The Geometric Structure of Externally Actuated Planar Locomoting Systems in Ambient Media

We now review the underlying geometric structure for locomoting systems like the three-link wheeled snake robot.

The usage of geometric mechanics in locomotion has traditionally assumed the existence of a principal fiber bundle structure in a system’s configuration space. This structure decomposes the space into two distinct subspaces, often called a fiber bundle, with a mapping called a connection relating trajectories from one to trajectories in the other.

In this work, we will consider the configuration spaces of a robot and its environment to comprise smooth fiber bundles, particularly principal fiber bundles.

Fig. 3 shows a typical visualization of the fiber bundle as a direct product of a base space $B$ and a fiber space $G$. The space $B$ corresponds to the system’s shape variables, which are traditionally assumed to be fully actuated. The space $G$ corresponds to the system’s configuration variables, which typically correspond to symmetries in the system, so that the dynamics of the system do not depend explicitly on the fiber variables. Since we will make use of symmetries in both of the examples presented, a system exhibits a continuous symmetry if its dynamics are invariant under the free action of a Lie group on its configuration manifold [4].

In this paper, the symmetry exploited for the three-link wheeled snake robot on a platform is given by the invariance of the system’s dynamics under the action of $\text{SE}(2)$ on its configuration manifold $Q=\text{SE}(2)\times\mathbb{S}^{1}\times\mathbb{S}^{1}$. The Chaplygin beanie on a platform system exhibits a symmetry arising from the action of the Lie group $G=\text{SE}(2)\times\mathbb{R}^{2}$ on the configuration manifold $Q=\text{SE}^{2}\times\mathbb{R}^{2}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$.

The local form of the connection $A(b)$, which depends only on base variables $b$, governs how the system’s fiber variables change due to changes in the base. The equation is typically of the form

where $\dot{g}$ are the velocities of the fibers, and $\dot{b}$ are the velocities of the shape variables. The mapping $T_{e}L_{g}$ for $g\in\text{SE}(2)$ is given by

with $\theta$ representing the orientation of the robot relative to an inertial reference frame in the plane. The mapping $T_{e}L_{g}$ transforms fiber velocities from a body-attached frame to a world frame; in practice, one can drop the $T_{e}L_{g}$ mapping by performing all analysis in the body frame. If we are able to express the fiber velocities $\dot{g}$ as body velocities $\xi$ via the transformation $\dot{g}=T_{e}L_{g}\xi$, then Eq. (1) can be recast into the kinematic reconstruction equation[32], given by

As an example, consider the wheeled robot of Fig. 1 in the absence of an underlying platform. Its configuration space can naturally be written as $Q_{w}=G_{w}\times B_{w}$, where $g_{w}=(x,y,\theta)^{T}\in G_{w}$ are the position and orientation fiber variables, and $b_{w}=(\alpha_{1},\alpha_{2})^{T}$ are the joint, or shape, variables.¹¹1We use the subscript $w$ for the configuration space of the wheeled snake robot. Similarly, we will use the subscript $c$ for the Chaplygin beanie. The fiber $G_{w}$ is the Lie group of planar translations and rotations SE$(2)$. The body velocities $\xi$ are related to the world velocities $\dot{g}_{w}$ through the mapping $T_{e}L_{g}$.

The structure of the connection form in Eq. (3) can be visualized in order to understand the response of $\xi$ to input trajectories without regard to time, according to [32]. By integrating each row of Eq. (3) over a given joint trajectory, one can obtain a measure of displacement corresponding to the body frame directions. For $\text{SE}(2)$, this measure provides the exact rotational displacement, i.e., $\dot{\theta}=\xi_{\theta}$ for the third row, and an approximation of the translational component for the first two rows.

