Motion and trajectory planning modeling for mobile landing mechanism systems based on improved genetic algorithm

In many traditional soft-landing missions, researchers design the lander and the rover as two separate individuals, which has its limitations. At present, research on landers mainly focuses on the performance analysis of those who cannot move, and the motion of legged mobile lander has not yet been studied. In this paper, a novel Mobile Landing Mechanism (MLM) is proposed. Firstly, the monte-Carlo method is used to solve the workspace, and the motion feasibility of the mechanism is verified. Secondly, combining with the constraints of velocity, acceleration and secondary acceleration of each driving joint of the MLM, the trajectory of its joint space is planned by using cubic spline curve. And based on the weighted coefficient method, an optimal time-jerk pedestal trajectory planning model is established. Finally, by comparing the genetic algorithm (GA) with the adaptive genetic algorithm (AGA), an optimization algorithm is proposed to solve the joint trajectory optimization problem of the MLM, which can obtain better trajectory under constraints. Simulation shows that the motion performance of the mechanism is continuous and stable, which proves the rationality and effectiveness of the foot trajectory planning method.

Lander and rover played a vital role in landing exploration, in previous missions [1], many different types of landers and rovers were launched onto the surface of the moon and other planets. Nowadays, with the increasing requirements of landing detection technology, the lander also needs to have more functions. In the past few decades, many important models have been developed, such as Luna 16 [2,3], Euro Moon 2000 [4], Altair Lander [5], The Insight Mars Lander [6], Chang e 4 [7], etc.. Most of them have four legs, each of which consists of a main buffer mechanism, an auxiliary buffer mechanism and a foot pad, can be folded and then unfolded, and absorbs shock on impact. However, it is not possible of them to realize functions like attitude adjustment and walking (or moving). Therefore, detection tasks, though very limited, cannot be accomplished without the aid of rover.
In order to expand the detection range of the extra-terrestrial galaxies' surface, some movable rovers with wheeled mechanisms have been developed, such as Sojourner rover [8], Mars Rover mission [9] Spirit and Opportunity rovers, Jade Rabbit rover [10] and so on. However, most wheeled rovers have limited ability to travel through complex and harsh terrain environments, and even basic functions like moving and adjusting directions are impossible.
The rover must be carried to the surface of the extra-terrestrial galaxies by the lander before the subsequent detection mission, which has some limitations [11][12][13][14]: (1) The explore range of motion of the rover is very limited. The rover cannot reach remote destinations remote from the landing site since they have to receive energy and other supplies from the lander after extravehicular activity, which means, even both lander and rover can keep working without damage, it is still impossible of the rover to explore areas beyond a certain safe range. (2) As the exploration mission became more complex, it will be a great challenge for the wheeled rover to pass through rough terrains full of obstacles and slopes.
To solve problems above, a novel legged mobile landing mechanism is in urgent need. At present, some configurations of the legged mobile lander have been proposed. However, they are still in stages of conception, and no in-depth study on their walking characteristics and trajectory planning is ever conducted. One key issue of designing such a lander is to design the structure of its leg, which should bear high payload and have high reliability at the same time. To cope with such demands, the parallel configuration is considered to be a good choice. Parallel mechanisms have been widely used in aircraft simulators, force/torque sensors and acceleration sensors [15,16]. In recent years, parallel mechanisms that both maintain the inherent advantages of parallel mechanisms and possess several other advantages in terms of the total cost reduction in manufacturing and wider workspace (the Delta robot [17], and Tricept robot and Trivariant robot [18]), are drawing increasing attention of researchers. Combined with the traditional parallel robot configuration features, and combine parallel robots with traditional lander configurations, such as PH-Robot [19], Prototypes of Octopus robot [20], and so on.
A novel design for the kinematic control structure of the wheeled mobile robot (WMR) path planning and path-following was presented [21]. A mobile robot path planning method in the visualize plane using an overhead camera based on interval type-2 fuzzy logic (IT2FIS) was proposed. It is necessary to determine the location of a mobile robot in an environment surrounding the robot [22]. The Mem-PBPF algorithm yields improved performance in terms of time execution by using a parallel implementation on a multi-core computer was proposed. Therefore, the Mem-PBPF algorithm achieves high performance yielding competitive results for autonomous mobile robot navigation in complex and real scenarios [23]. A novel proposal makes use of the Artificial Potential Field (APF) method with a Bacterial Evolutionary Algorithm (BEA) to obtain an enhanced flexible path planner method taking all the advantages of using the APF method, strongly reducing its disadvantages [24]. A navigation software called Ant Colony Test Center designed to teach the different stages involved in mobile robotics was presented [25]. A novel proposal to solve the problem of path planning for mobile robots based on Simple Ant Colony Optimization Meta-Heuristic (SACO-MH) was presented [26].
In this paper, we focus on type synthesis of an innovative legged mobile lander combining characteristics and capabilities of both the lander and the rover inspired by the configurations of existing landers and walking robots. The MLM works as a landing buffer during landing. After that, it has 3 working modes: in mode 1, the stationary legged mobile lander works as a base camp; in mode 2, it performs exploratory tasks using its legs, which can be seen as a legged rover; while in mode 3, the MLM adjust its attitude to prepare for launching.
The purpose of designing the MLM is to ensure that the lander has good motion stability and environmental adaptation. One of the difficulties in lander motion control is to optimize the foot motion trajectory of the MLM [27,28]. Researchers have made achievements on the problem. (Ysway and E-sway motions [29,30], a sinusoidal sway [31] and the trajectory planning method based on the quantic spline curve [32]). However, few attentions have been paid to methods of foot trajectory planning for quadruped parallel robots. When performing a walking task, the leg structure will inevitably produce an obvious mechanical vibration, which will greatly jeopardize the walking stability of the whole machine. To solve such problems, by taking characteristics of configuration and motion into consideration, a method of trajectory planning for the foot is proposed.
At present, the research direction in the field of robot research mainly dedicated to the study of trajectory planning, under certain conditions this also to solve nonlinear constrained optimization problem provides a new train of thought Liu et al. [33] proposed a time optimal rapid continuous motion constraints, robot trajectory planning method in order to solve the problem of the optimal trajectory planning of robot, Xu et al. [34] put forward a kind of environment -genetic evolution immune clone algorithm Liu et al. [35] weighting coefficient method is used to establish the industrial robot trajectory planning model, and puts forward an improved adaptive genetic algorithm to solve Saramago and Steffen [36] solved the problem of manipulator moving at the minimum cost on the specified geometric path. Since the working environment of MLM studied in this paper is different from that of industrial robots, not only efficiency but also stability should be considered. The acceleration of impact force can be effectively controlled by controlling impact. This paper is organized as follows, after discussing the structure configuration in sections I, the kinematics of the MLM is studied and analyzed in section II. The optimal time-jerk trajectory planning method is proposed in section III. Simulation and experiment results and discussions are presented in section IV. Finally, conclusions are given in section V.

