迈向机器人自主：运动规划与控制在现实任务中的关键

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《机器人运动规划与控制》是一份由Jean-Paul Laumond编辑的报告，出自法国国家科研中心LAAS（Laboratoire d'Analyse et d'Architecture des Systèmes）的研究成果，LAAS报告编号为97438。这份报告最初作为Springer出版的《控制与信息科学讲座笔记》系列中的第229卷，ISBN 3-540-76219-1，于1998年发行，共343页。报告的核心关注点在于探讨机器人如何在现实世界中通过决定执行何种动作来完成任务。当前的工业机器人编程系统在运动规划方面的能力仍然相当有限，这限制了它们在复杂环境下的自主性，如太空探索、深海作业、危险环境干预以及服务机器人等领域。为了实现这些应用中的必要自主性，运动规划成为了关键要素之一。此外，它对于帮助设计工作单元的软件模拟机器人执行特定任务时避免碰撞路径也是基础性问题。机器人运动规划与控制是一个跨学科领域，其研究历史可以追溯到20世纪60年代计算机控制机器人初期的发展阶段。早期的机器人运动规划主要关注机器人的轨迹规划，涉及到数学、控制系统理论、机械工程、人工智能等多个学科的融合。随着技术的进步，研究人员不断开发更为复杂和精细的算法，如路径规划算法（如A*搜索、Dijkstra算法）、势场法、模型预测控制等，以提高机器人在动态环境中的灵活性和适应性。报告中可能深入探讨了以下内容： 1. **运动规划的基本概念**：介绍运动规划的定义，包括静态和动态规划，以及它们在解决机器人路径寻找问题中的作用。 2. **搜索算法**：详细讨论各种搜索算法在机器人路径规划中的应用，比如启发式搜索策略和最优路径求解方法。 3. **环境感知与避障**：阐述机器人如何利用传感器数据进行环境建模，以及如何构建有效的避障机制。 4. **控制理论**：分析如何将规划出的运动转换为实际的控制器指令，确保机器人按照预期执行任务。 5. **仿真与实验**：描述如何通过模拟环境测试和优化运动规划策略，以及实验验证其性能。 6. **未来趋势与挑战**：讨论当前研究领域的热点问题，如在线规划、实时性要求、多机器人协作等，以及面临的理论和实际挑战。总结来说，《机器人运动规划与控制》报告为读者提供了深入理解机器人在物理世界中实现自主任务的关键技术，强调了跨学科合作在推动这一领域发展中的重要作用。

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Guidelines in Nonholonomic Motion Planning for Mobile Robots 5

minimum turning radius. Simple computation shows that the dynamic model

of the car is:







˙x

˙y

˙w













w cos ζ cos θ

w cos ζ sin θ

w sin ζ































(3)

with |u

| ≤ u

1,max

and |u

| ≤ u

2,max

. This 5-dimensional system is not small-

time controllable from everywhere.

A ﬁrst simpliﬁcation consists in considering w as a control; it gives a 4-

dimensional system:







˙x

˙y













cos ζ cos θ

cos ζ sin θ

sin ζ







w +













(4)

This new system is symmetric without drift; applying the LARC condition

shows that it is small-time controllable from everywhere. Notice that w should

be C

. Up to some coordinate changes, we may show that this system is equiv-

alent to the kinematic model of a two-driving wheel mobile robot pulling a

“trailer” which is the rear axle of the car (see below). The mechanical con-

straint |ζ| ≤ ζ

max

≤

appears as an “obstacle” in R

× (S

)

Let us assume that we do not care about the direction of the front wheels.

We may still simplify the model. By setting v = w cos ζ and ω = w sin ζ we get

a 3-dimensionated control system:





˙x

˙y









cos θ

sin θ





v +









ω (5)

By construction v and ω are C

and their values are bounded. This system

looks like the kinematic model of the two-driving wheel mobile robot. The

main diﬀerence lies on the admissible control domains. Here the constraints

on v and ω are no longer independent. Indeed, by setting w

max

√

2 and

max

we get: 0 ≤ |ω| ≤ |v| ≤ 1. This means that the admissible control

domain is no longer convex. It remains symmetric; we can still apply the LARC

condition to prove that this system is small-time controllable from everywhere.

The main diﬀerence with the two-driving wheel mobile robot is that the feasible

paths of the car should have a curvature lesser than 1.