We can consider the time integral to be a line integral over the trajectory $\psi:[0,T]\rightarrow B$ in the shape space, since the kinematics depend only on the gait executed in shape space. This can then be viewed as an integral over $\beta$, the region of the joint space enclosed by $\psi$. Assuming that we have periodic trajectories, or gaits, the integral can be realized by Stokes’ theorem as

The first term in the final integral is the exterior derivative of $A(b)$ and is computed as the curl of $A(b)$ in two dimensions, i.e.,

where $A_{i,j}$ is the element corresponding to the $i$th row and $j$th column of $A$. The second term is the Lie bracket of the two vector fields comprising the local connection form $A(b)$. In this work, we consider only the first term in our approximation of the displacement. Prior works on similar locomoting systems have shown that this approximation suffices for the study of locomotion, even in the absence of Lie bracket and higher-order terms when considering an appropriate choice of coordinates [31, 1, 33]. Eq. (5) can be extended to arbitrary dimensions [34], however, we define it for two-dimensional base spaces since the present paper focuses on planar systems.

The magnitudes of the three-link robot’s connection exterior derivative over the joint space are depicted in Fig. 4, along with a gait trajectory shown as a closed curve on the surfaces. The area integral over the enclosed region is the geometric phase, a measure of the displacement in the body $x$ and $\theta$ directions (the body $y$ plot is not shown because it is zero everywhere). The $x$ plot is positive everywhere, meaning that any closed loop will lead to net displacement along the $\xi_{x}$ direction. The $\theta$ plot is anti-symmetric about $\alpha_{1}=-\alpha_{2}$, meaning that gaits symmetric about this line will yield zero net reorientation while simultaneously moving the robot forward.

Identifying conserved quantities in mechanical systems permits the use of reduction tools we are ultimately interested in leveraging in our analysis of an externally actuated locomoting system with drift in its dynamics, e.g., the Chaplygin beanie on a movable platform. In particular, Noether’s theorem [35] states that continuous symmetries correspond to conserved quantities in a mechanical system. In this section, we introduce the terminology necessary to leverage symmetries in the reduction we employ for the Chaplygin beanie on a movable platform.

The Lie algebra elements $\xi$ associated with each vector field $\xi_{Q}$ are needed to determine $\dot{J}$ in Eq. (9). To compute each $\xi$, it is necessary to define the subspace on which the constrained dynamics evolve. Let $\mathcal{D}$ be the distribution spanned by all tangent vectors that annihilate the constraints, and let the group orbit of the Lie group $G$ be

Associated with every vector field $\xi_{Q}$ referenced in Eq. (8) is a Lie algebra element $\xi\in\mathfrak{g}$ where $\mathfrak{g}$ is the Lie algebra of the Lie group $G$. We define the tangent space to the group orbit through a point $q\in Q$ as

where $\xi_{Q}(q)$ is the vector field $\xi_{Q}$ evaluated at $q$. We define the intersection of $\mathcal{D}_{q}$ and $T_{q}(\text{Orb}(q))$ to be the reduced space in which the constrained dynamics evolve, which we write as

where $\mathcal{D}_{q}$ denotes the constraint distribution at point $q$. We have the freedom to choose the set of vector fields comprising $T_{q}(\text{Orb}(q))$, and our choice of vector fields determines the reduced space (Eq. (12)) and thus the resulting evolution equations in Eq. (9).

If we consider the example of the three-link wheeled robot of Fig. 1 on a moving platform, then the configuration of the platform would live in a new fiber space. Furthermore, this new fiber space corresponds to another set of symmetries in the system. That is, the system’s dynamics and energetics are invariant with respect to the robot’s $(x,y)$ position (two of its fibers) as well as the platform’s position (external fibers). A subtlety does arise regarding the robot’s orientation, $\theta$, relative to the platform; the dynamics of the platform depend on how the robot is oriented in the environment. A helpful simplification would be to continuously orient the platform’s frame of reference to the robot’s body frame, eliminating the dependency of the dynamics on $\theta$. In order for this arrangement to be practical, the platform would also ideally be able to move along any direction in the plane.

For this type of system, the fiber space takes on a stratified structure $G=G^{\text{int}}\times G^{\text{ext}}$. We separately identify fiber variables associated with the configuration of the system $g^{\text{int}}\in G^{\text{int}}$ making up an internal fiber, describing the configuration of the robot, from those describing the configuration of the platform $g^{\text{ext}}\in G^{\text{ext}}$, the external fiber. This is visualized in Fig. 5 as two differently colored fiber components.

