Keep Rollin' - Whole-Body Motion Control and Planning for Wheeled Quadrupedal Robots

• Wheeled locomotion is faster and more efficient, while legged locomotion is more versatile in challenging environments.Source: §I · p. 1
• Most prior wheeled-legged robots use their legs as active suspension rather than as a locomotion alternative to the wheels.Source: A. Related Work · p. 1
• Existing approaches often omit whole-body dynamics, wheel-related degrees of freedom, contact forces, and torque control needed for robust dynamic locomotion.Source: A. Related Work · p. 2
• The proposed framework integrates online ZMP-based planning with hierarchical whole-body control and nonholonomic wheel constraints.Source: B. Contribution · p. 2
• The framework enables dynamic motions over flat, inclined, and rough terrain while supporting both driving and walking.Source: B. Contribution · p. 2

• A wheeled-legged robot is modeled as a free-floating base with legs and wheels attached as end-effectors.Source: §II · p. 2
• The generalized velocity dimension is nu = 6 + nj, combining six base coordinates with the joint coordinates.Source: §II · p. 2
• The equations of motion include the mass matrix, Coriolis, centrifugal and gravity terms, generalized torques, support Jacobians, and contact forces.Source: §II · p. 2
• Wheel-fixed contact points have nonzero acceleration, unlike stationary point contacts, because rolling introduces wheel motion into the contact kinematics.Source: A. Nonholonomic Rolling Constraint · p. 3
• Wheel contact acceleration depends on the wheel-frame rotation, wheel radius r0, and wheel joint angle θi.Source: A. Nonholonomic Rolling Constraint · p. 3

• The robot blindly locomotes on terrain locally modeled as a three-dimensional plane estimated from recent wheel-frame contact locations.Source: B. Terrain and Contact Point Estimation · p. 3
• Leg-fixed and wheel-fixed contact frames are introduced to simplify the motion controller and planner.Source: B. Terrain and Contact Point Estimation · p. 3
• Both contact frames lie at the intersection of the wheel plane and estimated terrain plane, with z aligned to the terrain normal.Source: B. Terrain and Contact Point Estimation · p. 3
• The contact-frame x-axis is perpendicular to the terrain normal and aligned with the wheel rolling direction cx.Source: B. Terrain and Contact Point Estimation · p. 3
• The leg-fixed contact point may move during contact, whereas the wheel-fixed contact Jacobian depends on the wheel joint angle θi.Source: B. Terrain and Contact Point Estimation · p. 3

• The planner simplifies complex whole-body dynamics into COM and foothold motion optimization using a ZMP model.Source: §III · p. 3
• Foothold, motion, and whole-body-control modules solve separate optimization problems in parallel without interrupting one another.Source: §III · p. 3
• Contact schedules define periodic lift-off and touch-down events, while driving keeps every leg in contact without lift-off events.Source: A. Contact Scheduler · pp. 3–4
• Support polygons are generated from expected footholds and their time durations using the contact schedule.Source: B. Foothold Optimizer · p. 4
• During driving, wheels permit trajectories along the rolling direction, making the leg-fixed contact position a moving foothold.Source: B. Foothold Optimizer · p. 4

• The optimizer generates x, y, and z whole-body COM references from the support polygon sequence and robot state.Source: C. Motion Optimizer · p. 4
• The COM position, velocity, and acceleration are represented with quintic spline sequences.Source: 1) Motion plan parameterization · p. 4
• A foothold is defined as the contact position of a grounded leg.Source: 1) Motion plan parameterization · p. 4
• Each spline uses 18 coefficients collected in α_k, and all spline coefficient vectors are stacked into the optimization parameter vector.Source: 1) Motion plan parameterization · p. 4

• The optimization minimizes objective f(ξ) subject to equality constraints c(ξ) and inequality constraints h(ξ).Source: 2) Optimization problem · p. 4
• Sequential quadratic programming solves for optimal spline coefficients continuously over a horizon of τ seconds.Source: 2) Optimization problem · p. 4
• The ZMP constraint uses the gravito-inertial wrench computed from COM position, acceleration, angular momentum, robot mass, and gravity.Source: 3) ZMP inequality constraint · p. 4
• The formulation constrains the ZMP position to lie inside the support polygon.Source: 3) ZMP inequality constraint · pp. 4–5
• Table I lists the costs and constraints included in the motion optimization problem.Source: Table I · p. 4

