Review of wheeled mobile robot collision avoidance under unknown environment

APF-based

The APF-based method is first proposed by Khatib, ²⁴ where the robot is assumed to move in a virtual potential field consisting of the attractive potential field and the repulsive potential field. The target is described as an attractive force, and the obstacle is described as a repulsive force. The resultant force is used to decide the next direction of the robot in motion planning.

The APF-based method is simple, convenient to implement. Merely, it is also easy to fall into the local optimum and with many limitations. Many applications of the APF-based method to the mobile robot navigation have been reported. A large number of conference papers in the proceedings are explicitly devoted to application of the improved APF-based method. Aiming at the drawback that the original APF-based method is easy to trap into the local minimum, simulated annealing algorithm is together used with the basic APF method in Park et al. ²⁵ Only when the basic APF falls into the local minimum, the simulated annealing algorithm will come into force. In Shi et al., ²⁶ an improved APF algorithm is proposed for obstacle avoidance of mobile robots by building a new potential force function. The presented method can avoid the robot trapping into the local minimum and mitigate the vibrating problem arising in the original APF. An obstacle avoidance method based on a gravity chain is proposed in Tang et al. ²⁷ In this method, it is supposed that there is a rubber band whose beginning connects with its ending in potential field around obstacle. By putting effective obstacle avoidance information into potential field through a gravity chain, the problem that the original APF-based method often converges to local minima is solved. Different from the basic APF, the potential force does not directly act on the robot. Therefore, the problems that the robot can not reach the target when the obstacle is near the target, as well as oscillation are also solved. Focusing on the non-reachable problem in a situation that the obstacle is near the target, an improved APF is proposed in Li et al. ²⁸ Specially, a regulative agent is introduced into the potential field. The attraction will be reduced as a linear function when the robot is close to the target, the repulsion is decreased as a higher-order function.

The original APF-based method is devoted to give a feasible collision-free path from the source to the target. To generate the shortest/approximately shortest path, a regression search method is developed in Li et al. ²⁹ to optimize the path planned by the APF-based method. Potential functions are redefined, virtual local target and repulsive force disappearance as well as circle tangential line of circle are utilized to eliminate oscillations, local minima, and non-reachable problems. In this paper, environmental information is known completely.

Extended to multiple robots from a mobile robot, the study ³⁰ considers the formation control problem of a group of non-holonomic mobile robots where every WMR moves along the predefined trajectory while maintaining a geometric formation in a 2D environment. The APF method is applied to bypass the collision with the obstacles before the robots reach the goal point. To avoid falling into local minimum, rotating fields are defined around obstacles. For a rectangular obstacle, an ellipse is defined around it whose field matches the direction of approaching robot. However, whether the collision avoidance refers to the collision with the environmental obstacles or with other team members working in a shared environment is not mentioned clearly in this paper. In Yang et al., ³¹ leader-follower-based formation control problem with collision, obstacle avoidance, and connectivity maintenance is considered for a class of second-order nonlinear multi-agent systems under external disturbances, where the APF method is used to achieve collision, obstacle avoidance. Attractive field function takes formation maintenance problem of multi-agents during they avoid obstacles into account. Instead of dealing with nonlinear dynamics of agents and environmental disturbances separately, in this paper, neural network technology is employed to approximate both collectively by treating nonlinear dynamics and unknown disturbances as a dynamics set. In this way, triangle formation maintenance and collision-free motion planning are achieved, and the system is robust to external disturbances.

Among most works related to mobile robot obstacle avoidance based on the APF, the position of the obstacle is usually assumed to be fixed. For dynamic obstacles, the concept of collision time is introduced into the potential function in Li et al., ³² which takes the relative velocity of the obstacle to the robot and the collision angle into account. An artificial neural network is used to predict the position and velocity information of the obstacle, which is trained by previous positions of the obstacle.

