Design and Performance Analysis of Deformed Wheels for Dual-Mode Wheels on Complex Pavement

Design and Performance Analysis of Deformed Wheels for Dual-Mode Wheels on Complex Pavement

Xinyu Jia (Corresponding Author)

School of Mechanical Engineering Tianjin University of Technology and Education

Tianjin 300222, China

Shijie Guo

School of Mechanical Engineering Tianjin University of Technology and Education

Tianjin 300222, China

AbstractTo address the challenges of inefficient mode switching and inadequate dynamic stability in wheel-legged robots operating on complex terrains, this study proposes a novel dual- mode deformable wheel mechanism. The design employs a crank- rocker-guide rod tandem transmission system integrated with a self-locking structure, enabling rapid and reliable transitions between wheeled and legged operational modes. Through mechanical modeling, the forward and reverse obstacle-crossing characteristics of the wheel-leg mechanism are analyzed, and the optimal number of wheel-legs is determined. Furthermore, a quantitative relationship between the maximum obstacle-crossing height and the wheel diameter is derived. Kinematic models for both synchronized gait and diagonal gait are established to investigate the variations in displacement, velocity, and acceleration under these two gait patterns. The results demonstrate that the proposed deformable wheel mechanism exhibits significant enhancements in obstacle-crossing capability and motion stability compared to conventional designs, offering a promising technical solution for the development of mobile robots in complex terrain environments.

Keywords Deformed hub; wheel – leg robot; obstacle

performance; gait planning; kinematics analysis

1. INTRODUCTION

   Industrial mobile robots exhibit excellent environmental adaptability and operational reliability. They are progressively replacing manual labor in performing high-risk, high-intensity, and repetitive tasks (1). With the rapidly growing demand for operation in unstructured environments, mobile robot technology has ushered in new developmental opportunities (2). As a crucial branch within the field of intelligent robotics, mobile robots integrate core functionalities including environmental perception, dynamic decision-making, motion planning, and behavior control (3)(4)(5). They have demonstrated significant application value in domains such as disaster relief (6), military reconnaissance (7), space exploration (8), pipeline inspection (9), and livelihood services (10).

   In terms of adaptability to complex terrain, wheel-legged robots possess unique structural advantages. They combine the

   efficient mobility of wheeled mechanisms with the flexibility of legged mechanisms (11)(12)(13), making them a key research direction in mobile robotics. In recent years, scholars both domestically and internationally have achieved significant breakthroughs in bionic locomotion mechanisms (14), intelligent control systems (15), and novel actuation methods

   (16). Comparatively, traditional wheeled mobile mechanisms offer distinct advantages in terms of high movement speed and control simplicity (17). However, they exhibit significant limitations in unstructured environments (18). Research and exploration into novel mobile mechanism design are therefore of great importance for enhancing the motion efficiency and terrain adaptability of balancing robots (19).

   Currently, wheel-legged robots still face numerous challenges in balancing complex terrain adaptability with motion efficiency. A planetary wheel-leg architecture demonstrated by Zhang et al. employs an epicyclic gear train that allows a 100- mm-diameter wheel to climb vertical steps up to twice its own radius; nevertheless, the additional gear stages and locking mechanisms increase part count by 38% compared with a conventional rigid wheel, making compact integration difficult

   (20). To reduce mechanical intricacy, Kim et al. proposed a six- spoke deformable rim that folds via pin joints located at 60° intervals. Field trials on simulated lunar regolith showed reliable negotiation of 15-cm-high rocks, yet the folding sequence relies on high-resolution pressure sensors whose calibration drifts >5% after repeated thermal cycling between

