# An Approach to the Dynamics and Control of Uncertain Robot Manipulators
## 1 Introduction  
The main methods currently used for dynamic modeling of robot manipulators with motion constraints are the Newton-Euler method [1; 2; 3], [PERSON]'s method [4; 5], and [PERSON]'s method [6]. The Newton-Euler method describes motion and force through the use of vectors. In the modeling procedure, every component of the mechanism is isolated and the corresponding Newton and Euler equations are established. The calculations involved are quite efficient, but it is hard to use this approach when attempting to design control systems for manipulators. For a multibody system with motion constraints, [PERSON]'s method can be used, generally with the introduction of Lagrange multipliers, a widely used technique for constrained systems. However, controlling these multipliers is difficult, and the approach is not very well suited for symbolic considerations. [PERSON]'s method combines the advantages of vector mechanics and analytical mechanics, with a generalized rate being used as an independent variable in the equations of motion of the system. The fundamental vector projection of the main force and the inertial force of the system is extended directly to derive the equations of motion. However, with this method, these dynamical equations cannot be obtained in the appropriate analytical form for a constrained mechanical system. All of these approaches share the common fundamental problem that they cannot provide explicit expressions for the dynamical equations of the constrained system. As is well known, the generation of dynamical equations forconstrained systems in symbolic form has a number of advantages with regard to issues of both control and mechanical design [7]. To address the control of robot manipulators, valuable control approaches have been presented, such as Jacobian-Matrix-Adaption (JMA) Method [8], sliding mode control (SMC) [9], Robust adaptive control [10], and adaptive neural network control [11]. Unlike most control methods for the control problem of robot manipulators, the present paper does not make any linearizations or approximations. Our methodology is based on the Udwadia-Kalaba equation (UKE) [12; 13; 14; 15; 16; 17]. The Udwadia-Kalaba equation is obtained by using [PERSON]'s principle rather than the more commonly used principles of Lagrange, Hamilton, Gibbs, and Appell [18]. This equation takes constraints into account in the dynamical equation and involves the generalized Moore-Penrose inverse [19]. It provides a simple and general explicit equation of motion for constrained mechanical systems without the need for Lagrange multipliers [14; 20]. This relatively simple approach allows detailed dynamical analysis of such systems and should improve fundamental understanding of constrained motion in multibody dynamics. Based on UKE, many researchers have used this method to solve problems in the dynamics analysis of constrained systems. [PERSON] et al. [21] use UKE to study the slider-crank mechanism and compared the numerical efficiency with the coordinate partitioning (CPE) equation. [PERSON] et al. [22] applied the approach to the trajectory tracking control of the mobile robot, by solving UKE to obtain the control input not make any linearization or approximations. For additional applications using UKE see [23; 24; 25; 26; 27; 28]. In this paper, we regard the desired trajectory as constraints, by using the UK equation to obtain the control input. Considering the moment of inertial of the robot manipulators are uncertain, a nonlinear controller was adopted to make nonlinear system to track the given trajectory. In the control methodology, this nonlinear controller is augmented by an additional additive controller based on a generalization of the notion of sliding surfaces [15; 16; 17; 29]. It's important to notice that the control obtained relies on recent advances in analytical dynamics rather than on control theory. The paper is organized as follows. In Section 2, we introduce the Udwadia-Kalaba equation.