Hierarchical Progressive Parameter Identification Strategy

• The paper introduces a two-stage framework that first estimates hydraulic-cylinder friction and stiffness before identifying a minimal rigid-body parameter set.
• It combines Denavit–Hartenberg modeling, Newton–Euler dynamics, and a Stribeck friction model to reduce 78 parameters to a robust set of 18 base parameters.
• Experimental validation shows residual standard deviations below 0.4 Nm across all joints, demonstrating improved conditioning and estimation accuracy.

Searching arXiv for the specified paper and closely related uses of hierarchical/progressive parameter-identification terminology. Hierarchical Progressive Parameter Identification Strategy denotes, in the context of a hydraulically driven curtain wall installation robotic arm, a two-stage dynamic-identification methodology that first estimates hydraulic-cylinder friction and stiffness parameters and then identifies the robot’s minimal rigid-body parameter set under those previously identified friction terms. The strategy is formulated on a composite parametric system that combines a Denavit–Hartenberg model based on measured structural parameters, Newton–Euler rigid-body dynamics, hydraulic cylinder dynamics, and a Stribeck friction model. Its estimation pipeline couples high-signal-to-noise-ratio excitation design with least-squares estimation, and experimental validation reports residual standard deviations below $0.4\ \mathrm{Nm}$ between theoretical and measured joint torques for all six joints (Liu et al., 23 Jul 2025).

1. Problem setting and conceptual basis

The strategy was introduced for a curtain wall installation robotic arm intended for construction applications, where traditional methods are stated to fail to meet modern demands for efficiency and quality. Within that setting, curtain wall installation is treated as a critical component of construction projects, and the identification problem is framed around a hydraulically driven robotic arm whose dynamics are influenced jointly by rigid-body effects and hydraulic actuation nonlinearities (Liu et al., 23 Jul 2025).

The defining feature of the method is its hierarchy and progression. Stage 1 targets hydraulic-cylinder friction and stiffness parameters; Stage 2 targets rigid-body link parameters and joint friction contributions in a reduced robot model. The paper states that, in this progression, the separately identified hydraulic-cylinder friction reduces the unknowns in the joint-torque regression, lowers collinearity, and improves conditioning for the final least-squares solve. It also states that hierarchical progression yields smaller residuals and more robust convergence than a “one-shot” joint identification of all 78 parameters together (Liu et al., 23 Jul 2025).

This makes the strategy structurally distinct from monolithic identification pipelines. Rather than solving all coupled parameters simultaneously, it isolates the hydraulic-cylinder subproblem first and then carries forward the identified friction quantities as known constants when estimating the robot’s minimal inertial parameterization. The progression is therefore both physical, because it follows subsystem boundaries, and numerical, because it modifies the conditioning of the regression solved in the second stage.

2. Dynamic model and parameterization

The hydraulic-cylinder submodel is built from a force-balance equation with stiffness, damping, and Stribeck friction:

$$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$

The Stribeck friction is written in linearized form as

$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$

For Stage 1, the reported measurables are piston displacement $x(t)$, chamber pressures $p_1,p_2$, known damping $c$, known mass $m$, and designed load $F_l(t)$. The corresponding linear-in-parameters regression is reported as

$$y(t)=p_1A_1-p_2A_2-F_l(t)-c\cdot \dot{x}(t) = [\dot{x},x,\operatorname{sgn}(\dot{x}),\dot{x},\dot{x}^{1/3}] \cdot [m,K,f_c,f_v,f_s]^\top+d(t),$$

with $\theta_{\text{cyl}}=\{m,K,f_c,f_v,f_s\}$ identified via recursive least squares (Liu et al., 23 Jul 2025).

The 6-DOF robotic-arm dynamics are formulated from standard D–H parameters and Newton–Euler recursion:

$$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$0

where $$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$1 is the inertia matrix, $$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$2 is the Coriolis and centrifugal term, $$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$3 is the gravity term, and

$$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$4

For parameter reduction, the link-by-link model is first written in linear-in-parameters form,

$$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$5

with

$$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$6

Columns of the global regressor $$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$7 that are all zero or linearly dependent are removed. Using the SymPybotics tool, the total of 78 parameters is reduced to 18 independent base parameters $$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$8, yielding

$$m\,\ddot{x} + c\,\dot{x} + K\,x + F_d(\dot{x}) = p_1A_1 - p_2A_2 - F_l .$$9

This minimal-parameter formulation is central to the second stage because the previously identified friction terms do not appear in $$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$0 during final calibration (Liu et al., 23 Jul 2025).

3. Two-stage identification workflow

The strategy can be summarized as a staged estimation architecture.

Stage	Target parameters	Estimation method
1	$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$1 for each hydraulic cylinder	Recursive least squares
2	Minimal rigid-body parameter set $$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$2	Ordinary least squares

In Stage 1, the discrete regression at sample $$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$3 is

$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$4

The recursive least-squares updates are

$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$5

$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$6

$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$7

After convergence, the identified $$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$8 are stored and inserted into the robot dynamics. Stage 2 then assembles data across all joints and time samples into

$$F_d(\dot{x}) = f_c\cdot \operatorname{sgn}(\dot{x}) + f_v\cdot \dot{x} + f_s\cdot |\dot{x}|^{1/3}.$$9

with

$x(t)$0

The closed-form least-squares estimate is

$x(t)$1

The paper’s algorithmic outline is explicitly progressive: initialize $x(t)$2; identify cylinder parameters over the first dataset; store the friction parameters from $x(t)$3; generate Fourier-based excitation $x(t)$4; collect joint torques and kinematic data; build $x(t)$5 with known friction substitutions; solve for $x(t)$6; and return all identified parameters (Liu et al., 23 Jul 2025).

