Research on the Optimal Trajectory Planning Method for the Dual-Attitude Adjustment Mechanism Based on an Improved Multi-Objective Salp Swarm Algorithm

Abstract

In this study, an optimization method for the motion trajectory of attitude actuators was investigated in order to improve assembly efficiency in the automatic docking process of large components. The self-developed dual-attitude adjustment mechanism (2-PPPR) is used as the research object, and the structure is symmetrical. Based on the modified Denavit–Hartenberg (MDH) parameter description method, a kinematic model of the attitude mechanism is established, and its end trajectory is parametrically expressed using a five-order B-spline curve. Based on the constraints of the dynamics and kinematics of the dual-posture mechanism, the total posturing time, the degree of urgency of each joint, and the degree of difficulty of the mechanism’s posturing are selected as the optimization objectives. The Lévy flight and Cauchy variation algorithms are introduced into the salp swarm algorithm (SSA) to solve the parameters of the multi-objective trajectory optimization model. By combining the evaluation method of the multi-objective average optimal solution, the optimal trajectory of the dual-tuning mechanism and the motion trajectory of each joint are obtained. The simulation and experiment results show that the trajectory planning method proposed in this paper is effective and feasible and can ensure that the large-part dual-posture mechanism can complete the automatic docking task smoothly and efficiently.

1. Introduction

In the field of automated assembly, mechanisms with multi-degree-of-freedom motion capability are gradually being developed as automated devices to achieve large-component assembly docking [1]. When performing the assembly task, the alignment mechanism needs to transport the docked parts from the initial position to the end position to complete the docking with the target parts [2]. In recent years, planning the motion trajectory mode of the large-component attitude mechanism has been an area of excellent research significance.

Trajectory planning for large-part tuning mechanisms refers to the planning of the expected motion trajectories while satisfying the work requirements and hardware constraints of the tuning components [3,4]. In recent years, traditional trajectory planning methods have mainly utilized polynomial functions to interpolate between trajectory points or multiple linear functions for trajectory fitting [5]. Based on the different spaces where the planning is located, trajectory planning can be categorized into joint space and Cartesian space trajectory planning. Joint space refers to that constituted by the joint motion of the robotic arm, while Cartesian space refers to that constituted by the translational and rotational motion of the end of the robotic arm. Both planning methods need to solve the position information of each joint based on inverse kinematics for trajectory planning.

The trajectory planning of a robotic arm corresponds to point-to-point motion, which requires the joint variables of the arms to be represented as time-dependent functions only and then constrained in terms of their angles, angular velocities, and angular accelerations [6]. Sabarigirish et al. performed trajectory planning for a five-degree-of-freedom robotic arm using a cubic polynomial interpolation algorithm, which ensured smooth motion but suffered from discontinuous joint acceleration [7]. For the optimization of robot joint trajectory, Dai et al. proposed an improved quantum genetic algorithm but did not consider the robot dynamics constraints [8]. Pu proposed an optimal control trajectory planning based on an improved gravitational search algorithm; however, their use of weighting factors was too subjective and lacked a scientific basis [9]. In dealing with the time-optimal problem of three-axis robots, Zhang et al. used an improved simulated annealing algorithm. However, it did not consider the position dimension and was not very general [10]. Yu et al. used a fifth-degree polynomial function to establish the positional information of each joint and an improved adaptive genetic algorithm to optimize the time intervals of the interpolation points of each joint of the robotic arm, achieving time-optimal trajectory planning [11]. Lamini et al. used chromosomes of genetic algorithms to model the path planning problem, where the first node of each chromosome contains the robot’s starting position and the last node contains the robot’s goal position, and the optimization process aims to minimize the length of the path from the initial position to the goal position [12]. Wang et al. introduce three genetic operators and propose a multi-objective trajectory optimization method for robotic arms based on an improved elite non-dominated sorting genetic algorithm, which provides a more efficient and stable solution for point-to-point robotic arm tasks [13]. Lan et al. adopt the seven times B-spline curve method to construct the joint trajectory of the robotic arm, and then propose a trajectory competition multi-objective particle swarm optimization algorithm to solve the Pareto solution set of the optimal trajectory of the robotic arm in terms of time, energy, and impact, which effectively reduces the movement time of the robotic arm, the impact of the joints, and the energy consumption [14]. However, the interpolation based the on seven times B-spline curve greatly increases the time complexity of the algorithm.

In summary, the existing research on the trajectory planning of mechanisms seldom considers the combined optimization problem of multiple indexes such as the cost, efficiency, and smoothness, and the planned trajectories are prone to problems such as acceleration discontinuities at the bit position connection points, which lead to a reduction in trajectory smoothness. Based on the driving force constraints in the dynamics model, although only under certain conditions, it can be ensured that its planned trajectory is free of singularities; the complex dynamics modelling and high computational time greatly reduce its practicality [15]. Therefore, this paper considers the kinematic constraints and proposes the optimal trajectory planning algorithm of the dual-attitude mechanism based on an improved multi-objective salp swarm algorithm, combines the Lévy flight and Cauchy variation algorithms into the salp swarm algorithm (LC-SSA), improves the optimization ability of the trajectory planning algorithm, optimizes the running time of the cabin segments’ auto-docking, and ultimately achieves trajectory optimization under multiple objectives.

2. System Structure and Kinematic Modeling

2.1. System Structure

The overall structural model of the dual-posture mechanism (2-PPPR) and the docking trajectory mission schematic are shown in Figure 1, which consists of the target cabin, the moving cabin, and 2 four-degree-of-freedom postures in parallel, of which the posture mechanism 1,2 has the same mechanical structure and is symmetric, and both are composed of three moving joints and one rotating joint, in which the moving joints J1,J5 realize the segment feeding and docking; the mobile joints J2,J6 realize the segment forward and backward and yaw motion; the rotating joints J3,J7 realize the segment lifting and pitching motion; and the rotating joints J4,J8 have their rotating axes parallel to the Z-axis to realize the segment transverse roll motion. Taking the self-developed 2-PPPR dual-posture parallel mechanism as a research object, we can realize the six-dimensional attitude adjustment in the x, y, and z direction translation as the pitch, yaw, and roll in the docking segment.

