This paper examines the problem of simultaneously optimizing the spooling and cross-current flight trajectory of a tethered marine hydrokinetic kite using an analytic solution of its inverse dynamics. Tethered kites hold considerable promise for energy production, especially when undergoing cross-current motion. The novelty of this work lies in the use of an analytic solution of the inverse dynamics of the kite to solve the trajectory optimization problem. The term “inverse dynamics” refer to the process of obtaining an exact solution for the actuator inputs from the position, velocity, and acceleration of the kite. While the literature on tethered kites explores trajectory optimization in great detail, most of the work exploits the forward dynamics of the kite, and does not simultaneously optimize the spooling motion and cross-current trajectory. This paper formulates the co-optimization of the kite spooling and cross-current trajectory using a three degrees-of-freedom kite model, paired with an inelastic tether model. The analytic solution of the inverse dynamics is solved in terms of the roots of a fourth-order polynomial in terms of the angle of attack. A simulation study validates the optimization approach and shows that the kite is able to achieve significant energy production.
This paper presents the simultaneous optimization of the spooling motion and cross-current trajectory of an energy harvesting marine hydrokinetic kite using an analytic solution of the kite's inverse dynamics. Compared to towered turbines, tethered kites (i) have cheaper installation costs, (ii) can fly faster than the prevailing fluid, thereby generating power an order of magnitude higher than towered turbines, and (iii) can reach greater depths. For example, the Gulf Stream off the coast of North Carolina can be exploited to meet the energy demand of the entire state, but since the current with the highest power density is at a depth of 200 m, towered turbines cannot easily reach such depths .
Tethered kites can be operated in either (i) drag mode, where the generator is on board the kite, or in (ii) pumping mode, where the power take off system is on a platform tethered to the kite . The focus of this paper will be on a pumping mode marine tethered kite, as shown in Fig. 1. In such systems, the kite is reeled out at high tension to generate power, and reeled in at low tension, thereby utilizing power. The overall goal is to produce maximum net positive power, and since both the spooling motion and the cross-current trajectory play an important role in the quantity of power produced, they both need to be optimized.
It is widely known that tethered kites produce significantly more power when flying cross-current than when flying in stationary motion . Cross-current motion refers to when the kite moves perpendicular to the incoming fluid velocity. Different techniques have been used to maximize power production through trajectory optimization, for both airborne and marine hydrokinetic kites. Canale et al., for instance, introduce a model of a pulsing mode kite and use it to optimize the kite's trajectory through model predictive control [3–5]. Numerical optimization via a direct multiple shooting method is used by Houska and Diehl for both a power generating and a towing kite [6,7]. Denlinger uses water channel experiment data to develop a two degrees-of-freedom (2DOF) model of a tethered kite, whose trajectory is then optimized online through extremum seeking control . Taking into account the kite's added mass and inertia, Cobb et al. build a “unifoil” marine hydrokinetic (MHK) model, which is later extended to a 6DOF model and optimized using iterative learning control [9,10]. Other methods in the literature include the use of maximum power tracking control by Mademlis et al. , and passivity based control by Li et al. [12,13].
For kites operating in pulsing mode, i.e., kites with the motor/generator on a platform, the spooling motion also affects power production. In pulsing mode, the kite is spooled in and out at low and high tension, respectively. However, pulsing mode kite trajectory optimization has been traditionally done in one of the following three ways: (i) optimizing the spooling motion and the trajectory separately as in Refs. , , and  or (ii) optimizing just the spooling motion as in Ref. ; or (iii) optimizing the spooling motion and the path (as opposed to the trajectory) simultaneously, as in Ref. .
One of the features of the above papers is that they mostly focus on solving the forward dynamics, i.e., solving for the kite's acceleration given its position, velocity, and actuation inputs. This paper presents a novel approach that solves the optimization problem through the analytic inverse dynamics solution. Given the position, velocity, and acceleration of the kite, the inverse dynamics furnish the exact solution of the actuation inputs, i.e., the angle of attack, the induced roll angle, and the tether tension. A previous paper by the authors  presented a detailed solution to the inverse dynamics of the kite. Solving for the inverse dynamics is important for several reasons:
The inverse dynamics solution provides an easy method to vet the feasibility of a given kite position, velocity, and acceleration trajectory.
The exact solution to the inverse dynamics provides insights into the potential multiplicity of actuator inputs corresponding to a given kite position, velocity, and acceleration trajectory.
This paper furthers the authors' previous work in Ref.  by utilizing the inverse dynamics solution to the kite spooling and trajectory optimization problem. This optimization is performed using a Fourier expansion of the kite's state trajectories over time that transforms the trajectory optimization problem into a nonlinear program [20,21]. The novelty of the paper stems both from the fact that it co-optimizes the kite spooling and cross-current trajectories simultaneously, as well as the fact that it exploits inverse kite dynamics in doing so.
