Hamel’s Formalism and Variational Integrators of the Hydrodynamic Chaplygin Sleigh

Abstract

Modeling of dynamic systems with nonholonomic constraints usually involves constraint multipliers. Consequently, the dynamic equations in the laboratory coordinate system have a complex form, and as a result, the corresponding numerical algorithms need to be improved in terms of both efficiency and accuracy. This paper addresses establishing the mathematical model of the hydrodynamic sleigh in the Hamel framework. Firstly, the Lie symmetry and the Noether theorem conserved quantities of classic Chaplygin sleigh in which the inertial frame is reviewed. Based on the symmetries and the nonholonomic constraints, the frame of the sleigh can be directly realized in the algebraic space. Based on the mutual coupling mechanism between the fluid and the sleigh in a potential flow environment, the reduced equations in the moving frame are proposed in nonintegrable constraint distributions. The corresponding Hamel integrator is constructed based on the discrete variational principle. For the sleigh model in potential flow, the Hamel integrator is used to verify the feasibility of parameter control based on rotation angles and mass distribution, and to obtain the dynamic characteristics of the sleigh blade with both a rotational offset and translational offset. Numerical results indicate that the modeling method in the Hamel framework provides a more concise and efficient approach for exploring the dynamic behavior of the hydrodynamic sleigh.

1. Introduction

A nonholonomic mechanical system refers to a system with nonintegrable differential constraints, and their motion is typified by comparatively intricate motion equations. There are a large number of engineering problems with nonholonomic constraints in fields such as rigid body dynamics [1], vehicle dynamics [2], aerospace vehicle dynamics [3], celestial mechanics [4], and so on. It is particularly important to efficiently model and accurately simulate the dynamic behavior of a system. The traditional modeling method is to handle nonholonomic constraints by introducing constraint multipliers based on the d’Alembert-Lagrange principle in the inertial frame [5,6]. Nevertheless, the d’Alembert-Lagrange principle has flaws in the long-term numerical simulation and in preserving the intrinsic structure of continuous systems, as it is not a variational principle. Additionally, the introduction of constraint multipliers indirectly increases the number of system variables and leads to error accumulation [7,8,9]. Therefore, the traditional modeling method in the inertial frame is not the optimal choice for a long-term simulation of the dynamic behavior of nonholonomic systems.

Hamel’s formalism directly constructs dynamic equations in the space of degrees of freedom allowed by constraints. These equations have more intuitive physical meanings and facilitate the analysis of the system’s dynamic behaviors (such as stability, energy transfer, etc.) [10]. By means of non-material velocities and quasi-coordinates, Hamel equations can transform the dynamic equations of nonholonomic constraint systems into ones that are similar to unconstrained equations, simplifying the process of handling nonholonomic constraints [11,12,13]. They are particularly suitable for modeling engineering systems with nonholonomic constraints, such as those in robotics and vehicle dynamics. The Hamel integrator has inherent advantages in simulating nonholonomic systems [11]. These advantages are reflected in two aspects. On the one hand, they avoid the introduction of redundant variables such as constraint multipliers [12,13]. On the other hand, they prevent the error accumulation of traditional differential-algebraic mathematical models [14,15]. It can also preserve the geometric properties of nonholonomic systems (such as the energy conservation of autonomous systems and the time reversal invariance of the d’Alembert-Lagrange manifold in nonholonomic dynamics) [16,17].

The classical Chaplygin sleigh is a typical nonholonomic constraint system [18,19]. In the Hamel framework, the mathematical model of the Chaplygin sleigh is obtained in a moving frame [20,21,22]. The numerical simulation can more accurately and efficiently predict the trajectory and other dynamic behaviors of hydrodynamic sleds [23,24,25]. In the hydrodynamic version, the motion of underwater vehicles is inherently a coupled process of the hydrodynamic constraints and nonholonomic constraints. Their propulsion (propellers) and steering (fins) correspond to the “sleigh blade”-type nonholonomic constraints [26,27,28]. Kirchhoff proposed a hydrodynamic model for free rigid bodies in the potential flow [29]. The potential flow’s influence on objects is fully described by added mass and inertia terms, which depend only on the geometry shape of the bodies [30]. This method for a potential flow fluid–structure interaction has been extended to the hydrodynamic Chaplygin sleigh [31,32]. It was also considered in the presence of circulation [26,33]. Ref. [34] considered the asymptotic behavior and the trajectory generation problem for the Chaplygin sleigh interacting with a potential fluid. The objective of Ref. [35] was to analyze the parameters affecting the roll-angle stability of an autonomous fish-like underwater swimmer. These modeling forms provide a feasible method for the theoretical analysis of underwater vehicles based on hydrodynamic sleigh models.

However, the modeling of a hydrodynamic sleigh is mainly based on the d’Alembert-Lagrange principle, to establish DAE models. The approach to handling nonholonomic constraints still relies on introducing redundant variables, namely Lagrange multipliers, which increases the complexity of the system’s analytical analysis. To overcome the shortcomings of constraint multipliers and DAE models in numerical computation, we establish a hydrodynamic sleigh model with the Hamel framework. The Hamel integrator, based on the discrete variational principle, can improve the efficiency and stability of numerical computation.

The rest of the paper is organized as follows. In Section 2.1, the kinematic model of a classic Chaplygin sleigh is reviewed, and the Lie symmetries of the sleigh are derived in Section 2.2; Section 3.1 briefly introduces the characteristics of potential flow and the mechanism by which potential flow acts on the sleigh and derives the form of the sleigh’s Lagrangian function in a potential flow environment; in Section 3.2, the mathematical model without multipliers of the hydrodynamic Chaplygin sleigh is constructed in a moving frame; Section 4 obtains the Hamel integrator via the discrete variational principle, and uses the Hamel integrator to simulate the dynamic behaviors of Model I (where the blade contact point is at the sleigh’s geometric center) and Model II (where the blade contact point has an offset), so as to verify the advantages of the Hamel integrator in the process of hydrodynamic sleigh modeling. Some conclusions are obtained in Section 5.

