Designing Robots That Bend and Bounce: A New Approach to Locomotion

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Author: Denis Avetisyan

Researchers have developed an automated framework for creating soft-rigid hybrid robots optimized for dynamic movement, paving the way for more adaptable and efficient robotic designs.

A novel design framework for soft-rigid hybrid robots integrates compliant soft bodies with strong, 3D-printed skeletal structures-strategically positioned and connected via simple bolted joints-and time-dependent actuation forces, allowing for co-optimization of morphology, skeletal layout, and control to achieve adaptable yet robust locomotion.
A novel design framework for soft-rigid hybrid robots integrates compliant soft bodies with strong, 3D-printed skeletal structures-strategically positioned and connected via simple bolted joints-and time-dependent actuation forces, allowing for co-optimization of morphology, skeletal layout, and control to achieve adaptable yet robust locomotion.

This work presents a differentiable simulation and optimization pipeline for the automated design of soft-rigid hybrid robots capable of complex locomotion.

While fully rigid robots struggle with adaptability and fully soft robots with force, biological systems elegantly combine strength and flexibility through integrated hard-soft tissues. Inspired by this principle, we present an automated design framework, detailed in ‘Automated design of soft-rigid hybrid robots for dynamic locomotion’, which optimizes both the morphology and actuation of these hybrid robots. Our method leverages a differentiable simulator coupling the material point method and extended position-based dynamics to generate designs-specifically, soft bodies reinforced by stiff truss structures-capable of complex locomotion. This approach successfully creates robots exhibiting walking gaits predicted by optimization, raising the question of how such automated design can unlock more sophisticated and versatile robot morphologies.

Embodied Intelligence: The Pursuit of Natural Movement

Conventional robotics historically prioritizes precision through the implementation of rigid components and intricate control algorithms. While this approach allows for highly repeatable actions in structured environments, it simultaneously introduces significant limitations in adaptability and energy expenditure. These machines often struggle with unpredictable terrains or dynamic interactions, requiring substantial power to maintain stability and overcome even minor obstacles. The very rigidity that enables precise movements hinders their ability to absorb shocks, navigate uneven surfaces, or efficiently distribute force. Consequently, traditional robots frequently exhibit limited maneuverability and comparatively short operational durations, prompting researchers to explore alternative designs that prioritize efficiency and resilience over absolute positional control.

Nature presents a compelling alternative to conventional robotics through the elegance of biological locomotion. Animals demonstrate extraordinary agility and efficiency not with stiff mechanisms and intricate programming, but with inherently flexible bodies and remarkably simple control schemes. This inspires a shift in robotic design, moving away from rigid structures towards systems composed of soft, compliant materials. These bio-inspired robots leverage the principles of elasticity and distributed compliance – allowing them to absorb impacts, navigate complex terrains, and conserve energy. Researchers are now exploring how to replicate the streamlined neural control observed in creatures, seeking to minimize the computational burden on robotic systems and maximize their adaptability – ultimately aiming for robots that move with the same grace and efficiency as their biological counterparts.

The pursuit of bio-inspired robotic movement hinges on a delicate balance between rigidity and compliance, demanding novel material science and sophisticated optimization strategies. Traditional robotics favors stiff structures for precision, but biological systems demonstrate that flexibility-allowing bodies to deform and store energy-significantly enhances efficiency and adaptability. Researchers are now exploring materials like soft elastomers, shape memory alloys, and granular materials to create robots capable of nuanced movements and absorbing impacts. Beyond material selection, computational optimization techniques, including evolutionary algorithms and finite element analysis, are crucial for designing structures that maximize energy return and minimize wasted motion. This involves precisely tuning the distribution of stiffness within a robot’s body, allowing it to bend, twist, and recover dynamically, ultimately paving the way for robots that move with the grace and efficiency observed in nature.

The proposed optimization framework progressively develops locomotion by first establishing actuation, then skeletal structure, and finally refining soft body morphology, resulting in improved performance while satisfying all constraints, as demonstrated by the evolution of design components and a decreasing loss function over time [latex] (scale bar = 30 mm) [/latex].
The proposed optimization framework progressively develops locomotion by first establishing actuation, then skeletal structure, and finally refining soft body morphology, resulting in improved performance while satisfying all constraints, as demonstrated by the evolution of design components and a decreasing loss function over time [latex] (scale bar = 30 mm) [/latex].

Co-Designing Form and Function: A Differentiable Approach

Locomotion optimization forms the central design methodology, wherein both the robot’s morphology – encompassing link lengths, joint configurations, and body mass distribution – and its control parameters, such as gait parameters and joint gains, are treated as optimization variables. This approach deviates from traditional robotics design, which typically fixes the robot’s physical structure and focuses solely on control software. By simultaneously optimizing both morphology and control, the system seeks to discover designs that maximize locomotion performance – measured by metrics like speed, stability, and energy efficiency – across diverse terrains and operational requirements. The optimization process utilizes a cost function that quantifies locomotion performance, and iteratively adjusts both structural and control variables to minimize this cost, resulting in a co-designed robot-control system.

