Source-linked AI summary
Wheel-loader V-Cycle Automation with Deep Koopman MPC
Armin Abdolmohammadi, Navid Mojahed, Dinesh Kumar, Bahram Ravani, Shima Nazari
TL;DR
Wheel-loader V-cycles require planning and control across coupled articulated motion and changing travel directions, while vehicle and terrain dynamics are difficult to capture analytically. The paper combines a jointly optimized geometric planner with direction-specific deep bilinear Koopman models for MPC; in simulation, all 616 MPC solves completed without failure or exceeding the 50-ms control period.
Problem
Wheel-loader V-cycles require long-horizon planning and predictive control across coupled articulated motion and changing travel directions, while analytical models face nonlinear vehicle–terrain interactions.
Source: §I · p. 1Method
A reduced-order planner jointly optimizes forward and reverse paths through a shared switchback configuration, while direction-specific deep bilinear Koopman models provide dynamics for real-time MPC.
Source: §I · p. 1Results
616 MPC problems were solved with zero solver failures, and no solve exceeded the 50-ms control period during the complete V-cycle.
Source: B. V-Cycle Planning and Closed-Loop… · p. 7Takeaways & Limitations
The simulation results support the potential of the framework as a computationally efficient approach to wheel-loader V-cycle automation.
Source: §VI · p. 7
Abstract
from arXiv · showhide
The repeated forward-reverse maneuvers performed by wheel loaders during earthmoving operations make them well suited for automation. However, the nonlinear dynamics of articulated vehicles and complex vehicle-terrain interactions limit the effectiveness of conventional model-based approaches. This paper presents a hierarchical framework that combines long-horizon geometric planning with data-driven predictive control for autonomous wheel-loader operation. A reduced-order articulated kinematic model is used to generate the maneuver geometry, where the forward and reverse trajectories are jointly optimized through a shared intermediate state. To capture the vehicle dynamics, two data-driven deep bilinear Koopman models are learned for the forward and reverse motions using data generated from high-fidelity simulations in Algoryx Dynamics. The learned Koopman representations are subsequently incorporated into a computationally efficient model predictive control (MPC) formulation for trajectory tracking. The resulting controller operates in real time within a 50-ms execution loop. High-fidelity simulation results demonstrate that the proposed end-to-end framework enables accurate and computationally efficient execution of wheel-loader V-cycle maneuvers, providing a promising approach toward autonomous operation of articulated heavy-duty machinery.
I. INTRODUCTION
Wheel-loader V-cycles require both long-horizon planning and predictive control across coupled articulated motion and changing travel directions. This work combines a jointly optimized geometric planner with direction-specific learned dynamics, using a reduced-order planning model and a high-fidelity simulation plant.
I. INTRODUCTION
- Earlier approaches did not jointly optimize the reverse and forward trajectories in one nonlinear program with the switchback configuration as a shared decision variable.The paper also notes that articulated-machine MPC relies predominantly on analytical predictive models, while learning often augments a policy or value function rather than replacing the state-transition model.Source: §I · p. 1
- Separate deep bilinear Koopman models for forward and reverse motions are trained with multi-step rollout prediction and selected for predictive accuracy and computational cost.The framework also evaluates real-time Koopman MPC in closed-loop execution of the complete forward–reverse–forward V-cycle in high-fidelity simulation.Source: §I · p. 1
- A nonlinear V-cycle planner jointly optimizes the reverse and forward pile-to-truck paths through a shared continuous switchback configuration.The reduced-order planner supplies maneuver geometry, while direction-specific deep bilinear Koopman models provide predictive dynamics for short-horizon MPC.Source: §I · p. 1
II. SYSTEM MODEL
- The high-fidelity target is a Komatsu WA475 in Algoryx simulation, including an articulated multibody chassis, powertrain, compliant tires, and tire–terrain interaction.Source: §II · p. 1
- The vehicle state includes front-axle pose, body-frame velocities, and articulation; forward and reverse are treated as separate modes, with perfect articulation-rate actuation assumed.Source: §II · p. 2
- For long-horizon planning, lateral velocity is omitted by assuming vy = 0, and longitudinal acceleration replaces the effort command to simplify trajectory optimization.Source: §II · p. 2
A. Deep Bilinear Koopman Model
The paper represents nonlinear vehicle dynamics with learned observables and bilinear lifted-state prediction. The lifted representation supports physical-state reconstruction and predictive control.
