A coordinate-free theory of virtual holonomic constraints

This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.


Introduction.
A virtual holonomic constraint (VHC) for a Lagrangian control system is a collection of relations among the configuration variables of the system that can be made invariant via feedback control.The precise meaning of this terminology is clarified in what follows, but the key idea is to emulate via feedback control the presence of a holonomic constraint in the Lagrangian control system.By appropriate design of the VHC, the constrained system may display useful properties.
The notion of VHC can be traced back to early twentieth century work of P. Appell in [2] and H. Beghin in [4], but it has emerged prominently in the last fifteen years as a tool for control of biped robots (see, e.g., [21,23,33,32,7]), and as an approach to motion planning for general robotic systems (e.g., [26,27,28,11]).In VHC-based robot control, the motion one wants to induce is represented implicitly in terms of constraints on the robot's configuration variables, and the control loop is designed to asymptotically stabilize a subset of the state space, the so-called constraint manifold.This control philosophy stands in contrast to the standard technique of parametrizing a desired motion by time, and then stabilizing the resulting reference signals.The VHC control paradigm has proved particularly effective in inducing complex behaviours in underactuated robots, and gives rise to a feedback loop that is intrinsically robust because it is not driven by any exogenous signal.
For biped robots, Grizzle and collaborators (see, e.g., [32]) defined VHCs in terms of invariance of a submanifold of the state space.The paper [18] generalized Grizzle's notion of VHC to mechanical control systems whose generalized coordinates are linear displacements or angles (i.e., systems whose configuration manifold is a generalized cylinder) and whose degree of underactuation is equal to one.For this class of systems, the authors of [19,20] showed that, generically, the constrained dynamics in the presence of a VHC do not possess a Lagrangian structure.They then gave necessary and sufficient conditions for a Lagrangian structure to exist.The theory of [18,20] does not handle mechanical systems whose configuration space is not a generalized cylinder, or whose degree of underactuation is greater than one.To illustrate, the configuration manifold of a rigid body is SE(3), a manifold that cannot be handled by the theory in [18,20].Similarly, a double pendulum on a cart has degree of underactuation two, which again is not contemplated in the theory of [18,20].
Main contributions.This paper generalizes the theory in [18,20] by presenting a coordinate-free formulation of VHCs for arbitrary configuration manifolds and arbitrary degrees of underactuation.We give a new geometric definition of VHC, and define a regularity property of VHCs in terms of transversality of two subbundles.We show that a regular VHC induces on the constraint manifold an affine connection, the so-called induced connection.In the absence of a potential function, orbits of the constrained dynamics are geodesics of this induced connection.We give an explicit characterization of the constrained dynamics in coordinates with formulas for the Christoffel symbols of the induced connection.We show that the problem of determining whether or not the constrained dynamics are Lagrangian amounts in great part to determining whether or not the induced connection is metrizable, i.e., it is Riemannian for a suitable metric.Leveraging this insight, and using existing results from the theory of affine connections, we give conditions for the existence of a Lagrangian structure for the constrained dynamics arising from a regular VHC.These conditions are applicable to Lagrangian control systems with arbitrary degree of underactuation.In the special case when the subbundle associated with the control accelerations is orthogonal to the constraint manifold, the constrained dynamics are always Lagrangian, and we show that they coincide with the dynamics one would obtain in the presence of an ideal holonomic constraint.Thus, the classics mechanics notion of holonomic constraint is a special case of our theory.For systems with underactuation degree one, our results provide an elegant geometric insight for the results in [20].
The focus of this paper is on the case when all control inputs are used to enforce the VHC, so that the constrained dynamics are unforced.In the more general case when the constrained dynamics are forced, the question of existence of a Lagrangian structure for the constrained dynamics turns into the more general question of feedback equivalence of the constrained dynamics to a Lagrangian control system.A local version of the latter question has been investigated in [24] for general control systems on smooth manifolds near zero velocity points.
Organization of the paper.In Section 2 we review concepts of Riemannian geometry, and the definition from [6] of a Lagrangian (control) system on a Riemannian manifold.Section 3 reviews the definition of VHC from [18] and the Lagrangian properties of the constrained dynamics from [20], valid for the case of systems with degree of underactuation one.Section 4 formulates a new coordinate-free theory of VHCs, characterizing the regularity of VHCs in terms of transversality of the constraint manifold and the distribution induced by control forces.It is shown that a VHC induces an affine connection on the constraint manifold, and this connection is then used to characterize the constrained dynamics.In Section 5 we give necessary and sufficient conditions under which the constrained dynamics are Lagrangian.We also treat the special case when the distribution induced by control forces is orthogonal to the constraint manifold.In Section 6 we give a tutorial overview of holonomy groups and results on metrizability of affine connections, treating the special cases of flat connections, of simply connected constraint manifolds, and one and two-dimensional constraint manifolds.Here we show that the results of Section 3 are a special case of the general theory of this paper.In Section 7, we present three examples illustrating the theory, in order of increasing complexity.The last example is a double pendulum on a cart with a VHC that constrains the angle of the second pendulum to be a function of the angle of the first pendulum, in such a way that the second pendulum is always confined to the upper half plane.

