A Stair-Climbing Wheelchair with Novel Spoke Wheels for Smooth Motion

Abstract

With the aging population and rising demand for assistive devices, electric wheelchairs have garnered significant attention. However, existing stair-climbing wheelchairs often suffer from complex structural complexity and limited flexibility. Spoke-wheel mechanisms, known for their simple structure and strong obstacle-crossing capabilities, hold promise but experience oscillation on flat terrain. This paper proposes an improved spoke-wheel mechanism (Flexwheel), which integrates springs into the spokes. These springs compress to varying lengths under gravitational force during ground contact, while sliding grooves and pre-compression constraints regulate spoke length, ensuring a stable height. A novel selection method for the optimal spring constant is developed based on mass, spoke length, and the number of spokes. This mathematical framework is applicable to stable, smooth ground motion under varying friction conditions between the upper and lower spokes. A wheelchair prototype equipped with four Flexwheels, a self-balancing mechanism, and multi-sensor fusion technology is designed. The simulation results indicate that Flexwheel reduces the range in body height from 10.75 mm (traditional spoke wheels) to 3.39 mm on flat terrain, a 68.47% improvement. During stair climbing, Flexwheel significantly reduces body oscillation compared to traditional spoke or circular wheels. Physical experiments validate that Flexwheel exhibits a 6.28 mm height fluctuation vs. traditional spokes wheels’ 12.13 mm, a 48.28% improvement, demonstrating its effectiveness in enhancing wheelchair stability and adaptability.

1. Introduction

Independent mobility is crucial for the development of physical, cognitive, communication, and social skills [1]. With the global population aging at an accelerating rate [2] and an increasing incidence of cardiovascular diseases among younger individuals [3], the number of people with motor dysfunction has risen significantly. As a result, electric wheelchairs have become one of the most essential assistive devices. However, most commercially available wheelchairs rely on wheeled locomotion, which offers high efficiency on flat surfaces but exhibits limited adaptability to uneven terrain. This limitation poses significant challenges when navigating steps, stairs, and other architectural barriers, ultimately restricting users’ mobility and independence.

To enhance the practicality of wheelchairs, various stair-climbing designs have been developed, including planetary-wheel, tracked, legged, and hybrid mechanisms. Lixin Fang et al. [4] designed a planetary-wheel stair-climbing wheelchair that improves terrain adaptability through a planetary wheel cluster mechanism, but its control system is complex. Dean Kamen et al. developed the iBot [5], an intelligent wheelchair equipped with multiple wheel sets and a self-balancing system, providing limited obstacle-crossing capabilities. Nonetheless, the intermittent contact between the wheels and stairs results in user discomfort. Shigeo Hirose et al. [6] proposed a dual-track wheelchair driven by a dual continuously variable transmission (CVT) system, enhancing traction and maneuverability, yet its intricate control mechanism and high energy consumption pose practical challenges. Suyang Yu et al. [7] introduced a wheelchair with a Variable Geometry Single Tracked Mechanism (VGSTM) that improves obstacle-crossing performance by adjusting its shape and track tension. However, the design is prone to tipping and slipping, compromising stability. Baishya et al. [8] developed a three-wheeled stair-climbing robot featuring dual front wheels, a single rear wheel, and an adjustable connecting link. By actively controlling the link angle based on the real-time IMU feedback, the system mitigates tipping risks. However, its elongated body limits its application in narrow environments. Zhengyan Qi et al. [9] designed a quadruped/biped reconfigurable wheelchair utilizing a parallel leg mechanism to improve load capacity and ensure stable locomotion on both flat and stair terrains. Nevertheless, its complex control system increases maintenance difficulty and failure risks. Yusuke Sugahara et al. [10] developed a bipedal wheelchair based on a Stewart platform capable of ascending and descending stairs. Despite its advanced mobility, the intricate control requirements and elevated user position may induce psychological discomfort. In recent years, researchers have focused on hybrid mobility mechanisms. Weijun Tao et al. [11] proposed a wheel-track hybrid wheelchair that seamlessly transitions between flat surfaces and stairs via a mode-switching mechanism, improving terrain adaptability. Noriaki Imaoka et al. [12] designed a transformable wheel–leg hybrid wheelchair that enhances load response by positioning motors at the top. However, these hybrid designs tend to suffer from structural complexity, excessive weight, and high production costs, limiting their widespread adoption. Jinwoo Lee et al. [13] developed a variable-stiffness wheel inspired by the surface tension of liquid droplets, achieving both high-speed locomotion and obstacle-crossing. However, its complex manufacturing process hinders practical implementation. Overall, existing stair-climbing wheelchairs generally exhibit drawbacks such as structural complexity, inability to maintain user posture, and reliance on external assistance [14]. Therefore, a simple-control, highly adaptive wheelchair is needed to support users in independently completing daily activities.

