Many important applications involve a mixture of discrete and continuous variables. This results in a state space that is a Cartesian product of the C-space and a finite set called the mode space. The resulting space can be visualized as having layers of C-spaces that are indexed by the modes, as depicted in Figure 7.11. The main application given in this section is manipulation planning; many others exist, especially when other complications such as dynamics and uncertainties are added to the problem. The framework of this section is inspired mainly from hybrid systems in the control theory community , which usually model mode-dependent dynamics. The main concern in this section is that the allowable robot configurations and/or the obstacles depend on the mode.