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Versatile, Robust, and Explosive Locomotion with Rigid and Articulated Compliant Quadrupeds

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Parallel springs let a quadruped jump 25% farther and turn 15% wider

desk verdict Solid modeling/control pipeline for compliant quadrupeds, but the headline performance numbers rest on single trials and a confounded froggy jump; still worth refereeing. read the letter →

arxiv 2504.12854 v1 pith:A34ROZ6D submitted 2025-04-17 cs.RO

Jiatao Ding , Peiyu Yang , Fabio Boekel , Jens Kober , Wei Pan , Matteo Saveriano , Cosimo Della Santina This is my paper · ORCID
classification cs.RO
keywords quadrupedallocomotionparallelcompliancespring-loadedinvertedpendulumtrajectoryoptimizationquaternionsingularity-freecontrolwhole-bodyexplosivemotionrobust
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that adding parallel springs to a quadruped's legs and explicitly modeling those springs in planning and control turns a rigid robot into one that can jump farther, rotate further, and tolerate rougher terrain. The authors introduce the trunk dual actuated spring-loaded inverted pendulum (TD-aSLIP), a reduced-order model with two massless spring legs and a rotating trunk, and pair it with a dual-layer trajectory optimizer and a whole-body controller. In hardware, the spring-equipped robot increased maximal pronking distance by at least 25%, hop-turn yaw angle by at least 15%, and froggy-jumping distance by at least 25% compared with the same robot without springs, and it survived a 10 cm terrain height variation where the rigid version managed 5 cm. The paper's claim is that these gains come not from the springs alone, but from the springs plus a planner and controller built around configuration-aware compliance.

What carries the argument

The central object is the TD-aSLIP, or trunk dual actuated spring-loaded inverted pendulum: a reduced-order model with a single rigid trunk, two massless legs, and, on each leg, an actuation force and a parallel spring force acting in parallel. Its load-bearing move is a configuration-aware stiffness mapping that converts joint-space spring constants into an equivalent Cartesian leg stiffness through the leg Jacobian, then captures how that stiffness changes with leg length by sampling the robot's workspace and fitting a cubic polynomial. This makes the model's spring term match the actual hardware configuration rather than assuming a fixed constant. Around this model the paper builds a two-layer planner, first SLIP-based trajectory optimization and then quaternion kinodynamics optimization, and a controller made of a quaternion single-rigid-body MPC feeding a whole-body quadratic program that compensates spring torques and enforces safety constraints.

What would settle it

Run the same 90-degree hop-turn plan under motion capture that reconstructs whole-body angular momentum, then re-plan with a centroidal-dynamics model that includes leg inertia; if the yaw error stays around 20 degrees and the rigid-versus-compliant gap persists, the single-rigid-body assumption is not the limiting factor and the paper's explanation would need revision.

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Extended reading notes

Core claim

The central claim is that explosive quadruped motion with large trunk rotation can be planned and tracked using a two-legged, one-trunk template model that separates spring forces from motor forces, and that this explicit separation is what unlocks the benefits of parallel compliance. Concretely, the paper claims that the TD-aSLIP model, where each of two massless legs couples an actuation force with a unidirectional spring force whose stiffness varies with configuration, generates pronking, froggy jumping, and hop-turns without a hand-tuned reference trajectory. A dual-layer trajectory optimizer first solves a coarse SLIP problem and then refines it with quaternion kinematics and dynamics, avoiding Gimbal-lock singularities during large rotations. A quaternion-based model-predictive controller and a whole-body controller with explicit spring compensation track the plan in hardware. Comparison between a rigid quadruped and the same robot with parallel thigh and calf springs shows that the compliant version reaches larger pronks, larger yaw turns, and longer froggy jumps, and tolerates twice the terrain height variation when jumping from unknown uneven surfaces.

Load-bearing premise

The entire pipeline treats the robot as a single rigid body with massless legs, so it assumes leg swing and body-shape changes do not carry significant angular momentum; the authors concede this is why the rigid robot undershoots a 90-degree hop-turn by about 20 degrees.

