Optimal trajectory planning for industrial robots: Minimizing time,jerk,and energy consumption using LSTM for energy profile modeling

Abstract

Due to the continuous rise in energy costs for industrial robots (IRs), energy conservation has become one of the primary concerns in modern industry. This article presents a new, efficient approach for optimal trajectory planning of industrial robots in terms of time, jerk, and energy, while taking into consideration the kinematic constraints of the robot. A fifth-order B-spline interpolation method is adopted for curve fitting the trajectory in joint space to ensure smooth and continuous jerk in the robot’s articulation movements. The adjustable parameters of the trajectory are then optimized using the non-dominated sorting genetic algorithm II (NSGA-II) to minimize traveling time, jerk, and energy consumption (EC) throughout the trajectory. Unlike time and jerk, establishing a precise mathematical relationship between energy consumption and the dynamics of a robot across different trajectories is challenging and not easily applicable. This study uses the deep learning technique long short-term memory (LSTM) to accurately uncover the quantitative relationships between trajectory operational parameters and energy consumption. The main advantage of this approach, compared to other proposed optimizations, is that it can predict and optimize the robot’s energy consumption before the real-time execution of the task, and it does not require setting a priori the overall execution time of the trajectory. The results on a six degree of freedom industrial robot demonstrate that the suggested approach reduces energy consumption by 49.87% and average absolute jerk by 60.56% compared to chord length distribution method with the same trajectory execution time.

Keywords

Industrial robots trajectory planning B-spline energy consumption multi-optimization deep learning-LSTM

1. Introduction

Robot manipulators are nowadays increasingly being integrated into assembly and manufacturing lines due to their ability to increase efficiency, flexibility, and safety (Rezali et al., 2024). However, higher demands regarding the motion performance of the industrial robots are also emphasized, notably minimum time, minimum jerk, and minimum energy during the trajectory planning.

Trajectory planning plays a crucial role in improving the productivity of industrial robots for several reasons: trajectory planning minimizes unnecessary movements, reducing cycle time; well-planned trajectories result in smoother motion profiles for the robot, minimizing jerky movements and vibrations that can lead to wear and tear on mechanical components; and optimized trajectories can reduce energy consumption by minimizing unnecessary accelerations and decelerations, leading to more efficient operation and lower operating costs. Thus, one of the main concerns associated with the use of robot manipulators is the optimization of trajectory planning, considering the three mentioned criteria.

Time-minimum trajectory planning is one of the requirements to drive higher production rates of robotic cells by determine a minimal time required to traverse a predefined path of the task while adhering to physical constraints of robot and without compromising the quality of the task. In recent years, many researches have addressed this problem; in Oberherber et al. (2015), the trajectory is represented by splines, and then optimized using dynamic programming (DP) algorithm to provide time-optimal trajectory and smoothness. In Kaserer et al. (2018), the authors present a new DP algorithm to obtain a time-optimal trajectory considering torque and jerk boundaries, and then compare it to the sequential convex programming (SCP) method. Serdar, in his work Kucuk (2016), developed a numerical method to obtain a trajectory that avoids collisions and robot singularities. He then constructed the trajectory using cubic splines and optimized it with a particle swarm optimization algorithm for optimal execution time. Further research on time-optimal trajectory is addressed using numerical methods, such as Kunz and Stilman (2012); Pham (2014); Pham and Quang-Cuong (2017, 2018); and Wang et al. (2023), where the work focuses on finding an appropriate time scaling to locate the switches points between maximum acceleration and maximum deceleration, reducing time trajectory. However, considering only the time factor in robot trajectory planning is not sufficient, especially given the growing cost of energy resources.

The trajectory minimum time of a robot can have a significant influence on jerk, which is the rate of change of acceleration. Jerk has an important role in determining how smoothly and efficiently a robot can move from one point to another, but it is an undesirable factor. Several studies have addressed to minimize both time and jerk in trajectory planning. In the work of Gasparetto and Zanotto (2010), third- and fifth-order B-spline interpolations are used for trajectory planning. Their approach optimizes a single objective function that combines execution time and the integral of squared jerk. However, this method is not suitable for scenarios with a non-convex solution set. In Li et al. (2018), the motion profile is generated using quintic B-spline curves for pick-and-place parallel robot to achieve C⁴-continuity, considering two performance indices based on acceleration and jerk. Junsen et al. Huang et al. (2018) also applied a 5th-order B-spline curve to interpolate the trajectory in joint space and optimized it using the elitist non-dominated sorting genetic algorithm (NSGA-II), considering time and jerk as objectives. In Lu et al. (2020), the viapoints of trajectory in Cartesian space are interpolated using 5th-degree polynomials and then optimized using a sequential quadratic programming (SQP) algorithm, with the objective function aimed at minimizing joint jerks. Zhang et al. (2021) also applied SQP algorithm to obtain time–jerk optimal trajectory for a robotic excavator. In Zhao et al. (2022), to manage the trade-off issue between the jerk and the time, hybrid meta-heuristic algorithms are used to optimize the objective function, where particle swarm optimization and improved whale optimization algorithm (PSO-IWOA) are merged to enhance the convergence of the optimal solution. But as we know, the required time to accomplish tasks and the jerk profile of a robot’s movement can significantly influence its energy consumption. Optimizing motion profiles, reducing task execution delays, and ensuring the efficient operation of components are key strategies for minimizing energy consumption in robotic systems (Gultekin et al., 2021).

The energy consumption minimization of a robot manipulator is an essential requirement to reduce the operating costs of the robotic applications, particularly in light of the growing cost of energy resources (Soori et al., 2023), and is also significant to reduce the effects on the environment (Rocha et al., 2021; Javaid et al., 2021). This has motivated academic research groups to focus on research into energy consumption recently. The authors in Carabin and Scalera (2020) present a method for minimum-energy trajectory planning in industrial robotic systems based on the modeling of an electro-mechanical system with one degree of freedom. While this method may be suitable for 1-DOF systems, it is not ideal for high-DOF systems such as those with six degrees of freedom. In Vidussi et al. (2021), an energy analysis of robot manipulators is presented to demonstrate the strong correlation between the inertia ellipsoid index and the robot’s effective energy consumption, based on the dynamic and electro-mechanical models of a 3-DOF SCARA robot. However, the authors do not account for the power consumption of the electronic components in the system cabinet. In Gadaleta et al. (2019), a software tool is designed that can interact with offline programming simulators of robots used in industrial applications, allowing it to compute and optimize motion parameters in terms of energy. A new investigation into energy consumption is carried out using a data-driven method, where the robot’s energy profile is modeled using an artificial neural network to predict it (Zhang and Yan, 2021; Jiang et al., 2023). In Li et al. (2023), the energy is optimized based on the robot’s dynamics, where joint torque is considered an objective function and optimized using the SQP algorithm. However, most research focusing on robot energy consumption does not pay attention to time and jerk, despite the importance of these criteria.

