The leg telescopic structure is composed of two sets of fixed frames and 24 contact feet, as shown in Fig. 1a. Each set of fixed frames includes one fixed disk and 24 gears, racks, sliding support plates, and electromagnetic clutches, which are evenly distributed along the circumferential direction, as shown in Fig. 1b. Connect the same direction rack on the two sets of fixed frames with the contact foot through bolts to form a telescopic leg. The electromagnetic clutch cooperates with the gear and rack to realize the independent movement of each telescopic leg.

The overall structure of the wheel–leg deformation mechanism is shown in Fig. 2. The whole deformation mechanism includes the following: (1) control unit; (2) inertial measurement unit; (3) vision sensor; (4) laser radar; (5) lithium battery; (6) sealing cover; (7) drive motor; (8) leg contact foot; (9) rolling wheel; (10) slide rail support plate; (11) rack; and (12) enclosure. Relevant design parameters of wheel–leg deformation mechanism are shown in Table 1.

According to the application scenarios and design requirements of the wheel–leg deformation mechanism, the mechanism rolls forward on the flat road, as shown in Fig. 3a. When the rolling wheel cannot cross complex road conditions such as steps and gullies, it will switch to the leg type obstacle-crossing mode, as shown in Fig. 3b.

The following will introduce the working principles of the two motion modes in combination with the internal transmission system diagram of the mechanism, as shown in Fig. 4. Where I is the motor input shaft; A, B, C, and D are electromagnetic clutches; 1 is driving center wheel in leg mode; 2 is driving center wheel in rolling mode; 3, 4, 8, and 9 are gears; 5 is a rolling wheel; and 6 and 7 are timing pulleys.

When the wheel–leg deformation mechanism encounters steps and gullies, the contact feet extend alternately to push the body upward. According to the nomenclature principle of leg mechanism, the contact legs extending alternately are defined as support legs and swing legs. The contact foot that is far away from the step, in contact with the ground, and plays a supporting role is called the support leg, and the contact foot that is near the step, in the extended state, and is not in contact with the ground is called the swing leg. Take climbing steps as an example, the obstacle-crossing process is divided into four stages: (1) the stage of the support leg 1 extension. When the body of the deformation mechanism touches the step, the contact foot (in the opposite direction of movement) closest to vertical line through centroid extends and the contact foot (support leg 1) collides with the ground to trigger the extension of its adjacent inner contact foot (swing leg 1), as shown in Fig. 5a. (2) The support leg 1 supports and pushes the fuselage upward. Support leg 1 extends to the maximum elongation, and then swing leg 1 contacts the ground, as shown in Fig. 5b. (3) The swing leg 1 is converted into a support leg 2. After the swing leg 1 contacts the ground, the support leg 1 retracts and the swing leg 2 starts to extend. The swing leg 1 supports and pushes the fuselage upward as the support leg 2, as shown in Fig. 5c. (4) Repeat the above process until the center of mass of the body crosses the boundary line of the obstacle, the contact foot retracts and climbs up the obstacle with the body, as shown in Fig. 5d.

When the wheel–leg deformation mechanism goes down the step, the movement process of its leg structure is opposite to that of the upper step. When the mechanism goes over the gully, it can be regarded as the movement combination of the upper and lower steps.

The schematic diagram of the kth and k+1th movements of the support leg and the swing leg when the wheel–leg deformation mechanism crosses the obstacle is shown in Fig. 6. When the support leg just touches the ground, the extended length is OkAk=lsk, and the included angle between the support leg and the ground is αk. The centroid of the supporting leg extending out of the main body is Ck. The centroid of the body is Ok. The contact point between the mechanism body and the step is point B. The distance between the centroid of the main body and the contact point is BOk=R. The angle between the connecting line BOk and the ground is φk. When the extension of the support leg reaches the maximum, the length is Ok+1Ak=L0, and the included angle between the support leg and the ground is αk+1. At this time, the swinging leg contacts the ground, and the extension length is Ok+1Ak+1=lsk+1. The centroid of the supporting leg extending out of the main body is Ck+1. The centroid of the body is Ok+1. The angle between the connecting line BOk+1 and the ground is φk+1. The coordinate system is established, the motion direction is the x axis, the vertical ground direction is the z axis, the coordinate origin is the contact point between the support leg 1 and the ground.

