A Single Actuator Driven Two-Fold Symmetric Mechanism for Versatile Dynamic Locomotion

Abstract

Tumbling, rolling, and somersaults are alternate forms of locomotion used by animals and robots to navigate rough terrains. In this paper, we present a Two-Fold Symmetric (TFS) mechanism that demonstrates dynamic tumbling and leaping using a single actuator. The dynamics of the proposed mechanism are captured by a hybrid dynamic model with discrete states based on the nature of ground contact. By changing the shape parameters of a trapezoidal actuation signal, various dynamic responses and gaits are attained. Simulations and hardware experiments demonstrate tumbling and leaping/hopping. It is shown that the mechanism demonstrates gait versatility and attains speeds up to $3.0$ Body Lengths per second and can jump up to a height of $60\%$ of its total height, all using a single actuator that sets it apart from contemporary tumbling robots.

1. Introduction

Leg mechanisms can be designed to achieve a versatile array of dynamic behaviors such as walking, running, or even leaping over or on top of obstacles. While less versatile, wheeled vehicles are typically significantly simpler and easier to control. In this research, we combine the advantages of these two modes of locomotion by proposing a tumbling robot that is driven using a single actuator.

Although most animals use their legs to walk or run, some use their legs to perform tumbling or somersaults. For instance, the Namibian golden wheel spider Carparachne aureoflava [1,2] configures itself by retracting its legs inwards forming a passive wheel, which it then uses to escape from predators by rolling down sand dunes. Similarly, the Morrocan flic-flac spider Cebrennus rechenbergi [3] uses its limbs to propel itself by performing somersaults and ground rolls, as seen in Figure 1, to escape predators. Both of these spiders drastically increase their speed by performing these non-conventional locomotive maneuvers. In Figure 1, the foot-fall patterns of a tumbling flic-flac spider for tumbling and sommersaults are shown.

Tumbling usually does not assume a ‘right side up’. Tumbling robots intentionally tip over and their geometric configuration allows them to repeat the process over and over again to achieve stable locomotion. Thus, they can respond well to unplanned environmental interaction [5,6,7] and slopes [8]. Furthermore, under the right conditions, they may also perform leaping and somersaults to get them out of situations where they are stuck.

There are a variety of approaches to designing tumbling robots in the literature. One class of mechanisms has internal Flywheels [9,10], or sliding masses [11] inside a rigid body frame. The inertial components of these mechanisms need to decelerate and then re-accelerate between movements, causing them to pause and restricting their utility to slower gait applications. These robots can also demonstrate various types of motion based on their environmental interaction and actuation conditions [12].

Another class of mechanisms move by morphing their physical shape. This displaces their center of mass relative to their support geometry. These mechanisms can exhibit relatively more agile locomotion. However, they typically have complex structures and a large number of actuators [13,14,15,16,17,18].

A third class of mechanisms moves by pushing a rigid body frame using forces generated by methods other than rapid inertial change or shape morphing. These mechanisms use appendages [6,19,20,21,22] or propellers [23,24].

The aforementioned robots vary based on the number of actuators, the type of actuation method and the geometries of the robots. In existing work, either additional mechanisms or actuators are required to provide the thrust [25,26], or the mechanism has a significant recovery time between consecutive hops [27]. We propose an equilateral four-sided polygon to perform a series of repeatable ‘tip-over’ maneuvers. These tip-over maneuvers alternate the grounded link of the polygon due to symmetry and chirality, thus allowing us to use a single actuator.

Using fewer actuators has several advantages. Motivated from passive dynamic walkers [28], the usage of fewer actuators usually corresponds to higher energy efficiency [29]. Mechanisms with fewer actuators often utilize their natural dynamics instead of working against them to move ahead rhythmically. Additional benefits of using fewer actuators include reduced robot weight [30] and simplified control [31,32].

Compared to their wheeled and legged counterparts, the dynamics of tumbling is relatively less explored. Early work consists of a variety of models. Hemes et al. adopt a statically stable gait for the Adelpod tumbling robot [19,20] that uses a stability margins based approach. Lee et al. [13] motivate the use of a hybrid dynamics approach to describe the motion of an icosahedral robot by splitting the dynamics between a manipulation phase and a tipping phase; however, there is no description of a flight phase. In contrast, Allen et al. [9] and Kobashi et al. [10] utilize inverted pendulum based models for their proposed robots and consider both tumbling and flight phases. However, shape manipulation during edge contact, similar to the one by Lee et al. is warranted but missing.

The limitations of existing models motivate the development of a unique new model to explain the dynamics. In this paper, we introduce a two-fold symmetric mechanism that is driven by a single actuator and can perform tumbling, hopping, and quarter somersaults. Because of the aerial phases during these dynamic behaviors, we propose a hybrid reduced-order dynamic model, with discrete state based model templates [33] to capture the dynamics of the robot in each distinct ground contact phase, edge support, vertex support, and flight phase. Finally, we simulate the model and compare the predictions of the model with hardware results.

