The length of the links and is given by meters. We will restrict the first joint position of our arm model to be in the range of and the second to be in the range of .
We will first discuss how to construct a Bayesian Network where we can sample from. We want to reach our target within time steps, i.e. we get a dynamic Bayesian Network with one node per time step (11 nodes), see Figure 9.
Each node represents the joint positions at time . For simplicity, we will use a discrete representation of the joint positions. Therefore we use a uniform grid to discretize the joint space. We will use a Gaussian motion prior in order to define the transition probabilities of from the th discrete joint position at time to the th joint position at time . Let be the joint position vector (in radians), then , where equals . The motion prior encodes our laziness, meaning that, if not necessary, we do not want to move away from .
In order to plan a trajectory to a certain end-effector position we still need to define our kinematic task space mapping. We will also use a discrete representation for task space (Cartesian coordinates of the hand). Here, we use again a uniform grid over the range for and . The probability of reaching the th discrete task space position when being in the th discrete joint space position is given by , where is the non-linear mapping from the joint positions to the endeffector coordinates. The covariance matrix is set to .