dutoit11_probabilisticcollisionchecking

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IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 4, AUGUST 2011 809

Probabilistic Collision Checking With Chance Constraints

Noel E. Du Toit and Joel W. Burdick

AbstractObstacle avoidance, and by extension collision checking, isa basic requirement for robot autonomy. Most classical approaches tocollision-checking ignore the uncertainties associated with the robot and

obstacles geometry and position. It is natural to use a probabilistic de-scription of the uncertainties. However, constraint satisfaction cannot beguaranteed, in this case, and collisionconstraints must instead be convertedto chance constraints. Standard results for linear probabilistic constraintevaluation have been applied to probabilistic collision evaluation; however,this approach ignores the uncertainty associated with the sensed obstacle.An alternative formulation of probabilistic collision checking that accountsfor robot and obstacle uncertainty is presented which allows for depen-dent object distributions (e.g., interactive robot-obstacle models). In orderto efficiently enforce the resulting collision chance constraints, an approx-imation is proposed and the validity of this approximation is evaluated.The results presented here have been applied to robot-motion planning indynamic, uncertain environments.

Index TermsChance constraints, collision avoidance, probabilisticcollision checking.

I. INTRODUCTION

Obstacle avoidance is a basic requirement for robots operating in

practical environments. This is especially true for robots operating

in dynamic, uncertain environments (DUEs). In such environments,

robots must work in close proximity to many other moving agents

whose future actions and reactions are often not well known. The robot

is plagued by noise in its own state measurements and in the sensing

of static and moving obstacles. An example of a DUE application

is a service robot, which must move through a swarm of humans in

a cafeteria during a busy lunch hour in order to deliver food items.

This paper presents an obstaclerobot collision-checking method that

accounts for the probabilistic uncertainties associated with the robot

and obstacle position states.Classical approaches to obstacle avoidance react locally to the ob-

stacles, but often ignore state uncertainty [1][3]. Approaches include

the velocity-obstacle approach [4], the dynamic-window approach [5],

and inevitable-collision states (ICSs) [6]. ICS differs from collision

checking in that the goal of ICS is to identify robot configurations for

which no sequence of actions exist that allows the robot to avoid a

collision. Recent extensions of the ICS problem to the uncertain case

have been proposed [7], [8].

In this study, the collision-checking problem is considered as part

of a robot motion-planning implementation [9], [10]. Obstacle avoid-

ance is obtained by recursively solving the planning problem, while

enforcing collision constraints along the trajectory. The specific prob-

lemconsidered in this study is probabilistic collision checking between

uncertain configurations for two objects, which is referred to as col-

lision chance constraints. Blackmore et al. [11] extended results for

linear-chance constraints to the problem of polygonal obstacle avoid-

ance, but the uncertainty associated with the obstacle is ignored. The

Manuscript received June 17, 2010; revised November 19, 2010; acceptedFebruary 9, 2011. Date of publication March 24, 2011; date of current ver-sion August 10, 2011. This paper was recommended for publication by Asso-ciate Editor S. Carpin and Editor L. Parker upon evaluation of the reviewerscomments.

The authors are with the Department of Mechanical Engineering, Califor-nia Institute of Technology, Pasadena, CA, 91125 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2011.2116190

main contribution of this paper is an alternative formulation of the

probability of collision that accounts for both robot and obstacle uncer-

tainty. The resulting formulation is a generalization of the formulation

presented by Lambert et al. [12]. Finally, the motion-planning prob-

lem in DUE is inherently nonconvex. Computational considerations

for this optimization problem motivate the development of efficient

constraint-evaluation routines. An approximation scheme is presented

to efficiently enforce the collision chance constraints and the validityof the approximation is investigated.

The layout of the paper is as follows. Previous results on proba-

bilistic linear constraints and the application of these results to prob-

abilistic collision checking is discussed in Section II. An alternative

formulation of the problem is given in Section III. An approximation

is introduced and specialized for systems with normally distributed

states (for independent- and dependent-object distributions). The valid-

ity of the approximation is investigated for disk objects with Gaussian-

distributed position states in Section IV. The application of the results

in this paper to the robot motion-planning problem in DUEs [9], [10]

is discussed in Section V.

