Implicit Nonholonomic Mechanics with Collisions

A nonholonomic system is a mechanical system subject to constraint functions which are, roughly speaking, functions on the velocities that are not derivable from position constraints. They arise, for instance, in mechanical systems that have rolling or certain types of sliding contact. There are multiple applications in the context of wheeled motion, mobile robotics and robotic manipulation.

A geometrical formulation for mechanical systems with one-sided constraints was developed by Lacomba and Tulczyjew (see Lacomba and Tulczyjew (1990)). Ibort et al. studied the geometric aspects of Lagrangian systems subject to impulsive and one-sided constraints (Ibort et al. (1998)). This was extended to the Hamiltonian formalism by Cortés and Vinogradov (2006).

Mechanical systems subject to collisions are confined within a region of space with a boundary. Collision with the boundary for elastic impacts activates a constraint on the momentum and on the energy after and before the collision. The problem of collisions has been extensively treated in the literature since the early days of mechanics (see Brogliato (1999) for a comprenshive review and references therein). More recently, much work has been done on the rigorous mathematical foundation of impact problems (see Haddad et al. (2006) and Westervelt et al. (2018)). Nonholonomic systems subject to impacts or impulse effects have been previously studied in Clark and Bloch (2019) and Colombo et al. (2022).

In mechanics, implicit Lagrangian and Hamiltonian systems appear in controlled mechanical systems. An important class of implicit mechanical systems studied in Yoshimura and Marsden (2006b) are those with nonholonomic constraints. The aim of this paper is to take one step further and consider implicit mechanical systems subject to nonholonomic constraints and elastic collisions, which occurs when the nonholonomic system impacts the boundary of the configuration space under some suitable conditions. The goal of this paper is to provide a variational formulation for nonholonomic implicit mechanical systems with collisions. In particular, those collisions that preserve energy and momentum at the impact.

The remainder of the paper is structured as follows. Section 2 introduces nonholonomic systems from an explicit point of view via the Lagrange–d’Alembert principle and from an implicit point of view via the Lagrange–d’Alembert–Pontryagin principle. In Section 3 we define the configuration space and the phase space with the objective of introducing the action functional in Section 4, where we derive variationally, via the Hamilton–-d’Alembert–-Pontryagin principle, the equations of motion for implicit nonholonomic Lagrangian systems subject to elastic collisions. We extend the framework to the Hamiltonian side in Section 5, where we derive nonholonomic implicit Hamiltonian systems subject to collisions from a variational perspective. Finally, we study the vertical rolling disk hitting a wall in Section 6.

Let $Q$ be a differentiable manifold with $\hbox{dim}(Q)=n$. Throughout the text, $q^{i}$ will denote a particular choice of local coordinates on this manifold and $TQ$ denotes its tangent bundle, with $T_{q}Q$ denoting the tangent space at a specific point $q\in Q$. Usually $v_{q}$ denotes a vector at $T_{q}Q$ and, in addition, the coordinate chart $q^{i}$ induces a natural coordinate chart on $TQ$ denoted by $(q^{i},\dot{q}^{i})$ with $\hbox{dim}(TQ)=2n$. Let $T^{*}Q$ be its cotangent bundle, locally described by the positions and the momentum for the system, i.e., $(q,p)\in T^{*}Q$ with $\hbox{dim}(T^{*}Q)=2n$. The cotangent bundle at a point $q\in Q$ is denoted as $T_{q}^{*}Q$.

A $k$-dimensional distribution $\Delta_{Q}$ on a manifold $Q,$ is a $k$-dimensional subspace $\Delta_{Q}(q)$ of $T_{q}Q$ for each $q\in Q$. $\Delta_{Q}$ is smooth if for each $q\in Q$ there exist a neighborhood $U$ of $q$ and $k$ $C^{\infty}$ vector fields $X_{1},\ldots,X_{k}$ on $U$ that span $\Delta_{Q}$ at each point of $U$, that is, $\Delta_{Q}(q)=\hbox{span}\{X_{1}(q),\ldots,X_{k}(q)\}$. The rank of $\Delta_{Q}$ at $q\in Q$ is the dimension of the subspace $\Delta_{Q}(q)$, i.e. $\varrho:Q\to\mathbb{R},$ $\varrho(q)=\dim\Delta_{Q}(q).$ For any $q_{0}\in Q$ it is clear that $\varrho(q)\geq\varrho(q_{0})$ in a neighborhood of $q_{0}.$ If $\varrho$ is a constant function, then $\Delta_{Q}$ is called a regular distribution. In an analogous fashion as for distributions, it is possible to define codistributions.

