ICRA 2007 Workshop: Planning, Perception and Navigation for Intelligent Vehicles
Sliding Mode Control of a Mobile Robot for Dynamic Obstacle Avoidance Based on a Time-Varying Harmonic Potential Field Antonella Ferrara and Matteo Rubagotti
Abstract— In this paper, a harmonic potential field method for dynamic environments is proposed to generate an on-line reference trajectory for a wheeled mobile robot. A sliding mode controller is used to make the robot move along the prescribed trajectory determined by the gradient lines. The potential field is modified on-line, in order to make the robot avoid the collision with obstacles which move along non a-priori known trajectories with time-varying speed. The mechanism through which the field is modified is based on the so-called ‘collision cone’ concept.
I. INTRODUCTION Various methods for path control of autonomous mobile robots have been the object of many research works during the past years. The potential field method, first used with robotic manipulators [1], is one of the most commonly applied in mobile robotics, because of its simplicity and plain mathematical analysis. The basic idea of this method is to consider an artificial potential field in the robot workspace, so that the robot turns out to be attracted by the goal point and repulsed away from the obstacles. Following the gradient lines of the considered potential field, the robot will reach the goal point avoiding the obstacles. The potential field method can also be used in the so-called path deformation, in order to modify on-line the pre-determined desired trajectory when unexpected obstacles are detected (see [2] and the references therein). Clearly, local minima in the potential field could make the robot stop in an undesired position: to circumvent this problem, in [3] a harmonic potential field is proposed for path following, together with sliding mode control [4], [5], which makes the control system robust with respect to parameter variations and external disturbances. Harmonic potential fields enable to avoid local minima because the gradient lines of a harmonic potential field begin and terminate at an obstacle position, at the goal point, or at infinity. This idea has been developed in [6], [7], [8] and [9] for different cases, all concerning static obstacles. In order to solve the problem of motion planning in a dynamic environment, that is when the obstacles move in the robot workspace, different approaches have been used. For instance, in [10] and [11] time is considered as a dimension of the model world and the moving obstacles are regarded as stationary; this approaches require an a-priori knowledge of the obstacles trajectories, which makes this approach difficult to be used in real applications. Another Antonella Ferrara and Matteo Rubagotti are with the Department of Computer Engineering and Systems Science, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy; email contact authors:
[email protected],
[email protected]
possible approach is proposed in [12], where the potential field method is extended to the case of moving obstacles, taking into account, for the construction of the potential field, the velocity of the obstacle, but not that of the robot. In other papers, like [13], the construction of the potential field exploits the measure of the relative velocity between robot and obstacle, but the potential fields are quadratic functions, and the problem of local minima needs to be solved. Obviously, one can consider methods which do not use the potential field method: an approach of this kind can be found for instance in [14]. In this paper, a new harmonic potential field method for moving obstacles avoidance is proposed. The key idea is to modify the radius of the ‘security circle’ around each obstacle on the basis of the so-called ‘collision cone’ [15], this latter being, at any time instant, the set of the possible directions of the relative velocity vector for which a future collision is going to occur. More specifically, the on-line variation of the radius of the security circle is performed so that the relative velocity vector is steered outside the collision cone. Relying on this variation, the overall harmonic potential field becomes time-varying. However, basic properties analogous to those of the static method are proved in the paper. II. P ROBLEM FORMULATION The mobile robot considered in this paper can be described by a couple of wheels with a common axle and independent wheel motors, as indicated in Fig. 1. This model can be used to approximate three-wheeled vehicles, as shown in [6]. Assuming rolling contact without slip between the tires and the ground, each wheel can only move along its longitudinal direction, with velocity vR (t) and vL (t), respectively: this condition points out the obvious presence of a nonholonomic constraint. The vehicle configuration is represented by q = [x, y, φ]T ∈ R3 in the world coordinate frame [xW , yW ]. The resulting longitudinal velocity and the rotational velocity of the vehicle are vC (t) and ω(t), respectively, where vC (t) = (vR (t) + vL (t))/2 and ω = (vR (t) − vL (t))/2D (D being the distance between the two wheels). Then, the kinematic part of the model of the vehicle can be expressed as x(t) ˙ = vC (t) cos φ(t) y(t) ˙ = vC (t) sin φ(t) ˙ φ(t) = ω(t)
(1)
while the dynamic part of the model is represented by τt (t) = τr (t) =
mv˙ C (t) + Nt (t) J ω(t) ˙ + Nr (t)
(2) (3)
where the positive scalars m and J denote the mass and the rotational inertia about the vertical axis, τt (t) and τr (t) are the control inputs, while the scalar quantities Nt (t) and Nr (t) include additional dynamics and external disturbances. In this work, the following bounds on the system parameters yR
yW
vL xR
where v0 is the desired constant cruise velocity, while ζ(p) is the distance left to reach the goal point pG , i.e.
