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Steven M. LaValle

Department of Computer Science

Iowa State University

Ames, IA 50011

lavalle@cs.iastate.edu

James J. Kuffner, Jr.

Department of Mechano-Informatics

University of Tokyo

Bunkyo-ku, Tokyo, Japan

kuffner@jsk.t.u-tokyo.ac.jp

Randomized

Kinodynamic

Planning

Abstract

This paper presents the ﬁrst randomized approach to kinodynamic

planning (also known as trajectory planning or trajectory design).

The task is to determine control inputs to drive a robot from an initial

conﬁguration and velocity to a goal conﬁguration and velocity while

obeying physically based dynamical models and avoiding obstacles

in the robot’s environment. The authors consider generic systems

that express the nonlinear dynamics of a robot in terms of the robot’s

high-dimensional conﬁguration space. Kinodynamic planning is

treated as a motion-planning problem in a higher dimensional state

space that has both ﬁrst-order differential constraints and obstacle-

based global constraints. The state space serves the same role as the

conﬁguration space for basic path planning; however, standard ran-

domized path-planning techniques do not directly apply to planning

trajectories in the state space. The authors have developed a ran-

domized planning approach that is particularly tailored to trajectory

planning problems in high-dimensional state spaces. The basis for

this approach is the construction of rapidly exploring random trees,

which offer beneﬁts that are similar to those obtained by success-

ful randomized holonomic planning methods but apply to a much

broader class of problems. Theoretical analysis of the algorithm

is given. Experimental results are presented for an implementation

that computes trajectories for hovercrafts and satellites in cluttered

environments, resulting in state spaces of up to 12 dimensions.

KEY WORDS—motion planning, trajectory planning, non-

holonomic planning, algorithms, collision avoidance

1. Introduction

There is a strong need for a general purpose, efﬁcient plan-

ning technique that determines control inputs to drive a robot

The International Journal of Robotics Research

Vol. 20, No. 5, May 2001, pp. 378-400,

from an initial conﬁguration and velocity to a goal conﬁgu-

ration and velocity while obeying physically based dynami-

cal models and avoiding obstacles in the robot’s environment

(see Fig. 1). In other words, a fundamental task is to design

a feasible open-loop trajectory that satisﬁes both global ob-

stacle constraints and local differential constraints. We use

the words kinodynamic planning, introduced in Donald et al.

(1993), to refer to such problems. (In nonlinear control liter-

ature, kinodynamic planning for underactuated systems is en-

compassed by the deﬁnition of nonholonomic planning. Us-

ing control-theoretic terminology, we characterize our work

as open-loop trajectory design for nonlinear systems with drift

and nonconvex state constraints. Other terms include trajec-

tory planning and trajectory design.) These solutions would

be valuable in a wide variety of applications. In robotics, a

nominal trajectory can be designed for systems such as mo-

bile robots, manipulators, space robots, underwater robots,

helicopters, and humanoids. This trajectory can be used to

evaluate a robot design in simulation, or as a reference tra-

jectory for designing a feedback control law. In virtual pro-

totyping, engineers can use these trajectories to evaluate the

design of many mechanical systems, possibly avoiding the

time and expense of building a physical prototype. For ex-

ample, in the automotive industry, the planning technique can

serve as a “virtual stunt driver” that determines whether a

proposed vehicle can make fast lane changes or can enter a

dangerous state such as toppling sideways. In the movie and

game industries, advanced animations can be constructed that

automate the motions of virtual characters and devices while

providing realism that obeys physical laws. In general, such

trajectories may be useful in any application area that can be

described using control theoretic models, from analog circuits

to economic systems.

