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Passive Whole-body Control for Quadruped Robots: Experimental Validation over Challenging Terrain

Shamel Fahmi, Carlos Mastalli, Michele Focchi, Claudio Semini

arXiv:1811.00884v2cs.RO

TL;DR

Dynamic quadruped locomotion requires control that accounts for full robot dynamics and contact while maintaining compliant trunk balance. The paper formulates passive whole-body control as a constrained QP and validates it on HyQ across gaits and challenging terrain. Against a quasi-static controller, it reports lower trunk tracking error at higher forward speeds.

  • Problem

    Dynamic locomotion requires reasoning about robot dynamics, actuation limits, and environmental interaction, while full-dynamics optimization can be difficult to solve in real time.

    Source: §I · p. 2

  • Method

    A passive whole-body controller formulates motion tracking as a QP incorporating full robot dynamics, with planner references, state estimates, and terrain information supporting control.

    Source: §I · p. 2, §II · p. 3

  • Results

    Approximately 50% lower trunk tracking error was reported as forward speed increased from 0.04 m/s to 0.15 m/s, compared with the quasi-static controller.

    Source: E. Comparison with Previous Controller… · p. 8

  • Takeaways & Limitations

    Experiments on HyQ validated the controller over challenging terrain and across crawl and trot gaits, including tests of friction-estimation errors, unstable footholds, and torque-limit load redistribution.

    Source: §VI · p. 9

Abstract

from arXiv · show

We present experimental results using a passive whole-body control approach for quadruped robots that achieves dynamic locomotion while compliantly balancing the robot's trunk. We formulate the motion tracking as a Quadratic Program (QP) that takes into account the full robot rigid body dynamics, the actuation limits, the joint limits and the contact interaction. We analyze the controller's robustness against inaccurate friction coefficient estimates and unstable footholds, as well as its capability to redistribute the load as a consequence of enforcing actuation limits. Additionally, we present practical implementation details gained from the experience with the real platform. Extensive experimental trials on the 90 kg Hydraulically actuated Quadruped (HyQ) robot validate the capabilities of this controller under various terrain conditions and gaits. The proposed approach is superior for accurate execution of highly dynamic motions with respect to the current state of the art.

I. INTRODUCTION

Dynamic quadruped control must account for full robot dynamics, actuation limits, and environmental interaction, while real-time optimization and reduced models create trade-offs. The paper presents a passive whole-body controller and validates it on HyQ across dynamic gaits and challenging terrain.

I. INTRODUCTION

  • Full-dynamics optimization can be difficult to solve in real time, so locomotion research often uses lower-dimensional models, relaxed constraints, or decoupled planning and control.Quasi-static assumptions and lower-dimensional models can limit dynamic locomotion capabilities.Source: §I · p. 2
  • The controller extends prior quasi-static work by incorporating full robot dynamics, passive compliant interaction, and kinematic and torque limits.The authors also integrate terrain mapping and state estimation onboard.Source: §I · p. 2
  • Experiments on HyQ test the controller during crawling and trotting over slopes and rough terrain of different heights.The paper presents experimental contributions alongside simulation results.Source: §I · p. 3

II. WHOLE-BODY CONTROLLER (WBC)

  • The WBC receives planner references and state estimates, with terrain estimation, disturbance observation, and torque control supporting its locomotion framework.References include desired CoM and trunk-orientation trajectories and swing-leg motion.Source: §II · p. 3
  • The controller regulates trunk balance and swing-foot placement through impedance-based tasks, then computes accelerations and contact forces by QP while respecting full dynamics and robot constraints.Constraints include terrain friction, unilateral contacts, and actuator torque and kinematic limits.Source: §II · p. 3

A. Robot Model

  • The robot model represents full floating-base and joint dynamics, with stance feet assumed stationary on rigid terrain and contact forces acting at those feet.The model accounts for inertial, Coriolis, centrifugal, and gravitational terms.Source: A. Robot Model · pp. 3–4

B. Trunk and Swing Leg Control Tasks

Trunk and swing-foot tracking are formulated as compliant Cartesian impedance tasks with feed-forward terms. The trunk wrench additionally includes gravity compensation and external-disturbance compensation.