II. CHANCECONSTRAINTS

It is often desirable to impose constraints on the states of a robotic

system when solving a motion-planning problem (e.g., velocity con-

straints and collision constraints between the robot and other objects).

In general, these constraints are described as inequality functions

c(x) 0of the system state x X Rn x . The system state may referto the robot state, obstacle state, or the augmented state (i.e., robot and

obstacle). However, in a probabilistic formulation, the system states

are represented by unbounded distributions (e.g., normal distributions)

and constraint satisfaction cannot be guaranteed for all realizations of

the states. It is instead necessary to introduce chance constraints, or

probabilistic state constraints, which are of the form

P(x / Xfree ) or P(x Xfree ) . (1)The probability of constraint violation is bounded. and are lev-els of confidence and Xfree denotes the set of states where the exact,

deterministic constraints are satisfied (e.g., Xfree = {x: c(x) 0}).Chance constraints have been commonly applied in stochastic con-

trol and stochastic receding horizon control (SRHC) [13][17]. To date,

two approaches have been proposed to evaluate chance constraints

given by the following. 1) Assume linear constraints on Gaussian-

distributed state variables, and convert the chance constraints into

constraints on the means of the states [11], [16], and 2) evaluate the

constraints by Monte Carlo simulation (MCS) [18]. Approach 1) is

restrictive, whereas approach 2) can handle non-Guassian distributions

and nonlinear constraints but is computationally intensive. Standard

results for linear constraints on Gaussian-distributed system states (ap-

proach 1) are reviewed next to give intuition about the effects of chanceconstraints.

A. Individual Linear Chance Constraints

For Gaussian-distributed statevariables, linear chance constraints on

state space X Rn x take the form P(aT x b) , where aT x R, x X, b R. Let x= E[x] be the mean of the state, and let thecovariance matrix ofxbe

=E[(x x)(x x)T ].

The chance constraintP(aT x b) is satisfied iffaT x+F1 ()

aT a b (2)

where F1 ()is the inverse of the cumulative distribution function for

astandardscalar Gaussian variable (i.e., zero mean and unit standard

1552-3098/$26.00 2011 IEEE


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810 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 4, AUGUST 2011

Fig. 1. (a) (Blue) Robots position is propagated, thus resulting in an uncer-tain state (which is represented by multiple transparent robots). (Green) Staticobstacle is to be avoided. (b) (Dashed blue lines) Constraint is tightened bysome amount that accounts for the level of confidence and the predicted statecovariance.

deviation) [19]. The chance constraint is converted into a constraint

on the mean of the robot state. The constraint is tightened1 by some

amount that is a function of the level of confidence and covariance (see

Fig. 1).

B. Using Linear-Chance Constraints for Collision Checking

When using linear-chance constraints to impose collision constraints

between the robot and polygonal objects, sets of linear constraints must

be satisfied simultaneously (or jointly) [11]. Two approaches for im-

posing joint constraints have emerged: 1) Probability mass is allocated

to the different components of the joint constraints, each of which is

then implemented using the individual constraint results, and 2) use an

ellipsoidal approximation (for Gaussian-distributed variables).

1) Probability-Mass Allocation: To imposethe setof constraints, it

is convenient to formulatethe problem in terms of the probability of not

being in free space:P(x / Xfree ) = 1 P(x Xfree ) 1 =.A part of the confidence is then assigned to each ofm individualconstraints that are to be satisfied [11], [16], [20]. Let us define the

constraint violation event asAj ; then,x Xfree m

j = 1Aj , wherenegation is represented by the overbar. The negation of this expression

is given by De Morgans law

mj = 1

Aj =

mj = 1

Aj . (3)

From Booles inequality, the probability of any constraint violation isbounded by

P

m

j = 1

Aj

mj = 1

P(Aj ). (4)

Thus, if a portion j ofis assignedto each componentof theconstraint(P(Aj ) j ), then

mj = 1

j = = P(x Xfree ) . (5)

1Constraint tightening refers to growing the constraint to account for

uncertainty.

