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Trajectory Planning
PR 502: Robot
Dynamics & Control
„
Tasks
Task Plan
Action Plan
Path Plan
Robot Trajectory
Planning
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Trajectory
Plan
Controller
Robot
Sensor
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effector is going to be
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„ What is the velocity
of the motion
„ What are the forces at the joints
Trajectory Planning
„ The way the robot is
in controlled manner
Goals
Coordinate the trajectories of two robot
manipulators so as to avoid collisions and deadlock.
deadlock.
„ Minimize total execution time
„ What are the joint variables must be
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Trajectory Planning
Kinematics and Dynamics :
„ Where the end
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Trajectory Planning
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Motion Planning:
Path planning
„ Geometric path
„ Issues: obstacle avoidance,
shortest path
„ Trajectory planning,
„ “interpolate”
interpolate” or “approximate”
approximate”
the desired path by a class of
polynomial functions and
generates a sequence of timetimebased “control set points”
points” for the
control of manipulator from the
initial configuration to its
destination.
„
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moved from one location to another
Definitions
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Path–
– Curve in Coordinate space
Trajectory – Time history of positions along a path
Sequence of movements that must be made create a
controlled motion
Note : Trajectory planning needs both Kinematics and Kinetics
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Joint v. Cartesian Space
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Trajectory Planning
Joint space description
Trajectory planning of a 2 DOF robot arm
„ Description
of motion to be made by the robot by its
joint variables
„ The motion between the two point is unpredictable
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(x , y)
β
Cartesian space description
„ A description of sequence of
movements that a robot
makes
„ Advantages : Easy to visualize
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α
Disadvantages : Computationally extensive, require fast processing
processing
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Cartesian Space Trajectory Planning
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Cartesian Space Trajectory Planning
Possibility 1 : Straight line path between the point A and B
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Method : Draw a line and divide into segments
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Solve for the angles for each segment
Cartesian space representation
Interpolation between the point A and B
However, joint angles are not uniformly changed.
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Possibility 2 : Not a straight line but a different
curve
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Ex : a quadratic equation
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Normalized by a common factor such that the joint with
smaller motion will move proportionally slower and both
joints will start and stop their motion simultaneously
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Polynomials of different orders
Linear functions
„ Parabolic blends
Both joints are at different speeds
Move continuously together
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Polynomial trajectory planning
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Path is still irregular
Need to solve Inverse Kinematics equations
Joint space representation
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Possibility 3: motion planning with controlled
characteristics.
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Result : segments of the movement are similar to each
other
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Characteristics
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Joint space trajectory planning
Possibility 2 : Normalized joint motion
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Path is irregular
Need to solve inverse kinematics equations
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Joint space trajectory planning
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Possibility 1: both joints are at maximum speed
Result : segments of the movement are not
similar to each other
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Joint space trajectory planning
Cartesian Space Trajectory Planning
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More segments
High accuracy
Problem : Actuators should be strong enough to
provide large forces necessary to accelerate and
decelerate the joint as needed.
Solution : Smaller segments at the beginning and
in the end.
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Known : initial location : Orientation of the robot
Solve Inverse Kinematics equations and calculate
the final joint angles
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Joint space trajectory planning
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Problem :
Acceleration cannot be defined with a 3rd order
polynomial
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Solution :
Use a 5th order polynomial.
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Joint space trajectory planning
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Joint space trajectory planning
Linear segments with parabolic blends
„ Objective :
„
Problem :
At the beginning and the in the end acceleration
must be infinite in order to maintain a constant
velocity.
„
Solution :
Linear segments are blended with parabolic
sections.
To run the joint at constant speed between
two points
Linear function :
Velocity is constant
„ Acceleration is zero
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