Harpy: Thruster-Assisted Bipedal Robot

• Harpy is a thruster-assisted bipedal robot that integrates legged and aerial locomotion through precise impact corrections and thrust-augmented stabilization.
• The platform evolved from a 5-link planar model to a 12-DoF floating-base system, employing hybrid zero dynamics and reference governors for robust control.
• Extensive studies validate Harpy’s performance with improved stability, reduced slipping, and accurate force estimation across varied terrains and dynamic maneuvers.

Searching arXiv for recent and foundational Harpy papers to ground the article. Harpy is a thruster-assisted bipedal robot developed at Northeastern University as a research platform for multimodal legged-aerial locomotion, contact-force regulation, and thrust-augmented stabilization. In the literature, Harpy appears in several forms: as a planar 5-link hybrid system used to study hybrid zero dynamics and impact correction, as a 12-DoF floating-base biped used for capture-point and reference-governor control, and as a hardware platform for frontal stabilization, trotting, jumping, slope walking, thrust estimation, and propulsion system identification (Dangol et al., 2020, Sihite et al., 2021, Dhole, 2024, Pitroda et al., 2024).

1. Research lineage and platform variants

The earliest Harpy formulation in this corpus is a planar 5-link biped with 5 DOF in single support: torso angle $q_T$, two femur joints $q_{1R}, q_{1L}$, and two tibia joints $q_{2R}, q_{2L}$. Each of the four leg joints is driven by a torque actuator, the stance foot is an ideal, non-slipping pivot in single support, impacts at foot exchange are modeled as inelastic while conserving angular momentum, and double support is extended to a finite interval of approximately $10$ ms so that thruster-based corrections can be applied before the next single-support phase (Dangol et al., 2020).

Later publications describe a substantially richer 3D floating-base robot. One account presents Harpy as a $0.6$ m tall, $4.5$ kg biped with 12 DoF, consisting of a 6-DoF torso and 3 DoF per leg, with eight custom high-energy-density leg actuators and two electric ducted-fan thrusters mounted on the torso (Pitroda et al., 2024). Another describes a 12-DoF floating-base biped whose mass is concentrated in the rigid torso and the joint actuator housings, with point feet and idealized thrusters used to regulate ground reaction forces (Sihite et al., 2021). Hardware-oriented theses introduce additional numerical descriptions: an approximately $4$ kg robot of height $30$ in and width $24$ in with thrust-to-weight ratio greater than $1.0$ (Pitroda, 2023), and later a $q_{1R}, q_{1L}$0 kg, $q_{1R}, q_{1L}$1 bipedal robot with 3 DoF per leg and two integrated thrusters (Cahill, 28 Apr 2025). This suggests that “Harpy” denotes an evolving hardware and modeling lineage rather than a single immutable specification.

2. Mechanical architecture and propulsion layout

Across the 3D descriptions, Harpy is consistently a biped with three actuated joints per leg: a hip-frontal joint, a hip-sagittal joint, and a knee joint. The hardware implementation uses compact motor-plus-Harmonic-Drive assemblies, carbon-fiber tubes for the thigh and shank, and 3D-printed carbon-fiber-reinforced linkages; one implementation also includes a parallel four-bar mechanism in the lower leg to keep foot orientation. Reported drivetrain details include T-Motor brushless units with $q_{1R}, q_{1L}$2 Harmonic Drives controlled by Elmo Gold Twitter drives over EtherCAT, and earlier series-elastic pantograph-style legs with Antigravity 4006 motors and Harmonic Drive CSF-11-30-2A-R gearboxes (Dhole, 2024, Kumar, 25 Aug 2025, Pitroda, 2023).