• Driving makes contact locations and footholds nonstationary, requiring a time-varying support polygon sequence.Source: 4) Deformation of support polygons… · p. 5
• The method assumes the support polygon keeps a constant number of edges, allowing one spline to describe COM motion.Source: 4) Deformation of support polygons… · p. 5
• Expected foothold positions over horizon τ are computed from reference velocities assumed constant across that horizon.Source: 4) Deformation of support polygons… · p. 5
• Each deformed edge is obtained by interpolating coefficients of the current and expected support polygon edges.Source: 4) Deformation of support polygons… · p. 5
• The COM and wheel references are tracked by a hierarchical whole-body controller formulated as prioritized quadratic programs.Source: §IV · p. 5

Prioritized Tasks

• The highlighted tasks in Table II are specifically tailored to wheeled-legged robots.Source: A. Prioritized Tasks · p. 5, Table II · p. 5
• The optimization enforces consistency with the floating-base equations of motion.Source: A. Prioritized Tasks · p. 5
• Torque limits and friction-pyramid constraints keep actuator torques bounded and contact forces within the estimated surface friction cone.Source: A. Prioritized Tasks · p. 5
• The nonholonomic rolling constraint is imposed as an equality constraint, including centripetal accelerations of contact points.Source: A. Prioritized Tasks · p. 5
• COM, swing-leg, and grounded-leg operational-space references are tracked through dedicated motion tasks.Source: A. Prioritized Tasks · pp. 5–6
• Swing-wheel rotation is damped for each swing leg using a derivative-gain task based on wheel rotational speed.Source: A. Prioritized Tasks · p. 5
• Grounded-leg tracking constrains acceleration along the rolling direction, while contact-force minimization reduces slippage.Source: A. Prioritized Tasks · pp. 5–6

• The controller computes desired actuation torques from the optimal solution ξ∗ and sends them to the robot.Source: B. Torque Generation · p. 6
• The torque-generation equation uses the actuated-joint dynamics and subtracts the contribution of contact forces mapped through the relevant constraint matrix.The terms Mj(q), hj(q, u), and JSj are defined as lower rows of the equations of motion relative to the actuated joints.Source: B. Torque Generation · p. 6

• The experiments use a real quadrupedal robot with non-steerable, torque-controlled wheels, external joystick velocity inputs, and onboard computation.State estimation fuses IMU and actuator kinematics, including wheel measurements, while the WBC and estimator run at 400 Hz.Source: §V · p. 6
• The motion optimizer solves the nonlinear problem through sequential QP problems and runs between 100 and 200 Hz depending on the gait.Source: §V · p. 6

Indoor Environment: Flat Terrain

• ANYmal transitions between driving and walking during a 90 degrees turn, driving small curvatures by yawing its base despite its non-steerable wheels.For larger curvatures, the robot needs to step, demonstrating combined wheeled and legged motion.Source: A. Indoor Environment: Flat Terrain · p. 6, Figure 5 · p. 6

Indoor Environment: Inclined Terrain

• Negligible tracking error results from a motion optimizer updating at up to 200 Hz and reinitializing with the robot’s measured state after every iteration.The executed COM and right-front-leg trajectories are almost identical to the optimized desired trajectories.Source: §V · p. 6, Figure 6 · p. 7
• 0.7 m/s: ANYmal drives over two inclines while maintaining COM height and avoiding friction-constraint violations at the wheels.The optimized trajectories show that the COM motion is unaffected by the two obstacles.Source: Figure 4 · p. 6, B. Indoor Environment: Inclined Terrain · p. 7

C. Outdoor Environment: Crossing a Street

• ANYmal crosses a street by driving down one step, walking over another, and rotating its base around yaw to change driving direction.Source: C. Outdoor Environment: Crossing a… · p. 7
• 1 m/s: the robot drives down steps and stairs without terrain perception and drives down the stair without stepping.The experiment also illustrates walking over a step and changing direction by yawing the base.Source: C. Outdoor Environment: Crossing a… · p. 7, Figure 7 · p. 7

D. High Speed and Low Cost of Transport

• 4 m/s: ANYmal exceeds its previously reported maximum speed of 1.5 m/s, while its mechanical COT is 0.1 at 2 m/s.Source: D. High Speed and Low Cost of Transport · p. 7
• 83 %: driving lowers mechanical COT relative to the trotting gait, while the reduction relative to skating motions is 17 %.At 2 m/s, the reported mechanical power consumption is 63.64 W.Source: D. High Speed and Low Cost of Transport · p. 7

VI. CONCLUSIONS

• The framework integrates torque-controlled wheels into whole-body planning and control, using continuously updated 3D ZMP plans and hierarchical WBC with nonholonomic constraints.The paper reports dynamic motions over flat and inclined terrains and over steps.Source: §VI · p. 7
• Long-horizon perceptive motion planning in challenging environments remains unsolved for wheeled-legged and legged robots.Source: §VI · p. 8