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

Sriram S. K. S. Narayanan; Umesh Vaidya

cs.ROApr 5, 2026

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

Authors: Sriram S. K. S. Narayanan, Umesh Vaidya

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

Abstract

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.

Figures & tables

Fig. 1: Single rigid-body (SRB) model. The floating base evolves under stance ground reaction forces $f_{i}$ applied at the contact points with moment arms $r_{i}$ , generating the net centroidal wrench that propels the robot forward.

Parameter	Value
Mass $m$ [kg]	12.75
Inertia $I_{B}$ [kg m 2 ]	$10^{-3}\!\times\!\begin{bmatrix}160&0.12&-16\\ 0.12&470&-0.03\\ -16&-0.03&520\end{bmatrix}$

TABLE I: SRB physical parameters for Unitree Go1

Fig. 2: RK-MPC locomotion stack. A hierarchical pipeline converts high-level velocity commands into joint-level torque/position commands. A state estimator provides the feedback state to the finite-state machine and the convex residual Koopman MPC, which generates optimal ground reaction forces. The resulting commands are tracked by the joint controller and executed on hardware.

Parameter	Trot	Crawl
$T$ [s]	0.45	1.10
$\beta$	0.5	0.75
$T_{\text{st}}$ [s]	0.225	0.825
$T_{\text{sw}}$ [s]	0.225	0.275
Freq. [Hz]	2.22	0.91
Phase bias (LF,LH,RF,RH)	[0,0.5,0.5,0]	[0,0.25,0.5,0.75]

TABLE II: Gait and footstep parameters.

Parameter name	Symbol	Value
Swing Cartesian P gain	$K^{\mathrm{sw}}_{p}$	$\mathrm{diag}(400,\,400,\,400)$
Swing Cartesian D gain	$K^{\mathrm{sw}}_{d}$	$\mathrm{diag}(10,\,10,\,10)$
Joint PD gains (swing)	$(k_{p}^{\mathrm{sw}},\,k_{d}^{\mathrm{sw}})$	$(3,\,2)$
Joint PD gains (stance)	$(k_{p}^{\mathrm{st}},\,k_{d}^{\mathrm{st}})$	$(0.8,\,0.8)$
Torque saturation limit [Nm]	$\tau_{\max}$	$50$

TABLE III: Torque control parameters.

Fig. 3: Residual Koopman modeling framework

Fig. 4: Dataset generation for Koopman residual learning. Left: body-velocity samples $(v_{x},v_{y})$ with marginal histograms, colored by yaw rate $\omega_{z}$ , illustrating excitation coverage. Middle: Gazebo simulation setup with randomized off-road terrain with per-episode friction $\mu$ . Right: representative episode time histories of base linear velocity $v$ , angular velocity $\omega$ , vertical contact forces $f_{z}$ , and trot gait phase.

Linear velocity (m/s)
State	SRB	EDMD (mono)	EDMD-SE(3)	Res-Koopman
$v_{x}$	$0.007\!\pm\!0.003$	$4.05\!\pm\!1.92$	$0.003\!\pm\!0.001$	$0.003\!\pm\!0.001$
$v_{y}$	$0.008\!\pm\!0.002$	$3.19\!\pm\!1.63$	$0.003\!\pm\!0.001$	$0.003\!\pm\!0.001$
$v_{z}$	$0.016\!\pm\!0.003$	$1.22\!\pm\!0.55$	$0.009\!\pm\!0.002$	$0.007\!\pm\!0.002$
Angular velocity (rad/s)
$\omega_{x}$	$0.091\!\pm\!0.018$	$3.16\!\pm\!1.91$	$0.070\!\pm\!0.022$	$0.064\!\pm\!0.017$

TABLE IV: Open-loop prediction RMSE (mean

\pm

std).

Fig. 5: Prediction performance: Comparison of the proposed residual Koopman model (orange) against a nonlinear SRB (blue dashed), EDMD with monomials (green dashed) and EDMD with SE(3) basis (purple dashed against a test trajectory (black). (a) prediction of

x-y

trajectory over 100 time steps. (b–c) Per-channel RMSE box plots for linear velocities

(v_{x},v_{y},v_{z})

and angular velocities

(\omega_{x},\omega_{y},\omega_{z})

, (d) Mean

SO(3)

geodesic attitude error of EDMD-

SE(3)

over multi-step rollout horizon, (e) Overall residual Koopman RMSE versus the number of training samples used and (f) Overall RMSE error against monomial degree sweep for the residual Koopman model.


