Synergistic Motion Stability of a Scorpion-like Composite Robot

Abstract

In this paper, a compliant control scheme based on the optimization of the contact force of the robot leg is proposed to improve the stability of the whole moving process of the robot. Firstly, according to the motion state of the robot, the change of its center of gravity is analyzed, then the stable gait of the robot is determined by the stability margin, and the smooth control of the robot’s foot trajectory is realized. Finally, the compliant control model of the robot leg is established. In the process of moving, the contact force between the legs and the ground is optimized in real-time, so that the composite robot can walk steadily on uneven terrain. The 3-D model of the scorpion composite robot was built with ADAMS software, and dynamics simulation was carried out according to the compliant control scheme. This paper takes the robot’s walking speed and torso angle as performance evaluation indexes and verifies the effectiveness of the compliant control scheme. The cooperative motion stability test is carried out on the actual uneven terrain. The test results show that the robot’s pitch angle and roll angle are between ±0.5°, which meets the motion stability requirements of the robot and verifies the correctness of the compliant control scheme and control model proposed in this paper.

1. Introduction

Footed robots have become a hot topic in modern robotics research. Compared with wheeled robots, leg robots are better able to adapt to all kinds of rough terrain and obstacles. Therefore, since 1990 [1,2,3,4], leg robots have received wide attention from scientists and engineers and have broad application prospects in lunar exploration, military exploration, engineering exploration, and so on. The imitation scorpion composite robot was developed for the temperature measurement of a coke oven plant. Concerning the development of robot gait, it is currently divided into static gait and dynamic gait [5,6]. Simply put, static gait is slower but more stable and has higher terrain adaptability [7], while dynamic gait is faster but less stable with relatively low terrain adaptation [8]. Zhang et al. studied a static gait planning method based on complex terrain that does not require any machine vision system and can be used for footed robots walking on unknown rugged terrain. There is an exploratory gait with force sensing integrated into the foot [9], which is a planning method in which when the robot’s foot is in the swing phase encounters an obstacle [10], the robot can actively adjust its gait planning. A special leg structure is designed to stabilize the motion of the robot and reduce the energy consumption of the robot [11,12], and changes in the movement of robotic legs by optimizing gait parameters and foot contact were also studied, which, in turn, reduced energy consumption [13,14,15]. Three-dimensional force sensors are installed on the foot-end actuators of the footed robot to implement the proposed haptic sensing. Minati et al. Used FPAA-based oscillators to build a hierarchical CPG network for an ant-like hexapod robot. Six oscillators form a central pattern generator to generate global leg coordination patterns, and each node is coupled to a local pattern generator dedicated to generating the trajectory of one leg. Several recognizable gait patterns and continuous generalized gait patterns can be generated by varying the topology and strength of the coupled oscillator connections [16,17,18]. This method has achieved good results in hexapod robots with multi-joint limbs. Luo et al. proposed a method based on lateral body tuning to improve the static gait performance of footed robots [19]. A static walking gait is a better choice for walking on complex terrain. Despite the relative maturity of static walking gait, there is still less research on how to control a hexapod robot for walking on complex and unknown terrain.

