cs.ROFeb 18, 2026
SaveSystem Identification under Constraints and Disturbance: A Bayesian Estimation Approach
Organizations: Robot Motor Intelligence (RoMI) Lab, Heriot-Watt University, U.K. · Informatics School, University of Edinburgh, U.K.
Abstract
We introduce a Bayesian system identification (SysID) framework for jointly estimating robot's state trajectories and physical parameters with high accuracy. It embeds physically consistent inverse dynamics, contact and loop-closure constraints, and fully featured joint friction models as hard, stage-wise equality constraints. It relies on energy-based regressors to enhance parameter observability, supports both equality and inequality priors on inertial and actuation parameters, enforces dynamically consistent disturbance projections, and augments proprioceptive measurements with energy observations to disambiguate nonlinear friction effects. To ensure scalability, we derive a parameterized equality-constrained Riccati recursion that preserves the banded structure of the problem, achieving linear complexity in the time horizon, and develop computationally efficient derivatives. Simulation studies on representative robotic systems, together with hardware experiments on a Unitree B1 equipped with a Z1 arm, demonstrate faster convergence, lower inertial and friction estimation errors, and improved contact consistency compared to forward-dynamics and decoupled identification baselines. When deployed within model predictive control frameworks, the resulting models yield measurable improvements in tracking performance during locomotion over challenging environments.
Figures & tables
Fig. 1 : The Unitree B1 quadruped robot performing a step-up maneuver. The leg includes a reduction gear and a four-bar linkage, both explicitly modeled as constraints within our SysID framework. The highlighted cross-section reveals the internal linkage bars, whose inertial properties and actuator-side friction effects are jointly identified to improve physical consistency and model fidelity.To watch the video, click the picture or see https://youtu.be/ToTYp7mCkOQ .
Fig. 2 : Overview of the our Bayesian optimization pipeline for system identification. Proprioceptive measurements (e.g., encoders, IMU) and optional exteroceptive measurements (e.g., visual odometry/ICP) are synchronized and used to jointly estimate the state trajectory, disturbances, actuation effects (including friction), and physical parameters. Inertial parameters are represented with a physically consistent parametrization, and process noise is projected onto the constraint-consistent tangent space to respect implicit motion constraints. Dynamic consistency is enforced through explicit inverse-dynamics and motion constraints (contacts, closed-loop kinematics, hybrid/reset events), together with general parameter equality/inequality constraints. Energy-based observations provide additional identifiability of friction by enforcing consistency between actuation power, dissipated energy, and changes in mechanical energy. The resulting structured KKT system is solved efficiently with an equality-constrained Riccati approach, enabling scalable SysID over long, multi-rate datasets and deployment in control pipelines (e.g., MPC).To watch the video, click the picture or see https://youtu.be/ToTYp7mCkOQ .
Fig. 3 : Top: Unitree B1 robot traversing a rough terrain. Bottom: Kangaroo robot walking on a sidewalk with an identified model. Planning, control, and estimation for these legged systems require accurately capturing contact dynamics and closed-loop mechanism effects inherent to multibody systems.
Fig. 4 : Illustration of Kangaroo’s leg kinematic structure and closed-loop modeling. (a) The actual mechanical design of the leg, including the actuator assembly, ankle linkage, and knee bars. The corresponding kinematic graph highlights the multiple closed-loop constraints present in the mechanism, resulting in a connectivity graph with cycles that prevent it from being a tree. (b) Representation of the same leg using an equivalent spanning tree, obtained by opening the closed loops. This process ensures that standard tree-based rigid-body algorithms can be applied.
Fig. 5 : Illustration of implicit constraints: (a) Standard four-bar mechanism, showing the loop closure; (b) Mechanism opened at one joint, breaking the loop and reducing the system to a serial chain with two additional constraint forces required to maintain the original motion; (c) Mechanism with one bar split in half, preserving the original degrees of freedom but now requiring two linear constraint forces and an additional torque to enforce the closed-chain kinematics; (d) Planar leg in contact with the ground, the bilateral constraint is modeled exactly the same as in the closed loop mechanism but with the force affecting only one body. The contact frame expressed in local coordinates does not align with the normal of the surface, world coordinates .
Fig. 6 : Exponential–eigenvalue (EE) parametrization of a rigid body’s inertial parameters. The physical quantities (mass, first moment, and inertia) are illustrated by the body frame , CoM, and inertia ellipsoid with principal moments . The unconstrained coordinates (shown as a mesh) are mapped to a physically consistent inertia via the exponential map, guaranteeing , , and triangle inequalities on the principal moments; the inverse (logarithmic) map recovers from .
Fig. 7 : Visualization of friction models evaluated at different smoothing coefficients. The friction model remains smooth and differentiable while capturing the Coulomb, viscous, and Stribeck effects.
