1. Detecting the robot's body structure
The mechanical structure of the AEC_II robot is shown in Figure 1. The underground environment of coal mines is complex. In narrow tunnels, in addition to belt conveyors and mine car rails, there are often piles of materials and equipment such as wire mesh, anchor bolts, sleepers, rails, and transformers. Some areas also have accumulated water, trenches, inclines, and stone steps. Under such complex environmental conditions, even under normal circumstances, ordinary wheeled mobile mechanisms are difficult to operate, let alone after mine disasters such as gas explosions, water inrushes, and collapses. Using a tracked mobile mechanism provides good stability, strong climbing, obstacle-crossing, and trench-crossing capabilities, and strong environmental adaptability. With auxiliary tracks, the ability to cross obstacles and trenches is further enhanced.
The robot is powered primarily by three DC permanent magnet motors: a left-wheel drive motor, a right-wheel drive motor, and an auxiliary arm drive motor. The three motors are driven independently, and the drive wheels are connected to the motors via synchronous belts to ensure smooth movements and reduce operating noise. The auxiliary arm swing motor uses a worm gear drive, which increases the reduction ratio and torque.
Figure 1. External structure diagram of the robot
The mechanical properties of the robot body are as follows:
(1)Dead weight: 35.5kg
(2) Continuous working time: ≥3 hours (uniform speed driving @1/2Vmax)
(3) Maximum speed: 3.6 km/h
(4) Climbing ability:
Loose soil or coal: 30°
Hard slopes such as concrete: 40°
(5) Obstacle crossing ability: It can successfully cross steep obstacles less than 200mm.
(6) Trench crossing capability: With the auxiliary arm extended, it can successfully cross trenches up to 350mm wide.
(7) Protection rating: The body protection rating reaches IP67.
Electrical performance:
(1) Maximum continuous operating power: 400W
(2) Continuous operating current: 16A
(3) Maximum power continuous working time: 1.5 hours
(4) Power supply system: Two 24V/13Ah lithium batteries, equivalent to 0.62kWh
2. Establishment of the kinematic model
Intelligent robot systems are a typical type of multi-input multi-output control system, and the study of kinematic models of mobile robots is fundamental to controlling their trajectories. For tracked robots, they can be approximated as a synchronous belt four-wheeled robot, and the kinematic model of the AEC tracked robot can be studied based on this. Figure 2 shows the top-view kinematic model, i.e., the kinematic equations of the robot when moving in straight lines and curves on a horizontal surface.
Figure 2. Top view motion model of the robot
Where vL is the speed of the robot's left wheel.
vR is the speed of the robot's right wheel.
v is the velocity at the geometric center of the robot.
The motion curve of the robot model is shown in Figure 3. In the figure, XOY is the Earth-fixed reference coordinate system, which is a right-handed coordinate system; coordinate O is the starting point of the robot's motion; the robot moves from point O to point M (the movement of the robot's geometric center) after time t. From this, the following relationship can be derived:
Calculation of the robot's turning radius
Tracked mobile vehicles employ a slip-steer principle, achieving steering by controlling the differential speed of the tracks on both sides. The following analysis uses a right turn as an example to examine the theoretical steering kinematics (the situation is similar for a left turn). It is assumed that the slippage between the tracked walking mechanism and the ground is zero. When the mobile vehicle turns around the steering center O at a constant angular velocity, it is called stable steering. As shown in Figure 4, the distance from the steering center O to the longitudinal axis of the robot body is R, called the theoretical turning radius of the tracked walking mechanism; v<sub> L</sub> , v<sub> R</sub> , and v<sub>0</sub> are the velocities of the left wheel, right wheel, and the robot's geometric center, respectively; the wheelbase is W, and the angular velocity of the turn is ω. Then we have...
Transforming the above equation, we get:
From this formula, we can see that the robot has the following types of movement:
When the tracks on both sides move in the same direction and at the same speed, the robot moves in a straight line.
