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SCIENCE CHINA Physics, Mechanics & Astronomy

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*Corresponding author (email: [email protected])

• Article • September 2014 Vol. 57 No. 9: 1771–1777

doi: 10.1007/s11433-013-5244-9

Levenberg-Marquardt based artificial physics method for mobile robot oscillation alleviation

ZHANG XiangYin1, DUAN HaiBin1,2* & LUO QiNan2

1 State Key Laboratory of Virtual Reality Technology and Systems, Beihang University, Beijing 100191, China; 2 Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University,

Beijing 100191, China;

Received December 28, 2012; accepted April 15, 2013; published online July 3, 2014

This paper proposed a modified artificial physics (AP) method to solve the autonomous navigation problem for mobile robots in complex environments. The basic AP method tends to cause oscillations in the presence of obstacles and in narrow passages, which can result in time consumption. To alleviate oscillation, we modified the AP method using the Levenbery-Marquardt (LM) algorithm. In the modified AP method, we altered the original directions of AP forces to the Newton direction, and adjust the parameter by the LM algorithm. A series of comparative experimental results show that the modified AP method can achieve smoother trajectories with less time consumption. This demonstrates the feasibility and effectiveness of our proposed approach.

artificial physics (physicomimetics), Levenbery-Marquardt (LM) algorithm, autonomous navigation, obstacle avoid-ance

PACS number(s): 89.20.-a, 05.65.+b, 45.40.Ln, 45.20.D-

Citation: Zhang X Y, Duan H B, Luo Q N. Levenberg-Marquardt based artificial physics method for mobile robot oscillation alleviation. Sci China-Phys Mech Astron, 2014, 57: 17711777, doi: 10.1007/s11433-013-5244-9

1 Introduction

Artificial physics (AP), also named physicomimetics, is a novel framework to design and build rapidly deployable, scalable, adaptive, cost-effective, and robust networks of autonomous distributed vehicles [1]. Recent research thrusts suggest strongly the close connection between physics and distributed control. Using this development of alternative distributed forms of computing based on nature, such as quantum computing, molecular computing and DNA com-puting [2,3], Spears et al. [4] proposed AP force-based method for distributed control in 1999, which is originally utilized to control multi-agent. Although AP method is mo-tivated by natural physical forces, it is not confined to them

[5]. In this paper, we investigate the application of AP method to the navigation problem for mobile robots moving toward a goal through obstacles.

Over the past few years, developmental robotics has ex-tended from morphogenetic robotics, which is primarily concerned with the physical development of the body and neural control, to epigenetic robotics, which focuses on the cognitive and mental development [6,7]. As to the former, there has been a considerable amount of research interest in the mobile robot research because of the characteristics of terrain movement [8–10]. An existing approach to this problem is the well-known artificial potential field method first introduced by Khatib [11], which has been widely used in mobile robot navigation for its simplicity and elegance. In most cases, the navigation function of mobile robots is a complex nonlinear function. When an environment that is

1772 Zhang X Y, et al. Sci China-Phys Mech Astron September (2014) Vol. 57 No. 9

totally or partially unknown or even dynamically changing, local minima are usually encountered, where the mobile robot is trapped and cannot continue movement [12]. Moreover, artificial potential field-based methods tend to result in oscillations when vehicles move in a long narrow passage or near obstacles [13]. To overcome the drawbacks of artificial potential field method, Jing et al. [12] defined a special coordinating force and added to the conventional artificial potential field. Ren et al. [13,14] introduced the modified Newtonian method to improve the potential filed method from an optimization viewpoint.

According to the optimization theory, the Levenberg- Marquardt (LM) algorithm is an iterative technique that locates the minimum of a multivariate function and has be-come a standard technique for nonlinear least-squares prob-lems [15]. In this paper we apply the LM algorithm to mod-ify the basic AP method. A series of comparative simulation results show that the modified AP method can greatly alle-viate oscillations and achieve smoother trajectories.

2 AP method and mobile robot navigation

2.1 Basic AP framework

Virtual physics forces drive self-organize swarms of mobile robots to a desired configuration or state [4]. The desired configuration is a stable and equilibrium state that minimiz-es overall system potential energy. Although the forces are virtual, mobile robots act as if they really do exist [4,5]. Each mobile robot is considered as a point-mass with posi-tion T( , )x yp and velocity T( , )x yv vv . The movement

is controlled via the m F a control law. A discrete time approximation is employed to the continuous behavior of mobile robots within time step t. At each time step, the position perturbation of each mobile robot depends on the average of the two velocities at both ends of time step t, that is, 0( ) 2,t t p v v where 0 .t v v v The

change in each velocity of the robot is controlled by the force on it, that is, ,t m v F where m represents the

mass of the mobile robot and F is the virtual force. Moreo-ver, a frictional force is employed for self-stabilization.

