Systems with Nonholonomic Constraints

Control of Underactuated Driftless Nonlinear Systems

3.2 Mechanical Systems with Nonholonomic Constraints

3.2.1 Systems with Nonholonomic Constraints

Definition 22 The horizontal subspace of the space of vector fields onQ, is denoted byX^H(Q) ⊂ X(Q).

It is the subspace of vector fields which are horizontal for all points inQ. The vertical subspace of the space of vector fields onQis denoted byX^V(Q)⊂ X(Q). It is the subspace of vector field which are vertical for all points inQ.

Definition 23 [69] Given an Ehresmann connection and a vector field onR,X ∈ X(R), there is a uniquely defined horizontal vector field onQ, denoted byX^h ∈ X^H(Q), such that

T π◦X^h=X.

The horizontal vector field,X^his called the horizontal lift ofX. Conversely, given a horizontal vector field X^h ∈ X^H(Q), there exists a vector fieldX ∈ X(R), whose horizontal lift isX^h.

Typically the base space, R, will be directly actuated and fully controllable. Consequently, there exists a collection of control vector fields,{Y_a}^m1 , wherem= dim(R)andY_a∈ X(M). The horizontal lift of the control vector fields, in local coordinates, may be used to give the governing equations for an underactuated driftless affine system,

˙

r =Y_au^a

˙

s=−A_loc(r, s)·Y_au^a (3.49)

Therefore, under constraints defined by an Ehresmann connection, the standard form for an underactuated driftless affine control system, given in Equation (3.1), may be rewritten as,

˙

q=Y_a^hu^a. (3.50)

The involutive closure atq ∈QisC^∞(q), c.f. Equation (3.5), and consists of iterated Jacobi-Lie brackets of horizontally lifted control vector fields. Using the structure of the Ehresmann connection, it is possible to refine the Lie algebra rank condition from Equation (3.6).

The Lie Algebra Rank Condition. Since vertical vector fields inX^V(Q)areπ-related to the zero vector field inX(R), it is possible to show that

hX^h, Y^vi

∈ X^V(Q), (3.51)

whereX ∈ X(R) and Y^v ∈ X^V(Q). In other words, the Jacobi-Lie bracket between a horizontal and a vertical vector field is again vertical. Additionally, the curvature of the Ehresmann connection, denoted by B, is given by,

B(X^h, Y^h) =h

X^h, Y^hi

. (3.52)

Importantly, the curvature is vertical valued. Consequently, the only vector fields inC^∞with a nontrivial horizontal projection are the horizontal lifts of the control vector fields.

Theorem 20 Under the assumption that the control vector fields, Ya ∈ X(R), span the tangent space T_π(q)Rforq∈Q, the Lie algebra rank condition atq is equivalent to the condition that

spann h

Y_i^h_k. . . ,h Y_i^h₂,h

Y_i^h₁, Y_i^h₀iii (q)o

=V_qQ, (3.53)

fork ∈Z⁺\ {0}.

The Ehresmann connection can be used to define the notion of parallel transport. Parallel transport is given by flow along horizontal vector fields,

Pt^X_0,t= Φ^X_0,t^h, X ∈ X(R). (3.54)

Parallel transport naturally leads to the notion of a covariant derivative for elementsf ∈C^∞(R),

∇Xf ≡ LX^hf (3.55)

Corollary 7 [69] Under the assumption that the control vector fields,Y_a∈ X(R), span the tangent space T_π(q)Rforq∈Q, the Lie algebra rank condition atq is equivalent to the condition that

spann

∇Y_ik· · · ∇Yi2B(Y_i^h₁, Y_i^h₀)(q)o

=V_qQ, (3.56)

fork ∈Z⁺\ {0}.

Because the curvature and its higher order covariant derivative are all vertical, the local curvature form may be used instead. The local curvature form may be obtained from the complementary projection,

B_loc(X, Y)≡T π_S◦B(X^h, Y^h), ∀X, Y ∈ X(R). (3.57) The covariant derivative,∇, induces a covariant derivative, ∇e, on the local curvature form via the identity,

∇eXB_loc(Y, Z) =T π_S

∇XB(Y^h, Z^h)

, ∀X, Y, Z∈ X(R). (3.58) Corollary 8 Under the assumption that the control vector fields,Y_a∈ X(R), span the tangent spaceT_π(q)R forq ∈Q, the Lie algebra rank condition atqis equivalent to the condition that

spann

∇eY_ik· · ·∇eYi2B_loc(Y_i1, Y_i0)(q)o

=T_πS(q)S, (3.59)

fork ∈Z⁺\ {0}.