The classic approach in robotics research has been to de-

couple the general robotics problem by solving basic path

378

LaValle and Kuffner / Randomized Kinodynamic Planning 379

Fig. 1. We consider planning problems with dynamic con-

straints induced by physical laws. The above image shows

the state exploration trees computed for a rigid rectangular

object (bottom left). The goal location is represented by a

sphere (upper right).

planning and then ﬁnding a trajectory and controller that

satisﬁes the dynamics and tracks the path (Bobrow,

Dubowsky, and Gibson 1985; Latombe 1991; Shiller and

Dubowsky 1991). The vast majority of basic path-planning

algorithms considers only kinematics while ignoring the sys-

tem dynamics entirely. In this paper, we consider kinody-

namic planning as a generalization of holonomic and nonholo-

nomic planning in conﬁguration spaces by replacing popu-

lar conﬁguration-space notions (Lozano-Perez 1983) by their

state-space (or phase-space) counterparts. A point in the state

space may include both conﬁguration parameters and velocity

parameters (i.e., it is the tangent bundle of the conﬁguration

space).

It may be the case that the result of a purely kinematic

planner will be unexecutable by the robot in the environment

because of limits on the actuator forces and torques. Impreci-

sion in control, which is always present in real-world robotic

systems, may require explicitly modeling system dynamics to

guarantee collision-free trajectories. Robots with signiﬁcant

dynamics are those in which natural physical laws, along with

limits on the available controls, impose severe constraints on

the allowable velocities at each conﬁguration. Examples of

such systems include helicopters, airplanes, certain classes

of wheeled vehicles, submarines, unanchored space robots,

and legged robots with fewer than four legs. In general, it

is preferable to look for solutions to these kinds of systems

that naturally ﬂow from the physical models, as opposed to

viewing dynamics as an obstacle.

These concerns provide the general basis for kinodynamic

planning research. One of the earliest algorithms was pre-

sented in Sahar and Hollerbach (1986), in which minimum-

time trajectories were designed by tessellating the joint space

of a manipulator and performing a dynamic programming-

based search that uses time-scaling ideas to reduce the search

dimension. Algebraic approaches solve for the trajectory

exactly, although the only known solutions are for point

masses with velocity and acceleration bounds in one dimen-

sion (O’Dunlaing 1987) and two dimensions (Canny, Rege,

and Reif 1991). Dynamic programming-based algorithms

that provide approximately optimal solution trajectories were

introduced in Donald et al. (1993). Other papers have ex-

tended or modiﬁed this technique (Donald and Xavier 1995a,

1995b; Heinzinger et al. 1990; Reif and Wang 1997). A

dynamic programming-based approach to kinodynamic plan-

ning for all-terrain vehicles was presented in Cherif (1999).

Ferbach (1996) performed a state-space search using an incre-

mental, variational approach. An approach to kinodynamic

planning based on Hamiltonian mechanics was presented in

Connolly, Grupen, and Souccar (1995). An efﬁcient approach

to kinodynamic planning was developed by adopting a sensor-

based philosophy that maintains an emergency stopping path

that accounts for robot inertia (Shkel and Lumelsky 1997).

Although several kinodynamic planning approaches exist,

they are limited to either low-degree-of-freedom problems

or particular systems that enable simpliﬁcation. Randomized

techniques have led to efﬁcient, incomplete planners for basic

holonomic path planning; however, there appears to be no

equivalent technique for the broader kinodynamic planning

problem (or even nonholonomic planning in the conﬁguration

space). We present a randomized approach to kinodynamic

planning that quickly explores the state space and scales well

for problems with high degrees of freedom and complicated

system dynamics. Our ideas build on a large foundation of

related research, which is brieﬂy presented in Section 2.

Section 3 deﬁnes the problem. Our proposed planning

approach is based on the concept of rapidly exploring random

trees (RRTs) (LaValle 1998b) and is presented in Section 4.

To demonstrate the utility of our approach, a series of planning

experiments for hovercrafts in 

and spacecrafts in 

are

presented in Section 5. Theoretical analysis of the planner’s

performance is given in Section 6. Finally, some conclusions

and directions for future research are provided in Section 7.