Table 2 gives comparison between the above-mentioned APF-based literatures. In summary, the inherent drawbacks of the basic APF method are mainly improved by researchers from two aspects: (1) designing a new potential field function; (2) integrating it with other advanced methods. For the first method, different scenarios usually need to design different potential field functions. Apart from solving the problem of local optimum and oscillation arising in the traditional APF method by improving the potential field function, control algorithms that are based on a quantity called danger field are proposed to achieve safety control for manipulators in a human-robot interaction environment in Kulić and Croft ³³ and Lacevic et al. ³⁴ The main principle behind the danger field-based approach is to reduce the danger index during the robot motion to ensure safety by generalizing several danger indices into the robot safety-assessment ³⁴ considers the kinematic state of the manipulator as a whole and indicates how dangerous the current posture and velocity of the robot are to the objects in the environment in comparison with Kulić and Croft. ³³ Except the danger field based on field concept, another field method, that is, artificial coordinating field method,^35,36 has also been proposed and has been found its application in safety planning of mobile robots.

OPEN IN VIEWER

Table 2.

Comparison between the above-mentioned APF-based literatures.

Literatures	Problems to be solved	WMRs number	Methods	Advantages	Improvements
Park et al. 25	Navigation	Single	Simulated annealing &0x002B; the APF	Improve performance of the original APF that is easy to trap into the local minimum	Is a feasible path
Shi et al. 26	Navigation	Single	Improve potential function	Improve the vibrating problem	Is a feasible path
Tang et al. 27	Navigation	Single	Introduce gravity chain concept into the APF	Improve the target non-reachable problem	Is a feasible path
Li et al. 29	Navigation	Single	Regression search &0x002B; the APF	Is the shortest path from source to target	Environmental information is known completely.
Azhar and Bilal Kadri 30	Formation	Multiple	Introduce rotating fields into the APF	Improve local minimum performance	Not clearly mention collision avoidance objects
Yang et al. 31	Formation	Multiple	Improve potential function	Consider connectivity maintenance during collision avoidance	Not inter-robots consider collision avoidance
Li et al. 32	Navigation	Single	Introduce collision time concept into the APF	Avoid dynamic obstacles	Only consider single WMR

Population-involved meta heuristic-based

The APF-based method shows weakness in a dynamic environment with moving obstacles. Intelligent obstacle avoidance methods such as GA-based, neural network-based, fuzzy logic-based are receiving compelling attention to achieve collision-free of mobile robots in a dynamic environment. Population-based meta-heuristic search algorithms, inspired by the social behavior of nature, are of great concern for a long time. They are with several advantages: for example, the implement is simple and easy, and the algorithm can bypass local optimum, the shortest path generation, etc. ³⁷ Motivated by their advantages, nowadays population-based meta-heuristic algorithms have been found their application in the collision-free motion planning problem of mobile robots.

A fixed solution length bit sequence is adopted in the traditional GA. In Kala et al., ³⁸ representation of a graphical node that is considered as a chromosome is proposed, which is effective for all sorts of highly chaotic conditions with multiple obstacles. In Elhoseny et al., ³⁹ an GA-based dynamic path planning method is proposed where the environment is monitored by a wireless sensor network, the movement path of the robot is updated by receiving a signal from a base station based on alerts that are periodically triggered by sensors. Bezier Curve is used to refine the final path based on the control points identified by the GA-based dynamic path planning method. Apart from avoiding the environmental obstacles successfully, the method reduces the time required to get the optimum path by 6% up to 47%. The path smoothness is also improved in the range of 8% and 52% based on the reported results.