Note that for systems for which locomotion is governed by a principal connection, the original fiber bundle is unchanged and we retain our original connection $-A^{\text{int}}(b)$. Furthermore, a second connection $-A^{\text{ext}}(b)$ from the robot’s base configuration to the medium’s configuration can be derived in a manner analogous to $-A^{\text{int}}(b)$, and takes the form

The structures of the two connections $-A^{\text{int}}(b)$ and $-A^{\text{ext}}(b)$ can both be analyzed in the same visual manner as shown in Fig. 4. For example, if we care only about the robot’s fiber motion in response to certain joint trajectories without regard to how the platform moves, then we need only look at $-A^{\text{int}}(b)$, and the problem is unchanged from before. On the other hand, looking at $-A^{\text{ext}}(b)$ tells us how the robot’s joint trajectories lead to motion of the platform.

In this paper, and in particular for the three-link robot on a platform system, we will consider both the traditional problem of mapping base trajectories to the external fiber, as well as the problem of having the controls in the external fiber $G^{\text{ext}}$, with the base variables remaining passive. Once we know our trajectories in $B$, the original connection $-A^{\text{int}}(b)$ ultimately lifts them to the internal fiber $G^{\text{int}}$. The problem of finding platform inputs in $G^{\text{ext}}$ to produce a set of desired base trajectories in $B$ is more difficult, as the “inverse” mapping of $-A^{\text{ext}}(b)$ may not be well defined. In this paper we consider only cases for which the dimension of the external actuation is equivalent to the dimension of the base manifold, ensuring $A^{\text{ext}}(b)$ is invertible.

In the three-link robot system, the preservation of symmetries holds as long as all quantities are prescribed or expressed relative to the body-fixed frame of the robot. If this is not the case, e.g., the platform velocity inputs are prescribed relative to an inertial frame, then one of the fiber components, the robot’s orientation $\theta$, breaks this symmetry. The decision to keep all quantities relative to an inertial frame may be one of practicality, especially when considering multiple distinct locomoting robots in an ambient medium.

In all of these cases, the system’s internal dynamics will no longer be symmetric with respect to a subset of the fiber degrees of freedom (e.g., $(x,y)$ in the case of a three-link robot on a platform). However, we may still be able to define a principal bundle on the base variables and the fibers that are symmetric. For example, a falling, reorienting robot is affected by gravity in one fiber direction (or two, if orientation is also considered). A visual representation is shown in Fig. 6; like Fig. 5 we see fibers atop a base space, the former of which is subdivided into two distinct components. Trajectories in a non-symmetric fiber component evolve dynamically depending on trajectories in the base. This evolution is depicted as flow along a vector field in the semi-transparent yellow plane in Fig.  6. Both trajectories are then mapped through a connection together to obtain trajectories in the symmetric fiber component.

Mathematically, a new connection mapping $-A^{\text{sym}}$ from the base variables to all of the configuration variables would depend explicitly on the fiber variables that lack symmetries. That is, if $g^{\text{sym}}$ are the fiber variables of which the dynamics are independent, and $g^{\text{non}}$ are the external variables that are not (previously we had $g^{\text{int}}$ and $g^{\text{ext}}$ for internal and external fiber variables, respectively), then

This splitting between $g^{\text{sym}}$ and $g^{\text{non}}$ is conducive to studying motion planning for the three-link snake robot on a platform when considering the platform fibers to be relative to an inertial reference frame, and not coinciding with the body frame of the robot. The first involves being able to decouple the two lines of Eq. (14). If it is the case that $\dot{g}^{\text{sym}}$ depends very weakly on $g^{\text{non}}$ or if $g^{\text{non}}$ evolves on a significantly slower time scale relative to $b$ and $g^{\text{sym}}$, then one can approximate the first line of Eq. (14) as a form of Eq. (1), where any $g^{\text{non}}$ dependencies are made constant and inputs are prescribed in $b$. The second line of Eq. (14), which deals with the evolution of $g^{\text{non}}$, can then be treated separately from the first as it has no dependencies on $g^{\text{sym}}$.

If such a decoupling cannot be done, we would have to work with the full form of Eq. (14). In this case, we can define $B^{\text{non}}=G^{\text{non}}\times B$ as an extended base space in which we have partial control over the base variables. We treat $g^{\text{non}}$ as if they are passive base variables, and we can first analyze their kinematics from the second line of Eq. (14) using techniques that we have already described. Once we have solved for the motion involving base and non-symmetric fibers only, it is straightforward to push through the original connection in the first line of Eq. (14) to find the rest of the system’s fiber motion.