Spoke wheels integrate the functions of both wheels and legs into a single wheel–leg mechanism, offering simple control and excellent terrain adaptability. These advantages make them highly promising for applications in outdoor exploration, material transportation, medical rehabilitation, and home mobility. For instance, TERMES robots developed by Harvard University [15] mimic ant swarm behavior for post-disaster search and environmental monitoring. The Whegs series from Case Western Reserve University [16] demonstrates superior locomotion on uneven terrains, making it suitable for hazardous-area operations. The ASGUARD platform from the German Research Center for Artificial Intelligence [17] is designed for military and emergency rescue missions, facilitating efficient transport. Additionally, A soft-driven robot developed by Hanyang University [18] achieves gecko-like crawling through coordinated wheel-leg movements, enabling operations in confined spaces for rescue and inspection tasks. These applications highlight the broad potential of spoke-wheel mechanisms. Despite their strong obstacle-crossing performance, maintaining stable motion on flat surfaces remains a challenge for traditional spoke wheels. To address this, researchers have explored transformable spoke wheels that morph into circular wheels for smoother locomotion, such as Texas A&M’s α-WaLTR [19], Shanghai Jiao Tong University’s Trimode [20], and Tianjin University’s RHex-T3 [21], as well as designs from Hanyang University [22] and the University of the West of England [23]. While these approaches improve performance, they introduce additional structural and control complexity. Without altering spoke-wheel morphology, Virginia Tech’s IMPASS [24] employs extendable active spokes to partially mitigate stability issues, albeit at the cost of increased system complexity. Similarly, Hanbat University [25] incorporated springs between spokes and their terminals to enhance compliance, yet this modification does not fully resolve the oscillation problem.

In summary, while spoke-wheel mechanisms offer simple control and strong adaptability to uneven terrain, their inherent oscillation during flat-terrain movement limits their practical application in wheelchairs. To overcome this limitation, this paper proposes a novel spoke-wheel mechanism, Flexwheel, designed to achieve smooth wheeled motion while preserving obstacle-crossing advantages. Based on this mechanism, an intelligent stair-climbing wheelchair prototype is developed, featuring simple control, strong terrain adaptability, and the ability to maintain user posture stability (as shown in Figure 1). Software simulations and physical experiments demonstrate that the proposed wheelchair not only mitigates the oscillation issue of traditional spoke wheels on flat surfaces but also integrates multiple sensors to enhance autonomy and intelligence.

2. Mechanism Design

The spoke-wheel wheelchair (as shown in Figure 2) consists of a top-mounted seat and four Flexwheels symmetrically positioned on both sides of the chassis. The four-spoke wheel design not only ensures stable locomotion across various terrains but also enables maneuvering within confined spaces, such as stair platforms, by adjusting the speed differential between the left and right Flexwheels. To enhance user comfort and safety, the wheelchair incorporates a chair adjustment mechanism to ensuring posture stability.

In the Flexwheel design, the motor is fixed to the wheelchair chassis with its hub directly connected to the motor output shaft. Each wheel comprises six upper spokes that are fixed to the hub, each incorporating a compression spring. The lower spokes are slidably mounted onto the upper spokes, counteracting the force of the compression springs. The connection between the upper and lower spokes is secured using a locking screw, which passes through a fixed hole in the upper spoke and a sliding groove in the lower spoke, allowing for relative movement. At the ends of the lower spokes, spherical foot pads are rigidly mounted to facilitate ground contact. When the foot pads engage with the terrain, the lower spokes slide relative to the upper spokes, compressing the springs. For traditional spoke wheels, as the wheel rotates, the angle between the spokes and the ground continuously changes. Larger angles lead to an increase in hub height, inducing vertical oscillations. By optimizing the spring constant and the constraints of the sliding groove, the compression of the springs counteracts the increase in hub height, stabilizing the wheelchair body and ensuring smoother motion.

3. Theoretical Analysis

3.1. Spring Configuration

3.1.1. Spoke Deformation Analysis

Let the number of spokes be n, and the distance from the center of rotation to the center of the spherical foot pad is denoted as l. The angle between the spokes and the ground is represented by θ. As the Flexwheel rotates, θ varies within the range:

$${\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\pi }{n}}\le \theta \le {\displaystyle \frac{\pi }{2}}$$

(1)

As shown in Figure 3, the supporting force $F_{\mathrm{N}}$ acting on the foot pad can be decomposed into two components, one along the spoke direction ($F_1$) and the other perpendicular to it ($F_2$):

$$\left\{\begin{array}{c}{F}_1=mg\sin \theta \\ F_2=mg\cos \theta \end{array}\right.$$

(2)

In practical applications, friction $f$ is generated due to the relative motion between the upper and lower spokes, given by:

$$f={\mu }_0·F_2$$

(3)

where ${\mu }_0$ is the friction coefficient between 304 stainless-steel surfaces.

When the spring constant $k$ approaches infinity, the compression spring undergoes negligible deformation. In this case, when the spoke is perpendicular to the ground, the maximum height of the wheelchair body (which coincides with the hub center height) is:

$$h_{\mathrm{B}\mathrm{M}\mathrm{A}\mathrm{X}}=l$$

(4)

The target height of the body’s center of mass (CoM), denoted as $h_{\mathrm{B}}$, should satisfy:

$$h_{\mathrm{B}}=j·l$$

(5)

where j is the body height ratio. To maintain a stable body height while ensuring obstacle-crossing capability, the range of j is:

$$0.4\le j\le \sin \left({\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\pi }{n}}\right)$$

(6)

During Flexwheel rotation, in order to maintain the CoM height at $h_{\mathrm{B}}$, the deformation $∆l$ of the compression spring must satisfy:

$$\left(l-∆l\right)\sin \theta =h_{\mathrm{B}}$$

(7)

