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Three Dimensional Trajectory Optimization of a Homing Parafoil
Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India
Three Dimensional Trajectory Optimization of a Homing Parafoil A.V. Suresh Babu ∗ V. Chandran Suja ∗∗ Ch. Vinay Reddy ∗∗∗ ∗
Aerospace Department, Indian Institute of Science, Bangalore-560012, (e-mail: [email protected]) ∗∗ Mechanical Department, Indian Institute of Science,Bangalore-560012, (e-mail: [email protected]) ∗∗∗ Aerospace Department, Indian Institute of Science, Bangalore-560012, (e-mail: [email protected] )
Abstract: This paper addresses the trajectory optimization for precision landing of parafoil assisted high altitude recovery modules. The problem is approached from an optimal control perspective. A simplified kinematic model of the parafoil payload system is considered. The angular velocity of the parafoil in the horizontal plane serves as the control input to the system. A multi objective performance index is formulated so as to ensure that the system achieves flared landing as closely as possible to a target point or region from a specified initial state with minimum control effort. The resulting optimal control problem with control and terminal constraints is solved numerically using the direct multiple shooting method to get the optimal trajectories that minimizes the performance index function. Subsequently, the effect of terminal payload delivery objectives on the optimal trajectories is looked into. This study is performed with a broad objective for obtaining preliminary results for achieving autonomous precision landing for a real world implementation of a high altitude recovery system. Keywords: Optimal control, Constraint satisfaction problems, Optimal trajectory, Kinematic model, Multi objective optimal problem 1. INTRODUCTION Recovery of scientific data gathered by weather balloons and sounding rockets have always been a difficult task due to the uncertainty in the landing location of the payloads. The payloads are typically recovered through extensive search operations with the help of passive tracking systems onboard the payload. This method is highly tedious and unreliable as the payload often ends up either in inaccessible terrains or in adverse environments resulting in permanent damage to the payload. The use of a ram air parafoil with an embedded control system instead of a regular parachute with a tracker is an attractive alternative to ferry the payload back to earth. Parafoils, first introduced by D C Jalbert (1966) in 1966, differ from usual parachute in their ability to actively control their descent trajectory with the help of two sets of control lines and by changing the center of mass of the system (Slegers and Costello (2004)). Manipulation of the control lines result in change in the camber of the parafoil which is shaped as an airfoil. As one set of control surfaces can be used to change the pitch and yaw of the parafoil they are sometimes referred to as flaperons. A great deal of recent research on atmospheric flight systems focus on the development of navigation, guidance, and control systems for parafoils. See, for example, Carter et al. (2005), Jiao et al. (2011) and Slegers and Costello (2004). Carter et al. (2005) discuss the design and implementation of a guidance, navigation, and control system for precision airdrop delivery using large parafoils. 978-3-902823-60-1 © 2014 IFAC
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The study focuses on a developing a software package with the help of which precision landing can be achieved. The guidance algorithm is based on table driven trajectory information. Jiao et al. (2011) apply modern model predictive control technique to develop an autonomous trajectory tracking controller for a linearized model of a parafoil-payload system. The reference trajectory in the horizontal plane is correlated with the heading angle of a linearized model. Both Carter et al. (2005) and Jiao et al. (2011) present the flight test results of the respective implementations. Slegers and Costello (2004) also address the trajectory tracking aspects of a parafoil system. Trajectory tracking for a six degree of freedom model has been achieved through a heading controller that is based on Active Disturbance Rejection Control (ADRC). The work presented in this paper presents the optimal descent trajectories for a simplified parafoil model for different situations. This work constitutes a preliminary phase in the authors’ attempt to design and flight test a control system for a ram air parafoil assisted recovery of scientific equipments dropped from a high altitude. For such missions, it is highly desirable to achieve precision landing so as to ensure the timely and safe recovery of the gathered data and the hardware. The payload which consists of a camera and pressure measurement units are required to be delivered with in a specified radius of the launch site. If precision landing can be achieved with minimal control effort, lighter sources can be used to power the onboard motors, and thus, the payload requirements for the mission can be reduced. Also, flare landing needs 10.3182/20140313-3-IN-3024.00219
2014 ACODS March 13-15, 2014. Kanpur, India
to be achieved for smooth touchdown. Ideally, a successful flare landing requires the parafoil to approach the landing site in a direction opposite to that of the wind close to ground. Thus, the optimality of a trajectory is determined by the control effort involved in guiding the parafoil along that trajectory, the extent of precision landing achieved, and the efficiency of flare landing. Several researchers have addressed various aspects of path planning and trajectory optimization of parafoils through different approaches. For instance, Zhang et al. (2013) present the optimal trajectories for a simplified parafoil model for precision landing from various altitudes with the help of an optimal control formulation. Rademacher (2009) and Rademacher et al. (2009) also use optimal control theory to study the onboard trajectory planning of autonomous parafoils in order to approach a specified point at a particular altitude with a given heading. Zhao and Jianyi (2009) converts the trajectory planning problem for a parafoil into a parameter optimization problem, and subsequently solve it numerically using the particle swarm optimization algorithm. Cleminson (2013) solve the problem of path planning for kinematic model of a parafoil with the help of dynamic programming. The online trajectory planning algorithm developed in Luders et al. (2013) takes the scenario further by achieving robustness against time varying uncertainties, besides incorporating collision avoidance strategies and executing precision landing. This paper presents the trajectory optimization of an autonomous parafoil to achieve precision landing from a given position and heading with minimum control effort. A simplified kinematic model of the parafoil has been used for the study. It is assumed that the only control to the system is the lateral steering and thus the forward and descent speeds of the parafoil cannot be controlled. The lateral control is in the form angular velocity of the parafoil about the vertical direction. The magnitude of this input is limited by the physical limitations of the motors that drive the control wires. The problem is posed as a multi-objective optimal control formulation. A control constraint is imposed on this formulation by the bounds on the magnitude of the inputs discussed above. The motivation for this approach was the formulation of Zhang et al. (2013), who generated optimal trajectories for a similar model using Gauss pseudospectral method. Initially, we validate the results through a comparison of the results of this paper with the corresponding ones of Zhang et al. (2013). Then, we extend the problem to further cases that are relevant to the high altitude recovery scenario mentioned earlier. We also study the effect of wind on the optimal trajectories and the control effort for a given situation. The multi-objective optimal control formulation described above has been solved and the optimal trajectories obtained by using the numerical direct multiple shooting method for optimal control problems. A brief account of this method is given in a later section of the paper. The organization of the paper is as follows. Section 1 presented a brief introduction to the problem and an overview of the related literature. An account on the mathematical model of the parafoil is given in Section 2. Section 3 details on the necessary tools of optimal control needed for the understanding and the solution of the 848
