Integrated Vehicle Dynamics MPC

基于车辆动力学的轨迹跟踪控制 Trajectory Tracking Control Based on Vehicle Dynamics

将轨迹跟踪、车辆动力学、轮胎摩擦极限与连续精确 Corridor 统一纳入实时优化，在安全边界内释放车辆极限性能。 Bring trajectory tracking, vehicle dynamics, tire-friction limits, and exact continuous corridors together in one real-time optimization to unlock the vehicle's limit performance within safe boundaries.

本页内容On this page 01 行业挑战01 Challenge 02 MPC 解决思路02 MPC Approach 03 动力学建模03 Dynamics 04 技术收益04 Benefits

01 Industry Challenge

横纵分离控制难以兼顾物理极限与精确安全边界Decoupled Control Cannot Fully Balance Physical Limits and Exact Safety Boundaries

传统方案大多采用“横纵分离”架构，例如横向 LQR/MPC 配合纵向 PID/MPC，并将上层规划下发的复杂几何走廊进行粗粒度近似。这种架构面临三重瓶颈：

1. 物理极限无法充分释放：大角度转向并同时加减速时，轴荷转移会改变轮胎侧向能力。横向控制器无法感知纵向驱动或制动动作，容易突破摩擦椭圆并引发打滑。
2. 横纵耦合导致约束匹配与跟踪失真：横纵向的动态耦合不仅会导致横向跟踪参考位置发生错位，更会导致避障点的匹配错位。这使得静态简化的避障约束（如点半空间）无法真实反映 Corridor 的真实物理边界要求，在弯道和贴边工况下近似失真，留下实际越界的碰撞风险。
3. 跨车型与全速域标定繁琐：分离控制高度依赖按速度分段的增益表。从低速泊车到高速巡航，再到不同车型移植，均需大量人工标定与调校。

Conventional systems separate lateral LQR/MPC from longitudinal PID/MPC and coarsely approximate complex geometric corridors. This creates three fundamental bottlenecks:

1. Physical limits cannot be fully exploited: When large steering angles coincide with acceleration or braking, axle-load transfer changes the lateral capability of the tires. A lateral controller that cannot sense drive or brake actions easily exceeds the friction ellipse and induces skidding.
2. Lateral-longitudinal coupling distorts constraint matching and tracking: Dynamic coupling between lateral and longitudinal motion misaligns not only the lateral tracking reference position but also the matching of obstacle-avoidance points. Statically simplified avoidance constraints such as point half-spaces therefore cannot represent the true physical boundary of the corridor; the approximation breaks down on curves and close to boundaries, leaving a real risk of crossing the boundary and colliding.
3. Tedious calibration across platforms and the full speed range: Decoupled control relies heavily on speed-segmented gain tables. From low-speed parking to high-speed cruising, and when porting to a different vehicle platform, extensive manual calibration and tuning are required.

Exact Corridor Control

连续走廊内的耦合轨迹优化Coupled Optimization in a Continuous Corridor

Exact

Corridor

Coupled

横纵动力学Dynamics

ms-level

实时求解Solving

02MPC Approach

在同一优化问题中统一动力学、附着力与可行驶空间Unify Dynamics, Tire Grip, and Road Space in One Optimization Problem

控制器实时输入车辆质心侧偏角、横摆角速度、纵横向速度及加速度，以车辆动力学为统一预测模型，并将时空轨迹跟踪、轮胎摩擦椭圆极限与连续精确 Corridor 边界约束同时纳入优化问题。

借助 OPTIMake 专门设计的路径跟踪接口（Squared Distance to Path）与走廊避障接口（Point to Boundary），控制器能在每个控制周期内，自适应精准匹配动态避障点与真实边界，无需对通道边界进行粗暴的静态简化，即可自动求解既精确跟踪、又不越界，且最大化利用轮胎附着力的转向、驱动与制动指令。

The controller takes the sideslip angle at the center of mass, yaw rate, longitudinal and lateral velocity, and acceleration in real time, uses vehicle dynamics as the unified prediction model, and brings spatio-temporal trajectory tracking, the tire-friction ellipse, and exact continuous corridor boundaries into the same optimization problem.

With the path-tracking interface (Squared Distance to Path) and corridor avoidance interface (Point to Boundary) designed specifically for OPTIMake, the controller adaptively and precisely matches dynamic avoidance points to the true boundaries in every control cycle. Without any crude static simplification of the corridor boundary, it automatically solves for steering, drive, and braking commands that track accurately, stay inside the boundary, and make the most of available tire grip.