Hexapod robots are nonlinear, unstable, multi-input, and multi-output systems that are difficult to control. They also interact with the environment, so controllers using inverse dynamics were developed for bionic limb control, and a feedback/feedforward control structure using an inverse dynamics approach was proposed by a research group at the Italian Institute of Technology (IIT) in HyQ [20]. A similar idea of energy consumption optimization based on an inverse dynamics model was adopted by Mahapatra et al. and applied to a hexapod motion controller [21]. As there are more disturbances in the unknown environment, it is difficult to achieve accurate control and often the robot needs to be compensated and a large number of nonlinear calculations are not conducive to real-time controllers. Therefore, at a very early stage, Feldman proposed the equilibrium point hypothesis, which seems to be more in line with biological motion, to mimic the reflex system of human muscles, when the body is about to lose balance, the corresponding force of the leg changes in real-time to maintain the balance of the body. This hypothesis links motion and posture synergistically into a single mechanism in which many balance controllers have been successfully developed on different robots, and similar ideas inspired by balance controllers were used in the design of redundantly driven manipulators by Kim et al. [22]. Shi et al. proposed a balance controller for a quadruped robot. In this controller, the toe balancing point is obtained indirectly by solving a quadratic planning problem to obtain the required balancing contact force, and the effectiveness of the controller is verified by a series of experiments [23]. Compliance control is an effective method to reduce robot body vibration and maintain robot stability. To improve the environmental compliance of footed robots on rugged terrain, impedance control was investigated to reduce the effect of contact forces, which would be too large in high-precision position control [24,25,26,27]. Impedance control can be divided into position-based impedance control and force-based impedance control [28,29]. So far, compliance controllers are mostly applied to robots, and relatively few compliance controllers are for foot-based robots. When designing leg controllers for composite robots, it is necessary to consider not only the coordination between a single leg and other legs but also the synergy between the legs and actuators. The interaction between the composite robot and the environment also significantly affects the operating state of the robot, so obtaining a stable contact force is the leg flexibility the main development direction of the controller.

In this paper, the concept of co-balancing is introduced into the actuator and motion mechanism of a composite robot. Firstly, the stability margin of the robot’s behavior gait is analyzed during normal operation. Then, using the compliance controller, the mobile contact force optimization method is introduced into the robot’s leg control scheme. The robot can better adapt to the environment and realize stable movement under unknown conditions.

2. Materials and Methods

2.1. Gait Stability Analysis

The simulation model of the scorpion-like composite robot designed in this paper is shown in Figure 1, where each leg has three degrees of freedom: two degrees of freedom for the hip joint and one degree of freedom for the knee joint. Figure 2 shows the specific leg kinematics DH modeling.

The gait stability needs to be designed according to the movement of each actuator. The workspace of each leg is determined according to the joint angle, as shown in Equation (1).

$${\theta }_{min}\le \theta \le {\theta }_{max}$$

(1)

where $\theta$ is the current joint angle, ${\theta }_{max}$ and ${\theta }_{min}$ are the maximum and minimum angles allowed for the joint. The maximum angle is the angle between each leg that does not interfere. The minimum angle is the minimum angle at which the robot can walk. Only when within the allowable range, the parts between the robots do not interfere with each other and walk normally.

For the analysis of the robot gait, it is roughly divided into two categories: intermittent gait and synergic gait, with the difference between the two gaits being that the intermittent gait is that the swing of the legs and the movement of the center of gravity of the robot in the process of walking is separated; synergic gait means that the center of gravity of the robot changes while the legs swing during the movement of the robot. Therefore, the moving efficiency of the synergic gait is higher than the intermittent gait, and the robot walks faster, smoother, and more bionic relatively, but the increase in walking speed leads to a decrease in the stability margin [30].

The dynamic stability criterion is a key parameter in the gait research of scorpion robots [31]. The rationality of gait planning is judged by the stability margin. The foundation of robot walking stability is that the pressure center M of the robot always falls in the polygon area formed by the end-effector of the supporting leg. As shown in Figure 3, ABC is the position of the end of the robot, M is the pressure center of the robot, and the stability margin refers to the shortest distance between M and the three edges. The larger the stability margin, the more stable the gait, but once the center of pressure is not within the polygon formed by the supporting leg, the walking is unstable.

The formula for calculating the coordinates of the M point $\left(x_M,y_M\right)$ is:

$$\begin{array}{l}{x}_M=\frac{F_{Az}{x}_A+F_{Bz}{x}_B+F_{Cz}{x}_C}{mg}\\ y_M=\frac{F_{Az}{y}_A+F_{Bz}{y}_B+F_{Cz}{y}_C}{mg}\end{array}$$

(2)

where m is the mass of the robot, $A\left(x_A,y_A\right)$, $B\left(x_B,y_B\right)$, and $C\left(x_C,y_C\right)$ are the coordinates of the supporting leg of the robot, and $F_{Az}$, $F_{Bz}$, and $F_{Cz}$ are the forces in the vertical direction of the foot-end effector of the supporting leg. In simple terms, the projection of the center of mass onto the supporting surface can be used instead of the center of pressure M.

