Real-Time Trajectory Generation and Hybrid Lyapunov-Based Control for Hopping Robots

TL;DR

The paper tackles the limitation of ballistic aerial phases in rotor-based hopping robots by delivering a real-time, non-linear drag compensated trajectory generation framework and a hybrid Lyapunov-based controller to command liftoff-to-touchdown motion. It leverages differential flatness with flat outputs $\nu=[x,y,z,\psi]$ and a three-keyframe scheme ($\alpha_0, \alpha_1, \alpha_2$) to synthesize trajectories via precomputed $\mathbf{P}_l^{-1}$ and $\mathbf{N}_l$, avoiding online optimization. A hybrid control strategy provides stability across the aerial phase while modeling the stance phase as a discrete event with a stability-linked constraint $V(\mathbf{x}_{LO})-V(\mathbf{x}_{TD})\le 0$, and incorporates non-linear drag terms into the dynamics for accurate tracking. Results demonstrate drag-aware trajectory tracking on horizontal and vertical touchdown surfaces, with performance advantages amplified for aggressive maneuvers and free TD constraints, and the approach is argued to be broadly applicable to hopping robots and quadrotors alike.

Abstract

The advent of rotor-based hopping robots has created very capable hopping platforms with high agility and efficiency, and similar controllability, as compared to their purely flying quadrotor counterparts. Advances in robot performance have increased the hopping height to greater than 4 meters and opened up the possibility for more complex aerial trajectories (i.e., behaviors). However, currently hopping robots do not directly control their aerial trajectory or transition to flight, eliminating the efficiency benefits of a hopping system. Here we show a real-time, computationally efficiency, non-linear drag compensated, trajectory generation methodology and accompanying Lyapunov-based controller. The combined system can create and follow complex aerial trajectories from liftoff to touchdown on horizontal and vertical surfaces, while maintaining strick control over the orientation at touchdown. The computational efficiency provides broad applicability across all size scales of hopping robots while maintaining applicability to quadrotors in general.

Figures (4)

• Figure 1: Photo of the MultiMo-MHR with components labeled.
• Figure 2: Hopping trajectory generation and control (Tables \ref{['tab:hop_keyframes']} and \ref{['tab:traj_error']} show details) for constant initial conditions (keyframe $\alpha_0$) including roll, pitch, and yaw of $[\phi, \theta, \psi] = [0,30,0]$ degrees, velocity magnitude aligned with the body z-axis of $5$ m/s, trajectory time $t_m = 1.75$ seconds (LO-TD), and reorientation time $\delta_t = 0.05$ seconds; where total simulation time equals $6$ seconds. The TD keyframe $\alpha_2$ for all generated trajectories includes $v_{TD_d}=5$ m/s and the desired orientation $[\phi, \theta, \psi]$ is set as: a,b) $[0,30,0]$, c,d) $[0,-30,0]$, e,f) $[0,0,0]$, g,h) $[0,0,0]$ degrees.
• Figure 3: Hopping trajectory generation and control (Tables \ref{['tab:hop_keyframes']} and \ref{['tab:traj_error']} show details) for constant initial conditions (keyframe $\alpha_0$) including roll, pitch, and yaw of $[\phi, \theta, \psi] = [0,30,0]$ degrees, velocity magnitude aligned with the body z-axis of $5$ m/s, trajectory time $t_m = 1.75$ seconds (LO-TD), and reorientation time $\delta_t = 0.05$ seconds; where total simulation time equals $6$ seconds. The TD keyframe $\alpha_2$ for all generated trajectories includes $v_{TD_d}=5$ m/s and the desired orientation $[\phi, \theta, \psi]$ alternates between $[0,-30,0]$ and $[0,30,0]$ degrees. a) No drag compensation. b) Drag compensation.
• Figure 4: Hopping trajectory generation and control (Tables \ref{['tab:hop_keyframes']} and \ref{['tab:traj_error']} show details) for constant initial conditions (keyframe $\alpha_0$) including roll, pitch, and yaw of $[\phi, \theta, \psi] = [5,30,0]$ degrees, velocity magnitude aligned with the body z-axis of $5$ m/s, trajectory time $t_m = 1.75$ seconds (LO-TD), and reorientation time $\delta_t = 0.05$ seconds; where total simulation time equals $6$ seconds. The TD keyframe $\alpha_2$ for all generated trajectories includes $v_{TD_d}=5$ m/s and the desired orientation $[\phi, \theta, \psi]$ alternates between $[-5,-30,0]$ and $[5,30,0]$ degrees. The desired trajectories are drag compensated. a) Shows the z-x plane. b) Shows the y-x plane. c) Shows the individual states.