Preliminaries.
In this section we present the notation used in this paper, review notions of Riemannian geometry, and review the definition of a Lagrangian (control) system on a Riemannian manifold.All results are found in [16,10,5,6].
Smooth manifolds.If M is a smooth manifold, we denote by C ∞ (M) the ring of smooth real-valued functions on M, by X(M) the set of smooth vector fields on M, and by Ω(M) the set of smooth one-forms on M. The tangent space to M at p ∈ M is denoted by T p M, and it dual, the cotangent space, is denoted by T ⋆ p M. We denote by T M and T ⋆ M the tangent and cotangent bundles of M, and by π : T M → M the natural projection on T M.An element of T M will be denoted by v q , with the understanding that v q ∈ T q M. If N is a submanifold of M, T M| N denotes the restriction of T M to N , defined as T M| N = p∈N T p M.
If (U, (x 1 , . . ., x n )) is a coordinate chart of M, for each p ∈ U the basis for T p M induced by the chart is denoted by ∂/∂x 1 | p , . . ., ∂/∂x n | p .The vector fields {∂/∂x 1 , . . ., ∂/∂x n } form a local frame for T M. If F : M → N is a smooth function between smooth manifolds and p ∈ M, we let F p := F (p), and we denote by open sets, we denote by ∂ x i F the partial derivative F with respect to its i-th argument.The notation ∂ 2 x i x j F indicates second-order partial differentiation with respect to the i-th and j-th argument.More generally, if U ⊂ R n is an open set and F : U → F (U ) ⊂ M is smooth, then we denote ∂ x i F := dF x (∂/∂x i ), where ∂/∂x i denotes the i-th natural basis vector of T x R n .In the special case of a function of one variable in R or S 1 , we let F ′ (x) := ∂ x F and Riemannian manifolds and connections.A Riemannian manifold is a pair (M, g), where M is a smooth manifold, and g : T M × T M → R, the Riemannian metric, is a smooth function such that, for each p ∈ M, g p is a bilinear form T p M × T p M → R which is symmetric and positive definite, i.e., for each v p , w p ∈ T p M, g p (v p , w p ) = g p (w p , v p ), and the function v p → g p (v p , v p ) is positive definite.Thus, g p is an inner product on T p M which varies smoothly with p.In the language of tensors, g is a type (0, 2) symmetric and positive definite tensor field on M .A Riemannian metric induces two maps.The flat map is the function for any f, g ∈ C ∞ (M) and X, Y, Z ∈ X(M).The vector field ∇ X Y is called the covariant derivative of Y in the direction of X.The covariant derivative of vector fields induces a covariant derivative of tensor fields, also denoted ∇, enjoying the properties listed in [16,Lemma 4.6].Among them, we mention the following.If F is a tensor field on M of type (0, s), and X ∈ X(M), then ∇ X F is a type (0, s) tensor field satisfying the following identity for all Y 1 , . . ., Y s ∈ X(M).The total covariant derivative of a type (0, s) tensor field F is the type (0, s + 1) tensor field ∇F given by The connection ∇ is symmetric or, in terms of the total covariant derivative, The Fundamental Lemma of Riemannian Geometry (e.g., [16]) states that there is a unique affine connection ∇ on a Riemannian manifold (M, g) with the property of being symmetric and compatible with g.This connection is called the Riemannian connection or the Levi-Civita connection of g.The covariant derivative ∇ X Y may be viewed as a differentiation of the vector field Y along X.If γ(t) is a smooth curve on M and Y ∈ X(M), the restriction of Y to γ(t), V (t) := Y (γ(t)), is a vector field along γ.The derivative D t V := ∇ γ Y is called the covariant derivative of V along γ.Although the definition just given relies on expressing V as the restriction to γ of a vector field Y on M, D t V does not depend on the values of Y outside of γ(t), in that any smooth extension of V outside of γ gives the same value of D t V .The geometric intuition of the notion of covariant derivative is as follows (see, e.g., [5,Chapter VII,Section 2]).If M is an embedded submanifold of R n with Riemannian metric induced from an inner product on R n , D t V is the orthogonal projection of the time derivative of Y (γ(t)) onto the tangent space T γ(t) M. Thus, roughly speaking, D t V measures how much the vector field V (t) turns as seen from the point of view of M. In essence, covariant derivatives embody the notion of acceleration of a curve.More precisely, the acceleration of a curve γ on M is the vector field D t γ along γ, and γ is called a geodesic of ∇ if its acceleration is zero, i.e., ∇ γ γ(t) = D t γ(t) ≡ 0. We remark that this definition of geodesic curve does not require ∇ to be a Riemannian connection.
Coordinate representation of the covariant derivative.In coordinates, covariant derivatives associated with a Riemannian connection take on a familiar form, which we now review.Consider a coordinate chart (U, (x 1 , . . ., x n )) on M and the associated local frame {∂/∂x 1 , . . ., ∂/∂x n } for T M. Let X i := ∂/∂x i .Given an affine connection ∇ on M, not necessarily Riemannian, the Christoffel symbols of ∇ associated with the local frame {X 1 , . . ., X n } are the where Y (z k ) is the Lie derivative of z k along Y .The acceleration of a smooth curve γ : I → M, I ⊂ R, can be computed as follows.Letting γ i (t) := x i (γ(t)) denote the i-th component of the coordinate representation of γ, we have We see that, in local coordinates, geodesics are solutions of the system of secondorder differential equations If ∇ is Riemannian, the Christoffel symbols may be computed using a matrix representation of the metric g.Using again the local frame {X 1 , . . ., X n }, let g ij (p) := g(∂/∂x i | p , ∂/∂x j | p ), and let g kl be (k, l)-th element of the inverse of the matrix (g ij ).Then, Lagrangian control systems on manifolds.Having reviewed basic notions of Riemannian geometry, we are ready to present the class of mechanical systems considered in this paper.The definitions below are adapted from [6].
Definition 2.1 (Lagrangian system).A Lagrangian system is a triple (Q, g, P ), where (Q, g) is an n-dimensional Riemannian manifold called the configuration manifold, and P : Q → R is a smooth function called the potential function.The triple (Q, g, P ) is also called a Lagrangian structure.A smooth curve q : I → Q, where I is an open interval in R, is a base integral curve of the Lagrangian system if ∇ q(t) q(t) = − grad P (q(t)), (9) for all t ∈ I. △ For each q 0 ∈ Q and each v q0 ∈ T q0 Q, there exists a unique maximal base integral curve q(t) such that q(0) = q 0 and q(0) = v q0 , where maximality is defined in the same way as for integral curves of vector fields.We will call q(t) the maximal base integral curve of (9) with initial condition (q 0 , v q0 ).
Base integral curves have the property of being extremizers of the action functional I L(q(t), q(t))dt, I ⊂ R, where L : T Q → R is the Lagrangian function defined as L(q, q) := 1 2 g q ( q, q) − P (q).( In Lagrangian systems, controls appear by way of forces.On Riemannian manifolds, forces are modelled as one-forms because, under coordinate changes, they transform like the components of one-forms (see [6]).Thus, a force on Q is a one-form F ∈ Ω(Q).The corresponding vector field F ♯ ∈ X(Q) can be thought of as the portion of the acceleration due to F .With this in mind, we proceed to the definition of a Lagrangian control system.
Definition 2.2 (Lagrangian control system).A Lagrangian control system is a quadruple (Q, g, P, F), where (Q, g, P ) is a Lagrangian system and F = {F 1 , . . ., F m }, F i ∈ Ω(Q), are called the control forces.A curve q : I → Q, where I is an open interval in R, is a base integral curve of the Lagrangian control system if there exist m smooth functions τ i : for all t ∈ I. △ Let τ ⋆ = (τ ⋆ 1 , . . ., τ ⋆ m ) : T Q → R m be a smooth function.Then, for each q 0 ∈ Q and each v q0 ∈ T q0 Q, there exists a unique maximal solution q : I → Q of (11) with τ i (t) = τ ⋆ i (q(t), q(t)), i = 1, . . ., m.We call it the maximal base integral curve of (11) with feedback τ = τ ⋆ (q, q) and initial condition (q 0 , v q0 ).
It is shown in [6] that the equations of motion of the Lagrangian system in (9) can be equivalently expressed as a smooth vector field on T Q.Similarly, the equations of motion of the Lagrangian control system in (11) have an equivalent representation as a smooth control-affine system with state space T Q.
In coordinates, equations ( 9) and (11) take on the familiar form of the Euler-Lagrange equations.More precisely, let q : I → Q be a base integral curve of a Lagrangian system (Q, g, P ).Given a chart (U, φ) for Q, let x = φ • q be the coordinate representation of q, and let L(x, ẋ) := L(φ −1 (x), (dφ −1 ) x ẋ), where L is the Lagrangian function defined in (10).Then, x = (x 1 , . . ., x n ) : I → R n satisfies the Euler-Lagrange equations Vice versa, if x : I → R n satisfies the above Euler-Lagrange equations, then q : I → Q, q := φ −1 • x is a base integral curve of (Q, g, P ).An analogous property holds for Lagrangian control systems (Q, g, P, F), where now q : I → Q is a base integral curve of (Q, g, P, {F i } i=1,...,m ) if, and only if, x = φ • q is a solution of the forced Euler-Lagrange equations In the above ) is the i-th coefficient of the expansion of F j in the local frame {∂/∂x 1 , . . ., ∂/∂x n } induced by the chart.Let D(x) be the matrix with components D ij (x) = g ij (φ −1 (x)), and set C(x, ẋ) = D(x) C(x, ẋ), where Ckj (x, ẋ) = i Γ k ij (φ −1 (x)) ẋi .Let P := P • φ −1 , and let B(x) be the matrix with elements B ij (x).Then the Euler-Lagrange equations ( 12) take on the familiar form where τ is the vector whose components are the control inputs τ i in (12).
3. Review of Virtual Holonomic Constraints.The configuration manifold Q of a robot whose joints are revolute or prismatic is a generalized cylinder.In other words, an element of Q may be represented as an n-tuple (q 1 , . . ., q n ), where each q i is either in R if the i-th joint is prismatic, or in S 1 if the i-th joint is revolute.In this case, the Lagrangian control system (11) admits a global coordinate representation of the form (13): D(q)q + C(q, q) q + ∇ q P (q) = B(q)τ.(14) In this section, we review basic facts concerning VHCs for the class of mechanical control systems (14).The rest of this paper will be devoted to the generalization of these results to the coordinate-free setting.We assume, throughout, that the matrix B has full-rank m.Definition 3.1 (Virtual holonomic constraint in coordinates, [18]).A virtual holonomic constraint of order k for system ( 14) is a relation h(q) = 0, where h : Q → R k is a smooth function such that 0 is a regular value of h and the set Γ = {(q, q) ∈ T Q : h(q) = 0, dh q q = 0} (15) is controlled invariant for (14).The set Γ is called the constraint manifold associated with the VHC h(q) = 0. △ In the definition above, requiring Γ to be controlled invariant means requiring that there exists a smooth feedback τ = τ ⋆ (q, q) rendering Γ positively invariant for the closed-loop system.If we let C := h −1 (0), then the hypothesis that rank (dh q ) = k for all q ∈ h −1 (0) implies that C is a closed embedded submanifold of Q.Moreover, the constraint manifold Γ in (15) is the tangent bundle of C, Γ = T C.
In the context of nonlinear control, the constraint manifold associated with a VHC h(q) = 0 is the zero dynamics manifold of system (14) with output e = h(q) (see [12]).A special case of interest is when this output function yields a well-defined relative degree, as in the next definition.Definition 3.2 (Regular VHC in coordinates, [18]).A relation h(q) = 0 is a regular VHC of order k for system (14) if h : Q → R k is smooth, and system (14) with output e = h(q) has vector relative degree {2, . . ., 2} everywhere on the constraint manifold Γ in (15).In other words, rank (dh q D −1 (q)B(q)) = k for all q ∈ h −1 (0).△ Since the matrix dh q D −1 (q)B(q) has dimension k × m, for a regular VHC it must hold that the number of constraints, k, be less than or equal to the number of controls, m.The next result provides a geometric interpretation of the regularity condition.

