Inspired by the dynamic gait adopted by gecko, we had put forward GPL (Gecko-inspired mechanism with a Pendular waist and Linear legs) model with one passive waist and four active linear legs. To further develop dynamic gait and reduce energy consumption of climbing robot based on the GPL model, the gait design and trajectory planning are addressed in this paper. According to kinematics and dynamics of GPL, the trot gait and continuity analysis are executed. The effects of structural parameters on the supporting forces are analyzed. Moreover, the trajectory of the waist is optimized based on system energy consumption. Finally, a bioinspired robot is developed and the prototype experiment results show that the larger body length ratio, a certain elasticity of the waist joint, and the optimized trajectory contribute to a decrease in the supporting forces and reduction in system energy consumption, especially negative forces on supporting feet. Further, the results in our experiments partly explain the reasonability of quadruped reptile’s kinesiology during dynamic gait.
Wall-climbing robots can move and work on a vertical wall to complete various tasks, which have attracted much attention of researchers around the world and have wide application fields including antiterrorism, postdisaster rescue, engineering test, and maintenance and inspection for hazardous environment [
Generally, gaits of multilegged climbing robot could be classified into two kinds by leg raise sequence and stability: the quasistatic gait and dynamic gait. As the name implies, the quasistatic gait shows that the whole robot system maintains static balance during the climbing process, which is easy to ensure the behavior of robot [
With the development of biomimetics, the dynamic gait which is closer to actual biologic locomotion has been applied to improve vertical climbing efficiency and reduce power consumption. To describe the dynamic gait, some models are abstracted from the biologic locomotion pattern, including the spring-mass (SM) model [
Trajectory planning for multilegged climbing robots refers to two major steps. Firstly, the trajectory solution for a required motion task could be obtained based on the forward and inverse kinematics. Furthermore, it is to ensure that the final path satisfies the needs of the desired conditions, such as minimum path, time, and energy consumption. Wang et al. investigated optimal attaching and detaching trajectory for wall climbing which guarantee reliability of the climbing robot [
In our previous work, we have proposed the GPL (Gecko inspired mechanism with a Pendular waist and Linear legs) model and verified its feasibility [
The organization of this paper is as follows: firstly, kinematics and dynamics of GPL are derived systematically and the conditions of gait reuse are obtained in the second section. And the singularity and feasible region of the waist are analyzed in detail. Then, the effects of structural parameters on the supporting forces are explored, including body length ratio, driving angle, and elasticity of the waist joint. Moreover, the trajectory of the waist is optimized based on system energy consumption. To testify our analysis, a bioinspired robot is developed and the prototype experiments are performed. Finally, conclusions are drawn and the future work is described.
As illustrated in Figure
The GPL and its dynamic model derived from gecko. The red dotted line indicates the current adherent mechanism. The black dots on the foot mean the corresponding diagonal legs are attached with the climbing surface, while the gray dots mean the corresponding diagonal legs are swinging. Here,
The trot gait inspired by gecko’s climbing is adopted for the GPL model, as shown in Figure
The trot gait inspired by gecko’s climbing for the GPL model. Here, red dotted lines denote that the corresponding mechanisms are executing the active movement. (a)–(c) show the gait movement principle during half one gait. (a) The initial state of trot gait.
In view of the symmetry of gait, we choose the procedure of one phase for analysis and apply the result to the next symmetrical phase, as shown by red dotted lines in Figure
Let
Based on (
For further dynamic analysis of GPL, driving forces on the linear legs are analyzed, which significantly determine the control and performance of the robot. According to Autumn et al.’s research [
Let
Here, the
According to Lagrange dynamical equation, driving force
Therefore, the joint power of the robot and system energy consumption in actual application of the climbing robot can be obtained as follows:
Generally, the gait designed for the robot has obvious periodicity in view of motion stability and control complexity. When a symmetrical gait is adopted, the trajectory planning for a step could be reused in the entire regular motion trajectory, especially linear trajectory or circle trajectory. Unlike the other quadruped robots, there is only a controllable local degree of freedom on the foot for the GPL model. This means that when two diagonal legs are attached with the climbing surface, the remaining legs are in an approximately free state, leading to the uncontrollable poses of the corresponding feet. In other words, the floating feet poses cannot reach the necessary conditions needed for the trot gait reuse when the GPL model is in the transitional state from one phase to the other one in one stride (
Figures
Set the initial time
To realize the gait continuity, the following constraints need to be satisfied when the following phase repeats the trajectory of the previous phase in one stride.