2. Symmetries and Noether Conserved Quantities of Classic Chaplygin Sleigh

2.1. The Motion of a Chaplygin Sleigh

Studies on underwater sleigh models often simulate the movements of actual underwater equipment, such as underwater robots and submersibles. Most underwater equipment has specific dynamic needs. For example, it must maneuver in complex environments and resist currents. To meet these needs, the blades and main shaft are designed with a non-zero angle. This design aims to actively regulate the direction and distribution of hydrodynamic forces. In turn, it optimizes the model’s dynamic performance, including propulsion efficiency, stability, and maneuverability. Suppose the angle between the ellipse axes and the blade direction is $\theta$ (where $\theta$ is non-zero constant in this paper). The configuration manifold for the sleigh is $Q= \mathrm{SE}\left(2\right) =S^1⋉ ℝ$ with local coordinates $q=\left(\phi , x, y\right)$. The angular orientation of the sleigh is $\phi$, and (x, y) are the coordinates of the contact point. The nonholonomic constraint and Lagrangian become

$$\dot{y}\cos \phi -\dot{x}\sin \phi =0,$$

(1)

$$\begin{array}{l}L\left(q,\dot{q}\right)={\displaystyle \frac{1}{2}}\left(m\left({\dot{x}}^2+{\dot{y}}^2\right)+\left({\mathcal{I}}_0+m{a}^2\right){\dot{\phi }}^2\right)\\ +ma\dot{\phi }\left(\dot{y}\cos \phi -\dot{x}\sin \phi \right),\end{array}$$

(2)

where the constants ${\mathcal{I}}_0$, m, and a are the sleigh’s moment of inertia, the mass, and the distance between the center of mass and the knife edge. The nonholonomic equation has the form

$$\ddot{x}=a\cos \phi {\dot{\phi }}^2-\dot{\phi }\sin \phi (\dot{x}\cos \phi +\dot{y}\sin \phi ),$$

(3)

$$\ddot{y}=a\sin \phi {\dot{\phi }}^2+\dot{\phi }\cos \phi (\dot{x}\cos \phi +\dot{y}\sin \phi ),$$

(4)

$$\ddot{\phi }=-{\displaystyle \frac{ma\dot{\phi }(\dot{x}\cos \phi +\dot{y}\sin \phi )}{{\mathcal{I}}_0+m{a}^2}}.$$

(5)

with

$$\lambda ={\displaystyle \frac{m{\mathcal{I}}_0}{{\mathcal{I}}_0+m{a}^2}}\dot{\phi }(\dot{x}\cos \phi +\dot{y}\sin \phi ).$$

(6)

2.2. Lie Symmetries of Chaplygin Sleigh

Based on the theory of symmetries and conserved quantities [28,36], we review the Lie symmetry and Noether-conserved quantities for nonholonomic systems. Suppose $M=TQ$ denotes the tangent bundle with coordinates $\left\{q^{\nu },{\dot{q}}^{\nu }\right\}$, and a nonintegrable distribution, $\mathcal{D}$, describes the linear nonholonomic constraints. The distribution, $\mathcal{D}$, is a collection of linear subspaces of the tangent spaces of $M$; that is, ${\mathcal{D}}_q\subset T_{q}M$ one for each $q\in M$, the distribution, $\mathcal{D}$, can be locally written as

$$B_{\sigma }^i\left(q,t\right){\dot{q}}^{\sigma }+B_0^i\left(q,t\right)-{\dot{q}}^i=0, \sigma =1,\dots ,s,i=s+1,\dots ,n$$

(7)

From the d’Alembert-Lagrange principle, the nonholonomic equation has the form

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}({\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{\dot{q}}^{\nu }}})-{\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{q}^{\nu }}}=Q_{\nu }+{\lambda }_{\beta }{B}_{\sigma }^i.$$

(8)

The right-hand side of Equation (8) represents the force induced by the constraints. Suppose that the system is nonsingular, one can solve all generalized accelerations as

$${\ddot{q}}^{\nu }=F^{\nu }\left(t,q,\dot{q}\right).$$

(9)

Equation (9) can also be written as the dynamic vector field, $Z$, and

$$Z=\mathsf{\partial }/\mathsf{\partial }t+{\dot{q}}^{\nu }\mathsf{\partial }/\mathsf{\partial }{q}^{\nu }+F^{\nu }\mathsf{\partial }/\mathsf{\partial }{\dot{q}}^{\nu }.$$

(10)

The Lie algebra of the symmetry group is realized by vector fields of the form

$${\xi }_{\alpha }={\tau }_{\alpha }(t,q)\mathsf{\partial }/\mathsf{\partial }t+{\xi }_{\alpha }^{\nu }(t,q)\mathsf{\partial }/\mathsf{\partial }{q}^{\nu },\left(\alpha =1,\dots ,r\right),$$

(11)

where ${\tau }_{\alpha },{\xi }_{\alpha }^{\nu }\in C^{\infty }\left(M\right)$. Suppose $\mathsf{\Psi }:M\to M$ is the local group of the transformation group on M. A family of integral curves of Z is denoted as ${\mathsf{\Lambda }}^{non}$; the determining equations of Lie symmetry are

$${\mathsf{\Psi }}^{non}\cdot {\gamma }^{non}\in {\mathsf{\Lambda }}^{non},$$

(12)

where ${\mathsf{\Psi }}^{non}:TQ\to TQ$ is the local group of transformations of $TQ$, ${\mathsf{\Lambda }}^{non}$ is a family of integral curves of the dynamic vector field $TQ$, and ${\gamma }^{non}\in {\mathsf{\Lambda }}^{non}$.

The invariance of Equation (7) under the infinitesimal transformation leads to

$$i_{{\xi }_{\alpha }^{(1)}}\left(B_{\sigma }^i\left(t,q\right){\dot{q}}^{\sigma }+B_0^i\left(q,t\right)-{\dot{q}}^i\right)=0.$$

(13)

The definition and Noether theorem of Lie symmetries for the nonholonomic systems are as follows [37,38,39]. The symbol i denotes the inner product operator.

Definition 1. 

For generators${\tau }_{\alpha },{\xi }_{\alpha }^{\nu }\in C^{\infty }\left(M\right)$, if they satisfy Equations (12) and (13), then this kind of invariance is called Lie symmetry of nonholonomic systems (1) and (2).

Theorem 1. 

For the nonholonomic systems, if the generators ${\tau }_{\alpha },{\xi }_{\alpha }^{\nu }\in C^{\infty }\left(M\right)$ are the Lie symmetry, and there exists a gauge function ${\varpi }_{\alpha }={\varpi }_{\alpha }(t, q,\dot{q})$ satisfy

$$L_{{\xi }_{\alpha }^{(1)}}\vartheta (L)+i_{{\xi }_{\alpha }^{(1)}}\left((\chi +{\lambda }_{\nu }\vartheta (F^{\nu }))\wedge dt\right)=d{\varpi }_{\alpha },$$

(14)

then the corresponding holonomic system (3)–(5) leads to the following exact invariants

$$I={\varpi }_{\alpha }-i_{{\xi }_{\alpha }^{(1)}}\vartheta (L)={\varpi }_{\alpha }-\left[L\tau +{\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{\dot{q}}^{\nu }}}\left({\xi }_{\alpha }^{\nu }-{\dot{q}}^{\nu }{\tau }_{\alpha }\right)\right].$$

(15)

where $\vartheta \left(F^{\nu }\right)=F^{\nu }dt+{\displaystyle \frac{\mathsf{\partial }{F}^{\nu }}{\mathsf{\partial }{\eta }^{\nu }}}{\theta }^{\nu }$ is the Poincaré-Cartan 1-form of $F^{\nu }$, $\chi =Q_{\nu }{\theta }^{\nu }$ the 1-form force and $Q_{\nu }$ is the nonconservative forces.

