Murray at 01:21, 30 July 2009

2009-07-30T01:21:42Z

Murray: /* Chapter Summary */

2009-07-25T02:55:08Z

Chapter Summary

Murray at 02:49, 25 July 2009

2009-07-25T02:49:42Z

Murray at 02:32, 22 July 2009

2009-07-22T02:32:47Z

Murray at 02:14, 22 July 2009

2009-07-22T02:14:36Z

Murray at 02:09, 22 July 2009

2009-07-22T02:09:09Z

New page

{{chapter header|Hand Dynamics and Control|Nonholonomic Behavior in Robotic Systems|Nonholonomic Motion Planning}}

@@ Line 13: / Line 13: @@
 drawn both from some basic theorems in differential geometry and from
 nonlinear control theory.
 == Chapter Summary ==

← Older revision		Revision as of 02:55, 25 July 2009
Line 15:		Line 15:

	== Chapter Summary ==		== Chapter Summary ==
		+
		+	<ol>
		+	<li> ''Nonholonomic constraints'' are linear velocity constraints
		+	of the form
		+	<center><amsmath>
		+	\omega_i (q) \dot q = 0 \quad i =1, \ldots, k
		+	</amsmath></center>
		+	which cannot be integrated to give constraints on the configuration
		+	variables <amsmath>q</amsmath> alone. By choosing <amsmath>g_j(q), j = 1, \ldots, n-k =:m</amsmath> to
		+	be a basis for the null space of the linear velocity constraints, we
		+	get the associated control system
		+	<center><amsmath>
		+	\dot q = g_1 (q) u_1 + \cdots + g_m (q) u_m.
		+	</amsmath></center>
		+	The problem of ''nonholonomic motion planning'' consists of finding
		+	a trajectory <amsmath>q(\cdot): [0,T] \to {\mathbb R}^n</amsmath>, given <amsmath>q(0) = q_0</amsmath> and
		+	<amsmath>q(T) = q_f</amsmath>.
		+	</li>
		+
		+	<li> The ''Lie bracket'' between two vector fields <amsmath>f</amsmath> and <amsmath>g</amsmath> on
		+	<amsmath>{\mathbb R}^n</amsmath> is a new vector field <amsmath>[f,g]</amsmath> defined by
		+	<center><amsmath>
		+	[f,g](q) = \frac{\partial g}{\partial q} f(q) - \frac{\partial
		+	f}{\partial q} g(q).
		+	</amsmath></center>
		+	</li>
		+
		+	<li> A ''distribution'' <amsmath>\Delta</amsmath> is a smooth assignment of a
		+	subspace of the tangent space to each point <amsmath>q \in {\mathbb R}^n</amsmath>. One
		+	important way of generating it is as the span of a number of vector
		+	fields:
		+	<center><amsmath>
		+	\Delta = \mbox{span} \{ g_1 , \ldots, g_m \}.
		+	</amsmath></center>
		+	The distribution <amsmath>\Delta</amsmath> is said to be ''regular'' if the dimension
		+	of <amsmath>\Delta_q</amsmath> does not vary with <amsmath>q</amsmath>. The distribution <amsmath>\Delta</amsmath> is
		+	said to be ''involutive'' if it is closed under the Lie bracket,
		+	that is if for all <amsmath>f,g \in \Delta</amsmath>, we have <amsmath>[f,g] \in \Delta</amsmath>.
		+	</li>
		+
		+	<li> A distribution <amsmath>\Delta </amsmath> of dimension <amsmath>k</amsmath> is said to be ''integrable'' if there exist <amsmath>n-k</amsmath> independent functions whose
		+	differentials annihilate the distribution. ''Frobenius' theorem''
		+	asserts that a regular distribution is integrable if and only if it is
		+	involutive. A Pfaffian system or codistribution <amsmath>\Omega</amsmath>
		+	<center><amsmath>
		+	\Omega = \mbox{span} \{ \omega_1, \ldots, \omega_k \}
		+	</amsmath></center>
		+	is ''completely nonholonomic'' if the involutive closure of the
		+	distribution <amsmath>\Delta = \Omega^\perp</amsmath> spans <amsmath>{\mathbb R}^n</amsmath> for all <amsmath>q</amsmath>.
		+	</li>
		+
		+	<li> Consider the system
		+	<center><amsmath>
		+	\dot q = g_1 (q) u_1 + \cdots + g_m (q) u_m.
		+	</amsmath></center>
		+	The ''controllability Lie algebra'' is the Lie algebra generated by
		+	the vector fields <amsmath>g_1, \dots , g_m</amsmath>. It is the smallest Lie algebra
		+	containing <amsmath>g_1, \dots , g_m</amsmath>. ''Chow's theorem'' asserts that if
		+	the controllability Lie algebra is full rank, we can steer this system
		+	from any initial to any final point.
		+	</li>
		+
		+	<li> Given a distribution <amsmath>\Delta</amsmath>, the ''filtration'' associated
		+	with <amsmath>\Delta</amsmath> is defined by <amsmath>\Delta_1 = \Delta</amsmath> and
		+	<center><amsmath>
		+	\Delta_i = \Delta_{i-1} + [ \Delta_1, \Delta_{i-1}],
		+	</amsmath></center>
		+	where
		+	<center><amsmath>
		+	[ \Delta_1, \Delta_{i-1}] = \mbox{span} \{ [g,h]: g \in \Delta_1, h
		+	\in \Delta_{i-1} \}.
		+	</amsmath></center>
		+	The filtration is said to be ''regular'' if each of the <amsmath>\Delta_i</amsmath>
		+	are regular. For a regular filtration, the smallest integer <amsmath>\kappa</amsmath>
		+	at which rank <amsmath>\Delta_\kappa</amsmath> is equal to that of <amsmath>\Delta_{\kappa+1},
		+	\Delta_{\kappa+2}, \ldots </amsmath> is called the ''degree of nonholonomy''
		+	of the distribution. The ''growth vector'' <amsmath>r \in {\mathbb Z}^\kappa</amsmath>
		+	for a regular filtration is defined as <amsmath>r_i := \mbox{rank} \;
		+	\Delta_i</amsmath>. The ''relative growth vector'' <amsmath> \sigma \in
		+	{\mathbb Z}^\kappa</amsmath> is defined as <amsmath>\sigma_i = r_i - r_{i-1}</amsmath> with <amsmath>r_0 =
		+	0</amsmath>.
		+	</li>
		+
		+	<li> Given <amsmath>\Delta = \mbox{span} \{ g_1, \ldots, g_m \}</amsmath>, a ''Lie
		+	product'' is any nested set of Lie brackets of the generators <amsmath>g_i</amsmath>. A
		+	Lie algebra generated by <amsmath>\Delta</amsmath> is said to be ''nilpotent'' if
		+	there exists an integer <amsmath>k</amsmath> such that all Lie products of length
		+	greater than <amsmath>k</amsmath> are zero. A ''Philip Hall basis'' is an ordered
		+	set of Lie products chosen by a set of rules so as to keep track of
		+	the restrictions imposed by the properties of the Lie bracket, namely
		+	skew-symmetry and the Jacobi identity.
		+	</li>
		+	</ol>

