Introduction

The primary transformation in industrial manufacturing systems is prioritizing human well-being at the system’s center in the face of the new industrial revolution1. The theoretical framework, enabling technologies, and industrial applications of human-centered intelligent manufacturing have garnered extensive recognition within academic and industrial communities2. In this context, the relationship between humans and machines has advanced from mere coexistence to fruitful collaboration. It is on the verge of further evolving towards seamless integration of human-machine synergy. The robotic arms, a pivotal component of physical systems in intelligent manufacturing, can acquire skills through imitation learning and effectively generalize their behavior to adapt to new scenarios seamlessly. Consequently, autonomous trajectory planning for the robotic arm has emerged as a crucial research concern, representing an indispensable technological branch in intelligent manufacturing3.

Imitation learning (IL) is a class of algorithms whereby a robot acquires skills by acquiring teaching samples derived from either a towed robot arm or datasets containing tracking points. It is a new method of acquiring skills from human demonstration4. The learning process involves extracting motion features from one or a few demonstration trajectories and subsequently generalizing these features to new scenarios, enhancing the robot’s adaptability (Robot Adaptive Behavior). Currently, there are four typical methods for imitation learning, namely trajectory learning based on statistical models, B-splines, affine transformations, and dynamical movement primitives5. Among these approaches, dynamical movement primitives (DMP), introduced by Ijspeert6, effectively enable the generalization of point-to-point and periodic movements through a single taught trajectory7. The efficacy of this approach has been demonstrated in instructing robots on various human tasks, such as gracefully swinging a ping-pong paddle8, skillfully playing drums9, elegantly writing10, as well as facilitating collaborative object manipulation between humans and robots11 and enabling human-robot collaborative assembly operations12.

However, autonomous obstacle avoidance poses an inevitable research challenge in robot adaptive behavior. In recent years, there has been a remarkable surge of interest in robot obstacle avoidance based on Dynamic Movement Primitives (DMP), and the strategies derived from this algorithm can be broadly classified into two distinct types13. The first type involves incorporating coupling terms into the DMP framework. For instance, Hoffmann et al.14 introduced steering angle as an additional component to DMP, while Tan H et al.15 integrated variable attractor coupling within the framework. Michele et al.16 proposed an extension of DMP that facilitates volumetric obstacle avoidance through superquadric potential (potential function). Nevertheless, these methods are confined to scenarios with simplistic obstacle geometry and simple tasks. The second type entails learning the coupling term by combining intelligent control algorithms17 or training it using feedback information from sensors or neural networks18. However, these algorithms encounter planning failures when confronted with multiple obstacles.

Notably, robot path planning based on intelligent algorithms is a prominent research topic, encompassing particle swarm optimization (PSO)19, genetic algorithm (GA)20, ant colony optimization (ACO)21, artificial fish swarm algorithm (AFSA)22, artificial neural networks (ANN)23, and so on. Among them, the PSO is widely applied in the planning of robot motion trajectories, especially in obstacle avoidance trajectory planning, due to its advantages such as simple parameters, fast convergence speed and low computational complexity24– 25. Zhang et al.26 introduced an elite strategy for efficiently updating particles to obtain paths. Shao et al.27 improved the initial distribution of particles using chaos-based logic mapping and enhanced the convergence speed through mutation strategies. Song et al.28 proposed an optimization method combining modified PSO with a cubic spline algorithm. Tharwat et al.29 presented a chaotic particle swarm algorithm for optimizing curve control points. R Palm et al.30 optimized the behavior of mobile robots using particle swarm optimization (PSO). However, these approaches have limited runtime improvement and failed to satisfy online path planning.

In this paper, a novel framework (DMP-PSO) for trajectory planning in robotic arms is presented by integrating dynamical movement primitives (DMP) with particle swarm optimization (PSO) in this paper. The key contributions of this study are as follows. Firstly, DMPs are employed to learn and generalize motion trajectories. Secondly, the initial state and search region of PSO are enhanced based on the generalized trajectories to generate obstacle avoidance trajectories within the search region rapidly. Finally, the proposed DMP-PSO framework autonomously generates diverse trajectories for robotic arms encompassing obstacle avoidance paths. Furthermore, the effectiveness of this framework is validated through various means.