formulation. Numerical results for optimal trajectories for various conditions are presented in Section 4. An analysis of the simulation results has also been included. Finally, Section 5 concludes the discussion by explaining the scope for future work, and further applications of the results. 2. PARAFOIL MODEL Parafoils are typically modeled as dynamical systems having six degrees of freedom (Mortaloni et al. (2003); Slegers and Costello (2004)). A number of researchers have also treated the parafoil and the payload as separate rigid bodies and have modeled them as dynamical systems having eight (Iacomini and Cerimele (1999a,b)) and nine degrees of freedom (Slegers and Costello (2003)) under suitable assumptions. Interestingly, reduced order dynamical models of parafoils were developed by Jann (1993) having three and four degrees of freedom which are valid only under steady state conditions. For the purpose of trajectory planning, simple kinematic models are usually used rather than complex dynamic models. The higher order dynamical models of parafoils are highly non linear and complex and hence for the purpose of this paper we identify a reduced order kinematic model for the parafoil following the assumptions of Jann (1993). 2.1 Simplified System Model For the purpose of simulation, we assume that the change in the l/d ratio of the parafoil is negligible when the control lines are actuated. This implies that the horizontal and vertical speeds of the parafoil remain nearly constant. Moreover, we assume an isotropic atmosphere, implying ∂ρ that ∂X = 0 where ρ is the density of air at any location in the atmosphere and X = X(x, y, z). The parafoil and the payload are approximated by a single point mass model. Under these conditions the following assumptions hold: • Constant horizontal velocity Vs . • Constant sink rate Vz . • Yaw rate changes due to control inputs are achieved instantly. • The Euler angles φ, θ are identically equal to zero. • The wind velocity vector is constant and has only one component, say in the positive x direction. We follow a right handed inertial coordinate system with the XY plane parallel to the ground and the positive z axis pointing to the center of the earth. We consider two broad cases for trajectory optimization. In the first case, the payload is desired to be delivered at the origin O of the coordinate system while in the second case, it is desired to be deployed within a proximity circle of a specified radius centered at O. A simple schematic of a parafoil in the selected co-ordinate system is shown in Fig. 1. Wf indicates the magnitude of the wind velocity vector directed along the positive X axis, ψ denotes the heading of the parafoil with respect to the positive X axis, Vs represents the velocity of the parafoil in the horizontal plane and Vz is the constant descent rate of the parafoil. For the selected coordinate system under the stated assumptions, the non linear six degree of freedom model of a parafoil-payload system reduces to the following:
2014 ACODS March 13-15, 2014. Kanpur, India
y (t0 ) = y0 , z (t0 ) = z0 and ψ (t0 ) = ψ0 . (2) Terminal Constraints: We constrain the parafoil to land within a circle of radius rc centered at origin. This can be formally stated as: 2
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x (tf ) + y (tf ) − rc2 ≤ 0 (3) Control constraint Physical limitations of the actuators can be modelled as constraints on the maximum turn rate of the parafoil. This can be expressed as: ˙ ψ ≤ umax 2.3 Performance Index
Fig. 1. Schematic of Parafoil showing the axis and velocity definitions x˙ = Vs .cos(ψ) + Wf y˙ = Vs .sin(ψ) (1) z˙ = Vz ψ˙ = u where (x, y) denotes the position of the parafoil in the horizontal plane and z indicates the altitude of the parafoil above the ground plane.
The trajectory of the reduced parafoil model for t ∈ [t0 , tf ] has to be such that the control input is minimum so as to ensure minimal energy consumption. The landing should be sufficiently accurate and also, a flared landing is necessary for a smooth final approach. Based on these criteria, the following objective functions are chosen. Z
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J3 = min (ψ(tf ) − π))
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2.2 Constraints of the System The parafoil trajectory begins at (x0 , y0 , z0 ) at time t0 with a heading of ψ0 . The trajectory terminates at time tf . The terminal point (xf , yf , zf ) is to desired to be as close as to the origin in the first case and within a circle of specified radius r centered at the origin in the second case. Irrespective of the case, the terminal point should be within an area where the payload can be safely deployed and retrieved. We approximate this area as a circular region contained in a radius of rc . The idea is that, in any case the parafoil should land within the critical recovery distance rc , beyond which it is assumed that the recovery is impossible. Within this feasible region, the closer it lands to the inner circle of radius r, the more convenient is the recovery. Thus, this requirement is rather incorporated in the objective function, than imposing it as a constraint. This allows for analyzing the trade-offs between the payoffs for various objectives. The heading of the parafoil is controlled by actuators. The actuators have a physical limit beyond which they cannot increase their output. This directly limits the turn rate of the parafoil. Considering the above, we can formulate the following constraints. (1) Initial Constraints: The initial conditions at time t0 are known, x (t0 ) = x0 , 849
J1 (Eq:2) signifies the total control effort required by the parafoil over the course of its trajectory. Minimizing J1 reduces the control effort required to guide the parafoil from its initial position to its terminal location. J2 (Eq:3) denotes the positional accuracy of the payload delivery during touchdown. It is selected to be one of functions J21 or J22 depending on the selected case. For the first case where we desire the parafoil to land at the origin, J2 is selected to have the functional form of J21 . Clearly, for this particular case, J2 attains its minimum value at the origin. For the second case where we require the payload to be delivered within or as close as possible to a circle of specified radius r, we chose J2 to be equal to J22 . The radius of this circle is denoted by r and is chosen such that it is less than rc . This is because the convenient payload recovery area is usually smaller than the area critical payload recovery area. The minimum value of J22 is zero and it has this value for any terminal point within the circle of radius r. It should be noted that the requirements relating the two circles are not redundant. The constraint involving rc ensures that the payload lands within the critical recovery region, thus assuring recovery. The component J22 of the objective function rewards the system for making the recovery easier by landing within, or at least near to, the boundary of the inner circle.
