Dynamics and stability analysis on stairs climbing of wheel

Abstract

A transformable wheel–track robot with the tail rod whose winding will coordinate the center of gravity of the robot is researched, and a theoretical basis for the stable climbing of the robot is provided. After a general introduction of the research, firstly the mechanical hardware and control hardware composition of the wheel–track robot is provided and the principles of its mechanical structure are illustrated. Secondly, through studying the fundamental constrains during the process of the robot climbing the obstacles, a mathematical model based on classical mechanics method is built to help analyze the dynamic principles of a wheel–track mobile robot climbing stairs. Thirdly, the dynamic stability analysis is carried out by analyzing not only the interaction among forces of track, track edge, and stair step but also the different stabilities of the robot when the track and the stairs have different touch points. Finally, an experiment of the modeling track robot climbing the stairs has convinced the effectiveness of the dynamic theories researched, which will be a beneficial reference for the future mobile robots obstacle climbing studies.

Keywords

Mobile robot wheel–track dynamical model stability analysis crossing stairs

Introduction

Mobile robots have become increasingly present in human-related activities either to remove the hazards or to carry large and critical payloads safely. Mobile robot is capable of moving around in different environments including stairs and outdoor grounds. In this circumstances, researches are carried out to explore extensive applications for mobile robots in special situations such as rescue, anti-terrorism, removing explosive, and so on.^1,2 In the past decade, both industry and academia have become more responsive to develop new designs for mobile robot platforms with better functionality, quality, and features. At present, there is a focus on designing robots with the capability of traveling over a variety of surfaces, carrying a range of payloads, changing into different configurations by modularity of structure and interface, and fast recovery from accidental rollover or getting stuck. Today, most of the current mobile robot platforms still cannot fulfill the said features due to a fixed structure of its locomotion system.

During the past few years, relevant experts and scholars have made lots of research on the tracked mobile robot, as a result many structures of the robot have been generated. For instance, the “Talon” series of robots from the Foster-Miller company and the “PackBot” series of robots from the iRobot company, both of which have been applied in the battlefield to implement various missions, including anti-terror and anti-explosives, military reconnaissance, and so on.^3
–5 There are also certain customized tracked mobile robot for circumstances such as disaster relief and fire fighting.^6

–9 In these complicated situations, stair climbing has become a basic function for mobile robots. Other than this, a mobile robot also needs to be more flexible and is able to climb under complex terrains. In this case, continuous researches are required to discover new capabilities of the mobile robot in the stair-climbing process in different environments.^10
–12

A new kind of transformable wheel–track robot has been developed.^13

–17 To save energy, it walks in wheel mode on flat grounds and will transfer into track mode on rough road or stairs as shown in Figure 1: a wheel–track robot only consists of one simplified and compact coupling reconstruction mechanism, which enables the robot transfer easily from walking in wheel mode to track mode, combining both advantages of a wheel robot and a track robot.

Figure 1.

Robot mechanical structure. (a) Wheel mode and track mode. (b) System composition.

As a result, it is a better choice to have a wheel–track robot in complex environments such as those with either rough grounds or changeable stairs. With the flexible capacities, the wheel–track robot has got a wider application not only for anti-terrorism and fire rescue but also for anti-terrorism, removing explosive, and even help the disabled.^18

–22 Compared to the conventional robot,^23
–25 the wheel/track robot is more compact in structure and is more flexible with lighter weight and smaller volume which consumes less energy. What’s more, the wheel/track robot applies a coupling reconstruction mechanism and has a tail rod design that can adjust the centroid position. The dynamical model and stability analysis of the robot based on Euler-Lagarange-like systems^26
–28 are established, which lays the theoretical foundation for the autonomous control of the robot in the future.

A transformable wheel–track robot that is lighter, more simplified, and compact has been demonstrated in Figure 1, which shows a tail rod has been added and by winding the tail rod will coordinate the center of gravity of the wheel–track robot to improve the stability of the robot movement. The mechanism structure and the control system of the robot are explained in detail in “Robot system” section. The dynamical model of the robot in stairs climbing has been established and analyzed through applying the classical mechanics method in “Dynamic process analysis on climbing stairs” section. A dynamic stability analysis of the wheel–track mobile robot is carried out by analyzing not only the interaction among forces of track, track edge, and stair step but also the different stabilities of the robot when the track and the stairs have different touch points in “Dynamic stability analysis of stair-climbing process” section. The stability of the dynamic functions is examined in “Simulation analysis and experiment” section, and the examination results have proved the effectiveness of the theories proposed.