The constraint (1) affects the derivation of Lie symmetries and the corresponding Noether-conserved quantities. The invariance property of constraint Equation (1) under infinitesimal transformations satisfies Equation (13). If the Lie symmetry of a nonholonomic system is exhibited, it must simultaneously satisfy the Lie symmetry determining equation (Equation (12)) and the constraint restriction equation (Equation (13)). Under this condition, if the symmetry generator further satisfies Equation (14) in Theorem 1, the Noether-conserved quantity induced by the Lie symmetry can be derived.

For the classic Chaplygin sleigh, that is ${\mathcal{D}}_q\subset T_{q}M$, one for each $q=(\phi ,x,y)\in M$, the distribution $\mathcal{D}$ can be locally written as

$${\mathcal{D}}_q=\mathrm{span}\left\{\dot{q}\in TM\left|\dot{y}\cos \phi -\dot{x}\sin \phi =0\right.\right\}.$$

(16)

The invariance of Equation (16) under the infinitesimal transformation leads to

$$i_{{\xi }_{\alpha }^{(1)}}\left(\dot{y}\cos \phi -\dot{x}\sin \phi \right)=0.$$

(17)

The symmetry group associated with the Lie algebra spanned by the operators

$${\xi }_1=\mathsf{\partial }/\mathsf{\partial }t, {\xi }_2=\mathsf{\partial }/\mathsf{\partial }x, {\xi }_3=\mathsf{\partial }/\mathsf{\partial }y, {\xi }_4=t\mathsf{\partial }/\mathsf{\partial }t, {\xi }_5=-y\mathsf{\partial }/\mathsf{\partial }x+x\mathsf{\partial }/\mathsf{\partial }y+\mathsf{\partial }/\mathsf{\partial }\phi .$$

(18)

The commutators of the operators are arranged in a table, where the commutator [ξα, ξβ] is placed at the intersection of the αth row with the βth column.

Table 1. Commutators of group.

	ξ1	ξ2	ξ3	ξ4	ξ5
ξ1	0	0	0	ξ1	0
ξ2	0	0	0	ξ1	ξ3
ξ3	0	0	0	0	−ξ2
ξ4	−ξ1	0	0	0	0
ξ5	0	−ξ3	ξ2	0	0

shows the commutators of the operators. We can conclude that the operators are closed under the Lie bracket.

The generators of the invariant group ${\xi }_{\alpha }={\tau }_{\alpha }(t, q)\mathsf{\partial }/\mathsf{\partial }t+{\xi }_{\alpha }^{\nu }(t, q)\mathsf{\partial }/\mathsf{\partial }{q}^{\nu }$ construct a five-dimensional subalgebra spanned by the basis $\left\{{\xi }_{\alpha },\alpha =1,\dots ,5\right\}$, respectively. From Noether Theorem 1, it can produce the Noether-conserved quantity

$$I_{\mathrm{con}}={\displaystyle \frac{1}{2}}\left(m\left({\dot{x}}^2+{\dot{y}}^2\right)+\left({\mathcal{I}}_0+m{a}^2\right){\dot{\phi }}^2\right),$$

(19)

which is the energy of the system.

For the simplicity of symbols, variables are shown in bold only when they are expressed in terms of components in this paper.

3. The Motion of Hydrodynamic Chaplygin Sleigh in Hamel Framework

In the absence of a potential flow, the kinetic energy tensor of a rigid body is determined solely by its own inertia. After introducing a potential flow, the kinetic energy tensor becomes the sum of the rigid body inertia and the added fluid inertia. This change affects the manifestation of symmetry in the kinetic energy through the added fluid inertia tensor. However, the existence of the potential flow does not alter the uniformity, isotropy of the space of the sleigh itself, or the corresponding group symmetry structure. Therefore, in a potential flow environment, we construct a moving frame based on the aforementioned symmetry generators to realize the modeling of the hydrodynamic sleigh within the Hamel framework.

3.1. Hydrodynamic

Consider a Chaplygin sleigh moving in a potential fluid. Assume that the flow of a fluid is irrotational and inviscid, with a zero velocity at the fluid boundary; no cavities are produced in the fluid. The effect of adding the fluid to the system can be represented by a fluid inertia tensor added to that of the body. Most of the material covered in this section can be found, for instance, in [40], as well as in the classic works of Lamb [30].

The classic Chaplygin sleigh is essentially a flat rigid body of mass, m, moving in the plane, supported at three points, two of which slide freely without friction while the third is a knife-edge constraint which allows no motion perpendicular to its edge.

A simple yet interesting case is an elliptical planar body with semi-axes A > B > 0. We define a kinetic energy Lagrangian $L:T\left(SE(2)\right)\to ℝ$. This time ${\mathbb{I}}_{\mathcal{F}}$ is a scalar, ${\mathcal{K}}_{\mathcal{F}}$ is a two-dimensional row vector, and ${\mathcal{M}}_{\mathcal{F}}$ is a 2 × 2 matrix. Thus, the Lagrangian for the system is

$$L(\zeta )={\displaystyle \frac{1}{2}}{\zeta }^T\left({\mathbb{I}}_F+{\mathbb{I}}_B\right)\zeta .$$

(20)

First, assume that the origin is at the center of the ellipse, and the blade makes an angle θ with the major axis of the ellipse, as illustrated in Figure 1.

We consider an elliptic planar rigid body of mass m, major semi-axis A, and minor semi-axis B, immersed in a potential incompressible fluid. By applying Kirchhoff decomposition and added masse theory, the kinetic energy of the fluid can be expressed as the quadratic form

$${\mathbb{I}}_{\mathcal{F}}=\rho \pi \left(\begin{array}{ccc}{\displaystyle \frac{{\left(A^2-B^2\right)}^2}{4}}& 0& 0\\ 0& B^2{\cos }^2\theta +A^2{\sin }^2\theta & {\displaystyle \frac{A^2-B^2}{2}}\sin (2\theta )\\ 0& {\displaystyle \frac{A^2-B^2}{2}}\sin (2\theta )& A^2{\cos }^2\theta +B^2{\sin }^2\theta \end{array}\right),$$

and the elements of ${\mathbb{I}}_{\mathcal{F}}$ depend solely on the body shape. The body inertia tensor is

$${\mathbb{I}}_{\mathcal{B}}=\left(\begin{array}{ccc}{\mathcal{I}}_0+m\left(a^2+b^2\right)& -mb& ma\\ -mb& m& 0\\ ma& 0& m\end{array}\right),$$

(21)

where m is the mass of the body, (a, b) are the body coordinates of the center of mass, and ${\mathcal{I}}_0$ is the moment of inertia of the body about the center of mass.

