## Abstract

This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under non-holonomic constraints. For this purpose, we derive, analyse and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton–Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange–d’Alembert principle. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, non-holonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle.

## 1. Introduction

The derivation and analysis of equations of motion for non-holonomic deterministic systems has a long history and remains a topic of active research [1]. A new set of challenges arises when such systems become stochastic. We introduce stochasticity that represents uncertainty in the observations and/or simulations of the angular velocity of rolling, e.g. due to finite time steps between observations or computations. For this example, we investigate the effects of this stochasticity in the angular velocity, in the presence of the non-holonomic rolling constraint.

Stochastic Hamiltonian systems were introduced and analysed in the foundational work [2]. These considerations were updated and recast in the language of geometric mechanics in [3], inspiring new considerations such as symmetry reduction and Noether’s theorem in the presence of stochasticity. In this context, the papers [4,5] introduced mechanical systems subjected to stochastic forces while obeying non-holonomic constraints. The physical background for the systems considered there could be understood as the dynamics of a microscopic object bombarded by outside molecules, while preserving a non-holonomic constraint, such as rolling contact. This problem is highly non-trivial, as the reaction forces generated by the constraint require careful consideration.

A different case of stochasticity arising in non-holonomic systems was considered by Gay-Balmaz & Putkaradze [6], where no random forces were acting on the system itself, but the constraint was stochastic. A physical realization of such system would be the motion of a deterministic rolling ball on a rough plane, or experiencing a random slippage at the contact point. It was shown that depending on the type of stochasticity in the constraint itself, the system can preserve some integrals of motion, and the energy is preserved in a general non-holonomic system as long as the constraint remains homogeneous in velocities. This result contrasts with [5], where no integrals of motion were found to be preserved for general stochastic forces. The preservation of integrals of motion described in [6] was due to the nature of the forces introduced by the stochastic constraints.

In this paper, we consider another possible case of stochasticity in non-holonomic systems. The noise we consider arises from, for example, errors in observations of transport velocity for the mechanical systems. For example, imagine a rolling ball which is being recorded by a video camera. The measurement of the orientation and velocity of the ball will always be subject to errors, which will produce a deviation from the expected deterministic trajectory. Unseen irregularities in the surfaces in contact may also cause intermittent changes in angular velocity, without violating the rolling constraint. We shall refer to this class of problems as *stochastic dynamics with transport noise*. Stochastic transport (ST) noise for systems was introduced in the context of fluid dynamics in [7]. For other recent investigations of ST dynamics, we refer the reader to [8–16]. In this paper, we investigate the effect ST has on non-holonomically constrained systems, both analytically and numerically. As it turns out, ST for non-holonomic systems affords an elegant and easy consideration of stochastic non-holonomic mechanics for the case of rolling-ball type systems, with the appropriately generalized applications of the Hamilton–Pontryagin principle and the Lagrange–d’Alembert principle.

### (a) Main content of the paper

This paper formulates an approach for quantifying uncertainty in rolling motion by deriving, analysing and numerically simulating the equations for an unbalanced spherical ball with stochasticity caused by observation uncertainty. More precisely, the stochasticity represents uncertainty in the angular velocity at which the non-holonomic rolling constraint is imposed. The stochastic path is reconstructed from the solution of the dynamical system on which a noisy non-holonomic angular velocity constraint is imposed on the motion of the group

The paper proceeds as follows.

(1) Section 2 presents the derivation of stochastic evolution equations for a system with noisy variational systems with non-holonomic constraints, by using the Hamilton–Pontryagin and Lagrange–d’Alembert variational principles. The noise in the system models observation uncertainty (e.g. temporal resolution) rather than the effect of external random force. These variational principles are applied to the case of an unbalanced rolling ball, which is a classic example of a non-holonomic system.

(2) Section 3 studies the example of a vertically rolling disc, where analytical solutions for the dynamical quantities can be obtained in terms of Stratonovich integrals. Numerical simulations of the rolling disc are also performed to illustrate the theoretical results.