Robot system

Mechanical hardware components of the robot structure

The robot mainly consists of four parts as shown in Figure 1: the variable structure device, the walking device, the box body, and the tail rod. The variable structure device is achieved by a double four-bar linkage mechanism. The worm gear motors with double output shafts on both sides of the robot respectively drive the two sets of the main driving shafts on both left and right sides of the wheel/track transformation mechanism. The worm gear mechanism is self-locking, one side of the double output shaft is used to connect the main driving shaft of the wheel/track transformation mechanism, and the other side is to install the potentiometer to complete the angle detection. The walking wheel device is mainly composed of an internal and external gear ring devices, which is driven by a decelerated DC motor. In order to make the robot’s structure more compact, the output torque of the walking motor speed reducer is transmitted to the gear shaft through a pair of gears, so that the walking motor on both sides can be arranged in a staggered way. The two gears of the gear shaft are separately inner geared with the inner teeth of the two gears of the walking wheel device; as a result, the inner and outer gear rings are able to complete synchronous rotation. The tail rod motor can drive one-way wheel rod to swing to adjust the centroid of the robot.

Take one end point of the swing arm rod of the open mechanism shown in Figure 2 as the reference point, carry out kinematics simulation analysis on this point, and measure the displacement, velocity, and acceleration curve as shown in Figure 3. It is illustrated in Figure 3 that when the open mechanism operates, the velocity of the swing rod is slow, and the speed is reduced gently, which decreases the impact of the elastic track, and this can satisfy the requirement of the mechanism design. When the open mechanism opens, the velocity of the swing rod decreases gently from the maximum speed to zero, and the track is fully open; when it distracts, the velocity of the swing rod increases slowly to its maximum speed.

Figure 2.

Four-bar linkage variable structure device.

Figure 3.

Displacement, velocity, and acceleration curve of the end point of the swing rod.

The elastic track shown in Figure 4 will not only deliver the movement but also stretch or contract frequently to get adapt to variable working environment. The inner side of the elastic track has gears that participate in driving mesh and delivering driving torque, and the outer gears are mainly used for mobile anti-skidding.

Figure 4.

Important mechanical components of the robot.

Control hardware components of the robot

The robot control system displayed in Figure 5 adopts a hierarchical structure, and it is composed of intelligent plate and control panel which collaboratively complete the interaction of human and computer, the external environment detection, and motion control. According to the change of external environment, the operator will input the instruction to the intelligent board by the wireless terminal, and the instruction which will be analyzed in the control board controls the movements of 2 DC motor and 3 worm gear motor to realize the control of the robot’s attitude. The robot’s attitude information is monitored real time by internal sensors such as gyroscope that are connected on the control board. This information works not only as the feedback references for attitude control but also is transferred through the intelligent board to the operator which is reflected in the PC man–machine interface. According to the feedback information and the real-time external environment, the robot attitude requirements are reset and sent back again to the robot through PC interface or remote control. In the movement process of the robot, external sensors including infrared and ultrasonic will send some external environment information detected to the control board to assist the operator with some obstacle avoidance work. The tail rod is driven by the tail rod motor.

Figure 5.

Control hardware components of the robot.

Dynamic process analysis on climbing stairs

Process of climbing stairs

When the robot starts to climb the stairs, it will firstly touch the first step of the stairs. With its variable structure, the elastic track of the robot will expand when the driving motors on both sides push the driving heels and the whole robot forward at the same time step by step during the climbing process until the entire robot leaves the ground and keeps climbing up the stairs. This climbing process can be generally divided into six stages which is shown in Figure 6: (a) approaching the first step, (b) touch the first step expanding the elastic track, (c) climb up the first step, (d) touch the second step totally expanding the elastic track, (e) climb up the stairs and steady movement on the stairs, and (f) leave the stairs and steady movement on the ground. It should be explained that when the robot touch the step, its back swing rod wheel will touch the ground and when the robot climb up the stairs, the robot’s bottom track will climb up onto the step.

Figure 6.

Robot’s climbing stair process.

