Point‐to‐Point Motion Control for a Spherical Robot on an Inclined Plane via Sliding Modes

www.as‐se.org/sss Studies in System Science (SSS) Volume 2, 2014

Point‐to‐Point Motion Control for a Spherical Robot on an Inclined Plane via Sliding Modes Tao Yu*1, Hanxu Sun2 1

Faculty of Mechanical Engineering and Automation, Liaoning University of Technology, Jinzhou, Liaoning, China

2

School of Automation, Beijing University of Posts and Telecommunications, Haidian District, Beijing, China

*1

yutaolanjie@163.com; 2hxsun@bupt.edu.cn

Abstract In this study, a variable structure approach based on exponential reaching law is presented for point‐to‐point motion control of a spherical mobile robot on an inclined plane. The multibody dynamics of the spherical mobile robot rolling without slipping and spinning on an inclined plane are derived by using the constrained Lagrangian formulation. Based on the complete equations of motion of the robotic system, a sliding mode algorithm is developed for point‐to‐point motion control of the robot by using the theory of VSS. The proposed control approach consists of designing a nonlinear reaching law by using a switching component that dynamically adapts to the variations of the sliding surface. The effectiveness of the proposed controller is validated through numerical simulations and experiments.

Keywords Spherical Robot; Inclined Plane; Point‐to‐Point Motion; Sliding Mode Control; Exponential Reaching Law

Introduction Most of the mobile robots that we have today have wheels, this is an obvious choice as there is considerable amount of knowledge about this type of locomotion. However, more and more possible applications occur where wheeled mobile robots have some flaws. Spherical mobile robots, which are a novel type of mobile robots with a spherical exoskeleton and a propulsion mechanism that uses unbalanced masses, might be an effective solution to some of these problems. As a spherical mobile robot is encompassed in a ball, it is possible to seal everything to enable the robot to withstand exposure to dust, dangerous substances and other environmental threats. As we can understand, this could be very handy in such applications as planetary exploration, surveillance and others. The above mentioned situations often involve dealing with difficult terrain as well. While wheeled mobile robots can cope with it pretty good, the risk of falling over still persists. On the other hand, a spherical mobile robot can't fall over at all. Also, it is quite a task to deploy a wheeled mobile robot without direct human intervention ‐ the landing spot has to be carefully chosen, and the robot has to land with wheels down, etc. However, a spherical mobile robot can be simply dropped out above the desired location.

Over the last few decades, there has been considerable interests in the development of powerful methods for motion control of spherical mobile robots. The motion control problems addressed in the literature can be roughly classified into two groups: trajectory tracking [1‐4] and path following [5]. Besides the above mentioned two control problems, point‐to‐point motion control is usually viewed as another basic motion control problem of this type of mobile robots, which has not yet been investigated till now. On the other hand, current researches on motion control of spherical mobile robots usually assume that the robot remains strictly on a level plane [1‐5]. Due to some components of the gravitational forces acting on the system, the motion of a spherical mobile robot on an inclined plane differs significantly from those on horizontal planes. As a result, the dynamic models, which are developed on the basis that the spherical mobile robot travels over a level plane, fail to represent the actual motion when the robot rolls along a slope.

According to the above review, there have not been any well‐established methodologies to resolve the point‐to‐point motion control problem of spherical mobile robots. Therefore, this paper basically focuses on practical solutions to point‐to‐point motion control of a spherical mobile robot which rolls on an inclined plane. The main contributions of this paper include two parts. Firstly, the kinematics and dynamics of the spherical robot rolling over constant‐slope terrain without slipping and spinning are derived. Secondly, a sliding mode control scheme based on a new exponential reaching law is developed for the point‐to‐point motion of the spherical robot.

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Mathematical Model System Description Our robot, named BYQ‐VIII [6][7], is a classic spherical mobile robot using a pendulum‐based design. The prototype robot and mechanical structure of BYQ‐VIII are illustrated in Fig. 1 and Fig. 2, respectively. BYQ‐VIII is mainly composed of three rigid bodies: its spherical shell, its internal mechanism holding different internal components of the robot (motors, microcontroller and sensors), and its pendulum.