The propulsion system is likewise described in multiple versions. In the planar model, two coaxial thruster pairs are fixed to the torso and aligned along the torso link, with only the thrust magnitudes modulated (Dangol et al., 2020). In later 3D models, Harpy carries two torso-mounted electric ducted-fan thrusters that are rigidly mounted on the torso sides and can be directed in the sagittal or frontal planes by varying torso attitude (Pitroda et al., 2024). A control-oriented formulation separates the thruster action into a net force about the center of mass, denoted $q_{1R}, q_{1L}$3, and equal-and-opposite lateral forces for roll and yaw stabilization without altering net center-of-mass force (Sihite et al., 2021). Hardware reports identify several EDF configurations, including Schübeler DS-51 units mounted bilaterally near the center of mass, each producing up to approximately $q_{1R}, q_{1L}$4 kgf and yielding a system thrust-to-weight ratio of about $q_{1R}, q_{1L}$5 (Dhole, 2024), Schübeler DS-30-AXI-HDS fans with maximum thrust $q_{1R}, q_{1L}$6 N and maximum roll moment $q_{1R}, q_{1L}$7 Nm (Cahill, 28 Apr 2025), and a later experimental platform using DS-51 EDFs rated at $q_{1R}, q_{1L}$8 kg each at $q_{1R}, q_{1L}$9 V (Kumar, 25 Aug 2025). This suggests that the propulsion hardware, like the leg design, was revised across successive Harpy studies.

3. Dynamic models and reduced-order abstractions

Harpy’s full-order dynamics are generally written in Euler–Lagrange form with explicit joint, thruster, and ground-contact inputs. A representative expression is

$q_{2R}, q_{2L}$0

where $q_{2R}, q_{2L}$1 denotes leg actuation, $q_{2R}, q_{2L}$2 the thruster forces, and $q_{2R}, q_{2L}$3 the ground-reaction forces. In reduced-order form, the platform is modeled as a Variable-Length Inverted Pendulum (VLIP) about the center of mass, with stance-foot center of pressure $q_{2R}, q_{2L}$4, leg-length acceleration input $q_{2R}, q_{2L}$5, and net thruster force $q_{2R}, q_{2L}$6 (Sihite et al., 2021).

A central abstraction in the capture-point literature is the effect of vertical thrust on effective pendulum stiffness. Starting from sagittal-plane centroidal dynamics and imposing constant center-of-mass height, the horizontal equation reduces to

$q_{2R}, q_{2L}$7

This motivates the definition $q_{2R}, q_{2L}$8 and the modified natural frequency

$q_{2R}, q_{2L}$9

which yields the thruster-augmented capture point

$10$0

The interpretation given in this line of work is that upward thrust produces a “virtual-buoyancy” effect by reducing the effective gravitational stiffness of the inverted-pendulum model (Pitroda et al., 2024).

The 2020 hybrid-systems formulation emphasizes periodic locomotion with impacts. In that model, the supervisory layer enforces virtual holonomic constraints

$10$1

with $10$2, and applies

$10$3

to preserve the zero-dynamics manifold through the single-support dynamics. Because impacts and reference modifications create deviations from the desired periodic orbit, the paper replaces the standard impact-invariance condition $10$4 with a double-support correction objective $10$5, where $10$6 is the short double-support flow map (Dangol et al., 2020).

4. Control architectures

One major control line on Harpy combines hybrid zero dynamics, feedback linearization, an explicit reference governor, and short-horizon predictive correction. In single support, Harpy tracks virtual constraints using

$10$7

which is stated to guarantee exponential convergence of $10$8 under fast actuator dynamics. To enforce state and input limits without online quadratic programming, the same work inserts an intermediary filter based on the emerging idea of reference governors, using a Lyapunov level-set condition on a reduced VLIP model. During the finite double-support interval, a short-horizon nonlinear MPC exploits thruster overactuation to steer the post-impact state back to the target zero-dynamics manifold before the next single-support phase (Dangol et al., 2020).

A second line of work uses reference governors to enforce unilateral and friction-cone ground-contact constraints. In that framework, the applied reference $10$9 replaces the nominal reference $0.6$0 inside the low-level joint and thruster controllers, and the governor continuously adjusts $0.6$1 so that Coulomb-type inequalities and minimum normal-force constraints remain satisfied. The Lyapunov function

$0.6$2

is used to show forward-time decrease toward the feasible set, thereby avoiding expensive optimization-based whole-body control for this specific constraint-enforcement task (Sihite et al., 2021).