It is necessary to specify the level of certainty j for each of theconstraint components. A naive approach is to assign a constant value,

j =/m, but this has been shown to be very conservative [20]. Black-more and Ono [20] and Ono and Williams [21] introduced the notion

of risk allocation, where more probability mass is assigned to parts of

the trajectory where collisions are more likely.

2) Ellipsoidal Approximation: The probability mass contained in

thefree space can be approximatedby theprobabilitymass contained inthe uncertainty ellipse associated with the jointly Gaussian-distributed

states. Then, the uncertainty ellipse is confined to the free space [19].

Again, the ellipsoidal approximation is very conservative [20].

C. Discussion

These standard results for linear constraints have been applied to

probabilistic collision checking, but uncertainty in obstacle position or

shape is ignored. However, it is desirable to evaluate collision chance

constraints when both the obstacle and robot locations are uncertain.

An alternative formulation of the probability of collision is presented

next.

III. PROBABILISTICCOLLISIONCHECKING

Probabilistic collision checking can be formulated as a chance con-

straint of theform P(C) , where Cis the collision condition (whichis defined below). Let W Rn x be the workspace of the robot. As-sume a rigid-body2 robot. Let us assign a body-fixed reference frame

to the robot, which is located atxR in the global reference frame. LetXR (xR ) W be the set of points occupied by the robot. In a similarmanner, assume a rigid-body obstacle (the obstacle might be moving)

with a body-fixedreference frame located at xA in the global frame.LetXA (xA ) W be the set of points occupied by the obstacle. The colli-sion conditionis defined as C(xR , xA ) : XR (xR ) XA (xA ) = {}.The probability of collision is defined as

P(C) =

x R

x A

IC(xA , xR )p(xR , xA )dxR dxA (6)

whereICis the indicator function, which is defined as

IC(xA , xR ) =

1, ifXR (xR ) XA (xA ) = {}0, otherwise.

(7)

For obstacle avoidance, collision chance constraints must be simul-

taneously enforced over the trajectory. The results from Section II-B1

can be applied directly where the individual collision constraint viola-

tions are the events Aj . A fraction of the total level of confidence isassigned at each stage to the individual collision constraints.

The resulting formulation is a generalization of the result3 from

Lambertet al. [12] to account for dependence between the robot and

obstacle distributions (e.g., when the object and robot motions aredependent, such as in interactive scenarios). This integral function is

difficult to evaluate for general robot and obstacle geometries. One

strategy is to use MCS, but that method can be computationally inten-

sive [12] (see also Section IV-A). As an alternative, an approximate

solution to the probability of collision is presented next.

2An idealized collection of points with a fixed relative position.3The motivating formulation of the probability of collision in Section III

[12, Eq. (5)] is incorrect: The probability of collision requires a double integralover the space, and the obtained result does not account for object sizes, asnoted by the authors. In addition, the resulting function is not a probabilitydensity function since there is no uncertain parameter. Instead, it is simply adeterministic function in the form of a Gaussian distribution. Later results (e.g.,

see [12, Eq. (10) in Sec. IV]) are correct.


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A. Approximate Probability of Collision

For simplicity, let us assume a spherical robot of radius

(XR=S(xR , )) and a point obstacle.4 The collision condition is

defined as C: xA S(xR , ). Let VSbe the volume of this sphere.From (6), the probability of collision is rewritten as

P(C) =

x R

x AS(x R, ) p(x

A |xR )dxAp(xR )dxR . (8)Assuming that the robot has a small radius ( 1), the inner in-

tegral can be approximated with a constant value of the conditional

distribution of the obstacle evaluated at the robot location, multiplied

by the volume occupied by the robot VSx AS(x R, )

p(xA |xR )dxA VSp(xA =xR |xR ). (9)

The approximate probability of collision for small objects is, there-

fore, given by

P(C)

VS x R p(xA =xR |xR )p(xR )dxR . (10)

This integral can be evaluated in closed form for systems where the

robot and obstacles have Gaussian-distributed position states, which

are 1) independent and 2) dependent.

