Research on Dynamics and Stability in the Stairs-Climbing of a Tracked Mobile Robot

Abstract

Aiming at the functional requirement of climbing up the stairs, the dynamics and stability during a tracked mobile robot's climbing of stairs is studied. First, from the analysis of its cross-country performance, the mechanical structure of the tracked mobile robot is designed and the hardware composition of its control system is given. Second, based on the analysis to its stairs-climbing process, the dynamical model of stairs-climbing is established by using the classical mechanics method. Next, the stability conditions for its stairs-climbing are determined and an evaluation method of its stairs-climbing stability is proposed, based on a mechanics analysis on the robot's backwards tumbling during the stairs-climbing process. Through simulation and experiments, the effectiveness of the dynamical model and the stability evaluation method of the tracked mobile robot in stairs-climbing is verified, which can provide design and analysis foundations for the tracked mobile robots' stairs-climbing.

Keywords

Tracked mobile robot Dynamical model Stability evaluation Stairs-climbing

1. Introduction

Due to the relative advantages as against wheel driving in cross-country performance and obstacle negotiation capability, track-driven mobility is most suitable for mobile robots with higher requirements for ground adaptability and cross-country and obstacle negotiation. Because track-driven-based mobile robots can take movements by adapting to the countryside landscape and since they can climb up ordinary stairs and some ground obstacles, they can be extensively applied in various dangerous situations and special environments, like disaster relief, anti-terrorism, military inspection and unmanned combat, so as to accomplish some dangerous and special tasks instead of working people ^[1, ^2].

In recent years, some experts and scholars have paid great attention to the tracked mobile robot and they have developed a number of tracked mobile robots with different structures. For Example, the “Talon” series of robots developed by the Foster-Miller company and the “PackBot” series of robots developed by the iRobot company have been applied in the battlefield to implement various missions, including anti-terror and anti-explosives, military reconnaissance and so on ^[3-5]. In addition, many

researchers have developed certain special tracked mobile robots for disaster relief and fire fighting ^[6-8].

Because stairs often arise in their application to such environments and situations, the developed tracked mobile robots have mostly regarded stairs-climbing as a key index for evaluating robots' capability in relation to cross-country and obstacle negotiation. On the other hand, the complexity of stairs-climbing is far more than simply climbing over an obstacle or going up a slope. As a result, it is necessary to analyse and study the stairs-climbing process of tracked mobile robots. At present, and with regard to different tracked mobile robots' structures, there have already been some research findings on this technology ^[9-13]. Aiming at a kind of self-reconfigurable tracked mobile robot, a method for the autonomous climbing and descending of stairs was investigated in Ben-Tzvi's paper ^[9], and its tip-over stability evaluation and online tip-over prediction method was proposed based on the track-stair interaction in Liu's paper ^[10]. Liu ^[11] studied the stairs-climbing ability of another tracked reconfigurable mobile robot with a modular structure by analysing the robot's kinematics and dynamics in the stairs-climbing process. Vu ^[12] developed a stairs-climbing algorithm by analysing the CG (centre of gravity) of a four-tracked mobile robot and successfully applied it to the MACbot. In Guo's paper ^[13], the relationship between stability in stairs-climbing and the centroid position of a search and rescue robot was studied by using the geometric analysis method, though the dynamic change of stability in stairs-climbing still needed to be further analysed. As distinct from the research mentioned in the above papers, we will investigate the stability of a tracked mobile robot with a front approach angle and its evaluation method in the stairs-climbing process combined with a dynamics analysis. First, and from the analysis of its cross-country performance, the mechanical structure of the tracked mobile robot is designed and the hardware composition of its control system is given. Second, based on the analysis of its stairs-climbing process, we establish the dynamical model of stairs-climbing and obtain the stability conditions for its stairs-climbing. Next, we take a mechanics analysis on the robot's backward tumbling during the stairs-climbing process and get an evaluation method of its stairs-climbing stability. Furthermore, through simulation and experiments the correctness and effectiveness of the dynamical model and the stability evaluation method in stairs-climbing is verified. Finally, a summary is presented.