Assuming that the height of the step is H, the mass of the body is M, and the mass of the single leg is MT, the speed of leg extension is V. Let CkAk=rk=(lsk-R)/2 and Ck+1Ak=r0k=(L0-R)/2. When the wheel–leg deformation mechanism crosses the obstacle, Eqs. (1)–(4) shall be satisfied. 1L0=Ok+1Aklsk=OkAkR=OkBR=Ok+1Blsk+1=Ok+1Ak+12Ok+1Aksin⁡(π-αk+1)-OkAksin⁡(π-αk)=Ok+1Bsin⁡φk+1-OkBsin⁡φk+13Ok+1Ak+12+Ok+1Ak2-2Ok+1Ak+1Ok+1Akcos⁡θ=Ak+1Ak4Ok+1B2+OkB2-2R2cos⁡θ=Ok+1Ok2. Based on the instantaneous inelastic collision hypothesis, the energy equation and momentum moment equation are established when the support leg touches the ground and the support leg extends to the maximum. The equations are as follows: 512MR2ωk-12+12MVck-12+12MTV2+12MTrk2α˙k+2+Mlsk+MTrkgcos⁡αk=12MR2ωk2+12MVck2+12MTr0k2α˙k+1-2+ML0+MTr0kgcos⁡αk+1 6MTrk2α˙k+=MTr0k2α˙k+1-. It should be pointed out that when establishing the energy equation and momentum equation, only the main body and the supporting leg is considered, and the influence of the swinging leg is ignored. Use “-” to indicate before the impact between the contact foot and the ground occurs, “+” to indicate that the impact is just completed, and “k” to indicate the number of steps.

Assume that the step height is 100 mm and the leg extension speed is 50 mm s-1. Combined with the design parameters of the mechanism, the kinematic characteristics of the wheel–leg deformation mechanism over typical obstacles are studied using the above theory. At the same time, the dynamic simulation software ADAMS is used to simulate the actual movement process of the mechanical system to verify the correctness of the leg-type obstacle-climbing theory of the wheel–leg deformation mechanism and the rationality of the mechanical structure design. The comparison between theoretical calculation and simulation analysis results of motion parameters is shown in Fig. 7. It is shown that when climbing a step with a height of 100 mm, the mechanism shall be supported by three contact feet alternately. When the first supporting leg works, it is called the first step. When the second supporting leg works, it is called the second step. When the third supporting leg works, it is called the third step. The following parts are named in the same way.

The change of the displacement of the centroid of the main body during the climbing process is shown in Fig. 7a. It is shown that the displacement of the body's centroid is a continuous smooth curve, and the simulation curve is in good agreement with the theoretical curve. The alternation of the support legs do not affect the smooth climbing of the mechanism. The change of the angular velocity of the centroid of the main body with time is shown in Fig. 7b. It is shown that the alternation of the supporting legs will cause the step change of the angular velocity of the centroid. Even if the support leg is the same, the angular velocity changes with the extension length of the support leg. Different from the uniform rolling on the flat road, the climbing process of mechanism is a process of accelerating rolling. The angular velocity of simulation curve with some fluctuation is relatively consistent with the theoretical curve, which is caused by the collision between the leg extension and the ground. The change of the angle between the support leg and the ground with time is shown in Fig. 7c. It is shown that different support legs have different initial contact angles with the ground. The contact angle increases with the extension of the leg. The change of the rolling angle of the main body with time is shown in Fig. 7d. It is shown that the rolling angle increases gradually during rolling. The initial value of the roll angle is related to the step height. When the step height is equal to the body radius, the initial value of the roll angle is zero.

When the wheel-legged deformation mechanism climbs an obstacle, different climbing methods can be adopted according to the height of the obstacle. (1) Roll-over mode: when the radius of the rolling wheel is far greater than the height of the step, the mechanism can climb over the step by rolling. (2) Obstacle crossing mode using support legs: when the step height is greater than the height that the rolling wheel can climb, but less than the extended length of the support leg, the leg structure is used for climbing. (3) Climbing mode with mixed wheels and legs: when the step height is greater than the extended length of the support leg, the leg structure can be used to raise the body height first, and then the rolling wheel can be used to climb over the step.

Note that the obstacle-surmounting performance of the wheel–leg hybrid mode can be obtained from the previous two modes. Therefore, only roll-over mode and obstacle-crossing mode using support legs are analyzed.

