Tunable Leg Stiffness in a Monopedal Hopper for Energy-Efficient Vertical Hopping Across Varying Ground Profiles

We present the design and implementation of HASTA (Hopper with Adjustable Stiffness for Terrain Adaption), a vertical hopping robot with real-time tunable leg stiffness, aimed at optimizing energy efficiency across various ground profiles (a pair of ground stiffness and damping conditions). By adjusting leg stiffness, we aim to maximize apex hopping height, a key metric for energy-efficient vertical hopping. We hypothesize that softer legs perform better on soft, damped ground by minimizing penetration and energy loss, while stiffer legs excel on hard, less damped ground by reducing limb deformation and energy dissipation. Through experimental tests and simulations, we find the best leg stiffness within our selection for each combination of ground stiffness and damping, enabling the robot to achieve maximum steadystate hopping height with a constant energy input. These results support our hypothesis that tunable stiffness improves energy-efficient locomotion in controlled experimental conditions. In addition, the simulation provides insights that could aid in future development of controllers for selecting leg stiffness.

Tunable stiffness in robotics can be achieved through two primary methods: software-based virtual compliance and hardware-based variable stiffness actuators. Software-based virtual compliance, implemented via impedance control with force or positional feedback, allows real-time stiffness tuning through various control strategies, resulting in spring-like dynamics. Examples of this approach can be seen in robots like ANYmal [5], Minitaur [6], and MIT Cheetah [7]. These systems benefit from fast response times due to the low-gearing of direct-drive motors [8], [9], but they also suffer from rapid heating due to joule losses [10]. Moreover, software-based solutions lack the energy storage benefits of passive compliance and require higher computational resources [11], [12]. On the other hand, hardware-based tunable stiffness mechanisms use mechanical designs integrated with actuators to physically alter the system’s mechanical properties. These systems offer advantages such as reduced need for active energy input [13]–[15], enhanced stability and adaptability, and the ability to reduce high-impact forces [9], [16]. However, they tend to be larger and heavier due to their complex mechanisms and often have slower response times for realtime adjustments [17]. In this work, we introduce HASTA (Hopper with Adjustable Stiffness for Terrain Adaption), a vertical hopper with tunable stiffness achieved through a pneumatic bellows actuator developed in our previous work [18]. This system allows us to explore how stiffness adjustments in the compliant leg can enhance locomotion performance, offering energy storage benefits, improved dynamic response, and reduced weight and control complexity by eliminating the need for external air sources. In the context of vertical hopping, we aim to achieve energy-efficient locomotion, where energy efficiency is defined by the steady-state hopping height for a given fixed input energy.

For analytical purposes, the ground is often modeled as a network of springs and dampers [17], [19], [20]. While research on tuning leg stiffness for different ground conditions exists, it has primarily focused on modeling and addressing the spring component of the ground [17], [19], [21]–[23], with limited exploration of damping effects, particularly in real-world scenarios. We argue that incorporating the damping component is essential for programming energyefficient locomotion, as it influences energy dissipation and overall system performance. Building on the work in [24], we further investigate energy-efficient vertical hopping on ground profiles with programmable stiffness and damping, with a focus on tuning stiffness in physical hardware to optimize energy efficiency. In this paper, we hypothesize that during vertical hopping, softer legs are more effective on soft, damped ground, minimizing penetration and energy loss, while stiffer legs perform better on hard, less damped ground by reducing limb deformation and energy dissipation. To our knowledge, this is the first study to systematically explore hardware-based tunable stiffness in a monopedal hopping robot across a range of both ground stiffness and damping conditions.

Our contributions are:

1. Development of HASTA, a dynamic vertical hopper with real-time tunable stiffness for exploring various ground profiles. It is equipped with one actuated degree of freedom (DOF) for tendon-driven actuation and three actuated DOFs for adjusting leg stiffness.