2. The tracked mobile robot

2.1 Mechanical structure of the tracked mobile robot

In order to ensure its flexibility of movement, the tracked mobile robot is designed to adopt an independent driven mode on both sides. On each side of the robot, a DC servomotor is used to connect with the driving wheel through decelerators and, thereby, realize forward, backwards and steering motions through two-sided driving wheels' velocity matching. On the overall arrangement of the wheel system, the driving wheel is post-placed and the steering wheel is settled on the upper front of the robot and, therefore, comes into a front approach angle together with the front fixed wheel. This front approach angle helps the robot to realize obstacle negotiation. In addition, a suspension wheel with an elastic cushioning device is settled between the two fixed wheels to compact the track. The specific structure is shown by Figure 1.

Figure 1.

Structure of the tracked mobile robot

2.2 Composition of the control hardware

The control system of the tracked mobile robot is mainly required to realize the robots' motion control and the sensory perception of their state and their surroundings. In light of these requirements, the robot's control system hardware is constructed as shown in Figure 2. The controller is based on an embedded computer with high performance which is used to drive two servo motors through two digital servo controllers to thus accomplish the robots' forward, backwards and steering motions.

Figure 2.

Hardware composition of the control system

In terms of its state and the surroundings, the obstacle information during its operation is detected by using ultrasonic sensors and optoelectronic sensors, while the robot's navigation and posture information is obtained through a digital compass and a tilt sensor. In addition, there is a camera and image-capture card for gaining video information in front of the robot, such that video information is transmitted to the remote control module through a wireless transmission system and then the control command is sent to the robot from the remote control module.

3. Analysis of the stairs-climbing dynamics

3.1 The basic process of the robot's stairs-climbing

During stairs-climbing, the tracked mobile robot begins by touching the first step of the stairs through its front track while its two-sided driving motors simultaneously push the driving wheels and the whole robot forward, up to the first step and then the second step, whereby the entire robot leaves the ground and completely climbs upwards on the stairs. The whole process may be divided into six stages, as shown in Figure 3: a) touch the first step; b) climb up the first step; c) touch the second step; d) climb up the second step; e) leave the ground and then ascend the stairs; (f) leave the stairs and then access the ground. Here, touching the step means the track near the robot's front approach angle climbing onto the step, while climbing up the step means the bottom track of the robot climbing onto the step.

Figure 3.

The robot's stairs-climbing process

In order to climb the stairs, first the track near the robot's front approach angle is required to touch the first step. Besides this, and for correctly climbing up the stairs, the front approach angle and the transverse-distributed external track teeth can adjust the robot's posture when the tracked mobile robot touches the first step. Simultaneously, the speed-closed feedback of the two-sided driven motors plays a role in guaranteeing their constant speed motion to prevent the robot from taking a lateral deflection during stairs-climbing.

3.2 Stairs-climbing dynamics of the tracked mobile robot

Let the basic structural parameters of the tracked mobile robot be as follows: the radius of the driving wheel is R; the radius of the steering wheel and the fixed wheel are both r; the length of the bottom track is L; the angle between the bottom track and the ground is β; the front approach angle is α; and the width and the height of each step are, respectively, b and h. By taking the driving wheel's centre as the origin O of the coordinates and taking the parallel-to-the-bottom-track direction as the X axis, setting the CG (centre of gravity) coordinate of the robot as (x_G, y_G), the distance of the CG relative to the origin coordinate in the coordinate plane XOY and its angle made with the X axis are represented by ρ and φ, respectively, such that we have $ρ = \sqrt{x_{G}^{2} + y_{G}^{2}}$ and φ = arctan(y_G/x_G) The robot's total mass is m and it moves forward with constant speed v. From Figure 3, we can see that the tracked mobile robot touching the first step and then the second step has almost same process and that its motion status changes when climbing the first step and the second step are similar. Therefore, we can build the dynamic models from the four stages “touch the step”, “climb up the step”, “ascend on the stairs” and “leave the stairs”.

(1) Touching the step

Figure 4 shows the loading situation when the robot touches the i-th stair (i=1, 2). F_k and N_R are, respectively, the traction generated by the driving motor and the ground reaction force at the position of the driving wheel, while N_i and f_i are, respectively, the supportive force and the frictional forces on the i-th step.

Figure 4.