Suppose the total inertia tensor of the fluid-body system is then given by

$$\mathbb{I}=\left[\begin{array}{ccc}J& -L_2& L_1\\ -L_2& M& Z\\ L_1& Z& N\end{array}\right]$$

(22)

where

$$\begin{array}{l}J={\mathcal{I}}_0+m(a^2+b^2)+\rho \pi {\displaystyle \frac{{(A^2-B^2)}^2}{4}},\\ M=m+\rho \pi \left(B^2{\cos }^2\theta +A^2{\sin }^2\theta \right),\\ N=m+\rho \pi \left(A^2{\cos }^2\theta +B^2{\sin }^2\theta \right),\\ Z=\rho \pi \left({\displaystyle \frac{A^2-B^2}{2}}\right)\sin (2\theta ),\\ L_1=ma,\hspace{1em}{L}_2=mb.\end{array}$$

Among them: J is the angular diagonal component of the total inertia tensor, representing the total rotational inertia of the system around the origin of the body coordinate system; M is the translational diagonal component of the total inertia tensor, representing the total translational inertia of the system along the blade-edge direction; N stands for translational diagonal component of the total inertia tensor, representing the total translational inertia of the system perpendicular to the blade edge direction; L₁ and L₂ are the coupling components of the total inertia tensor, reflecting the dynamic coupling between angular velocity and translational velocity; and Z stands for the translational–translational cross component of the total inertia tensor, reflecting the dynamic coupling between the two translational directions.

Immersing a sleigh in a fluid will lead to the coupling of its motions in different axial directions, and its dynamic behaviors will change drastically. Due to the high nonlinearity of real fluids (which are viscous and rotational), the interaction can be very complicated. In this paper, we study the hydrodynamic Chaplygin sleigh in an ideal fluid. The interaction between a fluid and a rigid body depends only on the body, and the results may provide a “solvable basic framework” for complex problems in real fluids.

3.2. The Hamel Equations of Sleigh Without Multipliers

A local coordinate chart induces a frame, ${\left\{\mathsf{\partial }/\mathsf{\partial }{q}^i\right\}}_{i=1}^n$, that forms a basis of the tangent fiber, $T_{q}Q$, which we refer to as the coordinate-induced frame. Then, given a coordinate chart and a frame field, ${\left\{u_j\right\}}_{j=1}^n$, there exists $\psi (q)\in GL(n)$ at each q, such that, in coordinates,

$$u_j(q)={\psi }_j^i(q){\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{q}^i}}.$$

(23)

Then, the components of $\dot{q}(t)$ are likewise expressible in terms of the new frame, according to the equation

$$\dot{q}={\dot{q}}^i{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{q}^i}}={\zeta }^j{\psi }_j^i(q){\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{q}^i}}={\zeta }^j{u}_j(q).$$

(24)

So that ${\dot{q}}^i={\zeta }^j{\psi }_j^i(q).$ Next, we rewrite the Lagrangian in terms of the new velocity components, so that in coordinates

$$L\left(q^i,{\dot{q}}^i\right)=L\left(q^i,{\zeta }^j{\psi }_j^i(q)\right)=\mathcal{l}\left(q^i,{\xi }^j\right).$$

(25)

For nonholonomic systems, the distribution $\mathcal{D}$ can be regarded as a collection of linear subspaces with an assumption that the distribution is nonsingular, smooth, and of a constant rank smaller than dim Q. We assume that there exists a unique orthogonal complementary distribution, $\mathcal{U}$, such that ${\mathcal{D}}_q\oplus {\mathcal{U}}_q=T_{q}Q$ for each $q\in Q$. Select the velocity operators ${\mathsf{\Psi }}_q$ on $\mathcal{U}$; then, there exist two closed subspaces $W^{\mathcal{D}}$ and $W^{\mathcal{U}}\in W$, such that $W=W^{\mathcal{D}}\oplus W^{\mathcal{U}}$ and ${\mathsf{\Psi }}_q={\mathsf{\Psi }}_q^{\mathcal{D}}\oplus {\mathsf{\Psi }}_q^{\mathcal{U}}$, where ${\mathsf{\Psi }}_q^{\mathcal{D}}$ maps from $W^{\mathcal{D}}$ to ${\mathcal{D}}_q$ and ${\mathsf{\Psi }}_q^{\mathcal{U}}$ maps from $W^{\mathcal{U}}$ to the complementary space of ${\mathcal{D}}_q$ in $T_{q}Q$ [11,14]. Then, each velocity, $\dot{q}={\mathsf{\Psi }}_q\zeta$, can be uniquely decomposed as

$$\dot{q}={\mathsf{\Psi }}_q{\zeta }^{\mathcal{D}}+{\mathsf{\Psi }}_q{\zeta }^{\mathcal{U}}.$$

(26)

Then, the velocity constraints read $\zeta ={\zeta }^{\mathcal{D}}$ or ${\zeta }^{\mathcal{U}}=0$. Likewise, each variation $\delta q\in T_{q}Q$ can be decomposed as

$$\delta q={\mathsf{\Psi }}_q{\eta }^{\mathcal{D}}+{\mathsf{\Psi }}_q{\eta }^{\mathcal{U}}.$$

(27)

and the variation constraints read $\eta ={\eta }^{\mathcal{D}}$ or ${\eta }^{\mathcal{U}}=0$.

The d’Alembert-Lagrange principle in the combination of (26) proves the following theorem [11,21]:

Theorem 2. 

The dynamics of a nonholonomic system is represented by the constrained Hamel’s equations

$${\left({\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\displaystyle \frac{\mathsf{\partial }\mathcal{l}}{\mathsf{\partial }\zeta }}-{\left[{\zeta }^{\mathcal{D}},{\displaystyle \frac{\mathsf{\partial }\mathcal{l}}{\mathsf{\partial }\zeta }}\right]}_q^{*}-u[\mathcal{l}]\right)}_{\mathcal{D}}=0,{\zeta }^{\mathcal{U}}=0,.$$

(28)

coupled with the kinematic equation $\dot{q}=u(q)\cdot {\zeta }^{\mathcal{D}}$.

In order to more naturally handle the blade constraint, we utilize a moving frame for the sleigh. We write $\zeta \in \mathfrak{s}\mathfrak{e}(2)$ as $\zeta :=({\zeta }^1,{\zeta }^2,{\zeta }^3)\in ℝ\times {ℝ}^2$. Many references showed that the equations of motion of the classic Chaplygin sleigh have a simpler structure when it is written using velocity components measured against a frame that is unrelated to the system’s local configuration coordinates [11,12]. Next, we briefly discuss the Hamel equations. The frame is considered, based on these Lie symmetry vectors $\left\{\mathsf{\partial }/\mathsf{\partial }x,\mathsf{\partial }/\mathsf{\partial }y,-y\mathsf{\partial }/\mathsf{\partial }x+x\mathsf{\partial }/\mathsf{\partial }y+\mathsf{\partial }/\mathsf{\partial }\phi \right\}$ of (19) and the constraint distribution (16). The intersection of the group trajectory and the constraint space are spanned by

$$u_1=\mathsf{\partial }/\mathsf{\partial }\phi ,$$

(29)

$$u_2=\cos \phi \mathsf{\partial }/\mathsf{\partial }x+\sin \phi \mathsf{\partial }/\mathsf{\partial }y,$$

(30)

$$u_3=-\sin \phi \mathsf{\partial }/\mathsf{\partial }x+\cos \phi \mathsf{\partial }/\mathsf{\partial }y.$$

(31)

which is the frame of the system and consistent with the results of Refs. [11,21]. The moving frame consists of three vectors: a vector directed perpendicular to the sleigh’s plane to measure the angular position, a vector along the blade direction, and another vector perpendicular to the blade in the sleigh’s plane.