Considering the limited distance between the actual motion and the safe docking region of the alignment mechanism, the actual distance between the docking surface and the target surface of the mechanism is usually kept within 200 mm. The extent of acceleration and deceleration at the end of the attitude mechanism should not be too large within the short motion distance, so the influence of the dynamics on the trajectory is ignored.

2.2. Kinematic Model

The kinematic analysis is based on the modified Denavit–Hartenberg (MDH) parameters description method [16] to establish the kinematic model of the tuning mechanism, taking the posture mechanism 1 as an example, and the model parameters are shown in

Table 1. MDH parameters.

$\mathit{i}$	${\mathit{\alpha }}_{\mathit{i}-1}$	${\mathit{a}}_{\mathit{i}-1}$	${\mathit{\theta }}_{\mathit{i}}$	d	Movement Range
1	−90	0	0	$h_{0^{\prime }}$ (225 mm)	$d_{1^{\prime }}$ (−100~100 mm)
2	90	g1 (60 mm)	−90	0	$d_{2^{\prime }}$ (−50~50 mm)
3	−90	0	90	g2 (330 mm)	$d_{3^{\prime }}$ (−50~50 mm)
4	−90	0	${\theta }_4$ (±15°)	g3 (150 mm)	0

.

The base coordinate system is constructed for the dual-posture adjustment mechanism docking platform $\left\{S_L\right\}{S}_{X_L}{S}_{Y_L}{S}_{Z_L}$,$\left\{S_R\right\}{S}_{X_R}{S}_{Y_R}{S}_{Z_R}$. A coordinate system is established for the end of the left and right alignment platforms $\left\{J_4\right\}{L}_{X_L}{L}_{Y_L}{L}_{Z_L}$,$\left\{J_8\right\}{R}_{X_R}{R}_{Y_R}{R}_{Z_R}$. Taking the left attitude adjustment platform as an example, the transformation matrix of the end coordinate system $\left\{J_4\right\}$ to the base frame coordinate system$\left\{S_L\right\}$ is as follows:

$${}_{S_L}{}^{{\mathrm{J}}_4}T={\displaystyle \prod }_{i=1}^4{}_{i-1}{}^{i}T=\left[\begin{array}{cccc}c{\theta }_i& -s{\theta }_i& 0& a_{i-1}\\ s{\theta }_{i}c{\alpha }_{i-1}& c{\theta }_{i}c{\alpha }_{i-1}& -s{\alpha }_{i-1}& -d_{i}s{\alpha }_{i-1}\\ s{\theta }_{i}s{\alpha }_{i-1}& c{\theta }_{i}c{\alpha }_{i-1}& c{\alpha }_{i-1}& d_{i}c{\alpha }_{i-1}\\ 0& 0& 0& 1\end{array}\right]$$

(1)

Here,$c=\cos ,s=\sin ,{\alpha }_i$ is the linkage(i) joint angle, $d_i$ is the linkage(i) offset, and $a_i$ is the linkage(i) length.

The transformation matrix of the center coordinate system of the attitude segment $\left\{P\right\}$ to the world coordinate system $\left\{O\right\}$ is as follows:

$${}_O{}^{\mathrm{P}}T={}_{{\mathrm{J}}_4}{}^{\mathrm{P}}T{}_O{}^{{\mathrm{J}}_4}T={}_{{\mathrm{J}}_4}{}^{\mathrm{P}}T{}_{S_L}{}^{{\mathrm{J}}_4}T{}_O{}^{{\mathrm{S}}_L}T=\left[\begin{array}{cccc}c\alpha c\gamma +s\alpha s\beta s\gamma & s\alpha s\beta c\gamma -c\alpha s\gamma & s\alpha c\beta & P_1\\ c\alpha s\beta s\gamma -s\alpha c\gamma & s\alpha s\gamma +c\alpha s\beta c\gamma & c\alpha c\beta & P_2\\ -c\beta s\gamma & -c\beta c\gamma & s\beta & P_3\\ 0& 0& 0& 1\end{array}\right]$$

(2)

Here, $P_1=-d_{2^{\prime }}-L_{1^{\prime }}s\alpha c\beta ; P_2=d_{1‘}+h_{0‘}-L_{1^{\prime }}c\alpha c\beta ; P_3=d_{3^{\prime }}+g_1+g_2+g_3+g_4-L_{1^{\prime }}s\beta$.

To analyze the kinematics of the attitude adjustment mechanism, let the position vector of the component be the following:

$$\mathrm{U}={\left[P_x P_x P_x \alpha \beta \gamma \right]}^T$$

(3)

where ${\left[P_x P_x P_x\right]}^T$ is the vector of the segment coordinate system $\left\{P\right\}$ in the world coordinate system $\left\{O\right\}$, and ${\left[\alpha \beta \gamma \right]}^T$ is the rotational attitude of the segment coordinate system $\left\{O\right\}$ in the world coordinate system.

