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Evaluation of Time-Shifted Feedforward Control for Unmanned Helicopter Path Tracking

Sven Lorenz1 and Johann C. Dauer2 German Aerospace Center (DLR), Institute of Flight Systems, Braunschweig, Germany

A novel approach to improve the path tracking performance based on time shifted feed-forward signals is presented. It uses approximately linear closed-loop dynamics through feedback linearization of the ARTIS unmanned helicopter. The flight controller is currently in use to track nonlinear trajectories represented by piece-wise cubic splines. Feedforward signals shifted in time with varying and constant time-shifts are evaluated to improve the suboptimal tracking behavior. The problem is illustrated for a representative path and the process of determining the time shift is discussed. Having runtime performance advantages, the constant variant of time-shifted feedforward signals is chosen for onboard implementa-tion. To validate the approach, it is implemented onboard our flying rotorcraft test bed and flight test results showing the improved tracking performance are presented.

Nomenclature highest acceleration achieved by the vehicle

error vector gravity constant

J quality cost function proportional feedback gain

, , position vector in geodetic frame quaternion of unit length representing the attitude

radius of a space curve s arc length T sample time

system input vector speed vector in path-fixed frame , maximum velocity achievable by the vehicle

state vector system output vector

path elevation angle vector of actuator signals Δt time-shift

curvature of a space curve spline parameter vector of body-fixed turn rates

Indices c command r reference s state

1 Research Scientist, Unmanned Aircraft Department, sven.lorenz@dlr.de 2 Research Scientist, Unmanned Aircraft Department, johann.dauer@dlr.de

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I. Introduction ISSIONS of unmanned aerial vehicles in urban environments impose constraints (e.g. obstacles) which often require hover capabilities and very good maneuverability. These requirements often lead to a selec-tion of helicopters over fixed-wing aircrafts. Dynamic maneuvers resulting from fast flight in such envi-ronments require control systems to achieve very high performance. In order to avoid obstacles paths are determined using a path planner. These paths have to tracked by the vehicle.

Gates 1 defines the task to track a path exactly to be the problem of letting a vehicles’ position converge to a (pre-) defined path over time. Moreover, the simplest case with constant speed only is already a quite challenging problem. Within the context of the presented approach here, following a virtual point in the desired path, as e.g. applied in 2-4, is used to determine the desired position on the path. In contrast to those approaches, the virtual refer-ence point is determined not by using a distance measure but a time span. Similar to Gates 1, this paper is concerned with to the outer-loop guidance only. Methods for combining the inner and outer loop, as presented by Murray 5 or Johnson and Calise 6, are not considered here since the solution must be applicable within a decoupled framework presented by Lorenz and Adolf 4. Similar to Lorenz and Adolf 4, the path is defined as continuously differentiable interpolation of 3D-waypoints in space. The waypoints are support points of the space curve as presented in Figure 1 in a 2D scenario. In our applica-tion, it is required to track the space curve as precisely as possible. Exceeding a predefined path deviation can lead to collisions, especially in dense obstacle scenarios. However, unavoidable disturbances, e.g. wind, or the reduced order of the interpolation function will cause path-tracking error. A safety margin to obstacles will be utilized to prevent these collisions. As shown by Adolf and Andert 7, the resulting complexity of generating a collision-free path through narrow passages in 3D-space is extensive. An integrated optimization of a motion planning task com-prises a whole set of computations when aspects like task sequence ordering, route planning, reactive obstacle avoidance, and vehicle control are considered.

In our approach path segments are used consisting of cubic spline segments that are continuously differentiable up to the third derivative. Given a sufficiently low velocity, helicopters are able to fly along paths with arbitrarily sharp turns with sufficient tracking performance. Thus, the velocity is used to parameterize the path in time. From the curvature of the path and the thrust available, the largest admissible speed can be determined, as presented by Lorenz and Adolf 4. However, the path does not consider specific vehicle dynamics explicitly in the implementation which would be required for accurate feedforward signals. Exact trajectory following require paths to be as often differentiable as the system order for exact path tracking. However, the system order of the helicopter is higher, than the third order of the cubic spline segments. Thus, an approach based on flatness or controllability is not directly applicable.