Solving for $∆l$:

$$∆l=l-{\displaystyle \frac{h_{\mathrm{B}}}{\sin \theta }}$$

(8)

When the spring deforms by $∆l$, it generates an elastic force $F_{\mathrm{E}}$, which follows Hooke’s law:

$$F_{\mathrm{E}}=k∆l$$

(9)

Applying Newton’s second law and considering force equilibrium along the spoke direction, we obtain:

$$F=f+F_{\mathrm{E}}$$

(10)

By substituting Equations (2), (3) and (9) into (10), the relationship between body height $h_{\mathrm{B}}$, angle $\theta$, and spring constant $k$ is derived:

$$mg\sin \theta ={\mu }_{0}mg\cos \theta +k\left(l-{\displaystyle \frac{h_{\mathrm{B}}}{\sin \theta }}\right)$$

(11)

During Flexwheel rotation, the actual spoke length $l_{\mathrm{A}}$ can be expressed as $l_{\mathrm{A}}=h_{\mathrm{B}}/\sin \theta$. Rearranging, the required spring constant k is given by:

$$k={\displaystyle \frac{mg\left(\sin \theta -{\mu }_0\cos \theta \right)}{l-l_{\mathrm{A}}}}$$

(12)

Further transformation yields the final expression for the body height:

$$h_{\mathrm{B}}=l\sin \theta -{\displaystyle \frac{mg\left({\sin }^2\theta -{\mu }_0\sin \theta \cos \theta \right)}{k}}$$

(13)

3.1.2. Selection Method for Spring Constant $k$

The spring constant k of the compression spring can be determined through the following four steps:

1.

Determine the optimal body height ratio${ \mathit{j}}_{\mathbf{B}\mathbf{E}\mathbf{S}\mathbf{T}}$;

Select different body height ratios $j$, substitute the corresponding $h_{\mathrm{B}}$ into Equation (12), and calculate the variance of the $k$ over the range of $\theta$.

The height $h_{\mathrm{B}}$ that results in the minimum variance of $k$ is selected as the optimal target body height, denoted as $h_{\mathrm{B}\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$.

2.

Determine the range of$\mathit{k}$ when${ \mathit{h}}_{\mathbf{B}}={\mathit{h}}_{\mathbf{B}\mathbf{B}\mathbf{E}\mathbf{S}\mathbf{T}}$;

Substitute $h_{\mathrm{B}\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$ into Equation (12) and calculate the variation range of spring constant $k$ over the full range of $\theta$.

3.

Determine the optimal spring constant${ \mathit{k}}_{\mathbf{B}\mathbf{E}\mathbf{S}\mathbf{T}}$;

Within the $k$-range obtained in Step 2, select different $k$ values and substitute them into Equation (13) to compute the variance of $h_{\mathrm{B}}$ over the range of $\theta$.

The $k$ that results in the minimum variance of $h_{\mathrm{B}}$ is selected as the optimal spring constant, denoted as $k_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$.

4.

Determine the range of${ \mathit{h}}_{\mathbf{B}}$ when$\mathit{k}={\mathit{k}}_{\mathbf{B}\mathbf{E}\mathbf{S}\mathbf{T}}$;

Substituting $k_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$ into Equation (13), we can calculate the variation range of $h_{\mathrm{B}}$ over the range of $\theta$.

If the resulting variation is sufficiently small, the method is validated. Additionally, we record the $h_{\mathrm{B}}$ value corresponding to the minimum spring deformation as $h_{\mathrm{I}}$.

3.1.3. Pre-Compression Determination

When $\theta =\pi /2-\pi /n$, the spring deformation is minimized, corresponding to a special state where both foot pads simultaneously contact the ground. To prevent excessive deformation caused by sudden force changes, the spring must be pre-compressed by a displacement $l_{\mathrm{I}}$ during assembly. This pre-compression ensures that the spring neither extends nor compresses further under bipedal loading conditions. In this state, the body’s CoM height is $h_{\mathrm{I}}$. The required pre-compression $l_{\mathrm{I}}$ is determined as follows:

$$l_{\mathrm{I}}=l-{\displaystyle \frac{h_{\mathrm{I}}}{\sin \left(\frac{\pi }{2}-\frac{\pi }{n}\right)}}$$

(14)

3.2. Theoretical Validation

The number of spokes n plays a crucial role in influencing the wheelchair’s overall stability and stair-climbing capability. To investigate this relationship, theoretical calculations were conducted to assess the range of the body height during Flexwheel rotation, as well as the maximum achievable stair-climbing height (the detailed calculation process is described in Section 3.3). An even number of spokes is adopted to ensure symmetrical force distribution during rotation. Accordingly, a comparative analysis was performed on Flexwheel configurations with 4, 6, and 8 spokes. A schematic diagram of the maximum stair-climbing height is presented in Figure 4, and the corresponding results are summarized in

Table 1. The performance of Flexwheels with different spoke numbers.

Number of Spokes n	$\mathbf{The} \mathbf{Value} \mathbf{of} {\mathit{k}}_{\mathbf{B}\mathbf{E}\mathbf{S}\mathbf{T}}$	The Range of the Body Height	Maximum Climbing Height
4	56.88 N/m	2.3 mm	120.27 mm
6	61.52 N/m	0.5 mm	110.85 mm
8	62.82 N/m	0.1 mm	103.00 mm

. While increasing the number of spokes could enhance the continuity of ground contact, it simultaneously reduces the stair-climbing performance. In contrast, fewer spokes would undermine overall stability. After balancing these factors, six spokes were selected as the optimal configuration, achieving a balance between stability, climbing capability, and lightweight design.