$\displaystyle\min_{\{x_{k+i},\,u_{k+i}\}}\quad$	$\displaystyle\sum_{i=0}^{N-1}\Big((x_{k+i+1}-x_{k+i+1}^{\star})^{\top}Q(x_{k+i+1}-x_{k+i+1}^{\star})+u_{k+i}^{\top}R\,u_{k+i}\Big)$
	$\displaystyle\hskip 32.00002pt+(x_{k+N}-x_{k+N}^{\star})^{\top}Q_{f}(x_{k+N}-x_{k+N}^{\star})$	(37a)
s.t.	$\displaystyle x_{k+i+1}=A^{\mathrm{nom}}x_{k+i}+B^{\mathrm{nom}}u_{k+i}+C^{\mathrm{res}}(A^{\mathrm{res}}z_{k+i}+B^{\mathrm{res}}u_{k+i}),$	(37b)
	$\displaystyle A_{\mathrm{ineq}}(\sigma_{k+i})\,u_{k+i}\leq b_{\mathrm{ineq}}(\sigma_{k+i}),$	(37c)
	$\displaystyle z_{k}=\psi(e_{k}),\quad e_{k}=\big[(v_{k}-v_{k}^{\mathrm{nom}})^{\top}\;(\omega_{k}-\omega_{k}^{\mathrm{nom}})^{\top}\big]^{\top},$	(37d)

Table 10

Fig. 6: MPC tracking performance (sim): Comparison of the proposed RK-MPC framework (orange) against a nonlinear SRB-MPC (blue) and SE3-KMPC (purple) on a circular tracking task over 10 laps. (a)–(b) Executed base

x

–

y

trajectory of the SRB-MPC and the RK-MPC respectively. (c)–(d) Phase portraits for attitude stability where SE3-KMPC exhibits a failure mode with excursions away from the near-origin region. (e)–(f) Per-axis tracking RMSE distributions for linear velocities and angular rates across 10 laps of the circle. (g) MPC solve-time distributions (ms) for each controller. (h) Closed-loop

{SO}(3)

drift of the SE3-KMPC internal rotation state. (i)–(k) Tracking performance of RK-MPC for

(v_{x},v_{y},\omega_{z})

, showing close agreement with the commanded signals. (l) Cross-track error for RKMPC and SRB-MPC over time.

Linear velocity RMSE (m/s)
Metric	SRB-MPC	SE3-KMPC	RK-MPC
$v_{x}$	$0.0229\!\pm\!0.0201$	$0.0686\!\pm\!0.0949$	$0.0232\!\pm\!0.0201$
$v_{y}$	$0.0196\!\pm\!0.0167$	$0.0877\!\pm\!0.1237$	$0.0195\!\pm\!0.0165$
$v_{z}$	$0.0172\!\pm\!0.0126$	$0.1374\!\pm\!0.2109$	$0.0177\!\pm\!0.0129$
Angular-rate RMSE (rad/s)
$\omega_{x}$	$0.0910\!\pm\!0.0695$	$1.0646\!\pm\!2.1215$	$0.0925\!\pm\!0.0713$

TABLE V: Closed-loop tracking RMSE and MPC solve-time statistics on a 10-lap circular task in simulation.

Fig. 7: Experiment 1: Velocity tracking on flat ground. (a) Executed base

x

–

y

trajectory during randomized planar commands (start/end annotated). (b) Executed base

x

–

y

trajectory during smooth loop commands (start/end annotated). (c) Randomized segment: commanded (black dashed) and measured (solid)

(v_{x},\,v_{y},\,\omega_{z})

. (d) Loop segment: commanded (black dashed) and measured (solid)

(v_{x},\,v_{y},\,\omega_{z})

Fig. 8: Experiment 2: Debris traversal and push recovery. (a) Representative snapshots of blind locomotion over a debris field. (b)–(c) Attitude phase portraits comparing debris (orange) and flat ground (blue): roll versus $\omega_{x}$ and pitch versus $\omega_{y}$ . (d) External push trial with recovery during steady walking. (e) Measured planar velocities $(v_{x},v_{y})$ during the push sequence (push instants annotated). (f) Measured base angles (roll, pitch) during the push sequence (push instants annotated).