I
vC
where function Atan(·) produces angles in the four quadrants of the cartesian plane, with φd (p) ∈ [−π, π[. The desired longitudinal velocity vd (p, t) is required to vary dynamically: given the maximum possible acceleration a0 , suitable starting, cruise and goal-approaching phases are given by p vd (p, t) = min(a0 t, v0 , 2a0 ζ(p)) (8)
y
ζ(p) = kp(t) − pG k.
p Approaching the goal point with velocity 2a0 ζ(p), the vehicle will reach it in a finite time [9]. In [9] the obstacles, the positions of which are fixed and considered known, are protected by circular security zones. To construct the gradient line, a harmonic dipole potential in IR2 is considered, given by µ ¶ µ ¶ 1 1 + ρ1 ln U = −ρ0 ln δ0 δ1
vR
x
Fig. 1.
xW
The mobile robot
are considered 0 < m− ≤ m ≤ m+ , |Nt (t)| ≤ Nt+ ,
0 < J− ≤ J ≤ J+ |Nr (t)| ≤ Nr+
(4) (5)
where m− , m+ , J − , J + , Nt+ , Nr+ are known positive constants. Then, relying on system (1)-(3) with the bounds (4)-(5), the problem to be solved is to drive the vehicle to a fixed goal point in the world coordinate frame [xW , yW ] with the desired final orientation, avoiding the collision, during the motion, with moving objects. III. S OME PRELIMINARIES ON GRADIENT TRACKING A. The artificial potential field method It is a well-known method for on-line trajectory generation and obstacle avoidance. In [6], [7], [8], and [9], assuming to take into account a single obstacle at a time, an artificial potential field is considered, with a global minimum at the goal point and a global maximum at the obstacle position. In this section, like in the cited works, the assumption is that the obstacle does not move: its position consists of a fixed value. The motion of the vehicle is guided by the gradient of the artificial potential field: no predetermined trajectory is available. Let p = (x, y) be the position of the center of mass of the vehicle in the plane. A vector indicating the motion direction opposite to the gradient of the artificial potential field U (p) is E(p) = −∇U (p). (6) Following the anti-gradient direction, defined for each point p, a continuous trajectory, called gradient line, is obtained. Exact tracking is achieved by orienting the velocity vector p(t) ˙ collinear to E(p). The vector p(t) ˙ can be represented in polar coordinates (vC (t), φ(t)). Thus, the desired orientation of the vehicle corresponds to φd (p) = Atan
Ey (p) Ex (p)
(7)
where ρ0 and ρ1 denote the ‘strength’ of the singularities, while δ0 and δ1 are the distances of the mobile robot from the goal point and the obstacle, respectively. If ρ0 and ρ1 are chosen respectively equal to 1 and R/(R + D) (R being the security circle radius, and D the distance between the obstacle and the goal point) the harmonic dipole presents two important properties. First, all gradient lines beginning outside the security circle remain outside: a consequence of this property is that, if the controller could make the robot follow exactly the reference trajectory, it would never enter inside a security circle. Second, along any gradient line outside the security circle, the distance to the goal point decreases monotonically. The problem with this approach is that, if the cruise velocity is too high, the robot can enter inside the security circle and collide with the obstacle, because the robot orientation can be quite different from the reference one if φd varies too fast. To solve this problem one can try different approaches: in [9], v0 (p, t) is not a constant value, but it is determined by the solution to ° µ ¶° °∂ E(p) ° ° v˙ 0 (p, t) = a0 − v02 (p, t) ° (9) ° ∂p kE(p)k ° . Given a certain p in [xW , yW ], then E(p) is a fixed value, and its derivatives with respect to vector p are known in closed form: consequently the differential equation (9) can be solved on-line. Moreover, v0 (p, t) automatically decreases in areas of high curvature of the gradient E(p), that is near the obstacle, in order to make the vehicle be able to follow the reference orientation in a more precise way, not entering the ‘forbidden circle’. B. The sliding mode controller In [7] and [9], a gradient–tracking based sliding mode controller for the considered mobile robot is proposed. The task of making the robot move along the direction of the gradient vector (6) with the correct speed can be split into
two subtasks: orientation control using τr (t) and velocity control using τt (t). For this reason a componentwise control design, based on the definition of two sliding manifolds can be adopted. The first one refers to the translational component vC (t), i.e. st (p, t) = vC (t) − vd (p, t) = 0.