FIG. 1 PROTOTYPE ROBOT OF BYQ‐VIII

The actuation mechanism of BYQ‐VIII consists of two separate actuators: a drive motor and a tilt motor. BYQ‐VIII is propelled by the pendulum suspended from an axis inside the casing and controlled by the two motors. Moving the pendulum forwards causes the robot to roll along, and the pendulum can swing from side to side, giving the robot the ability to steer left and right.

A number of on‐board sensors are installed on BYQ‐VIII to measure the states of the robot. Two pulse encoders are installed to measure the rotation angles and speeds of the drive motor and tilt motor. Furthermore, we have a gyro mounted on the internal mechanism to calculate the angle and angular rate of the yaw, roll and pitch of the spherical shell.

FIG. 2 BASIC STRUCTURE OF THE SPHERICAL ROBOT BYQ‐VIII

Kinematic Constraints A complete dynamic model is necessary to develop motion control algorithms for the spherical robot BYQ‐VIII. In the following subsections, we will derive the nonholonomic kinematic constraints of the rolling sphere, as well as the dynamic model of the robotic system using the constrained Lagrangian formulation.

In derivation of the equations of motion, we first assume that the exoskeleton of the robot is a rigid, homogeneous, and thin‐walled spherical shell which rolls over a perfectly flat surface of an inclined plane without slipping and spinning. Fig. 3 illustrates the simplified model of the robot travelling over constant‐slope terrain within a predefined limit [8][9]. It represents the spherical shell with its center of mass B, the internal mechanism with its center of mass D, which coincides with that of the spherical shell, and the pendulum (composed of a massless link and a counterweight at its end) with its center of mass E and the axis attached at the center of the sphere.

Next, we assign four coordinate frames as follows: (1) the inertial frame O  X ,Y , Z , whose XY plane is anchored to the flat surface, and Z is the vertical position to the incline, (2) the body coordinate frame  B  X b ,Yb , Zb  , whose

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origin is located at the center of the sphere, (3) the coordinate frame of internal mechanism  D  X c ,Yc , Zc  , whose origin is located at the center of mass of the internal mechanism, and whose x‐axis is always parallel to X b , and (4) the pendulum coordinate frame  E  X a ,Ya , Za  , whose origin is located at the center of mass of the pendulum, and

whose x‐axis represents the axis of symmetry of the pendulum. Note that Ya is always parallel to Yc. The definition and configuration of system and variables are shown in Table 1 and Fig. 3.

Zb

Yb



Yc

Zc









Xc B(D)



Ya

Za

Xb







E Xa C (x, y) Y

Z



O

X

FIG. 3 DEFINITION OF COORDINATE FRAMES AND SYSTEM VARIABLES TABLE 1 VARIABLE DEFINITION OF THE ROBOTIC SYSTEM

ms, mi, mf R, l Is Ixxi, Iyyi, Izzi Ixxf, Iyyf, Izzf α β γ  ,  , 

mass of the spherical shell, the internal mechanism, and the pendulum, respectively radius of the spherical shell, and length of the pendulum, respectively moment of inertia of the spherical shell moment of inertia of the internal mechanism about Xc, Yc and Zc axes moment of inertia of the pendulum about Xa, Ya and Za axes angle of rotation of the internal mechanism about Xb axis angle of rotation of the pendulum about Yc axis inclination angle of the slope yaw angle, roll angle, and pitch angle of the spherical shell, respectively

  ,  

drive torque of the drive motor, and tilt torque of the tilt motor, respectively

Here we first derive the kinematic constraints of the rolling sphere, and then develop the dynamic model of the spherical robot based on these constraints.

