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Co-design of trajectory and morphology for a vertical jump-climbing robot

Christopher Y. Xu, Elliot W. Hawkes

arXiv:2609.31014v1cs.RO

TL;DR

Untethered vertical climbing with an aerial phase remains an open robotic capability. The authors co-design a robot's morphology and trajectory for jump-climbing, achieving repeated vertical climbs at 0.375 m/s (1.97 body lengths/s).

  • Problem

    The authors identify untethered vertical climbing with an aerial phase as an unmet capability, despite such bounding in animals.

    Source: §I · p. 1

  • Method

    A trajectory–morphology co-design workflow optimizes a robot for repeated launches and reattachments on a carpeted wall with spined grippers.

    Source: §I · p. 2, §IV · p. 7

  • Results

    0.375 m/s (1.97 body lengths/s) was measured over four consecutive jumps up a 12-inch-diameter tube.

    Source: B. Continuous jump-climbing and mixed… · p. 7

  • Takeaways & Limitations

    The resulting morphology suggests central mass concentration, short distal links, and relatively long reach in front of the central mass for sagittal-plane climbing robots.

    Source: §IV · p. 7

    Trajectory optimization is sensitive to dynamics-model errors and requires manual adjustment; the adjusted open-loop trajectory spends about 45% of a jump settling, and the spines are limited to carpet.

    Source: §IV · p. 7

Abstract

from arXiv · show

Animals such as squirrels and even bears have adapted to rapidly climb up trees and other complex vertical terrain, but achieving comparable agility has been a challenge for climbing robots. Existing robots often use walking gaits and move conservatively to stay in contact with the surface, which limits the range of dynamic maneuvers. In this paper we present a 290 g robot that, to our knowledge, is the first to climb vertically by bounding (with an aerial phase). We leverage a co-design workflow, in which the morphology and trajectory are jointly optimized for fast locomotion, subject to adhesion force limitations seen in spined grippers. The resulting trajectory includes a rapid maneuver that launches the robot vertically, and an aerial reorientation that brings the front grippers back to the surface using the rear leg as an inertial tail. We evaluate the resulting jump forces in 2D force space and demonstrate that the optimized morphology is capable of continuous climbing at a speed of 0.375 m/s (1.97 body lengths/s), and can also achieve ground locomotion and transition to a vertical surface. Our work proposes design insights for the jump-climbing maneuver and serves as an important step toward creating climbing robots with agility on par with that of animals.

I. INTRODUCTION

Jumping and climbing offer complementary ways to cross obstacles and reach steep terrain, but untethered vertical climbing with an aerial phase remains an open challenge. This work introduces a co-designed robot that repeatedly jump-climbs vertical surfaces.

I. INTRODUCTION

  • Jumping clears gaps and obstacles, while climbing provides sustained access to steep surfaces; combining them could help monitor arboreal ecosystems.Both behaviors use environmental contact forces to support and accelerate the robot.Source: §I · p. 1
  • Earlier robots demonstrated vertical walking and other dynamic climbing motions, but examples included continuous contact, special support conditions, or substantial constraints on repeated wall jumps.The cited examples include a 9º-angled air-bearing setup, a 70º slope, and tethered jumps of about 2.5 cm every 6.2 s.Source: §I · p. 1
  • No previous untethered climbing robot was known to scale a vertical wall using an aerial phase, a maneuvering capability seen in squirrel and bear bounding.The authors identify aerial-phase bounding in vertical environments as an open challenge for robots.Source: §I · p. 1
  • The robot repeatedly launches from and reattaches to vertical surfaces, reaching 1.97 body lengths/s and also performing ground jumps and a single-jump transition to a vertical surface.The authors report that this speed is faster by body length than previous spined climbers on a vertical surface.Source: §I · p. 2

II. DESIGN

The robot uses a symmetric four-link, three-joint design with spined grippers. Gripper statics and force tests inform a pivot choice and a conservative feasible-force region for repeated jump-climbing.