2. The Two-Fold Symmetric Mechanism

In this section, we discuss a Two-Fold Symmetric Mechanism as a candidate for simple, uni-articulate, shape-changing robots. The term p-Fold Symmetry is borrowed from group theory, where it describes geometric shapes that maintain shape identity after being rotated by ${\displaystyle \frac{2\pi }{p}}$ radians. In the present work, we use the term to describe mechanisms that maintain this property of rotational symmetry throughout their admissible configuration space. In particular, we introduce a 6-link closed mechanism as seen in Figure 2. This mechanism can deform its shape by changing the internal angle $\theta$ (or its supplementary pair ${\theta }^{\prime }$) and may tumble by rotating about a vertex $v_n$, where $n\in \{1,2,3,4\}$. When one of the edges is planted, it governs which if the two internal scissors linkages is ‘grounded’ and which is allowed to morph the shape of the robot. If a rotary actuator is used, as proposed in the following chapters, it can be said that depending on which edge is in contact, either the rotor is grounded or the stator is.

2.1. Supplementary Angles and Antogonistic Motion

Antagonistic motion appears quite commonly in locomotion, be it through the retraction of the swing leg while simultaneously extending the stance leg in bipedal walking, climbing or crawling. In a scissor-like linkage configuration, the internal supplementary angle pair formed by the linkages demonstrates antagonistic motion depending on which link of the pair is being grounded. We utilize this property to realize tumbling. In Figure 2, the angles $\theta$ and ${\theta }^{\prime }$ are supplementary angles. Thus,

$$\begin{array}{c}\hfill {\theta }^{\prime }=\pi -\theta \Rightarrow {\dot{\theta }}^{\prime }=-\dot{\theta }.\end{array}$$

(1)

Since the domain of $\theta$ is bounded by hard limits of the mechanism, as $\theta$ changes, ${\theta }^{\prime }$ exhibits an antagonistic motion. Revisiting Figure 2, this can be visualized an antagonistic motion of virtual ’leg lengths’, for instance, ${\zeta }_1$ and ${\zeta }_2$. This is analogous to reduced order models of bipedal walking and serves as a motivation for deriving inspiration for tumbling and jumping from models of dynamic legged locomotion.

2.2. Symmetry and Chirality

For an arbitrary quadrilateral shape using a discrete state modeling approach would require modeling the dynamic behavior based on which edge or vertex (if any) are in contact with the ground. Assuming then that the vertices do not overlap and that the ground is flat, we would require at least nine models for when the quadrulateral is sitting on its edges or its vertices or while in the air. However, after introducing design constraints, such as assuming that the geometry is an equilateral quadrilateral, we can simplify our analysis. This simplification of modeling is carried out using the underlying principles of symmtery and chirality. As Aparez et al. [34] determined, morphological symmetries in locomotion mechanisms can be utilized to reduce the number of equations required for a complete dynamic description. Additionally, it is observed that the mechanism also exhibits ’mirroring’ or chirality as it alternates between contact edges. An exhaustive analysis of tumbling locomotion from the lens of group theory is currently beyond the scope of this paper and will be explored in future research.

Consider the special case of a geometric shape that has four congruent sides. Such a rhombus typically exhibits two reflective and a half-rotation symmetry. In Figure 2, the line segments $\overline{v_1{v}_3}$ and $\overline{v_2{v}_4}$ represent reflection axes, whereas the angle $\alpha$ is the orientation angle.

In Figure 2, it is concluded that because of rotational symmetry, geometric relations based on Edge A and Edge C are identical if either is ’grounded’. Independently, this also applies to Edges B and D. Similarly, because of reflective symmetries or antisymmetry, ${\zeta }_1$ is equal in magnitude to ${\zeta }_3$ and ${\zeta }_2$ is equal in magnitude to ${\zeta }_4$.

Symmetry and chirality, enable us to minimize the number of states and therefore the dynamic equations of the system. Furthermore, the transformations provide us information about initial conditions for the hybrid dynamics model in Section 3. In this work, we will strictly be considering the two-fold symmetric mechanisms that exhibit two reflective and a half-rotation symmetry. However, it should be noted that the proposed approach may be extendable to cases beyond this criterion.

2.3. Overcoming Static Stability

For clarity, in this paper, we describe tumbling as follows:

Definition 1.

A planar terrestrial robot is said to tumble as it transitions from one statically stable phase to another through a statically unstable phase, and there is a net incremental change in the body’s overall angular orientation.

By this definition, tumbling is different from pure rolling as it requires the presence of statically stable states where energy is dissipated upon state transition; however, no state transitions occurs during pure rolling. Interestingly, this dissipation of energy is also an emergent outcome of other reduced order models like the Rimless-Wheel Model [35]. Now we define somersaults as follows:

Definition 2.

A terrestrial robot is said to perform a somersault if there is a net incremental change in the body’s overall angular orientation during flight phase.

In the present case, a brake in angular momentum is provided by a statically stable phase that appears when any of the edges of the mechanism are in contact with the ground. Inverting the purpose of traditional Static and Quasi-Static Stability Margins [36], faster tumbling can be achieved with actuation schemes that quickly but predictably destabilize the robot. With these definitions and state transition framework in place we can describe a dynamic model to capture the tumbling of Two-Fold Symmetric mechanisms.