The simulation model was established based on the physical prototype, with specific parameters listed in

Table 2. Model parameters of the physical prototype.

Description	Symbol	Value
Length	$l_{\mathrm{L}}$	380 mm
Width	$l_{\mathrm{W}}$	200 mm
Number of spokes	$n$	6
Original spoke length	$l$	150 mm
Spring free length	$l_{\mathrm{S}}$	100 mm
Rigid segment length	$l_{\mathrm{R}}$	50 mm
Sliding groove length	$l_{\mathrm{C}}$	10 mm
Body mass	$4m$	2 kg

. The original spoke length consists of two components: the rigid segment length $l_{\mathrm{R}}$ and the spring free length $l_{\mathrm{S}}$, i.e., $l=l_{\mathrm{R}}+l_{\mathrm{S}}$.

In the simulation environment, the friction coefficient was set to ${\mu }_0=0$. Based on the parameters in Table 2, the spring constant selection method outlined in Section 3.1.2 is employed. The results of each step are illustrated in Figure 5. According to Equation (6), the allowable range of the body height ratio $j$ was calculated as $0.4<j<0.866$. As shown in Figure 5a, when $j=0.47$ (corresponding to $h_{\mathrm{B}}=70.5\mathrm{m}\mathrm{m}$), the variance of the spring constant $k$ reaches its minimum. Therefore, $h_{\mathrm{B}\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$ is determined to be 70.5 mm. In Step 2, the variation of $k$ with respect to the spoke-ground angle $\theta$ was analyzed. Figure 5b reveals that the spring constant $k$ varies within $61.41\le k\le 61.86$. Substituting these values into Equation (13), the variance of $h_{\mathrm{B}}$ across the full range of $\theta$ was further calculated. The results indicate that the minimum variance occurs at $k=61.52\mathrm{N}/\mathrm{m}$, determining the optimal spring constant $k_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$ to be 61.52 N/m. Figure 5d demonstrates the variation of $h_{\mathrm{B}}$ with $\theta$, showing a range of $70.17\mathrm{m}\mathrm{m}\le h_{\mathrm{B}}\le 70.63\mathrm{m}\mathrm{m}$, with a total variation of 0.46 mm. These results confirm that the proposed method for selecting the spring constant $k$ effectively stabilizes the body’s CoM height under frictionless conditions. Finally, by substituting $h_{\mathrm{I}}=70.17\mathrm{m}\mathrm{m}$ into Equation (14), the required pre-compression displacement $l_{\mathrm{I}}$ was calculated as 69 mm.

In real-world conditions, friction arises due to the relative motion between the upper and lower spokes. Given that the friction coefficient between 304 stainless steel is $t{\mu }_0=0.5$, the selection process for the optimal spring constant $k$ was repeated under these conditions. The results are illustrated in Figure 6. From the analysis of the spring constant range in Figure 6b, it is evident that the variance of $h_{\mathrm{B}}$ decreases monotonically as $k$ increases. To further pinpoint the value of $k$ that minimizes the variance of $h_{\mathrm{B}}$, an expanded parameter space was employed for exhaustive iteration. This optimization yielded an optimal spring constant $k_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}=252\mathrm{N}/\mathrm{m}$. Under these conditions, the body’s CoM height varies within $111.6\mathrm{m}\mathrm{m}\le h_{\mathrm{B}}\le 131\mathrm{m}\mathrm{m}$, and the range is 19.4 mm. The presence of friction leads to a substantial increase in $k_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{T}}$, while the oscillation amplitude and variation trend of the body closely resemble those of traditional spoke wheels.

To enhance the damping performance, an optimized approach was implemented by integrating springs with a lower spring constant and a sliding groove constraint. When using springs with reduced $k$, the body height exhibits a monotonically decreasing trend, leading to significant oscillations. To mitigate this, the spoke length is constrained by the sliding groove, ensuring a minimum length of $l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}=l-l_{\mathrm{I}}-l_{\mathrm{C}}$. This constraint stabilizes the body height by limiting excessive compression of the spokes. As shown in Figure 7, the body height variation is divided into three phases: (a) Spoke A compresses to its minimum length $l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}$, causing the body height to decrease; (b) Spoke A remains at $l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}$ while moving, resulting in the body height first rising and then falling; and (c) both Spoke A and Spoke B contact the ground simultaneously. The body height is governed by Spoke B, leading to a subsequent increase in height. By substituting $l_{\mathrm{A}}=h_{\mathrm{B}}/\sin \theta$ into Equation (13), the body height during Phases (a) and (b) can be expressed as:

$$\left\{\begin{array}{c}{l}_{\mathrm{A}}=max\left(l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}},\hspace{1em}l-{\displaystyle \frac{mg(\mathrm{s}\mathrm{i}\mathrm{n}\theta -{\mu }_0\mathrm{c}\mathrm{o}\mathrm{s}\theta )}{k}}\right)\\ h_{\mathrm{B}}=l_{\mathrm{A}}·\sin \theta \end{array}\right.$$

