The problem of path planning for an unmanned aerial vehicle (UAV) under risks has ... The UAV is assumed to move at a constant linear velocity, with the path ...
ISSN 10642307, Journal of Computer and Systems Sciences International, 2012, Vol. 51, No. 2, pp. 328–338. © Pleiades Publishing, Ltd., 2012. Original Russian Text © M.A. Andreev, A.B. Miller, B.M. Miller, K.V. Stepanyan, 2012, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2012, No. 2, pp. 166–176.
CONTROL SYSTEMS OF MOVING OBJECTS
Path Planning for Unmanned Aerial Vehicle under Complicated Conditions and Hazards M. A. Andreev, A. B. Miller, B. M. Miller, and K. V. Stepanyan Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994 Russia Monash University, Melbourne, Australia Received July 5, 2011
Abstract—The flight of an unmanned aerial vehicle under complicated conditions and hazards is con sidered. Hazards are given in terms of 2D relief. To find the optimal 2D path minimizing the risk given constraints on flight time and velocity, the problem with the nongiven boundary condition is trans formed to the problem with the fixed flight time and the boundary problem solved numerically. The found 2D path is used to construct the polynomial approximation of the 3D path taking into account the local relief. DOI: 10.1134/S1064230712010030
INTRODUCTION The problem of path planning for an unmanned aerial vehicle (UAV) under risks has been known since long ago [1, 2] and has remained the subject of research in recent years, especially as applied to path plan ning for autonomous UAV [3–7]. Starting from original works on path planning, this problem is stated as the problem of determining the path of a dynamic system with the given initial and terminal conditions that minimizes some functional characterizing integral risk and terminal miss. Applying methods of opti mal control theory requires knowing hazard rate and its derivatives. At this stage, the problem of deter mining the optimal admissible path for the given stationary risk distribution and given local relief is solved. The UAV is assumed to move at a constant linear velocity, with the path chosen so that to minimize the integral risk. To solve the problem, we apply numerical methods based on solving the boundary problem resulted from the necessary optimality condition in the form of the maximum principle. 1. TYPICAL RISK DISTRIBUTION The typical risk distribution is characterized by the following set of parameters: coordinates of hazard centers, spatial risk distribution. For the given coordinates (xi , yi ) of the center of the ith source of hazards (i = 1, … , M ), the spatial risk distribution can be of different types such as Gaussian [3], i.e., the spatial risk distribution is described by the probability density of a Gaussian ran dom variable with the mathematical expectation at the point (xi , yi ) and some rootmeansquare devia tions, rational [4], which is closer to natural physical models and is given by the relation f i (x, y) =
Vi m , ρ i (x, y)2
where ρi (x, y) is the distance from the current point to the center of the ith source of hazards and Vi is the relative velocity of UAV and the ith source of hazards to the power of m, modified rational [7] f i (x, y) = C(θ)
Vi m , ρ i (x, y)2
where C (θ) depends on the object orientation and is equivalent to the efficient reflection coefficient. 328
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Each representation has its own advantages and drawbacks discussed in [8]. Based on preliminary study for the given coordinates (xi , yi ) of the center of the ith source of hazards (i = 1, … , M ), the spatial risk distribution is given by