The loading situation when the robot touches the i-th step (i=1, 2)

Based on a classical mechanics analysis, we can set up the robot's dynamic model in this process. From the robot's forces' equilibrium equations and its torque equilibrium equation, we can get Equations (1)–(3):

F_{k} - N_{i} \cdot \sin (α + β) + f_{i} \cdot \cos (α + β) = m {\ddot{x}}_{G}

(1)

- m g + N_{R} + N_{i} \cos (α + β) + f_{i} \sin (α + β) = m {\ddot{y}}_{G}

(2)

\begin{array}{l} - m g ρ \cos (φ + β) + N_{i} [L \cos α + \frac{(i \cdot h - L \sin β)}{\sin (α + β)}] \\ + f_{i} \cdot L \sin α - m {\ddot{x}}_{G} \cdot (y_{G} + R) - m {\ddot{y}}_{G} \cdot x_{G} = J_{S} \ddot{β} \end{array}

(3)

Here, J_s is the moment of inertia where the robot moves around the landing spot of the driving wheel. Besides this, we can calculate the frictional force f_i on the i-th (i=1, 2) step by Equation (4), in which u₁ is a frictional factor between the bottom tracks and the sharp corner of the steps and u₁=0.1~0.3 is advisable:

f_{i} = u_{1} \cdot N_{i}

(4)

The traction F_k in the process of the robot's touching on the step can be obtained by Equations (1)–(4) and the driving torque M_k can be further calculated by Equation (5):

F_{k} = (M_{k} - M_{f}) / R

(5)

The total resisting torque M_f in Equation (5) can be obtained 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.

(2) Climbing up the step

The robot's bottom tracks touch the i-th step when the robot is climbing at the i-th step (i=1,2), such that the loading situation during this process is shown by Figure 5. By using the classical mechanics analysis, we can get Equations (6)–(8) from the robot's forces' equilibrium equations and its torque equilibrium equation:

Figure 5.

The loading situation when the robot climbs up the i-th step (i=1, 2)

F_{k} - N_{i} \sin β + f_{i} \cos β = m {\ddot{x}}_{G}

(6)

- m g + N_{R} + N_{i} \cdot \cos β + f_{i} \sin β = m {\ddot{y}}_{G}

(7)

- m g ρ \cos (φ + β) + N_{i} \frac{i h}{\sin β} - m {\ddot{x}}_{G} \cdot (y_{G} + R) - m {\ddot{y}}_{G} \cdot x_{G} = J_{S} \ddot{β}

(8)

In a similar way, we can get the traction F_k in the process of climbing up the i-th step by Equations (6)~(8) and calculate the driving torque M_k further by Equation (5).

(3) Ascending on the stairs

The tracked mobile robot is supported by two or three stairs simultaneously after it leaves the ground and ascends the stairs whereby the supportive force from each step to the robot is N_i and the frictional force between its bottom tracks and the sharp corners of the stairs is equal to the traction F_k. The loading situation of the robot during this process is illustrated by Figure 6. Based on the classical mechanics analysis, we can get Equations (9)–(11) from the robot's forces' equilibrium equations and its torque equilibrium equation:

Figure 6.

The loading situation when the robot ascends on the stairs

F_{k} - m g \sin (\arctan (h / b)) = 0

(9)

- m g \cos (\arctan (h / b)) + N_{1} + N_{2} = 0

(10)

- m g ρ \cos (φ + \arctan \frac{h}{b}) + N_{1} \cdot S_{t} + N_{2} (S_{t} + \sqrt{h^{2} + b^{2}}) = 0

(11)

S_t in Equation (11) is the distance between the driving wheel and the supportive point on the first step along the slope of stairs. We can get the traction F_k for the robot to ascend the stairs by Equations (9)–(11) and, furthermore, the driving torque M_k by Equation (5).

(4) Leaving the stairs

During the process where the tracked mobile robot leaves the stairs and returns to the ground, it twirls around the sharp corner of the top step under the action of gravity until its bottom tracks touch the ground. The supportive force from the sharp corner of the top step to the robot is N₁ and the frictional force between its bottom tracks and the sharp corner of the top step is the traction F_k. The acceleration of the robot's twirling angel is ε and the forces during this process are presented by Figure 7.

Figure 7.