Algorithm construction

Dynamic motion primitives

Dynamic movement primitives, originally developed by Ijspeert5, are designed for trajectory learning. This framework employs an ordinary differential equation (ODE) of the spring-mass-damper type with a forcing term. The algorithm was subsequently optimized by Hoffmann et al.13 to address the issue of invalidating the state learning term when the target position coincides with the initial position. The dynamic movement primitives comprise a system of ordinary differential equations:

$$\:\tau\:\dot{\varvec{v}}=\varvec{K}\left(\varvec{g}-\varvec{x}\right)-\varvec{D}\varvec{v}-\varvec{K}\left(\varvec{g}-{\varvec{x}}_{0}\right)\varvec{s}+\varvec{K}\varvec{f}\left(\varvec{s}\right)$$
(1)
$$\:\tau\:\dot{\varvec{x}}=\varvec{v}$$
(2)

Vectors x, v ∈ \(\:{\varvec{R}}^{\varvec{d}}\) are the position and velocity of the system respectively; x0∈ \(\:{\varvec{R}}^{\varvec{d}}\) is the starting position, and \(\:\varvec{g}\)∈ \(\:{\varvec{R}}^{\varvec{d}}\) is the goal position. Matrices K, D ∈ \(\:{\varvec{R}}^{\varvec{d}\times\:\varvec{d}}\) are, respectively, the elastic and damping terms of the system. Scalar τ ∈ R is a temporal scaling factor which can be used to make the execution of the trajectory faster or slower. The vector valued function \(\:\varvec{f}\left(\varvec{s}\right)\in\:\:{\varvec{R}}^{\varvec{d}}\) is a “forcing term”, used to model a desired trajectory, which depends on the parameter s. Scalar s ∈ (0, 1] is a reparametrize-tion of time t ∈ [0, T ] governed by the so called canonical system:

$$\:\tau\:\dot{s}=-as$$
(3)

Where, a is a predefined constant that determines the exponential decay of the regular system (a > 0), and the initial state is s(0) = 1. The forcing term \(\:\varvec{f}\left(\varvec{s}\right)=[{\varvec{f}}_{1}\left(\varvec{s}\right),{\varvec{f}}_{2}\left(\varvec{s}\right),\dots\:,{\varvec{f}}_{\varvec{d}}\left(\varvec{s}\right){]}^{\varvec{T}}\)is written in term of basis functions. Each component \(\:{\varvec{f}}_{\varvec{p}}\left(\varvec{s}\right)\) has the form:

$$\:{\varvec{f}}_{\varvec{p}}\left(\varvec{s}\right)=\frac{{\sum\:}_{i=1}^{N}{\omega\:}_{i}{\phi\:}_{i}\left(s\right)}{{\sum\:}_{i=1}^{N}{\phi\:}_{i}\left(s\right)}s,\:p=\text{1,2},\dots\:,d$$
(4)

where \(\:{\omega\:}_{i}\) is called weight, and \(\:{\phi\:}_{i}\left(s\right)\) is a basis function. These functions are defined as:

$$\:{\phi\:}_{i}\left(s\right)=\text{e}\text{x}\text{p}(-{h}_{i}{\left(s-{c}_{i}\right)}^{2})$$
(5)

with centers \(\:{c}_{i}\) defined as:

$$\:{c}_{i}=\text{exp}\left(-ai\frac{T}{N}\right),i=\text{1,2},\dots\:,N.$$
(6)

and widths defined as:

$$\:{h}_{i}=\frac{\widehat{h}}{{({c}_{i+1}-{c}_{i})}^{2}}\:,\:{h}_{N}={h}_{N-1}$$
(7)

The weights \(\:{\omega\:}_{i}\) in (4) are acquired through techniques such as weighted linear regression31; subsequently, the desired trajectory can be reproduced by numerically integrating (1). This framework offers several advantages that make it highly suitable for robots. Firstly, it enables the learning and execution of any trajectory while allowing flexibility in changing starting and goal positions. Secondly, the executed trajectory consistently converges towards the goal while maintaining a similar shape to the learned trajectory. Thirdly, adjusting a single parameter allows for implementing the learned trajectory at different speeds. In robot trajectory learning, DMP can be effectively utilized in both joint and cartesian space representations. The generalization trajectories are achieved by training the DMP algorithm with an imitation trajectory, as shown in Fig. 1, and subsequently simultaneously modifying the target position, initial position, or both.