The center support points of the alignment mechanism are J₄ and J₈. The vector representation of the two points in the world coordinate system {O} is as follows:

$$q_i={\left(q_{ix} q_{iy} q_{iz}\right)}^T$$

(4)

The vector representation in the cabin segment coordinate system {P} is as follows:

$$r_i={\left(r_{ix } r_{iy } r_{iz }\right)}^T$$

(5)

where$q_{ix },q_{iy },q_{iz }$ can also denote the end linkage displacement of the dual-attitude mechanism. This can be obtained as follows:

$$q_i=R{r}_i+T$$

(6)

Here, T is the vector representation of the segment coordinate system {P} in the world coordinate system {O};$R$is the coordinate transformation matrix of the segment coordinate system {P} to the world coordinate system {O}. The velocity, acceleration, and jerk of the support point at the end of the attitude mechanism can be obtained through derivation of Equation (6).

$$\dot{q_i}=\dot{R}{r}_i+\dot{T}$$

(7)

Continuing the derivation yields the acceleration at the end support point and the accelerated acceleration relation:

$$\begin{gathered} \ddot{q_i}=\ddot{\mathrm{R}}{r}_i+\ddot{\mathrm{T}} \\ \stackrel{\dots }{q_i}= \stackrel{\dots }{\mathrm{R}}{r}_i+ \stackrel{\dots }{\mathrm{T}} \end{gathered}$$

(8)

Taking $q_i$as the motion input of the attitude mechanism and the segment position vector $\mathrm{U}={\left[P_x P_x P_x \alpha \beta \gamma \right]}^T$as the motion adjustment output, the relationship between $q_i$and $U$can be obtained.

$$\mathrm{U}=\mathrm{J}{q}_i$$

(9)

where $\det \left(J^{-1}\right)\ne 0,$and the individual joint velocities can be found from the terminal velocities, considering the limitations of the mechanism’s own physical conditions, $\det \left(J^{-1}\right)$ can be used to determine whether the mechanism’s motion position point in the workspace is singular. $\det \left(J^{-1}\right)$ can also be used to quantify the mechanism’s ability to adjust the posture. The larger its value, the larger the mechanism’s space of non-singular postures under the current posture and the stronger its ability to adjust the posture; when the value is reduced to zero, the mechanism is in the singular posture, and it cannot adjust it.

2.3. Motion Trajectory Planning

2.3.1. Optimization of Terminal Trajectory Based on B-Spline Curve

Based on the five-order B-spline curve [17] for planning the terminal motion trajectory of the cabin segment attitude mechanism, the kinematic boundary conditions can be used to simplify the control points, and node vectors and the B-spline parametric equations of the terminal position can be obtained.

$$p\left(u\right)={{\displaystyle \sum }}_{i=0}^{10}{c}_i^p{N}_{i,5}\left(u\right)$$

(10)

where $c_i^p$ is the control point of the ith six-dimensional attitude representation of the tuning mechanism, $u$is a node vector containing s node elements, s = 11 + 5 + 1 = 17, and $N_{i,5}\left(u\right)$ is the ith five-order B-spline basis function.

Since the assembly task requires the terminals of the dual-alignment mechanism to be able to reach the target position from an arbitrary initial position and the velocity and acceleration at the start and end positions to be zero, the B-spline curve should be clamped, i.e., the start and end terminals of the curve should be tangent to the start and end positions, respectively. It is necessary to make the coordinates of the first three control points equal to the initial position and those of the last three control points equal to the target position:

$$\begin{gathered} c_i=P_{ini} i=0, 1, 2 \\ {c}_j=P_{fin} j=8, 9, 10 \end{gathered}$$

(11)

Here, ${\mathrm{P}}_{\mathrm{ini}}$ and ${\mathrm{P}}_{\mathrm{fin}}$ denote the coordinate values of the initial and target poses. As the position is a six-dimensional vector, the above relationship is expressed in the form of generalized coordinates as follows:

$$\begin{gathered} {\left[c_i^1,c_i^2,c_i^3,c_i^4,c_i^5,c_i^6\right]}^T={\left[P_{ini}^x,P_{ini}^y,P_{ini}^z,P_{ini}^{\alpha },P_{ini}^{\beta }{P}_{ini}^{\gamma }\right]}^T i=0, 1, 2 \\ {\left[c_j^1,c_j^2,c_j^3,c_j^4,c_j^5, c_j^6\right]}^T={\left[P_{fin}^x,P_{fin}^y,P_{fin}^z,P_{fin}^{\alpha },P_{fin}^{\beta },P_{fin}^{\gamma }\right]}^T j=8, 9, 10 \end{gathered}$$

(12)

The remaining control points $\left\{c_3,c_4,c_5,c_6,c_7\right\}$will be randomly generated during the optimization solution. An interval proportional mapping method is used to ensure monotonic ordering of all the positional points on the planned B-spline trajectory, where the remaining control points are accumulated and then mapped onto the constraint intervals determined by the initial and target positions, and the control points are randomly generated with a binary coded format as shown in Figure 2.

Assuming that the decoded control point generalized coordinate is $s_n$, the new control point generalized coordinate $S_i$ is generated cumulatively as follows:

$$S_i={{\displaystyle \sum }}_{n=3}^i{s}_n i=3, 4,\cdots , 7$$

(13)

After mapping according to the interval scaling relationship, the coordinates of the ith ordered control point can be obtained as follows:

$$c_i=P_{ini}+S_i\left(P_{fin}-P_{ini}\right)/S_7 i=3, 4,\cdots , 7$$

(14)

In addition, clamped-type B-spline curves also require that the beginning and end node vectors have k + 1 repetitions, with k referring to the number of B-splines. Then, the node vector with 6-times repeatability can be expressed as follows:

$$\begin{gathered} t_i^p=\left\{0,0,0,0,0,0,t_1,t_2,t_3,t_4,t_5,t_6,t_6,t_6,t_6,t_6,t_6\right\} \\ 0<t_1\le t_2\le t_3\le t_4\le t_5<t_6 \end{gathered}$$

(15)

The node vector $u$ is converted to a time node vector according to a fixed number of interpolation points or a fixed interpolation interval, and the value of $t$ is computed in increments $\mathsf{\Delta }{t}_i$during the optimization solution:

$$\begin{gathered} t_i^P\left(u\right)=t_i+u_i^P\left(\Delta t\right) \\ \mathsf{\Delta }{t}_i=t_i-t_{i-1} i=1, 2,\cdots , 6 \end{gathered}$$

(16)

So far, the planned terminal trajectory of the dual-posture adjustment mechanism is a five-order B-spline parameterized trajectory with the ordered control point $\left\{d_3,d_4,d_5,d_6,d_7\right\}$ and the time node increment $\left\{\mathsf{\Delta }{t}_1,\mathsf{\Delta }{t}_2,\mathsf{\Delta }{t}_3,\mathsf{\Delta }{t}_4,\mathsf{\Delta }{t}_5,\mathsf{\Delta }{t}_6\right\}$ as variables.

