Online Task Planning and Control for Aerial Robots with Fuel Constraints in Winds

Transcription

1 Online Task Planning and Control for Aerial Robots with Fuel Constraints in Winds Chanyeol Yoo, Robert Fitch, and Salah Sukkarieh Australian Centre for Field Robotics, The University of Sydney, Australia, Abstract. Real-world applications of aerial robots must consider operational constraints such as fuel level during task planning. This paper presents an algorithm for automatically synthesising a continuous nonlinear flight controller given a complex temporal logic task specification that can include contingency planning rules. Our method is a hybrid controller where fuel level is treated continuously in the low-level and symbolically in the high-level. The low-level controller assumes the availability of a set of point-estimates of wind velocity and builds a continuous interpolation using Gaussian process regression. Fuel burn and aircraft dynamics are modelled under physically realistic assumptions. Our algorithm is efficient and we show empirically that it is feasible for online execution and replanning. We present simulation examples of navigation in a wind field and surveillance with fuel constraints. Introduction Autonomous aerial robots have real potential to replace human-piloted aircraft in important applications such as cargo flights, surveillance for border protection, environmental monitoring, and commercial aviation. In these applications, aerial robots must not only be able to navigate autonomously but also must be able to execute complex tasks that involve contingency planning and rules governing operation in controlled airspace. Recent work has pioneered the application of formal methods to automatically synthesise controllers for such complex tasks. We are interested in efficient automatic synthesis of controllers for unmanned aerial vehicles (UAVs) subject to fuel constraints. Fuel constraints are important for UAVs because violation can lead to catastrophic failure. We would like to specify tasks that guarantee safe operation such as returning to base when fuel level drops below a threshold, and ensuring that a suitable landing site is reachable at all times in the event of an emergency. Important work in robotics has explored hybrid controllers where rich tasks are specified as linear temporal logic (LTL) formulas at a high-level discrete layer, and continuous controllers are designed or synthesised that execute the highlevel behaviours [ ]. Fuel constraints introduce a challenging case because it is undesirable to model such continuous values discretely in the high-level [4], yet task specifications must be able to encode behavioural goals with respect to these values. It is possible to treat this as a reactive task, where change in fuel

2 level is viewed as a change in the environment to which the high-level controller must react. But synthesis of reactive controllers generally is computationally expensive [5], limiting its potential for online execution. Designing low-level controllers is also challenging in this case because UAV dynamics depend on the gross weight, which decreases as fuel is burned (for non-electrically powered UAVs). Further, the behaviour of the UAV strongly depends on wind conditions such as tail winds and head winds. This optimal control problem, known as Zermelo s problem, is a two-point boundary value problem typically solved using shooting methods. In this paper we address these challenges and present efficient algorithms that synthesise correct task-level behaviour from LTL formulas for a UAV under physically realistic assumptions. We define a reactive task-level controller that is coupled to a low-level flight controller through operational state variables. The operational state of the robot is modelled in continuous form in the flight control layer, and also represented symbolically in the task layer. The task layer reacts to changes in operational state, such as if the fuel level drops below a certain value, in a way that satisfies the given LTL task specification. The flight controller plans a path for the robot given wind velocity predictions interpolated from point estimates using Gaussian process regression [6]. Change in gross weight of the robot due to fuel burn over time is modelled analytically using the well-known Breguet range equation. UAV dynamics are modelled using a set of non-linear differential equations and solved numerically. Reactive task-level synthesis is performed using a Büchi automaton, but not by constructing a product of automata as is typical. This approach drastically improves the efficiency of synthesis for the purpose of enabling online execution during flight. The main limitation of this approach is that efficiency gain comes at the cost of completeness. However, correctness at the task level is preserved. We have implemented our algorithms and report results from two simulation examples. The first example shows navigation with obstacle avoidance and illustrates how the wind field influences the trajectory. The second example shows how fuel constraints are maintained during a persistent surveillance task, where the UAV must return to a base when fuel level drops below a given threshold. We report clock time results that indicate the feasibility of our method in practice. The two main contributions of this work are our novel method of coupling the continuous operational (fuel) state of the robot with discrete task-level synthesis, and its efficient application to UAVs with a realistic model of wind effects on UAV dynamics. To the best of our knowledge, this work is the first such application. Related work Temporal logic is a class of logic that extends propositional or predicate logic with temporal properties [7]. LTL is a widely used form of temporal logic that is suitable in specifying linear time properties [8]. Unlike a formula in classical logic which determines the truth value of a set of Boolean variables at a given time, an LTL formula returns the truth value of an infinite trace (or sequence)

