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Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot

Christine Chevallereau, Jessy W. Grizzle, Ching-Long Shih

arXiv:1002.3258v1cs.RO

TL;DR

The paper addresses stable walking for a five-link 3D biped with passive point feet, where natural dynamics must support balance. It extends virtual constraints and hybrid zero dynamics, reducing analysis to low-dimensional dynamics, and explores three stabilization strategies, including a fast stable gait.

  • Problem

    The paper seeks stable walking for a 3D biped with unactuated point feet, without relying on large feet or slow movement.

  • Method

    The authors extend virtual constraints and hybrid zero dynamics, using optimization and reduced-dimensional stability analysis to design periodic motions and feedback controllers.

  • Results

    T = 0.175 seconds, L = 0.144 m, and average walking speed = 0.82 m/sec for a nominal gait whose eigenvalues all have magnitude less than 1.0.

  • Takeaways & Limitations

    Stable walking depends on the choice of controlled variables and virtual constraints, with stability achievable through optimization, stride-to-stride control, or within-stride control.

Abstract

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This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3D (spatial) bipedal robot consisting of a torso, two legs, and passive (unactuated) point feet. The contact between the robot and the walking surface is assumed to inhibit yaw rotation. The studied robot has 8 DOF in the single support phase and 6 actuators. The interest of studying robots with point feet is that the robot's natural dynamics must be explicitly taken into account to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit. This method allows the computations to be carried out on a 2-DOF subsystem of the 8-DOF robot model. The stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincaré map of the hybrid zero dynamics. Three strategies are explored. The first strategy consists of imposing a stability condition during the search of a periodic gait by optimization. The second strategy uses an event-based controller. In the third approach, the effect of output selection is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.

I. Introduction

The paper targets stable, fast walking for a spatial five-link biped with passive point feet, using feedback control that accounts for underactuated natural dynamics. It extends virtual constraints to 3D robots and studies a model with unactuated point contact, alternating support, and simplified rigid-link assumptions.

  • Motivation: The study seeks a time-invariant feedback controller that creates an exponentially stable, periodic walking motion without relying on large feet or slow movement.Virtual constraints coordinate the robot’s links while reducing the effective degrees of freedom.
  • Related work: Related spatial-biped controllers either decompose sagittal and frontal motion, linearize dynamics along a periodic orbit, emphasize passive energy efficiency, or assume full actuation.The paper positions its approach as addressing spatial robots with unactuated point feet and closed-loop stability analysis.
  • Robot model: The five-link robot has a torso, two legs with independently actuated knees, two-DOF actuated hips, and passive point-foot contact with no yaw rotation.The model represents each link by a point mass at its center and uses a stance foot as a passive pivot.
  • Assumptions: The study assumes alternating single- and instantaneous double-support phases, no slip or rebound at impact, symmetric steady-state gait, and flat-surface walking.The swing and stance legs exchange roles at each impact.
  • 3D dynamics: Although explicit yaw rotation is excluded, sagittal–frontal rotational coupling can produce net vertical-axis rotation between steps.The model’s coordinate description tracks stance-foot position and yaw reference variables across leg exchanges.

B. Dynamic model

The dynamic model represents single support with Euler–Lagrange equations and models impact as an instantaneous hybrid transition involving velocity jumps and coordinate relabeling. Separate support-leg models are related by a coordinate transformation.

  • Single-support dynamics: The single-support dynamics use an 8×8 positive-definite mass-inertia matrix, Coriolis and gravity terms, and a full-rank 8×6 actuation matrix.The input is a six-dimensional vector of joint torques.
  • Impact model: The instantaneous double-support phase comprises a rigid impact followed by coordinate relabeling when the swing foot contacts the ground.Configuration variables remain unchanged during impact, while generalized velocities undergo a jump.
  • Impact model: The impact equations are derived using extended coordinates and angular-momentum conservation about the swing-leg end.The swing-foot reaction force, extended inertia matrix, and contact Jacobian enter the impact formulation.
  • Hybrid-system formulation: The complete walking motion is expressed as a nonlinear hybrid system with impulse effects, switching surfaces, continuous flows, and impact maps for each support leg.For leg-2 support, analogous sets of switching surface, flow, and impact-map equations are constructed.

III. Virtual constraints

Virtual constraints coordinate the actuated joints as functions of a monotonic phase variable, enabling feedback enforcement and reduction of the robot’s dynamics to a two-degree-of-freedom swing-phase zero dynamics.

  • Constraint design: Virtual constraints impose one holonomic output per actuator and drive the actuated coordinates toward qa = hd(θ) through feedback control.The phase variable θ is strictly monotonic over a typical gait and replaces time for parameterizing the periodic motion.
  • Feedback control: The feedback controller uses positive gains Kp, Kd, and ǫ chosen so the output dynamics are exponentially stable and converge rapidly within single support.The torque required to remain on the virtual-constraint surface can be computed, yielding an input-output linearizing controller.
  • Hybrid zero dynamics: Enforcing y = h(q) = 0 yields qa = hd(θ) and reduces the robot’s dynamics to a low-dimensional autonomous subsystem.Substitution of the constrained coordinates and their derivatives produces the reduced dynamics.
  • Coordinate selection: The actuated coordinates are qa = [q3, q4, q5, q6, q7, q8]′, while the unactuated joints are qu = [q1, θ]′.The controlled coordinates are selected as the actuated joints in this formulation.
  • Hybrid zero dynamics: The reduced single-support model is a 2-DOF swing-phase zero dynamics whose properties depend on the selected virtual constraint.Changing hd(θ) can therefore change the dynamic properties relevant to periodic walking.