Proposition 3.3 ([13]
).A relation h : Q → R k is a regular VHC for system (14) if, and only if, letting In light of the proposition above, the VHC h(q) = 0 is regular if, for each q ∈ C, the mechanical system can produce control accelerations D −1 Bτ in any direction transversal to T q C. In the special case k = m, when the number of constrains is equal to the number of controls, the subspace sum in ( 16) becomes direct.
Regular VHCs are important in two respects.First, since the output e = h(q) yields a well-defined vector relative degree on the constraint manifold Γ, one may use input-output linearization to asymptotically stabilize Γ.For this, some technical assumptions on h and its differential are required, see [18].Second, regular VHCs induce well-defined constrained dynamics, as stated in the next proposition 1 .In what follows, we let B ⊥ : Q → R 1×n \{0} be a smooth left annihilator of B with rank one everywhere on h −1 (0).

Proposition 3.4 ([18]
).Let m = n − 1, and let h(q) = 0 be a regular VHC of order n − 1 for system (14).Let ϕ : Θ → Q be a regular parametrization of the curve C = h −1 (0), where Θ = S 1 if C is a Jordan curve, and Θ = R otherwise.Letting (q, q) = (ϕ(s), ϕ ′ (s) ṡ), the dynamics on the set Γ in (15) are globally described by the second-order differential equation where (s, ṡ) ∈ Θ × R and where (Γ i ) jk = Γ i jk is the matrix containing the Christoffel symbols of the metric g q (v q , w q ) = v ⊤ q D(q)w q .The dynamics in (17) are called the constrained (or reduced) dynamics associated with the VHC h(q) = 0.The next result, taken from [20], characterizes when the constrained dynamics (17) possess a Lagrangian structure.

Theorem 3.5 ([20]
). Define a map π : R → Θ as Define smooth functions MC , PC : R → R as, and (generally multi-valued) functions M C , P C : Θ ⇒ R as Then the following statements are true.As will become apparent in the development that follows, the results reviewed in this section are a special case of the general theory developed in this paper.

4.
Coordinate-free formulation of virtual holonomic constraints.In this section we reformulate and generalize the theory of Section 3 in a coordinate-free context.We consider throughout a Lagrangian control system (Q, g, P, F) with equations of motion We assume that the one-forms F = {F 1 , . . ., F m } are independent, and define the acceleration distribution 4.1.VHC definitions and relationships.
that for each q 0 ∈ C and each v q0 ∈ T q0 C, the maximal base integral curve q : I → Q of ( 11) with feedback τ = τ ⋆ (q, q) and initial condition (q 0 , v q0 ) satisfies q(t) ∈ C for all t ∈ I. △ Definition 4.2 (Virtual holonomic constraint).A virtual holonomic constraint of order k for the Lagrangian control system (11) where D A is the acceleration distribution defined in (21).
Transversality condition in the definition of regular VHC.
The transversality condition (22), illustrated in Figure 1, generalizes ( 16) in the case when the number of constraints, k, is equal to the number of controls, m.Moreover, as we now show, the foregoing definition of regularity can be expressed in terms of relative degree.Proposition 4.4.A closed embedded submanifold C of Q is a regular VHC of order m for (20) if and only if for each q ∈ C there exists an open subset U of Q containing q and a smooth submersion h : U → R m such that the following property holds.For any base integral curve q(t) of (20) such that Im(q) ⊂ U , letting e(t) = h(q(t)) it holds that ëi = g(∇ q grad h i , q) − g(grad h i , grad where b ij = g(grad h i , (F j ) ♯ ), and the m×m matrix with components b ij is invertible on C ∩ U .
The reader will recognize that the identity (23) implies that system (20) with output e = h(q) has vector relative degree {2, . . ., 2} on C ∩ U .
Proof.Since C is a closed embedded submanifold of Q with codimension m, by [17,Proposition 5.16] for each q ∈ C there exists a smooth submersion h : U → R m , where U is an open subset of Q containing q, such that C ∩ U = h −1 (0).Let q(t) be a base integral curve of (20) such that Im(q) ⊂ U , and set e(t) = h(q(t)).We have e i (t) = h i (q(t)) and ėi (t) = (dh i ) q(t) q(t).Using the sharp operator, the latter identity can be rewritten as ėi = g(grad h i , q).By the compatibility property (4) of Riemannian connections, we have ëi = g(∇ q grad h i , q) + g(grad h i , ∇ q q).Using the equations of motion (20), we obtain the expression in (23).Using the definition of grad, the functions b ij in (23) can be expressed as Let B(q) be the matrix-valued function with components b ij (q), and let q ∈ C ∩U be arbitrary.It remains to be shown that B(q) is invertible if and only if T q C ⊕D A (q) = T q Q.To this end, we use the following fact from linear algebra.If V, W are finitedimensional vector spaces, A : V → W is a surjective linear map, and Z is a subspace of V, then A(Z) = W if and only if Ker(A) + Z = V.Now using the fact that Ker dh q = T q C and applying the foregoing linear algebra result with V := T q Q, W := R m , Z := D A (q), and A := dh q , we deduce that Im(B(q)) = R m (i.e., B is invertible) if and only if T q C + D A (q) = T q Q.The sum is direct because the dimensions of the two subspaces being summed are complementary.
Remark 4.5.When C may be globally described by the zero level set of a smooth submersion2 , Proposition 4.4 recovers the relative degree property of Definition 3.2.In this case, just as in Section 3, the constraint manifold T C can be asymptotically stabilized3 by an input-output linearizing feedback τ ⋆ deduced from ( 23) by assigning the error dynamics ëi Is a regular VHC of order m a VHC in the sense of Definition 4.2?The answer is positive, as stated in the next result.Proposition 4.6.If a closed embedded submanifold C of Q is a regular VHC of order m for system (20), then C is also a VHC in the sense of Definition 4.2, and the smooth feedback its invariance is characterized by a tangency criterion, as we shall see.To the Lagrangian control system (20), we associate an equivalent control-affine system on T Q (see [6, Section 4.6.3]),whose first-order equations will be written in the form A smooth feedback τ ⋆ = (τ ⋆ i ) m i=1 : T C → R m renders T C invariant for the Lagrangian control system (20) if, and only if, it renders T C invariant for the controlaffine system (24).In turn, since T C is a closed embedded submanifold of T Q, τ ⋆ renders T C invariant for the control-affine system (24) if and only if the closedloop vector field (24) with τ = τ ⋆ (ξ) is tangent to T C. In other words, letting f ⋆ := f 0 + j f j τ ⋆ j , it holds that, for all ξ ∈ T C, f ⋆ (ξ) ∈ T ξ (T C).We will construct local feedbacks inducing this tangency property, then glue them together to get a global feedback on T C. To this end, let {U i } i∈N be a collection of open subsets of Q such that C ⊂ i U i and, for each i ∈ N, there exists a smooth submersion . The existence of the family {(U i , h i )} i∈N is guaranteed by the second countability of Q and the fact that C is an embedded submanifold.For each i ∈ N, we have We seek a feedback τ i : T C| Ui → R m inducing the tangency property Using the identity (25), the tangency property ( 26) is equivalent to the following requirement.If ξ(t) is an integral curve of the control affine system (24) with feedback τ = τ i (ξ) such that Im(ξ) ⊂ T Q| Ui , then letting e i (t) := h i • π(ξ(t)), it holds that (e i (t) ≡ ėi (t) ≡ 0) =⇒ ëi (t) ≡ 0.
Using Proposition 4.4, the above implication holds if and only if τ i (ξ) = B(q) −1 (−g(∇ q grad h i , q) + g(grad h i , grad P )) ξ=(q, q)∈T C|U i , where B(q) is the matrix-valued function with components b ij (q) given in the proposition.We thus have, for each open set U i , a unique smooth feedback τ i : T C| Ui → R m such that the vector field (24) By construction, τ ⋆ induces the tangency property ( 26) on each open subset V i of C, and therefore also on the entire set T C. The uniqueness of τ ⋆ follows from the uniqueness of the maps τ i .
The uniqueness of the feedback τ ⋆ in the foregoing proposition implies that the constrained dynamics on T C are characterized by an unforced dynamical system.This is due to the fact that, since the codimension of C is m, all m control forces are assigned to enforce the invariance of the constraint manifold.Our next objective is to characterize the constrained dynamics.We would like a coordinate-free generalization of Proposition 3.4 valid for any m, not just m = n − 1.