ai (1.1) . bi + ci (x − xi )2 + di (y − yi )2 This representation have no singularities that hamper numerical solution of the problem (see, for instance, [4]) and is characterized by natural physically real velocity of decrease for big distances from the hazard center, unlike the Gaussian distribution [3]. Other hazard rates depend on mutual location of UAV and the source, and simulating them requires knowing the UAV form and spatial orientation and is rather difficult [5–7]. We can add (1.1) with the rotation of the coordinate axes for an individual hazard f i (x, y) =
ai , bi + x T RKx where the designations are f i (x, y) =
⎛ cos ϕ sin ϕ ⎞ R=⎜ ⎟, ⎝ − sin ϕ cos ϕ ⎠
⎛c i
0 ⎞⎟ ⎟⎟ , ⎝ 0 di ⎠
K = ⎜⎜⎜
− xi ⎞⎟ . ⎜ y − y ⎟⎟ ⎝ i⎠ ⎛x
x = ⎜⎜
1.1. Total Hazard Rate We assume that the total hazard rate at the given point of space ( x, y ) is the sum of individual hazard intensities M
f (x, y) =
∑ f (x, y) . i
i =1
From the probabilistic point, this assumption corresponds to independent hazard sources [3] and the Markovian model of total risk generation. In this model, the total probability of mission failure is a mono tone function of the integral along the path (x(t ), y(t), z(t )) t ∈ [0, T ], T
J =
∫ f (x(t), y(t))dt. 0
We also assume that the local relief is either described by a known function
h = h(x, y) , or given as a digital map with the altitudes h j = h(x j , y j ) at some set of points (x j , y j ), j = 1, …, N . Note that in this statement the hazard distribution is independent of the altitude and therefore we actu ally use the projection of the path onto the plane z = 0 rather than the path itself since the flight range is not assumed to be big and we do not have to take into account the curvature of the Earth. 2. MOTION MODEL AND QUALITY CRITERION 2.1. UAV Kinematic Model At the first stage, we use the simplest model of controlled motion described by the equations
x(t ) = V cos γ cos θ, y(t ) = V sin γ cos θ, z(t ) = V sin θ, where V ∈ [Vmin ,Vmax ] is the given constant velocity and the controls γ ∈ [−π, π], θ ∈ [−π/2, π/2] are the course (jaw) angle and pitch angle, respectively. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Fig. 1. The contour map of risk distribution for hazard sources (2.2).
The initial position A = (x(0), y(0), z(0)) = (x0, y0, z 0 ) and the final position B = (x(T), y(T), z(T)) = (xT, yT, zT) are known. Constraints on the admissible flight time T ∈ [Tmin , Tmax ] and velocity V ∈ [Vmin ,Vmax ] are also given and satisfy the relations
L = (xT − x(0)) + (yT − y(0)) + (zT − z(0)) ≤ VmaxTmin , 2
2
2
L = (xT − x(0)) + (yT − y(0)) + (zT − z(0)) ≤ VminTmax . 2
2
2
2.2. Quality Criterion The control problem is to find the path that satisfies the initial and terminal constraints, trivial con straint on the flight altitude z(t ) ≥ h(x(t ), y(t )) and, probably, the additional constraint z(t) ≤ h(x(t), y(t)) , and minimizes the quality criterion T
J =
∫ [ f (x(t), y(t)) + φ(x(t), y(t), z(t))]dt + k[(x(T ) − x
T)
2
]
+ (y(T ) − yT ) + (z(T ) − zT ) , 2
2
(2.1)
0
where the first term under the integral is the typical risk distribution characterized by the coordinates of hazard centers and spatial risk distribution. The function φ(x, y, z) is a penalty function that characterizes the penalty for deviation from the given altitude range. The terminal summand in quality criterion (2.1) with the value k Ⰷ 1 characterizes the penalty for deviation of the path from the final position B = (xT , yT , zT ).. Figure 1 shows the example of the contour map of the spatial risk distribution for the set of hazards 4. 0 2. 0 f (x, y) = + 2 2 2 2 1. 0 + ( x + 1.3) + ( y + 1.3) 1.0 + ( x − 1.9) + ( y − 1.6) (2.2) 1 . 0 1.0 + + 2 2 2 2 1.0 + ( x − 0.4) + ( y − 0.1) 0.5 + ( y − 0.1) + ( x − 0.6) JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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3. REDUCING TO THE PROBLEM OF FINDING 2D PATH Since the solution of the initial problem turned out to be very sensitive to auxiliary variables (the change by 10–3 at the initial point results in the change by unities at the final point, which makes it inapplicable from practical point [9]), we decompose the problem into two parts—forming a plane path and control ling by the altitude along the found plane path. 3.1. UAV Kinematic Model To solve the first subproblem, we use the model of plane motion with the constant linear velocity V x(t ) = V cos γ, y(t ) = V sin γ,