   150°C and +100°C, limiting long-term reliability (21). Drawing inspiration from cockroach locomotion, Case Western Reserve Universitys Mini-WHEGS v3 combines alternating spoke wheels with passive, compliant tarsi to achieve a running speed of 1.2 m/s while clearing obstacles 1.5 times the wheel diameter. The design, however, requires precision-machined aluminium flexures and miniature ball bearings that raise the Bill-of-Materials cost by a factor of four relative to standard rover wheels (22). The aforementioned studies indicate that existing wheel-legged mechanisms still encounter bottlenecks such as low mode-switching efficiency (23), insufficient parameter adaptability (24), and poor dynamic stability (25). To address these limitations, this study proposes a dual-mode deformable hub design enabling dynamic switching between

   wheeled and legged configurations. A series transmission scheme combining crank-rocker and guide rod mechanisms is introduced. This scheme, integrated with a self-locking mechanism, facilitates rapid mode switching. This approach aims to enhance the motion efficiency and environmental adaptability of wheel-legged robots, offering a novel concept for the design of high-dynamic mobile mechanisms.

2. STRUCTURAL AND MECHANICAL ANALYSIS OF DUAL-MODE DEFORMABLE HUB

   1. design of the dual-mode deformable hub scheme

      In order to ensure the applicability of the dual-mode deformable hub in mobile robots, the operational requirements for obstacle crossing and traversal on flat terrain were analyzed. A dual-mode deformable hub structure was designed to enable free switching between these two operational modes, thereby accommodating both high-speed movement and enhanced stability requirements. Through an analysis of the performance characteristics of various wheel-leg configurations, the specific wheel-leg mechanism investigated in this study was selected. A series of analyses and investigations were subsequently conducted on this mechanism.

      1. Non-Circular Design Based on the Slider-Crank Mechanism

         This scheme utilizes the slider-crank mechanism as the foundational principle to achieve the transformation between the wheeled and legged configurations, fulfilling the deformation objective. The stability of the transmission is enhanced by incorporating a parallel linkage combining a double-rocker mechanism and the slider-crank mechanism. A conceptual diagram of the scheme is shown in figure 1.

         Fig. 1. Slider-crank series double-rocker mechanism

      2. Quasi-Circular Design Based on Crank-Rocker-Guide Rod Mechanism

         Motion is transmitted from the motors rotating shaft to the hook-shaped claw through a series-connected crank-rocker and

         Fig. 2. Quasi-circular design scheme based on crank-rocker-guide rod mechanism

         As shown in figure 3, while the non-circular design offers deformation convenience, the quasi-circular scheme enables dual-directional obstacle-crossing strategies. Crucially, this design demonstrates superiority in maximum obstacle-crossing height and, owing to its engineered arcuate profile, delivers more stable performance when traversing complex irregular obstacles. Consequently, the quasi-circular crank-rocker-guide rod mechanism was selected as the dual-mode deformable hub for this study, with detailed analses performed accordingly.

         Fig. 3. Wheel-leg scheme obstacle-crossing height comparison

   2. Wheel-Leg Parameters and Force Analysis

   As the arc-shaped wheel-leg arm exhibits distinct cross- sectional profiles on its anterior and posterior sides, their obstacle-crossing capabilities differ significantly. Defining counterclockwise rotation as forward mode and clockwise rotation as reverse mode, kinematic analysis reveals:

   Forward rotation: The arc-profile end initiates ground contact, enabling smooth traversal of low-height obstacles.

   Reverse rotation: The linear-profile end contacts ground first. While this compromises stability, it achieves superior obstacle- crossing height, allowing terrain-adaptive gait planning.

   For maximum obstacle-height analysis, the reverse mode configuration is adopted (Figure 3). The obstacle height HS satisfies:

   crank-guide-rod mechanism. This configuration provides high hub-space-utilization efficiency and enhanced structural

   HS

   R sin s

   2

   cos

   s

   stability. The conceptual design is illustrated in figure 2.

