We propose a decentralized errorbounded sliding mode control mechanism that ensures the prescribed tracking performance of a robot manipulator. A tracking errortransformed sliding surface was constructed and the barrier Lyapunov function (BLF) was used to ensure the transient and steadystate time performance of the positioning function of a robot manipulator as well as satisfy the ordinary sliding mode control properties. Unknown nonlinear functions and approximation errors are estimated by the RBF network and adaptive compensator. The effectiveness of the proposed control scheme was determined by comparing the results of an experiment evaluation with those of the conventional sliding mode control (SMC) and finitetime terminal sliding mode control (FTSMC) methods.
Robotic technology is rapidly developing in the industrial medical areas, particularly, in constraining the robot’s motion and force. Many researchers have studied geometric constraints, such as holonomic and nonholonomic constraints of a mobile robot and of the endeffector of a manipulator [
However, restricting the motions of a working robot within the free space is relatively difficult because a systematic approach for this problem has not been developed yet. Motion constraints within the free space prevent unexpected collisions between the robot and the environment, as well as other hazards. Most of the traditional systems designed a controller to guarantee the stability of the control system for an infinite length of time. These controllers obtained the desired performance by trial and error. Although a control gain has already been decided for a specific control objective, changes in the design specification would require that the control gains be retuned or that the controller be redesigned. Therefore, systematic and general constraint control performance is difficult to obtain from these conventional control schemes.
Three methods have been proposed to guarantee a timedomain performance in the design step without depending on the trialanderror method: a funnel control method [
The SMC technique provides robust nonlinear control because it applies system dynamics with invariant properties to uncertainties after the system dynamics are controlled on sliding mode surface [
The decentralized control combined with the SMC [
The dynamic equation of an
An errortransformed sliding mode surface is obtained by setting the joint tracking error as
A proper error constraint function to prescribe the performance is selected by the following:
RBF networks have been widely applied to many engineering fields. An RBF network is a fully connected threelayered feedforward network with a single layer of hidden units, called RBFs. In RBF outputs, which have trained connected weights, the maximum value is shown at the center, and the output values decrease as the input moves away from the center. The Gaussian function is typically used for activation. The RBFs are
The control objectives of the robot manipulator are as follows.
Determine the control laws such that the system output
The prescribed constraints for the output tracking error,
The time derivative of (
On the other hand, if the dynamics of the robotic systems are generally unknown, the approximation method for a nonlinear unknown function using a RBF system is generally used to tackle this problem. The unknown function of the robotic dynamics can be expressed using the RBF networks as follows:
The BLF used in [
The time derivative of
If the conventional SMC with RBF networks is considered instead of the proposed SMC, the sliding surface is adopted as a type of (
Next, the continuous finitetime terminal SMC (FTSMC) with RBF networks was used to guarantee a rapid convergence time compared to the conventional SMC and the proposed barrier Lyapunov functionbased SMC (BSMC). FTSMC is defined as
The proposed control scheme was evaluated using an experimental application on the Scorbot robot system that is described in Figure
Manipulator parameters.
Symbol  Parameter  Quantity 


Mass of link1 and 2  12.1 kg, 3.59 kg 

Mass of link1 and 2  0.3 m, 0.41 m 

Stiction level of joint1 and 2  0.063 Nm, 0.0648 Nm 

Coulomb friction of joint1 and 2  0.061 Nm sec/rad, 0.06 Nm sec/rad 

Stibeck velocity of joint1 and 2  0.00075 rad/sec, 0.00063 rad/sec 

Bristle stiffness of joint1 and 2  5400 Nm/rad, 8700 Nm/rad 

Presliding damping of joint1 and 2  5.4 Nm sec/rad, 6.2 Nm sec/rad 

Sliding damping of joint1 and 2  10.2 Nm sec/rad, 10.8 Nm sec/rad 
Photograph of the Scorbot robot ERII system.
Diagram of the Scorbot robot control system.
The sinewave joint motion was chosen as the desired trajectory. The sinewave position command was
Position tracking outputs of the SMC (dashed) and FTSMC (solid): (a) link1. (b) link2.
Position tracking results. (a) Tracking output of BSMC in link1. (b) Tracking output of BSMC in link2. (c) Tracking errors of SMC (dashed), FTSMC (dashdot), and BSMC (solid) in link1. (d) Tracking errors of SMC (dashed), FTSMC (dashdot), and BSMC (solid) in link2.
Position tracking results of BSMC: (a)
Control inputs. (a) Link1, SMC (dashed) and FTSMC (soild). (b) Link2, SMC (dashed) and FTSMC (solid). (c) BSMC, link1 (dashed) and link2 (solid).
A BLFbased decentralized errorconstrained SMC was developed to guarantee the position tracking performance of a robotic manipulator in the presence of unknown nonlinear dynamics. The errortransformed sliding surface was proposed to ensure the prescribed tracking error bound and to effectively compensate for the decentralizing uncertainty as well as the sliding condition. A prototype of the Scorbot manipulator demonstrated that the proposed BSMC scheme satisfies the prescribed tracking performance with RBF network approximation for an unknown nonlinear function. Therefore, the designed controller can have a simple structure and can more conveniently control the positioning function of robotic manipulator.
This research was supported by the Ministry of Science, ICT and Future Planning (MSIP), Korea, under the Information Technology Research Center (ITRC) support program (NIPA2013H0301132006) supervised by the National IT Industry Promotion Agency (NIPA).