We have seen that a path specifies the set of configurations a robot achieves as it moves from one configuration to another, and thus path planning (e.g., finding collision-free paths) is a kinematic/geometric problem. A path is not a complete description of the motion of a robot system, however, as the timing of the motion is not specified. A trajectory is a path plus a specification of the time at which each configuration is achieved. Trajectory planning is not only a geometric problem, but also a dynamic problem. Finding feasible trajectories of a system obeying dynamics requires knowledge of the masses and inertias of the system, actuator limits, and forces such as gravity and friction. Since we are now dealing with system dynamics, we can pose optimal control problems such as finding minimum-time or minimum-energy motions.

Since trajectory planning requires a full dynamic model of the robotic system, in section 10.1 we review the Lagrangian approach to deriving equations of motion for a mechanical system such as a robot arm. Section 10.2 explores the structure of the equations of motion and gives standard forms for writing them. In section 10.3 we consider systems subject to velocity constraints, such as those imposed when maintaining rolling and sliding contacts. Finally, section 10.4 studies the particular case of a rotating and translating rigid body.