3. Dynamic Model of Locomotion

A hybrid dynamics approach based on template modeling [33] is adopted to model the dynamic behavior of the TFS mechanism, which can be split into three phases. Based on the nature of contact made with the environment, these phases are the Edge Support, Vertex Support, and Flight phases. Restricted initially to just a planar case, these phases represent whether the contact shape is a curve, a point or if there is no contact, respectively. Figure 3 represents the hybrid dynamic model. The discrete phases are represented with blocks and their transitions with arrows.

During Edge and Vertex support phases, a kinematic no-slip constraint is implemented at the points of contact. This implies that during both of these phases, the mechanism does not slide and this assumption simplifies contact modeling by ensuring a single degree of freedom pin-joint constraint at the contact vertices. Thus, the linear velocity at the contact point is zero. The forces are no longer calculated based on material properties but are actually Lagrange multipliers, required to ensure the constraints. Compared to dry-coulomb friction assumptions, this approach offers a reduction in degree of freedom and is also beneficial for eliminating the need for solving linear complementarity problems associated with the non-smooth coulomb friction models, thus trading-off physical fidelity for mathematical simplification and tractability. This assumption is commonly made in legged SLIP-like (Spring Loaded Inverted Pendulum like) models [37,38,39] that we have used for inspiration.

We represent the dynamics during each phase with reduced order models. In order to further simplify our analysis, the dynamics are represented as motion of point masses, rather than rigid bodies wherever possible. The models are derived using Lagrangian approaches following classical dynamics texts [40,41].

3.1. Geometric Transformations

The model in Figure 3 defines $\varphi$ as the angle of the virtual leg corresponding to the planted vertex with reference to horizontal ground reference, and $\zeta$ as the virtual leg length extending from the centroid of the mechanism to the planted vertex. However, keeping in mind the chirality shown in Figure 2, the kinematics, in the form of a geometric relationship between the internal actuated angle $\theta$ and the virtual leg parameter $\zeta$, varies depending on which vertex is used as a reference. Let ${\zeta }_n$ be the virtual leg length extending from the centroid of the mechanism to the nth vertex. The transformation from $\theta$ and $\dot{\theta }$ to the leg lengths is as follows:

$$\begin{array}{cc}\hfill & {\zeta }_n=2lcos{\displaystyle \frac{\theta +(1+{(-1)}^n)\frac{\pi }{2}}{2}}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \dot{{\zeta }_n}={(-1)}^{n}l\dot{\theta }sin{\displaystyle \frac{\theta -(1+{(-1)}^n)\frac{\pi }{2}}{2}}\Rightarrow \dot{\mathit{\zeta }}={\mathit{J}}_{\mathit{c}}\dot{\theta }\hfill \end{array}$$

(3)

where $n\in \{1,2,3,4\}$ is the vertex number that is in contact, ${\mathit{J}}_{\mathit{c}}$ is the Jacobian Transform between Configuration-space and Workspace velocities. Furthermore, given the coordinates of the body frame in a global frame of reference ${\mathit{q}}_{\mathit{B}}={[x_B,y_B]}^T$, the global Cartesian coordinates of the nth vertex ${\mathit{q}}_{\mathit{vn}}={[x_{vn},y_{vn}]}^T$ are given by the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left[\begin{array}{c}{x}_{vn}\\ y_{vn}\end{array}\right]=\left[\begin{array}{c}{x}_B-{\zeta }_{n}cos(\varphi +{\displaystyle \frac{(n-1)}{2}}\pi )\\ y_B-{\zeta }_{n}sin(\varphi +{\displaystyle \frac{(n-1)}{2}}\pi )\end{array}\right]\end{array}\end{array}$$

3.2. Edge Support Dynamics

If one of the edges of the robot is planted on the ground, this results in the application of a holonomic constraint and therefore reduction in dimensionality of the system. Therefore, the mechanism acts like a single degree of freedom system with the internal angle $\theta$ acting as the independent state variable. This can be seen in Figure 3. The constraint equations is $l=\sqrt{{(x_B-x_P)}^2+{(y_B-y_P)}^2}$ where l is a radius with constant value and $x_B$ and $y_B$ are the coordinates of the center of mass with reference to an inertial frame and $x_P$ and $y_P$ is the position of the pivot during Edge Support. Since l has a constant value, and an angle $\underline{\theta }$ can be defined such that $\underline{\theta }=arctan\left({\displaystyle \frac{y_B-y_P}{x_B-x_P}}\right)$, to represent the motion of the mechanism, a simple inverted pendulum can be used to represent the motion of the robot during this phase. The underlined variables above represents the states and their chiral pairs. As a reminder, $\theta$ is the only controllable angle between the two central linkages of the mechanism. The dynamic equation takes the following form:

$$\begin{array}{c}\hfill \ddot{\underline{\theta }}=-{\displaystyle \frac{g}{l}}cos\underline{\theta }+{\displaystyle \frac{Q_{nc}}{m{l}^2}}\end{array}$$

(4)

where m is the mass of the robot, and g represents the acceleration due to gravity. $Q_{nc}={\tau }_{motor}-b\dot{\underline{\theta }}$ represents the non-conservative moments acting on the mechanism, where ${\tau }_{motor}$ is the applied motor torque and b is coefficient of viscous damping. The system is modeled as a fully actuated system in this phase.