Dynamic analysis on climbing stairs of the robot in tracked mode

The basic structural parameters of the robot are shown as follows:

R—driving wheel radius,

r—swing wheel radius,

α—angle between the bottom track and the ground,

b—width of the step,

h—height of the step,

m—robot’s total mass,

v—robot moves forward with constant speed,

O—swing arm wheel’s center as the coordinate origin,

X—axis parallel to direction of bottom track, and

(x_G , y_G )—robot’s gravity center in coordinate system.

Under the XOY coordinate system, the distance of the center of gravity ρ relative to the origin coordinate whose angle generated with the X axis is ϕ. The two corresponding formulas are $ρ = \sqrt{x_{G}^{2} + y_{G}^{2}}$ and ϕ= arctan(y_G /x_G ). Regarding the point p coordinate that is the contact point between the track and the stair as (x_p , y_p ), under the XOY coordinate system, the distance of the point p relative to the origin coordinate k whose angle generated with the X axis is φ, and the two corresponding formulas are $k = \sqrt{x_{p}^{2} + y_{p}^{2}}$ and φ = arctan(y_p /x_p ).

Figure 6 shows the process that mobile robot climbs the stairs in the tracked mode. The dynamic model is established according to the four phases of the process which are “Attitude adjudgment before climbing” including Figure 6(b) and (c), “climb up the stair” in Figure 6(d), “steady movement on the stairs” in Figure 6(e), and “leave the stairs” in Figure 6(f).

Attitude adjudgment before climbing

Figure 7 describes that the elastic track of the robot starts to touch the first stair. The two values $F_{k_{1}}$ and $N_{R_{1}}$ at the right bottom of the Figure 7 represent the traction generated by the driving motor and the ground reaction force at the position of the back swing rod wheel; another two values N ₁ and f ₁ are separately the supportive force and the frictional forces on the first step. The angle α is included angle between the first step and the track, and β is included angle between the ground and the track.

Figure 7.

Robot touches the first step.

In this process, the robot’s dynamic model can be gained based on the classical mechanics analysis. The following equation (1) can be concluded by the forces’ equilibrium equations and its torque equilibrium equation

{\begin{cases} F_{k_{1}} - N_{1} \cdot sin (α + β) + f_{1} \cdot cos (α + β) = m {\ddot{x}}_{G} \\ N_{R_{1}} + N_{1} \cdot cos (α + β) + f_{1} \cdot sin (α + β) - m g = m {\ddot{y}}_{G} \\ Q \cdot (k sin (φ + β) + r) - W k cos (φ + β) - m g ρ cos (ϕ + β) \\ - m {\ddot{x}}_{G} \cdot (y_{G} + r) - m {\ddot{y}}_{G} \cdot x_{G} = J_{S} \ddot{β} \end{cases}

In the above equations

Q = f_{1} \cdot cos (α + β) - N_{1} \cdot sin (α + β), W = f_{1} \cdot sin (α + β) + N_{1} \cdot cos (α + β)

The J_S in the last equation illustrates the moment of inertia where the robot moves around the landing spot of the back swing arm wheel. Also, the frictional force f ₁ on the step can be calculated by equation (4), in which μ ₁ is a frictional factor between the bottom tracks and the sharp corner of the steps and μ ₁ = 0.1–0.3 is advisable

f_{1} = μ_{1} \cdot N_{1}

With equations (1) to (2), the traction F_k in the process when the robot touch the first step can be obtained, and with equation (3), the driving torque M_k can be calculated

F_{k} = (M_{k} - {M^{'}}_{f}) / R

In the equation (3), the total resisting torque ${M^{'}}_{f}$ can also be known by calculating the caterpillar meshing frictional force, the resisting torque from the drive chain, and the frictional force torques of the bearings on each wheel shaft.

Figure 8 provides the illustration of the process that the robot is climbing at the first step when the robot’s track touches the first step. Equation (4) can be got from the robot’s forces’ equilibrium equations and its torque equilibrium equation using the classical mechanics analysis

Figure 8.

Robot climbs up the first step.