A smooth regular codistribution $\widetilde{\Delta_{Q}}$ on $Q$ is a subbundle of $T^{*}Q$ with $k$-dimensional fiber. Given the concept of codistribution, it is possible to define the annihilator of a distribution. Let $\Delta_{Q}\subset TQ$ be a distribution, the annihilator of $\Delta_{Q}$ is the codistribution defined as

for every $q\in Q$.

Linear constraints on the velocities are locally given by equations of the form

depending, in general, on their configuration coordinates and their velocities. From an intrinsic point of view, the linear constraints are defined by a regular distribution $\Delta_{Q}$ on $Q$ of constant rank $n-m$ such that the annihilator of $\Delta_{Q}$ is locally given at each point of $Q$ by

where the $1$-forms $\mu^{a}$ are independent at each point of $Q$.

We will restrict ourselves to the case of linear nonholonomic constraints. In this case, the constraints are given by a nonintegrable distribution $\Delta_{Q}$. In addition to these constraints, we need to specify the dynamical evolution of the system, usually by fixing a Lagrangian function $L\colon TQ\to\mathbb{R}$. The central concepts permitting the extension of mechanics from the Newtonian point of view to the Lagrangian one are the notions of virtual displacements and virtual work. These concepts were formulated in the developments of mechanics and in their application to statics. In nonholonomic dynamics, the procedure is given by the Lagrange–d’Alembert principle. This principle allows us to determine the set of possible values of the constraint forces from the set $\Delta_{Q}$ of admissible kinematic states alone. The resulting equations of motion are

where $\delta q^{i}$ denotes the virtual displacements verifying

By using Lagrange multipliers, we obtain

The term on the right-hand side represents the constraint force or reaction force induced by the constraints and the functions $\lambda_{a}$ are the Lagrange multipliers which, after being computed using the constraint equations, allow us to obtain a set of second-order differential equations.

Alternatively to the use of Lagrange multipliers, the phase space may be enlarged to the Pontryagin bundle $TQ\oplus T^{*}Q$ and the Lagrange–d’Alembert–Pontryagin principle may be considered. This variational principle is given by

where $v(t)\in\Delta_{Q}(q(t))$ and the variations $(\delta q(t),\delta v(t),\delta p(t))$ are such that $\delta q(t)\in\Delta_{Q}(q(t))$ and vanishes at the endpoints. Then stationary condition for a curve $(q(t),v(t),p(t))$ yields the implicit Lagrange–d’Alembert equations on $TQ\oplus T^{*}Q$ (see Yoshimura and Marsden (2006b)):

Let $Q$ be a smooth manifold with boundary, denoted by $\partial Q$, $L:TQ\to\mathbb{R}$ be a (possibly degenerate) Lagrangian, and $\Delta_{Q}\subset TQ$ be a (possibly nonholonomic) constraint distribution. According to Section 2, the annihilator of $\Delta_{Q}$ is denoted by $\Delta_{Q}^{\circ}\subset T^{*}Q$.

Given $[\tau_{0},\tau_{1}]\subset\mathbb{R}$ and $\tilde{\tau}\in[\tau_{0},\tau_{1}]$, the path space with a unique collision (at $\tau=\tilde{\tau}$) is defined as $\Omega(Q,\tilde{\tau})=\mathcal{T}\times\mathcal{Q}(\tilde{\tau})$, where

and

We only consider one singularity at $\tau=\tilde{\tau}$ for brevity, but similar results hold for a finite amount of singularities, $\{\tilde{\tau}_{i}\mid 1\leq i\leq N\}\subset[\tau_{0},\tau_{1}]$.

Under these assumptions, our development is valid in a neighborhood of each collision.

The tangent space of $\mathcal{Q}(\tilde{\tau})$ at $\alpha_{Q}\in\mathcal{Q}(\tilde{\tau})$ is given by

where $\pi_{TQ}:TQ\to Q$ is the natural projection. In order to incorporate the constraint distribution, we define the following subspace at each $\alpha_{Q}\in\mathcal{Q}(\tilde{\tau})$,

As usual, we denote $T\mathcal{Q}(\tilde{\tau})=\bigsqcup_{\alpha_{Q}\in\mathcal{Q}(\tilde{\tau})}T_{\alpha_{Q}}\mathcal{Q}(\tilde{\tau})$ and $\Delta_{\mathcal{Q}(\tilde{\tau})}=\bigsqcup_{\alpha_{Q}\in\mathcal{Q}(\tilde{\tau})}\Delta_{\mathcal{Q}(\tilde{\tau})}(\alpha_{Q})$.