Unified Prediction Model

实时车辆状态Vehicle states

β, r, v_x, v_y, a_x, a_y

↓

横纵耦合动力学 MPCCoupled Vehicle-Dynamics MPC

Friction ellipse + Exact corridor

↓

转向Steer驱动Drive制动Brake

03Flexible Dynamics Modeling

从单一车辆模型到多构型动力学控制From a Single Vehicle Model to Multi-Configuration Dynamic Control

轨迹跟踪控制不仅要精确跟踪参考路径，更要回答“不同运载构型如何在真实动力学边界内稳定执行”。

Trajectory tracking control must not only track a reference path precisely, but also determine how each transport configuration executes it within real dynamic limits.

面对不同运载构型，转向机构、车身铰接关系、推进方式及执行器约束均存在显著差异。OPTIMake 通过符号化模型与自定义动力学约束，将不同构型的运动学、动力学、执行器边界及耦合关系直接纳入统一控制优化框架，无需为每一种构型重新设计一套控制求解器。

Steering mechanisms, articulation, propulsion, and actuator limits vary substantially across transport configurations. With symbolic models and custom dynamic constraints, OPTIMake places each system's kinematics, dynamics, actuator bounds, and coupling relationships in one control-optimization framework—without designing a different control solver for every platform.

Unified Motion Model

EnvironmentGeometry + Boundaries

DynamicsStates + Actuators

OptimizationOne Solver

↓

High-Performance Motion Control

Flexible Modeling

不同构型快速建模Rapid modeling across configurations

Physics-aware

真实动力学直接进入优化Physical dynamics in the optimization

Fast Iteration

模型、参数与约束快速调整Rapid model and constraint iteration

One Solver

统一求解内核跨车型复用One solver core across platforms

Beyond Fixed Vehicle Models

动力学模型由实际系统定义Dynamics Defined by the Physical System

OPTIMake 不限定车辆运载构型，也不预设固定动力学模型。工程师可根据实际受控系统，自主定义状态量、控制输入、动力学方程、物理约束及其耦合关系。无论是不同车辆构型，还是其他复杂动态系统，均可复用同一套建模、求解与代码生成流程。

OPTIMake neither restricts the vehicle configuration nor presets a fixed dynamics model. Engineers define states, control inputs, dynamic equations, physical constraints, and their coupling relationships according to the actual controlled system. Different vehicle configurations and other complex dynamic systems all reuse the same modeling, solving, and code-generation workflow.

典型构型Representative Configurations

转向角 / 角速度Steering angle / rate 曲率连续性Curvature continuity 纵横向加速度Longitudinal / lateral acceleration 动态安全距离Dynamic clearance

01 · Single Vehicle

单车：前轮 / 后轮转向Single Vehicle: Front / Rear Steering

对于四轮转向的车辆控制构型，无论采用前后轮同相还是反相协同等模式，OPTIMake 都能以统一动力学约束进行描述，将前后轮转角、转向速率与车身状态纳入同一优化问题，精细协调各轮控制量，实现连续、稳定且可执行的车辆运动。

For four-wheel-steering vehicle configurations, whether the front and rear wheels work in phase or counter-phase, OPTIMake describes them with unified dynamic constraints, bringing front and rear steering angles, steering rates, and body states into the same optimization problem to finely coordinate the control commands of each wheel for continuous, stable, and executable vehicle motion.

铰接角 / 角速度Articulation angle / rate 联合运动学Coupled kinematics 挂车扫掠区域Trailer swept area 最小转弯半径Minimum turning radius

02 · Articulated System

拖挂系统：牵引车 + 挂车Articulated System: Tractor + Trailer

牵引车转向会通过铰接机构传递至挂车，使挂车产生独立的横摆与侧向运动。OPTIMake 将牵引车—挂车的耦合动力学直接作为控制约束，同步考虑挂车扫掠空间、铰接角与组合体运动学，在统一预测中求解牵引车与挂车的控制量。

Steering propagates through the articulation and produces independent trailer yaw and lateral motion. OPTIMake includes tractor-trailer coupling directly in the optimization, accounting for swept area, articulation angle, and combined-system kinematics at once.

四轮转角 / 速率Four-wheel angles / rates 车轮瞬时运动学Instantaneous kinematics 轮胎侧偏 / 速度Tire slip / velocity 完整车身避障Full-body avoidance

03 · Independent Steering

四轮独立转向底盘Four-Wheel Independent Steering

四轮独立转向底盘包含四个车轮转角及其运动耦合关系，可进行横移、小半径转向和复杂姿态调整。OPTIMake 将四轮转向机构与轮胎运动约束统一纳入优化，充分利用底盘自由度，协同求解高精度、可执行的控制量与运动轨迹。

A four-wheel independent steering chassis has four wheel-steering angles and their motion coupling relationships, allowing lateral translation, small-radius turns, and complex pose adjustments. OPTIMake brings the four-wheel steering mechanism and tire motion constraints into one optimization, fully exploiting the chassis degrees of freedom to jointly solve for high-accuracy, executable control commands and motion trajectories.