As concluded in the above discussion, limited works have been conducted on the consideration of execution time, jerk, and energy consumption simultaneously, except for some works that take all of them into account. As in Wu et al. (2021), the trajectory is planned using the method of non-uniform rational B-splines (NURBS) which takes into account time, energy, and jerk. In Chen et al. (2023), the trajectory of robot is interpolated by a quintic B-spline and optimized with a quantum-behaved particle swarm optimization (QPSO) algorithm to obtain time–jerk–energy optimal trajectory for a parallel robot. However, in these works, the authors consider the joints accelerations to represent consumption of energy, which is insufficient to express the actual energy consumed by the robot, primarily related to the torque provided by the actuators. This is difficult due to the unavailability of the exact dynamic model of the robot.

Despite these advancements in the field of trajectory planning, several deficiencies still exist: (1) The absence of an explicit mathematical relationship for energy consumption in robotic systems often leads to the use of indirect indicators, such as acceleration, to estimate energy usage. (2) Important factors, such as jerk (the rate of change of acceleration), are often overlooked during the optimization process. Neglecting jerk constraints can result in increased vibrations when the planned trajectory is executed by the robot manipulator. (3) Limited research has been conducted on the simultaneous consideration of execution time, energy consumption, and jerk, all of which are critical factors for achieving optimal performance in robotic trajectory planning.

In light of these requirements, this work focuses on trajectory planning to optimize the time, jerk, and energy. The highlights of this study are as follows:

• A 5th-order B-spline is applied in conjunction with a multi-objective optimization technique to construct the robot trajectory to ensure a smooth and continuous trajectory up to the jerk, which is a crucial factor for achieving precise and efficient robot motion.

• In the absence of an explicit relationship with energy, a predictive model of the energy consumption profile is designed using a long short-term memory (LSTM) technique, which is used as objective function in the optimization. It also enables the prediction of the robot’s energy consumption prior to the real-time execution of a task.

• The trajectory is optimized using NSGA-II to minimize time, jerk, and energy along the trajectory planning.

• To demonstrate the effectiveness of our method, we compare it with a classical trajectory planning method.

The outline of this article is as follows. In Section 2, the optimization problem of trajectory planning is stated. In Section 3, LSTM network is described for designing a model for predicting energy consumption profile. In Section 4, the NSGA-II tool is applied to find the time–energy–jerk optimal trajectory for a 6-DOF industrial robot. In Section 5, simulation results are presented with relevant discussions, and then compared with the classical method. Finally, Section 6 concludes the work.

2. Problem statements

2.1. Time–energy–jerk optimization problem formulation

To realize the motion path of a manipulator robot, it is necessary for the robot to move through a sequence of waypoints (or viapoints) in Cartesian space. As Gasparetto mentions in Gasparetto et al. (2015), the robot trajectories generally are planned in the joint space, and in this work, the trajectory planned is carried out in the joint space. Consecutive waypoints in Cartesian space are transformed into joint space via the inverse kinematic model. From the waypoints in joint space, the trajectory would be made then optimized to minimize certain objectives.

The trajectory must be sufficiently smooth to avoid excessive mechanical vibrations, and the travel time should be minimized as much as possible, especially the consumed energy, which must be kept to a minimum. This consideration includes kinematic limits such as velocity, acceleration, and jerk.

Different performance indices are used to evaluate the planned trajectory. In Gasparetto and Zanotto (2008, 2007), sum of time and integral of square jerk has been defined as a term in the objective function. In Kyriakopoulos and Saridis (1988), maximum jerk’s value is considered. In this paper, the performance indices for execution time, jerk, and energy consumption are presented in equations (1)–(3).

The robot trajectory planning translates into multi-objective optimization problem and can be defined mathematically as follows.

Minimize:

f_{time} = \sum_{k = 0}^{n - 1} T_{k} = \sum_{k = 0}^{n - 1} (t_{k + 1} - t_{k})

(1)

f_{jerk} = \sum_{i = 1}^{N} \sqrt{\frac{1}{T} \int_{0}^{T} {({\overset{\dots}{s}}_{i} (t))}^{2} d t}

(2)

f_{energy} = m a x (| E C (t) |)

(3)

Subject to:

\{\begin{cases} 0 < T \leq T_{max} \\ | {\dot{s}}_{i} (t) | \leq {\dot{s}}_{max}, i = 1,2, \dots, N . \\ | {\ddot{s}}_{i} (t) | \leq {\ddot{s}}_{max} \\ | {\overset{\dots}{s}}_{i} (t) | \leq {\overset{\dots}{s}}_{max} \end{cases}

(4)

The symbols mentioned above are explained in Table 1.

Table 1.

Explanation of symbols in the optimization problem formulation.

Symbol	Definition
f _time	Objective function of execution time
f _jerk	Objective function of jerk
f _energy	Objective function of energy consuming
EC	Total energy consuming of the robot
s _i	Position of ith robot joint
${\dot{s}}_{i}$	Velocity of ith robot joint
${\ddot{s}}_{i}$	Acceleration of ith robot joint
${\overset{\dots}{s}}_{i}$	Jerk of the ith joint
${\dot{s}}_{max}$	Velocity constraint value
${\ddot{s}}_{max}$	Acceleration constraint value
${\overset{\dots}{s}}_{max}$	Jerk constraint value
T _k	The time between kth and (k + 1)th waypoints
T	Total travel time of the complete trajectory
T _max	Total travel time constraint
t _k	Time instant of kth node
n + 1	Number of the waypoints
N	Number of robot joints

2.2. Trajectory planning by 5th-order B-spline in joint space

In robot programming applications, it is necessary to plan smooth and continuous trajectories until third derivative (jerk) (Simon, 2004). For this purpose, it is recommended to use B-spline method, as it offers great flexibility in shaping the trajectory and also can be optimized to accomplish the best possible trajectory in terms of time, jerk, and energy efficiency. It’s expressed by the following equation:

s (u) = \sum_{j = 0}^{m} d_{j} B_{j}^{p} (u)

(5)

where d_j represents the control points of curve,

B_{j}^{p} (u)

denotes spline functions of degree p, defined according to knot vector

u = [u_{0}, \dots, u_{n_{knot}}]

, and s(u) represents the joint position at the instant u. The jth B-spline basis function of degree p is recursively defined using the Cox-de Boor recursion formula (De Boor, 1972).