A last simpliﬁcation consists in putting |v| ≡ 1 and even v ≡ 1; by ref-

erence to the work in [65] and [22] on the shortest paths in the plane with

6 J.P. Laumond, S. Sekhavat and F. Lamiraux

bounded curvature such systems will be called Reeds&Shepp’s car and Dubins’

car respectively (see Sou`eres–Boissonnat’s chapter for an overview of recent

results on shortest paths for car-like robots). The admissible control domain of

Reeds&Shepp’s car is symmetric; LARC condition shows that it is small-time

controllable from everywhere

. Dubins’ car is a system with drift; it is locally

controllable but not small-time controllable from everywhere; for instance, to

go from (0, 0, 0) to (1−cos , sin , 0) with Dubins car takes at least 2π − unity

of time.

The diﬀerence between the small-time local controllability of the car of

Reeds & Shepp and the local controllability of Dubins’ car may be illustrated

geometrically. Figure 1 shows the accessibility surfaces in R

×S

of both sys-

tems for a ﬁxed length of the shortest paths. Such surfaces have been computed

from the synthesis of the shortest paths for these systems (see [76,51,15] and

Sou`eres–Boissonnat’s chapter). In the case of Reeds&Shepp’s car, the surface

encloses a neighborhood of the origin; in the case of Dubins’ car the surface is

not connected and it does not enclose any neighborhood of the origin.

2.3 Kinematic model of mobile robots with trailers

Let us now introduce the mobile robot with trailers which has been the canoni-

cal example of the work in nonholonomic motion planning; it will be the leading

thread of the rest of the presentation.

Figure 2 (left) shows a two-driving wheel mobile robot pulling two trailers;

each trailer is hooked up at the middle point of the rear wheels of the previous

one. The distance between the reference points of the trailers is assumed to be

1. The kinematic model is deﬁned by the following control system (see [47]) :

X = f

(X)v + f

(X)ω (6)

with

X = (x, y, θ, ϕ

, ϕ

)

(X) = (cos θ, sin θ, 0, −sin ϕ

, sin ϕ

− cos ϕ

sin ϕ

)

and

(X) = (0, 0, 1, 1, 0)

Note that the ﬁrst body can be viewed as the front wheels of a car; the

system then appears as modeling a car-like robot pulling a trailer.

After noticing that [f

, [f

, f

]] = f

, one may check that the family com-

posed of {f

, f

, [f

, f

], [f

, [f

, f

]], [f

, [f

, f

]]]} spans the tangent space

at every point in R

× (S

)

verifying ϕ

6≡

(regular points). The family

, f

, [f

, f

], [f

, [f

, f

]], [f

, [f

, f

]]]]} spans the tangent space else-

where (i.e., at singular points). Thanks to the LARC, we conclude that the

A geometric proof appears in [44].

Guidelines in Nonholonomic Motion Planning for Mobile Robots 9

Reeds & SheppCar-likeTwo-driving wheels Dubins

Fig. 3. Kinematic mobile robot models: four types of control domains.

Remark 1: Due to the constraint |ω| < |v|, the admissible paths for the car-like,

Reeds&Shepp’s and Dubins’ robots have their curvature upper bounded by 1

everywhere. As a converse any curve with curvature upper bounded by 1 is an

admissible path (i.e., it is possible to compute an admissible trajectory from

it).

Remark 2: This geometric constraint can be taken into account by consid-

ering the four-dimensionated control system (4) with |ζ| ≤

; the inequality

constraint on the controls for the 3-dimensionated system is then transformed

into a geometric constraint on the state variable ζ. Therefore the original con-

trol constraint |ω| < |v| arising in system (5) can be addressed by applying

“obstacle” avoidance techniques to the system (4).

From paths to trajectories The goal of nonholonomic motion planning

is to provide collision-free admissible paths in the conﬁguration space of the

mobile robot system. Obstacle avoidance imposes a geometric point of view

that dominates the various approaches addressing the problem. The motion

planners compute paths which have to be transformed into trajectories.

In almost all applications, a black-box module allows to control directly the

linear and angular velocities of the mobile robot. Velocities and accelerations

are of course submitted bounds.

The more the kinematic model of the robot is simpliﬁed, the more the trans-

formation of the path into a trajectory should be elaborated. Let us consider

for instance an elementary path consisting of an arc of a circle followed by a

tangent straight line segment. Due to the discontinuity of the curvature of the

path at the tangent point, a two driving-wheel mobile robot should stop at

this point; the resulting motion is clearly not satisfactory. This critical point

may be overcome by “smoothing” the path before computing the trajectory.

For instance clothoids and involutes of a circle are curves that account for the

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