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Representation-Free Model Predictive Control for Dynamic Motions in Quadrupeds

Yanran Ding, Abhishek Pandala, Chuanzheng Li, Young-Ha Shin, Hae-Won Park

arXiv:2012.10002v1cs.RO

TL;DR

Dynamic quadruped motions require control methods that handle under-actuation, physical constraints, and orientation changes problematic for Euler angles or quaternions. The paper presents RF-MPC using rotation-matrix dynamics, variation-based linearization, and QP transcription. Experiments stabilize multiple gaits and a controlled backflip through an Euler-angle singularity, with control rates up to 250 Hz.

  • Problem

    Dynamic quadruped control must handle under-actuated dynamics, hardware and environmental constraints, and orientation representations that can suffer from Euler-angle singularities or quaternion unwinding.

  • Method

    RF-MPC directly represents orientation with rotation matrices, applies variation-based linearization and an affine orientation-error approximation, and transcribes the controller into a Quadratic Program.

  • Results

    Experiments covered pose and balance control, walking and running trots, bounding, and a controlled backflip through the Euler-angle singularity; the controller reached 250 Hz.

  • Takeaways & Limitations

    RF-MPC stabilized dynamic 3D motions involving Euler-angle singularities without decomposing the robot motion into in-plane and out-of-plane parts.

  • Takeaways & Limitations

    First-order rotation-matrix predictions leave the SO(3) manifold, making prediction error more pronounced for longer horizons and creating a horizon-length trade-off.

Abstract

from arXiv · show

This paper presents a novel Representation-Free Model Predictive Control (RF-MPC) framework for controlling various dynamic motions of a quadrupedal robot in three dimensional (3D) space. Our formulation directly represents the rotational dynamics using the rotation matrix, which liberates us from the issues associated with the use of Euler angles and quaternion as the orientation representations. With a variation-based linearization scheme and a carefully constructed cost function, the MPC control law is transcribed to the standard Quadratic Program (QP) form. The MPC controller can operate at real-time rates of 250 Hz on a quadruped robot. Experimental results including periodic quadrupedal gaits and a controlled backflip validate that our control strategy could stabilize dynamic motions that involve singularity in 3D maneuvers.

I. INTRODUCTION

The paper develops RF-MPC for dynamic quadrupedal motions that challenge conventional orientation representations, especially maneuvers passing near Euler-angle singularities. It combines rotation-matrix dynamics, variation-based linearization, and QP transcription to enable real-time control.

  • Motivation: Quadrupedal robots are increasingly being developed for agile maneuvers beyond flat-terrain locomotion, including obstacle negotiation and acrobatic motions.The motivation spans dynamic running, challenging-terrain navigation, stair climbing, jumping, and backflips.
  • Challenges: Dynamic quadruped control must account for under-actuated dynamics and constraints from hardware capacity and the environment.Existing approaches include heuristic, inverse-dynamics, hierarchical operational-space, reinforcement-learning, and optimization-based controllers.
  • Orientation representations: Euler-angle MPC frameworks can encounter representation singularities, while quaternions may introduce ambiguity that causes unwinding.These representation issues restrict motions involving large orientation changes or singular poses.
  • Proposed framework: RF-MPC uses rotation matrices and variation-based linearization to control dynamic 3D motions, including rotations through 90° pitch angles.The formulation is designed to maintain consistent performance during complex 3D rotations and acrobatic motions.
  • Proposed framework: An affine approximation of the orientation error enables transcription of the MPC problem into a standard Quadratic Program for real-time control.The paper also introduces an improved three-dimensional variation vector for angular dynamics and a positive-definite orientation objective.
  • MPC formulation: MPC repeatedly solves constrained finite-horizon optimization, applying the first control input while incorporating state, input, ground-reaction-force, and friction-cone constraints.The framework adopts a simplified model to reduce optimization dimensionality for execution on resource-limited embedded hardware.

A. 3D Single Rigid Body Model

The paper models the quadruped as a three-dimensional single rigid body whose orientation is represented directly by a rotation matrix. Variation-based linearization addresses the nonlinear rotational dynamics around an operating point for real-time MPC formulation.