2. Real-world characterization of the relationship between leg stiffness and performance across different ground profiles. This characterization shows that as ground stiffness increases or damping decreases, stiffer legs reduce energy loss and achieve higher hopping heights.

3. Validation of a simulator for the described characterization, demonstrating that the simulation can be used to guide controllers in selecting optimal leg stiffness.

The task is to vertical hopping using a monopedal robot with a tunable stiffness leg, as this setup captures the fundamental interaction between leg stiffness and ground properties while isolating it from complexities such as gait dynamics and multi-legged coordination. Our goal is to demonstrate that an optimal leg stiffness setting exists, allowing the robot to achieve the maximum steady-state hopping height on different terrain types, given a constant energy input.

Here are specific improvements to AI systems based on the principles and findings of the HASTA robot research:

1. Improved Energy-Aware Locomotion Control for Dynamic Environments: The core improvement is developing an AI controller that dynamically selects the optimal leg stiffness setting in real-time based on perceived ground properties (stiffness and damping).

2. Adaptive Control Strategy for Terrain Profiling: The system should incorporate a real-time perception module (e.g., using sensor data from the Ground Emulator concept) to estimate the current ground profile and automatically map this profile to the optimal stiffness setting derived from the simulation/experimental grid search (Problem 1).

3. Predictive Model for Energy Management: Implement a reinforcement learning (RL) agent or model predictive control (MPC) framework that uses the system's state (current position, velocity, input energy) and predicted terrain characteristics to optimize the sequence of leg stiffness adjustments to maximize the steady-state apex hopping height for a given energy budget.

4. Damping Compensation and Energy Dissipation Modeling: Enhance the simulation and control systems by explicitly modeling or estimating unmodeled damping effects (like those caused by rail friction or lateral oscillation observed in experiments). The AI should learn to adjust leg damping coefficients (if controllable) or stiffness in anticipation of these losses to maintain energy efficiency.

5. Optimized Control Gain Selection: Develop an automated system for selecting the ground emulator's control gains (Kp, Kd) based on the estimated terrain profile, ensuring that the physical interaction between the robot and its environment is optimally tuned for energy transfer rather than just stability.

These improved AI systems can achieve the following specific capabilities:

1. An autonomous vertical hopper could execute long-distance hopping across heterogeneous surfaces (e.g., transitioning from a hard floor to a soft carpet) while maintaining a constant, minimal energy expenditure per hop, significantly extending operational endurance compared to fixed-stiffness controllers.

2. The system could perform proactive terrain adaptation, anticipating changes in the ground profile moments before impact and adjusting leg stiffness preemptively to maximize the potential jump height for that specific ground condition.

3. It could optimize locomotion not just for maximum height, but for a multi-objective goal, such as maximizing hopping height while simultaneously minimizing lateral oscillation (as suggested by the experimental anomalies) to reduce energy waste.

4. The AI would be capable of operating in environments where precise physical parameters are unknown or non-linear, relying on learned mappings from the simulation/experimental data to select the best physical configuration for current conditions.

We present the design and implementation of HASTA (Hopper with Adjustable Stiffness for Terrain Adaptation), a vertical hopping robot with real-time tunable leg stiffness, aimed at optimizing energy efficiency across various ground profiles (a pair of ground stiffness and damping conditions). By adjusting leg stiffness, we aim to maximize apex hopping height, a key metric for energy-efficient vertical hopping. We hypothesize that softer legs perform better on soft, damped ground by minimizing penetration and energy loss, while stiffer legs excel on hard, less damped ground by reducing limb deformation and energy dissipation. Through experimental tests and simulations, we find the best leg stiffness within our selection for each combination of ground stiffness and damping, enabling the robot to achieve maximum steady-state hopping height with a constant energy input. These results support our hypothesis that tunable stiffness improves energy-efficient locomotion in controlled experimental conditions. In addition, the simulation provides insights that could aid in the future development of controllers for selecting leg stiffness.