2. Other Related Research

2.1. Dynamic Programming

For problems that involve low degrees of freedom, classical

dynamic programming ideas can be employed to yield nu-

merical optimal control solutions (Bellman 1957; Bertsekas

1975; Larson and Casti 1982; LaValle 1998a). Because con-

trol theorists have traditionally preferred feedback solutions,

the representation often takes the form of a mesh over which

cost-to-go values are deﬁned using interpolation, enabling in-

puts to be selected over any portion of the state space. If open-

loop solutions are the only requirement, then each cell in the

380 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / May 2001

mesh could be replaced by a vertex that represents a single

state within the cell. In this case, the control-theoretic numer-

ical dynamic programming technique can often be reduced to

the construction of a tree grown from an initial state (Larson

1967). This idea has been proposed in path-planning liter-

ature for nonholonomic planning (Barraquand and Latombe

1991a; Lynch and Mason 1996) and kinodynamic planning

(Cherif 1999; Donald et al. 1993). Because these methods are

based on dynamic programming and systematic exploration

of a grid or mesh, their application is limited to problems with

low degrees of freedom.

2.2. Steering Methods

The steering problem has received considerable attention in

recent years. The task is to compute an open-loop trajectory

that brings a nonholonomic system from an initial state to a

goal state without the presence of obstacles. Given the gen-

eral difﬁculty of this problem, most methods apply to purely

kinematic models (i.e., systems without drift or momentum).

For a kinematic car that has a limited turning radius and moves

forward only, it was shown that the shortest path between any

two conﬁgurations belongs to one of a family of six kinds

of curves comprising straight lines and circular arcs (Dubins

1957). For a car that can move forward or backward, opti-

mal solutions comprising 48 curve types have been obtained

(Boissonnat, Cérézo, and Leblond 1994; Reeds and Shepp

1990; Sussman and Tang 1991). For more complicated kine-

matic models, nonoptimal steering techniques have been in-

troduced, which include for example a car pulling trailers

(Murray and Sastry 1993) and ﬁre trucks (Bushnell, Tilbury,

and Sastry 1995). Techniques also exist for general system

classes, such as nilpotent (Lafﬁerriere and Sussman 1991),

differentially ﬂat (Murray, Rathinam, and Sluis 1995; Fliess

et al. 1993), and chained form (Bushnell, Tilbury, and Sastry

1995; Murray and Sastry 1993; Struemper 1997). For sys-

tems with drift and/or obstacles, the steering problem remains

a formidable challenge.

2.3. Nonholonomic Planning

The nonholonomic planning problem was introduced in Lau-

mond (1987) and has blossomed into a rich area of research

in recent years. Rather than surveying this large body of re-

search, we refer the reader to recent, detailed surveys (Duleba

1998; Laumond, Sekhavat, and Lamiraux 1998). Most cur-

rent approaches to nonholonomic planning rely on the exis-

tence of steering methods that can be used in combination with

holonomic motion-planning techniques. Other approaches

exploit particular properties or very special systems (espe-

cially kinematic car models). For most nonholonomic sys-

tems, it remains a great challenge to design efﬁcient path-

planning methods.

2.4. Lower Bounds

Kinodynamic planning in general is at least as difﬁcult as

the generalized mover’s problem, which has been proven to

be PSPACE-hard (Reif 1979). Hard bounds have also been

established for time-optimal trajectories. Finding an exact

time-optimal trajectory for a point mass with bounded ac-

celeration and velocity moving amid polyhedral obstacles in

three dimensions has been proven to be NP-hard (Donald et al.

1993). The need for simple, efﬁcient algorithms for kinody-

namic planning, along with the discouraging lower bound

complexity results, has motivated us to explore the develop-

ment of randomized techniques for kinodynamic planning.

This parallels the reasoning that led to the success of random-

ized planning techniques for holonomic path planning.

2.5. Randomized Holonomic Planning

It would certainly be useful if ideas could be borrowed or

adapted from existing randomized path-planning techniques

that have been successful for basic, holonomic path planning.