B. Objects With Independent, Gaussian-Distributed States

Lemma 1: Let us assume that the robots and objects position

uncertainties are described with independentGaussian distributions:

xR N(xR , R )andxA N(xA , A ); thenx R

p(xA =xR |xR )p(xR )dxR

= 1det(2C)

exp1

2(xR xA )T 1C (xR xA )

(11)

whereC= R + A is the combined-position covariance.

Proof:Using the independence assumptionx R


=

x R

Nx R(xA , A )Nx R(xR , R )dxR . (12)

It is known that the product of independent Gaussian distributions is a

weighted Gaussian [22]

Nx R(xA , A )Nx R(xR , R ) = Nx R(mc , c ) (13)

where

=

1det(2C)

exp1

2

xR xA

T1C (xR xA )

(14)

and

mc = c

1R xR + 1A xA

c =

1R +

1A

1. (15)

4This analysis can readily be extended to the case where both the robot and

obstacle have spherical geometries by considering the configuration space.

C= R + A is the combined position covariance. Equation (12)

becomesx R

p(xA =xR |xR )p(xR )dxR =

x R

Nx R(mc , c )dxR (16)

and since the integral evaluates to 1, the result is obtained.

This analysis provides intuition about the appropriate combinationof the covariances of the two distributions. The probability of collision

is approximately given by

P(C) VS 1det(2C)

exp1


. (17)

The objective is to convert the constraint P(C) into a constrainton the robot mean state in terms of the obstacle mean state. From (17),

the equivalent constraint is

(xR xA )T 1C (xR xA ) 2 ln

det(2C) VS

= (18)

where is a function of the level of certainty, the sizes of the robotand obstacle, and the combined covariance of the position states. This

constraint is in theform of an ellipsoidaround theobstaclethatthe robot

has to avoid to ensure that the chance collision constraint is satisfied.

C. Objects With Dependent, Gaussian-Distributed States

Lemma 2:Let us assume that the robot and agent state (augmented

state) are jointly Gaussian (e.g., when modeling interaction between

the robot and the agent [9])

p(x) = N

xRxA

,

R M

TM A

(19)

thenx R


= 1det(2C)

exp1


(20)

whereC

= R + A M T

M.Proof: The dependence condition implies that M= 0 . Then,the marginal distribution of the robot state is p(xR ) = N(xR , R ),and the conditional distribution of the agent state is p(xA |xR ) =N

xA (xR ), A

, where xA (xR ) = xA + TM

1R (xR xR ), and

A = A TM1R M.The product of the two Gaussian distributions has the functional

form

N(xR , R )N

xA (xR ), A

= exp(L(xR ,xR ,xA )) (21)

where

= 1

det(2R )1

det

2A (22)


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and

L(xR ,xR ,xA ) = 1

2(xR xR )T 1R (xR xR )

+1

2(xA xA (xR ))T 1A (xA xA (xR )) .

(23)

Let us partition this function asL(xR ,xR ,xA ) = L1 (xR ,xR ,xA ) +L2 (xR ,xA )so that the integral can be evaluated efficiently. With thispartition, we obtain

x R


=

x R

exp(L1 (xR ,xR ,xA )dxR

exp(L2 (xR ,xA )). (24)

By evaluating the first and second partial derivatives5 with respect to

xR ,L1 (xR ,xR ,xA ) can be written as a normal distribution in termsofxR

L(xR ,xR ,xA )xR= 0 xR = xR (25)

where

xR=

1R 1A

TM

1R +

1R M

1A

TM

1R

xR

+

1A 1R M

1A

xA (26)

and

=

2 L(xR ,xR ,xA )

x2R

1= R

(R

M) R + A M

TM R

TM . (27)

