Non - Adaptive Sliding Mode Controllers in Terms of Inertial Quasi - Velocities 14

Trajectory control problem arises if the manipulator is required to follow a desired trajectory. In the robotic literature mainly two approaches are used: computer torque (inverse dynamic control) and sliding mode control Sciavicco & Siciliano (1996); Slotine & Li (1991). The system under inverse dynamics controller is linear and decoupled with respect to the newly obtained input. In robotics literature very popular is the sliding mode method described by Slotine & Li (1987; 1991). The approach differs from the previous one because even if the parameters are exactly known, the manipulator equations of motion are not linearized by the control law. The sliding mode control strategies are used in the manipulator joint space as well as in its operational space Sciavicco & Siciliano (1996); Slotine & Li (1987; 1991). From the practical point of view to track the position of the end-effector of the manipulator is more convenient than the joint position tracking because the task is realized directly. The motion control problem in the manipulator joint space and the operational space is investigated also in newer references Kelly & Moreno (2005); Moreno & Kelly (2003); Moreno et al. (2003); Moreno-Valenzuela & Kelly (2006). Sometimes also a friction model is taken into account, e.g. Moreno et al. (2003); Moreno-Valenzuela & Kelly (2006). One of known applications of the sliding mode approach allows one to control a shape Mochiyama et al. (1999). In order to design various versions of control laws strict Lyapunov functions for a class of global regulators for robot manipulators are introduced Santibanez & Kelly (1997); Spong (1992) or in terms of the IQV also in Herman (2009b). Classical description leads to obtaining second-order nonlinear differential equations of motion. The equations involve both generalized position vector and velocity vector which represent a joint space of the manipulator. However, for control purposes first-order equations of motion with diagonal mass matrix seemmore convenient than the second-order equations. It is possible to consider the dynamics of mechanical systems using quasi-velocities and differential geometry Kwatny & Blankenship (2000). The obtained first-order equations of motion are the Poincaré’s form of the Lagrange’s equations. One of useful solutions which leads to the diagonal or the unit inertia matrix is introducing so called inertial quasi-velocities (IQV). There exist several methods which enable such decomposition (e.g.Hurtado (2004); Jain & Rodriguez (1995); Junkins & Schaub (1997); Loduha & Ravani (1995); Sovinsky et al. (2005)). The method presented in Hurtado (2004) is associated with the Cholesky decomposition Sovinsky et al. (2005). In the method described by Jain & Rodriguez (1995) the normalized quasi-velocities (NQV) and the unnormalized quasi-velocities (UQV) were introduced. The Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities 14