(3) Section 4 computes the evolution equations for the analogues of the first integrals of deterministic rolling: energy, Jellet and Routh. We show that none of these classical integrals of motion are preserved for the stochastic case. We also study a particular case of Chaplygin sphere, when the centre of mass coincides with the geometric centre, and derive the evolution equations for the quantities that are conserved in the analogous deterministic case, but are not conserved in the stochastic case. Numerical simulations of the rolling sphere are performed to compute the variability of the energy, as well as that of the Jellet and Routh quantities and to illustrate the motion of the ball in space.

(4) Section 5 summarizes the paper and discusses other problems treatable by our method.

We mention some nomenclature from the literature. The term ‘Chaplygin’s ball’ refers to a (possibly inhomogeneous) sphere whose centre of mass coincides with the geometric centre. A rolling sphere whose centre of mass does not lie at the geometric centre is often called a ‘Chaplygin top’ in the Russian literature, or a ‘Routh sphere’ in the British literature.

## 2. Non-holonomic stochastic variational principles

### (a) An unbalanced ball rolling with uncertain velocity

#### (i) Problem statement

This section applies the Hamilton–Pontryagin approach [17] to a class of constrained action integrals which includes the motion of an unbalanced spherical ball rolling stochastically on a horizontal plane in the presence of gravity. For deterministic non-holonomic systems that are affine in velocity, this section verifies the equivalence of the Hamilton–Pontryagin principle and the more standard Lagrange–d’Alembert principle by direct computation [17]. This equivalence can be also established for the problems considered here, or, more generally, for the problems formulated on semidirect-product Lie groups, such as the rolling ball. We shall not discuss the equivalence between these two principles in general, as this topic is quite complex and is beyond the scope of this paper. We will use the Hamilton–Pontryagin principle in the remainder of the paper.

The stochasticity in this formulation models the uncertainty in the observation of the velocity of the rolling ball and leads to the reconstruction of the rolling path as a stochastic curve in the semidirect-product Lie group of Euclidean motions *P* in the ball’s reference coordinates to a point in space, *Q*(*t*), at time *t*, according to
*g*(*t*),*x*(*t*))∈SE(3) parameterized by time, *t*. Here, *g*(*t*)∈SO(3) represents the orientation of the ball, and
*P*, chosen to be the ball’s centre of mass. We will choose the initial point *P* to be located in an equilibrium position below the ball’s geometric centre along the vertical direction in space, *e*_{3}.

#### (ii) Rolling constraint

The *spatial non-holonomic constraint distribution**re*_{3} is the spatial vertical vector from the point of contact *C* on the plane to the centre of the ball, ℓ*χ* is the vector displacement in the body pointing along the unit vector from the ball’s geometric centre to its centre of mass, *σ*(*t*) is their sum as spatial vectors, and *s*(*t*) is their sum as body vectors. Figure 1 sketches the configuration of the *spatial vectors* at some time, *t*.

As introduced in (2.1), the time-dependent vectors *s*(*t*) and *Γ*(*t*) in the body reference frame are defined by

We decompose the stochastic quantities *t*) and noise (d*W*^{i}) components (in the Stratonovich representation, denoted as ° d*W*^{i}) following the spatial representation of the rolling constraint (2.1) written in the body frame as

In (2.4), the quantities *fixed* Lie algebra elements, which may be identified with vectors in *W*^{i}(*t*) denotes a set of independent Brownian motions. The set *ξ*_{i} need not span the whole space, and may contain more than three elements. Thus, the *ξ*_{i} do not necessarily form a basis for

Note that the stochastic decomposition in equations (2.4) still satisfies the deterministic rolling condition. That is,

### (b) Hamilton–Pontryagin variational principle for a stochastically rolling ball

We shall introduce the relations (2.1)–(2.4) as constraint equations for the stochastic motion of the rolling ball, as determined from the Hamilton–Pontryagin variational principle, *δS*=0, applied to the following constrained stochastic action integral
*l*(*Ω*,*Y*,*Γ*) is the Lagrangian. When the *ξ*_{i} vanish, the action integral (2.6) reduces to the deterministic case with the standard constraints for rolling without slipping, as discussed in textbooks, e.g. [17]. If *Y* were absent and the middle line in equation (2.6) were missing, then this problem would reduce to the stochastic heavy top, studied in [14,15].

