Source-linked AI summary
Feedback Control of a Cassie Bipedal Robot: Walking, Standing, and Riding a Segway
Yukai Gong, Ross Hartley, Xingye Da, Ayonga Hereid, Omar Harib, Jiunn-Kai Huang, Jessy Grizzle
TL;DR
The paper addresses feedback control for standing and walking on the Cassie bipedal robot. It uses virtual constraints, decoupled PD control, and gait libraries, producing standing, walking across varied terrain, and Segway riding, with software intended for open-source release.
Problem
The paper responds to the need for a documented controller for the Cassie bipedal robot as teams develop their own control laws.
Method
The controller combines virtual constraints implemented through decoupled PD controllers with a velocity-interpolated gait library.
Results
The controller enabled Cassie to stand, walk across sidewalks, grass, sand, snow, waxed floors, and brush, and ride a Segway.
Takeaways & Limitations
The documented controller supports Cassie locomotion and operation in environments beyond the laboratory, with open-source release planned if accepted.
Abstract
from arXiv · showhide
The Cassie bipedal robot designed by Agility Robotics is providing academics a common platform for sharing and comparing algorithms for locomotion, perception, and navigation. This paper focuses on feedback control for standing and walking using the methods of virtual constraints and gait libraries. The designed controller was implemented six weeks after the robot arrived at the University of Michigan and allowed it to stand in place as well as walk over sidewalks, grass, snow, sand, and burning brush. The controller for standing also enables the robot to ride a Segway. A model of the Cassie robot has been placed on GitHub and the controller will also be made open source if the paper is accepted.
I. INTRODUCTION AND RELATED WORK
The paper documents an open-source feedback controller for Cassie, using virtual constraints and a gait library to support standing and walking across challenging environments. It also describes Cassie’s morphology, actuation, and modeling context.
- I. INTRODUCTION AND RELATED WORK: The paper introduces and documents a feedback controller for Cassie, with implementation intended for open-source release.The controller provides the same basic locomotion features as Cassie’s proprietary native controller.
- I. INTRODUCTION AND RELATED WORK: Unlike ZMP-based methods requiring a nonzero support polygon, the paper addresses Cassie’s line feet using virtual constraints and gait-library stabilization.The related-work discussion also identifies foot placement as a way to regulate center-of-mass velocity in pendulum models.
- I. INTRODUCTION AND RELATED WORK: The gait-library approach uses the full dynamic model to design periodic gaits while incorporating joint, force, torque, and power limits into optimization.The individual gaits are gain-scheduled with current velocity to create a continuum of walking gaits.
- I. INTRODUCTION AND RELATED WORK: Cassie has 20 degrees of freedom, seven joints per leg, five actuated joints per leg, and two passive spring-based joints.The actuated joints control leg orientation, knee angle, and foot pitch.
B. Floating-base Model
The floating-base model expresses Cassie’s dynamics with inertia, velocity and gravity, motor torques, spring torques, and contact wrench terms. Open-source tools generate the model terms, while the control-design model treats spring stiffness as infinite.
- B. Floating-base Model: Cassie’s floating-base dynamics are written in Lagrange form using generalized coordinates, motor torques, spring torques, and contact wrench.The model includes mass-inertia, velocity and gravitational, motor distribution, and spring Jacobian terms.
- B. Floating-base Model: The model’s spring and ground-contact effects enter through Jacobian-transformed torques and contact forces.The ground contact Jacobian is specified in the following subsection.
- B. Floating-base Model: FROST generates the terms of the dynamic model, and the floating-base model used for control design is available as open-source code.For control design, spring stiffness is taken to infinity so the spring-related constraint holds.
C. Hybrid Model for Walking
Walking is modeled as a hybrid system alternating right- and left-stance phases, with instantaneous double support and impact-induced velocity changes. Ground contact constrains stance-foot position, yaw, and pitch while leaving roll unconstrained.
- Hybrid walking phases: Walking alternates single-support domains for the right and left stance legs, while double support is treated as instantaneous.Periodic gaits are computed over one step on each leg.
- Ground contact: Five independent holonomic contact constraints maintain stance-foot position, yaw, and pitch while leaving foot roll unconstrained.The narrow width of Cassie’s feet permits stance-foot roll.
- Ground contact: The stance contact Jacobian has full row rank five, enabling elimination of five coordinates and stance-foot ground reaction forces from the model.The resulting reduced Lagrangian model has 15 degrees of freedom, although the paper later uses differential-algebraic equations directly.