In the moving frame (29)–(31), the Lagrangian in terms of non-material velocity for the sleigh is

$$l\left(\zeta \right)={\displaystyle \frac{1}{2}}\left[J{\left({\zeta }^1\right)}^2+M{\left({\zeta }^2\right)}^2+N{\left({\zeta }^3\right)}^2-2L_2{\zeta }^1{\zeta }^2+2L_1{\zeta }^1{\zeta }^3+2Z{\zeta }^2{\zeta }^3\right].$$

(32)

Based on Theorem 1, the continuous Hamel equation is

$$\left\{\begin{array}{l}{\dot{\zeta }}^1={\displaystyle \frac{1}{MJ-L_2^2}}\left(-M\left(Z{\zeta }^2+L_1{\zeta }^1\right){\zeta }^2+L_2\left(L_1{\zeta }^1+Z{\zeta }^2\right){\zeta }^1\right)\\ {\dot{\zeta }}^2={\displaystyle \frac{1}{MJ-L_2^2}}\left(-L_2\left(Z{\zeta }^2+L_1{\zeta }^1\right){\zeta }^2+J\left(L_1{\zeta }^1+Z{\zeta }^2\right){\zeta }^1\right)\end{array}\right..$$

(33)

This is a system of first-order ordinary differential equations with corresponding analytical solutions. It is simpler than the sleigh dynamics equations in the inertial frame (a system of second-order differential-algebraic equations). In terms of numerical computation, there are also various numerical algorithms that can be directly applied, such as single-step methods (Euler method, Runge–Kutta method, etc.) and linear multi-step methods (Adams–Bashforth method, Adams–Moulton method, etc.), as well as their corresponding extended algorithms.

Next, based on the Lagrangian function in the moving frame, we adopt the discrete variational principle to obtain the variational integrator in the Hamel framework. In addition to advantages in terms of accuracy and stability, this numerical algorithm, which operates on the system’s constraint distribution, also has obvious structure-preserving characteristics compared with traditional algorithms.

4. Hamel Integrator for Hydrodynamic Chaplygin Sleigh in $\mathfrak{s}\mathfrak{e}(2)$

For the hydrodynamic Chaplygin sleigh, the nonholonomic systems (8) can be written as the constrained Euler–Lagrange equation

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}({\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{\dot{q}}^{\nu }}})-{\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{q}^{\nu }}}=F_{\beta }$$

where $F_{\beta }={\lambda }_{\beta }{B}_{\sigma }^i$. Given a discrete Lagrangian $L_d:Q\times Q\to ℝ$ with $q_d={\left\{q_k\right\}}_{k=0}^N$ and ${\lambda }_d={\left\{{\lambda }_k\right\}}_{k=1}^{N-1}$, the discrete nonholonomic constraints $F_{\beta }^{+}:={\lambda }_{\beta ,k+1}{B}_{\sigma ,k+1}^i$ and $F_{\beta }^{-}:={\lambda }_{\beta ,k}{B}_{\sigma ,k}^i$, the augmented Lagrangian ${\overline{L}}_d(q_k,q_{k+1})=L_d(q_k,q_{k+1})+F_{\beta }^{+}(q_k,q_{k+1})$ and

$$\begin{array}{c}{\overline{L}}_d^{+}(q_k,{\lambda }_k,q_{k+1},{\lambda }_{k+1})=L_d(q_k,q_{k+1})+〈{\lambda }_{\beta ,k+1},B_{\sigma ,k+1}^i〉\\ {\overline{L}}_d^{-}(q_k,{\lambda }_k,q_{k+1},{\lambda }_{k+1})=L_d(q_{k-1},q_k)+〈{\lambda }_{\beta ,k},B_{\sigma ,k}^i.〉.\end{array}$$

Based on the discrete variational principle ${\displaystyle \sum _{k=0}^N\delta {\overline{L}}_d\left(q_k,q_{k+1}\right)}=0$, that is

$$\begin{array}{l}{\displaystyle \sum _{k=0}^N\left[\frac{\mathsf{\partial }{\overline{L}}_d^{+}(q_k,{\lambda }_k,q_{k+1},{\lambda }_{k+1})}{\mathsf{\partial }{q}_k}\delta q_k+\frac{\mathsf{\partial }{\overline{L}}_d^{-}(q_k,{\lambda }_k,q_{k+1},{\lambda }_{k+1})}{\mathsf{\partial }{q}_{k+1}}\delta q_{k+1}\right]}\\ ={\displaystyle \sum _{k=1}^{N-1}\left[\frac{\mathsf{\partial }\left(L_d(q_k,q_{k+1})+〈{\lambda }_{\beta ,k+1},B_{\sigma ,k+1}^i〉\right)}{\mathsf{\partial }{q}_k}+\frac{\mathsf{\partial }\left(L_d(q_{k-1},q_k)+〈{\lambda }_{\beta ,k},B_{\sigma ,k}^i〉\right)}{\mathsf{\partial }{q}_k}\right]}\delta q_k=0\end{array}$$

In the above derivation, we employed the discrete integration by parts. This method essentially involves rearranging the sum and conditions, $\delta q_0=\delta q_N=0$. Taking into account the independence of $\delta q_k$, we can obtain the discrete Euler–Lagrange equation

$$D_2{L}_d\left(q_{k-1},q_k,k\right)+D_1{L}_d\left(q_k,q_{k+1},k+1\right)+F_{\beta }^{-}\left(q_{k-1},q_k\right)+F_{\beta }^{+}\left(q_k,q_{k+1}\right)=0\hspace{0.33em},$$

where $D_2{L}_d\left(q_{k-1},q_k,k\right)={\displaystyle \frac{\mathsf{\partial }{L}_d\left(q_{k-1},q_k,k\right)}{\mathsf{\partial }{q}_k}}\hspace{0.33em}$ and $D_1{L}_d\left(q_k,q_{k+1},k+1\right)={\displaystyle \frac{\mathsf{\partial }{L}_d\left(q_k,q_{k+1},k+1\right)}{\mathsf{\partial }{q}_k}}.$