(3.1)
where the control γ ∈ [−π, π] is the course (jaw) angle. 3.2. Statement of the Problem We assume that the initial position A = ( x(0), y(0)) and the final position B = (xT , yT ) are known and the constraints on the admissible flight time T ∈ [Tmin , Tmax ] and velocity V ∈ [V min ,Vmax ] are given and satisfy the relations
L = (xT − x(0)) + (yT − y(0)) ≤ VmaxTmin , 2
2
L = (xT − x(0)) + (yT − y(0)) ≤ VminTmax . 2
2
The problem is to find the time T * ∈ [Tmin , Tmax ] and control γ(t ), t ∈ [0, T *] that minimize the quality criterion
min J (T *,V , γ() ⋅ , k)
T ∗, V , γ(⋅)
for the given fixed k. Since this problem belongs to the class of problems with nonfixed time and param eters, it is quite difficult to solve it because of additional boundary conditions it entails [10, sections 2.7 and 2.8]. We simplify the statement, reducing the problem to the problem of optimal control with fixed time. 3.3. Auxiliary Problem with Fixed Time We change the variables s = t / T *, in integral (2.1). We have 1
1
∫
∫
J = T * f (x(sT *), y(sT *))ds = T * f (x(s), y(s))ds, 0
0
where the variables ( x(s), y(s)) satisfy the boundary conditions (x(0), y(0)) = (x(0), y(0)), and equations
(x(1), y(1)) = (x(T ), y(T ))
x(s) = T * V cos γ(sT *) = V cos γ(s),
(3.2)
y(s) = T * V sin γ(sT *) = V sin γ(s).
(3.3)
Here, V is chosen from the condition of problem solubility
V ∈ [min{VmaxTmin ,VminTmax }, max{VmaxTmin ,VminTmax }]. The auxiliary problem with fixed time is stated as the problem of minimizing the criterion
(3.4)
1
J
aux
=
∫ f (x(s), y(s))ds, 0
on the paths that satisfy (3.3) and terminal conditions (3.2) with the chosen modified velocity V that actu ally equals the length of the path satisfying condition (3.4). The solution of the auxiliary problem yields JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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the optimal value of the modified velocity V and the optimal control γ(s), s ∈ [0, 1], and the minimal value aux of the quality criterion J . 3.4. Finding Optimal Control in the Auxiliary Problem by the Maximum Principle We solve the optimal control problem for the fixed parameter V . Since problems with terminal condi tions are rather difficult, we modify the quality criterion, adding the penalty for the terminal miss 1 aux J =
1
∫ f (x(s), y(s))ds + Φ(x(1), y(1)) = ∫ f (x(s), y(s))ds + k[(x(1) − x(T )) 0
2
+ (y(1) − y(T )) ] , 2
0
where the coefficient k Ⰷ 1 should be chosen sufficiently big. We introduce the Hamiltonian
ψ x , ψ y ) = ψ xV cos γ + ψ yV sin γ − f (x, y) . H (x, y, γ, We solve the problem based on the necessary condition in the form of the Pontryagin maximum prin ciple [10]: if (x, y, γ ) are the optimal path and control, there exist conjugate variables ψ x (s), ψ y (s) satisfying the system of equations x(s) = ψ
∂ f (x, y) , ∂ x ( x(s), y(s))
ψ y(t ) =
∂ f (x, y) . ∂ y ( x(s), y(s))
(3.5)
The optimal control maximizes the Hamiltonian, i.e., for any s ∈ [0, 1]
ψ x (s), ψ y (s)) , γ(s) = argmax H (x(s), y(s), γ,
(3.6)
γ
hence
ψ x (s) , 2 ψ x (s) + ψ 2y (s)
cos γ(s) =
sin γ(s) =
ψ y (s) ψ 2x (s)
+ ψ 2y (s)
.
Closing system of equations (3.3), (3.5) by optimal control (3.6), we obtain the boundary problem sat isfied by the path in the auxiliary problem x(s) = x(s) = ψ
V ψ x (s) ψ 2x (s) + ψ 2y (s)
y(s) =
,
∂ f (x, y) , ∂ x ( x(s), y(s))
y(t ) = ψ
V ψ y (s) ψ 2x (s) + ψ 2y (s)
,
∂ f (x, y) . ∂ y ( x(s), y(s))
Initial conditions x(0) = x(0),
y(0) = y(0) .
