Collision Avoidance of Redundant Robotic Manipulators Using Newton’s Method

Abstract

This study investigates the application of Newton method to the problems of collision avoidance and path planning for robotic manipulators, especially robots with high Degrees of Freedom (DOF). The proposed algorithm applies to the potential fields method, where the Newton technique is used for performing the optimization. As compared to classical gradient descent method this implementation is mathematically elegant, enhances the performance of motion generation, eliminates oscillations, does not require gains tuning, and gives a faster convergence to the solution. In addition, the paper presents a computationally efficient symbolic formula for calculating the Hessian with respect to joint angles, which is essential for achieving realtime performance of the algorithm in high DOF configuration spaces. The method is validated successfully in simulation environment. Results for different methods (Newton, gradient descent and gradient descent with momentum) are compared in terms of quality of the path generated, oscillations, minimum distance to obstacles and convergence rate.

Appendix: Gradient and Hessian Formulas Expressed in Joint Space

In this section, the formulas for the first and the second order derivatives, gradient (4) and Hessian (3), of the potential field with respect to manipulator’s joint angles are deduced. Let u_k(x) be a potential field, this potential field is described in Cartesian space. It could be an attraction potential that attracts the EEF to the goal, or a repulsion potential that repels the robot away from obstacles. Then the second order approximation of the potential field u_k(x) near the current position x₀ is given by:

$$ u_{k}({x})\approx u_{k}({x_{0}})+\triangle{x}^{\mathrm{T}}\nabla u_{k}+\triangle{x}^{\mathrm{T}}\nabla^{2}u_{k}\triangle{x} $$

(10)

Where:

• u_k(x) is the approximate value of the potential function at the Cartesian position x;

• u_k(x₀) is the value of the potential function at the Cartesian position x₀;

• ∇u_k is the gradient of the potential function u_k with respect to x taken at the point x₀;

• △x = x − x₀;

• ∇²u_k is the Hessian of the potential function u_k with respect to x taken at the point x₀.

In addition, differential kinematics gives the following relationship between elemental displacements in Cartesian space and elemental displacements in joint space:

$$ \triangle{x}={\mathrm{J}}\triangle{q} $$

(11)

Where J is the Jacobian associated with the point of the manipulator x₀, and △q = q − q₀. By substituting (11) in (10) and fixing:

$$ u_{k}({x})\approx u_{k}({x_{0}})+\triangle{q}^{\mathrm{T}}{\mathrm{J}}^{\mathrm{T}}\nabla u_{k}+\triangle{q}^{\mathrm{T}}{\mathrm{J}}^{\mathrm{T}}\nabla^{2}u_{k}{\mathrm{J}}\triangle{q} $$

(12)

From the previous equation the gradient and the Hessian formulas are deduced:

• The gradient is associated with the first order term of the approximation:

$$ {g}_{k}={\mathrm{J}}^{\mathrm{T}}\nabla u_{k} $$

(13)

• The Hessian is associated with the second order term of the approximation:

$$ {\mathrm{H}}_{k}={\mathrm{J}}^{\mathrm{T}}\nabla^{2}u_{k}{\mathrm{J}} $$

(14)

Finally, in case of a manipulator subjected to several potential fields simultaneously, the total potential field is the sum of all of the potentials, and the total gradient g is the sum of the individual gradients:

$$ {g}=\sum{g}_{k} $$

(15)

The total Hessian is the sum of the individual Hessians:

$$ {\mathrm{H}}=\sum{\mathrm{H}}_{k} $$

(16)