RAS 557: FOLDABLE ROBOTICSProject: Bio-Inspired Myriapod Robot

Motivation and Project Goals

The primary motivation of this work is to quantify the dynamic trade-offs inherent in compliant segmented locomotion. While hexapedal (6-legged) implementations benefit from inherently stable tripod gaits, they often mask the specific contribution of spinal compliance to stability. By reducing the morphology to a quadruped (4-legged) system, we create a more challenging dynamic environment where the spine's role in maintaining ground contact becomes critical.

• Synthesize a High-Clearance Morphology: Integrate single-degree-of-freedom Jansen linkages to generate foot trajectories with superior vertical clearance compared to simple rotary legs.
• Design a Novel Compliant Spine: Replace complex multi-stage backbones with a simplified Cross-Axis Flexure (X-Flexure) spine, designed to provide tunable lateral flexibility while maintaining torsional rigidity.
• Bridge Simulation and Reality: Formulate a physics-based model in MuJoCo to predict the gait dynamics of this quadrupedal system and validate these predictions against a physical SCM prototype.
• Optimize Passive Dynamics: Investigate how the stiffness of the X-flexure spine influences the robot's ability to maintain a stable trot gait without complex active feedback.

Project Evolution: First Team Assignment to Current Stage

This report marks a significant leap from the initial spine-focused scope of the First Team Assignment, scaling up to a fully integrated quadrupedal robot. The key improvements are summarized below:

Baseline Template & Modifications

The baseline template follows the myriapod concept: a central body with repeated leg modules and a compliant backbone that couples segment dynamics. In the original work, the backbone is implemented as a sequence of Sarrus linkages and flexures. We retain the overall idea of a compliant backbone linking leg pairs but replace the Sarrus linkage with a simpler Parallel Articulation Mechanism (PAM) spine while also reducing the number of legs and actuators.

Smart Composite Microstructures (SCM)

We utilized a 5-layer SCM laminate structure to create lightweight, monolithic kinematic chains. The stack-up consists of:

Manufacturing Sequence & Assembly

The fabrication followed a sequential process:

1. Laser Machining: "First Pass" pattern cut into individual rigid/adhesive layers.
2. Lamination: Five layers aligned using a pin-jig system and heat-pressed.
3. Release: "Final Cut" pattern applied to release the structure.
4. Folding: 2D laminates folded into 3D kinematic configurations.

Electronic Integration & Iteration

Motor Mounting: Custom motor mounts were designed in SolidWorks and fabricated on a Prusa i3 MK3 using PLA with a 30% Gyroid infill for isotropic strength. We iterated to address initial failures:

• Material Selection: We switched to a thicker flexure material for the rigid layers to prevent buckling under weight.
• Torque Limiting: We implemented software-based torque limiting in the motor control loop to prevent current spikes during stalled states, protecting the mechanical couplers.

The control logic is handled by an ESP32 microcontroller running MicroPython. The electronics stack, including L298N motor drivers and an IMU, is mounted on the dorsal surface of the chassis.

Analytical Kinematics of the Spine

The articulated spine is modeled as a planar three-branch parallel linkage (3R–2R–3R), corresponding to the kinematic diagram in the design section. A Python-based kinematic model was developed to evaluate the workspace and compute Jacobians for force analysis.

All motion is defined in an inertial frame $\mathcal{N}$ with basis $\{\hat{n}_x,\hat{n}_y,\hat{n}_z\}$. The model uses link lengths $l_0,\dots,l_6$ as constants and joint angles $\theta_1,\dots,\theta_6$ as state variables. The loop-closure constraints ensure the branches meet at the front chassis:

$$ \mathbf{c}_1 = P_3 - P_5 = \mathbf{0} $$

$$ \mathbf{c}_2 = P_5 - P_7 - \tfrac{1}{2}\vec{v}_3 = \mathbf{0} $$

Jacobian and Force Mapping

To map actuator torques to endpoint forces, we differentiate the position of the spine tip $P_7$ with respect to the actuated joints $\boldsymbol{\theta}_a = [\theta_4, \theta_5, \theta_6]^T$. The planar Jacobian is:

$$ J(\boldsymbol{\theta}) = \frac{\partial}{\partial\boldsymbol{\theta}_a} \begin{bmatrix} x_7(\boldsymbol{\theta}) \\ y_7(\boldsymbol{\theta}) \end{bmatrix} $$

This enables force mapping analysis ($ \boldsymbol{\tau}_a = J^{\mathsf{T}} \mathbf{F}_{\text{ee}} $). For a peak vertical load of $F_y \approx 4$N, the model predicts joint torques of approximately 1 Nm, validating the flexure stiffness design.