A classic application of AP method to multi-robot system is used to form a hexagonal lattice, thus creating an effec-tive sensing grid [4,16]. The hexagon can be created via overlapping circles with the same radius R. In the AP framework, mobile robots exert virtual forces upon others and respond to forces from other robots. The only forces of concern are from those nearby robots. Imitating the Newto-nian gravitational force law, the virtual force law between robots can be defined as:

2

,GFr

(1)

where maxF F is the magnitude of the force between

two robots, and r denotes the distance between them. The “gravitational constant” G affects the strength of the force, which must be set at initialization. The virtual force is re-pulsive if r<R and attractive if r>R. If r=R, the virtual force is 0. R is the desired separation between that mobile robot and neighbouring robots. Each robot has a sensor that de-tects the range and bearing to nearby robots. To ensure that the virtual forces are local in nature, the visual range of ro-bots is limited to 1.5R [4,5]. Similarly, because of the dis-crete-time nature of the model, it is important to introduce the maximum force Fmax [5], the proper value of which is determined through experiments.

Figures 1 and 2 show how multi-robot system that began in a state of disorder has evolved into a regular lattice, using above simple force law as eq. (1). Figure 1 illustrates the construction process for 7 mobile robots from a small, ran-dom cluster over 200 time steps. Figure 2 shows that 100 mobile robots evolve into a hexagonal lattice. Although the hexagonal lattice is not perfect because of several flaws, the overall performance can be considered acceptable.

In addition to Newtonian force law, Hettiarachchi and Spears [17] investigated a second force law, which is a gen-eralization of the Lennard-Jones (LJ) force law, which model forces between molecules and atoms, such that:

12 6

13 7

224 ,

dR cRFr r

(2)

where the parameters F, r and R are defined as the same as those in eq. (1). The variable affects the strength of the

Figure 1 (Color online) Motion traces of 7 mobile robots in 200 time steps (b), which has a random configuration (a) initially.

Figure 2 (Color online) Initial deployment configuration of 100 mobile robots is a fairly tight cluster (a). Using AP method, they finally transform to a hexagonal lattice (b).

Zhang X Y, et al. Sci China-Phys Mech Astron September (2014) Vol. 57 No. 9 1773

force, while d and c control the relative balance between the attractive and repulsive components. The values of these parameters can be determined by experience or using opti-mization algorithms such as evolutionary algorithm (EA) [17,18].

The Newtonian force law is generally used to create rigid formations that act as solids in uncertain environment, while the LJ force law can easily model crystalline solid for-mations, liquids, and gases [17].

2.2 Applying AP method to mobile robot navigation

For simplicity, this paper only discusses the mobile robot manoeuvring in two dimensions, but the results can be ap-plied to the more complex motion of mobile robots, such as unmanned aerial vehicles (UAV) [19,20] and satellites, in three dimensional space. The dynamics of each mobile ro-bot is described by two differential equations as:

,

,

p vv u

(3)

where the (acceleration) control input u is the navigation function that combines a solution to the global path- plan-ning problem along with a feedback controller for the mo-bile robot.

Motion behaviours of the mobile robot can be classified into two fundamental behaviours: obstacle-avoidance and goal-seeking [21]. The goal-seeking behaviour can be car-ried out by the virtual physics force law which attracts the mobile robot to the goal. Using the Newtonian gravitational force law, the virtual attractive force on the mobile robot towards the goal is defined as:

goalatt agent goal2

agent goal

ˆ ,G

F rp p

(4)

where agent goalp p is the Euclidean distance between the

mobile robot and the goal, and agent goalˆ r represents the unit

vector from pagent to pgoal. The value of Ggoal is the gravita-tional constant of the goal.

The behavior of obstacle-avoidance can be employed by the virtual repulsive force, which maintains the mobile ro-bot away from obstacles. The definition of the repulsive force also employs the Newtonian gravitational force law as follows:

obsobs agent agent obs rep2

rep agent obs

ˆ , ,

0, otherwise,

GRr p p

F p p (5)

where obs agentˆ r represents the unit vector from the obstacle

to the mobile robot. Rrep is the action range of the repulsive force. Only when the mobile robot moves sufficiently near

to the obstacle can the repulsive force take action and push the mobile robot away.