Editorial extensions

If this is right

  • The same dual-layer planner, with only waypoints and contact sequence changed, should generate further flight-phase maneuvers such as sideways jumps or larger rotations without hand-tuned references; the paper demonstrates pronking, froggy jumping, and hop-turn from one formulation.
  • If parallel compliance is doing the work, the robustness gain should persist under modest spring mistuning or surface uncertainty, and the paper reports the allowable support-surface height variation doubles from 5 cm to 10 cm for froggy jumping.
  • The singularity-free quaternion formulation means the planned motions transfer across platforms without Gimbal-lock issues during large rotation, and the paper reports the same control gains work for both the rigid and spring-equipped versions.
  • The modeling principle of decoupling spring force from actuation force in a reduced-order template should extend to other explosive motions, such as consecutive jumps, because the dynamics equations do not depend on a particular contact schedule.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I would expect the configuration-aware stiffness mapping to matter most in deeply crouched stances like froggy jumping, where the leg Jacobian changes substantially; rerunning the planner with a fixed stiffness constant should produce visibly worse landing tracking on those motions.
  • The four legs are symmetrically paired into two SLIP legs; I would expect an asymmetric front/rear version with different spring constants or rest lengths to handle the 50 cm froggy jump even better than the authors' workaround of removing the right-rear calf spring, because the rear legs carry much more load during the rear-leg stance phase.
  • Since the paper's stated limitation is the ignored centroidal angular momentum, I would expect that replacing the single-rigid-body MPC with a momentum-aware or centroidal-dynamics controller would reduce the yaw tracking error in hop-turns, potentially pushing the compliant robot beyond the reported 145-degree limit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes TD-aSLIP, a reduced-order quadruped template model with two massless legs, a trunk rotation state, and explicitly modeled parallel compliance obtained by decoupling motor actuation from spring forces. A configuration-aware stiffness mapping is introduced, and a dual-layer trajectory optimization (SLIP-level coarse planning followed by quaternion-based kinodynamic refinement) is combined with a quaternion MPC and a whole-body controller with parallel-compliance compensation. The pipeline is evaluated in PyBullet simulation and on hardware: a rigid Unitree Go1 and a new compliant variant E-Go-V2 with thigh and calf springs. The central claims are that the model enables versatile explosive motions (pronking, hop-turn, froggy jumping), that parallel elasticity improves explosive performance (at least 25%, 15%, and 25% gains in maximum pronking distance, hop-turn yaw, and froggy-jump distance), and that it improves robustness, including a 100% increase in allowable support-surface height variation.

Significance. If the central claims hold, this is a useful contribution to model-based dynamic legged locomotion: it extends the authors' prior aSLIP line to include trunk rotation, uses a singularity-free quaternion formulation in both planning and control, provides a reproducible compliant quadruped platform with open-source CAD, and offers an external baseline (springs disengaged on the same robot) rather than only comparing against the authors' previous model. The simulation comparison against dual-aSLIP is a genuine incremental validation, and the statement that all control tasks share the same gains is a strength. The main significance risk is that the headline experimental gains rest on very thin evidence: single trials, no error bars, and a froggy-jump comparison performed with an asymmetric spring configuration.