2.3.2. Multi-Objective Optimization Model for Trajectory Planning

The B-spline parameterized trajectory only conforms to the basic requirements of the assembly task and needs to be further optimized to give full performance to the mechanism and to satisfy different specifications of the assembly task. In this paper, the main considerations are the efficiency of component docking, the smoothness of the posing motion, and the ease of the posing process. Therefore, combined with the kinematic characteristics of the dual-posture adjustment mechanism, the total posturing time, the maximum absolute value of each joint’s jerk, and the ease of posturing are defined as objective functions to establish a multi-objective trajectory optimization model.

First, the total alignment time, which is closely related to the docking efficiency, is obtained by accumulating the time node increments with the following expression:

$$t_{total}=\Delta t_{J1,J5}+\Delta t_{J2}+\Delta t_{J3}+\Delta t_{J4,J8}+\Delta t_{J6}+\Delta t_{J7}$$

(17)

where $\Delta t_{J1,J5}$ and $\Delta t_{J4,J8}$ are the maximum time for the synchronized movement of joints J1 and J5, and joints J4 and J8, respectively.

Secondly, compared with the docking efficiency, the smoothness of the motion of the attitude mechanism and the safety of the docking process are more important, and the jerk of each driving joint is closely related to them. Therefore, when the maximum absolute value of each joint’s jerk is smaller, the stability of the mechanism’s motion is higher, and the friction loss is smaller. The expression is as follows:

$$J_{max}=max\left\{\left|j_i\right|\right\}$$

(18)

Finally, in order to reduce difficulty in posing while avoiding trajectory singularities, an attempt could be made to increase the posing ability of the mechanism by increasing the $\det \left(J^{-1}\right)$ value on the whole trajectory. The logarithm of the $\det \left(J^{-1}\right)$ average value of the n attitude points on the trajectory is chosen as the third objective evaluation function, which evaluates the degree of difficulty of the mechanism’s attitude adjustment when moving along the trajectory. The expression is as follows:

$$N_{act}=n/{\log }_{10}\left({{\displaystyle \sum }}_{i=1}^n(\det \left(J^{-1}\right)\right)$$

(19)

In addition, due to the structural limitations of the mechanism itself, the constraints on the displacements, velocities, accelerations, and jerks of each joint are as follows:

$$\begin{gathered} \left|q_i\right|\le {\left|q_i\left(P\right)\right|}_{max} \\ \left|v_i\right|\le {\left|\dot{q_i}\left(P,\dot{P}\right)\right|}_{max} \\ \left|a_i\right|\le {\left|\ddot{q_i}\left(P,\dot{P},\ddot{P}\right)\right|}_{max} \\ \left|j_i\right|=\left|\dot{a_i}\right|\le J_{max} i=1, 2,\cdots , 8 \end{gathered}$$

(20)

Associating the above equation, the trajectory optimization model under multi-objectives is as follows:

$$\begin{gathered} \min F\left(x\right)=\left(f_1\left(x\right),f_2\left(x\right),f_3\left(x\right)\right) \\ {f}_1\left(x\right)=t_{total}=\Delta t_{J1,J5}+\Delta t_{J2}+\Delta t_{J3}+\Delta t_{J4,J8}+\Delta t_{J5}+\Delta t_{J6}+\Delta t_{J7} \\ {f}_2\left(x\right)=max\left\{\left|j_i\right|\right\} \\ {f}_3\left(x\right)=N_{act}=n/\lg \left({{\displaystyle \sum }}_{i=1}^n(\det \left({\mathrm{J}}^{-1}\right)\right) \\ \mathrm{s}.\mathrm{t}. \left|q_i\right|\le {\left|q_i\left(P\right)\right|}_{max} \\ \left|v_i\right|\le {\left|\dot{q_i}\left(P,\dot{P}\right)\right|}_{max} \\ \left|a_i\right|\le {\left|\ddot{q_i}\left(P,\dot{P},\ddot{P}\right)\right|}_{max} \\ \left|j_i\right|=\left|\dot{a_i}\right|\le J_{max} i=1, 2,\cdots , 8 \end{gathered}$$

(21)

Due to the structural limitations of the mechanism, the maximum values of each of these parameters are explicitly limited. ${\left|{\mathrm{q}}_1\left(\mathrm{P}\right)\right|}_{\max }={\left|{\mathrm{q}}_5\left(\mathrm{P}\right)\right|}_{\max }$ = 100 mm,${\left|{\mathrm{q}}_2\left(\mathrm{P}\right)\right|}_{\max }={\left|{\mathrm{q}}_6\left(\mathrm{P}\right)\right|}_{\max }$ = 200 mm,${\left|{\mathrm{q}}_3\left(\mathrm{P}\right)\right|}_{\max }={\left|{\mathrm{q}}_7\left(\mathrm{P}\right)\right|}_{\max }$ = 100 mm,${\left|{\mathrm{q}}_4\left(\mathrm{P}\right)\right|}_{\max }={\left|{\mathrm{q}}_8\left(\mathrm{P}\right)\right|}_{\max }$ = 20°, ${\left|{\mathrm{v}}_{\mathrm{i}}\right|}_{\max }$ = 20 mm/s, ${\left|{\mathrm{a}}_{\mathrm{i}}\right|}_{\max }$ = 11 mm/s², ${\mathrm{J}}_{\max }$ = 9 mm/s³.