In summary, we have identified two different approaches for accommodating external actuation into the principal fiber bundle picture. In the first, all external degrees of freedom are themselves symmetries of the entire system, so that we can consider them to be a separate fiber with their own connection mapping. In the second, these new degrees of freedom actually break symmetries, in which case we consider a splitting of the configuration space to differentiate between symmetric and non-symmetric variables, dealing with them separately. It is also possible to have systems that exhibit both types of splittings simultaneously.

Recall that the configuration space of the robot can be written as $Q_{w}=G^{\text{int}}_{w}\times B_{w}$, where $g^{\text{int}}_{w}=(x,y,\theta)^{T}\in G^{\text{int}}_{w}$ are the fiber variables describing the robot’s position and orientation, and $b_{w}=(\alpha_{1},\alpha_{2})^{T}\in\mathbb{T}^{2}$ are the base variables describing the robot’s joint angles. We designate the constant parameter $R$ for the link lengths of the robot, which are identical across all links.

The kinematics of the system are described by the set of nonholonomic constraints on the wheels, which prohibit motion perpendicular to each of the links’ longitudinal directions. They can be written as three equations of the form

where $(\dot{x}_{i},\dot{y}_{i})$ is the velocity and $\theta_{i}$ is the inertial orientation of the $i$th link. These quantities can be computed recursively to express them as functions of $q_{w}$. Starting with the proximal link, we have $(x_{1},y_{1},\theta_{1})=(x,y,\theta)$; for $i=2,3$:

The constraint equations are symmetric with respect to the fiber part $G_{w}$ of the configuration, since the kinematics do not explicitly depend on where the system is positioned or how it is oriented in space. We can thus rewrite the constraints in a reduced Pfaffian form as

where $\omega_{\xi}\in\mathbb{R}^{3\times 3}$, $\omega_{b}\in\mathbb{R}^{3\times 2}$, and $\xi=(\xi_{x},\xi_{y},\xi_{\theta})^{T}\in\mathfrak{se}(2)$ are the body velocities of the system expressed in a frame attached to the proximal link, as shown in Fig. 1. Recall that the velocities $\dot{g}_{w}$ and $\xi$ are related through $\dot{g}_{w}=T_{e}L_{g}\xi$, where $T_{e}L_{g}$ is the lifted left action given by Eq. (2).

The reduced constraints of Eq. (17) can then be rearranged to derive the connection relationship of Eq. (3). For the three-link robot, this can be written as

where $D=\frac{2}{R}(-\sin\alpha_{1}-\sin(\alpha_{1}-\alpha_{2})+\sin\alpha_{2})$. The values of the second row, corresponding to $\xi_{y}$, are zero since this corresponds to the direction prohibited by the wheel of the proximal link. The mapping $-A^{\text{int}}_{w}(b_{w})$ corresponds to the connection form between the base (robot joints) and the internal fiber (robot position and orientation). Visualizations of the exterior derivative of this connection form were shown in Fig. 4.

We now add in the dynamics of an underlying movable platform to the kinematics of the three-link wheeled snake robot. The position of the platform, which we denote by $(x_{p},y_{p})$, itself constitutes a symmetry group $G^{\text{ext}}_{p}$ since the modified system’s properties do not depend on where the platform is located in space. The configuration manifold is now rewritten as $Q_{p}=G^{\text{int}}_{w}\times G^{\text{ext}}_{p}\times B_{w}=\mathrm{SE(2)}\times\mathbb{R}^{2}\times\mathbb{T}^{2}$, although it is important to note that the connection relationship of Eq. (18) still holds, as the nonholonomic constraints Eq. (17) depend only on the robot’s pose and velocities relative to the platform, rather than an inertial frame.