A chaos-GA-integrated hybrid algorithm with adaptive and floating-point code is proposed in Gao et al., ⁴⁰ and is applied to the path planning of a mobile robot. The hybrid algorithm has higher accuracy and faster convergence speed than the basic GA, and the generated path is the shortest path from the source to the target. In hybrid GA proposed in Zhang et al., ⁴¹ the grid method is used to model the working environment of a mobile robot, the digital potential field is used to generate initial path population, and different fitness functions of feasible and unfeasible paths are also adopted, accelerating the convergence of the algorithm and improving the accuracy. The deleted and inserted operators are added in the GA, achieving the requirement of collision avoidance with the obstacle. An improved GA-based path planning scheme is proposed in Shi and Cui ⁴² for collision-free navigation of a mobile robot under unknown environment. Compared to the conventional GA, the real coding with low computational complexity and the reduced search space, a fitness function considering the collision avoidance path, the shortest distance, and smoothness of the path, specific genetic operators are devised in the improved GA. The presented GA is verified that it is effective under various complex and dynamic environments.

PSO-based collision-free motion planning algorithm for single robot in a rough terrain environment is proposed in Wang et al. ⁴³ In the proposed algorithm, a crowding radius-based position updating method is adopted in the global optimal position updating formulation, and the non-uniformity factor is used to update the position of particles when the path collides with obstacles. This way contributes to the efficiency of the original PSO method and population diversity. However, the simulation test is conducted in a static and known rough terrain environment. In Dadgar et al., ⁴⁴ and Das et al. ⁴⁵ PSO is used for collision avoidance motion planning of multiple mobile robots. In Dadgar et al., ⁴⁴ PSO algorithm is employed in multiple robots target searching optimization task performing in an unknown environment. The algorithm has good performance for small robot population and single-target searching task. The multi-targets searching problem is not involved in this study ⁴⁵ proposes a hybrid optimization algorithm (IPSO-DV) that combines an improved PSO (IPSO) with differentially perturbed velocity (DV), to determine an optimal path for multi-robots in a clutter environment. The main idea of the IPSO-DV algorithm is to adjust the velocities of the particles in IPSO with a vector differential operator borrowed from DE family. In this study, the environment and obstacles are static, the dynamic environmental obstacles are not considered. Except GA, PSO, a combination of these algorithms or a combination of them and other obstacle avoidance algorithms have also been applied to the mobile robot navigation. An overview of application of these algorithms in mobile robot navigation can be found in Pandey. ⁴⁶

In above-mentioned biological inspired algorithms, a great number of particles are usually needed to generate the shortest path. Beetle antennae search (BAS) algorithm, ⁴⁷ is inspired by the foraging principle of beetle, incrementally arises as a promising solution. The algorithm is of great concern due to its fast convergence speed, and it only need a particle in the optimal path generation. In Khan et al., ⁴⁸ BAS-based control strategy is proposed for simultaneous tracking control and obstacle avoidance of a redundant manipulator, where the tracking control and obstacle avoidance problem is unified as a single optimization function by a penalty term. Utilizing the BAS algorithm, the KUKA LBR IIWA-14 seven degree of freedom (DOF) manipulator successfully avoids an arbitrarily shaped environmental obstacle in front of it while following the desired path. However, there is no application of this method in obstacle avoidance motion planning of mobile robot.

Neural network-based

Artificial Neural network is a kind of nonlinear computing model composed of a large number of inter-connected neurons, which can model the complex relationship between input and output variables. The neural network-based path planning method is usually to establish a neural network model related to the movement path of the robot from the source to the target. The input of the model usually is sensor information, previous position of the robot, or the previous movement direction. Commands of next position and direction are output by training the model. Neural networks are widely used in control of uncertain systems due to their high approximation capabilities. ⁴⁹

An application of ANN in mobile robot navigation under dynamic unknown environment is also achieved in Kala et al. ³⁸ It is concluded that compared to the ANN, the GA takes more time in terms of path generation. For the ANN-based path planning method, the overall path number traveled by the robot is less compared to the GA-based method. However, the ANN method will fail in a highly chaotic environment. Merely, the memory requirement for the GA is very high in comparison with the ANN.

