment

_{w}of 15.88 rad/s (20 km/h). If ω

_{w}can be maintained at a more stable speed without variation, the motor torque is assumed to assist the cyclist.

#### 5.1. NMT and CT Experiment

_{w}, and ω

_{w}waveforms under normal NMT. In this case, the corresponding pedaling torque condition can be used as a benchmark to compare the four different torque control methods.

_{w}and ω

_{w}. In this control, the motor torque is controlled to maintain a 45 Nm rated torque. With additional assisted torque, the resulting average pedaling ${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}}$ is reduced to 13.05 Nm. However, similar to the prior simulation, the pedaling variation ${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}\_\mathrm{r}}$ is increased to 63.86% due to the limitation on the constant motor torque regulation. The key differences in the performance of the torque are summarized in Section 5.4.

#### 5.2. SPPT and DPPT Experiment

_{w}and ω

_{w}are smaller, as shown in Figure 27 and Figure 29. A detailed comparison of α

_{w}and ω

_{w}will be explained in Section 5.5.

#### 5.3. Proposed CGPT Experiment

_{w}and ω

_{w}waveforms are included in Figure 31. As seen in Section 4.5, the CGPT-assisted torque is determined based on (17). For the actual experiment, ${\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is determined at 25 Nm, which is the average pedaling torque ${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}}$ on the rear wheel under normal NMT. When the pedaling torque transmission to the rear wheel is smaller than 25 Nm, ${\mathrm{T}}_{\mathrm{M}}$ should be enabled similarly to the DPPT control condition. Based on the simulation, it is expected that the average and maximum motor ${\mathrm{T}}_{\mathrm{M}}$ are the lowest among the four torque control methods. This leads to better E-bike battery usage.

#### 5.4. E-Bike Torque Performance Comparison

Assisted Method | NMT | CT | SPPT | DPPT | CGPT | |
---|---|---|---|---|---|---|

Parameter | ||||||

Average pedaling torque (Nm) | 29.81 | 13.05 | 20.83 | 21.05 | 25.63 | |

Average motor torque (Nm) | NA | 45 | 12.01 | 11.93 | 8.24 | |

Max pedaling torque (Nm) | 74.37 | 36.11 | 48.09 | 48.13 | 62.67 | |

Max motor torque (Nm) | NA | 45 | 27.05 | 27.07 | 25 | |

Pedaling torque variation (Nm) | 44.56 | 23.06 | 27.26 | 27.08 | 37.04 | |

Variation ratio (Nm/%) | 59.92% | 63.86% | 56.69% | 56.26% | 59.10% | |

Average total torque (Nm) | 29.81 | 58.05 | 32.84 | 32.98 | 33.87 | |

Max total torque (Nm) | 74.37 | 81.11 | 75.14 | 75.20 | 87.67 |

_{w}at 15.88 rad/s (20 km/h).

_{w}and ω

_{w}might be different due to different peak total torques with these two control methods.

_{w}and ω

_{w}of the E-bike is expected.

#### 5.5. E-Bike Speed and Acceleration Comparison

_{w}and speed ω

_{w}in Table 7 under the different proposed torque controls. It is noted that the average value and ripple of α

_{w}and ω

_{w}are both dependent on the total torque ${\mathrm{T}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ in Table 6. Since the CT-assisted control results in the highest variation in ${\mathrm{T}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, the highest ripples of both α

_{w}and ω

_{w}are shown in Table 7. This experimental result is consistent with the simulation in Table 3.

_{w}and ω

_{w}for the DPPT control are smaller than those with SPPT control. Finally, for the proposed CGPT method, the corresponding α

_{w}and ω

_{w}ripple is slightly higher than those with the DPPT methos. However, compared to CT and SPPT controls, the CGPT method still results in a better α

_{w}and ω

_{w}ripple performance for E-bike torque-assisted control.

#### 5.6. Simulation and Experiment Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Con sent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

θ_{crank} | Crank rotating angle |

F_{px} | Pedaling horizontal force |

F_{py} | Pedaling vertical force |

${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}}$ | Pedaling torque |

T_{M} | Motor-assisted torque |

T_{total} | Total synergetic torque |

T_{roll} | Friction-reflected torque |

T_{wind} | Windage torque |

T_{slope} | Climbing-reflected torque |

T_{dis} | External disturbance torque |

T_{drv} | Actual wheel driving torque |

${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}\_\mathrm{r}}$ | Pedaling torque variation |

${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}\_\mathrm{m}\mathrm{a}\mathrm{x}}$ | Maximum measured pedaling torque |

${\mathrm{T}}_{\mathrm{p}\mathrm{d}\mathrm{l}\_\mathrm{a}\mathrm{v}\mathrm{g}}$ | Average measured pedaling torque |

${\mathrm{R}}_{\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}}$ | Crank rotating radius |