   (1)

   At this time, if point A is in the unstressed state and point C is in the critical sliding state, then the friction force at point C can be expressed as:

   f N

   (4)

   Fig. 4. Stress analysis diagram of single spoke

   Where is the coefficient of static friction between the wheel leg and the stair surface, and the equilibrium equation of point C in the horizontal direction can be written as:

   In the expression, HS denotes the obstacle-crossing height

   corresponding to s wheel-legs, R represents the maximum radius of the wheel-leg, s signifies the included angle

   P Ncos sin

   (5)

   between adjacent wheel-leg spokes, and s indicates the angle

   between the wheel-leg-ground contact point and the vertical plane. Through analytical derivation, the following expression is obtained:

   H3 1.732R;

   It can be further simplified as:

   P cos sin N

   (6)

   H4 1.414R; H6 1.5R; H8 1.38R; H10 1.31R

   (2)

   Among them, when is arbitrary and P 0 , the friction

   N

   horizontal component force of the mechanism is used as the driving force to resist the horizontal component force from the body and the horizontal component force of the support force,

   and the obstacle crossing is completed. When P 0 , the

   Therefore, the maximum height of obstacle crossing decreases with the increase of s when rotating in the reverse direction.

   When analyzing the maximum forward obstacle-surmounting height, the maximum obstacle-surmounting height is limited by

   N

   horizontal component of the body is used as the driving force, and the body is crossing the obstacle. Take 0.4d , calculate

   the friction force due to the contact between the arc surface and the step surface. It is necessary to analyze the force of single-

   the corresponding P

   N

   value when 020406080 ,

   wheel leg obstacle crossing.

   As shown in figure 4, F and P are the vertical and horizontal portions of the robot body on the wheel legs, respectively, N and f are the bearing capacity and friction force of the step on the wheel leg, H is the height of the step, R0 is the torque of the robot acting on the mechanism, P is the angle of the friction provided by the step for the mechanism to the obstacle. The following equilibrium equation is given:

   and then calculate the critical angle of obstacle crossing. Draw the result as a curve as shown in figure 5:

   P fcos Nsin 0

   F fsin Ncos 0

   R M M 0

   0 N f

   (3)

   Fig. 7. Force analysis diagram of front wheel obstacle negotiation

   Fig. 5. Functional diagram of mode-switching limit angles for wheel-leg

   Among them,

   f1N1 and f2N2

   are the friction and pressure

   mechanism

   when the front and rear wheels of the robot contact with the

   And when

   P 0 , is about 21.804. This angle is called

   N

   pedal. The vertical and vertical components used by the

   P1F1

   the limit angle of the variable hub mechanism when the obstacle crossing and moving state are switched. When

   and

   P2F2

   robots on the front and rear wheels respectively ;

   < 21.804, the mechanism is in a moving state at this time, otherwise the mechanism is avoiding obstacles.

   P'F ' and

   P'F ' are the reaction force of the front and rear

   1 1

   2 2

   Based on the above analysis, the obstacle-crossing process of the wheel-legged robot is simulated by analyzing the sequential

   wheel legs to the hub center respectively;

   T1 and

   T2 are the

   T

   T

   posture variations of front and rear legs during different torque applied by the hub to the front and rear wheel legs,

   traversal stages (figure 6). The complete process can be classified into four characteristic phases according to the

   respectively.

   ' and

   ' are the reaction torque of the front

   1

   2

   dynamic interaction relationships between the front/rear wheel- leg assemblies and the obstacle.

   and rear wheel legs to the hub center. G is the gravity of the whole mechanism.

   The force analysis of the robot shows that:

   f1cos1 N1sin1 f2cos2 N2sin2 0

   G f sin N cos f sin N cos 0

   (7)

   1 1 1 1 2 2 2 2

   M M M M 0

   ON 2 Of 2 ON1 Of 1

   Where

   1 2

   MON MON

   is the torque of

   N1N2

   to point O ;

   Fig. 6. Analysis of obstacle crossing of front and rear wheel legs

   MOf MOf

   is the torque of

   f1f2

   to O point.