A backward ANN is applied to the movement control of a mobile robot under a dynamically changing environment with the moving obstacle in Zarate et al. ⁵⁰ The ANN structure considers the past position and the future position at the same time. Past positions provide the ANN with memories of the mobile robot previous positions. Future position provides the ANN with a goal that the robot would go next. The advantage is that the robot can adaptively predict the next coordinates.

In Hu et al., ⁵¹ pedestrian collision avoidance problem is considered during the mobile robot navigation. Two deep neural networks are used to achieve pedestrian avoidance and path following tasks respectively. Neural networks are trained with images labeled with movement decisions. The training process is end-to-end, and requires less time in labeling. For image labeling, computer vision-based labeling technology and a monocular RGB camera based one are used for comparison. In terms of static obstacles avoidance, ultrasonic sensor is used. Experiments results verify that the deep learning method using a RGB camera is more robust compared to the computer vision-based.

In Guan et al., ⁵² an interval type-2 fuzzy neural network is designed for the wheeled mobile robot to achieve obstacle avoidance smoothly and position stabilization. In the presented method, membership functions are redefined through the addition of the uncertain means and standard deviation, and fuzzy sets are used as membership values, reducing the effect of uncertainties. An WMR can follow a shorter path from the source to the target and achieve smoother movement during obstacle avoidance based on the presented neural network method.

In Huang et al., ⁵³ a neural network-based Q-learning architecture is developed for an autonomous mobile robot. Q-learning algorithm is used to learn obstacle avoidance experience online through samples collecting from interaction with the real environment. A three-layer Back-Propagation neural network trained by the error back propagation algorithm is used to store the Q values. The presented method does not need the complete knowledge around the external environment and can learn online. Therefore, the robot can adaptively tune itself behavior to react complex, unknown, and dynamic working environment.

A modified pulse-coupled neural network method is proposed in Qu et al. ⁵⁴ to generate a collision-free path for a mobile robot under a dynamic environment. The proposed neural network is topologically organized with only local lateral connections among neurons. The method requires no prior knowledge of target or obstacles, and can give the shortest path from the source to the target. The computational complexity of the proposed algorithm is only related to the length of the generated shortest path. However, a weakness is that global knowledge of the current environment is assumed to be available, which is not very realistic in real applications.

In Wang et al., ⁵⁵ a distance-based spiking neural network (SNN) behavior controller is designed for WMRs using ultrasonic sensory information, where unsupervised spike-based Hebbian learning algorithm is used to train the SNN. Compared with the classical NNs, the SNN has better robustness to noise and is easy to model. Merely, due to its output is pulses, the gradient-decent-based learning rules cannot be employed to SNN directly. In this study, SNN-based controller achieves collision avoidance motion planning with less neurons compared to the traditional NNs. The SNN is further extended to mobile robot navigation solution in Wang et al., ⁵⁶ a behavior-based modular navigation controller is proposed. The controller does not need accurate the environment information, and is suitable to unknown and unstructured environments.

