Kinematic performance analysis of a robot climbing steps without movable tracks on a swing arm

A lightweight tracked robot with no moving swing arm is designed, which is mainly used in exploring narrow spaces. The mathematical model of the position of the robot’s center of mass with the elevation Angle of the vehicle body and the Angle between the swinging arm is established. The kinematics of the robot in different typical terrains, such as ramp, ditch, and step, is analyzed. By analyzing the critical state of the change of the position of the center of mass, the maximum Angle and width of climbing and crossing are calculated under ideal conditions. The maximum height of the robot is also calculated when using different ways to overcome obstacles, corresponding to the position attitude under the maximum climbing height. It provides a data theory basis for robot motion control strategy.


Introduction
A tracked robot has good motion performance and can adapt to different terrain.Obstacle crossing ability can be effectively improved by adding a swing arm mechanism.According to the number of swing arms, the robot can be divided into four track double swing arm robots and six-track four swing arm robots [1] .
At present, domestic and foreign scholars have systematic and mature theoretical studies on the motion characteristics of the above-mentioned swing arm-tracked robots [2][3][4] .For example, Wang et al. [5] analyzed the stability performance of the double-swinging arm-tracked robot in crossing obstacles.Li et al. [6] analyzed the obstacle-crossing mechanism and obstacle-crossing ability of four swinging arms and two swinging-arms-tracked robots from the perspective of kinematics.Wang et al. [7] analyzed the traversability of a four-articulated-arm, six-track robot in single-sided obstacle terrains.Hagiwara et al. [8] analyzed the maximum height of a six-track four-swing arm.However, this type of articulated arm robot typically has tracks mounted on the arms, which increases the weight and size of the robot.This is not conducive to obstacle crossing in confined spaces.
This article presents a non-articulated arm-tracked robot, where tracks are mounted on the arm.With this design, the articulated arm-tracked robot can adapt to different terrains with flexibility, while maintaining high stability and obstacle-crossing capabilities.Furthermore, since the tracked system on the arm does not require additional moving components, the size and weight of the robot are effectively reduced.This enables the robot to better adapt to confined working environments, enhancing its operational flexibility and efficiency.It provides an effective solution for tasks requiring exploration in narrow spaces.By analyzing the motion process of the robot in typical obstacle terrains, the corresponding motion parameters are obtained.This provides a theoretical basis for data-driven robot control in relevant scenarios.

Robot structure design and parameters
As shown in Figure 1, the overall model of the robot has a symmetrical structure and a low design height of the pith, which supports the recovery of the pith into the interior of the car body, so as to achieve the purpose of protecting the camera.At the same time, in order to cope with the possible impact of climbing over special obstacles, the track vehicle can continue to perform reconnaissance tasks after the vehicle is turned 180°, thus greatly improving its flexibility.With the geometric center of the robot site as the origin and the robot's forward direction as the Xaxis, the coordinate system is established, as shown in Figure 2. Because there are many parts in the robot, non-homogeneous objects, but the centroid of each standard part such as the motor, camera, and other parts is known, so the centroid of the robot ( , , ) can be found through the combined moment theorem: where , , respectively represent the space positions of the standard parts inside the chassis in the robot coordinate system.According to the above method, the robot is divided into two parts, the car body and the swing arm, then the robot's center of mass can be represented: We can find the overall center of mass of the robot ( , , ) where θ represents the Angle between the robot's swing arm and its own coordinate system, through the elevation Angle α, swing arm Angle β can be expressed as:

= −
The relationship diagram between elevation Angle α, swing arm Angle β, and center of mass position can be obtained by using the MATLAB data processing tool.As shown in the figure below, with the change of swing arm Angle, the overall center of mass position of the robot will change in the area shown in Figure 3 and Figure 4.

Kinematics analysis of tracked robot
The road surface faced by the tracked robot is complicated, but most of the complex terrain can be superimposed by several typical terrains.It can be divided into three types: slopes, steps, and trenches [9]   .This paper mainly analyzes the kinematic characteristics of the tracked robot under typical terrain.