When the two tracks move in the same direction but at different speeds, the robot turns towards the lower-speed side.
When the two tracks move in different directions but at the same speed, the robot performs a zero-radius in-place turning motion, turning towards the reversing side;
When the two tracks move in different directions and at different speeds, the robot makes a small-radius turning motion, turning towards the side that is reversing.
It is easy to see that tracked vehicles have the following characteristics when steering:
Compared to wheeled vehicles with steering wheels, tracked vehicles have a much smaller minimum turning radius;
Tracked vehicles can turn in narrow areas and can turn with zero radius. The actual turning radius is half the diagonal length of the vehicle body (with the claws retracted).
Compared to synchronous four-wheeled vehicles, tracked vehicles have lower steering precision.
Based on equations (2-5) and (2-6), the speeds of the robot's two wheels can be derived (assuming the steering speed and steering radius are set):
Effect of slip ratio on robot kinematics
Figure 5 shows a schematic diagram of a track system. Under the action of the driving torque, the drive wheel teeth mesh with the track, driving the track to perform transmission around the drive wheel, causing the track to continuously lay tracks and move forward. Therefore, the tracked walking mechanism can be considered to be essentially a "self-contained track" wheeled walking mechanism. Under the action of the driving torque M, the meshing of the drive wheel teeth and the track generates tension in the track. The tension is transmitted along the drive section of the track to the support section, pulling the track backward and deforming the soil in the support section. At the same time, the soil exerts a forward horizontal reaction force Fd (i.e., thrust) on the track support area. The thrust Fd drives the track forward while overcoming the rolling resistance Ff of the track.
Because the road surface has limited deformation capacity, the track contact point moves backward relative to the ground, inevitably resulting in varying degrees of track slippage. This necessitates the definition of slip ratio, denoted by the letter δ.
Where: v <sub>r</sub> is the actual speed of the vehicle.
v is the ideal operating speed of the vehicle.
Different ground conditions result in different slip ratios, and even within the same ground environment, different locations may exhibit varying slip ratios due to factors such as terrain. Generally, under ideal road conditions, the slip ratio of tracked vehicles is relatively small compared to wheeled vehicles, but it still causes some speed loss. The permissible slip ratio is typically between 5% and 7%. However, on soft surfaces such as sand and snow, the slip ratio becomes a major factor affecting the robot's motion accuracy, and its uncertainty becomes an adverse effect on motion control.
In summary, tracked mobile mechanisms differ significantly from wheeled ones in their motion characteristics, particularly in directional motion, making motion control relatively more challenging. Furthermore, factors such as transmission characteristics, mechanical wear, load, external environment, and road surface conditions all contribute to the complexity. Strictly speaking, it's impossible to have an accurate mathematical model for the speed of a tracked mechanism; the formulas and derivations described above are not precise.
3. Establishment and Analysis of the Dynamic Model
Dynamics, a branch of theoretical mechanics, primarily studies the relationship between forces acting on an object and its motion. Dynamic analysis is the foundation and basis for robot mechanism design. In this paper, the robot's dynamic model comprises three views: top view, side view, and rear view.
3.1 Introduction to Robot Dynamics and Dynamic Models
Robot dynamics primarily studies the dynamics of robot mechanisms. Research on robot dynamics began with the very emergence of robots and has been continuously enriched and accumulated with the development of robotics technology. However, a relatively systematic and complete system of robot dynamics has only emerged in recent years. Its fundamental theory is the rapidly developing multibody system dynamics.
3.2 Top View Dynamics Analysis and Application
The top-view dynamic model of the robot, which represents the robot's dynamics when moving in straight lines and curves on a horizontal surface, is shown in Figure 6. Based on this model, the robot's driving force and the maximum load it can carry can be analyzed.