Moreover, the rate limiter is introduced and the velocity of the mobile robot is limited to max .vv A frictional

force is also introduced by the following equation:

.f f v (6)

Above all, control law (or the navigation function) for the basic AP method that drives the mobile robot moving consists of three virtual forces:

att rep , u F F f (7)

where the mass of a single mobile robot is set as m=1. Fig-ure 3 displays the virtual forces on the mobile robot when moving through obstacle fields.

3 LM-based modified AP method

3.1 Modified Newtonian method

The AP method can be regarded as a gradient descent ap-proach of the potential filed method. The AP forces are in combination with the negative gradient direction of corre-sponding gravitational potential field. When the mobile ro-bot is in a complex environment with multiple obstacles or long and narrow passages, the AP method can also result in serious oscillations. This phenomenon causes an extremely long task completion time because of too many unnecessary control manoeuvres.

To solve the problem of oscillation, we apply the modi-fied Newton method of the optimization theory [22] to AP forces. As in optimization theory, Newton methods adopt the most important search direction, the Newton direction. This direction is derived from the second-order Taylor se-ries approximation. Compared to the methods using the negative gradient such as the search direction, Newton methods have a good convergence, fast solution speed and high precision. Herein we modify the basic AP method us-ing Newton methods and change directions of AP forces

Figure 3 (Color online) AP forces on the robot.


toward the Newton directions. The dynamics of each mobile robot based on the modi-

fied AP method is given by the control law:

att att rep rep, , , u B p F F B p F F f (8)

where the matrix B(p, F) is based on the modified Newton method. As in optimization theory, B should be a posi-tive-definite matrix and is defined as:

1,v

B G I (9)

where the presence of v can guarantee the matrix invertible. The value of G is the Hessian matrix of corresponding con-servative force potential function, which is defined as:

,

x x

y y

F Fx yF Fx y

G (10)

where (Fx, Fy)T is the components of the force F along

x-axis and y-axis. We can intuitively analyze the basic and modified AP methods from the optimization point of view. The basic AP method is effectively a type of gradient de-scent method, which is inherently associated with the oscil-lation problem because of the orthogonal search directions such as most potential filed methods. Furthermore, the gra-dient descent method can only achieve linear convergence rate [22,23]. Therefore, using the basic AP method, the mo-bile robot will spend more time achieving the goal. The modified Newton method employs the quadratic derivation as well as the single gradient information, thus it can greatly alleviate oscillations with the quadratic convergence.

3.2 Levenberg-Marquardt algorithm

In order to ensure that the matrix B in eq. (8) is posi-tive-definite, the parameter =(p, F)0 is introduced ac-cording to the LM algorithm [15,22,23]. Thus, if is suffi-ciently large, B must be positive-definite. However, a too large can reduce the convergent rate and cause oscillations, because control laws of the mobile robot are more con-sistent to the basic AP forces.

The LM method can be derived by replacing the line search strategy with a trust-region strategy. In this section, we imitate the LM method to adjust the value of dynami-cally. The adjusting algorithm of is shown in Figure 4.

4 Results and comparison

In order to measure the performance of the proposed LM- based modified AP method, simulation experiments were carried out in following environments. The experimental

Algorithm 1 Pseudo-code for the modified AP method to adjust v

Step 1: IF G+v·I is not positive-definite, THEN v=4·, go to Step 2, ELSE v=/2, go to Step 3.

Step 2: IF G+v·I is positive-definite, THEN go to Step 4, ELSE go to Step 1;

Step 3: IF G+v·I is not positive-definite, THEN v=2·, go to Step 4, ELSE go to Step 1; Step 4: Output and end adjusting.

Figure 4 Pseudo-code for adjusting v.

platform is a personal computer with Intel(R) Core(TM) i3, 3.07 GHz CPU, 4 GB memory, and Windows 7. The algo-rithm is implemented by Matlab-2009a, and no commercial algorithm tools are used. To verify the effectiveness of the proposed method, the solutions found by the following methods are compared: 1) the basic AP method as described in sect. 2.2; 2) MNM-based AP method described in sect. 3.1, where the parameter is fixed; 3) LM-based AP meth-od described in sect. 3.2, where the parameter is dynami-cally adjusted; 4) the potential field method (PFM) [13,14], where the Gaussian potential function is adopted. For all scenarios, the parameters of AP methods are set as: Rrep=2, Ggoal=1000, Gobs=100, f=0.2, the sampling time t=0.1 s, and the maximum velocity of the mobile robot vmax=1 m/s.