Let v B and  denote respectively the velocity of the center of mass of the spherical shell and its angular velocity with respect to the inertia frame O . Then, we have v B  x i  y j  z k (1)

  x i   y j  z k (2) where  x , y , z  are the coordinates of the center of mass of the spherical shell with respect to the inertial frame O , and  l , m , n  are the unit vectors of the inertia frame O .

x   cos cos  sin

 y  cos   sin cos

z    sin

The non‐spinning constraint of the rolling sphere requires that the angular rate of the sphere around the Z‐axis is zero at any instant. According to Eq. (2), the non‐spinning constraint of the rolling sphere can be formulated as

  sin  0 (3) The constraint in Eq. (3) is nonintegerable and hence it represents a nonholonomic constraint.

The pure rolling and non‐slipping constraints of the sphere require that the velocity of the contact point between the sphere and the incline is zero at any instant, i.e., vC  0 (4)

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where vC is the velocity of the contact point C as shown in Fig. 3. Now, we can express v B as  v B    rBC  vC (5)  where rBC   Rk represents the vector from point C to point B. Substituting Eq. (2) and Eq. (4) into Eq. (5), and combining the result with Eq. (1), we have







x  R cos   sin cos  0



y  R sin  cos cos  0 (6) z  0 (7)

The constraints in Eq. (6) are nonholonomic, whereas the constraint in Eq. (7) is holonomic and can be integrated to obtain z  R (8)

Therefore, the configuration of the robotic system can be completely described by a generalized coordinates vector of seven independent variables,

q   x , y ,  ,  ,  ,  ,   (9) T

The nonholonomic kinematic constraints of the robotic system are now expressed in the following form A  q  q  0 (10)

where 1 0 0 Rcos A  q    0 1 0 Rsin  0 0 1 0

Rsin cos  Rcos cos sin

0 0 0 0  0 0 

Robot Dynamics In this subsection, we develop the equations of motion by calculating the Lagrangian L  T  P of the system, where T and P are the kinetic energy and potential energy of the system respectively. We divide the whole robotic system into three parts: 1) spherical shell, 2) internal mechanism, and 3) pendulum.

1) Spherical Shell The kinetic energy of the spherical shell is given by



1 2



1 2





s  ms x 2  y 2  z 2  I s x2   y2  z2 (11) Substituting Eq. (2) and Eq. (7) into Eq. (11) yields









1 1  sin (12) s  ms x 2  y 2  I s 2   2   2  2 2

2

According to Eq. (8), the potential energy of the spherical shell is Ps  ms g  y sin   R cos   (13)

where g denotes the gravitational acceleration.

2) Internal Mechanism Let ωB denote the angular velocity of the spherical shell with respect to the body coordinate frame  B . Then, we have ωB   xB l   yB m  zB n (14)

where  l , m , n  are the unit vectors of the body coordinate frame  B .

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 xB    sin   yB  cos  sin cos zB  cos cos  sin

Let ωD denote the angular velocity of the internal mechanism with respect to the coordinate frame of internal mechanism  D . Then, we have

 

T

 D  RDB

 B   l 

 xD l    yD m  zD n

(15)

where  l , m , n  are the unit vectors of the coordinate frame of internal mechanism  D , RDB is the transformation matrix from the body coordinate frame  B to the coordinate frame of internal mechanism  D .

0 1  B RD  0 cos  0 sin 

xD    sin  

0   sin   cos  

 yD  cos      cos sin      zD  cos cos      sin    

Let  xi , yi , zi  be the coordinates of the center of mass of the internal mechanism with respect to the inertia frame O . The transformation from the center of mass of the spherical shell to that of the internal mechanism is described by  xi   x       yi    y  (16)  zi   z 

Let Ti l and Ti r denote the translational kinetic energy and rotational kinetic energy of the internal mechanism respectively. Differentiating Eq. (16) and using Eq. (15), we can obtain the kinetic energy of the internal mechanism Ti .