2014 ACODS March 13-15, 2014. Kanpur, India
J3 (Eq:4) quantifies the heading accuracy of the parafoil during touchdown. As mentioned before, a successful flared landing requires the parafoil to approach the terminal phase in a direction opposite to the predominant wind direction near the ground. For the study of this paper, we have assumed the wind to be directed along the positive x axis. This requires the final heading angle ψf to equal to π for a successful flared landing. The functional form of J3 is such that it attains its minimum value when the final heading angle is π. It has to be noted that even though J3 is significant only when wind is present, we have utilized it even in cases where there is no wind. We justify this by pointing to cases where a strict final approach is required, for example, when deployment areas are accessible only through certain directions. Minimizing J1 reduces the energy consumption of the actuator while the minimization of J2 and J3 reduces the deviation of the landing location with respect to the the desired location and the attitude error in achieving a perfect flare landing, respectively. We can form a scalarized performance index from a linear combination of the multiple objectives J1 , J2 and J3 as given below. J =w1 .J1 + w2 .J2 + w3 .J3 where, w1 ≥ 0, w2 ≥ 0, w2 ≥ 0,
(5)
By assigning suitable values to the weights w1 , w2 and w3 , we can control the relative importance of the different objective functions in determining the final optimal trajectory subjected to the defined constraints. In cases where the touchdown has to be very smooth, the value of w3 is selected relatively higher than w1 and w2 . For precision landing w2 is tweaked. Increasing w1 results in trajectories requiring reduced control inputs. For the analysis of this paper, we select w1 : w2 : w3 = 10 : 1 : 5, assuming a case where considerable positional error within a bounded circle can be tolerated. 3. SOLUTION OF OPTIMAL CONTROL PROBLEMS 3.1 The Control Constrained Optimal Control Problem An optimal control problem deals with finding the optimal control input U to transfer a dynamic system from the initial state X(t0 ) at time t = t0 to the final state X(tf ) at time t = tf subject to some constraints while simultaneously minimizing certain performance index J.The trajectory optimization problem addressed in this paper gives rise to an optimal control problem with bounds on control. Such a control constrained optimal control problem can be mathematically posed as: Minimize, Z
tf
V (X(t), U, t) dt (6)
J = S(X(t0 ), t0 , X(tf ), tf ) + t0
subject to the dynamics of the system, ˙ X(t) = F(X(t), U, t),
(7)
and inequality constraints on control, GU (U(t)) ≤ 0,
(8) 850
and the boundary conditions, X(t0 ) = X0 and X(tf ) = Xf with tf and X(tf ) being fixed or free.
(9)
3.2 Numerical Methods for Optimal Control It is often difficult and mostly impossible to obtain closed form solutions to optimal control problems, especially state/control constrained problems. Several methods have been developed so far to tackle optimal control problems computationally. Betts (1998) gives a survey of various numerical tools for optimal control problems. Numerical methods for optimal control can be broadly classified into direct and indirect methods. In a direct method, the optimal state and control vectors are numerically determined. In an indirect method, the state and costate vectors along with the other Lagrangian multipliers, if any, are numerically obtained. In this paper, the direct multiple shooting method along with a non linear programming technique has been used for obtaining numerical solutions to the parafoil trajectory optimization problem. A detailed account on this method can be found in Betts (2001). However, a brief note on the numerical direct multiple shooting method is provided in the following section for clarity. 3.3 The Direct Multiple Shooting Method The various steps involved in the direct multiple shooting method are given below. • discretization of the time domain: The time domain is subdivided into N − 1 sub-intervals using N grid points. • formation of NLP variables: The value of the states and control variables at the grid points along with any time instants need to be optimized so as to minimize the performance index subject to various constraints. Thus, they form the set of NLP variables. • state equations: The values of the states at two successive grid points are related through the discretized state equations. These relations appear as constraints between the NLP variables. For example,the first order Euler discretization method generates constraints as: Xk+1 − Xk − Fk hk = 0; k = 0, 1, ...N − 1, and the second order trapezoidal discretization gives the constraints as: Xk+1 − Xk − (Fk + Fk+1 )hk /2 = 0; k = 0, 1, ...N − 1, where, hk = (tF − t0 )/(N − 1). • state and/or control constraints and other constraints: State and/control constraints are also transformed to the constraints between NLP variables. • solution of the NLP: The nonlinear program formulation is solved using a standard NLP solver after making an intelligent initial guess. • check for convergence: The optimized control values at the grid points are interpolated to get a continuous control. This resulting control is used to integrate the state equations and thus to calculate the performance index. This value is compared with the value of the optimized performance index given by the NLP solver to assess the convergence of the solution.
