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Linear Matrix Inequalities for Physically-Consistent Inertial Parameter Identification: A Statistical Perspective on the Mass Distribution

Patrick M. Wensing, Sangbae Kim, Jean-Jacques Slotine

arXiv:1701.04395v3cs.RO

TL;DR

Accurate inertial parameters are needed for model-based control, but identification must balance physical plausibility with optimization tractability. The paper introduces LMI constraints based on mass-distribution covariance and moment theory, then applies them to MIT Cheetah 3 leg identification, where tighter constraints yield accurate models more rapidly and reduce overfitting.

  • Problem

    Inertial-parameter identification needs constraints that ensure physical plausibility while retaining convex optimization and robustness to noisy data.

  • Method

    The paper formulates physical-consistency and ellipsoidal density-realizability constraints as LMIs using covariance of the mass distribution and classical moment theory.

  • Results

    Tighter constraints produced accurate models more rapidly as samples increased and yielded lower validation error, while MIT Cheetah 3 validation RMS errors were 1.48, 1.69, and 1.16 Nm for the ab/ad, hip, and knee joints.

  • Takeaways & Limitations

    The covariance-based LMI formulation combines physically realistic inertial parameters with convex identification and semidefinite-programming global optimality.

Abstract

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With the increased application of model-based whole-body control in legged robots, there has been a resurgence of research interest into methods for accurate system identification. An important class of methods focuses on the inertial parameters of rigid-body systems. These parameters consist of the mass, first mass moment (related to center of mass location), and rotational inertia matrix of each link. The main contribution of this paper is to formulate physical-consistency constraints on these parameters as Linear Matrix Inequalities (LMIs). The use of these constraints in identification can accelerate convergence and increase robustness to noisy data. It is critically observed that the proposed LMIs are expressed in terms of the covariance of the mass distribution, rather than its rotational moments of inertia. With this perspective, connections to the classical problem of moments in mathematics are shown to yield new bounding-volume constraints on the mass distribution of each link. While previous work ensured physical plausibility or used convex optimization in identification, the LMIs here uniquely enable both advantages. Constraints are applied to identification of a leg for the MIT Cheetah 3 robot. Detailed properties of transmission components are identified alongside link inertias, with parameter optimization carried out to global optimality through semidefinite programming.

I. INTRODUCTION

Accurate inertial models are important for model-based whole-body control, motivating identification methods that combine physical plausibility with convex optimization. This paper formulates physical-consistency constraints as LMIs and connects them to covariance and classical moment theory.

  • Accurate dynamic models remain important because whole-body control performance depends on them.
  • Prior identification work addressed trajectory design, noise robustness, floating-base systems, and constrained optimization for physically realistic parameters.
  • The paper expresses physical-consistency constraints as LMIs, reformulating prior manifold constraints into convex constraints that support globally optimal least-squares identification.
  • Physical consistency is interpreted through covariance of the mass distribution, enabling bounding-ellipsoid constraints derived from the classical problem of moments.
  • LMI-representable constraints can be enforced with semidefinite programming, which provides guarantees of global optimality.

B. Rigid-Body Dynamics

Rigid-body dynamics are linear in standard inertial parameters, enabling least-squares identification but not guaranteeing physical realizability without additional constraints. The paper characterizes realizability through density measures and convex cones, distinguishing rotational inertia from moments of the mass distribution.

  • Robot dynamics can be written linearly in the inertial parameters through a regressor matrix, enabling least-squares identification.
  • Each body's standard parameter vector contains mass, first mass moment, and six rotational-inertia entries.
  • Unconstrained least-squares optimization may achieve global optimality while producing parameters that do not correspond to any physical system.
  • Physical inertial parameters arise from a nonnegative spatial density, and density realizability requires that a density supported on a specified set reproduce the parameters.
  • The moments of inertia are not moments of the density, so physical plausibility is addressed more directly through mass-distribution moments.
  • The set of positive, density-realizable inertial parameters is a convex cone, with some cases admitting LMI representations without discretization.

III. PREVIOUS RESULTS

Physical consistency is studied for a single rigid body using the rotational inertia about the center of mass and its relation to inertia about a body-fixed origin.

  • The section focuses on physical consistency for one rigid body and relates center-of-mass rotational inertia to inertia about a body-fixed origin.

A. Physical Semi-consistency: An LMI Parameterization

Physical semi-consistency requires positive mass and positive-definite rotational inertia, but full physical consistency additionally imposes triangle inequalities on the principal moments. The earlier manifold parameterization enforces these conditions through nonlinear optimization without guaranteeing global optimality.