(15)

At the critical angle ${\theta }_{\mathrm{C}1}$, the following condition is satisfied:

$$l-{\displaystyle \frac{mg\left(\sin {\theta }_{\mathrm{C}1}-{\mu }_0\cos {\theta }_{\mathrm{C}1}\right)}{k}}=l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}$$

(16)

For ${\theta }_{\mathrm{C}1}<\theta <{\theta }_{\mathrm{C}2}$, the sliding groove restricts Spoke A’s minimum length to $l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}$, resulting in an initial rise in body height followed by a decline. During this descent, Spoke B contacts the ground prematurely. Based on the sine law, the critical angle ${\theta }_{\mathrm{C}2}$ satisfies:

$${\displaystyle \frac{l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}}{\sin \left({\theta }_{\mathrm{C}2}-\pi /3\right)}}={\displaystyle \frac{l-l_{\mathrm{I}}}{\sin \left(\pi -{\theta }_{\mathrm{C}2}\right)}}$$

(17)

For $\theta <{\theta }_{\mathrm{C}2}$, the body height is primarily determined by the length of Spoke A ($l_{\mathrm{A}}$). Conversely, for $\theta >{\theta }_{\mathrm{C}2}$, the height is governed by the length of Spoke B ($l_{\mathrm{B}}$). When Spoke B contacts the ground, its spoke–ground angle is $\theta -\pi /3$. This smaller angle leads to reduced spring force and deformation. Although theoretical predictions suggest that $l_{\mathrm{B}}$ should elongate, the pre-compression displacement $l_{\mathrm{I}}$ prevents excessive extension. Thus, the body height in Phase (c) is reformulated as:

$$\left\{\begin{array}{c}{l}_{\mathrm{B}}=min\left(l-{\displaystyle \frac{mg(\mathrm{s}\mathrm{i}\mathrm{n}(\theta -\pi /3)-{\mu }_0\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\pi /3\left)\right)}{k}},l-l_I\right)\\ h_{\mathrm{B}}=l_{\mathrm{B}}\cdot sin(\theta -\pi /3)\end{array}\right.$$

(18)

As illustrated in the figure, when Spoke A disengages from the ground, the body height seamlessly transitions its initial contact value, ensuring smooth motion without sudden discontinuities, in alignment with mechanical principles. During Phase (b), when Spoke A is perpendicular to the ground ($\theta =90°$), the body height reaches its peak value:

$$h_{\mathrm{A}\mathrm{M}\mathrm{A}\mathrm{X}}=l_{\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{N}}·\sin \left(90°\right)=l-l_{\mathrm{I}}-l_C$$

(19)

To minimize body height fluctuation, the peak height $h_{\mathrm{A}\mathrm{M}\mathrm{A}\mathrm{X}}$ must match the initial body height $h_{\mathrm{I}}$:

$$h_{\mathrm{A}\mathrm{M}\mathrm{A}\mathrm{X}}=h_{\mathrm{I}}$$

(20)

where $h_{\mathrm{I}}$ represents the body height at the moment Spoke A first makes contact with the ground. From Equation (14), it can be expressed as:

$$h_{\mathrm{I}}=\left(l-l_{\mathrm{I}}\right)·\sin \left({\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\pi }{n}}\right)$$

(21)

By substituting parameters from Table 2 into the above equations, the pre-compression displacement is calculated as $l_{\mathrm{I}}\approx 75\mathrm{m}\mathrm{m}$. Consequently, after installation, the effective spoke length without ground contact is $l-l_{\mathrm{I}}=75\mathrm{m}\mathrm{m}$. At $\theta =60°$, the optimal spring constant $k$ that minimizes the variance of $h_{\mathrm{B}}$ is derived from Equation (10):

$$k={\displaystyle \frac{F-f}{l_{\mathrm{I}}}}=40.25\mathrm{N}/\mathrm{m}$$

(22)

With this spring constant and the sliding groove structure, the body height varies within $58.6\mathrm{m}\mathrm{m}\le h_{\mathrm{B}}\le 64.6\mathrm{m}\mathrm{m}$, yielding a range of 6 mm. Compared to a traditional spoke wheel (spoke length $l-l_{\mathrm{I}}$), which exhibits a 10 mm height variation, our design achieves a 40% reduction in fluctuation. Furthermore, the proposed sliding groove mechanism improves height stabilization by 69% over a design that only optimizes the spring constant. Even under high-friction conditions, the combination of springs with lower-$k$ and the sliding groove maintains stability, underscoring the robustness of the design.

3.3. Maximum Climbing Height

In daily life, navigating complex terrains such as steps and stairs presents a major challenge for wheelchairs. The legged locomotion of spoke wheels offers significant advantages in overcoming these terrains. As shown in Figure 8, the maximum step height $h_{\mathrm{S}}$ that the Flexwheel can climb is determined when two adjacent foot pads make contact with the step, with the upper foot pad positioned at the edge of the step. The maximum climbing height consists of the hub height and the vertical projection of the spoke length. Among these, the hub height depends on the contact angle between the ground and the engaged spoke, which is 3π/n. By substituting $\theta =3\pi /n$ into Equation (13) to calculate $h_{\theta =3\pi /n}$, the maximum step height $h_{\mathrm{S}\mathrm{M}\mathrm{A}\mathrm{X}}$ can be expressed as:

$$h_{\mathrm{S}\mathrm{M}\mathrm{A}\mathrm{X}}=h_{\theta =3\pi /n}+\left(l-l_{\mathrm{I}}\right)·\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\pi }{n}}\right)$$

(23)

where under negligible friction, $h_{\theta =3\pi /n}=l\sin 3\pi /n-mg{\sin }^2\left(3\pi /n\right)/k$. Under high-friction conditions, depending on the value of $3\pi /n$, we substitute this into either Equation (15) or Equation (18).