2014 ACODS March 13-15, 2014. Kanpur, India
• mesh refinement: If the solution has not converged, the current NLP solution is used to generate the initial guess for the next step with a higher number of grid points. The above process is repeated with the new number of grid points until the solution converges.
Parafoil trajectory in the horizontal plane 500 Direct Multiple Shooting method Guass Pseudospectral method 0
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4. NUMERICAL RESULTS The trajectory optimization of the parafoil which was cast as an optimal control problem is converted into a Non-Linear Programming problem (NLP) using the direct multiple shooting method. The NLP is solved using R (2010) subject to the ‘fmincon’ routine of MATLAB the constraints using the effective and commonly used Sequential Quadratic Programming algorithm.
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4.1 Validation The numerical solution is validated against the following test case given by Zhang et al. (2013). The details are given in Table 1. (xo , yo , zo ) (m)
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From Fig.3, we see a close similarity in the results obtained by the Direct Numerical Shooting method and those obtained by Zhang et al. (2013) through the Guass Pseudospectral method. The slight difference in the results for identical cases is expected due to the difference in the algorithms used. Once the accuracy of the developed code is verified, we proceed to use it for a case study. The case study is aimed at evaluating the effect of the choice of the functional form of J2 and the effect of wind on the optimal trajectories. 4.2 Case Study We primarily consider two cases as mentioned earlier. The first case is one in which the parafoil is desired to land at 851
Fig. 3. Validation of current method with the Guass Pseudospectral method of Zhang et al. (2013). (a) Comparison of optimum Parafoil trajectory in the horizontal plane. (b) Comparison of optimum control input required over time. the origin. For the second case we desire the parafoil to land with in a circle of radius r. In addition to the above two cases, we consider two more cases to understand the effects of wind. The third case is identical to the first case but does not have wind while fourth case is identical to second case with no wind. The cases are summarized in Table 2. For all cases Vz and Vs were chosen to be equal to 4.6 and 15 m/s, respectively. The radius of the constraint circle rc is fixed at 60 m while the radius r for Case 2 and 4 are fixed at 50 m. Case
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Table 2. Selected simulation cases
2014 ACODS March 13-15, 2014. Kanpur, India
in which there was no effect of wind. This observation is supported by the fact that we have obtained similar results as Zhang et al. (2013) using models having no wind. See Fig.3.
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Fig. 4. (a) Comparison of optimum Parafoil trajectory in the horizontal plane. (b) Comparison of optimum control input required over time. Fig.4 shows the effect of wind on the optimal trajectory of the parafoil. Two identical cases were selected one of which (Case 1) was subjected to a constant wind of magnitude 10 m/s. The major difference in the trajectories lie in the initial phase during which the parafoil subjected to wind moved slightly against the wind. The terminal trajectories of both the cases are similar. The presence of wind increases the control effort as expected and demands control input through out the descent of the parafoil. This can be observed by comparing the relatively smooth control input required for the case with no wind and that required for the case with wind. At this point the authors wish to point out that Zhang et al. (2013) attempt a similar simulation to show the effect of wind on the trajectory of a parafoil. However, Zhang et al. (2013) perform a coordinate transformation on (1), changing it into a form that is consistent with a wind coordinate system. Further, while imposing constraints, a terminal constraint valid only in the inertial coordinate system is chosen. This, in effect, resulted in simulations 852
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(b) Fig. 5. Consequence of the functional form of J2 without wind. (a) Comparison of optimum parafoil trajectory in the horizontal plane without wind. (b) Comparison of optimum control input required over time without wind. Fig.5 shows the optimal trajectories for two cases differing only in the form of the function J2 . Clearly the optimal trajectory corresponding to Case 4 is smoother and requires less control input. It can also be seen that the terminal point corresponding to Case 4 is not on the origin but slightly away even though the scaling of the graphs makes this less obvious. Effect of function J2 with wind Fig.6 shows the consequence of the choice of the function J2 in cases where there is wind. Here the advantage of selecting the J22 as the functional form for J2 is more
2014 ACODS March 13-15, 2014. Kanpur, India
identical cases using J22 with the difference amplified in cases with wind.
Parafoil trajectory in the horizontal plane 0 Case 1 Case 2
5. CONCLUSION AND FUTURE WORK
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The trajectory optimization of the kinematic model of a parafoil-payload system was performed. The problem was cast as a multi-objective optimal control problem. Using the direct multiple shooting method, the problem was transformed into an NLP before solving it using R . We have validated the integrity of our soMATLAB lutions against a published test case before conducting a case study. The advantage of selecting a terminal objective desiring to deploy the parafoil in an area rather than at a point, especially in the presence of wind, is a key finding of the study. The results of this study are proposed to be used in the process of design and implementation of an autonomous high altitude payload recovery system.