For the purpose of discussion, we choose two different tech-

niques that have been successful in recent years: randomized

potential ﬁelds (e.g., Barraquand and Latombe 1991b; Chal-

lou et al. 1995) and probabilistic roadmaps (e.g., Amato and

Wu 1996; Kavraki et al. 1996). In the randomized potential

ﬁeld approach, a heuristic function is deﬁned on the conﬁg-

uration space that attempts to steer the robot toward the goal

through gradient descent. If the search becomes trapped in

a local minimum, random walks are used to assist in escape.

In the probabilistic roadmap approach, a graph is constructed

in the conﬁguration space by generating random conﬁgura-

tions and attempting to connect pairs of nearby conﬁgurations

with a local planner. Once the graph has been constructed,

the planning problem becomes one of searching a graph for

a path between two nodes. If an efﬁcient steering method

exists for a particular system, then it is sometimes possible

to extend randomized holonomic planning techniques to the

case of nonholonomic planning (Svestka and Overmars 1995;

Sekhavat et al. 1997; Sekhavat et al. 1998).

2.6. Drawing Inspiration from Previous Work

Inspired by the success of randomized path-planning tech-

niques and Monte Carlo techniques in general for addressing

high-dimensional problems, it is natural to consider adapt-

ing existing planning techniques to our problems of inter-

est. The primary difﬁculty with existing techniques is that

although they are powerful for standard path planning, they

do not naturally extend to general problems that involve dif-

ferential constraints. The randomized potential ﬁeld method

(Barraquand, Langlois, and Latombe 1992), while efﬁcient

for holonomic planning, depends heavily on the choice of

a good heuristic potential function, which could become a

daunting task when confronted with obstacles and differential

LaValle and Kuffner / Randomized Kinodynamic Planning 381

constraints. In the probabilistic roadmap approach (Amato

and Wu 1996; Kavraki et al. 1996), a graph is constructed in

the conﬁguration space by generating random conﬁgurations

and attempting to connect pairs of nearby conﬁgurations with

a local planner that will connect pairs of conﬁgurations. For

planning of holonomic systems or steerable nonholonomic

systems (see Laumond, Sekhavat, and Lamiraux 1998), the

local planning step might be efﬁcient; however, in general the

connection problem can be as difﬁcult as designing a non-

linear controller, particularly for complicated nonholonomic

and dynamical systems. The probabilistic roadmap technique

might require the connections of thousands of conﬁgurations

or states to ﬁnd a solution, and if each connection is akin to

a nonlinear control problem, it seems impractical for solving

problems with differential constraints. Furthermore, the prob-

abilistic roadmap is designed for multiple queries. The un-

derlying theme in that work is that it is worthwhile to perform

substantial precomputation on a given environment to enable

numerous path-planning queries to be solved efﬁciently.

In our approach, we are primarily interested in answering a

single query efﬁciently without any preprocessing of the envi-

ronment. In this case, the exploration and search are combined

in a single method without substantial precomputation that is

associated with a method such as the probabilistic roadmap.

This idea is similar to classical artiﬁcial intelligence search

techniques, the Ariadne’s clew algorithm for holonomic plan-

ning (Mazer, Ahuactzin, and Bessière 1998), and the related

holonomic planners in Hsu, Latombe, and Motwani (1999)

and Yu and Gupta (1998).

To directly handle differential constraints, we would like

to borrow some of the ideas from numerical optimal control

techniques while weakening the requirements enough to ob-

tain methods that can apply to problems with high degrees of

freedom. As is common in most of path-planning research,

we forgo an attempt to obtain optimal solutions and attempt to

ﬁnd solutions that are “good enough” as long as they satisfy

all of the constraints. This avoids the use of dynamic pro-

gramming and systematic exploration of the space; however,

a method is needed to guide the search in place of dynamic

programming. These concerns have motivated our develop-

ment of RRTs (LaValle 1998b) and the proposed planning

algorithm.