3 of a set of Boolean variables. Such expressivity allows for specifying a number of interesting behaviours such as functional correctness, liveness, safety, fairness, reachability and real-time properties. A system with a controller can be verified or model-checked [7] against a given LTL formula to show the absence of error. Verification methods can also be used to synthesise a controller that formally guarantees correct behaviour. In robotics, temporal logic is important for task-level planning with missions that can be specified with natural language [9 ]. Temporal logic is expressive enough to specify a number of tasks such as coverage, sequencing, conditioning and avoidance []. The time complexity of synthesis is doubly-exponential in the size of the formula in the general case [], limiting its use in practice. There are a number of techniques for faster synthesis by restricting the form of LTL formulas [, 4, 5]. For example in [, 4], a restricted class of LTL called generalised reactivity() is used to synthesise a reactive controller with polynomial time complexity. Synthesised controllers are often executed in continuous space [ ], typically with simple dynamics assuming no external factors such as wind. In [6, 7], optimal controllers are synthesised to minimise a cost function in a weighted transition system, but also do not consider external disturbances such as wind. Other work synthesises controllers that guarantee the execution of an LTL formula for a class of dynamical systems [8], but does not consider the coupling between low-level operational and task-level states of the robot. Our work considers this coupled case with continuous execution under the influence of a continuous wind field assuming realistic aircraft dynamics and fuel models. Problem Formulation Suppose we have a UAV in an environment with a complex mission such as surveillance and sequencing under the influence of wind and fuel constraints. The UAV is required to synthesise a task-level planner for the mission and a low-level controller for actuation. In this paper, we develop a two-layered online synthesis algorithm which consists of discrete synthesis and continuous execution. In discrete synthesis, the environment and the UAV dynamics are discretised and wind vectors are approximated for each discrete state. The algorithm is to find a sequence of discrete states satisfying the task-level mission specification. In particular, the sequence minimises the fuel consumption, with the presence of wind affecting the fuel consumption. We present an efficient algorithm to plan at the task-level with the given complex mission specification so that the planning can be done online. In continuous execution, the sequence of discrete synthesis is realised with continuous dynamics of the UAV and the influence of continuous wind. In this work, we focus on a practical flight mission where the UAV is to start a landing procedure when the fuel goes below a certain threshold. Therefore we have a mission of a form If the fuel level is above a threshold, mission φ is taken. If not, a landing procedure φ e starts.

4 In this section, we introduce LTL and Büchi automata for the purpose of expressing complex missions with natural language. We then state the fuel model of the UAV and the interpolation methods for the wind field. Lastly the UAV dynamics and the controller model are defined.. Linear Temporal Logic (LTL) LTL is an extension of a classical propositional logic that expresses and reasons about the behaviour of systems over time [7]. An LTL formula φ can be an atomic proposition a AP and can be formed with operators. The standard Boolean operators such as negation ( φ), conjunction (φ φ ) and disjunction (φ φ ) can be used. The operators such as implication (φ φ ) and equivalence (φ φ ) can be constructed. Temporal operators consist of in next ( φ) and until (φ U φ ). Additional operators such as in future ( φ) and always ( φ) can be constructed from the prior operators: φ = true U φ and ϕ = ϕ. LTL is used to express a variety of robotic tasks such as coverage, sequencing, conditions and avoidance. For example, p room room denotes that room, room and room are reachable in any order (coverage), (room (room room )) denotes that room, room and room are reached in order (sequencing), (room room ) denotes that room will be visited immediately if currently in room, and dangeruroom denotes that there is no danger until reaching room. More complex missions can be expressed with nesting, conjunction/disjunction and negation of multiple LTL formulas as defined in the syntax. For example, a surveillance mission can be written as ( room room ) which denotes rooms are visited infinitely often.. Deterministic Büchi Automaton A deterministic Büchi automaton B is a tuple < Q, q, Σ, δ, F >, where Q is a finite set of states, q Q is an initial state, A Σ = AP is a set of input alphabets, δ : Q Σ Q is a deterministic transition relation and F Q is a set of accepting states. In order to solve for the truth of an infinite sequence of states over an LTL formula, an equivalent Büchi automaton is built which accepts all and only the infinite sequences of words ω where ω i Σ satisfies the given formula. An infinite sequence is said to be accepted by a Büchi automaton if and only if the accepting states are visited infinitely often. We define an additional function trans : q Σ that returns a set of input alphabets that allows a transition from state q to q where q q. Similarly we define stay : q Σ that returns a set of input alphabets that leads q = q. Finally we have allowed : q Σ where allowed(q) = trans(q) stay(q). Note that trans(q) stay(q) =. A Büchi automaton for an LTL formula ( a b) is shown in Fig. where q is an initial state and q is an accepting state. Any word (or sequence) of infinite length that visits q would be an accepting word. For example, a word ω =

