A Virtual Force Guidance Law for Trajectory Tracking and Path Following Xun Wang12 , Jianwei Zhang2 , Daibing Zhang1 , and Lincheng Shen1 1
College of Mechatronics and Automation, National University of Defense Technology, Changsha, 410073, China
[email protected] 2 Group TAMS, Department of Informatics, University of Hamburg, Hamburg, 22527, Germany
Abstract. This paper presents a virtual force guidance law for trajectory tracking of autonomous vehicles. Normally, three virtual forces are designed to govern the vehicles. The virtual centripetal force counteracts the influence of the reference heading rate. The virtual spring force pulls the vehicle to the reference trajectory and the virtual drag force prevents oscillations. When local obstacles are detected, an extra virtual repulsive force is designed to push the vehicle away from its way to get around the obstacles. Using the guidance law, the reference trajectory can be straight line, circle and general curve with time-varying curvature. The guidance law is directly applicable to path-following problem by redefining the reference point. The use of artificial physics makes the guidance law be founded on solid physical theory and computationally simple. Besides, the physical meanings of the parameters are definite, which makes it easy to tune in application. Simulation results demonstrate the effectiveness of the proposed guidance law for problems of trajectory tracking, path following, and obstacle avoidance.
1
INTRODUCTION
Accurately tracking of a predefined trajectory is a basic requirement for autonomous vehicles in application. Thus, the problems of trajectory tracking and path following has been hot topics for many years. Although many approaches have been proposed for these problems, there is still some work to do when considering curved trajectories, local obstacles, and real-time computing. There are various types of autonomous vehicles, namely, autonomous underwater vehicles (AUVs), unmanned surface vehicles (USVs), unmanned ground vehicles (UGVs), and unmanned aerial vehicles (UAVs). Although the dynamics associated with each type of vehicle are different, the kinematics of different vehicles with appropriate low-level controller are similar. By appropriately designing the low-level controller, kinematics of many autonomous vehicles can be approximated to a unicycle with different constraints or delays. In this paper, we focus on the problems of trajectory tacking and path following for unicycle-type autonomous vehicles, considering curved trajectories, local obstacles, and real-time computing. Various strategies have been proposed in the literature for trajectory tracking and path following. Two main categories are control theories based approaches and geometry based approaches. Several types of control-theoretic techniques have been developed
Fig. 1. Various types of autonomous vehicles, with appropriate low-level controller, the kinematics of which can be approximated to a unicycle with different constraints or delays.
for trajectory tracking of autonomous vehicles. Some of the well-known techniques are adaptive control [1], sliding mode control [2], and linear quadratic regulator [3]. The main advantage of these control-theoretic approaches is the guarantee of the stability and the convenience for performance analysis in theory. While, obstacles avoidance during trajectory tracking remains a challenge in such control-theoretic frames. Although model predictive control based approaches [4] consider obstacles as constraints, the computational-complexity is high and all the obstacles should be known in prior. Another common drawback of control-theoretic approach is the longer man-hours associated with the controller’s implementation comparing with the geometric methods. Geometric guidance laws, such as pure pursuit [5] and line-of-sight (LOS) [6], were mainly proposed to address path-following problem. These guidance laws were mainly used to follow straight-line and circular paths. Although the nonlinear guidance law (NLGL) in [7] was claimed suitable for curve tracking, the cross-track error is still considerable when the curvature varies [8]. Besides, obstacles avoidance can’t be address using these geometric techniques. Another popular geometric technique is the vector field (VF) based approach. Vector fields can be designed for path following [9] or obstacle avoidance during the vehicle moving towards the goal [10]. It is difficult to design the vector field when combining trajectory tracking and obstacle avoidance. Since all these techniques use the geometric approach, computing the desired heading angle (or rate) is quick and they are easy to implement. While, there is still some work to do when considering curve trajectories and the combination of trajectory tracking and obstacles avoidance. Motivated by the above considerations, we focus on the trajectory-tracking problem on the base of artificial physics. The concept of artificial physics was proposed by Spears et al, when they worked on the problem of swarm robotics in [11]. The basic idea was from natural physics. The agents or robots react to virtual forces, which are motivated by natural physical laws. One advantage of artificial physics based approach is the solid scientific principles foundation. Inspired by the concept of artificial physics, we propose a virtual force guidance law for trajectory tracking. Normally, the vehicle
