Robust Precision Landing of a Quadrotor with Online Temporal Scaling Adaptation of Dynamic Movement Primitives

Kongkiat Rothomphiwat, Prakarn Jaroonsorn, Pakpoom Kriengkomolt, Poramate Manoonpong
Vidyasirimedhi Institute of Science & Technology, Rayong, Thailand
Paper
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Abstract

In this work, we address the challenges of robust precision landing maneuvers for a quadrotor on both stationary and moving ground targets in the presence of disturbances that can cause the quadrotor to deviate from its desired trajectory, leading to maneuver failure. To overcome this, we propose a novel online adaptive trajectory planning approach based on the online temporal scaling adaptation of dynamic movement primitives (DMPs). This adaptation enables the desired trajectory to be dynamically adjusted in response to tracking errors and the goal’s state. Consequently, our proposed approach enhances accuracy, precision, and safety during landing maneuvers. The effectiveness of the approach is evaluated through comprehensive experiments conducted in both physical simulations and real- world environments, covering various disturbance scenarios.

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Stability Analysis

The stability analysis is separated into two parts, which include:

1. Stability of Trajectory Adaptation: Prove that the adaptation mechanism adjusts the trajectory in response to errors and goal changes, ensuring that the trajectory remains effective and feasible.

2. Overall System Stability: Demonstrate that the entire system, including the trajectory adaptation, is stable and converges to the desired state, ensuring robustness and precision in performance.

System Description

The key equations governing the system are:

Transformation system:

\[ \tau\dot{z} = (1 - s) \alpha_y \left( \beta_y (g - y) + \tau ( \dot{g} - \dot{y} ) \right) + \Omega, \]

\[ \Omega = \text{diag} \left( g - y_0 \right) f(s), \]

\[ \tau \dot{ y} = z, \]

\[ \tau \dot{s} = - \alpha_x s, \]

Temporal scaling adaptation law:

\[ \dot{\tau} = -\kappa_{\tau}(\tau-\tau_{g}) + \dot{\tau_{g}}, \]

\[ \tau_{g} = \frac{\big\|g - y_{0} \big\|}{\big\|g_{d} - y_{0,d} \big\|}\tau_{d} + k_{c}e_{t}^2, \]

Error Dynamics:

\[ e_{t} = (1-\alpha_{e})\big|y_{s}-y\big| - \alpha_{e}e_{t-1}. \]

Stability Analysis Assumptions

We assume the controller is contracting, meaning that small deviations from the desired trajectory should decay over time. The adaptation mechanism slows down the evolution of the trajectory to allow for error correction.

1. Stability of Trajectory Adaptation

1.1 Tracking Error Dynamics

The tracking error \( e_t \) is governed by:

\[ e_{t} = (1-\alpha_{e})\big|y_{s}-y\big| - \alpha_{e}e_{t-1} \]

This equation is a low-pass filter that smooths out the effect of rapid changes in tracking error. The term \(\big|y_{s}-y\big|\) represents the deviation of the quadrotor’s state from the desired trajectory. As long as \(y_s \) deviates significantly from \(y \), \(e_t\) will increase.

1.2 Temporal Scaling and Stability

The temporal scaling adaptation law:

\[ \dot{\tau} = -\kappa_{\tau}(\tau-\tau_{g}) + \dot{\tau_{g}} \]

\[ \tau_{g} = \frac{\big\|g - y_{0} \big\|}{\big\|g_{d} - y_{0,d} \big\|}\tau_{d} + k_{c}e_{t}^2 \]

Shows that as \( e_t \) increases, \( \tau_g \) increases, which in turn increases \( \tau \). Since \( \tau \) represents the timescale of the trajectory evolution, a larger \( \tau \) means the system slows down, giving more time for the error \( e_t \) to decay.

1.3 Contraction Mapping

For stability, we need to show that the system forms a contraction mapping, meaning that the difference between two states in successive iterations decreases:

\[ \| e_{t+1} \| < \| e_t \| \]

Given the low-pass filter dynamics:

\[ \| e_{t+1} \| = \left\| (1 - \alpha_e) |y_{s}-y\big| - \alpha_e e_t \right\| \]

As long as \( \alpha_e \) is chosen such that \( 0 < \alpha_e < 1 \) and the controller is contracting, the error \( e_t \) will reduce over time.