   1 2

   Taking (3) as an example, further stress analysis is carried out. The overall force analysis is carried out as shown in figure 7:

   From the force balance analysis of the robot, it can be obtained:

   1 2

   P' P' 0

   1 2

   G F' F' 0

   (8)

   T' T' M' M' M' M' 0

   1 2 OP1 OP2 OF1 OF2

   Where

   M '

   OP1

   M '

   OP2

   is the torque of

   P'P'

   to point O ;

   1 1max , the wheel leg in contact with the front wheel can

   1 2

   OF1 OF2 1 2

   M' M' is the torque of F'F'

   to point O .

   cross the step.

   Similarly, when the current wheel is the driving wheel and

   It can be concluded from the analysis of the force balance of the front wheel:

   1 0 , the maximum driving force

   P1max of the front wheel is:

   P1 f1cos1 N1sin1 0

   P Gl

   T1 T2

   F f sin N cos 0

   1max

   l l 1 tan 2 cos

   1 1 1 1 1

   1 2 (14)

   T M M 0

   1 O1N1 O1f 1

   (9)

   The maximum value of

   2max is:

   It is obtained by the force balance of the rear wheel leg of the robot:

   2G

   P f cos N sin 0

   2max arctan 2

   2 2 2 2 2

   G 1

   P1max

   F f sin N cos 0

   (15)

   2 2 2 2 2

   T M M 0

   2 O2 N 2 O2f 2

   The torque applied to the robot is:

   MOP1' P1 'l1sin

   MOP 2' 2 2

   P 'l sin

   (10)

   In summary, under the condition of known parameters such as friction coefficient, stair size, robot weight and centroid position, the overturn condition of the mechanism under single wheel contact obstacle can be calculated.

   Therefore, as long as the angle between the tangent of the contact point of the wheel leg and the step and the step is less than 54.26°, the wheel leg may cross the obstacle, so the obstacle height with themaximum forward rotation of different

   M

   OF1

   F1 'l1cos

   wheel legs can be determined, as shown in figure 8:

   MOF2' F2 'l2cos

   (11)

   When

   2 0 , the maximum driving force provided by the rear

   wheel (driving wheel) to the body is

   P

   Gl

   T1 T2

   2max

   l l

   1 tan

   1 cos

   1 2

   The maximum values of

   1max arctan

   (12)

   1 is:

   2G

   2

   Fig. 8. Comparison of maximum obstacle-crossing height under different number of wheel legs

   Under the forward and reverse rotation of the analysis mechanism, the obstacle height values under the number of wheel legs are extracted and compared, and the broken line

   diagram shown in figure 9 is obtained. In the case of forward

   G 1

   P2max

   and reverse rotation, the function trend is basically the same. In

   (13)

   order to ensure that the mechanism has good obstacle crossing

   function in the dual mode, the number of wheel legs s 3 is

   It can be seen from the above formula that when the rear wheel

   selected as the final number of wheel legs of the deformation

   is the driving wheel and

   2 0 , if the mechanism satisfies

   mechanism.

   4

   3

   2

   1

   0

   3

   4

   5

   6

   8

   10

   HS

   hs2

   Fig. 9. Wheel-leg configuration vs. obstacle negotiation performance diagram

3. 3 DIAGONAL GAIT PLANNING AND SYNCHRONIZED GAIT PLANNING FOR DUAL-MODE

   DEFORMABLE HUB MECHANISM

   1. Synchronized Gait Planning and Analysis for Dual-Mode Deformable Hub

      The synchronized gait refers to the motion analysis when all four wheels maintain identical speed and phase

      Fig. 10. Four-wheel coordinate system

      According to the geometric relationship, the following conclusions can be drawn:

      synchronization, where all four wheel-legs remain in the stance

      XO YO ZO XO YO

      ZO

      phase at any given moment. The stance phase indicates the state

      i i i 0 0 0

      (18)

      where a robot's leg contacts the ground to provide support, and the average number of supporting legs during multi-legged

