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LQR-Assisted Whole-Body Control of a Wheeled Bipedal Robot with Kinematic Loops
Victor Klemm, Alessandro Morra, Lionel Gulich, Dominik Mannhart, David Rohr, Mina Kamel, Yvain de Viragh, Roland Siegwart
TL;DR
The paper addresses whole-body control of Ascento, whose wheeled balancing behavior and leg kinematic loops complicate robust operation on uneven terrain. It derives full rigid-body and loop dynamics, uses analytic rolling constraints and an LQR motion task, and regulates lean during curves. Experiments show recovery from disturbances and steadier curve driving with the proposed controller.
Problem
Whole-body control must handle Ascento’s kinematic loops, non-minimum-phase balancing dynamics, and rolling constraints for robust uneven-terrain operation.
Method
The paper combines full rigid-body dynamics with closed-form loop constraints, rotation-matrix rolling constraints, LQR-assisted balancing, and ZMP-based lean regulation in a hierarchical whole-body controller.
Results
The controller recovers from a horizontal impact with a T90 time of ca. 1 s and produces significantly steadier curve driving when lean is regulated through the ZMP.
Takeaways & Limitations
The proposed whole-body controller extends Ascento’s operation to uneven outdoor terrain while improving robustness to disturbances and curve-driving conditions.
Abstract
from arXiv · showhide
We present a hierarchical whole-body controller leveraging the full rigid body dynamics of the wheeled bipedal robot Ascento. We derive closed-form expressions for the dynamics of its kinematic loops in a way that readily generalizes to more complex systems. The rolling constraint is incorporated using a compact analytic solution based on rotation matrices. The non-minimum phase balancing dynamics are accounted for by including a linear-quadratic regulator as a motion task. Robustness when driving curves is increased by regulating the lean angle as a function of the zero-moment point. The proposed controller is computationally lightweight and significantly extends the rough-terrain capabilities and robustness of the system, as we demonstrate in several experiments.
I. INTRODUCTION
Wheeled-legged robots combine wheeled efficiency with legged mobility, but applying whole-body control to Ascento requires handling its kinematic loops and non-minimum-phase balancing dynamics. The proposed approach derives full dynamics, formulates rolling constraints analytically, and extends robustness to outdoor terrain and disturbances.
- Motivation: Wheeled-legged robots combine wheel efficiency with legged ability to traverse uneven terrain and obstacles.They can also support swift, cost-effective designs with fewer actuators and turning on the spot.
- Motivation: Ascento uses four actuators and four-bar linkages in both legs, reducing cost, weight, and mechanical complexity.The design is presented as desirable for inspection and search-and-rescue applications.
- Contribution: The proposed LQR-assisted whole-body controller extends Ascento to outdoor scenarios and improves robustness through active compliance with uneven terrain.The paper illustrates this capability with uneven-terrain experiments.
- Related Work: Prior work had not shown whole-body-control stabilization of wheeled bipedal robots with inherent non-minimum-phase dynamics.Earlier modeling treated Ascento as a standard two-wheeled inverted pendulum and neglected leg dynamics.
- Contribution: The paper derives full rigid-body dynamics for kinematic loops and introduces a compact closed-form rolling-constraint formulation using rotation matrices.The modeling approach opens loops and closes them using dynamic constraint forces, allowing closed-form solutions for non-trivial systems.
- Contribution: The controller addresses balancing with an LQR motion task and improves curve-driving robustness by shifting the ZMP toward the line-of-support center through lean control.These mechanisms form the paper’s whole-body-control scheme for wheeled bipedal robots.
II. MODELING
The modeling formulation represents Ascento with full open-loop rigid-body dynamics, then imposes kinematic-loop and wheel-ground constraints through analytically derived force and acceleration relations. The approach accounts for loop geometry, torsional springs, rotating frames, and changing ground contact conditions.
- A. Coordinates and Conventions: The model uses generalized coordinates, velocities, accelerations, and actuation torques for the wheeled bipedal system.The generalized coordinates include base and joint variables, with joint angles indexed for the left and right sides.
- B. Open-Loop Dynamics: The open-loop dynamics combine the mass matrix, Coriolis and centrifugal terms, gravity, knee-spring effects, and selected actuation torques.The system is first formulated without enforcing the kinematic loops, allowing loop closure forces to be introduced separately.
