Difference between revisions of "Nonholonomic Behavior in Robotic Systems"
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== 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 == |
Revision as of 02:55, 25 July 2009
Prev: Hand Dynamics and Control | Chapter 7 - Nonholonomic Behavior in Robotic Systems | Next: 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
- Nonholonomic constraints are linear velocity constraints
of the form
<amsmath> \omega_i (q) \dot q = 0 \quad i =1, \ldots, k
</amsmath>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
<amsmath> \dot q = g_1 (q) u_1 + \cdots + g_m (q) u_m.
</amsmath>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>.
- 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
<amsmath> [f,g](q) = \frac{\partial g}{\partial q} f(q) - \frac{\partial f}{\partial q} g(q).
</amsmath> - 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:
<amsmath> \Delta = \mbox{span} \{ g_1 , \ldots, g_m \}.
</amsmath>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>.
- 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>
<amsmath> \Omega = \mbox{span} \{ \omega_1, \ldots, \omega_k \}
</amsmath>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>.
- Consider the system
<amsmath> \dot q = g_1 (q) u_1 + \cdots + g_m (q) u_m.
</amsmath>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.
- Given a distribution <amsmath>\Delta</amsmath>, the filtration associated
with <amsmath>\Delta</amsmath> is defined by <amsmath>\Delta_1 = \Delta</amsmath> and
<amsmath> \Delta_i = \Delta_{i-1} + [ \Delta_1, \Delta_{i-1}],
</amsmath>where
<amsmath> [ \Delta_1, \Delta_{i-1}] = \mbox{span} \{ [g,h]: g \in \Delta_1, h
\in \Delta_{i-1} \}.
</amsmath>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>.
- 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.