1.4 Conclusion

By increasing \( \tau \) in response to large tracking errors (due to strong perturbations), the system effectively slows down the evolution of the desired trajectory, allowing time for the error to decay. The combination of the low-pass filter and temporal scaling adaptation contributes to the stability of the system. Thus, under the assumption that the controller is contracting and \( \alpha_e \) is properly tuned, the system can be shown to stabilize around the desired trajectory, ensuring that the error \( e_t \) decreases over time.

2. Overall System Stability

2.1 Lyapunov Function Definition

To analyze the stability of the system, we define a Lyapunov candidate function that encompasses the energy of the system. The Lyapunov function is chosen as:

\[ V(\tau, z, s, e_t) = \frac{1}{2} (\tau - \tau_g)^2 + \frac{1}{2} z^2 + \frac{1}{2} s^2 + \frac{1}{2} e_t^2. \]

This function captures the deviation of the temporal scaling \( \tau \) from its reference \( \tau_g \), the system state \( z \), the scaling variable \( s \), and the tracking error \( e_t \).

2.2 Time Derivative of the Lyapunov Function

To prove stability, we compute the time derivative of \( V(\tau, z, s, e_t) \):

\[ \dot{V}(\tau, z, s, e_t) = (\tau - \tau_g) \dot{\tau} + z \dot{z} + s \dot{s} + e_t \dot{e}_t. \]

Substituting the system dynamics into \( \dot{V} \):

\[ \dot{\tau} = -\kappa_\tau (\tau - \tau_g) + \dot{\tau}_g, \]

\[ \dot{z} =\frac{(1 - s) \alpha_y (\beta_y (g - y) + \tau ( \dot{g} - \dot{y} )) + \Omega}{\tau}, \]

\[ \dot{s} = -\frac{\alpha_x s}{ \tau}, \]

\[ \dot{e}_t \approx -\alpha_e e_t \]

The time derivative of \( V(\tau, y, e_t) \) becomes:

\[ \dot{V} = (\tau - \tau_g) \left( -\kappa_\tau (\tau - \tau_g) + \dot{\tau}_g \right) + z\frac{ (1 - s) \alpha_y (\beta_y (g - y) + \tau ( \dot{g} - \dot{y} )) + \Omega}{\tau} - \frac{\alpha_x s^2}{\tau} -\alpha_e e_t^2. \]

2.3 Negative Definiteness of \( \dot{V} \)

We now analyze the individual terms:

  • The term \( (\tau - \tau_g) (-\kappa_\tau (\tau - \tau_g)) \) is non-positive as long as \( \kappa_\tau > 0 \) and \( T_g \) must be bounded.
  • The term \( z \cdot \dot{z} \) depends on the system dynamics and can be non-positive under proper \( \alpha_y\) and \(\beta_y \).
  • The term \( -\frac{\alpha_x s^2}{ \tau} \) is non-positive as \( \alpha_x > 0 \) and \( \tau > 0 \).
  • The tracking error term \( e_t \cdot \dot{e}_t \) is also non-positive if \( 0 < \alpha_e < 1 \) and \( |y_s-y| \) contracting (from above assumption).

    • Thus, with appropriate design of the controller parameters, \( \dot{V} \) can be made negative definite. This implies that \( V(\tau, z, s, e_t) \) decreases over time, leading to the conclusion that the system is stable and converges to the desired trajectory. The negative definiteness of \( \dot{V} \) implies that the system's energy decreases over time, leading to global asymptotic stability.

    2.4 Conclusion

    Through the Lyapunov-based stability analysis, we have shown that the proposed online adaptive trajectory planning approach with temporal scaling is globally asymptotically stable. This implies that the system can effectively detect and correct deviations caused by disturbances, ensuring that the quadrotor follows the desired trajectory, and that disturbances are mitigated over time. This analysis confirms that the system's state \( y_s \) controlled by the controller will converge globally asymptotically, thereby solving the disturbance problem and ensuring robust precision landing maneuvers.