Firstly, we use randomly generated circular obstacles in an area (120 m×120 m). This test bed represents a complex, unknown, random environment. Set =0.2 for the MNM- based method in this scenario. Figure 5 shows trajectories of the four methods, where the ▲ represents the starting point, and ★ in the upper right represents the goal. It can be seen that all the methods can reach the goal through the complex obstacle area. However, a large amount of oscillation is caused by the basic AP method and the PFM when obsta-cles are nearby, and so the trajectories obtained by the

Figure 5 (Color online) Motion traces of the mobile robot with multiple circle obstacles for four methods.


MNM-based and the LM-based AP methods are smoother. Table 1 presents the number of time steps spent by the mo-bile robot seeking the goal point using different methods. It can be seen that, using the LM-based AP method, the mo-bile robot can reach the goal point more easily.

Figure 6 shows the velocity and turning angle response for the four methods. Using the proposed method, the mo-bile robot can keep a more steady velocity without frequent turning. As is shown in Table 2, the standard deviation of the turning angle is introduced to evaluate the smoothness of the trajectory. Thus, the smoother trajectory has a smaller standard deviation of turning angle. From Table 2, it can be concluded that the trajectory achieved by LM-based method is smoother than the other methods.

Secondly, we compare the four methods when passing through a long and narrow passage. Figure 7 shows the tra-jectory comparison of the basic AP method and the LM-

Table 1 Performance comparison for the number of steps

Case Method

Basic AP MNM AP LM AP PFM

1 4206 1519 1438 1397 2 3934 665 650 644 3 3751 1549 1549 1407

Table 2 Performance comparison for the standard deviation of turning angle ()

Case Method

Basic AP MNM AP LM AP PFM

1 0.3943 0.0508 0.0265 0.0648 2 1.5720 0.0145 0.0051 6.3e-7 3 0.1893 0.0304 0.0304 0.2138

based AP method. It can be seen that the proposed LM- based AP method can maintain a fast and smooth movement through the narrow passage, but the basic AP method ex-hibits serious oscillation. In addition, the quantifying com-parative results of the four methods are presented in Tables 1 and 2. In this scenario, we also set =0.2 for the MNM- based method. Compared to the basic and the MNM-based AP methods, the proposed modified method can reach the goal point in fewer steps with smoother movement.

Finally, we test the four methods on a task of passing through obstacles of different shapes. The parameter of the MNM-based method is set as =2. Figure 8 shows motion trajectories of the mobile robot using different methods. It can be seen that both the MNM-based and the LM-based AP methods can drive the mobile robot to reach the goal point with smoother trajectories. Although the PFM can reach the goal more quickly (1407 steps), the trajectory ob-tained has more oscillations than the LM-based AP method.

Compared with the MNM-based AP method, the LM- based AP method does not depend on the parameter , therefore, the proposed method has strong robustness. Also, compared with the general PFM, the AP method does not need to construct complex potential function but exerts vir-tual physics forces on the mobile robot directly. Thus it has more accurate, clear, and comprehensive physical meaning.

According to the above experimental results, it is readily apparent that our proposed LM-based AP method has the following advantages over the basic AP method when ap-plying to the mobile robot navigation in complex environ-ment: smoother trajectories without serious oscillations;

Figure 6 (Color online) Velocity and turning angle of the mobile robot through the circular obstacles field for the four methods.


Figure 7 (Color online) (a) Motion traces of the mobile robot through a narrow passage for the basic AP method and the LM-based AP method. (b) Parts of enlarged version of (a).

Figure 8 (Color online) Motion traces of the mobile robot through circle, triangle and rectangular obstacles for the four methods.

faster achieving the goal with fewer steps; longer step size or smaller sampling frequency.

5 Conclusion

In this paper we have presented an AP-based method to the problem of the mobile robot navigation, and proposed the LM algorithm-based modified AP method to solve the in-herent oscillation problem through complex obstacle fields. Some comparative experiments are implemented using dif-ferent methodologies. The experiment results show that our

proposed method can greatly improve the overall system performance when solving the mobile robot navigation problem, and the resulting trajectories are smoother. Our future work will focus on how to implement our proposed method to multi-agent systems.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61273054 and 61333004), the National Key Basic Re-search Program of China (Grant No. 2014CB046401), the Program for New Century Excellent Talents in University of China (Grant No. NCET-10-0021), the Top-Notch Young Talents Program of China, Gradu-ate Innovation Foundation for Beihang University (Grant No. YCSJ-01-201206), and Aeronautical Foundation of China (Grant No. 20135851042).

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