Linear walking dynamics analysis
The drive wheels generate tension in the track drive section under the driving force. In practice, the meshing losses and impact losses between the track and drive wheels must be considered for tension to be generated. As the tension is transmitted along the drive section to the track plates in the support area, losses also occur in the track drive section, including frictional losses and vibration losses due to the relative movement between the hinge pins of the drive section. Only after overcoming these losses can the tension pull the track backward, causing shear strain in the soil and generating thrust F.
Top-view dynamic model application
- Calculation of power of tracked traveling mechanism
When the tracked walking mechanism moves at a constant velocity in linear motion on level soil, the effective power of the motor is Pe , the power lost in the transmission system is Pc , ηc is the transmission system efficiency, ηq is the drive efficiency, the drive power generated on the drive wheel is Pd , the power lost in the track drive section is Pdc , and the slip loss power caused by the consumption of soil shear strain under the support section is Ps . δ is the slip ratio. From this, the following relationship is obtained.
(2) Selection of drive motor
Based on the design requirements, assume the following operating weight for the traveling mechanism: G = 295N (mass m = 29.5kg);
The driving radius of the drive wheel is rq = 0.08m;
Slope (including staircase slope): α = 40°;
The acceleration is 1 m/ s² , Fd = ma = 29.5 N;
When the slip ratio δ of the tracks is 7%, the speed on slopes (climbing ladders) is V = 3.2 km/h = 0.9 m/s. The theoretical speed on flat ground is 0.92 m/s.
Assume the efficiency of the transmission system ηc is 75%.
Taking the drag factor as f = 0.12, we get
The total power of the drive motor should not be less than 260W.
(3) Selection of battery capacity
This robot is powered by a lithium battery. Estimating the lithium battery capacity is an important indicator for battery selection, and the estimation formula is as follows:
Where: F is the total driving force
d represents the distance traveled.
Assuming good ground conditions, i.e., μ = 0.06 (see Table 3-1), the total thrust during steady motion is:
The robot travels at a medium speed of 2 km/h. According to equation 3-8, the robot's power is 180W. Adding the power of the electrical components, approximately 60W, the total power is 180 + 60 = 240W. The battery voltage ranges from 25.8 to 30.1V, with an average voltage of 27.5V. Therefore, the lithium battery capacity is 25Ah after 3 hours of operation.
3.3 Side View Dynamics Analysis and Application
The robot's side-view dynamics model is the equation of gravity shift when the robot moves on uneven surfaces or crosses obstacles. This is the main focus of this article. Based on this model, the robot's obstacle-crossing ability can be theoretically calculated.
Changes in the center of mass of the robotic arm during rotation
When a robot is moving, it sometimes cannot continue forward because a protruding part at the front or rear encounters an obstacle. There are two scenarios: one is that the bottom of the robot hits a convex obstacle, causing the robot to be suspended; the other is that the front of the robot hits a concave obstacle, causing the robot to get stuck. Because the movement of the forearm can change the shape of the robot's front, the articulated tracks greatly increase the robot's ability to traverse such obstacles compared to wheeled vehicles.
Meanwhile, the robot's forearm can change the position of the robot's overall center of gravity during movement, thus achieving stability when overcoming obstacles. In addition, raising the forearm can improve the robot's ability to traverse steep obstacles.
When traveling on a flat road, the robot retracts its swing arm (as shown in Figure 7), at which point the robot occupies the least space. At the same time, due to the use of a tracked movement mechanism, it can turn on the spot and is easy to operate in narrow environments. When encountering obstacles such as steep slopes, protrusions, or stairs, the robot raises its swing arm, as shown in the figure, to adjust its posture to adapt to changes in terrain.
During ladder climbing, the robot continuously adjusts its arm posture to adapt to changes in terrain. As the joint angles change, the position of the robot's center of gravity also changes, thus affecting the stability of the ladder climb.