### Theorem 2.1 (Hamilton–Pontryagin principle)

*The stationarity condition for the non-holonomically constrained Hamilton–Pontryagin principle defined in equation (2.6) under the rolling constraint in body coordinates given in (*2.3*) by
**implies the following stochastic equation of motion,
**where s=s(Γ)=rΓ+ℓχ is the vector in the body directed from the point of rolling contact to the centre of mass.*

*In addition, the Lagrange multipiers in the constrained stochastic action integral in (2.6) are given by
*

Before explaining the proof, we recall the definitions of the operations *ad** and

### Remark 2.2 (The *ad** and ⋄ operations)

The coadjoint action

The operation *ad** is defined as the dual of the ad operation,

For the diamond operation *Γ*∈*V* . By its definition in (2.11), the diamond operation for the (left) action

### Proof.

One evaluates the variational derivatives in the constrained Hamilton’s principle (2.6) from the definitions of the variables as
*η*:=*g*^{−1}*δg* and the last equation is computed from [17]

### Remark 2.3

In computing formulae (2.12), one must first take variations of the definitions, and only afterwards evaluate the result on the constraint distribution defined by

Expanding the variations of the Hamilton–Pontryagin action integral (2.6) using relations (2.12) and then integrating by parts yields
*g* using formulae (2.12) obtained from relation (2.13). Stationarity (*δS*=0) for the class of action integrals *S* in equation (2.6) for variations *η* that vanish at the endpoints now proves the formula for the constrained equation of motion (2.8) in the statement of the theorem, while it also evaluates the Lagrange multipliers *Π*, *κ* and *λ* in terms of variational derivatives of the Lagrangian, as

### Remark 2.4 (Stochastic volatility of the noise)

Theorem 2.1 for the Hamilton–Pontryagin principle persists and its proof still proceeds along the same lines, even if further uncertainty is introduced, in the form of volatility in the amplitude of the stochastic processes in the reconstruction relations. In particular, theorem 2.1 and its proof persists modulo small modifications when the stochasticity in the previous reconstruction relations (2.4) for *α*_{i}(*t*) and *β*_{i}(*t*) are, correspondingly, prescribed drift and diffusion terms that may depend upon time, but not any of the dynamical variables, and d*W*^{i}(*t*) is a set of three independent Brownian motions.

### Remark 2.5 (Explicit form of the motion equation)

Expanding out the equation of motion in (2.14) and using the definitions in (2.15) yields

### Remark 2.6 (Vector notation)

For *g*∈*SO*(3), the equation of constrained motion (2.8) arising from stationarity (*δS*=0) of the action in (2.14) may be expressed in ** s** computed from

**s**=

**s**(

**)=**

*Γ**r*

**+ℓ**

*Γ***, as**

*χ*

### Remark 2.7 (Standard form in vector notation)

In what follows, we will assume that the reduced Lagrangian in equation (2.6) for the Chaplygin top (Routh sphere) in body coordinates is given in vector notation as in Holm [17]
** Y**=

**×**

*Ω***s**.

### (c) Standard stochastic differential equation form

It is also useful to rewrite (2.21) in the standard stochastic differential equation (SDE) form, by separating noise and drift terms. For this, we separate ** Y**=

**×**

*Ω***s**:

**computed from**

*s***s**=

**s**(

**)=**

*Γ**r*

**+ℓ**

*Γ***, as**

*χ***x**

_{gc}of the ball later, the following formula will be useful

**e**

_{3}is the fixed vector in vertical direction in the spatial frame. One can see from (2.23) that the vertical coordinate of the geometric centre

**x**

_{gc}⋅

**e**

_{3}is preserved exactly, as expected.

Itô form of (2.22). It is also useful to write (2.22) in Itô form as this formulation is frequently used for numerical solutions of SDEs [19]. For notational convenience, we write this equation in the following compact form:
**b**^{i}_{Γ} does not depend upon the stochastic variables ** Ω** and

**. The Stratonovich-to-Itô conversion formula then yields**

*Γ***b**

_{Γ}(

**,**

*Ω***,**

*Γ**t*) on

**and**

*Ω***given by (2.24) to compute the derivatives in the new drift terms**

*Γ***a**

^{i}

_{Γ,Itô}.