Suppose that each of the links has identical mass $M^{l}$ and a moment of inertia $J$, while the platform has a mass $M^{p}$. Recalling that the inertial link positions and orientation are given by $(x_{i},y_{i},\theta_{i})$, the system’s Lagrangian is

Since the nonholonomic constraints are independent of $\dot{x}_{p}$ and $\dot{y}_{p}$, governing the trajectories of $(\dot{x},\dot{y},\dot{\theta})$, the free directions of motion are simply the degrees of freedom of the platform. The conserved momenta are given by

Noting that the form of Eq. (19) is the same as the non-reduced form of Eq. (17), we can rearrange to obtain

where $M=3M^{l}+M^{p}$.

Here, the matrix $-A^{\text{ext}}_{p}(b_{w})$ is the external connection of our now stratified bundle structure. It is also the local form of the mechanical connection, so named because it is derived from the conservation of momentum for the combined robot-platform system. In practice, we can drop the momentum drift terms if the platform starts at rest. The main complexity comes from the rotation matrix in front of the mechanical connection—the range of the connection is still the robot’s Lie algebra, or velocities expressed in the robot-fixed frame. In the next subsection we first bypass this problem by assuming that we can prescribe platform velocities in the robot’s body frame.

In deriving Eq. (20) we noted that the main difficulty in using it, even after dropping the drift terms using a start-from-rest assumption, is the presence of the rotation matrix. This relationship signifies a dependence on $\theta$, the orientation of the robot relative to the platform (and inertial frame). However, if we know or can prescribe the platform’s position relative to the robot’s body frame²²2Of course, this assumes that we have free control over the platform’s motion in all directions, rather than along a set of fixed axes., we can directly use the external, mechanical connection $-A^{\text{ext}}_{p}(b_{w})$ without accounting for the effect of $\theta$ on the robot-platform interaction. In other words, we can choose to specify the platform’s velocity using the variables

Again assuming that the system starts from rest, Eq. (20) then reduces to

Since $-A^{\text{ext}}_{p}(b_{w})$ plays a role identical to that of the kinematic connection for the robot, we can also plot its exterior derivative in the same way that we have done for the robot. Fig. 7 shows their shape for a set of chosen system parameters.

The main observation one finds from these plots is that the $u_{p}$ plot is in general a flipped version of the robot’s $x$ exterior derivative plot (Fig. 4). This is reasonable; as the robot moves in the forward or backward direction, we would expect the platform to move in the opposite direction, preserving net initial momentum of zero. The exterior derivative for $v_{p}$ (motion of the platform in a direction lateral to the robot’s heading) resembles the robot’s $\theta$ exterior derivative (second plot of Fig. 4), but reflected about the $\alpha_{1}=-\alpha_{2}$ axis. This can be physically interpreted as a counteraction of the platform due to rotation of the robot; as the platform does not have a rotation degree of freedom, the counteraction is projected onto the $v_{p}$ component.

With the derivation of Eq. (22) we are now also able to consider the problem of actuating the platform to induce a completely passive robot to move in a desired way. In other words, we can prescribe platform velocities $(\dot{u}_{p},\dot{v}_{p})$, which would produce trajectories $b_{w}(t)$ in the robot’s joint variables. These base trajectories (and the associated velocities) then result in the robot moving along its fiber degrees of freedom according to Eq. (18) and Fig. 4.

This problem can be approached in two ways. The first is to assume that we have access to a particular robot fiber motion, either desired or measured, for which we can find a corresponding gait in the base space through analysis of the robot’s original connection mapping in Eq. (18). The same gait can also then be pushed through the platform connection relationship shown in Eq. (22). This would yield the required platform velocity inputs $\dot{u}_{p}(t)$ and $\dot{v}_{p}(t)$ that would generate the robot’s overall desired motion. As an example, suppose we want to induce the robot to locomote as shown in the top figure of Fig. 8, in which the robot advances forward without reorientation. The gait that would achieve this is the gait shown as a closed loop in Fig. 7. Mapping this gait through the platform connection produces the required velocities as shown in the bottom figure of Fig. 8.

One drawback with the above approach is that there is no guarantee on the feasibility of platform input. This is reflected in Fig. 8 in that the required input requires a constant nonzero offset in $\dot{u}_{p}(t)$, meaning that the platform would essentially have to move on the plane with the robot. The second, more control-oriented, way to approach this problem is to invert the mechanical connection in Eq. (22) to find feasible $(\dot{u}_{p},\dot{v}_{p})$ trajectories to obtain a desired $\dot{b}_{w}$.