Let us define L1 (xR ,xR ,xA )= 1

2(xR xR )T 1 (xR xR ), and

note that x R

exp(L1 (xR ,xR ,xA )) dxR =

det(2) (28)

so that (24) becomesx R


=

det(2) exp(L2 (xR ,xA )). (29)

5For a generic normal distribution of random variable z , the mean, , andthe covariance,, of the distribution can be recovered from the first and secondderivatives of the functionL(z)[23]

N(z; , ) = 1det(2)

exp(L(z))

L(z) = 1

2(z )T 1 (z ).

Then

L(z)

z = 1 (z ) and L(z)

z = 0 z =

2 L(z)

z 2 = 1 .

Recall thatL2 (xR ,xA ) =L(xR ,xR ,xA ) L1 (xR ,xR ,xA ), whichis quadratic in xR and xA . A similar strategy is used to show that(29) is in the form of a normal distribution Nx R (m, ), where

m= xA , and = R + A M TM. Since

det(2) =

1/

det(2), and L2 (xR ,xA ) = (1/2)(xR m)T 1 (xR m);thus, the result is obtained.

Again, the objective is to convert the constraint P(C)

into a

constraint on the robot mean state in terms of the obstacle mean state,

thereby resulting in (18), with the exception that C= R + A

M TM.

IV. APPROXIMATIONVALIDATION

The approximation introduced in Section III-A assumes a constant

probability of collision over the volume occupied by the robot. This

approximation is only valid for a small robot, and it is of interest to

evaluate the validity of this approximation as the size of the robot is

increased. A scheme to evaluate the true probability of collision is

presented before a comparison with the approximation is performed by

investigating the effect of the relative size of the robot and the spread of

the distribution, as well as the shape of the distribution on the accuracy

of the approximation.

A. True Probability of Collision Evaluation

Let us consider the integral in (6). By introducing the augmented

state,x= [xR xA ]

T , the equation is rewritten as

P(C) =

x

IC(x)p(x)dx (30)

Using the central limit theorem, this integral can be approximated

by drawing N independent and identically distributed samples fromthe joint distribution p(xR , xA ), and then, evaluating the functionIC(xR , xA ), we have

P(C) 1N

Ni= 1

IC(xi ) (31)

wherexi is the sample drawn fromp(xR , xA ). This approach is knownas MCS. By writing the integral in terms of the joint distribution and

augmented state, a generalization of the algorithm, which was intro-

duced by Lambert et al. [12], to dependent distributions is obtained. By

assuming independent distributions, Lambertet al.formulate the colli-

sion probability in terms of the individual distributions. However, using

this formulation, a naive MCSapproach with a doublesummation is ob-

tained.6 In contrast, by formulating the probability of collision in terms

of the joint distribution between the robot- and the obstacle-position

states, dependent distributions can be handled, and the resulting algo-rithm requires evaluation of a single summation. The accuracy of the

estimate of the true probability of collision as a function of number of

samples is investigated in simulation in [12]. For small probabilities of

collision, it is concluded that1000samples are sufficient. A more for-mal analysis is presented here, which is based on the expected accuracy

of the estimate in terms of the coefficient of variation. The coefficient

of variation is a normalized measure of the dispersion. For theestimate

of the probability of collision, and noting that I2C(x) = IC(x), it isdefined as

=

N

(32)

6In an attempt to increase computational efficiency, the algorithm is modified

to use a single summation without justification.


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Fig. 2. Values of the probability of collision at different obstacle locationsfor a robot of radius 0.4m as calculated from the (a) true and (b) approximatedresults.

where

=

x

IC(x)p(x)dx (33)

and

= x I2C(x)p(x)dx 2 (34)

=(1 ). (35)

The coefficient of variation simplifies to

=

1

N . (36)

Assuming that a coefficient of variation of 0.1 (i.e., the standard de-

viation of the estimate is 10% of the mean) and a target probabilityof collision of1% (= 0.01), then at least N= 9900 samples arerequired. For a C++implementation7 of the MCS algorithm and usingN= 10000samples, the average runtime is 4.56 ms, which is on thesame order as presented by Lambert et al. [12].