VR
R
T
r
D T
(10) Fig. 2.
The other manifold refers to the orientation of the vehicle (the rotational component) and is defined as sr (p, t) = cφe (p, t) + φ˙ e (p, t)
E
VO
(11)
where φe (p, t) = φ(t) − φd (p), with φd (p) defined in (7), while c is a positive constant. In [9] it is proved that, by applying the two control variables (dimensionally a traction force and a rotational torque) defined as τt (p, t) = −˜ τt sign(st (p, t))
(12)
τr (p, t) = −˜ τr sign(sr (p, t))
(13)
where τ˜t and τ˜r are positive constants, then the two sliding manifolds are reached in finite time, provided that the amplitudes τ˜t and τ˜r are large enough. As a consequence, the vehicle follows the reference trajectory robustly, rejecting the disturbances Nt (t) and Nr (t). In order to make the problem solvable by means of finite amplitude control signals (12)-(13), the variables v˙ d (p, t), φ˙ e (p, t) and φ¨d (p, t) must be bounded. This condition is always verified, with the exception of the time instants when vd (p, t), defined in (8), switches from a velocity reference p to another (e.g. from v0 to 2a0 ζ(p)). In spite of this, the mobile robot reaches the goal point in finite time and then it stops. In the following, since the focus of this paper is on a modification of the potential field based method rather than on the control law design, the same control law used in [9], reported in (12)-(13), is adopted with the reference trajectory indicated in (7)-(8). IV. T HE COLLISION CONE CONCEPT The collision cone approach has his roots in aerospace literature, where it is used mainly for collision achievement, e.g. for the guidance of a missile [16]. In [15], this method is used for obstacle avoidance: in the present section some of the results obtained in [15] are presented; these results will be used to improve the gradient tracking method presented in the previous section, in order to efficiently avoid not only static obstacles, but dynamic objects too. Consider a mobile robot and another moving object; if we know the size of both, the robot can be modeled as a point, while the radius R of the security circle of the obstacle can be the sum of the obstacle radius and the radius of a circle around the robot. As represented in Fig. 2, r is the distance between the robot and the obstacle center, VR and VO are the velocities of the robot and the obstacle, respectively; α, β and θ are the angles that VR , VO , and the segment which connects the two centers form with the horizontal line, respectively. The relative speed between the obstacle and the robot can be split into two components: a radial one, called Vr , directed along
The quantities of equation (14)
the segment which connects the two centers, and another one Vθ perpendicular to Vr : Vr Vθ
= VO cos(β − θ) − VR cos(α − θ) = VO sin(β − θ) − VR sin(α − θ).
(14)
Now, assume that the robot and the obstacle are moving with constant velocities and Vr is negative, i.e. Vr (t) = V¯r < 0,
Vθ (t) = V¯θ .
