Online Omnidirectional Jumping Trajectory Planning for Quadrupedal Robots on Uneven Terrains

TL;DR

This paper proposes a general and complete cascade online optimization framework for omnidirectional jumping for quadruped robots that achieves jump trajectory generation in approximately 0.1 seconds with a warm start and has been successfully validated on two quadruped robots on uneven terrains.

Abstract

Natural terrain complexity often necessitates agile movements like jumping in animals to improve traversal efficiency. To enable similar capabilities in quadruped robots, complex real-time jumping maneuvers are required. Current research does not adequately address the problem of online omnidirectional jumping and neglects the robot's kinodynamic constraints during trajectory generation. This paper proposes a general and complete cascade online optimization framework for omnidirectional jumping for quadruped robots. Our solution systematically encompasses jumping trajectory generation, a trajectory tracking controller, and a landing controller. It also incorporates environmental perception to navigate obstacles that standard locomotion cannot bypass, such as jumping from high platforms. We introduce a novel jumping plane to parameterize omnidirectional jumping motion and formulate a tightly coupled optimization problem accounting for the kinodynamic constraints, simultaneously optimizing CoM trajectory, Ground Reaction Forces (GRFs), and joint states. To meet the online requirements, we propose an accelerated evolutionary algorithm as the trajectory optimizer to address the complexity of kinodynamic constraints. To ensure stability and accuracy in environmental perception post-landing, we introduce a coarse-to-fine relocalization method that combines global Branch and Bound (BnB) search with Maximum a Posteriori (MAP) estimation for precise positioning during navigation and jumping. The proposed framework achieves jump trajectory generation in approximately 0.1 seconds with a warm start and has been successfully validated on two quadruped robots on uneven terrains. Additionally, we extend the framework's versatility to humanoid robots.

Figures (25)

• Figure 1: The omnidirectional jumping task consists of Take-off, Flight, and Landing phases. The green, red, and blue dot lines show the jumping trajectory. The yellow plane shows the jumping plane. (a) Lateral view of the jumping task phases. (b) Top view of the jumping task phases.
• Figure 2: An overview of the proposed online cascading omnidirectional jumping framework, incorporating Latin Hypercube Sampling (LHS), Differential Evolution (DE), Offline initial guess library for Differential Evolution algorithm (Pre-Motion Library), configuration space (C-space), and navigation modules. The red dashed line represents the mapping and trajectory planning with navigation modules, while the blue blocks depict the navigation trajectory tracking controller. The green dashed line indicates the blocks corresponding to the jumping controller.
• Figure 3: Side view of the omnidirectional jump. The orange region denotes the jump plane $J$-$Z$. The green coordinate system represents the intersection between the jumping plane and the polygon formed by the robot's four feet projected onto a 2D plane. The coordinates of this intersection in the body frame are denoted as $\bm{p}_{j1}$ and $\bm{p}_{j2}$. The variables $\bm{u}_{j1}$ and $\bm{u}_{j2}$ represent the resultant forces along the jumping plane, while $\bm{u}_{z1}$ and $\bm{u}_{z2}$ denote the resultant forces along the z-axis in frame {J}. The positions of the feet in the body frame are indicated by $\bm{p}_{f1}, \bm{p}_{f2}, \bm{p}_{f3}$, and $\bm{p}_{f4}$. The components of the jumping-plane resultant forces acting on each foot are represented by $\bm{f}_{j1}, \bm{f}_{j2}, \bm{f}_{j3}$, and $\bm{f}_{j4}$. The forces $\bm{f}_{zc}$ and $\bm{f}_{jc}$ denote the resultant forces at the center of mass (CoM), with $\bm{\tau}_{c}$ indicating the resultant torque in the jumping direction. Detailed explanations and applications are provided in Sec. \ref{['sec:opt_variables']} and in Sec. \ref{['sec:application_omini']}.
• Figure 4: Top view of the omnidirectional jump. The green line denotes the jump plane $J$-$Z$, while $\bm{p}_{tg}$ and $\bm{\theta}_{tg}$ represent the target position within the body coordinate system and the angle of rotation of the jump plane relative to the X-axis, respectively. The angle $\bm{\theta}_f$ defines the foot's orientation around the z-axis in the body coordinate system, distinguishing the direction of the jump plane. Red arrows illustrate the resultant forces exerted by the jump plane on the center of mass. $\bm{u}_{J1}$ and $\bm{u}_{J2}$ are the components of the resultant force decomposed for each foot. Detailed explanations are provided in Sec. \ref{['sec:opt_variables']}.
• Figure 5: Illustration of the two-step take-off phase during a jump. The black curved arrow indicates the direction of rotation. (a) The sequence of foot departures for a left flip. (b) The front foot lifts off first, followed by the back foot, enabling obstacle crossing.