- C. Loop Closure: Kinematic loops are closed by applying equal and opposite loop closure forces at the opened hinge points within the loop plane.The forces are interpreted as bearing forces and their reactions acting on the two hinge points.
- C. Loop Closure: The loop position constraint is differentiated twice and projected onto the loop plane to obtain acceleration-level constraints for determining unknown closure forces.The resulting constraints are stacked for both loops and combined with the ground-contact constraints.
- D. Ground Contact: Wheel rolling is modeled by parameterizing the ground contact point on the wheel contour and deriving its constraint using rotation matrices and the current ground normal.The contact frame is aligned with the ground normal and wheel heading, while the contour parameter captures the changing contact location.
D. Ground Contacts
The ground-contact model represents wheel contact along the wheel contour and derives acceleration-level rolling constraints for both wheels. It accounts for changing contact geometry, friction, and an assumption about ground-surface motion.
- Rolling constraints: The rolling constraint is formulated using rotation matrices and differentiated to acceleration level so contact forces can be determined.The formulation is applied analogously to both wheels and stacked.
- Contact geometry: Wheel contact is parameterized along the entire circumference using a contour parameter σ rather than a single fixed contact point.The contact-point velocity is derived from the wheel’s forward differential kinematics.
- Constraint properties: The constraint formulation correctly captures dynamics for arbitrary values of the relative position vector and avoids numerical divergence associated with opening the kinematic loop.
- Assumptions: The ground-surface normal is used to calculate the contour parameter, while the derivation assumes zero ground-normal velocity unless that motion is appended to the generalized velocity.Formulating ground-surface estimation as constrained state estimation is beyond the scope of the letter.
- Rolling constraints: Perfect rolling enforces zero acceleration in the x- and z-directions of each contact frame.
- Friction: The model includes wheel-contour centripetal compensation and velocity-dependent friction, with slipping represented in the contact-frame y-directions.Friction is modeled with a differentiable tanh function using the sliding friction coefficient.
E. Solving for the Unknown Constraint Forces
The unknown constraint forces are obtained by stacking the constraints and solving the constrained equations of motion. Substitution then yields the final system dynamics.
- Force solution: Unknown constraint forces and their corresponding constraints are stacked before solving the constrained equations of motion.
- Force solution: The constrained equations are solved for generalized acceleration, which is inserted into the second constraint equation to recover the constraint forces.
- Force solution: The resulting force expression combines inverse mass, constraint Jacobians, generalized actuation, and Coriolis, gravity, and spring terms.
- Final dynamics: Substituting the solved forces produces the final equations of motion for the system.
A. Whole-Body Control
Whole-body control is posed as hierarchical quadratic optimization, with tasks added iteratively and converted into quadratic programs. The resulting torques are applied directly after the final iteration.
- Optimization formulation: The whole-body-control problem is formulated as a hierarchical quadratic optimization.
- Optimization formulation: At each iteration, a task is added as a linear equality Ai x = bi and satisfied as accurately as possible in the 2-norm.
- Quadratic program: Expanding the objective converts each iteration into a quadratic program with equality and inequality constraints.
- Actuation: After the final iteration, the computed actuation torques τ are directly applied to the robot.
B. Motion, Force and Torque Tasks
The controller prioritizes motion, force, and torque tasks while enforcing physically consistent dynamics. Motion tasks regulate base height and roll, including references adapted to robot attitude and curved driving.
- Task hierarchy: Motion, force, and torque tasks are ordered hierarchically from highest to lowest priority.
- Physical consistency: Optimization variables must satisfy the constrained equations of motion to keep the commanded motion physical.
- Base height: The base-height task uses a local frame aligned with the robot heading and controls operational-space acceleration through a PD reference trajectory.The proportional and derivative gains define the PD law.
- Base height: The height reference incorporates current roll and pitch so the robot behaves like an inverted pendulum with fixed length.
- Roll control: The roll task maintains a prescribed base orientation despite changing leg extensions, including changes caused by unmodeled uneven terrain.Roll acceleration is controlled along a reference trajectory using a PD law.
- Roll control: During tight-curve driving, the roll reference is computed so the zero-moment point lies at G to counteract reduced tipping robustness.