II. DESIGN

  • The robot has a symmetric sagittal-plane four-link, three-joint design, with spined grippers on two front arms and one rear leg.A low-power servo adjusts front-arm abduction/adduction to aid detachment and accommodate flat or cylindrical surfaces.Source: §II · p. 2

A. Gripper design and characterization

  • The study uses 3 mm polyester carpet and 2.54 mm-pitch male pin-header spines, choosing an easy-to-grip surface because the focus is robot design rather than spine design.Spines can grip soft natural substrates or interlock with asperities on rough hard surfaces, but those surfaces are not the test substrate here.Source: A. Gripper design and characterization · p. 2
  • The gripper model balances wall-contact forces against robot load, with a moment balance about the pivot determining the spar force and spine adhesion requirement.Moving the pivot closer to the wall, lowering it, or lengthening the spar can reduce the required spine adhesion, up to the point where spar force reaches zero.Source: A. Gripper design and characterization · p. 2
  • Tests of three pivot positions across 400–2000 g loads and angles up to about 135º informed a conservative safe-force region approximated by two linear inequalities.The chosen position's constraints are used for trajectory optimization.Source: A. Gripper design and characterization · pp. 2–3, Figure 4 · p. 3
  • Position 2 was chosen because it places the pivot as close to the wall as packaging allows while remaining high enough to release reliably.Position 1 achieved less negative normal force; position 3 sometimes held extreme-angle loads but varied across trials and could fail to release.Source: A. Gripper design and characterization · p. 3

B. Co-design of trajectory and morphology

Trajectory optimization can enforce adhesion limits, while morphology also affects motion and adhesion needs. The authors therefore jointly search morphology and trajectories for vertical bounding.

B. Co-design of trajectory and morphology

  • Prior work optimized climbing trajectories under adhesion limits and optimized leaping dynamics, while morphology affects inertial reorientation and the forces required for climbing.Co-design methods can jointly optimize motion and physical parameters, but prior examples did not verify trajectory–morphology co-design for completely vertical climbing on hardware.Source: B. Co-design of trajectory and… · p. 3
  • The nested co-design approach searches morphology outside and optimizes a trajectory inside for each candidate, adapting motion to each morphology's dynamics.The authors fix the four-link structure, gripper force constraints, and actuator selection, then optimize link lengths, mass distribution, and joint trajectories.Source: B. Co-design of trajectory and… · p. 3

C. Dynamics

The dynamics model represents the robot as a planar articulated system and derives quantities for its manipulator equation. Those quantities capture inertia, bias forces, and contact forces at the grippers.

C. Dynamics

  • The planar model has five point masses connected by four rigid links, with the rear foot as a floating base and a six-dimensional configuration vector.The configuration is q = [x5, y5, ϕ5, θ4, θ3, θ2].Source: C. Dynamics · p. 3
  • Symbolic kinematics and energies with automatic differentiation produce the mass matrix, bias-force vector, and front- and rear-gripper contact Jacobians.The Jacobians are evaluated at the gripper positions p1 and p5.Source: C. Dynamics · p. 3
  • The manipulator equation uses actuator torques mapped to the actuated joints, the active-contact set, and Cartesian contact forces acting at each gripper.The three torques act at θ4, θ3, and θ2.Source: C. Dynamics · p. 3

D. Trajectory optimization

Trajectory optimization prescribes a five-phase bounding contact schedule and optimizes one jump to maximize height gain per unit time. It enforces contact-force, clearance, periodicity, and impact constraints, then refines the numerical solution from coarse to higher resolution.

D. Trajectory optimization

  • The prescribed schedule progresses from both grippers contacting, to rear-only contact, flight, front-only contact, and both grippers reattached.Each phase has a fixed node count and an optimizable duration.Source: D. Trajectory optimization · p. 4
  • The objective maximizes height gain divided by total jump period, with gripper forces constrained to the safe region at each node.Direct collocation evaluates dynamics at fixed nodes across the five phases.Source: D. Trajectory optimization · p. 4
  • At attachment, the front gripper is constrained to stop, and an impulse decision variable updates the robot’s generalized momentum.The modeled attachment impulses occur at the phase 3–4 and 4–5 transitions.Source: D. Trajectory optimization · p. 4
  • Foot and inner-joint clearance, continuous-jump height periodicity, timestep, torque, and joint-angle bounds constrain feasible trajectories.The trajectory is solved with IPOPT in CasADi, using coarse exploration followed by higher-resolution refinement from interpolated variables.Source: D. Trajectory optimization · p. 4

E. Geometry optimization

The outer-loop search evaluates candidate link lengths and inner-mass distributions using a trajectory optimized for each morphology. The resulting design concentrates mass at the spine, shortens distal links, and favors longer front links; broader sampling reports matching trends.

E. Geometry optimization

  • The optimized morphology concentrates mass at the spine joint, uses short distal links, and has longer combined front than rear links.The search varied four link lengths and three inner masses, optimizing a trajectory for each candidate and using climbing speed as fitness.Source: E. Geometry optimization · p. 4
  • Approximately 3x improvement over the initial generation was achieved by the best morphology/trajectory combination across 50 CMA-ES generations.Progress was faster in earlier generations.Source: Figure 6 · p. 5
  • Across random morphology samples, higher spine mass, shorter distal links, and a bias toward longer front links were more likely to achieve higher climbing speeds.The sampling took over eight hours and did not identify a clear optimum; its trends agreed with the best CMA-ES result.Source: E. Geometry optimization · p. 5

F. Hardware realization

Hardware realization adapts the optimized mass distribution and structure to fabrication and impact demands, then uses updated CAD properties for a final trajectory optimization. The robot’s controller tracks reference joint trajectories without additional sensor feedback.