The loading situation when the robot leaves the stairs

Based on the classical mechanics analysis, we can get Equations (12)–(14) from the robot's forces' equilibrium equations and its torque equilibrium equation:

F_{k} - m g \sin γ = m {\ddot{x}}_{G}

(12)

- m g \cos α + N_{1} = m {\ddot{x}}_{G}

(13)

- m g v \cdot (t - t_{0}) \cos α = J_{N} \cdot ε

(14)

The variable t₀ in Equation (14) is the time by which the projective line of the robot's CG gets through the sharp corner of the top step, the angle between the robot's bottom tracks and the ground of the top step is γ, while J_N is moment of inertia when the robot twirls around the sharp corner of the top step. When the robot's bottom tracks touches the top step, γ is equal to zero; thus, the process of leaving the stairs ends. During this process, the relation between the time variable t and γ, ε is as below:

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

(15)

From Equations (12)–(15), we can get the traction F_k for the robot to leave the stairs and calculate the driving torque M_k further by Equation (5).

According to the phased dynamical model of the tracked mobile robot during the stairs-climbing process, we can reveal the change law of the required traction and driving torque during the stairs-climbing process, thus laying the basis for the tracked mobile robot's stability during stairs-climbing.

4. Evaluation of the robot's stairs-climbing stability

4.1 The basic conditions for stairs-climbing

According to the stairs-climbing process shown in Figure 3, we can get three basic conditions of stairs-climbing for the caterpillar track robot, as below.

(1) The geometrical condition

This is for guaranteeing that the robot's front approach angle can successfully touch the step in the stage shown in Figure 3(a). Thus, the central height of the robot's steering wheel is required to be higher than the height of a step. This geometrical condition can be represented by formula (16):

2 R - r \geq h

(16)

(2) The traction condition

This is so that the traction from the driving motor can provide enough power for the requirements of the robot in climbing the stairs. Let the driving torque and the resisting torque converted to the driving motor be M_k and M_f, respectively, and the required traction during the climbing be F_k. Thus, the traction condition can be expressed by formula (17):

(M_{k} - M_{f}) / R \geq \max (F_{k})

(17)

(3) The friction condition

This is for guaranteeing that no sliding down happens when the robot climbs up the stairs. Let the maximum required friction be f_max – from the analysis of the six stages during stairs-climbing shown in Figure 3, the maximum friction f_max should happen during the stage where the robot ascends the stairs. At this time, it is easy to slide down due to the component of the robot's gravity along the slope of stairs. Thus, the friction condition can be expressed by formula (18):

f_{\max} \geq m g b / \sqrt{h^{2} + b^{2}}

(18)

Among the just mentioned basic conditions, the geometrical condition is realized by the structure of the front approach angle, the traction condition is achieved by selecting the proper motors and reduction ratio, and the friction condition is guaranteed by the shape and material of the external track teeth of the rubber tracks.

4.2 Analysis of stairs-climbing stability

To let the robot steadily climb up the stairs – and besides the three mentioned basic conditions – it is also necessary to ensure that no backwards tumbling happens during the robot's climbing. During the stages in Fig.3(a)–(d), because the rotating fulcrum can only be on the robot's driving wheel and its position is relatively post-placed, it is relatively hard for it to tumble backwards. During the stage when it ascends the stairs – shown in Figure 3(e) – when the driving wheel leaves the ground and moves on the stairs, its rotating fulcrum suddenly changes from the point on the driving wheel to the corner of the first step, which more easily leads to the robot tumbling backwards. Since the tracked mobile robot moves forward at constant speed v and since the speed v is relatively lower when the robot climbs up the stairs, its percussive action cannot be considered during the process's analysis. The loading situation during ascending the stairs is shown by Figure 6. From Equations (9)–(11), we can calculate the support force N_i and the traction F_k during the stage when the robot ascends the stairs. Furthermore we can judge whether the robot will tumble backwards from the obtained values. When N₂>0, the robot will not tumble backwards; when N₂=0, the robot will tumble backwards on the stairs.