  • A. 3D Single Rigid Body Model: The single rigid-body model uses position, velocity, rotation matrix, and body-frame angular velocity as its state.The rotation matrix represents the body frame in the inertial frame and belongs to SO(3).
  • A. 3D Single Rigid Body Model: Ground-reaction forces at four foot locations provide the control inputs and generate the rigid body's external force and torque.The indexed contacts are front-left, front-right, hind-left, and hind-right.
  • A. 3D Single Rigid Body Model: The rotational dynamics are nonlinear because the rotation matrix evolves on the SO(3) manifold.The paper therefore introduces a variation-based linearization scheme.
  • A. 3D Single Rigid Body Model: Small rotation variations are used to approximate predicted orientations near the operating point, with higher-order variation terms neglected during linearization.The approximation uses a first-order Taylor expansion of the matrix exponential and assumes the variation is small.
  • A. 3D Single Rigid Body Model: The resulting linearized dynamics are used both as affine dynamics and in constructing the MPC objective function.The GRF input is expressed as a variation from its current operating-point value.

C. Vectorization

The vectorization procedure converts matrix-valued rotational dynamics into vector equations suitable for standard QP formulation. It represents tangent-space rotation variations with a three-dimensional vector and uses Kronecker products for matrix products.

  • C. Vectorization: Vectorization is introduced because the linearized rotation and angular-velocity dynamics remain difficult to express in standard QP form.The method transforms matrix-matrix products into matrix-vector products.
  • C. Vectorization: Kronecker products are used to rewrite the rotational matrix products as matrix-vector expressions.This procedure is applied to the vectorized rotation and angular-velocity dynamics.
  • C. Vectorization: A vector ξ represents the skew-symmetric rotation variation δR in the tangent space of SO(3).A constant matrix N maps ξ to vec(δR).
  • C. Vectorization: The identity vec(ˆv) = Nv provides the mapping used to vectorize the rotation variation.Here, N is a constant matrix in R9×3.
  • C. Vectorization: The discrete orientation dynamics are likewise propagated using forward Euler and mapped back through the left pseudo-inverse of N.This produces the discrete dynamics in ξ.
  • C. Vectorization: The discrete angular-velocity dynamics are propagated with forward Euler integration.The update is ωk+1 = ωk + dt ˙ωk.

D. Discrete-time Affine Dynamics

After linearizing and vectorizing the single rigid-body model, the paper defines variation-based state and control vectors and expresses the dynamics as a discrete-time affine state-space system.

  • D. Discrete-time Affine Dynamics: The new control input is each leg's GRF variation from its operating-point value.The variation is denoted δui for the ith leg.
  • D. Discrete-time Affine Dynamics: The assembled discrete-time dynamics take the affine form xt+k+1|t = A|op · xt+k|t + B|op · δut+k|t + d|op.The matrices and offset are constructed from measurements at the operating point.
  • D. Discrete-time Affine Dynamics: The linearized model is a locally valid linear time-varying system obtained by linearizing the nonlinear dynamics around the operating point.The paper states that this system can be stabilized to track reference trajectories.
  • D. Discrete-time Affine Dynamics: The discrete-time affine dynamics are imposed as equality constraints in the MPC problem.This connects the state-space model directly to the MPC optimization formulation.

E. Cost Function

The MPC cost penalizes state and control deviations from reference trajectories using quadratic terms. A linear orientation-error approximation makes the rotation contribution affine in the state variation.

  • E. Cost Function: The stage cost is quadratic and penalizes deviations from desired state and control trajectories.Positive-definite weighting matrices govern state and control penalties.
  • E. Cost Function: The state cost includes position, velocity, orientation, and angular-velocity error terms.The corresponding weighting matrices are diagonal and positive definite.
  • E. Cost Function: The original orientation error uses a matrix logarithm and is nonlinear in the optimization variables.This distinguishes it from the other error terms in the stage cost.
  • E. Cost Function: The proposed approximate orientation error is linear in ξk because the desired and operating-point rotation matrices are known.It combines the geodesic between the operating-point and desired rotations with the tangent-space variation ξk.
  • E. Cost Function: The orientation error function is positive definite when its weighting matrix QR is positive definite.The terminal cost is defined similarly with terminal gains.
  • E. Cost Function: The control cost penalizes predicted GRF variations from the operating-point input.The variation is denoted δuk at prediction step k.