2.3.3. Multi-Objective Trajectory Optimization Based on Improved Salp Swarm Algorithm

Due to the nature of the multi-objective optimization model, there are objective conflicts. A multi-objective optimization algorithm can optimize all the objectives simultaneously and then perform a search to obtain the Pareto-optimal solution set. The objective functions of the optimized solutions in the solution set cannot be compared with each other, and for each solution, the value of any sub-objective function may be weakened while the values of other objective functions are improved. Common optimization algorithms include particle swarm optimization (PSO) [18], butterfly optimization algorithm (BOA) [19], and bat algorithm (BA) [20]. This paper considers the diversity and convergence of solutions. Finally, this paper adopts the salp swarm algorithm (SSA) [21], as set out below. This is a new type of swarm intelligence algorithm proposed by Mirjalili et al. from Australia in 2017, which simulates the salp swarm foraging behavior and has the advantages of fewer control parameters, easier implementation, and lower computational effort.

• Population initialization:

A B-spline parameterized curve is defined; 36 variables are obtained from Equation (10), i.e., six-dimensional generalized coordinates of 5 control points and 6 time node increments; and binary coding is used to randomly generate N population individuals in the variable decision space to obtain an initial bottled sea squirt population X, i.e., a set of B-spline trajectories is initialized. The maximum number of iterations A and the population size E are also set.

2.

Constraint and iterative condition determination:

According to the ith generation of the salp swarm population, i.e., the ith generation of B-spline trajectory, the displacement, velocity, acceleration, and jerk of each joint are calculated from the kinematic equations to determine whether they satisfy the constraints of Equation (20). If the constraints are satisfied, then their corresponding fitness is calculated, i.e., the three objective functions. Otherwise, they are assigned to infinity and penalized out. Then, it is determined whether the number of iterations is maximum. If so, then the iteration is terminated, and the current optimal value is output. Otherwise, continue to update the population and iteration.

3.

Leader phase:

The traditional SSA in trajectory optimization encounters low search accuracy and can easily fall into local optimal solutions. In order to solve these problems, it is proposed to introduce the Levy flight algorithm [22] into the SSA; the randomness of its search is utilized to enhance the robustness of individual salp swarms and improve the overall search capability of the algorithm. The improved salp swarm individual possesses stronger vigor, can search the space where it is located sufficiently, improves the optimization accuracy, and effectively shortens the running time of the dual-posture adjustment mechanism. Levy(s) represents the randomly searched path, and its expression is as follows:

$$\mathrm{Levy}\left(s\right)={\displaystyle \frac{\lambda a\mathsf{\Gamma }\left(\lambda \right)\sin \left(\pi \lambda /2\right)}{\pi }}\otimes {\displaystyle \frac{1}{s^{\lambda +1}}}$$

(22)

where λ is an exponent parameter with value 1, $\otimes$ is the point multiplication, Γ is the gamma function, and s is the step size of the Levy flight, whose expression is as follows:

$$s=u/{\left|v\right|}^{{\beta }^{-1}}$$

(23)

where $\beta$ is a [0, 2] random number set to 1.5, u ~ N (0,${\sigma }_u^2$), v ~ N (0,${\sigma }_v^2$), ${\sigma }_u$ and ${\sigma }_v$ are standard deviations obeying a normal distribution with ${\sigma }_v$ = 1.

$${\sigma }_u={\left[{\displaystyle \frac{\mathrm{\Gamma }\left(1+\beta \right)·sin\left(\pi \beta /2\right)}{\mathrm{\Gamma }\left(1+\beta /2\right)·\beta ·2^{\beta -1/2}}}\right]}^{1/\beta }$$

(24)

Levy flight is applied to the global search phase of the SSA. The randomness of this algorithm can effectively prevent the SSA from falling into the local optimum, maximize the use of the speed and acceleration of the dual-posture adjustment mechanism, and greatly shorten the machine’s running time. Thus, the formula for the leader position update after introducing the Levy flight strategy is as follows:

$$x_{1,j}^t=\{\begin{array}{c}{F}_j+c_1\left(\left(u{b}_j-l{b}_j\right)\otimes levy+l{b}_j\right),c_3\ge 0.5\\ F_j-c_1\left(\left(u{b}_j-l{b}_j\right)\otimes levy+l{b}_j\right),c_3<0.5\end{array}$$

(25)

where $\otimes$ is the dot product, levy is the step size of the Levy flight, $x_{1,j}^t$denotes the position of the first salp swarm in the jth dimension, and the parameter $c_1$ is calculated as follows:

$$c_1=2e^{-{({\displaystyle \raisebox{1ex}{$4t$}\!\left/ \!\raisebox{-1ex}{$t_{max}$}\right.})}^2}$$

(26)

where $t$ is the current number of iterations and $t_{max}$ is the maximum number of iterations.

Location update method based on Cauchy variation.

The Cauchy variation algorithm [23] is introduced into the global search phase of the SSA. During the search process, the Cauchy variation algorithm is regarded as an optimization operator to prevent the algorithm from falling into a locally optimal solution at the later stage of iteration. The probability density of the one-dimensional Cauchy distribution is as follows:

$$f\left(x\right)={\displaystyle \frac{1}{\pi }}·{\displaystyle \frac{a}{a+x^2}} ,x\in \left(-\infty ,+\infty \right)$$

(27)

The Cauchy variation is introduced into the target position update method to exert the disturbance ability of the Cauchy operator, and the global optimization performance of the algorithm is improved.

$$x_{1,j}^t=x_b\left(t\right)+cauchy\left(0,1\right)\oplus x_b\left(t\right)$$

(28)

where $cauchy\left(0,1\right)$ is the standard Cauchy distribution, $x_b\left(t\right)$ represents the global best position, and the Cauchy distribution random variable generation function is $\eta =\tan \left[\left(\xi -0.5\right)\pi \right]$.