\{\begin{cases} B_{j}^{0} (u) = \{\begin{cases} 1 & if u_{j} \leq u < u_{j + 1} \\ 0 & otherwise \end{cases} \\ B_{j}^{p} (u) = \frac{(u - u_{j})}{(u_{j + p} - u_{j})} B_{j}^{p - 1} (u) + \\ \frac{(u_{j + p + 1} - u)}{(u_{j + p + 1} - u_{j + 1})} B_{j + 1}^{p - 1} (u) \end{cases}

(6)

The continuity of the jerk requires adopting a fifth-degree B-spline (p = 5). The trajectory in the Cartesian space is discretized into number of waypoints and two virtual (vp = (2) points are added at the second and second-last position of the waypoints sequence to obtain a smooth trajectory (no jerk) at either extremities (Gasparetto and Zanotto, 2008). These points are then mapped into joint space through the robot’s inverse kinematic model to find the viapoints q_k (k = 0, …, n), which are interpolated at times t_k. The goal is to find the control points d_j, j = 0, …, m that ensure

s (t_{k}) = q_{k}

(7)

It is first necessary to define the knot vector u, composed of sequence of t_k corresponding to the waypoints.

\begin{aligned} u = [\begin{matrix} \underset{p + 1}{\underset{⏟}{\begin{matrix} t_{0} & \dots & t_{0} \end{matrix}}} & t_{v p, 1} & t_{1} & \dots & t_{n - 1} \dots \end{matrix} \\ \begin{matrix} \dots t_{v p, 2} & \underset{p + 1}{\underset{⏟}{\begin{matrix} t_{n} & \dots & t_{n} \end{matrix}}} \end{matrix}] \end{aligned}

(8)

where t_vp,1 and t_vp,2 are time instants of the virtual points.

The length of knot vector is n + 2p + vp + 1 = n_knot + 1; in B-spline function calculations, the relation between n_knot, m, and p is n_knot − p − 1 = m; thus the control points number is n + vp + p = m + 1.

To determine the control point vector d, a linear system can be constructed by combining of n + 1 equations that obtain through interpolation conditions imposed for each viapoint q_k at instant t_k:

\begin{aligned} [\begin{matrix} q_{0} \\ q_{1} \\ ⋮ \\ q_{n - 1} \\ q_{n} \end{matrix}] = [\begin{matrix} B_{0}^{p} (t_{0}) & B_{1}^{p} (t_{0}) & \dots & B_{m - 1}^{p} (t_{0}) \\ B_{0}^{p} (t_{1}) & B_{1}^{p} (t_{1}) & \dots & B_{m - 1}^{p} (t_{1}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ B_{0}^{p} (t_{n - 1}) & B_{1}^{p} (t_{n - 1}) & \dots & B_{m - 1}^{p} (t_{n - 1}) \\ B_{0}^{p} (t_{n}) & B_{1}^{p} (t_{n}) & \dots & B_{m - 1}^{p} (t_{n}) \end{matrix} \dots \\ \dots \begin{matrix} B_{m}^{p} (t_{0}) \\ B_{m}^{p} (t_{1}) \\ ⋮ \\ B_{m}^{p} (t_{n - 1}) \\ B_{m}^{p} (t_{n}) \end{matrix}] [\begin{matrix} d_{0} \\ d_{1} \\ ⋮ \\ d_{m - 1} \\ d_{m} \end{matrix}] \end{aligned}

(9)

From equation (9), n + vp + p equations are required to build a square system of m + 1 equations and m + 1 unknown control points, and the extra p + 1 can be provided by imposing initial and final conditions of velocity, acceleration, and jerk of trajectory.

The formulas of velocity, acceleration, and jerk can be obtained by differentiating equation (5) up to the third order.

\dot{s} (u) = \sum_{j = 0}^{m - 1} c_{j} B_{j}^{p - 1} (u)

(10)

\ddot{s} (u) = \sum_{j = 0}^{m - 1} l_{j} B_{j}^{p - 2} (u)

(11)

\overset{\dots}{s} (u) = \sum_{j = 0}^{m - 1} w_{j} B_{j}^{p - 3} (u)

(12)

where c_j, l_j, and w_j are, respectively, the velocity curve, acceleration, and jerk control points and are computed as

\{\begin{cases} c_{j} = p \frac{d_{j + 1} - d_{j}}{u_{j + p + 1} - u_{j + 1}} \\ l_{j} = (p - 1) \frac{c_{j + 1} - c_{j}}{u_{j + p + 1} - u_{j + 1}} \\ w_{j} = (p - 2) \frac{l_{j + 1} - l_{j}}{u_{j + p + 1} - u_{j + 1}} \end{cases}

(13)

The velocity, acceleration, and jerk at waypoints can be described as following:

\begin{aligned} {\dot{q}}_{k} = [\begin{matrix} {B_{0}^{p}}^{(1)} (t_{k}) & {B_{1}^{p}}^{(1)} (t_{k}) & \dots & {B_{m - 1}^{p}}^{(1)} (t_{k}) \dots \end{matrix} \\ \begin{matrix} \dots {B_{m}^{p}}^{(1)} (t_{k}) \end{matrix}] [\begin{matrix} d_{0} \\ d_{1} \\ ⋮ \\ d_{m - 1} \\ d_{m} \end{matrix}] \end{aligned}

(14)

\begin{aligned} {\ddot{q}}_{k} = [\begin{matrix} {B_{0}^{p}}^{(2)} (t_{k}) & {B_{1}^{p}}^{(2)} (t_{k}) & \dots & {B_{m - 1}^{p}}^{(2)} (t_{k}) \dots \end{matrix} \\ \begin{matrix} \dots {B_{m}^{p}}^{(2)} (t_{k}) \end{matrix}] [\begin{matrix} d_{0} \\ d_{1} \\ ⋮ \\ d_{m - 1} \\ d_{m} \end{matrix}] \end{aligned}

(15)

\begin{aligned} {\overset{\dots}{q}}_{k} = [\begin{matrix} {B_{0}^{p}}^{(3)} (t_{k}) & {B_{1}^{p}}^{(3)} (t_{k}) & \dots & {B_{m - 1}^{p}}^{(3)} (t_{k}) \dots \end{matrix} \\ \begin{matrix} \dots {B_{m}^{p}}^{(3)} (t_{k}) \end{matrix}] [\begin{matrix} d_{0} \\ d_{1} \\ ⋮ \\ d_{m - 1} \\ d_{m} \end{matrix}] \end{aligned}

(16)

where

{B_{j}^{p}}^{(i)} (t_{k})

represent ith derivative of

B_{j}^{p} (t)

at t_k.