major comments (4)
  1. [Section VIII, Table 2 and the 'Robustness' subsection] The headline quantitative claims are not established as reproducible effects. Table 2 reports single maximum values for pronking distance (40 cm vs 50 cm), hop-turn yaw (125 vs 145 degrees), and froggy-jump distance (40 cm vs 50 cm), and the robustness section reports 5 cm vs 10 cm allowable height variation, all without repeated trials, ranges, medians, or success rates. As written, the abstract's 'at least 25%, 15%, 25%, 100%' statements are maxima from single trials and could reflect trial-to-trial variation or favorable conditions rather than a systematic benefit of parallel elasticity. Please provide repeated trials with spread or success counts for each extreme-motion comparison.
  2. [Section VIII, Remark 4 (Enhanced froggy jumping)] The froggy-jumping gain, one of the three explosive-motion percentages, is confounded. Remark 4 states that for jumps longer than 40 cm the RR calf spring is removed, so the 50 cm compliant result is achieved by an asymmetric compliant robot, not by the symmetric TD-aSLIP configuration used in planning. The Discussion also notes that unidirectional springs require extra flight-phase torque, so removing the RR spring may improve the result by removing a disturbance rather than by demonstrating the modeled parallel compliance. At minimum, the 25% froggy-jump gain in Table 2 should be separated from the symmetric-compliance claims, or the experiment should be repeated in a configuration that matches the planning model.
  3. [Section IV-B2 (Configuration-aware stiffness mapping)] The configuration-aware stiffness is obtained as a cubic polynomial fitted to sampled data (Fig. 4), but no fit error, cross-validation, or comparison against the direct Jacobian-based stiffness is reported. Since this polynomial is the mechanism by which 'configuration-aware' compliance enters the planner, its accuracy is load-bearing for the claim of improved model accuracy. In addition, Eq. (9) defines the scalar spring constant as the Euclidean norm of the diagonal of the equivalent stiffness matrix; this does not obviously equal the leg-direction stiffness used in Eq. (4), so the mapping needs a derivation or numerical validation.
  4. [Section I (Contribution 1) and Discussion ('Momentum-aware motion planning and control')] The paper states in Contribution 1 that the model captures 'the varying angular momentum' and uses this as part of its novelty relative to prior SLIP models, but the TD-aSLIP dynamics assume massless legs and only model trunk inertia, so the centroidal angular momentum is not captured. The Discussion explicitly acknowledges that this limits tracking, e.g., the rigid robot lands at about 70 degrees in a 90-degree hop-turn. This does not invalidate the compliant-vs-rigid comparison, but the contribution statement overstates what the model captures; the claim should be reformulated to 'trunk angular momentum' or the model should be extended with leg inertia.
minor comments (4)
  1. [Eq. (35)] The last row of the matrix T(Q) appears to be [-qz, qx, qw] as printed; for consistency with Eq. (34) and the quaternion derivative it should be [-qy, qx, qw].
  2. [Fig. 4] The caption refers to blue, green, and purple curves, but the figure appears in grayscale; please use distinguishable line styles or annotate the curves directly.
  3. [Section VIII (System setting)] The manual landmark-based distance measurement is described, but it is not stated whether measurements were taken by a single observer, whether images were processed automatically, or whether any inter-rater variability was considered; a brief statement would improve reproducibility.
  4. [Eq. (24)] The use of Euclidean distance with the min over Q and -Q is an acknowledged approximation to the quaternion distance; this is acceptable, but please state explicitly in the text that this is a heuristic rather than the geodesic metric, since it affects the cost landscape.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central claims are validated against an external spring-disengaged baseline, and the model equations are derived in this paper.

full rationale

The paper's derivation chain is self-contained. The TD-aSLIP dynamics (Eqs. 1-5) are derived explicitly, with spring forces defined by Eq. 4 and the configuration-aware stiffness mapping obtained by equating joint-space and Cartesian-space potential energies (Eqs. 6-9), then fitting a cubic polynomial to sampled workspace data (Section IV-B2). This calibration is an input to motion generation, not a fitted stand-in for the reported outcomes. The signature claims - pronking distance, hop-turn yaw, froggy jumping distance, and allowable terrain height variation - come from hardware comparisons on the same E-Go-V2 robot with springs engaged versus disengaged (Section VIII, Table 2), which is an external baseline rather than a quantity reconstructed from the model. Self-citations to Ding et al. (2024a,b) are contextual lineage references for the aSLIP family and the earlier E-Go platform; no uniqueness theorem or ansatz is imported from them to force the present results. The acknowledged limitations (Discussion: momentum-aware planning; Remark 4: RR calf spring removed for froggy jumps above 40 cm) are experimental-design confounds and modeling gaps, not circular reductions; they affect the reliability and interpretation of the headline percentages, but they do not make any prediction equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The model's predictive content relies on a fitted stiffness polynomial and manually selected cost weights, which are the main free parameters.