2 Equations of Motion
This section presents the 3DOF kite model used for this paper's optimization work, building on earlier kite modeling research by the authors . The kite in this model is a pulsing mode kite, i.e., the generator is located on a fixed platform to which the kite is tethered. In building the kite model, we make the following key assumptions:
The kite's motion can be controlled through its angle of attack, induced roll angle, and tether tension, an assumption that justifies this paper's representation of the kite as a point mass.
The kite is neutrally buoyant, and the hydrodynamic side forces are negligible because the kite “weathercocks.”
Figure 2 shows the coordinate frames used to model the kite and the forces acting on it. Two coordinate frames are defined: (i) a fluid frame and (ii) a base frame, with the origin at the tether anchor point.
where the coefficients in the above relationships, as well as other kite parameters, are summarized in Table 1.
|ρ||Density||1000 kg m−3|
|uw||Fluid speed||1 m s−1|
|ρ||Density||1000 kg m−3|
|uw||Fluid speed||1 m s−1|
3 Optimization Approach
The above optimization statement has multiple objectives:
To maximize average mechanical power produced over one time period, defined as the time taken for the kite to spool in and out once. Here ti is the initial time, and tf is the final time.
To maintain a smooth motion trajectory by penalizing the time rates of change of the three control inputs.
To ensure that the control inputs do not surpass reasonable minimum and maximum bounds.
In the above Fourier expansions, the vectors and are expansion coefficients corresponding to the nth harmonic of the Fourier expansion. In this work, n is chosen to be 3 for simplicity. The period of the Fourier series is . Expressing the state as a Fourier expansion allows for the optimization of the initial kite position and velocity, as part of trajectory optimization. Thus, the coefficient vectors , and the time period Tperiod become the optimization variables. These optimization variables must satisfy the state dynamics. In the authors' previous work, this was solved by representing both the states and the control inputs as Fourier series, and then imposing a penalty on the discrepancy between the kite forces produced by the assumed control input trajectories versus the forces required for achieving the desired state trajectories. In contrast, this paper uses an analytic solution of the inverse dynamics, allowing one to solve the trajectory optimization exactly. Since the kite's dynamics are nonlinear, the trajectory optimization problem is therefore nonconvex. Given this nonconvexity, a particle swarm algorithm is used for the initial exploration of the optimization space, followed by the use of the Nelder–Meade simplex algorithm for the local refinement of the particle swarm solution. Figure 3 shows a high level summary of the optimization approach.
Using higher harmonics leads to only a small increase in the power production. For example, as seen in Fig. 4, while the Pareto cost decreases by 25.69% from n = 4 to n = 5, the power production only increases by 1.65%. From n = 3 to n = 4, the Pareto cost decreases by 2.65% and power production increases by 6.31%. Increasing the number of Fourier coefficients also increases the computation time, and therefore, in this paper, n = 3 was chosen as the main case to be examined numerically.
Similarly, an analysis of different Pareto weights shows that the increase in power and reduction in Pareto cost is minimal with higher Pareto weight. Therefore, wp = 2500 is a reasonable choice for the subsequent simulation study (Fig. 5).
4 Analytical Solution to Inverse Dynamics
The inverse dynamics of the tethered kite, i.e., the exact solution of the control inputs given the position, velocity, and acceleration, can be obtained from the above equations through a fourth-order polynomial of the angle of attack. Building on previous work by the authors, this paper adopts the following approach for solving these inverse dynamics :
The unit vectors defining the fluid coordinate frame, , and , can be calculated given the kite's position, velocity, the freestream fluid velocity, and the tether orientation. Moreover, since the unit vector in the direction of the tether, , is orthogonal to can be defined as follows:
If is the total force required for achieving the desired kite acceleration, it can be calculated from the given kite velocity
The projection of the above force onto the fluid coordinate frame furnishes three required forces, f0, f1, and f2, in this frame
Squaring the subsequent expression leads to the following expression:
Equation (36) can finally be solved for α analytically in terms of the roots of a fourth-order polynomial. The values can then be substituted in Eq. (29) to solve for T. Moreover, Eqs. (30) and (31) can be used to solve for and , respectively, which makes it possible to uniquely determine .
One interesting outcome of the analytical solution is that there are four potential solutions for the kite's inverse dynamics. Since L/D represents a particular force direction, and because of the relationship between CL and CD and α, a particular L/D ratio can be achieved using a high or low combination of lift, drag, and α. Moreover, since each solution of α has a symmetric counterpart when the kite flies “upside down,” each of the above combinations also has a symmetric counterpart. In solving the above optimization problem, therefore, one needs to select the best inverse dynamics solution arc for every given kite trajectory.