5 a^b a a^b q a^ b b q a a^ b b q Fig.. Constructed Büchi automaton of LTL formula ( a b) is shown. Starting from initial Büchi state q, the accepting state q has to be visited infinitely often by the word ω of an infinite length. ababab... repeating ab is an accepting sequence of the formula since the accepting state q is visited infinitely often.. Breguet Range Equation Suppose a petrol-powered UAV is in operation at constant altitude and air speed, subject to wind currents. Since fuel is consumed over time, the change in the mass of the fuel affects the flight dynamics of the UAV significantly. The relationship between the ground distance travelled and the mass of fuel is represented by the Breguet Range equation d g = v g C a log M i M f () where d g is the ground distance travelled, v g is the ground velocity, and M i and M f are the initial and the final mass of the fuel respectively. We have C a = I sp L/D where I sp is the specific impulse, L/D is the lift-to-drag ratio. With the presence of a tail wind, the ground velocity is re-written as v g = v a + v w where v a is the air velocity and v w is the tail wind velocity. The mass after travelling an infinitesimal ground distance (dx) or time (dt) is shown as dx M f = M i exp( ) (v a + v w ) C a = M i exp( dt C a ). ().4 Wind Field Interpolation We use Gaussian process regression to interpolate wind field values given a number of observation points. The wind vector is assumed to be time-invariant and noise-free [6]. We use the typical squared exponential covariance function k(x, x ) = exp( λ x x ) where λ is a length scale. Suppose we are interested in a wind vector at a point x with a number of observation points X and the corresponding observed wind vectors Y. We have the following equation: V w (x ) = K(x, X)[K(X, X)] Y, ()

6 where K is the covariance matrix with components k(x, x ) for all x, x X. Note that x and y dimensions are independent and share the same λ. The value of the length scale is not optimised since it is not in the scope of this paper. Assuming that the wind field is smooth and does not vary rapidly, a large value is suitable for the interpolation. In particular, we assume that the wind field values are spatially correlated and that the length scale is approximately the distance between two nearest observations. Given the environment is discretised into a grid with a set of discrete states S, the mean wind vector for each discrete state is V w [s] = x X s V w (x) dx D, (4) d= Xd s where s S is a discrete state, X s is the bounds for the cell s where X d s is the length of the cell in the d-dimension..5 UAV Dynamics and Controller Model The UAV is assumed to maintain a constant altitude and a constant airspeed v a. Therefore the state vector for the UAV is X = [x, y, ψ] T where ψ is the heading angle. The control input is u = ψ U where U is a set of possible turn rates. If the UAV is moving in a wind field represented with a function V w (x, y) as interpolated in Sec..4, the dynamics of the UAV become x (t) = v a cos(ψ(t)) + V wx (x(t), y(t)) y (t) = v a sin(ψ(t)) + V wy (x(t), y(t)) ψ (t) = u(t) U (5) where V wx and V wy are the tail wind in the x and y axis respectively. Since the system of differential equations is non-linear, we solve them numerically: x[t + t] = (v a cos(ψ[t]) + V wx (x[t], y[t])) t + x[t] y[t + t] = (v a sin(ψ[t]) + V wy (x[t], y[t])) t + y[t] ψ[t + t] = u[t] t + ψ[t]. (6) 4 Controller Synthesis for Continuous Trajectories 4. Discrete Synthesis As mentioned in Sec., the normal flight mission is to be aborted when the fuel level is below a threshold and a landing procedure should then begin. The mission is expressed in LTL as: ( e α φ) (e α φ e ) (7)