Firstly, the obstacle-surmounting dynamic model of the mechanism in rolling mode is established. Taking the overall system of the mechanism as the research object, the coordinate system is established at the center of the rolling wheel. The force on the mechanism is shown in Fig. 8. Assuming x¨,y¨,θ¨ is the acceleration in the x, y direction and angular acceleration of the centroid of the main body, the equations of the force and moment are as follows: 7Ff1+Ff2cos⁡β-FN2sin⁡β=Mx¨8FN1+FN2cos⁡β-G+GT+Ff2sin⁡β=My¨9T1-G+GTL1-Ff1H+FN1L1=Jθ¨, where T1 is driving torque of rolling wheel, FN1 and FN2 are support forces of the ground and steps acting on the rolling wheel, Ff1, Ff2 is frictional force of the ground and steps acting on the rolling wheel, and β is the inclination angle between body and step, L1=Rsin⁡α, Ff2=μFN2Ff1=μFN1.

Assuming ψ is the internal friction angle, the conditions that the mechanism does not slip at the contact point are as follows: 10Fm1≤FN1tan⁡ψ,Fm2≤FN2tan⁡ψ,Fm2-Ff2cos⁡β=FN2sin⁡β. In addition, in order to ensure that the mechanism can climb the steps, the following conditions need to be met: x¨≥0; y¨≥0; θ¨≥0. Therefore, the driving torque of the roller can be obtained as follows: 11T1≥G+GTL1+G+GTsin⁡β-μcos⁡βμH-L1μ2sin⁡β+sin⁡β12T1≤G+GTL1+G+GTsin⁡β-tan⁡ψcos⁡βHtan⁡ψ-L1sin⁡β+μtan⁡ψsin⁡β, where 0≤β≤arctan⁡(tan⁡ψ-μ).

According to the geometric relationship in Fig. 8, the height range of the mechanism that can climb steps is 0≤H≤R-Rcos⁡β. Let ψ=35∘, μ=0.1. The stable boundary line when rolling over the steps is shown in Fig. 9. It is shown that when the inclination angle of the mechanism increases, the height of the rolling climbing step increases, and the driving torque of the wheels also increases. According to the constraint condition that no slip occurs, the maximum inclination of the mechanism can be obtained as 54.28∘, thus the maximum climbing height of roll-over mode can be obtained as 87.36 mm.

Secondly, dynamic model of obstacle-crossing mode using support legs is established. The force on the mechanism is shown in Fig. 10. According to the geometric relationship, it can be obtained that the height range of support leg climbing steps is as follows: 1387.36mm≤H≤L0cos⁡αk-π/2-L2tan⁡π/2-φk. The equations of the force and moment are as follows: 14Ff1+Ff2cos⁡π/2-φk-FN2sin⁡π/2-φk=Mx¨15FN1+FN2cos⁡π/2-φk+Ff2sin⁡π/2-φk-G-GT=My¨16FN1L1-GT×L1-lsk-R/2×sin⁡αk-π/2-GL2-Ff1H=Jθ¨. Let the height of the step be H=100 mm, the variation of force on the mechanism during climbing is shown in Fig. 11. It is shown that support force of the ground on the leg decreases with the increase of climbing height. When the climbing height is greater than 80 mm, the supporting force drops sharply. However, the supporting force of the steps on the main body increases with the increase of climbing height. When support leg 1 is switched to support leg 2, the supporting force and frictional force gradually increases or decreases. When support leg 2 is switched to support leg 3, the supporting force and frictional force change abruptly.

The supporting force of the ground on the leg changes with the rolling angle and contact angle, as shown in Fig. 12. It is shown that for the same support leg, the support force decreases with the increase of the rolling angle of the main body, while the change of the contact angle has little effect on the support force. When the rolling angle is 84.3∘, the support force on the leg is close to 0, and the wheel–leg deformation mechanism turns over the step. The supporting force of the step on the main body varies with the rolling angle and contact angle, as shown in Fig. 13. It is shown that the support force of the step increases with the increase of the rolling angle of the main body, and when the rolling angle is greater than 70∘, the support force increases sharply.

Let driving torque T2=FN1-GTL1, the driving torque changes with the rolling angle and contact angle, as shown in Fig. 14. It is shown that the driving torque decreases with the increase of the rolling angle of the main body. That is to say, when the wheel–leg deformation mechanism climbs over the steps, the driving torque required is gradually reduced, and finally the climbing can be completed with the help of gravity.

The wheel–leg deformation mechanism is often disturbed by external forces, environment, and other external factors when walking over obstacles, and may overturn. Therefore, the stability cone method is used to comprehensively evaluate the static and dynamic stability of the side line tipping and corner tipping of the wheel–leg deformation mechanism, in order to ensure the normal operation of the mechanism. The centroid P0 of the fuselage is taken as the apex of the stability cone, and the contact points Pi (i=1, … 3) between the deformation mechanism and the obstacle and the ground are regarded as the corners of the stabilizing cone. The coordinate system is established on the stable cone, take P0 as the origin, the vertical direction is the z axis, and the direction of mechanism movement is the x axis. The y axis is gotten by the right-hand rule. The stability cone model is shown in Fig. 15.