M_{e} | Mass of E-bike |

M_{c} | Mass of cyclist |

J_{w} | Wheel inertia |

R_{w} | Bike wheel radius |

g | Gravitational constant |

$\mathsf{\rho}$ | Air density |

${\mathrm{C}}_{\mathrm{d}}$ | Aerodynamic drag coefficient |

A | Frontal area |

${\mathrm{K}}_{\mathrm{g}\mathrm{e}\mathrm{a}\mathrm{r}}$ | Transmission gear ratio |

$\mathrm{P}$_{T} | Tire pressure |

ω_{w} | Wheel angular speed |

α_{w} | Wheel angular acceleration |

K_{roll} | Wheel resistance coefficient |

V_{wind} | Wind speed |

V_{bike} | Bike wheel speed |

θ_{slope} | Slope angle |

i_{d} | Direct-axis (d-axis) motor current |

i_{q} | Quadrature-axis (q-axis) motor current |

V_{d} | Direct-axis (d-axis) motor voltage |

V_{q} | Quadrature-axis (q-axis) motor voltage |

K_{pd} | Direct-axis (d-axis) proportional controller gain |

K_{pq} | Quadrature-axis (q-axis) proportional controller gain |

K_{id} | Direct-axis (d-axis) integral controller gain |

K_{iq} | Quadrature-axis (q-axis) integral controller gain |

$\widehat{\mathrm{L}}$ | Motor phase inductance |

$\widehat{\mathrm{R}}$ | Motor phase resistance |

${\widehat{\mathsf{\theta}}}_{\mathrm{k}}$ | Estimated motor position |

${\mathsf{\theta}}_{\mathrm{k}-1}$ | Last position measured by Hall sensors |

${\widehat{\mathsf{\omega}}}_{\mathrm{k}-1}$ | Estimated speed based on prior Hall sensor position |

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**Figure 1.**Relationship between crank position and pedaling torque: (

**a**) pedaling torque component ${\mathrm{F}}_{\mathrm{p}\mathrm{y}}$/${\mathrm{F}}_{\mathrm{p}\mathrm{x}}$ and crank position; (

**b**) pedaling torque with respect to the crank position (no horizontal pedal force is assumed for simplicity).

**Figure 23.**NMT-reflected E-bike response: (

**a**) wheel angular acceleration and (

**b**) wheel angular speed.

**Figure 25.**CT-reflected E-bike response: (

**a**) wheel angular acceleration and (

**b**) wheel angular speed.

**Figure 27.**SPPT-reflected E-bike response: (

**a**) wheel angular acceleration and (

**b**) wheel angular speed.

**Figure 29.**DPPT-reflected E-bike response: (

**a**) wheel angular acceleration and (

**b**) wheel angular speed.

**Figure 31.**CGPT-reflected E-bike response: (

**a**) wheel angular acceleration and (

**b**) wheel angular speed.

Category | References |
---|---|

Torque control for assisted E-bike | [1,2,24,38,39] |

Instantaneous pedaling torque waveform | [4,5,6,7] |

Pedaling torque component analysis | [8,9,10,11,12,13,14] |

Factors affecting riding | [15,16,17,18,19,20,21,22,23,25,26,27,28,29,30,31,32,33,34,35] |

Recharge control for assisted E-bike | [36,37] |

Parameter | Value |
---|---|

Mass of E-bike (M_{e}) | 25 kg |

Mass of cyclist (M_{c}) | 70 or 50 kg |

Wheel inertia (J_{w}) | 5.8 or 4.6 kg/${\mathrm{m}}^{2}$ |

Bike wheel radius (Rw) | 0.35 m |

Gravitational constant (g) | 9.81 $\mathrm{m}/{\mathrm{s}}^{2}$ |

Density of air ($\mathsf{\rho}$) | 1.2258 kg/${\mathrm{m}}^{3}$ |

Aerodynamic drag coefficient (${\mathrm{C}}_{\mathrm{d}}$) [40] | 0.4 |

Frontal area (A) | 0.645 ${\mathrm{m}}^{2}$ |

Maximum cadence per minute | 30 cpm |

Transmission gear ratio (${\mathrm{K}}_{\mathrm{g}\mathrm{e}\mathrm{a}\mathrm{r}}$) | 44/14 |