Above-mentioned works focus on the nonlinear control of single mobile robot. In Li et al., ⁵⁷ a neural-network-based approach is proposed for a multi-robots system with moving obstacles. Simulation results show that the developed intelligent controller is able to generate collision-free paths for multiple robots in a workspace with moving obstacles. Leader-follower based formation control problem of multiple mobile robots is considered in Dierks and Jagannathan. ⁵⁸ Treating other robots in a shared environment as obstacles, the whole formation is also asymptotically stable during the obstacle avoidance by applying robust integral of the sign of the error method. The region reaching formation control problem for multi-robot systems is considered in Yu et al., ⁵⁹ neural networks are trained online, and are use to approximate the robotic dynamics model uncertainties and external disturbances, without requiring any preliminary off-line training. A feed-forward neural network is used to learn the unknown dynamics. An adaptive control gain law combined with the RBF neural network is derived, which is used to adjust the weight of the control task. The developed NNs-based control scheme is a distributed control strategy, the desired formation does not entirely depend on the objective region and effect caused by the model uncertainties and external disturbances can be suppressed by the designed robust compensator. The formation control solution of multi-agent system in the presence of heterogeneous communication delays is discussed in Guo et al. ⁶⁰ A continuous repulsive APF is incorporated into agents’ velocities to avoid collision. The radial basis function neural networks (RBFNNs) based adaptive control method is proposed to ensure the robustness against model uncertainties, disturbances, and communication delays. In Yu et al., ⁵⁹ and Guo et al. ⁶⁰ illustrating example only gives the simulation result corresponding to the fixed environmental obstacle. In Wang et al., ⁶¹ reinforcement learning based duel neural network is employed to control multi-robot coordination behavior. Only image as input, the neural network is trained by learning the actions of each robot. The end-to-end deep reinforcement learning is used, where negative rewards are set if collisions between multiple robots and obstacles occur. If the robot does not collide or that the robot reaches the target point, the reward will be positive. Reinforcement learning algorithms are trained based on robots’ own experience, unlike supervised learning that learn from human expert knowledge. Moreover, a great quanlity of dataset samples are usually required for supervised learning method. The combination of reinforcement learning and the deep neural network shows a powerful performance, such as well-known AlphaGo ⁶² and AlphaGo Zero ⁶³ that win the human Go champion in Go games. Except reinforcement learning, the combination of broad learning and neural network is also showing a potentiality.^64,65

Neural network model can learn, and model linear, nonlinear and complex relationships. In the last years, robot skill learning is receiving the compelling attention. Through the accumulation of prior knowledge, robots are expected to generate new knowledge and experience so that it can autonomously make decisions or obtain behavior guidance in a complex environment. There are a lot of related works that have been reported, for example,^66–70 Neural network-based robot skill learning will be an important development direction of robots in the future. In addition, among NN-based methods, a fact that a quantity of data need to be trained is also indisputable.

Fuzzy logic-based

The concept of FL is introduced by Zadeh, ⁷¹ which is inspired by human reasoning. Fuzzy systems can handle uncertainty and imprecise information using linguistic rules. FL-based method provides an effective way for obstacle avoidance, which contains a fuzzy rule base that is constructed and tuned by a human expert. Fuzzy systems can handle uncertainty and imprecise information using linguistic rules. Fuzzy logic control is widely used for mobile robot navigation, this is mainly due to it can offer inference using environmental data, even under motion and sensor uncertainties. ⁷²

In Li and Yang, ⁷³ a vision-based landmark recognition system is developed for mobile robot navigation. A GA-based search method for pattern recognition of digital images is proposed to recognize artificial landmarks by searching all the predefined patterns. A combination of eight ultrasonic sensors is designed to achieve collision-free through a set of fuzzy rules. An FL-based online robot navigation method is investigated in Faisal et al. ⁷⁴ for mobile robots under a unknown dynamic environment. The authors introduces how to use two fuzzy logic controllers for achieving the target tracking control and collision avoidance of robots, respectively.

WMRs usually work a dynamic and unknown environment. The uncertainties of environment and the insufficient information on the environment impose challenges on robotic control. To addressing these challenges, controller design that combines both learning with reasoning abilities becomes popular in the few years. Neuro-fuzzy controllers integrate the advantages of neural network and fuzzy logic control. ⁶⁴

Behavior-based control approach for mobile robots navigation refers to that a complex navigation task is decomposed as several behaviors which are easy to design and perform. In Song and Lin, ⁷² the robot navigation task is decomposed as three primary behaviors that are obstacle avoidance, wall following, and goal seeking. These three behaviors are implemented by using fuzzy logic methods. A neural network architecture is proposed for determining the fuzzy weight of every behavior. Neural network is used to map the environmental information sensed by sensors to choose suitable fusion weights. However, it is observed from the experimental results that the robot does not reach the desired goal position with some errors.

