Stable and grasped configurations

Two important subsets of ${{\cal C}_{adm}}$ are used in the manipulation planning problem. See Figure 7.15. Let ${\cal C}^p_{sta}$ denote the set of stable part configurations, which are configurations at which the part can safely rest without any forces being applied by the manipulator. This means that a part cannot, for example, float in the air. It also cannot be in a configuration from which it might fall. The particular stable configurations depend on properties such as the part geometry, friction, mass distribution, and so on. These issues are not considered here. From this, let ${{\cal C}_{sta}}\subseteq {{\cal C}_{adm}}$ be the corresponding stable configurations, defined as

$${{\cal C}_{sta}}= \{ (q^a,q^p) \in {{\cal C}_{adm}}\;\vert\; q^p \in {\cal C}^p_{sta} \} .$$ (7.19)

The other important subset of ${{\cal C}_{adm}}$ is the set of all configurations in which the robot is grasping the part (and is capable of carrying it, if necessary). Let this denote the grasped configurations, denoted by ${{\cal C}_{gr}}\subseteq {{\cal C}_{adm}}$. For every configuration, $(q^a,q^p) \in {{\cal C}_{gr}}$, the manipulator touches the part. This means that ${\cal A}(q^a) \cap {\cal P}(q^p) \not = \emptyset$ (penetration is still not allowed because ${{\cal C}_{gr}}\subseteq {{\cal C}_{adm}}$). In general, many configurations at which ${\cal A}(q^a)$ contacts ${\cal P}(q^p)$ will not necessarily be in ${{\cal C}_{gr}}$. The conditions for a point to lie in ${{\cal C}_{gr}}$ depend on the particular characteristics of the manipulator, the part, and the contact surface between them. For example, a typical manipulator would not be able to pick up a block by making contact with only one corner of it. This level of detail is not defined here; see [681] for more information about grasping.

We must always ensure that either $x \in {{\cal C}_{sta}}$ or $x \in {{\cal C}_{gr}}$. Therefore, let ${\cal C}_{free}= {{\cal C}_{sta}}\cup {{\cal C}_{gr}}$, to reflect the subset of ${{\cal C}_{adm}}$ that is permissible for manipulation planning.

The mode space, $M$, contains two modes, which are named the transit mode and the transfer mode. In the transit mode, the manipulator is not carrying the part, which requires that $q \in {{\cal C}_{sta}}$. In the transfer mode, the manipulator carries the part, which requires that $q \in {{\cal C}_{gr}}$. Based on these simple conditions, the only way the mode can change is if $q \in {{\cal C}_{sta}}\cap {{\cal C}_{gr}}$. Therefore, the manipulator has two available actions only when it is in these configurations. In all other configurations the mode remains unchanged. For convenience, let ${{\cal C}_{tra}}= {{\cal C}_{sta}}\cap {{\cal C}_{gr}}$ denote the set of transition configurations, which are the places in which the mode may change.

Using the framework of Section 7.3.1, the mode space, $M$, and C-space, ${\cal C}$, are combined to yield the state space, $X = {\cal C}\times M$. Since there are only two modes, there are only two copies of ${\cal C}$, one for each mode. State-based sets, ${X_{free}}$, ${X_{tra}}$, ${X_{sta}}$, and ${X_{gr}}$, are directly obtained from ${\cal C}_{free}$, ${{\cal C}_{tra}}$, ${{\cal C}_{sta}}$, and ${{\cal C}_{gr}}$ by ignoring the mode. For example,

$${X_{tra}}= \{ (q,m) \in X \;\vert\; q \in {{\cal C}_{tra}}\} .$$ (7.20)

The sets ${X_{free}}$, ${X_{sta}}$, and ${X_{gr}}$ are similarly defined.

The task can now be defined. An initial part configuration, $q^p_{init} \in {{\cal C}_{sta}}$, and a goal part configuration, $q^p_{goal} \in {{\cal C}_{sta}}$, are specified. Compute a path $\tau: [0,1] \rightarrow {X_{free}}$ such that $\tau(0) = q^p_{init}$ and $\tau(1) = q^p_{goal}$. Furthermore, the action trajectory $\sigma : [0,1] \rightarrow U$ must be specified to indicate the appropriate mode changes whenever $\tau(s) \in {X_{tra}}$. A solution can be considered as an alternating sequence of transit paths and transfer paths, whose names follow from the mode. This is depicted in Figure 7.16.

Figure 7.16: The solution to a manipulation planning problem alternates between the two layers of $X$. The transitions can only occur when $x \in {X_{tra}}$.