Condiciones y consideraciones

We will first analyze how it performs planning from one point to another in free space. As we continue, remember that the shortest path between two points is the Euclidean shortest path, a straight line from one to the other with the length equivalent to the Euclidean distance.

In our framework, the paths are comprised of a sequence of motion primitives, so the arm’s motion is artificially constrained to moving along a specific set of headings. Unless the arm can move from the start to the goal by repeating the sameprimitive the required number of times, at least one change in heading will occur and the path will be automatically longer than the Euclidean distance. Since a relatively small set of motion primitives are used throughout this thesis, it is expected that many changes in heading will be needed to move the arm from one point to another in free space. Given that it’s important to the planner’s efficiency to keep the branching factor low, a solution of enlarging the set of motion primitives isn’t practical.

Note, in the last paragraph when we referred to the ”sameprimitive”, we meant having the same unit vector. Two primitives can share a unit vector but have different magnitudes and the same statement would ring true for those two as well. Note that in our framework, this is quite common given that the motion primitives we use are multi-resolution (details are in Section 4.2).

Bear in mind that the statement made earlier about our framework constraining the arm’s mo- tions to a discrete set of actions is true regarding the set of static motion primitives. However, if

adaptive motion primitives are used, then during the search when the pre-defined criteria are met, the arm isn’t artificially constrained by the predefined set of primitives but rather by the specifics of the motions theampis capable of generating at runtime (Section 4.2). For example, in single-arm planning, if a state sis expanded whose end-effector position, efxyz(s), is within a pre-defined distance to the goal end-effector position,dik, we use an inverse kinematics solver to generate an additional motion primitive,ampik(s, sgoal)for states. Ifampik(s, sgoal)is a valid edge, then the corresponding segment of the arm’s path will be along the straight line path fromstosgoal, uncon- strained by the static primitives. Note that all of the experiments in this section contain goals that are more thandikfrom the initial pose of the end-effector (and quite a bit more sincedik = 0.20m). In this set of experiments, we ran the single-arm planner on the 30 experiments presented in Section 5.3.2, however, this time without any obstacles. We configured the planner to use the same cost function as presented in the previous set of experiments with the same set of motion primitives.

M PlowD contains 8 static motion primitives, each rotating a single shoulder or elbow joint8◦ in either direction. M PhighD contains 14 static motions, 8 of them rotate a single shoulder or elbow joint4◦ and the remaining 6 joints each rotate a single forearm or wrist joint 36◦. Two adaptive motion primitives are also used, ampik and ampos. We also compared our solutions to RRT*, which is configured to minimize the joint space distance traveled and configure it RRT* optimize the path for 10 seconds. An initial solution is generally found in the first 0.3 seconds.

In Table 8.3, we present our findings using a couple of methods of comparing the planners’ out- put (with and without any post-processing) to the line-of-sight paths. Average ratios are presented that compare the 3D end-effector path in Cartesian space to the 3D Euclidean distance between the start and goal. We also compare the joint space path length of the planned paths to the Euclidean distance between the joint configurations at the start and goal. Some examples of start-goal pairs and the planned paths are shown in Figure 8.5.

meanplanned end-eff. path start-goal distance mean

shortcutted end-eff. path start-goal distance mean

planned joint path start-goal joint distance

planning time (s)

ε= 100 1.38 1.09 1.31 0.18

RRT* (10sec) 1.23 1.09 1.06 10.0

Figure 8.5: Shown above are the paths taken by the tip of the end-effector between four different start-goal pairs in free space, with the gripper shown at the goal. The planned paths (in red) exhibit

obvious discretization artifacts throughout. Notice the single joint motion for the majority of the path until the search entersdikand theampikis used. The green lines represent the end-effector’s

motion after shortcutting is performed, which safely removes all intermediate points between the start and goal.

Table 8.3 is a summary of the effects of planning on a manipulation lattice graph (a discretization of the C-space) on the solution quality in this set of experiments. The table clearly shows that the average joint path length is 31% longer than the Euclidean distance between the start and goal arm configurations. We believe that especially in free space, the end-effector distance traveled is a more meaningful measure because humans appear to take direct paths by the hand through space when performing reaching tasks in the absence of obstacles. In the first column of the table, we see that the discretization artifacts are responsible for a 38% longer motion by the end-effector than required by the line-of-sight path. However, the second column shows that a simple shortcutting process performed in 5-10ms would cut down the additional end-effector motion to just 9% which seems to be the distance traveled by the end-effector along the interpolated joint path between start and goal. In addition, we used RRT*, an algorithm that plans directly in C-space, as another point of comparison for planning in free space. We gave it 10 seconds to find and optimize the path. We found that the relative end-effector paths and joint paths are shorter than those computed by our planner in 0.18 seconds by approximately 15% and 25%, respectively. It undoubtedly follows that after a quick shortcutting step, the path lengths become the same because all intermediate points can be removed in free space.