  • A. Physical Semi-consistency: An LMI Parameterization: The positive-definite mass and inertia conditions alone do not ensure physical consistency.
  • A. Physical Semi-consistency: An LMI Parameterization: Physical semi-consistency requires m(π) > 0 and ¯I_C(π) ≻ 0, defining the set P of semi-consistent inertial parameters.
  • A. Physical Semi-consistency: An LMI Parameterization: Full physical consistency additionally requires the principal moments J1, J2, and J3 to satisfy all three triangle inequalities.
  • A. Physical Semi-consistency: An LMI Parameterization: The manifold parameterization represents physically consistent inertias using positive mass, a rotation, positive principal moments, triangle inequalities, and center of mass.
  • A. Physical Semi-consistency: An LMI Parameterization: Even for one rigid body, this parameterization produces nonlinear manifold optimization requiring custom solvers and offering no global-optimality guarantee.

IV. CONTRIBUTION: AN LMI FOR PHYSICALLY CONSISTENT INERTIAL PARAMETERS

The paper expresses physical-consistency conditions as an LMI over inertial parameters rather than using a manifold parameterization. This enables convex identification of physically plausible parameters.

  • IV. CONTRIBUTION: AN LMI FOR PHYSICALLY CONSISTENT INERTIAL PARAMETERS: Triangle inequalities can be expressed as an LMI over inertial parameters, avoiding manifold parameterization and enabling convex optimization for plausible identification.

A. A matrix inequality for triangle inequalities on ¯I C

The triangle inequalities on rotational inertia eigenvalues are rewritten as a trace-versus-maximum-eigenvalue condition and then as a matrix inequality. This reformulation is mathematically equivalent but initially lacks intuitive interpretation.

  • A. A matrix inequality for triangle inequalities on ¯I_C: The principal-moment triangle inequalities are rewritten as J1 + J2 + J3 ≥ 2Ji for each i = 1, ..., 3.
  • A. A matrix inequality for triangle inequalities on ¯I_C: Because Ji are the eigenvalues of ¯I_C, the rewritten condition can be expressed using the trace and maximum eigenvalue of ¯I_C.
  • A. A matrix inequality for triangle inequalities on ¯I_C: The resulting matrix inequality is mathematically equivalent to the rotational-inertia triangle inequalities but has limited intuitive meaning in this form.

B. The Density-Weighted Covariance of a Rigid Body

The paper interprets rotational-inertia consistency through the density-weighted covariance of the mass distribution. This covariance shares principal axes with rotational inertia, converts triangle inequalities into positive-semidefiniteness, and supports ellipsoidal and moment-based geometric constraints.

  • B. The Density-Weighted Covariance of a Rigid Body: ΣC and ¯I_C share principal axes, while each rotational-inertia eigenvalue equals the sum of the two covariance eigenvalues orthogonal to that axis.
  • B. The Density-Weighted Covariance of a Rigid Body: The covariance ellipsoid Eπ captures the mass distribution’s second-order shape and is invariant to uniform mass scaling because covariance is normalized by mass.
  • B. The Density-Weighted Covariance of a Rigid Body: For an infinitely thin plate, one covariance eigenvalue is zero, making the distribution degenerate and one triangle inequality tight.
  • B. The Density-Weighted Covariance of a Rigid Body: The covariance interpretation states that Σ ⪰ 0 is equivalent to positive rotational inertia satisfying the triangle inequalities.
  • B. The Density-Weighted Covariance of a Rigid Body: Physical consistency is equivalent to positive mass and positive-semidefinite density-weighted covariance ΣC, with positive definiteness for non-degenerate rigid bodies.
  • B. The Density-Weighted Covariance of a Rigid Body: The covariance connection is new in this formulation and links rigid-body mass measures to probability covariance and the classical problem of moments.

C. An LMI Representation of Physical Consistency

The paper reformulates physical consistency of rigid-body inertial parameters as LMIs by expressing the relevant conditions through a covariance-based moment matrix. This representation preserves the triangle-inequality structure while reducing the constraint dimension and enabling convex identification.