2.7. Recent Advances

This paper is an expanded version of LaValle and Kuffner

(1999). Since that time, several interesting developments

have occurred. The ideas contained in Frazzoli, Dahleh, and

Feron’s (1999) paper were applied to the problem of designing

trajectories for a helicopter ﬂying among polyhedral obsta-

cles. Substantial performance beneﬁts were obtained by using

a metric based on the optimal cost-to-go for a hybrid nonlin-

ear system that ignores obstacles. In Toussaint, Ba¸sar, and

Bullo (2000), H

∞

techniques and curve ﬁtting were applied

to yield an efﬁcient planner that builds on ideas from LaValle

and Kuffner (1999). A randomized kinodynamic planning

algorithm was proposed recently for the case of time-varying

environments in Kindel et al. (2000). A deterministic kinody-

namic planning algorithm based on collocation was presented

recently in Faiz and Agrawal (2000).

3. Problem Formulation: Path Planning in the

State Space

We formulate the kinodynamic planning problem as path

planning in a state space that has ﬁrst-order differential con-

straints. For kinodynamic planning, the state space will serve

the same purpose as the conﬁguration space for the classi-

cal path-planning problem. Let C denote the conﬁguration

space (C-space) that arises from a rigid or articulated body

that moves in a two- or three-dimensional world. Each con-

ﬁguration q ∈ C represents a transformation that is applied to

a geometric model of the robot. Let X denote the state space

in which a state, x ∈ X, is deﬁned as x = (q, ˙q). The state

could include higher order derivatives if necessary, but such

systems are not considered in this paper.

3.1. Differential Constraints

When planning in C, differential or nonholonomic constraints

often arise from the presence of one or more rolling contacts

between rigid bodies or from the set of controls that it is possi-

ble to apply to a system. When planning in X, nonholonomic

constraints also arise from conservation laws (e.g., angular

momentum conservation). Using Lagrangian mechanics, the

dynamics can be represented by a set of equations of the form

( ¨q, ˙q,q) = 0. Using the state-space representation, this

can be simply written as a set of m implicit equations of the

form g

(x, ˙x) = 0, for i = 1,... ,m and m<2n, in which

n is the dimension of C. It is well-known that under appro-

priate conditions, the implicit function theorem allows the

constraints to be expressed as

˙x = f (x, u), (1)

in which u ∈ U, and U represents a set of allowable controls

or inputs. Equation (1) effectively yields a convenient param-

eterization of the allowable state transitions via the controls

in U.

The proposed approach will require a numerical approxi-

mation to (1). Given the current state, x(t), and inputs applied

over a time interval, {u(t



) | t ≤ t



≤ t + t}, the task is to

compute x(t + t). This can be achieved using a variety

of numerical integration techniques. For example, assuming

constant input, u, over the interval [t, t+t ), a standard form

of fourth-order Runge-Kutta integration yields



= f



x(t) +

t

f (x(t), u), u



382 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / May 2001



= f



x(t) +

t







= f



x(t) +

t





and

x(t + t) ≈ x(t) +

t

(f (x(t), u) + 2x



+ 2x



+ x



For many applications, (1) might not be available. For

example, motions might be generated from a complicated dy-

namical simulation package that considers collisions, ﬂexible

parts, vehicle dynamics, ﬁnite element analysis, and so on.

For these cases, our techniques can be directly applied with-

out requiring (1). The only requirement is that an incremental

simulation of the system be generated for any current state and

input.

3.2. Obstacles in the State Space

Assume that the environment contains static obstacles and

that a collision detection algorithm can determine efﬁciently

whether a given conﬁguration is in collision. It may even be

assumed that an entire neighborhood around a conﬁguration is

collision free by employing distance computation algorithms

(Lin and Canny 1991; Mirtich 1997; Quinlan 1994). There are

interesting differences between ﬁnding collision-free paths in

C versus in the state space, X. When planning in C,itis

useful to characterize the set C

obst

of conﬁgurations at which

the robot is in collision with an obstacle (or itself) (Latombe

1991). The path-planning problem involves ﬁnding a contin-

uous path that maps into C

free

= C \ C

obst

. For planning in

X, this could lead to a straightforward deﬁnition of X

obst

declaring x ∈ X

obst

if and only if q ∈ C

obst

for x = (q, ˙q).

