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(Non-)Holonomic systems (constrains)

The Frobenius Theory

posted Nov 10, 2014, 9:47 PM by Javad Taghia   [ updated Nov 10, 2014, 9:47 PM ]

15.4.2.4 The Frobenius Theorem

The Lie bracket is the only tool needed to determine whether a system is completely integrable (holonomic) or nonholonomic (not integrable). Suppose that a system of the form (15.53) is given. Using the  system vector fields  there are Lie brackets of the form  for  that can be formed. A distribution  is called involutive [133] if for each of these brackets there exist  coefficients  such that

 (15.86)

In other words, every Lie bracket can be expressed as a linear combination of the system vector fields, and therefore it already belongs to . The Lie brackets are unable to escape  and generate new directions of motion. We did not need to consider all  possible Lie brackets of the system vector fields because it turns out that  and consequently . Therefore, the definition of involutive is not altered by looking only at the  pairs.

If the system is smooth and the distribution is nonsingular, then the Frobenius theorem immediately characterizes integrability:

A system is completely integrable if and only if it is involutive.
Proofs of the Frobenius theorem appear in numerous differential geometry and control theory books [133,156,478,846]. There also exist versions that do not require the distribution to be nonsingular.

Determining integrability involves performing Lie brackets and determining whether (15.86) is satisfied. The search for the coefficients can luckily be avoided by using linear algebra tests for linear independence. The  matrix , which was defined in (15.56), can be augmented into an  matrix  by adding  as a new column. If the rank of  is  for any pair  and , then it is immediately known that the system is nonholonomic. If the rank of  is  for all Lie brackets, then the system is completely integrable. Driftless linear systems, which are expressed as  for a fixed matrix , are completely integrable because all Lie brackets are zero.

Example 15..11 (The Differential Drive Is Nonholonomic)   For the differential drive model in (15.54), the Lie bracket  was determined in Example 15.9 to be . The matrix , in which , is
 (15.87)

The rank of  is  for all  (the determinant of  is ). Therefore, by the Frobenius theorem, the system is nonholonomic.
Example 15..12 (The Nonholonomic Integrator Is Nonholonomic)   We would hope that the nonholonomic integrator is nonholonomic. In Example 15.10, the Lie bracket was determined to be . The matrix  is
 (15.88)

which clearly has full rank for all
Example 15..13 (Trapped on a Sphere)   Suppose that the following system is given:
 (15.89)

for which  and . Since the vector fields are linear, the Jacobians are constant (as in Example 15.10):
 (15.90)

Using (15.80),
 (15.91)

This yields the matrix
 (15.92)

The determinant is zero for all , which means that  is never linearly independent of  and . Therefore, the system is completely integrable.15.10

The system can actually be constructed by differentiating the equation of a sphere. Let

 (15.93)

and differentiate with respect to time to obtain
 (15.94)

which is a Pfaffian constraint. A parametric representation of the set of vectors that satisfy (15.94) is given by (15.89). For each , (15.89) yields a vector that satisfies (15.94). Thus, this was an example of being trapped on a sphere, which we would expect to be completely integrable. It was difficult, however, to suspect this using only (15.89).

Steven M LaValle 2012-04-20

Example Unicycle

posted Nov 10, 2014, 6:21 PM by Javad Taghia   [ updated Nov 10, 2014, 6:24 PM ]

Lie Brackets

posted Nov 10, 2014, 6:06 PM by Javad Taghia   [ updated Nov 10, 2014, 6:11 PM ]

15.4.2.3 Lie brackets

The key to establishing whether a system is nonholonomic is to construct motions that combine the effects of two action variables, which may produce motions in a direction that seems impossible from the system distribution. To motivate the coming ideas, consider the differential-drive model from (15.54). Apply the following piecewise-constant action trajectory over the interval :

 (15.71)

The action trajectory is a sequence of four motion primitives: 1) translate forward, 2) rotate forward, 3) translate backward, and 4) rotate backward.