### (d) Lagrange–d’Alembert variational principle for a stochastically rolling ball

This section will show that the Lagrange–d’Alembert principle recovers precisely the same equation (2.17) for a stochastically rolling ball, as was obtained from the Hamilton–Pontryagin approach in the previous section.

The Lagrange–d’Alembert principle on the tangent space *TG* of a Lie group *G* acting on a vector space *V* with equations of motion on *T**(*G*×*V*) is equivalent to a constrained variational principle on *T*(*G*×*V*) with Euler–Poincaré equations on *L* on *T*(*G*×*V*) in the stationary principle is equal to that of the reduced Lagrangian *l* on *G* imply on the reduced space *η*=*g*^{−1}*δg*. As for the pure Euler–Poincaré theory with left-invariant Lagrangians, the proof of the variational formula for *Ω*=*g*^{−1}d*g* expressing *δΩ* in terms of *η* proceeds by direct computation, along the lines of (2.12) and (2.13).

Upon rearranging the stochastic rolling relations (2.4) in the body representation, we find
*s*(*Γ*)=*rΓ*+ℓ*χ* yields
*direct* computation of stationarity of the variation, *δS*, of the action

### Theorem 2.8 Hamilton–Pontryagin versus Lagrange–d’Alembert equivalenceHamilton--Pontryagin versus Lagrange--d'Alembert equivalence

*The non-holonomic motion equations obtained as extremal conditions for the Lagrange–d’Alembert variational principle in Euler–Poincaré form after left Lie group reduction are equivalent to those obtained from the corresponding Hamilton–Pontryagin variational principle.*

## 3. Analytically solvable case: the rolling vertical disc

In order to illustrate our methods, we present the even simpler case of a vertical rolling disc, where the solution can be written explicitly in terms of the stochastic integrals. We consider a flat, uniform disc of mass *m*, radius *r* and moment of inertia taken about the rotation normal to the flat part of the disc being *I* and any axis lying in the plane of the disc being *J*. The configuration space for the vertically rolling disc consists of four variables: *x*,*y*), the configuration is specified by the angle of rotation *ϕ*∈*S*^{1} with respect to a spatially vertical axis in the plane of rolling, and the angle of rotation about the axis of symmetry *θ*∈*S*^{1}. We define the angular velocities *ω* and *ν* and noise intensities *ξ*_{1} and *ξ*_{2} by
*ξ*_{1,2} to be constants, for simplicity in what follows. The rolling constraints are non-holonomic and require that the disc is moving tangent to its sharp edge without slipping. These constraints are written as
*δS* is written via the Hamilton–Pontryagin principle as
*μ*_{1},*μ*_{2}) impose constraints (3.2); (*p*_{1},*p*_{2}) define the velocities (*u*,*v*); and (*π*_{1},*π*_{2}) introduce stochasticity into the angular motion. The variations are given by
*p*_{1} and *p*_{2} are constants, and we will set them to be zero in what follows. The second equation of (3.4) yields equation for *μ*_{1} and *μ*_{2}
*π*_{1} and *π*_{2}, the third and fourth equations of (3.4) yield
*μ*_{1} and *μ*_{2}, then taking the stochastic time differential on the right side of the first equation of (3.6) and using the stochastic constraints for d*θ* and d*ϕ*, we obtain
*ξ*_{1} and *ξ*_{2} are constants, there is no distinction between Stratonovich and Itô noise, so the distribution function for the shifted variables *θ*−*ω*_{0}*t* and *ϕ*−*ν*_{0}*t* tends to the uniform distribution on (0,2*π*). Finally, the trajectory of the disk on the plane is given by integrating the constraint equations (*x*,*y*) given by (3.2), with *ϕ*(*x*) given by (3.7). Clearly, only the component *ξ*_{2} of the noise contributes to the spatial trajectory. We present the results of simulations of equations (3.7) and (3.2) in figure 2. All trajectories start at the origin at *t*=0, with *ξ*_{1}=*ξ*_{2}=0.1, and *ω*_{0}=*ν*_{0}=1. The trajectories initially stay close to the circle, which is the exact solution in the deterministic case, also presented in red in the figure. As time proceeds, the solution deviates further from the deterministic solution.