{\begin{matrix} F_{k_{2}} - {N^{'}}_{1} \cdot sin β^{'} + {f^{'}}_{1} \cdot cos β^{'} = m {\ddot{x}}_{G} \\ N_{R_{2}} + {N^{'}}_{1} \cdot cos β^{'} + {f^{'}}_{1} \cdot sin β^{'} - m g = m {\ddot{y}}_{G} \\ {N^{'}}_{1} \cdot {h / sin β}^{'} - m g ρ cos (ϕ + β^{'}) - m {\ddot{x}}_{G} \cdot (y_{G} + r) - m {\ddot{y}}_{G} \cdot x_{G} = J_{S} {\ddot{β}}^{'} \end{matrix}

Similarly, the traction $F_{k_{2}}$ in the process of climbing up the first step by equation (4) and the driving torque $M_{k_{2}}$ by equation (3) can both be calculated.

Steady movement on the stairs

In Figure 9, after the mobile robot leaves the ground and ascends the stairs, it is supported by two or three stairs at the same time, and the supportive force from each step to the robot is N_i,i and N_i,i+1 . The traction $F_{k_{3}}$ and $F_{k_{4}}$ are equal to the frictional force between its bottom tracks and the corners of the stairs. Figures 9 and 10 illustrate the loading situation of the robot in this climbing process. Equation (5) including forces’ equilibrium equations and torque equilibrium equation of the robot can be got through the classical mechanics analysis

Figure 9.

Robot ascends on the stairs.

Figure 10.

Robot climbs on the stairs.

{\begin{cases} F_{k_{3}} - m g sin (arctan (h / b)) = 0 \\ - m g cos (arctan (h / b)) + N_{11} + N_{22} = 0 \\ - m g ρ cos (φ + arctan (h / b)) + N_{11} \cdot S + N_{22} (S + \sqrt{h^{2} + b^{2}}) = 0 \end{cases}

{\begin{cases} F_{k_{4}} - m g sin (arctan (h / b)) = 0 \\ - m g cos (arctan (h / b)) + N_{12} + N_{23} + N_{34} = 0 \\ - m g ρ cos (φ + arctan (h / b)) + N_{12} \cdot S^{'} + \\ N_{23} (S^{'} + \sqrt{h^{2} + b^{2}}) + N_{34} (S^{'} + 2 \sqrt{h^{2} + b^{2}}) = 0 \end{cases}

In equations (5) and (6), S and S ^′ represent the distance between the back swing rod wheel and the supportive point on the first step along the slope of stairs. The traction $F_{k_{3}}$ and $F_{k_{4}}$ that created during the robot ascends the stairs when the track and the stairs contact in two points can be got by equation (5) and in three points by equation (6). The driving torque $M_{k_{3}}$ and $M_{k_{4}}$ can be gained by equation (3).

Leaving the stairs

Figure 11 shows the forces in the climbing process when the tracked robot leaves the stairs and touches to the ground. Under the action of gravity, the robot twirls around the sharp corner of the top step until its bottom tracks touch the ground. Among these forces, $N_{1}^{″}$ represents the supportive force between the corner of the top step and the traction. $F_{k_{5}}$ represents the friction force from its bottom tracks and the corner of the top step. ε is the acceleration of the robot’s twirling angle.

Figure 11.

Robot leaves the stairs.

Equation (7) including the robot’s forces’ equilibrium equations and its torque equilibrium equation based on the classical mechanics analysis can be concluded

{\begin{cases} F_{k_{5}} - m g sin γ = m {\ddot{x}}_{G} \\ {N^{″}}_{1} - m g cos γ = m {\ddot{y}}_{G} \\ - m g v \cdot (t - t_{0}) cos γ = J_{N} \cdot ε \end{cases}

In equation (7), there is a variable t ₀ which represents the time when the projective line of the robot’s center of gravity moves exactly to the corner of the top step. γ is the angle formulated between the robot’s bottom tracks and the ground of the top step, while J_N shows the moment of inertia after the robot twirls around the corner of the top step. The process of leaving the stairs completes when the robot’s bottom tracks touch the ground of the top step, and by this time γ is equal to zero. Equation (8) below explains the relation between the time variable t, γ, and ε during this process

γ = arctan (\frac{h}{b}) + \frac{ε}{2} {(t - t_{0})}^{2}

The equations (7) to (8) provide the way to get the traction $F_{k_{5}}$ as the robot leave the stairs. Equation (3) provides the driving torque $M_{k_{5}}$ . By referring to the dynamical model of the robot in different climbing stages, the change law of the required traction and driving torque can be summarized.