The Lie algebra rank condition has been simplified to the condition that the local form of the curvature and its covariant derivatives span the tangent to the fiber.

Controllability of Systems with Constraints. From Kelly and Murray [67], we may define the notion of strong and weak fiber controllability.

Definition 24 A system (3.50) is said to be strongly fiber controllable if, for anyq_i, q_f ∈Q, there exists a timeT >0and a curver(·)∈Rconnectingr_i=π(q_i)andr_f =π(q_f)such that the horizontal lift ofr(·) passing throughq_isatisfies,r^h(0) =q_iandr^h(T) =q_f.

Strong controllability is the standard definition of controllability interpreted using an Ehresmann connection form and its horizontal lift. For systems with constraints, trajectories in the configuration space must be horizontal lifts of trajectories in the base space. Kelly and Murray further noted that some locomotive systems did not require strong controllability, since the evolution of the base space was not of primary importance. By not imposing final conditions on the base space, weak controllability may be defined.

Definition 25 A system (3.50) is said to be weakly controllable if, for anyq_i ∈Qands_f ∈S, there exists a finite timeT >0and a curve in the base spacer(·)whose horizontal lift passing throughq_iat timet= 0 satisfies,r^h(T) =q_f, wheres_f =πS(q_f).

Weak controllability does not place restrictions on the final point of the base space. Mobile robots whose base space corresponds to the angles of wheels are systems that may be weakly controlled. By not restricting the final angle of the wheel, greater freedom is allowed. Most importantly, the need to stabilize or control the base space no longer holds for weakly controlled systems. Since controllability of only the fiber is sought, the control brackets may also be used to span the vertical space.

Theorem 21 For the system (3.50), the following are equivalent:

1. the system is small-time weakly controllable, 2. span

T_qπ_S Y_i^h

k. . . , Y_i^h₃,

Y_i^h₂, Y_i^h₁ (q)

=T_πS(q)S, 3. spann n

T_qπ_S

∇Y_ik· · · ∇Yi2B(Y_i^h₁, Y_i^h₀)(q) o S

T_qπ_S(Y_i^h₀) o

=T_πS(q)S, 4. spann n

∇eY_ik· · ·∇eYi2B_loc(Y_i1, Y_i0)(q)o S

T_qπ_S(Y_i^h₀) o

=T_πS(q)S, fork ∈Z⁺\ {0}anda= 1. . . m.

The theorem follows from the definition of weak controllability, Theorem 20, and Corollaries 7 and 8.

The Averaged Expansions. Suppose that the system (3.50) were to be weakly fiber controllable, and there was no need to stabilize the base space. Then the averaged expansions from Section 3.1 may be simplified. Suppose furthermore that controllability was obtained through the curvature and higher-order covariant derivatives.

Third-order averaging using the averaged expansion of Equation (3.19) is r˙

˙ s

=

( 0

1

2V^(α,β)_(1,0)(t)B_αβ(r, s) +¹₃V^(α,β,γ)_(1,1,0) (t)B_αβγ(r, s) )

(3.60) whereB_αβis the local form of the curvature form, defined as follows,

B_αβ ≡B_loc(Y_α, Y_β), andB_αβγ is the covariant derivative of the local curvature form,

B_αβγ ≡ ∇YαB_loc(Y_β, Y_γ).

Due to the structure of the vector fields, there will never be a contribution to the base space unless the first order average is non-vanishing. This means that all expansions will only affect the fiber space. In the average, the evolution of the base space is trivial, a very nice result when dealing with systems of this type.

Since, in the average, the base variables are constant, replacing them by their averages in the equations of motion reduces Equation (3.60) to

˙ s= 1

2V^(α,β)_(1,0)(t)B_αβ(r, s) +1

3V^(α,β,γ)_(1,1,0) (t)B_αβγ(r, s) (3.61) The equations have been simplified to evolution of the fiber only, possibly rendering further analysis tractable.

Higher-order averages have the same simplifying property.

Many of these results can be found in the literature when the system with nonholonomic constraints also involves Lie groups symmetries [137, 67]. The next section discuss the consequences of reduction under group symmetry.