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REFERENCES
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(b) Fig. 6. Consequence of the functional form of J2 with wind. (a) Comparison of optimum Parafoil trajectory in the horizontal plane with wind. (b) Comparison of optimum control input required over time with wind. Case
xf (m)
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−0.00029 −17.5554 −0.00077 17.7670
yo (m)
usum
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−0.00008 −46.8159 −0.00024 46.7349
6.0148 5.6786 4.1088 3.9666
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Table 3. Summary of simulation results evident from the optimal trajectories and especially from significant reduction in the required control input. Table 3 summarizes the results of the simulation for the four selected cases. In all the cases the terminal requirements are satisfied with the cases using J22 as the functional form of J2 having optimal touch down points close to the periphery of the preferred circle. The most significant result is the change in the magnitude of usum Rt (= tof |u| dt) depending on the choice of J2 . Clearly all cases using J22 require lower control inputs compared to 853
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Three Dimensional Trajectory Optimization of a Homing Parafoil A.V. Suresh Babu ∗ V. Chandran Suja ∗∗ Ch. Vinay Reddy ∗∗∗ ∗
Aerospace Department, Indian Institute of Science, Bangalore-560012, (e-mail: [email protected]) ∗∗ Mechanical Department, Indian Institute of Science,Bangalore-560012, (e-mail: [email protected]) ∗∗∗ Aerospace Department, Indian Institute of Science, Bangalore-560012, (e-mail: [email protected] )
Abstract: This paper addresses the trajectory optimization for precision landing of parafoil assisted high altitude recovery modules. The problem is approached from an optimal control perspective. A simplified kinematic model of the parafoil payload system is considered. The angular velocity of the parafoil in the horizontal plane serves as the control input to the system. A multi objective performance index is formulated so as to ensure that the system achieves flared landing as closely as possible to a target point or region from a specified initial state with minimum control effort. The resulting optimal control problem with control and terminal constraints is solved numerically using the direct multiple shooting method to get the optimal trajectories that minimizes the performance index function. Subsequently, the effect of terminal payload delivery objectives on the optimal trajectories is looked into. This study is performed with a broad objective for obtaining preliminary results for achieving autonomous precision landing for a real world implementation of a high altitude recovery system. Keywords: Optimal control, Constraint satisfaction problems, Optimal trajectory, Kinematic model, Multi objective optimal problem 1. INTRODUCTION Recovery of scientific data gathered by weather balloons and sounding rockets have always been a difficult task due to the uncertainty in the landing location of the payloads. The payloads are typically recovered through extensive search operations with the help of passive tracking systems onboard the payload. This method is highly tedious and unreliable as the payload often ends up either in inaccessible terrains or in adverse environments resulting in permanent damage to the payload. The use of a ram air parafoil with an embedded control system instead of a regular parachute with a tracker is an attractive alternative to ferry the payload back to earth. Parafoils, first introduced by D C Jalbert (1966) in 1966, differ from usual parachute in their ability to actively control their descent trajectory with the help of two sets of control lines and by changing the center of mass of the system (Slegers and Costello (2004)). Manipulation of the control lines result in change in the camber of the parafoil which is shaped as an airfoil. As one set of control surfaces can be used to change the pitch and yaw of the parafoil they are sometimes referred to as flaperons. A great deal of recent research on atmospheric flight systems focus on the development of navigation, guidance, and control systems for parafoils. See, for example, Carter et al. (2005), Jiao et al. (2011) and Slegers and Costello (2004). Carter et al. (2005) discuss the design and implementation of a guidance, navigation, and control system for precision airdrop delivery using large parafoils. 978-3-902823-60-1 © 2014 IFAC
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The study focuses on a developing a software package with the help of which precision landing can be achieved. The guidance algorithm is based on table driven trajectory information. Jiao et al. (2011) apply modern model predictive control technique to develop an autonomous trajectory tracking controller for a linearized model of a parafoil-payload system. The reference trajectory in the horizontal plane is correlated with the heading angle of a linearized model. Both Carter et al. (2005) and Jiao et al. (2011) present the flight test results of the respective implementations. Slegers and Costello (2004) also address the trajectory tracking aspects of a parafoil system. Trajectory tracking for a six degree of freedom model has been achieved through a heading controller that is based on Active Disturbance Rejection Control (ADRC). The work presented in this paper presents the optimal descent trajectories for a simplified parafoil model for different situations. This work constitutes a preliminary phase in the authors’ attempt to design and flight test a control system for a ram air parafoil assisted recovery of scientific equipments dropped from a high altitude. For such missions, it is highly desirable to achieve precision landing so as to ensure the timely and safe recovery of the gathered data and the hardware. The payload which consists of a camera and pressure measurement units are required to be delivered with in a specified radius of the launch site. If precision landing can be achieved with minimal control effort, lighter sources can be used to power the onboard motors, and thus, the payload requirements for the mission can be reduced. Also, flare landing needs 10.3182/20140313-3-IN-3024.00219
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to be achieved for smooth touchdown. Ideally, a successful flare landing requires the parafoil to approach the landing site in a direction opposite to that of the wind close to ground. Thus, the optimality of a trajectory is determined by the control effort involved in guiding the parafoil along that trajectory, the extent of precision landing achieved, and the efficiency of flare landing. Several researchers have addressed various aspects of path planning and trajectory optimization of parafoils through different approaches. For instance, Zhang et al. (2013) present the optimal trajectories for a simplified parafoil model for precision landing from various altitudes with the help of an optimal control formulation. Rademacher (2009) and Rademacher et al. (2009) also use optimal control theory to study the onboard trajectory planning of autonomous parafoils in order to approach a specified point at a particular altitude with a given heading. Zhao and Jianyi (2009) converts the trajectory planning problem for a parafoil into a parameter optimization problem, and subsequently solve it numerically using the particle swarm optimization algorithm. Cleminson (2013) solve the problem of path planning for kinematic model of a parafoil with the help of dynamic programming. The online trajectory planning algorithm developed in Luders et al. (2013) takes the scenario further by achieving robustness against time varying uncertainties, besides incorporating collision avoidance strategies and executing precision landing. This paper presents the trajectory optimization of an autonomous parafoil to achieve precision landing from a given position and heading with minimum control effort. A simplified kinematic model of the parafoil has been used for the study. It is assumed that the only control to the system is the lateral steering and thus the forward and descent speeds of the parafoil cannot be controlled. The lateral control is in the form angular velocity of the parafoil about the vertical direction. The magnitude of this input is limited by the physical limitations of the motors that