(15)
Then, at t = t0 , when the following inequality holds Vθ2 (t0 ) ≤
R2 V 2 (t0 ), r2 (t0 ) − R2 r
(16)
one has that the vector representing the relative speed between the robot and the obstacle belongs to a particular set called ‘collision cone’, which means that a collision between robot and obstacle is going to occur [15]. V. T HE PROPOSED MODIFIED POTENTIAL FIELD METHOD The aim of this paper is to modify the original gradient tracking method, adapting it to a case when the obstacle is not static, but it is moving following a non a-priori known trajectory with time-varying velocity. Even if the obstacle dimension does not vary, the idea is to modify the radius of the security circle around the obstacle dynamically. Roughly speaking, if the obstacle and the robot are pointing towards each other, it can happen that the robot is not able to avoid the collision if the obstacle is detected when the robot is already rather close to the security circle, and the obstacle is approaching at relatively high speed. The proposed solution is to make the value of R vary, so that the robot trajectory is affected by the presence of the obstacle before, and the collision avoidance task becomes less difficult. To make the value of R change as required, we have exploited the collision cone concept previously illustrated. From (16), one can obtain that |Vθ | R ≥ rp 2 . (17) Vθ + Vr2 When Vr < 0 inequality (17) can be used to define what can be called a ‘trend to collide’ of the robot: it is apparent that, if the assumption (15) holds, we can determine exactly if the two objects will collide or not; if it does not hold, but the velocities of the two objects are not varying too fast and Vr < 0, the robot is likely to collide with the obstacle if • R is large; • r is small;
•
√ |V2θ |
Vθ +Vr2
is small, that is the two objects are pointing
towards each other. Taking into account these considerations, the modified potential function introduced in this paper is µ ¶ µ ¶ R R 1 U= ln − ln (18) R+D δ1 δ0
h . Case 2: R = R + rβ In this case the potential function is the same as in (21), bur R is not a constant value. Like in Case 1, our objective is to prove that the radial component of the vector opposite to the gradient of U(r, ϕ) is positive for r < R. One can obtain that the radial component of the vector opposite to the gradient can be written as
where
µ ¶ h ¯ R = min R, R + , rβ
|Vθ | β=p 2 Vθ + Vr2
Er (r, ϕ) = E1 (r, ϕ) + E2 (r) (19)
where E1 (r, ϕ) =
if Vr < 0, while R=R
(20)
if Vr ≥ 0. R is the modified security radius, which increases its size relying on the trend to collide of the robot, h is a positive ¯ >R parameter that can be chosen by the designer, while R is a constant value representing the maximum security radius. The following results can be be proved: T HEOREM 1: Inside the security circle of radius R the gradient lines of (18) are directed away from the obstacle center. Proof: Rewrite the potential function with respect to the polar coordinates (r, ϕ). The obstacle center is located at the origin, i.e. r = 0, while the goal point is at a distance D from it with an orientation ϕ = 0. The potential function is given by µ ¶ µ ¶ R R 1 1 U(r, ϕ) = ln − ln . R+D r 2 r2 − 2Dr cos ϕ + D2 (21) Now, we have to consider two different cases: ¯ or R = R. Case 1: R = R ¯ one has that the radial In this case, from (21) if R = R, component of the vector opposite to the gradient of U(r, ϕ) is ¯ R r − D cos(ϕ) Er (r, ϕ) = − 2 , (22) ¯ r(R + D) r − 2Dr cos(ϕ) + D2 which is the same expression obtained in [8], even if it comes from a different potential function. Thus, from now on, the proof in Case 1 is the same as in [8]. To determine the set where Er > 0, one must examine the set where Er = 0, which corresponds to an equi-potential closed line. Due to geometric considerations, this line must encircle the origin; it is also known that close to the origin Er is positive. Solving Er = 0 we obtain µ ¶ s µ ¶2 ³r´ 1 D D D 1 = cos ϕ −1 ± cos2 ϕ −1 + . R 2 R 4 R R (23) Due to the fact that r > 0 and 0 ≤ ϕ < 2π, if R/D > 1, one can conclude that Er = 0 is solved only for values of r verifying r ≥ R. Case 1 is proved, since the line verifying Er = 0 lies outside the security circle. If instead R = R, because Vr has a positive value, the proof is exactly the same, because R is a constant value.
