Now consider defining tangent spaces on manifolds. Intuitively, the tangent space at a point on an -dimensional manifold is an -dimensional hyperplane in that best approximates around , when the hyperplane origin is translated to . This is depicted in Figure 8.8. The notion of a tangent was actually used in Section 7.4.1 to describe local motions for motion planning of closed kinematic chains (see Figure 7.22).

To define a tangent space on a manifold, we first consider a more complicated definition of the tangent space at a point in , in comparison to what was given in Section 8.3.1. Suppose that , and consider taking directional derivatives of a smooth function at a point . For some (unnormalized) direction vector, , the directional derivative of at can be defined as

(8.31) |

The directional derivative used here does not normalize the direction vector (contrary to basic calculus). Hence, , in which ``'' denotes the inner product or dot product, and denotes the gradient of . The set of all possible direction vectors that can be used in this construction forms a two-dimensional vector space that happens to be the tangent space , as defined previously. This can be generalized to dimensions to obtain

(8.32) |

for which all possible direction vectors represent the tangent space . The set of all directions can be interpreted for our purposes as the set of possible velocity vectors.

Now consider taking (unnormalized) directional derivatives of a smooth function, , on a manifold. For an -dimensional manifold, the tangent space at a point can be considered once again as the set of all unnormalized directions. These directions must intuitively be tangent to the manifold, as depicted in Figure 8.8. There exists a clever way to define them without even referring to specific coordinate neighborhoods. This leads to a definition of that is intrinsic to the manifold.

At this point, you may accept that is an -dimensional
vector space that is affixed to at and oriented as shown in
Figure 8.8. For the sake of completeness, however, a
technical definition of from differential geometry will be
given; more details appear in [133,872]. The construction
is based on characterizing the set of all possible directional
derivative operators. Let
denote the set of all smooth
functions that have domains that include . Now make the following
identification. Any two functions
are defined
to be *equivalent* if there exists an open set
such
that for any ,
. There is no need to
distinguish equivalent functions because their derivatives must be the
same at . Let
denote under this
identification. A directional derivative operator at can be
considered as a function that maps from
to
for some
direction. In the case of
, the operator appears as
for each direction . Think about the set of all directional
derivative operators that can be made. Each one must assign a real
value to every function in
, and it must obey two axioms
from calculus regarding directional derivatives. Let
denote a directional derivative operator at some (be
careful, however, because here is not explicitly represented since
there are no coordinates). The directional derivative operator must
satisfy two axioms:

It is helpful, however, to have an explicit way to express vectors in . A basis for the tangent space can be obtained by using coordinate neighborhoods. An important theorem from differential geometry states that if is a diffeomorphism onto an open set , then the tangent space, , is isomorphic to . This means that by using a parameterization (the inverse of a coordinate neighborhood), there is a bijection between velocity vectors in and velocity vectors in . Small perturbations in the parameters cause motions in the tangent directions on the manifold . Imagine, for example, making a small perturbation to three quaternion parameters that are used to represent . If the perturbation is small enough, motions that are tangent to occur. In other words, the perturbed matrices will lie very close to (they will not lie in because is defined by nonlinear constraints on , as discussed in Section 4.1.2).

Now consider different ways to express the tangent space at some point , other than the poles (a change of coordinates is needed to cover these). Using the coordinates , velocities can be defined as vectors in . We can imagine moving in the plane defined by and , provided that the limits and are respected.

We can also use the parameterization to derive basis vectors for the tangent space as vectors in . Since the tangent space has only two dimensions, we must obtain a plane that is ``tangent'' to the sphere at . These can be found by taking derivatives. Let be denoted as , , and . Two basis vectors for the tangent plane at are

(8.35) |

and

(8.36) |

Computing these derivatives and normalizing yields the vectors and . These can be imagined as the result of making small perturbations of and at . The vector space obtained by taking all linear combinations of these vectors is the tangent space at . Note that the direction of the basis vectors depends on , as expected.

The tangent vectors can now be imagined as lying in a plane that is tangent to the surface, as shown in Figure 8.8. The normal vector to a surface specified as is , which yields after normalizing. This could alternatively be obtained by taking the cross product of the two vectors above and using the parameterization to express it in terms of , , and . For a point , the plane equation is

(8.37) |

Steven M LaValle 2012-04-20