Design and validation of a dynamic parameter identification model for industrial manipulator robots

Abstract

This article presents the design and validation of a regression model for the identification of dynamic parameters in manipulator robots. The model exhibits implementation advantages as it is based on the acquisition of position, speed and voltage data from the actuator in each joint rather than on the calculation of acceleration and torque. Actuators can be direct current and/or servomotor type. The regression model developed is simulated using MATLAB/Simulink software to identify the parameters of 2-DoF (Degrees of Freedom) and 5-DoF manipulator robots. Additionally, the model is experimentally validated on a real 5-DoF redundant manipulator robot. The identification model has great advantages in terms of implementation.

Appendix A: Dynamic model for 5-DoF redundant manipulator robot

The components of the inertia matrix M(q), the centrifugal and Coriolis force vector C(q,q̇) and the gravity vector G(q) [79] are presented as follows.

$${\mathbf{M}}\left( {\mathbf{q}} \right) = \left( {\begin{array}{*{20}c} {M_{11} } & {M_{12} } & {M_{13} } & {M_{14} } & {M_{15} } \\ {M_{21} } & {M_{22} } & {M_{23} } & {M_{24} } & {M_{25} } \\ {M_{31} } & {M_{32} } & {M_{33} } & {M_{34} } & {M_{35} } \\ {M_{41} } & {M_{42} } & {M_{43} } & {M_{44} } & {M_{45} } \\ {M_{51} } & {M_{52} } & {M_{53} } & {M_{54} } & {M_{55} } \\ \end{array} } \right)$$

(67)

$$M_{11} = m_{1} + m_{2} + m_{3} + m_{4} + m_{5}$$

(68)

$$\begin{aligned} M_{12} & = M_{21} = M_{13} = M_{31} = M_{14} = M_{41} \\ & = \cdots M_{25} = M_{52} = M_{35} = M_{53} = M_{45} = M_{54} = 0 \\ \end{aligned}$$

(69)

$$M_{15} = M_{51} = - m_{5}$$

(70)

$$\begin{aligned} M_{22} & = l_{c2}^{2} m_{2} + \left( {l_{2}^{2} + l_{c3}^{2} + 2l_{2} l_{c3} {\text{cos}}\theta_{3} } \right)m_{3} \\ & \quad + \cdots I_{2} + I_{3} + I_{4} + \left( {l_{2}^{2} + l_{3}^{2} + l_{c4}^{2} + 2l_{2} l_{3} {\text{cos}}\theta_{3} } \right. \\ & \quad + \cdots 2l_{3} l_{c4} {\text{cos}}\theta_{4} + \left. {2l_{2} l_{c4} {\text{cos}}\left( {\theta_{3} + \theta_{4} } \right)} \right)m_{4} \\ & \quad + \cdots \left( {l_{2}^{2} + l_{3}^{2} + l_{4}^{2} } \right. + 2l_{2} l_{3} {\text{cos}}\theta_{3} \\ & \quad + \cdots \left. {2l_{3} l_{4} {\text{cos}}\theta_{4} + 2l_{2} l_{4} {\text{cos}}\left( {\theta_{3} + \theta_{4} } \right)} \right)m_{5} \\ \end{aligned}$$

(71)

$$\begin{aligned} M_{23} & = M_{32} = \left( {l_{c3}^{2} + l_{2} l_{c3} {\text{cos}}\theta_{3} } \right)m_{3} + l_{3}^{2} m_{4} \\ & \quad + \cdots I_{3} + I_{4} + \left( {l_{c4}^{2} + l_{2} l_{3} {\text{cos}}\theta_{3} + 2l_{3} l_{c4} {\text{cos}}\theta_{4} } \right. \\ & \quad + \cdots \left. {l_{2} l_{c4} {\text{cos}}\left( {\theta_{3} + \theta_{4} } \right)} \right)m_{4} \\ & \quad + \cdots \left( {l_{3}^{2} + l_{4}^{2} + l_{2} l_{3} {\text{cos}}\theta_{3} + 2l_{3} l_{4} {\text{cos}}\theta_{4} } \right. \\ & \quad + \cdots \left. {l_{2} l_{4} {\text{cos}}\left( {\theta_{3} + \theta_{4} } \right)} \right)m_{5} \\ \end{aligned}$$

(72)

$$\begin{aligned} M_{24} & = M_{42} = \left( {l_{c4}^{2} + l_{3} l_{c4} {\text{cos}}\theta_{4} } \right. + \cdots \left. {l_{2} l_{c4} {\text{cos}}\left( {\theta_{3} + \theta_{4} } \right)} \right)m_{4} \\ & \quad + \cdots \left( {l_{4}^{2} + l_{3} l_{4} {\text{cos}}\theta_{4} + l_{2} l_{4} {\text{cos}}\left( {\theta_{3} + \theta_{4} } \right)} \right)m_{5} + I_{4} \\ \end{aligned}$$

(73)

$$\begin{aligned} M_{33} & = l_{c3}^{2} m_{3} + \left( {l_{3}^{2} + l_{c4}^{2} + 2l_{3} l_{c4} {\text{cos}}\theta_{4} } \right)m_{4} \\ & \quad + \cdots \left( {l_{3}^{2} + l_{4}^{2} + 2l_{3} l_{4} {\text{cos}}\theta_{4} } \right)m_{5} + I_{3} + I_{4} \\ \end{aligned}$$

(74)

$$M_{34} = M_{43} = \left( {l_{c4}^{2} + l_{3} l_{c4} {\text{cos}}\theta_{4} } \right)m_{4} + \cdots \left( {l_{4}^{2} + l_{3} l_{4} {\text{cos}}\theta_{4} } \right)m_{5} + I_{4}$$