Structure parameter constraints It is mainly considered that the prototype design should keep bilateral symmetry. Thus,
Leg pose constraints These constraints include two aspects: feet angle and leg length. Here, the leg lengths can be actively controlled by motors.
Waist position constraints about central symmetry This is determined by the trot gait adopted by the GPL model. Thus,
Based on the geometric constraints, there exists a fixed relationship as follows:
According to ( From the above, it can be seen that the position of the waist has specific limitations to realize the gait continuity during the climbing process, rather than any position.
For the trajectory planning, singularity and workspace are both inevitable problems. Let
Obviously, the singularity of the robot will occur when the position of the waist is located on the diagonal anomaly line from (
As we know, trajectory planning must be restricted to the corresponding workspace. Here, the waist
From (
Given a large range
Based on (
As mentioned above, the feasible region of the waist directly affects trajectory planning of the climbing robot. Both driving forces of active legs and the feasible region of the waist are related to the structural parameters of the robot, including body length ratio, driving angle, and waist spring coefficient. Therefore, the structural parameters should be reasonably selected, which contributes to a decrease in energy consumption and improvement in the stability of the climbing robot.
To avoid the disturbance of unreasonable motion path to structural parameters, it is necessary to plan a smooth motion trail of the waist to highlight the effect of structural parameters on the climbing performance of the robot. In terms of control weaknesses of the single curve function, an acceleration function composed by piecewise continuous function is adopted, which combines advantages of trapezoidal function and sine function. In this case, the acceleration curve function transits smoothly. Thus, the inertia force of the system could be reduced, which effectively improves the climbing speed and stability.
Let
To ensure the smooth path and no impact velocity of the GPL model, the following boundary conditions are listed. Namely, the corresponding velocity and acceleration of legs
Define the trajectory interpolation functions
Give priority to acceleration, and set
Integrating (
According to (
Structural parameters of the climbing robot.
Parameters | ||||
---|---|---|---|---|
Value | 230 | 168 | 103 | 168 |
243 | 420 | 113 | 85 | 150 |
223 | 2 | 77 | 4 | 10.2 |
To study the effect of the length ratio of front and rear body on driving forces of the robot, the different body length ratios
Supporting forces of the GPL model with different body length ratios.
According to bionic research [
From Figure
The start and termination point space and the feasible region of the waist.
As we know, the reverse driving forces produce negative work, increasing system energy consumption. The driving forces are associated with the waist spring coefficient according to (
To optimize the energy consumption, we need to reduce thermal energy during robot climbing. It means that the negative work done by the supporting legs should be avoided. For a given waist trajectory or a climbing dynamic gait, the negative work will be as small as possible when the following (
Given the range of
The effect of the waist spring coefficient. (a) The supporting forces on front and rear feet with different waist spring coefficients. (b) The system energy consumption with different waist spring coefficients.
During practical applications, there are high requirements of energy supply to be put forward for the robot system. Generally, the load-bearing capability of multilegged climbing robots is limited, which cannot carry greater weight energy storage units. Thus, it is significant to reasonably control mechanical energy consumption and improve climbing efficiency of the robot during the design process of the climbing robot. In this section, we focus on trajectory optimization based on system energy consumption.
