Control of trajectory tracking of two-wheeled differential spherical mobile robot

WANG Wei, ZHANG Zhi-liang,, GAO Ben-wen, YI Ming

(1. School of Mechanical Engineering, Southwest Petroleum University, Chengdu 610500, China；2. Drilling Engineering Technology Institute of CNPC Xibu Drilling Engineering Company Limited, Urumqi 830011, China)

Abstract：This paper presents a two-wheeled differential spherical mobile robot in view of the problems that the motion of spherical robot is difficult to control and the sensor is limited by the spherical shell.The robot is simple in structure, flexible in motion and easy to control.The kinematics and dynamics model of spherical mobile robot is established according to the structure of spherical mobile robot.On the basis of the adaptive neural sliding mode control, the trajectory tracking controller of the system is designed.During the simulation of the s-trajectory and circular trajectory tracking control of the spherical mobile robot, it is concluded that the spherical mobile robot is flexible in motion and easy to control.In addition, the simulation results show that the adaptive neural sliding mode control can effectively track the trajectory of the spherical robot.The adaptive control eliminates the influence of unknown parameters and disturbances, and avoids the jitter of left and right wheels during the torque output.

Key words：mobile robot; adaptive neural sliding mode control; dynamics controller; trajectory tracking

0 Introduction

At present, robots play an increasingly important role in human life and work, and the mobile robot becomes an important branch in the field of robotics.During the development of the robot industry, the traditional wheeled and tracked mobile robots have been developed earlier, and the technology is more mature and widely used at present.Along with the continuous expansion of robot application, new spherical robots emerge.As a kind of mobile robot with special structure, it has a completely closed spherical shell which is composed of control system, power system, motion actuator, sensor and so on, realizing complex motion mainly according to centroid skewing, momentum conservation and other related principles.Compared with the traditional mobile robot, the spherical robot has many unique advantages.For example, it can achieve zero turning radius when it is steering.The spherical shell can effectively cushion the external impact force to protect the internal device.However, the spherical robot also has some defects.For example, because of the limitation of the spherical shell structure, the external information cannot be effectively transmitted to the internal controller via the sensor, which makes the spherical robot unable to achieve intelligent control.In addition, since the spherical robot is featured by non-integrity constraint, underactuation, non-chain, strong coupling and so on, its motion control becomes a difficult problem to solve.Therefore, it is very important to design a spherical robot with good structural characteristics to realize precise motion control, which has important theoretical significance and engineering application value.Jelassi et al.studied the optimization of scanning probe microscopy(SPM)for 3-RRR spherical parallel robot.In addition to the constrained workspace and dexterity, the distribution of single positions was also studied[1].Liu et al.has designed a new type of bionic spherical amphibious child-mother robot system, in which the spherical amphibious mother robot moves on land in a bionic four-legged crawling mode and uses a vector water jet motor to spray water for propulsion under water, featured by no noise, increased concealment and provision of control signals and energy for the micro child-robot.The micro child-robot is driven by wheels on land and designed with an amphibious impeller.The child-mother robot system realizes wireless communication through XBee communication module.Through land and underwater motion tests of the child-mother robot, the effectiveness of the designed child-mother robot system was verified[2].Huang et al.introduced a prototype of a spherical rolling robot with a new driving mechanism.The spherical robot has a momentum wheel(gyroscope)that rotates at high speed in an outer spherical shell.The test results were also provided to verify the feasibility of the mechanism[3].Fan et al.aiming at the uncertainty and nonlinearity of the mobile robot model, proposed a hybrid algorithm of PI-type sliding mode control(SMC)based on backstepping dynamic control and adaptive radial basis function neural network(RBFNN)to adjust sliding mode gain, so as to enhance the adaptability to stochastic uncertainty factors and eliminate the jitter of sliding mode control input[4].