Particle swarm optimization

The principle of particle swarm optimization simulates the collective behavior of birds in a flock by modeling massless particles with two attributes: speed V and position X. Where, velocity V represents the magnitude of movement, while position X indicates the direction of movement. The population size of the PSO is set to \(\:n\), and the dimension of the search area is set to \(\:j\). The current position of particle \(\:i（i=\text{1,2},\dots\:,n）\)is denoted as \(\:{X}_{ij}=({X}_{i1},{X}_{i2},\dots\:,{X}_{ij})\), and its current velocity of particle i is denoted as \(\:{V}_{ij}=({V}_{i1},{V}_{i2},\dots\:,{V}_{ij})\). The particle would tend to move towards its historical best position and the best position recorded by the swarm. The optimal position for each particle is represented by \(\:{P}_{ij}=({P}_{i1},{P}_{i2},\dots\:,{P}_{ij})\), while the optimal position searched by the entire swarm of particles is indicated as \(\:{P}_{gj}=({P}_{g1},{P}_{g2},\dots\:{,P}_{gj})\). The formulas for updating velocity and position are expressed as:

$$\:{V}_{ij}\left(k+1\right)=\omega\:{V}_{ij}\left(k\right)+{c}_{1}{r}_{1}\left({P}_{ij}\left(k\right)-{X}_{ij}\left(k\right)\right)+{c}_{2}{r}_{2}\left({P}_{gj}\left(k\right)-{X}_{ij}\left(k\right)\right)$$
(8)
$$\:{X}_{ij}\left(k+1\right)={X}_{ij}\left(k\right)+{V}_{ij}\left(k+1\right)$$
(9)

Where, \(\:{V}_{ij}\left(k+1\right)\) and \(\:{X}_{ij}\left(k+1\right)\) represent the jth dimensional components of particle i’s velocity and position at the (k + 1)th iteration, respectively; denotes the jth dimensional component of particle i’s optimal position solution after the kth iteration, while \(\:{P}_{gj}\left(k\right)\) represents the jth dimensional component of the optimal position solution for the entire particle swarm after the kth iteration. The parameter ω, representing the inertia weight, is initially set to 0.9 and then linearly decreases to 0.4 throughout the evolutionary process. The parameter \(\:{c}_{1}\) serves as the cognitive factor, and \(\:{c}_{2}\) acts as the social factor; both are consistently assigned a value of 2.0. Furthermore, \(\:{r}_{1}\) and \(\:{r}_{2}\) are random numbers selected from the range [0, 1]. The population size and the number of iterations (NI) are fine-tuned based on experimental insights.

Fig. 1
figure 1

Imitation and generalization trajectories of DMP in the 2D plane.

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In the PSO, each particle is dispersed throughout the solution space, occupying distinct positions and exhibiting unique adaptations. With inherent mobility and speed, particles continuously explore the predetermined solution space. The algorithm continually adjusts particle movement speed to optimize their position within the solution space. Through iterative refinement based on formula 8–9, individual particles’ optimal values (pbest) and those of the entire swarm (gbest) determine flight distance and direction for each particle. As iteration count increases, search region for all particles moves towards an increasingly better direction in pursuit of optimal solutions.

DMP-PSO framework construction

The DMP-PSO, an obstacle avoidance algorithm, is proposed in this paper by integrating particle swarm optimization into dynamic movement primitives for trajectory planning in the robot. The principle and flow of the framework are illustrated in Fig. 2. Initially, motion characteristic parameters of the system are obtained from an imitation trajectory (Demonstration Data) within the DMP framework as described in Sect. 2.1. Subsequently, new trajectories are generated based on the DMP to adapt to new scenarios by defining start and target positions for the robot; simultaneously, real-time machine vision is employed to capture obstacles and normalize their representation within the new scenarios. Finally, collision detection between the generalized trajectory and obstacles is performed in real-time. If no collision occurs, generalized trajectories are taken as working trajectories. Otherwise, local optimal paths in collision areas are searched using PSO.