Constrained dynamics.
A consequence of Proposition 4.4 is that, for a regular VHC of order m, the dynamics on the constraint manifold T C are governed by an unforced dynamical system.As we now show, the constrained dynamics are described by a special affine connection on C induced by the VHC.This so-called induced connection was originally developed in the context of affine differential geometry (see [22,Chapter 2]).We adopt it in the context of regular VHCs.
Before giving a formal definition of the induced connection, we present the basic idea behind it.If C is a regular VHC, the transversality condition (22) in Definition 4.3 states that, for each q ∈ C, the tangent space T q Q is the direct sum of T q C and D A (q).We may then define the projection σ q : T q Q → T q C of the vector space T q Q onto T q C along the subspace D A (q).The map σ q is uniquely determined by the following properties: (i) σ 2 q = σ q , (ii) Im σ q = T q C, (iii) Ker σ q = D A (q). Now consider the vector bundle map σ : T Q| C → T C, w q → σ q (w q ), illustrated in Figure 2.
Since the acceleration distribution D A (q) is smooth, so is σ.Using σ, we define a new connection on C as follows.Given two vector fields X, Y ∈ X(C), ∇ X Y is generally not a vector field on C, but its projection σ(∇ X Y ) is, and the next theorem shows that this operation identifies an affine connection on C.
Before presenting the theorem, we need to justify the notation ∇ X Y for vector fields X, Y ∈ X(C), since the affine connection ∇ accepts vector fields on Q.Consider arbitrary smooth extensions 4 X, Ỹ of X, Y on a neighbourhood of C in Q such that X| C = X and Ỹ | C = Y .Given any p ∈ C, by [16, Exercise 4.7, p.58], PSfrag replacements w q w q w q w q σ q (w q ) σ q (w q ) σ q (w q ) σ q (w q ) Theorem 4.7.Let C be a regular VHC of order m for the Lagrangian control system (20), and define the map where ∇ is the Riemannian connection of (Q, g).The map C ∇ is a symmetric affine connection on C.
Proof.Let X, Y ∈ X(C) be arbitrary.For each q ∈ C, σ(∇ X Y )(q) = σ q (∇ X Y (q)) ∈ T q C.Moreover, the map q → σ q (∇ X Y (q)) is smooth because it is the composition of two smooth maps.Thus, C ∇ X Y ∈ X(C).In order to prove that C ∇ is an affine connection we need to verify the identities in (1).Let f, g ∈ C ∞ (C) and X, Y, Z ∈ X(C).Using the fact that σ is fibrewise linear, we have We have thus proved that C ∇ is an affine connection on C. In order to prove that ∇ q q = −σ q (grad P (q)), (28) in the following sense.If q : I → Q is a maximal base integral curve of (20) such that q(I) ⊂ C, then q : I → C is a maximal base integral curve of system (28).Vice versa, if q : I → C is a maximal base integral curve of (28), then ι • q is a maximal base integral curve of (20).
By the uniqueness of the feedback τ ⋆ rendering T C invariant (see Proposition 4.6), it must hold that τ = τ ⋆ , a smooth feedback.This proves that ι(q) is an integral curve of (20).
We now prove maximality.Suppose, by way of contradiction, that q : I → Q a maximal base integral curve of (20) such that q(I) ⊂ C but the corresponding base integral curve q : I → C of ( 28) is not maximal.Let q : Ĩ → C, Ĩ ⊃ I, be the unique maximal base integral curve of (28) such that q| I = q.Then ι(q) is a base integral curve of (20) with a larger interval of existence than q, which contradicts the maximality of q.In an analogous way one shows that if q : I → C is a maximal base integral curve of (28) then ι(q) is maximal for (20).4.3.Constrained dynamics in coordinates.We now characterize the constrained dynamics in coordinates.Pick a coordinate chart for Q, (U, φ), with φ : U → Û ⊂ R n and C ∩ U = ∅.Letting x = φ(q) = (x 1 (q), . . ., x n (q)), the equations of motion (20) in x coordinates read as (cf.( 13)), Tangent space of C. The chart domain U can always be chosen small enough that the local representation of the constraint manifold, Ĉ = φ(C ∩ U ), is the image of a diffeomorphism ϕ : W ⊂ R n−m → Ĉ, s = (s 1 , . . ., s n−m ) → ϕ(s).Using this parametrization, we have T ϕ(s) Ĉ = Im(dϕ s ).Thus, letting Projection map.In coordinates, we have Letting B ⊥ be a full-rank left-annihilator of B, the coordinate representation of the projection map σ is the map σ : Indeed, one can readily verify that σ2 x = σx , Im(σ x ) = T x Ĉ, and Ker σ x = Im(D −1 (x)B(x)).These properties imply that σx is the projection onto T x Ĉ along the subspace Im(D −1 (x)B(x)), as required.
Induced connection C ∇.The coordinate chart (U, φ) induces Christoffel symbols Γ k ij , i, j, k ∈ {1, . . ., n}, of the Riemannian connection ∇.We now derive the Christoffel symbols of the induced connection, defined through the identity where Γ k ij are the symbols we are looking for.Using the definition of V i , V j , identity (6), and the expression for σ in (29), one gets where ϕ a denotes the a-th component of ϕ.Letting Γ a (x) be the matrix with components (Γ a ) bc = Γ a bc , we may rewrite the Christoffel symbols of the induced connection in the more economical form i, j, k ∈ {1, . . ., n − m}.
Constrained dynamics.The coordinate representation of grad P (q) is Using (29), the projection σ q (grad P (q)) in s-coordinates reads as . Letting e k denote the k-th natural basis vector of R n−m , the coordinate representation of the constrained dynamics (28) In the special case when C is diffeomorphic to a generalized cylinder, one may pick ϕ to be a global diffeomorphism (S 1 ) k × (R) n−m−k → C, in which case the ODEs in (31) constitute a global representation of the constrained dynamics.In particular, for systems with degree of underactuation one, i.e., when n − m = 1, ( 31) is always valid globally, and it reduces to The above is precisely the form of the constrained dynamics in ( 17)-(18).

4.4.
Examples of computation of constrained dynamics.We present two examples illustrating the formulas in Section 4.3.
Example 1.Consider the unit point-mass particle on the plane with inertial coordinates q = [q 1 q 2 ] ⊤ ∈ R 2 depicted in Figure 4.The particle is actuated by a force (R α q)τ , where and τ ∈ R is the control input.The equations of motion are q = (R α q)τ.
This is a Lagrangian system (Q, g, 0, F ), with Q = R 2 , g the Euclidean inner product, and F (q) = (q 1 cos α − q 2 sin α)dq 1 + (q 1 sin α + q 2 cos α)dq 2 .Let C be the unit circle centred at the origin.If α ∈ (−π/2, π/2), then C is a regular VHC since, for all q ∈ C, the vector R α q is transversal to C: The output function e = q ⊤ q − 1 yields vector relative degree {2, 2} everywhere on C, and the feedback asymptotically stabilizes the constraint manifold T C and renders it invariant.The map ϕ : S 1 → R 2 , ϕ(s) = [cos(s) sin(s)] ⊤ , is a parametrization of C. The Christoffel symbols Γ k ij of g are all zero.Letting B ⊥ (q) := q ⊤ R α+π/2 and using (30), the Christoffel symbol of the induced connection is C Γ 1 11 = tan α, so by (32) the constrained dynamics are given by s = −(tan α) ṡ2 .
One can also derive the constrained dynamics by multiplying both sides of (33) on the left by B ⊥ , and substituting q = ϕ(s), q = ϕ ′ (s)s + ϕ ′′ (s) ṡ2 in the resulting expression.The ODE one gets this way is the same as above.△ Example 2. Consider now a unit point-mass in R 3 with inertial coordinates q = [q 1 q 2 q 3 ] ⊤ ∈ R 3 , actuated by a control force (diag(1, 1, 2)q)τ , where τ ∈ R is the control input: q = B(q)τ, where B(q) = diag(1, 1, 2)q.This is a Lagrangian system (Q, g, 0, F ), where Q = R 3 , g is the Euclidean inner product, and F (q) = dq 1 + dq 2 + 2dq 3 .In this example, D A (q) = span{F ♯ (q)} = Im B(q).Let C be the unit sphere centred at the origin, C = {q ∈ R 3 : q ⊤ q = 1}.The set C is illustrated in Figure 5.For each q ∈ C, T q C PSfrag replacements is the orthogonal complement of span{q}.Since g(q, F ♯ (q)) = q ⊤ diag(1, 1, 2)q > 0, the control force is transversal everywhere to the sphere, and therefore, for any q ∈ C, Thus C is a regular VHC.For a parametrization of C, we use spherical coordinates: Letting W = (0, π) × (−π, π) and Ĉ = S 2 /{N, P }, where N and P are the north and south poles of C, the map ϕ : W → Ĉ is a diffeomorphism.To compute the Christoffel symbols of the induced connection on C, we define a left-annihilator of B(q) = diag(1, 1, 2)q: B ⊥ (q) = Im −q 2 q 1 0 −q 1 q 3 −q 2 q 3 (q 2 1 + q 2 2 )/2 .

