Modeling and Generative-AI-Based Design of Load-Adaptive Gravity Balancing Mechanisms

• g denotes gravitational acceleration, while l denotes the lever-arm length and spring mounting height.Source: NOMENCLATURE · p. 2
• M is payload mass; p and q denote passive and active system variables, respectively, and t is an alternative representation of p.Source: NOMENCLATURE · p. 2
• U denotes the LA-GBM potential field, with Ug for payload gravitational energy and Uc for the compensating field; Θ denotes the payload-mass range.Source: NOMENCLATURE · p. 2
• πx(Ω) denotes the projection of the feasible domain onto the active/passive-variable product space, and Ωx is its passive-variable fiber over q.Source: NOMENCLATURE · p. 2

1 INTRODUCTION

• GBMs can reduce actuator loads, improve energy efficiency, and enhance human–robot interaction safety; implementations commonly use counterweights, springs, or buoyancy.Source: §1 · p. 2
• Payload-adaptive GBMs vary spring preload, transmission ratio, or force-application point to accommodate different loads.Source: §1 · p. 2
• GBM design couples the kinematics needed for desired motion with static equilibrium, so elastic-element design variables generally cannot be chosen independently.Source: §1 · p. 2
• Prior approaches derive element arrangements for prescribed linkages, optimize joint arrangements, or optimize load-dependent elements and balancing springs for LA-GBMs.Source: §1 · p. 2
• Most existing design theories prescribe an architecture or available elements, restricting solutions; this is especially limiting for LA-GBMs with more functional constraints.Source: §1 · p. 2
• The study proposes generalized potential functions as intermediate representations and uses AI to explore LA-GBM designs without fixing the mechanism architecture in advance.Source: §1 · pp. 2–3

2 THEORY

• The theory treats LA-GBM potential fields as a general class satisfying affine and factorized forms, independent of mechanism architecture and constituent elements.These forms are derived from the defined mechanism scope and conditions.Source: §2 · p. 3

2.1 Definitions of Variables and Sets

• Active variables q are user-specified, passive variables p are determined by equilibrium, and the feasible domain Ω constrains their joint states.The payload-mass range Θ represents the masses to which the mechanism can adapt.Source: §2.1 · p. 3
• The projection πq(Ω) collects admissible active-variable values, while the fiber Ωq gives feasible passive states when q is fixed.These sets let the analysis treat active and passive variables independently.Source: §2.1 · p. 3

2.2 Definition and Scope of the Mechanisms Considered

• The considered LA-GBMs use locking elements to constrain selected variables during load adaptation and must satisfy both potential-field conditions.The first condition describes a local minimum reached by load adaptation and a constant-potential path during balancing; the second requires lifting the payload during balancing.Source: §2.2 · pp. 4–5
• In the lever-arm example, fixing q = 0 lets the slider move to p = Mg/k, after which fixing p keeps U constant as q varies.This example satisfies the stated conditions for load adaptation and gravity balancing.Source: §2.2 · pp. 5–6

2.3 Affine Formulation

• The affine form represents the potential as a superposition of payload gravitational energy and a compensating potential generated by mechanism elements.The compensating potential is described by a function of state variables only when mechanism parameters cannot be adjusted as payload changes.Source: §2.3 · p. 6
• The form extends quasi-static mechanical-energy conservation to account for payload mass, with arbitrary smooth functions f(q, p) and h(q, p).The cited passage identifies this extension with the conservation property satisfied by GBMs.Source: §2.3 · p. 6
• Payload gravitational potential is expressed as payload mass multiplied by a term corresponding to the payload’s vertical displacement.The passage identifies that displacement term as f(q, p).Source: §2.3 · p. 6
• Substituting the payload and compensation potentials yields the general potential form for an LA-GBM handling an unknown payload.Under that form, condition (4) can also be expressed in terms of f.Source: §2.3 · p. 6

2.4 Factorized Formulation

• Load adaptation is represented as a transition toward a potential equilibrium point, while gravity balancing is represented as a transition along a potential contour.The former holds the active variable fixed; the latter holds the passive variable fixed.Source: §2.4 · p. 6
• The factorized form is parameterized by functions ψ(t; M) and B(q, t; M), with t an alternative representation of passive variable p.The passage identifies ψ as real-valued and B as an m-by-n matrix-valued function.Source: §2.4 · p. 6
• The form’s function ψ depends on t and payload M, while B depends on active variables q, t, and payload M.Both functions are described as arbitrary.Source: §2.4 · p. 6