Consider a sequence of configurations, ${\left\{q_k\right\}}_{k=0}^N$. Define an interpolation point $q_{k+\tau }:=(1-\tau )q_k+\tau q_{k+1}, k=0,\dots ,N-1$ for the given parameter $\tau \in \left[0,1\right]$. A discrete analog of the kinematic is ${\displaystyle \frac{1}{\mathsf{\Delta }t}}\left(q_{k+1}-q_k\right)={\mathsf{\Psi }}_{q_{k+\tau }}{\zeta }_{k+\tau }$, the discrete constraints are ${\zeta }_{k,k+1}={\zeta }_{k,k+1}^{\mathcal{D}}$, the discrete Lagrangian $l_d:TQ\to ℝ$ is a function of $\left(q_{k+\tau },{\zeta }_{k+\tau }\right)$ and reads $l_d\left(q_{k+\tau },{\zeta }_{k+\tau }\right)$, which is constructed by the formula

$$l_d\left(q_{k+\tau },{\zeta }_{k+\tau }\right)=\mathsf{\Delta }tl\left(q_{k+\tau },{\zeta }_{k+\tau }\right).$$

(34)

Under the same assumptions for the frame selection as in the continuous systems, the variation in the discrete action of (43) is taken to obtain the integrator in the Hamel frame. We interpolate the variation ${\eta }_{k+\tau }:=\left(1-\tau \right){\eta }_k+\tau {\eta }_{k+1}$ and the discrete conjugate momentum ${\mu }_{k+\tau }:=D_2{l}_d\left(q_{k+\tau },{\zeta }_{k+\tau }\right)$ by

$$S_d:={\displaystyle \sum _{k=0}^{N-1}{l}_d\left(q_{k+\tau },{\zeta }_{k+\tau }\right)}=0$$

(35)

with ${\eta }_0={\eta }_N=0$.

Using the discrete variational principle, we have

$$\begin{array}{l}\delta s^d={\displaystyle \sum _0^{N-1}\frac{\mathsf{\partial }l}{\mathsf{\partial }{q}_{k+\tau }}\delta q_{k+\tau }+\frac{\mathsf{\partial }l}{\mathsf{\partial }{\xi }_{k,k+1}}\delta {\xi }_{k,k+1}}\\ =〈{\displaystyle \frac{\mathsf{\partial }l}{\mathsf{\partial }{q}_{k+\tau }}},\delta q_{k+\tau }〉+〈{\displaystyle \frac{\mathsf{\partial }l}{\mathsf{\partial }{\xi }_{k,k+1}}},\delta {\xi }_{k,k+1}〉\\ =〈{\displaystyle \frac{\mathsf{\partial }l}{\mathsf{\partial }{q}_{k+\tau }}},u_{k+\tau }\cdot {\eta }_{k+\tau }〉+〈{\displaystyle \frac{\mathsf{\partial }l}{\mathsf{\partial }{\xi }_{k,k+1}}},{\displaystyle \frac{{\eta }_{k+1}-{\eta }_k}{h}}+{\left[{\xi }_{k,k+1},{\eta }_{k+\tau }\right]}_{q_{k+\tau }}〉\\ =〈{\displaystyle \frac{\mathsf{\partial }l}{\mathsf{\partial }{q}_{k+\tau }}}{\psi }_j^i,{\eta }_{k+\tau }〉+〈{\mu }_{k,k+1},{\displaystyle \frac{{\eta }_{k+1}-{\eta }_k}{h}}〉+〈{\mu }_{k,k+1},{\left[{\xi }_{k,k+1},{\eta }_{k+\tau }\right]}_{q_{k+\tau }}〉\\ =〈u{[l]}_{k+\tau }+{\left[{\xi }_{k,k+1},{\mu }_{k,k+1}\right]}_{q_{k+\tau }}^{\ast },{\eta }_{k+\tau }〉+〈{\mu }_{k,k+1},{\displaystyle \frac{{\eta }_{k+1}-{\eta }_k}{h}}〉\\ =(1-\tau )〈u{[l]}_{k+\tau }+{\left[{\xi }_{k,k+1},{\mu }_{k,k+1}\right]}_{q_{k+\tau }}^{\ast },{\eta }_k〉+\tau 〈u{[l]}_{k+\tau }+{\left[{\xi }_{k,k+1},{\mu }_{k,k+1}\right]}_{q_{k+\tau }}^{\ast },{\eta }_{k+1}〉+{\displaystyle \frac{1}{h}}〈{\mu }_{k,k+1},{\eta }_{k+1}〉-{\displaystyle \frac{1}{h}}〈{\mu }_{k,k+1},{\eta }_k〉\end{array}$$

Considering the boundary term and the independence of ${\eta }_k$, we have

$${\displaystyle \frac{1}{h}}\left({\mu }_{k-1,k}-{\mu }_{k,k+1}\right)+\tau u{[l]}_{k-1+\tau }+(1-\tau )u{[l]}_{k+\tau }+\tau {\left[{\xi }_{k-1,k},{\mu }_{k-1,k}\right]}_{q_{k-1+\tau }}^{\ast }+(1-\tau ){\left[{\xi }_{k,k+1},{\mu }_{k,k+1}\right]}_{q_{k+\tau }}^{\ast }=0$$

In the above derivation, the following relation is used

$$\begin{array}{l}\delta q_{k+\tau }=u_{k+\tau }\cdot {\eta }_{k+\tau }={\psi }_b^i{\eta }_{k+\tau }^b,\delta {\xi }_{k,k+1}={\displaystyle \frac{{\eta }_{k+1}-{\eta }_k}{h}}+{\left[{\xi }_{k,k+1},{\eta }_{k+\tau }\right]}_{q_{k+\tau }}={\displaystyle \frac{{\eta }_{k+1}-{\eta }_k}{h}}+C_{ij}^b{\xi }_{k,k+1}{\eta }_{k+\tau },\\ {\eta }_{k+\tau }:=(1-\tau ){\eta }_k+\tau {\eta }_{k+1},〈p,{\left[v,w\right]}_q〉=〈{\left[v,p\right]}_q^{*},w〉\end{array}$$

Based on our modeling of constrained distributions, the dynamics of a discrete nonholonomic system is then given by the constrained Hamel equations

$$\begin{array}{l}{\displaystyle \frac{1}{h}}{\left({\mu }_{k-1,k}-{\mu }_{k,k+1}\right)}_{\mathcal{D}}+{\left(\tau u{[l]}_{k-1+\tau }+(1-\tau )u{[l]}_{k+\tau }\right)}_{\mathcal{D}}\\ +{\left(\tau {\left[{\zeta }_{k-1,k},{\mu }_{k-1,k}\right]}_{q_{k-1+\tau }}^{\ast }+(1-\tau ){\left[{\zeta }_{k,k+1},{\mu }_{k,k+1}\right]}_{q_{k+\tau }}^{\ast }\right)}_{\mathcal{D}}=0.\end{array}$$

(36)

The approximation order is one, in general, and becomes two when $\tau =1/2$. We refer the reader to Theorem 2.2 in [11] or Theorem 2.2 in [23] for more details. In this paper, $\tau =1/2$ is taken in numerical simulation. The discrete Lagrangian of the sleigh is

$$\begin{array}{l}{l}_d={\displaystyle \frac{1}{2}}{\left({\zeta }_{k+1/2}^1\right)}^{T}J\left({\zeta }_{k+1/2}^1\right)+{\displaystyle \frac{1}{2}}{\left({\zeta }_{k+1/2}^2\right)}^{T}M\left({\zeta }_{k+1/2}^2\right)+{\displaystyle \frac{1}{2}}{\left({\zeta }_{k+1/2}^3\right)}^{T}N\left({\zeta }_{k+1/2}^3\right)\\ -L_2{\zeta }_{k+1/2}^1{\zeta }_{k+1/2}^2+L_1{\zeta }_{k+1/2}^1{\zeta }_{k+1/2}^3+Z{\zeta }_{k+1/2}^2{\zeta }_{k+1/2}^3\end{array}$$