## 4. Conservation laws and particular cases

### (a) Conservation laws for the Routh sphere

Let us follow the classical results for the Routh sphere (Chaplygin ball), and investigate the preservation of integrals of motion. Mathematically, this particular case is obtained when two moments of inertia are equal, which we take to be *I*_{1}=*I*_{2}, and the axis of the third moment of inertia coincides with the direction ** χ**. In what follows, we shall assume cylindrical symmetry of the ball

*I*

_{1}=

*I*

_{2}, and take the centre of mass to be offset from the geometric centre along the

**E**

_{3}direction, so that

**=**

*χ***E**

_{3}. It is known that, in this case [1,17], the deterministic dynamics preserves three integrals of motion: energy, Jellet and Routh, whose precise expressions will be defined next.

### (b) Energy

Let us first consider the (full) energy defined as
** Ω**, we note that the stochastic evolutionary derivative or

*E*given by (4.1) can be formulated as

### (c) The Jellet integral

Let us turn our attention to Jellet integral *J*=**M**⋅**s**, where *J*≠0 as
*not* conserved in the stochastic rolling of a ball with axisymmetric mass distribution.

### (d) The Routh integral

Let us now turn our attention to the derivation of the Routh integral, which in our variables can be written as *J* is given by (4.3). Hence, even in the case when all *ξ*_{i}∥** χ**, the Jellet integral is

*not*conserved in the stochastic rolling of a ball with axisymmetric mass distribution, so the Routh integral is also

*not*conserved.

The non-conservation of energy, Jellet and Routh for stochastic rolling is verified in the numerical simulations shown in figure 3. This figure also displays the preservation in the numerical simulations of the modulus of the unit vector ** Γ**. The evolution of the projection of the geometric centre of the rolling ball is also shown in figure 4, computed from the formula (2.23) derived earlier. For simulations, we utilized the fully implicit Strong Stratonovich Euler–Heun numerical method for the computation of stochastic systems [21], implemented in Matlab. We refer the reader to that publication and also [6] for the details of numerical implementation of the method for the rolling sphere.

### Remark 4.1

For most initial conditions and parameter values, we have observed numerically a nearly affine relationship between Jellet and Routh integrals, *R*≃ *aJ*+*b*, where (*a*,*b*) depend on the parameter values. This means, in principle, that there may be a constant of motion which we have not been able to identify yet.

### (e) Stochastic rolling of Chaplygin’s ball

A simpler case that has been also well studied in the literature is known as the *Chaplygin ball*, which arises in the case when the centre of mass of the rolling ball coincides with its geometric centre, i.e. ℓ=0, for arbitrary moments of inertia (*I*_{1},*I*_{2},*I*_{3}). In this case, the right-hand side of equation (2.22) vanishes and one recovers the equations of motion for ** Chaplygin’s ball**:

**|**

*Γ*^{2}. The deterministic equations preserve all four of the quantities |

**|**

*Γ*^{2},

**⋅**

*M***, |**

*Γ***|**

*M*^{2}and energy with ℓ→0. By contrast, (4.6) preserves neither the magnitude of total momentum |

**M**|, nor the analogue of the Jellet integral

**M**⋅

**. Indeed, |**

*Γ***M**|

^{2}evolves according to

**M**⋅

**is computed to be**

*Γ***is arbitrary. For the deterministic solution behaviour of Chaplygin’s ball, see, e.g. [22]. We see from (4.8) that if**

*χ***is a time-independent vector in the body frame, the Jellet integral will not be conserved. However, as a test case for verifying our simulations, we may formally put**

*ξ***=0.1**

*ξ***. Technically, this choice is inconsistent as**

*Γ***cannot depend upon variables in the body frame. Nonetheless, this simulation is useful in illustrating the accuracy of our numerical schemes and, thereby, verifying the correctness of our analysis. We present the numerical solutions of Chaplygin’s ball in figure 5. As one can observe from (4.8), in this case, the analogue of Jellet integral should be conserved, which is indeed illustrated on figure 5**