1 2



 12  I

i  Ti l  Ti r  mi x i2  y i2  zi2 

xxi



2 2 2 xD  I yyi yD  I zzizD (17)

The potential energy of the internal mechanism is Pi  mi g  yi sin   zi cos   (18)

3) Pendulum Let ωE denote the angular velocity of the pendulum with respect to the pendulum coordinate frame  E . Then, we have

 

 E  RED

T

 D   m 

 xE l    yE m   zE n

(19)

where  l , m, n  are the unit vectors of the pendulum coordinate frame  E , RED is the transformation matrix from the coordinate frame of internal mechanism  D to the pendulum coordinate frame  E .  cos  D RE   0   sin 











 xE  cos    sin    sin  cos cos      sin      0 sin   1 0   yE  cos      cos sin      

0 cos    zE  sin    sin    cos  cos cos      sin     



Let x f , y f , z f be the coordinates of the center of mass of the pendulum with respect to the inertia frame O .

The transformation from the center of mass of the spherical shell to that of the pendulum is described by

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x f  yf   z f

 x  0  x   l sin       y   RO  l sin  cos   (20) O B D   0    y R R R    B D E   B      z      l z     l cos  cos     

where RBO is the transformation matrix from the inertia frame O to the body coordinate frame  B . Let s and c denote sin   and cos   , respectively. Then, the transformation matrix RBO is given by c c  R  Rz   Ry   Rx    c s   s  O B

s s  c c s   c s s  c s   c c 

c s s  c s c c  s s s c s

Here Rz   , Ry   and Rx   are rotational matrices in the following form.  cos Rz     sin  0

sin cos 0

0  cos 0  Ry     0  sin 1 

0 sin  0 1 1 0  Rx     0 cos  0 sin 0 cos 

0  sin  cos 

Let T fl and T fr denote the translational kinetic energy and rotational kinetic energy of the pendulum respectively.

Differentiating Eq. (20) and using Eq. (19), we can obtain the kinetic energy of the pendulum T f .

1 2



 12  I

 f  T fl  T fr  m f x 2f  y 2f  z 2f 

xxf



2 2 2 xE  I yyf  yE  I zzf zE (21)

The potential energy of the pendulum is given by





Pf  m f g y f sin   z f cos  (22)

4) Dynamic Equations The Lagrangian of the whole robotic system is then given by



 



L  Ts  Ti  T f  Ps  Pi  Pf (23)

Substituting Eq. (12), Eq. (13), Eq. (17), Eq. (18), Eq. (21) and Eq. (22) into Eq. (23), we can determine L. Note that there are only two control inputs available on the system. One is the drive torque   , and the other is the tilt torque   . Consequently, using the constrained Lagrangian method, the dynamic equations of the robotic system are given by M  q  q  N  q , q   E  q  u  AT  q   (24)

where M  q    77 is the inertia matrix, N  q , q    7 is the vector of position and velocity dependent forces, E  q  is the input transformation matrix, u is the input vector, A  q  as in Eq. (10) is the constraint matrix, and

 is the vector of constraint forces. 0 0 0 0 0 0 1  E q      0 0 0 0 1 1 0 

 1     2   3 

T

 u     u   1    u 2    

Equations of Motion In this subsection, we will eliminate the Lagrange multipliers in order to obtain a minimum set of differential equations in a manner similar to [10][11]. We first partition the constraint matrix A  q  into two parts, i.e., A   A1  A2  (25)

where

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 Rcos A2   Rsin  0

1 0 0  A1  0 1 0  0 0 1 

Rsin cos  Rcos cos sin

0 0 0 0  0 0 

Let   A1 A    A  C  q    1 2    2  (26)  I 44   I 44 

where   

 2

.

It is straightforward to verify that C  q  satisfies A  q  C  q   0 (27)

In view of Eq. (25), we adopt the following partition of the generalized coordinates vector q .

q  q   1  (28) q2  where q1   x , y ,   , q2   ,  ,  ,   .

T

T

According to Eq. (10), Eq. (25) and Eq. (28), we have A  q  q   A1

 q  A2   1   q1  A2 q2  0 (29) q2 

This implies q1   A2 q2 (30)

Differentiating Eq. (28) and using Eq. (30), we arrive at  q    A q    A1 A  q   1    2 2    1 2  q2  C  q  q2 (31) q2   q2   I 44 

If we choose q2 to be the four quasi‐coordinates, 1            t    2     (32) 3        4    

where   t   q2 is a velocity vector.