In Wen, ⁷⁵ Elman neural network and fuzzy logic integrated control algorithm is proposed to enhance the performance of the robot in obstacle avoidance. A virtual force field is built between the robot and obstacles by the hybrid force/position algorithm. Using Elman neural network to compensate uncertainty effect of the dynamic model of the WMA, and which is integrated with the fuzzy control method to self-tune the exact distance between the WMR and the obstacle online. Further application of the method proposed in Wen ⁷⁵ can be found in Yang et al. ⁷⁶ In Zhu and Yang, ⁷⁷ neuro fuzzy-based method is integrated for the WMR under unknown environment. A fuzzy logic system with both target tracking and obstacle avoidance performances is designed. Two neural network-based learning algorithms are developed, they are used to tune the parameters of membership functions and suppress redundant fuzzy rules, respectively. A neuro-fuzzy system with mixed supervised learning and reinforcement learning usage is proposed in Ye et al. ⁷⁸ Supervised learning method is used to determine the membership functions for the input and output variables simultaneously. After training, reinforcement learning algorithm is employed to fine-tune the membership functions for the output variables. The learning ability is enhanced by improving Sutton and Barto’s model. A virtual environment simulator is developed to obtain data that is used to train the neural fuzzy system.

In these above-mentioned works, they only reconsidered collision avoidance with environmental obstacles. In Pandey, ⁷⁹ the safe-navigation problem of multiple WMRs is investigated under unknown cluttered environment. An adaptive neuro-fuzzy control system integrated ANN and fuzzy logic technology is developed, in which the front, left, and right obstacle distances are chosen as control input. The steering angle is determined as the control variable to control the movement of the robot. Under the proposed control system, the generated path for the robot is smooth and optimal, and every robot could achieve the environmental obstacle avoidance, robot inner-collision avoidance, and the goal seeking behaviors. Literature ⁸⁰ proposes a hybrid rule-based-neuro-fuzzy controller with rules incorporating the desired motion and direction, distances between WMRs and obstacles/targets. The collision-free during motion planning is achieved by combining a repelling influence between WMRs and nearby obstacles with an attracting influence between WMRs and targets. As control variables, appropriate heading angle is output. The proposed controller improves navigation performance of multiple WMRs in complex and unknown environments. In Pradhan et al., ⁸¹ FL-based navigation performance is investigated for WMRs as many as 1000 in a totally unknown environment. For comparison, fuzzy logic controllers using different membership functions are developed. A conclusion is obtained that controller with Gaussian membership function is most efficient for WMRs navigation. For Pandey, ⁷⁹ and Parhi et al. ⁸⁰ the heading angle is used to as control variable. However, the considered environmental obstacle is static in Pandey, ⁷⁹ and every robot independently moves along the generated traveling path without considering the communication between robots. For online interaction, connectivity problem of multiple WMRs network should be addressed.

Constrained optimization-based

Recently, some scholars try to address the problem of mobile robot motion control from the perspective of constraint optimization.^82,83 The basic idea of this class of method can be described as: the collision avoidance strategy is formulated as an inequality equation that is related to the safety threshold,^84–86 which is treated as a constraint and is attached to the robot’s kinematic control scheme. By describing the control scheme as a quadratic programming (QP) minimization problem, the optimization technology is then used to resolve it online. An obvious advantages of this method is that multiple control goals such as collision avoidance, robot inherent physical limits, target tracking, and so on, can be achieved simultaneously. Physical constraints compliance can further ensure the safety and system stability of robots.