- Transitions and impacts: Single support transitions when the swing foot reaches ground height while descending, defining the right-to-left or left-to-right transition set.The swing foot’s vertical position is used to detect contact.
- Transitions and impacts: Impacts change system velocity discontinuously while positions remain unchanged, and the former swing foot then satisfies stance-foot constraints.The impact map projects the post-impact solution to velocity components.
3) Hybrid Model:
The hybrid model combines constrained continuous dynamics for each stance leg with discrete velocity resets at leg exchange. When both feet are flat, the model removes 10 degrees of freedom and can become fully actuated under stated assumptions.
- Right stance: For right stance, constrained dynamics use JR(q)¨q + ˙JR(q, ˙q) ˙q = 0, with a velocity reset at the right-to-left transition.The reset is represented by ∆R→L( ˙q−).
- Left stance: For left stance, constrained dynamics use JL(q)¨q + ˙JL(q, ˙q) ˙q = 0, with a velocity reset at the left-to-right transition.The reset is represented by ∆L→R( ˙q−).
- Standing: With both feet flat on the ground, 10 degrees of freedom are removed from the model.The standing model is given by (4) with the combined contact Jacobian.
- Standing: Assuming infinite spring stiffness and non-collinear foot x-axes, the standing model is fully actuated.This conclusion applies when the feet are flat on the ground.
III. WALKING ON VARIOUS TERRAINS
The walking controller uses virtual constraints, a sagittal-plane gait library, and leg-angle adjustment to regulate quantities chosen for their direct relation to bipedal gait outcomes. For Cassie, these quantities emphasize torso and virtual-leg behavior rather than raw actuated-joint angles.
- Controller design: The initial walking controller combines virtual constraints, a sagittal-plane gait library, and leg-angle adjustment in sagittal and frontal planes.This design targets feedback control of the walking gait.
- Virtual constraints: Virtual constraints are feedback-imposed functional relations among generalized coordinates, expressed as model outputs tracking desired evolution.The desired evolution is parameterized through h_d and spline coefficients α.
- Virtual constraints: The controller seeks to zero the outputs, although physical-robot implementation can achieve the virtual constraints only approximately and asymptotically.The output target is y ≡ 0.
- Choice of controlled quantities: Cassie regulates torso orientation, stance and swing leg lengths, swing leg orientation, and swing foot pitch instead of directly controlling actuated joints.These quantities are selected as universal across bipedal morphologies and directly relatable to gait outcomes.
- Choice of controlled quantities: The nine outputs include torso roll and pitch, stance hip yaw, stance leg length, swing leg roll and yaw, swing leg pitch and length, and swing foot pitch, while stance-foot pitch remains passive.Leg pitch and length are computed through the virtual leg’s forward kinematics.
C. Set of Gaits for Walking at Various Speeds
A library of seven optimized periodic gaits spans forward speeds from −0.5 m/s to +1.0 m/s, with desired trajectories interpolated across the library. Optimization enforces realizability and favors small torso and hip roll/yaw motions.
- Gait library: Seven gaits cover average sagittal velocities from −0.5 m/s to +1.0 m/s in 0.25 m/s increments.Each gait corresponds to a distinct constrained average velocity.
- Gait library: Each gait’s desired virtual-constraint trajectory uses 5th-order Bézier polynomials, and linear interpolation produces intermediate library behavior.All gaits use a constant step time.
- Optimization: Trajectory optimization solves one parameter matrix α_i independently for each gait across right-stance and left-stance domains.The hybrid problems are solved with FROST using direct collocation.
- Optimization: A weight c = 10,000 penalizes torso pitch/roll and hip roll/yaw to guide gaits toward minimal roll and yaw movement.The cost also includes motor effort and selected joint-angle terms.
- Optimization: Torque, joint-angle, joint-velocity, periodicity, and left-right symmetry constraints ensure the optimized gaits can be physically realized.The optimization minimizes the stated cost subject to these constraints.
D. Approximately Implementing the Virtual Constraints
Because the robot’s amplifiers, motors, network, and walking surface are not adequately characterized, the controller approximately enforces virtual constraints with decoupled PD feedback rather than model-based torque control.
- Model-based input-output linearization would require sufficiently accurate dynamics and joint angular-velocity estimates, with full-rank decoupling on the design model.
- Uncharacterized power amplifiers, motor dynamics, network delays, and walking surfaces prevent adequately characterized model-based torque control on the actual robot.The mechanical model itself is not identified as the main uncertainty source.