Then, we have q  C  q   t  (33)

The system represented by Eq. (24) is now transformed into a more appropriate representation for control purposes.

Differentiating Eq. (33), substituting the result into Eq. (24), and then premultiplying both sides by CT, we can eliminate the Lagrange multipliers  .

C T MC  C T MC  C T N  C T Eu (34)

The complete equations of motion of the nonholonomic spherical robot are now given by Eq. (33) and Eq. (34).

Through appropriate definitions we can rewrite Eq. (34) as follows M  q   V  q , q   N  q , q   E  q  u (35)

where

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M q  CT q M q C q

V  q , q   C T  q  M  q  C  q , q 

N  q , q   C T  q  N  q , q 

E q  CT q E q

Point-to-Point Motion Controller Design Controller Design In this subsection, we consider point‐to‐point motion control of the spherical robot and a sliding mode controller based on a new exponential reaching law is derived to asymptotically stabilize the robot around its desired target.

The control objective can be stated as follows: given a desired target location pd   xd

yd  , where xd and yd are T

some preset constants, determine a control law u , in the presence of parametric and other uncertainties, such that p  pd as t   , where p  q    x

y  is coordinate variables of the reference point of the robot. The sliding mode T

control algorithm proposed in this paper can be implemented in the following two steps [4][5].

Step 1: Based on the robot kinematics given by Eq. (33), we can select the desired velocity vector which ensures exponential convergence of the position error to zero. We define the position error as e  p  pd (36)

Let

e   e (37)

where    22 is a positive diagonal matrix.

Eq. (37) represents an exponentially stable error dynamics. Differentiating Eq. (36) and using Eq. (33), we arrive at D  q d   e (38)

where d respresents the desired velocity vector; D  q   J  q  C  q  , J  q  

p  q  q

are the decoupling matrix and the

Jacobian matrix, respectively.   Rcos D q     Rsin

1 0 0 0 0 0 0  J q    0 1 0 0 0 0 0

 Rsin cos Rcos cos

0 0 0 0 

Using Eq. (36) and Eq. (38), we have  1e1cos  2 e2 sin    d   R     (39) d   1e1sin  2 e2 cos    Rcos d h

where i and e j represent the i‐th diagonal element of the gain matrix  and the j‐th component of the error vector e , respectively.

By integrating Eq. (39), we can obtain the value of q2dh  t   d  t   d  t   at any given instant of time, where T

q2dh  t  represents the desired value of the two quasi‐coordinates q2h  t     t    t   .

T

Step 2: The control objective of the system is now transfered to properly choose the control input u , such that q2h  q2dh T

and  h   hd as t   , where  h  t     t    t   is a velocity vector. We rewrite the robot dynamics given by Eq. (35) in the following form m111  m122  m133  m144  V111  V122  V133  V144  n1  0 (40) m211  m222  m233  m244  V211  V222  V233  V244  n2  u 2 (41) m311  m322  m333  m344  V311  V32 2  V333  V344  n3  u 2 (42)

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m411  m422  m433  m444  V411  V422  V433  V444  n4  u1 (43)

where mij and Vij denote the element in the i‐th row and j‐th column of the matrix M  q  and V  q , q  respectively, and nk represents the k‐th component of the vector N  q , q  .

Adding Eq. (42) to both side of Eq. (41) gives  211  m  222  m  233  m  244  V211  V222  V233  V244  n 2  0 (44) m

where  21  m21  m31 m

 22  m22  m32 m

V21  V21  V31

 23  m23  m33 m

V22  V22  V32

 24  m24  m34 m

V23  V23  V33

n 2  n2  n3

V24  V24  V34

Combining Eq. (40) and Eq. (44), we have M1h  M 2l  V1 h  V2 l  N1  0 (45)

where m12  m M1   11   22  m21 m

V13 V14  m14  V11 V12  V1    V2          m24  V21 V22  V23 V24 

m M 2   13  23 m

n      N1   1   h   1   l   3   n2  2  4 

We put together Eq. (43) and Eq. (42) into the following form M 3h  M 4l  V3 h  V4 l  N 2  ur  u (46)

where ur   2 denotes bounded unknown disturbances including unstructured unmodeled dynamics.

