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A new closed-loop output error method for parameter identification of robot dynamics

Maxime Gautier, Alexandre Janot, Pierre-Olivier Vandanjon

arXiv:1009.4556v1cs.RO

TL;DR

Existing robot identification methods require joint position and force/torque measurements, whereas the paper addresses identification using only force/torque measurements. It introduces a closed-loop output-error method based on direct-dynamics simulation and an inverse-dynamics reformulation, and validates the approach experimentally on a 2-DOF direct-drive robot.

  • Problem

    Conventional inverse-dynamics and output-error identification methods require joint position together with joint force/torque measurements.

  • Method

    DIDIM replaces joint-position output with joint force/torque in a closed-loop output-error method using the direct dynamic model, shared control law, and reference trajectory, with inverse dynamics yielding a parameter-linear formulation.

  • Results

    A validation experiment on a 2-DOF direct-drive robot shows that the proposed method is efficient.

  • Takeaways & Limitations

    DIDIM provides a force/torque-only identification approach that combines closed-loop output-error and inverse-dynamic-model techniques.

Abstract

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Off-line robot dynamic identification methods are mostly based on the use of the inverse dynamic model, which is linear with respect to the dynamic parameters. This model is sampled while the robot is tracking reference trajectories that excite the system dynamics. This allows using linear least-squares techniques to estimate the parameters. The efficiency of this method has been proved through the experimental identification of many prototypes and industrial robots. However, this method requires the joint force/torque and position measurements and the estimate of the joint velocity and acceleration, through the bandpass filtering of the joint position at high sampling rates. The proposed new method requires only the joint force/torque measurement. It is a closed-loop output error method where the usual joint position output is replaced by the joint force/torque. It is based on a closed-loop simulation of the robot using the direct dynamic model, the same structure of the control law, and the same reference trajectory for both the actual and the simulated robot. The optimal parameters minimize the 2-norm of the error between the actual force/torque and the simulated force/torque. This is a non-linear least-squares problem which is dramatically simplified using the inverse dynamic model to obtain an analytical expression of the simulated force/torque, linear in the parameters. A validation experiment on a 2 degree-of-freedom direct drive robot shows that the new method is efficient.

I. INTRODUCTION

Robot dynamic identification commonly uses an inverse dynamic model with least squares, but requires force/torque and position measurements plus filtered velocity and acceleration estimates. The paper introduces a force/torque-only closed-loop output-error method and reports practical efficiency on a 2-DOF robot.

  • Existing identification methods: Inverse dynamic identification models support linear least-squares estimation of robot dynamic parameters from sampled trajectory data.The approach has been applied to inertial and friction parameter identification on robotic prototypes and industrial robots.
  • Output-error identification: Output-error identification minimizes the quadratic difference between actual and simulated outputs under a shared input, typically using nonlinear least-squares optimization.Its state-space output is usually joint position for mechanical systems.
  • Existing identification methods: The conventional method requires joint position and force/torque measurements, with velocity and acceleration obtained by bandpass filtering position data.The filtering must be well tuned for reliable derivative estimates.
  • Proposed method: The proposed method uses only joint force/torque measurements and simulates the robot in closed loop with the same control law and reference trajectory.It minimizes the 2-norm error between actual and simulated force/torque, while the inverse dynamic model provides a parameter-linear expression.
  • Validation: Experimental results on a 2-DOF direct-drive robot are presented to demonstrate the method’s practical efficiency.The prototype emphasizes nonlinear coupling relevant to studying the method.

II. IDIM: INVERSE DYNAMIC IDENTIFICATION MODEL TECHNIQUE

The IDIM technique expresses robot motor force/torque as a model linear in dynamic parameters and estimates them from trajectory measurements using least squares. Its implementation depends on measured position and force/torque data, filtered derivatives, excitation, and assumptions about noise and actuator dynamics.

  • Model formulation: The inverse dynamic model calculates motor torque from joint positions, velocities, accelerations, and dynamic parameters.The model includes inertia, Coriolis, gravitational, and friction contributions.
  • Model formulation: The inverse dynamic identification model represents measured torque as the model prediction plus an additive error and is linear in the base parameters.Base parameters remove dynamically ineffective or linearly dependent standard parameters.
  • Least-squares identification: IDIM least squares stacks joint equations across sampled reference-trajectory data into an over-determined linear system.Exciting trajectories are used to obtain a full-rank, well-conditioned observation matrix.
  • Data processing: The method requires off-line velocity and acceleration estimates obtained by filtering measured joint position, including zero-phase low-pass filtering and central differences.Parallel decimation low-pass filters torque and observation-matrix data to reduce high-frequency ripple and limit the modeled frequency range.
  • Assumptions: The identification formulation assumes known actuator-related gains and commonly treats the observation matrix as deterministic with independent zero-mean Gaussian measurement error.The current-loop transfer function is approximated by its static gain over the robot-dynamics frequency range.
  • Least-squares identification: The ordinary least-squares estimate minimizes the squared 2-norm of the residual and is computed from the observation matrix, typically using QR factorization.The formulation uses the measured torque vector and the basis-function matrix built from estimated joint states.