B. Trunk and Swing Leg Control Tasks

  • The desired CoM wrench combines Cartesian stiffness and damping with gravity compensation, feed-forward tracking, and external-disturbance compensation.The stiffness and damping matrices are positive definite.Source: B. Trunk and Swing Leg Control Tasks · p. 4
  • The trunk impedance acts on position and velocity tracking errors, while its feed-forward term uses the acceleration tracking error.The defined errors are desired minus actual CoM position, velocity, and acceleration.Source: B. Trunk and Swing Leg Control Tasks · p. 4
  • The swing-foot virtual force combines Cartesian stiffness and damping with a feed-forward term to track desired foot motion.The errors are desired minus actual swing-foot position and velocity.Source: B. Trunk and Swing Leg Control Tasks · p. 4

C. Optimization

The controller formulates motion tracking and stance-foot load distribution as a QP over generalized accelerations and contact forces, subject to equality and inequality constraints.

C. Optimization

  • The QP is formulated to fulfill motion tasks and distribute load across stance feet while respecting the controller’s constraints.Its decision variables are generalized accelerations and ground-reaction forces.Source: C. Optimization · p. 4
  • The cost minimizes trunk-task error and regularizes the solution, while equality constraints encode dynamics, stance conditions, and swing tasks.Inequality constraints encode friction, joint kinematic, and torque limits.Source: C. Optimization · p. 4
  • The QP’s inequality constraints impose friction, joint kinematic, and torque limits.These constraints complement the equality constraints for dynamic consistency and motion tasks.Source: C. Optimization · p. 4

1) Cost:

The optimization minimizes trunk-wrench tracking error while imposing dynamic consistency, stance-foot conditions, and swing-foot acceleration tasks.

1) Cost:

  • The cost includes the difference between actual and desired CoM wrenches, with the actual wrench computed from contact forces.The wrench-tracking term is part of the QP objective.Source: 1) Cost · p. 4

2) Physical consistency:

  • An equality constraint imposes the unactuated trunk dynamics to keep contact forces and generalized accelerations physically consistent.Source: 2) Physical consistency · p. 4

3) Stance condition:

  • Stationary stance feet are encoded at the acceleration level using the stance Jacobian and its time derivative.Differentiating the stance-foot velocity constraint gives J_st q̈ + J̇_st q̇ = 0.Source: 3) Stance condition · p. 4

4) Swing task:

  • The swing task is imposed as an equality constraint on desired swing-foot acceleration, with slacks allowing temporary task violations when needed.The text gives reaching kinematic limits as an example use for slacks.Source: 4) Swing task · p. 5

5) Friction cone constraints:

Friction constraints bound contact forces to avoid slippage and regulate leg loading; torque constraints enforce actuation limits that vary with joint position.

5) Friction cone constraints:

  • Contact forces are constrained within friction cones, with bounded normal forces to avoid slippage and regulate each leg’s loading.The cones are approximated by square pyramids, yielding linear constraints; the lower normal-force bound is almost zero for unilateral contact.Source: 5) Friction cone constraints · p. 5
  • The friction inequality constraints use a block-diagonal matrix encoding stance-leg cone boundaries and lower and upper bounds.The paper refers to for implementation details.Source: 5) Friction cone constraints · p. 5

6) Torque limits:

  • Joint torques are constrained between position-dependent lower and upper bounds using the actuated part of the robot’s full dynamics.For HyQ, the bounds are recomputed each control loop because sagittal-joint linkages produce a nonlinear torque profile over the joint range.Source: 6) Torque limits · p. 5

7) Joint kinematic limits:

Joint acceleration bounds are recomputed to bring a joint to its end-stop at zero velocity within a specified interval. The passivity analysis assumes a feasible QP solution achieving the motion tasks and omits feed-forward terms.

7) Joint kinematic limits:

  • Joint acceleration bounds are selected to reach an end-stop at zero velocity within Δt = 10dt and are recomputed each control loop.The bound depends on the joint’s distance from the end-stop and its velocity.Source: 7) Joint kinematic limits · p. 5

III. PASSIVITY ANALYSIS

  • The overall system is passive when the robot, environment, controller, and their interconnections are passive.Passivity is defined using a storage function bounded below whose derivative does not exceed the supply rate.Source: §III · p. 5
  • The passivity analysis assumes a feasible QP solution that achieves the motion tasks and leaves feed-forward terms for future developments.Source: §III · p. 5
  • The mapping uses forces exerted by the robot on the environment, so the mapping in equation (18) is negative.Source: §III · p. 5

A. Analysis

The analysis expresses controller energy through virtual impedances and tracks its time derivative, separating damping from non-damping terms.