m42  m M 3   41  m31 m32 

m M 4   43 m33

m44  m34 

V42  V V3   41  V31 V32 

V44  V V4   43  V33 V34 

n  N2   4   n3 

According to Eq. (45), we have

l   M 21 M1h  M 21V1 h  M 21V2 l  M 21 N1 (47) Substituting Eq. (47) into Eq. (46) yields    V   V   N  u  u (48) M 3 h 3 h 4 l 2 r

where   M  M M 1 M M 3 3 4 2 1

V3  V3  M 4 M 21V1

V4  V4  M 4 M 21V2

N 2  N 2  M 4 M 21 N1

Applying the following nonlinear feedback

u  M 3u  V3 h  V4 l  N 2 (49) where u  u1 u2  represents the new control inputs.

T

We can simplify Eq. (48) to the form    u  M  u (50) M 3 h r 3

We define the following PI sliding surface









S   h   hd   0t  h   hd dt   h   hd   q2h  qd2h (51)

where    22 is a positive diagonal matrix.

Differentiating Eq. (51) and using Eq. (50), we have









S  h  hd    h   hd  u    h   hd  hd  ur (52)

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T  1u represents the new perturbation term satisfying d   , d   and  ,  being where ur   d1 d2   M r 1 1 2 2 1 2 3 some positive constants.

The new ERL is realized based on the choice of a switching component that adapts to the variations of the switching function. This reaching law is given by si   N si  sgn si   kisi , i  1, 2 (53)

where si represents the i‐th component of the vector S , and ki is a positive constant.

N s i  

i

 i  1  1 si   i  e 

i

si

pi

 i

Here  i is a positive constant that is less than one, pi is a positive integer,  i and  i are also positive constants.

The proposed ERL given by Eq. (53) is composed of a variable rate reaching component revised from [12] and an exponential term. Comparing with the conventional exponential reaching law [13][14], we can see from Eq. (53) that if si increases, N si  converges to i  i   i , which is larger than i   i . This means that N si  increases in the reaching phase, and consequently the attraction to the sliding surface si is faster. On the other hand, if si decreases, then N si  converges to  i . This means that, when the system state approaches the sliding surface si , N si  gradually decreases to reduce the chattering. Therefore, the proposed ERL allows the controller to dynamically adapt to the variations of the switching function si by letting N si  vary between  i and i  i   i .

Remark 1: In a particular case, when there is no uncertainty included in the system, the control gains  i are set to be

zero. It then follows that, when the system state approaches the sliding surface si , N si  gradually decreases to zero, and the chattering phenomenon is completely removed.

According to Eq. (52) and Eq. (53), the sliding mode control law is designed as follows

u  hd    h   hd   N S  sgn S   K S (54) where  N s1  0  N S     N s 2    0

 sgn s1   sgn S     sgn s 2  

k K  1 0

0 k2 

Here 1   1  1 ,  2   2   2 ;  1 and  2 are positive constants.

Stability Analysis

Theorem 1: Considering the spherical robot system represented by Eq. (33) and Eq. (35), if the sliding mode control law is designed as in Eq. (54), then the sliding surface S given by Eq. (51) is asymptotically stable.

Proof: Substituting Eq. (54) into Eq. (52) yields si   N si  sgn si   kisi  di , i  1, 2 (55)

Define the following Lyapunov function candidates 1 Vi  si2 , i  1, 2 (56) 2

Differentiating Eq. (56) and using Eq. (55), we have Vi  sisi   N si  si  kisi2  si di  kisi2   i si  i si

(57)

 kisi2   i si  0

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Integrating both sides of Eq. (57), we have 1 Vi  t   si2  Vi  0    2





lim 0t  i si  kisi2 dt  Vi  0   

t 

(58)

According to Eq. (58), we have si  L , si  L2 . And according to (57), we have si  L . Consequently, by applying Babalat’s lemma we can conclude that the sliding surface S is asymptotically stable, i.e., lim S  0

t 

Then, the system can be guaranteed to be asymptotically stable.