纵横向速度 / 横摆Surge / sway / yaw rate 推力 / 喷嘴偏转边界Thrust / nozzle deflection 船体与码头边界Hull / dock boundaries 终端姿态与状态Terminal pose / state

04 · Marine Craft

游艇 / 小型船舶Yacht / Small Marine Craft

船舶在低速操纵时易受水动力、风浪流和执行器响应特性的影响。OPTIMake 基于扰动预测建立统一动力学模型，并通过高频实时优化动态分配推进器推力与矢量喷嘴角度，使控制系统能够快速抵消环境扰动，实现精准、平稳的轨迹与姿态控制。

During low-speed maneuvering, vessels are easily affected by hydrodynamics, wind, waves, currents, and actuator response characteristics. OPTIMake builds a unified dynamics model based on disturbance prediction and dynamically allocates thruster force and vector-nozzle angles through high-frequency real-time optimization, letting the control system quickly counteract environmental disturbances for precise, smooth trajectory and attitude control.

＋ Custom Dynamics

从计算级安全到低算力芯片量产落地From Computation-Level Safety to Mass Production on Low-Compute Chips

01 / SAFETY

底层自适应计算级安全，真正“不越界”Adaptive Computation-Level Safety That Truly Stays In Bounds

依托专用的 Corridor 避障接口，算法能够自适应匹配车身关键点与避障约束，彻底解决避障匹配错位问题。直接严密求解真实边界，在狭窄通道、高速避障和紧凑贴边时，车辆可以精细利用合法路面，既不盲目保守，也不突破安全底线。

With a dedicated corridor avoidance interface, the algorithm adaptively matches key points on the vehicle body to the avoidance constraints, completely resolving avoidance-matching misalignment. By solving the true boundaries directly and rigorously, the vehicle can make precise use of the legal road surface in narrow passages, high-speed avoidance, and close-boundary driving—neither blindly conservative nor crossing the safety line.

02 / PERFORMANCE

释放动力学能力，实现“高精跟踪 + 高稳定性”High Precision and Stability at the Physical Limit

专用的路径跟踪接口消除了路径与速度耦合带来的跟踪参考误差。控制器实时预判加减速对轮胎侧向附着力的影响，在毫秒级内最优分配转向与加减速指令。确保了车辆在动力学可行域内以最优位姿跟踪轨迹，实现轮胎附着力与道路空间利用率的双重最大化。

The dedicated path-tracking interface eliminates the tracking reference error caused by path-speed coupling. The controller predicts in real time how acceleration and braking affect lateral tire grip and optimally allocates steering and acceleration commands within milliseconds, so the vehicle tracks the trajectory with the best possible pose inside its dynamically feasible region, maximizing both tire adhesion and road-space utilization.

03 / ADAPTATION

全速域自适应，告别海量参数标定Full-Speed Adaptation Without Massive Calibration

统一物理模型天然适应不同车速和工况。新车型只需输入轴距、转动惯量、轮胎侧向刚度等物理参数，即可完成绝大部分移植，将数周标定缩短至天级。

A unified physical model adapts naturally across speeds and conditions. Entering wheelbase, inertia, and tire stiffness completes most vehicle migration, reducing weeks of tuning to days.

04 / DEPLOYMENT

低算力 MCU 运行高维复杂约束High-Dimensional Complex Constraints on Low-Compute MCUs

OPTIMake 通过数值结构优化与代码生成，将融合耦合动力学和精确 Corridor 的高维 MPC 求解时间压缩至毫秒级，使其可下沉到低算力车规芯片量产，同时释放高性能智驾芯片的安全控制算力。

Through numerical structure optimization and code generation, OPTIMake compresses the solve time of high-dimensional MPC that fuses coupled dynamics with exact corridors to the millisecond level, so it can move down to low-compute automotive-grade chips for mass production while freeing up safety-control compute on high-performance autonomous-driving chips.

让复杂动力学模型落地 Turn Complex Dynamic Models into Production-Ready Real-Time Control

了解 OPTIMake 如何帮助你将复杂动力学模型落地并实现实时控制 Learn how OPTIMake helps you bring complex dynamic models into production with real-time control

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