Stable and Robust SLIP Model Control via Energy Conservation-Based Feedback Cancellation for Quadrupedal Applications

Abstract: In this paper, we present an energy-conservation based control architecture for stable dynamic motion in quadruped robots. We model the robot as a Spring-loaded Inverted Pendulum (SLIP), a model well-suited to represent the bouncing motion characteristic of running gaits observed in various biological quadrupeds and bio-inspired robotic systems. The model permits leg-orientation control during flight and leg-length control during stance, a design choice inspired by natural quadruped behaviors and prevalent in robotic quadruped systems. Our control algorithm uses the reduced-order SLIP dynamics of the quadruped to track a stable parabolic spline during stance, which is calculated using the principle of energy conservation. Through simulations based on the design specifications of an actual quadruped robot, Ghost Robotics Minitaur, we demonstrate that our control algorithm generates stable bouncing gaits. Additionally, we illustrate the robustness of our controller by showcasing its ability to maintain stable bouncing even when faced with up to a 10% error in sensor measurements.

• The paper introduces an energy conservation-based feedback cancellation scheme that delivers robust quadrupedal locomotion using a reduced-order SLIP model.
• It employs analytical trajectory synthesis with a parabolic spline to compute desired apex heights and motion transitions, balancing computational efficiency and real-world dynamics.
• Simulations demonstrate stability and noise resilience (up to 10% sensor error), suggesting practical trade-offs for hardware deployment in bio-inspired robotics.

Stable and Robust SLIP Model Control via Energy Conservation-Based Feedback Cancellation for Quadrupedal Applications

Introduction and Contextualization

The paper presents a formalized approach to dynamic quadrupedal locomotion via a reduced-order spring-loaded inverted pendulum (SLIP) model, targeting bio-inspired, energy-conserving control for agile running and bouncing gaits. The motivation is drawn from animal biomechanics, with a particular focus on the characteristic compliant and efficient gaits of terrestrial quadrupeds that leverage elastic limb dynamics for energy storage and release during motion. The approach aims to abstract and algorithmically replicate such dynamics in legged robotic systems. The authors center their development on bridging the gap between computational tractability and real-world robustness by employing a reduced-order SLIP model paired with a feedback cancellation-based controller architecture.

(Figure 1)

Figure 1: (a) Real quadruped and (b) spring-loaded inverted pendulum (SLIP) reduced-order abstraction, used for control synthesis.

SLIP Model for Quadruped Dynamics

The model constitutes a simplified point-mass-on-spring abstraction, capturing the hybrid (stance/flight) mechanics central to high-speed quadrupedal gaits.

(Figure 2)

Figure 2: Stance and flight phase definitions within the SLIP model, with transitions governed by touchdown/liftoff guard surfaces.

The SLIP equations partition the dynamics into ballistic flight, where $m \ddot{x}=0$ and $m \ddot{y}=-mg$, and stance, where the ground leg acts as a massless spring, derived as: $$Flight:\quad \begin{bmatrix} m\ddot{x} \ m\ddot{y} \end{bmatrix} = \begin{bmatrix} 0 \ -mg \end{bmatrix}$$

$$Stance:\quad \begin{bmatrix} m\ddot{r} \ \frac{d}{dt} (m r^2 \dot{\theta}) \end{bmatrix} = \begin{bmatrix} m r \dot{\theta}^2 - m g \sin \theta - b \dot{r} - \frac{d}{dr} U(r) \ m g r \cos \theta \end{bmatrix}, \quad U(r) = \frac{1}{2} k (r_0 - r)^2$$

This representation aligns with empirical observations of legged animal COM trajectories and offers direct implementation relevance for platforms like Ghost Robotics Minitaur\textsuperscript{\texttrademark}.

(Figure 3)

Figure 3: Apex height and compression length mapping, fundamental to the controller’s energy-conserving reference trajectory computation.

Energy Conservation-Based Control Synthesis

The controller operates on energy conservation over full gait cycles, injecting non-conservative work (spring actuation) as required to regulate total system energy, primarily during stance. The trajectory for the robot's center of mass (COM) during stance is computed as a parabolic spline derived analytically from desired apex heights, using the following sequence:

1. Specify desired apex height and horizontal velocity.
2. Compute required compression at minimum stance height via

$$\Delta y_{\text{min}} = \sqrt{ \frac{2mg \Delta h_{\text{apex}} }{k} }$$

1. Define transition events (takeoff, touchdown) and duration.
2. Construct a time-parameterized quadratic ($y(t) = a x(t)^2 + b x(t) + c$) satisfying boundary constraints, solved via matrix inversion.