      Xi

      Yi Zi

      walking can be represented by the stability coefficient ,

      Among them,

      X Y Z

      0 0 0

      i i i

      represents the position

      p coordinates of the geometric center Oi

      of each wheel leg

      relative to the ground coordinate system ; the bias distance of

      expressed as:

      X0i

      Y Z

      0

      0

      i i

      relative to

      O0 in X, Y and Z directions

      p s

      p

      (16)

      has the following relationship:

      X1 X3 0

      In this context, p represents the duty cycle coefficient,

      X X 0

      2 4

      defined as the ratio of a single wheel-leg's support time to the complete gait cycle duration. When the robot moves in synchronized gait mode, since all four wheel-legs remain in the

      stance phase simultaneously, the duty cycle coefficient p 1

      Y1 Y3 0

      2 4

      Y Y 0

      Z1 Z3 0

      and the stability coefficient p 4 , indicating the robot

      Z Z 0

      achieves optimal stability under this configuration.

      Let O- XYZ denote the ground-fixed reference frame, while

      2 4

      (19)

      Oi – Xi Yi Zi

      (i = 1, 2, 3, 4) represent the wheel-attached frames.

      The body frame is

      Oo – Xo Yo Zo

      defined at the robot's

      The combination of Equation (17), Equation (18) and Equation

      geometric center. The initial configuration aligns axis OZ with vector Oo Zo under the rigid-body assumption. The robot's

      (19) can be obtained:

      centroid position w in global coordinates can then be expressed

      by Equation (17):

      w XO0

      O

      \ ;Y Z

      0

      O

      1 3

      O0

      w X Y Z

      XO1 \ Y 1

      ZO XO

      Y Z

      O

      3

      O

      3

      (20)

      O0 O0

      O0 (17)

      O

      XO2 Y 2

      2

      2 4

      ZO XO

      2

      Y Z

      O

      4

      O

      4

      ZT Zi (1

      3)Zxi

      (24)

      Among them,

      0.328 ;

      0.571 ;

      p k r R sin( 2 ) r sin

      The gait Cycle

      3

      Fig. 11. Determination of geometric parameters of wheel legs

      The kinematic behavior of the wheel-leg mechanism exhibits similarities to most legged systems, where critical parameters including the instantaneous rotation center height and velocity are not constant but demonstrate periodic variations corresponding to different motion phases. Taking the

      configuration presented in figure 11-a as the reference state, the

      T 2 / 3, k 0, 1, 2, 3

      The above velocity equation is differentiated by time t, and the law of the velocity of the center of mass in the horizontal and vertical directions with time can be derived when the robot moves in a synchronous gait:

      motion cycle of this mechanism can be categorized into two sequential phases: during the initial phase (corresponding to

      (1

      3)r cos (t kT) r

      figure 11-b), the curved rim maintains continuous dynamic contact with the ground surface while performing pure rolling motion, with the contact point progressively shifting along the wheel-leg profile; subsequently, in the secondary phase (as shown in figure 11-c), the motion transitions to fixed-point contact at the spoke tip, causing the mechanism to exhibit pendulum-like rotation about this stationary contact point. Through systematic geometric analysis, the time-dependent

      displacement functions Xi t and Zi t of the wheel-leg's

      VTx

      kT t kT

      (1 3)R cos t kT

      kT t kT 2

      3

      (25)

      rotation center along both horizontal and vertical directions can be mathematically derived, thereby establishing the trajectory

      1

      3 r sin (t kT)

      equation of the system's center of mass as follows:

      kT t kT

      V

      (26)

      p r sin (t kT)

      Tz 1 3 R sin (t kT)

      r(t kT) r sin kT t kT

      X

      (21)

      kT t kT

      2

      xi p r R sin (t kT)

      3

      r sin kT t kT 2

      Similarly, the above velocity equation is differentiated by time

      XT Xi (1

      3

      3)Xxi

      (22)

      t, and the law of acceleration of the center of mass in the horizontal and vertical directions with time is obtained:

      2r sin2 (t kT)

      kT t kT

      Zxi

      2

      (23)

      2

      R cos t kT

      kT

      t kT

      3

      (1

      3)2r sin (t kT)

      P k r R sin( ) r sin ,

      0.770 ,

      TD ,.