Description of the MLM
In this paper, the MLM is a parallel mechanism, as shown in Figure 1(a), R, U, P, and S denote revolute, universal, prismatic and spherical joint respectively. P pair is the actuated joint driven by an actuator. The MLM's structure includes: main body, swinging limb, RPR limb, upper leg, UPS limb-1, UPS limb-2, ball joint mounting bracket and foot. One end of the swinging limb is connected to the main body by a revolute joint, and the other connected to the upper leg by a Hooke joint. The middle position of the swinging limb is connected with the RPR limb by a revolute joint. The other end of the RPR limb is connected to the main body by a revolute joint, and the linear actuator installed on the RPR limb provides power for the swing rod installed on the body to swing up and down. The ball joint bearing is installed at the lower end of the upper leg. The two UPS limbs are of the same structure, whose one end is connected to the main body by a Hook joint, and the other connected to the ball joint mounting bracket on the upper leg by a ball joint. The middle part is the linear actuator.  The MLM is composed of three groups of buffer-drive mechanism, as shown in Figure 1(b). The buffer drive mechanism is mainly composed of the outer tube, compression tube, internal buffer materials, external buffer materials, piston rod, locking mechanism, screw drive mechanism, step motor and reset spring. Due to the complexity of landing environment, buffer-drive mechanism may be stretched or compressed during landing impact. When compressed, the internal buffer material is compressed by the piston rod. When stretched, the external buffer material is compressed by the piston rod through the locking mechanism and the outer tube. Since the compression length of the external buffer material does not affect the motion performance of MLM, only the compressed state of the internal buffer material is considered. Before compression, the initial distance of the internal buffer material is 1 v . Piston rod can move distance is 2 v . After compression, buffer material compression distance is v  . At this point, the piston rod can move distance is Therefore, in order to ensure the safety and feasibility of MLM mechanism. 2 v is used as the input conditions of the workspace.  , as follows:

by C S C C C B S S C C C B S bzC S ay B C S C C S by C C C S S B C bzS S az bzC B C S byS S by C S C C C B S S C C C B S bzC S ay B C S C C S by C C C S
Thus, the displacements of 2 L and 3 L can be represented as：