*ξ**b*. figure 5

*a*shows the evolution of energy, which is not conserved, as expected from the analysis. If we take

**to be an arbitrary vector in the body frame, which is either constant or has a prescribed dependence on time, but not on the dynamical variables, then neither the energy nor the Jellet integral is conserved.**

*ξ*## 5. Summary

This paper has shown that stochasticity representing uncertainty in the angular velocity at which a non-holonomic rolling constraint is applied can have dramatic effects on the ensuing dynamics. In particular, this sort of stochasticity destroys the corresponding deterministic conservation laws and thereby liberates the solution to produce large deviations in wandering paths. Thus, non-holonomic constraints can amplify the effects of this sort of noise and create large uncertainty. The Hamilton–Pontryagin approach taken here has been shown to possess an equivalent Lagrange–d’Alembert counterpart. Consequently, all of these results are also available from an alternative viewpoint in the more traditional Lagrange–d’Alembert approach. Moreover, all of the properties associated with geometric mechanics, such as reduction by symmetry, are retained in both approaches.

We expect that it will be interesting in future work to see how the underlying geometric framework for deterministic non-holonomic systems will be used to characterize the probabilistic aspects of their solution behaviour when stochasticity is introduced into the non-holonomic constraints. For example, the relationship between stochastic variational methods and other approaches to introducing stochasticity in mechanical systems offers opportunities for further development. To be more concrete, let us come back to the physical question of experimental observation of a ball rolling on a table. Suppose that the ball is traced using several features on its surface, e.g. bright dots, using a camera. To track the ball’s dynamics, one can either determine the position of the dots and infer its orientation, compute the velocity of the dots in space and infer the angular velocity of the ball, combine these methods with the information provided by the linear velocity of the ball and use the rolling condition, or perhaps even employ an approach combining all of the above methods. Each of these techniques will lead to a different stochasticity in equations, and some of these measurement methods may not be of ST type considered here. One may also be interested in combining stochastic extensions of non-holonomic rolling conditions as in [6] with the ST noise. Making this combination represents another interesting and challenging problem which should be treatable by our methods. Yet another conceivable endeavour would be to use the present formalism in developing control methods for non-holonomic systems with errors in location or velocity, e.g. moving on rough terrain and/or experiencing random slippage. This endeavour might be interesting for the development of practical rolling robots. We believe that further considerations and combinations of different types of stochastic dynamics in non-holonomic systems will be interesting and important. We will consider these endeavours in our future work.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

Both authors contributed equally to the derivation of theoretical models, their analysis and interpretation of numerical results.

## Competing interests

The authors have no competing interests.

## Funding

The work of D.D.H. was partially supported by the European Research Council Advanced Grant 267382 FCCA and EPSRC Standard grant no. EP/N023781/1. The work of V.P. was partially supported by the NSERC Discovery grant and University of Alberta’s Centennial Professorship.

## Acknowledgements

We are enormously grateful to our friends and colleagues whose remarks and responses have encouraged us in this work. We thank A. A. Bloch, F. Gay-Balmaz, M. Leok, T. S. Ratiu, D. V. Zenkov and many others who have offered their valuable suggestions in the course of this work.

## Appendix A. Derivation of evolution for Jellet and Routh integrals in the stochastic case

**(a) Jellet integral**

The evolution equations for Jellet integral *J*=**M**⋅**s** are obtained as follows.
**s**⋅(** χ**×

**)=0, and also noted that**

*Γ**I*

_{1}=

*I*

_{2}.

**(b) Routh integral**

We remind the reader that the Routh integral can be written in our notation as ** χ**-projection of equation (2.21) by computing the following:

*Y*

_{3}=

**⋅**

*χ***. We multiply (2.22) by**

*Y***and compute using (A 2) as follows.**

*χ**Ω*

_{3}, we obtain

*J*given by (A 1).

- Received July 13, 2017.
- Accepted December 1, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.