  • LMI construction: The moment matrix J(π) is linear in π, converting the covariance-based condition into an LMI over the inertial parameters.The covariance condition itself is not linear in π, motivating the second-moment matrix construction.
  • LMI representation: Theorem 3 establishes that the physically consistent parameter set is strictly LMI representable through the pseudo-inertia matrix J(π).The pseudo-inertia matrix contains moments through second order and is linear in the inertial parameters.
  • Covariance interpretation: Physical consistency is equivalent to positive semidefiniteness of the rigid-body mass-distribution covariance, linking inertial-parameter plausibility to first- and second-moment realizability.The covariance interpretation connects the mechanical condition to the classical moment conditions used in probability and statistics.
  • Physical consistency: The LMI on J(π) captures the triangle inequalities that distinguish full physical consistency from merely positive-definite kinetic-energy metrics.These additional inequalities impose structure beyond positive definiteness of the spatial inertia metric.
  • Computational form: Using a 4 × 4 pseudo-inertia matrix instead of a 6 × 6 spatial-inertia matrix provides computational benefits when enforcing the physical-consistency LMIs.The smaller representation retains the relevant moment structure while making the constraint lower dimensional.

V. CONTRIBUTION: LMI CONSTRAINTS FOR INERTIAL PARAMETER REALIZABILITY ON ELLIPSOIDS

The paper extends covariance-based moment LMIs from physical consistency to mass-distribution realizability inside bounding ellipsoids and unions of ellipsoids. These constraints account for both center-of-mass location and second moments, with tighter and more efficient conditions than a looser 4 × 4 LMI.

  • Ellipsoidal realizability: Center-of-mass containment alone is insufficient for ellipsoid realizability because large second moments can imply mass outside the bounding ellipsoid.As the center of mass approaches the ellipsoid boundary, the rigid body must degenerate to a point mass.
  • Ellipsoidal realizability: Theorem 4 characterizes density realizability on an ellipsoid through moment conditions and guarantees an equivalent representation by four point masses inside the ellipsoid.The point-mass representation provides a finite certificate for any realizable inertial-parameter set.
  • Ellipsoidal realizability: Second-moment information yields the tighter 1D linear constraint Tr(J(π) Q) ≥0, compared with the looser 4 × 4 LMI.The paper states that the tighter condition is also more computationally efficient to enforce.
  • Illustration: Figure 3 varies the covariance ellipsoid’s center and shape to illustrate their roles in determining whether a mass distribution is realizable within the bounding ellipsoid.When a distribution exists, the figure shows four point masses as guaranteed by Theorem 4.
  • Union of ellipsoids: For a body contained in a union of ellipsoids, realizability is equivalent to decomposing its inertial parameters into component parameters, each realizable on one ellipsoid.This extends the single-ellipsoid characterization to more complex bounding shapes.
  • Implications: The nested convex constraints are expected to improve convergence benefits for higher-degree-of-freedom robots because the relative volume of the physically consistent parameter set decreases exponentially with body count.The passage connects tighter constraints to faster convergence in online estimation and adaptive control.

VI. EXPERIMENTAL VALIDATION

The proposed constraints were used to identify a six-body MIT Cheetah 3 leg model including links, rotors, and transmission effects. Validation shows accurate torque prediction, while tighter constraints reduce the samples needed for accurate, physically realistic models.

  • Experimental setup: The identified model includes three links, three rotors, and diagonal viscous and Coulomb friction terms for the gearbox transmission.The leg has three 3-DoF joints, high-inertia rotors, and 10.6:1 gearboxes.
  • Validation results: 1.46 Nm overall RMS error was achieved when predicting motor torques on validation data distinct from training.The identified Coulomb friction was Bc = diag(3.12, 1.25, 0.95) Nm.
  • Validation results: Validation RMS errors were 1.48, 1.69, and 1.16 Nm for the ab/ad, hip, and knee joints, respectively.The errors were computed on the next 10,000 samples after identification.
  • Component contributions: The dominant contributions of link inertias, rotor inertias, and friction depend on both motion and joint, motivating identification of all modeled components.The relative effects are specific to the robot and transmission.
  • Sample-size analysis: Tighter constraints empirically produced accurate models more rapidly as training samples increased and yielded lower validation error through reduced overfitting.These constraints also produced physically realistic model parameters.

VII. CONCLUSIONS

The paper formulates physical consistency of rigid-body inertial parameters through LMIs based on mass-distribution moments and covariance. It also extends these ideas to density realizability within bounding ellipsoids while distinguishing the results from stochastic modeling.

  • The paper introduces LMIs that rigorously characterize physically consistent rigid-body inertial parameters.
  • The formulation uses a pseudo-inertia matrix and density-weighted covariance to characterize physical consistency.
  • Connections to the classical problem of moments yield LMI constraints for density realizability on an ellipsoid.
  • The resulting constraints define a cone containing physically consistent inertial parameters with certainty for a given rigid-body bounding ellipsoid.
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