Dynamic Modeling and MPC for Locomotion of Tendon-Driven Soft Quadruped

Saumya Karan; Neerav Maram; Suraj Borate; Madhu Vadali

cs.ROFeb 18, 2026

Dynamic Modeling and MPC for Locomotion of Tendon-Driven Soft Quadruped

Authors: Saumya Karan, Neerav Maram, Suraj Borate, Madhu Vadali

Organizations: IIT Gandhinagar, Department of Electrical Engineering, Gujarat, India · Sreenidhi Institute of Science and Technology, Department of Electronics and Computer Engineering, Telangana, India

Abstract

SLOT (Soft Legged Omnidirectional Tetrapod), a tendon-driven soft quadruped robot with 3D-printed TPU legs, is presented to study physics-informed modeling and control of compliant legged locomotion using only four actuators. Each leg is modeled as a deformable continuum using discrete Cosserat rod theory, enabling the capture of large bending deformations, distributed elasticity, tendon actuation, and ground contact interactions. A modular whole-body modeling framework is introduced, in which compliant leg dynamics are represented through physically consistent reaction forces applied to a rigid torso, providing a scalable interface between continuum soft limbs and rigid-body locomotion dynamics. This formulation allows efficient whole-body simulation and real-time control without sacrificing physical fidelity. The proposed model is embedded into a convex model predictive control framework that optimizes ground reaction forces over a 0.495 s prediction horizon and maps them to tendon actuation through a physics-informed force-angle relationship. The resulting controller achieves asymptotic stability under diverse perturbations. The framework is experimentally validated on a physical prototype during crawling and walking gaits, achieving high accuracy with less than 5 mm RMSE in center of mass trajectories. These results demonstrate a generalizable approach for integrating continuum soft legs into model-based locomotion control, advancing scalable and reusable modeling and control methods for soft quadruped robots.

Figures & tables

Parameter	Value	Description
$L$	0.190 m	Total leg length
$N$	31	Number of nodes
$dx$	$L/(N-1)$	Segment length
$w$	0.02 m	Constant leg width
$h_{\text{base}}$	0.0135 m	Thickness at fixed base
$h_{\text{tip}}$	0.0035 m	Thickness at tip

TABLE I : Simulation and Material Parameters

Fig. 2 : Single-leg deformation under tendon actuation.

Fig. 3 : Force application from four Cosserat rod legs onto the rigid cuboid torso.

Fig. 4 : Model Predictive Control block diagram.

\bm{x}_{k}=[\phi,\theta,\psi,\,p_{x},p_{y},p_{z},\,\omega_{x},\omega_{y},\omega_{z},\,v_{x},v_{y},v_{z}]

Algorithm 1 ROS2-Based Hardware-in-the-Loop MPC Locomotion Pipeline for SLOT

Fig. 5 : Comparison of experimental and simulated vertical position $z(t)$ at four locations along the single leg—Node 0 (base), Node 10 (one-third length), Node 20 (two-thirds length), and Node 30 (tip).

Metric	Fixed Base	1/3 Length	2/3 Length	Tip
RMSE (mm)	5.20	5.72	4.00	0.89
MAE (mm)	3.85	3.90	2.60	0.71
NRMSE (%)	5.87	6.94	7.88	19.22
Avg Error (mm)	2.23	2.87	2.32	0.15
Error (%)	2.51	3.47	4.56	3.21
Accuracy (%)	97.49	96.53	95.44	96.79

TABLE II : Single-Leg Validation Error Metrics

Fig. 6 : Comparison of simulated and experimental center-of-mass (CoM) trajectories for the forward walking gait in the $x$ and $z$ directions.

Fig. 7 : SLOT walking in a straight line in the simulation environment.

Fig. 8 : Comparison of simulated versus experimental center-of-mass (CoM) trajectories for the crawl gait.

Gait	RMSE in $x_{\text{CoM}}$ (mm)	RMSE in $z_{\text{CoM}}$ (mm)
Forward Walk	4.0	4.8
Crawl	5.1	4.6

TABLE III : Quantitative Validation Metrics for Whole-Body Locomotion

Panel of Figure 13 — Fig. 9 : Experimental and simulated results of quadruped MPC for the walking gait. Leg 1: FL, Leg 2: RL, Leg 3: RR, Leg 4: RL.

Fig. 10 : SLOT hardware robot walking in a straight line under MPC control, demonstrating stable forward locomotion.

Panel of Figure 15 — Fig. 11 : Simulation results for the $60^{\circ}$ omnidirectional gait under MPC.

Perturbation	Max Cost	Mean Cost	Settling Time (s)
Baseline	4.9739	1.1665	14.553
Roll	4.9719	1.1659	14.553
Pitch	4.7920	1.1120	13.695
Height	6.2248	1.5397	14.553
Velocity	2.3827	0.1720	10.098
Combined	3.6959	0.7873	12.606

TABLE IV : Stability Metrics for Perturbations (Walk gait)

Fig. 12 : Combined MPC cost trajectories under perturbations for Walk gait. Baseline and Roll overlap; Noise exhibits non-settling oscillations.

Panel of Figure 18 — Fig. 13 : Dynamic Constraint Map (DCM) illustrating the feasibility status of the MPC solution.

Quadruped Locomotion

Quadruped locomotion studies how four-legged robots move efficiently, safely, and reliably across varied terrain and tasks.

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RK-MPC: Residual Koopman Model Predictive Control for Quadruped Locomotion in Offroad Environments

Sriram S. K. S. Narayanan, Umesh Vaidya

Department of Mechanical Engineering, Clemson University, Clemson, SC 29630, USA

Model Predictive Control Quadruped Locomotion Residual Dynamics Learning Koopman Operator Learning Terrain-Aware Robot Locomotion

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This paper presents Residual Koopman MPC (RK-MPC), a Koopman-based, data-driven model predictive control framework for quadruped locomotion that improves prediction fidelity while preserving real-time tractability. RK-MPC augments a nominal template model with a compact linear residual predictor learned from data in lifted coordinates, enabling systematic correction of model mismatch induced by contact variability and terrain disturbances with provable bounds on multi-step prediction error. The learned residual model is embedded within a convex quadratic-program MPC formulation, yielding a receding-horizon controller that runs onboard at 500 Hz and retains the structure and constraint-handling advantages of optimization-based control. We evaluate RK-MPC in both Gazebo simulation and Unitree Go1 hardware experiments, demonstrating reliable blind locomotion across contact disturbances, multiple gait schedules, and challenging off-road terrains including grass, gravel, snow, and ice. We further compare against Koopman/EDMD baselines using alternative observable dictionaries, including monomial and

SE(3)

-structured bases, and show that the residual correction improves multi-step prediction and closed-loop performance while reducing sensitivity to the choice of observables. Overall, RK-MPC provides a practical, hardware-validated pathway for data-driven predictive control of quadrupeds in unstructured environments. See https://sriram-2502.github.io/rk-mpc for implementation videos.

Abstract

Quadruped Locomotion

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