4. Simulation Verification

To validate the performance of the proposed wheelchair design, a comprehensive simulation workflow was implemented, consisting of the following steps:

• 3D Modeling

A detailed 3D model of the wheelchair was constructed using SolidWorks 2023, including the Flexwheel mechanism and frame structure.

2.

Simulation Environment Integration

The model was exported and imported into V-REP (CoppeliaSim) to establish a dynamic simulation environment.

3.

Control Program Development

Control algorithms were developed in MATLAB R2023b to coordinate joint movements and record positional data. The V-REP environment was connected to MATLAB via the remote API, enabling real-time command execution and data exchange.

4.

Physics Engine and Joint Configuration

The ODE physics engine was selected for simulating physical interactions.

Revolute joints were set to dynamic mode with velocity control, where velocity inputs were governed by the MATLAB control script.

Prismatic joints were configured in spring control mode, with the spring constant (K) and damping coefficient (C) set to optimized values derived from theoretical analysis. Specifically, the damping coefficient C was determined to satisfy the critical damping condition. The condition follows the formula $\mathrm{C}=2\sqrt{\mathrm{K}\mathrm{m}}$.

Additionally, the Position Range (Pos. range) of each prismatic joint was set to the sliding groove length $l_{\mathrm{C}}$, while both Pos. min and the initial Position were configured as $l-l_{\mathrm{I}}$.

4.1. Flat Terrain Motion Simulation

To evaluate the influence of different spring constants on the wheelchair’s motion performance, we conducted simulations on wheelchair models equipped with various springs traversing flat terrain. The Flexwheel was driven at a rotational speed of 1.2 rad/s, and the fluctuation range of the body height was recorded as a function of the spring constant $k$, as illustrated in Figure 9. The results indicate that as $k$ increases, the fluctuation range of the body height initially decreases and then increases. When $k=61.7\mathrm{N}/\mathrm{m}$, the fluctuation range reaches its minimum, demonstrating that this spring configuration effectively reduces oscillations during flat terrain motion. These findings align well with theoretical predictions, further validating the feasibility of the method proposed in Section 3.1.2.

However, the simulation also revealed significant variations in the model’s pitch angle, suggesting an uneven load distribution between the front and rear wheels. During the rotation of the Flexwheel, the phase from the initial ground contact to the vertical position involves acceleration, where kinetic energy is converted into the spring’s elastic potential energy. As the spoke rotates past the vertical position and lifts off, the stored elastic potential energy is released back into kinetic energy. The cyclic energy conversion influences the wheelchair’s dynamic response, affecting load distribution. During acceleration, inertial forces induce a rearward thrust on the front wheel, causing slight lifting and shifting more load to the rear wheel [26]. Such variations in pitch angle could cause discomfort to the wheelchair user. To mitigate this issue, the front wheel should be equipped with a spring of lower $k$. Figure 10 illustrates the variance of the body’s pitch angle for different front-wheel spring constants. Through testing multiple front-wheel spring configurations, it was found that the pitch angle variance is minimized when $k_{\mathrm{F}\mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}}=40.5\mathrm{N}/\mathrm{m}$.

Therefore, the rear Flexwheels were assigned a spring constant of 61.7 N/m, while the front Flexwheels were set to 40.5 N/m. The parameter configuration of the simulation model is shown in

Table 3. Parameter configuration of the simulation model.

Joints	Parameter	Value
Revolute joints	Speed [rad/s]	1.2
Prismatic joints	Pos. min [m]	0.069
	Pos. range [m]	0.010
	Position [m]	0.069
Front prismatic joints	Spring constant K [N/m]	40.5
	Spring constant C [N·s/m]	9.0
Rear prismatic joints	Spring constant K [N/m]	61.7
	Spring constant C [N·s/m]	11.1

.

Figure 11 compares body height variations between a traditional spoke wheel model and a Flexwheel-equipped model. The results show that the traditional spoke wheels experience significant body height fluctuations, with a range of 10.75 mm. In contrast, the Flexwheel model reduces this range to 3.39 mm. Oscillations in pitch angle contribute to variations in body height, leading to minor discrepancies between the simulation results and theoretical expectations. However, the Flexwheel model achieves a 68.47% reduction in body height fluctuation range compared to the traditional design, highlighting its effectiveness in height stabilization. Additionally, the average body height of the Flexwheel model is 79.87 mm. Accounting for the radius of the Flexwheel’s foot pad, the average distance from the hub center to the foot pad’s spherical center was 70.87 mm, which closely aligns with the theoretical value of 70.4 mm. A demonstration can be viewed in the supplementary video via https://youtu.be/jJpFh1ejd24 (accessed on 7 May 2025)

4.2. Stair Climbing Simulation

To assess the wheelchair’s performance in stair climbing, stairs with a riser height of 3 cm and a tread width of 10 cm were constructed. Wheelchair models equipped with circular wheels, traditional spoke wheels, and Flexwheels were driven to climb the stairs, and the results are presented in Figure 12. The simulation results indicate that the Flexwheel-equipped wheelchair effectively reduces jolts and maintains a relatively stable motion state during stair climbing. In contrast, models with circular or traditional spoke wheels exhibit significant oscillations and instability.