3.3. Vertex Support Dynamics

During vertex support, the mechanism is considered to be pivoted at the vertex in contact with the ground. In this case, the motion can be completely described with a telescoping inverted pendulum model (see Figure 3). The constraint equation takes the form of $\underline{\zeta }=\sqrt{{(x_B-x_P)}^2+{(y_B-y_P)}^2}$ where $\underline{\zeta }$ is a variable known as the virtual leg length. Withing a domain of admissible internal angle $\underline{\theta }$ values, it is possible to find a unique mapping from $\underline{\theta }$ to $\underline{\zeta }$ through kinematic analysis. The angle $\underline{\varphi }=arctan\left({\displaystyle \frac{y_B-y_P}{x_B-x_P}}\right)$ is called the external angle. It is not possible to directly control $\underline{\varphi }$; however, changing the internal angle $\underline{\theta }$ changes the virtual leg length $\underline{\zeta }$, which in turn can affect $\dot{\underline{\varphi }}$ because of conservation of angular momentum. The dynamic equations are as follows:

$$\begin{array}{c}\hfill \left[\begin{array}{c}\ddot{\underline{\zeta }}\\ \ddot{\underline{\varphi }}\end{array}\right]=\left[\begin{array}{c}\underline{\zeta }{\dot{\underline{\varphi }}}^2\\ -{\displaystyle \frac{2}{\underline{\zeta }}}\dot{\underline{\zeta }}\dot{\underline{\varphi }}\end{array}\right]-\left[\begin{array}{c}gsin\varphi \\ {\displaystyle \frac{g}{\zeta }}cos\varphi \end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{{\mathbb{Q}}_{nc}}{m}}\\ 0\end{array}\right]\end{array}$$

(5)

Since the last matrix is not full rank, the mechanism is underactuated during vertex support. In terms of control, the system is not fully controllable. The variable ${\mathbb{Q}}_{nc}$ represents a component of the force when transformed from torque provided by the actuator. The Jacobian matrix calculated in the Section 2 can help us determine the mapping from $Q_{nc}$ to ${\mathbb{Q}}_{nc}$.

3.4. Flight Dynamics

During flight, the mechanism is treated as a point mass and that the vertices are massless. In our simplified approach, we assume that the rotational motion of the vertices does not contribute to the overall angular momentum of the mechanism and the rotational information is used to keep track of which vertex or vertices make contact first. The dynamic equations of the model can therefore be expressed as follows:

$$\begin{array}{c}\hfill {\left[\begin{array}{c}\ddot{x_B},\ddot{y_B},\ddot{{\zeta }_n},\ddot{{\varphi }_n}\end{array}\right]}^T={\left[\begin{array}{c}0,-g,0,0\end{array}\right]}^T\end{array}$$

(6)

where $x_B$, $y_B$, ${\zeta }_n$, and ${\varphi }_n$ are the horizontal, vertical displacements, the shortest distance of a reference n vertex from the center of mass (COM) and the angle of the reference vertex n with respect to the body, respectively. Since the vertices act as massless legs, the equations of ${\zeta }_n$ and ${\varphi }_n$ are assumed to be decoupled. ${\zeta }_n$ is controlled directly through zero-th order dynamics of a commanded input signal, which means that the reference vertex is tracked without lag while in flight.

$$\begin{array}{c}\hfill {\zeta }_n=2lcos{\displaystyle \frac{{\theta }_{Des}+(1+{(-1)}^n)\frac{\pi }{2}}{2}}\end{array}$$

(7)

3.5. Actuation Model

The forcing term ${\tau }_{motor}$ that appears in the equations of motion represents the actuator output. This is limited on the basis of the speed torque of the motor. Let ${\theta }_{Des}$ be the desired internal angle and ${\dot{\theta }}_{Des}$ is the desired angular velocity, then the desired torque is given by the following:

$$\begin{array}{c}\hfill {\tau }_{Des}=K_p({\theta }_{Des}-\theta )+K_v({\dot{\theta }}_{Des}-\dot{\theta })\end{array}$$

(8)

where $K_p$ and $K_v$ represent the position and velocity gains of a PD controller. In the present work, these gains are hand-tuned. The desired torque ${\tau }_{Des}$ is then mapped on the speed torque curve of the actual motor that is used. A saturation function is applied if the torque exceeds the speed torque curve’s threshold as seen in Figure 4. This torque ${\tau }_{Motor}$ forms part of Edge Support’s non-conservative forcing term $Q_{nc}$. The vertex force $F_{motor}$ which forms part of the vertex support’s forcing term ${\mathbb{Q}}_{nc}$ is dependent on the internal angle $\theta$ and the vertex $v_n$ that is in contact with the ground.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill F_{motor}={\displaystyle \frac{{\tau }_{motor}}{l}}sin{\displaystyle \frac{\theta -(1+{(-1)}^n)\frac{\pi }{2}}{2}}\end{array}\end{array}$$