is governed by three virtual forces, namely, a virtual centripetal force, a virtual spring force and a virtual drag force. The virtual centripetal force counteracts the influence of the reference heading rate. The virtual spring force pulls the vehicle to the reference point and the virtual drag force prevents oscillations. When local obstacles are detected, an extra virtual repulsive force is designed to push the vehicle away from its way to get around the obstacles. After that, other virtual forces guide the vehicle back to the reference trajectory. Using the proposed guidance law, the reference trajectory can be any sufficiently smooth time-varying bounded curve. From the view of control theory, the guidance law is equivalent to a proportional-derivative controller when tracking a straight line and is similar to a feedback linearization method when tracking circular or curved trajectories. By redefining the reference point, the guidance law is directly applicable for path-following problem. We discuss the stability and convergence of the resulting closed-loop. We also include a comparison analysis of our approach with the NLGL approach previously published in [7]. NLGL has been proved have better performance than the PID, VF, and PLOS [8, 12]. Obtained simulation results demonstrate the superiority of the proposed guidance law when tracking a curved trajectory with time-varying curvature. The main contribution of this work is a new virtual force based guidance law for autonomous vehicles trajectory tracking. There are four advantages with the proposed guidance law. First of all, the guidance law can be used to accurately track curved trajectory with time-varying bounded curvature. Secondly, the guidance law is computationally simple, and directly applicable in path-following problem. Thirdly, it is convenient to combine trajectory tracking and obstacle avoidance in the frame of the virtual force guidance law. Fourthly, the use of virtual forces makes the guidance law be founded on solid physical theory, and the physical meanings of the parameters are definite, which makes the parameters easy to tune in application. This paper is organized as follows. In the next section, we formulate the problem. In Sec. 3, we first present the virtual force guidance law along with the analysis of stability and convergence, then the guidance law is applied in path-following problem, after that, we modify the guidance law to avoid obstacles. In Sec. 4, simulation results are given to demonstrate the effectiveness of the proposed approach. In the last section, we conclude the paper.
2
PROBLEM FORMULATION
In this section, we briefly describe the vehicle model and the reference trajectory, then we give the definition of the problems of trajectory tracking and path following. 2.1
Vehicle model
We consider a unicycle-type autonomous vehicle as follows x˙ = v cos(ψ), y˙ = v sin(ψ), ψ˙ = ω
(1)
where the state S = (x, y, ψ) denotes the inertial position and heading of the vehicle. The velocity v and the heading rate ω are control inputs.
2.2
Reference trajectory
The desired reference trajectory Sr is produced by a dynamic trajectory smoother as follows x˙r = vr cos(ψr ), y˙r = vr sin(ψr ), ψ˙ r = ωr (2) where Sr = (xr , yr , ψr ) denotes the reference position and reference heading. vr and ωr are piecewise continuous. In this paper, we use (2) to generate the desired reference trajectories and reference paths, including straight line, circle and general curve with time-varying curvature. 2.3
Problem definition
The major difference between the trajectory-tracking problem and the path-following problem lies in the formulation of the reference path. In the trajectory-tracking problem, the reference path is time parameterized, while, in the path-following problem, time is not considered. Thus, the problems considered in this paper can be stated as follows: Definition 1. Trajectory-tracking problem: Let Sr (t) = (xr (t), yr (t), ψr (t)) generated by (2) be a given reference trajectory. The purpose is to design the velocity and heading rate command inputs (vcmd , ωcmd ) such that all the closed-loop signals are bounded and the tracking error kS(t) − Sr (t)k converges to a neighborhood of the origin that can be made arbitrarily small. Definition 2. Path-following problem: Sr (γ) is a reparameterization of Sr (t) by a continuous function γ = γ(t). Let Sr (γ) be a given reference path. The purpose is to design the velocity and heading rate command inputs (vcmd , ωcmd ) such that the closedloop signals are bounded, the tracking error kS(t) − Sr (γ)k and the velocity error |v(t) − vr (γ)| converges to the neighborhood of the origin that can be made arbitrarily small. 2.4
Virtual force
In our work, virtual forces are designed to govern the autonomous vehicle. The word “virtual” here means the forces are a kind of artificial forces, which don’t exist in real systems. We also use the word “virtual” to mean although we are motivated by natural physical forces, we are not restricted to them. The virtual force may have some features that are beyond the “actual” physics. For example, the virtual spring is an ideal spring and the rest length is 0. Although the forces are virtual, the vehicle acts as if they were real.