A. Deep Bilinear Koopman Model
- Koopman methods represent nonlinear dynamics through linear evolution of observables in a higher-dimensional function space, while a deep encoder learns those observables directly from data.Source: A. Deep Bilinear Koopman Model · pp. 2–3
- The lifted state concatenates the normalized physical state with learned observables, with normalization statistics computed from the training dataset.Source: A. Deep Bilinear Koopman Model · p. 3
- A linear projection recovers the normalized physical state from a predicted lifted state, which is then denormalized to physical units.Source: A. Deep Bilinear Koopman Model · p. 3
- The controlled lifted dynamics use a bilinear Koopman model whose matrices capture lifted-state interactions with longitudinal effort and articulation rate.Source: A. Deep Bilinear Koopman Model · p. 3
B. Loss function parametrization
The loss jointly trains the encoder and Koopman matrices to balance one-step consistency, recursive prediction, and regularized lifted dynamics. A spectral penalty discourages instability over sampled inputs but does not guarantee stability.
B. Loss function parametrization
- The encoder and Koopman matrices are jointly optimized from trajectory sequences for one-step consistency, recursive prediction, and stable, regularized lifted dynamics.They are optimized through backpropagation.Source: B. Loss function parametrization · p. 3
- Recursive rollout loss lifts only each sequence’s initial state, then propagates it without intermediate re-encoding.This trains multi-step prediction in lifted space.Source: B. Loss function parametrization · p. 3
- The spectral penalty discourages unstable lifted dynamics over a sampled operating-input region but provides no formal stability guarantee.The penalty is evaluated on a finite grid spanning that region.Source: B. Loss function parametrization · p. 3
- Parameter regularization is applied separately to the base model and bilinear coupling matrices, using Frobenius norms.The passage defines the Frobenius norm for the regularization terms.Source: B. Loss function parametrization · p. 3
C. Data Acquisition and Preprocessing
Forward and reverse datasets are collected independently from the simulated loader using randomized input excitation. Episode-level splits, local-frame sequences, and direction-specific normalization prepare the data for training and evaluation.
C. Data Acquisition and Preprocessing
- Each direction contributes 2,000 successful stop-to-stop episodes, with randomized piecewise-constant excitation over 8–15 seconds and states recorded at 0.05-second intervals.The effort command spans [-1, 1], and articulation rate spans [-0.30, 0.30] rad/s.Source: C. Data Acquisition and Preprocessing · p. 3
- Episodes are split 1,400/300/300 into training, validation, and test sets before extracting 5-second sequences at a 0.5-second stride.Each sequence has 101 state samples and 100 input samples and uses a local frame attached to its initial front-axle pose.Source: C. Data Acquisition and Preprocessing · p. 3
- Each sequence is translated and rotated so its initial front-axle position and heading are zero; body-relative velocities and articulation need no further transformation.Normalization statistics are calculated from each direction’s training split and applied unchanged to its validation and test sets; forward and reverse models retain independent normalization.Source: C. Data Acquisition and Preprocessing · pp. 3–4
D. Training Procedure
The learned forward and reverse models are trained independently with direction-specific hyperparameter selection. V-cycle planning combines an initial approach with a coupled pile-to-truck maneuver, while initial-to-pile planning imposes a stopped terminal state.
