Force/Motion Sliding Mode Control of Three Typical Mechanisms

[[abstract]]A number of papers [1-7] have been presented to address the issues of multi-body mechanisms. Examples of their applications are found in gasoline and diesel engines, where the gas force acts on the slider and the motion is transmitted through the links. Whether the connecting rod is assumed to be rigid or not, the steady-state and dynamic responses of the connecting rod of the mechanism with time-dependent boundary condition were obtained by Fung et al. [1-3]. In addition, a number of controllers, for example, repetitive control [4], adaptive control [5],computed torque control [6], and fuzzy neural network control [7] were designed for the multi-body mechanisms. Over the past 25 years, the SMC algorithm [8-10] has been taken into account for dynamic control problems. The main feature of the SMC is to allow the sliding mode to occur on a prescribed switching surface, so that the system is only governed by the sliding equation and remains insensitive to a class of disturbances and parameter variations [8]. It is noted that the SMC is a robust control method and has been well established in pure motion control [9]. Afterwards, in order to eliminate the chattering phenomenon, which is commonly found in simulation of discontinuous SMC systems, and to simplify a hybrid numerical method that incorporates benefits of both SMC and differential algebraic equations, the (DAE) stabilization method was developed and successfully used to simulate constrained multi-body systems (MBS) whether under holonomic constraint or not [10]. However, the development of a control law which has been induced by a constrained force has not been adequately developed consistently in the previous studies. Su et al. [11] attempted to use the SMC for simultaneous position and force control on a constrained robot manipulator. They asserted that the control law, along with inclusion of the constraint force error in the definition of the sliding surface, produces an asymptotically stable force tracking error. However, Grabbe and Bridges [12] addressed their formulation as being a departure from the typical definition of a sliding surface, which is a linear differential equation in one tracking error variable [13], and the errors in the separate force control law and stability analysis were presented in [11]. Recently, Lian and Lin [14] have proposed a new sliding surface in terms of motion error and force error, and claimed that the errors in [11] are improved; therefore, the asymptotic stability of the motion-tracking error and force- tracking error can be ensured. However, Dixon and Zergeroglu [15] pointed out an error in the sliding mode control stability analysis of [14]. In this chapter, our intent is to improve the errors in [11, 14] and simplify the control design and stability proof for the three typical mechanisms, including the slider-crank mechanism, the quick-return mechanism and the toggle mechanism as shown in Figs. 1~3 respectively, which are not seen in any references addressing the force/motion SMC. Here, a separate sliding surface is proposed using the measurements of the angular position and speed of the crank, but the SMC algorithm is derived as well in a simple manner using only the force tracking error to construct the controller. In these schemes, the force tracking error is shown to be arbitrarily small by changing the force control feedback gain. Then, by exploiting the structure of its dynamics, the fundamental properties of the dynamics are obtained to facilitate controller design, whereby the asymptotic stability of motion tracking error in sliding surface and force tracking error accumulated in controller can be ensured. The organization of this chapter is arranged as follows. In Section 2, the kinematic and dynamic analysis of the multi-body mechanism is investigated. A number of previous papers [4-7, 16-17] have shown the position and speed controllers for the regulation and tracking problems of the multi-body mechanism in the theoretical analysis and experimental results. However, control of the constrained force has not been investigated. The SMC laws are designed in Section 3. The simulated examples are shown in Section 4 and, finally, some conclusions are drawn