7 s {a} s 4 {c} s 7 a^b s s 5 s 8 a^ c a^b q a^ c^ b c^ b q a^ c a^ c^ b a^ c^b a^b q s s 6 s 9 {b} (a) Example environment (b) Büchi automaton Fig.. (a) Simple example environment shown discretised into a grid with continuous wind vector field and mean wind vector for each discrete state (bold arrows). States s, s 9 and s 4 are labelled with a, b and c respectively. (b) A deterministic Büchi automaton of LTL formula ( cua) ( cub) where a and b have to be visited infinitely often while avoiding c. where φ is an LTL formula for the normal flight mission, d land is a proposition to avoid and g land is a proposition to reach in the landing procedure. The symbol e is a signal produced by the low-level controller when the fuel goes below a threshold α and φ e = d land U g land. Note that we solve for φ and φ e separately. More details about solving the formula are shown in Sec. 4.. We discretise a continuous-space environment into a set of discrete position states S where X s is the geometric size of each cell in the environment. For each discrete state, we calculate the mean wind vector as in Eqn. 4. Each discrete state is labelled with symbolic propositions based on the mission. We propose a greedy Büchi algorithm (GBA) to find a sequence of discrete states that minimises fuel consumption. The algorithm is optimal in one Büchi horizon, where n Büchi horizons refers to n transitions in Büchi states. The sequence is minimum fuel consuming for one transition in the Büchi automaton. A Büchi automaton is generated from a given LTL formula φ. From a discrete state s and Büchi state q, we find a sequence of discrete states that produces a finite word ω to transit to the next Büchi state q. The sequence of discrete states generated is optimal in the discrete space with respect to fuel consumption with mean wind vectors. The advantage of our approach over the typical approaches [7, ] of building a product automaton is discussed in Sec. 4.. Consider an example environment shown in Fig. (a) and a Büchi automaton in Fig. (b) of formula ( cua) ( cub) (i.e., visit a and b infinitely often while avoiding c ). From the Büchi automaton with initial state q, the set of valid input alphabets are expressed as a b, a c b and a c, where the first two expressions allow transiting to the next available Büchi state (i.e. trans(q ) = a b a c b and stay(q ) = a c).

8 s V w [s] s' V w [s'] V wx[s] V wx[s'] d g d g Fig.. A transition from discrete state s to s is shown with wind vectors V w[s] and V w[s ]. In the approximation of fuel consumption, we assume that the UAV moves from centre of state s to another. The UAV has a tail wind (i.e., same direction) when in state s and a head wind (i.e., opposite direction) when in state s. The problem is then reduced to a Markov Decision Problem (MDP). Given a Büchi state q and a labelling function L : S Σ, the discrete states satisfying L(s) trans(q) are to be reached while moving through the states satisfying L(s) stay(q) while minimising fuel consumption. Solving the MDP provides an optimal sequence to transit from the Büchi state q to another. For example, starting from s and q, the goal is to reach s while avoiding s 4 with minimum fuel consumption. One possible accepting sequence would be s s s 5 s 7 s 4 s where the produced word a is accepted by the automaton. The mean wind vectors are computed as in Sec. 4. Suppose a transition is made from a discrete state s to adjacent state s as shown in Fig.. Since the UAV is moving horizontally, the wind vectors affecting the movement in the x direction are denoted as V wx [s] and V wx [s ]. Therefore, the fuel equation for travelling between the centres of s and s from Eqn. can be re-written as d g M = M exp( ) exp( (v a ± V wd [s]) C a (v a ± V wd [s ) ]) C a = M exp( d g ( C a v a ± V wd [s] + v a ± V wd [s ] )), where ±V wd [s] is the tail wind in the direction of UAV movement. If the direction of V wd [s] is opposite to the UAV movement, then the value becomes negative. For example, the wind vector in state s from Fig. is positive since it is a tail wind whereas the vector in state s is negative since it is blowing against the movement of the UAV. In order to synthesise an optimal sequence given a discrete state s, Büchi state q and a set of approximated wind vectors, we solve for the following equation with value iteration: F [s ] exp( d g ( C a v a ± V wd [s] + v a ± V wd [s ] )) if L(s) stay(q) F [s] = if L(s) trans(q) and q / Q seq \q otherwise (9) d g (8)

