 Research
 Open Access
Realtime obstacle avoidance of mobile robots using statedependent Riccati equation approach
 Seyyed Mohammad Hosseini Rostami^{1},
 Arun Kumar Sangaiah^{2},
 Jin Wang^{3} and
 Hyejin Kim^{4}Email author
https://doi.org/10.1186/s1364001803191
© The Author(s). 2018
 Received: 30 May 2018
 Accepted: 13 August 2018
 Published: 29 August 2018
Abstract
In this paper, statedependent Riccati equation (SDRE) methodbased optimal control technique is applied to a robot. In recent years, issues associated with the robotics have become one of the developing fields of research. Accordingly, intelligent robots have been embraced greatly; however, control and navigation of those robots are not easy tasks as collision avoidance of stationary obstacles to doing a safe routing has to be taken care of. A moving robot in a certain time has to reach the specified goals. The robot in each time step needs to identify criteria such as velocity, safety, environment, and distance in respect to defined goals and then calculate the proper control strategy. Moreover, getting information associated with the environment to avoid obstacles, do the optimal routing, and identify the environment is necessary. The robot must intelligently perceive and act using adequate algorithms to manage required control and navigation issues. In this paper, smart navigation of a mobile robot in an environment with certain stationary obstacles (known to the robot) and optimal routing through Riccati equation depending on SDRE is considered. This approach enables the robot to do the optimal path planning in static environments. In the end, the answer SDRE controller with the answer linear quadratic controller will be compared. The results show that the proposed SDRE strategy leads to an efficient control law by which the robot avoids obstacles and moves from an arbitrary initial point × 0 to a target point. The robust performance of SDRE method for a robot to avoid obstacles and reach the target is demonstrated via simulations and experiments. Simulations are done using MATLAB software.
Keywords
 Optimal control
 Nonlinear control
 Obstacle avoidance
 SDRE
1 Introduction
A mobile intelligent robot is a useful tool which can lead to the target and at the same time avoid an obstacle when faced with it. Obstacle avoidance means that the robot avoids colliding with obstacles such as fixed objects or moving objects. So when a robot encounters with an obstacle, it must decide to avoid it and at the same time, consider the most efficient path to the target with a good decision; this decision is performed in this research using the statedependent Riccati equation (SDRE) control method. The optimal control of nonlinear systems cannot be done similar to the methods of linear systems. One of the significant viewpoints for optimal control of nonlinear systems is using the SDRE. This method in addition to stability creates a proper functioning and robust for a wide range of nonlinear systems. Nonlinear optimal controller SDRE is a developed linear optimal control LQR, in which equations Riccati is statedependent. Strategy SDRE has been introduced in the past decade. This strategy is a very effective algorithm for feedback nonlinear analysis that its states are nonlinear; however, a flexibility idea of through the weight matrix is dependent on the state. This involves factoring (ie, parameterization) of nonlinear dynamics to state vector and produces a value function for the matrix which is dependent on its state.
There are conventional methods of obstacle avoidance such as the path planning method [1], the navigation function method [2], and the optimal regulator [3]. Hence, an SDRE regulator [4, 5] is used for this paper. In recent years, many researchers have investigated the obstacle avoidance problem from different perspectives. SDRE technique developed as a design method, which provides a systematic and effective design of nonlinear controllers, filters, and observers. Because of its versatile features, SDRE is broadly used for different cases [6–14]. SDRE can also be used in the field of medicine, for example, in [15, 16] the presentation of the optimal chemotherapy protocol for cancer treatment, considering the metastasis, has used the full optimal SDRE feedback. In [17], the SDRE algorithm is studied on motion design of the cablesuspended robot with uncertainties and moving obstacles. A method for controlling the tracing of a robot has been developed by the SDRE. Vibration control of flexiblelink manipulator in [18] is used in SDRE controller and Kalman filtering. The problem of estimating flexural states during the use of the SDRE for flexible control is the focus of this paper. Wanga et al. [19] present a novel H_{2} − H_{∞}SDRE control approach with the purpose of providing a more effective control design framework for continuoustime nonlinear systems to achieve a mixed nonlinear quadratic regulator and H_{∞}control performance criteria.
It should be noted that the algorithms discussed for avoiding an obstacle are different from each other, and each algorithm has proposed a separate and new method for avoiding obstacles. So far, novel methods to avoid obstacles in addition to the mentioned methods are also presented [20–23].
This paper is organized as follows: a description of the SDRE formulation is given in Section 2. Then, motion equations of the robot and its constraints are presented in Section 3. In Section 4, SDRE formulation of the robot is derived. And simulation results have been reported in Section 5. Finally, in Section 6, some conclusions are drawn from this research.
