Source-linked AI summary
Planning and Execution of Dynamic Whole-Body Locomotion for a Hydraulic Quadruped on Challenging Terrain
Alexander W. Winkler, Carlos Mastalli, Ioannis Havoutis, Michele Focchi, Darwin G. Caldwell, Claudio Semini
TL;DR
Foothold choice is crucial for quadrupedal locomotion over challenging terrain. The framework combines onboard mapping and action planning with ZMP-based whole-body motion generation and compliant control; trials report speeds up to 5.8 times those of previous work.
Problem
Dynamic quadrupedal locomotion over challenging terrain depends on choosing appropriate footholds, while legged robots are intended to address unstructured environments.
Source: §I · p. 1Method
The framework builds an onboard terrain map, plans body actions and footholds, optimizes body motion using ZMP stability, and executes it with inverse dynamics and virtual-model control.
Source: §I · p. 1, §II · p. 2Results
Up to 5.8 times the speed of previous work was achieved in the reported trials.
Source: §I · p. 1Takeaways & Limitations
Real-world trials demonstrate dynamically stable motion across irregular swing-leg sequences, with the ZMP kept inside the current support triangle.
Source: 5) Stability despite irregular… · p. 6, §VII · p. 7
Abstract
from arXiv · showhide
We present a framework for dynamic quadrupedal locomotion over challenging terrain, where the choice of appropriate footholds is crucial for the success of the behaviour. We build a model of the environment on-line and on-board using an efficient occupancy grid representation. We use Any-time-Repairing A* (ARA*) to search over a tree of possible actions, choose a rough body path and select the locally-best footholds accordingly. We run a n-step lookahead optimization of the body trajectory using a dynamic stability metric, the Zero Moment Point (ZMP), that generates natural dynamic whole-body motions. A combination of floating-base inverse dynamics and virtual model control accurately executes the desired motions on an actively compliant system. Experimental trials show that this framework allows us to traverse terrains at nearly 6 times the speed of our previous work, evaluated over the same set of trials.
I. INTRODUCTION
Legged robots can navigate unstructured terrain, but difficult terrain makes foothold selection important. This framework combines onboard perception and planning with dynamically stable whole-body motion, extending prior terrain-planning work for challenging environments.
I. INTRODUCTION
- Legged robots are suited to unstructured terrain, including disaster relief, search and rescue, forestry, and construction settings.Source: §I · p. 1
- The framework uses perception to map terrain, select online footholds, and optimize body trajectories by ZMP, producing dynamic whole-body motions up to 5.8 times previous-work speed, with compliant inverse-dynamics execution.Source: §I · p. 1
- The framework generates footholds online for pallet climbing, stepping stones with irregular swing-leg sequences, and traversing a 35 cm gap.Its contribution also includes optimized dynamic whole-body motion and inverse-dynamics/virtual-model control.Source: §I · p. 1
II. RELATED WORK
- Reactive control can handle lightly unstructured terrain, but large gaps and stairs require higher-level planning that considers terrain and selects footholds.Prior approaches either choose the next reachable foothold or plan a complete start-to-goal sequence.Source: §II · pp. 1–2
- Building on [1], the framework updates an onboard terrain cost map using real-time perception and replans actions and footholds with body-motion primitives and ARA*.It also chooses the CoG trajectory to comply with the ZMP stability metric, producing agile, fast, natural motions.Source: §II · p. 2
III. PERCEPTION AND (RE-)PLANNING
The planning pipeline builds and updates terrain information onboard, then searches feasible body actions to determine a route toward the goal. Its action costs combine terrain quality with action difficulty and potential collision or posture penalties.
III. PERCEPTION AND (RE-)PLANNING
- An onboard terrain-information server maintains the environment state, while the body-action and footstep planners determine movement direction and specific footholds.Source: §III · p. 2
A. Terrain Information
- A 3D occupancy grid from a scanning RGBD sensor represents terrain at 4 cm × 4 cm × 2 cm resolution, approximately matching the robot foot dimensions.The robot state estimate is built alongside the map using an Extended Kalman Filter.Source: A. Terrain Information · p. 2
- The terrain cost combines height variation, slope, and curvature, and is recomputed in a 2.5 m × 5 m area around the robot when the map changes.Source: A. Terrain Information · p. 2
B. Body Action Planner
- The body-action planner represents the robot by position and yaw, and searches feasible motion primitives toward a goal while accounting for terrain, action difficulty, reachability, and collisions.The primitives are empirically chosen to match the robot’s kinematics and dynamics.Source: B. Body Action Planner · p. 2
- ARA* searches a graph of body states and actions for a low-cost sequence, first returning a quick solution and then improving it as computation time permits.Its heuristic estimates remaining steps and a lower bound on future action costs.Source: B. Body Action Planner · p. 3, Figure 3 · p. 3
C. Footstep Sequence Planner
The footstep planner maps the selected body actions to footholds, adapting its search areas and swing-leg sequence to each action. It selects the lowest-cost foothold in each search area.