Terminal conditions
ψ x (1) = −k((x(1) − x(T )),
ψ y (1) = −k((y(1) − y(T )) .
We solve the boundary problem numerically for different values V satisfying constraints (3.4) and find aux the value V such that J attains its minimum. To do it, we need to solve the problem of parametric opti mization with respect to the scalar parameter V . 3.5. Describing the Algorithm That Finds Parameter We choose the coefficient k included in the criterion preliminary and fix it. S t e p 1. Find the limits of possible values of the parameter (T , V ) from the constraints on the pair V . JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Fig. 2. The family of optimal paths for different values V ∈ [9 . 2, 14 . 56] for the flight from the point A = (− 3, − 4) to the point B = (3, 3).
V: 9.220; J: 2.145; VJ: 19.771 V: 9.474; J: 1.435; VJ: 13.591 V: 13.289; J: 0.466; VJ: 6.192 V: 13.543; J: 1.107; VJ: 14.999 V: 13.798; J: 1.173; VJ: 16.191 V: 14.052; J: 0.417; VJ: 5.865 V: 14.561; J: 0.438; VJ: 6.371 Threats Fig. 3. The table of values of the parameter V , criteria and respective designations for the family of paths given in Fig. 2.
S t e p 2. Choose some admissible value V, solve the auxiliary problem min J aux (x(s), y(s), k) x(s ), y(s )
and calculate the product V min J (V ). S t e p 3. Repeat the previous step for another admissible value of the parameter V and find the subop timal value of the parameter by enumeration ⎡ ⎤ min ⎢V min J aux (V )⎥ . V ⎣ ( x(s), y(s )) ⎦ S t e p 4. For the obtained x(s), y(s), γ(s),V , determine the solution to the initial problem by inverse interchange. Figure 2 shows the family of optimal paths in the auxiliary problem for different values of the parameter V calculated using Matlab. This problem is enumerative, with no regular dependence on parameter (see Fig. 3), and is easyto parallelize [11]. aux
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3.6. Finding Optimal Control and Path in the Initial Problem To solve the initial problem, we need to give the flight time T*, find the flight velocity V = V / T * and the optimal control γ(t) = γ(t /T *). The value of the integral risk is J = T *J aux , therefore the optimal flight time and velocity can be found from the relations
T * = arg min{T : T ∈ [Tmin ,Tmax ],V ∈ [Vmin ,Vmax ],VT = V}. To obtain the optimal path and optimal control, we need to change the variables t = sT * and, hence, x(t ) = x(t / T *),
y(t ) = y(t / T *),
γ(t ) = γ(t / T *) .
We find the variables x(t ) = x(t / T *), y(t ) = y(t / T *) from system of equations (3.1), where cos γ(t ) = cos γ(t / T *) = sin γ(t ) = sin γ(t / T *) =
ψ x (t / T *) 2 ψ x (t / T *) + ψ 2y (t / T *) ψ y (t / T *) 2 ψ x (t / T *)
+
2 ψ y (t / T *)
= =
ψ x(t ) , 2 2 ψ x(t ) + ψ y(t ) ψ y(t ) 2 ψ x(t )
+ ψ2y(t )
.
The conjugate variables (ψ x(t ), ψ y(t )) = (ψ x (t / T *), ψ y (t / T *)) satisfy the system of equations x(t ) = 1 ∂ f (x, y) , ψ T * ∂ x ( x(t ), y(t ))
y(t ) = 1 ∂ f (x, y) . ψ T * ∂ y ( x(t ), y(t ))
4. CALCULATING 3D PATH GIVEN THE RELIEF If the local relief is represented as an analytical function, we can find the altitude at any point. However, generally we have only the set of altitudes given on some set of points. In this case, we have the problem of determining the altitude at intermediate points that do not belong to this set. 4.1. Relief Representation One of the ways to represent the relief is to give the set of altitudes on the rectangular grid and calculate altitudes at intermediate points by approximation. Suppose we have a uniform rectangular grid. The value of the altitude at the grid nodes is known or is calculated using the analytical function. Further, all points of the grid are connected by noncrossing segments so that new segments cannot be added without cross ing the existing ones. Connection is done using Delaunay triangulation. D e f i n i t i o n 1. The Delaunay triangulation [12] for the set of points S on the plane is the triangula tion DT(S) such that no point A from S is inside the circle circumscribed around any triangle from DT(S) such that the point A is not either of its vertices. The Delaunay triangulation allows approximating the relief even when the altitudes are given on an arbitrary nonuniform grid, for instance, as the coordinates of elevations and depressions. The respective Matlab program solves this problem for an arbitrary set of points on the plane. We use the uniform grid. 4.2. Calculating Altitudes along Optimal 2D Path On the given rectangular grid, we construct a plane path in the form ( x(t ), y(t )). For its arbitrary points, we find the points (xi , y j ) such that
x(t) ∈ [xi , xi +1],
y(t) ∈ [y j , y j +1],
where
xi +1 − xi = hx ,
y j +1 − y j = hy .