Dynamic stability analysis of stair-climbing process

Interaction between the track and the stair

The force analysis of track, track edge, and stair step is shown in Figure 12. F _f1 and F _y1 are the equivalent traction force and equivalent supporting force in coordinate system, respectively; ${F^{'}}_{f 1}$ and ${F^{'}}_{y 1}$ are the actual traction force and actual supporting force, respectively; μ is the static friction factor of the track and the stair; and α is the angle between the track and the stair step.

Figure 12.

Force analysis of track and track edge and stair step.

The necessary and sufficient conditions for the non-slipping of the track when the track contacts the stair step edge line

- μ < \frac{F_{f 1}}{F_{y 1}} < μ

When the track edge contacts the stair step, the necessary and sufficient conditions for firmly hooking stair step are

- μ \leq \frac{{F^{'}}_{f 1}}{{F^{'}}_{y 1}} = \frac{F_{f 1} cos α - F_{y 1} sin α}{F_{f 1} sin α + F_{y 1} cos α} \leq μ

If μ < cotα

\frac{sin α - μ cos α}{cos α + μ sin α} \leq \frac{F_{f 1}}{F_{y 1}} \leq \frac{sin α + μ cos α}{cos α - μ sin α}

Assuming $d = \frac{F_{f 1}}{F_{y 1}}$ , by equation (11)

d_{max} = {\begin{matrix} \frac{sin α + μ cos α}{cos α - μ sin α} μ < cot α \\ + \infty μ \geq cot α \end{matrix}

Stability analysis of the robot and the stairs

In the process of climbing the stairs, some assumptions are as follow: the friction between the driving wheel and the deformation of the track is not considered, and the movement state of both sides of the track is consistent.

Analysis of one point contact between robot and stair

The force analysis of contact in one point between robot and stair is shown in Figure 13. The robot coordinate system o ₁ xy is located in the center of the swing arm wheel of the robot; the total quality of the robot is m; (x ₀, y ₀) is the centroid coordinate of the robot in o ₁ xy coordinate system; the angle between the robot and the ground is α ₁; the radius of the drive wheel is R; the radius of the swing arm wheel is r; the height of the step is h; the angular velocities of back swing arm wheel, driving wheel, and front swing arm wheel are ω ₁, ω ₂, and ω ₃, respectively; and the geometric centers of back swing arm wheel, driving wheel, and front swing arm wheel are o ₁, o ₂, and o ₃, respectively.

Figure 13.

Force analysis of contact in one point between robot and stair.

The inertia forces of the direction x and y are

{\begin{matrix} F_{a x} = G a_{x} \\ F_{a y} = G a_{y} \end{matrix}

a_x and a_y are the acceleration of the x and y direction, respectively.

The traction force of the track and ground is composed of two parts: horizontal traction and vertical traction¹⁰

{\begin{matrix} F_{f 1}^{x} = μ_{x} N_{1} {1 - exp (\frac{- K | {\dot{s}}_{x} |}{max (| {\dot{s}}_{x} + r ω_{1} |, | r ω_{1} |)})} \times cos (π + arctan 2 ({\dot{s}}_{z}, {\dot{s}}_{x})) \\ F_{f 1}^{z} = μ_{z} N_{2} {1 - exp (\frac{- K | {\dot{s}}_{x} |}{max (| {\dot{s}}_{x} + r ω_{1} |, | r ω_{1} |)})} \times sin (π + arctan 2 ({\dot{s}}_{z}, {\dot{s}}_{x})) \end{matrix}

where μ _x and μ _z are the horizontal and vertical friction factor, K the test values for dragging and slipping, and ${\dot{s}}_{x}$ and ${\dot{s}}_{z}$ the slip velocity of horizontal and vertical direction.

When track edge hooks the first step edge, considering track slipping of in direction x, no sliding in direction z, the direction z is the vertical direction of the coordinate system o ₁ xy

{\begin{matrix} {\dot{s}}_{x} = r ω_{1} \\ {\dot{s}}_{z} = 0 \\ F_{f 1} = μ_{x} N_{1} (1 - exp (- K)) \end{matrix}

The following equilibrium equations are obtained by Newton Euler and Darren Bell’s principle

