General transformation equation

Comparing

Solution

Geometric approach

Path Vs TrajectoryPath is an ordered locus of points in the space(either joint or operational), which the robot shouldfollow.

Path provides a pure geometric description ofmotion.

Path is usually planned globally taking into accountobstacle avoidance, traversing a complicated maze,etc.

Path is an ordered locus of points in the space(either joint or operational), which the robot shouldfollow.

Path provides a pure geometric description ofmotion.

Path is usually planned globally taking into accountobstacle avoidance, traversing a complicated maze,etc.

TrajectoryTrajectory is a path plus velocities andaccelerations in its each point.

A design of a trajectory does not need globalinformation, which simplifies the tasksignificantly.

The trajectory is specified and designedlocally. Parts of a path are covered byindividual trajectories.

Trajectory is a path plus velocities andaccelerations in its each point.

A design of a trajectory does not need globalinformation, which simplifies the tasksignificantly.

The trajectory is specified and designedlocally. Parts of a path are covered byindividual trajectories.

Motion planning in robotics

Motion Planning an overview Path planning (global)

Geometric path. Issues: obstacle avoidance, shortest path.

Trajectory generating (local) The path planning provides the input the chunk of a path usuallygiven as a set of points defining the trajectory. Approximate the desired path chunk by a class of polynomialfunctions and generate a sequence of time-based control setpoints for the control of manipulator from the initial configurationto its destination.

Path planning (global) Geometric path. Issues: obstacle avoidance, shortest path.

Trajectory generating (local) The path planning provides the input the chunk of a path usuallygiven as a set of points defining the trajectory. Approximate the desired path chunk by a class of polynomialfunctions and generate a sequence of time-based control setpoints for the control of manipulator from the initial configurationto its destination.

Path PlanningThe path planning task: Find a collision free path for the robot from oneconfiguration to another configuration. Path planning is a difficult search problem. The involved task has an exponential complexity withrespect to the degrees of freedom (controllable joints). With industrial robots, path planning is not usually solvedby a computer as human operators plan the paths.

The path planning task: Find a collision free path for the robot from oneconfiguration to another configuration. Path planning is a difficult search problem. The involved task has an exponential complexity withrespect to the degrees of freedom (controllable joints). With industrial robots, path planning is not usually solvedby a computer as human operators plan the paths.

Trajectory planning Planned path is typically represented by via-points Via-points = sequence of points (or end-effector poses) along thepath. Trajectory generating = creating a trajectory connecting twoor more via

points. In industrial settings, a trajectory is performed by a human expertand later played back (by teach-and-playback). Recent research utilizes as the input several tens of trajectoriesperformed by human experts. They vary statistically. Machine learning techniques are used to create the final trajectory.

Planned path is typically represented by via-points Via-points = sequence of points (or end-effector poses) along thepath. Trajectory generating = creating a trajectory connecting twoor more via

points. In industrial settings, a trajectory is performed by a human expertand later played back (by teach-and-playback). Recent research utilizes as the input several tens of trajectoriesperformed by human experts. They vary statistically. Machine learning techniques are used to create the final trajectory.

Trajectory generation Path is a function of space whereas trajectoryis a function of time q(t)

q(t0) = qinit and q(tf) = qfinal tf-t0 represents the amount of time taken toexecute the trajectory

Path is a function of space whereas trajectoryis a function of time q(t)

q(t0) = qinit and q(tf) = qfinal tf-t0 represents the amount of time taken toexecute the trajectory

Point to Point Path is specified by initial and finalconfigurations

Best suited for material transfer applicationwhere the workspace is free from anyobstacles. Also common in teach and playbackmode

The basic of trajectory planning

Path is specified by initial and finalconfigurations

Best suited for material transfer applicationwhere the workspace is free from anyobstacles. Also common in teach and playbackmode

The basic of trajectory planning

Trajectory planning for point-to-point

Trajectory generation (PP)-I At time t0 the joint variable satisfies

The final values we wish to attain is

If needed, constraints on initial and finalacceleration can also be added

At time t0 the joint variable satisfies

The final values we wish to attain is

If needed, constraints on initial and finalacceleration can also be added

Trajectory generation (PP)-II

Cubic Polynomial Trajectories-I When we specify the initial point and finalpoint with time and velocity, there ispossibility for multiple solutions

One way to generate a smooth curve for giventwo points is to generate a polynomial andsolve for it

When we specify the initial point and finalpoint with time and velocity, there ispossibility for multiple solutions

One way to generate a smooth curve for giventwo points is to generate a polynomial andsolve for it

Cubic Polynomial Trajectories-II

We have four constraints to satisfy, so wegenerate a cube polynomial

The velocity component is given by the aboveequation

We have four constraints to satisfy, so wegenerate a cube polynomial

The velocity component is given by the aboveequation

Cubic Polynomial Trajectories-III Thus taking into account all the initial and finalconstraints

Cubic Polynomial Trajectories-IV It is easier to solve this in matrix form

Ma = b (or) b/M = a (or) M(inverse)b = a

It is easier to solve this in matrix form

Ma = b (or) b/M = a (or) M(inverse)b = a

Example

Quintic Polynomial Trajectory I

Quintic Polynomial Trajectory II

Quintic Polynomial Trajectory III Multiple segments with different accelerationprofiles will have issues

Discontinuity in acceleration

The derivative of this kind of acceleration iscalled jerk

Multiple segments with different accelerationprofiles will have issues

Discontinuity in acceleration

The derivative of this kind of acceleration iscalled jerk

Quintic Polynomial Trajectory IV Why is this a problem?

Less accurate Tracking is a problem

For this reason, we specify acceleration also asone of the constraint along with position andvelocity

Why is this a problem? Less accurate Tracking is a problem

For this reason, we specify acceleration also asone of the constraint along with position andvelocity

Polynomial including acceleration

Final Matrix form

LSPB trajectory I Linear Segments with Parabolic Bend Subtle variations in trajectory parameters Three parts

Constant acceleration i.e ramp velocity parabolicposition

Zero acceleration Constant velocity or linear position Constant deceleration ramp velocity parabolicposition

Has a trapezoidal profile

Linear Segments with Parabolic Bend Subtle variations in trajectory parameters Three parts

Constant acceleration i.e ramp velocity parabolicposition

Zero acceleration Constant velocity or linear position Constant deceleration ramp velocity parabolicposition

Has a trapezoidal profile

LSPB trajectory IIJoi

nt var

iable

linear Quadratic polynomial

tt=0 t=tft=tb t=tf-tb

tb = blend time

So the quadratic equations governing thetrajectory are..

LSPB trajectory III So between t=0 and t=tb

The equation describing the velocity is

From the above two equations

So between t=0 and t=tb

The equation describing the velocity is

From the above two equations

LSPB trajectory IV Now at tb, we want the velocity to remainconstant for a value V

Trajectory required from 0 to tb is

Now at tb, we want the velocity to remainconstant for a value V

Trajectory required from 0 to tb is

LSPB trajectory V Let us see the trajectory from tb to tf-tb

because of the linearity

Let us see the trajectory from tb to tf-tb

because of the linearitySymmetrical