D. Training Procedure
- Forward and reverse models are trained independently, with hyperparameters selected separately using validation prediction error, terminal-position RMSE, and spectral growth.Validation runs every 10 epochs; training uses early stopping and is capped at 300 epochs.Source: D. Training Procedure · p. 4
III. V-CYCLE MOTION PLANNING
- The V-cycle plan treats the initial-to-pile approach as a forward optimization and the pile-to-truck transfer as a coupled reverse–forward problem with an optimized switchback.Excavation is treated separately, and the return maneuver follows the initial-to-pile procedure and is omitted as redundant.Source: §III · p. 4
A. Initial-to-Pile Planning
- Initial-to-pile planning targets a specified pose in front of the pile and enforces the terminal longitudinal velocity as a hard full-stop constraint.The dynamics and input constraints span the planning horizon, and the model is discretized using fourth-order Runge–Kutta integration.Source: A. Initial-to-Pile Planning · p. 4
B. Pile-to-Truck Dual-Stage Planning
Pile-to-truck planning jointly optimizes reverse and forward trajectories around a shared switchback configuration. Its cost penalizes motion and control deviations, including large articulation-rate commands near a stop.
B. Pile-to-Truck Dual-Stage Planning
- The reverse and forward stages share a jointly optimized switchback configuration, with travel direction imposed through signed longitudinal-velocity bounds.The coupled optimization spans the reverse and forward intervals.Source: B. Pile-to-Truck Dual-Stage Planning · pp. 4–5
- The stage cost penalizes articulation, speed error, control effort, and variation; a low-speed term discourages large articulation-rate commands near a stop.The low-speed regularization constant prevents singular behavior near zero longitudinal velocity.Source: B. Pile-to-Truck Dual-Stage Planning · p. 4
- The coupled pile-to-truck optimization starts from the measured state at the end of the initial-to-pile maneuver and is solved once for that movement phase.The resulting piecewise trajectory supplies the long-horizon reference for Koopman MPC.Source: B. Pile-to-Truck Dual-Stage Planning · p. 5
IV. BILINEAR KOOPMAN MODEL PREDICTIVE CONTROL
The long-horizon V-cycle trajectory is tracked with short-horizon Koopman MPC, using separate learned dynamics for forward and reverse motion.
IV. BILINEAR KOOPMAN MODEL PREDICTIVE CONTROL
- Stages 1 and 3 use the forward model, while Stage 2 uses the independently identified reverse model to track the planned trajectory.The direction superscript is omitted in the subsequent derivation for compactness.Source: §IV · p. 5
A. Frozen-State Linearization of Bilinear Model
The controller freezes the lifted state in bilinear terms to form an affine prediction over each MPC horizon, and builds a moving path reference in the current vehicle-local frame.
A. Frozen-State Linearization of Bilinear Model
- Freezing the lifted state at its current measured value makes the bilinear prediction affine over the MPC horizon.The effective input matrix is recomputed at each control update and held fixed over that prediction horizon.Source: A. Frozen-State Linearization of… · p. 5
B. Moving Path Reference
- The planned path and stage goal are transformed into the vehicle-local frame, and the running reference omits longitudinal and lateral velocity.The geometric planner supplies long-horizon guidance; the front-axle pose defines the local-frame origin.Source: B. Moving Path Reference · p. 5
- The moving reference uses a path anchor searched from the complete stage path initially, then restricts later searches to prevent backward jumps in arclength.Reference points are sampled ahead of the anchor; position, heading, and articulation are interpolated, with heading unwrapped before interpolation.Source: B. Moving Path Reference · p. 5
C. Path Following MPC Formulation
The MPC tracks a moving geometric reference and a fixed stage goal, optimizing normalized inputs while applying constraints and penalties to physical commands.