Let $T_{\alpha_{Q}}^{\prime}\mathcal{Q}(\tilde{\tau})=\{\phi_{\alpha_{Q}}:T_{\alpha_{Q}}\mathcal{Q}(\tilde{\tau})\to\mathbb{R}\mid\phi_{\alpha_{Q}}\text{ is linear}\newline \text{ and continuous}\}$ be the topological dual of $T_{\alpha_{Q}}\mathcal{Q}(\tilde{\tau})$. Since $\mathcal{Q}(\tilde{\tau})$ is an infinite dimensional manifold, its topological cotangent bundle is too large to formulate mechanics. For that reason, we will restrict ourselves to the vector subbundle where the Legendre transform of the Lagrangian lie, i.e., we consider a vector subbundle $T^{\star}\mathcal{Q}(\tilde{\tau})\subset T^{\prime}\mathcal{Q}(\tilde{\tau})$ such that $\mathbb{F}L\circ\nu_{\alpha_{Q}}\in T^{\star}\mathcal{Q}(\tilde{\tau}))$ for each $\nu_{\alpha_{Q}}\in T_{\alpha_{Q}}\mathcal{Q}(\tilde{\tau})$, where $\mathbb{F}L:TQ\to T^{*}Q$ is the Legendre transform of $L$.

Observe that, in general,

as the Lagrangian is possibly degenerate. As a straightforward consequence of the previous lemma, the vector bundle

is a vector subbundle of the topological cotangent bundle of $\mathcal{Q}(\tilde{\tau})$.

In the same vein, for each $\nu_{\alpha_{Q}}\in T_{\alpha_{Q}}\mathcal{Q}(\tilde{\tau})$ and $\pi_{\alpha_{Q}}\in T_{\alpha_{Q}}^{\star}\mathcal{Q}(\tilde{\tau})$, the iterated bundles are given by

where $\pi_{T(TQ)}:T(TQ)\to TQ$ is the natural projection, and

where $\pi_{T(T^{*}Q)}:T(T^{*}Q)\to T^{*}Q$ is the natural projection. In particular, we consider the constrained iterated bundle,

Given a path $\alpha=(\alpha_{T},\alpha_{Q})\in\Omega(Q,\tilde{\tau})$, the associated curve is defined as

Similarly, given $\nu_{\alpha_{Q}}\in T_{\alpha_{Q}}\mathcal{Q}(\tilde{\tau})$ and $\pi_{\alpha_{Q}}\in T_{\alpha_{Q}}^{\prime}\mathcal{Q}(\tilde{\tau})$, we set

It is clear that $\pi_{TQ}\circ v_{\alpha}=\pi_{T^{*}Q}\circ p_{\alpha}=q_{\alpha}$.

By regarding $\Omega(Q,\tilde{\tau})$ as a trivial vector bundle over $\mathcal{Q}(\tilde{\tau})$ with the projection onto the second factor, the Lagrange–d’Alembert–Pontryagin action functional,

where $\times_{\mathcal{Q}(\tilde{\tau})}$ denotes the fibered product over $\mathcal{Q}(\tilde{\tau})$, is defined as

The equality between the first and the second expressions can be easily checked by considering the change of variable $t=\alpha_{T}(\tau)$. By recalling that the energy of the system, $E:TQ\oplus T^{*}Q\to\mathbb{R}$, is given by