The result of all four motion primitives in succession from  is shown in Figure 15.16. It is fun to try this at home with an axle and two wheels (Tinkertoys work well, for example). The result is that the differential drive moves sideways!15.9From the transition equation (15.54) such motions appear impossible. This is a beautiful property of nonlinear systems. The state may wiggle its way in directions that do not seem possible. A more familiar example is parallel parking a car. It is known that a car cannot directly move sideways; however, some wiggling motions can be performed to move it sideways into a tight parking space. The actions we perform while parking resemble the primitives in (15.71).

Algebraically, the motions of (15.71) appear to be checking for commutativity. Recall from Section 4.2.1 that a group  is called commutative (or Abelian) if  for any . A commutator is a group element of the form . If the group is commutative, then  (the identity element) for any . If a nonidentity element of  is produced by the commutator, then the group is not commutative. Similarly, if the trajectory arising from (15.71) does not form a loop (by returning to the starting point), then the motion primitives do not commute. Therefore, a sequence of motion primitives in (15.71) will be referred to as the commutator motion. It will turn out that if the commutator motion cannot produce any velocities not allowed by the system distribution, then the system is completely integrable. This means that if we are trapped on a surface, then it is impossible to leave the surface by using commutator motions.

Now generalize the differential drive to any driftless control-affine system that has two action variables:

 (15.72)

Using the notation of (15.53), the vector fields would be  and ; however,  and  are chosen here to allow subscripts to denote the components of the vector field in the coming explanation.

Suppose that the commutator motion (15.71) is applied to (15.72) as shown in Figure 15.17. Determining the resulting motion requires some general computations, as opposed to the simple geometric arguments that could be made for the differential drive. If would be convenient to have an expression for the velocity obtained in the limit as  approaches zero. This can be obtained by using Taylor series arguments. These are simplified by the fact that the action history is piecewise constant.

The coming derivation will require an expression for  under the application of a constant action. For each action, a vector field of the form  is obtained. Upon differentiation, this yields

 (15.73)

This follows from the chain rule because  is a function of , which itself is a function of . The derivative  is actually an  Jacobian matrix, which is multiplied by the vector . To further clarify (15.73), each component can be expressed as
 (15.74)

Now the state trajectory under the application of (15.71) will be determined using the Taylor series, which was given in (14.17). The state trajectory that results from the first motion primitive  can be expressed as

 (15.75)

which makes use of (15.73) in the second line. The Taylor series expansion for the second primitive is
 (15.76)

An expression for  can be obtained by using the Taylor series expansion in (15.75) to express . The first terms after substitution and simplification are
 (15.77)

To simplify the expression, the evaluation at  has been dropped from every occurrence of  and  and their derivatives.

The idea of substituting previous Taylor series expansions as they are needed can be repeated for the remaining two motion primitives. The Taylor series expansion for the result after the third primitive is

 (15.78)

Finally, the Taylor series expansion after all four primitives have been applied is
 (15.79)

Taking the limit yields
 (15.80)

which is called the Lie bracket of  and  and is denoted by . Similar to (15.74), the th component can be expressed as
 (15.81)

The Lie bracket is an important operation in many subjects, and is related to the Poisson and Jacobi brackets that arise in physics and mathematics.

Example 15..9 (Lie Bracket for the Differential Drive)   The Lie bracket should indicate that sideways motions are possible for the differential drive. Consider taking the Lie bracket of the two vector fields used in (15.54). Let  and. Rename  and  to  and  to allow subscripts to denote the components of a vector field.

By applying (15.81), the Lie bracket  is

 (15.82)

The resulting vector field is , which indicates the sideways motion, as desired. When evaluated at , the vector  is obtained. This means that performing short commutator motions wiggles the differential drive sideways in the  direction, which we already knew from Figure 15.16

Example 15..10 (Lie Bracket of Linear Vector Fields)   Suppose that each vector field is a linear function of . The  Jacobians  and  are constant.

As a simple example, recall the nonholonomic integrator (13.43). In the linear-algebra form, the system is

 (15.83)

Let  and . The Jacobian matrices are
 (15.84)

Using (15.80),
 (15.85)

This result can be verified using (15.81).

Steven M LaValle 2012-04-20

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