5.
Existence of a Lagrangian structure for the constrained dynamics.In this section we investigate this question: given the Lagrangian control system (20) and a regular VHC C of order m, determine whether there exists a Riemannian metric g C on C and a smooth potential function P C : C → R such that the constrained dynamics (28) are generated by the Lagrangian structure (C, g C , P C ).If this is the case, we say that the constrained dynamics are Lagrangian.The solution in the special case m = n − 1 was reviewed in Theorem 3.5.Here we investigate the problem from a more general perspective.
where grad C P C ∈ X(C) is the gradient vector field of P C induced by the metric g C , i.e., defined by the identity dP C (v q ) = g C (grad C P C , v q ) for all v q ∈ T C.
Moreover, if (i) and (ii) hold, the Lagrangian structure of the constrained dynamics is (C, g C , P C ).
Proof.(⇐=) If conditions (i) and (ii) hold, then it follows directly from the definition that the constrained dynamics (28) are generated by the Lagrangian system (C, g C , P C ). (=⇒) Suppose the constrained dynamics (28) are generated by a Lagrangian system (C, g C , P C ).Let ∇ be the Riemannian connection associated with g C .By Theorem 4.8, a curve q : I → C satisfies C ∇ q q + σ q (grad P (q)) = 0 (36) if and only if it satisfies ∇ q q + grad C P C (q) = 0. (37) For any q 0 ∈ C, let q : I → C be the maximal integral curve of the constrained dynamics (28) with initial condition (q 0 , 0).If {X 1 , . . ., X n−m } is any local frame for T C defined in a neighbourhood of q 0 , identity (7) and the fact that q| t=0 = 0 imply that C ∇ q q t=0 = ∇ q q t=0 .Since (36) and (37) hold, we deduce that σ q0 (grad P (q 0 )) = grad C (P (q 0 )), proving that P C satisfies condition (ii).Next, subtracting (36) from (37) we get Consider an arbitrary coordinate chart on C with associated local frame {X 1 , . . ., X n−m }.

C
∇ is symmetric and thus 38), then, expresses the equality of two symmetric quadratic forms, which can only hold if In turn, the equality of the Christoffel coefficients implies the equality of the two connections C ∇ and ∇ on the chart domain.
Since the coordinate chart is arbitrary, we conclude that C ∇ = ∇, and hence C ∇ is metrizable.This proves condition (i).
The considerations above also show that if conditions (i) and (ii) hold, then the Lagrangian structure of the constrained dynamics is (C, g C , P C ).

Case of orthogonal control accelerations.
Referring to the regularity condition (22), when the acceleration distribution D A is fibrewise orthogonal to T C (see Figure 6), the feedback rendering T C invariant produces a control force that does no work on base integral curves contained in C. In this setting, the control force is iden-PSfrag replacements . Illustration of the case when the control accelerations are orthogonal to C.
tical to the constraint force that would arise if C were a holonomic constraint 5 .Just like in classical mechanics, one expects the constrained dynamics to be Lagrangian, with Lagrangian structure given by the restriction of the original Lagrangian structure to C. The next proposition makes this intuition precise.Recall the inclusion map ι : C → Q.The metric g : T Q × T Q → R on Q gives rise to a metric on C via the pullback ι * g(v q , w q ) = g(dι q (v q ), dι q (w q )) for all v q , w q ∈ T q C.
The metric ι * g is called the induced metric on C.
Proposition 5.2.If C is a regular VHC of order m for the Lagrangian control system (20) such that ) with orthogonality holding with respect to the metric g, then the constrained dynamics (28) are Lagrangian with Lagrangian structure (C, g C , P C ), where g C = ι * g and for all X, Y, Z ∈ X(Q), and therefore also for all X, Y, Z ∈ X(C).Let X, Y, Z ∈ X(C) be arbitrary.In light of the regularity condition ( 16), we have where N X Y is a vector field in the control distribution span{(F i ) ♯ , i = 1, . . ., m}.
By the orthogonality hypothesis, we have The second identity in (41) is due to the fact that C ∇ X Y and Z are vector fields on C. Analogously to (41), we have Substituting ( 41) and ( 42) into (40) and using the fact that g(Y, Z) = g C (Y, Z), we get Next, we need to show that, for each q ∈ C, σ q (grad P (q)) = grad C P | C , where grad C P | C is the gradient vector field of P | C induced by the metric g C .Since, by assumption, the subspace D A (q) is orthogonal to T q C, the projection σ q : T q Q → T q C along D A (q) is a map whose kernel, D A (q), is orthogonal to its image, T q C.This fact implies that σ q is a self-adjoint map.Thus, for all v q ∈ T q C, g(σ q (grad P (q)), v q ) = g(grad P (q), σ q (v q )) = g(grad P (q), v q ). (43) Using the definition of grad, we have g(grad P (q), v q ) = dP q (v q ).(44) Since P | C = P • ι and, for all v q ∈ T q C, dι q (v q ) = v q , we may write Substituting ( 44) and ( 45) into (43) and using the definition of grad C , we get g(σ q (grad P (q)), v q ) = (dP for all v q ∈ T q C. Since σ q (grad P (q)) and v q lie in T q C, g(σ q (grad P (q)), v q ) = g C (σ q (grad P (q)), v q ), and so g C (σ q (grad P (q)), v q ) = g C (grad C P | C (q), v q ), for all v q ∈ T q C. In conclusion, σ q (grad P (q)) = grad C P | C (q), proving that P | C satisfies condition (ii) of Theorem 5.1.
Remark 5.3.As mentioned earlier, the foregoing proposition states that the constrained dynamics associated with a VHC satisfying condition (39) coincide with the constrained dynamics one would have if the Lagrangian system (Q, g, P ) (without control) were subjected to an ideal holonomic constraint.To gain further understanding of this concept, it is worth comparing Proposition 5.2 with Proposition 4.97 in [6].In [6], a holonomic constraint C is a maximal integral manifold of a distribution D representing a linear velocity constraint.This distribution is used to define a constrained connection D ∇ on Q. Proposition 4.97 in [6] states that the restriction of is the Riemannian connection of ι * g, and Proposition 4.85 in [6] implies that D ∇ X Y coincides with our C ∇ X Y for all X, Y ∈ X(C).Taken together, these results recover the proof of the first part of Proposition 5.2, illustrating the strong analogy between VHCs satisfying condition (39) and holonomic constraints in [6].As a caveat, we remark that, unlike the framework in [6], we do not require an integrable distribution to define the induced connection C ∇, for C ∇ is only required to be defined on X(C) × X(C).

6.
Conditions for metrizability of affine connections.Theorem 5.1 establishes that in the absence of a potential function, assessing whether or not the constrained dynamics induced by a regular VHC are Lagrangian amounts to assessing the metrizability of the induced connection.In this section we review the main results on metrizability of affine connections, presenting concrete results for the cases dim C = 1 (already covered in Theorem 3.5) and dim C = 2.To keep the notation simple, throughout the section we will consider a symmetric affine connection ∇ : X(C) × X(C) → X(C) with the understanding that all result will apply to the induced connection C ∇.We also assume throughout that the submanifold C is connected.
6.1.The holonomy group of an affine connection.A vector field X(t) along a smooth curve γ on C is said to be parallel if its covariant derivative along γ vanishes, i.e., D t X ≡ 0. Given a point q ∈ C and a tangent vector v q ∈ T q C, the equation D t X = 0 with initial conditions X(0) = v q uniquely determines a parallel vector field X(t) along γ (see [16, Chapter II, Proposition 3.3]).This vector field is called the parallel translation of v q along γ.In local coordinates, the equation D t X = 0 is the linear time-varying ODE where Γ k ij are the Christoffel symbols of ∇ and (X 1 , . . ., X n−m ) is the coordinate representation of X.
In what follows, if q, p ∈ C, a piecewise smooth curve in C starting at q and ending at p will be denoted γ p q .More precisely, γ p q : [0, T ] → C will be a piecewise smooth map such that γ p q (0) = q and γ p q (T ) = p.On the other hand, a loop at q, i.e., a piecewise smooth closed curve through q will be denoted by γ q .The set of all loops at q will be denoted by L q .For a curve γ p q , the parallel transport map along γ p q , denoted P γ p q : T q C → T p C, is defined as P γ p q (v q ) := X(T ), where X is the parallel translation of v q along γ p q .For γ q ∈ L q , P γq maps T q C onto itself.The parallel transport map associated with a loop is illustrated in Figure 7.