4.3 Evaluation method of stairs-climbing stability

To ensure the stability during stairs-climbing for the tracked mobile robot, it is extremely important to establish a quantitative evaluation method to allow for a certain stability margin for effective prevention against track deformation, vibration and percussive impact. Since the process where the robot tumbles during stairs-climbing is in fact a process of changing potential energy, when the robot turns around the fulcrum as compelled by an external force to the state where its CG becomes the highest point and its potential energy reaches its maximum, the robot reaches a critical state in turning to tumble. In fact, some stability evaluation methods based on analysing the change of potential energy – called the “Energy Stability Margin ^[14–15]”, the “Dynamic Energy Stability Margin ^[16]” and the “Normalized Energy Stability Margin ^[17]” – had been developed and applied to the legged robots and wheeled vehicles. Here, we also take the variations of potential energy required for the tracked mobile robot to achieve its critical status so as to estimate its stability. Moreover, to describe the tracked mobile robot quantitatively, we make a unitary processing and get the calculation formula for the degree of stability D_s during stairs-climbing:

D_{s} = E_{t} / E_{0}

(19)

In formula (19), E₀ is the maximum variation of potential energy required for tumbling on the flat ground and E_t is the variation of potential energy required for the robot's tumbling during stairs-climbing. Generally speaking, for the least possibility of a robot's tumbling on the flat ground, the variation of potential energy required under this situation should be correspondingly maximized. In formula (19), the degree of stability takes values in (-1,1). When D_s<0, only its own gravity can cause it to tumble and when it is on the flat ground, the degree of stability is the maximum 1. Therefore, in reference to Figure 8 and Figure 9, we analyse the maximum variation of potential energy E₀ and the variations of potential energy E_t for tumbling during climbing the stairs.

Figure 8.

The robot tumbles on the flat ground

Figure 9.

The robot tumbles during climbing stairs

From Figure 8, we can see that when the tracked mobile robot tumbles on the flat ground, it is necessary to provide enough energy to lift its CG from point G(x_G,y_G) to point G′(x′_G, y′_G). Thus, the calculation of the required variations of potential energy E₀ is obtained from the following equation:

E_{0} = m g ({y^{'}}_{G} - y_{G}) = m g (\sqrt{h^{2} + b^{2}} - y_{G})

(20)

In reference to Figure 6 and Equations (9)–(11), the distance S_t between the robot's driving wheel and the supporting point on the first step from the stair slope changes with time, while the rotating angle δ for a robot turning backwards to the critical status can be calculated by Equation (21). Meanwhile, the variation of potential energy E_t required for backwards tumbling during stairs-climbing can be achieved by calculating the energy to lift its CG from point G(x_G,y_G) to point G″(x″_G, y″_G), as shown in Equation (22):

δ = π / 2 - \arctan \frac{h}{b} - \arctan \frac{y_{G}}{x_{G} - S_{t}}

(21)

E_{t} = m g ({y^{″}}_{G} - y_{G}) = m g \sqrt{y_{G}^{2} + {(x_{G}^{​} - S_{t})}^{2}} \cdot (1 - \cos δ)

(22)

Having got the maximum variation of potential energy E₀ and the variation of potential energy E_t during stairs-climbing, we can put them into formula (19) and get the degree of stability D_s during the robot's stairs-climbing, as well as allowing a certain stability margin for effective prevention against track deformation, vibration and percussive impacts so as to ensure that the robot can stably climb up the stairs.

5. Simulation and experiments

For the purpose of proving the results of the dynamical model and the stability analysis for stairs-climbing, we perform a simulation calculation and experiments on the developed tracked mobile robot. The structural parameters of the robot are: the radius of the driving wheels is R=250mm; the radius of the steering wheels and the fixed wheels are both r=140mm; the length of the bottom tracks is L=600mm; the front approach angle is α=40°. The robot has two DC servo motors with 400 watts power and the total mass of the robot is m=62kg. The frictional factor between the tracks and the sharp corner of the steps is μ=0.2. The coordinates of the robot's CG are x_G=420mm and y_G=20mm while the width of the step is b=220mm and its height is h=150mm. During the simulation and the experiments, let the tracked mobile robot's motion be as following: firstly, it moves forward for 0.5s at a speed v=0.6m/s to the front of the stairs and then the robot climbs up the stairs with the same speed.