F. Force Constraints

The controller enforces physically feasible ground reaction forces using a conservative linear friction-pyramid approximation and normal-force bounds, allowing the MPC problem to become a standard QP. The section also defines an Euler-angle singularity measure used in simulation analysis.

  • Force feasibility: The force constraints require nonnegative normal force and tangential forces inside the contact friction cone.These constraints ensure that solved ground reaction forces are physically feasible when a foot contacts the ground.
  • QP-compatible constraints: The second-order friction-cone constraint is conservatively approximated by a friction pyramid so linear constraints can be used in the QP.Normal force is additionally bounded so commanded torque remains within actuator limits.
  • Contact scheduling: Swing legs receive zero lower and upper normal-force bounds, leaving foothold tracking to the swing-leg controller.The feasible force sets are polyhedral because the constraints form intersections of finitely many closed halfspaces.
  • QP formulation: The affine dynamics, convex quadratic cost, and linear force constraints reformulate the nonlinear MPC problem as a Quadratic Program.Equality constraints encode linear dynamics, while inequality constraints impose force feasibility.
  • Singularity analysis: Euler-angle singularity is quantified with the inverse condition number κ−1(TΘ), which tends toward 0 as the robot approaches a singular pose.The transformation TΘ is invertible except at the stated pitch singularities, and the measure supports comparisons near singular configurations.
  • Singularity experiment: A 0.5 s pose-control simulation varies the desired pose from the singular pose to a pose rotated 1 rad about the positive y axis for RF-MPC and EA-MPC.All feet are fixed and force constraints are lifted to isolate the effect of Euler-angle singularity.

B. Walking Trot

The walking-trot simulation accelerates the quadruped from rest to 0.5 m/s while monitoring velocity, orientation, and vertical ground reaction forces. RF-MPC maintains bounded tracking deviations during this motion.

  • Walking trot: 0.5 m/s is reached in the x-direction after acceleration from a stationary pose, with velocity deviation within ±0.1 m/s in all directions.The simulation applies a prescribed acceleration while the robot transitions from rest to the final walking-trot speed.
  • Singularity study: The pose-control singularity study plots 0.5 s CoM position deviations for RF-MPC and EA-MPC against the rotation error magnitude.The comparison removes force constraints by fixing all feet, focusing the test on orientation-representation singularity.
  • Walking trot: Orientation deviation remains within ±0.02 rad during the walking-trot simulation.The figure also reports vertical ground reaction forces for all four legs.
  • Simulation setup: The walking-trot gait is executed using a time-based schedule while the nonlinear model is integrated with MATLAB ode45 at each sampling time.The applied control input is evaluated on the original nonlinear dynamics rather than only on the linear prediction model.

D. Aperiodic Complex Dynamic Maneuver

The aperiodic maneuver study evaluates RF-MPC on a twist jump involving singular Euler-angle orientations and compares two linearization schemes under disturbance. RF-MPC tracks the maneuver and operating-point linearization recovers from disturbances more robustly.

  • Aperiodic maneuver: The twist jump uses a 45° inclined starting surface, 0.1 s of four-foot stance, 0.1 s of hind-leg stance, and a 0.3 s aerial phase.The reference poses, center-of-mass trajectory, and feed-forward forces define the complex backflip-with-twist maneuver.
  • Singularity comparison: RF-MPC successfully stabilizes the backflip through the Euler-angle singularity, whereas EA-MPC becomes unstable and diverges from the reference trajectory.The comparison includes orientation and CoM deviations as the robot approaches and passes through singularity.
  • Robustness: RF-MPC remains stable when the slope angle changes from 45° to 53.6° and the initial orientation is consequently perturbed.An open-loop simulation instead shows rapid orientation deviation under the perturbed initial condition.
  • Linearization schemes: Scheme 1 linearizes around the reference trajectory, while scheme 2 linearizes around the current operating state and control using constant prediction matrices.The two schemes use the same reference trajectory, gait timing, controller gain, and disturbance in the comparison.
  • Disturbance rejection: RF-MPC with scheme 2 recovers from the disturbance and tracks the reference, while scheme 1 fails at 1.5 s as velocity and orientation diverge.The disturbance has a maximum force of 27 N and is applied in the positive y direction during constant-speed bounding.
  • Prediction quality: Scheme 1 produces high rotation-matrix prediction error after deviation begins, supporting scheme 2’s empirical robustness in disturbance rejection.Prediction quality is evaluated by comparing predicted rotation matrices with their operating-point rotations and SO(3) projections.