Design Parameter Study

We performed a parametric study on the Inner Shaft Size ($S_{inner}$) vector to identify dimensions that maximize gait stability by minimizing lateral body oscillations (yaw) while maintaining forward velocity.

Design Selection

Based on the solver validation, the final design parameters were selected:

• Length ($L=50$ mm): Chosen because shorter spines ($L<40$ mm) introduce interference, while longer spines ($L>80$ mm) increase yaw inertia ($I_z \propto L^2$) and instability.
• Width ($W=30$ mm): Provides necessary lateral stiffness to resist buckling during the stance phase.

Experimental Validation

Experiments were conducted on a smooth whiteboard surface using overhead video tracking. We compared the physics model predictions against the physical reality.

Results: Sim-to-Real Gap

This section compares the idealized performance predicted by the MuJoCo simulation against the physical realities observed during testing.

Simulation Results

The simulation predicted stable locomotion with a monotonic increase in forward position ($v_x > 0$) for the baseline spine stiffness ($L=50$mm), assuming a friction coefficient of $\mu=0.5$.

Experimental Results

Contrary to the simulation, the physical prototype failed to achieve net forward locomotion ($\hat{v}_x \approx 0$). Quantitative analysis revealed significant instability:

• Trajectory Churning: The robot oscillated within a small bounding box (approx. 10cm), indicating stride displacement was canceled by slip.
• Heading Drift: A severe heading drift ($>40^\circ$ over 18s) was observed, confirming predicted "fishtailing" for asymmetric conditions.
• Roll Instability: Roll angle oscillated between $\pm 20^\circ$, suggesting failure of the support polygon.

Conclusion

This work presented the design, modeling, and fabrication of a myriapod-inspired quadruped, aiming to exploit a compliant Cross-Axis Flexure spine for passive gait stabilization. While our MuJoCo simulation successfully identified an optimized spine geometry ($L=50$ mm) that theoretically minimized yaw oscillation, the physical implementation highlighted severe limitations in current SCM fabrication for this morphology.

Experimental testing did not validate the efficacy of the spine in generating locomotion; rather, it exposed a qualitative mode switch from the predicted "walking" to "slipping". The theoretical benefits of spinal tuning were masked by first-order mechanical failures:

• Traction Limits: The assumption of high friction ($\mu=0.5$) proved invalid for cardstock end-effectors, leading to friction cone violation.
• Structural Stiffness: The idealized rigid bodies in MuJoCo did not account for the parasitic out-of-plane buckling observed in the physical laminate legs.

Future work must prioritize high-friction footpads (silicone) and stiffer laminate materials (e.g., FR4) to restore the traction model before the subtle dynamics of spinal compliance can be effectively analyzed or tuned.

Abstract

This project presents the design and physical prototyping of a bio-inspired mechanism intended to serve as the articulated spine of a small quadruped robot with Jansen mechanisms for its four legs. Taking inspiration from the classical Sarrus linkage, we develop a planar articulated spine composed of a three-branch parallel linkage (3R–2R–3R) that connects two rigid 3D-printed chassis plates and is intended to provide vertical support while allowing controlled in-plane bending. The spine is realized in a Smart Composite Microstructures (SCM) format using a five-layer laminate (rigid--adhesive--flexure--adhesive--rigid). Stiff outer layers form the links, while the flexure layer creates passive spring-like joints that generate a restoring torque as the spine bends. The overall morphology is scaled to roughly match the proportions of a juvenile cat while operating at a much lower mass of about 250g. We construct both a paper analogue of the mechanism and a laminate prototype integrated into a quadruped chassis. A planar kinematic model, implemented in Python, represents the 3R–2R–3R parallel spine in the sagittal plane and is used to compute representative joint torques and joint velocities for expected ground reaction forces and vertical motions of the front chassis. Using this model, we compute representative joint torques and joint velocities for expected ground reaction forces and vertical motions of the front chassis. The results demonstrate that a linkage-based, passively compliant spine is feasible within the constraints of laminate fabrication and provides a structured way to tune the directional stiffness of the robot torso for future integration with Jansen-leg quadruped locomotion.

8. Conclusion

We have designed, fabricated, and modeled a laminate articulated spine for a small quadruped robot inspired by both the directional stiffness of mammalian spines and the linkage-based backbones of myriapod-style robots. The mechanism uses a Sarrus-inspired parallel linkage implemented in a five-layer laminate stack, yielding stiff links and passive spring-like joints.

A Python-based kinematic model of the planar 3R–2R–3R parallel spine provides estimates of joint torques under expected ground reaction forces and joint velocities for prescribed body motions. These results establish the articulated spine as a feasible and tunable subsystem for future Jansen-leg quadruped prototypes.

References

Project Highlight Video