	== Additional Information ==		== Additional Information ==

← Older revision		Revision as of 02:49, 25 July 2009
Line 1:		Line 1:
	{{chapter header\|Hand Dynamics and Control\|Nonholonomic Behavior in Robotic Systems\|Motion Planning}}		{{chapter header\|Hand Dynamics and Control\|Nonholonomic Behavior in Robotic Systems\|Motion Planning}}
		+
		+	In this chapter, we study the effect of nonholonomic constraints on
		+	the behavior of robotic systems. These constraints arise in systems
		+	such as multifingered robot hands and wheeled mobile robots, where
		+	rolling contact is involved, as well as in systems where angular
		+	momentum is conserved. We discuss the problem of determining when
		+	constraints on the velocities of the configuration variables of a
		+	robotic system are integrable, and illustrate the problem in a variety
		+	of different situations. The emphasis of this chapter is on the basic
		+	tools needed to analyze nonholonomic systems and the application of
		+	those tools to problems in robotic manipulation. These tools are
		+	drawn both from some basic theorems in differential geometry and from
		+	nonlinear control theory.
		+
		+	== Chapter Summary ==
		+
		+	== Additional Information ==

← Older revision		Revision as of 02:32, 22 July 2009
Line 1:		Line 1:
−	{{chapter header\|Hand Dynamics and Control\|Nonholonomic Behavior in Robotic Systems\|~~Nonholonomic~~ Planning}}	+	{{chapter header\|Hand Dynamics and Control\|Nonholonomic Behavior in Robotic Systems\|Motion Planning}}