The initial and final values of velocity, acceleration, and jerk can be expressed as follows:

\{\begin{cases} \dot{s} (t_{0}) = {\dot{q}}_{0} \\ \dot{s} (t_{n}) = {\dot{q}}_{n} \\ \ddot{s} (t_{0}) = {\ddot{q}}_{0} \\ \ddot{s} (t_{n}) = {\ddot{q}}_{n} \\ \overset{\dots}{s} (t_{0}) = {\overset{\dots}{q}}_{0} \\ \overset{\dots}{s} (t_{n}) = {\overset{\dots}{q}}_{n} \end{cases}

(17)

This leads equation (9) to be square linear system with same number of equations and unknowns of m + 1.

A d = c

(18)

where

d = {[\begin{matrix} d_{0} & d_{1} & \dots & d_{m - 1} & d_{m} \end{matrix}]}^{T}

(19)

c = {[\begin{matrix} q_{0} & \dot{q_{0}} & \overset{\dots}{q_{0}} & q_{1} & \dots & q_{n - 1} & \overset{\dots}{q_{n}} & \ddot{q_{n}} & \dot{q_{n}} & q_{n} \end{matrix}]}^{T}

(20)

\begin{aligned} A & = [\begin{matrix} B_{0}^{p} (t_{0}) & B_{1}^{p} (t_{0}) & \dots & B_{m - 1}^{p} (t_{0}) \\ {B_{0}^{p}}^{(1)} (t_{0}) & {B_{1}^{p}}^{(1)} (t_{0}) & \dots & {B_{m - 1}^{p}}^{(1)} (t_{0}) \\ {B_{0}^{p}}^{(2)} (t_{0}) & {B_{1}^{p}}^{(2)} (t_{0}) & \dots & {B_{m - 1}^{p}}^{(2)} (t_{0}) \\ {B_{0}^{p}}^{(3)} (t_{0}) & {B_{1}^{p}}^{(3)} (t_{0}) & \dots & {B_{m - 1}^{p}}^{(3)} (t_{0}) \\ B_{0}^{p} (t_{1}) & B_{1}^{p} (t_{1}) & \dots & B_{m - 1}^{p} (t_{1}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ B_{0}^{p} (t_{n - 1}) & B_{1}^{p} (t_{n - 1}) & \dots & B_{m - 1}^{p} (t_{n - 1}) \\ {B_{0}^{p}}^{(3)} (t_{n}) & {B_{1}^{p}}^{(3)} (t_{n}) & \dots & {B_{m - 1}^{p}}^{(3)} (t_{n}) \\ {B_{0}^{p}}^{(2)} (t_{n}) & {B_{1}^{p}}^{(2)} (t_{n}) & \dots & {B_{m - 1}^{p}}^{(2)} (t_{n}) \\ {B_{0}^{p}}^{(1)} (t_{n}) & {B_{1}^{p}}^{(1)} (t_{n}) & \dots & {B_{m - 1}^{p}}^{(1)} (t_{n}) \\ B_{0}^{p} (t_{n}) & B_{1}^{p} (t_{n}) & \dots & B_{m - 1}^{p} (t_{n}) \end{matrix} \dots \\ \dots \begin{matrix} B_{m}^{p} (t_{0}) \\ {B_{m}^{p}}^{(1)} (t_{0}) \\ {B_{m}^{p}}^{(2)} (t_{0}) \\ {B_{m}^{p}}^{(3)} (t_{0}) \\ B_{m}^{p} (t_{1}) \\ ⋮ \\ B_{m}^{p} (t_{n - 1}) \\ {B_{m}^{p}}^{(3)} (t_{n}) \\ {B_{m}^{p}}^{(2)} (t_{n}) \\ {B_{m}^{p}}^{(1)} (t_{n}) \\ B_{m}^{p} (t_{n}) \end{matrix}] \end{aligned}

(21)

Thus, the control points can be obtained by equation (22):

d = A^{- 1} c

(22)

Considering that the control points vector relies on the times intervals between waypoints, alterations to the trajectory’s velocity, acceleration, and jerk may arise.

3. Prediction model of energy consumption using LSTM

As stated in the problem formulation, an energy consumption model is required to construct an optimization objective function. To determine the energy consumed by the manipulator robot, one must either analyze and extract the mathematical relationship between the current, voltage, and time of each servo motor of the robot to calculate power, which is difficult to resolve, or describe it by torque through robot dynamic model which is difficult to obtain in practice.

Methods based on deep learning have recently gained popularity for creating models to predict the robot’s energy consumption. This is due to their simple modeling approach, which avoids complex dynamic modeling processes (Yan and Zhang, 2021; Lin et al., 2024; He et al., 2020).

LSTM is an improved version of the theory of recurrent neural network (RNN) regression, due to its ability to identify non-linear and complex relationships (Graves, 2012). In this study LSTM is applied to establish the relationship between the executed trajectory and its corresponding consumed energy.

The components of the proposed LSTM network are data preprocessing module, input layer, three LSTM layers, dropout layer, fully connected layer, and regression layer. The data preprocessing module normalizes the training data and removes features with constant values that may have a negative impact on training. The LSTM layers can learn from sequences data and retain information over time steps and exploit deep relationships between energy consumption and input variables. To prevent LSTM over-fitting risk, a dropout layer is adopted. The fully connected layer is added for learning non-linear combinations of data features. The regression layer calculates the error loss in output layer for the regression task. Figure 1 illustrates the architecture of our proposed LSTM network for regression. The input sequence of our designing LSTM network can be represented as

x_{t} = {[\begin{matrix} s_{1} & \dots & s_{6} & {\dot{s}}_{1} & \dots & {\dot{s}}_{6} & {\ddot{s}}_{1} & \dots & {\ddot{s}}_{6} \end{matrix}]}^{T}

(23)

and the output sequence, representing the energy consumption indicated by the absolute value of the torque at each robot joint, can be expressed as

\sum_{i = 1}^{N} \int_{0}^{T} | τ_{i} (t) | d t

(24)

Figure 1.

Schematic of the proposed LSTM network.

3.1. Structure of LSTM cell

The LSTM cell is the alternative element of the neuron in RNNs, capable of modeling long-term dependencies through the use of memory cells to store information and gates to regulate its flow between them (see Figure 2).

Figure 2.

LSTM cell architecture.

The flow of information is regulated by the forget gate G_t, input gate I_t, output gate O_t, and also cell candidate $\tilde{C} t$ via following formulas.