2.2 Scope and Conditions

• The scope focuses on LA-GBMs employing locking elements.The passage points to Fig. 3.Source: §2.2 · p. 7
• Vertical mobility along the valley is identified as a condition in the scope.Source: §2.2 · p. 7

2.4 Factorized Formulation

• The design flow uses generative AI to propose candidate potential functions from the affine and factorized forms, then develops them into mechanism structures.Source: Figure 6 · p. 7
• The derivation assumes a positive-definite Hessian near p = p* satisfying t = 0, together with a regular variable transformation.These conditions are stated for the derivation of Eq. (16).Source: §2.4 · p. 8
• The potential is first expressed as a series in polynomial form in q and t.Source: §2.4 · p. 8
• The expansion separates the potential into ϕ(q; M), ψ(t; M), and a cross-term qᵀB(q, t; M)t.Source: §2.4 · p. 8
• Load adaptation is interpreted as movement toward a potential equilibrium with q fixed, while gravity balancing follows a contour with p fixed.At t = 0, the potential reaches the local minimum U*(M) = U(q, 0; M).Source: §2.4 · p. 8
• Imposing U(q, 0; M) = U*(M) eliminates the cross-term and requires ϕ(q; M) to be constant.Source: §2.4 · p. 8
• The constant ϕ(M) can be absorbed into ψ by mapping ϕ(M) + ψ(q; M) to ψ(q; M), yielding the factorized form.Source: §2.4 · p. 8

3.1 Design Flow

• The design flow gives generative AI the affine and factorized forms and payload-energy variation condition, then verifies its candidate functions analytically.The active and passive state dimensions m and n are design parameters; AI serves as a candidate-search tool, not the direct designer.Source: §3.1 · p. 8

3.2 Example of Potential-Field Generation

• In the example, Eq. (21), constructed with arbitrary constant matrix A and arbitrary constant vector b satisfying Ab ≠ 0, is confirmed to admit both the affine form of Eq. (11) and factorized form of Eq. (16).Source: §3.2 · pp. 8–9
• The AI-generated candidate satisfies the requirements of an LA-GBM, including the factorized-form condition and the required variation in payload potential energy.The analysis concludes that the function search successfully generated a potential field meeting the LA-GBM requirements.Source: §3.2 · p. 9

3.3 Example of Mechanism Design

• The example potential can be assembled from a spring for the p² term, a counterweight on a qp-producing linkage, and a payload on a q + p-producing linkage.The three elements are connected in parallel; the construction assumes the terms are converted to energy units using appropriate constants or coefficients.Source: §3.3 · p. 10, Figure 8 · p. 10
• Fabrication and experimental validation of the example design are left for future work.Source: §3.3 · p. 10

4 DISCUSSION

• Only a subset of mathematically generated fields can be reproduced with existing mechanical elements, and an even smaller subset is expected to meet practical implementation requirements.Those requirements include sufficiently large workspace and ease of fabrication.Source: §4 · pp. 10–11
• The qp linkage example is mechanically realizable, but its configuration and workspace are limited.Source: §4 · p. 11
• The function-first pipeline separates potential generation from mechanism construction; adding function-representation methods and practical constraints to AI design conditions may mitigate this challenge.Source: §4 · p. 11

5 CONCLUSION

• The method formulates necessary conditions for LA-GBM potential fields as affine and factorized forms, then uses them as generative-AI design requirements.The conditions are generalized for many existing LA-GBMs with locking function.Source: §5 · p. 11
• A potential field for a nontrivial LA-GBM with (m + n)-dimensional state variables was generated, and synthesis was demonstrated for q ∈ R and p ∈ R.The synthesis demonstration used a specific example based on the generated potential field.Source: §5 · p. 11
• Mathematical compliance does not guarantee a simple realizable mechanism; implementation constraints, fabrication, and experimental performance evaluation remain future work.Workspace and mechanism complexity are among the practical constraints to consider.Source: §5 · p. 11