      3 3

      kT t kT

      The change function of

      XDZD

      with respect to time t can be

      aTx

      (27)

      obtained:

      (1 3)2R sin t kT

      kT t kT 2

      3

      P r sin t kTD r t kTD r sin

      (1

      3) kT

      t kT

      D D

      1

      3 2 r cos (t kT)

      XD

      kP r Rsin t kT

      r sin

      D

      2

      kT t kT

      (1 3) kT t kT

      aTz

      (28)

      D

      D 3

      1

      3 2R cos (t kT)

      (29)

      kT t kT 2

      3

      2(1

      3)r sin2 t kTD

      2

   2. Diagonal Gait Planning and Analysis for Dual-Mode

   Deformable Hub

   The characteristics of the cross-symmetric gait are as follows: In the motion timing configuration, the two groups of wheels

   D

   Z kTD t kTD

   R cos t kT

   D

   and legs in the diagonal line maintain complete

   2

   kTD t kTD

   synchronization, while there is a

   3

   phase lag between the

   3

   (30)

   adjacent two groups of wheels and legs, and each wheel and leg maintains an equal angular velocity drive. When the supporting wheel leg in the supporting state enters the moment of transition from the ground, the opposite wheel leg, which was originally in the suspended state, synchronously establishes the ground contact, forming a persistent alternating mechanism of

   The above velocity equation is differentiated by time t, and the law of the velocity of the center of mass in the horizontal and vertical directions with time can be derived when the robot moves in a synchronous gait:

   double limb support and double limb swing. The action cycle

   (1

   3)r cos t kT

   ratio parameter

   p 0.5 , the motion stability parameter

   D

   p 2 .

   From the time dimension analysis, the support period and swing dynamics of the gait cycle account for half respectively.

   r kTD t kTD

   VDx

   R cos t kTD

   2

   kT t kT

   (31)

   This time phase distribution characteristic makes it show superior dynamic performance when moving at high speed on flat ground, but it is easy to cause instability under low speed conditions.

   In order to analyze the kinematics of diagonal gait, the same coordinate system as Figure 16 is established. It is now

   D

   D 3

   stipulated that: 0.129 ,

   (1 3)r sin t kTD

   (3) Force analysis reveals the influence of driving force and

   kT

   t kT

   friction coefficient on the critical conditions for obstacle

   V

   Dz

   (1

   D D

   3)R sin t kTD

   crossing. The established theoretical model is experimentally validated and matches practical requirements.

   (4) Gait performance analysis compares the dynamic

   kT t kT

   2

   D D

   characteristics of synchronous gait (stability coefficient q = 1) and diagonal gait, confirming the stability advantage of

   3 (32)

   synchronous gait at low speeds and the applicability of diagonal gait for high-speed scenarios.

   The proposed dual-mode deformable hub mechanism shows

   Similarly, the above velocity equation is differentiated by time t, and the law of the acceleration of the center of mass of the robot in the horizontal and vertical directions with time is obtained:

   significant improvements in obstacle-crossing ability, motion stability, and environmental adaptability compared to conventional designs, providing a new technical approach for mobile robots in complex terrains.

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4. CONCLUSIONS

This study focuses on addressing the issues of low mode- switching efficiency and insufficient dynamic stability in wheel-legged robots operating on complex terrains. A dual- mode deformable hub mechanism based on crank-rocker-guide rod series transmission and self-locking structure is proposed through theoretical analysis and experimental verification.

1. The quasi-circular crank-rocker-guide rod design demonstrates advantages over non-circular configurations in terms of space utilization and obstacle-crossing capability. The bidirectional switching mechanism enhances terrain adaptability.

2. The optimal number of wheel-legs is determined, and the maximum obstacle height is calculated based on the wheel diameter and contact angle.

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