In combination with the flexible motion of wheeled mobile robot and the protective motion of spherical mobile robot, a two-wheeled differential spherical mobile robot is proposed here.Unlike the traditional spherical shell type rolling robot, the driving system of the two-wheeled differential spherical mobile robot realizes the complex motion of the whole system mainly by two hemispherical rollers.The hemispherical differential spherical robot is a mobile robot designed by integrating the features of the differential drive of left and right wheels of two-wheeled self-balancing robot based on the existing spherical robot.This robot not only still has the advantages of flexible movement of spherical robot in two-dimensional plane, but also eliminates the restriction of the spherical shell on internal sensor and external environment via the design of the central connecting platform, which makes it better to obtain the external information and carry a variety of sensor devices to improve the autonomous motion ability of the spherical robot.Therefore, this robot may have great advantages and extensive application prospects in the fields of planet exploration, dangerous environment detection, etc.

1 Driving system and operating principle

As shown in Fig.1, the stable platform is driven by the left and right hemispherical wheels, the motion of which are independent from each other, and it keeps a vertical state under the action of the balancing weight.The system is composed of driving wheel, stable platform, drive motor, rack and pinion, bearing and other components.The stable platform is installed by the upper and lower units by connecting bolts, and it can be equipped with sensors, cameras and control units of the system, and with the counterweight slots at below for the installation of the power supply of the system.For the spherical robot, the motor installed on the stable platform is directly connected with the gear, and the circular rack is fixedly connected with the spherical shell to keep connection with the stable platform via the bearing.The driving motor drives the gear to rotate, and the gear meshes with the circular rack, so as to drive the spherical shell to rotate.The stable platform keeps stable under the action of gravity.The omnidirectional motion of the spherical robot will be realized under the action of two driving wheels at different speeds.

1—Top cap；2—Left gear；3—Right gear；4—Fixed plate of motor；5—Motor；6—Bearing seat；7—Driving wheel；8—Round internal gear；9—Bearing；10—Stable platform；11—Connecting bolt

When the two driving wheels rotate in the same direction at the same speed, the spherical robot can move in a straight line.When the two driving wheels rotate in the same direction at a differential speed, the curve and circular motion can be realized.If the two driving wheels rotate in the opposite direction at the same speed, the in-situ steering motion can be realized.Therefore, compared with the existing spherical robots[3-8], the differential spherical robot is easier to realize various forms of motion.The robot is simple and dexterous and free of restriction of spherical shell, and can be carried with sensors and cameras, so as to capture external information, realize intelligence of spherical robot and expand its range of application.

2 Differential equation of motion

The spherical mobile robot is one of nonlinear multi-input and multi-output systems that are featured by nonholonomic constraints and strong coupling[2].At present, in the theoretical study of motion control of the mobile robot, it is generally assumed that the nonholonomic constraint of the system is an ideal constraint, that is, the wheel is in point contact with the ground and only pure rolling occurs at the point of contact without relative sliding(including lateral and longitudinal sliding).The spherical mobile robot controls the speed and direction of motion of the robot by controlling the speed of two driving wheels.

2.1 Kinematic modeling

In our work, we simplify the physical model of the robot when analyzing the motion of a spherical mobile robot, so as to highlight the impact of key factors.In addition, a rectangular coordinate system is established in the plane of motion, and the simplified model is shown in Fig.2.

Fig.2 Simplified model of spherical mobile robot

The centroid point of the stable platform isO, the distance between the two driving wheels isL, the radius of the driving wheel isR, the heading angle of the whole system isθ, the turning angle of the left driving wheel isωL, and the turning angle of the right driving wheel isωR.

The speed at the center of mass of the stable platform of the spherical mobile robot isV0, in a direction perpendicular to two drive axles, thus the components in thexaxis andyaxis directions can be obtained as

(1)

By eliminatingV0in the above equation, the constraint equation can be obtained as

(2)

The relationship of the linear velocity, heading angle and driving wheel angle of the robot system is expressed as

(3)

From that, the kinematical equations of the differential spherical mobile robot can be obtained as

(4)

(5)

2.2 Dynamical modeling

The Lagrange multiplier equation[12-13]is usually selected to analyze the dynamical model of the spherical mobile robot due to nonholonomic constraints in motion.