To further evaluate the climbing capability and stability of the wheelchair on stairs with different specifications, the root mean square (RMS) of Z-axis acceleration was employed as an indicator to quantify the magnitude of vibration. The RMS acceleration reflects the overall level of vibrational energy—the smaller the value, the better the wheelchair’s stability during stair climbing and the greater the riding comfort. In addition to stairs with dimensions of 3 cm × 10 cm, two additional stair configurations—2 cm × 4 cm and 4 cm × 8 cm—were constructed for testing. The experimental results, as shown in

Table 4. Comparison of stair-climbing vibration performance between traditional spoke wheels and Flexwheels based on simulation.

Motion Mechanism	Stair Specifications	RMS of Z-Axis Acceleration
Traditional spoke wheels	2 cm × 4 cm	0.48 m/s2
Flexwheels	2 cm × 4 cm	0.39 m/s2
Traditional spoke wheels	3 cm × 10 cm	0.51 m/s2
Flexwheels	3 cm × 10 cm	0.42 m/s2
Traditional spoke wheels	4 cm × 8 cm	0.52 m/s2
Flexwheels	4 cm × 8 cm	0.47 m/s2

, indicate that the Flexwheel is capable of handling stairs of various sizes. Furthermore, the Flexwheel consistently exhibits lower vibration levels compared to the traditional spoke wheel during the climbing process. These findings further underscore the critical role of the Flexwheel in enhancing wheelchair performance across complex terrains.

5. Experimental Validation

5.1. Prototype Fabrication

In terms of prototype fabrication, different manufacturing processes were selected and customized based on the characteristics and functions of each component. The key parts of the spoke wheel mechanism were made of stainless steel and manufactured using CNC machining to ensure structural precision and durability. The seat and motor housings were fabricated through 3D printing, while the baseplate and motor mounting plates were cut from carbon fiber sheets, achieving a lightweight yet high-strength structure. The combination of multiple manufacturing techniques not only enhanced the overall structural stability of the design but also significantly improved the wheelchair’s reliability and service life.

Based on Equation (22), the rear wheels of the prototype should be equipped with springs of ${k\mathrm{\text{'}}}_{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{R}}=40.25\mathrm{N}/\mathrm{m}$. By maintaining the front-to-rear spring constant ratio from the simulation model, the front-wheel spring constant for the prototype was determined as follows:

$${k\mathrm{\text{'}}}_{\mathrm{F}\mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}}={\displaystyle \frac{k_{\mathrm{F}\mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}}}{k_{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{R}}}}·{k\mathrm{\text{'}}}_{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{R}}=26.42$$

(24)

where $k_{\mathrm{F}\mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}}$ and $k_{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{R}}$ represent the spring constants of the front and rear wheels in the simulation model, respectively, and ${k\mathrm{\text{'}}}_{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{R}}$ is the rear-wheel spring constant for the physical prototype. By testing prototypes with various front-wheel spring constants, the minimum pitch angle variance was achieved when ${k\mathrm{\text{'}}}_{\mathrm{F}\mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}}=26.42\mathrm{N}/\mathrm{m}.$ Therefore, the final prototype configuration adopted rear springs with ${k\mathrm{\text{'}}}_{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{R}}=40.25\mathrm{N}/\mathrm{m}\mathrm{m}$ and front springs with ${k\mathrm{\text{'}}}_{\mathrm{F}\mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}}=26.42\mathrm{N}/\mathrm{m}.$ To enhance the intelligence of the wheelchair, the control system integrates multiple sensors, as illustrated in the block diagram in Figure 13. The controller adopts the high-performance STM32F407VET6 microcontroller (Manufactured by STMicroelectronics, located in Geneva, Switzerland), which includes an onboard IMU for the attitude sensor. The wheelchair’s bus servos support two operational modes: servo mode and geared motor mode. In servo mode, the servo enables precise rotational positioning within a 240° range, while in geared motor mode, it allows continuous 360° rotation with controllable direction and speed. The selected servos are Hiwonder LX-224 Bus Servo (Manufactured by Hiwonder Technology Co., Ltd., located in Shenzhen, China), featuring a rated voltage of 7.4, maximum torque of 20 kg·cm, and a rotation speed of 50 r/min at no load. The bus servos are controlled via UART-based asynchronous serial communication. The control parameters include the servo ID, target position, or target speed, depending on the issued command. By configuring different operating modes of the servos, the power system ensures precise Flexwheel control and stable self-balancing operation. The wheelchair is equipped with four laser ranging modules (ATK-MS53L1M), which communicate with the microcontroller via UART in Modbus mode. These sensors enable real-time distance measurement, providing critical data for stair detection. Power is supplied by a 7.4 V rechargeable lithium battery, with an onboard 5 V regulator ensuring stable power distribution to actuators and sensor modules.