drive the control wires. The problem is posed as a multi-objective optimal control formulation. A control constraint is imposed on this formulation by the bounds on the magnitude of the inputs discussed above. The motivation for this approach was the formulation of Zhang et al. (2013), who generated optimal trajectories for a similar model using Gauss pseudospectral method. Initially, we validate the results through a comparison of the results of this paper with the corresponding ones of Zhang et al. (2013). Then, we extend the problem to further cases that are relevant to the high altitude recovery scenario mentioned earlier. We also study the effect of wind on the optimal trajectories and the control effort for a given situation. The multi-objective optimal control formulation described above has been solved and the optimal trajectories obtained by using the numerical direct multiple shooting method for optimal control problems. A brief account of this method is given in a later section of the paper. The organization of the paper is as follows. Section 1 presented a brief introduction to the problem and an overview of the related literature. An account on the mathematical model of the parafoil is given in Section 2. Section 3 details on the necessary tools of optimal control needed for the understanding and the solution of the 848
formulation. Numerical results for optimal trajectories for various conditions are presented in Section 4. An analysis of the simulation results has also been included. Finally, Section 5 concludes the discussion by explaining the scope for future work, and further applications of the results. 2. PARAFOIL MODEL Parafoils are typically modeled as dynamical systems having six degrees of freedom (Mortaloni et al. (2003); Slegers and Costello (2004)). A number of researchers have also treated the parafoil and the payload as separate rigid bodies and have modeled them as dynamical systems having eight (Iacomini and Cerimele (1999a,b)) and nine degrees of freedom (Slegers and Costello (2003)) under suitable assumptions. Interestingly, reduced order dynamical models of parafoils were developed by Jann (1993) having three and four degrees of freedom which are valid only under steady state conditions. For the purpose of trajectory planning, simple kinematic models are usually used rather than complex dynamic models. The higher order dynamical models of parafoils are highly non linear and complex and hence for the purpose of this paper we identify a reduced order kinematic model for the parafoil following the assumptions of Jann (1993). 2.1 Simplified System Model For the purpose of simulation, we assume that the change in the l/d ratio of the parafoil is negligible when the control lines are actuated. This implies that the horizontal and vertical speeds of the parafoil remain nearly constant. Moreover, we assume an isotropic atmosphere, implying ∂ρ that ∂X = 0 where ρ is the density of air at any location in the atmosphere and X = X(x, y, z). The parafoil and the payload are approximated by a single point mass model. Under these conditions the following assumptions hold: • Constant horizontal velocity Vs . • Constant sink rate Vz . • Yaw rate changes due to control inputs are achieved instantly. • The Euler angles φ, θ are identically equal to zero. • The wind velocity vector is constant and has only one component, say in the positive x direction. We follow a right handed inertial coordinate system with the XY plane parallel to the ground and the positive z axis pointing to the center of the earth. We consider two broad cases for trajectory optimization. In the first case, the payload is desired to be delivered at the origin O of the coordinate system while in the second case, it is desired to be deployed within a proximity circle of a specified radius centered at O. A simple schematic of a parafoil in the selected co-ordinate system is shown in Fig. 1. Wf indicates the magnitude of the wind velocity vector directed along the positive X axis, ψ denotes the heading of the parafoil with respect to the positive X axis, Vs represents the velocity of the parafoil in the horizontal plane and Vz is the constant descent rate of the parafoil. For the selected coordinate system under the stated assumptions, the non linear six degree of freedom model of a parafoil-payload system reduces to the following:
2014 ACODS March 13-15, 2014. Kanpur, India
y (t0 ) = y0 , z (t0 ) = z0 and ψ (t0 ) = ψ0 . (2) Terminal Constraints: We constrain the parafoil to land within a circle of radius rc centered at origin. This can be formally stated as: 2
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x (tf ) + y (tf ) − rc2 ≤ 0 (3) Control constraint Physical limitations of the actuators can be modelled as constraints on the maximum turn rate of the parafoil. This can be expressed as: ˙ ψ ≤ umax 2.3 Performance Index
Fig. 1. Schematic of Parafoil showing the axis and velocity definitions x˙ = Vs .cos(ψ) + Wf y˙ = Vs .sin(ψ) (1) z˙ = Vz ψ˙ = u where (x, y) denotes the position of the parafoil in the horizontal plane and z indicates the altitude of the parafoil above the ground plane.
The trajectory of the reduced parafoil model for t ∈ [t0 , tf ] has to be such that the control input is minimum so as to ensure minimal energy consumption. The landing should be sufficiently accurate and also, a flared landing is necessary for a smooth final approach. Based on these criteria, the following objective functions are chosen. Z
tf
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u dt 2
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J3 = min (ψ(tf ) − π))
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2.2 Constraints of the System The parafoil trajectory begins at (x0 , y0 , z0 ) at time t0 with a heading of ψ0 . The trajectory terminates at time tf . The terminal point (xf , yf , zf ) is to desired to be as close as to the origin in the first case and within a circle of specified radius r centered at the origin in the second case. Irrespective of the case, the terminal point should be within an area where the payload can be safely deployed and retrieved. We approximate this area as a circular region contained in a radius of rc . The idea is that, in any case the parafoil should land within the critical recovery distance rc , beyond which it is assumed that the recovery is impossible. Within this feasible region, the closer it lands to the inner circle of radius r, the more convenient is the recovery. Thus, this requirement is rather incorporated in the objective function, than imposing it as a constraint. This allows for analyzing the trade-offs between the payoffs for various objectives. The heading of the parafoil is controlled by actuators. The actuators have a physical limit beyond which they cannot increase their output. This directly limits the turn rate of the parafoil. Considering the above, we can formulate the following constraints. (1) Initial Constraints: The initial conditions at time t0 are known, x (t0 ) = x0 , 849
J1 (Eq:2) signifies the total control effort required by the parafoil over the course of its trajectory. Minimizing J1 reduces the control effort required to guide the parafoil from its initial position to its terminal location. J2 (Eq:3) denotes the positional accuracy of the payload delivery during touchdown. It is selected to be one of functions J21 or J22 depending on the selected case. For the first case where we desire the parafoil to land at the origin, J2 is selected to have the functional form of J21 . Clearly, for this particular case, J2 attains its minimum value at the origin. For the second case where we require the payload to be delivered within or as close as possible to a circle of specified radius r, we chose J2 to be equal to J22 . The radius of this circle is denoted by r and is chosen such that it is less than rc . This is because the convenient payload recovery area is usually smaller than the area critical payload recovery area. The minimum value of J22 is zero and it has this value for any terminal point within the circle of radius r. It should be noted that the requirements relating the two circles are not redundant. The constraint involving rc ensures that the payload lands within the critical recovery region, thus assuring recovery. The component J22 of the objective function rewards the system for making the recovery easier by landing within, or at least near to, the boundary of the inner circle.