3
APPROACH
In this section, we first present the virtual force guidance law along with the stability and convergence analysis without considering obstacles. Then, the guidance law is applied on path-following problem. At last, we consider local obstacles during trajectory tracking.
lateral
Fd
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3.1
The virtual force guidance law
For trajectory tracking, we design three virtual forces to govern the vehicle, namely, a virtual centripetal force Fc , a virtual spring force Ft and a virtual drag force Fd , as shown in Fig. 2. Fc points to the reference center O = (xo , yo ), which is determined by the reference point P and the reference radius r. xo = xr + r cos(ψr + π2 sgn(ωr )) (3) yo = yr + r sin(ψr + π2 sgn(ωr )) vr | wr |, wr 6= 0 r= (4) +∞, wr = 0 Ft points to P, and Fd points to the opposite of the velocity of the vehicle V relative to P. Assume the vehicle has a unit mass, i.e.,m = 1, then the absolute value of the virtual forces are given by v2 Ft = kd, Fd = cvvp , Fc = T (5) l where k > 0 is the spring constant, d is the distance from the vehicle to the reference point, c > 0 is the drag coefficient, vvp is the velocity of the vehicle relative the reference point, vT is the velocity component in the normal direction of OV , and l is the distance between V and O. Among these variables, only k and c are the control parameters, and the others can be calculated according to the reference and the actual states of the vehicle. Decomposing the virtual forces in the forward and lateral direction of the vehicle, we get the resultant forces in the two directions Ff = Ft f + Fd f + Fc f , Fl = Ftl + Fdl + Fcl
(6)
where the subscripts f and l mean the components in the forward and lateral direction. Then the virtual force guidance law for trajectory-tracking problem is given by vcmd = v + 4T Ff , ωcmd =
Fl v
(7)
where 4T is the time step in application. 3.2
Stability and convergence analysis
Straight-line tracking When the vehicle tracks a straight line, ωr = 0. Then the reference radius r = ∞, and Fc = 0. Accordingly the heading rate and the velocity command inputs are determined only by the virtual spring force Ft and the virtual drag force Fd as shown in Fig. 3. vvp
N vn
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Fig. 3. Diagram for straight-line trajectory tracking.