G_{t} = s i g (W_{G} {[y_{t - 1} x_{t}]}^{T} + b_{G})

(25)

I_{t} = s i g (W_{I} {[y_{t - 1} x_{t}]}^{T} + b_{I})

(26)

{\tilde{C}}_{t} = t a n h (W_{C} {[y_{t - 1} x_{t}]}^{T} + b_{C})

(27)

O_{t} = s i g (W_{O} {[y_{t - 1} x_{t}]}^{T} + b_{O})

(28)

where W_(.) are the weight vectors of gates, b_(.) are bias vectors, sig is sigmoid activation function, and tanh is hyperbolic tangent activation function. They are defined as follows:

s i g (v) = \frac{1}{1 + e^{- v}}

(29)

t a n h (v) = \frac{e^{v} - e^{- v}}{e^{v} + e^{- v}}

(30)

while cell output y_t and cell state C_t are given by

C_{t} = G_{t} \otimes C_{t - 1} + I_{t} \otimes {\tilde{C}}_{t}

(31)

y_{t} = O_{t} \otimes t a n h (C_{t})

(32)

where ⊗ denotes element-wise multiplication of vectors.

4. Time–jerk–energy optimization using NSGA-II

The NSGA-II is multi-optimization method that was developed by Deb et al. (2002) has proven its effectiveness in various fields where optimization problems involve multiple conflicting objectives. This method aims to efficiently explore the solution space, maintain diversity among solutions, and identify solutions for conflicting objectives called Pareto-optimal solutions.

In the NSGA-II process, the offspring population Q_t is initially formed by utilizing the parental population P_it, along with the standard genetic operators. Subsequently, P_it and Q_it are merged to form a new population R_it, of size 2P. Then, the R_it population is classified based on non-domination sorting method. In new generation, the parent population P_it+1 is filled by non-dominated fronts increasingly until the population size achieves P. Solutions are sorted based on crowding distance in descending order. This scenario is illustrated in Algorithm 1.

According to the section analysis, solving the problem of time–jerk–energy optimal trajectory planning consists of finding the control points of the B-spline curve, which relies on the time instants of the waypoints. In this study, the optimizer works to find appropriate time instants for the waypoints of the trajectory, minimizing the objectives described in equations (1)–(3) under the kinematic constraints described in equation (4).

In the optimization procedure, the optimizer defines the instants of time t_k corresponding to the predefined waypoints of the reference trajectory. The generated trajectory is tested against the constraints, then evaluated through the objective functions, and optimizes the sequence of t_k, repeating the process until the objective is achieved. Figure 3 illustrates the optimization flowchart.

Figure 3.

Flowchart of optimization.

5. Results and discussion

To demonstrate the feasibility of the proposed approach, simulation experiments are carried out on an industrial robot model the of 6-DOF Fanuc M710iC70 (see Figure 4).

Figure 4.

3D robot model of Fanuc M710iC70.

The proposed approach is implemented in the MATLAB/Simulink software environment. The simulation computer is configured with a 3.5 GHz dual-core processor and 8 GB of RAM. The computation time for training the LSTM model is approximately 27 minutes, while the computation time for the optimization process is 17 minutes. The simulation experiments focus on three main aspects:

Robot energy consumption model: This involves designing an energy consumption profile model for the robot, which is based on data collection, training, and validation using a test sample.

Process optimization: This aspect involves running the process optimization to obtain optimal time parameters, which results in an optimal trajectory in terms of time, jerk, and energy.

Method comparison: This provides a comparison between the proposed approach and one of the classical methods in trajectory planning of robots to evaluate the effectiveness of our approach.

5.1. Robot energy consumption model

For training the energy model, input/output data were extracted from the experimental simulations. The position, velocity, and acceleration are considered input data, and the integral of the absolute value of the corresponding torque, representing energy consumption, is taken as output data. In this study, the input data consist of the position, velocity, and acceleration of six robot joints, as the movement of a single joint can affect the robot’s overall energy consumption, while in the output data, we preferred to take the sum of energy consumption of all joints to avoid complexity in calculation, as well as the total energy being what matters to us for an IR system.

The trajectories were generated using the B-spline method, with random waypoints selected using MATLAB’s random function. The waypoints were constrained within the robot’s joints movement limits of [–90°, 90°]. This approach ensures diverse trajectory generation, enabling the robot to learn effectively across a wide range of motions. In future work, this method can be extended to comprehensively cover the entire robot workspace. The measurements of position, velocity, acceleration, and corresponding energy consumption values are collected into a training set. The other LSTM network parameters are listed in Table 2.

Table 2.

Parameters of proposed LSTM.

Parameters	Values
Number of input features	18
Number of layers	3
Number of hidden units of each layer	100
Number of responses	1
Epochs	400
Min batch size	20
Learning rate	0.01
Solver	Adam

The LSTM model was trained using different numbers of trajectories (50, 100, 150, and 200) to evaluate its performance in terms of root mean squared error (RMSE) and achieve maximum accuracy. As shown in Figures 5 –7, the RMSE decreases as the number of trajectories increases, indicating improved model performance with larger training sets. At 200 trajectories, the RMSE stabilizes at iteration 2000, suggesting that further increasing the number of trajectories may not significantly enhance accuracy beyond this point as depicted in Figure 8.

Figure 5.

Training model performance with 50 trajectories.

Figure 6.

Training model performance with 100 trajectories.

Figure 7.

Training model performance with 150 trajectories.

Figure 8.

Training model performance with 200 trajectories.

After training the adopted energy model, the result showed an improvement in the convergence speed of the EC model for the robot. This suggests that the model has reached a satisfactory level of accuracy in predicting power consumption more rapidly compared to previous iterations. Throughout the training process, the training loss stabilized smoothly. This implies that the model’s performance plateaued, indicating that further training iterations were unlikely to yield significant improvements or that the model had reached a satisfactory level of performance without over-fitting to the training data. Figure 9 shows the evolution of energy consumption along a random trajectory and shows that the trained model can predict the robot’s EC and has good consistency with the experimental data.

Figure 9.

Test samples of EC predicted versus EC measured.

5.2. Running the optimization process

After modeling the energy profile of the robot, this section focuses on the optimization process. The time–jerk–energy optimization is addressed using NSGA-II to determine the optimal time instants. The proposed method is examined on the trajectory of waypoints in the joint space presented in Table 3, subject to kinematic constraints, which are presented in Table 4. The initial and final conditions of velocity, acceleration, and jerk of the trajectory are initialized to zero. The NSGA-II algorithm is used for optimizing the objective functions with these settings: number of generation = 50, size of population = 100, and the crossover probability is randomly selected at each generation.