{\begin{cases} (F_{f 1} - F_{f 3}) cos α_{1} + N_{1} sin α_{1} + F_{f 2} - G sin α_{1} - F_{a x} = 0 \\ - (F_{f 1} - F_{f 3}) sin α_{1} + N_{1} cos α_{1} + N_{2} - G cos α_{1} - F_{a y} = 0 \\ N_{2} \frac{h}{sin α_{1}} - (G cos α_{1} + F_{a y}) x_{0} + (G sin α_{1} + F_{a x}) (y_{0} + r) \\ - J_{1} {\ddot{α}}_{1} - J_{2} {\dot{ω}}_{2} - J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3}) = 0 \end{cases}

where F _f1 and F _f2 are the actual traction force of direction x; N ₁ and N ₂ the actual traction force of direction y; F _f3 the driving resistance of direction x; G the gravity of the robot; and J ₁, J ₂, and J ₃ the moment of inertia of the robot, moment of inertia of driving wheel, and moment of inertia of swing arm wheel, respectively.

By equation (16)

\begin{array}{l} N_{1} = \frac{G cos α_{1} + F_{a y} - N_{2}}{cos α_{1} + μ_{x} exp (- K) sin α_{1}} \\ N_{2} = sin α_{1} [\begin{array}{l} (G cos α_{1} + F_{a y}) x_{0} - (G sin α_{1} + F_{a x}) (y_{0} + r) \\ + J_{1} {\ddot{α}}_{1} + J_{2} {\dot{ω}}_{2} + J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3}) \end{array}] / h \\ F_{f 2} = G sin α_{1} + F_{a x} - N_{1} sin α_{1} + μ_{x} N_{1} exp (- K) cos α_{1} \end{array}

The stability criterion of the contact in one point between the track and stair step is as follows.

{N_{1} > 0} \cap {N_{2} > 0} \cap {F_{f 2} > 0} \cap {0 < (\frac{F_{f 2}}{N_{2}}) \leq μ}

Analysis of two points contact between robot and stair

When the center of mass of the robot passes through the first step of the stairs, the robot enters into contact in two points with the stairs, and the forces are shown in Figure 14. The following equilibrium equations are obtained by Newton Euler and Darren Bell’s principle

Figure 14.

Force analysis of contact in two points between robot and stair.

{\begin{cases} F_{f 1} + F_{f 2} - G sin α - F_{a x} = 0 \\ N_{1} + N_{2} - G cos α - F_{a y} = 0 \\ N_{2} \sqrt{b^{2} + h^{2}} - F_{a y} (y_{0} + r) tan α + F_{a x} (y_{0} + r) \\ - J_{1} \ddot{α} - J_{2} {\dot{ω}}_{2} - J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3}) = 0 \end{cases}

where F _f1 and F _f2 are the equivalent traction force of direction x; N ₁ and N ₂ the equivalent traction force of direction y; b the width of the stair; and α the angle between the track and the stair.

By equation (19)

\begin{array}{l} N_{1} = - N_{2} + G cos α + F_{a y} \\ N_{2} = \frac{F_{a y} (y_{0} + r) tan α - F_{a x} (y_{0} + r) + J_{1} \ddot{α} + J_{2} {\dot{ω}}_{2} + J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3})}{\sqrt{b^{2} + h^{2}}} \end{array}

Assuming $F_{f 0} = F_{f 1} + F_{f 2}$ , μ ₀, μ ₁, and μ ₂ are the ratio of the traction force to the support force of the robot in coordinate system.

Two track edges are simultaneously on the stairs

μ_{0} = \frac{F_{f 0}}{N_{0}} = \frac{F_{f 1} + F_{f 2}}{N_{1} + N_{2}}

Two track edges are not simultaneously on the stairs

\begin{matrix} μ_{1} = \frac{F_{f 1}}{N_{1}} = \frac{F_{f 1} + F_{f 2} - μ N_{2}}{N_{1}} \\ μ_{2} = \frac{F_{f 2}}{N_{2}} = \frac{F_{f 1} + F_{f 2} - μ N_{1}}{N_{2}} \end{matrix}

The stability criterion of the contact in two points between the track and stair step is as follows

{N_{1} > 0} \cap {N_{2} > 0} \cap {0 \leq max (\frac{F_{f}}{N}) \leq d_{max}}

Analysis of three points contact between robot and stair

When there is the contact in three points between the robot and the stair step, the forces are shown in Figure 15.

Figure 15.

Force analysis of contact in three points between robot and stair.

The following equilibrium equations are obtained by Newton Euler and Darren Bell’s principle.