Kinematic analysis of climbing
The movement of the tracked vehicle on the slope mainly includes two kinds: longitudinal driving and transverse driving.Regardless of which operating posture, the force is as follows: Gravity : its centroid position can be obtained through the centroid calculation formula.Air resistance: Because the track car is small and the speed is not fast, the air resistance is ignored.Ground resistance : Generally speaking, the resistance of the ground is proportional to the normal pressure of the tracked vehicle on the ground.= : Normal pressure of the track vehicle on the slope : ground resistance coefficient of the tracked vehicle during running Tractive effort : It is mainly caused by the friction between the track and the ground.When the robot moves, there is a relative slide between the track and the ground, and the sliding resistance is the source of traction.

Longitudinal driving.
Longitudinal driving refers to the robot driving along the slope direction.At this time, it is necessary to consider whether the robot has the possibility of turning over with the front and rear heavy wheels of the track as the center respectively during the driving process [10] , and two situations of uphill and downhill need to be considered.
When moving upward at a uniform speed along the ramp, the motion state model is shown in Figure 5. Taking a moment at point A: To prevent vehicle rollover, it is necessary to satisfy ≥0: The maximum gradient angle can be calculated as: By substituting the data into Table 1, we can obtain the maximum gradient angle under ideal conditions.When traveling down the slope at a constant speed, the motion state is shown in Figure 6.Taking point B into consideration, the moment can be calculated as follows: Being similar to the downhill case, it is necessary to satisfy l ≥ 0, which allows us to determine the maximum slope angle in this situation.The motion model is shown in Figure 7.In case, the balance equation is satisfied: Taking the moment at point A, we obtain: To prevent rollover, it is necessary to satisfy N_2 ≥ 0. Therefore, we have: The maximum slope angle in this case can be determined as: igure 7. Robot motion model along slope contour.In conclusion, the maximum gradient angle that can be climbed satisfies = ( , , ), which is approximately 66.2°.

Kinematic analysis of trench crossing
With the aid of a swing arm mechanism, a tracked vehicle can more effectively cross a trench.As long as the center of mass and the front and rear contact points are positioned above the trench while crossing, the vehicle can pass through normally.The motion process is shown in Figure 8.In order to effectively cross the trench, it is necessary for the front end contact point of the swing arm to make contact with the ground.The angle of the swing arm at this moment can be calculated, as shown in Figure 9.According to Table 1 and the given data, it is calculated that ≈ 7.17 ∘ .At this angle, as long as we can ensure that the center of mass and the front and rear contact points are not simultaneously inside the trench, the maximum distance of the trench that can be crossed satisfies: represents the maximum distance from the center of mass to the left contact point.represents the maximum distance from the center of mass to the right contact point.According to the formula for calculating the center of mass, in this state, the center of mass of the robot is located at (9.35, −3.35, 3.86).Therefore, the maximum length of the trench that can be crossed is approximately 105.65 mm.

Kinematic analysis of obstacle traversal
During robot navigation, one of the most common obstacles encountered is a step.Ascending and descending steps are reversible processes, meaning that if the robot can climb up a step, it can also descend from it.Here, we will focus on analyzing the process of climbing up a step.The robot has two main climbing postures for ascending a step: reverse posture and forward posture.

Forward climbing motion kinematic analysis of ascending a step.
The robot climbs the step in the forward direction by rotating the swing arm to place it on the step.This lifts the main body of the robot, allowing the front end of the tracks to make contact with the outer edge of the step, thus completing the obstacle traversal.In this posture, the swing arm is positioned at the front and the gimbal is positioned on top.After the climbing action is completed, there is no change in posture.The motion model is shown in Figure 10.
Because ∈ (0,90°), the height H of the step is an increasing function with respect to the position of the center of mass and a decreasing function with respect to .Therefore, in the climbing process relative to the robot's own coordinate system, when the position of the center of mass is larger and is smaller.The center of mass is closer to the left and lower side, and the obstacle traversal capability of the robot is better.
Differentiating H(α) with respect to the angle α between the robot and the ground yields: .At the same time, the maximum value of α can be determined based on the position of the center of mass ( , , ): If the height of the step exceeds , it is not possible to climb the step.Even if ∈ (0,90°) if > , the robot will tilt backward in reverse due to the center of mass being positioned higher, causing the contact point of the tracks to lose contact with the outer edge of the step.The reverse posture for climbing stairs (a): The tracked robot has its swing arm located at the front end of the vehicle body, with the gimbal positioned on top.By rotating the swing arm, the robot lifts the body, allowing the tracks to make contact with the outer edge of the step's corner.After completing the climbing motion, the posture will change: the swing arm will be at the rear and the gimbal will be positioned downwards.The tracked robot has its swing arm located at the rear end of the vehicle body, with the gimbal positioned downwards.The principle and posture are the same as posture a.By supporting the robot with the swing arm, the tracks can make contact with the step.After climbing the step in this posture, the posture remains unchanged, and the robot can adjust its posture again by rotating the swing arm.Regardless of the specific posture used for reverse stair climbing, in the end, the robot will successfully climb the stairs and reach the state depicted in Figure 15.
Critical state: The robot's tracks make contact with the edge of the step and lift the body to climb onto the step.However, in the next instantaneous moment , the swing arm will lose contact with the ground.At this point, the robot will lose the support force from the ground.As long as the center of mass position is to the right of the outer edge line of the step, the robot will not tilt backward.In this critical state, the height H of the step can be determined as follows:

Conclusion
From a kinematic perspective, an analysis was conducted on the motion of a robot with a stationary swing arm on various typical surfaces.Based on specific data, the climbing, obstacle crossing, and trench crossing capabilities of the robot were calculated under ideal conditions.① Under ideal conditions, the robot is capable of climbing slopes with a maximum angle of 66.7°.② The maximum distance that the robot can cross a trench under ideal conditions is approximately 105.65 mm.
③In the forward climbing posture, the maximum climbing height is approximately 65.1 mm, while in the reverse climbing posture, the robot can achieve a higher climbing height of approximately 274.7 mm.
Based on the theoretical results from kinematic analysis, it is possible to design suitable control algorithms to guide the movement of tracked vehicles.These algorithms can effectively determine whether the vehicle can pass through specific terrains such as slopes and trenches.They can also identify the appropriate posture for overcoming obstacles of different heights.This provides a theoretical foundation and data basis for the design of control systems.By utilizing the theoretical results and desired functionalities, appropriate control algorithms can be developed to enable the tracked vehicle to navigate effectively and overcome obstacles.

Figure 3 .
Figure 3.The range of the center of mass in the x plane.

Figure 4 .
Figure 4.The range of the center of mass in the y plane The main structural parameters of the robot are shown in the following table:

Figure 6 .
Figure 6.Robot downhill motion model.3.1.2.Lateral driving.Lateral movement refers to the robot moving at a constant speed along the contour lines of the slope.In this case, it is necessary to consider the possibility of the robot overturning with the entire body supported by the left and right tracks.The motion model is shown in Figure7.In case, the balance equation is satisfied:

Figure 8 .
Figure 8.The kinematic model of the robot for trench crossing.

Figure 10 .
Figure 10.The forward posture of the robot for climbing stairs.

Figure 11 .
Figure 11.Robot forward climbing step motion model.The swing arm angle determines the position of the robot's center of mass( , , ).With the center of mass position fixed, the pitch angle determines the maximum obstacle height that the robot can overcome.Using MATLAB simulation, the relationship diagram between α, β, and H can be obtained as shown in Figure 12.The results align with the expected outcomes from the derived formula.At = 45°and = 315°, the maximum obstacle height achieved is 65.1 mm.

Figure 12 . 2 .
Figure 12.The relationship between the pitch angle , the swing arm angle , and the step height . .

Figure 13 .
Figure 13.The reverse posture of the robot for climbing stairs (a).

Figure 14 .
Figure 14.The reverse posture of the robot for climbing stairs (b).

Figure 15 .
Figure 15.Robot reverse climbing step motion model.By simulating with MATLAB, the relationship between α, β, and H in this state can be obtained as shown in Figure 16.This will provide us with specific numerical results.At = 35.15°and = 54.85, the maximum obstacle height achieved is 274.7 mm.

Figure 16 .
Figure 16.The relationship between the pitch angle , the swing arm angle , and the step height .
ArgumentNumerical value Robot body mass / 3.5 The total mass of the two swinging arms / 0.1 Distance between center of front and rear track wheels / 230 Distance of center before and after swinging arm / 230 Distance between center of mass of swing arm and center of the front track wheel / 95.96 Track wheel radius (including track thickness) /