According to the defined modes of motion, we make a simulation calculation of the robot's stairs-climbing and carry out stairs-climbing experiments with the developed tracked mobile robot. We achieved the results shown by Figures 10 ~14. Figure 10 shows the experimental scenes during the robot's stairs-climbing. In Figure 11, the thick line represents the simulation results of the robot's tilt angle β during stairs-climbing, while the thin line is the actual detected results from the tilt sensor during its stairs-climbing. The angle β in these both two lines increases from 0 until it reaches the inclined angle of the stairs and then it returns to 0 when the robot leaves the stairs, having a similar changing process.

Figure 10.

Experimental scenes during the robot's stairs-climbing

Figure 11.

Tilt angle during the robot's stairs-climbing

Figure 12.

Driving torque of the robot during the stairs-climbing

Figure 13.

Stability of the robot during the stairs-climbing

Figure 14.

The range of the CG under different D_s

In Figure 12, the thick line is the simulation result of the driving torque M_k required for the robot's stairs-climbing while the thin line is the experimental result of the driving torque M_k which is calculated based on the measured driving current in the process of robot's stairs-climbing. From Figure 12, we can see that the driving torque required for the robot to touch the step at the front approach angle (refer to Phases (a) and (c) shown in Figure 3) is significantly larger than that of climbing up the step (refer to Phases (b) and (d) shown in Figure 3). The reason is that the robot has to overcome the resistance against it from the step in the process where it is touching and climbing the step; the required driving torque obviously increases and stays stable after the robot ascends the stairs, with the reasons that the robot must completely depend upon its traction so as to overcome its gravity component along the slope of stairs after it climbs onto them. Accordingly, its required driving torque increases greatly. Furthermore, the output torque and its change trend detected during the stairs-climbing is almost the same as the simulation results, thus proving the correctness and effectiveness of the built dynamical model of the robot's stairs-climbing.

In addition, we calculate its stability by using the proposed evaluation method of the robot's stairs-climbing stability and got the results shown in Figure 13. The lowest point of stability during the robot's stairs-climbing occurs during those moments when the driving wheels leave the ground and the robot's driving wheels roll over the first and second steps, from which the main reason is that the driving wheels are suddenly suspended in the air. This has the result that the robot's rolling fulcrum moves forward substantially and thus it inclines to tumble backwards more easily. This conclusion is also proven in the tracked mobile robot's stairs-climbing experiments.

Furthermore, an analysis is carried out for the minimum value of D_s in Figure 13, and we can get the boundary curves of the robot's CG coordinates x_G and y_G when the degree of stability is D_s=0.03, D_s=0.05 and D_s=0.10. The areas enclosed by these boundary curves with straight lines MK and NK describing the CG's coordinates' extreme boundary is the value area of the CG coordinates required for no backwards tumbling when D_s≥0.03, D_s≥0.05 and D_s≥0.10, as shown in Figure 14. In Figure 14 is given the position of the robot's CG coordinates when x_G=450mm and y_G=0 while it is within the area that D_s≥0.03, which is consistent with the results shown in Figure 13.

6. Conclusions

Given that tracked mobile robots mostly regard stairs-climbing as a key index for evaluating robots' capability in cross-country and obstacle negotiation, the issue of how to ensure stability in stairs-climbing is an important problem which needed to be solved. Looking at this problem, the dynamical model and the question of stability during the tracked mobile robot's stairs-climbing are studied based on a mechanics analysis. These proposed methods for dynamics modelling and stability analysis in stairs-climbing can also be applied to the process of the tracked mobile robot's moving down stairs. The acquired achievements can provide design and analysis foundations for the tracked mobile robots' stairs-climbing and solve the problem, which can be summarized as follows:

(1)

According to the requirement for stairs-climbing, the mechanical structure of a tracked mobile robot is designed and the hardware composition of its control system is given.

(2)

Based on the analysis of the stairs-climbing process, the dynamics model of the tracked mobile robot during stairs-climbing is established, which can provide fundamental support for the stability analysis.

(3)

The stability conditions for its stairs-climbing are obtained. A quantitative evaluation method of the stairs-climbing stability of the tracked mobile robot is proposed and verified through simulation and experiments. Based on this method, the value area of the robot's CG coordinates can be calculated to avoid tumbling backwards.

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