The presented work is part of the ARTIS project (Autonomous Rotorcraft Testbed for Intelligent Systems) of the German Aerospace Center in Braunschweig, Germany, consisting of a family of different sized helicopters. The mid-size helicopter test bed, with about2 rotor diameter and up to approximately15 take-off weight, is shown in Figure 2. The path tracking method presented in this paper is applied to our rotorcraft ARTIS. It is desired to fly as fast as possible, while the absolute tracking error must not exceed a selected limit, e.g. in terms of the rotor diam-eter.

M

Figure 1. Path based on cubic splines, supported by three waypoints.

Figure 2. Unmanned helicopter testbed ARTIS (Autonomous Helicopter Testbed for Intelligent Systems), 2m rotor diameter, 1.5kW 2-Stroke-Engine, 15kg take-off weight

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Figure 3. Control architecture of the ARTIS family consisting of a cascade in attitude, velocity and path fol-lowing.

This paper is organized as follows: At first the used controller is explained resulting in a more precise problem

statement and objective definition. The improvement caused by the time-shifted feedforward signals is demonstrated by using a characteristic loop-like path. Finally, the approach is evaluated for multiple three dimensional space curves in simulation and flight tests.

II. Closed-Loop Dynamics of Enclosed Control Loops and Error Feedback In order to have a proper context for the path controller derived later in this section, introduces the control archi-

tecture used for the ARTIS family. A separation of the control system into inner and outer loops is a well-established design method in aerospace applications 3. One of the main reasons for the decoupling of position con-trol from velocity and attitude within the ARTIS project are the different levels of autonomy required for the opera-tion of the vehicle. It is desired to have different command sources on different level of abstraction, e.g. safety pilot via remote control, ground control station via a joystick, and path following control as treated in this paper.

Figure 3 gives a compact overview of the architecture of the control system. This architecture is a special case of a Model Reference Adaptive Controller (MRAC), as presented by Lorenz 8. The adaptive elements consist of single hidden layer neural networks (NN) supporting the velocity loop only. The impact on the adaptive parts of unconsid-ered actuator dynamics within the reference models are tackled using Pseudo Control Hedging (PCH) as often pre-sented, e.g. by Johnson and Calise 6. A feedback free design of the reference models as presented by Lorenz 8 has been used. The subsequent paragraphs describe the inner two loops before focusing on the path controller.

The MRAC controller uses a reference model in each loop to generate feasible reference trajectories (denoted us-ing the index “r”) for the helicopter which are then used for tracking. There is a reference model for the velocity and one for the attitude loop. The velocity reference model generates the command accelerations and the correspond-ing reference trajectory of the velocity . This reference trajectory is then used for error feedback, using linear PI controller and the single hidden layer neural network.

The attitude reference model generates a reference trajectory for the attitude , represented by a quaternion of

unit length. This attitude is required to ensure the correct thrust orientation in space for the commanded acceleration. For the tracking error, the reference attitude trajectory is calculated and compared to the state of the helicopter. The second derivative of the body-fixed turn rates as well as acceleration commands from the outer loop are fed into the dynamic inversion represented by the inverse helicopter in Figure 3. The dynamic inversion of the helicop-ter model is based on an approximated hover model, as often used for these kinds of helicopters, e.g. presented by Mettler 9. It generates swash-plate and tail rotor control inputs corresponding to the reference trajectories.

Each of the reference models has been designed linearly. If the dynamic inversion augmented by the neural net-work feedback to compensate for model uncertainties is accurate enough, the inversion cancels out the helicopter dynamics. The cancelling only works, if the required helicopter state remains within the state space where the inver-sion is valid for. While the reference models have been designed such that these limits are not exceeded, the helicop-ter dynamics are compensated. The resulting closed-loop dynamics are hence approximately the ones of the de-signed reference models.

Path AttitudeVelocity

PI + NN

P

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The outer cascade in Figure 3, the path following loop, is the focus of this paper. The path loop controller gener-ates a mapping of a pre-defined geometrical representation of the path as space curve to time using a velocity profile. The input to the underlying control system is a velocity command vector , , . It is calculated as the sum of a velocity profile (see following paragraph A) and a feedback signal (B), determined using the supposed reference position and a proportional feedback . Integral feedback of the position error is not implemented; however, the inner loop velocity control loop is equipped with integral feedback of the velocity error.