3.6. State Exit Conditions

3.6.1. Edge Support Exit

According to the state machine in Figure 3, it is only possible to exit from edge support into vertex support. During Edge Support, as the center of mass traverses the semicircular constraint curve, the force required to apply the constraint can be obtained from the force balance along the normal direction of the contact surface.

$$\begin{array}{c}\hfill F_N=mgsin\theta -ml{\dot{\theta }}^2\end{array}$$

(9)

where the left hand side term represents the normal force and the right hand side terms represent the force component due to gravity and the centripetal force term. We can imagine a cart on a dome as a dynamically similar example for ease of understanding. The cart stays on the dome as long as long as the centrifugal force balances the force components of gravity. This is the constraint force, required to keep the cart on the dome, or in our case, the COM on the semi-circular constraint arc. As soon as the COM leaves that arc, it exits edge support phase. An interpretation of whether the left or right vertex lifts off the ground can be obtained from

$$\begin{array}{c}\hfill v_n=\left\{\begin{array}{cc}{v}_{left}\hfill & \mathrm{if}{\displaystyle \frac{m}{2}}(g(1+{(-1)}^{N}cos\theta )-l{\dot{\theta }}^{2}sin\theta )<0\hfill \\ v_{right}\hfill & \mathrm{if}{\displaystyle \frac{m}{2}}(g(1+{(-1)}^{N-1}cos\theta )-l{\dot{\theta }}^{2}sin\theta )<0\hfill \end{array}\right.\end{array}$$

(10)

where $N\in \{1,2,3,4\}$ is the edge that was last in contact with the ground.

3.6.2. Vertex Support Exit

There are two possible outcomes at the end of vertex support. Firstly, if any of the two neighboring vertices makes ground contact while the present vertex $v_n$ is in contact with the ground, the vertex support mode transitions into edge support:

$$\begin{array}{c}\hfill {\theta }^{+}=\left\{\begin{array}{cc}\theta \hfill & \mathrm{if}{y}_{vn-1}\le 0\hfill \\ {\theta }^{\prime }\hfill & \mathrm{if}{y}_{vn+1}\le 0\hfill \end{array}\right.\end{array}$$

(11)

where $y_{vn\pm 1}$ represent the vertical components of the vertices to the right and left of the grounded vertex n, Due to the kinematics of the robot, another way to interpret this exit condition is when the center or mass hits the semicircle that forms the constraint curve during edge support as seen in Figure 5.

Secondly, if the vertical component of the ground reaction force goes to zero, then the model escapes from vertex support into flight state:

$$\begin{array}{c}\hfill {\mathbb{Q}}_{nc}\le 0\end{array}$$

(12)

3.6.3. Flight Exit

Flight ends when either one or two of the four free vertices make contact. If one vertex makes contact then the model exits to vertex support; otherwise, it exits to edge support state.

$$\begin{array}{c}\hfill \forall n\in \{1,2,3,4\},v_{nVertical}=0\end{array}$$

(13)

where $v_{nVertical}$ is the vertical component of the $n-th$ vertex.

3.7. State Transitions

The state transition conditions are shown in Figure 3. In this subsection, they are discussed in detail.

3.7.1. Edge Support and Vertex Support

The transition from edge to vertex support is given by the following:

$$\begin{array}{c}\hfill \left[\begin{array}{c}\zeta \\ \dot{\zeta }\\ \varphi \\ \dot{\varphi }\end{array}\right]=\left[\begin{array}{c}2lcos{\displaystyle \frac{\theta +(1+{(-1)}^n)\frac{\pi }{2}}{2}}\\ {(-1)}^{n}lsin{\displaystyle \frac{\theta -(1+{(-1)}^n)\frac{\pi }{2}}{2}}\\ {\displaystyle \frac{\theta }{2}}+{\displaystyle \frac{\pi (1+{(-1)}^n)}{4}}\\ \mathrm{sgn}\left(\dot{\theta }\right){\displaystyle \frac{\left(\sqrt{l{\dot{\theta }}^2-{\dot{\zeta }}^2}\right)}{\zeta }}\end{array}\right]\end{array}$$

(14)

In this work, an inelastic contact model is used for vertex to edge transition. The assumption is that the impact vibrations fade quickly when the edges make contact. The transition from vertex to edge support is given by

$$\begin{array}{c}\hfill \left[\begin{array}{c}\theta \\ \dot{\theta }\end{array}\right]=\left[\begin{array}{c}2\varphi -{\displaystyle \frac{\pi (1+{(-1)}^n)}{2}}\\ {(-1)}^n{\displaystyle \frac{\dot{\zeta }}{l}}sin\varphi +{(-1)}^{n-1}{\displaystyle \frac{\zeta }{l}}\dot{\varphi }cos\varphi \end{array}\right]\end{array}$$

