# A single link robotic manipulator

## 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 $u(t)=\tau(t)$.

4

## 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