B. Validity of Approximation

When considering the validity of the approximation presented in

Section III-A, the shapeand spreadof the distribution (which is gov-

erned by the covariance) and the size of the object are relevant. The ap-

proximation assumes a constant-valued joint probability-density func-

tion over the space occupied by the robot, multiplied by the volume of

the object. If the object is large relative to the spread of the distribution,

then the approximation will be inaccurate. As a result, the effects of

the spread and robot size are connected. Two cases are considered in

2-D state space: The covariance is fixed (with covariances resulting in

circular and oblong uncertainty ellipsoids, respectively), and the size

of the robot is varied.

1) Effect of Relative Scaling: The covariance for both the robot andobstacle are fixed at = diag(1, 1). The radius,r, of the disk robot isvaried from 0.3 to 1 m. The robot position is fixed at the origin, and the

obstacle position is varied over the space x [5, 5]andy [5, 5].The true [see (6)] and approximate [see (17)] probabilities of collision

are calculated (e.g., the values of the calculated probabilities of collision

are plotted in Fig. 2 for a robot of radius 0.4 m).

In order to compare the validity of the approximation, the chance

constraint is evaluated (P(C)> for = 0.01). From (18), an el-lipsoidal constraint is expected where the size of the ellipsoid scales

with

det(C)/r 2 . By plotting the collision chance constraint, as

72.66-GHzIntel Core2 Duoprocessor, 4-GBRAM, and using GNU Scientific

Library (GSL) for random-number generation.

Fig. 3. Collision chance constraint is evaluated for a robot of radius 0.4 m(a) using the true probability of collision and the scaling of an equivalentelliptical constraint, i.e.,t , is determined, as illustrated in (b), together with acontour plot of the data in (a).

TABLE IELLIPSOIDALSCALINGS TOENFORCECOLLISIONCHANCECONSTRAINTS FOR

VARYING ROBOTSIZES

Fig. 4. Graphical comparison of the elliptical constraints obtained from the(black) true and (red) approximate probabilities of collision for a robot of radius0.7 m.

evaluated from the true probability of collision, the scaling, i.e., t ,of the ellipsoid required to enforce the true constraint can be obtained

(see Fig. 3). These values are compared with the approximation value,

i.e.,a , in Table I.The threshold for the validity of the approximation is hard to quan-

tify. The areas of the resulting ellipses from the true and approxi-mate results are compared. For r = 0.7m, the error is 7.4%, which isdeemed as the maximum allowable (from visual inspection, see Fig. 4).

For larger robot radii, the approximation is not accurate, and it is con-

cluded that the approximation is valid for

det(C)/r 2 >1.3.2) Effect of Distribution Shape: In the previous section, it was de-

termined that the approximation is valid for

det(C)/r 2 >1.3.However, the shape of the distribution is expected to affect the accu-

racy of the approximation since the size of the robot relative to the

major/minor axes of the resulting uncertainty ellipse is different. To

evaluate the effect of shape, the covariance matrices of the robot and

obstacle are fixed at = diag(1, 0.1). The size of the robot is cho-

sen to yield det(C)/r 2 1.3, thereby givingr = 0.4m. FromFig. 5, it is concluded that the effect of the shape is small for the region


6/7


Fig. 5. Collision chance constraint for a robot of radius 0.4 m is plotted forthe true probability of collision (contour plot).

where the approximation is valid. For larger robot sizes, the effect of

the shape is more pronounced.

V. APPLICATION TOROBOT-MOTIONPLANNING

The results presented in this paper can be used to evaluate collision

chance constraints as they arise in the robot-motion-planning problem

in DUEs [9], [10]. This type of motion-planning problem can be posed

in a receding horizon (RH) framework where an approximate problem

is solved over a finite-planning horizon and a small part of the plan

is executed before new information is incorporated and the problem is

resolved. In this RH framework, the above results can be used to model

the constraints posed by dynamic and static obstacles.

