Now we also want to add the velocities of the joints to our planning scenario. Therefore, we will also incorporate controls of the robot in our model. The controls directly represent the accelerations of the joints. The control-dependent state transitions are now given by
where is set to . Now, in difference to the previous tasks we incorporated controls to our model. For each dimension we will use discrete actions , resulting in a action space of actions. The actions are unknown, and hence, like every unknown hidden variable, they can be integrated out : . The term denotes the action prior, similarly to the previous example we again use it to code our laziness, i.e. we prefer doing no action at all , where is set to .
As we can see the controls are excluded from the inference process, however, they can be easily calculated from an estimated trajectory . We will again use a discretization of the state space with a uniform grid. Valid velocities are in the range of