2014 ACODS March 13-15, 2014. Kanpur, India
J3 (Eq:4) quantifies the heading accuracy of the parafoil during touchdown. As mentioned before, a successful flared landing requires the parafoil to approach the terminal phase in a direction opposite to the predominant wind direction near the ground. For the study of this paper, we have assumed the wind to be directed along the positive x axis. This requires the final heading angle ψf to equal to π for a successful flared landing. The functional form of J3 is such that it attains its minimum value when the final heading angle is π. It has to be noted that even though J3 is significant only when wind is present, we have utilized it even in cases where there is no wind. We justify this by pointing to cases where a strict final approach is required, for example, when deployment areas are accessible only through certain directions. Minimizing J1 reduces the energy consumption of the actuator while the minimization of J2 and J3 reduces the deviation of the landing location with respect to the the desired location and the attitude error in achieving a perfect flare landing, respectively. We can form a scalarized performance index from a linear combination of the multiple objectives J1 , J2 and J3 as given below. J =w1 .J1 + w2 .J2 + w3 .J3 where, w1 ≥ 0, w2 ≥ 0, w2 ≥ 0,
(5)
By assigning suitable values to the weights w1 , w2 and w3 , we can control the relative importance of the different objective functions in determining the final optimal trajectory subjected to the defined constraints. In cases where the touchdown has to be very smooth, the value of w3 is selected relatively higher than w1 and w2 . For precision landing w2 is tweaked. Increasing w1 results in trajectories requiring reduced control inputs. For the analysis of this paper, we select w1 : w2 : w3 = 10 : 1 : 5, assuming a case where considerable positional error within a bounded circle can be tolerated. 3. SOLUTION OF OPTIMAL CONTROL PROBLEMS 3.1 The Control Constrained Optimal Control Problem An optimal control problem deals with finding the optimal control input U to transfer a dynamic system from the initial state X(t0 ) at time t = t0 to the final state X(tf ) at time t = tf subject to some constraints while simultaneously minimizing certain performance index J.The trajectory optimization problem addressed in this paper gives rise to an optimal control problem with bounds on control. Such a control constrained optimal control problem can be mathematically posed as: Minimize, Z
tf
V (X(t), U, t) dt (6)
J = S(X(t0 ), t0 , X(tf ), tf ) + t0
subject to the dynamics of the system, ˙ X(t) = F(X(t), U, t),
(7)
and inequality constraints on control, GU (U(t)) ≤ 0,
(8) 850
and the boundary conditions, X(t0 ) = X0 and X(tf ) = Xf with tf and X(tf ) being fixed or free.
(9)
3.2 Numerical Methods for Optimal Control It is often difficult and mostly impossible to obtain closed form solutions to optimal control problems, especially state/control constrained problems. Several methods have been developed so far to tackle optimal control problems computationally. Betts (1998) gives a survey of various numerical tools for optimal control problems. Numerical methods for optimal control can be broadly classified into direct and indirect methods. In a direct method, the optimal state and control vectors are numerically determined. In an indirect method, the state and costate vectors along with the other Lagrangian multipliers, if any, are numerically obtained. In this paper, the direct multiple shooting method along with a non linear programming technique has been used for obtaining numerical solutions to the parafoil trajectory optimization problem. A detailed account on this method can be found in Betts (2001). However, a brief note on the numerical direct multiple shooting method is provided in the following section for clarity. 3.3 The Direct Multiple Shooting Method The various steps involved in the direct multiple shooting method are given below. • discretization of the time domain: The time domain is subdivided into N − 1 sub-intervals using N grid points. • formation of NLP variables: The value of the states and control variables at the grid points along with any time instants need to be optimized so as to minimize the performance index subject to various constraints. Thus, they form the set of NLP variables. • state equations: The values of the states at two successive grid points are related through the discretized state equations. These relations appear as constraints between the NLP variables. For example,the first order Euler discretization method generates constraints as: Xk+1 − Xk − Fk hk = 0; k = 0, 1, ...N − 1, and the second order trapezoidal discretization gives the constraints as: Xk+1 − Xk − (Fk + Fk+1 )hk /2 = 0; k = 0, 1, ...N − 1, where, hk = (tF − t0 )/(N − 1). • state and/or control constraints and other constraints: State and/control constraints are also transformed to the constraints between NLP variables. • solution of the NLP: The nonlinear program formulation is solved using a standard NLP solver after making an intelligent initial guess. • check for convergence: The optimized control values at the grid points are interpolated to get a continuous control. This resulting control is used to integrate the state equations and thus to calculate the performance index. This value is compared with the value of the optimized performance index given by the NLP solver to assess the convergence of the solution.