At first, this method was mainly used to address the redundant resolution problem of the robot manipulator.^4,83,87–89 In the reported works, multi-objectives integrated hybrid tasks are usually unified as a time-varying QP minimization optimization problem, where the inverse kinematics of the manipulator is described as an equality equation, and the obstacle avoidance problem is formulated as an equality or inequality equation which is universally solved in velocity level or acceleration level. In references,^{22,84–86,90–93} the obstacle avoidance problem is formulated as an inequality constraints. An obstacle avoidance scheme that is formulated as an equality constraint is proposed in Tang et al. ⁹⁴ In their works, basic idea of the obstacle avoidance strategy is that the distance between the manipulator and an obstacle is described as point-to-point distance based on the mathematical geometric. By ensuring that the distance vector keeps outside a safety threshold, the collision-free is guaranteed. Obstacle avoidance inequality is solved in velocity level in references^22,84,85 compared to references^86,91–93 that are solved in acceleration level. In Guo and Zhang,^85,86 and Chen and Zhang ⁹⁰ inner and outer safe thresholds are considered. In addition, the manipulator is simplified as a set of points by uniformly choosing point in links of the manipulator in Xu et al. ²² For this method, one possibility is that the chosen point does not collide with the obstacle, in fact the nearest component on the manipulator to the obstacle has collided with the obstacle. To overcome this drawback, ⁹¹ gives an improved obstacle avoidance scheme which can return the nearest point on the manipulator to the obstacle by utilizing vector relations between the geometric elements.

The above mentioned works only consider collision avoidance between the single manipulator and the environmental obstacles. Extending from single manipulator to multiple manipulator increases the computational complexity of the control algorithm. How to communicate each other and share information in a shared environment imposes a challenge on robotic control. QP-based cooperative kinematic control problem of multiple redundant manipulator in a shared workspace are investigated in references.^4,89,95–97 To mitigate influence of environment inference or signal loss on accuracy of the desired task achieved by the manipulator, control schemes with inherent noise-resistant are introduced in references.^82,98,99 However, there are few works that are devoted to collision avoidance in QP-based multiple manipulators cooperative control.

Chen and Zhang ⁸² and Zhang et al. ⁸³ extend this method to motion control of mobile robot manipulator. In Zhang et al., ¹⁰⁰ mutual collision avoidance of dual redundant robot manipulators is considered. In summary, application of QP-based method in multiple mobile robots have rarely developed and need the further investigation. In Li et al., ¹⁰¹ extend inequality-based collision avoidance method to multiple WMRs collision-free path following at the first time, which considered WMRs’ mutual collision avoidance except static or dynamic environmental obstacles. In above works introduced in this part, different technologies based neural network model are constructed for solving the resultant unified QP problem. Comparison on works based on constrained optimization method is shown in Table 3. Although they are effective, some parameters in controllers need to be adjusted manually based on the experimental requirement. How to set these parameters to derive the optimal performance or weaken effect of them on system performance are underway.

OPEN IN VIEWER

Table 3.

Comparison on works based on constrained optimization method.

Literatures	Robots type	Robots number	Collision avoidance	Problem description	Level	Mathematical formulation
Xu et al. 22 Zhang and Wang84,a	Manipulator	Single	Yes	Inequality	Velocity	Optimization
Zhang et al. 83	Mobile Manipulator	Single	Yes	Inequality	Velocity	Optimization
Guo and Zhang86,b	Manipulator	Single	Yes	Inequality	Acceleration	Optimization
Li et al. 89	Manipulator	Multiple	No	Null	Null	Game-theory
Zhao et al.91,b	Manipulator	Single	Yes	Inequality	Acceleration	Optimization
Tang et al. 94	Manipulator	Single	Yes	Equality	Velocity	Optimization
Li et al. 95 and Jin et al.96,97	Manipulator	Multiple	No	Null	Null	Optimization
Li et al.101,c	WMRs	Multiple	Yes	Inequality	Velocity	Optimization

a

The difference between Xu et al. ²² and Zhang and Wang ⁸⁴ is that the constructed controller. Lagrangian-based controller and dual neural network controller are proposed in Xu et al. ²² and Wang ⁸⁴ respectively.

b

In Guo and Zhang, ⁸⁶ the inner and outer safe threshold is set. Literature ⁹¹ gives the nearest point’s coordinate of the manipulator distance from the obstacle.

c

This is the first paper extended inequality-based collision avoidance method to multiple WMRs collision-free path following.