The walking stage of the robot is shown in Figure 4. The walking of the compound robot is divided into two stages; 1–2 is the fast walking stage, and 2–3 is the preparation of temperature measurement stage. The fast walking phase is to improve efficiency, and the robot walks quickly to reach the temperature measurement location; in this stage, the robot’s limbs are in a contracted state so that the robot’s center of gravity is as much as possible in the center of the body. The robot is a regular six-legged robot, using a triangular gait for walking, where the triangular gait is designed for a cooperative gait. For the preparation for the temperature measurement stage, during the walk, each of the robot’s actuators starts to unfold into a working state, meaning it is ready for temperature measurement. The forward movement of the pliers and tail will inevitably lead to a forward imbalance of the center of gravity, and an improved triangular gait is proposed in this case. The improved triangular gait is intermittent, and the walking speed is relatively slow, but it can better cope with the forward movement of the center of gravity. The sequencing diagram of the robot legs is shown in Figure 5.

Fast walking stage: To keep the center of gravity of the robot in the center of the body, first determine the contraction state of the pliers as shown in Figure 5, and then go to determine the balance state of the robot by adjusting the angle $\theta$ of the robot’s tail joint and observe whether the torso is tilted; the specific simulation diagram is shown in Figure 6, where the center of the warp plate coincides with the center of the torso in the vertical direction.

The simulation results show that the angle $\theta$ tilts to the right when it is 76–120 degrees and tilts to the left when it is 0–24 degrees, so we obtain the mid value of 50 from 24 to 76, which makes the center of gravity stable in the middle of the torso. According to some previous studies, the compound cycloid trajectory can reduce the impact between the foot end and the ground when planning the trajectory of the robot, so the compound cycloid method is used to plan its trajectory, and the trajectory equations are derived based on the joint velocity and acceleration continuity requirements, and the robot gait parameters are set, where the step length S = 60 mm, step height h = 30 mm, gait period T = 8 s, and the swing phase and support phase are switched at T0 = T/2; then, in the 0–T0 period, the foot end is in the swing phase sw, and in the T0–T period, the foot end is in the support phase st. Let the horizontal direction be the x direction, the vertical direction be the y direction, the foot-end contact point with the ground be the starting point of the trajectory planning, which is directly below the hip joint center of mass, and the height of the body center of mass to the ground is H = 7 cm, and the single leg movement time is t. Establish the foot end trajectory in the body coordinate system Equation: The acceleration equation in the x-direction of the swing phase is:

$${\ddot{x}}_{sw}=A\sin (4\pi t/T)$$

(3)

Integrating the velocity in the x-direction can be obtained as (3).

$${\dot{x}}_{sw}=\frac{-AT}{4\pi }\cos (4\pi t/T)+C_1$$

(4)

To make the robot run smoothly and prevent rollover, the boundaries of the robot foot trajectory planning are constrained: ${\dot{x}}_{sw}=0$ for t = 0 and ${\dot{x}}_{sw}(T/2)=0$ for t = T/2. Bringing the boundary conditions into the velocity equation yields the specific velocity expression as:

$${\dot{x}}_{sw}=\frac{AT}{2\pi }(1-\cos (4\pi t/T))$$

(5)

To obtain the displacement process in the horizontal direction, the displacement equation is integrated again in the same way, the boundary conditions are: when t = 0, $x_{sw}=0$; when t = T/2, $x_{sw}(T/2)=S$. The boundary conditions are brought into the displacement equation in the x-direction of the swing phase as:

$$x_{sw}=S(2t/T-\frac{1}{2\pi }\sin (4\pi t/T))$$

(6)