4) LQR-Assisted Balancing:
The controller addresses wheeled balancing’s non-minimum-phase behavior by embedding LQR feedback as a whole-body motion task. A simplified inverted-pendulum model captures tilt–drive coupling, while ZMP-based leaning improves curve-driving robustness.
- LQR-Assisted Balancing: Standard whole-body control can command forward acceleration directly, whereas wheeled balancing requires an initial backward wheel motion to build pitch angle.This mismatch motivates adding feedback that respects the system’s non-minimum-phase dynamics.
- LQR-Assisted Balancing: During curve driving, the roll reference is computed so the ZMP moves toward the center G of the line of support, increasing robustness against sideways tipping.Without roll adjustment, higher linear and angular velocities move the ZMP outward toward the line-of-support endpoint.
- LQR-Assisted Balancing: The simplified model lumps the base and legs into a pendulum with pitch angle θ and average wheel velocity v, using desired pitch acceleration as the tracked input.The model is linearized at the current operating point before controller synthesis.
- LQR-Assisted Balancing: The discrete linearized model yields an infinite-horizon LQR gain through the discrete-time algebraic Riccati equation.The resulting feedback law combines pitch-angle, pitch-rate, and ground-velocity tracking errors.
- LQR-Assisted Balancing: The feedback produces the required motion sequence for positive ground velocity: a small backward drive followed by forward acceleration.This behavior is then incorporated into whole-body control by tracking the desired pitch acceleration through an approximated lumped rotational Jacobian.
5) Base Yaw Angle:
The base-yaw task regulates the robot’s heading through desired yaw acceleration while preserving a unique whole-body solution by minimizing actuation torques.
- Base Yaw Angle: The base-yaw task controls the robot’s heading direction through desired yaw acceleration feedback.The feedback law is analogous to the controller used for other base motion quantities.
- Base Yaw Angle: A unique solution is enforced by minimizing all actuation torques, which also minimizes the robot’s power consumption.The optimization uses the actuator-torque identity matrix in the objective formulation.
- Base Yaw Angle: The relevant inequality constraints contribute to the whole-body-control formulation and are enforced at every hierarchy level.Their ordering therefore does not affect the hierarchy.
1) Actuator Saturation:
The controller enforces actuator, contact, and friction constraints while combining state estimation, model-based dynamics, and real-time hierarchical optimization for hardware experiments.
- Actuator Saturation: Joint torques remain within bidirectional actuator saturation bounds.This constraint limits commanded torques to the available actuator range.
- Actuator Saturation: Contact constraints prevent the robot from pulling on the ground and keep rolling-direction friction within static friction limits.The friction model uses the coefficient of static friction μh.
- Actuator Saturation: The control approach was validated in MATLAB and Gazebo before being tested on the Ascento hardware.The reported evaluation includes the control pipeline, state estimation, and three selected experiments.
- Actuator Saturation: State reconstruction uses an IMU, four motor encoders, and left-wheel odometry while assuming continuous ground contact.Absolute position and velocity are referenced to the left-wheel contact point.
- Actuator Saturation: The dynamics model and whole-body controller were implemented in ROS/C++ with Eigen, using a custom formulation for Jacobian computation.The implementation also uses OSQP for the QPs and the Control Toolbox for solving the DARE.
- Actuator Saturation: Missing state quantities are not integrated because high IMU measurement noise produces poor results.This limits the state-estimation approach used in the reported system.
B. Experiments
Experiments show recovery from impacts, adaptation to changing ground heights, and steadier curve driving with dynamically regulated roll. Future work targets impulsive-contact modeling, state estimation, and dynamics identification.
- B. Experiments: A 2 kg impact produced recovery with the proposed controller in approximately 1 s, whereas the previous LQR-based controller failed to stabilize the system.The load was dropped from a relative height of 1 m to create a horizontal impact.
- B. Experiments: On uneven ground, the controller maintained upright posture by adapting leg extensions while balancing and driving.The left leg stayed near 1.8 rad while the right leg followed the manually varied disturbance.
- B. Experiments: Dynamic roll-reference computation produced significantly steadier curve driving than a constant 0 rad roll reference.The comparison regulated the ZMP toward the center G of the line of support and used wheel odometry for assessment.
- B. Experiments: Future work will extend the approach to impulsive contact dynamics to support motions such as jumping in model predictive control.The paper also identifies improved state estimation and dedicated system-identification experiments as future directions.