A. Velocity Profile Within the path following control system, the velocity is represented in polar coordinates with its magnitude . The maximal flight speed that can be achieved on a certain point on the space curve is denoted as , . It depends

amongst others on the curvature , where represents the curve radius. A relation between this maximal flight

speed, the maximum acceleration generated by the helicopter , the acceleration , the earth gravity con-stant , and the flight path elevation angle is introduced by Lorenz and Adolf 4:

, . (1)

Using equation (1), a profile of maximum velocity on the geometric representation can be determined. Figure 4

presents an example. The highest achievable velocity is shown solid. Due to the accelerationlimitation of the heli-copter, the dotted velocity profile is used for actual commanded to slow down the vehicle early enough to reach a velocity minimum later or stop at the path’s end. Using this or equivalent velocity profile a time-wise position on the geometric path has been found. Thus the relation between the curve parameter and time has been uniquely defined.

For the nonlinear path used within the scope of this paper, a tangent at a point on the curve represents the direc-tion of a velocity vector, obtainable by the first derivative of the space curve w.r.t. its parameter thus resulting in the velocity command:

| |

. (2)

Figure 4. Velocity profile for a spline curve.

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The progress of the reference point is calculated by the Euler integration of the velocity command using the command sample time. To prevent the reference point from moving far away from the vehicle’s real position, a feedback and a limitation of its progress is used.

For a curve, a significant problem of this approach is that a flight direction obtained from the tangent will guide the constantly away from the desired path, as noticeable in Figure 5. Nevertheless, the application of an error feed-back will reduce this behavior and guide the vehicle back to the desired path.

B. Feedback Signal An improvement of the closed-loop dynamics of the position feedback would lead to faster error reduction and smaller path deviation. For a pure proportional feedback, the root locus of the corresponding linear system deter-mines the critical gain for this feedback. The linear reference models, designed for the velocity and attitude loops, are used for the feedback gain determina-tion. Further improvements of the feedback signal could only be achieved by using a controller of higher order. This controller would generate signals containing more challenging velocity commands for the helicopter. However, the reference models have been designed, in order to avoid unfeasible commands for the helicopter. Thus, forcing the reference models to follow certain trajectories is not desirable. Instead, the modification of the velocity command is presented to improve the tracking accuracy, and will be described in section III.

The response to a slope command for the position in one axis is illustrated in Figure 6. It shows the response of the system to a ramp with and without the profile velocity signal, called feedforward in the figure. After about4 , the system output follows the command. An overshoot of about 10% is apparent at 10 , when the command is held constant. It takes another5 for a steady-state condition.

For constantly changing spline paths, the state will not completely converge to the command. Therefore, the sys-tem outputs will not achieve the commands and a persistent path error will occur. In other words, the previously presented velocity profile algorithm is not able to account for the phase lag of the subordinated control loops. At a turn for example, the corresponding change in direction of the command will occur when the system is already at the beginning of the maneuver. However, the system needs some time to adjust to the new command. The path error increases and the error feedback loops are not fast enough to compensate immediately.

The tracking accuracy can be improved by at least two methods. First of all, a reduction of the absolute velocity command reduces the path deviation. The lower the total velocity the lower is the covered distance while the states of the systems are not settled to the desired. Second, the velocity command can be modified to account for the phase lag. The following section will present a method to adopt the velocity profile commands by a time-shift accordingly.

Figure 5. Path control: Both parts for path control, veloc-ity command feedforward direction and path error, per-pendicular to the path.

Figure 6. Comparison of the response to a distance command.

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III. Path Tracking Using Time-Shifted Feedforward Signals Tracking improvement can be achieved using a geometry motivated guidance law, as presented by Park, Deyst,

and How 3. This paper however, uses a reference point related to a period of flight time ahead instead of a geomet-rical distance. During a controller update step the last velocity command is used to determine the reference point on the path for a certain time in the future. Therefore, the future flight speed must be predicted. As a first step, the ve-locity is assumed to be constant. The flight speed and the prediction-time are resulting in a time-shifted velocity profile signal. This time-shift is equivalent to a velocity dependent distance.

As presented in section II, the direction of the velocity command vector is parallel to the tangent on the path. Moving the reference point for a certain distance along the path, depending on the flight velocity, and using the direction of this future point results in a time-shift of the feedforward signal, see Figure 7. The reference position is not modified, thus the path error remains unaffected.