2 Methods
2.1 Problem formulation
Consider the above equation which is nonquadratic in x but quadratic in u. The state and input weighting matrices are assumed statedependent such that Q : R^{n} → R^{n × n} and R : R^{n} → R^{n × m}. These design parameters satisfy Q(x) ≥ 0 and R(x) > 0 for all x [24].
2.2 SDRE nonlinear regulator problem
SDC form is not unique and should be chosen in such a way that \( \left\{{Q}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}(x),A(x)\right\} \) and {A(x), B(x)} are pointwise observable and pointwise stabilizable, respectively.
The response obtained is locally asymptotically stable and is optimal [25].
2.3 SDRE nonlinear tracking problem
3 The fourwheeled car
3.1 Vehicle’s constraints
So far, dynamic equations of the robot and corresponding constraints on states and control input have been presented. In the following section, the proper formulation for utilizing SDRE approach for robot motion control will be presented.
4 Implementation of SDRE controller for the robot
4.1 SDC parameterization
4.2 Obstacle avoidance
In this paper, the concept of the APF method and the navigation function [29] is used to avoid collisions with obstacles during the robot’s movement. The navigation function that is similar to the APF, with the help of the sensor, will detect obstacles and avoid them and converge the robot toward the target. This function has the local minimum at the destination and a local maximum at the obstacles.
5 Results and discussion
Figure 2 shows the mesh plot of the obstacle avoidance term in which the function has the global minimum at the destination and a local maximum at the obstacle. It shows that the obstacles like the summit and the target act like the cavity and absorbs the robot into the inside itself.
6 Conclusions
This paper focuses on the SDRE nonlinear regulator for solving the nonlinear optimal control problems. The existence of solutions as well as optimality and stability properties associated with SDRE controllers are the main of this paper. The paper is organized as follows. In Section 2, the formulation of the nonlinear optimal control problem, the concept of extended linearization and the SDRE controller for nonlinear optimal regulation are presented, then the additional degrees of freedom provided by the nonuniqueness of the SDC parameterization is reviewed. The necessary and sufficient conditions on the existence of solutions to the nonlinear optimal control problem, in particular, by SDRE feedback control, are reviewed. A theoretical study on the stability and optimality properties of SDRE feedback controls is pursued. SDRE method is compared with LQR method. It is concluded for that SDRE is a semilinear method, give the best answer than linear methods for nonlinear systems, because the basis of this method is that takes nonlinearity of the system and creates the nonlinear system that its statedependent coefficient matrix structure has semilinear and minimum nonlinear performance index with semiquadratic structure. The result is that contrary to controllers like LQR that first, they linearize nonlinear controllers, then the laws are designed for stability that can cause to remove some of the important elements of nonlinear systems that have a key role in system stability. In the method of SDRE, the important elements are not removed from the nonlinear system and as a result, according to the results of the various articles, it is concluded that the system is robust against disturbances and uncertainties and achieves better performance than LQR. Our target in this research is finding an equation control using the method SDRE for routing a robot and avoid collisions with obstacles and path optimization for the robot to reach the target. In the end, the proper suggestion for each robot’s path planning is to create an optimal combination algorithm according to the specific structure of each robot so that each algorithm can be covered the constraints of the other algorithm.
Abbreviations
 LQR:

Linear quadratic regulator
 SDC:

Statedependent coefficient
 SDRE:

Statedependent Riccati equation
Notation
 R:

Weighting matrix inputs
 Q:

Weighting matrix of states
 W:

Matrix final cost
 J:

Cost function
 u:

Control signal
Declarations
Acknowledgements
Authors of this would like to express the sincere thanks to the National Natural Science Foundation of China for their funding support to carry out this project.
Funding
This work is supported by the National Natural Science Foundation of China (61772454, 6171171570).
Availability of data and materials
The data is available on request with prior concern to the first author of this paper.
Authors’ contributions
All the authors conceived the idea, developed the method, and conducted the experiment. SMHR contributed to the formulation of methodology and experiments. AKS contributed in the data analysis and performance analysis. JW contributed to the overview of the proposed approach and decision analysis. HjK contributed to the algorithm design and data sources. All authors read and approved the final manuscript.
Ethics approval and consent to participate
All the authors of this manuscript would like to declare that they mutually agreed no conflict of interest. All the results of this paper are with respect to the experiments on human subject.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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