Subsequently, different types of static obstacles will be set up to verify and analyze the effectiveness of the framework. The spatial position information of moving obstacles needs to be extracted in real time and utilized by the visual system, which is beyond the discussion.

Fig. 2
figure 2

Framework of DMP-PSO.

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Specifically, when a collision between the generalized trajectory and an obstacle is detected, the collision region is captured through machine vision detection. Subsequently, the sample points of the generalized trajectory are refined, and the distance (d) from each point to the center position of the collision region is calculated using a two-point distance equation. If this distance (d) exceeds the safety distance (S), the generalized trajectory lies outside the obstacle interference range; such points are retained and recorded as part of the generalization points. The remaining points are designated as the sample points of the optimized path, with their initial and final positions serving as start and end positions for the PSO, respectively. The smallest rectangle encompassing the collision region is defined as the search area for the PSO too. Based on these settings, optimization points are performed using optimized path samples as initial positions, with iteration results being recorded as search-path points. Finally, the obstacle avoidance trajectory is produced based on fitting the points to a smooth curve; those points are integrated from the generalization points and the search-path points in order of motion. Therefore, based on the DMP-PSO framework, independent planning of the robot’s motion path can be achieved. The pseudo-code for DMP-PSO can be found in Table 1.

Table 1 The pseudo-code of the DMP-PSO
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Numerical simulation results and analysis

Numerical simulation environment

The performance of the MDP-PSO for trajectory planning will be evaluated through numerical simulations. Additionally, the steering angle method proposed in the literature13, the potential function method proposed in the literature15, and the elementary particle swarm algorithm (PSO) will also be assessed under the same scenario. Based on their analytical results, a comparative analysis of these four methods will be conducted. The environmental configuration required for these numerical simulation experiments is presented in Table 2.

Table 2 Hardware and software configuration.
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Trajectory planning and analysis in two-dimensional space

In this section, a trajectory, generalized by the Dynamic Movement Primitive with an initial position of (0,0) and a target position of (30,20), is selected as the path for the robot’s end-effector. This path is depicted in Fig. 1. The normalized obstacle in this scenario is represented by circle with a center of (9,7) and a radius of 4 mm. The paths are generated under different algorithms in the scenario, which are depicted in Fig. 3. Specifically, Fig. 3a,b, c, and d represent path generation using steering angle13, potential function15, MDP-PSO, and PSO, respectively.

Fig. 3
figure 3

Paths planning under different methods.

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The characteristics of each algorithm are further analyzed and compared by establishing several groups of normalized obstacle combinations with varying relative positional relationships. Each group comprises two circular obstacles as follows: (a) a circle centered at (9,7) with a radius of 4 mm, and another circle centered at (21,13) with a radius of 3 mm; (b) a circle centered at (9,7) with a radius of 4 mm, and another circle centered at (18,16.2) with a radius of 3 mm; (c) a circle centered at (9,7) with a radius of 4 mm, and another circle centered at (18,10.8) with a radius of 3 mm. Figures 4, 5, 6 and 7 present the sequential path planning results of the steering angle method, potential function method, MDP-PSO, and PSO in these three combined scenarios.

Fig. 4
figure 4

Paths generated by the steering angle method.

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The principle of the steering angle method lies in incorporating the obstacle-induced change in velocity as an additional term into the dynamic motion equation13. As the particle approaches closer to the obstacle, the declination angle of its velocity becomes more pronounced, leading to a trajectory that deviates further from the generalized path. Conversely, when moving away from the obstacle, the declination angle decreases and aligns its trajectory more closely with the generalized path. Illustrated in Fig. 4, although the trajectory starts at point (0,0), due to the influence of the leftmost obstacle, it diverges from its original generalized trajectory resulting in an avoidance path that circumvents said obstacle. However, as depicted in Fig. 4b, if there is another obstacle ahead along the motion direction, it may not generate sufficient deflection for successful circumnavigation around subsequent obstacles simultaneously. This indicates that limitations exist within this method when dealing with successive multiple obstacles.

Fig. 5
figure 5

Paths generated by the potential function method.