9 Algorithm Synthesis of Optimal Sequence to Next Büchi State function seq GetSequence(s, q, B, Q { seq) if L(s) B.trans(q ) and q / Q seq\q : s S, F [s] otherwise : repeat : F F 4: for all s B.stay(q ) do 5: {F [s], π[s]} max d D F [s ] exp( dg ( C a v a ± V wd [s] + v a ± V wd [s ] ) 6: end for 7: until min( F F ) < ɛ 8: seq get sequence from s by following π 9: return seq where F [s] is the proportion of fuel remaining when entering the destination and q is a Büchi state in s. Based on Eqn. 9, we solve for F [s] = max d D F [s] and π [s] = arg max d D F [s] where d D is a head direction, π is an optimal control policy and Q seq is a set of visited Büchi states. Note that the set of Büchi states already visited, Q seq, is not to be re-visited until reaching an accepting state F Q. The optimal sequence is calculated by following the control policy from an initial discrete state. Pseudocode is listed as Alg.. 4. Continuous Execution In order to solve the non-linear system of differential equations shown in Eqn. 5, we introduce two assumptions that allow us to find a sub-optimal but reasonable solution. First, we assume a discrete number of available control inputs (turn rates) U = {, a, a,, a, a, } deg s. Second, the number of trajectories for finding the best trajectory is bounded by a constant limit N. Given an initial state of the UAV x = [x, y, ψ ] at a discrete state s S and the optimal sequence from discrete synthesis, we iteratively forward integrate all available control inputs u U to create a set of candidate trajectories that reach the boundary X s of the current discrete state. After each control propagation, a trajectory is pruned if the next discrete state is not the next state in the discrete sequence. Since the number of candidate trajectories grows after each iteration, we limit the number of those trajectories by selecting N least-fuel-consuming candidates and prune all others. Therefore we have at most N trajectories as opposed to U K where K is the total number of sequences throughout the mission. After each iteration, we select a trajectory with the least fuel consumption. If the fuel left is below the specified threshold, then a new discrete sequence following a landing procedure is synthesised. If not, each candidate trajectory starts a new iteration by executing all control inputs. Once a trajectory reaches the end of the discrete sequence, the next sequence is synthesised as shown in Sec. 4.. The algorithm for following the sequence is shown in Alg. where g G denotes a candidate trajectory with the UAV position, discrete state, Büchi state

10 Algorithm Continuous Execution from Sequence of Discrete States function G new ExecuteSequence(G, seq, U, N) : G G : for i to seq.length do : snext seq[i] 4: G new { } 5: for all g G do 6: for all u U do 7: g new ExecuteControlInput(g, u) 8: if g new terminates at seq[i + ] then 9: G new.add(g new) : end if : G new N-best g G new : end for : end for 4: end for 5: return G new Algorithm Overall Execution function g Execute(x, φ, φ e, α, N) : Q seq : B ConstructBuchiAutomaton(φ), B e ConstructBuchiAutomaton(φ e) : g init(x, s GetDiscreteState(x ), q B.q, fuel ) 4: G.add(g ), g g 5: repeat 6: if g.fuel α then 7: seq GetSequence(g.s, g.q, B, Q seq) 8: G ExecuteSequence(G, seq, U, N) 9: g arg min G.fuel : else : seq e GetSequence(g.s, B e.q, B e, Q seq) : G ExecuteSequence(G, seq e, U, N) : g arg min G.fuel 4: end if 5: if g.q B.F then 6: Q seq 7: else 8: Q seq.add(g.q) 9: end if : until g.fuel < α : return g and fuel level. The overall execution is presented in Alg. where φ is an LTL formula for normal operation, φ e is a formula for landing procedure, and α is a fuel level threshold to execute the landing procedure.