free parameters (5)
  • Stiffness polynomial coefficients = not reported
    Cubic polynomial fitted to samples from a workspace (Section IV-B2, Fig. 4) maps leg length to equivalent spring stiffness; coefficients not listed.
  • TO cost weights = not reported
    Weights in Eqs. (11) and (23) are hand-chosen; no values are given.
  • Waypoint slack variable xi = not reported
    Slack in Eq. (12) is 'turned by hand' to relax waypoint constraints.
  • Spring rest lengths = 0.38 m for hardware
    Chosen as a compromise between peak torque and energy cost (Section VII-B2).
  • Contact phase durations = not specified
    Time steps ts and tf are optimized with bounds; the specific bounds are not listed.
assumptions (5)
  • domain assumption Lumped mass, massless legs for TD-aSLIP and SRB models.
    Used in Eq. (1) and in the MPC model in Section VI-A; leg mass and its angular momentum are neglected.
  • domain assumption Small joint angle variations in the equivalent stiffness mapping.
    Potential energy comparison in Eqs. (6)-(8) assumes small delta q_j to justify a constant Jacobian per configuration.
  • domain assumption Unidirectional spring force.
    Spring force in Eq. (4) is max(l0 - |l|, 0), i.e., springs only push.
  • domain assumption Leg pairing symmetry.
    Front pair and rear pair share the same joint variations and Jacobian (Section IV-B1), which is an approximation for asymmetric motions.
  • ad hoc to paper Quaternion distance approximated by minimum Euclidean distance.
    Eq. (24) uses min(||Q1-Q2||, ||Q1+Q2||) instead of geodesic distance, acknowledged as non-ideal.

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Cite this review

Pith. "Pith review of Versatile, Robust, and Explosive Locomotion with Rigid and Articulated Compliant Quadrupeds." pith.science (2026). https://pith.science/paper/A34ROZ6D

@misc{pith2026250412854,
  author       = {Pith},
  title        = {Pith review of: Versatile, Robust, and Explosive Locomotion with Rigid and Articulated Compliant Quadrupeds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A34ROZ6D}},
  note         = {Machine review of arXiv:2504.12854}
}
read the original abstract

Achieving versatile and explosive motion with robustness against dynamic uncertainties is a challenging task. Introducing parallel compliance in quadrupedal design is deemed to enhance locomotion performance, which, however, makes the control task even harder. This work aims to address this challenge by proposing a general template model and establishing an efficient motion planning and control pipeline. To start, we propose a reduced-order template model-the dual-legged actuated spring-loaded inverted pendulum with trunk rotation-which explicitly models parallel compliance by decoupling spring effects from active motor actuation. With this template model, versatile acrobatic motions, such as pronking, froggy jumping, and hop-turn, are generated by a dual-layer trajectory optimization, where the singularity-free body rotation representation is taken into consideration. Integrated with a linear singularity-free tracking controller, enhanced quadrupedal locomotion is achieved. Comparisons with the existing template model reveal the improved accuracy and generalization of our model. Hardware experiments with a rigid quadruped and a newly designed compliant quadruped demonstrate that i) the template model enables generating versatile dynamic motion; ii) parallel elasticity enhances explosive motion. For example, the maximal pronking distance, hop-turn yaw angle, and froggy jumping distance increase at least by 25%, 15% and 25%, respectively; iii) parallel elasticity improves the robustness against dynamic uncertainties, including modelling errors and external disturbances. For example, the allowable support surface height variation increases by 100% for robust froggy jumping.

Figures

Figures reproduced from arXiv: 2504.12854 by the authors.