C. Footstep Sequence Planner
- The footstep sequence planner computes footholds corresponding to the desired body actions.Source: C. Footstep Sequence Planner · p. 3
- Search areas and swing-leg sequences vary with the action; for a leftward move, swinging a left leg can avoid small support areas.This action-dependent selection improves robustness over the previous fixed-sequence approach.Source: C. Footstep Sequence Planner · p. 3
- Each search area contributes its lowest-cost foothold, scored by terrain, support-triangle, leg-collision, and body-orientation costs.Source: C. Footstep Sequence Planner · p. 3
D. Re-planning/Updating during Execution
Efficient mapping and action-graph search allow online replanning as the environment changes. The motion-generation formulation uses continuous, double-differentiable CoG trajectories built from fifth-order polynomial splines.
D. Re-planning/Updating during Execution
- The smaller graph over feasible actions and ARA* intermediate solutions let the robot replan motions and footholds online in response to environmental changes.The approach combines these with an efficient voxel-based occupancy map.Source: D. Re-planning/Updating during Execution · p. 3
IV. WHOLE-BODY MOTION GENERATION
- The motion generator finds a CoG trajectory that respects dynamic stability without explicitly generating a ZMP trajectory, allowing swing-leg sequences in any order through four-leg support phases.Source: §IV · p. 3
A. Problem Formulation
- The CoG trajectory uses fifth-order polynomial splines that are continuous and double differentiable at segment junctions, avoiding acceleration steps and discontinuous torques.The optimization can therefore be reduced to finding polynomial coefficients for each segment.Source: A. Problem Formulation · p. 3
B. Dynamic Stability
For dynamic motion, the CoG-inside-support-polygon criterion used for slow movements is replaced by a cart-table ZMP stability check. A stability margin can make support triangles disjoint during diagonal leg switches, so the trajectory generator allows a four-leg transition phase and supports irregular step sequences.
B. Dynamic Stability
- Dynamic stability requires the cart-table-estimated ZMP to remain inside the current support triangle at every sampling interval.The cart-table model concentrates the robot’s total mass in a cart; the support area is determined by the current footholds.Source: B. Dynamic Stability · p. 4, Figure 5 · p. 4
- A stability margin shrinks support triangles, making them disjoint for diagonal swing-leg switches; a four-leg phase instead uses the support polygon of all stance feet.The ZMP is then restricted by the shrunk four-foot support polygon during the transition.Source: B. Dynamic Stability · p. 4, Figure 5 · p. 4
- The ZMP optimizer handles arbitrary swing-leg sequences without a predefined gait, inserting a four-leg phase when consecutive swing legs are diagonally opposite.This decouples trajectory generation from the footstep planner, which can select useful swing-leg orders.Source: B. Dynamic Stability · p. 4
C. Cost function
The body trajectory cost favors low acceleration, with greater penalty on sideways acceleration, and is solved as a convex quadratic program. The planned footholds then feed a whole-body motion and compliant execution pipeline using feedforward and feedback control.
C. Cost function
- The trajectory cost minimizes acceleration to support higher execution speed and lower joint torques, weighting sideways acceleration 1.5 times more than forward-backward motion.The resulting convex quadratic program uses the stated equality and inequality constraints and QuadProg++ to obtain spline coefficients and the desired stable body trajectory.Source: C. Cost function · p. 4
V. EXECUTION OF WHOLE-BODY MOTIONS
- Planned footholds feed whole-body motion generation, whose inverse-dynamics and virtual-model controller combines desired accelerations with feedback and low-gain joint-space correction.The base state includes CoG position and base orientation, represented by a rotation matrix relative to the world frame.Source: §V · pp. 4–5, Figure 6 · p. 4
A. Virtual Model
Virtual-model feedback treats deviations from the desired body trajectory as spring-and-damper errors. Its forces and torques are converted into body feedback acceleration and combined with feedforward acceleration for compliant tracking.
A. Virtual Model
- Virtual springs and dampers convert deviations from the desired body trajectory into forces and torques that pull the trunk toward its desired state.Desired values come from the motion generator, while the robot state is estimated on board; the gains have physical stiffness and damping meanings.Source: A. Virtual Model · p. 5
- The controller scales the virtual-model wrench by inverse composite rigid-body inertia and adds the resulting feedback acceleration to desired acceleration for inverse dynamics.Combining feedforward and body-feedback acceleration supports accurate tracking with compliant behavior.Source: A. Virtual Model · p. 5
B. Floating Base Inverse Dynamics
Floating-base inverse dynamics partitions the robot equations into unactuated base and actuated joint coordinates. The base equations yield contact forces, and the joint equations yield reference torques.