11 Suppose we have an example shown in Fig. 4 where the objective is to visit a and b infinitely often while avoiding c. The initial candidate trajectory starts at a state [.5,.5, deg] in which the discrete state is s. The initial sequence given from the discrete synthesis is s s 6 s 5 s s. Fig. 4(a) shows the execution of all possible turn rates U = { 6, 4,,,, 4, 6} deg s until the trajectory hits the boundary of the next discrete state. Note that the bold black line is the least-fuel-consuming trajectory. Since s 6 is preceded by s in the sequence given, all trajectories remain for the next iteration. We select the best control action that minimises fuel consumption and the best turn rate of 6 deg s is found for the first discrete state s. In the next iteration in Fig. 4(b), the candidate trajectories terminate at discrete state s 5 where all other trajectories terminating at other states are abandoned. Note that the number of candidate trajectories is limited to in this example. At the 5th iteration in Fig. 4(e), a new sequence is synthesised from the discrete synthesis after reaching the goal discrete state with the target input alphabet a (a trans(q )). From the 5th to 8th iterations, the optimal sequence is s s s s 6 s 9. The trajectories at 8th iteration are shown in Fig. 4(i). 4. Analysis The time complexity of constructing a Büchi automaton B from an LTL formula φ is O( φ ) [7]. The value iteration algorithm in Eqn. 9, known to have the complexity O(poly( S )) [9], is run to find an optimal sequence of discrete states for a transition in the Büchi automaton. Note that poly(n) means polynomial in n. Since transition to any visited Büchi state is prohibited before reaching an accepting state, the maximum number of Büchi state transitions to reach an accepting state is Q where Q is a set of Büchi states and Q < φ. The maximum number of candidate trajectories in continuous execution is restricted to N. Therefore the overall time complexity of solving for a single Büchi transition is O(poly( S ) + S U N). The space complexity is O( S + N U + φ ). We need S space to solve value iteration, N U to find the best trajectory and φ to construct a Büchi automaton. Note that typical synthesis algorithms require a construction of a product automata with size O( S φ ) [7]. As GBA does not construct a product automaton, a locally optimal sequence of discrete states can be acquired online as opposed to constructing a product automaton and searching exhaustively. Synthesis for a single Büchi transition is O(poly( S )+ S U N), so this synthesis can feasibly be performed during the execution of the previous transition in a plan-as-you-go manner. Although the size of the Büchi automaton is exponential in the size of a formula, the formula is often relatively small compared to the size of the discrete state space. If formula size is assumed to be constant, the space complexity is O(poly( S )).

12 s {a} s 4 {c} s s s 5 s s s 6 s 9 {b} (a) i =, to reach s (b) i =, to reach s (c) i =, to reach s (d) i = 4, to reach s (e) i = 5, to reach s 9 (f) i = 6, to reach s (g) i = 7, to reach s 9 (h) i = 8, to reach s 9 (i) i = 8, to reach s Fig. 4. The UAV is to visit s and s 9 infinitely often while avoiding s 4. The optimal sequence of discrete states are given prior to executing a continuous trajectory. The size of the environment is m m with 9 discrete states. The airspeed is 5ms and the available turn rates are { 6, 4,,,,, 6}deg s. The least fuel consuming trajectory is plotted with bold black line with candidate trajectories shown in red. The maximum number of candidate trajectories is limited to.