Figure 1
Figure 1. The compliant Go1 with parallel springs (we call it E-Go-V2) performs versatile and explosive motions. From top to bottom, the robot performs 1) 40 cm forward pronking, 2) 40 cm forward froggy jumping, 3) 145◦ clockwise hop-turn, and 4) robust froggy jumping from the uneven surface. At the bottom, the robot jumps from the non-coplanar surface (10 cm height variation) without knowing the terrain information in advanc… view at source ↗
Figure 2
Figure 2. Optimization-base motion planning and control for versatile, robust, and explosive quadrupedal locomotion. The singularity-free formulation is incorporated into trajectory optimization and compliant control. Methodology overview This section provides a first glance at the proposed approach, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The TD-aSLIP model with parallel elasticity, (a) temple model, and (b) quadrupedal motion mapping in the homing pose. TD-aSLIP dynamics For legged robots, we assume a lumped mass in the trunk center, with massless legs. Considering trunk rotation, we have the following stance dynamics for the TD-aSLIP model (see [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: The equivalent spring stiffness regarding leg length. The raw data (marked by the blue colour) is sampled from a pre-designed workspace. The green solid curve shows the fitness result, while the purple dash curve shows varying stiffness regarding the vertical height. T…
Figure 5
Figure 5. Figure 5: Contact sequence for (a) pronking and (b) froggy jumping . SLIP-driven motion optimization Versatile explosive motion can be generated by defining different contact sequences [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: 40 cm forward froggy jumping trajectory generated by the proposed TO scheme. ‘SLIP TO’ and ‘Quaternion TO’ separately represent the first layer TO with SLIP dynamics and the dual-layer singularity-free TO. The two dots on each curve separately mark the transition from …
Figure 7
Figure 7. Figure 7: Velocity profiles for 40 cm forward froggy jumping. ‘SLIP TO’ and ‘Quaternion TO’ separately represent the first layer TO with SLIP dynamics and the dual-layer singularity-free TO. The top and bottom rows separately plot the translational velocity and rotational veloci…
Figure 9
Figure 9. Figure 9: 40 cm forward pronking for rigid and articulated robots. ‘Rigid’, ‘Soft-constant’ and ‘Soft-variation’ separately denote the motion generated by the TD-aSLIP model without parallel compliance, with constant stiffness and with varying stiffness. From the top to bottom, …
Figure 10
Figure 10. Figure 10: Sagittal pronking motion with different spring settings. The first row demonstrates the x−z motion with different spring stiffness, of which the ‘homing stiffness’ is indicated by the legends. In this case, l 1 0 = l 2 0 = 0.32 m. The second row displays the motion wi…
Figure 13
Figure 13. Figure 13: Forward tracking errors (‘δx’) when jumping with SLIP models. ‘Landing error’ refers to the tracking error of the forward landing position and ‘CoM error’ refers to the tracking error of the forward CoM position when detecting landing [PITH_FULL_IMAGE:figures/full_fi…
Figure 14
Figure 14. Figure 14: The articulated compliant quadruped with the parallel springs: from Delft E-Go (left) to E-Go-V2 (right) [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 12
Figure 12. Figure 12: 40 cm pronking motion with different SLIP models. ‘Ref.’ and ‘Real’ separately denote the reference and measured trajectories. Note that the durations of two reference motions are different from each other. controlled with the torque controller in Section VI, sharing …
Figure 16
Figure 16. Figure 16: The engagement process for thigh spring (top) and calf spring (bottom). On the top, step 1 sets the spring’s rest length, while 2 and 3 separately denote the release and engagement configurations. On the bottom, 1 and 2 separately denote the release and engagement con…
Figure 17
Figure 17. Figure 17: 40 cm forward pronking motions under different template models. The top panel demonstrates the motion achieved with the dual-aSLIP model without considering body rotation, while the bottom panel is achieved with the TD-aSLIP model. In each snapshot, the red solid line…
Figure 18
Figure 18. Figure 18: Trajectory profiles for 40 cm forward pronking with the dual-aSLIP model and TD-aSLIP model. ‘0.636’ and ‘0.924’ separately mark the take-off and landing moments when using the TD-aSLIP model. Since the robot reached the kinematic limits before entering the flight pha…
Figure 20
Figure 20. Figure 20: Comparative explosive motion between the robot with/without parallel springs engaged. From top to bottom, we present the comparison motions for 50 cm forward pronking (top group), 90◦ clockwise hop-turn (middle group), 50 cm froggy jumping (bottom group). In each grou…
Figure 23
Figure 23. Figure 23: , in the stance phase, the rigid robot without springs engaged outputs the largest peak torque to push off the ground, that is, 25 Nm, in the FL calf joint. Although the smaller peak torque is found in the soft case, the compliant robot jumped higher, see the real hei…
Figure 24
Figure 24. Figure 24: 30 cm froggy jumping on a floating pad. The floating pad moves backwards after the robot takes off. The rigid robot (top panel) landed stably but fell behind the desired position. The compliant robot (bottom panel) landed at the desired location. The purple arrows mar…
Figure 25
Figure 25. Figure 25: The compliant quadruped with springs engaged performs agile pacing (top panel), trotting (middle panel) and trotting-running (bottom panel). The red lines mark the stance feet. The step periods for pacing, trotting and trotting-running separately are 0.3 s, 0.3 s and …

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