Constraints
The size and shape of the reachable workspace of the foot end is constrained by the following factors: (1) length of the limb; (2) limitation of universal joint. The constraints can be expressed as follows: where i is real angle of the universal joint; max i  is the allowable maximal angle of the universal joint; min i L is the minimum of the limb length; max i L is the maximum of the limb length. Table 1 shows parameters of dimension and kinematic pairs of the MLM.  Table 1, the reachable workspace of the foot end can be solved by Monte Carlo method [38], as shown in Figure 4.  In Figure 4 it can be found that the value and fluctuate for the foot end is smaller when the reachable workspace is in the ranges x ∈ [−50 ~ 2800] mm, y ∈ [−3200 ~ −750] mm, z ∈ [−1100 ~ 1100] mm, which is reasonably large to walk. Therefore, this area can be chosen as the optimal motion workspace of the leg mechanism.
With the mechanism meeting requirements of walking, its mobility stability becomes particularly important since the lander works in a complex working environment, which requires the end motion trajectory of the mechanism to be optimized.

Construction of the end trajectory
Joint trajectories of the MLM can be obtained by polynomial or cubic spline method. Those generated by cubic splines have continuous accelerations, comparing with the higher order polynomial method, cubic splines overcome problems like over-oscillation and overshoot between pairs of reference points. The knots in the path of the motion joint in Cartesian space are mapped to joint space [39]. Meanwhile, the cubic spline curve was used to interpolate the knots of each motor joint. We obtained a cubic polynomial that satisfies the following conditions.

Constraints
The motion of MLM actuators is constrained by velocity, accelerator and the second accelerator.

Velocity constraints
being quadratic (13), its maximum can be denoted as Max

Acceleration constraints
we then obtain:

Optimal objective
We notice that quadratic acceleration in Eq (15) is not continuous, namely, there are jerks when the MLM's end moves form the initial position to some desired final position.
Jerks not only accelerate the wear and tear of the parts, but also increase the end-effector positioning errors. As a result, the stability of the whole machine is reduced. In order to make the manipulator satisfied a certain work efficiency, and to ensure the movement is relatively stable, we established a time-jerk optimal trajectory planning model with respect to the total operation time and the square of the jerk for the components, that is: Where T W denotes the weight of time, J W denotes the weight of jerk, and The values of the weights T W and J W can be chosen to obtain the minimum time-jerk trajectory to some extent. By choosing J W =0 a minimum-time trajectory is found, while setting T W =0 enables one to obtain a minimum-jerk trajectory.

Analysis of algorithms
In order to optimize the Eq (21), we first map the optimized search space to search space of the genetic algorithm (GA), then the optimization parameter and the fitness function was determined. In order to make the optimization problem of objective function conforms to the operation rules of GA. Finally, the optimal objective function value is obtained by time segments. AGA not only can not consider specific meaning of parameters and their complicated relationship, the algorithm can deal with complicated problems, especially some issues of value concept has stronger global searching ability, at the same time as a result of the parallel search method, makes the genetic algorithm has a faster search at the same time also can apply to most, solving nonlinear large peak of discontinuous function more for function optimization problems show no expression can also apply. An adaptive fitness function is adopted in this paper, which can be automatically adjusted individual fitness gap according to the changes of the individual living environment, making the algorithm converges to its optimal solution. Thus, the blindness of the initial optimization can be greatly reduced, the amount of computation can be saved, and the real-time performance of the algorithm can be improved.
In this paper, the model of minimum time-jerk trajectory planning is presented with cubic splines connected the points. However, this model has the characteristics of complex coupling relationship and strong nonlinearity, so if traditional genetic algorithm is used, it is easy to fall into the local optimal solution. An adaptive fitness function is adopted in this paper, which can be automatically adjusted individual fitness gap according to the changes of the individual living environment, making the algorithm converges to its optimal solution. Thus, the blindness of the initial optimization can be greatly reduced, the amount of computation can be saved, and the real-time performance of the algorithm can be improved.
Specific implementation steps:

Local optimization
A small population with less evolutionary algebra was set up, the local optimal velocity can be obtained quickly. The landing leg structure first runs along the trajectory according to this optimal velocity.