As shown in Figure 7, a coordinate system is established with the center of mass C2 of the main vehicle body as the origin, C1 as the center of mass of the swing arm, and O as the axis of rotation of the guide wheel. According to the formula for center of mass coordinates, the coordinates of the center of mass of the whole machine are C( XC , YC ).
In the formula: m1 is the mass of the swing arms (including both left and right swing arms),
m2 is the main vehicle body mass.
M is the mass of the entire robot, which can be expressed as M = m1 + m2 .
d1 is the distance from the center of mass of the swing arm to the axis of the guide wheel, which is the radius of rotation of the center of mass of the swing arm.
α is the joint angle of the swing arm body, that is, the angle between the line connecting the axis of the swing arm wheel and the axis of the guide wheel and the horizontal plane.
Equations (1) and (2) are transformed to obtain:
It can be seen that as the joint angle changes, the range of change of the center of mass lies within a region centered at a radius of .
On the disc. This allows us to determine the robot's center of gravity adjustment range during obstacle crossing, ensuring good obstacle-crossing stability. This raises a question: is there a possibility that the robot might tip over when its arm rotates?
We will discuss two cases: one is when the arm falls forward (let's assume this rotation direction is clockwise), and the other is when the arm retracts (let's assume counterclockwise). Assume the robot moves on a horizontal surface, and that the robot, except for its joints, is a rigid body, as is the ground.
a) When the arm is rotated clockwise
Since the mass m1 of the robot's arm is much smaller than the mass m2 of the robot body, the robot will not tip over when the arm rotates clockwise on a horizontal surface. The change in the robot's center of mass is shown in the following equation:
b) Arm reversal situation
When the arm rotates to the position shown in Figure 8, this is the critical position to determine whether the robot will tip over. At this point, the horizontal coordinate of the robot's center of gravity is:
Where r1 and r2 are the radii of the small and large wheels of the robot arm, respectively, and d is the length of the robot arm, i.e., the distance between the centers of the two wheels. Based on the principle of torque balance, the robot can achieve the condition of tipping over by rotating its arm.
As shown in the above formula, whether a robot can tip over by rotating its arm depends solely on the robot's own parameters. If the robot needs to tip over on its own, the length of the robot arm can be increased, bringing the robot's center of gravity closer to the front of the robot; conversely, if the robot is not allowed to tip over, the arm length can be increased.
Robot climbs over steep obstacles
When faced with steep obstacles, robots have two options: detour, which increases the robot's travel time and reduces its overall obstacle-crossing performance; or climb over them, which saves time. However, in many cases, there is no time for the operator to choose a new path, or there are simply no other paths available. In such situations, the robot needs to have the ability to climb over steep obstacles.
Here we discuss two ways to overcome steep obstacles: one is for the robot to lift its forearm to overcome the obstacle, and the other is for the robot to lift its rear end using its forearm to overcome the obstacle. We will discuss them separately below.
(1) Conditions under which a robot can overcome obstacles
Figure 9 shows the process of the robot raising its forearm to overcome obstacles.
As shown in Figure 9, the robot eventually overcomes the obstacle, influenced by its center of mass constraint. The robot's ability to overcome the obstacle can be considered as the robot's center of mass ultimately passing through the edge of the obstacle during its propulsion. That is, as shown in Figure 10, the range of change in the robot's overall center of mass during obstacle-crossing exceeds the edge fulcrum of the obstacle. This point is considered the critical point for obstacle-crossing capability; once this critical point is passed, the robot considers itself to have passed the platform. It is evident that center of mass analysis plays a crucial role in analyzing a robot's obstacle-crossing capability.
The angle between the robot and the ground is α. A reference coordinate system OXY is established with the upper right point O of the platform as the origin. Based on geometric calculations, the coordinates of the robot's center of mass in the O coordinate system are:
Let the coordinates of a point in the Ob coordinate system be (y <sub>b</sub> , z<sub> b</sub> ), and the counterclockwise rotation angle from the Ob coordinate system to the Oc coordinate system be θ. Then the coordinates (y <sub> c</sub>, z<sub> c</sub> ) in the Oc coordinate system satisfy the following relationship:
(3-15)
Substituting equation (3-3) into equations (3-4) and (3-5), note that the transition from the Octane to the Octane system in the above equations is clockwise, and the angle in equation (3-4) should be taken as...