Differentiable simulation is utilized to enable gradient-based optimization of robot morphology and control policies. This approach allows for the computation of [latex]\frac{\partial Reward}{\partial Parameters}[/latex], where ‘Reward’ represents a performance metric and ‘Parameters’ encompass both the robot’s physical design and its control inputs. By propagating gradients through the physics simulation, iterative updates to these parameters are performed via gradient descent algorithms. This contrasts with traditional methods that rely on discrete search or manual tuning, and facilitates automated design exploration and performance improvement. The differentiability of the simulation environment is achieved through automatic differentiation techniques applied to the underlying physics engine, allowing the computation of derivatives without requiring explicit analytical expressions.

Checkpointing was implemented to mitigate the substantial memory demands of backpropagation through the robot’s simulation. This technique involves discarding intermediate activations during the forward pass and recomputing them only when needed during the backward pass. Specifically, rather than storing the activations of every layer, we selectively store a subset, strategically chosen to allow efficient reconstruction of the necessary values for gradient computation. This trade-off between computation and memory significantly reduces peak memory usage, enabling optimization of more complex robot morphologies and control policies without requiring hardware upgrades, while maintaining numerical accuracy equivalent to full activation storage.

Maintaining feasible physical constraints during morphological optimization necessitates the implementation of a smooth projection function. This function operates by mapping optimized design parameters – such as link lengths, joint limits, and actuator capacities – back into the allowable design space after each gradient update. A “smooth” projection, as opposed to a hard clipping or thresholding approach, avoids introducing discontinuities in the gradient, which are detrimental to the optimization process. Discontinuities can lead to unstable training and prevent convergence to optimal solutions. The projection function effectively enforces constraints on parameters like [latex]l_{min} \le l \le l_{max}[/latex] for link lengths l, and [latex]\theta_{min} \le \theta \le \theta_{max}[/latex] for joint angles θ, without disrupting the gradient flow required for iterative improvement.

Sustained locomotion is achieved by a fabricated robot with a skeletal structure, as demonstrated by similar motion patterns between simulation and experiment-including coordinated leg lifting and node trajectories [latex]P1[/latex] and [latex]P2[/latex]-which contrast with the diminished vertical displacement and reduced forward progression observed without the skeletal support.
Sustained locomotion is achieved by a fabricated robot with a skeletal structure, as demonstrated by similar motion patterns between simulation and experiment-including coordinated leg lifting and node trajectories [latex]P1[/latex] and [latex]P2[/latex]-which contrast with the diminished vertical displacement and reduced forward progression observed without the skeletal support.

Bridging the Digital and Physical: Material Realization

Solenoid actuators were selected for the robot’s locomotion system due to their advantageous power-to-weight ratio and precise control capabilities. These electromagnetic actuators generate linear motion through the interaction of a magnetic field and a current-carrying coil, allowing for repeatable and predictable movements essential for coordinated gait. Compared to alternative actuation methods like pneumatic or hydraulic systems, solenoids offer a simpler mechanical design, reducing weight and complexity. Furthermore, the controllability of solenoid actuators-specifically the ability to modulate force through current control-facilitates the implementation of dynamic locomotion strategies and precise positioning of the robot’s limbs.

The robot’s rigid skeletal structure was constructed utilizing fused filament fabrication (FFF), a layer-additive manufacturing process. This fabrication method allowed for the creation of complex geometries directly from digital models, facilitating a highly iterative design workflow. FFF enabled rapid prototyping – physical parts could be produced within hours of design modifications – and permitted cost-effective production of multiple iterations for testing and refinement of the robot’s mechanical performance. The chosen filament material provided sufficient structural integrity for supporting the robot’s internal components and transmitting forces generated by the actuation system.

Platinum-cure silicone elastomers were chosen for the robot’s soft body due to their inherent mechanical properties. These elastomers exhibit high compliance, allowing for significant deformation under load without permanent damage, and provide the necessary flexibility for complex movements and adaptation to uneven terrain. The platinum-cure process ensures a consistent and reliable curing mechanism, resulting in a material with predictable mechanical behavior and minimal shrinkage. Furthermore, silicone elastomers demonstrate good hysteresis characteristics, meaning they efficiently return to their original shape after deformation, which is critical for repeatable locomotion. Their chemical stability and resistance to environmental factors also contribute to the long-term durability of the robotic system.

To improve the robot’s ability to maintain contact with surfaces during locomotion, a patterned surface texture consisting of a regular array of raised dots was applied to the foot structure. This dot pattern increases the effective contact area and introduces mechanical interlocking with the substrate, enhancing adhesive forces. Testing demonstrated a measurable increase in the coefficient of friction on common test surfaces, specifically reducing slippage during both static and dynamic movements. The dot geometry – diameter and pitch – were optimized through iterative testing to balance adhesion with minimal impact on overall foot compliance and energy expenditure.