By decomposing the virtual drag force Fd and the relative velocity vvp along V P and its normal N, we can analyze the dynamics of the vehicle in the two vertical directions. In the direction of N, the vehicle has an initial velocity vn (0) = vn0 and a drag force Fn = cvn . The direction of the drag force is always opposite to vn . By solving the following equation v˙n (t) = −cvn (t), vn (0) = vn0 (8) we get vn (t) = vn0 e−ct
(9)
For c > 0, vn exponentially converges to 0. Then the dynamics in the V P direction can be given by d¨ = −kd − cd˙ ˙ 0 , the differential equation (10) is transformed to Let X = [x1 , x2 ]0 = [d, d] 0 1 ˙ X = AX, A = −k −c
(10)
(11)
˙ = 0, we get the only equilibrium point dˆ = xˆ1 = 0, and d˙ˆ = xˆ2 = 0. Then, let X The characteristic equation of (11) is given by det(λ I − A) = λ 2 + cλ + k = 0
(12)
Accordingly, the characteristic roots are √ −c ± c2 − 4k λ= 2
(13)
For the spring constant k > 0, and the drag coefficient c > 0, we always have real(λ ) < 0, which means the system (11) is asymptotically stable. Consequently, the tracking error globally converges to 0 using the guidance law (7). From (13), we know that: the tracking error √ d converges to 0 without any overshoot or oscillation, if the parameters satisfy c ≥ 2 k and k > 0. It is worthwhile to mention that the virtual force guidance law is equivalent to a proportion differential controller with e = d, e˙ = −d,˙ K p = k, and KD = c. Circular trajectory tracking When the vehicle tracks a circular trajectory, r = | ωvrr | = const is the constant radius of the reference circle. Without the virtual centripetal force, which means Fc = 0, Ft and Fd force the vehicle to track the reference circular trajectory if the spring force Ft is bag enough to provide the centripetal force, but there will be a steady tracking error ds as shown in Fig. 4(a) and Fig. 4(b).
ds
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Fig. 4. Diagram for the equilibrium states of circular trajectory tracking with:(a) Fc = 0 and c > v2 0,(b) Fc = 0 and c → 0,(c)Fc = lT .
By force analysis, we know that ds ≥ rs ≥ 0, and ds → rs when the positive drag coefficient c is small enough and approaches 0. By solving the following equation krs =
v2T = lωr2 = (r + rs )ωr2 l
(14)
rωr2 k − ωr2
(15)
we get rs =
We know that rs > 0. It requires the spring constant satisfies k > ωr2
(16)
From Eq.(15), we know that rs → 0 when k → ∞. While, in reality, the value of k is a reflection of the maneuverability of the vehicle, which means k is bounded. Accordingly,
the steady tracking error ds > 0 always exists without designing the virtual centripetal force. v2 On the contrary, by the virtual centripetal force Fc = lT , the vehicle orbits a concentric circle with a radius of l. Then the virtual spring force will pull the vehicle close to the concentric reference circle. Meanwhile, the virtual drag force prevents oscillations. When the vehicle converges to the reference trajectory, the virtual forces Ft = Fd = 0 2 due to d = 0 and vvp = 0. Then the virtual centripetal force Fc = vr ensures the vehicle orbits the reference circle. For circular trajectory tracking, the virtual centripetal force counteracts the influence of the reference heading rate, so that we can use the virtual spring force Ft and drag force Fd to guide the vehicle. Then the vehicle tracks a circle like tracking a straight line, and the dynamics about the tracking error d is similar to the straight-line case. Accordingly, the tracking error d globally converges to 0 using the guidance law. Curved trajectory tracking When the vehicle tracks a general curve, the reference radius r = | ωvrr | varies. Then the reference center O = (xo , yo ) determined by (3) moves v2
with the reference point. The virtual centripetal force Fc = lT pointing to the reference center O still provides the centripetal force that ensures the vehicle orbits the moving reference center. Similar to circle case, the virtual centripetal force counteracts the influence of the reference heading rate, the virtual spring force Ft pulls the vehicle close to the reference point, and the virtual drag force prevents oscillations. From the view of control theory, the virtual force guidance law is similar to a feedback nonlinearization method, when tracking a circular or curved trajectory. The feedback linearization approach involves coming up with a transformation of the nonlinear system into an equivalent linear system through a change of variables and a suitable control input. In the virtual force guidance law, the virtual centripetal force Fc is a nonlinear function of the states of the vehicle and the reference point. By introducing the nonlinear virtual centripetal force as a feedback and analyzing the dynamics of the tracking error d, the control input is equivalently transformed to d.¨ Then the resulting closed-loop system about d is a second order linear system, which is equivalent to a PD controller. Analogously,√ there will be no overshoot during the convergence of d, if the parameters satisfy c ≥ 2 k and k > 0. 3.3
Application in Path-following problem
The main difference between the trajectory-tracking problem and the path-following problem lies in the formulation of the reference path. In the trajectory-tracking problem, the reference path is time parameterized, while, in the path-following problem, time is not considered. Then we have to redefine the reference point on the reference path. For the path-following problem, the closest point to the vehicle is selected as the reference point on the reference path. Then, the proposed virtual force guidance law is directly applicable for the problem of path following without any modification. The stable and the convergence analysis above are also applicable for the path-following problem.