Similarly, the displacement equation of the foot end of the x-direction of the support phase can be obtained as:

$$x_{st}=-S(t/T-\frac{1}{2\pi }\sin (4\pi t/T))+S$$

(7)

y-direction for [0, T/4] and [T/4, T/2] for trajectory planning, the acceleration expression is:

$${\ddot{y}}_{sw}=\{\begin{array}{ll}{k}_1\sin \frac{8\pi }{T}t& (0\le t\le T/4)\\ k_2\sin \frac{8\pi }{T}(t-T/4)& (T/4\le t\le T/2)\end{array}$$

(8)

The constraints are ${\dot{y}}_{sw}(0)=0$, ${\dot{y}}_{sw}(T/2)=0$, $y_{sw}(0)=-H$, $y_{sw}(T/4)=-h$, $y_{sw}(T/2)=-H$, and so on.

$$y_{sw}=\{\begin{array}{ll}2(H-h)(2t/T-1/4\pi \sin (8\pi t/T))-H& (0\le t\le T/4)\\ 2(h-H)(2t/T-1/4\pi \sin (8\pi t/T))+H-2h& (T/4\le t\le T/2)\end{array}$$

(9)

The height of the hip joint of the support phase of y concerning the ground is constant, so the expression for the displacement in the y direction of the support phase is:

$$y_{st}=-H(T/2\le t\le T)$$

(10)

In summary, the equation of the composite cycloid trajectory is:

$$x=\{\begin{array}{ll}{x}_{sw}=S(2t/T-\frac{1}{2\pi }\sin (4\pi t/T))& (0\le t\le T/2)\\ x_{st}=-S(2t/T-\frac{1}{2\pi }\sin (4\pi t/T))+S& (T/2\le t\le T)\end{array}$$

(11)

$$y=\{\begin{array}{ll}{y}_{sw}=2(H-h)(2t/T-1/4\pi \sin (8\pi t/T))-H& (0\le t\le T/4)\\ y_{sw}=2(h-H)(2t/T-1/4\pi \sin (8\pi t/T))+H-2h& (T/4\le t\le T/2)\\ y_{st}=-H& (T/2\le t\le T)\end{array}$$

(12)

A simplified model of the gait design for the fast-walking phase of the robot is shown in Figure 7. The middle point of Figure 7 is the center of gravity of the robot in the fast-walking stage. When the gait is the landing of the leg ends 1, 4, 5 landing, or the landing of ends 2, 3, and 6, the center of gravity of the robot is basically in the center of the triangle, and the distance from the center of gravity to the 3 edges is the same, with a large stability margin and stable walking. However, if it is in the preparation stage for temperature measurement, the center of gravity moves forward, as shown in Figure 7b. The robot’s center of gravity is essentially at the edge of a triangle, and the stability margin is small, making the robot’s motion extremely unstable.

The following intermittent gait changes were made to the preparation temperature measurement stage, with the landing points changed to 1, 2, 5 and 3, 4, 6, and with 5 and 6 placed at the back end, called gait 1, as shown in Figure 8.

Figure 8 shows the location of the center of gravity when 1, 2, and 5 landed, at this time, 3, 4, and 6 legs raised, ready to enter gait 2, with gait 2 as shown in Figure 9, and the 3, 4, 6 landed and the 1, 2, 5 legs raised and then moved to gait 1, reciprocating the cycle.

The gait timing diagram of the two stages corresponding to the robot is shown in Figure 10.

2.2. Compliance Control Model

Under ideal conditions, the composite robot can walk according to the given trajectory, but the actual environment often does not meet the ideal experimental and simulation conditions, such as the uncertainty of contact between the end and the ground and the asymmetry of the mechanism itself, which leads to a series of factors such as dragging and sliding of the robot’s hind legs during the movement, seriously affecting the robot’s stability. To achieve stable motion on the plane, a closed-loop feedback compliance control is necessary to drive the robot joints with torque and ensure the robot’s walking stability by coupling the corresponding physical environment. A compliance control scheme based on foot-end contact force is proposed. This control scheme integrates robot motion and poses into a single mechanism, which avoids uncertain parameters and complex nonlinear computations compared to inverse dynamics control. At present, the controller is ideal for controlling robotic hands with similar limbs, but there are few balance point controllers designed for foot robots.