{\begin{cases} F_{f 1} + F_{f 2} + F_{f 3} - F_{f 4} - G sin α - F_{a x} = 0 \\ N_{1} + N_{2} + N_{3} - G cos α - F_{a y} = 0 \\ N_{2} \sqrt{b^{2} + h^{2}} + 2 N_{3} \sqrt{b^{2} + h^{2}} - (G cos α + F_{a y}) x_{0} + \\ (F_{a x} + G sin α) (y_{0} + r) - J_{1} \ddot{α} - J_{2} {\dot{ω}}_{2} - J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3}) = 0 \\ N_{3} \sqrt{b^{2} + h^{2}} - N_{1} \sqrt{b^{2} + h^{2}} + (G cos α + F_{a y}) (\sqrt{b^{2} + h^{2}} - x_{0}) + \\ (F_{a x} + G sin α) (y_{0} + r) - J_{1} \ddot{α} - J_{2} {\dot{ω}}_{2} - J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3}) = 0 \end{cases}

where F _f1, F _f2, and F _f3 are the equivalent traction force of direction x; N ₁, N ₂, and N ₃ the equivalent traction force of direction y; and F _f4 the driving resistance of direction x.

It is assumed that the difference between the equivalent support forces is consistent with the linear distribution law

\begin{array}{l} N_{1} = N_{2} - A \\ N_{2} = \frac{1}{3} (N_{1} + N_{2} + N_{3}) = \frac{1}{3} (G cos α + F_{a y}) \\ N_{3} = N_{2} + A \end{array}

By equation (24)

\begin{array}{l} A = - \frac{1}{2} (G cos α + F_{a y}) + \frac{(G cos α + F_{a y}) x_{0}}{2 \sqrt{b^{2} + h^{2}}} + \\ \frac{- (G sin α + F_{a x}) (y_{0} + r) tan α - F_{a x} (y_{0} + r) + J_{1} \ddot{α} + J_{2} {\dot{ω}}_{2} + J_{3} ({\dot{ω}}_{1} + {\dot{ω}}_{3})}{2 \sqrt{b^{2} + h^{2}}} \end{array}

Assuming $F_{f 0} = F_{f 1} + F_{f 2} + F_{f 3}$ , μ ₁, μ ₂, μ ₃, and μ ₄ are the ratios of the traction force to the support force of the robot in coordinate system.

Three track edges are simultaneously on the stairs

μ_{1} = \frac{F_{f 0}}{F_{y 0}} = \frac{F_{f 1} + F_{f 2} + F_{f 2}}{N_{1} + N_{2} + N_{3}}

Three track edges are not simultaneously on the stairs

\begin{array}{l} μ_{2} = \frac{F_{f 1}}{N_{1}} = \frac{F_{f 1} + F_{f 2} + F_{f 3} - μ N_{2} - μ N_{3}}{N_{1}} \\ μ_{3} = \frac{F_{f 2}}{N_{2}} = \frac{F_{f 1} + F_{f 2} + F_{f 3} - μ N_{1} - μ N_{3}}{N_{2}} \\ μ_{4} = \frac{F_{f 3}}{N_{3}} = \frac{F_{f 1} + F_{f 2} + F_{f 3} - μ N_{1} - μ N_{2}}{N_{3}} \end{array}

The stability criterion of the contact in three points between the track and stair step is as follows.

{N_{1} > 0} \cap {N_{2} > 0} \cap {N_{3} > 0} \cap {0 \leq max (\frac{F_{f}}{N}) \leq d_{max}}

Simulation analysis and experiment

Specification of the robot

The structural model parameters of wheel–track robot are shown in the Table 1.

Table 1.

Specification of the robot.

Content	Unit	Symbol	Value
Total mass of the robot	kg	m	26
Radius of driving wheel	m	R	0.175
Radius of two swing arm wheel	m	r	0.05
Moment of inertia of the robot	kg m²	J ₁	0.69
Moment of inertia of driving wheel	kg m²	J ₂	0.075
Moment of inertia of swing arm wheel	kg m²	J ₃	0.0013

Simulation experiment

When a = 0.0001–1.0 m/s², the relationship curve is shown in Figure 16 between the acceleration a of the robot and the traction F _f2, and the relationship curve is shown in Figure 17 between the acceleration a and F _f2/N ₂. By the curves: when F _f2 > 0, N ₁ > 0, N ₂ > 0, and 0 < F _f2/N ₂ < 0.5, the requirement of the dynamic stability of the climbing stair satisfies the dynamic stability criterion of the contact in one point between the track and the step of the stairs which is provided by the formula (18).