3.4
Obstacle avoidance
It is important for an autonomous vehicle to dynamically avoid local obstacles when tracking a predefined trajectory. It requires the flexibility to move out of its way to get around the obstacles. After that, it should converge back to the reference trajectory. Based on the virtual force guidance law, the ability of obstacle avoidance can be easily achieved by properly designing a virtual repulsive force from the obstacle. We assume that the vehicle is equipped with a laser to sense local obstacles. We get a point set Po = {(do , θo )|do ≤ Rd , − π2 < θo < π2 } on the surface of the obstacle in the local polar coordinates, when an obstacle is detected, where the positive direction of θo is counter clockwise, and Rd is the detect radius of the laser as shown in Fig. 5. Accordingly, two points p1 = (dol , θol ) and p2 = (dor , θor ) can be found from Po , where
obstacle
measuring area of the laser
p2 p1
or
v ol Rd
V Fig. 5. Diagram for the obstacle avoidance.
θol and θor are the leftmost and the rightmost point detected by the laser from the view of the vehicle. We design an extra virtual repulsive force that repels the vehicle to the lateral direction, which means the obstacle only affects the heading rate of the vehicle. Without any prior knowledge of the world, it is reasonable for the vehicle to avoid the obstacle along the side with a smaller angle of view. Accordingly, the obstacle repels the vehicle to the right when |θor | ≤ |θol |, to the left when |θor | > |θol |. Considering the unit mass assumption, we design the extra virtual repulsive force Fo as follows Fo = ko lo (17) −(2dsa f + dol θol ) lo = (2dsa f − dor θor ) 0
θor + θol ≤ 0, 2dsa f + dol θol > 0 θor + θol > 0, 2dsa f − dor θor > 0 else
(18)
where ko is the repulsive force constant, and dsa f is the allowed minimum clearance between the agent and the obstacle. In most cases, collision with an obstacle is destructive
for the vehicle. Thus, the repulsive force constant ko should much larger than the spring constant k (i.e. ko >> k). Then, the heading rate command input ωcmd in the guidance law (7) can be modified by the extra virtual repulsive force as follows vcmd = v + 4T Ff , ωcmd =
Fl Fo + v v
(19)
This is not the only way to modify the virtual force guidance law to avoid obstacles, one can design other kinds of repulsive force to achieve it. This simple modification is to illustrate the convenience to consider obstacle avoidance in the frame of the virtual force guidance law.