(37)

Based on the construction of (35), we take the variation

$$\delta {\displaystyle \sum }_{k=0}^{N}h{l}^d=h\langle w^{k+1/2},\delta v^{k+1/2}\rangle =0.$$

(38)

We have the

$$\left\{\begin{array}{l}{w}^n=w^{k+1/2}-{\displaystyle \frac{h}{2}}a{d}_{v^{k+1/2}}^{\ast }{w}^{k+1/2}\\ w^{k+1}=w^{k+1/2}+{\displaystyle \frac{h}{2}}a{d}_{v^{k+1/2}}^{\ast }{w}^{k+1/2}\end{array}\right.$$

(39)

where $w^{k+1/2}=\left[\begin{array}{l}J{\zeta }_{k+1/2}^1-L_2{\zeta }_{k+1/2}^2+L_1{\zeta }_{k+1/2}^3\\ M{\zeta }_{k+1/2}^2-L_2{\zeta }_{k+1/2}^1+Z{\zeta }_{k+1/2}^3\\ N{\zeta }_{k+1/2}^3+L_1{\zeta }_{k+1/2}^1+Z{\zeta }_{k+1/2}^2\end{array}\right],v^{k+1/2}=\left[\begin{array}{l}{\zeta }_{k+1/2}^1\\ {\zeta }_{k+1/2}^2\\ {\zeta }_{k+1/2}^3\end{array}\right]$.

Figure 2a shows the error of the total energy of the hydrodynamic Chaplygin sleigh with the Hamel integrator, the RK45, and the analytical results. The Hamel integrator has more precise numerical results than those from the standard RK45 under the initial conditions m = 1 kg, ${\mathcal{I}}_0$ = 0.5 kg·m², x₀ = y₀ = 0 m, a = 0.2 m, b = 0 m, h = 0.001, A = 0.5 m, B = 0.3 m, $\theta =\pi /3,{({\zeta }^1)}_0=1.5 \mathrm{rad}/\mathrm{s}$, and ${({\zeta }^2)}_0=-1.0 \mathrm{m}/\mathrm{s}$. Figure 2b shows the energy error with the Hamel integrator method. For the comparison of the Hamel integrator with simulations, use the RK45 and the analytical solution. Compared with traditional numerical algorithms, the Hamel integrator has higher accuracy. Figure 3 shows the 2-norm of angular velocity. It shows that the convergence of the Hamel integrator is better than that of the RK45.

The nonholonomic RATTLE method extends RATTLE to nonholonomic constraint scenarios by means of the framework of geometric nonholonomic integrators (GNI), so as to meet the discretization requirements of nonholonomic mechanical systems. As an extension of the RATTLE algorithm for nonholonomic constraint systems, this method was first proposed in Reference [41]. Based on this geometric method, this paper briefly verifies the variation trends of the energy and linear velocity. When compared with the Hamel numerical algorithm proposed in this paper, it can be seen from Figure 4 (a = 0.7 m, b = 0 m) that the Hamel numerical algorithm still has advantages in terms of accuracy and stability over the nonholonomic RATTLE method. Figure 4 shows the Euclidean norm error of the configuration. The error (on the vertical axis) continues to decrease and approaches 0 as the number of iterations increases, indicating that the Hamel product integrator is convergent.

Figure 5 shows the trajectory of the sleigh when the center of mass changes from (a, b) = (0.2 m, 0 m) to (a, b) = (0.2 m, 0.1 m). The arrows in the figure indicate the direction of the sled’s movement. The same applies to the following figures. The classical sleigh trajectory is an asymptotic straight line. In a potential flow environment, due to the coupling terms induced by added inertia, the trajectory of the elliptic sleigh displays asymptotic evolution from one circular motion to another one, in opposite directions. The Hamel integrator exhibits the unique dynamic phenomenon of asymptotic circular motion under potential flow.

The hydrodynamic sleigh has different motion trajectories when the angle between the blade and the main shaft is changed. In Figure 6, when a > 0 and $\theta \in \left[0,\pi /2\right]$, the turning direction is counterclockwise, and the sleigh eventually moves in a clockwise limit circular motion. When the angle is greater than π/2 (in red), it causes the direction of the limit circle radius to change, and the turning direction shifts from clockwise to counterclockwise. The movement path and direction of the sleigh can be controlled by adjusting the angle, $\theta$, from the simulation results of the Hamel integrator.

Figure 7 simulates the trajectories of the system with centers of mass (COM) (a, b) = (0.5 m, 0 m) and (a, b) = (0.5 m, 1 m), respectively. The Hamel integrator accurately yields the analytical results of Ref. [26]: that is, the spacing between the centers of the limit circles is related to the structural parameters of the inertia tensor. Selecting different mass distributions corresponds to different spacings between the centers of the limit circles.

For comparison, the trajectory of the contact point and the looping phenomenon of the classical Chaplygin sleigh are simulated in Figure 8, using the Hamel integrator. If the limiting difference in the sleigh’s rotation angle is 1.73π, the angle change does not reach 2π, which is insufficient to form a complete loop. As the angle between the limiting straight lines gradually increases, multiple loops will be formed. Therefore, there are significant differences between the dynamic behaviors of the classical sleigh and the hydrodynamic sleigh; the latter exhibits more diverse manifestations in the fluid environment.

Next, we consider the most general case, in which every possible entry of the total inertia tensor is able to vary independently of the rest. That is, the origin of the moving frame is not in the center of the ellipse. Assume that the origin has coordinates (r, s), with respect to the frame that is aligned with the principal axes of the ellipse, and that the coordinate axes u₂ and u₃ are not aligned with the axes of the ellipse, forming an angle, $\theta$ (measured counter-clockwise), as illustrated in Figure 9. Then, the tensor takes a more general form with ${({\mathbb{I}}_{\mathcal{F}})}_{13}$ and ${({\mathbb{I}}_{\mathcal{F}})}_{23}$ non-zero, which can be calculated explicitly and lead to the corresponding modification of the total tensor, $\mathbb{I}$. For this geometry, the kinetic energy of the fluid can be expressed as the quadratic form [30,31].