PSfrag replacements q vq
Pγ q (vq) γq TqC Figure 7.The parallel transport map at the north pole of the unit sphere in R 3 , with Riemannian connection induced by the Euclidean metric in R 3 .The loop γ q is a triangle on the sphere.
Denoting by (γ p q ) −1 the curve obtained by reversing the orientation of γ p q , and by γ p q • γ r p the concatenation of γ p q with γ r p , we have the following result.Proposition 6.1 ([14], Chapter II, Proposition 3.3).For each q ∈ C and any piecewise smooth curves γ p q , γ r p , the parallel transport map P γ p q : T q C → T p C is an isomorphism enjoying the following properties: (i) If γ q ∈ L q is the constant loop γ q (t) ≡ q, then P γq is the identity map on T q C. (ii) P (γ p q ) −1 = (P γ p q ) −1 .(iii) P γ p q • γ r p = P γ p q • P γ r p .In particular, the set of all isomorphisms {P γq : γ q ∈ L q } forms a group under composition.
The group Hol q (∇) := {P γq : γ q ∈ L q } of all parallel transport maps along loops at q is called the holonomy group of ∇ with reference point q, while the subgroup Hol 0 q (∇) := {P γq : γ q ∈ L q is contractible to q} is the restricted holonomy group.A remarkable property of the holonomy groups is that they possess a Lie group structure.Theorem 6.2 ([14], Chapter II, Theorem 4.2).Let C be a connected manifold, and let q ∈ C. Then the following are true: (i) Hol 0 q (∇) is a connected Lie subgroup of GL(n).(ii) Hol q (∇) is a Lie subgroup of GL(n) whose identity component is Hol 0 q (∇).
6.2.Schmidt's metrizability theorem.The significance of the holonomy group at it pertains to the metrizability of ∇ rests upon the following consideration.
If ∇ is Riemannian with respect to a metric g, then it is a basic fact of Riemannian geometry that for any two vector fields V, W that are parallel along a curve γ, g(V, W ) is constant along γ.In particular, for any γ q ∈ Hol q (∇), g q (P γq (v q ), P γq (w q )) = g q (v q , w q ).Also, by definition, g q is a positive definite, symmetric bilinear form on T q C. In conclusion, a necessary condition for ∇ to be metrizable is that there exists a symmetric, positive definite bilinear form T q C × T q C → R that is invariant under the holonomy group Hol q (∇).This condition is also sufficient.

Theorem 6.3 ([25]
).Let ∇ be a symmetric affine connection on a connected manifold C and let q ∈ C be arbitrary.Then ∇ is metrizable if and only there exists a symmetric positive definite bilinear form g q : T q C × T q C → R that is invariant under Hol q (∇), i.e., for all γ q ∈ Hol q (∇) and all v q , w q ∈ T q C, g q (P γq (v q ), P γq (w q )) = g q (v q , w q ).( Schmidt's theorem only requires one to determine whether or not a bilinear form on T q C × T q C exists which is invariant under Hol q (γ q ).It then guarantees that the form in question can be extended to a Riemannian metric defined on the whole of T C × T C. We have already outlined the necessity part of the proof.The idea of the sufficiency proof is to extend the bilinear form g q to the entire tangent bundle T C by parallel translation along curves connecting q to arbitrary points in C. Specifically, for arbitrary p ∈ C, pick an arbitrary piecewise smooth γ p q connecting p and q, and define g p (v p , w p ) := g q P γ p q (v p ), P γ p q (w p ) .
The invariance of g q under Hol q (∇) guarantees that g p is path-independent, giving rise to a Riemannian metric on C. One can easily show that the extension so obtained is a Riemannian metric associated with ∇.
The result in Theorem 6.3 is of difficult application because the group Hol q (∇) is generally hard to find.The Ambrose-Singer theorem [1] characterizes Hol 0 q (∇) in terms of the curvature form of the connection, but it requires the knowledge of the so-called holonomy bundle, an object which is not readily available.In special cases, however, the computations are more manageable, as we discuss next.

Flat connections. The curvature endomorphism induced by an affine connection
If ∇ is a flat connection, i.e., the curvature R induced by ∇ is zero, then the Ambrose-Singer theorem implies that Hol 0 q (∇) is trivial, and by [3, Theorem 2] there exists a surjective homomorphism π 1 (C, q) → Hol q (∇), where π 1 (C, q) is the first homotopy group of C with reference point q.The homomorphism in question sends an equivalence class of loops [γ q ] ∈ π 1 (C, q) to a parallel transport map P γq ∈ Hol q (∇).In this case, to apply Theorem 6.3 it suffices to compute the transport maps associated with the generators of π 1 (C, q), as stated next.Proposition 6.4.Let ∇ be a symmetric affine connection on a connected manifold C, and suppose that ∇ is flat.Let q ∈ C be arbitrary, and let S q be a set of generators of π 1 (C, q).Then ∇ is metrizable if and only if there exists a symmetric positive definite bilinear form g q : T q C × T q C → R such that, for each equivalence class E q ∈ S q , there exist a piecewise smooth curve γ q ∈ E q for which g q (P γq (v q ), P γq (w q )) = g q (v q , w q ), for any v q , w q ∈ T q C. 6.4.Simply connected manifolds.When C is simply connected, Hol q (∇) = Hol 0 q (∇) because, by definition, all loops at q in C are contractible to q.By Theorem 6.2, Hol 0 q (∇) is a connected Lie group, implying that it is entirely characterized by its Lie algebra, the so-called holonomy algebra of ∇.We will denote by h the holonomy algebra.For simply connected manifolds, one may express the invariance condition in (47) in infinitesimal form, giving rise to a Lie algebraic metrizability criterion.Lemma 6.5 ([30]).Let ∇ be a symmetric affine connection on a simply connected manifold C and let q ∈ C be arbitrary.A symmetric positive definite bilinear form g q : T q C × T q C → R is invariant under Hol q (∇) if and only if for all A ∈ h and all v q , w q ∈ T q C, g q (Av q , w q ) + g q (v q , Aw q ) = 0. (49) Remark 6.6.If ∇ is real analytic6 , the holonomy algebra h is entirely characterized by the curvature and its covariant derivatives (see [14, Chapter II, Proposition 10.4 and Theorem 10.8]).Therefore, h can be computed in local coordinates.Then, a consequence of Lemma 6.5 is that if C is simply connected and ∇ is a real analytic affine connection on C, ∇ is metrizable if and only if it is locally metrizable (i.e., metrizable in local coordinates).
Exploiting Lemma 6.5 and the de Rham decomposition of a Riemannian manifold, Kovalski in [15] gave an effective decision algorithm for metrizability of real analytic affine connections on simply connected manifolds.We will not review the algorithm here, but we refer the reader to the review in [30].6.5.One-dimensional manifolds.If C is one-dimensional, then it is diffeomorphic to either R or S 1 .This situation occurs in Lagrangian control systems with degree of underactuation one, when a regular VHC of codimension one is enforced.In Theorem 3.5 of Section 3, we reviewed necessary and sufficient conditions for the constrained dynamics to be Lagrangian.Now we show that the conditions of Theorem 3.5 have an elegant interpretation in the context of induced connections.We will recover Theorem 3.5 as a corollary of Theorem 5.1 and Proposition 6.4.
We begin with the observation that any affine connection on a one-dimensional manifold is flat, so if dim C = 1, we may apply Proposition 6.4 to assess the metrizability of the induced connection.By comparing ( 18) and ( 32), we deduce that where we recall that Θ, defined in Proposition 3.4, is R or S 1 depending on whether ) is trivial, and Proposition 6.4 is trivially satisfied.Thus a connection on R is always metrizable.Using x ∈ R as coordinate for C, the Riemannian metric on R will have the form g x (v, w) = (1/2)k(x)vw, with k(x) > 0. By Theorem 5.1, the constrained dynamics are Lagrangian if and only if there exists a function P C : R → R such that σ x (grad P (x)) = k −1 (x)P ′ C (x).This identity is satisfied by letting P C be an antiderivative of the function k(x)σ x (grad P (x)).Having established the metrizability of the induced connection and the existence of P C , by Theorem 5.1 the constrained dynamics are always Lagrangian.This recovers the result of Theorem 3.5 when C ≃ R. Now consider the case C ≃ S 1 , so that Θ = S 1 .Since π 1 (S 1 , 0) = (Z, +), π 1 (S 1 , 0) is generated by the loop γ 0 : [0 2π] → S 1 , t → t mod 2π.By Proposition 6.4, the induced connection is metrizable if and only if there exists a positive definite quadratic form that is invariant under the transport map P γ0 .Recall the coordinate representation of the parallel transport map in (46).Using t ∈ R as local coordinates for S 1 , we have that P γ0 (v) = X(2π), where X is the solution of the linear time- To obtain the above ODE, we substituted the first identity of (50) into (46), and used the fact that the coordinate representation of Ψ 2 (s) is Ψ 2 • π(t), where π(t) = t mod 2π.The solution of the above scalar linear system is We pick s = 0 as reference point on S 1 .Then, modulo a multiplicative positive scalar, the only positive definite bilinear form on T 0 S 1 × T 0 S 1 is g 0 (v 0 , w 0 ) = v 0 w 0 , and the invariance condition in Proposition 6.4 reads as exp 2 The above identity holds for arbitrary v 0 , w 0 if and only if or, equivalently, if the function MC (x) in Theorem 3.5 is 2π-periodic.Thus, the periodicity requirement on MC in part (b) of Theorem 3.5 is equivalent to the requirement, in part (i) of Theorem 5.1, that the induced connection be metrizable.
To find the Riemannian metric on S 1 (denoted g in what follows), we extend the inner product g 0 : T 0 S 1 × T 0 S 1 → R to the whole T S 1 × T S 1 through parallel transport, as in (48).For any s ∈ S 1 , set g s (v s , w s ) := g 0 (P γ 0 s (v s ), P γ 0 s (w s )), where γ 0 s is an arbitrary curve from s to 0 in S 1 .For instance, pick any x ∈ π −1 (s), and define γ 0 s : [0, x] → S 1 as γ 0 s (t) = s − (t mod 2π).Then, The above metric on S 1 gives the kinetic energy of the constrained dynamics in Theorem 3.5, since it can be expressed as g s (v s , w s ) = M C (s)v s w s .Next, we turn our attention to condition (ii) of Theorem 5.1, namely the existence of P C : S 1 → R such that σ(grad P ) = grad C P C , or Equivalently, we need to check when is it that the one-form on S 1 −Ψ 1 (s)M C (s)ds is exact.This is the case if and only if the integral of the form along S 1 is zero, This is precisely the condition that the function PC (x) in Theorem 3.5 be 2πperiodic.Thus, the periodicity requirement on PC in part (b) of Theorem 3.5 is equivalent to the requirement, in part (ii) of Theorem 5.1, that σ(grad P ) = grad C P C .We have thus shown that Theorem 3.5 is a corollary of Theorem 5.1 and Proposition 6.4.
6.6.Two-dimensional manifolds.When dim C = 2, the metrizability of an affine connection has a powerful characterization in terms of the Ricci tensor [29,31].We remark that the situation dim C = 2 arises in Lagrangian control systems with degree of underactuation two, when a regular VHC of codimension two is enforced.The Ricci curvature tensor (see, e.g., [16]) is the (0, 2) tensor defined as where trace(•) denotes the trace of a linear map.If the affine connection ∇ is Riemannian with respect to a metric g, then the Ricci tensor is proportional to the metric, Ric(X, Y ) = g(X, Y )K, (52) where K ∈ C ∞ (C) denotes the Gaussian curvature of C (see [16,Lemma 8.7]).Recall from (5) that the compatibility of ∇ with g means that ∇g = 0.If ∇ has nonvanishing curvature, then K is nonvanishing, and setting α = 1/K, we have For each X, Y, Z ∈ X(M ), we have and using (53) we deduce that The above may be rewritten concisely using the total covariant derivative and the tensor product as ∇ Ric = d(− ln |α|) ⊗ Ric .A tensor field F whose total covariant derivative satisfies ∇F = ω ⊗ F , where ω is a one-form, is said to be recurrent.Thus, a necessary condition for metrizability of ∇ is that the Ricci tensor induced by ∇ be recurrent, with a one-form ω given by the exact differential of a function in C ∞ (C).Further, in light of the fact that when ∇ is metrizable identity (52) holds, another necessary condition for metrizability is that the Ricci tensor be definite (positive definite if K > 0, negative definite if K < 0).Together, these conditions are also sufficient.The idea behind the proof of sufficiency rests upon the fact that if ∇ Ric = df ⊗ Ric, then ∇ exp(−f + b) Ric = 0, and therefore both type (0, 2) tensor fields given by ± exp(−f + b) Ric are compatible with ∇.Since ∇ is symmetric, so is Ric.Since Ric is definite, g in the theorem statement is positive definite.