Therefore, the coordinates of the robot's center of mass in the Oct reference frame are:
Rotational transformation:
Translation transformation:
(3-16)
In the formula,
The angle of the swing arm is defined as the positive direction as counterclockwise around the Xb axis.
If, in the O c coordinate system, the robot's center of mass appears in the first quadrant as the slope gradually increases, then it is considered that the robot can cross the boss after passing this critical point.
(2) The process of overcoming obstacles
Phase 1: Preparing to overcome obstacles.
As shown in Figure 11, establish a coordinate system with the origin at O. Through geometric relationships, we have:
Where α is the angle between the central axis of the swing arm and the horizontal line;
β is the angle between the center line of the swing arm and the radius of the side track perpendicular to the swing arm. This angle is a constant in robot design.
h represents the height of the obstacle.
The second stage involves climbing the obstacles.
When the robot begins to climb the obstacle, according to the geometric relationship in Figure 13, we can obtain:
Where M is the torque borne by the arm axis.
Since the calculations in the above formula are all derived under the assumption of near-static conditions, the robot should maintain a low speed and stable speed as much as possible during the climbing process.
The third stage involves lowering the arm to adjust the center of gravity.
This stage occurs when the lower edge of the robot's front wheels has just passed the height of the obstacle.
In the fourth stage, the center of mass overcomes the obstacle.
At this point, the robot's center of mass has just crossed the obstacle. Using geometric relationships, we can obtain:
If the previous center of gravity equation is satisfied, the robot can pass through the obstacle.
Phase 5, completed.
The robot eventually completed the flip and returned its forearm to the raised position.
Robot descends steep obstacles
The first stage is preparing to overcome obstacles.
As shown in Figure 17, move the robot in a direction away from the obstacle so that the extended line of the obstacle's edge lies between the robot's overall center of mass C and the center of the front wheel. This allows the robot to smoothly swing its forearms for support and prevents it from prematurely falling off the obstacle and causing injury. Once the robot reaches this position, it can proceed to the next action.
In the second stage, lower your arms.
Here, we will discuss three cases based on the different heights of the obstacles.
(1) h<d+rR
At this point, the obstacle is relatively low, and when the robot extends its arm forward, it can reach the ground.
At this point, the obstacle is too high, and the robot arm cannot reach the ground. The robot should begin to move slowly; as its center of mass moves past the edge of the obstacle, it should tilt forward, and the arm should touch the ground.
(3) h>d+r+L
At this point, the obstacle is too high, and forcing its way down would put excessive stress on the robot's forearm. Furthermore, it could easily tip over upon contact with the ground. Therefore, this obstacle-crossing method is not recommended in non-emergency situations or when alternative routes are available.
In the third stage, the robotic arm retracts to provide support.
In the fourth stage, move forward with arm support to leave the obstacle.
These two stages are a breakdown of the same action. This action is optional; in non-emergency situations, it can be used to ensure the robot's tail lands smoothly, avoiding impact. In emergency situations, this action can be ignored, allowing the robot to move forward directly and thus detach its tail from the obstacle.
Fifth stage: Extend your arms forward to return to the starting position.
Analysis of the robot's process of going up and down stairs
The process of a robot going up and down stairs can actually be broken down into three parts: going up a steep obstacle, going down a steep obstacle, and moving on the stairs.
Let's discuss the robot's movement on stairs. In fact, a robot's movement on stairs can be approximated as movement on a ramp. The difference is that on stairs, due to insufficient robot length or changes in the center of mass between steps, it may "bump" or "tail." The problem becomes more severe as the span of the stairs increases. We will discuss this situation below.
