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Optimal Tracking Controller Design for a Small Scale Helicopter
Agus Budiyono, Singgih S. Wibowo
TL;DR
The paper addresses the challenge of tracking predefined trajectories with a small-scale helicopter, whose dynamics and control constraints motivate a practical controller. It develops an optimal-control tracking design with bounded inputs. In simulations, higher state-tracking weights produced lower tracking error, and the controller maintained small errors within stipulated input bounds in the reported cases.
Problem
The paper addresses how to track predefined trajectories with a small-scale helicopter using practical multivariable control that accounts for physical input constraints.
Source: §1 · p. 1Method
The authors formulate a tracking controller using optimal control, augment the model with tracking error, and incorporate input bounds through Pontryagin’s minimum principle.
Source: §1 · p. 1, §3.1 · p. 3, §3.3 · p. 4Results
For position tracking of E and A, Case 3 (Q = I17) MSE is 0.0142 and 0.000233 of Case 1 (Q = 0.01I17); Cases 3 and 4 also maintained minimum tracking errors within stipulated control bounds.
Source: §5 · p. 7Takeaways & Limitations
The authors demonstrate that optimal control handled more complex design criteria than those traditionally afforded by classical control design.
Source: §6 · p. 8Whether the technique maintains acceptable performance in wind remains a question for future research.
Source: §6 · p. 8
Abstract
from arXiv · showhide
A model helicopter is more difficult to control than its full scale counterparts. This is due to its greater sensitivity to control inputs and disturbances as well as higher bandwidth of dynamics. This works is focused on designing practical tracking controller for a small scale helicopter following predefined trajectories. A tracking controller based on optimal control theory is synthesized as part of the development of an autonomous helicopter. Some issues in regards to control constraints are addressed. The weighting between state tracking performance and control power expenditure is analyzed. Overall performance of the control design is evaluated based on its time domain histories of trajectories as well as control inputs.
1 Introduction
Empirically tuned classical SISO control is not suited to the helicopter’s more complex MIMO performance criteria and physical constraints. This paper applies optimal control theory to design a practical tracking controller for a small-scale helicopter as part of an autonomous-system effort.
1 Introduction
- Classical SISO proportional-derivative controllers were usually tuned empirically, an approach the authors say does not suit complex MIMO systems with sophisticated criteria and physical constraints.Prior work cited includes optimal-control implementations on small helicopters and an adaptive high-bandwidth controller.Source: §1 · p. 1
- The paper applies optimal control theory to synthesize a practical tracking controller intended for an onboard computer in an autonomous helicopter.The authors report encouraging preliminary results for multivariable control synthesis of high-bandwidth helicopter dynamics.Source: §1 · p. 1
2 Dynamics of a Small Scale Helicopter
The R-50 helicopter model is assembled from rigid-body equations and simplified rotor, stabilizer-bar, and yaw dynamics. Its state-space description collects the dynamics and control inputs into matrix form.
2 Dynamics of a Small Scale Helicopter
- The R-50 model derives six-degree-of-freedom motion from Newton–Euler equations, representing aerodynamic and gravitational forces with stability derivatives and rotor forces with control derivatives.Its physical characteristics are summarized in Table 1.Source: §2 · p. 1, Table 1 · p. 2
2.1 Lateral and longitudinal fuselage dynamics
- Newton–Euler equations provide the lateral and longitudinal fuselage dynamics.The passage introduces the equations but does not specify their terms.Source: §2.1 · p. 2
2.2 Heaving (vertical) dynamics
- The heaving equation includes vertical velocity and collective input; in hovering, the model sets forward velocities to zero, making the parenthesized centrifugal-force terms relevant only in cruise.The equation is introduced as the Newton–Euler rigid-body representation of heaving dynamics.Source: §2.2 · p. 2
2.3 Yaw dynamics
- Yaw is modeled as first-order bare-airframe dynamics with yaw-rate feedback, while rotor and stabilizer-bar flapping are represented by first-order lateral and longitudinal dynamics in the state-space model.The state vector includes yaw rate, feedback, rotor-flapping, and stabilizer-bar states; the model is assembled as ẋ = Ax + Bu.Source: §2.3 · p. 2
3 Tracking Control Design
The controller uses a quadratic-cost linear regulator, recasts it for reference tracking, and incorporates hard input bounds into the synthesis. The design therefore weighs tracking against control effort while accounting for actuator limits.