Footed robots move over often uneven and unpredictable terrain. Because of unpredictable terrain conditions, the robot’s feet may have a greater impact on contact with the ground. Even if the foot’s trajectory is smooth enough, contact with the ground can disrupt the force balance of the system, leading to a deviation in the robot’s body posture. Without proper compensation and control, this problem degrades walking performance. To achieve the robot’s stable operation, the control system should be designed to take into account not only the movement of the legs but also the force of the feet. Therefore, it is essential to establish a suitable model that can bridge the gap between leg motion and foot force. To solve this problem and achieve the equilibrium of the system motion, a contact force optimization method is established to modify the stance trajectory in real-time. Modifying the method, the desired contact foot force can be obtained indirectly. This optimization is akin to stepping on a raised hard object, where the contact force at the end is large enough to support the body’s stability. The robot then maintains the leg moving at a curved angle, as shown in Figure 11 below.

PID controller is a kind of linear controller, according to the given value $y_d(t)$ and the actual output value $y(t)$ constitutes the deviation, by adjusting the deviation, to achieve the ideal control effect. The compound robot is a complex multiple coupling control system, here to make the control simple, the output decoupling is multivariate, the output of the system in which the control object for robot leg joint variables, the joint Angle of $y(t)$ for the actual output, $y_d(t)$ to compensate for the joints get the expectations of the joint degree deviation, so the trajectory tracking control law can be obtained as follows:

$$\tau =k_{p}e+k_d\dot{e}+k_i{\displaystyle \mathrm{\int }edt}$$

(13)

where $\tau$ represents the joint driving torque, $e$ is the deviation between the expected joint driving angle and the actual output angle, $k_p$, $k_d$, and $k_i$ are proportional differential, and integral constants, respectively. The desired joint angle and the actual output joint angle are obtained by the PID controller, to achieve the goal of robot joint driving.

The actual foot force $F_i^{\prime }$ and the desired foot force $F_i$. $\Delta F_i$ represents the foot force deviation, namely the foot position correction value. First of all, we need to know the position of the robot’s feet relative to the robot’s body $\Delta P_i$. It can be solved with Equation (13).

$$\Delta P_i=R_W{P}_i-P_i+\Delta Q$$

(14)

$$R_W\approx \left[\begin{array}{lll}1& \Delta \beta \Delta \gamma -\Delta \alpha & \Delta \beta +\Delta \alpha \Delta \gamma \\ \Delta \alpha & \Delta \alpha \Delta \beta \Delta \gamma +1& \Delta \alpha \Delta \beta -\Delta \gamma \\ -\Delta \beta & \Delta \gamma & 1\end{array}\right]$$

(15)

The actual foot force of the robot can be obtained directly from the foot-end force sensor.$\Delta \alpha$, $\Delta \beta$ and $\Delta \gamma$ represent the small yaw, pitch, and roll angle variations of the robot body, respectively. $\Delta Q={\left[\Delta Q_x,\Delta Q_{\mathrm{y}},\Delta Q_{\mathrm{z}}\right]}^T$, represents three small displacements of the robot body, and Equation (16) gives the required foot force at the foot end in the z-direction

$$F_{\mathit{iz}}=-P_{\mathit{ix}}\Delta \beta +P_{\mathit{iy}}\Delta \gamma +\Delta Q_z$$

(16)

Straight ahead is the most important motion state of the hexapod robot. It is mainly driven by tangential contact forces along the direction of robot motion, namely $F_{\mathit{ix}}$. To keep the robot walking stably, the foot sliding resistance is considered in the calculation of $F_{\mathit{ix}}$. To reduce the risk of foot sliding, the maximum ratio of tangential to normal forces should be as small as possible. $\mu$ represents the force ratio at each foot, that is, the ratio of the tangential component of the force to the normal component [32], and $\mu$ is also related to the desired body forces, as shown below:

$$F_{\mathit{ix}}=\mu F_{\mathit{iz}}$$

(17)

$$\mu =\frac{F_x}{F_z}$$

(18)

where $F_x$ is the sum of all $F_{\mathit{ix}}$ and $F_z$ is the sum of all $F_{\mathit{iz}}$.