III. THE OUTPUT ERROR METHOD (OE)

The output-error method estimates robot dynamic parameters by minimizing the discrepancy between measured and simulated outputs generated from a direct dynamic model. Its nonlinear least-squares optimization can require careful initialization and repeated sensitivity calculations.

  • Output-error formulation: OE identification minimizes a quadratic error between actual and simulated system outputs.For robotic systems, the simulated output is obtained by integrating the direct dynamic model.
  • Limitations: Open-loop OE simulation is sensitive to initial states and numerical integration errors, creating difficulties with multiple local solutions.These issues motivate choosing the closed-loop formulation for controlled robots.
  • Output-error formulation: In the usual robotic OE formulation, joint position is the measured output and simulated joint position comes from the direct dynamic model.The direct dynamic model is expressed as a nonlinear state-space model with a linear output equation.
  • Optimization: The parameter criterion is a nonlinear least-squares problem solved with gradient, Newton, Levenberg–Marquardt, or Gauss–Newton methods.Gauss–Newton uses the Jacobian of the simulated output with respect to the parameters.
  • Optimization: Gauss–Newton linearizes the simulated output around the current parameter estimate and solves an over-determined linear system at each iteration.The process repeats until parameter or residual changes satisfy convergence criteria.

A. Theoretical approach

The proposed DIDIM approach replaces joint position with force/torque as the output of a closed-loop simulation. It uses the direct and inverse dynamic models together to retain a linear parameter update while simplifying sensitivity computation.

  • Theoretical approach: DIDIM changes the output from measured joint position to measured joint force/torque.The actual robot tracks a reference trajectory under closed-loop control, while the simulation uses the same control law and reference.
  • Theoretical approach: The simulated force/torque is computed analytically with the inverse dynamic model from the simulated joint state and parameter estimate.This gives the simulated output as an inverse-dynamic-model expression linear in the parameters.
  • Theoretical approach: The method approximates sensitivity functions with an algebraic inverse-dynamic equation instead of integrating complicated differential equations.This simplifies force/torque-error minimization compared with position-error minimization in usual OE identification.
  • Theoretical approach: DIDIM minimizes the 2-norm error between actual and simulated force/torque over an observation window.The method is named DIDIM because it combines the Direct and Inverse Dynamic Identification Models.
  • Advantages: DIDIM requires only actuator force/torque measurement or estimation and avoids tuning the IDIM bandpass filter.Its closed-loop simulation uses the integrated direct dynamic model to define the relevant dynamic frequency range.
  • Limitations: A limitation is that the actual closed-loop control-law structure and tuning must be known for the simulated closed loop.The simulated control law is adjusted during iterations to preserve comparable closed-loop behavior.

B. Initialization of the algorithm

The initialization procedure is designed to make closed-loop simulation robust to unknown parameters and initial states. It uses regular inertial initialization, matched control-law structure, and transient-data removal.

  • Initialization: The algorithm is designed to reduce sensitivity to initial parameter values and avoid multiple local solutions.The initialization condition is intended to hold at every iteration, including the first.
  • Closed-loop tuning: The simulated robot uses the same control-law structure and desired closed-loop poles as the actual robot.Only the simulated derivative gain is updated with the current inertia estimate during iterations.
  • Closed-loop tuning: The simulated control gains are adjusted to preserve the actual closed-loop bandwidth, stability margin, or pole locations.The simulated controller changes with parameter estimates while the actual controller remains unchanged.
  • Initialization: A regular initialization chooses an initial inertia matrix that is the identity or another regular matrix.The identity initialization sets the actuator rotor and gear inertia terms to one while other initial parameters are zero.
  • State initialization: Because actual initial states are unknown, the simulated state is initialized from the reference position and velocity.Force/torque samples during the transient mismatch, lasting approximately 5 desired time constants, are removed from identification data.

V. CASE STUDY: MODELING OF THE SCARA ROBOT

The case study models a 2 degree-of-freedom planar direct-drive SCARA robot with an eight-parameter dynamic model and evaluates identification on a known prototype. The experiment uses PD closed-loop control and simulates the robot with the same reference trajectory and control-law structure.

