Hierarchical Sliding Mode Control for a 3D Ballbot that is a Class of Second-order Under-actuated System

This paper proposes a hierarchical sliding mode controller for a three-dimensional ballbot in an extremely complicated operation of three omni-directional wheels, a ball, and body. Three ball motors simultaneously drive four outputs comprising ball motion and two tilt angles of the body. Simulation tests are performed to investigate the controller qualities. The proposed controller asymptotically stabilizes and consistently maintains system response.


INTRODUCTION
Ball robot (ballbot) with an omnidirectional driving mechanism shows the capability of omnidirectionality, agility, and maneuverability on the ground surface. These features allow the ballbot to be applied in various fields of automatic control and robotics areas [1][2][3][4].
Numerous studies have utilized different methods to investigate ballbot control. These methods vary from classical control methods [6][7][8] to modern techniques with single loop control system [9,10] and two multi-loop control system [11][12][13][14]. Intelligent control such as fuzzy logic control [15][16][17][18] was utilized to control both balancing and transferring ballbots on the ground. Ching-Chih Tsai [19] presented an intelligent consensus-based cooperative formation control using recurrent fuzzy wavelet cerebellar-model-articulation controller for a team of uncertain multiple ballbots. However, experiments were not included in this paper.
The sliding mode control (SMC) which belongs to a kind of variable-structure control system have been shown to be a robust and effective approach for the nonlinear mechanical system. The application of SMC for ball robot control can be found in several previous studies. Ching-Wen Liao [20] and Cheng-Kai Chan [21] proposed SMC method based on back-stepping to accomplish robust balancing and agile path tracking of the robot with unknown friction, parameters variations, and exogenous disturbances. The computer simulations, these SMC controllers were shown to be capable of steering this kind of robot to the desired position.
3D ballbot is naturally an under-actuated mechanical system, in which the number of actuators is less than the degree of freedom of the system. Furthermore, Actuators directly drive the ball, whereas the body has no direct control. With the inverted pendulum structure of the ballbot the large mass and momentum inertia of the body cause difficulty in maintaining balance and transfer. Therefore, the hierarchical SMC is a suitable approach to use in this study.

DYNAMIC SYSTEM
A ballbot is considered a multi-body system. The physical system of the ballbot is shown in Figure 1. The ballbot system consists of three point masses, namely mk, mw and ma, representing the masses of the spherical wheel (ball), each omni-directional wheel and all rotating components including the individual body, three driving motors and three gear boxes. Correspondingly, four generalized coordinates contesting of ball position xk(t) and yk(t), and motion body x(t) and y(t) are selected. x and y torques of three driving motors acting on the ball denote the control inputs. In addition, the friction of the omni-directional wheel-ball surface and that of ball-ground surface are linearly characterized by damping coefficients brx, bry, bx and by. The control inputs x and y denote torques composed of three motors along the x-and y-axes.
The equations of motion derived with EulerLagrange's equations describe the ballbot dynamics and are expressed in matrix form as where     T  M q M q is the symmetric mass matrix,   , C q q  denotes the matrix of centrifugal and damping elements,   G q indicates the gravity vector,   F q is the non-square matrix of control torque of the driving motors and u is control input vector. These components

Hierarchical Sliding Mode Control for a 3D Ballbot that is a Class of Second-order
where ma, mk are the masses of body, ball, respectively, Ik refer to the momentum inertia of the ball, Iw is momentum inertia of each omnidirectional wheel, Ix, Iy and l are momentum inertia about the x-and y-axes and the center of mass along the vertical axis of the body, rk and rw are radiuses of the ball and each omnidirectional wheel, g is the gravity acceleration, and  is zenith angle. The coefficients of   , C q q  are determined in Appendix A.
The non-zero elements of  

 
system dynamics (1) should be decomposed into four second-order nonlinear sub-systems where fi(x) with i = (1  4) and bi1(x), bi2(x) are the continuous nonlinear functions.

Design of control system
The first-layer sliding mode surfaces (SMSs) are defined by here Ci is a positive constant, xkd = const and ykd = const are the reference position. Differentiating si with respect to time yields Substituting (2) into (4) yields The control input u1 or u2 are composited of three terms to be where ueqi is the equivalent control input, and uswj (j = 1  2) is the switch control part of SMC. Let The hierarchical structure of the SMSs is designed in the following manner. The second-layer SMSs of sub-systems are chosen as based on the first-layer SMSs as follows: where Ki is positive. Differentiating of Sk and Sa with respect to time obtains Substituting (6), (7) into (5), and (5) into (14),   eq eq eq eq k eq eq eq eq eq eq eq eq a eq eq eq eq where Kk and k are positive constants. Then, where Ka and a are positive constants. Then, Solving (19) and (22) yields

NUMERICAL RESULTS
The system parameters using for simulation involve ma = 116 kg, Ix Case 1: Balancing and station-keeping control. In the first simulation, the robot has to keep its location with initial values of 5  and 5  induced to the roll and pitch angles of the body. Figs. 2 to 6 are the system performance of the ballbot under the proposed control system with the initial condition. The movement of the ball on the ground is shown in Fig. 3. Figs. 5 and 6 show the first-and second-level SMSs. Moreover, control inputs are illustrated in Fig. 6. The numerical results show that the controller is able to reach the zero references and maintain the robot stability.
Case 2: Tracking control. This simulation aims to control tracking motion. In this test, the robot is recommended to track the desired circle trajectory with radius of 0.5 m within a period of 200 s. The simulation results in Figs. 7 and 8 show that the proposed controller is capable of steering the robot to the desired path.

CONCLUSIONS
A hierarchical SMC was improved for a 3D ballbot system in a complicated operation. Numerical tests were performed. Most of the system responses were asymptotically stabilized. The body tilt angles approached zero. The ball motion approached the desired trajectory on the floor. The controller guarantees system robustness.