The strategy adopted to update the target location is determined using the selection probability [24]. The formula is as follows:

$$P_t=-exp{\left(1-{\displaystyle \raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t_{max}$}\right.}\right)}^{20}+\theta$$

(29)

here, θ is the adjustment parameter with a value of 0.05.

The specific selection strategy is as follows: Take s at any value in [0, 1]. If s < $P_t$, select the Levy flight strategy to update the position; otherwise, select the Cauchy variation perturbation strategy to update the target position.

Enhance the ability of the above two strategies to jump out of the local space, introduce greedy rules, and determine whether to update the position by comparing the fitness values of the old and new positions. The greedy rule is shown below, and f(x) denotes the position fitness value of x.

$$X_b\left(t\right)=\{\begin{array}{c}{X}_b\left(t\right),f\left(x_{1,j}^t\right)\le f\left(x_b\right)\\ x_{1,j}^t,f\left(x_{i,j}^t \right)>f\left(x_b\right)\end{array}$$

(30)

4.

Follower phase:

In the improved SSA, the followers move with the guidance of the leader and the position update expression is as follows:

$$x_{i,j}^t={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(x_{i,j}^t+x_{i-1,j}^t\right) i\ge 2$$

(31)

where $x_{i,j}^t$ denotes the position of the ist follower salp swarm in the jth dimension.

The flow of the algorithm is shown in Figure 3. This algorithm further improves the performance of the system in searching for an optimal solution. Its main purpose is to shorten the running time of the dual-posture adjustment mechanism while optimizing all the objectives at the same time and to simulate the chain foraging behavior of a group of salp swarms to generate the optimal Pareto solution through the continuous searching of the individual.

3. Simulation Experiment and Analysis

3.1. Algorithm Comparison

To prove the superiority of the proposed method, the improved SSA (LC-SSA) proposed in this paper is simulated in comparison with the SSA; first, the kinematic constraints of the dual attitude mechanism are determined as v_max = 20 mm/s and a_max = 11 mm/s². In order to reduce the calculation time for trajectory planning, the maximum movement time of the end of the mechanism is set to t_max = 450 s. For the fairness of the algorithm comparison, the initialization parameters of both algorithms are set to the same, as follows: population size N = 300, maximum number of iterations it_max = 500, population dimension D = 30, and number of leaders N/2 = 15. In addition, the interpolation accuracy of each component of the position control point is P_pre = 0.01 mm, of each component of the attitude angle control point is a_pre = 0.001 rad, and of the time node increment is t_pre = 0.01 s.

The Pareto solution sets obtained by simulating the improved SSA (LC-SSA) proposed in this paper against the SSA are shown in Figure 4. The two sets of algorithms were simulated for 500 iterations, and 79 sets and 52 sets of Pareto solution sets were obtained, respectively. From Figure 4, it can be seen that the single-objective optimal solutions of the three objective functions simulated by the LC-SSA are {A, B, C}, and the single-objective optimal solution set of the SSA is {D, E, F}, and the two sets of data are listed in

Table 2. The solutions of the Pareto-optimal front comparison table.

Algorithm	Data Point	Ttotal (s)	$\left|{\mathit{J}}_{\mathit{m}\mathit{a}\mathit{x}}\right|$(mm/s3)	${\mathit{N}}_{\mathit{a}\mathit{c}\mathit{t}}$
LC-SSA	A	378.012	0.756	1.243
	B	350.892	1.300	1.201
	C	27.126	86.600	3.551
SSA	D	441.772	0.678	1.262
	E	404.583	1.226	1.221
	F	39.201	85.708	4.142

. When the total postural adjustment time is the smallest, the value of its LC-SSA objective function is 27.126 s, and the value of its SSA objective function is 39.201 s; when the value of maximum joint sharpness is highest, the LC-SSA objective function value is 0.756 mm/s³ and the SSA objective function value is 0.678 mm/s³; when the total posturing time is the smallest, the LC-SSA objective function value is 1.201 and the SSA objective function value is 1.221; from Figure 4, it can be seen that the objective function solution set based on the LC-SSA is better than that on the SSA, and the individual distribution of the solution set is better than that of the SSA. The distribution of individuals in the solution set is better, more uniform, and closer to the real value.

The objective function values in Table 2 were compared. It was found that there was a target conflict between the total pose adjustment time and the joint jerk. Since there is no constraint on the joint jerk before trajectory planning when the trajectory movement time reaches the minimum of 27.126 s, the posture adjustment efficiency reaches the maximum, but the maximum jerk value of the joint reaches 86.600 mm/s³. On the contrary, if the maximum jerk value of the joint is controlled at the minimum value of 0.756 mm/s³, the end movement is extremely stable, and the time consumption reaches 378.012 s, which is almost close to the set maximum movement time of 450 s. There are also some conflicts between the total posture adjustment time and the difficulty of posture adjustment. The results need to be processed in order to obtain the optimal motion trajectory that satisfies multiple targets and is suitable for assembly tasks. In this paper, an average optimal evaluation method is adopted. By evaluating the value of the objective function, the average optimal solution of each objective equilibrium is selected as the final solution in many Pareto solution sets. The expression of the evaluation index is as follows:

$$min F={\displaystyle \sum }_{i=1}^3{c}_i{\displaystyle \raisebox{1ex}{$f_{i}x-f_i{x}_{min}$}\!\left/ \!\raisebox{-1ex}{$f_i{x}_{max}-f_i{x}_{min}$}\right.}$$

(32)

where $f_i{\left(x\right)}_{min}$ and $f_i{\left(x\right)}_{max}$ are the maximum and minimum values of the objective function $f_i\left(x\right)$ in the solution set, and $c_i$ is a weight coefficient. The assembly task has the same requirements for the three objective functions, and the weight coefficient is 1.