Global optimization
Expand the population and increase the evolutionary algebra so that the global optimal velocity is obtained, ultimately ensuring that the landing leg structure runs at the optimal velocity trajectory.
The individual is selected by the individual's fitness degree, the roulette model is established, and the elite with the smallest probability value is directly replaced and copied to the next generation with the elite selection probability s p [40]. Crossover and mutation probability affect the individual diversity, robustness and convergence of GA. The sigmoid function of neural network was introduced into GA as an adaptive function of crossover probability and mutation probability, as shown in Eqs (22)

Experimental setup
Considering the constraints of landing leg structure parameters and motion parameters, we chose a safe workspace for the simulation experiment. in the ranges x ∈ (1000 ~ 2000 mm), y ∈ (−3000 ~ −2000 mm) and z ∈ (−400 ~ 400 mm). Based on the kinematics equation obtained in part 2, under the premise of displacement, velocity and acceleration of the joint space, the AGA program is designed.
Based on the weighted coefficient method, an optimal time-jerk pedestal trajectory planning model is established. By adjusting different weighting coefficients, the minimum parameters of time and jerk are obtained. The limits of Kinematic Limits are expressed in Table 3, and the displacement of knots in joint space are reported in Table 4. By using the genetic algorithm (GA) for the optimal solution, we obtained the optimal trajectories in various T W and J W . As shown in Figure 6, we set the trajectory of the end-effector of the MLM with respect to the inertial reference frame A XYZ  as follows:

Experiment analyses
Aiming at the motion characteristics of the active joint of the MLM, the time-jerk motion trajectory planning problem is proposed in this paper. By setting its weight parameter, the time-jerk mobility parameter in its motion process is optimized. Firstly, simulation analysis was carried out for two states, W 1 and W 0. It can be found from Figures 7 and 8 that when W =1, the trajectory planning time was less than 4.5 s, and the maximum jerk was 1000 / .When W 0 and W 0, the simulation time is 6 s and the maximum jerk degree is 68 / . Therefore, we need to find a good set of weight parameters so as to optimize the motion trajectory of the driving joint. Table 5 shows that the shortest execution time is 2.9885 s, and as the time weighted coefficient values ( T W ) decrease, the execution time of the MLM increase, and jerks of actuators decrease. That is to say, the reduction of the jerk is at the expense of the execution efficiency. When the value of T W is less than 0.999999, the total value of maximum jerks of joints does not significantly decrease as exaction time increases. And when T W is less than 0.9993, the total time taken to complete the motion trajectory by the MLM will be more than 5 s, which cannot meet the design requirements. When T W = 0.999995 and J W = 0.000005, the optimal solution is obtained at a certain extent. The results of the simulations are reported in Figures 9 and 10.   The convergence of the algorithm is shown in Figure 11.  It can be found from Figure 11 and Table 6 that the algorithm has a fast convergence speed, which is suitable for solving the optimal time-jerk problem and meeting the actual demand. In the experiment, in order to verify the effectiveness of the algorithm, GA and AGA were used to solve the optimal trajectory of MLM. The population size of each optimization algorithm was 50, and the number of iterations was 600. By comparing Figures 11 and 12, it is found that the convergence rate of the improved AGA is better than that of GA. And AGA has fewer values of objective function. Table 6 compares the precocity and convergence performance of AGA and GA algorithms. The experiments were repeated 30 times, and the evolution algebra of both algorithms was 600 generations. It can be seen from the Table 6 that AGA has significantly improved its global optimization ability and fast convergence ability.

Conclusions
In this paper, a novel mobile landing mechanism (MLM) is proposed. It realizes functions like landing on the lunar surface and walking in complex terrain environments, and greatly extends features of traditional landers. In order to verify the walking feasibility and reliability of the MLM. Firstly, the monte-Carlo method is used to solve the workspace, and the motion feasibility of the mechanism is verified. Secondly, combining with the constraints of velocity, acceleration and secondary acceleration of each driving joint of the MLM, the trajectory of its joint space is planned by using cubic spline curve. And based on the weighted coefficient method, an optimal time-jerk pedestal trajectory planning model is established. Finally, by comparing the genetic algorithm (GA) with the adaptive genetic algorithm (AGA), an optimization algorithm is proposed to solve the joint trajectory optimization problem of the MLM, which can obtain better trajectory under constraints. Simulation shows that the motion performance of the mechanism is continuous and stable, which proves the rationality and effectiveness of the foot trajectory planning method.
Future work has several directions, such as: (1) as for the working space scope of the lander, further optimization is needed in the future to improve the motion performance of the lander. (2) the designed AGA for the weighted coefficient of time weighted coefficient and shock of optimal experimental method is used to approximate values, only to solve the approximate solution, and the lack of rigorous mathematical deduction, therefore, an optimization algorithm is designed to automatically obtain the global optimal solution of the objective function is a difficult problem to study for the future and goals. (3) The method in this paper can be extended and applied to a variety of series and parallel robots, so as to realize the motion capability of traditional robots.