(75)

$$M_{44} = l_{c4}^{2} m_{4} + l_{4}^{2} m_{5} + I_{4}$$

(76)

$$M_{55} = m_{5}$$

(77)

$${\mathbf{C}}\left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right) = \left( {\begin{array}{*{20}c} {C_{11} } & {C_{12} } & {C_{13} } & {C_{14} } & {C_{15} } \\ {C_{21} } & {C_{22} } & {C_{23} } & {C_{24} } & {C_{25} } \\ {C_{31} } & {C_{32} } & {C_{33} } & {C_{34} } & {C_{35} } \\ {C_{41} } & {C_{42} } & {C_{43} } & {C_{44} } & {C_{45} } \\ {C_{51} } & {C_{52} } & {C_{53} } & {C_{54} } & {C_{55} } \\ \end{array} } \right)$$

(78)

$$\begin{aligned} C_{11} & = C_{12} = C_{13} = C_{14} = C_{15} = C_{21} = C_{25} \\ & = \cdots C_{31} = C_{35} = C_{41} = C_{44} = C_{45} = C_{51} = C_{52} \\ & = \cdots C_{53} = C_{54} = C_{55} = 0 \\ \end{aligned}$$

(79)

$$\begin{aligned} C_{22} & = - \left( {m_{3} l_{2} l_{c3} + m_{4} l_{2} l_{3} + m_{5} l_{2} l_{3} } \right){\text{sin}}\theta_{3} \dot{\theta }_{3} \\ & \quad + \cdots - \left( {m_{4} l_{2} l_{c4} + m_{5} l_{2} l_{4} } \right){\text{sin}}\left( {\theta_{3} + \theta_{4} } \right)\left( {\dot{\theta }_{3} + \dot{\theta }_{4} } \right) \\ & \quad + \cdots - \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \dot{\theta }_{4} \\ \end{aligned}$$

(80)

$$\begin{aligned} C_{23} & = - \left( {m_{3} l_{2} l_{c3} + m_{4} l_{2} l_{3} + \cdots } \right.\left. {m_{5} l_{2} l_{3} } \right){\text{sin}}\theta_{3} \left( {\dot{\theta }_{2} + \dot{\theta }_{3} } \right) \\ & \quad + \cdots - \left( {m_{4} l_{2} l_{c4} + m_{5} l_{2} l_{4} } \right){\text{sin}}\left( {\theta_{3} + \theta_{4} } \right)\left( {\dot{\theta }_{2} + \dot{\theta }_{3} + \dot{\theta }_{4} } \right) \\ & \quad + \cdots - \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \dot{\theta }_{4} \\ \end{aligned}$$

(81)

$$\begin{aligned} C_{24} & = - \left( {m_{4} l_{2} l_{c4} + m_{5} l_{2} l_{4} } \right){\text{sin}}\left( {\theta_{3} + \theta_{4} } \right)\left( {\dot{\theta }_{2} + \dot{\theta }_{3} + \dot{\theta }_{4} } \right) \\ & \quad + \cdots - \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \left( {\dot{\theta }_{2} + \dot{\theta }_{3} + \dot{\theta }_{4} } \right) \\ \end{aligned}$$

(82)

$$\begin{aligned} C_{32} & = \left( {m_{3} l_{2} l_{c3} + m_{4} l_{2} l_{3} + m_{5} l_{2} l_{3} } \right){\text{sin}}\theta_{3} \dot{\theta }_{2} \\ & \quad + \cdots \left( {m_{4} l_{2} l_{c4} + m_{5} l_{2} l_{4} } \right){\text{sin}}\left( {\theta_{3} + \theta_{4} } \right)\dot{\theta }_{2} \\ & \quad + \cdots - \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \dot{\theta }_{4} \\ \end{aligned}$$

(83)

$$C_{33} = - \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \dot{\theta }_{4}$$

(84)

$$C_{34} = - \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \left( {\dot{\theta }_{2} + \dot{\theta }_{3} + \dot{\theta }_{4} } \right)$$

(85)

$$\begin{aligned} C_{42} & = \left( {m_{4} l_{2} l_{c4} + m_{5} l_{2} l_{4} } \right){\text{sin}}\left( {\theta_{3} + \theta_{4} } \right)\dot{\theta }_{2} \\ & \quad + \cdots \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \left( {\dot{\theta }_{2} + \dot{\theta }_{3} } \right) \\ \end{aligned}$$

(86)

$$C_{43} = \left( {m_{4} l_{3} l_{c4} + m_{5} l_{3} l_{4} } \right){\text{sin}}\theta_{4} \left( {\dot{\theta }_{2} + \dot{\theta }_{3} } \right)$$

(87)

$${\mathbf{G}}\left( {\mathbf{q}} \right) = \left( {\begin{array}{*{20}c} {\left( {m_{1} + m_{2} + m_{3} + m_{4} + m_{5} } \right)g} \\ 0 \\ 0 \\ 0 \\ { - m_{5} g} \\ \end{array} } \right)$$

(88)

where: m₁, m₂, m₃, m₄, m₅ and l₁, l₂, l₃, l₄, l₅ represent the mass and lengths of the first, second, third, fourth and fifth link, respectively, l_c2, l_c3, l_c4 express the length of origin in the mass centers of the second, third and fourth link, and I₂, I₃ and I₄ represent the inertia moments of the second, third and fourth link with respect to the z axis of the corresponding joint.