(24)
and
R r − Dcos(ϕ) − r(R + D) r2 − 2Drcos(ϕ) + D2 ¡ ¢ D ln R r +R+D E2 (r) = h . r2 β(R + D)2
(25)
(26)
In analogy with Case 1, it can be proved that E1 (r, ϕ) is positive for r < R, assuming R < D. It is also easy to see that, for r < R, E2 (r) > 0. So, one can conclude that, assuming R < D, Er (r, ϕ) is always positive inside the security circle, which proves the theorem. ¤ T HEOREM 2: All the gradient lines of (18) terminate in the goal point. Proof: Considering a harmonic dipole, gradient lines start from the positive singularity (the obstacle) and end in the negative one (the goal point) or at infinity. In this case, it is easy to verify that limr→+∞ Er (r, ϕ) = 0− : far from the obstacle, the gradient is always attractive. As a consequence, all the gradient lines terminate in the goal point. ¤ Because of the fact that the reference velocity near the goal point decreases in order to stop the robot, it is convenient define an area in which the moving obstacle cannot enter, because the small reference speed of the robot would make a collision avoidance maneuver very hard to be done. This assumption is verified in most practical applications, because when the robot moves near the goal point and stops, it has reached a ‘safe zone’. Starting from (8), it is easy to obtain that every point in the safe zone satisfies the inequality ζ≤
v02 . 2a0
(27)
Note that the assumption R < D can be relaxed in practical applications: the problem to cope with is that there is a sub-area inside the variable security circle where the vector Er is negative; if the robot is in this zone it is pointing towards the obstacle. Yet, the collision is very difficult to happen for the following reasons: • if the obstacle is moving fast towards the robot, this area becomes very small, and it is included in the safe zone; • if the obstacle is not pointing towards the robot or it is moving slowly, this area becomes larger, but it is easy
Using the sliding mode controller previously described, the bounds on φ˙ e (p, t) and φ¨d (p, t) do not hold at the time instants when the value of Vr changes from a negative value to a positive one, or vice-versa. The sliding variable sr tends to diverge instantaneously, but, as soon as the considered time instant has passed, the control variable τr is again capable of steering sr to zero. A picture of the control scheme in the single obstacle case is reported in Fig. 3, where (xO , yO ) is the position of the obstacle in the (xW , yW ) plane, while (vOx , vOy ) is the associated velocity.
Fig. 3.
Robot trajectory t4
8
t5
t3
t3
4
t2
t2
0
(28)
where j = 1, ..., n, j 6= i, dj = δ1 (j) − R(j), δ1 (j) and R(j) being the distance of the robot from the j-th obstacle center and the variable radius of the security circle of the jth obstacle, respectively. This solution introduces a problem: if in a time interval the values of d for different obstacles are very close to each other, the switchings could be too frequent, and this could make the trajectory of the robot present a chattering-like behavior. To solve this problem, an algorithm to determine the anti-gradient of the potential function Echosen = −∇Uchosen can be defined, as follows: ¯ • define a tolerance ∆d; • order the obstacles so that d1 ≤ d2 ≤ ... ≤ dn ; • for k = 1 to n ∆dk = dk − d1 ; if ∆dk > ∆d¯ then µk = 0 ¯ else µk = 1 − (∆dk /∆d); Pn Pn • Echosen = ( k=1 µk Ek ) / ( k=1 µk ). ¯ obstacles with larger values of Increasing the value of ∆d, ∆dk are taken into account, while if ∆d¯ = 0, only one obstacle is considered and the proposed algorithm degenerates into the solution proposed at the beginning of this section.
t5
t6
t2
t1
t0
2
t7 t6
t1
t1
t0
t7
−6
t0
−8 −10 −15
−10
−5
x [m]
0
5
10
Fig. 4. The mobile robot trajectory in the plane: the robot is represented as a triangle, while the two obstacles are represented as a circle and a rectangle, respectively Velocity references 8 a0 t √ 2a0 ζ
6
v0
v [m/s]
Uchosen = Ui so that di < dj
t4
t4
−4
When dealing with multiple obstacles, it is necessary to assume that the distance of the center of each obstacle from the center of another obstacle cannot be less than 2R, i.e. the obstacles minimum security circles cannot overlap. The solution proposed in [8] is based on the idea of considering only the obstacle which is nearest to the actual robot position. This may require a switching between different potential fields. If the obstacles are moving, the approach suggested in [8] is not efficient, since the relative velocity of the obstacle with respect to the robot plays a crucial role. So, having n different potential fields Uj , one for each obstacle, our suggestion is to chose to consider the potential field
t5 t3
6
−2
VI. T HE MULTIPLE OBSTACLES CASE
Block diagram of the control system
10
y [m]
•
for the robot to reach the safe zone before a change of trajectory and speed of the obstacle, which would make a collision more likely; in both cases, the goal point is between the obstacle and the robot, and the obstacle cannot cross the safe zone, but it has to follow a longer trajectory, which decreases the trend to collide.