Fig. 9: Experiment 3: Gait generalization. RK-MPC trained on trot data is executed on hardware under both trot and crawl gait schedules. representative snapshots (a-b). and measured per-foot ground reaction forces $f_{x}$ , $f_{y}$ , and $f_{z}$ for trot (c) and crawl (d). Bottom: corresponding gait phase/contact schedule for each leg (FR, FL, RR, RL), showing consistency between stance phases and force generation.

Fig. 10: Experiment 4: Off-road generalization. (a)–(d) Representative snapshots of blind locomotion on grass, gravel, ice, and snow. (e)–(f) Attitude phase portraits (roll versus $\omega_{x}$ and pitch versus $\omega_{y}$ ) across terrains. (g) Ice: measured vertical ground-reaction forces $f_{z}$ approaching the stance limit during slip onset. (h) Snow: commanded (dashed) and measured (solid) planar velocities $(v_{x},v_{y})$ during a turning trial. (i)–(j) Grass and gravel: commanded (dashed) and measured (solid) $(v_{x},v_{y})$ time histories. (k) Ice contact schedule for the shown segment. (l) Snow: commanded (dashed) and measured (solid) yaw rate $\omega_{z}$ .

Model Predictive Control

Model Predictive Control optimizes future actions with a system model, repeatedly applying the first action as new information arrives.

See all 296 papers

DR-MPC: Fast and Feasible Dynamics-Relaxed Model-Predictive Control for Legged Locomotion
Sep 17, 2026Run Wang, Alapati Tuerxun, Shuo Liu +4Model Predictive Control Constrained Optimization
Department of Automation, Tsinghua University, Beijing, China · Department of Mechanical Engineering, Boston University, Brookline, MA, USA · School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore +1
Right Model, Right Time: Real-Time Cascaded-Fidelity MPC for Bipedal Walking
May 6, 2026Franek Stark, Felix Wiebe, Shubham Vyas +2Nonlinear MPC Whole-Body Control
Robotics Innovation Center at the German Research Center for Artificial Intelligence (DFKI), Bremen, Germany · University Bremen, Germany
RACER: Residual-Adaptive Closed-Loop Estimation for Sampling-Based Planning in Wheeled-Quadruped Racing
Oct 5, 2026Yuxiang Liu, Marla Eisman, Lizhi Yang +2Residual Dynamics Learning Sim-to-Real Transfer
University of California, Berkeley, CA, USA. · California Institute of Technology, Pasadena, CA, USA.

Sep 17, 2026J/K move · Enter open · S save

DR-MPC: Fast and Feasible Dynamics-Relaxed Model-Predictive Control for Legged Locomotion

Run Wang, Alapati Tuerxun, Shuo Liu, Wei Xiao+3

Department of Automation, Tsinghua University, Beijing, China · Department of Mechanical Engineering, Boston University, Brookline, MA, USA · School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore+1

Model Predictive Control Constrained Optimization Quadruped Locomotion

Open paper AskPDF arXiv

This paper presents dynamics-relaxed model predictive control (DR-MPC), a novel MPC formulation for legged locomotion, and a tailored interior-point method (IPM) solver. The formulation combines online optimization feasibility by construction with a contact-aware input parameterization. DR-MPC moves the dynamics equality and affine input constraints into quadratic penalties and retains only nonempty box constraints. The resulting box-constrained quadratic program (QP) has a block-arrow Hessian that enables the state and affine-output directions to be eliminated through a Schur complement. The solver factors only the reduced control system after swing-force elimination and contact-aligned move blocking. For the evaluated implementations using the same DR-MPC formulation, our method achieves median end-to-end MPC speedups of

16.0\times

over HPIPM and

4.4\times

over OSQP, with comparable locomotion performance in simulation. DR-MPC achieves a median onboard MPC end-to-end time of

4.4

ms and is validated on a Unitree Go1 quadruped. Open-source code will be made available after publication.

Abstract

Model Predictive Control

Explore similar work

DR-MPC: Fast and Feasible Dynamics-Relaxed Model-Predictive Control for Legged Locomotion

Right Model, Right Time: Real-Time Cascaded-Fidelity MPC for Bipedal Walking

RACER: Residual-Adaptive Closed-Loop Estimation for Sampling-Based Planning in Wheeled-Quadruped Racing