自动驾驶平台的轨迹跟踪与动力学控制研究

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"该文档是Alexander Liniger的硕士论文，主题是自动驾驶平台的自主漂移控制，由瑞士联邦理工学院（ETH Zurich）自动控制实验室的Nikolaos Kariotoglou教授和John Lygeros博士指导，完成于2012年12月。论文主要探讨了自动驾驶平台在轨迹跟踪和动力学控制方面的设计方法，涉及到硬件测试平台、车辆模型、轮胎模型以及PID路径跟踪控制器和模型预测轮廓控制等多个方面。" 在自动驾驶领域，这篇论文详细阐述了以下几个关键知识点： 1. **目标与背景** (Goals and Previous/Related Work): 论文首先介绍了研究目标，即解决自动驾驶平台的轨迹跟踪和动力学控制问题。同时，对已有的相关工作进行了回顾，为后续的分析和设计提供了基础。 2. **硬件部分** (Hardware): 讨论了现有的测试平台及其存在的问题。这包括对平台的结构、性能和限制的描述，为后续的模型建立和控制策略优化提供实际依据。 3. **车辆模型** (Drift Model): 阐述了自动驾驶平台的机械模型，分为平面内运动和平面外运动两个方面。这部分详细解析了车辆在不同运动状态下的行为，包括无滑移模型和滑移模型，以模拟真实道路条件下的行驶情况。 - **平面内运动** (In-Plane Motion): 主要关注车辆在前进、转向和制动时的行为。 - **平面外运动** (Out-of-Plane Motion): 涉及到车辆的侧滑和翻滚等复杂动态。 4. **轮胎模型** (Tire Models): 分析了无滑移模型和各种滑移模型，以更准确地描述轮胎与路面的相互作用，这对于动力学控制至关重要。 5. **非线性模型分析** (Nonlinear Model Analysis): 在理解车辆动态的基础上，对非线性模型进行了深入研究，这有助于开发更适应复杂环境的控制策略。 6. **PID路径跟踪控制器** (PID Path Tracking Controller): 提出了一种基于PID的转向状态反馈控制器，详细描述了控制合成、路径跟踪、实现方法以及控制器调整等步骤。还涵盖了错误计算、线性化点选择、控制器参数校准以及参考路径和速度剖面的设定。 7. **模型预测轮廓控制** (Model Predictive Contouring Control): 作为一种先进的控制策略，模型预测控制能够处理约束并优化长期性能，论文探讨了模型线性化、软约束应用等技术。通过这些内容，论文展示了如何将理论模型应用于实践，以实现自动驾驶平台的高效和安全控制。这些研究成果对于自动驾驶系统的设计和优化具有重要的理论和实际价值。

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Chapter 3

Drift Model

To successfully control the car while drifting, a mathematical model which can

represent the dNano car in all driving situations has to be derived. The deriva-

tion of this model is separated into multiple steps. First the generic car model

which can r e present drift is derived. The derivation has two main parts, the

mechanical model and the tir e model. In a next step the model is identiﬁed

such that it represents the used dNano car, and lastly the resulting model is

analyzed.

3.1 Mechanical Model

The mechanical model of a car can be derived by simplifying the real car to one

rigid body. Using this approximation most of the dynamics of a rea l c ar can

be captured and only the torsion of the car is neglected. The dynamics of the

rigid body can be separated into in-plane and out-of-plane motion. The in-plane

motion describes the movement of the car on a ﬂat surfac e (x and y directio n

as well as the yaw angle) and the out- of-plane motions are the movement in z

direction as well as the pitch and roll angles.

For drift and general mo tion control the in-plane motion is of interest. However

the o ut- of-plane motion can also be of interest because it can inﬂuence the in-

plane motion by changing the normal loads of the tires and by this the friction

force between the tire a nd the ground.

Nevertheless, the in-plane motion can be formulated without considering the

out-of-plane motion, because the normal load changes inﬂuence the tire model

and not directly the in-plane dynamics.

3.1.1 In-Plane Motion

The in-plane motion describes the r igid body moving on a ﬂat surface, where the

wheels a re described using gene ral forces. This forces generally have a lateral

and a longitudinal compone nt, relative to the wheel position.