However, another interesting possibility exists: the region of

inevitable collision. Let X

ric

denote the set of states in which

the robot is either in collision or, because of momentum, can-

not do anything to avoid collision. More precisely, a state

lies in X

ric

if there exist no inputs that can be applied over

any time interval to avoid collision. Note that X

obst

⊆ X

ric

Thus, it might be preferable to deﬁne X

free

= X \ X

ric

opposed to X \ X

obst

The region of inevitable collision, X

ric

, provides some

intuition about the difﬁculty of kinodynamic planning over

holonomic and purely kinematic nonholonomic planning.

Figure 2 illustrates conservative approximations of X

ric

for

a point mass robot that obeys Newtonian mechanics without

gravity. The robot is assumed to have L

-bounded accelera-

tion and an initial velocity pointing along the positive x-axis.

As expected intuitively, if the speed increases, X

ric

grows.

Ultimately, the topology of X \ X

ric

may be quite distinct

from the topology of X \ X

obst

, which intuitively explains

part of the challenge of the kinodynamic planning problem.

Even though there might exist a kinematic collision-free path,

a kinodynamic trajectory might not exist. For the remainder

of the paper, we assume that X

free

= X \ X

obst

(one could

alternatively deﬁne X

free

= X \ X

ric

3.3. A Solution Trajectory

The kinodynamic planning problem is to ﬁnd a trajectory

from an initial state x

init

∈ X to a goal state x

goal

∈ X

or goal region X

goal

⊂ X. A trajectory is deﬁned as a time-

parameterized continuous path τ :[0,T]→X

free

that sat-

isﬁes the nonholonomic constraints. Recall from (1) that the

change in state is expressed in terms of an input, u. A more

convenient way to formulate the problem is to ﬁnd an input

function u :[0,T]→U that results in a collision-free tra-

jectory that starts at x

init

and ends at x

goal

or X

goal

. The

trajectory, x(t) for t ∈[0,T], is determined through the inte-

gration of (1). It might also be appropriate to select a path that

optimizes some criterion, such as the time to reach x

goal

.Itis

assumed that optimal solutions are not required; however, in

this paper we assume that there is some loose preference for

better solutions. A similar situation exists in the vast majority

of holonomic planning methods.

3.4. Why Does the Problem Present Unique Challenges?

The difference between X and C is usually a factor of 2 in di-

mension. The curse of dimensionality has already contributed

to the success and popularity of randomized planning methods

for C-space; therefore, it seems that there would be an even

greater need to develop randomized algorithms for kinody-

namic planning. One reason that might account for the lack

of practical, efﬁcient planners for problems in X-space is that

attention is usually focused on obtaining optimal solutions

with guaranteed deterministic convergence. The infeasibility

of this requirement for generic high-dimensional systems has

led many researchers to adopt a decoupled approach in which

classical motion planning is performed and trajectory design

is optimized around a particular motion-planning solution.

Another reason randomized kinodynamic planning ap-

proaches have not appeared is that kinodynamic planning is

considerably harder owing to momentum. Consider adapting

randomized holonomic planning techniques to the problem of

ﬁnding a path in X

free

that also satisﬁes (1) instead of ﬁnd-

ing a holonomic path in C

free

. The potential ﬁeld method

appears to be well suited to the problem because a discrete-

time control can repeatedly be selected that reduces the poten-

tial. The primary problem is that dynamical systems usually

have drift, which could easily cause the robot to overshoot the

goal, leading to oscillations. Without a cleverly constructed

potential function (which actually becomes a difﬁcult nonlin-

ear control problem), the method cannot be expected to work

well. Imagine how often the system will be pulled into X

ric

The problem of designing a good heuristic function becomes

extremely complicated for the case of kinodynamic planning.

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