B. Floating Base Inverse Dynamics
- The method computes the left-hand dynamics term with Featherstone’s Recursive Newton-Euler Algorithm and transforms CoG acceleration into base spatial acceleration.The coordinate transform is applied because the CoG acceleration is defined in a frame aligned with the base but centered at the CoG.Source: B. Floating Base Inverse Dynamics · p. 5
- Partitioning the dynamics by unactuated base and actuated joint coordinates allows ground reaction forces to be estimated from the base equations and reference joint torques to be computed from the joint equations.The ground reaction forces are obtained in a least-squares sense using the Moore-Penrose generalized inverse.Source: B. Floating Base Inverse Dynamics · p. 5
VI. EXPERIMENTAL RESULTS
The experiments evaluate the framework on the HyQ quadruped across four challenging-terrain scenarios, comparing performance with prior results on the same benchmark tasks.
A. Experimental Setup
- Trials use the approximately 90 kg, torque-controlled HyQ with onboard state estimation and no external state-measuring system.Perception and planning run on an i7/2.8 GHz PC, while real-time-critical computation runs on a PC104 stack.Source: A. Experimental Setup · p. 5
- The four scenarios test initial foothold planning, climbing one or two pallets, crossing a 35 cm gap, and traversing stepping stones between pallets.The pallets are 1.2 m × 0.8 m × 0.15 m; the gap is approximately half the robot’s body length.Source: A. Experimental Setup · p. 5
- The framework plans initial footsteps, then executes dynamically toward a specified (x, y, θ) goal state in each experiment.Performance is compared with previous results on the same benchmark tasks.Source: A. Experimental Setup · p. 5
1) Perception and (re-)planning:
The framework updates an onboard terrain model and replans footholds online, while experimental sections describe dynamic stability, gap traversal, and model accuracy. The supplied passages identify the benchmark table’s scope but do not provide its cell values.
1) Perception and (re-)planning:
- 2 Hz terrain-cost-map updates and approximately 0.5 Hz foothold replanning support online planning for goals up to 5 m.The mapping computation focuses on an area around the robot body; Fig. 7 depicts the evaluated trials, and Table I is captioned as comparing forward speed and success rate averaged over 10 trials with prior results.Source: 1) Perception and (re-)planning · pp. 5–6, Figure 7 · p. 6, Table I · p. 6
2) Speed while dynamically stable:
- The pallet-climbing and gap experiments attribute faster motion while stable to dynamic stability, optimized accelerations, and shorter four-leg-support phases.Moving the ZMP between support triangles through acceleration is faster than moving the CoG, described as moving the entire body.Source: 2) Speed while dynamically stable · p. 6
3) Model accuracy:
- During a 35 cm gap crossing, the robot accelerates to 0.5 m/s and decelerates without losing balance, supporting the cart-table model as an approximation for large quadrupeds.The passage identifies the gap as approximately half the robot’s body length.Source: 3) Model accuracy · p. 6
4) Avoiding kinematic limits:
Dynamic motions avoid the leg overextension associated with statically stable gap crossing, while the controller maintains stability over irregular stepping-stone sequences. The conclusion summarizes onboard foothold planning, ZMP-stable motion generation, and compliant tracking, and identifies closer integration of planning and execution as ongoing work.
4) Avoiding kinematic limits:
- On the gap, a statically stable gait tends to overextend the legs, whereas dynamic motions can keep the CoG nearer the four feet and avoid kinematic limits.The passage attributes the static-gait issue to large body motions needed to reach statically stable positions.Source: 4) Avoiding kinematic limits · p. 6
- The motion generator inserts four-leg-support phases when it detects disjoint support triangles in the swing-leg sequence.The described transition is from the LH support triangle to the RF support triangle.Source: 4) Avoiding kinematic limits · p. 6
5) Stability despite irregular swing-leg sequences:
- The controller executes irregular stepping-stone swing-leg sequences dynamically while keeping the ZMP inside the current support triangle.A 6 cm margin from stability borders provides robustness to the stated CoG tracking error.Source: 5) Stability despite irregular… · p. 6, Figure 8 · p. 7
VII. CONCLUSION
- The conclusion describes onboard foothold execution, online replanning after environmental change, ZMP-stable trajectories for irregular sequences, and compliant tracking through virtual-model and floating-base inverse-dynamics control.Real-world trials on challenging terrain demonstrate the framework’s capability.Source: §VII · pp. 6–7
- The authors are working toward jointly optimizing desired states and torques while accounting for robot dynamics, footholds, friction, and torque and joint limits.They aim to produce motions such as jumping and rearing, with fewer or no legs in contact.Source: §VII · p. 7