  • The test platform is a 2 degree-of-freedom planar direct-drive prototype with two DC permanent-magnet motors supplied by PWM amplifiers.
  • The robot is selected because its known nominal parameters allow the physical meaning of identified parameters to be checked.
  • The dynamic model contains 8 minimal dynamic parameters, including 4 friction parameters.
  • The closed-loop controller is a PD control law whose desired natural frequencies are 1 rad/s and 10 rad/s for the two joints.
  • The control and measurement sample rates are both 200Hz, and torque data are obtained from the torque relation and current-reference data.
  • The simulated robot uses the same reference trajectory and control-law structure as the actual robot, with simulator gains calculated using the corresponding tuning procedure.

VI. EXPERIMENTAL IDENTIFICATION RESULTS

The experiments compare the established IDIM technique with DIDIM across initialization, sampling rate, filtering, and closed-loop tuning conditions. The study is conducted in SI units on the joint side.

  • The experiments compare IDIM and DIDIM while testing robustness to initialization, acquisition sampling rate, data filtering, and closed-loop tuning.
  • All reported results use SI units on the joint side.

A. Comparison of IDIM and DIDIM with good initial values,

With good initial values and filtered data, DIDIM reaches an optimal solution in two steps and produces torque predictions close to measured torques. The identified model has a relative torque-norm error below 3%.

  • 2 steps are needed to obtain the optimal solution with good initial values and filtered data.
  • The actual and identified-model joint torques are very close at the 200Hz measurement frequency.
  • <3% relative torque-norm error is obtained by both IDIM and DIDIM, indicating good accuracy for the model and identified values.
  • The friction parameters Fv1 and Fv2 have no significant estimates because their relative standard deviations exceed 30%.
  • The comparison retains all parameters despite the weak friction estimates to compare the IDIM and DIDIM identification methods.

B. DIDIM, validation of the regular initialization,

DIDIM remains robust under regular initialization, low sampling, unfiltered data, and closed-loop tuning errors. Its simulated trajectories stay close to actual trajectories, torque error converges rapidly, and parameter estimates remain close to reference estimates.

  • Regular initialization: 0.5% position, 5% velocity, and 10% acceleration relative norm errors remain nearly constant during regular-initialization validation.
  • Regular initialization: The simulated trajectory is 3 to 5 times closer to the actual trajectory than to the reference trajectory, with errors near 1.5%, 15%, and 30%.
  • Regular initialization: 3 steps are sufficient for the relative joint-torque norm error to decrease dramatically, demonstrating fast algorithm convergence.
  • Low sampling rate: At 0.5Hz measurement sampling, IDIM estimates are not good whereas DIDIM estimates remain good, showing robustness to measurement sampling rate.
  • Low sampling rate: IDIM fails at low sampling because central differences distort velocity and acceleration estimates, producing relative norm errors of 80% for both.
  • Without data filtering: Without data filtering, DIDIM remains good while IDIM becomes biased by noise in the observation matrix caused by differentiating position data.
  • Closed-loop tuning robustness: With half the closed-loop bandwidth, parameters converge in 6 steps to values very close to those obtained with full bandwidth.
  • Closed-loop tuning robustness: The relative torque error decreases in 6 steps, only twice more than with full bandwidth, and DIDIM is not very sensitive to bandwidth error when control-law structure is known.

VII. CONCLUSION

DIDIM combines closed-loop output-error identification with inverse and direct dynamic models, using force/torque output and closed-loop simulation. Experiments on a 2-DOF robot indicate agreement with IDIM and robustness to initialization and control-tuning errors.

  • Method: DIDIM replaces joint-position output with joint force/torque in a closed-loop output-error identification method.It uses the direct dynamic model, the same control-law structure, and the same reference trajectory for the actual and simulated robots.
  • Method: The nonlinear least-squares problem is solved by Gauss–Newton iterations, each simplified to a linear IDIM regression.This combines the closed-loop output-error and inverse dynamic identification techniques.
  • Implementation: DIDIM also addresses nonlinear least-squares initialization through regular parameter initialization and control-law gain updates.The initial state is set from the initial reference-trajectory values.
  • Validation: DIDIM gives the same results as IDIM with well-tuned filtering and is robust to parameter, state, and closed-loop performance-tuning initialization errors.The robustness to control-tuning errors assumes the actual and simulated robots use the same control-law structure.
  • Advantages: DIDIM avoids joint-position bandpass-filter tuning and requires only actuator force/torque measurement or estimation.Closed-loop simulation uses the direct dynamic model to define the corresponding frequency range for the actual system and model.
  • Scope: The method combines and validates inverse and direct dynamic models within one identification procedure for computed-torque control and simulation.The authors identify validation on a 6-DOF industrial robot as future work.
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