This section presents simulation results showing that the optimization algorithm that incorporates the inverse dynamics solution and the 3DOF kite model, successfully furnishes a trajectory with desirable power production. A kite of the area used in this study would theoretically produce a maximum of 83.9 kW of power during steady-state reel-out . The optimization algorithm was able to produce an average power of 21.3 kW within one time period, consistent with the authors' previous work that approximated the kite's dynamics during optimization instead of solving them exactly . This is important because it provides more confidence in the feasibility of achieving such a level of average power. Figure 6 further shows that the kite consumes very little power during reel-in, never exceeding 3 kW. Moreover, during reel-out, the kite is able to produce significant instantaneous power, reaching almost 60 kW. Therefore, the kite is able to achieve a Loyd factor, defined as the ratio of the generated power to the maximum theoretical steady-state reel-out power, of 25.38%.
The kite follows a simple trajectory of a short reel-out and reel-in episode, before being reeled-out for a longer period of time, as shown in Fig. 7. It is assumed that the kite's anchor point is sufficiently underwater such that the kite does not rise above the surface as it traverses this trajectory.
The full trajectory, as seen in Fig. 8, is similar to results obtained in Ref.  in that the optimal path shape is once again not Fig. 8, and can be approximated as a sequence of circles in the yz-plane. The non-Fig. 8 path shape eliminates the need for switching the direction of induced roll, which is attractive from the perspective of minimizing the rates of change of various actuation inputs. Moreover, this shape is acceptable due to the fact the kite is neutrally buoyant, and therefore does not need to exploit the potential safety benefits of a Fig. 8 trajectory where the kite turns upward (as opposed to downward) when it “tacks” (i.e., switches direction).
As shown in Fig. 9, the velocity of the kite in the yz-plane is larger than in the x-direction, thereby showing that the kite is exploiting cross-current motion substantially. The speed with which the kite reels in and out, , does not always equal the speed of the kite parallel to the freestream velocity, . This implies that the kite achieves a nontrivial portion of its reel-in/out motion by traveling either toward or away from the origin in the yz-plane. Such in-plane reel-in/out motion is intuitively attractive because it minimizes the total distance required for kite operation parallel to the freestream fluid velocity. According to Ref. , the optimal velocity for the kite to produce maximum power should be 1/3 of the freestream velocity, which in this case is 1/3 ms−1. The kite's transient reel-out speeds are slightly higher, enabling higher levels of power generation during transient reel-out.
During spool-out, the angle of attack reaches the maximum value ensuring a high lift-to-drag ratio, as shown in Figs. 10(a) and 10(b). This also ensures that during the spool-out, the tension is high, as seen in Fig. 11. During spool-in, the angle of attack becomes negative, and the tether tension at the corresponding time reaches a minimum value.
The roll angle does not change sign during the duration of the trajectory, and reaches a maximum value during spool-in, as shown in Fig. 12. This causes the tether tension and lift vector to be misaligned, and thereby ensures low tension spool-in.
The above results are appealing for at least four reasons. First, from a fundamental perspective, the results represent the first optimization study where both spooling and cross-current motion were co-optimized using direct transcription, with an underlying analytic solution of the inverse kite dynamics. Second, from a practical perspective, the results furnish a kite trajectory that confirms the attainability of significant energy harvesting through cross-current motion, confirming earlier findings by the authors based on an approximate solution of the kite dynamics. Third, the results provide new insights into the shape of the optimal kite trajectory. In particular, they highlight the degree to which circular trajectories can potentially provide energy harvesting benefits similar to those seen in the literature with Fig. 8 trajectories. Moreover, they highlight the degree to which kite reeling motion can take place, at least partially, in plane, as opposed to in alignment with the freestream fluid velocity. This is particularly attractive from the perspective of multikite spacing in a farm of kites, where the smaller the total distance taken up by each kite in the freestream fluid direction, the greater the potential for maximizing effective overall farm power density.
This paper presents the first effort to solve the co-optimization problem along with the inverse dynamics of a pumping mode MHK kite analytically. The study highlights how the inverse dynamics can provide useful insights into trajectory optimization, and highlight the degree to which cross-current trajectory optimization is beneficial for MHK systems. Future work can take into account actuator dynamics, lower-level controllers, tether dynamics, added/entrained mass, and fluid–structure interactions.
U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (DOE EERE), through DOE (Award No. #DE-EE0008635; Funder ID: 10.13039/100006134).
Data Availability Statement
The authors attest that all data for this study are included in the paper.