Numerical and Experimental Verification In order to illustrate the effectiveness of the proposed controller, numerical simulations and experiments were conducted on the spherical mobile robot BYQ‐VIII. The following numerical values of the dimensions and inertial parameters of the robot are used [7]. ms  1.2 kg

mi  1.85 kg

I s  0.018 kg  m 2

m f  2.05 kg

R  0.15 m

I xxi  0.0017 kg  m 2

I xxf  0.0006 kg  m 2

I yyi  0.0068 kg  m 2

I yyf  0.0024 kg  m 2

  12 deg

l  0.12 m

I zzi  0.0071 kg  m 2

I zzf  0.0021 kg  m 2

The simulation runs were conducted without any joint frictions and unknown perturbations, and with the following initial conditions and coordinates of the desired target.

 x0 ,

0 ,  0 , 0 , 0    0, 0, 0, 0 

y0 , 0 ,  0 ,  0 ,  0 ,  0    0, 0 ,   2 , 0 , 0, 0 , 0 

 xd ,

yd    0.5, 0.5  m

For the proposed sliding mode control algorithm, the control parameters are chosen as 2 0

4 0

3.9

    0 4 K   0 0 2   

0    1.4 2  1.3 1   2  0.1  1   2  10 4.2  1

p1  p2  1

The simulation results are depicted in Fig. 4 and Fig. 5. As it is theoretically expected, the reference point of the robot is asymptotically stabilized to the desired target.

0.8

0.8

actual desired

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

1

2

3

4 t/s

5

actual desired

0.7

y/m

x/m

0.7

6

7

0

8

0

1

2

3

4 t/s

5

6

7

8

FIG. 4 SIMULATION RESULTS OF THE COORDINATE VARIABLES OF THE ROBOT REFERENCE POINT

In addition, an experimental test has been conducted on the real‐life spherical robot, having both joint frictions and external disturbances. The experimental setup is shown in Fig. 6, where a non‐slip mat with rubber material is laid on the incline to ensure that the slope surface can provide enough friction to prevent the robot from slipping. The experimental performance is illustrated in Fig. 7, which shows that the robot is finally stabilized to a vicinity of the desired target within a finite time. Under the parameters tuned to be 1  1.5 and  2  2.1 , practical stability and

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good performance of the robotic system were obtained: the convergence time of the point‐to‐point motion is about 5 sec, and the distance between the final position of the robot and the desired target is about 0.048m.

0.8

target start

0.7 0.6

y/m

0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 x/m

0.5

0.6

0.7

0.8

FIG. 5 SIMULATION RESULTS OF THE TRAJECTORY OF THE ROBOT REFERENCE POINT

FIG. 6 EXPERIMENTAL TESTBED FOR POINT‐TO‐POINT MOTION CONTROL OF THE SPHERICAL ROBOT 0.8

target start

0.7 0.6

y/m

0.5 0.4 0.3 0.2 0.1 0 −0.1

0

0.1

0.2

0.3

0.4 x/m

0.5

0.6

0.7

0.8

FIG. 7 EXPERIMENTAL RESULTS OF THE TRAJECTORY OF THE ROBOT REFERENCE POINT

Conclusions In this paper, a sliding mode approach with exponential reaching law is presented for point‐to‐point motion control of a spherical mobile robot rolling on an inclined plane. We first studied the non‐slipping and non‐spinning constraints of the robot and established the dynamic model of the robot using the constrained Lagrangian method.

Then we investigated the point‐to‐point motion control algorithm and derived a sliding mode control law that guaranteed the asymptotic stability of the robotic system. The basic principle of the proposed methodology is the introduction of a novel ERL to the control mechanism in order to both reduce the chattering levels and guanrantee the regulation performances, which is impossible to be achieved with the conventional ERL. Capabilities of the proposed synthesis and its performance issues were illustrated in numerical and experimental studies.