$${\mathbb{I}}_{\mathcal{F}}=\rho \pi \left(\begin{array}{ccc}\begin{array}{l}{\displaystyle \frac{{(A^2-B^2)}^2}{2}}+s^2(B^2{\cos }^2\theta +A^2{\sin }^2\theta )+\\ +r^2(A^2{\cos }^2\theta +B^2{\sin }^2\theta )+\\ -rs(A^2-B^2)\sin (2\theta )\\ \end{array}& \begin{array}{l}s(B^2{\cos }^2\theta +A^2{\sin }^2\theta )+\\ -{\displaystyle \frac{1}{2}}r(A^2-B^2)\sin (2\theta )\end{array}& \begin{array}{l}-r(A^2{\cos }^2\theta +B^2{\sin }^2\theta )+\\ +{\displaystyle \frac{1}{2}}s(A^2-B^2)\sin (2\theta )\end{array}\\ \begin{array}{l}s(B^2{\cos }^2\theta +A^2{\sin }^2\theta )+\\ -{\displaystyle \frac{1}{2}}r(A^2-B^2)\sin (2\theta )\\ \end{array}& B^2{\cos }^2\theta +A^2{\sin }^2\theta & {\displaystyle \frac{A^2-B^2}{2}}\sin (2\theta )\\ \begin{array}{l}-r(A^2{\cos }^2\theta +B^2{\sin }^2\theta )+\\ +{\displaystyle \frac{1}{2}}s(A^2-B^2)\sin (2\theta )\end{array}& {\displaystyle \frac{A^2-B^2}{2}}\sin (2\theta )& A^2{\cos }^2\theta +B^2{\sin }^2\theta \end{array}\right)$$

So, every entry of the total inertia tensor in (22) is

$$\begin{array}{l}J={\mathcal{I}}_0+m(a^2+b^2)+\rho \pi \left[\begin{array}{l}{\displaystyle \frac{{(A^2-B^2)}^2}{2}}+s^2(B^2{\cos }^2\theta +A^2{\sin }^2\theta )+\\ +r^2(A^2{\cos }^2\theta +B^2{\sin }^2\theta )-rs(A^2-B^2)\sin (2\theta )\end{array}\right]\\ M=m+\rho \pi \left[B^2{\cos }^2\theta +A^2{\sin }^2\theta \right]\\ N=m++\rho \pi \left[A^2{\cos }^2\theta +B^2{\sin }^2\theta \right]\\ L_1=ma+\rho \pi \left[-r(A^2{\cos }^2\theta +B^2{\sin }^2\theta )+{\displaystyle \frac{1}{2}}s(A^2-B^2)\sin (2\theta )\right]\\ L_2=mb-\rho \pi \left[s(B^2{\cos }^2\theta +A^2{\sin }^2\theta )-{\displaystyle \frac{1}{2}}r(A^2-B^2)\sin (2\theta )\right]\\ Z=\rho \pi \left[{\displaystyle \frac{A^2-B^2}{2}}\sin (2\theta )\right]\end{array}$$

Figure 9 is an extension of Figure 1. Next, we employ the Hamel integrator to explore its dynamic behavior. The contact point (r, s), with respect to the frame, is determined by the principal axes of the ellipse. As the contact point deviates from the center of the ellipse, the off-diagonal terms of the fluid-added inertia tensor become more complex. This results in a more generalized form of the total inertia tensor, which contains more non-zero cross terms. These cross terms in the total inertia tensor act together, leading to stronger coupling between translational velocity and angular velocity. Such strong coupling complicates the transition process of the system from initial motion to asymptotic circular motion.

Figure 9. Arbitrary position and orientation of the body frame.

Figure 9. Arbitrary position and orientation of the body frame.

Specifically, Figure 1 shows the case where the contact point is at the center of mass (referred to as “Model I” in this paper), while Figure 9 depicts the scenario where the contact point does not coincide with the center of mass (designated as “Model II” herein). As shown in Figure 10, the trajectories of these two models are presented under the conditions A = 2, B = 1.5, $\theta =\pi /4$, (a, b) = (0.5, 0.2), and (r, s) = (0.5, 1.5). Through detailed trajectory analysis, this paper systematically illustrates the trajectory trends, variations in blade deflection angles, and key characteristics of contact points. This, in turn, verifies that the Hamel integrator is capable of accurately simulating the dynamic behavior of the target system.

It can be seen that (i) compared to Model I, the distance between the centers of the trajectory’s stable limit cycle and the unstable limit cycle of Model II is larger, and the transition path of the trajectory bends more sharply; (ii) Model I’s trajectory transition path has a relatively uniform “degree of curvature”. Its trajectory symmetry is relatively high (the two circles are approximately symmetric about an axis). But, for Model II, the line connecting the centers of the two circles no longer passes through the trajectory’s “geometric center”. It shows stronger asymmetry overall; for example, one circle has greater curvature and the transition path is biased toward one direction.

The reason for the above phenomenon is that the sleigh has non-zero components on both body coordinate axes simultaneously, and the path is affected by strong coupling from the center-of-mass offset and contact point offset.

The increased complexity of the total inertia tensor induced by the offset of the contact point leads to enhanced motion coupling, and also changes in asymptotic parameters (radius, spacing). However, the basic asymptotic law (the transition from one circular motion to the reverse circular motion) remains preserved. This coupling also breaks the system’s symmetry and conserved quantities, such as the conserved quantity of momentum. The conserved quantity of momentum in the classical Chaplygin sleigh system has its root of symmetric transformation in spatial translation symmetry. According to the symmetry-conserved quantity correspondence of Noether’s theorem, it ultimately manifests as the conservation of the total momentum. In a potential flow environment, the presence of fluid causes the system momentum (total momentum, including the momentum of the rigid body and the fluid-added momentum) to no longer be conserved. The core reason is related to the coupling effect of the fluid on the rigid body and non-holonomic constraints. The corresponding symmetries are broken by the added mass tensor.

5. Conclusions

This manuscript describes the motion of a hydrodynamic Chaplygin sleigh model in the Hamel framework. The advantages of Hamel equations of the hydrodynamic sleigh are as follows: (i) the equations are simplified with both the dimension (3 − 1 = 2) and order (2 − 1 = 1) reduced and (ii) the method avoids introducing constraint multipliers.

The Hamel integrator is not a direct discretization of the Hamel equations; it originates from the discrete variational principle and can preserve the geometric structure of the system during the discretization process. The advantages of the Hamel integrator for a hydrodynamic sleigh are as follows: (i) it can avoid the problem of error accumulation in traditional differential-algebraic equations, so it has higher computational accuracy; (ii) it avoids the disadvantage of reduced efficiency in traditional algorithms caused by adding redundant variables (constraint multipliers).

The mathematical model of a hydrodynamic sleigh in the Hamel framework is a simplified model of research fields such as trajectory planning and the control of wheeled robots, path tracking of autonomous vehicles, satellite attitude adjustment, and spacecraft orbital maneuvering. In a potential fluid, this model may provide a theoretical framework in the design of underwater vehicles and mechanisms.

If non-ideal conditions such as viscosity are taken into account, the problem will become more generalized and will also be capable of handling the characterization of more complex and realistic fluid interactions. References [42,43] studied the complex, non-conservative fluid forces. Applying the variational integrator to such a data-driven or non-parametric model would represent a far more novel contribution.