3.1 Linear Regulator Problem
- The linear regulator minimizes a quadratic cost on state and control, yielding a Riccati-based feedback law; the finite-horizon solution is time-varying, while the steady-state limit gives the algebraic Riccati equation.The paper states that the resulting gain can be computed in advance and stored for real-time state feedback.Source: §3.1 · pp. 2–3
3.2 Path tracking problem formulation
- Tracking is formulated using reference-minus-state error and an augmented state, with η weighting tracking performance in the cost alongside control expenditure.The error dynamics use the derivative of the tracking error; the formulation assumes perfect measurements and uses a predefined constant reference in the stated case.Source: §3.2 · p. 4
3.3 Control synthesis in the presence of input constraints
- The synthesis imposes hard bounds on helicopter control inputs by minimizing the modified Hamiltonian over the admissible input set, rather than simply clipping an unconstrained solution.The authors explicitly caution that bounded-control histories cannot be obtained by calculating the unconstrained history and saturating violations.Source: §3.3 · pp. 4–5
4 Numerical Simulation
The simulations compare tracking and control expenditure across reference trajectories and weight settings, focusing on cruise data. Position and velocity results are transformed into local-horizon coordinates for interpretation.
4 Numerical Simulation
- The simulation compares tracking performance across reference trajectories and weighting matrices, including Q = 0.01I17 and Q = I17.The analysis focuses on cruise data and examines the tradeoff between tracking performance and control expenditure.Source: §4 · p. 5
- A body-to-local-horizon coordinate transformation is used to present the simulated positions and velocities in local-horizon coordinates.The passage defines c(·) as cosine and s(·) as sine.Source: §4 · p. 5
- Four cases pair rectangular or rectangular-then-circular reference trajectories with low or high weighting-matrix settings.Cases 1 and 2 use η = 0.01 and 0.01I weights; Cases 3 and 4 use η = 5 and identity weights.Source: §4 · p. 5, Figure 2 · p. 5, Figure 3 · p. 5
5 Discussion on results
Higher tracking and control weights are associated with improved tracking while keeping inputs within stated bounds, including on a more complex trajectory. Low weights yield degrading tracking, and the discussion identifies weight assignment as influential but largely case-specific.
5 Discussion on results
- Under low weights and no control-input constraints, tracking degrades near the end of Case 1, especially vertically, and degradation is more pronounced for Case 2’s complex trajectory.The unbounded control expenditure in Case 1 does not provide acceptable tracking performance.Source: §5 · p. 6
- With increased tracking and control weights, the controller handles the more complex trajectory while keeping tracking errors minimal and inputs within stipulated boundaries.The bounded-input results are shown for Cases 3 and 4.Source: §5 · p. 7
- Case 3’s Q = I17 tracking-error MSE is reported as 0.0142 and 0.000233 of Case 1’s Q = 0.01I17 value; the comparison is summarized in Table 2.The Q-weight effect is reported as more pronounced for position tracking of E and A.Source: Table 2 · p. 7, §5 · p. 7
- The case study assigns equal weights to control elements but does not treat augmented states as equally important.The passage presents this as an assumption about the optimization setup.Source: §5 · p. 7
- Weight assignments significantly influence tracking performance, while their determination has primarily been ad hoc and case-by-case.The discussion also points to pole-placement and polynomial approaches for more formal weighting assignment.Source: §5 · pp. 7–8
6 Concluding Remarks
The paper applies optimal control to a small-scale helicopter and reports that it handles more complex design criteria than traditional control design. The criteria include waypoint objectives, while performance in wind remains a question for future research.
6 Concluding Remarks
- The optimal-control approach handled more complex design criteria than those traditionally afforded by classical control design.The design is part of ongoing research, design, and integration of an autonomous helicopter.Source: §6 · p. 8
- The criteria include flying from an arbitrary origin to a waypoint in minimum time, minimizing control effort, and minimizing final-state deviation from the desired waypoint.Source: §6 · p. 8
- Whether the proposed technique maintains acceptable performance in wind is identified as a future research question.Source: §6 · p. 8
I. The state-space model of R-50 helicopter
This section introduces the state-space equation describing the R-50 helicopter dynamics.
I. The state-space model of R-50 helicopter
- The R-50 helicopter dynamics are described by a state-space equation.Source: §I · p. 9
II. The model parameters of R-50 helicopter
The section points to model parameters for hover and cruise flight and a table of control derivatives and time constants.
II. The model parameters of R-50 helicopter
- Model parameters for the helicopter during hover and cruise flight are summarized in a table.Source: §II · p. 9
- Table 3 is identified as covering control derivatives and time constants.Source: Table 3 · p. 9