The position of the robot are described by the three-dimensional generalized coordinates , the mobile robot is regarded as a point, and pointOis the current position of the mobile robot.

Since there is a nonholonomic constraint in the system, as represented by Eq.(2), which is given in the form of formula as

(6)

Assuming that

(7)

whereS(q)represents the velocity transition matrix of the system as

v(t)=[ωRωL]Tmeans the speed matrix.The speed of the coordinate system of the robot,v(t)=[ωRωL]T, can be converted into the speed,V(t)=[vω]Tin the Cartesian coordinate system, wherevis the linear speed of the center of massC, andωis the angular speed thereof.Then the kinematical equation of the spherical mobile robot is

(8)

Total kinetic energy,T, of the system is

(9)

whereTVis the translation kinetic energy of the system;TJis the rotational kinetic energy of the system;mis the mass of the system; andIis rotational inertia.

A binding force is added as an input item to the dynamical equation of the system to prevent the driving wheel from sideslip.The Lagrange equation with multipliers is represented as

(10)

Substituting Eq.(9)into Eq.(10), we can obtain the dynamical model of the system as[9]

(11)

It can be seen that the dynamic equation of the spherical mobile robot is[11-12]

(12)

Eq.(12)can be further converted to

(13)

where

3 Design of trajectory tracking controller

Based on the kinematical and dynamical equations of the spherical mobile robot mentioned above, firstly, an appropriate control function is selected for the pose error system of kinematical Eq.(8)to design a reasonable kinematics controller, and the linear speedvand angular speedωoutput from the controller are used as auxiliary control inputs, so that the actual and planned trajectories of motion of the robot converge to zero.After that, according to the dynamical equations of the system, the torque controller of the spherical mobile robot is designed by using adaptive neural sliding mode control[9-10], so as to make its speed converge to the desired speed given by the motion controller.

3.1 Design of kinematical controller

According to the motion system of the spherical mobile robot, the motion trajectory is given asqd=[xdydθd]T, and the tracking error of the position and course of the system is

(14)

Then, the differential equation ofqeis

(15)

Then, according to Lyapunov function[10], the auxiliary speed control input,VF, is designed as

(16)

wherekx,ky,ks,αandλ(α+λ=1)are constant greater than zero.VFmakes the pose error of the spherical mobile robot converge to zero.

3.2 Design of dynamical controller

By using adaptive neural sliding mode control[11-13], the speed tracking error of spherical mobile robot is designed as

(17)

The sliding mode surface is selected by Eq.(13)as

(18)

whereη>0.

Then, the derivative ofS(t)can be got as

(19)

LetB=E0τ, the equivalent control lawBeqis

(20)

The control characteristic of the robot is realized by adjustingη.However, there are unknown parameters and disturbances in the whole system.And also, the above Eq.(20)cannot accurately describe the characteristics of the system and the stability of the entire system.Therefore, the adaptive neural network is used to approximate Eq.(19), the input of the network is 2, the number of hidden units is N, and the output of the network is 2.Taking the tracking error as the input of the neural network, i.e.xi=ec(i),(i=1,2), and the output of the neural network is

(21)

where the weight vector of the neural network isWi=[wi1wi2…wim]T;Hirefers to the radial basis vector,Hi=[hi1hi1…h(huán)im]T, andhijis a Gaussian function in the form as

(22)

(23)

wherej=1,2,…,m;c(∶,j)refers to the network center, andbjrefers to the base width.

It is assumed that the optimal output value of the network is

(24)

Because of the minimum deviation in the system model, its equivalent control is

(25)

In order to overcome the influence of uncertain parameters and disturbances in the system, the adaptive adjustment of neural network weights is adopted, and the equivalent control is

(26)

(27)

(28)

At that time, the law of neural sliding mode control is

(29)

In order to eliminate the jittering in the system, radial basis function neural network(RBFNN)is used to adjust the sliding mode gain,Γ, and the sliding mode surface is taken as the input of RBFNN, causingxi=si,i=1,2, so that the output of RBFNN[13]is

(30)

whereAi=[αi1αi2…αim]Tis the weight vector of the network, andψi=[φi1φi2…φim]T, in whichφijis Gauss function, namely

i=1,2,j=1,2,…,m,

(31)

wheredijis the center of thejth node of theith input in RBFNN, andδijis the base width, all of which are the constants greater than zero.