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• mesh refinement: If the solution has not converged, the current NLP solution is used to generate the initial guess for the next step with a higher number of grid points. The above process is repeated with the new number of grid points until the solution converges.
Parafoil trajectory in the horizontal plane 500 Direct Multiple Shooting method Guass Pseudospectral method 0
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4. NUMERICAL RESULTS The trajectory optimization of the parafoil which was cast as an optimal control problem is converted into a Non-Linear Programming problem (NLP) using the direct multiple shooting method. The NLP is solved using R (2010) subject to the ‘fmincon’ routine of MATLAB the constraints using the effective and commonly used Sequential Quadratic Programming algorithm.
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4.1 Validation The numerical solution is validated against the following test case given by Zhang et al. (2013). The details are given in Table 1. (xo , yo , zo ) (m)
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From Fig.3, we see a close similarity in the results obtained by the Direct Numerical Shooting method and those obtained by Zhang et al. (2013) through the Guass Pseudospectral method. The slight difference in the results for identical cases is expected due to the difference in the algorithms used. Once the accuracy of the developed code is verified, we proceed to use it for a case study. The case study is aimed at evaluating the effect of the choice of the functional form of J2 and the effect of wind on the optimal trajectories. 4.2 Case Study We primarily consider two cases as mentioned earlier. The first case is one in which the parafoil is desired to land at 851
Fig. 3. Validation of current method with the Guass Pseudospectral method of Zhang et al. (2013). (a) Comparison of optimum Parafoil trajectory in the horizontal plane. (b) Comparison of optimum control input required over time. the origin. For the second case we desire the parafoil to land with in a circle of radius r. In addition to the above two cases, we consider two more cases to understand the effects of wind. The third case is identical to the first case but does not have wind while fourth case is identical to second case with no wind. The cases are summarized in Table 2. For all cases Vz and Vs were chosen to be equal to 4.6 and 15 m/s, respectively. The radius of the constraint circle rc is fixed at 60 m while the radius r for Case 2 and 4 are fixed at 50 m. Case
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J21 J22 J21 J22
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Table 2. Selected simulation cases
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in which there was no effect of wind. This observation is supported by the fact that we have obtained similar results as Zhang et al. (2013) using models having no wind. See Fig.3.
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Fig. 4. (a) Comparison of optimum Parafoil trajectory in the horizontal plane. (b) Comparison of optimum control input required over time. Fig.4 shows the effect of wind on the optimal trajectory of the parafoil. Two identical cases were selected one of which (Case 1) was subjected to a constant wind of magnitude 10 m/s. The major difference in the trajectories lie in the initial phase during which the parafoil subjected to wind moved slightly against the wind. The terminal trajectories of both the cases are similar. The presence of wind increases the control effort as expected and demands control input through out the descent of the parafoil. This can be observed by comparing the relatively smooth control input required for the case with no wind and that required for the case with wind. At this point the authors wish to point out that Zhang et al. (2013) attempt a similar simulation to show the effect of wind on the trajectory of a parafoil. However, Zhang et al. (2013) perform a coordinate transformation on (1), changing it into a form that is consistent with a wind coordinate system. Further, while imposing constraints, a terminal constraint valid only in the inertial coordinate system is chosen. This, in effect, resulted in simulations 852
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(b) Fig. 5. Consequence of the functional form of J2 without wind. (a) Comparison of optimum parafoil trajectory in the horizontal plane without wind. (b) Comparison of optimum control input required over time without wind. Fig.5 shows the optimal trajectories for two cases differing only in the form of the function J2 . Clearly the optimal trajectory corresponding to Case 4 is smoother and requires less control input. It can also be seen that the terminal point corresponding to Case 4 is not on the origin but slightly away even though the scaling of the graphs makes this less obvious. Effect of function J2 with wind Fig.6 shows the consequence of the choice of the function J2 in cases where there is wind. Here the advantage of selecting the J22 as the functional form for J2 is more
2014 ACODS March 13-15, 2014. Kanpur, India
identical cases using J22 with the difference amplified in cases with wind.
Parafoil trajectory in the horizontal plane 0 Case 1 Case 2
5. CONCLUSION AND FUTURE WORK
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The trajectory optimization of the kinematic model of a parafoil-payload system was performed. The problem was cast as a multi-objective optimal control problem. Using the direct multiple shooting method, the problem was transformed into an NLP before solving it using R . We have validated the integrity of our soMATLAB lutions against a published test case before conducting a case study. The advantage of selecting a terminal objective desiring to deploy the parafoil in an area rather than at a point, especially in the presence of wind, is a key finding of the study. The results of this study are proposed to be used in the process of design and implementation of an autonomous high altitude payload recovery system.
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REFERENCES
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(b) Fig. 6. Consequence of the functional form of J2 with wind. (a) Comparison of optimum Parafoil trajectory in the horizontal plane with wind. (b) Comparison of optimum control input required over time with wind. Case
xf (m)
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−0.00029 −17.5554 −0.00077 17.7670
yo (m)
usum
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−0.00008 −46.8159 −0.00024 46.7349
6.0148 5.6786 4.1088 3.9666
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Table 3. Summary of simulation results evident from the optimal trajectories and especially from significant reduction in the required control input. Table 3 summarizes the results of the simulation for the four selected cases. In all the cases the terminal requirements are satisfied with the cases using J22 as the functional form of J2 having optimal touch down points close to the periphery of the preferred circle. The most significant result is the change in the magnitude of usum Rt (= tof |u| dt) depending on the choice of J2 . Clearly all cases using J22 require lower control inputs compared to 853
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