- Decoupled PD controllers approximately impose the virtual constraints after rewriting the outputs for implementation.
- The output ordering assigns four stance-leg actuators followed by five swing-leg actuators, while stance-foot torque is set to zero.
- Diagonal 9 × 9 KP and KD matrices provide the proportional and derivative gains for the classical PD controller.
E. Gait Library and Stabilization by Leg Angle Adjustment
The controller interpolates seven sagittal-plane gaits into a velocity-indexed continuum, then adds leg-angle feedback to stabilize longitudinal and lateral motion.
- The Gait Library interpolates seven discrete gaits into a continuum valid for −0.5 ≤vx ≤ 1.0 m/s.The robot’s filtered sagittal velocity is the interpolation parameter, and the implementation adds no new controller parameters.
- Without leg-angle adjustment, the closed-loop Cartesian velocity coordinates ( ˙x, ˙y) behave approximately as integrators and are unstable.
- Leg-angle adjustment stabilizes longitudinal and lateral velocity using controllers based on the cited gait-library equations.These controllers add four control parameters.
F. Parameter Tuning
The walking controller was hand-tuned in simulation and then on Cassie, progressing from walking in place to varied speeds and outdoor tests across multiple terrains.
- The controller uses shared left- and right-leg parameters, with 18 joint-level and 4 leg-angle PD parameters tuned through a posted robot-tuning process.Implementation was performed in Real-Time Simulink, with initial parameter tuning on a SimMechanics model.
- The first walking-controller version was implemented six weeks after Cassie Blue arrived and was later transferred from simulation to the robot.An overhead gantry was used during robot tuning to catch the robot in case of a fall.
- After walking in place was achieved, walking at various speeds followed, with a typical limit cycle shown in the knee phase portrait.
- Outdoor demonstrations covered sidewalks, grass, reduced-friction surfaces, snow, and controlled burns involving slopes, smoke, burning grass, branches, and leaves.During the controlled burn, Cassie Blue never fell.
- Figure 6 documents Cassie Blue walking on various unmodeled terrains.
5) Sand Volley Ball Court:
On uneven and soft terrain, Cassie’s standing and walking controllers use passive-foot behavior and posture feedback to maintain stability, while the standing controller supports crouching and Segway riding.
- 5) Sand Volley Ball Court:: On the sand volleyball court, Cassie’s narrow feet sank into the sand, yet the passive stance foot kept the gait quite stable.The robot walked more slowly than on grass, possibly because of foot slip, but traversed the entire course.
- A. Quiet Standing: The standing controller uses all ten actuators and enables Cassie to ride a Segway.Figure 7 shows standing on an uneven surface and Segway riding; Segway acceleration and direction are determined by body lean.
- A. Quiet Standing: Standing assumes Cassie reaches the pose from stepping in place or startup with flat feet beneath the torso.
- A. Quiet Standing: Static standing is maintained when the center of mass lies within the convex hull of the flat feet.
- A. Quiet Standing: The standing controller regulates sagittal center-of-mass position with foot pitch and indirectly regulates lateral position through zero roll and differential leg length.Roll feedback is used because it is less noisy than the kinematically calculated lateral center-of-mass position.
- A. Quiet Standing: With feet roughly 0.3 m apart, Cassie can squat almost flat and stand approximately one meter tall.
B. Riding a Segway
Cassie rides a Segway by using body lean, implemented through shifts in the target center-of-mass position, with modified foot feedback to prevent oscillation. In a crouched posture, the robot rode sidewalks and grass at roughly 4 m/s, although mounting and dismounting were not addressed.
- Control strategy: The Segway’s acceleration and direction are controlled by Cassie’s body lean through adjustments to the target center-of-mass position.Forward or backward shifts in pCoM_x accelerate or decelerate, while lateral shifting of pCoM_y enables turning by leaning into the center bar.
- Control strategy: Nominal standing control caused oscillations on the Segway, which ceased after the foot feedback gains were reduced.
- Demonstration: A crouched posture enabled Cassie Blue to ride on sidewalks and grassed areas at roughly 4 m/s.
- Demonstration: Mounting and dismounting the Segway were not addressed because an operator placed the robot on the platform.
- Controller context: The controller’s broader design uses virtual constraints, decoupled PD objectives, gain-scheduled gait libraries, and demonstrations across varied terrain.These methods supported seven gaits and demonstrations on sidewalks, grass, sand, waxed floors, snow, and short brush; data-driven feedforward torque design remained a next step.