A third family of controllers modifies capture-point regulation by explicitly incorporating thrust. One implementation computes a thruster-augmented capture point, uses foot-placement planning for the next step, and commands the net center-of-mass thrust through a simple proportional law on capture-point error, while joint-level PD controllers track the desired leg-length trajectory and torso attitude (Pitroda et al., 2024). A subsequent formulation embeds the discretized capture-point dynamics in a small-horizon QP with cost

$0.6$3

subject to foot-placement, friction-cone, and thruster-limit constraints, and reports implementation in MATLAB/Simulink with the qpSWIFT solver, horizon $0.6$4, and $0.6$5 s (Pitroda et al., 2024). For inclined walking, another QP solves at $0.6$6 Hz for

$0.6$7

under contact and thrust constraints, after which a whole-body mapping allocates the resulting generalized forces to stance-leg torques and planar thruster commands (Pitroda et al., 2024).

Hardware implementation adds a state-estimation and gait-execution layer. One closed-loop system fuses motion capture at $0.6$8 Hz and IMU data at $0.6$9 Hz through an EKF, uses Raibert foot placement for sagittal speed regulation, generates swing trajectories with 5th-order Bézier curves, stabilizes frontal roll with differential-thrust PID,

$4.5$0

and tracks joint references with high-bandwidth PD control at $4.5$1 kHz (Dhole, 2024).

5. Locomotion modes and validated behavior

Harpy has been used to study low-friction locomotion and direct contact-force regulation. In VLIP-level results for unilateral force regulation, the nominal reference alone violated $4.5$2 of the friction-cone constraints, whereas the combination of reference governor and thrusters kept all constraints in strict feasibility and the manipulated reference tracked the nominal reference with mean deviation less than $4.5$3. In the full 3D model, the baseline no-thruster system slipped repeatedly, while the RG-plus-thruster version eliminated slipping and maintained center-of-mass velocity at $4.5$4 m/s with less than $4.5$5 cm lateral tracking error. The same study reports stable gait only for $4.5$6 without thrusters, versus stable gait down to $4.5$7 with thrusters and reference governance (Sihite et al., 2021).

In capture-point-based walking, a Simscape/Matlab model on flat terrain showed nominal walking stabilization in approximately $4.5$8 s, or about $4.5$9 steps, with mean step time around $4$0 s and step length about $4$1 m. Thrusters supplied up to $4$2 N per side during transitions, peak knee torques were approximately $4$3 N·m immediately after initial impact and settled to $4$4 N·m in steady state, and an impulsive sagittal push of $4$5 N·s was rejected within two steps when thrusters were active. Without thrusters, the same impulse required roughly $4$6 larger step length and often led to tip-over (Pitroda et al., 2024). In the QP-based extension, the principal observations were that the center of mass stabilized around the desired position with drift approaching zero, whereas the no-QP case exhibited continuous drift; increasing thruster magnitude reduced the unstable eigenvalue $4$7, and capture-point lengths shrank as thruster support increased (Pitroda et al., 2024).

Harpy has also been studied on inclined terrain and in obstacle-clearing or jumping maneuvers. The slope-walking QP controller was evaluated on a $4$8 terrain slope in MATLAB/Simscape with RK4 at $4$9 Hz and a $30$0 Hz qpSWIFT solver. Reported performance includes center-of-mass position and velocity tracking within a few centimeters and $30$1 m/s around the setpoint, friction-cone compliance at all times, initial peak thrust around $30$2 N settling to around $30$3 N per side, stance hip and knee peaks around $30$4 Nm, and limit-cycle convergence in fewer than ten steps (Pitroda et al., 2024). Earlier rough-terrain simulations showed frontal angles remaining within $30$5 under $30$6 N lateral pushes and successful jumps over obstacles corresponding to approximately $30$7 leg length and $30$8 leg length, respectively (Liang et al., 2021).

Hardware-oriented validation focused first on frontal stabilization and dynamic walking. In trotting-in-place tests within a $30$9 motion-capture arena, activating the thrusters for frontal control reduced roll-angle RMS error from approximately $24$0 to approximately $24$1, increased time-to-fall from about $24$2 s to more than $24$3 s, reduced lateral center-of-mass standard deviation from $24$4 m to $24$5 m, and achieved stable trotting with $24$6 and $24$7. Foot-placement errors were reported below $24$8 cm in both $24$9 and $1.0$0 (Dhole, 2024).