One requirement of the RH approach is real-time solution of the

approximate-planning problem. In Section IV-A, it was noted that the

MCS implementation can evaluate constraints in a few milliseconds.

However, the general optimal motion-planning-problem formulationresults in a nonconvex optimization problem [9]. These problems are

notoriously hard to solve efficiently, regardless of the constraint check-

ing implementation. General approaches to nonconvex-optimization

problems can require thousands of constraint evaluations per plan-

ning stage. As a result, the MCS implementation of probabilistic colli-

sion checking (as presented in Section IV-A) can be insufficient for

real-time implementation when combined with a naive nonconvex-

optimization algorithm. This motivates the need for approximate prob-

abilistic collision-constraint evaluation.

The approximation presented in this study offers a viable method

for efficient constraints enforcement, noting that small errors arising

from approximations are compensated by the information feedback

when the problem is resolved in the RH framework. The range of

covariances where theapproximation is valid canbe extendedin certain

applications (e.g., the motion-planning problem with disk objects and

random walk models [9], [10]). This extension is possible when the

dynamic and measurement models for the robot and dynamic obstacles

do not alter the shape of the uncertainty, as it is propagated through

the system. The functional form of the constraints, as obtained from

the approximation, can be applied directly by adjusting the scaling

factor to account for the nonpoint geometry. A lookup table of scaling

factors can be constructed offline for different covariances by recording

the required scaling from the true probability of collision, i.e.,t , asobtained from MCS in Section IV-A. By using the lookup table, the

constraints can be evaluated very efficiently. Additionally, since the

constraints are based on the true probability of collision, the enforced

constraints lack the conservatism that plagued previous approaches.

VI. CONCLUSION

Probabilistic collision checking is a basic requirement for a robot

operating in an a priori unknown environment. Standard approaches

ignore uncertainty entirely, or account only for robot state uncertainty.

In this study, the state uncertainties of both the robot and object are

accounted for. Probabilistic collision checking is formulated for robots

and obstacles of arbitrary geometry with arbitrary probability distri-

butions for the robot and obstacle-position states. Evaluation of theresulting formula is cumbersome; however, intuition is gained when

using a small objects approximation. Closed-form solutions are ob-

tained for this approximate problem for the cases where the robot and

obstacle states are described by independent and dependent Gaussian-

distributed variables, respectively. The validity of the approximation is

analyzed in terms of the relative scale of object size and distribution

spread, as well as the shape of the distribution. The result has been

applied to the robot motion-planning problem in [9] and [10].

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Modeling and Evaluation of Low-Cost Force Sensors

C. Lebosse, P. Renaud, B. Bayle, and M. de Mathelin

AbstractLow-cost piezoresistive sensors can be of great interest inrobotic applications due not only to their advantageous cost but to theirdimension as well, which enables an advanced mechanical integration. Inthis paper, a comparison of two commercial piezoresistive sensors basedon different technologies is performed in the case of a medical roboticsapplication. The existence of significant nonlinearities in their dynamicbehavior is demonstrated, and a nonlinear modeling is proposed. A com-

pensation scheme is developed for the sensor with the largest nonlinearitiesbefore discussing the selection of a sensor for dynamic applications. It isshown that force control is achievable with these kinds of sensors, in spiteof their drawbacks. Experiments with both types of sensors are presented,including force control with a medical robot.

Index TermsForce control, nonlinear modeling, piezoresistive sensors.

I. INTRODUCTION

In the field of robotics, force control is needed to ensure safety andcomfort in human/robot interaction, for instance, in medical robotics

applications [2], [3]. An increasingly number of industrial [4] and ser-

vice [5] robotic applications are also concerned by close human/robot

interactions and force control. For these applications, low-cost, easy-

to-integrate force sensors are becoming very necessary.