That Eq. (22) is reduced, i.e., we have eliminated all robot fiber dependencies from the equation, allows us to consider only the interaction between the robot’s joints and the platform fiber variables. Furthermore, the equation bears a resemblance to the dynamics between a set of actuated and a set of unactuated joints on the same robot, as detailed in [1]; the only difference is that the actuated variables in this case are external to the robot.

For example, it can be numerically shown that increasing the relative phase between the two input directions will increase the offset of the resultant joint trajectories. Such a gait is shown in Fig. 10 (top); note that $\dot{v}_{p}$ lags $\dot{u}_{p}$ more and more over time. This has the effect of shifting the center of the gait in shape space (Fig. 10, bottom left) from the third quadrant to the first quadrant. As we know from the $\theta$ plot of Fig. 4, gaits with negative offsets in $\alpha_{1}$ and $\alpha_{2}$ acquire negative net orientation; gaits with positive offsets acquire positive net orientation. In addition, the robot continues to move forward in its body direction while rotating, since all gaits enclose a positive net area in the $x$ plot of Fig. 4. All of these observations are apparent in the robot’s fiber trajectory for this simulation (Fig. 10, bottom right). It starts at the origin moving forward on a trajectory rotating clockwise (negative curvature) and at around $t=60$ begins rotating counterclockwise (positive curvature), as the relative phase between $\dot{u}_{p}$ and $\dot{v}_{p}$ becomes large and the center of the resulting $\alpha_{1}$-$\alpha_{2}$ gait becomes positive.

In the previous subsection, the assumption that we can actuate the platform relative to the robot body’s longitudinal and lateral directions requires that the platform can freely move in all directions. An alternative scenario is that it may be limited to move only in two fixed directions, corresponding to two independent degrees of freedom. Assuming that the system starts from rest, this model is described exactly by Eq. (21).

Even if the system again starts from rest, dropping the momenta terms from the equation, the dependency of Eq. (21) on the robot’s orientation $\theta$ introduces an additional complexity. We assume that we know the robot’s orientation throughout the system’s operation, for example via an overhead camera, so that $\theta$ is not an unknown quantity. However, it is a symmetry-breaking fiber variable that changes the form of the connection $-A_{p}(b_{w})$ (shown in the general case by Eq. (14)) and its corresponding exterior derivative plots in Fig. 7.

Fig. 9 shows the effect of $\theta$ on $\text{d}A_{p}(b_{w})$; in other words, the plots show the exterior derivative, for different values of $\theta$, of a $\theta$-dependent connection

The plots shown correspond to $\theta$ at $45$, $90$, and $180$ degrees (of course, the nominal plots of Fig. 7 correspond to $\theta=0$). As we would expect, the nature of the interaction between the robot and platform changes with the orientation of the robot. As the robot rotates to $90$ degrees, its body $x$ axis points along the inertial $y$ axis, while its body $y$ axis points along the inertial $x$ axis. The $x_{p}$ plot is thus identical to $-v_{p}$, or an inverted version of the $v_{p}$ plot in Fig. 7, while $y_{p}$ is identical to $u_{p}$ (recall that the corresponding robot and platform components are inverted). At $180$ degrees, both $x_{p}$ and $y_{p}$ are completely inverted from $u_{p}$ and $v_{p}$, the latter again corresponding to $x_{p}$ and $y_{p}$ at $0$ degrees. Finally, as the robot reorients back to $0$ degrees, both plots smoothly deform back to their nominal forms (Fig. 7).

With a varying $\theta$, robot base trajectories no longer solely lie on the $\alpha_{1}$-$\alpha_{2}$ plane in interacting with the inertial platform velocities $\dot{x}_{p}$ and $\dot{y}_{p}$. However, we are able to simplify this problem and reduce motion planning to analysis of a single exterior derivative function as before if $\theta$ is known to be periodic. In other words, the robot may start out at some arbitrary orientation, but its net motion only moves it forward in the body direction and does not turn it away from its initial orientation. From the connection exterior derivative plot that depicts the mapping between internal joint changes and robot position changes in Fig. 4, we know that such gaits are centered about the origin.