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The essence of the potential function method lies in seamlessly integrating the perturbation term as an additional component into the dynamic motion equation15. In this case, the term solely relies on position rather than velocity. It represents the inverse gradient of the potential, thereby extending DMP to facilitate volumetric obstacle avoidance through a superquadric potential function. The direction is perpendicular to the quadratic curve (surface), and the closer the obstacle, the greater the potential field force. The particle is affected by this field force, producing an acceleration component perpendicular to the quadratic curve (surface), which drives the trajectory of motion around the obstacle. However, when two obstacles are close to each other, their potential fields overlap and mutually influence each other, leading to uncertainty in both magnitude and direction of the potential field in overlapping regions and causing random jumps in trajectory when particles pass through these regions as shown in Fig. 5c. These results also indicate that this method has limitations when dealing with multiple obstacles.

Fig. 6
figure 6

Paths generated by MDP-PSO.

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The results illustrated in Fig. 6 demonstrate that the path generated by employing DMP-PSO successfully accomplishes the obstacle avoidance task and effectively preserves the original trajectory’s inherent shape characteristics. Furthermore, this approach surpasses the limitations associated with the steering angle and potential function methods, thereby emphasizing its remarkable adaptability.

Fig. 7
figure 7

Paths generated by the PSO.

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The observation depicted in Fig. 7 reveals that particle swarm optimization can effectively achieve obstacle avoidance trajectories in three different scenarios. However, there are noticeable disparities in the shape characteristics between the resultant path and the original generalized trajectory, thereby hindering the complete utilization of the latter’s velocity and acceleration advantages, which are derived from a teaching trajectory. Moreover, when employing PSO for the afore mentioned three scenarios, the particle swarm dimension is set to 10, and the generation of an obstacle avoidance trajectory occurs after 1000 iterations. Regrettably, this process has an average duration of 6.8 s. In contrast, the initial position of the optimized particle swarm is refined, and the scope of path search is narrowed within the framework of DMP-PSO, thereby reducing the dimensionality of the particle swarm to 5 and limiting the number of iterations to 100. This refinement also enables the generation of local optimization paths, resulting in a remarkable reduction in time consumption to just 1.12 s and a significant enhancement in obstacle avoidance path planning efficiency.

In summary, based on the numerical simulation results and analysis in a two-dimensional space, it can be observed that the DMP-PSO proposed in this study exhibits superior consistency with the original generalized trajectory and demonstrates broader adaptability compared to both the steering angle method and the potential function method. Furthermore, in contrast to the utilization of PSO alone for path planning, DMP-PSO not only possesses remarkable learning and generalization capabilities for paths but also significantly amplifies the efficiency of path search.

Trajectory planning and analysis in three-dimensional space

The DMP-PSO described in Sect. 2.3 is also applicable for trajectory planning in three-dimensional space under Cartesian coordinates. An ellipsoid of minimal size is employed to represent obstacles with irregular geometry and arbitrary orientation, which are captured using a 3D camera like Kinect. Similarly, when the generalized trajectory does not intersect with the obstacle, it can be directly utilized as the motion trajectory of the robot arm. However, if the generalized trajectory passes through an obstacle region, its sample points become sparse. Based on safety distance considerations, this sparse point set is further divided into generalization points and optimization points. The optimization points play a crucial role in determining parameters such as starting position, target position, and initial particle position for the PSO while defining search region based on the least square method within collision area boundaries. Iterative calculations yield search-path points that record results based on these optimized settings. Finally, the obstacle avoidance trajectory of the robotic arm is generated by fitting a smooth curve to integrate both the optimization points and the search-path points into a motion sequence. The parameters of DMP-PSO and obstacles are shown in Table 1.

Table 3 Parameters of the DMP-PSO and Obstacles.
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In this section, the system employs the DMP to acquire the motion characteristics of the demonstration trajectories, and three distinct end-effector trajectories for a robotic arm are generalized by manipulating both the initial position and target position. These three trajectories are seamlessly interconnected in a head-to-tail fashion, forming a comprehensive work cycle for the robotic end-effector. Moreover, an obstacle is strategically positioned within the X-Y plane, while another spatial obstacle is situated within the XYZ coordinate system. The visually depicted path generated automatically using the DMP-PSO framework can be observed in Fig. 8. It can be seen from the figure that the DMP-PSO constructed in this paper possesses the capability to autonomously devise trajectories for avoiding obstacles in 3D space.