(15)

3.7.2. Vertex Support and Flight

The transition from vertex support to Flight is given by

$$\begin{array}{c}\hfill \left[\begin{array}{c}{x}_B\\ \dot{x_B}\\ y_B\\ \dot{y_B}\\ {\zeta }_n\\ \dot{{\zeta }_n}\\ {\varphi }_n\\ \dot{{\varphi }_n}\end{array}\right]=\left[\begin{array}{c}\zeta cos\varphi \\ \dot{\zeta }cos\varphi -\zeta \dot{\varphi }sin\varphi \\ \zeta sin\varphi \\ \dot{\zeta }sin\varphi +\zeta \dot{\varphi }cos\varphi \\ -\zeta \\ -\dot{\zeta }\\ -\varphi \\ -{\displaystyle \frac{{\zeta }^2+4l^2}{{\zeta }^2}}\dot{\varphi }\end{array}\right]\end{array}$$

(16)

The transition from flight to vertex stance is given by

$$\begin{array}{c}\hfill \left[\begin{array}{c}\zeta \\ \dot{\zeta }\\ \varphi \\ \dot{\varphi }\end{array}\right]={\left[\begin{array}{c}\sqrt{\delta x^2+\delta y^2},{\displaystyle \frac{\delta x\dot{x}+\delta y\dot{y}}{\sqrt{\delta x^2+\delta y^2}}},{tan}^{-1}{\displaystyle \frac{\delta y}{\delta x}},{\displaystyle \frac{\delta x\dot{y}-\delta y\dot{x}}{\delta x^2+\delta y^2}}\end{array}\right]}^T\end{array}$$

(17)

where $\delta x=x_B-x_{vn}$ and $\delta y=y_B-y_{vn}$.

3.7.3. Flight to Edge Support

In some special cases, the mechanism exits from flight phase into edge support. Although this interaction can be modeled with a brief period of vertex support between flight and stance, simulations using numerical solvers like ode45 fail if the angle is flat enough to fall within the gaps of the minimum step size. In this case, the transition is given by

$$\begin{array}{c}\hfill \left[\begin{array}{c}\theta \\ \dot{\theta }\end{array}\right]=\left[\begin{array}{c}2\varphi -{\displaystyle \frac{\pi (1+{(-1)}^{N+n-1})}{2}}\\ {(-1)}^n{\displaystyle \frac{\dot{\zeta }}{l}}sin\varphi +{(-1)}^{n-1}{\displaystyle \frac{\zeta }{l}}\dot{\varphi }cos\varphi \end{array}\right]\end{array}$$

(18)

4. Hardware and Experimental Setup

In order to validate the results of the simulation, a planar TFS mechanism was designed and built. The mechanism is driven by a direct drive motor which makes it back-drivable. The linkages of the mechanism are made of laser-cut ABS (Acrylonitrile Butadiene Styrene) and its vertices are made of compliant 3D-printed TPU (Thermoplastic polyurethane). The mechanism has an onboard moteus r4.11 motor controller that utilizes an internal hall-effect sensor for local control. The hardware specifications are presented in

Table 1. Hardware specifications.

Property	Value	Units
Mass (without Boom)	0.5	kg
Mass (with Boom)	0.7	kg
Link Length	13.5	cm
Maximum Angular Sweep	2.1	rad
Actuator Stall Torque	1.7	Nm

.

Through initial experiments, a minimum torque and power threshold for jumping was found. This was done by making the mechanism jump up from its rest position on a single vertex, while tethered to the boom. The Moteus motor was selected from commercial off-the-shelf options because it provided torques beyond the minimum threshold. Iteratively, the mass and mass distribution of the robot were also revised and improved.

The mechanism is attached to the end of a boom setup as seen in Figure 6. The boom has encoders in place to measure the polar and azimuthal angles. These translate to horizontal and vertical displacements, respectively. The boom-to-mechanism interface is composed of a single degree of freedom unactuated pivot joint. This is realized through an interface assembly that contains a thrust bearing and a slip-ring connector to allow the power and signal wires through without mechanical disconnection.

The robot carries a Moteus r4.11 motor controller board with an integrated hall-effect sensor communicating through FD-CAN to USB protocol with a desktop computer. A five wire slip-ring commutator is used as mentioned for power and smooth communication channeling without winding up the wiring. The Moteus board receives instructions from a python script running on the desktop computer.