7.
Examples.We now illustrate the results of Sections 5 and 6 with three examples.First, we revisit Examples 1 and 2. Then we investigate the Lagrangian structure of a double pendulum on a cart subject to a regular VHC of order 1.
Example 1 (Continued).Consider again the dynamics of the planar point-mass of Example 1, a Lagrangian control system with underactuation degree one.The constrained dynamics on the unit circle C depicted in Figure 4 are We want to determine the existence of a Lagrangian structure for these constrained dynamics.For this, we may use Theorem 3.5, with Ψ 1 (s) = 0 and Ψ 2 (s) = − tan α.We have The constrained dynamics are Lagrangian if and only if MC is 2π-periodic, or α = ±π/2 mod 2π.Thus the constrained point-mass is Lagrangian if and only if the control force is orthogonal to the circle C, in which case the Lagrangian is L(s, ṡ) = (1/2) ṡ2 .As predicted by Proposition 5.2, L(s, ṡ) is the restriction of the original Lagrangian to C, i.e., L(s, ṡ) = (1/2)g( q, q)| q=ϕ ′ (s) ṡ. △ Example 2 (Continued).We return now to the unit mass of Example 2, a Lagrangian control system with degree of underactuation two.We seek to determine whether or not the constrained dynamics with coordinate representation given in (35) are Lagrangian.Since this Lagrangian control system has no potential function, by Theorem 5.1 we only need to check whether or not the induced connection on C is metrizable.To this end, we will use Theorem 6.7.Since C is simply connected and the induced connection is real analytic, it suffices to check the recurrence condition of Theorem 6.7 in local coordinates (see Remark 6.6).
The one-form ω is exact, ω = df , with f (s) = −4 atanh(sin 2 (s 1 )/(sin 2 (s 1 ) − 4)).By Theorem 6.7, the constrained dynamics on T C are Lagrangian, and a Lagrangian function in local coordinates is given by where D C (s) is the matrix exp(−f (s))[Ric], with [•] denoting the matrix representation of the Ricci tensor.Specifically, we have In applying Theorem 6.7, we used the plus sign in the metric because the matrix [Ric] is positive definite.One may check that the Euler-Lagrange equation with L as above gives the constrained dynamics (35).We stress once again that although our computations are done in local coordinates, by the argument in Remark 6.6 the foregoing considerations imply that the constrained dynamics are globally Lagrangian.△ Example 3. Consider the double pendulum on a cart depicted in Figure 8, a Lagrangian control system with three degrees-of-freedom and one input.We investigate two cases.
• Case (a): control input is the force imparted to the cart.
Consider the embedded submanifold of Q, The configuration of the double pendulum on C is illustrated in Figure 9.The function ρ above was already used in [8] for path following control of a PVTOL aircraft, and in [18] for pendubot swing-up.In the context of the double pendulum on a cart of Figure 8, the function ρ induces the interesting property that, on C, the last link does not perform full revolutions and remains confined to the upper half-plane.Letting h(q) = q 3 − ρ(q 3 ), one may check that, in both cases (a) and (b), dh q D −1 (q)B(q) = 0 for all q ∈ C, so by the equivalence of Definitions 3.2 and 4.3 (see Remark 4.5), C is a regular VHC.The set C is diffeomorphic to a cylinder via the diffeomorphism R×S 1 → C, (s 1 , s 2 ) → (s 1 , s 2 , ρ(s 2 )).Using this global parametrization and the formulas in (30), one may show that, in both cases (a) and (b), the only nonzero Christoffel symbols of 22 (s 2 ), and they are functions of s 2 only.Similarly, the representation of σ(grad P ) in (s 1 , s 2 )-coordinates is a function of s 2 only.Therefore, in both cases (a) and (b) the constrained dynamics (31) have the form where (λ 1 (s 2 ), λ 2 (s 2 )) is the coordinates representation of σ(grad P ).The precise expressions are easy to determine with the formulas in Section 4.3, but they are too long to report here.The function ρ is odd, i.e., ρ(−s 2 ) = −ρ(s 2 ), and as a result, the functions 22 (s 2 ), λ i (s 2 ) are odd as well.Now we investigate the Lagrangian nature of the constrained dynamics.We will show that in case (a) (force on the cart), the constrained dynamics are not Lagrangian, while in case (b) (torque on the second revolute joint), they are.We begin by checking condition (i) of Theorem 5.1, i.e., the metrizability of C ∇.The coefficients of the curvature endomorphism in (s 1 , s 2 ) coordinates may be computed using the formula (54).Owing to the fact that only the symbols C Γ k 22 are nonzero, and that they are functions of s 2 only, we see from (54) that the curvature endomorphism is identically zero, i.e., the induced connection is flat.We can then use Proposition 6.4 to determine whether or not the induced connection is metrizable.Recall that (s 1 , s 2 ) ∈ R × S 1 .In what follows, the point (0, 0) ∈ R × S 1 will be denoted 0 in subscripts.
With reference to (46), to find P γ0 we solve the linear time-varying system and put P γ0 (v) = X(2π).The solution is X(t) = (X 1 (t), X 2 (t)) with Recall that the functions Γ k 22 are odd and 2π-periodic.The integral over one period of an odd periodic function is zero.Using this fact, we have I 1 (2π) = 0. Since the integral of an odd function is an even function, exp(I 1 (t)) is even, and its product with C Γ 1 22 (t) is odd.Thus I 2 (2π) = 0.In conclusion, P γ0 is the identity map, implying that any positive definite quadratic form on (T (0,0) R × S 1 ) × (T (0,0) R × S 1 ) is invariant under P γ0 .By Proposition 6.4, C ∇ is metrizable.This result holds for both cases (a) and (b).
Next, we find all Riemannian metrics on R × S 1 compatible with C ∇. Just like in Section 6.5, we will use the notation g (in place of g C ) for such a metric.Modulo scalar multiples, the generic positive definite quadratic form on (T (0,0) R × S 1 ) × (T (0,0) R × S 1 ) is with a, b ∈ R such that b > a 2 .As in Section 6.5, the Riemannian metric on R × S 1 is found by parallel transporting g 0 by means of (48).The construction is illustrated in Figure 10.Let s = (s 1 , s 2 ) ∈ R × S 1 be arbitrary, denote s := (0, s 2 ).Pick any x 2 ∈ π −1 (s 2 ) and define a path γ s 0 (t) as the concatenation of paths γ s 0 and γ s s , where γ s 0 : [0, x 2 ] → R × S 1 is defined as γ s 0 (t) := (0, t mod 2π), and γ s s : [0, s 1 ] → R × S 1 is defined as γ s s (t) := (t, s 2 ).Then, γ s 0 : [0, s 1 + x 2 ] → R × S 1 is a piecewise smooth path connecting (0, 0) to (s 1 , s 2 ).See Figure 10.By Proposition 6.1, we have P γ s 0 = P γ s 0 • P γ s s .Since γ s s is a translation along the real line, P γ s s is the identity map.On the other hand, P γ s 0 is given by (58) at time s 2 .In conclusion, the parallel transport map from (0, 0) to (s 1 , s 2 ) is .
By Proposition 6.1, the parallel transport map from (s 1 , s 2 ) to (0, 0) is P −1 γ s 0 .Now we define a Riemannian metric g on R × S 1 by transporting tangent vectors in T (s 1 ,s 2 ) (R × S 1 ) to T (0,0) (R × S 1 ) (cf. ( 48 (60) Such a function P C exists if and only if the one-form on the right-hand side of the above identity is exact.For this, we need to check whether this one-form is closed, and whether its integral over the loop γ 0 defined earlier is zero.As far as closedness of the form is concerned, we need to check whether or not there exists a ∈ R such that ∂ s 2 λ 1 (s 2 ) + exp(−I 1 (s 2 ))(I 2 (s 2 ) + a)λ 2 (s 2 ) = 0. (61) In case (a) when the control force is on the cart, one can show that there is no value of a for which (61) holds, whereas in case (b), when there is a control torque on the last joint, (61) is satisfied with a = −1/2.In the latter case, the one-form on the right-hand side of (60) is also exact because its components are odd functions of s 2 .We choose any b > a 2 , for instance b = 1, and obtain that the constrained dynamics in (57) are a Lagrangian system (R × S 1 , g, P C ), with g given in (59) and with dP C given in (60), and γ s 0 defined earlier.We summarize our results for this example in the following table.q 3 = ρ(q 2 ) regular? C ∇ metr'le?P C exists?
Force on cart yes yes no no Force on last joint yes yes yes yes Figure 11 depicts the (s 2 , ṡ2 ) orbits (equivalently, the (q 2 , q2 ) orbit) of a few solutions of the constrained system in cases (a) and (b).In both cases, we observe two types of behaviours: there are trajectories along which q 2 exhibits a rocking motion around π, and others along which q 2 performs full revolutions.The behaviour of q 1 , not shown in the figure, is a drifting motion with bounded, sign-definite speed.△ (q 2 , q2 ) orbits of a few solutions of the double pendulum on a cart subject to the VHC q 3 = ρ(q 2 ).On the left, case (a) (force on cart).On the right, case (b) (torque on last joint).
8. Conclusions.We introduced a coordinate-free framework of virtual holonomic constraints for underactuated Lagrangian control systems, exposing the role of induced connections in the characterization of constrained dynamics.In this framework, the classical mechanics notion of ideal holonomic constraint becomes the special case in which the acceleration distribution is orthogonal to the VHC.We showed that, generally, the constrained dynamics are not Lagrangian, and the metrizability of the induced connection is key for the existence of a Lagrangian structure.When the constrained dynamics are forced (i.e., when the order of the regular VHC is less than the number of control inputs), the problem remains open of determining when the constrained dynamics are feedback equivalent to a Lagrangian control system.One possible avenue of investigation for the solution of this latter problem is to globalize the local theory of [24] in the context of affine connection control systems.