In order to determine an appropriate time-shiftΔt, a linear approximation of the reference models of the velocity and attitude loops is considered. Using this model instead of the real helicopter model assumes that in real flight, the helicopter tracks this designed reference models very closely. In contrast to the helicopter’s overall dynamics, all parameters of the reference model are exactly known as it was specifically designed. An approximation is used to simplify the following calculations.

The approximated reference model is simulated over the whole path as if it describes the helicopter dynamics precisely. The path deviation is minimized by applying a number of time-shiftsΔt t varying over the simulation time of the whole path, withΔt ∈ 0, 0.01, 0.02,… , 2.5 s. Calculating a cost function at each time step and deter-mining the minimum of , results in the optimal time-shift

argmin , (3)

With the error vector the cost function , is defined as

, √ (4)

The sample time of the simulation isT = 0.05s. The optimal time-shift to minimize the Euclidian distance to

the path is calculated for an optimization horizon ofT 2.5s beyond the current time step .

Figure 7. Velocity command obtained by a timely-shifted point on the path.

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For the optimization process, a two dimensional loop with a changing curvature is selected as presented in Fig-ure 8. At the origin 0,0 , the helicopter model is initialized with the maximum velocity of , 20 / . Figure 9 shows the evolution of the time-shiftΔt, the velocity V , and the path error over time. The velocity command is calculated based on the curvature respectively the curve radius, which is presented in the corresponding figure as well. The dashed line shows the response of the linearly approximated reference model.

Figure 8 shows a significant reduction of the path deviation for a time-shifted feedforward signal. The maximum path tracking error is less than3 . The time-shift varies between0.8 and1.5 (see Figure 9). Unfortunately, a direct relation between time-shift, the flight speed and the curve radius is not apparent. Furthermore, the method to determine the time-shift is based on a simulation, which takes about a hundred times longer than the time to fly the path.

To reduce the necessary calculation time, the application of a constant time-shift is evaluated. Figure 9 presents the time shift evaluation as multiple of the sample time. Its mean is aboutΔ 1.1 , which is used as constant time shift in this simulation. Figure 10 shows the resulting path error for varying, constant and without time-shift (Δ 0 . It is apparent that for constant time-shifts, the tracking performance degrades not significantly compared to the path of variable time-shift. In both cases, a remarkable improvement of the path tracking is achieved. This confirms that the time lag motivated approach has significant impact on the overall tracking quality.

Motivated by these results, focusing on a constant time-shift becomes feasible using an overall system simula-tion. This simulation includes a proper flight mechanical model based on system identification around hover condi-tion, disturbance models, the complete nonlinear flight controller, sensor emulation as well as fusion and mission management. For the overall simulation, the resulting time-shift obtained by manual tuning, turns out to be in the order of 2 . In order to also compensate for system inherent delays also, the manual tuning was necessary within the final flight software architecture, which does not allow iterative optimization. The time-shift is in the same scope of magnitude as the one resulting from the optimization problem, however, the nonlinearities of the system as well as state estimation errors are important to consider for real flight application. Using the Δ 2 time-shift the flight results of the subsequent section have been achieved.

IV. Evaluation of Constant Time-Shift for a 3D-Spline Path in Flight Test and Simulation It is most likely that a simulation environment does not include all effects occurring in reality. Therefore, a flight

test as validation of the simulation environment is necessary. Objective of the flight test is the validation of the mag-nitude of the path error reduction observed in the simulation studies, as presented in the sections before.

Figure 8. Characteristic path deviation depending on a time-shift.

Figure 9. Time-shift, resulting velocity command and path error for path loop

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A. Test Setup The helicopter presented in Figure 2 is capable to fly a ground speed up to20 / , under calm wind conditions. The sensor suite currently available includes differential GPS, magnetometer, accelerometer and rate sensors, as well as state-of-the-art sensor fusion algorithms. Wind speed and direction are not monitored and, therefore, cannot be in-cluded in the control algorithms. Wind disturbances are resulting in external disturbances which are compensated by the control error feedback, similar to those resulting from uncertainties in the plant dynamics model or the initial conditions. The flight velocity, calculated by equation (1), is limited to15 / to have sufficient reserve for wind disturbance compensation.