4
vd
2 0
0
5
10
15
20
25
30
35
40
t [s]
Fig. 5. The three velocity profiles: the minimum of them (solid line) is the actual reference signal
VII. S IMULATION RESULTS To verify in simulation the proposed control law, a threewheeled vehicle is considered, with a nominal mass of 10 Kg (m− = 9 Kg and m+ = 11 Kg), and nominal moment of inertia of 10 Kg · m2 (J − = 9 Kg · m2 and J + = 11 Kg · m2 ). In this simulation, the two external disturbances, Nt (t) and Nr (t), which affect the system, are assumed to be sinusoidal signals with amplitude equal to 0.5 N and 0.5 N m, respectively. The vehicle starts at (−12, 2) in the [xW , yW ] plane, with an angle φ = 0 rad, and vC = ω = 0, and the parameters in (8) are a0 = 0.2 m/s2 and v0 = 1 m/s; from this two values is it immediate to obtain from (27) that the safe zone has a radius of 2.5 m. The mission is to
Distance from the first obstacle δ (1) [m]
15
1
10 5 0
0
5
10
15
20
25
30
35
40
t [s]
around the obstacles. Fig. 8 represents the sliding manifolds relative to the translational and the rotational component, respectively, which are both efficiently steered to zero by the controller.
Distance from the second obstacle δ (2) [m]
30
VIII. C ONCLUSIONS
1
20 10 0
0
5
10
15
20
25
30
35
40
t [s]
Fig. 6. Time evolution of the two distances, δ1 (1) and δ1 (2), between the mobile robot and the obstacles Security radius of the first obstacle 20
1
R [m]
15 10 5 0
0
5
10
15
20
25
30
35
40
30
35
40
t [s]
Security radius of the second obstacle 20
2
R [m]
15 10
R EFERENCES
5 0
0
5
10
15
20
25
t [s]
Fig. 7. Time evolution of the two time-varying security radii, R1 and R2 , of the obstacles Sliding manifold st 0.1
s
t
0.05 0
−0.05 −0.1
0
5
10
15
20
25
30
35
40
25
30
35
40
t [s]
Sliding manifold sr 20
s
r
0 −20 −40
In this paper, an original dynamic harmonic potential field is proposed, so as to generate an on-line trajectory for a mobile robot. The considered sliding mode controller can make the robot follow the prescribed trajectory, avoiding moving obstacles in its workspace, and reaching the goal point. The performances in simulation are satisfactory, putting into evidence the robustness of the control system with respect to external disturbances and parameter variations, as well as the ability of the reference trajectory to adapt to timevarying environments. Future works will be devoted to the experimental verification and validation of this proposal.
0
5
10
15
20
t [s]
Fig. 8.
Time evolution of st (t) and sr (t), respectively
reach the origin of the plane, avoiding two moving obstacles: the first one moves with a circular trajectory around the goal point, 6 m far from it, while the second one starts at (−14, −4) and moves with a constant speed of 0.75 m/s. Taking into account the bounds on the parameters (4) and (5), together with the bounds on the reference variables v˙ d (p, t), φ˙ e (p, t), and φ¨d (p, t), the chosen values of the control variables are τ˜t = 2.5 N and τ˜r = 10 N m, with c in (11) equal to 1. A switching mechanism between the two obstacles according to the proposed multiple obstacle algorithm is used, while the parameters in (19) are h = 2, ¯ = 20 m for both obstacles. Fig. 4 shows R = 1 m, and R the trajectory described by the robot (the robot is represented as a triangle, while the obstacles are represented by a circle and a rectangle, respectively); t0 , ..., t7 are time instants with increasing values. As it is apparent, the robot correctly avoids the obstacles and reaches the goal point. In Fig. 5 the three velocity profiles used in (8) to determine vd (p, t) and the resulting vd (p, t) are plotted. The time evolution of the two distances, δ1 (1) and δ1 (2), between the mobile robot and the obstacles are illustrated in Fig. 6, while Fig. 7 represents the time evolution of the two variable security radii, R1 and R2 ,
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