Table 3.

Kinematic constraints of the robot joints.

Number of joints	1	2	3	4	5	6
Velocity ${\dot{s}}_{max}$ (deg/s)	10	15	10	10	15	10
Acceleration ${\ddot{s}}_{max}$ (deg/s²)	15	20	20	10	15	10
Jerk ${\overset{⃛}{s}}_{max}$ (deg/s³)	18	25	20	15	20	12

Table 4.

Waypoints of trajectory in joint space (°).

Waypoints	J1	J2	J3	J4	J5	J6
1	19	70	76	−2	−17	18
2	Virtual
3	23	89	66	−6	−16	13
4	18	81	75	−10	−7	21
5	21	81	71	−8	−5	23
6	26	86	73	−5	0	24
7	29	88	77	−4	4	27
8	Virtual
9	26	90	75	−3	1	27

Figure 10 illustrates the optimal Pareto front of the time–jerk–energy optimization obtained by NSGA-II. In the Pareto front solutions, the trajectory time varies between 9.5 s and 11.99 s, and the jerk index ranges from 71.73 deg./s³ to 156.91 deg./s³ while the energy consumption index from 26534 N.m.s to 30238 N.m.s. The shortest execution time with the greatest value of jerk is for solution A, while the energy value lies intermediate between their limiting values; meanwhile, solution C requires a longer traveling time but exhibits the least jerk and is highly energy consuming. In terms of execution time, option B is better than solution C. It also performs better than solution A when it comes to jerk, while surpassing both solutions A and C in terms of energy consumption. Therefore, it can be considered a compromise solution that provides a good balance between the competing objectives. The final selection of solutions from the Pareto front depends on the specific problem and the preferences of the decision-maker. This process may involve choosing one or more solutions and considering factors such as computational cost.

Figure 10.

Pareto front of time–jerk–energy optimization.

Given that the optimal solutions on the Pareto front comply with the kinematic constraints and that the difference between the traveling time values corresponding to these solutions is not significant, the criterion for choosing the optimal solution is therefore the energy consumption, which in this case is solution B with a predicted energy consumption of 26534 N.m.s and with a traveling time of 11.49s; the corresponding time intervals of this solution are shown in Table 5. The position, velocity, acceleration, and jerk of trajectory with a 11.49s execution time are shown in Figures 11 –14. From the resulting trajectory, the first important observation is that the trajectory passes through the predefined waypoints, demonstrating the flexibility of the B-spline method.

Table 5.

Time intervals of solution 11.49s.

Interval	T ₁	T ₂	T ₃	T ₄	T ₅	T ₅	T ₇	T ₈
Interval of time	1.952s	1.877s	2.195s	1.645s	0.803s	1.093s	0.674s	1.249s

Figure 11.

The position of joints of robot trajectory: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 12.

The velocity of joints of robot trajectory: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 13.

The acceleration of joints of robot trajectory: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 14.

The jerk of joints of robot trajectory: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Additionally, all kinematic limits are respected, proving that NSGA-II selects solutions that best meet specific needs and constraints. In Figure 14, the jerk curve is smooth and continuous, which increases the accuracy of the trajectory movement, avoids actuator vibrations, and reduces energy consumption. Figure 15 represents the trajectory of robot’s end-effector, while p₁⋯p₇ are waypoints in Cartesian space. Figure 16 shows the measured and predicted energy consumption during the trajectory, both of which exhibit nearly the same growth. This demonstrates that our LSTM model maintained its effectiveness during the optimization process.

Figure 15.

Path of robot’s end-effector.

Figure 16.

Energy consumption for the trajectory with an execution time of 11.49s.

5.3. Comparison of the suggested approach with the classical method

This section compares the proposed approach with the classical method for trajectory planning in terms of energy and jerk. In trajectory planning using the B-spline method, determination the time instants corresponding to the interpolated waypoints is needed. This section presents a comparison in obtaining the time vector between the suggested approach and one of the most widespread methods found in the literature Piegl and Tiller (1995), known as the chord length distribution.

5.3.1. Chord length distribution method

In the chord length distribution method, which is referred to by the expression “the classic method” in this article, the time parameter vector $\bar{u}$ covering the interval [0, 1] is obtained based on computed distances between successive waypoints.

The waypoints of the robot’s trajectory are labeled as D₀, D₁, D₂,…,D_n. The length between two consecutive waypoints D_k and D_k+1 is written as

L_{k} = | D_{k + 1} - D_{k} |

(33)

and the total length of the trajectory is the sum of the lengths of these segments.

L = \sum_{k = 0}^{n - 1} | D_{k + 1} - D_{k} |

(34)

Therefore, $\bar{u}$ can be determined by equation (35).

\{\begin{cases} {\bar{u}}_{0} = 0 \\ {\bar{u}}_{k} = {\bar{u}}_{k - 1} + \frac{| D_{k + 1} - D_{k} |}{\sum_{k = 0}^{n - 1} | D_{k + 1} - D_{k} |} & k = 1,2, \dots, n - 1 \\ {\bar{u}}_{n} = 1 \end{cases}

(35)

It is required to set the total execution time of the robot trajectory a priori to determine the time instants of waypoints. For a fair comparison, the execution time must be the same for both methods. Therefore, an execution time of 11.49s is considered for calculating the time instants based on equation (33). The resulting time intervals are presented in Table 6.

Table 6.

Time intervals from the classic method.

Interval	T ₁	T ₂	T ₃	T ₄	T ₅	T ₅	T ₇	T ₈
Interval of time	0.905s	0.908s	2.233s	2.582s	1.43s	1.499s	0.969s	0.963s

The comparison of the trajectory is shown in Figures 17 –20 in regard to position, velocity, acceleration, and jerk. Figure 21 shows energy consuming comparison.

Figure 17.

Comparison of proposed method and chord length distribution for the position of the six joints of trajectory with an execution time of 11.49s: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 18.

Comparison of proposed method and chord length distribution for the velocity of the six joints of trajectory with an execution time of 11.49s: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 19.

Comparison of proposed method and chord length distribution for the acceleration of the six joints of trajectory with an execution time of 11.49s: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 20.

Comparison of proposed method and chord length distribution for the jerk of the six joints of trajectory with an execution time of 11.49s: (a) Joint 1, (b) Joint 2, (c) Joint 3, (d) Joint 4, (e) Joint 5, and (f) Joint 6.

Figure 21.

Comparison of energy consumption between proposed method and classical method with an execution time of 11.49s.