Fig. 8
figure 8

Obstacle avoidance trajectory of the DMP-PSO in 3D space.

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Synthetic experiment

Virtual simulation experiment

The DMP-PSO framework was further validated through the construction of a virtual simulation environment for conducting virtual experiments with the UR5 robotic arm model. CoppeliaSim, a widely used robot simulation software, was selected as the simulation environment in this study due to its extensive employment in various domains such as robot simulation, hardware control, algorithm development, and optimization design of automation equipment32. In accordance with Sect. 3.3, the UR5 machinery and its working scene were established in CoppeliaSim, as depicted in Fig. 9. Python was utilized to write control commands for simulating the robot’s movement process in real scenes while also illustrating corresponding movement trajectories shown in Fig. 9.

Fig. 9
figure 9

Trajectory planning in CoppeliaSim.

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As illustrated in Fig. 9, the motion path traced by the end-effector of the UR5 robotic arm within the CoppeliaSim environment exhibits remarkable congruence with the numerical simulation trajectory outlined in Sect. 3.3. Through a detailed examination of the motion trajectory coordinate data, it has been determined that the maximum deviation between the two trajectories is a minimal 6.7 mm. This observation validates that the DMP-PSO proposed in this study effectively controls the robotic arm to navigate from its initial position to its target location along a generalized trajectory. Moreover, it demonstrates that our system is capable of dynamically planning new motion paths based on terrain and obstacle information within its surroundings, ensuring smooth and stable movements without any sliding or loss of control phenomena. Consequently, these findings further substantiate the algorithm’s robustness and practicality.

Physical verification experiment

To validate the effectiveness of the DMP-PSO in real-world robotic arms and their working scenarios, we conducted experiments using the UR5e robotic arm. The experimental setup was designed to mimic the working environment described in Sect. 4.1 (Fig. 10). Specifically, during material handling tasks, the UR5e robotic arm had to navigate around material trays placed in the middle area of the workbench. After completing material handling, it also needed to avoid collision with a tool rack while returning to its original position. Additionally, Vision algorithms applied ArUco codes to the workspace for localization recognition. For control purposes within the ROS framework, a socket communication mechanism was employed to control the UR5e robotic arm. Python programming code instructions were utilized for end motion control of the UR5e robotic arm based on the inverse kinematics of the robot arm (Figs. 10 and 11).

Fig. 10
figure 10

Material transfer process of robotic arm.

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Fig. 11
figure 11

Process of returning to the initial position for robotic arm.

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It can be found from Figs. 10 and 11 that the control system, based on the DMP-PSO framework, effectively guides the robotic arm to navigate through multiple obstacles in a working scenario. Experimental results demonstrate that the proposed DMP-PSO not only achieves motion convergence even after target point changes but also successfully avoids multiple obstacles while planning motion trajectories. The approach ensures smooth transitions at both the beginning and end points of obstacle avoidance, resulting in minimal impact and jitter of the robotic arm. These findings further validate the algorithm’s stability and practicality.

Summary

In this paper, a novel framework for trajectory planning in robotic arms is developed by integrating dynamical movement primitives with particle swarm optimization (DMP-PSO). Firstly, the dynamic motion primitive algorithm is employed for learning and generalization of robotic arm motion trajectories. Secondly, the initial state and search region of the particle swarm algorithm is improved based on the generalized trajectory to realize the fast search of the local obstacle avoidance trajectory. Finally, dynamic motion primitives and particle swarm algorithms are integrated to generate autonomous trajectory planning for the robotic arm, including autonomous obstacle avoidance.

The limitations of two typical obstacle avoidance algorithms based on DMP and the efficiency of trajectory planning based on the PSO are analyzed in detail. The comparison of numerical simulation results reveals that trajectory planning based on DMP-PSO exhibits enhanced adaptability and strong consistency with generalized trajectories, and the efficiency of trajectory planning is significantly improved.

When compared to employing deep learning-based imitation models, the DMP-PSO framework eliminates the need for extensive data to train the model, enabling faster deployment and utilization across diverse application scenarios. Nevertheless, in future research endeavors, deep reinforcement learning techniques will be integrated into this framework to bolster the manipulator’s autonomous decision-making and operational capabilities.