Here, the waist joint coordinates can be expressed by the m-rank polynomial through the Hermite interpolation method, as follows:
The above equation can be written in matrix form, as follows:
As in Section
Combining the above equation constraints, the boundary conditions can be obtained as follows:
In addition, the waist joint coordinates are limited to the dimension constraint relation as in (
In terms of constraint condition for waist joint coordinate
According to the above theoretical analysis, the system energy consumption is mainly caused by active joints. So, the power consumption of the robot at any time can be represented by multinomial coefficient matrices
The optimization problem of system energy consumption can be described as
So the trajectory optimization of the waist can be transformed into a nonlinear programming problem in the nonlinear equality and inequality constraints. According to the structural parameters and gait conditions, the multinomial coefficient matrices
The optimal trajectory of waist during the whole gait. Here, the blue line and blue circle dot denote the optimal trajectory and the start and terminal points during the half gait, respectively.
The robot based on the GPL model is composed of two rigid parts and four identical leg modules. According to Figure
Prototype design based on the GPL model.
The configuration of the climbing prototype
Design of the leg module
In view of the overturning torque in the whole climbing process, it is necessary to reduce the torque to avoid drop damage. As we know, the overturning torque becomes larger as the thickness of the robot increases. So, the thickness of the robot is limited to 14.8 mm and the motor is placed on the rigid parts. Moreover, a thin flat tail acting as a support is designed to obtain a more stable climbing movement. In fact, the tail is ignored in dynamics of the GPL model due to its micro weight. In addition, the controller and battery are located in the upper part, which have the effect of counter weight by adjusting their corresponding position.
In order to verify the designed gait and condition of gait reuse, the test platform with linen is established and the actual climbing experiment is conducted. According to Figure
Actual climbing experiments on cloth-covered climbing surface and force measurement on supporting feet. (a) The sequence of climbing gait. Here, the red dotted line indicates the current adherent mechanism and the yellow circles indicate the adherent feet. (b) The test facilities of climbing robot and supporting force experiment.
In the supporting force measurement experiment, the planning trajectory of the waist in Section
The supporting forces on front and rear feet with different waist spring coefficients and body length ratios: (a)
To verify the effectiveness of the optimized trajectory in Section
Energy consumption experiments of the climbing robot.
According to Figure
Energy consumption of the climbing robot with different trajectories. (a) The total input power of the robot with unfaltering data; (b) the total input power of robot with faltering data.
Benefitting from dynamic gait, the GPL model inspired by gecko has been proposed in our previous work. In order to further develop dynamic gait and reduce energy consumption of the climbing robot based on the GPL model, the gait design and trajectory planning are analyzed in this paper. In view of the special construction of the GPL model, the trot gait is adopted and the conditions of gait reuse are presented. According to the kinematics and dynamics of the GPL model, the structural parameters have a significant effect on the trajectory planning of the robot. The prototype experiment results show that the larger body length ratio and a certain elasticity of the waist joint contribute to a decrease in the supporting forces and reduction in system energy consumption, especially negative forces on supporting feet. Moreover, it suggests that the driving angle plays an important role on obtaining a reasonable performance of gait continuity. From the perspective of gecko, the pendular waist with certain elasticity is beneficial to storing kinetic energy in fluctuation and reducing energy consumption in the process of climbing. Therefore, the optimal trajectory of the waist is planned based on system energy consumption. The energy consumption experiments about optimal trajectory and gecko’s sinusoidal curve are conducted, where a similar trend and less value of power suggest that the optimized trajectory is effective. Further, the results in our experiments partly explain the reasonability of the quadruped reptile’s kinesiology during dynamic gait.
In the future work, the deformation of the structure will be considered in dynamics of the GPL model. And the flexible material and foot structure with multiple degrees of freedom will be included in the future design of climbing robots.
The authors declare that there is no conflict of interest regarding the publication of this manuscript.
This study is supported by the National Natural Science Foundation of China (no. 51475018), Beijing Natural Science Foundation (no. 3162018), and Innovation Practice Foundation of BUAA for Graduates (no. YCSJ-01-201709).