Figure 2 .
Figure 2. The vector bundle map σ : T Q| C → T C.the value of ∇ X Ỹ (p) depends only on Xp (and thus X p ) and the value of Ỹ along any smooth curve γ : (−ε, ε) → Q such that γ(0) = p and γ(0) = X p .Since X p ∈ T p C, we may pick a curve γ contained in C, so that the value of Ỹ along γ coincides with that of Y .Therefore, on C the function ∇ X Ỹ is uniquely determined by X, Y .These considerations justify the slight abuse of notation ∇ X Y for vector fields X, Y ∈ X(C).

ThusC∇
is symmetric, as required.We call C ∇ the induced connection, or the connection induced by the regular VHC C. While C ∇ is symmetric, it is generally not a Riemannian connection with respect to the induced Riemannian metric on C.This fact is discussed in the next section.Now the main result of this section.In what follows, let ι : C → Q denote the inclusion map.Theorem 4.8.If C is a regular VHC of order m for the Lagrangian control system (20), then the constrained dynamics on T C are described by the equation of motion C

Figure 4 .
Figure 4.The set C in Example 1 and its parametrization.

Figure 5 .
Figure 5.The VHC C in Example 2 and its parametrization.

∇
is compatible with g C .Since, by Theorem 4.7, C ∇ is symmetric, C ∇ is Riemannian with respect g C , proving that condition (i) of Theorem 5.1 holds.

Theorem 6 . 7 (
[29],[31] 7 ).Let C be a two-dimensional connected manifold and ∇ a symmetric affine connection on C such that the curvature induced by C is nowhere zero.Then ∇ is metrizable if and only if the Ricci tensor induced by ∇ is definite and recurrent, with the corresponding one-form being exact.If this is the case, and ∇ Ric = df ⊗ Ric holds for some f ∈ C ∞ (C), then all Riemannian metrics compatible with ∇ are given by g = ± exp(−f + b) Ric, b ∈ R arbitrary, with plus sign if Ric is positive definite, and minus sign otherwise.

Figure 8 .
Figure 8.The double pendulum on a cart of Example 3. Case (a): control force on the cart.Case (b): control torque on the last joint.The orthogonal frame in the figure is the inertial reference frame.

Figure 9 .
Figure 9. Configurations of the double pendulum on the VHC C of Example 3. The missing configurations on the right-hand side are deduced by symmetry with respect to the vertical axis.

Figure 11 .
Figure 11.(q 2 , q2 ) orbits of a few solutions of the double pendulum on a cart subject to the VHC q 3 = ρ(q 2 ).On the left, case (a) (force on cart).On the right, case (b) (torque on last joint).
[17,pen subset of T C| Ui and the family {V i } i∈N is an open cover of T C. The smooth maps τ i : V i → R m agree on overlaps: τ i | Vi∩Vj = τ j | Vi∩Vj for all i, j.By the Gluing Lemma for Smooth Maps (see[17, Corollary 2.8]), there exists a unique smooth map τ