A common misunderstanding is to treat the method as merely a least-squares fit. The source instead presents least squares as the estimation engine inside a broader hierarchy: subsystem decomposition, parameter freezing, base-parameter reduction, and constrained excitation design are all integral to the identification strategy.

4. Excitation design and identifiability enhancement

The method does not rely on arbitrary motion commands. It designs high-signal-to-noise-ratio displacement excitation signals for hydraulic cylinders and combines Fourier series to construct optimal excitation trajectories that satisfy joint constraints and effectively excite the characteristics of each parameter in the minimal parameter set (Liu et al., 23 Jul 2025).

For each joint $x(t)$7, the trajectory is represented by an $x(t)$8-term truncated Fourier series:

$x(t)$9

$p_1,p_2$0

$p_1,p_2$1

The amplitudes are constrained to satisfy joint limits and smooth boundary conditions:

$p_1,p_2$2

together with

$p_1,p_2$3

The objective is to choose $p_1,p_2$4 so as to minimize $p_1,p_2$5, where $p_1,p_2$6 is the assembled observation matrix (Liu et al., 23 Jul 2025).

This design choice is significant because it ties identifiability directly to the regression conditioning. The paper therefore treats excitation synthesis not as a peripheral implementation detail but as a precondition for reliable recovery of the base parameter vector. The broader implication is that hierarchy in estimation is paired with hierarchy in excitation: the inputs are crafted to expose the parameters that each stage is intended to recover.

5. Experimental validation and reported accuracy

The hydraulic-cylinder identification experiments are reported at a sampling rate of 50 Hz, under no external load, with $p_1,p_2$7. Stage 1 yields Stribeck parameters in units of N·m, including the following reported values: Joint 1 has $p_1,p_2$8, $p_1,p_2$9, $c$0; Joint 2 has $c$1, $c$2, $c$3; Joint 6 has $c$4, $c$5, $c$6. The paper states that Stribeck model curves are obtained for each joint (Liu et al., 23 Jul 2025).

For Stage 2, the identified result is a minimal 18-parameter vector $c$7. The comparison between calculated $c$8 and measured $c$9 is reported as an excellent match. Residual standard deviations, in Nm, are given as follows: Joint 1, 0.226; Joint 2, 0.289; Joint 3, 0.378; Joint 4, 0.193; Joint 5, 0.179; Joint 6, 0.336. The paper emphasizes that all residual standard deviations are below $m$0, and interprets this as demonstrating high identification accuracy (Liu et al., 23 Jul 2025).

Two methodological claims are attached to these results. First, separating hydraulic-cylinder friction removes one major source of nonlinearity and strong coupling from the full regression, thereby lowering collinearity and improving conditioning. Second, reducing the model from 78 parameters to 18 base parameters reduces computational burden and over-fitting risk. These claims describe the reported mechanism by which the hierarchical progression attains smaller residuals and more robust convergence than non-hierarchical least-squares methods in the literature (Liu et al., 23 Jul 2025).

6. Position within the broader literature on hierarchical and progressive estimation

The phrase “hierarchical progressive” is used in several arXiv works across distinct technical domains, but with a recurring structural motif: coarse or foundational components are estimated first, then higher-order, residual, or more localized parameters are activated under constraints.

In learned lossless image compression, the framework built around the Hierarchical Parallel Autoregressive ConvNet uses “Spatially-Aware Rate-Guided Progressive Fine-tuning” to freeze pre-trained weights $m$1, introduce low-rank adapters $m$2, compute a per-patch rate map, and progressively expand a contiguous high-bitrate region $m$3 for adaptation. The paper explicitly states that early steps identify the minimal set of parameters needed to explain the most “surprising” regions, calling this a hierarchical progressive identification of $m$4 (Li et al., 14 Nov 2025).

In multi-qubit quantum noise characterization, the “Hierarchical Progressive Optimization” framework organizes Pauli-transfer parameters into a foundation layer and a residual layer, freezes low-weight topologies through a combinatorial projection mask, and optimizes only high-weight residual correlations. The reported effect is a complexity reduction from $m$5 to $m$6, with 96.3% parameter compression on a 5-qubit system and state-fidelity recovery from 0.7431 to 0.9381 in the 10-qubit HHL mitigation experiment (Ge et al., 19 Apr 2026).

In latent tree models, “progressive EM” estimates parameters bottom-up on small submodels, keeping previously fitted conditional probability tables fixed; this yields a reported 10× to 100× speed-up while preserving nearly the same held-out likelihood and topic quality in hierarchical topic detection (Chen et al., 2015). In multi-resolution online deterministic annealing, a progressive partitioning of the data space is coupled to online stochastic approximation and a tree structure so that complexity increases gradually and dense regions are refined first (Mavridis et al., 2022).

A plausible implication is that the robotic-arm strategy belongs to a broader methodological family rather than representing a universal standalone algorithm. Across these works, “hierarchical progressive parameter identification” consistently denotes staged estimation under parameter freezing, structural decomposition, or restricted activation. What varies is the organizing principle: physical subsystem in hydraulic robotics, spatial region in image compression, Pauli weight and Hamming neighborhood in quantum noise modeling, subtree locality in latent trees, and partition depth in deterministic annealing.