3. Results

3.1. The Simulation Results

To verify the effectiveness of the compliant control in terms of walking stability, ADAMS simulations and practical experiments were carried out. First, a common rigid flat terrain walking experiment was conducted to verify the effectiveness of the gait trajectory of the control scheme. Secondly, a natural walking experiment in the field was carried out to verify the advantages of the control scheme in maintaining a stable walking posture. To better explain the improvement of the robot’s walking control scheme, a typical inverse dynamics control scheme was used in the simulation experiments. The inverse dynamics control scheme is shown in Figure 12.

Figure 13, Figure 14 and Figure 15 show the simulation process at points 1, 2, and 3 corresponding to Figure 4. The drive curve of the joints can be obtained during the simulation as shown in Figure 16. During the 50 s to 60 s, the robot completed the gait transition, entering gait 2 from gait 1.

The velocity diagram at each leg support point derived from the compliance control model is shown in Figure 17.

The velocity profiles out of the contact points of leg 1, leg 4, and leg 6 derived from the inverse kinematic control model are shown in Figure 18.

From the simulation results shown in Figure 17 and Figure 18, it can be seen that at the moment when the foot lands, the contact point has a certain vibration with the ground, which causes the speed of the contact point to have a certain oscillation fluctuation, and the oscillation of the compliant control model is maintained at 1°/s, but the inverse motion control model is maintained at about 2–3°/s, and the relative speed control of the compliant control is better than that of the inverse motion control, and the walking performance is improved. Because the entire control scheme of the compliant control model proposed in this paper is completely linear, different from the nonlinear inverse dynamic model, the calculation process of the foot contact force in the supple control scheme is simpler and the response time is faster. Therefore, in the case of obtaining the same stable control performance, the proposed compliant control scheme is easier to adopt.

3.2. The Prototype Experiment

The imitation scorpion composite robot is composed of an imitation scorpion mechanical structure, 2500 mAH power supply, Raspberry Pi 4B control board, extended version, high-definition camera, 20 kg/cm LX-224HV serial bus servo, voltage display module, infrared distance sensor, MPU6050 sensor, and other modules; the main control board uses 11.1 V rechargeable lithium-ion battery for power supply, which can make the robot no-load continuous motion time of about 40 min. It is also equipped with a voltage display module that can detect the voltage of the robot at any time. In order to make the robot run more stably, the walking mechanism adopts an intelligent serial bus servo, which uses serial bus communication, and can directly use serial port commands to control multiple servos connected together. Each steering gear is assigned a different ID, and then the behavior of the entire robot is controlled by python programming. The entire control system is precisely controlled by this ID.

Through theoretical analysis and simulation results analysis, the superiority of the planar walking compliance control scheme is proved. To verify the applicability of compliance control to steady robot motion in the real environment, a natural field walking experiment was carried out. The compliance control scheme is only used in this experiment. The field site is shown in Figure 19. The soil of the field is hard, and the depth of foot sinking is about 2 mm, so it will cause the robot’s foot sinking depth to be disregarded during the walking process, and the robot’s body is always parallel to the ground when the robot walks. The process of robot walking is shown in Figure 20 and Figure 21.

The actual pitch angle and rolling angle of the robot are shown in Figure 22. In the process of walking, the pitch angle and roll angle hover between plus and minus 0.5 degrees, respectively. The proposed compliant control scheme enables the robot to achieve stable motion.

4. Conclusions

This article concludes through simulation that the foot-end vibration speed of the robot in contact with the ground under the flexible controller is reduced by 1–2°/s compared with the robot’s foot-end speed under the control of inverse kinematics. The actual pitch angle and rolling angle of the robot obtained by the experimental prototype walking experiment in the field hover at plus or minus 0.5 degrees can verify the effectiveness of the flexible controller we proposed, but the simulation and experiment at the current stage were carried out on a horizontal plane. In order to enable the robot to walk smoothly on various slopes in the future, we will optimize the flexible controller through a large number of experiments.