for the uniform grid. For the obtained set of points lying on the path, we find triangles that include these points. The triangles are colored yellow (see Fig. 4). JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Fig. 4. Giving the triangular representation along the optimal 2D path.
4.3. Calculating Approximated Altitude along Optimal 2D Path We use the altitude approximated by the method of triangles when the value at the intermediate point is a convex combination of altitudes at the triangle angles with weights corresponding to weights of the approximated point. For each given point (xi , y j ) , we find the triangle with altitudes (x A , y A ), (x B , y B ), and (xC , yC ) . Then, we calculate the coefficients of the convex combination of angular points that gives the point ( x(t ), y(t )), i.e., we solve the system of equations
⎧(x(t), y(t)) = α1(x A , y A ) + α 2(x B , y B ) + α3(xC , yC ) , ⎪ ⎨α1 + α 2 + α 3 = 1, ⎪⎩α1 ≥ 0, α 2 ≥ 0, α3 ≥ 0 . Finally, we obtain the system of equations
⎧x(t) = α1x A + α 2 x B + α3 xC , ⎪⎪y(t ) = α y + α y + α y , 1 A 2 B 3 C ⎨ ⎪α1 + α 2 + α 3 = 1, ⎪⎩α1 ≥ 0, α 2 ≥ 0, α 3 ≥ 0. We use the solution of the system to find the altitude at the point ( x(t ), y(t )) by the formula
h(x(t), y(t)) = α1h(x A , y A ) + α 2h(x B , y B ) + α3h(xC , yC ) . Enumerating the points of the path, we construct the profile of altitudes
z(t) = h(x(t), y(t)) . Figure 5 shows the calculated profile of altitudes in the given relief and its approximation. 4.4. Calculating 3D Path Having the profile of altitudes that bounds the path below, we choose points along the 2D path and give the altitudes at them. This is done manually by the task scheduler, with various options possible. The sim plest one is a horizontal flight at the fixed altitude. Another option is terrain following when UAV is to keep JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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0.1
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Fig. 5. The profile of altitudes along the 2D path.
to a certain altitude with respect to the relief. The coordinates of the chosen points and the altitudes form the set of threedimensional points used to construct the threedimensional polynomial approximation. For each coordinate, we construct its polynomial, for instance, the Lagrange polynomial. Knowing the polynomials, we can easily find the required derivatives for the velocity. All we need now is to ensure motion at the constant linear velocity. Thus, the 3D path is described by the system of equations
Vx(μ(t))
⎧x(t) = ⎪ ⎪ ⎪ ⎪y(t) = ⎪ ⎨ ⎪z(t) = ⎪ ⎪ ⎪μ (t) = ⎪⎩
2 2 x(μ(t)) + y(μ(t)) + z(μ(t )) Vy(μ(t)) 2
x(μ(t))2 + y(μ(t))2 + z(μ(t ))2 Vz(μ(t))
, ,
, 2 2 2 x(μ(t)) + y(μ(t)) + z(μ(t )) V , 2 2 2 x(μ(t)) + y(μ(t)) + z(μ(t ))
where T is the given flight time, the parameter V = L / T , and the value 1
L=
∫
2 2 2 x(s) + y(s) + z(s) ds
0
can be found using the known polynomials. 4.5. Example of Calculating 3D Path Figures 6 and 7 give the example of calculating the 3D path. Hazards are given by relations (2.2), with the expression
h(x, y) =
1 . 1 + 0. 5x + (y + 1)2 2
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Fig. 6. The graph of the polynomial approximation of the 3D path in the given local relief (• − • − •.. ) and the found 2D path.
