## Model description:

The dynamic equations governing the behavior of a single link robot with flexible joint are traditionally obtained from Lagrangian dynamics considerations. The simple robot system under study is shown in the attached image. Let $x_1=\theta_m$ be the motor angular position, the corresponding angular velocity $x_2 = d\theta/dt$, the elastic force $x_3 = k_s(\theta_t - \theta_m)$ and $x_4 = \{ d\theta_l/dt - d\theta_m/dt\}/\rho$, where $\rho^2=1/k_s$. Then the state variable representation is:

$$\begin{align*} \dot{x}_1(t) &= x_2(t)\\ \dot{x}_2(t) &= -a_5x_2(t)+a_1x_3(t)+a_1u(t)\\ \dot{x}_3(t) &= x_4(t)/\rho\\ \dot{x}_4(t) &= \{ -a_2a_3\sin{[\rho^2x_3(t)+x_1(t)]}-a_4x_3(t)-a_7x_2(t)-a_6\rho x_4(t) - a_1u(t)\}/\rho \end{align*}$$

with $a_1=1/J_m$, $a_2=1/J_l$, $a_3=mgl$,$a_4=a_1+a_2$,$a_5=B_m/J_m;a_6=B_l/J_l$,$a_7=a_6-a_5$ and $u(t)=\tau(t)$.

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## Publication details:

Title | Physical parameter estimation of the nonlinear dynamics of a single link robotic manipulator with flexible joint using the HMF method |

Publication Type | Conference Paper |

Year of Publication | 1997 |

Authors | Daniel-Berhe, S., and Unbehauen H. |

Conference Name | Proceedings of the 1997 American Control Conference, 1997. |

Date Published | 06/1997 |

Publisher | IEEE |

Conference Location | Albuquerque, NM |

ISBN Number | 0-7803-3832-4 |

Accession Number | 6016897 |

Keywords | algebra, continuous time systems, manipulator dynamics, Nonlinear dynamical systems, parameter estimation |

Abstract | The application of the Hartley modulating functions (HMF) method is investigated to estimate the physical parameters of a single link robotic manipulator with flexible joint. The approach uses a weighted least-squares algorithm in the frequency domain. Knowing the structure of a continuous-time system, the identification method will only focus on the estimation of the physically-based system parameters using input and noise-corrupted output signal records. The methodology facilitates the conversion of a system differential equation into an algebraic equation in the parameters. Numerical simulations for a single link robotic manipulator with flexible joint are reported, which illustrate the application and performance of the methodology. The HMF method shows promising results for the identification of physically-based continuous-time nonlinear systems in the presence of noticeable measurement noises |

DOI | 10.1109/ACC.1997.610763 |