A comparison of the two algorithms based on the multi-objective average optimization method to find the optimal solution of the objective function is shown in

Table 3. The optimal solutions of multi-objective functions comparison table.

Optimal Solution	Ttotal (s)	$\left|{\mathbf{J}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\right|$(mm/s3)	${\mathbf{N}}_{\mathbf{a}\mathbf{c}\mathbf{t}}$
SSA	47.231 s	5.276 mm/s3	1.272
LC-SSA	30.479 s	8.574 mm/s3	1.200

. From the table, it can be seen that the improved LC-SSA improves the total time for attitude adjustment by 35.57$\%$, the docking efficiency is higher, and the difficulty of attitude adjustment is smaller, which verifies the superiority of the method.

3.2. Simulation Analysis

The motion range of the dual-posture adjustment mechanism in each displacement direction is set to [−100 mm, 100 mm], and the angle range is set to [−10°, 10°]. In order to make full use of the motion of the mechanism, two non-singular position points are arbitrarily selected on both sides and away from the zero position of the mechanism as the starting and ending position points of the trajectory planning.

$$\begin{gathered} {\mathrm{P}}_{\mathrm{ini}}={\left[500.19, 158.33, 1220.00, -1.91\text{°}, 1.15\text{°}, 2.3\right]}^{\mathrm{T}} \\ {\mathrm{P}}_{\mathrm{final}}={\left[455.40, 128.25, 1210.00, -0.31\text{°}, 6.10\text{°}, 7.9\right]}^{\mathrm{T}} \end{gathered}$$

The optimal solution of the objective function of the multi-objective average optimization method is based on the LC-SSA. The values of 36 B-spline optimization parameters under the average optimal solution are as follows:

$$\begin{gathered} {\overline{d}}_3^{\mathrm{p}}=\left[498.392, 156.987, 1218.137,-1.862, 1.375, 3.124\right] \\ {\overline{d}}_4^{\mathrm{p}}=\left[482.392, 149.987, 1216.137,-1.662, 2.075, 4.319\right] \\ {\overline{d}}_5^{\mathrm{p}}=\left[476.610, 141.460, 1214.867,-1.204, 3.324, 5.022\right] \\ {\overline{d}}_6^{\mathrm{p}}=\left[472.029, 138.027, 1213.086,-0.972, 4.262, 5.961\right] \\ {\overline{d}}_7^{\mathrm{p}}=\left[468.052, 129.004, 1211.713,-0.428, 5.751, 6.083\right] \\ \Delta t_i=\left[10.164, 4.215, 3.920, 2.968, 2.392, 6.778\right],\left(i=1,2,\dots , 6\right) \end{gathered}$$

In the process of modeling the mechanical system of the attitude mechanism, according to its structural characteristics, based on the 3D model, we established a suitable coordinate system through the sw_urdf_exporter plug-in; set an independent coordinate system for each joint; determined the type of joints; and defined each parent–child linkage, limit parameters, velocity parameters, and inertia parameters, etc., to build the SimMechanics simulation platform to realize the visual expression of the simulation model.

The cabin segment docking simulation mainly includes a position fitting module, trajectory planning module, motion simulation module, etc. According to the position parameters provided by the position fitting module, the multi-objective singularity-free attitude trajectory planning algorithm writes the m program, which plans the space motion attitude trajectory of the attitude mechanism moving platform to realize the cabin segment docking. Figure 5 shows the simulation schematic of the completed simulation based on the multi-objective optimal trajectory dual attitude tuning cabin segment docking, the simulation process of the docking of the cabin segment from the initial position$P_{ini}$to the target position$P_{final}$can be obtained, and the three sub-objective function values corresponding to this trajectory are as follows: T_total: 30.479 s;$\left|J_{max}\right|$: 8.574 mm/s³; $N_{act}$: 1.200. All the objective function values have reached the range of the demand of the assembly task indexes and satisfy the assembly task indexes.

According to the optimal trajectory B spline parameters, the spatial attitude curve of the end can be obtained as shown in Figure 6; the space position x, y, z optimal trajectory of the terminal can be obtained as shown in Figure 6a; α, β, γ optimal trajectory curve as shown in Figure 6b, which indicates that the B spline trajectory has strong flexibility and trajectory smoothing; and the tuning trajectory meets the requirement of docking the cabin section from the initial attitude movement to the target attitude when the values of the objective function are average optimal.

For the dual-attitude mechanism in posture mechanism 1, joints 1−3 are moving joints, and joint 4 is the rotating joint; in posture mechanism 2, joints 5−7 are moving joints, and joint 8 is the rotating joint. Sub-graphs are shown due to the difference in the numerical units of the moving joint and the rotating joint. According to the end trajectory equations, the displacement, velocity, acceleration, and jerk of each joint are obtained from the inverse kinematic solution, respectively. Figure 7a shows the rotation angle of rotating joints 4 and 8, Figure 7b shows the rotation angle velocity, Figure 7c shows the rotation angle acceleration, and Figure 7d shows the rotation angle jerk.

Figure 8 shows the simulation curves of moving joints 1−3 and 5−7, in which the displacement, velocity, acceleration, and jerk of each joint change within the kinematic constraints set by the algorithm.

Combined with Figure 7 and Figure 8, it can be seen that the displacement, rotation angle, velocity, and acceleration of each joint change within the kinematic constraints set by the algorithm, which is in line with the requirement of smooth starting and stopping of the tuning mechanism. Only in joint 3, the maximum value of the sharpness is 4.3392 mm/s³, and the overall change of the sharpness of each branch chain is smooth, which shows that the posturing mechanism moves along the average optimal trajectory with less vibration and can realize the requirement of smooth posturing under the condition of balanced optimization of each objective function. According to the planned trajectory, the motion of the attitude adjustment mechanism should be controlled so that the moving cabin section reaches the target position after one adjustment. However, it should be pointed out in the displacement simulation curves that when the platform is located in the zero position, the elongation of each joint is not taken as the initial value of displacement, but the initial displacement is set as the elongation of each joint in the initial position. Therefore, in the displacement curves of each joint, the value of displacement, i.e., the elongation of each joint, changes from the initial state and increases as it approaches the terminal position.