13 s s s s s 4 s 5 s 6 s 7 s 8 s 9 s s s s s 4 s 5 s 6 s 7 s 8 s s s s s s 4 s 5 s 6 s 7 s 8 s s 4 s 4 s 4 s 4 s 44 s 54 s 64 s 74 s 84 s s 5 s 5 s 5 s 5 s 45 s 55 s 65 s 75 s 85 s 95 s 6 s 6 s 6 s 6 s 46 s 56 s 66 s 76 s 86 s 96 s 7 s 7 s 7 s 7 s 47 s 57 s 67 s 77 s 87 s s 8 s 8 s 8 s 8 s 48 s 85 s 68 s 78 s 88 s s 9 s 9 s 9 s 9 s 49 s 59 s 69 s 79 s 89 s (a) s s s s 4 s 5 s 6 s 7 s 8 s 9 s (b) Fig. 5. Wind vectors drawn on the environment sized m by m. The vector field is interpolated with Gaussian process regression from observation locations. 5 Examples In this section, we present examples with a petrol-powered UAV flying at fixed airspeed and constant altitude. The size of the environment is m m and the wind field is interpolated as in Sec..4. The environment is shown in Fig. 5 where wind observation points are demonstrated with bold arrows. The environment is discretised into a grid. The airspeed of the UAV is ms, the set of control inputs is U = { 4,,,, 4} deg s and the UAV-specific constant C a is.. 5. Reach goal while avoiding danger with direction constraint We consider an initial scenario where the UAV is to avoid danger regions until reaching goal which has to be approached from runway region. With no landing procedure (i.e., α = ), the mission specification is written in LTL as φ = (goal danger) U (runway goal). () The environment is labelled with symbolic propositions on the discrete position states appropriately and the synthesis algorithm is executed starting from [m, m, deg] T in Fig. 6. For the purpose of comparison, we demonstrate two trajectories with different algorithms: one with GBA with minimum fuel consumption is shown in Fig. 6(a) and the other that takes the sequence with the minimum number of discrete states is shown in Fig. 6(b). The numbers of discrete steps to accomplish the mission are 5 and steps for GBA and the other respectively, however the proportions of fuel left are % and %. The higher efficiency can be visually observed since the trajectory with GBA

14 (a) (b) Fig. 6. Comparing two different algorithms in the same problem environment. The goal of the UAV is avoid the danger regions while approaching the goal region from the runway region. (a) uses GBA to minimise fuel consumption with 5 discrete steps and (b) takes a path with the minimum number of sequence with discrete steps. The amount of fuel left at mission completion is %(a) and % (b). follows the wind flow to minimise the effective air distance whereas the other algorithm often goes against the wind. Note that the difference in efficiency could be greater in larger environments. 5. Surveillance mission In this scenario, we demonstrate a surveillance mission where a number of locations of interest must be visited infinitely often and a landing procedure begins when the fuel goes below %. The LTL formula is written as N g φ = ( e % ( goal i )) (e % land) () i where the goal regions are at s, s 7 and s 86, and the landing base is at s 9. Figure 7 shows the result of synthesis for this mission. In Table, we show the average time to perform discrete synthesis for this scenario with different numbers of discrete states. The environment is divided into grids from 4 4 (6 discrete states) up to ( discrete states). The average flight distance is 8m and average flight time is 8s. When the number of discrete states is below, the average time to synthesise a trajectory is less than the average flight time. Therefore in these cases, synthesis can be performed during execution of the previous trajectory. This is practical for UAV tasks because the size of a grid cell must be sufficiently large to accommodate the UAV s limited turn rate and minimum velocity. UAV navigation tasks can be well represented using a relatively coarse grid.

15 Fig. 7. A continuous execution of the surveillance mission encoded in Eqn. is shown where the UAV visits three regions and begins to reach the landing zone when the fuel goes below %. The sequence of discrete states between regions is optimal w.r.t. the fuel consumption based on the approximated wind vector. The continuous trajectory follows the sequence by selecting the best control action at each discrete state. Table. Task-level synthesis time for different numbers of discrete states is shown for the problem in Sec. 5.. The total flight distance and time are approximately 8m and 8s. Partial synthesis and partial flight time refer to synthesising and executing a state sequence that corresponds to a single Büchi transition. Number of Discrete States Average Time for Partial Synthesis (s) Average Partial Flight Time (s) Time for Total Synthesis (s) Conclusion This paper has presented an efficient synthesis algorithm for complex UAV tasks involving constraints on the operational state of the robot under realistic physical assumptions. We illustrated the behaviour of this algorithm through two examples where the UAV performs navigation and surveillance tasks in a static continuous wind field with fuel constraints. Our simulation results indicate that synthesis is fast enough to allow for replanning during long-duration tasks where wind estimates evolve over time.

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