# Trajectory planning for energy minimization of industry robotic manipulators using the Lagrange interpolation method

## Abstract

We propose to use Lagrange interpolation method to express each joint trajectory function to realize trajectory planning for energy minimization of industrial robotic manipulators. We give the position constraints to the industrial robotic manipulators and the stability of the industrial robotic manipulators should be satisfied. In order to avoid Runge’s phenomenon of Lagrange interpolation method, we use the Chebyshev interpolation points for our approach. Through derivation, the angular velocity functions and angular acceleration functions can be obtained. Lagrange interpolation method can satisfy position constraints, and ensure the smoothness of joint angular positions, velocities, accelerations, and joint torques. By taking these functions into the Performance Index (PI) of energy minimization, as well as the direct iteration method used for optimization of energy consumption, we can obtain the optimal trajectory for industrial robotic manipulators.

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## Abbreviations

θ :

vector of joint angles of the robotic manipulators

$$\dot \theta$$ :

vectors of joint angular velocities of the robotic manipulators

$$\ddot \theta$$ :

vectors of joint angular accelerations of the robotic manipulators

M(θ):

symmetric positive definite inertial matrix

V(θ, $$V(\theta ,\ddot \theta )$$):

component of the torque depending on centrifugal force and the Coriolis forces

G(θ):

component depending on gravity forces

τ :

vectors of joint torques of the robotic manipulators

n :

number of the joints of the robotic manipulators

j :

j th joint of the robotic manipulators

N :

number of time interpolation points

l i :

basis polynomials of the j th segment of time for Lagrange interpolation method

ω :

basis polynomial for error function of the Lagrange interpolation method

a j_k :

coefficients of the polynomial function

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Correspondence to Chang-Soo Han.

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Luo, LP., Yuan, C., Yan, RJ. et al. Trajectory planning for energy minimization of industry robotic manipulators using the Lagrange interpolation method. Int. J. Precis. Eng. Manuf. 16, 911–917 (2015). https://doi.org/10.1007/s12541-015-0119-9

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• DOI: https://doi.org/10.1007/s12541-015-0119-9

### Keywords

• Energy minimization
• Trajectory planning
• Industrial robotic manipulators
• Lagrange interpolation method
• Performance Index (PI)
• Iterative method