The following strategy is one which we referred to in Section IV, that of decoupling the dependency of the fiber trajectories on the evolution of the non-symmetric fiber, which in the present case is $g^{\text{non}}=\theta$. Assuming that the joint trajectories $\alpha_{1}$ and $\alpha_{2}$ are also periodic, we can apply harmonic balance on the third line of Eq. (18), a first-order differential equation $\dot{\theta}=f(b_{w},\dot{b}_{w})$, to obtain a solution for $\theta(t)$ in terms of the parameters of $\alpha_{1}(t)$ and $\alpha_{2}(t)$ [36]. In other words, if

then we can find

where $\Theta$, $\psi$, and $C$ are functions of $B_{1}$, $B_{2}$, and $\phi$. Taking this procedure one step further, we can apply trigonometric identities to write

where $a_{1}$ and $a_{2}$ are the solutions of the nonlinear equations

The unknowns can be analytically solved as

With the numerical values of $a_{1}$ and $a_{2}$ in hand, we can substitute Eq. (27) into Eq. (23) and obtain a reduced connection mapping solely between the joint variables $b_{w}$ and the platform fiber variables $x_{p}$ and $y_{p}$, without regard to $\theta$. Again, this new connection assumes that $\theta$ is periodic and is only valid for the given or known functional form of $\theta$, whether through analysis or visual tracking. With this connection form, we are able to reduce the previous exterior derivative plots from three dimensions ($\alpha_{1}$, $\alpha_{2}$, and $\theta$) back down to two ($\alpha_{1}$ and $\alpha_{2}$ only).

Two effective connection exterior derivatives over different ranges of $\theta$ are shown in Fig. 11 (one may be able to characterize such approximations if the robot has limited reorientation over its trajectory, for example). In the first row, $\theta$ has a range from about $\frac{\pi}{2}$ to about $\frac{2\pi}{3}$; in the second, $\theta$ ranges from about $\pi$ to about $\frac{7\pi}{6}$. In both rows, the representative exterior derivative functions corresponding to $x_{p}$ and $y_{p}$ are shown. Interestingly, the $x_{p}$ plot in the first row and $y_{p}$ plot in the second row are not very different from their constant $\theta$ counterparts at $90$ and $180$ degrees, respectively. This indicates that those particular surfaces are more stable and hold over large ranges of $\theta$. On the other hand, the other two plots are noticeably different, particularly on the edges. In both the $y_{p}$ plot in the first row and $x_{p}$ plot in the second, the edges are the first regions to deform as $\theta$ changes.

Given that we are able to find single representative connection derivative plots over a given range of $\theta$, we can use this visual tool for locomotion analysis and planning for the three-link wheeled snake robot on a platform system, whether the inputs are applied to the robot on a passive platform or to an actuated platform underneath a passive robot. A weakness of this approach is that the computed connection only holds for the specific $\theta$ range. If this range changes, whether by shifting or scaling, a new connection must be found.

Constrained to the platform via a wheel located at its rear, the Chaplygin beanie locomotes using a rotor sitting atop its body. The rotor can be driven actively, for example by a motor, or passively, via an elastic element coupling the cart to the rotor. In this subsection, we study the latter of these two configurations, and assume the elastic element to be a linear torsional spring. The total mass of the vehicle is represented by $m$, its rotational inertia about the center of mass as $C$, rotational inertia of the rotor about the center of mass as $B$, and the mass of the platform as $M$. The distance between the center of mass and the contact point at the wheel is denoted by $a$, and the stiffness of the spring coupling the rotor to the body denoted by $k$. The position of the vehicle relative to the platform is given coordinates $(x,y)$, its orientation $\theta$, the rotor angle relative to the vehicle heading by $\phi$, and the position of the platform in a laboratory frame by $(x_{p},y_{p})$.