the action functional may be rewritten as

Proof: Let $TQ\simeq Q\times V$ be a trivialization of the tangent bundle of $Q$, and consider the induced trivializations of the cotangent bundle of $Q$, $T^{*}Q\simeq Q\times V^{*}$, as well as of the iterated bundles $T(TQ)\simeq Q\times V\times V\times V$ and $T(T^{*}Q)\simeq Q\times V^{*}\times V\times V^{*}$. Locally, we may write $\alpha^{\prime}\simeq(\alpha_{T},\alpha_{Q},\alpha_{T}^{\prime},\alpha_{Q}^{\prime})$, $\nu_{\alpha_{Q}}\simeq(\alpha_{Q},\nu_{Q})$ and $\pi_{\alpha_{Q}}\simeq(\alpha_{Q},\pi_{Q})$ for some $\alpha_{Q}^{\prime},\nu_{Q}:[\tau_{0},\tau_{1}]\to V$ and $\pi_{Q}:[\tau_{0},\tau_{1}]\to V^{*}$. Moreover, the variations locally read $\delta\alpha\simeq(\alpha_{T},\alpha_{Q},\delta\alpha_{T},\delta\alpha_{Q})$, $\delta\nu_{\alpha_{Q}}\simeq(\alpha_{Q},\nu_{Q},\beta_{Q},\delta\nu_{Q})$ and $\delta\pi_{\alpha_{Q}}\simeq(\alpha_{Q},\pi_{Q},\gamma_{Q},\delta\pi_{Q})$ for some $\delta\alpha_{Q},\beta_{Q},\gamma_{Q},\delta\nu_{Q}:[\tau_{0},\tau_{1}]\to V$, $\delta\pi_{Q}:[\tau_{0},\tau_{1}]\to V^{*}$. In fact, equation (8) ensures that $\delta\alpha_{Q}=\beta_{Q}=\gamma_{Q}$. Moreover, by locally regarding $\Delta_{Q}(q)\subset V$ for each $q\in Q$, the conditions $\nu_{\alpha_{Q}}\in\Delta_{\mathcal{Q}(\tilde{\tau})}$ and $\delta\nu_{\alpha_{Q}}\in\Delta_{T\mathcal{Q}(\tilde{\tau})}(\nu_{\alpha_{Q}})$ read $\nu_{Q}(\tau)\in\Delta_{Q}(\alpha_{Q}(\tau))$ and $\delta\alpha_{Q}(\tau)\in\Delta_{Q}(\alpha_{Q}(\tau))$ for each $\tau\in[\tau_{0},\tau_{1}]$, respectively. At last, the condition $\delta\nu_{\alpha_{Q}}(\tilde{\tau})\in T_{\nu_{\alpha_{Q}}(\tilde{\tau})}(T\partial Q)$ yields the local condition $\delta\alpha_{Q}(\tilde{\tau})\in W$, where $W\subset V$ is a subspace of co-dimension one such that $T\partial Q\simeq\partial Q\times W$.

As a result, the variation of the action functional reads

where the Lagrangian, as well as its partial derivatives, are evaluated at $\left(\alpha_{Q},\nu_{Q}\right)$. After splitting the integration domain, $[\tau_{0},\tau_{1}]-\{\tilde{\tau}\}=[\tau_{0},\tilde{\tau})\cup(\tilde{\tau},\tau_{1}]$, as well as integrating by parts on each sub-interval, we may rewrite the previous expression as

Since the previous expression vanishes for free variations $(\delta\alpha,\delta\nu_{Q},\delta\pi_{Q})$ such that $\delta\alpha_{Q}\in\Delta_{Q}(\alpha_{Q})$, $\delta\alpha_{T}(\tau_{0})=\delta\alpha_{T}(\tau_{1})=0$ and $\delta\alpha_{Q}(\tau_{0})=\delta\alpha_{Q}(\tau_{1})=0$, we obtain the desired equations and impact conditions. $\square$

By using the change of variable $t=\alpha_{T}(\tau)$, we have $\dot{q}_{\alpha}=\alpha_{Q}^{\prime}/\alpha_{T}^{\prime}$ and $\dot{p}_{\alpha}=\pi_{Q}^{\prime}/\alpha_{T}^{\prime}$. Then, the implicit Euler–Lagrange equations for a (local) curve

take the form

on $\left[t_{0},\tilde{t}\right)\cup\left(\tilde{t},t_{1}\right]$. Similarly, the conditions for the elastic impact read

Energy balance: It may be shown that the conservation of the energy along the solutions, $\dot{E}(q_{\alpha},v,p)=0$, is redundant, as it may be obtained from the remaining equations.

For unconstrained systems, i.e., $\Delta_{Q}=TQ$, the Hamilton–d’Alembert–Pontryagin principle reduces to the Hamilton–Pontryagin principle, and the implicit Euler–Lagrange equations of motion read as

The results in the previous section may be obtained in the Hamiltonian side as well. Namely, given a (possibly degenerate) Hamiltonian, $H:T^{*}Q\to\mathbb{R}$, the Hamilton–d’Alembert–Pontryagin action functional,

is defined as