(Figure 4)

Figure 4: SLIP stance configuration, clarifying controlled variables $r$ and $\theta$.

This formulation allows encoding of terrain constraints and desired motion outcomes into reference trajectories, directly amenable to real-time feedback tracking.

Feedback Cancellation Tracking Procedure

For parabolic reference trajectory tracking, the control law is derived via output error dynamics analysis on the linearized system: $\dot{x} = A x + B u, \quad y = C x$ Error terms are defined as $e_1 = x - \hat{x}$ and $e_2 = y_d - y$, leading to the control law: $u = (CB)^{-1} ( \dot{y}_d - CA \hat{x} + K e_2 )$ Observer gains ($K_e$) and tracking gains ($K$) are allocated via pole placement to guarantee convergence of error dynamics: $$\dot{\mathbf{e}} = E \mathbf{e}, \quad \text{with}~ E \prec 0$$ This construction provides for explicit design of error convergence rates, robust tracking, and strict enforcement of reference trajectory adherence.

(Figure 5)

Figure 5: Control architecture illustrating interleaved stance and flight controllers, event listeners, and observer integration for real-world sensor/measurement emulation.

Simulation Results: Stability and Robustness

Simulations parameterized according to the Minitaur robot demonstrate the practical efficacy:

• Stable vertical dynamics: State trajectories are periodic and repeatable, with successive hops precisely overwriting prior gait cycles (see phase space depiction).
• Noise resilience: Controller maintains stability under sensor measurement error of up to 10%, effectively rejecting state estimation noise within each cycle and regaining the nominal trajectory before takeoff.

(Figure 6)

Figure 6: Phase plot with controller-stabilized SLIP dynamics: apexes (A, B), liftoff (C), touchdown (D) are invariant across cycles, signifying stability.

Figure 7: Controller response under 10% touchdown measurement error; stable trajectory is recovered prior to next takeoff.

Implementation Guidance and Trade-offs

Computational Requirements

• The reduced-order model significantly decreases computational cost, making full-state estimation and trajectory synthesis tractable for embedded robotic applications.
• Linearization around reference trajectories mitigates the challenges of hybrid system nonlinearity and facilitates real-time feedback implementation.

Sensor and Estimator Considerations

• The observer design accommodates sensor noise and partial observability, yielding robustness to real-world imperfections.
• Event listeners are strictly based on observable outputs, simulating practical deployment where ground truth state is unavailable.

Strategy Trade-offs

• By neglecting active modulation of leg stiffness (prevalent in biological counterparts but rare in robotic systems), the method is directly implementable on current hardware, at the expense of potential agility/efficiency in more advanced future platforms.
• Explicit energy shaping via ballistic reference trajectory enables reliable stability, but may underperform when the full-order dynamics diverge sharply from the SLIP abstraction, e.g., in extreme terrain or highly underactuated robots.

Implications and Future Directions

The proposed SLIP energy-conserving feedback-cancellation control provides a technically sound and reproducible scheme for stable, robust quadrupedal bouncing gaits. The controller’s insensitivity to 10% sensor error is a substantial claim, implying resilience to common real-world state estimation challenges. Alignment with biological paradigms frames the contribution as a meaningful advance in bio-inspired robotics, with direct applicability to existing quadrupedal platforms.

Practical implications include deployment on constrained hardware, rapid trajectory synthesis, and ease of extension to different robot morphologies adhering to SLIP-like COM dynamics. The approach could serve as a foundation for more advanced control schemes, such as those incorporating active stiffness control or 3D gait modulation, and would benefit from empirical trials on hardware to evaluate model fidelity and offer further refinements.

Theoretically, the underlying energy-shaping methodology invites future work on integrating adaptive or learning-based elements for terrain generalization, formal model-mismatch compensators, and global stability guarantees.

Conclusion

This work formalizes a bio-inspired, computationally efficient SLIP-based control architecture for dynamic quadruped running, validated in simulation with strong stability and sensor noise robustness claims. The controller design is analytically grounded and implementation-ready, offering a valuable blueprint for practitioners seeking robust, real-time dynamic gait control on compliant legged robots. The next steps comprise hardware deployment and further development towards 3D locomotion and model-mismatch compensation.