IV. Design for a Symmetric Periodic Gait

The periodic-gait design uses virtual constraints for a symmetric gait composed of alternating single-support phases and impacts, reducing analysis to one step plus a symmetry relation.

  • Gait construction: The gait is designed as single-support phases separated by impacts, with the legs exchanging roles from one step to the next.Symmetry allows the analysis to be limited to a single step.
  • Gait construction: A symmetry relation completes the periodic-gait design by linking the motion before and after leg exchange.The formulation uses the alternating support-leg structure of the hybrid walking cycle.

A. Virtual constraints and Bezier polynomials

The paper designs virtual constraints as Bezier-polynomial functions of a monotonic phase variable, then formulates periodic-gait generation as constrained optimization over terminal state parameters. The optimization integrates the unactuated zero dynamics while enforcing periodicity and physical constraints.

  • Bezier polynomials: Bezier polynomials connect the desired initial and final actuated configurations and velocities while incorporating boundary conditions.The coefficients are selected to join the actuated portions of qi and qf and their velocities as θ varies from θi to θf.
  • Virtual constraints: Virtual constraints prescribe the actuated coordinates as functions of a strictly monotonic phase variable, replacing time for periodic-motion parameterization.The phase variable increases or decreases strictly along a typical gait, while the desired actuated evolution is given by hd(θ).
  • Zero-dynamics integration: The unactuated variables are obtained by integrating the stance-phase zero dynamics from the initial state until θ reaches θf.The actuated evolution follows the virtual constraints, allowing the required torque and stance-foot reaction force to be calculated.
  • Constrained optimization: The search minimizes integral-squared torque per step length over 15 parameters prescribing the final configuration and velocity.The optimization is constrained by symmetry, monotonic phase evolution, no take-off, friction, and periodicity conditions.
  • Constrained optimization: A fixed point minimizing the criterion defines the desired periodic walking cycle, but the local optimization result depends on the initial optimization parameters.The criterion has many local minima, and the optimization technique is local.

C. An example periodic motion minimizing integral-squared torque

The torque-optimized example produces a periodic 3D walking gait, while the modified virtual-constraint controller creates a hybrid zero dynamics suitable for reduced-dimensional stability analysis. The resulting orbit converges toward periodic motion under the restricted Poincaré map.

  • Periodic motion: The nominal motion satisfies the reported inequality constraints, including the stance-foot reaction-force and swing-leg-tip conditions.The ground-reaction-force and swing-leg-tip profiles are shown over the gait steps.
  • Hybrid zero dynamics: The original feedback law creates stance-phase zero dynamics but not hybrid zero dynamics because virtual constraints need not remain compatible across impacts.Consequently, complete-model simulation is required to predict off-orbit behavior under that law.
  • Hybrid zero dynamics: A stride-to-stride correction term hc is added to the virtual constraints so they match the robot’s initial state at each step.The correction is updated at the beginning of each step and held constant throughout it.
  • Hybrid zero dynamics: The corrected output reaches the original virtual constraint by the end of each step, with initial output and derivative errors joined smoothly at mid-step.The reported effect is lower torque than a high-gain controller, reduced ground-reaction-force variation, and avoidance of sliding or take-off.
  • Reduced stability test: The restricted Poincaré map acts on S ∩ Z, and its 3 × 3 Jacobian Az determines local exponential stability when its eigenvalues have magnitude strictly less than one.Computing Az requires six evaluations of the restricted map, each including impact calculation and swing-phase zero-dynamics integration.

B. Example of the periodic motion minimizing integral-squared torque

Torque-optimized periodic motions are generally unstable under the within-stride control law, motivating three stabilization strategies: stability-aware optimization, stride-to-stride control, and output redesign.

  • The virtual constraints for the optimal periodic motion are implemented using the control law defined by (20).
  • The linearized restricted Poincaré map is evaluated around the periodic motion using specified perturbations in joint positions, velocities, and θ̇.
  • One eigenvalue has magnitude greater than one, so the gait is unstable under this controller.
  • Most periodic motions optimized for integral-squared torque per step length are unstable under the control law defined by (20).
  • The third strategy uses a judicious linear combination of configuration variables as controlled outputs to stabilize walking without supplemental event-based control.

VI. Periodic motion optimized with respect to a stability criterion

The authors optimize periodic motion using eigenvalue magnitudes as a stability criterion, obtaining a fast, locally exponentially stable gait whose perturbed trajectories converge toward periodic motion.