A. Analysis

  • Controller energy comprises virtual-impedance energy at the CoM and swing feet, plus gravitational potential energy associated with CoM compensation.Source: A. Analysis · p. 6
  • The analysis computes time derivatives of the stored energy and rewrites them using the defined velocity errors.Source: A. Analysis · p. 6
  • The derivative of stored energy is regrouped into non-damping and damping terms.Source: A. Analysis · p. 6

B. Proof

Under the stated assumption, the whole-body controller is passive when its storage function is bounded below and its derivative satisfies the passivity inequality.

B. Proof

  • Under Assumption 2, the controller maps joint-velocity error to torque as an impedance controller with gravity compensation, similar to a PD+.Source: B. Proof · p. 6
  • The controller is passive when its storage derivative satisfies the stated supply-rate inequality; the paper concludes the overall system is passive.Source: B. Proof · p. 6
  • The storage function is bounded below because its potentials are positive definite and gravitational potential is assumed bounded below.Source: B. Proof · p. 6

IV. IMPLEMENTATION DETAILS

The implementation combines a stance task for limiting foot slip under terrain-estimation uncertainty with selective constraint softening to manage conflicts in the QP. Torque and dynamics remain hard constraints, while swing-task tracking can be relaxed when joint limits conflict.

A. Stance task

  • A stance task adds Cartesian impedance to limit stance-foot motion when terrain-normal or friction estimates are uncertain, while preserving free GRF optimization in normal conditions.The task is formulated as a desired stance-foot acceleration referenced to the touchdown position.Source: A. Stance task · p. 6

B. Constraint Softening

  • Torque and dynamics constraints remain hard, while swing-task constraints may be relaxed when kinematic limits conflict with them.The authors report that adding slacks to friction constraints did not significantly improve results.Source: B. Constraint Softening · p. 6
  • Nonnegative slack variables bound swing-task constraint violations and receive a high-weight penalty so violations occur only when constraints conflict.Separate slacks by direction avoid coupling errors; a single slack could let an HFE limit affect Y-direction tracking unnecessarily.Source: B. Constraint Softening · pp. 6–7
  • With increasing kinematic-limit slack, the HFE joint remains clamped while foot tracking is affected mainly in X and Z.The figure plots desired and actual left-hind foot position by axis and the HFE joint position against its limits.Source: Figure 3 · p. 7

V. RESULTS

Experiments and simulations assess the controller across gaits, terrain, friction-estimation error, and joint torque constraints. The results show bounded slippage under friction mismatch, load redistribution under torque limits, and real-time QP execution with slack variables.

V. RESULTS

  • On HyQ, the WBC and torque-control loops ran at 250 Hz and 1 kHz, respectively, during validation across terrain conditions and gaits.Experimental trials also used a low-gain joint-space swing-task attractor to address instability from imprecise knee-torque tracking.Source: §V · p. 7

A. Constraint Softening through Slack Variables

  • Kinematic-limit clamping mainly affects X- and Z-direction foot tracking; adding slacks raises QP solve time 30% on average, from 90–110 to 120–150 µs.The reported solve times remain suitable for real-time implementation at 250 Hz.Source: A. Constraint Softening through Slack… · p. 7

B. Friction Constraints and Bounded Slippage

C. Torque Limits and Load Redistribution

D. Different Torque Regularization Schemes

E. Comparison with Previous Controller (Quasi-Static)

The dynamic whole-body controller improves trunk tracking over the quasi-static controller and maintains balance through an unstable foothold. Experiments also demonstrate terrain traversal and orientation tracking across challenging gaits.

E. Comparison with Previous Controller (Quasi-Static)

  • Tracking error is reduced by approximately 50% against the quasi-static controller as forward speed increases from 0.04 m/s to 0.15 m/s.The comparison uses l2-norms of linear and angular trunk-task errors; the figure notes that errors reset to zero at each replanning step.Source: E. Comparison with Previous Controller… · p. 8, Figure 6 · p. 8

F. Disturbance Rejection against Unstable Foothold

G. Locomotion over Slopes and Terrain Mapping

  • Experiments with online terrain mapping used terrain normals from onboard mapping and a friction coefficient conservatively set to 0.7.The terrain trials include crossing a 22 cm gap with a 7 cm step and traversing two ramps separated by a 15 cm gap.Source: G. Locomotion over Slopes and Terrain… · p. 8, Figure 8 · p. 9

H. Tracking Performance with Different Gaits

VI. CONCLUSION

  • Experiments on HyQ validate the controller over challenging terrain and crawl and trot gaits, and demonstrate superiority to quasi-static control.The conclusion also reports analysis of inaccurate friction estimates, unstable footholds, regularization changes, and load redistribution under restrictive torque limits.Source: §VI · p. 9
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