Eight-Spoke Double Rimless Wheel

• Eight-spoke double rimless wheel is a robotic model that integrates paired telescopic legs to create an eight-contact rolling gait.
• It employs classical rimless wheel physics with limit-cycle analysis to ensure dynamic stability through calculated impulse delivery and energy compensation.
• Deep reinforcement learning augments the model, uncovering novel gaits and enhancing adaptability across varying terrains and speed conditions.

The eight-spoke double rimless wheel is a model for robotic rolling locomotion, exemplified by the TARS3D robot platform. It reconfigures telescopic legs with multi-point curved feet into dual "wheels," each presenting eight evenly spaced ground contact points. This structure supports an efficient hybrid limit-cycle gait, analytically derived and augmented by reinforcement learning for expanded dynamic behavior.

1. Geometric Construction and Kinematics

Eight spokes are realized by pairing two telescopic legs on one side, each leg featuring a foot plate that spans a $45^\circ$ arc with radius $r = 0.124$ m. Each such foot delivers four distinct ground contact points ("spokes"), so two legs yield an eight-spoke pattern. The robot configures its four legs as two wheels, left and right, which act as mirror images and are phased by the hip joint. The mechanism adheres to hip angular constraints of $±150^\circ$ and alternates left-right contact without interference. Rather than employing a traditional axially rotating body, the architecture "stitches" together two passive contact cycles, creating an eight-step hybrid limit cycle—one full roll consists of eight consecutive ground contacts.

2. Analytical Rolling Dynamics and Limit-Cycle Conditions

Rolling dynamics adopt the classical rimless wheel’s reduced-order inverted pendulum model. The angular equation during stance is:

$\frac{d^2\theta}{dt^2} = \frac{g}{l} \sin \theta$

Here, $\theta$ is the angle from the vertical, $g$ is gravitational acceleration, and $l$ is the leg length. During rolling, the center of gravity (CoG) cyclically ascends from $-\alpha$ to $+\alpha$, where $2\alpha = 45^\circ$, converting gravitational potential energy to kinetic energy. At each ground contact (impact), angular momentum is nearly conserved, modeled by:

$\dot{\theta}^+ = \dot{\theta}^- \cos 2\alpha$

$\dot{\theta}^-$ and $\dot{\theta}^+$ denote angular velocities immediately before and after contact, respectively. To preserve dynamic feasibility, the post-impact velocity must satisfy:

$$\dot{\theta}^+ > \theta_\textrm{min} = \sqrt{(2g/l)(1-\cos\alpha)}$$

Impacts dissipate rotational energy by a constant proportion, necessitating energetic compensation. TARS3D accomplishes this via telescopic extension, imparting angular impulse $J$:

$$\dot{\theta}^- = \dot{\theta}_0^+ + \frac{J}{m l^2}$$

Coupling with the collision rule, the steady-state angular velocity is:

$$\dot{\theta}^+ = \cos 2\alpha \left(\dot{\theta}_0^+ + \frac{J}{m l^2}\right)$$

Solving for required impulse, one finds:

$$J^* = m l^2 \dot{\theta}^* \frac{1 - \cos 2\alpha}{\cos 2\alpha}$$

The required change in leg length for impulse delivery is:

$\Delta l \approx \frac{J^*}{2 m l \dot{\theta}^-}$

For TARS3D ($m ≈ 0.99~\textrm{kg}$, $l = 0.124~\textrm{m}$, $\dot{\theta}^- ≈ 3.1~\textrm{rad/s}$), this computes to $\Delta l \approx 5~\textrm{mm}$—attainable by the robot's actuators (dynamic extension at $78~\textrm{mm/s}$ over $\sim$30 ms).

3. Telescopic Leg Redundancy and Morphological Implications

The TARS3D platform uses telescopic legs with dual-functional redundancy. Each leg serves as both a source of multiple contact points and a reservoir for delivering targeted angular impulses through extension. This structure supports:

• Uniform ground contact distribution by providing four spokes per foot,
• Additional actuation reserves to counteract collision-based energy losses at each impact,
• Shifting the center of gravity for maintaining minimum angular velocity and sustaining limit-cycle rolling.

The redundancy enables the system to realize both analytically derived gait modes (walking and rolling) and further locomotion behaviors discovered by learning, expanding beyond configurations typical of legged robots. A plausible implication is increased adaptability to various terrain types and task requirements enabled by the hardware's gait-generating flexibility.

4. Hybrid Limit Cycle and Locomotion Stability

The rolling gait is comprised of an eight-step hybrid limit cycle. Each ground contact constitutes a discrete phase and together form a repeatable rolling trajectory. This approach assures that the system remains within hardware-imposed joint limits ($±150^\circ$ at the hip) and maintains strict alternation of left and right wheel contacts, avoiding mechanical interference or loss of stability. The system leverages the energy policy described above for continued motion, and the discrete, multi-spoke design offers increased stability compared to classical rimless wheel implementations.

5. Deep Reinforcement Learning for Expanded Gait Repertoire

Beyond analytically tractable gaits, the eight-degree-of-freedom (8-DoF) robot can exhibit a much larger set of dynamic behaviors. Deep reinforcement learning (DRL), using algorithms such as Proximal Policy Optimization (PPO), is applied in simulation, with frameworks including NVIDIA IsaacLab. When seeded with morphological priors (e.g., locking outer joints at $90^\circ$ to enforce the wheel configuration), DRL recapitulates the known analytic rolling gait. Policies learned in these conditions produce center-of-mass trajectories matching theoretical predictions.

DRL also uncovers non-analytic locomotion patterns. Examples include "frog-hop" behaviors at reduced speeds and asymmetric gaits in the presence of heading misalignments. Reward functions used in simulation balance metrics such as forward progress, energy efficiency, and stability, with penalties for joint limit violations. Thus, DRL acts as both validation for limit-cycle models and an engine for discovering new locomotion strategies not reachable by analytic reduced-order design.

6. Operational Metrics and Experimental Validation

TARS3D hardware validates the analytic predictions: rolling mode maintains the eight-contact hybrid limit cycle, stays within the $\pm 150^\circ$ hip joint envelope, and alternates side contacts as designed. The telescopic mechanism achieves the calculated $\Delta l$ of $\sim$5 mm during each rolling step, with actuator velocities matching specification ($\sim$78 mm/s and $\sim$30 ms actuation window).

DRL-discovered gaits were implemented and compared to analytic trajectories. Learned policies reliably recover limit-cycle rolling under proper priors while demonstrating additional modes that respect hardware and energetic constraints.

7. Significance and Prospects in Multimodal Robotics

The eight-spoke double rimless wheel paradigm illustrates how fiction-inspired, bio-transcending robot morphologies can deliver versatile locomotion within compact, redundant mechanistic frameworks. The integrative analytic and learning-driven approach gives rise to multiple locomotion styles, robust under varying energetic regimes and terrain types. A plausible implication is that combining limit-cycle synthesis with DRL will accelerate the discovery of novel gaits, particularly in systems where actuator or contact redundancy exists. The result is widened operational robustness and adaptability, exceeding that of conventional legged or wheel-based designs and informing future research directions in multimodal robotics (Sripada et al., 6 Oct 2025).