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Fig. 7. The graph of the polynomial approximation of the 3D path in the hazard environment. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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CONCLUSIONS In this work, we propose the approach to forming the reference path for the autonomous UAV taking into account the relief and hazards. The work is triggered by the need to construct simple methods appli cable in real situations of mission planning. We give the sequence of steps that allow forming such path, viz. setting the coordinates of centers of hazard sources and their characteristics, setting the initial and final point of the UAV path, solving the twodimensional boundary problem and determining a “plane” component called 2D path here, constructing the profile of altitudes along the found 2D path, choosing points on it and setting the alti tudes at the chosen points, using polynomial approximation to construct 3D path based on the existing set of threedimensional points. Since at the last stage only the set of threedimensional points is required to construct the 3D path, the software, which is now being debugged, can be used to compare the paths obtained without solving opti mization problem, i.e., based on some reasoning, with respect to criterion. ACKNOWLEDGMENTS This study was partially supported by the Australian Research Council (grant no. DP0988685) and the Russian Foundation for Basic Research (grant no. 100100710). A.B. Miller and K.V. Stepanyan partially held the research during their visit to the Monash University (Melbourne, Australia) as invited researches. REFERENCES 1. A. M. Letov, Flight and Control Dynamics (Nauka, Moscow 1969) [in Russian]. 2. J. Vian and J. Moore, “Trajectory Optimization with Risk Minimization for Military Aircraft,” AIAA J. Guid ance 12 (3), 311–317 (1989). 3. A. Dogan and U. Zengin, “Unmanned Aerial Vehicle Dynamic–Target Pursuit by Using Probabilistic Threat Exposure Map,” AIAA J. Guidance, Control and Dynamics 29 (4), 723–732 (2006). 4. A. Galayev, E. Maslov, and E. Rubinovich, “On a Motion Control Problem for An Objects in a Conflict Envi ronment,” J. Computers and Systems Sciences International 40 (3), 458–464 (2009). 5. M. Zabarankin, S. Uryasev, and P. Pardaios, “Optimal Risk Path Algorithm,” in Cooperative Control and Opti mization, Ed. by R. Murphay and P. Pardaios, (Kluwer, Dordrecht, 2002), pp. 271–303. 6. R. Murphey, S. Uryasev, and M. Zabarankin, “Trajectory Optimization in a Threat Environment,” Research Report, 20039. 7. M. Zabarankin, S. Uryasev, and R. Murphey, “Aircraft Routing Under the Risk of Detection,” Naval Research Logistics 53, 728–747 (2006). 8. B. M. Miller, K. V. Stepanyan, and A. B. Miller, “Simulation of Permissible UAV Trajectories,” in Proceedings of 8th International Conference on Nonequlibrium Processes in Nozzles and Jets NPNJ’2010, Alushta, Ukraine, 2010, pp. 321–323. 9. K. V. Stepanyan, A. B. Miller, and B. M. Miller, “Planning Trajectories of a Pilotless Vehicle in Complex Con ditions under Presence of Threats,” in Proceedings of 33rd Conference of Young Scientists and Specialists of IPPI RAS on Information Technologies and Systems, Gelendzhik, Russia, 2010, pp. 263–268. 10. A. Bryson and Ho YuChi, Applied Optimal Control (Hemisphere Washington, 1969). 11. B. M. Miller, K. V. Stepanyan, A. B. Miller, et al., “Parallel Implementation of the UAV Path Planning in a Haz ard Environment,” in Proceedings of 8th International Conference on Nonequlibrium Processes in Nozzles and Jets NPNJ’, Alushta, Ukraine, 2010, pp. 698–700. 12. B. N. Delone, “On Emptiness of the Sphere,” Izv. Akad. Nauk SSSR OMEN, No. 4, 793–800 (1934).
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