The flight path selected for the analysis and evaluation of the approach under real conditions is shown in Figure 11. The helicopter starts from the origin (0,0,0) by turning into flight direction first. The X-axis is pointing to the north, Y-axis is pointing to the east. The wind conditions were around 5 8 / from the south. The flight path was selected due to its characteristics of including sharp turns in left and right direction, as well as including ascent and descent parts that are observable from a safety pilot from a steady position on ground.

B. Test Results The nominal flight path from Figure 11 is first evaluated in simulation. Figure 14 shows the results. The flight time for both configurations, with and without time-shift, differs. Therefore, the time axis is normalized by the total flight time. The maximum path error without time-shift is close to25 . The approach including a time-shift of2 is able to reduce the error below7 . The total flight velocities as well as the velocity tracking error are similar for both configurations, as presented in the two subfigures of Figure 14. The velocity tracking error | | is low0.6 / for both, with and without time-shift.

Figure 15 presents a segment of the 3D-path following performance for the flight. Significant differences are ev-ident. Figure 12 shows the path error, flight velocity and velocity tracking error for the flight test in detail. Com-pared to the simulation results the absolute path error maximum is increased to 32 for the configuration without time-shift. By enabling the time-shift of 2 , the path following error is reduced to less than8 . Remarkably, the velocity tracking error is also increased, in the beginning up to about2 / , later below1 / . This indicates that the small increase of the path error compared to the simulation may result from the decreasing velocity tracking performance. The reduced velocity tracking performance is hence primarily caused by wind and plant modeling errors. Measurement and state estimation errors are not available, as the commands are compared to the state esti-mates rather than to the real, and not exactly known, states of the helicopter.

Figure 13 presents the path tracking error decoupled into horizontal and vertical direction. Using no time-shift, there is a significant difference between the horizontal and vertical error. Introducing the time-shift, results in the flight path error in the horizontal direction being within the same scope of the vertical error.

Figure 10. Nominal flight path selected for flight tests.

Figure 11. Nominal flight path selected for flight tests

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Concluding, the flight test directly confirms the simulation results. It is interesting to note, that the magnitude of the flight velocity do not differ significantly. However, as the flight path error is significantly reduced using the time-shift, thus reducing the duration of the flight. The flight path error is reduced by a factor of about 3.5 as was suggested in the simulation.

V. Conclusion The presented approach utilizes the linear reference dynamic model of an explicit model following control sys-

tem. The cubic splines used for the geometric path representation are continuously differentiable only up to the third derivative. Therefore, common approaches, e.g. providing a command obtained from the differentiation of the path, could not be applied for exact tracking. Certain path tracking error is accepted as finding a collision-free three times differentiable path in a densely populated 3D-environment is already a complex task.

Time-shift based feedforward signals on the other hand improve the tracking error significantly. This paper out-lines a tuning approach for the time-shift parameter based on the performance for cubic spline curves. Variable time-shifts improve the flight path error significantly. However, the computational effort does not allow the use of the

Figure 14. Simulation results: Path error, flight velocity and velocity tracking error for 3D-Spline path.

Figure 15. Flight test results: Snapshot of path following performance with and without time-shift.

Figure 12. Flight test results: Path error, flight velocity and velocity tracking error.

Figure 13. Flight test results: Decoupled path tracking error.

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necessary online optimization in real flight. Additionally, a relation between the necessary time-shift and geometry of the space curve is not directly apparent. In real flight, constant time-shifted feedforward is applied. The results show a marginally larger path tracking error compared to the variable time-shift. The overall path error is reduced by a factor of 3.5 for the presented flight test using constant time-shifted feedforward commands.

This paper does not treat the remaining degree of freedom, which is the helicopter’s heading. Treating the head-ing dynamics according to this approach can achieve a flight without sideslip angle. Furthermore, the vertical flight path error remains unaffected by the time-shift. It remains for future investigation, whether the reason is different dynamical properties of the helicopter’s vertical axis. A separation of the time-shift in vertical and horizontal direc-tions could thus be required for further improvement. The remaining flight path error might also reach a limitation now, caused by the achievable velocity tracking of the underlying control loops.

Acknowledgments The authors gratefully acknowledge the contribution of Timm Faulwasser and Markus Heiny of the Otto von

Guericke University of Magdeburg, Germany, and the team members of the unmanned aircraft department for the support and constructive discussions.

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