Residual Dynamics Learning for Trajectory Tracking for Multi-rotor Aerial Vehicles

TL;DR

A technique to model the residual dynamics between a high-level planner and a low-level controller by considering reference trajectory tracking in a cluttered environment as an example scenario and it is shown that the proposed approach outperformed the others with less flight time without losing computational efficiency.

Abstract

This paper presents a technique to cope with the gap between high-level planning, e.g., reference trajectory tracking, and low-level controlling using a learning-based method in the plan-based control paradigm. The technique improves the smoothness of maneuvering through cluttered environments, especially targeting low-speed velocity profiles. In such a profile, external aerodynamic effects that are applied on the quadrotor can be neglected. Hence, we used a simplified motion model to represent the motion of the quadrotor when formulating the Nonlinear Model Predictive Control (NMPC)-based local planner. However, the simplified motion model causes residual dynamics between the high-level planner and the low-level controller. The Sparse Gaussian Process Regression-based technique is proposed to reduce these residual dynamics. The proposed technique is compared with Data-Driven MPC. The comparison results yield that an augmented residual dynamics model-based planner helps to reduce the nominal model error by a factor of 2 on average. Further, we compared the proposed complete framework with four other approaches. The proposed approach outperformed the others in terms of tracking the reference trajectory without colliding with obstacles with less flight time without losing computational efficiency.

Figures (9)

• Figure 1: The high-level control command generation is based on Model Predictive Control (MPC). Residual dynamics ($y$) that arise between high-level control generated by MPC and low-level control generated by flight controller can not be estimated analytically. Hence, the Sparse Gaussian Process(SGP)-based learning technique is proposed to estimate $y$, where $\mathbf{x}_k$, $\bar{\mathbf{x}}$, and $\hat{\mathbf{x}}$ are denoted current state, next desired state, and actual state after applying the current control to the system, respectively.
• Figure 2: The high-level overview of the proposed framework for learning the residual dynamics. Design matrix $B_{\mathbf{z}}$ defines which states and control inputs should be trained. Learned residual dynamics $g(\mathbf{z}_{k+i-1})$ is added to nominal dynamics $f_{norm}(\mathbf{x}_{k+i-1}, \mathbf{u}_{k+i-1})$ when formulating NMPC by using the multiple shooting technique. In the training stage, collect the data $D = \{X, \mathbf{y}\} = \{(\mathbf{x}_i, \bar{\mathbf{u}}_i, \bar{\mathbf{x}}_i, \hat{\mathbf{x}}_i), y_i\}, i=0,...,n$ for n number of times, where $(\mathbf{x}_i, \bar{\mathbf{u}}_i, \bar{\mathbf{x}}_i$, and $\hat{\mathbf{x}}_i)$ denote current state, estimated near-optimal control inputs and state after applying NMPC, and actual system state after applying estimated control inputs, respectively. We used the same PD regulator that was proposed in kulathunga2021trajectory. This work focuses on the DJI M100 quadrotor as the representative real system.
• Figure 3: Epistemic uncertainty propagation of residual dynamics $\mathbf{y}_y$ for changing of input velocity $\mathbf{v}_y$, where subscript $_y$ indicates the y axis. Variable $\mathbf{y}_y$, defined the rate of the different between predicted velocity $\bar{\mathbf{v}}_y$ and actual velocity $\hat{\mathbf{v}}_y$. Inducing points provide an approximation for $\mathbf{y}_y$. A Gaussian process is formed using such inducing points, i.e., sparse Gaussian process, which can be used to predict residual dynamics for corresponding input velocity $\mathbf{v}_y$.
• Figure 4: The right sub-figure: A trajectory of motion along all the axes was used to collect the data for learning and testing the latent representation of residual dynamics (Fig. \ref{['fig:induced_points']}). The left sub-figure: DJI M100 quadrotor is used for real-world experiments and PX4-enabled quadrotor is used for simulated environments
• Figure 5: The relationship between residual dynamics $\mathbf{y}_y$ and input velocity $\mathbf{v}_y$. In left sub-figure: the first plot displays how $\mathbf{y}_y$ and $\mathbf{v}_y$ vary over time, while the second plot shows the approximation of the $\mathbf{y}_y$ using two different approaches: Sparse Gaussian Process-based (proposed) and cluster selection approach torrente2021data, in right sub-figure: residual dynamics before and after incorporating the learnt Sparse Gaussian process. Root Mean Square Error drops from 0.4012 to 0.0751 after introducing the residual dynamics into the nominal motion model