The used dNano cars o nly have front wheel steering and only the back wheels

are actuated in longitudinal direction. Thus the front wheels have a steering

angle input which changes the orientation of the wheel relative to the car, but

there is no longitudinal force component. Or in other words the wheels are

10 3.1. Mechanical Model

by integrating the transformed velocities

X = v

cos(ϕ) − v

sin(ϕ) (3.6)

Y = v

sin(ϕ) + v

cos(ϕ) (3.7)

˙ϕ = ˙ϕ (3.8)

Where X and Y is the position in the initial frame and ϕ the orientation relative

to this frame.

The in-plane motion is fully described by the last six diﬀerential equations,

governing the three velocities, position and orientation of the car.

3.1.2 Out of Plane Motion

The out of plane motion is mainly of interest due to its inﬂuence on the normal

load of the tires, which aﬀect the friction forces of the in-plane mo del.

A general friction force is g iven by:

= µ

(3.9)

Thus the frictio n force is described by the friction force curve µ

multiplied with

the normal lo ad, F

If the car is not moving this is not of interest, because the no rmal load is constant

and can thus be included into the friction force curve to one friction law. On

the other hand, if the car is moving this load can change based on the dynamics

of the system.

In the following subsection the diﬀerent possible models for such lo ad changes

are discussed. The re are two possible ways to model the motion, ﬁrst using

dynamic models and secondly using alg ebraic models.

Dynamic Suspension Model

One out-of-plane motion which can be of interest is the susp e nsion dynamics

which is normally modeled as a dynamic spring damper sys tem, this can be

done in pitch and r oll dir ection.

The roll suspe nsion dynamics of the car are modeled as an inverted pendulum,

attached to a spring damper system, rota ting around a ﬁx axis. The move ment

around the axis moves the CG of the car and so changes the normal load of e ach

wheel. This model of roll can be used only when using a four-wheel model.

The suspension dynamics in pitch direction are mode led allowing the rigid body,

which repr esents the car, to move up and down in z direction and to rotate

around the CG. The body and the ground are connected using a spring damper

element at the position of each wheel. The modeling of such a suspension can

be found in [20].

Algebraic Load C hange

The second approach to model the load changes us es an algebraic model a nd

not a dynamic model. The approach uses a force and moment balance in pitch

Chapter 3. Drift Model 11

direction. The force and moment bala nc e leads to the following normal forces

f,N

mg − hmgµ

r,x

+ l

) − h(µ

f,y

sin(δ) + µ

r,x

(3.10)

r,N

= mg − F

f,N

(3.11)

where the normal forces aﬀect the force acting on the bicycle model by

i,j

= F

i,N

i = r, f j = x, y (3.12)

and h the height from the ground to the CG. The rest of the notations are used

as in Equation 3.3.

Thus the normal nor mal loads depend on the control inputs δ and µ

r,x

, as well

as on velocities, because the force laws µ

i,j

are velocity dependent. This re sults

in the fact that the normal loa ds can be inﬂuenced by the control inputs.

Discussion

The dynamic suspension model adds more information to the model and helps

to more accur ately de scribe the real car, but the pitch model alone adds four

states to the model. This might be useful in a simulation, however fo r control

it is problematic. Moreover, the suspensio n dynamics are not of interest for the

used c ar, as the CG is very low and the suspe nsion very stiﬀ. The algebraic load

changes, which would not lead to mor e states, but increa sing the complexity of

the model and additional even more non linearities are also neglected.

A model including algebraic load changes could be interesting for oﬄine trajec-

tory calculation, like in [11] and [19] where such models are successfully used in

trajectory optimization problems.

3.1.3 Summary

In summary an in-plane dyna mic bicy cle model is suﬃcient for the car used in

the project and helps keeping the needed states to a minimum. The out-of-

plane eﬀects can be neglected because the CG of the car is very low and the

suspension very stiﬀ.

3.2 Tire Mod els

The dominant part of the res ulting car model is the tire force laws. T he shape

of the force law is a determining factor for whether and when the car drifts and

is therefore the main focus of interest.

There are three types of models which are investigated in mor e depth. The

ﬁrst tire model is a no slip model which has been used until now for control in

the ORCA project. This model is investigated to compare the new drift model

with the existing model. The second model is a lateral slip model which can

represent drift. However the model neglects the spinning of the wheels. The

spinning, which is a relative velocity between the ground and the tire , aﬀects

the lateral force. To include the eﬀect of spinning, a combined slip model has

to be used, which is the third kind of model which is investigated.

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