For the experimental set-up, a periodic continuous-time odd signal is used as a reference signal. Thinking about the reference actuation signal as an odd or anti-symmetric signal has certain benefits. Firstly, imagine a mechanism that started out by resting on its edge tips over its leading vertex: a single positive pulse may be enough to make it do so. However, once the mechanism has tumbled about a vertex, it reaches a chiral or antisymmetric state when it comes to rest in the next edge support phase. Ironically, the mechanism will have to be driven in reverse (relative to its initial actuation) to tip it again about its leading vertex. Another way to imagine this requirement of odd actuation is this: Imagine an arbitrary edge is in contact with a flat ground. Of the two internal linkages driven by the actuator in the hardware, the one parallel to the arbitrary edge, and therefore the ground, will remain grounded subject to constraint satisfaction; however, the other linkage will be driven, say counterclockwise. However, once the mechanism has tipped over to the next edge, now the internal linkage that is parallel to the ground has changed: what was previously the driven linkage has become parallel to the ground and vice versa. The actuator, however, only sees the motion of one internal linkage driven with respect to the other and is blind to the edge support state. Therefore, an anti-symmetric signal is sent to the actuator. For reference, for a single antisymmetric continuous-time signal, the mechanism tips over twice. To introduce nomenclature, we shall refer to the initial half-wave of the actuation signal as a propulsive phase and the consecutive one as recovery phase.

Some candidates of reference antisymmetric waves include sinusoidal, trapezoidal, triangular, or square waves. A trapezoidal wave was chosen as a reference signal for validating simulation and experimental results compared to the common sinusoidal function because of the independence of signal’s derivative from signal frequency which is used as a reference the local controller. Thus, the limitation that the slope of the sine as a function of its frequency is overcome. As seen in Figure 7, there are five parameters of the trapezoidal wave that can be varied; they are the amplitude, frequency, slope from low to high, slope from high to low, and the duty factor. Figure 7 also shows that these parameters can be tuned to yield triangular, square, sawtooth, and other trapezoidal waev forms. Although this is not a wholly continuous-time signal, the control system dynamics and the hardware’s response lend a low-pass nature to the over-all dynamics.

5. Results

As previously shown in Figure 4 and discussed in Section 3, a PD controller is used for tracking the reference trapezoidal signal and its derivative. With this control law in place, several unique dynamic behaviors were observed. By changing the parameters of the reference trapezoidal signal, versatile dynamic behaviors are observed and characterized by the periodic transitions between various phases of locomotion. We therefore refer to each unique periodic sequence of transitions between edge, vertex, and flight phases as a gait.

5.1. Lazy Tumble Gait

This gait is called as such because the mechanism ‘lazily’ collapses towards and tumbles about the front-vertex and repeats this as it collapses towards the new front-vertex. The motion of the mechanism does not have an aerial phase. This is a form of ‘low energy’ tumbling. One instance of this behavior can be seen in Figure 8, where the mechanism travels at roughly 1.2 body lengths per seconds when actuated by a 1 Hz trapezoidal wave with an amplitude of of 60 degrees and slopes of approximately 1.2 Hz.

5.2. Jump Tumble Gait

This gait is so called because the mechanism ‘jumps back’ to the back vertex as it tries to shoot forward but does not have the energy required to enter flight phase. Tumbling is achieved as the front vertex makes contact and the robot is ‘catapulted’ about the front vertex. One instance of this behavior can be seen in Figure 9. The trapezoidal actuation function that demonstrates this kind of behavior is marked by steeper slopes. The mechanism travels at roughly slightly above 1.25 body lengths per seconds when actuated by a 1 Hz trapezoidal wave with an amplitude of of 60 degrees and slopes of approximately 5 Hz. This is a faster gait compared to lazy tumbling.

5.3. Quarter Somersaults

Closely resembling the Jump Tumble gait, the robot demonstrates a quarter somersault as the mechanism does a partial rotation in the air before landing on the next vertex or edge. Because the angle of launch is determined by the amplitude and slope of the trapezoidal function, the distance covered by the robot can also be modulated. This gait can be seen in Figure 10 where the mechanism is traveling at a speed of 2.1 BL/s. This gait is characterized by a prominent flight phase and can be used to traverse distances that are not approximate multiples of the edge-length of the mechanism.

The presence of flight phase distinguishes the quarter somersault from our interpretation of a jump tumble gait. However, cases of jump tumble behavior. Correlatedly, these gaits are also faster and high energy gaits. In legged locomotors, gaits with lower duty factors, that is, the ratio of time spent in contact with the ground versus the total gait time tend to be faster than higher duty factor gaits. For instance, sprinting is faster than walking and galloping is faster than pacing. Since quarter somersaults have a lower duty factor than jump tumbling, these tend to be faster gaits.

Unlike the Jump tumble gait, the step length covered during the quarter somersault is not fixed. This means that a variable step length can be achieved adding a richness to the spectrum of dynamic behaviors that the proposed mechanism can demonstrate.

5.4. Vertex to Vertex Tumble Gait

This gait may be viewed as a transitional gait between the jump tumble gait and the quarter somersaults discussed later. The frequency of the trapezoidal wave is selected such that the mechanism does not have enough time to settle during edge support and starts expanding in the opposite direction. This results in the mechanism quickly transitioning from one vertex to another and on. This gait is highly sensitive to the timing of the vertex contact during the gait cycle and in hardware tests it was observed that although these gaits are some of the fastest, precisely achieving them is a non trivial endeavor and significant variations in behavior were seen around the conditions necessary for this gait. Since vertex to vertex tumbling is highly sensitive to initial conditions and since real world factors such as attack angle of the contact vertex upon landing and conditions of the ground surface may vary significantly for each interaction in hardware tests, a state based controller instead of the proposed trapezoidal wave approach may help in better convergence. Some of the fastest gaits were achieved using this approach with speeds around 3 BL/s. One instance of this behavior can be seen in Figure 11.