First, when the robot climbs the stairs, it lays its forearms flat, allowing its body to reach its maximum length. At this point, the robot is most stable when moving up the stairs. All the issues we discuss below are based on this principle.
When the robot's forearm extends, the overall effective length is:
L eff =L+dR
The position of the robot's center of mass at this moment can be obtained according to the formula for calculating the center of mass.
As shown in Figure 20, when there are two staircase edges under the robot's bottom, the robot's center of mass C is projected between the two edges, and the operation is stable. As it continues to move forward, two possible scenarios will occur:
(1) The projection of the robot's center of mass C has crossed point E, and the robot's arm has not yet reached point D.
At this point, the robot will "bump its head," as shown in Figure 21, a critical state. If the staircase span is any longer, the robot will tip forward as it continues to move, causing an impact to its front. The critical span of the staircase at this point is:
(3-25)
Where γ is the inclination angle of the staircase.
X'c is the distance from the robot's overall center of mass projection to the front wheel axle projection. When the span of the stairs is greater than Ls , the robot will "bump its head" phenomenon. In this case, the problem of climbing stairs should be transformed into climbing steep obstacles, and a strategy for climbing steep obstacles should be adopted.
(2) The projection of the robot's center of mass C has not yet passed point E, but the robot's tail has already passed point F.
At this point, the robot will exhibit a "tail-knocking" phenomenon. Because the robot's center of mass is shifted rearward, the robot's pitch angle will increase, but the tail will not actually suffer a significant impact. Conversely, when the center of mass projection crosses point E, it will flip forward and eventually "knock" again.
As shown in Figure 23, as the robot continues to move forward, the angle γ´ between the robot's chassis and the inclined plane formed by the stairs increases, forming an angle Δγ between the robot and the stair slope γ. This means the robot will no longer be in contact with the stair slope. When the projection of the robot's center of mass C crosses point E, the robot will lean forward to conform to the stair slope. Therefore, the larger Δγ is, the more severe the robot's "head-bumping" problem becomes. Equation 3-26 is the formula for calculating Δγ.
Where Xc” is the distance between the projection of the robot’s overall center of mass and the projection of the rear wheel center.
Analysis of robot tipping critical situation
The robot's anti-tilt capability, i.e., longitudinal stability. Generally speaking, tracked mobile mechanisms are less likely to tilt forward under normal external forces than wheeled mechanisms. The limiting slope angle θl is usually used as one of the basic evaluation indicators of its anti-tilt capability. When the robot climbs a slope as shown in Figure 24, if the slope angle θ ≥ θl , the robot will tip over. Based on geometric analysis and the principle of torque balance, we obtain...
3.4 Rear View Dynamics Analysis and Application
The robot rear view dynamics model is a model that observes the forces acting on the robot from the rear (or front) of the robot. It is mainly used for force analysis of the robot in a tilted state.
As shown in Figure 25, when a robot moves laterally on an incline with an angle of θ, the supporting force exerted on the robot by the incline is equal to the component of its gravity perpendicular to the incline. The frictional force acting on the robot is equal to the component of its gravity parallel to the incline.
When the robot is stationary on a slope or moving at a constant speed in a straight line, the coefficient of friction between the robot's tracks and the slope is the static friction coefficient. When the robot accelerates, decelerates, or turns on a slope, the coefficient of friction between the tracks and the ground is the dynamic friction coefficient. These two friction coefficients are distinguished here because the difference between the dynamic and static friction coefficients of the tracks and the ground is significant, as shown in Table 3-2.
4. Conclusion
The development of robot work platform systems is crucial for robot development. Taking the self-developed AEC_II detection robot as an example, this paper details the microscopic research of underground coal mine detection robots. This robot is specifically used in dangerous areas of coal mines and has certain reference value for the detection application in dangerous areas.
Contact person: He Xiaohu