F. Hardware realization

  • The 46 g battery is placed near the spine motor to concentrate mass there, while other links are hollowed to reduce mass.The optimized geometry and mass distribution are adjusted for fabrication constraints.Source: F. Hardware realization · p. 5
  • Critical joints use stiff carbon-fiber-reinforced nylon, while slender protrusions and electronics enclosures use impact-resistant polyurethane.This material combination balances rigidity for position control with robustness to falls from several body lengths.Source: F. Hardware realization · p. 5
  • The ESP32 commands reference joint angles without additional sensor feedback, and a final high-resolution trajectory is optimized using updated CAD link lengths, masses, and inertia.Motor ESCs track joint angles with a 1000 Hz proportional-derivative loop; the ESP32 sends commands at 500 Hz.Source: F. Hardware realization · pp. 5–6
  • A long spar on each front foot limits foot rotation to prevent spine-hooking failures during landing, while also reducing required spine adhesion force.A stronger return spring was avoided because it would oppose the jump motion.Source: F. Hardware realization · pp. 5–6

A. Single jump-climb

The optimized open-loop trajectory often failed to reattach reliably, so manual tuning traded jump speed for dependable wall catches. Force-plate measurements show phase-dependent loading and brief impact bounce, including associated joint oscillation.

A. Single jump-climb

  • 25 of 30 trials with the manually tuned trajectory caught both front and rear grippers, compared with 1 of 30 optimized open-loop trials landing ready for another jump.The optimized reference caught with the front grippers in 12 of 30 trials; tuning extended settling time and simplified hip motion to reduce rapid direction changes during aerial reorientation.Source: A. Single jump-climb · p. 6
  • Joint torques reached 0.25 Nm during pushoff, while shoulder, spine, and hip joints produced 0.43 J, 0.61 J, and 0.33 J of mechanical work, respectively.Joint tracking was generally underdamped but consistent across trials.Source: A. Single jump-climb · p. 6
  • Gripper forces stayed within the safe region except briefly before release, when spine unhooking dynamics and a conservative safe-region approximation may explain the excursions.During experiments, both grippers bounced at landing and exerted brief positive normal force, more pronounced at the rear gripper.Source: A. Single jump-climb · pp. 6–7
  • The landing bounce is also visible as shoulder and spine joint oscillation during phases 4 and 5.Figure 8 plots optimized and manually adjusted trajectories alongside a representative hardware tracking trial and average front and rear gripper forces from five trials.Source: Figure 8 · p. 6, A. Single jump-climb · p. 7

B. Continuous jump-climbing and mixed locomotion

The robot repeatedly jump-climbs flat and cylindrical vertical surfaces, while also demonstrating horizontal jumps and a single-jump transition onto a vertical surface.

B. Continuous jump-climbing and mixed locomotion

IV. DISCUSSION

The discussion identifies morphology features and actuator allocation relevant to climbing performance, while noting limitations in trajectory optimization, gripper scope, steering, and simulation fidelity.

IV. DISCUSSION

  • A concentrated central mass, short distal links, and relatively long reach ahead of the central mass are proposed as useful morphology features for sagittal-plane climbing.The authors suggest spine mass and longer front links may extend front-foot reach during flight, while spine mass leaves lighter limbs to accelerate; Fig. 10 illustrates the reach effect of mass distribution.Source: §IV · p. 7, Figure 10 · p. 7
  • Shorter distal links can increase gripper force for a given motor torque, while a shorter wall distance reduces pitch-back moment and required adhesion.Source: §IV · p. 7
  • The spine motor produced the most work and saturated in torque below 60 rad/s, suggesting future designs could allocate it more actuator mass and trade speed for torque.Possible approaches include gear reductions, linkage mechanisms, or larger motors.Source: §IV · p. 7
  • The current workflow is model-sensitive and manually adjusted, and its spines and force constraints are specific to carpet.The open-loop settling phases occupy about 45% of a jump period; the robot also tends to veer without steering, while higher-fidelity 3D simulation could improve real-world transfer.Source: §IV · p. 7
  • Further study of jump-climbing may reveal principles for dynamic vertical locomotion and inform robophysical models of arboreal animal locomotion.Source: §IV · p. 7
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