Figure 17 shows the position of the six joints using the proposed approach and the conventional method. The execution time is the same for both methods, and the passage of the trajectory of the six joints over the waypoints is respected, but at different time instants. The movement of the joint departs progressive from the initial point compared to the classical method, and, as is known, the startup phase is important in terms of energy consumption. The velocity is shown in Figure 18, which indicates that the variation in velocity resulting from the proposed approach is smoother than of the classical method. It also shows that the maximum velocity of each joint remains within the kinematic constraints for the proposed approach, whereas in the classical method, the velocity exceeds the kinematic limits in the second and third joints. For acceleration, as illustrated in Figure 19, in classical method the maximum acceleration value of the first, fourth, and fifth joints is within the kinematic constraint limits, while the values for the second, third, and sixth joints exceed the constraint limits. So far, we can say that this method does not meet our needs. In contrast, the maximum values of acceleration of the six joints are within the acceleration limits of the kinematic constraints in our proposed method. As is known, jerk is an important criterion in trajectory planning. In Figure 20, the jerk result from the classical method has peak values that also exceed the kinematic constraints, which confirms weakness of the classical method. In contrast, the jerk results of the proposed method respect the jerk constraints of the robot.

Table 7 presents the joints’ kinematic indices and the energy consumption for the two methods. The maximum absolute kinematic indices for the proposed method are well within their constraint limits, thus ensuring the safety of the robot during operation.

Table 7.

The joints’ kinematic indices for both methods.

		Number of joint
		J1	J2	J3	J4	J5	J6
Maximum absolute velocity (deg/s)	Proposed method	6.46	9.42	6.08	4.17	6.95	4.69
Maximum absolute velocity (deg/s)	Classic method	5.76	24.4	12.63	5.5	6.05	8.68
Maximum absolute acceleration (deg/s²)	Proposed method	8.32	9.3	8.22	4.52	9.48	4.55
Maximum absolute acceleration (deg/s²)	Classic method	10.08	37.71	23.04	7.09	9.5	13.44
Maximum absolute jerk (deg/s³)	Proposed method	16.15	18.13	16.31	10.04	18.17	9.32
Maximum absolute jerk (deg/s³)	Classic method	24.48	97.94	57.86	15.25	21.21	31.47

Comparing the energy consumption of the two methods shows that the proposed method consumes less energy than the classical method (see Figure 21), where there is reduction of 49.87% energy consumption. Table 8 shows the time, jerk, and energy indices of the trajectory for the two methods. These trajectory indices play a significant role in ensuring productivity, flexibility, and cost-effectiveness of energy resources during the operation of the manipulator.

Table 8.

The indices to optimal trajectory for the two methods.

	The proposed approach	The classical method
The execution time (s)	11.49	11.49
The energy consuming (N.m.s)	30513	60875
The average absolute jerk (deg/s³)	14.69	41.37

6. Conclusion

This study offers a significant view of approaches and real-world methods used for planning the trajectory of industrial robots before their real-time implementation. Execution time, jerk, and energy consumption are taken into account to find an optimal trajectory intended to achieve maximum performance and robot protection by reducing vibrations and energy consumption.

The challenge faced by this study is finding a compromise solution of trade-off between time, jerk, and energy in the absence of an explicit energy expression. First, the robot trajectory planning problem is translated into a problem of multi-objectives optimization which includes the objectives of time, jerk, and energy consumption, while constraints include the robot’s kinematic limits. Second, energy objective model is constructed using a deep learning approach (LSTM). After that, a multi-objective optimization tool called NSGA-II is employed to solve the optimization problem. Finally, our proposed approach is compared with the classical method of chord length distribution with the same execution time to prove the superiority of the proposed approach. However, a significant challenge facing this method is the extensive data requirements for network learning, as is the case with all techniques based on deep learning.

The outline of our future work is as follows:

• The present study considers a scenario in which the robot controller uses a B-spline method for building the trajectory and the trajectory planner has access to the controller for setting optimal parameters. However, most robot manufacturing companies do not provide access to their software. Our next research aim is to identify the method model used for trajectory planning in robots and then use it for optimization.

• We will conduct real-world experiments to examine the practicality and reliability of the proposed approach in real-world scenarios.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Baghdadi Rezali

Benaoumeur Ibari

Mourad Hebali

Redouane Ayad

References

Carabin

Scalera

(2020) On the trajectory planning for energy efficiency in industrial robotic systems. Robotics 9(4): 89.

Chen

Wang

Liu

, et al. (2023) Time-energy-jerk optimal trajectory planning for high-speed parallel manipulator based on quantum-behaved particle swarm optimization algorithm and quintic b-spline. Engineering Applications of Artificial Intelligence 126: 107223. DOI: 10.1016/j.engappai.2023.107223.

De Boor

(1972) On calculating with b-splines. Journal of Approximation Theory 6(1): 50–62. DOI: 10.1016/0021-9045(72)90080-9.

Deb

Pratap

Agarwal

, et al. (2002) A fast and elitist multiobjective genetic algorithm: nsga-ii. IEEE Transactions on Evolutionary Computation 6(2): 182–197. DOI: 10.1109/4235.996017.

Gadaleta

Pellicciari

Berselli

(2019) Optimization of the energy consumption of industrial robots for automatic code generation. Robotics and Computer-Integrated Manufacturing 57: 452–464. DOI: 10.1016/j.rcim.2018.12.020.

Gasparetto

Zanotto

(2007) A new method for smooth trajectory planning of robot manipulators. Mechanism and Machine Theory 42(4): 455–471. DOI: 10.1016/j.mechmachtheory.2006.04.002.

Gasparetto

Zanotto

(2008) A technique for time-jerk optimal planning of robot trajectories. Robotics and Computer-Integrated Manufacturing 24(3): 415–426. DOI: 10.1016/j.rcim.2007.04.001.

Gasparetto

Zanotto

(2010) Optimal trajectory planning for industrial robots. Advances in Engineering Software 41(4): 548–556. DOI: 10.1016/j.advengsoft.2009.11.001.

Gasparetto

Boscariol

Lanzutti

, et al. (2015) Path planning and trajectory planning algorithms: a general overview. In: Motion and Operation Planning of Robotic Systems: Background and Practical Approaches: 3–27. Berlin: Springer. DOI: 10.1007/978-3-319-14705-5_1.

10.

Graves

(2012) Long Short-Term Memory. Berlin, Heidelberg: Springer Berlin Heidelberg, 37–45. DOI: 10.1007/978-3-642-24797-2-4.

11.

