dual-envelope-mpc-vehicle-drift

name

dual-envelope-mpc-vehicle-drift

description

Dual-envelope constrained nonlinear MPC for autonomous vehicle drifting control. Methods for constructing stability envelopes in phase plane, model predictive control with envelope constraints, and handling bounded steering/yaw-moment control for distributed drive EVs. Triggers: vehicle drifting control, MPC envelope constraint, autonomous drifting, distributed drive EV, yaw moment control, saddle point stability, phase plane envelope.

Dual-Envelope Constrained NMPC for Vehicle Drifting

Model predictive control framework for autonomous drifting with stability envelope constraints.

Overview

Paper: "Dual-Envelope Constrained Nonlinear MPC for Distributed Drive Electric Vehicles Drifting Under Bounded Steering and Direct Yaw-Moment Control" (arXiv: 2604.07342v1, April 2026)

Key contribution: Extended dual envelope framework that accounts for control input coupling, enabling smoother drift convergence and improved tracking.

Vehicle Dynamics Model

Distributed Drive Configuration

• Independent torque control for each wheel
• Direct yaw moment generation via torque distribution
• Superior yaw control compared to conventional vehicles

Drift State Variables

Variable	Description
Slip angle (β)	Vehicle side slip angle
Yaw rate (r)	Rotational velocity around vertical axis
Velocity (v)	Longitudinal velocity
Steering angle (δ)	Front wheel steering

Tire Model

Nonlinear tire model captures:

• Force saturation at high slip angles
• Road adhesion coefficient effects
• Combined slip effects

Saddle Point Analysis

Phase Plane Construction

Build stability boundaries in (β, r) phase plane:

1. Identify drift equilibrium (saddle point)
2. Construct stability boundaries from equilibrium analysis
3. Account for control input effects on saddle location

Control Input Coupling Effects

Critical insight: Control inputs reshape phase plane:

• Steering angle shifts saddle point location
• Yaw moment modifies stability boundaries
• Open-loop envelopes may be invalid for closed-loop control

Saddle Point Coordinate Model

# Account for all relevant parameters
saddle_point = {
    'road_adhesion': μ,       # Road friction coefficient
    'velocity': v,            # Longitudinal velocity
    'steering': δ_f,          # Front wheel steering angle
    'yaw_moment': M_z,        # Additional yaw moment
}

Dual Envelope Framework

Outer Envelope

Purpose: Define recoverable set under bounded control inputs

Characteristics:

• States reachable from drift equilibrium
• Constrained by steering angle limits
• Constrained by yaw moment limits
• Defines maximum drift extent

Inner Envelope

Purpose: Characterize non-drifting stability region

Characteristics:

• States with unsaturated tire forces
• Normal driving stability region
• Transition boundary to drift regime

Envelope Construction

Phase Plane → Envelope Boundaries:
1. Analyze convergence tendency toward saddle points
2. Apply steering angle bounds
3. Apply yaw moment bounds
4. Compute reachable state sets
5. Define inner/outer envelope boundaries

NMPC Controller Design

Control Objective

Track reference trajectory while maintaining drift stability:

• Maintain desired drift angle
• Track velocity reference
• Control yaw rate
• Respect envelope boundaries

Constraint Implementation

# NMPC constraints
constraints = {
    'state_envelope': inner < β < outer,  # Slip angle bounds
    'yaw_envelope': r_min < r < r_max,    # Yaw rate bounds
    'steering_limit': |δ_f| < δ_max,      # Steering angle limit
    'yaw_moment_limit': |M_z| < M_z_max,  # Yaw moment limit
}

Optimization Problem

minimize: tracking_error + control_effort
subject to: envelope_constraints + input_limits + dynamics

Performance Results

Hardware-in-the-Loop Experiments

Compared NMPC with envelope constraints vs. unconstrained NMPC:

Metric	Improvement
Steady-state speed error	-33.07%
Steady-state sideslip error	-71.18%
Steady-state yaw rate error	-31.27%
Peak tracking error (friction mismatch)	-63.66%

Benefits

• Smaller convergence toward drift saddle point
• Reduced tracking errors across all state variables
• Robust to friction mismatch - Handles road condition changes

Implementation Guidance

Controller Tuning

Key parameters:

1. Envelope boundary margins
2. Control horizon length
3. Prediction step size
4. Weighting matrices for cost function

Real-time Considerations

• Optimize solver for real-time execution
• Use warm-starting for successive solves
• Pre-compute envelope boundaries offline

Reference

• Paper: arXiv:2604.07342v1
• PDF: https://arxiv.org/pdf/2604.07342v1
• Category: eess.SY (Systems and Control)