A collision-free path planning method for industrial robot manipulators considering safe human–robot interaction

Abstract

This paper introduces a new method to solve the collision-free path planning problem for industrial robots considering safe human–robot coexistence. The aim is to keep the robot at a safe distance away from the human and let the robot’s tip reach the target location through a smooth path. The proposed method is iterative, and each iteration provides random candidate waypoints for the robot’s tip. The waypoint that the robot will follow for each iteration is determined by solving the optimization problem. The objective function is formulated considering the distance between the human and the robot, as well as the criteria for the robot’s tip to reach the target by following a smooth path. The human and the robot in the environment are represented by the capsules, and the minimum distance calculation is performed between these capsules using the Gilbert–Johnson–Keerthi algorithm. The simulation results demonstrate the performance of the proposed method for different scenarios involving human–robot coexistence.

Appendix: Uniform distribution over a unit sphere surface

For the uniform distribution over a unit sphere surface with an area of 4π, the probability density should be constant, which is $p\left(x\right)= \frac{1}{4\pi }$. At the same time, the probability that a point lies in the differential surface area can be expressed as $p\left(x\right) dA$. In this case, Eq. (34) can be used to find the probability density function $p\left(\phi ,\ominus \right)$ that constitutes a uniform distribution on the sphere.

$$p\left(x\right) dA=p\left(\phi ,\ominus \right)d\phi d\ominus $$

(34)

Since surface element $dA=\mathrm{sin}\ominus d\ominus d\phi$, the probability density function becomes:

$$p\left(\phi ,\ominus \right)=\frac{\mathrm{sin}\ominus }{4\pi }$$

(35)

When the spherical coordinates $\ominus$ and $\phi$ are desired to be sampled independently of each other so that $p\left(\phi ,\ominus \right)=p\left(\phi \right)p\left(\ominus \right)$, the probability density functions are given by:

$$p\left(\phi \right)= \underset{0}{\overset{\pi }{\int }}p\left(\phi ,\ominus \right)d\ominus =\frac{1}{2\pi }$$

(36)

$$p\left(\ominus \right)= \underset{0}{\overset{2\pi }{\int }}p\left(\phi ,\ominus \right)d\phi =\frac{\mathrm{sin}\ominus }{2}$$

(37)

It is obvious that $\phi$ is uniformly distributed with a constant distribution, while probability density $p\left(\ominus \right)$ varies with $\mathrm{sin}\ominus$. Therefore, $p\left(\ominus \right)$ can be sampled by using the inverse transform sampling. The cumulative distribution functions $P\left(\ominus \right)$ and $P\left(\phi \right)$ are needed to generate $\ominus$ and $\phi$ in this method given by:

$$P\left(\phi \right)= \underset{0}{\overset{\phi }{\int }}p\left(\phi \right)d\phi =\frac{\phi }{2\pi }$$

(38)

$$P\left(\ominus \right)= \underset{0}{\overset{\ominus }{\int }}p\left(\ominus \right)d\ominus =\frac{1-\mathrm{cos}\ominus }{2}$$

(39)

If two random variables $u$ and $v$, which are both uniform on the interval $[\mathrm{0,1}]$, are chosen so that $P\left(\phi \right)=u$ and $P\left(\ominus \right)=v$, then inverse functions of the cumulative distribution functions gives:

$${\phi }_{i}=2\pi u$$

(40)

$${\ominus }_{i}={\mathrm{cos}}^{-1}(1-2v)$$

(41)