Assuming the vectors of contact points between the left and right rolling wheels and the obstacle are P1 and P3, and the contact points between the support leg and the ground are P2. The normal vector li of the tipping edge line ai is as follows:

17li=I-a^ia^iTPi+1(i=1,…n),Pn+1=P118ai=Pi+1-Pii=1,…n-1,an=P1-Pn19a^i=ai/ai, where I is identity matrix.

In the process of obstacle crossing, it is assumed that the total external force received by the mechanism is Fr, and the total external moment received by the mass center of the main body is τr. The equivalent force Fi∗ of the total external force Fri, and the total external moment τri acting on the tipping edge line i is as follows: 20Fi∗=Fri+lili2τri. When the wheel–leg deformation mechanism is under the action of equal force, the tipping angle of the sideline ai is defined as the angle γi, which is the angle between the equal force Fi∗ and the normal li. The tipping angle of the stable cone corner is defined as the angle ϕi, which is the angle between the equal force Fi∗ and Pi. The angle γi and the angle ϕi can be obtained by the following formula: 21γi=σiarccos⁡F^i∗⋅l^i,ϕi=σiarccos⁡F^i∗⋅P^i, where F^i∗=Fi∗/Fi∗, when (F^i∗×l^i)⋅ai<0, σi=1. Otherwise, σi=-1.

Considering the global stability of the system, let θi=min⁡(γi,ϕi,ϕi+1), then the minimum stability angle of the whole system is 22θs=min⁡(θi). Comprehensively considering the side line tipping angle and corner tipping angle, the stability index of the wheel–leg deformation mechanism is 23Sd=max⁡1γi+1ϕi+1ϕi+1. Without interference, the stability analysis results of the wheel–leg deformation mechanism are shown in Fig. 16. When support leg 1 works, the change of side line tipping angle and corner tipping angle with time is shown in Fig. 16a. It is shown that because of the symmetry of the structure, the tipping angles of the sidelines a1 and a2 are the same, and the corner P1 and P3 have the same tipping angle. The tipping angles of side line a3, corner P1, and P3 decrease with time. However, the tipping angle of the corner P3 increases with time. When support leg 2 works, the change of side line tipping angle and corner tipping angle with time is shown in Fig. 16b. It is shown that the tipping angle of sidelines a1 and a2 increases with time. The change of the tipping angle of the other sidelines and corners with time is similar to that of the support leg 1. When support leg 3 works, the change of side line tipping angle and corner tipping angle with time is shown in Fig. 16c. It is shown that the tipping angle of side line a3 gradually approaches 0∘ with the increase of time, and the tipping angles of other side lines and corners change little with time. The change of stability index with time during the climbing process of the wheel–leg deformation mechanism is shown in Fig. 16d. It is shown that when support leg 1 and support leg 2 work, the system is relatively stable. When the support leg 3 works, the system becomes extremely unstable. When t=2.95 s, the whole mechanism overturns along the side line a3; that is, the whole mechanism crosses the step.

To test the anti-interference ability of the wheel–leg deformation mechanism, the interference force is applied in the positive and negative directions along the x axis, the positive direction of the y axis, and the positive direction of the z axis, respectively. Note that because the roller is symmetrical to the center of mass, the interference effect in the positive and negative directions of y axis is the same, so only the interference force in the positive direction of y axis is discussed. The interference effect in the negative direction of the z axis is the same as that of gravity, and the overturning stability is similar to that without interference. Therefore, only the interference force in the positive direction of the z axis is discussed. Let F^x=±1, F^y=1, and F^z=1, the change of the minimum stability angle of the system with time is shown in Fig. 17a. It is shown that the minimum stability angle will jump when the support legs alternate. When the disturbing force is applied along the positive direction of the x axis and the positive direction of the z axis, the minimum stability angle will increase with time. When the disturbing force is applied along the negative direction of the x axis and the positive direction of the y axis, the minimum stability angle will decrease with time. The change of the stability index of the system with time is shown in Fig. 17b. It is shown that when the disturbing force is applied along the positive direction of the x axis, the stability index will decrease with time. Other disturbing forces have little change with time. The existence of interference force has little influence on the stability of the system. This shows that the mechanism has good anti-interference ability.