For the gain of switching control, the optimal parameters should be selected in the design process of sliding mode controller.Because of the uncertain parameters and unknown disturbances as well as the uncertain factors of approximation accuracy of neural network, the parameters obtained by mobile robot system are often imprecise.Therefore, the adaptive control law is used to estimate the optimal parameters[14-15].

Assuming that the switching controller of the optimal gain is

(32)

and the estimated gain switching controller is

(33)

(34)

(35)

whereγi>0, from which the adaptive neural sliding mode controller of the system is obtained as

(36)

3.3 Certificate of stability

Theorem: According to Eq.(8)of spherical mobile robot system, Lyapunov function is selected to design Eq.(16)of the kinematical controller, and the adaptive neural sliding mode is used to design Eq.(36)of the dynamical controller of the system.The adaptive law of parameters is Eqs.(27)and(35), which indicates that Eq.(12)of the whole system is asymptotically stable.

It is proved that Lyapunov function should be selected as

L=L1+L2,

(37)

(38)

(39)

Substituting the derivative ofL1in respect of time into Eq.(15), we can obtain

xe(vrcosθe-v-Ksθew)+vrsinθe(ye+Ksθe)+

((ye+Ksθs)Ks+sinθe/Ky)(wr-w).

(40)

Substituting Eq.(16)into Eq.(40), we can obtain

(41)

Substituting the derivative of Eq.(39)into Eqs.(19),(26),(27),(34),(35)and(36), we can obtain

(42)

Ifηi>0, the following equation can be obtained as

(43)

?≥0.

(44)

4 Simulation test

In this paper, a simulation model is built in Matlab/Simulink to verify the trajectory tracking control of the spherical mobile robot according to the control designed by the neural network adaptive sliding mode control.The control algorithm is shown in Fig.3.

Fig.3 Structural diagram of kinematical and dynamical control algorithm of spherical mobile robot

Fig.4 S-trajectory tracking of spherical mobile robot

For the circular trajectory,yr=sin(t),θr(t)=t,vr=1 m/s andωr=1 rad/s.The initial error between the actual trajectory and the reference trajectory of the spherical mobile robot is defined asqe=(0.5 0 0), and the initial reference pose isq=(1 0 π/2).The simulation of the robot in trajectory tracking of 40 s is shown in Fig.5.It can be seen from Figs.4 and 5 that the robot has strong flexibility in motion and can complete complex trajectory motion.Under the adaptive neural sliding mode control, the system can realize the trajectory tracking control well, and enters the steady state in 3 s, during which the trajectory and heading angle converge to the reference value gradually.

Fig.5 Tracking of circular trajectory of spherical mobile robot

It can be seen from shown Figs.6 and 7, the tracking error of the system gradually converges to zero in 3 s, and no jittering is found in the whole system in the process of the trajectory tracking, and the influence of unknown parameters and disturbances is eliminated.

Fig.6 System tacking error

Fig.7 Control curve of left and right wheels

5 Conclusion

In this paper, we discuss a two-wheeled differential spherical mobile robot.The robot body is composed of an internal support platform and two hemispherical rollers, which increases the loading space of the sensor.The robot is simple in structure, flexible in motion and easy to control.Based on the adaptive neural sliding mode control method, the trajectory tracker of the system is designed, and a model is built for simulation.The simulation test results show that the spherical robot can complete the control over the tracking along the s-trajectory and the circular trajectory and eliminate the influence of unknown parameters and disturbances in the motion of the spherical robot, and no jittering is found during the torque output of the whole system.Therefore, the system proposed is stable and reliable, meeting the requirements of mobile robot system for jittering prevention and anti-interference.