C. Path Following MPC Formulation
- The MPC tracking error compares decoded predicted physical states against the moving reference, while a separate terminal error uses the fixed stage goal.The goal is transformed into the current vehicle-local frame before each solve, and desired terminal velocities are zero because each stage ends at rest.Source: C. Path Following MPC Formulation · p. 5
- Normalized control inputs are optimization variables, but physical commands are recovered for control penalties and input constraints.The objective penalizes moving-reference error, terminal-goal error, control effort, and control variation; the lifted states are recursively substituted so only the input sequence is optimized.Source: C. Path Following MPC Formulation · pp. 5–6
- Fig. 3 compares full bilinear and fixed-state linearized predictions with ground truth across six physical states over a 5-s trajectory, marking the deployed 3-s horizon.Table I summarizes stage-specific Koopman MPC parameters.Source: Figure 3 · p. 6, Table I · p. 6
D. Stage Completion and Handover
Stage completion is determined from measured vehicle-state tolerances, and each handover starts from the current measured state; the framework evaluation spans model identification, planning, and closed-loop simulation.
D. Stage Completion and Handover
- A stage completes when measured position, heading, longitudinal-velocity, and articulation errors meet their allowable bounds; lateral velocity has no completion condition.The terminal cost provides only soft convergence, so completion is evaluated separately from the global vehicle state.Source: D. Stage Completion and Handover · p. 6
- At handover, the next stage starts from the current measured state, preventing residual tracking error from becoming an assumed initial condition.Stages 2 and 3 follow reverse and forward portions of the same dual-stage plan without replanning at the switchback.Source: D. Stage Completion and Handover · p. 6
- The controller updates its local frame, moving reference, and effective input matrix every 50 ms, applies only the first optimal input, and solves the MPC quadratic program with OSQP.The prediction horizon is 3.0 s with N = 60; input and articulation-rate bounds are specified for the deployed controller.Source: D. Stage Completion and Handover · p. 6
V. RESULTS AND DISCUSSION
- The framework is evaluated through Koopman model identification and selection, nonlinear V-cycle planning, and closed-loop bilinear Koopman MPC in high-fidelity simulation.Source: §V · p. 6
A. Deep Bilinear Koopman Model
The section evaluates direction-specific deep bilinear Koopman predictors and their use in a planned, closed-loop three-stage V-cycle. The complete maneuver met configured terminal and bound constraints, with MPC computation remaining within the 50-ms control period.
A. Deep Bilinear Koopman Model
- At the deployed 3-s horizon, the selected nz = 38 forward and reverse models averaged 24.0 and 27.2 ms per solve, while the nz = 70 forward model averaged 57.7 ms.The nz = 70 model exceeded the 50-ms control period in 89.5% of solves, motivating the compact nz = 38 selection for both directions.Source: A. Deep Bilinear Koopman Model · p. 7
- Over 300 held-out episodes per direction and a 3-s horizon, full bilinear-model RMSEs were lower than fixed-state approximation RMSEs for every reported forward and reverse state.For example, forward position RMSE was 0.128 m for the full model versus 1.327 m for the approximation; reverse position RMSE was 0.148 m versus 1.187 m.Source: A. Deep Bilinear Koopman Model · p. 7
B. V-Cycle Planning and Closed-Loop Koopman MPC Results
- Stages 2 and 3 were jointly optimized in 859.3 ms to determine the switchback and truck-approach path, and Stage 3 ended with 0.221 m position error and 6.325° heading error.The geometric reference is selected by current path arclength; the closed-loop truck approach used a less aggressive articulation profile than the plan’s ±35° bound.Source: B. V-Cycle Planning and Closed-Loop… · p. 7
- Across 616 MPC solves, all three stages reached their terminal vicinities within imposed bounds, with zero failures and no solve exceeding the 50-ms control period.Mean solution times remained near 25 ms; the maximum was 35.02 ms.Source: B. V-Cycle Planning and Closed-Loop… · p. 7
VI. CONCLUSION
- The framework combines a reduced-order articulated-vehicle planner with separate forward and reverse deep bilinear Koopman models in MPC, using high-fidelity Algoryx Dynamics simulation data.The conclusion identifies the approach as a potential foundation for data-driven predictive control in broader articulated construction-machinery tasks.Source: §VI · pp. 7–8