Fig. 8 : Sequential snapshots of the Kangaroo robot executing multiple walking gaits. Key leg components are highlighted: ankle differential drives (blue) transmitting through a second pair of differentials to the knee (green), then to hip-mounted actuators (orange), with a prismatic knee linkage forming an additional closure (magenta). The presence of three loop closures in series makes the mechanism highly constrained and the estimation/identification problem particularly challenging.
Fig. 9 : Optimal excitation motions executed on the Unitree B1 quadruped. Top: excitation of a single leg while the rest of the robot remains fixed. Bottom: excitation of the full floating-base system. These trajectories are generated by the proposed framework to maximize parameter observability while respecting actuation limits and kinematic constraints.
Fig. 10 : Cost, estimated trajectory errors, and inertial parameters errors for inverse and forward dynamics formulations. The inverse dynamics formulation shows faster convergence and lower estimation error of both the state trajectory and inertial parameters when compared with the forward dynamics formulation.
Fig. 11 : Evolution of the cost, trajectory estimation error, and inertial parameter identification error for both constrained and unconstrained SysID formulations. The addition of parameter constraints maintains similar convergence behavior and trajectory accuracy compared to the unconstrained case, while notably enhancing the precision of the identified inertial parameters. The improvement is especially pronounced in platforms exhibiting discrete symmetries, such as bipeds and quadrupeds.
Fig. 12 : Validation of the identified model on the Unitree B1 quadruped. Top: traversal of rough terrain. Middle: stair climbing. Bottom: climbing bigger stairs. The motions are executed using the model identified from optimal excitation trajectories, demonstrating transfer and generalization to locomotion tasks beyond the identification motions.
| Pendulum | Manipulator | Biped | Quadruped | |||||
|---|---|---|---|---|---|---|---|---|
| w/o | with | w/o | with | w/o | with | w/o | with | |
| Iterations | 20.4 | 31.2 | 7.3 | 11.4 | 15.6 | 24.1 | 9.2 | 15.5 |
| Cost | 0.09 | 1.62 | 0.29 | 3.28 | 0.47 | 4.72 | 0.41 | 3.67 |
| Traj. Err [ ] | 0.41 | 0.46 | 2.58 | 2.83 | 5.88 | 6.34 | 4.76 | 5.08 |
| Iner. Err [ ] | 0.022 | 0.018 | 0.56 | 0.46 | 1.86 | 1.62 | 1.47 | 1.28 |
| Fric. Err [ ] | 0.031 | 0.016 | 0.75 | 0.40 | 2.18 | 1.08 | 1.82 | 0.91 |
TABLE I : Effect of energy-based observations on convergence (100 random initial guesses). Entries are means over trials.
| Gaussian noise | ID (tuned SG) | Energy | Ours |
|---|---|---|---|
TABLE II : Identification errors under increasing Gaussian noise. Both regression baselines use the EE inertial parametrization to guarantee physical consistency. ID uses numerically differentiated accelerations filtered with a Savitzky–Golay window tuned to the best of our capabilities. Energy uses work–energy consistency over each interval.
Fig. 13 : Visualization of the four-bar mechanism with friction effects: (a) Colored internal mechanism of B1’s leg, highlighting the four-bar mechanism; (b) Our model of the four-bar mechanism; (c) Evolution of the kinematic Jacobian along the joint position, the highlighted area shows B1’s operating range; (d) Joint friction model evolution. A hypothetical friction originated in the motor’s gearbox is reflected nonlinearly in the output joint following the kinematic Jacobian.
Fig. 14 : Identified friction characteristics for the KFE joint in the Unitree B1 leg. The plots show the normalized estimated joint friction torque as a function of joint velocity for the actuated and passive (non-actuated) joints, present in the four-bar mechanism. Each joint exhibits distinct friction profiles, which are explained by a combination of static, viscous, and Stribeck friction effects.
Fig. 15 : Measured (blue) versus model-predicted (orange) joint torques for a representative trajectory of B1’s leg (HAA, HFE, and KFE joints). (a) Nominal model obtained from the initial URDF parameters: large torque prediction errors, especially at KFE (approximately one order of magnitude). (b) After identifying inertial and friction parameters on a reduced model, torque prediction improves for the actuated joints (HAA and HFE) but remains inaccurate at KFE due to the unmodeled four-bar closed-loop transmission. Nevertheless, the highlighted intervals show good agreement when the associated transmission mapping Jacobian is approximately unitary. (c) Identification with closed-loop kinematics enforces loop closure and resolves the KFE discrepancy, yielding consistent torque prediction across all joints.
Fig. 16 : Top: Unitree B1 with a Z1 arm tracing a circular end–effector path while standing in four–point contact. The arc highlights the commanded circle at the wrist; contact and loop–closure constraints are enforced during estimation, and the actuation/friction and energy terms are used to identify parameters without force/torque sensors. Bottom: End–effector tracking performance using the identified model (blue) versus the nominal URDF model (orange).
Fig. 17 : Top: Unitree B1 with a Z1 arm tracing a plane while climbing up the stairs. Bottom: tracking of the centroidal angular momentum.