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ACKNOWLEDGMENT

The authors wish to acknowledge the financial support provided by the National Natural Science Foundation of China (No. 51175048) and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708011).

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[2]

Zhan, Q., Liu, Z., and Cai, Y. “A Back‐stepping Based Trajectory Tracking Controller for a Non‐chained Nonholonomic Spherical Robot.” Chinese Journal of Aeronautics, 21(5), 472‐480, 2008.

[3]

Cai, Y., Zhan, Q., and Xi, X. “Path Tracking Control of a Spherical Mobile Robot.” Mechanism and Machine Theory, 51, 58‐73, 2012.

[4]

Zheng, M., Zhan, Q., Liu, J., and Cai, Y. “Trajectory Tracking of a Spherical Robot Based on a RBF Neural Network.”

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Zheng, M., Zhan, Q., Liu, J., and Cai, Y. “Control of a Spherical Robot: Path Following Based on Nonholonomic Kinematics and Dynamics.” Chinese Journal of Aeronautics, 24(3), 337‐345, 2011.

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Yu, T., Sun, H., Zhang, Y., and Zhao, W. “Control and Stabilization of a Pendulum‐driven Spherical Mobile Robot on an Inclined Plane.” International Symposium on Artificial Intelligence, Robotics and Automation in Space, Turin, Italy, 22‐28, 2012.

[7]

Yu, T. “Study on Control Methodology for the Slope Motion of a Spherical Robot.” PhD diss., Beijing University of Posts and Telecommunications, 2014.

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Yue, M. and Deng, Z. “Dynamic Modeling and Optimal Controller Design of a Spherical Robot in Climbing State.” Chinese Journal of Mechanical Engineering, 45(11), 46‐51, 2009.

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Sang, S. et al., “Electric Magnetic Driving Mechanism Design and Performance Analysis of a Spherical Robot.” Journal of East China University of Science and Technology, Natural Science Edition, 37(6), 775‐781, 2011.

[10] Bloch, A. M., Reyhanoglu, M., and McClamroch, N. H. “Control and Stabilization of Nonholonomic Dynamic Systems.” IEEE Transactions on Automatic Control, 37(11), 1746‐1757, 1992.

[11] Xu, Y. S., Ben, H. B., Au, K. W. “Dynamic Mobility with Single‐wheel Configuration.” The International Journal of Robotics Research, 18(7), 728‐738, 1999.

[12] Zhang, X., Sun, L., Zhao K., and Sun L. “Nonlinear Speed Control for PMSM System Using Sliding‐mode Control and Disturbance Compensation Techniques.” IEEE Transactions on Power Electronics, 28(3), 1358‐1365, 2013.

[13] Hung, J. Y., Gao, W., and Hung, J. C. “Variable Structure Control: A Survey.” IEEE Transactions on Industrial Electronics, 40(1), 2‐22, 1993.

[14] Gao, W. and Hung, J. C. “Variable Structure Control of Nonlinear Systems: A New Approach.” IEEE Transactions on Industrial Electronics, 40(1), 45‐55, 1993.

Tao Yu was born in Jinzhou, Liaoning, China, in 1980. He received the B.S. and M.S. degrees in mechanical engineering from Jilin University, Changchun, Jilin, China, in 2002 and 2005, respectively, and the Ph.D. degree in mechanical engineering from Beijing University of Posts and Telecommunications, Beijing, China, in 2014.

He is currently an Associate Professor with Faculty of Mechanical Engineering and Automation, Liaoning University of Technology, Jinzhou, Liaoning, China. His main research interests include sliding mode control, intelligent control, and robotics.

Hanxu Sun was born in Hanzhong, Shanxi, China, in 1960. He received the M.S. degree in mechanical engineering from Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin, China, in 1986, and the Ph.D. degree in mechanical engineering from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1989.

He is currently a Professor with School of Automation, Beijing University of Posts and Telecommunications, Beijing, China. His main research interests concern the dynamics and control of industrial robots, space robots and special robots.

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