F. Reference Trajectory Generation

Reference trajectories are generated with gait scheduling, heuristic force allocation, periodicity design, and offline trajectory optimization tailored to each motion. The resulting MPC system integrates these references with estimation, swing control, and torque control on the Panther platform.

  • Trotting: Trotting references distribute robot weight across contacting legs and assume constant acceleration from rest until maximal velocity.A binary contact variable from the finite state machine determines which legs receive reference forces; walking and running trot share this trajectory.
  • Aperiodic maneuver: The twist-jump reference uses offline single-shooting trajectory optimization with fixed contact timing, bounded final state, reachability, collision avoidance, and friction-cone constraints.The three phases are four-foot stance, hind-leg stance, and aerial motion, lasting 0.1 s, 0.1 s, and 0.3 s respectively.
  • Bounding: The bounding trajectory models the robot as a 2D single rigid body and uses pre-specified contact phases with Bézier-polynomial vertical-force and pitch-torque profiles.Scaling factors and initial conditions are selected to achieve periodicity in vertical position and pitch.
  • Controlled backflip: The controlled-backflip reference is generated with direct-collocation trajectory optimization on a 2D single-rigid-body model, penalizing large and nonsmooth ground reaction forces.Additional motion constraints supplement those used for the complex maneuver.
  • Control architecture: The finite state machine sends desired state and control trajectories to the MPC, while the QP solver returns force corrections that are combined with swing-leg control before torque actuation.State estimation, joint-level compensation, and BLDC motors complete the hardware control loop.
  • Phase-aware prediction: A prediction horizon spanning multiple gait phases lets RF-MPC account for upcoming contact changes during bounding and acrobatic jumps.This supports planning current control across stance, swing, and aerial transitions.
  • Hardware platform: The experiments use Panther, a 5.5 kg fully torque-controllable quadruped with an onboard computer, joint encoders, and an IMU.The platform combines custom BLDC motor units with a lightweight carbon-fiber-based body structure.

C. Computation Setup

The RF-MPC implementation runs at real-time rates, with sensing and estimation at high base frequencies and QP solution rates up to 250 Hz. Swing-leg control combines feed-forward and feedback terms, while foot placement combines velocity-based and capture-point feedback components.

  • Computation: The QP is solved at 160–250 Hz, while encoder and lower-level kinematics run at 4 kHz and IMU-based estimation at 1 kHz.Joystick input is updated at 23 Hz, and qpSWIFT solves the proposed QP for all experiments.
  • Swing-leg control: Swing-leg control uses feed-forward workspace inverse dynamics together with feedback based on position and velocity errors.The swing leg is modeled as a 3-link serial manipulator attached to a stationary base.
  • Swing-leg control: The feedback component uses position and velocity gain matrices for swing-leg control.These gains act on desired and measured foot position and velocity quantities.
  • Foot placement: Desired foot placement combines a velocity-based feed-forward term with capture-point-based feedback.The desired step location is defined on the ground plane using the projected hip position and velocity-related terms.

E. State Estimation

State estimation fuses kinematic and inertial measurements, while contact detection uses a proprioceptive generalized-momenta disturbance observer. Motor friction is modeled and compensated to improve force-control accuracy.