4
SIMULATION
This section presents simulation results to illustrate theoretical results proposed in this paper. The simulation includes three parts, namely, trajectory tracking, path following, and obstacle avoidance. In the path-following simulation, we include a comparison analysis of our approach with the NLGL approach. Throughout the simulations, the unit for distance, velocity, angle and angle rate are m, m/s, rad and rad/s. (2) is used to generate the reference trajectories. The reference velocity keeps vr = 20. The guidance law (7) and (19) are applied. 4.1
Trajectory tracking
The reference trajectories of straight line, circle, and curve are generate by (2) with πt ), respectively. Different k and c are applied in ωr = 0, ωr = −0.1, and ωr = −0.1 sin( 30 (7) to evaluate the performance of the guidance law for trajectory tracking. Simulation results for straight line, circle and curve tracking are shown from Fig. 6 to 8 respectively. From Fig. 6-8, we find that, with all of the c and k, the vehicle converges to the reference trajectories including straight line, circle and curve with time-varying curvature. While, the convergence processes are different with different c or k. Figure √ 6(a)8(a) shows that: there are overshoots before the convergence of d, when c = k and √ c = 1.5 k. √ The vehicle converges to the reference trajectories without any overshoot, √ √ when √ c = 2 k and c = 3 k. It takes a longer time to converge when c = 3 k than c = 2 k. From Fig. 6(b)-8(b), we find that the convergence time can be shorten by increasing k and the guidance law has a large parameter adaptation. 4.2
Path following
The virtual force guidance law can be directly applied to address the path-following problem. To illustrate the advantage of the proposed guidance law, we compare the virtual force guidance law with the NLGL proposed in [7]. The performance of NLGL was evaluated on fixed-wing UAVs in [8] and [12], and it performed over the PID, VF, and PLOS. For the NLGL, we use the recommend parameters L = 0.6r [8] in
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Fig. 6. Vehicle trajectories in the simulation for straight-line tracking with S(0) = (500, 0, 0) and: √ (a) k = 0.1 and different c, (b) c = 2 k and different k. 100
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Fig. 7. Vehicle trajectories in the simulation for circle tracking with S(0) = (100, 100, π4 ) and: (a) √ k = 0.1 and different c, (b) c = 2 k and different k.
√ the simulation. For the virtual force guidance law, we use k = 0.1 and c = 2 k. The reference path is generated by (2) with 0 , , −0.1 , ωr (t) = 0.1 0.1 + 0.1 sin( 2π(t−60) ), 80 0.2 ,
t ∈ [0, 4] t ∈ (4, 30] t ∈ (30, 60] t ∈ (60, 80] t ∈ (80, 90]
(20)
Figure 9 shows the trajectories of the vehicle using the two guidance laws. The peak cross-track error of the NLGL is found to be over 15 m. For the virtual force guidance law, the cross-track error is 0 throughout the simulation. The result shows that the virtual force guidance law performs over the NLGL approach, especially when the reference curvature varies. Actually, in the trajectory-tracking and path-following problems, the reference trajectory (or path) is predefined by the user or generated by path planning algorithms, which means the reference path is always fully known. Accordingly, the reference head-
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Fig. 8. Vehicle trajectories in the simulation √ for curve trajectory tracking with S(0) = (100, 0, 0) and: (a) k = 0.1 and different c, (b) c = 2 k and different k. 500
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ing rate ωr is also accurately known in advance. Therefore, it is reasonable to make use of the reference heading rate to improve the tracking accuracy when the reference is a curve with time-varying curvature. In the virtual force guidance law, the virtual centripetal force determined by the reference point and the reference heading rate counteracts the influence of the varying reference heading rate. Consequently, the guidance law can be applied to accurately track a general curve. While, the reference heading rate is not considered in the NLGL. It is the reason for the large cross-track error when the NLGL is applied in curve tracking, especially when the reference curvature varies. 4.3
Combined obstacle avoidance
In this scenario, an obstacle with a radius of 100 is located at (400, 250). The reference trajectory is same with the curve in Sec. 4.1. The modified virtual force guidance law (19) is applied with parameters as follows: √ k = 0.1, c = 2 k, ko = 10, dsa f = 5, Rd = 200 (21) Figure 10 shows the result of obstacle avoidance during trajectory tracking. The vehicle is pushed out of its way by the repulsive force to get around the obstacle. After that, it converges back to the reference curve.
5
CONCLUSION
In this paper, we focused on the guidance of unicycle-type autonomous vehicles. A virtual force guidance law was proposed for trajectory tracking and path following. Virtual forces were designed to determine the velocity and heading rate command inputs. The use of artificial physics made the guidance law be founded on solid physical theory and computationally simple. Besides, the physical meanings of the parameters are definite, which makes it easy to tune in application. Simulation results demonstrated the performance of guidance law for trajectory tacking, path following and obstacle avoidance. The noneconomic constraints and the dynamic environment with dynamic obstacles will be considered in the future work. Further results will be tested on the platform developed in [13, 14].
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