Figure 16.

Curve between acceleration and force in one point.

Figure 17.

Curve between acceleration and F_f ₂/N ₂ in one point.

When a > 0.005 m/s², N ₁ > 0, N ₂ > 0, and 0 < max(μ ₀, μ ₁, μ ₂) < 1.832 as shown in Figures 18 and 19. Based on formula (23), which is the dynamic stability criterion of the contact in two points between the track and steps, the condition that the maximum ratio between traction F_f and supporting force N is less than d _max satisfies the requirement of climbing stair.

Figure 18.

Curve between acceleration and force in two points.

Figure 19.

Curve between acceleration and F_f /N in two points.

When a > 0.005 m/s², N ₁ > 0, N ₂ > 0, N ₃ > 0, and 0 < max(μ ₁, μ ₂, μ ₃, μ ₄) < 1.832 as shown in Figures 20 and 21. Based on formula (29), which is the dynamic stability criterion of the contact in three points between the track and steps, the condition that the maximum ratio between traction F_f and supporting force N is less than d _max satisfies the requirement of climbing stair.

Figure 20.

Curve between acceleration and force in three points.

Figure 21.

Curve between acceleration and F_f /N in three points.

Prototype experiment

When the robot meets the step obstacle, it will switch from the wheel mode to the track mode, demonstrated in Figure 22(a) and (b). With the robot climbing up the steps, the inclination angle of the robot become gradually bigger and there is no sign of leaning forward, then swing the tail rod to help the robot to change robot attitude as is shown in Figure 22(d). Under the help of the tail rod, the robot stably climbs over the following steps and eventually completes the steps obstacle-surmounting climbing.

Figure 22.

Robot climbing stairs experiment.

The line in Figure 23 represents the driving torque M_k experiment result that the robot requires for climbing stairs. The line in Figure 24 shows the experiment result of the robot’s tilt angle β when the robot climbs stairs. As is shown, the angle β has a changing process which increases from zero until it reaches the angle which is the same as inclined angle of stairs and finally it is back to zero when the robot leaves the stairs.

Figure 23.

Relationship of driving torque and time.

Figure 24.

Relationship of tilt angle and time.

Conclusion

Wheel/track coupling type is a new type of mobile combination compared with the traditional wheel/track multiple robots which only contain one set of walking mechanisms. The structure of a wheel/track coupling type robot is simple and compact, and it can easily switch between wheeled mode and tracked mode. As climbing the stairs is considered as a critical important function to evaluate the capability of a mobile robot on crossing obstacles. On the other side, the stability of such a robot to climb the stairs needs also to be ensured. As far as we know, the achievements gained can provide great reference for designing and analyzing a mobile robot. Through researching on the requirements of the robot climbing stairs, the mechanical structure of a wheel–track mobile robot is designed and the hardware composition of its control system is explored. After studying the process of the robot climbing the stairs, the dynamics model of the robot which will provide fundamental support for the stability analysis can be established.

Although the extensive tests and experiments performed to demonstrate that the robot meets the research goals, some improvements need to be further addressed in future developments. The future work is summarized as follows:

Develop more add-on attachments and interfaces for expanding the robot’s functions and applications.

Develop an autonomous control system which can be switched on and off by remote control to assist the user’s operations.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by State Key Laboratory of Robotics, Shenyang Institute of Automation Chinese Academy of Sciences (grant no. 2015009).

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Dynamics and stability analysis on stairs climbing of wheel–track mobile robot

Abstract

Keywords

Introduction

Robot system

Mechanical hardware components of the robot structure

Control hardware components of the robot

Dynamic process analysis on climbing stairs

Process of climbing stairs

Dynamic analysis on climbing stairs of the robot in tracked mode

Attitude adjudgment before climbing

Steady movement on the stairs

Leaving the stairs

Dynamic stability analysis of stair-climbing process

Interaction between the track and the stair

Stability analysis of the robot and the stairs

Analysis of one point contact between robot and stair

Analysis of two points contact between robot and stair

Analysis of three points contact between robot and stair

Simulation analysis and experiment

Specification of the robot

Simulation experiment

Prototype experiment

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References