A later experimental analysis examined $1.0$1 trotting cycles over $1.0$2 s and $1.0$3 jump trials on a gantry-constrained platform. Trotting employed alternating left-right stance and swing phases of approximately $1.0$4 s each, with closed loops in the underactuated $1.0$5 and $1.0$6 phase portraits. Jumping reached peak center-of-mass heights of $1.0$7–$1.0$8 m, corresponding to $1.0$9–$q_{1R}, q_{1L}$00 m, with foot trajectory errors of $q_{1R}, q_{1L}$01 m in $q_{1R}, q_{1L}$02 and $q_{1R}, q_{1L}$03 m in $q_{1R}, q_{1L}$04, knee velocities of $q_{1R}, q_{1L}$05 rad/s, hip-sagittal velocities of $q_{1R}, q_{1L}$06 rad/s, step-to-step trotting symmetry within $q_{1R}, q_{1L}$07 variance, and jumping repeatability of $q_{1R}, q_{1L}$08 m in height and $q_{1R}, q_{1L}$09 in pitch. That analysis states that the legs provide primary propulsion, while the thrusters enable additional aerial-phase control (Kumar, 25 Aug 2025).

6. Force estimation, system identification, and unresolved issues

As Harpy’s control stack matured, thrust estimation became a dedicated topic. A conjugate-momentum observer was derived from the generalized momentum

$q_{1R}, q_{1L}$10

and the dynamics

$q_{1R}, q_{1L}$11

The estimator reconstructs generalized thruster forces through a first-order observer on the missing force term, then extracts body-frame thrust using a pseudo-inverse map. Two cases were studied: known ground-reaction forces and unknown terrain handled through a contact constraint. For the contact-constraint case, the reported normalized RMSE values for the generalized wrench $q_{1R}, q_{1L}$12 were $q_{1R}, q_{1L}$13, $q_{1R}, q_{1L}$14, $q_{1R}, q_{1L}$15, $q_{1R}, q_{1L}$16, $q_{1R}, q_{1L}$17, and $q_{1R}, q_{1L}$18, respectively; the paper summarizes these as all being under $q_{1R}, q_{1L}$19 (Pitroda et al., 2024).

Propulsion and actuator identification were later treated at the hardware level. One thesis models thrust using an energy-momentum approach,

$q_{1R}, q_{1L}$20

then augments it with empirical correction terms, reporting fitted values $q_{1R}, q_{1L}$21 and $q_{1R}, q_{1L}$22. CFD validation at $q_{1R}, q_{1L}$23 RPM yielded thrust errors of $q_{1R}, q_{1L}$24, $q_{1R}, q_{1L}$25, $q_{1R}, q_{1L}$26, and $q_{1R}, q_{1L}$27 across four propeller diameter/pitch combinations, while a static thrust stand measured peak thrust around $q_{1R}, q_{1L}$28 g, or about $q_{1R}, q_{1L}$29 kgf $q_{1R}, q_{1L}$30 N, matching CFD and theory within $q_{1R}, q_{1L}$31. The same work characterizes Harpy’s joint motors with a torque constant around $q_{1R}, q_{1L}$32 Nm/A and reports closed-loop torque response bandwidth greater than $q_{1R}, q_{1L}$33 Hz with less than $q_{1R}, q_{1L}$34 overshoot (Cahill, 28 Apr 2025).

Several unresolved issues recur across the literature. Control papers identify onboard computing limits and real-time QP solve times, thruster modeling errors including actuator delays and aerodynamic cross-coupling, sensor delays and noise in center-of-mass estimation and foot-contact detection, and the power and thermal constraints of high-energy-density thrusters (Pitroda et al., 2024). Hardware reports add that current control decouples sagittal and frontal loops, leaves aggressive cross-coupling unhandled, and often depends on external motion capture rather than onboard perception (Dhole, 2024). Estimation work further notes that unknown terrain complicates the recovery of ground forces, and that double-support can degrade contact-constraint conditioning because the relevant Jacobian term loses rank (Pitroda et al., 2024). Together, these limitations define Harpy’s current research frontier: not whether thrusters can augment bipedal locomotion, but how tightly thrust generation, contact mechanics, state estimation, and whole-body control can be integrated in real time.