The presence of a skeletal structure modulates stride frequency response, enhancing forward displacement amplitude and promoting stable lifting of the front limb at peak frequencies (around 10-12 Hz, with peak performance), while its absence leads to localized deformation and decreased locomotion speed, particularly at lower frequencies (6 Hz).
The presence of a skeletal structure modulates stride frequency response, enhancing forward displacement amplitude and promoting stable lifting of the front limb at peak frequencies (around 10-12 Hz, with peak performance), while its absence leads to localized deformation and decreased locomotion speed, particularly at lower frequencies (6 Hz).

Dynamic Harmony: Optimized Performance and Resonant Response

The robot’s ability to move effectively hinges on its response to different actuation frequencies, a relationship critical for both speed and maintaining balance. Lower frequencies severely limit locomotion, nearly halting movement, while higher frequencies can introduce instability if not carefully managed. This frequency-dependent behavior arises from the interplay between the robot’s physical properties-mass, stiffness, and damping-and the timing of its movements. Optimizing the actuation frequency allows for a resonant response, maximizing energy transfer to the environment and enhancing both propulsive force and the robot’s capacity to recover from disturbances, ultimately leading to a more robust and efficient gait.

The design optimization process incorporated an augmented Lagrangian formulation to rigorously enforce physical constraints, preventing the generation of unrealistic or unbuildable robotic designs. This method effectively transforms constrained optimization problems into a series of unconstrained sub-problems, iteratively refining the design while ensuring adherence to limitations on joint angles, actuator forces, and overall structural integrity. By penalizing constraint violations within the objective function, the augmented Lagrangian approach guides the optimization algorithm toward solutions that are not only efficient in terms of performance but also physically plausible and realizable in a robotic system, ultimately leading to a functional and robust design.

The optimization of the robot’s design relied heavily on the Adam optimizer, a sophisticated algorithm designed to efficiently navigate the complex landscape of possible configurations. Traditional optimization methods can become trapped in local minima or require excessive computation time, but Adam dynamically adjusts the learning rate for each design variable, allowing for quicker and more reliable convergence towards an optimal solution. By leveraging both momentum and adaptive estimation of first and second moments, the Adam optimizer significantly accelerated the process of refining the robot’s parameters-such as link lengths and actuator strengths-ultimately leading to a functional prototype capable of sustained locomotion at 69.6 mm/s. This efficient updating of design variables proved crucial in overcoming the computational challenges inherent in optimizing a complex, multi-parameter system.

The culmination of this research is a functional robotic prototype capable of sustained locomotion, achieving a measured speed of 69.6 mm/s with a standard deviation of ±1.118 s.d. This performance is critically linked to an optimized actuation frequency of 11 Hz; lower frequencies, such as 6 Hz, result in a dramatic reduction in locomotion speed, nearing complete cessation of movement. The demonstrated improvement highlights the efficacy of the integrated design and optimization process, showcasing a substantial leap in robotic agility and efficiency compared to designs operating at suboptimal frequencies. This result confirms the importance of frequency-dependent response in achieving robust and effective robotic movement.

The automated design of soft-rigid hybrid robots, as detailed in this work, echoes a fundamental principle of holistic systems. The framework doesn’t merely address individual components-morphology, actuation, or control-but rather optimizes the interplay between them to achieve dynamic locomotion. This resonates with Donald Davies’ observation, “What scales are clear ideas, not server power.” The computational power enabling this design process is significant, yet the true scalability lies in the elegance of the underlying concepts – a clear articulation of how structure dictates behavior within the robot’s biomechanical system. The successful integration of differentiable simulation and optimization techniques demonstrates that complex functionality arises not from brute force, but from thoughtfully designed interactions.

Beyond Morphology: The Path Forward

The automated design of robots, even those embracing the complexities of soft materials, ultimately reveals a familiar truth: optimization converges on local minima. This work demonstrates a capacity to generate plausible morphologies, but the true cost lies not in computation, but in the definition of ‘locomotion’ itself. Current metrics, however elegantly expressed through differentiable simulation, remain blunt instruments, failing to capture the nuanced interplay between robot, environment, and task. The field will inevitably move beyond simply optimizing for speed or energy efficiency, and towards designs that prioritize robustness, adaptability, and – critically – predictable failure modes.

The reliance on simulation, while currently necessary, introduces an unavoidable abstraction leak. Position-based dynamics and material point methods are approximations of reality, and the transfer of optimized designs to physical embodiments will always require iterative refinement. Future work must address this gap, perhaps through the development of more sophisticated co-simulation techniques or, more radically, by integrating real-world data directly into the optimization loop. A truly scalable solution will not seek to eliminate the physical world, but to embrace it as an inherent part of the design process.

The elegance of this framework lies in its generality, yet the potential for brittle specialization looms large. Each material, each actuator, each environment introduces new constraints and complexities. The long-term challenge is not simply to automate design, but to create systems that can learn to design themselves, adapting to unforeseen circumstances and evolving in response to changing needs. The architecture that survives will be the one that anticipates its own limitations.

Original article: https://arxiv.org/pdf/2605.29389.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-05-31 09:05