Robot Path Planning Idea and concept of planning

Robot Path Planning Idea and concept of planning & scheduling Robot path planning – interpolation at joints Robot planning 1

Problem Goal with constraint Mission determine or design activity make a plan and schedule run and control activity x o Planning refers to determining Path 1 or Path 2 or more so as to achieve the best goal under the constraint. o Scheduling puts time stamp on planning, e. g. , at time 3 pm, one arrives at X. Robot planning 2

o Not only the location or position but also the orientation is under consideration (left diagram) o Type of constraint may not be geometrical only Generalization of constraints Robot planning 3

The required motion of a robot is specified at the end-effector but the source of the motion (or motor) is at the joint. End-effector (e. g. , gripper) Robot planning 4

o The required motion of the end-effector mapped to the required motion of the motors at the joints. o Running of the motors at the joints produces the (actual) motion of the end-effector, which is expected to meet the required motion of the end-effector. A series of movements of the end-effector (1, 2, 3) θ 1 Motor 1 t θ 2 t Motor 2 Robot planning

How about the points between? θ 1 X Motor 1 X t θ 2 t X Motor 2 Why do we need to consider the points between? Robot planning 6

X Fast slow X A B Velocity profile X Interpolation x x Dynamic behavior at A and B and energy profile over the period of A to B In this course, path planning refers to the interpolation at the joint space Robot planning 7

Robot Path Planning Idea and concept of planning & scheduling Robot path planning – interpolation at joints Robot planning 8

Interpolation at joints q Criterion: make the motion as smooth as possible. q Smoothness is measured acceleration, jerk. Robot planning by velocity, 9

Interpolation at joints Problem definition for interpolation at joints: Given the initial and end goal points, there are different ways to interpolate, see Figure 2 Rotary motor Robot planning 10

Interpolation at joints General methodology: Step 1: Assume a form of polynomial function, e. g. , Step 2: Define the boundary condition in terms of velocity, acceleration, and jerk with the number of conditions being equal to the number of coefficients in the polynomial function. Robot planning 11

Interpolation at joints Example: cubic polynomials: There at least four conditions to constrain the interpolation: Stop at start and end, i. e. , velocity is zero. (1) (2) (3) (4) Robot planning 12

Robot Path Planning (6) Robot planning 13

Robot Path Planning A single-link robot with a rotary joint is motionless at It is desired to move the joint in a smooth manner to in 3 seconds. Find the coefficients of a cubic which accomplishes this motion and brings the manipulator to rest at the goal. Robot planning 14

Robot Path Planning Figure 3 shows the position Velocity Acceleration Robot planning 15

Robot Path Planning Stop or velocity is zero at both start and end, which is one work cycle Acceleration is infinitely large at both start and end Figure 3 Robot planning 16

Robot Path Planning Velocity at start and end may not be zero θ t Robot planning 17

Robot Path Planning In this case Robot planning 18

Robot Path Planning Robot planning 19

Robot Path Planning Robot planning 20

Robot Path Planning Application: start: 0, end: D, between there are several via points Robot planning 21

List of equations provided in classroom Robot planning 22

Constant Velocity Improvement Linear function with parabolic blends Robot planning 23

Robot Path Planning Robot planning 24

B A α Eight equations If α is given, there are nine equations Robot planning 25

Robot Path Planning Summary: 1. Path planning problem starts at the end-effector level but is converted to path planning at the joint level. 2. There are two categories of problems: (1) velocity at start and end is zero and (2) velocity at start and end is not zero. 3. Different paths will affect the smoothness of the motion of a robot. 4. Constant velocity path has the advantage of smooth motion in the period, but have infinitely large acceleration at the start and end points of the period. Local modification at the start and end is an effective means to trade-off the pros and cons of constant velocity plan Robot planning 26