5.2. Flat Terrain Motion Experiment

In this experiment, the bus servos were set to motor mode with a movement speed of 60 deg/s, driving the prototypes equipped with traditional spoke wheels and Flexwheels on flat terrain. The motion process was recorded on video and analyzed in Kinovea-2023.1.2 software to generate body height variation curves over time, as illustrated in Figure 14. The experimental results indicate that the wheelchair prototype equipped with traditional spoke wheels exhibited significant fluctuations in body height, with a fluctuation range of 12.13 mm. In contrast, the prototype equipped with Flexwheels demonstrated improved stability, reducing the fluctuation range to 6.28 mm—a 48.28% reduction.

During the initial ground contact of a single spoke, the spoke compresses to its minimum length, causing a temporary decrease in body height. As the spoke remains compressed, the body height initially rises and then declines. When a second spoke contacts the ground, the body height increases once more. Throughout the wheelchair’s motion, multiple factors contribute to the observed fluctuations, including pitch angle variations, friction between the upper and lower spokes, and the time required for the springs to fully compress. These dynamic interactions account for the discrepancies between the experimental results and theoretical predictions.

5.3. Stair-Climbing Experiment

To evaluate the stair-climbing performance of the Flexwheel-equipped wheelchair in real-world conditions, physical experiments were conducted on stairs with dimensions of 2 cm × 4 cm. Wheelchair prototypes equipped with traditional spoke wheels and Flexwheels were tested under identical conditions. As shown in Figure 15, the Flexwheel configuration significantly improved stability and reduced vibration during ascent, aligning closely with the results observed in simulation.

Additionally, comparative experiments were conducted on stairs measuring 4 cm × 8 cm to further evaluate the system’s robustness across a broader range of stair configurations. A summary of the results from the three stair-climbing experiments is presented in

Table 5. Comparison of stair-climbing vibration performance between traditional spoke wheels and Flexwheels based on physical prototyping.

Motion Mechanism	Stair Specifications	RMS of Z-Axis Acceleration
Traditional spoke wheels	2 cm × 4 cm	0.45 m/s2
Flexwheels	2 cm × 4 cm	0.36 m/s2
Flexwheels	4 cm × 8 cm	0.43 m/s2

. The Flexwheel configuration consistently exhibited lower RMS values of Z-axis acceleration compared to the traditional spoke wheel configuration. These outcomes confirm the Flexwheel’s superior performance in reducing oscillation and enhancing overall climbing stability, further validating the effectiveness of the design under practical conditions.

5.4. Multi-Sensor Fusion Stair Climbing Experiment

In practical indoor scenarios, staircases are often arranged in L-shaped or U-shaped patterns. To enable autonomous stair-climbing, the proposed control system integrates four laser ranging modules and one camera for environmental perception. The flowchart of the embedded controller system is shown in Figure 16.

During operation, distance measurements from four laser ranging modules are stored in an array, where each index corresponds to a specific sensor. When the front-left module detects a distance exceeding 500 mm, the system commands the left wheels to reverse while the right wheels move forward. This maneuver continues until the two right-side modules measure equal distances, indicating that the wheelchair is aligned parallel to the wall. Subsequently, the camera activates to identify the presence of stairs. If stairs are detected ahead, the system categorizes the environment as an L-left stair scenario and proceeds directly with stair climbing; otherwise, it identifies a U-left stair scenario, in which case the prototype executes a turn-left maneuver again before ascending the stairs. The same logic applies to right-turn scenarios. This hierarchical control strategy enables the wheelchair to flexibly adapt to complex environments, efficiently detect stairs, and dynamically plan its path in real-time.

To evaluate the wheelchair’s motion performance and adaptive capability on stairs, experiments were conducted on different stair types, each with a riser height of 2 cm and a tread width of 5 cm, as shown in Figure 17. Under identical control parameters, the wheelchair was tested on L-left, L-right, U-left, and U-right stair configurations. The experimental results demonstrated that the wheelchair not only successfully climbed stairs but also exhibited strong adaptability to various stair geometries.

6. Conclusions

This paper presents an optimized spoke-wheel mechanism (Flexwheel) and develops a stair-climbing wheelchair designed for stable motion. The wheelchair has the advantages of simple control and strong adaptability, enabling smooth motion on flat terrain and reliable performance across complex environments. Simulation analyses and prototype experiments demonstrate that the Flexwheel-equipped wheelchair significantly outperforms traditional spoke wheel designs in motion stability, both on flat surfaces and stairs. However, the simulation and physical test results did not fully align with theoretical expectations, primarily due to frictional losses and pitch angle fluctuations during motion. When friction is low, the damping effect is pronounced. As the friction coefficient increases, it becomes increasingly difficult to achieve smooth motion with springs alone. To address this, a sliding groove mechanism is introduced to restrict the minimum spoke length, thereby enhancing motion stability. Future research will focus on optimizing the mechanical structure design, particularly reducing friction between the upper and lower spokes, to further improve the wheelchair’s performance.

7. Patents

Patent Title: A Spoke-Wheel Mechanism for Achieving Smooth Wheeled Motion.

• Patent Number: CN115817066B.
• Applicant: Nanjing University of Information Science and Technology.
• Inventors: Yuting Li, Zhong Wei, Jiwen Zhang. et al.
• Filing Date: 24 November 2022.
• Publication Date: 16 August 2024.
• Status: Granted.