Tire pressure (P_{T}) | 32 psi |

Assisted Method | NMT | CT | SPPT | DPPT | CGPT | |
---|---|---|---|---|---|---|

Parameter | ||||||

Average pedaling torque (Nm) | 30 | 30 | 30 | 30 | 30 | |

Average motor torque (Nm) | N/A | 45 | 27.91 | 27.91 | 5.37 | |

Max motor torque (Nm) | N/A | 45 | 45 | 45 | 25 | |

Speed ripple (rad/s) | 0.28 | 0.60 | 0.55 | 0.15 | 0.10 | |

Average speed (rad/s) | 7.69 | 36.44 | 26.89 | 26.89 | 11.82 | |

Acceleration ripple (rad/s^{2}) | 2.38 | 2.41 | 5.09 | 1.15 | 0.84 | |

Average acceleration (rad/s^{2}) | 0.48 | 1.90 | 1.48 | 1.48 | 0.72 | |

Cycling time (s) | 30 | 30 | 30 | 30 | 30 |

Assisted Method | NMT | CT | SPPT | DPPT | CGPT | |
---|---|---|---|---|---|---|

Parameter | ||||||

Average motor torque (Nm) | N/A | 45 | 27.91 | 27.91 | 5.37 | |

Max motor torque (Nm) | N/A | 45 | 45 | 45 | 25 | |

Final speed (rad/s) | 15.00 | 15.00 | 15.00 | 15.00 | 15.00 | |

Average acceleration (rad/s^{2}) | 0.48 | 3.18 | 2.19 | 2.19 | 0.80 | |

Required time s) | 31.32 | 4.72 | 6.86 | 6.86 | 18.72 |

Parameter | Value |
---|---|

Rated voltage | 36 V |

Maximum torque | 45 Nm |

Rated power | 250 W |

Weight | 2.46 kg |

Outer radius | 129 mm |

Maximum speed | 3000 rpm |

Installation location | Rear wheel |

Assisted Method | NMT | CT | SPPT | DPPT | CGPT | |
---|---|---|---|---|---|---|

Parameter | ||||||

Speed ripple (rad/s) | 3.43 | 4.56 | 3.32 | 0.94 | 1.76 | |

Average speed (rad/s) | 16.84 | 15.56 | 16.85 | 16.59 | 16.14 | |

Acceleration ripple (rad/s^{2}) | 1.97 | 3.37 | 2.96 | 1.56 | 2.12 | |

Average acceleration (rad/s^{2}) | 0.16 | 1.20 | 0.16 | 0.16 | 0.07 | |

Cycling time (s) | 30 | 30 | 30 | 30 | 30 |

Assisted Method | NMT | CT | SPPT | ||||
---|---|---|---|---|---|---|---|

Parameter | (sim.) | (exp.) | (sim.) | (exp.) | (sim.) | (exp.) | |

Average pedaling torque (Nm) | 30 | 29.81 | 30 | 13.05 | 30 | 20.83 | |

Max pedaling torque (Nm) | 48.78 | 74.37 | 48.78 | 36.11 | 48.78 | 48.09 | |

Average motor torque (Nm) | N/A | N/A | 45 | 45 | 27.91 | 12.01 | |

Max motor torque (Nm) | N/A | N/A | 45 | 45 | 45 | 27.05 | |

Speed ripple (rad/s) | 0.28 | 3.43 | 0.60 | 4.56 | 0.55 | 3.32 | |

Acceleration ripple (rad/s^{2}) | 2.38 | 1.97 | 2.41 | 3.37 | 5.09 | 2.96 | |

Average speed (rad/s) | 7.69 | 16.84 | 36.44 | 15.56 | 26.89 | 16.85 | |

Average acceleration (rad/s^{2}) | 0.48 | 0.16 | 1.90 | 1.20 | 1.48 | 0.16 | |

Cycling time (s) | 30 | 30 | 30 | 30 | 30 | 30 | |

Assisted Methods | DPPT | CGPT | |||||

Parameters | (sim.) | (exp.) | (sim.) | (exp.) | |||

Average pedaling torque (Nm) | 30 | 21.05 | 30 | 25.63 | |||

Max pedaling torque (Nm) | 48.78 | 48.13 | 48.78 | 62.67 | |||

Average motor torque (Nm) | 27.91 | 11.93 | 5.37 | 8.24 | |||

Max motor torque (Nm) | 45 | 27.07 | 25 | 25 | |||

Speed ripple (rad/s) | 0.15 | 0.94 | 0.10 | 1.76 | |||

Acceleration ripple (rad/s^{2}) | 1.15 | 1.56 | 0.84 | 2.12 | |||

Average speed (rad/s) | 26.89 | 16.59 | 11.82 | 16.14 | |||

Average acceleration (rad/s^{2}) | 1.48 | 0.16 | 0.72 | 0.07 | |||

Cycling time (s) | 30 | 30 | 30 | 30 |

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## Share and Cite

**MDPI and ACS Style**

Ho, P.-J.; Yi, C.-P.; Lin, Y.-J.; Chung, W.-D.; Chou, P.-H.; Yang, S.-C.
Torque Measurement and Control for Electric-Assisted Bike Considering Different External Load Conditions. *Sensors* **2023**, *23*, 4657.
https://doi.org/10.3390/s23104657

**AMA Style**

Ho P-J, Yi C-P, Lin Y-J, Chung W-D, Chou P-H, Yang S-C.
Torque Measurement and Control for Electric-Assisted Bike Considering Different External Load Conditions. *Sensors*. 2023; 23(10):4657.
https://doi.org/10.3390/s23104657

**Chicago/Turabian Style**

Ho, Ping-Jui, Chen-Pei Yi, Yi-Jen Lin, Wei-Der Chung, Po-Huan Chou, and Shih-Chin Yang.
2023. "Torque Measurement and Control for Electric-Assisted Bike Considering Different External Load Conditions" *Sensors* 23, no. 10: 4657.
https://doi.org/10.3390/s23104657