The configuration manifold for this system is $Q_{c}=\text{SE}(2)\times\mathbb{S}^{1}\times\mathbb{R}^{2}$, with a subscripted $()_{c}$ to denote Chaplygin beanie. Adhering to our earlier notation that delineates between internal and external configurations, we let $G^{\text{int}}_{c}=\text{SE}(2)$, $B_{c}=\mathbb{S}^{1}$, and $G^{\text{ext}}_{c}=\mathbb{R}^{2}$. Furthermore, we let $g^{\text{int}}_{c}=(x,y,\theta)\in G^{\text{int}}_{c}$, $b_{c}=\phi\in B_{c}$, and $g^{\text{ext}}_{c}=(x_{p},y_{p})\in G^{\text{ext}}_{c}$. Note that the use of this notation does not assume the existence of a connection relating trajectories in $B_{c}$ to trajectories in $G^{\text{int}}_{c}$ or $G^{\text{ext}}_{c}$. Like the three-link snake robot, the Chaplygin beanie is constrained via a no-slip condition on its wheel. The nonholonomic constraint at the wheel is given by

Constraints of this kind can also be thought of as one forms lying in the codistribution on the configuration manifold $Q_{c}$, written equivalently as

Given an initial condition $g^{\text{int}}_{c}(0)=(0,0,-\pi/4)$, $b_{c}=\pi$, $g^{\text{ext}}_{c}(0)=(0,0)$, and setting the inertial parameters and $k$ equal to unity, we observe the trajectory in Fig 13.

Stable trajectories like the one in Fig. 13 illustrate fixed points in a reduced space. To analyze these fixed points, we employ nonholonomic reduction, and leverage the geometric structure of the passive system to guide our development of a formal understanding of the types of stable locomotive trajectories that can arise from external actuation.

The evolution equations arising from the reduction are the changes in linear and angular momentum permitted by the no-slip constraint at the wheel and will replace the equations describing the evolution of $\dot{x}$, $\dot{y}$, and $\dot{\theta}$. The presence of a platform gives rise to two additional evolution equations, one of which is the time evolution of forward translational momentum of both the Chaplygin beanie and the platform, the second of which is the time evolution of the momentum of both the Chaplygin beanie and the platform in the direction orthogonal to that allowed by the no-slip consraint at the wheel. We also refer to this momentum term as momentum lateral to the forward motion of the vehicle. The Lagrangian for the system is given by

We require that the one form describing the no-slip constraint be annihilated by the system’s generalized velocity, having coordinates $(x,y,\theta,\phi,x_{p},y_{p})$, at every point $q_{c}\in Q_{c}$. For the Chaplygin beanie on a platform with finite inertia, the manifold on which the dynamics evolve is described by the configuration manifold $Q_{c}=\text{SE}(2)\times\mathbb{S}^{1}\times\mathbb{R}^{2}$.

We can use the mapping of tangent vectors according to the above Jacobian to show that the Lagrangian, Eq. 31, is invariant under the tangent lifted action, Eq. 33. We must now verify the constraint one form, $\omega$, is invariant under the group action, $\Phi$. Given a point $q_{c}\in Q_{c}$ and a tangent vector $\dot{q}_{c}\in T_{q_{c}}Q_{c}$, we check for left-invariance by computing

We take an approach presented in [6], involving the choosing of appropriate left-invariant vector fields spanning the intersection of the constraint distribution and the space tangent to the orbit of the group action, and leverage [37] in computing the components of the nonholonomic momentum. We designate the distribution $\mathcal{D}_{q_{c}}$ as the space of all tangent vectors which annihilate the constraint one form, $\omega$, and is given by

We invoke the Einstein summation convention in the following definition of the momentum map and designate $q_{c}^{i}$ as the $i$th coordinate on the configuration manifold, $Q_{c}$. The nonholonomic momenta are computed following

The resulting momenta are given directly by Eq. (42). It is clear by inspection that $J_{LT}$ and $J_{RW}$ correspond to forward translational momentum and angular momentum about the contact point of the wheel, respectively. The quantities $J_{X}$ and $J_{Y}$ correspond to forward translational momentum and momentum lateral to the direction allowed by the nonholonomic constraint for both the Chaplygin beanie and the platform.

A formal understanding of the types of stable locomotive trajectories of a completely passive system for which $b_{c}(0)=\phi_{0}$ is nonzero can give us insight into how to actuate the platform to induce locomotion in a passive Chaplygin beanie. In this section, we provide proof for the following proposition.

This proposition effectively says that all of the rotational momentum in the system will be transformed into forward translational momentum. We now present a formal argument for Proposition 1.