5.5. Leaping and the Significance of Flight Phase

By increasing the gradient of the slopes of the actuation signals, the mechanism can quickly escape both edge and vertex support states and enter flight phase quickly. On hardware, the mechanism launches into the air and can jump over or onto an obstacle. In an experimental setup, seen in Figure 12, the mechanism was placed close to an elevated obstacle that had a height of about $55\%$ of the mechanisms link length. With a slope of around 30 Hz the mechanism performed a double hop and jump to about 60% of its center of mass height.

This ability of the proposed mechanism to control its flight phase and to hop and leap is distinct in the domain of contemporary tumbling robots. Aerial phase not only adds a degree of versatility in terms of agility and gait generation, as discussed in Section 5.2 and Section 5.3, but also the ability of the mechanism to make controlled jumps to clear obstacles lays the foundation for handling discontinuous terrain. In summary, with sufficient actuator power, leaping, jumping, and somersaulting is realizable, its dynamics can be captured by models of running and walking and demonstrated through hardware.

5.6. Discussion

Despite having a single actuator, the hardware and the simulations demonstrate a variety of dynamic and quasi-dynamic behaviors as seen in Figure 13. With state feedback independent control, the mechanism can be directed to achieve desirable and predictable tumbling and leaping over obstacles. An understanding of this is established through the reduced order model presented in Section 3. Parameters of the trapezoidal function play a key role in instantiating various types of behaviors. In summary, parameters that resulted in these behaviors are presented in

Table 2. Summary of parameters of the trapezoidal function for various gaits and behaviors.

Behavior	Amp. (deg)	Freq. (Hz)	Slopes A&B (Hz)
Lazy Tumble	60	1.0	1.2
Jump Tumble	60	1.0	5
Vertex to Vertex Tumble	70	1.5	30
Quarter Somersault	50	1	30

.

The examples shown in this section represent stable periodic dynamic behavior. There is also the prevalence of aperiodic or chaotic behaviors that the hardware demonstrates and the simulations predict. In picking up candidate parameters of the trapezoidal function, the maximum frequency of the reference signal and the PD gains play a significant role. At frequencies above 3 Hz, with slope parameters that cause abrupt jumps in the actuation signal or due to similar signals that may cause continuous abrupt motion reversal, the controller has difficulty mitigating the ground impact and also making the mechanism move at such high frequencies. However, where traditional legged mechanisms would become unstable and fall down, the two-fold symmetric mechanism either tumbles in arbitrary directions or hits its mechanical angular thresholds. The former is an example of chaotic behavior and the latter of unstable gaits. In the present study, both are avoided.

6. Conclusions and Future Work

This paper demonstrates that a single motor driven Two-Fold Symmetric mechanism can demonstrate versatile dynamic behaviors. With a fixed-trajectory tracking controller, we show that the mechanism can not only execute various forms of tumbling but also leaping over and onto obstacles. Inspired from reduced order models of legged locomotion, we develop a three phase, hybrid dynamic model with discrete state models for each contact phase: Edge support, vertex support, and flight phase. Through hardware experimentation, the mechanism demonstrates Lazy Tumbling, Jump Tumbling, Quarter Somersaults and Vertex to Vertex Tumbling as tumbling gaits to achieve speeds as high as 3.0 Body Lengths per second and can also jump up to a height of $60\%$ of its equilibrium state height.

The diverse dynamic behaviors demonstrated in the present work were found through visual analysis of the change of input parameters to the robot’s locomotion. In the future, we plan to explore the possibility of new gaits that can be achieved with the proposed mechanism. Our plan is to investigate these locomotion behaviors, possibly through the lens of agility, stability, and energetics.

The present mechanism is attached to a boom that serves two purposes. First, it restricts out-of-plane roll motion of the tumbling mechanism, and secondly, it measures the mechanism’s position. In the future, the addition of a second two-fold symmetric mechanism at an out-of-plane distance will provide a similar constraint, and additional onboard sensing using Inertial Measurement Units (IMUs) and contact sensors will provide positional information.

In the present work, a controller tracks the desired position and velocity signals. However, with sensor placement and state feedback, we plan to explore the development of state-based controllers. We also plan on investigating the role of controller gains on state-based control. This will be used to explore the locomotion of the two-fold symmetric mechanism in uneven or irregular terrains. Additionally, this will enable the development of posture control policies for different phases.

The model proposed in this paper broadly shows agreement with experimental data. However, the drift observed at higher speeds could be attributed to slippage. These results can be improved by using a friction based constraint model to replace the pin-joint constraint during edge and vertex control, and an improved contact model. This approach towards modeling, along with the state based controllers discussed earlier, will be used to study the locomotion of the proposed mechanism on various ground substrates and uneven terrains.