3.3. Experimental Analysis

In order to verify the feasibility of the multi-objective attitude planning method and the correctness of the kinematic model, this study builds a segment docking test platform. The size of the module to be attuned: D400 mm, L2600 mm. The base of the dual-attitude mechanism is fixed on the workbench, and the distance is fixed: 2300 mm. Figure 9a shows the system’s overall structure, including posture mechanism 1,2, attitude segment, motion controller, work-bench, and laptop. Motion control software(visual studio 2022) establishes the transmission link between the hardware and computer, mainly including the transmission module, single-axis control, multi−axis linkage trajectory planning module, position adjustment module, return to zero and emergency stop module, multi−axis information feedback, status information feedback, and alarm module.

The effective travels of the joints of the dual attitude mechanism are shown in

Table 4. The joints movement range.

Range	Posture Mechanism 1				Posture Mechanism 2
Joint No.	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6	Joint 7	Joint 8
Parameter	$\pm$100 mm	$\pm$50 mm	$\pm$50 mm	$\pm$10°	$\pm 1$00 mm	$\pm$50 mm	$\pm$50 mm	$\pm$10°

. Each moving platform of the posturing platform adopts KYCT125DPY57−M, equipped with 57 stepping motors and driven by precision ball screws; the repetition positioning accuracy and absolute positioning accuracy can be kept within 5 um. The angle-adjusting table adopts KYCT140DJW15-M, the effective load of the table surface is 0−40 kg, equipped with 57 stepping motors and driven by turbo and worm gears, and the repetition positioning error is not more than 0.1°. The moving joints of the posture platform are equipped with a high-precision grating scale, and the rotating joints are equipped with an absolute encoder for real-time position reading information, the precision of the grating scale is 10 um, and the precision of the absolute encoder is 2500 lines.

The attitude segment is fixedly connected to the dual-posture mechanism platform in the segment docking test platform. Before docking, there are unknown position and attitude deviations between the movement segment and the target position, and the relative distance between the two in the axial direction does not exceed the effective stroke of the attitude mechanism. The attitude adjustment mechanism realizes the initial attitude random state of the module docking, and the target attitude vector is consistent with the simulation experiment:

$$P_{final}={\left[455.40, 128.25, 1210.00, -0.31°, 6.10°, 7.90\right]}^{\mathrm{T}}$$

Based on the improved algorithm proposed in this paper, the automatic adjustment process of the postural position of the alignment mechanism is shown in Figure 9b. During the whole process of attitude adjustment, the joints move smoothly without abnormal jitter. To further verify the effectiveness of the algorithm, in the same experimental environment, the algorithm of this paper has carried out five times in repeated experiments; for each time, the adjustment time is shown in

Table 5. Posture adjustment experiments time statistics.

Experiments No.	Initial Position	Posture Position	Average Time
1	${\left[521.40, 136.25, 1227.66, -0.25\text{°}, 9.10\text{°}, 6.20\right]}^{\mathrm{T}}$	32.14 s	31.426 s
2	${\left[478.31, 146.72, 1198.62, -1.41\text{°}, 3.68\text{°}, -5.92\right]}^{\mathrm{T}}$	31.36 s
3	${\left[432.39, 118.64, 1185.53, 0.46\text{°}, -4.25\text{°}, 7.32\right]}^{\mathrm{T}}$	30.96 s
4	${\left[497.76, 102.45, 1235.41, 1.21\text{°}, -5.76\text{°}, -4.82\right]}^{\mathrm{T}}$	32.42 s
5	${\left[497.41, 126.46, 1217.73, -0.25\text{°}, 9.10\text{°}, 16.21\right]}^{\mathrm{T}}$	30.25 s

, and it can be seen that the average time consumed is 31.426 s, showing that this paper’s improved algorithm has great improvements in terms of planning time, and the simulation results are consistent. The experiment verifies the practicality of this paper’s algorithm.

Finally, taking Experiment 1 in Table 5 as an example, the joint states of each axis of the photoelectric coded disk are read to obtain the motion state of each joint during path planning. The displacement joints curves are shown in Figure 10a, and rotating joints feedback position curves are shown in Figure 10b. Analyzing the joint angle change graph in Figure 10, the change curves of each joint are smooth, and the joints run smoothly when the motion is planned.

According to the planned trajectory to control the motion of the attitude adjustment mechanism, for the mobile module section, after one adjustment to achieve the six-dimensional target attitude vector, the docking effect is in line with expectations, the docking process is smooth and continuous, and the trajectory tracking motion state is stable. The test results show the effectiveness and practicality of this paper’s method.

4. Conclusions

With regard to the task of assembling and docking the components of the dual-attitude mechanism, this study used a control-point-ordering method to achieve a non-singular trajectory of the B-spline curve planning mechanism while also ensuring that the initial and final poses were non-singular.

Based on the multi-objective requirements of assembly tasks, this study established a multi-objective trajectory optimization model based on the kinematics of the dual-posture adjustment mechanism and obtained the single-objective and average optimal trajectories satisfying the multi-objective requirements through the improved SSA.

Combined simulation and experimental results show that the proposed trajectory planning method is effective and feasible. In addition, the practicability and versatility of the method are strong, which can provide a reference for the non-singular trajectory planning of other types of structural robots.

In addition, in order to realize the real sense of automatic docking of the module segments, the sources of measurement, motion, control, and other errors in the docking process of the module segments should be comprehensively considered, the error transfer relationship should be studied, and the real-time position detection algorithm of the organization and the intelligent error compensation method should be further researched in future.