Over the past two decades, sensors such as Interlink Force Sensing

Resistor (FSR), Tekscan Flexiforce, or Peratech QTC have been de-

veloped using novel piezoresistive effects [6][8]. Even though these

low-cost sensors exhibit a lower accuracy than other types of sensors,

they combine interesting dimensions with a very small thickness and a

high flexibility that facilitates their integration, and it has been demon-

strated that some sensors like the Flexiforce are insensitive to magnetic

Manuscript received May 5, 2010; revised November 19, 2010; acceptedFebruary 17, 2011. Date of publication April 5, 2011; date of current versionAugust 10, 2011. Thispaper was recommended for publication by AssociateEd-itor S. Hirai and Editor G. Oriolo upon evaluation of the reviewers comments.This paper was presented in part at the 2008 IEEE International Conference onRobotics and Automation, Pasadena, CA.

C. Lebosse is with Luxscan Technologies, L-4384 Ehlerange, Luxembourg(e-mail: [email protected]).

P. Renaud, B. Bayle, and M. de Mathelin are with the Laboratoire desSciences de IImage de IInformation et de la Teledetection, Centre Na-tional de la Recherche Scientifique, Strasbourg University, 67412 Illkirch,France (e-mail: [email protected]; [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2011.2119850

Fig. 1. Robotic system and its end-effector with the embedded force sensor.

fields [9]. For a growing number of medical applications involving the

placement of the patient inside a Magnetic Resonance Imaging (MRI)

scanner [10] or in pulsed magnetic fields [3], these sensors seem to be

well adapted to force-control tasks and have already been considered

in control schemes [11][13].

Therefore, several researchers have proposed a characterization of

low-cost piezoresistive force sensors. Although static properties have

always been evaluated [14][16], only a few authors have considered

their response to dynamic loading. In [17], the hysteresis of the sensor

output is assessed when the sensor is submitted to a dynamic load.

Ottoet al.[18] have shown that Tekscan K-Scan sensors exhibit a very

strong time drift that is responsible for large errors when the sensors are

submitted to harmonic loading. A compensation scheme is then pro-

posed. It performs a deconvolution using Boltzmann heredity integral,

which was developed in the context of viscoelasticity of materials [19].

Komiet al.[20] analyzed the Tekscan 9811, the Interlink Flexiforce,

and the Peratech QTC sensors when submitted to sinusoidal loads. Theevolution of the sensors output is evaluated in terms of time drift. The

authors observed that this drift can be significant since the sensor out-

put decreases by 10% in 5 s. The duration of the experiments, however,

remains shorter than 1 min, which is not sufficient for many applica-

tions. For example, the analysis of prostheses behavior in biomechanics

or the compensation of breathing motion by force control in medical

robotics applications require much longer experiments.

In the following, robotized transcranial magnetic stimulation (TMS)

is considered [3]. In this medical application, a magnetic stimulation

coil located at the end-effector of the robotic system (see Fig. 1) has

to be placed in contact with the head of the patient. Therefore, force

control is needed, which prevents any air gap between the coil and

the patients head, guarantees the safety of the patient as well as his

comfort, and enables the compensation of the patients movements.

The maximum contact force must remain below 5 N during a typical

20-mintreatment. Thecontact of thecoil on thehead canbe continuous,

in the case of repetitive stimulation, or discontinuous, in the case of

some protocol like the cortex mapping using single-pulse stimulations.

Furthermore, the contact of the coil on the head can be discontinued

after sudden head movements. During the treatment, the sensor loading

will vary due to the patient movements. Low-cost piezoresistive sensors

offer an interesting solution for force measurement in this context,

because of their dimensions and their immunity to magnetic fields.

In this paper, the focus is put on two widespread commercial low-

cost piezoresistive sensors: the Tekscan Flexiforce and the Interlink

FSR, like in [1]. The existence of strong nonlinearities in their dy-

namic response is outlined, and the opportunity to compensate for

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