  • State estimation: A complementary filter combines low-frequency kinematic estimates with high-frequency accelerometer information to estimate CoM velocity and position.The CoM position estimate fuses leg-position kinematics with the estimated CoM velocity.
  • Contact detection: Contact detection uses proprioceptive measurements and a generalized-momenta disturbance observer rather than conventional force estimation.The method uses joint positions, velocities, and commanded torque, focusing on knee joints because contact changes their momentum most.
  • Contact detection: Contact is declared when the four-leg residual vector exceeds a threshold value.The residual accumulates previous residuals and commanded torque information.
  • Actuation compensation: Gearbox and bearing friction is modeled and compensated because the robot uses a relatively high 23.36:1 planetary gearbox ratio.The output torque adds the friction compensation term to the motor torque, with a saturation function used to relax the sign behavior near equilibrium.

2) Center of Mass Location:

The experiments evaluate RF-MPC across pose control, balancing, walking and running trots, bounding, and a controlled backflip. The results show tracking and stabilization across locomotion and a 3D maneuver passing through an orientation singularity.

  • 2) Center of Mass Location:: CoM location is estimated experimentally because small-robot CAD models may poorly capture electronics mass distribution.The robot is suspended from multiple known attachment points, and the CoM is obtained from the corresponding lines.
  • Experiments: RF-MPC is evaluated on pose control, moving-platform balancing, walking trot, running trot, bounding, and controlled backflip tasks.The framework is presented as a general motion-control system for multiple quadrupedal objectives.
  • Walking trot: Walking trot maintains small orientation deviations within ±0.06 rad while tracking commanded planar velocities.The experiment evaluates velocity tracking in both horizontal directions and vertical ground-reaction forces.
  • Running trot: Running trot includes a 40 ms free-fall aerial phase and vertical ground-reaction forces reaching as high as 60 N.The data also include vertical CoM velocity, vertical GRF, and knee torque during the experiment.
  • Bounding: Bounding uses full robot dynamics with extensive body-pitch oscillation, while RF-MPC stabilizes the robot toward desired state and ground-reaction-force trajectories.The reference trajectory enables a longer aerial phase, although an elaborate reference is optional for stabilization.
  • D. Controlled Backflip: During the backflip, RF-MPC controls a 3D robot through a singular pose without decomposing sagittal and out-of-plane motion.The robot flips backward around the y-axis, passes upright, and lands upside down.
  • D. Controlled Backflip: Ten backflip trials show that RF-MPC remains stabilizing under an initial-stage-height perturbation and the robot lands safely.The controller is active during the four-leg-contact and hind-leg-contact phases, while impedance control handles landing.

VI. DISCUSSION

RF-MPC stabilizes complex 3D quadruped motions by representing orientation with rotation matrices and converting the controller to a real-time QP. The experiments demonstrate controlled backflips through Euler-angle singularities, while the method remains constrained by local linearization accuracy and state-estimation quality.

  • Findings: Rotation-matrix orientation enables RF-MPC to stabilize 3D dynamic motions involving Euler-angle singularities without switching to in-plane and out-of-plane control.The controlled backflip passes through the singularity while RF-MPC stabilizes the full 3D motion.
  • Findings: Operating-point linearization provides more robust RF-MPC behavior than reference-trajectory linearization in the reported simulation case study.The operating-point scheme uses constant linearization matrices throughout the prediction horizon, whereas the reference-trajectory scheme uses time-varying matrices.
  • Limitations: First-order rotation-matrix linearization produces predicted matrices outside SO(3), making longer prediction horizons less accurate and creating a trade-off with predictive ability.A horizon that is too long can yield inaccurate rotational dynamics, while one that is too short can produce myopic behavior.
  • Limitations: Bounding deviations and tracking errors may be affected by simple state-estimation and contact-detection algorithms, including velocity measurements derived from state estimation.This limitation concerns the reported experimental interpretation rather than the rotation-matrix formulation itself.
  • Experiments: RF-MPC was demonstrated in pose and balance control, walking and running trot, bounding, and a controlled backflip on the Panther quadruped.The backflip experiment specifically demonstrated stabilization of a dynamic motion involving Euler-angle singularity.
  • Experiments: 250 Hz control frequency was achieved using the custom qpSWIFT quadratic-program solver.The formulation's QP structure supports real-time control implementation on the quadruped robot.
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