  • Stability is incorporated into periodic-motion optimization by requiring or minimizing eigenvalue magnitudes below one.
  • The resulting gait has period T = 0.175 seconds, step length L = 0.144 m, and average walking speed 0.82 m/sec.
  • The step width is 0.173 m, close to the hip width, and the maximum frontal-plane center-of-mass variation is less than 1.5 cm.
  • All eigenvalues have magnitude below 1.0, indicating that the nominal orbit is locally exponentially stable.
  • Under closed-loop simulation from a perturbed fixed point, uncontrolled variables converge toward the periodic motion, while phase-plane trajectories converge for controlled and uncontrolled variables.

VII. Stride-to-Stride controller

When a desired gait is unstable or converges too slowly, the paper augments continuous stance-phase control with impact-to-impact parameter updates that stabilize the hybrid fixed point.

  • Event-based control can be integrated with the continuous stance-phase controller when the desired gait is unstable or its convergence is insufficiently rapid.
  • Parameters β are held constant during stance and updated at each impact using either full-state information or hybrid-zero-dynamics state information.
  • The output is augmented with a term hs(θ, β), producing y = h(q, yi, ẏi, β) = qa − hd(θ) − hc(θ, yi, ẏi) − hs(θ, β).
  • The chosen augmentation permits mid-stance virtual-constraint updates while preserving the original hybrid zero dynamics when impact occurs near the end of the step.
  • Feedback gains are selected so the eigenvalues of Az − FK have magnitude strictly below one, which exponentially stabilizes the fixed point.

B. Example of the periodic motion minimizing the integral-squared torque

A stride-to-stride DLQR controller modifies the virtual constraints through impact-updated parameters and stabilizes the previously unstable periodic motion.

  • The periodic motion is stabilized by modifying the virtual constraints according to equation (25) on each step.
  • A sixth-order polynomial is used over the transition interval, ensuring continuity of position, velocity, and acceleration.
  • The matrix F is computed numerically, while Az is defined by equation (24).
  • The gain matrix K is calculated via DLQR using the state-feedback law δβk = −Kδxz_k.
  • The coefficient r trades convergence rate against virtual-constraint changes that affect the region of convergence.
  • The closed-loop eigenvalues indicate a stable gait with the stride-to-stride controller.
  • With the event-based DLQR controller, uncontrolled variables and phase-plane trajectories converge toward periodic motion.

VIII. Improved Selection of the Outputs to be Controlled

The paper shows that selecting controlled outputs can dramatically affect gait stability in a 3D biped with two degrees of underactuation. Replacing the swing-leg-width output with a frontal-distance output stabilizes the periodic motion without a supplemental event-based controller.

  • Effect of output selection: Output selection changes the zero dynamics and can therefore dramatically improve closed-loop gait stability.For systems with two degrees of underactuation, the choice of controlled output affects stability through the choice of M^-1.
  • Proposed output: The proposed output controls distance between the swing-leg end and the center of mass along the frontal direction instead of q6.The remaining actuated joints q3, q4, q5, q7, and q8 retain their original virtual-constraint control.
  • Stability evaluation: The resulting walking gait is stable, with restricted Poincaré-map eigenvalues λ1 = 0.7846 and λ2,3 = −0.028 ± 0.250i.The complex pair has magnitude |λ2,3| = 0.2512.
  • Closed-loop behavior: A simulation initialized with −1° joint-position errors and −5° s^-1 joint-velocity errors shows convergence of the uncontrolled variables toward the periodic motion.The perturbation is applied to every joint and joint velocity.
  • Closed-loop behavior: Phase-plane plots show convergence toward periodic motion for both controlled and uncontrolled variables.The plots include the first four variables and account for instantaneous impact phases.

IX. Conclusions

The paper develops feedback control for asymptotically stable walking of a five-link 3D biped with unactuated point feet. Its contributions include efficient periodic-motion optimization, low-dimensional hybrid-zero-dynamics stability analysis, stride-to-stride stabilization, and the finding that output selection is crucial in 3D.

  • Conclusions: The study develops a time-invariant feedback controller for asymptotically stable walking without relying on large feet.The robot has five links, knees, a torso, point feet, and no actuation between the feet and ground.
  • Conclusions: The paper computes periodic 3D motions efficiently and analyzes stability using a low-dimensional hybrid-zero-dynamics subsystem.The Poincaré return map is computed in dimension three for a robot with two degrees of underactuation.
  • Conclusions: The study finds that human-like periodic walking can be stable or unstable depending on the actuated variables and corresponding virtual constraints.It also identifies controlled-output selection as important for the stability of a given periodic motion.
  • Conclusions: A stride-to-stride controller stabilizes a walking motion that is not naturally stable for a given choice of controlled outputs.This contribution extends prior planar-walking approaches, while output-selection effects are identified as specific to the 3D setting.
  • Scope and limitations: The analysis is limited to a simplified spatial biped model, and extending the unactuated point-foot assumption to include yaw rotation is not known to be straightforward.The strategy may extend to actuated, non-trivial feet, but yaw-inclusive point-foot extension remains unclear.
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