Gultekin

Gürel

Taspinar

(2021) Bicriteria scheduling of a material handling robot in an m-machine cell to minimize the energy consumption of the robot and the cycle time. Robotics and Computer-Integrated Manufacturing 72: 102207. DOI: 10.1016/j.rcim.2021.102207.

12.

, et al. (2020) A generic energy prediction model of machine tools using deep learning algorithms. Applied Energy 275: 115402. DOI: 10.1016/j.apenergy.2020.115402.

13.

Huang

, et al. (2018) Optimal time-jerk trajectory planning for industrial robots. Mechanism and Machine Theory 121: 530–544. DOI: 10.1016/j.mechmachtheory.2017.11.006.

14.

Javaid

Haleem

Singh

, et al. (2021) Substantial capabilities of robotics in enhancing industry 4.0 implementation. Cognitive Robotics 1: 58–75. DOI: 10.1016/j.cogr.2021.06.001.

15.

Jiang

Wang

, et al. (2023) Energy consumption prediction and optimization of industrial robots based on lstm. Journal of Manufacturing Systems 70: 137–148. DOI: 10.1016/j.jmsy.2023.07.009.

16.

Kaserer

Gattringer

Müller

(2018) Nearly optimal path following with jerk and torque rate limits using dynamic programming. IEEE Transactions on Robotics 35(2): 521–528. DOI: 10.1109/TRO.2018.2880120.

17.

Kucuk

(2016) Maximal dexterous trajectory generation and cubic spline optimization for fully planar parallel manipulators. Computers & Electrical Engineering 56: 634–647. DOI: 10.1016/j.compeleceng.2016.07.012.

18.

Kunz

Stilman

(2012) Time-optimal trajectory generation for path following with bounded acceleration and velocity. Robotics 1: 209–216. DOI: 10.7551/mitpress/9816.003.0032.

19.

Kyriakopoulos

Saridis

(1988) Minimum jerk path generation. In: Proceedings. 1988 IEEE International Conference on Robotics and Automation. Piscataway: IEEE, 364–369. DOI: 10.1109/ROBOT.1988.12075.

20.

Huang

Chetwynd

(2018) An approach for smooth trajectory planning of high-speed pick-and-place parallel robots using quintic b-splines. Mechanism and Machine Theory 126: 479–490. DOI: 10.1016/j.mechmachtheory.2018.04.026.

21.

Wang

Yang

, et al. (2023) Energy-optimal planning of robot trajectory based on dynamics. Arabian Journal for Science and Engineering 48(3): 3523–3536. DOI: 10.1007/s13369-022-07185-7.

22.

Lin

Mandal

Wibowo

(2024) Bn-lstm-based energy consumption modeling approach for an industrial robot manipulator. Robotics and Computer-Integrated Manufacturing 85: 102629. DOI: 10.1016/j.rcim.2023.102629.

23.

Ding

(2020) Minimum-jerk trajectory planning pertaining to a translational 3-degree-of-freedom parallel manipulator through piecewise quintic polynomials interpolation. Advances in Mechanical Engineering 12(3): 1687814020913667. DOI: 10.1177/1687814020913667.

24.

Oberherber

Gattringer

Müller

(2015) Successive dynamic programming and subsequent spline optimization for smooth time optimal robot path tracking. Mechanical Sciences 6(2): 245–254. DOI: 10.5194/ms-6-245-2015.

25.

Pham

(2014) A general, fast, and robust implementation of the time-optimal path parameterization algorithm. IEEE Transactions on Robotics 30(6): 1533–1540. DOI: 10.1109/TRO.2014.2351113.

26.

Pham

Quang-Cuong

(2017) On the structure of the time-optimal path parameterization problem with third-order constraints. In: 2017 IEEE International Conference on Robotics and Automation (ICRA). Piscataway: IEEE, 679–686. DOI: 10.1109/ICRA.2017.7989084.

27.

Pham

Quang-Cuong

(2018) A new approach to time-optimal path parameterization based on reachability analysis. IEEE Transactions on Robotics 34(3): 645–659. DOI: 10.1109/TRO.2018.2819195.

28.

Piegl

Tiller

(1995) The NURBS Book. 2nd edition. New York, NY, USA: Springer-Verlag. DOI: 10.1007/978-3-642-97385-7.

29.

Rezali

Ibari

Hebali

, et al. (2024) Sensorless robot collision detection based on fuzzy momentum observer. Transactions of the Institute of Measurement and Control 47: 01423312241262538.

30.

Rocha

Freitas

Alemão

, et al. (2021) Event-driven interoperable manufacturing ecosystem for energy consumption monitoring. Energies 14(12): 3620. DOI: 10.3390/en14123620.

31.

Simon

(2004) Data smoothing and interpolation using eighth-order algebraic splines. IEEE Transactions on Signal Processing 52(4): 1136–1144. DOI: 10.1109/TSP.2004.823489.

32.

Soori

Arezoo

Dastres

(2023) Optimization of energy consumption in industrial robots, a review. Cognitive Robotics 3: 142–157. DOI: 10.1016/j.cogr.2023.05.003.

33.

Vidussi

Boscariol

Scalera

, et al. (2021) Local and trajectory-based indexes for task-related energetic performance optimization of robotic manipulators. Journal of Mechanisms and Robotics 13(2): 021018.

34.

Wang

Xie

Chen

, et al. (2023) Path-constrained time-optimal motion planning for robot manipulators with third-order constraints. IEEE 28: 3005–3016. DOI: 10.1109/TMECH.2023.3234584.

35.

Zhao

Zhang

(2021) Optimum time-energy-jerk trajectory planning for serial robotic manipulators by reparameterized quintic nurbs curves. Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science 235(19): 4382–4393. DOI: 10.1177/0954406220969734.

36.

Yan

Zhang

(2021) A transfer-learning based energy consumption modeling method for industrial robots. Journal of Cleaner Production 325: 129299. DOI: 10.1016/j.jclepro.2021.129299.

37.

Zhang

Yan

(2021) A data-driven method for optimizing the energy consumption of industrial robots. Journal of Cleaner Production 285: 124862. DOI: 10.1016/j.jclepro.2020.124862.

38.

Zhang

Sun

, et al. (2021) Time-jerk optimal trajectory planning of hydraulic robotic excavator. Advances in Mechanical Engineering 13(7): 16878140211034611. DOI: 10.1177/16878140211034611.

39.

Zhao

Zhu

Song

(2022) Serial manipulator time-jerk optimal trajectory planning based on hybrid iwoa-pso algorithm. IEEE Access 10: 6592–6604. DOI: 10.1109/ACCESS.2022.3141448.